Add files using upload-large-folder tool

.gitattributes

@@ -7782,3 +7782,60 @@ YdFLT4oBgHgl3EQfVC8l/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex
 R9E3T4oBgHgl3EQfDQnp/content/2301.04286v1.pdf filter=lfs diff=lfs merge=lfs -text
 OdAzT4oBgHgl3EQflP0i/content/2301.01543v1.pdf filter=lfs diff=lfs merge=lfs -text
 bNAzT4oBgHgl3EQfnP3n/content/2301.01579v1.pdf filter=lfs diff=lfs merge=lfs -text
+0dFQT4oBgHgl3EQf0Daw/content/2301.13415v1.pdf filter=lfs diff=lfs merge=lfs -text
+1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf filter=lfs diff=lfs merge=lfs -text
+OdAzT4oBgHgl3EQflP0i/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+2NE0T4oBgHgl3EQfdwA8/content/2301.02380v1.pdf filter=lfs diff=lfs merge=lfs -text
+8dAyT4oBgHgl3EQfp_jS/content/2301.00536v1.pdf filter=lfs diff=lfs merge=lfs -text
+jNFAT4oBgHgl3EQfaR1d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+d9E1T4oBgHgl3EQfLgNb/content/2301.02977v1.pdf filter=lfs diff=lfs merge=lfs -text
+5NFKT4oBgHgl3EQf-C45/content/2301.11956v1.pdf filter=lfs diff=lfs merge=lfs -text
+5tAzT4oBgHgl3EQf9_5e/content/2301.01927v1.pdf filter=lfs diff=lfs merge=lfs -text
+5dE1T4oBgHgl3EQfBAJm/content/2301.02846v1.pdf filter=lfs diff=lfs merge=lfs -text
+DNE4T4oBgHgl3EQfGAw7/content/2301.04890v1.pdf filter=lfs diff=lfs merge=lfs -text
+X9E1T4oBgHgl3EQfcARq/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+29E1T4oBgHgl3EQfAQI2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+B9FJT4oBgHgl3EQfsy1U/content/2301.11614v1.pdf filter=lfs diff=lfs merge=lfs -text
+TNAzT4oBgHgl3EQfJftl/content/2301.01080v1.pdf filter=lfs diff=lfs merge=lfs -text
+49E4T4oBgHgl3EQfBAv2/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+LdE1T4oBgHgl3EQfZAQe/content/2301.03144v1.pdf filter=lfs diff=lfs merge=lfs -text
+vtAzT4oBgHgl3EQfB_rN/content/2301.00953v1.pdf filter=lfs diff=lfs merge=lfs -text
+8tE2T4oBgHgl3EQfPwar/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+EtAyT4oBgHgl3EQf4_op/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+KNE3T4oBgHgl3EQfAQkY/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+0dFQT4oBgHgl3EQf0Daw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+bNAzT4oBgHgl3EQfnP3n/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+2NE0T4oBgHgl3EQfdwA8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+rtE0T4oBgHgl3EQfrQGn/content/2301.02564v1.pdf filter=lfs diff=lfs merge=lfs -text
+mNE2T4oBgHgl3EQfeQdZ/content/2301.03914v1.pdf filter=lfs diff=lfs merge=lfs -text
+jNFAT4oBgHgl3EQfaR1d/content/2301.08550v1.pdf filter=lfs diff=lfs merge=lfs -text
+m9E0T4oBgHgl3EQfpwHn/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+kb_27/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+nNE1T4oBgHgl3EQfOQP4/content/2301.03014v1.pdf filter=lfs diff=lfs merge=lfs -text
+m9E_T4oBgHgl3EQf7Ry5/content/2301.08369v1.pdf filter=lfs diff=lfs merge=lfs -text
+29E1T4oBgHgl3EQfAQI2/content/2301.02836v1.pdf filter=lfs diff=lfs merge=lfs -text
+1dAyT4oBgHgl3EQf1fkT/content/2301.00734v1.pdf filter=lfs diff=lfs merge=lfs -text
+WtFIT4oBgHgl3EQfiCtg/content/2301.11290v1.pdf filter=lfs diff=lfs merge=lfs -text
+1dAyT4oBgHgl3EQf1fkT/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+o9FKT4oBgHgl3EQfGS3Z/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+v9FKT4oBgHgl3EQf4i7d/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PtE3T4oBgHgl3EQfxwuM/content/2301.04714v1.pdf filter=lfs diff=lfs merge=lfs -text
+6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf filter=lfs diff=lfs merge=lfs -text
+WtFIT4oBgHgl3EQfiCtg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+6NFKT4oBgHgl3EQfTS23/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+PtE3T4oBgHgl3EQfxwuM/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+DNE4T4oBgHgl3EQfGAw7/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+GNAzT4oBgHgl3EQfHPvT/content/2301.01043v1.pdf filter=lfs diff=lfs merge=lfs -text
+Plough[[:space:]]Knowledge[[:space:]]Ocean[[:space:]]Intro/content/北斗知海系统用户指南.pdf filter=lfs diff=lfs merge=lfs -text
+6NE0T4oBgHgl3EQfewDQ/content/2301.02396v1.pdf filter=lfs diff=lfs merge=lfs -text
+ZNFRT4oBgHgl3EQfPjcf/content/2301.13517v1.pdf filter=lfs diff=lfs merge=lfs -text
+YNE0T4oBgHgl3EQf3wIQ/content/2301.02728v1.pdf filter=lfs diff=lfs merge=lfs -text
+YNE0T4oBgHgl3EQf3wIQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+v9FKT4oBgHgl3EQf4i7d/content/2301.11934v1.pdf filter=lfs diff=lfs merge=lfs -text
+ttAzT4oBgHgl3EQfBfrk/content/2301.00945v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf filter=lfs diff=lfs merge=lfs -text
+ztFJT4oBgHgl3EQfjCwq/content/2301.11572v1.pdf filter=lfs diff=lfs merge=lfs -text
+ZNFRT4oBgHgl3EQfPjcf/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+LNFRT4oBgHgl3EQf1jiU/content/2301.13657v1.pdf filter=lfs diff=lfs merge=lfs -text
+Plough[[:space:]]Knowledge[[:space:]]Ocean[[:space:]]Intro/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+m9E_T4oBgHgl3EQf7Ry5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text

0dFQT4oBgHgl3EQf0Daw/content/2301.13415v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:955ef5716a76012e0dd17cb4da62f5701c3509be5d9c8f4866f5f5d92e3780be
+size 685420

0dFQT4oBgHgl3EQf0Daw/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c1f5921c9ade8904b85baf248f8b850284b931528c2fd9b02f66e6077f1d5b5a
+size 3473453

0dFQT4oBgHgl3EQf0Daw/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6b6d94c514263772947439ff7be38357e618fe23756bcc3e65ceec438f97b724
+size 153274

1dAyT4oBgHgl3EQf1fkT/content/2301.00734v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:789ddf5df6b9fffcd8fb5ff1c6b5b36a1644f795d770f906371f3987ee42391e
+size 3341053

1dAyT4oBgHgl3EQf1fkT/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1d9a9c48aab675b808f9c94fa6c48fe50d8055fec39ec98750a24813e0277d35
+size 4063277

1dAyT4oBgHgl3EQf1fkT/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:518c71f7f6f9305b5159834e05a7574a08e2bfc42a2254805346b6b1e29cfd34
+size 138036

1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6151d09c731dec29a29e78550f2f648d010f28e48684f456e77e5b76df0cfdb0
+size 6048361

29E1T4oBgHgl3EQfAQI2/content/2301.02836v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1dfdf001d5b6611ffd732d9e5e862dba8c5bb11e773f0583991336bcd150e505
+size 3265144

29E1T4oBgHgl3EQfAQI2/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:dd60e9699a39d5e98748911aa33a20d5902c1bd95f41ef4d926afbe19e2f4502
+size 4653101

2NE0T4oBgHgl3EQfdwA8/content/2301.02380v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:450912517615b8ae9ce2d3e75386592bb970de22dd21c8aaa42d18d2c15b03d1
+size 13064830

2NE0T4oBgHgl3EQfdwA8/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ba3cbe47fd8fdababc85984fc9ba63542592212da856031cc3dea5d9d0854b17
+size 2097197

2NE1T4oBgHgl3EQfSAMg/content/tmp_files/2301.03059v1.pdf.txt

@@ -0,0 +1,581 @@
+arXiv:2301.03059v1  [math.CO]  8 Jan 2023
+Eggs in finite projective spaces
+and unitals in translation planes
+Giusy Monzillo∗
+[email protected]
+Dipartimento di Matematica, Informatica ed Economia
+Universit`a degli Studi della Basilicata
+Viale dell’Ateneo Lucano 10
+85100 Potenza, Italy
+Tim Penttila
+[email protected]
+School of Mathematical Sciences
+The University of Adelaide
+Adelaide, South Australia
+5005 Australia
+Alessandro Siciliano∗
+[email protected]
+Dipartimento di Matematica, Informatica ed Economia
+Universit`a degli Studi della Basilicata
+Viale dell’Ateneo Lucano 10
+85100 Potenza, Italy
+Abstract
+Inspired by the connection between ovoids and unitals arising from the
+Buekenhout construction in the Andr´e/Bruck-Bose representation of trans-
+lation planes of dimension at most two over their kernel, and since eggs of
+PG(4m − 1, q), m ≥ 1, are a generalization of ovoids, we explore the rela-
+tion between eggs and unitals in translation planes of higher dimension over
+their kernel. By investigating such a relationship, we construct a unital in the
+Dickson semifield plane of order 310, which is represented in PG(20, 3) by a
+∗The research was supported by the Italian National Group for Algebraic and Geometric Struc-
+tures and their Applications (GNSAGA-INdAM).
+1
+
+cone whose base is a set of points constructed from the dual of the Penttila-
+Williams egg in PG(19, 3). This unital is not polar; so, up to the knowledge
+of the authors, it seems to be a new unital in such a plane.
+Keywords: Unital, Blocking set, Egg, Projective plane
+1
+Introduction
+Field reduction has become a theme of finite geometry which turned out very fruit-
+ful in the last few decades. Given a construction of an interesting object from a
+configuration in a vector space of dimension r over a field of order qn, the question
+is raised as to which objects give rise similar configurations in a vector space of
+dimension rn over a field of order q.
+The Buekenhout-Metz construction of unitals in finite translation planes [15, 33]
+(which gives all known unitals in Desarguesian planes) can be recontextualized in
+this fashion, with cones projecting an ovoid as a base in the Andr´e/Bruck-Bose
+representation of such planes.
+It has long been known [14] that unitals are extremal in size among minimal
+blocking sets (at the other end than that most studied - large rather than small). The
+observation of Lunardon [29] at the turn of the millennium that changing the field
+gave access to many more subspaces, some of which were blocking sets, transformed
+the theory of blocking sets in the process giving rise to the idea of linear sets. Thus,
+the idea of Buekenhout and Metz was taken by Sz˝onyi et al. [40] and, later, by
+Mazzocca and Polverino [31] to provide further minimal blocking sets, using cones
+rather than subspaces.
+For the construction by Tits of generalized quadrangles from (ovals and) ovoids,
+the configurations that arise by applying a field reduction are eggs, and the similar
+objects are translation generalized quadrangles. Thus, changing the field for ovoids
+and studying eggs gave the possibility of new translation generalized quadrangles,
+first realized in work of Kantor [25] from three decades past; the result of field
+reduction applied to the concept of an ovoid is an egg.
+Motivated by the relationship between ovoids and unitals via the Buekenhout-
+Metz construction, and since eggs are generalization of ovoids, we explore possible
+relationships between eggs and unitals. Putting all the above ideas together, in this
+paper we construct a unital in the Dickson semifield plane of order 310, which is
+represented in PG(20, 3) by a cone whose base is a set of points constructed from
+the dual of the Penttila-Williams egg in PG(19, 3). This unital does not arise from
+a polarity; so it is a new unital, up to the knowledge of the authors.
+While field reduction is usually thought of in a projective setting, algebraic di-
+mensions are more amenable to an introductory discussion of it, so we will take a
+vector space approach along all the paper.
+2
+
+2
+Definitions and preliminary results
+A unital in a finite projective plane π of order n2 is a set U of n3 + 1 points such
+that every line of π meets U in 1 or n + 1 points. Therefore, U is equipped with a
+family of subsets, each of size n + 1, such that every pair of distinct points of U is
+contained in exactly one subset of the family; such subsets are usually called blocks,
+and U turns out to be a 2-(n3 + 1, n + 1, 1) design.
+In a computer search, Brouwer [10, 11] found a large number of mutually non-
+isomorphic 2-(28, 4, 1) designs. Only a few of these are embeddable in a projective
+plane of order 9 as unitals. One of the examples has been generalized by Gr¨uning
+[23], who constructed a unital of order q for any odd prime power in both the Hall
+plane and dual Hall plane of order q. An infinite family of non-Buekenhout unitals
+in the Hall planes of order q2 have been constructed in [19]. Other infinite families
+of unitals in various square order planes are known to exist; see e.g. [1], [5], [6], [18],
+[36], [37]. The only known 2-(n3 + 1, n + 1, 1) design with n not a prime power is
+the one found in [4] and [30] where n = 6. For more on 2-(n3 + 1, n + 1, 1) designs
+embeddable as unitals in projective plane, see [8].
+In the Desarguesian projective plane PG(2, q2), a unital can arise from a unitary
+polarity: the points of the unital are the absolute points, and the blocks are the
+intersections of the non-absolute lines of the polarity with U. These unitals are
+called classical or Hermitian unitals. By a result of Seib [39], the absolute points
+of a unitary polarity in any square order projective plane form the point-set of a
+unital. Such unitals are called polar unitals. So, classical unitals of PG(2, q2) are
+examples of polar unitals, and Ganley [21] showed that polar unitals exist in any
+Dickson commutative semifield plane of odd order.
+A finite semifield is a finite set S with two binary operations + and ∗, such that
+(S, +) is an abelian group and (S \ {0}, ∗) is a loop such that both distributive laws
+hold.
+Let π(S) be the point-line geometry whose points are the elements in S × S and
+in {(m) : m ∈ S ∪ {∞}}, and the lines are the sets
+[m, k] = {(y, x) ∈ S × S : m ∗ x + y = k} ∪ {(m)},
+[z] = {(y, z) : y ∈ S} ∪ {(∞)}
+and
+[∞] = {(m) : m ∈ S} ∪ {(∞)}.
+with m, k, z ∈ S, and ∞ a symbol not in S.
+It turns out that π(S) is a translation plane which is called the semifield plane
+coordinatized by S. We refer to [9] and [17] for basic information on semifields and
+translation planes.
+3
+
+For any semifield S, the subset Nl = {a ∈ S : a ∗ (x ∗ y) = (a ∗ x) ∗ y, ∀x, y ∈ S} is
+called the left nucleus of S. Similarly, the middle nucleus Nm and the right nucleus
+Nr are defined. The set K = {a ∈ Nl ∩ Nm ∩ Nr : a ∗ b = b ∗ a, ∀b ∈ S} is called the
+center of S. Each of these four structures is a field, and a finite semifield is a left
+vector space over its left nucleus and a two-sided vector space over its center [17].
+Here, K is isomorphic to the kernel of the translation plane π(S).
+For any element b of the semifield S with center K, the map φb : x ∈ S �→ xb ∈ S is
+a linear map when S is considered over its left nucleus Nl. It turns out that the set
+CS = {φb : b ∈ S} is a K-vector subspace of the vector space of the Nl-linear maps
+of S. Since S is finite, we may assume K = Fq, Nl = Fqn and S is an t-dimensional
+left vector space over Fqn, for some positive integers n and t.
+Under the previous indentification, the set CS satisfies the following properties: (i)
+CS has qnt elements; (ii) CS contains the zero and the identity maps; (iii) A − B is
+non-singular for all distinct A, B ∈ CS. A set of linear maps of V (t, qn) satisfying
+the above properties is called a spread set of V (t, qn).
+A (t−1)-spread of the (r −1)-dimensional projective space PG(r −1, q) over Fq is
+a set S of (t−1)-dimensional projective subspaces such that every point is contained
+in exactly one subspace of S. It is known that a (t−1)-spread of PG(r −1, q) exists
+if and only if t divides r [17].
+Let C be a spread set of V (t, qn) = F t
+qn. In PG(2t − 1, qn) consider the subspaces
+Sτ = {((x1, . . . , xt)τ, x1, . . . , xt) : xi ∈ Fqn},
+for all τ ∈ C. Then, the set S = {Sτ : τ ∈ C}∪{S∞}, with S∞ = {(x1, . . . , xt, 0, . . . , 0) :
+xi ∈ Fqn} forms a (t − 1)-spread of PG(2t − 1, qn).
+Conversely, let S be a (t − 1)-spread of PG(2t − 1, qn). Then, it is possible to
+choose homogeneus coordinates in PG(2t − 1, qn) such that there is a spread set C
+of V (t, qn) from which S is constructed as above. Thanks to the Andr´e/Bruck-Bose
+construction, the spread S defines a translation plane Π(S) [3, 12, 13]. If the set C
+is closed under the sum, then there is a (finite) semifield S that coordinatizes Π(S)
+such that C = CS; the left nucleus of S is Fqn and S can be viewed as a t-dimensional
+left vector space over Fqn [17]. In addition, if Fq is the largest subfield K of Fqn
+such that C is a K-vector subspace of the vector space of the Fqn-linear maps of
+V (t, qn), the center of S is Fq. Therefore, there exists a canonical correspondence
+between translation planes coordinatized over a semifield S with dimension t over its
+left nucleus Fqn and center Fq, and the (t−1)-spreads of PG(2t−1, qn) arising from
+a spread set of V (t, qn), that is closed under the sum. Moreover, it is well-known
+that the resulting plane is Desarguesian if and only if S is a Desarguesian spread
+[13].
+Buekenhout [15], and Metz [33] (by refining Buekenhout’s idea), constructed uni-
+tals in any translation planes with dimension at most two over their kernel by using
+the Andr´e/Bruck-Bose representation of such planes. These unitals are cones of
+4
+
+PG(4, q) projecting an ovoid in a 3-dimensional subspace of PG(4, q) from a point
+at infinity. These unitals are called Buekenhout-Metz unitals. Since classical unitals
+can be obtained in this way, they fall in the class of Buekenhout-Metz unitals which,
+so far, are the only known unitals of PG(2, q2).
+Many other authors have used the above representation of PG(2, qn) in PG(2n, q)
+to study objects in the Desarguesian plane in order to determine whether this higher
+dimensional representation provides additional information about those objects in
+the plane. In particular, the projective plane PG(2, q4), modelled in PG(8, q), has
+been considered in [7] to study the representation of classical unitals, and the rep-
+resentation of PG(2, q2m) in PG(4m, q), for m > 1, have been considered to study
+other geometric objects of the plane; see [31, 32, 38, 40] just to cite some.
+A blocking set in a projective plane π is a set of points such that every line of π
+has a non-empty intersection with the set. A blocking set is said to be minimal if
+through any of its points there is a line of π intersecting it precisely in that point.
+In the paper [14], Bruen and Thas proved that, when the order of the projective
+plane is a square, say n2, then the size of a minimal blocking set is bounded by
+n3 + 1. This size is reached if and only if the minimal blocking set is a unital.
+In [31] the following geometric setting was introduced to construct large minimal
+blocking sets of PG(2, q2m) from cones in its Andr´e/Bruck-Bose representation in
+PG(4m, q). Let z be a fixed element of a (2m−1)-spread S of Σ∞ and V an (m−1)-
+dimensional subspace of z. Let Γ be a (3m−1)-dimensional subspace of Σ∞ disjoint
+from V. For every x ∈ S, x ̸= z, let I(x) be the (2m − 1)-dimensional subspace
+⟨x, V⟩∩Γ. We denote by I(V) the set of all the subspaces I(x), x ∈ S. Let Γ′ be an
+affine 3m-dimensional subspace of PG(4m, q) through Γ, and denote by F(V) the
+set of all affine 2m-dimensional subspaces of Γ′ containing an element of I(V).
+Let F be a family of 2m-dimensional subspaces of Γ′. An F-blocking set of Γ′ is
+a set B of affine points such that every element of F has a non-empty intersection
+with B. The blocking set B is said to be minimal if through any point of B there is
+an element in F intersecting B precisely in that point.
+By keeping the above geometric setting in mind, the following result, which is a
+sharpening of Corollary 3.3 in [31], is crucial for our succeeding considerations.
+Proposition 2.1. Let B be a set of affine points of Γ′ and
+B∗ =
+�
+P ∈B
+⟨V, P⟩ ∪ {z}.
+(1)
+If B is a minimal F(V)-blocking set, then B∗ is a minimal blocking set of size |B∗| =
+qm|B| + 1 in the translation plane Π(S).
+Proof. Construction 2 in [31] works perfectly well under the milder hypothesis that
+S is any (2m − 1)-spread of Σ∞. The details are left to the reader.
+5
+
+By combining the above result of Bruen and Thas with Proposition 2.1, we get
+the following theorem.
+Theorem 2.2. Let B be a minimal F(V)-blocking set of size q2m. Then, the cone
+B∗ defined in Proposition 2.1 is a unital in Π(S).
+If S is a Desarguesian (2m−1)-spread of Σ∞, then there is a unique Desarguesian
+(m−1)-spread, say T , that fills every element of S, i.e., T induces a (m−1)-spread
+in each spread element of S [20]. The following result gives a characterization of
+Buekenhout-Metz unitals as cones in PG(4m, q).
+Proposition 2.3. [31] Let S be a Desarguesian (2m − 1)-spread of Σ∞ and B a
+minimal F(V)-blocking set of size q2m. Then, the cone B∗ is a Buekenhout-Metz
+unital in PG(2, q2m) if and only if V is an element of the spread T .
+3
+Unitals from eggs
+An egg in PG(4m − 1, q) is a set E of q2m + 1 pairwise disjoint (m − 1)−dimensional
+subspaces such that any three egg elements span a (3m−1)−dimensional subspace.
+When m = 1, this definition recovers indeed the notion of ovoid in PG(3, q). There-
+fore, since the notion of an egg, introduced by J.A. Thas in [41], generalizes that of
+an ovoid, it make sense to investigate whether it is possible to mimic Buekenhout’s
+construction to get unitals in translation planes with dimension over their kernel
+greater than two, by using eggs. Apart from the so-called elementary eggs, which
+are obtained by applying the field reduction to an ovoid in PG(3, qm), there are few
+other known examples of eggs, namely, the Kantor-Knuth eggs, the Cohen-Ganley
+eggs and the (sporadic) Penttila-Williams egg; see [27] for an explicit description of
+these objects.
+Let E be an egg in PG(4m − 1, q). For every egg element E there exists a unique
+(3m − 1)-dimensional subspace, denoted by E∗, containing E and disjoint from any
+other egg element; it is called the tangent space of E at E. Therefore, the egg E
+defines an egg in the dual space of PG(4m − 1, q), called the dual egg of E and
+denoted by ED.
+The following result is a corollary of Theorem 2.2.
+Theorem 3.1. Let E be an egg in PG(4m − 1, q), and E∞ a fixed egg element. Let
+Γ′ be a 3m-dimensional subsubspace of PG(4m − 1, q) containing the tangent space
+E∗
+∞ at E∞. In Γ′ we consider the sets:
+BE = {E ∩ Γ′ : E ∈ E, E ̸= E∞}
+and
+IE = {E∗ ∩ E∗
+∞ : E ∈ E, E ̸= E∞}.
+6
+
+Let FE be the family of all affine 2m-dimensional subspaces of Γ′ containing an
+element of IE, and assume that BE is a minimal FE-blocking set.
+Embed Γ′ in PG(4m, q) in such a way that E∗
+∞ is a subspace of the hyperplane at
+infinity Σ∞ of PG(4m, q), and Γ′ is an affine subspace.
+If there exist a (2m − 1)-spread S of Σ∞ and a (m − 1)-dimensional subspace V
+disjoint from E∗
+∞ and contained in a spread element z such that IE = I(V), then
+the cone
+B∗ =
+�
+P ∈BE
+⟨P, V⟩ ∪ {z}
+is a unital in Π(S).
+Proof. Here, BE is a set of q2m points of Γ′ \ E∗
+∞, and hence it consists of affine
+points of PG(4m, q). Furthermore, every element in IE is a (2m − 1)-dimensional
+subspace of E∗
+∞. By Theorem 2.2, if FE coincides with the family F(V) previously
+defined, then B∗ is a unital in the semifield plane Π(S). Since FE consists of all affine
+2m-dimensional subspaces of Γ′ through an element of IE, we get that FE = F(V)
+if and only if IE = I(V).
+An egg is said to be good at an element E if every (3m−1)-dimensional subspace
+containing E and at least two other egg elements, contains exactly qm + 1 egg
+elements [42].
+Let K be the quadratic cone in PG(3, qm) with equation X0X1 = X2
+2. A flock of
+K is a set of qm planes partitioning the cone minus its vertex V = ⟨(0, 0, 0, 1)⟩ into
+disjoint conics. In accordance with this choice of coordinates, the planes of a flock
+of K can be written as tX0 + f(t)X1 + g(t)X2 + X3 = 0, for all t ∈ Fqm, for some
+f, g : Fqm → Fqm. We denote this flock by F(f, g). If f and g are linear over a
+subfield of Fqm, then the flock is called a semifield flock. The maximal subfield with
+this property is called the kernel of the flock.
+From now on, we assume that the kernel of a semifield flock F(f, g) is Fq. This
+implies that the f and g are Fq-linearized polynomials, i.e.
+f(t) =
+m−1
+�
+i=0
+citqi,
+g(t) =
+m−1
+�
+i=0
+bitqi,
+for some bi, ci ∈ Fqm, i = 0, . . . , m − 1.
+If a basis of Fqm over Fq is fixed, then every r-ple (x1, . . . , xr) ∈ Fr
+qm can be
+viewed as a rm-ple over Fq, which will be denoted by (x1, . . . , xr)q. In the paper
+[27] it was shown that for every semifield flock F(f, g) there corresponds an egg in
+PG(4m − 1, q) whose dual, say E, is good at an element, which can be assumed to
+7
+
+be E∞. Then, the elements and the tangent spaces of E have the following form,
+respectively:
+E(a, b) = {(t, −g(a,b)(t), −at, −bt)q : t ∈ Fqm}, for all a, b ∈ Fqm,
+E∞ = {(0, t, 0, 0)q : t ∈ Fqm},
+E∗(a, b) = {(t, h(a,b)(r, s) + g(a,b)(t), r, s)q : t, r, s ∈ Fqm}, for all a, b ∈ Fqm,
+E∗
+∞ = {(0, t, r, s)q : t, r, s ∈ Fqm},
+(2)
+with
+g(a,b)(t) = a2t +
+m−1
+�
+i=0
+(biab + cib2)1/qit1/qi,
+and
+h(a,b)(r, s) = 2ar +
+m−1
+�
+i=0
+(bi(as + br) + 2cibs)1/qi.
+Because of the expression of the polynomials g(a, b) and h(a, b), such an egg will be
+denoted by E(b, c).
+Theorem 3.2. Let E = E(b, c) be a good egg of PG(4m − 1, q), which is good at
+E∞. Then, the set BE is a minimal FE-blocking set in Γ′ = PG(3m, q) if and only
+if X2 + �m−1
+i=0 (biXY + ciY 2)1/qi + c = 0 has a solution for all c ∈ Fqm.
+Proof. Let Γ′ = {(u, t, r, s)q : u ∈ Fq and r, s, t ∈ Fqm}. It is evident that Γ′ is
+a projective space of dimension 3m over Fq and it contains E∗
+∞. By taking into
+account the general form of the elements of E = E(b, c), we get
+BE = {⟨(1, −g(a,b)(1), −a, −b)q⟩ : a, b ∈ Fqm}
+and
+IE = {I(a, b) : a, b ∈ Fqm},
+where
+I(a, b) = E∗(a, b) ∩ E∗
+∞ = {(0, h(a,b)(r, s), r, s)q : r, s ∈ Fqm}.
+(3)
+All the affine 2m-dimensional subspaces of Γ′ through an I(a, b) are determined
+by joining it with an affine point of the affine m-dimensional subspace spanned by
+E∞ and O = ⟨(1, 0, 0, 0, 0)⟩. Therefore, the elements of FE have the form
+F(a, b, c) = {(u, uc + h(a,b)(r, s), r, s)q : u ∈ Fq and r, s ∈ Fqm},
+for all a, b, c ∈ Fqm.
+A point P(x, y) = ⟨(1, −g(x,y)(1), −x, −y)q⟩ ∈ BE lies in F(a, b, c) if and only if
+−g(x,y)(1) = −h(a,b)(x, y) + c
+8
+
+or, equivalently, if and only if (x, y) is a solution of
+X2 +
+m−1
+�
+i=0
+(biXY + ciY 2)1/qi − 2aX −
+m−1
+�
+i=0
+(bi(aY + bX) + 2cibY )1/qi + c = 0.
+(4)
+We refer to the polynomial on the left-hand side of the equation as H(a,b,c)(x, y).
+Since E is an egg, for any given a, b ∈ Fqm, the intersection of the tangent space
+E∗(a, b) with Γ′ is the 2m-dimensional subspace F(a, b, c′) ∈ FE, with c′ = g(a,b)(1).
+Therefore, through the point P(a, b) ∈ BE there is the element F(a, b, c′) ∈ FE
+intersecting BE precisely at P(a, b). This implies that BE is a minimal FE-blocking
+set if and only if Eq. (4) has a solution (x, y) ∈ Fqm × Fqm for any given elements
+a, b, c ∈ Fqm.
+From [26, Lemma 1.4], for any a, b ∈ Fqm, the linear collineation
+ψa,b :
+PG(4m − 1, q)
+���→
+PG(4m − 1, q)
+⟨(u, t, r, s)q⟩
+�→
+⟨(u, t + h(a,b)(r, s) − g(a,b)(u), r − ua, s − ub)q⟩
+fixes E∞ pointwise and maps E(a′, b′) to E(a′+a, b′+b). In addition, ψa,b fixes Γ′, and
+hence BE. A straightforward, though tedious, calculation shows that ψa,b acts also on
+the set of tangent spaces by fixing E∗
+∞ setwise and mapping E∗(a′, b′) to E∗(a+a′, b+
+b′). This implies that ψa,b fixes both IE and FE setwise; in particular, F(a′, b′, c) is
+mapped to F(a + a′, b + b′, c′), with c′ = c − ga,b(1) + h(a′+a,b′+b)(a, b). This means
+that, because of the linearity of the second sum in Eq. (4), H(a′+a,b′+b,c)(x, y) = 0
+has a solution for all c ∈ Fqm if and only if H(a,b,c)(x, y) = 0 has a solution for all
+c ∈ Fqm. Therefore, BE is a minimal FE-blocking set if and only if, for a fixed pair
+(a, b) ∈ Fqm × Fqm, Eq. (4) has at least one solution (x, y) ∈ Fqm × Fqm, for all
+c ∈ Fqm. In particular, we can chose (a, b) = (0, 0) so that Eq. (4) reduces to
+X2 +
+m−1
+�
+i=0
+(biXY + ciY 2)1/qi + c = 0.
+(5)
+4
+A new unital in a Dickson commutative semi-
+field plane
+In [35], Penttila and Williams constructed an ovoid of the parabolic quadric Q(4, 35)
+in PG(4, 35), i.e., a set O of 310 + 1 points having exactly one point on each gen-
+erator of the quadric. Moreover, O is a translation ovoid, meaning that the points
+of O can be coordinatized by using functions that are additive over F3. According
+to a construction given in [28], such a translation ovoid corresponds to a semifield
+9
+
+flock of the quadratic cone in PG(3, 35), which, in turn, corresponds to a generalized
+quadrangle with parameters (310, 35), whose point-line dual is a translation gener-
+alized quadrangle. By a result of Payne and Thas [34, 8.7.1], the latter generalized
+quadrangle is isomorphic to T(E) for some egg E in PG(19, 3). By Theorem 3.4 in
+[27], the dual egg of E forms a good egg ED in PG(19, 3). Whence, via the above
+correspondences, the Penttila-Williams ovoid of Q(4, 35) gives rise to a good (dual)
+egg in PG(19, 3). In order to simplify the notation, we will refer to it as E = E(b, c)
+with b = (0, 1, 0, 0, 0), c = (0, 0, 0, −1, 0); see [27].
+According to the expressions of the polynomials g(a,b)(t) and h(a,b)(r, s) in this case,
+the egg elements of E are defined by the polynomials
+g(a,b)(t) = a2t − (b2)32t32 + (ab)34t34
+(6)
+and
+h(a,b)(r, s) = −ar + b32s32 + (br + as)34,
+(7)
+for all a, b ∈ F35.
+Let p be an odd prime and ξ a non-square in Fpm. By [17, p.241], the multiplication
+defined by
+(x, y) ∗ (a, b) = (ax + ξbαyα, bx + ay)
+with α ∈ Aut(Fpm) not the identity, turns F2
+pm into a Dickson commutative semifield
+of order p2m which we denote by D = D(pm, ξ, α). In particular, its middle nucleus
+is Nm = {(a, 0) : a ∈ Fpm}, and its left nucleus is Nl = {(a, 0) : a ∈ Fix(α)},
+coinciding with its center K.
+Now, let p = 3 and m = 5. For any pair (a, b) ∈ F2
+35, we consider the following
+map
+τ(a,b) :
+(x, y)
+�→
+(bx + ay, −ax + b32y32),
+which defines the subspaces S(a, b) = {((x, y)τ(a,b), x, y)3 : x, y ∈ F35} of PG(19, 3).
+Set S = {S(a, b) : a, b ∈ F35} ∪ {S∞}, where S∞ = {(x, y, 0, 0)3 : x, y ∈ F35}.
+Let ϕ be the linear map ϕ : (x, y) �→ (−y, x). Then, the set {ϕτ(a,b) : a, b ∈ F35}
+is precisely the spread set of F10
+3 associated with the Dickson commutative semifield
+D = D(35, −1, 32).
+It turns out that S is a 9-spread of Σ∞ = PG(19, 3) and, by [2], the translation
+plane Π(S) is isomorphic to the Dickson commutative semifield plane π(D).
+Let V = {(t, −t34, 0, 0)3 : t ∈ F35}. Then, V is contained in the spread element
+z = S∞ and it intersects trivially the subspace Γ = E∗
+∞ = {(0, t, r, s)3 : t, r, s ∈ F35}.
+We also have
+⟨S(a, b), V⟩ = {(br + as + t, −ar + b32s32 − t34, r, s)3 : t, r, s ∈ F35},
+giving
+⟨S(a, b), V⟩ ∩ Γ = {(0, −ar + b32s32 + (br + as)34, r, s)3 : r, s ∈ F35}
+10
+
+which is precisely the subspace I(a, b) defined by expression (3), with h(a,b)(r, s) as
+in (7).
+Proposition 4.1. The set BE defined by the Penttila-Williams egg E = E(b, c) is a
+minimal F(E)-blocking set.
+Proof. By taking into account Theorem 3.2, BE is a minimal FE-blocking set if and
+only if
+X2 + (XY )34 − (Y 2)32 = −c
+(8)
+has a solution for all c ∈ Fqm.
+We distinguish two cases: −c is a square in Fqm or not. If −c is a square, then
+(±√−c, 0) are solutions of Eq.
+(8); if −c is not a square, then (0, ±
+√
+c33) are
+solutions of Eq. (8).
+By Theorem 3.1, the cone
+B∗
+E = {⟨(1, c, −g(a,b)(1) − c34, −a, −b)3⟩ : a, b, c ∈ F35} ∪ {S∞},
+with g(a,b)(t) as in (6), is a unital in the translation plane Π(S).
+Consider the collineation of PG(20, 3) defined as ϕ : ⟨(u, v, t, r, s)3⟩ �→ ⟨(u, −t, v, r, s)3⟩.
+Then, Π(S)ϕ represents the Dickson commutative semifield plane π(D). It turns out
+that the set
+U = {(g(a,b)(1) + c34, c, −a, −b) : a, b, c ∈ F35} ∪ {(∞)}
+is a unital in π(D). Note that U cannot be a Buekenhout-Metz unital since π(D) is
+a 10-dimensional translation plane over its kernel F3. On the other hand, as π(D)
+admits unitary polarities [21], U might be a polar unital. The following result shows
+that this is not the case.
+Theorem 4.2. The unital U is not a polar unital in π(D).
+Proof. Since the tangent space at the egg element E(0, 0) is E∗(0, 0), the tangent
+line of Π(S) at the point O = ⟨(1, 0, 0, 0, 0)3⟩ ∈ B∗
+E is the subspace spanned by
+S(0, 0) and O. Then, the tangent line of π(D) at (0, 0) ∈ U is [0, 0].
+From [24, Theorem 2.1], any unitary polarity of π(D) mapping (0, 0) to [0, 0] is
+given by
+ρa :
+(x1, x2, y1, y2)
+↔
+[ax1, −ax2, −y1, y2],
+(m1, m2)
+↔
+(a−1m1, −a−1m2)
+(∞)
+↔
+[∞].
+for some non-zero a ∈ F35.
+The unital U is a polar unital with respect to ρa, for some a ∈ F35, if and only if
+each of its points is an absolute point. Straightforward calculations show that the
+point (1, 1, 0, 0) ∈ U is not incident with ρa(1, 1, 0, 0) = [a, −a, 0, 0] for all non-zero
+a ∈ F35, showing that U is not a polar unital.
+11
+
+References
+[1] V. Abatangelo, B. Larato and L.A. Rosati, Unitals in planes derived from
+Hughes planes, J. Combin. Inform. System Sci. 15 (1990), 151–155.
+[2] A.A. Albert, Finite division algebras and finite planes, Proc. Sympos. Appl.
+Math. 10 (1960), 53–70.
+[3] J. Andr´e, ¨Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe,
+Math. Z. 60 (1954), 156–186.
+[4] B. Bagchi and S. Bagchi, Designs from pairs of finite fields, I. A cyclic unital
+U(6) and other regular Steiner 2-designs, J. Combin. Theory Ser. A 52 (1989),
+51–61.
+[5] A. Barlotti and G. Lunardon, Una classe di unitals nei ∆-piani, Riv. Mat. Univ.
+Parma (4) 5 (1979), 781–785.
+[6] S.G. Barwick, Unitals in the Hall plane, J. Geom. 58 (1997), 26–42.
+[7] S.G. Barwick, L.R.A. Casse and C.T. Quinn, The Andr´e/Bruck and Bose rep-
+resentation in PG(2h, q): unitals and Baer subplanes, Bull. Belg. Math. Soc.
+Simon Stevin 7 (2000), 173–197.
+[8] S. Barwick and G. Ebert, Unitals in projective planes, Springer, New York,
+2008.
+[9] M. Biliotti, V. Jha and N.L. Johnson, Foundations of Translation Planes, Pure
+and Applied Mathematics, Vol. 243, Marcel Dekker, New York, Basel, 2001.
+[10] A.E. Brouwer, Some unitals on 28 points and their embeddings in projective
+planes of order 9, Geometries and Groups, Springer Lecture Notes in Mathe-
+matics, 893 (1981), 183–188.
+[11] A.E. Brouwer, A unital in the Hughes plane of order nine, Discrete Math. 27
+(1989), 55–56.
+[12] R.H. Bruck and R.C. Bose, The construction of translation planes from projec-
+tive spaces, J. Algebra 1 (1964), 85–102.
+[13] R.H. Bruck and R.C. Bose, Linear representation of projective planes in pro-
+jective spaces, J. Algebra 4 (1966), 117–172.
+[14] A.A. Bruen and J.A. Thas, Blocking sets, Geometriae Dedicata 6 (1977), 193–
+203.
+[15] F. Buekenhout, Existence of unitals in finite translation planes of order q2 with
+a kernel of order q, Geometriae Dedicata 5 (1976), 189–194.
+12
+
+[16] J. Cannon and C. Playoust, An Introduction to MAGMA, University of Sydney
+Press, 193.
+[17] P. Dembowski, Finite geometries, Springer-Verlag, New York, 1968.
+[18] M. J. de Resmini and N. Hamilton, Hyperovals and unitals in Figueroa planes,
+European J. Combin. 19 (1998), 215–220.
+[19] J. Dover, A family of non-Buekenhout unitals in the Hall planes, in “Mostly
+finite geometries (Iowa City, IA, 1996)”, Lecture Notes in Pure and Appl. Math.
+190 (1997), 197–205.
+[20] K. Drudge, On the orbits of Singer groups and their subgroups, Electron. J.
+Combin. 9 (2002), 10 pp. (electronic).
+[21] M.J. Ganley, A class of unitary block designs, Math. Z. 128 (1972), 34–42.
+[22] K. Gr¨uning, Das kleinste Ree-Unital, Arch. Math. 46 (1986), 473–480.
+[23] K. Gr¨uning, A class of unitals of order q which can be embedded in two different
+planes of order q2, J. Geom. 29 (1987), 61–77.
+[24] A.M.W. Hui, H.F. Law, Y.K. Tai and P.P.W. Wong, A note on unitary polarities
+in finite Dickson semifield planes, J. Geom. 106 (2015), 175–183.
+[25] W.M. Kantor, Some generalized quadrangles with parameters q2, q, Math. Z.
+192 (1986), 45–50.
+[26] M. Lavrauw, Characterizations and properties of good eggs in PG(4m − 1, q),
+q odd, Discrete Math. 301 (2005), 106–116.
+[27] M. Lavrauw and T. Penttila, On eggs and translation generalised quadrangles,
+J. Combin. Theory Ser. A 96 (2001), 303–315.
+[28] G. Lunardon, Flocks, ovoids of Q(4, q) and designs, Geom. Dedicata 66 (1997),
+163–173.
+[29] G. Lunardon, Normal spreads, Geom. Dedicata 75 (1999), 245–261.
+[30] R. Mathon, Constructions for cyclic Steiner 2-designs, Ann. Disc. Math. 34
+(1987), 353–362.
+[31] F. Mazzocca and O. Polverino, Blocking sets in PG(2, qn) from cones of
+PG(2n, q), J. Algebraic Combin. 24 (2006), 61–81.
+[32] F. Mazzocca, O. Polverino and L. Storme, Blocking sets in PG(r, qn), Des.
+Codes Cryptogr. 44 (2007), 97–113.
+[33] R. Metz, On a class of unitals, Geom. Dedicata 8 (1979), 125–126.
+13
+
+[34] S.E Payne and J.A. Thas, Finite generalized quadrangles, Second edition, in:
+EMS Series of Lectures in Mathematics, Z¨urich, 2009.
+[35] T. Penttila and B. Williams, Ovoids in parabolic spaces, Geom. Dedicata, 82
+(2000), 1–19.
+[36] G. Rinaldi, Hyperbolic unitals in the Hall planes, J. Geom. 54 (1995), 148–154.
+[37] L.A. Rosati, Disegni unitari nei piani di Hughes, Geom. Dedicata 27 (1988),
+295–299.
+[38] S. Rottey,
+J. Sheekey and G. Van de Voorde,
+Subgeometries in the
+Andr´e/Bruck-Bose representation, Finite Fields Appl. 35 (2015), 115–138.
+[39] M. Seib, Unit¨are Polarit¨aten endlicher projektiver Ebenen, Arch. Math. 21
+(1970), 103–112.
+[40] T. Sz˝onyi, A. Cossidente, A. G´acs, C. Mengyn, A. Siciliano and Z. Weiner, On
+large minimal blocking sets in PG(2, q), J. Combin. Des. 13 (2005), 25–41.
+[41] J.A. Thas, Geometric characterization of the [n − 1]-ovaloids of the projective
+space PG(4n − 1, q), Simon Stevin 47 (1974), 97–106.
+[42] J.A. Thas, Generalized quadrangles of order (s, s2), II, J. Combin. Theory Ser.
+A 79 (1997), 223–254.
+14
+

2NE1T4oBgHgl3EQfSAMg/content/tmp_files/load_file.txt

@@ -0,0 +1,491 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf,len=490
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='03059v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='CO]  8 Jan 2023  Eggs in finite projective spaces  and unitals in translation planes  Giusy Monzillo∗  monzillo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='giusy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='com  Dipartimento di Matematica, Informatica ed Economia  Universit`a degli Studi della Basilicata  Viale dell’Ateneo Lucano 10  85100 Potenza, Italy  Tim Penttila  penttila86@msn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='com  School of Mathematical Sciences  The University of Adelaide  Adelaide, South Australia  5005 Australia  Alessandro Siciliano∗  alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='siciliano@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='it  Dipartimento di Matematica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Informatica ed Economia  Universit`a degli Studi della Basilicata  Viale dell’Ateneo Lucano 10  85100 Potenza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Italy  Abstract  Inspired by the connection between ovoids and unitals arising from the  Buekenhout construction in the Andr´e/Bruck-Bose representation of trans-  lation planes of dimension at most two over their kernel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' and since eggs of  PG(4m − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' m ≥ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' are a generalization of ovoids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' we explore the rela-  tion between eggs and unitals in translation planes of higher dimension over  their kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By investigating such a relationship, we construct a unital in the  Dickson semifield plane of order 310, which is represented in PG(20, 3) by a  ∗The research was supported by the Italian National Group for Algebraic and Geometric Struc-  tures and their Applications (GNSAGA-INdAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  1  cone whose base is a set of points constructed from the dual of the Penttila-  Williams egg in PG(19, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This unital is not polar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' so, up to the knowledge  of the authors, it seems to be a new unital in such a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Keywords: Unital, Blocking set, Egg, Projective plane  1  Introduction  Field reduction has become a theme of finite geometry which turned out very fruit-  ful in the last few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Given a construction of an interesting object from a  configuration in a vector space of dimension r over a field of order qn, the question  is raised as to which objects give rise similar configurations in a vector space of  dimension rn over a field of order q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  The Buekenhout-Metz construction of unitals in finite translation planes [15, 33]  (which gives all known unitals in Desarguesian planes) can be recontextualized in  this fashion, with cones projecting an ovoid as a base in the Andr´e/Bruck-Bose  representation of such planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  It has long been known [14] that unitals are extremal in size among minimal  blocking sets (at the other end than that most studied - large rather than small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The  observation of Lunardon [29] at the turn of the millennium that changing the field  gave access to many more subspaces, some of which were blocking sets, transformed  the theory of blocking sets in the process giving rise to the idea of linear sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thus,  the idea of Buekenhout and Metz was taken by Sz˝onyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' [40] and, later, by  Mazzocca and Polverino [31] to provide further minimal blocking sets, using cones  rather than subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  For the construction by Tits of generalized quadrangles from (ovals and) ovoids,  the configurations that arise by applying a field reduction are eggs, and the similar  objects are translation generalized quadrangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thus, changing the field for ovoids  and studying eggs gave the possibility of new translation generalized quadrangles,  first realized in work of Kantor [25] from three decades past;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' the result of field  reduction applied to the concept of an ovoid is an egg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Motivated by the relationship between ovoids and unitals via the Buekenhout-  Metz construction, and since eggs are generalization of ovoids, we explore possible  relationships between eggs and unitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Putting all the above ideas together, in this  paper we construct a unital in the Dickson semifield plane of order 310, which is  represented in PG(20, 3) by a cone whose base is a set of points constructed from  the dual of the Penttila-Williams egg in PG(19, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This unital does not arise from  a polarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' so it is a new unital, up to the knowledge of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  While field reduction is usually thought of in a projective setting, algebraic di-  mensions are more amenable to an introductory discussion of it, so we will take a  vector space approach along all the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  2  2  Definitions and preliminary results  A unital in a finite projective plane π of order n2 is a set U of n3 + 1 points such  that every line of π meets U in 1 or n + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Therefore, U is equipped with a  family of subsets, each of size n + 1, such that every pair of distinct points of U is  contained in exactly one subset of the family;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' such subsets are usually called blocks,  and U turns out to be a 2-(n3 + 1, n + 1, 1) design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  In a computer search, Brouwer [10, 11] found a large number of mutually non-  isomorphic 2-(28, 4, 1) designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Only a few of these are embeddable in a projective  plane of order 9 as unitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' One of the examples has been generalized by Gr¨uning  [23], who constructed a unital of order q for any odd prime power in both the Hall  plane and dual Hall plane of order q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' An infinite family of non-Buekenhout unitals  in the Hall planes of order q2 have been constructed in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Other infinite families  of unitals in various square order planes are known to exist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' [1], [5], [6], [18],  [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The only known 2-(n3 + 1, n + 1, 1) design with n not a prime power is  the one found in [4] and [30] where n = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' For more on 2-(n3 + 1, n + 1, 1) designs  embeddable as unitals in projective plane, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  In the Desarguesian projective plane PG(2, q2), a unital can arise from a unitary  polarity: the points of the unital are the absolute points, and the blocks are the  intersections of the non-absolute lines of the polarity with U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' These unitals are  called classical or Hermitian unitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By a result of Seib [39], the absolute points  of a unitary polarity in any square order projective plane form the point-set of a  unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Such unitals are called polar unitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' So, classical unitals of PG(2, q2) are  examples of polar unitals, and Ganley [21] showed that polar unitals exist in any  Dickson commutative semifield plane of odd order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  A finite semifield is a finite set S with two binary operations + and ∗, such that  (S, +) is an abelian group and (S \\ {0}, ∗) is a loop such that both distributive laws  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let π(S) be the point-line geometry whose points are the elements in S × S and  in {(m) : m ∈ S ∪ {∞}}, and the lines are the sets  [m, k] = {(y, x) ∈ S × S : m ∗ x + y = k} ∪ {(m)},  [z] = {(y, z) : y ∈ S} ∪ {(∞)}  and  [∞] = {(m) : m ∈ S} ∪ {(∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  with m, k, z ∈ S, and ∞ a symbol not in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  It turns out that π(S) is a translation plane which is called the semifield plane  coordinatized by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' We refer to [9] and [17] for basic information on semifields and  translation planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  3  For any semifield S, the subset Nl = {a ∈ S : a ∗ (x ∗ y) = (a ∗ x) ∗ y, ∀x, y ∈ S} is  called the left nucleus of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Similarly, the middle nucleus Nm and the right nucleus  Nr are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The set K = {a ∈ Nl ∩ Nm ∩ Nr : a ∗ b = b ∗ a, ∀b ∈ S} is called the  center of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Each of these four structures is a field, and a finite semifield is a left  vector space over its left nucleus and a two-sided vector space over its center [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Here, K is isomorphic to the kernel of the translation plane π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  For any element b of the semifield S with center K, the map φb : x ∈ S �→ xb ∈ S is  a linear map when S is considered over its left nucleus Nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' It turns out that the set  CS = {φb : b ∈ S} is a K-vector subspace of the vector space of the Nl-linear maps  of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Since S is finite, we may assume K = Fq, Nl = Fqn and S is an t-dimensional  left vector space over Fqn, for some positive integers n and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Under the previous indentification, the set CS satisfies the following properties: (i)  CS has qnt elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (ii) CS contains the zero and the identity maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (iii) A − B is  non-singular for all distinct A, B ∈ CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A set of linear maps of V (t, qn) satisfying  the above properties is called a spread set of V (t, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  A (t−1)-spread of the (r −1)-dimensional projective space PG(r −1, q) over Fq is  a set S of (t−1)-dimensional projective subspaces such that every point is contained  in exactly one subspace of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' It is known that a (t−1)-spread of PG(r −1, q) exists  if and only if t divides r [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let C be a spread set of V (t, qn) = F t  qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In PG(2t − 1, qn) consider the subspaces  Sτ = {((x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , xt)τ, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , xt) : xi ∈ Fqn},  for all τ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the set S = {Sτ : τ ∈ C}∪{S∞}, with S∞ = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , xt, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , 0) :  xi ∈ Fqn} forms a (t − 1)-spread of PG(2t − 1, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Conversely, let S be a (t − 1)-spread of PG(2t − 1, qn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, it is possible to  choose homogeneus coordinates in PG(2t − 1, qn) such that there is a spread set C  of V (t, qn) from which S is constructed as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thanks to the Andr´e/Bruck-Bose  construction, the spread S defines a translation plane Π(S) [3, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' If the set C  is closed under the sum, then there is a (finite) semifield S that coordinatizes Π(S)  such that C = CS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' the left nucleus of S is Fqn and S can be viewed as a t-dimensional  left vector space over Fqn [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In addition, if Fq is the largest subfield K of Fqn  such that C is a K-vector subspace of the vector space of the Fqn-linear maps of  V (t, qn), the center of S is Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Therefore, there exists a canonical correspondence  between translation planes coordinatized over a semifield S with dimension t over its  left nucleus Fqn and center Fq, and the (t−1)-spreads of PG(2t−1, qn) arising from  a spread set of V (t, qn), that is closed under the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Moreover, it is well-known  that the resulting plane is Desarguesian if and only if S is a Desarguesian spread  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Buekenhout [15], and Metz [33] (by refining Buekenhout’s idea), constructed uni-  tals in any translation planes with dimension at most two over their kernel by using  the Andr´e/Bruck-Bose representation of such planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' These unitals are cones of  4  PG(4, q) projecting an ovoid in a 3-dimensional subspace of PG(4, q) from a point  at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' These unitals are called Buekenhout-Metz unitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Since classical unitals  can be obtained in this way, they fall in the class of Buekenhout-Metz unitals which,  so far, are the only known unitals of PG(2, q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Many other authors have used the above representation of PG(2, qn) in PG(2n, q)  to study objects in the Desarguesian plane in order to determine whether this higher  dimensional representation provides additional information about those objects in  the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In particular, the projective plane PG(2, q4), modelled in PG(8, q), has  been considered in [7] to study the representation of classical unitals, and the rep-  resentation of PG(2, q2m) in PG(4m, q), for m > 1, have been considered to study  other geometric objects of the plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' see [31, 32, 38, 40] just to cite some.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  A blocking set in a projective plane π is a set of points such that every line of π  has a non-empty intersection with the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A blocking set is said to be minimal if  through any of its points there is a line of π intersecting it precisely in that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  In the paper [14], Bruen and Thas proved that, when the order of the projective  plane is a square, say n2, then the size of a minimal blocking set is bounded by  n3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This size is reached if and only if the minimal blocking set is a unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  In [31] the following geometric setting was introduced to construct large minimal  blocking sets of PG(2, q2m) from cones in its Andr´e/Bruck-Bose representation in  PG(4m, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let z be a fixed element of a (2m−1)-spread S of Σ∞ and V an (m−1)-  dimensional subspace of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let Γ be a (3m−1)-dimensional subspace of Σ∞ disjoint  from V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' For every x ∈ S, x ̸= z, let I(x) be the (2m − 1)-dimensional subspace  ⟨x, V⟩∩Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' We denote by I(V) the set of all the subspaces I(x), x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let Γ′ be an  affine 3m-dimensional subspace of PG(4m, q) through Γ, and denote by F(V) the  set of all affine 2m-dimensional subspaces of Γ′ containing an element of I(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let F be a family of 2m-dimensional subspaces of Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' An F-blocking set of Γ′ is  a set B of affine points such that every element of F has a non-empty intersection  with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The blocking set B is said to be minimal if through any point of B there is  an element in F intersecting B precisely in that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  By keeping the above geometric setting in mind, the following result, which is a  sharpening of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='3 in [31], is crucial for our succeeding considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let B be a set of affine points of Γ′ and  B∗ =  �  P ∈B  ⟨V, P⟩ ∪ {z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  (1)  If B is a minimal F(V)-blocking set, then B∗ is a minimal blocking set of size |B∗| =  qm|B| + 1 in the translation plane Π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Construction 2 in [31] works perfectly well under the milder hypothesis that  S is any (2m − 1)-spread of Σ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  5  By combining the above result of Bruen and Thas with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1, we get  the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let B be a minimal F(V)-blocking set of size q2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the cone  B∗ defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1 is a unital in Π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  If S is a Desarguesian (2m−1)-spread of Σ∞, then there is a unique Desarguesian  (m−1)-spread, say T , that fills every element of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=', T induces a (m−1)-spread  in each spread element of S [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The following result gives a characterization of  Buekenhout-Metz unitals as cones in PG(4m, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' [31] Let S be a Desarguesian (2m − 1)-spread of Σ∞ and B a  minimal F(V)-blocking set of size q2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the cone B∗ is a Buekenhout-Metz  unital in PG(2, q2m) if and only if V is an element of the spread T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  3  Unitals from eggs  An egg in PG(4m − 1, q) is a set E of q2m + 1 pairwise disjoint (m − 1)−dimensional  subspaces such that any three egg elements span a (3m−1)−dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  When m = 1, this definition recovers indeed the notion of ovoid in PG(3, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' There-  fore, since the notion of an egg, introduced by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thas in [41], generalizes that of  an ovoid, it make sense to investigate whether it is possible to mimic Buekenhout’s  construction to get unitals in translation planes with dimension over their kernel  greater than two, by using eggs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Apart from the so-called elementary eggs, which  are obtained by applying the field reduction to an ovoid in PG(3, qm), there are few  other known examples of eggs, namely, the Kantor-Knuth eggs, the Cohen-Ganley  eggs and the (sporadic) Penttila-Williams egg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' see [27] for an explicit description of  these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let E be an egg in PG(4m − 1, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' For every egg element E there exists a unique  (3m − 1)-dimensional subspace, denoted by E∗, containing E and disjoint from any  other egg element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' it is called the tangent space of E at E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Therefore, the egg E  defines an egg in the dual space of PG(4m − 1, q), called the dual egg of E and  denoted by ED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  The following result is a corollary of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let E be an egg in PG(4m − 1, q), and E∞ a fixed egg element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let  Γ′ be a 3m-dimensional subsubspace of PG(4m − 1, q) containing the tangent space  E∗  ∞ at E∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In Γ′ we consider the sets:  BE = {E ∩ Γ′ : E ∈ E, E ̸= E∞}  and  IE = {E∗ ∩ E∗  ∞ : E ∈ E, E ̸= E∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  6  Let FE be the family of all affine 2m-dimensional subspaces of Γ′ containing an  element of IE, and assume that BE is a minimal FE-blocking set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Embed Γ′ in PG(4m, q) in such a way that E∗  ∞ is a subspace of the hyperplane at  infinity Σ∞ of PG(4m, q), and Γ′ is an affine subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  If there exist a (2m − 1)-spread S of Σ∞ and a (m − 1)-dimensional subspace V  disjoint from E∗  ∞ and contained in a spread element z such that IE = I(V), then  the cone  B∗ =  �  P ∈BE  ⟨P, V⟩ ∪ {z}  is a unital in Π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Here, BE is a set of q2m points of Γ′ \\ E∗  ∞, and hence it consists of affine  points of PG(4m, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Furthermore, every element in IE is a (2m − 1)-dimensional  subspace of E∗  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='2, if FE coincides with the family F(V) previously  defined, then B∗ is a unital in the semifield plane Π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Since FE consists of all affine  2m-dimensional subspaces of Γ′ through an element of IE, we get that FE = F(V)  if and only if IE = I(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  An egg is said to be good at an element E if every (3m−1)-dimensional subspace  containing E and at least two other egg elements, contains exactly qm + 1 egg  elements [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let K be the quadratic cone in PG(3, qm) with equation X0X1 = X2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A flock of  K is a set of qm planes partitioning the cone minus its vertex V = ⟨(0, 0, 0, 1)⟩ into  disjoint conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In accordance with this choice of coordinates, the planes of a flock  of K can be written as tX0 + f(t)X1 + g(t)X2 + X3 = 0, for all t ∈ Fqm, for some  f, g : Fqm → Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' We denote this flock by F(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' If f and g are linear over a  subfield of Fqm, then the flock is called a semifield flock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The maximal subfield with  this property is called the kernel of the flock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  From now on, we assume that the kernel of a semifield flock F(f, g) is Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This  implies that the f and g are Fq-linearized polynomials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  f(t) =  m−1  �  i=0  citqi,  g(t) =  m−1  �  i=0  bitqi,  for some bi, ci ∈ Fqm, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  If a basis of Fqm over Fq is fixed, then every r-ple (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , xr) ∈ Fr  qm can be  viewed as a rm-ple over Fq, which will be denoted by (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' , xr)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In the paper  [27] it was shown that for every semifield flock F(f, g) there corresponds an egg in  PG(4m − 1, q) whose dual, say E, is good at an element, which can be assumed to  7  be E∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the elements and the tangent spaces of E have the following form,  respectively:  E(a, b) = {(t, −g(a,b)(t), −at, −bt)q : t ∈ Fqm}, for all a, b ∈ Fqm,  E∞ = {(0, t, 0, 0)q : t ∈ Fqm},  E∗(a, b) = {(t, h(a,b)(r, s) + g(a,b)(t), r, s)q : t, r, s ∈ Fqm}, for all a, b ∈ Fqm,  E∗  ∞ = {(0, t, r, s)q : t, r, s ∈ Fqm},  (2)  with  g(a,b)(t) = a2t +  m−1  �  i=0  (biab + cib2)1/qit1/qi,  and  h(a,b)(r, s) = 2ar +  m−1  �  i=0  (bi(as + br) + 2cibs)1/qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Because of the expression of the polynomials g(a, b) and h(a, b), such an egg will be  denoted by E(b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let E = E(b, c) be a good egg of PG(4m − 1, q), which is good at  E∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the set BE is a minimal FE-blocking set in Γ′ = PG(3m, q) if and only  if X2 + �m−1  i=0 (biXY + ciY 2)1/qi + c = 0 has a solution for all c ∈ Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Let Γ′ = {(u, t, r, s)q : u ∈ Fq and r, s, t ∈ Fqm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' It is evident that Γ′ is  a projective space of dimension 3m over Fq and it contains E∗  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By taking into  account the general form of the elements of E = E(b, c), we get  BE = {⟨(1, −g(a,b)(1), −a, −b)q⟩ : a, b ∈ Fqm}  and  IE = {I(a, b) : a, b ∈ Fqm},  where  I(a, b) = E∗(a, b) ∩ E∗  ∞ = {(0, h(a,b)(r, s), r, s)q : r, s ∈ Fqm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  (3)  All the affine 2m-dimensional subspaces of Γ′ through an I(a, b) are determined  by joining it with an affine point of the affine m-dimensional subspace spanned by  E∞ and O = ⟨(1, 0, 0, 0, 0)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Therefore, the elements of FE have the form  F(a, b, c) = {(u, uc + h(a,b)(r, s), r, s)q : u ∈ Fq and r, s ∈ Fqm},  for all a, b, c ∈ Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  A point P(x, y) = ⟨(1, −g(x,y)(1), −x, −y)q⟩ ∈ BE lies in F(a, b, c) if and only if  −g(x,y)(1) = −h(a,b)(x, y) + c  8  or, equivalently, if and only if (x, y) is a solution of  X2 +  m−1  �  i=0  (biXY + ciY 2)1/qi − 2aX −  m−1  �  i=0  (bi(aY + bX) + 2cibY )1/qi + c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  (4)  We refer to the polynomial on the left-hand side of the equation as H(a,b,c)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Since E is an egg, for any given a, b ∈ Fqm, the intersection of the tangent space  E∗(a, b) with Γ′ is the 2m-dimensional subspace F(a, b, c′) ∈ FE, with c′ = g(a,b)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Therefore, through the point P(a, b) ∈ BE there is the element F(a, b, c′) ∈ FE  intersecting BE precisely at P(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This implies that BE is a minimal FE-blocking  set if and only if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (4) has a solution (x, y) ∈ Fqm × Fqm for any given elements  a, b, c ∈ Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  From [26, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='4], for any a, b ∈ Fqm, the linear collineation  ψa,b :  PG(4m − 1, q)  −→  PG(4m − 1, q)  ⟨(u, t, r, s)q⟩  �→  ⟨(u, t + h(a,b)(r, s) − g(a,b)(u), r − ua, s − ub)q⟩  fixes E∞ pointwise and maps E(a′, b′) to E(a′+a, b′+b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In addition, ψa,b fixes Γ′, and  hence BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A straightforward, though tedious, calculation shows that ψa,b acts also on  the set of tangent spaces by fixing E∗  ∞ setwise and mapping E∗(a′, b′) to E∗(a+a′, b+  b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This implies that ψa,b fixes both IE and FE setwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' in particular, F(a′, b′, c) is  mapped to F(a + a′, b + b′, c′), with c′ = c − ga,b(1) + h(a′+a,b′+b)(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' This means  that, because of the linearity of the second sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (4), H(a′+a,b′+b,c)(x, y) = 0  has a solution for all c ∈ Fqm if and only if H(a,b,c)(x, y) = 0 has a solution for all  c ∈ Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Therefore, BE is a minimal FE-blocking set if and only if, for a fixed pair  (a, b) ∈ Fqm × Fqm, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (4) has at least one solution (x, y) ∈ Fqm × Fqm, for all  c ∈ Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In particular, we can chose (a, b) = (0, 0) so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (4) reduces to  X2 +  m−1  �  i=0  (biXY + ciY 2)1/qi + c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  (5)  4  A new unital in a Dickson commutative semi-  field plane  In [35], Penttila and Williams constructed an ovoid of the parabolic quadric Q(4, 35)  in PG(4, 35), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=', a set O of 310 + 1 points having exactly one point on each gen-  erator of the quadric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Moreover, O is a translation ovoid, meaning that the points  of O can be coordinatized by using functions that are additive over F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' According  to a construction given in [28], such a translation ovoid corresponds to a semifield  9  flock of the quadratic cone in PG(3, 35), which, in turn, corresponds to a generalized  quadrangle with parameters (310, 35), whose point-line dual is a translation gener-  alized quadrangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By a result of Payne and Thas [34, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1], the latter generalized  quadrangle is isomorphic to T(E) for some egg E in PG(19, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='4 in  [27], the dual egg of E forms a good egg ED in PG(19, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Whence, via the above  correspondences, the Penttila-Williams ovoid of Q(4, 35) gives rise to a good (dual)  egg in PG(19, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In order to simplify the notation, we will refer to it as E = E(b, c)  with b = (0, 1, 0, 0, 0), c = (0, 0, 0, −1, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' see [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  According to the expressions of the polynomials g(a,b)(t) and h(a,b)(r, s) in this case,  the egg elements of E are defined by the polynomials  g(a,b)(t) = a2t − (b2)32t32 + (ab)34t34  (6)  and  h(a,b)(r, s) = −ar + b32s32 + (br + as)34,  (7)  for all a, b ∈ F35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let p be an odd prime and ξ a non-square in Fpm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By [17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='241], the multiplication  defined by  (x, y) ∗ (a, b) = (ax + ξbαyα, bx + ay)  with α ∈ Aut(Fpm) not the identity, turns F2  pm into a Dickson commutative semifield  of order p2m which we denote by D = D(pm, ξ, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' In particular, its middle nucleus  is Nm = {(a, 0) : a ∈ Fpm}, and its left nucleus is Nl = {(a, 0) : a ∈ Fix(α)},  coinciding with its center K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Now, let p = 3 and m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' For any pair (a, b) ∈ F2  35, we consider the following  map  τ(a,b) :  (x, y)  �→  (bx + ay, −ax + b32y32),  which defines the subspaces S(a, b) = {((x, y)τ(a,b), x, y)3 : x, y ∈ F35} of PG(19, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Set S = {S(a, b) : a, b ∈ F35} ∪ {S∞}, where S∞ = {(x, y, 0, 0)3 : x, y ∈ F35}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let ϕ be the linear map ϕ : (x, y) �→ (−y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the set {ϕτ(a,b) : a, b ∈ F35}  is precisely the spread set of F10  3 associated with the Dickson commutative semifield  D = D(35, −1, 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  It turns out that S is a 9-spread of Σ∞ = PG(19, 3) and, by [2], the translation  plane Π(S) is isomorphic to the Dickson commutative semifield plane π(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Let V = {(t, −t34, 0, 0)3 : t ∈ F35}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, V is contained in the spread element  z = S∞ and it intersects trivially the subspace Γ = E∗  ∞ = {(0, t, r, s)3 : t, r, s ∈ F35}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  We also have  ⟨S(a, b), V⟩ = {(br + as + t, −ar + b32s32 − t34, r, s)3 : t, r, s ∈ F35},  giving  ⟨S(a, b), V⟩ ∩ Γ = {(0, −ar + b32s32 + (br + as)34, r, s)3 : r, s ∈ F35}  10  which is precisely the subspace I(a, b) defined by expression (3), with h(a,b)(r, s) as  in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The set BE defined by the Penttila-Williams egg E = E(b, c) is a  minimal F(E)-blocking set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' By taking into account Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='2, BE is a minimal FE-blocking set if and  only if  X2 + (XY )34 − (Y 2)32 = −c  (8)  has a solution for all c ∈ Fqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  We distinguish two cases: −c is a square in Fqm or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' If −c is a square, then  (±√−c, 0) are solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' if −c is not a square, then (0, ±  √  c33) are  solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1, the cone  B∗  E = {⟨(1, c, −g(a,b)(1) − c34, −a, −b)3⟩ : a, b, c ∈ F35} ∪ {S∞},  with g(a,b)(t) as in (6), is a unital in the translation plane Π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Consider the collineation of PG(20, 3) defined as ϕ : ⟨(u, v, t, r, s)3⟩ �→ ⟨(u, −t, v, r, s)3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Then, Π(S)ϕ represents the Dickson commutative semifield plane π(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' It turns out  that the set  U = {(g(a,b)(1) + c34, c, −a, −b) : a, b, c ∈ F35} ∪ {(∞)}  is a unital in π(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Note that U cannot be a Buekenhout-Metz unital since π(D) is  a 10-dimensional translation plane over its kernel F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' On the other hand, as π(D)  admits unitary polarities [21], U might be a polar unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The following result shows  that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' The unital U is not a polar unital in π(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Since the tangent space at the egg element E(0, 0) is E∗(0, 0), the tangent  line of Π(S) at the point O = ⟨(1, 0, 0, 0, 0)3⟩ ∈ B∗  E is the subspace spanned by  S(0, 0) and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Then, the tangent line of π(D) at (0, 0) ∈ U is [0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  From [24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='1], any unitary polarity of π(D) mapping (0, 0) to [0, 0] is  given by  ρa :  (x1, x2, y1, y2)  ↔  [ax1, −ax2, −y1, y2],  (m1, m2)  ↔  (a−1m1, −a−1m2)  (∞)  ↔  [∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  for some non-zero a ∈ F35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  The unital U is a polar unital with respect to ρa, for some a ∈ F35, if and only if  each of its points is an absolute point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Straightforward calculations show that the  point (1, 1, 0, 0) ∈ U is not incident with ρa(1, 1, 0, 0) = [a, −a, 0, 0] for all non-zero  a ∈ F35, showing that U is not a polar unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  11  References  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Abatangelo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Larato and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Rosati, Unitals in planes derived from  Hughes planes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' System Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 15 (1990), 151–155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Albert, Finite division algebras and finite planes, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 10 (1960), 53–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Andr´e, ¨Uber nicht-Desarguessche Ebenen mit transitiver Translationsgruppe,  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 60 (1954), 156–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bagchi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bagchi, Designs from pairs of finite fields, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A cyclic unital  U(6) and other regular Steiner 2-designs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A 52 (1989),  51–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Barlotti and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Lunardon, Una classe di unitals nei ∆-piani, Riv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Parma (4) 5 (1979), 781–785.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Barwick, Unitals in the Hall plane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 58 (1997), 26–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Barwick, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Casse and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Quinn, The Andr´e/Bruck and Bose rep-  resentation in PG(2h, q): unitals and Baer subplanes, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Belg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Simon Stevin 7 (2000), 173–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Barwick and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Ebert, Unitals in projective planes, Springer, New York,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Biliotti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Jha and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Johnson, Foundations of Translation Planes, Pure  and Applied Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 243, Marcel Dekker, New York, Basel, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Brouwer, Some unitals on 28 points and their embeddings in projective  planes of order 9, Geometries and Groups, Springer Lecture Notes in Mathe-  matics, 893 (1981), 183–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Brouwer, A unital in the Hughes plane of order nine, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 27  (1989), 55–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bruck and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bose, The construction of translation planes from projec-  tive spaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Algebra 1 (1964), 85–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bruck and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bose, Linear representation of projective planes in pro-  jective spaces, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Algebra 4 (1966), 117–172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Bruen and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thas, Blocking sets, Geometriae Dedicata 6 (1977), 193–  203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [15] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Buekenhout, Existence of unitals in finite translation planes of order q2 with  a kernel of order q, Geometriae Dedicata 5 (1976), 189–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  12  [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Cannon and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Playoust, An Introduction to MAGMA, University of Sydney  Press, 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dembowski, Finite geometries, Springer-Verlag, New York, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' de Resmini and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Hamilton, Hyperovals and unitals in Figueroa planes,  European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 19 (1998), 215–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dover, A family of non-Buekenhout unitals in the Hall planes, in “Mostly  finite geometries (Iowa City, IA, 1996)”, Lecture Notes in Pure and Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  190 (1997), 197–205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [20] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Drudge, On the orbits of Singer groups and their subgroups, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 9 (2002), 10 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' (electronic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Ganley, A class of unitary block designs, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 128 (1972), 34–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Gr¨uning, Das kleinste Ree-Unital, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 46 (1986), 473–480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [23] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Gr¨uning, A class of unitals of order q which can be embedded in two different  planes of order q2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 29 (1987), 61–77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Hui, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Law, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Tai and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Wong, A note on unitary polarities  in finite Dickson semifield planes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 106 (2015), 175–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [25] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Kantor, Some generalized quadrangles with parameters q2, q, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  192 (1986), 45–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Lavrauw, Characterizations and properties of good eggs in PG(4m − 1, q),  q odd, Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 301 (2005), 106–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Lavrauw and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Penttila, On eggs and translation generalised quadrangles,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' A 96 (2001), 303–315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [28] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Lunardon, Flocks, ovoids of Q(4, q) and designs, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dedicata 66 (1997),  163–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Lunardon, Normal spreads, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dedicata 75 (1999), 245–261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Mathon, Constructions for cyclic Steiner 2-designs, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 34  (1987), 353–362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [31] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Mazzocca and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Polverino, Blocking sets in PG(2, qn) from cones of  PG(2n, q), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Algebraic Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 24 (2006), 61–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Mazzocca, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Polverino and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Storme, Blocking sets in PG(r, qn), Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  Codes Cryptogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 44 (2007), 97–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [33] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Metz, On a class of unitals, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dedicata 8 (1979), 125–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  13  [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='E Payne and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thas, Finite generalized quadrangles, Second edition, in:  EMS Series of Lectures in Mathematics, Z¨urich, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [35] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Penttila and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Williams, Ovoids in parabolic spaces, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dedicata, 82  (2000), 1–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [36] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Rinaldi, Hyperbolic unitals in the Hall planes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 54 (1995), 148–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Rosati, Disegni unitari nei piani di Hughes, Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Dedicata 27 (1988),  295–299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Rottey,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Sheekey and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Van de Voorde,  Subgeometries in the  Andr´e/Bruck-Bose representation, Finite Fields Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 35 (2015), 115–138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Seib, Unit¨are Polarit¨aten endlicher projektiver Ebenen, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 21  (1970), 103–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [40] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Sz˝onyi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Cossidente, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' G´acs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Mengyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Siciliano and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Weiner, On  large minimal blocking sets in PG(2, q), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' 13 (2005), 25–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [41] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thas, Geometric characterization of the [n − 1]-ovaloids of the projective  space PG(4n − 1, q), Simon Stevin 47 (1974), 97–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  [42] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Thas, Generalized quadrangles of order (s, s2), II, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  A 79 (1997), 223–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}
+page_content='  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE1T4oBgHgl3EQfSAMg/content/2301.03059v1.pdf'}

3tAzT4oBgHgl3EQfuf3H/content/2301.01693v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ec1a47218fed0611463dc6c349e62808c09708cb8b5fe6da99f4aef6413c80aa
+size 426798

3tAzT4oBgHgl3EQfuf3H/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e26b9d545aca9de47119a533953dd6d4a31af2854d410ec927aed97c4312c168
+size 102665

49E4T4oBgHgl3EQfBAv2/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:2b4d334996a7ca22e4d9c19c53e8c05fcc43385b3b892bae53796d5865b32410
+size 3276845

5NFKT4oBgHgl3EQf-C45/content/2301.11956v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c64496cc345070c050d021bc0561cb1ac4da9036ed05703d8a53470f4124c2d1
+size 607482

5NFKT4oBgHgl3EQf-C45/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ca16430c9e92b4e5ac23af028aef965b82f86d47d994efe735e5b1d85bb7d187
+size 237595

5dE0T4oBgHgl3EQfegDz/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8246b0384503c5689734f4314f9021ad5ab9dc75e6156617b4caca92d1090266
+size 214345

5dE1T4oBgHgl3EQfBAJm/content/2301.02846v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6e14dcc0243d976bf69fabfdab952f1f11f69d93b4bfc91b4accd1b93001d0c7
+size 2663648

5tAzT4oBgHgl3EQf9_5e/content/2301.01927v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8850f2ef130ecf68869913340bd279d394c2073bcba6bd09cbda75f2af689e49
+size 12143416

6NE0T4oBgHgl3EQfewDQ/content/2301.02396v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b5791a0050ca7362f3e19a95f7b815d738fc67a6a2f302a8fed8d0633060d017
+size 1170487

6NE0T4oBgHgl3EQfewDQ/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8d188c8e652dd0f6f5d39de9a5b20a5f4dda3128e1a0232cb4b4d1b9a97ab401
+size 212673

6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:413b14e0a28485130ca8be383a798eae98af773b1266dee6bbd5ac98c0cb73a1
+size 313596

6NFKT4oBgHgl3EQfTS23/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5d18498fd861fb326f6618ed9a4d147cc639dbf839dd43d822dcb02ddde98dfd
+size 1179693

6NFKT4oBgHgl3EQfTS23/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fbb649e130a2c58d2a9bf8961734511c23c8fd7c3a54501a32ed3f7ffbd52b51
+size 46122

7tE3T4oBgHgl3EQfqQo3/content/tmp_files/2301.04649v1.pdf.txt

@@ -0,0 +1,1644 @@
+The alignment of galaxies at the Baryon Acoustic Oscillation scale
+Dennis van Dompseler, Christos Georgiou,∗ and Nora Elisa Chisari†
+Institute for Theoretical Physics, Utrecht University,
+Princetonplein 5, 3584 CC, Utrecht, The Netherlands.
+(Dated: January 12, 2023)
+Massive elliptical galaxies align pointing towards each other in the structure of the Universe. Such
+alignments are well-described at large scales through a linear relation with respect to the tidal field
+of the large-scale structure. At such scales, galaxy alignments are sensitive to the presence of baryon
+acoustic oscillations (BAO). The shape of the BAO feature in galaxy alignment correlations differs
+from the traditional peak in the clustering correlation function. Instead, it appears as a trough
+feature at the BAO scale. In this work, we show that this feature can be explained by a simple
+toy model of tidal fields from a spherical shell of matter. This helps give a physical insight for the
+feature and highlights the need for tailored template-based identification methods for the BAO in
+alignment statistics. We also discuss the impact of projection baselines and photometric redshift
+uncertainties for identifying the BAO in intrinsic alignment measurements.
+I.
+INTRODUCTION
+Baryon acoustic oscillations (BAO, [1]) are sound
+waves supported by the plasma present in the Universe
+before recombination. After the Universe became neu-
+tral, these waves could no longer travel and remained
+frozen at a comoving scale of ∼ 150 Mpc. In the late
+Universe, BAO manifest themselves as a subtle but signif-
+icant percent-level peak in the auto-correlation function
+of galaxies or matter. Because they constitute a standard
+ruler of an absolute distance scale, they are regularly used
+to probe the expansion of the Universe [2].
+Any cosmological observable that correlates with the
+matter field can have a manifestation of BAO. One
+such observable beyond galaxy clustering statistics is the
+alignments of galaxies.
+Galaxies are known to align
+themselves radially towards other galaxies [3], and this
+phenomenon can be described, when the alignment is
+weak, by a proportional response of the projected shape
+of a galaxy to the projected tidal field of matter [4]. This
+model is successful in describing the observed alignments
+of luminous red galaxies at large-scale from wide sur-
+veys [5–9]. Although intrinsic alignments are typically re-
+garded a contaminant to other cosmological observables
+[3, 10–13], there are examples of how they can be used
+for extracting cosmological information [14–18].
+In principle, a detection of BAO could be achieved
+in the correlation function of galaxy alignments around
+other galaxies. In [14], it was shown that such a detection
+was within the reach of existing surveys. For luminous
+galaxies in the Baryon Oscillation Spectroscopic Survey
+(BOSS, [19]), the signal-to-noise ratio (S/N) would be of
+the order of ∼ 2.7. For upcoming data sets such as the
+Dark Energy Spectroscopic Instrument (DESI, [20]), the
+expectation is for this to increase to S/N ∼ 12.
+Searches for BAO in galaxy statistics often adopt
+matched templates [21, 22], decompositions thereof [23,
+∗ [email protected]
+† [email protected]
+24] or remove the smooth (no BAO) component [25]. In
+[14], it was noticed that the shape of the BAO differs
+from the traditionally expected ‘peak’ at 150 Mpc. When
+looking at the alignment of galaxies with the matter field,
+it rather appears as a trough at a similar distance, fol-
+lowed by a peak at larger comoving separations. This
+behavior was recently confirmed by [26], who measured
+the alignment of massive (cluster-scale) halos with the
+underlying matter field in the DarkQuest N-body sim-
+ulations [27]. These authors also pointed out a similar
+behaviour for the correlation of halo alignments with the
+velocity field, with the BAO appearing as trough rather
+than a peak.
+In light of possible upcoming detections of this feature,
+we aim here to give an intuitive physical picture of the ori-
+gin of this trough pattern rooted in simple linear physics.
+We show that gravitational tides in and around a spher-
+ical shell of matter display exactly the trough pattern
+and justify its appearance in both matter- and velocity-
+alignment cross-correlations. We also discuss the impact
+of long projection baselines and photometric redshifts for
+identifying the BAO in observational data.
+This work is organised as follows. In Section II, we in-
+troduce the most widely used linear model for the shapes
+of galaxies and halos, we present the equations for cor-
+relations with matter, galaxies and velocity field, we ex-
+plain how we model BAO using two approaches which
+differ in complexity, and how tidal fields are calculated
+for the simple toy model. Section III gives our results
+and we conclude in Section IV.
+In the matter power spectrum, BAOs appear as a se-
+ries of successive peaks or ‘wiggles’ at different wavenum-
+bers.
+In real space, this corresponds to a peak in the
+three-dimensional correlation function of galaxies, at a
+comoving scale of ∼ 150 Mpc [28].
+In this work, we
+model a Universe with and without BAO ‘wiggles’ using
+the analytical approximation of [29] for a cosmology with
+σ8 = 0.8158, h = 0.6774, Ωm = 0.3089, Ωb = 0.0486 and
+ns = 0.9667, consistent with constraints from the Planck
+satellite [30].
+The matter power spectra at z = 0 are
+output by the nbodykit software [31].
+Other cosmo-
+arXiv:2301.04649v1  [astro-ph.CO]  11 Jan 2023
+
+2
+logical quantities were obtained via the Core Cosmology
+Library[32] [33]. In the following sections, we compare
+our predictions for the alignment correlation function for
+both models.
+II.
+MODELLING
+A.
+Linear alignment model
+In the linear alignment model [4], galaxies align their
+observed two-dimensional shapes proportionally to the
+projected tidal field of matter. This is mathematically
+described as:
+(γ+, γ×) = − C1
+4πG(∂2
+x − ∂2
+y, 2∂x∂y)φp
+(1)
+where C1 is an unknown proportionality constant, i.e.
+the alignment ‘bias’ and φp is the primordial gravita-
+tional potential (i.e.
+at some high redshift when the
+galaxy was formed). This gives a prescription for con-
+necting galaxy shapes to the underlying gravitational po-
+tential field and leaves C1 as a free parameter. As a re-
+sult, galaxy shapes are expected to be correlated with
+any observable that depends on the gravitational poten-
+tial, or the matter field which sources it.
+The linear alignment model is known to provide a
+good description of elliptical galaxies in both simulations
+[34–37] and observations [5–9] and it is widely used in
+cosmological studies which aim to extract information
+from gravitational lensing [e.g. 38–40].
+Here, intrinsic
+alignments act as a contaminant.
+(We will not cover
+blue/spiral galaxies in this work, to which different mod-
+els are thought to apply, namely based on tidal torque
+theory [41–43].)
+In general, the strength (or ‘bias’) of alignment C1 is
+constrained from observations [5–9]. We know from them
+that this constant is generally positive. In this context,
+this means that elliptical galaxies tend to point radi-
+ally towards peaks in the density field.
+[44] proposed
+a method for estimating C1 using the stellar distribution
+function of elliptical galaxies. Using this method, once
+again one expects that C1 > 0. However, the predicted
+alignment seemed to fall short of the observed one. This
+could be a consequence of alignments of galaxies being
+built-up over time, rather than instantaneously reacting
+to the tidal field.
+For our purposes, it suffices to em-
+phasise that the sign of C1 is at least observationally
+constrained for elliptical galaxies that are the subject of
+our work. A completely analogous model and arguments
+would apply for halos as well, where C1 is also known to
+be positive [45].
+The most commonly measured statistic of galaxy in-
+trinsic shapes is the projected correlation function of
+galaxy positions and the + component of the shape,
+wg+(rp), which is a function of the projected comoving
+separation between galaxies. At any given redshift, this
+is given by an integral along separation in comoving ra-
+dial distance (Π) of the three-dimensional correlation of
+positions and shapes, ξg+(rp, Π, z):
+wg+(rp, z) =
+� Πmax
+−Πmax
+dΠ ξg+(rp, Π, z).
+(2)
+Here, ξg+ is defined as:
+1 + ξg+ = ⟨[1 + δg(x1)]γ+(x2)⟩,
+(3)
+and r = x2 − x1.
+Because galaxy alignments only arise between galaxies
+that are physically close, Πmax is usually restricted to
+scales ≲ 100 h−1Mpc. This justifies assuming a separate
+dependence of ξg+ on Π and redshift.
+In the linear alignment model, ξg+(rp, Π) is given by
+ξg+(rp, Π, z) = bgC1ρcritΩm
+2π2D(z)
+� ∞
+0
+dkz
+� ∞
+0
+dk⊥
+k3
+⊥
+k2 P(k, z)J2(k⊥rp) cos(kzΠ)
+(4)
+where D(z) is the growth function, normalized to (z +
+1)D(z) = 1 during matter domination, P(k, z) is the
+matter power spectrum, ρcrit is the critical density today
+and J2 is the Bessel function of the first kind of order 2.
+This results in a projected correlation function
+wg+(rp, z) = bgC1ρcritΩm
+π2D(z)
+� ∞
+0
+dkz
+� ∞
+0
+dk⊥
+k3
+⊥
+k2kz
+P(k, z)J2(k⊥rp) sin(kzΠmax)
+(5)
+Notice that when bg = 1, wg+ reduces to the correlation
+between matter and the + component of galaxy shapes,
+
+3
+wm+. For simplicity, we will work with wm+ from here
+on. We also note that in the linear alignment model, wg×
+is expected to be zero, and we do not consider it further
+in this work.
+For comparison, the projected correlation function of
+the matter field, wmm, is given by
+wmm(rp, z) = 1
+π2
+� ∞
+0
+dkz
+� ∞
+0
+dk⊥
+k⊥
+kz
+P(k, z)J0(k⊥rp) sin(kzΠmax) .
+(6)
+This model was used in forecasts by [14], where it was
+proposed that a detection of BAO could be achieved in
+the projected alignment correlation function of galaxies.
+It is also commonly used in fits to the data [e.g. 5, 7].
+However, other works adopt larger projections lengths,
+effectively taking Πmax to infinity [e.g. 8].
+The corre-
+sponding projected correlation functions in those cases
+are
+wmm(rp, z) =
+� ∞
+0
+dk⊥
+2π k⊥P(k⊥, z)J0(k⊥rp) ,
+(7)
+wm+(rp, z) = ˜C1
+� ∞
+0
+dk⊥
+2π k⊥P(k⊥, z)J2(k⊥rp) . (8)
+where ˜C1 = C1ρcritΩm/D(z) for simplicity.
+Eq.
+8 is
+derived explicitly in the appendix.
+B.
+Shape-shape correlations
+Although more sensitive to shape noise, intrinsic shape
+auto-correlations have been derived in previous work in
+the context of the linear alignment model and also de-
+tected in spectroscopic survey observations [e.g. 5]. The
+projected correlation functions for shape-shape correla-
+tions take the form
+w(++,××)(rp, z) =
+1
+2π2
+�C1ρcritΩm
+D(z)
+�2 � ∞
+0
+dkz
+� ∞
+0
+dk⊥
+k5
+⊥
+k4kz
+P(k, z)[J4(k⊥rp) ± J0(k⊥rp)] sin(kzΠmax).
+(9)
+C.
+Modelling photometric redshifts
+The correlation functions presented in the sections
+above assume precise knowledge of the redshift informa-
+tion of our galaxy samples. This would be the case when
+data are taken from a spectroscopic survey, but such sur-
+veys require a predetermined target selection and long
+integration times which limit the size of galaxy samples
+that can be obtained.
+Photometric surveys can over-
+come this problem at the cost of significantly reduced
+accuracy in the determination of redshift information by
+using band photometry instead of spectra.
+The accuracy of the photometric redshifts depend on a
+number of factors, such as the signal-to-noise of the flux
+measurement of galaxies and the existence of a represen-
+tative calibration data-set. There are several techniques
+that can increase the typical accuracy of photometric
+redshifts. These include mapping of the galaxy red se-
+quence (limited to intrinsically red galaxies) [46, 47], us-
+ing machine learning techniques with representative over-
+lapping spectroscopic samples as training set (limited by
+the training set) [48–50] or using narrow band photome-
+try to resolve more features in a galaxy’s spectral energy
+distribution (which is more observationally costly com-
+pared to broad band photometry) [51, 52]. In light of
+these techniques, it is interesting to investigate how the
+projected correlation functions change when the galaxy
+samples used are obtained through photometric data.
+To compute the projected correlation functions in this
+context, we model the impact of redshift uncertainty fol-
+lowing [48]. The uncertainty is expressed in the proba-
+bility density function p(z|¯z), where z, ¯z is the true and
+observed redshift of a galaxy, respectively. We choose to
+model this with a generalized Lorentzian distribution,
+p(z|¯z) ∝
+�
+1 + ∆z2
+2as2
+�−a
+,
+(10)
+where ∆z = (z − ¯z)/(1 + z) and a, s are free parame-
+ters. In [53] it was shown that this distribution better
+describes the probability density function compared to
+a Gaussian one, especially the long tails away from the
+mean. We fix a = 2.613 as was found in [53] and vary
+s ∈ {0.0035, 0.015, 0.025} to mimic different photometric
+redshift precision scenarios. The precision is commonly
+expressed in terms of the scaled median absolute devi-
+ation (SMAD) of ∆z, given by ˆσ∆z = k · MAD, where
+k ≈ 1.4826 and p (|∆z| ≤ MAD) = 1/2 (using the fact
+that the median of (10) is at ∆z = 0). The SMAD is
+a way to quantify a standard deviation equivalent in the
+case where the distribution is different than a Gaussian.
+Assuming that the line-of-sight separation between two
+galaxy pairs is small compared to the comoving radial
+distance of their mean redshift, we can express their true
+redshifts z1 + z2 = 2zm as Π ≈ c(z1 − z2)/H(zm). The
+
+4
+matter-matter projected correlation function in the pres-
+ence of redshift uncertainty can be modelled by
+wphot
+mm (rp, zm) =
+� Πmax
+−Πmax
+dΠ
+� ∞
+0
+dℓ ℓ
+2π J0(ℓθ)Cmm(ℓ, ¯z1, ¯z2) ,
+(11)
+where rp ≈ θχ(zm) and Cmm is the matter-matter angu-
+lar power spectrum, computed using p(zm|¯z1,2) for the
+redshift distribution of its tracers.
+In a similar way,
+one can compute the projected matter-shape and shape-
+shape correlation functions, using the matter-intrinsic
+and intrinsic-intrinsic angular power spectra, via
+wphot
+m+ (rp, zm) = −
+� Πmax
+−Πmax
+dΠ
+� ∞
+0
+dℓ ℓ
+2π J2(ℓθ)CmI(ℓ, ¯z1, ¯z2)
+(12)
+and
+wphot
+(++,××)(rp, zm) =
+� Πmax
+−Πmax
+dΠ
+� ∞
+0
+dℓ ℓ
+2π [J4(ℓθ) ± J0(ℓθ)] CII(ℓ, ¯z1, ¯z2) .
+(13)
+D.
+Correlations with the velocity field
+Because intrinsic alignments are correlated with the
+matter field, we also expect them to be correlated with
+the velocity field of the large-scale structure [26].
+On
+linear scales, the velocity field and the matter den-
+sity are related by the continuity equation:
+∇ · ⃗v =
+−δ (1 + z)/H(z)f(z), where H is the Hubble factor, f =
+d ln D/d ln a is the logarithmic growth rate and ⃗v is the
+irrotational velocity field. This leads to a correlation be-
+tween the divergence of the velocity field and the + com-
+ponent of galaxy shapes. In practice, one expects to actu-
+ally measure the correlation between projected + shapes
+and radial velocities (along the line-of-sight) [54], which
+in Fourier space is vr ∝ (kz/k)δ/k. The wvr+ correlation
+function is thus modelled by
+wvr+(rp, z) = C1ρcritΩm(1 + z)
+π2D(z)H(z)f(z)
+� ∞
+0
+dkz
+� ∞
+0
+dk⊥
+k3
+⊥
+k4 P(k, z)J2(k⊥rp) sin(kzΠmax).
+(14)
+Since the radial velocity field often requires spectroscopic
+information to be constructed, we do not discuss the
+effect of photometric redshifts on the wvr+ correlation
+function (but see [54] for an alternative approach).
+E.
+Tidal field for a spherical mass distribution
+To give a qualitative explanation of how the BAO fea-
+tures in the wm+ correlation, we recall that the gravita-
+tional potential of a spherical mass distribution is given
+by
+φ(r) = −4πG
+�1
+r
+� r
+0
+dr1ρ(r1)r2
+1 +
+� ∞
+r
+dr1ρ(r1)r1
+�
+.
+(15)
+where ρ(r1) is the density of matter as a function of ra-
+dius.
+For an extended object, the difference between the
+force acting at any point and the force acting at the cen-
+ter of mass is the tidal force: T = F(x) − F(xCM). A
+small displacement from the center of mass gives rise to
+a differential change of the force of dTj = τijdxi, implic-
+itly summing over i and where τij = −∂i∂jφ is the tidal
+tensor. A spherically symmetric gravitational potential
+originates a tidal field given by [55]:
+τrr(r) = −∂2
+rφ(r),
+(16)
+τθθ(r) = τφφ(r) = −∂rφ(r)/r.
+(17)
+The explicit expression in terms of the density profile of
+the object is
+τrr(r) = 4πG
+� 2
+r3
+� r
+0
+dr1ρ(r1)r2
+1 − ρ(r)
+�
+,
+(18)
+τθθ(r) = τφφ(r) = 4πG
+r3
+� r
+0
+dr1ρ(r1)r2
+1,
+(19)
+and this can also be expressed in terms of the mean den-
+sity interior to a given radius, ¯ρ(r).
+For example, as
+τrr(r) = 4πG[2¯ρ(r)/3 − ρ(r)].
+F.
+A simple BAO model
+We model the BAO as a spherical shell of mass MBAO,
+with an inner radius RBAO, width ∆R and uniform den-
+sity ρBAO. We will neglect the smooth extended compo-
+nent that corresponds to the matter distribution inside
+and outside the shell, and focus only on how the tidal
+field changes when the BAO shell is added.
+
+5
+Looking at Figure 1, we first examine the tidal field of
+the mass configuration on the xy plane. This should be
+qualitatively representative of the projection along the
+line of sight, although we will discuss the impact of the
+projection in more detail below. We imagine taking a
+spherical coordinate system where φ = 0 is aligned with
+the projection (z) axis. According to Eq. 1, we would
+then have the change in shapes due to the presence of
+the BAO being γBAO
++
+= C1[τrr(r) − τθθ(r)]/(4πG).
+The radial and θ components of the tidal field for such
+configuration are
+τrr(r) =
+�
+�
+�
+0
+r ≤ RBAO
+(I)
+−4πGρBAO[1/3 + 2/3(RBAO/r)3]
+RBAO < r ≤ RBAO + ∆R
+(II)
+2GMBAO/r3
+r > RBAO + ∆R (III)
+(20)
+τθθ(r) =
+�
+�
+�
+0
+r ≤ RBAO
+(I)
+4πGρBAO[1 − (RBAO/r)3]/3
+RBAO < r ≤ RBAO + ∆R
+(II)
+GMBAO/(3r3)
+r > RBAO + ∆R (III)
+(21)
+respectively, where we have identified three regions of
+interest: inside the spherical shell (I), within the shell
+(II) and outside (III). Similarly, τφφ(r) = τθθ(r).
+In addition to this simple model, we also consider a
+slightly more realistic Gaussian form for the density pro-
+file of the shell, with a center at RBAO + ∆R/2 and a
+dispersion σBAO = ∆R/2. We obtain the tidal field in
+this scenario numerically integrating Eqs. 18 and 19. We
+then use the change of shapes (γBAO
++
+) to explain devia-
+tions in wmm from wm+ based on the definition given in
+Eq. 2.
+III.
+RESULTS
+Figure 1 illustrates the geometry of the problem.
+Stacking on as many galaxies as possible and measur-
+ing the matter (or galaxy) distribution around them,
+one would find it slightly enhanced at scales equal to
+or smaller than the BAO comoving distance scale due to
+projection over the line-of-sight. The wider the range in
+Π, the higher the dilution of the BAO peak in projection,
+and the further in it will move in rp.
+Figure 2 shows the projected matter correlation func-
+tion (top panel), computed at z = 0, for different val-
+ues of Πmax in a universe with and without wiggles.
+BAO feature as an enhancement of the correlation func-
+tion at a projected comoving separation of approximately
+∼ 110 h−1 Mpc. Because of projection effects, such a
+distance is slightly reduced compared to the comoving
+distance at which one would find the BAO peak for the
+three-dimensional correlation function of matter.
+The
+larger the projection baseline (Πmax), the further the
+peak moves towards smaller separations.
+In the bottom panel of Figure 2, we show the projected
+alignment correlation function, computed at z = 0, for
+different values of Πmax in a Universe with and without
+wiggles. Compared to wmm in the top panel of Figure 2,
+FIG. 1. A sketch showing the geometry of the problem. Ac-
+cording to observational constraints on the linear alignment
+model, galaxies (orange) align themselves radially towards
+density peaks. These constraints come from integrating the
+three-dimensional correlation function of galaxy positions and
+shapes along a line-of-sight baseline of Πmax (black cylinder),
+typically ≲ RBAO, the BAO scale. The BAO is represented
+as a spherical shell of matter around the center of the poten-
+tial. The reader should interpret that the smooth ‘no wiggles’
+component has been subtracted in this image.
+we see clearly that, at the location of the original BAO
+peak, there is now a trough, followed by a peak at a larger
+distance. This is indeed the feature that was seen in pre-
+vious theoretical predictions and numerical simulations.
+To explain why it differs so from wmm, we make the
+following simplification of the problem: we assume that
+
+4Z
+AR
+BAO
+RBAO
++max
+p
+-max
+X6
+60
+80
+100
+120
+140
+r
+p
+ [M c/h]
+−20
+−15
+−10
+−5
+0
+5
+10
+15
+20
+r
+p
+w
+mm
+ [(M c/h)
+2
+]
+Wiggles Π
+max
+=
+40 M c/h
+No wiggles Π
+max
+=
+40 M c/h
+Π
+max
+=
+80 M c/h
+Π
+max
+=
+200 M c/h
+Π
+max
+=
+∞
+BAO  eak
+60
+70
+80
+90
+100
+110
+120
+130
+140
+r
+p
+ [Mpc/h]
+0.1
+0.2
+0.3
+w
+ 
++
+[Mpc/h]
+Wiggles Π
+ ax
+=
+40 Mpc/h
+No wiggles Π
+ ax
+=
+40 Mpc/h
+Π
+ ax
+=
+60 Mpc/h
+Π
+ ax
+=
+80 Mpc/h
+Π
+ ax
+=
+100 Mpc/h
+Π
+ ax
+=
+200 Mpc/h
+Π
+ ax
+=
+∞
+BAO peak
+FIG. 2. Projected correlation functions for matter clustering,
+wmm (top) and alignments of galaxies with the matter field,
+wm+ (bottom), projected over different line-of-sight baselines,
+Πmax = [40, 60, 80] h−1 Mpc, for universes with (solid) and
+without (dashed) BAO. The BAO peak scale is indicated as
+a dotted vertical line. This corresponds to a peak in the case
+of wmm and a trough for wm+.
+the BAO is a spherical shell centered at the origin, and
+that we are interested in computing the tides produced
+by this shell in the radial direction: τrr and in the θ
+direction: τθθ, according to Eqs. 20 and 21, respectively.
+This can be combined to predict γBAO
++
+. Our assumptions
+are justified by our findings in Figure 2, in which we see
+the BAO feature appear as a peak in wmm (top panel).
+γBAO
++
+is shown in Figure 3.
+This should be inter-
+preted as the change in the intrinsic shapes of elliptical
+galaxies from a universe with BAO to a universe without
+BAO. (For illustration purposes, we adopt here C1 = 1.)
+It is calculated by fixing the outer rim of the shell to
+RBAO + ∆R = 150 Mpc and varying the choice of ∆R.
+The overall mass normalization, MBAO is arbitrary, but
+conserved, while varying ∆R. Consistently with Eq. 20
+we see that as a result of the BAO matter shell, the tidal
+field is unchanged inside the shell (Region I: R < RBAO),
+60
+70
+80
+90
+100
+110
+120
+130
+140
+r [M c/h]
+−3.0
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+[τ
+rr
+(r)
+−
+τ
+θθ
+(r)]/(4πG)
+1e−7
+R
+BAO
++
+ΔR
+ΔR
+=
+20 M c
+ΔR
+=
+30 M c
+ΔR
+=
+40 M c
+Gaussian
+FIG. 3. γBAO
++
+given the tidal field (normalized) of a spherical
+mass shell configuration spanning from RBAO to RBAO + ∆R
+and assuming C1 = 1 for illustration purposes. We plot the
+function for different BAO widths: ∆R = [20, 30, 40] Mpc
+in shades of red. In region III, where r > RBAO + ∆R, the
+increase in the tidal field is consistent with the addition of a
+point mass MBAO. In Region II, within the BAO shell, we see
+a suppression of the tidal field compared to the ‘no wiggles’
+case. In Region I, inside the BAO shell, the tidal field remains
+unchanged. We also plot γBAO
++
+as originated from a spherical
+mass shell with a Gaussian profile centered at RBAO + ∆R/2
+and with a dispersion of ∆R/2 (gray).
+75
+80
+85
+90
+95
+100
+105
+110
+115
+x [Mpc/h]
+−20
+−15
+−10
+−5
+0
+5
+10
+15
+20
+y [Mpc/h]
+I
+II
+III
+⃗
+g
+FIG. 4. The gravitational acceleration vector, ⃗g from a spheri-
+cal shell of matter. Three regions are indicated: region I inside
+the shell, region II within the shell and region III outside the
+shell. There is no gravity in region I. It builds up in region II
+and is the same as for a point mass with MBAO in region III.
+it decreases within the shell and it increases outside of
+it. The increase is originated by the addition of the mass
+MBAO, compared to the case where this is absent.
+The Gaussian model (gray curve) represents a slightly
+more realistic situation in which the BAO has no sharp
+edge. For this case, we only show one possible scenario
+
+7
+with a dispersion which corresponds to 10 Mpc.
+The
+behaviour of the curve is similar in general to the hard-
+edge model, although γBAO
++
+transitions from negative to
+positive values at larger separations, above RBAO + ∆R.
+A model galaxy represented by a sphere embedded in
+this tidal field is deformed in the following way. Inside
+the shell, in region I, there is no deformation. Within
+the shell, in region II, the gravitational force increases
+with separation. This can be seen in the two-dimensional
+representation of the gravitational acceleration vector
+(⃗g = −∇φ) shown in Figure 4. The tidal field is thus neg-
+ative in region II and thus compressive along the radial
+direction. Outside the shell, in region III, tidal forces are
+positive and thus disruptive, elongating the galaxy along
+the radial direction. This is due to the gravitational force
+decreasing outside the shell in the radial direction.
+A.
+Velocity-shape alignments
+We also obtained the line-of-sight velocity-intrinsic
+shape projected correlation function, shown in Figure 5.
+This shows very similar BAO behaviour to wm+ in the
+right panel of Figure 2.
+There is trough at the BAO
+scale, followed by an excess at larger scales compared to
+the ‘no wiggles’ case. This is justified by the fact that
+at these scales, the velocity field of the large-scale struc-
+ture follows the linear continuity equation, resulting in
+v(k) ∝ δ(k)/k. It is thus not surprising that the BAO
+would also follow qualitatively the tidal field of the spher-
+ical shell of mass as presented in Figure 3, confirming the
+findings of [26].
+B.
+Shape-shape correlations
+For completion, we also show in Figure 6 the impact
+of the BAO feature in shape-shape correlations. BAO
+appear as a peak in the w++ correlation function rather
+than a trough. The opposite is true for the w×× correla-
+tion.
+C.
+Long projection baselines and photometric
+redshifts
+The bottom panel of Figure 2 presents wm+ integrated
+over an infinite projection baseline. We see that as Πmax
+increases, BAO become progressively smeared. The evo-
+lution of the amplitude of wm+ is monotonic and informa-
+tion progressively saturates as Πmax → ∞. In practice,
+most observational works adopt 60 h−1 Mpc < Πmax <
+100 h−1 Mpc. The shape of the BAO is preserved with
+increasing Πmax. This is similar in the case of wmm in
+the top panel of Figure 2, though here the correlation
+amplitude does not change monotonically. For all cases,
+we notice the BAO feature (peak or trough depending on
+60
+70
+80
+90
+100
+110
+120
+130
+140
+r
+p
+ [Mpc/h]
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.10
+0.11
+w
+v
+r
++
+[Mpc/h]
+Wiggles Π
+ ax
+=
+40 Mpc/h
+No wiggles Π
+ ax
+=
+40 Mpc/h
+Π
+ ax
+=
+60 Mpc/h
+Π
+ ax
+=
+80 Mpc/h
+BAO peak
+FIG. 5.
+Projected correlation function for line-of-sight
+velocity-alignment statistics, wvr+, projected over different
+line-of-sight baselines, Πmax = [40, 60, 80] h−1 Mpc, for uni-
+verses with (solid) and without (dashed) BAO. The BAO peak
+scale is indicated as a dotted vertical line. This corresponds
+to a trough in wvr+.
+the fields considered) move inwards as a consequence of
+the increased projection length.
+Next, we address the impact of redshift uncertainty,
+such as in the case of photometrically obtained red-
+shift information (photo-z), on the projected correla-
+tion functions.
+We choose three different uncertainty
+scenarios: redshifts obtain by narrow-band photometry
+with ˆσ∆z ∼ 0.004 [52], redshifts obtained over a bright
+galaxy sample or using the galaxy red sequence with
+ˆσ∆z ∼ 0.018 [46, 47, 49] and redshifts obtained from an
+optimized gold sample from large photometric surveys
+with ˆσ∆z ∼ 0.03 [50, 56]. The last value is also equal
+to requirements in the uncertainty of the photo-z scat-
+ter for next generation photometric surveys, such as the
+Vera Rubin Observatory LSST [57].
+Figure 7 shows the projected correlation functions
+wmm, wm+ and w++ computed at z = 0.2, in a universe
+with and without BAO, for the three different redshift
+uncertainty scenarios. We see that, as the uncertainty
+gets larger, the BAO feature is less pronounced for all
+three functions.
+The behaviour of the clustering and
+alignment signals are similar to the case with accurate,
+spectroscopic redshifts (spec-z).
+It is also interesting to compare the projected corre-
+lation functions in the case of no redshift uncertainty
+and an infinite Πmax to functions with modelled redshift
+uncertainty. We show this in Figure 8 where the photo-
+z signal has ˆσ∆z ∼ 0.018. The signal obtained through
+photo-z’s is closer to zero in both the clustering and align-
+ment correlation. Since the clustering signal crosses zero
+at around 110 Mpc/h, the photometric clustering signal
+appears simply flatter. In the case of matter-shape cor-
+relations, the photo-z signal is about a factor of 2 smaller
+than the spec-z.
+
+8
+60
+70
+80
+90
+100
+110
+120
+130
+140
+r
+p
+ [Mpc/h]
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+w
++
++
+[Mpc/h]
+Wiggles Π
+ ax
+=
+40 Mpc/h
+No wiggles Π
+ ax
+=
+40 Mpc/h
+Π
+ ax
+=
+60 Mpc/h
+Π
+ ax
+=
+80 Mpc/h
+BAO peak
+60
+70
+80
+90
+100
+110
+120
+130
+140
+r
+p
+ [Mpc/h]
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+w
+×
+×
+[Mpc/h]
+FIG. 6. Projected shape-shape correlation functions for ++
+(top) and ×× (bottom), projected over different line-of-sight
+baselines, Πmax = [40, 60, 80] h−1 Mpc, for universes with
+(solid) and without (dashed) BAO. The BAO peak scale is
+indicated as a dotted vertical line.
+IV.
+CONCLUSION
+While BAO appear as a peak in the matter field pro-
+jected auto-correlation, in the correlation of matter with
+intrinsic galaxy shapes, the pattern is replaced by a
+trough at the same scale, followed by an excess at larger
+separations. We showed that this behavior is consistent
+with the response of galaxy shapes to the linear tidal
+field represented by a shell of matter with radius simi-
+lar to the location of the BAO peak. A similar behavior
+is observed for the correlation between intrinsic shapes
+and radial velocities. Our work highlights the need for
+dedicated templates for the BAO in such statistic, if a
+detection is to be attempted.
+This is, in fact, not far
+from the reach of current surveys [14].
+Progressively increasing projection baselines for the
+correlation function results in a smearing of the BAO
+peak. In the case of redshift uncertainty, such as for a
+sample where the redshift was obtained through photom-
+60
+70
+80
+90
+100
+110
+120
+130
+140
+rp [Mpc/h]
+5
+0
+5
+10
+rp wmm [Mpc/h]2
+z = 0.2
+Wiggles, 
+z = 0.004
+No wiggles, 
+z = 0.004
+z = 0.018
+z = 0.030
+BAO peak
+60
+70
+80
+90
+100
+110
+120
+130
+140
+rp [Mpc/h]
+6
+7
+8
+9
+rp wm +  [Mpc/h]2
+60
+70
+80
+90
+100 110 120 130 140
+rp [Mpc/h]
+1400
+1600
+1800
+2000
+2200
+2400
+(10
+3) rp w + +  [Mpc/h]2
+FIG. 7.
+Projected correlation functions for matter-matter
+(top), matter-shape (middle) and shape-shape (bottom) cor-
+relations computed in the case of redshift uncertainties, quan-
+tified by ˆσ∆z = [0.004, 0.018, 0.3], for universes with (solid)
+and without (dashed) BAO. The BAO peak scale is indicated
+as a dotted vertical line.
+etry, two effects take place in the projected correlation
+functions.
+Firstly, the correlation function is closer to
+zero, meaning the signal is lower. Note, however, that
+typically large photometric samples are easier to obtain
+compared to spectroscopic ones. The second effect is that
+the BAO feature is washed out by the redshift uncer-
+tainty. The higher the uncertainty, the less pronounced
+the BAO feature will be, across all correlation functions.
+
+9
+60
+70
+80
+90
+100
+110
+120
+130
+140
+rp [Mpc/h]
+10
+5
+0
+5
+10
+15
+rp wmm [Mpc/h]2
+z = 0.2
+Wiggles, spec-z
+No wiggles, spec-z
+Photo-z
+BAO peak
+60
+70
+80
+90
+100
+110
+120
+130
+140
+rp [Mpc/h]
+6
+8
+10
+12
+14
+16
+18
+rp wm +  [Mpc/h]2
+FIG. 8.
+Projected correlation functions for matter-matter
+(top) and matter-shape (bottom) correlations computed in
+the case of accurate, spectroscopic (grey) and photometric
+(maroon) redshift information, for universes with (solid) and
+without (dashed) BAO. The BAO peak scale is indicated as
+a dotted vertical line.
+ACKNOWLEDGMENTS
+This publication is part of the project “A rising
+tide:
+Galaxy intrinsic alignments as a new probe of
+cosmology and galaxy evolution” (with project number
+VI.Vidi.203.011) of the Talent programme Vidi which is
+(partly) financed by the Dutch Research Council (NWO).
+This work is also part of the Delta ITP consortium, a
+program of the Netherlands Organisation for Scientific
+Research (NWO) that is funded by the Dutch Ministry
+of Education, Culture and Science (OCW).
+Appendix A: Limber approximation
+In this appendix we explicitly show the derivation of Eq. 8 by making use of the Limber approximation [58]. For
+completeness, we will consider the g+ correlation instead of m+ and we will explicitly model the window functions
+for the galaxy populations used to trace the density and shape fields. These will be labelled qg(χ) = dNg/dχ and
+qγ(χ) = dNγ/dχ for number and shape tracers, respectively, and where χ is the comoving line-of-sight distance.
+First, we establish that our goal is to calculate Eq. 2 to the case where Πmax → ∞:
+wg+(rp) =
+�
+dχ qg(χ)
+�
+dχ′ qγ(χ′)⟨δ(⃗xp, χ)γ+(⃗x′
+p, χ′)⟩.
+(A1)
+The Limber approximation [58] consists of assuming that the galaxy positions and the intrinsic + component of
+the shape field are uncorrelated unless they are evaluated at the same redshift or line-of-sight distance. In other
+words, there is a coherence scale [59] over which the correlation is non-zero and this is much smaller than the infinite
+projection baseline we are using to project ξm+.
+Replacing the three-dimensional correlation function by its Fourier transform, we obtain
+wg+(rp) =
+�
+dχ qg(χ)
+�
+dχ′ qγ(χ′)
+�
+d3k
+(2π)3
+�
+d3k′
+(2π)3 ⟨ˆδ(⃗k, χ)ˆγ+(⃗k′, χ′)⟩e−i⃗k⊥·x⊥e−i⃗k′
+⊥·x′
+⊥e−ikzχe−ik′
+zΠ′.
+(A2)
+Here we have aligned the component of the wavevector that is perpendicular to the line-of-sight with the x axis
+
+10
+without loss of generality [5]. By explicitly modelling the power spectrum of the density and the shapes, we can write
+wg+(rp) =
+�
+dχ qg(χ)
+�
+dχ′ qγ(χ′)
+�
+d3k
+(2π)3
+�
+d3k′
+(2π)3 Pm+(⃗k, z)(2π)3δD(⃗k − ⃗k′)e−i⃗k⊥·x��e−i⃗k′
+⊥·x′
+⊥e−ikzχe−ik′
+zΠ′. (A3)
+and collapse one of the integrals in wavevector to obtain
+wg+(rp) =
+�
+dχ qg(χ)
+�
+dχ′ qγ(χ′)
+�
+d3k
+(2π)3 Pm+(⃗k, z)e−i⃗k⊥·(⃗x⊥−⃗x′
+⊥)e−ikzχe−ikzχ′.
+(A4)
+Applying the Limber approximation,
+wg+(rp) =
+�
+dχ qg(χ) qγ(χ)
+�
+dχ′
+�
+d3k
+(2π)3 Pm+(⃗k, z)e−i⃗k⊥·(⃗x⊥−⃗x′
+⊥)e−ikzχe−ikzχ′.
+(A5)
+From here onward, we will assume qg(χ) = qγ(χ) = δD(χ), which corresponds to correlations are evaluated at z = 0 for
+simplicity. The integral over χ′ can now be brought inside, resulting in a Dirac delta over the line-of-sight wavevector:
+wg+(rp) =
+�
+d3k
+(2π)3 Pg+(⃗k, z = 0)e−i⃗k⊥·(⃗x⊥−⃗x′
+⊥)e−ikzχ2πδD(kz).
+(A6)
+Before continuing we re-write Pg+ explicitly:
+wg+(rp) = −C1ρcritΩmbg
+D(z)
+�
+d3k
+(2π)3 P(k, z = 0)k2
+x − k2
+y
+k2
+e−i⃗k⊥·(⃗x⊥−⃗x′
+⊥)e−ikzχ2πδD(kz).
+(A7)
+where ⃗k⊥ = (kx, ky). The presence of the Dirac delta in kz simplifies the whole expression to
+wg+(rp) = −C1ρcritΩmbg
+D(z)
+�
+d2k⊥
+(2π)2 P(k⊥, z = 0)k2
+x − k2
+y
+k2
+⊥
+e−i⃗k⊥·(⃗x⊥−⃗x′
+⊥).
+(A8)
+If θk is the angle between ⃗k⊥ and the x axis and k⊥ = |⃗k⊥|, then
+wg+(rp) = −C1ρcritΩmbg
+D(z)
+� dk⊥dθk k⊥
+(2π)2
+P(k⊥, z = 0) cos(2θk)e−ik⊥rp cos θk.
+(A9)
+This makes the second order Bessel function appear and now the integral is over the absolute value of
+wg+(rp) = C1ρcritΩmbg
+D(z)
+� dk⊥
+2π k⊥P(k⊥, z)J2(k⊥rp).
+(A10)
+Similarly for the correlation with the matter field,
+wm+(rp) = C1ρcritΩm
+D(z)
+� dk⊥
+2π k⊥P(k⊥, z)J2(k⊥rp).
+(A11)
+This is in agreement with Eq. 8.
+[1] B. Bassett and R. Hlozek, Baryon acoustic oscillations, in
+Dark Energy: Observational and Theoretical Approaches,
+edited by P. Ruiz-Lapuente (2010) p. 246.
+[2] D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein,
+C. Hirata, A. G. Riess, and E. Rozo, Observational
+probes of cosmic acceleration, Phys. Rep. 530, 87 (2013),
+arXiv:1201.2434 [astro-ph.CO].
+[3] M. L. Brown, A. N. Taylor, N. C. Hambly, and S. Dye,
+Measurement of intrinsic alignments in galaxy elliptici-
+ties, MNRAS 333, 501 (2002), arXiv:astro-ph/0009499
+[astro-ph].
+[4] P. Catelan, M. Kamionkowski, and R. D. Blandford, In-
+trinsic and extrinsic galaxy alignment, MNRAS 320, L7
+(2001), arXiv:astro-ph/0005470 [astro-ph].
+
+11
+[5] J. Blazek, M. McQuinn, and U. Seljak, Testing the
+tidal alignment model of galaxy intrinsic alignment,
+J.
+Cosmology
+Astropart.
+Phys.
+2011,
+010
+(2011),
+arXiv:1101.4017 [astro-ph.CO].
+[6] B. Joachimi, R. Mandelbaum, F. B. Abdalla, and S. L.
+Bridle, Constraints on intrinsic alignment contamination
+of weak lensing surveys using the MegaZ-LRG sample,
+A&A 527, A26 (2011), arXiv:1008.3491 [astro-ph.CO].
+[7] S. Singh, R. Mandelbaum, and S. More, Intrinsic align-
+ments of SDSS-III BOSS LOWZ sample galaxies, MN-
+RAS 450, 2195 (2015), arXiv:1411.1755 [astro-ph.CO].
+[8] H. Johnston, C. Georgiou, B. Joachimi, H. Hoekstra,
+N. E. Chisari, D. Farrow, M. C. Fortuna, C. Heymans,
+S. Joudaki, K. Kuijken, and A. Wright, KiDS+GAMA:
+Intrinsic alignment model constraints for current and
+future weak lensing cosmology, A&A 624, A30 (2019),
+arXiv:1811.09598 [astro-ph.CO].
+[9] M. C. Fortuna, H. Hoekstra, H. Johnston, M. Vakili,
+A. Kannawadi, C. Georgiou, B. Joachimi, A. H. Wright,
+M. Asgari, M. Bilicki, C. Heymans, H. Hildebrandt,
+K. Kuijken, and M. Von Wietersheim-Kramsta, KiDS-
+1000: Constraints on the intrinsic alignment of luminous
+red galaxies, A&A 654, A76 (2021), arXiv:2109.02556
+[astro-ph.CO].
+[10] C. M. Hirata, Tidal alignments as a contaminant of
+redshift space distortions, MNRAS 399, 1074 (2009),
+arXiv:0903.4929 [astro-ph.CO].
+[11] D. Kirk, A. Rassat, O. Host, and S. Bridle, The cos-
+mological impact of intrinsic alignment model choice for
+cosmic shear, MNRAS 424, 1647 (2012), arXiv:1112.4752
+[astro-ph.CO].
+[12] E. Krause, T. Eifler, and J. Blazek, The impact of in-
+trinsic alignment on current and future cosmic shear sur-
+veys, MNRAS 456, 207 (2016), arXiv:1506.08730 [astro-
+ph.CO].
+[13] K. Zwetsloot and N. E. Chisari, Impact of intrinsic align-
+ments on clustering constraints of the growth rate, MN-
+RAS 516, 787 (2022), arXiv:2208.07062 [astro-ph.CO].
+[14] N. E. Chisari and C. Dvorkin, Cosmological informa-
+tion in the intrinsic alignments of luminous red galax-
+ies, J. Cosmology Astropart. Phys. 2013, 029 (2013),
+arXiv:1308.5972 [astro-ph.CO].
+[15] N. E. Chisari, C. Dvorkin, and F. Schmidt, Can weak
+lensing surveys confirm BICEP2?, Phys. Rev. D 90,
+043527 (2014), arXiv:1406.4871 [astro-ph.CO].
+[16] F. Schmidt, N. E. Chisari, and C. Dvorkin, Imprint of
+inflation on galaxy shape correlations, J. Cosmology As-
+tropart. Phys. 2015, 032 (2015), arXiv:1506.02671 [astro-
+ph.CO].
+[17] M. Biagetti and G. Orlando, Primordial gravitational
+waves from galaxy intrinsic alignments, J. Cosmology As-
+tropart. Phys. 2020, 005 (2020), arXiv:2001.05930 [astro-
+ph.CO].
+[18] A. Taruya and T. Okumura, Improving Geometric
+and Dynamical Constraints on Cosmology with In-
+trinsic Alignments of Galaxies, ApJ 891, L42 (2020),
+arXiv:2001.05962 [astro-ph.CO].
+[19] K. S. Dawson, D. J. Schlegel, C. P. Ahn, S. F. Anderson,
+´E. Aubourg, S. Bailey, R. H. Barkhouser, J. E. Bautista,
+A. Beifiori, A. A. Berlind, V. Bhardwaj, D. Bizyaev,
+C. H. Blake, M. R. Blanton, M. Blomqvist, et al., The
+Baryon Oscillation Spectroscopic Survey of SDSS-III, AJ
+145, 10 (2013), arXiv:1208.0022 [astro-ph.CO].
+[20] DESI
+Collaboration,
+A.
+Aghamousa,
+J.
+Aguilar,
+S. Ahlen, S. Alam, L. E. Allen, C. Allende Prieto,
+J. Annis, S. Bailey, C. Balland, O. Ballester, C. Baltay,
+L. Beaufore, C. Bebek, T. C. Beers, E. F. Bell, et al.,
+The DESI Experiment Part I: Science,Targeting, and
+Survey Design, arXiv e-prints , arXiv:1611.00036 (2016),
+arXiv:1611.00036 [astro-ph.IM].
+[21] H.-J. Seo and D. J. Eisenstein, Improved Forecasts for
+the Baryon Acoustic Oscillations and Cosmological Dis-
+tance Scale, ApJ 665, 14 (2007), arXiv:astro-ph/0701079
+[astro-ph].
+[22] H.-J. Seo, J. Eckel, D. J. Eisenstein, K. Mehta, M. Metch-
+nik, N. Padmanabhan, P. Pinto, R. Takahashi, M. White,
+and X. Xu, High-precision Predictions for the Acoustic
+Scale in the Nonlinear Regime, ApJ 720, 1650 (2010),
+arXiv:0910.5005 [astro-ph.CO].
+[23] P. Arnalte-Mur, A. Labatie, N. Clerc, V. J. Mart´ınez,
+J. L. Starck, M. Lachi`eze-Rey, E. Saar, and S. Pare-
+des, Wavelet analysis of baryon acoustic structures
+in the galaxy distribution, A&A 542, A34 (2012),
+arXiv:1101.1911 [astro-ph.CO].
+[24] H. J. Tian, M. C. Neyrinck, T. Budav´ari, and A. S.
+Szalay, Redshift-space Enhancement of Line-of-sight
+Baryon Acoustic Oscillations in the Sloan Digital Sky
+Survey
+Main-galaxy
+Sample,
+ApJ
+728,
+34
+(2011),
+arXiv:1011.2481 [astro-ph.CO].
+[25] W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol,
+J. A. Peacock, A. C. Pope, and A. S. Szalay, Measuring
+the Baryon Acoustic Oscillation scale using the Sloan
+Digital Sky Survey and 2dF Galaxy Redshift Survey,
+MNRAS 381, 1053 (2007), arXiv:0705.3323 [astro-ph].
+[26] T. Okumura, A. Taruya, and T. Nishimichi, Intrinsic
+alignment statistics of density and velocity fields at large
+scales: Formulation, modeling, and baryon acoustic os-
+cillation features, Phys. Rev. D 100, 103507 (2019),
+arXiv:1907.00750 [astro-ph.CO].
+[27] T. Nishimichi, M. Takada, R. Takahashi, K. Osato,
+M. Shirasaki, T. Oogi, H. Miyatake, M. Oguri, R. Mu-
+rata, Y. Kobayashi, and N. Yoshida, Dark Quest. I. Fast
+and Accurate Emulation of Halo Clustering Statistics
+and Its Application to Galaxy Clustering, ApJ 884, 29
+(2019), arXiv:1811.09504 [astro-ph.CO].
+[28] E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett,
+B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. Limon,
+L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut,
+S. S. Meyer, et al., Five-Year Wilkinson Microwave
+Anisotropy Probe Observations: Cosmological Interpre-
+tation, ApJS 180, 330 (2009), arXiv:0803.0547 [astro-
+ph].
+[29] D. J. Eisenstein and W. Hu, Baryonic Features in
+the Matter Transfer Function, ApJ 496, 605 (1998),
+arXiv:astro-ph/9709112 [astro-ph].
+[30] Planck Collaboration,
+P. A. R. Ade,
+N. Aghanim,
+M.
+Arnaud,
+M.
+Ashdown,
+J.
+Aumont,
+C.
+Bacci-
+galupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett,
+N. Bartolo,
+E. Battaner,
+R. Battye,
+K. Benabed,
+A. Benoˆıt, A. Benoit-L´evy, et al., Planck 2015 results.
+XIII. Cosmological parameters, A&A 594, A13 (2016),
+arXiv:1502.01589 [astro-ph.CO].
+[31] N. Hand, Y. Feng, F. Beutler, Y. Li, C. Modi, U. Seljak,
+and Z. Slepian, nbodykit: An Open-source, Massively
+Parallel Toolkit for Large-scale Structure, AJ 156, 160
+(2018), arXiv:1712.05834 [astro-ph.IM].
+[32] https://github.com/LSSTDESC/CCL.
+
+12
+[33] N. E. Chisari, D. Alonso, E. Krause, C. D. Leonard,
+P. Bull, J. Neveu, A. S. Villarreal, S. Singh, T. McClin-
+tock, J. Ellison, Z. Du, J. Zuntz, A. Mead, S. Joudaki,
+C. S. Lorenz,
+T. Tr¨oster,
+J. Sanchez,
+F. Lanusse,
+M. Ishak, R. Hlozek, J. Blazek, J.-E. Campagne, H. Al-
+moubayyed, T. Eifler, M. Kirby, D. Kirkby, S. Plaszczyn-
+ski, A. Slosar, M. Vrastil, E. L. Wagoner, and LSST Dark
+Energy Science Collaboration, Core Cosmology Library:
+Precision Cosmological Predictions for LSST, ApJS 242,
+2 (2019), arXiv:1812.05995 [astro-ph.CO].
+[34] A. Tenneti, S. Singh, R. Mandelbaum, T. di Matteo,
+Y. Feng, and N. Khandai, Intrinsic alignments of galaxies
+in the MassiveBlack-II simulation: analysis of two-point
+statistics, MNRAS 448, 3522 (2015), arXiv:1409.7297
+[astro-ph.CO].
+[35] N. Chisari, S. Codis, C. Laigle, Y. Dubois, C. Pichon,
+J. Devriendt, A. Slyz, L. Miller, R. Gavazzi, and K. Ben-
+abed, Intrinsic alignments of galaxies in the Horizon-
+AGN cosmological hydrodynamical simulation, MNRAS
+454, 2736 (2015), arXiv:1507.07843 [astro-ph.CO].
+[36] N. Chisari, C. Laigle, S. Codis, Y. Dubois, J. Devriendt,
+L. Miller, K. Benabed, A. Slyz, R. Gavazzi, and C. Pi-
+chon, Redshift and luminosity evolution of the intrinsic
+alignments of galaxies in Horizon-AGN, MNRAS 461,
+2702 (2016), arXiv:1602.08373 [astro-ph.CO].
+[37] S. Hilbert, D. Xu, P. Schneider, V. Springel, M. Vogels-
+berger, and L. Hernquist, Intrinsic alignments of galax-
+ies in the Illustris simulation, MNRAS 468, 790 (2017),
+arXiv:1606.03216 [astro-ph.CO].
+[38] C. Hikage, M. Oguri, T. Hamana, S. More, R. Man-
+delbaum,
+M.
+Takada,
+F.
+K¨ohlinger,
+H.
+Miyatake,
+A. J. Nishizawa, H. Aihara, R. Armstrong, J. Bosch,
+J. Coupon,
+A. Ducout,
+P. Ho,
+et al., Cosmology
+from cosmic shear power spectra with Subaru Hy-
+per Suprime-Cam first-year data, PASJ 71, 43 (2019),
+arXiv:1809.09148 [astro-ph.CO].
+[39] C. Heymans, T. Tr¨oster, M. Asgari, C. Blake, H. Hilde-
+brandt, B. Joachimi, K. Kuijken, C.-A. Lin, A. G.
+S´anchez, J. L. van den Busch, A. H. Wright, A. Amon,
+M. Bilicki, J. de Jong, M. Crocce, et al., KiDS-1000
+Cosmology: Multi-probe weak gravitational lensing and
+spectroscopic galaxy clustering constraints, A&A 646,
+A140 (2021), arXiv:2007.15632 [astro-ph.CO].
+[40] L. F. Secco, S. Samuroff, E. Krause, B. Jain, J. Blazek,
+M. Raveri, A. Campos, A. Amon, A. Chen, C. Doux,
+A. Choi,
+D. Gruen,
+G. M. Bernstein,
+C. Chang,
+J. DeRose, DES Collaboration, et al., Dark Energy Sur-
+vey Year 3 results: Cosmology from cosmic shear and
+robustness to modeling uncertainty, Phys. Rev. D 105,
+023515 (2022), arXiv:2105.13544 [astro-ph.CO].
+[41] C. Porciani, A. Dekel, and Y. Hoffman, Testing tidal-
+torque theory - I. Spin amplitude and direction, MNRAS
+332, 325 (2002), arXiv:astro-ph/0105123 [astro-ph].
+[42] C. Porciani, A. Dekel, and Y. Hoffman, Testing tidal-
+torque theory - II. Alignment of inertia and shear and the
+characteristics of protohaloes, MNRAS 332, 339 (2002),
+arXiv:astro-ph/0105165 [astro-ph].
+[43] S. Codis, C. Pichon, and D. Pogosyan, Spin align-
+ments within the cosmic web: a theory of constrained
+tidal torques near filaments, MNRAS 452, 3369 (2015),
+arXiv:1504.06073 [astro-ph.CO].
+[44] G. Camelio and M. Lombardi, On the origin of intrin-
+sic alignment in cosmic shear measurements: an analytic
+argument, A&A 575, A113 (2015), arXiv:1501.03014
+[astro-ph.CO].
+[45] E. van Uitert and B. Joachimi, Intrinsic alignment of
+redMaPPer clusters: cluster shape-matter density cor-
+relation, MNRAS 468, 4502 (2017), arXiv:1701.02307
+[astro-ph.CO].
+[46] E. Rozo, E. S. Rykoff, A. Abate, C. Bonnett, M. Crocce,
+C. Davis, B. Hoyle, B. Leistedt, H. V. Peiris, R. H. Wech-
+sler, T. Abbott, F. B. Abdalla, M. Banerji, A. H. Bauer,
+A. Benoit-L´evy, et al., redMaGiC: selecting luminous red
+galaxies from the DES Science Verification data, MNRAS
+461, 1431 (2016), arXiv:1507.05460 [astro-ph.IM].
+[47] M. Vakili, M. Bilicki, H. Hoekstra, N. E. Chisari,
+M. J. I. Brown, C. Georgiou, A. Kannawadi, K. Kui-
+jken, and A. H. Wright, Luminous red galaxies in the
+Kilo-Degree Survey: selection with broad-band photom-
+etry and weak lensing measurements, MNRAS 487, 3715
+(2019), arXiv:1811.02518 [astro-ph.CO].
+[48] B. Joachimi, R. Mandelbaum, F. B. Abdalla, and S. L.
+Bridle, Constraints on intrinsic alignment contamination
+of weak lensing surveys using the MegaZ-LRG sample,
+A&A 527, A26 (2011), arXiv:1008.3491 [astro-ph.CO].
+[49] M. Bilicki, H. Hoekstra, M. J. I. Brown, V. Amaro,
+C. Blake, S. Cavuoti, J. T. A. de Jong, C. Geor-
+giou, H. Hildebrandt, C. Wolf, A. Amon, M. Brescia,
+S. Brough, M. V. Costa-Duarte, T. Erben, et al., Pho-
+tometric redshifts for the Kilo-Degree Survey. Machine-
+learning analysis with artificial neural networks, A&A
+616, A69 (2018), arXiv:1709.04205 [astro-ph.CO].
+[50] A. H. Wright, H. Hildebrandt, J. L. van den Busch,
+and
+C.
+Heymans,
+Photometric
+redshift
+calibration
+with self-organising maps, A&A 637, A100 (2020),
+arXiv:1909.09632 [astro-ph.CO].
+[51] O. Ilbert, P. Capak, M. Salvato, H. Aussel, H. J. Mc-
+Cracken, D. B. Sanders, N. Scoville, J. Kartaltepe,
+S. Arnouts, E. Le Floc’h, B. Mobasher, Y. Taniguchi,
+F. Lamareille, A. Leauthaud, S. Sasaki, et al., Cosmos
+Photometric Redshifts with 30-Bands for 2-deg2, ApJ
+690, 1236 (2009), arXiv:0809.2101 [astro-ph].
+[52] M. Eriksen, A. Alarcon, E. Gaztanaga, A. Amara,
+L. Cabayol, J. Carretero, F. J. Castander, M. Crocce,
+M. Delfino, J. De Vicente, E. Fernandez, P. Fosalba,
+J. Garcia-Bellido, H. Hildebrandt, H. Hoekstra, et al.,
+The PAU Survey:
+early demonstration of photomet-
+ric redshift performance in the COSMOS field, MNRAS
+484, 4200 (2019), arXiv:1809.04375 [astro-ph.GA].
+[53] M. Bilicki, A. Dvornik, H. Hoekstra, A. H. Wright, N. E.
+Chisari, M. Vakili, M. Asgari, B. Giblin, C. Heymans,
+H. Hildebrandt, B. W. Holwerda, A. Hopkins, H. John-
+ston, A. Kannawadi, K. Kuijken, et al., Bright galaxy
+sample in the Kilo-Degree Survey Data Release 4. Se-
+lection, photometric redshifts, and physical properties,
+A&A 653, A82 (2021), arXiv:2101.06010 [astro-ph.GA].
+[54] I. R. van Gemeren and N. E. Chisari, Erratum: Prospects
+for detection and application of the alignment of galax-
+ies with the large-scale velocity field [Phys. Rev. D
+102, 123507 (2020)], Phys. Rev. D 104, 069902 (2021),
+arXiv:2011.07087 [astro-ph.CO].
+[55] M. Masi, On compressive radial tidal forces, Ameri-
+can Journal of Physics 75, 116 (2007), arXiv:0705.3747
+[astro-ph].
+[56] A. Carnero Rosell, M. Rodriguez-Monroy, M. Crocce,
+J. Elvin-Poole,
+A. Porredon,
+I. Ferrero,
+J. Mena-
+Fern´andez, R. Cawthon, J. De Vicente, E. Gaztanaga,
+A. J. Ross, E. Sanchez, I. Sevilla-Noarbe, O. Alves,
+
+13
+F. Andrade-Oliveira, DES Collaboration, et al., Dark En-
+ergy Survey Year 3 results: galaxy sample for BAO mea-
+surement, MNRAS 509, 778 (2022), arXiv:2107.05477
+[astro-ph.CO].
+[57] The LSST Dark Energy Science Collaboration, R. Man-
+delbaum, T. Eifler, R. Hloˇzek, T. Collett, E. Gawiser,
+D. Scolnic, D. Alonso, H. Awan, R. Biswas, J. Blazek,
+P. Burchat, N. E. Chisari, I. Dell’Antonio, S. Digel,
+J. Frieman, D. A. Goldstein, I. Hook, ˇZ. Ivezi´c, S. M.
+Kahn, S. Kamath, D. Kirkby, T. Kitching, E. Krause,
+P.-F. Leget, P. J. Marshall, J. Meyers, H. Miyatake,
+J. A. Newman, R. Nichol, E. Rykoff, F. J. Sanchez,
+A. Slosar, M. Sullivan, and M. A. Troxel, The LSST
+Dark Energy Science Collaboration (DESC) Science Re-
+quirements Document, arXiv e-prints , arXiv:1809.01669
+(2018), arXiv:1809.01669 [astro-ph.CO].
+[58] D. N. Limber, The Analysis of Counts of the Extragalac-
+tic Nebulae in Terms of a Fluctuating Density Field., ApJ
+117, 134 (1953).
+[59] M. Bartelmann and P. Schneider, Weak gravitational
+lensing,
+Phys.
+Rep.
+340,
+291
+(2001),
+arXiv:astro-
+ph/9912508 [astro-ph].
+

89FPT4oBgHgl3EQfYjQR/content/tmp_files/2301.13072v1.pdf.txt

@@ -0,0 +1,763 @@
+Guided Deep Reinforcement Learning for Articulated Swimming Robots
+Jiaheng Hu1 and Tony Dear1
+Abstract— Deep reinforcement learning has recently been
+applied to a variety of robotics applications, but learning
+locomotion for robots with unconventional configurations is still
+limited. Prior work has shown that, despite the simple modeling
+of articulated swimmer robots, such systems struggle to find
+effective gaits using reinforcement learning due to the hetero-
+geneity of the search space. In this work, we leverage insight
+from geometric models of these robots in order to focus on
+promising regions of the space and guide the learning process.
+We demonstrate that our augmented learning technique is able
+to produce gaits for different learning goals for swimmer robots
+in both low and high Reynolds number fluids.
+I. INTRODUCTION
+Articulated swimming robots have attracted much interest
+from researchers due to their effective locomotive capabilities
+as well as the richness of their geometric structure. The basis
+of their locomotion arises from the interaction between ac-
+tuation of their joints and the surrounding fluid environment.
+Such interactions depend highly on the nature of the fluid,
+but previous work has shown that in the cases of extremely
+low or extremely high Reynolds number fluids, a kinematic
+system can be approximated, leading to great insights into
+trajectory planning [1].
+Even for these idealized systems, however, it is still
+difficult to derive optimal trajectories analytically. These
+difficulties are compounded when dealing with robots with
+more complex morphologies or higher-dimensional joint
+spaces. Deep reinforcement learning (RL) has recently shown
+promise to be an effective search strategy, as algorithms
+have developed to make techniques feasible on physical
+systems. However, the heterogeneity of the search space and
+the sparsity of the corresponding reward functions introduce
+additional challenges for motion planning with RL.
+In this paper, we exploit the geometric structure of three-
+link swimmer systems in low and high Reynolds number
+fluids to restrict the search space of our reinforcement
+learning algorithm and learn effective locomoting gaits from
+a blank slate. We show that this approach is able to speed
+up training time, as the robot is less likely to be trapped into
+executing suboptimal gaits. At the same time, we show that
+the RL method is still flexible enough to be optimized for
+different objectives, such as energy and speed.
+To the best of our knowledge, this is the first attempt to
+confine RL policy search by utilizing the geometry of the
+system at hand. This is also one of the first attempts to the
+locomotion problem of articulated swimmers using model-
+free deep reinforcement learning.
+1Computer Science Department, Columbia University, New York, NY
+10027, USA {jh3916, tbd2115}@columbia.edu
+Fig. 1: A swimming snake robot comprised of three artic-
+ulated slender bodies. The coordinates (x, y, θ) denote the
+SE(2) inertial configuration of the proximal link, which also
+has velocities (ξx, ξy, ξθ) relative to a body-fixed frame. The
+relative angles of the joints are denoted α = (α1, α2).
+II. PRIOR WORK
+A. Geometric Structure
+In recent decades, techniques and methods from geometric
+mechanics have been a popular way to model and control
+mechanical systems. A key idea is that of symmetries in a
+system’s configuration space, which allow for the reduction
+of the equations of motion to a simpler form. This has been
+addressed for general mechanical systems by [2], as well
+as nonholonomic systems by [3]. For locomoting systems,
+geometric reduction is often leveraged in tandem with a
+decomposition of the configuration variables into actuated
+shape variables and unactuated position variables. If such
+a splitting is possible, then the configuration space often
+takes on a fiber bundle structure, whereby a mapping called
+the connection relates trajectories between each subspace.
+Analysis of the connection can then give us intuition into
+ways to perform motion planning and control of the system,
+as detailed by [4] and [5]. This mathematical structure also
+lends itself to visualization and design tools, detailed by [6].
+Much of the progress in the geometric mechanics of loco-
+motion is predicated on the assumption that the symmetries
+of a system coincide exactly with the position degrees of
+freedom. Robots that can be modeled with nonholonomic
+constraints are examples in which these symmetries occur.
+Nonholonomic wheeled snake robots have received consid-
+erable attention from researchers such as [7] and [8] treating
+them as kinematic systems, so named because constraints
+that eliminate the need to consider second-order dynamics
+when modeling its locomotion. This allows for the treatment
+of the system’s locomotion, and subsequent motion planning,
+as a result of geometric phase (see [9], [10], [11], [12]).
+Geometric methods have also examined systems locomot-
+ing in fluids. As with terrestrial systems, such a description
+arXiv:2301.13072v1  [cs.RO]  30 Jan 2023
+
+is most useful if the position degrees of freedom correspond
+to system symmetries and the rest to internal shape. For
+single bodies, motion may be achieved through temporal
+deformation of the body’s shape. For articulated swimmers
+like the three-link robot shown in Fig. 1, deformation occurs
+naturally when joints are moved relative to each other (see
+[1], [13], [14]), analogous to the terrestrial version of the
+system.
+Articulated swimmer robots belong to a family of gen-
+eral snake-like robots, which are characterized by a large
+number of degrees of freedom and locomotion patterns
+that exhibit cyclic motions through coordination of their
+joints. Therefore, snake-like robots are usually controlled
+through kinematics-based methods [15], [16]. These meth-
+ods, however, often rely on hand-tuning a number of different
+parameters, which can be costly as well as inflexible in new
+environments.
+B. Gait Optimization and Reinforcement Learning
+The problem of gait optimization has been approached
+through a variety of traditional optimization methods, such
+as evolutionary algorithms [17], gradient-based methods [18]
+and Bayesian optimization [19]. However, these methods
+often suffer from local optima, and while the resulting gaits
+appear effective in locomoting the robots, they are often
+still quite inefficient when compared to the natural motion
+achieved by animals.
+Reinforcement learning is a data-driven method that
+searches for a reward-maximizing policy under a given
+environment. As an algorithm based on trial-and-error, it
+has the advantage of not requiring a specific model of the
+environment or expert knowledge of the problem. With re-
+cent advancements in deep neural network and reinforcement
+learning algorithms, reinforcement learning has become a
+useful tool for solving robot control tasks such as walker’s lo-
+comotion [20], dexterous manipulation [21], and autonomous
+driving [22].
+There have been a few attempts to solve the problem of
+gait optimization through reinforcement learning. Bing et
+al. [23] used PPO to train a forward-locomotion controller
+for a wheeled snake robot and were able to generate gaits
+that out-perform those derived from Bayesian optimization
+and grid search. Sharma and Kitani [24] proposed phase-
+DDPG, where they explicitly trained a cyclic policies for a
+walker robot by oscillating the weight of the policy network
+with the phase of the robot. These methods were able to
+generate fairly natural gaits on certain robots, but often failed
+to converge to global optima as the robot environment grew
+more complex. For example, none of the methods were able
+to solve the swimmer environment [25].
+III. MODEL AND METHODS
+A. Swimmer Model
+As shown in Fig. 1, our swimmer robot consists of three
+rigid links, each of length R, which can rotate relative to
+one another. Its configuration is defined by q ∈ Q = G × B,
+where g = (x, y, θ)T ∈ G = SE(2) specifies the position
+and orientation of the first link in an inertial frame; we
+measure a link’s position at the center of the link. The
+joint angles α = (α1, α2)T ∈ B specify the links’ relative
+orientation. We can view Q as a principal fiber bundle,
+in which trajectories in the shape or base space B lift to
+trajectories in the group G (see [11]).
+1) Low Reynolds Number: Following the treatment of [1],
+we assume that the swimmer is comprised of three slender
+bodies and suspended in a planar fluid. In the low Reynolds
+number case, viscous drag forces dominate inertial forces.
+This allows us to approximate the drag forces as linear
+functions of the system’s body and shape velocities ξ and
+˙α; we also assume that net forces acting on the system are
+zero for all time due to damping out by drag forces. We can
+then derive a Pfaffian constraint on the swimming system’s
+velocities as
+F =
+�
+�
+Fx
+Fy
+Fθ
+�
+� =
+�
+�
+0
+0
+0
+�
+� = ω1(α)ξ + ω2(α) ˙α,
+(1)
+where ω1 ∈ R3×3 and ω2 ∈ R3×2. The variables ξ =
+(ξx, ξy, ξθ)T give us the body velocity of the system, as
+shown in Fig. 1. In SE(2), the mapping that takes body
+velocities to inertial velocities is given by ˙g = TeLgξ, where
+TeLg =
+�
+�
+cos θ
+− sin θ
+0
+sin θ
+cos θ
+0
+0
+0
+1
+�
+� .
+(2)
+The full forms of these components can be found in [1].
+The general approach would be to first compute local drag
+forces on each link, and then combine them to find the total
+force components for each of the body frame directions. In
+addition to the system link length R, the kinematics also
+utilize the drag constant of the fluid, characterized by k.
+Since the number of independent constraints is equal to
+the dimension of the group, these equations are sufficient
+to derive a kinematic connection for the system ([8]). In
+other words, the constraint equations fully describe the first-
+order dynamics of the group variables in terms of the shape
+variables only. Thus, Eq. (3) can be rearranged to show this
+explicitly as the kinematic reconstruction equation:
+ξ = −A(α) ˙α = −ω−1
+1 ω2 ˙α.
+(3)
+A(α) is called the local connection form, a mapping that
+depends only on the shape variables, in this case α1 and α2.
+2) High Reynolds Number: The high Reynolds number
+case is opposite from the low Reynolds number environment
+in that inertial forces dominate viscous forces. Despite the
+entirely different swimming conditions, the model of the
+swimmer robot can once again be approximated as kine-
+matic. A Lagrangian for the robot can be expressed in terms
+of its kinetic energy, as there is no means of storing energy
+or application of external forces:
+L = 1
+2
+�
+ξ
+˙α
+�
+M(α)
+�ξ
+˙α
+�
+.
+(4)
+
+The mass matrix M is a function of the system configuration
+α, and it can be decomposed into blocks containing the
+system’s local connection [8]:
+M(α) =
+�
+I(α)
+I(α)A(α)
+(I(α)A(α))T
+m(α)
+�
+.
+To derive the mass matrix M, we recognize that the
+Lagrangian of the three-link system is equal to the sum of the
+Lagrangians Li of each of the individual links. Each link has
+an associated inertia tensor Ii dependent on the shape that
+we use to model it. In addition, each link has an added mass
+Mi, which arises due to the inertia of a displaced fluid as a
+body moves through it; like the inertia tensor, Mi is solely
+a function of the body geometry. [1] gives an example of the
+added mass tensor for an elliptical body. The total effective
+inertia of a single link is then Ii + Mi, which gives us a
+Lagrangian of the form
+L =
+3
+�
+i=1
+Li =
+3
+�
+i=1
+1
+2ξT
+i (Ii + Mi)ξi
+(5)
+Once the total Lagrangian is written down, it can be
+rearranged into the form of Eq. (4), from which the local
+connection A(α) can then be extracted.
+3) Connection Visualization: The structure of the connec-
+tion form in Eq. (3) can be visualized in order to understand
+the response of ξ to input trajectories without regard to time,
+according to [6]. We can first integrate each row of Eq. (3)
+over time to obtain a measure of displacement corresponding
+to the body frame directions. In the world frame, this measure
+provides the exact rotational displacement, i.e., ˙θ = ξθ for
+the third row, and an approximation of the translational
+component for the first two rows. If our input trajectories
+are periodic, we can transform this “body velocity integral”
+into one over the trajectory ψ : [0, T] → B in the joint
+space, since the kinematics are independent of input pacing.
+Stokes’ theorem can then be applied to perform a second
+transformation into an area integral over β, the region of the
+joint space enclosed by ψ:
+−
+� T
+0
+A(α(τ)) ˙α(τ) dτ = −
+�
+ψ
+A(α) dα = −
+�
+β
+dA(α).
+(6)
+The integrand in the rightmost integral is the exterior deriva-
+tive of A, computed as the curl of A in two dimensions.
+For example, the connection exterior derivative of Eq. (3)
+has three components, one for each row i given by
+dAi(α) = ∂Ai,2
+∂α1
+− ∂Ai,1
+∂α2
+,
+(7)
+where Ai,j is the element corresponding to the ith row and
+jth column of A.
+The magnitudes of the body-x component (first row) of
+the connection exterior derivative of each swimmer over the
+α1-α2 joint space, for a fixed set of sample parameters,
+are depicted in Fig. 2. The area integral over an enclosed
+region is the geometric phase, a measure of the expected
+displacement in the corresponding direction. In particular, a
+Fig. 2: Visualizations of the body-x components of the
+local connection’s exterior derivative for the low and high
+Reynolds swimmers, respectively. Periodic trajectories can
+be represented as closed curves on these surfaces, and the
+robot’s associated displacement corresponds to the enclosed
+volume.
+trajectory that advances in a counter-clockwise direction over
+time in joint space will yield positive displacement, since that
+corresponds to a positive area integral; negative displacement
+is achieved with a clockwise trajectory.
+For both swimmers, we see that a high value of the
+body velocity integral, and thus a high displacement per
+gait cycle, is generally achieved by executing gaits that
+encircle a zero contour of these exterior derivative surfaces.
+However, the optimal parameters of these gaits differ for the
+two swimmers, with a larger range for the low Reynolds
+case and a smaller range for the high Reynolds case. In
+addition, the means of finding a gait is not obvious when
+the joint angles are restricted to be smaller than the zero
+contour. Finally, while we do not show them here we may
+also be concerned with the y and θ components as well.
+Analytically optimizing gaits is thus equivalent to solving
+a multi-objective constrained optimization problem over a
+continuous space, a task that becomes exponentially more
+difficult with increasing system complexity.
+B. Baseline-Guided Policy Search (BGPS)
+Based on the geometric models of the robots, we pro-
+pose an augmented reinforcement learning algorithm called
+
+Low Reynolds Number dAx
+2
+α2
+0
+2
+0.10
+0.05
+0.00
+2
+0
+-2
+α1High Reynolds Number dAx
+2
+α2
+0
+5
+0
+-5
+2
+0
+α1
+-2Baseline-Guided Policy Search (BGPS), in which we restrict
+the policy search space of the learning algorithm by utilizing
+a baseline policy approximated from the geometric structure.
+1) Robot Environment Setup: In this work, we focus on
+locomotion for three-link swimmer robots; the study of more
+complex robots will the subject of future work. The state of
+the robot at time t is st = (α1, α2, θ, t) , which contains
+both the joint angles and orientation of the swimmer. The
+action taken by the robot at time t is at = ( ˙α1, ˙α2), the
+velocities of the two joints. We investigate two choices of
+reward functions, which corresponding to two tasks with
+different optimization goals.
+The first task is to optimize the total distance the robot
+travels in a pre-determined direction in a given amount of
+time. The reward is therefore very straightforward: after the
+robot makes a transition (st, at, st+1), the value of the reward
+function Rt is set to be
+Rt = xt+1 − xt.
+(8)
+The second task is to simultaneously maximize the dis-
+tance travelled and minimize the energy spent. We use a
+kinetic energy metric and define the reward function as
+Rt = xt+1 − xt − β∥ ˙α∥,
+(9)
+where β is a coefficient that controls the weight of the energy
+penalty.
+2) Proximal Policy Optimization: A number of reinforce-
+ment learning algorithms have been shown to be effective
+for different physical systems, although the comparison of
+their various performances is not the focus of this paper. For
+this work, we choose the proximal policy optimization (PPO)
+algorithm by Schulman et al. [26], in which an agent seeks
+to optimize the surrogate objective within the trust region
+by clipping the probability ratio. PPO has been shown to
+outperform other online policy gradient methods, with the
+advantage of being easy to implement.
+3) Baseline from Geometric Structure: The key idea of
+this work is that we can exploit what we know about
+the system structure, e.g., as shown in Fig. 2, to help
+restrict the search space in which reinforcement learning
+operates. Specifically, the exterior derivative plots suggest
+that the optimal gaits for moving forward can be roughly
+approximated as single-frequency sinusoidal functions whose
+joint-space loops overlay the blue ridges and whose phases
+are large enough to encircle the widths of the same. Note
+that the actual optimal policies have no such restriction, e.g.
+as single-frequency sinusoidal functions. This is particularly
+the case if we have joint limits that prevent the joint angles
+from extending all the way out to the zero contour at the ends
+of the ridges. However, such an approximation is sufficient
+for formulating a baseline policy from which RL techniques
+can then improve upon with a large number of degrees of
+freedom.
+4) RL Policy from Baseline: Once we obtain a baseline
+policy πbase(s) through the method described above, we then
+use reinforcement learning to search for a separate policy
+Fig. 3: The training curve of different action ranges for opti-
+mizing the travelled distance for the low Reynolds swimmer.
+Red: 0.1, orange: 0.2, cyan: 0.3, blue: 0.6.
+πRL(s). Our eventual policy is then
+πfinal(s) = πbase(s) + πRL(s)
+(10)
+The most important reason for using a baseline is that we can
+now control the size of the policy search space by reducing
+the action range of our RL-learned policy, |πRL|. By doing
+so, we limit the policy search to be within the vicinity of our
+baseline policy, thus guiding the policy search. A properly
+small action range can shape the policy search space to be
+near convex, allowing gradient-based methods like RL to be
+particularly suitable.
+5) Action Range from Geometric Structure: Given an
+environment step length t, the amount of deviation δ that
+the robot is allowed from the baseline policy, and an action
+range α, we can relate these quantities as δ = αt. Thus, for
+each action cycle of length T, the maximum deviation per
+cycle is δtotal = αT = Nδ, where N is the number of steps
+per cycle.
+The choice of action range α is another parameter whose
+value can be informed by the system’s geometric structure.
+α can be interpreted as the maximum amount that we would
+allow the policy to “stray” away from the baseline. Since
+the baseline is just an approximation for the optimal policy,
+α needs to be sufficiently large to allow exploration of the
+policy space to occur. However, the exterior derivative plots
+can also give us an upper bound on the action range, as there
+is a finite distance away from our chosen baseline at which
+the effectiveness of an action would start to drop.
+IV. LEARNING AND RESULTS
+We implement BGPS with different action ranges, and
+compare the performances directly with PPO and phase-
+DDPG [24]. Our results show that our algorithm generally
+outperforms the other methods, and that a smaller action
+range is able to boost the performance of the learned policy,
+confirming the importance of confining the policy search
+space.
+A. Parameters
+We implemented both a low and high Reynolds three-link
+swimmer for our simulations. We used a link length of 0.3
+
+80
+Reward
+50
+20
+500k
+1.5M
+2.5M
+3.5M
+4.5M
+Training StepBFG
+PPO
+Phase-DDPG
+BGPS (0.6)
+BGPS (0.3)
+BGPS (0.2)
+BGPS (0.15)
+BGPS (0.1)
+Distance
+111.05
+31.79
+1.14
+32.08
+39.39
+117.6
+133.3
+130.8
+Energy
+75.08
+28.58
+0.08
+21.63
+15.61
+29.88
+37.58
+85.22
+BFG
+PPO
+Phase-DDPG
+BGPS (0.6)
+BGPS (0.3)
+BGPS (0.2)
+BGPS (0.15)
+BGPS (0.1)
+Distance
+94.71
+19.73
+13.27
+122.8
+116.5
+141.8
+126.2
+121.4
+Energy
+58.75
+13.20
+9.62
+9.04
+9.47
+72.87
+77.31
+76.47
+TABLE I: The average reward of the learned policy for the low Reynolds swimmer (top) and high Reynolds swimmer
+(bottom). BFG refers to the baseline policy that we observed from the robots’ geometric structures (no learning). PPO and
+phase-DDPG are the main algorithms to which we compared results. BGPS refers to Baseline-Guided Policy Search (our
+method), with results provided for several choices of action range for different trials.
+m for the low Reynolds case, a nod toward the prevalence of
+micro-swimmers in this category. For the high Reynolds case,
+we set the fluid density to ρ = 1 kg/m3, and treat the links as
+ellipses with semi-major axis a = 4 m and semi-minor axis
+b = 1 m. The exterior derivative plots of the swimmers in
+Fig. 2 were obtained using the same parameter values. Our
+environment step time was set to 0.04 s per step. For both
+the low and high Reynolds swimmer, we run separate trials
+for optimizing the speed with and without energy concern.
+We set β to 0.1 for the task of optimizing for energy usage.
+B. Network Architecture
+We followed the settings outlined in Schulman et al. [26]
+for implementing PPO. Our policy network, which maps
+from observation to action, consists of two hidden layers
+of size 64 and a linear output layer at the end. Rectified
+Linear units (ReLU) were used as the activation function for
+every layer except the output layer. Our value network has
+the same architecture as our policy network, except mapping
+from (observation, action) to value space. No parameter is
+shared between the two networks.
+C. Training Settings
+We run our experiments on a a computer with an i7-8650U
+CPU running at 1.90Ghz and an Nvidia GTX 1070 GPU. For
+each given algorithms and settings, we run for 2.5 million
+time steps. For each single trial, our algorithm takes about 3
+hours to run.
+D. Results
+Table
+I shows the results of different algorithms for
+learning locomotive gaits for each swimmer. The ”Distance”
+row refers to the task of maximizing the distance traveled
+per time in a given direction (the x axis), and the ”Energy”
+row refers to the task of locomoting the robot forward while
+simultaneously minimizing the energy spent.
+BFG refers to “baseline from geometry,” which is the
+baseline gait we estimated by looking at the geometric model
+of the robot. For both swimmers, we set a baseline of
+0.6cos(t) for each joint, with a phase difference of 1 rad
+between them. Baseline-Guided Policy Search (BGPS) is
+our method, and the accompanying number on each column
+header marks the action range for that trial. Both PPO and
+Phase-DDPG are learning from scratch without utilizing the
+geometric model, and both of them perform extremely poorly
+Fig. 4: Joint angle (top) and workspace (bottom) trajectories
+of the low Reynolds swimmer from the best learning trian
+(BGPS 0.15). The joint angle trajectories are similar to but
+improve upon the baseline derived from geometry.
+comparing to the other methods shown. In particular, they are
+unable to learn a gait that performs even close to the baseline
+gait derived from simple inspection.
+BGPS also performs poorly when the action range is too
+large, but beats all other baselines as the action range is
+reduced. Fig. 3 shows the training curve of optimizing the
+distance for the low Reynolds swimmer. We can clearly see
+from the plot that training tends to converge to a higher
+reward when the action range is between 0.1 and 0.2, but
+fails to converge when between 0.3 and 0.6. This shows that
+a smaller action range within the appropriate region is the key
+to our algorithm’s success at locomoting the swimmer. For
+both the low and high Reynolds swimmers, our algorithm
+produced the best result for both the task of optimizing
+distance and of minimizing energy spent, among all the
+methods we tested.
+
+0.8
+al
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Time [sec]0.10
+Robot Trajectory
+0.05
+0.00-
+>
+0.05
+-0.10
+0.15
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+XThe joint angle and workspace trajectories of the low
+Reynolds swimmer learned from the best trial (BGPS 0.15)
+are shown in Fig. 4. As expected, the joint angle trajectories
+are not entirely too different from the baseline that we
+wrote down from inspection of geometry. However, subtle
+differences, such as the varying of the relative phases and
+amplitudes of the two joints over time, suggest the existence
+of higher-frequency components that were not at all obvi-
+ous from simple inspection. The accompanying workspace
+trajectory maximizes the distance reward compared to the
+other learning trials, as shown in the first row of Table I.
+V. CONCLUSIONS AND FUTURE WORK
+We have leveraged traditional motion planning techniques
+from geometric mechanics to make deep reinforcement
+learning feasible for training articulated swimming robots.
+Such systems exhibit challenges, such as a policy search
+space with many local optima, that have previously made it
+difficult for common DRL approaches. Our approach, which
+combines intuition with learning, is able to produce superior
+results for different robot models and different environments.
+The fact that our algorithm is able to work across different
+tasks and robots suggests that this method may easily be
+generalized. Other robots with similar kinematics or even
+dynamics can benefit from initialization with an informed
+baseline. Since the baseline need not be exact, this also opens
+presents an opportunity for work with higher-dimensional
+systems for which pure optimization is very difficult. Visu-
+alization of geometry would not be necessary to determine
+the exact form of optimal gaits.
+The task of selecting a proper action range is still under
+investigation. In this work we had the ability to compare
+different values of this parameter and found the best one
+for the given robot and environment, and the interpretation
+of this parameter will certainly vary for other systems. Real
+systems would not have the luxury of trying different values
+until finding the one that works best. Thus, a direct line of
+future work would be to determine whether the action range
+can also be guided by system geometry.
+REFERENCES
+[1] R. L. Hatton and H. Choset, “Geometric swimming at low and high
+reynolds numbers,” IEEE Transactions on Robotics, vol. 29, no. 3, pp.
+615–624, 2013.
+[2] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symme-
+try: A Basic Exposition of Classical Mechanical Systems.
+Springer
+Science & Business Media, 2013, vol. 17.
+[3] A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and R. M. Murray,
+“Nonholonomic mechanical systems with symmetry,” Archive for
+Rational Mechanics and Analysis, vol. 136, no. 1, pp. 21–99, 1996.
+[4] S. D. Kelly and R. M. Murray, “The geometry and control of
+dissipative systems,” in Proceedings of the 35th IEEE Conference on
+Decision and Control, vol. 1.
+IEEE, 1996, pp. 981–986.
+[5] J. P. Ostrowski and J. W. Burdick, “The geometric mechanics of
+undulatory robotic locomotion,” The International Journal of Robotics
+Research, vol. 17, no. 7, pp. 683–701, 1998.
+[6] R. L. Hatton and H. Choset, “Geometric motion planning: The local
+connection, stokes’ theorem, and the importance of coordinate choice,”
+The International Journal of Robotics Research, vol. 30, no. 8, pp.
+988–1014, 2011.
+[7] J. Ostrowski, “Computing reduced equations for robotic systems
+with constraints and symmetries,” Robotics and Automation, IEEE
+Transactions on, vol. 15, no. 1, pp. 111–123, Feb 1999.
+[8] E. A. Shammas, H. Choset, and A. A. Rizzi, “Geometric motion plan-
+ning analysis for two classes of underactuated mechanical systems,”
+The International Journal of Robotics Research, vol. 26, no. 10, pp.
+1043–1073, 2007.
+[9] R. M. Murray and S. S. Sastry, “Nonholonomic motion planning:
+Steering using sinusoids,” IEEE Transactions on Automatic Control,
+vol. 38, no. 5, pp. 700–716, 1993.
+[10] R. Mukherjee and D. P. Anderson, “Nonholonomic motion planning
+using stoke’s theorem,” in Robotics and Automation, 1993. Proceed-
+ings., 1993 IEEE International Conference on. IEEE, 1993, pp. 802–
+809.
+[11] S. D. Kelly and R. M. Murray, “Geometric phases and robotic
+locomotion,” Journal of Robotic Systems, vol. 12, no. 6, pp. 417–431,
+1995.
+[12] F. Bullo and K. M. Lynch, “Kinematic controllability for decou-
+pled trajectory planning in underactuated mechanical systems,” IEEE
+Transactions on Robotics and Automation, vol. 17, no. 4, pp. 402–412,
+2001.
+[13] J. B. Melli, C. W. Rowley, and D. S. Rufat, “Motion planning for an
+articulated body in a perfect planar fluid,” SIAM Journal on applied
+dynamical systems, vol. 5, no. 4, pp. 650–669, 2006.
+[14] L. Burton, R. L. Hatton, H. Choset, and A. Hosoi, “Two-link swim-
+ming using buoyant orientation,” Physics of Fluids, vol. 22, no. 9, p.
+091703, 2010.
+[15] Z. Zuo, Z. Wang, B. Li, and S. Ma, “Serpentine locomotion of a
+snake-like robot in water environment,” 03 2009, pp. 25 – 30.
+[16] M. Tesch, K. Lipkin, I. Brown, R. Hatton, A. Peck, J. Rembisz,
+and H. Choset, “Parameterized and scripted gaits for modular snake
+robots,” Advanced Robotics, vol. 23, no. 9, pp. 1131–1158, 2009.
+[17] S. Chernova and M. Veloso, “An evolutionary approach to gait learning
+for four-legged robots,” in 2004 IEEE/RSJ International Conference
+on Intelligent Robots and Systems (IROS) (IEEE Cat. No.04CH37566),
+vol. 3, Sep. 2004, pp. 2562–2567 vol.3.
+[18] N. Kohl and P. Stone, “Machine learning for fast quadrupedal loco-
+motion,” in AAAI, 2004.
+[19] R. Calandra, N. Gopalan, A. Seyfarth, J. Peters, and M. Deisenroth,
+“Bayesian gait optimization for bipedal locomotion,” 02 2014, pp.
+274–290.
+[20] X. B. Peng, G. Berseth, K. Yin, and M. van de Panne, “Deeploco:
+Dynamic locomotion skills using hierarchical deep reinforcement
+learning,” ACM Transactions on Graphics (Proc. SIGGRAPH 2017),
+vol. 36, no. 4, 2017.
+[21] A. Rajeswaran, V. Kumar, A. Gupta, J. Schulman, E. Todorov,
+and S. Levine, “Learning complex dexterous manipulation with
+deep
+reinforcement
+learning
+and
+demonstrations,”
+CoRR,
+vol.
+abs/1709.10087, 2017. [Online]. Available: http://arxiv.org/abs/1709.
+10087
+[22] P. Long, T. Fan, X. Liao, W. Liu, H. Zhang, and J. Pan, “Towards
+optimally decentralized multi-robot collision avoidance via deep rein-
+forcement learning,” 2017.
+[23] Z. Bing, C. Lemke, Z. Jiang, K. Huang, and A. Knoll, “Energy-
+efficient slithering gait exploration for a snake-like robot based on
+reinforcement learning,” 2019.
+[24] A. Sharma and K. M. Kitani, “Phase-parametric policies for reinforce-
+ment learning in cyclic environments,” in AAAI, 2018.
+[25] R. Coulom, “Reinforcement learning using neural networks, with
+applications to motor control,” 2002.
+[26] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov,
+“Proximal
+policy
+optimization
+algorithms,”
+arXiv
+preprint
+arXiv:1707.06347, 2017.
+

89FPT4oBgHgl3EQfYjQR/content/tmp_files/load_file.txt

@@ -0,0 +1,499 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf,len=498
+page_content='Guided Deep Reinforcement Learning for Articulated Swimming Robots  Jiaheng Hu1 and Tony Dear1  Abstract— Deep reinforcement learning has recently been  applied to a variety of robotics applications, but learning  locomotion for robots with unconventional configurations is still  limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Prior work has shown that, despite the simple modeling  of articulated swimmer robots, such systems struggle to find  effective gaits using reinforcement learning due to the hetero-  geneity of the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In this work, we leverage insight  from geometric models of these robots in order to focus on  promising regions of the space and guide the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  We demonstrate that our augmented learning technique is able  to produce gaits for different learning goals for swimmer robots  in both low and high Reynolds number fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' INTRODUCTION  Articulated swimming robots have attracted much interest  from researchers due to their effective locomotive capabilities  as well as the richness of their geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The basis  of their locomotion arises from the interaction between ac-  tuation of their joints and the surrounding fluid environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Such interactions depend highly on the nature of the fluid,  but previous work has shown that in the cases of extremely  low or extremely high Reynolds number fluids, a kinematic  system can be approximated, leading to great insights into  trajectory planning [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Even for these idealized systems, however, it is still  difficult to derive optimal trajectories analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' These  difficulties are compounded when dealing with robots with  more complex morphologies or higher-dimensional joint  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Deep reinforcement learning (RL) has recently shown  promise to be an effective search strategy, as algorithms  have developed to make techniques feasible on physical  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' However, the heterogeneity of the search space and  the sparsity of the corresponding reward functions introduce  additional challenges for motion planning with RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  In this paper, we exploit the geometric structure of three-  link swimmer systems in low and high Reynolds number  fluids to restrict the search space of our reinforcement  learning algorithm and learn effective locomoting gaits from  a blank slate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We show that this approach is able to speed  up training time, as the robot is less likely to be trapped into  executing suboptimal gaits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' At the same time, we show that  the RL method is still flexible enough to be optimized for  different objectives, such as energy and speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  To the best of our knowledge, this is the first attempt to  confine RL policy search by utilizing the geometry of the  system at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' This is also one of the first attempts to the  locomotion problem of articulated swimmers using model-  free deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  1Computer Science Department, Columbia University, New York, NY  10027, USA {jh3916, tbd2115}@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='edu  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1: A swimming snake robot comprised of three artic-  ulated slender bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The coordinates (x, y, θ) denote the  SE(2) inertial configuration of the proximal link, which also  has velocities (ξx, ξy, ξθ) relative to a body-fixed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The  relative angles of the joints are denoted α = (α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' PRIOR WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Geometric Structure  In recent decades, techniques and methods from geometric  mechanics have been a popular way to model and control  mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' A key idea is that of symmetries in a  system’s configuration space, which allow for the reduction  of the equations of motion to a simpler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' This has been  addressed for general mechanical systems by [2], as well  as nonholonomic systems by [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For locomoting systems,  geometric reduction is often leveraged in tandem with a  decomposition of the configuration variables into actuated  shape variables and unactuated position variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' If such  a splitting is possible, then the configuration space often  takes on a fiber bundle structure, whereby a mapping called  the connection relates trajectories between each subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Analysis of the connection can then give us intuition into  ways to perform motion planning and control of the system,  as detailed by [4] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' This mathematical structure also  lends itself to visualization and design tools, detailed by [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Much of the progress in the geometric mechanics of loco-  motion is predicated on the assumption that the symmetries  of a system coincide exactly with the position degrees of  freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Robots that can be modeled with nonholonomic  constraints are examples in which these symmetries occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Nonholonomic wheeled snake robots have received consid-  erable attention from researchers such as [7] and [8] treating  them as kinematic systems, so named because constraints  that eliminate the need to consider second-order dynamics  when modeling its locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' This allows for the treatment  of the system’s locomotion, and subsequent motion planning,  as a result of geometric phase (see [9], [10], [11], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Geometric methods have also examined systems locomot-  ing in fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' As with terrestrial systems, such a description  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='13072v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='RO]  30 Jan 2023  is most useful if the position degrees of freedom correspond  to system symmetries and the rest to internal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For  single bodies, motion may be achieved through temporal  deformation of the body’s shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For articulated swimmers  like the three-link robot shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1, deformation occurs  naturally when joints are moved relative to each other (see  [1], [13], [14]), analogous to the terrestrial version of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Articulated swimmer robots belong to a family of gen-  eral snake-like robots, which are characterized by a large  number of degrees of freedom and locomotion patterns  that exhibit cyclic motions through coordination of their  joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Therefore, snake-like robots are usually controlled  through kinematics-based methods [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' These meth-  ods, however, often rely on hand-tuning a number of different  parameters, which can be costly as well as inflexible in new  environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Gait Optimization and Reinforcement Learning  The problem of gait optimization has been approached  through a variety of traditional optimization methods, such  as evolutionary algorithms [17], gradient-based methods [18]  and Bayesian optimization [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' However, these methods  often suffer from local optima, and while the resulting gaits  appear effective in locomoting the robots, they are often  still quite inefficient when compared to the natural motion  achieved by animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Reinforcement learning is a data-driven method that  searches for a reward-maximizing policy under a given  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' As an algorithm based on trial-and-error, it  has the advantage of not requiring a specific model of the  environment or expert knowledge of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' With re-  cent advancements in deep neural network and reinforcement  learning algorithms, reinforcement learning has become a  useful tool for solving robot control tasks such as walker’s lo-  comotion [20], dexterous manipulation [21], and autonomous  driving [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  There have been a few attempts to solve the problem of  gait optimization through reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Bing et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' [23] used PPO to train a forward-locomotion controller  for a wheeled snake robot and were able to generate gaits  that out-perform those derived from Bayesian optimization  and grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Sharma and Kitani [24] proposed phase-  DDPG, where they explicitly trained a cyclic policies for a  walker robot by oscillating the weight of the policy network  with the phase of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' These methods were able to  generate fairly natural gaits on certain robots, but often failed  to converge to global optima as the robot environment grew  more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For example, none of the methods were able  to solve the swimmer environment [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' MODEL AND METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Swimmer Model  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1, our swimmer robot consists of three  rigid links, each of length R, which can rotate relative to  one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Its configuration is defined by q ∈ Q = G × B,  where g = (x, y, θ)T ∈ G = SE(2) specifies the position  and orientation of the first link in an inertial frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' we  measure a link’s position at the center of the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The  joint angles α = (α1, α2)T ∈ B specify the links’ relative  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We can view Q as a principal fiber bundle,  in which trajectories in the shape or base space B lift to  trajectories in the group G (see [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  1) Low Reynolds Number: Following the treatment of [1],  we assume that the swimmer is comprised of three slender  bodies and suspended in a planar fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In the low Reynolds  number case, viscous drag forces dominate inertial forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  This allows us to approximate the drag forces as linear  functions of the system’s body and shape velocities ξ and  ˙α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' we also assume that net forces acting on the system are  zero for all time due to damping out by drag forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We can  then derive a Pfaffian constraint on the swimming system’s  velocities as  F =  �  �  Fx  Fy  Fθ  �  � =  �  �  0  0  0  �  � = ω1(α)ξ + ω2(α) ˙α,  (1)  where ω1 ∈ R3×3 and ω2 ∈ R3×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The variables ξ =  (ξx, ξy, ξθ)T give us the body velocity of the system, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In SE(2), the mapping that takes body  velocities to inertial velocities is given by ˙g = TeLgξ, where  TeLg =  �  �  cos θ  − sin θ  0  sin θ  cos θ  0  0  0  1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  (2)  The full forms of these components can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  The general approach would be to first compute local drag  forces on each link, and then combine them to find the total  force components for each of the body frame directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In  addition to the system link length R, the kinematics also  utilize the drag constant of the fluid, characterized by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Since the number of independent constraints is equal to  the dimension of the group, these equations are sufficient  to derive a kinematic connection for the system ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In  other words, the constraint equations fully describe the first-  order dynamics of the group variables in terms of the shape  variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' (3) can be rearranged to show this  explicitly as the kinematic reconstruction equation:  ξ = −A(α) ˙α = −ω−1  1 ω2 ˙α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  (3)  A(α) is called the local connection form, a mapping that  depends only on the shape variables, in this case α1 and α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  2) High Reynolds Number: The high Reynolds number  case is opposite from the low Reynolds number environment  in that inertial forces dominate viscous forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Despite the  entirely different swimming conditions, the model of the  swimmer robot can once again be approximated as kine-  matic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' A Lagrangian for the robot can be expressed in terms  of its kinetic energy, as there is no means of storing energy  or application of external forces:  L = 1  2  �  ξ  ˙α  �  M(α)  �ξ  ˙α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  (4)  The mass matrix M is a function of the system configuration  α, and it can be decomposed into blocks containing the  system’s local connection [8]:  M(α) =  �  I(α)  I(α)A(α)  (I(α)A(α))T  m(α)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  To derive the mass matrix M, we recognize that the  Lagrangian of the three-link system is equal to the sum of the  Lagrangians Li of each of the individual links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Each link has  an associated inertia tensor Ii dependent on the shape that  we use to model it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In addition, each link has an added mass  Mi, which arises due to the inertia of a displaced fluid as a  body moves through it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' like the inertia tensor, Mi is solely  a function of the body geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' [1] gives an example of the  added mass tensor for an elliptical body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The total effective  inertia of a single link is then Ii + Mi, which gives us a  Lagrangian of the form  L =  3  �  i=1  Li =  3  �  i=1  1  2ξT  i (Ii + Mi)ξi  (5)  Once the total Lagrangian is written down, it can be  rearranged into the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' (4), from which the local  connection A(α) can then be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  3) Connection Visualization: The structure of the connec-  tion form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' (3) can be visualized in order to understand  the response of ξ to input trajectories without regard to time,  according to [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We can first integrate each row of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' (3)  over time to obtain a measure of displacement corresponding  to the body frame directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In the world frame, this measure  provides the exact rotational displacement, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=', ˙θ = ξθ for  the third row, and an approximation of the translational  component for the first two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' If our input trajectories  are periodic, we can transform this “body velocity integral”  into one over the trajectory ψ : [0, T] → B in the joint  space, since the kinematics are independent of input pacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Stokes’ theorem can then be applied to perform a second  transformation into an area integral over β, the region of the  joint space enclosed by ψ:  −  � T  0  A(α(τ)) ˙α(τ) dτ = −  �  ψ  A(α) dα = −  �  β  dA(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  (6)  The integrand in the rightmost integral is the exterior deriva-  tive of A, computed as the curl of A in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  For example, the connection exterior derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' (3)  has three components, one for each row i given by  dAi(α) = ∂Ai,2  ∂α1  − ∂Ai,1  ∂α2  ,  (7)  where Ai,j is the element corresponding to the ith row and  jth column of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  The magnitudes of the body-x component (first row) of  the connection exterior derivative of each swimmer over the  α1-α2 joint space, for a fixed set of sample parameters,  are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The area integral over an enclosed  region is the geometric phase, a measure of the expected  displacement in the corresponding direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In particular, a  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 2: Visualizations of the body-x components of the  local connection’s exterior derivative for the low and high  Reynolds swimmers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Periodic trajectories can  be represented as closed curves on these surfaces, and the  robot’s associated displacement corresponds to the enclosed  volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  trajectory that advances in a counter-clockwise direction over  time in joint space will yield positive displacement, since that  corresponds to a positive area integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' negative displacement  is achieved with a clockwise trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  For both swimmers, we see that a high value of the  body velocity integral, and thus a high displacement per  gait cycle, is generally achieved by executing gaits that  encircle a zero contour of these exterior derivative surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  However, the optimal parameters of these gaits differ for the  two swimmers, with a larger range for the low Reynolds  case and a smaller range for the high Reynolds case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In  addition, the means of finding a gait is not obvious when  the joint angles are restricted to be smaller than the zero  contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Finally, while we do not show them here we may  also be concerned with the y and θ components as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Analytically optimizing gaits is thus equivalent to solving  a multi-objective constrained optimization problem over a  continuous space, a task that becomes exponentially more  difficult with increasing system complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Baseline-Guided Policy Search (BGPS)  Based on the geometric models of the robots, we pro-  pose an augmented reinforcement learning algorithm called  Low Reynolds Number dAx  2  α2  0  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='00  2  0  2  α1High Reynolds Number dAx  2  α2  0  5  0  5  2  0  α1  2Baseline-Guided Policy Search (BGPS), in which we restrict  the policy search space of the learning algorithm by utilizing  a baseline policy approximated from the geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  1) Robot Environment Setup: In this work, we focus on  locomotion for three-link swimmer robots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' the study of more  complex robots will the subject of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The state of  the robot at time t is st = (α1, α2, θ, t) , which contains  both the joint angles and orientation of the swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The  action taken by the robot at time t is at = ( ˙α1, ˙α2), the  velocities of the two joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We investigate two choices of  reward functions, which corresponding to two tasks with  different optimization goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  The first task is to optimize the total distance the robot  travels in a pre-determined direction in a given amount of  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The reward is therefore very straightforward: after the  robot makes a transition (st, at, st+1), the value of the reward  function Rt is set to be  Rt = xt+1 − xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  (8)  The second task is to simultaneously maximize the dis-  tance travelled and minimize the energy spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We use a  kinetic energy metric and define the reward function as  Rt = xt+1 − xt − β∥ ˙α∥,  (9)  where β is a coefficient that controls the weight of the energy  penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  2) Proximal Policy Optimization: A number of reinforce-  ment learning algorithms have been shown to be effective  for different physical systems, although the comparison of  their various performances is not the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For  this work, we choose the proximal policy optimization (PPO)  algorithm by Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' [26], in which an agent seeks  to optimize the surrogate objective within the trust region  by clipping the probability ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' PPO has been shown to  outperform other online policy gradient methods, with the  advantage of being easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  3) Baseline from Geometric Structure: The key idea of  this work is that we can exploit what we know about  the system structure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=', as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 2, to help  restrict the search space in which reinforcement learning  operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Specifically, the exterior derivative plots suggest  that the optimal gaits for moving forward can be roughly  approximated as single-frequency sinusoidal functions whose  joint-space loops overlay the blue ridges and whose phases  are large enough to encircle the widths of the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Note  that the actual optimal policies have no such restriction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  as single-frequency sinusoidal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' This is particularly  the case if we have joint limits that prevent the joint angles  from extending all the way out to the zero contour at the ends  of the ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' However, such an approximation is sufficient  for formulating a baseline policy from which RL techniques  can then improve upon with a large number of degrees of  freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  4) RL Policy from Baseline: Once we obtain a baseline  policy πbase(s) through the method described above, we then  use reinforcement learning to search for a separate policy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 3: The training curve of different action ranges for opti-  mizing the travelled distance for the low Reynolds swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Red: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='1, orange: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2, cyan: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3, blue: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  πRL(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Our eventual policy is then  πfinal(s) = πbase(s) + πRL(s)  (10)  The most important reason for using a baseline is that we can  now control the size of the policy search space by reducing  the action range of our RL-learned policy, |πRL|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' By doing  so, we limit the policy search to be within the vicinity of our  baseline policy, thus guiding the policy search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' A properly  small action range can shape the policy search space to be  near convex, allowing gradient-based methods like RL to be  particularly suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  5) Action Range from Geometric Structure: Given an  environment step length t, the amount of deviation δ that  the robot is allowed from the baseline policy, and an action  range α, we can relate these quantities as δ = αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Thus, for  each action cycle of length T, the maximum deviation per  cycle is δtotal = αT = Nδ, where N is the number of steps  per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  The choice of action range α is another parameter whose  value can be informed by the system’s geometric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  α can be interpreted as the maximum amount that we would  allow the policy to “stray” away from the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Since  the baseline is just an approximation for the optimal policy,  α needs to be sufficiently large to allow exploration of the  policy space to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' However, the exterior derivative plots  can also give us an upper bound on the action range, as there  is a finite distance away from our chosen baseline at which  the effectiveness of an action would start to drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' LEARNING AND RESULTS  We implement BGPS with different action ranges, and  compare the performances directly with PPO and phase-  DDPG [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Our results show that our algorithm generally  outperforms the other methods, and that a smaller action  range is able to boost the performance of the learned policy,  confirming the importance of confining the policy search  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Parameters  We implemented both a low and high Reynolds three-link  swimmer for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We used a link length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3  80  Reward  50  20  500k  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5M  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5M  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5M  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5M  Training StepBFG  PPO  Phase-DDPG  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='15)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='1)  Distance  111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='05  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='14  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='08  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='39  117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6  133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3  130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='8  Energy  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='08  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='08  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='63  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='61  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='88  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='58  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='22  BFG  PPO  Phase-DDPG  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='15)  BGPS (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='1)  Distance  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='71  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='73  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='27  122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='8  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5  141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='8  126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2  121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='4  Energy  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='75  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='20  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='62  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='04  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='47  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='87  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='31  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='47  TABLE I: The average reward of the learned policy for the low Reynolds swimmer (top) and high Reynolds swimmer  (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' BFG refers to the baseline policy that we observed from the robots’ geometric structures (no learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' PPO and  phase-DDPG are the main algorithms to which we compared results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' BGPS refers to Baseline-Guided Policy Search (our  method), with results provided for several choices of action range for different trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  m for the low Reynolds case, a nod toward the prevalence of  micro-swimmers in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For the high Reynolds case,  we set the fluid density to ρ = 1 kg/m3, and treat the links as  ellipses with semi-major axis a = 4 m and semi-minor axis  b = 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The exterior derivative plots of the swimmers in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 2 were obtained using the same parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Our  environment step time was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='04 s per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For both  the low and high Reynolds swimmer, we run separate trials  for optimizing the speed with and without energy concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  We set β to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='1 for the task of optimizing for energy usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Network Architecture  We followed the settings outlined in Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' [26]  for implementing PPO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Our policy network, which maps  from observation to action, consists of two hidden layers  of size 64 and a linear output layer at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Rectified  Linear units (ReLU) were used as the activation function for  every layer except the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Our value network has  the same architecture as our policy network, except mapping  from (observation, action) to value space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' No parameter is  shared between the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Training Settings  We run our experiments on a a computer with an i7-8650U  CPU running at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='90Ghz and an Nvidia GTX 1070 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For  each given algorithms and settings, we run for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5 million  time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For each single trial, our algorithm takes about 3  hours to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Results  Table  I shows the results of different algorithms for  learning locomotive gaits for each swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The ”Distance”  row refers to the task of maximizing the distance traveled  per time in a given direction (the x axis), and the ”Energy”  row refers to the task of locomoting the robot forward while  simultaneously minimizing the energy spent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  BFG refers to “baseline from geometry,” which is the  baseline gait we estimated by looking at the geometric model  of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For both swimmers, we set a baseline of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6cos(t) for each joint, with a phase difference of 1 rad  between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Baseline-Guided Policy Search (BGPS) is  our method, and the accompanying number on each column  header marks the action range for that trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Both PPO and  Phase-DDPG are learning from scratch without utilizing the  geometric model, and both of them perform extremely poorly  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 4: Joint angle (top) and workspace (bottom) trajectories  of the low Reynolds swimmer from the best learning trian  (BGPS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The joint angle trajectories are similar to but  improve upon the baseline derived from geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  comparing to the other methods shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In particular, they are  unable to learn a gait that performs even close to the baseline  gait derived from simple inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  BGPS also performs poorly when the action range is too  large, but beats all other baselines as the action range is  reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 3 shows the training curve of optimizing the  distance for the low Reynolds swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' We can clearly see  from the plot that training tends to converge to a higher  reward when the action range is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2, but  fails to converge when between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' This shows that  a smaller action range within the appropriate region is the key  to our algorithm’s success at locomoting the swimmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' For  both the low and high Reynolds swimmers, our algorithm  produced the best result for both the task of optimizing  distance and of minimizing energy spent, among all the  methods we tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='8  al  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='8  0  5  10  15  20  25  30  35  40  Time [sec]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='10  Robot Trajectory  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='00-  >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='0  XThe joint angle and workspace trajectories of the low  Reynolds swimmer learned from the best trial (BGPS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='15)  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' As expected, the joint angle trajectories  are not entirely too different from the baseline that we  wrote down from inspection of geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' However, subtle  differences, such as the varying of the relative phases and  amplitudes of the two joints over time, suggest the existence  of higher-frequency components that were not at all obvi-  ous from simple inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' The accompanying workspace  trajectory maximizes the distance reward compared to the  other learning trials, as shown in the first row of Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' CONCLUSIONS AND FUTURE WORK  We have leveraged traditional motion planning techniques  from geometric mechanics to make deep reinforcement  learning feasible for training articulated swimming robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Such systems exhibit challenges, such as a policy search  space with many local optima, that have previously made it  difficult for common DRL approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Our approach, which  combines intuition with learning, is able to produce superior  results for different robot models and different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  The fact that our algorithm is able to work across different  tasks and robots suggests that this method may easily be  generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Other robots with similar kinematics or even  dynamics can benefit from initialization with an informed  baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Since the baseline need not be exact, this also opens  presents an opportunity for work with higher-dimensional  systems for which pure optimization is very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Visu-  alization of geometry would not be necessary to determine  the exact form of optimal gaits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  The task of selecting a proper action range is still under  investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' In this work we had the ability to compare  different values of this parameter and found the best one  for the given robot and environment, and the interpretation  of this parameter will certainly vary for other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Real  systems would not have the luxury of trying different values  until finding the one that works best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Thus, a direct line of  future work would be to determine whether the action range  can also be guided by system geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  REFERENCES  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Hatton and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Choset, “Geometric swimming at low and high  reynolds numbers,” IEEE Transactions on Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 29, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  615–624, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Marsden and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Ratiu, Introduction to Mechanics and Symme-  try: A Basic Exposition of Classical Mechanical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  Springer  Science & Business Media, 2013, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Bloch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Krishnaprasad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Marsden, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Murray,  “Nonholonomic mechanical systems with symmetry,” Archive for  Rational Mechanics and Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 136, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 21–99, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Kelly and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Murray, “The geometry and control of  dissipative systems,” in Proceedings of the 35th IEEE Conference on  Decision and Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  IEEE, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 981–986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Ostrowski and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Burdick, “The geometric mechanics of  undulatory robotic locomotion,” The International Journal of Robotics  Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 683–701, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Hatton and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Choset, “Geometric motion planning: The local  connection, stokes’ theorem, and the importance of coordinate choice,”  The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  988–1014, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Ostrowski, “Computing reduced equations for robotic systems  with constraints and symmetries,” Robotics and Automation, IEEE  Transactions on, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 111–123, Feb 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Shammas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Choset, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Rizzi, “Geometric motion plan-  ning analysis for two classes of underactuated mechanical systems,”  The International Journal of Robotics Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  1043–1073, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Murray and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Sastry, “Nonholonomic motion planning:  Steering using sinusoids,” IEEE Transactions on Automatic Control,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 700–716, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Mukherjee and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Anderson, “Nonholonomic motion planning  using stoke’s theorem,” in Robotics and Automation, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Proceed-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=', 1993 IEEE International Conference on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' IEEE, 1993, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 802–  809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Kelly and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Murray, “Geometric phases and robotic  locomotion,” Journal of Robotic Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 417–431,  1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Bullo and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Lynch, “Kinematic controllability for decou-  pled trajectory planning in underactuated mechanical systems,” IEEE  Transactions on Robotics and Automation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 402–412,  2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Melli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Rowley, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Rufat, “Motion planning for an  articulated body in a perfect planar fluid,” SIAM Journal on applied  dynamical systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 650–669, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Burton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Hatton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Choset, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Hosoi, “Two-link swim-  ming using buoyant orientation,” Physics of Fluids, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  091703, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [15] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Zuo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Li, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Ma, “Serpentine locomotion of a  snake-like robot in water environment,” 03 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 25 – 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Tesch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Lipkin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Hatton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Peck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Rembisz,  and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Choset, “Parameterized and scripted gaits for modular snake  robots,” Advanced Robotics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 1131–1158, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Chernova and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Veloso, “An evolutionary approach to gait learning  for four-legged robots,” in 2004 IEEE/RSJ International Conference  on Intelligent Robots and Systems (IROS) (IEEE Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='04CH37566),  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 3, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 2562–2567 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [18] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Kohl and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Stone, “Machine learning for fast quadrupedal loco-  motion,” in AAAI, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Calandra, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Gopalan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Seyfarth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Peters, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Deisenroth,  “Bayesian gait optimization for bipedal locomotion,” 02 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  274–290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [20] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Peng, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Berseth, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Yin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' van de Panne, “Deeploco:  Dynamic locomotion skills using hierarchical deep reinforcement  learning,” ACM Transactions on Graphics (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' SIGGRAPH 2017),  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' 4, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Rajeswaran, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Gupta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Schulman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Todorov,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Levine, “Learning complex dexterous manipulation with  deep  reinforcement  learning  and  demonstrations,”  CoRR,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  abs/1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='10087, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Available: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='org/abs/1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  10087  [22] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Long, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Fan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Liao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Pan, “Towards  optimally decentralized multi-robot collision avoidance via deep rein-  forcement learning,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [23] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Bing, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Lemke, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Jiang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Huang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Knoll, “Energy-  efficient slithering gait exploration for a snake-like robot based on  reinforcement learning,” 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Sharma and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Kitani, “Phase-parametric policies for reinforce-  ment learning in cyclic environments,” in AAAI, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Coulom, “Reinforcement learning using neural networks, with  applications to motor control,” 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='  [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Schulman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Wolski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Dhariwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Radford, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content=' Klimov,  “Proximal  policy  optimization  algorithms,”  arXiv  preprint  arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}
+page_content='06347, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89FPT4oBgHgl3EQfYjQR/content/2301.13072v1.pdf'}

8dAyT4oBgHgl3EQfc_fQ/content/tmp_files/2301.00295v1.pdf.txt

@@ -0,0 +1,380 @@
+arXiv:2301.00295v1  [math.GT]  31 Dec 2022
+PACKING MEETS TOPOLOGY
+MICHAEL H. FREEDMAN
+ABSTRACT. This note initiates an investigation of packing links into a region of Euclidean space to
+achieve a maximal density subject to geometric constraints. The upper bounds obtained apply only
+to the class of homotopically essential links and even there seem extravagantly large, leaving much
+working room for the interested reader.
+1. INTRODUCTION AND THEOREMS
+Optimal packing of balls into Euclidean space has a long history and recent astonishing suc-
+cesses, including Hale’s resolution of the Kepler conjecture [H+17] and the optimality of the E8
+and Leech lattices [Via17,CKM+17], resulting in a 2022 Fields Medal to Maryna Viazovska.
+In this note, we introduce the idea of packing links, rather than points, again with the goal of
+achieving the highest possible density subject to the geometric constraints that certain link compo-
+nents must maintain a distance ≥ ε from certain other components. There will be an observation
+about higher dimensions but let us begin with packing classical links into Euclidean 3-space. In
+the classical sphere packing problem, all points are constrained to have distance ≥ ε from each
+other. The analogous stipulation for links, that all components must maintain a distance ≥ ε from
+each other, is also of potential interest, but in that case, the coarse outline of the subject is broadly
+similar to point packings. That is, in both cases, each component takes up a definite amount of
+volume so no more that O(ε−3) link components can be ε-embedded into the unit cube, where
+ε-embedded means no two components approach within ε of each other. While this coarse upper
+bound holds for all link types, complicated links almost surely have smaller upper bounds. For
+example, we conjecture that if Ln is the link type consisting of n-fibers of the Hopf map S3 → S2
+then n can grow no more quickly than O(ε−2).
+However, this note focuses on a regime, partial-ε-embeddings, where even the coarse answer
+can be quite mysterious. By a partial-ε-embedding we mean that only certain specified pairs of
+components must stay ε apart. In this context we are often left puzzled as to whether the number of
+link components that can fit into the unit cube is: (1) countably infinite1, (2) finite but unbounded,
+(3) exponential or super-exponential in ε−1, or (4) polynomial in ε−1.
+The theme of this note is well illustrated by our first example: Hn, which by definition is the
+2n-component link type formed by taking n-Hopf links r
+b each separated from the others by
+some smooth embedded 2-sphere. One may write the link as Hn :=
+∏
+n
+j=1Hj, where Hj = r j ∪bj.
+The partial-ε-embedding condition we study is that for all j, dist(r j,bj) ≥ ε. We call such an
+embedding a diagonal-ε-embedding.
+1This case, of course, would require slightly relaxing the definition of “embedding” to a 1-1 map which, when
+restricted to any finite collection of circles, is a smooth embedding of the expected link type.
+1
+
+2
+MICHAEL H. FREEDMAN
+We use Ω-notation, g(x) = Ω( f(x)), to mean that, for some a > 0 and sufficiently large x,
+g(x) ≤ a f(x).
+Theorem 1. If Hn has a diagonal-ε-embedding into the unit cube then n(ε) = Ω
+�
+eaε−3�
+for some
+a > 0.
+Proof. Let I3 be the unit cube. Tile I3 by cells dual to a triangulation of I3. The cells should have
+the property that they are somewhat regular: each cell should have an inscribed sphere of radius
+> ε
+20 and an excribed sphere of radius < ε
+2. These dual cells have the property that any union of
+them is a PL 3-manifold with boundary. We prefer not to use the obvious coordinate sub-cubes of
+I3, because they fail to have this property.2 The number of cells in this tiling τ is O(ε−3).
+Assume Hn is diagonally-ε-embedded. For each j, 1 ≤ j ≤ n, 3-color the tiling τ according
+to the rule that a cell is red if it meets r j, blue if it meets bj, and white otherwise. Call this
+coloring cj. Now we decorate cj with additional homological information. Let Rj (Bj) be the
+union of the red (blue) cells under cj. The first homology, H1(Rj;Z2) is a vector space over
+Z2 of dimension dj = Ω(ε−3), for which we choose a basis bj1,...,bjdj. Similarly, H1(Bj;Z2)
+has dimension ej = Ω(ε−3) with basis f j1,..., f je j. Let Ljpq be the djxej matrix of Z2-linking
+numbers, Lpq
+j = Link(bjp,ejq).
+Now, homologically we may express the class of r j in H1(Rj) (the class of bj in H1(Bj))
+as r j = ∑
+dj
+p=1xjpbjp (bj = ∑
+e j
+q=1 yjq f jq). Now the mod 2 linking numbers of the link Hj can be
+recovered as:
+(1)
+1 = Link(bj,r j) = Lpq
+j xjpyjq,
+Einstein summation convention in effect.
+Where cj was the jth coloring, let ˆcj be a decorated j-coloring where the decoration amounts
+to fixing the mod 2 numbers {xp}, 1 ≤ p ≤ dj, and {yq}, 1 ≤ q ≤ ej, which express, within the
+arbitrarily chosen bases, how bj and r j lie homologically in H1(Bj;Z2) and H1(Rj;Z2).
+How many possible decorated colorings, #DC(ε) can there be?
+(2)
+#DC(ε) = Ω
+�
+3a′ε−3 ·2a′′ε−3 ·2a′′ε−3�
+= Ω
+�
+eaε−3�
+,
+all constants > 0.
+The first factor bounds the number of 3-colorings and the second two factors the possible values
+of the binary strings {xjp} and {yjq}, respectively.
+Now by the pigeonhole principle if n(ε) were not Ω(eaε−3), two Hopf links Hi and Hj, i ̸= j,
+within Hn must determine the same decorated coloring ˆci = ˆcj. But with Hi and Hj having identical
+thickenings: Bi = Bj and Ri = Rj, and identical homological data. Line 1 can also be read as a
+computation for the off-diagonal linking number:
+(3)
+0 = Link(bj,ri) = Lpq
+j xjpyiq = Lpq
+i xjpyjq = Lpq
+i xipyiq = 1
+This contradiction proves the theorem.
+□
+2For this first proof, the manifold property is actually not necessary, but for Theorems 2 and 6 the manifold property
+is an added convenience.
+
+PACKING MEETS TOPOLOGY
+3
+Before leaving this example, what packings can we imagine to supply a lower bound on n(ε),
+for Hn? The simplest starting point would be to link two circles of radius ε into a small, rigid,
+Hopf link, and then throw copies of these “linked key rings” into a unit box, shaking gently until
+full. This seems to yield n = O(ε−3). But then we realize the box is not as full as we thought.
+We can sprinkle in a second generation of orthogonally linked pairs of radius 3ε circles, ignoring
+the presence of the first generation. By ignoring the first generation, we will create many linking
+number = 1 with the first generation, but these can be undone “finger moves” of length ≤ ε to the
+second generation. By the triangle inequality, the second generation will still satisfy the diagonal-
+ε-embedded condition after all finger moves. We are still not done; we can add a third generation
+of orthogonal radius = 7ε Hopf links, which will retain the diagonal-ε condition after length ≤ 3
+finger moves recovers the correct link type, Hn. We can of course itterate with Hopf links of radius
+{ri}, r0 = ε, ri+1 = 2ri +1, until ri approaches unit size. From scale considerations, but ignoring
+unimportant boundary effects, we see that if ni is the number of ith generation Hopf links in the
+box, then n0,n1,n2,... is dominated by the geometric series n0,2−3n0,4−3n0,.... So summing this
+series we find that the total number n = ∑ni of the Hopf links satisfies
+(4)
+n(ε) < 8
+7n0
+So, in the end, all our extra work only changed (slightly) the leading coefficient. Not being
+able to find anything more clever, this leaves the huge gap between O(ε−3) and O(eaε−3), in which
+the truth must lie. Our conjecture is that n = O(ε−3), but the proof calls out for a new idea.
+Before discussing other link types, let us make a quick remark regarding higher dimensions.
+If d = p +q +1, then two disjoint closed submanifolds of Rd have a well-defined mod 2 linking
+nmber if they have dimensions p and q respectively. Now in Rd let Hn denote any link of 2n
+component {r j,bj}, 1 ≤ j ≤ n, with mod 2 linking numbers given by:
+L(ri,r j) = 0, L(bi,bj) = 0, i ̸= j, and L(ri,bj) = δi j
+Identical reasoning shows that the maximum possible n(ε), nmax(ε), satisfies:
+(5)
+O(ε−d) ≤ nmax(ε) ≤ ea(d)ε−d
+for some a > 0, which actually generalizes Theorem 1 even when d = 3. nmax(ε) is the largest
+number such that a 2n(ε)-component link can be embedded in the unit d-cube with the specified
+linking and dist(r j,bj) ≥ ε, 1 ≤ j ≤ n.
+Returning to dimension d = 3, let us give a further example, which steps away, slightly, from
+linking number. Consider the problem of packing the disjoint union (again this means smoothly
+embedded spheres separating the copies of) of n copies Bn of a three component link B, such as the
+Borromean rings, which has all linking numbers 0 and Milnor’s µ-invariant µ123(L) ̸≡ 0 mod 3
+[Mil54]. B has components l1,l2,l3, Bj = (l j1,l j2,l j3). Again, colors r1,r2,r3 are associated to the
+3-components. We now enforce the diagonal-ε-condition: for each j, 1 ≤ j ≤ n, dist(l ji,l ji′) ≥ ε
+whenever i ̸= i′.
+Let nB(ε) be the largest n for which such an embedding exists, or ∞ is no such bound exists.
+Theorem 2. For all ε > 0, the Borromean packing number nB(ε) is indeed a finite integer, with
+nB(ε) = Ω(eaε−9).
+
+4
+MICHAEL H. FREEDMAN
+Proof. We begin, as before, with a generic tessellation of I3 of scale between ε
+20 and ε
+2. Now, for
+each j, 1 ≤ j ≤ n, make a 4-coloring cj of I3 by the rule that a cell gets the color ri, 1 ≤ i ≤ 3, of
+the component it meets; if it meets none then it is white. But now we proceed differently, for the
+decoration: homology is wholly insufficient. To motivate our new decoration recall a classic:
+Theorem 3 (Burnside). Any finitely generated group of exponent 3 is finite.
+□
+Note. To estimate nB(ε) in Theorem 2, we used the calculation of [LvdW33] that the order of
+the free, restricted Burnside group is |B(m,3)| = 3m+(m
+2)+(m
+3). This will imply the bound stated in
+Theorem 2.
+We create a bespoke invariant to exploit Burnside’s theorem.
+Definition. Define 3-link-homotopy to be Milnor’s classical link-homotopy [Mil54] (individual
+components may cross themselves during the homotopy but not other components) with the addi-
+tional ad hoc relation: at any moment during the homotopy, any component may be band summed
+to g3, where g is a free loop in the complement of the other components. The cube means wrap 3
+times around g.
+Whereas before, the coloring cj was decorated with homological information, now the deco-
+ration ˆcj assigns to the submanifold Cji colored ri (according to our rule for the jth coloring cj)
+the conjugacy class [l ji] of the component l ji in the Burnside group π3
+1(Cji), where by definition,
+π3
+1(X) means π1(X) with the additional relations that all elements cube to the identity.
+Lemma 4. The 3-link-homotopy class of a link Bj in I3 can be recovered from the decorated
+coloring ˆcj.
+Proof. Since Cji and Cji′ are disjoint for i ̸= i′, a homotopy of Lj in which each component l ji
+stays within its Cji′ is a link-homotopy. Furthermore, if each l ji is permitted to vary in Cji within
+its π3
+1(Cji) conjugacy class, this is a special case of 3-link-homotopy. Thus, if each l ji is rechosen
+within its π3
+1(Cji) conjugacy class, the 3-link homotopy class is preserved.
+□
+Lemma 5. For a 3-component link with vanishing linking numbers3, µ123(B) is conserved mod 3
+under 3-link-homotopy.
+Proof. We may assume that during the 3-link-homotopy only one component moves or is altered
+at any given time. The “cyclic symmetry” theorem ([Mil57] Theorem 6) says that w.l.o.g. we
+may assume that the active component is the one being Magnus-expanded in the link group of the
+others. To recall, for any k-component link L, µI(L), I = i1,...,ik distinct indices, is computed by
+expending, as below, the component lik in the polynomial ring denoted by R[xi1,...,xik−1] [Mil54].
+This is Milnor’s notation for the integers adjoined k −1 non-commuting variables which are also
+“non-repeating,” meaning that one divides out by the ideal generated by monomials in which any
+variable occurs more than once.
+3It is actually only necessary to assume 3 ∤ gcd(link(li,lj)), i ̸= j.
+
+PACKING MEETS TOPOLOGY
+5
+(6)
+[lik] ∈ π1(I3 \(li1 ∪···∪lik−1))
+։
+M(I3 \(li1 ∪···∪lik−1))
+։
+FMk−1(mi−1,...,mik−1)
+Magnus
+−−−−→ R[xi1,...,xik−1]
+mi j �→ 1+xi
+m−1
+i j �→ 1−xi
+where mi j are meridians to li j, M denotes the Milnor link group obtained by adding the relations
+that each meridian commutes with all its conjugates, and FM is the corresponding free Milnor
+group generated by mi1,...,mik−1 subject only to these commutation relations. As the diagram
+indicates, [lik] is first projected, then lifted to FMk−1, and finally expanded.
+Then by definition, µI(L) = the coefficient of xi1,...,xik−1 of Magnus[l jk]. Any ambiguity in
+the expansion due to the choice of lifting constitutes the indeterminancy of that µI. For general
+background on µ invariants see [Mil54,Mil57,Kru98].
+As the kth-component moves by link-homotopy the element and its expansion are constant.
+Adding the cube g3 of a loop g to l jk multiplies its Magnus expansion M by the Magnus expansion
+Mg3 of g3, M → MMg3 = M(Mg)3.
+Since B has 3 components, k − 1 = 2, (Mg)3 is the cube of some monoic polynomial in two
+variables x1 and x2: (Mg)3 = (1 + c1x1 + c2x2 + c12x1x2 + ...)3. A brute force consideration of
+the 27 possible coefficient values mod 3 shows that in all cases the coefficients of x1, x2, and x1x2
+in (Mg)3 are all divisible by 3. Multiplying out we see that µ123(B) mod 3 is invariant under
+3-link-homotopy.
+□
+The number #c(ε) of possible colors is #c(ε) = Ω(ea′ε−3) and the number of decorations pos-
+sible for a coloring c is bounded by the product of the order of the Burnside groups Ω(ea′′ε−9) for
+each of the colored (not white) regions. Thus the number of possible decorated coloring #ˆc(ε) has
+a similar bound as a function of ε. As in Theorem 1, the pigeonhole principle tells us that if we
+could place n > #ˆc(ε) copies of B in I3, obeying the diagonal-ε-condition then for 1 ≤ i < j ≤ n, Bi
+and Bj will determine identical decorated colorings.
+But Lemma 4 now tells us three things: Li has 3-link-homotopy type B, Bj has 3-link-homotopy
+type B, and Bi j has 3-link-homotopy type B, where Bi j is the link obtained by starting with Bi and
+then swapping out any one component of Bi for the corresponding component of Bj. The first two
+conclusions are as we expect, but the third sounds wrong. Because Bi and Bj are split (separated
+by a smoothly embedded 2-sphere), so Bi j is a split link and all its µ123-invariant must vanish. But
+this vanishing contradicts Lemma 5, which says any link (including Bi j) in the 3-link-homotopy
+class of B has its µ123-invariant not congruent to 0 mod 3. This proves Theorem 2.
+□
+Replacing Burnside groups with the mod p lower central series (p-lcs) quotients allows a joint
+extension of Theorems 1 and 2, although with an exponentially weaker upper bound.
+Theorem 6. Let E be any homotopically essential link of k-components, E = (e1,...,ek) and En
+be the disjoint union of n copies of E. For every ε > 0 there is a largest n, nmax, such that En
+
+6
+MICHAEL H. FREEDMAN
+embeds in the unit cube I3 with the property that for all j, 1 ≤ j ≤ n, dist(eji,eji′) ≥ ε for i ̸= j,
+and 1 ≤ i ̸= i′ ≤ k. nmax = Ω
+�
+eap((a′ε−3)k)�
+, where p is the smallest prime not dividing the first
+nontrivial non-repeating µ-invariant of E, and a,a′ > 0 are fixed constants.
+Proof. Begin in the familiar fashion by creating a (k+1)-coloring cj of a fixed ε-scale tessellation
+of I3 in which each cell meeting eji is colored ri and the remaining cells are colored white. Similar
+to Theorems 1 and 2, we need to specify some finite amount of data about eji in its ri-colored region
+Rji sufficient to (1) certify the homotopically essential nature of Ej and (2) create the contradiction
+that a related split link Ei j would also be homotopically essential.
+By induction, it suffices to consider the case that E is almost homotopically trivial, meaning all
+its sub-links are all homotopically trivial, or more algebraically, that all non-repeating µ-invariants
+of length < k vanish.
+As in the proof of Theorem 2, cyclic symmetry implies that we may focus on a single “active”
+component l jik (and going forward drop the j-index for the embedding and replace ik′ by a single
+index), project l jik, now denoted simply by lk, to the Milnor group, and choose a lift αk to the free
+Milnor group FMk−1. The finite data we consider is the image of αk in Qp
+k := FMk−1/[FMk−1]p
+k,
+where for any group G, [G]p
+n is the nth-term of the mod p lower central series of G. This is defined
+by saying G1 = G, and Gm is generated (= normally generated) by the words aua−1u−1vp, a ∈ G,
+and u,v ∈ Gm−1.
+Regarding the bound, its essential ingredient is that the order
+��Qp
+k
+�� = Ω
+�
+p((a′ε−3)k)�
+. π1(Rji)
+has g = Ω(ε−3) generators, so this also bounds the size of the free Milnor group under considera-
+tion. The quotient Qp
+k is (k −1)-stage nilpotent with at most gs new (twisted) 2p factors added by
+during the sth-central extension. Thus the total number of copies of Zp twisted together to make
+the p-group Qp
+k−1 is Ω(ε−3)k−1, giving the order bound.
+Returning to the main line of the proof, we need:
+Lemma 7. Suppose β ∈ [FM(m1,...,mk−1)]p
+i , 1 ≤ i ≤ k − 1, then Magnus(β) maps to (1 +
+monomials of degree ≥ i) under reduction of coefficients Z → Zp, inducing R[x1,...,xk−1]
+πp
+−→
+Rp[x1,...,xk−1] := Zp[x1,...,xk−1]/(repeating ideal), i.e. πp(Magnus(β)) = (1 +terms of degree
+≥ i).
+Proof. By induction. When i = 1 the statement is that pth power has no linear terms when expanded
+into Rp[x1,...,xk−1]. Now assume that Lemma 7 is true for i −1 and expand aua−1u−1vp, where
+a ∈ FM(m1,...,mk−1), and u,v ∈ [FM(m1,...,mk−1)]p
+i−1. The lowest positive degree (= i − 1)
+monomials in Magnus(u) and Magnus(u−1) are identical except for reversed signs; consequently,
+the aua−1u−1 factor expands to (1 + monomials of degree ≥ i) as the degree = i − 1 terms all
+cancel. The vp factor has the same form since the degree i − 1 terms are now repeated p times
+each. Consequently, the product aua−1u−1v2 also expands to this form.
+□
+
+PACKING MEETS TOPOLOGY
+7
+The p-lcs subgroups are characteristic: they map to each other under homomorphisms and if
+F ։ G is an epimorphism then [F]p
+k maps epimorphically to [G]p
+k. Apply these facts to the maps:
+(7)
+π1(Rk) → π1(I3 \e1 ∪···∪ek−1) → M(I3 \e1 ∪···∪ek−1) ← FM(I3 \e1 ∪···∪ek−1)
+Magnus
+−→
+Rp[x1,...,xk−1]
+−→
+β ∈ [π1(Rk)]p
+k
+and apply Lemma 7 to conclude that any β ∈ [π1(Rh)]p
+k−1 will Magnus expand to 1 in line 6.
+Now consider a second version of a bespoke link homotopy, (p,k)-link-homotopy, in which
+components homotope (while maintaining disjointness), for convenience only move one at a time,
+and finally the active component (which our notation treats as the last component) is permitted
+at any moment to form an ambient connected sum with any loop β ∈ [π1(Rk)]p
+k. The present
+analog of Lemma 5 is that the non-repeating length k µ invariants of L are invariant mod p under
+(p,k)-link-homotopy. The proof is parallel to that of Lemma 5, simply Magnus expand ek#β ⊂
+I3 \(e1∪,···∪ek−1) into Rp[x1,...,xk−1].
+The present analog of Lemma 4 is that any k-component E′ (p,k)-link-homotopic to E contin-
+ues to have nontrivial, non-repeating, µ invariants of length k. This follows from Lemma 7, again
+by expanding ek#β into Rp[x1,...,xk−1]. In particular, no such E′ can be a split link.
+The proof of Theorem 6 is completed, once again, by an application of the pigeonhole principle.
+If nmax(ε) exceeds the cardinality of the decorate colorings {ˆcj}, where now each colored region
+Rji is decorated by a conjugacy class of [π1(Rji)]p
+k representing the invariant information regarding
+the location of eji inside Rji, then for j ̸= j′, Ej and Ej′ will induce identical data. We have just
+argued that this data suffices to reconstruct the nontrivial (p,k)-link-homotopy classes of both Ej
+and Ej′. This is as it should be. But now define Ej j′ by starting with Ej and exchanging any one of
+its components with the corresponding component of Ej′. Exactly the same data now tells us that
+Ej j′ has a non-vanishing, µ-invariant of length k. This is a contradiction since Ej j′ is a split link,
+split by the 2-sphere separating Ej from Ej′.
+□
+2. DISCUSSION
+The use of Z2 coefficients in the initial homological disucssion was arbitrary, and any finite
+coefficient ring would suffice. However, for Theorem 2, the choice of the prime 3 was crucial.
+p = 2 would make the Burnside group abelian and provide no useful information. In this regard
+it is amusing to check that the Borromean rings is indeed 2-link-homotopy equivalent to the 3-
+component unlink (µ123 is not conserved mod 2 under 2-link-homotopy). The restricted Burnside
+groups B(n,k) are only known to be finite for k = 2,3,4, and 6. The most general Theorem 6
+exploits the interplay of the mod p-lcs with the µ-invariants. While broadest, the estimate there is
+exponentially worse.
+Our philosophy is that the upper bounds we offer, based on homology or µ, are terrible. Firstly,
+the estimates seem way too big, and second they only apply to links with easy algebraic features;
+boundary link and even the Whitehead link are left untouched. Our conjecture, a challenge to the
+reader, is that every non-trivial link L of two or more components has an ε-diagonal packing bound
+for the number of ε-diagonally embedded copies of the form #L(ε) = Ω(ε−3).
+
+8
+MICHAEL H. FREEDMAN
+3. ACKNOWLEDGEMENTS
+The question studied here arose while working with Michael Starbird on [FS22]. An Ω(ε−3)
+bound for the Hopf link problem might offer an alternative proof strategy for that paper’s main
+theorem. I would also like to thank Slava Krushkal for insightful discussions.
+REFERENCES
+[CKM+17] Henry Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko, and Maryna Viazovska, The sphere
+packing problem in dimension 24, Ann. Math. 185 (2017), no. 3, 1017–1033.
+[FS22] Michael Freedman and Michael Starbird, The geometry of the Bing involution (2022), available at
+arXiv:2209.07597.
+[H+17] Thomas Hales et al., A formal proof of the Kepler conjecture, Forum Math. Pi 5 (2017).
+[Kru98] Vyacheslav Krushkal, Additivity properties of Milnor’s µ-invariants, J. Knot Theory Ramif. 7 (1998),
+no. 5, 625–637.
+[LvdW33] Friedrich Levi and B.L. van der Waerden, ¨Uber eine besondere Klasse von Gruppen, Abh. Math. Semin.
+Univ. Hambg. 9 (1933), 154–158.
+[Mil54] John Milnor, Link groups, Ann. Math. 59 (1954), no. 2, 177–195.
+[Mil57]
+, Isotopy of links, Algebraic geometry and topology, 1957.
+[Via17] Maryna Viazovska, The sphere packing problem in dimension 8, Ann. Math. 185 (2017), no. 3, 991–1015.
+MICHAEL H. FREEDMAN, MICROSOFT RESEARCH, STATION Q, AND DEPARTMENT OF MATHEMATICS, UNI-
+VERSITY OF CALIFORNIA, SANTA BARBARA, SANTA BARBARA, CA 93106
+

8dAyT4oBgHgl3EQfc_fQ/content/tmp_files/load_file.txt

@@ -0,0 +1,271 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf,len=270
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='00295v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='GT]  31 Dec 2022  PACKING MEETS TOPOLOGY  MICHAEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' FREEDMAN  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This note initiates an investigation of packing links into a region of Euclidean space to  achieve a maximal density subject to geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The upper bounds obtained apply only  to the class of homotopically essential links and even there seem extravagantly large, leaving much  working room for the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' INTRODUCTION AND THEOREMS  Optimal packing of balls into Euclidean space has a long history and recent astonishing suc-  cesses, including Hale’s resolution of the Kepler conjecture [H+17] and the optimality of the E8  and Leech lattices [Via17,CKM+17], resulting in a 2022 Fields Medal to Maryna Viazovska.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  In this note, we introduce the idea of packing links, rather than points, again with the goal of  achieving the highest possible density subject to the geometric constraints that certain link compo-  nents must maintain a distance ≥ ε from certain other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' There will be an observation  about higher dimensions but let us begin with packing classical links into Euclidean 3-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' In  the classical sphere packing problem, all points are constrained to have distance ≥ ε from each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The analogous stipulation for links, that all components must maintain a distance ≥ ε from  each other, is also of potential interest, but in that case, the coarse outline of the subject is broadly  similar to point packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' That is, in both cases, each component takes up a definite amount of  volume so no more that O(ε−3) link components can be ε-embedded into the unit cube, where  ε-embedded means no two components approach within ε of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' While this coarse upper  bound holds for all link types, complicated links almost surely have smaller upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' For  example, we conjecture that if Ln is the link type consisting of n-fibers of the Hopf map S3 → S2  then n can grow no more quickly than O(ε−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  However, this note focuses on a regime, partial-ε-embeddings, where even the coarse answer  can be quite mysterious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' By a partial-ε-embedding we mean that only certain specified pairs of  components must stay ε apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' In this context we are often left puzzled as to whether the number of  link components that can fit into the unit cube is: (1) countably infinite1, (2) finite but unbounded,  (3) exponential or super-exponential in ε−1, or (4) polynomial in ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  The theme of this note is well illustrated by our first example: Hn, which by definition is the  2n-component link type formed by taking n-Hopf links r  b each separated from the others by  some smooth embedded 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' One may write the link as Hn :=  ∏  n  j=1Hj, where Hj = r j ∪bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  The partial-ε-embedding condition we study is that for all j, dist(r j,bj) ≥ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We call such an  embedding a diagonal-ε-embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  1This case, of course, would require slightly relaxing the definition of “embedding” to a 1-1 map which, when  restricted to any finite collection of circles, is a smooth embedding of the expected link type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  1  2  MICHAEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' FREEDMAN  We use Ω-notation, g(x) = Ω( f(x)), to mean that, for some a > 0 and sufficiently large x,  g(x) ≤ a f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' If Hn has a diagonal-ε-embedding into the unit cube then n(ε) = Ω  �  eaε−3�  for some  a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Let I3 be the unit cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Tile I3 by cells dual to a triangulation of I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The cells should have  the property that they are somewhat regular: each cell should have an inscribed sphere of radius  > ε  20 and an excribed sphere of radius < ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' These dual cells have the property that any union of  them is a PL 3-manifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We prefer not to use the obvious coordinate sub-cubes of  I3, because they fail to have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='2 The number of cells in this tiling τ is O(ε−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Assume Hn is diagonally-ε-embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' For each j, 1 ≤ j ≤ n, 3-color the tiling τ according  to the rule that a cell is red if it meets r j, blue if it meets bj, and white otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Call this  coloring cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Now we decorate cj with additional homological information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Let Rj (Bj) be the  union of the red (blue) cells under cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The first homology, H1(Rj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='Z2) is a vector space over  Z2 of dimension dj = Ω(ε−3), for which we choose a basis bj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',bjdj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Similarly, H1(Bj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='Z2)  has dimension ej = Ω(ε−3) with basis f j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=', f je j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Let Ljpq be the djxej matrix of Z2-linking  numbers, Lpq  j = Link(bjp,ejq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Now, homologically we may express the class of r j in H1(Rj) (the class of bj in H1(Bj))  as r j = ∑  dj  p=1xjpbjp (bj = ∑  e j  q=1 yjq f jq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Now the mod 2 linking numbers of the link Hj can be  recovered as:  (1)  1 = Link(bj,r j) = Lpq  j xjpyjq,  Einstein summation convention in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Where cj was the jth coloring, let ˆcj be a decorated j-coloring where the decoration amounts  to fixing the mod 2 numbers {xp}, 1 ≤ p ≤ dj, and {yq}, 1 ≤ q ≤ ej, which express, within the  arbitrarily chosen bases, how bj and r j lie homologically in H1(Bj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='Z2) and H1(Rj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  How many possible decorated colorings, #DC(ε) can there be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  (2)  #DC(ε) = Ω  �  3a′ε−3 ·2a′′ε−3 ·2a′′ε−3�  = Ω  �  eaε−3�  ,  all constants > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  The first factor bounds the number of 3-colorings and the second two factors the possible values  of the binary strings {xjp} and {yjq}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Now by the pigeonhole principle if n(ε) were not Ω(eaε−3), two Hopf links Hi and Hj, i ̸= j,  within Hn must determine the same decorated coloring ˆci = ˆcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' But with Hi and Hj having identical  thickenings: Bi = Bj and Ri = Rj, and identical homological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Line 1 can also be read as a  computation for the off-diagonal linking number:  (3)  0 = Link(bj,ri) = Lpq  j xjpyiq = Lpq  i xjpyjq = Lpq  i xipyiq = 1  This contradiction proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  2For this first proof, the manifold property is actually not necessary, but for Theorems 2 and 6 the manifold property  is an added convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  PACKING MEETS TOPOLOGY  3  Before leaving this example, what packings can we imagine to supply a lower bound on n(ε),  for Hn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The simplest starting point would be to link two circles of radius ε into a small, rigid,  Hopf link, and then throw copies of these “linked key rings” into a unit box, shaking gently until  full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This seems to yield n = O(ε−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' But then we realize the box is not as full as we thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  We can sprinkle in a second generation of orthogonally linked pairs of radius 3ε circles, ignoring  the presence of the first generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' By ignoring the first generation, we will create many linking  number = 1 with the first generation, but these can be undone “finger moves” of length ≤ ε to the  second generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' By the triangle inequality, the second generation will still satisfy the diagonal-  ε-embedded condition after all finger moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We are still not done;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' we can add a third generation  of orthogonal radius = 7ε Hopf links, which will retain the diagonal-ε condition after length ≤ 3  finger moves recovers the correct link type, Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We can of course itterate with Hopf links of radius  {ri}, r0 = ε, ri+1 = 2ri +1, until ri approaches unit size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' From scale considerations, but ignoring  unimportant boundary effects, we see that if ni is the number of ith generation Hopf links in the  box, then n0,n1,n2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' is dominated by the geometric series n0,2−3n0,4−3n0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='. So summing this  series we find that the total number n = ∑ni of the Hopf links satisfies  (4)  n(ε) < 8  7n0  So, in the end, all our extra work only changed (slightly) the leading coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Not being  able to find anything more clever, this leaves the huge gap between O(ε−3) and O(eaε−3), in which  the truth must lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Our conjecture is that n = O(ε−3), but the proof calls out for a new idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Before discussing other link types, let us make a quick remark regarding higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  If d = p +q +1, then two disjoint closed submanifolds of Rd have a well-defined mod 2 linking  nmber if they have dimensions p and q respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Now in Rd let Hn denote any link of 2n  component {r j,bj}, 1 ≤ j ≤ n, with mod 2 linking numbers given by:  L(ri,r j) = 0, L(bi,bj) = 0, i ̸= j, and L(ri,bj) = δi j  Identical reasoning shows that the maximum possible n(ε), nmax(ε), satisfies:  (5)  O(ε−d) ≤ nmax(ε) ≤ ea(d)ε−d  for some a > 0, which actually generalizes Theorem 1 even when d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' nmax(ε) is the largest  number such that a 2n(ε)-component link can be embedded in the unit d-cube with the specified  linking and dist(r j,bj) ≥ ε, 1 ≤ j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Returning to dimension d = 3, let us give a further example, which steps away, slightly, from  linking number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Consider the problem of packing the disjoint union (again this means smoothly  embedded spheres separating the copies of) of n copies Bn of a three component link B, such as the  Borromean rings, which has all linking numbers 0 and Milnor’s µ-invariant µ123(L) ̸≡ 0 mod 3  [Mil54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' B has components l1,l2,l3, Bj = (l j1,l j2,l j3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Again, colors r1,r2,r3 are associated to the  3-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We now enforce the diagonal-ε-condition: for each j, 1 ≤ j ≤ n, dist(l ji,l ji′) ≥ ε  whenever i ̸= i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Let nB(ε) be the largest n for which such an embedding exists, or ∞ is no such bound exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' For all ε > 0, the Borromean packing number nB(ε) is indeed a finite integer, with  nB(ε) = Ω(eaε−9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  4  MICHAEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' FREEDMAN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We begin, as before, with a generic tessellation of I3 of scale between ε  20 and ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Now, for  each j, 1 ≤ j ≤ n, make a 4-coloring cj of I3 by the rule that a cell gets the color ri, 1 ≤ i ≤ 3, of  the component it meets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' if it meets none then it is white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' But now we proceed differently, for the  decoration: homology is wholly insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' To motivate our new decoration recall a classic:  Theorem 3 (Burnside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Any finitely generated group of exponent 3 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' To estimate nB(ε) in Theorem 2, we used the calculation of [LvdW33] that the order of  the free, restricted Burnside group is |B(m,3)| = 3m+(m  2)+(m  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This will imply the bound stated in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  We create a bespoke invariant to exploit Burnside’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Define 3-link-homotopy to be Milnor’s classical link-homotopy [Mil54] (individual  components may cross themselves during the homotopy but not other components) with the addi-  tional ad hoc relation: at any moment during the homotopy, any component may be band summed  to g3, where g is a free loop in the complement of the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The cube means wrap 3  times around g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Whereas before, the coloring cj was decorated with homological information, now the deco-  ration ˆcj assigns to the submanifold Cji colored ri (according to our rule for the jth coloring cj)  the conjugacy class [l ji] of the component l ji in the Burnside group π3  1(Cji), where by definition,  π3  1(X) means π1(X) with the additional relations that all elements cube to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The 3-link-homotopy class of a link Bj in I3 can be recovered from the decorated  coloring ˆcj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Since Cji and Cji′ are disjoint for i ̸= i′, a homotopy of Lj in which each component l ji  stays within its Cji′ is a link-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Furthermore, if each l ji is permitted to vary in Cji within  its π3  1(Cji) conjugacy class, this is a special case of 3-link-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Thus, if each l ji is rechosen  within its π3  1(Cji) conjugacy class, the 3-link homotopy class is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' For a 3-component link with vanishing linking numbers3, µ123(B) is conserved mod 3  under 3-link-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We may assume that during the 3-link-homotopy only one component moves or is altered  at any given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The “cyclic symmetry” theorem ([Mil57] Theorem 6) says that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' we  may assume that the active component is the one being Magnus-expanded in the link group of the  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' To recall, for any k-component link L, µI(L), I = i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',ik distinct indices, is computed by  expending, as below, the component lik in the polynomial ring denoted by R[xi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xik−1] [Mil54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  This is Milnor’s notation for the integers adjoined k −1 non-commuting variables which are also  “non-repeating,” meaning that one divides out by the ideal generated by monomials in which any  variable occurs more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  3It is actually only necessary to assume 3 ∤ gcd(link(li,lj)), i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  PACKING MEETS TOPOLOGY  5  (6)  [lik] ∈ π1(I3 \\(li1 ∪···∪lik−1))  ։  M(I3 \\(li1 ∪···∪lik−1))  ։  FMk−1(mi−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',mik−1)  Magnus  −−−−→ R[xi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xik−1]  mi j �→ 1+xi  m−1  i j �→ 1−xi  where mi j are meridians to li j, M denotes the Milnor link group obtained by adding the relations  that each meridian commutes with all its conjugates, and FM is the corresponding free Milnor  group generated by mi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',mik−1 subject only to these commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' As the diagram  indicates, [lik] is first projected, then lifted to FMk−1, and finally expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Then by definition, µI(L) = the coefficient of xi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xik−1 of Magnus[l jk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Any ambiguity in  the expansion due to the choice of lifting constitutes the indeterminancy of that µI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' For general  background on µ invariants see [Mil54,Mil57,Kru98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  As the kth-component moves by link-homotopy the element and its expansion are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Adding the cube g3 of a loop g to l jk multiplies its Magnus expansion M by the Magnus expansion  Mg3 of g3, M → MMg3 = M(Mg)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Since B has 3 components, k − 1 = 2, (Mg)3 is the cube of some monoic polynomial in two  variables x1 and x2: (Mg)3 = (1 + c1x1 + c2x2 + c12x1x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=')3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' A brute force consideration of  the 27 possible coefficient values mod 3 shows that in all cases the coefficients of x1, x2, and x1x2  in (Mg)3 are all divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Multiplying out we see that µ123(B) mod 3 is invariant under  3-link-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  The number #c(ε) of possible colors is #c(ε) = Ω(ea′ε−3) and the number of decorations pos-  sible for a coloring c is bounded by the product of the order of the Burnside groups Ω(ea′′ε−9) for  each of the colored (not white) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Thus the number of possible decorated coloring #ˆc(ε) has  a similar bound as a function of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' As in Theorem 1, the pigeonhole principle tells us that if we  could place n > #ˆc(ε) copies of B in I3, obeying the diagonal-ε-condition then for 1 ≤ i < j ≤ n, Bi  and Bj will determine identical decorated colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  But Lemma 4 now tells us three things: Li has 3-link-homotopy type B, Bj has 3-link-homotopy  type B, and Bi j has 3-link-homotopy type B, where Bi j is the link obtained by starting with Bi and  then swapping out any one component of Bi for the corresponding component of Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The first two  conclusions are as we expect, but the third sounds wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Because Bi and Bj are split (separated  by a smoothly embedded 2-sphere), so Bi j is a split link and all its µ123-invariant must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' But  this vanishing contradicts Lemma 5, which says any link (including Bi j) in the 3-link-homotopy  class of B has its µ123-invariant not congruent to 0 mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This proves Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  Replacing Burnside groups with the mod p lower central series (p-lcs) quotients allows a joint  extension of Theorems 1 and 2, although with an exponentially weaker upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Let E be any homotopically essential link of k-components, E = (e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',ek) and En  be the disjoint union of n copies of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' For every ε > 0 there is a largest n, nmax, such that En  6  MICHAEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' FREEDMAN  embeds in the unit cube I3 with the property that for all j, 1 ≤ j ≤ n, dist(eji,eji′) ≥ ε for i ̸= j,  and 1 ≤ i ̸= i′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' nmax = Ω  �  eap((a′ε−3)k)�  , where p is the smallest prime not dividing the first  nontrivial non-repeating µ-invariant of E, and a,a′ > 0 are fixed constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Begin in the familiar fashion by creating a (k+1)-coloring cj of a fixed ε-scale tessellation  of I3 in which each cell meeting eji is colored ri and the remaining cells are colored white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Similar  to Theorems 1 and 2, we need to specify some finite amount of data about eji in its ri-colored region  Rji sufficient to (1) certify the homotopically essential nature of Ej and (2) create the contradiction  that a related split link Ei j would also be homotopically essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  By induction, it suffices to consider the case that E is almost homotopically trivial, meaning all  its sub-links are all homotopically trivial, or more algebraically, that all non-repeating µ-invariants  of length < k vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  As in the proof of Theorem 2, cyclic symmetry implies that we may focus on a single “active”  component l jik (and going forward drop the j-index for the embedding and replace ik′ by a single  index), project l jik, now denoted simply by lk, to the Milnor group, and choose a lift αk to the free  Milnor group FMk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The finite data we consider is the image of αk in Qp  k := FMk−1/[FMk−1]p  k,  where for any group G, [G]p  n is the nth-term of the mod p lower central series of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This is defined  by saying G1 = G, and Gm is generated (= normally generated) by the words aua−1u−1vp, a ∈ G,  and u,v ∈ Gm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Regarding the bound, its essential ingredient is that the order  ��Qp  k  �� = Ω  �  p((a′ε−3)k)�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' π1(Rji)  has g = Ω(ε−3) generators, so this also bounds the size of the free Milnor group under considera-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The quotient Qp  k is (k −1)-stage nilpotent with at most gs new (twisted) 2p factors added by  during the sth-central extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Thus the total number of copies of Zp twisted together to make  the p-group Qp  k−1 is Ω(ε−3)k−1, giving the order bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Returning to the main line of the proof, we need:  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Suppose β ∈ [FM(m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',mk−1)]p  i , 1 ≤ i ≤ k − 1, then Magnus(β) maps to (1 +  monomials of degree ≥ i) under reduction of coefficients Z → Zp, inducing R[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1]  πp  −→  Rp[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1] := Zp[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1]/(repeating ideal), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' πp(Magnus(β)) = (1 +terms of degree  ≥ i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' By induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' When i = 1 the statement is that pth power has no linear terms when expanded  into Rp[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Now assume that Lemma 7 is true for i −1 and expand aua−1u−1vp, where  a ∈ FM(m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',mk−1), and u,v ∈ [FM(m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',mk−1)]p  i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The lowest positive degree (= i − 1)  monomials in Magnus(u) and Magnus(u−1) are identical except for reversed signs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' consequently,  the aua−1u−1 factor expands to (1 + monomials of degree ≥ i) as the degree = i − 1 terms all  cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The vp factor has the same form since the degree i − 1 terms are now repeated p times  each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Consequently, the product aua−1u−1v2 also expands to this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  PACKING MEETS TOPOLOGY  7  The p-lcs subgroups are characteristic: they map to each other under homomorphisms and if  F ։ G is an epimorphism then [F]p  k maps epimorphically to [G]p  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Apply these facts to the maps:  (7)  π1(Rk) → π1(I3 \\e1 ∪···∪ek−1) → M(I3 \\e1 ∪···∪ek−1) ← FM(I3 \\e1 ∪···∪ek−1)  Magnus  −→  Rp[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1]  −→  β ∈ [π1(Rk)]p  k  and apply Lemma 7 to conclude that any β ∈ [π1(Rh)]p  k−1 will Magnus expand to 1 in line 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Now consider a second version of a bespoke link homotopy, (p,k)-link-homotopy, in which  components homotope (while maintaining disjointness), for convenience only move one at a time,  and finally the active component (which our notation treats as the last component) is permitted  at any moment to form an ambient connected sum with any loop β ∈ [π1(Rk)]p  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The present  analog of Lemma 5 is that the non-repeating length k µ invariants of L are invariant mod p under  (p,k)-link-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The proof is parallel to that of Lemma 5, simply Magnus expand ek#β ⊂  I3 \\(e1∪,···∪ek−1) into Rp[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  The present analog of Lemma 4 is that any k-component E′ (p,k)-link-homotopic to E contin-  ues to have nontrivial, non-repeating, µ invariants of length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This follows from Lemma 7, again  by expanding ek#β into Rp[x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=',xk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' In particular, no such E′ can be a split link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  The proof of Theorem 6 is completed, once again, by an application of the pigeonhole principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  If nmax(ε) exceeds the cardinality of the decorate colorings {ˆcj}, where now each colored region  Rji is decorated by a conjugacy class of [π1(Rji)]p  k representing the invariant information regarding  the location of eji inside Rji, then for j ̸= j′, Ej and Ej′ will induce identical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' We have just  argued that this data suffices to reconstruct the nontrivial (p,k)-link-homotopy classes of both Ej  and Ej′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This is as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' But now define Ej j′ by starting with Ej and exchanging any one of  its components with the corresponding component of Ej′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Exactly the same data now tells us that  Ej j′ has a non-vanishing, µ-invariant of length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' This is a contradiction since Ej j′ is a split link,  split by the 2-sphere separating Ej from Ej′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' DISCUSSION  The use of Z2 coefficients in the initial homological disucssion was arbitrary, and any finite  coefficient ring would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' However, for Theorem 2, the choice of the prime 3 was crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  p = 2 would make the Burnside group abelian and provide no useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' In this regard  it is amusing to check that the Borromean rings is indeed 2-link-homotopy equivalent to the 3-  component unlink (µ123 is not conserved mod 2 under 2-link-homotopy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The restricted Burnside  groups B(n,k) are only known to be finite for k = 2,3,4, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' The most general Theorem 6  exploits the interplay of the mod p-lcs with the µ-invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' While broadest, the estimate there is  exponentially worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Our philosophy is that the upper bounds we offer, based on homology or µ, are terrible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Firstly,  the estimates seem way too big, and second they only apply to links with easy algebraic features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  boundary link and even the Whitehead link are left untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Our conjecture, a challenge to the  reader, is that every non-trivial link L of two or more components has an ε-diagonal packing bound  for the number of ε-diagonally embedded copies of the form #L(ε) = Ω(ε−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  8  MICHAEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' FREEDMAN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' ACKNOWLEDGEMENTS  The question studied here arose while working with Michael Starbird on [FS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' An Ω(ε−3)  bound for the Hopf link problem might offer an alternative proof strategy for that paper’s main  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' I would also like to thank Slava Krushkal for insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  REFERENCES  [CKM+17] Henry Cohn, Abhinav Kumar, Stephen Miller, Danylo Radchenko, and Maryna Viazovska, The sphere  packing problem in dimension 24, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 185 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 3, 1017–1033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [FS22] Michael Freedman and Michael Starbird, The geometry of the Bing involution (2022), available at  arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='07597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [H+17] Thomas Hales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=', A formal proof of the Kepler conjecture, Forum Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Pi 5 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [Kru98] Vyacheslav Krushkal, Additivity properties of Milnor’s µ-invariants, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Knot Theory Ramif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 7 (1998),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 5, 625–637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [LvdW33] Friedrich Levi and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' van der Waerden, ¨Uber eine besondere Klasse von Gruppen, Abh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Semin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Hambg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 9 (1933), 154–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [Mil54] John Milnor, Link groups, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 59 (1954), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 2, 177–195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [Mil57]  , Isotopy of links, Algebraic geometry and topology, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  [Via17] Maryna Viazovska, The sphere packing problem in dimension 8, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 185 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' 3, 991–1015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content='  MICHAEL H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}
+page_content=' FREEDMAN, MICROSOFT RESEARCH, STATION Q, AND DEPARTMENT OF MATHEMATICS, UNI-  VERSITY OF CALIFORNIA, SANTA BARBARA, SANTA BARBARA, CA 93106' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dAyT4oBgHgl3EQfc_fQ/content/2301.00295v1.pdf'}

8dAyT4oBgHgl3EQfp_jS/content/2301.00536v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f7a368b7bd42f1e822330f4946bc1da94a8283e9aa479c929c0d3f1c63f43391
+size 500170

8dAyT4oBgHgl3EQfp_jS/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:73ef4b076bc583091e6deaae93f1d0051b5dfc67d072d43a05106b0cba1bb67c
+size 248920

8tE2T4oBgHgl3EQfPwar/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:eccdaac563382e65211abf7535e053f2d9418c4b94097293a64c26192b95d4e7
+size 3538989

9NE0T4oBgHgl3EQfwgHr/content/tmp_files/2301.02635v1.pdf.txt

@@ -0,0 +1,1144 @@
+arXiv:2301.02635v1  [math.AC]  6 Jan 2023
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+NAOKI ENDO, LAURA GHEZZI, SHIRO GOTO, JOOYOUN HONG, SHIN-ICHIRO IAI,
+TOSHINORI KOBAYASHI, NAOYUKI MATSUOKA, AND RYO TAKAHASHI
+Dedicated to the memory of Wolmer V. Vasconcelos
+Abstract. Let A be a Noetherian local ring with canonical module KA. We characterize A when
+KA is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3. If A is a Cohen-Macaulay
+ring, H.-B. Foxby proved in 1974 that the A-module KA is q-torsionfree if and only if the ring A
+is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary
+Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the
+canonical module is closely related to the ring being Gorenstein in low codimension, we also explore
+quasi-normal rings, introduced by W. V. Vasconcelos. We provide several examples as well.
+1. Introduction
+This paper investigates the question of the structure of a Noetherian local ring A if its canonical
+module KA is a torsionless, reflexive, or more generally, q-torsionfree A-module for an integer q ≥ 3.
+The notion of q-torsionfree modules was one of the central contributions of the famous research
+of M. Auslander and M. Bridger [1], which was succeeded by H.-B. Foxby [11] to be a striking
+study of q-Gorenstein rings. Among many interesting results, Foxby settled the above question
+in the case where A is a Cohen-Macaulay ring. More precisely, the A-module KA is q-torsionfree
+if and only if the ring A is q-Gorenstein, i.e., Ap is a Gorenstein ring for every p ∈ Spec A with
+depth Ap < q (see [10, Proposition 3.2]). It remains unclear what happens if we do not assume the
+ring A is Cohen-Macaulay. The theory of canonical modules nowadays has been developed mainly
+over Cohen-Macaulay rings in connection with the Gorenstein property; see e.g., [7, 13, 14, 15, 20].
+However, over Noetherian local (not necessarily Cohen-Macaulay) rings, there are also remarkable
+preceding researches on canonical modules, including the study of their endomorphism algebras;
+see [3, 4, 6]. Therefore, behaviors of canonical modules, even for non-Cohen-Macaulay rings, are
+interesting and the q-torsionfree property is well worth studying. The motivation for the present
+2020 Mathematics Subject Classification. 13H10, 13A02, 13A15.
+Key words and phrases. Canonical module, Gorenstein ring, Cohen-Macaulay ring, q-torsionfree module, q-Gorenstein
+ring, quasi-normal ring.
+N. Endo was partially supported by JSPS Grant-in-Aid for Young Scientists 20K14299. L. Ghezzi was partially
+supported by the Fellowship Leave from the New York City College of Technology-CUNY (Fall 2022-Spring 2023)
+and by a grant from the City University of New York PSC-CUNY Research Award Program Cycle 53. S. Goto was
+partially supported by JSPS Grant-in-Aid for Scientific Research (C) 21K03211. J. Hong was partially supported
+by the Sabbatical Leave Program at Southern Connecticut State University (Spring 2022). T. Kobayashi was partly
+supported by JSPS Grant-in-Aid for JSPS Fellows 21J00567. N. Matsuoka was partially supported by JSPS Grant-in-
+Aid for Scientific Research (C) 18K03227. R. Takahashi was partially supported by JSPS Grant-in-Aid for Scientific
+Research (C) 19K03443.
+1
+
+2
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+research started with this question that arose while the second and fourth authors were writing the
+last paper with Vasconcelos concerning (torsionless) canonical modules [5].
+To explain our results more precisely, let us start from definitions which we will use throughout
+this paper. For a Noetherian local ring A of dimension d with maximal ideal m, a canonical module
+K of A is a finitely generated A-module satisfying
+�A ⊗A K ∼= Hom �
+A(Hd
+�m( �A), �E)
+where Hd
+�m( �A) denotes the dth local cohomology module of the m-adic completion �A of A with respect
+to its maximal ideal �m and �E is the injective hull of the �A-module �A/�m ([14, Definition 5.6]).
+Equivalently, a finitely generated A-module K is a canonical module of A if HomA(K, E) ∼= Hd
+m(A),
+where Hd
+m(A) is the dth local cohomology module of A with respect to m and E is the injective hull
+of A/m ([6, Definition 12.1.2, Remarks 12.1.3]). The canonical module KA is uniquely determined
+up to isomorphisms ([3, (1.5)], see also [14, Lemma 5.8]) if it exists. Although the existence is
+not guaranteed even for Cohen-Macaulay local domains, provided A is Cohen-Macaulay, the ring
+A admits the canonical module if and only if A is a homomorphic image of a Gorenstein ring
+([17, 19]). The fundamental theory of canonical modules over Cohen-Macaulay rings was developed
+in the monumental book [14] of J. Herzog and E. Kunz. We shall in this paper freely refer to [14]
+for basic results on canonical modules (see [7, Chapter 3] also).
+We now continue to state our setup. Let R be a Noetherian (not necessarily local) ring. For an
+R-module M, we have a canonical homomorphism
+ϕ : M → M∗∗
+defined by
+�
+ϕ(x)
+�
+(f) = f(x) for each f ∈ M∗ and x ∈ M, where (−)∗ = HomR(−, R) denotes the
+R-dual functor. We say that M is torsionless (resp. reflexive) if ϕ is injective (resp. bijective).
+Torsionless modules are torsionfree, i.e., there is no nonzero torsion elements, and the converse holds
+if the total ring of fractions Q(R) of R is Gorenstein ([22, Theorem (A.1)]). Moreover, the R-module
+M is torsionless (resp. reflexive) if and only if Exti
+R(D(M), R) = (0) for i = 1 (resp. i = 1, 2),
+where D(M) denotes the Auslander transpose of M ([1]). From this point of view, Auslander and
+Bridger introduced a q-torsionfree module M to be Exti
+R(D(M), R) = (0) for all i = 1, 2, . . . , q. In
+addition, for an integer n, we say that
+• M satisfies (Sn) if depth Mp ≥ min {n, dim Rp} for every p ∈ Spec R,
+• M satisfies (�Sn) if depth Mp ≥ min {n, depth Rp} for every p ∈ Spec R,
+• R satisfies (Gn) if Rp is Gorenstein for every p ∈ Spec R with dim Rp ≤ n,
+• R satisfies (�Gn) if Rp is Gorenstein for every p ∈ Spec R with depth Rp ≤ n.
+The condition (Sn) is known as Serre’s condition. A Noetherian ring satisfying (�Gn) coincides with
+(n + 1)-Gorenstein ring in earlier publications such as [1, 11]. The condition (�Gn) is equivalent to
+saying that the ring satisfies both (Sn+1) and (Gn).
+Let us now state our results, explaining how this paper is organized. In Section 2, after recalling
+the necessary definitions and preliminaries, we give a criterion for a Noetherian local ring A to
+have the torsionless canonical module. We show that the A-module KA is torsionless if and only
+if Ap is Gorenstein for every p ∈ Assh A, where Assh A = {p ∈ Spec A | dim A/p = dim A} =
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+3
+AssA KA (Proposition 2.3).
+Section 3 is devoted to the characterizations of local rings A with
+reflexive canonical modules. When dim A = 1, this is exactly the case where A is a Gorenstein
+ring (Proposition 3.3). We elaborate on the one-dimensional case in Section 5. For the higher
+dimensional case, the reflexivity of KA is characterized by the local ring Ap being Gorenstein for
+every p ∈ SuppA KA with dim Ap ≤ 1 and Ass A ∩ V (U) = Assh A, where U denotes the unmixed
+component of (0) in A and V (U) is the set of all prime ideals in A containing U (Theorem 3.6).
+This lead us to obtain Corollary 3.8, which claims that KA is reflexive if and only if A satisfies (G1),
+provided Assh A = Ass A. This indicates that the reflexivity of canonical modules is deeply related
+to the ring being Gorenstein in low codimension.
+Thus Section 4 is dedicated to quasi-normal
+rings, i.e., rings with (S2) and (G1), which have been introduced by Vasconcelos. In Section 6, we
+generalize Foxby’s result on q-torsionfree canonical modules to arbitrary Noetherian local rings A
+admitting a canonical module. Our results of Sections 2 and 3 provide a complete generalization
+in case q = 1, 2. When q ≥ 3, Theorem 6.6 states that the A-module KA is q-torsionfree if and
+only if the ring A satisfies (Gq−1) and (Sq−1) on SuppA KA, provided that KA satisfies (Sq). In the
+final section we provide concrete examples of Cohen-Macaulay and q-Gorenstein rings in order to
+illustrate our theorems.
+2. Torsionless canonical modules
+Throughout the section, let (A, m) be a Noetherian local ring of dimension d. We begin with
+some preliminaries. Let (0) =
+�
+p∈Ass A
+Q(p) denote a primary decomposition of (0) in A. We set
+Assh A = {p ∈ Spec A | dim A/p = d}
+and
+U =
+�
+p∈Assh A
+Q(p)
+where U is called the unmixed component of (0) in A. Let V (U) denote the set of all prime ideals
+of A containing U.
+Lemma 2.1. There is an embedding 0 → A/U → A of A-modules.
+Proof. We may assume that U ̸= (0). Then Assh A ⊊ Ass A. Let
+L =
+�
+p∈Ass A\Assh A
+Q(p).
+We then have L ̸⊆
+�
+p∈Assh A
+p. Choose an element a ∈ L but a ̸∈
+�
+p∈Assh A
+p. Since a is a non-zerodivisor
+on A/U and aU ⊆ L∩U = (0), we obtain ((0) :A a) = U. Then U is the kernel of the homomorphism
+ϕ : A → A given by ϕ(1) = a. Thus, A/U ∼= Im(ϕ) ֒→ A.
+□
+In the rest of this section, we assume the ring A admits the canonical module KA. We recall
+several known facts about KA which we will use throughout this article; see [3, (1.6), (1.7), (1.8),
+(1.9), (1.10), Theorem 3.2] and [14, Korollar 6.3] (also [6, Chapter 12]) for the proofs.
+Proposition 2.2. The following assertions hold true.
+(1) The annihilator of KA is U. In particular, dimA KA = d and AssA KA = Assh A.
+
+4
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+(2) If a ∈ m is A-regular, then a is KA-regular.
+(3) V (U) = SuppA KA = {p ∈ Spec A | dim A = dim A/p + htA p}.
+(4) Both KA and HomA(KA, KA) satisfy (S2).
+(5) KAp =
+�
+KA
+�
+p for every p ∈ SuppA KA.
+(6) SuppA KA = Spec A if and only if Min A = Assh A.
+(7) Ass A = Assh A if and only if KA is a faithful A-module.
+(8) Suppose A is Cohen-Macaulay. If a ∈ m is A-regular, then KA/(a) exists and KA/(a) ∼= KA/aKA.
+(9) Suppose that KA/U exists. Then KA/U, as an A-module, is the canonical module of A.
+By Proposition 2.2-(2), the canonical module KA is torsionfree as an A-module.
+In general,
+torsionless modules are torsionfree, and the converse holds if and only if Ap is a Gorenstein local
+ring for every p ∈ Ass A ([22, Theorem (A.1)]). Therefore, if the total ring of fractions Q(A) of A is
+Gorenstein, then KA is torsionless. The following proposition shows the case where KA is torsionless
+without assuming Q(A) is Gorenstein. It is also a generalization of [5, Proposition 3.2].
+Proposition 2.3. The following conditions are equivalent :
+(1) KA is a torsionless A-module ;
+(2) Ap is a Gorenstein ring for every p ∈ Assh A ;
+(3) KA ∼= I for some ideal I of A.
+Proof. (1) ⇒ (2) Since KA is torsionless, there exists an exact sequence 0 → KA → F of A-modules,
+where F is a finitely generated free A-module. Let p ∈ Assh A ⊆ SuppA KA. Then Ap is Artinian
+and
+�
+KA
+�
+p is the canonical module of Ap. Therefore, we may assume that
+�
+KA
+�
+p is the injective hull
+of Ap/pAp. The splitting monomorphism 0 →
+�
+KA
+�
+p → Fp induces that
+�
+KA
+�
+p is a direct summand
+of the free Ap-module Fp. Since Ap is Artinian, by the Matlis duality, we have
+�
+KA
+�
+p ∼= Ap. Hence
+Ap is a Gorenstein ring.
+(2) ⇒ (3) Let W = A \ �
+p∈Assh A p. By assumption,
+�
+KA
+�
+p ∼= Ap for every p ∈ Assh A. Thus,
+W −1KA ∼= W −1A. Moreover, we have W −1A ∼= W −1(A/U) because W −1U = (0) by the proof of
+Lemma 2.1. Since every element of W is a non-zerodivisor on both KA and A/U, the isomorphism
+W −1KA ∼= W −1(A/U) induces the embedding KA ֒→ A/U. By Lemma 2.1, there is an embedding
+KA ֒→ A.
+(3) ⇒ (1) is clear.
+□
+If A is reduced, which means there are no nonzero nilpotents, then the local ring Ap is a field for
+every p ∈ Ass A. Hence we obtain the following.
+Corollary 2.4. If A is a reduced ring, then KA ∼= I for some ideal I of A.
+We recall that if A is Cohen-Macaulay, the canonical module KA has rank one if and only if the
+ring A is generically Gorenstein, i.e., Ap is a Gorenstein local ring for every p ∈ Min A. When
+one of the equivalent conditions of [7, Proposition 3.3.18] is satisfied, the canonical module can be
+identified with an ideal of A (see also [14, Satz 6.21]). The example below shows that the assumption
+p ∈ Assh A is necessary for Proposition 2.3.
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+5
+Example 2.5. Let S = k[[X, Y, Z]] be the formal power series ring over a field k and set A =
+S/[(X) ∩ (Y, Z)2]. Let x, y, z denote the images of X, Y, Z in A, respectively. Then we have
+U = (x) and KA ∼= A/(x) ∼= y2A.
+By Proposition 2.3, KA is torsionless. However, A is not generically Gorenstein. In fact, Aq is not
+a Gorenstein ring for q = (y, z) ∈ Min A.
+3. Reflexive canonical modules
+Let (A, m) be a Noetherian local ring of dimension d admitting the canonical module KA. We
+denote by U the unmixed component of (0) in A. In this section we will show how the reflexivity
+of the canonical module is related to the Gorensteinness of the ring. We begin with the following
+simple but effective lemma.
+Lemma 3.1. Suppose that KA is reflexive. Then Ass A ∩ V (U) = Assh A. In particular, if KA is
+reflexive and depth A = 0, then dim A = 0.
+Proof. The assertion follows from
+V (U) ∩ Ass A = SuppA K∗
+A ∩ Ass A = AssA HomA(K∗
+A, A) = AssA K∗∗
+A = AssA KA = Assh A.
+□
+The example below shows that the reflexivity of KA may require a rather strong restriction on A.
+Example 3.2. Let S = k[[X, Y ]] be the formal power series ring over a field k and set A =
+S/[(X) ∩ (X2, Y )]. Let x, y denote the images of X, Y in A, respectively. Let m = (x, y) be the
+maximal ideal of A. Then we have Assh A = {(x)}, U = (x), and KA = A/U.
+(1) Let p = (x). Since Ap is a field, by Proposition 2.3, KA is torsionless.
+(2) Since depth A = 0 and dim A = 1, by Lemma 3.1, KA is not reflexive.
+Proposition 3.3. Suppose d = 1. Then KA is a reflexive A-module if and only if A is a Gorenstein
+ring.
+Proof. Suppose that KA is reflexive. By Lemma 3.1, A is Cohen-Macaulay. Since KA is reflexive,
+there exists an exact sequence 0 → KA → F1 → F0, where F0, F1 are finite free A-modules [11,
+Proposition 2.1]. Let a ∈ m be an A-regular element. Since A is Cohen-Macaulay, KA/aKA is the
+canonical module of A/aA. Moreover, the embedding 0 → KA/aKA → F1/aF1 proves that KA/aA
+is torsionless. Therefore, by Proposition 2.3, A/aA is a Gorenstein ring. Thus, A is a Gorenstein
+ring. The converse is clear.
+□
+Remark 3.4. There exist non-Cohen-Macaulay local rings with reflexive canonical module. Exam-
+ple 6.2 shows a two-dimensional non-Cohen-Macaulay local ring A with KA reflexive. The example
+also shows that, even if KA is reflexive, the equality Ass A = Assh A does not hold true in general.
+Recall that a finitely generated A-module M is reflexive, i.e., the canonical map ϕ : M → M∗∗ is
+an isomorphism, if and only if there is at least one isomorphism M ∼= M∗∗ of A-modules.
+
+6
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+Lemma 3.5. Suppose that there is an exact sequence
+0 → KA → K∗∗
+A → C → 0
+of A-modules. If C ̸= (0), then Ap is a Cohen-Macaulay ring with dim Ap = 1 for every p ∈ AssA C
+with depth Ap ≥ 1. In particular, p ∈ V (U) and Up = (0).
+Proof. Let p ∈ AssA C such that depth Ap ≥ 1. Then p ∈ SuppA KA = V (U). Since
+�
+KA
+�
+p ∼= KAp,
+by passing to the ring Ap, we may assume depth A > 0 and depthA C = 0. Since AssA K∗∗
+A ⊆ Ass A,
+we have depthA K∗∗
+A ≥ 1. From the exact sequence 0 → KA → K∗∗
+A → C → 0, we obtain
+0 = depthA C ≥ min{depthA KA − 1, depthA K∗∗
+A }.
+Thus, depthA KA = 1. Since KA satisfies (S2), we have
+1 = depthA KA ≥ min{2, dim A}.
+Therefore, A is a Cohen-Macaulay ring of dimension 1. In particular, U = (0).
+□
+Now we aim to generalize Proposition 3.3.
+Theorem 3.6. The following conditions are equivalent :
+(1) KA is a reflexive A-module ;
+(2) Ass A ∩ V (U) = Assh A, and Ap is Gorenstein for every p ∈ SuppA KA with htA p ≤ 1.
+Proof. (1) ⇒ (2) By Lemma 3.1, we have Ass A∩V (U) = Assh A. Let p ∈ SuppA KA with htA p ≤ 1.
+If p ∈ Assh A, then Ap is Gorenstein by Proposition 2.3. Otherwise, we have dim Ap = 1. Since
+KAp is a reflexive Ap-module, the ring Ap is Gorenstein by Proposition 3.3.
+(2) ⇒ (1) Since Ap is Gorenstein for every p ∈ Assh A, by Proposition 2.3, KA is torsionless. Hence
+we have the exact sequence
+0 → KA
+ϕ
+−→ K∗∗
+A → C → 0
+of A-modules, where ϕ is the canonical homomorphism. Suppose that C ̸= (0). Let p ∈ AssA C.
+Note that p ∈ SuppA KA and
+�
+KA
+�
+p ∼= KAp. If htA p ≤ 1, then by assumption Ap is Gorenstein.
+Then Cp = (0), which is a contradiction. Thus, htA p ≥ 2. Since Ass A ∩ V (U) = Assh A, we have
+depth Ap ≥ 1. This shows, by Lemma 3.5, that Ap is a Cohen-Macaulay ring with dim Ap = 1,
+which is a contradiction. Therefore C = (0) and KA is a reflexive A-module.
+□
+We summarize some consequences of Theorem 3.6.
+Note that A satisfies (S1) if and only if
+Ass A = Min A, and the latter condition implies Ass A ∩ V (U) = Assh A.
+Corollary 3.7. If A satisfies (S1) and (G1), then KA is a reflexive A-module.
+Recall that if Assh A = Ass A, then Spec A = SuppA KA. Thus, we obtain the following as another
+direct consequence of Theorem 3.6.
+Corollary 3.8. Suppose that Assh A = Ass A. Then KA is a reflexive A-module if and only if A
+satisfies (G1).
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+7
+If A is a Cohen-Macaulay ring, the above corollary recovers [14, Korollar 7.29]. Recall that A is a
+generalized Cohen-Macaulay ring, if the ith local cohomology module Hi
+m(A) is a finitely generated
+A-module for every i ̸= d.
+Corollary 3.9. Suppose that A is a generalized Cohen-Macaulay ring and d > 0. Then KA is a
+reflexive A-module if and only if depth A > 0 and A satisfies (G1).
+Proof. By assumption, we have Ass A \ {m} ⊆ Assh A. If KA is reflexive, then by Lemma 3.1 we see
+that depth A > 0. Without loss of generality, we may assume depth A > 0. Hence Ass A = Assh A.
+The assertion follows from Corollary 3.8.
+□
+It seems natural to ask for the relation between the reflexivity of the A-module KA and that of
+the A/U-module KA/U. As for this question, we have the following.
+Theorem 3.10. The following conditions are equivalent :
+(1) KA is a reflexive A-module ;
+(2) KA/U is a reflexive A/U-module and Ass A ∩ V (U) = Assh A.
+Proof. Let B = A/U. Then KA = KB ([3, 1.8]). Also note that Ass B = Assh B.
+(1) ⇒ (2) By Lemma 3.1, we have Ass A ∩ V (U) = Assh A. By Corollary 3.8, it suffices to show
+that B satisfies (G1). Let P ∈ Spec B be a prime with htB P ≤ 1. We write P = p/U for some
+p ∈ V (U). Then htA p = htB P ≤ 1. Moreover, KAp is a reflexive Ap-module. By Propositions
+2.3 and 3.3, Ap is Gorenstein. Since Up = (0) :Ap KAp = (0), we obtain BP = Ap. Thus, BP is a
+Gorenstein ring.
+(2) ⇒ (1) Let p ∈ SuppA KA with htA p ≤ 1. By Theorem 3.6, it is enough to show that Ap is
+Gorenstein. Let P = p/U. Then by Corollary 3.8, BP is a Gorenstein ring. Since Ass A ∩ V (U) =
+Assh A, the ring Ap is Cohen-Macaulay. In particular, Up = (0) and Ap = BP. Therefore Ap is
+Gorenstein.
+□
+Corollary 3.11. Suppose that A/U is a Gorenstein ring. Then the following assertions hold true.
+(1) KA is a reflexive A-module if and only if Ass A ∩ V (U) = Assh A.
+(2) If A satisfies (S1), then KA is reflexive.
+Proof. Note that (1) follows directly from Theorem 3.10. To prove (2), it is enough to show that
+Ass A ∩ V (U) ⊆ Assh A. Let p ∈ Ass A ∩ V (U). Since A satisfies (S1), we have htA p = 0. Since
+p ∈ V (U), we have dim A = dim A/p + htA p = dim A/p. Therefore p ∈ Assh A.
+□
+Closing this section, we provide the examples of (not necessarily Cohen-Macaulay) local rings
+admitting reflexive canonical modules.
+Example 3.12. Let S = k[[X, Y1, Y2, . . . , Yn]] (n ≥ 2) be the formal power series ring over a field k
+and let A = S/[(Xm) ∩ J] where m ≥ 1 and J is a (Y1, Y2, . . . , Yn)-primary ideal of S. Let x denote
+the image of X in A. Then U = (xm), Assh A = {(x)}, and A/U is a Gorenstein ring. By Corollary
+3.11, the A-module KA is reflexive.
+
+8
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+Example 3.13. Let k be a field and R = k[∆] be the Stanley-Reisner ring of a simplicial complex
+∆. Since R is reduced, the graded canonical module KR (see [13, 21]) is torsionless. Moreover, if
+#(Assh R) = 1, the ring R/U is Gorenstein, so that KR is reflexive as an R-module, where U stands
+for the unmixed component of (0) in R and Assh R = {p ∈ Spec R | dim R/p = dim R}.
+4. Quasi-normal rings
+Quasi-normal rings were introduced by Vasconcelos ([22, Definition 1.2]) and they are exactly
+2-Gorenstein rings.
+Definition 4.1. A Noetherian ring R is said to be quasi-normal if R satisfies (S2) and (G1).
+The following is a direct consequence of [11, Proposition 2.3] (for a local ring case, see also [8,
+Theorem 3.8]). Here we include our alternative proof specifically for quasi-normal rings.
+Proposition 4.2. Let R be a quasi-normal ring and let M be a finitely generated R-module. If
+depthRp Mp ≥ min{2, dim Rp} for every p ∈ Spec R, then M is reflexive.
+Proof. Consider the exact sequence of R-modules
+0 → X → M
+ϕ
+−→ M∗∗ → C → 0,
+where ϕ denotes the canonical homomorphism. Suppose X ̸= (0) and choose p ∈ AssR X. By
+assumption, we have dim Rp = 0. Since R satisfies (G1), the local ring Rp is Gorenstein, whence Mp
+is reflexive. Hence Xp = (0), which is a contradiction. So X = (0), and we have the exact sequence
+0 → M → M∗∗ → C → 0.
+Suppose C ̸= (0). Let p ∈ AssR C. If dim Rp = 0, then Mp is reflexive, so Cp = (0). This is a
+contradiction. Thus dim Rp ≥ 1. As R satisfies (S2), we have depth Rp ≥ min{2, dim Rp}. Hence
+depth Rp ≥ 1. Since depthRp M∗∗
+p
+≥ min{2, depth Rp}, we then have depthRp M∗∗
+p
+≥ 1. The exact
+sequence
+0 → Mp → M∗∗
+p
+→ Cp → 0
+gives that
+0 = depthRp Cp ≥ min{depthRp Mp − 1, depthRp M∗∗
+p }.
+Hence depthRp Mp ≤ 1. By assumption, we have
+1 ≥ depthRp Mp ≥ min{2, dim Rp}.
+Therefore dim Rp = 1 and depthRp Mp = 1. Since R satisfies (G1), the ring Rp is Gorenstein. By
+[22, Corollary 2.3], we see that Mp is reflexive. Hence Cp = (0), which is a contradiction.
+□
+A finitely generated R-module ωR is a canonical module of R, if (ωR)m is the canonical module of
+Rm for all maximal ideals m of R. In contrast to the local case, the canonical module is in general
+not unique up to isomorphisms; see e.g., [7, Remark 3.3.17].
+Corollary 4.3. Let R be a Noetherian ring with d = dim R > 0. Suppose that there exists a
+canonical module ωR and Ass Rm = Assh Rm for every maximal ideal m. Then R is quasi-normal if
+and only if R satisfies (S2) and ωR is reflexive.
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+9
+Proof. Suppose R is quasi-normal. Since ωR satisfies (S2) and dimRp[ωR]p = dim Rp for every p ∈
+Spec R, by Proposition 4.2, we conclude that ωR is reflexive. For the converse, it remains to show
+that R satisfies (G1). Let A = Rm, where m is a maximal ideal of R. Then KA = (ωR)m is reflexive.
+Therefore we have Ass A = Assh A. By Corollary 3.8, A satisfies (G1). Thus R satisfies (G1).
+□
+We summarize some examples.
+First, we note examples of quasi-normal rings which are not
+normal. The simplest ones are non-normal Gorenstein rings.
+Example 4.4. Suppose that R is a Cohen-Macaulay ring with canonical module ωR.
+We set
+T = R ⋉ωR to be the idealization of ωR over R. Then T is a Gorenstein ring ([17]), but not normal
+because it is never a reduced ring.
+For a commutative ring R, we denote by R the integral closure of R in Q(R). We refer to [7, p.
+178] for background on numerical semigroups.
+Example 4.5. Let H = ⟨a1, a2, . . . , aℓ⟩ be a symmetric numerical semigroup. We consider R =
+k[s, ta1, ta2, . . . , taℓ], where s, t are indeterminates and k is a field. Then R is a two-dimensional
+Gorenstein ring with R = k[s, t], so that R is not normal if 1 ̸∈ H. As a special case, the ring
+R = k[s, t2, t3] is quasi-normal, but not normal.
+Next, we note examples of quasi-normal but non-normal Cohen-Macaulay rings which are more-
+over not Gorenstein.
+Example 4.6. Let k be a field and k[X, Y ] the polynomial ring over k. Let H = ⟨a1, a2, . . . , aℓ⟩
+be a symmetric numerical semigroup such that 1 ̸∈ H and let k[H] = k[ta1, ta2, . . . , taℓ] denote the
+semigroup ring of H over k, where t is an indeterminate. Let T = k[Xn, Xn−1Y, . . . , XY n−1, Y n],
+where n ≥ 3 is an integer. We set R = T ⊗k k[H]. Then R is a quasi-normal Cohen-Macaulay
+ring with dim R = 3, which is neither Gorenstein nor normal. Indeed, because R = T ⊗k k[t]
+and k[H] ̸= k[t], the ring R is not normal. As T is normal, we see that R is a quasi-normal ring
+(see Proposition 7.3 (2)). Moreover, R is not a Gorenstein ring because T is not Gorenstein. The
+simplest example in this class is R = k[X3, X2Y, XY 2, Y 3] ⊗k k[t2, t3].
+Example 4.7. Let T = k[X, Y, Z, V ] be the polynomial ring over a field k. We denote by I2(N) the
+ideal of T generated by all the 2×2 minors of a matrix N. Let I = I2(M) where M =
+�
+Xa Y b+V
+Zc
+Y b′
+Zc′
+Xa′
+�
+for some integers a, b, c, a′, b′, c′ ≥ 1.
+We set R = T/I.
+Then R is a Cohen-Macaulay ring of
+dimension 2. Let x, y, z, v denote the images of X, Y, Z, V in R, respectively. We first check the
+isomorphism ωR ∼= (xa, yb′)R. In fact, by setting
+f = Zc+c′ − Xa′(Y b + V ),
+g = Xa+a′ − Y b′Zc,
+and h = −XaZc′ + Y b′(Y b + V ),
+we can consider the exact sequence
+0
+� T 2
+tM � T 3 (f
+g
+h) � T
+� R
+� 0
+
+10
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+of T-modules. By taking the T-dual, we get the presentation of ωR of the form T 3 M
+→ T 2 → ωR → 0.
+Therefore, the complex of R-modules
+R3
+M
+� R2 ( Y b′ −Xa ) � (xa, yb′)R
+� 0
+induces a natural epimorphism
+ϕ : ωR ։ (xa, yb′)R
+of R-modules. Moreover, ϕ is an isomorphism because ωR is a torsionfree R-module of rank one
+and xa is a non-zerodivisor on R. Hence ωR ∼= (xa, yb′)R, as claimed. We similarly have
+ωR ∼= (yb + v, zc′)R ∼= (zc, xa′)R.
+We also note that the isomorphisms can be obtained by using the procedure of [23, Section 6.1.2].
+In particular, R is not a Gorenstein ring, since the type of R is two.
+Next, we show that R is a quasi-normal ring. Let p ∈ Spec R with htR p ≤ 1. If x ̸∈ p, then
+[ωR]p ∼= (xa, yb′)Rp = Rp, so that Rp is a Gorenstein ring. Assume that x ∈ p. Similarly, we may
+assume that y, z ∈ p. Then, v /∈ p, since htR p ≤ 1. Therefore, [ωR]p ∼= (yb + v, zc′)Rp = Rp, so that
+Rp is a Gorenstein ring. Hence R is a quasi-normal ring.
+Finally, we prove that R is a normal ring if and only if a′ = b′ = c = 1. Assume a′ = b′ = c = 1
+and consider the ideal
+J = I2
+
+
+
+
+Zc′
+(a + 1)Xa
+Y b + V
+−(b + 1)Y b + V
+−Z
+bXY b−1
+c′XaZc′−1
+−Y
+−(c′ + 1)Zc′
+−Y
+0
+X
+
+
+
+ .
+Then J + I/I is the Jacobian ideal of R over k.
+A direct computation shows that
+√
+J + I =
+(X, Y, Z, V ) and hence, by the Jacobian criterion, the local ring Rp is regular for every p ∈ Spec R \
+{(x, y, z, v)}. Hence R is a normal ring. Conversely, we assume a′ ≥ 2, or b′ ≥ 2, or c ≥ 2. By
+taking
+P =
+�
+(X, Y b + V, Z)
+(if c ≥ 2)
+(X, Y, Z)
+(if c = 1),
+we then have J ⊆ P. We set p = PR. Then htR p = 1, but Rp is not a DVR. Indeed, because
+ε = Y or ε′ = Y b + V is invertible in TP, we see that
+JTP =
+
+
+
+
+
+
+
+�
+Zc+c′ − Xa′(Y b + V ), Xa+a′
+εb′
+− Zc, −XaZc′
+ε
++ (Y b + V )
+�
+⊆ (Y b + V ) + (X, Z)2
+(if c ≥ 2)
+�Zc+c′
+ε′
+− Xa′, Xa+a′ − Y b′Zc, −XaZc′
+ε′
++ Y b′
+�
+⊆ (Y ) + (X, Z)2
+(if c = 1)
+in TP. Thus Rp = TP/JTP cannot be a DVR. Hence R is not a normal ring. As a special case,
+R = k[X, Y, Z, V ]/I2
+� X Y +V
+Z
+Y
+Z
+X2
+�
+is a quasi-normal ring, but not normal.
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+11
+5. Reflexive canonical modules in dimension one
+Let (A, m) be a Cohen-Macaulay local ring with dim A = 1 admitting the canonical module KA.
+In this section, we explore the question of when A has a reflexive canonical module. We denote
+by Q(A) the total ring of fractions of A. Throughout this section, we assume that there exists an
+A-submodule K of Q(A) such that A ⊆ K ⊆ A and K ∼= KA as an A-module, where A denotes
+the integral closure of A in Q(A). Note that the assumption is automatically satisfied if Q(A) is
+Gorenstein and the residue class field A/m is infinite; see [12, Corollaries 2.8, 2.9]. For A-submodules
+X and Y of Q(A), let X : Y = {a ∈ Q(A) | aY ⊆ X}. If we consider ideals I, J of A, we set
+I :A J = {a ∈ A | aJ ⊆ I}; hence I :A J = (I : J) ∩ A.
+Proposition 5.1. The following conditions are equivalent :
+(1) A is a Gorenstein ring ;
+(2) K2 : K = K ;
+(3) KA is a reflexive A-module.
+Proof. (1) ⇔ (3) See Proposition 3.3.
+(3) ⇔ (2) Since A : K = [K : K] : K = K : K2 ([14, Bemerkung 2.5]), we have
+A : (A : K) = (K : K) : (K : K2) = [K : (K : K2)] : K = K2 : K.
+Therefore, K2 : K = K if and only if A : (A : K) = K, that is KA is a reflexive A-module.
+□
+Recall that an ideal I of A is called a canonical ideal of A, if I ̸= A and I ∼= KA as an A-module.
+By [12, Corollary 2.8], there exists a canonical ideal I of A. We then have the following.
+Theorem 5.2. Let I be a canonical ideal of A. Then the following conditions are equivalent :
+(1) A is a Gorenstein ring ;
+(2) I2 :A I = I ;
+(3) I/I2 is a free A/I-module ;
+(4) I is a reflexive A-module.
+Proof. By Proposition 5.1, it suffices to show (2) ⇒ (1). Enlarging the residue class field A/m of A
+if necessary, we may assume that A/m is infinite. Let I = (x1, x2, . . . , xn) (n > 0) so that each (xi)
+is a reduction of I. We set Ki = x−1
+i I and choose a non-zerodivisor b of A so that bK2
+i ⊆ A for all
+1 ≤ i ≤ n. Let J = bI and yi = bxi for 1 ≤ i ≤ n. Then (yi) is a reduction of J. Notice that A/I
+and A/J are both Gorenstein rings, since I, J ∼= KA as A-modules.
+Claim 1. J2 :A J = J.
+Proof of Claim 1. Suppose that J2 :A J ⊋ J. Then, J :A m ⊆ J2 :A J. Since A/J is a Gorenstein
+ring, we have J :A m = J + Aϕ for some ϕ ∈ (J :A m) \ J. Hence, ϕ
+b ∈ Q(A) and m· ϕ
+b ⊆ I, so that
+ϕ
+b ∈ I : m. Because I ⊊ I :A m ⊆ I : m and ℓA ((I : m)/I) = 1 (since A/I is a Gorenstein ring),
+we get ϕ
+b ∈ I :A m, so that ϕ
+b ∈ A. On the other hand, ϕ
+b ·I ⊆ I2, since ϕ·bI = ϕJ ⊆ J2 = b2I2.
+Consequently, ϕ
+b ∈ I2 :A I = I, whence ϕ ∈ bI = J, which is impossible. Thus J2 :A J = J.
+□
+
+12
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+Let yi denote the image of yi in J/J2.
+We then have J/J2 = �n
+i=1(A/J)·yi, and therefore,
+(0) :A/J yi = (0) for some i, since A/J is a Gorenstein ring and (0) :A/J J/J2 = (0) by Claim 1.
+Without loss of generality, assume i = 1. Then, J2 :A y1 = J. On the other hand, since bK2
+1 ⊆ A
+and K1 = y−1
+1 J, we get b·(y−1
+1 J)2 ⊆ A , whence bJ2 ⊆ (bx1)2. Therefore, J2 ⊆ (bx2
+1) ⊆ (bx1) = (y1).
+Hence, J2 = y1·(J2 :A y1) = y1J. Thus A is a Gorenstein ring (see [12, Theorem 3.7]).
+□
+6. q-torsionfree canonical modules
+The purpose of this section is to give a generalization of Proposition 2.3 and Theorem 3.6, which
+characterize local rings with q-torsionfree canonical modules for q = 1, 2.
+Let R be a Noetherian (not necessarily local) ring and q an integer. Let M be a finitely generated
+R-module with a finite projective presentation P1
+σ→ P0 → M → 0. By applying the R-dual functor
+(−)∗ = HomR(−, R), we obtain the exact sequence
+0 −→ M∗ −→ P ∗
+0
+σ∗
+−→ P ∗
+1 −→ Dσ(M) −→ 0
+of R-modules. We set D(M) = Dσ(M) and call it the Auslander transpose of M. Note that D(M)
+is uniquely determined up to projective equivalence.
+Definition 6.1 ([1, Definition 2.15]). A finitely generated R-module M is said to be q-torsionfree
+if Exti
+R(D(M), R) = 0 for all i = 1, 2, . . . , q.
+By [1, Proposition 2.6], there exists an exact sequence
+0 −→ Ext1
+R(D(M), R) −→ M
+ϕ
+−→ M∗∗ −→ Ext2
+R(D(M), R) → 0
+of R-modules, where ϕ is the canonical homomorphism, and furthermore, we have
+Exti+2
+R (D(M), R) ∼= Exti
+R(M∗, R)
+for all i > 0.
+This shows M is torsionless (resp. reflexive) if and only if M is 1-torsionfree (2-torsionfree). When
+q ≥ 3, the R-module M is q-torsionfree if and only if M is reflexive and Exti
+R(M∗, R) = (0) for all
+i = 1, 2, . . . , q − 2.
+Example 6.2. Let S = k[[X, Y, Z]] be the formal power series ring over a field k and set A =
+S/[(X) ∩ (Y, Z)].
+Let x, y, z denote the images of X, Y, Z in A, respectively.
+Then we have
+Assh A = {(x)}, U = (x), and KA = A/U, where U denotes the unmixed component of (0) in A.
+By dualizing the exact sequence
+A
+·x
+−→ A → A/(x) = KA → 0,
+we obtain
+0 → K∗
+A → A
+·x
+−→ A → A/(x) → 0.
+Thus, D(KA) = KA. Consider the free resolution of D(KA) = D:
+A5
+τ4
+−→ A3
+τ3
+−→ A2
+τ2
+−→ A
+·x
+−→ A −→ D −→ 0,
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+13
+where τ2 = [y z], τ3 =
+�
+x
+0
+z
+0
+x
+−y
+�
+, and τ4 =
+
+
+y
+z
+0
+0
+0
+0
+0
+y
+z
+0
+0
+0
+0
+0
+x
+
+ . Dualize this free resolution to
+obtain
+0 −→ D∗ −→ A
+·x
+−→ A
+σ2
+−→ A2
+σ3
+−→ A3
+σ4
+−→ A5,
+where σ2, σ3, σ4 are the transposes of τ2, τ3, τ4, respectively. Let a ∈ Ker σ2. Then
+a ∈
+�
+(0) :A y
+�
+∩
+�
+(0) :A z
+�
+= (x) ∩ (x) = (x) = Im(·x).
+Thus, Ext1
+A(D, A) = (0).
+Let ( a1
+a2 ) ∈ Ker σ3.
+Then a1, a2 ∈ (0) :A x = (y, z) and a1z − a2y = 0.
+Hence ( −a2
+a1 ) ∈ Ker τ2 = Im(τ3). Let −a2 = c1x − c3z, and a1 = c2x + c3y for some c1, c2, c3 ∈ A.
+Then c1x = −a2 + c3z ∈ (y, z).
+Thus, c1 = 0 and a2 = c3z.
+Similarly, a1 = c3y.
+We obtain
+( a1
+a2 ) ∈ Im σ2. Then Ext2
+A(D, A) = (0). Hence KA is 2-torsionfree. Note that Ker σ4 is generated by
+� x
+0
+0
+�
+,
+� 0
+x
+0
+�
+,
+� 0
+0
+y
+�
+,
+� 0
+0
+z
+�
+. Then Ext3
+A(D, A) ̸= (0). Thus, KA is not 3-torsionfree.
+Definition 6.3 ([1, Definition 2.15]). A finitely generated R-module M is called q-syzygy, if there
+exist finite free R-modules F1, F2, . . . , Fq and an exact sequence 0 → M → F1 → F2 → · · · → Fq of
+R-modules.
+Note that (a) M is torsionless if and only if M is 1-syzygy, (b) every q-torsionfree R-module is
+q-syzygy, and (c) if M is q-syzygy and x is an R-regular element, then M/xM is (q �� 1)-syzygy as
+an R/xR-module.
+Although the following theorem has been proved by Foxby in a more general setting involving
+Gorenstein modules, we restate it and give its proof in our context for the sake of completeness.
+Recall that R is q-Gorenstein if Rp is Gorenstein for every prime p with depth Rp < q.
+Theorem 6.4 ([10, Proposition 3.2]). Let A be a Cohen-Macaulay local ring admitting the canonical
+module KA. Then the following conditions are equivalent :
+(1) A is q-Gorenstein ;
+(2) KA is q-torsionfree ;
+(3) KA is q-syzygy.
+Proof. Since A is Cohen-Macaulay, we have Spec A = SuppA KA and [KA]p = KAp for every p ∈
+Spec A. Notice that KAp is maximal Cohen-Macaulay as an Ap-module.
+(1) ⇒ (2) Since every A-regular sequence is KA-regular, the A-module KA is q-torsionfree by [11,
+Proposition 2.3].
+(2) ⇒ (3) This follows from [11, Proposition 2.1].
+(3) ⇒ (1) Let p ∈ Spec A with depth Ap < q. Set n = dim Ap. When n = 0, the ring Ap
+is Gorenstein.
+Assume n > 0 and choose a system f1, f2, . . . , fn of parameters of Ap.
+Then
+it is an Ap-regular sequence, so that KAp/(f1, f2, . . . , fn)KAp is 1-syzygy because n < q.
+Since
+KAp/(f1, f2, . . . , fn)KAp ∼= KAp/(f1,f2,...,fn)Ap, we conclude that Ap/(f1, f2, . . . , fn)Ap is Gorenstein by
+Proposition 2.3, whence so is the ring Ap. This completes the proof.
+□
+As a direct consequence of Theorem 6.4, we have the following.
+
+14
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+Corollary 6.5. Let A be a Cohen-Macaulay local ring with d = dim A admitting the canonical
+module KA. Then A is a Gorenstein ring if and only if KA is (d + 1)-torsionfree.
+Theorem 6.4 and Corollary 6.5 lead us to the question of the structure of a local ring A with
+q-torsionfree canonical module without the assumption that A is Cohen-Macaulay.
+For a subset Φ of prime ideals in a Noetherian ring R, we say that
+• R satisfies (Sn) on Φ if depth Rp ≥ min {n, dim Rp} for every p ∈ Φ,
+• R satisfies (Gn) on Φ if Rp is Gorenstein for every p ∈ Φ with dim Rp ≤ n.
+The main result of this section gives an answer to the above question.
+Theorem 6.6. Let A be a Noetherian local ring admitting the canonical module KA. Suppose that
+KA satisfies (Sq). Then the following conditions are equivalent :
+(1) A satisfies both (Gq−1) and (Sq−1) on SuppA KA ;
+(2) KA is q-torsionfree ;
+(3) KA is q-syzygy.
+To show this, we need some auxiliaries. Let A be a Noetherian local ring. For each integer i ≥ 1,
+let ΩiM denote the ith syzygy of a finitely generated A-module M with respect to a minimal free
+resolution
+· · · −→ Fi
+∂i
+−−→ Fi−1
+∂i−1
+−−→ · · ·
+∂1
+−−→ F0
+∂0
+−−→ M −→ 0.
+We set Ω0M = M for convention. The A-module ΩiM depends, up to isomorphisms, only on M.
+We note the following due to Okiyama. Here pdA M denotes the projective dimension of M.
+Theorem 6.7 ([16], cf. [2, Proposition 1.2.8]). Let A be a Noetherian local ring and M a finitely
+generated A-module with pdA M = ∞. Then the following assertions hold true.
+(1) depthA ΩiM ≥ depth A for every i > max{0, depth A − depthA M}.
+(2) If depthA M > depth A, then depthA ΩiM = depth A for every i > 0.
+(3) Let n > 0 and assume depthA ΩnM > depth A. Then n = (depth A − depthA M) + 1 and
+depthA ΩiM = depth A for every i ≥ 0 with i ̸= n.
+The result above yields the following, which plays an important role in our argument.
+Lemma 6.8. Let A be a Noetherian local ring and M a nonzero finitely generated A-module.
+Assume that q ≥ depth A + 2 and M is q-syzygy. Then depthA M = depth A.
+Proof. Since M is q-syzygy, we have M = ΩqN ⊕ F for some finitely generated A-module N and
+finite free A-module F. If pdA N < ∞, then ΩqN = 0 because pdA N ≤ depth A ≤ q − 2, and
+hence depthA M = depth A. We may assume pdA N = ∞. Since q ≥ depth A ≥ max{0, depth A −
+depthA N}, we have depthA ΩqN ≥ depth A (Proposition 6.7 (1)). If depthA ΩqN > depth A, then
+q = depth A−depthA N+1 (Proposition 6.7 (3)), so that depth A+2 ≤ q = depth A−depthA N+1 ≤
+depth A + 1. This is a contradiction. Hence depthA ΩnN = depth A, so depthA M = depth A.
+□
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+15
+Theorem 6.9. Let R be a Noetherian ring and M a finitely generated R-module. If M is (q + 1)-
+syzygy, then one has
+depth Rp ≥ min{q, depthRp Mp}
+for all p ∈ SuppR M.
+In particular, if M satisfies (Sq), then R satisfies (Sq) on SuppR M.
+Proof. By localizing at p ∈ SuppR M, it suffices to show depth R ≥ min{q, depthR M}. If depth R ≥
+q, the assertion is obvious. Otherwise, if depth R < q, the assertion follows from Lemma 6.8.
+□
+As consequences of Theorems 6.4, 6.9, we get the following.
+Corollary 6.10. Let A be a Noetherian local ring with d = dim A admitting the canonical module
+KA. Then the following conditions are equivalent :
+(1) A is Gorenstein ;
+(2) KA is a (d + 1)-torsionfree maximal Cohen-Macaulay A-module ;
+(3) KA is a (d + 1)-syzygy maximal Cohen-Macaulay A-module.
+Proof. We only need to show (3) ⇒ (1). By Theorem 6.9 we have that depth A = d, so that A is
+Cohen-Macaulay. Hence the assertion follows from Theorem 6.4.
+□
+Corollary 6.11. Let (A, m) be a Noetherian local ring with d = dim A admitting the canonical
+module KA. Furthermore, we assume one of the following conditions (i) and (ii).
+(i) Hi
+m(A) = (0) for every integer i ̸= 0, 1, d.
+(ii) d ≤ 2.
+Then the following conditions are equivalent :
+(1) A is Gorenstein ;
+(2) KA is (d + 1)-torsionfree ;
+(3) KA is (d + 1)-syzygy.
+Proof. (i) By passing to the m-adic completion, we may assume A is m-adically complete. In view
+of [18, (2.3) Satz], it follows that KA is maximal Cohen-Macaulay. Therefore the assertion follows
+from Corollary 6.10.
+(ii) Since KA satisfies (S2), the assertion follows from Corollary 6.10.
+□
+Remark 6.12. Let A be a Noetherian local ring admitting the canonical module KA. We say
+that A is quasi-Gorenstein if KA ∼= A as an A-module. When d ≥ 3, there exist non-Gorenstein
+quasi-Gorenstein local rings of dimension d (see e.g., [3, Theorem 2.11]). Notice that, in such a ring
+A, KA is q-torsionfree for all q ≥ 1. So, Corollary 6.11 fails without the condition (i) or (ii).
+Based on the above observation, it is natural to raise the following question.
+Question 6.13. Let A be a Noetherian local ring with d = dim A ≥ 3 admitting the canonical
+module KA. When are the following conditions equivalent?
+(i) A is a quasi-Gorenstein ring, i.e., KA ∼= A.
+(ii) KA is a (d + 1)-torsionfree A-module.
+
+16
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+In what follows, let R be a Noetherian ring and M a finitely generated R-module. The equivalence
+of (1) and (2) in the next theorem was essentially proved by Auslander and Bridger [1]. Notice that
+this is a qth version of [7, Proposition 1.4.1].
+Theorem 6.14. The following conditions are equivalent :
+(1) M is q-torsionfree ;
+(2) M satisfies the conditions below :
+(i) Mp is q-torsionfree for every p ∈ SuppR M with depth Rp < q ;
+(ii) M satisfies (�Sq) ;
+(3) M satisfies the conditions below :
+(i) Mp is q-torsionfree for every p ∈ SuppR M with depthRp Mp < q ;
+(ii) depthRp Mp = depth Rp for every p ∈ SuppR M with depth Rp < q − 1.
+Proof. Without loss of generality, we may assume q ≥ 1.
+(1) ⇒ (2) This follows from [11, Proposition 2.1].
+(2) ⇒ (3) (i) Let p ∈ SuppR M with depthRp Mp < q. Since M satisfies (�Sq), we have depthRp Mp ≥
+depth Rp. This implies depth Rp < q, and hence Mp is q-torsionfree. (ii) Let p ∈ SuppR M with
+depth Rp < q − 1. Then Mp is q-torsionfree, so that depthRp Mp = depth Rp by Lemma 6.8.
+(3) ⇒ (1) For each i ∈ {1, 2, . . . , q}, we set Ei = Exti
+R(D(M), R). Suppose Eq ̸= 0 and seek a
+contradiction. Take p ∈ AssR Eq. Since p ∈ SuppR M, by (i) we have depthRp Mp ≥ q. Then by
+(ii), depth Rp ≥ q − 1. By passing to the localization Rp at p, we may assume R is a local ring,
+depth R ≥ q − 1, depthR M ≥ q, and depthR Eq = 0.
+We proceed by induction on q. First, assume that q = 1. Since E1 is isomorphic to a submodule
+of M, it follows that depthR M = 0, a contradiction. Thus E1 = (0). Next, we assume q = 2.
+Applying the depth lemma to the exact sequence 0 → M → M∗∗ → E2 → 0 of R-modules, we get
+depthR M = 1, as depthR M∗∗ ≥ 1. This is impossible, whence E2 = (0). Suppose q ≥ 3 and the
+assertion holds for q − 1, i.e., M is (q − 1)-torsionfree. Hence E1 = · · · = Eq−1 = (0). Consider a
+free resolution (Fi, ∂i) of M∗. Applying the R-dual functor (−)∗, we get the exact sequence
+0 → M∗∗ → F ∗
+0
+∂∗
+1
+−→ F ∗
+1 → · · · → F ∗
+q−3
+∂∗
+q−2
+−−→ F ∗
+q−2
+of R-modules because E3 = · · · = Eq−1 = (0). Let C be the cokernel of ∂∗
+q−2. Since M is reflexive
+as an R-module, we obtain the exact sequence of the form
+0 → M → F ∗
+0 → · · · → F ∗
+q−2 → C → 0.
+Since Eq = Extq−2
+R (M∗, R) may be regarded as a submodule of C, we see that depthR C = 0. Hence
+depthR M = q − 1. This gives a contradiction. Hence we conclude that Eq = (0), which shows M
+is q-torsionfree.
+□
+Corollary 6.15. Suppose that the following conditions are satisfied :
+(a) Mp is q-torsionfree for every p ∈ SuppR M with dim Rp < q ;
+(b) M satisfies (Sq) ;
+(c) depth Rp ≥ min{q − 1, dim Rp − 1} for every p ∈ SuppR M.
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+17
+Then M is q-torsionfree.
+Proof. We will check condition (3) in Theorem 6.14. Let p ∈ SuppR M. (i) Assume that depthRp Mp <
+q. By (b), depthRp Mp = dim Rp, so that dim Rp < q. Therefore Mp is q-torsionfree by (a). (ii)
+Assume that depth Rp < q − 1. By (c), depth Rp ≥ dim Rp − 1, so that dim Rp < q. Therefore Mp
+is q-torsionfree by (a), whence depthRp Mp = depth Rp by Lemma 6.8.
+□
+Corollary 6.16. Suppose that the following conditions are satisfied :
+(a) M satisfies (Sq) ;
+(b) R satisfies both (Gq−1) and (Sq−1) on SuppR M.
+Then M is q-torsionfree.
+Proof. By Corollary 6.15, it suffices to show that Mp is q-torsionfree on {p ∈ SuppR M | dim Rp <
+q}. Let p ∈ SuppR M with dim Rp < q. Then by (b), Rp is Gorenstein. As M satisfies (Sq),
+depthRp Mp ≥ dim Rp. Hence Mp is maximal Cohen-Macaulay as an Rp-module. In particular, Mp
+is q-torsionfree.
+□
+We are now ready to prove Theorem 6.6.
+Proof of Theorem 6.6. (1) ⇒ (2) This follows from Corollary 6.16.
+(2) ⇒ (3) This follows from [11, Proposition 2.1].
+(3) ⇒ (1) By Theorem 6.9, the ring A satisfies (Sq−1) on SuppA KA. Let p ∈ SuppA KA with
+dim Ap ≤ q − 1. Then Ap is a Cohen-Macaulay ring of dimension at most q − 1. Hence Theorem
+6.4 implies that Ap is a Gorenstein ring.
+□
+As consequences of Theorem 6.6, we get the following corollaries.
+Corollary 6.17. Let A be a Noetherian local ring admitting the canonical module KA. Suppose
+that q ≥ 2 and KA satisfies (Sq). Consider the following conditions :
+(1) KA is (q + 1)-torsionfree ;
+(2) KA is (q + 1)-syzygy ;
+(3) A satisfies both (Sq) and (Gq−1) on SuppA KA ;
+(4) A satisfies both (Sq) and (Gq−1), that is, A is q-Gorenstein ;
+(5) A satisfies (Sq) and KA is q-torsionfree ;
+(6) A satisfies (Sq) and KA is q-syzygy.
+Then the implications (1)⇔(2)⇒(3)⇔(4)⇔(5)⇔(6) hold true.
+Proof. The implications (1) ⇒ (2) and (4) ⇒ (3) are clear.
+The equivalence of (3), (5), and
+(6) immediately follows from Theorem 6.6. Thus it suffices to check the implications (2) ⇒ (3),
+(3) ⇒ (4), and (2) ⇒ (1).
+(2) ⇒ (3) Theorem 6.6 shows the ring A satisfies (Gq−1) on SuppA KA. On the other hand, by
+Theorem 6.9, we deduce that A satisfies (Sq) on SuppA KA.
+(3) ⇒ (4) Since q ≥ 2, by [4, Lemma 1.1] we have SuppA KA = Spec A.
+(2) ⇒ (1) The implication (2) ⇒ (4) guarantees that A is q-Gorenstein. Hence KA is q-torsionfree
+by [1, Proposition 4.21].
+□
+
+18
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+Since KA satisfies (S2), from Corollary 6.17 we have the following.
+Corollary 6.18. Let A be a Noetherian local ring admitting the canonical module KA. Consider
+the following conditions :
+(1) KA is 3-torsionfree ;
+(2) KA is 3-syzygy ;
+(3) A satisfies both (S2) and (G1), that is, A is quasi-normal ;
+(4) A satisfies (S2) and KA is 2-torsionfree ;
+(5) A satisfies (S2) and KA is 2-syzygy.
+Then the implications (1)⇔(2)⇒(3)⇔(4)⇔(5) hold true.
+Corollary 6.19. Let A be a Noetherian local ring admitting the canonical module KA. Suppose
+that q ≥ 2 and KA satisfies (Sq+1). Then the following conditions are equivalent :
+(1) KA is (q + 1)-torsionfree ;
+(2) KA is (q + 1)-syzygy ;
+(3) A satisfies both (Sq) and (Gq) on SuppA KA ;
+(4) A satisfies both (Sq) and (Gq).
+Proof. This follows from Theorem 6.6 and the fact that SuppA KA = Spec A ([4, Lemma 1.1]).
+□
+Corollary 6.20. Let A be a Noetherian local ring with d = dim A which is a homomorphic image
+of a Gorenstein ring. Suppose that q ≥ d
+2 + 1 and KA is (q + 1)-syzygy satisfying (Sq). Then A is a
+Cohen-Macaulay ring.
+Proof. By Theorem 6.9, we see that A satisfies (Sq). We may assume d > 0. Then q ≥ 2, so that A
+is equidimensional by [4, Lemma 1.1]. Furthermore, either A is Cohen-Macaulay or depth A ≥ q.
+We assume depth A ≥ q. Since KA satisfies (Sq), every A-regular sequence of length at most q
+is KA-regular ([11, Proposition 2.1]). The assertion follows from [9, Corollary (2.6)] (see also [10,
+Proposition 4.2]).
+□
+7. Examples of q-Gorenstein rings
+Closing this paper, in order to illustrate our theorems, we provide additional examples of Cohen-
+Macaulay and q-Gorenstein rings, i.e., rings with (Sq) and (Gq−1) conditions, or equivalently, rings
+with (�Gq−1) condition.
+Theorem 7.1. Let A be a Gorenstein local ring with d = dim A ≥ 3 and let a1, a2, . . . , ad be a
+system of parameters of A. Let a = (a1, a2, . . . , aℓ) (3 ≤ ℓ ≤ d) and let
+R = A[a1t, a2t, . . . , aℓt] ⊆ A[t]
+be the Rees algebra of a, where t denotes an indeterminate. Then, R is not a Gorenstein ring, but
+it is a Cohen-Macaulay (ℓ + 1)-Gorenstein ring of dimension d + 1.
+Proof. Recall that R is a Cohen-Macaulay ring of dimension d + 1. Let S = A[X1, X2, . . . , Xℓ] be
+the polynomial ring over A and let ϕ : S → R denote the surjective homomorphism of A-algebras
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+19
+defined by ϕ(Xi) = ait for each 1 ≤ i ≤ ℓ.
+The homomorphism ϕ preserves the grading and
+Ker(ϕ) = I2
+� X1 X2 ... Xℓ
+a1 a2 ... aℓ
+�
+is the perfect ideal of S of grade ℓ − 1 generated by the 2 × 2 minors of
+the matrix
+� X1 X2 ... Xℓ
+a1 a2 ... aℓ
+�
+. We set I = Ker(ϕ). We then have the following.
+Claim 2. Let P ∈ Spec S such that I ⊆ P but (X1, X2, . . . , Xℓ) + (a1, a2, . . . , aℓ) ̸⊆ P. Then,
+SP/ISP is a Gorenstein ring.
+Proof of Claim 2. We may assume that X1 ̸∈ P. Let �S = S[ 1
+X1], �A = A[X1,
+1
+X1], and Yi =
+Xi
+X1
+for 2 ≤ i ≤ ℓ.
+Then, �S = �A[Y2, Y3, . . . , Yℓ] and I �S = (ai − a1Yi | 2 ≤ i ≤ ℓ)�S.
+Because
+a1 �S + (ai − a1Yi | 2 ≤ i ≤ ℓ)�S = (ai | 1 ≤ i ≤ ℓ)�S and a1, a2, . . . , aℓ is an �S-regular sequence, the
+sequence a2 − a1Y2, a3 − a1Y3, . . . , aℓ − a1Yℓ is �SP-regular, so that SP/ISP is a Gorenstein ring.
+□
+Let P ∈ Spec S and suppose that I ⊆ P. We set p = ϕ(P) ∈ Spec R. Then, (X1, X2, . . . , Xℓ) +
+(a1, a2, . . . , aℓ) ̸⊆ P if htS P < 2ℓ, while
+htR p = htS/I P/I = htS P − (ℓ − 1).
+Therefore, if htR p < ℓ+1, then htS P −(ℓ−1) < ℓ+1, that is htS P < 2ℓ, so that (X1, X2, . . . , Xℓ)+
+(a1, a2, . . . , aℓ) ̸⊆ P, whence Rp = SP/ISP is a Gorenstein ring by Claim 2. Thus, R is an (ℓ + 1)-
+Gorenstein ring.
+□
+Since the proofs of the following assertions are standard, we left them to the interested readers.
+Lemma 7.2. Let ϕ : A → B be a flat local homomorphism of Noetherian local rings and q ≥ 1 be
+an integer. Then the following conditions are equivalent :
+(1) B is a q-Gorenstein ring ;
+(2) A is a q-Gorenstein ring and BP/pBP is a Gorenstein ring for every P ∈ Spec B with depth BP <
+q, where p = ϕ−1(P).
+Proposition 7.3. Let R be a Noetherian ring. Then the following assertions hold true.
+(1) Let q ≥ 1 be an integer. Then R[t] is a q-Gorenstein ring if and only if R is a q-Gorenstein
+ring, where t is an indeterminate.
+(2) Let H be a symmetric numerical semigroup. If R is a q-Gorenstein ring, then the semigroup
+ring R[H] of H over R is a q-Gorenstein ring.
+(3) Let X = {Xij}1≤i≤ℓ,1≤j≤m be indeterminates where ℓ, m ≥ 2, and set T = R[X]. Let t be an
+integer such that 2 ≤ t ≤ min {ℓ, m} and let I = It(X) denote the ideal of S generated by the
+t × t minors of the matrix X. We set S = T/I.
+(a) Let ℓ = m. If R is a q-Gorenstein ring, then S is a q-Gorenstein ring.
+(b) Suppose that R is a field and let t = 2. Then S is a d-Gorenstein ring, where d = ℓ+m−1.
+References
+[1] M. Auslander and M. Bridger, Stable module theory, Amer. Math. Soc., Memoirs, 94, 1969. 1, 2, 12, 13, 16, 17
+[2] L. Avramov, Infinite free resolutions, Six Lecture on Commutative Algebra, Birkhuser Verlag, Basel, (2010),
+1–118. 14
+[3] Y. Aoyama, Some basic results on canonical modules, J. Math. Kyoto Univ, 23 (1983), no. 1, 85–94. 1, 2, 3, 7,
+15
+
+20
+N. ENDO, L. GHEZZI, S. GOTO, J. HONG, S.-I. IAI, T. KOBAYASHI, N. MATSUOKA, AND R. TAKAHASHI
+[4] Y. Aoyama and S. Goto, On the endomorphism ring of the canonical module. J. Math. Kyoto Univ., 25 (1985),
+no. 1, 21–30. 1, 17, 18
+[5] J. Brennan, L. Ghezzi, J. Hong and W. V. Vasconcelos, Generalization of bi-canonical degrees, S˜ao Paulo J.
+Math. Sci. (2022), https://doi.org/10.1007/s40863-022-00333-9. 2, 4
+[6] M. P. Brodmann and R. Y. Sharp, Local cohomology, An algebraic introduction with geometric applications,
+Cambridge Studies in Advanced Mathematics 136, 2nd edition, Cambridge University Press, 1993. 1, 2, 3
+[7] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University Press, 2013. 1, 2, 4, 8, 9, 16
+[8] E. G. Evans and P. Griffith, Syzygies, London Mathematical Society Lecture Note Series, 106, Cambridge
+University Press, Cambridge, 1985. 8
+[9] R. Fossum, H.-B. Foxby, P. Griffith, I. Reiten, Minimal injective resolutions with applications to dualizing
+modules and Gorenstein modules, Inst. Hautes ´Etudes Sci. Publ. Math., 45 (1975), 193–215. 18
+[10] H.-B. Foxby, Gorenstein modules and related modules, Math. Scand., 31 (1972), 267–284. 1, 13, 18
+[11] H.-B. Foxby, n-Gorenstein rings, Proc. Amer. Math. Soc., 42 (1974), 67–72. 1, 2, 5, 8, 13, 16, 17, 18
+[12] S. Goto, N. Matsuoka, and T. T. Phuong, Almost Gorenstein rings, J. Algebra, 379 (2013), 355–381. 11, 12
+[13] S. Goto and K.-i. Watanabe, On graded rings I, J. Math. Soc. Japan, 30 (1978), no. 2, 179–213. 1, 8
+[14] J. Herzog and E. Kunz, Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics,
+238, Springer-Verlag, Berlin-New York, 1971. 1, 2, 3, 4, 7, 11
+[15] E. Kunz, The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc., 25 (1970), 748–
+751. 1
+[16] S. Okiyama, A local ring is CM if and only if its residue field has a CM syzygy, Tokyo J. Math. 14 (1991),
+489–500. 14
+[17] I. Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc., 32 (1972),
+417–420. 2, 9
+[18] P. Schenzel, Zur lokalen Kohomologie des kanonischen Moduls. Math. Z. 165 (1979), no. 3, 223–230. 15
+[19] R. Y. Sharp, On Gorenstein modules over a complete Cohen-Macaulay ring, Quart. J. Math., 22, no. 3 (1971),
+425–434. 2
+[20] R. Stanely, Hilbert functions of graded algebras, Adv. Math., 28 (1978), no. 1, 57–83. 1
+[21] R. Stanley, Combinatorics and commutative algebra, Second Edition, Birkh¨auser, Boston, 1996. 8
+[22] W. V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc., 19 (1968), 1349–1355. 2,
+4, 8
+[23] W. V. Vasconcelos, Integral Closure. Rees algebras, multiplicities, algorithms. Springer Monographs in Mathe-
+matics. Berlin, Springer-Verlag, 2005. 10
+
+RINGS WITH q-TORSIONFREE CANONICAL MODULES
+21
+School of Political Science and Economics, Meiji University, 1-9-1 Eifuku, Suginami-ku, Tokyo
+168-8555, Japan
+Email address: [email protected]
+URL: https://www.isc.meiji.ac.jp/~endo/
+Department of Mathematics, New York City College of Technology and the Graduate Center,
+The City University of New York, 300 Jay Street, Brooklyn, NY 11201, U.S.A.; 365 Fifth Avenue,
+New York, NY 10016, U.S.A.
+Email address: [email protected]
+Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-
+mita, Tama-ku, Kawasaki 214-8571, Japan
+Email address: [email protected]
+Department of Mathematics, Southern Connecticut State University, 501 Crescent Street, New
+Haven, CT 06515-1533, U.S.A.
+Email address: [email protected]
+Mathematics laboratory, Sapporo College, Hokkaido University of Education, 1-3 Ainosato 5-3,
+Kita-ku, Sapporo 002-8502, Japan
+Email address: [email protected]
+Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-
+mita, Tama-ku, Kawasaki 214-8571, Japan
+Email address: [email protected]
+Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-
+mita, Tama-ku, Kawasaki 214-8571, Japan
+Email address: [email protected]
+Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi 464-
+8602, Japan
+Email address: [email protected]
+

9dFQT4oBgHgl3EQfJTUp/content/tmp_files/2301.13255v1.pdf.txt

@@ -0,0 +1,847 @@
+XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE 
+Wavelet Analysis for Time Series Financial Signals 
+via Element Analysis 
+Nathan Zavanelli  
+George W. Woodruff School of Mechanical Engineering,  
+College of Engineering, Georgia Institute of Technology 
+Atlanta, GA 30332, USA 
+[email protected] 
+Abstract— The method of element analysis is proposed here as an 
+alternative to traditional wavelet-based approaches to analyzing 
+perturbations in financial signals by scale.  In this method, the 
+processes that generate oscillations in financial signals are modelled 
+as scaled, shifted, and isolated events that produce ripples of various 
+frequencies across a sea of noise as opposed to a simple sinusoidal or 
+mixed frequency oscillation or an impulse. This allows one to directly 
+estimate the wavelet parameters derived only from the generating 
+functions, rejecting spurious perturbations driven by noise or 
+extraneous factors. Financial signals may then be reconstructed based 
+on a finite set of generators localized in time and frequency. This 
+method offers a marked advantage compared to traditional 
+econometric tools because it directly targets the generators of 
+oscillations. Furthermore, the choice of the Morse wavelet allows for 
+wide latitude in capturing a broad set of diverse generators. In this 
+work, the basic mathematical principles underlying element analysis 
+are presented, and the method is applied to the study of variance in 
+financial data, where the advantages of element analysis over 
+traditional wavelet techniques is demonstrated. Specifically, in the 
+example analysis of inflation expectations, element analysis shows a 
+clear ability to distinguish between oscillations formed by noise and 
+those formed by generators logically matched to historical events.  
+Keywords—econometrics, wavelet, element analysis, variance, 
+financial signals 
+I. INTRODUCTION  
+Wavelet transforms are a powerful tool for analyzing 
+financial data because they decompose the fluctuations in a 
+signal (like a graph of stock price vs time) into different 
+frequency scales. This multi-resolution analysis is increasingly 
+used to isolate trends by time scale, derive scale-based 
+assessments of data variance, and assess correlation between 
+signals by scale1-6. For instance, Crowley et al used a 
+continuous wavelet transform (CWT) to analyze growth cycles 
+of productivity in the European Union (EU), United States (US) 
+and United Kingdom (UK), and they discovered that cycles 
+occurred at various frequencies beyond those classically 
+studied7,8. Furthermore, they characterized the correlation 
+between each region’s productivity cycles by frequency scale, 
+enabling them to hypothesize how international and national 
+factors drive production volatility in short- and long-term 
+scales. Similar analysis has been conducted for high frequency 
+stock trading, analyzing market trends, assessing relations 
+between variables and the yield curve, and quantifying risk4,6,9.   
+However, frequency decomposition techniques, like the 
+wavelet transform, have not achieved their full potential in 
+finance because the mathematical tools have not been 
+sufficiently updated in conjunction with recent discoveries in 
+adjacent fields10,11. In order to better understand the problem of 
+frequency decomposition, let us consider the development of 
+suitable approaches from simplest to most complex. The 
+Fourier transform is the simplest frequency decomposition 
+technique, representing a signal as a sum of sinusoidal 
+variations at different frequencies.  However, this method is ill-
+suited for handling non-sinusoidal signals12.  On the other 
+extreme, the modulus maxima method can be used to analyze 
+signals that are nearly impulses13,14. However, almost all 
+financial signals fall at neither extreme, instead exhibiting 
+complex morphologies  positioned over a background of 
+noise15. These morphologies are well represented by a series of 
+events localized in time with varying spatial distributions and 
+oscillatory and non-oscillatory components15.  Thus, an 
+effective means for studying these signals is to model them as 
+a sum of various scaled orthogonal wavelets, or the wavelet 
+transform2,4,16. However, this transform does not sufficiently 
+separate signal from noise for two reasons. First, any waveform 
+component, be it noise or signal, is mapped to a wavelet scale 
+without any means of distinguishing the two. Second, the signal 
+almost always does not exactly match the chosen wavelet, so it 
+is itself dispersed across several scales. The result is a blurred 
+transform, where significant information may be lost due to the 
+presence of noise4. Several traditional methods are commonly 
+used to address this issue, like wavelet thresholding and 
+complex statistical tests6,10,11. These approaches, however, are 
+also limited. In the first case, statistically significant wavelet 
+coefficients are identified and maximized, but the underlying 
+limitations of the wavelet transform are never addressed8,12. In 
+the second,  one typically must make strong assumptions about 
+either the duration or form of a signal, which can lead to 
+significant biases in analysis and great difficulty in 
+application10.   
+Instead, a new method termed element analysis, 
+developed by Lilly, can produce a much clearer distinction 
+between signal and noise16. The key intuition is to model the 
+processes that generate perturbations in financial signals as 
+scaled, shifted, and isolated events that produce ripples of 
+
+various frequencies across a sea of noise as opposed to a simple 
+sinusoidal or mixed frequency oscillation or an impulse.  Here, 
+a time series signal x(𝑡) is modelled not as a sum of sine waves, 
+impulses, or wavelets, but instead as a baseline of stationary 
+and Gaussian noise upon which are added many individual 
+copies of a complex valued function Ψ(𝑡) with a morphology 
+and time localization that is simply controlled by a time-offset, 
+phase shift, and scaling.  
+ 
+                 x(t) =  ∑ ℜ {cnΨμ,γ (t − tn
+ρn
+)}
+n
+n=1
++ xe(t)                1.1 
+ 
+where the complex parameter cn = |cn|eiϕn sets the amplitude 
+|cn| and phase ϕn of the event tn and ρn sets the event scale. 
+xe(t) represents the aforementioned noise. This representation 
+(1.1) is referred to as the element model. Element analysis 
+based on this model is similar to the CWT, but it limits the 
+signal reconstruction only to isolated points in both time and 
+frequency that correspond to specific events, rejecting spurious 
+noise. In general, this method has three steps. First, the wavelet 
+transform maxima corresponding only to events are identified. 
+Second, the significant of these maxima is examined in relation 
+to the noise threshold. Third, the reconstruction is performed 
+based on the coefficients resulting from these maxima. Element 
+analysis is a distinct improvement over wavelet analysis 
+because its goal is not to faithfully capture all signal content, 
+like the CWT, but instead to infer properties of key signal 
+events over a noise threshold. In essence, element analysis 
+seeks to assess the significance of signal events over the null 
+hypothesis of white noise. This method allows for a clear 
+distinction of financial signals separate from the noise, marking 
+a strong improvement over traditional wavelet approaches. 
+Although element analysis has been successfully employed for 
+a variety of signal processing disciplines, it has not been 
+employed for econometrics to the author’s knowledge, marking 
+a large missed opportunity in financial data analysis17,18.  
+ 
+The remainder of the paper will consist of the 
+following sections: a brief discussion of essential wavelet 
+principles, a general summary of the element method, an 
+example relating to financial volatility analysis, and a 
+discussion. In conjunction with his seminal paper, Lilly created 
+a freely available toolbox of Matlab functions, called jLab, 
+available at http://www.jmlilly.net16. Furthermore, all software 
+and data relating to the econometrics techniques discussed here 
+is 
+made 
+available 
+by 
+the 
+author 
+at 
+https://github.com/nzavanelli/Element_Analysis_Financial_D
+ata 
+ 
+II. WAVELET ESSENTIALS 
+ 
+This section seeks to briefly cover several of the key 
+wavelet properties needs to understand element analysis. For 
+further details, please see the following references. These next 
+two sections will also represent a simplification of the material 
+presented in Lilly’s work, which the reader may also 
+reference16. This section is divided into 2 subsections: (a) 
+continuous wavelet transforms based on the Morse wavelet and 
+(b) additional Morse wavelet properties.  
+ 
+A. CWT approaches with the Morse Wavelet 
+ 
+The Morse wavelet Ψ𝛽,𝛾  is a complex function 
+represented for 𝛽 ≥ 0 𝑎𝑛𝑑 𝛾 > 0 as follows: 
+ 
+                          Ψ𝛽,𝛾 = 𝛼𝛽,𝛾𝜔𝛽𝑒−𝜔𝛾 × {
+1  𝜔 > 0
+1
+2   𝜔 = 0
+0  𝜔 < 0
+                  2.1 
+ 
+where 𝛽  is the order, which controls the low frequency 
+behavior, 𝛾 the family, controlling the high frequency decay, 𝜔 
+the frequency, and 𝛼𝛽,𝛾 the normalizing constant of  
+ 
+                                          𝛼𝛽,𝛾 = 2 (𝑒𝛾
+𝛽 )
+𝛽
+𝛾                                     2.2 
+ 
+With this definition, the Morse wavelet is strictly analytic, 
+meaning that it must contain both complex and real 
+components. Therefore, the wavelets may be naturally grouped 
+into odd and even pairs, allowing them to capture phase 
+information similar to sine and cosine representations. The 
+wavelet transform of a signal x(𝑡) is represented in the time 
+domain and frequency domain, respectively, as follows: 
+ 
+𝒲𝛽,𝛾(τ, s) = ∫
+1
+𝑠
+∞
+−∞
+ Ψ∗𝛽,𝛾 (𝑡 − 𝜏
+𝑠
+) 𝑥(𝑡)𝑑𝑡 
+                                            = 1
+2𝜋 ∫
+𝑒𝑖𝜋𝜏Ψ∗𝛽,𝛾(𝑠, 𝜔)𝑋(𝜔)𝑑𝜔          2.3
+∞
+−∞
+ 
+ 
+where 𝑋(𝜔) denotes the Fourier transform of x(t) defined as  
+ 
+                                 𝑥(𝑡) = 1
+2𝜋 ∫ 𝑒𝑖𝜋𝜏𝑋(𝜔)𝑑𝜔                       2.4
+∞
+−∞
+ 
+ 
+This transform in the time domain is simply the inner product 
+of the signal and shifted, time scaled versions of the Morse 
+wavelet. In the frequency domain, the scale variable s 
+represents the stretching or compression of the signal, and the 
+rescaled frequency domain wavelet will always be maximized 
+at 𝜔𝑠 = 
+𝜔𝛽,𝛾
+𝑠 , which is referred to as the scale frequency. Note 
+that normalization by
+1
+√𝑠 is typically performed to ensure the 
+wavelet maintains constant energy. However, 
+1
+𝑠 normalization 
+is employed here because it allows for the transform values to 
+be controlled by only cn and not ρn, greatly simplifying the 
+analytic calculations employed in element analysis. 
+ 
+ 
+ 
+
+B. Morse wavelet properties 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 1. Morse wavelet representations with various 𝜷 and 
+𝜸 values. Here, the real, imaginary, and envelope components 
+are illustrated as blue, red, and yellow, respectively. 
+ 
+ 
+One highly attractive feature of Morse wavelets is that 
+they can assume a wide range of morphologies, which is easily 
+controlled by the choice of 𝛽 and 𝛾. This is illustrated in Fig 1. 
+Increasing 𝛽 tends to make the signal more oscillatory, and 
+increasing 𝛽 with a fixed 𝛾 causes more oscillations to fit in the 
+same envelope. On the other hand, modifying 𝛾  tends to 
+modulate the overall function and envelope shape.  
+ 
+III. ELEMENT ANALYSIS 
+ 
+This section pertains to the method of element analysis 
+developed by Lilly, representing a summary treatment16. Here, 
+the Morse wavelet is introduced as a signal element in 3(a). 
+Next, it is shown in 3(b) that a wavelet transform of a Morse 
+function is in fact another Morse wavelet. This allows for the 
+derivation in 3(c) of the element analysis method to produce 
+transform maxima. Finally, the algorithm is completed in 3(d) 
+by reproducing the element properties based on these maxima.  
+ 
+A. Morse wavelet representations of signal elements 
+ 
+Consider the wavelet function in (1.1), where 𝜇 and 𝛾 
+determine the element function properties (as described in Fig 
+1) and 𝜌 serves as the scale s. Taking the wavelet transform of 
+(1.1) with a Morse wavelet Ψ∗
+𝛽,𝛾 (
+𝑡−𝜏
+𝑠 ) leads to  
+ 
+𝒲𝛽,𝛾(τ, s) = 1
+2 ∑ 𝑐𝑛
+𝑛
+𝑛=1
+∫
+1
+𝑠
+∞
+−∞
+ Ψ∗
+𝛽,𝛾 (𝑡 − 𝜏
+𝑠
+) Ψ𝜇,𝛾 (𝑡 − 𝜏
+𝜌𝑛
+) 𝑑𝑡 + 𝜀𝛽,𝛾(𝜏, 𝑠)        3.1 
+ 
+where 𝜀𝛽,𝛾(𝜏, 𝑠) represents the wavelet transform of the noise 
+process in (1.1). Now, let us define the wavelet maxima points 
+as the time and scale coordinates where the wavelet transform 
+modulus is maximized. This will occur when the following four 
+conditions are met.  
+ 
+                                       ∂
+𝜕τ |𝑤𝛽,𝛾(𝜏, 𝑠)| = 0                                  3.2 
+                                       ∂
+𝜕s |𝑤𝛽,𝛾(𝜏, 𝑠)| = 0                                  3.3 
+                                      𝜕2
+𝜕𝑡2 |𝑤𝛽,𝛾(𝜏, 𝑠)| < 0                                3.4 
+                                      𝜕2
+𝜕𝑠2 |𝑤𝛽,𝛾(𝜏, 𝑠)| < 0                                3.5 
+ 
+The goal of element analysis is to simply to use the values of 
+the wavelet transform at these maxima points to estimate the 
+coefficients cn, the scales ρn, and the times tn of the N signal 
+events that constitute the signal. From there, a highly denoised 
+scalogram containing only the event content may be produced.  
+ 
+B. Wavelet transform of a Morse function 
+ 
+When one performs a wavelet transform of a 𝜇 order 
+Morse wavelet Ψ𝜇,𝛾 (
+𝑡
+𝜌) with a 𝛽 order wavelet of the same family 
+𝛾, the result is a modified wavelet 𝜁(𝛽,𝜇,𝛾) (
+𝜏
+𝜌 ,
+𝑠
+𝜌), as shown in 
+3.1. This transform is defined as 
+ 
+    𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) =
+𝛼𝛽,𝛾𝛼𝜇,𝛾
+𝛼𝛽+𝜇,𝛾
+𝑠𝛽
+(√𝑠𝛾 + 1
+𝛾
+)
+𝛽+𝜇+1 𝜓𝛽+𝜇,𝛾 (
+𝜏
+√𝑠𝛾 + 1
+𝛾
+)     3.6 
+ 
+For a rigorous derivation, please refer to Lilly’s work. Briefly, 
+this result may be obtained by substituting the wavelet 
+definition, evaluating the triple integral, rescaling the wavelet, 
+and performing a simple change of variables. Notably, 3.6 
+shows that performing a wavelet transform of a Morse wavelet 
+modifies the time and scale of the original wavelet, but does not 
+affect the transform amplitude. The result is a wavelet of order 
+𝛽 + 𝜇, which follows because both 𝛽 and 𝜇 are powers of 𝜔 in 
+the frequency domain, where the wavelet transform 
+corresponds to multiplication.  
+ 
+This modified wavelet also has two intriguing 
+properties: First, the amplitude of the wavelet transform is 
+highly dependent on the scales s and 𝜌. Second, the wavelet’s 
+time argument can be effectively rescaled by the transform 
+scale s. To examine the scaling effect on 3.6 in more detail, 
+consider the following two cases: 
+ 
+    𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) = 𝛼𝛽,𝛾𝛼𝜇,𝛾
+𝛼𝛽+𝜇,𝛾 ×
+{  
+  𝜌
+𝑠
+𝜇+1
+Ψ𝛽+𝜇,𝛾 (
+𝑡
+𝑠)                   𝑠 ≫ 𝜌
+𝑠
+𝜌
+𝛽
+Ψ𝛽+𝜇,𝛾 (
+𝑡
+𝜌)                      𝑠 ≪ 𝜌
+        3.7 
+ 
+The result is that when 𝑠 ≫ 𝜌, the resultant wavelet  𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) 
+is smoothed, with the transform spread out over the scale s. This is 
+because the transform wavelet Ψ𝛽,𝛾 (
+𝑡
+𝑠) is much broader than the 
+
+β=1
+β = 1/5
+β=5
+y=1
+=4
+=8Morse wavelet being transformed Ψ𝜇,𝛾 (
+𝑡
+𝜌). In the opposite case, 
+the wavelet scale becomes fixed at 𝜌, decreasing in magnitude 
+with further decreases in s.  
+ 
+C. Transform maxima 
+ 
+The modified wavelet function 𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) can be 
+used to identify the wavelet transform values at the maxima. 
+First, consider the wavelet transform definition from 3.1 with 
+the modified wavelet: 
+ 
+                        𝒲𝛽,𝛾(τ, s) = 1
+2 ∑ 𝑐𝑛
+𝑛
+𝑛=1
+𝜁(𝛽,𝜇,𝛾) (𝜏 − 𝑡𝑛
+𝜌𝑛
+, 𝑠
+𝜌𝑛
+) + 𝜀𝛽,𝛾(𝜏, 𝑠)            3.8  
+ 
+The expected value of the squared modulus of this wavelet 
+transform thus may be approximated as: 
+ 
+𝐸 {|𝒲𝛽,𝛾(τ, s)|
+2} ≈ 1
+4 ∑|𝑐𝑛|2
+𝑛
+𝑛=1
+|𝜁(𝛽,𝜇,𝛾) (𝜏 − 𝑡𝑛
+𝜌𝑛
+, 𝑠
+𝜌𝑛
+)|
+2
++ 𝐸 {|𝜀𝛽,𝛾(𝜏, 𝑠) |
+2}    3.9 
+ 
+This approximation requires the assumption that cross-terms 
+within the summation may be neglected on the basis of the zero 
+mean and that events are well separated. The second 
+assumption is generally not fully valid for financial signals, and 
+the result is the potential for low level maxima amplitudes to 
+arise. Fortunately, these amplitudes are generally higher than 
+the noise floor, but still lesser than a pure signal with only one 
+generating function. However, care must be taken when 
+selecting 𝛽 and 𝛾 parameters to ensure strong maxima in the 
+case of most signals. In general, financial signals with many 
+complicated interactions should avoid large 𝛾  and small 𝛽 
+values to ensure strong monotonic decay and avoid sidelobe 
+maxima effects.  
+ 
+Now let us consider the scale locations and wavelet 
+transform values corresponding to the wavelet maxima. Note 
+that the maxima of 𝜁(𝛽,𝜇,𝛾) (
+𝜏
+𝜌 ,
+𝑠
+𝜌) with respect to time occurs at 
+𝜏 = 0 , at which point 𝜁(𝛽,𝜇,𝛾) (0,
+𝑠
+𝜌)  assumes the real and 
+positive value: 
+ 
+    𝜁(𝛽,𝜇,𝛾) (0, 𝑠
+𝜌) =
+𝛼𝛽,𝛾𝛼𝜇,𝛾
+2𝜋𝛾
+Γ (𝛽 + 𝜇 + 1
+𝛾
+)
+(𝑠
+𝜌)
+𝛾
+√((𝑠
+𝜌)
+𝛾
++ 1)
+𝛾
+β+𝜇+1   3.10 
+ 
+This value may be derived by combining 3.6 with the definition 
+of the wavelet function at 𝜏 = 0 . Defining  𝑠̃ ≡
+𝑠
+𝜌  and 
+differentiating 𝜁(𝛽,𝜇,𝛾)(0, 𝑠̃) with this new variable allows for 
+one to determine that the maximal value occurs at: 
+ 
+                               𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 = 𝜁(𝛽,𝜇,𝛾)(0, 𝑠̃𝑚𝑎𝑥)                             3.11 
+ 
+                                                      𝑠̃𝑚𝑎𝑥 = (
+𝛽
+𝜇 + 1)
+1
+𝛾
+                                            3.12 
+ 
+Inserting 3.11 into 3.10 allows for the determination of the maximum 
+value of the modified wavelet transform: 
+ 
+ 
+                     𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 =
+𝛼𝛽,𝛾𝛼𝜇,𝛾
+2𝜋𝛾
+Γ (𝛽 + 𝜇 + 1
+𝛾
+) 𝜂𝛽,𝜇,𝛾               3.13 
+ 
+where 𝜂𝛽,𝜇,𝛾 is the scale weighting function defined as: 
+ 
+          𝜂𝛽,𝜇,𝛾 ≡
+𝑠̃𝑚𝑎𝑥
+𝛾
+√(𝑠̃𝑚𝑎𝑥
+𝛾 + 1)
+𝛾
+β+𝜇+1  =
+(
+𝛽
+𝜇 + 1)
+𝛽
+𝛾
+(
+𝛽
+𝜇 + 1 + 1)
+𝛽+𝜇+1
+𝛾
+        3.14 
+ 
+Thus, is can readily seen that the maximum value 𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 is indeed 
+independent of the scale 𝜌.  
+ 
+D. Estimating element properties from the transform maxima  
+ 
+In the case of well-behaved signals with a proper 
+choice of wavelet parameters, we will have one maximum point 
+for each of the N generating events, and the nth maxima will be 
+located at time 𝑡𝑛 and scale 𝑠𝑛 = 𝜌𝑛𝑠̃𝑚𝑎𝑥. It is clear from 3.8 
+that the wavelet transform here is thus: 
+ 
+                                            𝒲𝛽,𝛾(𝑡𝑛,𝑠𝑛) = 1
+2𝑐𝑛𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥                           3.15 
+ 
+Now, one may use the equations in 3.2-3.5 to define 
+the set of observed time/scale maxima points, which will be 
+denoted as (𝜏̂𝑛, 𝑠̂𝑛). From these points, the element properties  
+(𝑡𝑛, 𝑐𝑛, 𝜌𝑛)  may be simply estimated. If one defines                 
+  𝒲𝑛 ≡ 𝒲𝛽,𝛾(𝜏̂𝑛,𝑠̂𝑛) as the wavelet transform at each observed 
+maximum, then these element properties become: 
+ 
+𝑡̂𝑛 = 𝜏̂𝑛                   𝑐̂𝑛 = 2
+  𝒲𝑛
+𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥
+                   𝜌̂𝑛 =
+𝑠̂𝑛
+𝑠̃𝑚𝑎𝑥
+         3.16 
+ 
+where the quantities are hatted to show that these values are 
+estimates of the true element properties. Before the method is 
+complete, one final modification is necessary. Here, the 
+frequency of the function is reported instead of the scale. This 
+may be rectified by substituting     𝑠 =
+ω𝛽,𝛾
+ω𝑠  and 𝜌 =
+ω𝜇,𝛾
+ω𝑠  into 
+𝜌̂𝑛 =
+𝑠̂𝑛
+𝑠̃𝑚𝑎𝑥 from 3.16 to yield:  
+ 
+                            ω𝜌̂𝑛 = ω𝑠̂𝑛
+ω𝜇,𝛾
+ω𝛽,𝛾
+𝑠̃𝑚𝑎𝑥 = 𝜔𝑠̂𝑛
+𝜔𝜇,𝛾
+𝜔𝛽,𝛾
+(
+𝛽
+𝜇 + 1)
+1
+𝛾
+                         3.17 
+ 
+which is the relationship between the frequency band 𝜔𝑠̂𝑛of the 
+observed wavelet maximum and that of the corresponding 
+element, ω𝜌̂𝑛. We now have all the parameters necessary to 
+reconstruct the signal transform as a scalogram containing the 
+information of the N elements, without the noise function. An 
+algorithm to do so with examples and all code used in this work 
+is 
+proved 
+by 
+the 
+author 
+at 
+https://github.com/nzavanelli/Element_Analysis_Financial_D
+
+ata. Furthermore, the reader is encouraged to consider the 
+original algorithms derived by Lilly et al, which are available 
+at  http://www.jmlilly.net and upon which the author’s 
+algorithms are heavily based.  
+ 
+IV. APPLICATION TO VARIANCE ANALYSIS 
+ 
+  
+As mentioned in the introduction, wavelet analysis is 
+a powerful, yet underutilized, tool in econometrics for 
+analyzing financial data by time scale. Although many complex 
+analyzes are possible, like assessing the correlation of variables 
+to the yield curve by scale, two very simple examples will be 
+shown here to demonstrate that the element method of wavelet 
+analysis offers a notable improvement over traditional wavelet 
+methods.  
+First, let us consider the expected 10 year inflation rate 
+in the United States (E10YRI) between July 2018 and July 
+2022. Fig 2(A) shows the E10YRI versus time over the period 
+described. A third order high-pass Butterworth infinite impulse 
+response filter with a cutoff of 1/3 years is then applied to 
+isolate only the higher frequency perturbations in the signal, 
+removing any longer-term trends. The result is the graph in Fig 
+2(B). Next, a traditional wavelet scalogram is produced from 
+the data in Fig 2(B) using a Morse wavelet with parameters 𝛽 =
+3 and 𝛾 = 1. The resultant scalogram is shown in Fig 2(C). The 
+scalogram appears to show a clear and persistent long-term 
+volatility on the order of multiple months (1/12 – 1/20 years), 
+which generally waxes and wanes with time. Furthermore, 
+several shocks are present in March 2020, 2021, and 2022. 
+Interestingly, the volatility associated with the shock in 2022 
+appears to be notably higher frequency than that of 2021.   
+The data can be extracted from each of these frequency 
+bands and analyzed separately for a variety of purposes. 
+Although this scalogram does offer a promising and useful tool 
+for analyzing volatility in the E10YRI, a comparison with an 
+element analysis scalogram will reveal several limitations. The 
+element analysis method described in this paper was used with 
+identical wavelet parameters to produce the scalogram shown 
+in Fig 2(D). From this scalogram, several additional high 
+volatility events that are not evident in Fig 2(C) are clearly 
+visible. For instance, an uptick in volatility associated with the 
+stock market contraction of December 2018 is easily visible, 
+and the volatility in this shock can be clearly delineated by 
+frequency scale. In Fig 2(C), this shock is not readily visible, as 
+it is drowned out by noise and the other generating variabilities. 
+However, a careful inspection of Fig 2(B) shows that this 
+volatility is, in fact, present in December 2018. Whereas the 
+traditional scalogram cannot capture this information, the 
+element analysis method can. Likewise, news regarding 
+currency conflict between the EU and US and anxiety over 
+trade with China fueled volatility in July 2019, which is 
+captured in the element analysis scalogram and not the 
+traditional wavelet scalogram. Finally, the twin events of 2021 
+and 2022 can be better studied via element analysis. Whereas 
+the traditional wavelet scalogram indicates that the volatility in 
+2022 is bifurcated into two separate modes, the element 
+analysis produces the more intuitive result that the volatility is 
+in fact continuous. Similarly, the start and end times in Fig 2(C) 
+are generally more spread out than those clearly delineated in 
+Fig 2(D). Finally, it is noteworthy that Fig 2(C) generally lacks 
+much of the changes in high frequency volatility that is present 
+in Fig 2(D). Overall, it should be clear from this example that 
+the element analysis method offers a promising alternative to 
+traditional wavelet scalogram methods for analyzing financial 
+data.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2. Wavelet scalogram analysis of expected 10 year 
+inflation rate (E10YRI)  in the United States. (A) Plot of 
+E10YRI vs time. (B) Plot of high frequency filtered 
+perturbations in E10YRI vs time. (C) Wavelet scalogram 
+produced from the data in B. (D) Element analysis scalogram 
+produced from the data in B.   
+ 
+V. DISCUSSION 
+ 
+Here, the element analysis method of Lilly has been 
+described and, and its application to econometrics has been 
+demonstrated in a simple example16. The key intuition is to 
+model the processes that generate perturbations in financial 
+signals as scaled, shifted, and isolated events that produce 
+ripples of various frequencies across a sea of noise as opposed 
+to a simple sinusoidal or mixed frequency oscillation. This 
+method is similar to the continuous wavelet transform and 
+based on the Morse wavelet, but it is unique in that it produces 
+a new transform for each generating event, allowing for a 
+
+A
+Bsignificant improvement in noise reduction and signal clarity. 
+The analysis created by Lilly et al marks a valuable addition to 
+the econometrist’s toolbox for analyzing financial signals 
+because it can more precisely capture generators of 
+perturbations in financial signals than traditional wavelet 
+methods. This was demonstrated in an analysis of the expected 
+10-year inflation rate in the United States (E10YRI) between 
+July 2018 and July 2022, where several clear events were 
+present in the element analysis that could not be studied using 
+traditional wavelet methods.  In addition, there are ample 
+opportunities for this method to be improved to better fit 
+financial data. For instance, assumptions regarding the 
+distribution of noise could effect the expected value determined 
+in 3.9, and variance and bias terms could also be introduced at 
+this juncture. There is also a lack of clear criteria to determine 
+the optimal wavelet parameters for the transform wavelet, 
+especially for financial data. Finally, this method could be 
+further generalized by modelling events as superpositions of 
+higher order wavelets as opposed to single functions.  
+ 
+VI. REFERENCES 
+ 
+1. 
+Connor, J., and Rossiter, R. (2005). Wavelet 
+Transforms and Commodity Prices. Studies in 
+Nonlinear Dynamics & Econometrics 9. 
+2. 
+Collard, F. (1999). Spectral and persistence properties 
+of cyclical growth. Journal of Economic Dynamics 
+and Control 23, 463– 488. 
+3. 
+Dalkir, M. (2004). A new approach to causality in the 
+frequency domain. Economics Bulletin 3 (4), 1-14. 
+4. 
+Armah, M., Amewu, G., and Bossman, A. (2022). 
+Time-frequency analysis of financial stress and global 
+commodities prices: Insights from wavelet-based 
+approaches. Cogent Economics & Finance 10. 
+10.1080/23322039.2022.2114161. 
+5. 
+Karaev, A.K., Gorlova, O.S., Sedova, M.L., 
+Ponkratov, V.V., Shmigol, N.S., and Demidova, S.E. 
+(2022). Improving the Accuracy of Forecasting the 
+TSA Daily Budgetary Fund Balance Based on 
+Wavelet Packet Transforms. Journal of Open 
+Innovation: Technology, Market, and Complexity 8. 
+10.3390/joitmc8030107. 
+6. 
+Li, X., and Tang, P. (2020). Stock index prediction 
+based on wavelet transform and FCD ‐ MLGRU. 
+Journal 
+of 
+Forecasting 
+39, 
+1229-1237. 
+10.1002/for.2682. 
+7. 
+Crowley, P. (2007). A Guide To Wavelets For 
+Economists. Journal of Economic Surverys 21, 207-
+267. 
+8. 
+Crowley, P.M., and Hallett, A.H. (2015). Correlations 
+Between Macroeconomic Cycles in the US and UK: 
+What Can a Frequency Domain Analysis Tell Us? 
+Italian Economic Journal 2, 5-29. 10.1007/s40797-
+015-0023-6. 
+9. 
+Aguiar-Conraria, L., Martins, M.M.F., and Soares, 
+M.J. (2012). The yield curve and the macro-economy 
+across time and frequencies. Journal of Economic 
+Dynamics 
+and 
+Control 
+36, 
+1950-1970. 
+10.1016/j.jedc.2012.05.008. 
+10. 
+Singh, S., Parmar, K.S., Kumar, J., and Makkhan, 
+S.J.S. (2020). Development of new hybrid model of 
+discrete wavelet decomposition and autoregressive 
+integrated moving average (ARIMA) models in 
+application to one month forecast the casualties cases 
+of COVID-19. Chaos Solitons Fractals 135, 109866. 
+10.1016/j.chaos.2020.109866. 
+11. 
+Adebayo, T.S. (2020). New Insights into Export-
+growth Nexus: Wavelet and Causality Approaches. 
+Asian 
+Journal 
+of 
+Economics, 
+Business 
+and 
+Accounting, 32-44. 10.9734/ajeba/2020/v15i230212. 
+12. 
+Arfaoui, S., Ben Mabrouk, A., and C., C. (2021). 
+Wavelet Analysis Basic Concepts and Applications. 
+Chapman and Hall/CRC 1. 
+13. 
+Liu, J., Enderlin, E.M., Marshall, H.-P., and Khalil, A. 
+(2021). Automated Detection of Marine Glacier 
+Calving Fronts Using the 2-D Wavelet Transform 
+Modulus Maxima Segmentation Method. IEEE 
+Transactions on Geoscience and Remote Sensing 59, 
+9047-9056. 10.1109/tgrs.2021.3053235. 
+14. 
+Ding, W., and Li, Z. (2018). Research on adaptive 
+modulus maxima selection of wavelet modulus 
+maxima denoising. The Journal of Engineering 2019, 
+175-180. 10.1049/joe.2018.8958. 
+15. 
+Das, P. (2019). Econometrics in Theory and Practice. 
+Springer 1, 247-259. 
+16. 
+Lilly, J.M. (2017). Element analysis: a wavelet-based 
+method for analysing time-localized events in noisy 
+time series. Proc Math Phys Eng Sci 473, 20160776. 
+10.1098/rspa.2016.0776. 
+17. 
+Zavanelli, N., Kim, H., Kim, J., Herbert, R., 
+Mahmood, M., Kim, Y.S., Kwon, S., Bolus, B., 
+Torstrick, F.B., Lee, C.S.D., and Yeo, W.H. (2021). 
+At-home wireless monitoring of acute hemodynamic 
+disturbances to detect sleep apnea and sleep stages via 
+a soft sternal patch. Sci Adv 7, eabl4146  
+18. 
+Lee, S.H., Kim, Y.S., Yeo, M.K., Mahmood, M., 
+Zavanelli, N., Chung, C., Heo, J.Y., Kim, Y., Jung, 
+S.S., and Yeo, W.H. (2022). Fully portable continuous 
+real-time auscultation with a soft wearable stethoscope 
+designed for automated disease diagnosis. Sci. Adv. 8, 
+eabo5867. 
+ 
+

9dFQT4oBgHgl3EQfJTUp/content/tmp_files/load_file.txt

@@ -0,0 +1,400 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf,len=399
+page_content='XXX-X-XXXX-XXXX-X/XX/$XX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='00 ©20XX IEEE Wavelet Analysis for Time Series Financial Signals via Element Analysis Nathan Zavanelli George W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Woodruff School of Mechanical Engineering, College of Engineering, Georgia Institute of Technology Atlanta, GA 30332, USA nzavanelli@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='edu Abstract— The method of element analysis is proposed here as an alternative to traditional wavelet-based approaches to analyzing perturbations in financial signals by scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In this method, the processes that generate oscillations in financial signals are modelled as scaled, shifted, and isolated events that produce ripples of various frequencies across a sea of noise as opposed to a simple sinusoidal or mixed frequency oscillation or an impulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This allows one to directly estimate the wavelet parameters derived only from the generating functions, rejecting spurious perturbations driven by noise or extraneous factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Financial signals may then be reconstructed based on a finite set of generators localized in time and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This method offers a marked advantage compared to traditional econometric tools because it directly targets the generators of oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Furthermore, the choice of the Morse wavelet allows for wide latitude in capturing a broad set of diverse generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  In this work, the basic mathematical principles underlying element analysis are presented, and the method is applied to the study of variance in financial data, where the advantages of element analysis over traditional wavelet techniques is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Specifically, in the example analysis of inflation expectations, element analysis shows a clear ability to distinguish between oscillations formed by noise and those formed by generators logically matched to historical events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Keywords—econometrics, wavelet, element analysis, variance, financial signals I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' INTRODUCTION Wavelet transforms are a powerful tool for analyzing financial data because they decompose the fluctuations in a signal (like a graph of stock price vs time) into different frequency scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This multi-resolution analysis is increasingly used to isolate trends by time scale, derive scale-based assessments of data variance, and assess correlation between signals by scale1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' For instance, Crowley et al used a continuous wavelet transform (CWT) to analyze growth cycles of productivity in the European Union (EU), United States (US) and United Kingdom (UK), and they discovered that cycles occurred at various frequencies beyond those classically studied7,8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Furthermore, they characterized the correlation between each region’s productivity cycles by frequency scale, enabling them to hypothesize how international and national factors drive production volatility in short- and long-term scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  Similar analysis has been conducted for high frequency stock trading, analyzing market trends, assessing relations between variables and the yield curve, and quantifying risk4,6,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, frequency decomposition techniques, like the wavelet transform, have not achieved their full potential in finance because the mathematical tools have not been sufficiently updated in conjunction with recent discoveries in adjacent fields10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In order to better understand the problem of frequency decomposition, let us consider the development of suitable approaches from simplest to most complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The Fourier transform is the simplest frequency decomposition technique, representing a signal as a sum of sinusoidal variations at different frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, this method is ill- suited for handling non-sinusoidal signals12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' On the other extreme, the modulus maxima method can be used to analyze signals that are nearly impulses13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, almost all financial signals fall at neither extreme, instead exhibiting complex morphologies  positioned over a background of noise15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' These morphologies are well represented by a series of events localized in time with varying spatial distributions and oscillatory and non-oscillatory components15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Thus, an effective means for studying these signals is to model them as a sum of various scaled orthogonal wavelets, or the wavelet transform2,4,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, this transform does not sufficiently separate signal from noise for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  First, any waveform component, be it noise or signal, is mapped to a wavelet scale without any means of distinguishing the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Second, the signal almost always does not exactly match the chosen wavelet, so it is itself dispersed across several scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The result is a blurred transform, where significant information may be lost due to the presence of noise4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Several traditional methods are commonly used to address this issue, like wavelet thresholding and complex statistical tests6,10,11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' These approaches, however, are also limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In the first case, statistically significant wavelet coefficients are identified and maximized, but the underlying limitations of the wavelet transform are never addressed8,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In the second,  one typically must make strong assumptions about either the duration or form of a signal, which can lead to significant biases in analysis and great difficulty in application10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Instead, a new method termed element analysis, developed by Lilly, can produce a much clearer distinction between signal and noise16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The key intuition is to model the processes that generate perturbations in financial signals as scaled, shifted, and isolated events that produce ripples of  various frequencies across a sea of noise as opposed to a simple sinusoidal or mixed frequency oscillation or an impulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Here, a time series signal x(𝑡) is modelled not as a sum of sine waves, impulses, or wavelets, but instead as a baseline of stationary and Gaussian noise upon which are added many individual copies of a complex valued function Ψ(𝑡) with a morphology and time localization that is simply controlled by a time-offset, phase shift, and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  x(t) =  ∑ ℜ {cnΨμ,γ (t − tn ρn )} n n=1 + xe(t)                1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1  where the complex parameter cn = |cn|eiϕn sets the amplitude |cn| and phase ϕn of the event tn and ρn sets the event scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' xe(t) represents the aforementioned noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1) is referred to as the element model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Element analysis based on this model is similar to the CWT, but it limits the signal reconstruction only to isolated points in both time and frequency that correspond to specific events, rejecting spurious noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In general, this method has three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' First, the wavelet transform maxima corresponding only to events are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Second, the significant of these maxima is examined in relation to the noise threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Third, the reconstruction is performed based on the coefficients resulting from these maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Element analysis is a distinct improvement over wavelet analysis because its goal is not to faithfully capture all signal content, like the CWT, but instead to infer properties of key signal events over a noise threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In essence, element analysis seeks to assess the significance of signal events over the null hypothesis of white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This method allows for a clear distinction of financial signals separate from the noise, marking a strong improvement over traditional wavelet approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Although element analysis has been successfully employed for a variety of signal processing disciplines, it has not been employed for econometrics to the author’s knowledge, marking a large missed opportunity in financial data analysis17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  The remainder of the paper will consist of the following sections: a brief discussion of essential wavelet principles, a general summary of the element method, an example relating to financial volatility analysis, and a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In conjunction with his seminal paper, Lilly created a freely available toolbox of Matlab functions, called jLab, available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='jmlilly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='net16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Furthermore, all software and data relating to the econometrics techniques discussed here is made available by the author at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='com/nzavanelli/Element_Analysis_Financial_D ata  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' WAVELET ESSENTIALS  This section seeks to briefly cover several of the key wavelet properties needs to understand element analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' For further details, please see the following references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' These next two sections will also represent a simplification of the material presented in Lilly’s work, which the reader may also reference16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This section is divided into 2 subsections: (a) continuous wavelet transforms based on the Morse wavelet and (b) additional Morse wavelet properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' CWT approaches with the Morse Wavelet  The Morse wavelet Ψ𝛽,𝛾  is a complex function represented for 𝛽 ≥ 0 𝑎𝑛𝑑 𝛾 > 0 as follows:  Ψ𝛽,𝛾 = 𝛼𝛽,𝛾𝜔𝛽𝑒−𝜔𝛾 × { 1  𝜔 > 0 1 2   𝜔 = 0 0  𝜔 < 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1  where 𝛽  is the order, which controls the low frequency behavior, 𝛾 the family, controlling the high frequency decay, 𝜔 the frequency, and 𝛼𝛽,𝛾 the normalizing constant of  𝛼𝛽,𝛾 = 2 (𝑒𝛾 𝛽 ) 𝛽 𝛾                                     2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2  With this definition, the Morse wavelet is strictly analytic, meaning that it must contain both complex and real components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Therefore, the wavelets may be naturally grouped into odd and even pairs, allowing them to capture phase information similar to sine and cosine representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The wavelet transform of a signal x(𝑡) is represented in the time domain and frequency domain, respectively, as follows:  𝒲𝛽,𝛾(τ, s) = ∫ 1 𝑠 ∞ −∞ Ψ∗𝛽,𝛾 (𝑡 − 𝜏 𝑠 ) 𝑥(𝑡)𝑑𝑡 = 1 2𝜋 ∫ 𝑒𝑖𝜋𝜏Ψ∗𝛽,𝛾(𝑠, 𝜔)𝑋(𝜔)𝑑𝜔          2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='3 ∞ −∞  where 𝑋(𝜔) denotes the Fourier transform of x(t) defined as  𝑥(𝑡) = 1 2𝜋 ∫ 𝑒𝑖𝜋𝜏𝑋(𝜔)𝑑𝜔                       2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='4 ∞ −∞  This transform in the time domain is simply the inner product of the signal and shifted, time scaled versions of the Morse wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In the frequency domain, the scale variable s represents the stretching or compression of the signal, and the rescaled frequency domain wavelet will always be maximized at 𝜔𝑠 = 𝜔𝛽,𝛾 𝑠 , which is referred to as the scale frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Note that normalization by 1 √𝑠 is typically performed to ensure the wavelet maintains constant energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, 1 𝑠 normalization is employed here because it allows for the transform values to be controlled by only cn and not ρn, greatly simplifying the analytic calculations employed in element analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Morse wavelet properties  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Morse wavelet representations with various 𝜷 and 𝜸 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Here, the real, imaginary, and envelope components are illustrated as blue, red, and yellow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  One highly attractive feature of Morse wavelets is that they can assume a wide range of morphologies, which is easily controlled by the choice of 𝛽 and 𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This is illustrated in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Increasing 𝛽 tends to make the signal more oscillatory, and increasing 𝛽 with a fixed 𝛾 causes more oscillations to fit in the same envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' On the other hand, modifying 𝛾  tends to modulate the overall function and envelope shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' ELEMENT ANALYSIS  This section pertains to the method of element analysis developed by Lilly, representing a summary treatment16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Here, the Morse wavelet is introduced as a signal element in 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Next, it is shown in 3(b) that a wavelet transform of a Morse function is in fact another Morse wavelet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This allows for the derivation in 3(c) of the element analysis method to produce transform maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Finally, the algorithm is completed in 3(d) by reproducing the element properties based on these maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Morse wavelet representations of signal elements  Consider the wavelet function in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1), where 𝜇 and 𝛾 determine the element function properties (as described in Fig 1) and 𝜌 serves as the scale s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Taking the wavelet transform of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1) with a Morse wavelet Ψ∗ 𝛽,𝛾 ( 𝑡−𝜏 𝑠 ) leads to  𝒲𝛽,𝛾(τ, s) = 1 2 ∑ 𝑐𝑛 𝑛 𝑛=1 ∫ 1 𝑠 ∞ −∞ Ψ∗ 𝛽,𝛾 (𝑡 − 𝜏 𝑠 ) Ψ𝜇,𝛾 (𝑡 − 𝜏 𝜌𝑛 ) 𝑑𝑡 + 𝜀𝛽,𝛾(𝜏, 𝑠)        3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1  where 𝜀𝛽,𝛾(𝜏, 𝑠) represents the wavelet transform of the noise process in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Now, let us define the wavelet maxima points as the time and scale coordinates where the wavelet transform modulus is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This will occur when the following four conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  ∂ 𝜕τ |𝑤𝛽,𝛾(𝜏, 𝑠)| = 0                                  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2 ∂ 𝜕s |𝑤𝛽,𝛾(𝜏, 𝑠)| = 0                                  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='3 𝜕2 𝜕𝑡2 |𝑤𝛽,𝛾(𝜏, 𝑠)| < 0                                3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='4 𝜕2 𝜕𝑠2 |𝑤𝛽,𝛾(𝜏, 𝑠)| < 0                                3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='5  The goal of element analysis is to simply to use the values of the wavelet transform at these maxima points to estimate the coefficients cn, the scales ρn, and the times tn of the N signal events that constitute the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' From there, a highly denoised scalogram containing only the event content may be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Wavelet transform of a Morse function  When one performs a wavelet transform of a 𝜇 order Morse wavelet Ψ𝜇,𝛾 ( 𝑡 𝜌) with a 𝛽 order wavelet of the same family 𝛾, the result is a modified wavelet 𝜁(𝛽,𝜇,𝛾) ( 𝜏 𝜌 , 𝑠 𝜌), as shown in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This transform is defined as  𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) = 𝛼𝛽,𝛾𝛼𝜇,𝛾 𝛼𝛽+𝜇,𝛾 𝑠𝛽 (√𝑠𝛾 + 1 𝛾 ) 𝛽+𝜇+1 𝜓𝛽+𝜇,𝛾 ( 𝜏 √𝑠𝛾 + 1 𝛾 )     3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='6  For a rigorous derivation, please refer to Lilly’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Briefly, this result may be obtained by substituting the wavelet definition, evaluating the triple integral, rescaling the wavelet, and performing a simple change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Notably, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='6 shows that performing a wavelet transform of a Morse wavelet modifies the time and scale of the original wavelet, but does not affect the transform amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The result is a wavelet of order 𝛽 + 𝜇, which follows because both 𝛽 and 𝜇 are powers of 𝜔 in the frequency domain, where the wavelet transform corresponds to multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  This modified wavelet also has two intriguing properties: First, the amplitude of the wavelet transform is highly dependent on the scales s and 𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Second, the wavelet’s time argument can be effectively rescaled by the transform scale s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' To examine the scaling effect on 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='6 in more detail, consider the following two cases:  𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) = 𝛼𝛽,𝛾𝛼𝜇,𝛾 𝛼𝛽+𝜇,𝛾 × { 𝜌 𝑠 𝜇+1 Ψ𝛽+𝜇,𝛾 ( 𝑡 𝑠)                   𝑠 ≫ 𝜌 𝑠 𝜌 𝛽 Ψ𝛽+𝜇,𝛾 ( 𝑡 𝜌)                      𝑠 ≪ 𝜌 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='7  The result is that when 𝑠 ≫ 𝜌, the resultant wavelet  𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) is smoothed, with the transform spread out over the scale s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This is because the transform wavelet Ψ𝛽,𝛾 ( 𝑡 𝑠) is much broader than the  β=1 β = 1/5 β=5 y=1 =4 =8Morse wavelet being transformed Ψ𝜇,𝛾 ( 𝑡 𝜌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In the opposite case, the wavelet scale becomes fixed at 𝜌, decreasing in magnitude with further decreases in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Transform maxima  The modified wavelet function 𝜁(𝛽,𝜇,𝛾)(𝜏, 𝑠) can be used to identify the wavelet transform values at the maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' First, consider the wavelet transform definition from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1 with the modified wavelet:  𝒲𝛽,𝛾(τ, s) = 1 2 ∑ 𝑐𝑛 𝑛 𝑛=1 𝜁(𝛽,𝜇,𝛾) (𝜏 − 𝑡𝑛 𝜌𝑛 , 𝑠 𝜌𝑛 ) + 𝜀𝛽,𝛾(𝜏, 𝑠)            3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='8  The expected value of the squared modulus of this wavelet transform thus may be approximated as:  𝐸 {|𝒲𝛽,𝛾(τ, s)| 2} ≈ 1 4 ∑|𝑐𝑛|2 𝑛 𝑛=1 |𝜁(𝛽,𝜇,𝛾) (𝜏 − 𝑡𝑛 𝜌𝑛 , 𝑠 𝜌𝑛 )| 2 + 𝐸 {|𝜀𝛽,𝛾(𝜏, 𝑠) | 2}    3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='9  This approximation requires the assumption that cross-terms within the summation may be neglected on the basis of the zero mean and that events are well separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The second assumption is generally not fully valid for financial signals, and the result is the potential for low level maxima amplitudes to arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Fortunately, these amplitudes are generally higher than the noise floor, but still lesser than a pure signal with only one generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, care must be taken when selecting 𝛽 and 𝛾 parameters to ensure strong maxima in the case of most signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In general, financial signals with many complicated interactions should avoid large 𝛾  and small 𝛽 values to ensure strong monotonic decay and avoid sidelobe maxima effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  Now let us consider the scale locations and wavelet transform values corresponding to the wavelet maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Note that the maxima of 𝜁(𝛽,𝜇,𝛾) ( 𝜏 𝜌 , 𝑠 𝜌) with respect to time occurs at 𝜏 = 0 , at which point 𝜁(𝛽,𝜇,𝛾) (0, 𝑠 𝜌)  assumes the real and positive value:  𝜁(𝛽,𝜇,𝛾) (0, 𝑠 𝜌) = 𝛼𝛽,𝛾𝛼𝜇,𝛾 2𝜋𝛾 Γ (𝛽 + 𝜇 + 1 𝛾 ) (𝑠 𝜌) 𝛾 √((𝑠 𝜌) 𝛾 + 1) 𝛾 β+𝜇+1   3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='10  This value may be derived by combining 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='6 with the definition of the wavelet function at 𝜏 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Defining  𝑠̃ ≡ 𝑠 𝜌  and differentiating 𝜁(𝛽,𝜇,𝛾)(0, 𝑠̃) with this new variable allows for one to determine that the maximal value occurs at:  𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 = 𝜁(𝛽,𝜇,𝛾)(0, 𝑠̃𝑚𝑎𝑥)                             3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='11  𝑠̃𝑚𝑎𝑥 = (  𝛽  𝜇 + 1)  1  𝛾  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='12  Inserting 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='11 into 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='10 allows for the determination of the maximum value of the modified wavelet transform:  𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 = 𝛼𝛽,𝛾𝛼𝜇,𝛾 2𝜋𝛾 Γ (𝛽 + 𝜇 + 1 𝛾 ) 𝜂𝛽,𝜇,𝛾               3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='13  where 𝜂𝛽,𝜇,𝛾 is the scale weighting function defined as:  𝜂𝛽,𝜇,𝛾 ≡ 𝑠̃𝑚𝑎𝑥 𝛾 √(𝑠̃𝑚𝑎𝑥 𝛾 + 1) 𝛾 β+𝜇+1  = ( 𝛽 𝜇 + 1) 𝛽 𝛾 ( 𝛽 𝜇 + 1 + 1) 𝛽+𝜇+1 𝛾 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='14  Thus, is can readily seen that the maximum value 𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 is indeed independent of the scale 𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Estimating element properties from the transform maxima  In the case of well-behaved signals with a proper choice of wavelet parameters, we will have one maximum point for each of the N generating events, and the nth maxima will be located at time 𝑡𝑛 and scale 𝑠𝑛 = 𝜌𝑛𝑠̃𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' It is clear from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='8 that the wavelet transform here is thus:  𝒲𝛽,𝛾(𝑡𝑛,𝑠𝑛) = 1 2𝑐𝑛𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥                           3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='15  Now, one may use the equations in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='5 to define the set of observed time/scale maxima points, which will be denoted as (𝜏̂𝑛, 𝑠̂𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' From these points, the element properties (𝑡𝑛, 𝑐𝑛, 𝜌𝑛)  may be simply estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' If one defines 𝒲𝑛 ≡ 𝒲𝛽,𝛾(𝜏̂𝑛,𝑠̂𝑛) as the wavelet transform at each observed maximum, then these element properties become:  𝑡̂𝑛 = 𝜏̂𝑛                   𝑐̂𝑛 = 2 𝒲𝑛 𝜁(𝛽,𝜇,𝛾)𝑚𝑎𝑥 𝜌̂𝑛 = 𝑠̂𝑛 𝑠̃𝑚𝑎𝑥 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='16  where the quantities are hatted to show that these values are estimates of the true element properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Before the method is complete, one final modification is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Here, the frequency of the function is reported instead of the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This may be rectified by substituting     𝑠 = ω𝛽,𝛾 ω𝑠  and 𝜌 = ω𝜇,𝛾 ω𝑠  into 𝜌̂𝑛 = 𝑠̂𝑛 𝑠̃𝑚𝑎𝑥 from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='16 to yield:  ω𝜌̂𝑛 = ω𝑠̂𝑛  ω𝜇,𝛾  ω𝛽,𝛾  𝑠̃𝑚𝑎𝑥 = 𝜔𝑠̂𝑛  𝜔𝜇,𝛾  𝜔𝛽,𝛾  (  𝛽  𝜇 + 1)  1  𝛾  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='17  which is the relationship between the frequency band 𝜔𝑠̂𝑛of the observed wavelet maximum and that of the corresponding element, ω𝜌̂𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' We now have all the parameters necessary to reconstruct the signal transform as a scalogram containing the information of the N elements, without the noise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' An algorithm to do so with examples and all code used in this work is proved by the author at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='com/nzavanelli/Element_Analysis_Financial_D  ata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Furthermore, the reader is encouraged to consider the original algorithms derived by Lilly et al, which are available at  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='jmlilly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='net and upon which the author’s algorithms are heavily based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' APPLICATION TO VARIANCE ANALYSIS  As mentioned in the introduction, wavelet analysis is a powerful, yet underutilized, tool in econometrics for analyzing financial data by time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Although many complex analyzes are possible, like assessing the correlation of variables to the yield curve by scale, two very simple examples will be shown here to demonstrate that the element method of wavelet analysis offers a notable improvement over traditional wavelet methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' First, let us consider the expected 10 year inflation rate in the United States (E10YRI) between July 2018 and July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Fig 2(A) shows the E10YRI versus time over the period described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' A third order high-pass Butterworth infinite impulse response filter with a cutoff of 1/3 years is then applied to isolate only the higher frequency perturbations in the signal, removing any longer-term trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The result is the graph in Fig 2(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Next, a traditional wavelet scalogram is produced from the data in Fig 2(B) using a Morse wavelet with parameters 𝛽 = 3 and 𝛾 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The resultant scalogram is shown in Fig 2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The scalogram appears to show a clear and persistent long-term volatility on the order of multiple months (1/12 – 1/20 years), which generally waxes and wanes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Furthermore, several shocks are present in March 2020, 2021, and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Interestingly, the volatility associated with the shock in 2022 appears to be notably higher frequency than that of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  The data can be extracted from each of these frequency bands and analyzed separately for a variety of purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Although this scalogram does offer a promising and useful tool for analyzing volatility in the E10YRI, a comparison with an element analysis scalogram will reveal several limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The element analysis method described in this paper was used with identical wavelet parameters to produce the scalogram shown in Fig 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' From this scalogram, several additional high volatility events that are not evident in Fig 2(C) are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' For instance, an uptick in volatility associated with the stock market contraction of December 2018 is easily visible, and the volatility in this shock can be clearly delineated by frequency scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In Fig 2(C), this shock is not readily visible, as it is drowned out by noise and the other generating variabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' However, a careful inspection of Fig 2(B) shows that this volatility is, in fact, present in December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Whereas the traditional scalogram cannot capture this information, the element analysis method can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Likewise, news regarding currency conflict between the EU and US and anxiety over trade with China fueled volatility in July 2019, which is captured in the element analysis scalogram and not the traditional wavelet scalogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Finally, the twin events of 2021 and 2022 can be better studied via element analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  Whereas the traditional wavelet scalogram indicates that the volatility in 2022 is bifurcated into two separate modes, the element analysis produces the more intuitive result that the volatility is in fact continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Similarly, the start and end times in Fig 2(C) are generally more spread out than those clearly delineated in Fig 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Finally, it is noteworthy that Fig 2(C) generally lacks much of the changes in high frequency volatility that is present in Fig 2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Overall, it should be clear from this example that the element analysis method offers a promising alternative to traditional wavelet scalogram methods for analyzing financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Wavelet scalogram analysis of expected 10 year inflation rate (E10YRI)  in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (A) Plot of E10YRI vs time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (B) Plot of high frequency filtered perturbations in E10YRI vs time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (C) Wavelet scalogram produced from the data in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (D) Element analysis scalogram produced from the data in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' DISCUSSION  Here, the element analysis method of Lilly has been described and, and its application to econometrics has been demonstrated in a simple example16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The key intuition is to model the processes that generate perturbations in financial signals as scaled, shifted, and isolated events that produce ripples of various frequencies across a sea of noise as opposed to a simple sinusoidal or mixed frequency oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This method is similar to the continuous wavelet transform and based on the Morse wavelet, but it is unique in that it produces a new transform for each generating event, allowing for a  A Bsignificant improvement in noise reduction and signal clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The analysis created by Lilly et al marks a valuable addition to the econometrist’s toolbox for analyzing financial signals because it can more precisely capture generators of perturbations in financial signals than traditional wavelet methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' This was demonstrated in an analysis of the expected 10-year inflation rate in the United States (E10YRI) between July 2018 and July 2022, where several clear events were present in the element analysis that could not be studied using traditional wavelet methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' In addition, there are ample opportunities for this method to be improved to better fit financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' For instance, assumptions regarding the distribution of noise could effect the expected value determined in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='9, and variance and bias terms could also be introduced at this juncture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' There is also a lack of clear criteria to determine the optimal wavelet parameters for the transform wavelet, especially for financial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Finally, this method could be further generalized by modelling events as superpositions of higher order wavelets as opposed to single functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' REFERENCES  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Connor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Rossiter, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Wavelet Transforms and Commodity Prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Studies in Nonlinear Dynamics & Econometrics 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Collard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Spectral and persistence properties of cyclical growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Journal of Economic Dynamics and Control 23, 463– 488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Dalkir, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' A new approach to causality in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Economics Bulletin 3 (4), 1-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Armah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Amewu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Bossman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Time-frequency analysis of financial stress and global commodities prices: Insights from wavelet-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Cogent Economics & Finance 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1080/23322039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2114161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Karaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Gorlova, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Sedova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Ponkratov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Shmigol, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Demidova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Improving the Accuracy of Forecasting the TSA Daily Budgetary Fund Balance Based on Wavelet Packet Transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Journal of Open Innovation: Technology, Market, and Complexity 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='3390/joitmc8030107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Tang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Stock index prediction based on wavelet transform and FCD ‐ MLGRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Journal of Forecasting 39, 1229-1237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1002/for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Crowley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' A Guide To Wavelets For Economists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Journal of Economic Surverys 21, 207- 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Crowley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Hallett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Correlations Between Macroeconomic Cycles in the US and UK: What Can a Frequency Domain Analysis Tell Us?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Italian Economic Journal 2, 5-29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1007/s40797- 015-0023-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Aguiar-Conraria, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Martins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Soares, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  The yield curve and the macro-economy across time and frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Journal of Economic Dynamics and Control 36, 1950-1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='jedc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Parmar, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Makkhan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Development of new hybrid model of discrete wavelet decomposition and autoregressive integrated moving average (ARIMA) models in application to one month forecast the casualties cases of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Chaos Solitons Fractals 135, 109866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='109866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Adebayo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' New Insights into Export- growth Nexus: Wavelet and Causality Approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Asian Journal of Economics, Business and Accounting, 32-44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='9734/ajeba/2020/v15i230212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Arfaoui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Ben Mabrouk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Wavelet Analysis Basic Concepts and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Chapman and Hall/CRC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Enderlin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Marshall, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Khalil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Automated Detection of Marine Glacier Calving Fronts Using the 2-D Wavelet Transform Modulus Maxima Segmentation Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' IEEE Transactions on Geoscience and Remote Sensing 59, 9047-9056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1109/tgrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='3053235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Ding, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Research on adaptive modulus maxima selection of wavelet modulus maxima denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' The Journal of Engineering 2019, 175-180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1049/joe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='8958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Das, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Econometrics in Theory and Practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Springer 1, 247-259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Lilly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Element analysis: a wavelet-based method for analysing time-localized events in noisy time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='  Proc Math Phys Eng Sci 473, 20160776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='1098/rspa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='0776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Zavanelli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Herbert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Mahmood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kwon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Bolus, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Torstrick, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Lee, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Yeo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' At-home wireless monitoring of acute hemodynamic disturbances to detect sleep apnea and sleep stages via a soft sternal patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Sci Adv 7, eabl4146 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Yeo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Mahmood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Zavanelli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Chung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Heo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Kim, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', Jung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=', and Yeo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Fully portable continuous real-time auscultation with a soft wearable stethoscope designed for automated disease diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}
+page_content=' 8, eabo5867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFQT4oBgHgl3EQfJTUp/content/2301.13255v1.pdf'}

AtAzT4oBgHgl3EQfhv1C/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:310681fdf028969e38e85c54cf6ede8eb8083e566b80e6ec0f2bbd5a65c17603
+size 196591

B9FJT4oBgHgl3EQfsy1U/content/2301.11614v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:fd6f7bb84382c780c8f6de6e68da3f9e77420b89406549ebe24512ce334d4652
+size 9753420

D9AzT4oBgHgl3EQfGvu-/content/tmp_files/2301.01034v1.pdf.txt

@@ -0,0 +1,1275 @@
+arXiv:2301.01034v1  [math.CT]  3 Jan 2023
+VARIETIES OF QUANTITATIVE OR CONTINUOUS
+ALGEBRAS (EXTENDED ABSTRACT)
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+Abstract. Quantitative algebras are algebras enriched in the cat-
+egory Met of metric spaces so that all operations are nonexpand-
+ing. Mardare, Plotkin and Panangaden introduced varieties (aka
+1-basic varieties) as classes of quantitative algebras presented by
+quantitative equations. We prove that they bijectively correspond
+to strongly finitary monads T on Met. This means that T is the
+Kan extension of its restriction to finite discrete spaces. An analo-
+gous result holds in the category CMet of complete metric spaces.
+Analogously, continuous algebras are algebras enriched in CPO,
+the category of ω-cpos, so that all operations are continuous. We
+introduce equations between extended terms, and prove that vari-
+eties (classes presented by such equations) correspond bijectively
+to strongly finitary monads T on CPO. This means that T is the
+Kan extension of its restriction to finite discrete cpos. (The two
+results have substantially different proofs.) An analogous result is
+also presented for monads on DCPO.
+We also characterize strong finitarity in all the categories above
+by preservations of certain weighted colimits. As a byproduct we
+prove that directed colimits commute with finite products in all
+cartesian closed categories.
+1. Introduction
+Quantitative algebraic reasoning was formalized in a series of arti-
+cles of Mardare, Panangaden and Plotkin [10, 21, 22, 11] as a tool for
+studying computational effects in probabilistic computation. Those pa-
+pers work with algebras in the category Met of metric spaces or CMet of
+complete metric spaces. Metrics are always understood to be extended:
+the distance ∞ is allowed; morphisms are the nonexpanding maps f
+which means that for x, y in the domain one has d(x, y) ≤ d(f(x), f(y)).
+Quantitative algebras are algebras acting on a (complete) metric space
+A so that every n-ary operation is a nonexpanding map from An (with
+the maximum metric) to A. Mardare et al. introduced quantitative
+equations, which are formal expressions t =ε t′ where t and t′ are terms
+and ε ≥ 0 is a rational number. A quantitative algebra A satisfies this
+equation iff for every interpretation of the variables the elements of A
+corresponding to t and t′ have distance at most ε. A variety (called
+J. Ad´amek and M. Dost´al acknowledge the support by the Grant Agency of the
+Czech Republic under the grant 22-02964S.
+1
+
+2
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+1-basic variety in [21]) is a class of quantitative algebras presented by
+a set of quantitative equations. Classical varieties of (non-structured)
+algebras are well known to correspond bijectively to finitary monads
+T = (T, µ, η) on Set, i.e. T preserves directed colimits: every variety
+is isomorphic to the category SetT of algebras for T, and vice versa.
+The question whether an analogous correspondence holds for quanti-
+tative algebras has been posed in two presentations of LICS 22, see [2]
+and [24]. We answer this by working with enriched (i.e. locally nonex-
+panding) monads on the category Met of metric spaces, and introduc-
+ing weighted colimits called precongruences. We prove that varieties
+of quantitative algebras bijectively correspond to categories MetT for
+strongly finitary monads T on Met. And we characterize these monads
+as precisely those that preserve directed colimits and colimits of pre-
+congruences (Theorem 3.21). Analogously for strongly finitary monads
+on the category CMet (Theorem 3.22).
+We also study closely related continuous algebras which are alge-
+bras acting on a cpo (a poset with joins of ω-chains) so that their
+operations are continuous. Here we use equations t = t′ between ex-
+tended terms which allow not only the formation of composite terms
+t = σ(t0, . . . , tn−1) for n-ary operations σ, but also the formation of for-
+mal joins t = �
+k∈N tk for countable collections of terms. A variety of
+continuous algebras is a class presented by a set of such equations. We
+again work with enriched (i.e. locally continuous) monads. We prove
+that varieties of continuous algebras bijectively correspond to categories
+CPOT for strongly finitary monads on CPO. And we characterize these
+monads as precisely those that preserve directed colimits and reflexive
+coinserters (Theorem 4.25). The proof substantially uses that (unlike
+Met and CMet) the category CPO is cartesian closed. We prove that in
+every cartesian closed category directed colimits commute with finite
+products (Theorem 2.12).
+Related Work The main tool of Mardare et at. ([21, 22]) are ω-
+basic equations: for a finite metric space M on the set of variebles of
+terms t and t′ one wries M ⊢ t =ε t′. An algebra A satisfies this equa-
+tion if every nonexpanding interpretation f : M → A of the variebles
+the elements corresponding to t and t′ have distance at most ε.
+A
+class of quantitative algebras presented by such equations is called an
+ω-variety. Unfortunately, the free-algebra monad of an ω-variety need
+not be finitary ([2], Example 4.1). However, when the category Met
+is substituted by its full subcategory UMet of ultrametric spaces, then
+ω-varieties were proved in [2] to correspond bijectively to enriched mon-
+ads preserving directed colimits of split monomorphisms and surjective
+morphisms.
+Full proofs of the results presented in this extended abstract can be
+found in [4] and [5].
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+3
+2. Strongly Finitary Monads
+Assumption 2.1. Throughout our paper we work with categories and
+functors enriched over a symmetric monoidal closed category (V , ⊗, I).
+We recall these concepts shortly and introduce strongly finitary monads
+on V , giving a characterization of them via preservation of certain
+colimits. Our leading examples of V are (complete) metric spaces and
+(complete) partially ordered sets.
+Definition 2.2 ([13], 6.12). A symmetric monoidal category is given by
+a category V , a bifunctor ⊗ : V × V → V and an object I. Moreover,
+natural isomorphisms are given expressing that ⊗ is commutative and
+associative, and has the unit I (all up to coherent natural isomorphism).
+Finally, for every object Y a right adjoint of the functor −⊗Y : V → V
+is given. We denote it by [Y, −] and denote the morphism corresponding
+to f : X ⊗ Y → Z by �f : Y → [X, Z].
+Often ⊗ is the categorical product and I the terminal object; then
+V is called cartesian closed.
+Examples 2.3.
+(1) V = Pos, the category of posets, is cartesian closed, [X, Y ] is the
+poset of all monotone maps f : X → Y ordered pointwise. Here
+�f = curryf is the curried form of f.
+(2) V = CPO, the category of cpos (more precisely: ω-cpos) which
+are posets with joins of ω-chains.
+Morphisms are the continu-
+ous maps: monotone maps preserving joins of ω-chains. It is also
+cartesian closed, [X, Y ] is the cpo of all continuous maps (ordered
+again pointwise). Analogously V = DCPO is the category of posets
+with directed joins (dcpos) where morphisms (also called continu-
+ous maps) preserve directed joins.
+(3) V = Met, the category of (extended) metric spaces and nonexpand-
+ing maps. Objects are metric spaces defined as usual, except that
+the distance ∞ is allowed. Nonexpanding maps are those maps
+f : X → Y with d(x, x′) ≥ d(f(x), f(x′)) for all x, x′ ∈ X. A
+product of metric spaces Ai (i ∈ I) is the metric space on �
+i∈I Ai
+with the supremum metric
+d((xi), (yi)) = sup
+i∈I
+d(xi, yi).
+This category is not cartesian closed: curryfication is not bijec-
+tive. However, Met is symmetric closed monoidal w.r.t. the tensor
+product X ⊗ Y which is the cartesian product with the addition
+metric
+d((x, y), (x′, y′)) = d(x, x′) + d(y, y′).
+Here [X, Y ] is the metric space Met(X, Y ) of all morphisms f :
+X → Y with the supremum metric: the distance of f, g : X → Y
+is supx∈X d(f(x), g(x)). And I is the singleton space.
+
+4
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+(4) The category CMet of complete metric spaces is the full subcategory
+of Met on spaces with limits of all Cauchy sequences. It has the
+same symmetric closed monoidal structure as above: if X and Y
+are complete spaces, then so are X ⊗ Y and [X, Y ].
+Notation 2.4.
+(1) Every set X is considered as a discrete cpo with x ⊑ x′ iff x = x′.
+This is the coproduct �
+X I in CPO (and also in DCPO). Anal-
+ogously, X is considered as a discrete metric space: all distances
+of x ̸= x′ are ∞. This is the coproduct �
+X I in Met (and also in
+CMet).
+(2) For the category Setf of finite sets and mappings we define a functor
+K : Setf → V ,
+X �→
+�
+X
+I.
+Thus for V = Met, CMet, CPO or DCPO it assigns to every fi-
+nite set the corresponding discrete metric space or discrete cpo,
+respectively.
+Convention 2.5. By a catgory C we always mean a category enriched
+over V . It is given by
+(1) a class obC of objects,
+(2) an object C (X, Y ) of V (called the hom-object) for every pair X, Y
+in obC ,
+(3) a ’unit’ morphism uX : I → C (X, X) in V for every object X ∈
+obC , and
+(4) ’composition’ morphisms
+cX,Y,Z : C (X, Y ) ⊗ C (Y, Z) → C (X, Z)
+for all X, Y, Z ∈ obC , subject to commutative diagrams expressing
+the associativity of composition and the fact that uX are units of
+composition. For details see [13], 6.2.1.
+Examples 2.6.
+(1) If V = Met then C is an ordinary category in which every hom-set
+C (X, Y ) carries a metric such that composition is nonexpanding.
+Analogously for V = CMet.
+(2) If V = CPO then each hom-set C (X, Y ) carries a cpo such that
+composition is continuous. Analogously for DCPO.
+Let us recall the concept of an enriched functor F : C → C ′ for
+(enriched) categories C and C ′. It assigns
+(1) an object FX ∈ obC ′ to every object X ∈ obC , and
+(2) a morphism FX,Y : C (X, Y ) → C ′(FX, FY ) of V to every pair
+X, Y ∈ obC so that the expected diagrams expressing that F pre-
+serves composition and identity morphisms commute.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+5
+Convention 2.7. By a functor we always mean an enriched functor.
+We use ’ordinary functor’ in the few cases where a non-enriched functor
+is meant.
+Examples 2.8.
+(1) For categories enriched over Met a functor F : C → C ′ is an ordi-
+nary functor which is locally nonexpanding: given f, g ∈ C (X, Y )
+we have d(f, g) ≥ d(Ff, Fg). Analogously for CMet.
+(2) For categories enriched over Pos functors F are the locally mono-
+tone ordinary functors: given f ⊑ g in C (X, Y ), we get Ff ⊑ Fg
+in C (FX, FY ).
+(3) If V = CPO, then F is an ordinary functor which is locally con-
+tinuous: it is locally monotone and for all ω-chains fn : X → Y
+in C (X, Y ) we have F(�
+n<ω fn) = �
+n<ω Ffn.
+Analogously for
+DCPO.
+Remark 2.9.
+(1) In general one also needs the concept of an enriched natural trans-
+formation between parallel (enriched) functors. However, if V is
+one of the categories of Example 2.3, this concept is just that of an
+ordinary natural transformation between the underlying ordinary
+functors.
+(2) Given two categories D, C , we denote by [D, C ] the category of all
+functors F : D → C enriched by assigning to every pair of functors
+F, G : D → C an appropriate object [F, G] of V of all natural
+transformations.
+In case V = Pos, CPO or DCPO the order of [F, G] is component-
+wise: given τ, τ ′ : F → G put τ ⊑ τ ′ iff τX ⊑ τ ′
+X holds in [FX, GX]
+of all X ∈ obD.
+In case V = Met or CMet the distance of τ, τ ′ is supX∈obD d(τX, τ ′
+X).
+Definition 2.10 ([13, 16]). A weighted diagram in a category C is
+given by a functor D : D → C together with a weight W : Dop → V .
+A weighted colimit is an object C = colimWD of C together with
+isomorphisms in V :
+ψX : C (C, X) → [Dop, C ](W, C (D−, X))
+natural in X ∈ obC . The unit of this colimit is the natural transfor-
+mation
+ν = ψC(id C) : W → C (D−, C).
+In all categories of Example 2.3 weighted colimits (for all D small)
+exist.
+Example 2.11. (Conical) directed colimits are the special case where
+D is a directed poset and the weight W is trivial: the constant functor
+with value 1 (the terminal object).
+
+6
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+(1) In Pos directed colimits are formed on the level of the underlying
+sets. They commute with finite products.
+(2) In CPO directed colimits exist, but are not formed on the level of the
+underlying sets. For example, the finite ordinals An = {0, . . . , n−1}
+form a directed diagram with inclusions An ֒→ An+1 as connecting
+maps for n < ω. The colimit of this diagram in CPO is
+N⊤ = colim
+n<ω An,
+the chain of natural numbers with a top element ⊤ added. Still,
+directed colimits commute with finite products in CPO:
+Theorem 2.12. In every cartesian closed category C directed colimits
+commute with finite products.
+Proof.
+(1) Suppose D, D′ : D → C are diagrams, where D is a directed poset.
+Given colimit cocones cd : Dd → C and c′
+d : D′d → C, it is our
+task to prove that for the diagram D × D′ : D → C (given by
+d �→ Dd × D′d) the cocone cd × c′
+d : Dd × D′d′ → C × C′ is a
+colimit, too.
+(2) Define a diagram D ∗ D′ : D × D → C by (d, d′) �→ Dd × D′d′. We
+shall prove that it has the colimit cd × c′
+d′ : Dd × D′d′ → C × C′.
+This proves the theorem: since D is directed, the diagonal ∆ :
+D → D × D is a cofinal functor, thus D ∗ D′ has the same colimit
+as D × D′ = (D ∗ D′) · ∆.
+(3) Given a cocone fd,d′ : Dd × D′d′ → E of D ∗ D′, we prove that it
+factorizes through cd ×c′
+d′; it is easy to verify that the factorization
+is unique. Fix an object d′ ∈ D and form the adjoint transposes
+�fd,d′ : D′d′ → [Dd, E] for all d ∈ D. They form a cocone of D,
+thus, there exists a unique factorization through the cocone cd.
+That is, we have a unique gd′ : D′d′ × C → E with �fd,d′ = �gd′ · cd
+(for all d ∈ D). For the isomorphism u : C × D′d′ → D′d′ × C
+put hd′ = gd′ · u : C × D′d′ → E and form adjoint transposes
+�hd′ : D′d′ → [C, E] for all d′ ∈ D. This is a cocone of D′, thus
+there exists a unique factorization through the cocone c′
+d′: we have
+a unique h : C × C′ → E with h′
+d′ = �h · c′
+d′ (for all d′ ∈ D). It
+follows that h is the desired factorization of fd,d′:
+h · (cd × c′
+d′) = h · (C × c′
+d′) · (cd × C′) = hd′ · (cd × D′d′) = fd.d′.
+□
+Example 2.13. (Conical) directed colimits in Met and CMet also ex-
+ist. Again, they are not formed on the level of the underlying sets.
+For example, consider the diagram of metric space An = {0, 1} with
+dn(0, 1) = 2−n, where the connecting maps are id : An → An+1 (n < ω).
+The colimit is a singleton space.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+7
+Theorem 2.14. Directed colimits in Met or CMet commute with finite
+products.
+Proof sketch.
+(1) For directed diagrams (Di)i∈I in Met, cocones ci : Di → C forming
+a colimit were characterized in [8], Lemma 2.4, by the following
+properties:
+(a) C = �
+i∈I ci[Di], and
+(b) for every i ∈ I, given y, y′ ∈ Di we have
+d(ci(y), ci(y′)) = inf
+j≥i d(fj(y), fj(y′))
+where fj : Di → Dj denotes the connecting map.
+Given another directed diagram (D′
+i)i∈I with a cocone c′
+i : D′
+i →
+C′ satisfying (a) and (b), it is our task to prove that the cocone
+ci ×c′
+i : Di ×D′
+i → C ×C′ satisfies (a), (b), too. Since I is directed,
+(a) is clear, and (b) needs just a short computation.
+(2) For directed colimits in CMet the characterization of colimit co-
+cones is analogous: (b) is unchanged, and in (a) one states that
+�
+i∈I ci[Di] is dense in C. The further argument is then analogous
+to (1).
+□
+Example 2.15.
+(1) For our next development an important type of a weighted colimit
+in Pos, CPO or DCPO is the coinserter. Let f0, f1 : A → B be
+an ordered parallel pair.
+Its coinserter is a universal morphism
+c : B → C w.r.t. the property c · f0 ⊑ c · f1. Universality means
+that
+(a) every morphism c′ with c′ · f0 ⊑ c′ · f1 factorizes through c and
+(b) given u, v : C → D with u · c ⊑ v · c, it follows that u ⊑ v.
+For Pos this is precisely the weighted colimit of the diagram D :
+D → Pos where D consists of a single parallel pair δ0, δ1 : d → d
+(where D(d, d) is a discrete poset) and Dδi = fi.
+The weight
+W : Dop → Pos is given by Wd = {0, 1} where 0 < 1, Wd = {∗}
+and Wδi(∗) = i. Analogously for CPO or DCPO.
+(2) A concrete example: every poset C is a coinserter of a parallel
+pair between discrete posets. Indeed, let |C| be the discrete poset
+underlying C and let C(2) ⊆ |C|×|C| be the set of all pairs x0 ⊑ x1
+in C. For the pair of projections π0, π1 : C(2) → |C| the coinserter
+is id : |C| → C. Analogously for CPO and DCPO.
+Definition 2.16. A coinserter c of f0, f1 is called surjective if c is a
+surjective map. It is called reflexive if f0, f1 is a reflexive pair, i.e. they
+are split epimorphisms with a joint splitting d (f0 · d = f1 · d = id).
+
+8
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+Example 2.17. The coinserter of Example 2.15 (2) is reflexive and
+surjective. In Pos, all coinserters are surjective, in CPO they are not in
+general.
+Proposition 2.18. In Pos, CPO and DCPO reflexive coinserters com-
+mute with finite products.
+The proof is similar to that of Theorem 2.12.
+Analogously to coinserters of discrete cpos yielding all cpos, we now
+introduce weighted colimits in Met of diagrams called precongruences
+(a name borrowed from [14]).
+They express every metric space as
+a colimit of discrete spaces. (The weight used for precongruence is,
+however, not discrete.)
+In the following definition |M| denotes the underlying set (a discrete
+metric space) of a metric space M.
+Definition 2.19.
+(1) We define the basic weight W0 : Dop
+0
+→ Met as follows. The cat-
+egory D0 has an object a and objects ε for every rational number
+ε > 0. The only non-trivial hom-spaces are the spaces
+D0(ε, a) = {λε, ρε} with d(λε, ρε) = ε
+Thus D0 consists of the discrete category of positive rationals to-
+gether with a pair of cocones (having codomain a). The values
+of W0 are W0a = {0} and W0ε = {l, r} with d(l, r) = ε.
+The
+morphisms W0λε, W0ρε : {0} → {l, r} are given by 0 �→ l, 0 �→ r,
+respectively.
+(2) For every metric space M we define its precongruence as the weighted
+diagram DM : D0 → Met with the basic weight W0, where DMa =
+|M| and DMε ⊆ |M|×|M| is the set of all pairs of distance at most
+ε. Here Dλε, Dρε : DMε → |M| are the left and right projections,
+respectively.
+Proposition 2.20. Every metric space M is the weighted colimit of
+its precongruence in Met.
+Proof. Given a space X, to give a natural transformation τ : W0 →
+[DM−, X] means to specify a map f = τa(0) : |M| → X together with
+τε(l), τε(r) : DMε → X with τε(l) = f · πl and τε(r) = f · πr. Thus τ is
+determined by f and the given equations are equivalent to f : M → X
+being nonexpanding. The desired isomorphism ψX of Definition 2.10
+is given by ψX(τ) = f.
+□
+Remark 2.21. Analogously we define precongruences in CMet: we just
+use the codomain restrictions W0 : Dop
+0
+→ CMet and DM : D0 → CMet.
+Again, every complete space is the weighted colimit of its precongruence
+in CMet.
+Remark 2.22.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+9
+(1) Let C = colimWD be a weighted colimit of D : D → C with unit
+ν : W → C (D−, C). Given an (enriched) functor F : C → C ′,
+it preserves the colimit provided that the diagram FD : D → C ′
+with weight W has the colimit colimWFD = FC with the unit
+ν : W → C ′(FD−, FC) having components νd = Fνd.
+(2) A functor is finitary if it preserves directed colimits.
+Example 2.23.
+(1) The endofunctor (−)n of Met or CMet preserves colimits of precon-
+gruences for every n ∈ N. This is easy to verify. By Example 2.13
+(−)n is finitary.
+(2) The endofunctor (−)n on CPO or Pos preserves reflexive coinsert-
+ers for every n ∈ N by Proposition 2.18, and is finitary by Theo-
+rem 2.12.
+Let us recall the concept of the (enriched) left Kan extension [16] of a
+functor F : A → C along a functor K : A → C : this is an endofunctor
+LanKF : C → C endowed with a universal natural transformation
+τ : F → (LanKF) · K. The universal property states that given a
+natural transformation σ : F → G·K for any endofunctor G : C → C ,
+there exists a unique natural transformation σ : LanKF → G with
+σ = σK · τ.
+Definition 2.24. An endofunctor F of V is strongly finitary if it is a
+left Kan extension of its restriction F · K to Setf:
+F = LanK(F · K)
+(see Notation 2.4).
+Examples 2.25.
+(1) An endofunctor of Set is strongly finitary iff it is finitary.
+(2) An endofunctor of Pos is strongly finitary iff it is finitary and pre-
+serves reflexive coinserters, see [3].
+In order to characterize strong finitarity for V = CPO, DCPO, Met
+and CMet, we apply Kelly’s concept of density presentation that we
+now recall.
+Notation 2.26. Let K : A → C be a functor. We denote by �K :
+C → [A op, V ] the functor �KC = C (K−, C).
+For example, the functor K : Setf → CPO yields �K : CPO →
+[Setop
+f , CPO] taking a cpo C to the functor C(−) : Setop
+f
+→ CPO of
+finite powers of C. Analogously for K : Setf → Met.
+Definition 2.27 ([16]). A density presentation of a functor K : A →
+C is a collection of weighted colimits in C such that
+(1) �K preserves those colimits, and
+(2) C is the (iterated) closure of the image K[A ] under those colimits.
+
+10
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+Example 2.28.
+(a) K : Setf → Pos has a density presentation consisting of directed
+colimits and reflexive coinserters.
+Indeed, Condition (2) in the
+above definition follows from Example 2.15 (2): every finite poset
+is a coinserter of a pair in Setf. And every poset is a directed col-
+imit of finite subposets. For Condition (1) observe that �K : Pos →
+[Setop
+f , Pos] assigns to every poset A the functor of its finite pow-
+ers A(−) : Setop
+f
+→ Pos. This functor preserves directed colimits
+(Proposition 2.12) and reflexive coinserters (Example 2.23).
+(b) K : Setf → CPO also has a density presentation consisting of
+directed colimits and reflexive coinserters. Here Condition (1) is
+verified as above, using Proposition 2.12 and Example 2.23. To
+verify Condition (2), we express every cpo in four steps, starting
+from Setf:
+(i) Every finite cpo is a reflexive coinserter of a parallel pair in
+Setf (Example 2.15 (2)).
+(ii) The cpo N⊤ is a directed colimit of finite cpos An (Exam-
+ple 2.11 (2)). Analogously, every copower r • N⊤ of r copies,
+r ∈ N, is a directed colimit of r • An for n < ω.
+(iii) In this step we create all reflexive coinserters C of pairs
+f0, f1 : r • N⊤ → r′ • N⊤ for all r, r′ ∈ N. Such cpos C are
+called basic.
+(iv) The proof is concluded by proving that every cpo A is a
+directed colimit of the diagram of all of its basic sub-cpos Ai
+(i ∈ I). In fact, that this diagram is directed follows from
+the fact that a coproduct of two basic cpos is clearly basic.
+Given a cocone si : Ai → S, we are to prove that there is
+a unique continuous map s : A → S extending each si. For
+each x ∈ A the subposet {x} is clearly basic. Thus given i
+with x ∈ Ai the value si(x) is independent of i. This follows
+easily from the compatibility of the cocone si. The desired
+map is defined by s(x) = si(x). The verification that s is
+continuous is a bit more subtle.
+Remark 2.29.
+(1) The reflexive coinserters used in steps (i) and (iii) of the last exam-
+ple are all surjective. Consequently, K : Setf → CPO also has the
+density presentation consisting of directed colimits and reflexive,
+surjective coinserters.
+(2) The functor K : Setf → DCPO also has the density presentation
+of all directed colimits and reflexive (surjective) coinserters. The
+proof is the same as for CPO: all cpos used in the last example are
+indeed dcpos.
+Corollary 2.30. An endofunctor of CPO or DCPO is strongly finitary
+iff it preserves directed colimits and reflexive (surjective) coinserters.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+11
+Indeed, by [16], Theorem 5.29, given a density presentation of K :
+Setf → CPO, strong finitarity means precisely preservation of all col-
+imits in that presentation. The same corollary also holds in DCPO.
+Example 2.31. The categories Met and CMet have a density pre-
+sentation of K consisting of all directed diagrams and precongruences
+of finite spaces.
+Indeed, Condition (1) follows from Examples 2.13
+and 2.23 (1). For Condition (2) observe that finite metric spaces are
+obtained from Setf as colimits of precongruences by Lemma 2.20, and
+every (complete) metric space is a directed colimit of all of its finite
+subspaces in Met (or CMet, resp.).
+Corollary 2.32. An endofunctor of Met or CMet is strongly finitary
+iff it preserves directed colimits and colimits of precongruences.
+Example 2.33. Every coproduct of endofunctors (−)n with n finite on
+Met, CMet, CPO or DCPO is strongly finitary. Indeed, strongly finitary
+functors are closed under coproducts, which follows directly from the
+definition.
+3. Varieties of Quantitative Algebras
+We now prove that varieties of quantitative algebras on the cate-
+gories Met and CMet bijectively correspond to strongly finitary monads.
+These are monads carried by a strongly finitary endofunctor. Through-
+out this section Σ = (Σn)n∈N denotes a signature, and V is a specified
+countable set of variables.
+Notation 3.1.
+(1) Following Mardare, Panangaden and Plotkin [21], a quantitative
+algebra is a metric space A endowed with a nonexpanding opera-
+tion σA : An → A for every σ ∈ Σn (w.r.t. the supremum metric
+(Example 2.3)). We denote by Σ-Met the category of quantitative
+algebras and nonexpanding homomorphisms. Its forgetful functor
+is denoted by UΣ : Σ-Met → Met.
+(2) Analogously, a complete quantitative algebra is a quantitative alge-
+bra carried by a complete metric space.
+The category Σ-CMet
+is the corresponding full subcategory of Σ-Met.
+We again use
+UΣ : Σ-CMet → CMet for the forgetful functor.
+(3) The underlying set of a metric space M is denoted by |M|.
+Example 3.2. (1) A free quantitative algebra on a metric space M is
+the usual algebra TΣM of terms on variables from M. That is, the
+smallest set containing |M| and such that for every n-ary symbol
+σ and every n-tuple of terms ti (i < n) we obtain a composite term
+σ(ti)i<n. To describe the metric, let us introduce the equivalence ∼
+on TΣM (similarity of terms): it is the smallest equivalence making
+all variables of |M| into one class, and such that σ(ti)i<n ∼ σ′(t′
+i)i<n
+
+12
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+holds iff σ = σ′ and ti ∼ t′
+i for all i < n. The metric of TΣM extends
+that of M as follows: d(t, t′) = ∞ if t is not similar to t′. For similar
+terms t = σ(ti) and t′ = σ(t′
+i) we put d(t, t′) = supi<n d(ti, t′
+i).
+(2) If M is a complete space, TΣM is also complete, and this is the free
+quantitative algebra on M in Σ-CMet.
+In particular, consider the specified set V of variables as a discrete
+metric space, then TΣV is the discrete algebra of usual terms. For every
+algebra A and every interpretation of variables f : V → A (in Met or
+CMet) we denote by f ♯ : TΣV → A the corresponding homomorphism:
+it interprets terms in A.
+Definition 3.3 ([21]). By a quantitative equation (aka 1-basic quanti-
+tative equation) is meant a formal expression t =ε t′ where t, t′ are
+terms in TΣV and ε ≥ 0 is a rational number.
+An algebra A in
+Σ-Met (or Σ-CMet) satisfies that equation if for every interpretation
+f : V → A we have d(f ♯(t), f ♯(t′)) ≤ ε. We write t = t′ in case ε = 0.
+By a variety, aka 1-basic variety, of quantitative (or complete quan-
+titative) algebras is meant a full subcategory of Σ-Met (or Σ-CMet,
+resp.) specified by a set of quantitative equations.
+Example 3.4.
+(1) Quantitative monoids are given by the usual signature: a binary
+symbol · and a constant e, and by the usual equations: (x · y) · z =
+x · (y · z), e · x = x, and x · e = x.
+(2) Almost commutative monoids are quantitative monoids in which
+the distance of ab and ba is always at most 1. They are presented
+by the quantitative equation x · y =1 y · x.
+Proposition 3.5 (See [21]). Every variety V of quantitative algebras
+has free algebras: the forgetful funtor UV : V → Met has a left adjoint
+FV : Met → V.
+Notation 3.6. We denote by TV the free-algebra monad of a variety
+V on Met. Its underlying functor is TV = UV · FV.
+Example 3.7. For V = Σ-Met we have seen the monad TΣ in Exam-
+ple 3.2: TΣM is the metric space of all terms over M. Observe that TΣ
+is a coproduct of endofunctors (−)n, one summand for each similarity
+class of terms on n variables over M (which is independent of the choice
+M). Thus TΣ is a strongly finitary monad: see Example 2.33.
+Remark 3.8.
+(1) Recall the comparison functor KV : V → MetTV: it assigns to every
+algebra A of V the algebra on UVA for TV given by the unique
+homomorphism α : FVUVA → A extending id UVA. More precisely:
+KVA = (UVA, UVα).
+(2) By a concrete category over Met is meant a category together with
+a faithful ’forgetful’ functor UV : V → Met. For example a variety,
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+13
+or MetT for every monad T. A concrete functor is a functor F :
+V → W with UV = UWF. For example, the comparison functor
+KV.
+Proposition 3.9. Every variety V of quantitative algebras is concretely
+isomorphic to the category MetTV: the comparison functor KV : V →
+MetTV is a concrete isomorphism.
+Proof. For classical varieties (over Set) this is proved in [20], Theo-
+rem VI.8.1. The proof for Met in place of Set is analogous.
+□
+Notation 3.10. (1) Given a natural number n denote by [n] the sig-
+nature of one n-ary symbol δ.
+If a term t ∈ TΣV contains at
+most n variables (say, x0, . . . , xn−1), we obtain a monad morphism
+t : T[n] → TΣ as follows. For every space M the function tM takes
+a term s using the single symbol δ and substitutes each occurence
+of δ by t(x0, . . . , xn−1).
+(2) Every metric space A defines a monad ⟨A, A⟩ on Met assigning to
+X ∈ Met the space ⟨A, A⟩X = [[X, A], A]. More precisely: the
+functor [−, A] : Met → Metop is self-adjoint, and ⟨A, A⟩ is the
+monad corresponding to that adjunction.
+(3) Let T be a monad on Met and α : TA → A an algebra for it.
+We denote by �αX : TX → ⟨A, A⟩X the morphism adjoint to the
+following composite
+[X, A] ⊗ TX
+T(−)⊗TX
+−−−−−−→ [TX, TA] ⊗ TX
+ev
+−→ TA
+α−→ A.
+Theorem 3.11 ([15]). Given an algebra α : TA → A for a monad T
+on Met, the morphisms �αX above form a monad morphism �α : T →
+⟨A, A⟩. Every monad morphism from T to ⟨A, A⟩ has that form for a
+unique α.
+Lemma 3.12. Let A be a Σ-algebra expressed by α : TΣA → A in
+MetTΣ. It satisfies a quantitative equation l =ε r iff the distance of
+�α · l, �α · r : T[n] → ⟨A, A⟩ is at most ε.
+Notation 3.13. The category of finitary monads on Met (and monad
+morphisms) is denoted by Monf(Met). Its full subcategory of strongly
+finitary monads by Monsf(Met).
+We also denote by Monc(Met) the
+category of monads preserving countably directed colimits.
+Lemma 3.14. The category Monf(Met) has weighted colimits, and
+Monsf(Met) is closed under them.
+Proof sketch. (1) The category Monc(Met) is locally countably presentable
+as an enriched category, thus it has weighted colimits, and
+(2) both Monf(Met) and Monsf(Met) are coreflective subcategories of
+Monc(Met). The coreflection of a countably accessible monad T
+in Monsf(Met) is given by the left Kan extension LanK(T · K),
+analogously for Monf(Met).
+□
+
+14
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+Lemma 3.15. Every monad morphism α : TΣ → S in the category
+Monf(Met) factorizes as a morphism TΣ → S with surjective compo-
+nents followed by a morphism S → S whose components are isometric
+embeddings.
+Theorem 3.16. For every variety V of quantitative algebras the free-
+algebra monad TV is strongly finitary.
+Proof sketch.
+(1) Let V be given by a signature Σ and quantitative equations li =ε ri
+(i ∈ I), each containing ni variables. For every i ∈ I we consider
+the signature [n(i)] of one symbol δi of arity n(i), then the terms
+li, ri yield the corresponding monad morphisms li, ri : T[n(i)] → TΣ
+of Notation 3.10. An algebra α : TΣA → A lies in V iff the distance
+of �α·li, �α·ri : T[n(i)] → ⟨A, A⟩ is at most εi for each i (Lemma 3.12).
+(2) We verify that TV is a weighted colimit of strongly finitary monads
+in Monf(Met). The domain D of the weighted diagram D : D →
+Monf(Met) is the discrete category I (indexing the equations) en-
+larged by a new object a, and by morphisms λi, ρi : i → a for every
+i ∈ I. Then put Di = T[n(i)] and Da = TΣ; further Dλi = li
+and Dρi = ri. The weight W : Dop → Met takes i to the space
+{l, r} with d(l, r) = εi and a to {0}. We define Wλi(0) = l and
+Wρi(0) = r. The monads TΣ and T[n(i)] are strongly finitary by
+Example 3.7. This will finish the proof by Lemma 3.14.
+We denote by T the weighted colimit T = colimWD in Monf(Met).
+The proof is concluded by proving that V is isomorphic, as a con-
+crete category, to the category MetT of algebras for T.
+Then
+T is the free-algebra monad of V.
+For T we have the unit γ :
+W → �D−, T� of the weighted colimit T = colimWD (Defini-
+tion 2.10).
+Its component νa assigns to 0 a monad morphism
+γ = νa(0) : TΣ → T, whereas for i ∈ I the component νi is given by
+l �→ γ ·li and r �→ γ ·ri. Since νi is nonexpanding, we conclude that
+γ ·λi, γ ·ρi : T[n(i)] → T have distance at most εi. We thus obtain a
+functor E : MetT → V assigning to every algebra α : TA → A the
+Σ-algebra corresponding to α · γA : TΣA → A: it satisfies li =εi ri
+due to d(γ · λi, γ · ρi) ≤ εi. Moreover, γ has surjective components,
+which can be derived from Lemma 3.15. Therefore, E is a concrete
+isomorphism, which concludes the proof.
+□
+Construction 3.17. In the reverse direction we assign to every strongly
+finitary monad T = (T, µ, η) on Met a variety VT, and prove that T is
+its free-algebra monad. For every morphism k : X → Y let us denote
+by k∗ = µY · Tk : TX → TY the corresponding homomorphism in
+MetT. Recall our fixed set V = {xi | i ∈ N} of variables, and form,
+for each n ∈ N, the finite discrete space Vn = {xi | i < n}.
+The
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+15
+signature we use has as n-ary symbols the elements of the space TVn:
+Σn = |TVn| for n ∈ N. The variety VT is given by the following quanti-
+tative equations, where each symbol σ ∈ Σn is considered as the term
+σ(x1, . . . , xn−1) and n, m below range over N:
+(1) σ =ε σ′ for all σ, σ′ ∈ Σn with d(σ, σ′) ≤ ε in TVn;
+(2) k∗(σ) = σ(k(xi))i<n for all σ ∈ Σn and all maps k : Vn → Σm;
+(3) ηVn(xi) = xi for all i = 0, . . . , n − 1.
+Lemma 3.18. Every algebra α : TA → A in MetT yields an algebra
+A in VT with operations σA : An → A defined by σA(a(xi)) = a∗(σ) for
+all σ ∈ Σn and a : Vn → A. Moreover, every homomorphism in MetT
+is also a Σ-homomorphism between the corresponding algebras in VT.
+Proof sketch. (1) The operation σA is nonexpanding because T is lo-
+cally nonexpanding. It satisfies 1) in Construction 3.17 because
+for every interpretation a : Vn → A we have d(a∗(l), a∗(r)) ≤ ε.
+Satisfaction of 2) follows from a∗ · k∗ = (a∗ · k)∗, and 3) is clearly
+satisfied. Thus the Σ-algebra A lies in VT.
+(2) Given a morphism h : (A, α) → (B, β) in MetT (i.e., h· α = β · Th)
+we are to prove that h · σA = σB · hn for all σ ∈ TVn. This follows
+easily from h · a∗ = (h · a)∗ for each a : Vn → A.
+□
+Theorem 3.19. Every strongly finitary monad T on Met is the free-
+algebra monad of the variety VT.
+Proof. For every metric space M we want to prove that the Σ-algebra
+associated with (TM, µM) in Lemma 3.18 is free in VT w.r.t. the uni-
+versal map ηM. Then the theorem follows from Proposition 3.9.
+We have two strongly finitary monads, T and the free algebra monad
+of VT (Theorem 3.16). Thus, it is sufficient to prove the above for finite
+discrete spaces M. Then this extends to all finite spaces because we
+have M = colimW0DM (Lemma 2.20) and both monads preserve this
+colimit. Since they coincide on all finite discrete spaces, they coincide
+on M. Finally, the above extends to all spaces M: we have a directed
+colimit M = colim
+i∈I
+Mi of the diagram of all finite subspaces Mi (i ∈ I)
+which both monads preserve.
+Given a finite discrete space M, we can assume without loss of gen-
+erality M = Vn for some n ∈ N.
+For every algebra A in VT and
+an interpretation f : Vn → A, we prove that there exists a unique
+Σ-homomorphism f : TVn → A with f = f · ηVn.
+Existence: Define f(σ) = σA(f(xi))i<n for every σ ∈ TVn. The equal-
+ity f = f · ηVn follows since A satisfies the equations ηVn(xi) = xi, thus
+the operation of A corresponding to ηVn(xi) is the i-th projection. The
+map f is nonexpanding: given d(l, r) ≤ ε, the algebra A satisfies l =ε r.
+
+16
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+Therefore given an n-tuple f : Vn → A we have
+d(lA(f(xi)), rA(f(xi))) ≤ ε.
+To prove that f is a Σ-homomorphism, take an m-ary operation symbol
+τ ∈ TVm. We prove f · τVm = τA · f
+m. This means that every k : Vm →
+TVn fulfils
+f · τVm(k(xj))j<m = τA · f
+m(k(xj))j<m.
+The definition of f yields that the right-hand side is τA(k(xj)A(f(xi))).
+Due to equation (2) in Construction 3.17 with τ in place of σ this is
+k∗(τ)A(f(xi)). The left-hand side yields the same result since
+f
+m(k(xj)) = (k(xj))A(f(xi)).
+Uniqueness: Let f be a nonexpanding Σ-homomorphism with f =
+f · ηVn. In TVn the operation σ asigns to ηVn(xi) the value σ. (Indeed,
+for every a : n → |TVn| we have σTVn(ai) = a∗(σ) = µVn · Ta(σ). Thus
+σTVn(ηVn(xi)) = µVn · TηVn(σ) = σn.) Since f is a homomorphism, we
+conclude
+f(σ) = σA(f · ηVn(xi)) = σA(f(xi))
+which is the above formula.
+□
+Corollary 3.20. Varieties of quantitative algebras correspond bijec-
+tively, up to isomorphism, to strongly finitary monads on Met.
+Indeed, a stronger result can be deduced from Theorems 3.16 and 3.19:
+let Var(Met) denote the category of varieties of quantitative algebras
+and concrete functors (Remark 3.8). Recall that Monsf(Met) denotes
+the category of strongly finitary monads.
+Theorem 3.21. The category Var(Met) of varieties of quantitative al-
+gebras is equivalent to the dual of the category Monsf(Met) of strongly
+finitary monads on Met.
+Proof. Morphisms ϕ : S → T between monads in Monsf(Met) bijec-
+tively correspond to concrete functors ϕ : MetT → MetS ([12], Theo-
+rem 3.3): ϕ assigns to an algebra α : TA → A of MetT the algebra
+α · ϕA : SA → A in MetS.
+We know that for every variety V the
+comparison functor is invertible (Proposition 3.9). This yields a func-
+tor Φ : Var(Met)op → Monsf(Met) assigning to a variety V the monad
+TV (Theorem 3.16). Given a concrete functor F : V → W between
+varieties, there is a unique monad morphism ϕ : TW → TV such that
+ϕ = KW · F · K−1
+V
+: MetTV → MetTW. We define ΦF = ϕ and get a
+functor which is clearly full and faithful. Thus Theorem 3.19 implies
+that Φ is an equivalence of categories.
+□
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+17
+The whole development of the present section works in Σ-CMet as
+well as in CMet. First observe that for every complete space M the
+space TΣM of Example 3.2 is also complete (being a coproduct of fi-
+nite powers of M). The resulting monad TΣ on the category CMet is
+strongly finitary (as in Example 3.7). More generally, every variety V
+of complete quantitative algebras yields a monad TV on CMet which
+is strongly finitary, and V is isomorphic to MetTV. The proof is anal-
+ogous to that of Theorem 3.16, just at the end we use, instead of the
+factorization system (surjective, isometric embedding) of Met the fac-
+torization system of CMet consisting of dense morphisms f : A → B
+(every element of B is a limit of a sequence in f[A]) followed by closed
+isometric embeddings. The proof that every strongly finitary monad
+on CMet is the free-algebra monad of a variety is completely analogous
+to that of Theorem 3.19. We thus obtain
+Theorem 3.22. The category Var(CMet) of varieties of complete quan-
+titative algebras is equivalent to the dual of the category Monsf(CMet)
+of strongly finitary monads on CMet.
+4. Varieties of Continuous Algebras
+For the categories Pos, CPO and DCPO we obtain here and in Sec-
+tion 5 the same result: varieties of algebras bijectively correspond to
+strongly finitary monads. For Pos we have proved this in [3]. The proof
+for CPO presented below is very different from the proofs in [3] and in
+the previous section. In fact, already the concept of equation is entirely
+different since it uses formal joins �
+k∈N tk of collections t0, t1, t2, . . . of
+terms. The idea of such formal joins stems from [7], but our concept is
+slightly more restrictive: we request that all the terms ti contain only
+a finite set of variables. We assume again that Σ is a finitary signa-
+ture, and that a countable set V of variables has been chosen. The
+underlying set of a cpo M is denoted by |M|.
+Definition 4.1. A continuous algebra is a cpo A endowed with con-
+tinuous operations σA : An → A for every n-ary symbol σ ∈ Σ (w.r.t.
+the coordinate-wise order on An). We denote by Σ-CPO the category
+of continuous algebras and continuous homomorphisms.
+Example 4.2. A free continuous algebra on a cpo M is the usual
+algebra TΣM of terms on variables from |M| (compare Example 3.2)
+with the following order ⊑ extending that of M: t ⊑ t′ iff t and t′ are
+similar, t = σ(ti)i<n, t′ = σ(t′
+i)i<n and such that ti ⊑ t′
+i for every i < n.
+In particular, considering V as a discrete cpo (no distinct elements
+are comparable), then TΣV is the discrete cpo of the usual terms. For
+every continuous algebra A and every interpretation f : V → A of
+variables we again denote by f ♯ : TΣV → A the corresponding homo-
+morphism. As already mentioned, usual terms are not sufficient for
+
+18
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+equational presentations: we need formal joins of terms. We use the
+symbol �
+k∈N for them, while �
+k∈N denotes ω-joins in a given poset.
+Definition 4.3.
+(1) The set TΣV of extended terms is the smallest set containing TΣV ,
+and such that for every countable collection tk (k ∈ N) of extended
+terms containing only finitely many variables we get an extended
+term �
+k∈N tk.
+(2) For every continuous algebra A and every interpretation f : V → A
+of variables we define the interpretations of extended terms as the
+following partial function f @ : TΣV ⇀ A:
+(a) f @ extends f ♯ (thus f @(t) is defined for all terms t ∈ TΣV ),
+and
+(b) f @ is defined in t = �
+k∈N tk iff each f @(tk) is defined and fulfils
+f @(tk) ⊑ f @(tk+1) in A; then f @(t) = �
+k∈N f @(tk).
+Example 4.4. Given a unary operation σ, the extended term �
+k∈N σk(x)
+is well formed, but �
+k∈N σ(xk) is not: it contains infinitely many vari-
+ables.
+Definition 4.5. By an equation we understand a formal expression
+t = t′, where t, t′ are extended terms in TΣV .
+A continuous algebra satisfies t = t′ if for every interpretation f :
+V → A of the variables both f @(t) and f @(t′) are defined and are
+equal.
+A variety of continuous algebras is a full subcategory of Σ-CPO pre-
+sented by a set of equations.
+Remark 4.6.
+(1) We do not need presentation by inequations t ≤ t′.
+Indeed, to
+satisfy such an inequation means precisely to satisfy t = �
+k∈N tk
+where t0 = t and tk = t′ for all k > 0.
+(2) A term t is definable in an algebra A iff for every interpretation
+f : V → A of variables f @(t) is defined.
+Instead of equations,
+we can use definability to introduce varieties. Indeed, an algebra
+satisfies t = t′ iff the term �
+k∈N sk where s0 = t, s1 = t′ and sk = t
+for k ≥ 2 is definable. Conversely, t is definable in A iff A satisfies
+t = t.
+Example 4.7.
+(1) Continuous monoids are monoids acting on cpos with continuous
+multiplication: for all ω-chains (ak), (bk) we have (� ak)(� bk) =
+�(akbk). This is a variety presented by the usual monoid equations
+(see Example 3.4).
+(2) Continuous monoids satisfying a ⊑ a2 are presented by the defin-
+ability of the term �
+k∈N xk.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+19
+(3) The equation �
+k∈N xk = e presents continuous monoids satisfying
+a ⊑ a2 and �
+k∈N ak = e for all a.
+Remark 4.8. In the classical universal algebra Birkhoff’s Variety The-
+orem states that varieties are precisely the HSP classes, i.e., closed un-
+der homomorphic images, subalgebras, and products. In Σ-CPO we
+have the corresponding constructions:
+(i) A product of algebras Ai (i ∈ I) is their cartesian product �
+i∈I Ai
+with operations and order given coordinate-wise.
+(ii) A subalgebra of an algebra A is a subobject m : B → A such that
+m is an embedding: x ⊑ y holds in B iff m(x) ⊑ m(y).
+(iii) A homomorphic image of an algebra A is a quotient object e :
+A → B such that e is surjective.
+Lemma 4.9. Every variety of continuous algebras is an HSP-class.
+The proof in [7] on pp. 339-340 works in our setting without any
+changes. The main result of [7] is the converse implication, but the
+proof does not work for our extended terms (more special than the
+terms in op. cit.):
+Open Problem 4.10. Is every HSP-class in Σ-CPO a variety of con-
+tinuous algebras?
+Definition 4.11. A subset X of a cpo C is dense if the only sub-cpo
+containing X is all of C.
+Example 4.12. Given a directed diagram D with a colimit as : As →
+A (s ∈ S) in CPO, the union �
+s∈S as[As] is dense in A. Indeed, if a sub-
+cpo A′ contains that union, then the codomain restrictions a′
+s : As → A′
+form a cocone of D in CPO. From the fact that this cocone factorizes
+through as it follows that A′ = A.
+Lemma 4.13. If X is dense in a cpo C, then Xn is dense in Cn for
+each n ∈ N.
+Proposition 4.14. Every variety V of continuous algebras has free
+algebras: the forgetful functor UV : V → CPO has a left adjoint FV :
+CPO → V.
+Proof sketch. For V = Σ-CPO we have described the free algebras in
+Example 4.2. Every variety V is closed under products and subalge-
+bras by Lemma 4.9. The category Σ-CPO is complete and wellpowered.
+It has the factorization system (E, M) where E consists of homomor-
+phisms e : A → B with e[A] dense in B and M consists of embed-
+dings of closed subalgebras. (Indeed, let a continuous homomorphism
+f : A → C have such a factorization f = m · e for e : A → X and
+m : X → C in Met. Then there is a unique algebra structure on X
+making e and m homomorphisms.)
+It follows that V is a reflective
+subcategory: the inclusion functor E : V → Σ-CPO has a left adjoint
+
+20
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+by Theorem 16.8 in [6]. Since UV = UΣ · E, we conclude that UV has a
+left adjoint.
+□
+Notation 4.15.
+(1) We denote by TV the free-algebra monad of a variety V on CPO.
+Its underlying functor is TV = UV · FV.
+(2) Concrete categories (and functors) over CPO are defined analo-
+gously to Remark 3.8.
+Example 4.16. For V = Σ-CPO the monad TΣ of Example 4.2 assigns
+to a cpo M the cpo TΣM. As in Example 3.7, TΣ is strongly finitary.
+Proposition 4.17. Every variety V of continuous algebras is con-
+cretely isomorphic to the category CPOTV:
+the comparison functor
+KV : V → CPOTV is a concrete isomorphism.
+This is, as Proposition 3.9, analogous to the classical case.
+Lemma 4.18 ([7], Proposition 3.5). Let h : A → B be a morphism in
+Σ-CPO and t an extended term. Given an interpretation f : V → A
+with f @(t) defined, then (hf)@ is also defined: (hf)@(t) = h(f @(t)).
+Proposition 4.19. The forgetful functor UΣ : Σ-CPO → CPO creates
+directed colimits: given a directed diagram D of continuous algebras
+with a colimit ci : UDi → C in CPO, there exists a unique algebra
+structure on C making all ci homomorphisms; moreover the resulting
+cocone ci : Di → C is a colimit of D in Σ-CPO.
+Proof sketch. This follows from CPO being cartesian closed. Thus di-
+rected colimits commute with finite products (Theorem 2.12). Given
+an n-ary symbol σ ∈ Σ, from the fact that cn
+i : UDn
+i → Cn is a directed
+colimit it follows that there is a unique morphism σC : Cn
+i → C such
+that the given operations σCi : Cn
+i → Ci fulfil σC · cn
+i = ci · σCi. That
+is, ci are homomorphisms. The verification that this yields a colimit of
+D in CPO is easy.
+□
+Proposition 4.20. The functor UΣ creates reflexive coinserters.
+Proof sketch. Indeed, if e : B → C is such a coinserter of f0, f1 :
+A → B, then for an n-ary symbol σ we have a coinserter en of f n
+0 , f n
+1
+(Example 2.23). We thus obtain a unique σC : Cn → C with σC ·
+en = e · σB, hence e is a homomorphism which is easily seen to be the
+coinserter in Σ-CPO.
+□
+Theorem 4.21. Every variety of continuous algebras is closed under
+directed colimits in Σ-CPO.
+Proof sketch. Let as : As → A (s ∈ S) be a directed colimit in Σ-CPO.
+Consider an extended term t ∈ TΣV such that in each of the algebras
+As t is definable. We prove that t is definable in A. This concludes the
+proof using Remark 4.6. We use structural induction: the statement
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+21
+is obvious if t is a classical term. Thus we need to prove the induction
+step: let t = �
+k∈N tk with each tk definable in all As (s ∈ S), then t
+is definable in A. By the definition of extended terms we know that
+there is a finite set V0 ⊆ V of variables with tk ∈ TΣV0 for all k ∈ N.
+Thus we can work with interpretations f : V0 → A. The colimit cocone
+as : As → A has the property that the set X = �
+s∈S as[As] is dense
+(Example 4.12).
+In the cpo [V0, A] of all interpretations the subset
+[V0, X] is thus dense, too (Lemma 4.13). We use this to prove that
+f @(t) is defined by structural induction on f:
+(i) if f[V0] ⊆ X then f @ is defined, and
+(ii) given f = �
+n∈N fn in [V0, A] with all f @
+n (t) defined, then f @(t) is
+defined.
+Step (i) is easy, using the fact that since V0 is finite, f : V0 → � as[As]
+factorizes through one of the subset as[As], as f = as · f.
+For the
+interpretaion f : V0 → As we know that f
+@(t) is defined. Then we
+apply Lemma 4.18 to h = as. Step (ii) is more involved since it works
+with double induction: for tk and fn.
+□
+Corollary 4.22. The monad TV on CPO is strongly finitary for every
+variety V of continuous algebras.
+Proof. We know that UΣ : Σ-CPO → CPO creates directed colimits and
+reflexive coinserters (Propositions 4.19 and 4.20).
+Given a variety V, the embedding E : V ֒→ Σ-CPO preserves directed
+colimits (Theorem 4.21) and reflexive surjective homomorphisms: in-
+deed, V is closed under homomorphic images (Lemma 4.9). Conse-
+quently, the forgetful functor UV = UΣ · E preserves directed colim-
+its and reflexive surjective coinserters.
+Its left adjoint FV preserves
+weighted colimits. Thus TV = UV · FV preserves directed colimits and
+reflexive coinserters. This finishes the proof by Corollary 2.30.
+□
+Construction 4.23. Analogously to Construction 3.17, to every strongly
+finitary monad T = (T, µ, η) on CPO we assign a variety VT of algebras
+of the signature Σn = |TVn| presented by equations as follows:
+(1) σ = �
+k∈N σk for every ω-chain (σk)k<ω in the cpo TVn with σ =
+�
+k∈N σk;
+(2) k∗(σ) = σ(k(xi))i<n for all σ ∈ Σn and all maps k : Vn → Σm;
+(3) ηVn(xi) = xi for all i = 0, . . . , n − 1.
+Lemma 4.24. Every algebra α : TA → A in CPOT defines an algebra
+in VT with σA(a(xi)) = a∗(σ) for all σ ∈ Σn and a : Vn → A. Moreover,
+every homomorphism in CPOT is also a Σ-homomorphism between the
+corresponding algebras in VT.
+The proof is analogous to that of Lemma 3.18.
+Theorem 4.25. Every strongly finitary monad T on CPO is the free-
+algebra monad of the variety VT.
+
+22
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+Proof. By Example 2.28 the functor K : Setf → CPO has the den-
+sity presentation consisting of directed colimits and surjective reflexive
+coinserters. Both the free-algebra monad T′ of VT and T are strongly
+finitary (Corollary 4.22). Therefore, it is sufficient to prove for every
+finite discrete cpo X that the Σ-algeba of VT corresponding to the free
+algebra (TX, µX) of CPOT is a free algebra on X in VT. In other words,
+T and T′ have the same free algebras on objects of Setf. Since they both
+preserve the colimits of the density presentation of K : Setf ֒→ CPO, it
+follows that for every cpo the free algebras of T and T′ are the same.
+Thus the monads T and T′ are isomorphic, which proves the theorem.
+We can assume X = Vn for some n ∈ N.
+The verification that for every algebra A in VT and every interpre-
+tation f : Vn → A, there is a unique Σ-homomorphism f : TΣVn → A
+with f = f ·ηVn is analogous to that in Theorem 3.19: In the ’existence’
+part, f is defined by the same formula. It is continuous because, given
+an ω-join σ = �
+k∈N σk in TV , the algebra A satisfies σ = �
+k∈N σk, thus
+f(σ) = σA(f(xi)) =
+�
+k∈N
+(σk)A(f(xi)) =
+�
+k∈N
+f(σk).
+The ’uniqueness’ part is identical.
+□
+Corollary 4.26. Varieties of continuous algebras correspond bijec-
+tively, up to isomorphism, to strongly finitary monads on CPO.
+The proof is analogous to that of Theorem 3.21.
+5. Varieties of ∆-Continuous Algebras
+We now turn from CPO to DCPO.
+Definition 5.1. A ∆-continuous algebra is a dcpo endowed with con-
+tinuous operations. We denote by Σ-DCPO the category of ∆-continuous
+algebras and continuous (directed-joins preserving) homomorphisms.
+We assume again that a signature Σ is given, and a countable set V
+of variables is chosen.
+Example 5.2. A free ∆-continuous algebra on a dcpo M is the algebra
+TΣM of terms on variables from |M|, see Example 4.2. Indeed, the
+underlying poset is a coproduct of copies of |M|n (Example 3.7), thus
+TΣM is a dcpo. We again use the notation f ♯ : TΣM → A for the
+homomorphism extending f : M → A.
+We are ready to define equations and varieties of ∆-continuous al-
+gebras. The extended terms here have the form �
+k<α tk for arbitrary
+ordinals α. We namely use the fact that a poset is a dcpo iff it has
+joins of ordinal-indexed chains, and a map preserves directed joins iff
+it preserves joins of ordinal-indexed chains ([9], Corollary 1.7). Re-
+call that an ordinal is the linearly ordered set of all smaller ordinals,
+α = {k ∈ Ord | k < α}.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+23
+Definition 5.3.
+(1) We define ∆-extended terms as the smallest set T ∆
+Σ V containing
+TΣV and such that for each ordinal α and every collection tk (k <
+α) of ∆-extended terms containing only finitely many variables we
+get a ∆-extended term �
+k<α tk.
+(2) For every ∆-continuous algebra A and every interpretation f : V →
+A we denote by f @ : T ∆
+Σ V ⇀ A the partial map extending f ♯
+which is defined in t = �
+i<α ti iff each f @(ti) is defined and fulfils
+f @(ti) ⊑ f @(tj) for all i < j < α; then f @(t) = �
+i<α f @(ti).
+(3) An equation is a pair of extended terms; we write again t = t′. An
+algebra A satisfies it iff for every interpretation f : V → A both
+f @(t) and f @(t′) are defined, and are equal.
+We thus obtain varieties of ∆-continuous algebras as the full subcat-
+egories of Σ-DCPO presented by a set of equations between ∆-extended
+terms. Please note that although T ∆
+Σ V is a proper class, every variety
+is presented by a set (not a proper class) of equations.
+Proposition 5.4. Every variety V of ∆-continuous algebras has free
+algebras, and for the ensuing monad TV it is concretely isomorphic to
+DCPOTV.
+Proof. The existence of free algebras is verified as in Proposition 4.14.
+We just need to understand density of a set X ⊆ C for a dcpo C to
+mean that the only sub-dcpo containing X is all of C. The rest is, like
+Proposition 3.9, analogous to the classical case.
+□
+Theorem 5.5. The monad TV on DCPO is strongly finitary for every
+variety of ∆-continuous algebras.
+The proof is analogous to that of Corollary 4.22.
+Construction 5.6. To every strongly finitary monad T on DCPO we
+assign a variety of ∆-continuous algebras of signature Σn = |TVn|. It
+is presented by the equations as in Construction 4.23, except that in
+Item 1) we choose arbitrary ordinals
+α ≤ card |TV |
+and then form the following equations:
+σ =
+�
+k<α
+σk
+for every α-chain (σk)k<α in the dcpo TVn with σ = �
+k<α σk. Observe
+that Items 1)-3) yield only a set of equations.
+Theorem 5.7. Every strongly finitary monad on DCPO is the free-
+algebra monad of the above variety.
+The proof is analogous to that of Theorem 3.19.
+Corollary 5.8. Varieties of ∆-continuous algebras correspond bijec-
+tively, up to isomorphism, to strongly finitary monads on DCPO.
+
+24
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+6. Conclusion and Open Problems
+Varieties (aka 1-basic varieties) of quantitative algebras of Mardare
+et al. [21, 22] correspond bijectively to strongly finitary monads on
+the category Met of metric spaces.
+This is the main result of our
+paper. It is in surprising contrast to the fact that ω-varieties in op. cit.
+(where distance restrictions on finitely many variables in equations are
+considered) do not even yield finitary monads in general, as shown
+in [2]. For varieties of complete quantitative algebras the same result
+holds: they correspond bijectively to strongly finitary monads on the
+category CMet of complete metric spaces. This relates the quantitative
+algebraic reasoning of Mardare et al. closely to the classical equational
+reasoning of universal algebra where varieties are known to correspond
+to finitary monads on Set [20].
+Open Problem 6.1. Characterize monads on Met or CMet corre-
+sponding to ω-varieties of quantitative algebras.
+In [2] a partial answer has been given: when moving from Met to its
+full subcategory UMet on all ultrametric spaces, then enriched monads
+on UMet corresponding to ω-varieties of quantitative algebras in UMet
+are characterized.
+Analogously to the famous Birkhoff Variety Theorem in classical
+algebra, varieties of quantitative algebras can, as proved in [23], be
+characterized as precisely the HSP-classes: closed in Σ-Met under ho-
+momorphic images, subalgebras, and products. However, in CMet that
+proof does not seem to work.
+Open Problem 6.2. Does an analogy of the Birkhoff Variety Theorem
+hold for varieties of complete quantitative algebras?
+We have also presented a parallel theory of varieties of continuous
+algebras. Here we worked in the category CPO of ω-cpos (or DCPO
+of dcpos), and proved that varieties correspond bijectively to strongly
+finitary monads on CPO (or DCPO). Although the result sounds the
+same as that for Met, the proof is substantially different. It relies on
+CPO and DCPO being cartesian closed.
+Open Problem 6.3. Does an analogy of the Birkhoff Variety Theorem
+hold for varieties of continuous algebras?
+In [7] an affirmative answer is presented, but the extended terms
+used there are more general than in our paper.
+Our work in CPO and DCPO is based on the surprising fact we have
+proved: in cartesian closed categories directed colimits commute with
+finite products.
+
+QUANTITATIVE AND CONTINUOUS ALGEBRAS
+25
+References
+[1] J. Ad´amek, Free algebras and automata realizations in the language of cat-
+egories, Comment. Math. Univ. Carolinae 15 (1974), 589–602
+[2] J. Ad´amek, Varieties of quantitative algebras and their monads, Proceedings
+of Logic in Computer Science (LICS 2022),1–12
+[3] J. Ad´amek, M. Dost´al and J. Velebil, A categorical view of varieties of or-
+dered algebras, Math. Struct. Comput. Sci. 32, no. 4 (2022), 349–373
+[4] J. Ad´amek, M. Dost´al and J. Velebil, Quantitative algebras and a classifica-
+tion of metric monads, arXiv:2210.01565.
+[5] J. Ad´amek, M. Dost´al and J. Velebil, Strongly finitary monads for continuous
+algebras, manuscript.
+[6] J. Ad´amek, H. Herrlich and G. Strecker, Abstract and concrete categories:
+The joy of cats, John Wiley and Sons, New York 1990
+[7] J. Ad´amek, E. Nelson and J. Reiterman, The Birkhoff variety theorem for
+continuous algebras, Algebra Universalis 20 (1985), 328-350
+[8] J. Ad´amek and J. Rosick´y, Approximate injectivity and smallness in metric-
+enriched categories, J. Pure Appl. Algebra 226 (2022), 1–30
+[9] J. Ad´amek and J. Rosick´y, Locally presentable and accessible categories,
+Cambridge University Press, 1994
+[10] G. Bacci, R. Mardare, P. Panaganden and G. D. Plotkin, An algebraic theory
+of Markov processes, Proceedings of Logic in Computer Science (LICS 2018)
+ACM (2018), 679–688
+[11] G. Bacci, R. Mardare, P. Panaganden and G. D. Plotkin, Tensors of quanti-
+tative equational theories, Proceedings of Coalgebraic and Algebraic Methods
+in Computer Science (CALCO 2021).
+[12] M. Barr and Ch. Wells, Toposes, triples and theories, Springer-Verlag, New
+York 1985
+[13] F. Borceux, Handbook of Categorical Algebra: Volume 2, Categories and
+Structures, Cambridge Univ. Press, 1994
+[14] J. Bourke and R. Garner, Monads and theories, Adv. Math. 351 (2019),
+1024–1071
+[15] E. J. Dubuc, Kan Extensions in Enriched Category Theory, Lecture Notes
+in Mathematics, vol. 145, Springer-Verlag 1970
+[16] G. M. Kelly, Basic concepts of enriched category theory, London Math. Soc.
+Lecture Notes Series 64, Cambridge Univ. Press, 1982, also available as Repr.
+Theory Appl. Categ. 10 (2005)
+[17] G. M. Kelly, Structures defined by finite limits in the enriched context I,
+Cah. Topol. G´eom. Diff´er. Cat´eg. XXIII (1982), 3–42
+[18] G. M. Kelly and S. Lack, Finite-product-preserving functors, Kan extensions
+and strongly-finitary 2-monads, Appl. Categ. Structures 1 (1993), 85–94
+[19] A. Kurz and J. Velebil, Quasivarieties and varieties of ordered algebras: reg-
+ularity and exactness, Math. Structures Comput. Sci. (2016), 1–42.
+[20] S. Mac Lane, Categories for the working mathematician, 2nd ed., Springer
+1998
+[21] R. Mardare, P. Panangaden and G. D. Plotkin, Quantitative algebraic rea-
+soning, Proceedings of Logic in Computer Science (LICS 2016), IEEE Com-
+puter Science 2016, 700–709
+[22] R. Mardare, P. Panangaden and G. D. Plotkin, On the axiomatizability
+of quantitative algebras, Proceedings of Logic in Computer Science (LICS
+2017), IEEE Computer Science 2017, 1–12
+[23] S. Milius and H. Urbat, Equational axiomatization of algebras with structure,
+arXiv 1812-02016v2
+
+26
+J. AD´AMEK, M. DOST´AL, AND J. VELEBIL
+[24] M. Mio and V. Vignudelli, Monads and Quantitative Equational Theories for
+Nondeterminism and Probability, Proceedings of CONCUR 2020, vol. 171 of
+LIPIcs.
+[25] J. Rosick´y, Metric monads, Math. Struct. Comput. Sci., 31(5) (2021), 535–
+552
+[26] J. Rosick´y, Discrete equational theories, arXiv 2204.02590v1
+Department of Mathematics, Faculty of Electrical Engineering,
+Czech Technical University in Prague, Czech Republic and Institute
+for Theoretical Computer Science, Technical University Braunschweig,
+Germany
+Email address: [email protected]
+Department of Mathematics, Faculty of Electrical Engineering,
+Czech Technical University in Prague, Czech Republic
+Email address: {dostamat,velebil}@fel.cvut.cz
+

DNE4T4oBgHgl3EQfGAw7/content/2301.04890v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6748ca9aaca05bae1a0ef14a0dcd368afced6f060d48255f9d4141d6911311f8
+size 516370

DNE4T4oBgHgl3EQfGAw7/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:1695e56035dee1c50cd99e10391afbb63af14be11ffe29e99927ae3e4e86acf6
+size 5898285

DNE4T4oBgHgl3EQfGAw7/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a9e9dd6ee7fb982fd987a7d9ddc8ce63f7bffac608171de480944a6c4714e4b1
+size 244753

DtE2T4oBgHgl3EQf9gmM/content/tmp_files/2301.04229v1.pdf.txt

@@ -0,0 +1,1373 @@
+1
+Abstract
+mm-Wave communication systems use narrow directional beams due to the spectrum’s characteristic
+nature: high path and penetration losses. The mobile and the base station primarily employ beams in
+line of sight (LoS) direction and when needed in non-line of sight direction. Beam management protocol
+adapts the base station and mobile side beam direction during user mobility and to sustain link during
+blockages. To avoid outage in transient pedestrian blockage of the LoS path, the mobile uses reflected
+or NLoS path available in indoor environments. Reflected paths can sustain time synchronization and
+maintain connectivity during temporary blockages. In outdoor environments, such reflections may not
+be available and prior work relied on dense base station deployment or co-ordinated multi-point access
+to address outage problem.
+Instead of dense and hence cost intensive network deployments, we found experimentally that
+the mobile can capitalize on ground reflection. We developed TERRA protocol to effectively handle
+mobile side beam direction during transient blockage events. TERRA avoids outage during pedestrian
+blockages 84.5 % of the time in outdoor environments on concrete and gravel surfaces. TERRA also
+enables the mobile to perform a soft handover to a reserve neighbor base station in the event of a
+permanent blockage, without requiring any side information unlike the existing works. Evaluations
+show that TERRA maintains received signal strength close to the optimal solution while keeping track
+of the neighbor base station.
+arXiv:2301.04229v1  [eess.SY]  10 Jan 2023
+
+2
+TERRA: Beam Management for Outdoor
+mm-Wave Networks
+Santosh Ganji, Jaewon Kim, Romil Sonigra and P. R. Kumar
+Texas A&M University
+I. INTRODUCTION
+High path loss in millimeter wave (mm-wave) bands necessitates both the mobile and the
+base station to communicate using narrow directional line-of-sight (LoS) beams [1]. The LoS
+direction changes as the user move. To remain connected, the base station and the mobile manage
+beam directions to counter user mobility. Another challenge to mm-wave communication is that
+the human body significantly attenuates radiation [2], [3]. Pedestrian blockers cause the narrow
+directional LoS beam inoperable. mm-wave device must adapt the beam direction to overcome
+the challenges caused by blockage and user mobility.
+When a pedestrian obstructs the mm-wave Line of Sight (LoS) link, the received signal strength
+drops by about 15 dB [4]. Even though such blockage events are temporary and last only about
+two hundred milliseconds [4], they create link outages as 96.8% of the signal is lost. The signal
+amplifier cannot improve SNR as both signal and noise are equally amplified. The mobile loses
+connectivity during such temporary blockages. After the temporary blockage, it has to reconnect
+like a new user, which in technologies like 5G NewRadio takes about a second [5], disrupting
+low latency applications like virtual reality and online gaming [6]. In an indoor environment, a
+usable non-line-of-sight (NLoS) path is available [4].
+Presuming that reflected NLoS beams are scarce in outdoor environments [7], [8], it has
+been suggested that the solution to avoid disruption is to handover to another base station.
+However, handover for every pedestrian blockage requires high deployment density. The study
+[9] suggests 200 base stations per Sq. KM is necessary to meet the latency requirements. Such
+high deployment density is expensive.
+Based on outdoor signal measurement studies under the pedestrian blockage, we show that
+there is frequently a ground reflection from hard surfaces, and propose a protocol called TERRA.
+
+3
+It employs such a ground-reflected NLoS path as a control channel to sustain critical time
+synchronization with the base station throughout a temporary blockage. This makes it possible
+to revert to the LoS beam as soon as the blockage concludes. TERRA efficiently maintains the
+beam identity of such a reserve NLoS path at all times, and refreshes it as needed, so that it
+is always ready for blockage since blockage can happen at any time. Experiments show that
+TERRA avoids outage events 84.5% time during pedestrian blockages. When either blockage
+does not disappear or the current base station is out of range, handover to another base station
+is the only viable choice for the mobile to continue communication. TERRA also addresses the
+challenges in a handover event that arise from the directional nature of beams.
+To avoid hard handover where the mobile needs to connect to the next base station as if
+it is a new user, TERRA must manage mobile beams both with the serving and the neighbor
+base station. TERRA first needs to determine neighboring base stations for potential handover
+targets. And for the entire transition process, the protocol must ensure that it has aligned beams
+at the detected neighbor base station. The mobile must also manage beams with the serving base
+station till the transfer of connection.
+Prior work [10], [11] corrects the beam misalignment that happens with user mobility. While
+the mobile can continually adapt its beam during user mobility with the serving base station with
+cooperation from the serving base station [10], [11], the neighbor base station neither adapts
+its beam nor provides any assistance to the mobile for receive beam adaptation, as the mobile
+is yet to establish communication with it. Terra uses information only in the radio domain,
+as is desirable, without requiring any additional sensors. Experiments on TERRA show that it
+maintains the received signal strength of a neighbor base station within 3 dB of an omniscient
+oracle. We have also evaluated the performance of TERRA under several pedestrian mobility
+patterns.
+The rest of this paper is organized as follows. We explain the challenges in outdoor beam
+management in Section II. In Section III, we present our outdoor experiments during pedestrian
+blockages. Section IV-B shows the search overheard to discover the neighbor base station. We
+present our protocol in Section V. Section VI presents TERRA’s efficacy in various user mobility
+scenarios.
+
+4
+II. THE CHALLENGES OF BEAM MANAGEMENT FOR BLOCKAGE RESILIENCY AND
+HANDOVER
+In this section, we elaborate on the challenges in managing beams during pedestrian block-
+age and in handover. We also provide the outlines of the solution which are further detailed,
+experimentally verified and validated in the subsequent sections.
+A. Pedestrian Blockage
+In the case of omnidirectional transmission, the environment scatters electromagnetic radiation
+in all directions and an omnidirectional receiver can capture the multi-path components of the
+transmitted signal. In contrast, due to directional transmission, there are fewer distinct multi-path
+components in the mm-wave bands. A narrow directional radio receiver beam can only receive
+signal components that arrive within a small angular spread of a beam direction. A 32x32 element
+uniform planar array can produce beamwidths as narrow as 4◦. To discover either an LoS or such
+NLoS paths, the base station sweeps beams within a sector, and the mobile receiver similarly
+performs a spatial scan.
+In the mm-wave band, an interposed pedestrian does obstruct the narrow directional LoS beam.
+The human body attentuates the signal by 15 dB, resulting poor RSS. With most of the signal
+energy lost and receiver’s amplifier cannot improve SNR. Since the mobile receives weak signal
+energy and amplifiers cannot improve SNR, link suffers from outage during pedestrian blockage
+event.
+The mobile is left with one of two choices to continue communication with the network – to
+switch to an NLoS path if such a path exists between the base station and mobile, or to perform
+handover to a neighboring base station (or otherwise employ a neighboring base station through,
+say, coordinated multipoint transmission).
+Pedestrian blockage is sudden and unpredictable [10]. To avoid outage by employing NLoS
+path, the mobile must therefore always have in hand an NLoS beam direction that it can quickly
+switch to. If the mobile has no backup NLoS path to use in the event of blockage, link outage
+occurs, and the mobile gets disconnected from the base station.
+The disconnected mobile will need to re-perform an initial network access procedure, as
+though it were a new user, which takes several seconds, due to the following. To acquire new
+users, base stations periodically sweep directional beams with reference signals and broadcast
+information such as cell and network identity. A mobile also sweeps through all its receive beams
+
+5
+to discover at least one of the base station’s beams when it is pointed towards it. The number
+of receive beams increases with the reciprocal of the beamwidth. To complete a bi-directional
+connection, the mobile transmits a random preamble in the same direction as the discovered
+base station’s beam, and awaits a response. After physical layer procedures to establish reliable
+data communication, the network authenticates the mobile before granting network access. This
+complete procedure takes several seconds [12].
+On the other hand, were the mobile to have an NLoS path in hand when blockage occurs,
+it can use that to sustain connectivity in the following way. The NLoS path has lesser RSS
+than the LoS path, not enough to sustain the earlier high data rate communication of the LoS
+path. However, the NLoS path can continue to sustain time synchronization between the mobile
+and the base station. This is critical since it allows the mobile to revert to LoS communication
+without delay as soon as the blockage disappears, and, typically, such blockage is temporary
+and only lasts a few hundred milliseconds [4]. For such recovery from temporary blockages, it
+is critical that it be performed without requiring any out of band communication, and this is
+exactly what the NLoS path makes possible.
+The critical issue is therefore: Do such NLoS paths exist outdoors? [13] Unlike indoor
+environments, there is not a multiplicity of surfaces when outdoors. One that is always there
+though is the ground. Indeed, ground reflections are used to shape glide paths for aircraft
+instrument landing systems [14]. There is also a report of ground reflections at mm-wave
+frequencies in [15] in an outdoor environment. However, the potential for this to help with
+mm-wave communications, especially blockage, has apparently not been pursued. Motivated by
+this possibility, we have conducted a measurement campaign in the 60GHz band to determine
+whether mm-wave signals are reflected from the ground, and whether they are usable during
+blockage events.
+In Section III, we report on the results of the measurement study that show that mm-wave
+signals are reflected from outdoor surfaces such as concrete and gravel with a loss of just 4-6
+dB over LoS. Also important for base station to hand-held mobile communications is that base
+stations are usually deployed with a slight downward tilt, as shown in Fig. 1, and are equipped
+with phased arrays that steer beams. The hand-held mobile’s receiver can therefore capture these
+ground reflections. We also report in Section III on the link measurements with human blocking
+of the LoS link between the base station and the mobile.
+In cellular communication systems, the base station schedules data transmission and reception
+
+6
+Base Station-
+Array
+LoS Path
+Ground 
+Reflection
+𝐻𝑇
+𝐻𝐵
+𝐻𝑅
+𝐷𝐵𝑅
+Figure 1: Ground Reflection
+opportunities. Time synchronization helps mobile adjust its timeline to that of base station’s. To
+align the mobile’s time line, the base station transmits synchronization signals. Through signal
+correlation, the mobile determines the temporal location of these signals in the captured over
+the air samples and adjusts its timeline. A good received signal strength and high enough SNR
+are necessary to improve the accuracy of signal processing algorithms and help achieve tight
+timing alignment with the base station. Any timing offset induces packet errors in both uplink
+and downlink [16] as the mobile’s transmissions fall out of the base station’s listening window
+and vice versa. The mobile must therefore continuously adapt its timeline to account for the
+propagation delay with user mobility. It is for this reason that TERRA switches to a NLoS beam
+to maintain time synchronization throughput the temporary blockage period. This allows it to
+revert to LoS communication as soon as the blockage ends.
+B. Beam Management for handover
+1) Brief overview of 5G mm-Wave handover process: When a mobile user moves to the
+boundary of the currently connected base station’s coverage region, called a “cell”, the mobile
+experiences degraded radio conditions. At the edge of the cell, the received signal strength is
+weak and hence Signal to Noise Ratio (SNR) is poor. When SNR is bad, packet decoding fails.
+Cellular technologies like 5G can use hybrid packet retransmissions to use previously transmitted
+bits to decode a message before sending the original message again with a reduced code rate.
+Either way, when a packet is decoded in error, the recovery mechanisms increase communication
+latency. A similar situation occurs when a LoS beam is permanently blocked by a building.
+Under such edge radio or permanent blockage conditions, improve link performance, the
+mobile searches for a neighbor base station. While omni-directional mobile receivers need to
+
+人7
+perform only a frequency scan to discover the neighbor base station and initiate the handoff
+process, in the mm-wave bands, the mobile uses narrow directional beams and therefore needs
+to perform a spatial scan to discover a neighbor base station. A 5G mm-wave base station period-
+ically sweeps broadcast information using narrow directional beams [5]. Broadcast messages help
+mobile discover the neighbor base station’s timeline. The mobile must adjust its timeline to align
+with the neighbor base station. This process called “time synchronization” is the foremost step
+a mobile performs before starting the initial access protocol. During initial access procedures,
+the mobile informs the neighbor base station of its presence in the coverage area. The mobile
+transmits an uplink preamble signal in a listening window of the base station and anticipates a
+response. Tight time synchronization at the mobile ensures that the sent preamble reaches the
+base station.
+Serving base station cannot help a mobile with the time schedules of a neighbor base station.
+Without strict time synchronization among the base stations in the network, a base station cannot
+have the knowledge of timing of another. For example, serving base station precisely time
+synchronize with neighbor base station to convey to its mobile when a beam is available from
+the neighbor base station to initiate communication.
+The broadcast messages also carry schedules of when the base station uses a particular beam
+to listen to a mobile’s transmissions. To complete handover, the mobile must transmit precisely
+at the instants when a neighbor base station is listening in the direction of the beam discovered
+after the spatial scan. Moreover, cellular standards require the mobile to choose time-frequency
+resources for preamble signals randomly. The neighbor base station listens to the sent preamble
+and responds when there is no resource collision. A response from the base station to a preamble
+is necessary for the mobile to advance further in the handover protocol. The mobile waits for
+a response for a pre-configured interval, after which it retransmits a new preamble [12] Upon
+receiving a response to the preamble signal, the neighbor base station and mobile exchange
+critical control plane messages for user authentication and connection transfer. The mobile must
+maintain a highly aligned beam throughout this process to avoid handover failure.
+In short, the mobile must search for a neighbor base station, time synchronize and perform
+initial access procedure to handover. Below we elaborate on the main stages in a handover
+protocol and implications of beam management on those stages.
+2) Neighbor Base Station Search: As mentioned earlier, the serving base station schedules
+persistent measurement occasions for the mobile to discover neighbor base stations [17]. In
+
+8
+the granted opportunities, mm-wave mobile performs directional search on frequencies that are
+communicated by the serving base station. Mobile first measures signal strength temporarily
+tuning the radio receiver to carrier frequencies of neighbor base station and attempts decoding
+the broadcast information that contains the network related information.
+The mobile searches for a neighbor base station using one receive beam at a time. As transmit
+beam schedules of neighbor base station are unknown, mobile uses the same receive beam for
+the entirety of one beam sweeping interval. In 5G mm-wave network, mobile holds each of its
+receive beams for 20 ms, the duration in which base station sweeps all its beams once. Mobile
+has complete freedom on the beamwidth of beams. Search concludes after discovering a base
+station and mobile must report signal measurements to the serving base station. Time to discover
+a base station is impacted with beamwidth, number of beams, and user mobility pattern.
+Based on the reported measurements, serving base station then makes final decision on
+handover. However, such a mechanism does not have any particular advantage. Irrespective of
+whether mobile or serving base station that decides on the switch, there is no way to evade
+measurements. Until the switching decision is made, the mobile’s receiver needs to toggle
+between serving and neighbor base stations. Mobile must keep track of the found neighbor
+base station beam until the handover is initiated. Also, the mobile must keep track of serving
+base station beam. Failure to track the serving base station beam results in hard handover whereas
+losing track of neighbor base station beam requires mobile to search again.
+3) Beam Tracking: Mere one-time discovery of a neighbor base station beam is insufficient to
+complete the handover. The mobile must maintain alignment with the found base station beam
+to overcome mobility impairments and maintain good received signal strength throughout the
+transition process. This adaptation step, called Beam Tracking, involves the mobile switching its
+receive beams to maintain high received signal strength. Beam tracking is essential to complete
+all the handover protocol message exchanges and avoid a hard handover.
+As the mobile is yet to establish connection with the neighbor base station before handover,
+at the time of handover neither does the neighbor base station adapt its beams to counter user
+mobility nor does it assist the mobile to adapt its beams to preserve beam alignment. The
+mobile can only rely on its own beam adaptation. Since this adaptation of its receive beam to
+the neighbor base station is done without any communication with the neighbor base station, we
+call it “silent tracking.” Silent Tracking is different from adaptation that is done with the serving
+base station, or which will be done with the neighbor base station after a connection handover
+
+9
+since both of the latter exploit two-way communication with the respective base stations. In
+connected state, the base station also adapts its beams to counter user mobility and aids the
+mobile in adapting its receive beams. This can done with mobile’s adaptation procedure in the
+companion Beamsurfer protocol [10] where the mobile shifts to one of its better adjacent beams
+when the RSS of the current beam drops by 3dB. The details can be found in [10].
+4) Time Synchronization, Random Access and Connection Transfer: After beam discovery
+and timing synchronization, the mobile transmits a preamble signal to announce its presence
+to the neighbor base station. The preamble and the time-frequency resources to transmit the
+signal are chosen randomly from a set known to the base station. This step in the initial access
+procedure is called random access. The base station listens for all possible preamble resources.
+After listening to the preamble signal, the base station responds and allocates resources for the
+mobile to complete the rest of the initial access procedure. The preamble must arrive within the
+base station’s listening window. As the communication at mm-wave bands is directional, the base
+station listens in a particular direction in each window. The mobile must therefore maintain tight
+time synchronization with the neighbor base station. Upon receiving a response to the preamble
+signal, both the mobile and the base station exchange several protocol messages to complete the
+transition. So, the mobile must still maintain a receive beam adapted to user mobility to continue
+the handover procedures. The neighbor base station neither adapts its beams nor provides any
+assistance to mobile in adapting beams during the initial connection.
+Figure 2: 60 GHz Transceiver System
+III. MEASUREMENTS UNDER PEDESTRIAN BLOCKAGE
+We performed signal strength measurements under pedestrian blockage in outdoor built envi-
+ronments with commonly found ground surfaces. The goal of the experiments is to study signal
+
+Digital
+Analog
+Baseband
+Baseband
+Processing
+Data
+Baseband
+Receiver
+FPGA
+mmWave
+Phased Array
+Interface
+Ctrl
+Receiver
+And Up
+Down
+Digital I/O FAM
+RF Ctrl FPGA
+Host
+mmWave
+converter
+Phased Array
+Board
+Transmitter
+Baseband
+Processing
+Transmitter
+FPGA
+Analog
+Data
+Digital
+Baseband
+Baseband10
+on LoS path during blockage and to explore NLoS paths, in particular ground reflections. NLoS
+paths can sustain the link and help mobile maintain time synchronization when the LoS path is
+blocked.
+We conducted experiments using National Instruments 60 GHz software-defined radios. Func-
+tional diagram of transceiver is shown in Fig. 2. Baseband IQ sample generation at transmitter
+and signal processing at the receiver are implemented in FPGA. An analog baseband signal
+of 2 GHz bandwidth is upconverted to 60 GHz carrier frequency. A 12 element phased array
+is used both at the transmitter and receiver. The phase weights for desired radiation patterns
+are calculated and stored as beam codebooks. The beam codebook has 25 beams, with narrow
+beams of azimuth width approximately 18◦, within a 120◦ azimuth sector. The zenith beamwidth
+is around 60◦. Figs. 3 and 4 present azimuth and elevation radiation patterns of the bore sight
+beam. Transmit power is fixed at 20 dBm. The directivity gain of the phased array is 17 dB.
+Further details on the transceiver design and implementation are available in [4], [18].
+On each surface under study, the transmitter array was positioned at a height of HT =2.5
+m above ground level, with the receiver antenna array held about HR =1 m above the the
+surface. The transmitter and receiver arrays were positioned facing each other, and are placed
+DTR = 6 m apart. For each scenario, experiments were repeated for two different cases where
+the transmitter antenna is tilted towards the ground by 10◦ or 20◦. This geometry corresponds
+to potential outdoor deployments where base stations are located higher than mobiles. This tilt
+is responsible for creating additional reflected directions towards the receiver. Moreover, most
+of the zenith beamwidth is directed towards the receiver.
+While the transmitter beam is in the LoS direction of the receiver, the RSS at the receiver,
+denoted by RSSLoS, is measured using a receive beam that is highly aligned with the transmitter
+beam. It serves as a reference to calculate total loss suffered by ground reflection. RSSLoS in the
+experiments was measured to be −60 dBm. When a human body obstructs the LoS direction by
+standing in between transmitter and receiver, the RSS drops to −78 dBm, which is the noise floor
+of the receiver, showing that pedestrian obstruction leads to LoS link loss. Although the pedestrian
+blockage is transient, an undesirable outage event occurs at the receiver if communication, and
+thereby time-synchronization, is not maintained as described in Section II.
+Let HB be the height of a pedestrian human blocker. The pedestrian blocker can obstruct the
+transmission only when she is close to the receiver. Using ray tracing, the maximum distance
+DBRmax between blocker and the receiver to obstruct LoS transmissions can be calculated as
+
+11
+Figure 3: Azimuth Cut
+Figure 4: Elevation Cut
+follows:
+DBRmax = DTR ∗ HB − HR
+HT − HR
+.
+(1)
+For HR = 1m, HT = 2.5m, HB= 1.78m, and DTR = 6m, DBRmax was found to be 3.12m in
+our experiments.
+Table I: Concrete Surface
+Transmitter Tilt
+RSSGR (dBm)
+DBR (m)
+0◦
+-66
+2
+0◦
+-66
+3
+10◦
+-64.7
+2
+10◦
+-64.5
+3
+20◦
+-64.1
+2
+20◦
+-64
+3
+Tables I and II present the RSS for outdoor reflections from Concrete and Gravel pathways.
+The extra loss incurred by the ground reflection is between 4 and 6 dB.
+When both transmitter and receiver phased arrays are parallel to the ground surface, the only
+ground reflection available to the receiver is from radiation in one-half of the 60◦ elevation
+beamwidth of the transmitter beam. To capture the reflection in this direction, the receiver needs
+
+90
+60
+120
+10
+0
+150
+20
+9°
+3
+30
+8.83
+180
+0°
+12.16
+-9°
+8.83
+-150
+-120
+-60
+-9090
+10
+09
+120
+0
+150
+10
+20
+30
+8
+0°
+12.16
+1
+-150
+5
+9
+-120
+-60
+-9012
+Table II: Gravel Surface
+Transmitter Tilt
+RSSGR (dBm)
+DBR (m)
+0◦
+-66.1
+2
+0◦
+-65.9
+3
+10◦
+-64.8
+2
+10◦
+-64.4
+3
+20◦
+-64.4
+2
+20◦
+-64.3
+3
+to tilt its beams towards the ground while maintaining LoS in azimuth. Indeed we measured
+slightly more RSS when the receiver is so tilted.
+When the transmitter array is also tilted towards ground, directions with stronger incident
+radiation get reflected, resulting in higher RSS. The highest RSSGR observed on all three surfaces
+under study is around -64 dBm. This implies that ground reflected radiation is just 4 dB less
+than LoS.
+Based on the experiments, the following are our main observations:
+• Pedestrian blockers can create mm-wave link outage, however, NLoS paths to preserve link
+are available in outdoors too.
+• Strong ground reflections are available on gravel and concrete built outdoor surfaces.
+• Ground reflections are available in the same azimuth LoS direction at the receiver.
+• Tilting the transmitter towards the ground helps the receiver with even stronger ground
+reflections.
+• Finally, when the ground reflected directed path is known to the mobile, there is no need to
+handover to a neighboring base station in outdoor environments during transient blockage
+events.
+We note as an aside that after discovering ground reflections outdoors, we also experimented
+with surfaces indoors, and discovered the presence of indoor ground reflections also when the
+floor is hard. Table III presents RSSGR averaged over 100 measurements from an indoor surface
+with concrete tiles. In fact, compared to the indoor NLoS paths reported in [10], the RSSGR is
+at least 6 dB higher than RSS of NLoS paths. Therefore it appears that even indoors one can
+
+13
+preferably use ground reflections when the floor is a hard surface. TERRA protocol harvests these
+ground reflected directions, stores in the memory and employs that NLoS path in a blockage
+event.
+Table III: Indoor Floor, Concrete Tiles
+Transmitter Tilt
+RSSGR (dBm)
+DBR (m)
+0◦
+-65.7
+2
+0◦
+-66
+3
+10◦
+-64.5
+2
+10◦
+-64.45
+3
+20◦
+-64.4
+2
+20◦
+-64.3
+3
+IV. MOBILITY EXPERIMENTS ON OVERHEAD OF BEAM SEARCH
+We now address the overhead of TERRA to maintain a potential neighbor base station to use
+in case handover is needed. We present the experimental results on the number of beam searches
+necessary to discover neighbor base station beams while a user is walking. We show that search
+overhead varies with user’s location, orientation of the antenna array, and user mobility pattern.
+During the pedestrian mobility experiments, the user moves with the phased array in hand.
+Additionally, to showcase how rotational motion impacts search, a phased array is rotated with
+angular velocities corresponding to the natural movement of a user’s hand. We also present the
+results in the scenario where two contending neighbor base stations are visible to the mobile.
+A. Experiment Setup:
+We use National Instruments’ software-defined mm-wave transceivers operating at 60 GHz
+for the experiments. One of the transceiver operates as base station, another as mobile.
+1) Data Collection:: The base station and mobile use 25 different beams, each of beam
+width approximately 15◦. These 25 beams cover −50◦ to 60◦ azimuth around the boresight
+of the array. On similar lines as existing 5G cellular standards [5], the base station transmits
+a reference signal known to the mobile in 25 different beams every 20 milli-seconds. Base
+
+14
+Figure 5: Beam patterns of a few beams from codebook used in the testbed
+stations transmits reference symbols in time slots, changing transmitter beams every 8 slots or
+800 micro-seconds.
+The mobile attempts to discover at least one base station beam. In doing so, the mobile
+performs an exhaustive search, switching the receive beams. The base station’s broadcast beam
+schedules are unknown to the mobile during the beam discovery phase, so the mobile holds a
+receive beam for 20 milli-seconds before switching to another. The duration to sweep all the
+25 receive beams at the mobile receiver is 500 milli-seconds. For beam discovery experiments,
+each trial lasts for 25 seconds during which the mobile repeats the search. Fig. 5 presents the
+azimuth cut of a few beams used in the experiment.
+The mobile’s baseband signal processor correlates the received samples with the reference
+signal and calculates the received signal strength if the reference signal is detected. We make a
+note of the number of searches needed to discover a base station beam and the received signal
+strength of found beam.
+B. Beam Search
+As part of search process, the mobile measures the received signal strength on each of its
+receive beams to discover a base station beam. The search terminates when one of the receive
+beams has sufficient received signal strength to decode broadcast information and discovers
+a neighbor. This beam direction is recorded for potential future handover, and the search is
+commenced anew.
+
+AzimuthCut,12elementURABeams
+06
+120
+60
+10
+150
+-10
+30
+20
+180
+P1
+14.31
+-150
+-30
+O Deg?
+-120
+-60
+44 Deg
+06-
+-30 Deg15
+Sensors providing tilt or pose information of a mobile can help with searches. They can be
+especially helpful in the case of searching in a purely zenith direction for a ground reflected beam.
+However, during initial search or base station discovery, additional side information from sensors
+gives no advantage. Also, the particular challenges of utilizing sensors such as gyroscopes, tilt
+sensors, and accelerometers, to obtain angle information like pose have been investigated in [5],
+[19], [20].
+1) Beam Search During Walk: We conducted mobility experiments to determine the search
+duration necessary to find a base station beam during pedestrian mobility. We present the
+following metrics for each mobility pattern; the number of searches required and the received
+signal strength. For the human walk experiments, we consider two mobility scenarios. In the first,
+a user with a phased array in her hand follows a linear trajectory walking at 1m/s. The linear
+trajectory captures the translational motion component of a user walking along a pavement. On
+the other hand, the free walk has both translational and rotational components, for example when
+playing games with virtual reality gear. The length of each trajectory is 2 m. We conducted 50
+trials for each scenario, and at five different locations.
+To perform a spatial scan with all the 25 beams holding each beam for 20 ms takes half a
+second. At four different locations, that are each half a second long along each trajectory, we
+present the variation in the number of searches to discover a base station beam and received
+signal strength. We number these locations as positions 1 to 4.
+A mm-wave base station has only a small coverage region, i.e., a small “cell”. The mobile
+can listen to transmissions of a base station when the mobile is in the coverage region, and the
+mobile’s receive beam aligns with the base station beam.
+Linear Trajectory: In this mobility experiment, the boresight of the mobile array faces the
+base station, so that the neighbor base station array and mobile array are facing each other. This
+experiment shows how the location of the mobile alone impacts the search process. The mobile
+does not know when the neighbor base station directs a beam towards it. Even though there is
+only one mobile beam that is in LoS direction at a particular location with the neighbor base
+station, number of beams to search to discover LoS vary with time.
+Along the trajectory, the number of searches necessary to detect the line of sight base station
+beam varies with position. In Fig. 6a, the median number of beam searches is 14 at positions 1
+and 2, and 13 at positions 3 and 4. As the perfect beam alignment may not occur all the time and
+due to slight difference in gains of beams in our code book, signal strength in our experiments
+
+16
+(a)
+Position 1
+Position 2
+Position 3
+Position 4
+11
+12
+13
+14
+15
+16
+17
+Number of Beam Searches
+Motion: Walk, Beam: Narrow, Boresight Angle: 0°
+(b)
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+Received Signal Strength (dBm)
+0
+0.2
+0.4
+0.6
+0.8
+1
+CDF
+Motion: Walk, Beam: Narrow, Boresight Angle: 0°
+Position 1
+Position 2
+Position 3
+Position 4
+Figure 6: Search during linear translational motion
+(a)
+Narrow, 120°
+Wide, 120°
+Wide, 60°
+5
+10
+15
+20
+25
+Number of Beam Searches
+Motion: Walk, Beam: Narrow, Wide, Boresight: 60°, 120°
+(b)
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+Received Signal Strength (dBm)
+0
+0.2
+0.4
+0.6
+0.8
+1
+CDF
+Motion: Walk, Beam: Narrow, Wide, Boresight: 60°, 120°
+Narrow, 120°
+Wide, 120°
+Narrow, 60°
+Figure 7: Impact of array boresight direction on search
+is not uniform across all trials. The median received signal strength observed from Fig. 6b at
+positions 1 and 2 is 2 dB less than that of positions 3 and 4. We find it reasonable to expect
+variation in signal strength due to mobility in directional mm-wave communication systems.
+Impact of Boresight Direction: The direction of the antenna array i.e., the array boresight,
+also has an impact on the spatial scan. As shown in Fig. 7, when we repeat the experiments
+with different boresight directions of the phased array, we observe the median number of beams
+to search when the array points towards 60◦ azimuth is 20, while it is 9 for 120◦. The median
+received signal strengths in Fig. 7b are slightly lower compared to Fig. 6b, the case where both
+
+17
+the base station and mobile are facing each other. The reason is the irregularities in the gains
+across beams in the codebook. The codebook needs careful design to have uniform gains.
+Rotational Mobility: Rotational mobility disrupts beam alignment between the base station
+and mobile faster than translation mobility [10], [21]. The alignment of the beams therefore lasts
+for a shorter duration, and the mobile side receive beam can listen to a base station beam only
+for a shorter period. We experimentally investigated the expected large variation in the number
+of beams searched and the received signal strength on the aligned beam. Before the experiments,
+we first observe angular velocity from the gyroscope on a commercial mobile device [22]. We
+logged data for one day. To name a few motion patterns during data collection are answering a
+phone call, walking with the mobile in hand, sitting on a chair, etc. Fig. 8 shows the CDF of
+the angular velocity of the mobile. We observe that angular velocity can reach up to 8 rad/s.
+We rotated phased array in our experiment with 40th and 60th percentile angular velocities from
+the observed data i.e., 90◦/s and 180◦/s and performed spatial scan during the mobility.
+In Fig. 9a, we present the number of beams searched for at four equidistant positions while a
+phased array rotates in a 120◦ sector with angular velocities of 90◦/s and 180◦/s. As observed in
+Figs. 6a, 7a, 9c, and d, the standard deviation of the number of beams searched during rotational
+mobility, 6, is higher than the corresponding number of 2 under the linear translational motion.
+0
+2
+4
+6
+8
+10
+12
+ rad/s
+0
+0.2
+0.4
+0.6
+0.8
+1
+F(x)
+Angular velocity during common human movements
+Figure 8: Gyroscope data during daily activities
+Walk: First we studied walking on a linear trajectory, next we looked at rotational motion, but
+human walk often has both rotational and translational components. So, we studied free walk,
+wherein a human walks casually, freely turns and changes direction of the motion.
+
+18
+(a)
+Position 1
+Position 2
+Position 3
+Position 4
+0
+5
+10
+15
+20
+25
+Number of Beam Searches
+Device mobility, Narrow Beam Width, 
+ = 90°/s
+(b)
+-74
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+-58
+Received Signal Strength (dBm)
+0
+0.2
+0.4
+0.6
+0.8
+1
+CDF
+Device mobility, Narrow Beam Width, 
+ = 90°/s
+Position 1
+Position 2
+Position 3
+Position 4
+(c)
+Narrow Beam, Fast
+Wide Beam, Slow
+Wide Beam, Fast
+0
+5
+10
+15
+20
+25
+Number of Beam Searches
+Device Mobility
+(d)
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+Received Signal Strength (dBm)
+0
+0.2
+0.4
+0.6
+0.8
+1
+CDF
+Device Mobility
+Wide, Slow
+Narrow, Fast
+Wide, Fast
+Figure 9: Impact of rotational mobility on search
+We repeated the search experiments during such “casual walks”. We have also repeated the
+such experiments near two base stations to see if search is any faster under such deployments.
+Fig. 10 a & b present search overhead and received signal strength during walk. We observe
+slight decrease in search overhead when two beams from beams are visible to the mobile.
+mobile discovers a base station quickly. Similar observations can be made from Fig. 11a where
+we repeated rotational mobility experiments near two base stations. Prior art [23] also found
+dense mm-wave network deployments helpful.
+C. Tracking the Neighbor Base Station Beam
+After completing the spatial scan, it is ideal for mobile to not to repeat search, as it is both
+time consuming and energy intensive. To accomplish that mobile must keep track of the transmit
+
+19
+(a)
+Narrow Beam
+Wide Beams
+0
+5
+10
+15
+20
+25
+Number of Beam Searches
+Motion: Walk, Beams: Narrow and Wide
+(b)
+-74
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+-58
+Received Signal Strength (dBm)
+0
+0.2
+0.4
+0.6
+0.8
+1
+CDF
+Motion: Walk, Beams: Narrow and Wide
+Narrow Beams
+Wide Beams
+Figure 10: Beam search while walking near one neighbor and two neighbor base stations a)
+number of beam searched b) received signal strength
+(a)
+ = 90°
+ = 180°
+0
+5
+10
+15
+20
+Number of Beam Searches
+Motion: Rotation, Two Neighbor Base stations
+(b)
+-74
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+-58
+Received Signal Strength (dBm)
+0
+0.2
+0.4
+0.6
+0.8
+1
+CDF
+Motion: Rotation, Two Neighbor Base stations
+ =90°
+ =180°
+Figure 11: Beam search while rotating mobile near one neighbor and two neighbor base stations
+a) number of beam searched b) received signal strength
+beam of the found base station until handover is complete. Tracking involves the mobile adapting
+its receive beam to maintain alignment with the base station beam. Prior works have established
+that the alignment of base station and mobile-side beams degrades over time as the user moves
+[24], [25], [21]. Unlike connected mode beam tracking, where the serving base station can help
+with beam adaptation, the mobile does not get any help from the neighbor station. The mobile
+must solely and continuously track the discovered base station beam until the connection transfer
+
+20
+is complete.
+We have conducted experimental investigations to study whether the highly aligned mobile-
+side receive beams follow any particular pattern as the user moves. Several mobility patterns
+are studied, with the mobile’s phased array’s rotated with angular velocity 60◦/s, 120◦/s and
+240◦/s as well as natural walk. For 100 experiment trials, we recorded the highly aligned mobile
+side receive beam directions. In each trail, there are 100 beam directions that provided highest
+receive signal strength at the mobile. These sequences represent the best aligned LoS directions
+in an experiment trial.
+Our observation is that at any given point in time in the experiment, the best beam direction
+is an angular neighbor to past receive beam. For better visualization, suppose if we replace
+each element of the sequence with the angular distance from the previous element, sequence
+is a simple random walk. To further illustrate this observation we performed statistical tests 1
+of randomness on the 100 sequences. The hypothesis tests failed to accept that sequences are
+generated from a deterministic source and hence might have random pattern. This implies that
+relying on the historical aligned beams to predict next beam is of no help. As sequence is a
+simple random walk, to track the base station beam it is sufficient for the mobile to check the
+signal strength on the angular neighbors of the current beam best beam direction.
+Following our experimental observations, below is the summary of Beam Management of
+TERRA protocol
+Blockage Beam Management: Contrary to the presumed requirement of the dense mm-wave
+base station deployment to mitigate blockage in outdoors, we found a simple alternative. By
+switching beam in the direction of ground reflection, mobile can maintain connectivity during
+temporary pedestrian blockage in outdoor environments. Received signal strength on the ground
+reflected beam direction is at least 6 dB less than LoS and is sufficient to maintain connectivity
+as well as low data rate control plane traffic.
+Handover Beam Management: Search for a neighbor base station during user mobility is
+delay prone, consequently mobile needs a head start in search. To perform soft handover, mobile
+must have found a reserve base station and keeping track of its beam prior to switching decision.
+To track the found beam i.e., to maintain an aligned receive beam with neighbor base station,
+1To test the randomness hypothesis, we performed runs-test on the sequences [26] with 95% confidence interval and the null
+hypothesis being the sequence is random.
+
+21
+the mobile can only rely on the receive beam adaptation. The history of aligned receive beams
+cannot help mobile predict the next beam to employ to track the neighbor base station beam.
+Given the currently aligned receive beam, mobile can only statistically keep track of a neighbor
+base station beam. We show that by following the approach proposed in [10], that reduces the
+beam search space to the neighbors of the currently aligned receive beam, mobile can keep track
+of neighbor base station until handover is complete.
+We present an in-band beam management protocol called TERRA in Section V that manages
+beams in outdoor environments. We present its efficacy both under temporary blockage and
+during handover in Section VI
+V. THE TERRA PROTOCOL FOR TRANSIENT BLOCKAGE AND HANDOVER
+LoS.Op: Line of Sight Operation
+NLoS.Op: Non Line of Sight 
+Operation
+SBA: Serving Base station beam 
+Adaptation
+NBA: Neighbor Base station beam 
+Adaptation
+N-A/R: Neighbor Acquisition/Re-
+acquisition
+GRD: Ground Reflection Discovery
+NBS: Neighbor Beam Search
+ES: Exhaustive Search
+A: Ground Reflection Discovery
+B: No pose information
+C: Pose information
+D: Reflected Beam Found
+E: Blockage event
+F: Normal Operation
+G: Neighbor Base Station Search
+H: Search Success
+I: Beam Alignment to counter 
+mobility
+GRD
+NBS
+ES
+LoS.
+Op
+NBA
+Reflected Path
+Discovery 
+I
+B
+C
+N-
+A/r
+NLoS
+.Op
+A
+E
+H
+G
+Handover Preparation
+D
+D
+F
+Figure 12: TERRA state machine
+In this section we present the TERRA protocol that maximizes connectivity in transient
+blockages as well as conducts soft handover over directional beams if needed. Fig. 12 presents
+the state transition diagram of the TERRA protocol.
+A mm-wave mobile must continuously adapt LoS beam with the serving base station to counter
+user mobility. In LoS Operation State, mobile adjusts its receive beam to maintain high degree
+
+22
+of alignment with serving base station beam. Mobile may use beam alignment method proposed
+in [10] to adjust LoS beam with serving base station to counter user motion. In addition to that,
+the mobile needs a backup NLoS beam to avoid disconnection during the temporary pedestrian
+blockages. TERRA protocol ensures this in Ground Reflected Beam Discovery (GRD) state
+wherein mobile identifies a ground reflected beam direction that we showed to have usable
+received signal strength. If pose of the mobile is available to the protocol, mobile moves to
+Neighbor Beam Search state to search zenith neighbors to current LoS beam and discovers the
+ground reflections. When the knowledge of pose of the mobile is not available, TERRA protocol
+initiates search using all the receive beams in Exhaustive Search (ES) state to discover backup
+NLoS ground reflected beam. Ground reflected direction changes whenever mobile adapts its LoS
+direction. Therefore, TERRA must rediscover ground reflected direction too. Protocol erases the
+stored reflected direction in memory after LoS beam adaptation and visits GRD state to identify
+the ground reflected path. Protocol detects the transient blockage event when the received signal
+strength suddenly decreases by 15 dB as shown in Fig. 13a, and in such an event TERRA
+employs ground reflection and keeps RSS within 6 dB of RSS on LoS beam.
+Unlike transient pedestrian blockage event where mobile can switch to LoS path after briefly
+operating on NLoS path, during a permanent blockage event, when the LoS beam is occluded
+by a building, tree or any other immovable obstacle, mobile must switch base station. Blockage
+either transient or permanent is sudden, only difference is that, it is possible to switch to LoS
+beam after transient blockage. In any case, if the mobile decides to switch to neighbor base
+station, TERRA provides mechanism to do so. TERRA protocol searches for a neighbor base
+station and keeps track of a neighbor base station to quickly perform soft-handover.
+To search for a neighbor base station, the mobile moves to the Neighbor Acquisition/Re-
+acquisition state (N-A/R). In this state, the mobile performs spatial scan and discovers at least
+one neighbor base station beam. Mobile also identifies a receive beam, which we call the “current
+receive beam of neighbor station” on which it can listen to the found base station beam.
+The mobile needs to adapt its receive beams to counter user mobility during handover and
+monitor the neighbor base station beam.
+Mobile employs the only viable choice, the receive beam adaptation, to track neighbor base
+station beam, in the Neighbor base station Receive Beam Adaptation (N-RBA) state. In this
+state, whenever the received signal strength of neighbor base station beam drops by 3 dB,
+mobile tests received signal strength on all the spatial neighbor beams to the current receive
+
+23
+beam and chooses the beam improves the RSS. While the mobile maintains connectivity with
+serving base station and tracks the neighbor base station beam, handover maybe initiated to a
+neighbor when the received signal strength exceeds hysteresis thresholds. This work focuses only
+on beam management during handover and there is extensive prior work on switching criteria
+for handover.
+VI. EVALUATION
+We evaluate TERRA protocol under pedestrian blockages and its ability to track a neighbor
+base station beam. The experiments and evaluation although performed on our 60 GHz National
+Instruments transceiver [27] can be reproduced on any available mm-wave hardware. We provide
+a python notebook [28] to control the beams of an off-the-shelf and cheaper mm-wave hardware
+to help researchers reproduce the work.
+A. Blockage Recovery
+Fig. 13a shows the received signal strength of the LoS dropping below the noise floor of
+NI’s 60 GHz receiver [27], i.e., -70 dBm, during a pedestrian blockage that lasts for about 200
+ms. Receiver experiences outage during this event as it cannot decode transmitted information.
+However, the ground reflected beam did not suffer outage. In this particular experiment trial on
+concrete surface, received signal strength is -64 dBm in ground reflected direction and -60 dBm
+in LoS. We repeated our pedestrian blockage experiments at different locations with concrete
+and gravel surfaces.
+Fig. 13b plots the CDFs obtained by employing TERRA’s blockage recovery mechanism
+during 50 blockage events. The colored region shows the outage region. The experiments show
+that TERRA employs beams that are outside the outage region 84.5% of time, and within 6 dB of
+normal operation 60% of the time. When pose information of receiver is available, TERRA either
+finds ground reflected radiation in just two measurements, or else it searches all the available
+25 beams till successful.
+B. Neighbor Base Station Beam Tracking
+Tracking the neighbor base station beam is crucial for successful soft handover. TERRA
+actively tracks the neighbor base station. In this section, we present the experimental performance
+evaluation of TERRA. We conducted 50 trials of each of the mobility pattern mentioned in
+
+24
+(a)
+0
+50
+100
+150
+200
+250
+300
+350
+Time (ms)
+-78
+-76
+-74
+-72
+-70
+-68
+-66
+-64
+-62
+-60
+-58
+Received Signal Strength (dBm)
+Link RSS in a Blockage Event: Concrete Surface
+LoS Path
+Ground Reflected Path
+(b)
+-80
+-78
+-76
+-74
+-72
+-70
+-68
+-66
+-64
+Received Signal Strength (dBm)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+CDF
+ Link RSS
+LoS path
+TERRA/GR path
+Blockage region
+X -69.85
+F(x) 0.6763
+X -69.77
+F(x) 0.1533
+X -66.38
+F(x) 0.4007
+Figure 13: (a) RSS in a blockage event (b) Performance of TERRA during pedestrain blockages
+Table IV. TERRA starts with the receive beam found after the initial search and monitors the
+received signal strength on that beam. Once the received signal strength goes down by 3 dB
+due to mobility, TERRA switches its receive beam direction to an adjacent beam that improves
+received signal strength. It then continues to monitor received signal strength using the new
+beam. Fig. 14a shows received the signal strength at the mobile in one of the trials of our
+experiments where the user is walking near the edge of the cell. The different beam directions
+are colored differently in Fig. 14a and b, thus showing the beam switching resulting from the
+TERRA protocol whose performance is indicated in dashed line.
+We also present in Fig. 14a and b the performance of an “Oracle” that has omniscient
+knowledge of the received signal strength of all the codewords and chooses the maximum. To
+measure Oracle received signal strength, we swept all the receive beams in the beam codebook
+and took the maximum. A parallel curve representing a 3dB loss over Oracle is also shown in
+Figs. 14a,b. As can be seen, for 3 seconds of the in the Walk experiment, and 2.4 seconds in the
+Rotational Motion experiment, the received signal strength observed using TERRA is within 3
+dB of the Oracle signal strength. Also, we can see that each receive beam of the TERRA takes
+advantage of the entire main lobe before switching to the next. Although Oracle signal strength
+is above TERRA’s, it is achievable only if the mobile exhaustively searches all its beams every
+time.
+
+25
+(a)
+0
+400
+800
+1200
+1600
+2000
+2400
+2800
+3200
+Time (ms)
+-64
+-63.5
+-63
+-62.5
+-62
+-61.5
+-61
+-60.5
+-60
+-59.5
+-59
+Received Signal Strength (dBm)
+Beam Tracking: Walk
+TERRA
+Oracle
+3dB from Oracle
+(b)
+0
+400
+800
+1200
+1600
+2000
+2400
+Time (ms)
+-64
+-63.5
+-63
+-62.5
+-62
+-61.5
+-61
+-60.5
+-60
+-59.5
+-59
+Received Signal Strength (dBm)
+Beam Tracking: Rotational Motion
+TERRA
+Oracle
+3dB from Oracle
+Figure 14: Beam tracking performance: a) Walking b) Rotation Motion ω = 120 deg/s
+Table IV: deviation of the TERRA’s received signal strength from oracle solution.
+Motion
+Root Mean Square Loss (in dB)
+Rotational motion at ω = 60 (deg/s)
+.4247
+Rotational motion at ω= 120 (deg/s)
+.71
+Walk: User holding phased array walks near base station
+.8343
+Table IV tabulates the deviation of TERRA’s received signal strength from the Oracle in all
+mobility experiments. TERRA can be seen to maintain received signal strength within 0.5 dB
+during slow rotational motion. In a faster rotational experiment with angular velocity 120 deg/s,
+it is 0.71 dB. For pedestrian mobility where a human walks near the neighbor base station
+holding phased array in hand, it is 0.83 dB. Overall, in all the mobility patterns that we have
+studied, tracking performance is very close to Oracle, within 1 dB.
+C. Simulation Study of Beam Tracking:
+Using computer simulations, we compare TERRA’s beam tracking complexity with state-of-
+the-art beam alignment schemes in terms of the number of measurements necessary to identify
+a receive beam.
+The simulation study uses a receiver with a uniform planar array with 32X32 elements,
+operating at 28 GHz, and is 10 m away from the transmitter. The total number of possible
+
+26
+receive beams is 1024. Exhaustive search needs 1024 measurements to identify a receive beam
+that is highly aligned with a transmitter beam. Receive beam adaptation is necessary when the
+current receive beam is no longer adequate.
+The Hierarchial Beam Alignment (HBA) [29] method first measures with an Omni-directional
+beam, and narrows down the beamwidth after each measurement. HBA [29] employs correlated
+bandit learning to narrow down the beam. In our study, HBA [29] needed 63 measurements to find
+an aligned receive beam. Given the sparse nature of the mm-wave channel in the spatial domain,
+compressive-sensing based approaches have been studied in literature to quickly identify aligned
+beams; fast beam alignment with low-resolution phase shifters (FALP) [30] uses a variation of
+the compressive sensing method.
+FALP [30] requires 70 measurements to align the receive beam. Another approach called Agile
+Link [11] uses carefully designed beam patterns that have multiple lobes to receive the signal
+from multiple directions. This design gives Agile Link [11] the ability to search in multiple
+directions in one measurement. In our simulation study Agile Link [11] took 110 measurements
+to find a receive beam that aligns with the transmit beam. In contrast, TERRA searches only the
+neighbors to the previously aligned beam, and yet finds a receive beam that provides received
+signal strength within 3 dB of the Oracle beam. While Oracle needs an exhaustive search,
+TERRA takes a maximum of 8 searches to find an aligned receive beam.
+State of the art
+Maximum number of measurements
+Terra
+8
+HBA
+63
+FALP
+70
+Agile Link
+110
+Exhaustive Search
+1024
+Table V: Tracking overhead
+VII. RELATED WORK
+When the LoS beam of mobile is blocked for prolonged duration or when the user moves to
+the edge of the serving cell, mobile switches base stations. Broadly, beam management methods
+proposed so far to handover rely on location of the user; Reinforcement Learning; Machine
+Learning. Much of the prior work used computer simulations and a thorough experimental work
+
+27
+addressing all the challenges in handling beams during a handover is missing, and TERRA
+protocol fills the void.
+Significant number of works in the literature proposed either dense deployment of base
+stations or co-ordinated multiple point access (CoMP) to address outdoor pedestrian blockage.
+Recently, using reflective surfaces near the base stations in mm-wave deployments have been
+under investigation to overcome mm-wave link impairments. Mezavilla et al. have studied the
+use of such surfaces [31]. To the best of our knowledge, we first evaluated ground reflections to
+address temporary blockage in outdoor environments.
+User location: Junshen et al. have proposed a method [32] to reduce the numbers of beams
+to search to discover the target base station for handover. Along with the geometry model of
+environment, the approach requires co-ordination among current and target base station. Parada et
+al. have proposed a method [33] using user’s direction of motion to predict the target base station
+for handover. Using a multitude of access points, Palacio et al. have developed a user localization
+algorithm [34] and proposed location aware methods for beam adaptation and handover decisions.
+TERRA doesn’t rely on user location, instead it performs a search to discover a neighbor base
+station and keeps track of the found beam with little overhead as shown in Section VI.
+Reinforcement Learning: To learn a good beam management policy i.e., which beam and
+the base station to switch to, reinforcement learning based methods require precise and high
+fidelity model of the environment. A learning agent using any of the popular reinforcement
+learning algorithms interacts with environment model and learns a beam management policy.
+Additionally, one must do fine reward engineering that helps agent evaluate the actions for every
+state of the mobile in a given environment. State of the mobile may include signal strength,
+signal-to-noise ratio, user location, and speed of the mobile.
+Zang et al. have employed model-based reinforcement learning [35] to learn a beam manage-
+ment policy for handover. The model uses the mobile’s location, velocity, and connection state
+as state information, and uses a Gauss-Markov mobility model for the transition kernel. Sun
+et al have explored multi-arm bandit approaches [36]. Adding location and direction of motion
+of the user as state information, Sun et al. also applied contextual bandits [37]. Other works
+[38], [39], [40], [41] have applied various reinforcement learning algorithms to make handover
+decisions with state information being the location and trajectory of users. However, through our
+experiments, we found that given the current best aligned beam, the next best beam is its angular
+neighbor. Also, in a random walk, the learned policy is stochastic and suggests Reinforcement
+
+28
+Learning agent to try the neighbor beams which is not different from TERRA protocol.
+Another shortcoming Reinforcement Learning approaches is that it is not clear how a learned
+policy in an environment may perform in other environments and developing such algorithms,
+called meta learning, is still an active area of research in reinforcement learning. To keep the
+protocol design simple, TERRA switches to first found base station.
+Machine Learning: Authors applied [42] a popular sequence prediction method in machine
+learning literature, Long-Short Term Memory (LSTM), to predict the mm-wave link quality. To
+that end authors simulated a base station deployment in 200x200 m2 area. Although not accurate,
+LSTM approached performed better than moving average in predicting blockage events. TERRA
+protocol can leverage link prediction methods and employ ground reflected path. The challenge
+with data driven machine learning methods is that they may or may not perform well outside the
+trained environment. Kaya et al. utilized ray tracing to construct radio environment for a video
+feed from a traffic intersection in a major city in the USA, and [43] applied LSTM to predict
+beams to aid pedestrian and vehicular mobility. LSTM predicts top-2 with beams that gives 96
+% prediction accuracy. From the plots in work, we found that algorithms predicts neighbor beam
+to past best aligned beam. TERRA achieves this prediction accuracy without the need of history
+of past beams.
+Conclusion: In this work, we propose TERRA protocol that effectively manages mobile side
+beam in outdoors environments. TERRA ensures an outdoor mobile act quickly in a transient
+blockage event and tracks a neighbor base station to perform soft-handover. We present both
+experimental and simulated evaluation to show efficacy of the protocol. While TERRA helps
+mobile avoid outage in outdoors, still careful base deployment is a must to address crowded
+environments. In future, we plan to study optimal network deployment that maximizes visibility
+of ground reflections to the mobiles.
+REFERENCES
+[1] S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,”
+Proceedings of the IEEE, vol. 102, no. 3, pp. 366–385, March 2014.
+[2] S. Collonge, G. Zaharia, and G. Zein, “Influence of the human activity on wide-band characteristics of the 60 ghz indoor
+radio channel,” IEEE Transactions on Wireless Communications, vol. 3, no. 6, pp. 2396–2406, 2004.
+[3] G. R. MacCartney, T. S. Rappaport, and S. Rangan, “Rapid fading due to human blockage in pedestrian crowds at 5g
+millimeter-wave frequencies,” in GLOBECOM 2017 - 2017 IEEE Global Communications Conference, 2017, pp. 1–7.
+
+29
+[4] V. Ganji, T.-H. Lin, F. A. Espinal, and P. R. Kumar, “Unblock: Low complexity transient blockage recovery for mobile
+mm-wave devices,” in 2021 International Conference on COMmunication Systems and NETworkS (COMSNETS), 2021,
+pp. 501–508.
+[5] 3GPP, “Base station (bs) radio transmission and reception,” 2017. [Online]. Available: https://portal.3gpp.org/
+desktopmodules/Specifications/SpecificationDetails.aspx?specificationId=3202
+[6] S. Vlahovic, M. Suznjevic, and L. Skorin-Kapov, “The impact of network latency on gaming qoe for an fps vr game,” in
+2019 Eleventh International Conference on Quality of Multimedia Experience (QoMEX), 2019, pp. 1–3.
+[7] S. Sur, X. Zhang, P. Ramanathan, and R. Chandra, “Beamspy: Enabling robust 60 ghz links under blockage,” in Proceedings
+of the 13th Usenix Conference on Networked Systems Design and Implementation, ser. NSDI’16.
+USA: USENIX
+Association, 2016, p. 193–206.
+[8] R. Baldemair, T. Irnich, K. Balachandran, E. Dahlman, G. Mildh, Y. Sel´en, S. Parkvall, M. Meyer, and A. Osseiran,
+“Ultra-dense networks in millimeter-wave frequencies,” IEEE Communications Magazine, vol. 53, no. 1, pp. 202–208,
+2015.
+[9] I. K. Jain, R. Kumar, and S. S. Panwar, “The impact of mobile blockers on millimeter wave cellular systems,” IEEE
+Journal on Selected Areas in Communications, vol. 37, no. 4, pp. 854–868, 2019.
+[10] V. S. S. Ganji, T.-H. Lin, F. A. Espinal, and P. R. Kumar, “Beamsurfer: Minimalist beam management of mobile mm-wave
+devices,” IEEE Transactions on Wireless Communications, vol. 21, no. 11, pp. 8935–8949, 2022.
+[11] H. Hassanieh, O. Abari, M. Rodriguez, M. Abdelghany, D. Katabi, and P. Indyk, “Fast millimeter wave beam
+alignment,” in Proceedings of the 2018 Conference of the ACM Special Interest Group on Data Communication, ser.
+SIGCOMM ’18.
+New York, NY, USA: Association for Computing Machinery, 2018, p. 432–445. [Online]. Available:
+https://doi.org/10.1145/3230543.3230581
+[12] S. Desai and F. Malandra, “Delay estimation of initial access procedure for 5g mm-wave cellular networks,” in 2021 17th
+International Conference on Distributed Computing in Sensor Systems (DCOSS), 2021, pp. 489–494.
+[13] R. T. Rakesh, G. Das, and D. Sen, “An analytical model for millimeter wave outdoor directional non-line-of-sight channels,”
+in 2017 IEEE International Conference on Communications (ICC), 2017, pp. 1–6.
+[14] F. W. Iden, “Glide-slope antenna arrays for use under adverse siting conditions,” IRE Transactions on Aeronautical and
+Navigational Electronics, vol. ANE-6, no. 2, pp. 100–111, 1959.
+[15] S. Rajagopal, S. Abu-Surra, and M. Malmirchegini, “Channel feasibility for outdoor non-line-of-sight mmwave mobile
+communication,” in 2012 IEEE Vehicular Technology Conference (VTC Fall), 2012, pp. 1–6.
+[16] Y. Mostofi and D. Cox, “Mathematical analysis of the impact of timing synchronization errors on the performance of an
+ofdm system,” IEEE Transactions on Communications, vol. 54, no. 2, pp. 226–230, 2006.
+[17] 3GPP, “Nr; physical layer measurements,” 2017. [Online]. Available: https://portal.3gpp.org/desktopmodules/Specifications/
+SpecificationDetails.aspx?specificationId=3217
+[18] S. K. Saha, T. Siddiqui, D. Koutsonikolas, A. Loch, J. Widmer, and R. Sridhar, “A detailed look into power consumption
+of commodity 60 ghz devices,” p. 1–10, 06 2017. [Online]. Available: https://ieeexplore.ieee.org/document/7974282
+[19] “Sensors overview-android developers.” [Online]. Available: https://developer.android.com/guide/topics/sensors/sensors
+overview
+[20] S. Odenwald, “Guide to smartphone sensors - nasa.” [Online]. Available: https://spacemath.gsfc.nasa.gov/Sensor/
+SensorsBook.pdf
+[21] B. Satchidanandan, S. Yau, P. R. Kumar, A. Aziz, A. Ekbal, and N. Kundargi, “Trackmac: An ieee 802.11ad-compatible
+beam tracking-based mac protocol for 5g millimeter-wave local area networks,” in 2018 10th International Conference on
+Communication Systems and Networks (COMSNETS), 2018, pp. 185–182.
+
+30
+[22] “Lsm6dso.” [Online]. Available: https://www.st.com/en/mems-and-sensors/lsm6dso.html
+[23] I. K. Jain, R. Kumar, and S. Panwar, “Driven by capacity or blockage? a millimeter wave blockage analysis,” in 2018 30th
+International Teletraffic Congress (ITC 30), vol. 01, 2018, pp. 153–159.
+[24] V. S. S. Ganji, T.-H. Lin, F. Espinal, and P. R. Kumar, “Beamsurfer: Simple in-band beam management for mobile mm-
+wave devices.”
+ACM Special Interest Group on Data Communication on the applications, technologies, architectures,
+and protocols for computer communication (SIGCOMM ’20 Demos and Posters), 08 2020, pp. 1–3.
+[25] V. Va, J. Choi, and R. W. Heath, “The impact of beamwidth on temporal channel variation in vehicular channels and its
+implications,” IEEE Transactions on Vehicular Technology, vol. 66, no. 6, pp. 5014–5029, June 2017.
+[26] J. V. Bradley, Distribution-free statistical tests, 1968.
+[27] National Instruments, “Introduction to the NI mmWave Transceiver System Hardware,” 02 2020. [Online]. Available:
+https://www.ni.com/en-us/innovations/white-papers/16/introduction-to-the-ni-mmwave-transceiver-system-hardware.html
+[28] S. Ganji, “Shotsan/60ghz-mikrotik-python-dash-api: Python script and notebook to visualize data from mikrotik
+60ghz router, control beams, read mcs, data rate and packet errors.” [Online]. Available: https://github.com/shotsan/
+60GHz-mikrotik-python-dash-api
+[29] W. Wu, N. Cheng, N. Zhang, P. Yang, W. Zhuang, and X. Shen, “Fast mmwave beam alignment via correlated bandit
+learning,” IEEE Transactions on Wireless Communications, vol. 18, no. 12, pp. 5894–5908, 2019.
+[30] N. J. Myers, A. Mezghani, and R. W. Heath, “FALP: Fast beam alignment in mmwave systems with low-resolution phase
+shifters,” IEEE Transactions on Communications, vol. 67, no. 12, pp. 8739–8753, 2019.
+[31] L. Jiao, P. Wang, A. Alipour-Fanid, H. Zeng, and K. Zeng, “Enabling efficient blockage-aware handover in ris-assisted
+mmwave cellular networks,” IEEE Transactions on Wireless Communications, pp. 1–1, 2021.
+[32] W. Junsheng, C. Yawen, L. Zhaoming, W. Xiangming, and W. Zifan, “A low-complexity beam searching method for fast
+handover in mmwave vehicular networks,” in 2019 IEEE Wireless Communications and Networking Conference Workshop
+(WCNCW), 2019, pp. 1–6.
+[33] R. Parada and M. Zorzi, “Context-aware handover in mmwave 5g using ue’s direction of pass,” in European Wireless
+2018; 24th European Wireless Conference, 2018, pp. 1–6.
+[34] J. Palacios, P. Casari, H. Assasa, and J. Widmer, “Leap: Location estimation and predictive handover with consumer-grade
+mmwave devices,” in IEEE INFOCOM 2019 - IEEE Conference on Computer Communications, 2019, pp. 2377–2385.
+[35] S. Zang, W. Bao, P. L. Yeoh, H. Chen, Z. Lin, B. Vucetic, and Y. Li, “Mobility handover optimization in millimeter
+wave heterogeneous networks,” in 2017 17th International Symposium on Communications and Information Technologies
+(ISCIT), 2017, pp. 1–6.
+[36] L. Sun, J. Hou, and T. Shu, “Optimal handover policy for mmwave cellular networks: A multi-armed bandit approach,”
+in 2019 IEEE Global Communications Conference (GLOBECOM), 2019, pp. 1–6.
+[37] L. Sun, J. Hou, and T. Shu, “Spatial and temporal contextual multi-armed bandit handovers in ultra-dense mmwave cellular
+networks,” vol. 20, no. 12, 2021, pp. 3423–3438.
+[38] S. Khosravi, H. S. Ghadikolaei, and M. Petrova, “Learning-based load balancing handover in mobile millimeter wave
+networks,” in GLOBECOM 2020 - 2020 IEEE Global Communications Conference, 2020, pp. 1–7.
+[39] S. Khosravi, H. Shokri-Ghadikolaei, and M. Petrova, “Learning-based handover in mobile millimeter-wave networks,”
+IEEE Transactions on Cognitive Communications and Networking, vol. 7, no. 2, pp. 663–674, 2021.
+[40] Y. Koda, K. Yamamoto, T. Nishio, and M. Morikura, “Reinforcement learning based predictive handover for pedestrian-
+aware mmwave networks,” in IEEE INFOCOM 2018 - IEEE Conference on Computer Communications Workshops
+(INFOCOM WKSHPS), 2018, pp. 692–697.
+
+31
+[41] M. Mezzavilla, S. Goyal, S. Panwar, S. Rangan, and M. Zorzi, “An mdp model for optimal handover decisions in mmwave
+cellular networks,” in 2016 European Conference on Networks and Communications (EuCNC), 2016, pp. 100–105.
+[42] S. H. A. Shah, M. Sharma, and S. Rangan, “Lstm-based multi-link prediction for mmwave and sub-thz wireless systems,”
+in ICC 2020 - 2020 IEEE International Conference on Communications (ICC), 2020, pp. 1–6.
+[43] A. O. Kaya and H. Viswanathan, “Deep learning-based predictive beam management for 5g mmwave systems,” in 2021
+IEEE Wireless Communications and Networking Conference (WCNC), 2021.
+