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+arXiv:2301.02842v1  [math.AG]  7 Jan 2023
+STRATIFIED BUNDLES ON THE HILBERT SCHEME OF n POINTS
+SAURAV HOLME CHOUDHURY
+Abstract. Let k be an algebraically closed field of characteristic p > 3 and S be a smooth projective
+surface over k with k-rational point x. For n ≥ 2, let S[n] denote the Hilbert scheme of n points on S. In
+this note, we compute the fundamental group scheme πalg(S[n], ˜
+nx) defined by the Tannakian category of
+stratified bundles on S[n].
+1. Introduction
+For a variety X over C, one has the classical notion of the fundamental group π1(Xan, x) defined using
+the analytic topology on X. Over arbitrary base fields k, one has several analogues of the fundamental group
+defined in terms of algebro-geometric information.
+In [SGA1], Grothendieck introduced the notion of ´etale fundamental group π´et(X, x), where X is a scheme
+and x is a geometric point of X, in terms of the finite etale covers of X. In [N76], Nori defined the Nori
+fundamental group scheme πN(X, x), where X is a connected, reduced and complete scheme over a perfect
+field k and x is a k-rational point, via Tannakian reconstruction using the category of essentially finite vector
+bundles on X. The definition of πN(X, x) was extended to the case of connected and reduced k-schemes in
+[N82]. Another analogue, the S-fundamental group scheme πS(X, x) was introduced and studied by Langer
+in [Lan11] and [Lan12] for smooth projective varieties X over an algebraically closed field k. It is defined via
+Tannakian reconstruction using the category of numerically flat vector bundles on X. The S-fundamental
+group scheme for a smooth projective curve C over an algebraically closed field k was already introduced
+and studied in [BPS06].
+The variant of the fundamental group scheme which is of prime importance in this note is the algebraic
+fundamental group πalg(X, x). In [Gie75], Gieseker defined πalg(X, x) as the fundamental group scheme
+corresponding to the Tannakian category of DX-modules, where DX is the sheaf of differential operators on
+X. For X smooth over a field of positive characteristic, Gieseker introduced the notion of stratified bundles
+and showed that the category of DX-modules is tensor equivalent to the category of stratified bundles on
+X. Stratifed bundles were further studied in [dS07] and [BHdS21]. Precise definitions and statements will
+be given in the next section.
+Let S be a smooth projective surface over k. For n ≥ 2, let S[n] denote the Hilbert scheme n points on S.
+It is well known that S[n] is a smooth projective variety of dimension 2n. In [PS20], the authors show that
+for char k > 3 and n ≥ 2, there is an isomorphism of affine group schemes over k
+π†(S, x)ab → π†(S[n], ˜
+nx)
+where † = S, N or ´et.
+In this note, we extend their results to the case of πalg and prove the following theorem.
+Theorem. Let char k > 3 and n ≥ 2. There is an isomorphism of affine group schemes over k
+f : πalg(S, x)ab → πalg(S[n], ˜
+nx)
+In section 2, we recall the definition of stratified bundles and some of their basic properties. The formalism
+of Tannakian reconstruction is recalled in section 3 and used to define the algebraic fundamental group
+πalg(X, x).
+1
+
+The geometrical properties of the Hilbert scheme of n points on a smooth projective surface are in section 4.
+In section 5, we prove a result about descent of stratified bundles which allows us to define the homomorphism
+f by defining the associated functor of Tannakian categories. The concluding section 6 establishes the main
+theorem by showing that f is an isomorphism.
+Acknowledgements. We would like to thank Indranil Biswas and Ronnie Sebastian for their comments
+on earlier drafts of this note.
+2. Stratified bundles
+Let k be a field of characteristic p and X be a noetherian scheme over k. Stratified bundles on X are
+sequences of coherent sheaves on X satisfying infinite Frobenius descent. More precisely, the category of
+stratified bundles on X, denoted S(X), consists of
+• Objects (Ei, αi) are sequences of coherent OX-modules Ei, i ∈ N along with isomorphisms
+αi : F ∗Ei+1 → Ei
+for all i ∈ N, where F is the absolute Frobenius on X.
+• Morphisms φ : (Ei, αi) → (Fi, βi) consists of a sequence of OX-module morphisms φi : Ei → Fi
+such that φi ◦ αi = βi ◦ F ∗ (φi+1)
+Let f : Y → X be a morphism and (Ei, αi) be a stratified bundle on X. Then we can define the pullback
+along f, denoted f ∗(Ei, αi), as consisting of the sequence of OY coherent sheaves f ∗Ei and isomorphisms
+are given by the composite maps
+F ∗f ∗Ei+1
+γEi+1
+−−−→ f ∗F ∗Ei+1
+f ∗(αi)
+−−−−→ f ∗Ei
+where γ : F ∗f ∗ → f ∗F ∗ is the natural isomorphism of functors.
+Thus S(X) is contravariant functor in X. One also has a tensor product on S(X) defined by taking term
+by term tensor product. Also S(X) is an abelian category [cf. [BHdS21], Proposition 4.4].
+We recall some well known results about stratified bundles [cf [dS07], [Gie75]].
+Proposition. If (Ei, αi) is a stratified bundle on X, then Ei is a locally free OX-module for all i ∈ N.
+This allows us to define duals of stratified bundles, making S(X) into an abelian rigid tensor category.
+The rank of a stratified bundle (Ei, αi) is defined to be the rank of E0. The trivial stratified bundles on X
+are of the form ⊕(OX, ...; F ∗, ...).
+Let DX be the sheaf of differential operators on X. The category of DX modules consists of
+• Objects coherent OX modules E equipped with a DX action i.e a morphism of OX-algebras
+DX → Endk(E)
+• Morphisms OX-linear maps E → F compatible with the DX action
+A theorem of Katz [[Gie75], Theorem 1.3] shows that for X smooth over k, then the category of stratified
+bundles on X and the category of DX modules are tensor equivalent to each other.
+We close this section with the definition of G equivariant stratified bundles on a variety X admitting
+action of a group G on it.
+Definition. A stratified bundle (Ei, αi) is said to be a G-equivariant stratified bundle if Ei are G-equivariant
+vector bundles and αi are G-equivariant OX module morphisms.
+3. Tannakian categories and fundamental group schemes
+In this section we recall the definition and basic properties of Tannakian categories.
+We then recall
+Gieseker’s definition of the fundamental group scheme πalg using the Tannakian formalism.
+2
+
+3.1. Tannakian Categories and affine group schemes. Tannakian categories were defined and studied
+in [DM82] to formalize the properties of Repk(G), the category of finite dimensional k-representations of G,
+an affine group scheme over k.
+Definition (Neutral Tannakian Categories). A rigid abelian tensor category C with End I = k is a neutral
+Tannakian category if it admits an exact faithful k-linear tensor functor ω : C → Veck. Any such functor is
+said to be a fiber functor for C.
+Given a neutral Tannakian category (C, ⊗, ω, I), we define the functor Aut⊗(ω) : k − algebra → Sets
+such that for k-algebra R, Aut⊗(ω)(R) consists of the families (λX) for X ∈ ob(C), where λX is a R-linear
+automorphism of X ⊗ R such that λX1⊗X2 = λX1 ⊗ λX2, λI = idR, and
+λY ◦ (α ⊗ 1) = (α ⊗ 1) ◦ λX : X ⊗ R → Y ⊗ R
+for all morphisms α : X → Y .
+Theorem (Main theorem for neutral Tannakian categories, [DM82], Theorem 2.11). Let (C, ⊗) be a rigid
+abelian tensor category such that k = End(I) and let ω : C → Veck be an exact faithful tensor functor. Then
+• The functor Aut⊗(ω) of k-algebras is represented by an affine group scheme G.
+• The functor C → Repk(G) is an equivalence of tensor categories.
+Theorem. Let (C, ⊗, ω, I) and (C′, ⊗, ω′, I′) be neutral Tannakian categories which correspond to the repre-
+sentation categories of the affine k group schemes G and G′ respectively. Then any functor of Tannakian
+categories from C → C′ is induced by a unique morphism of affine k group schemes G′ → G.
+This theorem allows us to define many variants of fundamental groups of a scheme X by considering
+different Tannakian categories naturally associated with X. The following result is very useful in establishing
+a given morphism between affine group schemes is an isomorphism.
+Theorem ([DM82], Theorem 2.21). Let f : G → G′ be a homomorphism of group schemes over k and
+Rep (f) : Rep (G′) → Rep (G) be the corresponding functor of Tannakian categories. Then
+• f is faithfully flat if and only if Rep (f) is fully faithful and has essential image closed under subobjects
+i.e for V ′ ∈ Rep (G′) and suboject W ⊂ Rep (f)(V ′), there is a subobject W ′ ⊂ V ′ in Rep (G′) such
+that Rep (f)(W ′) ≃ W in Rep (G)
+• f is closed immersion if and only if every object of Rep (G) is a subquotient of some object in the
+essential image of Rep (f).
+We finish by recalling a basic result on affine group schemes (we refer to section 4.1 in [PS20] for details).
+Let G be a affine group scheme over k, Gab be its abelianization (i.e the maximal abelian quotient of G)
+and α : G → Gab be the (faithfully flat) quotient morphism . We can then define the composite morphism
+φ : Gn
+αn
+−−→ Gn
+ab
+m
+−→ Gab
+where m is the multiplication homomorphism. As Sn acts on the k-group scheme Gn, we can define the
+notion of a Sn-invariant group morphism ψ : Gn → H for any k-group scheme H.
+Lemma 3.1. Let G and H be two group schemes over k. For an integer n ≥ 2, the set of Sn-invariant
+group morphisms Gn → H is in bijective correspondence with the set of group morphism Gab → H i.e any
+morphism of k-group schemes φ : Gn → H which is Sn-invariant factors uniquely through a morphism
+ψ : Gab → H such that φ = ψ ◦ h
+3.2. The group scheme πalg(X, x). Classically, over C, the Riemann-Hilbert correspondence identifies the
+category of vector bundles equipped with integrable connections on a smooth connected projective variety
+X/C with the category of representations of the topological fundamental group πtop(X, x) for some chosen
+base point x. Via GAGA, this gives a purely algebraic description of the category of representations of the
+topological fundamental group π(X, x). This category (equipped with the fiber functor (E, ∇) → Ex) is
+a neutral Tannakian category and can be identified, via the Tannakian formalism, with the representation
+3
+
+category of the proalgebraic completion of the topological fundamental group, denoted as πtop(X, x)alg.
+Over a field k of characteristic 0, the category of flat connections on a smooth variety X is tensor equiva-
+lent to the category of DX-modules. However over a field of characteristic p, the category of flat connections
+on X is not as well behaved as the category of DX-modules and one defines a fundamental group scheme for
+X by Tannakian formalism using the category of DX-modules. By Katz’s theorem mentioned before, the
+fundamental group coincides with the one defined using S(X) below.
+Let x ∈ X(k) be a k-rational point. Then the abelian rigid tensor category S(X) is neutralized by the
+fiber functor
+Tx : S(X) → V eck
+The fundamental group scheme defined by the neutral Tannakian category (S(X), ⊗, Tx, (OX, F ∗)) is
+called the algebraic fundamental group of X based at x and is denoted by πalg(X, x).
+The following basic properties of πalg are well known.
+• (Independence of basepoint) Let X be a geometrically connected, smooth projective k-scheme. Then
+for all x1, x2 ∈ X(k), one has
+πalg(X, x1) ≃ πalg(X, x2)
+• (Product rule) For X1, X2 geometrically connected and smooth over k and xi ∈ Xi(k), there is an
+isomorphism
+πalg(X1, x1) × πalg(X2, x2) → πalg(X1 × X2, (x1, x2))
+• For X smooth and open immersion U
+i−→ X such that the complement of U in X has codimension
+≥ 2 and x ∈ U(k), then the homomorphism
+πalg(U, x) → πalg(X, x)
+associated to the restriction functor i∗ : S(X) → S(U) is an isomorphism.
+4. Geometry of Hilbert Scheme of points
+Let S be a smooth projective surface over k. We fix notation as follows
+• Sn denotes the n-fold cartesian product of S with itself.
+• S(n) denotes the nth symmetric product of S defined as the quotient Sn/Sn, where Sn denotes the
+symmetric group on n letters.
+• S[n] denotes the Hilbert scheme of n points on S.
+Let ρ : Sn → S(n) be the quotient map and h : S[n] → S(n) be the Hilbert-Chow morphism. We write
+S(n)
+◦
+for the open subset of S(n) consisting of distinct points with S[n]
+◦
+:= h−1(S(n)
+◦
+) and Sn
+◦ := ρ−1(S(n)
+◦
+).
+The map hn,◦ : S[n]
+◦
+→ S(n)
+◦
+is an isomorphism. We have the diagram:
+S[n]
+Sn
+S(n)
+hn
+ρn
+In general, Hilbert schemes of points on a projective variety display a lot of pathological features. But in
+[Fog68] the author shows that, in the case of smooth projective surface S, S[n] is a smooth projective variety.
+Thus, in this case, the Hilbert-Chow morphism h : S[n] → S(n) is a resolution of singularities.
+4
+
+One can consider S(n) as the set of effective 0-cycles of degree n on S(n). In this case it is easy to see that
+S(n) admits a stratification by type, where the type of a 0-cycle y of degree n is a tuple (n1, . . . , nr) where y
+can be written as
+y = Σr
+j=1njxj
+where xj are distinct points of S with multiplicities n1 ≥ n2 ≥ · · · ≥ nr, where nj are positive integers.
+Let C(n1, . . . , nr) denote the subset of S(n) consisting of points of the type (n1, . . . , nr). Let S(n)
+∗
+=
+C(1, . . . , 1) � C(2, . . . , 1) denote the open subset of S(n) consisting of points of type (1, . . . , 1) and (2, . . . , 1).
+Let S[n]
+∗
+and Sn
+∗ denote the preimage of S(n)
+∗
+under h and ρ respectively.
+We recall some basic properties below which we will need later (we refer to [Fog68], [PS20] for details).
+• The subsets C(n1, . . . , nr) are nonsingular of dimension 2r.
+• The closed subset S(n) \ S(n)
+∗
+is of codimension ≥ 2 in S(n).
+• The closed subset S[n] \ S[n]
+∗
+is of codimension 2 in S[n].
+• The closed subset Sn \ Sn
+∗ is of codimension ≥ 4 in Sn.
+• The closed subset S(n)
+∗
+\ S(n)
+◦
+is of codimension 2 in S(n)
+∗
+.
+• When characteristic of k ̸= 2, for y ∈ C(2, 1, . . . , 1), the scheme theoretic fiber h−1(y) is isomorphic
+to P1
+k. In fact, S[n]
+∗
+is the blowup of S(n)
+∗
+along C(2, 1, . . . , 1).
+We end this section by recalling a result of Fogarty ([Fog77], Proposition 3.6).
+Proposition. If L is a Sn-invariant line bundle on Sn, there exists a line bundle L′ on S(n) such that
+h∗L′ ≃ L.
+It follows that L′ in the proposition is isomorphic to σ∗(L)Sn
+5. The functor between Tannakian categories
+Let S be a smooth projective surface over k and (Ei, αi) be a stratified bundle on S[n]. Restricting to S[n]
+∗
+gives us a functor
+i∗ : S(S[n]) → S(S[n]
+∗ )
+which is a equivalence of categories as S[n]
+∗
+is the complement of a codimension 2 closed subset of S[n].
+Next we show that a stratified bundle on S[n]
+∗
+can be pushed forward under h to get a stratified bundle on
+S(n)
+∗
+. First we begin by a result on descent of vector bundles along the morphism h : S[n]
+∗
+→ S(n)
+∗
+. Similar
+results have been established by authors in [Ish83] and [PS20].
+Proposition 1. Assume char k ̸= 2. Let E be a vector bundle on S[n]
+∗
+which restricts to trivial vector bundles
+on the fibers of h over S(n)
+∗
+. Then h∗E is a locally free OS(n)
+∗ -module. Moreover the natural map
+h∗h∗(E) → E
+is an isomorphism.
+Proof. Let x ∈ S(n)
+∗
+be a point of type (2, 1, . . ., 1). Then by assumption, the fiber of h over x is isomorphic
+to P1
+k. Let J be the ideal sheaf of the closed subscheme h−1(x) and Ix be the ideal sheaf of the closed point
+x. We have
+J = IxOS[n]
+∗
+For all n ≥ 1, let Yn denote the closed subscheme of S[n]
+∗
+corresponding to the ideal sheaf J n. Consider
+the following short exact sequence of sheaves on S[n]
+∗
+0 → J ⊗ E → E → E|Y1 → 0
+5
+
+Pushing forward by h, we get the following exact sequence of sheaves on S(n)
+∗
+h∗E → H0(Y1, E|Y1) → R1h∗(J ⊗ E)
+We claim that the completion of R1h∗(J ⊗ E) at the maximal ideal mx in OS(n)
+∗
+,x is 0. The proof uses
+the theorem of formal functions which says that
+(R1h∗(J ⊗ E))∧ ≃ lim
+←− H1(Yn, J ⊗ E ⊗ OS[n]
+∗ /J n)
+We prove by induction that H1(Yn, J ⊗E ⊗OS[n]
+∗ /J n) = 0. As Y1 ≃ P1
+k, the sheaves J n/J n+1 are locally
+free. These sheaves are also globally generated over Y1 as we have the surjection
+mn
+x/mn+1
+x
+⊗O
+S(n)
+∗
+,x OS[n]
+∗
+≃ In
+x /In+1
+x
+⊗O
+S(n)
+∗
+OS[n]
+∗
+։ J n/J n+1
+As J n/J n+1 is locally free on Y1 ≃ P1
+k and globally generated, it is a direct sum of line bundles each of
+which has degree ≥ 0. Thus one gets the base case of induction from degree considerations, as
+H1(Y1, J ⊗ E ⊗ OS[n]
+∗ /J = H1(Y1, J /J 2 ⊗ EY1) = 0
+Assume that the claim is true for n. Then the proof for n + 1 follows from the long exact sequence in
+cohomology attached to the short exact sequence of sheaves on Yn+1
+0 → J n+1/J n+2 ⊗ E → J /J n+2 ⊗ E → J /J n+1 ⊗ E → 0
+which gives us the exact sequence
+H1(Yn+1, J n+1/J n+2 ⊗ E) → H1(Yn+1, J /J n+2 ⊗ E) → H1(Yn+1, J /J n+1 ⊗ E)
+We know H1(Yn, J n+1/J n+2⊗E) = H1(Y1, J n+1/J n+2⊗E) = 0 (by degree consideration) and H1(Yn+1, J /J n+1⊗
+E) = H1(Yn, J /J n+1 ⊗ E) = 0 (by induction hypothesis), thus we get
+H1(Yn+1, J /J n+2 ⊗ E) = 0
+Thus the stalk of R1h∗(J ⊗ E) at x is 0.
+This shows that the natural map h∗E → H0(Y1, E|Y1) is surjective in a neighbourhood of x. Let f1, ..., fr
+be a basis of H0(Y1, E|Y1). Let Spec(R) be an affine neighbourhood of x where the natural map is surjective
+and let ˜fi ∈ Γ(Spec(R), h∗E) = Γ(h−1(Spec(R)), E) be lifts of fi. Using ˜fi one defines a homomorphism
+O⊕r
+S[n]
+∗ |h−1(Spec(R)) → E
+on h−1(Spec(R) which is a surjection (and hence an isomorphism) on Y1. As h is proper, there exists a
+smaller affine neighbourhood U of x over which there is an isomorphism
+O⊕r
+V
+≃ E
+where V = h−1(U). Applying h∗, we get
+(h∗OV )⊕r ≃ h∗E
+As S(n)
+∗
+is normal and h : S[n]
+∗
+→ S(n)
+∗
+is birational with connected fibers, by a form of Zariski’s main
+theorem [cf [Har77], Corollary 11.3 and 11.4], we have that h∗OV ≃ OU and thus h∗E is locally free. The
+natural morphism
+h∗h∗(E) → E
+is clearly an isomorphism.
+□
+6
+
+Let VBS(n)
+∗
+be the category of locally free sheaves on S(n)
+∗
+and VBh
+S[n]
+∗
+be the category of locally free
+sheaves on S[n]
+∗
+which restrict to trivial vector bundles on the fibers of h. Proposition 1 above gives us an
+equivalence of categories.
+Proposition 2. Assume char k ̸= 2. The pushforward functor
+h∗ : VBh
+S[n]
+∗
+→ VBS(n)
+∗
+is an equivalence of categories with the inverse given by
+h∗ : VBS(n)
+∗
+→ VBh
+S[n]
+∗
+Proof. We observe that if E′ ≃ h∗(E), then E ≃ h∗E′. This shows that h∗ is essentially surjective. The
+natural map
+HomS(n)
+∗ (h∗E, h∗F) → HomS[n]
+∗ (E, F)
+is bijective. Thus h∗ is an equivalence of categories.
+□
+Corollary. For all E ∈ VBh
+S[n]
+∗ , the natural map
+F ∗h∗(E) → h∗F ∗(E)
+is an isomorphism over S(n)
+∗
+.
+Proof. As F ∗E is also an object of VBh
+S[n]
+∗ , thus both sheaves are locally free of the same rank. Thus it
+suffices to show that the natural map
+F ∗h∗(E) → h∗F ∗(E)
+is surjective. As F is faithfully flat on the smooth locus of S(n)
+∗
+, the claim holds on the smooth locus. Let
+x ∈ S(n)
+∗
+be of type (2, 1, . . . , 1). Then the restriction of F ∗h∗(E) to x is naturally isomorphic to H0(Y1, E|Y1)
+and the restriction of h∗F ∗(E) to x is H0(Y1, F ∗(E|Y1). The restriction of the natural map to x is the map
+F ∗ : H0(Y1, E1) → H0(Y1, F ∗E1)
+which is surjective.
+□
+By Theorem 2.2 of [Gie75], we have that every stratified bundle on P1
+k is trivial. Thus the above results
+give us
+Proposition 3. Assume char k ̸= 2. Let (Ei, αi) be a stratified bundle on S[n]
+∗ . Then h∗(Ei) is locally free
+OS(n)
+∗ -module for all i ∈ N. Moreover the natural map
+h∗h∗(Ei) → Ei
+is an isomorphism. Furthermore the natural map
+F ∗h∗(Ei) → h∗F ∗(Ei)
+is an isomorphism over S(n)
+∗
+.
+This allows us to define the pushforward of a stratified bundle (Ei, αi) on S[n]
+∗ . The pushforward denoted
+h∗(Ei, αi) is given by the sequence of vector bundles h∗Ei for all i ∈ N and the isomorphisms are given by
+the composite
+F ∗h∗(Ei+1)
+ηEi+1
+−−−→ h∗F ∗(Ei+1)
+h∗(αi)
+−−−−→ h∗(Ei)
+where η : F ∗h∗ → h∗F ∗ is the natural transformation.
+7
+
+Thus we get a functor
+h∗ : S(S[n]
+∗ ) → S(S(n)
+∗
+)
+h∗ is additive tensor functor as on the smooth locus S(n)
+◦
+we have the isomorphisms
+h∗((Ei, αi) ⊕ (Fi, βi))|S(n)
+◦
+≃ h∗(Ei, αi)|S(n)
+◦
+⊕ h∗(Fi, βi))|S(n)
+◦
+h∗((Ei, αi) ⊗ (Fi, βi))|S(n)
+◦
+≃ h∗(Ei, αi)|S(n)
+◦
+⊗ h∗(Fi, βi))|S(n)
+◦
+which extend to S(n)
+∗
+due to codimension reasons.
+The following commutative diagram shows that h∗h∗(Ei, αi) is isomorphic to (Ei, αi) as stratified bundles
+with the isomorphism given by the natural morphisms h∗h∗Ei → Ei.
+F ∗h∗h∗Ei+1
+h∗F ∗h∗Ei+1
+h∗h∗F ∗Ei+1
+h∗h∗Ei
+F ∗Ei+1
+F ∗Ei+1
+Ei
+h∗ηEi+1
+h∗h∗αi
+αi
+γh∗Ei+1
+=
+Consider the pullback functor
+ρ∗ : S(S(n)
+∗
+) → S(Sn
+∗ )
+which takes values in the category of Sn-equivariant stratified bundles on Sn
+∗ . Also we have the extension
+functor
+j∗ : S(Sn
+∗ ) → S(Sn)
+which is an equivalence of categories. Composing these functors together, we get a functor
+T : S(S[n]) → S(Sn)
+given by
+T = j∗ ◦ ρ∗ ◦ h∗ ◦ i∗
+Clearly T is an additive tensor functor. Note that h∗ is fully faithful, ρ∗ : S(S(n)
+∗
+) → S(Sn
+∗ ) is fully faithful
+(as ρ : Sn
+◦ → S(n)
+◦
+is finite ´etale) and j∗ : S(Sn
+∗ ) → S(Sn) is an equivalence of categories (due to codimension
+reasons). Thus T is fully faithful.
+5.1. The homomorphism. Fix n distinct k-valued points x1, . . . , xk ∈ S(k).
+Let ˜x ∈ S[n] such that
+h(˜x) = σ(x1, . . . , xn) = z ∈ S(n)
+◦
+. Then the categories S(S[n]) and S(Sn) are neutralized by the respective
+fiber functors
+τ˜x : S(S[n]) → Veck
+(Ei, αi) �→ (E0)˜x
+τ(x1,...,xn) : S(Sn) → Veck
+(Fi, βi) �→ (F0)(x1,...,xn)
+If T ((Ei, αi)) = (Fi, βi) that we have natural isomorphisms (E0)˜x ≃ h∗(E0)z ≃ (F0)(x1,...,xn).
+Thus we have a functor of Tannakian categories
+T : (S(S[n]), ⊗, τ˜x, (OS[n], d)) → (S(Sn), ⊗, τ(x1,...,xn), (OSn, d))
+8
+
+which by the independence of basepoint property of S induces a functor of Tannakian categories
+T : (S(S[n]), ⊗, τ ˜
+nx, (OS[n], d)) → (S(Sn), ⊗, τ(x,...,x), (OSn, d))
+and hence a morphisms of the associated fundamental group schemes
+˜f : πalg(Sn, (x, . . . , x)) → πalg(S[n], ˜
+nx)
+Note that by proposition 3.2 we have
+πalg(Sn, (x, . . . , x)) ≃ πalg(S, x)n
+.
+As
+T : (S(S[n]), ⊗, T ˜
+nx, (OS[n], d)) → (S(Sn), ⊗, T(x,...,x), (OSn, d))
+takes stratified bundles on S[n] to Sn-equivariant stratified bundles on Sn and a Sn-equivariant stratified
+bundles on Sn corresponds to a Sn-invariant representation of πalg(S, x)n, by 3.1, ˜f factors uniquely through
+f : πalg(S, x)ab → πalg(S[n], ˜
+nx)
+6. Isomorphism of fundamental group schemes
+In this section, we show that f is an isomorphism of affine group schemes. We begin by proving a result
+about Sn-equivariant stratified line bundles on Sn.
+Proposition 4. Let (Li, αi) be a Sn-equivariant stratified line bundles on Sn. Then there exists a stratified
+line bundle (Li, βi) such that ρ∗(Li, βi) ≃ (Li, αi)
+Proof. By Fogarty’s result mentioned above, for any Sn-equivariant line bundle Li there exists line bundle
+Li ≃ ρ∗LSn
+i
+such that ρ∗Li ≃ Li. Pushing forward αi and taking Sn invariants we get the isomorphism
+ρ∗(F ∗Li+1)Sn
+ρ∗(αi)Sn
+−−−−−−→ ρ∗(Li)Sn
+We show that the natural homomorphism
+F ∗(ρ∗(Li)Sn) → (F ∗ρ∗(Li)Sn) → (ρ∗F ∗(Li)Sn)
+is an isomorphism. Pulling back the morphism under ρ, we get the commutative diagram
+ρ∗F ∗((ρ∗Li)Sn)
+ρ∗((ρ∗F ∗Li)Sn)
+F ∗Li
+F ∗Li
+=
+where the vertical morphisms are the natural morphism which are isomorphisms by Fogarty’s theorem.
+By pushing forward under ρ and taking Sn invariants we get that
+F ∗(ρ∗(Li)Sn) → (ρ∗F ∗(Li)Sn)
+is an isomorphism. We define βi to be the composite isomorphism
+F ∗(ρ∗(Li)Sn) → (ρ∗F ∗(Li)Sn)
+ρ∗(αi)Sn
+−−−−−−→ ρ∗(Li)Sn
+The commutative diagram also gives us that ρ∗(Li, βi) ≃ (Li, αi)
+□
+9
+
+6.1. Faithfully flat. Next we show that the morphism f is faithfully flat
+Proposition 5. The homomorphism
+f : πalg(S, x)ab → πalg(S[n], ˜
+nx)
+is faithfully flat.
+Proof. By [[DM82] Theorem 2.21], this is equivalent to showing that the functor
+T : S(S[n]) → S(Sn)
+is fully faithful and the essential image of T is closed under taking subobjects. We already know that T
+is fully faithful. Let E• = (Ei, αi) be a stratified bundle on S[n] and F• := T (E•) be the corresponding
+Sn-equivariant stratified bundle on Sn. If F′
+• ⊂ F• is a Sn-equivariant stratified subbundle, then we need
+to show there exists E′
+• ⊂ E• such that T (E′
+•) = F′
+•.
+The proof proceeds by induction on the rank of E•. If rank E• = 1, the proof is immediate. Let rank
+E• ≥ 2
+Then the stratified bundles F• and F′
+• correspond to the representations
+πalg(Sn, (x, . . . , x) → πalg(S, x)ab → GL(V )
+and
+πalg(Sn, (x, . . . , x) → πalg(S, x)ab → GL(V ′)
+respectively.
+As πalg(S, x)ab is an abelian affine group scheme over k, all its irreducible representations are one dimen-
+sional. Thus one gets that the πalg(S, x)ab-module V/V ′ has a one dimensional quotient W. Thus there is
+a πalg(S, x)ab-module surjection V → W such that the kernel contains V ′. Let L• be the Sn-equivariant
+stratified bundle corresponding to W. Thus we have a short exact sequence of Sn-equivariant stratified
+bundles
+0 → K• → F• → L• → 0
+where F′
+• ⊂ K•.
+By proposition 1 above, we know that Li := ρ∗LSn
+i
+is a line bundle on S(n) and ρ∗Li = Li
+We claim that the following complex of sheaves on S(n)
+∗
+is exact for all i ∈ N
+(1)
+0 → (ρ∗Ki)Sn|S(n)
+∗
+→ (ρ∗Fi)Sn|S(n)
+∗
+→ (ρ∗Li)Sn|S(n)
+∗
+→ 0
+It is enough to show that (ρ∗Fi)Sn|S(n)
+∗
+→ (ρ∗Li)Sn|S(n)
+∗
+is surjective. We note that (ρ∗Fi)Sn|S(n)
+∗
+=
+h∗(Ei|S[n]
+∗ ). Let C be the cokernel
+h∗(Ei|S[n]
+∗ ) → (ρ∗Li)Sn|S(n)
+∗
+→ C → 0
+Pulling back under ρ, we get the following commutative diagram on Sn
+∗
+ρ∗h∗(Ei|S[n]
+∗ )
+ρ∗((ρ∗Li)Sn|S(n)
+∗ )
+ρ∗C
+0
+Fi
+Li|Sn
+∗
+0
+=
+=
+10
+
+The rows are exact and hence ρ∗C = 0. As ρ is surjective, this implies C = 0. Thus Ki := (ρ∗Ki)Sn|S(n)
+∗
+is locally free on S(n)
+∗
+Pulling back the exact sequence (1) under h, we get a short exact sequence of locally free sheaves on S[n]
+∗
+0 → h∗Ki|S[n]
+∗
+→ Ei|S[n]
+∗
+→ ˜Li|S[n]
+∗
+→ 0
+where ˜Li := h∗Li.
+As the complement of S[n]
+∗
+in S[n] is of codimension ≥ 2 and Ei, L are locally free, the surjective morphism
+Ei|S[n]
+∗
+→ ˜Li|S[n]
+∗
+extends to a unique morphism τi : Ei → ˜Li. This is surjective as L is of rank 1 and τ := (τi) give a
+nonzero morphism of stratified bundles
+E• → ˜L•
+where ˜L• := h∗(ρ∗(L•)Sn). Let κ• be the kernel of the morphism E• → ˜L•. Then T (κ•) = K•. Thus, by
+the induction hypothesis on rank, there exists a stratified subbundle E′
+• ⊂ κ• ⊂ E• such that T (E′
+•) = F′
+•.
+□
+6.2. Closed immersion. We begin by recalling a result from [PS20].
+Let p ∈ S(n) be a point of type (n1, n2, . . . , nr). Let p′
+i, for i = 1, 2, . . . m be the points in the fiber h−1(p).
+Let A be the local ring OS(n),p and B be the semilocal ring OSn ⊗OS(n) A. Then B is a finite A module and
+BSn = A.
+Lemma 6.1. When char k > n1, any Sn-equivariant surjective B-module homomorphism f : M → N of
+finitely generated B modules descends to surjective A-module homomorphism of the Sn-invariants M Sn →
+N Sn
+This allows us to prove the following analogue of Proposition 5.3.6 in [PS20].
+Proposition. Let E• = (Ei, αi) be a Sn-equivariant stratified bundle on Sn
+(1) Let p ∈ S(n) be a point of type (n1, n2, . . . , nr). If char k > n1, then the sheaf ρ∗ESn
+i
+is locally free
+in a neighbourhood of p for all i.
+(2) Let U denote the largest open subset where ρ∗ESn
+i
+is locally free, then on ρ−1(U), the natural mor-
+phism
+ρ∗ρ∗ESn
+i
+→ Ei
+is an isomorphism for all i ∈ N
+Proof. The first assertion is proved by induction on the rank of E•. If E• is a Sn-equivariant stratified bundle
+of rank 1, then by proposition 1, ρ∗ESn
+i
+is locally free on S(n) for all i. In general, as E• corresponds to a
+representation of the abelian group scheme πalg(S, x)ab, there exists a Sn-equivariant short exact sequence
+of locally free sheaves on Sn
+0 → K• → E• → L• → 0
+Pushing forward by ρ and taking Sn-invariants we get the exact sequence for all i
+0 → ρ(Ki)Sn → ρ(Ei)Sn → ρ(Li)Sn
+We claim that the homomorphism on the right is surjective in the neighbourhood of a point p of type
+(n1, n2, . . . , nr). Surjectivity can be checked after passing to a formal neighbourhood of p and thus reduces to
+lemma 6.1. By induction hypothesis on rank, both ρ(Ki)Sn and ρ(Li)Sn are locally free on a neighbourhood
+of p and hence so is ρ(Ei)Sn.
+11
+
+The second assertion follows from the observation that the natural homomorphism
+ρ∗ρ∗ESn
+i
+→ Ei
+is an isomorphism on ρ−1(S(n)
+◦
+) as as ρ : Sn
+◦ → S(n)
+◦
+is finite ´etale. As the complement of Sn
+◦ in ρ−1(U) is of
+codimension ≥ 2 and both sheaves are locally free on ρ−1(U), thus the natural morphism is an isomorphism.
+□
+Proposition 6. Let char k > 3. The homomorphism
+f : πalg(S, x)ab → πalg(S[n], ˜
+nx)
+is faithfully flat.
+Proof. By [[DM82], Theorem 2.21], it is enough to show that the functor
+T : S(S[n]) → S(Sn)
+is essentially surjective. Thus we want to show that for any Sn-equivariant stratified bundle E• on Sn, there
+exists a stratified bundle F• on S[n] such that T (F•) = E•.
+Let U be the open subset of S(n) consisting of points of type (1, 1, . . . , 1), (2, 1, . . . , 1), (3, 1, . . ., 1) and
+(2, 1, 1, . . ., 1). By assumption on characteristic of k and the previous proposition, we get that ρ∗ESn
+i
+is
+locally free on U. Also we have on ρ−1(U), the natural morphism
+ρ∗ρ∗ESn
+i
+→ Ei
+is an isomorphism. Imitating proposition 1 above, this allows us to define a stratified bundle (ρ∗ESn
+i
+, βi) on
+U such that ρ∗(ρ∗ESn
+i
+, βi) ≃ E•. Pulling back under h to h−1(U) (whose complement in S[n] has codimension
+≥ 3) and extending to S[n], we get a stratified bundle F• such that T (F•) = E•
+□
+As f is both faithfully flat and a closed immersion, we get the following theorem
+Theorem 6.2. Let char k > 3. The homomorphism
+f : πalg(S, x)ab → πalg(S[n], ˜
+nx)
+is an isomorphism.
+References
+[BHdS21] Biswas, Indranil, Ph`ung Hˆo Hai, and Jo˜ao Pedro Dos Santos. ”On the fundamental group schemes of certain quotient
+varieties.” Tohoku Mathematical Journal 73, no. 4 (2021): 565-595.
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+bundle over a curve.” Duke Mathematical Journal 132, no. 1 (2006): 1-48.
+[DM82] Deligne, Pierre; Milne, James (1982), ”Tannakian categories”, in Deligne, Pierre; Milne, James; Ogus, Arthur; Shih,
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+[dS07] Dos Santos, Jo˜ao Pedro Pinto. ”Fundamental group schemes for stratified sheaves.” Journal of Algebra 317, no. 2 (2007):
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+23, no. 1 (1983): 73-83.
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+[PS20] Paul, Arjun, and Ronnie Sebastian. ”Fundamental group schemes of Hilbert scheme of n points on a smooth projective
+surface.” Bulletin des Sciences Math´ematiques 164 (2020): 102898.
+12
+
+[SGA1] Grothendieck, Alexander, and Michele Raynaud. ”Revˆetements´etales et groupe fondamental (SGA 1).” arXiv preprint
+math/0206203 (2002).
+The Institute of Mathematical Sciences, (HBNI), Chennai 600113.
+Email address: [email protected]
+13
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='02842v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='AG]  7 Jan 2023  STRATIFIED BUNDLES ON THE HILBERT SCHEME OF n POINTS  SAURAV HOLME CHOUDHURY  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let k be an algebraically closed field of characteristic p > 3 and S be a smooth projective  surface over k with k-rational point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' For n ≥ 2, let S[n] denote the Hilbert scheme of n points on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In  this note, we compute the fundamental group scheme πalg(S[n], ˜  nx) defined by the Tannakian category of  stratified bundles on S[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Introduction  For a variety X over C, one has the classical notion of the fundamental group π1(Xan, x) defined using  the analytic topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Over arbitrary base fields k, one has several analogues of the fundamental group  defined in terms of algebro-geometric information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  In [SGA1], Grothendieck introduced the notion of ´etale fundamental group π´et(X, x), where X is a scheme  and x is a geometric point of X, in terms of the finite etale covers of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In [N76], Nori defined the Nori  fundamental group scheme πN(X, x), where X is a connected, reduced and complete scheme over a perfect  field k and x is a k-rational point, via Tannakian reconstruction using the category of essentially finite vector  bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The definition of πN(X, x) was extended to the case of connected and reduced k-schemes in  [N82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Another analogue, the S-fundamental group scheme πS(X, x) was introduced and studied by Langer  in [Lan11] and [Lan12] for smooth projective varieties X over an algebraically closed field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' It is defined via  Tannakian reconstruction using the category of numerically flat vector bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The S-fundamental  group scheme for a smooth projective curve C over an algebraically closed field k was already introduced  and studied in [BPS06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The variant of the fundamental group scheme which is of prime importance in this note is the algebraic  fundamental group πalg(X, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In [Gie75], Gieseker defined πalg(X, x) as the fundamental group scheme  corresponding to the Tannakian category of DX-modules, where DX is the sheaf of differential operators on  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' For X smooth over a field of positive characteristic, Gieseker introduced the notion of stratified bundles  and showed that the category of DX-modules is tensor equivalent to the category of stratified bundles on  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Stratifed bundles were further studied in [dS07] and [BHdS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Precise definitions and statements will  be given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let S be a smooth projective surface over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' For n ≥ 2, let S[n] denote the Hilbert scheme n points on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  It is well known that S[n] is a smooth projective variety of dimension 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In [PS20], the authors show that  for char k > 3 and n ≥ 2, there is an isomorphism of affine group schemes over k  π†(S, x)ab → π†(S[n], ˜  nx)  where † = S, N or ´et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  In this note, we extend their results to the case of πalg and prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let char k > 3 and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' There is an isomorphism of affine group schemes over k  f : πalg(S, x)ab → πalg(S[n], ˜  nx)  In section 2, we recall the definition of stratified bundles and some of their basic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The formalism  of Tannakian reconstruction is recalled in section 3 and used to define the algebraic fundamental group  πalg(X, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  1  The geometrical properties of the Hilbert scheme of n points on a smooth projective surface are in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  In section 5, we prove a result about descent of stratified bundles which allows us to define the homomorphism  f by defining the associated functor of Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The concluding section 6 establishes the main  theorem by showing that f is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We would like to thank Indranil Biswas and Ronnie Sebastian for their comments  on earlier drafts of this note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Stratified bundles  Let k be a field of characteristic p and X be a noetherian scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Stratified bundles on X are  sequences of coherent sheaves on X satisfying infinite Frobenius descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' More precisely, the category of  stratified bundles on X, denoted S(X), consists of  Objects (Ei, αi) are sequences of coherent OX-modules Ei, i ∈ N along with isomorphisms  αi : F ∗Ei+1 → Ei  for all i ∈ N, where F is the absolute Frobenius on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Morphisms φ : (Ei, αi) → (Fi, βi) consists of a sequence of OX-module morphisms φi : Ei → Fi  such that φi ◦ αi = βi ◦ F ∗ (φi+1)  Let f : Y → X be a morphism and (Ei, αi) be a stratified bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then we can define the pullback  along f, denoted f ∗(Ei, αi), as consisting of the sequence of OY coherent sheaves f ∗Ei and isomorphisms  are given by the composite maps  F ∗f ∗Ei+1  γEi+1  −−−→ f ∗F ∗Ei+1  f ∗(αi)  −−−−→ f ∗Ei  where γ : F ∗f ∗ → f ∗F ∗ is the natural isomorphism of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Thus S(X) is contravariant functor in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' One also has a tensor product on S(X) defined by taking term  by term tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Also S(X) is an abelian category [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' [BHdS21], Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  We recall some well known results about stratified bundles [cf [dS07], [Gie75]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' If (Ei, αi) is a stratified bundle on X, then Ei is a locally free OX-module for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  This allows us to define duals of stratified bundles, making S(X) into an abelian rigid tensor category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The rank of a stratified bundle (Ei, αi) is defined to be the rank of E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The trivial stratified bundles on X  are of the form ⊕(OX, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' F ∗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let DX be the sheaf of differential operators on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The category of DX modules consists of  Objects coherent OX modules E equipped with a DX action i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='e a morphism of OX-algebras  DX → Endk(E)  Morphisms OX-linear maps E → F compatible with the DX action  A theorem of Katz [[Gie75], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='3] shows that for X smooth over k, then the category of stratified  bundles on X and the category of DX modules are tensor equivalent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  We close this section with the definition of G equivariant stratified bundles on a variety X admitting  action of a group G on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' A stratified bundle (Ei, αi) is said to be a G-equivariant stratified bundle if Ei are G-equivariant  vector bundles and αi are G-equivariant OX module morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Tannakian categories and fundamental group schemes  In this section we recall the definition and basic properties of Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  We then recall  Gieseker’s definition of the fundamental group scheme πalg using the Tannakian formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Tannakian Categories and affine group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Tannakian categories were defined and studied  in [DM82] to formalize the properties of Repk(G), the category of finite dimensional k-representations of G,  an affine group scheme over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Definition (Neutral Tannakian Categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' A rigid abelian tensor category C with End I = k is a neutral  Tannakian category if it admits an exact faithful k-linear tensor functor ω : C → Veck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Any such functor is  said to be a fiber functor for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Given a neutral Tannakian category (C, ⊗, ω, I), we define the functor Aut⊗(ω) : k − algebra → Sets  such that for k-algebra R, Aut⊗(ω)(R) consists of the families (λX) for X ∈ ob(C), where λX is a R-linear  automorphism of X ⊗ R such that λX1⊗X2 = λX1 ⊗ λX2, λI = idR, and  λY ◦ (α ⊗ 1) = (α ⊗ 1) ◦ λX : X ⊗ R → Y ⊗ R  for all morphisms α : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Theorem (Main theorem for neutral Tannakian categories, [DM82], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let (C, ⊗) be a rigid  abelian tensor category such that k = End(I) and let ω : C → Veck be an exact faithful tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then  The functor Aut⊗(ω) of k-algebras is represented by an affine group scheme G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The functor C → Repk(G) is an equivalence of tensor categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let (C, ⊗, ω, I) and (C′, ⊗, ω′, I′) be neutral Tannakian categories which correspond to the repre-  sentation categories of the affine k group schemes G and G′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then any functor of Tannakian  categories from C → C′ is induced by a unique morphism of affine k group schemes G′ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  This theorem allows us to define many variants of fundamental groups of a scheme X by considering  different Tannakian categories naturally associated with X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The following result is very useful in establishing  a given morphism between affine group schemes is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Theorem ([DM82], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let f : G → G′ be a homomorphism of group schemes over k and  Rep (f) : Rep (G′) → Rep (G) be the corresponding functor of Tannakian categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then  f is faithfully flat if and only if Rep (f) is fully faithful and has essential image closed under subobjects  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='e for V ′ ∈ Rep (G′) and suboject W ⊂ Rep (f)(V ′), there is a subobject W ′ ⊂ V ′ in Rep (G′) such  that Rep (f)(W ′) ≃ W in Rep (G)  f is closed immersion if and only if every object of Rep (G) is a subquotient of some object in the  essential image of Rep (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  We finish by recalling a basic result on affine group schemes (we refer to section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1 in [PS20] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let G be a affine group scheme over k, Gab be its abelianization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='e the maximal abelian quotient of G)  and α : G → Gab be the (faithfully flat) quotient morphism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We can then define the composite morphism  φ : Gn  αn  −−→ Gn  ab  m  −→ Gab  where m is the multiplication homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As Sn acts on the k-group scheme Gn, we can define the  notion of a Sn-invariant group morphism ψ : Gn → H for any k-group scheme H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let G and H be two group schemes over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' For an integer n ≥ 2, the set of Sn-invariant  group morphisms Gn → H is in bijective correspondence with the set of group morphism Gab → H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='e any  morphism of k-group schemes φ : Gn → H which is Sn-invariant factors uniquely through a morphism  ψ : Gab → H such that φ = ψ ◦ h  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The group scheme πalg(X, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Classically, over C, the Riemann-Hilbert correspondence identifies the  category of vector bundles equipped with integrable connections on a smooth connected projective variety  X/C with the category of representations of the topological fundamental group πtop(X, x) for some chosen  base point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Via GAGA, this gives a purely algebraic description of the category of representations of the  topological fundamental group π(X, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' This category (equipped with the fiber functor (E, ∇) → Ex) is  a neutral Tannakian category and can be identified, via the Tannakian formalism, with the representation  3  category of the proalgebraic completion of the topological fundamental group, denoted as πtop(X, x)alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Over a field k of characteristic 0, the category of flat connections on a smooth variety X is tensor equiva-  lent to the category of DX-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' However over a field of characteristic p, the category of flat connections  on X is not as well behaved as the category of DX-modules and one defines a fundamental group scheme for  X by Tannakian formalism using the category of DX-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' By Katz’s theorem mentioned before, the  fundamental group coincides with the one defined using S(X) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let x ∈ X(k) be a k-rational point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then the abelian rigid tensor category S(X) is neutralized by the  fiber functor  Tx : S(X) → V eck  The fundamental group scheme defined by the neutral Tannakian category (S(X), ⊗, Tx, (OX, F ∗)) is  called the algebraic fundamental group of X based at x and is denoted by πalg(X, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The following basic properties of πalg are well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  (Independence of basepoint) Let X be a geometrically connected, smooth projective k-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then  for all x1, x2 ∈ X(k), one has  πalg(X, x1) ≃ πalg(X, x2)  (Product rule) For X1, X2 geometrically connected and smooth over k and xi ∈ Xi(k), there is an  isomorphism  πalg(X1, x1) × πalg(X2, x2) → πalg(X1 × X2, (x1, x2))  For X smooth and open immersion U  i−→ X such that the complement of U in X has codimension  ≥ 2 and x ∈ U(k), then the homomorphism  πalg(U, x) → πalg(X, x)  associated to the restriction functor i∗ : S(X) → S(U) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Geometry of Hilbert Scheme of points  Let S be a smooth projective surface over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We fix notation as follows  Sn denotes the n-fold cartesian product of S with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  S(n) denotes the nth symmetric product of S defined as the quotient Sn/Sn, where Sn denotes the  symmetric group on n letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  S[n] denotes the Hilbert scheme of n points on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let ρ : Sn → S(n) be the quotient map and h : S[n] → S(n) be the Hilbert-Chow morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We write  S(n) for the open subset of S(n) consisting of distinct points with S[n] := h−1(S(n) ) and Sn  := ρ−1(S(n) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The map hn,◦ : S[n] → S(n) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We have the diagram:  S[n]  Sn  S(n)  hn  ρn  In general, Hilbert schemes of points on a projective variety display a lot of pathological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' But in  [Fog68] the author shows that, in the case of smooth projective surface S, S[n] is a smooth projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Thus, in this case, the Hilbert-Chow morphism h : S[n] → S(n) is a resolution of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  4  One can consider S(n) as the set of effective 0-cycles of degree n on S(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In this case it is easy to see that  S(n) admits a stratification by type, where the type of a 0-cycle y of degree n is a tuple (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr) where y  can be written as  y = Σr  j=1njxj  where xj are distinct points of S with multiplicities n1 ≥ n2 ≥ · · · ≥ nr, where nj are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let C(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr) denote the subset of S(n) consisting of points of the type (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let S(n)  ∗  =  C(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1) � C(2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1) denote the open subset of S(n) consisting of points of type (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1) and (2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let S[n]  ∗  and Sn  ∗ denote the preimage of S(n)  ∗  under h and ρ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  We recall some basic properties below which we will need later (we refer to [Fog68], [PS20] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The subsets C(n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr) are nonsingular of dimension 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The closed subset S(n) \\ S(n)  ∗  is of codimension ≥ 2 in S(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The closed subset S[n] \\ S[n]  ∗  is of codimension 2 in S[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The closed subset Sn \\ Sn  ∗ is of codimension ≥ 4 in Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The closed subset S(n)  ∗  \\ S(n) is of codimension 2 in S(n)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  When characteristic of k ̸= 2, for y ∈ C(2, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1), the scheme theoretic fiber h−1(y) is isomorphic  to P1  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In fact, S[n]  ∗  is the blowup of S(n)  ∗  along C(2, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  We end this section by recalling a result of Fogarty ([Fog77], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' If L is a Sn-invariant line bundle on Sn, there exists a line bundle L′ on S(n) such that  h∗L′ ≃ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  It follows that L′ in the proposition is isomorphic to σ∗(L)Sn  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The functor between Tannakian categories  Let S be a smooth projective surface over k and (Ei, αi) be a stratified bundle on S[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Restricting to S[n]  ∗  gives us a functor  i∗ : S(S[n]) → S(S[n]  ∗ )  which is a equivalence of categories as S[n]  ∗  is the complement of a codimension 2 closed subset of S[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Next we show that a stratified bundle on S[n]  ∗  can be pushed forward under h to get a stratified bundle on  S(n)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' First we begin by a result on descent of vector bundles along the morphism h : S[n]  ∗  → S(n)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Similar  results have been established by authors in [Ish83] and [PS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Assume char k ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let E be a vector bundle on S[n]  ∗  which restricts to trivial vector bundles  on the fibers of h over S(n)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then h∗E is a locally free OS(n)  ∗ -module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Moreover the natural map  h∗h∗(E) → E  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let x ∈ S(n)  ∗  be a point of type (2, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then by assumption, the fiber of h over x is isomorphic  to P1  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let J be the ideal sheaf of the closed subscheme h−1(x) and Ix be the ideal sheaf of the closed point  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We have  J = IxOS[n]  ∗  For all n ≥ 1, let Yn denote the closed subscheme of S[n]  ∗  corresponding to the ideal sheaf J n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Consider  the following short exact sequence of sheaves on S[n]  ∗  0 → J ⊗ E → E → E|Y1 → 0  5  Pushing forward by h, we get the following exact sequence of sheaves on S(n)  ∗  h∗E → H0(Y1, E|Y1) → R1h∗(J ⊗ E)  We claim that the completion of R1h∗(J ⊗ E) at the maximal ideal mx in OS(n)  ∗  ,x is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The proof uses  the theorem of formal functions which says that  (R1h∗(J ⊗ E))∧ ≃ lim  ←− H1(Yn, J ⊗ E ⊗ OS[n]  ∗ /J n)  We prove by induction that H1(Yn, J ⊗E ⊗OS[n]  ∗ /J n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As Y1 ≃ P1  k, the sheaves J n/J n+1 are locally  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' These sheaves are also globally generated over Y1 as we have the surjection  mn  x/mn+1  x  ⊗O  S(n)  ∗  ,x OS[n]  ∗  ≃ In  x /In+1  x  ⊗O  S(n)  ∗  OS[n]  ∗  ։ J n/J n+1  As J n/J n+1 is locally free on Y1 ≃ P1  k and globally generated, it is a direct sum of line bundles each of  which has degree ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus one gets the base case of induction from degree considerations, as  H1(Y1, J ⊗ E ⊗ OS[n]  ∗ /J = H1(Y1, J /J 2 ⊗ EY1) = 0  Assume that the claim is true for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then the proof for n + 1 follows from the long exact sequence in  cohomology attached to the short exact sequence of sheaves on Yn+1  0 → J n+1/J n+2 ⊗ E → J /J n+2 ⊗ E → J /J n+1 ⊗ E → 0  which gives us the exact sequence  H1(Yn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J n+1/J n+2 ⊗ E) → H1(Yn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J /J n+2 ⊗ E) → H1(Yn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J /J n+1 ⊗ E)  We know H1(Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J n+1/J n+2⊗E) = H1(Y1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J n+1/J n+2⊗E) = 0 (by degree consideration) and H1(Yn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J /J n+1⊗  E) = H1(Yn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J /J n+1 ⊗ E) = 0 (by induction hypothesis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' thus we get  H1(Yn+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' J /J n+2 ⊗ E) = 0  Thus the stalk of R1h∗(J ⊗ E) at x is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  This shows that the natural map h∗E → H0(Y1, E|Y1) is surjective in a neighbourhood of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=', fr  be a basis of H0(Y1, E|Y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let Spec(R) be an affine neighbourhood of x where the natural map is surjective  and let ˜fi ∈ Γ(Spec(R), h∗E) = Γ(h−1(Spec(R)), E) be lifts of fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Using ˜fi one defines a homomorphism  O⊕r  S[n]  ∗ |h−1(Spec(R)) → E  on h−1(Spec(R) which is a surjection (and hence an isomorphism) on Y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As h is proper, there exists a  smaller affine neighbourhood U of x over which there is an isomorphism  O⊕r  V  ≃ E  where V = h−1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Applying h∗, we get  (h∗OV )⊕r ≃ h∗E  As S(n)  ∗  is normal and h : S[n]  ∗  → S(n)  ∗  is birational with connected fibers, by a form of Zariski’s main  theorem [cf [Har77], Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='3 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='4], we have that h∗OV ≃ OU and thus h∗E is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The  natural morphism  h∗h∗(E) → E  is clearly an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  □  6  Let VBS(n)  ∗  be the category of locally free sheaves on S(n)  ∗  and VBh  S[n]  ∗  be the category of locally free  sheaves on S[n]  ∗  which restrict to trivial vector bundles on the fibers of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Proposition 1 above gives us an  equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Assume char k ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The pushforward functor  h∗ : VBh  S[n]  ∗  → VBS(n)  ∗  is an equivalence of categories with the inverse given by  h∗ : VBS(n)  ∗  → VBh  S[n]  ∗  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We observe that if E′ ≃ h∗(E), then E ≃ h∗E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' This shows that h∗ is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The  natural map  HomS(n)  ∗ (h∗E, h∗F) → HomS[n]  ∗ (E, F)  is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus h∗ is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  □  Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' For all E ∈ VBh  S[n]  ∗ , the natural map  F ∗h∗(E) → h∗F ∗(E)  is an isomorphism over S(n)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As F ∗E is also an object of VBh  S[n]  ∗ , thus both sheaves are locally free of the same rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus it  suffices to show that the natural map  F ∗h∗(E) → h∗F ∗(E)  is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As F is faithfully flat on the smooth locus of S(n)  ∗  , the claim holds on the smooth locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let  x ∈ S(n)  ∗  be of type (2, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then the restriction of F ∗h∗(E) to x is naturally isomorphic to H0(Y1, E|Y1)  and the restriction of h∗F ∗(E) to x is H0(Y1, F ∗(E|Y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The restriction of the natural map to x is the map  F ∗ : H0(Y1, E1) → H0(Y1, F ∗E1)  which is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  □  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='2 of [Gie75], we have that every stratified bundle on P1  k is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus the above results  give us  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Assume char k ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let (Ei, αi) be a stratified bundle on S[n]  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then h∗(Ei) is locally free  OS(n)  ∗ -module for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Moreover the natural map  h∗h∗(Ei) → Ei  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Furthermore the natural map  F ∗h∗(Ei) → h∗F ∗(Ei)  is an isomorphism over S(n)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  This allows us to define the pushforward of a stratified bundle (Ei, αi) on S[n]  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The pushforward denoted  h∗(Ei, αi) is given by the sequence of vector bundles h∗Ei for all i ∈ N and the isomorphisms are given by  the composite  F ∗h∗(Ei+1)  ηEi+1  −−−→ h∗F ∗(Ei+1)  h∗(αi)  −−−−→ h∗(Ei)  where η : F ∗h∗ → h∗F ∗ is the natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  7  Thus we get a functor  h∗ : S(S[n]  ∗ ) → S(S(n)  ∗  )  h∗ is additive tensor functor as on the smooth locus S(n) we have the isomorphisms  h∗((Ei, αi) ⊕ (Fi, βi))|S(n) ≃ h∗(Ei, αi)|S(n) ⊕ h∗(Fi, βi))|S(n) h∗((Ei, αi) ⊗ (Fi, βi))|S(n) ≃ h∗(Ei, αi)|S(n) ⊗ h∗(Fi, βi))|S(n) which extend to S(n)  ∗  due to codimension reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The following commutative diagram shows that h∗h∗(Ei, αi) is isomorphic to (Ei, αi) as stratified bundles  with the isomorphism given by the natural morphisms h∗h∗Ei → Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  F ∗h∗h∗Ei+1  h∗F ∗h∗Ei+1  h∗h∗F ∗Ei+1  h∗h∗Ei  F ∗Ei+1  F ∗Ei+1  Ei  h∗ηEi+1  h∗h∗αi  αi  γh∗Ei+1  =  Consider the pullback functor  ρ∗ : S(S(n)  ∗  ) → S(Sn  ∗ )  which takes values in the category of Sn-equivariant stratified bundles on Sn  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Also we have the extension  functor  j∗ : S(Sn  ∗ ) → S(Sn)  which is an equivalence of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Composing these functors together, we get a functor  T : S(S[n]) → S(Sn)  given by  T = j∗ ◦ ρ∗ ◦ h∗ ◦ i∗  Clearly T is an additive tensor functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Note that h∗ is fully faithful, ρ∗ : S(S(n)  ∗  ) → S(Sn  ∗ ) is fully faithful  (as ρ : Sn  → S(n) is finite ´etale) and j∗ : S(Sn  ∗ ) → S(Sn) is an equivalence of categories (due to codimension  reasons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus T is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Fix n distinct k-valued points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , xk ∈ S(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let ˜x ∈ S[n] such that  h(˜x) = σ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , xn) = z ∈ S(n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then the categories S(S[n]) and S(Sn) are neutralized by the respective  fiber functors  τ˜x : S(S[n]) → Veck  (Ei, αi) �→ (E0)˜x  τ(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=',xn) : S(Sn) → Veck  (Fi, βi) �→ (F0)(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=',xn)  If T ((Ei, αi)) = (Fi, βi) that we have natural isomorphisms (E0)˜x ≃ h∗(E0)z ≃ (F0)(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=',xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Thus we have a functor of Tannakian categories  T : (S(S[n]), ⊗, τ˜x, (OS[n], d)) → (S(Sn), ⊗, τ(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=',xn), (OSn, d))  8  which by the independence of basepoint property of S induces a functor of Tannakian categories  T : (S(S[n]), ⊗, τ ˜  nx, (OS[n], d)) → (S(Sn), ⊗, τ(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=',x), (OSn, d))  and hence a morphisms of the associated fundamental group schemes  ˜f : πalg(Sn, (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , x)) → πalg(S[n], ˜  nx)  Note that by proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='2 we have  πalg(Sn, (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , x)) ≃ πalg(S, x)n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  As  T : (S(S[n]), ⊗, T ˜  nx, (OS[n], d)) → (S(Sn), ⊗, T(x,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=',x), (OSn, d))  takes stratified bundles on S[n] to Sn-equivariant stratified bundles on Sn and a Sn-equivariant stratified  bundles on Sn corresponds to a Sn-invariant representation of πalg(S, x)n, by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1, ˜f factors uniquely through  f : πalg(S, x)ab → πalg(S[n], ˜  nx)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Isomorphism of fundamental group schemes  In this section, we show that f is an isomorphism of affine group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We begin by proving a result  about Sn-equivariant stratified line bundles on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let (Li, αi) be a Sn-equivariant stratified line bundles on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then there exists a stratified  line bundle (Li, βi) such that ρ∗(Li, βi) ≃ (Li, αi)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' By Fogarty’s result mentioned above, for any Sn-equivariant line bundle Li there exists line bundle  Li ≃ ρ∗LSn  i  such that ρ∗Li ≃ Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Pushing forward αi and taking Sn invariants we get the isomorphism  ρ∗(F ∗Li+1)Sn  ρ∗(αi)Sn  −−−−−−→ ρ∗(Li)Sn  We show that the natural homomorphism  F ∗(ρ∗(Li)Sn) → (F ∗ρ∗(Li)Sn) → (ρ∗F ∗(Li)Sn)  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Pulling back the morphism under ρ, we get the commutative diagram  ρ∗F ∗((ρ∗Li)Sn)  ρ∗((ρ∗F ∗Li)Sn)  F ∗Li  F ∗Li  =  where the vertical morphisms are the natural morphism which are isomorphisms by Fogarty’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  By pushing forward under ρ and taking Sn invariants we get that  F ∗(ρ∗(Li)Sn) → (ρ∗F ∗(Li)Sn)  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We define βi to be the composite isomorphism  F ∗(ρ∗(Li)Sn) → (ρ∗F ∗(Li)Sn)  ρ∗(αi)Sn  −−−−−−→ ρ∗(Li)Sn  The commutative diagram also gives us that ρ∗(Li, βi) ≃ (Li, αi)  □  9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Next we show that the morphism f is faithfully flat  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The homomorphism  f : πalg(S, x)ab → πalg(S[n], ˜  nx)  is faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' By [[DM82] Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='21], this is equivalent to showing that the functor  T : S(S[n]) → S(Sn)  is fully faithful and the essential image of T is closed under taking subobjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We already know that T  is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let E• = (Ei, αi) be a stratified bundle on S[n] and F• := T (E•) be the corresponding  Sn-equivariant stratified bundle on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' If F′  ⊂ F• is a Sn-equivariant stratified subbundle, then we need  to show there exists E′  ⊂ E• such that T (E′  ) = F′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  The proof proceeds by induction on the rank of E•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' If rank E• = 1, the proof is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let rank  E• ≥ 2  Then the stratified bundles F• and F′  correspond to the representations  πalg(Sn, (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , x) → πalg(S, x)ab → GL(V )  and  πalg(Sn, (x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , x) → πalg(S, x)ab → GL(V ′)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  As πalg(S, x)ab is an abelian affine group scheme over k, all its irreducible representations are one dimen-  sional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus one gets that the πalg(S, x)ab-module V/V ′ has a one dimensional quotient W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus there is  a πalg(S, x)ab-module surjection V → W such that the kernel contains V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let L• be the Sn-equivariant  stratified bundle corresponding to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus we have a short exact sequence of Sn-equivariant stratified  bundles  0 → K• → F• → L• → 0  where F′  ⊂ K•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  By proposition 1 above, we know that Li := ρ∗LSn  i  is a line bundle on S(n) and ρ∗Li = Li  We claim that the following complex of sheaves on S(n)  ∗  is exact for all i ∈ N  (1)  0 → (ρ∗Ki)Sn|S(n)  ∗  → (ρ∗Fi)Sn|S(n)  ∗  → (ρ∗Li)Sn|S(n)  ∗  → 0  It is enough to show that (ρ∗Fi)Sn|S(n)  ∗  → (ρ∗Li)Sn|S(n)  ∗  is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We note that (ρ∗Fi)Sn|S(n)  ∗  =  h∗(Ei|S[n]  ∗ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let C be the cokernel  h∗(Ei|S[n]  ∗ ) → (ρ∗Li)Sn|S(n)  ∗  → C → 0  Pulling back under ρ, we get the following commutative diagram on Sn  ∗  ρ∗h∗(Ei|S[n]  ∗ )  ρ∗((ρ∗Li)Sn|S(n)  ∗ )  ρ∗C  0  Fi  Li|Sn  ∗  0  =  =  10  The rows are exact and hence ρ∗C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As ρ is surjective, this implies C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus Ki := (ρ∗Ki)Sn|S(n)  ∗  is locally free on S(n)  ∗  Pulling back the exact sequence (1) under h, we get a short exact sequence of locally free sheaves on S[n]  ∗  0 → h∗Ki|S[n]  ∗  → Ei|S[n]  ∗  → ˜Li|S[n]  ∗  → 0  where ˜Li := h∗Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  As the complement of S[n]  ∗  in S[n] is of codimension ≥ 2 and Ei, L are locally free, the surjective morphism  Ei|S[n]  ∗  → ˜Li|S[n]  ∗  extends to a unique morphism τi : Ei → ˜Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' This is surjective as L is of rank 1 and τ := (τi) give a  nonzero morphism of stratified bundles  E• → ˜L•  where ˜L• := h∗(ρ∗(L•)Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let κ• be the kernel of the morphism E• → ˜L•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then T (κ•) = K•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus, by  the induction hypothesis on rank, there exists a stratified subbundle E′  ⊂ κ• ⊂ E• such that T (E′  ) = F′  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Closed immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' We begin by recalling a result from [PS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let p ∈ S(n) be a point of type (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let p′  i, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' m be the points in the fiber h−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let A be the local ring OS(n),p and B be the semilocal ring OSn ⊗OS(n) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Then B is a finite A module and  BSn = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' When char k > n1, any Sn-equivariant surjective B-module homomorphism f : M → N of  finitely generated B modules descends to surjective A-module homomorphism of the Sn-invariants M Sn →  N Sn  This allows us to prove the following analogue of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='6 in [PS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let E• = (Ei, αi) be a Sn-equivariant stratified bundle on Sn  (1) Let p ∈ S(n) be a point of type (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' If char k > n1, then the sheaf ρ∗ESn  i  is locally free  in a neighbourhood of p for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  (2) Let U denote the largest open subset where ρ∗ESn  i  is locally free, then on ρ−1(U), the natural mor-  phism  ρ∗ρ∗ESn  i  → Ei  is an isomorphism for all i ∈ N  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The first assertion is proved by induction on the rank of E•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' If E• is a Sn-equivariant stratified bundle  of rank 1, then by proposition 1, ρ∗ESn  i  is locally free on S(n) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' In general, as E• corresponds to a  representation of the abelian group scheme πalg(S, x)ab, there exists a Sn-equivariant short exact sequence  of locally free sheaves on Sn  0 → K• → E• → L• → 0  Pushing forward by ρ and taking Sn-invariants we get the exact sequence for all i  0 → ρ(Ki)Sn → ρ(Ei)Sn → ρ(Li)Sn  We claim that the homomorphism on the right is surjective in the neighbourhood of a point p of type  (n1, n2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Surjectivity can be checked after passing to a formal neighbourhood of p and thus reduces to  lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' By induction hypothesis on rank, both ρ(Ki)Sn and ρ(Li)Sn are locally free on a neighbourhood  of p and hence so is ρ(Ei)Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  11  The second assertion follows from the observation that the natural homomorphism  ρ∗ρ∗ESn  i  → Ei  is an isomorphism on ρ−1(S(n) ) as as ρ : Sn  → S(n) is finite ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' As the complement of Sn  in ρ−1(U) is of  codimension ≥ 2 and both sheaves are locally free on ρ−1(U), thus the natural morphism is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  □  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let char k > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The homomorphism  f : πalg(S, x)ab → πalg(S[n], ˜  nx)  is faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' By [[DM82], Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='21], it is enough to show that the functor  T : S(S[n]) → S(Sn)  is essentially surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Thus we want to show that for any Sn-equivariant stratified bundle E• on Sn, there  exists a stratified bundle F• on S[n] such that T (F•) = E•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Let U be the open subset of S(n) consisting of points of type (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1), (2, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' , 1), (3, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=', 1) and  (2, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' By assumption on characteristic of k and the previous proposition, we get that ρ∗ESn  i  is  locally free on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Also we have on ρ−1(U), the natural morphism  ρ∗ρ∗ESn  i  → Ei  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Imitating proposition 1 above, this allows us to define a stratified bundle (ρ∗ESn  i  , βi) on  U such that ρ∗(ρ∗ESn  i  , βi) ≃ E•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Pulling back under h to h−1(U) (whose complement in S[n] has codimension  ≥ 3) and extending to S[n], we get a stratified bundle F• such that T (F•) = E•  □  As f is both faithfully flat and a closed immersion, we get the following theorem  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' Let char k > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content=' The homomorphism  f : πalg(S, x)ab → πalg(S[n], ˜  nx)  is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  References  [BHdS21] Biswas, Indranil, Ph`ung Hˆo Hai, and Jo˜ao Pedro Dos Santos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
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+page_content=' 4 (2021): 565-595.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
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+page_content='  The Institute of Mathematical Sciences, (HBNI), Chennai 600113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='  Email address: sauravhc@imsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}
+page_content='in  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQfAwIb/content/2301.02842v1.pdf'}

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+Adversarial Networks and Machine Learning for
+File Classification
+Ken St. Germain1, Josh Angichiodo
+Department of Cyber Science
+United States Naval Academy
+Annapolis, MD
+[email protected]
+Abstract—Correctly identifying the type of file under exam-
+ination is a critical part of a forensic investigation. The file
+type alone suggests the embedded content, such as a picture,
+video, manuscript, spreadsheet, etc. In cases where a system
+owner might desire to keep their files inaccessible or file type
+concealed, we propose using an adversarially-trained machine
+learning neural network to determine a file’s true type even
+if the extension or file header is obfuscated to complicate its
+discovery. Our semi-supervised generative adversarial network
+(SGAN) achieved 97.6% accuracy in classifying files across
+11 different types. We also compared our network against a
+traditional standalone neural network and three other machine
+learning algorithms. The adversarially-trained network proved
+to be the most precise file classifier especially in scenarios
+with few supervised samples available. Our implementation of
+a file classifier using an SGAN is implemented on GitHub
+(https://ksaintg.github.io/SGAN-File-Classier/).
+I. INTRODUCTION
+M learning can be used to determine file types based on
+a file’s byte value distribution. In this work, we introduce an
+adversarial learning approach to accurately identify file types
+regardless of file extension, headers, or footers. By inspecting
+the histogram-based distribution of byte values in a file, we
+can greatly reduce the time and effort expended by subject
+matter experts during the course of a forensic investigation.
+Machine learning algorithms are designed to extract relevant
+information from data [1], and the field of deep learning has
+been shown effective in solving classification problems [2]. In
+this paper we use a generative adversarial network (GAN) to
+determine the type of file under investigation. Specifically, we
+employ a GAN model with semi-supervised learning known
+as a semi-supervised GAN (SGAN) [3] where only a small
+portion of the training dataset is labeled.
+A. Hiding files
+Privacy advocates [4] urge users to protect their private in-
+formation from criminal interception or unlawful government
+overreach, and protecting the digital data stored on users’
+computers, phones, and other devices can include denying
+physical access or employing encryption. While encryption has
+become more commonplace and accessible [5], users desiring
+more security against cryptographic weaknesses [6], [7] may
+apply additional measures to safeguard their information.
+By changing file extensions or removing them altogether, a
+user can obfuscate the true file type. While this rudimentary
+technique applied to a small number of files may not be a
+challenge to computer forensic investigators, it may be more
+effective if used across a large body of files composed of
+varying types.
+Many operating systems will select (or suggest) an appli-
+cation to open a file based on the file extension [8]. For
+example, Microsoft Windows will use the file extension, such
+as .docx to determine the application to open the file. A
+file named cat.docx suggests that the file is a document
+that can be opened by Microsoft Word. However, users can
+change the names and extensions of the file to any arbitrary
+string of characters. A file originally created as a bitmap file
+named cat.bmp and renamed to cat.docx will not open
+and render correctly using Microsoft Word.
+There are a variety of reasons to keep the nature of a
+file unknown to all but the user. By obfuscating file types,
+malware developers may hope to evade email filters or anti-
+virus software [9] [10]. A user engaged in illicit activities may
+desire to hinder law enforcement by complicating evidence
+discovery [11]. Whatever the user’s motivation, without the
+correct file extension and absent a brute-force approach, an
+investigator will require a tool to efficiently discover the
+appropriate program to open the file.
+B. Finding files
+Many file types can be determined by examining the file
+header and footer information, also known as a “magic num-
+ber”. The file header is the first few bytes in a file and the
+footer is the last few bytes in a file. Depending on file type,
+the file headers and footers will be of various lengths and have
+different values. Many file types will have unique headers and
+footers, yet some file types will share header and footer values,
+e.g., .xls, .doc, .ppt [12].
+File headers and footers can be analyzed through command-
+line tools that perform a binary or hexadecimal dump, or
+by using binary or hexadecimal readers/editors to provide
+insight to the file type. Alternatively, tools like Scalpel [13]
+search a chunk of data that may contain multiple files, and
+based on user-configured options, will perform file carving
+that allows the investigator to see the chunk’s number and file
+types within. Scalpel’s configurable options use header and
+footer values as well as common signatures within a file. For
+arXiv:2301.11964v1  [cs.LG]  27 Jan 2023
+
+example, although an html file is plaintext and will not have
+a header, it will likely include the text string <html>.
+Regardless of an investigator’s methods, specialized knowl-
+edge is required to conclude the type of file under examination.
+If the hexadecimal string D0 CF 11 E0 A1 B1 1A E1 is
+found in the header, this could be one of five Microsoft Office
+file types [12]. When several thousand or more files require
+classification, the time demand on the most experienced in-
+vestigator greatly increases.
+C. Contributions
+This work uses machine learning algorithms trained on
+extracted file features to identify the type of file under in-
+vestigation. We created histograms based on the frequency of
+byte-values (ranging from zero to 255) to train and then test
+our machine learning algorithms. Specifically, our contribution
+provides:
+• A classifier from a semi-supervised generative adversarial
+network designed to identify file types
+• Comparison of classifier accuracy with the performance
+of a traditionally-trained multi-layer perceptron (MLP)
+network
+• Comparison and analysis of the neural network method
+compared to the results from non-neural network machine
+learning algorithms, specifically Decision Tree, extreme
+gradient boosting (XGBoost), and k-Nearest Neighbor
+(kNN)
+To the best of our knowledge, no other work has used a
+classifier of an adversarially-trained neural network to conduct
+file type classification. We show improved accuracy over pre-
+viously explored methods can be achieved with reduced expert
+analysis required to create samples for a training dataset.
+This paper provides background and discussion of related
+works in Section II. We then discuss our dataset and how
+we derive our samples for machine learning in Section III.
+We present our SGAN architecture in Section IV and discuss
+other machine learning algorithms in Section V. The results
+of our work are summarized in Section VI and we provide our
+conclusions and future work in Section VII.
+II. BACKGROUND
+This section examines previous work in file classification
+and introduces the SGAN. We summarize the use of byte
+values within files to determine file types and we discuss
+the use of machine learning in file classification. Finally, we
+discuss the nature of adversarial networks and examine the
+SGAN model.
+A. Classification using byte values
+As an alternative to header and footer inspection, McDaniel
+and Heydari used the binary content of files to identify the
+type in [14]. They used several algorithms based on a byte
+frequency distribution fingerprint to determine a file type,
+showing that file classification can be accomplished by com-
+paring a candidate file’s byte distribution to the distribution of
+120 other files of known type. The accuracy of their proposed
+algorithms was just under 96% when they grouped together
+.acd, .doc, .xls, and .ppt file types into one class.
+When these files were separately classified, the accuracy rate
+dropped to 85%. Based on the binary frequency distribution
+in [14], several authors have extended the research on file
+classification.
+In [15], Li et al. were able to improve on McDaniel and
+Heydari’s accuracy in [14] using a centroid-based approach
+and saw improved accuracy when truncating the files. Li used
+the Manhattan Distance for each files’ byte distributions to
+compare files and determine the appropriate classification. Be-
+cause of file header similarity, Li created centroid models that
+combined file types similar to McDaniel’s approach in [14].
+Specifically, there was one model that combined .exe and
+.dll files into one class, and another model that combined
+.doc, .xls, and .ppt files together in another class.
+Moody and Erbacher introduced the Statistical Analysis
+Data Identification (SADI) algorithm in [16]. After calculating
+byte values for each file, a range of statistical information
+was gathered and subsequently used to determine file types.
+The accuracy of SADI had varying success with nine different
+file types, reaching 76% accuracy of all file types after initial
+analysis. A secondary assessment on file types that previously
+did not reach greater than 92% accuracy showed improvement
+when characteristic patterns were considered.
+Using fragments of .pdf, .rtf, and .doc files from a
+publicly-available dataset [17], Rahmet et al. leveraged longest
+common sub-sequences to identify file fragments in [18]. The
+authors’ algorithm successfully classified these file fragments
+with 92.91% overall accuracy.
+Our work extends the efforts discussed here, and we also
+made use of byte values and the frequency in which they arose
+in a file. The byte value distribution was provided to machine
+learning algorithms, and each file type was classified. While
+we also used file types that shared the same header strings
+and files that did not contain headers, we created models that
+differentiated the files uniquely instead of choosing to group
+them together.
+B. Machine learning for file classification
+In [19], Amirani et al. used principle component analysis
+(PCA) and neural networks to achieve file classification accu-
+racy of 98.33% against a pool of six different file types. The
+authors used two neural networks: a five-layer MLP network
+that uses PCA features as the input, and a second three-layer
+MLP network to conduct file classification. Each of their six
+file types were equally represented in the dataset, with 120
+files of each type.
+Konaray et al. conducted several experiments using a variety
+of machine learning algorithms in [20]. The dataset used by
+Konaray were composed of 13 text-based files (e.g., .html,
+.py, .bat, etc.). The authors were able to achieve an
+accuracy of 97.83% using the XGBoost algorithm [21].
+Comparing statistical classification algorithms such as sup-
+port vector machine (SVM) and kNN with commercially
+available tools, Gopal et al. showed that machine learning
+
+algorithms could outperform commercial products in [22]. The
+authors collected byte values for their experiments using an
+n-gram approach. They showed that kNN with 1-gram byte
+values and SVM with 2-gram byte values greatly outperformed
+commercial tools in terms of accuracy.
+Inspired by the efforts in machine learning research, we
+also hope to improve file classification accuracy. As we will
+discuss in Section III, the dataset we used provided access to
+more file types and of a wider variety than those mentioned
+in the works here. The present work uses 11 types of files,
+including some that are solely composed of ASCII characters
+such as .txt and .html. In order to further research in this
+domain, we investigated the SGAN-trained classifier.
+C. Semi-supervised GAN
+Semi-supervised learning requires that only a portion of the
+training data be labeled. Semi-supervised learning differs from
+supervised learning where all training data is labeled, and also
+unsupervised learning, where no labels exist and the networks
+must find their own way to organize the data. Semi-supervised
+learning is valuable for large training data sets when it would
+be laborious and time-intensive to manually label each file.
+When training an unsupervised GAN, the discriminator,
+D, is a two-class classifier that receives authentic samples
+from the training dataset or spoofed samples created by
+the generator, G. The generator uses random variable input
+to create the fake samples and the parameters in G. The
+discriminator assigns a probability from zero to one based on
+its assessment that the sample is fake (0.0) or authentic (1.0).
+The value function that describes this relationships from the
+original work by Goodfellow [23] is given by
+min
+G max
+D V (D, G) = Ex∼pdata(x)[log D(x)]
++ Ez∼pz(z)[log(1 − D(G(z)))]
+(1)
+where D(x) is the probability that x came from the data
+distribution pdata(x) containing authentic training samples,
+and D(G(z)) is the estimate of the probability that the dis-
+criminator incorrectly identifies the fake instance as authentic.
+The generator network attempts to maximize D(G(z)), while
+the discriminator network tries to minimize it. The generator
+creates samples, G(z), based on the parameter values in G
+and the random values z provided to the generator consistent
+with pz(z).
+With semi-supervised learning, a small percentage of the
+training data is labeled and the discriminator becomes a multi-
+class classifier. For N classes, the model requires N + 1
+outputs to account for all the authentic classes plus one
+additional class for the fake generated class. This can be
+implemented in a variety of ways. Following Salimans et
+al. [24], we can build an N-class classifier network, C, with
+output logits {l1, l2, . . . , lN} prior to a softmax activation for
+C. The logits vector is used as the input to a single perceptron
+followed by the sigmoid activation function for D. The sigmoid
+function is given as σ(z) =
+1
+1+e−z , where z is the output
+value of the discriminator output layer perceptron. Because
+D and C share the same input and hidden layer weights, both
+networks act as a single network, D/C, that is updated during
+backpropagation based on their respective loss functions, J(D)
+and J(C). The generator loss function is given by J(G).
+Figure 1 shows a functional depiction of an SGAN in
+training. The training dataset is partially labeled and provided
+to the D/C model for classification by C. The remainder of the
+training dataset as well as the generated samples from G are
+used as input to D/C for discrimination where D will predict
+whether the sample came from the training dataset or if it was
+created by G.
+III. DATASET
+A dataset containing a variety of different files was desired
+to ensure we could discern among a range of files. We used
+Govdocs1, a publicly-available repository of about one million
+files taken from webservers in the .gov domain [17]. The entire
+Govdocs1 corpus consists of 1,000 directories, however we
+only used the first three folders (000, 001, and 002) creating
+a total dataset of 2,946 files, totaling 1.56 GB. We chose to
+limit the dataset to ensure the processing demands would not
+require exceptional computational resources. This work was
+accomplished using a laptop computer with a 2.60 GHz Intel
+i7 processor and 32 GB RAM. Limiting the dataset also allows
+our work to be easily reproduced.
+The dataset contained many common file types to include
+.csv, .doc, .gif, .html, .jpg, .pdf, .txt, .xls,
+etc. We noted an unequal distribution of these files such as 28
+.csv files, 254 .doc files, and 726 .pdf files. Unfortunately
+there were some types that were especially underrepresented,
+including one .gls file and two .java files.
+A. Histograms
+To capture byte value distributions, every file was converted
+to a histogram. Each histogram contained 256 bins in the
+range [0 , 255], representing the decimal value of each byte
+in the file. For every bin, the frequency of that decimal value
+occurring in the file was recorded. Histogram examples are
+shown in Figure 2. In each plot, the bins are shown on the
+N
+classes
+Unlabeled
+samples
+Labeled samples
+Error
+Error
+[[Class1]
+[Class2]
+...
+[ClassN]] 
+[0,1]
+Noise
+Fake samples
+Fig. 1: Training a semi-supervised generative adversarial net-
+work with N classes.
+
+0
+128
+255
+0
+200000
+400000
+.log
+0
+128
+255
+0
+50
+100
+150
+.html
+0
+128
+255
+0
+20
+40
+60
+80
+.html
+0
+128
+255
+0
+200
+400
+600
+.html
+0
+128
+255
+0
+20
+40
+60
+80
+.jpg
+0
+128
+255
+0
+10000
+20000
+30000
+.doc
+0
+128
+255
+0
+200
+400
+600
+.xml
+0
+128
+255
+0
+20000
+40000
+.txt
+0
+128
+255
+0
+500
+1000
+1500
+2000
+.html
+0
+128
+255
+0
+500
+1000
+.txt
+Fig. 2: Sample of histograms showing byte value distribution for various files.
+horizontal axis while the frequency value is represented on the
+vertical axis.
+As Figure 2 shows, there are differences in the byte dis-
+tribution between both files of the same type and files of
+different types, but there are also similarities in different file
+types such as .txt and .html files. Machine learning is an
+appropriate tool to capture the histogram distributions and not
+only differentiate among the different file types but also group
+together files of matching type despite varying byte values.
+B. Samples
+After creating histograms for each file, we then processed
+the histograms into samples. To ensure consistency across our
+samples regardless of file size, we normalized each histogram,
+scaling each to a cumulative distribution of 1.0. Figure 3
+shows the same .pdf file where Figure 3a is the original
+and Figure 3b is normalized.
+Since insufficient sample sizes for each class can precipi-
+tate classification error [25], we removed the file types that
+appeared less than 20 times, representing less than 0.7% of
+the total. There were 14 different file types and 86 total
+files removed, leaving our dataset with 11 classes and 2860
+samples. Our dataset’s composition is shown in Table I. The
+sample order was then shuffled and finally split into training
+0
+50
+100
+150
+200
+250
+0
+200
+400
+600
+800
+(a)
+0
+50
+100
+150
+200
+250
+0.000
+0.025
+0.050
+0.075
+0.100
+0.125
+0.150
+0.175
+(b)
+Fig. 3: Example histogram samples showing (a) an unscaled
+.pdf file and (b) a normalized .pdf file.
+and testing datasets. The training dataset used 80% of the total
+samples, while the remaining 20% were reserved for testing.
+IV. SGAN ARCHITECTURE
+The adversarial competition in the SGAN is a minimax
+game described by (1) where the discriminative model at-
+tempts to correctly identify authentic training samples from
+a distribution produced by the scaled histograms representing
+the dataset files, pdata, and fake training samples created by
+the generator.
+While D and G adversarially train each other, they learn
+to improve their individual performances. Additionally, C is
+trained on labeled samples from the training dataset. Although
+C does not directly receive unlabeled authentic or fake sam-
+ples, the weights of C are affected by unsupervised training
+since it shares weights with D in the D/C implementation.
+The SGAN was implemented using the Python program-
+ming language, Keras [26] front-end, and Tensorflow [27]
+back-end. Additionally, Numpy, and Matplotlib Python li-
+braries were used. The overall SGAN design is summarized
+in Table II, with a total of 417,271 parameters for the dis-
+criminator and the generator, and 304,779 parameters for the
+classifier. The file size of the classifier was 3,634 KB.
+The discriminator/classifier network, D/C, is a densely or
+fully connected MLP deep neural network (DNN) with a single
+input for the file histograms. Four additional fully connected
+layers of size 512, 256, 128 and 64 are followed with rectified
+linear unit (ReLU) activation functions. The ReLU function,
+g is given by g(z) = max(0, z). The four hidden layers use
+Dropout of 0.3 to prevent overfitting. Prior to the output layers,
+a fully connected layer of size 11 is used to capture the number
+of file types to be classified. The discriminator output layer of
+size 1 is fully connected and uses a sigmoid activation function
+to provide values [0.0, 1.0] as discussed in Section II-C. The
+classifier output is a softmax activation connected to the 11
+TABLE I: Dataset file composition
+file type
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+samples
+28
+254
+40
+681
+229
+726
+207
+40
+486
+137
+32
+
+TABLE II: SGAN architecture
+Discriminator/Classifier:
+layer
+output size
+activation
+Input: x ∼ pdata(x)
+256
+Fully Connected
+512
+ReLU
+Dropout = 0.3
+Fully Connected
+256
+ReLU
+Dropout = 0.3
+Fully Connected
+128
+ReLU
+Dropout = 0.3
+Fully Connected
+64
+ReLU
+Dropout = 0.3
+Fully Connected
+11 ln = {l1, l2, . . . , l11}
+Discriminator Output
+1
+sigmoid
+Classifier Output
+11
+softmax
+Generator:
+layer
+output
+activation
+Input: z ∼ pz(z)
+100
+Fully Connected
+32
+ReLU
+Dropout = 0.3
+Fully Connected
+64
+ReLU
+Dropout = 0.3
+Fully Connected
+128
+ReLU
+Dropout = 0.3
+Fully Connected
+256
+ReLU
+Output
+256
+sigmoid
+node layer. The softmax function indicates the most likely class
+to which the input belongs. The learning rate for D/C was
+0.0005 using the Adam [28] optimizer and training was done
+with batches of 32 samples.
+The generator network, G, has a single input with 100 nodes
+fully connected to the first hidden layer of size 32. Two
+additional hidden layers of sizes 64 and 128 are again fully
+connected using ReLU activations. Finally, a layer of size 256
+is connected to the output layer and sigmoid activation that
+ultimately creates the fake histograms samples. The learning
+rate for G was 0.0005 using the Adam optimizer.
+V. MACHINE LEARNING ALGORITHMS
+In order to illustrate the SGAN’s performance when clas-
+sifying files, we used additional machine learning algorithms.
+We assessed another neural network, the decision trees learn-
+ing method, the XGBoost algorithm, and the nearest neighbors
+algorithm. The same training and testing dataset were used
+for each machine learning model. The SGAN was the most
+complex to train due to using multiple neural networks and
+no convergence to a global minima.
+In terms of structure, the closest model to the SGAN is a
+supervised learning-based neural network. We created an MLP
+network with identical architecture to our SGAN classifier.
+The standalone MLP network was trained in a fully supervised
+manner to accurately select the correct file type based on input.
+Both the SGAN and standalone MLP models were trained with
+a batch size of 32 samples, and training was limited to no more
+than 300 epochs. Following training, the best classifiers were
+selected based on their accuracy against the training dataset.
+These classifiers were then evaluated on the test dataset as
+reported in Section VI.
+Decision trees are a supervised learning approach that can
+be used to accomplish multi-class classification [29]. Using
+the features of the histograms, the decision tree algorithm
+examines the parametric values in each sample and attempts
+to accurately classify the file based on a series of decisions
+based on learned thresholds.
+The XGBoost algorithm was implemented as a classifier.
+XGBoost is a supervised learning tool that can be used to
+help us predict the correct file type category. With multiple
+classes, the multi-class logistic loss function was used to train
+the model.
+Finally, the nearest neighbors classification algorithm com-
+pares measurements of the input data and training data [29]
+based on previously stored training information. The classifi-
+cation result is determined by the number of samples selected,
+k, with the smallest Euclidean distance among the sample
+attributes. We iterated k from one to six to determine the most
+appropriate number of neighbors to consider when deciding
+the classification.
+VI. RESULTS
+Our results are summarized in Table III. The SGAN was
+most accurate among all other machine learning algorithms
+regardless of the number of supervised samples used in train-
+ing. When using the entirety of the training data for training
+the SGAN classifier, we achieved the highest classification
+performance with the SGAN at 97.552% accuracy. Figure 4
+shows the confusion matrix of the SGAN when the classifier
+had access to the entire training data. We see that the SGAN
+performed worst at identifying .xml files at 83% accuracy,
+confusing them with .html files. Looking over our dataset,
+we note that .xml files were among the fewest number of
+samples available for training. For some test samples, the
+SGAN confused .ppt files with .doc files, and some .pdf
+files were misidentified as .jpg files. The standalone MLP
+network was nearly as accurate, reaching 96.15%.
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+Predicted label
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+True label
+1.0
+1.0
+1.0
+0.97
+1.0
+0.96
+0.97
+1.0
+0.98
+1.0
+0.83
+SGAN File Classifier
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Fig. 4: Confusion matrix for fully-supervised SGAN.
+
+TABLE III: Classification Accuracy
+Number of
+supervised samples
+SGAN
+Standalone
+MLP
+Decision Tree
+XGBoost
+kNN, k = 1
+kNN, k = 2
+kNN, k = 3
+kNN, k = 4
+kNN, k = 5
+kNN, k = 6
+2288
+0.97552
+0.96154
+0.90734
+0.90384
+0.88986
+0.82692
+0.874126
+0.83042
+0.85490
+0.81293
+1144
+0.93357
+0.92132
+0.86363
+0.87413
+0.86713
+0.79720
+0.84091
+0.75350
+0.81469
+0.76049
+500
+0.91783
+0.9021
+0.82168
+0.77972
+0.84965
+0.71504
+0.76573
+0.62063
+0.74650
+0.65734
+100
+0.87413
+0.81469
+0.48252
+0.65559
+0.71504
+0.44406
+0.61189
+0.48427
+0.52800
+0.38990
+50
+0.81993
+0.62062
+0.26573
+0.56818
+0.66084
+0.38112
+0.54895
+0.43007
+0.30944
+0.08741
+If we reduce the number of supervised samples provided
+to our machine learning algorithms, we expect our testing
+accuracy will be somewhat reduced. In the course of a forensic
+investigation, subject matter expertise and a finite amount of
+time must be prioritized, and since creating a fully-labeled
+dataset is resource intensive, a worthy goal might be to balance
+diminishing returns from further training a machine learning
+algorithm against the time requirements needed for other tasks.
+When drastically reducing the training input down to a sample
+size of 50, only 2.2% of the training dataset, the SGAN
+achieved 81.99% accuracy while the standalone MLP dropped
+to 62.06% accuracy. The confusion matrices for this case are
+shown in Figure 5.
+Comparing Figure 5a and Figure 5b, we see that with
+fewer samples, both neural networks continued to struggle
+in categorizing .xml and .gif files. However, with 50
+supervised samples, both the SGAN and standalone MLP had
+more confusion between .xml files and .html files. The
+.gif files were incorrectly predicted to be .ppt files as
+before, but also as .jpg files. We also see .xls files were
+incorrectly categorized as .doc files, which is notable as
+they are both Microsoft products and share the same header
+information.
+The decision tree, XGBoost, and kNN algorithms performed
+relatively poorly with respect to classification accuracy com-
+pared to the neural networks, especially as the number of
+supervised samples were reduced. This is likely due to the
+number of dimensions under assessment with our training
+and testing samples. The “curse of dimensionality” [30] can
+sometimes be overcome with enough samples, so reducing
+a training dataset has a predictably deleterious effect on
+performance.
+The fully supervised SGAN model is implemented on
+GitHub (https://ksaintg.github.io/SGAN-File-Classier/). Re-
+searchers can make use of this implementation to test how
+altering file headers, changing byte values, deleting portions
+of files, etc. will affect classification accuracy. To determine if
+the file headers would change the SGAN accuracy, we tested
+our fully-supervised SGAN with a test dataset with altered file
+headers. Except for .xml, .html, and .txt files which do
+not make use of file headers, we replaced the first six bytes
+of each test dataset file with the hexadecimal string AA BB
+CC DD EE FF. The test accuracy for determining the file
+type between files with altered and unaltered file headers was
+nearly identical. Training for up to 300 epochs on a fully-
+supervised training dataset with unaltered headers, there was
+only a disparity in altered vs. unaltered test accuracy at epoch
+135, where the difference was 0.14%.
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+Predicted label
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+True label
+1.0
+0.8
+0.62
+0.81
+0.9
+0.93
+0.93
+0.85
+0.64
+0.65
+0.5
+SGAN File Classifier
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+Predicted label
+.csv
+.doc
+.gif
+.html
+.jpg
+.pdf
+.ppt
+.ps
+.txt
+.xls
+.xml
+True label
+1.0
+0.86
+0.62
+0.67
+0.67
+0.75
+0.86
+0.46
+0.3
+0.39
+0.5
+Standalone MLP File Classifier
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b)
+Fig. 5: Confusion matrices for (a) SGAN and (b) stan-
+dalone MLP trained with 50 supervised samples.
+.
+VII. CONCLUSION AND FUTURE WORK
+The adversarial training of a neural network produced
+encouraging results in terms of classification accuracy. While
+the neural networks were more complex to train than the
+other machine learning algorithms, the accuracy results were
+far superior. Though the SGAN was the most complex of all
+the models, its accuracy was the best at correctly classifying
+files based on their byte value distribution, especially with
+
+few supervised samples. Once trained, the time difference in
+classifying the dataset between any of the algorithms was
+indistinguishable. This work leads to future research using
+additional neural network architectures and using our spoofed
+histograms from the generator network to improve other
+machine learning algorithms.
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+Received 20 December 2022;
+Revised 06 January 2023;
+Accepted 06 January 2023
+DOI: xxx/xxxx
+PROCEEDINGS
+Equations of State for Dense Matter and Atrophysical Constraints
+Rafael Bán Jacobsen1 | Verônica Dexheimer2 | Ricardo Luciano Sonego Farias1
+1Universidade Federal de Santa Maria
+(UFSM), Santa Maria, Brazil
+2Department of Physics, Kent State
+University, Kent, OH 44243, USA
+ABSTRACT: This conference proceeding presents an overview of the modern
+approaches in the study of baryonic matter at high densities, focusing on the use of
+online repositories such as CompOSE and MUSES for the calculation of neutron
+star properties. In this context, relevant astrophysical constraints for the equations of
+state (mass-radius relation, speed of sound, tidal deformability) are discussed.
+KEYWORDS:
+Neutron Star EoS, Dense matter, Astrophysical constraints, CompOSE, MUSES
+1
+GENERAL ASPECTS OF THE
+EQUATION OF STATE FOR DENSE MATTER
+The study of the properties of compressed baryonic matter,
+or, more specifically, strongly interacting matter at high densi-
+ties, is a mostly relevant topic for current research in Physics,
+with implications both in the microscopic and in the large scale
+realms of nature. In the first domain, heavy-ion collision exper-
+iments, such as those carried out by the Relativistic Heavy
+Ion Collider (RHIC) at Brookhaven National Laboratory and
+the Large Hadron Collider (LHC) at CERN, provide numerous
+data on the behavior of baryonic matter at extreme condi-
+tions of density and temperature. Additionally, in the second
+domain, astronomic observations of neutron stars, from both
+orbiting and ground based observatories, spanning the electro-
+magnetic spectrum from 훾-rays to radio wavelengths and now
+also including gravitational waves, can unveil significant prop-
+erties of baryonic matter at high densities, since neutron stars
+contain compressed baryonic matter in their centers. These
+remnants of massive stars after core-collapse supernova explo-
+sions are typically about 12 kilometers across and may contain
+up to 2 solar masses (2푀⊙), implying core densities as high as
+10 times nuclear saturation density (∼ 1015푔∕푐푚3).
+In both cases, linking data to theoretical description of bary-
+onic matter depends on the equation of state (EoS) adopted.
+In a broad sense, an EoS is a thermodynamic equation relat-
+ing state variables (and usually including the pressure). In the
+specific field of nuclear astrophysics, it is also expected that
+an EoS provides a full thermodynamic list of variables (e.g.,
+chemical potentials, entropy per baryon), particle composition
+of the system (the proportion of the different types of lep-
+tons, nucleons, and hyperons), microscopic information (e.g.,
+effective masses and pairing gaps) and stellar properties (e.g.,
+maximum mass and radius, tidal deformability).
+EoS input tables for astrophysical simulations usually
+includes baryon number density (푛퐵), charge fraction (푌푄), and
+temperature (푇 ) as independent variables. A 1-dimensional
+EoS table depends only on the parameter 푛퐵 and may describe
+cold isospin-symmetric matter (푇 = 0 and 푌푄 = 0.5), cold
+neutron matter (푇 = 0 and 푌푄 = 0.0), or cold 훽-equilibrated
+matter (푇 = 0 and 푌푄 determined by the conditions of 훽-
+equilibrium and charge neutrality). A 2-dimensional EoS table
+depends on two of the three aforementioned independent vari-
+ables and may describe, for example, dense matter at zero tem-
+perature (varying 푛퐵 and 푌푄 with 푇 = 0), symmetric matter
+(varying 푛퐵 and 푇 with 푌푄 = 0.5), neutron matter (varying 푛퐵
+and 푇 with 푌푄 = 0), and 훽-equilibrated matter (varying 푛퐵 and
+푇 , and calculating 푌푄 according to 훽-equilibrium and charge
+neutrality). Nonetheless, a 3-dimensional EoS table depends
+on all three free parameters and serves for general purposes.
+Namely, a 3-dimensional EoS table is required for supernova
+and mergers simulations as long as, differently from neutron
+stars, the matter in proto-neutron stars and in hypermassive
+stars is hot and not 훽-equilibrated.
+A complete EoS for neutron stars is expected to describe
+a system with nuclei in the lower density regime, evolving to
+bulk hadronic matter (nucleons, hyperons, deconfined quarks)
+at higher densities. Inside neutron stars, this corresponds to the
+crust and core, respectively (see Fig. 1 ).
+arXiv:2301.13252v1  [astro-ph.HE]  30 Jan 2023
+
+2
+V. Dexheimer, R.B. Jacobsen, R.L.S. Farias
+FIGURE 1 Schematic structure of a neutron star. Figure
+modified from Weber et al. (2014).
+An EoS for dense and hot matter must be based on a quan-
+tum relativistic description, because this framework ensures
+respect to causality, as long as vector interactions are not too
+strong. A realistic dense and hot EoS must also obey a series
+of nuclear and quantum chromodynamics (QCD) constraints:
+• To reproduce chiral symmetry restoration, as demanded
+by QCD at large densities and temperatures (with a cor-
+respondent decrease in the overall baryonic masses);
+• To reproduce lattice QCD results at finite temperature
+(which are provided at any isospin and strangeness, but
+are restricted to low density relative to the temperature);
+• To be in agreement with the (nearly) isospin-symmetric
+and zero net strangeness heavy-ion collision physics at
+finite temperature;
+• To reproduce perturbative QCD results in the relevant
+regime.
+• To
+reproduce
+standard
+zero-temperature
+isospin-
+symmetric nuclear physics results around saturation
+density.
+2
+MODERN SOURCES FOR EQUATIONS
+OF STATE
+In order to face the challenge of finding an adequate EoS for
+dense matter in this variety of phenomena, online repositories
+of equations of state have been built in recent years. CompOSE
+and MUSES are among these modern sources for 1-, 2-, and
+3-dimensional EoS tables.
+2.1
+COMPOSE
+CompOSE (CompStar Online Supernovae Equations of
+State)1 is the largest repository of this kind, offering almost 300
+equations of state, divided in families (cold neutron star EoS,
+cold matter EoS, neutron matter EoS, general purpose EoS,
+and neutron star crust EoS) and their subgroups (models with
+hyperons and delta resonances, hybrid quark-hadron models,
+models with hyperons, models with kaon condensate, nucle-
+onic models, and quark models). The repository also provides
+a software to interpolate data, calculate additional quantities,
+and graph EoS dependencies. Data tables, associated software
+and the manual, can be freely downloaded, cf. Dexheimer et
+al. (2022); Typel et al. (2022).
+Paradigmatic examples of the usefulness of such a database
+can be found in studies that carry out comparisons of the
+predictions made by different models for the same phys-
+ical system. For instance, a set of microscopic, covariant
+density-functional, and non-relativistic Skyrme-type equations
+of state, obtained from CompOSE, has been employed to
+study the structure of purely nucleonic 훽-equilibrated neu-
+tron stars at finite temperature (Wei, Burgio, Raduta, &
+Schulze, 2021). Considering the agreement with presently
+available astrophysical observational constraints, this study
+showed that the magnitude of thermal effects depends on
+the nucleon effective mass as well as on the stiffness of
+the cold equation of state. Regarding the equations of state
+themselves, an appropriate quantity to analyze in this con-
+text is the relative thermal pressure, defined as 푝푟푎푡푖표
+=
+푝푡ℎ∕푝0 = [푝(휌퐵, 푥푇 , 푇 ) − 푝(휌퐵, 푥0, 0)] ∕푝(휌퐵, 푥0, 0), where 휌퐵
+is the baryonic density, 푇 is temperature and 푥0 and 푥푇 are
+the respective proton fractions of cold and hot matter. The
+ratio of thermal pressure as a function of density is shown
+in the upper panel of Fig. 2 for the different equations of
+state studied. Moreover, in order to appreciate the astrophysi-
+cal implications of these equations of state, the relative change
+of the maximum gravitational neutron-star mass, defined as
+푀푟푎푡푖표 = (푀ℎ표푡
+푚푎푥 − 푀푐표푙푑
+푚푎푥
+) ∕푀푐표푙푑, can be plotted as a function
+of the thermal pressure ratio. The result is shown in the lower
+panel of Fig. 2 .
+2.2
+MUSES
+MUSES (Modular Unified Solver of the Equation of State)2 is
+a large collaboration project that is developing a new cyber-
+infrastructure to provide novel tools to answer critical inter-
+disciplinary questions in nuclear physics, gravitational wave
+astrophysics and heavy-ion physics. The MUSES collabora-
+tion consists of many researchers and technical professionals
+1https://compose.obspm.fr
+2https://muses.physics.illinois.edu/
+
+NeutronStar
+Surface
+.Hydrogen/Heliumplasma
+Ironnuclei
+OuterCrust
+.lons
+.Electrongas
+InnerCrust
+Heavyions
+Relativisticelectrongas
+Superfluidneutrons
+OuterCore
+.Neutrons,protons
+.Electrons,muons
+InnerCore
+Neutrons
+Superconductingprotons
+Electrons,muons
+Hyperons(Z,A,三)
+Deltas (△)
+Boson(元,K)condensates
+Deconfined(u,d,s)quarks/color-
+superconductingquarkmatterV. Dexheimer, R.B. Jacobsen, R.L.S. Farias
+3
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+FIGURE 2 Upper panel: Ratio of thermal pressure as a func-
+tion of density for several equations of state. Lower panel:
+Relative change of the maximum gravitational mass as a func-
+tion of the pressure ratio at the center of the star. Figures
+modified from Fig.6 and Fig.8 in (Wei et al., 2021).
+across dozens of institutions spread across the globe who are
+building and using a collaborative platform which is modular
+because, while at low baryonic chemical potential the EoS is
+known from first principles, at high there will be different mod-
+els for the user to choose; besides, it is unified in as much as
+different modules will be merged together to ensure maximal
+coverage of the phase diagram. Building up MUSES, physi-
+cists and computer scientists will work together to develop the
+software that generates equations of state over large ranges of
+temperature and chemical potentials to cover the whole QCD
+phase diagram. The group of users is composed by interested
+scientists from different communities, who provide input to the
+future open-source cyberinfrastructure.
+3
+ASTROPHYSICAL CONSTRAINTS
+Any consistent EoS has to pass the test posed by the astrophys-
+ical constraints related to neutron stars, the most fundamen-
+tal being the mass-radius relation for these compact objects.
+Nonetheless, many relevant features cannot be appreciated on
+such a basis; for example, the possible existence of different
+exotic matter associated with different phase transitions inside
+a neutron star can easily be seen in the speed of sound (푐푆)
+behavior but not necessarily in the mass-radius relation. As a
+matter of fact, 2푀⊙ stars demand a stiff EoS (with 푐푆
+2 ←→ 1
+in natural units) at intermediate densities; on the other hand,
+푐푆
+2
+←→ 1∕3 from below at asymptotically large densities
+because of the conformal limit of massless free quarks. Thus,
+a non-monotonic behavior is expected for 푐푆, implying the
+occurrence of bumps related to the softening of the EoS due to
+new degrees of freedom, cf.Bedaque & Steiner (2015).
+Figure 3 , adapted from Tan, Dore, Dexheimer, Noronha-
+Hostler, & Yunes (2022), show how bumps (that also appear
+in realistic microscopic models) can be produced under a con-
+trolled 푐푠 parametrization, allowing a correlation between the
+density at which the bump appears and curves in the neutron
+star mass-radius diagram. Thus, this more systematic para-
+metric form for the speed of sound can help to determine
+neutron-star composition; besides, maximum stellar mass and
+radius can determine width, density, and height of the bumps.
+The non-smooth structure of the speed of 푐푆 related to phase
+transitions in dense matter makes feasible the constitution of
+ultra-heavy neutron stars (with masses larger than 2.5푀⊙).
+These stars pass all observational and theoretical constraints,
+including those imposed by recent LIGO/Virgo gravitational-
+wave observations and NICER X-ray observations.
+Another observational test that may be used to constrain
+equations of state is the evaluation of tidal deformabilities in
+neutron stars inferred from gravitational-wave measurements.
+In a coalescing binary of neutron stars, the gravitational field
+of one star perturbs the field of the other (and vice-versa), caus-
+ing an acceleration in their inspiral. This change in the inspiral
+rate shapes the gravitational-wave emitted, and this wave thus
+provides information about the tidal deformabilities Λ1,2 of
+the neutron stars. Considering a sequence of central densi-
+ties for a given EoS and a fixed mass ratio, one can construct
+the binary Love relations (BLRs) Λ푠 and Λ푎, definined with
+the symmetric and anti-symmetric tidal deformabilities Λ푠,푎 =
+(Λ1 ±Λ2)∕2. Due to phase transitions and the consequent non-
+smooth structure of the speed of sound 푐푆, which may tilt the
+mass-radius diagram, peculiar structures (such as slopes, hills,
+drops and swooshes) are created in the BLRs (Tan, Dexheimer,
+Noronha-Hostler, & Yunes, 2022), as shown in Figure 4 .
+The change in slope in the BLRs may be observable already
+during the fifth LIGO observing run if a sufficiently loud and
+
+4
+V. Dexheimer, R.B. Jacobsen, R.L.S. Farias
+ε=3p
+causal limit
+□
+●××
+▲
+□●××▲
+max central nB/nsat
+(a)
+eos1
+eos2
+eos3
+eos4
+0
+2
+4
+6
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+nB/nsat
+cs
+2
+J0740+6620
+GW190814
+GW170817
+J0030+0451
+(c)
+10
+11
+12
+13
+14
+15
+16
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R [km]
+M [M⊙]
+ε=3p
+causal limit
+□ ● ×× ▲
+□●××▲
+max central nB/nsat
+(a)
+eos1
+eos2
+eos3
+eos4
+0
+2
+4
+6
+8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+nB/nsat
+cs
+2
+J0740+6620
+GW190814
+GW170817
+J0030+0451
+(c)
+10
+11
+12
+13
+14
+15
+16
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+R [km]
+M [M⊙]
+FIGURE 3 Upper panels: Speed of sound (left) and mass-
+radius diagram (right) for a subfamily of equations of state with
+peaks in the speed of sound of different widths at the same
+location. Lower panels: Speed of sound (left) and mass-radius
+diagram (right) for a subfamily of equations of state with peaks
+in the speed of sound of the same width at different locations.
+Modified from Fig.4 in Tan, Dore, et al. (2022).
+low mass neutron-star binary is detected. The detection of
+drops and swooshes is more challenging, because both occur
+at very small Λ푎, and such detection would require very low
+uncertainties in the measurements, which are achievable only
+if an exceptionally loud signal is detected.
+4
+CONCLUSIONS
+From the recent developments here reported, one my infer that
+new tight constraints from experiment, observation and the-
+ory are slowly determining dense matter and neutron-star core
+properties. In this context, EoS repositories (such as Com-
+pOSE and MUSES) help speeding up the understanding of
+dense matter. Furthermore, astrophysical constraints must be
+taken into account and, in this context, gravitational waves are
+providing new ways to study the dense matter EoS. Besides
+the basic mass-radius relation of neutron stars, more specific
+and subtle quantities (such as the speed of sound and tidal
+deformabilities for these objects) can be used to probe differ-
+ent equations of state. Advances in this field can be expected
+shortly, since LIGO, Virgo, and KAGRA are coordinating a
+new observing run in March 2023. Thus, open questions in
+nuclear astrophysics may soon find their answers and induce
+further interrogations about the intimate structure of matter.
+EoS5
+EoS6
+EoS7
+EoS8
+0
+1
+2
+3
+4
+5
+6
+7
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+n/nsat
+cs
+2
+(a)
+J0740+6620
+J0030+0451
+Am
+IL/MD
+Am
+IL/MD
+10
+11
+12
+13
+14
+15
+16
+0.8
+1.2
+1.6
+2.
+2.4
+R [km]
+M [M⊙]
+(b)
+type (B.1)
+type (B.2)
+60
+80
+100
+120
+140
+-100
+-50
+0
+50
+100
+Λs
+Λa
+(c)
+type (A)
+type (C)
+0
+50
+100
+150
+200
+0
+2
+4
+6
+8
+10
+12
+Λs
+Λa
+(d)
+FIGURE 4 Upper panels: Speed of sound (left) and mass-
+radius diagram (right) for for different equations of state.
+First-order phase transitions (푐푆 = 0) introduce a second sta-
+ble branch in the mass-radius curves. Lower panels: BLRs
+between stars (with a mass ratio 0.75) in the same branch (types
+A and C) or in different branches (types B.1 and B.2) produce
+a slope, hill, drop, and swoosh. Figure modified from Fig.1
+in Tan, Dexheimer, et al. (2022).
+ACKNOWLEDGEMENTS
+V. D. acknowledges support from the National Science Foun-
+dation under grants PHY1748621, MUSES OAC-2103680,
+and NP3M PHY-2116686. R.L.S.F. acknowledges support
+from Conselho Nacional de Desenvolvimento Científico e
+Tecnológico (CNPq), Grant No. 309598/2020-6 and Fundação
+de Amparo à Pesquisa do Estado do Rio Grande do Sul
+(FAPERGS), Grants Nos. 19/2551- 0000690-0 and 19/2551-
+0001948-3.
+REFERENCES
+Bedaque, P., & Steiner, A. W. 2015, Phys. Rev. Lett., 114(3), 031103.
+Dexheimer, V., Mancini, M., Oertel, M., Providência, C., Tolos, L.,
+& Typel, S. 2022, Particles, 5(3), 346.
+Tan, H., Dexheimer, V., Noronha-Hostler, J., & Yunes, N.
+2022,
+Physical Review Letters, 128(16).
+Tan, H., Dore, T., Dexheimer, V., Noronha-Hostler, J., & Yunes, N.
+2022, Physical Review D, 105(2).
+Typel, S., Oertel, M., Klähn, T. et al. 2022, Eur. Phys. J. A, 58(11),
+221.
+Weber, F., Contrera, G. A., Orsaria, M. G., Spinella, W., & Zubairi,
+O. 2014, Modern Physics Letters A, 29, 1430022.
+Wei, J.-B., Burgio, G. F., Raduta, A. R., & Schulze, H.-J.
+2021,
+Physical Review C, 104(6).
+

8tFQT4oBgHgl3EQfITXl/content/tmp_files/load_file.txt

@@ -0,0 +1,344 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf,len=343
+page_content='Received 20 December 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Revised 06 January 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Accepted 06 January 2023  DOI: xxx/xxxx  PROCEEDINGS  Equations of State for Dense Matter and Atrophysical Constraints  Rafael Bán Jacobsen1 | Verônica Dexheimer2 | Ricardo Luciano Sonego Farias1  1Universidade Federal de Santa Maria  (UFSM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Santa Maria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Brazil  2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Kent State  University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Kent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' OH 44243,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' USA  ABSTRACT: This conference proceeding presents an overview of the modern  approaches in the study of baryonic matter at high densities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' focusing on the use of  online repositories such as CompOSE and MUSES for the calculation of neutron  star properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' In this context, relevant astrophysical constraints for the equations of  state (mass-radius relation, speed of sound, tidal deformability) are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  KEYWORDS:  Neutron Star EoS, Dense matter, Astrophysical constraints, CompOSE, MUSES  1  GENERAL ASPECTS OF THE  EQUATION OF STATE FOR DENSE MATTER  The study of the properties of compressed baryonic matter,  or, more specifically, strongly interacting matter at high densi-  ties, is a mostly relevant topic for current research in Physics,  with implications both in the microscopic and in the large scale  realms of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' In the first domain, heavy-ion collision exper-  iments, such as those carried out by the Relativistic Heavy  Ion Collider (RHIC) at Brookhaven National Laboratory and  the Large Hadron Collider (LHC) at CERN, provide numerous  data on the behavior of baryonic matter at extreme condi-  tions of density and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Additionally, in the second  domain, astronomic observations of neutron stars, from both  orbiting and ground based observatories, spanning the electro-  magnetic spectrum from 훾-rays to radio wavelengths and now  also including gravitational waves, can unveil significant prop-  erties of baryonic matter at high densities, since neutron stars  contain compressed baryonic matter in their centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' These  remnants of massive stars after core-collapse supernova explo-  sions are typically about 12 kilometers across and may contain  up to 2 solar masses (2푀⊙), implying core densities as high as  10 times nuclear saturation density (∼ 1015푔∕푐푚3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  In both cases, linking data to theoretical description of bary-  onic matter depends on the equation of state (EoS) adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  In a broad sense, an EoS is a thermodynamic equation relat-  ing state variables (and usually including the pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' In the  specific field of nuclear astrophysics, it is also expected that  an EoS provides a full thermodynamic list of variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=',  chemical potentials, entropy per baryon), particle composition  of the system (the proportion of the different types of lep-  tons, nucleons, and hyperons), microscopic information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=',  effective masses and pairing gaps) and stellar properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=',  maximum mass and radius, tidal deformability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  EoS input tables for astrophysical simulations usually  includes baryon number density (푛퐵), charge fraction (푌푄), and  temperature (푇 ) as independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' A 1-dimensional  EoS table depends only on the parameter 푛퐵 and may describe  cold isospin-symmetric matter (푇 = 0 and 푌푄 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5), cold  neutron matter (푇 = 0 and 푌푄 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0), or cold 훽-equilibrated  matter (푇 = 0 and 푌푄 determined by the conditions of 훽-  equilibrium and charge neutrality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' A 2-dimensional EoS table  depends on two of the three aforementioned independent vari-  ables and may describe, for example, dense matter at zero tem-  perature (varying 푛퐵 and 푌푄 with 푇 = 0), symmetric matter  (varying 푛퐵 and 푇 with 푌푄 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5), neutron matter (varying 푛퐵  and 푇 with 푌푄 = 0), and 훽-equilibrated matter (varying 푛퐵 and  푇 , and calculating 푌푄 according to 훽-equilibrium and charge  neutrality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Nonetheless, a 3-dimensional EoS table depends  on all three free parameters and serves for general purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Namely, a 3-dimensional EoS table is required for supernova  and mergers simulations as long as, differently from neutron  stars, the matter in proto-neutron stars and in hypermassive  stars is hot and not 훽-equilibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  A complete EoS for neutron stars is expected to describe  a system with nuclei in the lower density regime, evolving to  bulk hadronic matter (nucleons, hyperons, deconfined quarks)  at higher densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Inside neutron stars, this corresponds to the  crust and core, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='13252v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='HE]  30 Jan 2023  2  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Dexheimer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Jacobsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Farias  FIGURE 1 Schematic structure of a neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Figure  modified from Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  An EoS for dense and hot matter must be based on a quan-  tum relativistic description, because this framework ensures  respect to causality, as long as vector interactions are not too  strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' A realistic dense and hot EoS must also obey a series  of nuclear and quantum chromodynamics (QCD) constraints:  To reproduce chiral symmetry restoration, as demanded  by QCD at large densities and temperatures (with a cor-  respondent decrease in the overall baryonic masses);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  To reproduce lattice QCD results at finite temperature  (which are provided at any isospin and strangeness, but  are restricted to low density relative to the temperature);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  To be in agreement with the (nearly) isospin-symmetric  and zero net strangeness heavy-ion collision physics at  finite temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  To reproduce perturbative QCD results in the relevant  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  To  reproduce  standard  zero-temperature  isospin-  symmetric nuclear physics results around saturation  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  2  MODERN SOURCES FOR EQUATIONS  OF STATE  In order to face the challenge of finding an adequate EoS for  dense matter in this variety of phenomena, online repositories  of equations of state have been built in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' CompOSE  and MUSES are among these modern sources for 1-, 2-, and  3-dimensional EoS tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='1  COMPOSE  CompOSE (CompStar Online Supernovae Equations of  State)1 is the largest repository of this kind, offering almost 300  equations of state, divided in families (cold neutron star EoS,  cold matter EoS, neutron matter EoS, general purpose EoS,  and neutron star crust EoS) and their subgroups (models with  hyperons and delta resonances, hybrid quark-hadron models,  models with hyperons, models with kaon condensate, nucle-  onic models, and quark models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' The repository also provides  a software to interpolate data, calculate additional quantities,  and graph EoS dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Data tables, associated software  and the manual, can be freely downloaded, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Dexheimer et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Typel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Paradigmatic examples of the usefulness of such a database  can be found in studies that carry out comparisons of the  predictions made by different models for the same phys-  ical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' For instance, a set of microscopic, covariant  density-functional, and non-relativistic Skyrme-type equations  of state, obtained from CompOSE, has been employed to  study the structure of purely nucleonic 훽-equilibrated neu-  tron stars at finite temperature (Wei, Burgio, Raduta, &  Schulze, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Considering the agreement with presently  available astrophysical observational constraints, this study  showed that the magnitude of thermal effects depends on  the nucleon effective mass as well as on the stiffness of  the cold equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Regarding the equations of state  themselves, an appropriate quantity to analyze in this con-  text is the relative thermal pressure, defined as 푝푟푎푡푖표  =  푝푡ℎ∕푝0 = [푝(휌퐵, 푥푇 , 푇 ) − 푝(휌퐵, 푥0, 0)] ∕푝(휌퐵, 푥0, 0), where 휌퐵  is the baryonic density, 푇 is temperature and 푥0 and 푥푇 are  the respective proton fractions of cold and hot matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' The  ratio of thermal pressure as a function of density is shown  in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' 2 for the different equations of  state studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Moreover, in order to appreciate the astrophysi-  cal implications of these equations of state, the relative change  of the maximum gravitational neutron-star mass, defined as  푀푟푎푡푖표 = (푀ℎ표푡  푚푎푥 − 푀푐표푙푑  푚푎푥  ) ∕푀푐표푙푑, can be plotted as a function  of the thermal pressure ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' The result is shown in the lower  panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
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+page_content='2  MUSES  MUSES (Modular Unified Solver of the Equation of State)2 is  a large collaboration project that is developing a new cyber-  infrastructure to provide novel tools to answer critical inter-  disciplinary questions in nuclear physics, gravitational wave  astrophysics and heavy-ion physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
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+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='FIGURE 2 Upper panel: Ratio of thermal pressure as a func-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='tion of density for several equations of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Lower panel:  Relative change of the maximum gravitational mass as a func-  tion of the pressure ratio at the center of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Figures  modified from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='8 in (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  across dozens of institutions spread across the globe who are  building and using a collaborative platform which is modular  because, while at low baryonic chemical potential the EoS is  known from first principles, at high there will be different mod-  els for the user to choose;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' besides, it is unified in as much as  different modules will be merged together to ensure maximal  coverage of the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Building up MUSES, physi-  cists and computer scientists will work together to develop the  software that generates equations of state over large ranges of  temperature and chemical potentials to cover the whole QCD  phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' The group of users is composed by interested  scientists from different communities, who provide input to the  future open-source cyberinfrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  3  ASTROPHYSICAL CONSTRAINTS  Any consistent EoS has to pass the test posed by the astrophys-  ical constraints related to neutron stars, the most fundamen-  tal being the mass-radius relation for these compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Nonetheless, many relevant features cannot be appreciated on  such a basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' for example, the possible existence of different  exotic matter associated with different phase transitions inside  a neutron star can easily be seen in the speed of sound (푐푆)  behavior but not necessarily in the mass-radius relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' As a  matter of fact, 2푀⊙ stars demand a stiff EoS (with 푐푆  2 ←→ 1  in natural units) at intermediate densities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' on the other hand,  푐푆  2  ←→ 1∕3 from below at asymptotically large densities  because of the conformal limit of massless free quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Thus,  a non-monotonic behavior is expected for 푐푆, implying the  occurrence of bumps related to the softening of the EoS due to  new degrees of freedom, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='Bedaque & Steiner (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Figure 3 , adapted from Tan, Dore, Dexheimer, Noronha-  Hostler, & Yunes (2022), show how bumps (that also appear  in realistic microscopic models) can be produced under a con-  trolled 푐푠 parametrization, allowing a correlation between the  density at which the bump appears and curves in the neutron  star mass-radius diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Thus, this more systematic para-  metric form for the speed of sound can help to determine  neutron-star composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' besides, maximum stellar mass and  radius can determine width, density, and height of the bumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  The non-smooth structure of the speed of 푐푆 related to phase  transitions in dense matter makes feasible the constitution of  ultra-heavy neutron stars (with masses larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5푀⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  These stars pass all observational and theoretical constraints,  including those imposed by recent LIGO/Virgo gravitational-  wave observations and NICER X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Another observational test that may be used to constrain  equations of state is the evaluation of tidal deformabilities in  neutron stars inferred from gravitational-wave measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  In a coalescing binary of neutron stars, the gravitational field  of one star perturbs the field of the other (and vice-versa), caus-  ing an acceleration in their inspiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' This change in the inspiral  rate shapes the gravitational-wave emitted, and this wave thus  provides information about the tidal deformabilities Λ1,2 of  the neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Considering a sequence of central densi-  ties for a given EoS and a fixed mass ratio, one can construct  the binary Love relations (BLRs) Λ푠 and Λ푎, definined with  the symmetric and anti-symmetric tidal deformabilities Λ푠,푎 =  (Λ1 ±Λ2)∕2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Due to phase transitions and the consequent non-  smooth structure of the speed of sound 푐푆, which may tilt the  mass-radius diagram, peculiar structures (such as slopes, hills,  drops and swooshes) are created in the BLRs (Tan, Dexheimer,  Noronha-Hostler, & Yunes, 2022), as shown in Figure 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  The change in slope in the BLRs may be observable already  during the fifth LIGO observing run if a sufficiently loud and  4  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Dexheimer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Jacobsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Farias  ε=3p  causal limit  □  ××  ▲  □●××▲  max central nB/nsat  (a)  eos1  eos2  eos3  eos4  0  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2  nB/nsat  cs  2  J0740+6620  GW190814  GW170817  J0030+0451  (c)  10  11  12  13  14  15  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  R [km]  M [M⊙]  ε=3p  causal limit  □ ● ×× ▲  □●××▲  max central nB/nsat  (a)  eos1  eos2  eos3  eos4  0  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2  nB/nsat  cs  2  J0740+6620  GW190814  GW170817  J0030+0451  (c)  10  11  12  13  14  15  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  R [km]  M [M⊙]  FIGURE 3 Upper panels: Speed of sound (left) and mass-  radius diagram (right) for a subfamily of equations of state with  peaks in the speed of sound of different widths at the same  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Lower panels: Speed of sound (left) and mass-radius  diagram (right) for a subfamily of equations of state with peaks  in the speed of sound of the same width at different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  Modified from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='4 in Tan, Dore, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  low mass neutron-star binary is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' The detection of  drops and swooshes is more challenging, because both occur  at very small Λ푎, and such detection would require very low  uncertainties in the measurements, which are achievable only  if an exceptionally loud signal is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  4  CONCLUSIONS  From the recent developments here reported, one my infer that  new tight constraints from experiment, observation and the-  ory are slowly determining dense matter and neutron-star core  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' In this context, EoS repositories (such as Com-  pOSE and MUSES) help speeding up the understanding of  dense matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Furthermore, astrophysical constraints must be  taken into account and, in this context, gravitational waves are  providing new ways to study the dense matter EoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Besides  the basic mass-radius relation of neutron stars, more specific  and subtle quantities (such as the speed of sound and tidal  deformabilities for these objects) can be used to probe differ-  ent equations of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Advances in this field can be expected  shortly, since LIGO, Virgo, and KAGRA are coordinating a  new observing run in March 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Thus, open questions in  nuclear astrophysics may soon find their answers and induce  further interrogations about the intimate structure of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  EoS5  EoS6  EoS7  EoS8  0  1  2  3  4  5  6  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='0  n/nsat  cs  2  (a)  J0740+6620  J0030+0451  Am  IL/MD  Am  IL/MD  10  11  12  13  14  15  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='4  R [km]  M [M⊙]  (b)  type (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='1)  type (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2)  60  80  100  120  140  100  50  0  50  100  Λs  Λa  (c)  type (A)  type (C)  0  50  100  150  200  0  2  4  6  8  10  12  Λs  Λa  (d)  FIGURE 4 Upper panels: Speed of sound (left) and mass-  radius diagram (right) for for different equations of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  First-order phase transitions (푐푆 = 0) introduce a second sta-  ble branch in the mass-radius curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Lower panels: BLRs  between stars (with a mass ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='75) in the same branch (types  A and C) or in different branches (types B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='2) produce  a slope, hill, drop, and swoosh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' Figure modified from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='1  in Tan, Dexheimer, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' acknowledges support from the National Science Foun-  dation under grants PHY1748621, MUSES OAC-2103680,  and NP3M PHY-2116686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' acknowledges support  from Conselho Nacional de Desenvolvimento Científico e  Tecnológico (CNPq), Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' 309598/2020-6 and Fundação  de Amparo à Pesquisa do Estado do Rio Grande do Sul  (FAPERGS), Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
+page_content=' 19/2551- 0000690-0 and 19/2551-  0001948-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFQT4oBgHgl3EQfITXl/content/2301.13252v1.pdf'}
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+Introducing Variational Inference in
+Undergraduate Statistics and Data Science
+Curriculum
+Vojtech Kejzlar
+Department of Mathematics and Statistics, Skidmore College
+and
+Jingchen Hu
+Department of Mathematics and Statistics, Vassar College
+January 4, 2023
+Abstract
+Probabilistic models such as logistic regression, Bayesian classification, neural net-
+works, and models for natural language processing, are increasingly more present in
+the undergraduate statistics and data science curriculum due to their wide range of
+applications. In this paper, we present a one-week undergraduate course module on
+variational inference, a popular optimization-based approach for approximate infer-
+ence with probabilistic models. Our proposed module is guided by active learning
+principles: In addition to lecture materials on variational inference, we provide an
+accompanying class activity, an R shiny app, and a guided lab based on a real data
+application of clustering documents using Latent Dirichlet Allocation with R code.
+The main goal of our module is to expose undergraduate students to a method that
+facilitates statistical modeling and inference with large datasets. Using our proposed
+module as a foundation, instructors can adopt it and adapt to introduce more realistic
+use cases and applications in data science, Bayesian statistics, multivariate analysis,
+and statistical machine learning courses.
+Keywords: Active learning, Bayesian Statistics, Statistical Computing, Probabilistic Mod-
+els, Undergraduate Curriculum
+1
+arXiv:2301.01251v1  [stat.OT]  3 Jan 2023
+
+1
+Introduction
+With the recent and rapid expansion of undergraduate curricula with offerings in data
+science, Bayesian statistics, multivariate data analysis, and statistical machine learning,
+probabilistic models and Bayesian methods have grown to become more popular (Schwab-
+McCoy et al. 2021, Dogucu & Hu (2022)). In many settings, a central task in applications
+of probabilistic models is the evaluation of posterior distribution p(θ | y) of m model
+parameters θ ∈ Rm (m ≥ 1) conditioned on the observed data y = (y1, . . . , yn) provided
+by the Bayes’ theorem
+p(θ | y) = p(y | θ)p(θ)
+p(y)
+∝ p(y | θ)p(θ).
+(1)
+Here, p(y | θ) is the sampling density given by the underlying probabilistic model for data,
+p(θ) is the prior density that represents our prior beliefs about θ before seeing the data,
+and p(y) is the marginal data distribution. The posterior distribution p(θ | y), however,
+has closed form only in a limited number of scenarios (e.g., conjugate priors) and there-
+fore typically requires approximation. By far the most popular approximation methods
+are Markov chain Monte Carlo (MCMC) algorithms including Gibbs sampler, Metropo-
+lis, Metropolis-Hastings, and Hamiltonian Monte Carlo (Gelman et al. 2013), to name a
+few. See Albert & Hu (2020) for a review of these algorithms in undergraduate Bayesian
+courses. While useful for certain scenarios, these MCMC algorithms do not scale well with
+large datasets and can have a hard time approximating multimodal posteriors (Rudoy &
+Wolfe 2006, Bardenet et al. (2017)). Such challenges therefore limit the applications of
+probabilistic models that can be discussed in the classroom and restrict students’ exposure
+to more realistic use cases that include applying neural networks, pattern recognition, and
+natural language processing to massive datasets.
+Variational inference is an alternative to the sampling-based approximation via MCMC
+that approximates a target density through optimization. Statisticians and computer sci-
+entists (starting with Peterson & Anderson (1987), Jordan et al. (1999), Blei et al. (2017))
+have been using variational techniques in a variety of settings because these techniques tend
+to be faster and easier to scale to massive datasets. Despite its popularity among statistics
+and data science practitioners, variational inference is rarely discussed in undergraduate
+2
+
+courses as it is believed to be a rather advanced topic (Dogucu & Hu 2022). With this in
+mind, we have developed a one-week course module to disrupt this notion of being an ad-
+vanced topic and help instructors to introduce variational techniques in their undergraduate
+courses for more realistic use cases of probabilistic models. Our proposed one-week module
+is based on the best practices of active learning, which have been shown to improve student
+learning and engagement (Michael 2006, Freeman et al. (2014), Deslauriers et al. (2019)).
+Our main guiding principle in designing the module is to involve students in the learning
+process by introducing student-centered class activities and labs. The guiding principle
+also includes assigning open-ended questions, focusing on problem-solving, providing ap-
+propriate scaffolding for activities, and creating opportunities to work collaboratively with
+peers.
+Our module is designed for students to gain a fundamental understanding and practical
+experience with variational inference over the course of two class meetings. During the first
+meeting, students are exposed to the fundamentals of variational inferences including the
+Kullback-Leibler divergence, evidence lower bound, gradient ascent, and coordinate ascent.
+Additionally, they gain their first hands-on experience by applying variational inference to
+a simple probabilistic model for count data. To encourage and empower undergraduate
+instructors to adopt and adapt this variational inference module, we provide an accompa-
+nying in-class handout and an R Shiny app with details explained in the supplementary
+materials. During the second class meeting, students work on a guided R lab to apply
+variational inference to a realistic scenario of clustering documents with Latent Dirichlet
+Allocation (Blei et al. 2003). See Table 1 for the breakdown of the module.
+Content
+1st class
+Lecture: Fundamentals of variational inference
+Class activity: Probabilistic model for count data with variational inference
+2nd class
+Lab: Document clustering
+Table 1:
+Outline of the one-week variational inference module.
+As for the audience, we believe that the module can be seamlessly integrated into any
+intermediate- or upper-level undergraduate course in data science, Bayesian statistics, mul-
+tivariate data analysis, and statistical machine learning that covers topics on clustering,
+3
+
+classification, or text analysis. The prerequisites needed for the module are a basic under-
+standing of statistical modeling, probability distributions, and elementary calculus.
+The remainder of the paper is organized as the following. In Section 2, we provide an
+overview of variational inference essentials that can be readily used as a basis for a lecture
+instruction. Section 3 presents a motivating example and the Gamma-Poisson model for
+count data that serves as the first hands-on class activity with variational inference. In
+Section 4, we offer a realistic use case for variational inference based on a Latent Dirichlet
+Allocation model for document clustering with implementation details in R, which can be
+used as a computing lab. We end the paper in Section 5 with a few concluding remarks.
+2
+Lecture: Foundations of Variantional Inference
+In this section, we introduce concepts and definitions of variational inference in Section
+2.1, discuss the choices of variational families in Section 2.2, and present details of ELBO
+optimization in Section 2.3. We also include recommendations of variational families and
+ELBO optimization strategies with pedagogical considerations for an undergraduate audi-
+ence. Instructors can design their lecture based on these materials tailored to their needs.
+2.1
+Concepts and Definitions
+The main idea behind variational inference is to approximate the target probability density
+p(θ | y) by a member of some relatively simple family of densities q(θ | λ), indexed by
+the variational parameter λ ∈ Rm′ (m′ ≥ 1), over the space of model parameters θ. Note
+that λ = (λ1, . . . , λm) has m components of (potentially) varying dimensions. Variational
+approximation is done by finding the member of variational family that minimizes the
+Kullback-Leibler (KL) divergence of q(θ | λ) from p(θ | y):
+q∗ = arg min
+q(θ|λ)
+KL(q(θ | λ)||p(θ | y)),
+(2)
+with KL divergence being the expectation of the log ratio between the q(θ | λ) and p(θ | y)
+with respect to q(θ | λ):
+KL(q(θ | λ)||p(θ | y)) = Eq
+�
+log q(θ | λ)
+p(θ | y)
+�
+.
+(3)
+4
+
+Figure 1: Illustration of variational inference as the optimization-based approximation. The
+goal of variational inference is to find a member of the variational family that minimizes
+KL divergence with the target distribution.
+The KL divergence measures how different is the probability distribution q(θ | λ) from
+p(θ | y) (Kullback & Leibler 1951). Note that while we use the KL divergence to measure
+the similarity between two densities, it is not a metric because the KL divergence is not
+symmetric and does not satisfy the triangle inequality. In fact, the order of q(θ | λ) and
+p(θ | y) in Equation (2) is deliberate as it leads to taking the expectation with respect
+to the variational distribution q(θ | λ). One can naturally think of reversing the roles of
+q(θ | λ) and p(θ | y). However, this leads to a “different kind” of variational inference called
+expectation propagation (Minka (2001)), which loses computational efficiency of variational
+inference defined in Equation (2).
+In a nutshell, rather than sampling, variational inference approximates densities us-
+ing optimization. See Figure 1 for a graphical illustration, i.e., by finding the values of
+variational parameters from λinit to λ∗ through optimization which lead to a variational
+distribution q(θ | λ) that is close to the target posterior distribution p(θ | y) defined by
+the smallest KL divergence. Finding the optimal q∗ is done in practice by maximizing an
+equivalent objective function, L(λ), the evidence lower bound (ELBO), because the KL
+5
+
+q(0|入)
+Optimization
+init
+Smallest KL
+p(0ly)divergence is intractable as it requires the evaluation of the marginal distribution p(y):
+L(λ) =
+Eq[log p(y, θ) − log q(θ|λ)]
+=
+Eq[log p(y|θ)]
+�
+��
+�
+Expected log-likelihood of data
+−
+KL(q(θ|λ)||p(θ))
+�
+��
+�
+KL div. between the variational and prior densities
+.
+(4)
+The ELBO is the sum between the negative KL divergence of the variational density from
+the target density and the log of the marginal density p(y). Since the term log p(y) is
+constant with respect to q(θ | λ), the objective functions in Equation (3) and Equation (4)
+are equivalent. Examining the ELBO also reveals the intuition behind variational inference.
+On the one hand, the first term in Equation (4) encourages the variational approximation
+to place mass on parameter values that maximize the sampling density p(y | θ). On the
+other hand, the second term in Equation (4) prefers closeness of the variational density to
+the prior. Therefore, the ELBO shows a similar tension between the sampling density and
+the prior known in Bayesian inference.
+2.2
+Variational Families with Pedagogical Recommendations
+We now move on to the implementation details of variational inference starting with the
+selection of the variational family q(θ | λ). This choice is crucial as it affects the complexity
+of optimization outlined in Section 2.1 as well as the quality of variational approximation.
+Mean-field Variational Family
+By far the most popular is the mean-field variational family which assumes that all the
+unknown parameters are mutually independent, each approximated by its own univariate
+variational density:
+q(θ | λ) =
+m
+�
+i=1
+q(θi | λi).
+(5)
+For example, a typical choice for real-valued parameters is the normal variational family
+q(θ | µ, σ2) and the log-normal or Gamma for non-negative parameters. The main advan-
+tage of the mean-field family is in its simplicity as it requires only a minimum number
+of parameters to be estimated (no correlation parameters) and often leads to uncompli-
+cated optimization. However, the mutually independent parameter assumption comes at
+6
+
+a price because the mean-field family cannot capture relationships between model param-
+eters. To illustrate the pitfalls of mean-field approximation, consider a simple case of a
+two-dimensional normal target density with highly correlated components. Figure 2 shows
+the optimal mean-field variational approximation given by the product of two normal den-
+sities. One can clearly see that the optimal variational densities match well with the means
+of the target density, but the marginal variances are underestimated. To further understand
+this common flaw of mean-field approximation, consider the definition of KL divergence
+in Equation (3). The objective function penalizes more larger density in q(θ | λ) in areas
+where p(θ | y) has low density than the opposite direction (recall that the expectation is
+taken with respect to the variational density).
+Recommendation for Undergradudate Instruction
+It is worth noting that the development of new variational families which improves on the
+trade-off between complexity and expressiveness of variational approximations has been
+a fruitful and active area of research. To keep the scope of the undergraduate one-week
+variational inference module manageable to both the students and the instructors, we
+recommend solely focusing on the mean-field approximation. For interested students who
+want to explore further, we encourage the instructors to refer them to the recent work
+of Ambrogioni et al. (2021) that provides a detailed discussion on many state-of-the-art
+variational families and their associated implementation challenges.
+2.3
+ELBO Optimization with Pedagogical Recommendations
+Besides the choice of variational family, another key implementation detail to address is
+the way in which we find the member of the variational family that maximizes the ELBO.
+Since this is a fairly general optimization problem, one can in principle use any optimization
+procedure. In the variational inference literature, the coordinate ascent and the gradient
+ascent procedures are the most prominent and widely used (Blei et al. (2017)).
+7
+
+3
+2
+1
+0
+1
+2
+3
+1
+3
+2
+1
+0
+1
+2
+3
+2
+2
+0
+2
+1
+0.0
+0.2
+0.4
+0.6
+Density
+2
+0
+2
+2
+Target density
+M-F approximation
+Figure 2: Mean-field variational approximation of a two-dimensional normal target density.
+The figure illustrates the common pitfall of the mean-field approximation in situations with
+correlated model parameters.
+8
+
+Corrdinate Ascent
+The coordinate ascent approach is based on the simple idea that one can maximize ELBO,
+which is a multivariate function, by cyclically maximizing it along one direction at a time.
+Starting with initial values (denoted by superscript 0) of the m variational parameters λ0
+λ0 = (λ0
+1, . . . , λ0
+m),
+one obtains the (k + 1)th updated value of variational parameters by iteratively solving
+λk+1
+i
+= arg max
+x
+L(λk+1
+1
+, . . . , λk+1
+i−1 , x, λk
+i+1, . . . , λk
+m),
+which can be accomplished without using gradients (Blei et al. 2017).
+Gradient Ascent
+Variational inference via gradient ascent uses the standard iterative optimization algorithm
+based on the idea that the ELBO grows fastest in the direction of its gradient (Hoffman
+et al. 2013). In particular, the update of variational parameters λ at the (k +1)th iteration
+is given by
+λk+1 ← λk + η × ∇λL(λk),
+where ∇λL(λ) is the ELBO gradient, and η is the step size which is also called the learning
+rate. The step size controls the rate at which one updates the variational parameters.
+For both coordinate and gradient ascent, we typically declare convergence of variational
+parameters once the change in ELBO falls below some small threshold (Blei et al. 2017).
+Recommendation for Undergradudate Instruction
+Our recommendation for an undergraduate variational inference module is to take the
+route of gradient ascent. This pedagogical choice is guided by our combined experience of
+teaching statistical modeling, Bayesian statistics, and data science at various undergradu-
+ate levels to students with diverse statistical backgrounds. Our recommendation has also
+taken into account the pedagogical advantages and disadvantages of gradient ascent and
+coordinate ascent for undergraduates: Variational inference via coordinate ascent, while
+conceptually straightforward, requires non-trivial and model-specific derivations which can
+9
+
+easily obscure the overall goal of this one-week module to expand students’ exposure to
+the state-of-the-art approximate inference for probabilistic models; gradient-based varia-
+tional inference, in contrast, leads to a black-box optimization that does not require any
+model-specific derivations due to an extensive autodifferentiation capabilities of modern
+statistical software such as RStan (Stan Development Team 2022) and Python packages
+PyTorch (Paszke et al. 2019) and TensorFlow (Abadi et al. 2015), to name a few.
+We believe that from an undergraduate-level pedagogical perspective, gradient descent
+reflects better the current data science pipeline and allows the instruction to be focused
+on conceptual understanding of variational inference rather than technical details.
+Of
+course, using gradient-based optimization requires the students to be familiar with partial
+derivatives. Such a pre-requisite potentially restricts the audience for our module to a course
+with a multivariable calculus prerequisite. Nevertheless, we believe that an instructor with
+sufficient preparation can explain the basics behind gradient ascent to an audience with a
+minimal calculus background.
+3
+Class Activity:
+A Probabilistic Model for Count
+Data with Variational Inference
+In this section, we provide a fully developed hands-on class activity with variational in-
+ference for count data. Starting with a motivating example in Section 3.1, we give an
+overview of the Gamma-Poisson model in Section 3.2, and discuss details of the variational
+inference of this model in Section 3.3, illustrated with an R Shiny app we have developed
+for instruction purpose. Instructors can adopt and adapt this class activity based on these
+materials tailored to their needs.
+3.1
+A Motivating Example
+To illustrate how ELBO optimization leads to a good approximation of target posterior
+distribution, we consider Poisson sampling with a Gamma prior, which is a popular one-
+parameter model for count data (Gelman et al. 2013, Albert & Hu (2019), Johnson et al.
+(2022)). To get started, we provide the following motivating example:
+10
+
+Our task is to estimate the average number of active users of a popular mas-
+sively multiplier online role-playing game (mmorpg) playing between the peak
+evening hours 7 pm and 10 pm. This information can help game developers
+in allocating server resources and optimizing user experience. To estimate the
+average number of active users, we will consider the counts (in thousands) of
+active players collected during the peak evening hours over a two-week period in
+the past month.
+We have chosen the Gamma-Poisson model as the probabilistic model in this class ac-
+tivity for two reasons. First, the Gamma-Poisson model is relatively easy to understand for
+students with an elementary knowledge of probability distributions. Second, the Gamma
+is a conjugate prior for Poisson sampling which means that one can derive the exact poste-
+rior distribution (another Gamma) and check the fidelity of variational approximation by
+comparing to the analytical Gamma solution. The learning objective of this class activity
+is to get students familiarized with various aspects of variational inference presented in Sec-
+tion 2, such as ELBO and variational family, with a simple example. Afterwards, students
+are better prepared to move on to more realistic scenarios, such as document clustering,
+described in Section 4.
+3.2
+Overview of the Gamma-Poisson Model
+We now provide an overview of the Gamma-Poisson model which can be readily turned
+into a class lecture. Suppose that y = (y1, . . . , yn) represent the observed counts in n time
+intervals where the counts are independent, and each yi follows a Poisson distribution with
+the same rate parameter θ > 0. The joint probability mass function of y = (y1, . . . , yn) is
+p(y | θ) =
+n
+�
+i=1
+p(yi | θ) ∝ θ
+�n
+i=1 yie−nθ.
+(6)
+The posterior distribution for the rate parameter θ is our inference target as θ represents
+the expected number of counts that occur during the given time intervals. Note that the
+Poisson sampling relies on several assumptions about the sampling process.
+First, one
+assumes that the time interval is fixed. Second, the counts occurring during different time
+11
+
+intervals are independent. Lastly, the rate θ at which the counts occur is constant over
+time.
+The Gamma-Poisson conjugacy states that if θ follows a Gamma prior distribution with
+shape and rate parameters α and β, it can be shown that the posterior distribution p(θ | y)
+will also have a Gamma density. Namely, if
+θ ∼ Gamma(α, β),
+(7)
+then
+θ | y ∼ Gamma(α +
+n
+�
+i=1
+yi, β + n).
+(8)
+In other words, given α, β, and y, one can derive the analytical solution to the posterior
+of p(θ | y) and can subsequently sample from Gamma(α + �n
+i=1 yi, β + n) to get posterior
+samples of θ. While no approximation is needed, it serves as a good example of illustrating
+how variational inference works in such a setting and allows evaluations of the performance
+of variational inference.
+3.3
+Variational Inference of the Gamma-Poisson Model
+Recall from Section 2 that variational inference approximates the (unknown) posterior
+distribution of a parameter by a simple family of distributions. In this case, we will ap-
+proximate the posterior distribution p(θ | y) by a log-normal distribution with mean µ and
+standard deviation σ:
+q(θ | µ, σ) =
+1
+θσ
+√
+2πe− (ln θ−µ)2
+2σ2
+.
+(9)
+The log-normal distribution is a continuous probability distribution of a random variable
+whose logarithm is normally distributed. It is a popular variational family for non-negative
+parameters because it can be expressed as a (continuously) transformed normal distribu-
+tion, and therefore it is amenable to automatic differentiation. Automatic differentiation is
+a computation method for derivatives in computer programs that relies on the application
+of chain rule in differential calculus. It provides accurate and fast numerical derivative
+evaluations that leads to machine learning algorithms (such as variational inference) that
+do not require users to manually work out and code derivatives (Kucukelbir et al. 2017,
+Baydin et al. (2018)).
+12
+
+0.0
+0.1
+0.2
+0.3
+40
+50
+60
+θ
+Density
+Prior
+True posterior
+log−normal(3.7, 0.05)
+ELBO maximization: log−normal(3.9, 0.04)
+Figure 3: Variational approximation based on the motivating scenario of mmorpg’s player
+activity. The true Gamma(792, 100) posterior and the prior Gamma(100,2) distributions
+are included.
+In the supplementary materials, we provide an accompanying in-class handout and an R
+Shiny app based on the motivating scenario of mmorpg described in Section 3.1. The first
+two parts of the handout present the motivating example and the overview of the Gamma-
+Poisson model. In the third part of the handout, students carry out exact posterior inference
+for the unknown rate parameter θ using a small dataset of observed counts of mmorpg’s
+active players. In the fourth and final part, students find variational approximation of
+p(θ | y) and check how well their approximation matches the true posterior distribution.
+Figure 3 shows the final variational approximation compared to the true Gamma(792, 100)
+posterior distribution from the handout example. We can see, on the one hand, the resulting
+log-normal(3.9, 0.04) distribution (the black dash line) that maximizes the ELBO visually
+overlaps with the true posterior (ELBO = −42.52, KL divergence < 0.001). On the other
+hand, another member of the variational family, the log-normal(3.7, 0.05) distribution (the
+blue dot-dash line; with ELBO = −57.55 and KL divergence = 15.085), clearly differs from
+the target. This example illustrates the good performance of variational inference through
+optimization for the Gamma-Poisson count model.
+13
+
+The design of this class activity is guided by the active-learning principles listed in
+Section 1 and the goal is to give students their first hands-on experience with variational
+inference without the need of coding. Specifically, we include open-ended questions that
+focus on problem-solving and create opportunities for students to collaborate with peers.
+Moreover, the accompanying R Shiny app provides appropriate and sufficient scaffolding so
+that students can concentrate on conceptual understanding instead of the technical details,
+which follows our pedagogical recommendations in Section 2.
+We now turn to a guided R lab to illustrate the use of variational inference for a more
+realistic use case of document clustering, applied to a sample of Associated Press newspaper
+articles.
+4
+Lab: Document Clustering
+Among the many models approximated by variational inference techniques, Latent Dirichlet
+Allocation (LDA) might be one of the most popular (Blei et al. 2003). LDA is a mixed-
+membership clustering model, commonly used for document clustering. Specifically, LDA
+models each document to have a mixture of topics, where each word in the document is
+drawn from a topic based on the mixing proportions (Stan Development Team 2022). While
+the LDA model is relatively easy and straightforward to follow, using conventional MCMC
+estimation techniques has proven to be too computationally demanding due to the large
+number of parameters involved. Therefore, researchers and practitioners turn to variational
+inference techniques when using LDA for document clustering (Blei et al. 2003).
+In Section 4.1, we briefly introduce the LDA model following the presentation in Stan
+Development Team (2022). In Section 4.2, we present an LDA application to a collection
+of Associate Press newspaper articles where variational inference is implemented by the
+cmdstanr R package. For brevity, we focus on the interpretation of results and discuss
+pedagogical considerations and leave a Stan script for the LDA model and the details of
+the guided lab assignment with R code in the supplementary materials.
+14
+
+4.1
+Overview of the LDA model
+The LDA model considers K topics for M documents made up of words drawn from a
+vocabulary of V distinct words. For a document m, a topic distribution θm over K topics
+is drawn from a Dirichlet distribution,
+θm ∼ Dirichlet(α),
+(10)
+where �K
+k=1 θm,k = 1 (0 ≤ θm,k ≤ 1) and α is a vector of length K with positive values.
+Each of the Nm words {wm,1, . . . , wm,Nm} in document m is then generated indepen-
+dently conditional on θm. To do so, first, the topic zm,n for word wm,n in document m is
+drawn from
+zm,n ∼ categorical(θm),
+(11)
+where θm is the document-specific topic-distribution defined in Equation (14).
+Next, the word wm,n in document m is drawn from
+wm,n ∼ categorical(φz[m,n]),
+(12)
+which is the word distribution for topic zm,n. Note that z[m, n] in Equation (16) refers to
+zm,n.
+Lastly, a Dirichlet prior is given to distributions φk over words for topic k as
+φk ∼ Dirichlet(β),
+(13)
+where β is the prior a vector of length V (i.e., the total number of words) with positive
+values. Figure 10 shows a graphical model representation of LDA.
+4.2
+Clustering of Associated Press Newspaper Articles
+As a realistic application of variational inference, we consider a collection of 2246 Asso-
+ciated Press newspaper articles to be clustered using the LDA model.
+The dataset is
+(conveniently) part of the topicmodels R package. We believe this dataset is well suited
+to demonstrate the capabilities of variational inference in the classroom as it is too large
+for the MCMC approximation to be feasible but small enough for the variational inference
+to take just a few minutes to converge. For brevity, we highlight the results based on a
+15
+
+Figure 4: Graphical model representation of LDA. The largest box represents the docu-
+ments. On the left, the inner box represents the topics and words within each document.
+On the right, the box represents the topics.
+two-topic LDA model (i.e., K = 2) and leave the details to the guided lab in the sup-
+plementary materials. The number of topics is set to 2 for demonstration purposes and
+simplicity of interpretations. Comparing LDA with a different number of topics is often
+done with metrics such as semantic coherence or held-out data likelihood (Mimno et al.
+2011). While such a comparison is beyond the scope of this lab, interested students are
+encouraged to explore mentored by the instructors.
+Figure 5 shows the evolution of ELBO for the two-topic LDA model which converged
+after a little bit over 100 iterations of the gradient ascent algorithm described in Section
+2.3. On a standard laptop computer, this typically takes between 5-10 minutes depending
+on the CPU speed. We recommend running the algorithm repeatedly (i.e., 2-3 times) with
+a different random seed in the classroom and discussing the dependency of variational
+inference on initial values of variational parameters which can occur in practice.
+Figures 6 and 7 are examples of graphical displays of the topics that were extracted
+from the collection of articles based with the LDA. In particular, Figure 6 shows the 10
+most common words for each topic; that is, the parts of distribution φk, for k ∈ {1, 2},
+with the largest mass. Figure 7 displays similar information for the 20 most common words
+for each topic in the form of a word cloud. The most common words in topic 1 include
+people, government, president, police, and state, suggesting that this topic may represent
+political news. In contrast, the most common words in topic 2 include percent, billion,
+16
+
+N
+α
+m
+β
+K
+M−1120000
+−1100000
+−1080000
+−1060000
+−1040000
+0
+30
+60
+90
+Iteration
+ELBO
+Figure 5: The evolution of ELBO for the two-topic LDA model based on 2246 Associated
+Press newspaper articles.
+million, market, American, and states, hinting that this topic may represent news about
+the US economy.
+5
+Concluding Remarks
+In this paper, we present a newly-developed one-week course module that exposes un-
+dergraduate students to approximation via variational inference. The proposed module
+is self-contained in the sense that it encourages and empowers potential instructors to
+adopt and adapt the module as we provide an overview of variational inference, an active-
+learning-based class activity with an R Shiny app, and a guided lab based on a realis-
+tic application with R code (see the supplementary materials or https://github.com/
+kejzlarv/variational_inference_module). Its design is rooted in the best practices of
+active learning that have been demonstrated to improve student learning and engagement.
+The module can be integrated into any intermediate- or upper-level undergraduate
+course where students learn probabilistic models (including logistic regression, Bayesian
+classifiers, neural networks, or models for natural language processing), such as Bayesian
+statistics, multivariate data analysis, and data science courses. The applications discussed
+17
+
+Topic 1
+Topic 2
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+american
+billion
+first
+states
+united
+last
+million
+year
+new
+percent
+told
+officials
+soviet
+state
+government
+police
+two
+president
+people
+i
+Word distributions ( ϕ )
+Word
+Figure 6: Word distributions based on the two-topic LDA model. The 10 most common
+words are displayed.
+Figure 7: World clouds consisting of the 20 most common words for each of the two topics
+extracted by the LDA.
+18
+
+market company
+south house
+first
+t united
+partygovernment
+billion
+daylastyear
+I federal
+told
+people
+rs
+york
+eal
+percentreport
+twO I police city
+@national
+bush
+new
+news
+president
+states
+court
+soviet
+military
+million
+S
+week
+officials say
+american
+time
+thursdayin these courses are typically limited to scenarios with relatively small datasets, since
+the required use of MCMC does not scale well to large datasets. Given the popularity
+and scalability of variational inference, we hope that undergraduate instructors adopting
+and adapting this module will be able to integrate more realistic and fun use cases in
+their classrooms. Moreover, the references and further readings provided in this paper are
+readily available resources for a deeper dive of variational inference by interested students
+with appropriate mentoring by their undergraduate instructors.
+References
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+22
+
+Supplementary Materials for Introducing Variational Inference In
+Undergraduate Statistics and Data Science Curriculum
+The supplementary materials include: 1) Details of the class activity on probabilistic
+model for count data with variational inference, introduced in Section 3 in the main text;
+2) The manual of the R shiny app we have developed for the module, mentioned in Section
+3 in the main text; and 3) Details of the guided R lab of the LDA application to a sample
+of the Associated Press newspaper articles with variational inference, presented in Section
+4 in the main text.
+6
+Class Activity: Probabilistic Model for Count Data
+with Variational Inference
+The goal of this activity is to illustrate variational inference on a simple example of Gamma-
+Poisson conjugate model, which is a popular model for count data.
+6.1
+A Motivating Example
+Our task is to estimate the average number of active users of a popular massively multiplier
+online role-playing game (mmorpg) playing between the peak evening hours 7 pm and 10
+pm.
+This information can help the game developers in allocating server resources and
+optimizing user experience. To make this estimate, we will consider the following counts
+(in thousands) of active players collected during the peak evening hours over a two-week
+period past month.
+Sun
+Mon
+Tue
+Wed
+Thu
+Fri
+Sat
+Week 1
+50
+47
+46
+52
+49
+55
+53
+Week 2
+48
+45
+51
+50
+53
+46
+47
+6.2
+Overview of the Gamma-Poisson Model
+Sampling density:
+23
+
+Suppose that y = (y1, . . . , yn) represent the observed counts in n time intervals where
+the counts are independent, then each yi follows a Poisson distribution with rate θ > 0.
+Namely,
+yi | θ ∼ Poisson(θ)
+• E(yi | θ) = θ
+• Var(yi | θ) = θ
+Prior distribution:
+θ ∼ Gamma(α, β)
+• α > 0 is the shape parameter
+• β > 0 is the rate parameter
+• E(θ) = α
+β
+• Var(θ) =
+α
+β2
+Posterior distribution:
+θ | y1, . . . , yn ∼ Gamma(α +
+n
+�
+i=1
+yi, β + n)
+6.3
+Exact Inference with the Gamma-Poisson Model
+We will start by selecting a prior distribution for the unknown average number of active
+users. Suppose that we elicit an expert’s advice on the matter, and they tell us that a
+similar mmorpg has typically about 50,000 users during peak hours. However, they are
+not too sure about that, so the interval between 45,000 and 55,000 users should have a
+reasonably high probability. Suppose that we elicit an expert’s advice on the matter, and
+they tell us that a similar mmorpg has typically about 50,000 users during peak hours.
+However, they are not too sure about that, so the interval between 45,000 and 55,000 users
+should have a reasonably high probability. This reasoning leads to a Gamma(100, 2) as a
+reasonable prior for the average number of active users.
+24
+
+Task 1: Explain the reasoning behind using Gamma(100, 2) as the prior distri-
+bution.
+Task 2: Use the information above to find the exact posterior distribution for
+the average number of active users.
+Task 3: What are the mean and standard deviation of the posterior distribu-
+tion that you just obtained? What is your recommendation about the typical
+number of active users playing the mmorpg between the peak evening hours
+7pm and 10pm?
+6.4
+Variational Inference with the Gamma-Poisson Model
+Variational inference approximates the (unknown) posterior distribution of a parameter
+by a simple family of distributions. In this case, we will try to approximate the posterior
+distribution of the mmorpg’s average number of active users between the peak hours θ by
+a log-normal distribution with mean µ and standard deviation σ. Log-normal distribution
+is a continuous probability distribution of a random variable whose logarithm is normally
+distributed. It also happens to be a popular variational family for non-negative parameters
+as it is amenable to autodifferentiation. Since we know exactly how the posterior distri-
+bution for Gamma-Poisson model looks like, we will be able to check the fidelity of the
+variational approximation. Use the accompanying applet titled Variational Inference with
+Gamma-Poisson Model for count data to complete the following task.
+25
+
+Task 4: Use the sliders in the applet to manually find the member of a log-
+normal variational family that well approximates the posterior distribution of
+θ. What is your strategy?
+Task 5: Compare your approximation with a neighbor. Whose approximation
+is closer to the exact posterior distribution of θ? How are you deciding?
+Task 6: Check the Fit a variational approximation box in the applet to find
+the variational approximation using the gradient ascent algorithm. How close
+was the variational approximation that you found manually to the one found
+here?
+7
+Manual of the R shiny app
+This document describes the elements of R Shiny applet that accompanies the “Proba-
+bilistic Model for Count Data with Variational Inference” class activity. Note that the
+numbering in Section 7.1 and Section 7.2 corresponds to the numbered boxes in Figure 8
+and Figure 9.
+7.1
+Manual search for variational approximation
+1. Sliders to control the mean µ and the standard deviation σ of log-normal variational
+family.
+2. The ELBO and KL divergence values for variational approximation based on the
+mean and standard deviations selected in box 1.
+3. A plot that displays the true Gamma(792, 100) posterior distribution, the Gamma(100, 2)
+prior distribution, and the variational approximation based on the selection in box 1.
+4. A checkbox to display the results of ELBO maximization via gradient ascent algo-
+rithm. The resulting variational approximation is plotted in box 3.
+26
+
+Figure 8: The applet is based on the class activity presented in Section 1 of the supplemen-
+tary materials. The applet visual before checking the “Fit a variational approximation“
+checkbox is displayed.
+7.2
+Variational approximation based on ELBO maximization
+5. The resulting mean µ, standard deviation σ, and ELBO values of variational approx-
+imation based on ELBO maximization.
+6. A plot depicting ELBO values for each iteration of the gradient ascent algorithm.
+8
+Lab: Document Clustering
+The goal of this lab is to gain a practical experience with variational inference on a real-
+istic use case based on Latent Dirichlet Allocation (LDA) and implement the model in R
+applied to a dataset of documents. To do so, we consider a collection of 2246 Associated
+Press newspaper articles to be clustered using the LDA model. The dataset is part of the
+topicmodels R package. You can load the dataset AssociatedPress with the following R
+command.
+data("AssociatedPress", package = "topicmodels")
+27
+
+Variational Inference with Gamma-Poisson Model for count data
+Variational approximation using log-normal variational family:
+Prior
+True posterior-. VI- Manual
+3
+μ
+1
+0.3-
+3.5
+3.7
+4.2
+3.82
+3.9
+4.14 4.2
+0.2
+6
+PDF
+0.01
+0.05
+0.1
+0.1
+ELBO value:
+2
+-57.496
+0.0
+40
+50
+KL Divergence value:
+60
+0
+15.031
+OFit a variational approximation
+4Figure 9: The applet visual after checking the “Fit a variational approximation“ checkbox
+is displayed.
+28
+
+Variational Inference with Gamma-Poisson Model for count data
+Variational approximation using log-normal variational family:
+:Prior
+True posterior -: VI-Manual - VI -ELBO maximization
+μ
+0.3-
+3.5
+3.7
+4.2
+3.5
+ 3.58 3.66 3.74 3.82 3.9 3.984.064.14 4.2
+0.2
+PDF
+0.01
+0.05
+0.1
+0.1
+0.010.020.030.040.050.060.070.080.090.1
+ELBO value:
+-57.496
+0.0-
+40
+50
+60
+KL Divergence value:
+15.031
+ Fit a variational approximation
+0
+D
+Results of ELBo maximization via gradient ascent:
+5
+μ
+-50
+3.901
+0
+.BO
+-100
+0.035
+E
+ELBO value:
+-150
+-42.52
+-200
+0
+50
+100
+150
+200
+Iteration8.1
+Overview of the LDA model and Stan script
+The LDA is a mixed-membership clustering model, commonly used for document clustering.
+LDA models each document to have a mixture of topics, where each word in the document is
+drawn from a topic based on the mixing proportions. Specifically, the LDA model assumes
+K topics for M documents made up of words drawn from V distinct words. For document
+m, a topic distribution θm is drawn over K topics from a Dirichlet distribution,
+θm ∼ Dirichlet(α),
+(14)
+where �K
+k=1 θm,k = 1 (0 ≤ θm,k ≤ 1) and α is the prior a vector of length K with positive
+values.
+Each of the Nm words {wm,1, . . . , wm,Nm} in document m is then generated indepen-
+dently conditional on θm. To do so, first, the topic zm,n for word wm,n in document m is
+drawn from
+zm,n ∼ categorical(θm),
+(15)
+where θm is the document-specific topic-distribution defined in Equation (14).
+Next, the word wm,n in document m is drawn from
+wm,n ∼ categorical(φz[m,n]),
+(16)
+which is the word distribution for topic zm,n. Note that z[m, n] in Equation (16) refers to
+zm,n.
+Lastly, a Dirichlet prior is given to distributions φk over words for topic k as
+φk ∼ Dirichlet(β),
+(17)
+where β is the prior a vector of length V (i.e., the total number of words) with positive
+values. Figure 10 shows a graphical model representation of LDA.
+Below, we include the Stan script for the LDA model provided by Stan Development
+Team (2022). Note that Stan supports the calculation of marginal distributions over the
+continuous parameters by summing out the discrete parameters in mixture models (Stan
+Development Team 2022). This process corresponds to the gamma parameter in the Stan
+script below. We refer interested readers to Stan Development Team (2022) for further
+details.
+29
+
+Figure 10: Graphical model representation of LDA. The outer box represents the docu-
+ments. On the left, the inner box represents the topics and words within each document.
+On the right, the box represents the topics.
+data {
+int<lower=2> K;
+// number of topics
+int<lower=2> V;
+// number of words
+int<lower=1> M;
+// number of docs
+int<lower=1> N;
+// total word instances
+int<lower=1, upper=V> w[N];
+// word n
+int<lower=1, upper=M> doc[N];
+// doc ID for word n
+vector<lower=0>[K] alpha;
+// topic prior vector of length K
+vector<lower=0>[V] beta;
+// word prior vector of length V
+}
+parameters {
+simplex[K] theta[M];
+// topic distribution for doc m
+simplex[V] phi[K];
+// word distribution for topic k
+}
+model {
+for (m in 1:M)
+theta[m] ~ dirichlet(alpha);
+30
+
+N
+α
+m
+β
+K
+Mfor (k in 1:K)
+phi[k] ~ dirichlet(beta);
+for (n in 1:N) {
+real gamma[K];
+for (k in 1:K)
+gamma[k] = log(theta[doc[n], k]) + log(phi[k, w[n]]);
+target += log sum exp(gamma);
+// likelihood;
+}
+}
+8.2
+Variational inference with the LDA model
+For demonstration purposes, we shall start with a two-topic LDA model (i.e., K = 2).
+Before that, we recommend removing the words from AssociatedPress datasets that are
+rare using the function removeSparseTerms() from the tm package. These words have a
+minimal effect on the LDA parameter estimation. Nevertheless, they increase the compu-
+tational cost of variational inference and therefore should be removed using the following
+R command.
+dtm <- removeSparseTerms(AssociatedPress, 0.95)
+We are now ready to fit the LDA model using variational inference capabilities of the
+cmdstanr package. The following code achieves the goal:
+LDA model cmd <- cmdstan model(stan file = "LDA.stan")
+N TOPICS <- 2
+data <- list(K = N TOPICS,
+V = dim(dtm)[2],
+M = dim(dtm)[1],
+N = sum(dtm$v),
+w = rep(dtm$j,dtm$v),
+31
+
+doc = rep(dtm$i,dtm$v),
+#according to Griffiths and Steyvers(2004)
+alpha = rep(50/N TOPICS,N TOPICS),
+beta = rep(1,dim(dtm)[2])
+)
+vi fit <- LDA model cmd$variational(data = data,
+seed = 1,
+output samples = 1000,
+eval elbo = 1,
+grad samples = 10,
+elbo samples = 10,
+algorithm = "meanfield",
+output dir = NULL,
+iter = 1000,
+adapt iter = 20,
+save latent dynamics=TRUE,
+tol rel obj = 10^-4)
+The “LDA.stan” file contains the Stan script for the LDA model provided in Section
+8.1. We recommend the usage of the R help to get familiar with the variational() method
+of the cmdstan model() function. The variable vi fit contains the results of variational
+approximation of the LDA parameters. For example, one can obtain the word distributions
+for the each of the topics with vi fit$summary("phi").
+Finally, to access the ELBO values, use the following:
+vi diag <- utils::read.csv(vi fit$latent dynamics files()[1],
+comment.char = "#")
+ELBO <- data.frame(Iteration = vi diag[,1], ELBO = vi diag[,3])
+32
+
+Task 1: Use a graphical display to show the 10 most common words for each
+of the two-topics and their probabilities.
+Task 2: Use the function wordcloud() from the wordcloud package and display
+the most common words for each of the topics as a world clowd. What kinds
+of articles do these topics represent?
+Task 3: Fit a three-topic LDA model, display the most common words for each
+of the topics. How do the results differ from the two-topic LDA?
+Task 4 (Advanced): Use the three-topic LDA model and diplay the topic preva-
+lence among the 2246 Associated Press articles. That is, show what proportions
+of articles fall under each topic.
+All necessary R code for fitting the LDA model to the Associated Press sample, including
+the graphical displays shown in the main text, is included in a separate R script file called
+LDA LAB.R available as a part of the supplementary materials. We also include a printout
+of the R script below for interested readers.
+library(cmdstanr)
+# Checking integrity of installation of cmdstanr
+check cmdstan toolchain()
+install cmdstan(cores = 2)
+cmdstan path()
+cmdstan version()
+# Auxiliary packages
+library(tm)
+library(tidyverse)
+library(tidytext)
+library(topicmodels)
+33
+
+## Get data
+data("AssociatedPress", package = "topicmodels")
+## Removing rare words from the vocabulary
+dtm <- removeSparseTerms(AssociatedPress, 0.95)
+dim(dtm)
+## Input for stan model
+N TOPICS <- 2
+data <- list(K = N TOPICS,
+V = dim(dtm)[2],
+M = dim(dtm)[1],
+N = sum(dtm$v),
+w = rep(dtm$j,dtm$v),
+doc = rep(dtm$i,dtm$v),
+#according to Griffiths and Steyvers(2004)
+alpha = rep(50/N TOPICS,N TOPICS),
+beta = rep(1,dim(dtm)[2])
+)
+### VB fit
+LDA model cmd <- cmdstan model(stan file = "LDA.stan")
+LDA model cmd$print()
+vb fit <- LDA model cmd$variational(data = data,
+seed = 1,
+output samples = 1000,
+eval elbo = 1,
+grad samples = 10,
+34
+
+elbo samples = 10,
+algorithm = "meanfield",
+output dir = NULL,
+iter = 1000,
+adapt iter = 20,
+save latent dynamics=TRUE,
+tol rel obj = 10^-4)
+# Plotting ELBO
+vb diag <- utils::read.csv(vb fit$latent dynamics files()[1],
+comment.char = "#")
+ELBO <- data.frame(Iteration = vb diag[,1],
+ELBO = vb diag[,3])
+ggplot(data = ELBO, aes(x = Iteration, y = ELBO)) + geom line(lwd=1.5) +
+theme(text = element text(size = 20),
+panel.background = element rect(fill = "transparent",
+color = "lightgrey"),
+panel.grid.major = element line(colour = "lightgrey")) +
+xlim(0,110)
+## Accessing parameters
+vb fit$summary("theta") # dim: M-by-K
+vb fit$summary("phi") # dim: K-by-V
+## Word distribution per topic
+V <- dim(dtm)[2]
+odd rows <- rep(c(1,0), times = V)
+Topic1 <- vb fit$summary("phi")[odd rows == 1,]
+Topic2 <- vb fit$summary("phi")[odd rows == 0,]
+35
+
+word probs <- data.frame(Topic = c(rep("Topic 1", V),
+rep("Topic 2", V)),
+Word = rep(dtm$dimnames$Terms,N TOPICS),
+Probability = c(Topic1$mean, Topic2$mean))
+# Selecting top 10 words per topic
+top words <- word probs %>% group by(Topic) %>% top n(10) %>%
+ungroup() %>% arrange(Topic, -Probability)
+top words %>%
+mutate(Word = reorder within(Word, Probability, Topic)) %>%
+ggplot(aes(Probability, Word, fill = factor(Topic))) +
+geom col(show.legend = FALSE) +
+facet wrap(~ Topic, scales = "free") +
+scale y reordered() + theme(text = element text(size = 15)) + xlim(0,0.025) +
+xlab("Word distributions ( \u03d5 )")
+# Word Cloud display
+#install.packages("wordcloud")
+library(wordcloud)
+top words <- word probs %>% group by(Topic) %>% top n(20) %>%
+ungroup() %>% arrange(Topic, -Probability)
+mycolors <- brewer.pal(8, "Dark2")
+wordcloud(top words %>% filter(Topic == "Topic 1") %>% .$Word ,
+top words %>% filter(Topic == "Topic 1") %>% .$Probability,
+random.order = FALSE,
+color = mycolors)
+36
+
+mycolors <- brewer.pal(8, "Dark2")
+wordcloud(top words %>% filter(Topic == "Topic 2") %>% .$Word ,
+top words %>% filter(Topic == "Topic 2") %>% .$Probability,
+random.order = FALSE,
+color = mycolors)
+37
+

CdE3T4oBgHgl3EQfUQrt/content/tmp_files/2301.04450v1.pdf.txt

@@ -0,0 +1,720 @@
+Letter
+Optica
+1
+Subnanometer confinement and bundling of atoms in a
+Rydberg empowered optical lattice
+MOHAMMADSADEGH KHAZALI
+Institute for Research in Fundamental Sciences (IPM), Tehran 19538-33511, Iran
+Department of Physics, University of Tehran, North Kargar Ave., Tehran P.O. Box 14395-547, Iran
+Compiled January 12, 2023
+Optical lattices are the basic blocks of atomic quan-
+tum technology.
+The scale and resolution of these
+lattices are diffraction-limited to the light wavelength.
+Tight confinement of single sites in conventional lat-
+tices requires excessive laser intensity which in turn
+suppresses the coherence due to enhanced scattering.
+This article proposes a new scheme for atomic opti-
+cal lattice with sub-wavelength spatial structure. The
+potential is formed by the nonlinear optical response
+of the three-level Rydberg dressed atoms, which is
+not constrained by the diffraction limit of the driv-
+ing fields. The lattice consists of a 3D array of ultra-
+narrow Lorentzian wells with sub-nanometer widths.
+The scheme allows moving adjacent sites to close dis-
+tances with sub-nanometer resolution. These extreme
+scales are now optically accessible by a hybrid scheme
+deploying the dipolar interaction and optical twist of
+atomic eigenstates. The interaction-induced two-body
+resonance that forms the trapping potential, only oc-
+curs at a peculiar laser intensity, localizing the trap
+sites to ultra-narrow regions over the standing-wave
+driving field. The Lorentzian trapping potentials with
+2Å width and 30MHz depth are realizable with scatter-
+ing rates as low as 1Hz. The mentioned improvements
+allow quantum logic operations with Rydberg-Fermi
+interaction. These techniques are particularly demand-
+ing for the realization of atomtronics, quantum walks,
+Hubbard models, and neutral-atom quantum simula-
+tion.
+© 2023 Optical Society of America
+http://dx.doi.org/10.1364/ao.XX.XXXXXX
+1. INTRODUCTION
+The primary enabling technology in atomic quantum proces-
+sors is the coherent control of the position and motion of atoms
+by lasers. The underlying mechanism in conventional optical
+lattices is the ac-Stark shift of atomic levels formed by far-off-
+resonant laser fields. The diffraction limit, which is about the
+wavelength of the light, is what determines the scale and spatial
+resolution of such optical potential landscapes. This fundamen-
+tally limits the optical manipulation of atoms, affecting some of
+⟩
+|𝟓𝒔𝟐, 𝟏𝑺𝟎
+⟩
+|𝟓𝒔𝟓𝒑, 𝟑𝑷𝟏
+⟩
+|𝟓𝒔𝒏𝒔, 𝟑𝑺𝟏
+⟩
+|𝒈
+⟩
+|𝒑
+⟩
+|𝒆
+V
+−∆
+𝛀𝟏
+𝛀𝟐(x)
+(a)
+𝜃
+(b)
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.
+-1.2
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+Fig. 1. Ultra-tight confinement of atoms in an interaction-
+induced atomic lattice. (a) Rydberg dressing of ground state
+Sr atoms with standing-wave laser field (blue line) results
+in an interaction-induced periodic trapping potential that
+features sharp resonance at a narrow range of laser intensity.
+This would form ultra-narrow trapping wells (green line) that
+results in sub-nanometer atomic confinement. (b) The level
+scheme presents in-resonance two-photon Rydberg excitation.
+the quantum technology applications. For instance, in the re-
+cently proposed Rydberg-Fermi quantum simulator [2, 3], ultra-
+tight confinement of atoms within the single lobe of the Rydberg
+wave-function is required for high-fidelity scalable quantum
+processing. Tight confinement is also demanding for applica-
+tions that are based on distance selective interaction [4, 5] and
+controlled Rydberg anti-blockade operations [6]. Finally, tight
+confinement is demanded for improving the fidelity of neutral
+atom processors [7–10].
+This article presents the first scheme for ultra-tight sub-
+nanometer confinement of atoms in an optical lattice with dy-
+namic features to move pair of lattice sites close to each other
+at extreme scales of about 4Å which is the realm of solid-state
+crystals. The dynamic control of the lattice separation allows a
+new type of quantum gate operations powered by the remote
+Rydberg-Fermi spin-flip. The Rydberg-Fermi spin-flip has been
+observed in Bose Einstein Condensate (BEC) at inter-atomic
+distances of about 30nm [11]. However, the real application
+of this phenomenon in an atomic lattice quantum processor
+was elusive. The real application requires dynamic maneuver
+of the interatomic distance from micrometer scale during the
+laser-addressing of individual sites [12, 13] to 30nm over the
+interaction stage [11]. Furthermore, since the Rydberg-Fermi
+interaction is proportional to the Rydberg wave-function proba-
+bility amplitude, the interatomic distance must be fixed within
+nanometer-scale precision. The tight confinement and ultra-high
+precision of interatomic distances in the current lattice proposal
+arXiv:2301.04450v1  [quant-ph]  11 Jan 2023
+
+Letter
+Optica
+2
+opens new opportunities to develop nano-scale quantum tech-
+nologies of this type.
+Tight confinement of atoms in conventional optical lattices
+requires an extensive power i.e. the spatial width is inversely
+proportional to the quadruple root of the laser intensity. The
+drawback is the loss of coherence due to the enhanced scattering.
+In an alternative approach, this article deploys the nonlinear
+response of Rydberg-dressed atoms to the intensity of standing-
+wave driving field, as a means to form a lattice of ultra-narrow
+trapping potentials. The sub-wavelength resolution arises when
+the composition of eigenstates on two-atom basis twists rapidly
+at a specific light intensity to form interaction-induced reso-
+nance over a short length scale of the standing-wave. Unlike the
+conventional ac-Stark shift potentials, this interaction-induced
+potential is a quantum effect, with magnitude proportional to
+¯h. This effect forms 3D lattices with potential widths as small as
+the neutral atom radius. The proposed lattice features dynamic
+terms that bundle pairs of atoms and draw them near to sub-
+nanometer distances; the realm that used to be exclusive to solid
+state crystals.
+The recent advances in optical control of Rydberg atoms have
+opened a wide range of applications in quantum technology [14–
+21]. The required dipolar interaction in this proposal is formed
+by the in-resonance dressing of ground-state atoms with the
+highly excited Rydberg state [22, 23]. Rydberg dressing of a
+BEC with homogeneous laser lights could form triangular and
+quasi-ordered droplet crystals [23, 24]. However, this periodic
+structure would not be fixed in space. The spatial pattern of the
+driving field and intensity dependence of the potential would
+spatially pin the lattice sites to the nodes of standing wave.
+Therefore, the lattice structure would be fixed in the space. This
+feature is required for addressing individual sites in atomic
+processors.
+The atomic lattice scheme is based on dressing 88Sr atoms
+with the highly excited Rydberg level, see Fig. 1. In the two-
+photon in-resonance dressing scheme [22, 23], the single atom
+Hamiltonian is given by
+Hi/¯h = Ω1
+2 (σi
+gp + σi
+pg) + Ω2(x)
+2
+(σi
+ep + σi
+pe) − ∆σpp,
+(1)
+where σα,β = |α⟩⟨β| is the transition operator. The two Rabi
+frequencies Ω1,2 are applied by 689nm and 318nm lasers that are
+detuned from the intermediate state |p⟩ by ∆. With negligible
+Rydberg decay rates, the system would follow the dark eigen-
+state |d⟩ ∝ Ω2|g⟩ − Ω1|e⟩ with zero light-shift. In the limit of
+Ω1 ≪ Ω2, ground state atoms will be partially dressed by Ryd-
+berg states with the population of Pe = (Ω1/Ω2)2. The van-der
+Waals interaction between Rydberg atoms Vij = ¯hC6/r6
+ijσieeσj
+ee
+is a function of interatomic distance rij.
+The interaction of
+|5sns 3S1⟩ Rydberg atoms is repulsive. This strong interaction
+could exceed atom-light coupling over several micrometers of
+interatomic separations.
+The dynamic of the system under Rydberg interaction is gov-
+erned by the master equation of two-body density matrices. The
+two-body density matrices ρij = Tr¯i,¯jρ are obtained by tracing
+over all but i and j particles. The corresponding master equation
+would be given by
+∂tρij = − i
+¯h [Hi + Hj + Vij, ρij] + Li(ρij) + Lj(ρij)
+(2)
+The internal state dynamics are governed by single-particle dissi-
+pation described by Li operator acting on ith atom. The Liouvil-
+lian term Li(ρ) = ∑β D(cβ)ρi with D(c)ρi = cρic† − 1/2(c†cρi +
+0
+0.5
+1
+1.5
+x/Rc
+-1
+-0.5
+0
+U/|U0|
+2/2
+(a)
+-5
+-1 0 1
+5
+-1
+-0.5
+0
+Num
+Ana
+(b)
+-5
+-1 0 1
+5
+-1
+-0.5
+0
+Num
+Ana
+(c)
+Fig. 2. Interaction-induced atomic lattice. (a) The red line
+shows the spatial profile of the Rabi frequency Ω2(x) =
+Ω2c + Ω2sw| sin(kx sin(θ/2))|, where k = 2π/λ . The blue
+line shows the interaction of two atoms as a function of in-
+teratomic distance. When two atoms are within the soft-core
+radius and are both located at the nodes of the standing wave
+they experience a strong trapping potential. With a single
+atom per lattice site, the effective trapping interaction would
+be the sum of two-body interactions of all the sites within the
+±Rc distance. (b) The interaction induced resonance occurs at
+Ω2 = 2|∆|, with maximum depth of U0 = 3¯hΩ4
+1
+8∆γ2 and a HWHM
+of Ω2 − 2|∆| = ±γp. The analytical form of Eq. 5 and numerical
+calculation of the interaction potential (Eq. 3) presents a per-
+fect match. (c) Spatial form of the interaction-induced trap at
+the position of the ith standing wave node. The Lorentzian po-
+tential of Eq. 6 with the width w (Eq. 7) shows a perfect match
+with the numeric calculation of Eq. 3. Chosen parameters in (a)
+are Ω2c = 2∆ = 2π × 10MHz, Ω2sw = ∆/2 loss limited to 1Hz,
+n = 100, θ = π.
+ρic†c) in the Lindblad form governs the dissipative time evolu-
+tion. Lindblad terms encounter spontaneous emission from Ry-
+dberg cpe = √γe|p⟩⟨e| and intermediate level cgp = √γp|g⟩⟨p|.
+The spontaneous emission rates are γp/2π = 7.6kHz and γe can
+be found in [25].
+Considering the steady state ¯ρij of Eq. 2, the effective interac-
+tion would be given by
+U(rij) = Tr[ ¯ρij(Hi + Hj + Vij)].
+(3)
+For homogeneous lasers, a plateau-type interaction profile
+would be formed with constant interaction within the soft-core
+as depicted by the dotted line in Fig. 2a. In Rydberg-dressing
+the interaction region is defined by Rc; the interatomic distance
+within which interaction-induced laser detuning equals the ef-
+fective laser bandwidth PrV(Rc) = Ω1Ω2/2∆ [23]. The soft-
+core interaction features a sharp peak at Ω2 = 2|∆|, due to an
+interaction-induced resonance, see Fig. 2b.
+To form the optical lattice with the mentioned interaction-
+induced resonance, a space-dependent variation of the upper
+laser is deployed. Using different intensities for the counter-
+propagating 318nm lasers, results in the desired spatial pattern
+
+Letter
+Optica
+3
+|𝛽"〉
+𝛽$
+|𝛽$〉
+|𝑔𝑔〉
+|𝜆"〉
+(b)
+-∆
+−2∆
+2Ω1
+⟩
+|𝑒𝑒
+2Ω1
+2Ω1
+𝑉
+2Ω2
+2Ω2
+-∆
+⟩
+|𝑔𝑒$
+⟩
+|𝑔𝑝$
+⟩
+|𝑔𝑔
+⟩
+|𝑝𝑝
+⟩
+|𝑝𝑒$
+(a)
+|𝜆$〉
+|𝜆0〉
+𝛽"
+Ω2
+Fig. 3. The origin of the trapping potential is the interaction-
+induced resonance at Ω2 = 2|∆|. (a) With Ω1 ≪ Ω2 the two-
+atom Hilbert space would be organized in three subspaces
+of ground state, single-excitation (green box) and double-
+excitations (yellow box) that are coupled by weak Ω1 laser.
+(b) The effects of strong coupling Ω2 and interaction V could
+be observed by diagonalizing the green and yellow subspaces
+with eigen-states of |β±⟩ and |λ0,±⟩ respectively. At the nodes
+of the standing wave Ω2(x) = −2∆, the interaction-induced
+level shift, makes the |λ−⟩ in-resonance with the ground state,
+significantly enhancing the interaction and forming the trap-
+ping potential.
+of the Rabi frequency
+Ω2(x) = Ω2c + Ω2sw| sin(kx sin(θ/2))|,
+(4)
+where k = 2π/λ is the laser wave-vector and θ is the angle
+between counter propagating lights, see Fig. 1a.
+Adjusting
+Ω2c = −2∆ in Eq. 4 forms periodic trapping potentials at the
+nodes of standing wave upon the presence of at least two atoms
+within the core radius Rc, see Fig. 2a,c. In a 1D lattice with single
+atom ocupation per site, the effective potential experienced by
+each site is the sum of two-body interactions of neighboring
+lattice sites within the interacting range of ±Rc. Considering the
+isotropic Rydberg interaction of the S orbital, extension to the
+3D lattice is trivial.
+Here we analytically formulate the interaction-induced reso-
+nance peak around Ω2(x) = 2|∆|. Considering the level scheme
+of Fig. 3a in two-atom basis, the doubly excited Rydberg state
+asymptotically decouples within the interaction region Rc as
+V → ∞. Taking into account the remaining states, in the limit of
+Ω1 ≪ Ω2c the steady state density could be obtained by adding
+three orders of perturbative corrections to the initial ground
+state. In the limit of γp ≪ ∆ the dressing interaction of the
+steady state simplifies to
+U(x) =
+¯hΩ4
+1
+Ω2(x)2
+4∆[2∆2 + Ω2(x)2]
+[4∆2 − Ω2(x)2]2 + 16γ2p∆2 .
+(5)
+The maximum interaction occurs at Ω2 = 2|∆| with the value
+of U0 = 3¯hΩ4
+1
+8∆γ2 . Note that the attractive or repulsive nature of
+the potential peak is determined by the sign of detuning ∆.
+The half-width at half-maximum of interaction peak occurs at
+Ω2(x) − |2∆| = ±γp. The presented analytic model of Eq. 5
+perfectly resemble the numerical results, see Fig. 2. Considering
+the spatial variation of the Ω2 in Eq. 4 over the narrow area of
+the potential peak k.(x − xi) ≪ 1 with Ω2c = −2∆, the spatial
+profile of the ith trapping site has a Lorentzian form
+Ui(x) =
+U0
+1 + (x − xi)2/w2
+(6)
+where the half-width at half-maximum and the depth of the
+spatial trap well are given by
+w =
+γp
+k sin(θ/2)Ω2sw
+;
+U0 = 3¯hΩ4
+1
+8∆γ2 .
+(7)
+Figure 2c compares this analytical form of Eq. 6 with the numer-
+ical results. The scale of the trap width as a function of Ω2sw is
+plotted in Fig. 4c. Remarkably, with Ω2sw/2π = 1.7MHz the
+trap width would be as tight as the radius of 88Sr atoms.
+0
+5
+10
+2c/2  (MHz)
+0
+0.5
+1
+1.5
+2
+U0 (kHz)
+(a)
+0
+2
+4
+6
+8
+10
+2c/2  (MHz)
+0
+0.01
+0.02
+0.03
+1/
+2c
+(b)
+102
+104
+2sw/2  (kHz)
+100
+101
+trap width w (nm)
+(c)
+Fig. 4. (a) The scale of trap depth U0 for two atoms located
+within the core distance of Rc is plotted as a function of Ω2c
+for the constant scattering rate of 1Hz. Having N lattice sites
+within the interaction distance Rc, the trapping potential ex-
+perienced by an atom would add up to NU0. (b) The decoher-
+ence rate is adjusted to 1Hz by controlling the ratio of Ω1/Ω2c.
+(c) The width of Lorentzian traps w (Eq. 7) is plotted as a func-
+tion of Ω2sw for θ = π.
+The origin of the enhanced interaction at Ω2 = 2|∆| can be
+traced to two-atom resonance that occurs in the presence of
+strong interaction [22]. Considering the laser coupling on the
+two-atom basis, for Ω1 ≪ Ω2 the two-atom Hilbert space would
+be organized in three subspaces that are coupled by weak Ω1
+laser, see Fig. 3. These subspaces are the ground state |gg⟩, one
+atom excitation {|gp⟩+, |ge⟩+}, and two atom excitation states
+{|pp⟩, |pe⟩+, |ee⟩}, with |αβ⟩+ = (|α⟩ + |β⟩)/
+√
+2 represents sym-
+metric two-particle states. The strong coupling Ω2, mixes the
+states in each subspace. Pre-diagonalizing the subsystems quan-
+tifies the light-shifts experienced by the eigen-states, see Fig. 3b.
+For the second subspace with single excitation, the coupling
+Hamiltonian in the {|gp⟩+, |ge⟩+} basis is given by
+S2/¯h =
+�
+� −∆
+Ω2/2
+Ω2/2
+0
+�
+� .
+(8)
+The eigen-energies in this subspace β±/¯h
+=
+−∆/2 ±
+1/2
+�
+∆2 + Ω2
+2 does not get resonant with the ground state. In
+the third subspace with double excitations, the coupling Hamil-
+
+Letter
+Optica
+4
+tonian in the {|pp⟩, |pe⟩+, |ee⟩} basis is given by
+S3/¯h =
+�
+�
+�
+�
+�
+−2∆
+Ω2/
+√
+2
+0
+Ω2/
+√
+2
+−∆
+Ω2/
+√
+2
+0
+Ω2/
+√
+2
+V
+�
+�
+�
+�
+� .
+(9)
+For large interaction inside the softcore V → ∞, the eigen-
+energies are λ0 = V and λ± = − 3¯h
+2 ∆ ± ¯h/2
+�
+∆2 + 2Ω2
+2. The
+doubly excited Rydberg state |λ0⟩ ≈ |ee⟩ decouples asymp-
+totically. At Ω2 = 2|∆| one of |λ±⟩ eigen-states couples res-
+onantly with the ground state, generating an enhanced light-
+shift. As discussed above, small deviation of laser intensity
+Ω2 − 2|∆| = ±γp makes the λ− eigen-state out of resonance.
+Hence, the interaction-induced resonant peaks would be local-
+ized at very narrow areas of the Ω2(x) standing-wave.
+The main source of decoherence in this system is the spon-
+taneous emission from the intermediate state. Rydberg inter-
+action disturbs individual atom’s dark state, populating the
+intermediate state |p⟩, and hence increases the loss rate per atom
+Γ = Tr[ρi(γpσpp + γeσee)] at the trapping potential teeth. The
+loss rate spatial profile is approximately given by γpU(x)/∆.
+The maximum loss for a given Ω2(x) profile could be controlled
+by adjusting the intensity of Ω1 laser. The scale of trap depth
+for two atoms located within the soft-core is plotted in Fig. 4a
+as a function of Ω2c for the constant scattering rate of 1Hz. The
+interaction-to-loss ratio enhances by applying stronger laser
+driving of Ω2c. Having N single-atom-occupied trapping sites
+within the ±Rc interaction distance, the trapping potential expe-
+rienced by an atom would add up to NU0. In a case study, con-
+sidering the lattice constant of λ/2, dressing ground state atoms
+to |5s100s 3S1⟩ with Ω2c/2π = 10MHz and limiting the loss rate
+to 1Hz, the collective trapping potential experienced by a single
+atom in 1D (3D) lattice interacting by neighboring cites within
+the soft-core would be N1DU0 = 77kHz, (N3DU0=37MHz).
+An important feature of the proposed Rydberg-empowered
+optical lattice is the possibility to move the pair lattice sites close
+to each other by manipulating the laser intensity. The inter-
+atomic distance could approach the extreme scales that used
+to be limited to solid-state crystals. As mentioned above, the
+resonance trapping potential occurs at the points of standing
+waves that fulfill the Ω2c + Ω2sw| sin(kx sin(θ/2))| = 2|∆|. Ac-
+cordingly, by adjusting the Ω2c and Ω2sw the resonance could
+occur at positions other than the nodes of the standing-wave
+driving-field, making a lattice of dimers as shown in Fig. 5b.
+To find the precision in adjusting the minimum interatomic
+distance, consider the case that the pair sites are very close
+to the position of the nodes |x − xn| ≪ (k sin(θ/2))−1 where
+xn is the position of a random node. At this regime, the sep-
+aration of two lattice sites from a node would be given by
+x − xn = ±(2|∆| − Ω2c)/(Ω2swk sin(θ/2)). Therefore, having
+larger Ω2sw would enhance the adjustment precision of intra-
+dimer lattice spacing. A sample laser parameters for trapping
+two lattice sites at 4Å distance are Ω2sw/2π = 10MHz and
+2|∆| − Ω2c = 2π × 25kHz and θ = π which are experimentally
+realizable.
+Outlook- The sub-nanoscale resolution in trapping and
+bundling of pair sites demonstrated here extends the toolbox of
+neutral atom quantum technology. Ultra-narrow wells in this
+proposal allow significant suppression of the lattice constant
+with a time-sharing approach [26]. In this approach, the applied
+standing wave is stroboscopically shifted in space by λ/2N and
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+(a)
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+-1
+-0.5
+0
+0.5
+(b)
+Fig. 5. Moving lattice sites by adjusting the relative intensity
+of counter-propagating lasers. The lattice constant is originally
+Λ/4 in (a). By changing the laser intensity, the resonance con-
+dition Ω2(x) = 2|∆| fulfills at positions other than the nodes of
+standing-wave, forming a lattice of dimers. The distance of the
+atomic pairs could be made as small as 4Å.
+hence the effective lattice constant would be smaller by a factor
+of N−1. This compaction of the atomic lattice is quite demanding
+for scaling the lattice sites with the current limited laser powers.
+Furthermore, in quantum simulation with optical lattices, the en-
+ergy scale of Hubbard models for both hopping and interaction
+of atoms is set by the minimum lattice constant which used to be
+limited to λ/2, leading to challenging temperature requirements
+to observe quantum phases of interest [1].
+A distinct research avenue looks at the applications of the
+presented scheme with ultra-narrow repulsive peaks. Equation
+5 shows that changing the detuning sign would preserve the
+interaction profile but only flip the potential sign from attrac-
+tive to repulsive. These ultra-narrow barriers are ideal for the
+realization of the Kronig-Penney (KP) lattice model [27]. Fur-
+thermore, the three dimensional repulsive δ-function peaks form
+nearly perfect box-traps [28]. These repulsive narrow peaks also
+realize thin tunnel junctions for atomtronic devices [29, 30]. The
+potential is easily generalizable to other geometries in three
+dimensions using the holographically designed laser intensity
+[31].
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+

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+Classification of Cross-cultural News Events 
+Abdul Sittar
+∗ 
+[email protected] 
+Jožef Stefan Institute and Jožef Stefan International 
+Postgraduate School 
+Jamova cesta 39 
+Ljubljana, Slovenia 
+ABSTRACT 
+We present a methodology to support the analysis of culture 
+from text such as news events and demonstrate its usefulness 
+on categorising news events from different categories (society, 
+business, health, recreation, science, shopping, sports, arts, com- 
+puters, games and home) across different geographical locations 
+(different places in 117 countries). We group countries based on 
+the culture that they follow and then filter the news events based 
+on their content category. The news events are automatically 
+labelled with the help of Hofstede’s cultural dimensions. We 
+present combinations of events across different categories and 
+check the performances of different classification methods. We 
+also presents experimental comparison of different number of 
+features in order to find a suitable set to represent the culture. 
+KEYWORDS 
+cultural barrier, news events, text classification 
+ 
+1 INTRODUCTION 
+Culture is defined as a collective programming of the mind which 
+distinguishes the members of one group or category of people 
+from another [9]. It has a huge impact on the lives of people and 
+in result it influences events that involve cross-cultural stake- 
+holders. News spreading is one of the most effective mechanisms 
+for spreading information across the borders. The news to be 
+spread wider cross multiple barriers such as linguistic, economic, 
+geographical, political, time zone, and cultural barriers. Due to 
+rapidly growing number of events with significant international 
+impact, cross-cultural analytics gain increased importance for 
+professionals and researchers in many disciplines, including digi- 
+tal humanities, media studies, and journalism. The most recent 
+examples of such events include COVID-19 and Brexit [1]. There 
+are few determinants that have significant influence on the pro- 
+cess of information selection, analysis and propagation. These 
+include cultural values and differences, economic conditions and 
+association between countries. For instance, if two countries are 
+culturally more similar, there are more chances that there will 
+be a heavier news flow between them [10], [3]. In this paper, 
+we focus on classification of news events across different cul- 
+tures. We select some of the most read daily newspapers and 
+collect information using Event Registry about the news they 
+have published. Event Registry is a system which analyzes news 
+articles, identifies groups of articles that describe the same event 
+and represent them as a single event [7]. The description of the 
+ 
+Permission to make digital or hard copies of part or all of this work for personal 
+or classroom use is granted without fee provided that copies are not made or 
+distributed for profit or commercial advantage and that copies bear this notice and 
+the full citation on the first page. Copyrights for third-party components of this 
+work must be honored. For all other uses, contact the owner/author(s). 
+Information Society 2021, 4–8 October 2021, Ljubljana, Slovenia 
+© 2021 Copyright held by the owner/author(s). 
+Dunja Mladenić 
+[email protected] 
+Jožef Stefan Institute and Jožef Stefan International 
+Postgraduate School 
+Jamova cesta 39 
+Ljubljana, Slovenia 
+meta data of an event is shown in the Table 1. The main scientific 
+contributions of this paper are the following: 
+(1) A novel perspective of aligning news events across dif- 
+ferent cultures through categorising countries and news 
+events. 
+(2) A cross-cultural automatically annotated dataset in several 
+different domains (Business, Science, Sports, Health etc.). 
+(3) Experimental comparison of several classification mod- 
+els adopting different set of features (character ngrams, 
+GLOVE embeddings and word ngrams). 
+Table 1: The description of the meta data of an event. 
+  Attributes    Description 
+ 
+title 
+title of the event 
+summary 
+summary of the event 
+source 
+event reported by a news source 
+categories 
+list of DMOZ categories 
+location 
+location of the event 
+ 
+ 
+ 
+2 RELATED WORK 
+In this section, we review the related literature about the influ- 
+ence of culture, its representation and classification in different 
+fields. 
+Countries that share a common culture are expected to have 
+heavier news flows between them when reporting on similar 
+events [10]. There are many quantitative studies that found de- 
+mographic, psychological, socio-cultural, source, system, and 
+content-related aspects [2]. 
+Cross-cultural research and understanding the cultural influences 
+in different fields have competitive advantages. The goal of re- 
+searching the impact of culture might be to draw conclusions 
+in which way the cultural factors influence a specific corporate 
+action. There are many type of cultures such as societal, organi- 
+zational, and business culture etc [8]. 
+The hidden nature of cultural behavior causes some difficulties 
+in measurement and defining these. To cope with difficulties, 
+researchers have developed measurements that measure culture 
+on a general scale to compare differences among cultures and 
+management styles. These results can be used to find similarities 
+within a region and differences to other regions. There are many 
+models that have tried to explain cultural differences between 
+societies. Hofstede’s national culture dimensions (HNCD) have 
+been widely used and cited in different disciplines [6, 5]. Hofst- 
+ede’s dimensions are the result of a factor analysis at the level 
+of country means of comprehensive survey instrument, aimed 
+at identifying systematic differences in national cultural. Their 
+purpose is to measure culture in countries, societies, sub-groups, 
+and organizations; they are not meant to be regarded as psycho- 
+logical traits. 
+There is a plethora of research studies that were conducted to un- 
+derstand the cultural influences such as cross-culture privacy and 
+
+Information Society 2021, 4–8 October 2021, Ljubljana, Slovenia 
+Abdul and Dunja, et al. 
+ 
+ 
+attitude prediction, and cultural influences on today’s business. 
+[4] explores how culture affects the technological, organizational, 
+and environmental determinants of machine learning adoption 
+by conducting a comparative case study between Germany and 
+US. Rather than looking at the influence of cultural differences 
+within one domain, we intend to understand association between 
+news events belonging to different domains (society, business, 
+health, recreation, science, shopping, sports, arts, computers, 
+games and home) and different cultures (117 countries from all 
+the continents). We conduct this research to find an appropriate 
+representation and classification of culture across different do- 
+mains. 
+ 
+3 DATA DESCRIPTION 
+3.1 Dataset Statistics 
+We choose the top 10 daily read newspapers in the world in 2020 1 
+and collect the events reported by these newspapers using Event 
+Registry [7] over the time period of 2016-2020. Approximately 
+8000 events belongs to each newspaper with exception of “Za- 
+man” that has only 900 events. Figure 1 shows the number of 
+events reported by the selected newspapers on a yearly basis. 
+This dataset can be found on the Zenodo repository (version 
+1.0.0) 2 
+ 
+ 
+ 
+Figure 1: Each color in a bar represents the total number 
+of events per year by a daily newspaper and a complete 
+bar shows the total number of events per year by all the 
+newspapers. 
+ 
+The attributes of an event with description are displayed in 
+Table 1. Few attributes are self-explanatory such as title, summary, 
+date, and source. DMOZ-categories are used to represent topics 
+of the content. The DMOZ project is a hierarchical collection of 
+web page links organized by subject matters 3. Event Registry use 
+top 3 levels of DMoz taxonomy which amount to about 50,000 
+categories 4. 
+4 MATERIAL AND METHODS 
+4.1 Problem Definition 
+There are two main parts of the problem that we are addressing. 
+The first part is to label the examples by assigning a culture C to a 
+news event E using its location L. The second part is a multi-class 
+classification task where we predict the culture C of a news event 
+E using its summary description S and its content category G as 
+ 
+1 https://www.trendrr.net/ 
+2 https://zenodo.org/record/5225053 
+3 https://dmoz-odp.org/ 
+4 https://eventregistry.org/documentation?tab=terminology 
+ 
+provided by the Event Registry. This task can be formulated as: 
+𝐶 = 𝑓 (𝑆, 𝐺) 
+C donates the culture of the news event, f is the learning function, 
+S donates summary of a news event and G donates category of a 
+news event (see Table 1). 
+4.2 Methodology 
+ 
+ 
+ 
+ 
+ 
+Figure 2: Classification of cross-cultural news events. 
+ 
+4.2.1 
+Data labeling. Each news event has information about the 
+type of categories to which it belongs and the location where it 
+happened (see Table 1). Each event has many categories and each 
+category has a weight reflecting its relevance for the event. We 
+only keep the most relevant categories and group the news events 
+based on their categories. For each group of events, we estimate 
+the cultural characteristic of each event through the country of 
+the place where the event occurred. We cluster the countries 
+based on their culture. We utilize the Hofstede’s national culture 
+dimensions (HNCD) to represent the culture of a country. We take 
+average of cultural dimensions and call it average cultural score. 
+Based on this score, we find optimal number of clusters using 
+popular clustering algorithm k-means (see Figure 4). Finally, we 
+label each news event with one of the six cultural clusters. 
+ 
+ 
+ 
+Figure 3: The pie chart depicts the percentage of the news 
+events that occurred in six different clusters (each cluster 
+consists of a list of countries with similar culture). 
+ 
+4.2.2 
+Data representation. Each news event in Event Registry 
+has associated categories with it along with a weight (see Table 
+1), we take the top categories based on their weight. In case of 
+multiple categories with equal weight, we sort them alphabeti- 
+cally and keep the first one. We represent each news event by a 
+short summary S and a set of content categories G. 
+Clusters of Countries 
+Char Ngrams 
+News Events 
+Dataset Annotation 
+Glove Embeddings 
+Classification 
+Category of Events 
+Word Ngrams 
+
+Newspaper
+14k
+asahi.com
+chinadally.com.cn
+dawn.com
+12k
+nytimes.com
+smh.com.au
+10k
+theguardian.com
+timesofindia.indiatimes.com
+8k
+washingtonpost.com
+wsj.com
+6k
+zaman.com.tr
+4k
+2k
+0
+2016
+2017
+2018
+2019
+2020
+2021ranhnza
+ZambiaThaland,Jord
+Bangladesh, Lithuaria, Indones
+Dominican Republic
+Estonia,IndiaChina,Buga
+MoccoAigria,
+Romania, Serbia, Azerbajan,
+Croatia,
+Portugat,Boivia, Liby
+Chile, Sovenia,Philippin
+Amenia, Belarus,SouthKorea,
+Malaysia, Uruguay,
+24-
+Taiwan,Abania,Urae,Mod
+Georgia, Argentina, lraq
+25%
+Montenegro, Czesh Republic
+Spain,
+Kazakhstan, Bosnia and Herzegovina
+Turkey, Brazi, Grece
+C3
+C1
+Russia, Slovakia, Japan
+Saudi Arabia, Poland
+Trindad and Tobago, Mczambique
+Colonbia
+Ghana,AngolaPueoRi
+heral,emkceia
+C5
+C2
+Noay,SwnLat
+C6
+8.0496
+Niger,  Salvador, Venezuela
+Finland, ireland,New
+Zealand,Nethertands,
+21:4%
+C4
+Cota ca,cuadrunisia,Egy
+Canada, SouthAfrica,
+Kuwait,Panaa,Guaa
+Australia.
+UnitdAbes,Ca
+United States,Aistia
+8.93%
+Suriname
+Kingdom,Gemany,taly
+Luovembourg,Unitd
+12.5%
+Malaw,Jamaica,Nepal,SierraLeon
+Swtt
+Fij,HonurasKya,Bhuta
+Hungary, Belgium
+Nambia Sianka,Senegal,Burki
+FasoSvnaLebanonClassification of Cross-cultural News Events 
+Information Society 2021, 4–8 October 2021, Ljubljana, Slovenia 
+ 
+ 
+ 
+ 
+ 
+Figure 4: In word cloud, the color of each word shows cluster to whom it belongs (see Figure 3). Radial dendrograms 
+illustrate the shared categories of news events between the pair of six clusters. 
+ 
+4.2.3 
+Data Modeling. For multi-class classification task, we use 
+simple classification models (SVM, Decision Tree, KNN, Naive 
+Bayes, Logistic Regression) as well as neural network. For sim- 
+ple classification models, we input character and word ngrams 
+varying the number of ngrams and compare the results. We also 
+use pre-trained Glove embeddings. 
+5 EXPERIMENTAL EVALUATION 
+5.1 Evaluation Metric 
+For multi-class classification task, we use following most com- 
+monly used evaluation measures: accuracy, precision, recall, and 
+F1 score. 
+6 RESULTS AND ANALYSIS 
+6.1 Annotation Results 
+The results of annotation are six clusters where almost 50% news 
+events belong to the two clusters (shown with red and blue colors) 
+and remaining 50% belong to the other four clusters 3. Looking 
+in each group, we find that clusters do not lies in a specific 
+geographic area or a continent. Rather all the countries in a 
+cluster belong to the different continents. Similarly, these clusters 
+do not have all the countries that are economically rich or poor. 
+There are more categories in green and red colors in the word 
+cloud (see Figure 4) which represent to the cluster with that colors. 
+Radial dendrograms in Figure 4 present the shared categories 
+between the clusters. In the figure, root of the tree is data and 
+then there are ten pair of clusters that share the same categories. 
+The objective of this whole process was to keep news events 
+according to the category to whom they belongs. Moreover, we 
+can only observe the cultural differences when we have same 
+type of news events from different places. 
+6.2 Classification Results 
+Fro the experimental results we can see that the best performance 
+is achieved by Logistic Regression, kNN and Decision Tree. The 
+performance of SVM varies depending on the number of selected 
+features: the highest F1-score is achieved with the top 10K or 20K 
+word ngrams using 1 to 3 word ngrams (see Figure 5). Looking at 
+the character ngrams, the highest F1-score is achieved when we 
+select the top 15K characters for all the tested algorithms except 
+Naive Bayes which declines in performance with the growing 
+set of features. Based on these settings, we achieve the highest 
+accuracy (0.85) using Logistic Regression. Using Glove embed- 
+dings, we experiment with and without using the category of 
+event. The highest F1-score with and without the category is 0.80 
+and 0.79 respectively. 
+ 
+ 
+ 
+7 CONCLUSIONS AND FUTURE WORK 
+For researchers and professionals, it is very important to anal- 
+yse the cross-cultural differences in different disciplines. As the 
+international impact is increasing and international events are 
+becoming popular, the need to develop some automatic methods 
+is significantly increasing and leaving a blank space. We con- 
+ducted experiments on news events related to different fields 
+to have a broader look on data and machine learning methods. 
+Further research would be helpful in examining the impact of 
+specific socio-cultural factors on news events. In this research 
+work, we estimate the culture of a specific place by its country, 
+use basic features and simple classification models. To continue 
+this work further, we would like to improve feature set such as 
+by including part of speech tagging (POS) as well as other state 
+of the art embeddings. 
+ 
+ 
+ 
+ACKNOWLEDGMENTS 
+The research described in this paper was supported by the Slove- 
+nian research agency under the project J2-1736 Causalify and 
+by the European Union’s Horizon 2020 research and innovation 
+programme under the Marie Skłodowska-Curie grant agreement 
+No 812997. 
+
+Shopping.Clothing
+echnolog
+condt
+oistic:
+SOrtS
+Softwar
+Recreation Collecting
+ortsBow
+Shopping
+sports
+Societylssues
+science
+Environment
+Business Financial
+Services
+ArtsMoviess
+portsGolfArts_Music
+soclety_M
+SocietyIssue
+Sports
+Socce
+Society
+Eguestrian
+Society
+Manai
+Society GayELesbian andBisexua
+creation
+Business Food and Related Products
+orts-Mart
+Soorts
+ycilno
+Home
+am
+Collectibles
+soorts
+BasketbalColads
+Envr
+"20
+Sports
+sety
+Equestrian
+Strength.Sports.
+Team Spinit-
+Sports
+-Cycling
+Rope Skipping
+Soccer-
+C3C4
+Sports+
+MartialAts
+Society*
+-Religion and Spintualty
+Hockey
++Sports
+C2C
+Science-
+Astronomy
+Equestrian
+Cycing
+Soci
+ntsInformation Society 2021, 4–8 October 2021, Ljubljana, Slovenia 
+Abdul and Dunja, et al. 
+ 
+ 
+ 
+ 
+Figure 5: First two line charts illustrate the variations in 
+F1 score by simple classification models after varying the 
+number of features. The first line chart depicts the results 
+of word ngrams whereas the second one shows the results 
+for character ngrams. The last line graph presents com- 
+parison between Glove embeddings (with and without cat- 
+egory feature). 
+ 
+REFERENCES 
+[1] 
+Sara Abdollahi, Simon Gottschalk, and Elena Demidova. 
+2020. Eventkg+ click: a dataset of language-specific event- 
+centric user interaction traces. arXiv preprint arXiv:2010.12370. 
+[2] 
+Hosam Al-Samarraie, Atef Eldenfria, and Husameddin 
+Dawoud. 2017. The impact of personality traits on users’ 
+information-seeking behavior. Information Processing & 
+Management, 53, 1, 237–247. 
+[3] 
+Tsan-Kuo Chang and Jae-Won Lee. 1992. Factors affecting 
+gatekeepers’ selection of foreign news: a national survey 
+of newspaper editors. Journalism Quarterly, 69, 3, 554–561. 
+[4] 
+Verena Eitle and Peter Buxmann. 2020. Cultural differences 
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+[5] 
+Meihan He and Jongsu Lee. 2020. Social culture and in- 
+novation diffusion: a theoretically founded agent-based 
+model. Journal of Evolutionary Economics, 1–41. 
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+Mahmood Khosrowjerdi, Anneli Sundqvist, and Katriina 
+Byström. 2020. Cultural patterns of information source use: 
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+[7] 
+Gregor Leban, Blaz Fortuna, Janez Brank, and Marko Gro- 
+belnik. 2014. Event registry: learning about world events 
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+[8] 
+Björn Preuss. 2017. Text mining and machine learning to 
+capture cultural data. Technical report. working paper, 2. 
+doi: 10.13140/RG. 2.2. 30937.42080. 
+[9] 
+Giselle Rampersad and Turki Althiyabi. 2020. Fake news: 
+acceptance by demographics and culture on social media. 
+Journal of Information Technology & Politics, 17, 1, 1–11. 
+[10] 
+H Denis Wu. 2007. A brave new world for international 
+news? exploring the determinants of the coverage of for- 
+eign news on us websites. International Communication 
+Gazette, 69, 6, 539–551. 
+
+Top K features versus Accuracy (Word Ngrams, 1-3)
+1.0
+SVM
+Decision
+0.8
+Tree
+KNN
+Naive
+0.6
+Bayes
+Logistic
+0.4
+Regres..
+0.2
+0.0 
+5,000
+10,000
+15,000
+20,000
+Top K
+Top K features versus F1 (Char Ngrams, 2-6)
+1.0
+SVM
+Decision
+0.8
+Tree
+KNN
+Naive
+0.6
+Bayes
+Logistic
+0.4
+Regres..
+0.2
+0.0 
+2,500
+5,000
+7,500
+10,000
+12,500
+15,000
+Top K
+Categories vs. Without Categories
+0.85
+Glove
+(with
+category)
+0.80
+Glove
+(without
+category)
+0.75
+0.70
+0.65
+2
+4
+6
+8
+10
+Epochs

DtE5T4oBgHgl3EQfUQ_q/content/tmp_files/load_file.txt

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+page_content='Classification of Cross-cultural News Events  Abdul Sittar  ∗  abdul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='sittar@ijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='si  Jožef Stefan Institute and Jožef Stefan International  Postgraduate School  Jamova cesta 39  Ljubljana, Slovenia  ABSTRACT  We present a methodology to support the analysis of culture  from text such as news events and demonstrate its usefulness  on categorising news events from different categories (society,  business, health, recreation, science, shopping, sports, arts, com-  puters, games and home) across different geographical locations  (different places in 117 countries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We group countries based on  the culture that they follow and then filter the news events based  on their content category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The news events are automatically  labelled with the help of Hofstede’s cultural dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We  present combinations of events across different categories and  check the performances of different classification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We  also presents experimental comparison of different number of  features in order to find a suitable set to represent the culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  KEYWORDS  cultural barrier, news events, text classification  1 INTRODUCTION  Culture is defined as a collective programming of the mind which  distinguishes the members of one group or category of people  from another [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' It has a huge impact on the lives of people and  in result it influences events that involve cross-cultural stake-  holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' News spreading is one of the most effective mechanisms  for spreading information across the borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The news to be  spread wider cross multiple barriers such as linguistic, economic,  geographical, political, time zone, and cultural barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Due to  rapidly growing number of events with significant international  impact, cross-cultural analytics gain increased importance for  professionals and researchers in many disciplines, including digi-  tal humanities, media studies, and journalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The most recent  examples of such events include COVID-19 and Brexit [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' There  are few determinants that have significant influence on the pro-  cess of information selection, analysis and propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' These  include cultural values and differences, economic conditions and  association between countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' For instance, if two countries are  culturally more similar, there are more chances that there will  be a heavier news flow between them [10], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' In this paper,  we focus on classification of news events across different cul-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We select some of the most read daily newspapers and  collect information using Event Registry about the news they  have published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Event Registry is a system which analyzes news  articles, identifies groups of articles that describe the same event  and represent them as a single event [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The description of the  Permission to make digital or hard copies of part or all of this work for personal  or classroom use is granted without fee provided that copies are not made or  distributed for profit or commercial advantage and that copies bear this notice and  the full citation on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Copyrights for third-party components of this  work must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' For all other uses, contact the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Information Society 2021, 4–8 October 2021, Ljubljana, Slovenia  © 2021 Copyright held by the owner/author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Dunja Mladenić  dunja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='mladenic@ijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='si  Jožef Stefan Institute and Jožef Stefan International  Postgraduate School  Jamova cesta 39  Ljubljana, Slovenia  meta data of an event is shown in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The main scientific  contributions of this paper are the following:  (1) A novel perspective of aligning news events across dif-  ferent cultures through categorising countries and news  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  (2) A cross-cultural automatically annotated dataset in several  different domains (Business, Science, Sports, Health etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  (3) Experimental comparison of several classification mod-  els adopting different set of features (character ngrams,  GLOVE embeddings and word ngrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Table 1: The description of the meta data of an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Attributes    Description  title  title of the event  summary  summary of the event  source  event reported by a news source  categories  list of DMOZ categories  location  location of the event  2 RELATED WORK  In this section, we review the related literature about the influ-  ence of culture, its representation and classification in different  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Countries that share a common culture are expected to have  heavier news flows between them when reporting on similar  events [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' There are many quantitative studies that found de-  mographic, psychological, socio-cultural, source, system, and  content-related aspects [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Cross-cultural research and understanding the cultural influences  in different fields have competitive advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The goal of re-  searching the impact of culture might be to draw conclusions  in which way the cultural factors influence a specific corporate  action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' There are many type of cultures such as societal, organi-  zational, and business culture etc [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  The hidden nature of cultural behavior causes some difficulties  in measurement and defining these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' To cope with difficulties,  researchers have developed measurements that measure culture  on a general scale to compare differences among cultures and  management styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' These results can be used to find similarities  within a region and differences to other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' There are many  models that have tried to explain cultural differences between  societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Hofstede’s national culture dimensions (HNCD) have  been widely used and cited in different disciplines [6, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Hofst-  ede’s dimensions are the result of a factor analysis at the level  of country means of comprehensive survey instrument, aimed  at identifying systematic differences in national cultural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Their  purpose is to measure culture in countries, societies, sub-groups,  and organizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' they are not meant to be regarded as psycho-  logical traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  There is a plethora of research studies that were conducted to un-  derstand the cultural influences such as cross-culture privacy and  Information Society 2021, 4–8 October 2021, Ljubljana, Slovenia  Abdul and Dunja, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  attitude prediction, and cultural influences on today’s business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  [4] explores how culture affects the technological, organizational,  and environmental determinants of machine learning adoption  by conducting a comparative case study between Germany and  US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Rather than looking at the influence of cultural differences  within one domain, we intend to understand association between  news events belonging to different domains (society, business,  health, recreation, science, shopping, sports, arts, computers,  games and home) and different cultures (117 countries from all  the continents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We conduct this research to find an appropriate  representation and classification of culture across different do-  mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  3 DATA DESCRIPTION  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='1 Dataset Statistics  We choose the top 10 daily read newspapers in the world in 2020 1  and collect the events reported by these newspapers using Event  Registry [7] over the time period of 2016-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Approximately  8000 events belongs to each newspaper with exception of “Za-  man” that has only 900 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Figure 1 shows the number of  events reported by the selected newspapers on a yearly basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  This dataset can be found on the Zenodo repository (version  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='0) 2  Figure 1: Each color in a bar represents the total number  of events per year by a daily newspaper and a complete  bar shows the total number of events per year by all the  newspapers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  The attributes of an event with description are displayed in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Few attributes are self-explanatory such as title, summary,  date, and source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' DMOZ-categories are used to represent topics  of the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The DMOZ project is a hierarchical collection of  web page links organized by subject matters 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Event Registry use  top 3 levels of DMoz taxonomy which amount to about 50,000  categories 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  4 MATERIAL AND METHODS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='1 Problem Definition  There are two main parts of the problem that we are addressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  The first part is to label the examples by assigning a culture C to a  news event E using its location L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The second part is a multi-class  classification task where we predict the culture C of a news event  E using its summary description S and its content category G as  1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='trendrr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='net/  2 https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='org/record/5225053  3 https://dmoz-odp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='org/  4 https://eventregistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='org/documentation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='tab=terminology  provided by the Event Registry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' This task can be formulated as:  𝐶 = 𝑓 (𝑆, 𝐺)  C donates the culture of the news event, f is the learning function,  S donates summary of a news event and G donates category of a  news event (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2 Methodology  Figure 2: Classification of cross-cultural news events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='1  Data labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Each news event has information about the  type of categories to which it belongs and the location where it  happened (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Each event has many categories and each  category has a weight reflecting its relevance for the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We  only keep the most relevant categories and group the news events  based on their categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' For each group of events, we estimate  the cultural characteristic of each event through the country of  the place where the event occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We cluster the countries  based on their culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We utilize the Hofstede’s national culture  dimensions (HNCD) to represent the culture of a country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We take  average of cultural dimensions and call it average cultural score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Based on this score, we find optimal number of clusters using  popular clustering algorithm k-means (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Finally, we  label each news event with one of the six cultural clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Figure 3: The pie chart depicts the percentage of the news  events that occurred in six different clusters (each cluster  consists of a list of countries with similar culture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2  Data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Each news event in Event Registry  has associated categories with it along with a weight (see Table  1), we take the top categories based on their weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' In case of  multiple categories with equal weight, we sort them alphabeti-  cally and keep the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We represent each news event by a  short summary S and a set of content categories G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Clusters of Countries  Char Ngrams  News Events  Dataset Annotation  Glove Embeddings  Classification  Category of Events  Word Ngrams  Newspaper  14k  asahi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  chinadally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='cn  dawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  12k  nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  smh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='au  10k  theguardian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  timesofindia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='indiatimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  8k  washingtonpost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  wsj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com  6k  zaman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='tr  4k  2k  0  2016  2017  2018  2019  2020  2021ranhnza  ZambiaThaland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Jord  Bangladesh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Lithuaria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Indones  Dominican Republic  Estonia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='IndiaChina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Buga  MoccoAigria,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Romania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Serbia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Azerbajan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Croatia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Portugat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Boivia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Liby  Chile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Sovenia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Philippin  Amenia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Belarus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='SouthKorea,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Malaysia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Uruguay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  24-  Taiwan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Abania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Urae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Mod  Georgia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Argentina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' lraq  25%  Montenegro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Czesh Republic  Spain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Kazakhstan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Bosnia and Herzegovina  Turkey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Brazi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Grece  C3  C1  Russia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Slovakia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Japan  Saudi Arabia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Poland  Trindad and Tobago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Mczambique  Colonbia  Ghana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='AngolaPueoRi  heral,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='emkceia  C5  C2  Noay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='SwnLat  C6  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='0496  Niger,  Salvador, Venezuela  Finland, ireland,New  Zealand,Nethertands,  21:4%  C4  Cota ca,cuadrunisia,Egy  Canada, SouthAfrica,  Kuwait,Panaa,Guaa  Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  UnitdAbes,Ca  United States,Aistia  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='93%  Suriname  Kingdom,Gemany,taly  Luovembourg,Unitd  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='5%  Malaw,Jamaica,Nepal,SierraLeon  Swtt  Fij,HonurasKya,Bhuta  Hungary, Belgium  Nambia Sianka,Senegal,Burki  FasoSvnaLebanonClassification of Cross-cultural News Events  Information Society 2021, 4–8 October 2021, Ljubljana, Slovenia  Figure 4: In word cloud, the color of each word shows cluster to whom it belongs (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Radial dendrograms  illustrate the shared categories of news events between the pair of six clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='3  Data Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' For multi-class classification task, we use  simple classification models (SVM, Decision Tree, KNN, Naive  Bayes, Logistic Regression) as well as neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' For sim-  ple classification models, we input character and word ngrams  varying the number of ngrams and compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We also  use pre-trained Glove embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  5 EXPERIMENTAL EVALUATION  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='1 Evaluation Metric  For multi-class classification task, we use following most com-  monly used evaluation measures: accuracy, precision, recall, and  F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  6 RESULTS AND ANALYSIS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='1 Annotation Results  The results of annotation are six clusters where almost 50% news  events belong to the two clusters (shown with red and blue colors)  and remaining 50% belong to the other four clusters 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Looking  in each group, we find that clusters do not lies in a specific  geographic area or a continent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Rather all the countries in a  cluster belong to the different continents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Similarly, these clusters  do not have all the countries that are economically rich or poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  There are more categories in green and red colors in the word  cloud (see Figure 4) which represent to the cluster with that colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Radial dendrograms in Figure 4 present the shared categories  between the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' In the figure, root of the tree is data and  then there are ten pair of clusters that share the same categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  The objective of this whole process was to keep news events  according to the category to whom they belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Moreover, we  can only observe the cultural differences when we have same  type of news events from different places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2 Classification Results  Fro the experimental results we can see that the best performance  is achieved by Logistic Regression, kNN and Decision Tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The  performance of SVM varies depending on the number of selected  features: the highest F1-score is achieved with the top 10K or 20K  word ngrams using 1 to 3 word ngrams (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Looking at  the character ngrams, the highest F1-score is achieved when we  select the top 15K characters for all the tested algorithms except  Naive Bayes which declines in performance with the growing  set of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Based on these settings, we achieve the highest  accuracy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='85) using Logistic Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Using Glove embed-  dings, we experiment with and without using the category of  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The highest F1-score with and without the category is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='80  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='79 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  7 CONCLUSIONS AND FUTURE WORK  For researchers and professionals, it is very important to anal-  yse the cross-cultural differences in different disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' As the  international impact is increasing and international events are  becoming popular, the need to develop some automatic methods  is significantly increasing and leaving a blank space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' We con-  ducted experiments on news events related to different fields  to have a broader look on data and machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Further research would be helpful in examining the impact of  specific socio-cultural factors on news events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' In this research  work, we estimate the culture of a specific place by its country,  use basic features and simple classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' To continue  this work further, we would like to improve feature set such as  by including part of speech tagging (POS) as well as other state  of the art embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The research described in this paper was supported by the Slove-  nian research agency under the project J2-1736 Causalify and  by the European Union’s Horizon 2020 research and innovation  programme under the Marie Skłodowska-Curie grant agreement  No 812997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Clothing  echnolog  condt  oistic:  SOrtS  Softwar  Recreation Collecting  ortsBow  Shopping  sports  Societylssues  science  Environment  Business Financial  Services  ArtsMoviess  portsGolfArts_Music  soclety_M  SocietyIssue  Sports  Socce  Society  Eguestrian  Society  Manai  Society GayELesbian andBisexua  creation  Business Food and Related Products  orts-Mart  Soorts  ycilno  Home  am  Collectibles  soorts  BasketbalColads  Envr  "20  Sports  sety  Equestrian  Strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='Sports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Team Spinit-  Sports  Cycling  Rope Skipping  Soccer-  C3C4  Sports+  MartialAts  Society*  Religion and Spintualty  Hockey  +Sports  C2C  Science-  Astronomy  Equestrian  Cycing  Soci  ntsInformation Society 2021, 4–8 October 2021, Ljubljana, Slovenia  Abdul and Dunja, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Figure 5: First two line charts illustrate the variations in  F1 score by simple classification models after varying the  number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The first line chart depicts the results  of word ngrams whereas the second one shows the results  for character ngrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' The last line graph presents com-  parison between Glove embeddings (with and without cat-  egory feature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
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+page_content=' Technical report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' working paper, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='13140/RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' 30937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='42080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  [9]  Giselle Rampersad and Turki Althiyabi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' Fake news:  acceptance by demographics and culture on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  Journal of Information Technology & Politics, 17, 1, 1–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content='  [10]  H Denis Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' A brave new world for international  news?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' exploring the determinants of the coverage of for-  eign news on us websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
+page_content=' International Communication  Gazette, 69, 6, 539–551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtE5T4oBgHgl3EQfUQ_q/content/2301.05543v1.pdf'}
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+Uptrendz: API-Centric Real-time
+Recommendations in Multi-Domain Settings
+Emanuel Lacic1, Tomislav Duricic1,2, Leon Fadljevic1,
+Dieter Theiler1, and Dominik Kowald(�)1,2
+1 Know-Center GmbH, Graz, Austria
+{elacic,tduricic,lfadljevic,dtheiler,dkowald}@know-center.at
+2 Graz University of Technology, Graz, Austria
+Abstract. In this work, we tackle the problem of adapting a real-time
+recommender system to multiple application domains, and their underly-
+ing data models and customization requirements. To do that, we present
+Uptrendz, a multi-domain recommendation platform that can be cus-
+tomized to provide real-time recommendations in an API-centric way.
+We demonstrate (i) how to set up a real-time movie recommender us-
+ing the popular MovieLens-100k dataset, and (ii) how to simultaneously
+support multiple application domains based on the use-case of recom-
+mendations in entrepreneurial start-up founding. For that, we differenti-
+ate between domains on the item- and system-level. We believe that our
+demonstration shows a convenient way to adapt, deploy and evaluate a
+recommender system in an API-centric way. The source-code and doc-
+umentation that demonstrates how to utilize the configured Uptrendz
+API is available on GitHub.
+Keywords: Uptrendz, API-centric recommendations, multi-domain rec-
+ommendations, real-time recommendations
+1
+Introduction
+Utilizing recommender systems is nowadays recognized as a necessary feature to
+help users discover relevant content [15,14]. Most industry practitioners [3], when
+they build a recommender system, adapt existing algorithms to the underlying
+data and customization requirements of the respective application domain (e.g.,
+movies, music, news, etc.). However, the focus of the research community has
+recently shifted towards building recommendation systems that simultaneously
+support multiple application domains [4,7,16] in an API-centric way.
+In this work, we demonstrate Uptrendz3, an API-centric recommendation
+platform, which can be configured to simultaneously provide real-time recom-
+mendations in an API-centric way to multiple domains. Uptrendz supports pop-
+ular recommendation algorithms, e.g., Collaborative Filtering (CF), Content-
+based Filtering (CBF, or Most Popular (MP), that are applied across different
+3 https://uptrendz.ai/
+arXiv:2301.01037v1  [cs.IR]  3 Jan 2023
+
+2
+E. Lacic, T. Duricic, L. Fadljevic, D. Theiler, and D. Kowald
+RECOMMENDER
+CUSTOMIZATION
+SERVICE 
+ISOLATION
+DATA 
+HETEROGENEITY
+FAULT
+TOLERANCE
+MULTI-DOMAIN RECOMMENDER SYSTEM
+Fig. 1. Aspects that need to be addressed when building a recommender system for a
+multi-domain environment [10].
+application domains. The focus of this demonstration is to show how domain-
+specific data-upload APIs can be created to support the customization of the
+respective recommendation algorithms. Using the MovieLens-100k dataset [6]
+and a real-world use-case of entrepreneurial start-up founding4, we show how
+such an approach allows for a highly customized recommendation system that
+can be used in an API-centric way. The source-code and documentation for this
+demonstration is available via GitHub5.
+2
+The Uptrendz Platform
+The Uptrendz platform is built on top of the ScaR recommendation framework
+[11]. As shown in [10] and Figure 1, the microservice-based system architecture
+addresses four distinctive requirements of a multi-domain recommender system,
+i.e., (i) service isolation, (ii) data heterogeneity, (iii) recommender customization,
+and (iv) fault tolerance. Uptrendz provides a layer on top of the framework to
+dynamically configure an application domain and to instantly provide an API
+to (i) upload item, user and interaction data, and (ii) request recommendations.
+Domain-specific data model. As discussed by [1], different domains may em-
+ploy the same recommender algorithm but can differ with respect to what kind
+of data is utilized to build the model (e.g., interaction types, context, etc.). Given
+an API-centric approach, we show that in order to support the customization of
+recommender algorithms with domain-specific parameters, the underlying plat-
+form needs to unambiguously know which source of information should be used
+to calculate the recommendations. To do that, the Uptrendz platform first allows
+generating customized data upload APIs for multiple item and user entities (see
+Table 1). Second, with respect to interaction data, both user-item and user-user
+interactions can be configured. The interaction API is further customized in ac-
+cordance to what kind of interactions the respective application domain actually
+supports, i.e., (i) registered users, anonymous sessions or both, (ii) interaction
+timestamp tracking, and (iii) type of interaction (explicit or implicit).
+4 https://cogsteps.com/
+5 https://github.com/lacic/ECIR2023Demo
+
+API-Centric Real-time Recommendations in Multi-Domain Settings
+3
+Table 1. Supported attributes to configure the data upload API for items and users.
+Type
+Sub-Type
+Description
+Categorical
+Text
+Single
+Value
+String value, which usually represents a category. Used
+for post-filtering recommendation results.
+Multiple
+Values
+List of string values, which usually represent an array
+of categories. Used for post-filtering recommendation
+results.
+Free Text
+English
+English
+text, which is processed and utilized for
+content-based recommendations.
+German
+German text, which is processed and utilized for
+content-based recommendations.
+Numeric
+Integer
+Used for post-filtering recommendations (e.g., user
+age).
+Real
+Used for post-filtering recommendations (e.g., price).
+Date
+-
+Date information for the respective entity (e.g., creation
+date)
+Recommender customization. The Uptrendz platform fosters the notion of
+defining personalization scenarios (i.e., use-cases) when creating recommenda-
+tion APIs. The available selection of real-time recommendation models [11] for
+a given scenario depends on (i) what should be recommended (e.g., item or user
+entities), (ii) for whom the recommendations are targeted (e.g., registered or
+anonymous users) and, (iii) what kind of context is given [2] (e.g., item ID to
+recommend relevant content for). As we adopt a non-restricted configuration
+with respect to the number of freely defined user interaction types, algorithms
+that use this kind of data (e.g., Collaborative Filtering) can be customized to
+utilize any subset of the list of available interactions as well as to define how
+much weight a particular interaction type should have. With respect to post-
+filtering recommendation results, each model can use categorical (e.g., tags [12]
+or other semantic representations [8]) or numerical data attributes to ensure that
+the resulting recommendations either contain or exclude a particular value (see
+Table 1 for complete list of attributes).
+3
+Multi-Domain Support
+In order to provide a multi-domain recommender platform, we support the no-
+tions of a system-level and item-level domain in accordance with [5]. For the
+former, items and users belong to distinct systems (e.g., Netflix and Amazon).
+For the latter, individual domains have different types of items and users which
+may share some common attributes (e.g., movies and books).
+Demo Walkthrough: System-level domain. When a domain is created on
+a system level, the underlying data is physically stored in a different location
+than the data of other domains. Hence, domains do not share any data between
+themselves and the underlying services are isolated so that the performance of
+
+4
+E. Lacic, T. Duricic, L. Fadljevic, D. Theiler, and D. Kowald
+Fig. 2. Example of supporting multiple domains on the item-level (up) and configuring
+a hybrid recommendation algorithm (below) with previously created APIs.
+one domain does not impact the performance of another domain (e.g., during
+request load peaks). We demonstrate how to create a movie recommender on a
+system level. To utilize the MovieLens-100k dataset [6], we first need to configure
+the respective data services to upload (i) movie, (ii) user, and (iii) interaction
+data. Each entity needs to be separately created in the Uptrendz platform in
+order to generate an API that can be used to upload the MovieLens-specific
+data attributes. This allows creating recommendation scenarios for (i) similar
+movies (CBF), (ii) popular horror movies (MP with post-filtering), (iii) movies
+based on ratings (CF), (iv) their weighted hybrid combination (e.g., for cold-start
+settings [13], and (v) a user recommender for a given movie.
+Demo Walkthrough: Item-level domain. To showcase how to configure Up-
+trendz to support multiple-domains on an item-level, we present the use-case of
+entrepreneurial start-up founding. Here, we recommend experts that can provide
+
+Available attributes for entity:
+news
+Field Name
+Field Type
+Field Subtype
+id
+→
+Categorical Text
+[ Single Value
+content
+→
+Free Text
+→
+English
+name
+Free Text
+→
+English
+active
+Categorical Text
+→
+Single Value
+categories
+→
+Categorical Text
+→
+Multiple Values
+Available attributes for entity:
+innovation
+Field Name
+Field Type
+Field Subtype
+id
+→
+Categorical Text
+Single Value
+author
+→
+ Categorical Text
+→
+Single Value
+description
+→
+Free Text
+→
+[English
+name
+Free Text
+English
+headline
+ Free Text
+→
+English
+location
+→
+ Categorical Text
+Single Value
+development_phase
+→
+ Categorical Text
+→
+Single Value
+patent_description
+Free Text
+→
+English
+help_time
+→
+Numeric
+→
+Integer
+active
+→
+ Categorical Text
+→
+Single Value
+categories
+ Categorical Text
+Multiple Values
+Mutiple Values
+fields_of_interest
+→
+ Categorical Text
+→General Settings
+Scenario name
+ discover innovations
+Scenario ID: discover-innovations
+What will be recommended?
+Recommendation Model
+innovation
+>
+HybridRoundRobinWeightedSum
+V
+Items
+innovation
+ItemContext
+institution
+Choose Context
+education
+news
+Model Specific Settings
+Users
+user
+Select all desired scenarios which you would like to include into this hybrid scenario.
+In order to prioritize between reference scenarios, for each selected scenario you must assign a proper weight with an integer value
+Available profiles
+ Connect People Innovation Content
+ Invite People Brainstorm Content
+ Discover Innovations Personalized
+10
+ Discover Innovations Popular
+1
+ Discover Innovations Content History
+5API-Centric Real-time Recommendations in Multi-Domain Settings
+5
+Fig. 3. Uptrendz requires the specification of (i) the item types that should be recom-
+mended (e.g., products or users, depending on the domain - left figure), and (ii) the
+user types for which recommendations should be generated (e.g., registered users or
+session users - right figure).
+feedback to an innovation idea, support co-founder matching, help incubators,
+innovation hubs and accelerators to discover innovations but also provide rel-
+evant educational materials until the innovation idea matures enough to form
+a start-up. In this case, each recommendable entity has a separate data model
+and can be viewed as part of a standalone application domain. Figure 2 depicts
+how adding multiple item entities in the data catalog allows customizing data
+attributes for the respective domain. While configuring a recommendation al-
+gorithm, the respective item-level domain can be selected to be recommended.
+Here, via the example of a hybrid algorithm, only pre-configured algorithms can
+be utilized that belong to the same domain (i.e., innovation recommendations).
+Finally, in Figure 3, we show how Uptrendz allows the specification of (i)
+different item types that can be recommended, and (ii) different user types for
+which recommendations should be generated. Our demo application includes
+different specification examples.
+4
+Conclusion
+In this paper, we present Uptrendz, an API-centric recommendation platform
+that can be customized to provide real-time recommendations for multiple do-
+mains. To do that, we support the notions of a system-level and item-level do-
+main. We demonstrate Uptrendz using the popular MovieLens-100k dataset and
+the use-case of entrepreneurial start-up founding.
+In future work, we plan to support even more use cases from other domains,
+e.g., music recommendations [9]. Here, we also want to integrate fairness-aware
+recommendation algorithms for mitigating e.g., popularity bias effects.
+Acknowledgements. This research was funded by CogSteps and the “DDAI”
+COMET Module within the COMET – Competence Centers for Excellent Tech-
+nologies Programme, funded by the Austrian Federal Ministry for Transport,
+Innovation and Technology (bmvit), the Austrian Federal Ministry for Digital
+and Economic Affairs (bmdw), the Austrian Research Promotion Agency (FFG),
+the province of Styria (SFG) and partners from industry and academia.
+
+What would you like to
+围
+recommend?
+product, article, job...
+Add Item Entity
+网What kind of users do
+you have?
+user
+Add User Entity
+T6
+E. Lacic, T. Duricic, L. Fadljevic, D. Theiler, and D. Kowald
+References
+1. Adomavicius, G., Tuzhilin, A.: Context-aware recommender systems. In: Proceed-
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+

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+page_content='Uptrendz: API-Centric Real-time  Recommendations in Multi-Domain Settings  Emanuel Lacic1, Tomislav Duricic1,2, Leon Fadljevic1,  Dieter Theiler1, and Dominik Kowald(�)1,2  1 Know-Center GmbH, Graz, Austria  {elacic,tduricic,lfadljevic,dtheiler,dkowald}@know-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=' In this work, we tackle the problem of adapting a real-time  recommender system to multiple application domains, and their underly-  ing data models and customization requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' To do that, we present  Uptrendz, a multi-domain recommendation platform that can be cus-  tomized to provide real-time recommendations in an API-centric way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  We demonstrate (i) how to set up a real-time movie recommender us-  ing the popular MovieLens-100k dataset, and (ii) how to simultaneously  support multiple application domains based on the use-case of recom-  mendations in entrepreneurial start-up founding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' For that, we differenti-  ate between domains on the item- and system-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' We believe that our  demonstration shows a convenient way to adapt, deploy and evaluate a  recommender system in an API-centric way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' The source-code and doc-  umentation that demonstrates how to utilize the configured Uptrendz  API is available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Keywords: Uptrendz, API-centric recommendations, multi-domain rec-  ommendations, real-time recommendations  1  Introduction  Utilizing recommender systems is nowadays recognized as a necessary feature to  help users discover relevant content [15,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Most industry practitioners [3], when  they build a recommender system, adapt existing algorithms to the underlying  data and customization requirements of the respective application domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=',  movies, music, news, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' However, the focus of the research community has  recently shifted towards building recommendation systems that simultaneously  support multiple application domains [4,7,16] in an API-centric way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  In this work, we demonstrate Uptrendz3, an API-centric recommendation  platform, which can be configured to simultaneously provide real-time recom-  mendations in an API-centric way to multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Uptrendz supports pop-  ular recommendation algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Collaborative Filtering (CF), Content-  based Filtering (CBF, or Most Popular (MP), that are applied across different  3 https://uptrendz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='ai/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='01037v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='IR]  3 Jan 2023  2  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Lacic, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Duricic, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Fadljevic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Theiler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Kowald  RECOMMENDER  CUSTOMIZATION  SERVICE  ISOLATION  DATA  HETEROGENEITY  FAULT  TOLERANCE  MULTI-DOMAIN RECOMMENDER SYSTEM  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Aspects that need to be addressed when building a recommender system for a  multi-domain environment [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  application domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' The focus of this demonstration is to show how domain-  specific data-upload APIs can be created to support the customization of the  respective recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Using the MovieLens-100k dataset [6]  and a real-world use-case of entrepreneurial start-up founding4, we show how  such an approach allows for a highly customized recommendation system that  can be used in an API-centric way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' The source-code and documentation for this  demonstration is available via GitHub5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  2  The Uptrendz Platform  The Uptrendz platform is built on top of the ScaR recommendation framework  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' As shown in [10] and Figure 1, the microservice-based system architecture  addresses four distinctive requirements of a multi-domain recommender system,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', (i) service isolation, (ii) data heterogeneity, (iii) recommender customization,  and (iv) fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Uptrendz provides a layer on top of the framework to  dynamically configure an application domain and to instantly provide an API  to (i) upload item, user and interaction data, and (ii) request recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Domain-specific data model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' As discussed by [1], different domains may em-  ploy the same recommender algorithm but can differ with respect to what kind  of data is utilized to build the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', interaction types, context, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Given  an API-centric approach, we show that in order to support the customization of  recommender algorithms with domain-specific parameters, the underlying plat-  form needs to unambiguously know which source of information should be used  to calculate the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' To do that, the Uptrendz platform first allows  generating customized data upload APIs for multiple item and user entities (see  Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Second, with respect to interaction data, both user-item and user-user  interactions can be configured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' The interaction API is further customized in ac-  cordance to what kind of interactions the respective application domain actually  supports, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', (i) registered users, anonymous sessions or both, (ii) interaction  timestamp tracking, and (iii) type of interaction (explicit or implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  4 https://cogsteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='com/  5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='com/lacic/ECIR2023Demo  API-Centric Real-time Recommendations in Multi-Domain Settings  3  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Supported attributes to configure the data upload API for items and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Type  Sub-Type  Description  Categorical  Text  Single  Value  String value, which usually represents a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Used  for post-filtering recommendation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Multiple  Values  List of string values, which usually represent an array  of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Used for post-filtering recommendation  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Free Text  English  English  text, which is processed and utilized for  content-based recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  German  German text, which is processed and utilized for  content-based recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Numeric  Integer  Used for post-filtering recommendations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', user  age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Real  Used for post-filtering recommendations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', price).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Date Date information for the respective entity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', creation  date)  Recommender customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' The Uptrendz platform fosters the notion of  defining personalization scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', use-cases) when creating recommenda-  tion APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' The available selection of real-time recommendation models [11] for  a given scenario depends on (i) what should be recommended (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', item or user  entities), (ii) for whom the recommendations are targeted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', registered or  anonymous users) and, (iii) what kind of context is given [2] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', item ID to  recommend relevant content for).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' As we adopt a non-restricted configuration  with respect to the number of freely defined user interaction types, algorithms  that use this kind of data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Collaborative Filtering) can be customized to  utilize any subset of the list of available interactions as well as to define how  much weight a particular interaction type should have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' With respect to post-  filtering recommendation results, each model can use categorical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', tags [12]  or other semantic representations [8]) or numerical data attributes to ensure that  the resulting recommendations either contain or exclude a particular value (see  Table 1 for complete list of attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  3  Multi-Domain Support  In order to provide a multi-domain recommender platform, we support the no-  tions of a system-level and item-level domain in accordance with [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' For the  former, items and users belong to distinct systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Netflix and Amazon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  For the latter, individual domains have different types of items and users which  may share some common attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', movies and books).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Demo Walkthrough: System-level domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' When a domain is created on  a system level, the underlying data is physically stored in a different location  than the data of other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Hence, domains do not share any data between  themselves and the underlying services are isolated so that the performance of  4  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Lacic, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Duricic, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Fadljevic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Theiler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Kowald  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Example of supporting multiple domains on the item-level (up) and configuring  a hybrid recommendation algorithm (below) with previously created APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  one domain does not impact the performance of another domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', during  request load peaks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' We demonstrate how to create a movie recommender on a  system level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' To utilize the MovieLens-100k dataset [6], we first need to configure  the respective data services to upload (i) movie, (ii) user, and (iii) interaction  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Each entity needs to be separately created in the Uptrendz platform in  order to generate an API that can be used to upload the MovieLens-specific  data attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' This allows creating recommendation scenarios for (i) similar  movies (CBF), (ii) popular horror movies (MP with post-filtering), (iii) movies  based on ratings (CF), (iv) their weighted hybrid combination (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content='  Demo Walkthrough: Item-level domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' To showcase how to configure Up-  trendz to support multiple-domains on an item-level, we present the use-case of  entrepreneurial start-up founding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content='→General Settings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='Scenario name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='discover innovations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='Scenario ID: discover-innovations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='What will be recommended?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Recommendation Model  innovation  >  HybridRoundRobinWeightedSum  V  Items  innovation  ItemContext  institution  Choose Context  education  news  Model Specific Settings  Users  user  Select all desired scenarios which you would like to include into this hybrid scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  In order to prioritize between reference scenarios, for each selected scenario you must assign a proper weight with an integer value  Available profiles  Connect People Innovation Content  Invite People Brainstorm Content  Discover Innovations Personalized  10  Discover Innovations Popular  1  Discover Innovations Content History  5API-Centric Real-time Recommendations in Multi-Domain Settings  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Uptrendz requires the specification of (i) the item types that should be recom-  mended (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', products or users, depending on the domain - left figure), and (ii) the  user types for which recommendations should be generated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', registered users or  session users - right figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  feedback to an innovation idea, support co-founder matching, help incubators,  innovation hubs and accelerators to discover innovations but also provide rel-  evant educational materials until the innovation idea matures enough to form  a start-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In this case, each recommendable entity has a separate data model  and can be viewed as part of a standalone application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Figure 2 depicts  how adding multiple item entities in the data catalog allows customizing data  attributes for the respective domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' While configuring a recommendation al-  gorithm, the respective item-level domain can be selected to be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Here, via the example of a hybrid algorithm, only pre-configured algorithms can  be utilized that belong to the same domain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', innovation recommendations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Finally, in Figure 3, we show how Uptrendz allows the specification of (i)  different item types that can be recommended, and (ii) different user types for  which recommendations should be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Our demo application includes  different specification examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  4  Conclusion  In this paper, we present Uptrendz, an API-centric recommendation platform  that can be customized to provide real-time recommendations for multiple do-  mains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' To do that, we support the notions of a system-level and item-level do-  main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' We demonstrate Uptrendz using the popular MovieLens-100k dataset and  the use-case of entrepreneurial start-up founding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  In future work, we plan to support even more use cases from other domains,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', music recommendations [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Here, we also want to integrate fairness-aware  recommendation algorithms for mitigating e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', popularity bias effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' This research was funded by CogSteps and the “DDAI”  COMET Module within the COMET – Competence Centers for Excellent Tech-  nologies Programme, funded by the Austrian Federal Ministry for Transport,  Innovation and Technology (bmvit), the Austrian Federal Ministry for Digital  and Economic Affairs (bmdw), the Austrian Research Promotion Agency (FFG),  the province of Styria (SFG) and partners from industry and academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  What would you like to  围  recommend?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  product, article, job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  Add Item Entity  网What kind of users do  you have?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='  user  Add User Entity  T6  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Lacic, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Duricic, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Fadljevic, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Theiler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Kowald  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Adomavicius, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=' In: Proceedings of the 10th ACM conference on recommender  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': Cross-market  product recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In: Proceedings of the 30th ACM International Confer-  ence on Information & Knowledge Management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=' Cantador, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Fern´andez-Tob´ıas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': Cross-domain  recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In: Recommender Systems Handbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Springer (2015)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=' : The movielens datasets: History and context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Acm  transactions on interactive intelligent systems (tiis) 5(4), 1–19 (2015)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': Does a one-size recommendation system fit all?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' the effectiveness  of collaborative filtering based recommendation systems across different domains  and search modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' ACM Transactions on Information Systems (TOIS) 26(1), 4–es  (2007)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Kowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': The social se-  mantic server a framework to provide services on social semantic network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In:  Proceedings of I-SEMANTICS 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=' Kowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': Support  the underground: characteristics of beyond-mainstream music listeners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' EPJ Data  Science 10(1), 1–26 (2021)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Lacic, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Kowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': Tailoring recommendations for a multi-domain en-  vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Workshop on Intelligent Recommender Systems by Knowledge Transfer  & Learning (RecSysKTL’2017) co-located with the 11th ACM Conference on Rec-  ommender Systems (RecSys’2017) (2017)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Lacic, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Kowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Parra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Kahr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Trattner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=': Towards a scalable social  recommender engine for online marketplaces: The case of apache solr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In: Workshop  Proceedings of WWW’2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=' Lacic, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Kowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Seitlinger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=', Parra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=': Recommending items  in social tagging systems using tag and time information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Proceedings of the 1st  International Workshop on Social Personalisation (SP’2014) co-located with Hy-  pertext’2014 (2014)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Lacic, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Kowald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
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+page_content=': Tackling cold-  start users in recommender systems with indoor positioning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In: Poster  Proceedings of the 9th {ACM} Conference on Recommender Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Association  of Computing Machinery (2015)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' McNee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Riedl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Konstan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' : Being accurate is not enough: how accuracy  metrics have hurt recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In: CHI’06 extended abstracts on Human  factors in computing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' 1097–1101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' ACM (2006)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Resnick, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Varian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' : Recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Communications of the ACM  40(3), 56–58 (1997)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' Roitero, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Carterette, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Mehrotra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=', Lalmas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=': Leveraging behavioral het-  erogeneity across markets for cross-market training of recommender systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' In:  Companion Proceedings of the Web Conference 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}
+page_content=' 694–702 (2020)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfG_tj/content/2301.01037v1.pdf'}

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