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0dAzT4oBgHgl3EQftv1g/content/tmp_files/2301.01680v1.pdf.txt

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+arXiv:2301.01680v1  [math.NT]  3 Jan 2023
+CM ELLIPTIC CURVES AND VERTICALLY ENTANGLED 2-ADIC GROUPS
+NATHAN JONES
+Abstract. Consider the elliptic curve E given by the Weierstrass equation y2 = x3 − 11x − 14, which has
+complex multiplication by the order of conductor 2 inside Z[i]. It was recently observed in a paper of Daniels
+and Lozano-Robledo that, for each n ≥ 2, Q(µ2n+1) ⊆ Q(E[2n]). In this note, we prove that this (a priori
+surprising) “tower of vertical entanglements” is actually more a feature than a bug: it holds for any elliptic
+curve E over Q with complex multiplication by any order of even discriminant.
+1. Main result and proof
+Let E be an elliptic curve over Q with complex multiplication by the order OK,f ⊆ OK of conductor f
+inside the imaginary quadratic field K. Since every endomorphism of E defined over Q commutes with the
+action of Gal(Q/Q), it follows that the image of the Galois representation
+ρE,m : Gal(Q/Q) −→ Aut(E[m]) ≃ GL2(Z/mZ),
+(which is defined by letting Gal(Q/Q) act on E[m], the m-torsion subgroup of E, and fixing a Z/mZ-basis
+thereof) lies inside a certain subgroup Nδ,φ(m) ⊆ GL2(Z/mZ), which we now specify, following [2]. First,
+let us set
+φ = φ(OK,f, m) :=
+�
+0
+if ∆Kf 2 ≡ 0 mod 4 or if m is odd,
+f
+if ∆Kf 2 ≡ 1 mod 4 and m is even,
+δ = δ(OK,f, m) :=
+�
+∆Kf 2/4
+if ∆Kf 2 ≡ 0 mod 4 or if m is odd,
+(∆K − 1)f 2/4
+if ∆Kf 2 ≡ 1 mod 4 and m is even.
+Next, we define the associated Cartan subgroup Cδ,φ(m) by
+Cδ,φ(m) :=
+��
+a + bφ
+b
+bδ
+a
+�
+: a, b ∈ Z/mZ, a2 + φab − δb2 ∈ (Z/mZ)×
+�
+.
+(1)
+Finally, we define Nδ,φ(m) ⊆ GL2(Z/mZ) by
+Nδ,φ(m) :=
+��
+−1
+0
+φ
+1
+�
+, Cδ,φ(m)
+�
+.
+(2)
+If E is any elliptic curve over Q with CM by OK,f, then, for an appropriate choice of Z/mZ-basis of E[m],
+we have ρE,m(GQ) ⊆ Nδ,φ(m). For more details, see [2].
+Let E−16 be the elliptic curve defined by the Weierstrass equation y2 = x3 − 11x − 14 (i.e. the elliptic
+curve with Cremona label 32a3). The curve E−16 has CM by the order O := Z + 2iZ of conductor 2 inside
+the field Q(i). Furthermore, as observed in [1, Theorem 1.5], we have
+n ∈ N≥2 =⇒ Q(ζ2n+1) ⊆ Q(E−16[2n]).
+(3)
+The authors also observed that the elliptic curves E−4,1 and E−4,2, given, respectively, by the Weierstrass
+equations y2 = x2 + x and y2 = x3 + 2x satisfy
+Q(E−4,1[2]) = Q(ζ4)
+and
+Q(ζ8) ⊆ Q(E−4,2[4]).
+The purpose of this note is to show that the two elliptic curves E−4,1 and E−4,2 also satisfy (3). More
+generally, we will prove the following theorem.
+Theorem 1.1. Let E be any elliptic curve over Q with complex multiplication by an order OK,f in an
+imaginary quadratic field K. Assuming that the discriminant ∆Kf 2 of OK,f is even, we have that, for each
+n ∈ N≥2, Q(ζ2n+1) ⊆ Q(E[2n]).
+1
+
+Proof. Let us denote by G(2n) := ρE,2n(GQ) ⊆ Nδ,φ(2n) the mod 2n image associated to E.
+We will
+establish that, for each n ∈ N≥2, there is a surjective homomorphism δ : G(2n) ։ (Z/2n+1Z)× for which
+det |G(2n+1) = δ ◦ π, where π : G(2n+1) → G(2n) denotes the projection map. In other words, the following
+diagram will commute:
+G(2n+1)
+(Z/2n+1Z)×
+G(2n)
+(Z/2nZ)×.
+det
+π
+det
+δ
+(4)
+Once established, it will follow that
+Q(ζ2n+1) = Q(E[2n+1])ker det = Q(E[2n+1])π−1(ker δ) = Q(E[2n])ker δ ⊆ Q(E[2n]).
+(5)
+The key observation is that, since ∆Kf 2 is assumed even, it follows that φ = 0. Considering (1) and (2), we
+may then see that
+n ∈ N≥2 =⇒ ker
+�
+Nδ,φ(2n+1) → Nδ,φ(2n)
+�
+⊆ SL2(Z/2n+1Z).
+(6)
+(The reason we require n > 1 is that otherwise we do not have ker
+�
+Nδ,φ(2n+1) → Nδ,φ(2n)
+�
+⊆ Cδ,φ(2n+1),
+since
+�−1
+0
+φ
+1
+�
+≡ I mod 2; the consequent of (6) is false for n = 1.) We now define the map δ by
+δ(g) := det(g′),
+where g′ ∈ π−1(g).
+By virtue of (6), this is independent of the choice of g′ ∈ π−1(g), and thus defines a map δ : G(2n) →
+(Z/2n+1Z)×.
+It is surjective since det : G(2n+1) → (Z/2n+1Z)× is, and the diagram (4) commutes by
+definition of δ. Thus, by (5), we deduce that
+∀n ∈ N≥2,
+Q(ζ2n+1) ⊆ Q(E[2n]),
+as asserted.
+□
+The proof of Theorem 1.1 applies to a more general situation, as follows. Given an algebraic group G and
+a Galois representation ρ : Gal(Q/Q) → G(ˆZ), let us denote by ρm : Gal(Q/Q) → G(Z/mZ) the composition
+of ρ with the natural projection map G(ˆZ) → G(Z/mZ) and define Q(E(ρ[m])) := Q
+ker ρm.
+Definition 1.2. Suppose G is any algebraic group that admits a homomorphism δ : G → Gm to the mul-
+tiplicative group. We say that a Galois representation ρ : Gal(Q/Q) → G(ˆZ) extends the cyclotomic
+character if δˆZ ◦ ρ : Gal(Q/Q) → G(ˆZ) → ˆZ× agrees with the cyclotomic character. For any prime number
+p, we say that G(Zp) form a vertically entangled p-adic group if there exists n0 ∈ N so that, for each
+n ∈ N≥n0, we have ker
+�
+G(Z/pn+1Z) → G(Z/pnZ)
+�
+⊆ ker δpn+1, where δpn+1 : G(Z/pn+1Z) → (Z/pn+1Z)×
+denotes the group homomorphism associated to δ on the mod pn+1 points of G.
+Remark 1.3. Let ρ : Gal(Q/Q) → G(ˆZ) be any Galois representation that extends the cyclotomic character
+and suppose that, for some prime number p, the group G(Zp) is a vertically entangled p-adic group. Then
+the proof of Theorem 1.1 shows that, in this more general context, we have
+∀n ∈ N≥n0,
+Q(µpn+1) ⊆ Q(ρ[pn]).
+2. Acknowledgement
+The author gratefully acknowledges Harris Daniels for bringing the phenomenon (3) to his attention, and
+also Ken McMurdy for subsequent stimulating conversations.
+References
+[1] H. Daniels and A. Lozano-Robledo, Coincidences of division fields, preprint. To appear in Ann. Inst. Fourier. Available at
+https://arxiv.org/abs/1912.05618
+[2] A. Lozano-Robledo, Galois representations attached to elliptic curves with complex multiplication, preprint. To appear in
+Algebra and Number Theory. Available at https://arxiv.org/abs/1809.02584
+Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan
+St, 322 SEO, Chicago, 60607, IL, USA
+Email address: [email protected]
+2
+

0dAzT4oBgHgl3EQftv1g/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf,len=70
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='01680v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='NT]  3 Jan 2023  CM ELLIPTIC CURVES AND VERTICALLY ENTANGLED 2-ADIC GROUPS  NATHAN JONES  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Consider the elliptic curve E given by the Weierstrass equation y2 = x3 − 11x − 14, which has  complex multiplication by the order of conductor 2 inside Z[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' It was recently observed in a paper of Daniels  and Lozano-Robledo that, for each n ≥ 2, Q(µ2n+1) ⊆ Q(E[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' In this note, we prove that this (a priori  surprising) “tower of vertical entanglements” is actually more a feature than a bug: it holds for any elliptic  curve E over Q with complex multiplication by any order of even discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Main result and proof  Let E be an elliptic curve over Q with complex multiplication by the order OK,f ⊆ OK of conductor f  inside the imaginary quadratic field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Since every endomorphism of E defined over Q commutes with the  action of Gal(Q/Q), it follows that the image of the Galois representation  ρE,m : Gal(Q/Q) −→ Aut(E[m]) ≃ GL2(Z/mZ),  (which is defined by letting Gal(Q/Q) act on E[m], the m-torsion subgroup of E, and fixing a Z/mZ-basis  thereof) lies inside a certain subgroup Nδ,φ(m) ⊆ GL2(Z/mZ), which we now specify, following [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' First,  let us set  φ = φ(OK,f, m) :=  �  0  if ∆Kf 2 ≡ 0 mod 4 or if m is odd,  f  if ∆Kf 2 ≡ 1 mod 4 and m is even,  δ = δ(OK,f, m) :=  �  ∆Kf 2/4  if ∆Kf 2 ≡ 0 mod 4 or if m is odd,  (∆K − 1)f 2/4  if ∆Kf 2 ≡ 1 mod 4 and m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  Next, we define the associated Cartan subgroup Cδ,φ(m) by  Cδ,φ(m) :=  ��  a + bφ  b  bδ  a  �  : a, b ∈ Z/mZ, a2 + φab − δb2 ∈ (Z/mZ)×  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  (1)  Finally, we define Nδ,φ(m) ⊆ GL2(Z/mZ) by  Nδ,φ(m) :=  ��  −1  0  φ  1  �  , Cδ,φ(m)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  (2)  If E is any elliptic curve over Q with CM by OK,f, then, for an appropriate choice of Z/mZ-basis of E[m],  we have ρE,m(GQ) ⊆ Nδ,φ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' For more details, see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  Let E−16 be the elliptic curve defined by the Weierstrass equation y2 = x3 − 11x − 14 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' the elliptic  curve with Cremona label 32a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' The curve E−16 has CM by the order O := Z + 2iZ of conductor 2 inside  the field Q(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Furthermore, as observed in [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='5], we have  n ∈ N≥2 =⇒ Q(ζ2n+1) ⊆ Q(E−16[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  (3)  The authors also observed that the elliptic curves E−4,1 and E−4,2, given, respectively, by the Weierstrass  equations y2 = x2 + x and y2 = x3 + 2x satisfy  Q(E−4,1[2]) = Q(ζ4)  and  Q(ζ8) ⊆ Q(E−4,2[4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  The purpose of this note is to show that the two elliptic curves E−4,1 and E−4,2 also satisfy (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' More  generally, we will prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Let E be any elliptic curve over Q with complex multiplication by an order OK,f in an  imaginary quadratic field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Assuming that the discriminant ∆Kf 2 of OK,f is even, we have that, for each  n ∈ N≥2, Q(ζ2n+1) ⊆ Q(E[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Let us denote by G(2n) := ρE,2n(GQ) ⊆ Nδ,φ(2n) the mod 2n image associated to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  We will  establish that, for each n ∈ N≥2, there is a surjective homomorphism δ : G(2n) ։ (Z/2n+1Z)× for which  det |G(2n+1) = δ ◦ π, where π : G(2n+1) → G(2n) denotes the projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' In other words, the following  diagram will commute:  G(2n+1)  (Z/2n+1Z)×  G(2n)  (Z/2nZ)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  det  π  det  δ  (4)  Once established, it will follow that  Q(ζ2n+1) = Q(E[2n+1])ker det = Q(E[2n+1])π−1(ker δ) = Q(E[2n])ker δ ⊆ Q(E[2n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  (5)  The key observation is that, since ∆Kf 2 is assumed even, it follows that φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Considering (1) and (2), we  may then see that  n ∈ N≥2 =⇒ ker  �  Nδ,φ(2n+1) → Nδ,φ(2n)  �  ⊆ SL2(Z/2n+1Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  (6)  (The reason we require n > 1 is that otherwise we do not have ker  �  Nδ,φ(2n+1) → Nδ,φ(2n)  �  ⊆ Cδ,φ(2n+1),  since  �−1  0  φ  1  �  ≡ I mod 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' the consequent of (6) is false for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=') We now define the map δ by  δ(g) := det(g′),  where g′ ∈ π−1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  By virtue of (6), this is independent of the choice of g′ ∈ π−1(g), and thus defines a map δ : G(2n) →  (Z/2n+1Z)×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  It is surjective since det : G(2n+1) → (Z/2n+1Z)× is, and the diagram (4) commutes by  definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Thus, by (5), we deduce that  ∀n ∈ N≥2,  Q(ζ2n+1) ⊆ Q(E[2n]),  as asserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  □  The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='1 applies to a more general situation, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Given an algebraic group G and  a Galois representation ρ : Gal(Q/Q) → G(ˆZ), let us denote by ρm : Gal(Q/Q) → G(Z/mZ) the composition  of ρ with the natural projection map G(ˆZ) → G(Z/mZ) and define Q(E(ρ[m])) := Q  ker ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Suppose G is any algebraic group that admits a homomorphism δ : G → Gm to the mul-  tiplicative group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' We say that a Galois representation ρ : Gal(Q/Q) → G(ˆZ) extends the cyclotomic  character if δˆZ ◦ ρ : Gal(Q/Q) → G(ˆZ) → ˆZ× agrees with the cyclotomic character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' For any prime number  p, we say that G(Zp) form a vertically entangled p-adic group if there exists n0 ∈ N so that, for each  n ∈ N≥n0, we have ker  �  G(Z/pn+1Z) → G(Z/pnZ)  �  ⊆ ker δpn+1, where δpn+1 : G(Z/pn+1Z) → (Z/pn+1Z)×  denotes the group homomorphism associated to δ on the mod pn+1 points of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Let ρ : Gal(Q/Q) → G(ˆZ) be any Galois representation that extends the cyclotomic character  and suppose that, for some prime number p, the group G(Zp) is a vertically entangled p-adic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Then  the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='1 shows that, in this more general context, we have  ∀n ∈ N≥n0,  Q(µpn+1) ⊆ Q(ρ[pn]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Acknowledgement  The author gratefully acknowledges Harris Daniels for bringing the phenomenon (3) to his attention, and  also Ken McMurdy for subsequent stimulating conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='  References  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Daniels and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Lozano-Robledo, Coincidences of division fields, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' To appear in Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Fourier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Available at  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='org/abs/1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='05618  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Lozano-Robledo, Galois representations attached to elliptic curves with complex multiplication, preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' To appear in  Algebra and Number Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content=' Available at https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='org/abs/1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='02584  Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S Morgan  St, 322 SEO, Chicago, 60607, IL, USA  Email address: ncjones@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}
+page_content='edu  2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQftv1g/content/2301.01680v1.pdf'}

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+MNRAS 000, 1–5 (2018)
+Preprint 23 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The Effect of the Peculiar Motions of the Lens, Source and the Observer on
+the Gravitational Lensing Time Delay
+Gihan Weerasekara,1★ Thulsi Wickramasinghe,2 Chandana Jayaratne1
+1Department of Physics, University of Colombo, Sri Lanka
+2Department of Physics, The College of New Jersey, Ewing, NJ 08628, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+An intervening galaxy acts as a gravitational lens and produces multiple images of a single source such as a remote galaxy.
+Galaxies have peculiar speeds in addition to the bulk motion arising due to the expansion of the universe. There is a difference in
+light arrival times between lensed images. We calculate more realistic time delays between lensed images when galaxy peculiar
+motions, that is the motion of the Lens, the Source and the Observer are taken into consideration neglecting the gravitomagnetic
+effects.
+Key words: gravitational lensing: strong – galaxies: peculiar
+1 INTRODUCTION
+A remote galaxy S at redshift 𝑧𝑠 (Shown in Figure 1) is lensed by an
+intervening galaxy L at redshift 𝑧𝑑. A light ray from S bends by an
+angle 𝛼 before arriving at the observer O. The image I of S forms at
+an angle 𝜃 while S is at 𝛽. The distances 𝐷𝑑, 𝐷𝑠 and 𝐷𝑑𝑠 shown are
+the angular diameter distances. Walsh (1979), Chen (1995)
+From the theory of lensing, we can derive the angular positions 𝜃1
+and 𝜃2 of the two lensed images formed due to a single point lens.
+There is a delay Δ𝜏 of light arrival times from these two images.
+This delay is arising due to both geometrical path difference and the
+fact that two light rays are traveling in two different potential wells
+on either side of the lens. The total time delay is given by, Schneider
+(1992), Bradt (2008)
+Δ𝜏 =
+𝐷 𝑓
+𝑐 (1 + 𝑧𝑑)
+� 1
+2 (𝜃2
+1 − 𝜃2
+2) + |𝜃1𝜃2| ln
+����
+𝜃1
+𝜃2
+����
+�
+(1)
+where,
+𝐷 𝑓 = 𝐷𝑑𝐷𝑠
+𝐷𝑑𝑠
+(2)
+We calculate analytically a more realistic time delay between the
+two images when the peculiar speeds of the lens, the source and the
+observer are considered. These peculiar speeds are random speeds
+With respect to the cosmic microwave background radiation - Hubble
+flow.
+But as we already know a point mass lens is a highly idealized
+and less practical lensing model for a real lensing system, in the next
+part of the paper we will be considering a more practical Singular
+Isothermal Sphere (SIS) lensing model to calculate the time delay
+difference when the peculiar speeds of the objects are considered.
+★ E-mail: [email protected]
+Figure 1. Gravitational Lensing Diagram. The peculiar speed 𝑣 of the lens L
+is measured with respect to a freely falling observer with the Hubble flow at
+the location of the lens. The angle 𝜖 is measured from the optic axis OL.
+© 2018 The Authors
+arXiv:2301.08622v1  [astro-ph.CO]  20 Jan 2023
+
+10
+S
+Dds
+So
+α
+Ds
+β;
+PO
+;02
+Weerasekara et al.
+2 THEORY
+The angular diameter distance D of a source having no peculiar
+motion at a red shift 𝑧 is given by, Weinberg (1972), Hobson (2006)
+𝐷(𝑧, ΩΛ,0) = 𝑐
+𝐻0
+1
+1 + 𝑧
+1
+∫
+1
+1+𝑧
+𝑑𝑥
+√︃
+𝑥4 ΩΛ,0 + 𝑥 Ωm,0 + Ωr,0
+(3)
+where Ωi,0 is the density parameter of the substance 𝑖 of the cosmic
+fluid measured at the present time 𝑡0. We assume a flat universe
+(𝑘 = 0) for which Perlmutter (1999),
+Ωm,0 + Ωr,0 + ΩΛ,0 = 1
+(4)
+The red shift 𝑧𝑑𝑠 of S as measured by L is given by,
+1 + 𝑧𝑠 = (1 + 𝑧𝑑)(1 + 𝑧𝑑𝑠)
+(5)
+Thus, from the equations (3), (4) and (5), neglecting Ωr,0 and elimi-
+nating Ωm,0 and expressing everything with the dark energy, we can
+derive the value of 𝐷𝑑𝑠, the angular diameter distance of the source
+as measured by an observer on the lens as,
+𝐷𝑑𝑠
+�𝑧𝑑, 𝑧𝑠, ΩΛ,0
+� =
+𝑐
+𝐻0
+1
+√︁ΩΛ,0
+1 + 𝑧𝑑
+1 + 𝑧𝑠
+1
+∫
+1+𝑧𝑑
+1+𝑧𝑠
+𝑑𝑥
+√︂
+𝑥4 + 𝑥
+�
+1
+ΩΛ,0 − 1
+�
+(1 + 𝑧𝑑)3
+(6)
+By evaluating the integral analytically, the value of 𝐷𝑑𝑠 can be
+written as
+𝐷𝑑𝑠
+�𝑧𝑑, 𝑧𝑠, ΩΛ,0
+� = 𝑐
+𝐻0
+1
+1 + 𝑧𝑠
+�
+Ψ �𝑧𝑠, ΩΛ,0
+� − Ψ �𝑧𝑑, ΩΛ,0
+��
+(7)
+where in terms of hypergeometric function 2𝐹1
+Ψ �𝑧, ΩΛ,0
+� =
+1 + 𝑧
+√︁ΩΛ,0
+2𝐹1
+� 1
+3, 1
+2; 4
+3;
+�
+1 −
+1
+ΩΛ,0
+�
+(1 + 𝑧)3
+�
+(8)
+In the theory of lensing, the source S, lens L, and the observer O in
+Fig. 1 are all freely falling with the smooth expansion of the universe;
+that is, experiencing no peculiar motions. The angular diameter dis-
+tances 𝐷𝑠, 𝐷𝑑 and 𝐷𝑑𝑠 are then measured between these objects
+which are freely falling with the Hubble flow. Thus, the redshifts
+entering Eq (8) should be associated with the freely falling objects.
+However, all galaxies are subjected to peculiar or random motions,
+for an example in the scenario given here the Source S, the Lens L
+and the Observer O are having peculiar motions. Thus, the redshift
+of the lens we measure includes this peculiar motion. Therefore, the
+redshifts entering Eq (7), which should be the redshifts of freely
+falling objects, must be corrected for random peculiar motions. For
+this, consider initially the random motion of L neglecting the random
+motions of S and O. This is similar to OS axis being fixed and L
+having a peculiar motion with respect to this axis. An observer freely
+falling with the Hubble flow at the location of L will see a Doppler
+shift of L arising due to the random (peculiar) speed 𝜈. In addition
+to this shift, we have the cosmological redshift of that freely falling
+observer arising due to the bulk expanding motion of the universe.
+Thus, the redshift z of the freely falling observer, from special theory
+of relativity, becomes (see Figure. 1)
+1 + 𝑧 =
+√︁
+1 − 𝛽2
+1 − 𝛽 cos 𝜖 (1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑)
+(9)
+where 𝑣 = 𝛽𝑐 is the peculiar speed of the object as seen by the freely
+falling observer and 𝜖 is the angle between the peculiar velocity vector
+and the line-of-sight to L (see Fig. 1). It is this redshift 𝑧 (Eq. 9) that
+should enter in (7) for the angular diameter distance calculation. If
+𝜖 = 0, L is approaching a freely falling observer and if 𝜖 = 𝜋 it is
+receding. Inserting (9) in (8) and expanding to first order in 𝛽 we get,
+Ψ �𝑧, ΩΛ,0
+� ∼ 1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑
+√︁ΩΛ,0
+×
+2𝐹1
+�
+1 +
+�
+1 + 3
+8
+�
+1 −
+1
+ΩΛ,0
+� �
+1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑�3�
+𝛽 cos 𝜖
+�
+(10)
+where the hypergeometric function is the one appearing in (8) with
+𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑. Now that we have an expression to account for the
+peculiar motion of L, we can employ the same in our code to calculate
+the time delay taking all the peculiar motions into consideration. That
+is including the peculiar motions of S, L and O. while doing so, we
+find that the other higher order terms are very small and the time
+delay is linear to first order in 𝛽. Then the form of the observed time
+delay becomes,
+Δ𝜏 ≈ Δ𝜏0 (1 + 𝜅 𝛽 cos 𝜖)
+(11)
+where Δ𝜏0 is when the peculiar motions are neglected.
+As we now have an equation for the gravitational time delay differ-
+ence when the peculiar speeds are considered for a point mass lens
+model, let us now proceed to the Singular Isothermal Sphere lensing
+model and derive the time delay difference equation for that.
+According to the theory of lensing the time delay difference for a
+SIS model is given by the equation, Schneider (1992)
+𝑐Δ𝜏 =
+�
+4𝜋
+� 𝜎𝑣
+𝑐
+�2�2 𝐷𝑑𝐷𝑑𝑠
+𝐷𝑠
+(1 + 𝑧𝑑)2𝑦
+(12)
+further by making use of the following equations,
+𝑦 = 𝜂
+𝜂0
+(13)
+𝜉0 = 4𝜋
+� 𝜎𝑣
+𝑐
+�2 𝐷𝑑𝐷𝑑𝑠
+𝐷𝑠
+(14)
+we can arrive at the following equation that gives us the required
+time delay.
+Δ𝜏 = 4𝜋
+𝑐
+� 𝜎𝑣
+𝑐
+�2
+𝐷𝑑(1 + 𝑧𝑑)2𝛽
+(15)
+we do a realistic assumption for 𝛽 by making use of the point mass
+lens model as,
+𝛽 = 𝜃1 + 𝜃2
+(16)
+In this equation when we consider the peculiar speeds of the ob-
+jects, we have to use 𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑 in accordance with (9) similar
+to the calculation we have carried out with the point mass lens.
+MNRAS 000, 1–5 (2018)
+
+Gravitational Lensing Time Delay with Peculiar Motions
+3
+Figure 2. Lensing image. The Optical and Radio delay for this system has
+been measured. Koopmans (1998)
+3 RESULTS AND DISCUSSION
+The example we have used is the lensing system illustrated in the
+Figure 2. Koopmans (1998) This lens is referred to as B1600+434
+and it has the following characteristics.
+Optical time delay
+= 51 ± 2 Days
+𝑧𝑠
+= 1.59
+𝜃1
+= +1.14"
+𝑧𝑑
+= 0.42
+𝜃2
+= -0.25"
+According to the given set of angular distances and angles assum-
+ing the non-realistic assumption that the lens is a point mass, we
+can calculate a theoretical lensing delay time of 73.92 days for the
+WMAP cosmological parameters. When we compare the theoretical
+time delay and the observed time delays it is clear that they are not
+matching. We believe that the discrepancy is arising due to the lens
+point-mass assumption and that we have not taken peculiar speeds
+into account. However we would like to illustrate the effect of the
+peculiar motions on the time delay assuming initially a point-mass
+lens here.
+We simulated 1000 scenarios with the above given particular set
+of lensing parameters (𝑧𝑠 = 1.59, 𝑧𝑑 = 0.42, 𝜃1 = +1.14" and 𝜃2 =
+-0.25" ). For each scenario the lens and the observer have random
+peculiar speeds in random directions with respect to the back ground
+radiation. In the simulations of Figure 3/4/5. the peciliar speeds are
+non relativistic and they range from 0 to 0.01𝑐.
+for this lensing system Eq (11) can be written as,
+Δ𝜏 ≈ 73.92 (1 + 4.69 𝛽 cos 𝜖)
+(17)
+The observer, that is the Milky Way has an estimated peculiar
+speed of 600𝑘𝑚𝑠−1 Kogut (1993) with respect to the back ground
+radiation. The directions of the peculiar motions are taken to be
+random in relation to the OL axis. We have taken ΩΛ,0 = 0.73.
+The simulated time delays as shown in Figure 3. are showing a time
+delay range of 8 days with the contribution of the peculiar motions
+Figure 3. Point Mass lens. The Source, the Lens, and the Observer all are
+having peculiar speeds in the range of 0 to 0.01c in any random direction
+Figure 4. Point Mass lens. The Source and the Observer are having peculiar
+speeds in the range of 0 to 0.01c in any random direction. The Lens is
+stationary
+while no peculiar motion time delay being 73.9 days. Therefore the
+maximum time delay when all three objects are moving is nearly 4
+days and it is a significant value. Therefore the peculiar motions will
+give rise to a measurable and significant difference in the gravitational
+lensing time delay.
+In the second simulation given in Figure 4 we have excluded only
+the peculiar motion of the Lens. In this case it is seen that the maxi-
+mum time delay difference is about 1 day. From this result it is clear
+that the peculiar motions of the source and the Observer alone when
+the lens is not moving is not creating a significant gravitational lens-
+ing time delay. To further enhance this fact we have taken another
+simulation with only the Lens having peculiar motions and the ob-
+server and the source are stationary. That result is given in the Figure
+5.
+MNRAS 000, 1–5 (2018)
+
+1000TimeDelaysinDays
+NopeculiarmotionDelay=73.9days
+TheSource,theLensandtheObserverhavePeculiarSpeeds
+140
+120
+100
+ber
+Num
+80
+40F
+20
+70
+72
+74
+78
+78
+TimeDelayinDays1000TimeDelavsinDays
+Nopeculiar.motionDelay=73.9days
+OnlySourceandObserverhavePeculiarSpeeds,Lensisnotmoving
+140
+120
+100
+ber
+unN
+80
+60
+40
+20
+73.0
+73.5
+74.0
+74.5
+75.0
+TimeDelayinDays4
+Weerasekara et al.
+Figure 5. Point Mass lens. The Lens is having peculiar speeds in the range of
+0 to 0.01c in any random direction. The source and the observer are stationary
+The result we have obtained in Figure 5 is almost identical to the
+result we have obtained in the Figure 3.
+From these results it is clear that the gravitational lensing time
+delay is highly sensitive to the peculiar speeds of the lens. An-
+other interesting result of the simulation is the peculiar speeds of
+the observer and the source is not having a significant effect on the
+gravitational lensing time delay.
+As we have figured out by now, the gravitational lensing time delay
+is mostly affected by the peculiar motions of the Lens. Thus we can
+neglect the peculiar motions of the Observer and the Source.
+In the next simulation given in Figure 6, we have taken a lensing
+system with only the lens moving. In that we have taken the speed
+and the direction of the lens separately. The lens in the simulation
+is having speeds from 0 to 0.005𝑐 and the direction is 0 (The lens
+is approaching the observer) to 𝜋 (The lens is receding from the
+observer). If the 𝜖 is 𝜋/2 then the Lens is moving in a transverse
+direction.
+From Figure 6, we can identify that when the lens is moving
+towards the observer the gravitational lensing time delay is increasing
+and it is attaining larger values directly in proportion with the peculiar
+speed of the lens. That is, when the lens is having larger approaching
+peculiar speeds the gravitational lensing time delay is also larger.
+In contrast to that when the lens is receding from the observer the
+gravitation lensing time delay is decreasing. It can be also seen that
+when the receding peculiar speed is becoming larger the gravitational
+lensing time delay is becoming smaller.
+If the lens is moving in a transverse direction then there is no
+measurable effect in the gravitational lensing time delay as the effect
+is in second order.
+The lenses we have considered so far are having small velocities.
+But if we consider lenses having relativistic speeds then the effect
+become more prominent. That is the measurable gravitational lensing
+time delay becomes much larger. Results are illustrated in the Figure
+7, where the peculiar speeds of the lens are relativistic.
+In the example we have taken, the Lens B1400+434 is having an
+measured optical time delay of 51 days and a theoretical time delay
+of 73.92 days, assuming a point-mass lens. From our results we can
+account for the difference of this time delay. That is we can have
+this particular observed optical time delay difference if the lens is
+Figure 6. Point Mass lens. The lens is having different peculiar speeds in
+different directions
+Figure 7. Point Mass lens. The Lens is having relativistic peculiar speeds
+having a relativistic peculiar speed in the range of 0.05𝑐 to 0.06𝑐 in
+a receding direction from us provided that we model the lens as a
+point mass, which is not exact.
+As we now have a clear idea on gravitational lensing time delays
+when the peculiar speeds of the objects are considered while using
+a point mass lensing model, let us now investigate the same effect
+when a more realistic Singular Isothermal Sphere lensing model is
+used for the calculations.
+For this also we employ the same simulation with 1000 scenarios
+where random peculiar speeds are in random directions. when using
+Eq. (15) average velocity dispersion 𝜎𝑣 will be taken as 150𝑘𝑚𝑠−1
+Koopmans (1998). With this average velocity dispersion value and
+using Singular Isothermal Sphere model we have a very interesting
+result for the non peculiar motion lensing time delay, which is 51.45
+days. this value is almost identical to the observed lensing time delay
+value of 51 ± 2 Days.
+MNRAS 000, 1–5 (2018)
+
+1000TimeDelaysinDays
+NopeculiarmotionDelay=73.9days
+OnlyLensishavingPeculiarSpeeds,ObserverandSourcenotmoving
+200
+150
+ber
+unN
+100
+72
+73
+74
+75
+78
+77
+81
+TimeDelayinDaysUpperCurveforApproachingLens
+LowerCurveforRecedingLens
+MiddleCurveforLensMovinginTransverseDirection
+75.5
+75.0
+Days
+ApproachingLense
+74.5
+TransverseLense
+74:0
+Dela
+Receding Lense
+73.5
+73.0
+72.5
+0.000
+0.001
+0.002
+0.003
+0:004
+0.005
+β=IIRelativisticLens
+UpperCurveforApproachingLens
+LowerCurveforRecedingLens
+MiddleCurveforLensMovinginTransverseDirection
+120
+110
+Days
+100
+ApproachingLense
+90
+Transverse Lense
+80
+Receding Lense
+Del
+Time
+70
+60
+50F
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1%1=sGravitational Lensing Time Delay with Peculiar Motions
+5
+Figure 8. Singular Isothermal Sphere lens model. The Lens is having non
+relativistic peculiar speeds in the range of 0 to 0.01c in any random direction
+The simulation for the non relativistic peculiar speeds is given in
+the Figure 8. In that the non relativistic peculiar speeds are from 0
+to 0.01𝑐. further it can be noted in this simulation the time delays
+are ranging from 50.5 - 52.5 days while having a maximum delay
+difference of 1 day from the no peculiar motion instance. therefore
+even with non relativistic peculiar speeds it is clear that we can have
+measurable and significant time delay difference from the no peculiar
+motion instance when peculiar speeds of the lens is considered.
+In the next simulation given in the Figure 9. we consider a rela-
+tivistic peculiar speed distribution from 0 to 0.05𝑐. it can be noted
+in this figure when there is a relativistic peculiar speed distribution
+for the lens, the lensing time delays can range from 46-56 days with
+a maximum delay difference of 5 days from the no peculiar motion
+instance. therefore it is apparent from this simulation when there is a
+relativistic peculiar speed for the lens there can be a very significant
+gravitational lensing time difference from the non peculiar speed
+instance while using a more realistic Singular Isothermal Sphere to
+model the lens.
+4 CONCLUSIONS
+From the above simulations we have found out that in fact there is a
+significant measurable time delay difference arising from the peculiar
+speeds of the lens using both non realistic point mass lens and more
+realistic Singular Isothermal Sphere as the lensing model.
+The important observation is that an approaching lens results in
+an increase of the time delay while a receding lens gives rise to a
+decrease in the delay.
+We find that the time delay is not significantly affected by the
+source or observer peculiar motions.
+We see from Figure 7. and Figure 9. that a relativistically moving
+lens in any direction can significantly affect the lensing time delays.
+Figure 9. Singular Isothermal Sphere lens model. The Lens is having rela-
+tivistic peculiar speeds in the range of 0 to 0.05c in any random direction
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable request
+to the corresponding author.
+REFERENCES
+Bradt H., 2008, Astrophysics Processes, Cambridge University Press, UK,
+437, 482
+Chen G.H., Kochanek C.S. and Hewitt J.N., 1995, Astrophys. J. 447, 62
+Hobson M. P. , Efstathiou G. P. and Lasenby A. N. , 2006, General Relativity
+An Introduction for Physicists, Cambridge University Press, UK, 355,
+427
+Kogut A., Lineweaver C., Smoot G.F., Bennett C. L., Banday A., et al, 1993,
+Astrophysical Journal 419, 1 (1993)
+Koopmans L.V.E, de Bruyn A.G, Jackson N., et al, 1998, MNRAS, vol. 295,
+534 (1998)
+Perlmutter S. et al, 1999, Astrophys. J. 517, 565
+Schneider P. , Ehlers J. and Falco E.E , 1992 Gravitational Lenses, Springer-
+Verlag
+Walsh D., Carswell R.F. and Weyman R.J, 1979, Natwe 279, 381
+Weinberg S. 1972, , Gravitation & Cosmology, Wiley, New York, 407, 633,
+Weinberg S. , 2008, Cosmology, Oxford
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–5 (2018)
+
+1000TimeDelaysinDays
+NopeculiarmotionDelay=51.45days
+sigma =150
+lensmoving
+150
+100
+Number
+50
+50.5
+51.0
+51.5
+52.0
+52.5
+TimeDelayinDays1000TimeDelaysinDays
+NopeculiarmotionDelay=51.45days
+sigma=150
+lensmoving
+120
+100
+Number
+80
+60
+40
+20
+0上
+46
+48
+50
+52
+54
+56
+TimeDelay inDays

49FAT4oBgHgl3EQfmR1P/content/tmp_files/load_file.txt

@@ -0,0 +1,273 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf,len=272
+page_content='MNRAS 000, 1–5 (2018)  Preprint 23 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  The Effect of the Peculiar Motions of the Lens, Source and the Observer on  the Gravitational Lensing Time Delay  Gihan Weerasekara,1★ Thulsi Wickramasinghe,2 Chandana Jayaratne1  1Department of Physics, University of Colombo, Sri Lanka  2Department of Physics, The College of New Jersey, Ewing, NJ 08628, USA  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  An intervening galaxy acts as a gravitational lens and produces multiple images of a single source such as a remote galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Galaxies have peculiar speeds in addition to the bulk motion arising due to the expansion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' There is a difference in  light arrival times between lensed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' We calculate more realistic time delays between lensed images when galaxy peculiar  motions, that is the motion of the Lens, the Source and the Observer are taken into consideration neglecting the gravitomagnetic  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Key words: gravitational lensing: strong – galaxies: peculiar  1 INTRODUCTION  A remote galaxy S at redshift 𝑧𝑠 (Shown in Figure 1) is lensed by an  intervening galaxy L at redshift 𝑧𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' A light ray from S bends by an  angle 𝛼 before arriving at the observer O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The image I of S forms at  an angle 𝜃 while S is at 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The distances 𝐷𝑑, 𝐷𝑠 and 𝐷𝑑𝑠 shown are  the angular diameter distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Walsh (1979), Chen (1995)  From the theory of lensing, we can derive the angular positions 𝜃1  and 𝜃2 of the two lensed images formed due to a single point lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  There is a delay Δ𝜏 of light arrival times from these two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  This delay is arising due to both geometrical path difference and the  fact that two light rays are traveling in two different potential wells  on either side of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The total time delay is given by, Schneider  (1992), Bradt (2008)  Δ𝜏 =  𝐷 𝑓  𝑐 (1 + 𝑧𝑑)  � 1  2 (𝜃2  1 − 𝜃2  2) + |𝜃1𝜃2| ln  ����  𝜃1  𝜃2  ����  �  (1)  where,  𝐷 𝑓 = 𝐷𝑑𝐷𝑠  𝐷𝑑𝑠  (2)  We calculate analytically a more realistic time delay between the  two images when the peculiar speeds of the lens, the source and the  observer are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' These peculiar speeds are random speeds  With respect to the cosmic microwave background radiation - Hubble  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  But as we already know a point mass lens is a highly idealized  and less practical lensing model for a real lensing system, in the next  part of the paper we will be considering a more practical Singular  Isothermal Sphere (SIS) lensing model to calculate the time delay  difference when the peculiar speeds of the objects are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  ★ E-mail: contactgihan@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='com  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Gravitational Lensing Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The peculiar speed 𝑣 of the lens L  is measured with respect to a freely falling observer with the Hubble flow at  the location of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The angle 𝜖 is measured from the optic axis OL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  © 2018 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='08622v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='CO]  20 Jan 2023  10  S  Dds  So  α  Ds  β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  PO  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='02  Weerasekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  2 THEORY  The angular diameter distance D of a source having no peculiar  motion at a red shift 𝑧 is given by, Weinberg (1972), Hobson (2006)  𝐷(𝑧, ΩΛ,0) = 𝑐  𝐻0  1  1 + 𝑧  1  ∫  1  1+𝑧  𝑑𝑥  √︃  𝑥4 ΩΛ,0 + 𝑥 Ωm,0 + Ωr,0  (3)  where Ωi,0 is the density parameter of the substance 𝑖 of the cosmic  fluid measured at the present time 𝑡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' We assume a flat universe  (𝑘 = 0) for which Perlmutter (1999),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0 + Ωr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0 + ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0 = 1  (4)  The red shift 𝑧𝑑𝑠 of S as measured by L is given by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  1 + 𝑧𝑠 = (1 + 𝑧𝑑)(1 + 𝑧𝑑𝑠)  (5)  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' from the equations (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' (4) and (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' neglecting Ωr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0 and elimi-  nating Ωm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0 and expressing everything with the dark energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' we can  derive the value of 𝐷𝑑𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' the angular diameter distance of the source  as measured by an observer on the lens as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  𝐷𝑑𝑠  �𝑧𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 𝑧𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  � =  𝑐  𝐻0  1  √︁ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  1 + 𝑧𝑑  1 + 𝑧𝑠  1  ∫  1+𝑧𝑑  1+𝑧𝑠  𝑑𝑥  √︂  𝑥4 + 𝑥  �  1  ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0 − 1  �  (1 + 𝑧𝑑)3  (6)  By evaluating the integral analytically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' the value of 𝐷𝑑𝑠 can be  written as  𝐷𝑑𝑠  �𝑧𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 𝑧𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  � = 𝑐  𝐻0  1  1 + 𝑧𝑠  �  Ψ �𝑧𝑠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  � − Ψ �𝑧𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  ��  (7)  where in terms of hypergeometric function 2𝐹1  Ψ �𝑧,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  � =  1 + 𝑧  √︁ΩΛ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  2𝐹1  � 1  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 1  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 4  3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  �  1 −  1  ΩΛ,0  �  (1 + 𝑧)3  �  (8)  In the theory of lensing, the source S, lens L, and the observer O in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 1 are all freely falling with the smooth expansion of the universe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  that is, experiencing no peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The angular diameter dis-  tances 𝐷𝑠, 𝐷𝑑 and 𝐷𝑑𝑠 are then measured between these objects  which are freely falling with the Hubble flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Thus, the redshifts  entering Eq (8) should be associated with the freely falling objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  However, all galaxies are subjected to peculiar or random motions,  for an example in the scenario given here the Source S, the Lens L  and the Observer O are having peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Thus, the redshift  of the lens we measure includes this peculiar motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Therefore, the  redshifts entering Eq (7), which should be the redshifts of freely  falling objects, must be corrected for random peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' For  this, consider initially the random motion of L neglecting the random  motions of S and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' This is similar to OS axis being fixed and L  having a peculiar motion with respect to this axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' An observer freely  falling with the Hubble flow at the location of L will see a Doppler  shift of L arising due to the random (peculiar) speed 𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' In addition  to this shift, we have the cosmological redshift of that freely falling  observer arising due to the bulk expanding motion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Thus, the redshift z of the freely falling observer, from special theory  of relativity, becomes (see Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 1)  1 + 𝑧 =  √︁  1 − 𝛽2  1 − 𝛽 cos 𝜖 (1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑)  (9)  where 𝑣 = 𝛽𝑐 is the peculiar speed of the object as seen by the freely  falling observer and 𝜖 is the angle between the peculiar velocity vector  and the line-of-sight to L (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' It is this redshift 𝑧 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' 9) that  should enter in (7) for the angular diameter distance calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' If  𝜖 = 0, L is approaching a freely falling observer and if 𝜖 = 𝜋 it is  receding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Inserting (9) in (8) and expanding to first order in 𝛽 we get,  Ψ �𝑧, ΩΛ,0  � ∼ 1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑  √︁ΩΛ,0  ×  2𝐹1  �  1 +  �  1 + 3  8  �  1 −  1  ΩΛ,0  � �  1 + 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑�3�  𝛽 cos 𝜖  �  (10)  where the hypergeometric function is the one appearing in (8) with  𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Now that we have an expression to account for the  peculiar motion of L, we can employ the same in our code to calculate  the time delay taking all the peculiar motions into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' That  is including the peculiar motions of S, L and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' while doing so, we  find that the other higher order terms are very small and the time  delay is linear to first order in 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Then the form of the observed time  delay becomes,  Δ𝜏 ≈ Δ𝜏0 (1 + 𝜅 𝛽 cos 𝜖)  (11)  where Δ𝜏0 is when the peculiar motions are neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  As we now have an equation for the gravitational time delay differ-  ence when the peculiar speeds are considered for a point mass lens  model, let us now proceed to the Singular Isothermal Sphere lensing  model and derive the time delay difference equation for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  According to the theory of lensing the time delay difference for a  SIS model is given by the equation, Schneider (1992)  𝑐Δ𝜏 =  �  4𝜋  � 𝜎𝑣  𝑐  �2�2 𝐷𝑑𝐷𝑑𝑠  𝐷𝑠  (1 + 𝑧𝑑)2𝑦  (12)  further by making use of the following equations,  𝑦 = 𝜂  𝜂0  (13)  𝜉0 = 4𝜋  � 𝜎𝑣  𝑐  �2 𝐷𝑑𝐷𝑑𝑠  𝐷𝑠  (14)  we can arrive at the following equation that gives us the required  time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Δ𝜏 = 4𝜋  𝑐  � 𝜎𝑣  𝑐  �2  𝐷𝑑(1 + 𝑧𝑑)2𝛽  (15)  we do a realistic assumption for 𝛽 by making use of the point mass  lens model as,  𝛽 = 𝜃1 + 𝜃2  (16)  In this equation when we consider the peculiar speeds of the ob-  jects, we have to use 𝑧 = 𝑧𝑜𝑏𝑠𝑒𝑟 𝑣𝑒𝑑 in accordance with (9) similar  to the calculation we have carried out with the point mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  MNRAS 000, 1–5 (2018)  Gravitational Lensing Time Delay with Peculiar Motions  3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Lensing image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Optical and Radio delay for this system has  been measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Koopmans (1998)  3 RESULTS AND DISCUSSION  The example we have used is the lensing system illustrated in the  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Koopmans (1998) This lens is referred to as B1600+434  and it has the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Optical time delay  = 51 ± 2 Days  𝑧𝑠  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='59  𝜃1  = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='14"  𝑧𝑑  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='42  𝜃2  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='25"  According to the given set of angular distances and angles assum-  ing the non-realistic assumption that the lens is a point mass, we  can calculate a theoretical lensing delay time of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='92 days for the  WMAP cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' When we compare the theoretical  time delay and the observed time delays it is clear that they are not  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' We believe that the discrepancy is arising due to the lens  point-mass assumption and that we have not taken peculiar speeds  into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' However we would like to illustrate the effect of the  peculiar motions on the time delay assuming initially a point-mass  lens here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  We simulated 1000 scenarios with the above given particular set  of lensing parameters (𝑧𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='59, 𝑧𝑑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='42, 𝜃1 = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='14" and 𝜃2 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='25" ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' For each scenario the lens and the observer have random  peculiar speeds in random directions with respect to the back ground  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' In the simulations of Figure 3/4/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' the peciliar speeds are  non relativistic and they range from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='01𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  for this lensing system Eq (11) can be written as,  Δ𝜏 ≈ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='92 (1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='69 𝛽 cos 𝜖)  (17)  The observer, that is the Milky Way has an estimated peculiar  speed of 600𝑘𝑚𝑠−1 Kogut (1993) with respect to the back ground  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The directions of the peculiar motions are taken to be  random in relation to the OL axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' We have taken ΩΛ,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  The simulated time delays as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' are showing a time  delay range of 8 days with the contribution of the peculiar motions  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Source, the Lens, and the Observer all are  having peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='01c in any random direction  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Source and the Observer are having peculiar  speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='01c in any random direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Lens is  stationary  while no peculiar motion time delay being 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='9 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Therefore the  maximum time delay when all three objects are moving is nearly 4  days and it is a significant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Therefore the peculiar motions will  give rise to a measurable and significant difference in the gravitational  lensing time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  In the second simulation given in Figure 4 we have excluded only  the peculiar motion of the Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' In this case it is seen that the maxi-  mum time delay difference is about 1 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' From this result it is clear  that the peculiar motions of the source and the Observer alone when  the lens is not moving is not creating a significant gravitational lens-  ing time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' To further enhance this fact we have taken another  simulation with only the Lens having peculiar motions and the ob-  server and the source are stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' That result is given in the Figure  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  MNRAS 000, 1–5 (2018)  1000TimeDelaysinDays  NopeculiarmotionDelay=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='9days  TheSource,theLensandtheObserverhavePeculiarSpeeds  140  120  100  ber  Num  80  40F  20  70  72  74  78  78  TimeDelayinDays1000TimeDelavsinDays  Nopeculiar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='motionDelay=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='9days  OnlySourceandObserverhavePeculiarSpeeds,Lensisnotmoving  140  120  100  ber  unN  80  60  40  20  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  TimeDelayinDays4  Weerasekara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Lens is having peculiar speeds in the range of  0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='01c in any random direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The source and the observer are stationary  The result we have obtained in Figure 5 is almost identical to the  result we have obtained in the Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  From these results it is clear that the gravitational lensing time  delay is highly sensitive to the peculiar speeds of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' An-  other interesting result of the simulation is the peculiar speeds of  the observer and the source is not having a significant effect on the  gravitational lensing time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  As we have figured out by now, the gravitational lensing time delay  is mostly affected by the peculiar motions of the Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Thus we can  neglect the peculiar motions of the Observer and the Source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  In the next simulation given in Figure 6, we have taken a lensing  system with only the lens moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' In that we have taken the speed  and the direction of the lens separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The lens in the simulation  is having speeds from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='005𝑐 and the direction is 0 (The lens  is approaching the observer) to 𝜋 (The lens is receding from the  observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' If the 𝜖 is 𝜋/2 then the Lens is moving in a transverse  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  From Figure 6, we can identify that when the lens is moving  towards the observer the gravitational lensing time delay is increasing  and it is attaining larger values directly in proportion with the peculiar  speed of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' That is, when the lens is having larger approaching  peculiar speeds the gravitational lensing time delay is also larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  In contrast to that when the lens is receding from the observer the  gravitation lensing time delay is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' It can be also seen that  when the receding peculiar speed is becoming larger the gravitational  lensing time delay is becoming smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  If the lens is moving in a transverse direction then there is no  measurable effect in the gravitational lensing time delay as the effect  is in second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  The lenses we have considered so far are having small velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  But if we consider lenses having relativistic speeds then the effect  become more prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' That is the measurable gravitational lensing  time delay becomes much larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Results are illustrated in the Figure  7, where the peculiar speeds of the lens are relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  In the example we have taken, the Lens B1400+434 is having an  measured optical time delay of 51 days and a theoretical time delay  of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='92 days, assuming a point-mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' From our results we can  account for the difference of this time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' That is we can have  this particular observed optical time delay difference if the lens is  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The lens is having different peculiar speeds in  different directions  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Point Mass lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Lens is having relativistic peculiar speeds  having a relativistic peculiar speed in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='05𝑐 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='06𝑐 in  a receding direction from us provided that we model the lens as a  point mass, which is not exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  As we now have a clear idea on gravitational lensing time delays  when the peculiar speeds of the objects are considered while using  a point mass lensing model, let us now investigate the same effect  when a more realistic Singular Isothermal Sphere lensing model is  used for the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  For this also we employ the same simulation with 1000 scenarios  where random peculiar speeds are in random directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' when using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' (15) average velocity dispersion 𝜎𝑣 will be taken as 150𝑘𝑚𝑠−1  Koopmans (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' With this average velocity dispersion value and  using Singular Isothermal Sphere model we have a very interesting  result for the non peculiar motion lensing time delay, which is 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='45  days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' this value is almost identical to the observed lensing time delay  value of 51 ± 2 Days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  MNRAS 000, 1–5 (2018)  1000TimeDelaysinDays  NopeculiarmotionDelay=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='9days  OnlyLensishavingPeculiarSpeeds,ObserverandSourcenotmoving  200  150  ber  unN  100  72  73  74  75  78  77  81  TimeDelayinDaysUpperCurveforApproachingLens  LowerCurveforRecedingLens  MiddleCurveforLensMovinginTransverseDirection  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  Days  ApproachingLense  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  TransverseLense  74:0  Dela  Receding Lense  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='003  0:004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='005  β=IIRelativisticLens  UpperCurveforApproachingLens  LowerCurveforRecedingLens  MiddleCurveforLensMovinginTransverseDirection  120  110  Days  100  ApproachingLense  90  Transverse Lense  80  Receding Lense  Del  Time  70  60  50F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='10  1%1=sGravitational Lensing Time Delay with Peculiar Motions  5  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Singular Isothermal Sphere lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Lens is having non  relativistic peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='01c in any random direction  The simulation for the non relativistic peculiar speeds is given in  the Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' In that the non relativistic peculiar speeds are from 0  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='01𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' further it can be noted in this simulation the time delays  are ranging from 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5 - 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5 days while having a maximum delay  difference of 1 day from the no peculiar motion instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' therefore  even with non relativistic peculiar speeds it is clear that we can have  measurable and significant time delay difference from the no peculiar  motion instance when peculiar speeds of the lens is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  In the next simulation given in the Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' we consider a rela-  tivistic peculiar speed distribution from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='05𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' it can be noted  in this figure when there is a relativistic peculiar speed distribution  for the lens, the lensing time delays can range from 46-56 days with  a maximum delay difference of 5 days from the no peculiar motion  instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' therefore it is apparent from this simulation when there is a  relativistic peculiar speed for the lens there can be a very significant  gravitational lensing time difference from the non peculiar speed  instance while using a more realistic Singular Isothermal Sphere to  model the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  4 CONCLUSIONS  From the above simulations we have found out that in fact there is a  significant measurable time delay difference arising from the peculiar  speeds of the lens using both non realistic point mass lens and more  realistic Singular Isothermal Sphere as the lensing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  The important observation is that an approaching lens results in  an increase of the time delay while a receding lens gives rise to a  decrease in the delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  We find that the time delay is not significantly affected by the  source or observer peculiar motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  We see from Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' and Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' that a relativistically moving  lens in any direction can significantly affect the lensing time delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' Singular Isothermal Sphere lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content=' The Lens is having rela-  tivistic peculiar speeds in the range of 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='05c in any random direction  DATA AVAILABILITY  The data underlying this article will be shared on reasonable request  to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
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+page_content='45days  sigma =150  lensmoving  150  100  Number  50  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
+page_content='5  TimeDelayinDays1000TimeDelaysinDays  NopeculiarmotionDelay=51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49FAT4oBgHgl3EQfmR1P/content/2301.08622v1.pdf'}
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+Time-aware Hyperbolic Graph Attention Network
+for Session-based Recommendation
+Xiaohan Li∗†, Yuqing Liu∗‡, Zheng Liu‡, Philip S. Yu‡
+†Walmart Global Tech, Sunnyvale, CA, USA
+[email protected]
+‡University of Illinois at Chicago, Chicago, IL, USA
+{yliu363, zliu212, psyu}@uic.edu
+Abstract—Session-based Recommendation (SBR) is to predict
+users’ next interested items based on their previous browsing
+sessions. Existing methods model sessions as graphs or sequences
+to estimate user interests based on their interacted items to
+make recommendations. In recent years, graph-based methods
+have achieved outstanding performance on SBR. However, none
+of these methods consider temporal information, which is a
+crucial feature in SBR as it indicates timeliness or currency.
+Besides, the session graphs exhibit a hierarchical structure and
+are demonstrated to be suitable in hyperbolic geometry. But few
+papers design the models in hyperbolic spaces and this direction
+is still under exploration.
+In this paper, we propose Time-aware Hyperbolic Graph
+Attention Network (TA-HGAT) — a novel hyperbolic graph
+neural network framework to build a session-based recommenda-
+tion model considering temporal information. More specifically,
+there are three components in TA-HGAT. First, a hyperbolic
+projection module transforms the item features into hyperbolic
+space. Second, the time-aware graph attention module models
+time intervals between items and the users’ current interests.
+Third, an evolutionary loss at the end of the model provides
+an accurate prediction of the recommended item based on the
+given timestamp. TA-HGAT is built in a hyperbolic space to learn
+the hierarchical structure of session graphs. Experimental results
+show that the proposed TA-HGAT has the best performance
+compared to ten baseline models on two real-world datasets.
+Index Terms—recommender system, graph neural network,
+hyperbolic embedding
+I. INTRODUCTION
+Recommender systems have been an effective solution to
+help users overcome the information overload on the Internet.
+Many applications are developed based on this rationale,
+including online retail [1], music streaming [2], and con-
+tent sharing [3]. To better understand users, modeling their
+browsing sessions is a useful solution as sessions indicate
+their current interests. Session-based recommendation (SBR)
+predicts the users’ next interested items by modeling users’
+sessions. Deep learning models, including Recurrent Neural
+Networks (RNNs) [4], [5], Memory Networks [6], and Graph
+Neural Networks (GNNs) [7], [8] are applied to this problem
+and have achieved state-of-the-art performance.
+Recently, the most influential works on dealing with SBR
+are GNN-based methods. The GNN-based methods [7]–[12]
+∗Both authors contributed equally to this research.
+take each session as a graph to learn the items’ internal rela-
+tionship and their complex transitions. The most representative
+model is SR-GNN [7], which is the first work to apply GNN
+on session-based recommendation and achieve state-of-the-
+art performance. Based on SR-GNN, [10], [12] improve SR-
+GNN with attention layers. [8], [11] consider the item order
+in the session graph to build the models. [9], [13], [14] take
+additional information such as global item relationship, item
+categories, and user representations into account to devise
+more extensive models. HCGR [15] models session graphs
+into a hyperbolic space to extract hierarchical information.
+Although the existing GNN-based methods have achieved
+satisfactory performance, they still suffer from two limitations.
+First, unlike sequence-based models, graph structure cannot
+explicitly show the temporal information between items. Time
+interval is a crucial feature and can significantly improve
+the recommendation performance [16], [17], but it is ignored
+in the existing graph-based SBR models. Moreover, though
+modeling sessions into graphs has the advantage of learning
+items complex transitions [7], the sequential relation between
+items is unclear in the session graph because the beginning
+and end of a session are ambiguous under the graph structure.
+Second, according to [18], [19], graph data exhibits an un-
+derlying non-Euclidean structure, and therefore, learning such
+geometry in Euclidean spaces is not a proper choice. As a
+result, some recent studies [15], [20], [21] reveal that the
+real-world datasets of recommender systems usually exhibit
+tree-like hierarchical structures, and hyperbolic spaces can
+effectively capture such hierarchical information. Therefore,
+it is worth trying to learn session graphs in hyperbolic spaces.
+Hyperbolic spaces have the ability to model hierarchical
+structure data because they expand faster than Euclidean
+spaces. They can expand exponentially, but Euclidean spaces
+only expand polynomially. Existing work [15] demonstrates
+the hierarchical structure of session graphs. However, model-
+ing session graphs in hyperbolic spaces is still under explo-
+ration. First, time intervals indicate the correlation between
+two items. Since hyperbolic embedding is a better match to
+session graphs, it is necessary to define a new framework
+to identify the time intervals in the edges of session graphs.
+Second, learning users’ current interests in the graph is crucial,
+but it is difficult to realize in hyperbolic spaces. Previous works
+arXiv:2301.03780v1  [cs.IR]  10 Jan 2023
+
+[6], [7] devise models in Euclidean space based on the last
+item in the session. The last item plays an important role
+in predicting the next item because it represents the users’
+current interest. However, it is more challenging to model this
+feature in a hyperbolic space as the operations in hyperbolic
+spaces are more complicated than the Euclidean space. Third,
+when taking the time information into consideration, we can
+not only make next-item recommendations, but also provide
+recommendations based on a specific timestamp.
+To tackle the above challenges, we propose Time-aware Hy-
+perbolic Graph Attention Network (TA-HGAT), a hyperbolic
+GNN considering the comprehensive time-relevant features.
+Specifically, we project the item’s original features into a
+Poincar´e ball space via a hyperbolic projection layer. Then,
+we design a time-aware hyperbolic attention mechanism to
+learn the time intervals and users’ current interests together
+in a hyperbolic space. It includes two modules: hyperbolic
+self-attention with time intervals and hyperbolic soft-attention
+with users’ current interests. Finally, the model is trained via
+an evolutionary loss to predict which item the user may be
+interested in at a specific timestamp. All these three compo-
+nents are based on a fully hyperbolic graph neural network
+framework.
+Here, we summarize our contributions as follows:
+• To the best of our knowledge, this is the first paper that
+models temporal information in a hyperbolic space to
+improve the performance of the recommender system. We
+go beyond the conventional Euclidean machine learning
+models to model users’ time-relevant features in a more
+delicate manner.
+• We propose TA-HGAT, a hyperbolic GNN-based frame-
+work with three main components: hyperbolic projection,
+time-aware hyperbolic attention, and evolutionary loss.
+These three components work together in an end-to-end
+GNN to model items’ time intervals and users’ current
+interests. In the end, our model provides a time-specific
+recommendation.
+• We conduct experiments on two real-world datasets and
+compare our model with ten baseline models. The ex-
+periment results demonstrate the effectiveness of the TA-
+HGAT in MRR and Precision.
+II. PRELIMINARY
+A. Graph neural network
+GNNs [22], [23] are designed to handle the structural graph
+data. In GNNs, aggregation is the core operation to extract
+structural knowledge. By aggregating neighboring informa-
+tion, the central node can gain knowledge from its neighbors
+passed through edges and learn the node embedding. GNNs
+have been demonstrated to be powerful in learning node
+embeddings, so they are widely used on many node-related
+tasks such as node classification [23], graph classification [24],
+and link prediction [22].
+Based on the aggregation operation, the forward propagation
+of a GNN on graph G = (V, E) is to learn the embedding
+of node vi ∈ V via aggregating its neighboring nodes. We
+suppose that the initial node embedding of each node i is h(0)
+i ,
+which generally is the feature of the node. In each hidden layer
+of a GNN, the embedding of the central node h(l)
+i
+is learned
+from the aggregated embedding of the neighboring nodes in
+the previous hidden layer h(l−1)
+i
+. The process is described in
+math as follows:
+h(l)
+i
+= σ
+�
+W(l)(AGG
+j∈Ni (h(l−1)
+j
+)
+�
+,
+(1)
+where Ni represents the set of all neighbors of node i in the
+graph, including the node i itself. The aggregation function
+AGG(·) integrates the neighboring information together. A
+non-linear activation function σ, e.g., sigmoid or LeakyReLU,
+is applied to generate the embedding of node i in the layer l.
+Based on the vanilla GNN we mentioned above, GAT [25]
+is proposed to improve GNNs with self-attention mechanism
+[26]. Specifically, for all the neighbors of node i, we need to
+learn the attention coefficients for all its neighbors to calculate
+the importance of each neighbor node in the aggregation.
+Suppose the attention coefficient of the node pair (i, j) is αij,
+the process of learning αij is
+αij = softmax(dij) =
+exp(dij)
+�
+k∈Ni exp(dik),
+(2)
+where dij is the correlation between node i and j. dij here can
+be the joint embeddings of node i and j, e.g., concatenation
+of node embeddings or similarity of the node pair.
+B. Hyperbolic spaces
+In definition, hyperbolic space is a homogeneous space
+with negative curvature. It is a smooth Riemannian manifold,
+which can be modeled in several hyperbolic geometric models,
+including Poincar´e ball model [27], Klein model [28], Lorentz
+model [29], etc. In this paper, we choose the Poincar´e ball
+model because the distance between two points grows expo-
+nentially, which fits well with the hierarchical structure of
+the session graph. Formally, the space of the d-dimensional
+Poincar´e ball Pd
+c is defined as
+Pd
+c = {x ∈ Rd, c∥x∥<1},
+(3)
+where c is the radius of the ball and x is any point in manifold
+P. If c = 0, then Pd
+c = Rd and the ball is equal to the
+Euclidean surface. In this paper, we set c = 1. The tangent
+space TxP is a d-dimensional vector space approximating P
+around x, which is isomorphic to the Euclidean space. With
+the exponential map, a vector in the Euclidean space can be
+mapped to the hyperbolic space. The logarithmic map is the
+inverse of the exponential map, which projects the vector back
+to the Euclidean space.
+In hyperbolic spaces, the fundamental mathematical oper-
+ations of neural networks (e.g., addition and multiplication)
+are different from those in Euclidean space. In this paper, we
+choose M¨obius transformation as an algebraic operation for
+studying hyperbolic geometry. For a pair of random vectors
+(a, b), we list the operations that will be used in our model
+as follows:
+
+• M¨obius addition ⊕ [30] is to perform addition operation
+of a and b.
+a ⊕ b = (1 + 2⟨a, b⟩ + ∥b∥2)a + (1 − ∥a∥2)b
+1 + 2⟨a, b⟩ + ∥a∥2∥b∥2
+.
+(4)
+• M¨obius matrix-vector multiplication ⊗ [31] is employed
+to transform a with matrix W.
+W ⊗ a = tanh(∥Wa∥
+∥a∥
+tanh−1(∥a∥)),
+(5)
+• M¨obius scalar multiplication ⊗ is the multiplication of a
+scalar α with a vector b.
+α ⊗ b = tanh(α tanh−1(∥b∥)) b
+∥b∥
+(6)
+• Exponential map transforms a from the Euclidean space
+to a chosen point x in a hyperbolic space.
+expx(a) = x ⊕ (tanh(λx∥a∥
+2
+) a
+∥a∥),
+(7)
+• Logarithmic map projects the vector a back to the Eu-
+clidean space.
+logx(a) = 2
+λx
+arctanh(∥ − x ⊕ a∥)
+−x ⊕ a
+∥ − x ⊕ a∥
+(8)
+• λx is the conformal factor.
+λx =
+2
+1 − ∥x∥2 .
+(9)
+III. MODEL
+In this section, we present the framework of our pro-
+posed Time-aware Hyperbolic Graph Attention Network (TA-
+HGAT), which is designed to model the temporal information
+in the hyperbolic session graph. First, we define the session-
+based recommendation task. Then we illustrate the three main
+components of the model: hyperbolic projection, time-aware
+hyperbolic attention, and hyperbolic evolutionary loss. These
+three components train the model with time-relevant features
+and provide the recommendation results given a specific
+timestamp. The overall structure of TA-HGAT is shown in
+Figure1.
+A. Problem definition
+Session-based recommendation (SBR) is to predict the item
+a user will click next based on the user-item interaction
+sessions. Generally, it models the user’s short-term browsing
+session data to learn the user’s current interest. Here we
+formulate the SBR problem mathematically as below.
+In the SBR problem, a session is denoted as S = {v1, v2, ··
+·, vn} ordered by timestamps. Each v in S is an item, and
+the item set is Vs, which consists of all unique items in this
+session. To model the session into a directed graph, we take
+all items as nodes and the item-item sequential dependency as
+the edges to construct the session graph. The graph is denoted
+as Gs = (Vs, Es), where Vs, Es are the node and edge sets,
+respectively. Each edge connects two consecutive items, which
+is formulated as e = (vt−1, vt). Our target is to learn the
+embeddings of items and the session and generate the ranking
+of the items that the user may be interested in at the next
+timestamp.
+B. Hyperbolic projection
+In GNN, each node needs input as the initial embedding.
+Accommodated to SBR, the input of a GNN is the feature of
+items such as category or description. The initial embedding
+of item i is h0
+i . However, most feature embedding methods
+are based on the Euclidean space. To make the item features
+available in the hyperbolic space, we use the exponential map
+defined in Eq. 7 to project the initial item embeddings to
+the hyperbolic space. Specifically, the projection process is
+formulated as
+mi = expx(h0
+i ),
+(10)
+where mi is the mapped embedding in the hyperbolic space
+and x is the chosen point in the tangent space.
+To achieve a high-level latent representation of the node
+features, we also add a linear transformation parameterized by
+a weight matrix W1 ∈ Rd′×d, where d′ is the dimension of mi
+and d is the dimension of the node’s final embedding. Please
+note that W1 is a shared weight matrix for all nodes. M¨obius
+matrix-vector multiplication defined in Eq. 4 is employed to
+transform mi and the process is
+h1
+i = W1 ⊗ mi,
+(11)
+where h1
+i is the transformed embedding, which is also used
+as the initial node embedding in the following steps.
+C. Time-aware hyperbolic attention
+According to [15], [20], [32], [33], embedding users and
+items in hyperbolic spaces is a significant improvement of
+graph-based recommender systems. However, none of these
+works model the time intervals and users’ current interests in
+hyperbolic spaces. Our proposed model TA-HGAT is the first
+attempt to solve the problem, in which time-aware hyperbolic
+attention is the core component. It is composed of two
+attention layers: 1) Hyperbolic self-attention in the aggregation
+process, which considers time intervals between items; 2)
+Hyperbolic soft-attention in the session embedding learning,
+which models the user’s current interest.
+1) Hyperbolic self-attention with time intervals: According
+to Section II-A, a key step in graph attention is to learn the
+attention coefficient αij for each node pair (i, j). αij means
+the importance of the neighbors to the central node. To learn
+the αij, unlike the traditional attention networks which apply
+linear transformation [25] or inner product [26], here we use
+the distance of the node embeddings in the hyperbolic space.
+Specifically, we denote the distance of node pair (i, j) as
+(hi, hj), which is calculated as
+d(hl
+i, hl
+j) = arcosh(1 + 2
+∥hl
+i − hl
+j∥2
+(1 − ∥hl
+i∥2)(1 − ∥hl
+j∥2)).
+(12)
+
+...
+v5
+v1
+v6
+v7
+t'
+t'
+t'
+v2
+v1
+v3
+v7
+v4
+v5
+v6
+v1
+v2
+v3
+v5
+v6
+v4
+v7
+Hyperbolic Projection
+...
+Time-aware Hyperbolic Attention
+v2
+v1
+v3
+t'
+t'
+v1
+v2
+v5
+v4
+t'
+t'
+t'
+Hyperbolic Self-attention 
+with Time Intervals
+Hyperbolic Soft-attention 
+with Users' Current Interests
+Hyper bolic Attention 
+Networ k 
+s
+vn
+vn-1
+Hyperbolic 
+Evolutionary Loss
+v1
+v2
+v3
+v5
+v6
+v4
+v7
+v7
+s
+Fig. 1. Illustration of TA-HGAT. First, it builds directed session graphs based on the session sequences, and then projects the embeddings from the Euclidean
+space to the hyperbolic space. Next, hyperbolic self-attention is adopted to aggregate neighboring information and time intervals t′. After that, each session
+graph is represented as a session embedding using a hyperbolic soft-attention mechanism. Finally, TA-HGAT predicts top-k items that are most likely to be
+clicked at the next timestamp for each session.
+Then with the node distances, we further learn the attention
+coefficient αij of node i with all its neighbors (including itself)
+Ni as
+αij = softmax(dij) =
+exp(dij)
+�
+k∈Ni exp(dik),
+(13)
+The reason that we use distance in the hyperbolic space to
+calculate attention coefficients is because of two advantages.
+First, attention coefficients in Euclidean spaces are usually
+calculated by linear transformation [25] or inner product [26],
+which fail to meet the triangle inequality. In hyperbolic space,
+the learned attention coefficients are able to meet this criterion
+and preserve the transitivity among nodes. Second, the atten-
+tion coefficient of the node i with itself is αii = d(hi, hi) = 0,
+so the effect of the central node itself will not affect the
+calculation of attention coefficients.
+After we achieve attention coefficients, the next step is
+to aggregate the node embeddings to learn the central node
+embedding of the next layer. Here the learned attention co-
+efficients serve as the weights applied to the embeddings of
+neighbor nodes. The process is formulated as
+hl+1
+i
+= σ(
+⊕
+�
+j∈Ni
+αij ⊗ hl
+j),
+(14)
+where �⊕ is the M¨obius addition of the weighted neighbor
+node embeddings and σ is a nonlinear function such as
+sigmoid and LeakyReLU. Different from Eq. 11, the ⊗ in
+Eq. 14 is M¨obius scalar multiplication defined in Eq. 6.
+To integrate the temporal information into the attention
+layer, the core idea is to incorporate the time intervals into
+the aggregation process. Specifically, we transform the time
+intervals to the vectors in the hyperbolic space and combine
+the time vectors with the neighbor node embeddings for ag-
+gregation. As time intervals are continuous values, we project
+the time interval values into vectors with a mapping function.
+The mapping process is
+ht′ = wt ⊗ (t+ − t),
+(15)
+where t′ = t+ − t is the time interval, ⊗ here is M¨obius
+matrix-vector multiplication, and wt is the transition vector
+to project the time interval to a vector. In this paper, if two
+items have multiple time intervals between them, we choose
+the closest one. This process is done in the data preprocessing
+part before modeling.
+Motivated by TransE [34], time-aware hyperbolic attention
+translates the neighbor node embedding to the central node
+embedding via temporal information, so the joint embedding
+of nodes embedding and time embedding is generated by
+M¨obius addition, which is represented as hl
+j ⊕ ht′.
+In Eq. 14, all neighbors of the central node i are aggregated
+by M¨obius addition. As the M¨obius addition is complicated
+and consumes more computation resources than the addition
+in the Euclidean space, here we simplify the calculation in
+Eq. 14 using the logarithmic map to project the embeddings
+into a tangent space (Euclidean space) to conduct aggregation
+operation. Then the embeddings are projected back to the
+hyperbolic manifold with the exponential map. Therefore, we
+can re-write the aggregation process in Eq. 14 as
+hl+1
+i
+= exp
+�
+σ
+� �
+j∈Ni
+log(αij ⊗ (hl
+j ⊕ ht′))
+��
+.
+(16)
+2) Hyperbolic soft-attention with users’ current interests:
+In the process above, we update the embedding of node i with
+its neighbors and time intervals. To make recommendations
+based on the learned node embeddings, we also need to know
+the global embedding of the session graph by aggregating
+all node embeddings. Instead of simply adding all node
+embeddings together, we also provide another solution to learn
+the graph embedding while considering users’ current interests
+based on the most recent interacted items.
+
+Understanding users’ current interests are one of the main
+tasks in SBR. In the previous studies [6], [7], [12], the last
+item in the session is the most related feature in this task.
+To learn from the correlation of the last item p with each
+of the other items in the session, we adopt a soft-attention
+mechanism to generate attention coefficients for item p with all
+other items, which represent the importance of items w.r.t. the
+current timestamp. The learning process of the global session
+embedding hs is
+βpq = x⊺ ⊗ σ
+�
+W2 ⊗ hp) ⊕ (W3 ⊗ hq) ⊕ c
+�
+,
+(17)
+hs = exp
+�
+σ
+� �
+q∈Vs
+log(βpq ⊗ hq)
+��
+,
+(18)
+where βpq is the attention coefficient of item p to another
+item q in the session S. x ∈ Rd and W2, W3 ∈ Rd×d are
+weight matrices. hs is the session embedding that contains
+the session graph structure, temporal information, and user’s
+current intent, so we can use hs to infer the user’s next
+interaction in our next step.
+D. Hyperbolic evolutionary loss
+Here we introduce how to leverage evolutionary loss to
+provide recommendations given a specific timestamp. Unlike
+other works [3], [35], our evolutionary loss is also fully
+hyperbolic.
+1) Evolution formulas: The core idea of evolutionary loss
+is to predict the future session and next-item embeddings
+given a future timestamp and then make recommendations.
+The prediction results of evolutionary loss do not rely on the
+sequences like RNN-based models [4], [5] but are based on
+the final embeddings learned by the TA-HGAT.
+As hs is the predicted session embedding in the future, we
+also need an estimated future session embedding to measure
+whether the predicted embedding is accurate. Assume that the
+growth of the session embedding is smooth. The embedding
+vector of the session evolves in a contiguous space. Therefore,
+we devise a projection function to infer the future session
+embedding based on the element-wise product of the previous
+embedding and the time interval. The embedding projection of
+session S after current time t to the future time t+ is defined
+as follows:
+�ht+
+s
+= σ
+�
+ht
+s ⊙ (1 ⊕ ht′)
+�
+,
+(19)
+where 1 ∈ Rd is a vector with all elements 1 and ⊙ is M¨obius
+element-wise product. ht′ is the time interval vector, which
+is learned in the same way as Eq. 15. The 1 vector is to
+provide the minimum difference between the last and next
+session embeddings. With this projection function, the future
+session embedding grows in a smooth trajectory w.r.t. the time
+interval.
+After learning the projected embedding �ht+
+s
+of the session
+S, the next step is to apply another projection function to gen-
+erate the future embedding of the next item v, which is denoted
+as �ht+
+v . The projected future item embedding is composed of
+three components: the projected session embedding, the last
+item embedding, and the time interval, which are learned in
+the previous steps. Here, we define the projection formula of
+next item v as
+�ht+
+v = σv
+�
+(W4 ⊗ �ht+
+s ) ⊕ (W5 ⊗ hvn) ⊕ ht′
+�
+,
+(20)
+where W4 and W5 denote the weight matrix.
+2) Loss function: With the above projection functions, we
+can achieve the estimated future embeddings of the session and
+the next item. They are utilized as ground truth embeddings
+in our loss function. To train the model, the loss function is
+designed to minimize the distances between model-generated
+embeddings ht
+s, hvn and estimated ground truth embeddings
+�ht+
+s , �ht+
+v
+at each interaction time t. Also, another constraint
+for the item embeddings is necessary to avoid overfitting. We
+constrain the distance between the embeddings of the most re-
+cent two items vn−1 and vn to ensure the last item embeddings
+are consistent with the previous one. This constraint assumes
+that the last and next items reflect similar user intent, and the
+session embedding tends to be stable in a short time. Finally,
+the loss function is as follows:
+L =
+�
+(s,v,t)∈{Si}I
+i=0
+d(�ht+
+v , hvn) ⊕
+�
+λs ⊗ d(�ht+
+s , ht
+s)
+�
+⊕
+�
+λv ⊗ d(hvn, hvn−1)
+�
+,
+(21)
+where {St}I
+i=0 denotes all sessions in the datasets, and λs
+and λv are smooth coefficients, which are used to prevent
+the embeddings of the session and items from deviating too
+much during the update process. d(·) is the hyperbolic distance
+function which is described in Eq. 12.
+To make recommendations for a user, we calculate the
+hyperbolic distances between the predicted item embedding
+obtained from the loss function and all other item embeddings.
+Then the nearest top-k items are what we predict for the user.
+Compared with traditional BPR loss [36], the evolutionary
+loss is more suitable for time-aware recommendations because
+it takes time intervals into account. As a result, the changing
+trajectories are modeled by this loss [3], and it can make more
+precise recommendations for the next item given a specific
+timestamp.
+IV. EXPERIMENTS
+In this section, we describe the experimental results on two
+public datasets and compare our proposed TA-HGAT with ten
+state-of-the-art baseline models. Our experiments are designed
+to solve the following research questions:
+• RQ1: How does TA-HGAT compare with other state-of-
+the-art session-based recommendation models?
+• RQ2: How do the two modules of time-aware hyperbolic
+attention, i.e., hyperbolic self-attention with time intervals
+and hyperbolic soft-attention with users’ current interests,
+affect the performance of TA-HGAT?
+• RQ3: How does the hyperbolic evolutionary loss compare
+with other loss functions?
+• RQ4: How is the influence of different hyper-parameters,
+i.e. embedding dimensions?
+
+TABLE I
+THE NUMBER OF ITEMS, TRAINING SESSIONS, TESTING SESSIONS, THE
+AVERAGE LENGTH, AND CLICKS FOR EACH DATASET.
+Datasets
+Items
+train sessions
+test sessions
+Avg. len
+clicks
+Diginetica
+43,097
+719,470
+60,858
+5.12
+982,961
+Yoochoose1/64
+16,766
+369,859
+55,898
+6.16
+557,248
+Yoochoose1/4
+29,618
+5,917,746
+55,898
+5.71
+8,326,407
+A. Experiment settings
+1) Datasets: We conduct our experiments on two widely
+used public datasets: Yoochoose and Diginetica. The statistics
+of these datasets are listed in Table I.
+• Yoochoose1 is a public dataset released by the RecSys
+Challenge 2015, which contains click streams from yoo-
+choose.com within 6 months.
+• Diginetica2 is obtained from the CIKM Cup 2016. We
+use the item categories to initialize the item embeddings.
+2) Evaluation Metrics: We evaluate the performance of our
+model with Mean Reciprocal Rank (MRR@K) and Precision
+(P@K) in the comparison experiments.
+MRR@K considers the position of the target item in the
+list of recommended items. It is set to 0 if the target item is
+not in the top-k of the ranking list, or otherwise is calculated
+as follows:
+MRR@K = 1
+N
+N
+�
+i=1
+1
+Rank(vt),
+(22)
+where vt is the target item and N is the number of test
+sequences in the dataset.
+P@K measures whether the target item is included in the
+top-k list of recommended items, which is calculated as
+P@K = nhit
+N
+(23)
+3) Implementation: Our model 3 is implemented with Py-
+Torch 1.12.1 [37] and CUDA 10.2. In the testing phase, we
+take the interval between the session’s last timestamp and
+the testing item’s timestamp as a part of the input to obtain
+the recommendation list. This setting is different from other
+baseline models as they cannot deal with temporal information.
+In fact, this setting meets the actual situation in the industry
+because our model can provide recommendations as soon as
+the user logs into the website, and we can easily obtain the
+real-time time interval.
+B. Performance comparison (RQ1)
+To demonstrate the effectiveness of TA-HGAT, we conduct
+experiments on two public datasets and compare the model
+with ten state-of-the-art baseline models.
+1https://www.kaggle.com/datasets/chadgostopp/recsys-challenge-2015
+2https://competitions.codalab.org/competitions/11161
+3The datasets and codes will be available after accepted
+TABLE II
+EXPERIMENTS ON DIGINETICA AND YOOCHOOSE DATASETS COMPARE
+TA-HGAT WITH TEN BASELINE MODELS BASED ON THE TOP-20 OF THE
+RANKING LIST IN MEAN RECIPROCAL RANK (MRR@20) AND PRECISION
+(P@20). THE BOLD AND UNDERLINED NUMBERS ON EACH DATASET AND
+METRIC REPRESENT THE BEST AND SECOND-BEST RESULTS,
+RESPECTIVELY. ”IMPROV.” REFERS TO THE MINIMUM IMPROVEMENT
+AMONG ALL BASELINES.
+Models
+Diginetica
+Yoochoose 1/64
+Yoochoose 1/4
+MRR@20
+P@20
+MRR@20
+P@20
+MRR@20
+P@20
+S-POP
+13.68
+21.06
+18.35
+30.44
+17.75
+27.08
+FPMC
+8.92
+31.55
+15.01
+45.62
+-
+-
+GRU4REC
+8.33
+29.45
+22.89
+60.64
+22.60
+59.53
+NARM
+16.17
+49.70
+28.63
+68.32
+29.23
+69.73
+STAMP
+14.32
+45.64
+29.67
+68.74
+30.00
+70.44
+SR-GNN
+17.59
+50.73
+30.94
+70.57
+31.89
+71.36
+TAGNN
+18.03
+51.31
+31.12
+71.02
+32.03
+71.51
+HCGR
+18.51
+52.47
+31.46
+71.13
+32.39
+71.66
+NISER+
+18.72
+53.39
+31.61
+71.27
+31.80
+71.80
+SGNN-HN
+19.45
+55.67
+32.61
+72.06
+32.55
+72.85
+TA-HGAT
+19.73
+56.28
+32.90
+72.75
+32.94
+73.56
+Improv.
+1.44%
+1.10 %
+0.89%
+0.96%
+1.20%
+0.97%
+1) Baseline models:
+• S-POP takes the most popular items of each session as
+the recommended list.
+• FPMC [38] is a Markov chain-method for sequential
+recommendation, which only takes the item sequences
+in session-based recommendation since user features are
+unavailable.
+• GRU4REC [39] is the first work that applies RNN to
+the session-based recommendation to learn the sequential
+dependency of items.
+• NARM [5] utilizes an attention mechanism to model the
+sequential behaviors and the user’s primary purpose with
+global and local encoders.
+• STAMP [6] employs an attention and memory mecha-
+nism to learn the user’s preference and takes the last item
+as recent intent in the session to make recommendations.
+• SR-GNN [7] is the first work that model a session into
+a graph. It resorts to the gated graph neural networks to
+learn the complex item transitions in the sessions.
+• TAGNN [12] improves SR-GNN by learning the interest
+representation vector with different target items to im-
+prove the performance of the model.
+• NISER+ [40] handles the long-tail problem in SBR with
+L2 normalization and dropout to alleviate the overfitting
+problem.
+• SGNN-HN [41] applies a star graph neural network to
+consider the items without direct connections.
+• HCGR [15] models the session graphs in hyperbolic
+space and makes use of multi-behavior information to
+improve performance. In our experiments, we don’t use
+the behavior information as the datasets didn’t provide it
+and we are modeling a more general scenario.
+2) Result analysis: The complete experimental results of
+the comparison study are shown in Table II. From the results,
+we have the following observations:
+• Our proposed TA-HGAT outperforms all baseline models
+
+on all datasets and metrics, which demonstrates the
+effectiveness of the model. Besides, HCGR, which is
+another hyperbolic graph-based SBR model, has achieved
+better performance than the graph-based SBR model SR-
+GNN but worse than our model. Compared to HCGR, our
+model improves 4.3% and 4.1% on average over three
+datasets on metrics MRR@20 and P@20, respectively.
+HCGR is better than SR-GNN, indicating that hyperbolic
+embeddings match session graphs. And the improvement
+of TA-HGAT over HCGR shows the importance of tem-
+poral information in the SBR task.
+• In Table II, we also observe that our model has a better
+performance on dataset Diginetica than Yoochoose. On
+average, the performance of TA-HGAT on Diginetica
+outperforms Yoochoose for 37.8% and 14.0% on metrics
+MRR@20 and P@20, respectively. This phenomenon
+may result from the initial features of items. In Diginetica,
+each item has its category label, and we transform this
+feature into a one-hot vector as the initial embedding of
+the item. In HCGR, we model the initial feature to a
+feature vector in the hyperbolic space, which is shown
+in Eq. 10 and 11. Differences in performance between
+Diginetica and Yoochoose indicate that the hyperbolic
+embeddings have a better expression ability on the item
+features.
+C. Ablation study (RQ2)
+In the TA-HGAT, we have two main modules in time-aware
+hyperbolic attention: hyperbolic self-attention with time inter-
+vals and hyperbolic soft-attention with users’ current interests.
+In this section, we evaluate their effectiveness separately to
+show the improvement compared with the ablation models
+without these two modules.
+We set up four separate ablation models to compare the
+effectiveness of each attention layer. The first ablation model
+is no-att, in which we remove both the attention layers and
+only conduct the aggregation operations directly. The second
+and third ones are self-att and soft-att, and these two ablation
+models only include the self-attention and soft-attention layers,
+respectively. The fourth one is TA-HGAT, which is the com-
+plete model. The comparison results of the ablation models on
+datasets Diginetica and Yoochoose are illustrated in Figure 2.
+From Figure 2, we observe the following results:
+• On both Diginetica and Yoochoose datasets, the non-att
+performs worst, and TA-HGAT performs best. The results
+show the effectiveness of the attention layers. This is
+because the TA-HGAT makes full use of the temporal
+information. Compared to the GNNs without temporal
+information, our model builds the relations between items
+with time intervals and also considers users’ current
+interests. Hence, the rich information helps the model to
+achieve better results.
+• Self-att performs better than soft-att, which means time
+intervals are relatively more meaningful than users’ cur-
+rent interests. This phenomenon may be due to the fact
+that users’ current interests are more complicated, so the
+TABLE III
+COMPARISON OF PERFORMANCE FOR DIFFERENT LOSS FUNCTIONS.
+Loss
+Diginetica
+Yoochoose 1/64
+Yoochoose 1/4
+MRR@20
+P@20
+MRR@20
+P@20
+MRR@20
+P@20
+Softmax
+19.38
+55.81
+32.67
+72.10
+32.72
+72.95
+BPR
+19.43
+55.97
+32.53
+72.28
+32.66
+72.84
+TA-HGAT
+19.73
+56.28
+32.90
+72.75
+32.94
+73.56
+last item cannot fully represent them. In contrast, the time
+interval is a more straightforward feature, so our proposed
+hyperbolic self-attention layer can handle this information
+effectively.
+D. Comparison of loss functions (RQ3)
+In this section, we compare our proposed hyperbolic evolu-
+tionary loss to conventional loss functions, i.e., BPR [36] and
+softmax loss [7]. Because the learned session embeddings in
+the output of our model are in the hyperbolic space, we need
+to use the logarithmic map to project the embeddings back to
+the Euclidean space before applying BPR and softmax loss.
+The comparison results are shown in Table III. The hy-
+perbolic evolutionary loss is denoted as TA-HGAT in the
+table. From this table, we can find that the performance of
+BPR and softmax loss is similar, but our proposed hyperbolic
+evolutionary loss has a clear improvement compared to the
+other losses. This observation demonstrates that considering
+the specific timestamp is effective for the SBR task models
+designed in hyperbolic space.
+E. Hyperparameter analysis (RQ4)
+The embedding dimension is the hyperparameter in our pro-
+posed model, so we test the influence of different embedding
+dimensions in this section. The embedding dimensions range
+from 20 to 100. The results of the hyperparameter analysis are
+illustrated in Figure 3.
+It is observed that a proper embedding dimension is essen-
+tial for learning the item and session representations. From
+Figure 3, we can see that the Diginetica and Yoochoose
+1/64 all achieve the best performance when the embedding
+dimension is 60, and the best result of Yoochoose 1/4 is 80.
+Because Yoochoose 1/4 is much larger than the other two
+datasets, it indicates that larger datasets need larger embedding
+space.
+V. RELATED WORKS
+A. Hyperbolic spaces
+Recent research has shown that many types of complex
+data exhibit a highly non-Euclidean structure [19]. In many
+domains, e.g., natural language [42], computer vision [43],
+and healthcare [44], data usually has a tree-like structure
+or can be represented hierarchically. Since this type of data
+contains an underlying hierarchical structure, capturing such
+representations in Euclidean space is difficult. To solve this
+problem, current studies are increasingly attracted by the idea
+of building neural networks in Riemannian space, such as
+
+P@20
+MRR@20
+Metric
+0
+10
+20
+30
+40
+50
+Value
+non-att
+Self-att
+Soft-att
+TA-HGAT
+(a) Diginetica
+P@20
+MRR@20
+Metric
+0
+10
+20
+30
+40
+50
+60
+70
+Value
+non-att
+Self-att
+Soft-att
+TA-HGAT
+(b) Yoochoose 1/64
+P@20
+MRR@20
+Metric
+0
+10
+20
+30
+40
+50
+60
+70
+Value
+non-att
+Self-att
+Soft-att
+TA-HGAT
+(c) Yoochoose 1/4
+Fig. 2.
+The ablation study of TA-HGAT. ’non-att’ is our model without attention layers. ’Self-att’ and ’Soft-att’ are composed of only self-attention and
+soft-attention layers, respectively. TA-HGAT is the complete model.
+(a) Diginetica MRR
+(b) Yoochoose MRR
+(c) Diginetica Precision
+(d) Yoochoose Precision
+Fig. 3. The hyperparameter analysis of the embedding dimensions.
+the hyperbolic space, which is a homogeneous space with
+constant negative curvature [27]. Compared with Euclidean
+space, hyperbolic space in which the volume of a ball grows
+exponentially with radius instead of growing polynomially.
+Because of its powerful representation ability, hyperbolic
+space has been applied in many areas. For instance, [45]
+learns word and sentence embeddings in hyperbolic space in
+an unsupervised manner from text corpora. [46] demonstrates
+that hyperbolic embeddings are beneficial for visual data. [47]
+proposes Hyperbolic Graph Convolutional Neural Networks,
+which combines the expressiveness of GCNs and hyperbolic
+geometry to learn graph representations. These works show
+the potential and advantages of hyperbolic space in learning
+hierarchical structures of complex data.
+Based on the performance of hyperbolic space in these
+fields, it is natural for researchers to think of applying hyper-
+bolic learning to recommender systems. [48] justifies the use
+of hyperbolic representations for neural recommender systems.
+[49] proposes HyperML to bridge the gap between Euclidean
+and hyperbolic geometry in recommender systems through a
+metric learning approach. [21] proposes a hyperbolic GCN
+model for collaborative filtering. [50] presents HyperSoRec, a
+novel graph neural network (GNN) framework with multiple-
+aspect learning for social recommendation.
+B. Session-based Recommendation
+Session-based Recommendation (SBR) has increasingly en-
+gaged attention in both industry and academia due to its
+effectiveness in modeling users’ current interests. In the recent
+SBR studies, there are mainly three types of methods that
+apply deep learning to SBR and have achieved state-of-the-art
+performance, which are sequence-based [4], [51], attention-
+based [5], [6] and graph-based models [7]. GRU4REC [4]
+is the most representative work in the sequence-based SBR
+models. It employs GRU, a variant of RNN, to model the
+item sequences and make the next-item prediction. Follow-
+ing GRU4REC, some other papers [39], [51], [52] improve
+it with data augmentation, hierarchical structure, and top-k
+gains. Attention-based models aim to learn the importance of
+different items in the session and make the model focus on the
+important ones. NARM [5] utilizes an attention mechanism to
+model both local and global features of the session to learn
+users’ interests. STAMP [6] combines the attention model and
+memory network to learn the short-term priority of sessions.
+Graph-based models connect the items in a graph to learn their
+complex transitions. SR-GNN [7] is the first work that models
+the sessions into graphs. It leverages the Gated Graph Neural
+Network (GGNN) to model the session graphs and achieve
+state-of-the-art performance. Based on SR-GNN, [10], [12]
+improve SR-GNN with attention layers. [8], [11] consider the
+item order in the session graph in the model. [9], [13], [14]
+take additional information such as global item relationship,
+item categories, and user representations into account to design
+more extensive models. However, all these methods fail to
+consider the hierarchical geometry of the session graphs and
+the temporal information.
+
+19.73
+Dataset
+Diginetica
+19.72
+19.71
+19.70
+Value
+19.69
+19.68
+19.67
+19.66
+19.65
+20
+40
+60
+80
+100
+DimensionDataset
+32.94
+Yoochoose1/4
+Yoochoose1/64
+32.92
+32.90
+32.88
+Value
+32.86
+32.84 
+32.82
+32.80
+20
+40
+60
+80
+100
+Dimension56.28
+Dataset
+Diginetica
+56.26
+Value
+56.24
+56.22
+56.20
+20
+40
+60
+80
+100
+Dimension73.6
+73.4
+73.2
+Dataset
+alue
+Yoochoose1/4
+Yoochoose1/64
+73.0
+72.8
+72.6
+20
+40
+60
+80
+100
+DimensionC. GNN-based Recommendation Models
+GNNs have proven to be useful in different research fields
+[53]–[58]. There also exist many works considering the graph
+structures in data modeling of recommender systems. Gener-
+ally, there are two main ways regarding the graph structure
+and embedding space. One way is to model the user-item
+interaction graph in Euclidean spaces. Among them, [59]–[61]
+perform graph convolution on the user-item graph to explore
+their interactions. [62]–[64] utilize layer-to-layer neighbor-
+hood aggregation in GNNs to capture the high-order connec-
+tions. [65] pre-trains user-user and item-item graphs separately
+to learn the initial embeddings of the user-item interaction
+graph. [35] models the changes in user-item interactions with a
+dynamic graph and evolutionary loss. These works apply GNN
+to learn from high-dimensional graph data and generate low-
+dimensional node embeddings without feature engineering, but
+the learned embeddings are all in Euclidean spaces, while
+some graph data may be more suitable to other geometries
+in the representation learning.
+The other way is to model the recommendation graph in hy-
+perbolic space to learn the hierarchical geometry. HGCF [21]
+applies hyperbolic GCN to learn the node embeddings using
+a user-item graph. Wang et al. [20] propose a fully hyperbolic
+GCN where all operations are conducted in hyperbolic space.
+Xu et al. [32] model the product graph in a knowledge graph
+and learn the node embeddings in hyperbolic space. HCGR
+[15] is a novel hyperbolic contrastive graph representation
+learning method to make session-based recommendations.
+None of these models utilize the time-relevant information in
+the session graphs to improve the recommendation accuracy.
+In this paper, we propose a novel framework incorporating
+a time-aware graph attention mechanism in hyperbolic space,
+which is specifically devised for the session-based recommen-
+dation.
+VI. CONCLUSION
+Session-based Recommendation (SBR) is to predict users’
+next interested items based on their previous sessions. Existing
+works model the graph structure in the sessions and have
+achieved state-of-the-art performance. However, they fail to
+consider the hierarchical geometry and temporal information
+in the sessions. In this paper, we propose TA-HGAT, a
+hyperbolic GNN-based model that considers the time interval
+between items and users’ current interests. Experiment results
+demonstrate that TA-HGAT outperforms other SBR models on
+two real-world datasets.
+For future work, we will extend our model to more general
+recommender systems. The time intervals are not only in the
+SBR problem, but also in almost all recommender systems.
+As a result, we want to test how our model performs on other
+recommendation problems, e.g., next-basket recommendation
+and point-of-interest recommendation, where temporal infor-
+mation plays a crucial role in providing recommendations.
+VII. ACKNOWLEDGEMENT
+This work is supported in part by NSF under grants III-
+1763325, III-1909323, III-2106758, and SaTC-1930941.
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+Surface acoustic wave generation and detection in quantum paraelectric regime of
+SrTiO3-based heterostructure
+Dengyu Yang1,2, Muqing Yu1,2, Yun-Yi Pai1,2, Patrick Irvin1,2,
+Hyungwoo Lee3, Kitae Eom3, Chang-Beom Eom3, and Jeremy Levy1,2∗
+1.
+Department of Physics and Astronomy,
+University of Pittsburgh, Pittsburgh,
+Pennsylvania 15260, USA
+2.
+Pittsburgh Quantum Institute,
+Pittsburgh, Pennsylvania 15260, USA
+and
+3.
+Department of Materials Science and Engineering,
+University of WisconsinMadison, Madison, Wisconsin 53706, USA
+(Dated: January 16, 2023)
+Strontium titanate (STO), apart from being a ubiquitous substrate for complex-oxide heterostruc-
+tures, possesses a multitude of strongly-coupled electronic and mechanical properties. Surface acous-
+tic wave (SAW) generation and detection offers insight into electromechanical couplings that are
+sensitive to quantum paraelectricity and other structural phase transitions.
+Propagating SAWs
+can interact with STO-based electronic nanostructures, in particular LaAlO3/SrTiO3 (LAO/STO).
+Here we report generation and detection of SAW within LAO/STO heterointerfaces at cryogenic
+temperatures (T ≥ 2 K) using superconducting interdigitated transducers (IDTs). The temper-
+ature dependence shows an increase in the SAWs quality factor that saturates at T ≈ 8 K. The
+effect of backgate tuning on the SAW resonance frequency shows the possible acoustic coupling
+with the ferroelastic domain wall evolution. This method of generating SAWs provides a pathway
+towards dynamic tuning of ferroelastic domain structures, which are expected to influence electronic
+properties of complex-oxide nanostructures. Devices which incorporate SAWs may in turn help to
+elucidate the role of ferroelastic domain structures in mediating electronic behavior.
+I.
+INTRODUCTION
+Strontium titanate holds a unique place among the
+growing family of complex-oxide heterostructures and
+nanostructures [1].
+Apart from possessing a wealth of
+physical phenomena–ferroelectricity [2, 3], ferroelasticity
+[4, 5], superconductivity [6–8], high spin-to-charge inter-
+conversion [9], large third-order optical susceptibility [10]
+– STO also exhibits fascinating transport properties at
+interfaces and within conductive nanostructures [11, 12].
+These latter properties arise when STO is capped with
+a thin layer, often but not exclusively LaAlO3, which
+results in electron doping near the STO interface [13].
+Conductive nanostructures of many types have been cre-
+ated by “sketching” with a conductive atomic force mi-
+croscope (c-AFM) tip [14] or ultra-low-voltage focused
+electron beam [15]. The properties of these devices are
+profoundly affected by the intrinsic behavior of STO and
+are in many aspects not well understood.
+One of the least well-understood property interrela-
+tionships concerns the coupling between electronic, ferro-
+electric, and ferroelastic degrees of freedom. STO is cen-
+trosymmetric at room temperature with ABO3 cubic per-
+ovskite structure. Upon cooling below ∼105 K [16, 17],
+STO undergoes a cubic-to-tetragonal antiferrodistortive
+(AFD) phase transition [18]. Further cooling to ∼35 K
+∗ [email protected]
+[19] gives rise to an incipient ferroelectric or “quantum
+paraelectric” phase transition (QPE) in which the dielec-
+tric constant ε saturates at ∼10 K [20].
+At cryogenic
+temperature (T < 10 K), STO shows giant piezoelec-
+tricity even larger than the best well-known piezoelectric
+material such as quartz [21]. At microscopic scales, the
+coupling between polar phases in STO and ferroelastic
+domains [22] is quite strong and can be directly observed
+using scanning single electron transistor microscopy [5].
+Piezoelectric distortions were found to be the result of
+reorienting tetragonal domains, whose in-plane and out-
+of-plane lattice constants differ by ∼ 10−3.
+Surface acoustic waves (SAW), also known as Rayleigh
+waves [23], arise from linear piezoelectric coupling, and
+propagate parallel to the sample surface with its depth
+comparable to the SAW wavelength. SAW propagation is
+sensitive to both mechanical and electrical changes at the
+sample surface, making it surface-sensitive and useful for
+radio-frequency (RF) signal processing. A common tech-
+nique to generate and detect SAW is to apply RF signals
+to a pair of metallic inter-digitated transducers (IDT).
+However, the complexity and subtlety of the STO struc-
+ture with multiple phases make SAW generation and de-
+tection difficult to achieve. With STO, DC fields have
+been used to break cubic symmetry and generate polar-
+ization above 150 K [24, 25] via electrostrictive effect.
+To generate SAW, an extra piezoelectric layer, PZT, was
+deposited on top of LAO [26], and SAW was observed
+down to T =110 K. Below this temperature, the signal
+disappeared and SAW generation and detection has not
+arXiv:2301.05324v1  [cond-mat.str-el]  12 Jan 2023
+
+2
+to our knowledge been reported in STO or LAO/STO or
+at temperatures lower than T =105 K.
+In this paper, we demonstrate direct SAW generation
+and detection on LAO/STO surface at cryogenic temper-
+atures using superconducting IDTs. The linearly-coupled
+SAW shows an ultra-low phase velocity, indicating soft-
+ening of STO crystal at low temperature and consistent
+with earlier reports of large piezoelectric and electrostric-
+tion coefficients [21]. The temperature at which the qual-
+ity factor of the SAW resonator saturates coincides with
+the quantum-paraelectric (QPE) transition temperature
+(TQPE), showing that the quality factor Q is coupled to
+the dielectric constant and can be used to identify the
+onset of the quantum paraelectric phase. The resonance
+frequency can be tuned with a backgate voltage.
+The
+tunability with applying the backgate at negative side
+but not at the positive backgate side coincides with the
+tuning effect of ferroelastic domain with the backgate
+showing the coupling between ferroelastic domains and
+surface phonon. The applied DC bias confirms the elec-
+trostrictive effect from STO by showing the quadratic
+tuning behavior.
+II.
+EXPERIMENT
+LaAlO3 epitaxial films were grown on TiO2-terminated
+STO (001) substrates by pulsed laser deposition [13]. The
+thickness of LAO is fixed to 3.4 u.c., close to the critical
+thickness of metal-insulator transition [27]. To form a
+uniform-type single electrode IDT, an 80 nm thick film
+of NbTiN is deposited on top of the LAO/STO, with
+IDT fingers oriented along the (010) direction. Supercon-
+ducting NbTiN is chosen as the IDT material for three
+principal reasons: to help with impedance matching; to
+maximize the transmission; and to minimize ohmic losses
+and heating. A metallization ratio (m), defined as the fin-
+ger width divided by the finger spacing, m ≡ w/(w + d),
+is fixed such that m = 0.5 in all devices. SAW-related
+experiments are carried out in a physical property mea-
+surement system (PPMS) at temperatures T ≥ 2 K. Each
+IDT is grounded on one side, and the other side is con-
+nected to an input port of a vector network analyzer
+(VNA) to enable two-port scattering parameter measure-
+ments (Fig. 1(a)). Between the IDT and the VNA, a bias
+tee is inserted on each side to allow a DC bias to be (Vbias)
+applied between the IDT fingers. SAWs are generated by
+an IDT, transmitted along the (100) direction, and de-
+tected by the second IDT pair. To reduce contributions
+from bulk acoustic waves, the LAO/STO sample bottom
+surface is roughened and coated with silver epoxy as a
+“soft conductor” [28]. The bottom conducting electrode
+is also used to apply a voltage Vbg from the back of the
+STO substrate.
+Using a P = −10 dBm signal applied to the IDT, a
+clear resonant feature can be seen at 127.5 MHz in the
+reflection spectrum S11 (Fig. 1(c)), defined as the center
+frequency fc. By contrast, in a control device in which
+one side of the two comb structures in the IDT is miss-
+ing there is no resonance (Fig. 1 (d)), demonstrating that
+the resonance feature is a result of the paired comb struc-
+tures patterned with NbTiN, and not due to bulk acous-
+tic wave transmission or an electrical resonance from the
+cable or other parts of the instrument. Meanwhile, the
+SAW phase velocity is obtained from the measured fc by
+v = fcλ. The wavelength λ is determined by the distance
+between a pair of nearest IDT fingers with the same po-
+larity. Here we have λ = 8 µm, giving a SAW velocity
+on LAO/STO of v = 1, 020 m/s. The IDT comb struc-
+ture generates SAW by converting the electrical energy
+to elastic energy, causing a resonance dip in the reflection
+signal.
+The total quality factor Q is defined as
+Q ≡ fc/B,
+(1)
+where fc is the center resonance frequency and B is the
+half-power (-3 dB) bandwidth. The resonance spectrum
+shows a quality factor Q = 17.5, which is consistent with
+previous reports [25] on STO-based acoustic resonators
+without resonance-enhanced structures (e.g., Bragg mir-
+rors). Theoretically the bandwidth B can be determined
+from the IDT geometry according to Ref. [29],
+B ∼ 0.9fc/Np,
+(2)
+where Np = 16 is the number of comb pairs in the IDT.
+Here the calculated bandwidth is 7.2 MHz is close to the
+expected value of 7.3 MHz.
+The < 2% difference can
+come from the imperfect edge of IDT geometry related
+to the mask-less photolithography precision.
+The transmission spectrum S21 (Fig. 2 (a)) shows a
+resonance peak at a frequency fc that coincides with the
+reflection dip in S11, supporting the scenario that energy
+is transmitted efficiently via SAW from the transmitting
+IDT to the receiving IDT. When we sweep the temper-
+ature, the resonance peak disappears sharply at temper-
+atures larger than 13.7 K, corresponding to the temper-
+ature Tc above which the NbTiN is no longer super-
+conducting (see Supplementary material). The NbTiN
+normal resistance 1.57 kΩ gives an impedance mismatch
+which causes most of the power to be reflected and dis-
+sipated both internally in the IDT and externally into
+the transmission line leading to a sharp cut-off on the
+transmission signal.
+Notably,
+the resonance frequency is temperature-
+dependent. A quadratic scaling is observed between the
+center frequency and temperature, with a lower fc at a
+lower temperature. To understand this scaling, we may
+consider a Helmholtz free energy of phonons, F, descrip-
+tion of its equilibrium state,
+F(t, ψ) = a(t) + b(t)ψ2 + c(t)ψ4 + · · · ,
+(3)
+where ψ is the order parameter, and t = (T − Tc)/Tc
+is the reduced temperature. When we only consider the
+equilibrium states, we obtain
+b(t)ψ + 2c(t)ψ3 = 0
+
+3
+|ψ| ≈ (b1/2c0)|t|1/2.
+The asymptotic expression for F becomes:
+F(t, ψ) ≈ a0 − b2
+1
+4c0
+t2 + · · ·
+(4)
+Therefore, the free energy is expected to scale quadrat-
+ically with temperature.
+The resonance frequency fc,
+depending linearly on F, scales quadratically with tem-
+perature (Fig. 2 (b)).
+With a constant IDT geometry, such that the wave-
+length λ is kept fixed, a smaller fc corresponds to a lower
+SAW phase velocity v. We observe that lowering the tem-
+perature reduces the SAW velocity. This trend contrasts
+with results on most piezoelectric materials such as PZT,
+in which SAW have been reported to increase as tem-
+perature is lowered, because of the decreasing thermal
+fluctuations and increasing stiffness of the sample [30].
+Both the piezoelectric coefficient (d311) and the elec-
+trostriction coefficient (R311) of STO increase signifi-
+cantly with decreasing temperature, especially for T <
+10 K [21]. This finding implies both a softer crystal and
+a more efficient electro-mechanical energy conversion at
+low temperature. The quality factor, plotted versus tem-
+perature in Figure 2 (c) first increases as temperature
+is decreased, and then saturates at T ≈ 8 K. The sat-
+uration temperature coincides with the STO quantum
+paraelectric phase transition (TQPE) where the dielectric
+permittivity ε saturates, described by Barrett’s formula
+[20]. This correspondence indicates that SAW is sensitive
+to the quantum paraelectric phase transition and its Q
+factor is related to the ε variance. When T > Tc, Q drops
+quickly to near zero due to the increasing resistance R
+for the IDT.
+To verify the linear dispersion of the SAW (v = fλ),
+two different pairs of IDTs with different electrode spac-
+ing are compared, keeping the metallization ratio fixed
+such that m = 0.5. The IDT geometry is as shown in
+Fig. 3 (a). The IDT finger widths w = d = 2 µm and
+w = d = 3 µm correspond to wavelengths λ = 8 µm and
+λ = 12 µm, respectively. The measured resonance fre-
+quency fc, labeled with black arrows in Fig. 3 (b,c) shows
+the expected inverse linear scaling with wavelength, pro-
+viding further confirmation of the SAW origin of the res-
+onance feature.
+The transmission resonances in both devices show an
+appreciable hardening as a function of back gate volt-
+age Vbg (Fig. 3 (b,c)). The rise in acoustic velocity is
+associated with induced ferroelectric displacement which
+breaking the inversion symmetry of the crystal structure
+and couples to the strain field S. This phenomenon can
+be modeled using a Landau-Ginsburg-Devonshire (LGD)
+free-energy expression, expanding in powers of the dis-
+placement D up to the second order (Eq. 5) [25, 31–33].
+For STO when it is paraelectric, the dielectric electro-
+mechanical response is described within LGD theory [34].
+F − F0 = −pSD + 1
+2χ−1D2 + 1
+2GSD2 + · · ·
+(5)
+In Eq. 5, p is the piezoelectric tensor, χ is the dielectric
+permittivity tensor and G is the electrostrictive tensor.
+Surprisingly, the dependence of fc on Vbg is much
+stronger when Vbg < 0 compared to Vbg > 0 (Fig. 4).
+This dependence contrasts with pure electrical tuning of
+the dielectric constant through Vbg, in which the tun-
+ing effect is symmetric across zero bias [35]. The LAO
+thickness is below the critical thickness for spontaneous
+formation of a conductive LAO/STO interface [27]. The
+interface remains insulating during the experiment, even
+for the maximum backgate voltage that has been applied,
+and thus carrier screening or other effects associated with
+a gate-induced insulator-to-metal transition can be ruled
+out. One is left with explanations that are tied to the
+formation and evolution of ferroelastic domains. The mo-
+tion of such domains under back gate bias is consistent
+with prior imaging from Honig et al. [5] which showed
+that under large negative backgate voltage, tetragonal
+ferroelastic domains are observed, leading to the anoma-
+lously large piezoelectricity at low temperature. The for-
+mation and drifting of the ferroelastic domain under neg-
+ative backgate voltages play an important role coupling
+with the SAW.
+To investigate how the magnitude of the SAW coupling
+depends on the static bias across the IDT, we incorporate
+a bias tee between the VNA port and IDT connection to
+apply a dc bias Vbias between IDT neighboring fingers
+with opposite polarity, and measure the change in the
+resonance amplitude as a function of Vbias. The result
+(Fig. 5 (a)) shows a quadratic relationship for S11 ampli-
+tude. The quadratic dependence can be understood as
+an electrostriction effect in which electric field couples to
+strain up to the second order (Eq. 5). The Vbias induced
+first-order electric field breaks the inversion symmetry of
+STO, yielding a linear coupling. The scaling indicates
+a built-in STO polarization that can be modulated with
+an applied bias across the IDT. For comparison, port 2 is
+not subject to a dc bias, and as a result no tuning of the
+S22 amplitude (Fig. 5 (a)) is observed. When the applied
+Vbias cancels the built-in polarization, the resonance am-
+plitude is minimized, which happens Vbias ∼ −1 V. Sim-
+ilar tuning is observed for the transmission signals S12
+and S21, as expected.
+III.
+DISCUSSION AND CONCLUSION
+With an acoustic speed five orders lower than the speed
+of light, a relatively short acoustic wavelength, and high
+degree of surface sensitivity, SAW generation, propaga-
+tion, and detection can be regarded as useful building
+blocks for manipulating electronic and lattice degrees of
+freedom in complex-oxide heterostructures and nanos-
+tructures.
+Specifically, SAW has the potential to con-
+tribute to quantum information processing architectures
+[29] both in superconducting qubits [36–38] and elec-
+tron spin-based quantum computing architectures [39–
+41]. Coupling the superconducting qubits with SAW can
+
+4
+help control and measure quantum states [42]. Further-
+more, SAW generates moving potential wells with meso-
+scopic scale which transport electron charges with spin
+information propagating at speed of sound in a confined
+one-dimensional channel, helping to meet architectural
+challenges of long-range transport of spin information
+[43–45]. At the same time, SAW manipulation of elec-
+tronic properties may help provide insight into the nature
+of correlated electronic phases such as superconductivity.
+In conclusion, we demonstrate the direct generation
+and detection of SAW on LAO/STO at cryogenic tem-
+perature using superconducting IDTs. Spurious contri-
+butions arising from possible bulk acoustic wave compo-
+nents and electronic resonances from the instrument are
+carefully ruled out. The SAW shows an ultra-low phase
+velocity which reveals the coupling to the high permit-
+tivity of the STO at low temperatures. The SAW quality
+factor saturates at quantum paraelectric phase transition
+temperature corroborating the related Q-factor with di-
+electric permittivity. This method can thus be used to
+probe behavior near the quantum phase transition. The
+behavior is consistent with a linear electro-mechanical
+coupling that is tightly coupled with the ferroelastic do-
+main evolution.
+Supporting Information
+Supporting Information is available as Supplementary
+information.pdf.
+Acknowledgements
+JL acknowledges support from the Vannevar Bush Fac-
+ulty Fellowship program sponsored by the Basic Research
+Office of the Assistant Secretary of Defense for Research
+and Engineering and funded by the Office of Naval Re-
+search through grant N00014-15-1-2847.
+The work at
+University of Wisconsin-Madison was supported by the
+National Science Foundation under DMREF Grant No.
+DMR-1629270, AFOSR FA9550-15-1-0334 and AOARD
+FA2386-15-1-4046. This research is funded by the Gor-
+don and Betty Moore Foundations EPiQS Initiative,
+grant GBMF9065 to C.B.E., Vannevar Bush Faculty Fel-
+lowship (N00014-20-1-2844 (C.B.E.).
+Transport mea-
+surement at the University of WisconsinMadison was
+supported by the US Department of Energy (DOE), Of-
+fice of Science, Office of Basic Energy Sciences (BES),
+the Materials Sciences and Engineering (MSE) Division
+under award number DE-FG02-06ER46327.
+Conflict of Interest
+The authors declare no conflict of interest.
+Data Availability Statement
+Data generated or analysed during this study are in-
+cluded in this published article (and its supplementary
+information).
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+7
+VNA
+DC
+DC
+SAW
+(a)
+(b)
+(c)
+(d)
+FIG. 1. Surface Acoustic Wave (SAW) generated and detected by Interdigitated Transducers (IDTs). (a) Schematic diagram of
+the experiment setup. The orange parts are IDTs. Blue circuits denote the bias tee inserted between vector network analyzer
+(VNA) port and IDT. (b) Optical image of an IDT patterned with NbTiN. The black scale bar is 20 µm. (c) S11 from an
+experiment device with normal IDT comb structure in pair. The resonances is observable. (d) Reflection signal S11 from a
+control device without one side of IDT comb structure. There is no resonance observed from this control device.
+T = 2K
+(a)
+(b)
+(c)
+TQPE
+FIG. 2. Temperature dependence of resonance. (a) The upper intensity plot shows transmission signals with respect to the
+temperature between 2 K and 16 K. The lower figure is a transmission signal line cut through 2 K temperature showing
+the resonance peak.
+(b) Temperature dependence of resonance center frequency and calculated SAW phase velocity (blue
+diamonds). The red dashed line is a quadratic fit. (c) Quality factor Q = fc/B plotted with respect to the temperature. The
+red arrow shows the STO quantum paraelectric saturation temperature.
+
+SSZSS20
+41.0
+100
+200
+f (MHz)13.8
++
+-
++
+-
+w
+d
+l
+LAO
+STO
+(a)
+(b)
+(c)
+FIG. 3. Resonance frequency shift due to different wavelengths and negative backgate voltages. (a) Schematic diagram of
+NbTiN IDT geometry, where w is the finger width and d is the gap distance.
+Wavelength λ is determined by the center
+distance between two nearest same polarity fingers. (b) Transmission spectrum of Device “A” (λ = 8 µm) as a function of
+Vbg. Black arrow denotes the resonance frequency. (c) Transmission spectrum of Device “B” (λ = 12 µm) as a function of Vbg.
+Black arrow denotes the resonance frequency. All data taken at T = 2 K
+(a)
+(b)
+FIG. 4. Positive and negative Vbg dependence of reflection spectrum and resonance frequency. (a) Reflection spectrum as a
+function of Vbg. There is a resonance dip and a second harmonic dip observed in the spectrum. (b) Resonance center frequency
+fc as a function of Vbg. The green region highlights where the resonance frequency can be tuned with negative Vbg.
+
+入=12μm^=8μm9
+(a)
+(b)
+FIG. 5. Reflection scattering parameters (S11, S22) measured as a function of Vbias applied across the IDT fingers in port
+1, with zero bias at port 2. (a) Reflection amplitude for ports 1 (S11) and 2 (S22), referenced against the zero-bias value
+Sii (Vbias = 0). (b)Transmission amplitudes (S12, S21), measured as a function of Vbias.
+

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+arXiv:2301.01496v1  [math.DS]  4 Jan 2023
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC RANDOM
+WALKS ON THE CIRCLE
+KLAUDIUSZ CZUDEK
+Abstract. Fix an irrational number α and a smooth, positive, real function
+p on the circle. If current position is x ∈ R/Z then in the next step jump to
+x + α with probability p(x) or to x − α with probability 1 − p(x). In 1999
+Sinai has proven that if p is asymmetric (in certain sense) or α is Diophantine
+then the Markov process possesses a unique stationary distribution. Next year
+Conze and Guivarc’h showed the uniqueness of stationary distribution for an
+arbitrary irrational angle α.
+In this note we present a new proof of latter
+result.
+1. Introduction
+Fix an irrational number α ∈ R, and consider the family of Markov processes
+with the evolution governed by the transition kernel
+(1.1)
+p(x, ·) = p(x)δx+α + q(x)δx−α,
+p : T × B(T) → [0, 1],
+where B(S1) stands for the σ-algebra of Borel subsets of S1 and q(x) = 1 − p(x),
+x ∈ T. We call the function p symmetric if
+�
+T
+f(x)dx = 0,
+where
+(1.2)
+f(x) = ln p(x)
+q(x),
+x ∈ T,
+and asymmetric otherwise. We call a measure µ invariant for transition kernel (1.1)
+if distributing the starting point according to µ makes the Markov process with this
+transition kernel stationary (thus µ is called also often a stationary measure). Since
+T is compact, the Krylov-Bogoliubov technique yields existence of an invariant
+distribution for (1.1) for every choice of continuous p. However, it is far from being
+obvious if there exists more than one invariant distribution.
+The earliest paper known to the author dealing with similar (but still slightly
+different) system was by Sine [Sin79].
+More recently it was proven by Sinai in
+[Sin99] that if p ∈ C∞(T) is asymmetric or p ∈ C∞(T) is symmetric and α is
+Diophantine then the uniqueness follows.
+One year later Conze and Guivarc’h
+proved in [CG00] that in the symmetric case
+p(x)
+q(x+α) ∈ BV implies uniqueness no
+matter if α is Diophantine or not. The present paper contains another proof of the
+latter statement assuming p ∈ C1 is symmetric. The advantage of the new proof is
+that it gives more insight to the problem of mixing and the problem of uniqueness
+2020 Mathematics Subject Classification.
+Primary 37A50, 60F05.
+Key words and phrases. random rotations, Diophantine approximation, random walk, circle.
+1
+
+2
+KLAUDIUSZ CZUDEK
+in higher dimensional analogs (where T is replaced by Td). See Section 5 for more
+details.
+The strategy is based on Sinai’s. To explain it, fix x ∈ T and consider a Markov
+process (Xn) started at x with transition kernel (1.1). It is evident that the process
+can achieve only the points of the form x+jα, j ∈ Z. Thus to learn the distribution
+of (Xn) on T we consider a Markov chain (ξn) on Z, started at 0, with
+P(ξn+1 = k + 1|ξn = k) = p(x + kα)
+and
+P(ξn+1 = k − 1|ξn = k) = q(x + kα)
+for n ≥ 0 and k ∈ Z. Let us now restrict to the symmetric case, which is in our scope
+of interest. In that case the system on Z is recurrent. If p ∈ C∞(T) is symmetric
+and α is Diophantine then the cohomological equation f(x) = g(x + α) − g(x),
+where f is defined in (1.2), possesses a solution. Using the solution g we can easily
+check that the measure with density h(z)/q(z) is invariant, where h = exp(g). Now
+the whole difficulty in Sinai’s approach was to show the local limit theorem for (ξn)
+on Z. More precisely, in the symmetric case Sinai has proven that
+P(ξn = k) ∼ h(x + kα)
+p(x + kα)
+1
+√
+2πσ2n
+exp −k2
+2nσ2 ,
+for some σ > 0 and all x ∈ T, where ∼ means the ratio of both sides tends to one.
+With this fact one can show that
+Eϕ(Xn) →
+�
+T
+ϕ(z)h(z)
+q(z) dz,
+which easily implies the unique ergodicity (in fact it’s even a stronger property
+called mixing or stability).
+Unfortunately we cannot follow exactly the same path when generalizing result
+to all irrational α. Recently Dolgopyat, Fayad and Saprykina [DFS21] have proven
+that if α is Liouville then the behaviour of (ξn) on Z is erratic for the generic
+choice of smooth and symmetric p (see Theorems A-E therein).
+In particular,
+neither annealed, nor quenched central limit theorem holds (see Corollary D and
+G therein). However, we can still modify something in Sinai’s idea to get desired
+assertion. The main result of this work is the following.
+Theorem 1. If p ∈ C1(T) is symmetric and separated from 0 and 1 (i.e. 0 <
+p(x) < 1 for each x ∈ T) then there exists exactly one invariant measure for the
+transition kernel (1.1).
+As it was mentioned, the proof is some sense is in the spirit of Sinai’s. We still
+concentrate on the process (ξn) on Z but instead of proving the local limit theorem
+we focus on the limits
+lim
+n→∞
+P(ξ0 = k) + · · · + P(ξn−1 = k)
+P(ξ0 = m) + · · · + P(ξn−1 = m),
+where k, m ∈ Z are two states. The problem of existence of such limits for general
+(including countable space, null recurrent) Markov chains was raised by Kolmogorov
+in 1936 and answered two years later by Doeblin [Doe38] without identification of
+the value of the limit. It has been done only later by Chung [Chu50]. It turns out
+we can define certain infinite measure on Z, k �−→ ax,k (depending on x ∈ T since
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+3
+(ξn) depends on x ∈ T) such that the limit above tends to ax,k/ax,m for arbitrary
+two states k and m.
+In Section 2 we identify the measure k �−→ ax,k on Z and reproduce the proof
+of Doeblin ratio limit theorem. In Section 3 it is proved that if one takes a large
+interval of integers A of length q and projects the measure k �−→ ax,k to the circle
+(by identifying k with x + kα) then what we obtain is, after normalization and up
+to ε, independent of the choice of the interval A and the point x, provided q is
+sufficiently large. Section 4 contains how to complete the proof of Theorem 1 using
+the above results. Section 5 contains some final remarks.
+2. Acknowledgments and personal remarks
+When I proved the main theorem I wasn’t aware of Conze, Guivarc’h result.
+After discovering it, I started thinking if my proof can be used to show something
+more.
+I realized the advantage of mine is it can be modified to obtain mixing
+(assuming p is C1 and symmetric, no matter if α is Diophantine or not). Then
+I gave several talks about it, e.g. in the conference “Probabilistic techniques in
+random and time-varying dynamical systems”, Luminy 3-7.10.2022 or in the KTH
+dynamical systems seminar, where I announced “mixing” result. Although I still
+think this result is true, I didn’t predicted certain difficulties in the proof and I
+need more time and effort to complete it. Meanwhile I’m publishing the proof of
+uniqueness. It’s not going to be submitted to any journal.
+The research was supported by the Polish National Science Center grant Pre-
+ludium UMO-2019/35/N/ST1/02363.
+3. Basic facts about symmetric random walks on Z
+Fix x ∈ T and define (ξn) to be the Markov process on Z, started at 0, with
+P(ξn+1 = k + 1|ξn = k) = p(x + kα)
+and
+P(ξn+1 = k − 1|ξn = k) = q(x + kα)
+for n ≥ 0 and k ∈ Z. In present section we are going to prove recurrence of this
+random walk and some related results. We say that (ξn) is recurrent if almost
+surely there exists n > 0 with ξn = 0. We say (ξn) is null recurrent if it is recurrent
+and the expected time of the first return to 0 is infinite.
+Proposition 1. If p is of bounded variation, symmetric and separated from 0 and
+1 then the process (ξn) is recurrent. Moreover, for every r > 0 there exists m0 that
+can be chosen uniformly in x ∈ T such that the expected number of returns of (ξn)
+to zero until m0 is greater than r, i.e.
+P(ξ1 = 0) + · · · + P(ξn = 0) = E
+�
+1{0}(ξ1) + · · · + 1{0}(ξn)
+�
+> r
+for n ≥ m0, whatever x ∈ T.
+Proof. To show the recurrence of (ξn), we reproduce the analysis from [DFS21],
+Section 3.2. Let us define a function M : Z → R by M(0) = 0, M(1) = 1,
+M(n) = 1 +
+n−1
+�
+k=1
+k
+�
+j=1
+q(x + jα)
+p(x + jα)
+for n ≥ 2,
+
+4
+KLAUDIUSZ CZUDEK
+and
+M(−n) = −
+n
+�
+k=0
+k
+�
+j=0
+p(x − jα)
+q(x − jα)
+for n ≥ 1.
+To avoid complicated notation, we do not stress the dependence of M on x. It can
+be checked that (M(ξn)) is a martingale. Let a < 0 < b and let us define τ to be
+the first moment when (ξn) hits a or b. By Doob’s theorem EM(ξτ) = M(ξ0) = 0.
+On the other hand
+EM(ξτ) = M(a)P(ξτ = a) + M(b)P(ξτ = b)
+= M(a)P(ξτ = a) + M(b)(1 − P(ξτ = a)),
+which combined with EM(ξτ) = 0 yields
+P(ξτ = a) =
+M(b)
+M(b) − M(a).
+If ξτ = a then (ξn) returns to 0 before hitting b. Setting a = −1 above we get
+therefore
+(3.1)
+P
+�
+(ξn) returns to 0 before hitting b
+�
+≥
+M(b)
+M(b) − M(−1).
+Similarly
+(3.2)
+P
+�
+(ξn) returns to 0 before hitting a
+�
+≥
+−M(a)
+M(1) − M(a).
+This easy implies the random walk (ξn) is recurrent provided M(n) → ∞ as n → ∞
+and M(n) → −∞ as n → −∞. The latter is implied by the following consequence
+of the Denjoy-Koksma inequality.
+Lemma 1. For every A > 0 there exists n0 > 0 that is independent of x ∈ T such
+that M(n) > A for n ≥ n0 and M(n) < −A for n ≤ n0.
+Proof. Take n > 0. The function M(n) is a sum of expressions of the form
+k
+�
+j=1
+q(x + jα)
+p(x + jα)
+for k < n,
+therefore to show the assertion it is sufficient to find δ > 0 such that the product
+above is greater than δ for infinitely many k’s. Define f(x) = ln q(x) − ln p(x),
+x ∈ T, and observe we can write
+k
+�
+j=1
+q(x + jα)
+p(x + jα) = exp
+�
+k
+�
+j=1
+f(x + jα)
+�
+.
+The function f is of bounded variation and �
+T f(t)dt = 0 so the Denjoy-Koksma
+inequality (Theorem 3.1 in [Her79], p. 73) yields
+����
+q
+�
+j=1
+f(x + jα)
+���� =
+����
+q
+�
+j=1
+f(x + jα) − q
+�
+T
+f(t)dt
+���� < var(f)
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+5
+for an arbitrary x ∈ T and an arbitrary closest return time q. But this means for
+an arbitrary closest return time q we have
+exp
+�
+k
+�
+j=1
+f(x + jα)
+�
+> e−var(f) > 0.
+Thus the assertion follows with δ = e−var(f).
+□
+To show the remaining part of Proposition, fix r > 0 and take ε > 0 so small
+that (1 − ε)2r > 1/2. By (3.1), (3.2) and Lemma 1 there exists a > 0 (suitable for
+all x ∈ T) such that
+P
+�
+(ξn) returns to 0 before hitting −a or a
+�
+≥ 1 − ε
+2.
+Since p and q are separated from 0, there exists n0 so large (suitable for all x ∈ T)
+such that probability that (ξn) stays in (−a, a) for the first n0 steps is less than
+ε/2. Combining these two facts yields
+P
+�
+(ξn) returns to 0 before n0
+�
+> 1 − ε.
+By the strong Markov property
+P
+�
+(ξn) returns 2r-times to 0 before 2rn0
+�
+> (1 − ε)2r > 1/2,
+by the choice of ε. The assertion follows with m0 = 2rn0 since the expected number
+of returns to 0 before m0 is greater than 2r with probability 1/2.
+□
+It is advantageous to use the following notation in the remaining part of this
+section. Let pn
+i,j denote the probability of transition from state i to state j in n
+steps. We simply write pi,j instead of p1
+i,j. Let kpn
+i,j stands for the probability of
+transition from state i to state j in n steps under the restriction that state k is
+visited in neither of steps 1, . . . , n − 1. Again, these values depend on chosen point
+x ∈ T but we refrain from stressing that in the notation.
+Clearly jpn
+k,j is the probability of the first visit in j starting at k occurring in
+step n and kpn
+k,j is the probability of transition to j from k in n steps with the
+restriction that the state k is not visited in steps 1, . . . , n. The series �∞
+n=1 kpn
+k,j
+is interpreted as the expected number of visits in j starting at k before the first
+return to k. It is not difficult to show the convergence of this series.
+Lemma 2. If p is of bounded variation, symmetric and separated from 0 and 1 then
+the series �∞
+n=1 kpn
+i,j is convergent. Moreover, for any q ≥ 1 its sum is uniformly
+bounded over all k, i, j with |k − i|, |k − j|, |j − i| < q, x ∈ T. For every ε > 0
+and natural q ≥ 1 there exists N with �∞
+n=N kpn
+i,j < ε whatever x ∈ T, provided
+|k − j| ≤ q.
+Proof. Let m ∈ N be such that kpm
+j,k > η for some η > 0 and all j, k with the same
+parity and |j − k| ≤ q (remember the Markov chain is periodic with period two).
+It is clear m and η can be chosen uniformly in x ∈ T since p is separated from 0
+and 1. We have
+kpn
+i,j · kpm
+j,k ≤ kpn+m
+i,k
+
+6
+KLAUDIUSZ CZUDEK
+for n ∈ N, hence
+∞
+�
+n=N
+kpn
+i,j ≤
+1
+kpm
+j,k
+∞
+�
+n=N
+kpn+m
+i,k
+≤ 1
+η
+∞
+�
+n=N
+kpn+m
+i,k
+.
+The last series represents the probability that the first transition to k starting at i
+occurs at earliest at the step N + m. This number is bounded from above by ε if
+N is sufficiently large. Moreover, N can be chosen to be suitable for all x ∈ T by
+a reasoning similar to the proof of Lemma 1.
+□
+It is not difficult also to recover the value of �∞
+n=1 kpn
+k,j, which represents the
+expected value of appearances in state j of the process started at k before it returns
+to k.
+Lemma 3. If p is of bounded variation, symmetric and separated from 0 and 1 and
+ax,n is defined by1 ax,0 = 1 and
+(3.3)
+ax,n =
+q(x)
+q(x + nα)
+n−1
+�
+j=0
+p(x + jα)
+q(x + jα)
+and
+(3.4)
+ax,−n =
+p(x)
+p(x − nα)
+n−1
+�
+j=0
+q(x − jα)
+p(x − jα)
+for n > 0. Then
+∞
+�
+n=1
+kpn
+k,j = ax,j
+ax,k
+for any two states k, j ∈ Z.
+Proof. Fix k. First of all, the aim is to show the assertion for j = k + 1. Notice
+if the process started at k visits k − 1 in the first step then it necessarily visits k
+before ever reaching k + 1. Thus the probability of exactly one appearance in k + 1
+before returning to k is p(x + kα) · q(x + (k + 1)α) and the probability of exactly r
+appearances is p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α) (since after the first
+r − 1 visits it “jumps” to the state k + 2 with probability p(x + (k + 1)α) and right
+after r-th to k with probability q(x + (k + 1)α)). Hence the expected number of
+appearances is
+∞
+�
+n=1
+kpn
+k,j =
+∞
+�
+r=1
+r · p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α)
+= p(x + kα)q(x + (k + 1)α)
+∞
+�
+r=1
+rp(x + (k + 1)α)r−1
+= p(x + kα)q(x + (k + 1)α)
+(1 − p(x + (k + 1)α))2
+= p(x + kα)q(x + (k + 1)α)
+q(x + (k + 1)α)2
+=
+p(x + kα)
+q(x + (k + 1)α),
+where in the passing from the second line to the third one the formula �∞
+r=1 rzr−1 =
+1
+(1−z)2 was used. Since the last equals ax,k+1
+ax,k , this completes the proof for j = k+1.
+1In contrast to other symbols here we stress the dependence on x ∈ T. That is because this
+symbol appears in the next section where the dependence on x is significant.
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+7
+To end the proof we proceed by induction. Let us assume the assertion holds
+for k + 1, k + 2, ..., j for some j > k. Let us consider the process started at k. Take
+r > 0. It is easy to conclude the expected number of appearances of this process in
+j + 1 under the condition the number of appearances in k + 1 is r equals, by the
+induction assumption, to r · ax,j+1
+ax,k+1 . In turn, the probability of exactly r visits in k
+before returning to k is, as before, p(x+kα)·p(x+(k +1)α)r−1 ·q(x+(k +1)α). In
+the view of foregoing, the expected number of appearances in j + 1 of the process
+started at k before returning to k equals
+∞
+�
+r=1
+r · ax,j+1
+ax,k+1
+p(x + kα) · p(x + (k + 1)α)r−1 · q(x + (k + 1)α)
+= ax,j+1
+ax,k+1
+·
+p(x + kα)
+q(x + (k + 1)α) = ax,j+1
+ax,k+1
+· ax,k+1
+ax,k
+= ax,j+1
+ax,k
+.
+This completes the proof of Lemma 3 in the case of any two integers with j > k.
+The case j < k is symmetric.
+□
+The last result of this section is basically the Doeblin ratio limit theorem (cf.
+Corollary 2 to Theorem 4 in Section I.9, p. 48, in [Chu60]). However, reproducing
+the proof is necessary because we need a kind of uniform convergence result over
+all x ∈ T and states j, k that are sufficiently close to each other.
+Proposition 2. If p is of bounded variation, symmetric and separated from 0 and
+1 then for every ε > 0 and q ≥ 1 there exists N such that
+����
+P(ξ1 = j) + · · · + P(ξn = j)
+P(ξ1 = k) + · · · + P(ξn = k) − ax,j
+ax,k
+���� < ε
+for every n ≥ N, x ∈ T, provided |k|, |j| ≤ q and |k − j| ≤ q.
+Proof. Take ε > 0. By Lemma 2 there exists B > 0 such that �N
+n=1 kpn
+0,j ≤ B for
+every N and states k, j satisfying the assumptions. The number B can be chosen
+also such that
+max
+|j|,|k|≤q max
+x∈T
+ax,j
+ax,k
+≤ B.
+Apply Lemma 2 and 3 to get N0 so large that
+(3.5)
+����
+N−ν
+�
+n=1
+kpn
+k,j − ax,j
+ax,k
+���� < ε
+3
+for N ≥ N0.
+The number N ′
+0 > N0 should be so large that
+(3.6)
+2B
+�N
+n=1 pn
+0,k
+< ε
+3
+for N ≥ N ′
+0 and
+(3.7)
+BN0
+�N
+n=1 pn
+0,k
+< ε
+3
+The easily proven decomposition formula
+pn
+0,j = kpn
+0,j +
+n−1
+�
+ν=1
+pν
+0,k · kpn−ν
+k,j
+
+8
+KLAUDIUSZ CZUDEK
+yields
+N
+�
+n=1
+pn
+0,j =
+N
+�
+n=1
+kpn
+0,j +
+N−1
+�
+ν=1
+pν
+0,k
+N−ν
+�
+n=1
+kpn
+k,j.
+We have
+����
+P(ξ1 = j) + · · · + P(ξN = j)
+P(ξ1 = k) + · · · + P(ξN = k) − ax,j
+ax,k
+���� =
+����
+�N
+n=1 pn
+0,j
+�N
+n=1 pn
+0,k
+− ax,j
+ax,k
+����
+=
+����
+�N
+n=1 kpn
+0,j + �N−1
+ν=1 pν
+0,k
+�N−ν
+n=1 kpn
+k,j
+�N
+n=1 pn
+0,k
+−
+�N
+n=1
+ax,j
+ax,k pn
+0,k
+�N
+n=1 pn
+0,k
+����
+≤
+����
+�N
+n=1 kpn
+0,j − ax,j
+ax,k pN
+0,k + �N−1
+ν=1 pν
+0,k
+� �N−ν
+n=1 kpn
+k,j − ax,j
+ax,k
+�
+�N
+n=1 pn
+0,k
+����
+≤
+����
+�N
+n=1 kpn
+0,j − ax,j
+ax,k pN
+0,k
+�N
+n=1 pn
+0,k
+���� +
+����
+�N−1
+ν=N−N0+1 pν
+0,k
+� �N−ν
+n=1 kpn
+k,j − ax,j
+ax,k
+�
+�N
+n=1 pn
+0,k
+����
++
+����
+�N−N0
+ν=1
+pν
+0,k
+� �N−ν
+n=1 kpn
+k,j − ax,j
+ax,k
+�
+�N
+n=1 pn
+0,k
+����
+By (3.5) the third term is less than ε
+3. By the very definition of B, the numerator
+of the first term is less that 2B and the numerator of the second expression is less
+than BN0. Thus (3.6) and (3.7) complete the proof.
+□
+Remark 1. Let us consider an interval A ⊆ Z of length q. Let (ξn) be as usually
+the process started at 0, and let τ be the moment of the first visit of (ξn) in A.
+If N is given in Proposition 2. Since N was independent of x ∈ T, a conditional
+argument easily implies
+����
+P(ξ0 = j|Fτ) + · · · + P(ξn−1 = j|Fτ)
+P(ξ0 = k|Fτ) + · · · + P(ξn−1 = k|Fτ) − ax,j
+ax,k
+���� < ε
+almost surely on {τ < n − N} for any two states k, j ∈ A.
+Remark 2. Let us now consider certain function ϕ : Z → R with support contained
+in an interval A, as above, and ∥ϕ∥∞ ≤ 1. An easy argument using Remark 1 yields
+����
+E
+�
+ϕ(ξ0) + · · · + ϕ(ξn−1)
+��Fτ
+�
+E
+�
+1A(ξ0) + · · · + 1A(ξn−1)
+��Fτ
+� −
+�
+i∈A ϕ(i)ax,i
+�
+i∈A ax,i
+���� < ε
+almost surely on {τ < n − N}. It is clear that N can be chosen uniformly over all
+intervals A of fixed length q, x ∈ T and function ϕ as far as ∥ϕ∥∞ ≤ 1.
+4. Projection of measures
+Put
+ax,k = exp
+�
+Φ(x) + · · · + Φ(x + (k − 1)α)
+�1 + exp Φ(x + kα)
+1 + exp Φ(x)
+for k ≥ 1 and ax,0 = 1. Define
+µx,n =
+1
+Mx,n
+n−1
+�
+k=0
+ax,kδx+kα
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+9
+for x ∈ T and n ≥ 1, where Mx,n is the normalizing constant,
+Mx,n =
+n−1
+�
+k=0
+ax,k.
+Lemma 4. If x ∈ T, k1, k2 ∈ N, then
+ax,k1+k2 = ax,k1 · ax+k1α,k2
+and
+µx,k1+k2 =
+Mx,k1
+Mx,k1+k2
+µx,k1 + ax,k1
+Mx+k1α,k2
+Mx,k1+k2
+µx+k1α,k2.
+The proof is straightforward.
+Lemma 5. For every ε > 0 there exists N such that if q ≥ N is the closest return
+time then (1 − ε)ay,n ≤ ax,n ≤ (1 + ε)ay,n for every natural n ≤ q and x, y ∈ T
+with |x − y| < 2
+q .
+Proof. Take δ > 0. We can find n0 so large that
+(4.1)
+1
+n
+�
+f ′�
+x
+�
++ · · · + f ′�
+x + (n − 1)α
+��
+< δ
+for n ≥ n0 and every x ∈ T.
+Indeed, this is the consequence of the Birkhoff ergodic theorem applied to the
+rotation by angle α and the Lebesgue measure (uniform convergence in x follows
+from unique ergodicity and continuity of f ′, see e.g. Proposition 4.1.13 in [KH95]).
+Let q ≥ n0 be so large that
+(4.2)
+1
+q
+�
+f ′�
+x
+�
++ · · · + f ′�
+x + jα
+��
+< δ
+for j ≤ n0 and every x ∈ T.
+Finally, by uniform continuity, let us assume q to be so large that
+(4.3)
+1 − δ ≤ 1 + exp f(x)
+1 + exp f(y) ≤ 1 + δ
+for x, y ∈ T, |x − y| ≤ 2/q.
+Take x, y ∈ T with |x − y| ≤ 2/q, a natural n ≤ q. By the mean value theorem
+there exists z in the shorter arc joining x and y such that
+ax,n
+ay,n
+= exp
+��
+f ′(z) + · · · + f ′(z + (n − 1)α)
+�
+|x − y|
+�
+×1 + exp f(x)
+1 + exp f(y) · 1 + exp f(x + (n + 1)α)
+1 + exp f(y + (n + 1)α).
+If n ≥ n0 then apply (4.1) and the fact that |x − y| ≤ 2/q to get
+�
+f ′(z)+· · ·+f ′(z +(n−1)α)
+�
+|x−y| ≤ 1
+n
+�
+f ′�
+x
+�
++· · ·+f ′�
+x+(n−1)α
+��
+· n
+q ≤ 2δ,
+as n ≤ q. This combined with (4.3) yields
+e−2δ(1 − δ)2 ≤ ax,n
+ay,n
+≤ e2δ(1 + δ)2.
+Using (4.2) and (4.3) we can deduce similar statement in the case n < n0. If δ → 0
+then the values on the left and right above tend to 1, thus the assertion follows.
+□
+
+10
+KLAUDIUSZ CZUDEK
+Proposition 3. Let ϕ ∈ C(T). For every ε > 0 there exists N such that if q ≥ N
+is a closest return time then����
+�
+T
+ϕdµx,q −
+�
+T
+ϕdµy,q
+���� < ε
+for every x, y ∈ T.
+Proof. Take η > 0 and ϕ ∈ C(T). Choose δ > 0 small (to be determined), and let
+q be the closes return time such that Lemma 5 is satisfied with ε replaced by δ. As
+a consequence
+(4.4)
+1 − δ < az1,n
+az2,n
+< 1 + δ
+and
+1 − δ < Mz1,n
+Mz2,n
+< 1 + δ
+for n ≤ q and z1, z2 ∈ T with |z1 − z2| < 2/q. Further, using Lemma ?? we easily
+see az,qn → 1 uniformly in z, when (qn) is the sequence of closest return times.
+Thus q can be chosen so large that 1 − δ ≤ az,q ≤ 1 + δ for all z ∈ T. Using the
+first assertion in Lemma 4 it implies
+(4.5)
+1 − δ ≤ az,naz+nα,n−q ≤ 1 + δ
+for n < q and z ∈ T.
+The last thing we want to assume on q it is so large that
+(4.6)
+sup
+z∈T
+sup
+|h|≤ 2
+q
+|ϕ(z + h) − ϕ(z)| < δ.
+Let us take x, y ∈ T. Denote xj = x + jα, yj = y + jα, j ∈ [0, q]. Let t be
+the smallest natural number with d(xt, y) ≤ 1
+q . Since rotation is an isometry we
+immediately see d(xt+j, yj) < 1
+q for j = 0, 1, · · · q − t. In particular d(xq, yq−t) < 1
+q ,
+hence d(yq−t, x) ≤ d(yq−t, xq) + d(xq, x) < 1/q + 1/q = 2/q and, since the rotation
+is isometry, d(yq−t+j, xj) < 2
+q for j = 0, · · · , t.
+The measure µx,q is an atomic measure with atoms at the points x, x+α, . . . , x+
+(q −1)α. The idea is to represent µx,q as a convex combination of measures concen-
+trated on two disjoint subsets {x, x+α, . . . , x+(t−1)α} and {x+tα, . . . , x+(q−1)α}
+and, similarly, represent µy,q and a convex combinations of measures concentrated
+on two disjoint subsets {y, y +α, . . ., y +(q −t−1)α} and {y +(q −t)α, . . . , y +qα}.
+Namely, it is easy to check using Lemma 4 that
+µx,q = Mx,t
+Mx,q
+µx,t + ax,t
+Mxt,q−t
+Mx,q
+µxt,q−t
+and
+µy,q = My,q−t
+My,q
+µy,q−t + ay,q−t
+Myq−t,t
+My,q
+µyq−t,t.
+Since d(xt, y) ≤ 1/q, in view of (4.4) we expect the second measure in the decom-
+position of µx,q to be close to the first measure in decomposition of µy,q. Similar
+reasoning applies to two remaining terms since d(yq−t, x) < 2/q. We have
+����
+�
+T
+ϕdµx,q −
+�
+T
+ϕdµy,q
+���� ≤
+����
+Mx,t
+Mx,q
+�
+T
+ϕdµx,t − ay,q−t
+Myq−t,t
+My,q
+�
+T
+ϕdµyq−t,t
+����
+(4.7)
++
+����ax,t
+Mxt,q−t
+Mx,q
+�
+T
+ϕdµxt,q−t − My,q−t
+My,q
+�
+T
+ϕdµy,q−t
+����.
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+11
+Let us now focus on the second term on the right hand side. The analysis of the
+first term proceeds analogously. We have
+����ax,t
+Mxt,q−t
+Mx,q
+�
+T
+ϕdµxt,q−t − My,q−t
+My,q
+�
+T
+ϕdµy,q−t
+����
+(4.8)
+≤
+����ax,t
+Mxt,q−t
+Mx,q
+− My,q−t
+My,q
+����
+�
+T
+|ϕ|dµxt,q−t
++My,q−t
+My,q
+����
+�
+T
+ϕdµxt,q−t −
+�
+T
+ϕdµy,q−t
+����.
+We are going to show the first term in (4.8) is bounded by ∥ϕ∥∞η and the second
+by δ + ∥ϕ∥∞η. Since exactly the same estimates can be derived for the first term
+on the right-hand side of (4.7), it will give
+����
+�
+T
+ϕdµx,q −
+�
+T
+ϕdµy,q
+���� ≤ 2δ + 4∥ϕ∥∞η
+and will complete the proof. Thus what remains to do is to find the desired bounds
+on the right-hand side of (4.8).
+A. Analysis of the first term on the right-hand side of (4.8)
+We have
+(4.9)
+����ax,t
+Mxt,q−t
+Mx,q
+− My,q−t
+My,q
+���� = My,q−t
+My,q
+����ax,t · My,q
+Mx,q
+· Mxt,q−t
+My,q−t
+− 1
+����.
+Since d(y, xt) < 1/q ≤ 2/q we can apply (4.4) to get that 1 − δ ≤ Mxt,q−t
+My,q−t ≤ 1 + δ.
+Further, d(yq−t, x) ≤ 2/q, thus Lemma 4 and (4.4) give
+My,q = My,q−t + ay,q−tMyq−t,t ≤ (1 + δ)Mxt,q−t + (1 + δ)2axt,q−tMx,t.
+From (4.5) we have axt,q−t ≤ 1+δ
+ax,t . Finally
+My,q ≤ (1 + δ)Mxt,q−t + (1 + δ)2axt,q−tMx,t ≤ (1 + δ)Mxt,q−t + (1 + δ)3
+ax,t
+Mx,t
+≤ (1 + δ)3
+�
+Mxt,q−t +
+1
+ax,t
+Mx,t
+�
+= (1 + δ)3
+ax,t
+�
+ax,tMxt,q−t + Mx,t
+�
+= (1 + δ)3 Mx,q
+ax,t
+.
+So far we used only the bounds from above in (4.4 ) and (4.5 ). Applying the same
+reasoning with estimates from below we see that
+My,q ≥ (1 − δ)3 Mx,q
+ax,t
+.
+Going back to (4.9) we have
+(1 − δ)4 ≤ ax,t · My,q
+Mx,q
+· Mxt,q−t
+My,q−t
+≤ (1 + δ)4.
+Take η > 0. If δ was chosen sufficiently small then
+����ax,t · My,q
+Mx,q
+· Mxt,q−t
+My,q−t
+− 1
+���� < η.
+
+12
+KLAUDIUSZ CZUDEK
+Since My,q−t
+My,q
+≤ 1 it leads to the estimate
+����ax,t
+Mxt,q−t
+Mx,q
+− My,q−t
+My,q
+���� < η.
+Thus the first term on the right-hand side of (4.8) is bounded by η∥ϕ∥∞.
+B. Analysis of the second term on the right-hand side of (4.8)
+To deal with the second expression we have clearly My,q−t
+My,q
+≤ 1 and
+����
+�
+T
+ϕdµxt,q−t −
+�
+T
+ϕdµy,q−t
+���� =
+����
+q−t−1
+�
+k=0
+axt,k
+Mxt,q−t
+ϕ(xt +kα)−
+q−t−1
+�
+k=0
+ay,k
+My,q−t
+ϕ(y+kα)
+����
+≤
+����
+q−t−1
+�
+k=0
+axt,k
+Mxt,q−t
+ϕ(xt + kα) −
+q−t−1
+�
+k=0
+ay,k
+My,q−t
+ϕ(xt + kα)
+����
++
+����
+q−t−1
+�
+k=0
+ay,k
+My,q−t
+ϕ(xt + kα) −
+q−t−1
+�
+k=0
+ay,k
+My,q−t
+ϕ(y + kα)
+����
+≤
+q−t−1
+�
+k=0
+axt,k
+Mxt,q−t
+��ϕ(xt + kα)
+��
+����1 − ay,k
+axt,k
+· Mxt,q−t
+My,q−t
+����
++
+q−t−1
+�
+k=0
+ay,k
+My,q−t
+��ϕ(xt + kα) − ϕ(y + kα)
+��.
+Since d(xt, y) < 1/q, (4.4) yields
+(1 − δ)2 ≤ ay,k
+axt,k
+· Mxt,q−t
+My,q−t
+≤ (1 + δ)2,
+thus
+����1 − ay,k
+axt,k
+· Mxt,q−t
+My,q−t
+���� < η
+if δ is sufficiently small. This leads us to the estimate
+q−t−1
+�
+k=0
+axt,k
+Mxt,q−t
+��ϕ(xt + kα)
+��
+����1 − ay,k
+axt,k
+· Mxt,q−t
+My,q−t
+���� ≤ ∥ϕ∥η.
+Clearly,
+q−t−1
+�
+k=0
+ay,k
+My,q−t
+��ϕ(xt + kα) − ϕ(y + kα)
+�� ≤ δ
+by (4.6), which completes the proof.
+□
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+13
+5. Proof of Theorem 1
+We shall use the following criterion for the uniqueness of the stationary distri-
+bution.
+If for every ε > 0 and nonnegative ϕ ∈ C(T) with 1/2 < ∥ϕ∥∞ < 1 there exist
+β ∈ R and N > 0 such that
+����
+ϕ(x) + · · · + P n−1ϕ(x)
+n
+− β
+���� < ε
+for every x ∈ T and n ≥ N, then there exists exactly one stationary distribution.
+Let us take ε > 0 and ϕ ∈ C(T) as stated in the criterion. Let y ∈ T be arbitrary,
+and let β =
+�
+T ϕdµy,q, where q is chosen so large that Proposition 3 holds with ε
+replaced by ε/3.
+Take x ∈ T. Set Ak = [kq, (k + 1)q), k ∈ Z, and define
+ϕk(j) = 1Ak(j) · ϕ(x + jα),
+ϕk : Z → R, k ∈ Z.
+Observe that
+�
+i∈Ak ϕk(i)ax,i
+�
+i∈Ak ax,i
+=
+�
+T
+ϕdµx+kα,q
+for every k, thus Proposition 3 gives
+(5.1)
+����
+�
+i∈Ak ϕk(i)ax,i
+�
+i∈Ak ax,i
+− β
+���� < ε
+3,
+for an arbitrary k ∈ Z. For k ∈ Z denote by τk the moment of the first visit of (ξn)
+in Ak. Fix n sufficiently large and set Γ ⊆ Z to be the set of k’s such that Ak is
+visited with positive probability till n. Apply Proposition 2 and Remark 2 to get a
+number N such that
+(5.2)
+����
+E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+� − β
+���� < ε
+a.s. on {τk < n − N}.
+Let (Xn) be the process with transition kernel (1.1) started at x ∈ T. We have
+(5.3)
+|E
+�
+ϕ(X0) + · · · + ϕ(Xn)
+�
+− βn|
+=
+����E
+� �
+k∈Γ
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+�
+− βE
+� �
+k∈Γ
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+�����
+≤
+�
+k∈Γ
+E
+����E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+− βE
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�����
+=
+�
+k∈Γ
+E
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+·
+����
+E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+� − β
+����
+�
+.
+Let us fix k ∈ Γ and split the expectation above as follows.
+E
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+·
+����
+E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+� − β
+����
+�
+
+14
+KLAUDIUSZ CZUDEK
+= E1{τk<n−N}
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+·
+����
+E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+� − β
+����
+�
++E1{τk≥n−N}
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+·
+����
+E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+� − β
+����
+�
+By (5.2) the first expectation does not exceed
+εE1{τk<n−N}E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+(5.4)
+≤ εEE
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+= εE
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+�
+.
+To deal with the second expectation we use the fact that ∥ϕk∥∞ ≤ 1 and the
+support of ϕk is contained in Ak. These facts combined imply easily
+E
+�
+ϕk(ξ0) + · · · + ϕk(ξn−1)
+��Fτk
+�
+≤ E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+for every n and k ∈ Γ. Furthermore,
+E
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
+��Fτk
+�
+≤ N
+almost surely on {τk ≥ n − N}. Summarizing we get the second expectation does
+not exceed
+(5.5) P(τk ≥ n − N) · N · (1 + β) ≤ N(1 + β)P
+�
+{ξn−N ∈ Ak} ∪ · · · ∪ {ξn−1 ∈ Ak}
+�
+.
+We can combine now (5.4), (5.5) and (5.3) to get
+����
+E
+�
+ϕ(X0) + · · · + ϕ(Xn−1)
+�
+n
+− β
+����
+≤ 1
+n
+�
+k∈Γ
+εE
+�
+1Ak(ξ0) + · · · + 1Ak(ξn−1)
++ 1
+n
+�
+k∈Γ
+N(1 + β)P
+�
+{ξn−N ∈ Ak} ∪ · · · ∪ {ξn−1 ∈ Ak}
+�
+≤ 1
+n · ε · n + 1
+nN(1 + β)N.
+This is less than 2ε if n is sufficiently large.
+6. Final remarks
+(1) Theorem 1 is less general than the result of Conze, Guivarc’h. It was proven
+that the assumption there is optimal by Br´emont [Bre99].
+(2) One can replace the investigated system (random circle rotation) by higher
+dimensional analog, namely toral rotation, and ask the same question again
+about the uniqueness of stationary distribution. Sinai [Sin99] considered it
+on the same footing with circle rotations, which means that Sinai’s result
+holds also there with the correct definition of Diophantine vector α. Conze,
+Guivarc’h [CG00] ideas cannot be generalized to that case. Moreover, it
+has been proven by Nicolas Chevallier [Che04] that given Diophantine α ∈
+Rd there exists a Lipschitz p on Td for which one can find two different
+stationary distributions.
+
+UNIQUE ERGODICITY OF SIMPLE SYMMETRIC...
+15
+When we try to generalize the proof of present paper to higher dimen-
+sional tori an obstacle is revealed just on the very beginning in the part
+devoted to recurrence. Indeed, one can define a martingale as in the proof
+of Proposition 1 on state that recurrence is equivalent to M(n) → ∞ when
+n → ∞ and M(n) → −∞ when n → −∞. In one-dimensional setting it
+was the consequence of symmetry and Denjoy-Koksma inequality applied
+to f(x) = ln p(x)−ln q(x). The question therefore is if a higher dimensional
+analog of Denjoy-Koksma inequality holds. A counterexample (with ana-
+lytic observable!) was given by J.-C. Yoccoz in his paper [Yoc95], Appendix
+1. In my opinion it suggests a conjecture that for any d ≥ 2 there exits
+α ∈ Rd and analytic p such that the corresponding system has at least two
+different stationary measures.
+(3) In [DFS21] the authors asked about mixing (or stability) of investigated
+system. The reasoning of Conze, Guivarc’h doesn’t give any hopes to obtain
+this stronger property. However, in our paper one can replace Doeblin ratio
+limit theorem by strong ratio limit property (see [Ore61]) saying
+����
+P(ξ2n = j)
+P(ξ2n = k) − ax,j
+ax,k
+���� → 0
+as n → ∞ provided j, k are both even (the same should be true with odd
+states and epochs). Analogs of Proposition 2 and 3 are still valid. However,
+the estimates from Section 5. get much more troublesome and delicate, and
+require much more work than I expected.
+(4) A similar system was investigated in a sequence of papers by Dolgopyat
+and Goldsheid, see [Gol08], [DG13], [DG18], [DG19], [DG20], [DG21].
+(5) One can replace the circle rotation by any automorphisms of any space and
+ask about the properties of this system. A general nonsymmetric system
+with ergodic authomorphims where considered in [KS00b]. In [KS00a] the
+authors investigated typical behavior for Anosov diffeomorphisms.
+References
+[Bre99]
+J. Bremont. Comportement des sommes ergodiqtles pour les rotations et des fonctions
+peu r´eguli`eres. Publications des S´eminaires de Rennes, 1999.
+[CG00]
+Jean-Pierre Conze and Yves Guivarc’h. Marches en milieu al´eatoire et mesures quasi-
+invariants pour un syst`eme dynamique. volume 84/85, pages 457–480. 2000. Dedicated
+to the memory of Anzelm Iwanik.
+[Che04]
+Nicolas Chevallier. Mesures quasi-invariantes sur le tore Td. J. Anal. Math., 92:371–383,
+2004.
+[Chu50] K. L. Chung. An ergodic theorem for stationary markov chains with a countable number
+of states. volume Vol. 1, page p. 568. 1950.
+[Chu60] Kai Lai Chung. Markov chains with stationary transition probabilities. Die Grundlehren
+der mathematischen Wissenschaften, Band 104. Springer-Verlag, Berlin-G¨ottingen-
+Heidelberg, 1960.
+[DFS21] D. Dolgopyat, B. Fayad, and M. Saprykina. Erratic behavior for 1-dimensional random
+walks in a Liouville quasi-periodic environment. Electron. J. Probab., 26, 2021.
+[DG13]
+D. Dolgopyat and I. Goldsheid. Limit theorems for random walks on a strip in subdiffusive
+regimes. Nonlinearity, 26(6):1743–1782, 2013.
+[DG18]
+D. Dolgopyat and I. Goldsheid. Central limit theorem for recurrent random walks on a
+strip with bounded potential. Nonlinearity, 31(7):3381–3412, 2018.
+[DG19]
+D. Dolgopyat and I. Goldsheid. Invariant measure for random walks on ergodic environ-
+ments on a strip. Ann. Probab., 47(4):2494–2528, 2019.
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+16
+KLAUDIUSZ CZUDEK
+[DG20]
+D. Dolgopyat and I. Goldsheid. Local limit theorems for random walks in a random
+environment on a strip. Pure Appl. Funct. Anal., 5(6):1297–1318, 2020.
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+D. Dolgopyat and I. Goldsheid. Constructive approach to limit theorems for recurrent
+diffusive random walks on a strip. Asymptot. Anal., 122(3-4):271–325, 2021.
+[Doe38] W. Doeblin. Sur deux problemes de m.kolmogoroff concernant les chaines denombrables.
+Bulletin de la S. M. F., 66:210–220, 1938.
+[Gol08]
+Ilya Ya. Goldsheid. Linear and sub-linear growth and the CLT for hitting times of a
+random walk in random environment on a strip. Probab. Theory Related Fields, 141(3-
+4):471–511, 2008.
+[Her79]
+Michael-Robert Herman. Sur la conjugaison diff´erentiable des diff´eomorphismes du cercle
+`a des rotations. Inst. Hautes ´Etudes Sci. Publ. Math., (49):5–233, 1979.
+[KH95]
+A. Katok and B. Hasselblatt. Introduction to the modern theory of dynamical systems,
+volume 54 of Encyclopedia of Mathematics and its Applications. Cambridge University
+Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Men-
+doza.
+[KS00a] V. Yu. Kaloshin and Ya. G. Sinai. Nonsymmetric simple random walks along orbits of
+ergodic automorphisms. In On Dobrushin’s way. From probability theory to statistical
+physics, volume 198 of Amer. Math. Soc. Transl. Ser. 2, pages 109–115. Amer. Math.
+Soc., Providence, RI, 2000.
+[KS00b] V. Yu. Kaloshin and Ya. G. Sinai. Simple random walks along orbits of Anosov diffeo-
+morphisms. Tr. Mat. Inst. Steklova, 228(Probl. Sovrem. Mat. Fiz.):236–245, 2000.
+[Ore61]
+Steven Orey. Strong ratio limit property. Bull. Amer. Math. Soc., 67:571–574, 1961.
+[Sin79]
+R. Sine. On invariant probabilities for random rotations. Israel J. Math., 33(3-4):384–388
+(1980), 1979. A collection of invited papers on ergodic theory.
+[Sin99]
+Y. Sinai. Simple Random Walks on Tori. Journal of Statistical Physics, 94(3-4):695–708,
+1999.
+[Yoc95]
+Jean-Christophe
+Yoccoz.
+Centralisateurs
+et
+conjugaison
+diff´erentiable
+des
+diff´eomorphismes du cercle. Number 231, pages 89–242. 1995. Petits diviseurs en
+dimension 1.
+Klaudiusz Czudek, Nicolaus Copernicus University, Chopin Street 12/18, 87-100 Toru´n,
+Poland
+Email address: [email protected]
+

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+arXiv:2301.08663v1  [math.AP]  20 Jan 2023
+Uniqueness of the inverse conductivity problem
+once-differentiable complex conductivities in three dimensions
+Ivan Pombo
+June 2022
+Abstract
+We prove uniqueness of the inverse conductivity problem in three dimensions for complex
+conductivities in W 1,∞.
+We apply quaternionic analysis to transform the inverse problem into an inverse Dirac
+scattering problem, as established in two dimensions by Brown and Uhlmann.
+This is a novel methodology that allows to extend the uniqueness result from once-differentiable
+real conductivities to complex ones.
+1
+Introduction
+Let γ ∈ W 1,∞(Ω) be the complex-valued conductivity defined in a bounded Lipschitz domain
+Ω ⊂ R3 and given by γ = σ +iωǫ where σ is the electrical conductivity and satisfies σ ≥ c > 0,
+ǫ is the electrical permittivity and ω is the current frequency.
+Given a boundary value f ∈ H1/2(∂Ω) we can determine the respective electrical potential
+u ∈ H1(Ω) by uniquely solving
+�
+∇ · (γ∇u) = 0, in Ω,
+u|∂Ω = f.
+(1)
+This is the so-called conductivity equation which describes the behavior of the electrical
+potential, u, in a conductive body when a voltage potential is applied on the boundary, f.
+In 1980, A.P. Calder´on, [11] introduced the problem of whether one can recover uniquely
+a conductivity σ ∈ L∞(Ω) from the boundary measurements, i.e., from the Dirichlet-to-
+Neumann map
+Λσ : H1/2(∂Ω) → H−1/2(∂Ω),
+(2)
+f �→ σ ∂u
+∂ν
+����
+∂Ω
+which connects the voltage and electrical current at the boundary. The normal derivative
+exists as an element of H−1/2(∂Ω) by
+⟨Λσf, g⟩ =
+�
+Ω
+σ∇u · ∇v dx
+(3)
+where v ∈ H1(Ω) with v|∂Ω = g and u solves (1).
+In the same paper, Calder´on was able to prove that the linearized problem at constant
+real conductivities has a unique solution. Thereafter, many authors extended is work into
+global uniqueness results. Sylvester and Uhlmann [31] used ideas of scattering theory, namely
+the exponential growing solutions of Faddeev [14] to obtain global uniqueness in dimensions
+n ≥ 3 for smooth conductivities.
+Using this foundations the uniqueness for lesser regular
+conductivities was further generalized for dimensions n ≥ 3 in the works of ([1], [7], [8],
+[12], [13], [18], [22], [24], [27]). Currently, the best know result is due to Haberman [17] for
+conductivities γ ∈ W 1,3(Ω). The reconstruction procedure for n ≥ 3 was obtained in both [22]
+and [25] independently. As far as we are aware, there seems to be no literature concerning
+reconstruction for conductivities with less than two derivatives.
+In two dimensions the problem seems to be of a different nature and tools of complex analy-
+sis were used to establish uniqueness. Nachman [23] obtained uniqueness and a reconstruction
+method for conductivities with two derivatives. The uniqueness result was soon extend for
+once-differentiable conductivities in [9] and a corresponding reconstruction method was ob-
+tained in [20]. In 2006, Astala and P¨aiv¨arinta [3] gave a positive answer Calder´on’s problem
+1
+
+for σ ∈ L∞(Ω), σ ≥ c > 0, by providing the uniqueness proof through the reconstruction
+process.
+All of this definitions can be extended to the complex-conductivity case with the assump-
+tion Re γ ≥ c > 0, in particular, we can define the Dirichlet-to-Neumann as above Λγ.
+In this scenario, the first works was done in two-dimensions by Francini [15], by extending
+the work of Brown and Uhlmann [9] in two-dimensions proving uniqueness for small frequencies
+ω and γ ∈ W 2,∞. Afterwards, Bukgheim influential paper [10] proved the general result in
+two-dimensions for complex-conductivities in W 2,∞.
+He reduced the (1) to a Schr¨odinger
+equation and shows uniqueness through the stationary phase method (based on is work many
+extensions followed in two-dimensions [2], [4], [26]).
+Recently, by mixing techniques of [9]
+and [10], Lakshtanov, Tejero and Vainberg obtained [21] uniqueness for Lipschitz complex-
+conductivities in R2. In [28], the author followed up their work to show that it is possible to
+reconstruct complex-conductivity with a jump at least in a certain set of points.
+In three dimensions, the uniqueness results presented in [31] and [24] hold for twice-
+differentiable complex-conductivities in W 2,∞, but there was no reconstruction process pre-
+sented. Nachman’s reconstruction method in three dimensions [22] was used in [19] to re-
+construct complex conductivities from boundary measurements. Even though the Nachman’s
+proof was presented only for real conductivities, the paper [29] structures the proof in order
+to show the result holds for complex-conductivities. As far as we aware, the works with lower
+regularity require real-conductivities.
+In this paper our interest resides in Calder´on’s problem for once-differentiable complex-
+conductivities in three-dimensions. The aim is to prove the following theorem:
+Theorem 1.1. Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two
+complex-valued conductivities with Re γi ≥ c > 0.
+If Λγ1 = Λγ2, then γ1 = γ2.
+Our work basis itself on a transformation of (1) into a Dirac system of equation in three-
+dimensions with the help of quaternions. In this scenario, we obtain a potential q that we want
+to determine from boundary data. The main ideas follow the work of Brown and Uhlmann [9]
+for real conductivities in two-dimensions and Lakshtanov, Vainberg and Tejero [21], as well as
+the authors work [28], for complex-conductivities.
+In this paper, we provide a novel reconstruction of the bounded potential q from the
+boundary data, but we are yet to be able to establish a relation between this boundary
+data and the Dirichlet-to-Neumann map. This is essentially to answer Calder´on problem for
+Lipschitz complex conductivities, but the lack of a well-suited Poincar´e lemma that fits the
+quaternion structure does not allow such a simple work as in 2D.
+2
+Minimalistic lesson of Quaternions
+We present the basis of the quaternionic framework we will use for our work. Let R(2) be the
+real universal Clifford Algebra over R2. By definition, it is generated as an algebra over R by
+the elements {e0, e1, e2}, where e1, e2 is a basis of R2 with eiej + ejei = −2δij, for i, j = 1, 2
+and e0 = 1 is the identity and commutes with the basis elements. This algebra has dimension
+4 and is identified with the algebra of the quaternions, H. As such it holds e3 = e1e2. In the
+following, we refer to this algebra as the quaternions. An element of the quaternions can be
+written as:
+x = x0 + x1e1 + x2e2 + x3e3,
+(4)
+where xj, j = 0, ..., 3 are real. We define the quaternionic conjugate ¯x of an element x as
+¯x = x0 − x1e1 − x2e2 − x3e3.
+(5)
+Let x, y ∈ H, we write xy for the resulting quaternionic product. The product ¯xy defines
+a Clifford valued inner product on H. Further, we have xy = ¯y¯x and the conjugate of the
+conjugate of quaternion is that same quaternion. Let x ∈ H then Sc x = x0 denotes the scalar
+of x and Vec x = x − Sc x. The scalar of a Clifford inner product Sc(¯xy) is the usual inner
+product in R4 for x, y identified as vectors.
+With this inner product H is an Hilbert space and the resulting norm is the usual Euclidean
+norm.
+In order to introduce some of the concepts we also extend the real quaternions to complex
+quaternions C2 = C ⊗ H. Here, we use the same generators (e1, e2) as above, with the same
+2
+
+multiplication rules, however, the coefficients of the quaternion can be complex-valued. That
+is, λ ∈ C ⊗ H may be written as
+λ = λ0 + λ1e1 + λ2e2 + λ3e3,
+λj ∈ C, j = 0, ..., 3
+(6)
+or still as
+λ = x + iy,
+x, y ∈ H.
+(7)
+Due to the complexification we can still take another conjugation, to which we define has
+Hermitian conjugation and denote it by ·†. Explicitly, for λ ∈ C ⊗ H one has
+¯λ† = λc
+0 − λc
+1e1 − λc
+2e2 − λc
+3e3,
+(8)
+where ·c denotes complex conjugation, or
+¯λ† = ¯x − i¯y.
+(9)
+Similarly, one can introduce an associated inner product and norm in C ⊗ H by means of
+this conjugation:
+⟨λ, µ⟩ = Sc
+�
+¯λ†µ
+�
+;
+|λ|C2 =
+�
+Sc
+�¯λ†λ
+�
+.
+(10)
+For ease of notation, we also define for λ ∈ C2 the complex conjugation as
+¯λc = λc
+0 + λc
+1e1 + λc
+2e2 + λc
+3e3.
+(11)
+Now, we can also introduce Quaternion-valued functions f : R3 → C2 written as f =
+f0 + f1e1 + f2e2 + f3e3, where fj : R3 → C.
+The Banach spaces Lp, W n,p of C2-valued functions are defines by requiring that each
+component is in such space. On L2(Ω) we introduce the C2-valued inner product
+⟨f, g⟩ =
+�
+Ω
+¯f †(x)g(x)dx.
+(12)
+Analogously to the Wirtinger derivatives in complex analysis, we have the Cauchy-Riemann
+operators under (x0, x1, x2) coordinates of R3 defined as
+D = ∂0 + e1∂1 + e2∂2,
+(13)
+where ∂j is the derivative with respect to the xj, j = 0, 1, 2 variable; and
+¯D = ∂0 − e1∂1 − e2∂2.
+(14)
+The vector part of the Cauchy-Riemann operator is designated as Dirac operator. It holds
+that D ¯D = ∆ where ∆ is the Laplacian.
+We designate any function f fulfilling Df = 0 as a monogenic function, analogous to the
+holomorphic functions in complex analysis.
+2.1
+A bit of Operator Theory
+Let Ω be a bounded domain and f : Ω → C2. All the results in this subsection were taken out
+from the classical book on quaternionic analysis of G¨urlebeck and Spr¨ossig [16]
+The Cauchy-Riemann operator has a right-inverse in the form
+(T f) (x) = − 1
+ω
+�
+Ω
+y − x
+|y − x|3 f(y) dy, for x ∈ Ω,
+(15)
+where E(x, y) = − 1
+ω
+y−x
+|y−x|3 is the generalized Cauchy kernel and ω = 4π stands for the surface
+area of the unit sphere in R3, that is, DT f = f. This operator acts from W k,p(Ω) to W k+1,p(Ω)
+with 1 < p < ∞ and k ∈ N0.
+Furthermore, we introduce the boundary integral operator for x /∈ ∂Ω
+(F∂Ωf) (x) = 1
+ω
+�
+∂Ω
+y − x
+|y − x|3 α(y)f(y) dS(y),
+(16)
+where α(y) is the outward pointing normal unit vector to ∂Ω at y. We get the well-known
+Borel-Pompeiu formula
+(F∂Ωf) (x) + (T Df) (x) = f(x) for x ∈ Ω.
+Obviously, DF∂Ω = 0 holds through this formula it it holds that F∂Ω acts from W k− 1
+p ,p(∂Ω)
+into W k,p(Ω), for k ∈ N and 1 < p < ∞.
+One of the other well-known results we will need for our work is the Plemelj-Sokhotzki
+formula is obtaining by taking the trace of the boundary integral operator.
+First we introduce an operator over the boundary of Ω.
+3
+
+Proposition 2.1. If f ∈ W k,p(∂Ω), then there exists the integral
+(S∂Ωf) = 1
+2π
+�
+∂Ω
+y − x
+|y − x|3 α(y)f(y) dS(y)
+(17)
+for all points x ∈ Ω in the sense of Cauchy principal value.
+Furthermore, the operator S∂Ω is continuous in W k,p(∂Ω), for 1 < p < ∞, k ∈ N.
+From this the Plemelj-Sokhotzki formula is given as:
+Theorem 2.2. Let f ∈ W k,p(∂Ω) where by taking the non-tangential limit we have:
+lim
+x→x0,
+x∈Ω, x0∈∂Ω
+(F∂Ωf) (x) = 1
+2 (f(x0) + (S∂Ωf) (x0)) .
+One of the corollaries concerns the limit to the boundary acting as a projector. That is,
+Corollary 2.3. The operator P∂Ω denoting the projection onto the space of all H−valued
+functions which may be monogenicaly extended into the domain Ω.
+Then, this projection may be represented as
+P∂Ω = 1
+2 (I + S∂Ω) .
+The proofs of this results and others to follow in our proofs may be found in [16].
+Now we are ready to start constructing our work on the inverse conductivity problem.
+3
+Inverse Dirac scattering problem
+Transforming our conductivity equation into another type of equation also changes the in-
+verse problem we are concerned. We transform it into a system of equations based on the
+Cauchy-Riemann operator D (also called Dirac operator in some contexts) and thus we need
+to solve the inverse Dirac scattering problem first and only afterwards we care about the
+inverse conductivity problem.
+Let u be a solution to (1) for some boundary function. We define
+φ = γ1/2 � ¯Du, Du
+�T ,
+remark that γ1/2 is well-defined since it is contained in C+. Then, φ solves the system
+�
+Dφ1
+= φ2q1,
+¯Dφ2
+= φ1q2,
+in R3.
+(18)
+where q1 = − 1
+2
+¯
+Dγ
+γ
+and q2 = − 1
+2
+Dγ
+γ .
+This transformation arises as follows:
+Dφ1 = D
+�
+γ1/2 ¯Du
+�
+= Dγ1/2 ¯Du + γ1/2∆u
+= Dγ1/2 ¯Du − γ−1/2∇γ · ∇u
+= Dγ1/2 ¯Du − 1
+2γ−1/2 �
+Dγ ¯Du + Du ¯Dγ
+�
+= −1
+2
+�
+γ1/2Du
+� ¯Dγ
+γ
+= φ2q1
+Carefully, we can extend our potential to the outside by setting γ ≡ 1 outside of Ω, which
+lead us to treat the study the equation in R3.
+3.1
+Exponentially Growing Solutions
+We devise new exponentially growing solutions from the classical ones used in three dimensions.
+In most literature works, the exponential behavior is defined through the function ex·ζ, with
+ζ ∈ C3 fulfilling ζ · ζ = 0. However, in our scenario this function does not fulfill Deix·ζ = 0,
+which brings the simplicity in all of the literature works.
+Since we know that it is harmonic we can generate a monogenic function through it. Let
+ζ ∈ C3 such that ζ · ζ := ζ2
+0 + ζ2
+1 + ζ2
+2 = 0, then it holds
+∆ex·ζ = 0 ⇔ D
+�
+¯Dex·ζ�
+= 0 ≡ D
+�
+ex·ζ ¯ζ
+�
+4
+
+where now ζ is also defined as a quaternion through ζ = ζ0 + e1ζ1 + e2ζ2 ∈ C2. Thus
+the function E(x, ζ) = ex·ζ ¯ζ is monogenic. This also arises from the choice of ζ, since ζ ¯ζ =
+ζ2
+0 + ζ2
+1 + ζ2
+2 = 0.
+We make a clear statement of when ζ is a complex-quaternion or complex-a vector, but
+in most cases it is clear from context: it is a vector if it is in the exponent and a quaternion
+otherwise.
+We assume the following asymptotic behaviour for φ:
+φ1 = ex·ζ ¯ζµ1,
+(19)
+φ2 = ex·¯ζc ¯ζcµ2
+(20)
+Setting ˜µ1 = ¯ζµ1 and ˜µ2 = ¯ζcµ2 we have the equations:
+�
+D˜µ1
+= e−x·(ζ− ¯ζc)˜µ2q1
+¯D˜µ2
+= ex·(ζ− ¯ζc)˜µ1q2
+(21)
+Further, we assume ˜µ →
+�
+1
+0
+�
+as |x| → ∞. These system of equations will lead us to an
+integral equation from which we can extract interesting behaviour for ζ → ∞.
+The main point of this subsection is to demonstrate how we can obtain the system of
+integral equations related with (21). Here, the approach is similar to [21], but we need to be
+careful due to the non-commutative nature of quaternions.
+Recall, that DT = ¯D ¯T = I (in appropriate spaces). Hence, applying this to (21) it holds:
+
+
+
+
+
+
+
+˜µ1 = 1 + T
+�
+e−x·(ζ−¯ζc)˜µ2q1
+�
+˜µ2 = T
+�
+ex·(ζ−¯ζc)˜µ1q2
+�
+Thus, we can obtain two integral equations with respect to their function:
+
+
+
+
+
+
+
+˜µ1 = 1 + T
+�
+e−x·(ζ−¯ζc) ¯T
+�
+ex·(ζ−¯ζc)˜µ1q2
+�
+q1
+�
+˜µ2 = ¯T
+�
+ex·(ζ−¯ζc)q2
+�
++ ¯T
+�
+ex·(ζ−¯ζc)T
+�
+e−x·(ζ−¯ζc)˜µ2q1
+�
+q2
+�
+
+
+
+˜µ1 = 1 + M 1˜µ1
+˜µ2 = T
+�
+e
+x·
+�
+ζ−ζC�
+q2
+�
++ M 2˜µ2
+⇔
+�
+[I − M 1](˜µ1 − 1) = M 11
+[I − M 2](˜µ2) = ¯T
+�
+ex·(ζ−¯ζC)q2
+�
+(22)
+Our objective now is to study the uniqueness and existence of this equations, we approach
+this task by proving that M j, j = 1, 2 are contractions.
+Instead of working with all possible ζ ∈ C(2) fulfilling ζ ¯ζ = 0, we choose them for k ∈ R3
+as
+ζ = k⊥ + ik
+2 ,
+k⊥ · k = 0
+and k⊥ can be algorithmically found.
+We now describe our space of functions in terms of the space variable and k ∈ R3 as
+S = L∞
+x (Lp
+k(|k| > R))
+(23)
+where R > 0 is a constant. In this space we prove that the operators M 1, M 2 are indeed
+contractions:
+Lemma 3.1. Let p > 2. Then
+lim
+R→∞ ∥M j∥S = 0.
+To further study the system (22), we also need to show that the right-hand side is in S for
+an R large enough:
+Lemma 3.2. Let p > 2. Then there exists R > 0 such that
+M 11 ∈ S,
+(24)
+¯T
+�
+ex·(ζ−¯ζC)q2
+�
+∈ S
+(25)
+The above Lemmas imply the existence and uniqueness of (˜µ1, ˜µ2) solving the system (22)
+with respect to the potential q. This is essential for the reconstruction procedure we show up
+next.
+5
+
+3.2
+Reconstruction from scattering data
+In this section, we are mixing ideas from [21] and [22] with quaternionic theory to obtain the
+potential from the scattering data.
+Starting from Clifford-Green theorem
+�
+Ω
+�
+g(x)
+� ¯Df(x)
+�
++
+�
+g(x) ¯D
+�
+f(x)
+�
+dx =
+�
+∂Ω
+g(x)η(x)f(x) dSx
+and using g(x; iξ + ζ) = (iξ + ζ)e−x·(iξ+ζ) for ξ ∈ R3 such that (iξ + ζ) · (iξ + ζ) = 0. This
+implies that g ¯D = 0. Thus we define the scattering data as:
+h(ξ, ζ) = (iξ + ζ)
+�
+∂Ω
+e−x·(iξ+ζ)η(x)φ2(x, ζ) dx
+(26)
+Applying now Clifford-Green theorem we obtain another form for the scattering data:
+h(ξ, ζ) = (iξ + ζ)
+�
+Ω
+e−x·(iξ+ζ) ¯Dφ2(x, ζ) dx
+= (iξ + ζ)
+�
+Ω
+e−ix·ξ �
+e−x·ζφ1(x, ζ)
+�
+q2(x) dx,
+by Dφ2 = φ1q2
+= (iξ + ζ)
+�
+Ω
+e−ix·ξ (ζµ1(x, ζ)) q2(x) dx
+= iξ
+�
+Ω
+e−ix·ξ ˜µ1(x, ζ)q2(x) dx,
+since ¯ζζ = 0
+= iξˆq2(ξ) + iξ
+�
+Ω
+e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx.
+Thus, we have:
+ˆq2(ξ) = h(ξ, ζ)
+iξ
+−
+�
+Ω
+e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx
+(27)
+This is yet not enough to reconstruct the potential, since the integral acts as a residual in
+the reconstruction and requires data that we technically do not have. Therefore, we integrate
+everything over an annulus in k
+�
+R<|k|<2R
+ˆq2(ξ)
+|k|3 dk = 1
+iξ
+�
+R<|k|<2R
+h(ξ, ζ(k))
+|k|3
+dk
+−
+�
+R<|k|<2R
+1
+|k|3
+�
+Ω
+e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx,
+(28)
+since the potential does not depend on k it can be taken out of the integral and taking the limit
+as R → ∞ leads the second integral on the right to decay to zero, obtaining a reconstruction
+formula.
+Theorem 3.3. Let Ω ⊂ R3 a bounded Lipschitz domain, q ∈ L∞(Ω) be a complex-valued po-
+tential obtained through a conductivity γ ∈ W 1,∞(Ω), Re γ ≥ c > 0. Then, we can reconstruct
+the potential from
+ˆq2(ξ) = lim
+R→∞
+C
+iξ
+�
+R<|k|<2R
+h(ξ, ζ(k))
+|k|3
+dk,
+(29)
+where C =
+1
+4π ln(2).
+Proof. The scattering data is defined from the solutions of the Dirac system (22) and therefore
+it holds that ˜µ1 − 1 ∈ S. Starting from (28) we obtain by integrating the right-hand side for
+any ξ ∈ R3:
+4π ln 2 ˆq2(ξ) = 1
+iξ
+�
+R<|k|<2R
+h(ξ, ζ(k))
+|k|3
+dk
+−
+�
+R<|k|<2R
+1
+|k|3
+�
+Ω
+e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx
+6
+
+Let p > 2 and 1/p + 1/q = 1. We estimate the last integral:
+�����
+�
+R<|k|<2R
+1
+|k|3
+�
+Ω
+e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx
+����� ≤
+≤
+�
+R<|k|<2R
+1
+|k|3
+�
+Ω
+���e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x)
+��� dx
+≤ CΩ∥q∥∞
+�
+R<|k|<2R
+1
+|k|3 sup
+x
+|˜µ1(x, ζ(k)) − 1| dk
+≤ CΩ∥q∥∞
+��
+R<|k|<2R
+1
+|k|3q dk
+�1/q ��
+R<|k|<2R
+sup
+x |˜µ1(x, ζ(k)) − 1|p dk
+�1/p
+≤ CΩ∥q∥∞∥˜µ1 − 1∥S
+��
+R<|k|<2R
+1
+|k|3q dk
+�1/q
+Taking the limit as R → 0 the integral that is left goes to zero which implies the desired decay
+to zero and leaves us with our reconstruction formula.
+Now, in order to connect the functions that solve the electrical conductivity equation (1)
+and the solutions to the Dirac equation (18), which are exponential growing, we introduce the
+following result:
+Proposition 3.4. Let Ω be a bounded domain in R3.
+Let φ = (φ1, φ2) be a solution of
+the Dirac system (18) for a potential q ∈ L∞(Ω) associated with the complex-conductivity
+γ ∈ W 1,∞(Ω).
+If φ1 = ¯φ2 then there exists a unique solution u of:
+� ¯Du = γ−1/2φ1,
+Du = γ−1/2φ2.
+(30)
+Further, this function fulfills the conductivity equation
+∇ · (γ∇u) = 0 in Ω.
+Let us recall the main theorem, that we are now able to prove with all these pieces we
+assembled.
+Theorem 1.1 Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two
+complex-valued conductivities with Re γi ≥ c > 0.
+If Λγ1 = Λγ2, then γ1 = γ2.
+Proof. Due to Theorem 3.3, one only needs to show that the scattering data h for |k| >> 1
+is uniquely determined by the Dirichlet-to-Neumann map Λγ. Due to the lack of Poincar´e
+Lemma in our current framework in quaternionic analysis with the D and ¯D operator, than a
+new technique is required to obtain a similar proof to [9], for example.
+For such, let us start with two conductivities γ1, γ2 in W 1,∞(Ω) for Ω a bounded domain.
+By hypothesis Λγ1 = Λγ2 and thus by [29] we have γ1|∂Ω = γ2|∂Ω.
+Further, we can extend γj, j = 1, 2 outside Ω in such a way that in R3 \ Ω and γj − 1 ∈
+W 1,∞
+comp(R3). Let qj, φj, µj, hj, j = 1, 2 be the potential and the solution in (18), the function
+in (19), and the scattering data in (26) all associated with the conductivity γj.
+Due to the scattering formulation at the boundary ∂Ω, then we just want to know if φ1 = φ2
+on ∂Ω when |k| >> 1.
+First, by Proposition 3.4, we know that there exists an u1 such that
+φ1 = γ1/2
+1
+( ¯Du1, Du1)T ,
+which is a solution to
+∇ · (γ1∇u1) = 0 in R3.
+Now, let us define u2 by
+u2 =
+�
+u1
+in R3 \ Ω,
+u
+in Ω.
+7
+
+where ˆu is the solution to the Dirichlet problem
+�
+∇ · (γ2∇u) = 0
+in Ω,
+u = u1
+on ∂Ω.
+Let g ∈ C∞
+c (R3). Then,
+�
+R3 γ2∇u2∇g dx =
+�
+R3\Ω
+γ1∇u1∇g dx +
+�
+Ω
+γ2∇ˆu∇g dx
+= −
+�
+∂Ω
+Λγ1
+�
+u1|∂Ω
+�
+g dsx +
+�
+∂Ω
+Λγ2
+�
+u|∂Ω
+�
+g dsx
+= 0.
+Hence, u2 is the solution of ∇ · (γ2∇u2) = 0 in R3. Further, the following function
+ψ2 = γ1/2
+2
+� ¯Du2, Du2
+�T
+is the solution of (18) where the potential is given by γ2.
+Furthermore, ψ2 has the asymptotics of φ1 in R3 \ Ω, thus by Lemma 3.1 and 3.2 it will
+be the unique solution of the respective integral equation of (18). Thus, ψ2 will be equal φ2
+when |k| > R. Since, on the outside ψ2 ≡ φ1. Then we obtain:
+φ1 = φ2 in R3 \ Ω.
+In particular, we have equality at the boundary ∂Ω. So, this implies that if the Dirichlet-
+to-Neumann maps are equal the respective scattering data will also be the same. Thus, the
+Dirichlet-to-Neumann map uniquely determines the potential q.
+From the definition of q, we can uniquely determine the conductivity γ up to a constant,
+which in the end is defined by γ|∂Ω which is uniquely determined by the Dirichlet-to-Neumann
+map Λγ.
+4
+Auxiliary Proofs
+Proof of Lemma 3.1.
+Let us assume, without loss of generality, that f is a scalar function. Further, we present
+the proof for M 1, since for M 2 it follows analogously.
+Recall, that we choose ζ ∈ C(2) with respect to k ∈ R(2) as
+ζ = k⊥ + ik
+2 ,
+k⊥ · k = 0.
+In vector form, this leads to ζ − ζc = ik which implies the following deductions:
+M 1f(x) =
+�
+R3 e−w·(ζ−¯ζc) x − w
+|x − w|3
+�
+R3 ey·(ζ−¯ζc) w − y
+|w − y|3 f(y)q2(y) dy q1(w) dw
+=
+�
+R3
+�
+R3 e−iw·k x − w
+|x − w|3 eiy·k w − y
+|w − y|3 f(y)q2(y)q1(w) dwdy
+=
+�
+R3 A(x, y; k)f(y) dy,
+where
+A(x, y; k) =
+�
+R3 e−i(w−y)·k x − y
+|x − y|3
+w − y
+|w − y|3 q2(y)q1(w) dw.
+Due to the compact support of the potential q2, it holds that A has compact support on the
+second variable.
+Let us now apply the norm in terms of k to it:
+∥Mf(x, ·)∥Lp(|k|>R) =
+��
+|k|>R
+|Mf(x, ζ)|p dσζ
+�1/p
+=
+��
+|k|>R
+����
+�
+Ω
+A(x, y; k)f(y) dy
+����
+p
+dσk
+�1/p
+≤
+�
+Ω
+��
+|k|>R
+|A(x, y; k)f(y)|p dσk
+�1/p
+dy
+≤
+�
+Ω
+sup
+k
+|A(x, y; k)| dy ∥f∥S.
+8
+
+In order to complete the proof we show that the first integral goes to zero as R → ∞.
+Let As be given with the extra factor α(s|x−w|)α(s|w−y|) in the integrand, where α ∈ C∞
+is 1 outside a neighborhood of the origin and 0 inside a smaller neighborhood of it.
+Since,
+�
+B1(0)
+�
+B1(0)
+1
+|w|2
+1
+|w − y|2 dw dy,
+it holds that for any ǫ > 0 there exists an s > 0 such that:
+�
+Ω
+|A − As| dy < ǫ.
+Further, we denote As0,n the function As0 with potentials q1, q2 replaced by their L1
+smooth approximation Qn
+1 , Qn
+2 ∈ C∞. Since the other factors are bounded it holds
+�
+Ω
+|As0 − As0,n| dy < ǫ.
+Now it is enough to show that As0,n → 0 as |k| → 0 uniformly!
+All integrands inside of it will be in C∞ and uniformly bounded, thus by Riemann-Lebesgue
+the result follows.
+Proof of Lemma 3.2. Once again recall that ζ =
+�
+k⊥ + i k
+2
+�
+for k ∈ R3. First we show
+that M 11 ∈ S. We have
+M 11 =
+�
+Ω
+�
+Ω
+e−iw·k x − w
+|x − w|3
+w − y
+|w − y|3 eiy·kq2(y)q1(w) dy dw,
+and applying the Lp norm in k followed with Minkowski integral inequality we obtain
+��
+|k|>R
+|M 11|pdk
+�1/p
+≤
+�
+Ω
+|q1(w)|
+|x − w|2
+��
+|k|>R
+����
+�
+Ω
+eiy·k w − y
+|w − y|3 q2(y)dy
+����
+p
+dk
+�1/p
+dw
+The inner most integral resembles a Fourier transform, hence, applying the Hausdorff-Young
+inequality for p > 2 we have
+��
+|k|>R
+����
+�
+Ω
+eiy·k w − y
+|w − y|3 q2(y) dy
+����
+p
+dk
+�1/p
+≤
+��
+Ω
+|q2(y)|p′
+|w − y|2p′ dy
+�1/p′
+< C∥q2∥∞,
+where the last inequality follows quickly by Young’s convolution inequality and Riesz type
+estimate of the kernel.
+Therefore, by the same Riesz type estimate it holds:
+��
+|k|>R
+|M 11|p dk
+�1/p
+≤ C∥q2∥∞
+�
+Ω
+|q1(w)|
+|x − w|2 dw ≤ C′∥q2∥∞∥q1∥∞.
+To complete the proof we need to show statement (25). Similarly, to the above proof, we
+have by Hausdorff-Young Inequality, Young’s convolution inequality and a Riesz type estimate
+the following:
+��
+|k|>R
+����
+�
+R3 eiy·k x − y
+|x − y|3 q2(y) dσy
+����
+p
+dσk
+�1/p
+≤
+��
+R3
+����
+x − y
+|x − y|3 q2(y)
+����
+p′
+dσy
+�1/p′
+≤ C∥q2∥∞
+We need the following auxiliary result for the proof of Proposition 3.4.
+Lemma 4.1. Let Ω be a bounded Lipschitz domain in R3.
+If h is a scalar-valued and harmonic function that fulfills
+Vec(S∂Ωh) = 0,
+then h|∂Ω is constant.
+9
+
+Proof. First, note that I + S∂Ω = P∂Ω is a projector and by Proposition 2.5.12 and Corollary
+2.5.15 of [16] it holds that P∂Ωh is the boundary value of a monogenic function in Ω.
+Since h is a scalar-valued function it holds that
+P∂Ωh = Sc(P∂Ωh) + Vec(P∂Ωh)
+= (h + Sc∂Ωh) + Vec(S∂Ωh).
+Let w = (h + Sc∂Ωh) and v = Vec(S∂Ωh). Now, we denote f as the monogenic extension
+of P∂Ωh in Ω, as such, the boundary values of f fulfill trf = w + v. Note that by hypothesis
+we have that v|∂Ω = 0.
+Hence, f is also an harmonic function, which implies that the scalar and vector components
+are harmonic.
+�
+∆(Vecf) = 0,
+Vec f|∂Ω = 0.
+By a mean value theorem or a maximum principle it holds that Vecf = 0. Due to this
+and f being monogenic we obtain that Df = 0 ⇔ D(Ref) = 0. Thus, Ref = c since D is a
+quaternionic operator.
+Consequently, the boundary values are also constant, which means that w = c in ∂Ω.
+Since, Sc(S∂Ωh) is an averaging operator it holds that h = c.
+Proof of Proposition 3.4
+Suppose that (u, v) are solutions to the following equations:
+� ¯Du = γ−1/2φ1
+Dv = γ−1/2φ2.
+From applying the operator D and ¯D to the first and second equation respectively, we
+obtain from φ2 = φ
+H
+1 and q2 = qH
+1 the following:
+∆u = D(γ−1/2φ1) = D(γ−1/2)φ1 + γ−1/2Dφ1
+= −1
+2γ−3/2(Dγφ1) + γ−1/2φ2q1
+= γ−1/2 [q2φ1 + φ2q1] = γ−1/2 �
+qH
+1 φ1 + φ
+H
+1 q1
+�
+= γ−1/2Sc (φ
+H
+1 q1).
+and
+∆v = ¯D(γ−1/2φ2) = ¯D(γ−1/2)φ2 + γ−1/2 ¯Dφ2
+= −1
+2γ−3/2( ¯Dγ)φ2 + γ−1/2φ1q2
+= γ−1/2 [q1φ2 + φ1q2] = γ−1/2 �
+q1φ
+H
+1 + φ1qH
+1
+�
+= γ−1/2Sc (φ1qH
+1 ).
+The first thing to notice is that both equations imply that u and v must be scalar-valued
+functions.
+Further, notice that
+∆(u − v) = γ−1/2 �
+Sc (φ
+H
+1 q1) − Sc (φ1qH
+1 )
+�
+= γ−1/2
+�
+Sc (φ
+H
+1 q1) − Sc (q1φ
+H
+1 )
+�
+= 0.
+Therefore, h = u − v is an harmonic function. Our objective is to show that h ≡ 0, thus
+showing that u = v.
+For such, let us consider the theory of integral transforms in quaternionic analysis. We
+have
+u = ¯T(γ−1/2φ1) + F ∂Ω(γ−1/2φ1) and
+u = ¯T(γ−1/2φ1) + F ∂Ω(u),
+which implies that
+F ∂Ω(γ−1/2φ1) = F ∂Ωu.
+10
+
+Analogously, we obtain
+F∂Ω(γ−1/2φ2) = F∂Ωv.
+Here, we can extrapolate from the first equation and from u being scalar-valued that
+γ−1/2φ1F∂Ω = F∂Ωu
+⇔ γ−1/2φ2F∂Ω = F∂Ωu.
+Applying the operator F∂Ω on the other side, we obtain:
+F 2
+∂Ωu = F∂Ω(γ−1/2φ2)F∂Ω and
+F 2
+∂Ωv = F∂Ω(γ−1/2φ2)F∂Ω
+⇒ F 2
+∂Ωh = F 2
+∂Ω(u − v) = 0.
+If we take the trace on both sides, the operator becomes a projector thus we obtain
+tr F∂Ωh = 0.
+Now, through the Sokhotski-Plemelj formula we obtain:
+tr F∂Ωh = h|∂Ω + S∂Ωh = 0, at ∂Ω.
+Since h is a scalar-valued function that we decompose this formulation with the scalar and
+vector part to obtain two conditions:
+�
+h + Sc(S∂Ωh) = 0
+Vec(S∂Ωh) = 0.
+Through the second condition and Lemma 4.1 we have that h is constant over ∂Ω.
+Now, given that h is a scalar constant, the first condition reduces to:
+h(1 + Sc(S∂Ω1)) = 0
+By [16] we obtain that 1 + Sc(S∂Ω1) = 1/2 in ∂Ω. Therefore, we conclude that h ≡ 0 in
+∂Ω. Given that h is harmonic, this immediately implies that h = 0 in Ω.
+Therefore, we obtain u = v, and therefore there exists a unique solution to the initial
+system through the T and F∂Ω operators in Ω.
+To finalize, we only need to show that u fulfills the conductivity equation in Ω.
+Bringing the first equation to light
+¯Du = γ−1/2φ1,
+changing the side of the conductivity we get γ1/2 ¯Du = φ1 and applying the D operator to
+both sides now brings
+D
+�
+γ1/2 ¯Du
+�
+= Dφ1
+⇔
+D
+�
+γ1/2�
+¯Du + γ1/2∆u = φ2q1
+⇔
+D
+�
+γ1/2�
+¯Du + γ1/2∆u = γ−1/2Du1
+2
+¯Dγ
+γ
+⇔
+1
+2γ1/2Dγ ¯Du + γ1/2∆u + 1
+2Du
+¯Dγ
+γ1/2 = 0
+⇔
+∇γ · ∇u + γ∆u = 0 ⇔ ∇ · (γ∇u) = 0
+As such, we conclude our proof of uniqueness for complex-conductivities in W 1,∞(Ω) from
+the Dirichlet-to-Neumann map Λγ. Notice that (29) even provides a reconstruction formula,
+but as mentioned in the previous section it is very unstable for computational purposes.
+References
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+13
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='08663v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='AP]  20 Jan 2023  Uniqueness of the inverse conductivity problem  once-differentiable complex conductivities in three dimensions  Ivan Pombo  June 2022  Abstract  We prove uniqueness of the inverse conductivity problem in three dimensions for complex  conductivities in W 1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  We apply quaternionic analysis to transform the inverse problem into an inverse Dirac  scattering problem, as established in two dimensions by Brown and Uhlmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  This is a novel methodology that allows to extend the uniqueness result from once-differentiable  real conductivities to complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  1  Introduction  Let γ ∈ W 1,∞(Ω) be the complex-valued conductivity defined in a bounded Lipschitz domain  Ω ⊂ R3 and given by γ = σ +iωǫ where σ is the electrical conductivity and satisfies σ ≥ c > 0,  ǫ is the electrical permittivity and ω is the current frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Given a boundary value f ∈ H1/2(∂Ω) we can determine the respective electrical potential  u ∈ H1(Ω) by uniquely solving  �  ∇ · (γ∇u) = 0, in Ω,  u|∂Ω = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (1)  This is the so-called conductivity equation which describes the behavior of the electrical  potential, u, in a conductive body when a voltage potential is applied on the boundary, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In 1980, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Calder´on, [11] introduced the problem of whether one can recover uniquely  a conductivity σ ∈ L∞(Ω) from the boundary measurements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', from the Dirichlet-to-  Neumann map  Λσ : H1/2(∂Ω) → H−1/2(∂Ω),  (2)  f �→ σ ∂u  ∂ν  ����  ∂Ω  which connects the voltage and electrical current at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The normal derivative  exists as an element of H−1/2(∂Ω) by  ⟨Λσf, g⟩ =  �  Ω  σ∇u · ∇v dx  (3)  where v ∈ H1(Ω) with v|∂Ω = g and u solves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In the same paper, Calder´on was able to prove that the linearized problem at constant  real conductivities has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Thereafter, many authors extended is work into  global uniqueness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Sylvester and Uhlmann [31] used ideas of scattering theory, namely  the exponential growing solutions of Faddeev [14] to obtain global uniqueness in dimensions  n ≥ 3 for smooth conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Using this foundations the uniqueness for lesser regular  conductivities was further generalized for dimensions n ≥ 3 in the works of ([1], [7], [8],  [12], [13], [18], [22], [24], [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Currently, the best know result is due to Haberman [17] for  conductivities γ ∈ W 1,3(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The reconstruction procedure for n ≥ 3 was obtained in both [22]  and [25] independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' As far as we are aware, there seems to be no literature concerning  reconstruction for conductivities with less than two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In two dimensions the problem seems to be of a different nature and tools of complex analy-  sis were used to establish uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Nachman [23] obtained uniqueness and a reconstruction  method for conductivities with two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The uniqueness result was soon extend for  once-differentiable conductivities in [9] and a corresponding reconstruction method was ob-  tained in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In 2006, Astala and P¨aiv¨arinta [3] gave a positive answer Calder´on’s problem  1  for σ ∈ L∞(Ω), σ ≥ c > 0, by providing the uniqueness proof through the reconstruction  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  All of this definitions can be extended to the complex-conductivity case with the assump-  tion Re γ ≥ c > 0, in particular, we can define the Dirichlet-to-Neumann as above Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In this scenario, the first works was done in two-dimensions by Francini [15], by extending  the work of Brown and Uhlmann [9] in two-dimensions proving uniqueness for small frequencies  ω and γ ∈ W 2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Afterwards, Bukgheim influential paper [10] proved the general result in  two-dimensions for complex-conductivities in W 2,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  He reduced the (1) to a Schr¨odinger  equation and shows uniqueness through the stationary phase method (based on is work many  extensions followed in two-dimensions [2], [4], [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Recently, by mixing techniques of [9]  and [10], Lakshtanov, Tejero and Vainberg obtained [21] uniqueness for Lipschitz complex-  conductivities in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In [28], the author followed up their work to show that it is possible to  reconstruct complex-conductivity with a jump at least in a certain set of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In three dimensions, the uniqueness results presented in [31] and [24] hold for twice-  differentiable complex-conductivities in W 2,∞, but there was no reconstruction process pre-  sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Nachman’s reconstruction method in three dimensions [22] was used in [19] to re-  construct complex conductivities from boundary measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Even though the Nachman’s  proof was presented only for real conductivities, the paper [29] structures the proof in order  to show the result holds for complex-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' As far as we aware, the works with lower  regularity require real-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In this paper our interest resides in Calder´on’s problem for once-differentiable complex-  conductivities in three-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The aim is to prove the following theorem:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two  complex-valued conductivities with Re γi ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  If Λγ1 = Λγ2, then γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Our work basis itself on a transformation of (1) into a Dirac system of equation in three-  dimensions with the help of quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In this scenario, we obtain a potential q that we want  to determine from boundary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The main ideas follow the work of Brown and Uhlmann [9]  for real conductivities in two-dimensions and Lakshtanov, Vainberg and Tejero [21], as well as  the authors work [28], for complex-conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In this paper, we provide a novel reconstruction of the bounded potential q from the  boundary data, but we are yet to be able to establish a relation between this boundary  data and the Dirichlet-to-Neumann map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' This is essentially to answer Calder´on problem for  Lipschitz complex conductivities, but the lack of a well-suited Poincar´e lemma that fits the  quaternion structure does not allow such a simple work as in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  2  Minimalistic lesson of Quaternions  We present the basis of the quaternionic framework we will use for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let R(2) be the  real universal Clifford Algebra over R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' By definition, it is generated as an algebra over R by  the elements {e0, e1, e2}, where e1, e2 is a basis of R2 with eiej + ejei = −2δij, for i, j = 1, 2  and e0 = 1 is the identity and commutes with the basis elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' This algebra has dimension  4 and is identified with the algebra of the quaternions, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' As such it holds e3 = e1e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In the  following, we refer to this algebra as the quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' An element of the quaternions can be  written as:  x = x0 + x1e1 + x2e2 + x3e3,  (4)  where xj, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', 3 are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We define the quaternionic conjugate ¯x of an element x as  ¯x = x0 − x1e1 − x2e2 − x3e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (5)  Let x, y ∈ H, we write xy for the resulting quaternionic product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The product ¯xy defines  a Clifford valued inner product on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Further, we have xy = ¯y¯x and the conjugate of the  conjugate of quaternion is that same quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let x ∈ H then Sc x = x0 denotes the scalar  of x and Vec x = x − Sc x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The scalar of a Clifford inner product Sc(¯xy) is the usual inner  product in R4 for x, y identified as vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  With this inner product H is an Hilbert space and the resulting norm is the usual Euclidean  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In order to introduce some of the concepts we also extend the real quaternions to complex  quaternions C2 = C ⊗ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Here, we use the same generators (e1, e2) as above, with the same  2  multiplication rules, however, the coefficients of the quaternion can be complex-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' That  is, λ ∈ C ⊗ H may be written as  λ = λ0 + λ1e1 + λ2e2 + λ3e3,  λj ∈ C, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', 3  (6)  or still as  λ = x + iy,  x, y ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (7)  Due to the complexification we can still take another conjugation, to which we define has  Hermitian conjugation and denote it by ·†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Explicitly, for λ ∈ C ⊗ H one has  ¯λ† = λc  0 − λc  1e1 − λc  2e2 − λc  3e3,  (8)  where ·c denotes complex conjugation, or  ¯λ† = ¯x − i¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (9)  Similarly, one can introduce an associated inner product and norm in C ⊗ H by means of  this conjugation:  ⟨λ, µ⟩ = Sc  �  ¯λ†µ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  |λ|C2 =  �  Sc  �¯λ†λ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (10)  For ease of notation, we also define for λ ∈ C2 the complex conjugation as  ¯λc = λc  0 + λc  1e1 + λc  2e2 + λc  3e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (11)  Now, we can also introduce Quaternion-valued functions f : R3 → C2 written as f =  f0 + f1e1 + f2e2 + f3e3, where fj : R3 → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  The Banach spaces Lp, W n,p of C2-valued functions are defines by requiring that each  component is in such space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' On L2(Ω) we introduce the C2-valued inner product  ⟨f, g⟩ =  �  Ω  ¯f †(x)g(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (12)  Analogously to the Wirtinger derivatives in complex analysis, we have the Cauchy-Riemann  operators under (x0, x1, x2) coordinates of R3 defined as  D = ∂0 + e1∂1 + e2∂2,  (13)  where ∂j is the derivative with respect to the xj, j = 0, 1, 2 variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' and  ¯D = ∂0 − e1∂1 − e2∂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (14)  The vector part of the Cauchy-Riemann operator is designated as Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' It holds  that D ¯D = ∆ where ∆ is the Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  We designate any function f fulfilling Df = 0 as a monogenic function, analogous to the  holomorphic functions in complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1  A bit of Operator Theory  Let Ω be a bounded domain and f : Ω → C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' All the results in this subsection were taken out  from the classical book on quaternionic analysis of G¨urlebeck and Spr¨ossig [16]  The Cauchy-Riemann operator has a right-inverse in the form  (T f) (x) = − 1  ω  �  Ω  y − x  |y − x|3 f(y) dy, for x ∈ Ω,  (15)  where E(x, y) = − 1  ω  y−x  |y−x|3 is the generalized Cauchy kernel and ω = 4π stands for the surface  area of the unit sphere in R3, that is, DT f = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' This operator acts from W k,p(Ω) to W k+1,p(Ω)  with 1 < p < ∞ and k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Furthermore, we introduce the boundary integral operator for x /∈ ∂Ω  (F∂Ωf) (x) = 1  ω  �  ∂Ω  y − x  |y − x|3 α(y)f(y) dS(y),  (16)  where α(y) is the outward pointing normal unit vector to ∂Ω at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We get the well-known  Borel-Pompeiu formula  (F∂Ωf) (x) + (T Df) (x) = f(x) for x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Obviously, DF∂Ω = 0 holds through this formula it it holds that F∂Ω acts from W k− 1  p ,p(∂Ω)  into W k,p(Ω), for k ∈ N and 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  One of the other well-known results we will need for our work is the Plemelj-Sokhotzki  formula is obtaining by taking the trace of the boundary integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  First we introduce an operator over the boundary of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  3  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' If f ∈ W k,p(∂Ω), then there exists the integral  (S∂Ωf) = 1  2π  �  ∂Ω  y − x  |y − x|3 α(y)f(y) dS(y)  (17)  for all points x ∈ Ω in the sense of Cauchy principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Furthermore, the operator S∂Ω is continuous in W k,p(∂Ω), for 1 < p < ∞, k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  From this the Plemelj-Sokhotzki formula is given as:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let f ∈ W k,p(∂Ω) where by taking the non-tangential limit we have:  lim  x→x0,  x∈Ω, x0∈∂Ω  (F∂Ωf) (x) = 1  2 (f(x0) + (S∂Ωf) (x0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  One of the corollaries concerns the limit to the boundary acting as a projector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' That is,  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The operator P∂Ω denoting the projection onto the space of all H−valued  functions which may be monogenicaly extended into the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Then, this projection may be represented as  P∂Ω = 1  2 (I + S∂Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  The proofs of this results and others to follow in our proofs may be found in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Now we are ready to start constructing our work on the inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  3  Inverse Dirac scattering problem  Transforming our conductivity equation into another type of equation also changes the in-  verse problem we are concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We transform it into a system of equations based on the  Cauchy-Riemann operator D (also called Dirac operator in some contexts) and thus we need  to solve the inverse Dirac scattering problem first and only afterwards we care about the  inverse conductivity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let u be a solution to (1) for some boundary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We define  φ = γ1/2 � ¯Du, Du  �T ,  remark that γ1/2 is well-defined since it is contained in C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Then, φ solves the system  �  Dφ1  = φ2q1,  ¯Dφ2  = φ1q2,  in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (18)  where q1 = − 1  2  ¯  Dγ  γ  and q2 = − 1  2  Dγ  γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  This transformation arises as follows:  Dφ1 = D  �  γ1/2 ¯Du  �  = Dγ1/2 ¯Du + γ1/2∆u  = Dγ1/2 ¯Du − γ−1/2∇γ · ∇u  = Dγ1/2 ¯Du − 1  2γ−1/2 �  Dγ ¯Du + Du ¯Dγ  �  = −1  2  �  γ1/2Du  � ¯Dγ  γ  = φ2q1  Carefully, we can extend our potential to the outside by setting γ ≡ 1 outside of Ω, which  lead us to treat the study the equation in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1  Exponentially Growing Solutions  We devise new exponentially growing solutions from the classical ones used in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In most literature works, the exponential behavior is defined through the function ex·ζ, with  ζ ∈ C3 fulfilling ζ · ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' However, in our scenario this function does not fulfill Deix·ζ = 0,  which brings the simplicity in all of the literature works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Since we know that it is harmonic we can generate a monogenic function through it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let  ζ ∈ C3 such that ζ · ζ := ζ2  0 + ζ2  1 + ζ2  2 = 0, then it holds  ∆ex·ζ = 0 ⇔ D  �  ¯Dex·ζ�  = 0 ≡ D  �  ex·ζ ¯ζ  �  4  where now ζ is also defined as a quaternion through ζ = ζ0 + e1ζ1 + e2ζ2 ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Thus  the function E(x, ζ) = ex·ζ ¯ζ is monogenic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' This also arises from the choice of ζ, since ζ ¯ζ =  ζ2  0 + ζ2  1 + ζ2  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  We make a clear statement of when ζ is a complex-quaternion or complex-a vector, but  in most cases it is clear from context: it is a vector if it is in the exponent and a quaternion  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  We assume the following asymptotic behaviour for φ:  φ1 = ex·ζ ¯ζµ1,  (19)  φ2 = ex·¯ζc ¯ζcµ2  (20)  Setting ˜µ1 = ¯ζµ1 and ˜µ2 = ¯ζcµ2 we have the equations:  �  D˜µ1  = e−x·(ζ− ¯ζc)˜µ2q1  ¯D˜µ2  = ex·(ζ− ¯ζc)˜µ1q2  (21)  Further, we assume ˜µ →  �  1  0  �  as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' These system of equations will lead us to an  integral equation from which we can extract interesting behaviour for ζ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  The main point of this subsection is to demonstrate how we can obtain the system of  integral equations related with (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Here, the approach is similar to [21], but we need to be  careful due to the non-commutative nature of quaternions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Recall, that DT = ¯D ¯T = I (in appropriate spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' applying this to (21) it holds:  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  ˜µ1 = 1 + T  �  e−x·(ζ−¯ζc)˜µ2q1  �  ˜µ2 = T  �  ex·(ζ−¯ζc)˜µ1q2  �  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' we can obtain two integral equations with respect to their function:  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  ˜µ1 = 1 + T  �  e−x·(ζ−¯ζc) ¯T  �  ex·(ζ−¯ζc)˜µ1q2  �  q1  �  ˜µ2 = ¯T  �  ex·(ζ−¯ζc)q2  �  + ¯T  �  ex·(ζ−¯ζc)T  �  e−x·(ζ−¯ζc)˜µ2q1  �  q2  �  \uf8f1  \uf8f2  \uf8f3  ˜µ1 = 1 + M 1˜µ1  ˜µ2 = T  �  e  x·  �  ζ−ζC�  q2  �  + M 2˜µ2  ⇔  �  [I − M 1](˜µ1 − 1) = M 11  [I − M 2](˜µ2) = ¯T  �  ex·(ζ−¯ζC)q2  �  (22)  Our objective now is to study the uniqueness and existence of this equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' we approach  this task by proving that M j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' 2 are contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Instead of working with all possible ζ ∈ C(2) fulfilling ζ ¯ζ = 0, we choose them for k ∈ R3  as  ζ = k⊥ + ik  2 ,  k⊥ · k = 0  and k⊥ can be algorithmically found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  We now describe our space of functions in terms of the space variable and k ∈ R3 as  S = L∞  x (Lp  k(|k| > R))  (23)  where R > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In this space we prove that the operators M 1, M 2 are indeed  contractions:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Then  lim  R→∞ ∥M j∥S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  To further study the system (22), we also need to show that the right-hand side is in S for  an R large enough:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Then there exists R > 0 such that  M 11 ∈ S,  (24)  ¯T  �  ex·(ζ−¯ζC)q2  �  ∈ S  (25)  The above Lemmas imply the existence and uniqueness of (˜µ1, ˜µ2) solving the system (22)  with respect to the potential q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' This is essential for the reconstruction procedure we show up  next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='2  Reconstruction from scattering data  In this section, we are mixing ideas from [21] and [22] with quaternionic theory to obtain the  potential from the scattering data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Starting from Clifford-Green theorem  �  Ω  �  g(x)  � ¯Df(x)  �  +  �  g(x) ¯D  �  f(x)  �  dx =  �  ∂Ω  g(x)η(x)f(x) dSx  and using g(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' iξ + ζ) = (iξ + ζ)e−x·(iξ+ζ) for ξ ∈ R3 such that (iξ + ζ) · (iξ + ζ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' This  implies that g ¯D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Thus we define the scattering data as:  h(ξ, ζ) = (iξ + ζ)  �  ∂Ω  e−x·(iξ+ζ)η(x)φ2(x, ζ) dx  (26)  Applying now Clifford-Green theorem we obtain another form for the scattering data:  h(ξ, ζ) = (iξ + ζ)  �  Ω  e−x·(iξ+ζ) ¯Dφ2(x, ζ) dx  = (iξ + ζ)  �  Ω  e−ix·ξ �  e−x·ζφ1(x, ζ)  �  q2(x) dx,  by Dφ2 = φ1q2  = (iξ + ζ)  �  Ω  e−ix·ξ (ζµ1(x, ζ)) q2(x) dx  = iξ  �  Ω  e−ix·ξ ˜µ1(x, ζ)q2(x) dx,  since ¯ζζ = 0  = iξˆq2(ξ) + iξ  �  Ω  e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Thus, we have:  ˆq2(ξ) = h(ξ, ζ)  iξ  −  �  Ω  e−ix·ξ [˜µ1(x, ζ) − 1] q2(x) dx  (27)  This is yet not enough to reconstruct the potential, since the integral acts as a residual in  the reconstruction and requires data that we technically do not have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Therefore, we integrate  everything over an annulus in k  �  R<|k|<2R  ˆq2(ξ)  |k|3 dk = 1  iξ  �  R<|k|<2R  h(ξ, ζ(k))  |k|3  dk  −  �  R<|k|<2R  1  |k|3  �  Ω  e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx,  (28)  since the potential does not depend on k it can be taken out of the integral and taking the limit  as R → ∞ leads the second integral on the right to decay to zero, obtaining a reconstruction  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let Ω ⊂ R3 a bounded Lipschitz domain, q ∈ L∞(Ω) be a complex-valued po-  tential obtained through a conductivity γ ∈ W 1,∞(Ω), Re γ ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Then, we can reconstruct  the potential from  ˆq2(ξ) = lim  R→∞  C  iξ  �  R<|k|<2R  h(ξ, ζ(k))  |k|3  dk,  (29)  where C =  1  4π ln(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' The scattering data is defined from the solutions of the Dirac system (22) and therefore  it holds that ˜µ1 − 1 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Starting from (28) we obtain by integrating the right-hand side for  any ξ ∈ R3:  4π ln 2 ˆq2(ξ) = 1  iξ  �  R<|k|<2R  h(ξ, ζ(k))  |k|3  dk  −  �  R<|k|<2R  1  |k|3  �  Ω  e−ix·ξ [˜µ1(x, ζ(k)) − 1] q2(x) dx  6  Let p > 2 and 1/p + 1/q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We estimate the last integral:  �����  �  R<|k|<2R  1  |k|3  �  Ω  e−ix·ξ [˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' ζ(k)) − 1] q2(x) dx  ����� ≤  ≤  �  R<|k|<2R  1  |k|3  �  Ω  ���e−ix·ξ [˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' ζ(k)) − 1] q2(x)  ��� dx  ≤ CΩ∥q∥∞  �  R<|k|<2R  1  |k|3 sup  x  |˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' ζ(k)) − 1| dk  ≤ CΩ∥q∥∞  ��  R<|k|<2R  1  |k|3q dk  �1/q ��  R<|k|<2R  sup  x |˜µ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' ζ(k)) − 1|p dk  �1/p  ≤ CΩ∥q∥∞∥˜µ1 − 1∥S  ��  R<|k|<2R  1  |k|3q dk  �1/q  Taking the limit as R → 0 the integral that is left goes to zero which implies the desired decay  to zero and leaves us with our reconstruction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Now, in order to connect the functions that solve the electrical conductivity equation (1)  and the solutions to the Dirac equation (18), which are exponential growing, we introduce the  following result:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let Ω be a bounded domain in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let φ = (φ1, φ2) be a solution of  the Dirac system (18) for a potential q ∈ L∞(Ω) associated with the complex-conductivity  γ ∈ W 1,∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  If φ1 = ¯φ2 then there exists a unique solution u of:  � ¯Du = γ−1/2φ1,  Du = γ−1/2φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (30)  Further, this function fulfills the conductivity equation  ∇ · (γ∇u) = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let us recall the main theorem, that we are now able to prove with all these pieces we  assembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1 Let Ω ⊂ R3 a bounded Lipschitz domain, γi ∈ W 1,∞(Ω), i = 1, 2 be two  complex-valued conductivities with Re γi ≥ c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  If Λγ1 = Λγ2, then γ1 = γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='3, one only needs to show that the scattering data h for |k| >> 1  is uniquely determined by the Dirichlet-to-Neumann map Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Due to the lack of Poincar´e  Lemma in our current framework in quaternionic analysis with the D and ¯D operator, than a  new technique is required to obtain a similar proof to [9], for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  For such, let us start with two conductivities γ1, γ2 in W 1,∞(Ω) for Ω a bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  By hypothesis Λγ1 = Λγ2 and thus by [29] we have γ1|∂Ω = γ2|∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Further, we can extend γj, j = 1, 2 outside Ω in such a way that in R3 \\ Ω and γj − 1 ∈  W 1,∞  comp(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let qj, φj, µj, hj, j = 1, 2 be the potential and the solution in (18), the function  in (19), and the scattering data in (26) all associated with the conductivity γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Due to the scattering formulation at the boundary ∂Ω, then we just want to know if φ1 = φ2  on ∂Ω when |k| >> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  First, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='4, we know that there exists an u1 such that  φ1 = γ1/2  1  ( ¯Du1, Du1)T ,  which is a solution to  ∇ · (γ1∇u1) = 0 in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Now, let us define u2 by  u2 =  �  u1  in R3 \\ Ω,  u  in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  7  where ˆu is the solution to the Dirichlet problem  �  ∇ · (γ2∇u) = 0  in Ω,  u = u1  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let g ∈ C∞  c (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Then,  �  R3 γ2∇u2∇g dx =  �  R3\\Ω  γ1∇u1∇g dx +  �  Ω  γ2∇ˆu∇g dx  = −  �  ∂Ω  Λγ1  �  u1|∂Ω  �  g dsx +  �  ∂Ω  Λγ2  �  u|∂Ω  �  g dsx  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Hence, u2 is the solution of ∇ · (γ2∇u2) = 0 in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Further, the following function  ψ2 = γ1/2  2  � ¯Du2, Du2  �T  is the solution of (18) where the potential is given by γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Furthermore, ψ2 has the asymptotics of φ1 in R3 \\ Ω, thus by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='2 it will  be the unique solution of the respective integral equation of (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Thus, ψ2 will be equal φ2  when |k| > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Since, on the outside ψ2 ≡ φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Then we obtain:  φ1 = φ2 in R3 \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In particular, we have equality at the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' So, this implies that if the Dirichlet-  to-Neumann maps are equal the respective scattering data will also be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Thus, the  Dirichlet-to-Neumann map uniquely determines the potential q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  From the definition of q, we can uniquely determine the conductivity γ up to a constant,  which in the end is defined by γ|∂Ω which is uniquely determined by the Dirichlet-to-Neumann  map Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  4  Auxiliary Proofs  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let us assume, without loss of generality, that f is a scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Further, we present  the proof for M 1, since for M 2 it follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Recall, that we choose ζ ∈ C(2) with respect to k ∈ R(2) as  ζ = k⊥ + ik  2 ,  k⊥ · k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  In vector form, this leads to ζ − ζc = ik which implies the following deductions:  M 1f(x) =  �  R3 e−w·(ζ−¯ζc) x − w  |x − w|3  �  R3 ey·(ζ−¯ζc) w − y  |w − y|3 f(y)q2(y) dy q1(w) dw  =  �  R3  �  R3 e−iw·k x − w  |x − w|3 eiy·k w − y  |w − y|3 f(y)q2(y)q1(w) dwdy  =  �  R3 A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' k)f(y) dy,  where  A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' k) =  �  R3 e−i(w−y)·k x − y  |x − y|3  w − y  |w − y|3 q2(y)q1(w) dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Due to the compact support of the potential q2, it holds that A has compact support on the  second variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let us now apply the norm in terms of k to it:  ∥Mf(x, ·)∥Lp(|k|>R) =  ��  |k|>R  |Mf(x, ζ)|p dσζ  �1/p  =  ��  |k|>R  ����  �  Ω  A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' k)f(y) dy  ����  p  dσk  �1/p  ≤  �  Ω  ��  |k|>R  |A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' k)f(y)|p dσk  �1/p  dy  ≤  �  Ω  sup  k  |A(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' k)| dy ∥f∥S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  8  In order to complete the proof we show that the first integral goes to zero as R → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let As be given with the extra factor α(s|x−w|)α(s|w−y|) in the integrand, where α ∈ C∞  is 1 outside a neighborhood of the origin and 0 inside a smaller neighborhood of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Since,  �  B1(0)  �  B1(0)  1  |w|2  1  |w − y|2 dw dy,  it holds that for any ǫ > 0 there exists an s > 0 such that:  �  Ω  |A − As| dy < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Further, we denote As0,n the function As0 with potentials q1, q2 replaced by their L1  smooth approximation Qn  1 , Qn  2 ∈ C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Since the other factors are bounded it holds  �  Ω  |As0 − As0,n| dy < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Now it is enough to show that As0,n → 0 as |k| → 0 uniformly!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  All integrands inside of it will be in C∞ and uniformly bounded, thus by Riemann-Lebesgue  the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Once again recall that ζ =  �  k⊥ + i k  2  �  for k ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' First we show  that M 11 ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We have  M 11 =  �  Ω  �  Ω  e−iw·k x − w  |x − w|3  w − y  |w − y|3 eiy·kq2(y)q1(w) dy dw,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  and applying the Lp norm in k followed with Minkowski integral inequality we obtain  ��  |k|>R  |M 11|pdk  �1/p  ≤  �  Ω  |q1(w)|  |x − w|2  ��  |k|>R  ����  �  Ω  eiy·k w − y  |w − y|3 q2(y)dy  ����  p  dk  �1/p  dw  The inner most integral resembles a Fourier transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' applying the Hausdorff-Young  inequality for p > 2 we have  ��  |k|>R  ����  �  Ω  eiy·k w − y  |w − y|3 q2(y) dy  ����  p  dk  �1/p  ≤  ��  Ω  |q2(y)|p′  |w − y|2p′ dy  �1/p′  < C∥q2∥∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  where the last inequality follows quickly by Young’s convolution inequality and Riesz type  estimate of the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Therefore, by the same Riesz type estimate it holds:  ��  |k|>R  |M 11|p dk  �1/p  ≤ C∥q2∥∞  �  Ω  |q1(w)|  |x − w|2 dw ≤ C′∥q2∥∞∥q1∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  To complete the proof we need to show statement (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Similarly, to the above proof, we  have by Hausdorff-Young Inequality, Young’s convolution inequality and a Riesz type estimate  the following:  ��  |k|>R  ����  �  R3 eiy·k x − y  |x − y|3 q2(y) dσy  ����  p  dσk  �1/p  ≤  ��  R3  ����  x − y  |x − y|3 q2(y)  ����  p′  dσy  �1/p′  ≤ C∥q2∥∞  We need the following auxiliary result for the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Let Ω be a bounded Lipschitz domain in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  If h is a scalar-valued and harmonic function that fulfills  Vec(S∂Ωh) = 0,  then h|∂Ω is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' First, note that I + S∂Ω = P∂Ω is a projector and by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='12 and Corollary  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='15 of [16] it holds that P∂Ωh is the boundary value of a monogenic function in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Since h is a scalar-valued function it holds that  P∂Ωh = Sc(P∂Ωh) + Vec(P∂Ωh)  = (h + Sc∂Ωh) + Vec(S∂Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Let w = (h + Sc∂Ωh) and v = Vec(S∂Ωh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Now, we denote f as the monogenic extension  of P∂Ωh in Ω, as such, the boundary values of f fulfill trf = w + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Note that by hypothesis  we have that v|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Hence, f is also an harmonic function, which implies that the scalar and vector components  are harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  �  ∆(Vecf) = 0,  Vec f|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  By a mean value theorem or a maximum principle it holds that Vecf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Due to this  and f being monogenic we obtain that Df = 0 ⇔ D(Ref) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Thus, Ref = c since D is a  quaternionic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Consequently, the boundary values are also constant, which means that w = c in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Since, Sc(S∂Ωh) is an averaging operator it holds that h = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='4  Suppose that (u, v) are solutions to the following equations:  � ¯Du = γ−1/2φ1  Dv = γ−1/2φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  From applying the operator D and ¯D to the first and second equation respectively, we  obtain from φ2 = φ  H  1 and q2 = qH  1 the following:  ∆u = D(γ−1/2φ1) = D(γ−1/2)φ1 + γ−1/2Dφ1  = −1  2γ−3/2(Dγφ1) + γ−1/2φ2q1  = γ−1/2 [q2φ1 + φ2q1] = γ−1/2 �  qH  1 φ1 + φ  H  1 q1  �  = γ−1/2Sc (φ  H  1 q1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  and  ∆v = ¯D(γ−1/2φ2) = ¯D(γ−1/2)φ2 + γ−1/2 ¯Dφ2  = −1  2γ−3/2( ¯Dγ)φ2 + γ−1/2φ1q2  = γ−1/2 [q1φ2 + φ1q2] = γ−1/2 �  q1φ  H  1 + φ1qH  1  �  = γ−1/2Sc (φ1qH  1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  The first thing to notice is that both equations imply that u and v must be scalar-valued  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Further, notice that  ∆(u − v) = γ−1/2 �  Sc (φ  H  1 q1) − Sc (φ1qH  1 )  �  = γ−1/2  �  Sc (φ  H  1 q1) − Sc (q1φ  H  1 )  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Therefore, h = u − v is an harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Our objective is to show that h ≡ 0, thus  showing that u = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  For such, let us consider the theory of integral transforms in quaternionic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' We  have  u = ¯T(γ−1/2φ1) + F ∂Ω(γ−1/2φ1) and  u = ¯T(γ−1/2φ1) + F ∂Ω(u),  which implies that  F ∂Ω(γ−1/2φ1) = F ∂Ωu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  10  Analogously, we obtain  F∂Ω(γ−1/2φ2) = F∂Ωv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Here, we can extrapolate from the first equation and from u being scalar-valued that  γ−1/2φ1F∂Ω = F∂Ωu  ⇔ γ−1/2φ2F∂Ω = F∂Ωu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Applying the operator F∂Ω on the other side, we obtain:  F 2  ∂Ωu = F∂Ω(γ−1/2φ2)F∂Ω and  F 2  ∂Ωv = F∂Ω(γ−1/2φ2)F∂Ω  ⇒ F 2  ∂Ωh = F 2  ∂Ω(u − v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  If we take the trace on both sides, the operator becomes a projector thus we obtain  tr F∂Ωh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Now, through the Sokhotski-Plemelj formula we obtain:  tr F∂Ωh = h|∂Ω + S∂Ωh = 0, at ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Since h is a scalar-valued function that we decompose this formulation with the scalar and  vector part to obtain two conditions:  �  h + Sc(S∂Ωh) = 0  Vec(S∂Ωh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Through the second condition and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='1 we have that h is constant over ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Now, given that h is a scalar constant, the first condition reduces to:  h(1 + Sc(S∂Ω1)) = 0  By [16] we obtain that 1 + Sc(S∂Ω1) = 1/2 in ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Therefore, we conclude that h ≡ 0 in  ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Given that h is harmonic, this immediately implies that h = 0 in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Therefore, we obtain u = v, and therefore there exists a unique solution to the initial  system through the T and F∂Ω operators in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  To finalize, we only need to show that u fulfills the conductivity equation in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Bringing the first equation to light  ¯Du = γ−1/2φ1,  changing the side of the conductivity we get γ1/2 ¯Du = φ1 and applying the D operator to  both sides now brings  D  �  γ1/2 ¯Du  �  = Dφ1  ⇔  D  �  γ1/2�  ¯Du + γ1/2∆u = φ2q1  ⇔  D  �  γ1/2�  ¯Du + γ1/2∆u = γ−1/2Du1  2  ¯Dγ  γ  ⇔  1  2γ1/2Dγ ¯Du + γ1/2∆u + 1  2Du  ¯Dγ  γ1/2 = 0  ⇔  ∇γ · ∇u + γ∆u = 0 ⇔ ∇ · (γ∇u) = 0  As such, we conclude our proof of uniqueness for complex-conductivities in W 1,∞(Ω) from  the Dirichlet-to-Neumann map Λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Notice that (29) even provides a reconstruction formula,  but as mentioned in the previous section it is very unstable for computational purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  References  [1] Alessandrini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Stable determination of conductivity by boundary measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Applicable Analysis, 27(1-3), 153-172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [2] Astala, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Faraco, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Rogers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' Unbounded potential recovery in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  11  [3] Astala, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' Calder´on’s inverse conductivity problem in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  Annals of Mathematics, 265-299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=', Imanuvilov, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Yamamoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Stability and uniqueness for  a two-dimensional inverse boundary value problem for less regular potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Inverse  Problems & Imaging, 9(3), 709-723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [5] Beals, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' Multidimensional inverse scatterings and nonlinear partial differential  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' 43, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' 45-70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' Electrical impedance tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Inverse problems, 18(6), R99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [7] Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Global uniqueness in the impedance-imaging problem for less  regular conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' SIAM Journal on Mathematical Analysis, 27(4), 1049-1056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [8] Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Torres, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Uniqueness in the inverse conductivity problem  for conductivities with 3/2 derivatives in Lp, p > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Journal of Fourier Analysis and  Applications, 9(6), 563-574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [9] Brown, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Uhlmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Uniqueness in the inverse conductivity problem  for non-smooth conductivities in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Communications in partial differential  equations, 22(5-6), 1009-1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [10] Bukhgeim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Recovering a potential from Cauchy data in the two-dimensional  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [11] Calder´on, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' On an inverse boundary value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Computational & Ap-  plied Mathematics, 25, 133-138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [12] Caro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Rogers, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Global uniqueness for the Calder´on problem with Lips-  chitz conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' In Forum of Mathematics, Pi (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [13] Chanillo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' A problem in electrical prospection and an n-dimensional Borg-  Levinson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Proceedings of the American Mathematical Society, 108(3), 761-767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [14] Faddeev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' Growing solutions of the Schr¨odinger equation, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Nauk  SSSR, 165, 514–517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [15] Francini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Recovering a complex coefficient in a planar domain from the  Dirichlet-to-Neumann map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Inverse Problems, 16(1), 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=', Spr¨ossig, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Quaternionic Analysis and Elliptic Boundary Value  Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' nternational Series of Numerical Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Birkh¨auser Basel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Uniqueness in Calder´on’s problem for conductivities with un-  bounded gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Communications in Mathematical Physics, 340(2), 639-659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [18] Haberman, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Tataru, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Uniqueness in Calder´on’s problem with Lipschitz  conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Duke Mathematical Journal, 162(3), 497-516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [19] Hamilton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Isaacson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Kolehmainen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Muller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Toivainen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Bray, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' 3D Electrical Impedance Tomography reconstructions from simulated electrode  data using direct inversion texp and Calder´on methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Inverse Problems & Imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [20] Knudsen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Tamasan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Reconstruction of less regular conductivities in the  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Communications in Partial Differential Equations, 1, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [21] Lakshtanov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=', Vainberg, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Uniqueness in the inverse conductivity  problem for complex-valued Lipschitz conductivities in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' SIAM Journal on  Mathematical Analysis, 49(5), 3766-3775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [22] Nachman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Reconstructions from boundary measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Annals of Math-  ematics, 128(3), 531-576.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [23] Nachman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Global uniqueness for a two-dimensional inverse boundary value  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Annals of Mathematics, 71-96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [24] Nachman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Sylvester, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Uhlmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' An n-dimensional Borg-Levinson the-  orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Communications in Mathematical Physics, 115(4), 595-605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Multidimensional inverse spectral problem for the equation  −∆ψ + (v(x) − Eu(x))ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Functional Analysis and Its Applications, 22(4), 263-272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [26] Novikov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Santacesaria, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Global uniqueness and reconstruction for the  multi-channel Gelfand–Calder´on inverse problem in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Bulletin des Sci-  ences Mathematiques, 135(5), 421-434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [27] P¨aiv¨arinta, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Panchenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Uhlmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Complex geometrical optics solu-  tions for Lipschitz conductivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Revista Matematica Iberoamericana, 19(1), 57-72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  12  [28] Pombo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' CGO-Faddeev approach for complex conductivities with regular jumps  in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Inverse Problems, 36(2), 024002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [29] Pombo, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Reconstructions from boundary measurements: complex conductivi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='09894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [30] Ros´en, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Geometric multivector analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Springer International Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content='  [31] Sylvester, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=', Uhlmann, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' A global uniqueness theorem for an inverse boundary  value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}
+page_content=' Annals of mathematics, 153-169  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtFAT4oBgHgl3EQftR52/content/2301.08663v1.pdf'}

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+Typeset using LATEX twocolumn style in AASTeX631
+Molecular Mapping of DR Tau’s Protoplanetary Disk, Envelope, Outflow, and Large-Scale Spiral Arm
+Jane Huang,1, ∗ Edwin A. Bergin,1 Jaehan Bae,2 Myriam Benisty,3 and Sean M. Andrews4
+1Department of Astronomy, University of Michigan, 323 West Hall, 1085 S. University Avenue, Ann Arbor, MI 48109, United States of
+America
+2Department of Astronomy, University of Florida, Gainesville, FL 32611, United States of America
+3Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+4Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA
+ABSTRACT
+DR Tau has been noted for its unusually high variability in comparison with other T Tauri stars.
+Although it is one of the most extensively studied pre-main sequence stars, observations with millimeter
+interferometry have so far been relatively limited. We present NOEMA images of 12CO, 13CO, C18O,
+SO, DCO+, and H2CO toward DR Tau at a resolution of ∼ 0.5′′ (∼ 100 au). In addition to the
+protoplanetary disk, CO emission reveals an envelope, a faint asymmetric outflow, and a spiral arm with
+a clump. The ∼ 1200 au extent of the CO arm far exceeds that of the spiral arms previously detected
+in scattered light, which underlines the necessity of sensitive molecular imaging for contextualizing
+the disk environment. The kinematics and compact emission distribution of C18O, SO, DCO+, and
+H2CO indicate that they originate primarily from within the Keplerian circumstellar disk. The SO
+emission, though, also exhibits an asymmetry that may be due to interaction with infalling material or
+unresolved substructure. The complex environment of DR Tau is reminiscent of those of outbursting
+FUor sources and some EXor sources, suggesting that DR Tau’s extreme stellar activity could likewise
+be linked to disk instabilities promoted by large-scale infall.
+Keywords: protoplanetary disks—ISM: molecules—stars: individual (DR Tau)
+1. INTRODUCTION
+In the classic schema of low-mass star formation,
+young stellar objects (YSOs) are divided into four
+classes (0, I, II, and III) based on their spectral energy
+distributions (e.g., Lada & Wilking 1984; Lada 1987;
+Andre et al. 1993). These classes are generally thought
+to correspond to different evolutionary stages, such that
+a Class 0 YSO has an envelope mass comparable to or
+greater than that of the protostar (and its possible disk),
+a Class I YSO has an envelope that is less massive than
+the protostar but still comparable to its disk, a Class II
+YSO has a disk with negligible envelope material, and
+a Class III YSO has negligible amounts of remaining
+circumstellar material (e.g., Wilking et al. 1989; Andre
+& Montmerle 1994; Dunham et al. 2014). The corre-
+spondence between SED class, evolutionary stage, and
+morphology, though, is known to be imperfect (e.g., Ro-
+bitaille et al. 2006).
+Corresponding author: Jane Huang
+[email protected]
+∗ NASA Hubble Fellowship Program Sagan Fellow
+Planet formation models often adopt the characteris-
+tics of envelope-free Class II disks as a starting point
+(e.g., ¨Oberg et al. 2011a; Lambrechts & Johansen 2012;
+Zhang et al. 2018). However, scattered light and molec-
+ular imaging have yielded identifications of a number of
+Class II disks that appear to be interacting either with
+(remnant) envelopes or ambient cloud material (e.g.,
+Grady et al. 1999; Garufi et al. 2020; Ginski et al. 2021;
+Huang et al. 2022). The pace of these identifications has
+increased with the advent of instruments such as ALMA
+and VLT/SPHERE. Detections of gaps and rings in the
+millimeter continuum of some Class II disks that ap-
+pear to have remnant envelope material, and even a few
+embedded Class I disks, offer evidence that planet for-
+mation can take place under more dynamically complex
+conditions than typically assumed (e.g., ALMA Part-
+nership et al. 2015; Segura-Cox et al. 2020; Huang et al.
+2021; Kanagawa et al. 2021). Moreover, Currie et al.
+(2022) recently detected a protoplanet in the disk of
+AB Aur, a system that appears to still be undergo-
+ing infall from a remnant envelope (e.g., Tang et al.
+2012). Simulations suggest that accretion of cloud or
+envelope material by the disk can influence its thermal
+structure, surface density profile, stability, and degree
+of misalignment (e.g., Bae et al. 2015; Dullemond et al.
+arXiv:2301.02674v1  [astro-ph.SR]  6 Jan 2023
+
+2
+Huang et al.
+2019; Kuznetsova et al. 2022). These disk conditions, in
+turn, are expected to influence where, when, and how
+planets form and migrate, as well as their composition
+(e.g., Stevenson & Lunine 1988; Boss 1997; Kokubo &
+Ida 2002). Hence, observations of the immediate envi-
+ronments of young stars are essential to establish the
+range of circumstances under which planet formation
+might proceed.
+Recent
+observations
+of
+DR
+Tau
+(J2000
+04:47:06.215+16:58:42.81), a T Tauri star located at
+a distance of 192 ± 1 pc in the Taurus star-forming re-
+gion (Gaia Collaboration et al. 2021; Bailer-Jones et al.
+2021), have suggested that its disk is being externally
+perturbed. Mesa et al. (2022) detected spiral arms in
+scattered light images of the DR Tau protoplanetary
+disk and hypothesized that one of them was triggered
+by infalling material. Meanwhile, Sturm et al. (2022)
+detected non-Keplerian emission in ALMA observations
+of 13CO and [C I] toward DR Tau, attributing this
+component to an infalling stream of gas.
+DR Tau is perhaps best known for its unusual degree
+of stellar variability.
+The star has faded and bright-
+ened in B-band by several magnitudes over the course
+of almost a century (Chavarria-K. 1979). Most notably,
+DR Tau brightened in B-band by about five magni-
+tudes between 1970 to 1979, an event that Chavarria-
+K. (1979) compared to the outbursts of FUor (also
+known as FU Ori) sources. DR Tau also exhibits sig-
+nificant short-term spectroscopic and photometric vari-
+ability—on timescales of a few days, DR Tau has been
+observed to change by up to a couple magnitudes in B-
+band and by up to a factor of a few in its optical line
+fluxes (e.g., Bertout et al. 1977; Guenther & Hessman
+1993; Alencar et al. 2001). DR Tau has a high stellar
+mass accretion rate of 4.8 × 10−7 M⊙ yr−1 (McClure
+2019). This high accretion level leads to significant con-
+tinuum veiling, which poses a challenge for determining
+its spectral type (e.g., Cohen & Kuhi 1979). Spectral
+type estimates have ranged from M0 to K4 (e.g. Imhoff
+& Appenzeller 1987; Herczeg & Hillenbrand 2014; Mc-
+Clure 2019; Gangi et al. 2022).
+DR Tau was part of the original list of outbursting
+EXor variables by Herbig (1989), although it has not
+always been included in subsequent compilations of EX-
+ors (e.g., Audard et al. 2014). DR Tau is unique among
+the EXors listed in the Herbig (1989) catalog in that
+the 18-year rise time to its outburst was much longer
+than those of the other EXors, which were typically on
+the order of a couple hundred days. EXors are usually
+distinguished from outbursting FUor sources insofar as
+EXor outbursts tend to be more modest in magnitude
+and duration, and EXors have T Tauri-like spectra dur-
+ing outbursts rather than the supergiant-like spectra
+of FUors (e.g., Audard et al. 2014). Several hypothe-
+ses have been proposed to account for the outbursts of
+young stars, including disk instabilities driven by mass
+buildup through infall from envelopes or cloud material,
+binary interactions, and stellar flybys (e.g., Bonnell &
+Bastien 1992; Vorobyov & Basu 2005; Zhu et al. 2010;
+Forgan & Rice 2010; Bae et al. 2014; Dullemond et al.
+2019). Because these outbursts affect the disk thermal
+structure, they may significantly affect how planet for-
+mation proceeds by altering molecular abundances, dust
+properties, and snowline locations (e.g., Juh´asz et al.
+2012; Banzatti et al. 2012; Cieza et al. 2016; van ’t Hoff
+et al. 2018; Jørgensen et al. 2022). The hypothesized
+connection between outbursts and environmental inter-
+actions further motivates an examination of DR Tau’s
+surroundings.
+Although DR Tau has been a popular target for obser-
+vations ranging from infrared to ultraviolet wavelengths
+(e.g., Kenyon et al. 1994; Ardila et al. 2002; Salyk et al.
+2008; Pontoppidan et al. 2011; Banzatti et al. 2014, and
+references above), relatively few observations with mil-
+limeter interferometry have been reported.
+The mil-
+limeter continuum, which traces the distribution of large
+dust grains, has been imaged on several occasions (e.g.,
+Kitamura et al. 2002; Andrews & Williams 2007; Taz-
+zari et al. 2016; Long et al. 2019). The millimeter con-
+tinuum emission is fairly compact, with 95% of the flux
+contained within a 53 au radius (Long et al. 2019). Al-
+though no substructures are immediately apparent in
+the highest resolution image to date (tracing scales down
+to ∼ 20 au), modeling of the visibilities suggests the
+presence of gaps and rings that may be associated with
+planet-disk interactions (Jennings et al. 2020). Other
+than 13CO, C18O, and [C I] (Braun et al. 2021; Sturm
+et al. 2022), no interferometric line observations of DR
+Tau have previously been published.
+The upgraded wideband capabilities of the Northern
+Extended Millimeter Array (NOEMA) provided an op-
+portunity to observe a number of lines simultaneously
+toward DR Tau. We obtained sensitive observations of
+12CO, 13CO, C18O, SO, DCO+, and H2CO at a resolu-
+tion of ∼ 0.5′′ (∼ 100 au) to map DR Tau’s structure.
+The observations and data reduction are summarized
+in Section 2. The molecular detections are analyzed in
+Section 3, and the implications of DR Tau’s complex
+structures are discussed in Section 4. The summary and
+conclusions are presented in Section 5.
+2. OBSERVATIONS AND DATA REDUCTION
+DR Tau was observed with the NOEMA PolyFiX
+correlator in dual polarization mode during program
+W20BE (PI: J. Huang). The correlator setup covered
+frequencies from 213.9-221.6 GHz and 229.4-237.2 GHz
+at a resolution of 2 MHz. Within these frequency ranges,
+we placed a series of chunks, each with a resolution of
+62.5 kHz and width of 64 MHz, in order to resolve molec-
+ular lines of interest (detailed further in Section 3 and
+Appendix A).
+The first set of observations was executed in C config-
+uration on 2021 January 08, with baseline lengths rang-
+ing from 24 to 328 m. The second set of observations
+
+Molecular Mapping of DR Tau
+3
+was executed in A configuration on 2021 March 03, with
+baseline lengths ranging from 32 to 760 m. Each con-
+figuration used eleven antennas. For each set of obser-
+vations, LkHα 101 served as the flux calibrator, 3C 84
+served as the bandpass calibrator, and 0446+112 and
+0507+179 served as the phase calibrators. The on-source
+time was 3.0 hours in C configuration and 3.4 hours in
+A configuration.
+The raw data were calibrated with the NOEMA
+pipeline in CLIC, which is part of the GILDAS package
+(Pety 2005; Gildas Team 2013).
+Then, the following
+steps were performed with the GILDAS MAPPING software.
+The calibrated visibilities were written out to separate
+uv-tables corresponding to the low spectral resolution,
+wide bandwidth data and the high spectral resolution,
+narrow bandwidth spectral windows. After flagging of
+channels with strong line emission, the wide bandwidth
+uv-tables were spectrally averaged to produce contin-
+uum uv-tables.
+For each of the four basebands, the
+continuum was imaged with the CLEAN algorithm and
+three phase self-calibration loops were performed using
+solution intervals of 180, 90, and 45 seconds. The self-
+calibration solutions were then applied to the uv-tables
+for the narrow spectral windows that fell within the cor-
+responding basebands. Continuum subtraction was per-
+formed for each spectral window separately in the uv
+plane by fitting a linear baseline.
+The self-calibrated, continuum-subtracted uv tables
+were converted to measurement sets to enable imag-
+ing with the Common Astronomy Software Applica-
+tions (CASA) 6.4 (CASA Team et al. 2022). Because
+GILDAS outputs frequencies in the rest frame of the
+source (i.e., the frequency that corresponds to the source
+systemic velocity input by the observer is the rest fre-
+quency of the line of interest), we had to manually cor-
+rect the frequencies in the measurement sets so that
+CASA would output image cubes with the appropri-
+ate LSRK velocities.
+Each line was imaged with the
+tclean implementation of the multi-scale CLEAN al-
+gorithm (Rau & Cornwell 2011).
+We set the robust
+value to 0.5 and and the image cube channel spacing
+to 0.2 km s−1.
+To accommodate the irregular mor-
+phology of the 12CO and 13CO J = 2 − 1 emission,
+we employed the auto-multithresh algorithm (Kep-
+ley et al. 2020) to define the CLEAN masks, choos-
+ing the following parameter values after some experi-
+mentation: sidelobethreshold=2.0, noisethreshold=4.0,
+minbeamfrac=0.3, and negativethreshold=7.0.
+Initial
+imaging tests yielded prominent striping artifacts due
+to the poor uv sampling of the spatially extended cloud
+emission, so we re-imaged these lines without baselines
+shorter than 20 kλ.
+For the other molecules, where
+only compact emission was detected, we used a circu-
+lar CLEAN mask with a radius of 2.6′′ and included all
+baselines. A Gaussian uv taper of 1.0′′ was used to in-
+crease sensitivity to weaker lines (i.e., lines other than
+12CO, 13CO, C18O, SO, DCO+, and H2CO JKaKc =
+303 − 202). After CLEANing, a primary beam correc-
+tion was applied to each image cube.
+Calibrated visibilities and images can be down-
+loaded
+at
+https://zenodo.org/record/7370498#
+.Y7U-qezMKeB.
+3. RESULTS
+3.1. Overview of Line Observations
+The primary line targets were 12CO, 13CO, C18O, SO,
+DCO+, and H2CO. The CO isotopologues serve as gas
+tracers, SO is a potential shock tracer (e.g., Pineau des
+Forˆets et al. 1993), and H2CO and DCO+ are common
+cold disk gas tracers (e.g., Huang et al. 2017; Pegues
+et al. 2020). The synthesized beam and per-channel rms
+(estimated from line-free channels) for the primary line
+targets are listed in Table 1, and channel maps are pre-
+sented in Appendix B. Spectra for the detected lines,
+which were extracted using circular masks with diam-
+eters listed in Table 1, are shown in Figure 1.
+Since
+the spatial extent of 12CO and 13CO are ambiguous due
+to spatial filtering and cloud contamination, we used
+extraction masks approximately equal to the primary
+beam FWHM at 1.3 mm (21′′). The mask sizes for the
+other lines were chosen based on the approximate radial
+extent of the 3σ emission in the image cubes. Fluxes
+were measured by integrating each spectrum within the
+velocity ranges listed in Table 1. The velocity integra-
+tion ranges for the CO isotopologues were selected based
+on where emission above the 3σ level is detected. For
+the weaker lines, the C18O velocity integration range
+was adopted. The 1σ flux uncertainties were estimated
+as ∆v ×
+√
+N × σspec, where ∆v is the channel width (in
+km s−1), N is the number of channels spanned by the
+line, and σspec is the standard deviation (in Jy) mea-
+sured from a signal-free portion of the spectrum (this is
+not to be confused with the per-channel rms value listed
+in Table 1 (in mJy beam−1), which is calculated from the
+image cube). However, the statistical uncertainties do
+not capture the true uncertainty of the fluxes for 12CO
+and 13CO, which are affected by cloud contamination
+and spatial filtering.
+We categorize a line as detected if emission is above
+the 5σ level within 2′′ of DR Tau in at least one channel
+of the image cube and above the 3σ level in at least two
+adjacent channels. By these criteria, 12CO, 13CO, C18O,
+SO, DCO+, and H2CO 303 − 202 are firmly detected.
+While H2CO 321 − 220 does not meet these criteria, its
+integrated flux is ⪆ 4σ when extracted over the same ve-
+locity range and emitting region as the strong 303 − 202
+transition, so this line is considered to be tentatively de-
+tected. The channel maps for the 322 − 221 transition
+(Appendix B) show 4σ emission at 10.2 kms−1 that is
+cospatial with the stronger 303 − 202 transition, but the
+velocity-integrated flux from the spectrum is < 2σ. Fur-
+thermore, the peak of the spectrum occurs at a velocity
+
+4
+Huang et al.
+Table 1. Imaging Summary for Primary Line Targets
+Transition
+Synthesized beam
+Per-channel RMS noisea
+Velocity rangeb
+Extraction Mask Diameter
+Fluxc
+(arcsec × arcsec (◦))
+(mJy beam−1)
+(km s−1)
+(arcsec)
+(mJy km s−1)
+12CO J = 2 − 1
+0.79 × 0.47 (18.2◦)
+7
+[−2, 17]
+21
+37900 ± 200d
+13CO J = 2 − 1
+0.84 × 0.49 (17.3◦)
+6
+[7.6, 11.4]
+21
+5730 ± 70d
+C18O J = 2 − 1
+0.85 × 0.50 (17.3◦)
+6
+[9.0, 10.8]
+4
+622 ± 10
+SO JN = 65 − 54
+0.85 × 0.50 (17.2◦)
+6
+[9.0, 10.8]
+3
+195 ± 9
+SO JN = 55 − 44
+0.86 × 0.50 (17.1◦)
+6
+[9.0, 10.8]
+3
+96 ± 10
+DCO+ J = 3 − 2
+0.86 × 0.50 (17.1◦)
+6
+[9.0, 10.8]
+3
+40 ± 10
+H2CO JKaKc = 303 − 202
+0.86 × 0.50 (17.2◦)
+6
+[9.0, 10.8]
+3
+248 ± 11
+H2CO JKaKc = 322 − 221
+1.20 × 0.93 (17.1◦)
+7
+[9.0, 10.8]
+3
+< 30
+H2CO JKaKc = 321 − 220
+1.20 × 0.93 (17.1◦)
+7
+[9.0, 10.8]
+3
+38 ± 8
+aWith channel widths of 0.2 km s−1.
+b LSRK velocity range over which moment maps are produced and the flux is estimated.
+c The 1σ error bars do not include the systematic flux uncertainty (∼ 10%).
+dThese lines are significantly affected by spatial filtering, so the statistical uncertainty does not reflect the true uncertainty in the fluxes.
+well offset from the peak of the 303 − 202 and 321 − 220
+lines. Therefore, we do not consider the 322 − 221 tran-
+sition to be detected.
+Integrated intensity maps of the primary line targets
+are presented in Figure 2, using the velocity integration
+ranges listed in Table 1. The intensity-weighted veloc-
+ity maps of the stronger lines are presented in Figure
+3. For 12CO and 13CO, the integrated intensity maps
+excluded pixels in the image cube below the 3σ level
+and the intensity-weighted velocity map excluded pix-
+els below the 6σ level in order to reduce contributions
+from cloud contamination and artifacts from spatial fil-
+tering. For all other lines, no clipping was used for the
+integrated intensity maps, and a 4σ clip was adopted for
+the intensity-weighed velocity maps.
+A summary of the auxiliary line observations (none of
+which yielded a detection) is presented in Appendix C.
+3.2. Structures traced by CO isotopologues
+Due to their differing optical depths, the three de-
+tected CO isotopologues reveal different components of
+the DR Tau system, including the circumstellar disk, an
+arm, an envelope, and an outflow.
+An overhead car-
+toon schematic of the system is shown in Figure 4. We
+describe each component in further detail below.
+3.2.1. The circumstellar disk
+C18O is the least optically thick of the three detected
+CO isotopologues and therefore best traces the Keple-
+rian rotation of the circumstellar disk (see Figure 3).
+The southern side is blueshifted and the northern side is
+redshifted relative to the systemic velocity, which Braun
+et al. (2021) estimated to be vsys = 9.9+0.08
+−0.09 km s−1 from
+ALMA observations of 13CO and C18O J = 2−1. Signs
+of Keplerian rotation are visible in the inner regions of
+the NOEMA 13CO intensity-weighted velocity map and
+coincide with the bright, compact emission component
+in the integrated intensity map, but the disk edge is
+not well-defined due to the presence of extended, non-
+Keplerian emission. From visual inspection of the 13CO
+emission, we estimate that the Keplerian disk has a ra-
+dial extent of ∼ 300 au, but this should only be consid-
+ered a lower bound for the disk size because the abun-
+dance of 13CO is generally too low in the outer disk
+to recover the disk size robustly (e.g., Trapman et al.
+2019). Finally, 12CO is dominated by large-scale, non-
+Keplerian structures.
+The NOEMA observations do not strongly constrain
+the disk orientation, since the C18O emission is spanned
+by only a few synthesized beams. However, Long et al.
+(2019) measured a position angle (P.A.) of 3.4+8.2
+−8.0 de-
+grees east of north and an inclination angle of 5.4+2.1
+−2.6
+degrees from ALMA millimeter continuum observations
+at an angular resolution of ∼ 0.1′′, which corresponds
+to ∼ 20 au.
+We adopt these values for our analysis.
+Although our new 12CO and 13CO observations show
+significant non-disk emission, Sturm et al. (2022) found
+that their ALMA C18O observations could be largely re-
+produced by a Keplerian disk model employing the disk
+orientation derived from Long et al. (2019). Because the
+disk is nearly face-on, the C18O spectrum only exhibits
+a single peak at the systemic velocity rather than the
+double-peak characteristic of more inclined disks.
+
+Molecular Mapping of DR Tau
+5
+5
+0
+5
+10
+15
+20
+25
+LSRK Velocity (km s
+1)
+0
+5
+10
+15
+20
+25
+Flux (Jy)
+12CO 2-1
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.0
+2.5
+5.0
+7.5
+Flux (Jy)
+13CO 2-1
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.00
+0.25
+0.50
+0.75
+Flux (Jy)
+C18O 2-1
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.0
+0.1
+0.2
+Flux (Jy)
+SO 65
+54
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.05
+0.00
+0.05
+0.10
+0.15
+Flux (Jy)
+SO 55
+44
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.04
+0.00
+0.04
+0.08
+Flux (Jy)
+DCO +  3
+2
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.0
+0.1
+0.2
+0.3
+0.4
+Flux (Jy)
+H2CO 303
+202
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.050
+0.025
+0.000
+0.025
+0.050
+Flux (Jy)
+H2CO 322
+221
+5.0
+7.5
+10.0
+12.5
+15.0
+LSRK Velocity (km s
+1)
+0.025
+0.000
+0.025
+0.050
+Flux (Jy)
+H2CO 321
+220
+Figure 1. Source-integrated spectra of the primary line targets toward DR Tau. The vertical red dotted line marks the systemic
+velocity. The gray bars denote regions where cloud contamination is apparent.
+
+6
+Huang et al.
+9
+6
+3
+0
+3
+6
+9
+9
+6
+3
+0
+3
+6
+9
+12CO 2
+1
+200 au
+5 100 300
+900 2100
+9
+6
+3
+0
+3
+6
+9
+9
+6
+3
+0
+3
+6
+9
+13CO 2
+1
+200 au
+5
+25
+75
+150 300
+Integrated Intensity (mJy beam
+1 km s
+1)
+9
+6
+3
+0
+3
+6
+9
+9
+6
+3
+0
+3
+6
+9
+C18O 2
+1
+200 au
+0
+25 50
+100
+160
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+SO 65
+54
+200 au
+0
+10 20 30 40 50
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+SO 55
+44
+200 au
+0
+7
+14
+21
+28
+35
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+DCO +  3
+2
+200 au
+0
+5
+10
+15
+20
+25
+2
+1
+0
+1
+2
+ ['']
+2
+1
+0
+1
+2
+ ['']
+H2CO 303
+202
+200 au
+0
+15
+30
+45
+60
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+H2CO 322
+221
+200 au
+0
+4
+8
+12
+16
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+H2CO 321
+220
+200 au
+0
+6
+12
+18
+24
+Figure 2. Integrated intensity maps of primary line targets observed toward DR Tau. The synthesized beam is drawn in the
+lower left corner of each panel. Black crosses mark the disk center. The axes show offsets from the disk center in arcseconds.
+For the CO isotopologues, the color scale uses an arcsinh stretch to make faint extended features more visible. Note that the
+size scales are different between the top row and the other rows.
+
+Molecular Mapping of DR Tau
+7
+8
+4
+0
+4
+8
+8
+4
+0
+4
+8
+12CO 2
+1
+200 au
+8.4
+9.4
+10.4
+11.4
+8
+4
+0
+4
+8
+8
+4
+0
+4
+8
+13CO 2
+1
+200 au
+8.4
+9.4
+10.4
+11.4
+Intensity-weighted velocity
+(km s
+1)
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+C18O 2
+1
+200 au
+9.3
+9.6
+9.9 10.2 10.5
+2
+1
+0
+1
+2
+ ['']
+2
+1
+0
+1
+2
+ ['']
+SO 65
+54
+200 au
+9.3
+9.6
+9.9 10.2 10.5
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+SO 55
+44
+200 au
+9.3
+9.6
+9.9 10.2 10.5
+2
+1
+0
+1
+2
+2
+1
+0
+1
+2
+H2CO 303
+202
+200 au
+9.3
+9.6
+9.9 10.2 10.5
+Figure 3. Intensity-weighted velocity maps of strong lines detected toward DR Tau. The synthesized beam is drawn in the lower
+left corner of each panel. The purple cross denotes the disk center. The axes show offsets from the disk center in arcseconds.
+Note that the velocity ranges and size scales are not the same for all imags.
+3.2.2. Blueshifted spiral arm
+The intensity-weighted velocity maps for 12CO and
+13CO (Figure 3) both show an arm that is blueshifted
+with respect to the systemic velocity.
+To isolate the
+emission from the CO arm, we produced integrated in-
+tensity maps between 8.4 and 9.0 km s−1 (Figure 5).
+The arm is connected to the south side of the disk and
+curves around the western side, terminating at a pro-
+jected distance of ∼ 1200 au from DR Tau at a P.A. of
+∼ 330◦. The 12CO emission also shows a clump along
+the arm at a projected distance of ∼ 500 au southwest
+from DR Tau. This arm was not detected in previously
+published high-resolution ALMA 13CO images of DR
+Tau (Sturm et al. 2022), presumably due to some com-
+bination of lack of sensitivity and spatial filtering. How-
+ever, low angular resolution ALMA ACA observations of
+[C I] from Sturm et al. (2022) show extended blueshifted
+emission, which may originate from the arm traced by
+CO in our NOEMA observations.
+In order to estimate the pitch angle of the arm, we
+transformed the integrated intensity map of the arm into
+a polar coordinate map (i.e., as a function of deprojected
+radius R and polar angle θ), assuming that the arm is in
+the plane of the disk (Figure 6). We then measured the
+position of the spiral arm by searching for local radial
+maxima in the polar coordinate map for fixed values of θ
+in steps of 8◦ from 124◦ to 260◦. The arm was modelled
+as an Archimedean spiral of the form R(θ) = a + cθ3,
+where θ is in radians. (We found that logarithmic spirals
+and Archimedean spirals with smaller exponents did not
+fit the data well). The log-likelihood function was spec-
+ified as log L = −0.5 �
+n
+�
+(Rdata−Rmodel)2
+σ2
++ log(2πσ2)
+�
+,
+where σ is the standard deviation of the major axis of
+the synthesized beam. Uniform priors of [0, 2000] and
+[−2000, 0] were used for a and c, respectively. Posteriors
+were explored using the affine-invariant sampler emcee
+(Goodman & Weare 2010; Foreman-Mackey et al. 2013)
+with 40 walkers and 1000 steps. After discarding the
+first 500 steps as burn-in, we computed the 50th per-
+
+8
+Huang et al.
+Outflow
+Disk
+Blueshifted 
+arm
+Line of Sight
+EAST
+WEST
+Envelope
+Figure 4. A proposed cartoon schematic of the DR Tau system from an overhead perspective (i.e., perpendicular to the line
+of sight). The components are not drawn to scale. Note that while the disk is drawn such that the east side is tilted toward the
+observer in order to show that the disk is slightly inclined, the observations do not constrain which side is closer to the observer.
+The 3-dimensional orientations of the envelope and the arm are not known in detail, but the former is drawn in front of the
+disk and the latter is drawn behind the disk (from the perspective of the observer) under the assumption of infalling motion.
+However, the observations may also be explained by other configurations of the structures.
+6
+3
+0
+3
+6
+ ['']
+6
+3
+0
+3
+6
+ ['']
+12CO J = 2
+1 (8.4-9.0 km s
+1)
+Clump
+200 au
+0.0
+50.0
+100.0
+150.0
+200.0
+250.0
+6
+3
+0
+3
+6
+6
+3
+0
+3
+6
+13CO J = 2
+1 (8.4-9.0 km s
+1)
+200 au
+0.0
+6.0
+12.0
+18.0
+24.0
+Integrated Intensity
+(mJy beam
+1 km s
+1)
+Figure 5. Integrated intensity maps of 12CO (left) and 13CO, summed up between 8.4 and 9.0 km s−1 to highlight DR Tau’s
+blueshifted spiral arm. The blue cross marks the center of the disk. The synthesized beam is shown as a white ellipse in the
+lower left corner of each panel. The 12CO color scale is saturated in order to show the fainter arm emission more clearly.
+
+Molecular Mapping of DR Tau
+9
+0
+300
+600
+900
+1200
+Deprojected radius (au)
+0
+60
+120
+180
+240
+300
+360
+ (degrees)
+6
+4
+2
+0
+2
+4
+6
+ ['']
+6
+4
+2
+0
+2
+4
+6
+ ['']
+120 150 180 210 240 270
+ (degrees)
+0
+15
+30
+45
+60
+75
+Pitch angle (degrees)
+Figure 6. Left: Integrated intensity map of the 12CO arm, replotted as a function of deprojected radius and polar angle θ.
+Center: Integrated intensity map of the 12CO arm, overplotted with the spiral function defined by the posterior median values
+of the spiral parameters. Right: Pitch angle of the arm as a function of the polar angle θ. The black curve corresponds to the
+values derived from the median of the spiral arm parameter posteriors, while the blue curves correspond to 1000 random draws
+from the posterior.
+
+10
+Huang et al.
+centile of the marginal posterior distribution to obtain a
+point estimate and the 16th and 84th percentiles to ob-
+tain error estimates: a = 1060±30 au and c = −7.8±0.6
+au. We computed the pitch angles (φ = arctan
+��� 1
+R
+dR
+dθ
+��)
+�
+corresponding to the median values of a and c, then
+also computed pitch angles for spiral curves defined by
+1000 random draws of (a, c) from the posterior. Figure
+6 shows the median spiral plotted over the integrated
+intensity map and a plot of the derived pitch angles as
+a function of polar angle θ. The pitch angles range from
+6 to 56 degrees between polar angle values of 124 to
+260 degrees (corresponding to deprojected radius values
+between 980 and 330 au).
+In other words, the pitch
+angle appears to decrease with distance from the star,
+although the true values may differ if the assumption
+that the arm is in the plane of the disk is incorrect.
+We computed the escape velocity, vesc =
+�
+2GM∗
+r
+, at
+the tip of the arm to assess whether it is gravitationally
+bound to DR Tau.
+The dynamical mass of DR Tau
+has been measured to be 1.2 M⊙ (Braun et al. 2021).
+Emission from the arm is detected up to r ∼ 1200 au
+at an LSRK velocity of 8.8 km s−1, which is offset from
+the systemic velocity by 1.1 km s−1. The corresponding
+escape velocity at r = 1200 au is 1.3 km s−1. Thus the
+arm appears to be compatible with being gravitationally
+bound to DR Tau, but not definitively so, since there
+may also be a transverse velocity component.
+Mesa et al. (2022) recently identified two spiral arms
+in SPHERE H-band Qφ observations of DR Tau. Figure
+7 compares the arms identified in the SPHERE image
+to the CO arm. The CO arm is much more extended
+than the scattered light arms, which are only detected
+up to ∼ 220 au in projection from the star. Because the
+NOEMA synthesized beam is comparable in scale to the
+SPHERE spiral arms, it is not clear whether the CO arm
+is an extension of one of the arms detected in scattered
+light or a separate structure. Mesa et al. (2022) mea-
+sured pitch angles of 11◦ and 26◦ for the two scattered
+light arms, which are smaller than the pitch angle mea-
+sured for the inner region of the CO arm.
+However,
+since the pitch angles appear to change along the arm,
+the differing values do not necessarily imply that they
+are separate structures. Mesa et al. (2022) also noted
+that the northeastern spiral in the SPHERE image had
+a clump-like feature, which they hypothesized was asso-
+ciated with a protoplanet embedded in a dusty envelope.
+While this compact feature is well below the resolution
+limits of our NOEMA observations, the presence of a dif-
+ferent clump in the 12CO arm suggests that the clumps
+could be intrinsic features of the arms themselves.
+3.2.3. Envelope
+DR Tau shows envelope emission in 12CO up to ∼ 5′′
+(1000 au) in projection from the star (Figure 8). En-
+velope emission is detected between 9.8 and 12 km s−1,
+i.e., mostly redshifted with respect to the systemic ve-
+locity. In most of these channels, the envelope emission
+is more spatially extended and brighter on the northern
+side.
+As with 12CO, the 13CO emission is more extended
+north of the star compared to south of the star for
+LSRK velocities above 10.4 km s−1.
+In contrast to
+12CO, though, the 13CO maps show features that ap-
+pear more streamer-like than envelope-like.
+However,
+since this is the velocity range where cloud contamina-
+tion is most significant, spatial filtering of large-scale
+emission may be artificially creating the appearance of
+streamers. Sturm et al. (2022) identified a possible in-
+falling stream in ALMA observations of 13CO toward
+DR Tau, but those observations were likewise affected
+by spatial filtering. Observations of other lines that are
+bright but less susceptible to cloud contamination (e.g.,
+species with higher critical densities like HCO+ or CO
+transitions with higher upper energy levels) might help
+to clarify the nature of these apparent streamers.
+The channels where envelope emission is detected in
+12CO overlap with the channels where [C I] exhibits a
+redshifted non-Keplerian component that Sturm et al.
+(2022) attributed to an infalling stream. However, since
+the beam FWHM of the [C I] observations is ∼ 3′′, most
+of the emission is spatially unresolved. Given the simi-
+lar velocities to the 12CO envelope, it is likely that the
+redshifted non-Keplerian [C I] emission also originates
+from the envelope.
+3.2.4. Outflow
+DR Tau’s 12CO spectrum (Figure 1) exhibits a faint
+blueshifted line wing without a corresponding redshifted
+line wing, suggesting the presence of an asymmetric out-
+flow.
+The channel maps (Figure 10.1) show compact
+emission at LSRK velocities lower than 8.4 km s−1. To
+highlight this compact outflow emission more clearly, we
+extracted a new 12CO spectrum using a smaller circu-
+lar aperture with a diameter of 4′′ (Figure 9). Because
+DR Tau is nearly face-on, it is not straightforward to
+separate the outflow emission from the line wings of the
+Keplerian disk. However, the asymmetry of the line pro-
+file allows us to estimate the velocities at which outflow
+emission dominates by mirroring the outflow spectrum
+about the systemic velocity and taking the ratio of the
+original and mirrored spectrum. We assume that the
+outflow emission on the blueshifted side dominates when
+the ratio exceeds 10, which occurs at 7.4 km s−1.
+While the outflow emission is weak in individual
+channels (Figure 10.1), the spatial distribution of the
+blueshifted side can be better seen by producing an in-
+tegrated intensity map between −2.0 and 7.4 km s−1.
+The lower bound of the velocity integration range was
+determined by where the emission in individual chan-
+nels drops below 3σ. For comparison, we also produced
+an integrated intensity map from 12.4 to 21.8 km s−1,
+corresponding to the redshifted channels at the opposing
+offsets from the systemic velocity. The two integrated in-
+
+Molecular Mapping of DR Tau
+11
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+ [′′]
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+ [′′]
+Southern spiral
+Northeastern
+spiral
+H-band Q
+Clump
+H-band Q
+50 au
+6
+4
+2
+0
+2
+4
+6
+6
+4
+2
+0
+2
+4
+6
+H-band Q  vs. 12CO arm
+200 au
+Figure 7. A comparison between the SPHERE H-band Qφ image of DR Tau from Mesa et al. (2022) and the 12CO NOEMA
+observations from this work. Left: H-band Qφ image of DR Tau. The arrows point to the northeastern spiral, southern spiral,
+and clump identified in Mesa et al. (2022). The gray circle shows the extent of the SPHERE coronagraph. Right: A contour
+plot of the 12CO arm overlaid atop the H-band Qφ image. Note that the size scale is different from the image on the left. The
+contours, drawn at 50, 100, and 150 mJy beam−1 km s−1, correspond to the 12CO integrated intensity map from Figure 5.
+tensity maps are presented in Figure 9. The blueshifted
+map shows relatively compact emission with a radial ex-
+tent of ∼ 2′′ (∼ 400 au). Although the opening angle
+of the outflow cannot be computed because the disk is
+nearly face-on, the compactness of the emission suggests
+that the outflow is quite collimated. The redshifted map
+shows emission near the stellar position, but given that
+the redshifted emission is fainter and much more com-
+pact than the blueshifted outflow, it seems likely that
+the compact redshifted emission originates from the line
+wing of the Keplerian disk emission. While a redshifted
+outflow component is not readily visible in the 12CO
+spectrum, the redshifted map shows a faint ring with a
+radius of ∼ 4.5′′ (∼ 900 au), which is significantly wider
+than the blueshifted outflow component.
+3.3. SO, DCO+, and H2CO emission
+SO, DCO+, and H2CO emission all originate from a
+relatively compact region within 300 au of DR Tau. The
+SO 65 − 54, SO 55 − 44, and H2CO 303 − 202 intensity-
+weighted velocity maps (Figure 3) show velocity gradi-
+ents similar to that of C18O, indicating that they like-
+wise (largely) originate from the Keplerian disk. The
+kinematics of DCO+ are not well-defined due to the low
+signal-to-noise ratio, but the compactness of the emis-
+sion suggests that it also primarily traces the Keplerian
+disk.
+That said, whereas C18O and H2CO 303−202 both ex-
+hibit relatively axisymmetric emission in the integrated
+intensity maps (Figure 2) and line profiles that are sym-
+metric about the systemic velocity (Figure 1), SO 65−54
+and 55 − 44 are both asymmetric.
+Their emission is
+stronger on the northern (redshifted) side of the disk.
+In addition, their spectra both peak at an LSRK veloc-
+ity of 10.2 km s−1, which is redshifted by 0.3 km s−1
+with respect to the systemic velocity. The DCO+ spec-
+trum also appears stronger on the redshifted side, but
+given that its SNR is lower than that of the SO lines,
+more sensitive observations will be necessary to deter-
+mine whether the DCO+ asymmetry is genuine.
+4. DISCUSSION
+4.1. The evolutionary stage of DR Tau
+DR Tau is traditionally considered to have a Class II
+SED, with stellar age estimates ranging from 0.9 to 3.2
+Myr (Kenyon & Hartmann 1995; McClure 2019; Long
+et al. 2019). The presence of the envelope, if primordial,
+would suggest that the younger end of the age range is
+more likely. DR Tau’s chemistry also appears to point
+to a younger age. Although SO is detected in DR Tau,
+it has otherwise rarely been detected in Class II disks,
+
+12
+Huang et al.
+9.8
+200 au
+10.0
+10.2
+10.4
+10.6
+10.8
+6
+3
+0
+3
+6
+ [′′]
+6
+3
+0
+3
+6
+ [′′]
+11.0
+11.2
+11.4
+11.6
+11.8
+12.0
+0
+50
+100
+250
+500
+12CO J = 2
+1 intensity (mJy beam
+1)
+9.8
+200 au
+10.0
+10.2
+10.4
+10.6
+10.8
+6
+3
+0
+3
+6
+ [′′]
+6
+3
+0
+3
+6
+ [′′]
+11.0
+11.2
+11.4
+11.6
+11.8
+12.0
+0
+50
+100
+200
+13CO J = 2
+1 intensity (mJy beam
+1)
+Figure 8. Channel maps of 12CO J = 2 − 1 and 13CO J = 2 − 1 over the velocity range where envelope emission is present.
+The black contours denote the 5, 15, 25, and 35σ contours of C18O J = 2 − 1 to serve as a visual reference for the kinematics of
+the Keplerian disk. (Note that because C18O is less abundant than 12CO and 13CO, the Keplerian line wings of C18O are not
+detected out to as high velocities as the other two isotopologues).
+especially those hosted by T Tauri stars (e.g., Guilloteau
+et al. 2016; Semenov et al. 2018; Le Gal et al. 2021). It
+is commonly detected, though, in younger, embedded
+Class 0 and I systems (e.g., Sakai et al. 2014; Le Gal
+et al. 2020; Garufi et al. 2022; Mercimek et al. 2022).
+In addition, Sturm et al. (2022) found that gas-phase
+carbon is not as severely depleted in DR Tau as other
+Class II disks that have been observed, although it is
+still more depleted than Class 0/I systems.
+However, simulations have suggested that pre-main se-
+quence stars might be able to form second-generation en-
+velopes through interaction with cloud material, a pro-
+cess sometimes referred to as “late infall” (e.g., Dulle-
+mond et al. 2019; Kuffmeier et al. 2020). Indeed, Mesa
+et al. (2022) hypothesized that DR Tau was undergo-
+ing late infall based on the detection of spiral arms
+in scattered light.
+Given the range of ages estimated
+for DR Tau, it is ambiguous whether infall onto DR
+Tau should be considered “late.” As noted above, DR
+
+Molecular Mapping of DR Tau
+13
+0
+10
+20
+LSRK Velocity (km s
+1)
+2
+0
+2
+4
+6
+8
+10
+12
+Flux (Jy)
+12CO inner 4′′
+9
+6
+3
+0
+3
+6
+9
+ ['']
+9
+6
+3
+0
+3
+6
+9
+ ['']
+-2.0 to 7.4 km s
+1
+0
+100
+200 300400
+Integrated Intensity (mJy beam
+1 km s
+1)
+9
+6
+3
+0
+3
+6
+9
+9
+6
+3
+0
+3
+6
+9
+12.4 to 21.8 km s
+1
+0
+50
+100
+150
+Figure 9. Overview of DR Tau’s outflow emission. Left: 12CO spectrum extracted from a circular aperture with a 4′′ diameter,
+showing a blueshifted outflow wing. The approximate velocity range of the blueshifted outflow is shaded in blue. The purple
+dotted line marks the system velocity. Middle:
+12CO integrated intensity map covering velocities from −2.0 to 7.4 km s−1.
+Compact emission from the blueshifted side of the outflow is visible. An arcsinh stretch is used on the color scale to make faint
+emission more readily visible. The faint vertical striping is due to the sidelobes of the point spread function. The pink cross
+marks the position of the disk center. Right: 12CO integrated intensity map covering velocities from 12.4 to 21.8 km s−1. The
+map shows a faint ring with a radius of ∼ 4.5′′ (∼ 900 au) and compact emission located at the stellar position. The redshifted
+compact emission may be from a line wing of the Keplerian disk rather than the outflow.
+Tau’s chemistry seems to suggest that the disk is rela-
+tively young. This appearance of chemical youthfulness,
+though, stems from comparisons of Class 0/I sources to
+isolated Class II disks. Based on molecular observations
+of GM Aur, a Class II disk with large-scale spiral arms
+suggestive of ongoing late infall, Huang et al. (2021)
+speculated that accretion of cloud material could par-
+tially reset disk chemistry such that it bears greater
+resemblance to that of Class 0/I sources. The impact
+of late infall on disk chemistry will need to be examined
+through astrochemical modeling to determine the extent
+to which chemical properties can be used to sort disks
+by relative age.
+Given that DR Tau is commonly included in sur-
+veys of Class II disks because of its relatively large
+disk mass and bright line emission (e.g., Salyk et al.
+2011; Long et al. 2019; Arulanantham et al. 2020; Sturm
+et al. 2022), an erroneous classification of its evolution-
+ary stage may skew interpretations of disk observations.
+Huang et al. (2022) remarked that a similar problem
+exists for DO Tau, another commonly observed Class II
+disk that also shows signatures of being partially embed-
+ded. Interestingly, among the twelve single star systems
+that Long et al. (2019) identified as having “smooth”
+disks in millimeter continuum emission, at least three
+of them (DR Tau, DO Tau, and Haro 6-13) exhibit ev-
+idence of an envelope in spatially resolved CO emission
+(e.g., Fern´andez-L´opez et al. 2020; Garufi et al. 2021;
+Huang et al. 2022, and this work). Most of the remain-
+ing sources (including both the “smooth” and structured
+disks) lack high quality interferometric CO observations,
+so it is unknown whether they might be embedded as
+well.
+Analyses of where and when disk substructures
+tend to emerge will require sensitive, spatially resolved
+molecular line observations to provide context about the
+evolutionary stages of the objects being studied.
+4.2. Origin of SO in DR Tau
+The detection of SO in DR Tau is notable given that
+SO detections have thus far been uncommon in Class II
+disks, which has been attributed to high gas-phase C/O
+ratios (> 1) disfavoring SO production (e.g., Guilloteau
+et al. 2016; Semenov et al. 2018; Le Gal et al. 2021). In
+two of the disks where SO has been detected, AB Aur
+and Oph IRS 48, the gas-phase C/O ratio has been esti-
+mated to be less than 1 (Rivi`ere-Marichalar et al. 2020;
+Booth et al. 2021). Based on thermochemical model-
+ing of [C I] and CO isotopologue emission, Sturm et al.
+(2022) estimated that DR Tau has a gas-phase C/O ra-
+tio of 0.47. The detection of SO toward DR Tau is thus
+qualitatively consistent with SO production in disks be-
+ing favored in gas with C/O ratios less than 1.
+DR Tau’s SO emission exhibits a mild asymmetry that
+is not seen in C18O. This suggests that the SO asym-
+metry is not merely tracing the underlying gas surface
+density, but could instead be due to some dynamical
+process locally favoring SO production.
+SO has been
+proposed to be enhanced by outflows, winds, gravita-
+
+14
+Huang et al.
+tional instabilities, or accretion shocks (e.g., Pineau des
+Forˆets et al. 1993; Sakai et al. 2014; Tabone et al. 2017;
+Ilee et al. 2017). We discuss these possibilities in turn
+for DR Tau.
+Our NOEMA observations have shown that DR Tau
+has a molecular outflow, and past CO ro-vibrational
+spectroscopy indicates that DR Tau has a wide-angle
+molecular wind (Pontoppidan et al. 2011). An outflow
+shock does not appear to be a likely major contributor
+to SO in DR Tau, since SO is not detected at the same
+high velocities as CO. At the spatial and spectral resolu-
+tion of our NOEMA observations, it is unclear whether
+the SO kinematics are consistent with those expected
+for a disk wind (e.g. Haworth & Owen 2020), but DR
+Tau’s bright emission makes it an excellent target for
+more detailed follow-up.
+Chemical modeling of gravitationally unstable disks
+suggests that gas-phase SO can be enhanced either by
+spiral shocks or within warm disk fragments (Ilee et al.
+2011, 2017). While DR Tau does feature spiral struc-
+ture, Mesa et al. (2022) argued that DR Tau’s spiral
+arms are unlikely to be due to gravitational instability
+given that its disk-to-stellar mass ratio and stellar accre-
+tion rate are a factor of a few lower than hydrodynamical
+simulations suggest would be necessary to induce grav-
+itational instabilities. Nevertheless, disk mass is notori-
+ously difficult to measure (e.g., Miotello et al. 2022, and
+references therein), and studies of other spiral-armed
+disks have often disagreed on whether they are massive
+enough to be gravitationally unstable (e.g., P´erez et al.
+2016; Cleeves et al. 2016; Veronesi et al. 2019; Sierra
+et al. 2021).
+Higher spatial resolution would help to
+determine if the SO asymmetry traces spiral structure
+and/or a disk fragment.
+Accretion shocks in protoplanetary disks might oc-
+cur due to cloud or envelope material being accreted
+by the disk or disk material being accreted by an em-
+bedded planet (e.g., Bodenheimer 1974; Boss & Graham
+1993; Yorke & Bodenheimer 1999; Szul´agyi & Mordasini
+2017).
+Accretion streamers traced by SO have been
+observed in several Class I protostellar systems (e.g.,
+Garufi et al. 2022; Artur de la Villarmois et al. 2022).
+Given that DR Tau is now known to be partially embed-
+ded, its asymmetric SO emission may arise in a manner
+similar to Class I systems. Booth et al. (2022) proposed
+that an SO asymmetry in the HD 100546 disk could
+be due to shocks from gas accreting onto an embedded
+planet. This likely does not account for DR Tau’s SO
+asymmetry, since high-contrast imaging from Mesa et al.
+(2022) rules out the presence of a companion above sev-
+eral Jupiter masses at separations greater than 50 au
+from DR Tau.
+4.3. Origin of DR Tau’s molecular spiral arm
+Mesa et al. (2022) hypothesized that the northeastern
+spiral arm detected in scattered light toward DR Tau is
+due to planet-disk interactions, while the southern arm
+is due to infall from cloud material. As noted in Section
+3.2.2, the angular resolution of our NOEMA observa-
+tions does not allow us to determine whether the CO
+spiral arm is an extension of either scattered light spiral
+arm, but the very large extent of the molecular arm sug-
+gests that it is unlikely to be generated by interactions
+with a bound planet. Mesa et al. (2022) placed an upper
+limit of several Jupiter masses on any companion farther
+out than 50 au from DR Tau. Given that the millimeter
+continuum appears smooth down to a resolution of 20
+au (Long et al. 2018), it is unlikely that the disk harbors
+massive (super-Jovian) companions within 50 au. More-
+over, hydrodynamical simulations indicate that external
+companions exceeding several MJ should create a pair
+of (nearly) symmetric spiral arms (e.g., Zhu et al. 2015;
+Dong et al. 2016), contrary to what is observed for DR
+Tau. Thus, a stellar companion is also unlikely to be
+responsible for the arm.
+An infalling stream is a plausible explanation for the
+molecular arm, given that similar large-scale structures
+have been detected in association with a number of em-
+bedded Class 0/I sources as well as Class II disks pro-
+posed to be undergoing late infall (e.g., Tang et al. 2012;
+Yen et al. 2019; Pineda et al. 2020; Huang et al. 2021;
+Garufi et al. 2022; Valdivia-Mena et al. 2022). One pos-
+sible difference of note is that the structures proposed
+to be infalling streams in the other systems have tended
+to be open, whereas DR Tau’s pitch angle exhibits a
+marked decrease with distance from the star. However,
+this apparent difference may simply be a projection ef-
+fect, since we do not know their three-dimensional ori-
+entations.
+As noted in the previous subsection, it is uncertain
+whether DR Tau is gravitationally unstable. The possi-
+bility that DR Tau’s arm arises from gravitational in-
+stabilities remains intriguing given that clumpy arms
+are a hallmark of simulations of fragmenting disks (e.g.,
+Zhu et al. 2012; Basu & Vorobyov 2012). Furthermore,
+migration of clumps onto stars has been proposed as
+a trigger for FUor outbursts (e.g., Boley et al. 2010).
+Clump migration might likewise explain DR Tau’s ex-
+treme brightening event in the 1970s. If DR Tau’s disk
+mass has been estimated correctly, then the presence
+of a clump along DR Tau’s arm raises the question of
+whether fragmentation can occur under less stringent
+conditions than models demand.
+Close stellar encounters can also generate large-scale
+arm-like structures with pitch angles comparable to that
+observed for the DR Tau molecular arm (e.g., Dai et al.
+2015; Cuello et al. 2019, 2020). However, Shuai et al.
+(2022) inferred from an analysis of Gaia EDR3 data
+(Gaia Collaboration et al. 2021) that the closest ex-
+pected approach between DR Tau and a neighboring
+star in the past 10,000 years is ∼ 105 au, which would
+be too distant to meaningfully perturb the known cir-
+cumstellar environment of DR Tau. Mesa et al. (2022)
+found that DQ Tau may have passed within 5100 au of
+
+Molecular Mapping of DR Tau
+15
+DR Tau 0.23 Myr ago, but considered such an encounter
+unlikely to be responsible for DR Tau’s spiral arms be-
+cause flyby-induced arms are only expected to survive on
+timescales of several thousand years (e.g., Cuello et al.
+2022).
+4.4. Connections to EXor and FUor phenomena
+In recent years, the circumstellar environments of a
+number of FUors and EXors have been spatially resolved
+with millimeter interferometry and high-contrast scat-
+tered light imaging.
+FUors are often associated with
+envelopes, outflows, and arm-like structures (e.g., Liu
+et al. 2016; Zurlo et al. 2017; Ru´ız-Rodr´ıguez et al. 2017;
+K´osp´al et al. 2017), similar to the structures associated
+with DR Tau.
+In FUor systems, the envelopes sup-
+ply infalling material that may help to activate gravita-
+tional instabilities (and then possibly magnetorotational
+instabilities), the arms may form as a consequence of
+gravitational instabilities, and instabilities may trigger
+outbursts that subsequently drive outflows (e.g., Evans
+et al. 1994; Vorobyov & Basu 2005; Zhu et al. 2010).
+With the exception of EX Lup and V1647 Ori, the latter
+of which is sometimes considered to be an FUor source,
+the EXor sources imaged so far have generally lacked
+analogous features (e.g., Principe et al. 2018; Hales et al.
+2018; Cieza et al. 2018; Hales et al. 2020). However, DR
+Tau exhibits striking similarities to EX Lup in that they
+both feature outflows, non-Keplerian spiral-like struc-
+tures, and (remnant) envelopes. The circumstellar en-
+vironments of DR Tau and EX Lup also share similar-
+ities with that of RU Lup, which is not classified as
+an EXor source but is nevertheless an exceptionally ac-
+tive T Tauri star (Joy 1945; Gahm et al. 1974; Huang
+et al. 2020). Hales et al. (2018) suggested that the pres-
+ence of complex structures associated with EX Lup but
+not other EXors is an indication that EX Lup occupies
+an intermediate evolutionary stage between FUors and
+most EXors. The same may hold true for DR Tau (and
+perhaps RU Lup). Alternatively, the differences in EXor
+circumstellar environments may indicate that EXors are
+a heterogeneous group of objects, only some of which are
+closely related to the FUor phenomenon. In any case,
+the observations of EX Lup, DR Tau, and RU Lup moti-
+vate more spatially resolved imaging of extremely active
+T Tauri stars to elucidate the connection between cir-
+cumstellar environments and stellar properties.
+4.5. A changing view of Class II disks
+In the past decade, the introduction of high angular
+resolution imaging at millimeter wavelengths has trans-
+formed our understanding of planet formation by show-
+ing that dust substructures on scales of several au are
+common (e.g., ALMA Partnership et al. 2015; Andrews
+et al. 2018). Meanwhile, sensitive molecular imaging at
+more modest resolution has highlighted a deficit in our
+understanding of disk environments on scales of tens to
+thousands of au. With single-dish telescopes and ear-
+lier generations of interferometers, signs of large-scale
+non-Keplerian emission towards Class II disks were of-
+ten ascribed to foreground contamination (e.g., Thi et al.
+2001; Hughes et al. 2009; ¨Oberg et al. 2011b). Even in
+the era of more powerful millimeter interferometers, in-
+sufficient integration times or insufficient uv coverage
+at larger spatial scales can lead to key structures being
+missed.
+High-quality molecular mapping, though, has demon-
+strated that there are indeed large-scale tails, spirals,
+streams, and/or remnant envelopes associated with a
+number of Class II systems (e.g., Akiyama et al. 2019;
+Huang et al. 2021; Paneque-Carre˜no et al. 2021; Huang
+et al. 2022).
+Scattered light imaging has also played
+an important role in uncovering examples of Class II
+disks that appear to be interacting with surrounding
+material (e.g., Grady 2004; Garufi et al. 2018; Ginski
+et al. 2021), although as demonstrated by the examples
+of DR Tau from this work and RU Lup from Huang et al.
+(2020), molecular observations can reveal structures far
+beyond the detected extent of scattered light features.
+Infall from these larger-scale structures is increasingly
+being invoked to explain certain disk structures observed
+at smaller scales, such as misalignments or spiral arms
+(e.g., Ginski et al. 2021; Paneque-Carre˜no et al. 2021;
+Mesa et al. 2022). These observations thus imply an in-
+triguing link between dynamical processes operating on
+disparate size scales.
+For individual systems, though,
+infall is only one of several possible explanations for the
+observed disk phenomena. More systematic molecular
+line observations will be key for establishing patterns of
+association between large- and small-scale properties.
+5. SUMMARY
+We present new NOEMA observations of 12CO, 13CO,
+C18O, SO, DCO+, and H2CO toward the T Tauri star
+DR Tau, representing the highest-quality millimeter line
+observations of this source to date. Our findings are as
+follows:
+1. CO emission shows that the DR Tau protoplane-
+tary disk is associated with an envelope, a faint
+asymmetric outflow, and a large non-Keplerian
+spiral arm with a clump.
+2. The molecular spiral arm resembles a scaled-up
+version of the spiral arms detected in scattered
+light, although the angular resolution of NOEMA
+is not sufficient to determine whether the molecu-
+lar arm is an extension of one of the scattered light
+arms or a separate feature. Whereas the scattered
+light arms are only detected up to ∼ 220 au in
+projection from DR Tau, the molecular arm is de-
+tected up to ∼ 1200 au in projection from the star.
+3. We report detections of SO, DCO+, and H2CO in
+the DR Tau disk for the first time. Their kinemat-
+
+16
+Huang et al.
+ics and compact emission extent suggest that they
+primarily trace the Keplerian circumstellar disk.
+4. SO emission is stronger on the northern, redshifted
+side of the disk. This asymmetry might be linked
+to infall from an asymmetric envelope or to un-
+resolved spiral substructure associated with the
+arms detected in scattered light. Higher angular
+resolution observations of SO will be needed to
+clarify the origins of the asymmetry.
+DR Tau’s envelope, outflow, and arm are reminiscent
+of the structures that have been observed in association
+with various FUor sources as well as the EXor source EX
+Lup. Given that FUor and EXor outbursts have been
+linked to instabilities driven by envelope accretion, a
+similar mechanism may account for DR Tau’s dramatic
+stellar brightness changes. The NOEMA observations
+of DR Tau highlight the utility of sensitive, spatially re-
+solved molecular line observations for providing context
+about the conditions under which young stars and their
+protoplanetary disks evolve.
+This work is based on observations carried out un-
+der project number W20BE with the IRAM NOEMA
+Interferometer.
+IRAM is supported by INSU/CNRS
+(France), MPG (Germany) and IGN (Spain). This work
+is also based on observations collected at the Euro-
+pean Southern Observatory under ESO programme(s)
+0102.C-0453(A). We thank our NOEMA local contact,
+Ana Lopez-Sepulcre, for setting up the observing scripts
+and assisting with data reduction. We also thank Arthur
+Bosman, Ke Zhang, Joel Bregman, Lee Hartmann,
+Merel van’t Hoff, Ardjan Sturm, Melissa McClure, and
+Ewine van Dishoeck for helpful discussions. We thank
+the referee, Ruobing Dong, for helpful comments im-
+proving the clarity of the manuscript. Support for J.
+H. was provided by NASA through the NASA Hubble
+Fellowship grant #HST-HF2-51460.001-A awarded by
+the Space Telescope Science Institute, which is operated
+by the Association of Universities for Research in As-
+tronomy, Inc., for NASA, under contract NAS5-26555.
+This project has received funding from the European
+Research Council (ERC) under the European Union’s
+Horizon 2020 research and innovation programme (grant
+agreement No. 101002188).
+Facilities: NOEMA
+Software: analysisUtils (https://casaguides.nrao.
+edu/index.php/Analysis Utilities),
+AstroPy
+(Astropy
+Collaboration et al. 2013), CASA (CASA Team et al.
+2022), cmasher (van der Velden 2020), emcee (Foreman-
+Mackey
+et
+al.
+2013),
+GILDAS
+(Pety
+2005;
+Gildas
+Team 2013), matplotlib (Hunter 2007), pandas (pan-
+das development team 2022; Wes McKinney 2010),
+scikit-image (van der Walt et al. 2014), SciPy (Vir-
+tanen et al. 2020)
+APPENDIX
+A. SPECTROSCOPIC PARAMETERS OF TARGETED LINES
+The spectroscopic parameters of the targeted lines, taken from the Cologne Database for Molecular Spectroscopy
+(M¨uller et al. 2001, 2005) via Splatalogue1, are listed in Table 2. Primary line targets are marked in bold.
+1 https://splatalogue.online//
+
+Molecular Mapping of DR Tau
+17
+Table 2.
+Spectroscopic Parameters of All Targeted
+Lines
+Transition
+Rest frequency
+Eu
+(GHz)
+(K)
+13C17O J = 2 − 1
+214.5738730
+15.4
+SO JN = 55 − 44
+215.2206530
+44.1
+DCO+ J = 3 − 2
+216.1125822
+20.7
+H2S JKaKc = 220 − 211
+216.7104365
+84.0
+c-C3H2 JKaKc = 330 − 221
+216.2787560
+19.5
+SiO J = 5 − 4
+217.1049190
+31.3
+DCN J = 3 − 2
+217.2385378
+20.9
+c-C3H2 JKaKc = 514 − 423
+217.9400460
+35.4
+H2CO JKaKc = 303 − 202
+218.2221920
+21.0
+HC3N J = 24 − 23
+218.3247230
+131.0
+H2CO JKaKc = 322 − 221
+218.4756320
+68.1
+H2CO JKaKc = 321 − 220
+218.7600660
+68.1
+C18O J = 2 − 1
+219.5603541
+15.8
+SO JN = 65 − 54
+219.9494420
+35.0
+13CO J = 2 − 1
+220.3986842
+15.9
+12CO J = 2 − 1
+230.5380000
+16.6
+OCS J = 19 − 18
+231.0609934
+110.9
+N2D+ J = 3 − 2
+231.3218283
+22.2
+13CS J = 5 − 4
+231.2206852
+33.3
+C2S JN = 1918 − 1817
+233.9384580
+109.6
+PN J = 5 − 4
+234.9356940
+33.8
+HC3N J = 26 − 25
+236.5127888
+153.2
+H2CS JKaKc = 717 − 616
+236.7270204
+58.6
+B. CHANNEL MAPS
+Channel maps of the primary line targets (listed in
+Table 1) are presented in Figure 10.1.
+C. AUXILIARY LINE TARGETS
+Table 3 lists the beam sizes, per-channel rms of the
+image cubes, and 3σ flux upper limits for the auxiliary
+line targets. The flux upper limits were estimated as-
+suming the same velocity range and aperture used to
+measure the C18O flux (see Table 1). Flux upper lim-
+its may be underestimated if the molecule is primarily
+present in the envelope or outflow rather than the disk.
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+Molecular Mapping of DR Tau
+23
+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
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+ [′′]
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+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
+2
+0
+2
+ [′′]
+10.0
+200 au
+10.2
+10.4
+10.6
+10.8
+0
+15
+30
+45
+60
+SO JN = 65
+54 intensity (mJy beam
+1)
+Figure 10.4. Channel maps of SO JN = 65 − 54 toward DR Tau. Contours are drawn in pink at the 3, 5, 10σ levels.
+
+24
+Huang et al.
+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
+2
+0
+2
+ [′′]
+10.0
+200 au
+10.2
+10.4
+10.6
+10.8
+0
+15
+30
+45
+SO JN = 55
+44 intensity (mJy beam
+1)
+Figure 10.5. Channel maps of SO JN = 55 − 44 toward DR Tau. Contours are drawn in pink at the 3, 5σ levels.
+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
+2
+0
+2
+ [′′]
+10.0
+200 au
+10.2
+10.4
+10.6
+10.8
+0
+9
+18
+27
+36
+DCO +  J = 3
+2 intensity (mJy beam
+1)
+Figure 10.6. Channel maps of DCO+ J = 3 − 2 toward DR Tau. Contours are drawn in pink at the 3, 5σ levels.
+
+Molecular Mapping of DR Tau
+25
+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
+2
+0
+2
+ [′′]
+10.0
+200 au
+10.2
+10.4
+10.6
+10.8
+0
+25
+50
+75
+100
+H2CO JKaKc = 303
+202 intensity (mJy beam
+1)
+Figure 10.7. Channel maps of H2CO JKaKc = 303 −202 toward DR Tau. Contours are drawn in pink at the 3, 5, 10, 15σ levels.
+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
+2
+0
+2
+ [′′]
+10.0
+200 au
+10.2
+10.4
+10.6
+10.8
+0
+10
+20
+30
+H2CO JKaKc = 322
+221 intensity (mJy beam
+1)
+Figure 10.8. Channel maps of H2CO JKaKc = 322 − 221 toward DR Tau. Contours are drawn in pink at the 3, 4σ levels.
+
+26
+Huang et al.
+9.0
+9.2
+9.4
+9.6
+9.8
+2
+0
+2
+ [′′]
+2
+0
+2
+ [′′]
+10.0
+200 au
+10.2
+10.4
+10.6
+10.8
+0
+5
+10
+15
+20
+25
+H2CO JKaKc = 321
+220 intensity (mJy beam
+1)
+Figure 10.9. Channel maps of H2CO JKaKc = 321 − 220 toward DR Tau. Contours are drawn in pink at the 3σ level.
+Table 3. Imaging Summary for Auxiliary Line Targets
+Transition
+Synthesized beam
+Per-channel RMS noisea
+3σ Flux Upper Limit
+(arcsec × arcsec (◦))
+(mJy beam−1)
+(mJy km s−1)
+13C17O J = 2 − 1
+1.21 × 0.93 (18.0◦)
+8
+< 40
+H2S JKaKc = 220 − 211
+1.21 × 0.93 (16.6◦)
+7
+< 40
+c-C3H2 JKaKc = 330 − 221
+1.21 × 0.93 (16.7◦)
+7
+< 30
+SiO J = 5 − 4
+1.21 × 0.93 (16.4◦)
+7
+< 30
+DCN J = 3 − 2
+1.21 × 0.93 (16.5◦)
+7
+< 40
+c-C3H2 JKaKc = 514 − 423
+1.21 × 0.93 (16.7◦)
+7
+< 30
+HC3N J = 24 − 23
+1.20 × 0.94 (17.0◦)
+7
+< 30
+OCS J = 19 − 18
+1.18 × 0.91 (17.3◦)
+7
+< 30
+N2D+ J = 3 − 2
+1.18 × 0.91 (17.3◦)
+10
+< 70
+13CS J = 5 − 4
+1.18 × 0.91 (17.3◦)
+8
+< 50
+C2S JN = 1918 − 1817
+1.17 × 0.90 (16.6◦)
+10
+< 50
+PN J = 5 − 4
+1.17 × 0.90 (16.8◦)
+9
+< 40
+HC3N J = 26 − 25
+1.17 × 0.90 (16.7◦)
+10
+< 50
+H2CS JKaKc = 717 − 616
+1.17 × 0.90 (16.6◦)
+11
+< 80
+aFor channel widths of 0.2 km s−1.
+
+Molecular Mapping of DR Tau
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+

KtE3T4oBgHgl3EQfvgte/content/tmp_files/2301.04694v1.pdf.txt

@@ -0,0 +1,361 @@
+ 
+1 
+ 
+ 
+Science Priorities for the Extraction of the Solid MSR Samples  
+from their Sample Tubes 
+ 
+ 
+ 
+ 
+NASA-ESA Mars Rock Team 
+Nicolas Dauphas, Sara S. Russell, David Beaty, Fiona Thiessen, 
+Jessica Barnes, Lydie Bonal, John Bridges, Thomas Bristow, John Eiler, Ludovic Ferrière, Teresa 
+Fornaro, Jérôme Gattacceca, Beda Hoffman, Emmanuelle J. Javaux, Thorsten Kleine, Harry Y. 
+McSween, Manika Prasad, Liz Rampe, Mariek Schmidt, Blair Schoene, Kirsten L. Siebach, Jennifer 
+Stern, Nicolas Tosca.  
+ 
+ 
+ 
+ 
+ 
+Requestor: NASA-ESA MCSG1 team 
+Date: January 11, 2023 
+ 
+ 
+ 
+ 
+Citation to this report: 
+Dauphas N., Russell S.S., Beaty D., Thiessen F., Barnes J., Bonal L., Bridges J., Bristow T., Eiler J., 
+Ferrière L., Fornaro T., Gattacceca J., Hoffman B., Javaux E.J., Kleine T., McSween H.Y., Prasad M., 
+Rampe L., Schmidt M., Schoene B., Siebach K.L., Stern J., Tosca N. (2023) Science priorities for the 
+extraction of the solid MSR samples from their sample tubes.  NASA-ESA Mars Rock Team Report 1 
+ 
+ 
+
+ 
+2 
+Background: The NASA-ESA Mars Rock Team is an outgrowth of the MCSG1 team. It is composed of 
+scientists with expertise in handling and analyses of both terrestrial and extraterrestrial samples, rock 
+physics, and contamination mitigation. Two online meetings were organized in the Fall of 2022 where 
+Oscar Rendon Perez (JPL) and Paulo Younse (JPL) described the engineering options for opening the 
+tubes that will contain the samples returned from Mars' Jezero crater. This prompted discussions 
+between the Rock Team members (during online meetings and through emails). The Rock Team 
+leadership met online with the team focused on gas analysis (Gas Team) to understand their 
+constraints and make sure that the solutions envisioned for headspace gas extraction would not 
+compromise solid core retrieval. This report summarizes the consensus view of the Rock Team. It was 
+written by the Rock Team leadership with input from all team members.  
+Summary:  Preservation of the chemical and structural integrity of samples that will be brought back 
+from Mars is paramount to achieving the scientific objectives of MSR. Given our knowledge of the 
+nature of the samples retrieved at Jezero by Perseverance, at least two options need to be tested for 
+opening the sample tubes: (1) One or two radial cuts at the end of the tube to slide the sample out. 
+(2) Two radial cuts at the ends of the tube and two longitudinal cuts to lift the upper half of the tube 
+and access the sample. Strategy 1 will likely minimize contamination but incurs the risk of affecting 
+the physical integrity of weakly consolidated samples. Strategy 2 will be optimal for preserving the 
+physical integrity of the samples but increases the risk of contamination and mishandling of the sample 
+as more manipulations and additional equipment will be needed. A flexible approach to opening the 
+sample tubes is therefore required, and several options need to be available, depending on the nature 
+of the rock samples returned. Both opening strategies 1 and 2 may need to be available when the 
+samples are returned to handle different sample types (e.g., loosely bound sediments vs. indurated 
+magmatic rocks). This question should be revisited after engineering tests are performed on analogue 
+samples. The MSR sample tubes will have to be opened under stringent BSL4 conditions and this 
+aspect needs to be integrated into the planning. 
+Introduction:  NASA-ESA are planning to collect and transport from Mars to Earth a set of samples of 
+martian materials for the purpose of scientific investigation (Kminek et al. 2022).  The samples are 
+currently collected by the Perseverance Rover (Farley and Stack, 2022) and consist of rocks, regolith, 
+and at least one dedicated sample of atmospheric gas. In addition, for the rock and regolith samples, 
+the process of sealing the sample tubes at the martian surface will result in the volume above the solid 
+samples (referred to as the head space) being occupied by martian atmospheric gas.  The samples will 
+be contained within titanium sample tubes, which will be sealed at the martian surface with a 
+compression-style cap.  
+The rocks sampled thus far by the Perseverance Rover comprise magmatic rocks like basalt and olivine 
+cumulates that experienced various degrees of secondary water alteration, water-laid detrital 
+sedimentary rocks that show various levels of induration, and unconsolidated Mars regolith that could 
+contain grains from afar transported to the Jezero crater. Two main considerations weigh on the 
+strategy that should be adopted for opening the samples: 
+(1) Important information is contained in the vertical successions and textural characteristics of layers 
+in sediments, which can provide important clues for interpreting the depositional setting (Fig. 1). For 
+example, in terrestrial lakes, vertical gradation in grain size can reflect the relative density of 
+depositional and lacustrine fluids or gradations in organic matter content can reflect seasonal changes 
+in biological productivity. Fine laminations can sometimes reflect the presence of microbial mats. The 
+method used for opening the tubes must imperatively preserve those fine structures. 
+
+ 
+3 
+ 
+Fig. 1. Examples of possible fine-scale laminations in terrestrial environments (left; seasonal varves from Lake 
+Belau, Northern Germany; Dörfler et al. 2012; right Microbially-Induced Sedimentary Structures-MISS in the 
+middle neoproterozoic Chuar Group, Grand Canyon, Arizona; Bohacs and Junium 2007). 
+(2) Some critical measurements are sensitive to contamination either from the tube, the apparatus 
+used for cutting the tubes, or surrounding contaminants present in the isolator. Organic matter is of 
+particular concern given the high stakes involved in any claim for the presence of any form of biotic or 
+prebiotic chemistry on Mars. Inorganic trace element isotopes may provide dates on when Mars was 
+habitable, and these are also prone to contamination. 
+Beginning in 2022, an engineering team was tasked with developing the processes needed to open 
+the sample tubes and to extract the solid and gaseous samples.  The engineering team was asked to 
+develop engineering priorities associated with this process.  Two science teams were asked to develop 
+parallel science priorities:  A group we call the “Gas Team” evaluated the priorities related to the 
+science associated with all returned gaseous sample, and a second group called the “Rock Team” (the 
+authors of this report) evaluated the priorities associated with solid materials contained within the 
+sample tubes.  Both the "Gas Team" and "Rock Team" work under the oversight of a third committee, 
+the Mars Campaign Science Group (MCSG1). 
+The solid samples returned from the martian surface are certain to include sedimentary rocks (most 
+important for the search for biosignatures), igneous rocks, and regolith, and they may also include 
+other kinds of rocks, such as hydrothermal rocks, or impact breccia.  The samples will be the basis for 
+answering the main scientific questions of Mars Sample Return (iMOST, 2018). 
+The rock samples at Mars will all have been collected from various outcrops (or perhaps very large 
+blocks of coherent rock).  However, at least some of the rocks are relatively weak (i.e. a low 
+compressive strength), and are vulnerable to fracturing during drilling and during several dynamic 
+events associated with spacecraft operations during the return phase (most importantly, at Earth 
+landing).  It is anticipated that the mechanical state of each sample, as received in the laboratory on 
+Earth, will be assessed by a method like computer tomography (CT) scanning prior to opening.  The 
+decision on how to open each sample tube can therefore be based on geological data from the field 
+(collected by the M2020 science team), tests done on analogue samples, as well as the penetrative 
+imaging data obtained on Earth. 
+The engineering team has proposed a 2-phase process for opening the sample tubes:  First, puncture 
+the tube in a way that will allow any gas present to be extracted and captured, then second, cut the 
+metal of the tube in a way that would allow the solid materials to be removed.  Regarding cutting the 
+metal of the tubes, three primary mechanisms have been proposed (Fig. 2): 
+• 
+A single radial cut to the end of the tube, so that the sample could be tipped out. 
+• 
+A radial cut at each end of the tube, which would enable the sample to be pushed out from 
+one end 
+
+9
+belowtopof core segment
+10
+12
+(cm)
+13
+14
+151cm 
+4 
+• 
+Two radial cuts and two longitudinal cuts, to reveal the whole sample during cutting. 
+An option frequently used on Earth to access core samples, for example used with deep sea drill cores, 
+is to cut the core tube and the core together with something like a band saw.  This is not an option for 
+samples returned from Mars as this would have the effect of driving contamination from both the 
+metallic core tube and band saw into the interior of the rock core. 
+ 
+ 
+ 
+Figure 2. Proposed protocols for opening the sample tubes. Drawings courtesy of Oscar Rendon Perez. In the one 
+radial cut approach, a sharp hard metal wheel shears through the tube by slowly rotating and tightening it 
+around the tube (bottom panel; left). The sample is extracted from the tube by inclining it and controlling the 
+rate of descent with a piston. The second approach involves doing a second cut to push the sample outwards. A 
+virtue of this approach is that it allows for a more controlled extraction, and it minimizes the risk of the sample 
+getting jammed in the tube. Both options 1 and 2 involve the sample sliding out of the tube and incur the risk of 
+losing the chemical and structural layering of the sample. The third approach involves doing two longitudinal 
+cuts on the side of the tube to expose the whole sample within the tube. It is least likely to disturb the physical 
+integrity of the sample, which stays in place in the tube, but it involves cutting the tube along its length through 
+a white alumina coating (deposited on the tubes to reduce their heat absorption while seating on Mars' surface) 
+possibly using a circular blade (bottom panel; right). The chance of contamination is higher with this third 
+approach as more tube manipulations are involved, more tube material is cut, and the setup to remove or cut 
+the alumina coating will be more involved than the wheel cutter used in approaches 1 and 2. 
+Approach: 
+The issue of how to open the tubes was discussed by the team over two telecons. Presentations by 
+engineers Oscar Rendon Perez and Paolo Younse were delivered to explain the design of the tubes 
+and different options for opening them (Fig. 2).  
+The Rock Sample Team concluded there are three main considerations: 
+• 
+Need to minimise (and have knowledge of) contamination 
+• 
+Need to preserve stratigraphy and other textural relationships  
+• 
+Need to maximise the amount of sample material that ends up in a scientifically useful state 
+from the tubes. For some samples like the detrital sediments or the regolith sample, each 
+
+BUEHLER
+DIAMOND
+WAFERING BID
+BUEHILER 
+5 
+small grain may provide a unique record of Mars' surface history, so dust adhering to the tube 
+surface should be recovered to the greatest extent possible. However, such dust will likely 
+represent a small fraction of the total mass and its retrieval could be done later. Or it could be 
+used for quickly surveying the petrography and mineralogy of the core as part of a preliminary 
+examination phase as this material will be of lesser value for other tasks and could be sterilized.  
+Minimal cutting (i.e., a single radial cut) was considered optimal to minimise potential contamination 
+of trace elements, especially metals, and organic material from the tubes and cutting tools. The 
+structural integrity of the sample would, however, be best preserved with radial and longitudinal cuts; 
+this is considered especially important for sedimentary rocks that may be friable but contain internal 
+stratigraphic structures. The yield may be maximised by at least two radial cuts. These considerations 
+may conflict with each other and the approach to be used will depend on the exact nature of each 
+returned sample. Magnetic contamination should also be minimized during cutting operation and 
+sample handling. 
+The preferred opening strategies are summarized in Table 1, which ponders each criterion (structure 
+integrity, chemical integrity, and yield) for three categories of samples (consolidated rocks, friable 
+rocks, and loose regolith). We summarize the Rock Team recommendations at the bottom of each 
+column. The rationale for each entry is summarized below: 
+Consolidated rocks (example microgabbro). To minimize the risk of contamination, one radial cut is 
+preferred as cutting by shearing with a hard metal solid wheel will generate little dust, cause little 
+heating, involve no use of fluid, and involve the least amount of tube material of all considered options. 
+To get the sample out of the tube, putting it on a vertical incline and lowering the sample in a 
+controlled manner with a piston would preserve the structural integrity of the sample. One radial cut 
+is likely to preserve the structural integrity of the sample. The cutting wheel will create a metal lip that 
+will protrude in the tube, so provision should be ready to straighten that lip so that the sample can be 
+extracted without rubbing against the lip. With a consolidated sample, there is however a concern 
+that jamming could occur, as a fragment might be trapped in compression between the solid core and 
+the tube wall. A second cut might be needed to push/pull the sample from the other side and free it 
+from such entrapment. Fine dust adhering to the inner tube surface might be difficult to retrieve with 
+a single radial cut. A second radial cut would allow one to get the fine dust out by pushing it out with 
+an appropriate instrument. The Rock Team favours 1 radial cut, with 2 radial cuts possibly needed for 
+sample retrieval in case of jamming and to recover fine dust adhering to the interior tube surface. 
+Friable rocks (example detrital sediments). These rocks are the ones for which preserving the 
+stratigraphy is of upmost importance. The rationale is the same as with consolidated rocks that a single 
+radial cut would be preferred from the point of avoiding contamination. To extract the sample, a single 
+radial cut might be sufficient as the less consolidated nature of those rocks means that they are less 
+likely to be hard jammed in the tube. A possible approach would be to put place a piston against the 
+sample on the opening side with the tube horizontal. The sample tube and piston would then be 
+rotated to a vertical position, and the piston would be lowered in a controlled manner to transfer the 
+sample core in a transparent sample holder (quartz or sapphire) with predesigned longitudinal 
+openings. The reason to transfer the sample vertically is to minimize shear on the tube surface. After 
+vertical transfer of the sample from the tube to the holder, the holder would be rotated back to 
+horizontal to be then opened, giving access to the sample. 
+Alternatively, it might be possible to 2 radial cuts, and one piston to push the sample out in a slightly 
+inclined orientation and another piston at the open side against the sample to prevent collapse, so 
+the sample keeps its integrity but we can avoid the longitudinal cuts to avoid more risk of 
+
+ 
+6 
+contamination. If too friable, the sample could be gently pushed this way into a transparent sample 
+holder with predesigned longitudinal openings, allowing visible inspection of the enclosed protected 
+sample 
+Letting the sample slide out from one side incurs the risk however that rock fragments will be moved 
+out of sequence, that the sample will disaggregate, and that important chemical features be smeared 
+throughout the core. The latter point could include, for instance, organic distribution. If a layer is highly 
+enriched in organics, sliding the whole sample along the sides may smear the signature throughout 
+the entire core surface. For preserving the stratigraphy, it may therefore be advantageous to make 2 
+radial cuts and 2 longitudinal cuts to access the core without disturbing it. The constraints on fine dust 
+recovery are the same as with other sample types. 
+Regolith. There is no stratigraphic information to preserve in that sample and little risk of jamming, 
+so a single radial cut is preferred as this minimizes the risk of contamination. The fine dust in the 
+sample may come from afar and each grain will likely tell a story, so complete recovery of dust 
+adhering to the tube inner surface is important. 
+Table 1. Preferred opening strategies depending on rock cohesiveness and criteria considered. 
+ 
+ 
+The Rock Sample Team finds that a single approach will not be appropriate for all the rock samples 
+returned, but instead a flexible and bespoke approach will be needed for each sample tube opening, 
+with all three of the above options available.  As a general principle, minimal cutting is favoured as 
+this will also minimise potential contamination issues. However, an overriding consideration is that 
+Consolidated rocks
+Example: microgabbro
+Friable rocks
+Example: detrital 
+sediments, igneous 
+cumulate rocks
+Regolith
+Trace element and 
+organic contamination
+1 radial cut
+1 radial cut
+1 radial cut
+Structural integrity of 
+the sample
+1 radial cut likely OK
+Maybe 2 radial cuts in 
+case of jamming
+1 radial cut or
+2 radial cuts and 2 
+longitudinal cuts
+1 radial cut
+Complete retrieval of the 
+sample (including dust)
+1 or 2 radial cuts
+1 or 2 radial cuts
+1 or 2 radial cuts
+Rock Team 
+recommendation
+1 OR 2 radial cuts
+1 radial cut OR
+2 radial cuts and 2 
+longitudinal cuts
+1 OR 2 radial cuts
+FINDING:  There is not one single approach for opening the sample tubes that will 
+work sufficiently well for all MSR rock samples.  Multiple options need to be available. 
+
+ 
+7 
+the structural integrity of the rock sample is key to understanding its petrology, and this should remain 
+intact, even if this requires more processing. 
+For regolith samples, a single radial cut followed by tipping out the grains is likely to be appropriate, 
+since this will minimise contamination and there is no need to preserve spatial relationships within 
+the tube. For well consolidated (e.g., some igneous rock) samples, a radial cut perhaps followed by a 
+second radial cut may be required to extract the sample completely. For sedimentary rocks, and any 
+friable igneous rocks, the decision is more complex because a longitudinal cut may be necessary to 
+observe and preserve structural relationships, but this must be weighed against potentially 
+contributing more contamination. One possible solution to test for sedimentary samples could be to 
+make one or two radial cuts, then push the sample or let it slide down while keeping its stratigraphy 
+in place (possibly with high inclination to minimize shear along tube surface, with a sliding stopper 
+against the sample to control the sliding rate) into another tube with a closed longitudinal aperture 
+that allows longitudinal opening later. 
+The physical state of each core (consolidated or friable) will not be known for certain until the samples 
+are bought back to Earth, where CT-scanning will reveal the fine structure of the samples and guide 
+the strategy that adopted for tube opening.   
+Future Work: 
+The team suggests areas which require more work prior to sample return. These include:  
+• 
+Investigate how/whether analogue sedimentary samples and aqueously altered cumulate 
+rocks can be removed in a manner that preserves their structural integrity with only one radial 
+cut. 
+• 
+Investigate ways to efficiently remove the fines left behind after core extraction. 
+• 
+Impurities in all tube materials, coatings, and opening contraption (e.g., materials used in the 
+saw) must be characterized with appropriate techniques (e.g., ICP-MS). We suggest that a task 
+group be established to undertake an in-depth contaminant characterization campaign. 
+• 
+Investigate if it is possible to remove the alumina coating without compromising the sample, 
+and without causing damage (e.g. by vibration) to the martian sample inside the core tube. 
+• 
+Investigate the degree to which the different cutting protocols can introduce contamination.  
+• 
+Integrate these studies with CT and related scanning techniques. 
+• 
+Investigate how the cutting and related techniques can be performed in a Biological Hazard 
+Level BSL4 environment. 
+ 
+A concept that is not discussed in this report, but that has been considered elsewhere, is that the 
+opportunity exists to do penetrative imaging/mineralogical characterization of the sample-bearing 
+Mars sample tubes once they make it to Earth, so that we can obtain data on the mechanical state of 
+each sample as received prior to tube opening.  This eliminates the need to make guesses based on 
+pre-sampling field data, or accelerations measured by the return spacecraft, etc.  That imaging data 
+will give us the opportunity to help make decisions on how to open each tube.  We know that for 
+samples with different kinds of mechanical integrity, different tube-opening strategies may be 
+required to avoid the risk of damage that unnecessarily affects the scientific usefulness of the sample. 
+A component of the technology program is needed to develop the datasets for what happens when 
+tubes containing samples with different degrees of mechanical integrity are opened by each of the 
+three methods described.  This will become the basis for future decision-making.  We also need data 
+
+ 
+8 
+on the real contamination implications of making the horizontal cuts, and what kind of science is 
+affected by such contamination. 
+ 
+References. 
+iMOST (2018), The Potential Science and Engineering Value of Samples Delivered to Earth by Mars 
+Sample Return, (co-chairs D. W. Beaty, M. M. Grady, H. Y. McSween, E. Sefton-Nash; documentarian 
+B.L. Carrier; plus 66 co-authors), 186 p. white paper.  Posted August, 2018 by MEPAG at 
+https://mepag.jpl.nasa.gov/reports.cfm. 
+Bohacs K.M., Junium (2007) Microbial mat sedimentary structures and their relation to organic-carbon 
+burial in the Middle Neoproterozoic Chuar Group, Grand Canyon, Arizona, USA Microbial-Induced 
+Sedimentary Structures-MISS in the middle neoproterozoic Chuar Group, Grand Canyon, Arizona. In 
+Atltas of microbial mat features within the clastic rock record, Schieber J. et al. (Eds). Elsevier 208-213. 
+Dörfler W et al. (2012) A high-quality annually laminated sequence from Lake Belau, Northern 
+Germany: revised chronology and its implications for palynological and tephrochronological studies. 
+The Holocene 22, 1413-1426. 
+Farley KA, Stack K. (2022) Mars 2020 Initial Reports, Crater Floor Campaign, August 11, 2022.  
+Kminek G., Meyer M.A., Beaty D.W., Carrier B.L., Haltigin T., Hays L.E. (2022) Mars Sample Return 
+(MSR): 
+Planning 
+for 
+Returned 
+Sample 
+Science. 
+Astrobiology.Jun 
+2022.S-1-S-
+4.http://doi.org/10.1089/ast.2021.0198 
+ 
+

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+page_content=', Bonal L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=', McSween H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=', Rampe L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', Schmidt M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', Schoene B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', Siebach K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', Stern J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', Tosca N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' (2023) Science priorities for the extraction of the solid MSR samples from their sample tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' NASA-ESA Mars Rock Team Report 1  2 Background: The NASA-ESA Mars Rock Team is an outgrowth of the MCSG1 team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' It is composed of scientists with expertise in handling and analyses of both terrestrial and extraterrestrial samples, rock physics, and contamination mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=" Two online meetings were organized in the Fall of 2022 where Oscar Rendon Perez (JPL) and Paulo Younse (JPL) described the engineering options for opening the tubes that will contain the samples returned from Mars' Jezero crater." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' This prompted discussions between the Rock Team members (during online meetings and through emails).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The Rock Team leadership met online with the team focused on gas analysis (Gas Team) to understand their constraints and make sure that the solutions envisioned for headspace gas extraction would not compromise solid core retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' This report summarizes the consensus view of the Rock Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' It was written by the Rock Team leadership with input from all team members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Summary:  Preservation of the chemical and structural integrity of samples that will be brought back from Mars is paramount to achieving the scientific objectives of MSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Given our knowledge of the nature of the samples retrieved at Jezero by Perseverance, at least two options need to be tested for opening the sample tubes: (1) One or two radial cuts at the end of the tube to slide the sample out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' (2) Two radial cuts at the ends of the tube and two longitudinal cuts to lift the upper half of the tube and access the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  Strategy 1 will likely minimize contamination but incurs the risk of affecting the physical integrity of weakly consolidated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Strategy 2 will be optimal for preserving the physical integrity of the samples but increases the risk of contamination and mishandling of the sample as more manipulations and additional equipment will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' A flexible approach to opening the sample tubes is therefore required, and several options need to be available, depending on the nature of the rock samples returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Both opening strategies 1 and 2 may need to be available when the samples are returned to handle different sample types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', loosely bound sediments vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' indurated magmatic rocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' This question should be revisited after engineering tests are performed on analogue samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The MSR sample tubes will have to be opened under stringent BSL4 conditions and this aspect needs to be integrated into the planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Introduction:  NASA-ESA are planning to collect and transport from Mars to Earth a set of samples of martian materials for the purpose of scientific investigation (Kminek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The samples are currently collected by the Perseverance Rover (Farley and Stack, 2022) and consist of rocks, regolith, and at least one dedicated sample of atmospheric gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  In addition, for the rock and regolith samples, the process of sealing the sample tubes at the martian surface will result in the volume above the solid samples (referred to as the head space) being occupied by martian atmospheric gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The samples will be contained within titanium sample tubes, which will be sealed at the martian surface with a compression-style cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The rocks sampled thus far by the Perseverance Rover comprise magmatic rocks like basalt and olivine cumulates that experienced various degrees of secondary water alteration, water-laid detrital sedimentary rocks that show various levels of induration, and unconsolidated Mars regolith that could contain grains from afar transported to the Jezero crater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Two main considerations weigh on the strategy that should be adopted for opening the samples: (1) Important information is contained in the vertical successions and textural characteristics of layers in sediments, which can provide important clues for interpreting the depositional setting (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' For example, in terrestrial lakes, vertical gradation in grain size can reflect the relative density of depositional and lacustrine fluids or gradations in organic matter content can reflect seasonal changes in biological productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Fine laminations can sometimes reflect the presence of microbial mats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The method used for opening the tubes must imperatively preserve those fine structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Examples of possible fine-scale laminations in terrestrial environments (left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' seasonal varves from Lake Belau, Northern Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Dörfler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' right Microbially-Induced Sedimentary Structures-MISS in the middle neoproterozoic Chuar Group, Grand Canyon, Arizona;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Bohacs and Junium 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' (2) Some critical measurements are sensitive to contamination either from the tube, the apparatus used for cutting the tubes, or surrounding contaminants present in the isolator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Organic matter is of particular concern given the high stakes involved in any claim for the presence of any form of biotic or prebiotic chemistry on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Inorganic trace element isotopes may provide dates on when Mars was habitable, and these are also prone to contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Beginning in 2022, an engineering team was tasked with developing the processes needed to open the sample tubes and to extract the solid and gaseous samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The engineering team was asked to develop engineering priorities associated with this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Two science teams were asked to develop parallel science priorities:  A group we call the “Gas Team” evaluated the priorities related to the science associated with all returned gaseous sample, and a second group called the “Rock Team” (the authors of this report) evaluated the priorities associated with solid materials contained within the sample tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Both the "Gas Team" and "Rock Team" work under the oversight of a third committee, the Mars Campaign Science Group (MCSG1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  The solid samples returned from the martian surface are certain to include sedimentary rocks (most important for the search for biosignatures), igneous rocks, and regolith, and they may also include other kinds of rocks, such as hydrothermal rocks, or impact breccia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The samples will be the basis for answering the main scientific questions of Mars Sample Return (iMOST, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The rock samples at Mars will all have been collected from various outcrops (or perhaps very large blocks of coherent rock).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' However, at least some of the rocks are relatively weak (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' a low compressive strength), and are vulnerable to fracturing during drilling and during several dynamic events associated with spacecraft operations during the return phase (most importantly, at Earth landing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' It is anticipated that the mechanical state of each sample, as received in the laboratory on Earth, will be assessed by a method like computer tomography (CT) scanning prior to opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The decision on how to open each sample tube can therefore be based on geological data from the field (collected by the M2020 science team), tests done on analogue samples, as well as the penetrative imaging data obtained on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The engineering team has proposed a 2-phase process for opening the sample tubes:  First, puncture the tube in a way that will allow any gas present to be extracted and captured, then second, cut the metal of the tube in a way that would allow the solid materials to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  Regarding cutting the metal of the tubes, three primary mechanisms have been proposed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' 2): • A single radial cut to the end of the tube, so that the sample could be tipped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • A radial cut at each end of the tube, which would enable the sample to be pushed out from one end  9 belowtopof core segment 10 12 (cm) 13 14 151cm 4 • Two radial cuts and two longitudinal cuts, to reveal the whole sample during cutting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' An option frequently used on Earth to access core samples, for example used with deep sea drill cores, is to cut the core tube and the core together with something like a band saw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' This is not an option for samples returned from Mars as this would have the effect of driving contamination from both the metallic core tube and band saw into the interior of the rock core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Proposed protocols for opening the sample tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Drawings courtesy of Oscar Rendon Perez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' In the one radial cut approach, a sharp hard metal wheel shears through the tube by slowly rotating and tightening it around the tube (bottom panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The sample is extracted from the tube by inclining it and controlling the rate of descent with a piston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The second approach involves doing a second cut to push the sample outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' A virtue of this approach is that it allows for a more controlled extraction, and it minimizes the risk of the sample getting jammed in the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Both options 1 and 2 involve the sample sliding out of the tube and incur the risk of losing the chemical and structural layering of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The third approach involves doing two longitudinal cuts on the side of the tube to expose the whole sample within the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=" It is least likely to disturb the physical integrity of the sample, which stays in place in the tube, but it involves cutting the tube along its length through a white alumina coating (deposited on the tubes to reduce their heat absorption while seating on Mars' surface) possibly using a circular blade (bottom panel;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The chance of contamination is higher with this third approach as more tube manipulations are involved, more tube material is cut, and the setup to remove or cut the alumina coating will be more involved than the wheel cutter used in approaches 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  Approach: The issue of how to open the tubes was discussed by the team over two telecons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Presentations by engineers Oscar Rendon Perez and Paolo Younse were delivered to explain the design of the tubes and different options for opening them (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The Rock Sample Team concluded there are three main considerations: • Need to minimise (and have knowledge of) contamination • Need to preserve stratigraphy and other textural relationships • Need to maximise the amount of sample material that ends up in a scientifically useful state from the tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=" For some samples like the detrital sediments or the regolith sample, each  BUEHLER DIAMOND WAFERING BID BUEHILER 5 small grain may provide a unique record of Mars' surface history, so dust adhering to the tube surface should be recovered to the greatest extent possible." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' However, such dust will likely represent a small fraction of the total mass and its retrieval could be done later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Or it could be used for quickly surveying the petrography and mineralogy of the core as part of a preliminary examination phase as this material will be of lesser value for other tasks and could be sterilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Minimal cutting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', a single radial cut) was considered optimal to minimise potential contamination of trace elements, especially metals, and organic material from the tubes and cutting tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The structural integrity of the sample would, however, be best preserved with radial and longitudinal cuts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' this is considered especially important for sedimentary rocks that may be friable but contain internal stratigraphic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The yield may be maximised by at least two radial cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' These considerations may conflict with each other and the approach to be used will depend on the exact nature of each returned sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Magnetic contamination should also be minimized during cutting operation and sample handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The preferred opening strategies are summarized in Table 1, which ponders each criterion (structure integrity, chemical integrity, and yield) for three categories of samples (consolidated rocks, friable rocks, and loose regolith).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  We summarize the Rock Team recommendations at the bottom of each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The rationale for each entry is summarized below: Consolidated rocks (example microgabbro).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' To minimize the risk of contamination, one radial cut is preferred as cutting by shearing with a hard metal solid wheel will generate little dust, cause little heating, involve no use of fluid, and involve the least amount of tube material of all considered options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' To get the sample out of the tube, putting it on a vertical incline and lowering the sample in a controlled manner with a piston would preserve the structural integrity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' One radial cut is likely to preserve the structural integrity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The cutting wheel will create a metal lip that will protrude in the tube, so provision should be ready to straighten that lip so that the sample can be extracted without rubbing against the lip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' With a consolidated sample, there is however a concern that jamming could occur, as a fragment might be trapped in compression between the solid core and the tube wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' A second cut might be needed to push/pull the sample from the other side and free it from such entrapment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Fine dust adhering to the inner tube surface might be difficult to retrieve with a single radial cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' A second radial cut would allow one to get the fine dust out by pushing it out with an appropriate instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  The Rock Team favours 1 radial cut, with 2 radial cuts possibly needed for sample retrieval in case of jamming and to recover fine dust adhering to the interior tube surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Friable rocks (example detrital sediments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' These rocks are the ones for which preserving the stratigraphy is of upmost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The rationale is the same as with consolidated rocks that a single radial cut would be preferred from the point of avoiding contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' To extract the sample, a single radial cut might be sufficient as the less consolidated nature of those rocks means that they are less likely to be hard jammed in the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' A possible approach would be to put place a piston against the sample on the opening side with the tube horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The sample tube and piston would then be rotated to a vertical position, and the piston would be lowered in a controlled manner to transfer the sample core in a transparent sample holder (quartz or sapphire) with predesigned longitudinal openings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The reason to transfer the sample vertically is to minimize shear on the tube surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' After vertical transfer of the sample from the tube to the holder, the holder would be rotated back to horizontal to be then opened, giving access to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  Alternatively, it might be possible to 2 radial cuts, and one piston to push the sample out in a slightly inclined orientation and another piston at the open side against the sample to prevent collapse, so the sample keeps its integrity but we can avoid the longitudinal cuts to avoid more risk of  6 contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' If too friable, the sample could be gently pushed this way into a transparent sample holder with predesigned longitudinal openings, allowing visible inspection of the enclosed protected sample Letting the sample slide out from one side incurs the risk however that rock fragments will be moved out of sequence, that the sample will disaggregate, and that important chemical features be smeared throughout the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The latter point could include, for instance, organic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' If a layer is highly enriched in organics, sliding the whole sample along the sides may smear the signature throughout the entire core surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' For preserving the stratigraphy, it may therefore be advantageous to make 2 radial cuts and 2 longitudinal cuts to access the core without disturbing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The constraints on fine dust recovery are the same as with other sample types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Regolith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' There is no stratigraphic information to preserve in that sample and little risk of jamming, so a single radial cut is preferred as this minimizes the risk of contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The fine dust in the sample may come from afar and each grain will likely tell a story, so complete recovery of dust adhering to the tube inner surface is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Preferred opening strategies depending on rock cohesiveness and criteria considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  The Rock Sample Team finds that a single approach will not be appropriate for all the rock samples returned, but instead a flexible and bespoke approach will be needed for each sample tube opening, with all three of the above options available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' As a general principle, minimal cutting is favoured as this will also minimise potential contamination issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' an overriding consideration is that Consolidated rocks Example: microgabbro Friable rocks Example: detrital sediments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content='cuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='longitudinal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='cuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='OR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='radial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='cuts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='FINDING:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='There is not one single approach for opening the sample tubes that will work sufficiently well for all MSR rock samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Multiple options need to be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  7 the structural integrity of the rock sample is key to understanding its petrology, and this should remain intact, even if this requires more processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' For regolith samples, a single radial cut followed by tipping out the grains is likely to be appropriate, since this will minimise contamination and there is no need to preserve spatial relationships within the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' For well consolidated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', some igneous rock) samples, a radial cut perhaps followed by a second radial cut may be required to extract the sample completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' For sedimentary rocks, and any friable igneous rocks, the decision is more complex because a longitudinal cut may be necessary to observe and preserve structural relationships, but this must be weighed against potentially contributing more contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' One possible solution to test for sedimentary samples could be to make one or two radial cuts, then push the sample or let it slide down while keeping its stratigraphy in place (possibly with high inclination to minimize shear along tube surface, with a sliding stopper against the sample to control the sliding rate) into another tube with a closed longitudinal aperture that allows longitudinal opening later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The physical state of each core (consolidated or friable) will not be known for certain until the samples are bought back to Earth, where CT-scanning will reveal the fine structure of the samples and guide the strategy that adopted for tube opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  Future Work: The team suggests areas which require more work prior to sample return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' These include: • Investigate how/whether analogue sedimentary samples and aqueously altered cumulate rocks can be removed in a manner that preserves their structural integrity with only one radial cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • Investigate ways to efficiently remove the fines left behind after core extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • Impurities in all tube materials, coatings, and opening contraption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', materials used in the saw) must be characterized with appropriate techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=', ICP-MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' We suggest that a task group be established to undertake an in-depth contaminant characterization campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • Investigate if it is possible to remove the alumina coating without compromising the sample, and without causing damage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' by vibration) to the martian sample inside the core tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • Investigate the degree to which the different cutting protocols can introduce contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • Integrate these studies with CT and related scanning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' • Investigate how the cutting and related techniques can be performed in a Biological Hazard Level BSL4 environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  A concept that is not discussed in this report, but that has been considered elsewhere, is that the opportunity exists to do penetrative imaging/mineralogical characterization of the sample-bearing Mars sample tubes once they make it to Earth, so that we can obtain data on the mechanical state of each sample as received prior to tube opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' This eliminates the need to make guesses based on pre-sampling field data, or accelerations measured by the return spacecraft, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' That imaging data will give us the opportunity to help make decisions on how to open each tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' We know that for samples with different kinds of mechanical integrity, different tube-opening strategies may be required to avoid the risk of damage that unnecessarily affects the scientific usefulness of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' A component of the technology program is needed to develop the datasets for what happens when tubes containing samples with different degrees of mechanical integrity are opened by each of the three methods described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' This will become the basis for future decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' We also need data  8 on the real contamination implications of making the horizontal cuts, and what kind of science is affected by such contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='  References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' iMOST (2018), The Potential Science and Engineering Value of Samples Delivered to Earth by Mars Sample Return, (co-chairs D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=' Sefton-Nash;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' documentarian B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Carrier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' plus 66 co-authors), 186 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' white paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Posted August, 2018 by MEPAG at https://mepag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='jpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='gov/reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='cfm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Bohacs K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=', Junium (2007) Microbial mat sedimentary structures and their relation to organic-carbon burial in the Middle Neoproterozoic Chuar Group, Grand Canyon, Arizona, USA Microbial-Induced Sedimentary Structures-MISS in the middle neoproterozoic Chuar Group, Grand Canyon, Arizona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' In Atltas of microbial mat features within the clastic rock record, Schieber J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' (Eds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Elsevier 208-213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Dörfler W et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' (2012) A high-quality annually laminated sequence from Lake Belau, Northern Germany: revised chronology and its implications for palynological and tephrochronological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' The Holocene 22, 1413-1426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Farley KA, Stack K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' (2022) Mars 2020 Initial Reports, Crater Floor Campaign, August 11, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Kminek G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content=' (2022) Mars Sample Return (MSR): Planning for Returned Sample Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content=' Astrobiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
+page_content='Jun 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfvgte/content/2301.04694v1.pdf'}
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+arXiv:2301.04263v1  [math.AP]  11 Jan 2023
+Existence of solutions to fractional semilinear
+parabolic equations in Besov-Morrey spaces
+Erbol Zhanpeisov∗
+Okinawa Institute of Science and Technology
+1919-1 Tancha, Onna-son, Kunigami-gun
+Okinawa, Japan 904-0495
+Abstract
+In this paper, we establish the existence of solutions to fractional semilinear
+parabolic equations in Besov-Morrey spaces for a large class of initial data
+including distributions other than Radon measures. We also obtain sufficient
+conditions for the existence of solutions to viscous Hamilton-Jacobi equations.
+1
+Introduction and main results
+Consider a semilinear parabolic equation
+�
+∂tu + (−∆)
+θ
+2 u = |u|γ−1u,
+x ∈ RN, t ∈ (0, T),
+u(x, 0) = ϕ(x),
+x ∈ RN
+(1.1)
+and a viscous Hamilton-Jacobi equation
+�
+∂tu + (−∆)
+θ
+2u = |∇u|γ,
+x ∈ RN, t ∈ (0, T),
+u(x, 0) = ϕ(x),
+x ∈ RN,
+(1.2)
+where γ > 1, N ≥ 1, T > 0 and θ > 0 (resp. θ > 1) for problem (1.1) (resp.
+problem (1.2)). The purpose of this paper is to obtain sufficient conditions for the
+existence of solutions to the Cauchy problem (1.1) and (1.2) for a large class of
+initial data by introducing inhomogeneous Besov-Morrey spaces. This enables us to
+take distributions other than Radon measures as initial data.
+Let us consider the Cauchy problem for the semilinear parabolic equation (1.1)
+with θ > 0 and γ > 1. The solvability of problem (1.1) has been studied in many
+papers, see e.g., [3, 7, 9–17, 19–21, 23–27]. (See also the monograph [22].) Among
+2010 AMS subject classification. 35K58,35K25
+∗E-mail: [email protected]
+1
+
+others, Ishige, Kawakami, and Okabe [17] developed the arguments in [16] and ob-
+tained sufficient conditions for the existence of solutions to problem (1.1) for general
+θ > 0. As corollaries of their main results, they proved the following properties:
+(a) Let 1 < γ < 1 + θ/N. Then problem (1.1) possesses a local-in-time solution if
+sup
+x∈RN ∥ϕ∥L1(B(x,1)) < ∞;
+(b) Let γ = 1 + θ/N. Then there exists c > 0 such that, if
+|ϕ(x)| ≤ c|x|−N
+����log
+�
+e + 1
+|x|
+�����
+− N
+θ −1
+,
+x ∈ RN,
+then probolem (1.1) possesses a local-in-time solution;
+(c) Let γ > 1 + θ/N. Then there exists c > 0 such that, if
+|ϕ(x)| ≤ c|x|−
+θ
+γ−1 ,
+x ∈ RN,
+then probolem (1.1) possesses a local-in-time solution.
+In the case of either 0 < θ ≤ 2 or θ ∈ {4, 6, . . . }, it is shown in [13] and [16] that
+sufficient conditions in (b) and (c) are sharp. More precisely, there exists c′ > 0
+such that, if
+ϕ(x) ≥
+
+
+
+
+
+
+
+c′|x|−N
+����log
+�
+e + 1
+|x|
+�����
+− N
+θ −1
+if
+γ = 1 + θ
+N ,
+c′|x|−
+θ
+γ−1
+if
+γ > 1 + θ
+N ,
+x ∈ B(0, 1),
+then problem (1.1) possesses no local-in-time nonnegative solutions.
+On the other hand, in the case of (a), distributions other than Radon measures
+such as the derivative of the Dirac distribution can be considered as the initial data
+to problem (1.1) with θ = 2. For instance, problem (1.1) with θ = 2 is well-posed
+in certain negative order inhomogeneous Besov-Morrey spaces Ns
+p,q,r(RN), see [19]
+and Remark 1.1. The arguments in [19] are based on delicate decay estimates of
+the heat kernel in inhomogeneous Besov-Morrey spaces and the power nonlinearity
+of the semilinear parabolic equation. It seems difficult to apply their arguments
+directly to the Cauchy problem (1.1) and problem (1.2), in particular, the case
+of fractional diffusion θ ̸= 2 and the case of the nonlinearity depending on ∇u.
+In this paper, we develop the arguments in [19] and prove the unique existence
+of the solution to problem (1.1) (resp.
+problem (1.2)) in inhomogeneous Besov-
+Morrey spaces Ns
+p,q,r(RN) for general θ > 0 (resp. θ > 1). This enables us to take
+2
+
+distributions other than Radon measures as initial data and the results in the case
+(a) is extended for more general initial data.
+For viscous Hamilton-Jacobi equations (1.2), the solvability has been studied
+in [1, 2, 4, 8, 18].
+Using the majorant kernel, Ishige, Kawakami, and Okabe [17]
+obtained the same results for problem (1.2) as for problem (1.1). That is, when
+1 < γ < 1 + (θ + 1)/(N + 1), there exists a solution to problem (1.2) if the initial
+measure satisfies
+sup
+x∈RN ∥ϕ∥L1(B(x,1)) < ∞.
+We extend these results to more general initial data. See Remark 1.2 for more details
+on the relation to previous studies.
+We recall the definition of local Morrey spaces and introduce inhomogeneous
+Besov-Morrey spaces.
+Definition 1.1 (local Morrey spaces) Let 1 ≤ q ≤ p < ∞. The local Morrey
+space Mp
+q (RN) is defined to be the set of measurable functions u in RN such that
+∥u |Mp
+q ∥ :=
+sup
+x∈RN, 0<ρ≤1
+ρ
+N
+p − N
+q ∥u |Lq(B(x, ρ))∥ < ∞.
+The local measure space of the Morrey type Mp(RN) is defined as the sets of the
+Radon measures µ on RN such that
+∥µ|Mp∥ :=
+sup
+x∈RN,0<ρ≤1
+ρ
+N
+p −N|µ|(B(x, ρ)) < ∞,
+where |µ| denotes the total variation of the measure µ.
+Let ζ(t) be a smooth function on [0, ∞) such that 0 ≤ ζ(t) ≤ 1, ζ(t) ≡ 1 for
+t ≤
+3
+2 and supp ζ ⊂ [0, 5
+3). For j ∈ Z, put ϕj(ξ) := ζ(2−j|ξ|) − ζ(21−j|ξ|) and
+ϕ(0)(ξ) := ζ(|ξ|). Then we have ϕj(ξ), ϕ(0)(ξ) ∈ C∞
+0 (RN) and
+ϕ(0)(ξ) +
+∞
+�
+j=1
+ϕj(ξ) = 1
+for any
+ξ ∈ RN.
+Definition 1.2 (inhomogeneous Besov-Morrey space) Let 1 ≤ q ≤ p < ∞,
+1 ≤ r ≤ ∞ and s ∈ R. The local Besov-Morrey space is defined as the sets of
+distributions u ∈ S′(RN) such that F −1ϕ(0)(ξ)Fu ∈ Mp
+q and F −1ϕj(ξ)Fu ∈ Mp
+q for
+every positive integer j, and that
+∥u|Ns
+p,q,r∥ := ∥F −1ϕ(0)(ξ)Fu|Mp
+q ∥ + ∥{2sj∥F −1ϕj(ξ)Fu|Mp
+q ∥}∞
+j=1|ℓr∥ < ∞,
+where F denotes the Fourier transform on RN.
+3
+
+For every t > 0 and every u ∈ S′(RN), put S(t)u := F −1 exp(−t|ξ|θ)Fu.
+We
+formulate a solution to problem (1.1) and (1.2) .
+Definition 1.3 Let T > 0 and ϕ ∈ Ns
+p,q,r for some s ∈ R, 1 ≤ q ≤ p < ∞ and
+1 ≤ r ≤ ∞. We say that u is a solution to problem (1.1) in RN × [0, T) if
+u ∈ BC(RN × (τ, T))
+for τ ∈ (0, T), and u satisfies
+u(x, t) = [S(t)ϕ](x) +
+� t
+0
+[S(t − τ)|u(·, τ)|γ−1u(·, τ)](x) dτ
+for (x, t) ∈ RN × (0, T).
+Definition 1.4 Let T > 0 and ϕ ∈ Ns
+p,q,r for some s ∈ R, 1 ≤ q ≤ p < ∞ and
+1 ≤ r ≤ ∞. We say that u is a solution to problem (1.2) in RN × [0, T) if
+u , ∇u ∈ BC(RN × (τ, T))
+for τ ∈ (0, T), and u satisfies
+u(x, t) = [S(t)ϕ](x) +
+� t
+0
+[S(t − τ)|∇u(·, τ)|γ](x) dτ
+for (x, t) ∈ RN × (0, T).
+We are ready to state the main results of this paper.
+Theorem 1.1 Let γ > 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ N/p − θ/(γ − 1).
+Then there exist δ > 0 and M > 0 such that for every ϕ(x) ∈ Ns
+p,q,∞ satisfying
+lim sup
+j→∞
+2sj∥F −1ϕjFϕ|Mp
+q ∥ < δ,
+(1.3)
+problem (1.1) possesses the unique solution u(x, t) on RN × [0, T) for some T > 0
+with a bound sup0<t≤T t−s/θ∥u(·, t) |Mp
+q ∥ ≤ M.
+Remark 1.1 To see the relation of these results with previous studies, we remark
+here that inhomogeneous Besov-Morrey spaces under the assumption of Theorem 1.1
+includes the following functions and function spaces. Let p0 = N(γ − 1)/θ.
+• Let γ > 1+θ/N and take p as max{γ, p0} < p < p0γ, then by Proposition 2.1
+and Proposition 2.2, we have
+|x|−
+θ
+γ−1 ∈ Mp0
+p0γ/p,∞ ⊂ N0
+p0,p0γ/p,∞ ⊂ NN/p−θ/(γ−1)
+p,γ,∞
+.
+4
+
+Since the assumption of Theorem 1.1 is satisfied with NN/p−θ/(γ−1)
+p,γ,∞
+for above
+p, we see by (1.3) that there exists c > 0 such that, if
+|ϕ(x)| ≤ c|x|−
+θ
+γ−1,
+x ∈ RN,
+then probolem (1.1) possesses a local-in-time solution. This result is consistent
+with that of [17] and thus the condition (1.3) is necessary.
+• Let γ = 1 + θ/N. Then by Proposition 2.1 and Proposition 2.2, for any p > 1
+we have
+Lp = Mp
+p ⊂ N0
+p,p,∞ ⊂ N−N/p+N/γ
+γ,γ,∞
+.
+Since the assumption of Theorem 1.1 is satisfied with N−N/p+N/γ
+γ,γ,∞
+, we see that
+if ϕ ∈ Lp with p > 1, then probolem (1.1) possesses a local-in-time solution.
+Note that in the case of θ = 2, problem (1.1) is not well-posed in L1 (See for
+example, [5,6]).
+• Let 1 < γ < 1 + θ/N. Then by Proposition 2.1 and Proposition 2.2, we have
+δ(x) ∈ M1 ⊂ N0
+1,1,∞ ⊂ N−N+N/γ
+γ,γ,∞
+.
+Since the assumption of Theorem 1.1 is satisfied with N−N+N/γ
+γ,γ,∞
+, we see that if
+ϕ is a Radon measure, then probolem (1.1) possesses a local-in-time solution,
+which is consistent with the result of [17]. Furthermore, since
+∂|α|δ(x) ∈ N−N+N/γ−|α|
+γ,γ,∞
+,
+we see that probolem (1.1) possesses a local-in-time solution for ϕ = ∂[θ]δ and
+γ <
+N+θ
+N+[θ] if θ is not an integer, and for ϕ = ∂θ−1δ and γ <
+N+θ
+N+θ−1 if θ is an
+integer.
+Theorem 1.2 Let 1 < γ < θ, γ ≤ q ≤ p < ∞, p > N(γ−1)/(θ−1), 1−θ/γ < s < 0
+and s ≥ N/p + (γ − θ)/(γ − 1). Then there exist δ > 0 and M > 0 such that for
+every ϕ(x) ∈ Ns
+p,q,∞ satisfying
+lim sup
+j→∞
+2sj∥F −1ϕjFϕ|Mp
+q ∥ < δ,
+problem (1.2) possesses the unique solution u(x, t) on RN × [0, T) for some T > 0
+with a bound sup0<t≤T t−s/θ∥u(·, t) |Mp
+q ∥ ≤ M and sup0<t≤T t−s/θ+1/θ∥∇u(·, t) |Mp
+q ∥ ≤
+M.
+Remark 1.2 To see the relation of these results with previous studies, we remark
+here that inhomogeneous Besov-Morrey spaces under the assumption of Theorem 1.2
+includes the following functions and function spaces. Let p1 = N(γ − 1)/(θ − γ).
+5
+
+• Let (N + θ)/(N + 1) < γ < θ and take p as max{γ, p1} < p < p1γ, then by
+Proposition 2.1 and Proposition 2.2, we have
+|x|− θ−γ
+γ−1 ∈ Mp1
+p1γ/p,∞ ⊂ N0
+p1,p1γ/p,∞ ⊂ NN/p−(θ−γ)/(γ−1)
+p,γ,∞
+.
+Since the assumption of Theorem 1.2 is satisfied with NN/p−(θ−γ)/(γ−1)
+p,γ,∞
+for above
+p, we see that there exists c > 0 such that, if
+|ϕ(x)| ≤ c|x|− θ−γ
+γ−1 ,
+x ∈ RN,
+then probolem (1.2) possesses a local-in-time solution. This result is consistent
+with that of [17].
+• Let γ = (N + θ)/(N + 1). Then by Proposition 2.1 and Proposition 2.2, for
+any p > 1 we have
+Lp = Mp
+p ⊂ N0
+p,p,∞ ⊂ N−N/p+N/γ
+γ,γ,∞
+.
+Since the assumption of Theorem 1.2 is satisfied with N−N/p+N/γ
+γ,γ,∞
+, we see that
+if ϕ ∈ Lp with p > 1, then probolem (1.2) possesses a local-in-time solution.
+• Let 1 < γ < (N + θ)/(N + 1). Then by Proposition 2.1 and Proposition 2.2,
+we have
+δ(x) ∈ M1 ⊂ N0
+1,1,∞ ⊂ N−N+N/γ
+γ,γ,∞
+.
+Since the assumption of Theorem 1.2 is satisfied with N−N+N/γ
+γ,γ,∞
+, we see that if
+ϕ is a Radon measure, then probolem (1.2) possesses a local-in-time solution.
+This result is consistent with that of [17]. Furthermore, since
+∂|α|δ(x) ∈ N−N+N/γ−|α|
+γ,γ,∞
+,
+we see that probolem (1.2) possesses a local-in-time solution for ϕ = ∂[θ]−1δ
+and γ <
+N+θ
+N+[θ] if θ is not an integer, and for ϕ = ∂θ−2δ and γ <
+N+θ
+N+θ−1 if θ is
+an integer.
+We explain the idea of the proof of Theorem 1.1 and Theorem 1.2. Let S(t)u :=
+F −1 exp(−t|ξ|θ)Fu. By modifying the arguments in [19], we first prove the heat
+kernel estimates of the fractional Laplacian in inhomogeneous Besov-Morrey spaces
+and obtain the estimate
+∥S(t)u|Nσ
+p,q,1∥ ≤ C(1 + t(s−σ)/θ)∥u|Ns
+p,q,∞∥,
+∥∇S(t)u|Nσ
+p,q,1∥ ≤ C(1 + t(s−σ−1)/θ)∥u|Ns
+p,q,∞∥,
+for t > 0 and σ > s.
+Here, one of the main difficulties comes from the non-
+smoothness of the function exp(−t|ξ|θ), see Lemma 2.2 and Remark 2.1.
+6
+
+Then we show that the approximate solutions converge in some Banach space
+based on the local Morrey spaces with a bound near t = 0.
+The rest of this paper is organized as follows. In Sections 2, we obtain the heat
+kernel estimates of the fractional Laplacian in inhomogeneous Besov-Morrey spaces.
+In section 3, we prove Theorem 1.1. In section 4, we prove Theorem 1.2.
+2
+Preliminaries
+In this section, we recall some preliminary facts about Besov-Morrey spaces and give
+estimates of heat kernel of fractional Laplacian in these function spaces.
+The following two propositions collect basic facts about Morrey spaces and Besov-
+Morrey spaces.
+Proposition 2.1 ( [19, Theorem 2.5]) Let 1 ≤ q ≤ p < ∞, r ∈ [1, ∞] and
+s ∈ R. Then the following embeddings are continuous:
+Ns
+p,q,r ⊂ Bs−N/p
+∞,r
+,
+(2.1)
+Ns
+p,q,r ⊂ Ns−N(1−l)/p
+p/l,q/l,r
+for any
+l ∈ (0, 1).
+(2.2)
+Proposition 2.2 ( [19, Proposition 2.11]) Let 1 ≤ q ≤ p < ∞. Then the fol-
+lowing embeddings are continuous:
+N0
+p,q,1 ⊂ Mp
+q ⊂ N0
+p,q,∞,
+(2.3)
+Mp ⊂ N0
+p,1,∞.
+We modify the arguments in [19, Theorem 2.9 (2)] and prepare the following two
+lemmas for the estimates of heat kernel of fractional Laplacian in inhomogeneous
+Besov-Morrey spaces. Here, we denote by ⌊x⌋ the greatest integer less than or equal
+to x ∈ R.
+Lemma 2.1 Let m ∈ R, 1 ≤ q ≤ p < ∞ and P(ξ) ∈ C⌊N/2⌋+1(RN \ {0}). Assume
+that there is A > 0 such that
+����
+∂αP
+∂ξα (ξ)
+���� ≤ A|ξ|m−|α|
+for all α ∈ (N ∪ {0})N with |α| ≤ ⌊N/2⌋ + 1 and for all ξ ̸= 0. Then the multiplier
+operator P(D)u := F −1P(ξ)Fu satisfies the estimate
+��F −1ϕjF(P(D)u)|Mp
+q
+�� ≤ CA2mj ��F −1ϕjFu|Mp
+q
+��
+for every positive integer j and u ∈ S′(RN) such that F −1ϕj(ξ)Fu ∈ Mp
+q , where
+C > 0 is a constant independent of j, A, and u.
+7
+
+Proof. Put Φj := ϕj−1 + ϕj + ϕj+1 and K(x) := F −1Φj(ξ)P(ξ) for j ∈ Z. Note
+that supp ϕj(ξ) ⊂ {ξ ∈ RN; 2j+1/3 ≤ |ξ| ≤ 2j+1} and Φj ≡ 1 on supp ϕj(ξ).
+Putting also N0 := ⌊N/2⌋ + 1, we have
+∥K|L1(RN)∥ =
+�
+|x|≤2−j |K(x)| dx +
+�
+|x|≥2−j |K(x)| dx
+≤
+��
+|x|≤2−j dx
+�1/2 ��
+|x|≤2−j |K(x)|2 dx
+�1/2
++
+��
+|x|≥2−j |x|−2N0 dx
+�1/2 ��
+|x|≥2−j |x|2N0|K(x)|2 dx
+�1/2
+≤ C
+
+2−Nj/2∥K(x)|L2(RN)∥ + 2(N0−N/2)j �
+|α|=N0
+∥xαK(x)|L2(RN)∥
+
+
+= C
+
+2−Nj/2∥Φj(ξ)P(ξ)|L2(RN)∥ + 2(N0−N/2)j �
+|α|=N0
+����
+∂|α|
+∂ξα(Φj(ξ)P(ξ))|L2(RN)
+����
+
+
+≤ C(2−Nj/22(m+N/2)jA + 2(N0−N/2)j2(m−N0+N/2)jA) = C2mjA
+for some constant C > 0, depending on N, m, ∥ζ|BCN0(R)∥, but not on j and A.
+Since F −1ϕjF(P(D)u) = K ∗ (F −1ϕjFu), we see by [19, Lemma 1.8] that
+∥F −1ϕjF(P(D)u)|Mp
+q ∥ ≤ CA2mj∥F −1ϕjFu|Mp
+q ∥
+for every positive integer j, and the proof is complete. ✷
+Lemma 2.2 Let m > 0, 1 ≤ q ≤ p < ∞ and P(ξ) ∈ C⌊N/2⌋+1(RN \ {0}). Assume
+that there is A > 0 such that
+����
+∂αP
+∂ξα (ξ)
+���� ≤ A|ξ|m−|α|
+for all α ∈ (N ∪ {0})N with |α| ≤ ⌊N/2⌋ + 1 and for all ξ ∈ B(0, 4) \ {0}. Then the
+multiplier operator P(D)u := F −1P(ξ)Fu satisfies the estimate
+∥F −1ϕ(0)F(P(D)u)|Mp
+q ∥ ≤ CA∥F −1ϕ(0)Fu|Mp
+q ∥
+for every u ∈ S′(RN) such that F −1ϕ(0)(ξ)Fu ∈ Mp
+q , where C > 0 is a constant
+independent of A and u.
+Proof. Put Kj(x) := F −1ϕj(ξ)P(ξ) and Φ(0) := ϕ(0) + ϕ1. In the same way as in
+Lemma 2.1, we have
+∥F −1Φ(0)(ξ)P(ξ)|L1(RN)∥ ≤
+1
+�
+j=−∞
+∥Kj|L1(RN)∥
+≤
+1
+�
+j=−∞
+C2mjA ≤ CA
+8
+
+with some constant C > 0 independent of A. This implies in the same way as in
+Lemma 2.1
+∥F −1ϕ(0)F(P(D)u)|Mp
+q ∥ ≤ CA∥F −1ϕ(0)Fu|Mp
+q ∥,
+and the proof is complete. ✷
+Remark 2.1 Note that we do not assume the smoothness of P(ξ) at ξ = 0, which
+is useful for the estimates of the derivative of heat kernel of fractional Laplacian
+since P(ξ) = exp(−t|ξ|θ) is not smooth at ξ = 0 in general. In this respect, we
+improved [19, Theorem 2.9 (2)] where the smoothness at ξ = 0 is needed.
+In the following theorem, we obtain estimates of heat kernel of fractional Laplacian
+in inhomogeneous Besov-Morrey spaces.
+Theorem 2.1 Let s ≤ σ, 1 ≤ q ≤ p < ∞ and r ∈ [1, ∞]. Then there exists C > 0
+such that the estimate
+∥S(t)u|Nσ
+p,q,r∥ ≤ C(1 + t(s−σ)/θ)∥u|Ns
+p,q,r∥
+for
+t > 0
+(2.4)
+holds. Furthermore, if s < σ, the estimate
+∥S(t)u|Nσ
+p,q,1∥ ≤ C(1 + t(s−σ)/θ)∥u|Ns
+p,q,∞∥
+for
+t > 0
+(2.5)
+holds.
+Proof. By induction we see that for every α ∈ NN there exist homogeneous poly-
+nomials Pα,k(ξ) of degree |α| for k = 1, 2, . . . , |α| such that for ξ ̸= 0
+∂|α| exp(−t|ξ|θ)
+∂ξα
+= exp(−t|ξ|θ)|ξ|−2|α|
+|α|
+�
+k=1
+Pα,k(ξ)tk|ξ|kθ.
+(2.6)
+We have for m = s − σ
+|ξ|−m+|α|∂|α| exp(−t|ξ|θ)
+∂ξα
+≤ Ct
+m
+θ exp(−t|ξ|θ)
+|α|
+�
+k=1
+(t
+1
+θ |ξ|)kθ−m
+≤ Cαt
+m
+θ .
+This together with Lemma 2.1 implies
+∥F −1ϕj(ξ)F(S(t)u)|Mp
+q ∥ ≤ Ct
+m
+θ 2mj∥F −1ϕj(ξ)Fu|Mp
+q ∥
+(2.7)
+for every positive integer and every t > 0. On the other hand, since
+∥F −1ϕ(0)(ξ)F(S(t)u)|Mp
+q ∥
+≤ ∥F −1Φ(0) ∗ F −1 exp(−t|ξ|θ)|L1(RN)|∥∥F −1ϕ(0)(ξ)Fu|Mp
+q ∥
+≤ C∥F −1ϕ(0)(ξ)Fu|Mp
+q ∥,
+9
+
+where Φ(0) is as in Lemma 2.2. This together with (2.7) implies the inequality (2.4).
+The inequality (2.5) follows exactly in the same way as in [19, Theorem 3.1] from
+the inequality (2.4), and the proof is complete. ✷
+In the following lemma, we obtain another estimate of the heat kernel of frac-
+tional Laplacian by using the smallness condition on the initial data.
+Lemma 2.3 Let 1 ≤ q ≤ p < ∞ and s < σ. Then there exists A > 0 such that, for
+every u ∈ Ns
+p,q,∞ and every B > 0, satisfying
+A lim sup
+j→∞
+2sj∥F −1ϕjFu|Mp
+q ∥ < B,
+there exists T > 0 such that
+sup
+0<t≤T
+t(σ−s)/θ∥S(t)u|Nσ
+p,q,1∥ < B.
+Proof. Let C0 be a positive constant satisfying the estimate
+∥S(t)u|Nσ
+p,q,1∥ ≤ C0(1 + t(s−σ)/θ)∥u|Ns
+p,q,∞∥,
+and put C1 = max{1, 2∥F −1ϕ0|L1(RN)∥} and A = C0C1.
+Take δ > 0 such that
+lim sup
+j→∞
+2sj∥F −1ϕjFu|Mp
+q ∥ < δ < B/A,
+then for some m ∈ N, the estimate
+2sj∥F −1ϕjFu|Mp
+q ∥ ≤ δ < B/A
+holds for every j ≥ m. Put u1 = F −1ϕ(0)(2−m·)Fu and u2 = u − u1. Since
+supp ϕ(0)(2−mξ) ⊂
+�
+ξ ∈ RN; |ξ| ≤ 5
+32m
+�
+,
+supp ϕj(ξ) ⊂
+�
+ξ ∈ RN; 2j+1
+3
+≤ |ξ| ≤ 2j+1
+�
+,
+ϕ(0)(2−mξ) ≡ 1
+on
+{ξ ∈ RN; |ξ| ≤ 3 · 2m−1},
+we have
+F −1ϕjFu1 =
+
+
+
+
+
+F −1ϕjFu
+for
+j ≤ m − 1,
+F −1(ϕm−1 + ϕm)ϕjFu
+for
+j = m, m + 1,
+0
+for
+j ≥ m + 2,
+and
+F −1ϕjFu2 =
+
+
+
+
+
+0
+for
+j ≤ m − 1,
+F −1(ϕm+1 + ϕm+2)ϕjFu
+for
+j = m, m + 1,
+F −1ϕjFu
+for
+j ≥ m + 2.
+10
+
+It follows from the the definition of the constant C1 and the fact ∥F −1ϕj|L1∥ =
+∥F −1ϕ0|L1∥ that ∥u2|Ns
+p,q,∞∥ ≤ C1δ. Therefore, we have
+t(σ−s)/θ∥S(t)u2|Nσ
+p,q,1∥ ≤ C0(1 + t(σ−s)/θ)∥u2|Ns
+p,q,∞∥
+≤ C0C1δ(1 + T (σ−s)/θ) = Aδ(1 + T (σ−s)/θ) < Aδ + B
+2
+(2.8)
+for every t ∈ (0, T], by taking T > 0 sufficiently small. On the other hand, since
+u1 ∈ N(σ+s)/2
+p,q,∞
+, we have the estimate
+t(σ−s)/θ∥S(t)u1|Nσ
+p,q,1∥ ≤ C0(t(σ−s)/2θ) + t(σ−s)/θ))∥u1|N(s+σ)/2
+p,q,∞
+∥
+≤ C0T (σ−s)/2θ(1 + T (σ−s)/2θ)∥u1|N(s+σ)/2
+p,q,∞
+∥ < B − Aδ
+2
+(2.9)
+for every t ∈ (0, T], by taking T > 0 sufficiently small. We obtain the conclusion
+from (2.8) and (2.9), and the proof is complete. ✷
+3
+Proof of Theorem 1.1.
+In this section, we prove Theorem 1.1 by using Theorem 2.1. Let XT denote the set
+of Lebesgue measurable functions u(x, t) on RN × (0, T) such that
+∥u|XT∥ := sup
+0<t<T
+t−s/θ∥u(·, t) |Mp
+q ∥ < ∞.
+Set u0(x, t) = [S(t)ϕ](x). Define un(x, t) (n = 1, 2, . . .) inductively by
+un(x, t) := u0(x, t) +
+� t
+0
+[S(t − τ)|un−1(·, τ)|γ−1un−1(·, τ)](x) dτ.
+(3.1)
+We prepare the following three lemmas for the proof of Theorem 1.1.
+Lemma 3.1 Let γ > 1, T ≤ 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ���
+N/p − θ/(γ − 1). Then there exists C2 > 0 independent of T such that
+∥un+1|XT∥ ≤ ∥u0|XT∥ + C2∥un|XT∥γ
+for n = 0, 1, . . ..
+11
+
+Proof. By (2.2), (2.3), (2.5) and (3.1), we see that
+∥un+1(·, t) − u0(·, t)|Mp
+q ∥ ≤ C∥un+1(·, t) − u0(·, t)|N0
+p,q,1∥
+≤ C
+� t
+0
+∥S(t − τ)|un(·, τ)|γ−1un(·, τ)|N0
+p,q,1∥ dτ
+≤ C
+� t
+0
+∥S(t − τ)|un(·, τ)|γ−1un(·, τ)|NN(γ−1)/p
+p/γ,q/γ,1 ∥ dτ
+≤ C
+� t
+0
+{1 + (t − τ)−N(γ−1)/pθ}∥|un(·, τ)|γ|N0
+p/γ,q/γ,∞∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ∥|un(·, τ)|γ|Mp/γ
+q/γ ∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ∥un(·, τ)|Mp
+q ∥γ dτ
+≤ C∥un|XT∥γ
+� t
+0
+(t − τ)−N(γ−1)/pθτ sγ/θ dτ
+≤ Ct−N(γ−1)/pθ+sγ/θ+1∥un|XT∥γ.
+Therefore, we have
+∥un+1 − u0|XT∥ ≤ Ct1+(γ−1)(s/θ−N/pθ)∥un|XT∥γ
+≤ C∥un|XT∥γ
+for T ≤ 1, and the proof is complete. ✷
+Lemma 3.2 Let γ > 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ N/p −
+θ/(γ − 1). Then there exists C3 > 0 such that for every ϕ(x) ∈ Ns
+p,q,∞ satisfy-
+ing lim supj→∞ 2sj∥F −1ϕjFϕ|Mp
+q ∥ < δ for some δ > 0, we can choose a positive
+number T ≤ 1 so small that the inequality ∥u0|XT∥ < C3δ holds. Furthermore, we
+can choose δ so small that supn ∥un|XT∥ ≤ M for some M > 0.
+Proof. By Lemma 2.3 with B = Aδ, we can take T ≤ 1 such that the estimate
+sup
+0<t≤T
+t−s/θ∥u0|N0
+p,q,1∥ < Aδ
+holds. This together with (2.3) implies ∥u0|XT∥ < C3δ for some constant C3 > 0.
+For δ > 0 satisfying
+2γC2Cγ
+3 δγ−1 < 1,
+we see by induction that
+sup
+n ∥un|XT∥ ≤ 2C3δ =: M,
+and the proof is complete. ✷
+12
+
+Lemma 3.3 Let γ > 1, γ ≤ q ≤ p < ∞, −θ/γ < s < 0 and s ≥ N/p − θ/(γ − 1).
+Suppose that δ and T ≤ 1 are small enough so that the assertion of Lemma 3.2
+holds. Then there exists a positive constant C independent of T such that
+∥un+2 − un+1|XT∥ ≤ CMγ−1∥un+1 − un|XT∥
+for n = 0, 1, . . ..
+Proof. By (2.2), (2.3), (2.5) and (3.1), we see that
+∥un+2(·, t) − un+1(·, t)|Mp
+q ∥ ≤ C∥un+2(·, t) − un+1(·, t)|N0
+p,q,1∥
+≤ C
+� t
+0
+∥S(t − τ)(|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ))|N0
+p,q,1∥ dτ
+≤ C
+� t
+0
+∥S(t − τ)(|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ))|NN(γ−1)/p
+p/γ,q/γ,1 ∥ dτ
+≤ C
+� t
+0
+(t − τ)− N(γ−1)
+pθ
+∥|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ)|N0
+p/γ,q/γ,∞∥ dτ
+≤ C
+� t
+0
+(t − τ)− N(γ−1)
+pθ
+∥|un+1(·, τ)|γ−1un+1(·, τ) − |un(·, τ)|γ−1un(·, τ)|Mp/γ
+q/γ ∥ dτ
+≤ C
+� t
+0
+(t − τ)− N(γ−1)
+pθ
+∥|un+1(·, τ) − un(·, τ)|(|un+1(·, τ)|γ−1 + |un(·, τ)|γ−1)|Mp/γ
+q/γ ∥ dτ
+≤ CMγ−1
+� t
+0
+(t − τ)− N(γ−1)
+pθ
+∥un+1(·, τ) − un(·, τ)|Mp
+q ∥ dτ
+≤ CMγ−1∥un+1 − un|XT∥
+� t
+0
+(t − τ)−N(γ−1)/pθτ sγ/θ dτ
+≤ CMγ−1t−N(γ−1)/pθ+sγ/θ+1∥un+1 − un|XT∥.
+We used here [19, Lemma 1.4]. Therefore, we have
+∥un+2 − un+1|XT∥ ≤ CMγ−1t1+(γ−1)(s/θ−N/pθ)∥un+1 − un|XT∥
+≤ CMγ−1∥un+1 − un|XT∥,
+and the proof is complete. ✷
+Proof of Theorem 1.1.
+Take δ and T so small that
+∥un+2 − un+1|XT∥ ≤ 1
+2∥un+1 − un|XT∥
+for n = 0, 1, . . ., and we see that un(x, t) converges in XT. Set u(x, t) as a limit of
+un(x, t) in XT and we see that
+u(x, t) := [S(t)ϕ](x) +
+� t
+0
+[S(t − τ)|u(·, τ)|γ−1u(·, τ)](x) dτ.
+(3.2)
+13
+
+We next prove that u(x, t) ∈ L∞([ε, T] × RN) for every ε > 0. Let n be the
+smallest integer greater than Nγ/θp.
+Then we can take an increasing sequence
+of positive numbers {pj}n
+j=1 such that p1 = p, N/pj+1 > N/pj − θ/γ for every
+j = 1, 2, · · · , n − 1 and N/pn < θ/γ. We also define {qj}n
+j=1 and {sj}n
+j=1 as q1 = q,
+qj+1 = pj+1qj/pj, s1 = s and sj+1 = N/pj+1 − N/pj.
+By the obtained result, we see that the solution u(x, t) belongs to the spaces
+L∞ �� ε
+2n, T
+�
+, Mp
+q
+�
+⊂ L∞ �� ε
+2n, T
+�
+, N0
+p1,q1,∞
+�
+⊂ L∞ �� ε
+2n, T
+�
+, Ns2
+p2,q2,∞
+�
+.
+Since γ ≤ q2 ≤ p2, −γ/θ < s2 < 0 and s2 ≥ N/p2 − θ/(γ − 1), we can apply the
+obtained result to see u(x, t) ∈ L∞ �� 2ε
+2n, T
+�
+, Mp2
+q2
+�
+. In the same way, since
+L∞
+�� jε
+2n, T
+�
+, Mpj
+qj
+�
+⊂ L∞
+�� jε
+2n, T
+�
+, N0
+pj,qj,∞
+�
+⊂ L∞
+�� jε
+2n, T
+�
+, Nsj
+pj+1,qj+1,∞
+�
+,
+where γ ≤ qj+1 ≤ pj+1, −γ/θ < sj+1 < 0 and sj+1 ≥ N/pj+1 − θ/(γ − 1), we
+have u(x, t) ∈ L∞ ��
+(j+1)ε
+2n , T
+�
+, M
+pj+1
+qj+1
+�
+for j = 1, 2, · · · , n − 1. Therefore, we have
+u(x, t) ∈ L∞ �� ε
+2, T
+�
+, Mpn
+qn
+�
+, where pn > Nγ/θ. It follows from (2.1) that
+����
+� t
+ε/2
+S(t − τ)|u(·, τ)|γ−1u(·, τ) dτ|L∞
+����
+≤ C
+� t
+ε/2
+��S(t − τ)|u(·, τ)|γ−1u(·, τ)|B0
+∞,1
+�� dτ
+≤ C
+� t
+ε/2
+���S(t − τ)|u(·, τ)|γ−1u(·, τ)|NNγ/pn
+pn/γ,qn/γ,1
+��� dτ
+≤ C
+� t
+ε/2
+�
+1 + (t − τ)−Nγ/θpn� ��|u(·, τ)|γ−1u(·, τ)|N0
+pn/γ,qn/γ,∞
+�� dτ
+≤ C
+� t
+ε/2
+(t − τ)−Nγ/θpn
+���|u(·, τ)|γ|Mpn/γ
+qn/γ
+��� dτ
+≤ C
+� t
+ε/2
+(t − τ)−Nγ/θpn ��u(·, τ)|Mpn
+qn
+��γ dτ
+≤ C
+�
+t − ε
+2
+�1−Nγ/θpn
+sup
+ε/2≤τ≤t
+��u(·, τ)|Mpn
+qn
+��γ dτ
+≤ CT 1−Nγ/θpn
+sup
+ε/2≤τ≤t
+��u(·, τ)|Mpn
+qn
+��γ dτ < ∞
+(3.3)
+for ε/2 ≤ t ≤ T ≤ 1. On the other hand, we have
+∥S(t − ε/2)u(·, ε/2)|L∞∥ ≤ C∥S(t − ε/2)u(·, ε/2)|B0
+∞,1∥
+≤ C∥S(t − ε/2)u(·, ε/2)|NN/p
+p,q,1∥
+≤ C
+�
+1 + (t − ε/2)−N/θp� ��|u(·, ε/2)||Mp
+q
+��
+≤ C(ε/2)−N/θp ��|u(·, ε/2)||Mp
+q
+�� < ∞
+(3.4)
+14
+
+for ε ≤ t ≤ T ≤ 1. Since
+u(x, t) =
+�
+S
+�
+t − ε
+2
+�
+u
+�
+·, ε
+2
+��
+(x) +
+� t
+ε/2
+�
+S(t − s)|u(·, τ)|γ−1u(·, τ)
+�
+(x) dτ,
+this together with (3.3) and (3.4) implies that u(x, t) ∈ L∞([ε, T] × RN) for every
+ε > 0.
+Finally, we prove the uniqueness of the solution.
+Assume that u(1)(x, t) and
+u(2)(x, t) are solutions to (3.2) satisfying sup0≤t≤T t−s/θ∥u(j)(·, t)|Mp
+q ∥ < ∞.
+Let
+u = u(1) − u(2) and h(t) = ∥u(·, t)|Mp
+q ∥. Then exactly in the same way as in the
+proof of Lemma 3.3, we have
+sup
+0<t≤T
+t−s/θh(t) ≤ CMγ−1 sup
+0<t≤T
+t−s/θh(t) ≤ 1
+2 sup
+0<t≤T
+t−s/θh(t).
+Therefore, we see that u ≡ 0, and the proof is complete. ✷
+4
+Proof of Theorem 1.2.
+In this section, we prove Theorem 1.2. Let T > 0 be small and consider the Banach
+space
+YT := {u(x, t) on (0, T) × RN : ∥u |YT∥ < ∞},
+where
+∥u |YT∥ := sup
+0<t<T
+{t−s/θ∥u(·, t) |Mp
+q ∥ + t(−s+1)/θ∥∇u(·, t) |Mp
+q ∥}.
+Set u0(x, t) = [S(t)ϕ](x). Define un(x, t) (n = 1, 2, . . .) inductively by
+un(x, t) := u0(x, t) +
+� t
+0
+[S(t − τ)|∇un−1(·, τ)|γ](x) dτ.
+(4.1)
+For every t > 0 and every u ∈ S′, put Sj(t)u := F −1(iξj) exp(−t|ξ|θ)Fu for
+1 ≤ j ≤ N.
+As in Section 2, we prove the derivative estimate for S(t) in the
+following theorem.
+Theorem 4.1 Let s ≤ σ, 1 ≤ q ≤ p < ∞ and r ∈ [1, ∞]. Then there exists C > 0
+such that the estimate
+∥Sj(t)u|Nσ
+p,q,r∥ ≤ C(1 + t(s−σ−1)/θ)∥u|Ns
+p,q,r∥
+for
+t > 0
+(4.2)
+holds. Furthermore, if s < σ, the estimate
+∥Sj(t)u|Nσ
+p,q,1∥ ≤ C(1 + t(s−σ−1)/θ)∥u|Ns
+p,q,∞∥
+for
+t > 0
+(4.3)
+holds.
+15
+
+Proof. By (2.6) we see that for every α ∈ NN there exists a homogeneous poly-
+nomial Pα,k(ξ) of degree |α| for k = 1, 2, . . . , |α| and Pα−ej,k(ξ) of degree |α| − 1 for
+k = 1, 2, . . . , |α| − 1 such that for ξ ̸= 0
+∂|α|(iξj) exp(−t|ξ|θ)
+∂ξα
+= iξj exp(−t|ξ|θ)|ξ|−2|α|
+|α|
+�
+k=1
+Pα,k(ξ)tk|ξ|kθ
++ iαj exp(−t|ξ|θ)|ξ|−2|α|+2
+|α|−1
+�
+k=1
+Pα−ej,k(ξ)tk|ξ|kθ.
+We have for m = s − σ
+|ξ|−m+|α|∂|α|(iξj) exp(−t|ξ|θ)
+∂ξα
+≤ Ct
+m−1
+θ
+exp(−t|ξ|θ)
+|α|
+�
+k=1
+(t
+1
+θ |ξ|)kθ−m+1
+≤ Cαt
+m−1
+θ .
+This together with Lemma 2.1 implies
+∥F −1ϕj(ξ)F(Sj(t)u)|Mp
+q ∥ ≤ Ct
+m
+θ 2mj∥F −1ϕj(ξ)Fu|Mp
+q ∥
+(4.4)
+for every positive integer and every t > 0. On the other hand, by (2.6) we have
+����
+∂|α|(iξj) exp(−t|ξ|θ)
+∂ξα
+���� ≤ Cα|ξ|1−|α|.
+for every ξ ∈ B(0, 4) \ {0}. This together with Lemma 2.2 implies
+∥F −1ϕ(0)(ξ)F(S(t)u)|Mp
+q ∥ ≤ C∥F −1ϕ(0)(ξ)Fu|Mp
+q ∥.
+(4.5)
+The inequality (4.2) follows from (4.4) and (4.5). The inequality (4.3) follows exactly
+in the same way as in [19, Theorem 3.1] from the inequality 4.2, and the proof is
+complete. ✷
+Lemma 4.1 Let 1 ≤ q ≤ p < ∞ and s < σ. Then there exists A > 0 such that, for
+every u ∈ Ns
+p,q,∞ and every B > 0, satisfying
+A lim sup
+j→∞
+2sj∥F −1ϕjFu|Mp
+q ∥ < B,
+there exists T > 0 such that
+sup
+0<t≤T
+t(σ−s+1)/θ∥Sj(t)u|Nσ
+p,q,1∥ < B.
+16
+
+Proof. Let C0 be a positive constant satisfying the estimate
+∥Sj(t)u|Nσ
+p,q,1∥ ≤ C0(1 + t(s−σ−1)/θ)∥u|Ns
+p,q,∞∥,
+and put C1 = max{1, 2∥F −1ϕ0|L1(RN)∥} and A = C0C1. Then there exists m ∈ N
+such that the estimate 2sj∥F −1ϕjFu∥ ≤ δ < B/A holds for every j ≥ m. Put
+u1 = F −1ϕ(0)(2−mξ)Fu and u2 = u − u1. Take δ > 0, m ∈ N, u1 and u2 as in the
+proof of Lemma 2.3. Then we have
+t(σ−s+1)/θ∥Sj(t)u2|Nσ
+p,q,1∥ ≤ C0(1 + t(σ−s+1)/θ)∥u2|Ns
+p,q,∞∥
+≤ C0C1δ(1 + T (σ−s+1)/θ) = Aδ(1 + T (σ−s+1)/θ) < Aδ + B
+2
+(4.6)
+for every t ∈ (0, T], by taking T > 0 sufficiently small. On the other hand, since
+u1 ∈ N(σ+s)/2
+p,q,∞
+, we have the estimate
+t(σ−s+1)/θ∥Sj(t)u1|Nσ
+p,q,1∥ ≤ C(t(σ−s)/2θ) + t(σ−s)/θ))∥u1|N(s+σ)/2
+p,q,∞
+∥
+≤ CT (σ−s)/2θ(1 + T (σ−s)/2θ)∥u1|N(s+σ)/2
+p,q,∞
+∥ < B − Aδ
+2
+(4.7)
+for every t ∈ (0, T], by taking T > 0 sufficiently small. We obtain the conclusion
+from (4.6) and (4.7). The proof is complete. ✷
+We prepare the following three lemmas for the proof of Theorem 1.2.
+Lemma 4.2 Let 1 < γ < θ, T ≤ 1, γ ≤ q ≤ p < ∞, p > N(γ − 1)/(θ − 1),
+1 − θ/γ < s < 0 and s ≥ N/p + (γ − θ)/(γ − 1).
+Then there exists C2 > 0
+independent of T such that
+∥un+1|YT∥ ≤ ∥u0|YT∥ + C2∥un|YT∥γ
+for n = 0, 1, . . ..
+17
+
+Proof. By (2.2), (2.3), (2.5) and (4.1) we see that
+∥un+1(·, t) − u0(·, t)|Mp
+q ∥ ≤ C∥un+1(·, t) − u0(·, t)|N0
+p,q,1∥
+≤ C
+� t
+0
+∥S(t − τ)|∇un(·, τ)|γ|N0
+p,q,1∥ dτ
+≤ C
+� t
+0
+∥S(t − τ)|∇un(·, τ)|γ|NN(γ−1)/p
+p/γ,q/γ,1 ∥ dτ
+≤ C
+� t
+0
+{1 + (t − τ)−N(γ−1)/pθ}∥|∇un(·, τ)|γ|N0
+p/γ,q/γ,∞∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ∥|∇un(·, τ)|γ|Mp/γ
+q/γ ∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ∥∇un(·, τ)|Mp
+q ∥γ dτ
+≤ C∥un|YT∥γ
+� t
+0
+(t − τ)−N(γ−1)/pθτ (s−1)γ/θ dτ
+≤ Ct−N(γ−1)/pθ+(s−1)γ/θ+1∥un|YT∥γ.
+In the same way, by (2.2), (2.3), (4.1) and (4.3) we see that
+∥∂jun+1(·, t) − ∂ju0(·, t)|Mp
+q ∥ ≤ C
+� t
+0
+∥Sj(t − τ)|∇un(·, τ)|γ|N0
+p,q,1∥ dτ
+≤ C
+� t
+0
+∥Sj(t − τ)|∇un(·, τ)|γ|NN(γ−1)/p
+p/γ,q/γ,1 ∥ dτ
+≤ C
+� t
+0
+{1 + (t − τ)−N(γ−1)/pθ−1/θ}∥|∇un(·, τ)|γ|N0
+p/γ,q/γ,∞∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ−1/θ∥|∇un(·, τ)|γ|Mp/γ
+q/γ ∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ−1/θ∥∇un(·, τ)|Mp
+q ∥γ dτ
+≤ C∥un|YT∥γ
+� t
+0
+(t − τ)−N(γ−1)/pθ−1/θτ (s−1)γ/θ dτ
+≤ Ct−N(γ−1)/pθ−1/θ+(s−1)γ/θ+1∥un|YT∥γ.
+Therefore, we have
+∥un+1 − u0|YT∥ ≤ Ct1+(γ−1)(s/θ−N/pθ)−γ/θ∥un|YT∥γ
+≤ C∥un|YT∥γ
+for T ≤ 1, and the proof is complete. ✷
+18
+
+Lemma 4.3 Let 1 < γ < θ, T ≤ 1, γ ≤ q ≤ p < ∞, p > N(γ − 1)/(θ − 1),
+1 − θ/γ < s < 0 and s ≥ N/p + (γ − θ)/(γ − 1).
+Then there exists C3 > 0
+such that for every ϕ(x) ∈ Ns
+p,q,∞ satisfying lim supj→∞ 2sj∥F −1ϕjFϕ|Mp
+q ∥ < δ for
+some δ > 0 we can choose a positive number T ≤ 1 so small that the inequality
+∥u0|YT∥ < C0δ holds. Furthermore, we can choose δ so small that ∥un|YT∥ ≤ M for
+some M > 0.
+Proof. By Lemma 4.1 with B = Aδ, we can take a positive number T ≤ 1 such
+that the estimate
+sup
+0<t≤T
+(t−s/θ∥u0|N0
+p,q,1∥ + t(−s+1)/θ∥∇u0|N0
+p,q,1∥) < Aδ
+holds. This together with (2.3) implies ∥u0|YT∥ < C3δ for some constant C3 > 0.
+For δ > 0 satisfying
+2γC2Cγ
+3 δγ−1 < 1,
+we see by induction that
+sup
+n ∥un|XT∥ ≤ 2C3δ =: M,
+and the proof is complete. ✷
+Lemma 4.4 Let 1 < γ < θ, T ≤ 1, γ ≤ q ≤ p < ∞, p > N(γ − 1)/(θ − 1),
+1 − θ/γ < s < 0 and s ≥ N/p + (γ − θ)/(γ − 1). Suppose that δ and T ≤ 1 are
+small enough so that the assertion of Lemma 4.3 holds. Then there exists a positive
+constant C independent of T such that
+∥un+2 − un+1|YT∥ ≤ CMγ−1∥un+1 − un|YT∥
+for n = 0, 1, . . ..
+Proof. By (2.2), (2.3), (2.5) and (4.1) we see that
+∥un+2(·, t) − un+1(·, t)|Mp
+q ∥ ≤ C∥un+2(·, t) − un+1(·, t)|N0
+p,q,1∥
+≤ C
+� t
+0
+∥S(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|N0
+p,q,1∥ dτ
+≤ C
+� t
+0
+∥S(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|NN(γ−1)/p
+p/γ,q/γ,1 ∥ dτ
+≤ C
+� t
+0
+{1 + (t − τ)−N(γ−1)/pθ}∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|N0
+p/γ,q/γ,∞∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|Mp/γ
+q/γ ∥ dτ
+≤ CMγ−1∥un+1 − un|YT∥
+� t
+0
+(t − τ)−N(γ−1)/pθτ (s−1)γ/θ dτ
+≤ CMγ−1t−N(γ−1)/pθ+(s−1)γ/θ+1∥un+1 − un|YT∥.
+19
+
+In the same way, by (2.2), (2.3), (4.1) and (4.3) we see that
+∥∂j(un+2(·, t) − un+1(·, t))|Mp
+q ∥ ≤ C∥∂j(un+2(·, t) − un+1(·, t))|N0
+p,q,1∥
+≤ C
+� t
+0
+∥Sj(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|N0
+p,q,1∥ dτ
+≤ C
+� t
+0
+∥Sj(t − τ)(|∇un+1(·, τ)|γ − |∇un(·, τ)|γ)|NN(γ−1)/p
+p/γ,q/γ,1 ∥ dτ
+≤ C
+� t
+0
+{1 + (t − τ)−N(γ−1)/pθ−1/θ}∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|N0
+p/γ,q/γ,∞∥ dτ
+≤ C
+� t
+0
+(t − τ)−N(γ−1)/pθ−1/θ∥|∇un+1(·, τ)|γ − |∇un(·, τ)|γ|Mp/γ
+q/γ ∥ dτ
+≤ CMγ−1∥un+1 − un|YT∥
+� t
+0
+(t − τ)−N(γ−1)/pθ−1/θτ (s−1)γ/θ dτ
+≤ CMγ−1t−N(γ−1)/pθ−1/θ+(s−1)γ/θ+1∥un+1 − un|YT∥.
+We used here [19, Lemma 1.4]. Therefore, we have
+∥un+2 − un+1|YT∥ ≤ CMγ−1t1+(γ−1)(s/θ−N/pθ)−γ/θ∥un+1 − un|YT∥
+≤ CMγ−1∥un+1 − un|YT∥,
+and the proof is complete. ✷
+Proof of Theorem 1.2. Take δ and T so small that
+∥un+2 − un+1|YT∥ ≤ 1
+2∥un+1 − un|YT∥
+for n = 0, 1, . . ., and we see that un(x, t) converges in YT. Set u(x, t) as a limit of
+un(x, t) in YT and we see that
+u(x, t) := u0(x, t) +
+� t
+0
+[S(t − τ)|∇u(·, τ)|γ](x) dτ.
+(4.8)
+We next prove that u(x, t) ∈ L∞([ε, T] × RN) and ∇u(x, t) ∈ L∞([ε, T] × RN)
+for every ε > 0. Let n be the smallest integer greater than Nγ/(θ − γ)p. Then
+we can take an increasing sequence of positive numbers {pj}n
+j=1 such that p1 = p,
+N/pj+1 > N/pj − (θ − γ)/γ for every j = 1, 2, · · · , n − 1 and N/pn < (θ − γ)/γ.
+We also define {qj}n
+j=1 and {sj}n
+j=1 as q1 = q, qj+1 = pj+1qj/pj, s1 = s and sj+1 =
+N/pj+1 − N/pj.
+By the obtained result, we see that the solution u(x, t) and ∇u(x, t) belong to
+the spaces
+L∞ �� ε
+2n, T
+�
+, Mp
+q
+�
+⊂ L∞ �� ε
+2n, T
+�
+, N0
+p1,q1,∞
+�
+⊂ L∞ �� ε
+2n, T
+�
+, Ns2
+p2,q2,∞
+�
+.
+20
+
+Since γ ≤ q2 ≤ p2, p2 > N(γ − 1)/(θ − 1), 1 − γ/θ < s2 < 0 and s2 ≥ N/p2 − (θ −
+γ)/(γ − 1), we can apply the obtained result to see u(x, t) ∈ L∞ �� 2ε
+2n, T
+�
+, Mp2
+q2
+�
+and
+∇u(x, t) ∈ L∞ �� 2ε
+2n, T
+�
+, Mp2
+q2
+�
+. In the same way, since
+L∞
+�� jε
+2n, T
+�
+, Mpj
+qj
+�
+⊂ L∞
+�� jε
+2n, T
+�
+, N0
+pj,qj,∞
+�
+⊂ L∞
+�� jε
+2n, T
+�
+, Nsj
+pj+1,qj+1,∞
+�
+,
+where γ ≤ qj+1 ≤ pj+1, pj+1 > N(γ − 1)/(θ − 1), 1 − γ/θ < sj+1 < 0 and
+sj+1 ≥ N/pj+1 − (θ − γ)/(γ − 1), we have u(x, t) ∈ L∞ ��
+(j+1)ε
+2n , T
+�
+, M
+pj+1
+qj+1
+�
+for
+j = 1, 2, · · · , n − 1. Therefore, we have u(x, t) ∈ L∞ �� ε
+2, T
+�
+, Mpn
+qn
+�
+and ∇u(x, t) ∈
+L∞ �� ε
+2, T
+�
+, Mpn
+qn
+�
+, where pn > Nγ/(θ − γ). It follows that
+����
+� t
+ε/2
+S(t − τ)|∇u(·, τ)|γ dτ|L∞
+����
+≤ C
+� t
+ε/2
+��S(t − τ)|∇u(·, τ)|γ|B0
+∞,1
+�� dτ
+≤ C
+� t
+ε/2
+���S(t − τ)|∇u(·, τ)|γ|NNγ/pn
+pn/γ,qn/γ,1
+��� dτ
+≤ C
+� t
+ε/2
+�
+1 + (t − τ)−Nγ/θpn� ��|∇u(·, τ)|γ|N0
+pn/γ,qn/γ,∞
+�� dτ
+≤ C
+� t
+ε/2
+(t − τ)−Nγ/θpn
+���|∇u(·, τ)|γ|Mpn/γ
+qn/γ
+��� dτ
+≤ C
+� t
+ε/2
+(t − τ)−Nγ/θpn ��∇u(·, τ)|Mpn
+qn
+��γ dτ
+≤ C
+�
+t − ε
+2
+�1−Nγ/θpn
+sup
+ε/2≤τ≤t
+��∇u(·, τ)|Mpn
+qn
+��γ dτ
+≤ CT 1−Nγ/θpn
+sup
+ε/2≤τ≤t
+��∇u(·, τ)|Mpn
+qn
+��γ dτ < ∞
+(4.9)
+for ε/2 ≤ t ≤ T ≤ 1. On the other hand, we have
+∥S(t − ε/2)u(·, ε/2)|L∞∥ ≤ C∥S(t − ε/2)u(·, ε/2)|B0
+∞,1∥
+≤ C∥S(t − ε/2)u(·, ε/2)|NN/p
+p,q,1∥
+≤ C
+�
+1 + (t − ε/2)−N/θp� ��|u(·, ε/2)||Mp
+q
+��
+≤ C(ε/2)−N/θp ��|u(·, ε/2)||Mp
+q
+�� < ∞
+(4.10)
+for ε ≤ t ≤ T ≤ 1. Since
+u(x, t) =
+�
+S
+�
+t − ε
+2
+�
+u
+�
+·, ε
+2
+��
+(x) +
+� t
+ε/2
+[S(t − τ)|∇u(·, τ)|γ] (x) dτ,
+21
+
+this together with (4.9) and (4.10) implies that u(x, t) ∈ L∞([ε, T] × RN) for every
+ε > 0. We next prove that ∇u(x, t) ∈ L∞([ε, T] × RN)for every ε > 0. It follows
+that
+����
+� t
+ε/2
+Sj(t − τ)|∇u(·, τ)|γ dτ|L∞
+����
+≤ C
+� t
+ε/2
+��Sj(t − τ)|∇u(·, τ)|γ|B0
+∞,1
+�� dτ
+≤ C
+� t
+ε/2
+���Sj(t − τ)|∇u(·, τ)|γ|NNγ/pn
+pn/γ,qn/γ,1
+��� dτ
+≤ C
+� t
+ε/2
+�
+1 + (t − τ)−Nγ/θpn−1/θ� ��|∇u(·, τ)|γ|N0
+pn/γ,qn/γ,∞
+�� dτ
+≤ C
+� t
+ε/2
+(t − τ)−Nγ/θpn−1/θ ���|∇u(·, τ)|γ|Mpn/γ
+qn/γ
+��� dτ
+≤ C
+� t
+ε/2
+(t − τ)−Nγ/θpn−1/θ ��∇u(·, τ)|Mpn
+qn
+��γ dτ
+≤ C
+�
+t − ε
+2
+�1−Nγ/θpn−1/θ
+sup
+ε/2≤τ≤t
+��∇u(·, τ)|Mpn
+qn
+��γ dτ
+≤ CT 1−Nγ/θpn−1/θ
+sup
+ε/2≤τ≤t
+��∇u(·, τ)|Mpn
+qn
+��γ dτ < ∞
+(4.11)
+for ε/2 ≤ t ≤ T ≤ 1. On the other hand, we have
+∥Sj(t − ε/2)u(·, ε/2)|L∞∥ ≤ C∥Sj(t − ε/2)u(·, ε/2)|B0
+∞,1∥
+≤ C∥Sj(t − ε/2)u(·, ε/2)|NN/p
+p,q,1∥
+≤ C
+�
+1 + (t − ε/2)−N/θp−1/θ� ��|u(·, ε/2)||Mp
+q
+��
+≤ C(ε/2)−N/θp−1/θ ��|u(·, ε/2)||Mp
+q
+�� < ∞
+(4.12)
+for ε ≤ t ≤ T ≤ 1. Since
+∇u(x, t) =
+�
+Sj
+�
+t − ε
+2
+�
+u
+�
+·, ε
+2
+��
+(x) +
+� t
+ε/2
+[Sj(t − τ)|∇u(·, τ)|γ] (x) dτ,
+this together with (4.11) and (4.12) implies that ∇u(x, t) ∈ L∞([ε, T] × RN) for
+every ε > 0.
+Finally, we prove the uniqueness of the solution.
+Assume that u(1)(x, t) and
+u(2)(x, t) are solutions to (4.8) satisfying
+sup
+0≤t≤T
+t−s/θ∥u(j)(·, t)|Mp
+q ∥ + t(−s+1)/θ∥∇u(j)(·, t)|Mp
+q ∥ < ∞.
+22
+
+Let u = u(1) − u(2) and h(t) = ∥u(·, t)|Mp
+q ∥. Then exactly in the same way as in the
+proof of Lemma 4.4, we have
+sup
+0<t≤T
+{t−s/θh(t) + t(−s+1)/θh(t)} ≤ CMγ−1 sup
+0<t≤T
+{t−s/θh(t) + t(−s+1)/θh(t)}
+≤ 1
+2 sup
+0<t≤T
+{t−s/θh(t) + t(−s+1)/θh(t)}.
+Therefore, we see that u ≡ 0, and the proof is complete. ✷
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