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+arXiv:2301.00518v1  [math.NT]  2 Jan 2023
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER
+GLOBAL FUNCTION FIELDS
+KI-SENG TAN
+Abstract. For an elliptic curve A defined over a global function field K of char-
+acteristic p > 0, the p-Selmer group of the Frobenius twist A(p) of A tends to have
+larger order than that of A. The aim of this note is to discuss this phenomenon.
+1. Introduction
+For an elliptic curve A defined over a global function field K of characteristic
+p > 0, the p-Selmer group of the Frobenius twist of A tends to have larger order
+than that of A. The aim of this note is to discuss this phenomenon.
+The Frobenius twist A(p) is the base change A ×K K of A over the absolute
+Frobenius Frobp : K −→ K, x �→ xp.
+1.1. The main results. We assume that A/K is ordinary, having semi-stable re-
+duction everywhere. Let ∆A/K denote the divisor of the global minimal discriminant
+of A/K. Comparing the defining equation for both curves yields
+∆A(p)/K = p · ∆A/K.
+(1)
+Let Apν be the the kernel of the multiplication by pν on A and let
+Selpν(A/K) ⊂ H1(K, Apν)
+denote the pν-Selmer group of A/K. If p = 2, let ð′ be the set of places of K at
+which A has non-split multiplicative reduction and has the group of components of
+even order; otherwise, put ð′ = ∅. Let Sb denote the set of bad reduction places of
+A/K. Write
+Sb = ð′ ⊔ ð.
+Let k = K(A(p)
+p ( ¯Ks)) and let ð0 ⊂ ð be the subset of places splitting completely
+over k. Let ℏ denote the p-rank of the subgroup of Hom(Gal(¯ks/k), Z/pZ) consisting
+of homomorphisms unramified everywhere and locally trivial at every places of k
+sitting over ð. Our main results are as follow. Let q denote the order of the constant
+field of K.
+Proposition 1. There exists an integer ǫ1, 2ℏ + 1 + |ð0| ≥ ǫ1 ≥ −|ð0|, such that
+logp | Selp(A(p)/K)| = (p − 1) deg ∆A/K
+12
+· logp q + ǫ1.
+Acknowledgement: This research was supported in part by Ministry of Science and Technol-
+ogy of Taiwan, MOST 109-2115-M-002-008-MY2. The author thanks F. Trihan for many valuable
+suggestions especially for helping him with the proof of Lemma 3.1.1.
+1
+
+2
+KI-SENG TAN
+Proposition 2. There exists an integer ǫ2, 2ℏ + 1 + |ð0| ≥ ǫ2 ≥ −2ℏ − 3|ð0|, such
+that for each positive integer ν,
+logp | Selpν+1(A(p)/K)| = (p − 1) deg ∆A/K
+12
+· logp q + logp | Selpν(A/K)| + ǫ2.
+Note that
+deg ∆A/K
+12
+is a non-negative integer (see [LLTT16, §2.2.1]), it is zero, if
+and only if A/K is isotrivial. Let Xp∞(A/K) denote the p-primary part of the
+Tate-Shafarevich group of A/K, let Xdiv(A/K) be its p-divisible subgroup, and
+denote the p-cotorsion
+X(A/K) := Xp∞(A/K)/Xdiv(A/K) = Selp∞(A/K)/ Seldiv(A/K),
+where Seldiv(A/K) is the p-divisible subgroup of Selp∞(A/K).
+Let r denote the
+Zp-co-rank of Seldiv(A/K).
+If ν is greater than the exponents of X(A/K) and
+X(A(p)/K), then
+| Selpν(A/K)| = |X(A/K)| · prν and | Selpν+1(A(p)/K)| = |X(A(p)/K)| · pr(ν+1).
+It follows from Proposition 2 that
+logp |X(A(p)/K)| + r = (p − 1) deg ∆A/K
+12
+· logp q + logp |X(A/K)| + ǫ2.
+Such kind of formulae is suggested by the conjectured Birch and Swinnerton-Dyer
+formulae (see [Ta66, Tan95]) for both A(p)/K and A/K.
+Next, let L/K be a Zd
+p-extension ramified only at a finite number of ordinary
+places of A/K.
+Write Γ := Gal(L/K) and ΛΓ := Zp[[Γ]].
+Let Z be a finitely
+generated torsion ΛΓ-module, so that there is an exact sequence
+0
+� �m
+i=1 ΛΓ/(pαi) ⊕ �n
+j=1 ΛΓ/(η
+βj
+j )
+� Z
+� N
+� 0,
+(2)
+where α1, ..., αm, β1, ..., βn are positive integers, η1, ..., ηn ∈ ΛΓ are irreducible, rela-
+tively prime to p, and N is pseudo-null. Although the exact sequence is not canon-
+ical, the modules �m
+i=1 ΛΓ/(pαi) and �n
+j=1 ΛΓ/(η
+βj
+j ) are uniquely determined by Z,
+we call them the p part and the non-p part of Z, call pα1, ..., pαm the elementary
+µ-invariants and m the µ-rank of Z. If Z is non-torsion, define the µ-rank to be ∞.
+Consider the Pontryagin dual X, X(p) of Selp∞(A/L), Selp∞(A(p)/L). They are
+finitely generated over ΛΓ (see [Tan14]). Put
+ð1 := {v ∈ ð | v splits completely over kL}.
+Proposition 3. The µ-rank of X(p) is at least
+(p−1) deg ∆A/K
+12
+· logp q − |ð1|.
+If L contains K(∞)
+p
+, the constant Zp-extension over K, then X and X(p) are torsion
+[OT09, Tan14], in this case |ð1| = 0.
+Proposition 4. If L contains K(∞)
+p
+, then the µ-rank m of X(p) equals
+(p−1) deg ∆A/K
+12
+·
+logp q. Moreover, if pα1, ..., pαm, α1 ≥ · · · ≥ αm, are the elementary µ-invariants
+of X(p), then those of X are pα1−1, ..., pαm′−1, where m′ is the greatest i such that
+αi > 1.
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+3
+For a finite extension F/K, let wF denote the p-completion of the divisor class
+group of of kF and for a Ze
+p sub-extension M/K of L/K, put wM := lim
+←−K⊂F ⊂M wF,
+which is finitely generated torsion over ΛGal(kM/k). Actually, by [Crw87], the char-
+acteristic ideal of wM has a generator ΘM := lim
+←−F ΘF, where basically for each
+F, ΘF ∈ Zp[Gal(kF/k)] is the Stickelberger element defined in [Ta84, §V.1.1], in
+particular, we have
+pL/M(ΘL) = ΘM · ∗,
+(3)
+where pL/M : ΛGal(Lk/k) −→ ΛGal(Mk/k) is the continuous Zp-algebra homomorphism
+extending the quotient map Gal(Lk/k) −→ Gal(Mk/k) and ∗ ∈ ΛGal(Mk/k) is a
+fudge factor not divisible by p.
+For simplicity, we shall identify ΛGal(Mk/k) with ΛGal(M/K), and view ΘL, wL as
+objects over ΛΓ. In the special case where L = K(∞)
+p
+, the module wL has trivial
+µ-rank, hence ΘL is not divisible by p. To see this, let Y be the complete smooth
+curve defined over the constant field of k, having k as its function field. For every
+finite sub-extensions F/K of L/K, we have the exact sequence
+0
+� w0
+F
+� wF
+deg � Zp
+� 0
+and w0
+F[p] is contained in the subgroup of p-division points of the Jacobian variety
+of Y. Therefore, the order of w0
+F[p] is bounded, and hence wL[p] = 0. In general,
+(3) says that if L contains K(∞)
+p
+, then ΘL is not divisible by p. Also, in this case,
+ð1 = ∅. Hence Proposition 4 is a special case of the following proposition.
+Proposition 5. If ΘL is not divisible by p and ð1 = ∅, then the µ-rank m of
+X(p) equals
+(p−1) deg ∆A/K
+12
+· logp q. Moreover, if pα1, ..., pαm, α1 ≥ · · · ≥ αm, are the
+elementary µ-invariants of X(p), then those of X are pα1−1, ..., pαm′−1, where m′ is
+the greatest i such that αi > 1.
+Since the Frobenius and Verschiebung induce pseudo isomorphisms between the
+non-p parts of X and X(p), the proposition implies the characteristic ideal of X(p)
+is the q
+(p−1) deg ∆A/K
+12
+multiple of that of X. If L = K(∞)
+p
+, this is also a consequence of
+the main theorem of [LLTT16].
+1.2. Notation. For a field F, let ¯F and ¯F s denote its algebraic closure and separable
+closure, and denote GF = Gal( ¯F s/F). For a place v, let Ov, πv and Fv denote the
+ring of integers, an uniformizer and the residue field. Write qv for |Fv|.
+Let Sss denote the set of place v of K at which A has supersingular reduction.
+For a set S of places of K and an algebraic extension F, let S(F) denote the set of
+places of F sitting over S. For an endomorphism ϕ of an abelian group H, let H[ϕ]
+denote the kernel. Use ∨ for Pontryagin dual, ∼ for pseudo isomorphism.
+In this note we use flat or Galois cohomology. Let
+F : A −→ A(p) and V : A(p) −→ A
+be the Frobenius and the Verschiebung homomorphisms. We have the exact se-
+quences
+
+4
+KI-SENG TAN
+0
+� Cp
+� Ap
+F
+� E(p)
+p
+� 0,
+(4)
+as well as
+0
+� E(p)
+p
+� A(p)
+p
+V
+� Cp
+� 0,
+(5)
+where Cp = ker F is connected and E(p)
+p
+= ker V, ´etale (see [LSc10]). For a field F
+containing K, we have A(p)
+p (F) = E(p)
+p (F). In particular, k = K(E(p)
+p ( ¯Ks)). Note
+that for p = 2, k = K, because the non-trivial point of E(p)
+p ( ¯Ks) is the only Galois
+conjugate of itself.
+1.3. The proofs. The key ingredient in the proof of Proposition 1 is local, concern-
+ing the kernel of the natural map H1(Kv, E(p)
+p ) −→ H1(Kv, A(p)), especially when v
+is a place of supersingular reduction. Lemma 2.2.1, Lemma 2.2.2 and Lemma 2.3.3
+treat different types of reduction and provide us criteria, in terms of the the con-
+ductor of the corresponding cyclic extension over kv, for determining if an element
+of H1(Kv, E(p)
+p ) is in such kernel.
+With the local criteria available and with the help of global class field, in Propo-
+sition 3.2.5, we determine the order of Sel(E(p)
+p /K), the E(p)
+p -part of Selp(A(p)/K).
+In doing so, we find an interesting phenomena that the discrepancy between the
+two discriminants as described in (1) is solely contributed by supersingular places
+(see (27)). Next, in Proposition 3.2.6 we determine the order of Sel(Cp/K), the
+Cp-part of Selp(A/K), by using the Poitou-Tate duality [Ces15]. Then Proposition
+1 is proved at the end of §3.2, as a consequence of the above two propositions.
+Proposition 2 is proved in §3.3 by applying Cassels-Tate duality.
+In §3.4, using a theorem of Monsky [Mon81], we establish a method of reducing
+the proofs of Proposition 3, 5 to the d = 1 case. Since the method can be applied
+to more general situations, we loosen the condition in that subsection by allowing
+A to be an ordinary abelian variety defined over a global field K. The main result
+is summarized in Lemma 3.4.4. As a consequence of this lemma and Proposition 1,
+2, the proofs of Proposition 3, 5 are given in the final subsection §3.5.
+2. Local fields
+At each place v of K, we have the long exact sequence
+· · ·
+� A(p)(Kv)
+V∗ � A(Kv)
+∂
+� H1(Kv, E(p)
+p )
+jv
+� H1(Kv, A(p))
+� · · ·
+(6)
+derived from 0
+� E(p)
+p
+j
+� A(p)
+V
+� A
+� 0 . The aim of this section is to
+determine ker(jv) = coker(V∗). For a place w of k sitting over v, the abelian group
+H1(kw, E(p)
+p ) = Hom(Gkw, Z/pZ) = Hom(k∗
+w/(k∗
+w)p, Z/pZ),
+(7)
+so each ξ ∈ H1(kw, E(p)
+p ) determines a degree p cyclic extension kw,ξ/kw.
+Let ordv denote the valuation on ¯Kv having ordv(πv) = 1.
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+5
+2.1. Frobenius and Verschiebung. Let F (resp. F (p)) denote the formal group
+law associated to A (resp. A(p)) over Kv. Since A has semi-stable reduction, F is
+stable under local field extensions. Let ¯A denote the reduction of A at v. For a
+place w of an algebraic extension F of K, sitting over v, the pre-image Ao(Fw) of
+0 ∈ ¯A(Fw) under the reduction map A(Fw) −→ ¯A(Fw) is identified with F(mw) via
+a bijection ι. Let ι(p) be the corresponding bijection associated to A(p)
+o (Fw).
+Let P ∈ Ov[[t]] be the unique power series fitting into the commutative diagram
+F(mw)
+Frobp �
+ι
+�
+F (p)(mw)
+P
+�
+ι(p)
+�
+F(mw)
+ι
+�
+Ao(Fw)
+F
+� A(p)
+o (Fw)
+V
+� Ao(Fw).
+(8)
+An element ξ ∈ mw satisfies P(ξ) = 0 if and only if ι(p)(ξ) ∈ E(p)
+p (Fw) ∩ A(p)
+o (Fw).
+Suppose the formal group law of F (resp. F (p)) is given by the Ov-coefficient
+power series f(X, Y ) (resp. f (p)(X, Y )).
+If ti is a solution to P(t) = 0, then P(f (p)(t, ti)) = P(t), furthermore, since
+A/Kv is ordinary, by [Sil86, §IV.7.2], P′(0) ̸= 0, and hence P′(ti) ̸= 0, too.
+Lemma 2.1.1. We have P(t) = u(t) · P(t), where u(t) is a unit in Ov[[t]] and
+P(t) is the associated distinguished polynomial. The polynomial P(t) is separable
+with P(0) = 0. If v is a supersingular place, then deg P = p; if v is ordinary, then
+P(t) = t.
+Proof. The first assertion follows from the claim that πv does not divides P(t). For
+ordinary v, because E(p)
+p ( ¯Kv) ∩ A(p)
+o ( ¯Kv) = {0}, we know that 0 is the only root of
+P(t) in ¯Kv. This implies that the distinguished polynomial P(t) = t. For supersin-
+gular v, the group E(p)
+p ( ¯Kv) ∩ A(p)
+o ( ¯Kv) ≃ Z/pZ, so deg P(t) = p, furthermore, since
+P(ti) = 0 and P′(ti) ̸= 0, we have P(ti) = 0 and P ′(ti) ̸= 0.
+To prove the claim, we first consider the case where v is a place of good reduction.
+The formal group law associated to ¯A is given by ¯f(X, Y ) := f(X, Y ) (mod (πv)),
+which has height 1 or 2, so
+¯
+P := P (mod (πv)) is non-zero. This proves the claim.
+For a multiplicative place v, we prove the claim by showing that the Verschiebung
+gives rise to an isomorphism F (p)(mv) −→ F(mv). If v is a split-multiplicative place
+and ˜Q is the local Tate period of A, then ˜Qp is the local Tate period of A(p) and
+the Verschiebung is given by K∗
+v/ ˜QpZ −→ K∗
+v/ ˜QZ, induced from the identity map
+on K∗
+v. This implies F (p)(mv) −→ F(mv) is an isomorphism, and hence the claim
+follows.
+If v is non-split multiplicative, then A/Kv is the twist of a split multiplicative
+elliptic curve B/Kv by the unramified quadratic extension Lw/Kv. Write Zv for the
+kernel of the norm map NLw/Kv : O∗
+w −→ O∗
+v. Then F (p)(mv) −→ F(mv) is given
+by the identity map Zv −→ Zv, so it is an isomorphism.
+□
+
+6
+KI-SENG TAN
+2.2. Ordinary places. The proof of Lemma 2.1.1 shows that if v is a split multi-
+plicative place, then V∗ is given by K∗
+v/ ˜QpZ −→ K∗
+v/ ˜QZ, and hence surjective, so by
+(6), ker(jv) = 0.
+Lemma 2.2.1. Let v be a multiplicative place. Then ker(jv) = 0, unless v ∈ ð′, in
+which case ker(jv) is of order 2 = p, consisting of ξ ∈ H1(Kv, E(p)
+p ) with Kv,ξ/Kv
+unramified.
+Note that if p = 2, then k = K, so Kv,ξ is defined.
+Proof. Suppose v is non-split multiplicative and let B/Kv and Lw/Kv be as in the
+proof of Lemma 2.1.1. Because A/Lw is split-multiplicative, we have the injection
+H1(Lw, E(p)
+p ) −→ H1(Lw, A(p)), and hence
+ker(jv) = ker(H1(Lw/Kv, E(p)
+p (Lw)) −→ H1(Lw/Kv, A(p)(Lw))).
+For p ̸= 2, we have H1(Lw/Kv, E(p)
+p (Lw)) = 0, so ker(jv) = 0.
+Denoting G =
+Gal(Lw/Kv), we have the commutative diagram
+H1(Lw/Kv, E(p)
+p (Lw))
+�
+≃
+�
+H1(Lw/Kv, A(p)(Lw))
+≃
+�
+Hom(G, Z/2Z)
+� B(p)(Kv)/ NLw/Kv(B(p)(Lw)).
+The non-trivial element of Hom(G, Z/2Z), sending the generator of G to the point of
+B(p)(Kv) obtained by the Tate local period Qv of B/Kv, corresponds to an element
+of ker(jv) if and only if Qv ∈ NLw/Kv(L∗
+w), or equivalently ordv Qv is even.
+□
+Lemma 2.2.2. Suppose v is a good ordinary place and w is a place of k sitting over
+v. Then ker(jw) is of order p, consisting of ξ ∈ H1(kw, E(p)
+p ) with kw,ξ/kw unramified.
+If Kv ̸= kw, then ker(jv) is trivial.
+Proof. In view of the diagram (8), Lemma 2.1.1 says A(p)
+o (kw)
+∼
+V
+� Ao(kw) . We
+have to determine the cokernel of the induced ¯V : ¯A(p)(Fw) −→ ¯A(Fw). The Frobe-
+nius ¯F identifies ¯A(p)(Fw) with ¯A(Fw) and under this, ¯V is identified with the mul-
+tiplication by p. The cokernel in question is isomorphic to
+¯A(Fw)/p ¯A(Fw) ≃ ¯Ap(Fw) = ¯E(p)
+p (Fw).
+The snake lemma applied to the diagram
+0
+� A(p)
+o (kw)
+�
+V
+≃
+�
+A(p)(kw)
+�
+V
+�
+¯A(p)(Fw)
+�
+¯V
+�
+0
+0
+� Ao(kw)
+� A(kw)
+� ¯A(Fw)
+� 0
+implies that the reduction map E(p)
+p (kw) −→ ¯E(p)
+p (Fw) is an isomorphism, so ker(jw)
+is of order p, and by [Mil06, §I.3.8], it is formed by all unramified ξ. If Kv ̸= kw,
+then ¯E(p)
+p (Fv) = E(p)
+p (Kv) = 0, and a similar argument shows ker(jv) is trivial.
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+7
+□
+2.3. Supersingular places. Suppose v is supersingular. Choose a nonzero t1 ∈ mw
+(in some Fw) with ι(p)(t1) ∈ E(p)
+p (Fw). Let [u] denote the multiplication by u on A(p).
+Because tu := ι(p)−1◦[u]◦ι(p)(t1) = ut1+higher terms, if p ∤ u, then ordv tu = ordv t1.
+Denote
+nv :=
+�
+i∈F∗p
+ordv ti = (p − 1) ordv(tu), for (u, p) = 1.
+(9)
+Write
+P(t) = tp + zp−1tp−1 + · + z1t,
+with
+ordv z1 = nv.
+(10)
+For s, t ∈ m write s ⊞ t for F (p)(s, t). Then ι(p)(s ⊞ t) = ι(p)(s) + ι(p)(t). Diagram
+(8) shows that for a given b ∈ mv, if a0 ∈ mw is a root of P(t) − b = 0, then all
+other roots equal au := a0 ⊞ tu = F (p)(a0, tu), u = 1, ..., p − 1. Let Q(t) be the
+distinguished polynomial associated to P(t) − b, whose roots are also a0, ..., ap−1.
+Since Q(0) = −b · ξ, for some ξ ∈ O∗
+v,
+p−1
+�
+u=0
+ordv au = ordv b.
+(11)
+Since F (p)(X, 0) = F (p)(0, X) = X, we can write F (p)(X, Y ) = X + Y + XY ·
+g(X, Y ). It follows that au = a0 + tu + higher terms. Hence
+ordv(au − au′) =
+nv
+p − 1.
+(12)
+Lemma 2.3.1. For every b ∈ mv with ordv b > pnv
+p−1, there exists an element a ∈ mv,
+with ordv a >
+nv
+p−1, such that P(a) = b. Conversely, for a ∈ mv, with ordv a >
+nv
+p−1,
+the element b = P(a) ∈ mv has ordv b = ordv a + nv > pnv
+p−1.
+Proof. If b ∈ mv and a0 is a solution to P(t) = b, with ordv(a0) >
+nv
+p−1, then by (12),
+other solutions au have ordv au =
+nv
+p−1. Therefore, if a ∈ mv, with ordv a >
+nv
+p−1, and
+b = P(a), then by (11), ordv b = ordv a + nv. Conversely, if b ∈ mv, ordv b > pnv
+p−1,
+by (11), there is a solution a to P(t) = b, such that ordv a >
+nv
+p−1. Comparing the
+valuations, we deduce that a is the only Galois conjugate of itself, whence a ∈ mv.
+□
+Lemma 2.3.2. If v is a supersingular place of A/K, then the cokernel of
+A(p)(Kv)
+V∗ � A(Kv)
+is of order pǫv · qnv
+v , where if kv = Kv, ǫv = 1; otherwise, ǫ = 0.
+Proof. Since ¯A(Fv) has order prime to p, by Diagram (8) we need to show the
+cokernel of F (p)(mv)
+P
+� F(mv) has the desired order.
+
+8
+KI-SENG TAN
+Let β = [
+1
+p−1nv]+1. Lemma 2.3.1 implies that P sends F (p)(mβ) onto F(mβ+nv).
+Therefore, it is sufficient to check the co-kernel of the induced homomorphism
+¯
+P : F (p)(m)/F (p)(mβ) −→ F(m)/F(mβ+nv).
+Since the kernel of
+¯
+P is of order pǫv, the proof is completed by counting.
+□
+Lemma 2.3.2 says
+| ker(jv)| = pǫv · qnv
+v .
+(13)
+Lemma 2.3.3. Let v be a place of K and w a place of k sitting over v. The group
+ker(jw) consists of all ξ with kw,ξ/kw having conductor at most pnw
+p−1.
+Proof. An element ξ ∈ ker(jw) can be written as ∂x for some x ∈ A(kw). Since
+¯A(Fw) has order prime to p, we may choose x = ι(b) ∈ Ao(kw), for some b ∈ mw.
+Let a0, ..., ap−1 be solutions to P(t) = b. Then all au are integral over Ow and
+kw,ξ = kw(a0). It follows from (12) that if Disc is the discriminant of kw,ξ/kw, then
+ordw(Disc) ≤ p · nw.
+This implies the conductor of kw,ξ/kw is at most pnw
+p−1. Classes ξ ∈ H1(kw, E(p)
+p ) with
+kw,ξ/kw unramified are in ker(jw) (see [Mil06, I.3.8]). They form a subgroup of order
+p. By the local class field theory, ramified cyclic extensions of kw of degree p and
+conductor at most m are characterized by the group Dm/Dp
+m, Dm := O∗
+w/1 + πm
+w Ow.
+In our case m = pnw
+p−1 is an integer divisible by p (by (9), because each tu ∈ kw). Since
+Ow = Fw[[πw]], the map
+D m
+p −→ Dp
+m, x �→ xp,
+is an bijection. Hence |Dm/Dp
+m| = qm−1
+w
+· (qw − 1)/q
+m
+p −1
+w
+· (qw − 1) = qnw
+w . In view of
+(13), the proof is completed by counting.
+□
+The lemma actually says that by (7),
+ker(jw) = Hom(k∗
+w/(1 + π
+pnw
+p−1
+w
+Ow) · (k∗
+w)p, Z/pZ).
+(14)
+3. Global fields
+Let X /Fq be the complete smooth curve having K as its function field. Let Xg,
+Xgo denote the open sets consisting of places where A has good reduction, good
+ordinary reduction respectively.
+3.1. Poitou-Tate duality. We first recall the following.
+Lemma 3.1.1. Let f : B/K −→ B′/K be a given isogeny of elliptic curves having
+good reductions at all v ∈ Xg and let f : B −→ B′ be the homomorphism extending f
+to N´eron models over Xg. Then N := ker [f] is a finite flat group scheme over Xg.
+Furthermore, if ˆ
+N denotes the kernel of the homomorphism ˆf : B′ −→ B extending
+the dual isogeny ˆf : B′ −→ B, then N and ˆ
+N are Cartier dual to each other.
+Note that the existence and the uniqueness of f and ˆf are due to the N´eron
+mapping property, see [BLR90, §1.2].
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+9
+Proof. Since B/Xg is an abelian scheme, the morphism f is proper. It follows that
+N /Xg is proper and quasi-finite, whence finite [EGA IV, part 4, 8.12.4] and flat
+[Mil06, III.C.8]. The exact sequence
+0
+� N
+� B
+f
+� B′
+� 0
+together with the isomorphism (see [SGA 7I, VIII.7.1], [BBM82] or [Mil06, III.C.14])
+B/Xg ≃ Ext 1
+Xg(B, Gm)
+induce the exact sequence
+· · ·
+� HomXg(B, Gm)
+� HomXg(N , Gm)
+� B′
+ˆf
+� B
+� · · ·.
+Here we apply the commutative diagram
+Ext 1
+Xg(B′, Gm)
+f ∗
+� Ext 1
+Xg(B, Gm)
+B′
+ˆf
+� B
+that extends the already known diagram on the generic fibre. Then we check the
+equality HomXg(B, Gm) = 0 fibre-wise by using the fact that over a field every
+homomorphism from an abelian variety to Gm is trivial.
+□
+Let notation be as in Lemma 3.1.1 and let N denote the generic fibre of N . For
+v ∈ Xg, we have (see [Mil06, §III.7])
+H1(Ov, N )� �
+� H1(Kv, N) .
+(15)
+Lemma 3.1.2. Let notation be as above. For v ∈ Xg, we have
+H1(Ov, N ) = ker(H1(Kv, N) −→ H1(Kv, B)).
+Proof. The lemma follows from the fact that H1(Ov, B) = 0 (see [Mil06, §III.2.1])
+and the commutative diagram of exact sequences
+B(Ov)
+� B′(Ov)
+� H1(Ov, N )
+�
+� �
+�
+H1(Ov, B)
+�
+B(Kv)
+� B′(Kv)
+� H1(Kv, N)
+� H1(Kv, B).
+□
+Let U ⊂ X be an open subscheme. Define
+S(N/U) := ker(H1(K, N) −→
+�
+v∈U
+H1(Kv, B)).
+Denote Sel(N/K) := S(N/X ), it is the kernel of
+S(N/U)
+� �
+v̸∈U H1(Kv, B) .
+
+10
+KI-SENG TAN
+Let Q(N/U) denote the cokernel of the localization map
+S(N/U)
+LN/K
+� �
+v̸∈U H1(Kv, N) .
+Lemma 3.1.3. If U ⊂ Xg, then H1(U, N ) = S(N/U).
+Proof. Let V ⊂ U be an open affine subscheme. By the localization sequence [Mil06,
+III.0.3(c)] and the computation at the beginning of [Mil06, III.7], we have the exact
+sequence
+0
+� H1(U, N )
+� H1(V, N )
+� �
+v∈U\V H1(Kv, N)/ H1(Ov, N ).
+[Gon09, Lemma 4.2] says the natural map H1(V, N ) −→ H1(K, N) is injective. The
+exact sequence implies H1(U, N ) ⊂ S(N/U). By [Gon09, Lemma 2.3], an element
+in S(N/U) can be obtained from H1(V, N ) for some V ⊂ U, and the exact sequence
+implies it is in H1(U, N ).
+□
+For U ̸= X , apply the local duality [Mil06, §III.6.10] and consider the composition
+S(Cp/U)� �
+LCp/U
+� �
+v̸∈U H1(Kv, Cp)
+∼
+� �
+v̸∈U H1(Kv, E(p)
+p )∨,
+(16)
+where the injectivity of LCp/U is due to [GoT12, Main Theorem].
+Lemma 3.1.4. If U ⊂ Xg and U ̸= X , then under (16), the group S(Cp/U) is the
+Pontryagin dual of Q(E(p)
+p /U).
+Proof. Extend F and V to F : A −→ A(p) and V : A(p) −→ A over Xg. Denote
+Cp = ker(F) and E(p)
+p
+= ker(V).
+They are Cartier dual to each other. By Lemma 3.1.3, we identify H1(U, Cp) with
+S(Cp/U). Then the lemma follows from Poitou-Tate duality [Ces15, (5.1.2)].
+□
+In view of Lemma 3.1.2, the following lemme generalizes the fact that for any place
+v ∈ Xg, the local duality identifies H1(Ov, Cp) ⊂ H1(Kv, Cp) with the annihilator of
+H1(Ow, E(p)
+p ) ⊂ H1(Kv, E(p)
+p ) [Mil06, §III, Theorem 7.1].
+Lemma 3.1.5. At each place v of K, under the duality H1(Kv, Cp) = H1(Kv, E(p)
+p )∨
+[Mil06, §III. Theorem 6.10], the kernel of j′
+v : H1(Kv, Cp) −→ H1(Kv, A), as a
+subgroup of H1(Kv, Cp), is exactly the annihilator of ker(jv) ⊂ H1(Kv, E(p)
+p ).
+We abbreviate the above as
+ker(j′
+v) = ker(jv)⊥.
+(17)
+Proof. Let ∂ denote the connecting homomorphism in the long exact sequence
+· · ·
+� A(Kv)
+F
+� A(p)(Kv)
+∂
+� H1(Kv, Cp)
+j′
+v
+� H1(Kv, A)
+� · · · ,
+(18)
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+11
+that, since p = F ◦ V, gives rise to the homomorphism ¯∂ in the diagram
+A(p)(Kv)/p · A(p)(Kv)
+¯∂
+�
+� �
+i
+�
+H1(Kv, Cp)
+H1(Kv, A(p)
+p )
+V∗
+� H1(Kv, Cp),
+(19)
+where the bottom left-arrow is due to (5) and the left down-arrow i is induced from
+the long exact sequence
+A(p)
+p (Kv)
+� A(p)(Kv)
+[p] � A(p)(Kv)
+� H1(Kv, A(p)
+p )
+� · · · .
+We claim that the diagram (19) is commutative, so that the commutative diagram
+H1(Kv, A(p)
+p ) × H1(Kv, A(p)
+p )
+V∗
+�
+(−,−)v
+� Q/Z
+H1(Kv, Cp) × H1(Kv, E(p)
+p )
+jv
+�
+(−,−)v
+� Q/Z,
+where the left-arrows are local pairings, yields the commutative diagram (see [Mil06,
+§III, Theorem 7.8])
+A(p)(Kv) × H1(Kv, A(p))
+∂
+�
+(−,−)A(p)/Kv
+� Q/Z
+H1(Kv, Cp) × H1(Kv, E(p)
+p )
+jv
+�
+(−,−)v
+� Q/Z.
+This shows that ker(jv)⊥ = Im(∂v) = ker(j′
+v). To prove the claim, we use ˇCech
+cocycles (see [Mil80, §III.2.10]).
+Let x ∈ A(p)(Kv) and denote ¯x its image in
+A(p)(Kv)/p · A(p)(Kv). Let y ∈ A(p)(K′
+w) = Hom(Spec(K′
+w), A(p)) be a p-division
+point of x over a finite extension K′
+w. Let prl, l = 1, 2, be the projection
+Spec(K′
+w) ×Spec(Kv) Spec(K′
+w) −→ Spec(K′
+w)
+to the l’th factor. Then ξ := y ◦ pr1 − y ◦ pr2 is a 1-cocycle representing the class
+i(¯x). Let z = V(y) ∈ Hom(Spec(K′
+w), A) so that F(z) = x. Then V ◦ ξ is a 1-cocycle
+representing both ¯∂(¯x) and V∗(i(¯x)).
+□
+3.2. The conductor. Recall that k = K(E(p)
+p ( ¯Ks)) so that E(p)
+p ( ¯Ks) = E(p)
+p (k),
+on which the action of the Galois group Φ := Gal(k/K) is given by an injective
+character
+c : Φ −→ F∗
+p
+such that x
+g
+= c(g) · x, for g ∈ Φ, x ∈ E(p)
+p (k). Since the order of Φ is prime to p,
+the Hochschild-Serre spectral sequence (see [Mil80, §II.2.21(a)]) yields
+H1(K, E(p)
+p )
+∼
+� H1(k, E(p)
+p )Φ
+Hom(Gk, E(p)
+p (k))Φ .
+(20)
+
+12
+KI-SENG TAN
+For w ∈ Sss(k), let ι(p)(t1) be a non-zero element of E(p)
+p (kw) as in §2.3. Put
+Mw :=
+
+
+
+
+
+(1 + tp
+1Ow) · (O∗
+w)p,
+if w ∈ Sss(k);
+k∗
+w,
+if w ∈ ð(k);
+O∗
+w,
+otherwise.
+Let A∗
+k denote the group of ideles of k and W the p-completion of k∗\A∗
+k/ �
+w Mw.
+Lemma 3.2.1. We have Sel(E(p)
+p /k) = Hom(W , E(p)
+p (k)).
+Proof. By the global class field theory, Hom(W , E(p)
+p (k)) ⊂ Hom(Gk, E(p)
+p (k)) con-
+sists of elements which are locally trivial at w ∈ ð(k), having conductors not greater
+than ordw(tp
+1) at supersingular places w, unramified at others. The lemma follows
+from Lemma 2.2.1, 2.2.2 and 2.3.3.
+□
+Every pro-p Φ-module Y can be decomposed as Y = �
+χ∈ˆΦ Y χ, where for each χ,
+Y χ denote the χ-eigenspace {y ∈ Y |
+y
+g
+= χ(g) · y} . By (20) and Lemma 3.2.1,
+Sel(E(p)
+p /K) = Hom(W , E(p)
+p (k))Φ = Hom(W c, Z/pZ).
+(21)
+For v ∈ Sss, put Wv := �
+w|v k∗
+w/((k∗
+w)p · Mw), and Uv := �
+w|v O∗
+w/Mw regarded
+as a subgroup of Wv.
+Lemma 3.2.2. If v ∈ Sss, then |Uc
+v| = qnv
+v .
+Proof. Again, by the Hochschild-Serre spectral sequence
+H1(Kv, E(p)
+p )
+∼
+� (�
+w|v H1(kw, E(p)
+p ))Φ
+(�
+w|v Hom(Gkw, E(p)
+p (k)))Φ.
+(22)
+Therefore, Lemma 2.3.3 implies ker(jv) ≃ Hom(W c
+v, Z/pZ). Then the lemma follows
+from (13), because if kw ̸= Kv, then W c
+v = Uc
+v; if kw = Kv, then we have the splitting
+exact sequence
+0 −→ Uc
+v −→ W c
+v −→ Z/pZ −→ 0.
+□
+3.2.1. An idelic computation. In this subsection only, we consider a general situation
+in which for w ∈ Sss(k), the Mw in the previous subsection is replaced by
+Mw := (1 + πav
+w Ow) · (O∗
+w)p,
+where v is a place of K sitting below w and av is a chosen integer depending only on
+v, and we keep Mw unchanged for other w. Denote a := (av)v∈Sss. Then let Wa be
+the p-completion of k∗\A∗
+k/ �
+w Mw so that W = Wo, where o := (p · ordw tv)v∈Sss.
+Put Ua := �
+w∈Sss O∗
+w/Mw, Wa := �
+w∈Sss k∗
+w/((k∗
+w)p · Mw). Assume that
+|Uc
+a| =
+�
+v∈Sss
+qρv
+v .
+(23)
+Let ¯Uc
+a denote the image of the natural map
+ςc
+a : Uc
+a −→ W c
+a .
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+13
+Let V be the group of ð(k)-units of k. Then ker(ςc) equals the image of
+̺c
+a : (V/pV )c −→ Uc
+a
+induced by the localization map V −→ �
+w∈Sss(k) O∗
+w. The torsion part of V is finite
+of order prime to p. The map V −→ �
+v∈ð Rv, where Rv := �
+w|v k∗
+w/O∗
+w, is injective
+on the free part of V . The module Rv is fixed by the decomposition subgroup Φv
+and is the regular representation of Φ/Φv. If k = K, the Zp-rank of (Zp ⊗Z V )c is
+max{|ð0| − 1, 0}; otherwise it is |ð0|, so
+| ker(ςc
+a)| = | Im ̺c
+a| ≤ p|ð0|.
+(24)
+If ð = ∅, letג denote the p-completion of the divisor class group of k; otherwise, let
+ג be the ð(k)-class group. Then ℏ is the p-rank ofג. The exact sequence
+0
+� ¯Uc
+a
+� W c
+a
+�גc
+� 0
+yields
+0
+� ¯Uc
+a
+� W c
+a [p]
+κc �גc[p]
+(25)
+Put
+τ :=
+�
+logp | Im(κc)| + 1,
+if ð = ∅ and k = K;
+logp | Im(κc)|,
+otherwise.
+(26)
+Via (20), we identify H1(K, E(p)
+p ) with Hom(Gk, E(p)
+p (k))Φ, so that the conductor
+of its element at each place v is defined. Let Xo ⊂ X be the complement of Sss.
+Definition 3.2.3. Define Sela(E(p)
+p /K) to be the subgroup of S(E(p)
+p /Xo) consisting
+of elements having conductors not greater than av at each v ∈ Sss.
+Lemma 3.2.4. Assuming (23), we have
+logp | Sela(E(p)
+p /K)| =
+�
+v∈Sss
+ρv · deg v + ˜ε1 − ˜ε2,
+where ˜ε1 = τ ≤ ℏ + 1, ˜ε2 = logp | ker(ςc
+a)| ≤ |ð0| ≤ |Sb|.
+Proof. Similar to (21), Sela(E(p)
+p /K) = Hom(W c
+a , Z/pZ). We shall write Wa addi-
+tively. If ð ̸= ∅ or k ̸= K, then W c
+a is finite with |W c
+a /pW c
+a | = |W c
+a [p]|; otherwise, W c
+a
+is finitely generated over Zp of rank 1, hence |W c
+a /pW c
+a | = p · |W c
+a [p]|. The lemma
+follows from (23), (24), (25) and (26).
+□
+Proposition 3.2.5. We have
+logp | Sel(E(p)
+p /K)| = (p − 1) deg ∆A/K
+12
+· logp q + ˜ε1 − ˜ε2,
+where ˜ε1 = τ ≤ ℏ + 1, ˜ε2 = logp | ker(ςc
+o)| ≤ |ð0| ≤ |Sb|.
+Proof. In view of (21), Lemma 3.2.2, and Lemma 3.2.4, we need to show that
+�
+v∈Sss
+nv · deg v = (p − 1) deg ∆A/K
+12
+.
+(27)
+
+14
+KI-SENG TAN
+Let ω be an invariant differential of A/K and for each place v let ω0,v and ω(p)
+0,v be
+respectively local N��eron differentials of A and A(p). By [Sil86, §IV. Corollary 4.3],
+ordv(V∗ω0,v/ω(p)
+0,v) = ordv
+dP(t)
+dt
+|t=0,
+which together with Lemma 2.1.1 and (10) yield
+�
+v∈Sss
+nv · deg v =
+�
+all v
+ordv(V∗ω0,v/ω(p)
+0,v) · deg v.
+(28)
+Now the formula (8) in [Tan95] implies
+deg ∆A/K
+12
+=
+�
+all v
+ordv( ω
+ω0,v
+) · deg v =
+�
+all v
+ordv(V ∗( ω
+ω0,v
+)) · deg v,
+and
+p · deg ∆A/K
+12
+= deg ∆A(p)/K
+12
+=
+�
+all v
+ordv(V ∗ω
+ω(p)
+0,v
+) · deg v.
+These and (28) lead to the desired equality.
+□
+3.2.2. Computing S(Cp/U). Next, we investigate S(Cp/U) for U = Xgo, Xg, or X .
+Let Sngo be the complement of Xgo in X , S′
+ngo := U ∩ Sngo, S′
+ss := U ∩ Sss, and † the
+complement of U in X . Write ‡ for † ∪ ð. We first treat the case in which Xgo ̸= X ,
+or equivalently, A/K is not isotrivial1. Put
+¯W :=
+�
+w∈Sngo(k)
+k∗
+w/(k∗
+w)p
+and
+¯
+W := k∗\A∗
+k/(
+�
+w∈Xgo(k)
+O∗
+w ×
+�
+w∈Sngo(k)
+(k∗)p).
+If ℓc and ℓc−1 denote the c and c−1 eigenspaces of the regular representation of Φ on
+Fp[Φ], then E(p)
+p (k) = ℓc. Hence
+�
+w∈Sngo(k)
+H1(kw, E(p)
+p ) = Hom( ¯W, ℓc) = Hom( ¯W ⊗Fp ℓc−1, Z/pZ) = ( ¯W ⊗Fp ℓc−1)∨,
+and
+S(E(p)
+p /Xgo × Spec k) = Hom( ¯
+W , ℓc) = (( ¯
+W c/p ¯
+W ) ⊗Fp ℓc−1)∨.
+In view of (16) and Lemma 3.1.4,
+S(Cp/Xgo × Spec k) = ker( ¯W −→
+¯
+W /p ¯
+W ).
+Therefore, by the Hochschild-Serre spectral sequence again,
+S(Cp/Xgo) = S(Cp/Xgo × Spec k)Φ = ker( ¯W c −→
+¯
+W c/p ¯
+W c).
+For each w, put
+Mw := Mw/Mw ∩ (k∗
+w)p = Mw · (k∗
+w)p/(k∗
+w)p.
+1Recall that A/K is assumed to be ordinary, having semi-stable reduction everywhere.
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+15
+Lemma 3.1.5 gives rise to the left block of the following commutative diagram
+H1(kw, Cp)
+∼
+� H1(kw, E(p)
+p )∨
+∼
+� Hom(k∗
+w/(k∗
+w)p, Z/pZ)∨
+∼
+� k∗
+w/(k∗
+w)p
+ker(j′
+w)
+∼
+�
+��
+�
+ker(jw)⊥
+∼
+�
+��
+�
+Hom(k∗
+w/Mw · (k∗
+w)p, Z/pZ)⊥
+∼
+�
+��
+�
+M w.
+��
+�
+The middle block is obtained by taking the dual of the commutative diagram
+H1(kw, E(p)
+p )
+∼
+� Hom(k∗
+w/(k∗
+w)p, Zp/pZp)
+ker(jw)
+��
+�
+∼
+� Hom(k∗
+w/(k∗
+w)p · Mw, Zp/pZp),
+��
+�
+(29)
+which is due to Lemma 2.3.3 together with the local class field theory (see (7) and
+(14)) while the right block is a direct consequence of duality.
+shows that if
+¯
+M = �
+w∈†(k) k∗
+w/(k∗
+w)p × �
+w∈S′ngo(k) M w, then
+S(Cp/U) = ker(
+¯
+M c −→
+¯
+W c/p ¯
+W c).
+(30)
+To proceed further, we introduce
+˜
+M =
+�
+w∈†(k)
+k∗
+w/(O∗
+p)p ×
+�
+w∈S′ngo(k)
+Mw · (O∗
+p)p/(O∗
+p)p
+and
+˜
+W ; = k∗\A∗
+k/(
+�
+w∈Xgo(k)
+O∗
+w ×
+�
+w∈Sngo(k)
+(O∗
+w)p).
+Then
+¯
+M =
+˜
+M /p
+˜
+M , and since the kernel of
+˜
+W
+� �
+¯
+W
+is inside p ˜
+W , we also have
+¯
+W /p ¯
+W =
+˜
+W /p ˜
+W . Hence (30) implies
+S(Cp/U) = ker(
+˜
+M c/p
+˜
+M c −→
+˜
+W c/p ˜
+W c).
+(31)
+Let V and M denote the kernel and image of the natural map ˜ς :
+˜
+M −→
+˜
+W .
+Let ˜ξ ∈ V and let ξ be a lift of ˜ξ to �
+w∈†(k) k∗
+w × �
+w∈S′ngo(k) Mw · (O∗
+p)p. Then there
+are α ∈ k∗ and θ ∈ �
+w∈Xgo(k) O∗
+w × �
+w∈Sngo(k)(O∗
+w)p such that
+ξ = α · θ.
+(32)
+Let V‡ denote the group of ‡(k)-units of k. The equality (32) implies that α is in
+the subgroup V ′
+‡ ⊂ V‡ consisting of those elements which are inside Mw · (O∗
+w)p, for
+all w ∈ S′
+ss(k). Suppose there is another expression ξ = α′ · θ′. Then α′α−1 actually
+belongs to the group F∗
+k of global units. Since Sngo = † ⊔ S′
+ngo, the correspondence
+˜ξ ↔ α (mod F∗
+k) gives rise to an isomorphism
+V ≃ V ′
+‡/F∗
+k.
+The exact sequence 0
+� V
+�
+˜
+M
+� M
+� 0 induces the exact sequence
+˜
+M [p]
+� M [p]
+∂
+� V /pV
+�
+˜
+M /p
+˜
+M
+� M /pM
+� 0 .
+
+16
+KI-SENG TAN
+For an h ∈ M [p], let ˜η be one of its preimage in
+˜
+M and let η be a lift of ˜η to
+�
+w∈†(k) k∗
+w × �
+w∈S′ngo(k) Mw · (O∗
+p)p. Put ξ = ηp, so that (32) holds for some α and
+θ, and ∂(h) is represented by α. In this case, since ξ is a p’th power, α ∈ (k∗
+w)p at
+all w ∈ Sngo(k), which is non-empty, so by the local Leopoldt’s conjecture [Kis93],
+α = βp, for some β ∈ V‡. Since pV‡ ⊂ V ′
+‡, we conclude that
+ker(
+˜
+M c/p
+˜
+M c −→ M c/pM c) = (V ′
+‡/pV‡)c.
+(33)
+Denote
+ℶ :=
+˜
+W /M = k∗\A∗
+k/(
+�
+w∈Xgo(k)
+O∗
+w ×
+�
+w∈†(k)
+k∗
+w ×
+�
+w∈S′ngo(k)
+Mw · (O∗
+p)p.
+Then we have the exact sequence
+˜
+W [p]
+� ℶ[p]
+� M /pM
+�
+˜
+W /p ˜
+W .
+(34)
+If y ∈
+˜
+W [p] is represented by an idele ζ = (ζw)w, then there are α ∈ k∗ and
+θ ∈ �
+w∈Xgo(k) O∗
+w · �
+w∈Sngo(k)(O∗
+w)p such that
+ζp = α · θ.
+Then at w ∈ Sngo(k), α ∈ (k∗
+w)p, so by the local Leopoldt’s conjecture again, α = βp,
+β ∈ k∗. Since y is also represented by ζ · β−1, we have the isomorphism
+�
+w∈Sngo(k) O∗
+w/(O∗
+w)p
+∼
+�
+˜
+W [p].
+Since theג equals the cokernel of �
+w∈Sngo(k) O∗
+w/(O∗
+w)p −→ ℶ, the above isomor-
+phism and (34) together imply
+ker(M c/pM c −→
+˜
+W c/p ˜
+W c) = Im(ℶ[p]c −→ג[p]c) ⊂ג]p]c
+(35)
+We have ℏ =ג[p] and
+logp |(V ′
+‡/p · V‡)c| ≤ logp |Zp ⊗Z V c
+‡ /p · Zp ⊗Z V c
+‡ | =
+
+
+
+
+
+0,
+if ‡ = ∅;
+|‡0 − 1|,
+if ‡ ̸= ∅ and k = K;
+|‡0|,
+otherwise,
+so by (33) and (35),
+logp |S(Cp/U)| ≤ logp |(V ′
+‡/p · V‡)c| + logp |גc[p]| ≤ |‡0| + ℏ.
+(36)
+Suppose A/K is isotrivial. Then Sb = Sss = ∅. If k ̸= K, choose a place v not
+spitting completely over k/K; otherwise, choose any v. Set ♮ := {v}. Let U be the
+complement of ♮. Put
+¯W :=
+�
+w∈♮(k)
+k∗
+w/(k∗)p
+and denote
+¯
+W := k∗\A∗
+k/(
+�
+w∈U(k)
+O∗
+w ·
+�
+w∈♮(k)
+(k∗)p),
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+17
+so that H1(Kv, E(p)
+p ) = Hom( ¯W c, Z/pZ) and S(E(p)
+p /U) = Hom( ¯
+W c/p ¯
+W c, Z/pZ).
+Thus, in view of (16) and Lemma 3.1.4, if
+¯
+M := �
+w∈♮(k) O∗
+w/(O∗
+w)p, then
+S(Cp/X ) = ker(
+¯
+M c −→
+¯
+W c/p ¯
+W c).
+The kernel of
+¯
+M −→
+¯
+W is in the image of V♮, the ♮(k)-units of k. Because of our
+choice, (Zp ⊗Z V♮)c = 0, and hence
+¯
+M c −→
+¯
+W c is injective.
+Similar to the previous case, the local Leopoldt’s conjecture at w ∈ ♮(k) implies
+�
+w∈♮(k) k∗
+w/(k∗
+w)p
+� �
+¯
+W [p].
+Since (�
+w∈♮(k) k∗
+w/(k∗
+w)p)c = 0, we have
+¯
+W [p] = 0. Now, ( ¯
+W /
+¯
+M )[p] =ג[p], whereג
+is the divisor class group of k. It follows from the exact sequence
+¯
+W c[p]
+�גc[p]
+�
+¯
+M c
+�
+¯
+W c/p ¯
+W c
+that
+|S(Cp/X )| = |גc[p]|.
+Thus, the following proposition is proved.
+Proposition 3.2.6. For U = Xgo, Xg, or X ,
+logp |S(Cp/U)| ≤ |‡0| + ℏ.
+In particular,
+logp | Sel(Cp/K)| ≤ |ð0| + ℏ.
+Proof of Proposition 1. The proposition is a consequence of Proposition 3.2.5 and
+Proposition 3.2.6, because we have the commutative diagram of exact sequences
+0
+� Sel(E(p)
+p /K)
+�
+� �
+�
+Selp(A(p)/K)
+�
+� �
+�
+Sel(Cp/K)
+� �
+�
+Cp(K)
+� H1(K, E(p)
+p )
+�
+�
+H1(K, A(p)
+p )
+V∗
+�
+�
+H1(K, Cp)
+�
+�
+v H1(Kv, A(p))
+�
+v H1(Kv, A(p))
+V∗ � �
+v H1(Kv, A),
+where the middle long exact sequence is induced from (5).
+□
+3.3. The Cassels-Tate duality. The Cassels-Tate pairing induces the perfect pair-
+ing (see [Mil06, III.9.5])
+⟨−, −⟩A/K : X(A/K) × X(A/K) −→ Qp/Zp.
+If ϕ : A −→ B is an isogeny with dual isogeny ϕt, then the commutative diagram
+X(A/K) × X(A/K)
+ϕ♮
+�
+⟨−,−⟩A/K
+� Qp/Zp
+X(B/K) × X(B/K)
+ϕt
+♮
+�
+⟨−,−⟩B/K
+� Qp/Zp
+
+18
+KI-SENG TAN
+yields the duality between X(A/K)/ ker(ϕ♮) and X(B/K)/ ker(ϕt
+♮). In particular
+|(X(A/K)/ ker(ϕ♮))[pν]| = |(X(B/K)/ ker(ϕt
+♮))[pν]|,
+(37)
+since |G/pνG| = |G[pν]| for a finite abelian group G.
+Proof of Proposition 2. Consider the exact sequence
+0
+� ker(F♮)
+� X(A/K)[pν]
+� (X(A/K)/ ker(F♮))[pν]
+·pν
+�✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐
+ker(F♮) ∩ pνX(A/K)
+� 0,
+where the morphism ·pν is induced from the multiplication by pν. This implies
+logp |X(A/K)[pν]| = logp |(X(A/K)/ ker(F♮))[pν]| + δ1,
+(38)
+with logp | Sel(Cp/K)| ≥ logp | ker(F♮)| ≥ δ1 ≥ 0. Also, consider the exact sequence
+0
+� ker(V♮)
+� (X(A(p)/K)/ ker(V♮))[pν]
+�❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣
+(X(A(p)/K)/X(A(p)/K)[p])[pν]
+·pν
+� T
+� 0,
+where T = (X(A(p)/K)[p]/ ker(V♮)) ∩ pν(X(A(p)/K)/ ker(V♮)). This together with
+(37) and (38) lead to
+logp |X(A/K)[pν]| = logp |(X(A(p)/K)/X(A(p)/K)[p])[pν]| + δ1 + δ2,
+(39)
+with
+logp | Sel(Cp/K)| ≥ logp | ker(F♮)| ≥ logp |V♮(X(A(p)/K)[p])| ≥ δ2 ≥ 0.
+Now
+(X(A(p)/K)/X(A(p)/K)[p])[pν] = X(A(p)/K)[pν+1]/X(A(p)/K)[p].
+Recall that r denote the co-rank of Seldiv(A/K) ≃ Seldiv(A(p)/K), so
+logp |X(A/K)[pν]| = logp | Selpν(A/K)| − rν,
+and a similar formula for A(p). Therefore,
+logp | Selpν(A/K)| = logp | Selpν+1(A(p)/K)| − logp | Selp(A(p)/K)| + δ1 + δ2.
+Then we apply Proposition 1 and Proposition 3.2.6.
+□
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+19
+3.4. The method of specialization. In this section only, we assume that A/K
+an ordinary abelian variety defined over a global field. As before, let L/K be a Zd
+p-
+extension unramified outside a finite set of places of K. Let Γ be the Galois group,
+ΛΓ the Iwasawa algebra.
+Endow lµ.. p∞ with the discrete topology and write ˆΓ for the group of all continuous
+characters Γ −→ lµ.
+. p∞. Let O be the ring of integers of Qp(lµ.
+. p∞). Every character
+χ ∈ ˆΓ extends to a unique continuous Zp-algebra homomorphism χ : ΛΓ −→ O.
+For g ∈ ΛΓ, not divisible by p, by Monsky [Mon81, Theorem 2.3], the set
+∇g := {χ ∈ ˆΓ | χ(g) ∈ p · O}
+is either ∅ or is contained in T1∪· · ·∪Tn, where for each i, there are ζi,1, ..., ζi,νi ∈ lµ.
+. p∞,
+νi > 0, and σi,1, ..., σi,νi ∈ Γ, extendable to a Zp-basis of Γ, such that
+Ti := {χ ∈ ˆΓ | χ(σi,j) = ζi,j, for j = 1, ..., νi}.
+For an element ψ ∈ Γ, extendable to a Zp-basis of Γ, set
+Tψ := {χ ∈ ˆΓ | χ(ψ) = 1}.
+Then Tψ ⊂ Ti can not hold, unless νi = 1, ζi,1 = 1, and σi,1, ψ topologically generate
+the same closed subgroup of Γ, in such case, we actually have Tψ = Ti. Therefore,
+there are finitely many rank one Zp-submodules of Γ such that if ψ is chosen away
+from them, then
+Tψ ⊊ ∇g.
+(40)
+For a Ze
+p-subextension L′/K of L/K. Denote Γ′ = Gal(L′/K). The quotient map
+Γ −→ Γ′ extends uniquely to a continuous Zp-algebra homomorphism
+pL/L′ : ΛΓ −→ ΛΓ′.
+Write Ψ := Gal(L/L′) so that Γ′ = Γ/Ψ. Put IΨ := ker(pL/L′).
+Let b be a finite set of places of K. For a subextension M of L/K, let bM ⊂ b
+denote the subset consisting of places splitting completely over M.
+Lemma 3.4.1. Suppose d ≥ 2. Let Zℓ, ℓ = 1, .., s, be finitely generated torsion
+ΛΓ-modules, θ an element in ΛΓ, not divisible by p, and b a finite set of places of K.
+There is an element ψ ∈ Γ extendable to a Zp-basis such that if L′ is the fixed field of
+ψ, then bL′ = bL, pL/L′(θ) is not divisible by p, and each ΛΓ′-module Z′
+ℓ := Zℓ/IΨZℓ
+is torsion, having the same elementary µ-invariants as those of Zℓ over ΛΓ.
+Proof. For each v ∈ b with non-trivial decomposition subgroup Γv, if ψ ̸∈ Γv or Γv
+is of Zp-rank greater than one, then v does not split completely over L′. To have
+bL′ = bL, we only need to choose ψ away from all rank one Γv, v ∈ b.
+Similar to (2), we have
+0
+� �mℓ
+i=1 ΛΓ/(pαℓ,i) ⊕ �nℓ
+j=1 ΛΓ/(η
+βℓ,j
+ℓ,j )
+� Zℓ
+� Nℓ
+� 0,
+(41)
+
+20
+KI-SENG TAN
+where Nℓ is pseudo-null and every ηℓ,j is not divisible by p. Choose for each ℓ, an
+annihilator hℓ ∈ ΛΓ of Nℓ, not divisible by p, and put
+g := θ ·
+s�
+ℓ=1
+hℓ · ηℓ,1 · · · · · ηℓ,nℓ.
+We also choose ψ satisfying (40). Since ˆΓ′ = Tψ, there exists χ ∈ ˆΓ′ such that
+χ(pL/L′(g)) ̸∈ p · O, so pL/L′(g) ̸∈ p · ΛΓ′. Because pα · gβ · Zℓ = 0, for some α, β ∈ Z,
+we have pα · pL/L′(gβ) · Z′
+ℓ = 0. Hence Z′
+ℓ is torsion over ΛΓ′.
+To compare the elementary µ-invariants of Zℓ and Z′
+ℓ, we apply the maps of
+multiplication by ψ−1 to (41) and use the snake lemma to obtain the exact sequence
+Nℓ[ψ − 1] −→
+mℓ
+�
+i=1
+ΛΓ′/(pαℓ,i) ⊕
+nℓ
+�
+j=1
+ΛΓ′/(pL/L′(ηℓ,j)βℓ,j) −→ Z′
+ℓ −→ Nℓ/IΨNℓ. (42)
+Because Nℓ[ψ − 1] is annihilated by pL/L′(g) which is relatively prime to p, the
+second arrow in the exact sequence is injective on �mℓ
+i=1 ΛΓ′/(pαℓ,i). We complete
+the proof by comparing the pth power factors of the characteristic ideals of items in
+the sequence.
+□
+In Lemma 3.4.1, the field L′ is a Zd−1
+p
+-extension of K. By repeatedly applying
+the lemma, we obtain sequences
+L ⊃ L′ ⊃ · · · ⊃ L(d−1),
+(43)
+and, for each ℓ,
+Zℓ −→ Z′
+ℓ −→ · · · −→ Z(d−1)
+ℓ
+.
+(44)
+Put Ψ0 = Ψ, Ψi = Gal(L/L(i+1)), and Γ(i+1) = Gal(L(i+1)/K) = Γ/Ψi. Then L(d−1)
+is a Zp-extension of K with bL(d−1) = bL, pL/L(d−1)(θ) not divisible by p, and for each
+ℓ, the ΛΓ(d−1)-module Z(d−1)
+ℓ
+= Zℓ/IΨd−2Zℓ has the same elementary µ-invariants as
+those of Zℓ over ΛΓ. These elementary µ-invariants pαℓ,1, ..., pαℓ,mℓ can be recovered
+by using the counting formula below. For each ν, define
+αℓ,i,ν = min{ν, αℓ,i}.
+Let σ ∈ Γ(d−1) be a topological generator and put x = σ − 1. Let Jν,n denote the
+ideal of ΛΓ(d−1) generated by pν and (x + 1)pn − 1.
+Lemma 3.4.2. Let the notation be as above. Then
+logp |Z(d−1)
+ℓ
+/Jν,nZ(d−1)
+ℓ
+| = pn ·
+mℓ
+�
+i=1
+αℓ,i,ν + O(1).
+Proof. Taking Z = Z(d−1)
+ℓ
+/pνZ(d−1)
+ℓ
+in (2), we deduce the following exact sequence,
+0
+� �mℓ
+i=1 ΛΓ(d−1)/(pαℓ,i,ν)
+� Z(d−1)
+ℓ
+/pνZ(d−1)
+ℓ
+� Nℓ,ν
+� 0,
+(45)
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+21
+where Nℓ,ν is finite, since the other two items are pseudo isomorphic. Write Rα for
+Zp/pαZp. The lemma is a consequence of the exact sequence induced from (45):
+N0[σpn − 1]
+� �mℓ
+i=1 Rαℓ,i,ν[x]/((x + 1)pn − 1)
+� Z(d−1)
+ℓ
+/Jν,nZ(d−1)
+ℓ
+�❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤
+N0/(σpn − 1)N0.
+□
+To apply the above to dual Selmer groups, we need the following simplified control
+lemma. For K ⊂ F ⊂ L, consider the restriction maps
+res(ν)
+L/F : Selpν(A/F) −→ Selpν(A/L)Gal(L/F ).
+Let K(n) denote the nth layer of the Zp-extension L(d−1)/K. Let r denote the
+ramification locus of L/K, which is assumed to be finite.
+Lemma 3.4.3. Let L(d−1)/K be an intermediate Zp-extension of L/K. Suppose r
+contains only places where A has good ordinary reduction or multiplicative reduction,
+bL(d−1) = bL and b contains r as well as all places where A has bad reduction. For a
+given ν, the orders of ker(res(ν)
+L/K(n)) and coker(res(ν)
+L/K(n)) are bounded, as n varies.
+Proof. Let M = Apν(L) = Apν(K′) for some finite sub-extension K′/K of L/K.
+Write K′(n) for K(n)K′. [Tan10, Lemma 3.2.1] says that for i = 0, 1, 2,
+| Hi(L/K′(n), M)| ≤ |M|di.
+Since [K′(n) : K(n)] ≤ [K′ : K], by counting the number of co-chains, we deduce
+| Hi(K′(n)/K(n), M)| ≤ |M|[K′:K]i.
+This bounds | ker(res(ν)
+L/K(n))|. To bound | coker(res(ν)
+L/K(n))|, by the Hochschild-Serre
+spectral sequence, we need to bound (see the proof of [Tan10, Theorem 4])
+|
+�
+all v
+�
+w|v
+H1(Lw/K(n)
+w , A(Lw))[pν]|.
+Write H (ν)
+w
+for H1(Lw/K(n)
+w , A(Lw))[pν]. If v ̸∈ b, then H (ν)
+w
+= 0 [Mil06, I.3.8],
+for all w | v. Also, H (ν)
+w
+= 0, for all w sitting over bL, because K(n)
+w = Lw.
+Suppose v ∈ b but v ̸∈ bL = bL(d−1). The number of places of K(n) sitting over v
+is bounded as n varies. We need to bound the order of H (ν)
+w , for all w | v. If A has
+good ordinary reduction at v, by [Tan10, (3) and Theorem 2], the order of H (ν)
+w
+is
+bounded by p2ν(d+1) dim A. It is well-known (for instance, see the last two paragraphs
+of [Tan10]) that if A has split multiplicative reduction at v, the order of H (ν)
+w
+is
+bounded by pνd dim A. In general, if A has multiplicative reduction at v, then over
+some unramified extension K′
+v/Kv, the reduction of A becomes split multiplicative.
+
+22
+KI-SENG TAN
+Since H (ν)
+w
+⊂ H1(LwK′
+v/Kv, A(LwK′
+v)), writing K′
+v, L′
+w for KvK′
+v, LwK′
+v, we end the
+proof by using the exact sequence
+H1(K′
+v/Kv, A(K′
+v))� �
+� H1(L′
+w/Kv, A(L′
+w))
+� H1(L′
+w/K′
+v, A(L′
+w))
+and the fact that the component group of A/K′
+v has p-rank bounded by dim A
+(see [BX96, Proposition 5.2]) so that by [Mil06, Proposition I.3.8] the order of
+H1(K′
+v/Kv, A(K′
+v)) is bounded.
+□
+Lemma 3.4.4. Let θ ∈ ΛΓ be an element not divisible by p. Let Aℓ, ℓ = 1, ..., s,
+be ordinary abelian varieties defined over K such that all Zℓ := Selp∞(Aℓ/L)∨ are
+torsion over ΛΓ and the ramification locus r contains only places where each Aℓ has
+either good ordinary reduction or multiplicative reduction. Let b be a finite set of
+places of K containing r and all places where some Aℓ has bad reduction. Assume
+that pαℓ,1, ..., pαℓ,mℓ are elementary µ-invariants of Zℓ. There exists an intermediate
+Zp-extension L(d−1)/K of L/K such that the following holds:
+(a) pL/L(d−1)(θ) ̸∈ pΛΓ(d−1), where Γ(d−1) = Gal(L(d−1)/K).
+(b) bL(d−1) = bL.
+(c) For each ℓ, the elementary µ-invariants of Selp∞(Aℓ/L(d−1))∨ over ΛΓ(d−1) are
+the same as those of Zℓ over ΛΓ.
+(d) In particular, if L/K is a Zp-extension and K(n) denote the nth layer, then
+logp | Selpν(Aℓ/K(n))| = pn ·
+mℓ
+�
+i=1
+αℓ,i,ν + O(1).
+(46)
+Proof. (a) and (b) are from Lemma 3.4.1. Observe that Z(d−1)
+ℓ
+/Jν,nZ(d−1) is nothing
+but the Pontryagin dual of Selpν(Aℓ/L)Gal(L/K(n)), so by Lemma 3.4.2, 3.4.3, we have
+logp | Selpν(Aℓ/K(n))| = pn ·
+mℓ
+�
+i=1
+αℓ,i,ν + O(1).
+(47)
+In the situation of (d), L = L(d−1), and hence, (46) holds. To show (c), we assume
+that the elementary µ-invariants of Selp∞(Aℓ/L(d−1))∨ over ΛΓ(d−1) are pα′
+ℓ,1, ..., pα′
+ℓ,wℓ.
+Apply (d) to L(d−1)/K and obtain
+logp | Selpν(Aℓ/K(n))| = pn ·
+wℓ
+�
+i=1
+α′
+ℓ,i,ν + O(1),
+where, as before, α′
+ℓ,i,ν := min{α′
+ℓ,i, ν}. This and (47) leads to
+wℓ
+�
+i=1
+α′
+ℓ,i,ν =
+mℓ
+�
+i=1
+αℓ,i,ν,
+for all ν. We conclude that wℓ = mℓ and pα′
+ℓ,1, ..., pα′
+ℓ,wℓ are the same as pαℓ,1, ..., pαℓ,mℓ.
+□
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+23
+3.5. The elementary µ-invariants. In this section, we complete the proof of
+Proposition 3 and Proposition 5. For each Zp-subextension F/K of L/K, write
+Sb(F) = ð′
+F ⊔ ðF,
+where if p = 2, ð′
+F is the set of places at which A/K has non-split multiplicative
+reduction such that the group of components is of even order; if p ̸= 2, ð′
+F = ∅. For
+w ∈ Sb(F) sitting on v ∈ Sb, if w ∈ ð′
+F and v ∈ ð, or w ∈ ðF and v ∈ ð′, then
+Fw ̸= Kv, and hence there is only finitely many places of F sitting over v.
+Proof of Proposition 3. We apply Lemma 3.4.4, taking s = 1, Z1 = X(p), b = r ∪Sb.
+Let K(n) be the nth layer of L(d−1)/K. Since the degrees of k/K and K(n)/K are
+relatively prime, one see that a place v of K splits completely in k, if and only if
+all places of K(n) sitting over v split completely in kK(n), because both assertions
+are equivalent to that the decomposition subgroup Gal(kK(n)/K)v contains no non-
+trivial element in Gal(kK(n)/K(n)), and hence contained in Gal(kK(n)/k). Put
+ð0,n := {w ∈ ðK(n) | w splits completely over kK(n)}.
+Then, by the discussion at the beginning of this section,
+|ð0,n| = |ð0(K(n))| + O(1) = pn · |ð1| + O(1).
+If Fq(n) denote the constant field of K(n), then
+deg ∆A/K(n) · logp q(n) = pn · deg ∆A/K · logp q.
+Therefore, Lemma 3.4.4(d) and Proposition 1 say if m is the µ-rank of A(p)/L, then
+pn · m = logp | Selp(A(p)/K(n))| + O(1) ≥ pn · ((p − 1) deg ∆A/K
+12
+· logp q − |ð1|) + O(1).
+This proves the proposition.
+□
+Proof of Proposition 5. Take s = 2, Z1 = X(p), Z2 = X, b = Sb ∪ r, θ = ΘL, , so
+that m = m1, and αi = α1,i, for i = 1, ..., m. By Lemma 3.4.4, bL(d−1) = bL = ∅ and
+ΘL(d−1) is not divisible by p. Thus, we may assume that d = 1.
+Let K(n) denote the nth layer of L/K.
+If L = K(∞)
+p
+, we have shown in §1.1
+that |wK(n)[p]| = O(1); otherwise, for n sufficiently large, L/K(n) is totally ramified
+at certain place, so that Hom(Gal(L/K(n)), Qp/Zp) ∩ Hom(wK(n), Qp/Zp) = {0}.
+Hence, Hom(wK(n), Qp/Zp) −→ Hom(wL, Qp/Zp) is injective, or equivalently, the
+map wL −→ wK(n) is surjective, for sufficiently large n. Since p ∤ ΘL, wL has trivial
+p-part, the order of wK(n)[p] must be bounded. The assumption says
+|Sb(K(n))| = O(1).
+Therefore, by Lemma 3.4.4(d) and Proposition 1, we obtain
+pn · m
+=
+pn · �m
+i=1 α1,i,1
+=
+logp | Selp(A(p)/K(n))| + O(1)
+=
+pn ·
+(p−1) deg ∆A/K
+12
+· logp q + O(1),
+
+24
+KI-SENG TAN
+that proves the first assertion. Then Lemma 3.4.4(d) and Proposition 2 lead to
+pn · �m2
+i=1 α2,i,ν
+=
+logp | Selpν(A/K(n))| + O(1)
+=
+pn · (logp | Selpν+1(A(p)/K(n))| −
+(p−1) deg ∆A/K
+12
+· logp q) + O(1)
+=
+pn · �m
+i=1(α1,i,ν+1 − 1) + O(1),
+which holds for every ν, so the proposition is proved.
+□
+References
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+fields of characteristic p > 0, Math. Proc. Camb. Philos. Soc. 146 (2009), 23-43.
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+1984).
+
+THE FROBENIUS TWISTS OF ELLIPTIC CURVES OVER GLOBAL FUNCTION FIELDS
+25
+Department of Mathematics, National Taiwan University, Taipei 10764, Taiwan
+Email address: tan@math.ntu.edu.tw
+

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+arXiv:2301.05076v1  [math.SP]  12 Jan 2023
+ROBUSTNESS OF FLAT BANDS ON THE PERTURBED KAGOME
+AND THE PERTURBED SUPER-KAGOME LATTICE
+JOACHIM KERNER, MATTHIAS T¨AUFER, AND JENS WINTERMAYR
+Abstract. We study spectral properties of perturbed discrete Laplacians on two-dimen-
+sional Archimedean tilings. The perturbation manifests itself in the introduction of non-
+trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian
+exhibits flat bands, namely the (3.6)2 Kagome lattice and the (3.12)2 “Super-Kagome”
+lattice. We characterize all possible choices for edge weights which lead to flat bands.
+Furthermore, we discuss spectral consequences such as the emergence of new band gaps.
+Among our main findings is that flat bands are robust under physically reasonable as-
+sumptions on the perturbation and we completely describe the perturbation-spectrum
+phase diagram. The two flat bands in the Super-Kagome lattice are shown to even ex-
+hibit an “all-or-nothing” phenomenon in the sense that there is no perturbation which
+can destroy only one flat band while preserving the other.
+1. Introduction
+This paper is about discrete Schr¨odinger operators on Archimedean tilings, a class of
+periodic two-dimensional lattices that were already investigated by Johannes Kepler in
+1619 [Kep19]. They are natural candidates for the geometry of two-dimensional nanoma-
+terials and due to advances in this field, most prominently represented by graphene, they
+have become increasingly a focus of attention.
+Much work has been devoted to understanding physical properties of such (new) materi-
+als [SYY22, TFGK22, dLFM19]. Most importantly, it can be expected that the underlying
+geometry, that is the particular lattice, is a key feature determining physical properties
+of the system. In fact, in particular in the mathematical physics literature, investiga-
+tions of the connection between the geometry (or topology) of a system and the spectral
+properties of the associated Hamiltonian have become ubiquitous. Classical examples in
+this context are so-called quantum waveguides [EK15, Exn20, Exn22] as well as quantum
+graphs [BK13, BE22]; see also [KP07] for a relatively recent reference relevant in our
+context.
+A closely related research direction is superconductivity: the existence of a boundary
+leads to boundary states in a superconductor with a higher critical temperature than the
+one of the bulk [SB20, SB21, HRS]. In this spirit, it seems very promising to also study
+the interplay of geometry and many-particle phenomena on Archimedean tilings. Yet
+another related investigation can be found in [JBT21, SYY22] where another important
+quantum phenomenon, namely Bose-Einstein condensation, is examined. It turns out
+that so-called flat bands, that are infinitely degenerate eigenvalues of the Hamiltonian,
+play an important role in understanding such many-particle effects, and for other physical
+phenomena [KFSH19]. One of the central motivations for this paper is to study robustness
+of flat bands under certain natural perturbations.
+Two Archimedean tilings, the (3.6)2 Kagome lattice and the (3.122) tiling 1, which
+we shall dub Super-Kagome lattice for reasons that will become clear over the course
+Date: January 13, 2023.
+1We explain the notation for the lattices in Section 2.
+1
+
+2
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+of the article, stand out: they are the only Archimedean lattices on which the discrete,
+unweighted Laplacian has flat bands. In particular the Kagome lattice is a prominent
+model in physics that has recently enjoyed increasing interest [BM18, MDY22, Dia21].
+From a mathematical point of view, our paper is motivated by [PT21] where flat bands
+for the discrete, unweighted Laplacian on Archimedean tilings have been studied in great
+detail, in combination with an explicit calculation of the integrated density of states.
+A priory, the flat-band phenomena on the Kagome and Super-Kagome lattice seem
+very sensitive to perturbations: if one replaces the adjacency matrix or the Laplacian by
+a variant with periodically chosen edge weights, one will generically destroy flat bands.
+However, the results of this paper suggest that, if one looks at proper, meaningful variants
+of the discrete Laplacian which respect certain, natural symmetries of the tiling (we
+call them monomeric Laplacians in Definition 3), then flat bands will persist.
+Since
+monomericity is a physically justifiable assumption, this makes a strong case that flat
+bands are a robust phenomenon, caused by the geometry of the lattice alone and specific
+to these two lattices, see Theorems 6, and 10.
+Other questions of interest on periodic graphs concern existence, persistence and esti-
+mates on the width of spectral bands and the gaps between them [KS19, KS19, MW89].
+We will completely identify the spectra as a function of the perturbation in these cases,
+see Theorems 8, and 11 as well as Figures 3, and 5. This provides an exhaustive descrip-
+tion of all nanomaterials based on Archimedean tilings on which discrete Laplacians can
+exhibit flat bands.
+Our paper is organized as follows: Sections 2, and 3 are of introductory nature, intro-
+ducing the notion of and arguing for the relevance of Archimedean tilings, and defining
+a proper notion of a discrete Laplace operator with non-uniform edge weights. Section 3
+also introduces the notion of flat bands and argues why it suffices to restrict our attention
+to the (3.6)2 Kagome and the (3.122) Super-Kagome lattice. Sections 4, and 5 contain our
+main results on the Kagome and Super Kagome lattice, respectively. The contributions
+of this paper are:
+(i) We identify the Kagome and Super-Kagome lattice as the only Archimedean lat-
+tices on which a natural class of periodic, weighted Laplacians can have flat bands
+(Proposition 5).
+(ii) We describe all periodic edge weights which lead to the maximal possible number of
+bands on the Kagome and Super-Kagome lattice, and prove that this is equivalent
+to so-called monomericity of the edge weights (Theorems 6 and 10).
+(iii) We completely describe the spectrum in the monomeric Kagome and Super-Kagome
+lattice (Theorems 8 and 11). In particular, the monomeric Super-Kagome lattice
+has a surprisingly rich spectrum-perturbation phase diagram (Figure 5) which might
+bear relevance for various applications.
+(iv) In the Super-Kagome lattice, under a weaker condition than monomericity, namely
+constant vertex weight, we explicitely describe all remaining “spurious” edge
+weights which have only one flat band. We describe the topology of this set within the
+parameter space and show in particular that it is disconnected from the monomeric
+two-band set (Theorem 12).
+2. Archimedean tilings
+Archimedean, Keplerian or regular tilings are edge-to-edge tesselations of the Euclidean
+plane by regular convex polygons such that every vertex is surrounded by the same pattern
+of adjacent polygons. We will adopt the notation of [GS89] and use the (counterclockwise)
+order of polygons arranged around a vertex as a symbol for a tiling (this is unique up to
+
+ROBUSTNESS OF FLAT BANDS
+3
+cyclic permutations), see Figure 1 for the (3.6)2 Kagome lattice and the (3.122) Super-
+Kagome lattice which will be investigated in this paper.
+(3.6)2 Kagome lattice
+(3.122) Super-Kagome lattice
+Figure 1. The two Archimedean tilings primarily investigated in this article.
+The first systematic investigation from 1619 is due to Kepler who identified all 11 such
+tilings [Kep19]2. Most importantly, Archimedean tilings provide natural candidates for
+geometries of two-dimensional nanomaterials since they form natural, symmetric arrange-
+ments of a single buiding block, positioned at every vertex. And indeed, these lattices
+can be observed in many naturally occurring materials [FK58, FK59, KHZ+20].
+From a physical point of view, two-dimensional materials such as graphene are inter-
+esting since they feature so-called Dirac points which are related to a specific behaviour
+of the electronic band structure of the material [FW12, LWL13, HC15].
+Also note that there are deep connections between Laplacians on these lattices, perco-
+lation, and self-avoiding walks which have also been studied extensively [SE64, Kes80,
+Nie82, SZ99, Ves04, Par07, Jac14, JSG16].
+An important quantity in this context is
+the so-called connective constant, which is known only in few cases, for example on the
+hexagonal lattice [DCS12].
+3. Defining a suitable Hamiltonian
+Every Archimedean tiling can be regarded as an infinite discrete graph G = (V, E) with
+(countable) vertex set V and (countable) edge set E. We write v ∼ w if the vertices v
+and w are joined by an edge and denote by
+|v| := #{w ∈ V : v ∼ w}
+the vertex degree of v (which in the case of Archimedean lattice graphs is v-independent).
+Archimedean lattices are Z2-periodic, and there exists a cofinite Z2-action
+Z2 ∋ β �→ Tβ : V → V ,
+that is a group of graph isomorphisms (intuitively understood as a group of shifts) iso-
+morphic to the group Z2. Let Q ⊂ V be a minimal (in particular finite) fundamental
+domain of this action, i.e. the quotient of V under the equivalence relation generated by
+the group of isomorphisms (Tβ)β∈Z2.
+2All 11 Archimedean tilings are: the (44) rectangular tiling, the (36) triangular tiling, the (63) hexa-
+gonal tiling, the (3.62) Kagome lattice, the (3.122) Super-Kagome lattice, the (33.42) tiling, the (4.82)
+tiling, the (32.4.3.4) tiling, the (3.4.6.4) tiling, the (4.6.12) tiling, and the (34.6) tiling.
+
+4
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+In the unweighted case, a natural, normalized choice for the Hamiltonian is the discrete
+Laplacian
+(∆f)(v) := 1
+|v|
+�
+w∼v
+(f(v) − f(w)) = f(v) − 1
+|v|
+�
+w∼v
+f(w) ,
+(1)
+as used for instance in [PT21].
+It can be written as ∆f = Id − 1
+|v|Π where Π is the
+adjacency matrix, that is Π(v, w) = 1 if v ∼ w and 0 else. The following is standard:
+Lemma 1. The unweighted, normalized Laplacian (1) with a uniformly bounded vertex
+degree boasts the following properties:
+(i) All restrictions of ∆ to finitely many vertices are real-symmetric M-matrices, that
+is, their off-diagonal elements are non-positive¸ and all their eigenvalues are non-
+negative.
+(ii) The infimum of the spectrum of ∆ is 0.
+(iii) All rows and columns of ∆ sum to zero.
+Furthermore, the spectrum is always contained in the interval [0, 2].
+Introducing non-trivial edge weights, we would like to keep a form of the Laplacian that
+preserves properties (i) to (iii). A natural candidate, similar to formula (2.11) in [KS], is
+(∆γf)(v) :=
+1
+�
+µ(v)
+�
+w∼v
+γvw
+�
+f(v)
+�
+µ(v)
+−
+f(w)
+�
+µ(w)
+�
+(2)
+where the edge weights γvw = γwv > 0 and vertex weights µ(v) satisfy the relation
+�
+w∼v
+γvw = µv
+for every v ∈ V .
+(3)
+As long as the vertex weights µ(v) (and thus also the γvw) are uniformly bounded, this
+will lead to an operator with properties (i) to (iii) and spectrum contained in [0, 2].
+Remark 2. In the literature, one often finds the definition
+(∆γf)(v) =
+1
+µ(v)
+�
+w∼v
+γvw (f(v) − f(w))
+as a normalized, discrete Laplacian. Note that, whenever µ(v) ̸= µ(w) for some v ∼ w,
+then this will not lead to a self-adjoint operator, but it can be made self-adjoint on a
+suitably weighted ℓ2(V )-space, cf. [KLW21]. If all µ(v) are the same, then this definition
+coincides with (2), and can be simplified to
+(∆γf)(v) = f(v) − 1
+µ
+�
+w∼v
+γwvf(w) .
+(4)
+Now, one can prescribe various degrees of the symmetry of the underlying Archimedean
+lattice to be respected by the Laplacian:
+Definition 3. Consider an Archimedean tiling (V, E) with periodic edge weights γvw =
+γwv > 0, that is γvw = γTβvTβw for all v, w ∈ V and β ∈ Z2, and corresponding vertex
+weights µ(v) = �
+w∼v γvw. Define the Laplacian ∆γ as in (2). Then, we say that the
+Archimedean tiling with Laplacian ∆γ
+(1) has constant vertex weight, if there is µ > 0 such that µ(v) = µ for all v ∈ V .
+(2) is monomeric if for all vertices v ∈ V the list of edge weights, arranged cyclically
+around v, coincides (up to cyclic permutations).
+
+ROBUSTNESS OF FLAT BANDS
+5
+Clearly, (2) is stronger than (1). However, in either case, the Laplacian reduces to (4).
+The term “monomeric” is inspired by the fact that the associated operators can be
+interpreted as describing properties of nanomaterials formed from one type of monomeric
+building block, positioned at every vertex of an Archimedean tiling. Clearly, monomeric
+Laplacians on Archimedean lattices have constant vertex weights, but the converse is not
+true in general. However, we will see in Theorems 6 and 10 that on the Kagome and
+Super-Kagome lattice, the validity of the converse implication is equivalent to existence
+(or persistence) of all flat bands. Also, monomericity seems a physically reasonable as-
+sumption for nanomaterials, which suggests that the emergence of flat bands, while a
+priori very sensitive to perturbations of coefficients in the operator, might nevertheless be
+robust within the class of physically relevant operators.
+Next, let T2 = R2/Z2 be the flat torus and define for every θ ∈ T2 the |Q|-dimensional
+Hilbert space
+ℓ2(V )θ := {f : V → C | f(Tβv) = ei⟨θ,β⟩f(v)}
+with inner product
+⟨f, g⟩θ :=
+�
+v∈Q
+f(v)g(v) .
+Given the Laplacian (4) on ℓ2(V ) with properties described in Definition 3, we define on
+ℓ2(V )θ the operator
+(∆θ
+γf)(v) := f(v) − 1
+µ
+�
+w∼v
+γwvf(w) .
+(5)
+Clearly, (5) can be represented as a |Q|-dimensional Hermitian matrix. Due to Floquet
+theory, we have
+σ(∆γ) =
+�
+θ∈T2
+σ(∆θ
+γ) ,
+and the following statement holds.
+Proposition 4 (See [PT21] and references therein). Let E ∈ R. Then, the following are
+equivalent:
+(i) E ∈ σ(∆θ
+γ) for all θ ∈ T2.
+(ii) E ∈ σ(∆θ
+γ) for a positive measure subset of θ ∈ T2.
+(iii) There is an infinite orthonormal family eigenfunctions of ∆γ to the eigenvalue E.
+Each of them can be chosen to be supported on a finite number of vertices.
+If any of (i) to (iii) is satisfied, we say that ∆γ has a flat band (at energy E).
+Note that, in the ℓ∞(V ) setting instead of the ℓ2(V ) setting, such infinitely degenerate
+eigenvalues are also referred to as “black hole eigenvalues” in [BL09]. Also, the existence of
+flat bands can be interpreted as a breakdown of the unique continuation principle [PTV17].
+In the Hilbert space ℓ2(V ) setting, is known that for constant edge weights, the discrete
+Laplacian has flat bands only on two of the 11 Archimedean lattices, namely the (3.6)2
+Kagome lattice and the (3.122) Super-Kagome lattice [PT21]. Before turning to perturbed
+versions of those two lattices, one should verify that there won’t be any surprises on the
+other lattices:
+Proposition 5. On the Archimedean lattices (44), (36), (63), (33.42), (4.82), (32.4.3.4),
+(3.4.6.4), (4.6.12), (34.6), there is no choice of periodic (with respect to the fundamental
+cell on the lattice) edge weights γvw = γwv > 0 which will make the weighted adjacency
+
+6
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+matrix
+Πγ(v, w) =
+�
+γvw
+if v ∼ w,
+0
+else
+have a flat band.
+Consequently, also the Laplacian with constant or monomeric edge
+weights has no flat bands on these lattices.
+Proposition 5 is proved by a series of straightforward but somewhat lengthy calculations
+in which one calculates the associated characteristic polynomials, and shows that there
+are no θ-independent roots, employing Proposition 4 (this should be compared to the
+proofs of Theorems 6 and 10 below). We omit them here for the sake of conciseness. In
+any case, Proposition 5 justifies to restrict our attention to the (perturbed) Kagome and
+Super-Kagome lattices from now on.
+4. The perturbed Kagome lattice
+In this section we discuss the Kagome lattice with non-uniform (periodic) edge weights.
+The elementary cell of the Kagome lattice contains three vertices and six edges (one can
+think of the edges as arranged around a hexagon). A priori, periodicity allows for six
+γ1
+γ2
+γ3
+γ4
+γ5
+γ6
+γ4
+γ5
+γ6
+γ5
+γ1
+γ6
+v1
+v2
+v3
+v2 + ω1
+v3 + ω1
+v3 + ω2
+v1 − ω2
+v1 − ω1
+v2 − ω2
+Figure 2. Fundamental domain of the Kagome lattice with edge weights.
+In the monomeric case, all edge weights around downwards pointing tri-
+angles are γ2 = γ4 = γ6 =: α and all edge weights on upwards pointing
+triangles are γ1 = γ3 = γ5 =: β, where 2α + 2β = µ.
+edge weights γ1, ..., γ6 > 0, and the Floquet Laplacian ∆θ
+γ can be written as the Hermitian
+matrix
+∆θ
+γ = Id −1
+µ
+
+
+0
+γ3 + wγ6
+wγ4 + zγ1
+γ3 + wγ6
+0
+γ2 + zγ5
+wγ4 + zγ1
+γ2 + zγ5
+0
+
+ ,
+(6)
+where w := eiθ1 and z := eiθ2. We denote the three real eigenvalues of ∆θ
+γ by λ1(θ, γ) ≤
+λ2(θ, γ) ≤ λ3(θ, γ).
+Note that the six degrees of freedom are to be further reduced, depending on the
+following symmetry conditions:
+• If we merely assume a constant vertex weight µ > 0, then identity (3) will impose
+the three additional linearly independent conditions
+γ1 + γ4 = γ2 + γ5 ,
+γ3 + γ6 = γ2 + γ5 ,
+γ1 + γ3 + γ4 + γ6 = µ ,
+(7)
+
+ROBUSTNESS OF FLAT BANDS
+7
+and we end up with three degrees of freedom.
+• If we also assume monomericity, then it is easy to see that the only choice is
+the breathing Kagome lattice, cf. [HKdP+22], with an edge weight α > 0 on all
+edges belonging to upwards pointing triangles and edge weight β > 0 on all edges
+belonging to downwards pointing triangles, where 2(α + β) = µ. After fixing the
+vertex weight µ, this amounts to only one degree of freedom.
+4.1. Flat bands in the perturbed Kagome lattice.
+Theorem 6. Consider the perturbed Kagome lattice with Laplacian (4), fixed vertex
+weight µ > 0 and periodic edge weights γ1, ..., γ6 > 0, satisfying the condition (3) on
+vertex and edge weights. Then, the following are equivalent:
+(i) There exists a flat band.
+(ii) The vertex weights are monomeric. More explicitly, there are α, β > 0 with 2(α +
+β) = µ such that
+γ2 = γ4 = γ6 := α,
+γ1 = γ3 = γ5 := β.
+The rest of this subsection is devoted to the proof of Theorem 6. We start with identi-
+fying flat bands using the weighted adjacency matrix
+Πθ
+γ :=
+
+
+0
+γ3 + wγ6
+wγ4 + zγ1
+γ3 + wγ6
+0
+γ2 + zγ5
+wγ4 + zγ1
+γ2 + zγ5
+0
+
+
+(8)
+which is spectrally equivalent to ∆θ
+γ up to scaling and shifting via the relation
+∆θ
+γ = Id −1
+µΠθ
+γ.
+In order to find flat bands, we will identify conditions for θ-independent eigenvalues of Πθ
+γ
+and therefore calculate
+det(λ Id −Πθ
+γ) = −λ3 + λ(|A|2 + |B|2 + |C|2) + 2ℜ(ABC)
+where A := γ3 + wγ6, B := wγ4 + zγ1 and C := γ2 + zγ5. Rearranging the terms yields
+det(λ Id −Πθ
+γ) =(w + w)(λγ6γ3 + γ3γ2γ4 + γ6γ5γ1)
++(z + z)(λγ5γ2 + γ6γ5γ4 + γ1γ3γ2)
++(wz + zw)(λγ1γ4 + γ3γ5γ4 + γ6γ2γ1)
++(−λ3 + λ(γ2
+1 + ... + γ2
+6) + 2(γ4γ6γ2 + γ3γ5γ1)) .
+The prefactors
+w + w = 2 cos θ1 ,
+z + z = 2 cos θ2 ,
+and
+wz + zw = 2 cos(θ1 − θ2) ,
+are linearly independent as measurable functions of θ on T2. Consequently, since all γi
+are positive, θ-independent eigenvalues exist if and only if the w and z-independent terms
+in every line are zero. This is only possible for negative λ, which (possibly after scaling
+the γi and µ for the moment) can be assumed to equal −1. Therefore, we obtain the
+conditions
+γ3γ6 = γ2γ3γ4 + γ1γ5γ6 ,
+γ2γ5 = γ4γ5γ6 + γ1γ2γ3 ,
+γ1γ4 = γ3γ4γ5 + γ1γ2γ6 ,
+(9)
+
+8
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+and
+1 − (γ2
+1 + · · · + γ2
+6) + 2 (γ2γ4γ6 + γ1γ3γ5) = 0 .
+(10)
+Lemma 7. The only positive solutions (meaning all γi are non-zero) of (7), (9), (10) are
+γ2 = γ4 = γ6 = x
+γ1 = γ3 = γ5 = y
+(11)
+with x, y ∈ (0, 1) and x + y = 1.
+Proof. By a direct calculation (11) solves (7), (9), (10).
+Conversely, assume that there are positive solutions γ1, ..., γ6 > 0. From (9) we obtain
+γ3 =
+γ1γ5γ6
+γ6 − γ2γ4
+,
+γ1 =
+γ3γ4γ5
+γ4 − γ2γ6
+,
+and this implies γ6 > γ2γ4 and γ4 > γ2γ6.
+Hence, combining both equations yields
+γ2
+2γ6 < γ6 which shows that γ2 < 1. In the same way one proves γi < 1 for every other i.
+Next, let γ2 + γ5 := Λ. By (7) one immediately concludes γ1 + γ4 = γ3 + γ6 = Λ. Now,
+we add (9) and (10) and rearrange the equations to obtain
+1
+2
+�
+γ2
+1 + · · · + γ2
+6 − 1
+�
++ γ3γ6 + γ2γ5 + γ1γ4 =γ2γ4γ6 + γ1γ3γ5
++ γ2γ3γ4 + γ1γ5γ6 + γ4γ5γ6
++ γ1γ2γ3 + γ3γ4γ5 + γ1γ2γ6 .
+By repeated factorization, the right hand side simplifies to
+γ1γ3(γ2 + γ5) + γ3γ4(γ2 + γ5) + γ1γ6(γ2 + γ5) + γ4γ6(γ2 + γ5) = Λ3,
+(12)
+and since for the left hand side one has
+1
+2
+�
+γ2
+1 + · · · + γ2
+6 − 1
+�
++ γ3γ6 + γ2γ5 + γ1γ4 = 3Λ2 − 1
+2
+,
+we arrive at the polynomial Λ3 − 3Λ2
+2 + 1
+2 = 0 the only positive solution of which is Λ = 1.
+Finally, adding the first the two equations of (9) yields
+γ3γ6 + γ2γ5 = (γ6γ5 + γ2γ3)(γ1 + γ4) = γ6γ5 + γ2γ3
+and this implies γ5 = γ3. Furthermore, adding the last two equations gives
+γ2γ5 + γ1γ4 = (γ4γ5 + γ1γ2)(γ3 + γ6) = γ4γ5 + γ1γ2
+giving γ4 = γ2.
+Conditions (7) hence give γ1 = γ5 and γ6 = γ2.
+This proves the
+statement.
+□
+We are now in the position to prove Theorem 6.
+Proof of Theorem 6. Comparing Πθ
+γ with ∆θ
+γ we conclude that ∆θ
+γ has a flat band with
+edge weights γ1, ..., γ6 if and only if there exists δ > 0 such that Πθ
+γ has a flat band for edge
+weights δγ1, ..., δγ6. From this observation the statement follows directly taking Lemma 7
+into account.
+□
+
+ROBUSTNESS OF FLAT BANDS
+9
+4.2. The spectrum and band gaps in the monomeric Kagome lattice. In the case
+where the perturbed Kagome lattice has a flat band, we further study the structure of the
+rest of the spectrum. We reiterate that, due to Theorem 6, the existence of a flat band is
+equivalent to the weights being monomeric.
+As shown for instance in [PT21], in the case where all edge weights are equal, the two
+other spectral bands, generated by the two other θ-dependent eigenvalues of ∆θ
+γ, touch
+at E = 3/4, and the derivative of the integrated density of states at E = 3/4 vanishes
+– an indication that the spectral density at 3/4 is sufficiently thin for a gap to form
+under perturbation. And indeed, this is the statement of the next theorem, which also
+characterises the width of the gap.
+Theorem 8 (Band gaps in the perturbed Kagome lattice). Consider the perturbed Kago-
+me lattice with fixed vertex weight µ > 0, and monomeric edge weights α, β > 0, satisfying
+2(α + β) = µ as characterized in Theorem 6. Then, the spectrum is given by
+I1 ∪ I2 :=
+�
+0, 3
+4 −
+����
+3α
+µ − 3
+4
+����
+� � �3
+4 +
+����
+3α
+µ − 3
+4
+���� , 3
+2
+�
+.
+Furthermore, there is always a flat band at 3
+2.
+Remark 9. Theorem 8 states that, as soon as α ̸= β, or alternatively, α ̸= µ
+4, a spectral
+gap of width
+����
+6α
+µ − 3
+2
+���� = 3
+µ|α − β|
+will form around 3
+4, see also Figure 3. The flat band at 3
+2 will always be connected to the
+energy band below it which means that the “touching” of the flat band at 3
+2 is protected in
+the class of monomeric perturbations.
+I1
+I2
+α = µ
+2
+α = 0
+α = µ
+4
+3
+2
+3
+4
+Flat band
+σ(∆γ)
+Figure 3. Spectrum of the monomeric (32.62) Kagome lattice with vertex
+weight µ > 0 as a function of the parameter α ∈ (0, µ
+2), describing the edge
+weights on edges adjacent to downwards pointing triangles.
+Proof. A calculation shows that the eigenvalues of ∆θ
+γ with the choice 2(α + β) = µ as in
+Theorem 6 are given by
+λ1,2(θ, γ) = 3
+4 ± 1
+4
+�
+1 + 8
+�
+1 + (F(θ) − 3)
+�2α
+µ − 4α2
+µ2
+��
+and
+λ3(θ, γ) = 3
+2
+
+10
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+where F(θ) := cos(θ1) + cos(θ2) + cos(θ1 − θ2). The function T2 ∋ θ �→ F(θ) takes all
+values in [−3/2, 3], see Lemma 3.1 in [PT21], whence λ1(θ, γ) and λ2(θ, γ) take all values
+in the intervals
+�
+0, 3
+4 −
+����
+3α
+µ − 3
+4
+����
+�
+,
+and
+�3
+4 +
+����
+3α
+µ − 3
+4
+���� , 3
+2
+�
+,
+respectively.
+□
+5. The perturbed Super-Kagome lattice
+In this section, we investigate the Archimedean tiling (3.122) which we call Super-
+Kagome lattice. Its minimal elementary cell contains six vertices and nine edges: three
+edges on upwards pointing triangles, three edges on downwards pointing triangles, and
+three edges bordering two dodecagons, see Figure 4.
+v4
+v3
+v5
+v6
+v2
+v1
+v1 − ω2
+v2 − ω1
+v6 + ω1
+v5 + ω2
+γ7
+γ3
+γ2
+γ1
+γ5
+γ6
+γ4
+γ9
+γ8
+γ8
+γ9
+Figure 4. Fundamental domain of the (3.122) tiling with edge weights. In
+the monomeric case, all edge weights around triangles triangles are γ1 =
+· · · = γ6 =: α and the remaining weights are γ7 = γ8 = γ9 =: β.
+Given a constant vertex weight µ > 0, the Floquet Laplacian (5) is a 6×6-matrix given
+by
+∆θ
+γ = Id −1
+µ
+
+
+
+
+
+
+
+0
+γ4
+γ6
+0
+zγ9
+0
+γ4
+0
+γ5
+0
+0
+wγ8
+γ6
+γ5
+0
+γ7
+0
+0
+0
+0
+γ7
+0
+γ3
+γ2
+zγ9
+0
+0
+γ3
+0
+γ1
+0
+wγ8
+0
+γ2
+γ1
+0
+
+
+
+
+
+
+
+,
+(13)
+where w := eiθ1, z := eiθ2.
+• If we fix a constant vertex weight µ > 0, the condition �
+w∼v γvw = µ for all v ∈ V
+leads to
+µ = γ2 + γ3 + γ7 = γ5 + γ6 + γ7 = γ1 + γ2 + γ8 = γ4 + γ5 + γ8
+= γ1 + γ3 + γ9 = γ4 + γ6 + γ9.
+(14)
+This can be seen to be a linear system of 6 linearly independent equations with 9
+unknowns, so the solution space is 3-dimensional. More precisely, by appropriate
+additions, we infer the three identities
+2γ1 + γ8 + γ9 = 2γ7 + γ2 + γ3,
+2γ4 + γ8 + γ8 = 2γ7 + γ5 + dγ6,
+γ2 + γ3 = γ5 + γ6
+(15)
+
+ROBUSTNESS OF FLAT BANDS
+11
+which imply γ1 = γ4. The identities γ2 = γ5, and γ3 = γ6 follow by completely
+analogous calculations. This leaves us with 6 independent variables γ1, γ2, γ3, and
+γ7, γ8, γ9 which are however still subject to the three conditions
+γ2 + γ3 + γ7 = γ1 + γ2 + γ8 = γ1 + γ3 + γ9 = µ
+from (14). Therefore, we are left with three degrees of freedom.
+• If we additionally prescribe monomericity, it is easy to see that there is only one
+degree of freedom: All edges around triangles carry the weight α > 0, and all
+remaining edges (separating two dodecagons) carry the weight β > 0 under the
+condition 2α + β = µ.
+5.1. Flat bands in the perturbed Super-Kagome lattice.
+Theorem 10. Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed
+vertex weight µ > 0, and periodic edge weights γ1, . . . , γ9 > 0 satisfying the condition (3)
+on vertex and edge weights. Then, the following are equivalent:
+(i) There exist exactly two flat bands.
+(ii) The Super-Kagome lattice is monomeric. More explicitly, there are α, β > 0 such
+that 2α + β = µ together with
+γ1 = γ2 = γ3 = γ4 = γ5 = γ6 = α ,
+γ7 = γ8 = γ9 = β .
+Proof. Recall that in the constant vertex weight case, we have
+γ1 = γ4 ,
+γ2 = γ5 ,
+and
+γ3 = γ6 ,
+and consider the weighted adjacency matrix
+Πθ
+γ :=
+
+
+
+
+
+
+
+0
+γ4
+γ6
+0
+zγ9
+0
+γ4
+0
+γ5
+0
+0
+wγ8
+γ6
+γ5
+0
+γ7
+0
+0
+0
+0
+γ7
+0
+γ3
+γ2
+zγ9
+0
+0
+γ3
+0
+γ1
+0
+wγ8
+0
+γ2
+γ1
+0
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+0
+γ1
+γ3
+0
+zγ9
+0
+γ1
+0
+γ2
+0
+0
+wγ8
+γ3
+γ2
+0
+γ7
+0
+0
+0
+0
+γ7
+0
+γ3
+γ2
+zγ9
+0
+0
+γ3
+0
+γ1
+0
+wγ8
+0
+γ2
+γ1
+0
+
+
+
+
+
+
+
+(16)
+which is a shifted and scaled version of ∆θ
+γ. We calculate
+det(λ Id −Πθ
+γ) = λ6 − λ4 �
+2γ2
+1 + 2γ2
+2 + 2γ2
+3 + γ2
+7 + γ2
+8 + γ2
+9
+�
+− 4λ3γ1γ2γ3
++ λ2�
+γ4
+1 + γ4
+2 + γ4
+3 + 2γ2
+1γ2
+2 + 2γ2
+2γ2
+3 + 2γ2
+3γ2
+1 + 2γ2
+1γ2
+7 + 2γ2
+2γ2
+9 + 2γ2
+3γ2
+8+
++ γ2
+7γ2
+8 + γ2
+8γ2
+9 + γ2
+9γ2
+7
+�
++ 4λγ1γ2γ3
+�
+γ2
+1 + γ2
+2 + γ2
+3
+�
+− γ4
+1γ2
+7 − γ4
+2γ2
+9 − γ4
+3γ2
+8 − γ2
+7γ2
+8γ2
+9 + 4γ2
+1γ2
+2γ2
+3
+− (w + w)
+�
+λ2γ2
+2γ7γ8 + 2λγ1γ2γ3γ7γ8 + γ2
+1γ2
+3γ7γ8 − γ2
+2γ7γ8γ2
+9
+�
+− (z + z)
+�
+λ2γ2
+3γ7γ9 + 2λγ1γ2γ3γ7γ9 + γ2
+1γ2
+2γ7γ9 − γ2
+3γ7γ2
+8γ9
+�
+− (wz + wz)
+�
+λ2γ2
+1γ8γ9 + 2λγ1γ2γ3γ8γ9 + γ2
+2γ2
+3γ8γ9 − γ2
+1γ2
+7γ8γ9
+�
+.
+Since w + w = 2 cos(θ1), z + z = 2 cos(θ2), and wz + wz = 2 cos(θ1 − θ2) are linearly on
+T2, λ is a θ-independent eigenvalue if and only if the conditions
+
+12
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+λ2γ2
+2 + 2λγ1γ2γ3 + γ2
+1γ2
+3 − γ2
+2γ2
+9 = 0,
+λ2γ2
+3 + 2λγ1γ2γ3 + γ2
+1γ2
+2 − γ2
+3γ2
+8 = 0,
+λ2γ2
+1 + 2λγ1γ2γ3 + γ2
+2γ2
+3 − γ2
+1γ2
+7 = 0,
+(17)
+as well as
+λ6 − λ4 �
+2γ2
+1 + 2γ2
+2 + 2γ2
+3 + γ2
+7 + γ2
+8 + γ2
+9
+�
+− 4λ3γ1γ2γ3
++ λ2�
+γ4
+1 + γ4
+2 + γ4
+3 + 2γ2
+1γ2
+2 + 2γ2
+2γ2
+3 + 2γ2
+3γ2
+1 + 2γ2
+1γ2
+7 + 2γ2
+2γ2
+9 + 2γ2
+3γ2
+8+
++ γ2
+7γ2
+8 + γ2
+8γ2
+9 + γ2
+9γ2
+7
+�
++4λγ1γ2γ3
+�
+γ2
+1 + γ2
+2 + γ2
+3
+�
+−γ4
+1γ2
+7 − γ4
+2γ2
+9 − γ4
+3γ2
+8 − γ2
+7γ2
+8γ2
+9 + 4γ2
+1γ2
+2γ2
+3 = 0
+(18)
+hold.3 Conditions (17) imply that any θ-independent eigenvalue of the matrix Πθ
+γ must
+satisfy
+λ = −γ1γ3
+γ2
+± γ9,
+λ = −γ1γ2
+γ3
+± γ8,
+and
+λ = −γ2γ3
+γ1
+± γ7.
+Since all γi are positive, the only way for these three equations to have the same set of
+solutions, that is for two flat bands to exist, is therefore
+− γ1γ3
+γ2
++ γ9 = −γ1γ2
+γ3
++ γ8 = −γ2γ3
+γ1
++ γ7
+(19)
+together with
+− γ1γ3
+γ2
+− γ9 = −γ1γ2
+γ3
+− γ8 = −γ2γ3
+γ1
+− γ7.
+(20)
+This implies that the matrix Πθ
+γ can only have two θ-independent eigenvalues if there are
+α, β > 0 with
+α = γ7 = γ8 = γ9
+and
+β = γ1 = γ2 = γ3,
+that is the monomeric case, and the only candidates for these eigenvalues are −β ±
+α. To see that they are indeed eigenvalues, one verifies by an explicit calculation that
+condition (18) is also fulfilled. This shows the stated equivalence.
+□
+Next, we further describe the spectrum of the monomeric Super-Kagome lattice.
+Theorem 11 (Band gaps in the perturbed Super-Kagome lattice). Consider the perturbed
+Super-Kagome lattice with Laplacian (4) with fixed vertex weight µ > 0 and monomeric
+edge weights α, β > 0, satisfying 2α + β = µ as characterized in Theorem 10. Then, the
+spectrum is given by
+I1 ∪ I2 :=
+�
+0,
+�
+1 − α
+2µ
+�
+− |3α − 2β|
+2µ
+� � ��
+1 − α
+2µ
+�
++ |3α − 2β|
+2µ
+, 2 − α
+µ
+�
+with flat bands at 3α
+µ and 2 − α
+µ.
+The spectrum and the position of the flat bands have been plotted in Figure 5. The
+spectrum generically consists of two distinct intervals (bands) except for the case 3α = 2β,
+that is α = 2µ
+7 , in which the two bands touch and the spectrum consists of one interval
+with an embedded flat band in the middle as well as a flat band at its maximum. This
+case α = 2µ
+7 connects two regimes with different spectral pictures:
+3As we will see later, despite its complexity, (18) will not impose further restrictions and hold in all
+relevant cases. This appears to be a consequence of symmetries of the lattice and the operator.
+
+ROBUSTNESS OF FLAT BANDS
+13
+• If α > 2µ
+7 the spectrum consists of two intervals the upper one of which has two
+flat bands at its endpoints. In the special case of uniform edge weights (that is
+α = µ
+3, this has already been observed, for instance in [PT21].
+• If α < 2µ
+7 , the spectrum will again consist of two intervals each of which will have
+a flat band at its maximum. Somewhat surprisingly, the lower flat band has now
+attach itself to the lower interval I2 upon passing the critical parameter α = 2µ
+7 .
+Another noteworthy observation is that no gap opens within the intervals I1 and I2,
+despite them being generated by two distinct Floquet eigenvalues and the density of
+states measure vanishing at a point in the interior of the bands, see again [PT21] for plots
+of the integrated density of states in the case of constant edge weights. In particular, this
+distinguishes the monomeric Super-Kagome lattice from the monomeric Kagome lattice
+where such a gap indeed opens within the spectrum at points of zero spectral density.
+I1
+I2
+α = µ
+2
+µ
+3 2µ
+7
+α = 0
+2
+Flat bands
+σ(∆γ)
+Constant edge weights
+Figure 5. Spectrum of the monomeric (3.122) “Super-Kagome” lattice
+with vertex weight µ > 0 as a function of the parameter α ∈ (0, µ
+2), describ-
+ing the edge weights on edges adjacent to triangles.
+Proof of Theorem 11. In the monomeric case, the characteristic polynomial det(λ Id −Πθ
+γ)
+of the matrix Πθ
+γ simplifies to
+((α + λ)2 − β2)·
+(λ4 − 2αλ3 − (3α2 + 2β2)λ2 + (4α3 + 2αβ2)λ + 4α4 + α2β2 + β4 − 2α2β2F(θ1, θ2)) ,
+where F(θ1, θ2) = cos(θ1) + cos(θ2) + cos(θ1 + θ2). Its six roots are
+�
+−α ± β, 1
+2
+�
+α ±
+�
+9α2 + 4β2 ± 4αβ
+�
+3 + 2F(θ1, θ2)
+��
+,
+whence the eigenvalues of ∆θ
+γ are given by
+λ1(θ, γ) = 1 − 1
+2µ
+�
+α +
+�
+9α2 + 4β2 + 4αβ
+�
+3 + 2F(θ1, θ2)
+�
+,
+λ2(θ, γ) = 1 − 1
+2µ
+�
+α +
+�
+9α2 + 4β2 − 4αβ
+�
+3 + 2F(θ1, θ2)
+�
+,
+λ3(θ, γ) = 1 + α − β
+µ
+= 3α
+µ =
+�
+1 − α−|3α−2β|
+2µ
+if 3α ≥ 2β ,
+1 − α−|3α−2β|
+2µ
+if 3α < 2β ,
+
+14
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+λ4(θ, γ) = 1 − 1
+2µ
+�
+α −
+�
+9α2 + 4β2 − 4αβ
+�
+3 + 2F(θ1, θ2)
+�
+,
+λ5(θ, γ) = 1 − 1
+2µ
+�
+β −
+�
+9α2 + 4β2 + 4αβ
+�
+3 + 2F(θ1, θ2)
+�
+,
+λ6(θ, γ) = 1 + α + β
+µ
+= 2 − α
+µ .
+Using that the map T2 ∋ (θ1, θ2) �→ F(θ1, θ2) takes all values in the interval (−3/2, 3), we
+conclude that the bands, generated by λ1(θ, γ) and λ2(θ, γ), as well as the bands generated
+by λ4(θ, γ) and λ5(θ, γ) always touch, and the spectrum consists of the two intervals
+�
+min
+θ∈T2 λ1(θ, γ), max
+θ∈T2 λ2(θ, γ)
+� � �
+min
+θ∈T2 λ4(θ, γ), max
+θ∈T2 λ5(θ, γ)
+�
+=
+�
+0, 1 − α + |3α − 2β|
+2µ
+� � �
+1 − α − |3α − 2β|
+2µ
+, 2 − α
+2µ
+�
+=
+�
+0,
+�
+1 − α
+2µ
+�
+− |3α − 2β|
+2µ
+� � ��
+1 − α
+2µ
+�
++ |3α − 2β|
+2µ
+, 2 − α
+µ
+�
+.
+□
+One might now wonder under which conditions only one flat band exists. The next
+theorem completely identifies all parameters for which one flat band exists:
+Theorem 12. Consider the perturbed Super-Kagome lattice with Laplacian (4), fixed
+vertex weight µ > 0, and periodic edge weights γ1, . . . , γ9 > 0 satisfying the condition
+(3) on vertex and edge weights. The set of (γi) such that exactly one flat band exists
+consists of six connected components which have no mutual intersections and have
+no intersection with the two-flat-band parameter set, identified in Theorem 10.
+The solution space is invariant under those permutations of the γi which correspond
+to rotations of the lattice by 2π
+3 , and 4π
+3 . Modulo these permutations, the two connected
+components can be described as follows
+• A one-dimensional submanifold, isomorphic to an interval, and explicitely descibed
+in equation (26),
+• Two one-dimensional submanifolds each isomorphic to an interval, explicitely de-
+scribed in (28), and (30), which intersect in a single point.
+Proof of Theorem 12. Recall that due to the reductions made at the beginning of the
+section, after fixing the constant vertex weight µ > 0, the space of edge weights is a
+3-dimensional manifold in the 6-dimensional parameter space {γ1, γ2, γ3, γ7, γ8, γ9 > 0},
+subject to the conditions
+γ1 + γ3 + γ9 = γ1 + γ2 + γ8 = γ2 + γ3 + γ7 = µ.
+(21)
+Furthermore, from the proof of Theorem 10 we infer that ∆γ has a flat band at λ if and
+only if the weighted adjacency matrix Πθ
+γ has the θ-independent eigenvalue ˜λ := µ(1−λ).
+This requires in particular that
+˜λ = −γ1γ3
+γ2
+± γ9 = −γ1γ2
+γ3
+± γ8 = −γ2γ3
+γ1
+± γ7
+(22)
+holds with a certain combination of plus and minus signs. Now, if equality in (22) holds
+with all three signs positive or all three signs negative, respectively, then the argument
+in the proof of Theorem 10 shows that this already implies that the edge weights are
+monomeric, the identities also hold with the opposite sign, the additional condition (18) is
+fulfilled, and there are two flat bands. As a consequence, the only chance for the existence
+
+ROBUSTNESS OF FLAT BANDS
+15
+of exactly one flat band is (22) to hold with different signs in front of γ7, γ8, γ9. Also, it
+is immediately clear that (22) with different signs does not allow for a monomeric and
+non-zero solution and hence the solution space consists of at most six mutually disjoint
+components which have no intersection with the two-flat-band manifold, identified in
+Theorem 10.
+By symmetry, it suffices to investigate two out of these six cases:
+Case(- + +):
+− γ1γ3
+γ2
+− γ9 = −γ1γ2
+γ3
++ γ8 = −γ2γ3
+γ1
++ γ7 = ˜λ ,
+(23)
+and
+Case(+ - -):
+− γ1γ3
+γ2
++ γ9 = −γ1γ2
+γ3
+− γ8 = −γ2γ3
+γ1
+− γ7 = ˜λ .
+(24)
+To solve Case(- + +), combine the second identities in in (21) and (23), to deduce
+γ3 − γ1 =
+γ2
+γ1γ3
+(γ2
+1 − γ2
+3)
+which, recalling γi > 0, is only possible if γ1 = γ3. But then, by (23), γ7 = γ8. Calling
+α′ := γ2, and β′ := γ9, we can use (21), to further express
+γ1 = γ3 = µ − β′
+2
+,
+and
+γ7 = γ8 = µ + β′
+2
+− α′.
+(25)
+Next, we eliminate β′ by resolving the yet unused first identity in (23), which yields
+− (µ − β′)2
+4α′
+− β′ = −α′ + µ + β′
+2
+− α′
+⇔
+β′ = µ − 3α′ ±
+�
+17α′2 − 8α′µ.
+This only has real solutions if α′ >
+8
+17µ > 1
+3µ, thus only
+β′ = µ − 3α′ +
+�
+17α′2 − 8α′µ.
+can be a positive solution. Furthermore, we need β′ ∈ (0, µ), which is the case if and only
+if
+γ2 = α′ ∈
+�µ
+2 , µ
+�
+.
+We therefore find the one-parameter solution set
+Case (- + +)
+
+
+
+
+
+
+
+
+
+γ1 = γ3
+= µ−β′
+2 ,
+γ2 = α′
+∈
+� µ
+2, µ
+�
+,
+γ7 = γ8
+= µ+β′
+2
+− α′,
+γ9 = β′
+:= µ − 3α′ +
+�
+17α′2 − 8αµ
+(26)
+with energy
+˜λ = −γ2 + γ7 = −2α′ + µ + β′
+2
+= −2α′ + 2µ − 3α′ +
+�
+17α′2 − 8α′µ
+2
+.
+Finally, an explicit calculation shows that with these parameters, (18) is indeed fulfilled.
+As for Case(+ - -), we combine the second identity in (21) with the second identity
+in (24) to deduce
+γ3 − γ1 =
+γ2
+γ1γ3
+(γ2
+3 − γ2
+1) .
+(27)
+Identity (27) has two types of solutions:
+Case(+ - -)(a): γ1 = γ3.
+
+16
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
+As before we find γ7 = γ8. Let α′ := γ2, β′ := γ9, and combine the remaining first identity
+in (24) with (25) to solve for β′, finding
+− (µ − β′)2
+4α′
++ β′ = −µ + β′
+2
+⇔
+β′ = µ + 3α′ ±
+�
+9α′2 + 8α′µ.
+Only the solution
+β′ = µ + 3α′ −
+�
+9α′2 + 8α′µ
+has a chance to be in (0, µ), and, indeed, this is the case if and only if
+γ2 = α′ ∈
+�
+0, µ
+2
+�
+.
+We obtain the one-parameter solution set
+Case(+ - -)(a)
+
+
+
+
+
+
+
+
+
+γ1 = γ3
+= µ−β′
+2 ,
+γ2 = α′
+∈
+�
+0, µ
+2
+�
+,
+γ7 = γ8
+= µ+β′
+2
+− α′,
+γ9 = β′
+:= µ + 3α′ −
+�
+9α′2 + 8α′µ
+(28)
+with energy
+˜λ = −γ2 − γ7 = −µ + β′
+2
+= −2µ + 3α′ −
+�
+9α′2 + 8α′µ
+2
+.
+Again, an explicit calculation shows that (18) is fullfilled.
+Case(+ - -)(b): The other solution of (27) is
+γ1γ3 = γ2(γ1 + γ3).
+We set α′′ := γ1, β′′ := γ3, whence
+γ2 =
+α′′β′′
+α′′ + β′′,
+and use (21) to infer
+γ7 = µ − 2α′′β′′ + β′′2
+α′′ + β′′
+,
+γ8 = µ − α′′2 + 2α′′β′′
+α′′ + β′′
+,
+γ9 = µ − α′′ − β′′.
+(29)
+Plugging (29) into the yet unused first identity in (24), we arrive at
+− (α′′ + β′′) + µ − α′′ − β′′ = −
+α′′2
+α′′ + β′′ − µ + α′′2 + 2α′′β′′
+α′′ + β′′
+⇔
+β′′ = µ − 3α′′ ±
+�
+(µ − 3α′′)2 + 4α′′(µ − α′′)
+2
+= µ − 3α′′ ±
+�
+µ2 − 2α′′µ + 5α′′2
+2
+We observe that only the solution with a plus has a chance to be positive and it is easy to
+see that this solution takes values in (0, µ) for all α′′ ∈ (0, µ). We obtain the one-parameter
+solution set
+Case (+ - -) (b)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+γ1 = α′′
+∈ (0, µ) ,
+γ2
+=
+α′′β′′
+α′′+β′′,
+γ3 = β′′
+:=
+µ−3α′′+√
+µ2−2α′′µ+5α′′2
+2
+,
+γ7
+= µ − 2α′′β′′+β′′2
+α′′+β′′
+,
+γ8
+= µ − α′′2+2α′′β′′
+α′′+β′′
+,
+γ9
+= µ − α′′ − β′′
+(30)
+
+ROBUSTNESS OF FLAT BANDS
+17
+at energy
+˜λ = −γ1γ3
+γ2
++ γ9 = µ − 2α′′ − 2β′′ = α −
+�
+µ2 − 2α′′µ + 5α′′2.
+Again, an explicit calculation verifies that with these choices, (18) is fullfilled.
+Finally, to conclude the claimed topological properties of the manifolds, we need to
+verify that the solution space (28) in Case(+ - -)(a) intersects the solution space (30)
+in Case(+ - -)(b) if and only if
+γ1 = γ3 = γ7 = γ8 = 2µ
+5 ,
+γ2 = γ9 = µ
+5.
+□
+X2
+X1
+One flat band, Case(- + +)
+One flat band, Case(+ - -) (a)
+One flat band, Case(+ - -) (b)
+Monomeric edge weights,
+two flat bands
+Extremal cases, not belonging
+to the parameter space
+Figure 6. Schematic overview of the topology of the six “spurious” one-
+flat-band solution sets, and the monomeric two-flat-band manifold within
+the constant-vertex weight parameter space.
+Case(- + +) solutions
+asymptotically meet the limit points of the two-flat-band manifold at one
+end of the parameter range, whereas Case(+ - -) (a) solutions asymptot-
+ically meet it at both ends of the parameter range.
+Remark 13. Theorems 10 and 12 imply that the six one-flat-band components and the
+two-flat-band component are mutually disjoint. However, a closer analysis of the extremal
+cases in Formulas (26), (28), and (30), as well as of the monomeric case, implies that
+when sending the parameters to their extremal values, the three one-dimensional manifolds
+corresponding to Case(+ - -) (a), and the two-flat-band-manifold of solutions converge
+to the two points
+X1 :=
+�
+0, 0, 0, µ
+2, µ
+2 , µ
+2
+�
+and
+X2 :=
+�µ
+2, µ
+2 , µ
+2 , 0, 0, 0
+�
+,
+which themselves do no longer belong to the space of admissible parameters. Likewise, the
+limit of solutions of Case(+ - -) in (26) corresponding to α′ = µ
+2 corresponds to the the
+point X2, see also Figure 6.
+Acknowledgement. JK would like to thank the Bergische Universit¨at Wuppertal where
+parts of this project were done while being on leave from the FernUniversit¨at in Hagen.
+JK and MT also acknowledge support by the Cost action CA18232 through the summer
+school “Heat Kernels and Geometry: From Manifolds to Graphs” held in Bregenz. MT
+would like to thank the Mittag-Leffler Institute where parts of this work were initiated
+during the trimester Program “Spectral Methods in Mathematical Physics”.
+
+18
+J. KERNER, M. T¨AUFER, AND J. WINTERMAYR
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+

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+arXiv:2301.00753v1  [cs.IT]  2 Jan 2023
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND
+NEW QUANTUM CODES
+REZA DASTBASTEH AND KHALIL SHIVJI
+Abstract. We give a polynomial representation for additive cyclic codes over Fp2. This repre-
+sentation will be applied to uniquely present each additive cyclic code by at most two generator
+polynomials. We determine the generator polynomials of all different additive cyclic codes. A
+minimum distance lower bound for additive cyclic codes will also be provided using linear cyclic
+codes over Fp. We classify all the symplectic self-dual, self-orthogonal, and nearly self-orthogonal
+additive cyclic codes over Fp2. Finally, we present ten record-breaking binary quantum codes
+after applying a quantum construction to self-orthogonal and nearly self-orthogonal additive
+cyclic codes over F4.
+Keywords: additive cyclic codes, quantum code, self-orthogonal codes, self-dual codes
+1. Introduction
+Quantum error-correcting codes, or simply quantum codes, are used in quantum computation
+to protect quantum information from corruption by noise (decoherence). A general framework
+of quantum codes is provided in [9, 13]. Throughout this paper, Fp2 is the finite field of p2
+elements, where p is a prime number. The parameters of a quantum code over Fp that encodes
+k logical qubits to n physical qubits and has minimum distance d is denoted by [[n, k, d]]p. An
+important family of quantum codes with many similar properties as classical block codes is
+the family of quantum stabilizer codes. In particular, quantum stabilizer codes are constructed
+using additive codes which are self-orthogonal with respect to a certain symplectic inner product.
+Several constructions of quantum stabilizer codes from various classical codes are given in [18].
+An interesting modification of the original definition of quantum stabilizer codes is by relaxing
+its self-orthogonality constraint [5, 19]. This method enables us to construct good quantum
+codes using not necessarily self-orthogonal additive codes over F4. Previously, this modification
+was applied for the construction of new quantum codes from different families of linear codes
+[6, 10, 20].
+Additive cyclic codes are of interest due to their rich algebraic properties and application
+in the construction of quantum codes. There have been several works in the literature toward
+the classification of additive cyclic codes for different applications [1, 4, 7, 16, 17, 21], and also
+due to their connection to other families of block codes such as quasi-cyclic codes [15].
+In
+[16], a canonical decomposition of additive cyclic code over F4 was introduced using certain
+finite field extensions of F4. This decomposition was applied to determine self-orthogonal and
+self-dual additive cyclic codes over F4 with respect to the trace inner product. In [3], it was
+shown that each additive cyclic code over F4 of length n can be generated by F2-span of at
+most two polynomials in F4[x]/⟨xn − 1⟩ and their cyclic shifts. Moreover, a criterion for the
+self-orthogonality of such codes with respect to the trace inner product was provided. Another
+interesting construction for a subclass of additive cyclic code, namely twisted codes, was provided
+in [1]. This construction is analogous to the way linear cyclic codes are constructed. In spite of
+many useful properties of twisted codes, all additive cyclic codes cannot be described using the
+theory of additive twisted codes.
+1
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+2
+In this work, we first give a canonical representation of all Fp-additive cyclic codes over Fp2
+using at most two generator polynomials. Our representation is more computationally friendly
+than the canonical representation of [16].
+This representation allows us to give a minimum
+distance lower bound for additive cyclic codes over Fp2 using the minimum distance of linear
+cyclic codes over Fp.
+Moreover, we provide a unique set of generator polynomials for each
+additive cyclic code over Fp2.
+This representation of generator polynomials will be used to
+characterize all self-orthogonal and self-dual additive cyclic codes with respect to the symplectic
+inner product. We also determine the generator polynomials of the symplectic dual of a given
+additive cyclic code over Fp2, and compute nearly the self-orthogonality of each additive cyclic
+code using only its generator polynomials. This allows us to apply the nearly self-orthogonal
+construction of quantum codes developed in [5, 19]. In particular, we provide a list of eleven
+record-breaking binary quantum codes after applying the mentioned quantum construction to
+nearly self-orthogonal additive cyclic codes. Furthermore, applying secondary constructions to
+our new quantum codes produce many more record-breaking binary codes. Note that such new
+quantum codes cannot be constructed using self-orthogonal additive cyclic codes of the same
+length.
+This paper is organized as follows. Section 2 briefly recalls the essential terminologies used in
+this work. Section 3 gives a canonical representation of additive cyclic codes over Fp2. In fact,
+we follow a module theory approach to decompose each additive cyclic code using its polynomial
+representation in Fp2[x]/⟨xn −1⟩. In Section 4, we compute the symplectic dual of each additive
+cyclic code. We provide the necessary and sufficient conditions for an additive cyclic code to
+be self-orthogonal, self-dual, or nearly self-orthogonality with respect to the symplectic inner
+product. Finally, in Section 5, we present the parameters of our record-breaking quantum codes.
+2. Preliminaries
+Let ω be a primitive element of Fp2. Then the set {1, ω} forms a basis for Fp2 over Fp. Let
+a + bω and a′ + b′ω ∈ Fn
+p2, where a, a′, b, b′ ∈ Fn
+p. The symplectic inner product of a + bω and
+a′ + b′ω is defined by
+⟨a + bω, a′ + b′ω⟩s = a′ · b − a · b′.
+(2.1)
+An Fp-linear subspace C ⊆ Fn
+p2 is called a length n additive code over Fp2.
+We denote the
+Fp-dimension of an additive code C over Fp2 with dimFp(C). Let C ⊆ Fn
+p2 be an additive code
+over Fp2 such that dimFp(C) = k. Then we call C an (n, pk) code. The set
+C⊥s = {x ∈ Fn
+p2 : ⟨x, y⟩s = 0 for all y ∈ C}.
+is called the symplectic dual of C. One can easily see that C⊥s is an (n, p2n−k) additive code
+over Fp2. The code C is called self-orthogonal (respectively self-dual) if C ⊆ C⊥s (respectively
+if C = C⊥s). For each x ∈ Fn
+p2, we denote the number of non-zero coordinates of x by wt(x).
+Moreover, the minimum weight among non-zero vectors of an additive code C is denoted by
+d(C). The connection between quantum stabilizer codes and classical additive codes was initially
+formulated by the independent works of Calderbank, Rains, Shor, and Sloane [3] and Gottesman
+[11]. A non-binary version of this connection is provided below.
+Theorem 2.1. [18, Corollary 16] Let C be an (n, pn−k) additive code over Fp2. Then there exists
+an [[n, k, d]]p quantum stabilizer code if C is symplectic self-orthogonal, where d = min{wt(x) :
+x ∈ C⊥
+s \ C} if k > 0 and d = min{wt(x) : x ∈ C} if k = 0.
+The quantum code of Theorem 2.1 is called pure if d = d(C⊥s). There are several secondary
+constructions of quantum code. A short list of such constructions is provided below.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+3
+Theorem 2.2. [18, Section XV] Let C be an [[n, k, d]]p quantum code.
+(1) If k > 0, then an [[n + 1, k, d]]p quantum code exists.
+(2) If C is pure and n, d ≥ 2, then an [[n − 1, k + 1, d − 1]]p pure quantum code exists.
+(3) If k > 1, then there exists an [[n, k − 1, d]]p quantum code.
+3. Additive cyclic codes over Fp2
+Throughout this section, we assume that n is a positive integer such that (n, p) = 1 and
+Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree two irreducible polynomial over
+Fp. In this section, we provide a canonical representation of additive cyclic codes over the field
+Fp2. In particular, we give a unique representation of each additive cyclic code over Fp2 using
+at most two generator polynomials. Moreover, we determine the generator polynomials of all
+different additive cyclic codes over Fp2. In particular, each additive cyclic code over F2
+p is a linear
+combination of cyclic shifts of its generator polynomials. Such representation is also suitable
+for practical computations of additive cyclic codes, especially using Magma computer algebra
+system [2]. More particularly, there exists a built-in function in Magma which forms additive
+cyclic codes generated by two given generator polynomials. At the end of this section, we give
+a minimum distance lower bound for the minimum distance of additive cyclic codes over Fp2
+using the minimum distance of linear cyclic codes over Fp.
+Definition 3.1. An Fp-subspace C ⊆ Fn
+p2 is called an additive cyclic code of length n over Fp2,
+if for every (a0, a1, . . . , an−1) ∈ C, the vector (an−1, a0, . . . , an−2) is also a codeword of C.
+We will use the following concepts of module theory frequently in this section, and for more
+details one, for example, can see [8, Chapter 12]. Let R be a principal ideal domain and M be an
+R-module. The annihilator of M is an ideal of R defined by {r ∈ R : rm = 0 for any m ∈ M}.
+An element m ∈ M is called a torsion element, if there exists 0 ̸= r ∈ R such that rm = 0.
+The module M is called a torsion module if all of its elements are torsion.
+The following
+theorem, known as the primary decomposition theorem of modules, plays an important role in
+our representation of additive cyclic codes.
+Theorem 3.2. [8, Chapter 12, Theorem 7] Let R be a principal ideal domain and M be a torsion
+R-module with the annihilator ⟨a⟩ ̸= 0. Let a = u
+n
+�
+i=1
+pai
+i , where u is a unit and pi is a prime
+element for each 1 ≤ i ≤ n. Then we can decompose M as a direct sum of its submodules in the
+form
+M =
+n
+�
+i=1
+Ni,
+(3.1)
+where Ni = {x ∈ M : xpai
+i = 0} for each 1 ≤ i ≤ n.
+Each element (a0, a1, . . . , an−1) ∈ Fn
+p2 can be represented uniquely as a polynomial in Fp2[x]/⟨xn−
+1⟩ in the form
+n−1
+�
+i=0
+aixi. One can easily verify that, under this correspondence, a length n additive
+cyclic codes over Fp2 is an Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩.
+Notation 3.3. Let f and g ∈ Fp2[x]/⟨xn −1⟩. We fix the following notations for the rest of this
+paper.
+(1) The ideal generated by f in Fp2[x]/⟨xn − 1⟩ is denoted by ⟨f⟩Fp2[x]. Equivalently it is
+the Fp2[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial f.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+4
+(2) The Fp[x]-submodule of Fp2[x]/⟨xn − 1⟩ generated by the polynomial g is denoted by
+⟨g⟩Fp[x].
+A straightforward computation shows that the annihilator of Fp2[x]/⟨xn − 1⟩ as an Fp[x]-
+module is the ideal ⟨xn−1⟩. Moreover, we can decompose xn−1 over Fp[x] as xn−1 =
+s
+�
+i=1
+fi(x),
+where each fi(x) is an irreducible polynomial corresponding to a p-cyclotomic coset modulo n.
+Next, we apply Theorem 3.2 to Fp2[x]/⟨xn − 1⟩. It is straightforward to see that
+Fp2[x]/⟨xn − 1⟩ =
+s
+�
+i=1
+Ni,
+(3.2)
+where Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x] for each 1 ≤ i ≤ s. We call a non-zero length n additive cyclic
+code C over Fp2 irreducible if for any additive cyclic code D ⊆ C, then D = {0} or D = C. The
+next lemma shows that each Ni can be decomposed as a direct sum of two irreducible additive
+cyclic codes. We determine the generator polynomial of all irreducible additive cyclic codes
+inside Ni and provide other useful information about additive cyclic codes inside each Ni.
+Lemma 3.4. Let f(x) be an irreducible divisor of xn − 1 over Fp[x] with deg(f) = k and
+N = ⟨(xn − 1)/f(x)⟩Fp2[x].
+(1) Let 0 ̸= r(x) ∈ N, then the set L = {r(x), xr(x), . . . , xk−1r(x)} forms a basis for
+⟨r(x)⟩Fp[x] as an Fp vector space.
+(2) Let 0 ̸= C ⊊ N be an additive cyclic code. The code C has Fp-dimension k and C =
+⟨r(x)⟩Fp[x] for any 0 ̸= r(x) ∈ C.
+(3) The additive cyclic code N can be decomposed as
+N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x].
+Moreover, dimFp(N) = 2k and N is linear over Fp2.
+(4) The number of irreducible additive cyclic codes inside N is 2k + 1. In particular, the
+following set gives all the different generator polynomials of such additive cyclic codes.
+A = {
+�
+(xn − 1)/f(x)
+��
+ω + g(x)
+�
+: g(x) ∈ Fp[x], deg(g(x)) < k} ∪ {(xn − 1)/f(x)}.
+(3.3)
+Proof. (1) Obviously L ⊆ ⟨r(x)⟩Fp[x]. Suppose, on the contrary, that L is linearly dependent
+over Fp. Hence we can find a polynomial 0 ̸= s(x) ∈ Fp[x] of degree less than k such that
+r(x)s(x) ≡ 0 (mod xn − 1). Since (xn − 1)/f(x) | r(x) and f(x) is irreducible, we conclude that
+f(x) | s(x). However, it is a contradiction with the fact that deg(s(x)) < k. This shows that L
+is linearly independent over Fp. Note that the set L ∪ {xkr(x)} is linearly dependent over Fp
+as this new set generates f(x)r(x) ≡ 0 (mod xn − 1). In a similar fashion, one can show that
+{xir(x)} for k < i < n − 1 can be written as a linear combination of elements of L over Fp.
+Therefore, L forms a basis for ⟨r(x)⟩Fp[x].
+(2) Let 0 ̸= r(x) ∈ C.
+Suppose in contrary that ⟨r(x)⟩Fp[x] ⊊ C.
+Then there exists a
+polynomial s(x) ∈ C such that s(x) ̸∈ ⟨r(x)⟩Fp[x]. Note that ⟨r(x)⟩Fp[x] ∩ ⟨s(x)⟩Fp[x] = {0} as
+otherwise, by part (1), for any polynomial a(x) in the intersection, we have
+⟨r(x)⟩Fp[x] = ⟨a(x)⟩Fp[x] = ⟨s(x)⟩Fp[x],
+which is a contradiction. Thus C = ⟨r(x)⟩Fp[x] and has dimension k over Fp.
+(3) It is easy to see that ⟨(xn − 1)/f(x)⟩Fp[x] ∩ ⟨ω((xn − 1)/f(x)⟩Fp[x] = {0} and
+N = ⟨(xn − 1)/f(x)⟩Fp[x] ⊕ ⟨ω((xn − 1)/f(x))⟩Fp[x].
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+5
+Hence N has dimension 2k over Fp. The linearity part follows immediately from the structure
+of its generator polynomials.
+(4) In order to find an additive cyclic code with Fp-dimension k, we need to choose a nonzero
+polynomial r(x) ∈ N to be its generator. Also, any non-zero elements of ⟨r(x)⟩Fp[x] generates
+the same code. Hence the number of additive cyclic codes with one non-zero generator inside
+N is 22k−1
+2k−1 = 2k + 1.
+Let C1 and C2 be two k-dimensional additive cyclic codes inside N. If C1 ∩ C2 ̸= {0}, then
+C1 = C2 by part (1). Equivalently, if C1 + C2 = N, then C1 ∩ C2 = {0}. Now we show that
+different elements of the set A generate different codes. Let g(x) ∈ Fp[x] such that deg(g(x)) < k.
+Clearly the additive cyclic code C1 = ⟨(xn − 1)/f(x), ((xn − 1)/f(x))(g(x) + ω)⟩Fp[x] contains
+(xn − 1)/f(x) and ω(xn − 1)/f(x). Therefore C1 = N. So ⟨(xn − 1)/f(x)⟩Fp[x] and ⟨((xn −
+1)/f(x))(g(x) + ω)⟩Fp[x] are different additive cyclic codes.
+Let g1(x) and g2(x) ∈ Fp[x] be two different polynomials of degree less than k. The code
+C = ⟨((xn − 1)/f(x))(ω + g1(x)), ((xn − 1)/f(x))(ω + g2(x))⟩Fp[x] contains (xn − 1)/f(x) and
+ω(xn − 1)/f(x). It is mainly because
+⟨
+�
+(xn − 1)/f(x)
+�
+(g1(x) − g2(x))⟩Fp[x] = ⟨(xn − 1)/f(x)⟩Fp[x].
+Thus C = N. This implies that the additive cyclic codes ⟨((xn − 1)/f(x))(ω + g1(x))⟩Fp[x] and
+⟨((xn−1)/f(x))(ω+g2(x))⟩Fp[x] are different. This proves that the set A contains all the different
+generators of irreducible additive cyclic codes inside N.
+□
+As we mentioned in part (1) of Lemma 3.4, each additive cyclic code inside ⟨(xn−1)/f(x)⟩Fp2[x]
+can have many different generator polynomials. Through the next remark, we fix a canonical
+representation for each additive cyclic code inside N.
+Remark 3.5. For each additive code 0 ̸= C ⊊ ⟨(xn −1)/f(x)⟩Fp2[x], we fix its generator polyno-
+mial inside the set A, introduced in (3.3), to be “the” generator polynomial of C. Similarly, the
+additive cyclic code C′ = ⟨(xn−1)/f(x)⟩Fp2[x] can be generated by the polynomials (xn−1)/f(x)
+and ω((xn − 1)/f(x)). We call them “the” generator polynomials of C′.
+This representation helps to uniquely identify each additive cyclic code inside N and avoid
+considering the same code more than once. Next, we use the result of Lemma 3.4 and characterize
+all the additive cyclic codes of length n over Fp2. Recall that xn − 1 =
+s
+�
+i=1
+fi(x), where fi(x) is
+an irreducible polynomial over Fp[x] for each 1 ≤ i ≤ s and Ni = ⟨(xn − 1)/fi(x)⟩Fp2[x].
+Theorem 3.6. Let C be a length n additive cyclic code over Fp2. Then
+(i) we can decompose the code C as C =
+s
+�
+i=1
+Ci, where each Ci is an additive cyclic code
+inside Ni.
+(ii) we have C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x], where
+(a) g(x) + ωk(x) =
+s
+�
+i=1
+gi(x) + ωki(x),
+(b) h(x) =
+s
+�
+i=1
+hi(x),
+(c) and Ci has the generator polynomial(s) gi(x) + ωki(x) and ωhi(x) selected as dis-
+cussed in Remark 3.5.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+6
+(iii) dimFp(C) =
+s
+�
+i=1
+(deg(fi) × # of non-zero generators of Ci).
+Proof. (i) As we mentioned in (3.2), the following decomposition holds
+Fp2[x]/⟨xn − 1⟩ =
+s
+�
+i=1
+Ni.
+So we can express C as C = �s
+i=1 Ci, where each Ci is an additive cyclic codes inside Ni.
+(ii) We show that the additive cyclic codes C =
+s
+�
+i=1
+Ci and ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] are the
+same. First note that g(x) + ωk(x), ωh(x) ∈ C and thus ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] ⊆ C. Let
+1 ≤ i ≤ s be a fixed integer. Since
+�
+(xn − 1)/fi(x)
+�
+| gi(x), ki(x), hi(x) and
+�
+(xn − 1)/fi(x)
+�
+gj(x) ≡
+�
+(xn − 1)/fi(x)
+�
+kj(x) ≡
+�
+(xn − 1)/fi(x)
+�
+hj(x) ≡ 0
+(mod xn − 1)
+for any j ̸= i, we have
+�
+(xn − 1)/fi(x)
+��
+g(x) + ωk(x)
+�
+≡
+�
+(xn − 1)/fi(x)
+��
+gi(x) + ωki(x)
+�
+(mod xn − 1)
+and
+�
+(xn − 1)/fi(x)
+�
+ωh(x) ≡
+�
+(xn − 1)/fi(x)
+�
+ωhi(x)
+(mod xn − 1).
+Moreover, we have
+Ci = ⟨gi(x)+ωki(x), ωhi(x)⟩Fp[x] = ⟨
+�
+(xn −1)/fi(x)
+��
+g(x)+ωk(x)
+�
+,
+�
+(xn −1)/fi(x)
+�
+ωh(x)⟩Fp[x].
+Thus
+Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x].
+This show that
+s
+�
+i=1
+Ci ⊆ ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] and completes the proof.
+(iii) Note that dimFp(C) =
+s
+�
+i=1
+dimFp(Ci). Moreover, by Lemmas 3.4, dimFp(Ci) = 0, ki, or
+2ki if Ci = 0, Ci is generated by one generator polynomial, or Ci has two generator polynomials,
+respectively. Combining these facts with the result of part (i) completes this proof.
+□
+Through the next corollary, we characterize all the length n irreducible additive cyclic codes
+over Fp2.
+Proposition 3.7. Let C be an additive cyclic code of length n over Fp2. Then C is irreducible
+if and only if C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s. Moreover, there are
+s
+�
+i=1
+(2deg(fi) + 1) many different irreducible additive cyclic codes.
+Proof. Let C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Ni and 1 ≤ i ≤ s. The result of part (1) in
+Lemma 3.4 shows that C is irreducible. Conversely, let C be an irreducible additive cyclic code.
+Then by part (i) of Theorem 3.6 we have C =
+s
+�
+i=1
+Ci. Since C is irreducible, we have C = Cj
+for some 1 ≤ j ≤ s. Moreover, since Nj is not irreducible by Lemma 3.4 part (3), we conclude
+that C = ⟨r(x)⟩Fp[x] for some 0 ̸= r(x) ∈ Nj.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+7
+Inside each Ni, there are 2deg(fi)+1 many different one generator additive cyclic codes. Hence
+the total number of irreducible codes is
+s
+�
+i=1
+(2deg(fi) + 1).
+□
+Remark 3.8. Henceforth, we always represent each additive cyclic code with its generator
+polynomials g(x) + ωk(x) and ωh(x) introduced in part (ii) of Theorem 3.6. Moreover, the way
+we generate these polynomials is unique, and therefore each additive cyclic code has a unique
+set of generators.
+From now on, we call Fp2-linear cyclic codes simply linear cyclic codes. Let C = ⟨g(x) +
+ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2. Note that Theorem 3.6 and part
+(3) of Lemma 3.4 imply that C is linear if and only if g(x) = h(x) and k(x) = 0. Hence linear
+cyclic codes can be easily distinguished from non-linear cyclic codes.
+Next, we provide a minimum distance bound for additive cyclic codes using linear cyclic
+codes over Fp. In general, the minimum distance computation for linear codes is faster than the
+additive codes. Hence the following result can speed up the minimum distance computation for
+additive cyclic codes. We denote the minimum distance of a code C with d(C).
+Theorem 3.9. Let C = ⟨g(x) + ωk(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2.
+Let G(x) =
+xn−1
+gcd(xn−1,g(x)), and let S(x) be the generator polynomial of the intersection of the
+length n linear cyclic code generated by k(x) and the linear cyclic code generated by h(x) over
+Fp. Suppose that D1, D2, D3, and D3 are the length n linear cyclic codes over Fp generated by
+g(x), gcd(k(x), h(x)), gcd(G(x)k(x), h(x)), and
+g(x)S(x)
+gcd(xn−1,k(x)), respectively. Then
+min{d(D3), d(D4), max{d(D1), d(D2)}} ≤ d(C).
+(3.4)
+Proof. Only the following three types of codewords may appear in the code C.
+T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]},
+T2 = {ωb(x) ∈ C : 0 ̸= b(x) ∈ Fp[x]},
+T3 = {a(x) + ωb(x) ∈ C : 0 ̸= a(x), 0 ̸= b(x) ∈ Fp[x]}.
+We bound the minimum distance of C by considering the minimum distance in each of these
+sets. Let f(x) ∈ T1. Then we can write it as f(x) = a1(x)(g(x) + ωk(x)) + b1(x)ωh(x) for some
+a1(x), b1(x) ∈ Fp[x]. Hence f(x) = a1(x)g(x) and a1(x)k(x) + b1(x)h(x) ≡ 0 (mod xn − 1).
+This implies that a1(x)k(x) is an element of the length n linear cyclic code over Fp generated
+by S(x). Hence
+S(x)
+gcd(xn−1,k(x)) | a1(x). In other words, f(x) = a(x)g(x) ∈ D4.
+Next, let ωf1(x) ∈ T2. Then ωf1(x) = a1(x)(g(x)+ωk(x))+b1(x)ωh(x) for some a1(x), b1(x) ∈
+Fp[x].
+Then a1(x)g(x) ≡ 0 (mod xn − 1) or equivalently G(x) | a1(x).
+This implies that
+f1(x) = a1(x)k(x) + b1(x)h(x). Therefore, f1(x) ∈ D3.
+Finally, let a(x) + ωb(x) ∈ T3. Then a(x) + ωb(x) = l(x)(g(x) + ωk(x)) + m(x)ωh(x) for some
+l(x), m(x) ∈ Fp[x]. Hence a(x) ∈ D1 and b(x) ∈ D2. This implies that wt(a(x) + ωb(x)) ≥
+max{d(D1), d(D2)}.
+□
+Note that if Di = 0 for any value 1 ≤ i ≤ 4, then we simply discard this code in the minimum
+distance lower bound of (3.4). For instance if D1 = 0, then the minimum distance lower bound
+of (3.4) becomes
+min{d(D3), d(D4), d(D2)} ≤ d(C).
+The following corollary gives a modification of this result to additive cyclic codes, which are
+generated by only one generator. In this result, the cyclic codes Ci are obtained from Di after
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+8
+substituting h(x) with 0 in Theorem 3.9 for 1 ≤ i ≤ 3.
+However, the code C4 is obtained
+differently by considering a more direct observation.
+Corollary 3.10. Let C = ⟨g(x) + ωk(x)⟩Fp[x] be a length n additive cyclic code over Fp2. Let
+C1, C2, C3, and C4 be the length n linear cyclic codes over Fp generated by polynomials g(x),
+k(x),
+xn−1
+gcd(xn−1,g(x))k(x), and
+xn−1
+gcd(xn−1,k(x))g(x), respectively. Then
+min{d(C3), d(C4), max{d(C1), d(C2)}} ≤ d(C).
+(3.5)
+Proof. As we mentioned above, the code Ci all are obtained after applying the condition h(x) = 0
+in the structure of the codes Di for 1 ≤ i ≤ 3 in Theorem 3.9. Since the code D4 in Theorem
+3.9 is applied to bound the minimum weight of the set
+T1 = {a(x) ∈ C : 0 ̸= a(x) ∈ Fp[x]},
+we compute the minimum weight of T1 directly in this proof. Let f(x) ∈ T1. Then we can write
+it as f(x) = a(x)(g(x)+ωk(x)) for some a(x) ∈ Fp[x]. Hence f(x) = a(x)g(x) and a(x)k(x) ≡ 0
+(mod xn − 1). This implies that
+xn−1
+gcd(xn−1,k(x)) | a(x). Hence
+xn−1
+gcd(xn−1,k(x))g(x) | a(x) and we
+have f(x) ∈ C4.
+□
+Next, we consider the restriction of the mentioned minimum distance bound to linear cyclic
+codes with the generator polynomials g(x) + ωk(x) and h(x), where k(x) = 0.
+Corollary 3.11. Let C = ⟨g(x), ωh(x)⟩Fp[x] be a length n additive cyclic code over Fp2. Let
+E1 and E2 be the length n linear cyclic codes over Fp generated by polynomials g(x) and h(x),
+respectively. Then
+min{d(E1), d(E2)} ≤ d(C).
+(3.6)
+Proof. Applying the condition k(x) = 0 to Theorem 3.9 implies that D1 = D4 = E1 and
+D2 = D3 = E2. Now the result follows from the minimum distance bound of Theorem 3.9.
+□
+4. Symplectic inner product and dual of additive cyclic codes
+In this section, we determine generator polynomials of the symplectic dual of a given additive
+cyclic code over Fp2. Moreover, we give the generator polynomials of all self-orthogonal and self-
+dual codes. We also measure how close is a given additive cyclic code from being symplectic self-
+orthogonal. Recall that p is a prime number and n is a positive integer coprime to p. Moreover,
+elements of Fp2 are represented by Fp2 = {α + βω : α, β ∈ Fp}, where ω is a root of a degree 2
+irreducible polynomial over Fp. Recall that in (2.1) we defined the symplectic inner product of
+two elements in Fn
+p2. We define the symplectic inner product of two polynomials analogously. In
+particular, for c(x) =
+n−1
+�
+i=0
+(ai + ωbi)xi and c′(x) =
+n−1
+�
+i=0
+(a′
+i + ωb′
+i)xi ∈ Fp2[x]/⟨xn − 1⟩, we define
+c(x) ∗ c′(x) =
+n−1
+�
+i=0
+(aib′
+i − a′
+ibi).
+Here we use a different notation for the symplectic inner product to differentiate between the
+vectors and polynomials as different objects.
+Remark 4.1. Let c(x) = g1(x) + ωg2(x) and c′(x) = g′
+1(x) + ωg′
+2(x) be two polynomials
+of Fp2[x]/⟨xn − 1⟩, where g1(x), g2(x), g′
+1(x), g′
+2(x) ∈ Fp[x]/⟨xn − 1⟩. Then c(x) ∗ c′(x) is the
+constant term of g1(x)g′
+2(x−1) − g2(x)g′
+1(x−1) (mod xn − 1). A similar argument shows that
+c(x) ∗ xic′(x) is the coefficient of xi in g1(x)g′
+2(x−1) − g2(x)g′
+1(x−1) (mod xn − 1).
+Thus if
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+9
+g1(x)g′
+2(x−1) − g2(x)g′
+1(x−1) ≡ 0 (mod xn − 1), then the code generated by c′(x) lies in the
+symplectic dual of the code generated by c(x). We use this property very frequently through
+this section.
+One can easily verify that the symplectic dual of an additive cyclic code C over Fp2 is also
+an additive cyclic code over Fp2. Recall that by Theorem 3.6 part (ii), each additive cyclic
+code of length n over Fp2 can be represented uniquely as C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x], where
+g1(x), g2(x), h(x) ∈ Fp[x]/⟨xn − 1⟩. Our next theorem gives a criterion for the self-orthogonality
+of additive cyclic codes. The proof is very similar to that of [3, Theorem 14 part c].
+Theorem 4.2. Let C = ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] be a length n additive cyclic code over Fp2.
+The code C is self-orthogonal if and only if the following conditions are satisfied:
+(1) g2(x)h(x−1) ≡ 0 (mod xn − 1),
+(2) g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1).
+Proof. ⇒: Suppose that C is self-orthogonal. For each 0 ≤ i ≤ n − 1, the inner product of
+g1(x) + ωg2(x) and xih(x) is the coefficient of xi in −g2(x)h(x−1) (mod xn − 1). Since C is self-
+orthogonal, we have g2(x)h(x−1) ≡ 0 (mod xn − 1). Moreover,
+�
+xi(g1(x) + ωg2(x))
+�
+∗
+�
+g1(x) +
+ωg2(x)
+�
+is the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1). Hence, for each
+0 ≤ i ≤ n − 1, the coefficient of xi in g1(x)g2(x−1) − g2(x)g1(x−1) (mod xn − 1) is zero. Thus
+g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1).
+⇐: Conversely, the fact that g1(x)g2(x−1) ≡ g2(x)g1(x−1) (mod xn − 1) implies that all
+the vectors inside ⟨g1(x) + ωg2(x)⟩Fp[x] are self-orthogonal. Moreover, since g2(x)h(x−1) ≡ 0
+(mod xn − 1), we conclude that h(x) is orthogonal to all the cyclic shifts of g1(x) + ωg2(x).
+Finally h(x) ∗ xih(x) = 0 for each 0 ≤ i ≤ n − 1. So ⟨g1(x) + ωg2(x), h(x)⟩Fp[x] is a symplectic
+self-orthogonal code.
+□
+Recall that xn − 1 =
+s
+�
+i=1
+fi(x), where each fi(x) is an irreducible polynomial in Fp[x].
+Moreover, as we mentioned earlier in (3.2), we have Fp2[x]/⟨xn − 1⟩ =
+s
+�
+i=1
+Ni, where Ni =
+⟨(xn − 1)/fi(x)⟩Fp2[x].
+Let α be a primitive n-th root of unity in a finite filed extension of
+Fp. We denote the p-cyclotomic cosets modulo n by Zi for each 1 ≤ i ≤ s in the way that
+fi(x) =
+�
+a∈Zi
+(x − αi). This gives a one-to-one correspondence between the sets Ni and all the
+p-cyclotomic cosets modulo n. Our first goal in this section is to find the symplectic dual of a
+given additive cyclic code. In order to achieve this goal, we need a few preliminary results. In
+the next lemma, we find the symplectic dual of each Ni.
+Lemma 4.3. Let 1 ≤ i ≤ s and C = Ni. Then C⊥s =
+s
+�
+k=1
+k̸=j
+Nk, where Zj = −Zi.
+Proof. First note that by Lemma 3.4 part (3) we have C = ⟨(xn−1)/fi(x), ω((xn−1)/fi(x))⟩Fp[x].
+If Zk ̸= −Zi, then fi(x) | (xn − 1)/fk(x−1) and fk(x) | (xn − 1)/fi(x−1). So we have
+•
+�
+(xn − 1)/fi(x)
+��
+(xn − 1)/fk(x−1)
+�
+≡ 0 (mod xn − 1) and
+•
+�
+(xn − 1)/fk(x)
+��
+(xn − 1)/fi(x−1)
+�
+≡ 0 (mod xn − 1).
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+10
+Hence the symplectic inner product of each element of Ni and each element of Nk is zero by
+definition. This proves that
+s
+�
+k=1
+k̸=j
+Nk ⊆ C⊥s. Note that both of Ni and Nj have Fp-dimension
+2 deg(fi). Now, the facts that dimFp(C) + dimFp(C⊥s) = 2n and dimFp(C) = 2 deg(fi) implies
+the other inclusion.
+□
+Next, we find the symplectic dual of each irreducible additive cyclic code inside Ni for 1 ≤
+i ≤ s.
+Lemma 4.4. Let C ⊊ Ni be a non-zero additive cyclic code for some 1 ≤ i ≤ s. Then
+C⊥s = (
+s
+�
+k=1
+k̸=j
+Nk)
+�
+⟨g1(x) + ωg2(x)⟩Fp[x],
+(4.1)
+where Zj = −Zi and
+g1(x) + ωg2(x) =
+�
+((xn − 1)/fj(x))(s(x−1) + ω)
+if C = ⟨
+�
+(xn − 1)/fi(x)
+��
+ω + s(x)
+�
+⟩Fp[x]
+(xn − 1)/fj(x)
+if C = ⟨(xn − 1)/fi(x)⟩Fp[x]
+.
+Proof. By Lemma 4.3, one can see that
+s
+�
+k=1
+k̸=j
+Nk ⊆ C⊥s. Note that dimFp(⟨g1(x)+ωg2(x)⟩Fp[x]) =
+dimFp(C). So it is sufficient to show that C is orthogonal to g1(x) + ωg2(x) and all its cyclic
+shifts. We prove the latter statement in two steps. First suppose that C = ⟨
+�
+(xn−1)/fi(x)
+��
+ω+
+s(x)
+�
+⟩Fp[x] for some s(x) ∈ Fp[x].
+To show that the codes C and ⟨g1(x) + ωg2(x)⟩Fp[x] are
+orthogonal, we apply Remark 4.1. In particular,
+�
+(xn − 1)/fi(x)
+�
+s(x)g2(x−1) −
+�
+(xn − 1)/fi(x)
+�
+g1(x−1) ≡
+�
+(xn − 1)/fi(x)
+�
+s(x)
+�
+(xn − 1)/fi(x)
+�
+−
+�
+(xn − 1)/fi(x)
+��
+(xn − 1)/fi(x)
+�
+s(x) ≡ 0
+(mod xn − 1).
+Next, suppose that C = ⟨(xn − 1)/fi(x)⟩Fp[x]. Then
+�
+(xn − 1)/fi(x)
+�
+g2(x−1) −
+�
+(xn − 1)/fi(x)
+�
+g1(x−1) ≡
+�
+(xn − 1)/fi(x)
+�
+0 − 0
+�
+(xn − 1)/fj(x)
+�
+s(x)
+≡ 0
+(mod xn − 1).
+This shows that the code C is orthogonal to the additive cyclic code generated by g1(x)+ωg2(x)
+and completes the proof.
+□
+Note that as we showed in Lemma 4.4, when C = ⟨
+�
+(xn − 1)/fi(x)
+��
+ω + s(x)
+�
+⟩Fp[x], its
+symplectic inner product contains the code C′ = ⟨(xn − 1)/fj(x))(s(x−1) + ω)⟩Fp[x]. The code
+C′ is not in one of the forms given in Lemma 3.4 part (4). In order to express the code C′
+using the standard notation introduced in 3.4 part (4), we choose its generator to be g(x) =
+(xn − 1)/fj(x))(t(x) + ω), where t(x) ≡ s(x−1) (mod fj(x)). Now it is easy to see that g(x)
+belongs to the set A introduced in Lemma 3.4 part (4) and C′ = ⟨(xn −1)/fj(x))(t(x)+ω)⟩Fp[x].
+Next, we combine the results of the previous two lemmas and the result of Theorem 3.6 to
+determine generator polynomials of the symplectic dual for any additive cyclic code.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+11
+Theorem 4.5. Let C be a length n additive cyclic code over Fp2 such that C =
+s
+�
+i=1
+Ci, where
+Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s.
+Then C⊥s = ⟨
+s
+�
+i=1
+gi(x) +
+ωki(x),
+s
+�
+i=1
+ωhi(x)⟩Fp[x], where for each 1 ≤ i ≤ s we have Zj = −Zi and
+• gi(x) = ki(x) = hi(x) = 0 if Cj = Nj,
+• gi(x) = hi(x) = (xn − 1)/fi(x) and ki(x) = 0 if Cj = 0,
+• gi(x)+ωki(x) =
+�
+(xn −1)/fi(x)
+��
+ω +ti(x)
+�
+and hi(x) = 0 if Cj = ⟨
+�
+(xn −1)/fj(x)
+��
+ω +
+sj(x)
+�
+⟩Fp[x] and ti(x) ≡ sj(x−1) (mod fj(x)),
+• gi(x) = (xn − 1)/fi(x) and ki(x) = hi(x) = 0 if Cj = ⟨(xn − 1)/fj(x)⟩Fp[x].
+Proof. We apply Lemmas 4.3 and 4.4 to prove the statement. If Cj = Nj, then C⊥s ∩ Ni = {0}
+by Lemma 4.3. Moreover, by the same lemma, if Cj = 0, then Ni ⊆ C⊥s. This proves the first
+two bullets. Finally, Lemma 4.4 implies that
+• ⟨
+�
+(xn − 1)/fi(x)
+��
+ω + ti(x)
+�
+⟩Fp[x] ⊆ C⊥s if Cj = ⟨
+�
+(xn − 1)/fj(x)
+��
+ω + sj(x)
+�
+⟩Fp[x], and
+• ⟨(xn − 1)/fi(x)⟩Fp[x] ⊆ C⊥s if Cj = ⟨(xn − 1)/fi(x)⟩Fp[x].
+This proves the statements of the last two bullets.
+□
+To determine self-orthogonal and self-dual additive cyclic codes over Fp2, we need more infor-
+mation about irreducible factors of xn − 1 over Fp. Let Z1, Z2, . . . , Zr and Z′
+1, −Z′
+1, . . . , Z′
+t, −Z′
+t
+be all the p-cyclotomic cosets modulo n, where Zi = −Zi and r + 2t = s.
+Each Zi is in
+correspondence to an irreducible polynomial fi(x) and (Z′
+j, −Z′
+j) are in correspondence with an
+irreducible pair of polynomials (fj1(x), fj2(x)) over Fp. Therefore, we can rewrite the irreducible
+decomposition of xn − 1 as
+xn − 1 =
+r
+�
+i=1
+fi(x)
+t�
+j=1
+fj1(x)fj2(x).
+We use the above representation of cyclotomic cosets in the upcoming results. Next, we classify
+self-orthogonal and self-dual additive codes over Fp2.
+Theorem 4.6. Let C be a length n additive cyclic code over Fp2 such that C =
+s
+�
+i=1
+Ci, where
+Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. Then C is symplectic self-orthogonal
+if and only if
+(1) for all 1 ≤ k ≤ r only one of the following holds.
+(a) Ck = 0.
+(b) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x], where fk | s(x−1) − s(x).
+(c) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x].
+(2) for all 1 ≤ j ≤ t only one of the following holds.
+(a) Cj1 = 0 or Cj2 = 0.
+(b) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) +
+ω)⟩Fp[x].
+(c) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x].
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+12
+Proof. First, let 1 ≤ k ≤ r. By Lemma 4.3, if Ck = Nk, then C⊥s ∩ Nk = {0}. So Ck cannot
+have two generator polynomials. Moreover, by Lemma 4.4, if 0 ̸= Ck = ⟨((xn −1)/fk(x))(s(x)+
+ω)⟩Fp[x], then ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x] ⊆ C⊥s. Thus Ck is self-orthogonal if and only
+if
+Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] = ⟨((xn − 1)/fk(x))(s(x−1) + ω)⟩Fp[x].
+Note that the above equality holds if and only if fk | s(x−1) − s(x). Thus Ck is self-orthogonal
+if and only if one of the conditions of Part (1) follows.
+Next, let 1 ≤ j ≤ t. By Lemma 4.3, if Cj1 = Nj1, then C⊥s ∩ Nj2 = {0}. So if one of Cj1 or
+Cj2 has two generator polynomials, the other code should be zero. Moreover, the same lemma
+shows that if Cj1 = 0 or Cj2 = 0, then Cj1 + Cj2 is self-orthogonal. So we concentrate only
+on the case when both Cj1 and Cj2 have exactly one non-zero generator. By Lemma 4.4, if
+Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x], then
+C⊥s ∩ Nj2 = ⟨((xn − 1)/fj2(x))(s(x−1) + ω)⟩Fp[x].
+In this case, the code Cj1 ⊕ Cj2 is self-orthogonal if and only if condition (2)(b) is satisfied.
+Condition (2)(c) follows similarly by applying Lemma 4.4.
+□
+Next, we use the above conditions to characterize all the symplectic self-dual additive cyclic
+codes over Fp2.
+Corollary 4.7. Let C be a length n additive cyclic code over Fp2 such that C =
+s
+�
+i=1
+Ci, where
+Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. Then C is symplectic self-dual if and
+only if
+(1) for all 1 ≤ k ≤ r only one of the following holds.
+(a) Ck = ⟨((xn − 1)/fk(x))(s(x) + ω)⟩Fp[x] where fk | s(x−1) − s(x).
+(b) Ck = ⟨(xn − 1)/fk(x)⟩Fp[x].
+(2) for all 1 ≤ j ≤ t only one of the following holds.
+(a) Cj1 = 0 and Cj2 = Nj2.
+(b) Cj2 = 0 and Cj1 = Nj1.
+(c) Cj1 = ⟨((xn − 1)/fj1(x))(s(x) + ω)⟩Fp[x] and Cj2 = ⟨((xn − 1)/fj2(x))(s(x−1) +
+ω)⟩Fp[x].
+(d) Cj1 = ⟨(xn − 1)/fj1(x)⟩Fp[x] and Cj2 = ⟨(xn − 1)/fj2(x)⟩Fp[x].
+Proof. Note that all the self-dual additive cyclic codes over Fp2 satisfy the conditions of Theorem
+4.6 and have maximal dimension.
+Thus the result easily follows by implying the maximal
+property into the conditions of theorem 4.6.
+□
+Our next goal is to compute the parameter e = dimFp(C) − dimFp(C ∩ C⊥s) for all additive
+cyclic codes. The parameter e determines how close an additive cyclic code C is from being
+self-orthogonal. This parameter plays an important role in the quantum construction that we
+are applying in the next section.
+Theorem 4.8. Let C be a length n additive cyclic code over Fp2 such that C =
+s
+�
+i=1
+Ci, where
+Ci is an additive cyclic codes inside Ni for each 1 ≤ i ≤ s. Let
+(1) B1 = {α1, α2, . . . , αt1} ⊆ {1, 2, . . . , r} such that Cαl = Nαl for all 1 ≤ l ≤ t1,
+(2) B2 = {β1, β2, . . . , βt2} ⊆ {1, 2, . . . , r} such that Cβl = ⟨((xn − 1)/fβl(x))(sβl(x) + ω)⟩Fp[x]
+and fβl ∤ sβl(x−1) − sβl(x) for all 1 ≤ l ≤ t2,
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+13
+(3) B3 = {γ1, γ2, . . . , γt3} ⊆ {1, 2, . . . , t} such that one of Cγl1 and Cγl2 is generated by two
+polynomials and the other one has only one generator polynomial for all 1 ≤ l ≤ t3,
+(4) B4 = {κ1, κ2, . . . , κt4} ⊆ {1, 2, . . . , t} such that both of Cκl1 and Cκl2 are generated by
+two polynomials for all 1 ≤ l ≤ t4,
+(5) B5 = {σ1, σ2, . . . , σt5} ⊆ {1, 2, . . . , t} such that both of Cσl1 and Cσl2 are generated by
+one polynomial for all 1 ≤ l ≤ t5 and
+(a) if Cσl1 = ⟨((xn−1)/fσl1(x))(sσl(x)+ω)⟩Fp[x], then Cσl2 ̸= ⟨((xn−1)/fσl2(x))(sσl(x−1)+
+ω)⟩Fp[x].
+(b) if Cσl1 = ⟨(xn − 1)/fσl1(x)⟩Fp[x], then Cσl2 ̸= ⟨(xn − 1)/fσl2(x))⟩Fp[x].
+Then
+e = dimFp(C)−dimFp(C∩C⊥s) =
+t1
+�
+l=1
+2|Zαl|+
+t2
+�
+l=1
+|Zβl|+
+t3
+�
+l=1
+2|Zγl|+
+t4
+�
+l=1
+4|Zκl|+
+t5
+�
+l=1
+2|Zσl|. (4.2)
+Proof. By Theorem 4.6, an additive cyclic code is not symplectic self-orthogonal if and only if at
+least one of the sets B1 −B5 is non-empty. Next, we consider all scenarios (1)-(5) independently.
+(1) Let j ∈ B1. In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩
+C⊥s) = 2|Zj|.
+(2) Let j ∈ B2. In this case, C⊥s ∩ Cj = {0} which implies that dimFp(Cj) − dimFp(Cj ∩
+C⊥s) = |Zj|.
+(3) Let j ∈ B3.
+Without loss of generality we assume that Cj1 = Nj1 and Cj2 is an
+irreducible subcode of Nj2. In this case, the intersection C⊥s∩(Cj1⊕Cj2) is an irreducible
+subcode of Nj1 which implies that dimFp(Cj)−dimFp((Cj1⊕Cj2)∩C⊥s) = 3|Zj|−|Zj| =
+2|Zj|.
+(4) Let j ∈ B4. In this case, C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that dimFp(Cj) −
+dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 4|Zj|.
+(5) Let j ∈ B5. In both parts (a) and (b), C⊥s ∩ (Cj1 ⊕ Cj2) = {0} which implies that
+dimFp(Cj) − dimFp((Cj1 ⊕ Cj2) ∩ C⊥s) = 2|Zj|.
+Now, the result follows by combining the above observations.
+□
+Note that the case (2) of Theorem 4.8 never happens for Ci with deg(fi(x)) = 1. Moreover,
+for each 1 ≤ i ≤ r, the cyclotomic coset Zi either is a singleton or it has an even size. This is
+mainly because for each 0 ̸= a ∈ Zi, if a ≡ −a (mod n), then n | 2a, which implies that n is
+even. Hence in this case p ̸= 2 (we assumed that gcd(n, p) = 1) and Zi = {a}. Therefore, if
+Zi satisfies the case (2) of Theorem 4.8 and |Zi| > 1, then for any a ∈ Zi, we have −a ∈ Zi
+and −a ̸≡ a (mod n). This implies that |Zi| is an even integer. This fact and the formula in
+(4.2) imply that the nearly self-orthogonality parameter e of an additive cyclic code is always
+an even integer. Next, we classify additive cyclic codes with small values of e. First, we need
+the following preliminary result.
+Lemma 4.9. Let p be a prime number and gcd(n, p) = 1 for some positive number n.
+(i) If gcd(n, p − 1) = d, then there are d singleton p-cyclotomic cosets modulo n and all of
+their coset leaders are {k n
+d : 0 ≤ k ≤ d − 1}.
+(ii) If gcd(n, p − 1) = d and gcd(n, p2 − 1) = d′. Then there are d′−d
+2
+p-cyclotomic cosets
+modulo n of size two.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+14
+Proof. (i) The proof easily follows from the fact that {a} is a singleton coset if and only if a ≡ pa
+(mod n) or equivalently if and only if a(p − 1) ≡ 0 (mod n). By elementary number theory, if
+gcd(n, p − 1) = d, then the latter equation has d solutions in the forms {k n
+d : 0 ≤ k ≤ d − 1}.
+(ii) A p-cyclotomic coset modulo n containing a has size two if and only if a ≡ p2a (mod n)
+and a ̸≡ pa (mod n). So we get d′ candidate for the size two cosets by solving a ≡ p2a (mod n).
+Moreover, each singleton cyclotomic coset is formed by a solution of the latter equation. Note
+also that the p-cyclotomic coset of size two containing a and pa is counted twice in our previous
+observation. Hence there are d′−d
+2
+many different cosets.
+□
+For example, for an odd n, the only singleton p-cyclotomic coset modulo n is {0} when p = 2
+or p = 3. If n is even, then {n
+2 } and {0} are the only singleton cyclotomic cosets for p = 3.
+The next theorem classifies all the additive cyclic codes with e = 2. Note that the case e = 0
+happens if an additive cyclic code is symplectic self-orthogonal, and this case was characterized
+in Theorem 4.6.
+Theorem 4.10. Let C =
+s
+�
+i=1
+Ci be an additive cyclic code of length n over Fp2. Then
+e = dimFp(C) − dimFp(C ∩ C⊥s) = 2
+if and only if all Ci satisfy the conditions of Theorem 4.6 except one which is in correspondence
+to
+(1) a singleton coset and satisfies condition (1) of Theorem 4.8,
+(2) a size two coset and satisfies condition (2) of Theorem 4.8.
+Proof. The result follows from considering the formula (4.2) and considering all conditions of
+Theorem 4.8.
+□
+Many of our record-breaking quantum codes provided in the next section have e = 2. In
+general, the total number of all additive cyclic codes can be a very large number.
+So the
+classification of e values significantly helps to prune the search algorithm for quantum codes
+with good parameters.
+5. New binary quantum codes
+In this section, we first recall a construction of binary quantum codes from additive codes,
+which does not require the symplectic self-orthogonality condition of Theorem 2.1. Then we
+apply this construction to several nearly self-orthogonal additive cyclic codes over F4 and con-
+struct new binary quantum codes. In the rest of this section, we show the quaternary filed by
+F4 = {0, 1, ω, ω + 1}, where ω2 = ω + 1.
+Theorem 5.1. [5, Corollary 3.3.7],[19] Let C be an (n, 2k) additive code over F4 and
+r = 2n − k − dimFp(C ∩ C⊥s)
+2
+Then there exists a binary quantum code with parameters [[n + r, k − n + r, d]]2, where
+d ≥ min{d(C), d(C + C⊥s) + 1}.
+Note that we take advantage of the result of Theorem 4.8 in the computation of Theorem 5.1.
+In particular, the value of r in Theorem 5.1 is
+dimFp(C⊥s)−dimFp(C∩C⊥s)
+2
+, where the numerator
+measures the nearly self-orthogonality of the code C⊥s. Next, we briefly describe two of our new
+binary quantum codes. The rest of our new binary quantum codes presented in Table 1 can be
+constructed in a similar way.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+15
+Example 5.2. Let n = 21 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where
+g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1
+and
+k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1.
+The code C is a (21, 220) additive code. Moreover, our computation using the result of Theorem
+4.8 shows that C has nearly self-orthogonality parameter e = 2. Moreover,
+7 = min{d(C⊥s), d(C + C⊥s) + 1}.
+So, applying the construction of Theorem 5.1 to the code C⊥s gives a new quantum code with
+parameters [[22, 2, 7]]2. It has a better minimum distance than the previous best-known quantum
+code with the same length and dimension, which had minimum distance 6.
+Example 5.3. Let n = 35 and C = ⟨g(x)+ωk(x)⟩F2[x] be an additive cyclic code over F4, where
+g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1
+and
+k(x) = x34 +x33 +x31 +x30 +x29 +x27 +x25 +x23 +x22 +x20 +x19 +x18 +x15 +x12 +x8 +x3 +x.
+The code C has parameters (35, 220) as an additive cyclic code over F4. Also, the result of
+Theorem 4.8 shows that C has nearly self-orthogonality parameter e = 4. Moreover,
+6 = min{d(C⊥s), d(C + C⊥s) + 1}.
+So, applying the construction of Theorem 5.1 to the code C⊥s gives a record-breaking quantum
+code with parameters [[37, 17, 6]]2. The previous best-known binary quantum code with the same
+parameters had minimum distance 5.
+In general, in order to apply the quantum construction given in Theorem 5.1, we target
+additive cyclic codes with the nearly self-orthogonality e ≤ 4.
+Because it is more likely to
+get a new quantum code when e value is small. In Table 1, we present the parameters of our
+new binary quantum codes. In the table, we start with an additive cyclic code C over F4 and
+compute its nearly self-orthogonality. Then we apply the quantum construction of Theorem 5.1
+to the code C⊥s. The parameters of the corresponding quantum code are given in the fourth
+column. Moreover, the minimum distance of the previous quantum code with the same length
+and dimension is provided in the last column of the table. The previous minimum distance is
+taken from Grassl’s code table [12]. We record the generator polynomials of the additive cyclic
+codes of Table 1 in Table 2.
+Note also that applying the secondary constructions presented in Theorem 2.2 to the new
+codes of Table 1 produces many more record-breaking quantum codes. In particular, the new
+[[52, 24, 8]]2 quantum codes alone produces the following new quantum codes:
+[[52, 21, 8]]2, [[52, 22, 8]]2, [[52, 23, 8]]2, [[53, 21, 8]]2, [[53, 22, 8]]2, [[53, 23, 8]]2, [[53, 24, 8]]2.
+Around the same time as us, authors of [14] independently found several new binary quantum
+codes by applying the connection between quasi-cyclic codes and additive cyclic codes.
+In
+particular, three of our new quantum codes, namely [[45, 6, 10]], [[45, 45, 10, 9]], and [[51, 8, 11]],
+are also among the new quantum codes of [14].
+Acknowledgement
+The authors would like to thank Petr Lisonˇek and Markus Grassl for many interesting dis-
+cussions and comments.
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+16
+No
+Length
+e value
+Parameters
+Previous distance
+1
+n = 21
+2
+[[22, 2, 7]]2
+6
+2
+n = 35
+4
+[[37, 17, 6]]2
+5
+3
+n = 45
+0
+[[45, 6, 10]]2
+9
+4
+n = 45
+0
+[[45, 10, 9]]2
+8
+5
+n = 51
+0
+[[51, 8, 11]]2
+10
+6
+n = 51
+2
+[[52, 16, 10]]2
+9
+7
+n = 51
+2
+[[52, 24, 8]]2
+7
+8
+n = 63
+2
+[[64, 33, 8]]2
+7
+9
+n = 63
+2
+[[64, 34, 8]]2
+7
+10
+n = 63
+2
+[[64, 35, 8]]2
+7
+Table 1. Parameters of new binary quantum codes.
+References
+[1] J. Bierbrauer and Y. Edel. Quantum twisted codes. Journal of Combinatorial Designs, 8(3):174–188, 2000.
+[2] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I: The user language. Journal of Symbolic
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+IEEE Transactions on Information Theory, 44(4):1369–1387, 1998.
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+arXiv:2211.00891, 2022.
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+encyclopedia of coding theory, chapter 2. Chapman and Hall/CRC, 2021.
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+Review A, 54(3):1862, 1996.
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+[13] M. Grassl. Algebraic quantum codes: Linking quantum mechanics and discrete mathematics. Int. J. Comput.
+Math. Comput. Syst. Theory, 6(4):243–259, 2021.
+[14] C. Guan, R. Li, and Z. Ma. Symplectic self-orthogonal quasi-cyclic codes. arXiv preprint arXiv:2212.14225,
+2022.
+[15] C. G¨uneri, F. ¨Ozdemir, and P. Sole. On the additive cyclic structure of quasi-cyclic codes. Discrete Mathe-
+matics, 341(10):2735–2741, 2018.
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+[17] W. C. Huffman. Additive cyclic codes over F4. Adv. Math. Commun., 2(3):309–343, 2008.
+[18] A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli. Nonbinary stabilizer codes over finite fields.
+IEEE transactions on information theory, 52(11):4892–4914, 2006.
+[19] P.
+Lisonˇek
+and
+R.
+Dastbasteh.
+Constructions
+of
+quantum
+codes.
+Presented
+at
+The
+3rd
+International
+Workshop
+on
+Boolean
+Functions
+and
+their
+Applications,
+loen,
+norway.
+https://people.uib.no/chunlei.li/workshops/BFA2018/Slides/Lisonek.pdf, 2018.
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+and Cryptography, 73(2):417–424, 2014.
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+Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
+Email address:
+rdastbas@sfu.ca, kh411@protonmail.com
+
+POLYNOMIAL REPRESENTATION OF ADDITIVE CYCLIC CODES AND NEW QUANTUM CODES
+17
+No
+Generator polynomials as in Theorem 3.6 part (II)
+1
+g(x) = x20 + x17 + x15 + x13 + x11 + x8 + x7 + x6 + x5 + x4 + x3 + 1
+k(x) = x19 + x18 + x17 + x16 + x14 + x10 + x5 + x4 + x3 + x2 + x + 1
+h(x) = 0
+2
+g(x) = x33 + x29 + x28 + x24 + x19 + x18 + x15 + x13 + x12 + x11 + x6 + x4 + x + 1
+k(x) = x34+x33+x31+x30+x29+x27+x25+x23+x22+x20+x19+x18+x15+x12+x8+x3+x
+h(x)=0
+3
+g(x) = x44 + x43 + x41 + x40 + x39 + x38 + x34 + x33 + x30 + x26 + x24 + x20 + x19 + x18 +
+x17 + x16 + x15 + x14 + x11 + x9 + x5 + x3 + 1
+k(x) = x43 + x42 + x41 + x40 + x36 + x33 + x32 + x31 + x30 + x28 + x26 + x25 + x17 + x16 +
+x15 + x13 + x11 + x10 + x2 + x
+h(x) = 0
+4
+g(x) = x44 + x43 + x40 + x38 + x37 + x34 + x31 + x27 + x22 + x21 + x20 + x19 + x18 + x17 +
+x14 + x12 + x7 + x6 + x5 + x3 + x + 1
+k(x) = x44 + x41 + x40 + x37 + x36 + x35 + x33 + x30 + x29 + x27 + x26 + x25 + x22 + x20 +
+x15 + x14 + x12 + x11 + x10 + x7 + x5
+h(x) = 0
+5
+g(x) = x50 + x49 + x48 + x46 + x45 + x43 + x42 + x41 + x40 + x37 + x36 + x35 + x30 + x29 +
+x28 + x26 + x23 + x19 + x18 + x17 + x16 + x15 + x14 + x13 + x9 + x7 + x6 + x
+k(x) = x50 + x47 + x44 + x43 + x42 + x41 + x40 + x38 + x36 + x35 + x33 + x32 + x28 + x26 +
+x24 + x21 + x20 + x16 + x14 + x12 + x9 + x8 + x7 + x + 1
+h(x)=0
+6
+g(x) = x48 + x40 + x37 + x36 + x33 + x31 + x30 + x24 + x23 + x21 + x19 + x15 + x11 + x10 +
+x9 + x8 + x7 + x4 + x3 + x + 1
+k(x) = x41 + x40 + x36 + x35 + x34 + x33 + x30 + x29 + x27 + x23 + x22 + x21 + x19 + x18 +
+x16 + x13 + x12 + x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x
+h(x) = x50 + x49 + x48 + x47 + x46 + x45 + x44 + x41 + x40 + x39 + x33 + x31 + x30 + x28 +
+x25 + x22 + x20 + x19 + x18 + x17 + x16 + x14 + x13 + x11 + x9 + x5 + x4
+7
+g(x) = x49 + x48 + x46 + x44 + x43 + x41 + x38 + x37 + x36 + x33 + x32 + x31 + x30 + x29 +
+x27 + x25 + x21 + x20 + x18 + x17 + x15 + x11 + x10 + x7 + x2
+k(x) = x43 + x42 + x41 + x40 + x38 + x37 + x33 + x32 + x30 + x26 + x24 + x22 + x19 + x18 +
+x16 + x15 + x13 + x9 + x5 + x4 + x2 + 1
+h(x) = x50 +x49 +x48 +x47 +x46 +x45 +x44 +x43 +x42 +x41 +x40 +x39 +x38 +x37 +x36 +
+x35 +x34 +x33 +x32 +x31 +x30 +x29 +x28 +x27 +x26 +x25 +x24 +x23 +x22 +x21 +x20 +x19 +
+x18 +x17 +x16 +x15 +x14 +x13 +x12 +x11 +x10 +x9 +x8 +x7 +x6 +x5 +x4 +x3 +x2 +x+1
+8
+g(x) = x61+x60+x59+x57+x56+x53+x52+x51+x42+x41+x38+x36+x34+x32+x31+x28+
+x27 +x26 +x24+x20 +x19+x16 +x14+x13 +x12+x11 +x9+x8+x7+x6+x5 +x4+x3+x2+x
+k(x) = x61 + x59 + x57 + x56 + x55 + x54 + x52 + x51 + x50 + x49 + x47 + x44 + x37 + x36 +
+x35 + x33 + x32 + x31 + x29 + x28 + x27 + x26 + x24 + x22 + x21 + x16 + x8 + x5 + x3 + x2
+h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 +
+x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x
+9
+g(x) = x60 + x59 + x58 + x55 + x54 + x53 + x52 + x51 + x48 + x47 + x45 + x44 + x40 + x38 +
+x37 + x36 + x35 + x34 + x33 + x32 + x31 + x30 + x29 + x28 + x27 + x24 + x23 + x22 + x21 + x15 +
+x13 + x10 + x9 + x7 + x6 + x3 + x + 1
+k(x) = x62 + x59 + x56 + x55 + x54 + x53 + x49 + x47 + x46 + x42 + x41 + x40 + x37 + x35 +
+x33 + x31 + x29 + x28 + x27 + x24 + x20 + x19 + x16 + x15 + x14 + x7 + x4 + x2 + x
+h(x) = 0
+10
+g(x) = x61+x60+x59+x58+x57+x53+x52+x49+x44+x41+x38+x37+x36+x35+x34+x32+
+x31 +x30+x27+x26+x24+x23+x21+x20 +x19+x13+x12+x11+x8+x6+x5+x4+x3+x+1
+k(x) = x60+x58+x57+x56+x52+x48+x47+x46+x44+x42+x40+x39+x38+x36+x35+x34+
+x32+x31+x30+x26+x25+x24+x22+x19+x18+x17+x13+x12+x9+x7+x6+x5+x4+x3+x2+1
+h(x) = x62 + x61 + x60 + x59 + x58 + x51 + x49 + x47 + x44 + x43 + x40 + x36 + x33 + x31 +
+x30 + x28 + x27 + x23 + x22 + x21 + x19 + x14 + x13 + x11 + x9 + x8 + x7 + x4 + x3 + x2 + x
+Table 2. Generator polynomials of additive cyclic codes of Table 1
+

6tAzT4oBgHgl3EQfgPwz/content/tmp_files/2301.01464v1.pdf.txt

@@ -0,0 +1,1472 @@
+Low-frequency shear Alfv´en waves at DIII-D: theoretical
+interpretation of experimental observations
+Ruirui Ma,1, 2, ∗ W.W. Heidbrink,3 Liu Chen,4, 3, 2 Fulvio Zonca,2, 4 and Zhiyong Qiu4, 2
+1Southwestern Institute of Physics, P.O. Box 432, Chengdu, 610041, China
+2Center for Nonlinear Plasma Science and C.R.
+ENEA Frascati, C.P. 65, 00044 Frascati, Italy
+3Department of Physics and Astronomy,
+University of California, Irvine, CA 92697-4574, USA
+4Institute for Fusion Theory and Simulation and Department of Physics,
+Zhejiang University, Hangzhou, 310027, People’s Republic of China
+(Dated: January 5, 2023)
+Abstract
+The linear properties of the low-frequency shear Alfv´en waves such as those associated with
+the beta-induced Alfv´en eigenmodes (BAEs) and the low-frequency modes observed in reversed-
+magnetic-shear DIII-D discharges (W. Heidbrink, et al 2021 Nucl. Fusion 61 066031) are theoret-
+ically investigated and delineated based on the theoretical framework of the general fishbone-like
+dispersion relation (GFLDR). By adopting representative experimental equilibrium profiles, it is
+found that the low-frequency modes and BAEs are, respectively, the reactive-type and dissipative-
+type unstable modes with dominant Alfv´enic polarization, thus the former being more precisely
+called low-frequency Alfv´en modes (LFAMs). More specifically, due to different instability mech-
+anisms, the maximal drive of BAEs occurs, in comparison to LFAMs, when the minimum of the
+safety factor (qmin) deviates from a rational number. Meanwhile, the BAE eigenfunction peaks
+at the radial position of the maximum energetic particle pressure gradient, resulting in a large
+deviation from the qmin surface.
+Moreover, the ascending frequency spectrum patterns of the
+experimentally observed BAEs and LFAMs can be theoretically reproduced by varying qmin and
+also be well interpreted based on the GFLDR. The present analysis illustrates the solid predictive
+capability of the GFLDR and its practical usefulness in enhancing the interpretative capability of
+both experimental and numerical simulation results.
+∗ corresponding author. Email address: rrma@swip.ac.cn
+1
+arXiv:2301.01464v1  [physics.plasm-ph]  4 Jan 2023
+
+I.
+INTRODUCTION AND MOTIVATION
+The low-frequency Alfv´en wave spectrum in the kinetic thermal-ion (KTI) gap frequency
+range [1] has been of research interest since the first observations of beta-induced Alfv´en
+eigenmodes (BAEs) [2, 3]. These modes are characterized with frequencies comparable to
+thermal ion transit and/or bounce frequencies, and can interact with both thermal and
+fast particles [4–9], with possible (positive/negative) impact on the corresponding transport
+processes resulting from finite fluctuation and zonal field structures levels [1, 9, 10]. The
+effects of energetic particles (EPs) on low-frequency shear Alfv´en waves (SAWs) ranging
+from kinetic ballooning mode (KBM) [11–13] to BAE are one of areas widely studied in
+the magnetic fusion literature. Recent papers on this topic cover the interpretation and
+modeling of experimental measurements by currently developed innovative diagnostics [14–
+18], as well as latest progress in comparing numerical investigation and/or simulation results
+with observed phenomena [19–24].
+A series of dedicated experiments have been recently conducted on DIII-D to investigate
+the stability of the low-frequency SAWs [16–18]. The experiments show that the observed
+low-frequency mode1, which was previously misidentified as ‘beta-induced Alfv´en acoustic
+eigenmode (BAAE)’ [25, 26], is actually a lower-frequency reactive unstable KBM which
+favors high thermal electron temperature but almost has no coupling with energetic ions
+[16]; while the BAE is resonantly excited by energetic ions with its stability depending
+sensitively on the beam power and injection geometry [17], consistent with earlier theoretical
+predictions [27] based on the GFLDR theoretical framework [28, 29]. These instabilities are
+also found to occur when the minimum of the safety factor (qmin) approaches rational values
+and the modes in ascending pattern of higher frequency BAEs and LFAMs are separated by
+approximately the toroidal rotation frequency (frot). However, the subtle differences between
+them are that, for LFAMs, the maximum frequency appears at rational values of qmin and
+the detected modes are radially localized near qmin, while BAEs occur at times near rational
+qmin values but the timing of unstable modes is less precise than that for LFAMs. In addition,
+compared with the LFAMs, the BAE eigenfunction shows more deviation from the radial
+position of qmin spatially. Although dedicated numerical simulations of the linear properties
+1We will refer from now on only to the low frequency Alfv´en mode (LFAM) which belongs to low-frequency
+SAWs predominantly Alfv´enic polarization, keeping in mind that this terminology is the same as the low-
+frequency mode observed in recent DIII-D experiments [16].
+2
+
+of the BAEs and LFAMs [24, 30] have been carried out, the above experimental phenomena
+have not been fully explained.
+Motivated by this, the present work aims to provide an
+in-depth theoretical understanding of the linear properties of low-frequency SAWs, with
+particular attention to the effects of energetic ions on their stability. The analysis is carried
+out based on the theoretical framework of the generalized fishbone-like dispersion relation
+(GFLDR) [28, 29, 31–35], and provides qualitative and quantitative interpretation of the
+main instability mechanisms underlying the numerical simulation results and experimental
+observations.
+As a result, our analysis provides yet another evidence of the predictive
+strength of the GFLDR theoretical framework and of its enhanced “interpretative capability
+for both experimental and numerical simulation results” [28, 29].
+In this work, unlike the previous paper not considering effects due to energetic particles
+(EPs) [36], we focus on the BAE excitation via transit resonance with passing fast ions
+created by NBI heating [17]. In this case, the dynamics of various species enter the dispersion
+relation of low-frequency SAW, and affect its behavior linearly at different pressure gradient
+scale lengths. For DIII-D discharge #178631, Fig. 1 shows the radial dependence of different
+scale lengths of thermal and energetic particle pressure (LPth and LPE), as well as the
+estimated radial mode width (∆m) for weak and/or vanishing magnetic shear range, i.e.,
+|s| = |(r/q)(dq/dr)| ≲ 0.05. More specifically, the EP pressure profiles are given by the
+following two limits.
+One is the relaxed EP profile provided with EFIT reconstruction
+[37], where the fast-ion pressure is the difference between the equilibrium pressure and the
+thermal pressure. The other is the “classical” EP profile obtained by TRANSP/NUBEAM
+[38] in the absence of fast-ion transport by instabilities. The pressure scale lengths of EPs
+are denoted by LPE;rel and LPE;cl for these two cases (respectively). The true EP profile when
+the modes are destabilized likely lies between these two limits. The actual pressure is closest
+to the EFIT-based one but this is measured after the unstable modes have (presumably)
+caused the gradients to flatten. Meanwhile, for the weak and/or vanishing magnetic shear
+region and given toroidal and poloidal mode numbers (n, m), the normalized parallel wave
+vector is ΩA,m = k∥n0qminR0 = nqmin − m, and the radial width of the mode can then
+be estimated by ∆m ≃ 1/|nq′′|1/2 [39, 40]. Here, k∥n0 represents the parallel wave-vector
+at r0, where q has a minimum given by qmin, q′′ denotes the second derivative of q in the
+radial direction, and R0 is the torus major radius. It can be found that in this region,
+LPth ≫ ∆m, which yields the usual local limit of the mode dispersion relation. This is the
+3
+
+case for the reactive unstable LFAM in the absence of EPs already studied in Ref. [36].
+However, for the energetic ion-driven BAEs, there are two distinct cases: the moderate
+EP pressure gradient case with LPE;rel > ∆m, which also approximately yields the usual
+local GFLDR [4, 28, 29, 32, 33, 35, 39, 40]; and the strong EP pressure gradient case with
+LPE;rel ≃ ∆m, for which the global dispersion relation of low-frequency SAWs is needed
+and will be discussed in Sec. II. Performing detailed numerical investigations of the two
+FIG. 1.
+The radial dependences of the typical scale lengths of thermal and energetic particle
+pressure (LPth and LPE), as well as the estimated radial mode width (∆m).
+cases, it is found that the LFAMs and BAEs can both be driven unstable, however, due to
+different instability mechanisms, these modes yield different experimental observations. All
+these features can be, quantitatively and qualitatively, interpreted theoretically based on the
+GFLDR. Moreover, it is also confirmed that the stability of BAAE is not affected by EPs,
+even though it becomes weakly damped after coupling with KBM, consistent with theoretical
+predictions by Chen and Zonca [27] as well as numerical simulation results reported in Refs.
+[20, 23, 24].
+The paper is structured as follows. Local and global dispersion relations for the low-
+frequency SAWs near weak and/or vanishing magnetic shear are introduced and discussed
+in Sec. II in different parameter regimes, depending on the relative magnitude of LPE and
+∆m. Detailed numerical investigations and theoretical analysis of the low-frequency SAWs
+in the presence of EPs are discussed in Sec. III, where comparisons between theory and
+experiments are also made. Finally, conclusions and further discussions are given in Sec.
+IV.
+4
+
+1.5
+th
+E;cl
+length (m)
+E;rel
+m
+0.5
+0
+0.2
+0.24
+0.28
+0.32
+r/aII.
+THE GENERAL FISHBONE-LIKE DISPERSION RELATION FOR LOW-
+FREQUENCY SAWS
+In this Section, we will present analytical dispersion relations for low-frequency SAW
+excitation in weakly reversed-shear DIII-D discharges. As stated in the previous Section,
+two cases determined by the relative magnitude of LPE and ∆m will be used to investigate the
+low-frequency SAW stability: case I, the local GFLDR model corresponding to LPE > ∆m;
+and case II, the global GFLDR corresponding to LPE ≃ ∆m.
+Consider case I first. For LPE;rel > ∆m, the scales of LPE and ∆m can be separated,
+and the vorticity equation [4, 9, 28, 29, 32, 33] which governs shear Alfv´en waves (SAWs)
+can yield the low-frequency electromagnetic fluctuation dispersion relation in the usual local
+limit, as derived and discussed in great details in Refs. [9, 28, 29, 32, 33, 35]. We just note
+that, for DIII-D case of interest, the reversed magnetic shear configuration and thermal
+plasma compression effects should be accounted for properly [36]. Thus, for s = 0 at r0 but
+with finite S ≡ (r/q)[q
+′′]1/2, the local GFLDR for low-frequency SAWs can be written as
+[27–29, 35, 40]
+iS(Λ2
+n − k2
+∥n0q2
+minR2
+0)1/2(1/n)1/2�
+k∥n0qminR0 − i(Λ2
+n − k2
+∥n0q2
+minR2
+0)1/2�1/2 = δ ˆWnf + δ ˆWnk(ω),
+(1)
+where the generalized inertia term Λn(ω) here, including both diamagnetic effects as well as
+kinetic effects of circulating and trapped particle dynamics, has been derived explicitly in
+Ref. [7] and the main results are summarized in Appendix A. The right hand side of Eq.
+(1) contains both “fluid” (δ ˆWnf) and “kinetic” (δ ˆWnk) contributions to the potential energy
+in the “regular” ideal region. In the low-frequency limits (|Λ2
+n| ≪ 1), δ ˆWnf is independent
+of the frequency and the explicit expression, specialized to the (s, α) model equilibrium [41]
+with circular flux surfaces, reads,
+δ ˆWnf ≃ π
+4
+�S2k∥0qminR0
+n
+− 3
+2α2S
+��k∥0qminR0
+n
+��1/2 + 9
+32α4
+�
+(2)
+where α = αc + αE, αc = −R0q2
+mindβ/dr and αE = − 1
+2R0q2
+mind(βE∥ + βE⊥)/dr. Note that
+Eq. (2) includes the contribution of the energetic particle adiabatic and convective responses
+as well [31].
+The term δ ˆWnk is always a function of the mode frequency ω, as it reflects resonant
+as well as non-resonant wave-particle interactions. For simplicity but still relevant to the
+5
+
+DIII-D case, we take F0E to be a single pitch angle (λ = µ/ε) slowing-down beam ion
+equilibrium distribution function; i.e., F0E =
+B0βE(r)
+25√
+2π2mEεb
+�
+(1 − λ0B0)ε−3/2δ(λ − λ0). Here,
+βE(r) ≡ 8πPE(r)/B2
+0 is the ratio of EP kinetic and magnetic pressures and B0 the on-
+axis equilibrium magnetic field, δ(x) is the Dirac function, µ is the magnetic moment and
+ε = υ2/2 ≤ εb with εb being the EP birth energy per unit mass. Then the explicit expression
+of non-adiabatic contribution δ ˆWnku for the passing energetic ions is given by [32, 33]
+δ ˆWnku ≃ παE
+25/2 (1 − λ0B0/2)¯ω
+�
+2 − ¯ω ln
+� ¯ω + 1
+¯ω − 1
+��
+,
+(3)
+where ¯ω = ω/ωtEm and ωtEm ≡ √2εb/qR0 is the EP transit frequency at the maximum
+particle energy.
+It is worthwhile emphasizing that the finite k∥n0qminR0 in Eq. (1) plays an important
+stabilizing role since it represents the finite line bending effect at r = r0 [28, 29, 35]. Further-
+more, the expression of Λn depends on the mode polarization via Sf ≡ (iδE∥/k∥)a.c.
+�
+δφd.c.,
+where a.c. and d.c. refer to the sinusoidal and nearly constant (flute-like) components of the
+parallel electric field, wave vector, and scalar potential fluctuation [21, 27]. The detailed
+expression of Sf, again, is given in the Appendix A. Here, we just note that |Sf| is much
+smaller than unity for shear Alfv´en wave and order of unity for ion acoustic wave [7, 21, 27].
+We remark here that, in the moderate pressure gradient case, the local GFLDR for
+the low-frequency SAWs is enough to delineate the underlying physics of the experimental
+and simulation results. However, the local GFLDR for the low-frequency SAWs, given by
+Eq. (1), will fail in the presence of strong EP pressure gradient, i.e., case II. In this case,
+two typical scale lengths LPE,cl and ∆m can not be separated anymore and, thus, a global
+dispersion relation is needed which can be derived from the vorticity equation, i.e., Eq. (1)
+of Ref. [40]. Noting that the mode structure is dominated by single toroidal and poloidal
+mode numbers, (n, m), the governing equation reads
+(eθ − erξ) ·
+�
+Λ2 − Ω2
+A,m
+�
+1 +
+x2
+ΩA,m
++
+x4
+4Ω2
+A,m
+��
+(eθ − erξ)δφm − (F + K)δφm = 0,
+(4)
+where k⊥/kθ = −(eθ − erξ) with er and eθ being, respectively, the radial and poloidal unit
+vectors, x2 = nq′′
+min(r − r0)2, ξ ≡ (i/n1/2)S(∂/∂x), and δφm is the mth poloidal harmonic
+of the scalar field perturbation. It is worth noting that, toroidal coupling among different
+poloidal harmonics is typically not important for modes in the reversed magnetic shear
+region, consistent with the mode being dominated by single m and n. The terms F and K
+6
+
+in Eq. (4) represent, respectively, the fluid-like particle and energetic ion contributions with
+their explicit form reading
+F ≃ D2
+S − 4α2DS + 2αD2
+S − (α + 1)α + 2α3,
+K ≃ 2πq2
+Eq2R2
+0ω
+mEc2
+�Ω2
+dEQF0E
+ω2
+tE − ω2
+�
+υ
+= 2
+πδ ˆWnku,
+(5)
+where DS = S
+�
+ΩA,m/n, qE and mE are the electric charge and mass of energetic ions, ΩdE =
+(υ2
+E⊥/2+υ2
+E∥)/ωcER0, ωtE = υE∥/qR0, QF0E = (ω∂ε+ˆω∗E)F0E, ˆω∗EF0E = ω−1
+cE (k×b)·∇F0E,
+ωcE = qEB/mEc, ⟨(...)⟩υ =
+�
+d3υ(...), and the subscripts ∥ and ⊥ represent the parallel and
+perpendicular components with respect to the equilibrium magnetic field b.
+Equation (4) is an ordinary differential equation and, generally, requires a numerical
+approach to be solved. However, for DIII-D case, the radial dependence of the normalized
+pressure gradient of energetic ions with the classical profile, as is shown by black curve in Fig.
+2, can be well fitted by the analytic formula αE(ρ) = c1 (1 − (ρ − c2)2/c2
+3), with c1 = 0.7099,
+c2 = 0.3018 and c3 = 0.2944. This allows us to obtain simple analytical dispersion relations
+for low-frequency SAWs excitation. We just note that the maximum drive of energetic ions
+is located around ρ = c2 = 0.3018, which deviates from the radial position of qmin. Then
+αE(r) in Eq. (3) can be rewritten as
+αE(r) = δaαE0
+�
+1 − (r − r0 + δb)2
+δ2
+cL2
+PE;cl
+�
+,
+(6)
+where δa = c1/αE0, δb = r0 − c2a and δc = c3a/LPE;cl, a is the minor radius, αE0 and LPE;cl
+are evaluated at r = r0. Introducing the notation x = r − r0 = σz − δb, Eq. (4) is readily
+cast into the form
+∂2
+∂z2δφm − nσ2
+S2
+�
+1 − F + 2δa
+π δ ˆWnku0
+ϵA0
+�
+δφm − 1
+4z2δφm = 0,
+2nσ4δaδ ˆWnku0
+ϵA0πS2δ2
+cL2
+PE;cl
+= 1
+4,
+(7)
+where ϵA0 = Λ2 − Ω2
+A,m, δ ˆWnku0 = παE0
+4
+√
+2
+�
+2 − ¯ω ln
+� ¯ω+1
+¯ω−1
+��
+. Then, Eq. (7) yields the following
+global dispersion relation for low-frequency SAWs,
+−n1/2π1/2δcLPE;clϵ1/2
+A0
+2
+√
+2Sδ1/2
+a δ ˆW 1/2
+nku0
+�
+1 − F + 2δa
+π δ ˆWnku0
+ϵA0
+�
+= 2L + 1,
+L = 0, 1, 2, 3 ...
+(8)
+7
+
+FIG. 2. The radial dependence of the normalized pressure gradient of EPs with the classical profile.
+Here, the normalized radial position of qmin is ρ0 ≡ r0/a = 0.28.
+Here, the integer L is the radial eigenmode number. The corresponding eigenfunction reads
+δφm(r) = HL(z)e−z2 ∝ exp
+�
+−(r − r0 + δb)2
+4σ2
+�
+,
+(9)
+where HL(z) represents Lth order Hermite polynomials and the causality constraints upon
+the discrete bound modes requiring Re(σ2) > 0, where σ2 is solved for from the second of
+Eqs. (7) consistently with the dispersion relation, Eq. (8). The typical radial width, w, of
+δφm(r) is determined by w2 = 4σ2.
+Equations (1) and (8) constitute the results of the present section, i.e., the local and
+global GFLDR for the low-frequency SAWs excited by energetic ions. With their explicit
+form, we can compute the individual terms involved in equations and investigate the linear
+properties of the experimentally observed low-frequency SAWs.
+III.
+THE LOW-FREQUENCY SAW INSTABILITIES NUMERICAL RESULTS
+AND ANALYSIS
+In this Section, we separately present numerical results for the local and global low-
+frequency SAW stability properties in the presence of energetic ions, for which the dispersion
+relation is given by Eqs. (1) and (8). The numerical investigations use experimental equilib-
+rium and profiles as shown in Fig. 3 for the DIII-D shot #178631 at the time t = 1200 ms
+[16], where the q-profile has a reversed shear configuration with qmin = 1.37 at r0/a = 0.28
+8
+
+0.72
+0.7
+0.68
+a
+0.66
+αE;cl;exp.
+vs. p
+0.64
+fit
+-Prediction bounds -99%
+0.62
+0.2
+0.25
+0.3
+0.35
+p=r/aFIG. 3. Radial profiles of (a) temperature and q and (b) density and toroidal rotation frequency
+frot of DIII-D shot #178631 used for numerical studies.
+and qmin decreases from 1.49 to 1.18 in the time window 1050 ms < t < 1350 ms, as shown
+in Fig. 6 (b) in Ref. 16.
+A.
+The local low-frequency SAW stability properties
+We first consider the linear properties of the low-frequency SAW with relaxed energetic ion
+profile, i.e., case I. The local equilibrium parameters used in the numerical studies evaluated
+at r0/a = 0.28 are S = 0.5895, τ = Te/Ti =3.86 keV/2.37 keV=1.62, ne = 3.80 × 1019
+m−3, ni = 3.19 × 1019 m−3, ϵr = r0/R = 0.10, βi ≃ 0.01, ϵni = Lni/R0 = 0.414, ηi =
+Lni/LTi = 0.8324, ω∗ni/ωti = 0.1919, (m, n) = (8, 6), kθρLi = 0.2555 and kθρLe = 0.0054.
+Other fixed equilibrium parameters are a = 0.64 m, R0 = 1.74 m, B0 = 1.8 T. Here, kθ
+is the poloidal wavenumber, ρLi and ρLe are the Larmor radii of thermal ions and thermal
+electrons, respectively.
+Dependencies of the (a) mode frequencies, (b) growth rates and (c) mode polarization
+predicted by Eq.
+(1) are shown in Fig.
+4 as a function of the normalized thermal ion
+diamagnetic frequency Ω∗pi ≡ ω∗pi/ωti for the cases without and with the consideration of
+EP effects. According to the scaling of mode frequencies with physical parameters and the
+value of the |Sf| [21], three branches in Fig. 4 can be classified as: (i) the KBM (red curves
+marked with circles), with a frequency scaling with ω ∼ ω∗pi; (ii) the BAE (blue curves),
+with the frequency being close to the well-known estimate ω/ωti = qmin
+�
+7/4 + τ ≃ 2.51;
+9
+
+(a)
+T。 (keV)
+6
+.T. (keV)
+q
+TE:cl (keV/10)
+4
+2
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+r/a(b)
+5
+n(1019m-3)
+ - -n, (1019m3)
+4
+4 × nE:cl (1019m=3)
+-4 × nE:rel (1019m-3)
+3
+-.0.5× frot (kHz)
+2
+1
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+r/aFIG. 4. Dependence of the (a) real frequencies, (b) growth rates and (c) polarization of the low-
+frequency SAWs on Ω∗pi ≡ ω∗pi/ωti for the cases without (w/o) and with (w/) EP effects. Here, a
+dashed vertical line represents the experimental value of Ω∗pi;exp of about 0.35.
+and (iii) the BAAE (green curves marked with diamonds), with a frequency of about half
+of the BAE and experiencing strong damping. The EP effects on the low-frequency SAW
+stabilities are apparent in the region highlighted by the purple curve of Fig. 4 (b), where the
+KBM is the only unstable mode in the absence of EPs, while both the KBM and BAE are
+unstable in the low-frequency region in the presence of EPs. In particular, the diamagnetic
+ion frequency calculated on the basis of experimental parameters is Ω∗pi;exp = 0.3517, as
+shown by the dashed vertical line. In this case, both KBM and BAE are unstable with the
+frequencies in the plasma frame being 5.6 kHz and 63.7 kHz, respectively, which are in good
+agreement with the experimental observations. Meanwhile, the polarization plot of Fig. 4
+(c) shows that KBM and BAE have small values for |Sf| ≲ 0.1, which indicates that the
+KBM and BAE are essentially of Alfv´enic polarization. Moreover, in order to exclude the
+10
+
+(a)
+KBMw/oEP
+BAAE w/o EP
+BAE w/o EP
+4
+KBM w/ EP
+Re(wlwti
+BAAE w/ EP
+BAE w/ EP
+0
+2
+4
+6
+2
+*
+pi:(b)
+m(w/wti
+KBM w/o EP
+BAAE w/o EP
+BAEw/o EP
+KBMw/EP
+BAAE w/ EP
+BAEw/EP
+0
+*pi;exp
+2
+4
+6
+U
+pi
+*100
+(c)
+KBM w/o EP
+BAAE w/o EP
+BAEw/oEP
+S
+KBM w/ EP
+BAAE w/ EP
+BAEw/EP
+0
+2
+4
+6
+* pispurious nonzero solutions produced by singularities of the transcendental function of the
+local GFLDR (D), the Nyquist diagram in the complex D plane presented in Fig. 5 shows
+that in the presence of EPs, the path encircles the origin twice (see Fig. 5 (b)) but only once
+without EPs (see Fig. 5 (a)), thus confirming there are two unstable modes with EPs. It
+FIG. 5. The Nyquist diagram in the complex D(ω) plane for the cases (a) without and (b) with
+EP effects.
+should be noted that, compared with the frequency insensitive to the EP effects, the growth
+rate of the KBMs changes significantly in the cases with and without EP effects.
+This
+occurs because in our theoretical model the adiabatic and convective contribution of EPs
+modifies the value of δ ˆWf via α, as is shown in Eq. (2). At this point, in order to obtain more
+convincing comparison of theoretical prediction and experimental observation, it is necessary
+to provide a more precise theoretical model and also a more comprehensive experimental
+analysis. We also note here that, in this case, the stability/property of the BAAE is not
+affected by energetic ions — as is shown by the green dashed lines with symbols (without
+EP effects) and solid lines with symbols (with EP effects) which are apparently overlaying
+in all three graphs — even though it becomes weakly damped by coupling with the KBM
+due to diamagnetic and trapped particle effects for sufficiently strong Ω∗pi. The numerical
+results are consistent with the numerical simulation results reported in Refs. [20, 23, 24]
+and the theoretical prediction in Ref. [27], that is, “EPs preferentially excite the BAE over
+the BAAE branch due to the stronger wave-EP interaction”.
+We now investigate the underlying instability mechanisms of the ascending spectrum of
+the higher frequency BAEs and LFAMs observed in DIII-D (see Fig. 8 of Ref. [17]) by using
+11
+
+×10-3
+(a)
+10
+5
+0
+-5
+-5
+0
+5
+Re(D)
+×10-3X10-3
+10
+(b)
+5
+Im(D)
+0
+.5
+-5
+0
+5
+Re(D)
+×10-3qmin as the scanning parameter. Figure 6 shows the dependence of the mode frequencies
+(solid curves with markers) and growth rates (dashed curves with markers) on qmin of the
+KBMs (red curves) and the BAEs (blue, green, purple and orange curves) for different
+poloidal and toroidal mode numbers (m, n). It is shown that the modes in ascending pattern
+FIG. 6. Dependence of mode frequencies (solid curves with markers) and growth rates (dashed
+curves with markers) on qmin of the KBMs (red curves) and the BAEs (blue, green, purple and
+orange curves) for different (m, n). The experimentally observed frequencies are also shown. For
+the BAE, since the modes span a range of frequencies, the lines indicate the upper and lower limits
+of the unstable bands; for the LFAM, the experimental frequency variation is < 0.5 kHz. In the
+abscissa, the experimentally measured qmin(t) fit shown in Fig. 8 of [17] is used to convert time to
+qmin, with an associated uncertainty of ∆qmin ≃ 0.01. In the ordinate, the theoretical lab-frame
+frequency incorporates a Doppler shift to the calculated plasma-frame frequency of nfrot, with an
+associated uncertainty of ∼ 0.5 × n kHz.
+of higher frequency BAEs and lower frequency KBMs are both separated by approximately
+12
+
+200
++7,5
+f: in the lab-frame
+(10,8)
+KBM
+f8,6
+KBM
+9,7
+f9,7
+KBM
+8.6
+f10,8
+150
+KBM
+(7,5)
+KBM
+KBM
+KBM
+ZH)
+.10,8
+100
+/2元
+“KBM
+Expi..data
+7,5
+BAE
+(range)
+8,6
+(lines with ★)
+(9,7)
+(10,8)
+(8,6)
+BAE
+(7,5)
+9,7
+BAE
+10,8
+50
+BAE
+/27
+BAE
+“BAE
+BAE
+10,8
+BAE
+1.45
+1.4
+1.35
+1.3
+1.25
+1.2
+minfrot of about 7.5 kHz. More specifically, for KBMs, the instabilities peak exactly at the
+rational values of qmin; while the BAEs occur at times near rational values of qmin but the
+timing of unstable modes is less precise than for KBMs. In addition, the low-n BAEs deviate
+more from rational qmin crossings than higher n modes. The comparison of the theoretically
+predicted frequencies with the experimentally measured values can also be seen clearly from
+Fig. 6. As discussed in more detail in the next section, these numerical results are in good
+agreement with the experimental observations.
+In order to gain insight into the different excitation mechanisms of the instabilities pre-
+sented in Fig. 6, let us further analyze the GFLDR in the high-frequency (|ω| ≫ ωti) and
+low-frequency |ω| ≪ ωbi limits.
+For |ω| ≫ |ωti|, the corresponding inertia term of the BAE can be reduced to the simplified
+expression with Λ2 ≃
+ω2−ω2
+BAE
+ω2
+A
+[4, 35, 42]. Here, ω2
+BAE = (7/4 + τ)υ2
+i /R2
+0 is the fluid limit
+expression of the BAE frequency. Taking ω = ωr + iγ and δ ˆWku = Reδ ˆWku + iImδ ˆWku, and
+assuming |γ/ωr|, we have |Imδ ˆWku/Reδ ˆWku| ≪ 1. Then, for the gap mode, the existence
+condition is δ ˆWnf + Re(δ ˆWnk(ωr)) < 0 and the real mode frequency is given by
+ω2
+r = ω2
+BAE
+�
+��1 +
+ω2
+A
+ω2
+BAE
+�
+�
+�k2
+∥n0q2
+minR2
+0 −
+n
+��k∥n0qminR0
+��
+�
+δ ˆWnf + Re(δ ˆWnk(ωr))
+�2
+S2
+�
+�
+�
+�
+�� ,
+(10)
+while the growth rate is obtained from
+γ = −Im(δ ˆWnk(ωr))ω2
+A
+ωr
+n
+�
+δ ˆWnf + Re(δ ˆWnk(ωr))
+�
+��k∥n0qminR0
+�� S2
+,
+(11)
+It can be readily obtained from Eq. (10) that the BAE frequency is positively correlated with
+��k∥n0qminR0
+��. Therefore, the more deviation from the rational qmin surface is, the larger the
+BAE frequency is, as is shown in Fig. 6. Note also that the BAE has a positive frequency.
+Equation (11) imposes Im(δ ˆWnk(ωr)) > 0 for BAE excitation by EPs via resonant wave-
+particle interaction. It can be concluded that the duration of BAEs is influenced by the
+associated resonances with the EPs, as well as by the value of qmin [17].
+Similarly, for KBM with |ω| ≪ |ωbi|, we have Λ2 ≃ c0
+q2
+min
+√ϵ
+(ω−¯ωdi)(ω−ω∗pi)
+ω2
+A
+[7, 16, 21, 35, 43].
+Here, ¯ωdi is the average thermal-ion precession frequency, c0 ≃ 1.6 due to trapped and barely
+13
+
+circulating particles [44, 45]. Thus, the real mode frequency is given by
+ω = 1
+2(¯ωdi+ω∗pi)±1
+2
+�
+��(ω∗pi − ¯ωdi)2 − 4ω2
+A
+√ϵ
+q2
+minc0
+�
+�
+�
+n
+�
+δ ˆWnf + Re(δ ˆWnk(ωr))
+�2
+��k∥n0qminR0
+�� S2
+− k2
+∥n0q2
+minR2
+0
+�
+�
+�
+�
+��
+1/2
+,
+(12)
+and the system is reactively unstable if
+|ω∗pi − ¯ωdi|2
+ω2
+A
+<
+4√ϵ
+q2
+minc0
+�
+�
+�
+n
+�
+δ ˆWnf + Re(δ ˆWnk(ωr))
+�2
+��k∥n0qminR0
+�� S2
+− k2
+∥n0q2
+minR2
+0
+�
+�
+� .
+(13)
+Note that δ ˆWf + Reδ ˆWku < 0, due to, again, the causality constraint. Therefore, for the
+reactive-type instability, the maximum drive sets in when k∥n0qminR0 → 0, which corre-
+sponds to the unstable KBM exactly peaking at the rational values of qmin.
+The above numerical results and theoretical analyses have explained the experimental
+observations that the BAEs deviate more from the rational qmin values temporally, com-
+pared with the KBM. To further delineate this deviation and its impact on the radial mode
+structure, numerical investigation of the global model for low-frequency SAWs is needed.
+B.
+The global low-frequency SAW stability properties
+In this part, we consider the case II and apply Eq. (8) to investigate the global low-
+frequency SAW stability properties with the classical energetic ion profile.
+Figure 7 shows (a) the dependence of the real frequencies (blue markers) and growth rates
+(red markers) of the KBM (triangle markers) and BAE (line with markers) on the radial
+mode number L; and (b) the radial mode structure δφm(r) for the L = 0 BAE. It can be
+found that (i) the ground eigenstate with L = 0 is most unstable for the BAE and KBM;
+(ii) for BAE, the frequency and growth rate in the plasma frame is (80.7 + 15.2i) kHz with
+the ratio of the growth rate to real frequency γ/ω ≃ 0.19, which is the typical feature of the
+marginally unstable gap mode excited by EPs; and (iii) for KBM, the frequency and growth
+rate in the plasma frame is (−3.2 + 5.7i) kHz with γ/ω ≃ 1.8, which is the typical feature
+of the reactive-type instability, consistent with the results reported in Ref. [24].
+Correspondingly, the radial eigenfunction plot of the BAE for L = 0, as shown in Fig.
+7 (b), presents that δφm has a Gaussian form with a shape similar to the experimentally
+14
+
+FIG. 7. (a) Dependence of the real frequencies (blue markers) and growth rates (red markers) of
+the KBM (triangle markers) and BAE (line with markers) on the radial mode number L; (b) the
+radial mode structure δφm(r) for the L = 0 BAE. The approximate experimental measurement of
+the mode structure of BAE is also shown.
+measured radial mode structure. In this case, the radial width of δφm by theory is w =
+0.2107, is comparable to the scale length of energetic-ion pressure, i.e., LPE;cl = 0.1773;
+consistent with the analysis of Fig. 1. Note that determined by the EP distribution, the
+BAE eigenfunction peaks at the radial position of the maximum energetic particle pressure
+gradient, resulting in a large deviation from the qmin surface. It can also be expected that
+the KBM eigenfunction should peak at the rational values of qmin where the instability drive
+is maximum.
+Finally, the continuous spectra plots for low-frequency shear Alfv´en and acoustic waves
+given by Λ2
+n(ω) = k2
+∥nq2R2
+0 = (nq−m)2 [4, 6, 28, 29, 42, 46, 47] are shown in Fig. 8. Here, the
+inertia term includes the diamagnetic effects and thermal ion compressibility as well as drift
+Alfv´en wave and drift wave sideband coupling via the wave-thermal-passing-ion interaction
+and diamagnetic effect [6]. The figure shows that based on the GFLDR, the nature of various
+branches can be clearly classified via their frequencies (a), growth rates (b) and polarizations
+(c). Here, the short notation “e-KBM” represents the branch of the KBM propagating in
+the thermal-electron diamagnetic drift direction. The unstable continuum spectrum of the
+e-KBM is due to the inclusion of the kinetic dynamics of thermal particles in inertia term. In
+addition, the frequencies of the (m, n) = (8, 6) BAE and the (m, n) = (8, 6) KBM calculated
+by the local and global cases are, respectively, in the gaps of the BAE and KBM continua,
+15
+
+4
+(a)
+3
+△ Re(wlwt) of KBM
+A
+Im(w/wti) of KBM
+2
+-Re(wlwt:) of BAE
+-- Im(w/wti) of BAE
+0
+2
+3(m,n)=(8,6)
+0.75
+analytic
+-ECE-measured
+m,n
+0.5
+0.25
+0
+0
+0.25
+0.5
+0.75
+1
+p=r/awhich is consistent with the numerical simulation results reported in Refs. [16, 24].
+FIG. 8. The continuous spectra of low-frequency shear Alfv´en and acoustic branches for n=6,
+m=8-15. The equilibrium profiles of DIII-D #178631 at 1200 ms are adopted.
+IV.
+SUMMARY AND DISCUSSIONS
+The present work has addressed linear properties of the low-frequency shear Alfv´en waves
+(SAWs) with the consideration of energetic ions in DIII-D reversed magnetic shear tokamak
+experiments. By analyzing the experimental equilibrium profiles, the local and global models
+for low-frequency SAWs for weak and/or vanishing magnetic shear are discussed based on the
+unified theoretical framework of the generalize fishbone-like dispersion relation (GFLDR).
+Resorting to numerical and theoretical analyses, the dependences of mode frequency, growth
+rate and polarization on the minimum of the safety factor (qmin), as well as the instability
+mechanisms are delineated.
+16
+
+200
+(a)
+n=6;
+BAE
+pi;exp
+BAAE
+米
+KBM; rel
+150
+m=8~15
+KBM
+BAE; rel
+KBM1
+米
+KBM; cl
+LFM
+BAE; cl
+(ZH>)
+e-KBM
+100
+p
+50
+0
+米
+米
+0.2
+0.4
+0.6
+0.8
+r/a10
+(b)
+0
+(kHz)
+-10
+-20
+2元
+BAE
+-30
+BAAE
+KBM
+n=6;
+KBM1
+-40
+m=8~15
+LFM
+e-KBM
+0.2
+0.4
+0.6
+0.8
+r/a102
+(c)
+BAE
+KBM1
+BAAE
+LFM
+KBM
+e-KBM
+Sf100
+0.2
+0.4
+0.6
+0.8
+r/aThe main results of this work are that the LFAMs and BAEs observed in DIII-D ex-
+periments are, respectively, the reactive-type and dissipative-type unstable modes with pre-
+dominantly Alfv´enic polarization. Due to the different instability mechanisms, BAE peak
+occurs further away from the rational qmin than LFAM peak does. The BAE eigenfunction
+is localized at the radial position with the strongest energetic-ion-drive spatially, which leads
+to deviation from the radial position of qmin.
+The theoretical analysis explains many experimental observations.
+1. The theory successfully explains the temporal pattern of two bands of instability, the
+BAE band and the LFAM band, that both appear near rational values of qmin but
+with distinctly different stability properties.
+2. The predicted values of KBM frequency are in excellent agreement with the experi-
+mental LFAM frequencies. The KBM can be unstable even in the absence of energetic
+particles (EPs).
+3. The predicted values of BAE frequency span the same range as the experimentally
+observed values.
+4. The theory also successfully explains the absence of a third branch of instability at
+BAAE frequencies, as that branch is predicted to be stable.
+5. Experimentally, an individual unstable BAE spans a much larger range of frequencies
+than an unstable LFAM, another feature successfully reproduced by theory.
+6. Experimentally, unstable LFAMs only persist for a few milliseconds. The short du-
+ration of the LFAM is consistent with the very strong qmin dependence of the KBM
+growth rate.
+7. In experiment, unstable BAEs persist longer than LFAMs, which is consistent with
+the weaker dependence of the BAE growth rate on qmin in theory.
+8. Temporally, in experiment, LFAMs occur at rational values of qmin; BAEs also occur
+near rational values but less precisely. This feature is also reproduced by the theoretical
+stability predictions: the KBM growth rate peaks sharply at rational qmin values but
+the peak of the BAE growth rate deviates slightly.
+17
+
+9. In experiment, for both the LFAM and the BAE, unstable modes with higher values of
+toroidal mode number n are of shorter duration than lower values of n. The narrower
+growth rate curves as n increases successfully explains this feature.
+10. Experimentally, the BAE radial eigenfunction has an approximately gaussian shape,
+consistent with the theoretical prediction that the L = 0 radial harmonic is most
+unstable.
+11. Experimentally, the LFAM is more unstable in plasmas with hydrogen than in pure
+deuterium plasmas [18], a feature explained by the higher value of ωA in hydrogen
+plasmas. As Eq. (13) shows, a larger value of ωA lowers the instability threshold.
+On the other hand, there are three discrepancies between theory and experiment.
+1. Although the predicted KBM growth rate correctly peaks sharply for rational values
+of qmin, it remains positive for a much longer duration than the LFAMs are observed
+experimentally. Evidently, an additional damping mechanism is missing in the theory.
+2. Although the predicted KBM growth rate has changed significantly for the cases with
+and without EPs, there is no apparent dependence of LFAM stability on EPs ex-
+perimentally. Therefore, a more precise theoretical model and more comprehensive
+experimental analysis are needed for meaningful comparison.
+3. Although the predicted BAE frequency spans the observed values, the predicted fre-
+quency has a parabolic shape with time, while the experimental frequency has a less
+regular shape. A likely explanation for this discrepancy is imprecise modeling of the
+fast-ion distribution function.
+Finally, there is one theoretical prediction that is inconclusive experimentally: the mode
+polarization. Theory predicts predominately Alfv´enic polarization for both the KBM and
+the BAE. In experiment, low toroidal mode number (n ≤ 3) BAEs are usually observed on
+external magnetic coils; LFAMs are never detected, but the inferred toroidal mode numbers
+typically span a larger range than those normally detected for RSAEs or BAEs. DIII-D is
+equipped with one diagnostic that can detect internal magnetic fields, a radial interferometer-
+polarimeter (RIP) [48] that measures the line integral of the density and radial magnetic
+field,
+�
+neBrdl. This diagnostic clearly detects RSAEs and BAEs, which is consistent with
+18
+
+their expected shear-wave polarization. Fluctuations are observed by RIP for some LFAMs,
+indicating that there is at least some magnetic component, but the signal is weaker than
+for RSAEs and BAEs. It is not presently known if this difference is due to a line-integral
+effect associated with the mode structure or if the LFAM polarization is less Alfv´enic than
+the other modes.
+ACKNOWLEDGMENTS
+One of authors (R.R. Ma) would like to acknowledge Dr. Lei Yang and Dr. Yunpeng Zou
+for their useful discussions and the DIII-D team for providing the experimental data. The
+authors thank Dr. Xiaodi Du for helpful comments concerning the mode polarization. R.R
+Ma is also grateful to the Center for Nonlinear Plasma Science (CNPS) for its enlightening
+academic discussion, which provides a valuable sources of scientific stimuli.
+This work has been supported in part by the National key R&D Program of China under
+Grant Nos. 2022YFE03040002 and 2018YFE0304103, by the National Science Foundation
+of China under Grant Nos. 12261131622 and 12175053 and Natural Science Foundation of
+Sichuan under Grant No. 2022NSFSC1814 and Sichuan Science and Technology Program
+under Grant No. 2022ZYD0019. This work has also been carried out within the framework
+of the EUROfusion Consortium, funded by the European Union via the Euratom Research
+and Training Programme (Grant Agreement No. 101052200 – EUROfusion). Views and
+opinions expressed are however those of the author(s) only and do not necessarily reflect
+those of the European Union or the European Commission. Neither the European Union
+nor the European Commission can be held responsible for them. This material is based
+upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion
+Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user
+facility, under Awards DE-FC02-04ER54698 and DE-SC0020337.
+This report was prepared as an account of work sponsored by an agency of the United States
+Government.
+Neither the United States Government nor any agency thereof, nor any of their
+employees, makes any warranty, express or implied, or assumes any legal liability or responsibility
+for the accuracy, completeness, or usefulness of any information, apparatus, product, or process
+disclosed, or represents that its use would not infringe privately owned rights. Reference herein to
+any specific commercial product, process, or service by trade name, trademark, manufacturer, or
+19
+
+otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring
+by the United States Government or any agency thereof.
+The views and opinions of authors
+expressed herein do not necessarily state or reflect those of the United States Government or any
+agency thereof.
+Appendix A: Detailed Expressions of Λ2
+n and Sf
+Detailed derivations of the generalized inertia, Λ2
+n and wave polarization, Sf, can be found
+in Ref. 7. Here, we only present the results. In low-β (β = 8πP/B2
+0 ≈ ϵ2) axisymmetric
+tokamak plasmas,
+Λ2
+n = Iφ
+� ω2
+ω2
+A
+�
+1 − ω∗pi
+ω
+�
++ Λ2
+cir + Λ2
+tra
+�
+,
+(A1)
+where Λ2
+cir and Λ2
+tra represent, respectively, the modified circulating and trapped ion re-
+sponses, and Iφ describes the non-vanishing ‘flute-like’ component of the parallel elec-
+tric field (δE∥) due to the effect of trapped thermal particle precession resonance [7, 21].
+Meanwhile, ωA = υA/qR0 is the Alfv´en frequency with υA being the Alfv´en velocity, and
+ω∗ps = (Tsc/esB)(k×b)·(∇ns/ns+∇Ts/Ts) ≡ ω∗ns+ω∗Ts is the thermal particle diamagnetic
+drift frequency due to density and temperature gradients.
+For Λ2
+n, the various terms involved in Eq. (A1) are given by [7]
+Λ2
+cir = q2ωωti
+ω2
+A
+��
+1 − ω∗ni
+ω
+��
+F
+� ω
+ωti
+�
++ ∆F
+� ω
+ωti
+��
+− ω∗Ti
+ω
+�
+G
+� ω
+ωti
+�
++ ∆G
+� ω
+ωti
+��
++ ωωti
+4¯ω2
+Di
+�
+N1
+� ω
+ωti
+�
++ ∆N1
+� ω
+ωti
+��
+Sf(ω, ¯ωDi, ωbi, ωti)
+�
+,
+(A2)
+Λ2
+tra = ω2ω2
+bi
+ω2
+A¯ω2
+Di
+q2
+√
+2ϵ
+�
+P3 + (P2 − P3)Sf(ω, ¯ωDi, ωbi, ωti)
+�
+,
+(A3)
+Iφ = 1 +
+√
+2ϵ(L(ω/¯ωDi) + τ −1L(ω/¯ωDe))
+1 + τω∗ni/ω +
+√
+2ϵτ[1 − ω∗ni/ω − M(ω/¯ωDi) − τ −1M(ω/¯ωDe)],
+(A4)
+and, as to Sf ≡ (iδE∥/k∥)a.c.
+�
+δφd.c., it is given by [7]
+Sf = −
+N1
+�
+ω
+ωti
+�
++ ∆N1
+�
+ω
+ωti
+�
++
+√
+2ϵP2
+1 + 1
+τ + D1
+�
+ω
+ωti
+�
++ ∆D1
+�
+ω
+ωti
+�
++
+√
+2ϵ (P1 − P2)
+(A5)
+where the functions F(x), ∆F(x), G(x), ∆G(x), N1(x), ∆N1(x), D1(x), ∆D1(x), P1, P2, P3,
+L(ω/¯ωDs) and M(ω/¯ωDs) with x = ω/ωti, and using the plasma dispersion function Z(x),
+20
+
+are defined as
+Z(x) = π−1/2
+� ∞
+−∞
+e−y2
+y − xdy,
+F(x) = x(x2 + 3/2) + (x4 + x2 + 1/2)Z(x),
+∆F(x) =
+1
+π1/2
+� ∞
+0
+e−y ln
+�x + √2ϵy
+x − √2ϵy
+�y2
+4 dy,
+G(x) = x(x4 + x2 + 2) + (x6 + x4/2 + x2 + 3/4)Z(x),
+∆G(x) =
+1
+π1/2
+� ∞
+0
+e−y ln
+�x + √2ϵy
+x − √2ϵy
+�y2
+4
+�
+y − 3
+2
+�
+dy,
+N1(x) = 2 ¯ωDi
+ωti
+��
+1 − ω∗ni
+ω
+�
+[x + (1/2 + x2)Z(x)] − ω∗Ti
+ω [x(1/2 + x2) + (1/4 + x4)Z(x)]
+�
+,
+∆N1(x) = ¯ωDi/ωti
+π1/2
+� ∞
+0
+ye−y ln
+�x + √2ϵy
+x − √2ϵy
+� �
+1 − ω∗ni
+ω
+− ω∗Ti
+ω
+�
+y − 3
+2
+��
+dy,
+D1(x) = x
+�
+1 − ω∗ni
+ω
+�
+Z(x) − ω∗Ti
+ω [x + (x2 − 1/2)Z(x)],
+∆D1(x) = ¯ωDi/ωti
+π1/2
+� ∞
+0
+e−y ln
+�x + √2ϵy
+x − √2ϵy
+� �
+1 − ω∗ni
+ω
+− ω∗Ti
+ω
+�
+y − 3
+2
+��
+dy,
+P1 = −2 ω2
+¯ω2
+Di
+��
+1 − ω∗ni
+ω
++ 3
+2
+ω∗Ti
+ω
+�
+G2 − ω∗Ti
+ω G4
+�
+,
+P2 = −2 ω
+¯ωDi
+��
+1 − ω∗ni
+ω
++ 3
+2
+ω∗Ti
+ω
+�
+G4 − ω∗Ti
+ω G6
+�
+,
+P3 = −2
+��
+1 − ω∗ni
+ω
++ 3
+2
+ω∗Ti
+ω
+�
+G6 − ω∗Ti
+ω G8
+�
+,
+Gn =
+1
+π1/2
+� ∞
+−∞
+e−x2xn
+(ω/¯ωDi − x2)2 − (ωbi/¯ωDi)2x2dx,
+M
+� ω
+¯ωDs
+�
+= −2 ω
+¯ωDs
+� �
+1 − ω∗ni
+ω
++ 3
+2
+ω∗Ti
+ω
+� �
+1 +
+� ω
+¯ωDs
+Z
+�� ω
+¯ωDs
+��
+− ω∗Ti
+ω
+�
+1
+2 + ω
+¯ωDs
++
+� ω
+¯ωDs
+�3/2
+Z
+�� ω
+¯ωDs
+�� �
+,
+L
+� ω
+¯ωDs
+�
+= −2
+� �
+1 − ω∗ni
+ω
++ 3
+2
+ω∗Ti
+ω
+� �
+1
+2 + ω
+¯ωDs
++
+� ω
+¯ωDs
+�3/2
+Z
+�� ω
+¯ωDs
+��
+− ω∗Ti
+ω
+�
+3
+4 + 1
+2
+ω
+¯ωDs
++
+� ω
+¯ωDs
+�2
++
+� ω
+¯ωDs
+�5/2
+Z
+�� ω
+¯ωDs
+�� �
+.
+(A6)
+Here the magnetic drift orbit precession frequency ¯ωds = ¯ωDsmsυ2/2Ts for deeply
+trapped particles (s = i, e) with ¯ωDs = (nq/r)Ts/msR0ωcs and ωcs = esB/msc; the
+bounce frequency of deeply trapped ions ωbi ≡ (r/R0)1/2(Ti/mi)1/2/(qR0) ≈ ϵ1/2ωti with
+21
+
+ωti = (2Ti/mi)1/2/qR0; and τ ≡ Te/Ti.
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+24
+

7NAzT4oBgHgl3EQf-f4D/content/tmp_files/2301.01933v1.pdf.txt

@@ -0,0 +1,1373 @@
+ 
+ 
+Abstract—Surface electromyogram (SEMG) decomposition 
+provides a promising tool for decoding and understanding neural 
+drive information non-invasively. In contrast to previous SEMG 
+decomposition methods mainly developed in offline conditions, 
+there are few studies on online SEMG decomposition. A novel 
+method for online decomposition of SEMG data is presented using 
+the progressive FastICA peel-off (PFP) algorithm. The online 
+method consists of an offline prework stage and an online 
+decomposition stage. More specifically, a series of separation 
+vectors are first initialized by the originally offline version of the 
+PFP algorithm from SEMG data recorded in advance. Then they 
+are applied to online SEMG data to extract motor unit spike trains 
+precisely. The performance of the proposed online SEMG 
+decomposition method was evaluated by both simulation and 
+experimental approaches. It achieved an online decomposition 
+accuracy of 98.53% when processing simulated SEMG data. For 
+decomposing experimental SEMG data, the proposed online 
+method was able to extract an average of 12.00 ± 3.46 MUs per 
+trial, with a matching rate of 90.38% compared with results from 
+the expert-guided offline decomposition. Our study provides a 
+valuable way of online decomposition of SEMG data with 
+advanced applications in movement control and health. 
+ 
+Index Terms—Surface electromyography, motor unit, online 
+decomposition, progressive FastICA peel-off 
+ 
+I. INTRODUCTION 
+lectromyogram (EMG) is an electrophysiological signal 
+generated by muscular activation, reflecting motor control 
+commands of the neuromuscular system [1]. It can be used to 
+analyze movement behaviors, intentions and health [2]-[4]. 
+Surface EMG (SEMG) refers to the EMG signals recorded by 
+electrodes placed on the skin surface. Due to its noninvasive 
+manner, SEMG has been widely applied in human-machine 
+interfaces [5]-[7], sports medicine [8]-[9] and rehabilitation 
+[10]-[12]. Ideally, an EMG signal is composed of multiple 
+action potentials generated by activated motor units (MUs), 
+transmitted and superimposed temporally and spatially at a 
+recording electrode [13]. Specifically, each MU consists of the 
+cell body and dendrites of an alpha motor neuron, the multiple 
+ 
+This work was supported by the National Natural Science Foundation of 
+China under Grant No. 61771444. 
+H. Zhao and X. Zhang are with the School of Information Science and 
+Technology at University of Science and Technology of China, Hefei, Anhui, 
+230026, China (email: xuzhang90@ustc.edu.cn). 
+branches of its axon, and the muscle fibers that are innervated 
+[14]. The MU is regarded as the basic component of the 
+peripheral neuromuscular system to describe the neural control 
+of muscular contraction and movement formation [15]. 
+Compared with the global features such as SEMG amplitude, 
+the MU activities can reflect the information of neural drives to 
+the muscle at a microscopic level. Therefore, it is valuable to 
+examine the MU activities and properties. EMG decomposition 
+enables resolving the composite EMG signal into its constituent 
+MU spike trains (MUSTs) and MU action potential (MUAP) 
+waveforms. The availability of these individual MU activities 
+can provide a promising way of decoding motor neural 
+commands of a neurophysiological nature [16]-[22].  
+Many efforts have been made toward EMG decomposition, 
+mainly relying on blind source separation (BSS) algorithms 
+which are aimed to solve the difficult math problem of 
+separating sources from observed signals without prior 
+knowledge of the source signals [23]. Besides, it brings huge 
+challenges to the SEMG decomposition due to its special 
+characteristics such as low signal-to-noise ratio, high similarity 
+and severe superposition of the MUAP waveforms, caused by 
+the low-pass filtering effect of the subcutaneous skin and fat 
+tissues. With the recent development of electronic and sensing 
+technologies, the use of high-density SEMG (HD-SEMG) by 2-
+dimensional flexible electrode arrays provides abundant spatial 
+information simultaneously recorded from dozens or even 
+hundreds of SEMG channels, facilitating implementing the 
+BSS algorithms in general, and the SEMG decomposition in 
+particular [24]. Convolution kernel compensation (CKC) [25] 
+and progressive FastICA peel-off (PFP) [26] are both 
+representative HD-SEMG decomposition methods, inspired by 
+the advanced BSS techniques [23], [27]. The CKC estimates 
+and updates cross-correlation vectors between the observed 
+SEMG signals and MUSTs in an iterative way [23]. The PFP 
+applies a classic FastICA algorithm [27] to the SEMG signals 
+to calculate the separation vectors and introduces a “peel-off” 
+procedure to progressively remove the separated MUAP 
+waveforms from the original SEMG signals. Such a procedure 
+mitigates the effect of the already identified MUs on the 
+M. Chen and P. Zhou are with Faculty of Biomedical and Rehabilitation 
+Engineering, University of Health and Rehabilitation Sciences, Qingdao, 
+Shandong, 266024, China (email: dr.ping.zhou@outlook.com).  
+ 
+Online Decomposition of Surface 
+Electromyogram into Individual Motor Unit 
+Activities Using Progressive FastICA Peel-off 
+Haowen Zhao, Xu Zhang, Maoqi Chen and Ping Zhou 
+E 
+
+ 
+FastICA convergence and effectively increase the number of 
+obtained MUs. The performance of both CKC and PFP has been 
+extensively validated [28]-[32]. Variations of both methods 
+have been developed to extract a relatively large number of 
+MUs at high muscle contraction levels, with successful 
+applications mainly in offline conditions [33]-[37].  
+Considering the application prospects of SEMG in many 
+fields, there are substantial demands for robust online SEMG 
+decomposition. Glaser et al. [38] conducted a pilot study on the 
+real-time SEMG decomposition based on the CKC algorithm 
+and demonstrated its feasibility. Afterwards, more relevant 
+studies were reported [39]-[44]. The development of these 
+online decomposition algorithms mainly relies on a basic 
+assumption that SEMG signals are quasi-stationary, and the 
+MU behaviors do not change in pattern over a short period of 
+time. This assumption has served as a primary basis of 
+conventional offline SEMG decomposition [25], [26]. On this 
+basis, these online decomposition algorithms were always 
+designed to use results from an offline decomposition as prior 
+knowledge, thus saving computational resources and allowing 
+the feasibility of online signal processing. Specifically, most 
+previous studies conducted online SEMG decomposition using 
+modified versions of the CKC method, whereas the online 
+version of the PFP method has not been investigated yet. 
+Considering the advantages of the PFP method in extracting a 
+great many MUs with high precision, it is necessary and 
+promising to develop its online version. 
+Accordingly, this paper presents an online SEMG 
+decomposition method based on the PFP algorithm, evolving 
+the key techniques of the PFP algorithm to meet the 
+requirements for its real-time usability. To avoid the time-
+consuming complexity from the offline decomposition methods, 
+the proposed method utilized a two-stage approach consisting 
+of an offline prework stage and an online decomposition stage. 
+Furthermore, an adaptive threshold selection algorithm was 
+developed to make it more suitable for precisely determining 
+each MUST while processing in real time. The performance of 
+the proposed online decomposition method was validated on 
+both simulated and experimental SEMG datasets. 
+II. RELATED WORK 
+A. SEMG Observation 
+Each MU has a unique and stable MUAP waveform 
+distribution pattern in different channels of a 2-dimensional 
+array, which can be used to distinguish and identify the MU. 
+The SEMG signal can be observed by a convolutional mixing 
+model expressed as [45]:  
+��(�) = � � ���(�)��(� − �) + ��(�)
+���
+���
+�
+���
+  
+ 
+(1)
+ 
+where � = 1,2,3 … … �  and  � = 1,2, … … � , ��(�) is the � th 
+SEMG channel and ��(�) represents the additive noise in the 
+�th channel. ���(�) denotes the waveform vector of length L, 
+which represents the waveform of the �th MU in the �th channel. 
+��(�) = ∑ �(� − ��(�))
+�
+ is the MUST expressed as a 0-1 
+impulse sequence indicating every spike firing timing at ��(�) 
+for the �th firing of the �th MU, whereas � is Dirac Delta 
+function. For each �, ��(� + 1) − ��(�) > � can be assumed. 
+Define the expansion vector of EMG signals and MUSTs as 
+ ��(�) = [��(�), ��(� − 1), … , ��(�), … , ��(� − � + 1)]
+ and 
+ ��(�) = [��(�), ��(� − 1), … , ��(�), … , ��(� − � + 1)]. 
+Thus, the equation can be rewritten in matrix form: 
+��(�) = ����(�) + ��(�)  
+(2)
+where ��(�)  represents noise. ��  is a matrix containing all 
+waveform vectors ���. For the mixing model analyzed above, 
+the task of EMG decomposition is to find a suitable separation 
+matrix �
+��� that consists of many separation vectors to extract the 
+MU firing events. As a result, the source signals of all MUs can 
+be estimated by ��(�) = �
+�����(�). 
+ 
+B. Automatic PFP (APFP) 
+The PFP algorithm has been automated, but it is suitable just 
+for offline data processing. More details of the algorithm and 
+the corresponding parameters can be found in [33] and the 
+APFP method was used in this study with the same settings as 
+reported in [33]. Below is a brief introduction to the APFP 
+method. 
+If a whitened observed signal � has been obtained and we 
+need to find an independent component � = ��� from it using 
+the ICA algorithm [23], [27], the following maximum negative 
+entropy problem needs to be optimized:  
+max      ��(�) = [�{�(���)} −  �{�(�)}]� 
+   �. �.        ���(�) = �{��} − 1 = �|�|�
+� − 1 = 0  
+ 
+(3)
+ 
+where �  is a non-polynomial function, and �  is a random 
+variable with standard normal distribution. 
+The problem above can be solved using the procedure of the 
+fix-point algorithm [46] to obtain a series of MU source signals 
+and their corresponding separation vectors. The spike trains can 
+be precisely extracted from these source signals using the initial 
+threshold determined by the Otsu algorithm [47]. However, the 
+spikes from one source signal often do not just belong to one 
+MU due to heavy MUAP superimposition or high MU 
+synchronization levels. Thus, a valley-seeking clustering 
+approach [48] is used to distinguish the spikes from the same 
+source signal based on their morphological features. On this 
+basis, the spikes belonging to each cluster are most likely from 
+the same MU [33]. After the valley-seeking clustering approach, 
+the constrained FastICA algorithm [49] is performed using the 
+extracted and clustered spike trains as constraints to converge. 
+Therefore, the MU source signals can be effectively updated 
+and meanwhile the possible firing errors are corrected. To 
+assess the reliability of the constrained FastICA outputs and 
+their corresponding MUSTs representing true MU activities, 
+some metrics are employed from the perspective of the 
+significance of correlation constrain [49], including the 
+consistency of spike amplitudes and inter-spike intervals [50], 
+and the physiologically reasonable firing rate [51]. In the APFP 
+method, the correlation coefficient between the output of 
+constrained FastICA and the testing spike trains (denoted as ξ), 
+the coefficient of variation of spike amplitudes and inter-spike 
+intervals (denoted as ������ and ������ ), and the firing rate 
+
+ 
+(denoted as FR) are employed. Moreover, a two-step criterion 
+describing a reasonable range of the above four metrics is 
+employed to judge the MU reliability comprehensively [33].  
+A “peel-off” procedure is performed later to subtract the 
+obtained MUAP waveforms from the original signals. The 
+MUAP waveforms of the identified MUs were estimated by a 
+straightforward approach following a least squares problem 
+[26], [52] instead of the conventional high-resolution alignment 
+algorithm [53]. More MUs can emerge when processing the 
+residual signals again with the FastICA algorithm. The 
+framework of the offline APFP method is summarized as 
+follows: 
+(1) Initialize the residual signal to the original EMG signal, 
+and make the MUST set γ empty. 
+(2) Apply the FastICA algorithm to the expanded residual 
+signal and obtain a series of source signals. 
+(3) Extract non-repetitive spike trains by Otsu algorithm and 
+use valley-seeking clustering to distinguish these spikes to 
+separate spike trains from different MUs. 
+(4) Use MUSTs obtained in step (2) as a reference signal, and 
+apply the constrained FastICA algorithm on the expanded 
+original EMG signal to detect the reliability of the MUSTs 
+and to correct possible erroneous or missing discharges.  
+(5) Judge whether the MUs obtained are reliable through 
+metrics calculation. Put reliable results in set γ. 
+(6) Estimate the waveforms of the reliable MUs, subtract the 
+estimated MUAP waveforms from the original signal and 
+update the residual signal. 
+(7) If no new reliable MU is found in the above steps, or the 
+APFP method reaches the preset termination condition, 
+the algorithm ends. Otherwise, go back to step (2). 
+ 
+III. METHODOLOGY 
+A. Experimental SEMG Data Collection and Preprocessing 
+1) Subjects and Experiments 
+Eight subjects (26.13±4.29 years) without any known history 
+of muscular or neural disorder participated in this study. The 
+study was approved by the Ethics Review Board of the 
+University of Science and Technology of China (Hefei, China). 
+All subjects signed consent prior to any procedure of the 
+experiments. 
+In this work, the HD-SEMG data were recorded from 
+abductor pollicis brevis (APB) muscle due to its wide 
+explorations and applications in SEMG studies [19]-[21]. Here, 
+a home-made, multi-channel signal acquisition system with a 
+force sensor and a set of 3D-printed apparatuses was used to 
+collect data, as shown in Fig. 1a. The subject’s hand was placed 
+on the fixed 3D-printed apparatus to prevent muscular 
+movement interferences from the wrist and other fingers, and 
+the muscle force was recorded by a load cell (LDST-V-HY, 
+Luckly Inc., Beijing, China) connected to a ring around the 
+thumb. Multiple electrodes were arranged in the form of 8 rows 
+× 8 columns to form a 2-dimensional electrode array. Each 
+electrode probe had a diameter of 2 mm, and the inter-electrode 
+distance between consecutive electrodes was 4 mm. Each 
+electrode was designed in a monopolar manner relative to a 
+round common reference electrode placed on the back of the 
+tested hand.  
+During the experiments, subjects were asked to sit and place 
+the tested hand in a relaxed and comfortable way. Before data 
+collection, the maximum voluntary contraction (MVC) of the 
+thumb abduction muscle was tested and recorded. Then, in each 
+trial of the task performance, subjects were instructed to 
+perform isometric muscle contractions with the muscle force 
+gradually increasing from 0 to a targeted force level (quantified 
+by MVC percentage) in 2s and then maintained at the targeted 
+level for around 3s, as shown in Fig. 1b. According to this force 
+generation pattern, the designed force curve was shown on the 
+screen to facilitate the subject’s task performance in each trial. 
+The targeted force level in this experiment was set to 30% MVC 
+and the trial was repeated at least nine times to acquire a 
+sufficient amount of data. The force and SEMG data were 
+digitized via a 16-bit A/D converter (ADS1198, Texas 
+Instruments, TX) at a sample rate of 2 kHz, and the data were 
+stored into the hard disk of a computer and imported into the 
+MATLAB software (version R2020a, MathWorks, Natick, MA, 
+USA) for further analyses. 
+ 
+2) Data Preprocessing 
+All channels of the recorded HD-SEMG signals were 
+inspected, and a few channels (3.75 ± 1.28 channels across all 
+subjects in this study) with low quality were discarded (due to 
+their excessive noise contamination resulting from motion 
+artifacts, occasional electrode drop, or environmental 
+interferences from surrounding electronic devices). The 
+channel deletion remained consistent within the EMG signals 
+of the same subject. The HD-SEMG signals within the 
+remaining channels were filtered through a 10-order 
+Butterworth band-pass filter to reduce possible low-frequency 
+or high-frequency interference. The bandwidth of the filter was 
+20-500Hz. Finally, the power line interference was removed 
+through a 50Hz second-order notch filter. The deleted channels 
+were not considered in the subsequent process of SEMG 
+ 
+Fig. 1. The experimental setup and protocol. (a) Apparatuses for 
+simultaneously recording thumb abduction force by a load cell and HD-
+SEMG data by a piece of 2-dimensional electrode array arranged in an 8×8 
+formation. (b) The illustration of the force generation pattern with both the 
+designed force curve (blue line) and an actual recorded force curve (red 
+line) in one trial of task performance. 
+
+Force:C8
+30%MVC
+C1C57
+C642
+5Time(s)(b)
+(a) 
+decomposition, but they were filled in by interpolation from 
+neighboring channels and considered during the estimation of 
+MUAP waveforms. In order to facilitate the data analysis, all of 
+the SEMG data were divided into a series of non-overlapping 
+data segments corresponding to the force generation task 
+repetitions over time. Therefore, the length of every SEMG data 
+segment was around 5 seconds. 
+ 
+B. SEMG Data Simulation 
+A data simulation approach was conducted to generate HD-
+SEMG data with known MU activities, which were used as the 
+ground-truth for validating the performance of the developed 
+online SEMG decomposition method. In the current study, this 
+approach was based on simulation models well described by 
+previous studies, including the motoneuron pool model [54], 
+the model describing the MUAP waveforms of different MUs, 
+and a tripole model [55] considering the generation and 
+extinction of the action potentials at the fiber end-plate and 
+tendon. 
+Here a cylindrical muscle with a radius of 8 mm was 
+simulated and the fat and skin layers of the muscle were set to 
+2.5 mm thickness. 120 MUs were set and distributed in parallel 
+in the muscle fibers. Most of the MUs had low recruitment 
+thresholds and a few had high thresholds. When the excitation 
+exceeded the threshold, every MU discharged at 8 Hz and its 
+firing rate increased as the excitation increased. All the relevant 
+parameters are listed in Table I. 
+The simulated SEMG signals were also set to be recorded by 
+a 64-channel surface electrode array arranged in an 8×8 grid 
+form. The inter-electrode distance was set at 4 mm for both 
+horizontal and vertical directions. The electrode array was 
+placed parallel to the muscle fiber direction and its center 
+electrodes were set to approximately over the innervation zones.  
+To be consistent with the force generation pattern of the 
+actual experiments, the excitation was set to increase from 0 to 
+a specific excitation level in the first 2 seconds, and maintained 
+for another 3 seconds with several repetitions. The maximum 
+excitation level was set to be 3%, corresponding to 33 active 
+MUs. In addition, zero-mean Gaussian noises were added to the 
+simulated EMG signals, generating three levels of SNR (signal-
+to-noise ratio) at 10 dB, 20 dB and 30 dB, respectively. Thus, 
+we considered four noise levels, three SNR levels and the level 
+without any additional noise. For each noise level, 21 
+repetitions were simulated to ensure data diversity, as shown in 
+Fig. 2. Therefore, 84 data segments (4 noise levels × 21 
+repetitions) were simulated in total.  
+C. Online Decomposition 
+The overall whole block diagram summarizing the proposed 
+online decomposition method is described in Fig. 3. 
+TABLE I 
+PARAMETERS FOR SEMG SIMULATION 
+ 
+Distribution 
+Mean 
+SD 
+Range 
+Fiber number 
+Uniform 
+70000 
+ 
+±0.5 mean 
+MU fiber endplate 
+center position 
+Uniform 
+0 
+ 
+±8 mm 
+Fiber endplate 
+position variation 
+Uniform 
+0 
+ 
+±2 mm 
+Half fiber length 
+Gaussian 
+40mm 
+4mm 
+±2 SD 
+Mean fiber 
+diameter for a MU 
+Gaussian 
+55μm 
+10μm 
+±2 SD 
+Fiber diameter 
+variation within a 
+MU 
+Gaussian 
+0 
+1μm 
+±2 SD 
+ISI variation 
+Gaussian 
+0 
+0.2*instant 
+mean ISI 
+±2 SD 
+ 
+ 
+Fig. 2. (a). The contraction condition of simulated signals. (b). Multi-
+channel simulated SEMG signals. 
+ 
+ 
+Fig. 3.  Block diagram of the proposed method for online SEMG decomposition 
+ 
+
+Online
+Divided
+extraction
+extraction
+data input
+Separation
+1vectors
+4
+Whitening
+calculation
+Vector set
+1
+and
+Offlin
+MUST
+MUST(a)
+Maximum
+excitationExtending
+Offline PFP
+Φ=
+W1, W2 -.. Wn?
+data
+connection
+indno
+decomposition
+-0 2
+5
+5s
+100s(b)
+ChannelMUST
+--
+Preprocessing
+Offline prework
+Online Decomposition
+extraction#1#64
+Data
+#1
+#2
+#3
+#20
+#21
+SegmentTime window
+Peak 
+With full consideration of the real-time usability of the 
+proposed online method, a two-stage approach was designed to 
+avoid considerable computational complexity caused by the 
+repeated operation of the FastICA algorithm and the iterations 
+of the constrained FastICA algorithm. More specifically, the 
+reliable separation vectors were initialized in the offline 
+prework stage and saved to accelerate the subsequent online 
+data processing. In the online decomposition stage, the data 
+stream of the input EMG signals was divided into a series of 
+temporally overlapping windows with window length and 
+increment set at 1 s and 0.2 s, respectively. Both settings helped 
+to facilitate online processing. 
+During the offline prework stage, several 5-s segments of 
+EMG signals were separately decomposed offline using the 
+APFP method and all of the resultant separation vectors were 
+put into the set �. The quality of these vectors was evaluated by 
+both criteria employed in the offline APFP method [33]: if the 
+coefficient of variation of spike amplitudes ������ was higher 
+than 0.3, and the coefficient of variation of inter-spike intervals 
+������ was higher than 0.4, the corresponding separation vector 
+was considered to be low-quality and it was removed from the 
+set � . Furthermore, any duplicated separation vector 
+corresponding to the same MU was removed as well. 
+In the online decomposition stage, every 0.2 s of data input 
+was combined with 0.8 s of historical data to form a 1-s window 
+for decomposition. The decomposed results from consecutive 
+windows were connected, while their overlapping portion was 
+used to align the obtained MUSTs. This ensured continuity of 
+ 
+Fig. 4.  Illustration of the online SEMG decomposition process using the proposed method. 
+ 
+
+#14#15
+0
+5
+10
+15
+20Channel
+Channel
+#1
+#1Time(s)#64
+#64Spike extraction & ConnectionWindow sliding
+ vectors
+X
+山 = {W1.W? ... WNMUST
+ Experimental Muscle Force
+MU
+DAWDecompose sEMG signals
+window by window#1
+30%Window
+Source signal of MU1
+Source signal of MU2开2DecomposeOffline Prework
+Online Decomposition
+0.2s 
+the decomposition results along with the original SEMG data 
+stream. The SEMG data in each window were first whitened 
+and extended. Then, the multiplication procedure was directly 
+applied to the extended EMG signals with separation vectors in 
+set � to estimate different MU source signals, from which 
+individual MUSTs were consequently identified. 
+For extracting MUSTs from the MU source signals, the 
+original offline APFP method employs repeated iterations of 
+the constrained FastICA algorithm, involving complex 
+computations as described above. This process was unsuitable 
+for online processing and therefore it was removed to avoid 
+heavy computational burden. To maintain high-precision 
+MUST extraction, the simple amplitude-thresholding process 
+by the Otsu algorithm had to be updated. A new algorithm was 
+designed for our online PFP method. First, this algorithm needs 
+to determine an initial threshold that is applied to each source 
+signal, using the Otsu algorithm in the same way as conducted 
+in the offline APFP method. Then, a group of spikes beyond 
+this threshold is detected and the corresponding amplitudes can 
+be ranked from small to large. Next, a series of successively 
+increasing thresholds that are a little higher than these 
+amplitudes are adopted to estimate a series of different spike 
+trains. Each resultant spike train can be further evaluated by 
+both ������ and ������ metrics, and the spike train with the 
+minimal summation of both metrics is finally considered the 
+most appropriate MUST. This algorithm for adaptive threshold 
+selection was termed the successive multi-threshold Otsu 
+algorithm.  
+A k-means clustering algorithm was usually used in some 
+offline decomposition methods [36]-[37] for extracting MUSTs 
+from the source signals. It was also implemented in this study 
+as an alternative threshold selection algorithm, in comparison 
+to the successive multi-threshold Otsu algorithm used in our 
+method. By applying the k-means clustering algorithm, all 
+sample amplitudes of the source signal time series can be 
+classified into 2-4 groups (2 in this work), so that the group with 
+the largest amplitudes of samples is selected as the extracted 
+MUST. 
+After the spike trains of all MUs were appropriately detected, 
+they were connected over windows to form the resultant MUST 
+for each MU, and its MUAP waveforms that spanned over all 
+channels were correspondingly estimated. Fig. 4 illustrates an 
+example of the online decomposition results. The pseudocode 
+of the proposed online decomposition method is shown in 
+Algorithm 1. 
+ 
+D. Performance Evaluation 
+For processing the experimental SEMG data, the proposed 
+online decomposition method was conducted in a user-specific 
+manner. Four segments were used in the offline prework stage 
+and the remaining 4 segments were processed in the online 
+decomposition stage. For processing the simulated SEMG data, 
+the first segment was used in the offline prework stage and the 
+remaining 20 segments were processed in the online 
+decomposition stage. All SEMG segments tested in the online 
+decomposition stage were sequentially arranged in the form of 
+a data stream to be processed continuously using our proposed 
+method. For comparison purposes, all of the SEMG segments 
+to be processed online was also decomposed by the offline 
+APFP method as well. 
+To evaluate the performance of online decomposition and 
+assess the decomposition results more comprehensively, we 
+Algorithm 1 The proposed online decomposition 
+method 
+1: 
+Decompose the SEMG signals offline. Extract 
+the MUSTs and calculate the corresponding 
+separation vectors. 
+2: 
+Remove the duplicated separation vectors and 
+vectors that are not well-decomposed. 
+3: 
+Save all the separation vectors ��, ��, ��…�� 
+for the online decomposition stage. 
+4: 
+while Acquiring SEMG signals do 
+5: 
+       Load and extend the EMG signals (��). 
+6: 
+       for j = 1; j < N + 1; j ++ do 
+7: 
+             Calculate the source signal, �� = ��
+���. 
+8: 
+             Estimate the initial threshold through the 
+Otsu algorithm and extract the spike train. 
+9: 
+             Successively increase the threshold and 
+extract a series of spike trains ����, ����, ����… 
+10: 
+             Find the spike train with the lowest  
+������ and ������ as the  �th MUST ���. 
+11: 
+        end for 
+12: 
+        Connect the MUSTs over the sliding 
+windows. 
+13: 
+end while 
+ 
+Fig. 5. The results for decomposing simulated SEMG data in terms of MR(a), FDR(b) and FNR(c) averaged over all data segments using the offline APFP 
+method, the proposed online PFP method and the online PFP method with k-means clustering at four noise levels, respectively. The error bar represents 
+standard deviations. N in the horizontal axis denotes the condition without any additional noise. 
+
+0.10.05
+1N
+30
+20
+10
+N
+30
+20
+10
+N
+30
+20
+10
+SNR (dB)
+SNR (dB)
+SNR (dB) The offline APFP method
+ The online PFP method with k-means clustering
+ The proposed online PFP method(a)
+(0)
+()
+MR(%)
+FDR
+FNR
+001
+0.250.3
+80
+0.2 
+used a series of metrics: matching rate (MR) can be calculated 
+as [33]: 
+�� = 
+2 ∙ �������
+�������  + ����������
+ 
+(4)
+where ������� denotes the number of firing events of the online 
+decomposition results, and ���������� denotes the number of 
+the reference spike trains. In the simulated data, the reference 
+spike train indicates the ground-truth firing events. However, 
+the actual MUSTs are not known a priori in the experimental 
+data. Therefore, the decomposition results of the experimental 
+data processed by the offline APFP method were used to define 
+���������� . �������  indicates the number of common 
+discharges appearing in both the online decomposition result 
+and the reference. The MR measures the matching degree and 
+it is able to quantify the precision of an online decomposition 
+method.  
+ 
+Fig. 6.  A representative example of validating the decomposition results from the online PFP method in terms of all decomposed MUSTs (in blue) with 
+respect to the reference (in red) derived from summarized offline decomposition results, using a data segment from one subject. The position of the black 
+dot indicates the missing or fault discharges and MR values are computed and shown on the right side of these spike trains. 
+ 
+Fig. 7. Two MUAPs of matched MUs with time-varying waveform shapes. Here we illustrate 64 electrode channels arranged in an 8×8 grid form. Blue and 
+red lines indicate the MUAP shapes from online PFP and the reference of offline decomposition, respectively. 
+ 
+ 
+Fig. 8. The relationship between the matching rate and the composite 
+decomposability index. 
+ 
+
+T Online PFP
+10
+MR (%)111194.87111 Online PFP
+600μv
+The reference (Offline decomposition
+30msnumb
+10098.95MU1
+MU2
+1
+2
+3
+4
+5
+6
+7
+8
+1
+2
+3
+4
+5
+6
+7
+81002
+23
+3
+4
+45
+5
+67
+8
+80.95
+Matching Rate
+0.9
+0.85
+0.8
+0
+10
+20
+30
+40
+50
+Composite Decomposability Index0
+1
+2
+4
+5Time(s) 
+Besides MR, both false negative rate (FNR) and false 
+discovery rate (FDR) were used to reveal the cause of the error 
+discharges. They are defined as 
+                     ��� = ���������� − �������
+����������
+ 
+ 
+(5) 
+               ��� = ������� −  �������
+�������
+ 
+ 
+ 
+They count the proportion of the number of unmatched 
+discharges to the total number of their respective discharges. 
+Specifically, the FNR measures the rate of “missing” discharges 
+with respect to the reference, and the FDR quantifies the rate of 
+“faulty” discharges appearing in the online decomposition 
+results. Generally speaking, the MR of a reliable MUST is close 
+to 1 but the FNR and FDR are close to 0. 
+For a more comprehensive view of the decomposition results, 
+we also calculated the mean discharge rate (MDR) and the 
+coefficient of variation (CoV) of the online identified MUSTs 
+with respect to the reference spike trains. It should be noted that 
+the CoV refers to the coefficient of variation of the inter-spike 
+intervals ������ to better understand the MU firing behaviors. 
+In addition, we calculated the decomposability index (DI) for 
+each common MU of experimental EMG data to precisely 
+quantify the proposed method’s performance [56]: 
+ �� = min {‖���‖, ‖��� − ��∗�‖}
+��
+���
+ 
+ 
+(6)
+ 
+where ��� is the MUAP of the �th MU in the �th channel and 
+��∗�  is the MUAP most similar to ���  among the other 
+MUAPs in the � th channel. ��
+���  is the root mean square 
+amplitude (RMS) of the � th channel and the operator ‖∙‖ 
+denotes the Euclidean norm. The DI measures the separation 
+between ��� and the template of MUAP nearest to it (or the 
+baseline), normalized by the standard deviation of the noise 
+component (interference plus baseline noise) projected along 
+their vector difference. The overall decomposability of the �th 
+MU was measured by the composite DI (CDI), defined as the 
+norm of the individual DIs [56].  
+For developing a real-time decomposition method, it is 
+necessary to evaluate the processing time delay which is 
+expected to be as short as possible. The time delay for 
+processing one single time window was recorded, and all these 
+time delay values were averaged across all windows and all 
+subjects to indicate the computational complexity. All of the 
+algorithms were implemented on a desktop computer with an 
+Intel Core i5-10400 processor (2.90 GHz) and 16 GB of 
+memory.  
+IV. RESULTS 
+A. Results of Simulated Data 
+As an offline decomposition method for validation, 21 MUs 
+were identified using offline APFP and the number was 22 
+using online PFP when no additional noise was added. Further, 
+the number of MUs correctly decomposed using online PFP 
+decreased to 11, 7, and 6 when noise was added at 30 dB, 20 dB 
+and 10 dB SNR, respectively.  
+The results for decomposing simulated SEMG data are 
+reported in Fig. 5. As compared with the offline APFP method, 
+the proposed online PFP method achieved comparable 
+performance in terms of a high MR over 90%, and a low FNR 
+below 0.05. The proposed online PFP method had a fluctuated 
+and relatively higher FDR than the offline APFP method under 
+three SNR levels. Specifically, a decreasing trend of the MR 
+was found from 99.29% to 94.13% for the offline APFP method 
+and from 98.53% to 92.79% for the online PFP method, 
+respectively, when the noise was successively added to generate 
+four noise levels. The ANOVAs revealed no significant 
+difference in either MR, FDR or FNR, between the offline 
+APFP method and the proposed online PFP method (p > 0.05).  
+When both threshold selection algorithms were compared, it 
+was evidently found that the successive multi-threshold Otsu 
+algorithm in the proposed online PFP method significantly 
+outperformed the K-means clustering algorithm in terms of 
+higher MR (p = 0.025) and lower FNR (p =0.022). Both 
+algorithms did not exhibit a significant difference in the FDR 
+metrics (p = 0.273).  
+Table II reports both MDR and CoV values calculated for all 
+common MUs between the decomposition results achieved by 
+the proposed online method and the ground truth. The ANOVA 
+revealed no difference in MDR (p = 0.217) or CoV (p = 0.105) 
+at no presence of noise. However, the MDR and CoV of online 
+decomposition results became significantly different from those 
+of the ground-truth (p < 0.05) when the noises were added. 
+B. Results of Experimental Data 
+When implementing online decomposition of experimental 
+data, the offline decomposition method was applied to establish 
+the reference for validation, and 10.31±1.79 MUs were 
+obtained, averaged across all subjects. 
+Fig. 6 is an example of an online decomposition result using 
+the proposed method, showing the decomposed MUSTs with 
+respect to the reference. It can be observed that almost all the 
+MU discharges derived from the online PFP method are well 
+matched with those in the reference, with sporadic missing or 
+erroneous ones. Fig. 7 illustrates the MUAP waveforms of two 
+matched MUs derived from both the online PFP method and the 
+reference, which demonstrate a very consistent waveform shape 
+in each channel and almost the same distribution pattern across 
+the electrode array. Fig. 8 plots the relationship between the 
+matching rate and composite decomposability index (CDI), 
+which displays the overall trend of the matching rates varying 
+TABLE II 
+COMPARISON OF MDR AND COV OF THE SIMULATED EMG SIGNALS 
+ 
+SNR 10dB 
+Online PFP/  
+Ground-truth 
+SNR 10dB 
+Online PFP/ 
+Ground-truth 
+SNR 30dB 
+Online PFP/ 
+Ground-truth  
+No adding noise 
+Online PFP/ 
+Ground-truth 
+MDR 
+9.86±1.99 
+8.77±0.18 
+9.55±1.54 
+8.75±0.23 
+10.47±1.81 
+8.74±0.22 
+8.77±0.51 
+8.70±0.18 
+CoV 
+0.245±0.053 
+0.199±0.003 
+0.257±0.032 
+0.202±0.005 
+0.231±0.044 
+0.201±0.005 
+0.211±0.024 
+0.199±0.007 
+ 
+
+ 
+with the CDIs. It contains the common MUs of all of the 
+collected SEMG segments.  
+Table III reports both the number of MUs decomposed by the 
+online PFP method and the number of common MUs matched 
+those in the reference (offline decomposition) for 8 subjects, 
+respectively. An average of 12.00±3.46 MUs were successfully 
+identified by the online PFP method, with an average of 
+6.69±1.84 MUs correctly matched. Besides, three metrics are 
+also computed from those common MUs and reported in Table 
+III. Averaged over all data segments to be decomposed and all 
+subjects, the MR was (90.38±2.80) %, the FDR was 
+0.091±0.022, and the FNR was 0.089±0.041. The estimated 
+MDR (p = 0.872) and CoV (p = 0.503) of online decomposition 
+results were not significantly different from the offline 
+decomposition reference. 
+ 
+C. Time Delay 
+The time delay for decomposing a 1-s window of SEMG data 
+using the proposed method in the online decomposition stage 
+was 0.084±0.028 s, averaged over all data segments and all 
+subjects; it was less than a 0.2-s time increment. For 
+comparison purposes, the offline APFP method costs 60.07 ± 
+9.82 s to decompose SEMG data in a single time window, much 
+longer than that of the proposed online decomposition method.   
+V. DISCUSSION 
+As a promising SEMG decomposition method, the PFP 
+algorithm has been reported recently and, therefore, it is 
+necessary and promising to develop an online version. This 
+study sought to propose an online SEMG decomposition 
+method based on the PFP algorithm. The results of both 
+simulated and experimental SEMG data analyses demonstrated 
+the feasibility of the proposed online PFP method in 
+decomposing a large number of MUs with high precision in the 
+context of isometric muscle contractions. Our study offers a 
+valuable tool for online SEMG decomposition with great 
+applications in biomechanics and rehabilitation. 
+In the results of processing simulated data, the proposed 
+online PFP method decomposed a similar number of MUs as 
+the offline APFP method, illustrating comparable performance. 
+Due to the use of initial separation vectors provided by the 
+APFP method in the offline prework stage, the proposed online 
+PFP method is expected to inherit a good capability of 
+decomposing a great number of MUs from its original offline 
+version. In terms of MR, the proposed online PFP method got a 
+slightly lower value compared with the offline APFP method. 
+This can be explained by the fact that the source signals were 
+calculated by directly multiplying previously initialized 
+separation vectors with the SEMG signals for the purpose of 
+real-time processing. In addition, the MUSTs were estimated 
+without the examination of iterative constrained FastICA, thus 
+increasing the negative influence of noise. The result 
+demonstrates that online decomposition was speeded up at the 
+cost of a little bit of decrease in precision. This is the main and 
+common difficulty in generalizing an offline decomposition 
+method to its online version [38]-[43]. However, it has been 
+found that the MDR and CoV of online decomposition were 
+significantly different from those of the ground-truth when the 
+noise was added. This can be partly explained by the limitations 
+of 
+the 
+online 
+decomposition 
+method 
+such 
+as 
+MU 
+synchronization [26] and firing events drift [33] that previous 
+studies have faced. 
+When some noises were successively added to EMG signals 
+to be decomposed, both the number of correctly identified MUs 
+and the precision of determining their firing timings were 
+reported to decrease substantially. This could partly explain that 
+the decrease of SNR resulted in more serious noise interference 
+to some small MUs and thus caused a negative influence on the 
+calculation of separation vectors as well as the performance of 
+the online decomposition method. On the other hand, it became 
+much harder to precisely extract MUSTs from source signals in 
+the online decomposition stage at a low SNR level, reflecting 
+the decline of the MR. As a consequence, it can be inferred that 
+the quality of SEMG signals significantly influenced the 
+performance of the decomposition method, as reported in [33], 
+[40]. 
+It is worth mentioning that the proposed online PFP method 
+introduced a progressive multi-thresholding process for 
+extracting MUSTs. The successive multi-threshold Otsu 
+TABLE III 
+SUMMARY OF DECOMPOSITION RESULTS FOR EXPERIMENTAL EMG SIGNALS. 
+Subject 
+Number of motor units 
+ 
+MDR (Hz) 
+ 
+CoV (%) 
+MR (%) 
+FDR 
+FNR 
+The 
+reference 
+Online PFP  
+ 
+The 
+reference 
+Online 
+PFP  
+ 
+The 
+reference 
+Online 
+PFP  
+All 
+Matched  
+1 
+12.75±1.50 
+19 
+9.00±1.41  
+19.76±5.08 19.71±4.43  
+22.92±7.71 
+24.31±8.53 
+92.06±5.91 
+0.084±0.082 
+0.051±0.057 
+2 
+9.50±1.29 
+8 
+4.50±0.58  
+22.00±4.76 20.63±4.00  
+27.44±6.59 
+22.46±5.95 
+89.92±7.21 
+0.093±0.075 
+0.106±0.086 
+3 
+9.00±0.82 
+14 
+6.00±0.81  
+15.79±3.22 14.65±3.30  
+23.44±3.78 
+25.44±4.59 
+93.20±6.02 
+0.065±0.068 
+0.067±0.074 
+4 
+11.00±1.41 
+13 
+8.75±1.71  
+20.15±3.93 21.28±3.84  
+24.08±6.98 
+24.67±6.25 
+91.17±3.35 
+0.076±0.033 
+0.056±0.029 
+5 
+8.50±0.57 
+9 
+5.50±0.58  
+20.29±3.99 20.62±3.00  
+26.09±4.19 
+28.48±4.70 
+85.18±4.04 
+0.116±0.051 
+0.175±0.068 
+6 
+9.50±1.29 
+10 
+6.25±1.71  
+20.35±4.25 19.67±4.30  
+23.87±3.05 
+24.18±3.73 
+91.51±6.45 
+0.084±0.071 
+0.082±0.076 
+7 
+11.75±1.71 
+11 
+7.00±0.82  
+23.03±3.60 24.66±3.94  
+24.46±3.54 
+24.82±2.78 
+87.26±5.47 
+0.131±0.073 
+0.108±0.028 
+8 
+10.50±1.29 
+12 
+6.50±1.73  
+18.57±2.72 18.73±1.86  
+18.74±2.96 
+19.41±1.66 
+92.70±4.26 
+0.080±0.058 
+0.064±0.040 
+Average 10.31±1.79 12.00±3.46 
+6.69±1.84  
+19.99±2.18 19.99±2.79  
+23.88±2.55 
+24.22±2.56 
+90.38±2.80 
+0.091±0.022 
+0.089±0.041 
+ 
+
+ 
+algorithm outperformed the conventional k-means clustering 
+algorithm especially in the condition of noise interference, 
+proving the potential to extract more precise discharges at low 
+SNR levels. The successive multi-threshold algorithm based on 
+the Otsu algorithm was inspired from the common Otsu 
+algorithm [48] used in the offline APFP method [33]. It was 
+able to successively increase multiple thresholds to overcome 
+the effect of noise interferences and find the most appropriate 
+one to extract MUSTs that followed the physiological 
+properties of MUs. The successive multi-threshold Otsu 
+algorithm takes consideration into the interval and waveform 
+information to ensure the result to be much more reliable, 
+depending on ������  and ������ . By contrast, the k-means 
+clustering algorithm only focuses on the amplitude information 
+of EMG source signals. As a result, it makes it much more 
+difficult to remove the noise interferences and leads to 
+decomposition performance degradation. The proposed online 
+PFP method replaced the complex iterative calculation of 
+constrained FastICA with the successive multi-threshold Otsu 
+algorithm 
+to 
+extract 
+MUSTs, 
+showing 
+a 
+significant 
+improvement in reducing the calculation complexity while 
+maintaining its high precision. 
+To evaluate the real-time performance, this study recorded 
+the processing time of online decomposition. The time delay 
+was effectively reduced from 60 seconds for the offline APFP 
+method to less than 0.08 seconds for the online decomposition. 
+The acceleration of data processing is attributed to reasons in 
+two respects. The first is that the repeated iteration of FastICA 
+was put in the offline prework stage, which initialized the 
+separation vectors for online decomposition. On the other hand, 
+some complex calculation procedures were adaptively 
+simplified. For example, the constrained FastICA algorithm in 
+the APFP method was replaced with the successive multi-
+threshold algorithm, as discussed above. 
+In the experimental SEMG data, a large number of MUs 
+decomposed by offline PFP can be correctly identified with 
+high precision in the online decomposition process, 
+demonstrating that the separation vectors used in the online 
+decomposition process were comprehensive and precise. In 
+addition, the MDR and CoV of online decomposition showed 
+no significant difference with the offline reference. These 
+findings indicate that the performance of the online 
+decomposition method is very close to that of the original 
+offline method, proving the feasibility and effectiveness of the 
+proposed online PFP method. In addition, it illustrates that the 
+advantages of the offline APFP method were still maintained in 
+the proposed online decomposition method. 
+There are still some limitations in this work. First, the online 
+decomposition process relied too much on the separation 
+vectors provided by the offline prework, proving the feasibility 
+that the separation vectors obtained from offline decomposition 
+can be used for online decomposition. However, the conditions 
+of muscle contraction change over time and the initialization 
+process needs to update the separation vectors, which has not 
+been validated in this work. In other words, the online process 
+was verifying whether the MUs corresponding to the separate 
+vectors were activated and the newly recruited MUs couldn’t be 
+captured. Moreover, the initial MU information and spike drift 
+needs to be corrected over time. Second, the experimental EMG 
+data were collected only from isometric contraction and most 
+muscle contractions in daily life are non-isometric and dynamic. 
+More contraction patterns will be added to the experimental 
+data for analysis. Third, the peel-off procedure needs to be 
+adopted in a real-time way to find more MUs and fully take 
+advantage of the offline PFP method. Further research will be 
+devoted to overcoming the limitations above. 
+VI. CONCLUSION 
+A new online SEMG decomposition method based on the 
+Progressive FastICA Peel-off procedure was proposed in this 
+paper, including offline prework and online decomposition 
+process. The proposed decomposition method took advantage 
+of offline PFP algorithms and demonstrated high precision with 
+the most identified MUs both on simulated and experimental 
+EMG signals. These results offer a new tool for precisely 
+identifying individual MU activities in a real-time way with the 
+potential applications of high-density EMG as a neural interface 
+in the fields of biomechanics, sports and rehabilitation. 
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+(1+1) dimensional scalar field theory on q-deformed
+space
+Poula Tadros
+Department of Applied Physics, Aalto University School of Science, FI-00076
+Aalto, Finland.
+email:poulatadros9@gmail.com
+Abstract
+We study scalar field theory in one space and one time dimensions on
+a q-deformed space with static background. We write the Lagrangian and
+the equation of motion and solve it to the first order in q − 1 where q is
+the deformation parameter of the space.
+1
+Introduction
+Non-commutative geometry was first introduced in string theory in ref-
+erence [1], where it was shown that the coordinates of the endpoints of
+strings on D-branes in the presence of a Neveu-Schwartz field are non-
+commutative. Non-commutative field theories have also been defined, as
+they can be derived from string theories and have interesting features, as
+described in references [2] and [3].
+The introduction of non-commutative spacetime in field theory is mo-
+tivated by the Heisenberg uncertainty principle in quantum mechanics,
+which states that at small distance scales, there is a large uncertainty in
+momentum measurement. This means that energy can reach very high
+values in a small spatial distance, approaching the Planck scale. However,
+according to the general theory of relativity, high energy in a small spatial
+distance creates a black hole, which prevents the position from being fully
+certain. To reconcile these two phenomena, it is necessary to introduce
+non-commutativity in spacetime, which implies non locality in the theory.
+This is explained in references [4] and [5].
+In this paper we study (1+1) dimensional classical scalar field theory
+with static spacetime on a q-deformed space, we present both analytical
+and numerical analysis of the resulting theory. In section 2, we review
+the some types of non-commutativity on space times and motivate the
+choice of q-deformation non-commutativity as the subject of the study .
+In section 3. we study the scalar field theory on q-deformed space time, we
+write the Lagrangian and deduce the equation of motion, we also truncate
+the equation of motion to the linear order in q −1 and solved the resulting
+equation. In section 4, we study the numerical solutions of the truncated
+equation of motion showing that the solutions grow exponentially with x
+and t meaning that the equation is stiff and there are instabilities in the
+theory. In section 5, we conclude the study and suggest topics for further
+research.
+1
+arXiv:2301.03106v1  [hep-th]  8 Jan 2023
+
+2
+Types of non-commutativity
+Here, we briefly review three of the most popular types of non-commutativity
+relations and justify our motivation to use the q-deformation type
+1. Canonical non-commutativity It is the simplest type which used in
+physics literature, it was introduced in [6], it is defined by imposing
+the following commutation relations
+[xµ, xν] = iθµν,
+where xµ are the spacetime coordinates and θµν is a constant, anti-
+symmetric matrix.
+The idea of canonical non-commutativity involves smearing the struc-
+ture of space-time in a particular way, regardless of the specific
+mathematical details of the space.
+In order to incorporate non-
+commutative geometry capturing the mathematical structures on
+the manifold, it is necessary to consider more complex forms of non-
+commutativity beyond just this basic version.
+2. Lie-type non commutativity
+In this case the coordinates has a Lie algebra structure i.e.
+the
+commutation relations can capture a Lie algebra structures [7]. The
+commutation relations are given by
+[xµ, xν] = if µν
+ρ xρ,
+where f µν
+ρ
+are the structure constants of the defined Lie algebra.
+However, this type is not useful because Lie structures are rigid i.e.
+any small deformation of a Lie algebra is isomorphic to the original
+Lie algebra.
+3. q-deformations
+A solution to the rigidity problem for Lie algebras is to replace Lie
+group with a flexible structure called quantum groups [8-10]. The
+term quantum group used in this context refers to the deformations
+of the universal enveloping algebra of a given group, these objects
+have Hopf algebra structures which are flexible structures unlike Lie
+groups and algebras.
+The commutation relations are given by
+xµxν = 1
+q Rµν
+στxσxτ,
+where q is a parameter and Rµν
+στ is the R-matrix of the quantum
+group defined on the space.
+In this space a Lie algebra is replaced by a non-commutative Hopf
+algebra with deformation parameter q. The resulting space is de-
+formed according to the Lie group on the space and on the parame-
+ter q, this is the simplest way to deform a space time while capturing
+the full algebraic structure of the space.
+2
+
+3
+Lagrangian and the equation of motion
+We begin with the Lagrangian of the scalar field on the commutative man-
+ifold then introduce non-commutativity by replacing the derivatives by
+Jackson derivatives, since the symmetry group is U(1), the deformations
+of its universal enveloping algebra gives a commutative algebra. Thus, we
+do not have to worry about defining a product of functions on the new
+space. The Lagrangian is then
+L = 1
+2∂µφ∂µφ − 1
+2m2φ2 → Lq = DµqφDµ
+q φ − m2φ2,
+where µ = 0, 1 with x0 = t and x1 = x.
+Assuming the field is defined everywhere and is infinitely differentiable
+and the deformations are small i.e. q ≈ 1, we can relate the theory on
+the non commutative topological space to the theory on the commuta-
+tive manifold (i.e. transforming the non-commutative theory back to the
+commutative manifold) using the formulae
+Dxq(f(x)) = ∂xf +
+∞
+�
+k=1
+(q − 1)k
+(k + 1)! xkf (k+1)(x),
+where f (k) is the k-th ordinary derivative of f with respect to x.
+Dtq(f(x)) = ∂tf +
+∞
+�
+k=1
+(q − 1)k
+(k + 1)! xkf (k+1)(x),
+where f [k] is the k-th ordinary derivative of f with respect to t.
+The resulting Lagrangian on the commutative manifold is
+Lq = 1
+2∂φ∂φ − 1
+2m2φ2 + 2∂φ
+∞
+�
+k=1
+(q − 1)k
+(k + 1)! xkφ(k+1)
++
+∞
+�
+l,m=1
+(q − 1)(l+m)
+(m + 1)!(l + 1)!xk+lφ(l+1)φ(m+1) + (x → t).
+where (x → t) means the same terms but with x replaced by t includ-
+ing in the derivatives.
+The Lagrangian has an infinite series of derivatives, in this case the
+Euler-Lagrange equation will be
+∂Lq
+∂φ +
+∞
+�
+k=1
+(−1)k dk
+dxk ( ∂Lq
+∂φ(k) ) +
+∞
+�
+k=1
+(−1)k dk
+dtk ( ∂Lq
+∂φ[k] ) = 0,
+(1)
+where k = 2, 3, ....
+The Lagrangian is clearly non local as expected from a non-commutative
+theory.
+The derivatives of the Lagrangian are given by
+∂Lq
+∂φ = −mφ,
+3
+
+∂Lq
+∂(∂φ) = ∂xφ + 2
+∞
+�
+n=1
+(q − 1)n
+(n + 1)! xnφ(n+1)
+(2)
+→ d
+dx( ∂Lq
+∂(∂xφ))
+= ∂x∂xφ + 2
+∞
+�
+n=1
+(n(q − 1)n
+(n + 1)! xn−1φ(n+1)) + 2
+∞
+�
+n=1
+((q − 1)n
+(n + 1)! xnφ(n+2)), (3)
+∂Lq
+∂(φ(k)) = 2(q − 1)k−1xk−1
+k!
+∞
+�
+n=0
+(q − 1)n
+(n + 1)! xnφ(n+1)
+→ d
+dx(
+∂Lq
+∂(φ(k)))
+= 2(q − 1)k−1x2k−1
+k!
+∞
+�
+m,n=0
+�
+k
+m
+�
+(q − 1)n
+(n + 1)!
+(n + k + 1)!
+(n + 2k − m − 1)!xn−mφ(n+k+1),
+(4)
+with similar formulae for derivatives with respect to t.
+Putting all together from (2), (3), (4) in (1) we get
+−∂µ∂µφ − m2φ − 2
+∞
+�
+n=1
+n(q − 1)n
+(n + 1)! xn−1φ(n+1) − 2
+∞
+�
+n=1
+(q − 1)n
+(n + 1)! xnφ(n+2)
++
+∞
+�
+k=2
+(−1)k 2(q − 1)k−1x2k−1
+k!
+∞
+�
+n=0
+k
+�
+m=0
+�
+k
+m
+�
+(q − 1)n(n + k − 1)!
+(n + 1)!(n + 2k − m − 1)!xn−mφ(n+k+1)
++ (x → t) = 0.
+(5)
+This is a partial differential equation of infinite order with variable
+coefficients.
+If we consider only small deformations i.e. q ≈ 1, then we can only
+keep terms up to the linear order in q − 1, the first order equation will be
+−∂µ∂µφ−m2φ−(q−1)[φ(2)+xφ(3)− x3
+6 φ(3)−x2φ(3)−xφ3+(x → t)] = 0.
+This equation is a stiff equation i.e. it is numerically unstable, this
+may indicate an instability in the theory due to the linear approximation
+used, but as seen from the full equation of motion the full theory is stable.
+The solution is φ = F(t)G(x) where
+F(t) = c1eiAt/√q+c2e−iAt/√q+(q−1)eiAt/√q
+2iA√q [ iA
+24q t4+(iA3
+3 −
+1
+12√q − i
+8A)t3
++(A + q
+2q
+− A√q
+2
+−
+i
+8A)t2 + (i(A + q)
+2A√q
+− iqA
+4
++
+√q
+8A2 )t
+4
+
++(A + q
+4A2
++ q
+√
+A
+4
++
+iq
+16A3 )] + O((q − 1)2),
+G(x) = c3eikx/√q+c4e−ikx/√q+(q−1)eikx/√q
+2ik√q [ ik
+24q x4+(ik3
+3 −
+1
+12√q − i
+8A)x3
++(k + q
+2q
+− k√q
+2
+− i
+8k )x2 + (i(k + q)
+2k√q
+− iqk
+4
++
+√q
+8k2 )x
++(k + q
+4k2
++ q
+√
+k
+4
++
+iq
+16k3 )] + O((q − 1)2),
+where c1, c2, c3, c4, A are normalisation constants and k = ±
+√
+A2 + m2.
+When q = 1, it reduces to the solution to the Klein Gordon equation as
+expected.
+4
+Numerical results
+Here, we present numerical solutions to the equation of motion to the
+first order in q − 1, we focus on G(x) only since the remaining part is
+similar. The solutions are exponentially growing in time establishing that
+the equation of motion was stiff.
+We set c3 = c4 = k = 1, A2 = 1
+2 and we plot the solution for different
+values of the parameter q.
+Figure 1: At q − 1 = 0.1 the solution grows exponentially with |x|. This is a
+feature of a stiff equation with unstable numerical solution. In the vicinity of
+x = 0 it is close to the usual Klein-Gordon solution but as we go further it
+becomes more and more distant
+5
+
+200000
+100000
+0
+100000
+-200000
+100
+75
+50
+25
+25
+50
+75
+100Figure 2: At q − 1 = 0.001 the solution still grows exponentially but slower.
+Figure 3: At q − 1 = 10−6 the solution resembles the Klein-Gordon solution up
+to |x| = 50 then decays for a bit but eventually blows up.
+Figure 4: At q − 1 = 10−9 the solution has the same behaviour as the previous
+graph but the decay happens at larger |x|, all smaller q − 1 values follow this
+pattern.
+6
+
+2000
+1500 
+1000 -
+500
+0
+500 
+1000
+1500
+2000
+100
+7550
+-25
+0
+25
+50
+75
+10030
+20 
+10
+0
+10
+20
+30
+-200
+150
+100
+50
+0
+50
+100
+150
+200E
+2
+1
+0
+-1
+-2
+-3
+600
+400
+200
+0
+200
+400
+600The above results shows an instability in the theory leading to di-
+vergent solutions to the equations of motion as x → ∞. To remove the
+instability we must add infinite terms corresponding to an infinite series
+of higher derivatives i.e. we have to consider the full theory. However,
+this approximation gives us an intuition on how the q-deformation affects
+the space, small q-deformations beside leading to non local effects appear
+to affect the space irregularly with only small effects locally.
+5
+Conclusion and outlook
+In conclusion, we showed that defining a field theory on a q-deformed
+space leads to an infinite series of higher derivatives in the Lagrangian
+even with static background. In the case presented the algebra was com-
+mutative so no new product of functions is needed. We also demonstrated
+that any approximation or truncation to the theory will lead to stiff equa-
+tions of motion resulting from instabilities in the theory.
+While we made a progress in the field, much more is to be studied, fu-
+ture research in this direction should focus on defining more complicated
+theories on q-deformed spaces with non-commutative function algebras
+and with dynamical spacetimes, also to define higher spin fields on such
+space and study the new symmetries of the theories as well as the types
+of instabilities arise if the Lagrangian is truncated.
+Acknowledgments
+We would like to thank Dr.Ivan Kolar for the useful discussions on the
+topic
+References
+[1] Seiberg, N. and Witten, E. (1999) “String theory and noncommu-
+tative geometry,” Journal of High Energy Physics, 1999(09), pp.
+032–032.
+[2] Szabo, R. (2003) “Quantum field theory on noncommutative spaces,”
+Physics Reports, 378(4), pp. 207–299.
+[3] Sheikh-Jabbari, M.M. (1999) “Super Yang-Mills theory on noncom-
+mutative torus from open strings interactions,” Physics Letters B,
+450(1-3), pp. 119–125.
+[4] Doplicher, S., Fredenhagen, K. and Roberts, J.E. (1995) “The quan-
+tum structure of spacetime at the Planck scale and Quantum Fields,”
+Communications in Mathematical Physics, 172(1), pp. 187–220.
+[5] Ahluwalia, D.V. (1994) “Quantum measurement, gravitation, and
+locality,” Physics Letters B, 339(4), pp. 301–303.
+[6] C. S. Chu and P. M. Ho, Noncommutative open string and D-brane,
+Nucl. Phys. B 550, 151 (1999) [hep-th/9812219].
+7
+
+[7] B. Jurco, S. Schraml, P. Schupp and J. Wess, Enveloping alge-
+bra valued gauge transformations for non-Abelian gauge groups on
+non-commutative spaces, Eur.
+Phys.
+J. C17, 521 (2000) [hep-
+th/0006246].
+[8] Chaichian, M. and Demichev, A.P. Introduction to quantum groups.
+Singapore: World Scientific (1996).
+[9] Bonneau, P. et al. (2004) “Quantum groups and deformation quanti-
+zation: Explicit approaches and implicit aspects,” Journal of Math-
+ematical Physics, 45(10), pp. 3703–3741.
+[10] A. Klimyk and K. Schmudgen, Quantum Groups and Their Repre-
+sentations, Springer (1997).
+8
+

B9E1T4oBgHgl3EQfVwR_/content/tmp_files/load_file.txt

@@ -0,0 +1,157 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf,len=156
+page_content='(1+1) dimensional scalar field theory on q-deformed  space  Poula Tadros  Department of Applied Physics, Aalto University School of Science, FI-00076  Aalto, Finland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  email:poulatadros9@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='com  Abstract  We study scalar field theory in one space and one time dimensions on  a q-deformed space with static background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' We write the Lagrangian and  the equation of motion and solve it to the first order in q − 1 where q is  the deformation parameter of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  1  Introduction  Non-commutative geometry was first introduced in string theory in ref-  erence [1], where it was shown that the coordinates of the endpoints of  strings on D-branes in the presence of a Neveu-Schwartz field are non-  commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' Non-commutative field theories have also been defined, as  they can be derived from string theories and have interesting features, as  described in references [2] and [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The introduction of non-commutative spacetime in field theory is mo-  tivated by the Heisenberg uncertainty principle in quantum mechanics,  which states that at small distance scales, there is a large uncertainty in  momentum measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' This means that energy can reach very high  values in a small spatial distance, approaching the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' However,  according to the general theory of relativity, high energy in a small spatial  distance creates a black hole, which prevents the position from being fully  certain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' To reconcile these two phenomena, it is necessary to introduce  non-commutativity in spacetime, which implies non locality in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  This is explained in references [4] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  In this paper we study (1+1) dimensional classical scalar field theory  with static spacetime on a q-deformed space, we present both analytical  and numerical analysis of the resulting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' In section 2, we review  the some types of non-commutativity on space times and motivate the  choice of q-deformation non-commutativity as the subject of the study .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' we study the scalar field theory on q-deformed space time, we  write the Lagrangian and deduce the equation of motion, we also truncate  the equation of motion to the linear order in q −1 and solved the resulting  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' In section 4, we study the numerical solutions of the truncated  equation of motion showing that the solutions grow exponentially with x  and t meaning that the equation is stiff and there are instabilities in the  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' In section 5, we conclude the study and suggest topics for further  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='03106v1  [hep-th]  8 Jan 2023  2  Types of non-commutativity  Here, we briefly review three of the most popular types of non-commutativity  relations and justify our motivation to use the q-deformation type  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' Canonical non-commutativity It is the simplest type which used in  physics literature, it was introduced in [6], it is defined by imposing  the following commutation relations  [xµ, xν] = iθµν,  where xµ are the spacetime coordinates and θµν is a constant, anti-  symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The idea of canonical non-commutativity involves smearing the struc-  ture of space-time in a particular way, regardless of the specific  mathematical details of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  In order to incorporate non-  commutative geometry capturing the mathematical structures on  the manifold, it is necessary to consider more complex forms of non-  commutativity beyond just this basic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' Lie-type non commutativity  In this case the coordinates has a Lie algebra structure i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  the  commutation relations can capture a Lie algebra structures [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' The  commutation relations are given by  [xµ, xν] = if µν  ρ xρ,  where f µν  ρ  are the structure constants of the defined Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  However, this type is not useful because Lie structures are rigid i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  any small deformation of a Lie algebra is isomorphic to the original  Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' q-deformations  A solution to the rigidity problem for Lie algebras is to replace Lie  group with a flexible structure called quantum groups [8-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' The  term quantum group used in this context refers to the deformations  of the universal enveloping algebra of a given group, these objects  have Hopf algebra structures which are flexible structures unlike Lie  groups and algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The commutation relations are given by  xµxν = 1  q Rµν  στxσxτ,  where q is a parameter and Rµν  στ is the R-matrix of the quantum  group defined on the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  In this space a Lie algebra is replaced by a non-commutative Hopf  algebra with deformation parameter q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' The resulting space is de-  formed according to the Lie group on the space and on the parame-  ter q, this is the simplest way to deform a space time while capturing  the full algebraic structure of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  2  3  Lagrangian and the equation of motion  We begin with the Lagrangian of the scalar field on the commutative man-  ifold then introduce non-commutativity by replacing the derivatives by  Jackson derivatives, since the symmetry group is U(1), the deformations  of its universal enveloping algebra gives a commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' Thus, we  do not have to worry about defining a product of functions on the new  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' The Lagrangian is then  L = 1  2∂µφ∂µφ − 1  2m2φ2 → Lq = DµqφDµ  q φ − m2φ2,  where µ = 0, 1 with x0 = t and x1 = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Assuming the field is defined everywhere and is infinitely differentiable  and the deformations are small i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' q ≈ 1, we can relate the theory on  the non commutative topological space to the theory on the commuta-  tive manifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' transforming the non-commutative theory back to the  commutative manifold) using the formulae  Dxq(f(x)) = ∂xf +  ∞  �  k=1  (q − 1)k  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xkf (k+1)(x),  where f (k) is the k-th ordinary derivative of f with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Dtq(f(x)) = ∂tf +  ∞  �  k=1  (q − 1)k  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xkf (k+1)(x),  where f [k] is the k-th ordinary derivative of f with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The resulting Lagrangian on the commutative manifold is  Lq = 1  2∂φ∂φ − 1  2m2φ2 + 2∂φ  ∞  �  k=1  (q − 1)k  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xkφ(k+1)  +  ∞  �  l,m=1  (q − 1)(l+m)  (m + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' (l + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='xk+lφ(l+1)φ(m+1) + (x → t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  where (x → t) means the same terms but with x replaced by t includ-  ing in the derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The Lagrangian has an infinite series of derivatives, in this case the  Euler-Lagrange equation will be  ∂Lq  ∂φ +  ∞  �  k=1  (−1)k dk  dxk ( ∂Lq  ∂φ(k) ) +  ∞  �  k=1  (−1)k dk  dtk ( ∂Lq  ∂φ[k] ) = 0,  (1)  where k = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='.  The Lagrangian is clearly non local as expected from a non-commutative  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The derivatives of the Lagrangian are given by  ∂Lq  ∂φ = −mφ,  3  ∂Lq  ∂(∂φ) = ∂xφ + 2  ∞  �  n=1  (q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xnφ(n+1)  (2)  → d  dx( ∂Lq  ∂(∂xφ))  = ∂x∂xφ + 2  ∞  �  n=1  (n(q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xn−1φ(n+1)) + 2  ∞  �  n=1  ((q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xnφ(n+2)), (3)  ∂Lq  ∂(φ(k)) = 2(q − 1)k−1xk−1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  ∞  �  n=0  (q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xnφ(n+1)  → d  dx(  ∂Lq  ∂(φ(k)))  = 2(q − 1)k−1x2k−1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  ∞  �  m,n=0  �  k  m  �  (q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  (n + k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  (n + 2k − m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='xn−mφ(n+k+1),  (4)  with similar formulae for derivatives with respect to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Putting all together from (2), (3), (4) in (1) we get  −∂µ∂µφ − m2φ − 2  ∞  �  n=1  n(q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xn−1φ(n+1) − 2  ∞  �  n=1  (q − 1)n  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' xnφ(n+2)  +  ∞  �  k=2  (−1)k 2(q − 1)k−1x2k−1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  ∞  �  n=0  k  �  m=0  �  k  m  �  (q − 1)n(n + k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  (n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' (n + 2k − m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='xn−mφ(n+k+1)  + (x → t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  (5)  This is a partial differential equation of infinite order with variable  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  If we consider only small deformations i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' q ≈ 1, then we can only  keep terms up to the linear order in q − 1, the first order equation will be  −∂µ∂µφ−m2φ−(q−1)[φ(2)+xφ(3)− x3  6 φ(3)−x2φ(3)−xφ3+(x → t)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  This equation is a stiff equation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' it is numerically unstable, this  may indicate an instability in the theory due to the linear approximation  used, but as seen from the full equation of motion the full theory is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  The solution is φ = F(t)G(x) where  F(t) = c1eiAt/√q+c2e−iAt/√q+(q−1)eiAt/√q  2iA√q [ iA  24q t4+(iA3  3 −  1  12√q − i  8A)t3  +(A + q  2q  − A√q  2  −  i  8A)t2 + (i(A + q)  2A√q  − iqA  4  +  √q  8A2 )t  4  +(A + q  4A2  + q  √  A  4  +  iq  16A3 )] + O((q − 1)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  G(x) = c3eikx/√q+c4e−ikx/√q+(q−1)eikx/√q  2ik√q [ ik  24q x4+(ik3  3 −  1  12√q − i  8A)x3  +(k + q  2q  − k√q  2  − i  8k )x2 + (i(k + q)  2k√q  − iqk  4  +  √q  8k2 )x  +(k + q  4k2  + q  √  k  4  +  iq  16k3 )] + O((q − 1)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  where c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' c4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' A are normalisation constants and k = ±  √  A2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  When q = 1, it reduces to the solution to the Klein Gordon equation as  expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  4  Numerical results  Here, we present numerical solutions to the equation of motion to the  first order in q − 1, we focus on G(x) only since the remaining part is  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' The solutions are exponentially growing in time establishing that  the equation of motion was stiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  We set c3 = c4 = k = 1, A2 = 1  2 and we plot the solution for different  values of the parameter q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Figure 1: At q − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='1 the solution grows exponentially with |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' This is a  feature of a stiff equation with unstable numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' In the vicinity of  x = 0 it is close to the usual Klein-Gordon solution but as we go further it  becomes more and more distant  5  200000  100000  0  100000  200000  100  75  50  25  25  50  75  100Figure 2: At q − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='001 the solution still grows exponentially but slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Figure 3: At q − 1 = 10−6 the solution resembles the Klein-Gordon solution up  to |x| = 50 then decays for a bit but eventually blows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Figure 4: At q − 1 = 10−9 the solution has the same behaviour as the previous  graph but the decay happens at larger |x|, all smaller q − 1 values follow this  pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  6  2000  1500  1000 -  500  0  500  1000  1500  2000  100  7550  25  0  25  50  75  10030  20  10  0  10  20  30  200  150  100  50  0  50  100  150  200E  2  1  0  1  2  3  600  400  200  0  200  400  600The above results shows an instability in the theory leading to di-  vergent solutions to the equations of motion as x → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' To remove the  instability we must add infinite terms corresponding to an infinite series  of higher derivatives i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' we have to consider the full theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' However,  this approximation gives us an intuition on how the q-deformation affects  the space, small q-deformations beside leading to non local effects appear  to affect the space irregularly with only small effects locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  5  Conclusion and outlook  In conclusion, we showed that defining a field theory on a q-deformed  space leads to an infinite series of higher derivatives in the Lagrangian  even with static background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' In the case presented the algebra was com-  mutative so no new product of functions is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' We also demonstrated  that any approximation or truncation to the theory will lead to stiff equa-  tions of motion resulting from instabilities in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  While we made a progress in the field, much more is to be studied, fu-  ture research in this direction should focus on defining more complicated  theories on q-deformed spaces with non-commutative function algebras  and with dynamical spacetimes, also to define higher spin fields on such  space and study the new symmetries of the theories as well as the types  of instabilities arise if the Lagrangian is truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Acknowledgments  We would like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='Ivan Kolar for the useful discussions on the  topic  References  [1] Seiberg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
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+page_content=' Wess, Enveloping alge-  bra valued gauge transformations for non-Abelian gauge groups on  non-commutative spaces, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
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+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
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+page_content='  Singapore: World Scientific (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  [9] Bonneau, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' (2004) “Quantum groups and deformation quanti-  zation: Explicit approaches and implicit aspects,” Journal of Math-  ematical Physics, 45(10), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' 3703–3741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
+page_content=' Klimyk and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9E1T4oBgHgl3EQfVwR_/content/2301.03106v1.pdf'}
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+Diving Deep into Modes of Fact Hallucinations in Dialogue Systems
+Souvik Das Sougata Saha Rohini K. Srihari
+{souvikda, sougatas, rohini}@buffalo.edu
+Department of Computer Science and Engineering, University at Buffalo, NY.
+Abstract
+Knowledge Graph(KG) grounded conversa-
+tions often use large pre-trained models and
+usually suffer from fact hallucination.
+Fre-
+quently entities with no references in knowl-
+edge sources and conversation history are in-
+troduced into responses, thus hindering the
+flow of the conversation—existing work at-
+tempt to overcome this issue by tweaking the
+training procedure or using a multi-step refin-
+ing method. However, minimal effort is put
+into constructing an entity-level hallucination
+detection system, which would provide fine-
+grained signals that control fallacious content
+while generating responses.
+As a first step
+to address this issue, we dive deep to iden-
+tify various modes of hallucination in KG-
+grounded chatbots through human feedback
+analysis.
+Secondly, we propose a series of
+perturbation strategies to create a synthetic
+dataset named FADE (FActual Dialogue Hal-
+lucination DEtection Dataset)1.
+Finally, we
+conduct comprehensive data analyses and cre-
+ate multiple baseline models for hallucination
+detection to compare against human-verified
+data and already established benchmarks.
+1
+Introduction
+Knowledge-grounded conversational models often
+use large pre-trained models (Radford et al., 2019;
+Brown et al., 2020). These models are notorious for
+producing responses that do not comply with the
+provided knowledge; this phenomenon is known
+as hallucination (Dziri et al., 2022b; Rashkin et al.,
+2021b). Faithfulness to the supplementary knowl-
+edge is one of the prime designing factors in these
+knowledge-grounded chatbots. If a response is
+unfaithful to some given knowledge, it becomes
+uninformative and risks jeopardizing the flow of
+the conversation. Despite retaining strong linguis-
+tics abilities, these large language models(LM) in-
+adequately comprehend and present facts during
+1https://github.com/souvikdgp16/FADE
+conversations. LMs are trained to emulate distribu-
+tional properties of data that intensify its hallucina-
+tory attributes during test time.
+Figure 1: Hallucination manifested by generated responses
+using GPT2(Radford et al., 2019) trained on KG triples can
+be more nuanced.
+On the one hand, many prior works (Wiseman
+et al., 2017; Parikh et al., 2020; Tuan et al., 2019)
+have suggested training these models on external
+data to ensure faithfulness may lead to a source-
+reference divergence problem, where the reference
+contains additional factual information.
+To ad-
+dress this problem holistically, Dziri et al. has
+proposed a two-step generate-then-refine approach
+by augmenting conventional dialogue generation
+with a different refinement stage enabling the di-
+alogue system to correct potential hallucinations
+by querying the KG. Also, this work employs a
+token-level hallucination classifier trained on a syn-
+thetic dataset constructed using two perturbation
+strategies 2. Though this method has clear benefits,
+the hallucination perturbation strategies proposed
+in this work might fail to capture some of the sub-
+tle attributions of a factual generative model. As
+illustrated in Figure 1, neural models can inject hal-
+lucinated entities into responses that are present in
+the k-hop KG and are deceptively similar to what
+is expected. Also, if we cannot detect these elusive
+hallucinations beforehand, it will cause a cascad-
+ing effect and amplify hallucinations in subsequent
+turns (See and Manning, 2021).
+2(1) Extrinsic perturbation: Dziri et al. have swapped an
+entity with a different entity of the same type and not present in
+1-hop subgraph. (2) Intrinsic perturbation: they have swapped
+an entity with its object or vice versa, taken from the golden
+1-hop subgraph.
+arXiv:2301.04449v1  [cs.CL]  11 Jan 2023
+
+Path(s): T, :['Outlander', 'written_by','Diana Gabaldon'] [Gold]
+T, :['Outlander','publication_date','1st June'] [Retrieved from
+1-hop KG]
+T, :['Outlander', 'published_by','Dell Publishing'l [Retrieved from
+1-hop KG]
+History: ['Do you like the book Outlander ?']
+GPT2 Response: “I've never read it, but I know it was written by Dell
+PublishingOn the other hand, relying on human annotations
+is challenging due to error-prone collection proto-
+cols and human ignorance to complete the tasks
+with care (Smith et al., 2022). Prior research (Dziri
+et al., 2022c) shows that knowledge-grounded con-
+versational benchmarks contain hallucinations pro-
+moted by a design framework that encourages infor-
+mativeness over faithfulness. As studied by Dziri
+et al., when the annotators are asked to identify
+hallucination in a response, there is a high chance
+of error due to lack of incentive, personal bias, or
+poor attention to the provided knowledge.
+See and Manning have studied different short-
+comings in a real-time neural model. In this work,
+based on some of the findings of See and Manning,
+like repetitive and unclear utterances promoting
+hallucination, we extend the already defined modes
+of hallucinations (Maynez et al., 2020; Dziri et al.,
+2021a). Our contributions to this work are three-
+fold:
+• We extend fact hallucination in KG-grounded
+dialogue systems into eight categories. To
+understand the degree to which our defined
+classes exist in real-life data, we conduct a sys-
+tematic human evaluation of data generated
+by a state-of-the-art neural generator.
+• Since human annotation is expensive and of-
+ten inaccurate, we design a series of novel
+perturbation strategies to simulate the de-
+fined ways of fact hallucinations and build
+a set of synthetic datasets collectively named
+as FADE (FActual Dialogue Hallucination
+DEtection Dataset).
+• We create multiple pre-trained model-based
+baselines and compare the performances on
+several constituent and mixed datasets. To
+assess our dataset’s generalization capability,
+we perform zero-shot inference on BEGIN
+(Dziri et al., 2021b), and FaithDial (Dziri et al.,
+2022a) datasets, which encompasses all cate-
+gories of hallucinated responses.
+2
+Different Modes of Hallucination in
+KG-grounded Dialogue Systems
+2.1
+Background
+We focus on the task of detecting halluci-
+nated spans in dialogues that are factually
+grounded on factoids derived from multi-relational
+graphs G = (V, E, R), termed as Knowledge-
+Graphs(KG). Each KG consists of an directed edge
+triples t = ⟨[SBJ], [PRE], [OBJ]⟩, where
+[SBJ], [OBJ] ∈ V are nodes denoting subject
+and object entities and [PRE] ∈ R is a predicate
+which can be understood as a relation type. Primar-
+ily, a neural dialogue system is guilty of generating
+hallucinated text when a valid path in the k-hop
+sub-graph Gk
+c ∈ G of the original KG anchored
+around a context entity c does not support it.
+Our study extends the work of (Dziri et al.,
+2021a) where they specifically explore two broad
+circumstances – extrinsic and intrinsic to the pro-
+vided KG, under which LMs are likely to exhibit
+unfaithful behavior. Though this categorization is
+beneficial for detecting hallucinations, these cate-
+gories can be further subdivided into subcategories,
+which are described in §2.3.
+2.2
+Base Dataset
+We use OpenDialKG (Moon et al., 2019), a
+crowded-sourced English dialogue dataset where
+two workers are paired to chat about a particular
+topic(mainly movie, music, sport, and book). We
+use this dataset for training a GPT2-based model
+for generating data for human feedback analysis
+and creating the perturbed datasets. More details
+about the dataset can be found in §C
+2.3
+Definitions
+We define below several categories of fact halluci-
+nation, comprehensive illustrations of each types
+are provided in Figure 2. In addition we have in-
+cluded detailed descriptions of each definitions in
+§A
+(a) (Extrinsic-Soft). An extrinsic-soft hallucina-
+tion corresponds to an utterance that brings a new
+span of text which is similar to the expected span
+but does not correspond to a valid triple in Gk
+c .
+(b) (Extrinsic-Hard). An extrinsic-hard halluci-
+nation corresponds to an utterance that brings a
+new span of text which is different from the expected
+span and does not correspond to a valid triple in
+Gk
+c .
+(c) (Extrinsic-Grouped). An extrinsic-grouped
+hallucination corresponds to an utterance that
+brings a new span of text which is different from the
+expected span but is of a specific predefined type
+and does not correspond to a valid triple in Gk
+c .
+(d) (Intrinsic-Soft). An intrinsic-soft hallucina-
+tion corresponds to an utterance that misuses any
+triple in Gk
+c such that there is no direct path be-
+tween the entities but they are similar to each other.
+(e) (Intrinsic-Hard). An intrinsic-hard hallucina-
+tion corresponds to an utterance that misuses any
+
+Figure 2: Illustration of our defined categories of fact hallucinations in KG-grounded dialogue systems
+triple in Gk
+c such that there is no direct path be-
+tween the entities and they are not related in any
+form.
+(f) (Intrinsic-Repetitive). An intrinsic-repetitive
+hallucination corresponds to an utterance that ei-
+ther misuses [SBJ] or [OBJ] in Gk
+c such that
+there is no direct path between the entities but the
+entity has previously occurred in conversational
+history..
+(g) (History Corrupted- Intrinsic/ Extrinsic). A
+history corrupted(intrinsic/extrinsic) hallucination
+corresponds to an utterance that is subjected to
+intrinsic or extrinsic hallucination which is influ-
+enced by hallucinated entities in conversational
+history.
+2.4
+Human Feedback Analysis
+To study the extent to which the previously de-
+scribed modes of hallucination exist in a real-world
+system, we did human feedback analysis on re-
+sponses generated using a GPT2-based generative
+model fine-tuned on OpenDialKG as described
+by Dziri et al.. We sampled 200 responses each
+from four different decoding strategies, Greedy,
+Beam Search, and Nucleus Sampling, with a prob-
+ability of 0.9 and 0.5.
+For each dialogue in-
+stance, we crowd-source human judgment by solic-
+iting evaluations from 2 different annotators(with
+a high approval rating) from Amazon Mechanical
+Turk(AMT)(Details in §B). One computer science
+graduate student additionally verified the Human
+Intelligence Task (HITS). For examples where hal-
+lucination was present, we asked the workers to
+identify the type of hallucination(examples of dif-
+ferent types of hallucinations were shown in the
+GPT2-KG
+Greedy
+Beam Search
+Nucleus 0.9
+Nucleus 0.5
+Extrinsic-Soft
+10.91
+8.8
+15.5
+14.77
+Extrinsic-Hard
+3.45
+4.22
+8.3
+9.8
+Extrinsic-Grouped
+1.12
+1
+0.44
+1.6
+History Corrupted-Extrinsic
+3.3
+3.1
+2.33
+1.1
+Intrinsic-Soft
+1.2
+1.38
+0.8
+0.3
+Intrinsic-Hard
+0.2
+0.8
+1.1
+2
+Intrinsic-Repetitive
+0.2
+0.8
+1.8
+4
+History Corrupted-Intrinsic
+0.7
+0.5
+1.33
+3.3
+Extrinsic Total
+18.78
+17.12
+26.57
+27.27
+Intrinsic Total
+2.3
+3.48
+5.03
+9.6
+Total
+21.08
+20.6
+31.6
+36.87
+Table 1: Fine-grain human feedback analysis
+instruction). The result of the human feedback is
+exhibited in Table 1. We rejected 21% of the HITS
+because of poor quality; we reported the average
+Krippendorf alpha coefficient to be 0.74 on the
+remaining annotations, indicating a moderate to
+a high agreement. Using Table 1 we made these
+observations:
+• Extrinsic-soft hallucination is the dominant
+form of hallucination. Also, this bolsters our
+prior observation that LMs generate entities
+similar to the golden entity.
+• Comparatively less amount of hallucinations
+was seen in responses generated using beam
+search decoding scheme, though the percent-
+age of extrinsic-hard hallucination was higher
+than greedy decoding.
+• Intrinsic-hard hallucination appears to be the
+least among all types. This suggests LM will
+always try to learn something from the given
+KG triples; generating something dissimilar
+will have a very low probability.
+3
+Dataset Creation
+FADE is a collection of datasets consisting of com-
+ponent datasets created using several perturbations
+
+Anchor Entity(c)
+HISTORY
+GOLDEN RESPONSE.
+HISTORY (CORRUPTED)
+1-Hop KG
+A: Could you recommend movies
+B: Christopher Nolan was the director .
+A: Could you recommend movies similar to
+similar to The Dark Knight ?
+He also directed Insomnia and
+The Dark Knight ?
+Inception.
+B: The sequel to [The Dark Knight -→ The
+B: The sequel to Batman Begins is The
+GOLD TRIPLE(S)
+Dark Knight Rises(Int.)] [The Dark Knight -
+Dark Knight .
+Spider-Man(Ext.)] is Batman Begins .
+['The Dark Knight', 'directed_by',
+A: Okay . Who is the director of The
+Christopher Nolan']
+A: Okay . Who is the director of The Dark
+Dark Knight and any other movies from
+DarkKnight
+['Christopher Nolan', 'is-a', 'Film
+Knight and any other movies from him not
+him not related to Batman ?
+director'l
+related to Batman ?
+Perturbed Entity
+PERTURBED RESPONSE(Soft)
+PERTURBED RESPONSE(Soft)
+PERTURBED RESPONSE(Intrinsic)
+B:Steven Spielberg was the director
+B: The Dark Knight RisesWas the
+B: The Dark Knight Rises was the
+He also directed insomnia and
+director . He also directed insomnia
+director . He also directed insomnia
+inception
+and inception
+and inception.
+PERTURBED RESPONSE(Hard)
+PERTURBED RESPONSE(Hard)
+PERTURBED RESPONSE(Extrinsic)
+B: Joe Biden was the director . He also
+B: United States of America was the
+B: Steven Spielberg was the director .
+directedinsomnia and inception
+director . He also directed insomnia
+He also directed insomnia and
+and inception
+inception :
+PERTURBED RESPONSE(Grouped)
+PERTURBEDRESPONSE(Repetitive)
+B: Warner Bros. was the director . He
+B: Batman Begins was the director . He
+also directed insomnia and inception
+also directed insomnia and inception 
+(a) Extrinsic Hallucination Types
+(b) Intrinsic Hallucination Types
+(c) History Corrupt Hallucination TypesHallucination Type
+Index Type
+Selection Criteria
+Soft
+Same as original entity
+ei with max document score
+Hard
+Same as original entity
+ei with min document score
+Grouped
+Same as one predefined type, selected randomly
+same as soft
+Table 2: Extrinsic hallucination perturbed entity selection
+criteria
+and a set of mixed datasets constructed using the
+component datasets.
+3.1
+Perturbation Strategies
+Extrinsic Hallucination All the entities present in
+OpenDialKG undergo a indexing process. At first,
+using Spacy we determine the named entity type 3
+for each entity, and create BM25 indexes4 for each
+entity type. Each KG triple corresponding to an en-
+tity is represented in this format – "[SBJ] [PRE]
+[OBJ]" and denoted as ti. Now, for an entity(ei)
+we create a document di = concat(t1, t2, ..tn), n
+is the number of KG-triples for that entity. Af-
+ter this, we index di and ei in the index corre-
+sponding to the entity type. During the perturba-
+tion process, we retrieve all the KG-triples for the
+entity we want to perturb and form 3 queries for
+each triple by permuting ([SBJ],[PRE],[OBJ]).
+Then based on the type of extrinsic halluci-
+nation, we query the indices to get the docu-
+ment scores in the following way:
+scores =
+average({BM25(qi, dj)}i∈(s,r,o),j∈(0,n)), the se-
+lection criteria of the perturbed entities are pro-
+vided in table 2.
+The groups for extrinsic-grouped hallucination
+are mentioned in Table 10. During the selection
+process, we iteratively check whether the perturbed
+entity exists in the conversation history, matches
+with the actual entity, and has appeared in the 1-hop
+sub-graph of the original entity. If an occurrence is
+found, we proceed to the following best entity.
+Intrinsic Hallucination Here, we dynamically
+create a BM25 index and index all the KG triples
+in the 1-hop sub-graph of the original entity. Again,
+a KG triple is represented in the same fashion as in
+extrinsic hallucination – "[SBJ] [PRE] [OBJ]".
+The goal here is to select entities that are similar
+or dissimilar to the original entities and present in
+the 1-hop graph. To achieve that, we follow a hy-
+brid triple retrieval approach to score each triple
+associated with the original entity. First, we use the
+final hidden layer of a pre-trained GPT2 to obtain
+initial embeddings for each node in Gk
+c (for details,
+check §D.3). A query is formed by using Equa-
+3https://spacy.io/api/entityrecognizer
+4https://solr.apache.org/
+Hallucination Type
+Selection Criteria
+Soft
+[SUB] or [OBJ] with max triple score
+Hard
+[SUB] or [OBJ] with min triple score
+Repetitive
+same as soft, should be occurring in the conversation history
+Table 3: Intrinsic hallucination perturbed entity selection cri-
+teria
+tion 1 each triple in Gk
+c is scored using a similarity
+scoring system as described in Equation 3.
+q =
+�
+i∈{s,r,o}
+ε
+p(qi) + ε vqi
+(1)
+Here ε is a free term parameter (§D.2), p(qi) is
+unigram probability of the query term and vqi is the
+embedding for each query term(here query terms
+are [SBJ], [PRE] ,[OBJ] of the original entity).
+ni =
+ε
+p(s) + ε vs +
+ε
+q(r) + ε vr +
+ε
+p(o) + ε vo
+(2)
+ni in Equation 2 represents a triple embedding
+in Gk
+c , when q(r) represents the rarity of the rela-
+tionship term in the subgraph, high occurrence is
+penalized, rest terms are analogous to Equation 1.
+EntitySimilarity(Q, t) = cos(q, ni)
+(3)
+Now, we query the BM25 index that we have
+created before with a simple query using the orig-
+inal triple: "[SBJ] [PRE] [OBJ]" and get the
+score for each of the triple(t). Finally, we get the
+final scores using Equation 4.
+Score(Q, t) = βEntitySimilarity(Q, t)
++(1 − β)BM25(Q, t)
+(4)
+Here 0 < β < 1.
+We select the perturbed entities based on the
+scores and selection criteria as defined in Table 3.
+Like extrinsic hallucinations, we iteratively filter
+the best-scored entity until it does not match the
+original entity or appears in history.
+History Corrupted Hallucination Conversa-
+tional history is corrupted using intrinsic or extrin-
+sic corruption strategy. We select the last k turns of
+the conversation and randomly perturb the entities.
+We also ensure that at least 50% of the previous k
+turns are corrupted.
+3.2
+Dataset Analysis
+Below we provide data statistics and character-
+ize the composition and properties of the datasets
+that are generated using our proposed perturbation
+strategies.
+
+Type
+Perturbed
+Non-perturbed
+Turn with
+perturbation>2
+soft
+12752
+64634
+558
+hard
+8540
+68872
+8254
+grouped
+22858
+54542
+11296
+history-corrupt
+8534
+68878
+8247
+Table 4: Extrinsic hallucination data statistics
+Type
+Perturbed
+Non-perturbed
+Turn with
+perturbation>2
+soft
+18560
+58558
+5
+hard
+18605
+58534
+6
+repetitive
+9712
+67560
+0
+history-corrupt
+18597
+58542
+6
+Table 5: Intrinsic hallucination data statistics
+3.2.1
+Data Statistics
+Table 4 and 5 shows the statistics of datasets created
+using different perturbation strategies. The base
+dataset contains 77,430 data points. However, the
+perturbed turns in each of these datasets are quite
+low in comparison. This low number is because
+not every entity in an utterance has a valid KG path.
+For extrinsic hallucination, ∼12,000 to ∼23,000
+utterances were perturbed, and ∼550 to ∼11,300
+utterances have multiple perturbations. The num-
+ber of perturbed data points for intrinsic hallucina-
+tion is less than extrinsic(∼9,000 to ∼18,000). The
+number of utterances with multiple perturbations
+is negligible due to the many checks the perturbed
+entities go through(for example, whether the KG
+path is present, has already occurred or not, etc.)
+To train and evaluate models, we vary the size of
+the train split in this range of 10% to 30%5 with a
+step of 2.5%, keeping in mind to avoid overfitting.
+The remaining data is split into equal halves for
+validation and testing.
+3.2.2
+Parsing Features
+In Figure 3 we show the top 10 Named Entity
+Recognition(NER) tags as identified by the Spacy
+library in extrinsic hallucinations. For extrinsic-
+soft hallucination, most NER tags are of type PER-
+SON. This corresponds to the fact that the original
+entities in the base dataset are primarily related to
+movies, books, and music. In extrinsic-soft halluci-
+nation, the associated PERSON name is changed
+to a closely affiliated person, or a movie name is
+changed to its director’s name. In contrast, the dis-
+tribution of NER tags is uniform for extrinsic-hard
+hallucination. Figure 4 and 5 shows the top-10 rela-
+tions of the perturbed entity with the original entity
+in both intrinsic-soft and hard hallucinations and
+the corresponding value in their counterparts. In
+intrinsic-soft hallucination, more relevant relations
+are selected like "release year", "starred actors",
+"written by", etc. On the other hand, in intrin-
+5sequential split
+Figure 3: NER distribution in Extrinsic-soft and hard halluci-
+nation
+sic hard hallucination, more unusual relations like
+"Country of Origin", and "Country of Nationality"
+were among the top relations.
+Figure 4: Top 10 relation in perturbed KG triples in intrinsic-
+soft hallucination
+Figure 5: Top 10 relation in perturbed KG triples in intrinsic-
+hard hallucination
+3.3
+Mixing Datasets
+Since in actual data, all kinds of hallucinations are
+expected to occur. We mix the previously con-
+structed datasets in specific proportions to create a
+more challenging dataset. Table 11 shows the dif-
+ferent mixing ratios for four types of datasets is as
+follows: Observed: We try to mimic the observed
+data, which is shown in §2.4, we take an average of
+percentages in for all the decoding strategies. Bal-
+anced: Goal here is to create a balanced dataset
+between hallucinated and non-hallucinated turns,
+each type of hallucination is also balanced. Extin-
+sic+: In this scenario, we increase the percentages
+
+PERSON
+1%1%
+4%
+5%
+Extrinsic Soft
+■ORG
+2%1%4%
+2% R
+2%
+6%
+DATE
+7%
+Extrinsic Hard
+GPE
+39%
+11%
+■CARDINAL
+■WORK OF
+18%
+ART
+■NORP
+12%
+65%
+EVENT
+ORDINAL
+23%
+■LOCrelease_year
+3%
+2%2%
+3%
+Intrinsic Soft
+21%
+starred actors
+4%
+12%
+8%
+written_by
+Intrinsic Hard
+5%
+13%
+21%
+has_genre
+11%
+ is-a
+AdaptedFrom
+4%
+18%
+Gender
+15%
+16%
+in_language
+23%
+ Subject
+18%
+ProducedbyCountry of origin
+Ihas_genre
+Intrinsic Soft
+21%
+5%
+starred actors
+26%
+7%
+17%
+Intrinsic Hard
+Country of
+7%
+nationality
+Iwritten_by
+8%
+15%
+- Produced by
+3%
+8%
+Original language
+3%
+23%
+13%
+10%
+AwardWon
+10%
+in_language
+22%
+2%
+■release_yearof extrinsic-soft, hard, and grouped by a factor of
+2, 1.5, and 1.5, respectively. Intrinsic+: here we
+increase the percentages of intrinsic-soft, hard and
+repetitive by a factor of 1.5. More details in §D.4.
+3.4
+Human Verification
+To verify whether our proposed perturbation strate-
+gies inject hallucinations in the original data, we
+randomly sample 150 examples from each of the
+mixed dataset’s test splits. Subsequently, these sam-
+ples were randomly ordered to form a consolidated
+sample of 600 data points annotated by at least
+three AMT workers, with the same setting as de-
+scribed in §2.4. Additionally, the graduate student
+verified where the hallucinations adhere to the per-
+turbation norms. Krippendorff’s alpha were 0.88
+and 0.76 among workers, and workers with per-
+turbed data(average), indicating a very high agree-
+ment. Since our perturbation strategies are purely
+deterministic, we kept a large-scale human verifi-
+cation of the automatically annotated data outside
+the scope of this work. We create a human-verified
+dataset of 500 samples, 300 taken from this set and
+200 from the human feedback study 2.4.
+4
+Task
+To identify utterances that contain hallucinations
+and to locate the entities of concern. We create two
+tasks:
+1. Utterance classification: Given the dialog
+history D, knowledge triples Kn and the cur-
+rent utterance xn+1 we classify xn+1 is hallu-
+cinated or not.
+2. Token classification: Given D, Kn and xn+1,
+we need to perform sequence labelling on
+xn+1 and identify the hallucinated spans.
+5
+Baseline Models
+As an initial effort toward tackling the suggested
+hallucination detection task, we create several
+baseline detection models based on pre-trained
+transformer models, including BERT, XLNet, and
+RoBERTa. These transformer-based models repre-
+sent the state-of-the-art and can potentially better
+leverage context or embedded world knowledge
+to detect self-contradictory or anti-commonsense
+content.
+For training the utterance classifier, given D, Kn
+and xn+1, we fine tune a pre-trained model M to
+predict binary hallucinated label y for xn+1 . Here,
+D and Kn are considered as sequence A with token
+type ids as 0 and xn+1 is considered as sequence B
+with token type ids as 1. During inference, from the
+last hidden states H ∈ Rl×h (h, l are hidden size
+and sequence length, respectively), then we obtain
+the representation w ∈ Rh by max pooling(i.e.,
+w = max_pool(H)). We then pass w through
+a MLP layer with a tanh activation to get the bi-
+nary label y ∈ {0, 1}. During training time, we
+fine-tune the model using cross entropy objective
+between the predicted labels and the actual labels.
+Similarly, for training the sequence classifier, we
+fine-tune a pre-trained model Ms. At first, we
+encode D, Kn and xn+1 using Ms to get the last
+hidden states H ∈ Rl×h, (h, l are hidden size and
+sequence length, respectively). Instead of doing
+a binary classification of each token, we adopt a
+BILOU encoding scheme. The hidden states are
+passed through an MLP layer with a tanh activa-
+tion to get the 5-way label y ∈ {B, I, L, O, U}.
+During training time, we fine-tune the model us-
+ing a cross-entropy objective between the predicted
+and actual labels.
+6
+Experimental Setup
+Baseline configurations we experiment with
+a
+variety
+of
+pre-trained
+models
+via
+Hug-
+ging Face Transformers, including BERT-base-
+uncased(110M), RoBERTa-base(125M) and XL-
+Net-base-cased(110M). Though using large or
+medium versions of these models will produce bet-
+ter results, we refrain from using those models as
+scaling large models in production is costly. More
+details about training parameters can be found in
+§E
+We also experimented with model architecture as
+follows: (i) Varied the length of the history (ii) Ex-
+perimented with max/ mean pooling. (iii) Whether
+to concatenate the hidden states corresponding to
+Kn with the hidden states corresponding to xn+1
+before passing them through the MLP layer. (iv)
+Using a CRF layer instead of MLP for predicting
+labels in the sequence tagger. The best configu-
+ration uses 4 turns of conversational history, max
+pooling, it does not concatenate hidden states of
+Kn with hidden states of xn+1 and uses a 2-layer
+MLP.
+Evaluation metrics We evaluate the baselines
+with formal classification metrics, including preci-
+sion, recall, and F1 for the hallucination sequence
+tagger. For the utterance-level hallucination classi-
+fier, we report accuracy, precision, recall, F1, and
+
+Dataset
+Best Model
+Token Level
+Utterance Level
+F1
+P
+R
+F1
+P
+R
+G-Mean(↑)
+BSS(↓)
+AUC
+extrinsic-grouped
+BERT(base-uncased)
+80.69
+80.56
+80.82
+91.30
+91.80
+90.81
+93.58
+5.29
+93.62
+extrinsic-hard
+XLNet(base-cased)
+72.12
+71.98
+72.25
+87.36
+87.13
+87.60
+92.80
+2.93
+92.96
+extrinsic-history-corrupt
+XLNet(base-cased)
+72.38
+72.35
+72.40
+88.10
+87.86
+88.34
+93.24
+2.75
+93.38
+extrinsic-soft
+BERT(base-uncased)
+64.09
+69.22
+59.67
+74.80
+81.96
+68.80
+81.62
+8.03
+82.81
+intrinsic-hard
+XLNet(base-cased)
+84.44
+85.08
+83.81
+90.88
+92.88
+88.97
+93.24
+4.48
+93.34
+intrinsic-history-corrupt
+XLNet(base-cased)
+83.67
+82.27
+85.11
+91.30
+91.86
+90.74
+93.97
+4.34
+94.02
+intrinsic-repetitive
+RoBERTa(base)
+82.70
+82.76
+82.64
+88.01
+89.51
+86.55
+92.31
+3.15
+92.50
+intrinsic-soft
+RoBERTa(base)
+78.80
+80.19
+77.45
+87.10
+90.54
+83.92
+90.26
+6.22
+90.50
+Table 6: Test benchmark (numbers in percentages (%)) for component datasets, models trained on 25% of the total dataset.
+Dataset
+Best Model
+Token Level
+Utterance Level
+F1
+P
+R
+F1
+P
+R
+G-Mean(↑)
+BSS(↓)
+AUC
+balanced
+RoBERTa-base
+73.41
+68.75
+78.74
+88.24
+83.85
+93.12
+86.21
+13.14
+86.47
+observed
+XLNet(base-cased)
+63.44
+57.98
+70.03
+77.71
+71.05
+85.73
+85.40
+14.73
+85.40
+intrinsic+
+RoBERTa-base
+75.05
+71.11
+79.44
+90.16
+86.52
+94.12
+84.51
+12.78
+85.00
+extrinsic+
+XLNet(base-cased)
+75.59
+70.79
+81.10
+90.75
+86.77
+95.11
+83.21
+12.65
+83.95
+Table 7: Test benchmark (numbers in percentages (%)) for mixed datasets, models trained on 25% of the total dataset.
+AUC (Area Under Curve) for ROC. We also use
+the G-Mean metric (Espíndola and Ebecken, 2005),
+which measures the geographic mean of sensitiv-
+ity and specificity. We also employ the Brier Skill
+Score (BSS) metric (Center, 2005), which com-
+putes the mean squared error between the reference
+distribution and the hypothesis probabilities.
+7
+Results and Discussion
+Baseline performance Table 6 and Table 7 show
+the baseline performance for the component
+datasets and mixed datasets. In both the settings,
+the utterance level hallucination classifier performs
+better than the token tagger in terms of F1. It can be
+inferred from Table 6 that, on average, it is compar-
+atively easier to detect intrinsic hallucinations than
+extrinsic hallucinations; due to grounding on exter-
+nal knowledge, which indicates the validity of our
+perturbation techniques. However, comparing the
+occurrence statistics from Table 1, it is noticed that
+extrinsic-soft hallucination, which has the least F1
+score among all types, has the highest occurrences.
+In extrinsic-grouped and extrinsic-soft hallucina-
+tions, it is interesting that BERT performs better
+than the other pre-trained models. Now for mixed
+datasets, we ran inference on the test set of ob-
+served dataset, as expected F1 scores(for utterance
+classifier and token level tagger) of the observed
+dataset are low as compared to other datasets due
+to high percentage of extrinsic-soft hallucination.
+Among other mixed datasets, the XLNet model
+fine-tuned on extrinsic+ dataset performs best in
+terms of F1 scores.
+Performance on human-verified data We test
+the best performing models fine-tuned on our
+mixed datasets on human-veri���ed data as de-
+Fine-tuned on
+Pretrain Model
+F1
+(Utterance-level)
+F1
+(Token-level)
+MNLI
+RoBERTa-large
+12.5
+-
+BEGIN
+RoBERTa-large
+15.4
+-
+FaithDial
+RoBERTa-large
+22.1
+-
+Intrin-Extrin(Dziri et al., 2021a)
+RoBERTa-large
+83.81
+68.2
+balanced
+RoBERTa-base
+92.27
+78.61
+observed
+XLNet(base-cased)
+90.15
+70.27
+extrinsic+
+XLNet(base-cased)
+93.97*
+85.7*
+intrinsic+
+RoBERTa-base
+93.01
+84.33
+Table 8: Performance of several benchmark models and mod-
+els trained on FADE on the 500 human-verified data( *p-value
+< 0.001))
+Fine-tuned on
+Model
+BEGIN
+FaithDial
+MNLI(3-way)(Dziri et al., 2021b)
+T5
+49.5
+-
+MNLI(Dziri et al., 2022a)
+RoBERTa-large
+61.1
+81.6
+intrinsic_hard
+RoBERTa-base
+37.12
+51.34
+intrinsic_history_corrupt
+RoBERTa-base
+43.23
+63.11
+intrinsic_hard
+RoBERTa-large
+44.42
+64.1
+intrinsic_history_corrupt
+RoBERTa-large
+55.11
+71.43
+Table 9: Zero-sort inference F1 scores on BEGIN and Faith-
+Dial benchmarks using utterance classification models trained
+on FADE
+scribed in §3.4. Using the existing benchmark and
+baseline models, we also perform a zero-shot in-
+ference on the human-verified data. From Table
+8, it is clear that the models fine-tuned on existing
+benchmark data cannot understand fact hallucina-
+tion, especially when entities are misplaced. On the
+other hand, models trained on our datasets have F1
+scores over 90% and outperform the current base-
+line by 10.16% and 17.5% in the two tasks using
+a pre-trained model with fewer parameters. This
+suggests that identifying abrupt fact hallucination
+is more challenging than other types of halluci-
+nation(like presenting more data than expected),
+which are more commonly exhibited in the bench-
+mark datasets.
+Generalisability We make zero-shot inference
+on BEGIN and FaithDial datasets’ test splits. To
+make a fair comparison with the benchmark mod-
+els, we further fine-tune roberta-large model
+on our datasets. Table 9 shows that F1 scores ob-
+tained from our best models underperform the best
+
+Figure 6: Positive and negative model predictions
+Figure 7: Generalisation capability of RoBERTa-large model
+fine-tuned using multiple splits of intrinsic-history-corrupt
+dataset
+performing baseline by 6% in BEGIN dataset and
+10.17% in the FaithDial dataset. Even though the
+performance is low, we have to understand that
+the benchmark datasets contain hallucinations that
+are fundamentally very different from fact hallu-
+cinations. Also, we notice that models trained on
+intrinsic hallucination perform the best because the
+hallucinatory responses in the benchmark dataset
+do not deviate much from the evidence. To estimate
+how much training data is optimum for generalis-
+ability, we ran inference on benchmark datasets
+using models fine-tuned to 10% to 30% (with a
+step of 2.5%) data in train split. As shown in Fig-
+ure 7 approximately 25% is found to be optimum.
+Model Predictions We visualized the predic-
+tions on different datasets in Figure 6. Our models
+were able to easily identify the hallucinated entities
+as shown in Figure 6a here "The Departed" is a
+movie in which "Mark Wahlberg" has acted but is
+not related to the movie discussed in the context,
+i.e., "The Italian Job". Similarly, predictions made
+on the FaithDial dataset(Figure 6c) show that our
+models could produce accurate predictions when
+the response is generating something that is not
+expected, but the hallucination has similarities with
+the evidence. Our model sometimes fails to under-
+stand when the history is convoluted(Figure 6b)).
+8
+Related Work
+Hallucination in Dialogue Systems Hallucination
+in knowledge-grounded dialogue generation sys-
+tem is an emerging area of research (Roller et al.,
+2021; Mielke et al., 2020; Shuster et al., 2021;
+Rashkin et al., 2021b; Dziri et al., 2021a). Prior
+work addressed this issue by conditioning genera-
+tion on control tokens (Rashkin et al., 2021b), by
+training a token level hallucination critic to identify
+troublesome entities and rectify them (Dziri et al.,
+2021a) or by augmenting a generative model with
+a knowledge retrieval mechanism (Shuster et al.,
+2021). Though beneficial, these models are trained
+on noisy training data (Dziri et al., 2022b) which
+can amplify the hallucinations further. Closest to
+our work (Dziri et al., 2021a) has created a hallu-
+cination critic using extrinsic-intrinsic corruption
+strategies. In contrast, we create more fine-grained
+corruption strategies so that hallucinated data mim-
+ics the attributions of a neural chat module.
+Hallucination Evaluation Recently several
+benchmarks have been introduced, such as BE-
+GIN(Dziri et al., 2021b), DialFact(Gupta et al.,
+2022), FaithDial(Dziri et al., 2022a) and At-
+tributable to Identified Sources (AIS) (Rashkin
+et al., 2021a) framework. Though these methods
+can serve as a decent benchmarking system, their
+performance in detecting entity-level hallucination
+is unknown. In this work, we further contribute to
+this problem by proposing an entity-level halluci-
+nation detector trained on data created by various
+fine-grained perturbation strategies.
+9
+Conclusion
+In this work, we have analyzed the modes of entity-
+level fact hallucination, which is an open problem
+in KG-grounded dialogue systems. Through a hu-
+man feedback analysis, we demonstrate that these
+KG-grounded neural generators manifest more nu-
+anced hallucinations than straightforward studied
+approaches. We have proposed fine-grained per-
+turbation strategies to create a dataset that mimics
+the real-world observations and create a series of
+datasets collectively known as FADE. Our entity-
+level hallucination detection model can predict hal-
+
+Knowledge Triples: ['Mike Zimmer', 'Sport coached', 'American
+football'j]
+Evidence: Dylan's Candy Bar is a chain of boutique candy
+History: ['Can you tell me some information about the Minnesota
+shops and candy supplier currently located in New York City
+Knowledge Triples: ['The Italian Job', 'starred_actors',
+'Mark Wahlberg'J]
+Vikings?','TheMinnesota Vikingsarecoached byMikeZimmerand
+East Hampton, New York; Los Angeles, Chicago and Miami
+apart of theNational Football League.Not a bigfan though.,'Me
+Beach, as well as in wholesale venues around the globe.
+either . Which team do you like ?', 'My most favorite American
+History: ['Do you knows who stars in The Italian Job ?']
+Football team is the Seattle Seahawks , I meant I was not a big fan of
+History: ["I love candy, what's a good brand?"]
+the Mlnnesota Vikings . Do you like American Football ?'l
+Response: Certainly! it stars Seth Green and The
+Departed. are you familiar with either?
+Response:Ido like Mike Zimmer.i like the washington redskins
+Response: I don't know how good they are, but Dylan's
+vikings have our old qb , kurt cousins .
+Candy Bar has a chain of candy shops in various cities.
+Tagged Response(RoBERTa.)
+): Certainly! it stars
+trinsic_sofi
+Seth Green and The Departed. Are you familiar with either?
+Tagged Response(RoBERTa.,
+ve): I do like Mike Zimmer . i
+):Hallucination
+like the washington redskins . vikings' have our old qb , kurt cousins .
+a)Intrinsic-soft:Correct
+(c) FaithDial: CorrectFaithDial
+BEGIN
+60
+score
+40
+20
+0
+10
+15
+20
+25
+30
+Training Data Size (%)lucinated entities with an F1 score of 75.59% and
+classify whether an utterance is hallucinated or not
+with an F1 score of 90.75%. Our models can gener-
+alize well when zero-shot predictions are made on
+benchmarks like BEGIN and FaithDial, indicating
+our perturbation strategies’ robustness. This work
+can be extended by devising more sophisticated per-
+turbation mechanisms, which can simulate other
+types of hallucinations.
+Limitations
+The major limitations of this work are as follows:
+• The token-level hallucination classifier and
+utterance-level hallucination classifier can
+have contradictory results; however, this hap-
+pens in a small percentage of data.
+• Models trained on extrinsic datasets do not
+generalize well on the benchmark datasets, as
+the benchmark dataset contains hallucination
+mostly related to the evidence provided.
+Acknowledgements
+We thank the anonymous reviewers for provid-
+ing valuable feedback on our manuscript. This
+work is partly supported by NSF grant number IIS-
+2214070. The content in this paper is solely the
+responsibility of the authors and does not neces-
+sarily represent the official views of the funding
+entity.
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+System Demonstrations, pages 38–45, Online. Asso-
+ciation for Computational Linguistics.
+A
+Definition Details
+Figure 8: Extrinsic Hallucination
+(a) (Extrinsic-Soft). An extrinsic-soft hallucina-
+tion corresponds to an utterance that brings a new
+span of text which is similar to the expected span
+but does not correspond to a valid triple in Gk
+c .
+A hallucination is considered extrinsic when
+knowledge is injected which is not authentically
+captured by Gk
+c . However, the injected knowledge
+
+HISTORY
+A: Could you recommend movies similar to
+the The Dark Knight ?
+B: The sequel to Batman Begins is The Dark
+Knight.
+A: Okay . Who is the director of The Dark
+Knight and any other movies from him not
+related to Batman ?
+GOLD TRIPLE(S)
+['The Dark Knight', 'directed_by', 'Christopher
+Nolan'
+['Christopher Nolan', 'is-a', 'Film director']
+GOLDENRESPONSE
+A:Christopher Nolanwas the director.He
+also directed Insomnia and Inception .
+CORRUPTEDRESPONSE(Soft)
+A:Steven Spielbergwas the director .He also
+directed insomnia and inception.
+CORRUPTED RESPONSE(Hard)
+A:Joe Bidenwas the director .He also
+directed insomniaand inception.
+CORRUPTEDRESPONSE(Grouped)
+A:Warner Bros.was the director . He also
+directed insomnia and inception .Group
+Definition
+Groups
+1
+A person, organization, political party, or part
+of a religious group can be related to each other.
+"PERSON", "ORG", "NORP
+2
+Location, building, airports, infrastructure
+elements, countries, cities, and states can be interrelated
+"LOC", "GPE", "FAC"
+3
+A product, work of art, or law can be interrelated.
+"PRODUCT", "WORK_OF_ART", "LAW"
+Table 10: Defined groups for extrinsic-grouped hallucination
+is similar to the expected entity. Identifying this
+type of hallucination can be challenging due to
+the high similarity between the injected and gold
+knowledge. For example, in Figure 8 the dialogue
+sample contains an extrinsic-soft hallucination as
+the entity in response – "Steven Spielberg" is simi-
+lar to "Christopher Nolan", and it is not supported
+within 1-hop sub-graph.
+(b) (Extrinsic-Hard). An extrinsic-hard hallucina-
+tion corresponds to an utterance that brings a new
+span of text which is different from the expected
+span and does not correspond to a valid triple in
+Gk
+c .
+An extrinsic-hard hallucination occurs when in-
+jected knowledge is dissimilar to the expected en-
+tity and is not supported within the 1-hop sub-graph.
+It is easier to detect extrinsic-hard than extrinsic-
+soft as the entities are fundamentally different from
+the entities present in the 1-hop sub-graph. How-
+ever, the entity type is retained, like an entity with a
+type "person" will be replaced by the same type of
+entity. Figure 8 shows an example of extrinsic-hard
+hallucination, where the golden entity "Christopher
+Nolan" is replaced by a different category of entity,
+"Joe Biden", but the type of entity is retained.
+(c) (Extrinsic-Grouped). An extrinsic-grouped hal-
+lucination corresponds to an utterance that brings
+a new span of text which is different from the ex-
+pected span but is of a specific predefined type and
+does not correspond to a valid triple in Gk
+c .
+Like an extrinsic-hard hallucination, extrinsic-
+grouped hallucination introduces an entity that is
+functionally different from the original entity and
+not supported by the 1-hop sub-graph. The only
+difference is that the corrupted entity is not of the
+same type; instead, it is replaced by an entity of
+a similar type, defined in Table 10. For example,
+Figure 8 shows "Christopher Nolan" which is of
+type "person" is replaced by "Warner Bros." of
+type "organization". Here, the types "person" and
+"organization" are placed in the same group.
+(d) (Intrinsic-Soft). An intrinsic-soft hallucination
+corresponds to an utterance that misuses any triple
+Figure 9: Intrinsic Hallucination
+in Gk
+c such that there is no direct path between the
+entities, but they are similar to each other.
+Intrinsic hallucinations occur when the KG
+triples are misused, especially in intrinsic-soft hal-
+lucination an entity is selected from Gk
+c which is
+very similar or closely related to the original entity.
+For example, in Figure 9, "Christopher Nolan" is
+replaced with "The Dark Knight Rises" which is
+retrieved from the 1-hop sub-graph and has close re-
+lation with the original entity "Christopher Nolan".
+(e) (Intrinsic-Hard). An intrinsic-hard hallucina-
+tion corresponds to an utterance that misuses any
+triple in Gk
+c such that there is no direct path be-
+tween the entities, and they are not related in any
+form.
+Similar to intrinsic-soft hallucination, it also mis-
+uses the information in KG triples. However, the
+similarity of the corrupted entity with the original
+entity is relatively tiny. For example, in Figure
+9, "Christopher Nolan" is replaced with "United
+States of America". Although the corrupted en-
+tity is drawn from Gk
+c , it is very different from the
+original entity.
+(f) (Intrinsic-Repetitive). An intrinsic-repetitive
+hallucination corresponds to an utterance that ei-
+ther misuse [SBJ] or [OBJ] in Gk
+c such that there
+is no direct path between the entities but the entity
+
+HISTORY
+A: Could you recommend movies similar to
+the The Dark Knight ?
+B:The sequel toBatman Beginsis The Dark
+Knight .
+A: Okay . Who is the director of The Dark
+Knight and any other movies from him not
+related to Batman ?
+GOLD TRIPLE(S)
+['The Dark Knight', 'directed_by','Christophel
+Nolan']
+['Christopher Nolan', "is-a','Film director'
+GOLDEN RESPONSE
+A: Christopher Nolan was the director . He
+also directed Insomnia and Inception .
+CORRUPTEDRESPONSE(Soft)
+A: The Dark Knight Riseswas the director
+He also directed insomnia and inception .
+CORRUPTED RESPONSE(Hard)
+A: United States of America was the director .
+He also directed insomnia and inception .
+CORRUPTEDRESPONSE(Repetitive)
+A:Batman Beginswas the director . He also
+directed insomnia and inception .has previously occurred in conversational history..
+An entity from the conversational history is of-
+ten repeated in the current utterances, which corre-
+sponds to intrinsic-repetitive hallucination. Here,
+an entity from the history which also occurs in Gk
+c
+and of high relatedness, is swapped with the origi-
+nal entity. Figure 9 shows "Batman Begins" which
+is supported by Gk
+c is replaced with "Christopher
+Nolan".
+Figure 10: History Corrupted Hallucination
+(g) (History Corrupted- Intrinsic/ Extrinsic). A
+history corrupted(intrinsic/extrinsic) hallucination
+corresponds to an utterance subjected to intrin-
+sic or extrinsic hallucination influenced by halluci-
+nated entities in conversational history.
+Sometimes conversational agents are driven into
+a perplexed state, and we can witness hallucina-
+tions in most turns. So, this hallucinated history
+can trigger hallucination in the current utterance.
+This phenomenon can be seen both in extrinsic and
+intrinsic forms of hallucination. Figure 10 depicts
+extrinsic/intrinsic hallucination occurring in his-
+tory – "The Dark Knight" is changed to "The Dark
+Knight Rises" for intrinsic hallucination; similarly,
+"The Dark Knight" is changed to "Spider-Man"
+for extrinsic hallucination. Hallucinations in the
+current utterance happen as described in previous
+sections.
+B
+AMT Instructions
+We present the screenshot of the annotation inter-
+face in Figure 12, 12 and 13. Workers were paid an
+average of $7-8 per hour across all tasks. We agree
+that this annotation process has a high learning
+curve. Even workers with high approval rates made
+errors in the initial rounds of annotation. A grad-
+uate computer science student manually verified
+randomly selected samples and provided feedback
+to the workers. Feedback was given to the workers,
+especially when they selected the same answers
+for ten consecutive HITS. After sending feedback
+three times, all spammed HITS were discarded.
+C
+OpenDialKG
+We use OpenDialKG (Moon et al., 2019), a
+crowded-sourced English dialogue dataset where
+two workers are paired together to chat about a par-
+ticular topic. The first speaker is requested to start
+the conversation about a given entity. The second
+speaker is assigned to write an accurate response
+based on facts extracted from an existing KG, Free-
+base (Bast et al., 2014). The facts represent paths
+from the KG that are either 1-hop or 2-hop from the
+initial entity. Once the second speaker responds,
+the first speaker continues discussing the topic en-
+gagingly, and new multi-hop facts from the KG are
+shown to the second speaker. The dialogue can
+be considered as traversing multiple paths in the
+KG. However, not all utterances within the same
+conversation are grounded on facts from the KG.
+The second speaker can decide not to select a path
+from the KG to form an answer and instead forms
+a "chit-chat" response. Overall, the dataset consists
+of four domains: movie, music, sport, and book,
+where each second speaker’s utterance is annotated
+with paths from the KG. The KG corresponds to an
+extensive subgraph extracted from Freebase with
+∼ 1.2M triples (subject, predicate, object), ∼ 101k
+distinct entities, and 1357 distinct relations. We use
+77,430 data points in the dataset for constructing
+FADE.
+D
+Perturbation Hyper-parameters
+D.1
+Search Index Details
+We use Solr in case of extrinsic hallucination.
+We use the BM25 index, defined by the class
+solr.BM25SimilarityFactory. We man-
+ually labeled 50 data points(for the entity type
+PERSON) for tuning the indexes through grid
+
+HISTORY (CORRUPTED)
+A:Couldyou recommendmovies similarto
+the The Dark Knight ?
+B: The sequel to [The Dark Knight →The Dark
+Knight Rises(Int.)] [The Dark Knight →
+Spider-Man(Ext.)] is Batman Begins .
+A: Okay . Who is the director of The Dark
+Knight and any other movies from him not
+related to Batman ?
+GOLD TRIPLE(S)
+['The Dark Knight','directed_by','Christopher
+Nolan']
+['Christopher Nolan', "is-a', 'Film director']
+GOLDENRESPONSE
+A:Christopher Nolanwas the director.He
+also directed Insomnia and Inception 
+CORRUPTED RESPONSE(Intrinsic)
+A:United States of Americawas the director.
+He also directed insomnia and inception .
+CORRUPTED RESPONSE(Extrinsic)
+A:Joe Bidenwasthedirector.He also
+directed insomnia and inception .Figure 11: Annotation interface for human feedback analysis(Instructions, part 1)
+Figure 12: Annotation interface for human feedback analysis(Instructions, part 2)
+
+Please state if the response contains irrelevant phrase(s) or not. If yes,
+then, please select its type and note down the phrase
+We have provided you with some knowledge paths and conversational history. In the given response,
+phrase. Examples of each type of error are provided below:
+Conversation history:
+Speaker A: Could you recommend movies similar to The Dark Knight?
+Speaker B: The sequel to Batman Begins is The Dark Knight.
+Speaker A: Okay . Who is the director of The Dark Knight and any other movies from him not related to Batman?
+Knowledge paths:
+Path 1: ['The Dark Knight', 'directed_by', 'Christopher Nolan']
+Path 2: ['Christopher Nolan', 'is-a', 'Film director']
+Golden Response(this is for reference, it does not appear in the real data):
+Speaker B: Christopher Nolan was the director. He also directed Insomnia and Inception.
+Extrinsic Hallucinations:
+Extrinsic soft: When an irrelevant phrase is introduced which is similar to the expected phrase but the
+phrase does not appear in the knowledge paths. For example,
+Speaker B: Steven Spielberg was the director. He also directed Insomnia and Inception
+Extrinsic hard: When an irrelevant phrase is introduced which is not similar to the expected phrase and the
+phrase does not appear in the knowledge paths. For example,
+Speaker B: Joe Biden was the director. He also directed Insomnia and Inception.
+Extrinsic grouped: When an irrelevant phrase is introduced which is related to the expected phrase but the
+phrasedoesnotappearintheknowledgepaths.Forexample,
+SpeakerB:WarnerBros_wasthedirector.HealsodirectedInsomniaandInception
+Valid relations:
+·
+Aperson,organization,political party,or part of a religiousgroup can be related to each other
+Location, building, airports, infrastructure elements, countries, cities, and states can be interrelated.
+Aproduct,workofart,orlaw canbeinterrelated.Intrinsic Hallucinations:
+Intrinsic soft: When an irrelevant phrase is introduced which is similar to the expected phrase and the
+Speaker B: The Dark Knight was the director. He also directed Insomnia and Inception.
+Intrinsic hard: When an irrelevant phrase is introduced which is not similar to the expected phrase and the
+phrase does appear/ or is related to the knowledge paths. For example,
+Speaker B: United States of America was the director. He also directed Insomnia and Inception.
+(Christopher Nolan is a citizen of the United States of America)
+Intrinsic repetitive: When an irrelevant phrase is introduced which is related to the expected phrase,
+appears in conversational history, and the phrase does appear/ or is related to the knowledge paths. For
+example,
+SpeakerB:BatmanBegins_wasthedirector.Healsodirected InsomniaandInception
+History Corrupted Hallucinations:
+Corrupted Conversation history:
+Speaker A: Could you recommend movies similar to The Dark Knight?
+Speaker B: The sequel to The Dark Knight Rises is Spider-Man.
+Speaker A: Okay . Who is the director of The Dark Knight and any other movies from him not related to Batman?
+Now consider this conversation history, if you look closely, the second turn is corrupted with irrelevant
+entities.
+History corrupt intrinsic: When an irrelevant phrase is introduced which is of any type of intrinsic
+hallucination AND the conversation history is corrupted. For example,
+Speaker B: The Dark Knight was the director. He also directed Insomnia and Inception.
+History corrupt extrinsic: When an irrelevant phrase is introduced which is of any type of intrinsic
+hallucination AND the conversation history is corrupted. For example,
+SpeakerB:WarnerBroswasthedirector.He also directed Insomnia and Inception.Figure 13: Annotation interface for human feedback analysis(example annotation, workers were ask to find up to 3 spans if
+hallucinations are found in the data)
+search. Grid-search conditions were as follows:
+b was varied from 0.3 to 0.9 with a step of
+0.1 and k1 was varied from 0.8 to 2.0 with a
+step of 0.2. Following grid search, an optimum
+MAP score of 0.789 was found, with b = 0.9
+and k1= 1.6. For the dynamic indexes that were
+created in the case of intrinsic hallucination, we
+use the python library https://github.com/
+dorianbrown/rank_bm25 with default con-
+figurations.
+D.2
+Free parameter & β optimization
+We use a free term weight parameter(ε) in in-
+trinsic hallucination to represent the queries and
+nodes. Similar to extrinsic hallucination we man-
+ually annotated 50 data-points and ran grid search
+for ε ∈ {10−i, 2 × 10−i; i ∈ {1, 5}}, and found
+ε = 2×10−4 to be the optimum value. We used the
+same technique for optimizing β, and the search
+space ranged from 0.1 to 0.7 with a step of 0.05.
+D.3
+KG embeddings
+We follow the same approach (Dziri et al., 2021a)
+for generating the KG embeddings. OpenDialKG
+Dataset Type
+Ext-Soft(%)
+Ext-Hard(%)
+Ext-Grp(%)
+Int-Soft(%)
+Int-Hard(%)
+Int-Rep(%)
+HC-Ext(%)
+HC-Int(%)
+N-Halluc(%)
+Observed
+12.495
+6.4425
+1.04
+0.92
+1.025
+1.7
+2.4575
+1.4575
+72.4625
+Balanced
+6.25
+6.25
+6.25
+6.25
+6.25
+6.25
+6.25
+6.25
+50
+Extrinsic+
+12.5
+9.375
+9.375
+6.25
+6.25
+6.25
+6.25
+6.25
+37.5
+Intrinsic+
+6.25
+6.25
+6.25
+9.375
+9.375
+9.375
+6.25
+6.25
+40.625
+Table 11: Mixing ratios for different datasets
+triples are also represented using a textual term
+called "render". For the triples containing this term,
+we pass it through to GPT2 and then extract hidden
+state representations for each entity’s word piece
+and finally obtain a final representation by applying
+a MaxPool over the hidden representations. For
+entity mentions not described in “render”, we get
+their representations directly from the last hidden
+states in GPT2.
+D.4
+Mixing Ratios
+Mixing ratios for creating the mixed datasets are
+defined in Table 11. Perturbed and non-perturbed
+samples are drawn randomly from component
+datasets.
+E
+Implementation Details
+The utterance and token level classifier are imple-
+mented using the Pytorch Huggingface Transform-
+ers library (Wolf et al., 2020). The following con-
+
+Now complete the following task:
+Knowledge paths:
+Path 1: ['Gautam Gambhir', 'is-a', 'Athlete']
+Path 2: ['Athlete', '~is-a', 'Venus Williams']]
+Conversation history:
+Speaker A: What do you think about Gautam Gambhir Indian cricketer ?
+Response:
+Speaker B: to be honest, I don't really know anything about him. I'm more of a tennis fan . one of my favorite players is Gautam
+Gambhir
+Does the response contain irrelevant phrase(s)?
+O Yes
+O No
+If yes, then write down the irrelevant phrases(s) and select their type(up to 3):
+irrelavant phrase(s)
+Type:
+O extrinsic_soft
+Oextrinsic_hard
+Oextrinsic_grouped
+O intrinsic_soft
+O intrinsic_hard
+Oextrinsic_history_corrupt
+O intrinsic_history_corrupt
+irrelavant phrase(s)
+Type:
+O extrinsic_soft
+O extrinsic_hard
+Oextrinsic_grouped
+O intrinsic_soft
+O intrinsic_hard
+Oextrinsic_history_corrupt
+O intrinsic_history_corrupt
+irrelavant phrase(s)
+Type:
+O extrinsic_soft
+O extrinsic_hard
+O extrinsic_grouped
+O intrinsic_soft
+O intrinsic_hard
+Oextrinsic_history_corrupt
+O intrinsic_history_corrupt
+SubmitHyperparameter
+Value
+train_batch_size
+12
+gradient_accumulation_steps
+2
+num_train_epochs
+4(Token)/10(Utt)
+weight_decay
+0.01
+warmup_proportion
+0.1
+learning_rate
+1e-5
+adam_epsilon
+1e-8
+max_grad_norm
+1
+eval_batch_size
+18
+Table 12: RoBERTa-base hyper parameters
+Hyperparameter
+Value
+train_batch_size
+12
+gradient_accumulation_steps
+2
+num_train_epochs
+4(Token)/10(Utt)
+weight_decay
+0.01
+warmup_proportion
+0.1
+learning_rate
+2e-5
+adam_epsilon
+1.5e-8
+max_grad_norm
+1
+eval_batch_size
+18
+Table 13: RoBERTa-large hyper parameters
+figuration were found to be best performing for
+each models, as shown in Table 12, 13, 14 and
+15. The models were trained in a single NVIDIA
+A5000 GPU, the average running time for the base
+models were 2.5 hours, and for the large model was
+∼ 5 hours.
+F
+Supplementary results
+We report metrics for all the models trained using
+25% of the dataset, for component datasets in Table
+16 and mixed datasets in Table 17.
+Hyperparameter
+Value
+train_batch_size
+12
+gradient_accumulation_steps
+2
+num_train_epochs
+4(Token)/10(Utt)
+weight_decay
+0.01
+warmup_proportion
+0.1
+learning_rate
+5e-5
+adam_epsilon
+1e-8
+max_grad_norm
+1
+eval_batch_size
+18
+Table 14: BERT-base-uncased hyper parameters
+Hyperparameter
+Value
+train_batch_size
+12
+gradient_accumulation_steps
+2
+num_train_epochs
+4(Token)/10(Utt)
+weight_decay
+0.01
+warmup_proportion
+0.1
+learning_rate
+5e-5
+adam_epsilon
+1e-8
+max_grad_norm
+1
+eval_batch_size
+18
+Table 15: XLNet-base hyper parameters
+
+Dataset
+Best Model
+Token Level
+Utterance Level
+F1
+P
+R
+F1
+P
+R
+G-Mean
+BSS
+AUC
+extrinsic_hard
+roberta-base
+0.70613382
+0.68956357
+0.72352004
+0.86181139
+0.83985441
+0.88494727
+0.93029609
+0.03277357
+0.93145803
+extrinsic_grouped
+roberta-base
+0.7986706
+0.77534593
+0.82344214
+0.90499405
+0.89090483
+0.91953606
+0.93487266
+0.0589842
+0.93500056
+intrinsic_hard
+roberta-base
+0.84409519
+0.84717262
+0.84104004
+0.90789771
+0.92741563
+0.8891844
+0.93192336
+0.04522725
+0.9329505
+intrinsic_soft
+roberta-base
+0.78797921
+0.80193163
+0.774504
+0.87102229
+0.90540109
+0.83915877
+0.90255779
+0.06217348
+0.90495271
+intrinsic_repetitive
+roberta-base
+0.82702178
+0.82759578
+0.82644857
+0.88005638
+0.89506881
+0.86553923
+0.92305012
+0.03146957
+0.92496078
+intrinsic_history_corrupt
+roberta-base
+0.83406626
+0.82763636
+0.84059684
+0.90857229
+0.92340555
+0.89420804
+0.93381877
+0.04511612
+0.93469609
+extrinsic_history_corrupt
+roberta-base
+0.72010547
+0.71212516
+0.72826667
+0.87400219
+0.85486834
+0.89401217
+0.93612638
+0.02971357
+0.93711831
+extrinsic_soft
+roberta-base
+0.60045426
+0.60811376
+0.59298532
+0.72017689
+0.74873563
+0.69371672
+0.81231271
+0.09344656
+0.82245014
+extrinsic_hard
+bert-base-uncased
+0.71146832
+0.72259569
+0.70067846
+0.88285121
+0.88299233
+0.88271013
+0.93232489
+0.02705296
+0.93371925
+extrinsic_grouped
+bert-base-uncased
+0.80688364
+0.8056026
+0.80816875
+0.91302235
+0.9180408
+0.90805848
+0.93577473
+0.05285693
+0.93619772
+intrinsic_hard
+bert-base-uncased
+0.83328471
+0.82308025
+0.84374538
+0.91416629
+0.92395896
+0.90457903
+0.93917074
+0.04259417
+0.93983215
+intrinsic_soft
+bert-base-uncased
+0.75277325
+0.79087205
+0.71817644
+0.85483616
+0.91836735
+0.79952621
+0.88349437
+0.06794858
+0.88790364
+intrinsic_repetitive
+bert-base-uncased
+0.7481198
+0.71392596
+0.78575388
+0.84134941
+0.82295256
+0.86058758
+0.91436157
+0.04330141
+0.9160416
+intrinsic_history_corrupt
+bert-base-uncased
+0.82318199
+0.8229997
+0.82336435
+0.90891209
+0.9316067
+0.8872969
+0.93164021
+0.04459424
+0.93274826
+extrinsic_history_corrupt
+bert-base-uncased
+0.67029785
+0.69294369
+0.64908533
+0.87358552
+0.88214169
+0.86519372
+0.92312672
+0.02886461
+0.9250663
+extrinsic_soft
+bert-base-uncased
+0.64089366
+0.6922167
+0.59665579
+0.7480315
+0.81958894
+0.68796592
+0.81616138
+0.08033967
+0.82810534
+extrinsic_hard
+xlnet-base-cased
+0.72115512
+0.71982018
+0.72249502
+0.8736255
+0.8712651
+0.87599872
+0.92800607
+0.02926739
+0.92954989
+extrinsic_grouped
+xlnet-base-cased
+0.78452923
+0.77288925
+0.79652518
+0.89920345
+0.8915677
+0.90697112
+0.92895654
+0.06212166
+0.92922301
+intrinsic_hard
+xlnet-base-cased
+0.84443122
+0.85082459
+0.83813322
+0.90878914
+0.92875867
+0.88966027
+0.93238499
+0.04477944
+0.93341088
+intrinsic_soft
+xlnet-base-cased
+0.76722735
+0.80484632
+0.73296801
+0.85379657
+0.90941058
+0.80459259
+0.88491991
+0.06892207
+0.88892969
+intrinsic_repetitive
+xlnet-base-cased
+0.7941989
+0.79135701
+0.79706127
+0.86978508
+0.88154897
+0.85833102
+0.91820183
+0.03428001
+0.9202899
+intrinsic_history_corrupt
+xlnet-base-cased
+0.83667247
+0.82269807
+0.85112982
+0.91298209
+0.91864812
+0.90738552
+0.9396723
+0.04337198
+0.94024672
+extrinsic_history_corrupt
+xlnet-base-cased
+0.72378159
+0.72354039
+0.72402294
+0.88100942
+0.87862377
+0.88340807
+0.93239789
+0.02749991
+0.93375627
+extrinsic_soft
+xlnet-base-cased
+0.60896216
+0.63207547
+0.58747961
+0.73844753
+0.79296016
+0.69094782
+0.81535862
+0.08484401
+0.82655921
+Table 16: All models benchmark (numbers in fractions) for component datasets, models trained on 25% of the total dataset.
+Dataset
+Best Model
+Token Level
+Utterance Level
+F1
+P
+R
+F1
+P
+R
+G-Mean
+BSS
+AUC
+balanced
+roberta-base
+0.73405875
+0.68751809
+0.78735795
+0.882424
+0.83853553
+0.93116042
+0.86213807
+0.131385
+0.86469621
+observed
+roberta-base
+0.62554537
+0.59004757
+0.66558773
+0.77904114
+0.73266454
+0.83168565
+0.85077041
+0.14126728
+0.85098938
+extrinsic_plus
+roberta-base
+0.74849152
+0.71339648
+0.78721816
+0.90921175
+0.87804878
+0.94266814
+0.84332203
+0.12278872
+0.84855698
+intrinsic_plus
+roberta-base
+0.75045075
+0.71112613
+0.79437919
+0.90157054
+0.86518353
+0.94115257
+0.84511316
+0.12778319
+0.85001331
+balanced
+bert-base-uncased
+0.6570643
+0.57930535
+0.7589345
+0.85119497
+0.78285516
+0.9326075
+0.81309735
+0.17265032
+0.82075474
+observed
+bert-base-uncased
+0.59965325
+0.52847854
+0.6929832
+0.76124302
+0.67629046
+0.87060443
+0.84589508
+0.16352531
+0.84624573
+extrinsic_plus
+bert-base-uncased
+0.72993044
+0.663004
+0.81188563
+0.90179749
+0.84940317
+0.96108049
+0.8086632
+0.1365238
+0.82074909
+intrinsic_plus
+bert-base-uncased
+0.71653573
+0.65640721
+0.78879093
+0.89301716
+0.84373548
+0.94841293
+0.82126564
+0.14130552
+0.82978853
+balanced
+xlnet-base-cased
+0.71863497
+0.66214437
+0.78566356
+0.87222741
+0.81850039
+0.93350331
+0.84619893
+0.14481173
+0.85028143
+observed
+xlnet-base-cased
+0.63436089
+0.57976023
+0.70031519
+0.77706573
+0.71053723
+0.85733951
+0.8540217
+0.14730018
+0.85402812
+extrinsic_plus
+xlnet-base-cased
+0.75593757
+0.7079124
+0.81095307
+0.90747949
+0.86768256
+0.95110254
+0.83209459
+0.12649216
+0.8395401
+intrinsic_plus
+xlnet-base-cased
+0.74488988
+0.68995602
+0.80932808
+0.90141776
+0.85748704
+0.95009285
+0.83869733
+0.129222
+0.84522772
+Table 17: All model benchmark (numbers in fractiom) for mixed datasets, models trained on 25% of the total dataset.
+

CtFJT4oBgHgl3EQfAiym/content/tmp_files/2301.11421v1.pdf.txt

@@ -0,0 +1,887 @@
+Non-invasive and noise-robust light focusing using confocal wavefront
+shaping
+Dror Aizik, Anat Levin
+Department of Electrical and Computer Engineering, Technion, Haifa, Israel
+Abstract
+One of the hardest barriers to our ability to see inside
+biological tissue is the fact that light is highly aberrated
+when scattering by the tissue sub-components. One of
+the promising approaches for overcoming such aberrations
+is wavefront-shaping, where one modulates the incoming
+and/or the outgoing wavefront in a way that will allow
+it to focus into one spot despite scattering in the tissue.
+Wavefront modulations are specific to the tissue sample
+being imaged and need to be estimated based on a non-
+invasive feedback from a camera collecting back-scattered
+light. Such modulations have been successively estimated
+using feedback from strong fluorescent beads which have
+been manually added to a sample.
+However, in a real
+biomedical application, such feedback should be provided
+by the fluorescent components of the tissue itself, whose
+emission is orders of magnitude lower than the one pro-
+vided by beads.
+When such a low number of photons
+is spread over multiple sensor pixels, the image is highly
+susceptible to noise, and the feedback signal required for
+previous algorithms cannot be detected.
+In this work we suggest a wavefront shaping approach
+that works with a confocal modulation of both illumina-
+tion and imaging arms. The advantage of this approach is
+that as part of the optimization, aberrations are corrected
+in the optics, before the detector. Hence the low photon
+budget can be directed into a single sensor spot and de-
+tected with high SNR. We derive the optimization prob-
+lem from mathematical principles and show why it favors
+modulations that actually correct the aberrations and fo-
+cus all light into one spot. We successfully demonstrate
+wavefront-shaping correction on EGFP neurons sliced out
+of a mouse brain, despite scattering through thick tissue.
+1
+Introduction
+One of the hardest barriers to light-based approaches to
+tissue imaging is the fact that light is heavily scattered
+due to variations in the refractive index of tissue struc-
+tures. A promising approach for overcoming the scatter-
+ing challenge is a wavefront-shaping based correction. By
+using a spatial light modulator (SLM) device, one can
+reshape the coherent wavefront illuminating the sample,
+such that its aberration is conjugate to the aberration
+that will happen inside the tissue. When such a wavefront
+propagates through the sample, all incoming light can be
+brought (focused) into a small spot. In the same way, by
+using a wavefront modulation element between the tissue
+and the sensor, we can correct the outgoing wavefront,
+so that light photons emerging from a single target point
+are brought into a single sensor point, despite the tissue
+aberration. The main advantage of this approach results
+from the fact that unlike ballistic-filtering approaches, all
+light photons are used.
+Earlier wavefront shaping approaches, formally known
+as adaptive optics [5, 12, 15], were first used to correct
+modest aberrations, for example due to imperfect optics
+or refractive index variations in the tissue [6,16,31]. More
+recently, wavefront shaping techniques [11, 13, 37] have
+shown that it is possible to focus light through thick,
+highly-scattering layers [28–30,35].
+Despite the large potential of the idea, finding the de-
+sired shape of the modulation correction is rather chal-
+lenging. The desired modulation varies between different
+tissue samples and even varies spatially between different
+positions of the same tissue. For thick tissue, the modu-
+lation is a complex pattern containing a large number of
+free modes.
+Earlier proof-of-concept demonstrations have used a
+validation camera behind the tissue to provide feed-
+back to the algorithm [7, 8, 24, 29, 30, 35], and other
+approaches have relied on the existence of a guiding-
+star [10, 13, 14, 18–20, 27, 28, 32–34].
+In the absence of
+such a guiding star, and when only non-invasive feed-
+back is available, determining whatever a wavefront has
+focused inside the tissue is not straightforward. The dif-
+ficulty results from the fact that even if we can find an
+illumination wavefront that actually focuses into a small
+spot inside the tissue, the light back-scattering from this
+1
+arXiv:2301.11421v1  [physics.optics]  26 Jan 2023
+
+spot is aberrated again on its way to the camera, forming
+yet another scattered pattern.
+The simplest way to evaluate whatever a wavefront
+modulation has focused is to use multi-photon fluores-
+cence feedback. In this way, the light emitted from a flu-
+orescence spot is a non-linear function of the excitation
+intensity arriving it, so when all light is focused into a sin-
+gle spot the total emission energy is maximized [15, 18].
+However, obtaining feedback using single-photon fluores-
+cence is highly desired as the process is significantly sim-
+pler and cheaper than the multi-photon one. The single-
+photon case cannot be evaluated using the simple score
+function applied in the multi-photon case, since the emis-
+sion energy is a linear function of the excitation energy
+and thus the amount of emission energy does not increase
+when all excitation is focused into a spot.
+Recently, progress has been made on non-invasive wave-
+front shaping using single-photon feedback [1, 3]. First,
+Boniface et al. [3] have suggested that one can evalu-
+ate whatever an incoming wavefront modulation has fo-
+cused by computing the variance of the emitted speckle
+pattern. More recently, Aizik et al. [1] has suggested a
+rapid approach that can find a wavefront shaping mod-
+ulation using a small number of iterative phase conjuga-
+tion iterations. Both approaches were only demonstrated
+when the fluorescent feedback was provided by synthetic
+fluorescent beads, which emit a relatively strong signal.
+However, the ultimate goal is to apply wavefront shap-
+ing modulation using feedback from biological samples,
+such as neurons. The signal emitted from such samples
+is orders of magnitude weaker than the one provided by
+fluorescent beads, and bleaching is reached much earlier.
+Both algorithms [1,3] inherently assume that the speckle
+pattern emitted from a single fluorescent spot can be mea-
+sured. However, the number of fluorescent photons emit-
+ted from a neuron spot is so low that when these photons
+are aberrated and spread over multiple sensor pixels, no
+speckle pattern can be observed and one can mostly mea-
+sure noise, see visualization in Fig. 2. With such a low
+photon count no speckle variance can be estimated as is
+required by [3], and no phase retrieval process can be ro-
+bustly carried as in [1].
+This work proposes a wavefront shaping framework
+that can apply in low-light scenarios and use feedback
+from realistic biological data. To this end, we propose to
+use a simultaneous wavefront shaping modulation both
+on the incoming excitation wavefront and on the outgo-
+ing emitted light. The advantage is that since scattered
+photons are corrected in the optical path and we attempt
+to bring all photons emitted from a single spot into a sin-
+gle detector, we can measure them with a much higher
+signal-to-noise (SNR) ratio.
+To quantify the quality of a candidate modulation cor-
+rection, we do not attempt to maximize the total energy
+emitted from the target. Rather, we seek to maximize the
+energy of the corrected wavefront in a single pixel. We
+show that despite the fact that we use linear single-photon
+fluorescence, due to the double correction on both illumi-
+nation and imaging arms, our score function scales non-
+linearly with the intensity arriving at the fluorescence tar-
+get. Thus, the returning energy at a single pixel is max-
+imized by a focusing modulation that manages to bring
+all light into a single spot. We show that effectively, this
+score function is equivalent to the one used by previous
+two-photon fluorescence wavefront-shaping work [18].
+2
+Problem formulation
+Imaging setup:
+In Fig. 1 we visualize a wavefront-
+shaping imaging setup. A laser beam illuminates a tis-
+sue sample via a microscope objective. A phase SLM in
+the illumination arm modulates the illumination pattern.
+We wish to image a fluorescent target at the back of the
+tissue layer. The light returning from the target is col-
+lected via the same objective, and reflected at a dichroic
+beam-splitter. A second phase SLM at the imaging arm
+modulates the emitted light. Lastly, the modulated light
+is measured by the front main camera. In our setup the
+SLMs (holoeye-pluto) are placed in the Fourier plane of
+the system.
+The setup includes a second validation camera behind
+the tissue sample to assess focusing quality and image an
+undistorted reference of the target. While earlier research
+demonstrations of wavefront-shaping use this camera to
+provide feedback to the algorithm, we emphasize that our
+goal in this research is to develop non-invasive techniques
+that can only use feedback by the main (front) camera.
+The validation camera cannot provide any input to the
+algorithm.
+We note that some research on wavefront shaping and
+adaptive optics modulates only the illumination or imag-
+ing arms. For generality, our problem formulation below
+will consider modulations at both arms.
+Image formation model: Consider a set of K fluores-
+cent particles inside a sample, and denote their positions
+by o1, . . . , oK. We assume the SLM in the illumination
+arm is illuminated with a spatially uniform plane wave
+and use the SLM to display a complex 2D electric field
+that we denote by u. Although u is a 2D field, we reshape
+it as a 1D vector. We also use ν to denote a K ×1 vector
+of the field propagating through the sample at each of the
+K fluorescent sources.
+The relation between u and ν is linear and can be
+described as a multiplication by a (very large) matrix
+ν = T iu. T i is the incoming transmission matrix de-
+scribing coherent light propagation in the tissue. We note
+that T i is specific to the tissue sample being tested, and
+2
+
+Fig. 1: Our wavefront correction fluorescent microscope setup: A laser beam is exciting fluorescent beads at the back of a
+tissue layer, and fluorescent emission is scattered again through the tissue, reflects at a dichroic beam-splitter and is collected
+by a main (front) camera. We place two SLMs in the Fourier planes of both illumination and imaging arms to allow reshaping
+these wavefronts. A validation camera views the beads at the back of the tissue directly. This camera is not actually used
+by the algorithm, and is only assessing its success.
+LP=linear polarizer, BS=beam-splitter, DBS=dichroic beam-splitter,
+BPF=bandpass filter, L1 . . . L7=lenses, Obj=Objective.
+different tissue samples are described by very different
+transmission matrices.
+For thick tissue T i can be an
+arbitrarily complex matrix incorporating multiple scat-
+tering events in the tissue. Likewise, the light returning
+from the particles to the SLM of the imaging arm can
+be described as T oν, where T o is the back-propagation
+transmission matrix.
+The propagation of light from the illumination SLM
+plane to the particles and back to the SLM of the imag-
+ing arm is then modeled using the combined transmission
+matrix
+T a ≡ T o · T i.
+(1)
+We denote by ζ the wavefront placed on the SLM of
+the imaging arm, and by D(ζ) a diagonal matrix with ζ
+on its diagonal. We denote by F the Fourier transform of
+the wavefront from the SLM plane to the camera sensor
+plane.
+In the fluorescent case, the emissions from different
+points are incoherent. The recorded intensity can be ex-
+pressed as
+I =
+�
+k
+|FD(ζ)T o
+↓,k|2|νk|2α,
+(2)
+where |νk|2 is the energy of the excitation light arriving
+at particle ok, and T o
+↓,k is the k-th column of T o, so that
+FD(ζ)T o
+↓,k is the wavefront arriving to the sensor from
+ok.
+Since light emitted from different fluorescent par-
+ticles changes phase incoherently, effectively the sensor
+sums the intensity of the wavefronts emitted by differ-
+ent particles, and their phases do not interfere. In (2) α
+denotes the type of fluorescent excitation. The simplest
+case α = 1 is known as single-photon fluorescence where
+the emission is linear in the excitation energy |νk|2. In
+two-photon fluorescence, α = 2, namely the emission is
+proportional to the squared excitation.
+Linear fluores-
+cence is significantly simpler and cheaper to achieve, but
+as we explain below, a non-linear (two-photon) fluores-
+cence feedback simplifies modulation estimation.
+Phase conjugation:
+For coherent imaging, the
+Helmholtz reciprocity principle leads to wave conjuga-
+tion, namely if we record the wavefront emitted from a
+source point inside the tissue and play it back in the re-
+verse direction, the wavefront will focus at the same point.
+This implies that the returning transmission matrix is the
+transpose of the incoming one [25]:
+T o = T i⊤.
+(3)
+Note that this is just a transpose and not the Hermi-
+tian (conjugate) transpose. In the fluorescent case, we
+should note that T i, T o describe propagation at a dif-
+ferent wavelength hence they cannot be completely the
+same. However, for linear single-photon fluorescence the
+excitation and emission wavelengths are relatively similar
+and we still assume T o ≈ T i⊤.
+Normalization: we assume for simplicity that our trans-
+mission matrices are normalized such that every column
+or row has a unit energy, that is for every k
+�
+x
+|T o
+x,k|2 = 1,
+�
+x
+|T i
+k,x|2 = 1.
+(4)
+3
+
+Illumination
+SLM
+Validation Arm
+Imaging Arm
+Laser
+Illumination ArmThis means that the total amount of energy that can ar-
+rive to particle ok or emerge from it is fixed. As the laser
+energy is fixed, we also assume w.l.o.g. that all illumina-
+tion vectors have a unit norm ∥u∥ = 1. As propagation
+through the tissue does not generate new energy, every
+incoming vector u should satisfy ∥T iu∥ ≤ 1 and thus the
+energy at the target is also bounded
+�
+k
+|νk|2 ≤ 1.
+(5)
+3
+Scoring wavefront shaping mod-
+ulations
+The first challenge when coming to design a wavefront
+shaping modulation is coming up with a score function
+that can actually evaluate the focusing quality facilitated
+by a candidate modulation mask, using a noninvasive
+feedback alone. We start by reviewing scores that were
+previously introduced in the literature and then propose
+our new, noise-robust confocal score.
+Image quality scores: Modulation evaluation is a sim-
+pler task when the same modulation can correct a suffi-
+ciently large isoplanatic image region. This assumption
+was made by adaptive optics research [2, 4, 5, 12, 15] and
+also by wavefront shaping approaches [9,26,36]. When the
+same modulation can correct a large image region, one of-
+ten evaluates the quality of the resulting image, either in
+terms of contrast [15], sharpness, or variance [36]. How-
+ever, for thick tissue, wavefront shaping correction can
+vary quickly between nearby pixels, and a modulation
+may only explain a very local region. This case makes
+the above image quality scores less applicable, as inher-
+ently they evaluate the quality of an image region rather
+then a pixel. For spatially varying modulations, ideally,
+we would like to be able to evaluate the success of the
+modulation based on a per-pixel criteria.
+The total intensity score: Consider a configuration
+where we only try to correct the illumination arm, and
+the SLM in the imaging arm of Fig. 1 is not used (equiv-
+alently, D(ζ) in (2) is the identity matrix). The easiest
+score that was considered in the literature [15,18] is just
+the total intensity measured over the entire sensor plane.
+Using (2) and (4) it is easy to show that this total inten-
+sity score reduces to
+MTI(u) ≡
+�
+x
+I(x) =
+�
+k
+|νk|2α.
+(6)
+Since the energy at the target is bounded (see (5)), for the
+case α > 1 this score is maximized when ν is a one-hot
+vector, which equals 1 at a single entry and zero at all
+the others. Therefore, in two-photon fluorescence finding
+a good modulation is easy. If we manage to modulate the
+illumination such that it focuses all the excitation energy
+in a single spot, the emitted power is maximized.
+Two-photon fluorescence is however more expensive
+and harder to implement, and solutions that can use
+a single-photon excitation feedback are highly desired.
+However, in the single-photon case where α = 1, (6) re-
+duces to the total power in ν, MTI(u) = �
+k |νk|2, and
+since this power is fixed, the same amount of energy re-
+turns whether we spread the excitation power over mul-
+tiple fluorescence sources or bring all of it into one spot.
+Therefore, wavefront shaping using single-photon fluores-
+cence has remained an open challenge in the literature
+until recently.
+The variance maximization score: Following on a
+setup that modulates only the illumination and not the
+imaging arm, Boniface et al. [3] have recently suggested
+that to evaluate focusing with linear single-photon feed-
+back, one should maximize the variance of the intensity
+measured by the sensor. The idea is that if we manage to
+focus all the excitation light at a single spot, the emitted
+light scattered through the tissue will generate a highly
+varying speckle pattern on the sensor plane. If the excita-
+tion is not focused, multiple sources emit simultaneously.
+The light emitted by these sources sums incoherently, and
+hence the variance of the speckle pattern on the sensor
+decays. A short calculation shows
+MVar(u) ≡ Var[I] ≡
+≡ 1
+n
+�
+x
+|I(x)|2 −
+�
+1
+n
+�
+x
+I(x)
+�2
+=
+�
+k
+|νk|4,
+(7)
+where n is the number of image pixels. Hence, as before,
+the score is a non-linear function of the power at differ-
+ent fluorescent particles and is maximized by a one-hot
+vector.
+This score was an important advance of the state-of-
+the-art, but it may be hard to evaluate it with suffi-
+cient noise robustness using weak biological sources. To
+demonstrate this, Fig. 2 visualizes two types of fluores-
+cent emissions, when excitation light is correctly focused
+into a single spot. Fig. 2(a) demonstrates an invitrogen
+bead (ThermoFisher Fluo-Spheres dark red) that was at-
+tached to a chicken breast tissue layer. This is a strong
+source, and a clear speckle pattern is imaged. The au-
+thors of [3] have demonstrated their approach on similar
+beads. However, biological samples are often significantly
+weaker than such beads. For example, in Fig. 2(c) we
+image EGFP neurons sliced out of a mouse brain. The
+fluorescent emission here is orders of magnitudes weaker,
+and the amount of laser power we can apply before the
+neuron bleaches is also limited. One can see that rather
+than a real speckle pattern, we mostly image noise. The
+variance of this image is dominated by the noise variance
+rather than the actual speckle variance.
+4
+
+(a) scattering, bead
+(b) focusing, bead
+(c) scattering, neuron
+(d) focusing, neuron
+Fig. 2:
+Types of fluorescent data:
+(a,b) emission from invitrogen fluorescent microspheres (excitation/emission at
+640/680nm).
+A single bead is excited and the emitted light scatters through the tissue to generate a wide speckle pat-
+tern in (a). In (b) we use an aberration correction in the imaging arm so that the sensor measures a sharp spot. With such
+synthetic sources we can image a speckle pattern at high SNR, but this is not always the case with real biological samples.
+For example, (c,d) demonstrate fluorescent emission from EGFP neurons (excitation/emission at 490/510nm), which is orders
+of magnitude weaker. When the aberrated wavefront propagates to the sensor a limited number of photons are spread over
+multiple pixels and noise is dominant.
+In (d) we have applied aberration correction in the optics and as all photons are
+collected by a single pixel, SNR is drastically improved. Note that images (c,d) are taken under equal exposure and equal
+excitation power.
+Confocal energy score: In this research we suggest a
+new score for evaluating a wavefront shaping modulation.
+While the previous score corrected only the illumination
+arm, we suggest to put the same correction at both illu-
+mination and imaging arms.
+The idea is that if we find a modulation focusing all ex-
+citation light into one spot, due to reciprocity, the same
+modulation also corrects the emitted light, bringing all of
+it into a single sensor pixel (assuming the excitation and
+emission wavelengths are sufficiently similar). To score
+the focusing quality of each modulation we will use the
+intensity at the central pixel, rather than the total inten-
+sity throughout the sensor.
+Assuming the central pixel is measuring the DC compo-
+nent of the Fourier transformation from the SLM plane to
+the image plane, the central row of Psens is just a simple
+averaging. Thus we can express its value as the product
+of the SLM modulation (at the imaging arm) with the
+outgoing transmission matrix:
+F0,→D(ζ)T o
+↓,k = ζT T o
+↓,k.
+(8)
+When the same modulation u is used in both illumination
+and imaging arm, we can express the energy of the central
+pixel as:
+MConf(u) ≡ I(0) =
+=
+�
+k
+|uT T o
+↓,k|2|T i
+k,→u|2 =
+�
+k
+|νk|4.
+(9)
+As before, this score favors one-hot ν vectors and the
+score is maximized when all light is focused at a single
+spot.
+While this score is equivalent to the variance maxi-
+mization score above, it is significantly less susceptible to
+noise. This is due to the fact that the small number of
+photons we have at hand are collected by one detector,
+rather than being spread over multiple pixels. Fig. 2(c-
+d) shows the images emitted from a single neural spot
+with and without modulation in the imaging arm, and
+the significant noise reduction.
+In this work we have explicitly optimized the confo-
+cal score ((9)) using standard Hadamard basis optimiza-
+tion [23], detailed in the supplement. This optimization
+is significantly slower than [1]. As emission is very weak,
+the fact that the SLM correction is applied before imaging
+helps collect all photons at one sensor pixel and improve
+SNR.
+Iterative phase conjugation: Recently [1] has pro-
+posed an incoherent iterative phase conjugation algorithm
+that can rapidly estimate a wavefront shaping modula-
+tion. This algorithm is not explicitly maximizing a cost
+function. It uses fast power iteration to seek an excitation
+wavefront which is an eigenvector of the transmission ma-
+trix of the tissue, although the definition of transmission
+matrices with incoherent light is a bit challenging. In-
+tuitively the confocal cost of (9) is maximized when the
+wavefront T o
+↓,k emerging from the system is correlated
+with the illuminating wavefront u. This means that the
+optimum can also be thought of as an eigenvector. The
+algorithm of [1] was successfully applied on fluorescent
+beads, which are both strong and sparse. In this research
+we aim to apply the confocal score of (9) to real bio-
+logical data such as EGFP neurons. This data is signif-
+icantly weaker, and also the fluorescent target exhibits
+a continuous area rather than sparse isolated dots. The
+algorithm of [1] heavily relies on the existence of some
+speckle variation in the input image due to the need to
+retrieve the phase of the wavefront arriving the sensor
+from the measured intensity. Thus, it does not directly
+apply to continuous fluorescent sources, where not much
+speckle variation can be measured.
+5
+
+10μm0.644
+0.53
+0.416
+0.302
+0.188
+0.07410 μm8.264
+6.649
+5.034
+3.419
+1.804
+0.18910gem0.207
+0.171
+0.135
+0.099
+0.063
+0.02710 gem1.23
+0.992
+0.753
+0.515
+0.276
+0.0384
+Results
+We image slices of mice brain with EGFP neurons, ex-
+cited at 488nm and imaged at 510nm. The mouse brain
+slices are 50nm thick, so neurons exhibit some 3D varia-
+tion and modest scattering. For more challenging scatter-
+ing we place these slices behind a layer of chicken breast
+tissue (200−300µm thick) or parafilm. We image fluores-
+cent emission with a sensitive sCMOS sensor prime BSI
+express.
+In Fig. 3 we visualize some results of our algorithm.
+Before starting the optimization we focus the objective
+such that the excitation light observed by the validation
+camera illuminates the smallest possible area. Fig. 3(a)
+shows an image of this excitation pattern from the val-
+idation camera behind the tissue. As can be observed,
+the tissue exhibits significant scattering. We made our
+best attempts to reduce the diameter of this pattern by
+adjusting the distance of the objective and the sample,
+but even at the best focus position, the light scatters to
+cover a wide sample area. In Fig. 3(b) we visualize the
+excitation light after optimizing the wavefront shaping
+modulation, which is nicely focused into a sharp spot. In
+Fig. 3(c-d) we have placed a band-pass filter on the val-
+idation camera to show the emission light. Before opti-
+mization a wide area is excited and we can see the neuron
+shape. At the end of the optimization a single point is
+excited. In Fig. 3(e-f) we visualize the views of the front
+main camera, providing the sole input our algorithm can
+access. Before optimization the emitted light is scattered
+over a wide sensor area. As a low number of photons is
+spread over multiple sensor pixels, the captured imaged
+is very noisy. Despite this low SNR, at the end of the
+optimization the aberration is corrected and all the pho-
+tons are brought into a single sensor pixel, leading to a
+high quality image where noise is much less visible, see
+Fig. 3(f). In Fig. 3(g) we demonstrate the actual point
+spread function of the tissue aberration. For that we have
+used the correction only at the illumination arm and fo-
+cused the illumination to excite a single spot. We used a
+blank SLM at the imaging arm so the emitted light is not
+corrected. One can see that the aberration of a single flu-
+orescent target is not negligible. We emphasize that each
+of the images in Fig. 3 is normalized so that its maximum
+is 1, but clearly the spot at the focused images received a
+much higher number of photons than the wide scattering
+images of unfocused light, despite the fact that all images
+were captured under equal exposure and equal excitation
+power.
+In Fig. 4 we use the recovered wavefront shaping mod-
+ulation to image a wide area rather than a single spot.
+For that we excite a wide area and use a correction only
+at the imaging arm. Due to the memory effect [17, 22],
+the same modulation can allow us to image a small lo-
+cal patch rather than a single spot. With the correction,
+the neuron is observed with a much higher contrast and
+even the axons (thin lines around the neuron) emerging
+from it, whose emission is much weaker, can be partially
+observed. As our SLMs are placed in the Fourier plane
+of the system and not at a plane conjugate to the sam-
+ple itself as suggested by [21], we tilt-shift the modulated
+pattern to image a somewhat wider area, as explained
+in [1].
+5
+Discussion
+In this research we have analyzed score functions for wave-
+front shaping correction using non-invasive feedback at
+the absence of a guiding star. Obtaining such feedback is
+challenging, because even if excitation light is corrected
+and focused at a single object spot, the light returning to
+the sensor is undergoing another aberration process while
+propagating through the tissue, leading to yet another
+scattered pattern. Moreover, real biological fluorescent
+sources are weak emitting a limited number of photons.
+When these photons are spread over multiple sensor pix-
+els the detectable signal is highly contaminated by noise.
+To assess focusing quality, we need a score function that
+can measure a non-linear function of the light emitted by
+different sources. This is naturally achieved when using
+two-photon fluorescent feedback, but is harder to achieve
+with linear fluorescence. We show that by using a confo-
+cal correction at both illumination and imaging arms we
+can measure such a non-linear feedback, which is maxi-
+mized when all excitation light is brought into one spot.
+Moreover, the fact that our system uses a correction of
+the emitted light as part of the optical path allows us to
+bring the limited number of emitted photons into a single
+sensor spot, facilitating a high SNR measurement.
+The drawback of our current approach is that it uses
+a slow Hadamard basis optimization [23].
+In our cur-
+rent implementation it takes about 30 min to optimize
+for one modulation pattern. Some of this can be largely
+optimized by better hardware such as a faster SLM. How-
+ever, due to the large number of iterations required by the
+simple coordinate descent optimization, this approach is
+inherently slower than the power iterations of [1]. We are
+exploring ways to accelerate our current optimization by
+extending the phase retrieval framework of [1] to model
+the full image formation model of the incoherent case.
+References
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+rescent wavefront shaping using incoherent iterative
+phase conjugation. Optica 9, 7 (Jul 2022), 746–754.
+6
+
+(a) Valid. laser
+(b) Valid. laser
+(c) Valid. fluor.
+(d) Valid. fluor.
+(e) Main fluor.
+(f) Main fluor.
+(g) Main fluor.
+No modulation
+With modulation
+No modulation
+With modulation
+No modulation
+With modulation
+Point aberration
+Fig. 3: Wavefront shaping results: we visualize views from the validation and main cameras, at the beginning of the algorithm
+where no correction is applied, compared to the modulated image at the end of the optimization. (a-b) The excitation light as
+viewed by the validation camera at the back of the tissue. Due to significant scattering, at the beginning a wide speckle pattern
+is generated, but after optimization, the modulated wavefront is brought into a single spot. (c-d) By placing a band-pass filter
+on the validation camera we visualize the emitted light with and without correction. (e-f) Views of the emitted light at the
+main front camera with and without correction. Note that this is the only input used by our algorithm. Without correction,
+light is scattered over a wide image area and is being measured with a very low SNR. A sharp clean spot can be imaged when
+the limited number of photons is corrected in the optical path and brought into a single sensor pixel. (g) By correcting the
+emission such that a single spot is excited and leaving the imaging path uncorrected, we can visualize the actual aberration
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+
+10μm10 μm10 μm10μm10μm0.176
+0.147
+0.119
+0.09
+0.061
+0.03210μm2.002
+1.614
+1.225
+0.837
+0.448
+0.0610 μm0.276
+0.228
+0.179
+0.131
+0.082
+0.03410μm10μm10 μm10μm10 μm0.222
+0.184
+0.146
+0.108
+0.07
+0.03210μm3.191
+2.57
+1.948
+1.326
+0.704
+0.08210μm0.436
+0.356
+0.277
+0.197
+0.117
+0.03710μm10 μm10μm10μm10μm0.265
+0.218
+0.171
+0.125
+0.078
+0.03110μm1.004
+0.812
+0.62
+0.427
+0.235
+0.04310μm0.271
+0.223
+0.176
+0.128
+0.081
+0.034(a) Uncorrected
+(b) Corrected
+(c) Reference
+Main Camera
+Main Camera
+Validation Camera
+Fig. 4: Wide area imaging: We use an unmodulated illumination to excite a wide fluorescent region, but place the recovered
+modulation the imaging SLM to correct the emitted light. We compare this to an undistorted reference viewed by the validation
+camera, and to the initial uncorrected image of the main camera. Notice how the neuron contrast is improved and even some
+of the axons are revealed.
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+Analysis of a subsolar-mass black hole trigger
+from the second observing run of Advanced LIGO
+Gonzalo Morr´as,1 Jos´e Francisco Nu˜no Siles,1 Alexis Men´endez-V´azquez,2 Christos
+Karathanasis,2 Katarina Martinovic,3 Khun Sang Phukon,4, 5, 6, 7 Sebastien Clesse,8 Juan
+Garc´ıa-Bellido,1 Mario Mart´ınez,2, 9 Ester Ruiz Morales,1, 10 and Mairi Sakellariadou3
+1Instituto de F´ısica Te´orica UAM/CSIC, Universidad Aut´onoma de Madrid, Cantoblanco 28049 Madrid, Spain
+2Institut de F´ısica d’Altes Energies (IFAE), Barcelona Institute of Science and Technology, E-08193 Barcelona, Spain
+3Theoretical Particle Physics and Cosmology Group,
+Physics
+Department,
+King’s College London,
+University
+of London,
+Strand,
+London
+WC2R
+2LS,
+UK
+4Nikhef - National Institute for Subatomic Physics,
+Science Park, 1098 XG Amsterdam, The Netherlands
+5Institute for High-Energy Physics, University of Amsterdam,
+Science Park, 1098 XG Amsterdam, The Netherlands
+6Institute for Gravitational and Subatomic Physics, Utrecht University,
+Princetonplein 1, 3584 CC Utrecht, The Netherlands
+7School of Physics and Astronomy and Institute for Gravitational Wave Astronomy,
+University of Birmingham, Edgbaston, Birmingham, B15 9TT, United Kingdom
+8Service de Physique Th´eorique, Universit´e Libre de Bruxelles (ULB),
+Boulevard du Triomphe, CP225, B-1050 Brussels, Belgium
+9Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA), Barcelona, Spain
+10Departamento de F´ısica, ETSIDI, Universidad Polit´ecnica de Madrid, 28012 Madrid, Spain
+(Dated: January 30, 2023)
+We perform an exhaustive follow-up analysis of a subsolar-mass black hole candidate from the
+second observing run of Advanced LIGO, reported by Phukon et al. in 2021. The origin of this trigger
+is unclear, because the reported signal-to-noise ratio (SNR) of 8.6 and inverse false alarm rate of
+about 0.5 yr are too low to claim a gravitational-wave origin, but large enough to be intriguing. When
+using more precise waveforms, extending the frequency range down to 20 Hz, removing a prominent
+blip glitch and marginalizing over all the model parameters, we find that the network signal-to-noise
+ratio posterior distribution lies mostly below the search value, with the 90% confidence interval
+being 7.94+0.70
+−1.05. If one assumes that the signal comes from a real gravitational-wave merger event,
+we find a light component m2 = 0.76+0.50
+−0.14M⊙, suggesting a compact object of mass below one
+solar mass at 83.8% confidence level.
+Such a low mass for a compact object would suggest an
+unexpectedly light neutron star or a black hole of primordial or exotic origin. The primary mass
+would be m1 = 4.71+1.57
+−2.18M⊙, likely in the lower mass gap, for a mass ratio of q =0.16+0.34
+−0.06, at a
+distance of DL =124+82
+−48Mpc. The improved sensitivity of the next observing runs would make it
+possible to observe similar signals with a higher SNR and to distinguish a sub-solar mass component.
+I.
+INTRODUCTION
+The development of gravitational wave (GW) astron-
+omy, with about 90 binary black hole (BBH) coalescence
+events detected so far [1–6] by the LIGO-Virgo-KAGRA
+(LVK) collaboration [7], is driving a true revolution in
+astrophysics and cosmology. As the number of detected
+events grows with successive observing catalogs, prop-
+erties of the progenitors seem to challenge prior expec-
+tations for a population of astrophysical objects.
+Re-
+cent examples are BBH events like GW190521 [8, 9] with
+its most massive component in the pair-instability mass
+gap [10], as well as events like GW190814 which has a
+very low mass ratio and a secondary in the lower mass
+gap [11]. Evidence for misaligned spins in the black hole
+population has been found [12], suggesting a dynamical
+binary formation.
+The frequency range of the LIGO [13] and Virgo [14]
+detectors makes them sensitive to compact object bina-
+ries with masses below 1M⊙.
+There is no compelling
+stellar evolution model that can produce neutron stars
+or black holes below 1 M⊙.
+Therefore, the detection
+of a subsolar-mass (SSM) black hole directly points to
+a new black hole formation mechanism operating in the
+Universe, an alternative to the astrophysical evolution
+and collapse of ordinary matter. Primordial black holes
+(PBHs) are natural candidates since they can be pro-
+duced with a wide mass spectrum in the early Universe
+through the collapse of highly overdense regions [15]. An
+SSM compact object detection provides the cleanest sig-
+nature for a PBH, though there are some proposals of
+dark matter with exotic properties that could also pro-
+duce subsolar-mass objects [16–29].
+Before the advent of GW astronomy, the only way to
+detect SSM black holes was via X-ray binaries [30] or
+microlensing [31]. At present, some hints of the existence
+of such light black holes come from microlensing events
+towards the bulge [32], from Andromeda [33] and lensed
+quasars [34, 35], although the mass, the nature and the
+abundance of the lenses remain uncertain.
+Complementary to these astrophysical searches, com-
+arXiv:2301.11619v1  [gr-qc]  27 Jan 2023
+
+2
+pact binary coalescences (CBCs) with at least one sub-
+solar component have been searched for in the first (O1),
+second (O2) and third (O3) observing runs of LVK, with-
+out convincing evidence [36–40]. Nevertheless, a further
+search for SSM black holes with low mass ratio in the
+O2 data has recently revealed four potential candidate
+events1 (we refer the reader to Table I of [41]) with a
+false alarm rate smaller than 2 yr−1.
+In this paper, we follow up this search and perform pa-
+rameter inference of the four events. Our primary goal is
+to further investigate these SSM triggers using the stan-
+dard parameter estimation (PE) methods. These allow
+us to extend the frequency range of the search and use
+more accurate waveforms including spin precession, and
+higher order modes, as well as the merger and ringdown
+phases. We also can visually inspect the quality of the
+data and subtract non-gaussianities using standard tools
+such as BayesWave [42–44].
+We focus on the third candidate event reported in Ta-
+ble I of [41], observed by both LIGO Hanford and LIGO
+Livingston interferometers. It is the most significant two
+detector trigger of the search and the only one having
+significant support for an SSM component after further
+inspection with PE. We analyse in detail the data and
+perform a careful PE around this trigger, observed on
+April 1st 2017 and referred here as SSM170401.
+Fur-
+thermore, we discuss the impact of a prominent glitch
+removal. As a by-product, the PE allows us to infer the
+component masses, spins, distance and sky locations. In
+particular, we infer the probability of an SSM compo-
+nent, if one assumes that the signal is coming from a
+BBH merger event.
+II.
+PROPERTIES OF THE CANDIDATE
+To obtain the properties of the candidate we interpret
+the signal as coming from the coalescence of two com-
+pact objects. The signal was found in data taken on 2017
+April 1, 01:43:34 UTC during the O2 LIGO-Virgo observ-
+ing run. This candidate was not reported by any of the
+LVK searches, both generic [3] and SSM specific [45], but
+it was reported as part of a search for SSM candidates
+in asymmetric binaries using the GstLAL pipeline [41].
+The trigger had detector frame masses of 4.897 M⊙ and
+0.7795 M⊙, with a false-alarm-rate (FAR) of 0.4134 yr−1
+and a combined network signal-to-noise ratio (SNR) of
+∼ 8.67.
+The strain in Hanford presents a glitch 14 s before co-
+alescence, as shown in Fig. 1. The search presented in
+Ref. [41] uses templates starting at f=45 Hz. The loud-
+est template, in this case, is only 10 s long and so should
+be unaffected by the glitch. However, PE was performed
+1 Three additional subsolar-mass triggers were recently reported
+in [40] which could be the object of future analysis.
+FIG. 1. Figure showing the Hanford original whitened strain
+˜hwhitened(f) = ˜h(f)/
+�
+Sn(f), the whitened glitch model and
+the whitened clean data after subtracting the glitch. Times
+are shown relative to the trigger time.
+with templates starting at 20 Hz, which are roughly 100
+s long for the component masses discussed. In this sit-
+uation the glitch must be removed.
+Using BayesWave
+[42, 43] we fit to the data a combined glitch, signal and
+Gaussian noise model. We then subtract the glitch part
+of the model from the original data and obtain the clean
+data to be used for PE. The same procedure was used by
+the LVK collaboration for GW170817 to clean the data
+from a large glitch [46].
+We infer the CBC parameters of the signal using a
+Bayesian analysis of the data from LIGO Livingston
+and LIGO Hanford, following the methodology outlined
+in Appendix B of [3].
+In analysing the data, we fit
+two different waveform models: IMRPhenomPv2 [47] and
+IMRPhenomXPHM [48], the latter including higher order
+modes.
+We then compare the posterior samples from
+each of these, and find consistency between the two mod-
+els, noting that both of them take into account precessing
+spins. The TaylorF2 [49] waveform model has also been
+tested and, despite it providing compatible results, it fails
+to reach the same level of significance.
+We use a low-frequency cutoff of 20 Hz in both detec-
+tors for the likelihood evaluation and choose uninforma-
+tive and wide priors (see Supplemental Material). The
+primary tool used for sampling the posterior distribution
+is the LALInference Markov Chain Monte Carlo imple-
+mentation as described in [55]. The power spectral den-
+sity used in the calculations of the likelihood is estimated
+using BayesWave [42, 43]. The study uses the O2 open
+access data [56] with a sampling frequency of 4096 Hz;
+however the likelihood is integrated up to 1600 Hz.
+The signal is present in the detector for ∼ 3000 cycles,
+allowing us to constrain the source properties. The esti-
+mated parameters are reported in Table I. The marginal-
+ized posterior for the absolute value of the matched filter
+SNR is 7.98+0.62
+−1.03
+for IMRPhenomPv2 and 7.94+0.70
+−1.05
+for
+
+Hanford
+original data
+15
+clean data
+glitch
+10
+Whitened Strain
+5
+-5
+14.19-14.16-14.13 -14.1 -14.07-14.04-14.01-13.98-13.95-13.92
+Time [seco0nds1 from 2017-04-01 01:43:34.677 UTC (1175046232.677)3
+Parameter
+IMRPhenomPv2 IMRPhenomXPHM
+Signal to Noise Ratio
+7.98+0.62
+−1.03
+7.94+0.70
+−1.05
+Primary mass (M⊙)
+4.65+1.21
+−2.15
+4.71+1.57
+−2.18
+Secondary mass (M⊙)
+0.77+0.50
+−0.12
+0.76+0.50
+−0.14
+Primary spin magnitude
+0.32+0.47
+−0.26
+0.36+0.46
+−0.30
+Secondary spin magnitude
+0.48+0.46
+−0.43
+0.47+0.46
+−0.42
+Total mass (M⊙)
+5.42+1.10
+−1.65
+5.47+1.43
+−1.68
+Mass ratio (m2/m1 ≤ 1)
+0.17+0.34
+−0.05
+0.16+0.34
+−0.06
+χeff [50, 51]
+−0.06+0.17
+−0.32
+−0.05+0.22
+−0.35
+χp [52]
+0.28+0.34
+−0.21
+0.33+0.33
+−0.26
+Luminosity Distance (Mpc)
+119+82
+−48
+124+82
+−48
+Redshift
+0.028+0.018
+−0.010
+0.028+0.017
+−0.011
+Ra (◦)
+−2+34
+−35
+−1+34
+−37
+Dec (◦)
+47+14
+−26
+46+14
+−29
+Final mass (M⊙)
+5.34+1.11
+−1.70
+5.40+1.45
+−1.73
+Final spin
+0.39+0.24
+−0.07
+0.42+0.22
+−0.10
+P(m2 < 1 M⊙)
+85.5%
+83.8%
+P(m2 < 1.2 M⊙)
+92.7%
+92.7%
+TABLE I. Parameters of SSM170401. All masses are in the
+source frame.
+We assume Planck15 Cosmology [53].
+The
+statistical uncertainty of all the parameters is quantified by
+the equal-tailed 90% credible intervals about the median of
+the marginalized one-dimensional posteriors. Right ascension
+(Ra) and declination (Dec) are measured in the International
+Celestial Reference System (ICRS) [54].
+IMRPhenomXPHM. The median value of the SNR is lower
+than that found by the search, which was 8.67. How-
+ever, these two quantities are not directly comparable.
+The SNR from the search is obtained by maximizing the
+ranking statistic over a discrete template bank [41, 57–
+59], while the quoted SNR from the PE is the median
+value over the samples. Since the ranking statistic and
+the SNR are closely related, the SNR that is more compa-
+rable to that of the search would be the maximum SNR
+as found by the PE. The values of this maximum PE SNR
+are 9.09 for IMRPhenomPv2 and 9.18 for IMRPhenomXPHM.
+These values are slightly larger than that of the search,
+which is consistent with what would happen if the signal
+was astrophysical. However, this is also expected in the
+noise case due to the larger parameter space that allows
+more flexibility for the PE analysis to fit the data. We
+also notice the maximum value of the SNR to be larger
+for IMRPhenomXPHM than for IMRPhenomPv2. In a simi-
+lar way, this is expected for an astrophysical signal but
+also for noise, since the waveform includes Higher Order
+Modes and thus has more flexibility to fit the data.
+The signal is then compatible with a compact binary
+system having an unequal mass ratio q =0.17+0.34
+−0.05
+(all
+uncertainties are quoted at 90% C.L.), a source frame
+1
+2
+3
+4
+5
+6
+7
+msource
+1
+[M ]
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+msource
+2
+[M ]
+IMRPhenomPv2
+IMRPhenomXPHM
+FIG. 2. Posterior distributions for the primary and secondary
+mass in the source frame. The 90% credible regions are in-
+dicated by the solid contour in the joint distribution, and by
+the dashed vertical and horizontal lines in the marginalized
+distributions.
+primary mass m1 = 4.65+1.21
+−2.15M⊙ and a source frame
+secondary mass m2 = 0.77+0.50
+−0.12M⊙ as shown in Fig. 2.
+The marginalised posterior distribution for the secondary
+mass favors a mass lower than 1M⊙ (85.5% C.L.) and
+provides strong support for a mass lower than 1.2M⊙
+(92.7% C.L.). Using the IMRPhenomXPHM waveform, we
+find almost identical results, with a mass lower than 1M⊙
+at 83.8%C.L.
+The left panel of Fig. 3 shows the posterior distribu-
+tions for the magnitude and tilt angle of the individual
+spins, measured at a reference frequency of 20 Hz. All
+pixels in this plot have an equal prior probability. The
+spin of the secondary BH is largely unconstrained, as ex-
+pected for very unequal masses.
+The primary spin, if
+present, is likely to be misaligned with the orbital an-
+gular momentum with a preference for small spin mag-
+nitude (a1 =0.32+0.47
+−0.26).
+As can be seen in the right
+panel of Fig. 3, this leads to a χeff compatible with 0
+(χeff =−0.05+0.22
+−0.35) and an uninformative posterior in χp
+(χp =0.33+0.33
+−0.26).
+The luminosity distance and inclination angle θJN pos-
+terior distributions are shown together in the left panel of
+Fig. 4, since these two quantities are correlated. We find
+a luminosity distance of dL =119+82
+−48Mpc. We identify
+a bimodal distribution for θJN due to the fact that we
+can not distinguish whether the system is being observed
+face-on (θJN ∼ 0) or face-away (θJN ∼ π), but it being
+edge-on (θJN ∼ π/2) is disfavoured. In the face-on(off)
+configuration, the effects of precession [60] and higher or-
+der modes in the signal are suppressed [61], as is the case
+
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+/(
+)
+/(
+)
+×
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+posterior probability per pixel
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+p
+1.00
+0.75
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+eff
+IMRPhenomPv2
+IMRPhenomXPHM
+Prior
+FIG. 3. Left: posterior distribution for the individual spins of SSM170401 according to the IMRPhenomXPHM waveform model.
+The radial coordinate in the plot denotes the dimensionless spin magnitude, while the angle denotes the spin tilt, defined as
+the angle between the spin and the orbital angular momentum of the binary at a reference frequency of 20 Hz. A tilt of 0°
+indicates that the spin is aligned with the orbital angular momentum. A nonzero magnitude and a tilt away from 0° and 180°
+imply a precessing orbital plane. All bins have an equal prior probability. Right: posterior distributions for the effective spin
+and effective in-plane spin parameters. The black lines in the right panel show the prior distributions for the effective spin
+parameters. The 90% credible regions are indicated by the solid contour in the joint distribution, and by dashed vertical and
+horizontal lines in the marginalized distributions. The large density for tilts close to 90° leads to non-zero values for χp and
+low values for χeff.
+here.
+From the right panel of Fig. 4, it can be seen that
+the signal came from a position in the sky for which the
+LIGO network has good sensitivity to the two GW polar-
+izations [62]. This, however, does not represent the area
+with the best sensitivity, which would be located on top
+of the continental US and its antipodes.
+In summary, the PE shows that the chirp and compo-
+nent masses, the effective spin, the luminosity distance
+and the sky location can be reconstructed, and are con-
+sistent with that of a BBH merger event. However, it
+is known that Gaussian noise can also mimic such a sig-
+nal [63], and given the low values of the SNR and iFAR,
+it is not possible to ascertain the origin of the signal, as
+it could very well have been generated by detector noise.
+III.
+DISCUSSION
+Even if the significance of the trigger did not improve
+with the present analysis and PE since it remains at the
+threshold limit, it is interesting to speculate on the possi-
+ble origin of the secondary component (with a preferred
+mass below 1 M⊙ for such a compact object, if inter-
+preted as a GW event).
+The neutron star nature of the light compact object
+seems disfavored. Indeed, neutron stars have relatively
+well-determined masses from observations of binary sys-
+tems, including pulsars or X-ray binaries involving an
+accreting neutron star from a companion. Their masses
+are contained within a narrow range 1.25-1.45 M⊙ [64],
+further confirmed by the observation of GW170817 [46].
+Even though there is a recent claim [65] for a neutron star
+of mass around 0.7 M⊙, modern core-collapse supernova
+simulations [66, 67] indicate it is difficult to form neutron
+stars with masses below one solar mass. Such a small
+mass for a neutron star probably requires a QCD equa-
+tion of state that is beyond theoretical predictions [68].
+Although the neutron star interpretation of the hypo-
+thetical trigger SSM170401 is observationally disfavored,
+given our current limited knowledge of the equation of
+state we cannot exclude a neutron star origin.
+On the other hand, PBHs [69–72], formed by the grav-
+itational collapse of large inhomogeneities in the early
+Universe are already considered as a possible explana-
+tion of LVK GW detections, see e.g. [73–85]. Depending
+
+5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+JN [radians]
+50
+100
+150
+200
+250
+300
+350
+400
+DL [Mpc]
+Prior
+IMRPhenomPv2
+IMRPhenomXPHM
+-135°
+-90°
+-45°
+0°
+45°
+90°
+135°
+0°
+30°
+60°
+60°
+30°
+0°
+-30°
+-60°
+-60°
+-30°
+50% area: 711 deg²
+90% area: 2,658 deg²
+FIG. 4. Left: posterior distributions for the luminosity distance and the inclination angle of SSM170401, according to the
+IMRPhenomXPHM and IMRPhenomPv2 waveform models. The inclination angle indicates the angle between the line of sight and
+the total angular momentum of the binary. For nonprecessing binaries, this is equal to the angle between the orbital angular
+momentum and the line of sight. The solid lines and the central contour denote 90% credible regions. Right: sky position of
+the event as evaluated from the Greenwich meridian according to the IMRPhenomXPHM waveform model.
+on the model, they may explain anything from a tiny
+fraction of Dark Matter to its entirety. PBHs have been
+the main motivation to conduct searches of SSM black
+holes in the LVK data [38, 39, 41, 45, 86–88], in partic-
+ular, the extended subsolar search with low-mass ratios
+in O2 which reported SSM170401 as a possible candi-
+date [41]. If some of the observed binary coalescences
+are indeed due to PBHs, they must have a relatively ex-
+tended mass distribution that would have been imprinted
+by the thermal history of the Universe [78, 89].
+This
+would lead to a peak in the mass distribution around a
+solar mass which is naturally produced at the QCD tran-
+sition [78, 89–94], and SSM170401 could be an example
+of a subsolar PBH around the QCD-induced peak. The
+spin posterior is quite broad and the spin is compatible
+with zero, although a slight preference for a primary spin
+around 0.3 is observed. In this case, the non-zero but
+relatively low spin of the primary component may have
+been acquired by matter accretion, previous mergers or
+hyperbolic encounters [84, 95, 96].
+Dark Matter with very special particle composition
+could also be at the origin of solar-mass black holes if
+it can accumulate inside neutron stars and lead to their
+collapse into a black hole. Several scenarios have been
+proposed [22, 26, 97–100], but they all require very par-
+ticular conditions, in order not to change the correspond-
+ing Chandrasekhar limit. Such transmutations, if leading
+to SSM black holes, would be accompanied by violent ex-
+plosions that would have been observed. It is therefore
+still unclear if such scenarios are realistic and compati-
+ble with observations. In an alternative scenario, with
+complex and dissipative Dark Matter composition, SSM
+black holes could form through the cooling and gravita-
+tional collapse of Dark Matter halos [23]. This model was
+constrained by the LVK data in [39, 101, 102].
+Another possibility, if considered as a real GW event,
+could be that the secondary component of SSM170401 is
+a boson star, a hypothetical horizonless compact object
+formed by an ultralight bosonic field. If the mass of the
+bosonic particle is larger than 10−10eV/c2, the boson star
+can have subsolar mass [103]. Note that due to Beken-
+stein’s bound, any object of a given mass which is as
+compact as a black hole can only be a black hole. Boson
+stars necessarily must be larger, implying a lower ISCO
+frequency in the middle or lower than the LIGO sensitiv-
+ity band. So the viable parameter space for a boson star
+is probably very limited.
+Finally, we comment on the mass of the primary com-
+ponent, which would preferably lie in the low mass gap
+between 2.5 and 5M⊙ (61%C.L.). Assuming a real GW
+merger event, it would most probably be a black hole. A
+neutron star origin with mass above 2.5M⊙ is strongly
+disfavoured. Other black hole candidates in the low mass
+gap were observed in the GWTC-3 catalog, which may
+bring additional support for a primordial origin since
+PBHs should not have a mass gap.
+IV.
+CONCLUSIONS
+In this work, we investigate the most significant can-
+didate reported in [41], removing a prominent blip glitch
+
+6
+in the data and estimating the CBC parameters with
+the state-of-the-art waveform families IMRPhenomPv2 and
+IMRPhenomXPHM, taking into account contributions from
+higher order modes and extending the frequency range
+down to 20 Hz. However, with this improved modelling
+with respect to the search, we find a 90% confidence level
+network SNR of 7.94+0.70
+−1.05, which is lower than the SNR
+of 8.6 obtained in the template-bank-based search.
+Nevertheless, if one would assume that it is coming
+from a real GW event, the trigger observed on the 1st of
+April, 2017, is identified consistently in both LIGO de-
+tectors with a light mass component, m2 = 0.76+0.50
+−0.14M⊙
+(90% credible interval). Such low mass is below one solar
+mass and below 1.2 solar masses at 83.8% and 92.7% con-
+fidence level, respectively. The compact binary coales-
+cence presents a total mass of 5.47+1.43
+−1.68M⊙, correspond-
+ing to a mass ratio of q =0.16+0.34
+−0.06, and a luminosity
+distance of 124+82
+−48Mpc.
+The values quoted here are a
+result of using IMRPhenomXPHM but consistent results are
+obtained with different analysis pipelines and waveform
+families. At this point, the observational data and the
+search from [41] do not show enough significance to claim
+a firm GW observation. Nevertheless, our analysis shows
+that the signal, if coming from a GW event, is consistent
+with an SSM black hole. We discuss several scenarios for
+the production of such a possible SSM black hole candi-
+date and conclude that a neutron star origin is disfavored
+or at least requires a non-standard matter equation of
+state. Other possibilities could include primordial black
+holes, black holes formed from the accretion of hypothet-
+ical Dark Matter particles onto neutron stars, or boson
+stars. Given that PBHs can also explain some intriguing
+properties of other compact binary coalescences without
+being restricted by the Chandrashekar mass, they can be
+considered as our preferred hypothesis.
+The data from the third observing run O3, as well as
+data from the future planned runs with improved sensi-
+tivity, O4 and O5, offer a great opportunity for discover-
+ing additional SSM candidate events, and could increase
+the statistical significance for the existence of a new class
+of SSM compact objects.
+ACKNOWLEDGMENTS
+S.C. acknowledges support from the Francqui Foun-
+dation
+through
+a
+Starting
+Grant.
+K.M.
+is
+sup-
+ported by King’s College London through a Postgrad-
+uate International Scholarship.
+M.S. is supported in
+part by the Science and Technology Facility Council
+(STFC), United Kingdom, under the research grant
+ST/P000258/1.
+This work is partially supported by
+the Spanish grants PID2020-113701GB-I00, PID2021-
+123012NB-C43 [MICINN-FEDER], and the Centro de
+Excelencia Severo Ochoa Program CEX2020-001007-S
+through IFT, some of which include ERDF funds from
+the European Union.
+IFAE is partially funded by the
+CERCA program of the Generalitat de Catalunya. We
+acknowledge the use of IUCAA LDG cluster Sarathi
+for the computational/numerical work. This material is
+based upon work supported by NSF’s LIGO Laboratory
+which is a major facility fully funded by the National
+Science Foundation.
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@@ -0,0 +1,2048 @@
+arXiv:2301.11996v1  [math.AP]  27 Jan 2023
+L2 DIFFUSIVE EXPANSION FOR NEUTRON TRANSPORT EQUATION
+YAN GUO AND LEI WU
+Abstract. Grazing set singularity leads to a surprising counter-example and breakdown [24] of the classical
+mathematical theory for L∞ diffusive expansion (1.9) of neutron transport equation with in-flow boundary
+condition in term of the Knudsen number ε, one of the most classical problems in the kinetic theory. Even
+though a satisfactory new theory has been established by constructing new boundary layers with favorable
+ε-geometric correction for convex domains [24, 7, 8, 22, 23], the severe grazing singularity from non-convex
+domains has prevented any positive mathematical progress. We develop a novel and optimal L2 expansion
+theory for general domain (including non-convex domain) by discovering a surprising ε
+1
+2 gain for the average
+of remainder.
+Contents
+1.
+Introduction
+1
+2.
+Asymptotic Analysis
+5
+3.
+Remainder Equation
+7
+4.
+Remainder Estimate
+10
+5.
+Proof of Main Theorem
+13
+References
+13
+1. Introduction
+1.1. Problem Formulation. We consider the steady neutron transport equation in a three-dimensional
+C3 bounded domain (convex or non-convex) with in-flow boundary condition. In the spatial domain Ω ∋
+x = (x1, x2, x3) and the velocity domain S2 ∋ w = (w1, w2, w3), the neutron density uε(x, w) satisfies
+
+
+
+w · ∇xuε + ε−1�
+uε − uε
+�
+= 0 in Ω × S2,
+uε(x0, w) = g(x0, w) for w · n < 0 and x0 ∈ ∂Ω,
+(1.1)
+where g is a given function denoting the in-flow data,
+uε(x) := 1
+4π
+�
+S2 uε(x, w)dw,
+(1.2)
+n is the outward unit normal vector, with the Knudsen number 0 < ε ≪ 1. We intend to study the asymptotic
+behavior of uε as ε → 0.
+Based on the flow direction, we can divide the boundary γ :=
+�
+(x0, w) :
+x0 ∈ ∂Ω, w ∈ S2�
+into the
+incoming boundary γ−, the outgoing boundary γ+, and the grazing set γ0 based on the sign of w · n(x0). In
+particular, the boundary condition of (1.1) is only given on γ−.
+2020 Mathematics Subject Classification. Primary 35Q49, 82D75; Secondary 35Q62, 35Q20.
+Key words and phrases. non-convex domains, transport equation, diffusive limit.
+Y. Guo was supported by NSF Grant DMS-2106650.
+L. Wu was supported by NSF Grant DMS-2104775.
+1
+
+2
+Y. GUO, L. WU
+1.2. Normal Chart near Boundary. We follow the approach in [8, 23] to define the geometric quantities,
+and the details can be found in Section 2.2. For smooth manifold ∂Ω, there exists an orthogonal curvilinear
+coordinates system (ι1, ι2) such that the coordinate lines coincide with the principal directions at any x0 ∈
+∂Ω. Assume ∂Ω is parameterized by r = r(ι1, ι2). Let the vector length be Li := |∂ιir| and unit vector
+ςi := L−1
+i ∂ιir for i = 1, 2.
+Consider the corresponding new coordinate system (µ, ι1, ι2), where µ denotes the normal distance to the
+boundary surface ∂Ω, i.e.
+x = r − µn.
+(1.3)
+Define the orthogonal velocity substitution for w := (ϕ, ψ) as
+−w · n = sin ϕ,
+w · ς1 = cos ϕ sin ψ,
+w · ς2 = cos ϕ cos ψ.
+(1.4)
+Finally, we define the scaled normal variable η = µ
+ε , which implies ∂
+∂µ = 1
+ε
+∂
+∂η .
+1.3. Asymptotic Expansion and Remainder Equation. We seek a solution to (1.1) in the form
+uε =U + U B + R =
+�
+U0 + εU1 + ε2U2
+�
++ U B
+0 + R,
+(1.5)
+where the interior solution is
+U(x, w) := U0(x, w) + εU1(x, w) + ε2U2(x, w),
+(1.6)
+and the boundary layer is
+U B(η, ι1, ι2, w) := U B
+0 (η, ι1, ι2, w).
+(1.7)
+Here U0, U1, U2 and U B
+0 are constructed in Section 2.1 and Section 2.2, and R(x, v) is the remainder.
+1.4. Literature. The study of the neutron transport equation in bounded domains, has attracted a lot
+of attention since the dawn of the atomic age.
+Besides its significance in nuclear sciences and medical
+imaging, neutron transport equation is usually regarded as a linear prototype of the more important yet more
+complicated nonlinear Boltzmann equation, and thus, is an ideal starting point to develop new theories and
+techniques. We refer to [10, 11, 12, 13, 14, 15, 16, 17, 18] for the formal expansion with respect to ε and explicit
+solution. The discussion on bounded domain and half-space cases can be found in [5, 4, 3, 1, 2, 19, 20, 21].
+The classical boundary layer of neutron transport equation dictates that U B
+0 (η, ι1, ι2, w) satisfies the Milne
+problem
+sin ϕ∂U B
+0
+∂η
++ U B
+0 − U B
+0 = 0.
+(1.8)
+From the formal expansion in ε (see (2.6)), it is natural to expect the remainder estimate [5]
+∥R∥L∞ ≲ ε.
+(1.9)
+Even though this is valid for domains with flat boundary, a counter-example is constructed [24] so that (1.9)
+is invalid for a 2D disk. This is due to the grazing set singularity.
+To be more specific, in order to show the remainder estimates (1.9), the higher-order boundary layer
+expansion U B
+1 ∈ L∞ is necessary, which further requires ∂ιiU B
+0 ∈ L∞. Nevertheless, though U B
+0 ∈ L∞, it is
+shown that the normal derivative ∂ηU B
+0 is singular at the grazing set ϕ = 0. Furthermore, this singularity
+∂ηU B
+0
+/∈ L∞ will be transferred to ∂ιiU B
+0
+/∈ L∞. A careful construction of boundary data [24] justifies this
+invalidity, i.e. both the method and result of the boundary layer (1.8) are problematic.
+A new construction of boundary layer [24] based on the ε-Milne problem with geometric correction for
+�
+U B
+0 (η, ι1, ι2, w)
+sin ϕ∂�
+U B
+0
+∂η
+−
+ε
+1 − εη cos ϕ∂�
+U B
+0
+∂ϕ + �
+U B
+0 − �
+U B
+0 = 0
+(1.10)
+has been shown to provide the satisfactory characterization of the L∞ diffusive expansion in 2D disk domains.
+With more detailed regularity analysis and boundary layer decomposition techniques for (1.10), such result
+has been generalized to 2D/3D smooth convex domains [7, 8, 22, 23] and even 2D annulus domain [25].
+
+DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION
+3
+In non-convex domains, the boundary layer with geometric correction is essentially
+sin ϕ∂�
+U B
+0
+∂η
+−
+ε
+1 + εη cos ϕ∂�
+U B
+0
+∂ϕ + �
+U B
+0 − �
+U B
+0 = 0
+(1.11)
+Compared to (1.10), this sign flipping dramatically changes the characteristics.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+−1.5
+−1
+−0.5
+0
+0.5
+1
+1.5
+η
+φ
+Figure 1. Characteristics in Convex Domains
+0
+1
+2
+3
+4
+5
+6
+7
+8
+−1.5
+−1
+−0.5
+0
+0.5
+1
+1.5
+η
+φ
+Figure 2. Characteristics in Non-
+Convex Domains
+In Figure 1 and Figure 2 [25], the horizontal direction represents the scaled normal variable η and the
+vertical direction represents the velocity ϕ. There exists a “hollow” region in Figure 2 that the characteristics
+may never track back to the left boundary η = 0 and ϕ > 0, making the W 1,∞ estimates impossible and
+thus preventing higher-order boundary layer expansion.
+In this paper, we will employ a fresh approach to design a cutoff boundary layer without the geometric
+correction and justify the L2 diffusive expansion in smooth non-convex domains.
+1.5. Notation and Convention. Let ⟨ · , · ⟩w denote the inner product for w ∈ S2, ⟨ · , · ⟩x for x ∈ Ω,
+and ⟨ · , · ⟩ for (x, w) ∈ Ω × S2. Also, let ⟨ · , · ⟩γ± denote the inner product on γ± with measure dγ :=
+|w · n| dwdSx = |sin ϕ| cos ϕdwdSx. Denote the bulk and boundary norms
+∥f∥L2 :=
+���
+Ω×S2 |f(x, w)|2 dwdx
+� 1
+2
+,
+|f|L2
+γ± :=
+��
+γ±
+|f(x, w)|2 dγ
+� 1
+2
+.
+(1.12)
+Define the L∞ norms
+∥f∥L∞ :=
+ess sup
+(x,w)∈Ω×S2
+��f(x, w)
+��,
+|f|L∞
+γ± := ess sup
+(x,w)∈γ±
+��f(x, w)
+��.
+(1.13)
+Let ∥·∥W k,p
+x
+denote the usual Sobolev norm for x ∈ Ω and |·|W k,p
+x
+for x ∈ ∂Ω, and ∥·∥W k,p
+x
+Lq
+w denote W k,p
+norm for x ∈ Ω and Lq norm for w ∈ S2. The similar notation also applies when we replace Lq by Lq
+γ. When
+there is no possibility of confusion, we will ignore the (x, w) variables in the norms.
+Throughout this paper, C > 0 denotes a constant that only depends on the domain Ω, but does not
+depend on the data or ε. It is referred as universal and can change from one inequality to another. We write
+a ≲ b to denote a ≤ Cb and a ≳ b to denote a ≥ Cb. Also, we write a ≃ b if a ≲ b and a ≳ b. We will use
+o(1) to denote a sufficiently small constant independent of the data.
+1.6. Main Results.
+Theorem 1.1. Under the assumption
+|g|W 3,∞L∞
+γ− ≲ 1,
+(1.14)
+there exists a unique solution uε(x, w) ∈ L∞(Ω × S2) to (1.1). Moreover, the solution obeys the estimate
+∥uε − U0∥L2 ≲ ε
+1
+2 .
+(1.15)
+
+4
+Y. GUO, L. WU
+Here U0(x) satisfies the Laplace equation with Dirichlet boundary condition
+�
+∆xU0(x) = 0 in Ω,
+U0(x0) = Φ∞(x0) on ∂Ω,
+(1.16)
+in which Φ∞(ι1, ι2) = Φ∞(x0) for x0 ∈ ∂Ω is given by solving the Milne problem for Φ(η, ι1, ι2, w)
+
+
+
+
+
+
+
+
+
+
+
+sin ϕ∂Φ
+∂η + Φ − Φ = 0,
+Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for
+sin ϕ > 0,
+lim
+η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2).
+(1.17)
+Remark 1.2. In [24, 22, 23] for 2D/3D convex domains, as well as [25] for 2D annulus domain, it is justified
+that for any 0 < δ ≪ 1
+���uε − �
+U0 − �
+U B
+0
+���
+L2 ≲ ε
+5
+6 −δ,
+(1.18)
+where �
+U B
+0 (η, ι1, ι2, w) is the boundary layer with geometric correction defined in (1.10), and �
+U0 is the cor-
+responding interior solution.
+[21, Theorem 2.1] reveals that the difference between two types of interior
+solutions
+����
+U0 − U0
+���
+L2 ≲ ε
+2
+3 .
+(1.19)
+Due to the rescaling η = ε−1µ, for general in-flow boundary data g, the boundary layer �
+U B
+0 ̸= 0 satisfies
+����
+U B
+0
+���
+L2 ≃ ε
+1
+2 .
+(1.20)
+Hence, we conclude that
+∥uε − U0∥L2 ≃ ε
+1
+2 .
+(1.21)
+Therefore, this indicates that (1.15) in Theorem 1.1 achieves the optimal L2 bound of the diffusive approxi-
+mation.
+1.7. Methodology. It is well-known that the key of the remainder estimate is to control R. In a series of
+work [24, 25, 7, 8, 22, 23] based on a L2 → L∞ framework, it is shown that
+��R
+��
+L2 ≲ ε−1 ��R − R
+��
+L2 ≲ 1
+(1.22)
+combined from the expected energy (entropy production) bound for ε−1 ��R − R
+��
+L2. This bound requires
+the next-order ε expansion of boundary layer approximation, which is impossible for non-convex domains,
+and barely possible by the new boundary layer theory with the ε-geometric correction. The key improvement
+in our work is
+��R
+��
+L2 ≲ ε
+1
+2
+(1.23)
+which is a consequence of the following conservation law for test function ξ(x) satisfying −∆xξ = R and
+ξ
+��
+∂Ω = 0:
+−
+�
+R, w · ∇xξ
+�
+= −
+�
+R − R, w · ∇xξ
+�
+=
+�
+S, ξ
+�
+,
+(1.24)
+where
+�
+R, w ·∇xξ
+�
+= 0 thanks to the oddness. This conservation law exactly cancels the worst contribution
+of ε−1 ��R − R
+��
+L2 in
+��R
+��
+L2 estimate, which comes from taking test function w · ∇xξ
+�
+γ
+R
+�
+w · ∇xξ
+�
+(w · n) −
+�
+R, w · ∇x
+�
+w · ∇xξ
+��
++ ε−1�
+R − R, w · ∇xξ
+�
+=
+�
+S, w · ∇xξ
+�
+.
+(1.25)
+Such a key cancellation produces an extra crucial gain of ε
+1
+2 . We then conclude the remainder estimate
+without any further expansion of the (singular) boundary layer approximation.
+
+DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION
+5
+In addition, we construct a new cut-off boundary layer near ϕ = 0 to avoid the singularity, and are able
+to perform delicate and precise estimates to control the resulting complex forcing term S (see (3.7)–(3.10)),
+in terms of the desired order ε for closure.
+2. Asymptotic Analysis
+2.1. Interior Solution. Inserting (1.6) into (1.1) and comparing the order of ε, following the analysis in
+[8, 23], we deduce that
+U0 = U 0,
+∆xU 0 = 0,
+(2.1)
+U1 = U 1 − w · ∇xU0,
+∆xU 1 = 0,
+(2.2)
+U2 = U 2 − w · ∇xU1,
+∆xU 2 = 0.
+(2.3)
+We need the boundary layer to determine the boundary conditions for U0, U 1 and U 2.
+2.2. Boundary Layer.
+2.2.1. Geometric Substitutions. The construction of boundary layer requires a local description in a neigh-
+borhood of the physical boundary ∂Ω. We follow the procedure in [8, 23]:
+Substitution 1: Spacial Substitution. Following the notation in Section 1.2, under the coordinate system
+(µ, ι1, ι2), we have
+w · ∇x = −(w · n) ∂
+∂µ −
+w · ς1
+L1(κ1µ − 1)
+∂
+∂ι1
+−
+w · ς2
+L2(κ2µ − 1)
+∂
+∂ι2
+,
+(2.4)
+where κi(ι1, ι2) for i = 1, 2 is the principal curvature.
+Substitution 2: Velocity Substitution. Under the orthogonal velocity substitution (1.4) for ϕ ∈
+�
+−π
+2 , π
+2
+�
+and
+ψ ∈ [−π, π], we have
+w · ∇x = sin ϕ ∂
+∂µ −
+� sin2 ψ
+R1 − µ + cos2 ψ
+R2 − µ
+�
+cos ϕ ∂
+∂ϕ + cos ϕ sin ψ
+L1(1 − κ1µ)
+∂
+∂ι1
++ cos ϕ cos ψ
+L2(1 − κ2µ)
+∂
+∂ι2
+(2.5)
++
+sin ψ
+R1 − µ
+�R1 cos ϕ
+L1L2
+�
+ς1 ·
+�
+ς2 ×
+�
+∂ι1ι2r × ς2
+���
+− sin ϕ cos ψ
+� ∂
+∂ψ
+− cos ψ
+R2 − µ
+�R2 cos ϕ
+L1L2
+�
+ς2 ·
+�
+ς1 ×
+�
+∂ι1ι2r × ς1
+���
+− sin ϕ sin ψ
+� ∂
+∂ψ,
+where Ri = κ−1
+i
+represents the radius of curvature. Note that the Jacobian dw = cos ϕdϕdψ will be present
+when we perform integration.
+Substitution 3: Scaling Substitution. Considering the scaled normal variable η = ε−1µ, we have
+w · ∇x =ε−1 sin ϕ ∂
+∂η −
+� sin2 ψ
+R1 − εη + cos2 ψ
+R2 − εη
+�
+cos ϕ ∂
+∂ϕ + R1 cos ϕ sin ψ
+L1(R1 − εη)
+∂
+∂ι1
++ R2 cos ϕ cos ψ
+L2(R2 − εη)
+∂
+∂ι2
+(2.6)
++
+sin ψ
+R1 − εη
+�R1 cos ϕ
+L1L2
+�
+ς1 ·
+�
+ς2 ×
+�
+∂ι1ι2r × ς2
+���
+− sin ϕ cos ψ
+� ∂
+∂ψ
+−
+cos ψ
+R2 − εη
+�R2 cos ϕ
+L1L2
+�
+ς2 ·
+�
+ς1 ×
+�
+∂ι1ι2r × ς1
+���
+− sin ϕ sin ψ
+� ∂
+∂ψ .
+2.2.2. Milne Problem. Let Φ(η, ι1, ι2, w) be the solution to the Milne problem
+sin ϕ∂Φ
+∂η + Φ − Φ =0,
+Φ(η, ι1, ι2) = 1
+4π
+� π
+−π
+�
+π
+2
+− π
+2
+Φ(η, ι1, ι2, w) cos ϕdϕdψ,
+(2.7)
+with boundary condition
+Φ(0, ι1, ι2, w) = g(ι1, ι2, w) for
+sin ϕ > 0.
+(2.8)
+
+6
+Y. GUO, L. WU
+We are interested in the solution that satisfies
+lim
+η→∞ Φ(η, ι1, ι2, w) = Φ∞(ι1, ι2)
+(2.9)
+for some Φ∞(ι1, ι2). Based on [8, Section 4], we have the well-posedness and regularity of (2.7).
+Proposition 2.1. Under the assumption (1.14), there exist Φ∞(ι1, ι2) and a unique solution Φ to (2.7)such
+that Ψ := Φ − Φ∞ satisfies
+
+
+
+
+
+
+
+
+
+
+
+sin ϕ∂Ψ
+∂η + Ψ − Ψ = 0,
+Ψ(0, ι1, ι2, w) = g(ι1, ι2, w) − Φ∞(ι1, ι2),
+lim
+η→0 Ψ(η, ι1, ι2, w) = 0,
+(2.10)
+and for some constant K > 0 and any 0 < r ≤ 3
+|Φ∞|W 3,∞
+ι1,ι2 +
+��eKηΨ
+��
+L∞ ≲1,
+(2.11)
+����eKη sin ϕ∂Ψ
+∂η
+����
+L∞ +
+����eKη sin ϕ∂Ψ
+∂ϕ
+����
+L∞ +
+����eKη ∂Ψ
+∂ψ
+����
+L∞ ≲1,
+(2.12)
+����eKη ∂rΨ
+∂ιr
+1
+����
+L∞
++
+����eKη ∂rΨ
+∂ιr
+2
+����
+L∞
+≲1.
+(2.13)
+Let χ(y) ∈ C∞(R) and �χ(y) = 1 − χ(y) be smooth cut-off functions satisfying χ(y) = 1 if |y| ≤ 1 and
+χ(y) = 0 if |y| ≥ 2. We define the boundary layer
+U B
+0 (η, ι1, ι2, w) := �χ
+�
+ε−1ϕ
+�
+χ(εη)Ψ(η, ι1, ι2, w).
+(2.14)
+Remark 2.2. Due to the cutoff in (2.14), we have
+U B
+0 (0, ι1, ι2, w) = �χ
+�
+ε−1ϕ
+��
+g(ι1, ι2, w) − Φ∞(ι1, ι2)
+�
+= �χ
+�
+ε−1ϕ
+�
+Ψ(0, ι1, ι2, w),
+(2.15)
+and
+sin ϕ∂U B
+0
+∂η
++ U B
+0 − U B
+0 = −�χ
+�
+ε−1ϕ
+�
+χ(εη)Ψ + Ψ�χ(ε−1ϕ)χ(εη).
+(2.16)
+2.3. Matching Procedure. We plan to enforce the matching condition for x0 ∈ ∂Ω and w · n < 0
+U0(x0) + U B
+0 (x0, w) =g(x0, w) + O(ε).
+(2.17)
+Considering (2.15), it suffices to require
+U0(x0) = Φ∞(x0) := Φ∞(ι1, ι2),
+(2.18)
+which yields
+U0(x0) + Ψ(x0, w) =g(x0, w).
+(2.19)
+Hence, we obtain
+U0(x0, w) + U B
+0 (x0, w) = g(x0, w) + χ
+�
+ε−1ϕ
+�
+Ψ(0, ι1, ι2, w).
+(2.20)
+Construction of U0. Based on (2.1) and (2.18), define U0(x) satisfying
+U0 = U0,
+∆xU 0 = 0,
+U0(x0) = Φ∞(x0).
+(2.21)
+From standard elliptic estimates [9] and Proposition 2.1, we have for any s ∈ [2, ∞)
+∥U0∥W 3+ 1
+s ,s + |U0|W 3,s ≲ 1.
+(2.22)
+Construction of U1. Based on (2.2), define U1(x, w) satisfying
+U1 = U 1 − w · ∇xU0,
+∆xU 1 = 0,
+U 1(x0) = 0.
+(2.23)
+From (2.22), we have for any s ∈ [2, ∞)
+∥U1∥W 2+ 1
+s ,sL∞ + |U1|W 2,sL∞ ≲ 1.
+(2.24)
+
+DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION
+7
+Construction of U2. Based on (2.2), define U2(x, w) satisfying
+U2 = U 2 − w · ∇xU1,
+∆xU 2 = 0,
+U 2(x0) = 0.
+(2.25)
+From (2.24), we have for any s ∈ [2, ∞)
+∥U2∥W 1+ 1
+s ,sL∞ + |U2|W 1,sL∞ ≲ 1.
+(2.26)
+Summarizing the above analysis, we have the well-posedness and regularity estimates of the interior
+solution and boundary layer:
+Proposition 2.3. Under the assumption (1.14), we can construct U0, U1, U2, U B
+0 as in (2.21)(2.23)(2.25)(2.14)
+satisfying for any s ∈ [2, ∞)
+∥U0∥W 3+ 1
+s ,s + |U0|W 3,s ≲1,
+(2.27)
+∥U1∥W 2+ 1
+s ,sL∞ + |U1|W 2,sL∞ ≲1,
+(2.28)
+∥U2∥W 1+ 1
+s ,sL∞ + |U2|W 1,sL∞ ≲1,
+(2.29)
+and for some constant K > 0 and any 0 < r ≤ 3
+��eKηU B
+0
+��
+L∞ +
+����eKη ∂rU B
+0
+∂ιr
+1
+����
+L∞
++
+����eKη ∂rU B
+0
+∂ιr
+2
+����
+L∞
+≲1.
+(2.30)
+3. Remainder Equation
+Denote the approximate solution
+ua :=
+�
+U0 + εU1 + ε2U2
+�
++ U B
+0 .
+(3.1)
+Inserting (1.5) into (1.1), we have
+w · ∇x
+�
+ua + R
+�
++ ε−1�
+ua + R
+�
+− ε−1�
+ua + R
+�
+= 0,
+�
+ua + R
+����
+γ−
+= g,
+(3.2)
+which yields
+w · ∇xR + ε−1�
+R − R
+�
+= −w · ∇xua − ε−1�
+ua − ua
+�
+,
+R
+���
+γ− =
+�
+g − ua
+����
+γ−.
+(3.3)
+3.1. Formulation of Remainder Equation. We consider the remainder equation
+
+
+
+w · ∇xR + ε−1�
+R − R
+�
+= S in Ω × S2,
+R(x0, w) = h(x0, w) for w · n < 0 and x0 ∈ ∂Ω,
+(3.4)
+where R(x) = 1
+4π
+�
+S2 R(x, w)dw. Here the boundary data h is given by
+h := −εw · ∇xU0 − ε2w · ∇xU1 − χ
+�
+ε−1ϕ
+�
+Ψ(0),
+(3.5)
+and the source term S is given by
+S := S0 + S1 + S2 + S3,
+(3.6)
+
+8
+Y. GUO, L. WU
+where
+S0 := − ε2w · ∇xU2,
+(3.7)
+S1 :=
+� sin2 ψ
+R1 − εη + cos2 ψ
+R2 − εη
+�
+cos ϕ∂U B
+0
+∂ϕ ,
+(3.8)
+S2 :=ε−1 sin φ�χ
+�
+ε−1ϕ
+�∂χ(εη)
+∂η
+Ψ + R1 cos ϕ sin ψ
+L1(R1 − εη)
+∂U B
+0
+∂ι1
++ R2 cos ϕ cos ψ
+L2(R2 − εη)
+∂U B
+0
+∂ι2
+(3.9)
++
+sin ψ
+R1 − εη
+�R1 cos ϕ
+L1L2
+�
+ς1 ·
+�
+ς2 ×
+�
+∂ι1ι2r × ς2
+���
+− sin ϕ cos ψ
+�∂U B
+0
+∂ψ
+−
+cos ψ
+R2 − εη
+�R2 cos ϕ
+L1L2
+�
+ς2 ·
+�
+ς1 ×
+�
+∂ι1ι2r × ς1
+���
+− sin ϕ sin ψ
+�∂U B
+0
+∂ψ ,
+S3 :=ε−1
+�
+�χ
+�
+ε−1ϕ
+�
+χ(εη)Ψ − Ψ�χ
+�
+ε−1ϕ
+�
+χ(εη)
+�
+.
+(3.10)
+3.2. Weak Formulation.
+Lemma 3.1 (Green’s Identity, Lemma 2.2 of [6]). Assume f(x, w), g(x, w) ∈ L2(Ω × S2) and w · ∇xf, w ·
+∇xg ∈ L2(Ω × S2) with f, g ∈ L2
+γ. Then
+��
+Ω×S2
+��
+w · ∇xf
+�
+g +
+�
+w · ∇xg
+�
+f
+�
+dxdw =
+�
+γ
+fg(w · n) =
+�
+γ+
+fgdγ −
+�
+γ−
+fgdγ.
+(3.11)
+Using Lemma 3.1, we can derive the weak formulation of (3.4). For any test function g(x, w) ∈ L2(Ω×S2)
+with w · ∇xg ∈ L2(Ω × S2) with g ∈ L2
+γ, we have
+�
+γ
+Rg(w · n) −
+��
+Ω×S2 R
+�
+w · ∇xg
+�
++ ε−1
+��
+Ω×S2
+�
+R − R
+�
+g =
+��
+Ω×S2 Sg.
+(3.12)
+3.3. Estimates of Boundary and Source Terms.
+Lemma 3.2. Under the assumption (1.14), for h defined in (3.5), we have
+|h|L2
+γ− ≲ ε.
+(3.13)
+Proof. Based on Proposition 2.3, we have
+|εw · ∇xU0|L2
+γ− +
+��ε2w · ∇xU1
+��
+L2γ− ≲ ε.
+(3.14)
+Noting the cutoff χ
+�
+ε−1ϕ
+�
+restricts the support to |ϕ| ≲ ε and dγ measure contributes an extra sin ϕ, we
+have
+��χ
+�
+ε−1ϕ
+�
+Ψ(0)
+��
+L2γ− ≲ ε.
+(3.15)
+Hence, our result follows.
+□
+Lemma 3.3. Under the assumption (1.14), for S0 defined in (3.7), we have
+∥S0∥L2 ≲ ε2.
+(3.16)
+Proof. This follows from Proposition 2.3.
+□
+Lemma 3.4. Under the assumption (1.14), for S1 defined in (3.8), we have
+���
+1 + η
+�
+S1
+��
+L2 ≲ 1.
+(3.17)
+Also, for the boundary layer U B
+0 defined in (2.14), we have
+���
+1 + η
+�
+U B
+0
+��
+L2 ≲ ε
+1
+2 ,
+���
+1 + η
+�
+U B
+0
+��
+L2xL1w ≲ ε
+1
+2 ,
+(3.18)
+and
+���
+��
+1 + η
+�
+S1, g
+���� ≲
+���
+�
+1 + η
+�
+⟨v⟩2 U B
+0
+���
+L2 ∥∇wg∥L2 ≲ ε
+1
+2 ∥∇wg∥L2 .
+(3.19)
+
+DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION
+9
+Proof. We split
+S1 = S11 + S12 :=
+� sin2 ψ
+R1 − εη + cos2 ψ
+R2 − εη
+�
+cos ϕ∂Ψ
+∂ϕ �χ
+�
+ε−1ϕ
+�
+χ(εη)
+(3.20)
++
+� sin2 ψ
+R1 − εη + cos2 ψ
+R2 − εη
+�
+cos ϕ∂�χ
+�
+ε−1ϕ
+�
+∂ϕ
+χ(εη)Ψ.
+Note that S11 is nonzero only when |ϕ| ≥ ε and thus based on Proposition 2.1, we know
+����
+∂Ψ
+∂ϕ
+���� ≤ |sin ϕ|−1 |Ψ| ≲
+ε−1. Hence, using dµ = εdη, we have
+∥S11∥L2 ≲
+���
+|ϕ|≥ε
+����
+∂Ψ
+∂ϕ
+����
+2
+dϕdµ
+� 1
+2
+≲
+���
+|ϕ|≥ε
+|sin ϕ|−2 |Ψ|2 dϕdµ
+� 1
+2
+(3.21)
+≲
+���
+|ϕ|≥ε
+|sin ϕ|−2 e−2Kηdϕdµ
+� 1
+2
+≲
+�
+ε
+��
+|ϕ|≥ε
+|sin ϕ|−2 e−2Kηdϕdη
+� 1
+2
+≲
+�
+εε−1� 1
+2 = 1.
+Noticing ∂�χ
+�
+ε−1ϕ
+�
+∂ϕ
+= ε−1�χ′�
+ε−1ϕ
+�
+, and �χ′�
+ε−1ϕ
+�
+is nonzero only when ε < |ϕ| < 2ε, based on Proposition
+2.1, we have
+∥S12∥L2 ≲ε−1
+���
+ε<|ϕ|<2ε
+|Ψ|2 dϕdµ
+� 1
+2
+≲ ε−1
+���
+ε<|ϕ|<2ε
+e−2Kηdϕdµ
+� 1
+2
+(3.22)
+≲ε−1
+�
+ε
+��
+ε<|ϕ|<2ε
+e−2Kηdϕdη
+� 1
+2
+≲ ε−1 (εε)
+1
+2 = 1.
+Collecting (3.21) and (3.22), we have (3.17). Note that e−Kη will suppress the growth from the pre-factor
+1 + η.
+(3.18) comes from Proposition 2.1.
+Then we turn to (3.19).
+The most difficult term in
+�� ⟨S1, g⟩
+�� is
+essentially
+����
+�∂U B
+0
+∂ϕ , g
+�����. Integration by parts with respect to ϕ implies
+����
+�∂U B
+0
+∂ϕ , g
+����� ≲
+����
+�
+U B
+0 , ∂g
+∂ϕ
+����� ≲
+��U B
+0
+��
+L2
+����
+∂g
+∂ϕ
+����
+L2 .
+(3.23)
+From (1.4) and ∂x
+∂ϕ = 0, we know the substitution (µ, ι1, ι2, w) → (µ, ι1, ι2, w) implies
+−∂w
+∂ϕ · n = cos ϕ,
+∂w
+∂ϕ · ς1 = − sin ϕ sin ψ,
+∂w
+∂ϕ · ς2 = − sin ϕ cos ψ.
+(3.24)
+Hence, we know
+����
+∂w
+∂ϕ
+���� ≲ 1, and thus
+����
+∂g
+∂ϕ
+���� ≲ |∇wg|
+����
+∂w
+∂ϕ
+���� ≲ |∇wg| .
+(3.25)
+Hence, we know that
+����
+�∂U B
+0
+∂ϕ , g
+����� ≲
+��U B
+0
+��
+L2 ∥∇wg∥L2 ≲ ε
+1
+2 ∥∇wg∥L2 .
+(3.26)
+□
+Lemma 3.5. Under the assumption (1.14), for S2 defined in (3.9), we have
+���
+1 + η
+�
+S2
+��
+L2 ≲ ε
+1
+2 ,
+���
+1 + η
+�
+S2
+��
+L2xL1w ≲ ε
+1
+2 .
+(3.27)
+
+10
+Y. GUO, L. WU
+Proof. Notice that
+����ε−1 sin φ�χ
+�
+ε−1ϕ
+�∂χ(εη)
+∂η
+���� ≲ 1. Based on Proposition 2.1 and Proposition 2.3, we directly
+bound
+∥S2∥L2 ≲
+��� �
+|Φ|2 +
+����
+∂Φ
+∂ι1
+����
+2
++
+����
+∂Φ
+∂ι2
+����
+2
++
+����
+∂Φ
+∂ψ
+����
+2 �
+dϕdµ
+� 1
+2
+(3.28)
+≲
+���
+e−2Kηdϕdµ
+� 1
+2
+≲
+�
+ε
+��
+e−2Kηdϕdη
+� 1
+2
+≲ ε
+1
+2 .
+Then the L2
+xL1
+w estimate follows from a similar argument noting that there is no rescaling in w variables.
+□
+Lemma 3.6. Under the assumption (1.14), for S3 defined in (3.10), we have
+���
+1 + η
+�
+S3
+��
+L2 ≲ 1,
+���
+1 + η
+�
+S3
+��
+L2xL1w ≲ ε
+1
+2 .
+(3.29)
+Proof. Using χ = 1 − �χ, we split
+S3 = S31 + S32 :=ε−1Ψχ
+�
+ε−1ϕ
+�
+χ(εη) − ε−1χ
+�
+ε−1ϕ
+�
+χ(εη)Ψ.
+(3.30)
+Noting that S31 is nonzero only when |ϕ| ≤ ε, based on Proposition 2.1, we have
+∥S31∥L2 ≲
+���
+|ϕ|≤ε
+��ε−1Ψ
+��2 dϕdµ
+� 1
+2
+≲
+�
+ε−2
+��
+|ϕ|≤ε
+e−2Kηdϕdµ
+� 1
+2
+(3.31)
+≲
+�
+ε−1
+��
+|ϕ|≤ε
+e−2Kηdϕdη
+� 1
+2
+≲
+�
+ε−1ε
+� 1
+2 ≲ 1.
+Analogously, noting that S32 contains w integral, we have
+∥S32∥L2 ≲
+���� ���ε−1Ψχ(ε−1ϕ)
+���
+2
+dϕdµ
+� 1
+2
+≲
+
+ε−2
+�� �����
+�
+|ϕ|≤ε
+Ψdϕ
+�����
+2
+dϕdµ
+
+
+1
+2
+(3.32)
+≲
+
+ε−2
+�� �����
+�
+|ϕ|≤ε
+e−Kηdϕ
+�����
+2
+dϕdµ
+
+
+1
+2
+≲
+�
+ε−2
+��
+ε2e−2Kηdϕdµ
+� 1
+2
+≲
+���
+e−2Kηdϕdµ
+� 1
+2
+≲
+�
+ε
+��
+e−2Kηdϕdη
+� 1
+2
+≲ ε
+1
+2 .
+Collecting (3.31) and (3.32), we have the L2 estimate. Similarly, we derive the L2
+xL1
+w bound:
+∥S31∥L2xL1w ≲
+�� � �
+|ϕ|≤ε
+��ε−1Ψ
+�� dϕ
+�2
+dµ
+� 1
+2
+≲
+��
+e−2Kηdµ
+� 1
+2
+≲
+�
+ε
+�
+e−2Kηdη
+� 1
+2
+≲ ε
+1
+2 ,
+(3.33)
+∥S32∥L2xL1w ≲
+�� � � ���ε−1Ψχ(ε−1ϕ)
+��� dϕ
+�2
+dµ
+� 1
+2
+≲
+�
+ε−2
+� � � �����
+�
+|ϕ|≤ε
+Ψdϕ
+����� dϕ
+�2
+dµ
+� 1
+2
+(3.34)
+≲
+�
+ε−2
+� � �
+εe−Kηdϕ
+�2
+dµ
+� 1
+2
+≲
+��
+e−2Kηdµ
+� 1
+2
+≲
+�
+ε
+�
+e−2Kηdη
+� 1
+2
+≲ ε
+1
+2 .
+□
+4. Remainder Estimate
+4.1. Basic Energy Estimate.
+Lemma 4.1. Under the assumption (1.14), we have
+ε−1 |R|2
+L2γ+ + ε−2 ��R − R
+��2
+L2 ≲ o(1)ε−1 ��R
+��2
+L2 + 1.
+(4.1)
+
+DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION
+11
+Proof. Taking g = ε−1R in (3.12), we obtain
+ε−1
+2
+�
+γ
+|R|2 (w · n) + ε−2�
+R, R − R
+�
+= ε−1�
+R, S
+�
+.
+(4.2)
+Then using the orthogonality of R and R − R, we have
+ε−1
+2
+|R|2
+L2γ+ + ε−2 ��R − R
+��2
+L2 = ε−1�
+R, S
+�
++ ε−1
+2
+|h|2
+L2γ− .
+(4.3)
+Using Lemma 3.2, we know
+ε−1 |R|2
+L2γ+ + ε−2 ��R − R
+��2
+L2 ≲ ε + ε−1�
+R, S0 + S1 + S2 + S3
+�
+.
+(4.4)
+Using Lemma 3.3, we have
+���ε−1�
+R, S0
+���� ≲ ε−1 ∥R∥L2 ∥S0∥L2 ≲ ε ∥R∥L2 ≲ o(1) ∥R∥2
+L2 + ε2.
+(4.5)
+Using Lemma 3.4, Lemma 3.5 and Lemma 3.6, we have
+���ε−1�
+R − R, S1 + S2 + S3
+���� ≲ε−1 ��R − R
+��
+L2 ∥S1 + S2 + S3∥L2
+(4.6)
+≲ε−1 ��R − R
+��
+L2 ≲ o(1)ε−2 ��R − R
+��2
+L2 + 1.
+Finally, we turn to ε−1�
+R, S1 + S2 + S3
+�
+. For S1, we integrate by parts with respect to ϕ and use Lemma
+3.4 to obtain
+���ε−1�
+R, S1
+���� =ε−1
+����
+�
+R,
+� sin2 ψ
+R1 − εη + cos2 ψ
+R2 − εη
+�
+cos ϕ∂U B
+0
+∂ϕ
+�����
+(4.7)
+=ε−1
+����
+�
+R,
+� sin2 ψ
+R1 − εη + cos2 ψ
+R2 − εη
+�
+U B
+0 sin ϕ
+�����
+≲ε−1 ��R
+��
+L2
+��U B
+1
+��
+L2xL1w ≲ ε− 1
+2 ��R
+��
+L2 ≲ o(1)ε−1 ��R
+��2
+L2 + 1.
+Also, Lemma 3.5 and Lemma 3.6 yield
+���ε−1�
+R, S2 + S3
+���� ≲ε−1 ��R
+��
+L2
+�
+∥S2∥L2xL1w + ∥S3∥L2xL1w
+�
+≲ ε− 1
+2 ��R
+��
+L2 ≲ o(1)ε−1 ��R
+��2
+L2 + 1.
+(4.8)
+Collecting (4.5)(4.6)(4.7)(4.8), we obtain
+���ε−1�
+R, S0 + S1 + S2 + S3
+���� ≲ o(1)ε−2 ��R − R
+��2
+L2 + o(1)ε−1 ∥R∥2
+L2 + 1.
+(4.9)
+Combining (4.9) and (4.4), we have (4.1).
+□
+4.2. Kernel Estimate.
+Lemma 4.2. Under the assumption (1.14), we have
+��R
+��2
+L2 ≲
+��R − R
+��2
+L2 + |R|2
+L2γ+ + ε.
+(4.10)
+Proof. Denote ξ(x) satisfying
+�
+−∆xξ = R in Ω,
+ξ(x0) = 0 on ∂Ω.
+(4.11)
+Based on standard elliptic estimates and trace estimates, we have
+∥ξ∥H2 + |ξ|H
+3
+2 ≲
+��R
+��
+L2 .
+(4.12)
+Taking g = ξ in (3.12), we have
+�
+γ
+Rξ(w · n) −
+�
+R, w · ∇xξ
+�
++ ε−1�
+R − R, ξ
+�
+=
+�
+S, ξ
+�
+.
+(4.13)
+Using oddness, orthogonality and ξ
+��
+∂Ω = 0, we obtain (1.24).
+Then taking g = w · ∇xξ in (3.12), we obtain (1.25).
+
+12
+Y. GUO, L. WU
+Adding ε−1×(1.24) and (1.25) to eliminate ε−1�
+R − R, w · ∇xξ
+�
+, we obtain
+�
+γ
+R
+�
+w · ∇xξ
+�
+(w · n) −
+�
+R, w · ∇x
+�
+w · ∇xξ
+��
+=ε−1�
+S, ξ
+�
++
+�
+S, w · ∇xξ
+�
+.
+(4.14)
+Notice that
+−
+�
+R, w · ∇x
+�
+w · ∇xξ
+��
+= −
+�
+R, w · ∇x
+�
+w · ∇xξ
+��
+−
+�
+R − R, w · ∇x
+�
+w · ∇xξ
+��
+,
+(4.15)
+where (4.12) and Cauchy’s inequality yield
+−
+�
+R, w · ∇x
+�
+w · ∇xξ
+��
+≃
+��R
+��2
+L2 ,
+(4.16)
+���
+�
+R − R, w · ∇x
+�
+w · ∇xξ
+����� ≲
+��R − R
+��2
+L2 + o(1)
+��R
+��2
+L2 .
+(4.17)
+Also, using (4.12) and Lemma 3.2, we have
+����
+�
+γ
+R
+�
+w · ∇xξ
+�
+(w · n)
+���� ≲
+�
+|R|L2γ+ + |h|L2γ−
+�
+|∇xξ|L2 ≲ o(1)
+��R
+��2
+L2 + |R|2
+L2γ+ + ε2.
+(4.18)
+Inserting (4.15)–(4.18) into (4.14), we obtain
+��R
+��2
+L2 ≲ε2 +
+��R − R
+��2
+L2 + |R|2
+L2γ+ +
+���ε−1�
+S, ξ
+���� +
+���
+�
+S, w · ∇xξ
+���� .
+(4.19)
+Then we turn to the estimate of source terms in (4.19). Cauchy’s inequality and Lemma 3.3 yield
+���ε−1�
+S0, ξ
+���� +
+���
+�
+S0, w · ∇xξ
+���� ≲ ε−1 ∥S0∥L2 ∥ξ∥H1 ≲ ε
+��R
+��
+L2 ≲ o(1)
+��R
+��2
+L2 + ε2.
+(4.20)
+Similar to (4.7), we first integrate by parts with respect to ϕ in S1. Using ξ
+��
+∂Ω = 0, (4.12), Hardy’s inequality
+and Lemma 3.4, Lemma 3.5, Lemma 3.6, we have
+���ε−1�
+S1 + S2 + S3, ξ
+���� ≲
+����ε−1�
+U B
+0 + S2 + S3,
+� µ
+0
+∂ξ
+∂µ
+����� =
+����
+�
+ηU B
+0 + ηS2 + ηS3, 1
+µ
+� µ
+0
+∂ξ
+∂µ
+�����
+(4.21)
+≲
+��ηU B
+0 + ηS2 + ηS3
+��
+L2xL1w
+����
+1
+µ
+� µ
+0
+∂ξ
+∂µ
+����
+L2
+≲
+��ηU B
+0 + ηS2 + ηS3
+��
+L2xL1w
+����
+∂ξ
+∂µ
+����
+L2
+≲ ε
+1
+2 ∥ξ∥H1
+≲ε
+1
+2 ��R
+��
+L2 ≲ o(1)
+��R
+��2
+L2 + ε.
+Analogously, we integrate by parts with respect to ϕ in S1. Then using (4.12), fundamental theorem of
+calculus, Hardy’s inequality and Lemma 3.4, Lemma 3.5, Lemma 3.6, we bound
+���
+�
+S1 + S2 + S3, w · ∇xξ
+���� ≲
+�����
+�
+U B
+0 + S2 + S3, ∇xξ
+���
+µ=0 +
+� µ
+0
+∂
+�
+∇xξ
+�
+∂µ
+������
+(4.22)
+≲
+����
+�
+U B
+0 + S2 + S3, ∇xξ
+���
+µ=0
+����� +
+�����ε
+�
+ηU B
+0 + ηS2 + ηS3, 1
+µ
+� µ
+0
+∂
+�
+∇xξ
+�
+∂µ
+������
+≲
+��U B
+0 + S2 + S3
+��
+L2xL1w |∇xξ|L2 + ε
+��ηU B
+0 + ηS2 + ηS3
+��
+L2
+�����
+∂
+�
+∇xξ
+�
+∂µ
+�����
+L2
+≲ε
+1
+2 |∇xξ|L2
+∂Ω + ε ∥ξ∥H2 ≲ ε
+1
+2 ��R
+��
+L2 ≲ o(1)
+��R
+��2
+L2 + ε.
+Hence, inserting (4.20), (4.21) and (4.22) into (4.19), we have shown (4.10).
+□
+4.3. Synthesis.
+Proposition 4.3. Under the assumption (1.14), we have
+ε− 1
+2 |R|L2γ+ + ε− 1
+2 ��R
+��
+L2 + ε−1 ��R − R
+��
+L2 ≲ 1.
+(4.23)
+
+DIFFUSIVE EXPANSION OF NEUTRON TRANSPORT EQUATION
+13
+Proof. From (4.1), we have
+ε−1 |R|2
+L2γ+ + ε−2 ��R − R
+��2
+L2 ≲ o(1)ε−1 ��R
+��2
+L2 + 1.
+(4.24)
+From (4.10), we have
+��R
+��2
+L2 ≲
+��R − R
+��2
+L2 + |R|2
+L2γ+ + ε.
+(4.25)
+Inserting (4.25) into (4.24), we have
+ε−1 |R|2
+L2γ+ + ε−2 ��R − R
+��2
+L2 ≲ 1.
+(4.26)
+Inserting (4.26) into (4.25), we have
+��R
+��2
+L2 ≲ ε.
+(4.27)
+Hence, adding ε−1×(4.27) and (4.26), we have
+ε−1 |R|2
+L2γ+ + ε−1 ��R
+��2
+L2 + ε−2 ��R − R
+��2
+L2 ≲ 1.
+(4.28)
+Then our result follows.
+□
+5. Proof of Main Theorem
+The well-posedness of (1.1) is well-known [5, 4, 24]. The construction of U0, Φ and Φ∞ follows from
+Proposition 2.1 and Proposition 2.3, so we focus on the derivation of (1.15).
+Based on Proposition 4.3 and (1.5), we have
+��uε − U0 − εU1 − ε2U2 − U B
+0
+��
+L2 ≲ ε
+1
+2 .
+(5.1)
+Using Proposition 2.3, we have
+��εU1 + ε2U2
+��
+L2 ≲ ε.
+(5.2)
+Using Proposition 2.3 and the rescaling η = ε−1µ, we have
+��U B
+0
+��
+L2 ≲ ε
+1
+2 .
+(5.3)
+Then (1.15) follows from inserting (5.2)(5.3) into (5.1).
+References
+[1] C. Bardos, F. Golse, and B. Perthame, The Rosseland approximation for the radiative transfer equations, Comm. Pure
+Appl. Math., 40 (1987), pp. 69–721.
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+and Rosseland approximation, J. Funct. Anal., 77 (1988), pp. 434–460.
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+[7] Y. Guo and L. Wu, Geometric correction in diffusive limit of neutron transport equation in 2D convex domains, Arch.
+Rational Mech. Anal., 226 (2017), pp. 321–403.
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+, Regularity of Milne problem with geometric correction in 3D, Math. Models Methods Appl. Sci., 27 (2017), pp. 453–
+524.
+[9] N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces., American Mathematical Society, Providence,
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+tering, Comm. Pure Appl. Math., 27 (1974), pp. 523–545.
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+(1974), pp. 299–305.
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+Y. GUO, L. WU
+[14] E. W. Larsen and J. D’Arruda, Asymptotic theory of the linear transport equation for small mean free paths I, Phys.
+Rev., 13 (1976), pp. 1933–1939.
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+Appl. Math., 26 (1973), pp. 525–537.
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+pp. 1987–1997.
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+(1976), pp. 1812–1820.
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+2169.
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+, Diffusive limit of transport equation in 3D convex domains, Peking Math. J., 4 (2021), pp. 203–284.
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+Phys., 336 (2015), pp. 1473–1553.
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+(Y. Guo)
+Division of Applied Mathematics, Brown University
+Email address: yan guo@brown.edu
+(L. Wu)
+Department of Mathematics, Lehigh University
+Email address: lew218@lehigh.edu
+

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