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It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M\_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part., 15 pages

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics., Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:2012.00341

We prove that the spaces $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee \otimes_R\mathcal{T}_{\operatorname{poly}}^{\bullet}\big)$ and $\operatorname{tot}\big(\Gamma(\Lambda^\bullet A^\vee)\otimes_R\mathcal{D}_{\operatorname{poly}}^{\bullet}\big)$ associated with a Lie pair $(L,A)$ each carry an $L_\infty$ algebra structure canonical up to an $L_\infty$ isomorphism with the identity map as linear part. These two spaces serve, respectively, as replacements for the spaces of formal polyvector fields and formal polydifferential operators on the Lie pair $(L,A)$. Consequently, both $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{T}_{\operatorname{poly}}^{\bullet})$ and $\mathbb{H}^\bullet_{\operatorname{CE}}(A,\mathcal{D}_{\operatorname{poly}}^{\bullet})$ admit unique Gerstenhaber algebra structures. Our approach is based on homotopy transfer and the construction of a Fedosov dg Lie algebroid (i.e. a dg foliation on a Fedosov dg manifold)., Comment: [v2] 50 pages, paper was expanded; [v1] Paper arXiv:1605.09656v1 was expended and split into two papers. The first part is arXiv:1605.09656v2. The second part is the present paper. A new result addressing uniqueness of the constructed structures has been added

When the reduced twisted $C^*$-algebra $C^*_r(\mathcal{G}, c)$ of a non-principal groupoid $\mathcal{G}$ admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r(\mathcal{G}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid $\mathcal{S}$ of $\mathcal{G}$. In this paper, we study the relationship between the original groupoids $\mathcal{S}, \mathcal{G}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum $\mathfrak{B}$ of the Cartan subalgebra $C^*_r(\mathcal{S}, c)$. We then show that the quotient groupoid $\mathcal{G}/\mathcal{S}$ acts on $\mathfrak{B}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly we show that, if the quotient map $\mathcal{G}\to\mathcal{G}/\mathcal{S}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$., 32 pages

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the 195 826 352 conjugacy classes of transitive subgroups of the symmetric group S 48 of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive groups of degree 48, of which there are 1 538 868 366 , all but 0.1625% of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published as a separate paper.

This paper proves the conjectures (previously proved in the case n = 0 ) that for any Coxeter system ( W , S ) and any natural number n, the set of n-small roots for W is bipodal and the set of n-low elements of W is a Garside shadow. The proof here uses special cases ( k = 3 ) of properties (notably, their existence) of certain maximal rank k reflection subgroups of W, for non-negative integers k.

Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for the affine Artin group of type A ˜ discovered by Shi and Corran–Lee–Lee, I will construct interval Garside structures related to this group. This construction is the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups.

In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations., Comment: 132 pages

There is an embedding of affine vertex algebras $V^k(\mathfrak{gl}_n) \hookrightarrow V^k(\mathfrak{sl}_{n+1})$, and the coset $\mathcal{C}^k(n) = \text{Com}(V^k(\mathfrak{gl}_n), V^k(\mathfrak{sl}_{n+1}))$ is a natural generalization of the parafermion algebra of $\mathfrak{sl}_2$. It was called the algebra of generalized parafermions by the third author and was shown to arise as a one-parameter quotient of the universal two-parameter $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,3,\dots)$. In this paper, we consider an analogous structure of orthogonal type, namely $\mathcal{D}^k(n) = \text{Com}(V^k(\mathfrak{so}_{2n}), V^k(\mathfrak{so}_{2n+1}))^{\mathbb{Z}_2}$. We realize this algebra as a one-parameter quotient of the two-parameter even spin $\mathcal{W}_{\infty}$-algebra of type $\mathcal{W}(2,4,\dots)$, and we classify all coincidences between its simple quotient $\mathcal{D}_k(n)$ and the algebras $\mathcal{W}_{\ell}(\mathfrak{so}_{2m+1})$ and $\mathcal{W}_{\ell}(\mathfrak{so}_{2m})^{\mathbb{Z}_2}$. As a corollary, we show that for the admissible levels $k = -(2n-2) + \frac{1}{2} (2 n + 2 m -1)$ for $\widehat{\mathfrak{so}}_{2n}$ the simple affine algebra $L_k(\mathfrak{so}_{2n})$ embeds in $L_k(\mathfrak{so}_{2n+1})$, and the coset is strongly rational. As a consequence, the category of ordinary modules of $L_k(\mathfrak{so}_{2n+1})$ at such a level is a braided fusion category., Comment: Minor corrections, final version to appear in J. Algebra

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups., 22 pages, 2 figures; V3: Final version, to appear in Izvestiya: Mathematics

Throughout this paper, all groups are finite and G always denotes a finite group. Let σ = { σ i | i ∈ I } be a partition of the set of all primes P . The group G is said to be: σ-primary if G is a σ i -group for some i = i ( G ) ; σ-nilpotent if G = G 1 × … × G n for some σ-primary groups G 1 , … , G n ; σ-soluble if every chief factor of G is σ-primary; σ-full if G possesses a Hall σ i -subgroup for all i such that σ i ∩ π ( G ) ≠ ∅ . A subgroup A of G is said to be σ-permutable in G provided G is σ-full and A permutes with every Hall σ i -subgroup H of G, that is, A H = H A for all i; G is said to be a PσT-group if σ-permutability is a transitive relation in G, that is, if K is a σ-permutable subgroup of H and H is a σ-permutable subgroup of G, then K is a σ-permutable subgroup of G. Let F be a class of group. Then a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F . We prove that: If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then Σ is a G-covering subgroup system for the classes of all σ-soluble groups, all σ-nilpotent groups, and all σ-soluble PσT-groups. This result gives positive answers to Questions 19.87 and 19.88 in the Kourovka Notebook and, also, allows us to obtain the following characterization of σ-soluble PσT-groups: G is a σ-soluble PσT-group if and only if each maximal subgroup of every Sylow subgroup of G has a supplement T in G such that T is a σ-soluble PσT-group.

Let G be a finite group, and T ( G ) be the sum of all complex irreducible character degrees of G. In this paper, we aim to characterize the structure of finite groups in terms of T ( G ) . We show that if | G | / T ( G ) ( p + 1 ) / 2 , then G has a normal Sylow p-subgroup; and that if | G | / T ( G ) 3 p 2 / ( p 2 + 2 ) , then G is p-supersolvable, where p is a prime.

The congruence subgroup problem for a finitely generated group Γ and for G ≤ Aut ⁢ ( Γ ) G\leq\mathrm{Aut}(\Gamma) asks whether the map G ^ → Aut ⁢ ( Γ ^ ) \hat{G}\to\mathrm{Aut}(\hat{\Gamma}) is injective, or more generally, what its kernel C ⁢ ( G , Γ ) C(G,\Gamma) is. Here X ^ \hat{X} denotes the profinite completion of 𝑋. In the case G = Aut ⁢ ( Γ ) G=\mathrm{Aut}(\Gamma) , we write C ⁢ ( Γ ) = C ⁢ ( Aut ⁢ ( Γ ) , Γ ) C(\Gamma)=C(\mathrm{Aut}(\Gamma),\Gamma) . Let Γ be a finitely generated group, Γ ¯ = Γ / [ Γ , Γ ] \bar{\Gamma}=\Gamma/[\Gamma,\Gamma] , and Γ * = Γ ¯ / tor ⁢ ( Γ ¯ ) ≅ Z ( d ) \Gamma^{*}=\bar{\Gamma}/\mathrm{tor}(\bar{\Gamma})\cong\mathbb{Z}^{(d)} . Define Aut * ⁢ ( Γ ) = Im ⁡ ( Aut ⁢ ( Γ ) → Aut ⁢ ( Γ * ) ) ≤ GL d ⁢ ( Z ) . \mathrm{Aut}^{*}(\Gamma)=\operatorname{Im}(\mathrm{Aut}(\Gamma)\to\mathrm{Aut}(\Gamma^{*}))\leq\mathrm{GL}_{d}(\mathbb{Z}). In this paper we show that, when Γ is nilpotent, there is a canonical isomorphism C ⁢ ( Γ ) ≃ C ⁢ ( Aut * ⁢ ( Γ ) , Γ * ) . C(\Gamma)\simeq C(\mathrm{Aut}^{*}(\Gamma),\Gamma^{*}). In other words, C ⁢ ( Γ ) C(\Gamma) is completely determined by the solution to the classical congruence subgroup problem for the arithmetic group Aut * ⁢ ( Γ ) \mathrm{Aut}^{*}(\Gamma) . In particular, in the case where Γ = Ψ n , c \Gamma=\Psi_{n,c} is a finitely generated free nilpotent group of class 𝑐 on 𝑛 elements, we get that C ⁢ ( Ψ n , c ) = C ⁢ ( Z ( n ) ) = { e } C(\Psi_{n,c})=C(\mathbb{Z}^{(n)})=\{e\} whenever n ≥ 3 n\geq 3 , and C ⁢ ( Ψ 2 , c ) = C ⁢ ( Z ( 2 ) ) = F ^ ω C(\Psi_{2,c})=C(\mathbb{Z}^{(2)})=\hat{F}_{\omega} is the free profinite group on countable number of generators.

Let O be the ring of integers of a number field, and let n ≥ 3 . This paper studies bi-interpretability of the ring of integers Z with the special linear group SL n ( O ) , the general linear group GL n ( O ) and the subgroup T n ( O ) of GL n ( O ) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories.

Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fej\'er points, defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of $\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some $\beta >0$ that depends on $p >1$., Comment: 16 pages. 7 figures and 3 tables. Keywords:Marcinkiewicz-Zygmund inequalities; $A_p$ weight; Lebesgue constants. J. Geom. Anal. (2021)

A subset $S$ of a group $G$ invariably generates $G$ if, when each element of $S$ is replaced by an arbitrary conjugate, the resulting set generates $G.$ An invariable generating set $X$ of $G$ is called minimal if no proper subset of $X$ invariably generates $G.$ We will address several questions related to the behaviour of minimal invariable generating sets of a finite group., This is a belated posting of a 2019 paper. 24 pp

We prove a complexity classification for Holant problems defined by an arbitrary set of complex-valued symmetric constraint functions on Boolean variables. This is to specifically answer the question: Is the Fisher-Kasteleyn-Temperley (FKT) algorithm under a holographic transformation (Valiant, SIAM J. Comput. 37(5), 1565–1594 2008) a universal strategy to obtain polynomial-time algorithms for problems over planar graphs that are intractable on general graphs? There are problems that are #P-hard on general graphs but polynomial-time solvable on planar graphs. For spin systems (Kowalczyk 2010) and counting constraint satisfaction problems (#CSP) (Guo and Williams, J. Comput. Syst. Sci. 107, 1–27 2020), a recurring theme has emerged that a holographic reduction to FKT precisely captures these problems. Surprisingly, for Holant, we discover new planar tractable problems that are not expressible by a holographic reduction to FKT. In particular, a straightforward formulation of a dichotomy for planar Holant problems along the above recurring theme is false. A dichotomy theorem for #CSPd, which denotes #CSP where every variable appears a multiple of d times, has been an important tool in previous work. However the proof for the #CSPd dichotomy violates planarity, and it does not generalize to the planar case easily. In fact, due to our newly discovered tractable problems, the putative form of a planar #CSPd dichotomy is false when d ≥ 5. Nevertheless, we prove a dichotomy for planar #CSP2. In this case, the putative form of the dichotomy is true. (This is presented in Part II of the paper.) We manage to prove the planar Holant dichotomy relying only on this planar #CSP2 dichotomy, without resorting to a more general planar #CSPd dichotomy for d ≥ 3. A special case of the new polynomial-time computable problems is counting perfect matchings (#PM) over k-uniform hypergraphs when the incidence graph is planar and k ≥ 5. The same problem is #P-hard when k = 3 or k = 4, which is also a consequence of our dichotomy. When k = 2, it becomes #PM over planar graphs and is tractable again. More generally, over hypergraphs with specified hyperedge sizes and the same planarity assumption, #PM is polynomial-time computable if the greatest common divisor (gcd) of all hyperedge sizes is at least 5. It is worth noting that it is the gcd, and not a bound on hyperedge sizes, that is the criterion for tractability.

Let λ ∈ P+ be a level-zero dominant integral weight, and w the coset representative of minimal length for a coset in W/Wλ, where Wλ is the stabilizer of λ in a finite Weyl group W. In this paper, we give a module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ over the negative part of a quantum affine algebra whose graded character is identical to the specialization at t = ∞ of the nonsymmetric Macdonald polynomial Ewλ(q, t) multiplied by a certain explicit finite product of rational functions of q of the form (1 − q−r)−1 for a positive integer r. This module $$ {\mathbbm{K}}_w^{-}\left(\uplambda \right) $$ (called a level-zero van der Kallen module) is defined to be the quotient module of the level-zero Demazure module $$ {V}_w^{-}\left(\uplambda \right) $$ by the sum of the submodules $$ {V}_z^{-}\left(\uplambda \right) $$ for all those coset representatives z of minimal length for cosets in W/Wλ such that z > w in the Bruhat order < on W.

Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $p>3$ be a prime where $E$ has good ordinary reduction, and let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis. In 1987, Perrin-Riou formulated an Iwasawa main conjecture for the Tate-Shafarevich group of $E$ over the anticyclotomic $\mathbf{Z}_p$-extension of $K$ in terms of Heegner points. In this paper, we give a proof of Perrin-Riou's conjecture under mild hypotheses. Our proof builds on Howard's theory of bipartite Euler systems and Wei Zhang's work on Kolyvagin's conjecture. In the case when $p$ splits in $K$, we also obtain a proof of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna., 22 pages. Revised version, to appear in Algebra & Number Theory

Traditional evidential clustering tends to build clusters where the number of data for each cluster fairly close to each other. However, it may not be suitable for imbalanced data. This paper proposes a new method, called credal clustering (CClu), to deal with imbalanced data based on the theory of belief functions. Consider a dataset with \(\mathcal {C}\) wanted classes, the credal c-means (CCM) clustering method is employed at first to divide the dataset into some (i.e., \(\mathcal {S}~(\mathcal {S}>\mathcal {C})\)) clusters. Then these clusters are gradually merged following a given principle based on the density of meta-clusters and the associated singleton clusters. The merging is finished when \(\mathcal {C}\) singleton wanted classes are obtained. During this merging procedure, the objects in each singleton cluster will be assigned to one new singleton class. Moreover, a weighted mean vector rule is developed to classify the objects in the unmerged meta-cluster to the associated new classes using the K-Nearest neighbor technique. Two experiments show that CClu can handle imbalanced datasets with high accuracy, and the errors are reduced by properly modeling imprecision.

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued., 17 pages

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{\lambda_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\phi$ on $\D$ for which $\phi(\lambda_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $\phi$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $\phi$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space., Comment: 27 pages

An orthogonality space is a set X together with a symmetric and irreflexive binary relation ⊥, called the orthogonality relation. A block partition of X is a partition of a maximal set of mutually orthogonal elements of X, and a decomposition of X is a collection of subsets of X each of which is the orthogonal complement of the union of the others. ( X , ⊥ ) is called normal if any block partition gives rise to a unique decomposition of the space. The set of one-dimensional subspaces of a Hilbert space equipped with the usual orthogonality relation provides the motivating example. Together with the maps that are, in a natural sense, compatible with the formation of decompositions from block partitions, the normal orthogonality spaces form a category, denoted by N O S . The objective of the present paper is to characterise both the objects and the morphisms of N O S from various perspectives as well as to compile basic categorical properties of N O S .

In this paper we give an affirmative answer to a conjecture proposed by Danny Neftin, that is, if the commutator of two permutations has at least n-4 fixed points where two permutations are in degree n symmetric group, then there exists a permutation in this symmetric inverting both of them., Comment: 16 pages,no figures

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\Delta$, there is a bijection from the Coxeter group $W$ of type $\Delta$ to the set of semibricks over the preprojective algebra $\Pi$ of type $\Delta$, which is restricted to a bijection from the set of join-irreducible elements of $W$ to the set of bricks over $\Pi$. This paper is devoted to giving an explicit description of these bijections in the case $\Delta=\mathbb{A}_n$ or $\mathbb{D}_n$. First, for each join-irreducible element $w \in W$, we describe the corresponding brick $S(w)$ in terms of "Young diagram-like" notation. Next, we determine the canonical join representation $w=\bigvee_{i=1}^m w_i$ of an arbitrary element $w \in W$ based on Reading's work, and prove that $\bigoplus_{i=1}^n S(w_i)$ is the semibrick corresponding to $w$., Comment: 37 pages

Inspired by a recent paper due to Jos\'{e} Luis Garc\'{i}a, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding $F\hookrightarrow M_n(G)$ exists provided that $n, Comment: 11 pages; slightly revised (e.g. new Lemma 1.3 added), minor typos corrected

Our goal in this paper is to investigate four conjectures, proposed by Daqing Wan, about the stable behavior of a geometric Z p -tower of curves X ∞ / X . Let h n be the class number of the n-th layer in X ∞ / X . It is known from Iwasawa theory that there are integers and such that the p-adic valuation v p ( h n ) equals to for n sufficiently large. Let Q p , n be the splitting field (over Q p ) of the zeta-function of n-th layer in X ∞ / X . The p-adic Wan-Riemann Hypothesis conjectures that the extension degree [ Q p , n : Q p ] goes to infinity as n goes to infinity. After motivating and introducing the conjectures, we prove the p-adic Wan-Riemann Hypothesis when λ ( X ∞ / X ) is nonzero.

We study the statistics and the arithmetic properties of the Robin spectrum of a rectangle. A number of results are obtained for the multiplicities in the spectrum, depending on the Diophantine nature of the aspect ratio. In particular, it is shown that for the square, unlike the case of Neumann eigenvalues where there are unbounded multiplicities of arithmetic origin, there are no multiplicities in the Robin spectrum for sufficiently small (but nonzero) Robin parameter except a systematic symmetry. In addition, uniform lower and upper bounds are established for the Robin-Neumann gaps in terms of their limiting mean spacing. Finally, that the pair correlation function of the Robin spectrum on a Diophantine rectangle is shown to be Poissonian., Comment: 3 figures. Arguments in section 2 simplified thanks to a referee. Accepted for publication in the Journal of Mathematical Physics, special collection of papers honoring Freeman Dyson

In this paper, we propose a class of growth models, named Fibonacci trees F ( t ) , with respect to the nature of Fibonacci sequence { F t } . First, we show that models F ( t ) have power-law degree distribution with exponent greater than 3. Then, we analytically study two significant topological indices, i.e., optimal mean first-passage time ( O M F P T ) and mean first-passage time ( M F P T ), for random walks on Fibonacci trees F ( t ) , and obtain the analytical expressions using some combinatorial approaches. The methods used are widely applied for other network models with self-similar feature to derive analytical solution to O M F P T or M F P T , and we select a candidate model to validate this viewpoint. In addition, we observe from theoretical analysis and numerical simulation that the scaling of M F P T is linearly correlated with vertex number of models F ( t ) , and show that Fibonacci trees F ( t ) possess more optimal topological structure than the classic scale-free tree networks.

Let M be a type II factor and let τ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type II ∞ case and normalized by τ ( I ) = 1 in the type II 1 case. Given A ∈ M + , we denote by A + = ( A − I ) χ A ( 1 , ‖ A ‖ ] the excess part of A and by A − = ( I − A ) χ A ( 0 , 1 ) the defect part of A. In [6] , V. Kaftal, P. Ng and S. Zhang provided necessary and sufficient conditions for a positive operator to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal) in type I and type III factors. For type II factors, V. Kaftal, P. Ng and S. Zhang proved that τ ( A + ) ≥ τ ( A − ) is a necessary condition for an operator A ∈ M + which can be written as the sum of a finite or infinite collection of projections and also sufficient if the operator is “diagonalizable”. In this paper, we prove that if A ∈ M + and τ ( A + ) ≥ τ ( A − ) , then A can be written as the sum of a finite or infinite collection of projections. This result answers affirmatively Question 5.4 of [6] .

The well-known Thompson theorem on character degrees states that if a prime p divides the degree of every nonlinear irreducible character of a finite group G, then G is p-nilpotent. In this paper, we give a strengthened version of Thompson's theorem in terms of ∑ χ ∈ Irr p ′ ( G ) χ ( 1 ) and ∑ χ ∈ Irr p ′ ( G ) χ ( 1 ) 2 , where Irr p ′ ( G ) denotes the set of all ordinary irreducible characters of G of degree coprime to p.