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In this paper, we present an algorithm on vanishing coefficients with arithmetic progressions in the sum of two generalized eta-quotients by using the theory of modular forms, and utilize our algorithm to prove five conjectures on matching coefficients in series expansions of certain q -products and their reciprocals. One of which was provided by Baruah and Das, and the others were found by Schlosser. For instance, we prove that for any n ≥ 0 , λ 12 (10 n + r) = λ 12 ′ (10 n + r − 6) , r ∈ { 7 , 9 } , where the sequences { λ 12 (n) } n ≥ 0 and { λ 12 ′ (n) } n ≥ 0 are defined by ∑ n = 0 ∞ λ 12 (n) q n = 1 R (q) R (q 2) R (q 4) R (q 8) = (∑ n = 0 ∞ λ 12 ′ (n) q n) − 1 , and where R (q) is the Rogers–Ramanujan continued fraction, which has the following celebrated product representation: R (q) = ∏ n = 0 ∞ (1 − q 5 n + 1) (1 − q 5 n + 4) (1 − q 5 n + 2) (1 − q 5 n + 3). Finally, we find that this phenomenon also exists in other infinite q -series expansions. [ABSTRACT FROM AUTHOR]

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato–Tate groups given a small number of Euler factors for the L -function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato–Tate group SU (2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato–Tate group USp (4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato–Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato–Tate distributions and may be able to classify curves efficiently. [ABSTRACT FROM AUTHOR]

The Super-SAT (SSAT) problem was introduced in [1,2] to prove the NP-hardness of approximation of two popular lattice problems - Shortest Vector Problem and Closest Vector Problem. SSAT is conjectured to be NP-hard to approximate to within a factor of n c (c is positive constant, n is the size of the SSAT instance). In this paper we prove this conjecture assuming the Projection Games Conjecture (PGC) [3]. This implies hardness of approximation of these lattice problems within polynomial factors, assuming PGC. We also reduce SSAT to the Nearest Codeword Problem and Learning Halfspace Problem [4]. This proves that both these problems are NP-hard to approximate within a factor of N c ′ / log ⁡ log ⁡ n (c ′ is positive constant, N is the size of the instances of the respective problems). Assuming PGC these problems are proved to be NP-hard to approximate within polynomial factors. [ABSTRACT FROM AUTHOR]

Let f be any arithmetic function and define S f (x) : = ∑ n ⩽ x f ([ x / n ]). When f equals the Euler totient function φ , several authors studied the upper and lower bounds of S φ (x). In this paper we shall prove that S φ (x) has an asymptotic formula by the method of exponential sums. This result proves a conjecture proposed by Bordellés, Dai, Heyman, Pan and Shparlinski. Some other asymptotic formulas for arbitrary f are also given in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we prove a supercongruence via the Wilf–Zeilberger method and symbolic summation algorithms in the setting of difference rings. That is, for any prime p > 3 , ∑ n = 0 (p − 1) / 2 3 n + 1 (− 8) n ( 2 n n ) 3 ≡ p (− 1 p) + p 3 4 (2 p) E p − 3 (1 4) (mod p 4) , where (⋅ p) stands for the Legendre symbol, and E n (x) are the Euler polynomials. This confirms a special case of a recent conjecture of Z.-W. Sun (Sun, 2019 , (2.18)). [ABSTRACT FROM AUTHOR]

Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial pair such that the polynomial is non-negative on the variety associated with the ideal if and only if the conjecture is true for this graph class. Using semidefinite programming we obtain numeric sum-of-squares certificates, which we then manage to transform into symbolic certificates confirming non-negativity of our polynomials. Specifically, we obtain exact low-degree sparse sum-of-squares certificates for particular classes of graphs. The obtained certificates allow generalizations for larger graph classes. Besides computational verification of these more general certificates, we also present theoretical proofs as well as conjectures and questions for further investigations. [ABSTRACT FROM AUTHOR]

Abstract Let G be an infinite group and let X be a finite generating set for G such that the growth series of G with respect to X is a rational function; in this case G is said to have rational growth with respect to X. In this paper a result on sizes of spheres (or balls) in the Cayley graph Γ (G , X) is obtained: namely, the size of the sphere of radius n is bounded above and below by positive constant multiples of n α λ n for some integer α ≥ 0 and some λ ≥ 1. As an application of this result, a calculation of degree of commutativity (d. c.) is provided: for a finite group F , its d. c. is defined as the probability that two randomly chosen elements in F commute, and Antolín, Martino and Ventura have recently generalised this concept to all finitely generated groups. It has been conjectured that the d. c. of a group G of exponential growth is zero. This paper verifies the conjecture (for certain generating sets) when G is a right-angled Artin group or, more generally, a graph product of groups of rational growth in which centralisers of non-trivial elements are "uniformly small". [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to present some contributions to the theory of finite transformation monoids. The dominating influence that permutation groups have on transformation monoids is used to describe and characterise transitive transformation monoids and primitive transitive transformation monoids. We develop a theory that not only includes the analogs of several important theorems of the classical theory of permutation groups but also contains substantial information about the algebraic structure of the transformation monoids. Open questions naturally arising from the substantial paper of Steinberg (2010) [11] have been answered. Our results can also be considered as a further development in the hunt for a solution of the Černý conjecture. [ABSTRACT FROM AUTHOR]

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with n G vertices and domination number k G and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of k G , n G , k H , and n H. In this paper, we consider their approach for k G = k H = 1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (n G + n H − 1) / 2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with n G + n H − 1 = d for general d , which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with k G = k H = 1 and n G + n H ≤ 15. [ABSTRACT FROM AUTHOR]

A singular k -tuple of a tensor T of format (n 1 , ... , n k) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n 1 , ... , n k) of rank at most one. A generic tensor has finitely many complex singular k -tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k − 1 dimensions n i , then the number of singular k -tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable n k , that stabilizes when (n 1 , ... , n k) reaches a boundary format. In this paper, we study the linear span of singular k -tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order- k tensors of format (2 , ... , 2 , n). As a consequence, if again we fix the first k − 1 dimensions n i and let n k increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k > 3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format. [ABSTRACT FROM AUTHOR]

The new identifiable case appeared in Angelini et al. (2018) , together with the analysis on simultaneous identifiability of pairs of ternary forms recently developed in Beorchia and Galuppi (2022) , suggested the following conjecture towards a complete classification of all simultaneous Waring identifiable cases: for any d ≥ 2 , the general polynomial vectors consisting of d − 1 ternary forms of degree d and a ternary form of degree d + 1 , with rank d 2 + d + 2 2 , are identifiable over C. In this paper, by means of a computer-aided procedure inspired to the one described in Angelini et al. (2018) , we obtain that the case d = 4 contradicts the previous conjecture, admitting at least 36 complex simultaneous Waring decompositions (of length 11) instead of 1. [ABSTRACT FROM AUTHOR]

We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to 2 it has recently been proven that the Fuglede conjecture holds (see [6] ). In this paper we study this question in higher dimensions over prime fields and provide some results and counterexamples. In particular we prove the existence of spectral sets which do not tile in Z p 5 for all odd primes p and Z p 4 for all odd primes p such that p ≡ 3 mod 4 . Although counterexamples in low dimensional groups over cyclic rings Z n were previously known they were usually for non-prime n or a small, sporadic set of primes p rather than general constructions. This paper is a result of a Research Experience for Undergraduates program ran at the University of Rochester during the summer of 2015 by A. Iosevich, J. Pakianathan and G. Petridis. [ABSTRACT FROM AUTHOR]

In [10] , Conjecture 6.6, Migliore, the first author, and Nagel conjectured that, for all n ≥ 4 , the artinian ideal I = (L 0 d , ... , L 2 n + 1 d) ⊂ R = k [ x 0 , ... , x 2 n ] generated by the d -th powers of 2 n + 2 general linear forms fails to have the weak Lefschetz property if and only if d > 1. This paper is entirely devoted to prove partially this conjecture. More precisely, we prove that R / I fails to have the weak Lefschetz property, provided 4 ≤ n ≤ 8 , d ≥ 4 or d = 2 r , 1 ≤ r ≤ 8 , 4 ≤ n ≤ 2 r (r + 2) − 1. [ABSTRACT FROM AUTHOR]

It has been known since Erdős that the sum of 1 / (n log ⁡ n) over numbers n with exactly k prime factors (with repetition) is bounded as k varies. We prove that as k tends to infinity, this sum tends to 1. Banks and Martin have conjectured that these sums decrease monotonically in k , and in earlier papers this has been shown to hold for k up to 3. However, we show that the conjecture is false in general, and in fact a global minimum occurs at k = 6. [ABSTRACT FROM AUTHOR]

The linear arboricity of a graph G , denoted by la (G) , is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in G whose union covers all the edges of G. A famous conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that la (G) ≤ ⌈ (Δ (G) + 1) / 2 ⌉ , where Δ (G) denotes the maximum degree of G. This conjectured upper bound would be best possible, as is easily seen by taking G to be a regular graph. In this paper, we show that for every graph G , la (G) ≤ Δ 2 + O (Δ 2 / 3 − α) for some α > 0 , thereby improving the previously best known bound due to Alon and Spencer from 1992. For graphs which are sufficiently good spectral expanders, we give even better bounds. Our proofs of these results further give probabilistic polynomial time algorithms for finding such decompositions into linear forests. [ABSTRACT FROM AUTHOR]

In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved. To cover levels that are important for elliptic curves, namely those that are not square-free, we establish results about L -functions with additive twists that are of independent interest. [ABSTRACT FROM AUTHOR]

Twenty years ago Bondy and Vince conjectured that for any nonnegative integer k , except finitely many counterexamples, every graph with k vertices of degree less than three contains two cycles whose lengths differ by one or two. The case k ≤ 2 was proved by Bondy and Vince, which resolved an earlier conjecture of Erdős et al. In this paper we confirm this conjecture for all k. [ABSTRACT FROM AUTHOR]

In 1998, Reed conjectured that every graph G satisfies χ (G) ≤ ⌈ 1 2 (Δ (G) + 1 + ω (G)) ⌉ , where χ (G) is the chromatic number of G , Δ (G) is the maximum degree of G , and ω (G) is the clique number of G. As evidence for his conjecture, he proved an "epsilon version" of it, i.e. that there exists some ε > 0 such that χ (G) ≤ (1 − ε) (Δ (G) + 1) + ε ω (G). It is natural to ask if Reed's conjecture or an epsilon version of it is true for the list-chromatic number. In this paper we consider a "local version" of the list-coloring version of Reed's conjecture. Namely, we conjecture that if G is a graph with list-assignment L such that for each vertex v of G , | L (v) | ≥ ⌈ 1 2 (d (v) + 1 + ω (v)) ⌉ , where d (v) is the degree of v and ω (v) is the size of the largest clique containing v , then G is L -colorable. Our main result is that an "epsilon version" of this conjecture is true, under some mild assumptions. Using this result, we also prove a significantly improved lower bound on the density of k -critical graphs with clique number less than k / 2 , as follows. For every α > 0 , if ε ≤ α 2 1350 , then if G is an L -critical graph for some k -list-assignment L such that ω (G) < (1 2 − α) k and k is sufficiently large, then G has average degree at least (1 + ε) k. This implies that for every α > 0 , there exists ε > 0 such that if G is a graph with ω (G) ≤ (1 2 − α) mad (G) , where mad (G) is the maximum average degree of G , then χ ℓ (G) ≤ ⌈ (1 − ε) (mad (G) + 1) + ε ω (G) ⌉. It also yields an improvement on the best known upper bound for the chromatic number of K t -minor free graphs for large t , by a factor of.99982. [ABSTRACT FROM AUTHOR]

The Hall–Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group J 4). We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent. Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type. [ABSTRACT FROM AUTHOR]

We report the results of our empirical investigations on the Bateman-Horn conjecture. This conjecture, in its commonly known form, produces rather large deviations when the polynomials involved are not monic. We propose a modified version of the conjecture which empirically demonstrates remarkable accuracy even for modest values of primes. For a video summary of this paper, please visit https://youtu.be/mINg-0n7nOY. [ABSTRACT FROM AUTHOR]

Denote by P the set of all primes and by P + (n) the largest prime factor of integer n ⩾ 1 with the convention P + (1) = 1. For each η > 1 , let c = c (η) > 1 be some constant depending on η and P a , c , η : = { p ∈ P : p = P + (q − a) for some prime q with p η < q ⩽ c (η) p η }. In this paper, under the Elliott-Halberstam conjecture we prove, for y → ∞ , π a , c , η (x) : = | (1 , x ] ∩ P a , c , η | ∼ π (x) or π a , c , η (x) ≫ a , η π (x) according to values of η. These are complement for some results of Banks-Shparlinski [1] , of Wu [12] and of Chen-Wu [2]. [ABSTRACT FROM AUTHOR]

In this paper, by inputting the Bessel identities over the complex field in previous work of the authors, the Waldspurger formula of Baruch and Mao is extended from totally real fields to arbitrary number fields. This is applied to give a non-trivial bound towards the Ramanujan conjecture for automorphic forms on the metaplectic group SL ˜ 2 for the first time in the generality of arbitrary number fields. [ABSTRACT FROM AUTHOR]

Under some supportive evidences of known results about non-spectral self-affine measures, Li proposed a conjecture in [18] (J. Funct. Anal. 255, 2008), which is so-called "non-spectral conjecture". In this paper, we prove that Li's non-spectral conjecture is true under some suitable condition. [ABSTRACT FROM AUTHOR]

Bollobás and Scott [5] conjectured that every graph G has a balanced bipartite spanning subgraph H such that for each v ∈ V (G) , d H (v) ≥ (d G (v) − 1) / 2. In this paper, we show that every graphic sequence has a realization for which this Bollobás-Scott conjecture holds, confirming a conjecture of Hartke and Seacrest [8]. On the other hand, we give an infinite family of counterexamples to this Bollobás-Scott conjecture, which indicates that ⌊ (d G (v) − 1) / 2 ⌋ (rather than (d G (v) − 1) / 2) is probably the correct lower bound. We also study bipartitions V 1 , V 2 of graphs with a fixed number of edges. We provide a (best possible) upper bound on e (V 1) λ + e (V 2) λ for any real λ ≥ 1 (the case λ = 2 is a question of Scott [12]) and answer a question of Scott (Problem 3.9 in [12]) on max ⁡ { e (V 1) , e (V 2) }. [ABSTRACT FROM AUTHOR]

Let G be a graph with possible multiple edges but no loops. The density of G , denoted by ρ (G) , is defined as max H ⊂ G , | V (H) | ≥ 2 ⁡ ⌈ | E (H) | ⌊ | V (H) | / 2 ⌋ ⌉. Goldberg (1973) and Seymour (1974) independently conjectured that if the chromatic index χ ′ (G) satisfies χ ′ (G) ≥ Δ (G) + 2 then χ ′ (G) = ρ (G) , which is commonly regarded as Goldberg's conjecture. An equivalent conjecture, usually credited to Jakobsen, states that for any odd integer m ≥ 3 , if χ ′ (G) ≥ m Δ (G) m − 1 + m − 3 m − 1 then χ ′ (G) = ρ (G). The Tashkinov tree technique, a common generalization of Vizing fans and Kierstead paths for multigraphs, has emerged as the main tool to attack these two conjectures. On the other hand, Asplund and McDonald recently showed that there is a limitation to this method. In this paper, we will go beyond Tashkinov trees and provide a much larger extended structure, using which we see hope to tackle the conjecture. Applying this new technique, we show that the Goldberg's conjecture holds for graphs with Δ (G) ≤ 39 or | V (G) | ≤ 39 and the Jakobsen Conjecture holds for m ≤ 39 , where the previously known best bound is 23. We also improve a number of other related results. [ABSTRACT FROM AUTHOR]

In this paper we study a Landis-type conjecture for fractional Schrödinger equations of fractional power s ∈ (0 , 1) with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns out for differentiable potentials with some a priori bounds, if a solution decays at a rate e − | x | 1 + , then this solution is trivial. On the other hand, for s ∈ (1 / 4 , 1) and merely bounded non-differentiable potentials, if a solution decays at a rate e − | x | α with α > 4 s / (4 s − 1) , then this solution must again be trivial. Remark that when s → 1 , 4 s / (4 s − 1) → 4 / 3 which is the optimal exponent for the standard Laplacian. For the case of non-differentiable potentials and s ∈ (1 / 4 , 1) , we also derive a quantitative estimate mimicking the classical result by Bourgain and Kenig. [ABSTRACT FROM AUTHOR]

In this paper, we study Iwasawa theory for p = 2. First of all, we show that the classical Iwasawa main conjecture holds true even for p = 2 over a totally real field k assuming μ = 0 and Leopoldt's conjecture. Using the Iwasawa main conjecture, we study the 2-component of the ideal class group of a CM-field K of finite degree as a Galois module. More precisely, for a CM-field K which is cyclic over the base field k , we determine the Fitting ideal of the minus quotient of the 2-component of the ideal class group. In particular, when k = Q and K / Q is imaginary and cyclic, we prove that the Fitting ideal coincides with the Stickelberger ideal. [ABSTRACT FROM AUTHOR]

The paper generalizes the method of Zhao for an infinite family of Q -curves and establishes some analytic results on the 2-part of the Birch and Swinnerton-Dyer conjecture. We give lower bounds on the 2-adic valuations of the algebraic part of the L -values at s = 1 for those Q -curves. Moreover, we also discuss 2-descent on Q -curves and compute their Mordell-Weil group and Tate-Shafarevich group. [ABSTRACT FROM AUTHOR]

We develop a new method for studying the Baum–Connes conjecture, which we call the direct splitting method. We introduce what we call property (γ) for G -equivariant Kasparov cycles. We show that the existence of a G -equivariant Kasparov cycle with property (γ) implies the split-injectivity of the assembly map μ A G for any separable G - C ⁎ -algebra A. We also show that if such a cycle exists, the assembly map μ A G is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K ⁎ (A ⋊ r G) of the Baum–Connes conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, we use this method to give a finite-dimensional proof of the Baum–Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space. [ABSTRACT FROM AUTHOR]

To extend Iwasawa's classical theorem from Z p -towers to Z p d -towers, Greenberg conjectured that the exponent of p in the n -th class number in a Z p d -tower of a global field K ramified at finitely many primes is given by a polynomial in p n and n of total degree at most d for sufficiently large n. This conjecture remains open for d ≥ 2. In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on p -adic stability of zeta functions in a p -adic Lie tower of function fields. [ABSTRACT FROM AUTHOR]

The families F 1 , ... , F s ⊂ 2 [ n ] are called q-dependent if there are no pairwise disjoint F 1 ∈ F 1 , ... , F s ∈ F s satisfying | F 1 ∪ ... ∪ F s | ≤ q. We determine max ⁡ | F 1 | + ... + | F s | for all values n ≥ q , s ≥ 2. The result provides a far-reaching generalization of an important classical result of Kleitman. The well-known Erdős Matching Conjecture suggests the largest size of a family F ⊂ ( [ n ] k ) with no s pairwise disjoint sets. After more than 50 years its full solution is still not in sight. In the present paper we provide a Hilton–Milner-type stability theorem for the Erdős Matching Conjecture in a relatively wide range, in particular, for n ≥ (2 + o (1)) s k with o (1) depending on s only. This is a considerable improvement of a classical result due to Bollobás, Daykin and Erdős. We apply our results to advance in the following anti-Ramsey-type problem, proposed by Özkahya and Young. Let a r (n , k , s) be the minimum number x of colors such that in any coloring of the k -element subsets of [ n ] with x (non-empty) colors there is a rainbow matching of size s , that is, s sets of different colors that are pairwise disjoint. We prove a stability result for the problem, which allows to determine a r (n , k , s) for all k ≥ 3 and n ≥ s k + (s − 1) (k − 1). Some other consequences of our results are presented as well. [ABSTRACT FROM AUTHOR]

Let X be a reduced closed subscheme in P n , π : X → π (X) ⊂ P n − 1 be a projection from a point outside X and Z i (X) ⊂ π (X) be the closed subscheme defined by the i -th partial elimination ideal K i (I X) , which is supported on the (i + 1) -th multiple points of π. In this paper, motivated from projection methods to prove Eisenbud-Goto conjecture on regularity in many cases, we describe the syzygetic behaviors and Castelnuovo normality of the projection with a viewpoint of the regularity of the partial elimination ideal K i (I X) , i ≥ 1 (or that of the multiple locus Z i (X) of π). We also give some applications to the syzygies and Castelnuovo normality of successive projections, which recover and generalize some known results in [1,3,15,16]. [ABSTRACT FROM AUTHOR]

This paper deals with an extremal problem for bounded harmonic functions in the unit ball B n. We solve the Khavinson conjecture in R 3 , an intriguing open question since 1992 posed by D. Khavinson, later considered in a general context by Kresin and Maz'ya. Precisely, we obtain the following inequality: | ∇ u (x) | ≤ 1 ρ 2 ( (1 + 1 3 ρ 2) 3 2 1 − ρ 2 − 1) sup | y | < 1 ⁡ | u (y) | , with ρ = | x | , thus sharpening the previously known with | 〈 ∇ u (x) , n x 〉 | instead of | ∇ u (x) | , where n x = x | x |. [ABSTRACT FROM AUTHOR]

Each (equigenerated) squarefree monomial ideal in the polynomial ring S = K [ x 1 , ... , x n ] represents a family of subsets of [ n ] , called a (uniform) clutter. In this paper, we introduce a class of uniform clutters, called decomposable clutters, whose associated ideal has linear quotients and hence linear resolution over all fields. We show that chordality of these clutters guarantees the correctness of a conjecture raised by R.S. Simon [23] on extendable shellebility of d -skeletons of a simplex 〈 [ n ] 〉 , for all d. We then prove this conjecture for d ≥ n − 3. [ABSTRACT FROM AUTHOR]

In this paper, we provide an effective and practical method to find all three term arithmetic progressions in a given Lucas sequence of the first or second kind. Our interest is the case when the sequence has a negative discriminant, since the case of positive discriminant has recently been resolved by Hajdu et al. [11]. We present a conjecture on the maximal number and length of such arithmetic progressions based on computational evidence. [ABSTRACT FROM AUTHOR]

The binomial Eulerian polynomials, introduced by Postnikov, Reiner, and Williams, are γ -positive polynomials and can be interpreted as h -polynomials of certain flag simplicial polytopes. Recently, Athanasiadis studied analogs of these polynomials for colored permutations. In this paper, we generalize them to s -inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of Ma, Ma, and Yeh. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain s -inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes. [ABSTRACT FROM AUTHOR]

Abstract In their study of truncated theta series, Andrews-Merca and Guo-Zeng made a conjecture on Jacobi's triple product identity, which is settled by Mao and Yee independently. In this paper, we reprove this ex-conjecture by providing an explicit series form with nonnegative coefficients, which is reminiscent of the results of Andrews-Merca and Guo-Zeng. We also provide a companion theorem to this ex-conjecture. [ABSTRACT FROM AUTHOR]

We clarify that one of the main results from “A local global principle for regular operators in Hilbert C ⁎ -modules” is already contained in an earlier paper by François Pierrot. Moreover, in the paper by Pierrot this result is proved in a stronger version that was only stated as a conjecture in our local global principle paper. We also present a short proof, based on Pierrot's argument, of the core issue in both of the two papers namely a result on the separation of a point from a closed submodule. [ABSTRACT FROM AUTHOR]

Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants T V r of hyperbolic 3-manifolds. We show that if the r -growth of | T V r (M) | for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work [11] we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose S O (3) -Turaev-Viro invariants have exponential r -growth. As a result, for any g > n ⩾ 2 , we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. [ABSTRACT FROM AUTHOR]

Using the equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape λ / μ. Morales, Pak and Panova found two q -analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape λ / μ and reverse plane partitions of shape λ / μ. When λ and μ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape λ / μ can be expressed as a determinant whose entries are related to q -analogues of the Euler numbers. The objective of this paper is to settle this conjecture. [ABSTRACT FROM AUTHOR]

Abstract Let G be a finite p -group. The breadth b (x) of an element x of a group G is defined as log p of the size of the conjugacy class of x in G. M.R. Vaughan-Lee proved that if the breadths of all elements in G are bounded by some natural number n , then | G ′ | ≤ p n (n + 1) 2 . In the present paper, we prove that if | G ′ | > p n (n + 1) 2 , then G can be generated by the elements of breadths > n , thereby confirming the hypothesis of James Wiegold from the "Kourovka notebook". [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems. [ABSTRACT FROM AUTHOR]

Abstract In 1971, Tomescu conjectured that every connected graph G on n vertices with chromatic number k ≥ 4 has at most k ! (k − 1) n − k proper k -colorings. Recently, Knox and Mohar proved Tomescu's conjecture for k = 4 and k = 5. In this paper, we complete the proof of Tomescu's conjecture for all k ≥ 4 , and show that equality occurs if and only if G is a k -clique with trees attached to each vertex. [ABSTRACT FROM AUTHOR]

Abstract The main purpose of this paper is to find double blocking sets in PG (2 , q) of size less than 3 q , in particular when q is prime. To this end, we study double blocking sets in PG (2 , q) of size 3 q − 1 admitting at least two (q − 1) -secants. We derive some structural properties of these and show that they cannot have three (q − 1) -secants. This yields that one cannot remove six points from a triangle, a double blocking set of size 3 q , and add five new points so that the resulting set is also a double blocking set. Furthermore, we give constructions of minimal double blocking sets of size 3 q − 1 in PG (2 , q) for q = 13 , 16, 19, 25, 27, 31, 37 and 43. If q > 13 is a prime, these are the first examples of double blocking sets of size less than 3 q. These results resolve two conjectures of Raymond Hill from 1984. [ABSTRACT FROM AUTHOR]

Abstract Let A an abelian variety of dimension r , defined over Q. For p a rational prime, we denote by F p the finite field of cardinality p. If A has good reduction at p , let A ¯ p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A (Q) , and let Γ p be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p , with p ≤ x , for which the quotient A ¯ p (F p) Γ p has at most 2 r − 1 cyclic components. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove that Wright's equation y ′ (t) = − α y (t − 1) { 1 + y (t) } has a unique slowly oscillating periodic solution for parameter values α ∈ (π 2 , 1.9 ] , up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α > π 2. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations. [ABSTRACT FROM AUTHOR]

Abstract Let G be a graph with a vertex colouring α. Let a and b be two colours. Then a connected component of the subgraph induced by those vertices coloured either a or b is known as a Kempe chain. A colouring of G obtained from α by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of G are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. A conjecture of Mohar (2007) asserts that, for k ≥ 3 , all k -colourings of a k -regular graph that is not complete are Kempe equivalent. It was later shown that all 3-colourings of a cubic graph that is neither K 4 nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each k ≥ 4. We also report the implications of this result on the validity of the Wang–Swendsen–Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature. [ABSTRACT FROM AUTHOR]

Abstract Let R be a noncommutative ring with unity. The commuting graph of R , denoted by Γ (R) , is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if x y = y x. Let F be a finite field and n ≥ 2. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if Γ (R) ≅ Γ (M n (F)) , then R ≅ M n (F). In this paper, we prove the conjecture whenever n is of the form 2 k 3 l with k ≠ 0. [ABSTRACT FROM AUTHOR]

Abstract For any sequences u = { u (n) } n ≥ 0 , v = { v (n) } n ≥ 0 , we define u v : = { u (n) v (n) } n ≥ 0 and u + v : = { u (n) + v (n) } n ≥ 0. We say an integer sequence u = { u (n) } n ≥ 0 is a polynomial generated sequence if there exists an integer k ≥ 1 and polynomials f i (x) (0 ≤ i < k) whose coefficients are integer sequences such that { u (k n + i) } n ≥ 0 = f i (u) , (0 ≤ i < k). In this paper, we study the polynomial generated sequences. Assume k ≥ 2 and f i (x) = a i x + b i (0 ≤ i < k). If a i are k -automatic and b i are k -regular for 0 ≤ i < k , then we prove that the corresponding polynomial generated sequences are k -regular. As a application, we prove that the Hankel determinant sequence { det ⁡ (p (i + j)) i , j = 0 n − 1 } n ≥ 0 is 2-regular, where { p (n) } n ≥ 0 = 0110100010000 ⋯ is the characteristic sequence of powers of 2. Moreover, we give a proof of a conjecture of Cigler about the Hankel determinants. [ABSTRACT FROM AUTHOR]

Abstract A shortest circuit cover F of a bridgeless graph G is a family of circuits that covers every edge of G and is of minimum total length. The total length of a shortest circuit cover F of G is denoted by S C C (G). For ordinary graphs (graphs without sign), the subject of shortest circuit cover is closely related to some mainstream areas, such as, Tutte's integer flow theory, circuit double cover conjecture, Fulkerson conjecture, and others. For signed graphs G , it is proved recently by Máčajová, Raspaud, Rollová and Škoviera that S C C (G) ≤ 11 | E | if G is s-bridgeless, and S C C (G) ≤ 9 | E | if G is 2-edge-connected. In this paper this result is improved as follows, S C C (G) ≤ | E | + 3 | V | + z where z = min ⁡ { 2 3 | E | + 4 3 ϵ N − 7 , | V | + 2 ϵ N − 8 } and ϵ N is the negativeness of G. The above upper bound can be further reduced if G is 2-edge-connected with even negativeness. [ABSTRACT FROM AUTHOR]