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Bernstein problem for affine maximal type equation (0.1) u i j D i j w = 0 , w ≡ det ⁡ D 2 u − θ , ∀ x ∈ Ω ⊂ R N has been a core problem in affine geometry. A conjecture proposed firstly by Chern (1977) [6] for entire graph and then extended by Trudinger-Wang (2000) [14] to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniformly convex C 4 -hypersurface in R N + 1 must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N = 2 and θ = 3 / 4 , and later extended by Jia-Li (2009) [12] to N = 2 , θ ∈ (3 / 4 , 1 (see also Zhou (2012) [16] for a different proof). On the past twenty years, many efforts were done toward higher dimensional issues but not really successful yet, even for the case of dimension N = 3. In this paper, we will construct non-quadratic affine maximal type hypersurfaces which are Euclidean compete for N ≥ 3 , θ ∈ (1 / 2 , (N − 1) / N). [ABSTRACT FROM AUTHOR]

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the following simple real exceptional Lie groups: EI = E 6 (6) , EIV = E 6 (− 26) , FI = F 4 (4) , FII = F 4 (− 20). Along the way, we find an irreducible unitary representation of F 4 (4) whose Dirac index vanishes, while its Dirac cohomology is non-zero. This disproves a conjecture raised in 2015 asserting that there should be no cancellation between the even part and the odd part of the Dirac cohomology. [ABSTRACT FROM AUTHOR]

In this paper characters of the normalizer of d -split Levi subgroups in SL n (q) and SU n (q) are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer. These results are applied to verify the Alperin-McKay conjecture for primes ℓ with ℓ ∤ 6 (q 2 − 1) and the Alperin weight conjecture for ℓ -blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of SL n (q) and SU n (q). [ABSTRACT FROM AUTHOR]

A well known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K 5 or K 3 , 3. Wagner proved in 1937 that if a graph other than K 5 does not contain any subdivision of K 3 , 3 then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K 5 then it is planar or it admits a cut of size at most 4. In this paper, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems. [ABSTRACT FROM AUTHOR]

In this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part for the neutral Tannakian category mixed Tate motives over Z. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown's matrix conjecture and the nondegeneracy conjecture about depth-graded motivic Lie subalgebra generated by the depth one part. [ABSTRACT FROM AUTHOR]

In this paper we consider iterated integral representations of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple zeta values in terms of unit-exponent alternating multiple zeta values. In particular, we prove several conjectures given by Borwein-Bradley-Broadhurst [3] , and give some general results. [ABSTRACT FROM AUTHOR]

For an irrational number θ , let 0 = a 0 < a 1 < a 2 < ... < a m < a m + 1 = 1 be the sequence of points { ℓ θ } , 1 ≤ ℓ ≤ m (where { x } denotes x − x , the fractional part of x) and define d θ (m) = max ⁡ { (a i − a i − 1) , 1 ≤ i ≤ m + 1 } Sarnak conjectured that sup ⁡ m d θ ⁎ m → ∞ (m) ≤ sup ⁡ m d θ m → ∞ (m) where θ ⁎ = 1 + 5 2 is the golden mean and θ is an arbitrary irrational number. In this paper we establish the conjecture, and we determine exactly sup ⁡ m d θ ⁎ (m). Four special properties of the golden mean are crucial to the proof. [ABSTRACT FROM AUTHOR]

The paper presented here focuses on the classification of trivial source Specht modules. We completely classify the trivial source Specht modules labelled by hook partitions. We also classify the trivial source Specht modules labelled by two-part partitions in the odd characteristic case. Moreover, in the even characteristic case, we prove a result for the classification of the trivial source Specht modules labelled by partitions with 2-weight 2, which justifies a conjecture of [16]. [ABSTRACT FROM AUTHOR]

Let C ⊂ P r be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps H 0 (O P r (m)) → H 0 (O C (m)) are of maximal rank; this determines the Hilbert function of C. In this paper, we prove an analogous statement for the union of hyperplane sections of general curves. More specifically, if H ⊂ P r is a general hyperplane, and C 1 , C 2 , ... , C n are general curves, we show H 0 (O H (m)) → H 0 (O (C 1 ∪ C 2 ∪ ⋯ ∪ C n) ∩ H (m)) is of maximal rank, except for some counterexamples when m = 2. As explained in [5] , this result plays a key role in the author's proof of the Maximal Rank Conjecture [7]. [ABSTRACT FROM AUTHOR]

In 1927, Artin hypothesized that for any given non-zero integer a other than 1, −1, or a perfect square, there exists infinitely many primes p for which a is a primitive root modulo p. In 1967, Hooley proved it under the assumption of the generalized Riemann hypothesis. Since then, there are many analogues and generalization of this conjecture. In this paper, we work on its generalization to composite moduli in the function fields setting. Let A = F q t be the ring of polynomials over the finite field F q and 0 ≠ a ∈ A. Let C be the A -Carlitz module. Let a be a fixed element in A. For n ∈ A , C (A / n A) is a finite A -module. The set of all annihilators of C (A / n A) is an ideal and generated by a monic polynomial, denoted by λ (n). Similarly, The set of all annihilators of the submodule of C (A / n A) generated by a is an ideal and let l a (n) be its monic generator. We say that a is a primitive root of n , if λ (n) = l a (n). Define N a (x) : = | { n ∈ A | deg ⁡ n = x , n is monic , a is a primitive root of n } | We prove that for a given non-constant a ∈ A , a ∉ E , an exceptional set, there exists an unbounded set V of integers such that lim inf x ∈ V N a (x) / q x = 0 This result is analogous to Li's theorem for Artin's conjecture on composite moduli. It is the first time that this kind of results holds in the setting of the function fields. [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]

It has been an open conjecture in the theory of non-cooperative games that Nash equilibrium is universal for the collection of (non-empty) compact semi-algebraic sets, meaning that for every such set there is a game whose set of Nash equilibria is homeomorphic to the given set. In this paper we prove this conjecture. [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]

In this paper, all of the conjectures in Aydin et al. (2022) [6] are settled: some proven correct, some proven correct with a modification, and some disproven, involving the equivalence and duality of polycyclic codes associated with trinomials. We also give methods to construct isodual and self-dual polycyclic codes, and study the self-orthogonal and dual-containing polycyclic codes over F 2. [ABSTRACT FROM AUTHOR]

In this paper, we establish the following two results: (1) a skew translation generalized quadrangle of even order is a translation generalized quadrangle, (2) a generalized quadrangle of even order does not admit a point regular group of automorphisms. The first result confirms a conjecture of Payne (1975) based on earlier work of Ott (2021), and the second result confirms a conjecture of Ghinelli (1992). [ABSTRACT FROM AUTHOR]

Recently Marcus, Spielman and Srivastava proved Weaver's KS r conjecture, which gives a positive solution to the Kadison-Singer problem. In [13,10] , Cohen and Brändén independently extended this result to obtain the arbitrary-rank version of Weaver's KS r conjecture. In this paper, we present a new bound in Weaver's KS r conjecture for the arbitrary-rank case. To do that, we introduce the definition of (k , m) -characteristic polynomials and employ it to improve the previous estimate on the largest root of the mixed characteristic polynomials. For the rank-one case, our bound agrees with the Bownik-Casazza-Marcus-Speegle bound when r = 2 [8] and with the Ravichandran-Leake bound when r > 2 [21]. For the higher-rank case, we sharpen the previous bounds from Cohen and Brändén. [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]

In our previous paper with Maulik, we proposed a conjectural Gopakumar-Vafa (GV) type formula for the generating series of stable pair invariants on Calabi-Yau (CY) 4-folds. The purpose of this paper is to give an interpretation of the above GV type formula in terms of wall-crossing phenomena in the derived category. We introduce invariants counting LePotier's stable pairs on CY 4-folds, and show that they count certain stable objects in D0-D2-D8 bound states in the derived category. We propose a conjectural wall-crossing formula for the generating series of our invariants, which recovers the conjectural GV type formula. Examples are computed for both compact and toric cases to support our conjecture. [ABSTRACT FROM AUTHOR]

In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we consider analogous objects in the category of higher secant varieties. Our main theorems say that there is a matryoshka structure among those basic objects including a generalized K p , 1 theorem, syzygetic and geometric characterizations of higher secant varieties of minimal degree and del Pezzo higher secant varieties, defined in this paper. For our purpose, we prove a weak form of the generalized Bronowski's conjecture raised by C. Ciliberto and F. Russo that relates the identifiability for higher secant varieties to the geometry of tangential projections. [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]