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In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether St r ⊗ L (λ) is a tilting module for L (λ) an irreducible representation of p r -restricted highest weight, and (iv) whether Ext G 1 1 (L (λ) , L (μ)) (− 1) is a tilting module where L (λ) and L (μ) have p -restricted highest weight. The authors establish affirmative answers to each of these questions with a new uniform bound, namely p ≥ 2 h − 4 where h is the Coxeter number. Notably, this verifies these statements for infinitely many more cases. Later in the paper, questions (i)-(iv) are considered for rank two groups where there are counterexamples (for small primes) to these questions. [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

The G -representation variety R G (Σ g) parametrizes the representations of the fundamental group π 1 (Σ g) of a closed surface into an algebraic group G. For G the groups of n × n upper triangular matrices or unipotent upper triangular matrices, we compare two methods for computing algebraic invariants of R G (Σ g). Using the geometric method initiated by González-Prieto, Logares and Muñoz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of R G (Σ g) in the Grothendieck ring of varieties for n = 1 , ... , 5. Introducing the notion of algebraic representatives we are able to efficiently compute the TQFT. Using the arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the E -polynomials of R G (Σ g) for n = 1 , ... , 10. For both methods, we describe how the computations can be performed algorithmically. Furthermore, we discuss how the representation varieties for the groups of unipotent upper triangular matrices are related to Higman's conjecture. The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Potočnik and S. Wilson (2014) [12] , as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p -adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m ∈ N such that p ℓ α is an infinite loop mod p m , for all ℓ ∈ N ∪ { 0 }. [ABSTRACT FROM AUTHOR]

In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in P N. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in P N for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces. [ABSTRACT FROM AUTHOR]

Let N k (m , n) denote the number of partitions of n with Garvan k -rank m. It is well-known that Andrews–Garvan–Dyson's crank and Dyson's rank are the k -rank for k = 1 and k = 2 , respectively. In this paper, we prove that the sequences (N k (m , n)) | m | ≤ n − k − 71 are log-concave for all sufficiently large integers n and each integer k. In particular, we partially solve the log-concavity conjecture for Andrews–Garvan–Dyson's crank and Dyson's rank, which was independently proposed by Bringmann–Jennings-Shaffer–Mahlburg and Ji–Zang recently. [ABSTRACT FROM AUTHOR]

Let r 1 , ... , r t be positive integers and let R 2 (r 1 , ... , r t) be the set of positive odd integers that can be represented as p + 2 k 1 r 1 + ⋯ + 2 k t r t , where p is a prime and k 1 , ... , k t are positive integers. It is easy to see that if r 1 − 1 + ⋯ + r t − 1 < 1 , then the set R 2 (r 1 , ... , r t) has asymptotic density zero. In this paper, we prove that if r 1 − 1 + ⋯ + r t − 1 ≥ 1 , then the set R 2 (r 1 , ... , r t) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research. [ABSTRACT FROM AUTHOR]

A simple graph G with maximum degree Δ is overfull if | E (G) | > Δ ⌊ | V (G) | / 2 ⌋. The core of G , denoted G Δ , is the subgraph of G induced by its vertices of degree Δ. Clearly, the chromatic index of G equals Δ + 1 if G is overfull. Conversely, Hilton and Zhao in 1996 conjectured that if G is a simple connected graph with Δ ≥ 3 and Δ (G Δ) ≤ 2 , then χ ′ (G) = Δ + 1 implies that G is overfull or G = P ⁎ , where P ⁎ is obtained from the Petersen graph by deleting a vertex. Cariolaro and Cariolaro settled the base case Δ = 3 in 2003, and Cranston and Rabern proved the next case, Δ = 4 , in 2019. In this paper, we give a proof of this conjecture for all Δ ≥ 4. [ABSTRACT FROM AUTHOR]

Let F be a totally real number field. Dasgupta conjectured an explicit p -adic analytic formula for the Gross–Stark units of F. In a later paper, Dasgupta–Spieß conjectured a cohomological formula for the principal minors and the characteristic polynomial of the Gross regulator matrix associated to a totally odd character of F. Dasgupta–Spieß conjectured that these conjectural formulas coincide for the diagonal entries of Gross regulator matrix. In this paper, we prove this conjecture when F is a cubic field. For a video summary of this paper, please visit https://youtu.be/hlBRUIOke04. [ABSTRACT FROM AUTHOR]

In this paper, the fractional problem of the Ambrosetti-Prodi type involving the critical Sobolev exponent is taken into account in a bounded domain of R N { A α u = u + 2 α ⁎ − 1 + λ u − s ¯ φ 1 , u > 0 , in Ω , u = 0 , on ∂ Ω , where A α is the spectral fractional operator, λ and s ¯ are real numbers, Ω ⊂ R N is bounded, 2 α ⁎ = 2 N N − 2 α is a critical exponent, 0 < α < 1 , φ 1 is the first eigenfunction of −Δ with zero Dirichlet boundary condition. We will construct bubbling solutions when the parameter is large enough, and the location of the bubbling point is near the boundary of the domain. [ABSTRACT FROM AUTHOR]

Given a finite group G and an irreducible complex character χ of G , the codegree of χ is defined as the integer cod (χ) = | G : ker (χ) | / χ (1). It was conjectured by G. Qian in [16] that, for every element g of G , there exists an irreducible character χ of G such that cod (χ) is a multiple of the order of g ; the conjecture has been verified under the assumption that G is solvable ([16]) or almost-simple ([13]). In this paper, we prove that Qian's conjecture is true for every finite group whose Fitting subgroup is trivial, and we show that the analysis of the full conjecture can be reduced to groups having a solvable socle. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a conjecture made by T. Fukaya in 2008. This conjecture concerns the exact value of the Z 2 -cup-length of the Grassmann manifold G ˜ n , 3 of oriented 3-planes in R n. Along the way, we calculate the heights of the Stiefel–Whitney classes of the canonical vector bundle over G ˜ n , 3. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms 2 a 3 b and 2 c 5 d. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every graph with average degree at least s + t + 2 has a vertex partition into two parts, such that one part has average degree at least s , and the other part has average degree at least t. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan. [ABSTRACT FROM AUTHOR]

The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree δ ≥ ln 25 ⁡ Δ , the following holds: for every assignment L of lists of colours to the edges of G , such that | L (e) | ≥ (1 + o (1)) ⋅ max ⁡ { deg ⁡ (u) , deg ⁡ (v) } for each edge e = u v , there is an L -edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k -uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR]

In this paper we show that the Poisson analogue of the Noether's Problem has a positive solution for essentially all symplectic reflection groups — the analogue of complex reflection groups in the symplectic world. Our proofs are constructive, and generalize and refine previously known results. The results of this paper can be thought as analogues of the Noncommutative Noether Problem and the Gelfand-Kirillov Conjecture for rational Cherednik algebras in the quasi-classical limit. An abstract framework to understand these results is introduced. As a consequence for complex reflection groups, we obtain the Poisson rationality of the Calogero-Moser spaces associated to any of them, and we verify the Gelfand-Kirillov Conjecture for trigonometric Cherednik algebras and the Poisson rationality of their corresponding Calogero-Moser spaces. [ABSTRACT FROM AUTHOR]

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional p -groups of maximal nilpotency class, where p ≥ 5. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on 2-groups of maximal class and for p = 3 , we prove block-exoticity of two exotic fusion systems described by Diaz–Ruiz–Viruel. [ABSTRACT FROM AUTHOR]

A combinatorial property that characterizes Cohen-Macaulay binomial edge ideals has long been elusive. A recent conjecture ties the Cohen-Macaulayness of a binomial edge ideal J G to special disconnecting sets of vertices of its underlying graph G , called cut sets. More precisely, the conjecture states that J G is Cohen-Macaulay if and only if J G is unmixed and the collection of the cut sets of G is an accessible set system. In this paper we prove the conjecture theoretically for all graphs with up to 12 vertices and develop an algorithm that allows to computationally check the conjecture for all graphs with up to 15 vertices and all blocks with whiskers where the block has at most 11 vertices. This significantly extends previous computational results. [ABSTRACT FROM AUTHOR]

In the paper, we give an affirmative answer to the conjecture in [1]. We prove that a Shamsuddin derivation D is simple if and only if Aut (K [ x , y 1 , ... , y n ]) D = { i d }. In addition, we calculate the isotropy groups of the Shamsuddin derivations D = ∂ x + ∑ j = 1 r (a (x) y j + b j (x)) ∂ j of K [ x , y 1 , ... , y r ]. We also prove that Im D is a Mathieu-Zhao subspace if and only if a (x) ∈ K. [ABSTRACT FROM AUTHOR]

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k × k) -wall or the line graph of a subdivision of the (k × k) -wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For t ≥ 2 , a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v , a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B , and each of length at least t. We prove that for all k , t and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t -theta, or a t -pyramid, or the line graph of a subdivision of the (k × k) -wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t -theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T , every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph. [ABSTRACT FROM AUTHOR]

The minimal degree of a permutation group G is the minimum number of points not fixed by non-identity elements of G. Lower bounds on the minimal degree have strong structural consequences on G. Babai conjectured that if a primitive coherent configuration with n vertices is not a Cameron scheme, then its automorphism group has minimal degree ≥ c n for some constant c > 0. In 2014, Babai proved the desired lower bound on the minimal degree of the automorphism groups of strongly regular graphs, thus confirming the conjecture for primitive coherent configurations of rank 3. In this paper, we extend Babai's result to primitive coherent configurations of rank 4, confirming the conjecture in this special case. The proofs combine structural and spectral methods. Recently (March 2022) Sean Eberhard published a class of counterexamples of rank 28 to Babai's conjecture and suggested to replace "Cameron schemes" in the conjecture with a more general class he calls "Cameron sandwiches". Naturally, our result also confirms the rank 4 case of Eberhard's version of the conjecture. [ABSTRACT FROM AUTHOR]

Applying a recent extension (2015) of a structure theorem of Robertson, Seymour and Thomas from 1993, in this paper we establish Robertson's magic-tree conjecture from 1997. [ABSTRACT FROM AUTHOR]

This paper aims to give an affirmative answer to a conjecture raised by Soave (2020) [25] and considers the qualitative properties of normalized solutions to Sobolev critical/subcritical Schrödinger equations with combined nonlinearities. Precisely, we establish the mass threshold a ¯ such that the mountain pass type normalized solution exists for the Sobolev critical/subcritical Schrödinger equation with combined mass critical and mass supercritical nonlinearities. We then show that a ¯ is also a threshold of the limit behavior of the mountain pass type normalized solution of the Schrödinger equation with combined nonlinearities as the exponent of lower order term tending to the mass critical exponent. Among which, the results in the case that the mass small than the threshold a ¯ give an affirmative answer to the conjecture raised by Soave. [ABSTRACT FROM AUTHOR]

Berenstein and Zelevinsky introduced quantum cluster algebras [3] and the triangular bases [4]. The support conjecture in [12] asserts that the support of a triangular basis element for a rank-2 cluster algebra is bounded by an explicitly described region that is possibly concave. In this paper, we prove the support conjecture for all skew-symmetric rank-2 cluster algebras. [ABSTRACT FROM AUTHOR]

A conjecture raised by Cossey in 2007 asserts that if G is a finite p -solvable group and φ is an irreducible p -Brauer character of G with vertex Q , then the number of lifts of φ is at most | Q : Q ′ |. This conjecture is now known to be true in several situations for p odd, but there has been little progress for p even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift might be neither linear nor conjugate. In this paper we show that if χ is a lift of an irreducible 2-Brauer character in a solvable group, then χ has a linear Navarro vertex if and only if all the vertex pairs of χ are linear, and in that case all of the twisted vertices of χ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for p = 2 "one vertex at a time". [ABSTRACT FROM AUTHOR]

In 2006 Bonato and Tardif posed the Tree Alternative Conjecture (TAC): the equivalence class of a tree under the embeddability relation is, up to isomorphism, either trivial or infinite. In 2022 Abdi, et al. provided a rigorous exposition of a counter-example to TAC developed by Tetano in his 2008 PhD thesis. In this paper we provide a positive answer to TAC for a weaker type of graph relation: the topological minor relation. More precisely, letting [ T ] denote the equivalence class of T under the topological minor relation we show that 1. | [ T ] | = 1 or | [ T ] | ≥ ℵ 0 and 2. ∀ r ∈ V (T) , | [ (T , r) ] | = 1 or | [ (T , r) ] | ≥ ℵ 0. In particular, by means of curtailing trees, we show that for any tree T with at least one ray with infinitely many vertices with degree at least 3: | [ T ] | ≥ 2 ℵ 0 . [ABSTRACT FROM AUTHOR]

Can we find a self-similar set on the line with positive Lebesgue measure and empty interior? Currently, we do not have the answer for this question for deterministic self-similar sets. In this paper we answer this question negatively for random self-similar sets which are defined with the construction introduced in the paper by Jordan et al. (2007) [6]. For the same type of random self-similar sets we prove the Palis-Takens conjecture which asserts that at least typically the algebraic difference of dynamically defined Cantor sets is either large in the sense that it contains an interval or small in the sense that it is a set of zero Lebesgue measure. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to present an answer to Bressoud's conjecture for the case j = 0 , resulting in a complete solution to Bressoud's conjecture. The case for j = 1 has been recently resolved by Kim. Using the connection established in our previous paper between the ordinary partition function B 0 and the overpartition function B ¯ 1 , we found that the proof of Bressoud's conjecture for the case j = 0 is equivalent to establishing an overpartition analogue of the conjecture for the case j = 1. By generalizing Kim's method, we obtain the desired overpartition analogue of Bressoud's conjecture for the case j = 1 , which eventually enables us to confirm Bressoud's conjecture for the case j = 0. [ABSTRACT FROM AUTHOR]

In this article we study semi-abelian analogues of Schanuel conjecture. As showed by the first author, Schanuel conjecture is equivalent to the Generalized Period conjecture applied to 1-motives without abelian part. Extending her methods, the second, the third and the fourth authors have introduced the abelian analogue of Schanuel conjecture as the Generalized Period conjecture applied to 1-motives without toric part. As a first result of this paper, we define the semi-abelian analogue of Schanuel conjecture as the Generalized Period conjecture applied to 1-motives. C. Cheng et al. proved that Schanuel conjecture implies the algebraic independence of the values of the iterated exponential and the values of the iterated logarithm, answering a question of M. Waldschmidt. The second, the third and the fourth authors have investigated a similar question in the setup of abelian varieties: the Weak Abelian Schanuel conjecture implies the algebraic independence of the values of the iterated abelian exponential and the values of an iterated generalized abelian logarithm. The main result of this paper is that a Relative Semi-abelian conjecture implies the algebraic independence of the values of the iterated semi-abelian exponential and the values of an iterated generalized semi-abelian logarithm. [ABSTRACT FROM AUTHOR]

In a paper of Kedlaya and Medvedovsky [KM19] , the number of distinct dihedral mod 2 modular representations of prime level N was calculated, and a conjecture on the dimension of the space of level N weight 2 modular forms giving rise to each representation was stated. In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

A famous conjecture of Parkin-Shanks predicts that p (n) is odd with density 1/2. Despite the remarkable amount of work of the last several decades, however, even showing this density is positive seems out of reach. In a 2018 paper with Judge, we introduced a different approach and conjectured the "striking" fact that, if for any A ≡ ± 1 (mod 6) the multipartition function p A (n) has positive odd density, then so does p (n). Similarly, the positive odd density of any p A (n) with A ≡ 3 (mod 6) would imply that of p 3 (n). Our conjecture was shown to be a corollary of an earlier conjecture of the same paper. In this brief note, we provide an unconditional proof of it. An important tool will be Chen's recent breakthrough on a special case of our earlier conjecture. [ABSTRACT FROM AUTHOR]

• In this paper, we characterize Hamming and Johnson graphs by their spectral gap. • We confirm Babai's conjecture on the motion of distance-regular graphs of bounded diameter. • Our characterization of Hamming graphs involves only inequality constraints. One of the central results in the representation theory of distance-regular graphs classifies distance-regular graphs with μ ≥ 2 and second largest eigenvalue θ 1 = b 1 − 1. In this paper we give a classification under the (weaker) approximate eigenvalue constraint θ 1 ≥ (1 − ε) b 1 for the class of geometric distance-regular graphs. As an application, we confirm Babai's conjecture on the minimal degree of the automorphism group of distance-regular graphs of bounded diameter. This conjecture asserts that if X is a primitive distance-regular graph with n vertices, and X is not a Hamming graph or a Johnson graph, then the automorphism group Aut (X) has minimal degree ≥ c n for some constant c > 0. It follows that Aut (X) satisfies strong structural constraints. [ABSTRACT FROM AUTHOR]

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure. [ABSTRACT FROM AUTHOR]

This paper presents an E -derivation analogue of a result on derivations due to van den Essen, Wright and Zhao. We prove that the image of a locally finite K - E -derivation of polynomial algebras in two variables over a field K of characteristic zero is a Mathieu subspace. This result together with that of van den Essen, Wright and Zhao confirms the LFED conjecture in the case of polynomial algebras in two variables. [ABSTRACT FROM AUTHOR]

In this paper, we present some partial results for the geometrically m -step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves over fields finitely generated over the prime field of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

For any positive integer t , a t-broom is a graph obtained from K 1 , t + 1 by subdividing an edge once. In this paper, we show that, for graphs G without induced t -brooms, we have χ (G) = o (ω (G) t + 1) , where χ (G) and ω (G) are the chromatic number and clique number of G , respectively. When t = 2 , this answers a question of Schiermeyer and Randerath. Moreover, for t = 2 , we strengthen the bound on χ (G) to 7 ω (G) 2 , confirming a conjecture of Sivaraman. For t ≥ 3 and { t -broom, K t , t }-free graphs, we improve the bound to o (ω t). [ABSTRACT FROM AUTHOR]

Carsten Thomassen in 1989 conjectured that if a graph has minimum degree much more than the number of atoms in the universe (δ (G) ≥ 10 10 10 ), then it contains a pillar , which is a graph that consists of two vertex-disjoint cycles of the same length, s say, along with s vertex-disjoint paths of the same length3 which connect matching vertices in order around the cycles. Despite the simplicity of the structure of pillars and various developments of powerful embedding methods for paths and cycles in the past three decades, this innocent looking conjecture has seen no progress to date. In this paper, we give a proof of this conjecture by building a pillar (algorithmically) in sublinear expanders. [ABSTRACT FROM AUTHOR]

We consider nonautonomous master equations of finite-state, continuous-time Markovian jump processes with uniformly continuous and bounded transition matrix functions. The Earnshaw–Keener conjecture states that if the omega-limit set of the transition matrix function contains at least one matrix which is neither decomposable nor splitting, then the difference of any two probability distribution solutions tends to zero at infinity. The conjecture has been confirmed under the additional assumption that the transition matrix function is almost-automorphic. In this paper, we prove the conjecture in its full generality. [ABSTRACT FROM AUTHOR]

For a finite group G , we denote by d (G) and by E (G) , respectively, the small Davenport constant and the Gao constant of G. Let C n be the cyclic group of order n and let G m , n , s = C n ⋊ s C m be a metacyclic group. In [2, Conjecture 17] , Bass conjectured that d (G m , n , s) = m + n − 2 and E (G m , n , s) = m n + m + n − 2 provided ord n (s) = m. In this paper, we show that the assumption ord n (s) = m is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for G m , n , s and the mn -product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G 2 m , 2 n , r , where r 2 ≡ s (mod n). [ABSTRACT FROM AUTHOR]

In this paper, we formally introduce the notion of Apéry-like sums and we show that every multiple zeta values can be expressed as a Z -linear combination of them. We even describe a natural way to do so. This allows us to put in a new theoretical context several identities scattered in the literature, as well as to discover many new interesting ones. We give in this paper new integral formulas for multiple zeta values and Apéry-like sums. They enable us to give a short direct proof of Zagier's formulas for ζ (2 , ... , 2 , 3 , 2 , ... , 2) as well as of similar ones in the context of Apéry-like sums. The relations between Apéry-like sums themselves still remain rather mysterious, but we get significant results and state some conjectures about their pattern. [ABSTRACT FROM AUTHOR]

Hajós conjectured in 1968 that every Eulerian n -vertex graph can be decomposed into at most ⌊ (n − 1) / 2 ⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986) [1] , Dean (1986) [7] , and Bollobás and Scott (1996) [2] it was analogously conjectured that every directed Eulerian graph can be decomposed into O (n) cycles. In this paper, we show that every directed Eulerian graph can be decomposed into O (n log ⁡ Δ) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least 1, there exists a cycle with weight at least Ω (log ⁡ log ⁡ n / log ⁡ n) , thus resolving a question by Bollobás and Scott. [ABSTRACT FROM AUTHOR]

Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for them. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, that is, a bridgeless cubic graph which is not 3-edge-colourable. In this paper we deal with the fact that the family of potential counterexamples to many interesting conjectures can be narrowed even further to the family S ≥ 5 of bridgeless cubic graphs whose edge set cannot be covered with four perfect matchings. The Cycle Double Cover Conjecture, the Shortest Cycle Cover Conjecture and the Fan–Raspaud Conjecture are examples of statements for which S ≥ 5 is crucial. In this paper we study parameters which have the potential to further refine S ≥ 5 and thus enlarge the set of cubic graphs for which the mentioned conjectures can be verified. We show that S ≥ 5 can be naturally decomposed into subsets with increasing complexity, thereby producing a natural scale for proving these conjectures. More precisely, we consider the following parameters and questions: given a bridgeless cubic graph, (i) how many perfect matchings need to be added, (ii) how many copies of the same perfect matching need to be added, and (iii) how many 2-factors need to be added so that the resulting regular graph is Class I? We present new results for these parameters and we also establish some strong relations between these problems and some long-standing conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we consider the homogeneous coordinate rings A (Y n , d) ≅ K [ Ω n , d ] of monomial projections Y n , d of Veronese varieties parameterized by subsets Ω n , d of monomials of degree d in n + 1 variables where: (1) Ω n , d contains all monomials supported in at most s variables and, (2) Ω n , d is a set of monomial invariants of a finite diagonal abelian group G ⊂ GL (n + 1 , K) of order d. Our goal is to study when K [ Ω n , d ] is a quadratic algebra and, if so, when K [ Ω n , d ] is Koszul or G-quadratic. For the family (1), we prove that K [ Ω n , d ] is quadratic when s ≥ ⌈ n + 2 2 ⌉. For the family (2), we completely characterize when K [ Ω 2 , d ] is quadratic in terms of the group G ⊂ GL (3 , K) , and we prove that K [ Ω 2 , d ] is quadratic if and only if it is Koszul. We also provide large families of examples where K [ Ω n , d ] is G-quadratic. [ABSTRACT FROM AUTHOR]

Triple correlation sums problem concerns the non-trivial power-saving bounds for the correlation of three objects. It is conjectured that these sums are non-trivial in any fixed but arbitrarily given ranges. In this paper, the uniform non-trivial bounds for the triple correlation sums ∑ m ≥ 1 , n ≥ 1 λ π (1 , m) λ ⋆ (n) λ f (m + p n) U (m / X) V (n / H) in the level aspect are derived, where π is any G L 3 -Maaß cuspidal form, f ∈ B k ⁎ (p) , any Hecke newform of prime level p and weight k ∈ N + , and λ ⋆ (n) , n ≥ 1 , are certain coefficients of arithmetic interest. As a result, we show that sums of this type follow the 1/3- significance level. We study the strength of the result by specifying ⋆ being the cusp forms on G L 3 and G L 2 , respectively, and further obtain the more significant cancellations in these sums. [ABSTRACT FROM AUTHOR]

A classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that there is a constant C such that every graph with average degree at least C k 2 has a subdivision of K k , the complete graph on k vertices. We study two directions extending this result. • Verstraëte conjectured that a quadratic bound guarantees in fact two vertex-disjoint isomorphic copies of a K k -subdivision. • Thomassen conjectured that for each k ∈ N there is some d = d (k) such that every graph with average degree at least d contains a balanced subdivision of K k. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on d (k) remains open. In this paper, we show that a quadratic lower bound on average degree suffices to force a balanced K k -subdivision. This gives the right order of magnitude of the optimal d (k) needed in Thomassen's conjecture. Since a balanced K m k -subdivision trivially contains m vertex-disjoint isomorphic K k -subdivisions, this also confirms Verstraëte's conjecture in a strong sense. [ABSTRACT FROM AUTHOR]

A hole in a graph is an induced subgraph which is a cycle of length at least four. A hole is called even if it has an even number of vertices. An even-hole-free graph is a graph with no even holes. A vertex of a graph is bisimplicial if the set of its neighbours is the union of two cliques. In an earlier paper [1] , Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial vertex, but we have recently been shown that the "proof" has a serious error. Here we give a proof using a different approach. [ABSTRACT FROM AUTHOR]

Blickle, Mustaţă and Smith proposed two conjectures on the limits of F -pure thresholds. One conjecture asks whether or not the limit of a sequence of F -pure thresholds of principal ideals on regular local rings of fixed dimension can be written as an F -pure thresshold in lower dimension. Another conjecture predicts that any F -pure threshold of a formal power series can be written as the F -pure threshold of a polynomial. In this paper, we prove that the first conjecture has a counterexample but a weaker statement still holds. We also give a partial affirmative answer to the second conjecture. [ABSTRACT FROM AUTHOR]