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In this paper we verify Navarro's refinement of the McKay conjecture for quasi-simple groups of Lie type in their defining characteristic. Navarro's refinement takes into account the action of specific Galois automorphisms on the characters presents in the McKay conjecture [12]. Our proof of this case of the conjecture relies on a character correspondence constructed by Maslowski in [11]. Building on this we verify the inductive condition for Navarro's refinement from [14] for most groups of Lie type in defining characteristic. [ABSTRACT FROM AUTHOR]

We report here on the computational verification of Zhi-Wei Sun's "1-3-5 conjecture" for all natural numbers up to 105 103 560 126. This, together with a result of two of the authors, completes the proof of that conjecture. Furthermore, the computations made in the verification process of the 1-3-5 conjecture revealed a refinement, which we state as a separate conjecture at the end of the paper. [ABSTRACT FROM AUTHOR]

Text Let N be the set of all nonnegative integers and k ≥ 2 be a fixed integer. For a set A ⊆ N , let r k ( A , n ) denote the number of solutions of a 1 + ⋯ + a k = n with a 1 , … , a k ∈ A . In this paper, we prove that for given positive integer u , there is a set A ⊆ N such that r k ( A , n ) ≥ 1 for all n ≥ 0 and the set of n with r k ( A , n ) = k ! u has density one. This generalizes recent results of Chen and Yang. Video For a video summary of this paper, please visit http://youtu.be/2fbKtDAOqQ0 . [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers Levy's related result on Noetherian rings [23, Theorem] and Matlis' related result on Prüfer domains [26, Theorem] . It also globalizes Couchot's related result on chained rings [10, Theorem 11] . New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions. [ABSTRACT FROM AUTHOR]

This article determines all the solutions in the finite field F 2 4 n of the equation x 2 3 n + 2 2 n + 2 n − 1 + (x + 1) 2 3 n + 2 2 n + 2 n − 1 = b. Specifically, we explicitly determine the set of b 's for which the equation has i solutions for any positive integer i. Such sets, which depend on the number of solutions i , are given explicitly and expressed nicely, employing the absolute trace function over F 2 n , the norm function over F 2 4 n relatively to F 2 n and the set of (2 n + 1) st roots of unity in F 2 4 n . The equation considered in this paper comes from an article by Budaghyan et al. ([2]) in which the authors have investigated novel approaches for obtaining alternative representations for functions from the known infinite APN families. In particular, they have been interested in determining the differential spectrum of some power functions among them is the one F (x) = x 2 3 n + 2 2 n + 2 n − 1 defined over F 2 4 n . The problem of the determination of such spectrum has led to a conjecture (Conjecture 27 in the preprint (2020) [2] for which an updated version will appear in 2022 at the IEEE Transactions Information Theory) stated by Budaghyan et al. As an immediate consequence of our results, we prove that the above equation has 2 2 n solutions for one value of b , 2 2 n − 2 n solutions for 2 n values of b in F 2 4 n and has at most two solutions for all remaining points b , leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in [9] gives the complete differential spectrum of F and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows to determine ultimately the set of b 's for which the considered equation has solutions as well as the solutions of the equation for any b in F 2 4 n . [ABSTRACT FROM AUTHOR]

Füredi conjectured that the Boolean lattice 2 [ n ] can be partitioned into ( n ⌊ n / 2 ⌋ ) chains such that the size of any two differs in at most one. In this paper, we prove that there is an absolute constant α ≈ 0.8482 with the following property: for every ϵ > 0 , if n is sufficiently large, the Boolean lattice 2 [ n ] has a chain partition into ( n ⌊ n / 2 ⌋ ) chains, each of them of size between ( α − ϵ ) n and O ( n / ϵ ) . We deduce this result by looking at the more general setup of unimodal normalized matching posets. We prove that a unimodal normalized matching poset P of width w has a chain partition into w chains, each of size at most 2 | P | w + 5 , and it has a chain partition into w chains, where each chain has size at least | P | 2 w − 1 2 . [ABSTRACT FROM AUTHOR]

A ( c 1 , c 2 , … , c k ) -coloring of a graph G is a mapping φ : V ( G ) ↦ { 1 , 2 , … , k } such that for every i , 1 ≤ i ≤ k , G [ V i ] has maximum degree at most c i , where G [ V i ] denotes the subgraph induced by the vertices colored i . Borodin and Raspaud conjecture that every planar graph with neither 5 -cycles nor intersecting triangles is 3 -colorable. We prove in this paper that every planar graph with neither 5 -cycles nor intersecting triangles is (2, 0, 0)-colorable. [ABSTRACT FROM AUTHOR]

The squared modulus of every real-valued on R function f from the Laguerre–Pólya class L – P obeys a MacLaurin-type series representation | f ( x + i y ) | 2 = ∑ k = 0 ∞ L k ( f ; x ) y 2 k , x , y ∈ R . If f is a polynomial with only real roots, then the sum becomes finite. The coefficients { L k } are representable as non-linear differential operators acting on f , and by a classical result of Jensen L k ( f ; x ) ≥ 0 for f ∈ L – P and x ∈ R . A conjecture of M. Patrick from 1971 states that for f = P n ( α , β ) , the n -th Jacobi polynomial, with α ≥ β > − 1 , the functions L k ( f ; x ) , 1 ≤ k ≤ n − 1 , attain their maxima in [ 0 , 1 ] at x = 1 . The aim of this paper is to validate Patrick's conjecture. Moreover, we prove a refined version of this conjecture, showing that { L k ( f ; x ) } k = 1 n − 1 are strictly monotonically increasing functions on the positive semi-axis. Towards our proof of Patrick's conjecture we extend the Sonin–Pólya majorization approach to all coefficient functions { L k ( f ; x ) } k = 0 n , f = P n ( α , β ) . [ABSTRACT FROM AUTHOR]

We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the following property of F : if the image p = F ( l ) of some light leaf l under F is bigger than the Coxeter number of the corresponding Weyl group, then there is a counterexample to Lusztig's conjecture in characteristic p . We also introduce the “double leaves basis” which is an improvement of the light leaves basis that has already found interesting applications. In particular it forms a cellular basis of Soergel bimodules that allows us to produce an algorithm to find “the bad primes” for Lusztig's conjecture. [ABSTRACT FROM AUTHOR]

Let P G ( q ) denote the chromatic polynomial of a graph G on n vertices. The ‘shameful conjecture’ due to Bartels and Welsh states that, P G ( n ) P G ( n − 1 ) ≥ n n ( n − 1 ) n . Let μ ( G ) denote the expected number of colors used in a uniformly random proper n -coloring of G . The above inequality can be interpreted as saying that μ ( G ) ≥ μ ( O n ) , where O n is the empty graph on n nodes. This conjecture was proved by F.M. Dong, who in fact showed that, P G ( q ) P G ( q − 1 ) ≥ q n ( q − 1 ) n for all q ≥ n . There are examples showing that this inequality is not true for all q ≥ 2 . In this paper, we show that the above inequality holds for all q ≥ 36 D 3 / 2 , where D is the largest degree of G . It is also shown that the above inequality holds true for all q ≥ 2 when G is a claw-free graph. [ABSTRACT FROM AUTHOR]

A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the periods of the zig–zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms which correspond to multiple zeta values ζ ( 2 , … , 2 , 3 , 2 , … 2 ) and using the method of graphical functions. The zig–zag graphs are the only infinite family of primitive graphs in ϕ 4 4 theory (in fact, in any renormalisable quantum field theory in four dimensions) whose periods are now known. [ABSTRACT FROM AUTHOR]

In 1969 Harold Widom published his seminal paper (Widom, 1969) which gave a complete description of orthogonal and Chebyshev polynomials on a system of smooth Jordan curves. When there were Jordan arcs present the theory of orthogonal polynomials turned out to be just the same, but for Chebyshev polynomials Widom’s approach proved only an upper estimate, which he conjectured to be the correct asymptotic behavior. In this note we make some clarifications which will show that the situation is more complicated. [ABSTRACT FROM AUTHOR]

In this paper, we prove two conjectures on truncated series. The first conjecture made by G.E. Andrews and M. Merca is related to Jacobi's triple product identity, while the second conjecture by V.J.W. Guo and J. Zeng is related to Jacobi's identity. [ABSTRACT FROM AUTHOR]

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be disjoint subsets of { 0 , 1 , ⋯ p − 1 } , where p is a prime and F = { F 1 , F 2 , … , F m } be a family of subsets of [ n ] such that | F i | (mod p ) ∈ K for all F i ∈ F and | F i ∩ F j | (mod p ) ∈ L for i ≠ j . In 1991 Alon, Babai and Suzuki conjectured that if n ≥ s + max 1 ≤ i ≤ r k i , then | F | ≤ n s + n s − 1 + ⋯ + n s − r + 1 . In this paper we prove this conjecture. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues. [Copyright &y& Elsevier]

A sequence e = e 1 e 2 ⋯ e n of natural numbers is called an inversion sequence if 0 ≤ e i ≤ i − 1 for all i ∈ { 1 , 2 , ... , n }. Recently, Martinez and Savage initiated an investigation of inversion sequences that avoid patterns of relation triples. Let ρ 1 , ρ 2 and ρ 3 be among the binary relations { < , > , ≤ , ≥ , = , ≠ , − }. Martinez and Savage defined I n (ρ 1 , ρ 2 , ρ 3) as the set of inversion sequences of length n such that there are no indices 1 ≤ i < j < k ≤ n with e i ρ 1 e j , e j ρ 2 e k and e i ρ 3 e k. In this paper, we will prove a curious identity concerning the ascent statistic over the sets I n (> , ≠ , ≥) and I n (≥ , ≠ , >). This confirms a recent conjecture of Zhicong Lin. [ABSTRACT FROM AUTHOR]