Representations by quaternary quadratic forms whose coefficients are 1, 4, 9 and 36

Explicit formulae are determined for the number of representations of a positive integer by the quadratic forms ax²+by²+cz²+dt² with a,b,c,d∈{1,4,9,36}, gcd(a,b,c,d)=1 and a?b?c?d.     

Partition Statistics and Quasiharmonic Maass Forms

Andrews recently introduced k-marked Durfee symbols, which area generalization of partitions that are connected to momentsof Dyson's rank statistic. He used these connections to findidentities relating to their generating functions as well asto prove Ramanujan-type congruences for these objects and findrelations between them. In this paper, we show that the hypergeometricgenerating functions for these objects are natural examplesof quasimock theta functions, which are defined as the holomorphicparts of harmonic Maass forms and their derivatives. In particular,these generating functions may be viewed as analogs of Ramanujan'smock theta functions with arbitrarily high weight. We use theautomorphic properties to prove the existence of infinitelymany congruences for the Durfee symbols. Furthermore, we showthat as k varies, the modularity of the k-marked Durfee symbolsis precisely dictated by the case k = 2. Finally, we use thisrelation in order to prove the existence of general congruencesfor rank moments in terms of level one modular forms of boundedweight.     

Subnormality of Bergman-like weighted shifts

For a,b,c,d?0 with ad−bc>0, we consider the unilateral weighted shift S(a,b,c,d) with weights . Using Schur product techniques, we prove that S(a,b,c,d) is always subnormal; more generally, we establish that for every p?1, all p-subshifts of S(a,b,c,d) are subnormal. As a consequence, we show that all Bergman-like weighted shifts are subnormal.     

Any Diophantine quintuple contains a regular Diophantine quadruple

A set {a₁,…,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1?i<j?m. It is conjectured that if {a,b,c,d} is a Diophantine quadruple with a<b<c<d, then d=d₊, where d₊=a+b+c+2abc+2rst and , , . In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a<b<c<d<e, then d=d₊.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

On the signless Laplacian spectral characterization of the line graphs of T-shape trees

A graph is determined by its signless Laplacian spectrum if no other nonisomorphic graph has the same signless Laplacian spectrum (simply G is DQS). Let T (a, b, c) denote the T-shape tree obtained by identifying the end vertices of three paths P _a+2, P _b+2 and P _c+2. We prove that its all line graphs L(T(a, b, c)) except L(T(t, t, 2t+1)) (t ? 1) are DQS, and determine the graphs which have the same signless Laplacian spectrum as L(T(t, t, 2t + 1)). Let µ₁(G) be the maximum signless Laplacian eigenvalue of the graph G. We give the limit of µ₁(L(T(a, b, c))), too.     

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

Heilbronn's conjecture on Waring's number (mod p)

Let p be a prime k|p−1, t=(p−1)/k and γ(k,p) be the minimal value of s such that every number is a sum of s kth powers . We prove Heilbronn's conjecture that γ(k,p)?k1/2 for t>2. More generally we show that for any positive integer q, γ(k,p)?C(q)k1/q for ?(t)?q. A comparable lower bound is also given. We also establish exact values for γ(k,p) when ?(t)=2. For instance, when t=3, γ(k,p)=a+b−1 where a>b>0 are the unique integers with a²+b²+ab=p, and when t=4, γ(k,p)=a−1 where a>b>0 are the unique integers with a²+b²=p.     

Uniformly optimal graphs in some classes of graphs with node failures

The uniformly optimal graph problem with node failures consists of finding the most reliable graph in the class Ω(n,m) of all graphs with n nodes and m edges in which nodes fail independently and edges never fail. The graph G is called uniformly optimal in Ω(n,m) if, for all node-failure probabilities q∈(0,1), the graph G is the most reliable graph in the class of graphs Ω(n,m). This paper proves that the multipartite graphs K(b,b+1,…,b+1,b+2) are uniformly optimal in their classes Ω((k+2)(b+1),(k²+3k+2)(b+1)²/2−1), where k is the number of partite sets of size (b+1), while for i>2, the multipartite graphs K(b,b+1,…,b+1,b+i) are not uniformly optimal in their classes Ω((k+2)b+k+i,(k+2)(k+1)b²/2+(k+1)(k+i)b+k(k+2i−1)/2).     

Efficient algorithms for decomposing graphs under degree constraints

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d(v)?a(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that d_A(v)?a(v) for every v∈A and d_B(v)?b(v) for every v∈B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d(v)?a+b (a,b?1) or just d(v)?a+b-1 (a,b?2) if G contains no cycles shorter than 4 or 5, respectively.The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.     

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form un=an(n+t)+b with (a,t,b)∈Z³, a?5, t?0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{¹²+1,²²+1,…,n²+1}?0,32n(1,442) (for all n?1). In the last part of this article, we study the integer lcm(n,n+1,…,n+k) (k∈N, n∈N^∗). We show that it has a divisor dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k).     

Wide diameter of Cartesian graph bundles

Fault tolerance and transmission delay of networks are important concepts in network design. The notions are strongly related to connectivity and diameter of a graph, and have been studied by many authors. Wide diameter of a graph combines studying connectivity with the diameter of a graph. Diameter with width k of a graph G, k-diameter, is defined as the minimum integer d for which there exist at least k internally disjoint paths of length at most d between any two distinct vertices in G. Denote by D_c(G) the c-diameter of G and κ(G) the connectivity of G. In the context of computer networks, wide diameters of Cartesian graph products have been recently studied by many authors. Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a Cartesian graph bundle with fiber F over base B, 0<a≤κ(F), and 0<b≤κ(B). We prove that D_a+b(G)≤D_a(F)+D_b(B)+1. Moreover, if G is a graph bundle with fiber F≠K₂ over base B≠K₂, then D_a+b(G)≤D_a(F)+D_b(B). The bounds are tight.     

The structure of rigid functions

A function f:R→R is called vertically rigid if graph(cf) is isometric to graph(f) for all c≠0. We prove Jankovi?'s conjecture by showing that a continuous function is vertically rigid if and only if it is of the form a+bx or a+bekx (a,b,k∈R). We answer the question of Cain, Clark and Rose by showing that there exists a Borel measurable vertically rigid function which is not of the above form. We discuss the Lebesgue and Baire measurable case, consider functions bounded on some interval and functions with at least one point of continuity. We also introduce horizontally rigid functions, and show that a certain structure theorem can be proved without assuming any regularity.     

Left division in the free left distributive algebra on one generator

Let A be the free algebra on one generator satisfying the left distributive law a(bc)=(ab)(ac). Using a division algorithm for elements of an extension P of A, we prove some facts about left division in A, one consequence of which is a conjecture of J. Moody: If a,b,c,d∈A,ab=cd,a and b have no common left divisors, and c and d have no common left divisors, then a=c and b=d.     

Exact stability sets for a linear difference system with diagonal delay

This paper is devoted to the stability analysis of a delay difference system of the form xn+1=axn−k+byn, yn+1=cxn+ayn−k, where a, b and c are real numbers and k is a positive integer. We establish some exact conditions for the zero solution of the system to be asymptotically stable.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

On solutions of “equations in symmetric groups”

Let x?S_n, the symmetric group on n symbols. Let θ? Aut(S_n) and let the automorphim order of x with respect to θ be defined by

$\text{γθ(x)=}\text{min}\text{{k:x xθ xθ}{}^2\text{? xθ}{}^{\mathrm{k?1}}\text{=1}}$

where xθ is the image of x under θ. Let α_g? Aut(S_n) denote conjugation by the element g?S_n. Let where s and k are positive integers and $\text{a}\text{b}$ denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?S_n let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let g_s denote the product of all cycles in g whose lengths do not divide s. Then g_s moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then $\text{b(g; s, 1:n)=∑}{}_{\mathrm{<mtext>i</mtext><mtext>s</mtext>}}\text{b(g; s, 1:n?1)(}{}^{\mathrm{t}}{}_{\mathrm{i?1}}\text{(i?1)!}$ (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z⁺ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in g_s^s of any given length is smaller than the smallest prime dividing s iff b(g_s; s, 1 : c) = 1. If g = (12 … p^m)^t and $\text{sk}\text{p}{}^{\mathrm{m}}$ then $\text{b(g;s,k:p}{}^{\mathrm{m}}\text{) {}{}^0{}_{\mathrm{\pm 1}}\text{(}\text{mod}\text{p)}$.     

Maximal periods of (Ehrhart) quasi-polynomials

A quasi-polynomial is a function defined of the form q(k)=cd(k)kd+cd−1(k)kd−1+?+c₀(k), where c₀,c₁,…,cd are periodic functions in k∈Z. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the cj(k) for Ehrhart quasi-polynomials. For generic polytopes, McMullen's bounds seem to be sharp, but sometimes smaller periods exist. We prove that the second leading coefficient of an Ehrhart quasi-polynomial always has maximal expected period and present a general theorem that yields maximal periods for the coefficients of certain quasi-polynomials. We present a construction for (Ehrhart) quasi-polynomials that exhibit maximal period behavior and use it to answer a question of Zaslavsky on convolutions of quasi-polynomials.     

Analysis of sharp polynomial upper estimate of number of positive integral points in a five-dimensional tetrahedra

Let a?b?c?d?e?1 be real numbers and P₅ be the number of positive integral solutions of . In this paper we show that 120P₅?(a-1)(b-1)(c-1)(d-1)(e-1). This confirms a conjecture of Durfee for the dimension 5 case. We show also that the upper estimate of P₅ given by Lin and Yau is strictly sharper than that suggested by Durfee conjecture if , but is not sharper than that suggested by Durfee conjecture if .