Colourings of graphs with two consecutive odd cycle lengths

In 1992 Gyárfás showed that a graph G having only k odd cycle lengths is (2k+1)-colourable, if it does not contain a K_2k+2. In this paper, we will present the results for graphs containing only odd cycles of length 2m−1 and 2m+1 as done in [S. Matos Camacho, Colourings of graph with prescribed cycle lengths, diploma thesis, TU Bergakademie Freiberg, 2006. [3]]. We will show that these graphs are 4-colourable.     

Non-isochronicity of the center in polynomial Hamiltonian systems

In 2002 Jarque and Villadelprat proved that planar polynomial Hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial Hamiltonian systems of even degree. Recently, it was proved that a planar polynomial Hamiltonian system is non-isochronous if a quantity, denoted by M_2m−2, can be computed such that M_2m−2≤0. As a corollary of this criterion, the open question was answered for those systems with only even degree nonlinearities. In this paper we consider the case of M_2m−2>0 and give a new criterion for non-isochronicity. Applying the new criterion, we also answer the open question for some cases in which some terms of odd degree are included.     

(m)-covering of a triangulation

Let G be a graph triangularly imbedded into a surface S, G_(m) is the graph constructed from G by replacing each vertex x by m vertices (xx,0), (x, 1), ..., (x, m − 1) and joining two vertices (x, i) and (y, j) by an edge if and only if x and y are joined in G. The main result is that the construction of G_(m) is possible whenever n is an odd prime and a well separating cycle (mod m) can be determined.     

Arc-regular cubic graphs of order four times an odd integer

A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4m where m is an odd integer. Using the Gorenstein?CWalter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of automorphisms that is isomorphic to a subgroup of Aut(PSL(2,q)) containing PSL(2,q) for some odd prime-power?q. Also a construction is given for infinitely many such graphs??namely a family of Cayley graphs for the groups PSL(2,p ³) where p is an odd prime; the smallest of these has order?9828.     

The number of K m,m -free graphs

A graph is called H-free if it contains no copy of H. Denote by f _n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f _n (H) ≤ 2^{(1+o(1))ex(n,H)}. This was first shown to be true for cliques; then, Erdős, Frankl, and R?dl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log₂ f _n (H) is not known. We prove that f _n (K _m,m ) ≤ 2 ^O (n ^2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K _m,m ) is conjectured to be Θ(n ^2−1/m ). Our method also yields a bound on the number of K _m,m -free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K _3,3-free graphs of order n have more than 1/20·ex(n,K _3,3) edges.     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

Generalized congruence properties of the restricted partition function p(n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let ? be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,?) modulo any power of that prime ? ^α . In addition, we establish general congruence relations for p(n,?) modulo ? ^α for any n.     

On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup H of a group G is pronormal if the subgroups H and H ^g are conjugate in 〈H,H ^g 〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL _n (q), PSU _n (q), E ₆(q), ² E ₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n (q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ (q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n (q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2 ^m or 2 ^m (2^2k +1), this group has a nonpronormal subgroup of odd index. If n = 2 ^m , then we show that all subgroups of P Sp_2n (q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n (q) is still open when n = 2 ^m (2^2k + 1) and q ≡ ±3 (mod 8).     

On Maillet determinant

For a positive integer m, let $\text{A = {1 ≤ a <}\text{m}\text{2}\text{| (a, m) = 1}}$ and let n = |A|. For an integer x, let R(x) be the least positive residue of x modulo m and if (x, m) = 1, let x′ be the inverse of x modulo m. If m is odd, then |R(ab′)|_a,b∈A = ?2^1?n(∏_χ(Σ_{a = 1}^{m ? 1}aχ(a))), where χ runs over all the odd Dirichlet characters modulo m.     

On D. Hägele’s approach to the Bessis-Moussa-Villani conjecture

The reformulation of the Bessis-Moussa-Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient α_m,k(A,B) of t^k in the polynomial Tr(A+tB)^m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd.     

A Radon-Nikodym type theorem for functionals

Let ($\text{Ω, τ, m}$) be a finite, nonatomic, separable measure space. This paper extends the Radon-Nikodym theorem to odd, disjointly additive, m-continuous functionals whose domain consists of all differences of characteristic functions which belong to a given subspace of L^∞(m). Such a functional will possess a density in L¹(m) provided that the subspace is weak¹-closed and separates sets; the conclusion can fail if the latter hypothesis is removed. Analogous results are obtained for functionals which are not necessarily odd.     

Sur la quasi-convexité des arcs trinomiaux

From the equationp ^n?k sinnθ?ρ ⁿ sin(n?k)θ=sinkθ we will show that the function σ=σ(θ) is increasing for the arcsA _m , obtained when one putsn=m, k=m?1 andm=3,4,5,… Next, we will study the arcsB _m obtained whenn=m, k=m?2 andm an odd integer larger than 3. In this case, σ(θ) will be shown to be a decreasing function. Finally, the Farey arcsF(p,q;r,s) are obtained whenn=s, k=q, s andq relatively prime. It will be proved that the function σ(θ) is strictly quasi-convex.     

3-Designs from the Z 4-Goethals Codes via a New Kloosterman Sum Identity

Recently, active research has been performed on constructing t-designs from linear codes over Z ₄. In this paper, we will construct a new simple 3 – (2 ^m , 7, 14/3 (2 ^m – 8)) design from codewords of Hamming weight 7 in the Z ₄-Goethals code for odd m 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.     

Permutation functions and (n,m)-commutativity of semigroups

By [4], a semigroupS is called an (n, m)-commutative semigroup (n, m ∈ ?⁺, the set of all positive integers) if $$x_1 x_2 \cdot \cdot \cdot x_n y_1 y_2 \cdot \cdot \cdot y_m = y_1 y_2 \cdot \cdot \cdot y_m x_1 x_2 \cdot \cdot \cdot x_n $$ holds for allx ₁,...,x _n ,y ₁,...,y _m ∈S It is evident that ifS is an (n, m)-commutative semigroup then it is (n′,m′)-commutative for alln′≥n andm′≥m. In this paper, for an arbitrary semigroupS, we determine all pairs (n, m) of positive integersn andm for which the semigroupS is (n, m)-commutative. In our investigation a special type of function mapping ?⁺ into itself plays an important role. These functions which are defined and discussed here will be called permutation functions.     

On Finite Groups in Which Cyclic Subgroups of the Same Order are Conjugate

We consider finite groups G for which any two cyclic subgroups of the same order are conjugate in G. We prove various structure results and, in particular, that any such group has at most one non-abelian composition factor, and this is isomorphic to PSL(2, p ^m ), with m odd if p is odd, or to Sz(2^2m+1), or to one of the sporadic groups M ₁₁, M ₂₃, or J ₁.     

On entire (0,m1,…,mq)-interpolation on equidistant nodes

In this paper, we study the so-called entire (0,m₁,m₂,…m_q)-interpolation on equidistant nodes for q=2r-1 case, obtain that it has unique solution B _rσ ² ,σ>0. if and only if E-O=−1, where E and O denote the number of even and odd integers in the {m₁,…,m_q} respectively, and give the explicit formulae of this kind of interpolation functions if they exist. The Project Supported by National Natural Science Foundation of China     

New exact solitary-wave special solutions for the nonlinear dispersive K(m, n) equations

In this paper, we study the nonlinear dispersive K(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.     

Generalized bent functions and their properties

Jet J_q^m denote the set of m-tuples over the integers modulo q and set $\text{i=}\text{?1}$, $\text{w = e}{}^{\mathrm{<mtext>i(</mtext><mtext>2\pi </mtext><mtext>q</mtext><mtext>)</mtext>}}$. As an extension of Rothaus' notion of a bent function, a function f, f: J_q^m → J_q¹ is called bent if all the Fourier coefficients of w^f have unit magnitude. An important feature of these functions is that their out-of-phase autocorrelation value is identically zero. The nature of the Fourier coefficients of a bent function is examined and a proof for the non-existence of bent functions over J_q^m, m odd, is given for many values of q of the form q = 2 (mod 4). For every possible value of q and m (other than m odd and q = 2 (mod 4)), constructions of bent functions are provided.     

Blocker sets, orthogonal arrays and their application to combination locks

Let A denote a set of order m and let X be a subset of A^k+1. Then X will be called a blocker (of A^k+1) if for any element say (a₁,a₂,…,a_k,a_k+1) of A^k+1, there is some element (x₁,x₂,…,x_k,x_k+1) of X such that x_i equals a_i for at least two i. The smallest size of a blocker set X will be denoted by α(m,k)and the corresponding blocker set will be called a minimal blocker. Honsberger (who credits Schellenberg for the result) essentially proved that α(2n,2) equals 2n² for all n. Using orthogonal arrays, we obtain precise numbers α(m,k) (and lower bounds in other cases) for a large number of values of both k and m. The case k=2 that is three coordinate places (and small m) corresponds to the usual combination lock. Supposing that we have a defective combination lock with k+1 coordinate places that would open if any two coordinates are correct, the numbers α(m,k) obtained here give the smallest number of attempts that will have to be made to ensure that the lock can be opened. It is quite obvious that a trivial upper bound for α(m,k) is m² since allowing the first two coordinates to take all the possible values in A will certainly obtain a blocker set. The results in this paper essentially prove that α(m,k) is no more than about m²/k in many cases and that the upper bound cannot be improved. The paper also obtains precise values of α(m,k) whenever suitable orthogonal arrays of strength two (that is, mutually orthogonal Latin squares) exist.     

Solutions for the Hyperbolic Dirac Equation on R1, m

In this article the hyperbolic unit ball in R ^m will be identified with the manifold of rays in the future null cone in R ^m+1. By means of the induced Clifford algebra structure there, one can introduce a definition of Dirac operators on sections of homogeneous line bundles. An infinite class of solutions for the resulting hyperbolic Dirac-equation will be constructed, in case of an odd dimension. In order to obtain these solutions a geometrical picture will be used, because each ray in the future cone will be identified with a point on a surface Σ in the future cone. The original Dirac-equation can then be rewritten in terms of the coordinates on this surface, and the resulting equation will be solved by means of Frobenius' method.