A refined bound for sum-free sets in groups of prime order

Improving upon earlier results of Freiman and the present authors,we show that if p is a sufficiently large prime and A is a sum-freesubset of the group of order p, such that n: = |A| > 0.318p,then A is contained in a dilation of the interval [n, p –n](mod p).     

On the maximum number of fixed points in automorphisms of prime order of 2-(v,k,1) designs

In this paper, we study 2-(v, k, 1) designs with automorphisms of prime orderp, having the maximum possible number of fixed points. We prove an upper bound on the number of fixed points, and we study the structure of designs in which this bound is met with equality (such a design is called ap-MFP(v, k)). Several characterizations and asymptotic existence results forp-MFP(v, k) are obtained. For (p, k)=(3,3), (5,5), (2,3) and (3,4), necessary and sufficient conditions onv are obtained for the existence of ap-MFP(v, k). Further, for 3≤k≤5 and for any primep≡1 modk(k−1), we establish necessary and sufficient conditions onv for the existence of ap-MFP(v, k).     

Asymptotics for the number of n-quasigroups of order 4

The asymptotic form of the number of n-quasigroups of order 4 is $3^{n + 1} 2^{2^n + 1} (1 + o(1))$ .     

Bosonic Formulas for (k,l)-Admissible Partitions

Bosonic formulas for generating series of partitions with certain restrictions are obtained by solving a set of linear matrix q-difference equations. Some particular cases are related to combinatorial problems arising from solvable lattice models, representation theory and conformal field theory.     

Translation invariance in groups of prime order

We prove that there is an absolute constant c>0 with the following property: if Z/pZ denotes the group of prime order p, and a subset A⊂Z/pZ satisfies 1<|A|<p/2, then for any positive integer there are at most 2m non-zero elements b∈Z/pZ with |(A+b)?A|?m. This (partially) extends onto prime-order groups the result, established earlier by S. Konyagin and the present author for the group of integers. We notice that if A⊂Z/pZ is an arithmetic progression and m<|A|<p/2, then there are exactly 2m non-zero elements b∈Z/pZ with |(A+b)?A|?m. Furthermore, the bound c|A|/ln|A| is best possible up to the value of the constant c. On the other hand, it is likely that the assumption can be dropped or substantially relaxed.     

The Noether numbers for cyclic groups of prime order

The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p−3 conjecture.”     

Signed total (k, k)-domatic number of digraphs

Let D be a finite and simple digraph with vertex set V(D), and let f: V(D) → {?1, 1} be a two-valued function. If k ≥?1 is an integer and ${\sum_{x \in N^-(v)}f(x) \ge k}$ for each ${v \in V(G)}$ , where N ^?(v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f ₁, f ₂, . . . , f _d } of signed total k-dominating functions on D with the property that ${\sum_{i=1}^df_i(x)\le k}$ for each ${x \in V(D)}$ , is called a signed total (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed total (k, k)-dominating family on D is the signed total (k, k)-domatic number on D, denoted by ${d_{st}^{k}(D)}$ . In this paper we initiate the study of the signed total (k, k)-domatic number of digraphs, and we present different bounds on ${d_{st}^{k}(D)}$ . Some of our results are extensions of known properties of the signed total domatic number ${d_{st}(D)=d_{st}^{1}(D)}$ of digraphs D as well as the signed total domatic number d _st (G) of graphs G, given by Henning (Ars Combin. 79:277–288, 2006).     

Upper bounds on the signed (k, k)-domatic number

Let G be a graph with vertex set V(G), and let f : V(G) → {?1, 1} be a two-valued function. If k ≥ 1 is an integer and ${\sum_{x\in N[v]} f(x) \ge k}$ for each ${v \in V(G)}$ , where N[v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f ₁,f ₂, . . . ,f _d } of distinct signed k-dominating functions on G with the property that ${\sum_{i=1}^d f_i(x) \le k}$ for each ${x \in V(G)}$ , is called a signed (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed (k, k)-dominating family on G is the signed (k, k)-domatic number of G. In this article we mainly present upper bounds on the signed (k, k)-domatic number, in particular for regular graphs.     

The number of maximal independent sets of (k + 1)-valent trees

We classify maximal independent sets of (k + 1)-valent trees into two groups, namely, type A and type B maximal independent sets and consider specific independent sets of these trees. We study relations among these three types of independent sets. Using the relations, we count the number of all maximal independent sets of (k + 1)-valent trees withn vertices of degreek + 1.     

On the number of derangements and derangements of prime power order of the affine general linear groups

A derangement is a permutation that has no fixed points. In this paper, we are interested in the proportion of derangements of the finite affine general linear groups. We prove a remarkably simple and explicit formula for this proportion. We also give a formula for the proportion of derangements of prime power order. Both formulae rely on a result of independent interest on partitions: we determine the generating function for the partitions with m parts and with the kth largest part not k, for every \(k\in \mathbb {N}\).     

The number of harmonic frames of prime order

Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed between inequivalent harmonic frames and the orbits of a particular set. Secondarily, the symmetry group of prime order harmonic frames is shown to contain a subgroup consisting of a diagonal matrix as well as a permutation matrix, each of which is dependent on the particular harmonic frame in question.     

(k, l)-Algebraic Stability of Runge-Kutta Methods

For ordinary differential equations satisfying a one-sided Lipschitzcondition with Lipschitz constant v, the solutions satisfy with l=hv, so that, in the case of Runge-Kutta methods, estimatesof the form ||y_n||²k(l)||y_n–1||² are desirable. Burrage(1986) has investigated the behaviour of the error-boundingfunction k for positive l for the family of s-stage Gauss methodsof order 2s, and has shown that k(l)=exp 2l+O(l³) (l0) for s3.In this paper, we extend the analysis of k to any irreduciblealgebraically stable Runge-Kutta method, and obtain resultsabout the maximum order of k as an approximation to exp 2l.As a particular example, we investigate the function k for allalgebraically stable methods of order 2s–1.     

(k, l)-Algebraic Stability of Gauss Methods

It is well known that the s-stage Gauss Runge-Kutta methodsof order 2s are algebraically stable, or equivalently (1, 0)-algebraicallystable. In this paper, we show that there exists some l_s >0 such that the Gauss methods are (k, l) algebraically stablefor l [0, l_s) with k(l)=e^2l+O(l^p+1, where p=2s if s=1 or s=2,and p=2 if s>3.     

Difference sets and sum-free sets in groups of order 16

Transversals for sum-free sets in the nine nonabelian order 16 groups are given. It is shown that exactly eight of the order 16 groups have difference sets Dwithλ = 2and D = -D. It is proven that the only (υ, k, λ) difference sets with υ = 2^t and λ = 2, have parameters (16, 6, 2). It is shown that exactly four of the order 16 groups can be partitioned into three sum-free sets.     

Normal edge-transitive Cayley graphs on non-abelian groups of order 4p, where p is a prime number

We determine all connected normal edge-transitive Cayley graphs on non-abelian groups with order 4p,where p is a prime number.As a consequence we prove if |G|=2δp,δ=0,1,2 and p prime,then Γ=Cay(G,S) is a connected normal 1/2 arc-transitive Cayley graph only if G=F4p,where S is an inverse closed generating subset of G which does not contain the identity element of G and F 4p is a group with presentation F4p = a,b|ap=b4=1,b-1ab=aλ,where λ2≡-1(mod p).     

Automorphisms fixing elements of prime order in finite groups

Let σ be an automorphism of a finite group G and suppose that σ fixes every element of G that has prime order or order 4. The main result of this paper shows that the structure of the subgroup H=[G, σ] is severely limited in terms of the order n of σ. In particular, H has exponent dividing n and it is nilpotent of class bounded in terms of n.