Reduction of the number of associate classes of hypercubic association schemes

The reduction of the number of associate classes of some hypercubic association schemes by clubbing certain associate classes has been studied in the paper. It has been found that the reduction of anm-class hypercubic association scheme forv=2 ^m treatments into a 2-class association scheme is always possible. Further it is proved herein that them-class hypercubic association scheme forv=s ^m treatments is reducible (i) to a 3-class association scheme, whens=3 and (ii) to a 2-class association scheme, whens=4, which really hasp ₁₁ ¹ =p ₁₁ ² and hence leads to a series of balanced incomplete block designs.     

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et al. proposed an association scheme of 64 points in R¹⁴ which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal collections of real mutually unbiased bases. These schemes are also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E.R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.     

Association Schemes Related to Kasami Codes and Kerdock Sets

Two new infinite series of imprimitive 5-class association schemes are constructed. The first series of schemes arises from forming, in a special manner, two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code). The second series of schemes is formally dual to the first. The construction applies vector space duality to obtain a fission scheme of a subscheme of the Cameron-Seidel 3-class scheme of linked symmetric designs derived from Kerdock sets and quadratic forms over GF(2).     

Difference set constructions of DRADs and association schemes

Doubly Regular Asymmetric Digraphs (DRAD) with rank 4 automorphism groups were previously thought to be rare. We exhibit difference sets in Galois Rings that can be used to construct an infinite family of DRADs with rank 4 automorphism groups. In addition, we construct difference sets in groups for all r?2 that can be used to construct DRADs and nonsymmetric 3-class imprimitive association schemes. Finally, we prove a new product construction for difference sets so that the resulting difference sets can be used to build nonsymmetric 3-class imprimitive association schemes.     

New semifields, PN and APN functions

We describe a method of proving that certain functions ${f:F\longrightarrow F}$ defined on a finite field F are either PN-functions (in odd characteristic) or APN-functions (in characteristic 2). This method is illustrated by giving short proofs of the APN-respectively the PN-property for various families of functions. The main new contribution is the construction of a family of PN-functions and their corresponding commutative semifields of dimension 4s in arbitrary odd characteristic. It is shown that a subfamily of order p ^4s for odd s > 1 is not isotopic to previously known examples.     

Sufficient conditions for the Lebesgue integrability of Fourier transforms

We give sufficient conditions for the Lebesgue integrability of the Fourier transform of a function f ∈ L ^p (?) for some 1 < p ≤ 2. These sufficient conditions are in terms of the L ^p integral modulus of continuity of f; in particular, they apply for functions in the integral Lipschitz class Lip(α, p) and for functions of bounded s-variation for some 0 < s < p. Our theorems are nonperiodic versions of the classical theorems of Bernstein, Szász, Zygmund and Salem, and recent theorems of Gogoladze and Meskhia on the absolute convergence of Fourier series.     

Linking systems in nonelementary abelian groups

Linked systems of symmetric designs are equivalent to 3-class Q-antipodal association schemes. Only one infinite family of examples is known, and this family has interesting origins and is connected to important applications. In this paper, we define linking systems, collections of difference sets that correspond to systems of linked designs, and we construct linking systems in a variety of nonelementary abelian groups using Galois rings, partial difference sets, and a product construction. We include some partial results in the final section.     

A strict maximum modulus theorem for certain Banach spaces

A strict maximum modulus theorem is proved for certain Banach spaces. As an application it is shown that ifAX=XA ^* withA hyponormal andX a member of any Schattenp-class withp≥1, thenA ^* X=XA.     

PBIB designs with m associate classes induced by full rank matrices modulo pm

This paper presents series of PBIB designs with m associate classes in which the treatment set is a subset of the Z(p^m)-module of n × 1 vectors over the ring of integers modulo p^m, p any prime. The association scheme of this series of designs is determined by the Fuller canonical form under row equivalence of n × 2 matrices [a,b] for vectors a and b in the treatment set. The blocking procedure utilizes full rank s × n matrices over Z(p^m), 1 ? s ? n ? 2, n ? 3. For m = 2, n = 3, s =1 and for each prime p, each PBIB is regular divisible and yields a finite proper uniform projective Hjelmslev plane with parameters j = p and k = p(p + 1).     

On a class of wreath products of hypergroups and association schemes

We characterize finite hypergroups S (in the sense of Frédéric Marty in Huitième Congres des Mathématiciens, pp. 45–59, 1934) satisfying |pq|=1 for any two elements p and q in S with p≠q ^? in terms of wreath products. The result applies to association schemes of finite valency and provides a corresponding characterization in scheme theory. For association schemes S of finite valency satisfying the above condition, we provide a second characterization, a characterization in terms of the subconstituent algebra of S.     

Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L ²-norm error over a finite time interval converges optimally as O(h ^p+1+??t ^s ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and ??t the time step.     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.      

Variational Problems with Unilateral Pointwise Functional Constraints in Variable Domains

We consider a sequence of convex integral functionals F_s: W¹,^p(Ω_s) → ? and a sequence of weakly lower semicontinuous and generally nonintegral functionals G_s: W¹,^p(Ω_s) → ?, where {Ω_s} is a sequence of domains in ?ⁿ contained in a bounded domain Ω ? ?ⁿ (n ≥ 2) and p > 1. Along with this, we consider a sequence of closed convex sets V_s = {v ∈ W¹,^p(Ω_s): v ≥ K_s(v) a.e. in Ω_s}, where K_s is a mapping from the space W¹,^p(Ω_s) to the set of all functions defined on Ω_s. We establish conditions under which minimizers and minimum values of the functionals F_s + G_s on the sets V_s converge to a minimizer and the minimum value of a functional on the set V = {v ∈ W¹,^p(Ω): v ≥ K(v) a.e. in Ω}, where K is a mapping from the space W¹,^p(Ω) to the set of all functions defined on Ω. These conditions include, in particular, the strong connectedness of the spaces W¹,^p(Ω_s) with the space W¹,^p(Ω), the condition of exhaustion of the domain Ω by the domains Ω_s, the Γ-convergence of the sequence {F_s} to a functional F: W¹,^p(Ω) → ?, and a certain convergence of the sequence {G_s} to a functional G: W¹,^p(Ω) → ?. We also assume some conditions characterizing both the internal properties of the mappings K_s and their relation to the mapping K. In particular, these conditions admit the study of variational problems with irregular varying unilateral obstacles and with varying constraints combining the pointwise dependence and the functional dependence of the integral form.     

Lp-Solutions of Vector Refinement Equations with General Dilation Matrix

The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius.     

Jensen inequalities for <Emphasis Type="Italic">P</Emphasis>-class functions

In this paper, we first give some characterizations for P-class functions. Then giving a Hermite–Hadamard type inequality for P-class functions, we prove equivalency of some significant metrics in normed linear spaces. We also obtain an operator version of the Jensen inequality for P-class functions. Introducing operator (mid) P-class functions, we present some characterizations for such functions.     

A Primitive Non-symmetric 3-Class Association Scheme on 36 Elements with p11 = 0 Exists and is Unique

In this paper we construct a primitive, non-symmetric 3-class association scheme with parameters v = 36, v₁ = 7, p¹₁₁ = 0 and p²₁₁ = 4 and show that such a scheme is determined by its parameters.     

On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal <Emphasis Type="Italic">d</Emphasis>-Sets

The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s∈(0,1] the smoothness parameter, the sharp upper bound min{d+1?s,d/s} is obtained. In particular, when passing from d≥s to d<s there is a change of behaviour from d+1?s to d/s which implies that even highly nonsmooth functions defined on cubes in ? ⁿ have not so rough graphs when restricted to, say, rarefied fractals.     

New Partial Difference Sets in Ztp2 and a Related Problem about Galois Rings

We generalize a construction of partial difference sets (PDS) by Chen, Ray-Chaudhuri, and Xiang through a study of the Teichmüller sets of the Galois rings. Let R=GR(p², t) be the Galois ring of characteristic p² and rank t with Teichmüller set T and let π:R→R/pR be the natural homomorphism. We give a construction of PDS in R with the parameters ν=p^2t, k=r(p^t−1), λ=p^t+r²−3r, μ=r²−r, where r=lp^{t−s(p, t)}, 1≤l≤p^{s(p, t)}, and s(p, t) is the largest dimension of a GF(p)-subspace W⊂R/pR such that π⁻¹(W)∩T generates a subgroup of R of rank <t. We prove that s(p, t) is the largest dimension of a GF(p)-subspace W of GF(p^t) such that dim W^p<t, where W^p is the GF(p)-space generated by {∏^p_i=1w_i∣w_i∈W, 1≤i≤p}. We determine the values of s(p, t) completely and solve a general problem about dim_EW^r for an E-vector space W in a finite extension of a finite field E. The PDS constructed here contain the family constructed by Chen, Ray-Chaudhuri, and Xiang and have a wider range of parameters.     

On regular graphs. IV

Let G be a connected combinatorial graph of valency p + 1 where p is an odd prime. Assume that there exists a group of automorphisms A of G whose induced action on the s-arcs of G is regular (sharply transitive). If s ≥ 2 we prove that p must be a Mersenne prime, i.e., of the form p = 2ⁿ ? 1. In general we know that s ≤ 5 or s = 7. We obtain some partial results when s = 7.     

Tetravalent s-Transitive Graphs of Order 4p 2

Let s be a positive integer. A graph is s -transitive if its automorphism group is transitive on s-arcs but not on (s?+?1)-arcs. Let p be a prime. Zhou (Discrete Math 309:6081?C6086, 2009) classified tetravalent s-transitive graphs of order 4p. In this article a complete classification of tetravalent s-transitive graphs of order 4p ² is given.