On p-covalenced association schemes

Let (X,S) denote an association scheme where X is a finite set. For a prime p we say that (X,S) is p-covalenced (p-valenced) if every multiplicity (valency, respectively) of (X,S) is a power of p. In the character theory of finite groups Ito's theorem states that a finite group G has a normal abelian p-complement if and only if every character degree of G is a power of p. In this article we generalize Ito's theorem to p-valenced association schemes, i.e., a p-valenced association scheme (X,S) has a normal p-covalenced p-complement if and only if (X,S) is p-covalenced.     

Embeddability of restricted linear spaces

In this note we examine the problem of embedding into finite projective planes finite linear spaces with p points and q lines satisfying (q ? p)² ? p and q ? 2.     

The L q spectra and Rényi dimension of generalized inhomogeneous self-similar measures

Very recently bounds for the L ^q spectra of inhomogeneous self-similar measures satisfying the Inhomogeneous Open Set Condition (IOSC), being the appropriate version of the standard Open Set Condition (OSC), were obtained. However, if the IOSC is not satisfied, then almost nothing is known for such measures. In the paper we study the L ^q spectra and Rényi dimension of generalized inhomogeneous self-similar measures, for which we allow an infinite number of contracting similarities and probabilities depending on positions. As an application of the results, we provide a systematic approach to obtaining non-trivial bounds for the L ^q spectra and Rényi dimension of inhomogeneous self-similar measures not satisfying the IOSC and of homogeneous ones not satisfying the OSC. We also provide some non-trivial bounds without any separation conditions.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)³ = S ². We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.     

Classifying a family of edge-transitive metacirculant graphs

A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups \mathbbZ_p^d:\mathbbZ_q\mathbb{Z}_{p^{d}}{:}\mathbb{Z}_{q} with p,q odd prime, of valency coprime to p. This characterization is then used to study an isomorphism problem regarding Cayley graphs, and to construct new families of half-arc-transitive graphs.     

Skew linear recurring sequences of maximal period over galois rings

Let p be a prime number, and R = GR(q ^d , p ^d ) be a Galois ring of q ^d = p ^rd elements and of characteristic p ^d . Denote by S = GR(q ^nd , p ^d ) a Galois extension of the ring R of dimension n and by ? the ring of all linear transformations of the module _R S. We call a sequence v over the ring S with the law of recursion $$ {\mathrm{for}\ \mathrm{all}\ }i \in {\mathbb{N}_0}:v\left( {i + m} \right) = {\psi_{m - 1}}\left( {v\left( {i + m - 1} \right)} \right) + \cdots + {\psi_0}\left( {v(i)} \right),\quad {\psi_0}, \ldots, {\psi_{m - 1}} \in \textit{\v{S}} $$ (i.e., a linear recurring sequence of order m over the module _? S) a skew LRS over S. It is known that the period T(v) of such a sequence satisfies the inequality T(v) ?? ?? = (q ^nm ?1)p ^d?1. If T(v) = ?? , then we call v a skew LRS of maximal period (a skew MP LRS) over S. A new general characterization of skew MP LRS in terms of coordinate sequences corresponding to some basis of a free module _R S is given. A simple constructive method of building a big enough class of skew MP LRS is stated, and it is proved that the linear complexity of some of them (the rank of the linear recurring sequence) over the module _S S is equal to mn, i.e., to the linear complexity over the module _R S.     

Relative hemisystems on the Hermitian surface

Let S be a generalized quadrangle of order (q ²,q) containing a subquadrangle S′ of order (q,q). Then any line of S either meets S′ in q+1 points or is disjoint from S′. After Penttila and Williford (J. Comb. Theory, Ser. A 118:502–509, 2011), we call a subset H of the lines disjoint from S′ a relative hemisystem of S with respect to S′, provided that for each point x of S?S′, exactly half of the lines through x disjoint from S′ lie in H. A new infinite family of relative hemisystems on the generalized quadrangle $\mathcal{H}(3,q^{2})$ admitting the linear group PSL(2,q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems.     

Quadratic residues and the combinatorics of sign multiplication

If S is a nonempty finite set of positive integers, we find a criterion both necessary and sufficient for S to satisfy the following condition: if q is a fixed nonnegative integer, then there exists infinitely many primes p such that S contains exactly q quadratic residues of p. This result simultaneously generalizes two previous results of the author, and the criterion used is expressed by means of a purely combinatorial condition on the prime factors of the elements of S of odd multiplicity.     

Indecomposable decompositions of modular standard modules for two families of association schemes

We investigate the indecomposable decomposition of the modular standard modules of two families of association schemes of finite order. First, we show that, for each prime number p, the standard module over a field F of characteristic p of a residually thin scheme S of p-power order is an indecomposable FS-module. Second, we describe the indecomposable decomposition of the standard module over a field of positive characteristic of a wreath product of finitely many association schemes of rank 2.     

On sets of vectors of a finite vector space in which every subset of basis size is a basis II

This article contains a proof of the MDS conjecture for k ≤?2p ? 2. That is, that if S is a set of vectors of ${{\mathbb F}_q^k}$ in which every subset of S of size k is a basis, where q?=?p ^h , p is prime and q is not and k ≤ 2p ? 2, then |S| ≤ q?+?1. It also contains a short proof of the same fact for k?≤ p, for all q.     

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GL_n(q) with defect group C_p^α × C_p^α, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GL_n(q) when p = 3, α = 1, in a different way.     

Regularity for semigroups of Ornstein–Uhlenbeck processes

Under mild conditions on the characteristic exponent or the symbol of Lévy process, we derive explicit estimates for L ^p (dx) → L ^q (dx) (1 ≤ p ≤ q ≤ ∞) norms of semigroups and their gradients of the associated Lévy driven Ornstein–Uhlenbeck process. Our result efficiently applies to the class of Lévy driven Ornstein–Uhlenbeck processes, where the asymptotic behaviour near infinity for the symbol of Lévy process is known.     

Characterization and recognition of generalized clique-Helly graphs

Let p?1 and q?0 be integers. A family of sets F is (p,q)-intersecting when every subfamily F^′⊆F formed by p or less members has total intersection of cardinality at least q. A family of sets F is (p,q)-Helly when every (p,q)-intersecting subfamily F^′⊆F has total intersection of cardinality at least q. A graph G is a (p,q)-clique-Helly graph when its family of (maximal) cliques is (p,q)-Helly. According to this terminology, the usual Helly property and the clique-Helly graphs correspond to the case p=2,q=1. In this work we present a characterization for (p,q)-clique-Helly graphs. For fixed p,q, this characterization leads to a polynomial-time recognition algorithm. When p or q is not fixed, it is shown that the recognition of (p,q)-clique-Helly graphs is NP-hard.     

Clarkson–McCarthy Inequalities for l p -Spaces of Operators in Schatten Ideals

In this paper we obtain generalized Clarkson–McCarthy inequalities for spaces l _q (S ^p ) of operators from Schatten ideals S ^p . We show that all Clarkson–McCarthy type inequalities are, in fact, some estimates on the norms of operators acting on the spaces l _q (S ^p ) or from one such space into another. We also extend some inequalities for partitioned operators and for Cartesian decomposition of operators.     

AC 1 make or break lemma

Mañé suggested the following question: Consider aC ^r flow on a compact manifold without boundary and suppose that the ω-limit set of a pointp intersets the α-limit set ofq, i.e. ω(p)∩α(q)≠Ø. Can the flow beC ^r-perturbed so that either (a)p is connected toq (p andq in the same orbit) or (b) ω(p)∩α(q)=Ø for the new flow? Here we solve positively a stronger version of this problem forC ¹ small perturbations of the original flow.     

A Class of Imprimitive Association Schemes

From Burnside's p^αq^β-Theorem, it follows that any nonabelian group of order p^αq^β, where p and q are primes, cannot be simple. As a main result of this article, we state and prove an analog of the mentioned theorem for commutative association schemes.     

The converse of the Solid Torus Theorem

Given positive integers p and q, a (p,q)-solid torus is a manifold diffeomorphic to Dp+1×Sq while a (p,q)-torus in a closed manifold M is the image of a differentiably embedding Sp×Sq→M. We prove that if n=p+q+1 with p=q=1 or p≠q, then M is homeomorphic to Sn whenever every (p,q)-torus bounds a (p,q)-solid torus. We also prove for p=q that every closed n-manifold for which every (p,p)-torus bounds an irreducible manifold is irreducible. Consequently, every closed 3-manifold for which every torus bounds an irreducible manifold is irreducible.     

The diameter of bipartite distance-regular graphs

It is shown that any bipartite distance-regular graph with finite valency k and at least one cycle is finite, with diameter d and girth g satisfying $\text{d≤}\text{(k?1)(g?2)}\text{2+1}$. In particular, the number of bipartite distance-regular graphs with fixed valency and girth is finite.