On left Hadamard transversals in non-solvable groups

Let G be a group of order 4n and t an involution of G. A 2n-subset R of G is called a left Hadamard transversal of G with respect to 〈t〉 if G=R〈t〉 and for some subsets S₁ and S₂ of G. Let H be a subgroup of G such that G=[G,G]H, t∈H, and t^G⊄H, where t^G is the conjugacy class of t and [G,G] is the commutator subgroup of G. In this article, we show that if R satisfies a condition , then R is a (2n,2,2n,n) relative difference set and one can construct a v×v integral matrix B such that BB^T=B^TB=(n/2)I, where v is a positive integer determined by H and t^G (see Theorem 2.6). Using this we show that there is no left Hadamard transversal R satisfying (*) in some simple groups.     

On ‐Cocyclic Hadamard Matrices

A characterization of ‐cocyclic Hadamard matrices is described, depending on the notions of distributions, ingredients, and recipes. In particular, these notions lead to the establishment of some bounds on the number and distribution of 2‐coboundaries over to use and the way in which they have to be combined in order to obtain a ‐cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in A. Baliga, K. J. Horadam, Australas. J. Combin., 11 (1995), 123–134 is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries defining ‐cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, define representatives for them, and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way, in terms of diagrams. Let be the set of cocyclic Hadamard matrices over having a symmetric diagram. We also prove that the set of Williamson‐type matrices is a subset of of size .     

New Pólya-Schoenberg type theorems

In this paper the theory of Hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so-called disk-like domains, i.e. such domains which are the union of disks or half-planes, all containing the origin. In such a domain, say Ω, we define (the class of) generalized prestarlike functions of order α?1 and ask for Hadamard multipliers g analytic at z=0 for which implies . We prove that such a multiplier necessarily has to be analytic in

     

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P(z)=(1−z)^−b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Reb<2. Let where μ is a complex-valued measure on the closed unit disk Such sequences are shown to be multipliers of H^p for 1?p?∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that {μ_n} is a multiplier of H^p for every p>0. When the support of μ is [0,1] we get the multiplier sequence which provides more concrete applications. We show that if the sequences {μ_n} and {ν_n} are related by an asymptotic expansion

     

On Gibbs measures of P-adic Potts model on the cayley tree

We consider a nearest-neighbor p-adic Potts (with q ≥ 2 spin values and coupling constant J ? _p) model on the Cayley tree of order k ≥ 1. It is proved that a phase transition occurs at k = 2, q ? p and p ≥ 3 (resp. q ? 2², p = 2). It is established that for p-adic Potts model at k ≥ 3 a phase transition may occur only at q ? p if p ≥ 3 and q ? 2² if p = 2.     

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

The topology of the monodromy map of a second order ODE

We consider the following question: given A∈SL(2,R), which potentials q for the second order Sturm-Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q∈L²([0,2π]) to , the lift to the universal cover of SL(2,R) of the fundamental matrix map ,

     

Weighted-Hardy functions with Hadamard gaps on the unit ball

We prove that an analytic function f on the unit ball B with Hadamard gaps, that is, (the homogeneous polynomial expansion of f) satisfying n_k+1/n_k?λ>1 for all k∈N, belongs to the space if and only if . Moreover, we show that the following asymptotic relation holds . Also we prove that lim_r→1(1-r²)^α‖Rf_r‖_p=0 if and only if . These results confirm two conjectures from the following recent paper [S. Stevi?, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)].     

Representing (0, 1)-matrices by boolean circuits

A boolean circuit represents an n by n(0,1)-matrix A if it correctly computes the linear transformation over GF(2) on all n unit vectors. If we only allow linear boolean functions as gates, then some matrices cannot be represented using fewer than Ω(n²/lnn) wires. We first show that using non-linear gates one can save a lot of wires: any matrix can be represented by a depth-2 circuit with O(nlnn) wires using multilinear polynomials over GF(2) of relatively small degree as gates. We then show that this cannot be substantially improved: If any two columns of an n by n(0,1)-matrix differ in at least d rows, then the matrix requires Ω(dlnn/lnlnn) wires in any depth-2 circuit, even if arbitrary boolean functions are allowed as gates.     

An extension to overpartitions of the Rogers-Ramanujan identities for even moduli

We study a class of well-poised basic hypergeometric series , interpreting these series as generating functions for overpartitions defined by multiplicity conditions on the number of parts. We also show how to interpret the as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases (a,q)→(1/q,q), (1/q,q²), and (0,q), where some of the functions become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.     

Semi-regular relative difference sets with large forbidden subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters (m,n,m,m/n) in groups of non-prime-power orders. Let p be an odd prime. We prove that there does not exist a (2p,p,2p,2) relative difference set in any group of order 2p², and an abelian (4p,p,4p,4) relative difference set can only exist in the group . On the other hand, we construct a family of non-abelian relative difference sets with parameters (4q,q,4q,4), where q is an odd prime power greater than 9 and . When q=p is a prime, p>9, and , the (4p,p,4p,4) non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters.     

Approximating reals by sums of two rationals

We generalize Dirichlet's diophantine approximation theorem to approximating any real number α by a sum of two rational numbers with denominators 1?q₁,q₂?N. This turns out to be related to the congruence equation problem with 1?x,y?q1/2+?.     

ε-Regularity theorem and its application to the blow-up solutions of Keller-Segel systems in higher dimensions

Let us consider m(KS) below for all N?2 and general exponents m and q. In particular, the 2-D semi-linear case such as N=2, m=1 and q=2 is included. We establish an ε-regularity theorem for weak solutions. As an application, we give an extension criterion in which coincides with a scaling invariant class of weak solutions associated with m(KS). In addition, the Hausdorff dimension of its singular set is zero if and .     

Tensor ranks for the inversion of tensor-product binomials

The main result reads: if a nonsingular matrix A of order n=pq is a tensor-product binomial with two factors then the tensor rank of A⁻¹ is bounded from above by min{p,q}. The estimate is sharp, and in the worst case it amounts to .     

The fluctuations in the number of points on a hyperelliptic curve over a finite field

The number of points on a hyperelliptic curve over a field of q elements may be expressed as q+1+S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that is distributed as the trace of a random 2g×2g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values ±1 with probabilities 1/2(1+q⁻¹) and the value 0 with probability 1/(q+1). When both the genus and the finite field grow, we find that has a standard Gaussian distribution.