Properties of submatrices of Sylvester Hadamard matrices

We investigate the algebraic behaviour of leading principal submatrices of Hadamard matrices being powers of 2. We provide analytically the spectrum of general submatrices of these Hadamard matrices. Symmetry properties and relationships between the upper left and lower right corners of the matrices in this respect are demonstrated. Considering the specific construction scheme of this particular class of Hadamard matrices (called Sylvester Hadamard matrices), we utilize tensor operations to prove the respective results. An algorithmic procedure yielding the complete spectrum of leading principal submatrices of Sylvester Hadamard matrices is proposed.     

On Hadamard Inequality for Log-Concave Functions

Some improvements and generalizations of Finks results about Hadamards inequality for log-concave functions are given.AMS Subject Classification: 26D07, 26D15, 26A51.     

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 the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

凸函数的Hadamard不等式的若干推广

本文获得两个定理 ,它们均是不等式f a +b2 1b -a∫baf (x) dx f (a) +f (b)2(其中 f是 [a,b]上的连续凸函数 )的推广 .     

On the sharp effect of attaching a thin handle on the spectral rate of convergence

Consider two domains connected by a thin tube: it can be shown that the resolvent of the Dirichlet Laplacian is continuous with respect to the channel section parameter. This in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation. Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain. As an application, we prove that, again under an explicit nondegeneracy condition, the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions.     

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 .     

Ryser's embedding problem for Hadamard matrices

What is the minimum order of a Hadamard matrix that contains an a by b submatrix of all 1's? Newman showed that where c^? denotes the smallest order greater than or equal to c for which a Hadamard matrix exists. It follows that if 4 divides both a and b, and if the Hadamard conjecture is true, then . We establish the improved bounds for min {a,b} ≥ 2. The Hadamard conjecture therefore implies that if 4 divides both 2ab and ?a/2? ?b/2?, then (a, b) = 2 · max {?a/2?b, ?b/2?a}. Our lower bound comes from a counting argument, while our upper bound follows from a sub‐multiplicative property of : Improvements in our upper bound occur when suitable conference matrices or Bush‐type Hadamard matrices exist. We conjecture that any (1,?1)‐matrix of size a by b occurs as a submatrix of some Hadamard matrix of order at most . © 2005 Wiley Periodicals, Inc. J Combin Designs     

On Circulant Complex Hadamard Matrices

We show that a circulant complex Hadamard matrix of order n is equivalent to a relative difference set in the group C ₄×C _n where the forbidden subgroup is the unique subgroup of order two which is contained in the C ₄ component. We obtain non-existence results for these relative difference sets. Our results are sufficient to prove there are no circulant complex Hadamard matrices for many orders.     

On ‐Cocyclic Hadamard Matrices

In this paper, we describe some necessary and sufficient conditions for a set of coboundaries to yield a cocyclic Hadamard matrix over the dihedral group . Using this characterization, new classification results for certain cohomology classes of cocycles over are obtained, extending existing exhaustive calculations for cocyclic Hadamard matrices over from order 36 to order 44. We also define some transformations over coboundaries, which preserve orthogonality of ‐cocycles. These transformations are shown to correspond to Horadam's bundle equivalence operations enriched with duals of cocycles.     

Cauchy-type integral formula and Plemelj formula of hypermonogenic functions on unbounded domains

In this paper, we discuss the Cauchy-type integral formula of hypermonogenic functions on unbounded domains in real Clifford analysis, then we extend the Plemelj formula and Cauchy–Pompeiu formula of hypermonogenic functions on bounded domains to unbounded domains. We also deal with the Green-type formula on unbounded domains and get several important corollaries.     

On the trace of symmetric stable processes on Lipschitz domains

It is shown that the second term in the asymptotic expansion as t→0 of the trace of the semigroup of symmetric stable processes (fractional powers of the Laplacian) of order α, for any 0<α<2, in Lipschitz domains is given by the surface area of the boundary of the domain. This brings the asymptotics for the trace of stable processes in domains of Euclidean space on par with those of Brownian motion (the Laplacian), as far as boundary smoothness is concerned.     

Some remarks on the second variation formula for the area

Iff is a minimal, isometric immersion of the Riemannian manifoldM of dimensionn in a Riemannian manifold of dimensionm and ifI _N is the differential Jacobi operator acting on the cross sections of the normal bundleN(M), we obtain some information on the Morse index and on the stability ofM through a detailed geometric analysis of the immersionf obtained when considering the higher fundamental forms off.     

On the classification of Hadamard matrices of order 32

All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard matrices. We find all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13, 680, 757 Hadamard matrices of one type and 26, 369 such matrices of another type. Based on experience with the classification of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insignificant. © 2009 Wiley Periodicals, Inc. J Combin Designs 18:328–336, 2010     

Structure and automorphism groups of Hadamard designs

Let n be the order of a Hadamard design, and G any finite group. Then there exists many non-isomorphic Hadamard designs of order 2^{12|G| + 13} n with automorphism group isomorphic to G.This research was supported in part by the National Science Foundation.     

On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order ²10+tq whenever . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order ²4N−2q.