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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

关于Hadamard乘积矩阵的一些性质的注记

对文 [1 ]的主要结论作了说明 ,给出 Hadamard乘积矩阵有关性质的更一般的结果 .     

On the automorphisms of Paley's type II Hadamard matrix

In this paper we determine the automorphism group of Paley's type II Hadamard matrix.     

On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives

In this paper, a class of nonlinear fractional order differential impulsive systems with Hadamard derivative is discussed. First, a reasonable concept on the solutions of fractional impulsive Cauchy problems with Hadamard derivative and the corresponding fractional integral equations are established. Second, two fundamental existence results are presented by using standard fixed point methods. Finally, two examples are given to illustrate our theoretical results.     

On GMW Designs and Cyclic Hadamard Designs

We generalise results of Jackson concerning cyclic Hadamard designs admitting SL(2,2ⁿ) as a point transitive automorphism group. The generalisation concerns the designs of Gordon, Mills and Welch and we characterise these as designs admitting GM(m,qⁿ) acting in a certain way. We also generalise a construction given by Maschietti, using hyperovals, of cyclic Hadamard designs, and characterise these amongst the designs of Gordon, Mills and Welch.     

M-矩阵Hadamard积的新不等式(英文)

设A和B是非奇异M-矩阵,给出了关于A和B-1的Hadamard积的最小特征值下界τ(A°B-1)的一个新估计式,该结果改进了文献[4]的结果.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in Cn given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in Cn, which is different from the well-known Leray formula, This new formula eliminates the term that contains the parameter λ from the classical Leray formula, and especially on some domains the uniform estimates for the -equation are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in Cn, which are different from the classical ones, when we properly select the vector function W.