Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

Representations for partially exchangeable arrays of random variables

Consider an array of random variables (X_i,j), 1 ≤ i,j < ∞, such that permutations of rows or of columns do not alter the distribution of the array. We show that such an array may be represented as functions f(α, ξ_i, η_j, λ_i,j) of underlying i.i.d, random variables. This result may be useful in characterizing arrays with additional structure. For example, we characterize random matrices whose distribution is invariant under orthogonal rotation, confirming a conjecture of Dawid.     

Hadamard Matrices and Dihedral Groups

Let D _2p be a dihedral group of order 2p, where p is an odd integer. Let ZD _2p be the group ring of D _2p over the ring Z of integers. We identify elements of ZD _2p and their matrices of the regular representation of ZD _2p . Recently we characterized the Hadamard matrices of order 28 ([6] and [7]). There are exactly 487 Hadamard matrices of order 28, up to equivalence. In these matrices there exist matrices with some interesting properties. That is, these are constructed by elements of ZD ₆. We discuss relation of ZD _2p and Hadamard matrices of order n=8p+4, and give some examples of Hadamard matrices constructed by dihedral groups.     

Adjacency preserving bijective maps on triangular matrices over any division ring

Let 𝕋 _n (D) be the set of n × n upper triangular matrices over a division ring D. We characterize the adjacency preserving bijective maps in both directions on 𝕋 _n (D) (n ≥ 3). As applications, we describe the ring semi-automorphisms and the Jordan automorphisms on upper triangular matrices over a simple Artinian ring.     

Some Asymptotic Formulae for Gaussian Distributions

This paper considers asymptotic expansions of certain expectations which appear in the theory of large deviation for Gaussian random vectors with values in a separable real Hilbert space. A typical application is to calculation of the “tails” of distributions of smooth functionals,p(r)=P{Φ(r⁻¹ξ)0},r→∞, e.g., the probability that a centered Gaussian random vector hits the exterior of a large sphere surrounding the origin. The method provides asymptotic formulae for the probability itself and not for its logarithm in a situation, where it is natural to expect thatp(r)=c′r^D exp{−c″r²}. Calculations are based on a combination of the method of characteristic functionals with the Laplace method used to find asymptotics of integrals containing a fast decaying function with “small” support.     

A new characterization of Bergman–Schatten spaces and a duality result

Let denote the space of all upper triangular matrices A such that lim_r→1⁻(1−r²)(A*C(r))^′_B(ℓ²)=0. We also denote by the closed Banach subspace of consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between and the Bergman–Schatten spaces . We also give a characterization of the more general Bergman–Schatten spaces , 1p<∞, in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L^p-behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219–237] for classical Bergman spaces.     

Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices

We consider C=A+B where A is selfadjoint with a gap (a,b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV)^d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.     

Sign reversing and matrix classes

The concept of sign reversing is a useful tool to characterize certain matrix classes in linear complementarity problems. In this paper, we characterize the sign-reversal set of an arbitrary square matrixM in terms of the null spaces of the matricesI–⁁+⁁M, where ⁁ is a diagonal matrix such that 0⁁I. These matrices are used to characterize the membership ofM in the classes P₀, P, and the class of column-sufficient matrices. A simple proof of the Gale and Nikaido characterization theorem for the membership in P is presented.We also study the class of diagonally semistable matrices. We prove that this class is contained properly in the class of sufficient matrices. We show that to characterize the diagonally semistable property is equivalent to solving a concave Lagrangian dual problem. For 2×2 matrices, there is no duality gap between a primal problem and its Lagrangian problem. Such a primal problem is motivated by the definition of column sufficiency.The author would like to thank Professor Richard W. Cottle for his helpful comments on earlier versions of this paper. The author would also like to thank an anonymous referee for numerous helpful comments on this paper.     

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S _n (𝔽) denote the vector space of all n?×?n symmetric matrices over 𝔽. We characterize i. subspaces of S _n (𝔽) all whose elements have rank at most two where n???3,

ii. linear maps from S _m (𝔽) to S _n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n???3 and |𝔽|?≠?2.

     

W(R)-splines

In [3] Golomb describes, for 1 < p < ∞, the H^r,p(R)-extremal extension F_* of a function ƒ:E → R (i.e., the H^r,p-spline with knots in E) and studies the cone H_*E^r,p of all such splines. We study the problem of determining when F_* is in W^r,p ≡ H^r,p ∩ L^p. If F_* ε W^r,p, then F_* is called a W^r,p-spline, and we denote by W_*E^r,p the cone of all such splines. If E is quasiuniform, then F_* ε W^r,p if and only if {ƒ(t_i)}_tiεE ε l^p. The cone W_*E^r,p with E quasiuniform is shown to be homeomorphic to l^p. Similarly, H_*E^r,p is homeomorphic to h^r,p. Approximation properties of the W^r,p-splines are studied and error bounds in terms of the mesh size ¦ E ¦ are calculated. Restricting ourselves to the case p = 2 and to quasiuniform partitions E, the second integral relation is proved and better error bounds in terms of ¦ E ¦ are derived.     

Global nonparametric estimation of conditional quantile functions and their derivatives

Let (X, Y) be a random vector such that X is d-dimensional, Y is real valued, and θ(X) is the conditional αth quantile of Y given X, where α is a fixed number such that 0 < α < 1. Assume that θ is a smooth function with order of smoothness p > 0, and set r = (p − m)/(2p + d), where m is a nonnegative integer smaller than p. Let T(θ) denote a derivative of θ of order m. It is proved that there exists estimate of T(θ), based on a set of i.i.d. observations (X₁, Y₁), …, (X_n, Y_n), that achieves the optimal nonparametric rate of convergence n^−r in L_q-norms (1 ≤ q < ∞) restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists estimate of T(θ) that achieves the optimal rate (n/log n)^−r in L_∞-norm restricted to compacts.     

Rational Approximation with Varying Weights III

Approximation by weighted rationals of the form wⁿr_n, where r_n=p_n/q_n, p_n and q_n are polynomials of degree at most [αn] and [βn], respectively, and w is an admissible weight, is investigated on compact subsets of the real line for a general class of weights and given α0, β0, with α+β>0. Conditions that characterize the largest sets on which such approximation is possible are given. We apply the general theorems to Laguerre and Freud weights.     

New fast divide‐and‐conquer algorithms for the symmetric tridiagonal eigenvalue problem

In this paper, two accelerated divide‐and‐conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N²r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy‐like matrices and are off‐diagonally low‐rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low‐rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off‐diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. Copyright © 2016 John Wiley & Sons, Ltd.     

Parameterized matching with mismatches

The problem of approximate parameterized string searching consists of finding, for a given text t=t₁t₂…t_n and pattern p=p₁p₂…p_m over respective alphabets Σ_t and Σ_p, the injection π_i from Σ_p to Σ_t maximizing the number of matches between π_i(p) and t_it_i+1…t_i+m−1 (i=1,2,…,n−m+1). We examine the special case where both strings are run-length encoded, and further restrict to the case where one of the alphabets is binary. For this case, we give a construction working in time O(n+(r_p×r_t)α(r_t)log(r_t)), where r_p and r_t denote the number of runs in the corresponding encodings for y and x, respectively, and α is the inverse of the Ackermann's function.     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

Sharp estimates for Jacobi matrices and chain sequences, II

Chain sequences are positive sequences {c_n} of the form c_n=g_n(1−g_n−1) for a nonnegative sequence {g_n}. They are very useful in estimating the norms of Jacobi matrices and for localizing the interval of orthogonality for orthogonal polynomials. We give optimal estimates for the chain sequences which are more precise than the ones obtained in the paper (Constructive Approx. 6 (1990) 363) and in our earlier paper (J. Approx. Theory 118 (2002) 94).     

Values of the Euler Function in Various Sequences

Let ϕ(n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation ϕ(n)^r = λ(n)^s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, not equal to a prime or twice a prime, such that ϕ(n) = p − 1 holds with some prime p, as well as those positive integers n such that the equation ϕ(n) = f(m) holds with some integer m, where f is a fixed polynomial with integer coefficients and degree degf > 1.     

Toeplitz and Wiener-Hopf determinants with piecewise continuous symbols

The asymptotics for determinants of Toeplitz and Wiener-Hopf operators with piecewise continuous symbols are obtained in this paper. If W_α(σ) is the Wiener-Hopf operator defined on L₂(0, α) with piecewise continuous symbol σ having a finite number of discontinuities at ξ_r, then under appropriate conditions it is shown that det W_α(σ) ˜ G(σ)^α α^Σλ_r2K(σ), where

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is a completely determined constant. An analogous result is obtained for Toeplitz operators. The main point of the paper is to obtain a result in the Wiener-Hopf case since the Toeplitz case had been treated earlier. In the Toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if, for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known. Fortunately in the Toeplitz case such asymptotics were known. The difficulty in the Wiener-Hopf case is that there was not a single singular case where the determinant was explicitly known. This problem was overcome by using the fact that Wiener-Hopf determinants when discretized become Toeplitz determinants whose entries depend on the size of the matrix. No theorem on Toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results.