Matrix positivity preservers in fixed dimension. I

A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with non-negative Taylor coefficients. Despite the great deal of interest generated by this theorem, a characterization of functions preserving positivity for matrices of fixed dimension is not known. In this paper, we provide a complete description of polynomials of degree $N$ that preserve positivity when applied entrywise to matrices of dimension $N$. This is the key step for us then to obtain negative lower bounds on the coefficients of analytic functions so that these functions preserve positivity in a prescribed dimension. The proof of the main technical inequality is representation theoretic, and employs the theory of Schur polynomials. Interpreted in the context of linear pencils of matrices, our main results provide a closed-form expression for the lowest critical value, revealing at the same time an unexpected spectral discontinuity phenomenon. Tight linear matrix inequalities for Hadamard powers of matrices and a sharp asymptotic bound for the matrix-cube problem involving Hadamard powers are obtained as applications. Positivity preservers are also naturally interpreted as solutions of a variational inequality involving generalized Rayleigh quotients. This optimization approach leads to a novel description of the simultaneous kernels of Hadamard powers, and a family of stratifications of the cone of positive semidefinite matrices.Comment: Changed notation for extreme critical value from $\mathfrak{C}$ to $\mathcal{C}$. Addressed referee remarks to improve exposition, including Remarks 1.2 and 3.3. Final version, 39 pages, to appear in Advances in Mathematic

Efficient implementation of selective recoupling in heteronuclear spin systems using Hadamard matrices

We present an efficient scheme which couples any designated pair of spins in heteronuclear spin systems. The scheme is based on the existence of Hadamard matrices. For a system of $n$ spins with pairwise coupling, the scheme concatenates $cn$ intervals of system evolution and uses at most $c n^2$ pulses where $c \approx 1$. Our results demonstrate that, in many systems, selective recoupling is possible with linear overhead, contrary to common speculation that exponential effort is always required.Comment: 7 pages, 4 figures, mypsfig2, revtex, submitted April 27, 199

Linear codes and error-correction

The process of encoding information for transmission from one source to another is a vital process in many areas of science and technology. Whenever coded information is sent, there arises a certain possibility that an error will occur, either during transmission or in decoding. Therefore, it is imperative to develop methods to detect and correct errors in a code. The study of coding theory is a new area of mathematics which is relatively undeveloped. This paper focuses on the properties of linear codes -and their corresponding methods of error-correction. To simplify the issue, only binary block codes are studied; hence all digits are either 0 or 1. The operation of addition is defined over modulo 2. In the decoding process, the principle of maximum likelihood decoding is used. This principle assumes that a minimum number of errors will occur in each codeword, since the overall probability of error decreases exponentially with the total number of errors. Whenever a string of digits is encoded, the digits are multiplied by a generator matrix, which returns the original digits and a specified number of check digits. The check digits are helpful in detecting and correcting errors. The parity check matrix, H, is also determined from the generator matrix. If the product of Hand the transpose of the received word is 0, then the received word is indeed a codeword. If the product is not 0, but is in fact the ith column of H, then an error occurs in the ith digit of the received string. The Hamming codes are specific linear codes which contain a maximum number of distinguishable columns. Therefore, the Hamming codes are ideal for error-correction, provided that only one error occurs in each codeword. It has been speculated that Hadamard matrices are ideal for the coding process, due to the mutual distinguishability of every row and column of the matrix. An Hadamard matrix is a square matrix of order n whose entries are 1 and -1 , and which satisfies the equation HHT = nl, where I is the identity matrix of order n. It is known that Hadamard matrices exist only for orders n = 1, n = 2 or n=0(mod 4). The rows and columns of the Hadamard matrix are orthogonal and linearly independent, which makes them ideal generator matrices. Hadamard matrices can be constructed using several different methods. In his paper Hadamard Matrices and Doubly Even Self-Dual Error Correcting Codes, Michio Ozeki proposed that if the rows of the generator matrix for a binary [n, k] code C all have weights divisible by 4 and are also orthogonal, then C is a doubly even self-dual code. Furthermore, when C is generated by a Hadamard matrix, the result is a doubly even self-dual linear [2n, n] code. It is now necessary to determine whether two codes will be equivalent if their corresponding Hadamard matrices are equivalent. The remainder of the paper will be devoted finding unique [56, 28] Hadamard codes. It is not known how many different Hadamard matrices exist of order 28. The method of integral equivalence will be used to determine the relationship between two distinct Hadamard matrices. A computer program will generate all of the individual codes

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

AbstractAll Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291–303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,pm) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71–87]

Chebyshev type inequalities for Hilbert space operators

We establish several operator extensions of the Chebyshev inequality. The main version deals with the Hadamard product of Hilbert space operators. More precisely, we prove that if $\mathscr{A}$ is a $C^*$-algebra, $T$ is a compact Hausdorff space equipped with a Radon measure $\mu$, $\alpha: T\rightarrow [0, +\infty)$ is a measurable function and $(A_t)_{t\in T}, (B_t)_{t\in T}$ are suitable continuous fields of operators in ${\mathscr A}$ having the synchronous Hadamard property, then \begin{align*} \int_{T} \alpha(s) d\mu(s)\int_{T}\alpha(t)(A_t\circ B_t) d\mu(t)\geq\left(\int_{T}\alpha(t) A_t d\mu(t)\right)\circ\left(\int_{T}\alpha(s) B_s d\mu(s)\right). \end{align*} We apply states on $C^*$-algebras to obtain some versions related to synchronous functions. We also present some Chebyshev type inequalities involving the singular values of positive $n\times n$ matrices. Several applications are given as well.Comment: 18 pages, to appear in J. Math. Anal. Appl. (JMAA