Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling of Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita [2] via a natural tiling construction. Then we find so...

In this paper, the properties of the WalshHadamard codes used as spreading codes in QCDMA cellular systems are analyzed. It is shown that among Walsh-Hadamard codes, obtained from Hadamard Matrix , only k 2 ) , 2 ( k H k 1 + k codes are orthogonal to each other in all phases. Also, the performance of the Walsh Hadamard codes is compared with that of the Gold codes from the average cross-correla...

In this paper, an equivalent ‘virtual’ SIMO (Single Input Multiple Output) radar representation of the MIMO radar system is established which enables the analysis of the full MIMO system geometry. Analytical expressions are derived to describe the key geometrical properties of the virtual array’s manifold, which fully characterises the entire transmit-receive system geometry. A new family of su...

The Hadamard conjecture is that Hadamard matrices exist for all orders 1,2, 4t where t 2 1 is an integer. We have obtained the following results which strongly support the conjecture: (i) Given any natural number q, there exists an Hadamard matrix of order 2 q for every s 2 [2log2 (q 3)]. (ii) Given any natural number q, there exists a regular symmetric Hadamard matrix with constant diagonal of...

Recently, Feng and Xiang [10] found a new construction of skew Hadamard difference sets in elementary abelian groups. In this paper, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the Paley difference sets. As a consequence, we show th...

A Hadamard matrix of side n is an n×n matrix with every entry either 1 or −1, which satisfies HHT = nI. Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when n increases. In this paper, a new a...

This paper presents a proximal point approach for computing the Riemannian or intrinsic Karcher mean of, n×n, symmetric positive definite (SPD) matrices. Our method derives from proximal point algorithm with Schur decomposition developed to compute minimum points of convex functions on SPD matrices set when it is seen as a Hadamard manifold. The main idea of the original method is preserved. Ho...

Quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis. We discuss a condition on the wave front set for the corresponding n-point distributions, called “microlocal spectrum condition” (μSC). On Minkowski space, this condition is satisfied as a consequence of the usual spectrum condition. Based on Radzikowski’s determination of the wave front set of th...

Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for several first-order algorithms on Hadamard manifolds. Specifically, we prove ...

From this result one can easily deduce I (H) = n 1. For a detailed discussion of Cheeger’s constant and related results, one can consult [Cha, Chapter 6]. In general, it is very di¢ cult to know if the isoperimetric constant is positive or not and it is almost impossible to compute it explicitly if it is known to be positive. In this short note, we prove that the isoperimetric constant is posit...