We associate to any given circulant complex matrix C another one E(C) such that E(E(C)) = C∗ the transpose conjugate of C. All circulant Hadamard matrices of order 4 satisfy a condition C4 on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by 4. We prove by a “descent” that uses our operator E that the only circulant Hadamard matrices of order n > 4,...

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inve...

We design heuristic algorithms to construct Hadamard matrices with two circulant cores. This hard combinatorial problem can be formulated in terms of objective functions of several binary variables, so that heuristic methodologies can be used. Our algorithms are based on local and tabu search and they use information on the geometry of the objective function landscapes. In addition, we use the ...

This paper proposes a simple detection scheme for a single carrier block transmission (SCBT) without guard interval (GI). Although the SCBT without the GI can significantly increase bandwidth efficiency, the received signal suffers from interblock interference (IBI) and inter-symbol interference (ISI), which cannot be effectively equalized by a frequency domain equalizer (FDE). In the proposed ...

We describe the results of a computer-based search for 5 and 6dimensional lattice rules of specified trigonometric degree. In this search only lattice rules that can be generated by a circulant or skew-circulant generator matrix are considered, which makes this approach significantly faster than earlier approaches. The drawback is that we do not necessarily obtain optimal lattice rules. We also...

Kirchhoo's Matrix Tree Theorem permits the calculation of the number of spanning trees in any given graph G through the evaluation of the determinant of an associated matrix. In the case of some special graphs Boesch and Prodinger 9] have shown how to use properties of Chebyshev polyno-mials to evaluate the associated determinants and derive closed formulas for the number of spanning trees of g...

Circulant matrix embedding is one of the most popular and efficient methods for the exact generation of Gaussian stationary univariate series. Although the idea of circulant matrix embedding has also been used for the generation of Gaussian stationary random fields, there are many practical covariance structures of random fields where classical embedding methods break down. In this work, we pro...

In this paper, we gives an upper bound estimation of the spectral norm for matrices A and B such that the entries in the first row of n×n r-circulant matrix A = Circr(a1, a2, . . . , an) and n×n symmetric r-circulant matrix B = SCircr(a1, a2, . . . , an) are ai = Pi or ai = P 2 i or ai = Pi−1 or ai = P 2 i−1, where {Pi}i=0 is Padovan sequence. At the last section, some illustrative numerical ex...

We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2qubit entangling gates we generate in the present work, we produce in particula...