Showing papers on "Circulant matrix published in 1998"

Symmetric functional differential equations and neural networks with memory

Abstract: We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

Adaptive detection schemes in compound-Gaussian clutter

Abstract: Radar detection of coherent pulse trains embedded in compound-Gaussian disturbance with partially known statistics is discussed. We first give a thorough derivation of two recently proposed adaptive detection structures. Next, we derive a different detection scheme exploiting the assumption that the clutter is wide-sense stationary. Resorting to the theory of circulant matrices, in fact, we demonstrate that the estimation of the structure of the clutter covariance matrix can be reduced to the estimation of its eigenvalues, which in turn can be (efficiently) done via fast Fourier transform codes. After a thorough performance assessment, mostly carried on via computer simulations, the results show that the newly proposed detector achieves better performance than the two previously introduced adaptive detectors. Moreover, a sensitivity analysis shows that, even though this detector does not strictly guarantee the constant false alarm rate property with respect to the clutter covariance matrix, it is robust, in the sense that its performance is only slightly affected by variations in the clutter temporal correlation.

High-resolution image reconstruction with multisensors

Abstract: This article considers the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors. This leads to a formulation involving a periodically shift-variant system model. The maximum a posteriori (MAP) estimation scheme is used subject to the assumption that the original high-resolution image is modeled by a stationary Markov-Gaussian random field. The resulting MAP formulation is expressed as a complex linear matrix equation, where the characterizing matrix involves the periodic block Toeplitz with Toeplitz block (BTTB) blur matrix and banded-BTTB inverse covariance matrix associated with the original image. By approximating the periodic-BTTB and the banded-BTTB matrices with, respectively, the periodic block circulant with circulant block (BCCB) and the banded-BCCB matrices, it is shown that the computation-intensive MAP formulation can be decomposed into a set of smaller matrix equations by using the two-dimensional discrete Fourier transform. Exact solutions are also considered through the use of the preconditioned conjugate gradient algorithm. Computer simulations are given to illustrate the procedure. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 294–304, 1998

Reliability analysis of circulant graphs

Abstract: The circulant graphs are of particular interest as models of communication networks. In this work, we present new reliability analysis results for circulants based on the concept of restricted edge connectivity, which generalizes the super-λ property of a graph. We evaluate the restricted edge connectivity λ′ and the number of i-cutsets Ni(G), λ ≤ i < λ′, for any circulant graph explicitly. This improves the previous results on the subject. © 1998 John Wiley & Sons, Inc. Networks 31:61–65, 1998

Fault-tolerant routing in circulant networks and cycle prefix networks

Abstract: Reliability and efficiency are important criteria in the design of interconnection networks. Connectivity is a widely used measurement for network fault-tolerance capacities, while diameter determines routing efficiency along individual paths. In practice, we are interested in high-connectivity, small-diameter networks. Recently, Hsu introduced the notion ofw-wide diameter, which unifies diameter and connectivity. This paper investigates thew-wide diameterd w (G) and two related parameters:w-fault diameterD w (G) andw-Rabin numberr w (G). In particular, we determined w (G) andD w (G) for 2≤w≤K(G) andG is a circulant digraphG(d n ; ∈1,d,...,d n−1∉) or a cycle prefix digraph.

Classification of Extremal Double Circulant Self-Dual Codesof Lengths 64 to 72

Abstract: Recently extremal double circulant self-dual codes have been classified for lengths n ≤ 62. In this paper, a complete classification of extremal double circulant self-dual codes of lengths 64 to 72 is presented. Almost all of the extremal double circulant singly-even codes given have weight enumerators for which extremal codes were not previously known to exist.

Model for particle masses, flavor mixing, and CP violation, based on spontaneously broken discrete chiral symmetry as the origin of families

Abstract: We construct extensions of the standard model based on the hypothesis that Higgs bosons also exhibit a family structure and that the flavor weak eigenstates in the three families are distinguished by a discrete Z{sub 6} chiral symmetry that is spontaneously broken by the Higgs sector. We study in detail at the tree level models with three Higgs doublets and with six Higgs doublets comprising two weakly coupled sets of three. In a leading approximation of S{sub 3} cyclic permutation symmetry the three-Higgs-doublet model gives a {open_quotes}democratic{close_quotes} mass matrix of rank 1, while the six-Higgs-doublet model gives either a rank-1 mass matrix or, in the case when it spontaneously violates {ital CP}, a rank-2 mass matrix corresponding to nonzero second family masses. In both models, the CKM matrix is exactly unity in the leading approximation. Allowing small explicit violations of cyclic permutation symmetry generates small first family masses in the six-Higgs-doublet model, and first and second family masses in the three-Higgs-doublet model, and gives a nontrivial CKM matrix in which the mixings of the first and second family quarks are naturally larger than mixings involving the third family. Complete numerical fits are given for both models, flavor-changing neutral current constraintsmore » are discussed in detail, and the issues of unification of couplings and neutrino masses are addressed. On a technical level, our analysis uses the theory of circulant and retrocirculant matrices, the relevant parts of which are reviewed. {copyright} {ital 1998} {ital The American Physical Society}« less

Primitivity of Positive Matrix Pairs: Algebraic Characterization, Graph Theoretic Description, and 2D Systems Interpretation

Abstract: In this paper the primitivity of a positive matrix pair (A,B) is introduced as a strict positivity constraint on the asymptotic behavior of the associated two-dimensional (2D) state model. The state evolution is first considered under the assumption of periodic initial conditions. In this case the system evolves according to a one-dimensional (1D) state updating equation, described by a block circulant matrix. Strict positivity of the asymptotic dynamics is equivalent to the primitivity of the circulant matrix, a property that can be restated as a set of conditions on the spectra of $A + e^{i \omega} B$, for suitable real values of $\omega$. The theory developed in this context provides a foundation whose analytical ideas may be generalized to nonperiodic initial conditions. To this purpose the spectral radius and the maximal modulus eigenvalues of the matrices $e^{i \theta} A + e^{i \omega} B$, $\theta$ and $\omega \in \hbox{{\bbb R}},$ are related to the characteristic polynomial of the pair (A,B) as well as to the structure of the graphs associated with A and B and to the factorization properties of suitable integer matrices. A general description of primitive positive matrix pairs is finally derived, including both spectral and combinatorial conditions on the pair.

Recognizing circulant graphs of prime order in polynomial time

Abstract: A circulant graph $G$ of order $n$ is a Cayley graph over the cyclic group ${\bf Z}_n.$ Equivalently, $G$ is circulant iff its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration ${\cal A}$ and, in particular, a Schur ring ${\cal S}$ isomorphic to ${\cal A}$. ${\cal A}$ can be associated without knowing $G$ to be circulant. If $n$ is prime, then by investigating the structure of ${\cal A}$ either we are able to find an appropriate ordering of the vertices proving that $G$ is circulant or we are able to prove that a certain necessary condition for $G$ being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in $n$.

Asymptotic eigenvalue distribution of block-Toeplitz matrices

Abstract: The need for approximating block-Toeplitz with Toeplitz block matrices by means of block-circulant with circulant block matrices with the objective of transforming an inherently ill-posed image deconvolution problem to a well-posed one motivated a surge of papers on the analysis of the quality as well as the speed of convergence of the algorithms that produce such approximants to serve as preconditioners for conjugate-gradient methods. This correspondence contributes to that surge by giving a simple proof of the fact that the sequence of eigenvalues of the Hermitian block Toeplitz with Toeplitz-block matrices are asymptotically equidistributed. To do this, Weyl's (1916) results on the distribution properties of multidimensional sequences are exploited. Inferences to related results are made.

Isomorphism Problem for Metacirculant Graphs of Order a Product of Distinct Primes

Abstract: In this paper, we solve the isomorphism problem for metacirculant graphs of order pq that are not circulant. To solve this problem, we first extend Babai's characterization of the CI-property to non-Cayley vertex- transitive hypergraphs. Addi- tionally, we find a simple characterization of metacirculant Cayley graphs of order pq, and exactly determine the full isomorphism classes of circulant graphs of order pq.

A total least squares method for Toeplitz systems of equations

Abstract: A Newton method to solve total least squares problems for Toeplitz systems of equations is considered. When coupled with a bisection scheme, which is based on an efficient algorithm for factoring Toeplitz matrices, global convergence can be guaranteed. Circulant and approximate factorization preconditioners are proposed to speed convergence when a conjugate gradient method is used to solve linear systems arising during the Newton iterations.

Routing in Recursive Circulant Graphs: Edge Forwarding Index and Hamiltonian Decomposition

Abstract: The recursive circulant graphs G(2 m ,4) were described in [13] as a concurrent to the hypercube considered as topology for multicomputer networks. In this paper we give the exact value of the edge forwarding index and bisection width of the generalize recursive circulant graphs G(cd m ,d) with d > c > 0. Moreover we prove that they admit a Hamiltonian decomposition.

Double circulant self-dual codes over Z/sub 2k/

Abstract: Recently there has been tremendous interest in self-dual codes over finite rings, specifically the rings Z/sub 2k/. In this paper, we investigate double circulant self-dual codes over Z/sub 2k/ for small k and short lengths. In particular, we give a classification of the extremal double circulant codes. Using invariant theory, a basis for the space of invariants to which the Hamming weight enumerators belong is given for self-dual codes over Z/sub 6/, Z/sub 8/, Z/sub 10/, and Z/sub 12/.

A new modification of the Rojo method for solving symmetric circulant five-diagonal systems of linear equations☆

Abstract: A new, effective, and stable modification of the Rojo method [1] for solving of real symmetric circulant five-diagonal systems of linear equations is proposed. This special kind of system appears in many applications: spline approximation, difference solution of partial differential equations, etc. The method presented in this paper a very efficient and stable method.

Hamiltonian Decomposition of Recursive Circulants

Abstract: We show that recursive circulant G(cdm, d) is hamiltonian decomposable. Recursive circulant is a graph proposed for an interconnection structure of multicomputer networks in [8]. The result is not only a partial answer to the problem posed by Alspach that every connected Cayley graph over an abelian group is hamiltonian decomposable, but also an extension of Micheneau's that recursive circulant G(2m, 4) is hamiltonian decomposable.

Inversion of Circulant Matrices over Zm

Abstract: In this paper we consider the problem of inverting an n x n circulant matrix with entries over Z m . We show that the algorithm for inverting circulants, based on the reduction to diagonal form by means of FFT, has some drawbacks when working over Z m . We present three different algorithms which do not use this approach. Our algorithms require different degrees of knowledge of m and n, and their costs range -roughly - from n log n log log n to n log 2 n log log n log m operations over Z m . We also present an algorithm for the inversion of finitely generated bi-infinite Toeplitz matrices. The problems considered in this paper have applications to the theory of linear Cellular Automata.

Circulant matching method for multiplexing ATM traffic applied to video sources

Abstract: In this paper a method is proposed, called circulant matching method, to approximate the superposition of a number of discrete-time batch Markovian arrival sources by a circulant batch Markovian process, while matching the stationary cumulative distribution and the autocorrelation sequence of the input rate process. Special attention is paid to periodic sources. The method is applied to the superposition of MPEG video sources and the obtained results are validated through experiments.

Circulant preconditioners for ill‐conditioned boundary integral equations from potential equations

Abstract: In this paper, we consider solving potential equations by the boundary integral equation approach. The equations so derived are Fredholm integral equations of the first kind and are known to be ill-conditioned. Their discretized matrices are dense and have condition numbers growing like O(n) where n is the matrix size. We propose to solve the equations by the preconditioned conjugate gradient method with circulant integral operators as preconditioners. These are convolution operators with periodic kernels and hence can be inverted efficiently by using fast Fourier transforms. We prove that the preconditioned systems are well conditioned, and hence the convergence rate of the method is linear. Numerical results for two types of regions are given to illustrate the fast convergence. © 1998 John Wiley & Sons, Ltd.

Minimal critical sets for some small Latin squares

Abstract: A general algorithm for finding a minimal critical set for any latin square is presented. By implementing this algorithm, minimal critical sets for all the latin squares of order six have been found. In addition, this algorithm is used to prove that the size of the minimal critical set for a back circulant latin square of order seven is twelve, and for order nine is twenty. These results provide further support for the conjecture that the back circulant latin square of odd order n has minimal critical set of size (n - 1)/4.

A reduction theorem for circulant weighing matrices.

Abstract: Circulant weighing matrices of order n with weight k, denoted by WC(n, k), are investigated. Under some conditions, we show that the existence of WC(n, k) implies that of WCG, ~). Our results establish the nonexistence of WC(n,k) for the pairs (n,k) = (125,25), (44,36), (64,36), (66,36), (80,36), (72,36), (118,36), (128,36), (136,36), (128,100), (144,100), (152,100), (88,36), (132,36), (160,36), (166,36), (176,36), (198, 36), (200, 36), (200,100). All these cases were previously open.

Commutation criterion for Toeplitz matrices

Abstract: It is proved that two Toeplitz matrices commute if and only if one of the matrices is the value of a linear function at the other one or both matrices belong to the same algebra of generalized circulant matrices.