Showing papers on "Circulant matrix published in 1979"
TL;DR: This paper proves the conjecture true if n = pq where p and q are distinct primes true, and shows that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n isdivisible by 24.
132 citations
TL;DR: The size of combinational switching networks required to synthesize monotone Boolean functions using only operations from the functionally incomplete set of primitives {disjunction, conjunction} is considered.
Abstract: In this paper, we consider the size of combinational switching networks required to synthesize monotone Boolean functions using only operations from the functionally incomplete set of primitives {disjunction, conjunction}. A general methodology is developed which is used to derive Q(n log n) lower bounds on the size of monotone switching circuits for certain bilinear forms (including Toeplitz and circulant matrix-vector products, and Boolean convolution), certain routing networks (including cyclic and logical shifting), and sorting and merging. A homomorphic mapping technique is also given whereby the lower bounds derived on the sizes of monotone switching networks for Boolean functions can be extended to a larger class of problem domains.
37 citations
TL;DR: In this article, the inverse of a circulant matrix having only three nonzero elements in each row (located in cyclically adjacent columns) is derived analytically from the solution of a recurrence equation.
35 citations
18 citations
TL;DR: In this paper, a class Σ of matrices is studied which contains, as special subclasses, p -circulant matrices (p ⩾ 1), Toeplitz symmetric matrices and the inverses of some special tridiagonal matrices.
16 citations
TL;DR: The generalized q-circulant as discussed by the authors is a generalization of circulants, which is a commutative algebra, and the determinant and a formula for the Moore-Penrose inverse of a generalized q -circularant are given.
14 citations
TL;DR: The set of ordered partitions of n into m parts acted upon by the cyclic permutation is considered and the resulting family of orbits P(n, m) is shown to have cardinality p( n,m) = 1n∑dmφ(d)(ndmd), where φ is Euler's φ-function.
10 citations
TL;DR: V. K. Bhargava recently made the following suggestion, which is sometimes possible to find a code equivalent to C which has a generator matrix of the form [I|A_{1}| \cdots |A_{t-1}] where each A_{i} is a p \times p circulant matrix.
Abstract: V. K. Bhargava recently made the following suggestion. Let C he a cyclic code of dimension p , and block length n , where n = tp, p is a prime. It is sometimes possible to find a code equivalent to C which has a generator matrix of the form [I|A_{1}| \cdots |A_{t-1}] where each A_{i} is a p \times p circulant matrix. Such a decomposition is useful in encoding and decoding and also in finding the weight distribution of C . Some general theory as to
8 citations
TL;DR: Methods for a fast and consistent computation of quadratic Toeplitz forms in a finite observation sequence of a stationary narrowband process are described.
7 citations
5 citations
TL;DR: In this paper, a procedure based on differences is used to yield an estimator of an unknown parameter occurring in the autocorrelation structure of a stationary Gaussian vector process.
Abstract: SUMMARY A procedure based on differences is used to yield an estimator of an unknown parameter occurring in the autocorrelation structure of a stationary Gaussian vector process. Various properties of the estimator including its existence, consistency and asymptotic normality are shown. Necessary and sufficient conditions are given for the estimator to be efficient. LET {y(t), t > 0} be a stationary Gaussian vector process having m components with E{y(t)} = 0, var {y(t)}- E{y(t) y(t)T} = I and E{y(t) y(t +s)T} = Po(s), s > O, where T denotes transpose, I is the m x m identity matrix, and P0(s) is the autocorrelation matrix, of {y(t), t> 0}, which depends on some real unknown parameter 0e E). It is desired to estimate 0 from a realization x(O),...,x(k) of the sampling process, {x(j)_ y(jr), 0 =o. Nevertheless, Whittle (1951, 1952, 1953) in a series of classical articles circumvented this problem by (asymptotically) approximating {x(j), j = 0,1, ...} by its corresponding circulant process, and succeeded in showing the following Theorem.
01 Jan 1979
01 Jan 1979
TL;DR: In this article, it was shown that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime and q is a distinct prime.
Abstract: Let Sc{l,...,n-1) satisfy -S =S mod n. The circulant graph G(n, S) with vertex set IQ, IJ l,“‘, u,_r} and edge set E satisfies “iiIj E E if and only if i - i E S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = -S. Adarn conjectured that G(n, S)=G(n, S’) if and only if S = US’ for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p2, and determine exact conditions on S that it be tme in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n is divisible by 24.
01 Apr 1979
TL;DR: A new method is investigated for a fast and consistent estimation of the spectral density of stationary processes which are narrowband, exploiting the Toeplitz structure of the classical smoothed periodogram estimates and their similarity to circulant forms.
Abstract: A new method is investigated for a fast and consistent estimation of the spectral density of stationary processes which are narrowband. The method exploits the Toeplitz structure of the classical smoothed periodogram estimates and their similarity to circulant forms. The new estimates compare favourably with the classical estimates and, using a fast Fourier transform algorithm, are more rapidly computable than many frequently used estimates when the process is sufficiently narrowband.
01 Jan 1979
TL;DR: In this paper, a family of weakly self-dual codes, C, over GF(3), with (m×2m) generating matrix was studied, where J is the identity matrix and A the circulant incidence matrix.
Abstract: This note looks at a family of weakly self-dual codes, C, over GF(3), with (m×2m) generating matrix
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, where J is the (m×m) matrix with every entry being +1, I the (m×m) identity matrix, and A the circulant incidence matrix of the (4t−1, 2t−1, t−1) block design, with first row being formed from the quadratic residues, mod(4t−1) (so m=4t−1, and is prime).
TL;DR: In this paper, the solution with least Euclidean norm of a subdefinite system of linear equations with an incomplete circulant matrix of dimensions m × n is described.
Abstract: AN ALGORITHM is described for finding the solution with least Euclidean norm of a subdefinite system of linear equations with an incomplete circulant matrix of dimensions m × n , in which the calculation of the pseudo-inverse matrix is replaced by some discrete Fourier transformations of vectors of dimension n and the inverse of a positive-definite matrix of order n - m .