On the asymptotic existence of Hadamard matrices

It is conjectured that Hadamard matrices exist for all orders 4t (t>0). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers k, there is a Hadamard matrix of order k²[a+blog₂k], where a and b are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take a=2 and b=0. Since Seberry's ground-breaking result, which showed that we may take a=0 and b=2, there have been several improvements where b has been by stages reduced to 3/8. In this paper, we show that for all ?>0, the set of odd numbers k for which there is a Hadamard matrix of order k²2+[?log₂k] has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result.     

On the asymptotic existence of cocyclic Hadamard matrices

Let q be an odd natural number. We prove there is a cocyclic Hadamard matrix of order ²10+tq whenever . We also show that if the binary expansion of q contains N ones, then there is a cocyclic Hadamard matrix of order ²4N−2q.     

On the asymptotic existence of complex Hadamard matrices

Let N = N(q) be the number of nonzero digits in the binary expansion of the odd integer q. A construction method is presented which produces, among other results, a block circulant complex Hadamard matrix of order 2^αq, where α ≥ 2N - 1. This improves a recent result of Craigen regarding the asymptotic existence of Hadamard matrices. We also present a method that gives complex orthogonal designs of order 2^α+1q from complex orthogonal designs of order 2^α. We also demonstrate the existence of a block circulant complex Hadamard matrix of order 2^βq, where © 1997 John Wiley & Sons, Inc. J Combin Designs 5:319–327, 1997     

Spectral properties of Jacobi matrices by asymptotic analysis

We consider two classes of Jacobi matrix operators in l² with zero diagonals and with weights of the form n^α+c_n for 0<α1 or of the form n^α+c_nn^α−1 for α>1, where {c_n} is periodic. We study spectral properties of these operators (especially for even periods), and we find asymptotics of some of their generalized eigensolutions. This analysis is based on some discrete versions of the Levinson theorem, which are also proved in the paper and may be of independent interest.     

On the spectra of a family of positive matrices

It is shown that every positive matrix A can be embedded in an analytic family of positive matrices {A(ν) : ν∈$\text{R}$} in such a way that A(1)=A, $\text{A(0)≡}\text{A}\text{?}$ is symmetric, and A(-1)=A^T. A necessary and sufficient condition that A and Å have the same maximal eigenvalue and that their ergodic limits have the same diagonal elements is stated and proved.     

On the asymptotic distribution of eigenvalues of banded matrices

We consider the abstract measures, known as thedensity- of- states measures, associated with the asymptotic distribution of eigenvalues of infinite banded Hermitian matrices. Two widely used definitions of these measures are shown to be equivalent, even in the unbounded case, and we prove that the density of states is invariant under certain, possibly unbounded, perturbations. Also considered are measures associated with the asymptotic distribution of eigenvalues of rescaled unbounded matrices. These measures are associated with the so-called contracted spectrum when the matrices are tridiagonal. Finally, we produce several examples clarifying the nature of the density of states.Communicated by Paul Nevai.     

On the computation of a versal family of matrices

The aim of this article is to present algorithms for the computation of versal deformations of matrices. A deformation of a matrixA ₀ is a holomorphic matrix-valued function whose value at a pointt ₀C ^p is the matrixA ₀. We want to study the properties of these matrices in a neighbourhood oft ₀. One could, for each valuet in this neighbourhood, compute the Jordan form as well as the change of basis matrix; but, generally, the results will not be analytic. So, we want to construct a deformation of the matrixA ₀ into which any deformation can be transformed by an invertible deformation of the matrixId. After having introduced the notion of versal deformation, we shall provide computer algebra algorithms to computer these normal forms. In the last section, we shall show that a one-parameter deformation can be transformed into a simpler form than the general versal deformation.     

On the asymptotic properties of a priori minimax estimates

Properties are studied of the a priori minimax estimates /1,2/ of an unknown initial state of a linear dynamic object under the condition that the measurements are taken at discrete instants, while noise exists only in the measurement channel and is modelled as an independent random variable. The paper continues the investigations begun in /3/.     

On the asymptotic properties of non-autonomous systems

By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system ${du/dt +Au\ni f}By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system du/dt +Au ' f{du/dt +Au\ni f}, where A is an accretive (possibly multivalued) operator in X × X, and for some f_￥ ? X{f_{\infty}\in X} and 1 ≤ p < ∞ we have g ? L^p((1,+￥);X){g\in L^p((1,+\infty);X)}, so that g(θ) = (f(θ) − f _∞)/θ, ${(\forall \theta > 1)}${(\forall \theta > 1)}. Our results extend and improve many previously known results. Moreover, we answer an open question raised by B. Djafari Rouhani.     

On the asymptotic properties of the Rees powers of a module

Let R be a commutative ring and G a free R-module with finite rank e>0. For any R-submodule E⊂G one may consider the image of the symmetric algebra of E by the natural map to the symmetric algebra of G, and then the graded components E_n, n≥0, of the image, that we shall call the n-th Rees powers of E (with respect to the embedding E⊂G). In this work we prove some asymptotic properties of the R-modules E_n, n≥0, which extend well known similar ones for the case of ideals, among them Burch’s inequality for the analytic spread.     

On the properties of a new tensor product of matrices

Previously, the author introduced a new tensor product of matrices according to which the matrix of the discrete Walsh-Paley transform can be represented as a power of the second-order discrete Walsh transform matrix H with respect to this product. This power is an analogue of the representation of the Sylvester-Hadamard matrix in the form of a Kronecker power of H. The properties of the new tensor product of matrices are examined and compared with those of the Kronecker product. An algebraic structure with the matrix H used as a generator element and with these two tensor products of matrices is constructed and analyzed. It is shown that the new tensor product operation proposed can be treated as a convenient mathematical language for describing the foundations of discrete Fourier analysis.     

On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks

We study the asymptotic behaviour of the eigenvalues of Hermitian block Toeplitz matrices , with Toeplitz blocks. Such matrices are generated by the Fourier coefficients of an integrable bivariate function , and we study their eigenvalues for large and , relating their behaviour to some properties of as a function; in particular we show that, for any fixed , the first eigenvalues of tend to , while the last tend to , so extending to the block case a well-known result due to Szegö. In the case the 's are positive-definite, we study the asymptotic spectrum of , where is a block Toeplitz preconditioner for the conjugate gradient method, applied to solve the system , obtaining strict estimates, when and are fixed, and exact limit values, when and tend to infinity, for both the condition number and the conjugate gradient convergence factor of the previous matrices. Extensions to the case of a deeper nesting level of the block structure are also discussed.

     

On the asymptotic properties of some canonical products

Siberian Mathematical Journal -     

On the asymptotic properties of multivariate sample autocovariances

We show that if a process can be obtained by filtering an autoregressive process, then the asymptotic distribution of sample autocovariances of the former is the same as the asymptotic distribution of linear combinations of sample autocovariances of the latter. This result is used to show that for small lags the sample autocovariances of the filtered process have the same asymptotic distribution as estimators utilizing more information (observations on the associated autoregression process and knowledge of the parameters of the filter). In particular, for a Gaussian ARMA process the first few sample autocovariances are jointly asymptotically efficient.     

On properties of cell matrices

In this paper properties of cell matrices are studied. A determinant of such a matrix is given in a closed form. In the proof a general method for determining a determinant of a symbolic matrix with polynomial entries, based on multivariate polynomial Lagrange interpolation, is outlined. It is shown that a cell matrix of size n>1 has exactly one positive eigenvalue. Using this result it is proven that cell matrices are (Circum-)Euclidean Distance Matrices ((C)EDM), and their generalization, k-cell matrices, are CEDM under certain natural restrictions. A characterization of k-cell matrices is outlined.     

On some properties of singular matrices

Some results of Ostrowski in [5] are generalized to the case of monotonic norms.