where Par denotes the set of all partitions, |λ| denotes the size (sum of the parts) of λ, θ(λ) denotes the number of odd parts in the partition λ, and θ(λ′) denotes the number of odd parts in the conjugate of λ. In this paper, we generalize this result and provide a bijective proof of our generalization. This provides a simple combinatorial proof of Andrews’ result. Other combinatorial proofs ...

This paper establishes converses to the well-known result: for any vector ũ such that the sine of the angle sin θ(u, ũ) = O( ), we have ρ(ũ) def = ũ∗Aũ ũ∗ũ = λ+O( ), where λ is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and “∗” denotes complex conjugate transpose. It shows that if ρ(ũ) is close to A’s largest eigenvalue, then ũ is close to the corresponding ei...

where ∗ denotes both the conjugate of a complex number and the conjugate of a matrix or an operator. The first result of this type is the well-known Fejer-Riesz theorem (Riesz and SzNagy, 1955), which states that if Π(iω) is a (scalar) polynomial, then we can choose a polynomial M(z) which, additionally, does not have zeros in the right half-plane. Extensions to polynomial, hence also to ration...

in exploration seismology quality factor is widely used as a seismic attribute to identify anomalies related to attenuation, especially those caused by hydrocarbon. previous studies have indicated that seismic energy loss known as attenuation is greater for high frequency components of seismic data compared to the low frequency components. here the continuous wavelet transform is used to study ...

Recent results on reduced-order approximation in the ν-gap metric, characterised in terms of a non-convex feasibility problem, are investigated further. A detailed analysis of the properties of the constituent rank constrained Linear Matrix Inequalities, when the nominal system has an LQGbalanced state-space realisation, reveals that it is possible to construct a feasible point directly for a p...

The only non-trivial step in evaluation of the maximum entropy spectral estimate is the computation of the vectorB−1 ·e0. Generic algorithms with run time O(m) can be used for this purpose (e.g. a Cholesky decomposition). However, if run time is critical, the special structure of the Toeplitz matrix should be exploited. Levinson’s algorithm provides a simple and highly efficient way to do this....

A complex square matrix A is called an orthogonal projector if A2 = A = A∗, where A∗ denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish v...

We consider the normwise condition number and backward error of eigenvalues of matrix polynomials having ⋆-palindromic/antipalindromic and ⋆-even/odd structure with respect to structure preserving perturbations. Here ⋆ denotes either the transpose T or the conjugate transpose ∗. We show that when the polynomials are complex and ⋆ denotes complex conjugate, then to each of the structures there c...