On the elliptic generating region of a tsunami

The surface elevation is calculated for the three-dimensional motion of waves in a fluid of constant depth subject to a given bottom velocity. An example, modeling tsunami generation, with antisymmetric bottom velocity, is considered in detail. The amplitude of the wave front is found to decay much more rapidly than the main wave. The distribution of amplitude with wave number and with angular position is computed for some cases

The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

The enhanced principal rank characteristic sequence (epr-sequence) of an n x n matrix is a sequence l(1) l(2) . . .l(n), where each l(k) is A, S, or N according as all, some, or none of its principal minors of order k are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric matrices, and incidental remarks have been made about results extending (or not extending) to (complex) Hermitian matrices. A systematic study of epr-sequences of Hermitian matrices is undertaken; the differences with the case of symmetric matrices are quite striking. Various results are established regarding the attainability by Hermitian matrices of epr-sequences that contain two Ns with a gap in between. Hermitian adjacency matrices of mixed graphs that begin with N A N are characterized. All attainable epr-sequences of Hermitian matrices of orders 2, 3, 4, and 5, are listed with justifications

The Enhanced Principal Rank Characteristic Sequence for Hermitian Matrices

The enhanced principal rank characteristic sequence (epr-sequence) of an n\x n matrix is a sequence $\ell_1 \ell_2 \cdots \ell_n$, where each $\ell_k$ is ${\tt A}$, ${\tt S}$, or ${\tt N}$ according as all, some, or none of its principal minors of order $k$ are nonzero. There has been substantial work on epr-sequences of symmetric matrices (especially real symmetric matrices) and real skew-symmetric matrices, and incidental remarks have been made about results extending (or not extending) to (complex) Hermitian matrices. A systematic study of epr-sequences of Hermitian matrices is undertaken; the differences with the case of symmetric matrices are quite striking. Various results are established regarding the attainability by Hermitian matrices of epr-sequences that contain two ${\tt N}$s with a gap in between. Hermitian adjacency matrices of mixed graphs that begin with ${\tt NAN}$ are characterized. All attainable epr-sequences of Hermitian matrices of orders $2$, $3$, $4$, and $5$, are listed with justifications

Reproduction numbers of infectious disease models

This primer article focuses on the basic reproduction number, â0, for infectious diseases, and other reproduction numbers related to â0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of â0 determining whether or not the disease dies out. The next generation matrix method of calculating â0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate â0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity. Keywords: Basic reproduction number, Disease control, West Nile virus, Cholera, Anthrax, Zika viru

Modifying the power method in max algebra

Elsner L, van den Driessche P. Modifying the power method in max algebra. In: Linear Algebra and its Applications. Linear Algebra and its Applications. Vol 332-334. ELSEVIER SCIENCE INC; 2001: 3-13.In the max algebra system, the eigenequation for an n x n irreducible nonnegative matrix A = [a(ij)] is A circle times x = mu (A)x. Here (A circle times x)(i) = max(j) a(ij)x(j) and mu (A) is the maximum circuit geometric mean. The complexity of the power method given in [L. Elsner, P. van den Driessche, Linear Algebra Appl., to appear] to compute mu (A) and x is considered. Under some assumptions on the critical matrix, it is shown that the algorithm may have time complexity O(n(4)), A modified power method, based on Karp's formula, is presented. For this new algorithm, with no assumptions on the critical matrix, mu (A) and x can be computed in O(n(3)) time. Furthermore, this algorithm can be used to compute all linearly independent eigenvectors corresponding to mu (A). (C) 2001 Elsevier Science Inc. All rights reserved

Max-algebra and pairwise comparison matrices

Elsner L, van den Driessche P. Max-algebra and pairwise comparison matrices. Linear Algebra and its Applications. 2004;385:47-62.The max-eigenvector of a symmetrically reciprocal matrix A can be used to construct a transitive matrix that is closest to A in a relative error measure. As an alternative to the Perron eigenvector, the max-eigenvector can be used successfully for ranking in the analytical hierarchy process. When either one measurement is corrected or a new alternative is added, the max-eigenvector gives more consistent rankings. Some properties of the max-eigenvector that are important in this process are discussed, and an O (n(3)) procedure to calculate the max-eigenvector is detailed. (C) 2003 Elsevier Inc. All rights reserved

On the power method in max algebra

Elsner L, van den Driessche P. On the power method in max algebra. In: Linear Algebra and its Applications. Linear Algebra and its Applications. Vol 302-303. ELSEVIER SCIENCE INC; 1999: 17-32.The eigenvalue problem for an irreducible nonnegative matrix A = [a(ij)] in the max algebra system is A x x = lambda x, where (A x x)(i) = max(j)(a(ij)x(j)) and lambda turns out to be the maximum circuit geometric mean, mu(A). A power method algorithm is given to compute mu(A) and eigenvector x. The algorithm is developed by using results on the convergence of max powers of A, which are proved using nonnegative matrix theory. In contrast to an algorithm developed in [4], this new method works for any irreducible nonnegative A, and calculates eigenvectors in a simpler and more efficient way. Some asymptotic formulas relating mu(A), the spectral radius and norms are also given. (C) 1999]Elsevier Science Inc. All rights reserved