The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)
01 Oct 1961-The Mathematical Gazette (Cambridge University Press (CUP))-Vol. 45, Iss: 353, pp 267-267
About: This article is published in The Mathematical Gazette.The article was published on 1961-10-01. It has received 2618 citations till now.
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TL;DR: In this article, a measure called the effective number of species is developed from a nonparametric probability inequality and is shown to have a simple interpretation in terms of comparing linear experiments.
Abstract: The diversity of a set of species refers to the joint dissimilarity of the species in the set. This paper discusses the measurement of diversity from the set of pairwise distances between the species in the set. A measure called the effective number of species is developed from a non-parametric probability inequality and is shown to have a simple interpretation in terms of comparing linear experiments.
2,957 citations
Book•
15 Jun 2001
TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.
Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index
2,581 citations
Book•
01 Jan 1977TL;DR: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toepler matrices with absolutely summable elements are derived in a tutorial manner in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject.
Abstract: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.
2,404 citations
TL;DR: The spatial component of environment, often neglected in modeling of ecological interactions, in general operates to increase species diversity due to the heterogeneity of the environment, but such heterogeneity can arise in an initially homogeneous environment due to what may be random initial events (e.g., colonization patterns).
Abstract: The spatial component of environment, often neglected in modeling of ecological interactions, in general operates to increase species diversity. This arises due to the heterogeneity of the environment, but such heterogeneity can arise in an initially homogeneous environment due to what may be random initial events (e.g., colonization patterns), effects of which are magnified by species interactions. In this way, homogeneous environments may become heterogeneous and heterogeneous environments even more so. In patchy environments, distinct patches are likely to be colonized initially by different species, and thereby a kind of founder effect results whereby individual patches evolve along different paths simply as a consequence of initial colonization patterns. Species which would be unable to invade may nevertheless survive by establishing themselves early and will moreover be found in lower densities in other areas as overflow from their "safe" areas. Spatially continuous environments may evolve toward es...
1,097 citations
TL;DR: The theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates has been studied in this paper, where a necessary and sufficient frequency domain condition for dissipativeness is derived.
Abstract: This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.
1,061 citations
References
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TL;DR: In this article, a measure called the effective number of species is developed from a nonparametric probability inequality and is shown to have a simple interpretation in terms of comparing linear experiments.
Abstract: The diversity of a set of species refers to the joint dissimilarity of the species in the set. This paper discusses the measurement of diversity from the set of pairwise distances between the species in the set. A measure called the effective number of species is developed from a non-parametric probability inequality and is shown to have a simple interpretation in terms of comparing linear experiments.
2,957 citations
Book•
15 Jun 2001
TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.
Abstract: Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index
2,581 citations
Book•
01 Jan 1977TL;DR: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toepler matrices with absolutely summable elements are derived in a tutorial manner in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject.
Abstract: The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.
2,404 citations
TL;DR: The spatial component of environment, often neglected in modeling of ecological interactions, in general operates to increase species diversity due to the heterogeneity of the environment, but such heterogeneity can arise in an initially homogeneous environment due to what may be random initial events (e.g., colonization patterns).
Abstract: The spatial component of environment, often neglected in modeling of ecological interactions, in general operates to increase species diversity. This arises due to the heterogeneity of the environment, but such heterogeneity can arise in an initially homogeneous environment due to what may be random initial events (e.g., colonization patterns), effects of which are magnified by species interactions. In this way, homogeneous environments may become heterogeneous and heterogeneous environments even more so. In patchy environments, distinct patches are likely to be colonized initially by different species, and thereby a kind of founder effect results whereby individual patches evolve along different paths simply as a consequence of initial colonization patterns. Species which would be unable to invade may nevertheless survive by establishing themselves early and will moreover be found in lower densities in other areas as overflow from their "safe" areas. Spatially continuous environments may evolve toward es...
1,097 citations
TL;DR: The theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates has been studied in this paper, where a necessary and sufficient frequency domain condition for dissipativeness is derived.
Abstract: This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.
1,061 citations