dorsal/arxiv
View SchemaSimultaneously Dissipative Operators And The Infinitesimal Moore Effect In Interval Spaces
| Authors | A. N. Gorban, Yu. I. Shokin, V. I. Verbitskii |
|---|---|
| Categories | |
| ArXiv ID | physics/9702021 |
| URL | https://arxiv.org/abs/physics/9702021 |
| License | http://creativecommons.org/licenses/by/3.0/ |
Abstract
One of shortcomings of stepwise interval methods is the following. The intervals determining the solution of a system are often expanded in the course of time irrespective of the method and step used (the {\em Moore effect}). We introduce the notion of general {\em interval spaces} and study the infinitesimal Moore effect (IME) in these spaces. We obtain the local conditions of absence of the IME in terms of Jacobi matrices field. The relation between the absence of IME and simultaneous dissipativity of the Jacobi matrices is established. We study simultaneously dissipative operators in $\Bbb{R}^n$. A linear operator $A$ is {\em dissipative} with respect to a norm $\|...\|$ if $\| \exp (At) \| \leq 1$ at all $t \geq 0$. For each norm, the dissipative operator form a closed convex cone. An operator $A$ is {\em stable dissipative} if it belongs to the interior of this cone. The family of linear operators $\{A_\alpha \}$ is called {\em simultaneously dissipative}, if there exists a norm with respect to which all the operators are dissipative. We studied general properties of such families. For example, let the family $\{A_\alpha \}$ be finite and generate a nilpotent Lee algebra and let for each $A_\alpha$ there exist a norm with respect to which it is dissipative. Then $\{A_\alpha \}$ is simultaneously dissipative. Let the family $\{A_\alpha \}$ be compact and generate solvable Lee algebra, and let the spectrum of each operator $A_\alpha $ lie in the open left half-plane. Then $\{A_\alpha \}$ is simultaneously stable dissipative, i.e. there exists a norm with respect to which all $A_\alpha $ are stable dissipative. We study the conditions of simultaneous dissipativity of the matrices of rank one and discussed their application to equations of {\em mass action law} kinetics.
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"abstract": "One of shortcomings of stepwise interval methods is the following. The\nintervals determining the solution of a system are often expanded in the course\nof time irrespective of the method and step used (the {\\em Moore effect}). We\nintroduce the notion of general {\\em interval spaces} and study the\ninfinitesimal Moore effect (IME) in these spaces. We obtain the local\nconditions of absence of the IME in terms of Jacobi matrices field. The\nrelation between the absence of IME and simultaneous dissipativity of the\nJacobi matrices is established. We study simultaneously dissipative operators\nin $\\Bbb{R}^n$. A linear operator $A$ is {\\em dissipative} with respect to a\nnorm $\\|...\\|$ if $\\| \\exp (At) \\| \\leq 1$ at all $t \\geq 0$. For each norm,\nthe dissipative operator form a closed convex cone. An operator $A$ is {\\em\nstable dissipative} if it belongs to the interior of this cone. The family of\nlinear operators $\\{A_\\alpha \\}$ is called {\\em simultaneously dissipative}, if\nthere exists a norm with respect to which all the operators are dissipative. We\nstudied general properties of such families. For example, let the family\n$\\{A_\\alpha \\}$ be finite and generate a nilpotent Lee algebra and let for each\n$A_\\alpha$ there exist a norm with respect to which it is dissipative. Then\n$\\{A_\\alpha \\}$ is simultaneously dissipative. Let the family $\\{A_\\alpha \\}$\nbe compact and generate solvable Lee algebra, and let the spectrum of each\noperator $A_\\alpha $ lie in the open left half-plane. Then $\\{A_\\alpha \\}$ is\nsimultaneously stable dissipative, i.e. there exists a norm with respect to\nwhich all $A_\\alpha $ are stable dissipative. We study the conditions of\nsimultaneous dissipativity of the matrices of rank one and discussed their\napplication to equations of {\\em mass action law} kinetics.",
"arxiv_id": "physics/9702021",
"authors": [
"A. N. Gorban",
"Yu. I. Shokin",
"V. I. Verbitskii"
],
"categories": [
"physics.comp-ph"
],
"license": "http://creativecommons.org/licenses/by/3.0/",
"title": "Simultaneously Dissipative Operators And The Infinitesimal Moore Effect In Interval Spaces",
"url": "https://arxiv.org/abs/physics/9702021"
},
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