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2,292
THE COMPLEX INVERSION FORMULA REVISITED
"... Abstract. We give a simplified proof of the complex inversion formula for semigroups and — more generally — solution families for scalar-type Volterra equations, including the stronger versions on UMD spaces. Our approach is based on (elementary) Fourier analysis. 1. ..."
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Abstract. We give a simplified proof of the complex inversion formula for semigroups and — more generally — solution families for scalar-type Volterra equations, including the stronger versions on UMD spaces. Our approach is based on (elementary) Fourier analysis. 1.
Unifying Evolutionary Dynamics
, 2002
"... Darwinian evolution is based on three fundamental principles, reproduction, mutation and selection, which describe how populations change over time and how new forms evolve out of old ones. There are numerous mathematical descriptions of the resulting evolutionary dynamics. In this paper, we show th ..."
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Cited by 315 (33 self)
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that apparently very different formulations are part of a single unified framework. At the center of this framework is the equivalence between the replicator–mutator equation and the Price equation. From these equations, we obtain as special cases adaptive dynamics, evolutionary game dynamics, the Lotka-Volterra
Matrix valued polynomials generated by the scalar-type Rodrigues’ formulas
"... Abstract. The properties of matrix valued polynomials generated by the scalartype Rodrigues ’ formulas are analyzed. A general representation of these polynomials is found in terms of products of simple differential operators. The recurrence relations, leading coefficients, completeness are establis ..."
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Cited by 3 (2 self)
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of scalar type is proved for Q = x and Q = x 2 −1 in dimension two, and for any dimension under genericity assumptions. Commutative classes of quasi-orthogonal polynomials are found, which satisfy all the properties usually associated to orthogonal polynomials. 1.
Model-checking algorithms for continuous-time Markov chains
- IEEE TRANSACTIONS ON SOFTWARE ENGINEERING
, 2003
"... Continuous-time Markov chains (CTMCs) have been widely used to determine system performance and dependability characteristics. Their analysis most often concerns the computation of steady-state and transient-state probabilities. This paper introduces a branching temporal logic for expressing real-t ..."
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Cited by 235 (48 self)
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steady-state probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steady-state operator) and a Volterra integral equation system (for time-bounded until). We then show that the problem of model-checking timebounded
On Miura Transformations and Volterra-Type Equations Associated with the Adler–Bobenko–Suris Equations
- SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2008
"... We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund ..."
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Cited by 15 (8 self)
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We construct Miura transformations mapping the scalar spectral problems of the integrable lattice equations belonging to the Adler–Bobenko–Suris (ABS) list into the discrete Schrödinger spectral problem associated with Volterra-type equations. We show that the ABS equations correspond to Bäcklund
NONTRIVIAL PERIODIC SOLUTIONS OF SOME VOLTERRA INTEGRAL EQUATIONS
"... I. Introductory Remarks. My main purpose in this paper is to prove a bifurcation theorem for nontrivial periodic solutions of a general system of Volterra integral equations. The motivation for considering this problem can be found in models which arise in population dynamics, epidemiology and econo ..."
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Cited by 1 (0 self)
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I. Introductory Remarks. My main purpose in this paper is to prove a bifurcation theorem for nontrivial periodic solutions of a general system of Volterra integral equations. The motivation for considering this problem can be found in models which arise in population dynamics, epidemiology
Volterra Integral Equations
, 2000
"... We discuss the properties and numerical treatment of various types of Volterra and Abel-Volterra integral and integro-differential equations. ..."
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We discuss the properties and numerical treatment of various types of Volterra and Abel-Volterra integral and integro-differential equations.
WITH APPLICATIONS TO PARABOLIC VOLTERRA EQUATIONS
"... An integration calculus for stochastic processes with stationary increments and spectral density with applications to parabolic Volterra equations ..."
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An integration calculus for stochastic processes with stationary increments and spectral density with applications to parabolic Volterra equations
STABILIZATION OF VOLTERRA EQUATIONS BY NOISE
"... The paper studies the stability of an autonomous convolution Itô-Volterra equation where the linear diffusion term depends on the current value of the state only, and the memory of the past fades exponentially fast. It is shown that the presence of noise can stabilize an equilibrium solution which ..."
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The paper studies the stability of an autonomous convolution Itô-Volterra equation where the linear diffusion term depends on the current value of the state only, and the memory of the past fades exponentially fast. It is shown that the presence of noise can stabilize an equilibrium solution which
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