Nonlinear semigroup approach to transport equations with delayed neutrons

This paper deal with a nonlinear transport equation with delayed neutron and general boundary conditions. We establish, via the nonlinear semigroups approach, the existence and uniqueness of the mild solution, weak solution, strong solution and local solution on L~p-spaces(1 ≤ p +∞). Local and non local evolution problems are discussed.     

Two-time scales in spatially structured models of population dynamics: A semigroup approach

The aim of this work is to provide a unified approach to the treatment of a class of spatially structured population dynamics models whose evolution processes occur at two different time scales. In the setting of the C₀-semigroup theory, we will consider a general formulation of some semilinear evolution problems defined on a Banach space in which the two-time scales are represented by a parameter ε>0 small enough, that mathematically gives rise to a singular perturbation problem. Applying the so-called aggregation of variables method, a simplified model called the aggregated model is constructed. A nontrivial mathematical task consists of comparing the asymptotic behaviour of solutions to both problems when ε→₀₊, under the assumption that the aggregated model has a compact attractor. Applications of the method to a class of two-time reaction-diffusion models of spatially structured population dynamics and to models with discrete spatial structure are given.     

A semigroup approach to age-dependent population dynamics with time delay

We study the McKendrick type models of population dynamics with instantaneous time delay in the birth rate. The models involve first order partial differential equations with nonlocal and delayed boundary conditions. We show that a semigroup can be associated

to it and identify the infinistimal generator. Its spectral properties are analyzed yielding large time behaviour. An interesting result is that if the total population converges to an equilibrium it will converge to it in an oscillatory fashion. Further, we consider a logistic ara age-dependent model with delay. A nonlinear semigroup is constructed to describe the evolution of the population. Existence and uniqueness of the nonlinear equation are proved.     

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions

This work deals with a mathematical model of an age‐cycle length structured cell population. Each cell is distinguished by its age and its cycle length. The cellular mitosis is mathematically described by non‐compact boundary conditions. We prove then that this mathematical model is governed by a positive C₀‐semigroup. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear boundary conditions derived by singular pertubation in age structured population dynamics model

In this article, we derive Ricker’s [22, 23] type nonlinear boundary condition for an age structured population dynamic model by using a singular perturbation. The question addressed in this paper is the convergence of the singularly perturbed system. We first obtain a finite time convergence for a fixed initial distribution. Then we focus on the convergence uniformly of the singularly perturbed system with respect to the initial distribution in bounded sets.     

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

This work is a natural continuation of two other works in which a mathematical model has been studied. This model is based on age‐cycle length structured cell population. The cellular mitosis is mathematically described by a non‐compact boundary condition. We investigate the asymptotic behavior of the generated semigroup, and we prove that the cell population possesses Asynchronous Growth Property. Copyright © 2015 John Wiley & Sons, Ltd.     

On a mathematical model of age–cycle length structured cell population with non‐compact boundary conditions (II)

This work is a natural continuation of an earlier one in which a mathematical model has been studied. This model is based on an age–cycle length structured cell population. The cellular mitosis is mathematically described by a non‐compact boundary condition. We investigate the spectral properties of the generated semigroup, and we give an explicit estimation of its type. Copyright © 2015 John Wiley & Sons, Ltd.     

Aggregation in age and space structured population models: an asymptotic analysis approach

In this paper we describe how techniques of asymptotic analysis can be used in a systematic way to perform ‘aggregation’ of variables, based on a separation of different time scales, in a population model with age and space structure. The main result of the paper is proving the convergence of the formal asymptotic expansion to the solution of the original equation. This result improves and clarifies earlier results of Arino et al. (SIAM J Appl Math 60(2):408–436, 1999), Auger et al. (Structured population models in biology and epidemiology. Springer Verlag, Berlin, 2008), Lisi and Totaro (Math Biosci 196(2):153–186, 2005).     

Nonlinear semigroup approach to a class of evolution equations arising from percolation in sandbanks

Summary In the framework of the theory of nonlinear contraction semigroups we study a class of parabolic variational inequalities arising from a model of percolation in gently sloping beaches. Some existence and uniqueness results are proven and the asymptotic behavior of the solutions is investigated.This work was done while the first author was visiting the «Laboratoire d'Analyse Numérique, Université P. et M. Curie, Paris», in 1985–86.     

A semigroup approach to American options

In this paper we give new conditions for the existence and uniqueness of the value of an American option by using semigroup theory. Some open questions given by Gozzi et al. and Kholodnyi are solved and improved, specifically, that the Black-Scholes operator is degenerate and that it is sectorial. Moreover, our results are extended to the original American problem. Finally, we obtain an integral equation which gives the value of an American option, providing an alternative method to calculate the value of American options.     

A semigroup approach to fractional powers

We say that A has fractional powers {A _t } _t≥0 if there exists a nondegenerate C-regularized semigroup {W(t)} _t≥0 such that A=C ⁻¹ W(1); then A _t ≡C ⁻¹ W(t). We show that this generalizes the usual definition of fractional powers for nonnegative operators, and enables many operators with spectrum containing the entire unit disc to have fractional powers. Our definition gives clear, simple proofs of the basic properties of fractional powers. We show that, for nonnegative operators, the fractional powers with the property that, if A is of type θ, then A _t is of type t θ, whenever t θ<π, are unique. More generally, for injective G∈B(X) commuting with A, we show that an operator A of G-regularized type θ has a unique family of fractional powers with the property that A _t is of G-regularized type t θ whenever t θ<π. This leads to a construction of fractional powers of operators with polynomially bounded resolvent outside of an appropriate sector. We show that an operator is of regularized type if and only if it has exponentially bounded regularized imaginary powers. This work was done while the second author was visiting Ohio University, with funding from Universitat de València. He would like to thank Ohio University and Professor deLaubenfels for their hospitality and support.     

Two-sex age structured dynamics in a fixed sex-ratio population

An age structured model is considered in order to analyze the growth of a two sex population with a fixed age-specific sex ratio. The model is intended to give an insight into the dynamics of a population where the mating process takes place at random and the proportion between females and males is not influenced by environmental or social factors, but only depends on a differential mortality or on a possible transition from one sex to the other (e.g. in sequential hermaphrodite species).First a basic model, asymptotically linear, is considered and its ergodicity is studied. Survival thresholds and their dependence on the sex ratio are then analyzed, in connection with the optimal sex ratio to guarantee survival.A further model including logistic effect is also considered and discussed in connection with existence and stability of steady states.     

Quiver approach to some ordered semigroup algebras

The aim of this paper is to research the relation among generalized path algebras, pseudo-admissible ordered semigroup algebras and path algebras over algebras. First, the Gabriel theorem for contract pseudo-admissible ordered semigroup algebras is given. Second, a family of pseudo-admissible ordered semigroups is mixed together via the method of generalized path algebras to construct a new pseudo-admissible ordered semigroup as the extended version. Finally, we characterize the path algebra over an algebra in two ways through normal and non-normal generalized path algebras, respectively, over a field.     

The asymptotic behavior of a structured cell population

This paper deals with a mathematical model of an age-cycle structured proliferating cell population. Individual cells are distinguished by age and cell cycle length. We consider a general biological rule corresponding to a non-compact boundary condition. We give the asymptotic behavior of the generated semigroup in the uniform topology.     

Oscillations in a molecular structured cell population model

We consider a nonlinear cyclin content structured model of a cell population divided into proliferative and quiescent cells. We show, for particular values of the parameters, the existence of solutions that do not depend on the cyclin content. We make numerical simulations for the general case obtaining, for some values of the parameters convergence to the steady state, but for others oscillations of the population.     

A groupoid approach to discrete inverse semigroup algebras

Let K be a commutative ring with unit and S an inverse semigroup. We show that the semigroup algebra KS can be described as a convolution algebra of functions on the universal étale groupoid associated to S by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal C^∗-algebra. It provides a convenient topological framework for understanding the structure of KS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.Using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup S that can be induced from associated groups as precisely those satisfying a certain “finiteness condition.” This “finiteness condition” is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.     

Three-page approach to knot theory. Universal semigroup

An explicit construction of a finitely presented semigroup whose central elements are in a one-to-one correspondence with the isotopy classes of unoriented links in the three-space is given, together with a finite presentation for the group of invertible elements of the semigroup. The group is presented by two generators and three relations. The commutator subgroup of the group is isomorphic to the braid group of infinite index. A similar construction is given for band-links. The kauffman theorems on the existence of polynomial band-link invariants satisfying some skein-relations are stated algebraically. This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090. Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 29–40, January–March, 2000. Translated by I. a. Dynnikov     

Numerical analysis to discontinuous Galerkin methods for the age structured population model of marine invertebrates

In this article we consider the age structured population growth model of marine invertebrates. The problem is a nonlinear coupled system of the age‐density distribution of sessile adults and the abundance of larvae. We propose the semidiscrete and fully‐discrete discontinuous Galerkin schemes to the nonlinear problem. The DG method is well suited to approximate the local behavior of the problem and to easily take the locally refined meshes with hanging nodes adaptively. The simple communication pattern between elements makes the DG method ideal for parallel computation. The global existence of the approximation solution is proved for the nonlinear approximation system by using the broken Sobolev spaces and the Schauder's fixed point theorem, and error estimates are obtained for both the semidiscrete scheme and the fully‐discrete scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009