Applications Of Semigroups Research Articles

Kahane’s Upper Density and Syndetic Sets in LCA Groups

Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.

Mean value theorem and semigroups of operators for interval-valued functions on time scales

In this paper, a new version of mean value theorem for interval-valued functions on time scales is established. Meantime, some basic concepts and results associated with semigroups of operators for interval-valued functions on time scales are presented. As an application of semigroups of operators, under certain conditions, we consider the initial value problem for interval-valued differential equations on time scales. Finally, two issues worthy of further discussion are presented.

SOME IMPORTANT APPLICATIONS OF SEMIGROUPS

This Paper deals with the some important applications of semigroups in general and regular semigroups in particular.The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In probability theory, semigroups are associated with Markov process. In section 2 we have seen different areas of applications of semigroups. We identified some Applications in biology, Partial Differential equation, Formal Languages etc whose semigroup structures are nothing but regular.

Semigroup applications everywhere.

Most dynamical systems arise from partial differential equations (PDEs) that can be represented as an abstract evolution equation on a suitable state space complemented by an initial or final condition. Thus, the system can be written as a Cauchy problem on an abstract function space with appropriate topological structures. To study the qualitative and quantitative properties of the solutions, the theory of one-parameter operator semigroups is a most powerful tool. This approach has been used by many authors and applied to quite different fields, e.g. ordinary and PDEs, nonlinear dynamical systems, control theory, functional differential and Volterra equations, mathematical physics, mathematical biology, stochastic processes. The present special issue of Philosophical Transactions includes papers on semigroups and their applications. This article is part of the theme issue 'Semigroup applications everywhere'.

Maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations.

This paper presents a survey of maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and their applications to stochastic evolution equations. This article is part of the theme issue 'Semigroup applications everywhere'.

Ornstein-Uhlenbeck semigroups in infinite dimension.

This is a survey paper about Ornstein-Uhlenbeck semigroups in infinite dimension and their generators. We start from the classical Ornstein-Uhlenbeck semigroup on Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for Ornstein-Uhlenbeck semigroups on Banach spaces. This article is part of the theme issue 'Semigroup applications everywhere'.

Semi-uniform stability of operator semigroups and energy decay of damped waves.

Only in the last 15 years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of C0-semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems. This article is part of the theme issue 'Semigroup applications everywhere'.

Feedback theory to the well-posedness of evolution equations.

In this paper, we cross the boundary between semigroup theory and general infinite-dimensional systems to bridge the isolated research activities in the two areas. Indeed, we first give a chronological history of the development of the semigroup approach for control theory. Second, we use the feedback theory to prove the well-posedness of a class of dynamic boundary problems. Third, the obtained results are applied to the well-posedness of neutral equations with non-autonomous past. We will also see that the strong connection between semigroup and control theories lies in feedback theory, where different kinds of perturbations appear. This article is part of the theme issue 'Semigroup applications everywhere'.

Form-perturbation theory for higher-order elliptic operators and systems by singular potentials.

We give a form-perturbation theory by singular potentials for scalar elliptic operators on [Formula: see text] of order 2m with Hölder continuous coefficients. The form-bounds are obtained from an L1 functional analytic approach which takes advantage of both the existence of m-gaussian kernel estimates and the holomorphy of the semigroup in [Formula: see text] We also explore the (local) Kato class potentials in terms of (local) weak compactness properties. Finally, we extend the results to elliptic systems and singular matrix potentials. This article is part of the theme issue 'Semigroup applications everywhere'.

On operator semigroups arising in the study of incompressible viscous fluid flows.

This article concentrates on various operator semigroups arising in the study of viscous and incompressible flows. Of particular concern are the classical Stokes semigroup, the hydrostatic Stokes semigroup, the Oldroyd as well as the Ericksen-Leslie semigroup. Besides their intrinsic interest, the properties of these semigroups play an important role in the investigation of the associated nonlinear equations. This article is part of the theme issue 'Semigroup applications everywhere'.

First-order evolution equations with dynamic boundary conditions.

In this paper, we introduce a general framework to study linear first-order evolution equations on a Banach space X with dynamic boundary conditions, that is with boundary conditions containing time derivatives. Our method is based on the existence of an abstract Dirichlet operator and yields finally to equivalent systems of two simpler independent equations. In particular, we are led to an abstract Cauchy problem governed by an abstract Dirichlet-to-Neumann operator on the boundary space ∂X. Our approach is illustrated by several examples and various generalizations are indicated. This article is part of the theme issue 'Semigroup applications everywhere'.

Semigroups for dynamical processes on metric graphs.

We present the operator semigroups approach to the first- and second-order dynamical systems taking place on metric graphs. We briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, we show two applications to biological models. This article is part of the theme issue 'Semigroup applications everywhere'.

Positive irreducible semigroups and their long-time behaviour†.

The notion Perron-Frobenius theory usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as t → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large t, while being absent for smaller times. This article is part of the theme issue 'Semigroup applications everywhere'.

Growth-fragmentation-coagulation equations with unbounded coagulation kernels.

In this paper, we prove the global in time solvability of the continuous growth-fragmentation-coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth-fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue 'Semigroup applications everywhere'.

On convergence and asymptotic behaviour of semigroups of operators.

The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed. This article is part of the theme issue 'Semigroup applications everywhere'.

The Ornstein-Uhlenbeck semigroup in finite dimension.

We gather the main known results concerning the non-degenerate Ornstein-Uhlenbeck semigroup in finite dimension. This article is part of the theme issue 'Semigroup applications everywhere'.

Towards a Koopman theory for dynamical systems on completely regular spaces

The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article, we look at dynamics given by continuous semiflows on completely regular spaces, which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato's equality). Finally, we demonstrate-using the example of attractors-how this Koopman approach can be used to examine properties of dynamical systems. This article is part of the theme issue 'Semigroup applications everywhere'.

On the identification of finite non-group semigroups of a given order

Identifying finite non-group semigroups for every positive integer is significant because of many applications of such semigroups are functional in various branches of sciences such as computer science, mathematics and finite machines. The finite non-commutative monoids as a type of such semigroups were identified in 2014, for every positive integer. We here attempt to identify the finite commutative monoids and finite commutative non-monoids of a given integer n=p^alpha q^beta, for every integers alpha , beta ge 2 and different primes p and q. In order to recognize the commutative monoids, we present a class of 2-generated monoids of a given order, and for the commutative non-monoids of order n=p^alpha q^beta, we give the minimal generating set. Moreover, we prove that there are exactly (p^{alpha }-2)(q^{beta }-2) non-isomorphic commutative non-monoids of order p^alpha q^beta. The identification of non-group semigroups for the integers p^{2alpha } and 2p^alpha is achieved. The automorphism groups of these groups are specified as well. As a result of this study, an interesting difference between the abelian groups and the commutative semigroups of order p^2 is presented.

Applications of amenable semigroups in operator theory

The paper deals with continuous homomorphisms $S \ni s \mapsto T_s \in L(E)$ of amenable semigroups $S$ into the algebra $L(E)$ of all bounded linear operators on a Banach space $E$. For a closed linear subspace $F$ of $E$, sufficient conditions are given under which there exists a projection $P \in L(E)$ onto $F$ that commutes with all $T_s$. And when $E$ is a Hilbert space, sufficient conditions are given for the existence of an invertible operator $R \in L(E)$ such that all $R T_s R^{-1}$ are isometries. Also certain results on extending intertwining operators, renorming as well as on operators on hereditarily indecomposable Banach spaces are offered.

Applications of stochastic semigroups to cell cycle models

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.