Generator Of Semigroup Research Articles

On perturbations of dynamical semigroups defined by covariant completely positive measures on the semi-axis

We consider perturbations of dynamical semigroups on the algebra of all bounded operators in a Hilbert space generated by covariant completely positive measures on the semi-axis. The construction is based upon unbounded linear perturbations of generators of the preadjoint semigroups on the space of nuclear operators. As an application we construct a perturbation of the semigroup of non-unital *-endomorphisms on the algebra of canonical anticommutation relations resulting in the flow of shifts.

M-ideals and its generators of ternary semigroups

The purpose of this paper is to introduce some new classes of m-bi ideals and m-quasi ideals in ternary semigroups and give some characterizations in terms of bi ideals and quasi ideals. Let U1 be the m-bi ideal of T and U2 be the m-bi ideal of U1 such that U3 2 = U2, shown that U2 is a m-bi ideal of T. Let U1, U2 and U3 be the three ternary subsemigroups of T, it has been shown that U1U2U3 is a t-bi ideal if at least one of U1, U2, U3 is l-RI or m-LATI or n-LI of T. Also we discuss m-bi ideal generated by B is < B > sub>m= B ∪ P finite

STOCHASTIC DIFFERENTIAL GAMES IN DISTRIBUTED SYSTEMS WITH DELAY

We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.

Strong stabilization of small water waves in a pool

This paper is about the strong stabilization of small amplitude gravity water waves in a vertical rectangle. The control imposes the horizontal acceleration of the water along one vertical boundary segment, as a multiple of a scalar input function u, times a function h of the position along the active boundary. The state z of the system is a vector with two components: one is the water level ζ along the top boundary and the other is its time derivative ζ. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u = Fz leads to a strongly stable closed-loop system. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1 + t) − 1/6. Our approach uses careful estimates on certain partial Dirichlet to Neumann and Neumann to Neumann operators associated to the rectangular domain, as well as non-uniform stabilization results due to Chill, Paunonen, Seifert, Stahn and Tomilov (2019).

SEMIGROUPS OF BINARY OPERATIONS AND MAGMA-BASED CRYPTOGRAPHY

In this article, algebras of binary operations as a special case of finitary homogeneous relations algebrasare investigated. The tools of our study are based on unary and associative binary operations acting on theset of ternary relations. These operations are generated by the converse operation and the left-composition ofbinary relations. Using these tools, we are going to define special kinds of ternary relations that correspondto functions, injections, right- and left-total binary relations. Then we obtain criteria for these properties interms of ordered semigroups. Note, that there is an embedding of the semigroup of quasigroups operationsin the semigroup of magmas operation and further in the semigroup of ternary relations. This is similar toembedding the semigroup of bijections in the semigroup of functions and then in the semigroup of binaryrelations. Taking a binary operation as the generator of a cyclic semigroup, we can apply an exponentialsquaring method for the fast computation of its positive integer powers. Given that this is the main methodof public key cryptography, we are adapting the Diffie-Hellman-Merkle key exchange algorithm for magmasas a result.

Complete positivity and self-adjointness

We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order structure and extreme points in some previously studied cases.

Recurrence for measurable semigroup actions

<p style='text-indent:20px;'>We study qualitative properties of the set of recurrent points of finitely generated free semigroups of measurable maps. In the case of a single generator the classical Poincare recurrence theorem shows that these properties are closely related to the presence of an invariant measure. Curious, but otherwise it turns out to be possible that almost all points are recurrent, while there is an wandering set of positive (non-invariant) measure. For a general semigroup the assumption about the common invariant measure for all generators looks somewhat unnatural (despite being widely used). Instead we give abstract conditions (of conservativity type) for this problem and propose a weaker version of the recurrent property. Technically, the problem is reduced to the analysis of the recurrence of a specially constructed Markov process. Questions of inheritance of the recurrence property from the semigroup generators to the entire semigroup and vice versa are studied in detail and we demonstrate that this inheritance might be rather unexpected.

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

Inverse source problem for the abstract fractional differential equation

Abstract In a Banach space, the inverse source problem for a fractional differential equation with Caputo–Dzhrbashyan derivative is considered. The initial and observation conditions are given by elements from D ⁢ ( A ) {D(A)} , and the operator function on the right side is sufficiently smooth. Two types of the observation operator are considered: integral and at the final point. Under the assumptions that operator A is a generator of positive and compact semigroup the uniqueness, existence and stability of the solution are proved.

Fractional Abstract Differential Equations and Applications

Boundary value problems for fractional elliptic equations with parameter in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for elliptic equations in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operators generated by these equations are positive and also are generators of the analytic semigroups. Moreover, maximal regularity properties of the fractional abstract parabolic equation are established. As an application, the parameter-dependent anisotropic fractional differential equations and the system of fractional differential equations are studied.

Infinitesimal generators of semigroups with prescribed boundary fixed points

We study infinitesimal generators of one-parameter semigroups in the unit disk $$\mathbb {D}$$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein–Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning. We also give a new proof of the classical Cowen–Pommerenke inequalities for univalent self-maps of $$\mathbb {D}$$ .

Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operators Iγ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators b,Iγ generated by b and Iγ are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for Iγ and b,Iγ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator Mγ and generalized fractional integrals ℒ−γ/2 in the context of weighted amalgam spaces, where 0&lt;γ&lt;n and L is the infinitesimal generator of an analytic semigroup on L2Rn with Gaussian kernel bounds.

Separable Differential Operators with Parameters

In this paper, we study boundary value problems for parameter-dependent elliptic differential-operator equations with variable coefficients in smooth domains. Uniform regularity properties and Fredholmness of this problem are obtained in vector-valued $$ {\text{L}}_{p} $$ -spaces. We prove that the corresponding differential operator is positive and is a generator of an analytic semigroup. Then, via maximal regularity properties of the linear problem, the existence and uniqueness of the solution to the nonlinear elliptic problem is obtained. As an application, we establish maximal regularity properties of the Cauchy problem for abstract parabolic equations, Wentzell–Robin-type mixed problems for parabolic equations, and anisotropic elliptic equations with small parameters.

Variants of epigroups and primary conjugacy

In a semigroup S with fixed one can construct a new semigroup called a variant by defining Elements are primarily conjugate if there exist such that This coincides with the usual conjugacy in groups, but is not transitive in general semigroups. Araújo et al. proved that transitivity holds in a variety of epigroups containing all completely regular semigroups and their variants, and asked if transitivity holds for all variants of semigroups in We answer this affirmatively as part of a study of varieties and variants of epigroups.

Construction of dynamical semigroups by a functional regularization à la Kato

A functional version of the Kato one-parametric regularisation for the construction of a dynamical semigroup generator of a relative bound one perturbation is introduced. It does not require that the minus generator of the unperturbed semigroup is a positivity preserving operator. The regularisation is illustrated by an example of a boson-number cut-off regularisation.

Scalar Poincaré implies matrix Poincaré

We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian d×d matrix valued functions, with the same Poincaré constant. This generalizes recent results [ABY19, Kat20] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.

Nonlocal Fractional Differential Equations and Applications

The regularity properties of nonlocal fractional differential equations in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional differential equations are studied.

Schur Complement and Continuous Spectrum in a Kinetic Plasma Model

The goal of this paper is spectral analysis of an evolutionary semigroup generator describing the dynamics of a rarefied two-component plasma subjected to a self-consistent electromagnetic field. For the problem in question, the spectrum is given in terms of the dispersion relationship and an effective approach to the calculation of the instability index is developed.

Stabilizability properties of a linearized water waves system

We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function u, times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ζ̇. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u=Fz renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1+t)−16. Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill et al. (2019).

Radii Problems for Starlike Functions and Semigroup Generators

In this paper, we discover new connections between topics in classical geometric function theory and complex dynamics. In particular, we study some classes of starlike functions and their embeddings in the classes of semigroup generators. In addition, we find explicit formulas for the radii of starlikeness for those classes and uniform rates of convergence of semigroups to their Denjoy–Wolff points.