Generator Of Semigroup Research Articles

Functional calculus for generators of symmetric contraction semigroups

We prove that every generator of a symmetric contraction semigroup on a $\sigma$-finite measure space admits, for $1<p<\infty$, a H\"ormander-type holomorphic functional calculus on $L^p$ in the sector of angle $\phi^*_p=\arcsin|1-2/p|$. The obtained angle is optimal.

An Entropic Gradient Structure for Lindblad Equations and Couplings of Quantum Systems to Macroscopic Models

We show that all Lindblad operators (i.e. generators of quantum semigroups) on a finite-dimensional Hilbert space satisfying the detailed balance condition with respect to the thermal equilibrium state can be written as a gradient system with respect to the relative entropy. We discuss also thermodynamically consistent couplings to macroscopic systems, either as damped Hamiltonian systems with constant temperature or as GENERIC systems. In particular we discuss the coupling of a quantum dot coupled to macroscopic charge carriers.

Equational theories of unstable involution semigroups

It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.

Asymptotic Behavior in a Cell Proliferation Model with Unequal Division and Random Transition using Translation Semigroups

Objectives: We provide a theoretical framework for the mathematical analysis of a cell cycle model described by a delay integral equation to get properties on the asymptotic behavior of the solutions. Methods: We relate the model to the class of translation semigroups of operators that are associated with a core operator φ and are solutions of equations of the type m (t) = Φ(m t ). Then by using the theory developed for such class of semigroups we establish results on the existence, uniqueness, positivity, compactness and spectral properties of the solution semigroup in order to conclude the asynchronous exponential growth (AEG) property for the model. Findings: The framework yields an innovative analysis method for the model where only conditions on the parameters of its associated core operator are considered. It allows better control of the parameters for getting the AEG property and the derivation in an automatic way of characterizations of associated generator, spectral properties and AEG property only in terms of the core operator φ Indeed the Malthusian parameter λ 0 is characterized as the only solution of the equation )r(Φ ˜ λ )= 1 where(Φ ˜ λ ) :=Φ( e λ ⊗), coincides with the spectral bound of the generator of the solution semigroup and is a dominant eigen value of this generator. Application/ Improvements: The provided framework will pave the way for the study of other aspects such as oscillations, bifurcation and chaos to get better insights of the dynamics of the model solutions.

A note on generators of the endomorphism semigroup of an infinite countable chain

In this note, we consider the semigroup [Formula: see text] of all order endomorphisms of an infinite chain [Formula: see text] and the subset [Formula: see text] of [Formula: see text] of all transformations [Formula: see text] such that [Formula: see text]. For an infinite countable chain [Formula: see text], we give a necessary and sufficient condition on [Formula: see text] for [Formula: see text] to hold. We also present a sufficient condition on [Formula: see text] for [Formula: see text] to hold, for an arbitrary infinite chain [Formula: see text].

Resolvent representations for functions of sectorial operators

We obtain integral representations for the resolvent of ψ(A), where ψ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and A is a sectorial operator on a Banach space. As a corollary, for a wide class of functions ψ, we show that the operator −ψ(A) generates a sectorially bounded holomorphic C0-semigroup on a Banach space whenever −A does, and the sectorial angle of A is preserved. When ψ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for A can be described, at least on Hilbert spaces, in terms of the existence of a bounded H∞-calculus for A. As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt operators.

Functional Analytic (Ir-)Regularity Properties of SABR-type Processes

The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller-properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach - and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.

Microspectral analysis of quasinilpotent operators

We develop a microspectral theory for quasinilpotent linear operators Q (i.e., those with σ(Q) = {0}) in a Banach space. When such Q is not compact, normal, or nilpotent, the classical spectral theory gives little information, and a somewhat deeper structure can be recovered from microspectral sets in C. Such sets describe, e.g., semigroup generation, resolvent properties, power boundedness as well as Tauberian properties associated to zQ for z ∈ C.

Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations

Bernstein functions of several semigroup generators on Banach spaces under bounded perturbations

Specification of Fractional Poincare’ Inequalities for the Sequence Measures Generalization

The Fractional Poincare’ inequalities in Rⁿ are endowed with a fairly general sequence measure. We show a control of L² norm by a non–Local quantity. The assumption on the sequence measure is that it satisfies the classical Poincare’ inequality, with general results. We also verify quantity of the tightness at infinity provided by the control on the fractional derivative in terms of a sequence of a weight growing at infinity. The illustration goes to the generator of the Ornstein-Uhlenbeck semi group and some estimates of its powers.

Inverses of semigroup generators: a survey and remarks

Inverses of semigroup generators: a survey and remarks

Semigroup Actions on Sets and the Burnside Ring

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gr\"othendieck group.

Exact controllability of a class of nonlinear distributed parameter systems using back-and-forth iterations

ABSTRACTWe investigate the exact controllability of a nonlinear plant described by the equation , where t ≥ 0. Here A is the infinitesimal generator of a strongly continuous group on a Hilbert space X, B and , defined on Hilbert spaces U and , respectively, are admissible control operators for and the function is continuous in t and Lipschitz in x, with Lipschitz constant independent of t. Thus, B and can be unbounded as operators from U and to X, in which case the nonlinear term in the plant is in general not Lipschitz in x. We assume that there exist linear operators F and Fb such that the triples and are regular and A + BFΛ and −A + BFb, Λ are generators of operator semigroups and on X such that decays to zero exponentially. We prove that if is sufficiently small, then the nonlinear plant is exactly controllable in some time τ > 0. Our proof is constructive, i.e. given an initial state x0 ∈ X and a final state xτ ∈ X, we propose an approach for constructing a control signal u of class L2 for the nonlinear plant which ensures that if x(0) = x0, then x(τ) = xτ. We illustrate our approach using two examples: a sine-Gordon equation and a nonlinear wave equation. Our main result can be regarded as an extension of Russell's principle on exact controllability to a class of nonlinear plants.

On exponent matrices of tiled orders

We describe two methods to determine all generators of the additive semigroup of the non-negative exponent [Formula: see text]-matrices, and illustrate them finding all generating [Formula: see text]-exponent matrices. The generating [Formula: see text]-exponent matrices are found using a computer. We consider the Hasse diagram [Formula: see text] of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix [Formula: see text] there exists an oriented path in [Formula: see text] starting in some matrix unit and ending in [Formula: see text] which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix [Formula: see text] there exists a chain of non-negative exponent matrices [Formula: see text] such that [Formula: see text] is a [Formula: see text]-matrix, and each [Formula: see text] is obtained from [Formula: see text] by adding a [Formula: see text]-matrix.

Stabilization of the wave equation with Ventcel boundary condition

We consider a damped wave equation on a open subset of Rn or a smooth Riemannian manifold with boundary, with Ventcel boundary conditions, with a linear damping, acting either in the interior or at the boundary. This equation is a model for a vibrating structure with a layer with higher rigidity of thickness δ>0. By means of a proper Carleman estimate for second-order elliptic operators near the boundary, we derive a resolvent estimate for the wave semigroup generator along the imaginary axis, which in turn yields the logarithmic decay rate of the energy. This stabilization result is obtained uniformly in δ.

Contact points and fractional singularities for semigroups of holomorphic self-maps of the unit disc

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit disc. We introduce “regular” fractional singularities and characterize them in terms of the behavior of the associated semigroups and Kœnigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Kœnigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc.

A class of quasicontractive semigroups acting on Hardy and weighted Hardy spaces

This paper studies semigroups of operators on Hardy and Dirichlet spaces whose generators are differential operators of order greater than one. The theory of forms is used to provide conditions for the generation of semigroups by second order differential operators. Finally, a class of more general weighted Hardy spaces is considered and necessary and sufficient conditions are given for an operator of the form f mapsto Gf^{(n_0)} (for holomorphic G and arbitrary n_0) to generate a semigroup of quasicontractions.

Group key management based on semigroup actions

In this work, we provide a suite of protocols for group key management based on general semigroup actions. Construction of the key is made in a distributed and collaborative way. Examples are provided that may in some cases enhance the security level and communication overheads of previous existing protocols. Security against passive attacks is considered and depends on the hardness of the semigroup action problem in any particular scenario.

Hypercyclic abelian semigroups of matrices on [formula omitted

In this paper, we bring together results about the existence of a somewhere dense (resp. dense) orbit and the minimal number of generators for abelian semigroups of matrices on Rn. We solve the problem of determining the minimal number of matrices in normal form over R which form a hypercyclic abelian semigroup on Rn. In particular, we show that no abelian semigroup generated by [n+12] matrices on Rn can be hypercyclic ([] denotes the integer part).

NORMAL EIGENVALUES IN EVOLUTIONARY PROCESS

Firstly, we establish spectral mapping theorems for normal eigenvalues (due to Browder) of a -semigroup and its generator. Secondly, we discuss relationships between normal eigenvalues of the compact monodromy operator and the generator of the evolution semigroup on associated with the -periodic evolutionary process on a Banach space X, where stands for the space of all -periodic continuous functions mapping to X.