Opial Type Inequalities for Semigroups

Various L_p form Opial type inequalities are given for semigroups with applications.     

Proportionally Modular Diophantine Inequalities and Full Semigroups

A proportionally modular numerical semigroup is the set of nonnegative integer solutions to a Diophantine inequality of the type ax mod b ≤ cx. We give a new presentation for these semigroups and we relate them with a type of affine full semigroups. Next, we describe explicitly the minimal generating system for the affine full semigroups we are considering. As a consequence, we obtain generating systems for proportionally modular numerical semigroups and we exhibit several families of these semigroups in terms of their generators. Finally, we use the concept of fundamental gap to study when a proportionally modular numerical semigroup is symmetric and we propose some open problems.     

Dimension-Independent Harnack Inequalities for Subordinated Semigroups

Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power α satisfies a dimension-free Harnack inequality provided \(\alpha \in \left(\frac{1}{2},1 \right)\), and it satisfies the log-Harnack inequality for all α?∈?(0, 1). Some infinite-dimensional examples are also presented.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.     

Functional Inequalities for the Decay of Sub-Markov Semigroups

A general functional inequality is introduced to describe various decays of semigroups. Our main result generalizes the classical one on the equivalence of the L ²-exponential decay of a sub-Markov semigroup and the Poincaré inequality for the associated Dirichlet form. Conditions for the general inequality to hold are presented. The corresponding isoperimetric inequality is studied in the context of diffusion and jump processes. In particular, Cheeger's inequality for the principal eigenvalue is generalized. Moreover, our results are illustrated by examples of diffusion and jump processes.     

Unitary Numerical Semigroups and Perfect Bricks

This article completes a previous investigation of balanced and unitary numerical semigroups. The main result establishes the equivalence of unitary numerical semigroups and perfect 2 × 2 bricks.     

On γ-hyperelliptic Numerical Semigroups

This work was realized while the author was ICTP (Trieste - Italy) with a grant from the International Atomic Energy Agency and UNESCO.     

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {a^p, a^p?1b, a^p?2b²,…, b^p} to allow the p copies of a (resp. b) to all be different. We call the sequence {a₁a₂a₃… a_p, b₁a₂a₃…a_p, b₁b₂a₃…a_p,…, b₁b₂b₃…b_p} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.     

Apery Sets of Numerical Semigroups #

Let S be a numerical semigroup. We examine a particular subset of the Apery set of S and establish a correspondence between this subset and the holes of S . This correspondence allows us to establish conditions for S to be almost symmetric.     

一些半群在加权Morrey空间上的振荡与变差不等式

研究了一些半群在加权Morrey空间上的振荡与变差不等式.通过向量值理论,建立了相关于Laplacian算子及Hermite算子的热半群与Poisson半群的振荡算子和变差算子在加权Morrey空间上的有界性.     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

Irreducible Numerical Semigroups Having Toms Decomposition

In this article we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.     

Representation of Numerical Semigroups by Dyck Paths

We introduce square diagrams that represent numerical semigroups and we obtain an injection from the set of numerical semigroups into the set of Dyck paths.     

Weak Poincaré Inequalities, Decay of Markov Semigroups and Concentration of Measures

For a reasonable class of weak Poincaré inequalities, the decay of the corresponding Markov semigroups obtained earlier by Röckner and the first named author is improved by removing an extra L ²–norm. Next, a concentration estimate of the reference measure is presented for the weak Poincaré inequality, which is sharp as illustrated by some examples of one–dimensional diffusion processes.     

The Isometric Representation Theory of Numerical Semigroups

We study representations of numerical semigroups ∑ by isometries on Hilbert space with commuting range projections. Our main theorem says that each such representation is unitarily equivalent to the direct sum of a representation by unitaries and a finite number of multiples of particular concrete representations by isometries. We use our main theorem to identify the faithful representations of the C*-algebra C*(∑) generated by a universal isometric representation with commuting range projections, and also prove a structure theorem for C*(∑). This research was supported by the Australian Research Council.     

A Note about Parabolic Systems and Analytic Semigroups

We investigate the question whether certain parabolic systems in the sense of Petrovskii fulfill the resolvent estimate required for the generation of an analytic semigroup and apply the result to a problem concerning the diffusion of gases.