A Semigroup Approach to Automaticity

We present an abstract, yet rather natural concept of automaticity which is based on semigroup actions. The standard notion of k-automaticity as well as many, if not all of its generalizations are recovered by specifying appropriate semigroup actions on $ \mathbb{N}_0 $ or $ \mathbb{N}^N_0 $ or other similar objects—e.g., the standard notion of k-automaticity is recovered by considering a certain "natural" action of the free semigroup with k generators on $ \mathbb{N}_0 $. We show that the main results characterizing automatic sequences (Cobham, 1972, Math. Systems Theory 6) hold almost verbatim in the general context, too.AMS Subject Classification: 11B85, 68Q45, 16W22, 18DXX.     

A Semigroup Approach to Harmonic Maps

We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (p_t)_t>0. Typical examples come from elliptic or parabolic second-order operators on Rⁿ, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (P_t^*)_t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (p_t)_t can be extended to the nonlinear semigroup (P_t^*)_t, for instance, L_p–L_q smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L₁-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L₁-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer     

A Semigroup Approach to Queueing Systems

We prove asymptotic stability of the solutions of equations describing a simple queueing system consisting of two machines separated by a finite storage buffer. Following an approach by G. Gupur, we apply the theory of C₀-semigroups and spectral theory of positive operators.     

A Semigroup Approach to Boundary Feedback Systems

In this paper we convert a (linear abstract) initial boundary value problem into an abstract Cauchy problem on some product space and use semigroup methods to solve it. In particular, we apply spectral theory in order to discuss stability under boundary feedback.     

A Semigroup Approach to Dynamic Boundary Value Problems

Dynamic boundary value problems occur in many situations and have been studied intensively with various techniques and goals. In this paper we present an abstract treatment of such problems and characterise wellposedness using semigroup methods and results on operator matrices.     

A Semigroup Approach to Nonlinear Equation of Age-dependent Population Dynamics

This paper deals with nonlinear model, of age-dependent population dynamics, with nonautonomous time dependence. Using a nonlinear semigroup approach, the authors prove the existence of the problem under general assumptions.     

A Weighted Composition Semigroup on BMOA

We study the weighted composition semigroup $$T_t(f)(z)=\varphi_t'(z)f(\varphi_t(z))$$ on the space BMOA. We identify the maximal subspace of strong continuity and we find conditions under which this maximal subspace coincides with VMOA.     

自由单演逆半群上的核—迹算子半群

对于逆半群上的同余ρ,在它的迹类中存在最大元ρT和最小元ρt.相应地,在它的核类中有最大元ρK和最小元ρk.因此,我们在S的同余格上得到四个算子Г={T,t,K,k}.本文将给出自由单演逆半群上,由算子半群Г生成的半群,即自由单演逆半群上的核一迹算子半群.     

A Regularity Criterion for Semigroup Rings

An analogue of the Kunz–Frobenius criterion for the regularity of a local ring in a positive characteristic is established for general commutative semigroup rings.     

A Semigroup Version of the Isoperimetric Inequality

In this article we extend the heat semigroup version of the isoperimetric inequality in Rⁿ established by M. Ledoux. To this purpose, the notion of the perimeter of a Caccioppoli set introduced by E. De Giorgi yields the appropriate measure theoretical background. In the proofs we use properties of the heat semigroup as well as its explicit integral representation.     

A Semigroup Approach to Stochastic Delay Equations in Spaces of Continuous Functions

We present a semigroup approach to stochastic delay equations of the form

Dreamland with classic higgs field, dimensional reduction and all that

This text, on the one hand, is related to the talk delivered at the conference “Gauge Fields. Yesterday, Today, Tomorrow” dedicated to the 70th anniversary of Andrei Slavnov; on the other—in the form of a fairy tale—it summarizes some results of the research performed after this Fest, mainly due to discussion around the talk.     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

The Zero-Divisor Graph Associated to a Semigroup

The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent if the product of the corresponding elements is zero. New criteria to identify zero-divisor graphs are derived using both graph-theoretic and algebraic methods. We find the lowest bound on the number of edges necessary to guarantee a graph is a zero-divisor graph. In addition, the removal or addition of vertices to a zero-divisor graph is investigated by using equivalence relations and quotient sets. We also prove necessary and sufficient conditions for determining when regular graphs and complete graphs with more than two triangles attached are zero-divisor graphs. Lastly, we classify several graph structures that satisfy all known necessary conditions but are not zero-divisor graphs.     

A Tight Inverse Semigroup Which Is Not Bisimple

We give an example of a tight inverse semigroup which is not bisimple and not congruence-free.     

Semigroup approach to a doubly degenerate parabolic equation

1.IntroductionWeareconcernedwiththesemigroupapproachtotheinitialvalueproblemfordoublynonlineardegenerateparabolicequationoftheformwhicharisesfromdifferentphysicalbackgroundssuchasthemodelingofthemotionofnon-Newtonianfluids.Inthepastyears,thenonlinear...     

A Semigroup Crossed Product Arising in Number Theory

Recently Bost and Connes have studied an interesting C*-algebraicHecke algebra arising in number theory. Here it is shown thatthis algebra can be realised as a semigroup crossed product,and be profitably studied using methods developed by the authorsfor analysing Toeplitz algebras. One main result is a characterisationof faithful representations of the Hecke algebra.