Gamma-type operators and the Black–Scholes semigroup

We study Gamma-type operators from the analytic and probabilistic viewpoint in the setting of weighted continuous function spaces and estimate the rate of convergence of their iterates towards their limiting semigroup, providing, in this way, a quantitative version of the classical Trotter approximation theorem. The semigroup itself has some interest, since it is generated by the Black–Scholes operator, frequently occurring in the theory of option pricing in mathematical finance.     

Analyticity of the global attractor for damped forced periodic Korteweg–de Vries equation

We consider here a weakly damped forced periodic KdV equation. We prove that if the forcing term is analytic in space, then the global attractor is also contained into a space of analytic functions. This result was conjectured in [23].     

Analyticity of the Schr?dinger propagator on the Heisenberg group

We discuss the analytic extension property of the Schr?dinger propagator for the Heisenberg sublaplacian and some related operators. The result for the sublaplacian is proved by interpreting the sublaplacian as a direct integral of an one parameter family of dilated special Hermite operators.     

A presentation for the singular part of the full transformation semigroup

The (full) transformation semigroup T_n\mathcal{T}_{n} is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group S_n í T_n{\mathcal{S}_{n}\subseteq \mathcal{T}_{n}} is the group of all permutations on {1,…,n} and is the group of units of T_n\mathcal{T}_{n}. The complement T_n\S_n\mathcal{T}_{n}\setminus \mathcal{S}_{n} is a subsemigroup (indeed an ideal) of T_n\mathcal{T}_{n}. In this article we give a presentation, in terms of generators and relations, for T_n\S_n\mathcal{T}_{n}\setminus \mathcal{S}_{n}, the so-called singular part of T_n\mathcal{T}_{n}.     

Analog of the Löwner–Kufarev equation for the semigroup of conformal mappings of the disk into itself with fixed points and invariant diameter

Mathematical Notes -     

On the ergodic theorem in the Kozlov–Treshchev form for an operator semigroup

We study nonuniform ergodic averages of the Kozlov – Treshchev type for operator semigroups and obtain estimates for the corresponding maximal functions.     

Fiat categorification of the symmetric inverse semigroup $$\textit{IS}_n$$ and the semigroup $$F^*_n$$Fn∗

Starting from the symmetric group \(S_n\), we construct two fiat 2-categories. One of them can be viewed as the fiat “extension” of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other 2-category can be viewed as the fiat “extension” of the 2-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.     

On non–symmetric generalized schrodinger semigroup

In this paper some general phenomena are described for not necessarily systemeric so–called generalized Schrödinger semigroups (or generalized absorption/exciatation semigroups). These results are also applicable in case we consider Schrödinger semigroups on R ^v. In particular we describe some results on integral kernels: continuity, pointwise inequalities, ultracontractivity etc.For these inequalities we use a kind of stochastic bridge measure. The operator H is a closed linear extension of the operator H ⁰ + V in the space C ⁰(E) Here E is a locally compact second countable Hausdorff space and –H ₀ is supposed to generate a Feller semigroup in C ₀(E). Results in L^p (E,m) are also availale. Some examples are given     

A note on the approximate amenability of semigroup algebras

It is known that the bicyclic semigroup S ₁ is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra ℓ ¹(S ₁) is not approximately amenable.     

Nonpresentability of the semigroup ω ω over ℍmathbb{F}(ℝ)

We prove that the semigroup of all functions from the set of natural numbers to natural numbers by composition has no one-valued Σ-presentations with parameters over the hereditarily finite superstructure over the real numbers.     

A Generalization of C—semigroup

§ 1.ＴｈｅＧｅｎｅｒａｌＣ ｓｅｍｉｇｒｏｕｐ　Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｌｅｔＸｂｅａＢａｎａｃｈｓｐａｃｅ,ａｎｄＢ(Ｘ)ｔｈｅｓｐａｃｅｏｆａｌｌｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｓｆｒｏｍＸｉｎｔｏＸ .　Ｄｅｆｉｎｉｔｉｏｎ 1 .1　ＳｕｐｐｏｓｅＣｉｓａｂｏｕｎｄｅｄｉｎｊｅｃｔｉｖｅｏｐｅｒａｔｏｒｆｒｏｍＸｉｎｔｏＸ .Ｔｈｅｆａｍｉｌｙｏｆｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｓ (Ｓ(ｔ) ) ｔ≥ 0 ｉｓａｇｅｎｅｒａｌＣ ｓｅｍｉｇｒｏｕｐｉｆｉｔｈａｓｔｈｅｆｏｌｌｏｗｉｎｇ…     

A proof of Devadze’s theorem on generators of the semigroup of Boolean matrices

In 1968 Devadze described, without a proof, minimal sets of generators of the semigroup of n×n Boolean matrices. We provide a proof of Devadze’s theorem.     

Hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup

We prove hypercontractivity for a quantum Ornstein–Uhlenbeck semigroup on the entire algebra of bounded operators on a separable Hilbert space h. We exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain. Then we deduce a logarithmic Sobolev inequality for the semigroup and gain an elementary estimate of the best constant.     

On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems

An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field, which in turn defines a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, 2007). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008), we give a necessary and sufficient condition for when an off-central path is analytic as a function of at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition.