Erratum to the paper “An Integral Delay Inequality and its Applications to Boundary Value Problems”

We consider the infinitesimal generator of the time reversal of a time-homogeneous one-dimensional gap diffusion with state space [0, 1] given by the infinitesimal generator A^o = dD_s⁺ - dk / dm with nonlocal boundary conditions of Feller-Wentzell-type. This leads to an infinitesimal generator which belongs to a class of generators introduced in the earlier note [W1].     

Über Die Konvergenz Operatorwertiger Cosinusfunktionen mit Gestörtem Infinitesimalen Erzeuger

On convergence of operator cosine functions with perturbed infinitesimal generator. The question under what kind of perturbations a closed linear operatorA remains of the class of infinitesimal generators of operator cosine functions seems to be a rather difficult one and is unsolved in general. In this note we give bounds for the perturbation of operator cosine functions caused byA-bounded perturbationsT ofA under the assumption thatT + A is also a generator.

     

Infinitesimal Generators of C *-Algebras

We develop the method introduced previously, to construct infinitesimal generators on locally compact group C ^*-algebras and on tensor product of C ^*-algebras. It is shown in particular that there is a C ^* -algebra A such that the C ^*-tensor product of A and an arbitrary C ^*-algebra B can have a non-approximately inner strongly one parameter group of ^*-automorphisms.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

K-Theory of Algebraic Tori and Toric Varieties

It is proved that under certain conditions the group K _n (X) of a smooth projective variety X over a field F is a natural direct summand of K _n (A) for some separable F-algebra A. As an application we study the K-groups of toric models and toric varieties. A presentation in terms of generators and relations of the groupK ₀(T) for an algebraic torus T is given.     

Limit theorems for a class of diffusion processes

We consider a diffusion process {x(t)} on a compact Riemannian manifold with generator δ/2 + b. A current‐valued continuous stochastic process {X _t} in the sense of Itô [8] corresponds to {x(t)} by considering the stochastic line integral X _t(a) along {x(t)} for every smooth 1-form a. Furthermore {X _t} is decomposed into the martingale part and the bounded variation part as a current-valued continuous process. We show the central limit theorems for {X _t} and the martingale part of {X _t}. Occupation time laws for recurrent diffusions and homogenization problems of periodic diffusions are closely related to these theorems     

Quantum homogeneous spaces with faithfully flat module structures

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB. In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB ⁺ is cosemisimple.     

Remarks on intermediate C-rings of C(X)

Let X be a Tychono? space and A(X) be a subring of C(X) containing C^?(X). We introduce the notion of -ideal in A(X). It is observed that the class of -ideals contains the class of z_A-ideals and is contained in the class of z-ideals of A(X). These containments may be proper. It turns out that coincidence of z-ideals of A(X) with -ideals characterizes intermediate C-rings of C(X).     

A note on the rank of semigroups

The rank of a semigroup $\mathcal{A}The rank of a semigroup A\mathcal{A} of functions from a finite set X to X is the minimum of |f(X)| over f ? Af\in \mathcal{A}. Given a finite set X and a subset Y of X, we show that if A\mathcal{A} is a semigroup of functions from X to X and ℬ a transitive semigroup of functions from Y to Y, then the rank of A\mathcal{A} divides that of ℬ provided that f(X)⊆Y for some f ? Af\in \mathcal{A} and that each function in ℬ is the restriction of a function in A\mathcal{A} to Y. To prove this, we generalize a result of Friedman which says that one can partition Y into q subsets of equal weight where q is the rank of ℬ. When one extends a transitive automaton by adding new states and letters, a similar condition guarantees that the rank of the extension divides the original rank.     

A generalization of d’Alembert formula

In this paper we find a closed form of the solution for the factored inhomogeneous linear equation

On the First-Passage Area of a One-Dimensional Jump-Diffusion Process

For a one-dimensional jump-diffusion process X(t), starting from x?>?0, it is studied the probability distribution of the area A(x) swept out by X(t) till its first-passage time below zero. In particular, it is shown that the Laplace transform and the moments of A(x) are solutions to certain partial differential-difference equations with outer conditions. The distribution of the maximum displacement of X(t) is also studied. Finally, some explicit examples are reported, regarding diffusions with and without jumps.     

Multiplicative perturbation of nonlinear m-accretive operators

Criteria are obtained for when an accretive product (i.e., composition) BA of nonlinear m-accretive operators A and B in a Banach space X will be itself m-accretive; and, in particular, when a monotone product of two maximal monotone operators in a Hilbert space will be maximal monotone. This extends the theory of multiplicative perturbation of infinitesimal generators of contraction semigroups to the nonlinear case. Also obtained as a biproduct are existence theorems for certain Hammerstein integral equations.     

Space $$ \mathcal{L} $$( X) of local dynamical systems and the Filippov space Aceu( X)

In the present paper, we establish a relationship between continuous local dynamical systems and spaces of the class A _ceu(X) of the Filippov theory. We suggest a construction method for a space of the class A _ceu(X) on the basis of a locally given dynamical system and conversely, a dynamical system is constructed locally in a specific way on the basis of a given space of the class A _ceu(X). The suggested construction method provides a homeomorphism between the space of all local dynamical systems on a locally compact metric space X and the space A _ceu(X). The obtained results generalize the Filippov theory to locally dynamical systems.     

Sums and commutators of generators of noncommuting strongly continuous groups

We consider two infinitesimal generators A, B of strongly continuous groups in a general Banach space X, such that either the commutator between A and B commutes both with e^tA and with e^tB, or the commutator is a multiple of A. We prove that under suitable assumptions the sum A+B and the commutator [A,B] are closable, and their closures generate strongly continuous groups. We give explicit representation formulas for such groups, in terms of e^tA and e^tB.     

On Short Time Asymptotic Behavior of Some Symmetric Diffusions on General State Spaces

For conservative symmetric diffusions on a general state space (X,m), the short time asymptotic behavior of tlog _X 1 _A T _t 1 _B dm is investigated, where T _t is the associated semigroup and A and B are measurable subsets of X. It is proved that the superior limit is dominated by the inferior limit up to some absolute constant. When ₂ of the associated Dirichlet form is lower bounded, it is shown that the limit exists for any A and B, and is described by the intrinsic metric between them. Applications to infinite-dimensional spaces and fractals are given.     

Determination of Spherical Area Measures by Means of Dilation Volumes

For a large class of compact sets X in ℝ^d which can be represented as finite unions of sets of positive reach, the spherical area measure of order d — 1 is determined by the volumes of dilations of X with infinitesimal multiples of all rotations of a suitable convex test set. For spherical area measures of lower orders, a similar result is obtained by approximating the set X by the closures of the complements of close parallel bodies.     

C0-estimates and smoothness of solutions to the parabolic equation defined by Kimura operators

Kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics. Their infinitesimal generator is an elliptic differential operator whose second-order coefficients matrix degenerates on the boundary of the domain. In this article, we consider the inhomogeneous initial-value problem defined by generators of Kimura diffusions, and we establish $C^0$-estimates, which allows us to prove that solutions to the inhomogeneous initial-value problem are smooth up to the boundary of the domain where the operator degenerates, even when the initial data is only assumed to be continuous.     

There do not exist minimal algebras between C*(X) and C(X) with prescribed real maximal ideal spaces

Let C(X) be the algebra of all real-valued continuous functions on a completely regular Hausdorff space X, and C*(X) the subalgebra of bounded functions. We prove that for any intermediate algebra A between C*(X) and C(X), other than C*(X), there exists a smaller intermediate algebra with the same real maximal ideals as in A. The space X is called A-compact if any real maximal ideal in A corresponds to a point in X. It follows that, for a noncompact space X, there does not exist any minimal intermediate algebra A for which A is A-compact. This completes the answer to a question raised by Redlin and Watson in 1987.     

Transformations of diffusions with jumps

The paper deals with some transformations of diffusions with jumps. We consider the class of diffusions with jumps that is closed with respect to composition with invertible, twice continuously differentiable functions. A special random time change gives us again a diffusion with jumps. A result on transformation of a measure is valid for this class of diffusions with jumps. Bibliographty: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 79–100.     

Well-posedness of non-autonomous linear evolution equations for generators whose commutators are scalar

We prove the well-posedness of non-autonomous linear evolution equations for generators \({A(t): D(A(t)) \subset X \to X}\) whose pairwise commutators are complex scalars, and in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the onefold commutators only the p-fold commutators of the operators A(t) are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: Indeed, stability in X and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators A(t) by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrödinger operators for particles in external electric fields.