On a Cameron–Martin Type Quasi-Invariance Theorem and Applications to Subordinate Brownian Motion

We present a Cameron–Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion.     

The Osofsky–Smith Theorem for Modular Lattices and Applications (I)

The aim of this article is to present a latticial version of the renown module theoretical Osofsky–Smith Theorem.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

The Pythagorean Theorem(二)

By translate the triangle we can get a figure like that right.The angle measures J. and y add to 90°, so the angles of the     

The Pythagorean Theorem(一)

From the area of a square you can determine the length ofany of its sides.If the length of the side is √A.That is why√A is called the square root of A. Just as a plant rests on itsroots,a square can rest on its(positive)square root.side=√AThus if the area of a square is 400 square feet,its side has length 20cm.If the area is 13square feet,its side has length √13 feet,You can verify the latter of these with a calculator:     

The Pythagorean Theorem(三)

ProofThe original triangle withits translation and rota-tion images is below.Itslegs are a and b and thehypotenuse is c.The areaof the shaded square is     

The FRT-type Theorem for the Hom–Long Equation

In this article, we introduce the concepts of Hom–Long equation and Hom–Long dimodule. We discuss how Hom–Long dimodules are connected to the Hom–Long equation. As the main result, we study the FRT-type theorem for the Hom–Long equation.     

The Brauer–Ostrowski Theorem for Matrices of Operators

The classical Brauer-Ostrowski Theorem gives a localization of the spectrum of a matrix by a union of Cassini ovals. In this paper we prove a corresponding result for operator matrices.     

The Erdős–Ko–Rado Theorem for Integer Sequences

n , q, t we determine the maximum number of integer sequences which satisfy for , and any two sequences agree in at least t positions. The result gives an affirmative answer to a conjecture of Frankl and Füredi. Received: January 26, 1998     

The Atiyah–Bott–Lefschetz Theorem for Manifolds with Conical Singularities

We establish an Atiyah–Bott–Lefschetz formula for elliptic operators on manifolds with conical singular points.     

The Loomis–Sikorski Theorem revisited

We prove, constructively, that the Loomis–Sikorski Theorem for σ-complete Boolean algebras follows from a representation theorem for Archimedean vector lattices and a constructive representation of Boolean algebras as spaces of Carathéodory place functions. We also prove a constructive subdirect product representation theorem for arbitrary partially ordered vector spaces. Received August 10, 2006; accepted in final form May 30, 2007.     

On Comparison Theorem and Solutions of BSDEs for Levy Processes

In this paper, we consider backward stochastic differential equations driven by a Levy process. A comparison theorem and an existence and uniqueness theorem of BSDEs with non-Lipschitz coefficients are obtained.     

The Eyring–Kramers Law for Markovian Jump Processes with Symmetries

We prove an Eyring–Kramers law for the small eigenvalues and mean first-passage times of a metastable Markovian jump process which is invariant under a group of symmetries. Our results show that the usual Eyring–Kramers law for asymmetric processes has to be corrected by a factor computable in terms of stabilisers of group orbits. Furthermore, the symmetry can produce additional Arrhenius exponents and modify the spectral gap. The results are based on representation theory of finite groups.     

A Decomposition Theorem for Lévy Processes on Local Fields

The study of Lévy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Lévy processes on local fields is given in terms of a structure result for measures on local fields and a Lévy–Khinchine representation. It is shown that a measure on a local field can be decomposed into three parts: a spherically symmetric measure, a totally non-spherically symmetric measure and a singular measure. We show that if the Radon–Nikodym derivative of the absolutely continuous part of a Lévy measure on a local field is locally constant, the Lévy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Lévy processes with single balls as support of their Lévy measure, end a singular Lévy process. These processes are independent. Explicit formulae for the transition function are obtained.     

Karhunen–Loève expansion for additive Slepian processes

For Gaussian random fields defined as additive processes based on Slepian processes, we study their Karhunen–Loève expansions and obtain the Pythagorean type distribution identities. As applications, the corresponding small deviation estimates are given.     

Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.     

Mackey–Ahrens Theorem for Spatial Scales

The Mackey–Ahrens theorem on topologies which are congruent in duality is extended to the duality of inductive scales of spaces.