态R_0代数

本文研究了R_0代数上有关态算子的问题.利用MV-代数上内态的引入方法引入了态算子,定义了态R_0代数,它是R_0代数的一般化.给出了一些非平凡态R_0代数的例子并讨论了态R_0代数的一些基本性质.在此基础上给出了态滤子和态局部R_0代数的概念,并利用态滤子刻画了态局部R_0代数.推广了局部R_0代数的相关理论.     

Quantum differential operators on the quantum plane

The universal enveloping algebra of a Lie algebra acts on its representation ring R through D(R), the ring of differential operators on R. A quantised universal enveloping algebra (or quantum group) is a deformation of a universal enveloping algebra and acts not through the differential operators of its representation ring but through the quantised differential operators of its representation ring. We present this situation for the quantum group of sl₂.     

Convergence of Newton–Kantorovich Approximations to an Approximate Zero

In this paper, we prove an a posteriori and an a priori convergence theorem for Newton–Kantorovich approximations starting from an initial point x ₀. We apply these results to operators that are analytic at interior points of a closed ball centered at x ₀ and of radius R. We obtain some theorems on approximate zeros and on approximate zeros of second kind for these operators, which improve previous results.     

Global solutions for operator Riccati equations with unbounded coefficients: A non-linear semigroup approach

Let X be a Banach space of real-valued functions on [0, 1] and let ?(X) be the space of bounded linear operators on X. We are interested in solutions R:(0, ∞) → ?(X) for the operator Riccati equation where T is an unbounded multiplication operator in X and the B_i(t)'s are bounded linear integral operators on X. This equation arises in transport theory as the result of an invariant embedding of the Boltzmann equation. Solutions which are of physical interest are those that take on values in the space of bounded linear operators on L¹(0, 1). Conditions on X, R(0), T, and the coefficients are found such that the theory of non-linear semigroups may be used to prove global existence of strong solutions in ?(X) that also satisfy R(t) ? ?(L¹(0,1)) for all t ≥ 0.     

Differentiability of a convex semigroup on Rn

It is proved that each continuous semigroup {P(t)}_t≥0 of convex operators P(t):Rⁿ→Rⁿ is continuously differentiable with respect to t. This note represents a first step towards a better understanding of semigroups formed by convex operators. We establish the differentiability of a convex semigroup in the finite dimensional case, generalizing a basic result from linear semigroup theory. Our motivation for the study of semigroups of convex operators comes from the theory of Markov decision processes. In [1] and in [2] it was shown that the maximum reward of these processes can be described by a certain nonlinear semigroup. The nonlinear operators are defined as suprema of linear operators (plus a constant), hence they are convex operators. It seems that the convexity assumption keeps its smoothing influence even in the infinite dimensional situation. We hope to discuss this in a future paper.     

Multiplicative perturbations of the Laplacian and related approximation problems

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on \mathbbR^N,  N 3 1{{\mathbb{R}}^{N},\; N\geq1} . It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C ₀-semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on \mathbbR^N{{\mathbb{R}}^{N}} . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

Note on theK0 of rings with Zariskian filtration

For a noncommutative Zariski ringR with associated graded ringG(R) of finite global dimension there is an injectionK _o(R) K _o (G(R)). The result applies to certain rings of differential operators, in particular to the ring of germs of microlocal differential operators that also provides an example of a nonpositively filtered Zariski ring.Research fellow at UIA, supported by a grant from the Province of Antwerp.Supported by an NFWO grant.     

ON THE SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

Sufficient conditions are established for the solvability of the boundary value problem where p : C(I; R ⁿ) × C(I; R ⁿ) L(I; R ⁿ), q : C(I; R ⁿ) L(I; R ⁿ), l : C(I, R ⁿ) × C(I; R ⁿ) R ⁿ, and c _n : C(I, R ⁿ) R ⁿ are continuous operators, and p(x, ) and l(x, ) are linear operators for any fixed .     

Abstract Riccati equations in an L1 space of finite measure and applications to transport theory

We consider operator-valued Riccati initial-value problems of the form R′(t) + TR(t) + R(t)T = TA(t) + TB(t)R(t) + R(t)TC(t) + R(t)TD(t)R(t), R(0) = R₀. Here A to D and R₀ have values as non-negative bounded linear operators in L¹ (μ), where μ is a finite measure, and T is a closed non-negative operator in L¹ (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.     

Sobolev spaces associated to a polyhedron and Fourier integral operations inR n

The theory of the Sobolev spacesH _m ^p (R ⁿ ) (mR,p polyhedron in R ²ⁿ )of [BG]is revisited here in the frame of new classes of pseudodifferential operators related to the same polyhedron p.These operators generalize to corresponding classes of Fourier integral operators, for which we present the main lines of a symbolic calculus and results of continuity on the H _m ^p (R ⁿ ) spaces.     

Factoring and decomposing a class of linear functional systems

Within a constructive homological algebra approach, we study the factorization and decomposition problems for a class of linear functional (determined, over-determined, under-determined) systems. Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module M over an Ore algebra to another one M′, where M (resp., M′) is a module intrinsically associated with the linear functional system Ry = 0 (resp., R′z = 0). These morphisms define applications sending solutions of the system R′z = 0 to solutions of R y = 0. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module M is equivalent to the existence of a non-trivial factorization R = R₂R₁ of the system matrix R. The corresponding system can then be integrated “in cascade”. Under certain conditions, we also show that the system Ry = 0 is equivalent to a system R′z = 0, where R′ is a block-triangular matrix of the same size as R. We show that the existence of idempotents of the endomorphism ring of the module M allows us to reduce the integration of the system Ry = 0 to the integration of two independent systems R₁y₁ = 0 and R₂y₂ = 0. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system Ry = 0, i.e., they allow us to compute an equivalent system R′z = 0, where R′ is a block-diagonal matrix of the same size as R. Applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in the Maple package Morphisms based on the library oremodules.     

Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory

A basic integral equation of random fields estimation theory by the criterion of minimum of variance of the estimation error is of the form Rh = f, where and R(x, y) is a covariance function.The singular perturbation problem we study consists of finding the asymptotic behavior of the solution to the equation as 0.$$" align="middle" border="0"> The domain D can be an interval or a domain in Rⁿ, n > 1. The class of operators R is defined by the class of their kernels R(x,y) which solve the equation Q(x, D_x)R(x, y) = P(x, D_x)δ(x − y), where Q(x, D_x) and Px, D_x) are elliptic differential operators.     

Solvability of systems of partial differential equations for functions defined on nonconvex sets

We consider systems of partial differential equations with constant coefficients of the form ( R(D_x, D_y)f = 0, P(D_x)f = g), f,g ? C^￥(W),\big ( R(D_x, D_y)f = 0, P(D_x)f = {g}\big ), f,g \in {C}^{\infty}(\Omega),, where R (and P) are operators in (n + 1) variables (and in n variables, respectively), g satisfies the compatibility condition R(D_x, D_y)g = 0  and  W ì \Bbb Rⁿ⁺¹R(D_x, D_y){g} = 0 \ {\rm and} \ \Omega \subset {\Bbb R}^{n+1} is open. Let R be elliptic. We show that the solvability of such systems for certain nonconvex sets W\Omega implies that any localization at ￥\infty of the principle part P_m of P is hyperbolic. In contrast to this result such systems can always be solved on convex open sets W\Omega by the fundamental principle of Ehrenpreis-Palamodov.     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

Generalized affine transformation monoid of a free module of finite rank over finite chain rings

Let R be a finite commutative chain ring, n a positive integer and R ⁿ the free R-module of rank n consisting of column vectors over R. The generalized affine transformation monoid Gaff _n (R) of R ⁿ is introduced, then Schützenberger groups of -classes, principal factors and group -classes of the monoid Gaff _n (R) are investigated. As corollaries, basic numerical information of Gaff _n (R) is given.     

Circulant integral operators as preconditioners for Wiener-Hopf equations

In this paper, we study the solutions of finite-section Wiener-Hopf equations by the preconditioned conjugate gradient method. Our main aim is to give an easy and general scheme of constructing good circulant integral operators as preconditioners for such equations. The circulant integral operators are constructed from sequences of conjugate symmetric functions {C }. Letk(t) denote the kernel function of the Wiener-Hopf equation and be its Fourier transform. We prove that for sufficiently large if {C } is uniformly bounded on the real lineR and the convolution product of the Fourier transform ofC with uniformly onR, then the circulant preconditioned Wiener-Hopf operator will have a clustered spectrum. It follows that the conjugate gradient method, when applied to solving the preconditioned operator equation, converges superlinearly. Several circulant integral operators possessing the clustering and fast convergence properties are constructed explicitly. Numerical examples are also given to demonstrate the performance of different circulant integral operators as preconditioners for Wiener-Hopf operators.Research supported in part by HKRGC grant no. 221600070.     

Unified characterizations for modifications ofR 0 andR 1 topological spaces

Cammaroto and Noiri [14] introduced a separation axiom calledm-R ₀ in anm-space (X, m). In this paper, we introduce the notion ofm-R ₁ spaces and offer many characterizations ofm-R ₀ (resp.m-R ₁) spaces which enable us to obtain unified characterizations of separation axiomsR ₀, semi-R ₀, pre-R ₀,α-R ₀,δ-semiR ₀, (δ, p)-R ₀ (resp.R ₁, semi-R ₁, pre-R ₁,α-R ₁,δ-semiR ₁, (δ, p)-R ₁).     

Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.