multiplicity one theorems the archimedean case let g be one of the classical lie groups gln 1 r gln 1 c u p q 1 o p q 1 on 1 c so p q 1 son 1 c and let g 0 be respectively the subgroup gln r gln c u p q o p q on c so p q son c embedded in g in the standard way we show that every irreducible casselman wallach representation of g 0 occurs with multiplicity at most one in every irreducible casselman wallach representation of g similar results are proved for the jacobi groups gln r nh2n 1 r gln c n h2n 1 c u p q nh2p 2q 1 r sp2n r nh2n 1 r sp2n c nh2n 1 c with their respective subgroups gln r gln c u p q sp2n r and sp2n c

on axiomatic characterizations of three pairs of covering based approximation operators rough set theory is a useful tool for dealing with inexact uncertain or vague knowledge in information systems the classical rough set theory is based on equivalence relations and has been extended to covering based generalized rough set theory this paper investigates three types of covering generalized rough sets within an axiomatic approach concepts and basic properties of each type of covering based approximation operators are first reviewed axiomatic systems of the covering based approximation operators are then established the independence of axiom set for characterizing each type of covering based approximation operators is also examined as a result two open problems about axiomatic characterizations of covering based approximation operators proposed by zhu and wang in ieee transactions on knowledge and data engineering 19 8 2007 1131 1144 proceedings of the third ieee international conference on intelligent systems 2006 pp 444 449 are solved

neighborhood systems and approximate retrieval an approximate retrieval model is proposed based on the notion of neighborhood systems the knowledge used in the model consists of an information table in which each object is represented by its values on a finite set of attributes and neighborhood systems on attribute values which provide semantic similarity or closeness of different values an information table can be used for exact retrieval with the introduction of neighborhood systems to information tables one is able to perform approximate retrieval operations on neighborhood systems are introduced based on power algebras an ordering relation representing the information of a neighborhood system is suggested and examined approximate retrieval is carried out by the relaxation of the original query using neighborhood systems and the combination of intermediate results using neighborhood system operations the final retrieval results are presented according to the proposed ordering relation in contrast to many existing systems a main advantage of the proposed model is that the retrieval results are a non linear ordering of objects

rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

defining rough sets using tolerances compatible with an equivalence abstract we consider tolerances t compatible with an equivalence e on u meaning that the relational product e t is included in t we present the essential properties of e compatible tolerances and study rough approximations defined by such e and t we consider rough set pairs xe xt where the lower approximation xe is defined as is customary in rough set theory but xt allows more elements to be possibly in x than xe motivating examples of e compatible tolerances are given and the essential lattice theoretical properties of the ordered set of rough sets xe xt x u are established

relational interpretations of neighborhood operators and rough set approximation operators this paper presents a framework for the formulation interpretation and comparison of neighborhood systems and rough set approximations using the more familiar notion of binary relations a special class of neighborhood systems called 1 neighborhood systems is introduced three extensions of pawlak approximation operators are analyzed properties of neighborhood and approximation operators are studied and their connections are examined

multiplicity one theorems for fourier jacobi models for every genuine irreducible admissible smooth representation pi of the metaplectic group widetilde rm sp 2n over a p adic field and every smooth oscillator representation omega psi of widetilde rm sp 2n we prove that the tensor product pi otimes omega psi is multiplicity free as a smooth representation of the symplectic group rm sp 2n similar results are proved for general linear groups and unitary groups

a generalization of zubov s method to perturbed systems a generalization of zubov s theorem on representing the domain of attraction via the solution of a suitable partial differential equation is presented for the case of perturbed systems with a singular fixed point for the construction it is necessary to consider solutions in the viscosity sense as a consequence maximal robust lyapunov functions can be characterized as viscosity solutions

data analysis based on discernibility and indiscernibility rough set theory models similarities and difierences of objects based on the notions of indiscernibility and discernibility with respect to any subset of attributes one can deflne two pairs of dual relations the strong indiscernibility and weak discernibility relations and the weak indiscernibility and strong discernibility relations the similarities of objects are examined by the indiscernibility relations and the difierence by the discernibility relations respectively alternatively one can construct an indiscernibility matrix to represent the family of strong indiscernibility or weak discernibility relations one also can construct a discernibility matrix to represent the family of strong discernibility or weak indiscernibility relations the consideration of the matrix counterpart of relations and the relation counterpart of matrices brings more insights into rough set theory based on indiscernibility and discernibility three difierent types of reducts can be constructed keeping the indiscernibility discernibility and indiscernibility anddiscernibility relations respectively although the indiscernibility reducts have been intensively studied in the literature the other two types of reducts are relatively new and require more attention the existing methods for constructing the indiscernibility reducts also can be applied to construct the other two types of reducts an empirical experiment for letter recognition is reported for demonstrating the usefulness of the discussed relations and reducts

maximal consistent block technique for rule acquisition in incomplete information systems in this paper the concept of a maximal consistent block is applied to formulate a new approximation to an object set in incomplete information systems with higher level of accuracy similar to the method in inform sci 112 1998 39 113 1999 271 the proposed rough set based rule acquisition method does not require change in the size of the original incomplete system it however has the additional advantage of using a set of simpler discernibility functions of an incomplete system this means that it can provide a more efficient computation for knowledge acquisition especially in large incomplete systems

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

vanishing of certain equivariant distributions on spherical spaces we prove vanishing of mathfrak z eigen distributions on a split real reductive group which change according to a non degenerate character under the left action of the unipotent radical of the borel subgroup and are equivariant under the right action of a spherical subgroup this is a generalization of a result by shalika that concerned the group case shalika s result was crucial in the proof of his multiplicity one theorem we view our result as a step in the study of multiplicities of quasi regular representations on spherical varieties as an application we prove non vanishing of spherical bessel functions

on the nonvanishing of the central value of the rankin selberg l functions first published in journal of the american mathematical society in volume 17 issue 3 published by the american mathematical society

almost lyapunov functions for nonlinear systems abstract we study convergence of nonlinear systems in the presence of an almost lyapunov function which unlike the classical lyapunov function is allowed to be nondecreasing and even increasing on a nontrivial subset of the phase space under the assumption that the vector field is free of singular points away from the origin and that the subset where the lyapunov function does not decrease is sufficiently small we prove that solutions approach a small neighborhood of the origin a nontrivial example where this theorem applies is constructed

neighborhood operator systems and approximations this paper presents a framework for the study of generalizing the standard notion of equivalence relation in rough set approximation space with various categories of k step neighborhood systems based on a binary relation on a finite universe six families of binary relations are obtained and the corresponding six classes of k step neighborhood systems are derived extensions of pawlak s rough set approximation operators based on such neighborhood systems are proposed properties of neighborhood operator systems and rough set approximation operators are investigated and their connections are examined

z finite distributions on p adic groups abstract for a real reductive group g the center z u g of the universal enveloping algebra of the lie algebra g of g acts on the space of distributions on g this action proved to be very useful see e g 17 18 32 8 over non archimedean local fields one can replace the action of z u g by the action of the bernstein center z of g i e the center of the category of smooth representations however this action is not well studied in this paper we provide some tools to work with this action and prove the following results the wavefront set of any z finite distribution on g over any point g g lies inside the nilpotent cone of t g g g let h 1 h 2 g be symmetric subgroups consider the space j of h 1 h 2 invariant distributions on g we prove that the z finite distributions in j form a dense subspace in fact we prove this result in wider generality where the groups h i are spherical subgroups of certain type and the invariance condition is replaced by equivariance further we apply those results to density and regularity of relative characters the first result can be viewed as a version of howe s expansion of characters the second result can be viewed as a spherical space analog of a classical theorem on density of characters of finite length representations it can also be viewed as a spectral version of bernstein s localization principle in the archimedean case the first result is well known and the second remains open

brief a new lyapunov design approach for nonlinear systems based on zubov s method the present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of lyapunov design in particular it relies on a short horizon model based prediction and optimization of the rate of energy dissipation of the system as it is realized through the time derivative of an appropriately selected lyapunov function the latter is computed by solving zubov s partial differential equation based on the system s drift vector field a nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned lyapunov function and closed loop performance characteristics a set of system theoretic properties of the proposed control law are examined as well finally the proposed lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum phase behaviour

relationship among basic concepts in covering based rough sets uncertainty and incompleteness of knowledge are widespread phenomena in information systems rough set theory is a tool for dealing with granularity and vagueness in data analysis rough set method has already been applied to various fields such as process control economics medical diagnosis biochemistry environmental science biology chemistry psychology and conflict analysis covering based rough set theory is an extension to classical rough sets in covering based rough sets there exist several basic concepts such as reducible elements of a covering minimal descriptions unary coverings and the property that the intersection of any two elements is the union of finite elements in this covering these concepts appeared in the literature of covering based rough sets separately in this paper we study the relationships between them in particular we establish the equivalence of the unary covering and the covering with the property that the intersection of any two elements is the union of finite elements in this covering we also investigate the relationship between the covering lower approximation operation and the interior operator a characterization of the interior operator by the covering lower approximation operation is presented in this paper correspondingly we study the relationship between the covering upper approximation operation and the closure operator in addition we explore the conditions under which the covering upper approximation operation is monotone the study of the relationships between these concepts will help us have a better understanding of covering based rough sets

stability analysis of nonlinear systems using higher order derivatives of lyapunov function candidates abstract the lyapunov method for stability analysis of an equilibrium state of a nonlinear dynamic system requires a lyapunov function v t x having the following properties 1 v is a positive definite function and 2 v is at least a negative semi definite function finding such a function is a challenging task the first theorem presented in this paper simplifies the second property for a lyapunov function candidate i e this property is replaced by negative definiteness of some weighted average of the higher order time derivatives of v this generalizes the well known lyapunov theorem the second theorem uses such weighted average of the higher order time derivatives of a lyapunov function candidate to obtain a suitable lyapunov function for nonlinear systems stability analysis even if we have a suitable lyapunov function then this theorem can be used to prove a bigger region of attraction the approach is illustrated by some examples

covering based rough set approximations we propose a framework for the study of covering based rough set approximations three equivalent formulations of the classical rough sets are examined by using equivalence relations partitions and s algebras respectively they suggest the element based the granule based and the subsystem based definitions of approximation operators covering based rough sets are systematically investigated by generalizing these formulations and definitions a covering of universe of objects is used to generate different neighborhood operators neighborhood systems coverings and subsystems of the power set of the universe they are in turn used to define different types of generalized approximation operators within the proposed framework we review and discuss covering based approximation operators according to the element granule and subsystem based definitions

mgrs a multi granulation rough set the original rough set model was developed by pawlak which is mainly concerned with the approximation of sets described by a single binary relation on the universe in the view of granular computing the classical rough set theory is established through a single granulation this paper extends pawlak s rough set model to a multi granulation rough set model mgrs where the set approximations are defined by using multi equivalence relations on the universe a number of important properties of mgrs are obtained it is shown that some of the properties of pawlak s rough set theory are special instances of those of mgrs moreover several important measures such as accuracy email protected quality of email protected and precision of email protected are presented which are re interpreted in terms of a classic measure based on sets the marczewski steinhaus metric and the inclusion degree measure a concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in mgrs as well finally we discuss how to extract decision rules using mgrs unlike the decision rules and rules from pawlak s rough set model the form of decision rules in mgrs is or several pivotal algorithms are also designed which are helpful for applying this theory to practical issues the multi granulation rough set model provides an effective approach for problem solving in the context of multi granulations

generalized harish chandra descent gelfand pairs and an archimedean analog of jacquet rallis s theorem in the first part of the paper we generalize a descent technique due to harish chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field f of characteristic zero our main tool is the luna slice theorem in the second part of the paper we apply this technique to symmetric pairs in particular we prove that the pairs gln k f gln f glk f and gln e gln f are gelfand pairs for any local field f and its quadratic extension e in the non archimedean case the first result was proven earlier by jacquet and rallis and the second by flicker we also prove that any conjugation invariant distribution on gln f is invariant with respect to transposition for non archimedean f the latter is a classical theorem of gelfand and kazhdan

a new asymptotic stability criterion for nonlinear time variant differential equations a new sufficient condition for asymptotic stability of ordinary differential equations is proposed unlike classical lyapunov theory the time derivative along solutions of the lyapunov function may have positive and negative values the classical lyapunov approach may be regarded as an infinitesimal version of the present theorem verification in practical problems is harder than in the classical case an example is included in order to indicate how the present theorem may be applied