axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

serrin s overdetermined problem and constant mean curvature surfaces for all n 9 we find smooth entire epigraphs in rn namely smooth domains of the form x rn xn f x1 xn 1 which are not half spaces and in which a problem of the form u f u 0 in has a positive bounded solution with 0 dirichlet boundary data and constant neumann boundary data on this answers negatively for large dimensions a question by berestycki caffarelli and nirenberg in 1971 serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball in analogy to a famous result by alexandrov that states that an embedded compact surface with constant mean curvature cmc in euclidean space must be a sphere in lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given cmc surface where serrin s overdetermined problem is solvable

a remark on liouville type theorems for the stationary navier stokes equations in three space dimensions abstract consider the 3d homogeneous stationary navier stokes equations in the whole space r 3 we deal with solutions vanishing at infinity in the class of the finite dirichlet integral by means of quantities having the same scaling property as the dirichlet integral we establish new a priori estimates as an application we prove the liouville theorem in the marginal case of scaling invariance

layer solutions in a half space for boundary reactions r n 1 are denoted by y y1 yn 1 our main goal is to study bounded solutions of 1 1 that are monotone increasing say from 1 to 1 in one of the y variables we call them layer solutions of 1 1 and we study their existence uniqueness symmetry and variational properties as well as their asymptotic behavior the interest in such increasing solutions comes from some models of boundary phase transitions when the nonlinearity f is given by f u sin cu for some constant c problem 1 1 in a half plane is called the peierls nabarro problem and it appears as a model of dislocations in crystals see 21 36 the peierls nabarro problem is also central to the analysis of boundary vortices in the paper 28 which studies a model for soft thin films in micromagnetism recently derived by kohn and slastikov 26 see also 27 our main result theorem 1 2 characterizes the nonlinearities f for which there exists a layer solution of 1 1 in dimension n 2 we prove that the necessary and sufficient condition is that the potential g defined by g f has two and only two absolute minima in the interval 1 1 located at 1 under the additional hypothesis g 1 0 we also establish the uniqueness of a layer solution up to translations in the y variable the proofs of both the necessity and the sufficiency of the condition on g for existence use new ingredients which we develop in this article a first one is a nonlocal estimate as well as a conserved or hamiltonian quantity satisfied by every layer solution in dimension 2 see theorem 1 3 the estimate can be seen as

effect of a protection zone in the diffusive leslie predator prey model abstract in this paper we consider the diffusive leslie predator prey model with large intrinsic predator growth rate and investigate the change of behavior of the model when a simple protection zone 0 for the prey is introduced as in earlier work y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 y du x liang a diffusive competition model with a protection zone j differential equations 244 2008 61 86 we show the existence of a critical patch size of the protection zone determined by the first dirichlet eigenvalue of the laplacian over 0 and the intrinsic growth rate of the prey so that there is fundamental change of the dynamical behavior of the model only when 0 is above the critical patch size however our research here reveals significant difference of the model s behavior from the predator prey model studied in y du j shi a diffusive predator prey model with a protection zone j differential equations 229 2006 63 91 with the same kind of protection zone we show that the asymptotic profile of the population distribution of the leslie model is governed by a standard boundary blow up problem and classical or degenerate logistic equations

extensions and intentions in the rough set theory abstract the approach to rough set theory proposed in this paper is based on the mutual correspondence of the concepts of extension and intension it is different from the well known approaches in the literature in that the upper approximations and the lower approximations of unknown sets are considered as certain families of known sets this approach makes it possible to formulate necessary and sufficient conditions for the existence of operations on rough sets which are analogous to classical operations on sets the basic results presented in this paper based on certain ideas of the second author were formulated by the first author in his doctoral dissertation prepared under the supervision of the third author

second order estimate on transition layers abstract in this paper we establish a uniform c 2 estimate for level sets of stable solutions to the singularly perturbed allen cahn equation in dimensions n 10 which is optimal the proof combines two ingredients one is a reverse application of the infinite dimensional lyapunov schmidt reduction method which enables us to reduce the c 2 estimate for these level sets to a corresponding one on solutions of toda system the other one uses a small regularity theorem on stable solutions of toda system to establish various decay estimates which gives a lower bound on distances between different sheets of solutions to toda system or level sets of solutions to allen cahn equation

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

parabolic evolution equations and nonlinear boundary conditions on etudie le probleme aux valeurs limites et initiales u 7b a t u f t u 7b b t u g t u 0 t t u 0 u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

large sets at infinity and maximum principle on unbounded domains for a class of sub elliptic operators abstract maximum principles on unbounded domains play a crucial role in several problems related to linear second order pdes of elliptic and parabolic type in this paper we consider a class of sub elliptic operators l in r n and we establish some criteria for an unbounded open set to be a maximum principle set for l we extend some classical results related to the laplacian by deny hayman and kennedy and to the sub laplacians on stratified lie groups by bonfiglioli and the second named author

the sliding method for the nonlocal monge ampere operator abstract we develop a sliding method for the nonlocal monge ampere operator we first establish a narrow region principle in bounded domains which plays an important role in the sliding method then we consider the equation with the nonlocal monge ampere operator and derive the monotonicity of solutions in both bounded domains and the whole space we use a new idea estimating the singular integrals defining the nonlocal monge ampere operator along a sequence of approximate maximum points

radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry let n n with n 2 a 1 0 0 1 and f 0 1 0 r such that for each u 0 r 1 ar2 n 2 2f r 1 ar2 n 2 2u 0 1 r is nonincreasing we show that each positive solution of u f x u 0in b u 0on b is radially symmetric where b is the open unit ball in rn

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

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

characterization of f extremal disks abstract we show uniqueness for overdetermined elliptic problems defined on topological disks with c 2 boundary i e positive solutions u to u f u 0 in m 2 g so that u 0 and u c t e along the unit outward normal along under the assumption of the existence of a candidate family to do so we adapt the galvez mira generalized hopf type theorem 19 to the realm of overdetermined elliptic problem when m 2 g is the standard sphere s 2 and f is a c 1 function so that f x 0 and f x x f x for any x r we construct such candidate family considering rotationally symmetric solutions this proves the berestycki caffarelli nirenberg conjecture in s 2 for this choice of f more precisely this shows that if u is a positive solution to u f u 0 on a topological disk s 2 with c 2 boundary so that u 0 and u c t e along then must be a geodesic disk and u is rotationally symmetric in particular this gives a positive answer to the schiffer conjecture d cf 33 35 for the first dirichlet eigenvalue and classifies simply connected harmonic domains cf 28 also called serrin problem in s 2

on the hopf boundary lemma for singular semilinear elliptic equations abstract in this paper we consider positive solutions to semilinear elliptic problems with singular nonlinearity we provide a hopf type boundary lemma via a suitable scaling argument that allows to deal with the lack of regularity of the solutions up to the boundary

asymptotic estimates for the least energy solution of a planar semi linear neumann problem abstract in this work we study the asymptotic behavior of the l norm of the least energy solution u p of the following semi linear neumman problem u u u 0 in u u p on where is a smooth bounded domain in r 2 our main result shows that the l norm of u p remains bounded and bounded away from zero as p goes to infinity more precisely we prove that lim p u l e

rough sets induced by ideals in lattices this paper is to study the rough sets within the context of lattices we study the special properties of the rough sets which can be constructed by means of the congruences determined by ideals of lattice also the properties of the generalized rough sets with respect to ideals of lattice are investigated finally we give an example of their application in formal concept analysis

fuzzy rough sets based on generalized residuated lattices this paper is devoted to propose generalized l fuzzy rough sets as a further generalization of the notion of l fuzzy rough sets a quadruple of approximation operators are defined to suit the situation when generalized residuated lattices are non commutative generalized l fuzzy rough sets are characterized from both constructive and axiomatic approaches in the constructive approach various classes of generalized l fuzzy rough sets are investigated moreover the relationship between generalized l fuzzy rough sets and l topologies on an arbitrary universe is discussed with generalized lower and upper sets as an application of generalized l fuzzy rough sets fuzzy rough sets are proposed and studied on the unit interval 0 1 which are based on generalized residuated lattices induced by left continuous pseudo t norms

rough approximations on a complete completely distributive lattice with applications to generalized rough sets the generalizations of rough sets considered with respect to similarity relation covers and fuzzy relations are main research topics of rough set theory however these generalizations have shown less connection among each other and have not been brought into a unified framework which has limited the in depth research and application of rough set theory in this paper the complete completely distributive ccd lattice is selected as the mathematical foundation on which definitions of lower and upper approximations that form the basic concepts of rough set theory are proposed these definitions result from the concept of cover introduced on a ccd lattice and improve the approximations of the existing crisp generalizations of rough sets with respect to similarity relation and covers when t similarity relation is considered the existing fuzzy rough sets are the special cases of our proposed approximations on a ccd lattice thus these generalizations of rough sets are brought into a unified framework and a wider mathematical foundation for rough set theory is established

generalized fuzzy rough sets this paper presents a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized approximation operators is defined the connections between fuzzy relations and fuzzy rough approximation operators are examined in axiomatic approach various classes of fuzzy rough approximation operators are characterized by different sets of axioms axioms of fuzzy approximation operators guarantee the existence of certain types of fuzzy relations producing the same operators

finite energy of generalized suitable weak solutions to the navier stokes equations and liouville type theorems in two dimensional domains abstract introducing a new notion of generalized suitable weak solutions we first prove validity of the energy inequality for such a class of weak solutions to the navier stokes equations in the whole space r n although we need certain growth condition on the pressure we may treat the class even with infinite energy quantity except for the initial velocity we next handle the equation for vorticity in 2d unbounded domains under a certain condition on the asymptotic behavior at infinity we prove that the vorticity and its gradient of solutions are both globally square integrable as their applications loiuville type theorems are obtained

unbounded periodic solutions to serrin s overdetermined boundary value problem we study the existence of nontrivial unbounded domains omega in mathbb r n such that the overdetermined problem delta u 1 quad rm in omega quad u 0 quad partial nu u rm const quad rm on partial omega admits a solution u by this we complement serrin s classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in mathbb r n the domains we construct are periodic in some variables and radial in the other variables and they bifurcate from a straight generalized cylinder or slab we also show that these domains are uniquely self cheeger relative to a period cell for the problem

on axiomatic characterisations of crisp approximation operators abstract in rough set theory one can find the result that the upper approximation defines a closure operator on the power set of the universe fixed in the paper presented first we have solved the problem under which additional assumptions a given closure operator can be represented by an upper approximation within the concepts of rough set theory second we have solved this problem within modal logic for a diamond operator represented by an arbitrary binary relation on the universe finally we have generalized these problems to lower approximations and modal box operators to rough approximations of fuzzy sets rough fuzzy sets to fuzzy approximations of crisp sets fuzzy rough sets and fuzzy approximations of fuzzy sets fuzzy diamond and fuzzy box

on the generalization of fuzzy rough sets rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets the existing generalizations of fuzzy rough sets are all based on special fuzzy relations fuzzy similarity relations t similarity relations it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches in this paper from the viewpoint of constructive approach we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined in axiomatic approach we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms the lattice and topological structures of fuzzy rough sets are also proposed in order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets a special lower approximation operator is applied to a fuzzy reasoning system which coincides with the mamdani algorithm

liouville theorems in unbounded domains for the time dependent stokes system in this paper we characterize bounded ancient solutions to the time dependent stokes system with zero boundary value in various domains including the half space

on fuzzy rough sets based on tolerance relations dubois and prade 1990 1 introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets which was originally proposed by pawlak 1982 8 later radzikowska and kerre introduced the so called i t fuzzy rough sets where i is an implication and t is a triangular norm in the present paper by using a pair of implications i j we define the so called i j fuzzy rough sets which generalize the concept of fuzzy rough sets in the sense of radzikowska and kerre and that of mi and zhang basic properties of i j fuzzy rough sets are investigated in detail

on rough approximations via ideal abstract in this paper we generalize the rough set model by defining new approximation operators in more general setting of a complete atomic boolean lattice by using an ideal an ideal on a set x is a non empty collection of subsets of x with heredity property which is also closed under finite unions we introduce the concept of lower and upper approximations via ideal in a lattice theoretical setting these decrease the upper approximation and increase the lower approximation and hence increase the accuracy properties of these approximations are studied also properties of the ordered set of the lower and upper of an element of a complete atomic boolean lattice via ideal are investigated we also study the connections between the rough approximations defined by jarvinen 10 11 and our new approximations various examples are given finally we give a new approach for defining the rough approximations w r t the induced map by using an ideal we study the connections between the rough approximations defined with respect to the induced map by using an ideal and the rough approximations defined with respect to the considered map under certain conditions of the map

symmetry and uniqueness of the solitary wave solution for the ostrovsky equation we consider herein the ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water or the capillary waves in a plasma with the effects of rotation using the modified sliding method we prove that the solitary wave moving to the left to the ostrovsky equation is symmetric about the origin and unique up to translations we also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed weak rotation and dispersive parameter

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

the fourth type of covering based rough sets as a technique for granular computing rough sets deal with the vagueness and granularity in information systems covering based rough sets have been proposed to generalize this theory for wider application three types of covering based rough sets have been studied for different situations to make the theory more complete this paper proposes a fourth type of covering based rough sets compared with the existing ones the new type shows its special characteristic in the interdependency between its lower and upper approximations we carry out a systematical study of this new theory first we discuss basic properties such as normality contraction and monotone then we investigate the conditions for this type of covering based rough sets to satisfy the properties of pawlak s rough sets and study the interdependency between the lower and upper approximation operations in addition axiomatic systems for the lower and upper approximation operations are established lastly we address the relationships between this type of covering based rough sets and the three existing ones

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

a qualitative phragmen lindelof theorem for fully nonlinear elliptic equations abstract we establish qualitative results of phragmen lindelof type for upper semicontinuous viscosity solutions of fully nonlinear partial differential inequalities of the second order in general unbounded domains of r n

stationary convection diffusion equation in an infinite cylinder abstract we study the existence and uniqueness of a solution to a linear stationary convection diffusion equation stated in an infinite cylinder neumann boundary condition being imposed on the boundary we assume that the cylinder is a junction of two semi infinite cylinders with two different periodic regimes depending on the direction of the effective convection in the two semi infinite cylinders we either get a unique solution or one parameter family of solutions or even non existence in the general case in the latter case we provide necessary and sufficient conditions for the existence of a solution

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

subharmonic functions on carnot groups many results of classical potential theory are extended to sub laplacians on carnot groups some characterizations of subharmonicity representation formulas of poisson jensen s kind and nevanlinna type theorems are proved we also characterize the riesz measure related to bounded above subharmonic functions in n

a pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature abstract we prove a pointwise gradient bound for bounded solutions of u f u 0 in possibly unbounded proper domains whose boundary has nonnegative mean curvature we also obtain some rigidity results when equality in the bound holds at some point

linear non degeneracy of the 1 d blow up limit in the phase segregation of bose einstein condensates abstract we show that the kernel of the linearization of the blow up problem at the regular part of the interface that separates segregated becs is one dimensional generated by translations in the normal direction to the interface this useful non degeneracy property was previously known only in one and two dimensions

positive solutions of some semilinear elliptic equations in r n with neumann boundary conditions abstract in this paper we consider positive classical solutions of u f u x x t r n n 3 u t h u on r n where f s is locally bounded in 0 and f s s 1 4 is non increasing in 0 for some n 2 h s is locally bounded in 0 and h s s 1 2 is non decreasing in 0 for the same we get that the possible solution only depends on t and several corollaries that include previous results of various authors are established

geometry and topology of some overdetermined elliptic problems we study necessary conditions on the geometry and the topology of domains in r2 that support a positive solution to a classical overdetermined elliptic problem u f u 0in u 0on u constanton the ideas and tools we use come from constant mean curvature surface theory in particular we obtain a partial answer to a question posed by h berestycki l caffarelli and l nirenberg in 1997 we investigate also some boundedness properties of the solution u some of our results generalize to higher dimensions

interaction between nonlinear diffusion and geometry of domain let be a domain in rn where n 2 and is not necessarily bounded we consider nonlinear diffusion equations of the form tu u let u u x t be the solution of either the initial boundary value problem over where the initial value equals zero and the boundary value equals 1 or the cauchy problem where the initial data is the characteristic function of the set rn we consider an open ball b in whose closure intersects only at one point and we derive asymptotic estimates for the content of substance in b for short times in terms of geometry of also we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to berestycki caffarelli and nirenberg these results tell us about interactions between nonlinear diffusion and geometry of domain

constructive and algebraic methods of the theory of rough sets this paper reviews and compares constructive and algebraic approaches in the study of rough sets in the constructive approach one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation different classes of rough set algebras are obtained from different types of binary relations in the algebraic approach one defines a pair of dual approximation operators and states axioms that must be satisfied by the operators various classes of rough set algebras are characterized by different sets of axioms axioms of approximation operators guarantee the existence of certain types of binary relations producing the same operators

approximation operators on complete completely distributive lattices rough set a tool for data mining deals with the vagueness and granularity in information systems in 2006 chen et al initiated the study of rough approximations on a complete completely distributive lattice ccd lattice for short and brought generalizations of rough sets into a unified framework in this paper we discuss the approximation operators on a ccd lattice based on the concept of neighborhood three kinds of upper approximation operators and a kind of lower approximation operator are constructed basic properties of lower and upper approximation operators are examined furthermore the relationships among these operators are analyzed

reasoning with rough description logics the current research progress and the existing problems of uncertain or imprecise knowledge representation and reasoning in description logics are analyzed in this paper approximate concepts are introduced to description logics based on rough set theory and a kind of new rough description logic rdl ac rough description logic based on approximate concepts is proposed based on approximate concepts the syntax semantics and properties of the rdl ac are given it is proved that the approximate concept satisfiability definitely satisfiability and possibly satisfiability reasoning problem and approximate concepts rough subsumption reasoning problem w r t rough tbox in rdl ac may be reduced to the concept satisfiability reasoning problem in almost standard alc the description logic that provides the boolean concept constructors plus the existential and universal restriction constructors the works of this paper provide logic foundations for approximate ontologies and theoretical foundations for reasoning algorithms of more expressive rough description logics including approximate concepts number restrictions nominals inverse roles and role hierarchies

on partially and globally overdetermined problems of elliptic type we consider some elliptic pdes with dirichlet and neumann data prescribed on some portion of the boundary of the domain and we obtain rigidity results that give a classication of the solution and of the domain in particular we nd mild conditions under which a partially overdetermined problem is in fact globally overdetermined this enables to use several classical results in order to classify all the domains that admit a solution of suitable general partially overdetermined problems these results may be seen as solutions of suitable inverse problems that is to say given that an overdetermined system possesses a solution we nd the shape of the admissible domains models of these type arise in several areas of mathematical physics and shape optimization

rough operations on boolean algebras in this paper we introduce two pairs of rough operations on boolean algebras first we define a pair of rough approximations based on a partition of the unity of a boolean algebra we then propose a pair of generalized rough approximations on boolean algebras after defining a basic assignment function between two different boolean algebras finally some discussions on the relationship between rough operations and some uncertainty measures are given to provide a better understanding of both rough operations and uncertainty measures on boolean algebras

rough sets based on complete completely distributive lattice in this paper a pair of rough approximation operators on a complete completely distributive ccd lattice based on an ordinary binary relation is defined this kind of rough sets can be seen as a unified framework for the study of rough sets based on ordinary binary relations rough fuzzy sets and interval valued rough fuzzy set moreover depending on classes of binary relations this paper defines several classes of rough sets on ccd lattices and investigates properties of these classes finally two generalized rough set models on two ccd lattices are given at the end of this paper

some symmetry results and liouville type theorems for solutions to semilinear equations abstract we study the one dimensional symmetry and the monotonicity of positive and bounded solutions to semilinear elliptic equations in euclidean half spaces we also prove some liouville type theorems for stable and finite morse index entire solutions

rough set analysis of a general type of fuzzy data using transitive aggregations of fuzzy similarity relations abstract in this paper we initially present and compile the tools and theories that are necessary for a rough set analysis of a general type of fuzzy data from this study we have developed two original contributions that can enhance the application of fuzzy rough sets techniques to data that may be affected by several sources of uncertainty in its measurement process for this type of analysis we have first considered the construction of homogeneous granules in which reflexivity symmetry and a certain transitivity property are satisfied this has encouraged us to study the application of t similarity relations in this framework and to examine the available theorems that permit us to aggregate this type of relations as a part of this study we have also developed a new theorem that allows us to use the dual of the generalized means as an aggregation operator in the fuzzy rough sets context another theoretical development that tries to respond to a practical problem is the concept of precision aggregation we use this new notion to export ziarko s variable precision rough set model to the analysis of fuzzy data a generalization that becomes a necessity for the fuzzy rough sets analysis of large databases the utility of this approach is demonstrated with an application example in the vehicle database donated by the turing institute

the dirichlet problem for δφ e φ in an infinite sector application to plasma equilibria abstract we consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane we prove the existence of a minimal solution to this equation and study its properties we infer from this analysis some asymptotics for the stationary solution of an equation arising in plasma physics

reduction and axiomization of covering generalized rough sets this paper investigates some basic properties of covering generalized rough sets and their comparison with the corresponding ones of pawlak s rough sets a tool for data mining the focus here is on the concepts and conditions for two coverings to generate the same covering lower approximation or the same covering upper approximation the concept of reducts of coverings is introduced and the procedure to find a reduct for a covering is given it has been proved that the reduct of a covering is the minimal covering that generates the same covering lower approximation or the same covering upper approximation so this concept is also a technique to get rid of redundancy in data mining furthermore it has been shown that covering lower and upper approximations determine each other finally a set of axioms is constructed to characterize the covering lower approximation operation

on the maximum principle for complete second order elliptic operators in general domains abstract this paper is concerned with the maximum principle for second order linear elliptic equations in a wide generality by means of a geometric condition previously stressed by berestycki nirenberg varadhan cabre was very able to improve the classical abp estimate obtaining the maximum principle also in unbounded domains such as infinite strips and open connected cones with closure different from the whole space now we introduce a new geometric condition that extends the result to a more general class of domains including the complements of hypersurfaces as for instance the cut plane the methods developed here allow us to deal with complete second order equations where the admissible first order term forced to be zero in a preceding result with cafagna depends on the geometry of the domain

flattening results for elliptic pdes in unbounded domains with applications to overdetermined problems semilinear elliptic pdes are dealt with either in the half space or in overdetermined settings we obtain several new results and also give new proofs of celebrated theorems by exploiting some geometric analysis of level sets a geometric inequality and a pointwise gradient estimate

asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains two approaches abstract in this paper we study the asymptotic behavior as x 1 of solutions of semilinear elliptic equations in quarter or half spaces for which the value at x 1 0 is given we prove the uniqueness and characterize the one dimensional or constant profile of the solutions at infinity to do so we use two different approaches the first one is a pure pde approach and it is based on the maximum principle the sliding method and some new liouville type results for elliptic equations in the half space or in the whole space r n the second one is based on the theory of dynamical systems

a uniform estimate for positive solutions of semilinear elliptic equations we consider the semilinear elliptic equation u w u with dirichlet boundary condition in a lipschitz possibly unbounded domain c n under suitable assumptions on the potential w we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate here uniform means that the estimate is independent of under some geometric restrictions on the domain we extend the analysis to the case of mixed dirichlet neumann boundary conditions as an application of our estimate we give a proof of the existence of potentials such that independent of the choice of and of the value of 0 the equation u w u has infinitely many positive solutions

axiomatic approaches to rough approximation operators on complete completely distributive lattices we proposed a pair of rough approximation operators on a complete completely distributive lattice ccd lattice for short in 2015 in this paper we further discuss its properties and study the axiomatic approaches to the rough approximation operators through these axioms fuzzy rough approximation operators can be seen as special cases of rough approximation operators on a ccd lattice we also discuss the axiomatic approaches to generalized rough sets on ccd lattices

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

concentrating solutions in a two dimensional elliptic problem with exponential neumann data abstract we consider the elliptic equation u u 0 in a bounded smooth domain in r 2 subject to the nonlinear neumann boundary condition u e u here 0 is a small parameter we prove that any family of solutions u for which e u is bounded develops up to subsequences a finite number m of peaks i in the sense that e u 2 k 1 m i as 0 reciprocally we establish that at least two such families indeed exist for any given m 1

symmetry of traveling wave solutions to the allen cahn equation in mathbb r 2 in this paper we prove even symmetry of monotone traveling wave solutions to the balanced allen cahn equation in the entire plane related results for the unbalanced allen cahn equation are also discussed

rough fuzzy functions in classification this paper generalizes the concept of rough membership functions in pattern classification tasks to rough fuzzy membership functions and rough fuzzy ownership functions unlike the rough membership value of a pattern which is sensitive only towards the rough uncertainty associated with the pattern the rough fuzzy membership or ownership value of the pattern signifies the rough uncertainty as well as the fuzzy uncertainty associated with the pattern in this paper various set theoretic properties of the rough fuzzy functions are exploited to characterize the concept of rough fuzzy sets these properties are also used to measure the rough fuzzy uncertainty associated with the given output class finally a few possible applications of the rough fuzzy functions are mentioned

the uniqueness of nondecaying solutions for the navier stokes equations the uniqueness of solutions of the navier stokes equations in the whole space is established when the velocity field is bounded and the pressure field is a bmo valued locally integrable in time function for bounded initial data here the velocity field may not decay at space infinity although there are a few results concerning uniqueness without the decay assumption our result is new and applicable for solutions constructed by solving the integral equations

one dimensional symmetry of bounded entire solutions of some elliptic equations this paper is about one dimensional symmetry properties for some entire and bounded solutions of u f u 0 in ir we consider solutions u such that 1 u 1 and u x1 xn 1 as xn uniformly with respect to x1 xn 1 under some conditions on f we prove that the solutions only depend on the variable xn we also discuss more general elliptic operators the qualitative properties then strongly depend on the coefficients of the operator these results extend to higher dimensions and to more general operators a result of ghoussoub and gui 21 proved for n 3 ams classification 35b05 35b40 35b50 35j60