existence and non existence of global solutions of the cauchy problem for higher order semilinear pseudo hyperbolic equations abstract we consider the cauchy problem for a higher order pseudo hyperbolic equation using the l p l q type estimation for the corresponding linear problem the existence and non existence criteria of global solutions are found the existence and uniqueness of smooth global solutions are also investigated we also establish the behavior of solutions and their derivatives as t

critical exponents for semilinear dissipative wave equations in rn abstract we shall present new critical exponents 1 2m n with m 1 2 to the cauchy problem utt u ut u p 1u with the initial data u 0 u 1 h 1 r n l m r n l 2 r n l m r n that is the small data global existence property can be derived to the cauchy problem above with power 1 2m n

uncertainty measures for rough formulae in rough logic an axiomatic approach rough set theory initiated by pawlak is a mathematical tool in dealing with inexact and incomplete information various types of uncertainty measure such as accuracy measure roughness measure etc which aim to quantify the imprecision of a rough set caused by its boundary region have been extensively studied in the existing literatures however a few of these uncertainty measures are explored from the viewpoint of formal rough set theory which however help to develop a kind of graded reasoning model in the framework of rough logic to solve such a problem a framework of uncertainty measure for formulae in rough logic is presented in this paper unlike the existing literatures we adopt an axiomatic approach to study the uncertainty measure in rough logic concretely we define the notion of rough truth degree by some axioms such a notion is demonstrated to be adequate for measuring the extent to which any formula is roughly true then based on this fundamental notion the notions of rough accuracy degree roughness degree for any formula and rough inclusion degree rough similarity degree between any two formulae are also proposed in addition their properties are investigated in detail these obtained results will be used to develop an approximate reasoning model in the framework of rough logic from the axiomatic viewpoint

three way group decisions with decision theoretic rough sets gdm based three way decisions as an extension model of dtrss is proposed gdm provides a new semantic interpretation for three way decisions our method explains the imprecise origin of the existing literatures on consideration of the effectiveness of group decision making gdm in practical complex decision problems we introduce gdm into three way decisions with decision theoretic rough sets dtrss and propose gdm based three way decisions gdm based three way decisions extend the range of applications of three way decisions with dtrss and provide a novel interpretation of the determination of loss functions based on dtrss we firstly focus on analyzing the determination for the loss functions under the gdm environment with the aid of the principle of justifiable granularity we adopt the important and majority suggestions of experts to measure each loss function which supports a coherent way of designing information granules in presence of numerics in this case the loss functions are determined in the form of interval valued information granule by using the interval comparison method we further deduce the three way decisions and design a corresponding decision procedure of gdm based three way decisions then an example of strategy supply selection is given to elaborate the gdm based three way decisions finally we validate the performance of our proposed method by experimental analysis

reducts within the variable precision rough sets model a further investigation abstract one fundamental aspect of the variable precision rough sets vprs model involves a search for subsets of condition attributes which provide the same information for classification purposes as the full set of available attributes such subsets are labelled approximate reducts or reducts being defined for a specified classification error denoted by this paper undertakes a further investigation of the criteria for a reduct within vprs certain anomalies and interesting implications are identified an additional condition is suggested for finding reducts which assures a more general level knowledge equivalent to that of the full set of attributes

three way decision model with two types of classification errors abstract for an uncertain target set three disjoint regions i e positive negative and boundary regions are established in a rough approximation space to improve the fault tolerance ability of the rough set model a probabilistic rough set model with a pair of thresholds is proposed and two types of classification errors are introduced from the viewpoint of probability and statistics a decision theoretic rough set model is proposed to minimize the decision risk and a pair of thresholds of the probabilistic rough set model can be calculated by the given cost parameters in this paper first two types of classification errors and two types of uncertain classifications are defined in the probabilistic rough set model then by considering four cost parameters that characterize the cost of two types of classification errors and two types of uncertain classifications a pair of thresholds of the probabilistic rough set model can again be obtained finally with the increase in new attributes the change regularities of the total cost with changing rough approximation spaces are discussed in detail and experimental results show that the proposed model for three way decision is reasonable and effective these results further enrich three way decision theory to effectively deal with uncertain classification problems

a novel and better fitness evaluation for rough set based minimum attribute reduction problem the minimum attribute reduction mar problem in the context of rough set theory is known to be np hard one popular way of dealing with this problem is to first transform it into a fitness maximization problem over a multi dimensional boolean space and to then solve this problem using population based stochastic optimization algorithms it is therefore important to have an appropriate fitness function in this paper two examples are presented to show that existing fitness functions either do not guarantee optimality equivalence between the mar problem and the transformed fitness maximization problem or may produce the so called overemphasis phenomenon that affects the performance of population based stochastic optimization algorithms to overcome these drawbacks we propose a new fitness function that we prove both guarantees the optimality equivalence and reduces the overemphasis phenomenon experimental results show that the proposed fitness function is better than existing fitness functions in terms of solution quality

the relationship between attribute reducts in rough sets and minimal vertex covers of graphs the problems to find attribute reduction in rough sets and to obtain the minimal vertex cover for graphs are both np hard problems this paper studies the relationship between the two problems the vertex cover problem for graphs from the perspective of rough sets is first investigated the attribute reduction of an information system is then studied in the framework of graph theory the results in this paper show that finding the minimal vertex cover of a graph is equivalent to finding the attribute reduction of an information system induced from the graph conversely the attribute reduction computation can be translated into the calculation of the minimal vertex cover of a derivative graph finally a new algorithm for the vertex cover problem based on rough sets is presented furthermore experiments are conducted to verify the effectiveness of the proposed method

classification of global phase portraits and bifurcation diagrams of hamiltonian systems with rational potential in this paper we study the global dynamics of the hamiltonian systems x hy x y y hx x y where the hamiltonian function h has the particular form h x y y2 2 p x q x h x y y2 2 p x q x p x q x r x p x q x r x are polynomials in particular h is the sum of the kinetic and a rational potential energies firstly we provide the normal forms by a suitable symplectic change of variables then the global topological classification of the phase portraits of these systems having canonical forms in the poincare disk in the cases where degree p 0 1 2degree p 0 1 2 and degree q 0 1 2degree q 0 1 2 are studied as a function of the parameters that define each polynomial we use a blow up technique for finite equilibrium points and the poincare compactification for the infinite equilibrium points finally we show some applications

reduction target structure based hierarchical attribute reduction for two category decision theoretic rough sets abstract attribute reduction is an essential subject in rough set theory but because of quantitative extension it becomes a problem when considering probabilistic rough set prs approaches the decision theoretic rough set dtrs has a threshold semantics and decision feature and thus becomes a typical and fundamental prs based on reduction target structures this paper investigates hierarchical attribute reduction for a two category dtrs and is divided into five parts 1 the knowledge preservation property and reduct are explored by knowledge coarsening 2 the consistency preservation principle and reduct are constructed by a consistency mechanism 3 region preservation is analyzed and the separability between consistency preservation and region preservation is concluded thus the double preservation principle and reduct are studied 4 structure targets are proposed by knowledge structures and an attribute reduction is further described and simulated thus general reducts are defined to preserve the structure targets or optimal measures 5 the hierarchical relationships of the relevant four targets and reducts are analyzed and a decision table example is provided for illustration this study offers promotion rationality structure hierarchy and generalization and it establishes a fundamental reduction framework for two category dtrs the relevant results also provide some new insights into the attribute reduction problem for prs

attribute reduction of data with error ranges and test costs in data mining applications we have a number of measurement methods to obtain a data item with different test costs and different error ranges test costs refer to time money or other resources spent in obtaining data items related to some object observational errors correspond to differences in measured and true value of a data item in supervised learning we need to decide which data items to obtain and which measurement methods to employ so as to minimize the total test cost and help in constructing classifiers this paper studies this problem in four steps first data models are built to address error ranges and test costs second error range based covering rough set is constructed to define lower and upper approximations positive regions and relative reducts a closely related theory deals with neighborhood rough set which has been successfully applied to heterogeneous attribute reduction the major difference between the two theories is the definition of neighborhood third the minimal test cost attribute reduction problem is redefined in the new theory fourth both backtrack and heuristic algorithms are proposed to deal with the new problem the algorithms are tested on ten uci university of california irvine datasets experimental results show that the backtrack algorithm is efficient on rational sized datasets the weighting mechanism for the heuristic information is effective and the competition approach can improve the quality of the result significantly this study suggests new research trends concerning attribute reduction and covering rough set

quick attribute reduction in inconsistent decision tables this paper focuses on three types of attribute reducts in inconsistent decision tables assignment reduct distribution reduct and maximum distribution reduct it is quite inconvenient to judge these three types of reduct directly according to their definitions this paper proposes judgment theorems for the assignment reduct the distribution reduct and the maximum distribution reduct which are expected to greatly simplify the judging of these three types of reducts on this basis we derive three new types of attribute significance measures and construct the q ara quick assignment reduction algorithm the q dra quick distribution reduction algorithm and the q mdra quick maximum distribution reduction algorithm these three algorithms correspond to the three types of reducts we conduct a series of comparative experiments with twelve uci machine learning data repository university of california at irvine data sets including consistent and inconsistent decision tables to evaluate the performance of the three reduction algorithms proposed with the relevant algorithm quickreduct 9 34 the experimental results show that quickreduct possesses weak robustness because it cannot find the reduct even for consistent data sets whereas our proposed three algorithms show strong robustness because they can find the reduct for each data set in addition we compare the q dra quick distribution reduction algorithm with the cebarknc conditional entropy based algorithm for reduction of knowledge without a computing core 43 because both find the distribution reduct by using a heuristic search the experimental results demonstrate that q dra runs faster than cebarknc does because the distribution function of q dra has a lower calculation cost instructive conclusions for these reduction algorithms are drawn from the perspective of classification performance for the c4 5 and rbf svm classifiers last we make a comparison between discernibility matrix based methods and our algorithms the experimental results indicate that our algorithms are efficient and feasible

relative reducts in consistent and inconsistent decision tables of the pawlak rough set model a relative reduct can be considered as a minimum set of attributes that preserves a certain classification property this paper investigates three different classification properties and suggests three distinct definitions accordingly in the pawlak rough set model while the three definitions yield the same set of relative reducts in consistent decision tables they may result in different sets in inconsistent tables relative reduct construction can be carried out based on a discernibility matrix the study explicitly stresses a fact that the definition of a discernibility matrix should be tied to a certain property regarding the three classification properties we can define three distinct definitions accordingly based on the common structure of the specific definitions of relative reducts and discernibility matrices general definitions of relative reducts and discernibility matrices are suggested

minimum cost attribute reduction in decision theoretic rough set models in classical rough set models attribute reduction generally keeps the positive or non negative regions unchanged as these regions do not decrease with the addition of attributes however the monotonicity property in decision theoretic rough set models does not hold this is partly due to the fact that all regions are determined according to the bayesian decision procedure consequently it is difficult to evaluate and interpret region preservation attribute reduction in decision theoretic rough set models this paper provides a new definition of attribute reduct for decision theoretic rough set models the new attribute reduction is formulated as an optimization problem the objective is to minimize the cost of decisions theoretical analysis shows the meaning of the optimization problem both the problem definition and the objective function have good interpretation a heuristic approach a genetic approach and a simulated annealing approach to the new problem are proposed experimental results on several data sets indicate the efficiency of these approaches

attribute reduction in decision theoretic rough set models rough set theory can be applied to rule induction there are two different types of classification rules positive and boundary rules leading to different decisions and consequences they can be distinguished not only from the syntax measures such as confidence coverage and generality but also the semantic measures such as decision monotocity cost and risk the classification rules can be evaluated locally for each individual rule or globally for a set of rules both the two types of classification rules can be generated from and interpreted by a decision theoretic model which is a probabilistic extension of the pawlak rough set model as an important concept of rough set theory an attribute reduct is a subset of attributes that are jointly sufficient and individually necessary for preserving a particular property of the given information table this paper addresses attribute reduction in decision theoretic rough set models regarding different classification properties such as decision monotocity confidence coverage generality and cost it is important to note that many of these properties can be truthfully reflected by a single measure c in the pawlak rough set model on the other hand they need to be considered separately in probabilistic models a straightforward extension of the c measure is unable to evaluate these properties this study provides a new insight into the problem of attribute reduction

probabilistic model criteria with decision theoretic rough sets in dealing with risk in real decision problems decision theoretic rough sets with loss functions aim to obtain optimization decisions by minimizing the overall risk with bayesian decision procedures two parameters generated by loss functions divide the universe into three regions as the decision of acceptance deferment and rejection in this paper we discuss the semantics of loss functions and utilize the differences of losses replace actual losses to construct a new four level approach of probabilistic rules choosing criteria ten types of probabilistic rough set models can be generated by the four level approach and form two groups of models two way probabilistic decision models and three way probabilistic decision models a reasonable decision with these criteria is demonstrated by an illustration of oil investment

an expanded double quantitative model regarding probabilities and grades and its hierarchical double quantitative attribute reduction a double quantitative model on probabilities and grades is proposed by integration the model with quantitative semantics expands dtrs model grs model pawlak model model regions are hierarchically and optimally calculated via two algorithms three reduct types srp reduct crp reduct app reduct are hierarchically studied the double quantitative reducts expand basic qualitative and quantitative reducts probabilities and grades serve as relative and absolute measures respectively they are used to establish the decision theoretic rough set dtrs and graded rough set grs two basic quantitative models the double quantification of probabilities and grades exhibits systematicness and completeness in view of the two dimensional feature of the approximate space however double quantitative construction becomes a problem and double quantitative reduction is rarely reported thus this paper mainly constructs an expanded double quantitative model by logically integrating probabilities and grades it further studies relevant double quantitative reduction by hierarchically preserving specific regions 1 first a novel model is established via the logic integration and expansion requirement and its regional system and granular hierarchy are studied via granular computing thus regional semantics is extracted via basic semantics granules regional calculation is realized by two algorithms and the algorithm regarding calculation granules exhibits optimization according to algorithm analyses 2 second three types of model regional preservation reducts and their hierarchy are discussed in the two category case thus srp reduct crp reduct and app reduct are studied by exploring four region preservation properties constructing two region classification regions and preserving four original approximations respectively furthermore a relevant reduction hierarchy is thoroughly achieved 3 moreover the model and its reduction are illustrated by two examples of decision tables the constructional model conducts double quantification regarding probabilities and grades thus it exhibits double quantitative semantics and benignly expands dtrs model grs model and pawlak model furthermore its hierarchical reduction reflects some double quantitative reduction essence thus its reduction expands qualitative pawlak reduction while guides quantitative dtrs reduction and grs reduction

rough set model based on formal concept analysis this paper proposes a rough set model based on formal concept analysis in this model a solution to an algebraic structure problem is first provided in an information system a lattice structure is inferred from the information system and corresponding nodes are called rough concepts how to deal with common problems in rough set theory based on rough concepts is then explored such as upper and lower approximation operators reducts and cores decision dependency has become a common form of knowledge representation owing to its properties of expressiveness and ease of understanding so it has been widely used in practice finally application of rough concepts to the extraction of decision dependencies from a decision table is studied a complete and non redundant set of decision dependencies can be obtained from a decision table examples demonstrate that application of the method presented in this paper is valid and practicable the results not only provide a better understanding of rough set theory from the perspective of formal concept analysis but also demonstrate a new way of combining rough set theory and formal concept analysis

nested structure in parameterized rough reduction abstract in this paper by strict mathematical reasoning we discover the relationship between the parameters and the reducts in parameterized rough reduction this relationship named the nested reduction shows that the reducts act as a nested structure with the monotonically increasing parameter we present a systematic theoretical framework that provides some basic principles for constructing the nested structure in parameterized rough reduction some specific parameterized rough set models in which the nested reduction can be constructed are pointed out by strict mathematical reasoning based on the nested reduction we design several quick algorithms to find a different reduct when one reduct is already given here different refers to the reducts obtained on the different parameters all these algorithms are helpful for quickly finding a proper reduct in the parameterized rough set models the numerical experiments demonstrate the feasibility and the effectiveness of the nested reduction approach

comparison of the probabilistic approximate classification and the fuzzy set model abstract it is shown that the generalized notion probabilistic approximate classification of rough sets can be conveniently described by the concept of fuzzy sets a discussion of the proper choice of the definition for the membership function of the intersection union of fuzzy sets is also presented however from the point of view of the probabilistic approximation space it is argued that there does not exist a universal definition for the fuzzy intersection union operation

global phase portraits and bifurcation diagrams for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis abstract this paper studies the global dynamics for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis by linear changes of variables it can be written as four systems one of them has been characterized in llibre 2018 in this paper the remaining three systems are investigated firstly the normal forms of these systems are given by an algebraic classification of their infinite singular points then we classify the global phase portraits of these systems having canonical forms on the poincare disk moreover we obtain the bifurcation diagrams for the corresponding global phase portraits combining this work with llibre 2018 we provide the complete classification of global phase portraits for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis

existence non existence and asymptotic behavior of global solutions to the cauchy problem for systems of semilinear hyperbolic equations with damping terms abstract we consider the cauchy problem for systems of semilinear hyperbolic equations using the l p l q type estimation for the corresponding linear parts the existence and uniqueness of weak global solutions are investigated we also established the behavior of solutions and their derivatives as t using the method of test functions developed in the works mitidieri and pokhozhaev 2001 11 veron and pohozaev 2001 12 and caristi 2000 23 we obtain the analogue of the fujita hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping it follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense

rough set based logics for multicriteria decision analysis in this paper we propose some decision logic languages for rule representation in rough set based multicriteria analysis the semantic models of these logics are data tables each of which is comprised of a finite set of objects described by a finite set of criteria attributes the domains of the criteria may have ordinal properties expressing preference scales while the domains of the attributes may not the validity support and confidence of a rule are defined via its satisfaction in the data table

phase portraits of separable hamiltonian systems abstract we study a generalization of potential hamiltonian systems h x y y 2 f x with one degree of freedom namely those with hamiltonian functions of type h x y f x g y which will be denoted by x h we present an algorithm to obtain the phase portrait including the behaviour at infinity of x h when f and g are arbitrary polynomials indeed from the graphs of the one variable functions f and g we are able to give the full description on the poincare disk therefore extending the well known method to obtain the phase portrait of potential systems in the finite plane the fact that the phase portraits can be fully described in terms of the two one variable real functions f and g allows as well a complete study of the bifurcation diagrams in complete families of vector fields the algorithm can be applied to study separable hamiltonian systems with one degree of freedom which include a vast amount of examples in physical applications

hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields we provide normal forms and the global phase portraits in the poincare disk for all hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields

decision region distribution preservation reduction in decision theoretic rough set model abstract in the pawlak rough set model the positive region the boundary region and the non negative region are monotonic with respect to the set inclusion of attributes however the monotonicity property of the decision regions positive region boundary region or non negative region with respect to the set inclusion of attributes does not hold in the decision theoretic rough set model therefore the decision regions may be changed after attribute reduction based on quantitative preservation or qualitative preservation of decision regions this effect is observed partly because three decision regions are defined by introducing the probabilistic threshold values in addition heuristic reduction algorithms based on decision regions may find super reducts because of the non monotonicity of decision regions to address the above issues this paper proposes solutions to the attribute reduction problem based on decision region preservation in the decision theoretic rough set model first the positive region distribution preservation reduct the boundary region distribution preservation reduct and the negative region distribution preservation reduct are introduced into the decision theoretic rough set model second three new monotonic measures are constructed by considering variants of the conditional information entropy from which we can obtain the heuristic reduction algorithms the results of the experimental analysis validate the monotonicity of new measures and verify the effectiveness of decision region distribution preservation reducts

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

a novel method for attribute reduction of covering decision systems attribute reduction has become an important step in pattern recognition and machine learning tasks covering rough sets as a generalization of classical rough sets have attracted wide attention in both theory and application this paper provides a novel method for attribute reduction based on covering rough sets we review the concepts of consistent and inconsistent covering decision systems and their reducts and we develop a judgment theorem and a discernibility matrix for each type of covering decision system furthermore we present some basic structural properties of attribute reduction with covering rough sets based on a discernibility matrix we develop a heuristic algorithm to find a subset of attributes that approximate a minimal reduct finally the experimental results for uci data sets show that the proposed reduction approach is an effective technique for addressing numerical and categorical data and is more efficient than the method presented in the paper d g chen c z wang q h hu a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets information sciences 177 17 2007 3500 3518

quantitative qualitative region change uncertainty certainty in attribute reduction knowledge coarsening is investigated to describe attribute deletion granule merging and its region distribution are used to gain region change functions region change with certainty monotonicity is analyzed in qualitative pawlak model region change with uncertainty drifting is analyzed in quantitative dtrs model attribute reduction is a fundamental research theme in rough sets and granular computing grc its scientific construction originally depends on the region change law at present only region change non monotonicity monotonicity is mined in the quantitative qualitative model the in depth region change truth and its grc mechanism have significance especially for follow up attribute reduction this paper commences probing region change essence mainly from a novel uncertainty certainty viewpoint concretely we make comparative region change analyses based on grc by resorting to the qualitative pawlak model and quantitative dtrs model the decision theoretic rough set model 1 knowledge coarsening is investigated to describe attribute deletion 2 granule merging and its region distribution are studied to probe region change functions 3 region change is analyzed in pawlak model to mine qualitative region change certainty and its relevant properties 4 region change is analyzed in dtrs model to mine quantitative region change uncertainty and its relevant properties 5 comparative region change analyses are summarized and further experiment verification is provided knowledge coarsening and granule merging establish grc mechanisms for extensive region change analyses quantitative qualitative region change uncertainty certainty and relevant principles are discovered via dtrs model pawlak model by virtue of the grc technology and comparative strategy this study reveals region change uncertainty certainty to deepen region change non monotonicity monotonicity furthermore it underlies attribute reduction especially with regard to quantitative models

quantitative information architecture granular computing and rough set models in the double quantitative approximation space of precision and grade because precision and grade act as fundamental quantitative information in approximation space they are used in relative and absolute quantifications respectively at present the double quantification regarding precision and grade is a novel and valuable subject but quantitative information fusion has become a key problem thus this paper constructs the double quantitative approximation space of precision and grade pg approx space and tackles the fusion problem using normal logical operations it further conducts double quantification studies on granular computing and rough set models 1 first for quantitative information organization and storage we construct space and plane forms of pg approx space using the cartesian product and for quantitative information extraction and fusion we establish semantics construction and semantics granules of pg approx space 2 second by granular computing we investigate three primary granular issues quantitative semantics complete system and optimal calculation accordingly six types of fundamental granules are proposed based on the semantic microscopic and macroscopic descriptions their semantics forms structures calculations and relationships are studied and the granular hierarchical structure is achieved 3 finally we investigate rough set models in pg approx space accordingly model regions are proposed by developing the classical regions model expansion is systematically analyzed some models are constructed as their structures are obtained and a concrete model is provided based on the quantitative information architecture this paper systematically conducts and investigates double quantification and establishes a fundamental and general exploration framework

z2 equivariant linear type bi center cubic polynomial hamiltonian vector fields abstract we study the global dynamical behavior of z 2 equivariant cubic hamiltonian vector fields with a linear type bi center at 1 0 by using a series of symbolic computation tools we obtain all possible phase portraits of these z 2 equivariant hamiltonian systems

damped wave equation with a critical nonlinearity we study large time asymptotics of small solutions to the cauchy problem for nonlinear damped wave equations with a critical nonlinearity 2 t u t u u u 1 2 n 0 x r n t 0 u 0 x eu 0 x t u 0 x eu 1 x x r n where e 0 and space dimensions n 1 2 3 assume that the initial data uo h 0 n h 0 u 1 e h 1 0 n h 1 where n 2 weighted sobolev spaces are h l m e l2 m l x l 2 1 x 2 also we suppose that 2 n 0 u 0 x dx 0 where then we prove that there exists a positive e 0 such that the cauchy problem above has a unique global solution u c 0 oo h 0 satisfying the time decay property u t e g t x e t lp ce 1 2 n g 1 n 2 t n 2 1 1 p for all t 0 1 p where e 0 e 0

three way decisions with probabilistic rough sets the rough set theory approximates a concept by three regions namely the positive boundary and negative regions rules constructed from the three regions are associated with different actions and decisions which immediately leads to the notion of three way decision rules a positive rule makes a decision of acceptance a negative rule makes a decision of rejection and a boundary rule makes a decision of abstaining this paper provides an analysis of three way decision rules in the classical rough set model and the decision theoretic rough set model the results enrich the rough set theory by ideas from bayesian decision theory and hypothesis testing in statistics the connections established between the levels of tolerance for errors and costs of incorrect decisions make the rough set theory practical in applications

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two

hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields abstract we provide normal forms and the global phase portraits in the poincare disk for all the hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields

bifurcation diagrams for hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields as a natural continuation of the work done in 7 we provide the bifurcation diagrams for the global phase portraits in the poincare disk of all the hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields

modeling rough granular computing based on approximation spaces the results reported in this paper create a step toward the rough set based foundations of data mining and machine learning the approach is based on calculi of approximation spaces in this paper we present the summarization and extension of our results obtained since 2003 when we started investigations on foundations of approximation of partially defined concepts see e g 2 3 7 37 20 21 5 42 39 38 40 we discuss some important issues for modeling granular computations aimed at inducing compound granules relevant for solving problems such as approximation of complex concepts or selecting relevant actions plans for reaching target goals the problems discussed in this article are crucial for building computer systems that assist researchers in scientific discoveries in many areas such as biology in this paper we present foundations for modeling of granular computations inside of system that is based on granules called approximation spaces our approach is based on the rough set approach introduced by pawlak 24 25 approximation spaces are fundamental granules used in searching for relevant complex granules called as data models e g approximations of complex concepts functions or relations in particular we discuss some issues that are related to generalizations of the approximation space introduced in 33 34 we present examples of rough set based strategies for the extension of approximation spaces from samples of objects onto a whole universe of objects this makes it possible to present foundations for inducing data models such as approximations of concepts or classifications analogous to the approaches for inducing different types of classifiers known in machine learning and data mining searching for relevant approximation spaces and data models are formulated as complex optimization problems the proposed interactive granular computing systems should be equipped with efficient heuristics that support searching for semi optimal granules

a general model of decision theoretic three way approximations of fuzzy sets based on a heuristic algorithm abstract a model of decision theoretic three way approximations of fuzzy sets that can calculate a pair of interpretable thresholds by combining shadowed sets with three way decisions was established by deng in that model the significant contribution is that the value 0 5 which denotes a membership grade with the highest uncertainty is used to replace the uncertain region with the interval 0 1 in shadowed sets from the principle of the minimum decision cost although the lowest overall cost can be achieved precisely in some data distributions based on the value 0 5 there is a difference between the obtained overall cost and the least cost in some cases therefore in this paper based on deng s model the concept of a general three way approximation of a fuzzy set is proposed to replace 0 5 with a variable value c 0

approaches to knowledge reduction based on variable precision rough set model this paper deals with approaches to knowledge reduction based on variable precision rough set model the concepts of lower distribution reduct and upper distribution reduct based on variable precision rough sets vprs are first introduced their equivalent definitions are then given and the relationships among lower and upper distribution reducts and alternative types of knowledge reduction in inconsistent systems are investigated it is proved that for some special thresholds lower distribution reduct is equivalent to the maximum distribution reduct whereas upper distribution reduct is equivalent to the possible reduct the judgement theorems and discernibility matrices associated with the lower and upper distribution reducts are also established from which we can obtain the approaches to knowledge reduction in vprs