rough prime ideals and rough fuzzy prime ideals in semigroups in this paper we shall introduce the notions of rough prime ideals and rough fuzzy prime ideals in a semigroup and give some properties of such ideals also we will discuss the relations between the upper lower rough prime ideals and the upper lower approximations of their homomorphism images

affine automata and related techniques for generation of complex images abstract in this paper we introduce probabilistic affine automata paa which are probabilistic finite generators having transitions labeled with affine transformations it is shown that paa are capable of generating highly complex images barnsley s 1988 ifs method to generate fractals is a special case of paa when the automaton happens to have only a single state a number of theoretical results on paa are shown the relationship between paa mutually recursive function systems mrfs and affine regular sets is investigated

succinct representation of regular languages by boolean automata ii abstract boolean automata are a generalization of finite automata in the sense that the next state the result of the transition function given a state and a letter is not just a single state deterministic automata or a set of states nondeterministic automata but a boolean function of the states boolean automata accept precisely the regular languages also they correspond in a natural way to certain language equation involving complementation as well as to sequential networks in a previous note we showed that for every n 1 there exists a boolean automaton b n with n states such that the smallest deterministic automaton for the same language has 2 2 n states in the present note we will show a precisely attainable lower bound on the succinctness of representing regular languages by boolean automata namely we will show that for every n 1 there exists a reduced automaton d n with n states such that the smallest boolean automaton accepting the same language has also n states

quantitative analysis for covering based rough sets through the upper approximation number coverings are a useful form of data while covering based rough sets provide an effective tool for dealing with this data covering based rough sets have been widely used in attribute reduction and rule extraction however few quantitative analyses for covering based rough sets have been conducted while many advances for classical rough sets have been obtained through quantitative tools in this paper the upper approximation number is defined as a measurement to quantify covering based rough sets and a pair of upper and lower approximation operators are constructed using the approximation number the operators not only inherit some important properties of existing approximation operators but also exhibit some new quantitative characteristics it is interesting to note that the upper approximation number of a covering approximation space is similar to the dimension of a vector space or the rank of a matrix

roughness in cayley graphs in this paper rough approximations of cayley graphs are studied and rough edge cayley graphs are introduced furthermore a new algebraic definition for pseudo cayley graphs containing cayley graphs is proposed and a rough approximation is expanded to pseudo cayley graphs in addition rough vertex pseudo cayley graphs and rough pseudo cayley graphs are introduced some theorems are provided from which properties such as connectivity and optimal connectivity are derived this approach opens new research fields such as data networks

existence and mann iterative approximations of nonoscillatory solutions of nth order neutral delay differential equations abstract in this paper we consider the following nth order neutral delay differential equation d n d t n x t c x t 1 n 1 f t x t 1 x t 2 x t k g t t t 0 where n is a positive integer c r 0 i 0 for i 1 k f c t 0 r k r and g c t 0 r by employing the contraction mapping principle we obtain several existence results of nonoscillatory solutions for the above equation construct a few mann type iterative approximation schemes for these nonoscillatory solutions and establish several error estimates between the approximate solutions and the nonoscillatory solutions in addition we obtain some sufficient conditions for the existence of infinitely many nonoscillatory solutions these results presented in this paper extend improve and unify many known results due to cheng and annie j f cheng z annie existence of nonoscillatory solution to second order linear neutral delay equation j systems sci math sci 24 2004 389 397 in chinese graef yang and zhang j r graef b yang b g zhang existence of nonoscillatory and oscillatory solutions of neutral differential equations with positive and negative coefficients math bohem 124 1999 87 102 kulenovic and had iomerspahic m r s kulenovic s had iomerspahic existence of nonoscillatory solution of second order linear neutral delay equation j math anal appl 228 1998 436 448 m r s kulenovic s had iomerspahic existence of nonoscillatory solution for linear neutral delay equation fasc math 32 2001 61 72 zhang and yu b g zhang j s yu on the existence of asymptotically decaying positive solutions of second order neutral differential equations j math anal appl 166 1992 1 11 zhang b g zhang on the positive solutions of a kind of neutral equations acta math appl sinica 19 1996 222 230 and zhou and zhang y zhou b g zhang existence of nonoscillatory solutions of higher order neutral differential equations with positive and negative coefficients appl math lett 15 2002 867 874 and others some nontrivial examples are given to illustrate the advantages of our results

the existence of positive solutions and oscillation of solutions of higher order difference equations with forcing terms sufficient conditions are established for the existence of positive solutions and oscilla tion of all bounded solutions of the neutral difference equation ap zn can l t qnf zn k hn n no where a is the forward difference operator azn xn l xn i and k are integers c i is a real number and qn and hn are real sequences it is also shown that some of these sufficient conditions are necemary 1998 elsevier science ltd all rights reserved

rough ideals in semigroups abstract we shall introduce the notion of a rough left right two sided bi ideal in a semigroup which is an extended notion of a left right two sided bi ideal in a semigroup and give some properties of such ideals

on the hopcroft s minimization technique for dfa and dfca we show that the absolute worst case time complexity for hopcroft s minimization algorithm applied to unary languages is reached only for deterministic automata or cover automata following the structure of the de bruijn words a previous paper by berstel and carton gave the example of de bruijn words as a language that requires o nlogn steps in the case of deterministic automata by carefully choosing the splitting sets and processing these sets in a fifo mode for the list of the splitting sets in the algorithm we refine the previous result by showing that the berstel carton example is actually the absolute worst case time complexity in the case of unary languages for deterministic automata we show that the same result is valid also for the case of cover automata and an algorithm based on the hopcroft s method used for minimization of cover automata we also show that a lifo implementation for the splitting list will not achieve the same absolute worst time complexity for the case of unary languages both in the case of regular deterministic finite automata or in the case of the deterministic finite cover automata as defined by s yu

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

on the structure of rough prime primary ideals and rough fuzzy prime primary ideals in commutative rings this paper is a continuation of ideas presented by davvaz roughness in rings inform sci 164 2004 147 163 roughness based on fuzzy ideals inform sci 176 2006 2417 2437 we introduce the notions of rough prime primary ideals and rough fuzzy prime primary ideals in a ring and give some properties of such ideals also we discuss the relations between the upper and lower rough prime primary ideals and the upper and lower approximations of their homomorphism images

necessary and sufficient conditions for oscillations of second order neutral equations consider the neutral delay differential equation d2dt2 y t py t qy t 0 1 where p q r and r we proved that eq 1 oscillates if and only if the characteristic equation 2 2pe qe 0 has no real roots

a fast approach to attribute reduction in incomplete decision systems with tolerance relation based rough sets efficient attribute reduction in large incomplete decision systems is a challenging problem existing approaches have time complexities no less than o c 2 u 2 this paper derives some important properties of incomplete information systems then constructs a positive region based algorithm to solve the attribute reduction problem with a time complexity no more than o c 2 u log u furthermore our approach does not change the size of the original incomplete system numerical experiments show that the proposed approach is indeed efficient and therefore of practical value to many real world problems the proposed algorithm can be applied to both consistent and inconsistent incomplete decision systems

a short note on algebraic t rough sets our aim in this paper is to introduce the concept of a set valued homomorphism for groups which is a generalization of ordinary homomorphism then using the definitions of lower inverse and upper inverse we introduce the definition of uniform set valued homomorphism and prove that every set valued homomorphism is uniform finally the concept of probabilistic t rough set is defined

roughness in rings in this paper basic notions of the rough set theory will be given in fact the paper concerns a relationship between rough sets and ring theory we shall introduce the notion of rough subring resp ideal with respect to an ideal of a ring which is an extended notion of a subring resp ideal in a ring and we shall give some properties of the lower and the upper approximations in a ring

rough approximation operators on r0 algebras nilpotent minimum algebras with an application in formal logic l an abstract axiomatization to pawlak rough set theory in the context of r0 algebras equivalently nm algebras has been proposed in the present paper more precisely by employing the conjunction operator and the disjunction operator in r0 algebras the notions of rough upper approximation operator u and rough lower approximation operator l on r0 algebras are proposed respectively owing to the logical properties of and any r0 algebra equipped with l and u forms an abstract approximation space in the sense of g cattaneo a duality relationship between the set of lower crisp elements and the set of upper crisp elements is established and some important properties are examined moreover its connection with tarski closure interior approximation space and halmos closure interior approximation is studied such a pair of rough approximations on r0 algebras can naturally induce a pair of rough upper lower truth degrees for formulae in l some uncertainty measures such as roughness degree and accuracy degree are subsequently presented and two kinds of approximate reasoning methods merging rough approximation and fuzzy logic are eventually established

roughness in mv algebras in this paper by considering the notion of an mv algebra we consider a relationship between rough sets and mv algebra theory we introduce the notion of rough ideal with respect to an ideal of an mv algebra which is an extended notion of ideal in an mv algebra and we give some properties of the lower and the upper approximations in an mv algebra

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

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

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 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

roughness and fuzziness in quantales abstract rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups rings and lattices in this paper we introduce first the notions of fuzzy prime semi prime primary ideals of quantales and investigate their properties several characterizations of such ideals are presented then we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales and establish relationships between these ideals and fuzzy prime ideals of quantales by applying rough set theory to fuzzy ideals of quantales we furthermore define rough fuzzy prime semi prime primary ideals of quantales generalizing yang and xu s work on quantales to the fuzzy environment finally relationships between the upper resp lower rough fuzzy ideals of quantales and the upper resp lower approximations of their homomorphic images are also discussed

rough sets and intelligent data analysis rough set based data analysis starts from a data table called an information system the information system contains data about objects of interest characterized in terms of some attributes often we distinguish in the information system condition and decision attributes such information system is called a decision table the decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table with every decision table a set of decision rules called a decision algorithm can be associated it is shown that every decision algorithm reveals some well known probabilistic properties in particular it satisfies the total probability theorem and the bayes theorem these properties give a new method of drawing conclusions from data without referring to prior and posterior probabilities inherently associated with bayesian reasoning

roughness in quantales rough set theory is a useful mathematical tool for dealing with the uncertainty and granularity in information systems in this paper rough approximations are introduced into quantales a kind of partially ordered algebraic structure with an associative binary multiplication the notions of upper lower rough prime semi prime primary ideals of quantales are proposed and verified to be the extended notions of usual prime semi prime primary ideals of quantales respectively global order properties of all lower upper rough ideals of a quantale are also investigated the rough radical of an upper rough ideal in a quantale is defined and the prime radical theorem of rings is generalized to upper rough ideals of quantales some results about homomorphic images of rough ideals and rough prime ideals of semigroups and rings are also generalized and improved in the field of quantales

reference points and roughness in this paper we introduce the notion of a reference point and provide local approximations for a subset of the universe the notion of a reference point naturally gives rise to a rough approximations framework wherein several approximations are possible on the same set also we present an extension to the decision theoretic rough set model by using reference points

roughness in modules the theory of rough sets was proposed by pawlak in 1982 the relations between rough sets and algebraic systems have been already considered by many mathematicians important algebraic structures are groups rings and modules rough groups and rough rings have been investigated by biswas and nanda kuroki and wang and davvaz in this paper we consider an r module as a universal set and we introduce the notion of rough submodule with respect to a submodule of an r module which is an extended notion of a submodule in an r module we also give some properties of the lower and the upper approximations in an r module

the existence of positive solution and bounded oscillation for even order neutral differential equations with unstable type abstract in this paper we consider the n order neutral delay differential equation with unstable type x t p t x t n q t x t 1 x t t t 0 where n 2 is an even integer 1 0 0 and p q c t 0 r we first prove that this equation always has an unbounded positive solution then some new bounded oscillation and nonoscillation criteria are obtained

existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations this paper studies a few existence results of nonoscillatory solutions for the following higher order nonlinear neutral delay differential equation dndtnx t cx t 1 n i 1didtih t x h1 t x h2 t x hk t 1 n 1f t x f1 t x f2 t x fk t g t t t0 where n is a positive integer 0 i n 1 c r 1 0 h ci t0 rk r f c t0 rk r g c t0 r hl ci t0 r and fl c t0 r with limt hl t limt fl t l 1 k constructs several mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions in addition sufficient conditions for the existence of infinitely many nonoscillatory solutions for the above equation are given three nontrivial examples are given to illustrate the advantages of our results

oscillation theorems for second order neutral functional differential equations some new oscillation criteria are given for second order neutral functional differential equations of the form d 2dt 2 x t lx t t mx t s qx t a px t b 0 where l t m s p and q are real numbers and a and b are real positive constants

on rough set and fuzzy sublattice let l be a lattice with the least element 0 and the greatest element 1 and let be a full congruence relation on l in this paper the notion of upper and lower approximations of a fuzzy subset of l is introduced and some important properties will be studied

efficient implementation of regular languages using reversed alternating finite automata alternating finite automata afa provide a natural and succinct way to denote regular languages we introduce a bit wise representation of reversed afa r afa transition functions and describe an efficient implementation method for r afa and their operations using this representation experiments have shown that this implementation is much more efficient than for instance the grail dfa implementation in both space and time

the lower and upper approximations in a quotient hypermodule with respect to fuzzy sets this paper provides a continuation of ideas presented by davvaz and mahdavipour b davvaz m mahdavipour roughness in modules inform sci 176 2006 3658 3674 the notion of hypermodule is a generalization of the notion of module in this paper we consider the quotient hypermodule m a and interpret the lower and upper approximations as subsets of the quotient hypermodule m a then we introduce the concept of quotient rough sub hypermodule also using the concept of fuzzy sets we introduce and discuss the concept of fuzzy rough hypermodules and then we obtain the relation between fuzzy rough sub hypermodules and level rough sets this relation is characterized as a necessary and sufficient condition

some properties of l fuzzy approximation spaces based on bounded integral residuated lattices in this paper we present a general framework for the study of l fuzzy rough sets based on bounded integral residuated lattices we introduce and discuss the notions of l fuzzy topological spaces left right lower and left right upper l fuzzy approximation operators based on a complete bounded integral residuated lattice investigate union intersection and composition operations of l fuzzy approximation spaces and study some properties of the l fuzzy topologies generated by these l fuzzy rough approximation operators and the l relations induced by an l fuzzy topology

existence of nonoscillatory solution of second order linear neutral delay equation consider the neutral delay differential equation with positive and negative coefficients formula wherep rand formula some sufficient conditions for the existence of a nonoscillatory solution of the above equation expressed in terms of sqi s ds i 1 2 and certain technical conditions implying thatq1 s dominatesq2 s are obtained for values ofp 1

topological and lattice structures of l fuzzy rough sets determined by lower and upper sets this paper builds the topological and lattice structures of l fuzzy rough sets by introducing lower and upper sets in particular it is shown that when the l relation is reflexive the upper resp lower set is equivalent to the lower resp upper l fuzzy approximation set then by the upper resp lower set it is indicated that an l preorder is the equivalence condition under which the set of all the lower resp upper l fuzzy approximation sets and the alexandrov l topology are identical however associating with an l preorder the equivalence condition that l interior resp closure operator accords with the lower resp upper l fuzzy approximation operator is investigated at last it is proven that the set of all the lower resp upper l fuzzy approximation sets forms a complete lattice when the l relation is reflexive

oscillation theorems for second order nonlinear neutral differential equations the purpose of this paper is to study the oscillation of the second order neutral differential equations of the form a t z t c q t x b s t 0 where z t x t p t x t t we explore properties of given equations by examining those of associated first order delay equations new comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second order delay differential equation by applying the oscillation criteria obtained to the first order delay equations the results obtained are easy to verify

oscillation theorems for certain neutral functional differential equations in this paper we shall offer sufficient conditions for the oscillation of all solutions to neutral functional differential equations of mixed type of the form dndtn x t ax t bx t q t x t g p t x t h 0 1 here p and q are periodic functions

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

oscillation for higher order superlinear delay differential equations with unstable type abstract we first show that the n order superlinear delay differential equation with unstable type x n t p t x t 1 x t t t 0 always has an unbounded and nonoscillatory solution where n 1 is an even integer p c t0 0 0 and 1 then an almost sharp bounded oscillation and nonoscillation criterion is obtained

solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation the paper is concerned with the higher order nonlinear neutral delay differential equation r t x t p t x email protected m n m 1 n m 1ft email protected 1 t x s l t g t t t 0 using the banach fixed point theorem we establish five existence results of uncountably many bounded nonoscillatory solutions for the above equation construct five mann iterative sequences with mixed errors for approximating these nonoscillatory solutions and discuss five error estimates between the approximate solutions and these nonoscillatory solutions to dwell the importance and applications of our results five nontrivial examples are constructed

oscillation theorems for second order neutral differential equations the aim of this paper is to study the oscillation of the second order neutral differential equations e r t x t p t x t t q t x s t 0 the obtained results are based on the new comparison theorems that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation the obtained comparison principles essentially simplify the examination of the studied equations

a characterization of l fuzzy prime ideals abstract in this paper the concepts of a l fuzzy prime and a l fuzzy completely prime ideal are given and some fundamental lemmas are proved in particular bijections among some of the l fuzzy prime ideals of two homomorphic rings are constructed also a characterization of a l fuzzy prime ideal of a ring is given

sufficient conditions for oscillation and nonoscillation of neutral equations received july 9 1986 consider the neutral delay differential equation x r p f x f r q t x r 0 o tat 1 where p q e c i co w and t d e iw we obtain sutlicient conditions for all solutions of eq 1 to oscillate our conditions are sharp in the sense than when the coefficients p and q are constants the conditions are also necessary we also obtain sufficient conditions for eq 1 to have a nonoscillatory solution when p r is a constant in the interval

nonoscillatory solutions for system of neutral delay equation abstract consider the system of neutral delay differential equations d d t x t p x t q t x t 0 where p r 0 0 and q is continuous matrix and the system d d t x t b x t q t x t 0 where b is a matrix we obtain the sufficient condition for the existence of certain types of solutions of the above equation to be q s d s for p 1

generalized rough sets over fuzzy lattices this paper studies generalized rough sets over fuzzy lattices through both the constructive and axiomatic approaches from the viewpoint of the constructive approach the basic properties of generalized rough sets over fuzzy lattices are obtained the matrix representation of the lower and upper approximations is given according to this matrix view a simple algorithm is obtained for computing the lower and upper approximations as for the axiomatic approach a set of axioms is constructed to characterize the upper approximation of generalized rough sets over fuzzy lattices

rough sets and boolean reasoning abstract in this article we discuss methods based on the combination of rough sets and boolean reasoning with applications in pattern recognition machine learning data mining and conflict analysis

necessary and sufficient conditions for oscillations of neutral equations with several coefficients consider the neutral equation with several positive and negative coefficients dd tx t ipix t i jrjx t j kqkx t k 0 1 where i j and k are initial segments of natural numbers pi i rj j qk are positive numbers and k 0 for i i j j and k k then a necessary and sufficient condition for the oscillation of all solutions of 1 is that its characteristic equation ipie i irje j kqke k 0 has no real roots

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

infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations abstract in this paper we consider the second order nonlinear neutral delay differential equation a t x t b t x t h t x h 1 t x h 2 t x h k t f t x f 1 t x f 2 t x f k t g t t t 0 where 0 a b g c t 0 r with a t 0 for t t 0 h c 1 t 0 r k r f c t 0 r k r h l c 1 t 0 r and f l c t 0 r with lim t h l t lim t f l t l 1 k under suitable conditions by making use of the banach fixed point theorem we show the existence of infinitely many nonoscillatory solutions which are uncountable for the above equation suggest several mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions five nontrivial examples are given to illustrate the advantages of our results

rough set based 1 v 1 and 1 v r approaches to support vector machine multi classification support vector machines svms are essentially binary classifiers to improve their applicability several methods have been suggested for extending svms for multi classification including one versus one 1 v 1 one versus rest 1 v r and dagsvm in this paper we first describe how binary classification with svms can be interpreted using rough sets a rough set approach to svm classification removes the necessity of exact classification and is especially useful when dealing with noisy data next by utilizing the boundary region in rough sets we suggest two new approaches extensions of 1 v r and 1 v 1 to svm multi classification that allow for an error rate we explicitly demonstrate how our extended 1 v r may shorten the training time of the conventional 1 v r approach in addition we show that our 1 v 1 approach may have reduced storage requirements compared to the conventional 1 v 1 and dagsvm techniques our techniques also provide better semantic interpretations of the classification process the theoretical conclusions are supported by experimental findings involving a synthetic dataset

the lower and upper approximations in a fuzzy group in this paper we shall introduce the notion of a rough subgroup with respect to a normal subgroup of a group and give some properties of the lower and the upper approximations in a group also we will discuss a rough subgroup with respect to a t level subset of a fuzzy normal subgroup

oscillation criteria for nth order neutral functional differential equations abstract in this paper we establish some new criteria for the oscillation of nth order neutral functional differential equations of the form d n dt n x t cx t h c x t h qx t g px t g 0 n is even where c c h h p and q are real numbers and g and g are positive constants

oscillations of mixed neutral functional differential equations abstract in this paper we establish several sufficient conditions for the oscillation of solutions of mixed neutral functional differential equations of the form d n dt n x t cx t h cx t h qx t g qx t g 0 where c c g g h and h are real constants and q and q are nonnegative real constants

on the existence of asymptotically decaying positive solutions of second order neutral differential equations abstract the existence of positive solutions of second order neutral differential equation x t cx t p t x g t 0 t t0 is investigated some sufficient conditions are given for the existence of positive solutions with asymptotic decay of for the cases of constant sign and nonconstant sign of coefficients it also is shown that one of these sufficient conditions is necessary

minimal nfa problems are hard finite automata fa s are of fundamental importance in theory and in applications the following basic minimization problem is studied given a dfa deterministic fa find a minimum equivalent no

a new measure of uncertainty based on knowledge granulation for rough sets in rough set theory accuracy and roughness are used to characterize uncertainty of a set and approximation accuracy is employed to depict accuracy of a rough classification although these measures are effective they have some limitations when the lower upper approximation of a set under one knowledge is equal to that under another knowledge to overcome these limitations we address in this paper the issues of uncertainty of a set in an information system and approximation accuracy of a rough classification in a decision table an axiomatic definition of knowledge granulation for an information system is given under which these three measures are modified theoretical studies and experimental results show that the modified measures are effective and suitable for evaluating the roughness and accuracy of a set in an information system and the approximation accuracy of a rough classification in a decision table respectively and have a much simpler and more comprehensive form than the existing ones