canonical and differential realizations of osp 3 2 superalgebra the osp 3 2 superalgebra is realized first by means of two supervectorial operators canonically conjugated and then by differential operators acting on functions defined on e 3 2 as for the vectorial representation of osp 1 2 the eigenvalue problem yields to the superspherical harmonics

application of generating function techniques to lie superalgebras the construction of generating functions for multiplicities of irreducible representations of lie superalgebras from generating functions for characters is examined and applied in order to obtain polynomial tensors and branching rules problems arising because of the existence of typical and atypical representations are discussed in detail the techniques are applied to osp 1 2 spl 1 2 osp 3 2 and osp 4 2

structure and perimeters of percolation clusters site percolation clusters are simulated at the percolation threshold on the square lattice an algorithm for walks around each cluster is used to obtain information on its fractal geometry the fractal dimensionality of the external perimeter is found to depend on the size of adsorbent particles used to measure it if the vacant perimeter sites have only nearest neighbour connectivity then the perimeter has dimension de 1 37 or 0 03 instead of that of the hull dh 1 75 the authors also measured the mass of internal dangling sites surrounded by the backbone s blobs singly and doubly connected sites the number of blobs and the average linear distance between entry into the exit from a blob

invariant operators for the n dimensional super poincare algebra and the decomposition of the scalar superfield an analysis of supersymmetric kaluza klein theories is begun by obtaining the casimir operators for the super poincare algebra in any number of dimensions the knowledge of these operators is used to decompose the general scalar superfield in 11 dimensions into its irreducible parts the irreducible superfields are expressed as products of grassmann hermite functions and grassmann bargmann wigner multispinor fields some lagrangians for these superfields are written down the formulation is off shell but global

reducts and constructs in classic and dominance based rough sets approach abstract the idea of the reduct as defined in the classic rough sets approach crsa has proven to be inspiring enough to get into closely related theories including the dominance based rough sets approach drsa the procedure of reduction is generally similar to that of feature selection but narrower as it is the descriptive rather than the predictive aspect of data exploration that constitutes its principal goal crsa reducts are thus defined as minimal subsets of attributes that retain sufficiently high quality of object description developed within crsa the crsa reducts have given rise to the generalized notion of crsa constructs which have turned out to be superior to reducts in numerous practical experiments with real life data sets the generalization process is continued in this paper in which a definition of constructs in the context of drsa is introduced the definition fully analogous to that of crsa constructs differs only in that it is context based in drsa while context free in crsa consequently the presented drsa constructs are expected to have analogous properties to that of crsa constructs including superiority to drsa reducts in experiments with real life data sets

representations of low rank orthosymplectic superalgebras by superfield techniques representations of lie superalgebras may be realised as functions superfields on graded manifolds suitably chosen onset spaces the technique is illustrated for osp 1 2 and osp 2 2 with a review of well known results and applied in the cases of osp 3 2 and osp 4 2 to construct some classes of irreducible representations

a fast heuristic attribute reduction approach to ordered decision systems abstract rough set theory has shown success in being a filter based feature selection approach for analyzing information systems one of its main aims is to search for a feature subset called a reduct which preserves the classification ability of the original system in this paper we consider ordered decision systems where the preference order a fundamental concept in dominance based rough set approach plays a critical role in recent literature based on the greedy hill climbing method many heuristic attribute reduction algorithms are proposed by utilizing significance measures of attributes and they are extended to deal with ordered decision systems unfortunately they are often time consuming especially when applied to deal with large scale data sets with high dimensions to reduce the complexity a novel accelerator is introduced in heuristic algorithms from the perspectives of objects and criteria based on the new accelerator the number of objects and the dimension of criteria are lessened thus making the accelerated algorithms faster than their original counterparts while maintaining the same reducts experimental analysis shows the validity and efficiency of the proposed methods

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

branching rules for typical and atypical representations of gl n 1 the gl n 1 down arrow gl n gl 1 branching rules are determined for all finite dimensional irreducible typical and atypical representations of gl n mod 1 using a recently introduced induced module construction for atypical modules and confirm those found recently by palev 1988 by another method the lowest weights and characters of the irreducible representations are found and the validity for gl n mod 1 of the character formula conjectured by hughes and king 1987 is confirmed some generalisations and extensions of the method are discussed

stochastic dominance based rough set model for ordinal classification in order to discover interesting patterns and dependencies in data an approach based on rough set theory can be used in particular dominance based rough set approach drsa has been introduced to deal with the problem of ordinal classification with monotonicity constraints also referred to as multicriteria classification in decision analysis however in real life problems in the presence of noise the notions of rough approximations were found to be excessively restrictive in this paper we introduce a probabilistic model for ordinal classification problems with monotonicity constraints then we generalize the notion of lower approximations to the stochastic case we estimate the probabilities with the maximum likelihood method which leads to the isotonic regression problem for a two class binary case the approach is easily generalized to a multi class case finally we show the equivalence of the variable consistency rough sets to the specific empirical risk minimizing decision rule in the statistical decision theory

new superfields for n supersymmetry with central charges the authors show that a judicious choice of nonlinear realisation of centrally extended n supersymmetry permits the construction of chiral like superfields with a 2n dimensional a number parameter at the expense of an external superspin index which labels multiplets of several irreducible representations of the lorentz group the generalized superfields considered here have 4n n2 component fields altogether as opposed to the 16n expected conventionally the n 2 case is examined in detail and the decomposition of arbitrary spin superfields into irreducible parts is given the connection with other n 2 superfield analyses is pointed out and the structure of spin reducing representations with p2 mod z mod 2 0 is exhibited

dominance based rough fuzzy set approach and its application to rule induction abstract theories of fuzzy sets and rough sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness respectively marrying both leads to the hybrid notion of rough fuzzy sets in order to get a more accurate account of imperfect information in this paper our attention is paid to ordered fuzzy decision systems where condition criteria are preference ordered and decision classes are not only ordered but also fuzzy first the dominance based rough fuzzy approximations of an upward or downward cumulated fuzzy set are introduced in ordered fuzzy decision systems second lower and upper reducts relative to a certain cumulated fuzzy set are proposed to eliminate redundant criteria in the system then two approaches to attribute reduction are presented based on the discernibility matrix and the heuristic strategy respectively also decision rules are extracted directly from these approximations and some applicable and simplified decision rules are obtained according to requirements of decision makers finally a case study in bankruptcy risk analysis is used to illustrate the mechanism of the proposed methods

young diagrams supercharacters of osp m n and modification rules the supercharacter of osp m n associated with an arbitrary young diagram is defined the distinction is made between osp m n standard and non standard supercharacters the corresponding modification rule which may be used to express a non standard supercharacter in terms of standard supercharacters is presented exemplified and proved this rule involves the removal of n 2 1 boundary strips from the young diagram in the case n 0 the rule reduces to the well known rule appropriate to o m for a non standard supercharacter corresponding to a typical irreducible representation of osp m n the modification yields a single typical standard supercharacter on the other hand for a non standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters

generation of rough sets reducts and constructs based on inter class and intra class information the reduct originating from the classic rough set approach crsa is an inclusion minimal subset of attributes that provides discernibility between objects from different classes in at least the same degree as the set of all attributes it can be thus referred to as being consistent and minimal which are the two important characteristics of filter based feature selection these two characteristics have been also utilized to define reducts within the dominance based rough set approach drsa further the classic reduct here referred to as an inter class reduct has evolved into what is known as intra class reduct and construct in crsa the idea is that while inter class reducts utilize only one part of information generated from all pairs of objects intra class reducts utilize the remaining part while constructs utilize both the paper delivers a final unification of inter class reducts intra class reducts and constructs across crsa and drsa showing how they can be both defined and computed uniformly i e using basically the same concepts and algorithms it also presents an exact algorithm capable of generating all exact reduced subsets but of considerable complexity as well as a simple and fast heuristic designed to generate a single reduced subset finally it illustrates the computation process with examples and some experimental evaluation of crsa constructs which show how the use of both the inter class and the intra class information can assist the attribute reduction process and help obtaining useful insights into the analyzed data set

rough set approach to multi attribute decision analysis abstract we review the methodology of the rough set analysis of multi attribute decision problems rough set theory proved to be a useful tool for analysis of a vague description of decision situations it answers two basic questions related to multi attribute decision problems one about explanation of a decision situation and another about prescription of some decisions basing on analysis of a decision situation we define four classes of multi attribute decision problems depending on the structure of their representation its interpretation and the kind of questions related then we characterize the rough set methodology for each particular class of decision problems we use simple practical examples to illustrate this presentation a review of related literature is made throughout the paper

irreducible so 2 extended superfields all the irreducible so 2 extended scalar superfields are obtained in so 2 invariant form these superfields can be labelled by the eigenvalues of three casimir operators of the enlarged algebra containing chiral transformation they may be still more simply classified by introducing two invariant simple superspaces lagrangians and supersymmetry transformation rules are obtained from the explicit form of the irreducible superfields the highest superspin multiplet is identified as the multiplet for the linearized so 2 weyl supergravity the linearized so 2 poincare supergravity is obtained by coupling this multiplet with the chiral superfields

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

the decomposition of spin reducing representations of n 2 supersymmetry the authors decompose any spin reducing representation of n 2 supersymmetry into a direct integral of irreducible representations in a unique manner in which each irrep has standard form a single real number is required to classify the irreps in the general representations

casimir operators for massless representations of the super poincare algebra and the reduction of the ten dimensional massless scalar superfield invariant operators for the massless little algebra of spd super poincare algebra in d dimensions are given they are used to decompose the scalar massless superfield in ten dimensions explicit expressions for the irreducible pieces are obtained after exploiting a relevant cartan subalgebra

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

stochastic resonance in overdamped two coupled anharmonic oscillators abstract we consider the overdamped version of two coupled anharmonic oscillators with the external periodic force f sin t and gaussian noise term t added to one of the two state variables of the system linear stability analysis is carried out in the absence of external periodic force and noise then the system with external periodic force only and noise term only are studied in the noise free system four period t orbits are found to co exist one in each of the four wells for small values of forcing amplitude f for fixed values of other parameters at a critical value of f f c cross well periodic orbit appears f c is found to increase with increase in the coupling strength in the absence of forcing the system is found to exhibit intermittent jumping motion between two wells above a particular value of noise strength d d c the noise induced jumping behaviour is characterized by first passage time and residence time distributions next the influence of noise in the presence of the external periodic force is numerically studied the system is found to exhibit stochastic resonance behaviour the observed stochastic resonance dynamics is characterized using power spectrum signal to noise ratio and residence time distribution the plot of maximal lyapunov exponent versus noise strength has shown a stochastic resonance profile the occurrence of stochastic resonance is studied by varying the forcing frequency and the coupling strength finally the influence of addition of external periodic force and noise term to different state variables is also presented

how to tune the system parameters to realize stochastic resonance the paper presents a numerical method of realizing stochastic resonance sr by tuning system parameters firstly a simple and effective method of evaluating the system response speed 1 is introduced then the parameter induced sr problem can be reduced to the parameter optimization problem under the condition 1 const to solve this optimization problem we put forward the following techniques i compensate the deviation from the prescribed system response speed 1 at each step ii take the curve length increment on the parameter plane as the step size in searching the maximal signal to noise ratio snr gain finally a numerical simulation confirms the effectiveness of this method

structure of atypical representations of the lie superalgebras sl m n in contrast to lie algebras lie superalgebras contain typical and atypical irreducible representations character formulae for typical representations of classical lie superalgebras have been known for a long time but the structure of atypical representations remained unsolved the author constructs the character formula for atypical representations of the special linear lie superalgebras sl m n

numerical verification of bona fide stochastic resonance the notion of stochastic resonance as an input output synchronization mechanism is reestablished quantitatively based on accurate numerical simulation of a simple two state model namely the schmitt trigger the corresponding phenomenological criterion known as bona fide stochastic resonance is thus proven applicable throughout the entire range of the input parameters claims to the contrary are briefly discussed

how to integrate over central charges the authors construct dimensionless actions for fields which have non trivial central charge behaviour by means of optimisation methods for systems with constraints they show how such actions must be defined in order to give the correct four dimensional equations of motion this is considered initially for a single real scalar free field with one central charge and extended to successively more complicated situations in particular they discuss fields with more than one central charge with suitable constraint conditions they also consider actions for superfields with central charges and obtain full superspace actions for the degenerate cases fields without central charges are also described by integration over central charge dimensions they conclude that their four dimensional world may be embedded as the boundary of a high dimensional manifold whose extra dimensions are not directly accessible on shell

dominance based rough set approach for group decisions the objective of this paper is to propose an approach to support group multicriteria classification the approach is composed of three phases the first phase exploits the knowledge provided by each decision maker to individually approximate the decision classes using rough approximation the second phase seeks to combine the outputs of individual approximation phase into a collective decision table by using an appropriate aggregation procedure the third phase uses the collective decision table in order to infer a set of collective decision rules which synthesize the judgements and perspectives of the different decision makers and to permit the classification of all decision objects the proposed approach relies on the dominance based rough set approach drsa which is used at two different levels first the drsa is used during the first phase to approximate the input data relative to each decision maker second the drsa is used during the third phase to approximate the collective decision table and generate the collective decision rules this paper presents the theoretical foundation of the proposed approach three case studies using real world data and a comparative study of recent similar proposals

class specific attribute reducts in rough set theory abstract the concept of attribute reducts plays a fundamental role in rough set analysis there are at least two possibilities to define an attribute reduct a classification based or global attribute reduct is a minimal subset of condition attributes that preserves the positive region of the decision classification namely the positive regions of all decision classes in a decision table a class specific class dependent or local attribute reduct is a minimal subset of condition attributes that preserves the positive region of a particular decision class while a classification based reduct may not work equally well for every decision class a class specific attribute reduct is optimally tailored to a particular decision class however studies in rough set theory are dominated by classification based reducts there is very limited investigation on class specific reducts an objective of this paper is to draw attention to class specific reducts we systematically compare the two types of reducts and investigate their relationships with respect to both individual reducts and families of all reducts it is possible to derive a class specific reduct from a classification based reduct and to derive a classification based reduct from a family of class specific reducts the families of all class specific reducts provide a pair of lower and upper bounds of the family of all classification based reducts based on a three way classification of attributes into the pair wise disjoint sets of core marginal and nonuseful attributes we examine relationships between the corresponding classes of classification based and class specific attributes the union of the sets of class specific core attributes is the set of classification based core attributes it is only possible to obtain an upper bound for the set of classification based marginal attributes and a lower bound for the set of classification based nonuseful attributes from the family of the class specific corresponding sets of attributes

decomposition of the scalar superfield in ten dimensions explicit expressions for the irreducible superfields included in the ten dimensional scalar superfield are derived by using a method based in a cartan subalgebra

reduced matrix elements of the orthosymplectic lie superalgebra we utilize characteristic identities to construct eigenvalue formulae for invariants and reduced matrix elements corresponding to irreducible representations of in presenting these results we further develop our programme of constructive representation theory via characteristic identities

superfield and matrix realizations of highest weight representations for osp m 2n a differential representation of the classical lie superalgebra osp m 2n acting on superfield functions is given this representation is used to construct matrix representations of the finite dimensional irreducible representations of the algebra inner products on the irreducible spaces are discussed and classes of star and grade star equivalent representations are identified

superunitary reduction of the orthosymplectic supertableaux the supertableaux of the orthosymplectic groups osp 2 2p and osp 2 1 2p are decomposed in sums of supertableaux of the superunitary group su p

representations of n 2 extended supergravity and unitarity conditions in osp n 4 the structure of unitary irreducible representations of the lie superalgebra osp 2 4 the algebra of n 2 extended supergravity is investigated in detail in particular four new classes of shortened multiplets are found and the complete unitarity conditions are given the shortened multiplets are shown to correspond to atypical infinite dimensional representations finally unitarity conditions for osp n 4 3 n 8 are constructed

irreducible representations of the superalgebras type ii the representation of the orthosymplectic algebras and the other members of their class are built explicitly with simple techniques

dimension and character formulas for lie supergroups a character formula is derived for lie supergroups the basic technique is that of symmetrization and antisymmetrization associated with young tableaux generalized to supergroups we rewrite the characters of the ordinary lie groups u n o n and sp 2n in terms of traces in the fundamental representation it is then shown that by simply replacing traces with supertraces the characters of certain representations for u n m and osp n 2m are obtained dimension formulas are derived by calculating the characters of a special diagonal supergroup element with 1 and 1 eigenvalues formulas for the eigenvalues of the quadratic casimir operators are given as applications the decomposition of a representation into representations of subgroups is discussed examples are given for the lie supergroup su 6 4 which has physical applications as a dynamical supersymmetry in nuclei

an irreducible representation of n 4 susy with two central charges using a method of dimensional reduction from an on shell multiplet in ten space time dimensions the authors give the supersymmetry transformation laws of a general irreducible representation of n 4 susy with central charges present with more than one central charge an extra condition is used to give a finite number of components and they give the field transformations and lagrangian for the case of two central charges

a no go theorem for off shell extended supergravities it is proved that there is no possible set of field redefinitions that give linearised off shell versions of n or 3 extended supergravity assuming that the associated global supersymmetry algebras have no central charges and that fermionic auxiliary fields are always monogamous the results also prove the absence of off shell verions of eleven dimensional simple supergravity

representations of global supersymmetries for all n the authors present the global supersymmetry rules for the fundamental representation of n extended supersymmetry for all n and an associated invariant quadratic lagrangian this is achieved by the use of a suitable set of basis functions defined in terms of the spinor generators acting on the vacuum state of the representation these basis functions allow the action of the generators on them to be obtained explicitly as well as their internal symmetry properties specific examples are given of this for n 3 4 6 and 8

self duality in super yang mills theories the self dual condition in superspace is analysed forn 1 2 4 super yang mills theories a complete solution of all the constraints in terms of a light cone superfieldj is presented where the only equation thatj satisfies is a susy generalization of the yang equation by reduction of that equation we obtain various two dimensional susy models we introduce the associated linear problem in terms ofj whose integrability condition gives us back the super yang equation and allows us to obtain the kac moody algebra structure of the theory

irreducible representations of the exceptional lie superalgebras d 2 1 α the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the exceptional lie superalgebrasd 2 1 for all cases the star or grade star conditions for the algebra are investigated among the finite dimensional representations there are no star and only a few grade star representations but an infinite class of infinite dimensional star representations is found explicit expressions are given for the doublet representation of d 2 1 the one missing label problem d 2 1 su 2 su 2 su 2 is discussed in detail and solved explicitly

young supertableaux of the basic lie superalgebras among the simple lie superalgebras lsa the basic classical ones are well known together with many of their finite dimensional representations this is particularly true for the typical representations which are known to be completely reducible it is well known that reducible but indecomposable representations of the basic lsa are not completely characterised by their highest weight hw or the eigenvalues of their casimir operators in fact there may be several different finite representations characterised by the same hw such representations are called non typical or atypical the aim of this paper is to establish a complete characterisation of the typical and non typical representations by means of properly defined young supertableaux yst therefore the definition we are going to use for our yst differs slightly from the standard definitions the examples given throughout this paper help the illustration of this new interpretation for the yst which makes possible the complete classification of the indecom posable representations characterised by a generalised notion of hw

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

new characterization of stochastic resonance in bistable square potential well the features of first passage time density function is analysed theoretically in a symmetric double square well system modulated periodically with a signal of arbitrary amplitude and frequency resonance is demonstrated as a maximum synchronization between periodic signal and noise resonance is characterized as a linear relation between noise strength at resonance and frequency this characterization is shown to hold good for amplitude lesser or greater than the depth of the unmodulated potential well the mean first passage time of the process at resonance is also shown to decrease linearly with the strength of the noise for high amplitude of the signal while it increases linearly with inverse of the noise strength for low amplitude

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

modification rules for orthosymplectic superalgebras ii atypical representations for pt i see ibid vol 19 no 3 p 321 7 1986 modification rules are presented for all finite dimensional atypical representations of osp m n with the representations specified by young supertableaux these rules relate non standard supertableaux to standard supertableaux via the removal of continuous boundary hooks

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

finite and infinite dimensional representations of the orthosymplectic superalgebra osp 3 2 the shift operator technique is used to give a complete analysis of all finite and infinite dimensional irreducible representations of the orthosymplectic superalgebra osp 3 2 for all cases the star or grade star conditions for the algebra are investigated only two finite dimensional representations are grade star representations if the representation space is required to be a graded hilbert space when the even part is so 3 sp 2 r su 2 su 1 1 an infinite class of infinite dimensional star representations is found one of them can be realized in terms of two valued functions of a complex variable this representation reduces to the sum of two metaplectic representations of sp 2 we show that it is precisely this metaplectic representation for osp 3 2 which gives the spin energy eigenstates for the one dimensional harmonic oscillator with spin 1 2 states