all degree 6 local unitary invariants of k qudits we give explicit index free formulae for all the degree 6 and also degree 4 and 2 algebraically independent local unitary invariant polynomials for finite dimensional k partite pure and mixed quantum states we carry this out using graph technical methods which provide illustrations for this abstract topic

statistical analysis of airport network of china through the study of airport network of china anc composed of 128 airports nodes and 1165 flights edges we show the topological structure of anc conveys two characteristics of small worlds a short average path length 2 067 and a high degree of clustering 0 733 the cumulative degree distributions of both directed and undirected anc obey two regime power laws with different exponents i e the so called double pareto law in degrees and out degrees of each airport have positive correlations whereas the undirected degrees of adjacent airports have significant linear anticorrelations it is demonstrated both weekly and daily cumulative distributions of flight weights frequencies of anc have power law tails besides the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight it is also shown the diameter of each subcluster consisting of an airport and all those airports to which it is linked is inversely proportional to its density of connectivity efficiency of anc and of its subclusters is measured through a simple definition in terms of that the efficiency of anc s subclusters increases as the density of connectivity does anc is found to have an efficiency of 0 484

on approximate reasoning with graded rules this contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy if then rules formalized by approximating formulas of a special type two alternatives that follow from the dual character of approximating formulas are developed in parallel the link to the theory of fuzzy control systems is also explained

on polynomial invariants of several qubits it is a recent observation that entanglement classification for qubits is closely related to local sl 2 c invariants including the invariance under qubit permutations dur et al phys rev a 62 062314 2000 osterloh a and siewert j phys rev a 72 012337 2005 o chterental and d okovic linear algebra research advances nova science hauppauge n y 2007 chap 4 p 133 which has been termed sl invariance in order to single out the sl invariants we analyze the sl 2 c invariants of four five qubits and decompose them into irreducible modules for the symmetric group s4 s5 of qubit permutations a classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of sl invariants vanishing on arbitrary product states we find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by osterloh and siewert phys rev a 72 012337 2005 int j quant inf 4 531 2006 our analys

on the geometry of four qubit invariants the geometry of four qubit entanglement is investigated we replace some of the polynomial invariants for four qubits introduced recently by new ones of direct geometrical meaning it is shown that these invariants describe four points six lines and four planes in complex projective space cp3 for the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables moreover suitable powers of their magnitudes are entanglement monotones that fit nicely into the geometric set of n qubit ones related to grassmannians of l planes found recently we also show that in terms of these invariants the hyperdeterminant of order 24 in the four qubit amplitudes takes a more instructive form than the previously published expressions available in the literature finally in order to understand two three and four qubit entanglement in geometric terms we propose a unified setting based on cp3 furnished with a fixed quadric

fuzzy control on the basis of equality relations with an example from idle speed control the way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications in most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment which reflects the indistinguishability of measurements or control values in this paper the authors show that equality relations turn out to be the natural way to represent such vague environments and they develop suitable interpolation methods to obtain a control function as a special case of our approach the authors obtain mamdani s model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations and they can explain why typical fuzzy partitions are preferred the authors also obtain a criterion for reasonable defuzzification strategies the fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the volkswagen golf gti

lattice valued approach to closed sets under fuzzy relations theory and applications motivated by fuzzy control problems and by some investigations of eigen fuzzy sets we deal with a closedness of fuzzy sets under fuzzy relations in two ways in one sense by directly analyzing fuzzy concepts and in the other by investigating the corresponding crisp problems in the cutworthy framework our main task is to investigate particular fuzzy functional equations and inequations appearing in this context which turn out to be essentially connected with fuzzy control problems we analyze procedures and find solutions of these equations and inequations pointing to important applications

a complete axiomatization of fuzzy functional dependencies using fuzzy function the notion of functional dependency in classical relational databases corresponds to a partial function therefore it appears natural that the notion of fuzzy functional dependency should correspond to the notion of partial fuzzy function following this approach we introduce the notion of fuzzy functional dependency in fuzzy relational databases and show that the classical inference rules for functional dependencies form a sound and complete formal system in our case

stochastic local operations and classical communication equations and classification of even n qubits for any even n qubits we establish four stochastic local operations and classical communication slocc equations and construct four slocc polynomials not complete of degree 2n 2 which can be exploited for the slocc classification not complete of any even n qubits in light of the slocc equations we propose several different genuine entangled states of even n qubits and show that they are inequivalent to the ghz w or l n the symmetric dicke states with l excitations under slocc via the vanishing or not of the polynomials the absolute values of the polynomials can be considered as entanglement measures

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

dissecting the qutrit to visualize a higher dimensional object it is convenient to consider its two dimensional cross sections the set of quantum states for a three level system has eight dimensions we supplement a recent paper by goyal et al arxiv 1111 4427 by considering the set of all possible two dimensional cross sections of the qutrit each such cross section is bounded by a plane cubic curve

smooth subgroups and smooth homomorphisms abstract in this paper as a continuation of demirci fuzzy sets and systems 117 2001 431 437 the concept of smooth subgroups and smooth homomorphisms are introduced and their basic properties are investigated

fuzzy functions with support vector machines a new fuzzy system modeling fsm approach that identifies the fuzzy functions using support vector machines svm is proposed this new approach is structurally different from the fuzzy rule base approaches and fuzzy regression methods it is a new alternate version of the earlier fsm with fuzzy functions approaches svm is applied to determine the support vectors for each fuzzy cluster obtained by fuzzy c means fcm clustering algorithm original input variables the membership values obtained from the fcm together with their transformations form a new augmented set of input variables the performance of the proposed system modeling approach is compared to previous fuzzy functions approaches standard svm lse methods using an artificial sparse dataset and a real life non sparse dataset the results indicate that the proposed fuzzy functions with support vector machines approach is a feasible and stable method for regression problems and results in higher performances than the classical statistical methods

slocc polynomial invariants of degree 6 for even n qubits we give a practical necessary and sufficient condition for two pure n qubit states to be stochastic local operations and classical communication slocc equivalent we also construct the slocc polynomial invariants of degree 6 for even n qubit n 4 states the polynomials are in the explicit form of 3 3 determinants whose entries are polynomials of degree 2 in the coefficients of the states further the polynomials satisfy an slocc equation and therefore are invariant under slocc the slocc equations can be used for the slocc classification of even n qubits

quantifying entanglement resources we present an overview of the quantitative theory of single copy entanglement in finite dimensional quantum systems in particular we emphasize the point of view that different entanglement measures quantify different types of resources which leads to a natural interdependence of entanglement classification and quantification apart from the theoretical basis we outline various methods for obtaining quantitative results on arbitrary mixed states this article is part of a special issue of journal of physics a mathematical and theoretical devoted to 50 years of bell s theorem

three qubit pure state canonical forms in this paper we analyse the canonical forms into which any pure three qubit state can be cast the minimal forms i e the ones with the minimal number of product states built from local bases are also presented and lead to a complete classification of pure three qubit states this classification is related to the values of the polynomial invariants under local unitary transformations by a one to one correspondence

hyperdeterminants as integrable discrete systems we give the basic definitions and some theoretical results about hyperdeterminants introduced by a cayley in 1845 we prove integrability understood as 4d consistency of a nonlinear difference equation defined by the 2 2 2 hyperdeterminant this result gives rise to the following hypothesis the difference equations defined by hyperdeterminants of any size are integrable we show that this hypothesis already fails in the case of the 2 2 2 2 hyperdeterminant

geometry of entangled states bloch spheres and hopf fibrations we discuss a generalization of the standard bloch sphere representation for a single qubit to two qubits in the framework of hopf fibrations of highdimensional spheres by lower dimensional spheres the single qubit hilbert space is the three dimensional sphere s 3 t hes 2 base space of a suitably oriented s 3 hopf fibration is nothing but the bloch sphere while the circular fibres represent the overall qubit phase degree of freedom for the two qubits case the hilbert space is a seven dimensional sphere s 7 which also allows for a hopf fibration with s 3 fibres and a s 4 base the most striking result is that suitably oriented s 7 hopf fibrations are entanglement sensitive the relation with the standard schmidt decomposition is also discussed

fuzzy homomorphisms of algebras in this paper we consider fuzzy relations compatible with algebraic operations which are called fuzzy relational morphisms in particular we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations called uniform fuzzy relational morphisms and those which are partially uniform f functions called fuzzy homomorphisms both uniform fuzzy relations and partially uniform f functions were introduced in a recent paper by us uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras we give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms we establish certain relationships between them and fuzzy congruences and we prove homomorphism and isomorphism theorems concerning them we also point to some applications of uniform fuzzy relational morphisms

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

invariants and polynomial identities for higher rank matrices we exhibit explicit expressions in terms of components of discriminants determinants characteristic polynomials and polynomial identities for matrices of higher rank we define permutation tensors and in terms of them we construct discriminants and the determinant as the discriminant of order d where d is the dimension of the matrix analogues of the characteristic polynomials and the cayley hamilton theorem are obtained therefrom for higher rank matrices

weak bisimulations for fuzzy automata abstract forward and backward bisimulations have been introduced recently by ciric et al 2012 9 as a means for modeling the equivalence between states of fuzzy automata and approximating the language equivalence as well as for use in the state reduction of fuzzy automata the main aim of the present paper is to introduce two new kinds of bisimulations weak forward and backward bisimulations which provide better approximations of the language equivalence than forward and backward bisimulations and when employed in the state reduction they provide better reductions we give procedures for deciding whether there exist weak forward and backward simulations and bisimulations and for computing the greatest ones whenever they exist using weak bisimulations in conjunction with the concept of a uniform fuzzy relation we determine necessary and sufficient conditions under which two fuzzy automata are weak bisimulation equivalent we also characterize uniform weak backward and forward bisimulations between two fuzzy automata in terms of isomorphisms between their nerode and reverse nerode automata

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

singularity of type d_4 arising from four qubit systems an intriguing correspondence between four qubit systems and simple singularity of type d4 is established we first consider the algebraic variety x of separable states within the projective hilbert space then cutting x with a specific hyperplane h we prove that the x hypersurface defined from the section x h x has an isolated singularity of type d4 it is also shown that this is the worst possible isolated singularity that one can obtain by this construction moreover it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of x which is nothing but the cayley hyperdeterminant of type 2 2 2 2 can be expressed in terms of the stochastic local operation and classical communication invariant polynomials as the discriminant of the miniversal deformation of the d4 singularity as a consequence of this correspondence one obtains a finer grained classification of entanglement classes of four qubit systems

approximation of extensional fuzzy relations over a residuated lattice we investigate a relationship between extensionality of fuzzy relations and their lipschitz continuity on generalized metric spaces the duality of these notions is shown and moreover two particular applications of the extensionality property in the field of approximation are given

plethysms replicated schur functions and series with applications to vertex operators specializations of schur functions are exploited to define and evaluate the schur functions s x and plethysms s s x for any integer real or complex plethysms are then used to define pairs of mutually inverse infinite series of schur functions m and l specified by arbitrary partitions these are used in turn to define and provide generating functions for formal characters s of certain groups h thereby extending known results for orthogonal and symplectic group characters each of these formal characters is then given a vertex operator realization first in terms of the seriesm m 0 and various l dual to l and then more explicitly in the exponential form finally the replicated form of such vertex operators are written down the characters of the orthogonal and symplectic groups have been found by schur 34 and weyl 35 respectively the method used is transcendental and depends on integration over the group manifold these characters however may be obtained by purely algebraic methods this algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration littlewood d e 1944 phil trans r soc london ser a 239 809 392 pacs numbers 02 10 v 02 10 de 02 20 a 02 20 hj mathematics subject classification 05e05 17b69 11e57 16w30 20e22 33d52 43a40 1751 8113

nondeterministic automata equivalence bisimulations and uniform relations in this paper we study the equivalence of nondeterministic automata pairing the concept of a bisimulation with the recently introduced concept of a uniform relation in this symbiosis uniform relations serve as equivalence relations which relate states of two possibly different nondeterministic automata and bisimulations ensure compatibility with the transitions initial and terminal states of these automata we define six types of bisimulations but due to the duality we discuss three of them forward backward forward and weak forward bisimulations for each of these three types of bisimulations we provide a procedure which decides whether there is a bisimulation of this type between two automata and when it exists the same procedure computes the greatest one we also show that there is a uniform forward bisimulation between two automata if and only if the factor automata with respect to the greatest forward bisimulation equivalences on these automata are isomorphic we prove a similar theorem for weak forward bisimulations using the concept of a weak forward isomorphism instead of an isomorphism we also give examples that explain the relationships between the considered types of bisimulations

bisimulations for weighted automata over an additively idempotent semiring weshowthatthemethodologyfordecidingtheexistenceandcomputationofthegreatestsimulationsandbisimulations developed in the framework of fuzzy automata m ciric et al fuzzy sets and systems 186 2012 100 139 208 2012 22 42 can be applied in a similar form to weighted automata over an additively idempotent semiring we define two types of simulations and four types of bisimulations for weighted automata over an additively idempotent semiring as solutions to particular systems of matrix inequalities and we provide polynomial time algorithms for deciding whether there is a simulation or bisimulation of a given type between two weighted automata and for computing the greatest one if it exists the algorithms are based on the concept of relative boolean residuation for matrices over an additively idempotent semiring that we introduce here we also prove that two weighted automata a and b are forward bisimulation equivalent i e there is a row and column complete forward bisimulation between them if and only if the factor weighted automata with respect to the greatest forward bisimulation equivalence matrices on a and b are isomorphic in addition we show that the factor weighted automaton with respect to the greatest forward bisimulationequivalence matrix on aweighted automaton a is the unique upto an isomorphism minimal automaton in the class of all weighted automata which are forward bisimulation equivalent to a

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

the mixed two qutrit system local unitary invariants entanglement monotones and the slocc group sl 3 mathbb c we consider local unitary invariants and entanglement monotones for the mixed two qutrit system character methods for the local su 3 su 3 transformation group are used to establish the count of algebraically independent polynomial invariants up to degree 5 in the components of the density operator these are identified up to quartic degree in the standard basis of gell mann matrices with the help of the calculus of f and d coefficients next investigating local measurement operations we study a slocc qutrit group which plays the role of a relativistic transformation group analogous to that of the lorentz group for the qubit case this is the group presented as a group of real 9 9 matrices acting linearly on the nine dimensional space of projective coordinates for the qutrit density matrix the counterpart for qutrits of the invariant 4 4 minkowski metric of the qubit case proves to be a certain 9 9 9 totally symmetric three fold tensor generalizing the gell mann d coefficient using this structure we provide a count of the corresponding local special linear polynomial invariants using group character methods finally we give an explicit construction of the lowest degree quantity the cubic invariant and its expansion in terms of su 3 su 3 invariants and we indicate how to construct higher degree analogues these quantities are proven to yield entanglement monotones this work generalizes and partly extends the paper of king et al 2007 j phys a math theor 40 10083 on the mixed two qubit system which is reviewed in an appendix

the normal form of a granular fuzzy function we consider here a particular kind of fuzzy relations f x y on x y such that the cut f is an ordinary function for each 0 1 from the underlying level granular structure on x to that on y which are determined by a given pair of similarity relations x x x and y y y on x and y respectively this type of fuzzy relations have a close connection with rough sets and functions between them and are called here granular fuzzy functions we show that each granular fuzzy function f x y has a unique normal form f x y f x y which preserves the values f x for each 0 1 where x is the equivalence class of x at the level for x the normal form functions f x y are in many ways more well behaved than the non normal forms in terms of function composition unique inverse function etc

fuzzy groups fuzzy functions and fuzzy equivalence relations to each fuzzy subgroup a fuzzy equivalence relation is associated and it is proved that a fuzzy subgroup is normal if and only if the operation of the group is compatible with its associated fuzzy equivalence relation in the definition of fuzzy subgroup only the subset is fuzzy whilst the group operation remains crisp in an opposite direction another structure can be constructed maintaining crisp the basic set x of the group and fuzzifying the group operation when in addition it is imposed to the fuzzy operation to be compatible with a given fuzzy equivalence relation this structure is called a vague group the results of the paper allow us to associate a vague group to every fuzzy subgroup in such a way that it can be interpreted as the fuzzy quotient group x special attention is paid to normal fuzzy groups and vague groups of the additive group r of the real numbers two examples related to triangular numbers illustrating the results of the paper are provided

operations fitting triangular norm based biresiduation abstract given a t norm t and the corresponding biresiduation t an operation k fits the latter or shortly is fitting if t is respected by k in all its arguments in this paper the structure of fitting operations is investigated special attention is paid to the case of basic t norms the connection between the fitting property and the lipschitz property is stressed some examples are given

complex multi projective variety and entanglement in this paper we will show that a vanishing generalized concurrence of a separable state can be seen as an algebraic variety called the segre variety this variety defines a quadric space which gives a geometric picture of separable states for pure bi and three partite states the variety equals the generalized concurrence moreover we generalize the segre variety to a general multipartite state by relating to a quadric space defined by two by two subdeterminants

a complete set of local invariants for a family of multipartite mixed states we study the equivalence of quantum states under local unitary transformations by using the singular value decomposition a complete set of invariants under local unitary transformations is presented for several classes of tripartite mixed states in composite systems two density matrices in the same class are equivalent under local unitary transformations if and only if all these invariants have equal values for these density matrices

a study of two qubit density matrices with fermionic purifications we study 12 parameter families of two qubit density matrices arising from a special class of two fermion systems with four single particle states or alternatively from a four qubit state with amplitudes arranged in an antisymmetric matrix we calculate the wootters concurrences and the negativities in a closed form and study their behavior we use these results to show that the relevant entanglement measures satisfy the generalized coffman kundu wootters formula of distributed entanglement an explicit formula for the residual tangle is also given the geometry of such density matrices is elaborated in some detail in particular an explicit form for the bures metric is given

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

a natural interpretation of fuzzy mappings in this paper we present a new and natural interpretation of fuzzy mappings which is a direct extension of the natural interpretation of fuzzy sets and fuzzy relations shown in our previous paper a natural interpretation of fuzzy sets and fuzzy relations fuzzy sets and systems 128 2002 135 we interpret fuzzy mappings as mappings in a heyting valued model for intuitionistic set theory and present a characterization of fuzzy mappings with membership functions which seems to be different from all kinds of known definitions this interpretation is quite natural for it deduces naturally various properties of fuzzy mappings and it makes clear the meaning of the famous extension principle by zadeh which is said to be very important in fuzzy theory

gradation of being fuzzy function abstract for crisp sets x and y considering two fuzzy equalities e x e y on x and y respectively the concept of fuzzy function and several types of it were studied in m demirci fuzzy sets and systems 106 1999 239 246 in this paper for a fuzzy relation f on x y considering the fuzzy equalities e x on x and e y on y the degree of f of being fuzzy function being surjective being injective and being bijective is defined then setting the connections between several types of degree of being fuzzy function and various types of fuzzy function the essential properties of these degrees are investigated

the mixed two qubit system and the structure of its ring of local invariants the local invariants of a mixed two qubit system are discussed these invariants are polynomials in the elements of the corresponding density matrix they are counted by means of group theoretic branching rules which relate this problem to one arising in spin isospin nuclear shell models the corresponding molien series and a refinement in the form of a four parameter generating function are determined a graphical approach is then used to construct explicitly a fundamental set of 21 invariants relations between them are found in the form of syzygies by using these the structure of the ring of local invariants is determined and complete sets of primary and secondary invariants are identified there are 10 of the former and 15 of the latter

on the algebra of local unitary invariants of pure and mixed quantum states we study the structure of the inverse limit of the graded algebras of local unitary invariant polynomials using its hilbert series for k subsystems we show that the inverse limit is a free algebra and the number of algebraically independent generators with homogenous degree 2m equals the number of conjugacy classes of index m subgroups in a free group on k 1 generators similarly we show that the inverse limit in the case of k partite mixed state invariants is free and the number of algebraically independent generators with homogenous degree m equals the number of conjugacy classes of index m subgroups in a free group on k generators the two statements are shown to be equivalent to illustrate the equivalence using the representation theory of the unitary groups we obtain all invariants in the m 2 graded parts and express them in a simple form both in the case of mixed and pure states the transformation between the two forms is also derived analogous invariants of higher degree are also introduced

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

entanglement of four qubit systems a geometric atlas with polynomial compass i the finite world we investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory which allows us to interpret certain entangled states as algebraic varieties more precisely we describe the nullcone i e the set of states annihilated by all invariant polynomials and also the so called third secant variety which can be interpreted as the generalization of ghz states for more than three qubits all our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper these 47 varieties correspond to 47 non equivalent entanglement patterns which reduce to 15 different classes if we allow permutations of the qubits

fuzzy functions and their fundamental properties in this paper it is noticed that the definition of fuzzy equality of sasaki 1993 is not concordant with that of es and coker 1995 and sostak 1985 1988 furthermore the construction of fuzzy equality and fuzzy function is far from the way of natural fuzzification for this purpose the fuzzy equality of two points of a crisp set which is more convenient than that in es and coker 1995 and sostak 1985 1988 and the fuzzy function having weaker conditions than that is given in sasaki 1993 are redefined in much more natural way then various types of fuzzy functions are introduced and their fundamental properties are established

vague groups and generalized vague subgroups on the basis of 0 1 le in this paper various elementary properties of vague groups and some properties of vague binary operations related with their associativity aspects are obtained furthermore the concept of vague isomorphism is defined and some basic properties of this concept are studied the concept of external direct product of vague groups is established later the definition of generalized vague subgroup which is a generalization of the vague subgroup defined by demirci is introduced and the validity of some classical results in this setting is investigated on the basis of the particular integral commutative complete quasi monoidal lattice 0 1

generating synthetic time series from bak sneppen co evolution model mixtures abstract the bak sneppen model of co evolution is used to derive synthetic time series with a priori specified fractal dimension or hurst exponent through a mixing of processes in various lattice dimensions both theoretical and numerical analyses concern the avalanches at the critical threshold and provide a model for time series reconstruction that can be tested as an alternative to the classical fractional brownian motion fbm because of differences in properties new results on critical threshold and avalanche structure are obtained up to euclidean dimension d 6 the method involves a lattice based structure and therefore is suitable for the application of parallel computing

fuzzy relations an fuzzy functions abstract in this paper we apply the standard element free mathematical definitions of equivalence relation and function as relations to the case of fuzzy relations one of the surprising results of this approach is that fuzzy functions take their values from the complemented elements of a brouwerian lattice thus in the common case of the values being taken from the unit interval the only fuzzy functions of this type are the ordinary functions this might suggest that fuzzy set theory is more than just a generalization of ordinary set theory

enhanced fuzzy system models with improved fuzzy clustering algorithm although traditional fuzzy models have proven to have high capacity of approximating the real world systems they have some challenges such as computational complexity optimization problems subjectivity etc in order to solve some of these problems this paper proposes a new fuzzy system modeling approach based on improved fuzzy functions to model systems with continuous output variable the new modeling approach introduces three features i an improved fuzzy clustering ifc algorithm ii a new structure identification algorithm and iii a nonparametric inference engine the ifc algorithm yields simultaneous estimates of parameters of c regression models together with fuzzy c partitioning of the data to calculate improved membership values with a new membership function the structure identification of the new approach utilizes ifc instead of standard fuzzy c means clustering algorithm to fuzzy partition the data and it uses improved membership values as additional input variables along with the original scalar input variables for two different choices of regression methods least squares estimation or support vector regression to determine ldquofuzzy functionsrdquo for each cluster with novel ifc one could learn the system behavior more accurately compared to other fsm models the nonparametric inference engine is a new approach which uses the alike nearest neighbor method for reasoning empirical comparisons indicate that the proposed approach yields comparable or better accuracy than fuzzy or neuro fuzzy models based on fuzzy rules bases as well as other soft computing methods

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

new s function series and non compact lie groups a number of new infinite series of s functions are described in terms of their generating functions and s function content applications to the character theory of non compact lie groups are noted

fuzzy relation equations and inequalities with two unknowns and their applications abstract in this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise we show that solutions to each of these systems form a complete lattice and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations we also demonstrate the application of solutions to these systems in data reduction

fuzzy modifiers based on fuzzy relations in this paper we introduce a new type of fuzzy modifiers i e mappings that transform a fuzzy set into a modified fuzzy set based on fuzzy relations we show how they can be applied for the representation of weakening adverbs more or less roughly and intensifying adverbs very extremely in the inclusive and the non inclusive interpretation we illustrate their use in an approximate reasoning scheme

weakly linear systems of fuzzy relation inequalities the heterogeneous case new types of systems of fuzzy relation inequalities and equations called weakly linear have been recently introduced in j ignjatovic et al on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations fuzzy sets syst 161 2010 3081 3113 the mentioned paper dealt with homogeneous weakly linear systems composed of fuzzy relations on a single set and a method for computing their greatest solutions has been provided this method is based on the computing of the greatest post fixed point contained in a given fuzzy relation of an isotone function on the lattice of fuzzy relations here we adapt this method for computing the greatest solutions of heterogeneous weakly linear systems where the unknown fuzzy relation relates two possibly different sets we also introduce and study quotient fuzzy relational systems and establish relationships between solutions to heterogeneous and homogeneous weakly linear systems besides we point out to applications of the obtained results in the state reduction of fuzzy automata and computing the greatest simulations and bisimulations between fuzzy automata as well as in the positional analysis of fuzzy social networks

analytic results for scaling function and moments for a different type of avalanche in the bak sneppen evolution model starting from the master equation for the hierarchical structure of avalanches of a different kind within the frame of the bak sneppen evolution model we derive the exact formula of the scaling function describing the probability distribution of avalanches the scaling function displays features required by the scaling ansatz and verified by simulations using the scaling function we investigate the avalanche moment denoted by s k ensuremath delta overline f it is found that for any non negative integer k s k ensuremath delta overline f diverges as ensuremath delta f ensuremath k which gives an infinite group of exact critical exponents simulation outcomes of avalanche moments with k 1 2 3 are found to be consistent with the corresponding analytical results

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

visualizing two qubits the notions of entanglement witnesses and separable and entangled states for a two qubit system can be visualized in three dimensions using the slocc equivalence classes this visualization preserves the duality relations between the various sets and allows us to give proof by inspection of a nonelementary result of horodecki et al phys lett a 223 1 8 1996 that for two qubits peres ppt positive partial transpose test for separability is iff we then show that the chsh bell inequalities can be visualized as circles and cylinders in the same diagram this allows us to give a geometric proof of yet another result of horodecki et al phys lett a 200 340 344 1995 which optimizes the violation of the chsh bell inequality finally we give numerical evidence that remarkably allowing alice and bob to use three rather than two measurements each does not help them to distinguish any new entangled slocc equivalence class beyond the chsh class

reduction of fuzzy automata by means of fuzzy quasi orders abstract in our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations in that paper we have also studied reductions by means of those solutions which are fuzzy equivalences in this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi orders generally fuzzy quasi orders and fuzzy equivalences are equally good in the state reduction but we show that right and left invariant fuzzy quasi orders give better reductions than right and left invariant fuzzy equivalences we also show that alternate reductions by means of fuzzy quasi orders give better results than alternate reductions by means of fuzzy equivalences furthermore we study a more general type of fuzzy quasi orders weakly right and left invariant ones and we show that they are closely related to determinization of fuzzy automatons we also demonstrate some applications of weakly left invariant fuzzy quasi orders in conflict analysis of fuzzy discrete event systems

different hierarchy of avalanches observed in the bak sneppen evolution model a quantity overline f denoting the average fitness of an ecosystem is introduced in the bak sneppen model through this quantity a different hierarchy of avalanches f 0 avalanche is observed in the evolution of bak sneppen model an exact gap equation governing the self organization of the model is presented in terms of overline f it is found that self organized threshold f c can be exactly obtained two basic exponents of the new avalanche ensuremath tau avalanche distribution and d avalanche dimension are given through simulations of one and two dimensional bak sneppen models it is suggested that overline f may be a good quantity in determining the emergence of criticality

local unitary invariants for multipartite quantum systems a method is presented to obtain local unitary invariants for multipartite quantum systems consisting of fermions or distinguishable particles the invariants are organized into infinite families in particular the generalization to higher dimensional single particle hilbert spaces is straightforward many well known invariants and their generalizations are also included

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

a bipartite class of entanglement monotones for n qubit pure states we construct a class of algebraic invariants for n qubit pure states based on bipartite decompositions of the system we show that they are entanglement monotones and that they differ from the well known linear entropies of the sub systems they therefore capture new information on the non local properties of multipartite systems

new branching rules induced by plethysm we derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor t of young symmetry type where is an integer partition the branchings and fixing a vector vi a symmetric tensor gij gji and an antisymmetric tensor fij fji respectively are obtained as special cases all new branchings are governed by schur function series obtained from plethysms of the schur function s by the basic m series of complete symmetric functions and the l m 1 series of elementary symmetric functions our main technical tool is that of hopf algebras and our main result is the derivation of a coproduct for any schur function series obtained by plethysm from another such series therefrom one easily obtains generalized newell littlewood formulae and the algebra of the formal group characters of these subgroups is established concrete examples and extensive tabulations are displayed for and showing their involved and nontrivial representation theory the nature of the subgroups is shown to be in general affine and in some instances non reductive we discuss the complexity of the coproduct formula and give a graphical notation to cope with it we also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2 cocycles the algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters modification rules are beyond the scope of the present paper but are briefly discussed

on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations in this paper we study systems of fuzzy relation inequalities and equations of the form u v i v i u i i where u is an unknown and v i i i are given fuzzy relations the dual systems v i u u v i i i their conjunctions the systems of the form u v i v i u i i and certain special types of these systems we call them weakly linear systems for each weakly linear system with a complete residuated lattice as the underlying structure of truth values we prove the existence of the greatest solution and we provide an algorithm for computing the greatest solution which works whenever the underlying complete residuated lattice is locally finite otherwise we determine some sufficient conditions under which the algorithm works the algorithm is iterative and each its single step can be viewed as solving of a particular linear system weakly linear systems emerged from the fuzzy automata theory but we show that they also have important applications in other fields e g in the concurrency theory and social network analysis

on the geometry of a class of n qubit entanglement monotones a family of n qubit entanglement monotones invariant under stochastic local operations and classical communication slocc is defined this class of entanglement monotones includes the well known examples of the concurrence the 3 tangle and some of the four five and n qubit slocc invariants introduced recently the construction of these invariants is based on bipartite partitions of the hilbert space in the form with l 2n n l 2n such partitions can be given a nice geometrical interpretation in terms of grassmannians gr l l of l planes in cl that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the plucker embedding the invariants are neatly expressed in terms of the plucker coordinates of the grassmannians

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

fuzzy function as an approximate solution to a system of fuzzy relation equations this paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations however we put a new light on this problem connecting it with the interpolation of a fuzzy function we introduce a notion of fuzzy function and its representation by fuzzy relation we demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations first we explain the general framework and then we prove some results related to the problem of the best approximation in particular we have shown that fuzzy relations introduced by sanchez and mamdani are the best approximations in certain approximation spaces

system of fuzzy relation equations as a continuous model of if then rules we claim that a fuzzy relation which models a system of fuzzy if then rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable in view of this connection it is important to consider the problem of solvability of a system of fuzzy relation equations with respect to an unknown fuzzy relation this problem is analyzed in the present paper we propose several new necessary and sufficient criteria for the so called mamdani relation to be a solution to the system we prove that in the general case these same criteria are sufficient but not always necessary for the solvability of the system a new easy to check criterion is proposed for the solvability of the system with special fuzzy parameters

the hopf algebra structure of the character rings of classical groups the character ring of covariant irreducible tensor representations of the general linear group admits a hopf algebra structure isomorphic to the hopf algebra of symmetric functions here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions we show that and also admit natural hopf algebra structures that are isomorphic to that of and hence to the isomorphisms are determined explicitly along with the specification of standard bases for and analogous to those used for a major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic schur hall scalar products significantly the adjoint with respect to multiplication no longer coincides as it does in the case with a foulkes derivative or skew operation the adjoint and foulkes derivative now require separate definitions and their properties are explored here in the orthogonal and symplectic cases moreover the hopf algebras and are not self dual the dual hopf algebras and are identified finally the hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified

functions defined on a set of z numbers abstract a wide class of problems in science and engineering requires mathematical formalization one of the important aspects in this regard is that we should account for various types of uncertainties which are simultaneously present in real world problems and result from imprecise and partially reliable information this mandates a need for a theoretical basis of uncertain functional dependencies the theories of interval valued functions functions of random variables and fuzzy functions are well developed the main disadvantage of these theories is that each of them can deal with only a single type of uncertainty in order to ground the formal basis for dealing with real world information zadeh suggested the concept of a z number as an ordered pair z a b of fuzzy numbers used to describe a value of a random variable x where a is a fuzzy constraint on values of x and b is a fuzzy value of probability measure of a unfortunately there is no study on functions of z numbers in existence in this paper for the first time we propose a general methodology for a construction of such functions on the basis of the extension principle applied to z numbers the proposed methodology is helpful in restricting the increase of uncertainty in computation the values of z valued functions some properties of functions of z numbers are considered we construct typical and commonly encountered functions such as addition of z numbers minimum of z numbers square root of z numbers and others a spectrum of examples is provided to illustrate the proposed approach

a theory of vague lattices based on many valued equivalence relations i general representation results the present work introduces a new and general theory of ordering relations and lattices based on many valued equivalence relations under the name vague ordering relations and vague lattices respectively representations and constructions of vague ordering relations and vague lattices are the main subjects of this paper and various desirable results in this direction are established furthermore algebraic characterizations of vague lattices are studied

geometry of a two qubit state and intertwining quaternionic conformal mapping under local unitary transformations in this paper the geometry of two qubit systems under a local unitary group so 2 su 2 is discussed it is shown that the quaternionic conformal map intertwines between this local unitary subgroup of sp 2 and the quaternionic mobius transformation which is rather a generalization of the results of lee et al 2002 quantum inf process 1 129

fuzzy relational inequalities and equations fuzzy quasi orders closures and openings of fuzzy sets the paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations as well as with systems of related fuzzy relation inequalities our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations the systems are studied in the framework of complete residuated lattices as the structure of membership values by means of fuzzy quasi orders and closures and openings on a collection of fuzzy sets

a graded notion of functionality abstract we will introduce the notion of functionality as a graded property of a fuzzy relation we will extend known results relating to functional fuzzy relations to their graded counterparts and provide new properties w r t various kinds of relational compositions as a contribution to the theory of fuzzy control we will moreover provide an example of use of graded properties in establishing implicative models of fuzzy rule bases where the non graded property fails

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given