rough games modeled via l fuzzy ambiguous representations of semilattices abstract monotonic predicates and l fuzzy ambiguous representations of continuous semilattices with bottom elements are applied to describe rough normal form games and rough extensive form two player zero sum games with imperfect information payoff predicates are introduced to find best guaranteed gains of the players these predicates are calculated recursively via backward payoff predicate transformers which are also studied it is shown how payoff predicates determine almost optimal strategies an approximate minimax theorem is also stated

bulk universality for generalized wigner matrices consider n n hermitian or symmetric random matrices h where the distribution of the i j matrix element is given by a probability measure ij with a subexponential decay let sigma ij 2 be the variance for the probability measure ij with the normalization property that sum i sigma 2 ij 1 for all j under essentially the only condition that c le n sigma ij 2 le c 1 for some constant c 0 we prove that in the limit n the eigenvalue spacing statistics of h in the bulk of the spectrum coincide with those of the gaussian unitary or orthogonal ensemble gue or goe we also show that for band matrices with bandwidth m the local semicircle law holds to the energy scale m 1

partial ordering in l underdeterminate sets abstract the purpose of this paper is to improve results on fuzzy partial orderings obtained by zadeh in 9 we overcome the difficulties connected with the axioms of antisymmetry and linearity moreover if the underlying lattice l is a complete residuated lattice we establish a szpilrajn theorem i e any l fuzzy partial ordering has a linear extension in opposition to zadeh s our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless therefore we first introduce the category lus cf 2 which can be considered as a mathematical model of fuzzy equality and subsequently we specify the axioms of l fuzzy partial orderings with respect to the frame given by lus the axioms we use clearly display the usefulness of having a zadeh like complementation and as a consequence the usefulness of a positivistic and nonintuitionistic frame of study an example concerning l rn which we give clearly shows that the lus version of the szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem but provides us with additional information

central limit theorem for linear eigenvalue statistics of random matrices with independent entries we consider n x n real symmetric and hermitian wigner random matrices n 1 2 w with independent modulo symmetry condition entries and the null sample covariance matrices n 1 x x with independent entries of m x n matrix x assuming first that the 4th cumulant excess 4 of entries of w and x is zero and that their 4th moments satisfy a lindeberg type condition we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem clt as n m m n c 0 oo with the same variance as for gaussian matrices if the test functions of statistics are smooth enough essentially of the class c 5 this is done by using a simple interpolation trick from the known results for the gaussian matrices and the integration by parts presented in the form of certain differentiation formulas then by using a more elaborated version of the techniques we prove the clt in the case of nonzero excess of entries again for essentially 5 test function here the variance of statistics contains an additional term proportional to 4 the proofs of all limit theorems follow essentially the same scheme

rough approximation operators on two universes of discourse and their fuzzy extensions based on a crisp binary relation between two universes of discourse one crisp covering and three crisp binary relations on the single universe of discourse are induced in this paper based on these induced notions four pairs of rough approximation operators are formulated so that the approximating sets and the approximated sets are on the same universe of discourse furthermore basic properties of the new approximation operators are investigated and the relationships among the operators are also examined subsequently conditions under which some or all of these approximation operators are equivalent are obtained on the other hand the generalizations of these rough approximation operators are also made in a fuzzy approximation space via a fuzzy implicator i and a triangular norm t consequently four pairs of fuzzy rough approximation operators are constructed basic properties and comparison of the fuzzy rough approximation operators are discussed if i is an r implicator with respect to a t norm t then it is shown that three pairs of the fuzzy rough approximation operators are identical iff the induced fuzzy covering is a fuzzy t partition

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

a note on the hypergraph functor the purpose of this note is to show that in the case of spatial locales l the hypergraph functor has a right adjoint functor

ambiguous representations as fuzzy relations between sets crisp and l fuzzy ambiguous representations of closed subsets of one space by closed subsets of another space are introduced it is shown that for each pair of compact hausdorff spaces the set of crisp or l fuzzy ambiguous representations is a lattice and a compact hausdorff lawson upper semilattice the categories of ambiguous and l ambiguous representations are defined and investigated

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

fuzzy congruences in groups the purpose of this paper is to introduce some basic concepts of fuzzy congruence and to prove some fundamental properties we obtain several results which are analogs of some basic theorems of group theory in particular we prove that in a group g the lattice of some fuzzy normal subgroups is modular

solution of a quadratic nonlinear oscillator by the method of harmonic balance abstract this paper deals with the quadratic nonlinear oscillator qno x x e x 2 0 this oscillator is compared with the nonlinear oscillators x x e x x 0 which are solved by the method of harmonic balance therefore we obtain the first approximate solution to the qno this solution is more accurate than the second approximate solution obtained by the lindstedt poincare method for qnos it has been pointed out that the angular frequency for x 0 should be different from the angular frequency for x 0 if solutions of higher degree of accuracy are required

exact solution of the quadratic mixed parity helmholtz duffing oscillator abstract in this paper the exact solution of the quadratic mixed parity helmholtz duffing oscillator is derived by using jacobi elliptic functions it is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind at the end of the paper we examine the stability of the system and determine the regions for which periodic and unbounded motions take place

fuzzy normal subgroups and fuzzy quotients abstract in case of groups a mormal subgroup n of a group g can be defined with three equivalent approaches namely i as a kernel of some group homomorphism ii as a subgroup commuting with every element of g and iii as the subgroup n closed with respect to conjugates of elements of n in case of generalizations of this concept to fuzzy sets the elementwise third approach is found more suitable and hence is widely adopted by many authors in this paper we further study this concept of fuzzy subgroups and introduce fuzzy quotients called fuzzy quotient groups for all 0 1 two different definitions of these fuzzy quotients have been given which are shown to lead to isomorphic structures we also study the direct products of fuzzy normal subgroups min product as per the terminology of sherwood 1983 and the problem of writing a fuzzy normal subgroups of a direct product of some groups as a direct product of certain fuzzy normal subgroups has been discussed in detail

fuzzy sets and sheaves part ii sheaf theoretic foundations of fuzzy set theory with applications to algebra and topology this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

product structure of the fuzzy factor groups abstract in this paper first we prove that the fuzzy factor group a b has the sup property if a has then we study the product structure of the fuzzy factor groups

reduction of fuzzy strict order relations abstract fuzzy strict order relations and the notion of their reduction are defined a necessary and sufficient condition is obtained for the transitive closure of the reduction to coincide with the strict order itself some possible graph theoretic significances of these results are discussed in the conclusion

fixed energy universality for generalized wigner matrices we prove the wigner dyson mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and hermitian wigner matrices previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for hermitian matrices if the energy parameter is fixed we develop a homogenization theory of the dyson brownian motion and show that microscopic universality follows from mesoscopic statistics 2015 wiley periodicals inc

asymptotic expansions based on smooth functions in the central limit theorem stein s method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables together with lyapounov estimates of the error in the approximation

dyson brownian motion for general beta and potential at the edge in this paper we compare the solutions of the dyson brownian motion for general beta and potential v and the associated mckean vlasov equation near the edge under suitable conditions on the initial data and the potential v we obtain optimal rigidity estimates of particle locations near the edge after a short time t mathrm o 1 our argument uses the method of characteristics along with a careful estimate involving an equation of the edge with the rigidity estimates as an input we prove a central limit theorem for mesoscopic statistics near the edge which as far as we know has been done for the first time in this paper additionally combining our results with landon and yau edge statistics of dyson brownian motion arxiv 1712 03881 2017 we give a proof of the local ergodicity of the dyson brownian motion for general beta and potential at the edge i e we show the distribution of extreme particles converges to the tracy widom beta distribution in a short time

generalized t norm structures abstract this is a continuation of the work in drossos and navara eufit 96 aachen germany 1996 in this paper we investigate the possibility of defining a quantitative structure non idempotent on the top of a usual qualitative structure these generalized monoidal structures are very much related to the notion of t norms and t conorms defined on the closed interval 0 1 we would then like to define an appropriate generalization of the notion of t norm as a generalized monoidal structure which will reflect a minimum of properties of t norms defined on 0 1 considered as absolutely basic

solutions of a quadratic nonlinear oscillator iteration procedure a modified iteration procedure is applied to a quadratic nonlinear oscillator qno when the solutions of the two auxiliary equations are available we obtain the first and second analytical approximate solutions to the qno our second approximation result significantly improves on the accuracy of the approximate solution obtained by using the first order harmonic balance method

spectral statistics of erdős renyi graphs i local semicircle law we consider the ensemble of adjacency matrices of erd s renyi random graphs that is graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one we prove that as long as pn with a speed at least logarithmic in n the density of eigenvalues of the erd s renyi ensemble is given by the wigner semicircle law for spectral windows of length larger than n 1 up to logarithmic corrections as a consequence all eigenvectors are proved to be completely delocalized in the sense that the l norms of the l2 normalized eigenvectors are at most of order n 1 2 with a very high probability the estimates in this paper will be used in the companion paper spectral statistics of erd s renyi graphs ii eigenvalue spacing and the extreme eigenvalues 2011 preprint to prove the universality of eigenvalue distributions both in the bulk and at the spectral edges under the further restriction that pn n2 3

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

isotropic self consistent equations for mean field random matrices we present a simple and versatile method for deriving an isotropic local laws for general random matrices constructed from independent random variables our method is applicable to mean field random matrices where all independent variables have comparable variances it is entirely insensitive to the expectation of the matrix in this paper we focus on the probabilistic part of the proof the derivation of the self consistent equations as a concrete application we settle in complete generality the local law for wigner matrices with arbitrary expectation

mesoscopic eigenvalue density correlations of wigner matrices we investigate to what extent the microscopic wigner gaudin mehta dyson wgmd or sine kernel statistics of random matrix theory remain valid on mesoscopic scales to that end we compute the connected two point spectral correlation function of a wigner matrix at two mesoscopically separated points in the mesoscopic regime density correlations are much weaker than in the microscopic regime our result is an explicit formula for the two point function this formula implies that the wgmd statistics are valid to leading order on all mesoscopic scales that in the real symmetric case there are subleading corrections matching precisely the wgmd statistics while in the complex hermitian case these subleading corrections are absent we also uncover non universal subleading correlations which dominate over the universal ones beyond a certain intermediate mesoscopic scale the proof is based on a hierarchy of schwinger dyson equations for a sufficiently large class of polynomials in the entries of the green function the hierarchy is indexed by a tree whose depth is controlled using stopping rules a key ingredient in the derivation of the stopping rules is a new estimate on the density of states which we prove to have bounded derivatives of all order on all mesoscopic scales

membership functions for a fuzzy group from similarity relations abstract we show that the membership function x of a fuzzy group g satisfies the equation x e x where x y is a similarity relation on g which is invariant under left translation we also show that under some natural assumptions one can represent the elements x g as 1 1 onto transformations px on a suitable universe such that x equals the proportion of points in which are fixed points of px these results provide a deeper insight on the impact of group operation on the membership values x

gaussian fluctuations of eigenvalues in wigner random matrices we study the fluctuations of eigenvalues from a class of wigner random matrices that generalize the gaussian orthogonal ensemble we begin by considering an n n matrix from the gaussian orthogonal ensemble goe or gaussian symplectic ensemble gse and let xk denote eigenvalue number k under the condition that both k and n k tend to infinity as n w e show thatxk is normally distributed in the limit we also consider the joint limit distribution of eigenvalues xk1 x km from the goe or gse where k1 n km and ki 1 ki 1 i m 1 tend to infinity with n the result in each case is an m dimensional normal distribution using a recent universality result by tao and vu we extend our results to a class of wigner real symmetric matrices with non gaussian entries that have an exponentially de caying distribution and whose first four moments match the gaussian moments

spectral statistics of erdős renyi graphs ii eigenvalue spacing and the extreme eigenvalues we consider the ensemble of adjacency matrices of erd s renyi random graphs i e graphs on n vertices where every edge is chosen independently and with probability p p n we rescale the matrix so that its bulk eigenvalues are of order one under the assumption p n gg n 2 3 we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum more precisely we prove 1 that the eigenvalue spacing of the erd s renyi graph in the bulk of the spectrum has the same distribution as that of the gaussian orthogonal ensemble and 2 that the second largest eigenvalue of the erd s renyi graph has the same distribution as the largest eigenvalue of the gaussian orthogonal ensemble as an application of our method we prove the bulk universality of generalized wigner matrices under the assumption that the matrix entries have at least 4 e moments

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

a hereditarily indecomposable mathcal l _ infty space that solves the scalar plus compact problem we construct a hereditarily indecomposable banach space with dual space isomorphic to l1 every bounded linear operator on this space is expressible as i k with a scalar and k compact

topological aspects of non convergent sequences a comment on burgin s concept of fuzzy limits abstract it is well known that metric spaces can be provided with the structure of 0 1 valued sets on this background this paper explains an application of topologies on 0 1 valued sets to metric spaces and shows that every bounded sequence has always local limit points represented by singletons of the underlying 0 1 valued set

on the 1 n corrections to the green functions of random matrices with independent entries we propose a general approach to the construction of 1 n corrections to the green function gn z of the ensembles of random real symmetric and hermitian n n matrices with independent entries hk l by this approach we study the correlation function cn z1 z2 of the normalized trace n 1trgn assuming that the average of mod hk l5 is bounded we found that to leading order cn z1 z2 n 2f z1 z2 where f z1 2 only depends on the second and the fourth moments of hk l for the correlation function of the density of energy levels we obtain an expression which in the scaling limit only depends on the second moment of hk l this can be viewed as supporting the universality conjecture of random matrix theory

exact solution of a quadratic nonlinear oscillator abstract the exact solution of a quadratic nonlinear oscillator qno is presented by using the jacobian elliptic sine function the constants in the solution are expressed in terms of the initial conditions the formulas for the period of oscillation t and amplitude b for x 0 are also given this exact solution can serve as a benchmark solution for various approximate solutions it also can be utilized for educational purpose

fuzzy congruence relations abstract if a x f is an universal algebra and if r x denotes the set of all fuzzy subsets of x x i e fuzzy relations on x then the algebraic operations fg f on x are extended to algebraic operations f on r x f is an algebra similar to a it is called relational algebra a fuzzy equivalence relation which is at the same time a fuzzy subalgebra of r x f is called a fuzzy congruence relation on x in this paper we study fuzzy congruence relations amongst other things we prove that the set of all fuzzy congruence relations on an algebra is a complete lattice and an algebraic closure system

a fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions a new algorithm is proposed to combine the split frequency harmonic balance method sf hbm with arc length continuation alc for accurate tracing of the frequency response of oscillators with non expansible nonlinearities alc is incorporated into the sf hbm in a two stage procedure stage i involves finding a reasonably accurate response frequency and solution using a relatively large number of low frequency harmonics this step is achieved using the sf hbm in conjunction with alc stage ii uses the sf hbm to obtain a very accurate solution at the frequency obtained in stage i to guarantee rapid path tracing the frequency axis is appropriately subdivided this gives high chance of success in finding a globally optimum set of harmonic coefficients when approaching a turning point however arc lengths are adaptively reduced to obtain a very accurate solution the combined procedure is tested on three hardening stiffness examples a duffing model an oscillator with non expansible stiffness and single harmonic forcing and an oscillator with non expansible stiffness and multiple harmonic forcing the results show that for non expansible nonlinearities and multiple harmonic forcing the proposed algorithm is capable of tracing out frequency response functions with high accuracy and efficiency

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory

asymptotic properties of large random matrices with independent entries we study the normalized trace gn z n 1 tr h zi 1 of the resolvent of n n real symmetric matrices h 1 jk wjk n j k 1n assuming that their entries are independent but not necessarily identically distributed random variables we develop a rigorous method of asymptotic analysis of moments of gn z for iz 0 where 0 is determined by the second moment of wjk by using this method we find the asymptotic form of the expectation e gn z and of the connected correlator e gn z1 gn z2 e gn z1 e gn z2 we also prove that the centralized trace ngn z e ngn z has the gaussian distribution in the limit n based on these results we present heuristic arguments supporting the universality property of the local eigenvalue statistics for this class of random matrix ensembles