universal microscopic correlation functions for products of independent ginibre matrices we consider the product of n complex non hermitian independent random matrices each of size n n with independent identically distributed gaussian entries ginibre matrices the joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single ginibre matrix but with a more complicated weight given by a meijer g function depending on n using the method of orthogonal polynomials we compute all eigenvalue density correlation functions exactly for finite n and fixed n they are given by the determinant of the corresponding kernel which we construct explicitly in the large n limit at fixed n we first determine the microscopic correlation functions in the bulk and at the edge of the spectrum after unfolding they are identical to that of the ginibre ensemble with n 1 and thus universal in contrast the microscopic correlations we find at the origin differ for each n 1 and generalize the known bessel law in the complex plane for n 2 to a new hypergeometric kernel 0fn 1

probability of all eigenvalues real for products of standard gaussian matrices with xi independent n n standard gaussian random matrices the probability that all eigenvalues are real for the matrix product pm xmxm 1 x1 is expressed in terms of an n 2 n 2 n even and n 1 2 n 1 2 n odd determinant the entries of the determinant are certain meijer g functions in the case m 2 high precision computation indicates that the entries are rational multiples of 2 with the denominator a power of 2 and that to leading order in n decays as we are able to show that for general m and large n with an explicit bm an analytic demonstration that as m is given

convergence of the largest singular value of a polynomial in independent wigner matrices for polynomials in independent wigner matrices we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables under fourth moment hypotheses we actually prove a more general result of the form no eigenvalues outside the support of the limiting eigenvalue distribution we build on ideas of haagerup schultz thorbjornsen on the one hand and bai silverstein on the other we refine the linearization trick so as to preserve self adjointness and we develop a secondary trick bearing on the calculation of correction terms instead of poincare type inequalities we use a variety of matrix identities and lp estimates the schwinger dyson equation controls much of the analysis

max min solutions for fuzzy multiobjective matrix games abstract in this paper we consider two person zero sum games with fuzzy multiple payoff matrices we assume that each player has a fuzzy goal for each of the payoffs a degree of attainment of the fuzzy goal is defined and the max min strategy with respect to the degree of attainment of the fuzzy goal is examined if all of the membership functions both for the fuzzy payoffs and for the fuzzy goals are linear the formulated mathematical programming problem which yields the max min strategy can be reduced to the linear programming problem by making use of sakawa s method the variable transformations and the relaxation procedure

on the concentration of random multilinear forms and the universality of random block matrices the circular law asserts that if mathbf x n is a n times n matrix with iid complex entries of mean zero and unit variance then the empirical spectral distribution of frac 1 sqrt n mathbf x n converges almost surely to the uniform distribution on the unit disk as n tends to infinity answering a question of tao we prove the circular law for a general class of random block matrices with dependent entries the proof relies on an inverse type result for the concentration of linear operators and multilinear forms

singular value correlation functions for products of wishart random matrices we consider the product of m quadratic random matrices with complex elements and no further symmetry where all matrix elements of each factor have a gaussian distribution this generalizes the classical wishart laguerre gaussian unitary ensemble with m 1 in this paper we first compute the joint probability distribution for the singular values of the product matrix when the matrix size n and the number m are fixed but arbitrary this leads to a determinantal point process which can be realized in two different ways first it can be written as a one matrix singular value model with a nonstandard jacobian or second for m 2 as a two matrix singular value model with a set of auxiliary singular values and a weight proportional to the meijer g function for both formulations we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct they are given in terms of the hypergeometric and meijer g functions generalizing the laguerre polynomials for m 1 our investigation was motivated from applications in telecommunication of multi layered scattering multiple input and multiple output channels we present the ergodic mutual information for finite n for such a channel model with m 1 layers of scatterers as an example

products of rectangular random matrices singular values and progressive scattering we discuss the product of m rectangular random matrices with independent gaussian entries which have several applications including wireless telecommunication and econophysics for complex matrices an explicit expression for the joint probability density function is obtained using the harish chandra itzykson zuber integration formula explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two matrix model and the method of biorthogonal polynomials this generalizes the classical result for the so called wishart laguerre gaussian unitary ensemble or chiral unitary ensemble at m 1 and previous results for the product of square matrices the correlation functions are given by a determinantal point process where the kernel can be expressed in terms of meijer g functions we compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices the location of the end points of support for the latter are analyzed in detail for general m finally we consider the so called ergodic mutual information which gives an upper bound for the spectral efficiency of a mimo communication channel with multifold scattering

equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs and fuzzy goals when the game theory is applied to real world problems such as decision making in public and managerial problems on occasions it is difficult to assess payoffs exactly because of inaccuracy of information and fuzzy understanding of situations by experts in such cases games with fuzzy payoffs in which payoffs are represented as fuzzy numbers are often considered in this paper we consider equilibrium solutions in multiobjective bimatrix games with fuzzy payoffs we introduce fuzzy goals for payoffs in order to incorporate ambiguity of a player s judgements and assume that the player tries to maximize degrees of attainment of the fuzzy goals the fuzzy goals for payoffs and the equilibrium solution with respect to the degree of attainment of the fuzzy goal are defined two basic methods one by weighting coefficients and the other by a minimum component are employed to aggregate multiple fuzzy goals when membership functions of fuzzy payoffs and fuzzy goals are all linear and the fuzzy decision in terms of the intersection is employed the necessary conditions that pairs of strategies be the equilibrium solutions is obtained when membership functions of fuzzy payoffs are quadratic functions those of fuzzy goals are linear and the fuzzy decision in terms of the convex combination is employed we also derive the necessary conditions that pairs of strategies be the equilibrium solutions

the circular law for random matrices we consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance we prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries we assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices the results are based on previous work of bai rudelson and the authors extending results to a larger class of sparse matrices

fuzzy approaches to the game of chicken game theory deals with decision making processes involving two or more parties also known as players with partly or completely conflicting interests decision makers in a conflict must often make their decisions under risk and under unclear or fuzzy information in this paper two distinct fuzzy approaches are employed to investigate an extensively studied 2 spl times 2 game model the game of chicken the first approach uses a fuzzy multicriteria decision analysis method to obtain optimal strategies for the players it incorporates subjective factors into the decision makers objectives and aggregates objectives using a weight vector the second approach applies the theory of fuzzy moves tfm to the game of chicken the theory of moves tom is designed to bring a dynamic dimension to the classical theory of games by allowing decision makers to look ahead for one or several steps so that they can make a better decision tom is the crisp counterpart of tfm the approach we implement here to deal with games that include fuzzy and uncertain information the application of fuzzy approaches to the game of chicken demonstrates their effectiveness in manipulating subjective uncertain and fuzzy information and provides valuable insights into the strategic aspects of chicken

products of independent elliptic random matrices for fixed m 1 we study the product of m independent n times n elliptic random matrices as n tends to infinity our main result shows that the empirical spectral distribution of the product converges with probability 1 to the m th power of the circular law regardless of the joint distribution of the mirror entries in each matrix this leads to a new kind of universality phenomenon the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves our result also generalizes earlier results of gotze tikhomirov on the asymptotic spectrum of products of independent random matrices available at http arxiv org abs 1012 2710 and o rourke soshnikov j probab 16 81 2219 2245 2011 concerning the product of independent iid random matrices

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

existence of equilibrium solution for a non cooperative game with fuzzy goals and parameters in this paper we introduce a concept of equilibrium for a non cooperative game with fuzzy goals involving fuzzy parameters this equilibrium is based on zimmermann s approach for solving linear multiobjective problems with fuzzy goals and the concept of n s equilibrium introduced by zhukovskii for a non cooperative game with payoffs involving unknown parameters in the case of complete ignorance of their behavior we also provide a theorem of existence of this equilibrium based on the fan inequality

nash equilibrium strategy for fuzzy non cooperative games in this article two person zero sum games are investigated in the fuzzy environment several models constructed by maeda in the symmetrical fuzzy environment are extended to the models in the asymmetric fuzzy environment the existence of equilibrium strategies for these extended models is proposed in the asymmetric fuzzy environment however in some cases nash equilibrium strategies may not exist therefore two special cases are presented for which nash equilibrium strategies do exist in order to investigate the existence of weak pareto nash equilibrium strategies for fuzzy matrix games we introduce the concept of crisp bi matrix games with parameters by solving the parametric bi matrix games we obtain the weak pareto nash equilibrium strategies for the fuzzy matrix games

a fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers the aim of this paper is to develop an effective method for solving matrix games with payoffs of triangular fuzzy numbers tfns which are arbitrary in this method it always assures that players gain floor and loss ceiling have a common tfn type fuzzy value and hereby any matrix game with payoffs of tfns has a tfn type fuzzy value based on duality theorem of linear programming lp and the representation theorem for fuzzy sets the mean and the lower and upper limits of the tfn type fuzzy value are easily computed through solving the derived lp models with data taken from 1 cut set and 0 cut set of fuzzy payoffs hereby the tfn type fuzzy value of any matrix game with payoffs of tfns can be explicitly obtained moreover we can easily compute the upper and lower bounds of any alfa cut set of the tfn type fuzzy value for any matrix game with payoffs of tfns and players optimal mixed strategies through solving the derived lp models at any specified confidence level alfa the proposed method in this paper is demonstrated with a numerical example and compared with other methods to show the validity applicability and superiority

fuzzy linear programming models to solve fuzzy matrix games abstract in this paper a zero sum two person game with imprecise values in its matrix of payoffs is considered we propose a method for its solution based on the establishment of a fuzzy linear programming f l p problem for each player the method is shown as a generalization of that conventionally used in the solution of a classical game to solve such f l p problems we propose the auxiliary models resulting from the application of some of the methods for ranking fuzzy numbers hence according to the kind of ranking method that players want to use different solutions to the former game can be obtained we show that these solutions are of the same nature as the parameters defining the game and corresponding to a fuzzy predicate that can be established as the value of the game is around

sharp commutator estimates via harmonic extensions abstract we give an alternative proof of several sharp commutator estimates involving riesz transforms riesz potentials and fractional laplacians our methods only involve harmonic extensions to the upper half space integration by parts and trace space characterizations the commutators we investigate are jacobians more generally coifman rochberg weiss commutators chanillo s commutator with the riesz potential coifman mcintosh meyer kato ponce vega type commutators the fractional leibniz rule and the da lio riviere three term commutators we also give a new limiting l 1 estimate for a double commutator of coifman rochberg weiss type and several intermediate estimates the beauty of our method is that all those commutator estimates which are originally proven by various specific methods or by general para product arguments can be obtained purely from integration by parts and trace theorems another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way

limiting distributions of spectral radii for product of matrices from the spherical ensemble abstract consider the product of m independent n n random matrices from the spherical ensemble for m 1 the spectral radius is defined as the maximum absolute value of the n eigenvalues of the product matrix when m 1 the limiting distribution for the spectral radii has been obtained by jiang and qi 2017 in this paper we investigate the limiting distributions for the spectral radii in general when m is a fixed integer we show that the spectral radii converge weakly to distributions of functions of independent gamma random variables when m m n tends to infinity as n goes to infinity we show that the logarithmic spectral radii have a normal limit

eigenvalues of block structured asymmetric random matrices we study the spectrum of an asymmetric random matrix with block structured variances the rows and columns of the random square matrix are divided into d partitions with arbitrary size linear in n the parameters of the model are the variances of elements in each block summarized in g r d d using the hermitization approach and by studying the matrix valued stieltjes transform we show that these matrices have a circularly symmetric spectrum we give an explicit formula for their spectral radius and a set of implicit equations for the full density function we discuss applications of this model to neural networks

random matrices universality of esds and the circular law given an n x n complex matrix a let mu a x y 1 n vertical bar 1 i n re lambda i x im lambda i y vertical bar be the empirical spectral distribution esd of its eigenvalues lambda i is an element of c i l n we consider the limiting distribution both in probability and in the almost sure convergence sense of the normalized esd mu 1 root n an of a random matrix a n a ij 1 i j n where the random variables a ij e a ij are i i d copies of a fixed random variable x with unit variance we prove a universality principle for such ensembles namely that the limit distribution in question is independent of the actual choice of x in particular in order to compute this distribution one can assume that x is real or complex gaussian as a related result we show how laws for this esd follow from laws for the singular value distribution of 1 root n a n zi for complex z as a corollary we establish the circular law conjecture both almost surely and in probability which asserts that mu 1 root n an converges to the uniform measure on the unit disc when the a ij have zero mean

fuzzy nash equilibriums in crisp and fuzzy games in this paper we introduce fuzzy nash equilibrium to determine a graded representation of nash equilibriums in crisp and fuzzy games this interpretation shows the distribution of equilibriums in the matrix form of a game and handles uncertainties in payoffs in addition a new method to rank fuzzy values with the user s viewpoint is investigated by this means the definition of satisfaction function which provides the result of comparison in the form of real value is developed when users have preferences regarding the payoffs

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

eigenvalues of large chiral non hermitian random matrices we study a non hermitian chiral random matrix of which the eigenvalues are complex random variables the empirical distributions and the radius of the eigenvalues are investigated the limit of the empirical distributions is a new probability distribution defined on the complex plane the graphs of the density functions are plotted the surfaces formed by the density functions are understood through their convexity and their gaussian curvatures the limit of the radius is a gumbel distribution the main observation is that the joint density function of the eigenvalues of the chiral ensemble after a transformation becomes a rotation invariant determinantal point process on the complex plane then the eigenvalues are studied by the tools developed by jiang and qi j theor probab 30 326 2017 32 353 2019 most efforts are devoted to deriving the central limit theorems for distributions defined by the bessel functions via the method of steepest descent and the estimates of the zero of a non trivial equation as the saddle point

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

solutions for fuzzy matrix games in this paper we deal with two person zero sum games with fuzzy payoffs and fuzzy goals we have presented two models for studying two person zero sum matrix games with fuzzy payoffs and fuzzy goals we assume that each player has a fuzzy goal for each of the payoffs we obtained that the fuzzy relation approach and the max min solution are equivalent

eigenvector statistics of the product of ginibre matrices we develop a method to calculate left right eigenvector correlations of the product of m independent n n complex ginibre matrices for illustration we present explicit analytical results for the vector overlap for a couple of examples for small m and n we conjecture that the integrated overlap between left and right eigenvectors is given by the formula o 1 m 2 n 1 and support this conjecture by analytical and numerical calculations we derive an analytical expression for the limiting correlation density as n for the product of ginibre matrices as well as for the product of elliptic matrices in the latter case we find that the correlation function is independent of the eccentricities of the elliptic laws