solving bimatrix games with fuzzy payoffs by introducing nature as a third player the paper elaborates on the fuzzy bimatrix games the payoffs are assumed to be fuzzy intervals with bounded support as defined by dubois and prade a new equilibrium concept is defined for this game by introducing nature as a third player and using a cuts sufficient conditions for its existence and a procedure for its computation are provided

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

two dimensional spectral estimation in this paper effective methods for generating two dimensional quarter plane causal autoregressive ar and autoregressive moving average arma spectral estimation models are developed these procedures are found to provide super resolution capabilities when compared to other more classical methods such as the fourier transform the arma method involves manipulation of the model equation sum min k 0 max p 1 sum min k 0 max p 2 a km x n 1 k n 2 m sum min k 0 max q 1 sum min k 0 max q 2 b km epsilon n 1 k n 2 m and utilizes the given finite set of observations x n 1 n 2 for 1 leq n 1 leq n 1 1 leq n 2 leq n 2 in the above relationship the random excitation epsilon n 1 n 2 is taken to be white this arma model s autoregressive a km coefficients are selected to minimize a weighted least squares criterion composed of error elements while the moving average b km coefficients are obtained using an alternative approach the spectral estimation performance of the ar and arma methods will be empirically demonstrated by considering the problem of resolving two sinusoids embedded in noise

a piecewise quadratic maximum entropy method for the statistical study of chaos abstract let the frobenius perron operator p s l 1 0 1 l 1 0 1 related to a nonsingular transformation s 0 1 0 1 have an invariant density f we propose a piecewise quadratic maximum entropy method for the numerical approximation of f the role of the partition of unity property of the quadratic functions for the numerical recovery of f has been depicted the proposed algorithm overcomes the ill conditioning shortage of the traditional maximum entropy method which only employs polynomials so that any number of moments can be used to increase the accuracy of the computed invariant density the convergence rate of the method is shown to be of order 3 numerical results are presented to justify the theoretical analysis of the method

maximum entropy power spectrum estimation with uncertainty in correlation measurements the purpose of this paper is to present a multidimensional mem algorithm valid for nonuniformly sampled arrays which satisfies a correlation approximating constraint to this end the correlation matching equality constraints of the usual mem are replaced by a single inequality constraint whose form is based on a measure of the noise in the given autocovariance function acf in this way one can incorporate into the model knowledge of the noisy nature of the given acf since the given acf is usually estimated from the samples of the wavefield specifically the covariance matrix of the correlation estimates is used in a quadratic form that weights the difference between the given acf and the one matched by the power spectrum the maximization of entropy under this inequality constraint leads ultimately to a steepest descent algorithm the algorithm has been tested with 1 d synthetic data representing multiple sinusoids buried in additive white noise the performance of this modified mem algorithm is compared to a traditional mem algorithm for extendible acf s and for different snr s examples of the mem spectrum are given for the case of nonextendible acf s

post lie algebra structures on pairs of lie algebras abstract we study post lie algebra structures on pairs of lie algebras g n which describe simply transitive nil affine actions of lie groups we prove existence results for such structures depending on the interplay of the algebraic structures of g and n we consider the classes of simple semisimple reductive perfect solvable nilpotent abelian and unimodular lie algebras furthermore we consider commutative post lie algebra structures on perfect lie algebras using lie algebra cohomology we can classify such structures in several cases we also study commutative structures on low dimensional lie algebras and on nilpotent lie algebras

on the convergence of moment problems we study the problem of estimating a nonnegative density given a finite number of moments such problems arise in numerous practical applications as the number of moments increases the estimates will always converge weak as measures but need not converge weakly in l this is related to the existence of functions on a compact metric space which are not essentially riemann integrable in some suitable sense we characterize the type of weak convergence we can expect in terms of riemann integrability and in some cases give error bounds when the estimates are chosen to minimize an objective function with weakly compact level sets such as the bolzmann shannon entropy they will converge weakly in l when an lp norm 1 p is used as the objective the estimates actually converge in norm these results provide theoretical support to the growing popularity of such methods in practice

power spectrum estimation with randomly spaced correlation samples three algorithms are developed to reproduce uniformly spaced correlation measurements for power spectrum estimation some general relations between the randomly spaced correlation samples and the power spectrum are derived the details of the three algorithms that can be used to reproduce uniformly spaced correlation measurements are included simulation examples using sinusoidal signals are given to examine the adverse effects due to the correlation randomness as well as to demonstrate improvements achieved by the suggested algorithms

spectral estimation via minimum energy correlation extension it is well known that the discrete fourier transform of a truncated correlation sequence the blackman tukey estimate can become negative at certain frequencies since the true power spectrum must in fact be positive negative estimates are undesirable in this paper we obtain the positive spectral estimate which is as close to the blaekman tukey estimate as possible in the l 2 norm while still matching the correlation constraints in the time domain this estimate yields the correlation extension of least energy for the specified truncated correlation sequence algorithms and numerical examples will be provided multidimensional minimum energy correlation extension is discussed in multidimensions the minimum energy estimate is shown to exist in certain cases when the autoregressive maximum entropy estimate fails to exist

extendability tests for multidimensional covariance sequences the problem of extendibility of multidimensional covariance sequences and the equivalent problem of the existence of maximum entropy me spectral estimates are analyzed using some recent results on the valid parameter space of gaussian markov random fields gmrfs for several nontrivial examples explicit conditions for extendability and using those conditions sketches of the space of extendible covariance sequences are obtained for the general case a cutting plane algorithm is proposed as an alternative to the two existing numerical procedures for ascertaining extendibility namely the linear programming procedure and the expanding hull algorithm the duality between the valid parameter space and the space of extendible covariances and the relationship between those two spaces and the space of admissible covariances for finite size data sequences are examined the connection between extendability and maximum likelihood ml estimation is established and some properties extendibility of covariances specified over increasing window sizes are presented

a hybrid approach to high resolution two dimensional spectrum analysis the 2 d spectrum estimation is divided into the two consecutive 1 d spectrum estimations a multichannel spectrum estimation in the time domain and a scalar spectrum estimation in the frequency domain the first is a purely parametric estimation while the second is a mixture of parametric and nonparametric estimation the maximum entropy method is used in each estimation in order to utilize the high resolution performance of the maximum entropy method on the one hand and to avoid the highly nonlinear nature of the maximum entropy method for 2 d problems on the other hand the estimate obtained by this procedure satisfies the covariance matching property and exhibits remarkably high resolution a simplified version of the multichannel levinson durbin algorithm is derived based on the special structure of the covariance data of homogeneous random fields the related extendibility of the covariance array is discussed it is shown that this method is particularly suitable for a class of random fields having some shift invariance

spectral estimation for sensor arrays the array processing problem is briefly discussed and an abstract spectral estimation problem is formulated this problem involves the estimation of a multidimensional frequency wave vector power spectrum from measurements of the correlation function and knowledge of the spectral support the investigation of correlation matching spectral estimates leads to the extendibility question does there exist any positive spectrum on the spectral support that exactly matches a given set of correlation samples in answering this question a mathematical framework is developed in which to analyze and design spectral estimation algorithms pisarenko s method of spectral estimation which models the spectrum as a sum of impulses plus a noise component is extended from the time series case to the more general array processing case pisarenko s estimate is obtained as the solution of a linear optimization problem which can be solved using a linear programming algorithm such as the simplex method

etale representations for reductive algebraic groups with one dimensional center abstract a complex vector space v is a prehomogeneous g module if g acts rationally on v with a zariski open orbit the module is called etale if dim v dim g we study etale modules for reductive algebraic groups g with one dimensional center for such g even though every etale module is a regular prehomogeneous module its irreducible submodules have to be non regular for these non regular prehomogeneous modules we determine some strong constraints on the ranks of their simple factors this allows us to show that there do not exist etale modules for g gl 1 s s with s simple

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

a classification of 2 simple prehomogeneous vector spaces of type i let p g gl v be a rational representation of a connected linear algebraic group g on a finite dimensional vector space v all defined over an algebraically closed field k of characteristic zero if v has a zariskidense g orbit we call a triplet g p v a prehomogeneous oector space abbrev p v when p is irreducible such p v s have been classified in 11 since then it has turned out gradually that the complete classification of reductive p v s i e p v s with reductive groups g is an extremely laborious task therefore it is natural to classify some restricted class of p v s e g 2 to get some insight into the general situation a p v g p v is called a 2 simple p v when 1 g gl 1 x g x g with simple algebraic groups g and gz 2 p is the composition of a rational representation p of g x g of the form p p1 0 p pkol a a l l z t with k s t l where pi tv resp pi t are nontrivial irreducible representations of g resp g and the scalar multiplications gz 1 on each irreducible component v for i 1 f where v i v we say that a 2 simple p v g p v is of type zif k l and at least one of gl l xg xg pi pi i 1 k is a nontrivial p v see definition 5 p 43 in 11 on the other hand if k l and all gl l xg xg p op i l k are trivial p v s it is called a 2 simple p v of type ii in 3 all 2 simple p v s of type ii has been already classified in this paper we shall classify all 2simple p v s of type i thus together with 3 we complete a classification of all 2 simple p v s for example the fact that all irreducible p v s are castling equivalent to 2 simple p v s or to x m x y m x g 2 1 1 i i n 1 with m 2 3 see 1 indicates the importance of 2 simple p v s for simplicity we write g p v or g p instead of g p v

characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff in this paper we consider fuzzy bimatrix games which mean bimatrix games with fuzzy payoff matrices we define two types of concepts of equilibrium strategy and investigate their relationships moreover we prove that these nash equilibrium strategies exist in any fuzzy bimatrix games

a procedure for ordering fuzzy subsets of the unit interval abstract we introduce a function to help in the ordering of fuzzy subsets of the unit interval this function is the integral of the mean of the level sets associated with the fuzzy subsets various properties of this function are studied

on characterization of equilibrium strategy of two person zero sum games with fuzzy payoffs in this paper we consider fuzzy matrix games namely two person zero sum games with fuzzy payoffs based on fuzzy max order for such games we define three kinds of concepts of minimax equilibrium strategies and investigate their properties first we shall show that these equilibrium strategies are characterized as nash equilibrium strategies of a family of parametric bi matrix games with crisp payoffs second we investigate properties of values of fuzzy matrix games by means of possibility and necessity measures in addition we give a numerical example to illustrate utility of our approaches

duality in linear programming with fuzzy parameters and matrix games with fuzzy pay offs abstract a dual for linear programming problems with fuzzy parameters is introduced and it is shown that a two person zero sum matrix game with fuzzy pay offs is equivalent to a primal dual pair of such fuzzy linear programming problems further certain difficulties with similar studies reported in the literature are discussed

extension of the pisarenko method to sparse linear arrays when applied to array processing the pisarenko harmonic decomposition phd method is limited to linear equispaced arrays we present an approach that allows us to extend it to general arrays although for the ease of exposition we consider only sparse linear arrays we exploit the fact that the phd can be seen as a deconvolution or model fitting approach that minimizes an t sub 1 norm and can thus be implemented as a standard linear program looking at the phd from this point of view has two advantages it allows us to extend its applicability to arbitrary arrays and by diverging slightly from the basic philosophy it allows us to improve its performance which is often quite poor in its original version

some p v equivalences and a classification of 2 simple prehomogeneous vector spaces of type a classification of 2 simple prehomogeneous vector spaces is completed by using some p v equivalences together with 3 some part is very different from the previous classification of the irreducible or simple cases 1 2 and some new method is necessary this result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces introduction in m sato and t kimura 1 all irreducible prehomogeneous vector spaces abbreviated irreducible p v s are classified up to castling equivalence and it is proved that any irreducible p v is castling equivalent to a 2 simple p v or to sl m x sl m x gl 2 a1 x a1 x a1 with m 2 3 therefore as a step to a classification of the general case it is natural to classify all nonirreducible 2 simple p v s in this paper we shall classify all 2 simple p v s of type ii and give the complete list of them up to strong equivalence in the sense of definition 4 p 36 in 1 however we need various p v equivalences cf propositions 1 7 1 12 1 34 1 36 1 37 theorem 1 16 and its corollary corollary 1 26 definition 1 31 remark 1 32 to carry out the classification and hence first we shall prove them we also investigate the regularity in some cases since all 2 simple p v s of type i are already classified in 3 this completes the classification of all 2 simple p v s here we say that g p v or simply g p is a 2 simple p v if it is a prehomogeneous vector space of the following form g gl 1 k s t xg x g2 where g1 and g2 are simple algebraic groups p the composition of the scalar multiplications gl l k s on each irreducible component and the representation p p1 p 1 pk p k a1 s 1 1 t1 t ofthegroupg1 x g2 where pi aj resp p tj are nontrivial irreducible representations of g1 resp g2 we write g p v instead of g p v for simplicity to avoid confusion we write instead of clearly each gl 1 x g1 x g2 pi p i 1 k is an irreducible 2 simple p v if at least one of them is a nontrivial p v see definition 1 1 we call it a 2 simple p v of type i on the other hand if all of them are trivial p v s we call it a 2 simple p v of type ii in this case we have g2 sl n received by the editors march 3 1986 and in revised form july 30 1986 1980 mathematics subject classification 1985 revision primary 11r20 11r29 secondary 11n25 11r18 01988 american mathematical society 0002 9947 88 1 00 25 per page

left invariant affine structures on reductive lie groups we describe left invariant affine structures that is left invariant flat torsion free affine connections on reductive linear lie groupsg they correspond bijectively to lsa structures on the lie algebra g ofg here lsa stands for left symmetric algebra if g has trivial or one dimensional center z then the affine representation 1 of g induced by any lsa structure g on g isradiant i e the radiance obstructionc h1 g g vanishes if dim z 1 we prove that g s z where s is split simple admits lsa structures if and only if s is of typeal that is g gln here we have the associative lsa structure given by ordinary matrix multiplication corresponding to the bi invariant affine structure on gl n which was believed to be essentially the only possible lsa structure on gln we exhibit interesting lsa structures different from the associative one they arise as certain deformations of the matrix algebra then we classify all lsa structures on glnusing a result of baues forn 2 we compute all structures explicitly over the complex numbers

pareto optimal security strategies in matrix games with fuzzy payoffs we present a new methodology for the analysis of fuzzy payoff matrix games the main difficulty that appears in the study of these games is the comparison between the payoff values associated to the strategies of the players because these payoffs are fuzzy quantities our approach does not transform the fuzzy payoffs to crisp numbers via standard defuzzyfication but we use standard fuzzy orders which allows us to find solutions within the same space of fuzzy numbers moreover we provide a method to solve these games finding equivalent fuzzy linear programs whose maximal solutions give the solutions of the games

existence and uniqueness of band limited positive semidefinite extrapolations the authors investigate the extrapolation of a one dimensional sequence subject to the contraints that its spectrum be bandlimited to a given frequency band and that it be real valued and nonnegative on that band a new and easily computable matrix test for the existence of a bandlimited positive semidefinite extrapolation is derived and an algorithm is presented for the recursive computation of such an extrapolation

left symmetric algebras for mathfrak gl n we study the classification problem for left symmetric algebras with commutation lie algebra gr n in characteristic 0 the problem is equivalent to the classification of etale affine representations of g n algebraic invariant theory is used to characterize those modules for the algebraic group sl n which belong to affine etale representations of gr n from the classification of these modules we obtain the solution of the classification problem for gl n as another application of our approach we exhibit left symmetric algebra structures on certain reductive lie algebras with a one dimensional center and a non simple semisimple ideal

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

counterexample to a conjecture about braces abstract we find an example of a finite solvable group in fact a finite p group in which is not possible to define a left brace structure or equivalently which is not an iyb group this answers a question posed by cedo jespers and del rio related to the yang baxter equation our argument is an improvement of an argument of rump using results about hopf galois extensions and lsa structures we explain explicitly the relation between these two areas of mathematics and brace theory hoping that it will be useful in the future

two person zero sum game approach for fuzzy multiple attribute decision making problems the multiple attribute decision making madm problem deals with candidate priority alternatives with respect to various attributes madm techniques are popularly used in diverse fields such as engineering economics management science transportation planning etc several approaches have been developed for assessing the weights of madm problems e g the eigenvector method electre topsis etc in this paper we obtained weights of a madm problem with a fuzzy decision matrix by formulating it as a two person zero sum game with an uncertain payoff matrix moreover the equilibrium solution and the resolution method for the madm game have also been developed these results are validated by a product development example of nano materials our study is an innovative approach for applying the game theory on madm problems with management of technology mot

correlation matching by finite length sequences the problem of matching a finite length data sequence to a set of not necessarily uniformly spaced correlation lags is considered a characterization of the set of correlations that can be derived from sequences of a given length is presented using this characterization an algorithm called the expanding hull algorithm is presented for determining the minimum sequence length and a sequence of this length which matches a given set of correlation values is obtained this sequence has a z transform which is the lowest order correlation matching moving average model the sequence also generates the minimum length correlation extension the expanding hull algorithm also provides a method for extendibility testing of missing lag and multidimensional correlation sequences numerical examples are provided

multidimensional mem spectral estimation the problem of multidimensional maximum entropy method mem spectral estimation from nonuniformly spaced correlation measurements is investigated a necessary and sufficient condition is derived for the existence and uniqueness of the mem spectral estimate in its usual form it is shown that this condition is not satisfied in many multidimensional problems of interest although it is satisfied in the important practical case of spectral supports composed of a finite number of points when the existence condition is satisfied calculation of the mem estimate reduces to the solution of a finite dimensional convex optimization problem the application of standard optimization techniques to this problem results in iterative computational algorithms which are guaranteed to converge the algorithms so obtained are compared to those previously proposed and a spectral estimation example is presented