computing real solutions of fuzzy polynomial systems abstract this paper presents an efficient algorithm called solvefuzzysystem or sfs for finding real solutions of polynomial systems whose coefficients are symmetrical l r fuzzy numbers with bounded support for which the spread functions l and r are bijective the real solutions of such a system are deduced from solutions of some polynomial systems with real coefficients this algorithm is based on new results that are universal because they are independent from the spread functions these theoretical results include the management of the fuzzy system s solutions signs an implementation in the fuzzy package of the free computer algebra software sagemath and a parallel version of the algorithm are described

iterative solution of fuzzy linear systems abstract linear systems have important applications to many branches of science and engineering in many applications at least some of the parameters of the system are represented by fuzzy rather than crisp numbers so it is immensely important to develop numerical procedures that would appropriately treat general fuzzy linear systems will be denoted by fls and solve them in this paper firstly a general fuzzy linear system using the embedding approach has been investigated and then several well known numerical algorithms for solving system of linear equations such as richardson extrapolated richardson jacobi jor gaus seidel egs sor aor esor ssor ussor ema and msor are extended for solving fls the iterative methods are followed by convergence theorems and the presented algorithms are tested by solving some numerical examples hackbusch noticed that gauss spelling is less correct than gaus w hackbusch iterative solution of large sparse systems of equations springer verlag inc new york 1994 9

on the polar decomposition of the quantum su 2 algebra polar decomposition of the generators of the su 2 q algebra is performed and its contraction limit to the deformed oscillator algebra is considered using the phase operator occurring in the decomposition and its conjugate the authors write a realisation of the su 2j 1 algebra into which they embed the deformed su 2 realisation finally they discuss the classical limit of the polar decomposition and the case of the quantum su 2 dynamics

exact solution of a reaction diffusion process with three site interactions the one dimensional reaction diffusion process aa a and a a aaa is exactly solvable through the empty interval method if the diffusion rate equals the coagulation rate independently of the particle production rate the model is always in the universality class of diffusion annihilation this allows us to check analytically the universality of finite size scaling in a non equilibrium critical point

symplectic geometry of suq h 0 2 and suq h 2 algebras the authors give a complete description of q deformation of su 2 algebra in terms of symplectic geometry the geometric meanings of such a deformation are manifestly shown at classical as well as at quantum level as a model they study the classical and quantum dynamics of a particle with q spin moving in electromagnetic fields in detail

fuzzy symmetric solutions of fuzzy linear systems in this paper we propose a simple and practical method to solve a fuzzy linear system email protected email protected where email protected and email protected are fuzzy triangular vectors with non zero spreads and matrix a is nonsingular with real coefficients the aim of this paper is twofold first we obtain the crisp solution of a fuzzy linear system to this end we solve the 1 cut of a fuzzy linear system second we allocate some unknown symmetric spreads to any rows of fuzzy linear system in 1 cut position thus a fuzzy linear system in 1 cut will be transformed to a system of interval equations the symmetric spreads of each element of a fuzzy vector solution are derived by solving such an interval system moreover based on the obtained symmetric spreads we derive three types of solutions however one of the mentioned spreads has pessimistic optimistic attitude that is determined with a decision maker it seems that such a solution is a connection solution between tolerable solution set tss and controllable solution set css also we derive the maximal solution and the minimal solution of an original fuzzy linear system which are placed in tss and css respectively finally some numerical examples are given to illustrate the efficiency and ability of the proposed method

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

maximal and minimal symmetric solutions of fully fuzzy linear systems in this paper we shall propose a new method to obtain symmetric solutions of a fully fuzzy linear system ffls based on a 1 cut expansion to this end we solve the 1 cut of a ffls in the present paper we assumed that the 1 cut of a ffls is a crisp linear system or equivalently the matrix coefficient and right hand side have triangular shapes then some unknown symmetric spreads are allocated to each row of a 1 cut of a ffls so after some manipulations the original ffls is transformed to solving 2n linear equations to find the symmetric spreads however our method always give us a fuzzy number vector solution moreover using the proposed method leads to determining the maximal and minimal symmetric solutions of the ffls which are placed in a tolerable solution set and a controllable solution set respectively however the obtained solutions could be interpreted as bounded symmetric solutions of the ffls which are useful for a large number of multiplications existing between two fuzzy numbers finally some numerical examples are given to illustrate the ability of the proposed method

hopf algebraic structures of suq h cross to 0 2 and suq h cross 2 algebras monopoles and symplectic geometry on 2d manifolds from the symplectic structure on the q deformed manifold sq2 of s2 the classical and quantum q deformed algebras suq h cross to 0 2 and suq h cross 2 of su 2 are discussed in detail and their hopf structures are constructed it is also shown that the symplectic geometry of s2 describes the properties of monopoles for the su 2 algebra the q deformed monopoles are studied from the symplectic geometry of sq2 and a relation between generic 2d manifold and algebraic structures is established

on similarity invariants of matrix commutators and jordan products abstract denote by x y the additive commutator xy yx of two square matrices x y over a field f in a previous paper the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of a x 1 x 2 x k when a is a fixed matrix and x 1 x k vary were studied moreover given any expression g x 1 x k obtained from distinct noncommuting variables x 1 x k by applying recursively the lie product and without using the same variable twice the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of g x 1 x k when one of the variables x 1 x k takes a fixed value in f n n and the others vary were studied the purpose of the present paper is to show that analogous results can be obtained when additive commutators are replaced with multiplicative commutators or jordan products

numerical methods for fuzzy system of linear equations in this paper numerical algorithms for solving fuzzy system of linear equations fsle are considered schemes based on the iterative jacobi and gauss sidel methods in detail are discussed and there are followed by convergence theorems algorithms are illustrated by solving some numerical examples

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

duality in fuzzy linear systems according to fuzzy arithmetic dual fuzzy linear system can not be replaced by a fuzzy linear system in this paper we investigate the existence of a solution of duality fuzzy linear equation systems two necessary and sufficient conditions for the solution existence are given

comment on the q analogues of the harmonic oscillator the heisenberg relation for the q analogues of the quantum harmonic oscillator introduced independently by macfarlane and biedenharn 1989 for the quantum group su 2 q is derived by a method analogous to that used by schwinger 1968 for the su 2 case the author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra iii when q is a root of unity for pt ii see ibid vol 23 p 5371 1990 the q oscillator of type one and the classical and the quantum q deformations of su 2 algebra realized through the q oscillators are studied in the case of q being roots of unity i e qk 1 the q excitations are found to be non bosonic in particular when k is of rank 4 and 6 the q excitations are shown to be fermionic and parafermionic respectively

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

multiparameter solutions of the yang baxter equation the multiparameter solutions 4 4 r matrices of the quantum yang baxter equation have been systematically investigated and more than thirty selected solutions are presented by using wu elimination the free parameters and non zero eigenvalues of these r matrices vary from one to five and zero to four respectively they are classified according to the corresponding algebraic structures

generalized fuzzy linear programming for decision making under uncertainty feasibility of fuzzy solutions and solving approach in this study a generalized fuzzy linear programming gflp method is developed for dealing with uncertainties expressed as fuzzy sets the feasibility of fuzzy solutions of the gflp problem is investigated a stepwise interactive algorithm sia based on the idea of design of experiment is then advanced to solve the gflp problem this sia method was implemented through i discretizing membership grade of fuzzy parameters into a finite number of a cut levels ii converting the gflp model into an interval linear programming ilp submodel under every a cut level iii solving the ilp submodels through an interactive algorithm and obtaining the associated interval solutions iv acquiring the membership functions of fuzzy solutions through statistical regression methods a simple numerical example is then proposed to illustrate the solution process of the gflp model through sia a comparison between the solutions obtained though sia and monte carlo method is finally conducted to demonstrate the robustness of the sia method the results indicate that the membership functions for decision variables and objective function are reasonable and robust

on the number of invariant polynomials of the matrix xax 1 b abstract let f be a field if a is an n n matrix over f we denote by i a the number of invariant polynomials of a different from 1 we shall prove that if a b are n n matrices over f and t 1 n then i a i b n t if and only if there exists a nonsingular matrix x over f such that i xax 1 b t except in a few cases

the q hermite polynomial and the representations of heisenberg and quantum heisenberg algebras the authors introduce a pair of canonical conjugate q deformed operators d x and discuss the relations between the deformed operators d x and q series the realizations of some lie symmetries heisenberg and quantum heisenberg algebras are given for the operators d and x they show that the q analogous hermite polynomials are representations of heisenberg and quantum heisenberg algebras realized in this way when q is a root of unity the properties of the q analogous hermite polynomials are also discussed

suq h 0 2 and suq h 2 the classical and quantum q deformations of the su 2 algebra 4 geometric quantization and hopf algebraic structures for pt iii see ibid vol 24 p 1427 1991 in the q deformed oscillator system the hopf algebraic structures of suq inf not 0 2 and suq 2 are constructed in the scheme of geometric quantization the classical and quantum representation spaces are analyzed with respect to whether or not q is a root of unity

reaction diffusion processes of hard core particles we study a 12 parameter stochastic process involving particles with two site interaction and hard core repulsion on ad dimensional lattice in this model which includes the asymmetric exclusion process contact processes and other processes the stochastic variables are particle occupation numbers taking valuesn x 0 1 we show that on a ten parameter submanifold thek point equal time correlation functions n x1 n xk satisfy linear differential difference equations involving no higher correlators in particular the average density n x satisfies an integrable diffusion type equation these properties are explained in terms of dual processes and various duality relations are derived by defining the time evolution of the stochastic process in terms of a quantum hamiltonianh the model becomes equivalent to a lattice model in thermal equilibrium ind 1 dimensions we show that the spectrum ofh is identical to the spectrum of the quantum hamiltonian of ad dimensional anisotropic spin 1 2 heisenberg model in one dimension our results hint at some new algebraic structure behind the integrability of the system

the number of invariant polynomials of a matrix with prescribed off diagonal blocks abstract let a be an n n matrix over an arbitrary field f of the form a a 1 2 a 1 2 a 2 1 a 2 2 where a1 1 fp p and a2 2 fq q and p q n we characterize the possible number of nontrivial invariant polynomials of a when the submatrices a1 2 and a2 1 are prescribed

fuzzy linear systems of the form a1x b1 a2x b2 linear systems of equations with uncertainty on the parameters play a major role in several applications in various areas such as economics finance engineering and physics this paper investigates fuzzy linear systems of the form a 1x b 1 a 2x b 2 with a 1 a 2 square matrices of fuzzy coefficients and b 1 b 2 fuzzy number vectors the aim of this paper is twofold first we clarify the link between interval linear systems and fuzzy linear systems second a generalization of the vector solution of buckley and qu solving systems of linear fuzzy equations fuzzy sets and systems 43 1991 33 43 to the fuzzy system a 1x b 1 a 2x b 2 is provided in particular we give the conditions under which the system has a vector solution and we show that the linear systems ax b and a 1x b 1 a 2x b 2 with a a 1 a 2 and b b 2 b 1 have the same vector solutions moreover in order to find the vector solution a simple algorithm is proposed

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

the pauli matrices in n dimensions and finest gradings of simple lie algebras of type a n 1 properties of the lie algebra gl n c are described for a basis which is a generalization of the 2 2 pauli matrices the 3 3 case is described in detail the remarkable properties of that basis are the grading of the lie algebra it offers each grading subspace is one dimensional and the matrix group it generates it is a finite group with the center of sl n c as its commutator group

conjugate gradient method for fuzzy symmetric positive definite system of linear equations in this paper the conjugate gradient method for solving fuzzy symmetric positive definite system of linear equation is considered the method in detail is discussed and followed by convergence theorem and illustrated by solving some numerical examples

q phase coherent states and their squeezing properties some kinds of q phase coherent states of a q harmonic osscillator in a finite dimensional hilbert space are constructed some properties of these states are discussed second order squeezing properties of these states with respect to the phase quadrature operators are studied the number phase squeezing and number phase uncertainty relations are also studied in detail for a two state system some new number phase minimum uncertainty states are found

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

quantum and classical mechanics of q deformed systems quantum and classical mechanics of a system of q deformed bosonic oscillators are considered the q deformed heisenberg weyl algebra of creation and destruction operators is realized by differential operators in a space of functions of real commutative variables the corresponding hilbert space is constructed in this approach the deformation parameter turns out to be a function of the planck constant an oscillator frequency and a parameter with dimension of length the hamiltonian path integral is derived and its semiclassical approximation is investigated to obtain the corresponding q deformed classical theory the phase space spanned by the usual commutative coordinates is shown to be a cylinder in classical theory the dimensional parameter introduced determines its radius it is argued that the q deformation can be associated with a special non canonical transformation the principle of least action for the classical q deformed system is formulated a representation of uq m in a space of functions on a phase space spanned by commutative coordinates is constructed

solutions of algebraic equations involving generalized fuzzy numbers abstract this paper deals with algebraic equations involving generalized fuzzy numbers with continuous membership functions the generalized fuzzy number defined in this paper is a general name for fuzzy numbers fuzzy intervals crisp numbers and interval numbers three important properties of the generalized fuzzy numbers with continuous membership functions are introduced as the basis for deriving the solvability criterion the sufficient and necessary condition for solving a x c is derived the sufficient and necessary condition for solving a x c and a x c when a and c are nonzero generalized fuzzy numbers is also derived in this paper instead of the inverse operation method which fails for fuzzy sets inverse operations on the boundary points of level set intervals for continuous membership functions are used when the derived criteria of solvability are satisfied a unique solution rather than a solution set is obtained

on the eigenvalues of jordan products abstract this paper studies the possible eigenvalues of the jordan product xa ax when a is fixed and x varies

inconsistent fuzzy linear systems the mxn inconsistent fuzzy linear system is studied the fuzzy least squares solution and therefore the weak fuzzy least squares solution to the fuzzy system are expressed by using the generalized inverses of the coefficient matrix the existence of the strong fuzzy least squares solution to the fuzzy system is also discussed some examples are presented to illustrate the theory

the adomian decomposition method for fuzzy system of linear equations in this paper the application of the adomian method for solving fuzzy system of linear equations fsle is considered for a fsle has been shown that the adomian decomposition method is equivalent to the jacobi iterative method appl math comput in press the algorithm is illustrated by solving some numerical examples

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

on boson algebras as hopf algebras certain types of generalized undeformed and deformed boson algebras which admit a hopf algebra structure are introduced together with their fock type representations and their corresponding r matrices it is also shown that a class of generalized heisenberg algebras including those underlying physical models such as that of calogero sutherland is isomorphic with one of the types of boson algebra proposed and can be formulated as a hopf algebra

approximate solution of dual fuzzy matrix equations in this paper we propose a simple and practical method to solve the dual fuzzy matrix equation ax b cx d in which a c are mxn matrices and b d are mxp lr fuzzy numbers matrices by means of the arithmetic operations on lr fuzzy numbers space the dual fuzzy matrix equation could be converted into two classical matrix equations and the lr minimal fuzzy solution and the strong weak lr minimal fuzzy solutions of the dual fuzzy matrix equation are derived by solving two classical matrix equations based on the generalized inverses of matrices meanwhile as a special case of the dual fuzzy matrix equation the fuzzy solutions of the dual fuzzy linear systems are investigated finally some numerical examples are given to illustrate the effectiveness of the proposed method

solution of non linear fuzzy systems by decomposition of incremental fuzzy numbers this paper presents a proposal for decomposing large uncertainties associated with fuzzy numbers which can be useful for solving non linear fuzzy problems with crisp coefficients the proposed procedure is based on a property of incremental fuzzy numbers and improves the quality of solution of the linearization process the methodology is illustrated by some numeric studies

integrable stochastic ladder models a general way to construct ladder models with certain lie algebraic or quantum lie algebraic symmetries is presented these symmetric models give rise to a series of integrable systems it is shown that corresponding to these su 2 symmetric integrable ladder models there are exactly solvable stationary discrete time resp continuous time markov processes with transition matrices resp intensity matrices having spectra which coincide with the ones of the corresponding integrable models

brst cohomology for uq sl 2 representations a nilpotent brst operator qq is constructed for the q deformation of the universal enveloping algebra of sl 2 an associated operator rq such that qq rq cq is introduced where cq is the q deformed casimir invariant cq fe 1 2h q 1 2h 1 q the related operator qq conjugate to qq is also considered the resulting brst cohomology of uq sl 2 representations is discussed

solving systems of linear fuzzy equations abstract in this paper we wish to construct solutions for x to the fuzzy matrix equation ax b when the elements in a and b are triangular fuzzy numbers a is square and always non singular we argue that the previous method of solving for x based on the extension principle and regular fuzzy arithmetic should be abandoned since it too often fails to produce a solution we present six new solutions of which we show that five are identical we then adopt the common value x of these five new solutions as our solution to ax b we show that x is a fuzzy vector which is the generalization to rn of real fuzzy numbers we also show how these results pertain to and extend previous research on this problem within the area of interval analysis

solving linear and quadratic fuzzy equations abstract we present necessary and sufficient conditions for some linear and quadratic equations to have a solution when the parameters are either real or complex fuzzy numbers the solution will be a real or complex fuzzy number applications in chemistry economics finance and physics are presented for these types of equations

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra ii the hopf algebra the yang baxter equation and multi deformed algebraic structures for pt i see ibid vol 23 p 4185 1990 the suq 2 algebra is realized by means of both the poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q deformed oscillators of two different types the structures of the hopf algebra and the quantum yang baxter equation are also discussed on a quantum level a set of j representations of the quantum algebra suq 2 is constructed based on the type ii q oscillators multi deformations of the oscillators of the two types and multi deformed algebras expressed in poisson brackets as well as in lie brackets are proposed

on the number of invariant polynomials of partially prescribed matrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 a 1 2 a 2 2 are prescribed

the number of invariant polynomials of a matrix with prescribed complementary principal submatrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 and a 2 2 are prescribed

q deformed oscillator algebra as a quantum group it is proved that the q deformed oscillator algebra is a quantum group the hopf algebraic structure is set up and verified to be consistent and compatible the universal r matrix and yang baxter equation are constructed the q differential operator algebra is briefly discussed

on the invariant polynomials of jordan products let f be an algebraically closed field this paper describes the possible numbers of nonconstant invariant polynomials of the jordan product xa ax when a is fixed and x varies

the construction of the q analogues of the harmonic oscillator operators from ordinary oscillator operators the q analogues of boson operators are constructed from ordinary boson operators by an embedding approach accordingly the q deformed quantum algebras su n q can be obtained straightforwardly from corresponding universal enveloping algebras of lie algebra su n

solution of the fully fuzzy linear systems using the decomposition procedure abstract this paper tries to extend the adomian decomposition method for solving fully fuzzy linear systems shown as ffls for finding a positive fuzzy vector x that satisfies a x b where a and b are respectively a fuzzy matrix and a fuzzy vector we employ dubois and prade s approximate arithmetic operators on lr fuzzy numbers we also transform the ffls and use the adomian decomposition method for solving an ffls then we compare this method with the jacobi iterative technique additionally we give some numerical examples

iteration algorithms for solving a system of fuzzy linear equations abstract in this paper we discuss the solution of a system of fuzzy linear equations x ax u and its iteration algorithms where a is a real n n matrix the unknown vector x and the constant u are all vectors consisting of n fuzzy numbers and the addition scale multiplication are defined by zadeh s extension principle after introducing a metric between two fuzzy vectors we prove that the system has unique solution if a we also give the convergence and the error estimation for using simple iteration to obtain the solution finally we give the convergence and the error estimation of successive iteration sequence for obtaining the solution

lu decomposition method for solving fuzzy system of linear equations in this paper lu decomposition method for solving fuzzy system of linear equations is considered we consider the method in spatial case when the coefficient matrix is symmetric positive definite the method in detail is discussed and followed by convergence theorem and illustrated by solving some numerical examples

successive over relaxation iterative method for fuzzy system of linear equations in this paper the gauss sidel iterative method in allahviranloo appl math comput in press for solving fuzzy system of linear equations fsle is transformed to the successive over relaxation sor method this method is discussed in detail and followed by convergence theorem algorithm is compared with the gauss sidel method by solving a numerical example

perturbation analysis of fuzzy linear systems this paper deals with the perturbation analysis of fuzzy linear systems three cases of perturbation are considered a the right hand side is perturbed while the coefficient matrix remains unchanged b the coefficient matrix is perturbed while the right hand side remains unchanged and c both the coefficient matrix and the right hand side are perturbed for all of the three cases the respective relative error bounds for solutions of fuzzy linear system are derived the results are illustrated by numerical examples

matrices with prescribed submatrices and number of invariant polynomials abstract we give necessary and sufficient conditions for the existence of a matrix a b over a field where a is a square matrix such that a b has a prescribed submatrix and the pencil xi a b has a prescribed number of invariant factors different from 1 as a corollary we give a necessary and sufficient condition for the existence of a completely controllable pair a b such that a b has a prescribed submatrix

q exponential and q gamma functions i q exponential functionsa some properties of the q exponential functions standard and symmetric are investigated for general complex nonzero q in particular for q 1 the asymptotic behavior of the q exponential functions of a complex argument z is studied as z approaches through an infinite geometric sequence zn qn n z of complex numbers with common ratio q a generalized q exponential function is defined and some of its properties are also discussed for q 0 including the singular case of q a root of unity or more generally q 1

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

integrable poisson algebras and two dimensional manifolds the relations between integrable poisson algebras with three generators and two dimensional symplectic manifolds are investigated it is shown that for a given integrable poisson algebra there exists a two dimensional symplectic manifold such that the poisson algebra generated by the coordinates of m coincides with the algebra vice versa the coordinates of a given smooth two dimensional symplectic manifold m embedded in generate an integrable poisson algebra moreover smooth poisson algebraic maps between two integrable poisson algebras are governed by equations involving the symplectic manifolds corresponding to these algebras

equivalences between stochastic systems time dependent correlation functions of unstable particles undergoing biased or unbiased diffusion coagulation and annihilation are studied this is achieved by similarity transformations between different stochastic models and between stochastic and soluble nonstochastic models for special cases we obtain exact results which are in good agreement with experiments on one dimensional annihilation coagulation processes

suq h 0 2 and suq h 2 the classical and quantum q deformations of the su 2 algebra by means of classical harmonic oscillators and their canonical quantization we show that the q deformation of the su 2 lie algebra can be realized at a classical level in the sense of poisson brackets denoted by suq inf not 0 2 and that the canonical quantization of the system leads to the quantum q deformation of the su 2 algebra denoted by suq 2 this means that the q deformation and the h cross quantization are in general two different concepts

tensor operators for quantum groups and applications tensor operators are defined for hopf algebras if the hopf algebra is quasitriangular the properly defined tensor product of tensor operators is a tensor operator in one application the problem of defining invariant correlation functions for the uq sl 2 symmetric spin 1 2 heisenberg chain is considered in another application it is shown that suitably modified bosonic and fermionic q creation and annihilation operators are tensor operators of uq sl n

the brst operator of uq sl 2 and real forms in a previous letter a nilpotent brst operator qq was constructed for the q deformation of the universal enveloping algebra of sl 2 the present addendum resolves issues of non uniqueness in the construction of the brst operator by imposing additional structure associated with real forms and corrects a misprint in the previous work

on similarity invariants of matrix commutators abstract we study the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of a x 1 x 2 x k when a is a fixed matrix and x 1 x k vary then we generalize these results in the following way let g x 1 x k be any expression obtained from distinct noncommuting variables x 1 x k by applying recursively the lie product and without using the same variable twice we study the possible eigenvalues ranks and numbers of nonconstant invariant polynomials of g x 1 x k when one of the variables x 1 x k takes a fixed value in f n n and the others vary

the quantum q deformed symmetric top an exactly solved model by means of symplectic geometry the author discusses the quantum q deformed symmetric top system quantum group suq 2 is realized in the system he shows that the quantum q deformed symmetric top system in which quantum group is realized and the quantum undeformed symmetric top system in which lie group is realized correspond to the same heisenberg equations and then the quantum q deformed symmetric top is an exactly solved model

on the number of invariant polynomials of matrix commutators we study the possible numbers of noneonstant invariant polynomials of the matrix commutator xa ax when x varies elsevier science inc 1997

general fuzzy linear systems we study the general mxn fuzzy linear system using the embedding method the solution and therefore the weak fuzzy solution to the fuzzy system are expressed by using the generalized inverses of the coefficient matrix the existence of the strong fuzzy solution to the fuzzy system is also discussed some examples are presented to illustrate the theory

on the number of invariant polynomials of the product of matrices with prescribed similarity classes abstract we study the possibilities for the number of nontrivial invariant polynomials of the product of two nonsingular matrices with prescribed similarity classes over an algebraically closed field