random walks on combs we develop techniques to obtain rigorous bounds on the behaviour of random walks on combs using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length we also calculate exactly the spectral dimension of some fixed non translationally invariant combs we relate the spectral dimension to the critical exponent of the mass of the two point function for random walks on random combs and compute mean displacements as a function of walk duration we prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension

spectral dimension of trees with a unique infinite spine using generating functions techniques we develop a relation between the hausdorff and spectral dimension of trees with a unique infinite spine furthermore it is shown that if the outgrowths along the spine are independent and identically distributed then both the hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n we apply this new method to both critical non generic trees and the attachment and grafting model which is a special case of the vertex splitting model resulting in a simplified proof for the values of the hausdorff and spectral dimension for the former and novel results for the latter

least squares orthogonal polynomials abstract in this paper the new notion of least squares orthogonal polynomials is introduced their existence and uniqueness is studied and some methods for their recursive computation are given the location of their zeros in the case of a positive definite functional is studied the paper ends with applications to pade type approximation and gaussian quadrature methods

multiple orthogonal polynomials and a counterexample to the gaudin bethe ansatz conjecture jacobi polynomials are polynomials whose zeros form the unique solution of the bethe ansatz equation associated with two irreducible modules we study sequences of polynomials whose zeros form the unique solution of the bethe ansatz equation associated with two highest weight irreducible modules with the restriction that the highest weight of one of the modules is a multiple of the first fundamental weight we describe the recursion which can be used to compute these polynomials moreover we show that the first polynomial in the sequence coincides with the jacobi pineiro multiple orthogonal polynomial and others are given by wronskian type determinants of jacobi pineiro polynomials as a byproduct we describe a counterexample to the bethe ansatz conjecture for the gaudin model

bi orthogonality and zeros of transformed polynomials let d and e be two real intervals we consider transformations that map polynomials with zeros in d into polynomials with zeros in e a general technique for the derivation of such transformations is presented it is based on identifying the transformation with a parametrised distribution f x x e d and forming the bi orthogonal polynomial system with respect to f several examples of such transformations are given

differently implicational universal triple i method of 1 2 2 type as a generalization of the triple i method differently implicational universal triple i method of 1 2 2 type universal triple i method for short is investigated first the concepts of residual operators and strongly residual operators are given and then related conclusions of residual pairs are provided second the related universal triple i solutions including fmp solutions fmt solutions and so on are strictly defined by the infimum where such solutions are divided into two parts respectively corresponding to the minimum and infimum then we put emphasis on the fmp solutions in which the unified forms of fmp solutions w r t strongly residual operators and a new idea for getting fmp solutions w r t infimum are achieved third as a result of analyzing the logic basis of a sort of cri compositional rule of inference method it is found that their cri solutions can be regarded as special cases of fmp solutions lastly the response functions of fuzzy systems via universal triple i method are discussed which demonstrates that the universal triple i method can provide bigger choosing space and get better fuzzy controllers by contrast with the triple i method and cri method

triple i method of fuzzy reasoning abstract the theory of the triple i method with total inference rules of fuzzy reasoning is investigated by using zadeh s implication operator rz the computational formulae for both fuzzy modus ponens fmp and fuzzy modus tollens fmt are obtained the reversibility properties for fmp and fmt are analyzed and the reversibility criteria are given we also investigated the generalized problem of the triple i method and obtained the formulae for the triple i fmp and the triple i fmt

uniform infinite planar triangulations the existence of the weak limit as n of the uniform measure on rooted triangulations of the sphere with n vertices is proved some properties of the limit are studied in particular the limit is a probability measure on random triangulations of the plane

trees and spatial topology change in causal dynamical triangulations generalized causal dynamical triangulations generalized cdt is a model of two dimensional quantum gravity in which a limited number of spatial topology changes is allowed to occur we solve the model at the discretized level using bijections between quadrangulations and trees in the continuum limit scaling limit the amplitudes are shown to agree with known formulas and explicit expressions are obtained for loop propagators and two point functions it is shown that from a combinatorial point of view generalized cdt can be viewed as the scaling limit of planar maps with a finite number of faces and we determine the distance function on this ensemble of planar maps finally the relation with planar maps is used to illuminate the mysterious identity of certain continuum cylinder amplitudes

a shortcut to asymptotics for orthogonal polynomials abstract we consider the asymptotic behavior of the ratios qn 1 z qn z of polynomials orthonormal with respect to some positive measure let the recurrence coefficients n and n converge to 0 and 1 2 respectively then qn 1 z qn z z for n locally uniformly for z c supp where maps c 1 1 conformally onto the exterior of the unit disc nevai 1979 we provide a new and direct proof for this and some related results due to nevai and apply it to convergence acceleration of diagonal pade approximants

two step methods and bi orthogonality on etudie l ordre et la zero stabilite des methodes a deux pas du type d obrechkoff pour les equations differentielles on etablit une relation entre l ordre et les polynomes orthogonaux du m ieme degre on obtient des formules du type de rodrigues et les lieux des racines pour ces polynomes

continuity of triple i methods based on several implications this paper focuses on the continuity problem of fully implicational triple i methods for fuzzy reasoning based on the residual implications generated by continuous triangular norms the residual implication generated by nilpotent minimum with the standard negation and the zadeh implication respectively continuity and uniform continuity properties of triple i methods are examined in hamming and uniform metrics we also investigate the continuity of the model of a fuzzy if then rule corresponding to the triple i methods

continuity in zadeh s compositional rule of inference abstract in this paper the continuity property of the generalized modus ponens in the approximate reasoning is discussed it is shown that the continuity in the uniform metric and the hausdorff metric is equivalent on the rl type intervals continuity of the generalized modus ponens is examined in hausdorff hausdorff and uniform metric

on the theory of biorthogonal polynomials let x u be a distribution in x e r for every u in a real parameter set q subject to additional technical conditions we study mth degree monic polynomials pm that satisfy the biorthogonality conditions roo pm z d9 x az o i 1 2 m m 1 j oo for a distinct sequence u1 u2 e q necessary and sufficient conditions for existence and uniqueness are established as well as explicit determinantal and integral representations we also consider loci of zeros existence of rodrigues type formulae and reducibility to standard orthogonality the paper is accompanied by several examples of biorthogonal systems

bi orthogonality in rational approximation the well known connection between pade approximants to stieltjes functions and orthogonal polynomials is crucial in locating zeros and poles and in convergence theorems in the present paper we extend similar types of analysis to more elaborate forms of approximation it transpires that the link with orthogonal polynomials remains valid with regard to rational interpolants whereas simultaneous pade and levin sidi approximants yield themselves to analysis with bi orthogonal polynomials

correct models of fuzzy if then rules are continuous we show that a system of fuzzy if then rules being modeled correctly works as a partially given fuzzy function its behavior is determined by a chosen structure for if then rules which assigns meaning to fuzzy sets in the if and then parts as well as to basic connectives this partial function can be extended to a whole domain of fuzzy sets with the help of the compositional rule of inference which plays a role of a mechanism for computing the dependent functional values we investigate the question whether this function is continuous two new notions are introduced a correct model of fuzzy if then rules in a structure and a continuous model of fuzzy if then rules with respect to given data we prove that correctness of models of fuzzy if then rules is connected with the problem of solvability of the respective system of fuzzy relation equations in this and only this case a model of fuzzy if then rules is continuous as a side result we have established a new criterion of a solvability of a system of fuzzy relation equations with sup composition

spectral transformation chains and some new biorthogonal rational functions a discrete time chain associated with the generalized eigenvalue problem for two jacobi matrices is derived various discrete and continuous symmetries of this integrable equation are revealed a class of its rational elementary and elliptic functions solutions appearing from a similarity reduction are constructed the latter lead to large families of biorthogonal rational functions based upon the very well poised balanced hypergeometric series of three types the standard hypergeometric series 9 f 8 basic series 10 9 and its elliptic analogue 10 e 9 for an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly

the potential for parallelism in runge kutta methods part 1 rk formulas in standard form the authors examine the potential for parallelism in runge kutta rk methods based on formulas in standard one step form both negative and positive results are presented many of the negative results are based on a theorem that bounds the order of an rk formula in terms of the minimal polynomial associated with its coefficient matrix the positive results are largely examples of prototypical formulas that offer a potential for effective coarse grain parallelism on machines with a few processors

orthogonal polynomials through the invariant theory of binary forms abstract we present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case our bottom line is a straightforward connection between apolarity of binary forms and the inner product provided by a linear functional defined on a polynomial ring explicit determinantal formulae and multivariable extension of the heine integral formula are stated moreover a general family of covariants that includes transvectants is introduced such covariants turn out to be the average value of classical basis of symmetric polynomials over a set of roots of suitable orthogonal polynomials

polya schur master theorems for circular domains and their boundaries we characterize all linear operators on finite or infinite dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region omega subset of c for arbitrary

polynomials biorthogonal to dilations of measures and their asymptotics we analyze polynomials pn that are biorthogonal to dilates of a positive measure supported on 0 0 pn x d n jx 0 1 j n we establish representations for pn in terms of the associated dilation polynomial rn y j 1n y 1 n j in the case where d t t e t dton 0 we show that strong asymptotics for rn in the complex plane as n lead to strong asymptotics for pn via the method of steepest descent

a note on the value similarity of fuzzy systems variables abstract the notion of similarity of fuzzy variables is discussed in arbitrary set in this paper it is shown that the property of similarity is preserved when the sup min compositional rule of inference is applied to similar values of fuzzy variables and fuzzy relations which extends the results of pappis 1991

perturbation of fuzzy sets and fuzzy reasoning based on normalized minkowski distances in this paper we extend the concept of the perturbation of fuzzy sets based on normalized minkowski distances and present some new conclusions on perturbation raised by various operations of fuzzy sets these operations are induced by triangular norms and conorms furthermore we discuss the perturbation of fuzzy reasoning

a restricted dimer model on a two dimensional random causal triangulation we introduce a restricted hard dimer model on a random causal triangulation that is exactly solvable and generalizes a model recently proposed by atkin and zohren 2012 phys lett b 712 445 50 we show that the latter model exhibits unusual behaviour at its multicritical point in particular its hausdorff dimension equals 3 and not as would be expected from general scaling arguments when viewed as a special case of the generalized model introduced here we show that this behaviour is not generic and therefore is not likely to represent the true behaviour of the full dimer model on a random causal triangulation

unified forms of triple i method first the present paper provides unified forms of triple i method for fuzzy modus ponens and fuzzy modus tollens of which diverse implication operators can be employed second it is clarified that in a sense zadeh s cri method for fuzzy modus ponens can be brought into line with the unified form of the triple i method lastly a unified form of a triple i method is established as well and a duality result concerning a triple i solutions of fuzzy modus ponens and fuzzy modus tollens is obtained

some explicit formulae in simultaneous pade approximation abstract using old results on the explicit calculation of determinants formulae are given for the coefficients of p 0 z and p 0 z f i z p i z where p i z are polynomials of degree i i 0 1 n p 0 z f i z p i z are power series in which the terms with z k 0 k vanish i 1 2 n 0 1 n is an n 1 tuple of nonnegative integers 0 1 n and f i n i 1 is the set of hypergeometric functions 1 f 1 1 c i z n i 1 c i z z drule n c i c j z or 2 f 0 a i 1 z n i 1 a i z n a i a j z under the condition 0 i 1 i 1 2 n

on the spectral dimension of causal triangulations we introduce an ensemble of infinite causal triangulations called the uniform infinite causal triangulation and show that it is equivalent to an ensemble of infinite trees the uniform infinite planar tree it is proved that in both cases the hausdorff dimension almost surely equals 2 the infinite causal triangulations are shown to be almost surely recurrent or equivalently their spectral dimension is almost surely less than or equal to 2 we also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely the triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two dimensional quantum gravity and therefore our results apply to that model

local limit of labeled trees and expected volume growth in a random quadrangulation exploiting a bijective correspondence between planar quadrangulations and well labeled trees we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume the limit random surface can be described in terms of a birth and death process and a sequence of multitype galton watson trees as a consequence we find that the expected volume of the ball of radius r around a marked point in the limit random surface is r4

the spectral dimension of generic trees we define generic ensembles of infinite trees these are limits as n of ensembles of finite trees of fixed size n defined in terms of a set of branching weights among these ensembles are those supported on trees with vertices of a uniformly bounded order the associated probability measures are supported on trees with a single spine and hausdorff dimension dh 2 our main result is that the spectral dimension of the ensemble average is ds 4 3 and that the critical exponent of the mass defined as the exponential decay rate of the two point function along the spine is 1 3