explicit forms for ergodicity coefficients of stochastic matrices abstract motivated by explicit expressions appearing in the work of a rhodius 1993 for n n stochastic matrices p it is shown that ordinary matrix norms on r n 1 for n 1 n 1 matrices of the form apb can be used to generate results of this kind

social balance as a satisfiability problem of computer science reduction of frustration was the driving force in an approach to social balance as it was recently considered by antal t antal p l krapivsky and s redner phys rev e 72 036121 2005 we generalize their triad dynamics to k cycle dynamics for arbitrary integer k we derive the phase structure determine the stationary solutions and calculate the time it takes to reach a frozen state the main difference in the phase structure as a function of k is related to k being even or odd as a second generalization we dilute the all to all coupling as considered by antal to a random network with connection probability w 1 interestingly this model can be mapped to a satisfiability problem of computer science the phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem in common to the cases we study the ideal solution without any frustration always exists but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time the answer depends on the choice of parameters after establishing the mapping between the two classes of models we generalize the social balance problem to a diluted network topology for which the satisfiability problem is usually studied on the other hand in connection with the satisfiability problem we generalize the random local algorithm to a p random local algorithm including a parameter p that corresponds to the propensity parameter in the social balance problem the qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time

quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

on the exponent of a primitive digraph abstract we characterize the equality case of the upper bound d n s n 2 for the exponent of a primitive digraph in the case s 2 where n is the number of the vertices of the digraph d and s is the length of the shortest elementary circuit of d we also answer a question about the equality case when s 1 the exponent of an n n primitive nonnegative matrix a is the same as the exponent of the associated digraph d a of a so every theorem in this paper can be translated into a theorem about nonnegative matrices

on the index of convergence of an irreducible boolean matrix abstract first we give a new proof of the sharp upper bound for the indices of convergence of n x n irreducible boolean or nonnegative matrices with period p then we characterize the matrices with the largest index

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

quasi exactly solvable models from finite dimensional matrices a new method of obtaining many dimensional quasi exactly solvable models is suggested it is based on constructing the generating function with the help of coefficients which obey a finite difference equation the structure of this equation is selected to obtain the closed second order differential equation for the generating function under some conditions this equation can be thought of as the schrodinger equation in curved space for the two dimensional case the many parametric class of solution is found explicitly the spherically symmetrical case is investigated in detail it is shown that this case contains spaces of a constant riemann curvature of both signs

on explicit forms for ergodicity coefficients abstract explicit forms for ergodicity coefficients are known only when the l1 norm or the l norm is used 11 15 the purpose of this paper is to give a new approach to ergodicity coefficients and their functional forms realizations of this general procedure provide further ergodicity coefficients their explicit forms and new eigenvalue bounds which are compared with the deutsch zenger numerical radius 2 8

generalized competition index of a primitive digraph for positive integers k and m and a digraph d the k step m competition graph cmk d of d has the same set of vertices as d and an edge between vertices x and y if and only if there are distinct m vertices v1 v2 vm in d such that there are directed walks of length k from x to vi and from y to vi for 1 i m in this paper we present the definition of m competition index for a primitive digraph the m competition index of a primitive digraph d is the smallest positive integer k such that cmk d is a complete graph we study m competition indices of primitive digraphs and provide an upper bound for the m competition index of a primitive digraph

scrambling index set of primitive digraphs abstract the scrambling index k d of a primitive digraph d is the smallest positive integer k such that for every pair of vertices x and y there exists a vertex v such that there exist directed walks of length k from x to v and from y to v in this paper we study the scrambling index set of primitive digraphs

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

the scrambling index of symmetric primitive matrices abstract a nonnegative square matrix a is primitive if some power a k 0 that is a k is entrywise positive the least such k is called the exponent of a in 2 akelbek and kirkland defined the scrambling index of a primitive matrix a which is the smallest positive integer k such that any two rows of a k have at least one positive element in a coincident position in this paper we give a relation between the scrambling index and the exponent for symmetric primitive matrices and determine the scrambling index set for the class of symmetric primitive matrices we also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

quasiexactly solvable problems and the dual q hahn polynomials a second order differential q difference eigenvalue equation is contructed whose solutions are generating functions of the dual q hahn polynomials the fact is noticed that these generating functions are reduced to the little q jacobi polynomials and implications of this for quasiexactly solvable problems are studied a connection with the azbel hofstadter problem is indicated

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

a bound on the scrambling index of a primitive matrix using boolean rank abstract the scrambling index of an n n primitive matrix a is the smallest positive integer k such that a k a t k j where a t denotes the transpose of a and j denotes the n n all ones matrix for an m n boolean matrix m its boolean rank b m is the smallest positive integer b such that m ab for some m b boolean matrix a and b n boolean matrix b in this paper we give an upper bound on the scrambling index of an n n primitive matrix m in terms of its boolean rank b m furthermore we characterize all primitive matrices that achieve the upper bound

characterization of irreducible boolean matrices with the largest generalized competition index abstract for a positive integer m the m competition graph of an irreducible boolean matrix a of order n denoted by c m a is the graph that has the same vertex set as its digraph d a and there is an edge between vertices x and y x y if and only if there exist m distinct vertices v 1 v 2 v m such that x v i and y v i for 1 i m in d the smallest positive integer q such that c m a q i c m a q r i for some positive integer r and every nonnegative integer i is called the m competition index generalized competition index of a in this paper we characterize irreducible boolean matrices with the largest generalized competition index

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

coefficients of ergodicity and the scrambling index for a primitive stochastic matrix s upper bounds on the second largest modulus of an eigenvalue of s are very important because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix in this paper we introduce the definition of the scrambling index for a primitive digraph the scrambling index of a primitive digraph d is the smallest positive integer k such that for every pair of vertices u and v there is a vertex w such that we can get to w from u and v in d by directed walks of length k it is denoted by k d we investigate the scrambling index for primitive digraphs and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph by doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

bounds on the base of primitive nearly reducible sign pattern matrices abstract in j y shao l h you bound on the base of irreducible generalized sign pattern matrices discrete math in press shao and you extended the concept of the base from powerful sign pattern matrices to non powerful and generalized sign pattern matrices in this paper we study the base for primitive non powerful nearly reducible sign pattern and generalized sign pattern matrices we obtain sharp upper bounds together with complete characterization of the equality cases of the base for primitive nearly reducible sign pattern and generalized sign pattern matrices we also show that there exist gaps in the base set of the classes of such matrices

rogers ramanujan and the baker gammel wills pade conjecture in 1961 baker gammel and wills conjectured that for functions f meromorphic in the unit ball a subsequence of its diagonal pade approximants converges uniformly in compact subsets of the ball omitting poles of f there is also apparently a cruder version of the conjecture due to pade himself going back to the early twentieth century we show here that for carefully chosen q on the unit circle the rogers ramanujan continued fraction

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

continued fractions and the partially asymmetric exclusion process we note that a tridiagonal matrix representation of the algebra of the partially asymmetric exclusion process pasep lends itself to interpretation as the transfer matrix for weighted motzkin lattice paths a continued fraction j fraction representation of the lattice path generating function is particularly well suited to discussing the pasep for which the paths have height dependent weights we show that this not only allows a succinct derivation of the normalization and correlation lengths of the pasep but also reveals how finite dimensional representations of the pasep algebra valid only along special lines in the phase diagram relate to the general solution that requires an infinite dimensional representation

convergence of inhomogenous products of matrices and coefficients of ergodicity abstract given a square matrix a and a norm the coefficient of ergodicity of a with respect to is defined as max x t all x r n x 1 xtf 0 with f as a matrix satisfying af 0 we demonstrate that for a bounded set of such matrices with all coefficients of ergodicity of the matrices in the set below 1 all sequences constructed through inhomogenous products of matrices from the set converge geometrically

the scrambling index set of primitive minimally strong digraphs abstract denote the scrambling index set of primitive minimally strong digraphs of order n by si n for n 13 we show that si n contains all integers in the interval 3 k 1 n where k 1 n n 2 12 n 13 8 if both n and n 1 2 are odd n 2 12 n 5 8 if n is odd and n 1 2 is even n 2 10 n 16 8 if n is even and n 2 is odd n 2 10 n 8 8 if both n and n 2 are even we also identify all elements of si n which are larger than n 2 6 n 13 2 n is odd or n 2 6 n 12 2 n is even it is shown that there exist more gaps in the set si n

dynamics of social balance on networks we study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes the network is endowed with dynamics in which the sense of a link in an imbalanced triad a triangular loop with one or three unfriendly links is reversed to make the triad balanced with this dynamics an infinite network undergoes a dynamic phase transition from a steady state to paradise all links are friendly as the propensity p for friendly links in an update event passes through 1 2 a finite network always falls into a socially balanced absorbing state where no imbalanced triads remain if the additional constraint that the number of imbalanced triads in the network not increase in an update is imposed then the network quickly reaches a balanced final state i introduction in this work we investigate the role of friends and enemies on the evolution of social networks we represent individuals as nodes in a graph and a relationship between individuals as a link that joins the corresponding nodes to quantify a relationship we assign the binary variable sij 1 to link ij with sij 1 if nodes i and j are friends and sij 1 ifi and j are enemies fig 1 a basic characterization of relationships between mutual acquaintances is the notion of social balance 1 2 the triad ijk is defined as balanced if the sign of the product of the links in the triad sijsjkski equals 1 while the triad is imbalanced otherwise we define a triad to be of type k if it contains k negative links thus 0 and 2 are balanced while 1 and 3 are imbalanced a balanced triad therefore fulfills the adage that i a friend of my friend is my friend ii an enemy of my friend

zeros and orthogonality of the askey wilson polynomials for q a root of unity we study some properties of the askey wilson polynomials awp when q is a primitive nth root of unity for general four parameter awp zeros of the nth polynomial and the orthogonality measure are found explicitly special subclasses of the awp e g the continuous q jacobi and big q jacobi polynomials are considered in detail a set of discrete weight functions positive on a real interval is described some new trigonometric identities related to the awp are obtained normalization conditions of some polynomials are expressed in terms of the gauss sums

a bound on the generalized competition index of a primitive matrix using boolean rank abstract for a positive integer m where 1 m n the m competition index generalized competition index of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y there exist m distinct vertices v 1 v 2 v m such that there are directed walks of length k from x to v i and from y to v i for 1 i m the m competition index is a generalization of the scrambling index and the exponent of a primitive digraph in this study we determine an upper bound on the m competition index of a primitive digraph using boolean rank and give examples of primitive boolean matrices that attain the bound

a bound on the exponent of a primitive matrix using boolean rank we present a bound on the exponent exp a of an n x n primitive matrix a in terms of its boolean rank b b a namely exp a b 1j2 2 further we show that for each 2 q b q n 1 there is an n x n primitive matrix a with b a b such that exp a b 1 2 2 and we explicitly describe all such matrices the new bound is compared with a well known bound of dulmage and mendelsohn and with a conjectured bound of hartwig and neumann several open problems are posed

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

discrete time volterra chain and classical orthogonal polynomials a non isospectral discrete time volterra chain dtvc is derived from a set of spectral transformations for symmetric orthogonal polynomials op such dtvc is a natural finite difference analogue of the well known factorization chain for the schrodinger equation a class of meromorphic solutions of dtvc is found from an ansatz of semiseparation of variables the latter yields the very general explicitly known systems of op the askey wilson and askey ismail polynomials

generalized scrambling indices of a primitive digraph from the background of a memoryless communication system we introduce the definition of the generalized scrambling indices for a primitive digraph which are natural extensions of the scrambling index and m competition index moreover we provide some exact lower and upper bounds for the generalized scrambling indices of various classes of primitive digraphs respectively

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

the generalized competition indices of primitive minimally strong digraphs abstract let d be a primitive digraph of order n and 1 m n the m competition index of d is the smallest positive integer k such that every pair of vertices x and y of d have at least m common preys in d k in this paper the upper bound of m competition indices 1 m n 1 for primitive minimally strong digraphs of order n is obtained furthermore it is shown that for 1 m n 1 there exist gaps in the m competition index set of primitive minimally strong digraphs of order n

quasi exactly solvable potentials on the line and orthogonal polynomials in this paper we show that a quasi exactly solvable normalizable or periodic one dimensional hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three term recursion relation in particular we prove that normalizable exactly solvable one dimensional systems are characterized by the fact that their associated polynomials satisfy a two term recursion relation we study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one dimensional quasi exactly solvable hamiltonian showing in particular that its associated stieltjes measure is supported on a finite set from this we deduce that the corresponding moment problem is determined and that the kth moment grows like the kth power of a constant as k tends to infinity we also show that the moments satisfy a constant coefficient linear difference equation and that this property actually characterizes weakly orthogonal polynomial systems

a bound of generalized competition index of a primitive digraph abstract for a positive integer m where 1 m n the m competition index generalized competition index of a primitive digraph d is the smallest positive integer k such that for every pair of vertices x and y there exist m distinct vertices v 1 v 2 v m such that there exist directed walks of length k from x to v i and from y to v i for 1 i m the m competition index is a generalization of the scrambling index and the exponent of a primitive digraph in this paper we study the upper bound of the m competition index of a primitive digraph using its order and girth

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

primitive digraphs with the largest scrambling index abstract the scrambling index of a primitive digraph d is the smallest positive integer k such that for every pair of vertices u and v there is a vertex w such that we can get to w from u and v in d by directed walks of length k it is denoted by k d in m akelbek s kirkland coefficients of ergodicity and the scrambling index preprint we gave the upper bound on k d in terms of the order and the girth of a primitive digraph d in this paper we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound

universality class of triad dynamics on a triangular lattice we consider triad dynamics as it was recently considered by antal et al phys rev e 72 036121 2005 as an approach to social balance here we generalize the topology from all to all to the regular one of a two dimensional triangular lattice the driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state the dynamics is parametrized by a so called propensity parameter p that determines the tendency of negative links to become positive as a function of p we find a phase transition between different kinds of absorbing states the phases differ by the existence of an infinitely connected percolated cluster of negative links that forms whenever p ensuremath leqslant p c moreover for p ensuremath leqslant p c the time to reach the absorbing state grows powerlike with the system size l while it increases logarithmically with l for pg p c from a finite size scaling analysis we numerically determine the static critical exponents ensuremath beta and ensuremath nu ensuremath perp together with ensuremath gamma ensuremath tau ensuremath sigma and the dynamical critical exponents ensuremath nu ensuremath vert and ensuremath delta the exponents satisfy the hyperscaling relations we also determine the fractal dimension d f that satisfies a hyperscaling relation as well the transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents we generalize the triad dynamics to four cycle dynamics on a square lattice in this case again there is a transition between different absorbing states going along with the formation of an infinite cluster of negative links but the usual scaling and hyperscaling relations are violated

generalized competition index of an irreducible boolean matrix abstract for a positive integer m where 1 m n the m competition graph of an irreducible boolean matrix a of order n denoted by c m a is the graph that has the same vertex set as its digraph d a and there is an edge between vertices x and y x y if and only if there exist m distinct vertices v 1 v 2 v m such that x v i and y v i for 1 i m in d a the smallest positive integer q such that c m a q i c m a q r i for some positive integer r and every nonnegative integer i is called the m competition index generalized competition index of a the m competition index is a generalization of the competition index and the index of an irreducible boolean matrix in this study we determine the upper bound on the m competition index of an irreducible boolean matrix

the scrambling index of primitive digraphs in 2009 akelbek and kirkland introduced a useful parameter called the scrambling index of a primitive digraph d which is the smallest positive integer k such that for every pair of vertices u and v there is a vertex w such that we can get to w from u and v in d by directed walks of length k in this paper we obtain some new upper bounds for the scrambling index of primitive digraphs moreover the maximum index problem the extremal matrix problem and the index set problem for the scrambling index of various classes of primitive digraphs e g primitive digraphs with d loops minimally strong digraphs nearly decomposable digraphs micro symmetric digraphs etc are settled respectively