on the structure and probabilistic interpretation of askey wilson densities and polynomials with complex parameters abstract we give equivalent forms of the askey wilson polynomials expressing them with the help of the al salam chihara polynomials after restricting parameters of the askey wilson polynomials to complex conjugate pairs we expand the askey wilson weight function in the series similar to the poisson mehler expansion formula and give its probabilistic interpretation in particular this result can be used to calculate explicit forms of q hermite moments of the askey wilson density hence enabling calculation of all moments of the askey wilson density on the way by setting certain parameter q to 0 we get some formulae useful in the rapidly developing so called free probability

very simple methods for all pairs network flow analysis a very simple algorithm for the classical problem of computing the maximum network flow value between every pair of nodes in an undirected capacitated n node graph is presented as in the well known gomory hu method the method given here uses only n 1 maximum flow computations our algorithm is implemented by adding only five simple lines of code to any program that produces a minimum cut a program to produce an equivalent flow tree which is a compact representation of the flow values is obtained by adding only three simple lines of code to any program producing a minimum cut a very simple version of the gomory hu cut tree method that finds one minimum cut for every pair of nodes is also derived and it is shown that the seemingly fundamental operation of that method node contraction is not needed nor must crossing cuts be avoided as a result this version of the gomory hu method is implemented by adding less than ten simple lines of code to any program that produces a minimum cut the algor

one dimensional double hecke algebra and gaussian sums we introduce and describe one dimensional cyclotomic gauss selberg sums generalizing the classical gaussian sums they correspond to irreducible self dual unitary spherical representations of the one dimensional double affine hecke algebra

holomorphic curves with shift invariant hyperplane preimages if f c p n is a holomorphic curve of hyper order less than one for which 2n 1 hyperplanes in general position have forward invariant preimages with respect to the translation z z c then f is periodic with period c 2 c this result which can be described as a difference analogue of m green s picard type theorem for holomorphic curves follows from a more general result presented in this paper the proof relies on a new version of cartan s second main theorem for the casorati determinant and an extended version of the difference analogue of the lemma on the logarithmic derivatives both of which are proved here finally an application to the uniqueness theory of meromorphic functions is given and the sharpness of the obtained results is demonstrated by examples

signless laplacians of finite graphs we survey properties of spectra of signless laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs the whole existing theory of spectra of the adjacency matrix and of the laplacian matrix transfers directly to the signless laplacian and so we consider arbitrary graphs with special emphasis on the non regular case the results which we survey old and new are of two types a results obtained by applying to the signless laplacian the same reasoning as for corresponding results concerning the adjacency matrix b results obtained indirectly via line graphs among other things we present eigenvalue bounds for several graph invariants an interpretation of the coefficients of the characteristic polynomial a theorem on powers of the signless laplacian and some remarks on star complements

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

askey wilson polynomials quadratic harnesses and martingales we use orthogonality measures of askey wilson polynomials to construct markov processes with linear regressions and quadratic conditional variances askey wilson polynomials are orthogonal martingale polynomials for these processes

dynamic programming and graph optimization problems abstract several classes of graph optimization problems which can be solved using dynamic programming are known to have more efficient tailor made algorithms this paper discusses four such classes and the underlying constraints on their subproblem interrelationships that yield these efficient algorithms these classes are also extended to handle more general cost functions

the minimum spectral radius of graphs with a given independence number abstract the independence number g of g is defined as the maximum cardinality of a set of pairwise non adjacent vertices which is called an independent set in this paper we characterize the graphs which have the minimum spectral radius among all the connected graphs of order n with independence number 1 2 n 2 n 2 1 n 3 n 2 n 1

triangulations tilings and certain block triangular matrices for a convex polygonp withn sides a partitioning ofp inton 2 nonoverlapping triangles each of whose vertices is a vertex ofp is called a triangulation or tiling and each triangle is a tile each tile has a given cost associated with it which may differ one from another this paper considers the problem of finding a tiling ofp such that the sum of the costs of the tiles used is a minimum and explores the curiosity that an abstract formulation of it can be cast as a linear program further the special structure of the linear program permits a recursive o n 3 algorithm

second main theorem in the tropical projective space abstract tropical nevanlinna theory introduced by halburd and southall as a tool to analyze integrability of ultra discrete equations studies the growth and complexity of continuous piecewise linear real functions the purpose of this paper is to extend tropical nevanlinna theory to n dimensional tropical projective spaces by introducing a natural characteristic function for tropical holomorphic curves and by proving a tropical analogue of cartan s second main theorem it is also shown that in the 1 dimensional case this result implies a known tropical second main theorem due to laine and tohge

an efficient parallel dynamic programming algorithm abstract antonio tsai and huang proposed a scheme in 1991 to parallelize the standard dynamic programming approach to solve combinatorial multistage problems however their dynamic programming approach is restricted to those multistage problems where the decision made at each stage depends only on decisions made in the stage immediately preceding it for many interesting problems the decision at each stage depends on the decisions made at all the previous stages and therefore their approach doesn t apply the matrix chain multiplication problem longest common subsequence problem and optimal polygon triangulation problem are some examples of such problems we also present techniques for parallelizing the dynamic programming solution to such problems the parallel algorithm we develop for a pram has complexity n employing n 2 processors since the traditional sequential algorithm for such problems is n 3 our parallel algorithm is an optimal parallel algorithm based on this traditional algorithm we also describe the results of our experiments that are in conformity with our theoretical complexity results we also compare and contrast our result with results obtained by earlier researchers and show that our parallel algorithm has optimal efficiency of 100 with respect to the traditional dynamic programming algorithm

difference analogue of the lemma on the logarithmic derivative with applications to difference equations the lemma on the logarithmic derivative of a meromorphic function has many applications in the study of meromorphic functions and ordinary dierential equa tions in this paper a dierence analogue of the logarithmic derivative lemma is presented and then applied to prove a number of results on meromorphic solutions of complex dierence equations these results include a dierence analogue of the clunie lemma as well as other results on the value distribution of solutions

the minimum signless laplacian spectral radius of graphs with given independence number abstract the independence number g of g is defined as the maximum cardinality of a set of pairwise non adjacent vertices which is called an independent set in this paper we characterize the graphs which have the minimum spectral radius among all the connected graphs of order n with independence number g 1 2 n 2 n 2 1 n 2 n 3

cramer and cayley hamilton in the max algebra abstract cramer s rule and the cayley hamilton theorem are formulated and provided in the so called max algebra which consists of the set of reals provided with two operations maximization and addition it is surprising to see that these well known theorems carry over to the max algebra almost without any changes provided that the conventional addition and multiplication are replaced by maximization and addition respectively the definitions of determinant and eigenvalue have to be adjusted to this new formulation the role of the determinant is taken over by the permanent and a refinement of this latter concept called the dominant

nevanlinna theory of the askey wilson divided difference operator abstract this paper establishes a version of nevanlinna theory based on askey wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane c a second main theorem that we have derived allows us to define an askey wilson type nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero pole scarce that is their zeros poles are indeed deficient in the sense of difference nevanlinna theory a natural consequence is a version of askey wilson type picard theorem we also give an alternative and self contained characterisation of the kernel functions of the askey wilson operator in addition we have established a version of unicity theorem in the sense of askey wilson this paper concludes with an application to difference equations generalising the askey wilson second order divided difference equation

unoriented laplacian maximizing graphs are degree maximal abstract a connected graph is said to be unoriented laplacian maximizing if the spectral radius of its unoriented laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges a graph is said to be threshold maximal if its degree sequence is not majorized by the degree sequence of any other graph and in addition the graph is connected it is proved that an unoriented laplacian maximizing graph is maximal and also that there are precisely two unoriented laplacian maximizing graphs of a given order and with nullity 3 our treatment depends on the following known characterization a graph g is threshold maximal if and only if for every pair of vertices u v of g the sets n u v n v u where n u denotes the neighbor set of u in g are comparable with respect to the inclusion relation and in addition the graph is connected a conjecture about graphs that maximize the unoriented laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

suffridge s convolution theorem for polynomials and entire functions having only real zeros abstract we present a q extension of polya s and schur s characterization of multiplier sequences and a q extension and a converse of newton s inequalities

on graphs whose signless laplacian index does not exceed 4 5 abstract let a g and d g be respectively the adjacency matrix and the degree matrix of a graph g the signless laplacian matrix of g is defined as q g d g a g the q spectrum of g is the set of the eigenvalues together with their multiplicities of q g the q index of g is the maximum eigenvalue of q g the possibilities for developing a spectral theory of graphs based on the signless laplacian matrices were discussed by cvetkovic et al d cvetkovic p rowlinson s k simic signless laplacians of finite graphs linear algebra appl 423 2007 155 171 in the latter paper the authors determine the graphs whose q index is in the interval 0 4 in this paper we investigate some properties of q spectra of graphs especially for the limit points of the q index by using these results we characterize respectively the structures of graphs whose the q index lies in the intervals 4 2 5 2 5 2 and 2 4 5 where 1 3 54 6 33 1 3 54 6 33 1 3 2 382975767

on meromorphic functions with finite logarithmic order by using a slow growth scale the logarithmic order with which to measure the growth of functions we obtain basic results on the value distribution of a class of meromorphic functions of zero order

computation of matrix chain products part ii this paper considers the computation of matrix chain products of the form m 1 times m 2 times cdots times m n 1 if the matrices are of different dimensions the order in which the product is computed affects the number of operations an optimum order is an order which minimizes the total number of operations we present some theorems about an optimum order of computing the matrices based on these theorems an o n log n algorithm for finding an optimum order will be presented in part ii

finite order meromorphic solutions and the discrete painleve equations let w z be a finite order meromorphic solution of the second order difference equation w z 1 w z 1 r z w z where r z w z is rational in w z and meromorphic in z then either w z satisfies a difference linear or riccati equation or else equation can be transformed to one of a list of canonical difference equations this list consists of all known difference painleve equation of the form together with their autonomous versions this suggests that the existence of finite order meromorphic solutions is a good detector of integrable difference equations