spectra of complex networks we propose a general approach to the description of spectra of complex networks for the spectra of networks with uncorrelated vertices and a local treelike structure exact equations are derived these equations are generalized to the case of networks with correlations between neighboring vertices the tail of the density of eigenvalues ensuremath rho ensuremath lambda at large ensuremath lambda is related to the behavior of the vertex degree distribution p k at large k in particular as p k ensuremath sim k ensuremath ensuremath gamma ensuremath rho ensuremath lambda ensuremath sim ensuremath lambda 1 ensuremath 2 ensuremath gamma we propose a simple approximation which enables us to calculate spectra of various graphs analytically we analyze spectra of various complex networks and discuss the role of vertices of low degree we show that spectra of locally treelike random graphs may serve as a starting point in the analysis of spectral properties of real world networks e g of the internet

autocorrelation in meter induction the role of accent structure the performance of autocorrelation based meter induction was tested with two large collections of folk melodies consisting of approximately 13 000 melodies for which the correct meters were available the performance was measured by the proportion of melodies whose meter was correctly classified by a discriminant function furthermore it was examined whether including different melodic accent types would improve the classification performance by determining the components of the autocorrelation functions that were significant in the classification it was found that periodicity in note onset locations was the most important cue for the determination of meter of the melodic accents included thomassen s melodic accent was found to provide the most reliable cues for the determination of meter the discriminant function analyses suggested that periodicities longer than one measure may provide cues for meter determination that are more reliable than shorter periodicities overall the method predicted notated meter with an accuracy reaching 96 for binary classification and 75 for classification into nine categories of meter

the laguerre and legendre polynomials from an operational point of view the use of the monomiality principle and of the associated operational calculus allows to reformulate the theory of laguerre and legendre polynomials from a unified point of view they are indeed shown to be particular cases of more general polynomials whose usefulness in purely mathematical and applied contexts is discussed the concepts introduced in the paper are also exploited to develop an elementary point of view on the theory of legendre and laguerre bessel functions

a high resolution fundamental frequency determination based on phase changes of the fourier transform the constant q transform described recently j c brown and m s puckette an efficient algorithm for the calculation of a constant q transform j acoust soc am 92 2698 2701 1992 has been adapted so that it is suitable for tracking the fundamental frequency of extremely rapid musical passages for this purpose the calculation described previously has been modified so that it is constant frequency resolution rather than constant q for lower frequency bins this modified calculation serves as the input for a fundamental frequency tracker similar to that described by brown j c brown musical fundamental frequency tracking using a pattern recognition method j acoust soc am 92 1394 1402 1992 once the fast fourier transform fft bin corresponding to the fundamental frequency is chosen by the frequency tracker an approximation is used for the phase change in the fft for a time advance of one sample to obtain an extremely precise value for this frequency graphical examples are given for musical passages by a violin executing vibrato and glissando where the fundamental frequency changes are rapid and continuous

topological properties of citation and metabolic networks topological properties of scale free networks are investigated by determining their spectral dimensions d s which reflect a diffusion process in the corresponding graphs data bases for citation networks and metabolic networks together with simulation results from the growing network model a l barabasi and r albert science 286 509 1999 are probed for completeness and comparisons lattice random and small world models are also investigated we find that d s is around 3 for citation and metabolic networks which is significantly different from the growing network model for which d s is approximately 7 5 this signals a substantial difference in network topology despite the observed similarities in vertex order distributions in addition the diffusion analysis indicates that the citation networks are treelike in structure whereas the metabolic networks contain many loops

structural properties of planar graphs of urban street patterns recent theoretical and empirical studies have focused on the structural properties of complex relational networks in social biological and technological systems here we study the basic properties of twenty 1 square mile samples of street patterns of different world cities samples are turned into spatial valued graphs in such graphs the nodes are embedded in the two dimensional plane and represent street intersections the edges represent streets and the edge values are equal to the street lengths we evaluate the local properties of the graphs by measuring the meshedness coefficient and counting short cycles of three four and five edges and the global properties by measuring global efficiency and cost we also consider as extreme cases minimal spanning trees mst and greedy triangulations gt induced by the same spatial distribution of nodes the measures found in the real and the artificial networks are then compared surprisingly cities of the same class e g grid iron or medieval exhibit roughly similar properties the correlation between a priori known classes and statistical properties is illustrated in a plot of relative efficiency vs cost

on a new family related to truncated exponential and sheffer polynomials abstract in this article the truncated exponential and sheffer polynomials are combined to introduce the 2 variable truncated exponential based sheffer polynomials 2vtesp by using operational methods examples of certain special polynomials belonging to this family are considered operational correspondence between the 2vtesp and sheffer polynomials is established which is applied to derive the results for some members belonging to the 2vtesp family

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

a note on truncated polynomials truncated polynomials have been shown to play a role of crucial importance in the evaluation of integrals involving products of special functions the properties of this family of polynomials are relatively little known and this paper is devoted to their study and to a deeper understanding of their role

vertex similarity in networks we consider methods for quantifying the similarity of vertices in networks we propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar this leads to a self consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network we test our similarity measure on computer generated networks for which the expected results are known and on a number of real world networks

operational rules and arbitrary order hermite generating functions operational identities often exploited in the theory of exponential operators are used to derive generating functions of hermite and kampe de feriet polynomials the method we propose allows us to obtain unilateral bilinear and bilateral generating functions by using the same procedure possible extensions of the technique are also discussed

operational methods and differential equations with applications to initial value problems the authors present and develop a general method of operational nature in order to explore and investigate such families of orthogonal polynomials and functions as the classical and generalized hermite and laguerre polynomials as well as their higher order and multi index cases they also consider several applications of these operational techniques to the solutions of a wide class of partial differential equations including for example the heat wave and transport equations some of which involve second order laguerre derivatives

graph laplacians nodal domains and hyperplane arrangements eigenvectors of the laplacian of a graph g have received increasing attention in the recent past here we investigate their so called nodal domains i e the connected components of the maximal induced subgraphs of g on which an eigenvector psi does not change sign an analogue of courant s nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of psi in the spectrum this bound however is not sharp in general in this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph the class of boolean hypercubes is discussed in detail we find that despite the simplicity of this graph class for which complete spectral information is available the computations are still non trivial nevertheless we obtained some new results and a number of conjectures author s abstract

on the metric topological and functional structures of urban networks abstract axial graphs are networks whose nodes are linear axes in urban space and whose edges represent intersections of such axes these graphs are used in urban planning and urban morphology studies in this paper we analyse distance distributions between nodes in axial graphs and show that these distributions are well approximated by rescaled poisson distributions we then demonstrate a correlation between the parameters governing the distance distribution and the degree of the polynomial distribution of metric lengths of linear axes in cities this correlation provides topological support to the metrically based categorisation of cities proposed in r carvalho a penn scaling and universality in the micro structure of urban space physica a 332 2004 539 547 finally we attempt to explain this topologico metric categorisation in functional terms to this end we introduce a notion of attraction cores defined in terms of aggregations of random walk agents we demonstrate that the number of attraction cores in cities correlates with the parameters governing their distance and line length distributions the intersection of all the three points of view topological metric and agent based yields a descriptive model of the structure of urban networks

hermite based appell polynomials properties and applications abstract by employing certain operational methods the authors introduce hermite based appell polynomials some properties of hermite appell polynomials are considered which proved to be useful for the derivation of identities involving these polynomials the possibility of extending this technique to introduce hermite based sheffer polynomials for example hermite laguerre and hermite sister celine s polynomials is also investigated

differential and integral equations for the 2 iterated appell polynomials in this article a set of differential equations of finite order k th order k n for the 2 iterated appell polynomials are derived particular cases k 1 and k 2 are also considered the integral equations for the appell and 2 iterated appell polynomials are established further as an illustration the differential and integral equations for the 2 iterated generalized bernoulli polynomials are derived this article is first attempt in the direction of deriving integral equations for the appell and 2 iterated appell families and for some members belonging to these families

the appell interpolation problem a general linear interpolation problem is considered we will call it the appell interpolation problem because the solution can be expressed by a basis of appell polynomials some classical and non classical examples are also considered finally numerical calculations are given

multi variable hermite matrix polynomials properties and applications abstract in this paper the multi variable hermite matrix polynomials are introduced by algebraic decomposition of exponential operators their properties are established using operational methods the matrix forms of the chebyshev and truncated polynomials of two variable are also introduced which are further used to derive certain operational representations and expansion formulae

finding mixed families of special polynomials associated with appell sequences abstract in this paper certain mixed special polynomial families associated with appell sequences are introduced and their properties are established further operational rules providing connections between these families and known special polynomials are established which are used to derive the identities and results for the members of these new families determinantal definitions of the polynomials associated with appell family are also derived the approach presented is general

a determinantal approach to sheffer appell polynomials via monomiality principle abstract in this article the sheffer and appell polynomials are combined to introduce the family of sheffer appell polynomials by using operational methods the determinantal definition and other properties of the sheffer appell polynomials are established as particular cases of these polynomials the sheffer bernoulli and sheffer euler polynomials are introduced and their determinantal definitions are obtained the operational correspondence between the appell and sheffer appell polynomials is used to derive the results for the sheffer appell polynomials certain results for the hermite appell and laguerre appell polynomials are also obtained

determinantal approach to certain mixed special polynomials related to gould hopper polynomials in this article the gould hopper sheffer polynomials are introduced by means of generating function and determinantal definition certain properties of these polynomials are established and operational relations between the sheffer and gould hopper sheffer polynomials are derived examples of some members belonging to the gould hopper sheffer and gould hopper appell polynomials families are considered and the corresponding numbers for certain polynomials are also obtained

spectral measures of bipartivity in complex networks we introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity it is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks trophic interactions in food webs construction principles in metabolic networks or communities in social networks

a study on q appell polynomials from determinantal point of view this research is aimed to give a determinantal definition for the q appell polynomials and show some classical properties as well as find some interesting properties of the mentioned polynomials in the light of the new definition

binomial convolution and transformations of appell polynomials abstract we show that the set a of appell sequences is an abelian group under the binomial convolution this is essentially equivalent to other approaches considered in the literature in particular the determinantal approach we also define a scale transformation t a a which is an isomorphism and a transformation r y a a based on expectations with respect to a random variable y using these tools we give explicit formulas for the determinantal form of appell sequences as well as for the formula of representation of powers and the srivastava pinter addition theorem various illustrative examples mainly referring to various generalizations of bernoulli and euler polynomials are discussed in detail

a matrix approach to sheffer polynomials abstract this paper deals with a unified matrix representation for the sheffer polynomials the core of the proposed approach is the so called creation matrix a special subdiagonal matrix having as nonzero entries positive integer numbers whose exponential coincides with the well known pascal matrix in fact sheffer polynomials may be expressed in terms of two matrices both connected to it as we will show one of them is strictly related to appell polynomials while the other is linked to a binomial type sequence consequently different types of sheffer polynomials correspond to different choices of these two matrices

certain results for the 2 variable apostol type and related polynomials in this article the 2 variable general polynomials are taken as base with apostol type polynomials to introduce a family of 2 variable apostol type polynomials these polynomials are framed within the context of monomiality principle and their properties are established certain summation formulae for these polynomials are also derived examples of some members belonging to this family are considered and numbers related to some mixed special polynomials are also explored

efficient sampling of complex network with modified random walk strategies abstract we present two novel random walk strategies choosing seed node csn random walk and no retracing nr random walk different from the classical random walk sampling the csn and nr strategies focus on the influences of the seed node choice and path overlap respectively three random walk samplings are applied in the erdos renyi er barabasi albert ba watts strogatz ws and the weighted usair networks respectively then the major properties of sampled subnets such as sampling efficiency degree distributions average degree and average clustering coefficient are studied the similar conclusions can be reached with these three random walk strategies firstly the networks with small scales and simple structures are conducive to the sampling secondly the average degree and the average clustering coefficient of the sampled subnet tend to the corresponding values of original networks with limited steps and thirdly all the degree distributions of the subnets are slightly biased to the high degree side however the nr strategy performs better for the average clustering coefficient of the subnet in the real weighted usair networks some obvious characters like the larger clustering coefficient and the fluctuation of degree distribution are reproduced well by these random walk strategies

spatiotemporal analysis of complex signals theory and applications we present a space time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos this permits the introduction of several characteristics of the signal three characteristic energies and entropies one temporal one spatial and one global and a characteristic dimension although the technique is general we concentrate on its applications to hydrodynamic problems specifically the transition to turbulence we consider two cases of application a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk in the latter we show a direct relation between the global entropy and the different instabilities that the flow undergoes as reynolds number increases

a determinantal approach to appell polynomials a new definition by means of a determinantal form for appell 1880 1 polynomials is given general properties some of them new are proved by using elementary linear algebra tools finally classic and non classic examples are considered and the coefficients calculated by an ad hoc mathematica code for particular sequences of appell polynomials are given

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

some relationships between the apostol bernoulli and apostol euler polynomials recently srivastava and pinter 1 investigated several interesting properties and relationships involving the classical as well as the generalized or higher order bernoulli and euler polynomials they also showed among other things that the main relationship proven earlier by cheon 2 can easily be put in a much more general setting the main object of the present sequel to these earlier works is to derive several general properties and relationships involving the apostol bernoulli and apostol euler polynomials some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the generalized bernoulli and generalized euler polynomials other relationships associated with the stirling numbers of the second kind are also considered

on a generalization of hermite polynomials abstract we consider a new generalization of hermite polynomials to the case of several variables our construction is based on an analysis of the generalized eigenvalue problem for the operator a x d acting on a linear space of polynomials of n variables where a is an endomorphism of the euclidean space r n and d is a second order differential operator our main results describe a basis for the space of hermite jordan polynomials

2 variable laguerre matrix polynomials and lie algebraic techniques the authors introduce 2 variable forms of laguerre and modified laguerre matrix polynomials and derive their special properties further the representations of the special linear lie algebra sl 2 and the harmonic oscillator lie algebra g 0 1 are used to derive certain results involving these polynomials furthermore the generating relations for the ordinary as well as matrix polynomials related to these matrix polynomials are derived as applications

random walks along the streets and canals in compact cities spectral analysis dynamical modularity information and statistical mechanics different models of random walks on the dual graphs of compact urban structures are considered analysis of access times between streets helps to detect the city modularity the statistical mechanics approach to the ensembles of lazy random walkers is developed the complexity of city modularity can be measured by an informationlike parameter which plays the role of an individual fingerprint of genius loci global structural properties of a city can be characterized by the thermodynamic parameters calculated in the random walk problem

analytical solutions for heat diffusion beyond fourier law we obtain solutions for differential equations describing a broad range of physical problems by the operational method with recourse to inverse differential operators integral transforms and operational exponent generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions the evolutional type problems for heat transfer in various heat conduction models are studied exact analytical solutions for guyer krumhansl hyperbolic heat equation are obtained and compared with those of fourier and cattaneo equations modelling heat pulse propagation from a laser source is performed in the framework of fourier cattaneo and guyer krumhansl heat transfer models compliance of obtained solutions with the maximum principle is studied

operational methods and laguerre gould hopper polynomials abstract in this paper the authors introduce the laguerre gould hopper polynomials by combining the operational methods with the principle of monomiality generating function series definition differential equation and certain other properties of laguerre gould hopper polynomials are derived further operational representations of these polynomials are established which are used to get integral representations and expansion formulae for these polynomials

laguerre type exponentials and generalized appell polynomials general classes of two variables appell polynomials are introduced by exploiting properties of an iterated isomorphism related to the so called laguerre type exponentials further extensions to the multi index and multivariable cases are mentioned

comments on monomiality ordinary polynomials and associated bi orthogonal functions we use laguerre and hermite polynomials to show that the monomiality principle can be exploited to study the properties of the polynomials and of the associated biorthogonal functions

orthogonality properties of the hermite and related polynomials the authors present a general method of operational nature with a view to investigating the orthogonality properties of several different families of the hermite and related polynomials in particular the classical hermite polynomials and some of their higher order and multi index generalizations are considered here

product of two hypergeometric functions with power arguments abstract the hypergeometric product formula describes the product of two hypergeometric functions with arguments xt and yt respectively the result is a power series in t whose coefficients are hypergeometric polynomials brychkov has extended the product formula such that in one argument the variable t can be squared we obtain a further extension such that in both arguments the variable t can be raised to positive integer powers

the network of commuters in london abstract we have studied the directed and weighted network in which the wards of london are vertices and two vertices are connected whenever there is at least one person commuting to work from one ward to another remarkably the in strength and in degree distribution tail is a power law with exponent around 2 while the out strength and out degree distribution tail is exponential we propose a simple square lattice model to explain the observed empirical behaviour

summation formulae for gould hopper generalized hermite polynomials in this article we derive summation formulae for the gould hopper generalized hermite polynomials g n m x y further we derive the summation formulae for polynomials related to g n m x y as applications furthermore we establish a summation formula for m variable generalized hermite polynomials h n 1 2 m x 1 x 2 x m

2 iterated appell polynomials and related numbers this article deals with the introduction of appell type sets of polynomials and associated numbers the 2 iterated appell polynomials are introduced and the bernoulli and euler based appell polynomials are deduced as their particular cases different sets of polynomials namely 2 iterated bernoulli and euler bernoulli euler or euler bernoulli and their related numbers are considered the operational relations between these families and appell polynomials are used on the results of the bernoulli and euler polynomials to obtain the results for the corresponding mixed polynomials

operational solution for some types of second order differential equations and for relevant physical problems abstract we present an operational method to obtain solutions for differential equations describing a broad range of physical problems including ordinary non integer order and high order partial differential equations inverse differential operators are proposed to solve a variety of differential equations integral transforms and the operational exponent are used to obtain the solutions generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions examples of solutions of physical problems related to propagation of the heat and other quantities are demonstrated by the developed operational technique in particular the evolution type problems the generalizations of the black scholes of the heat conduction of the fokker planck equations are considered as well as equations involving the laguerre derivative operator

a mathematical analysis of the effects of hebbian learning rules on the dynamics and structure of discrete time random recurrent neural networks we present a mathematical analysis of the effects of hebbian learning in random recurrent neural networks with a generic hebbian learning rule including passive forgetting and different timescales for neuronal activity and learning dynamics previous numerical work has reported that hebbian learning drives the system from chaos to a steady state through a sequence of bifurcations here we interpret these results mathematically and show that these effects involving a complex coupling between neuronal dynamics and synaptic graph structure can be analyzed using jacobian matrices which introduce both a structural and a dynamical point of view on neural network evolution furthermore we show that sensitivity to a learned pattern is maximal when the largest lyapunov exponent is close to 0 we discuss how neural networks may take advantage of this regime of high functional interest

efficiency of informational transfer in regular and complex networks we analyze the process of informational exchange through complex networks by measuring network efficiencies aiming to study nonclustered systems we propose a modification of this measure on the local level we apply this method to an extension of the class of small worlds that includes declustered networks and show that they are locally quite efficient although their clustering coefficient is practically zero unweighted systems with small world and scale free topologies are shown to be both globally and locally efficient our method is also applied to characterize weighted networks in particular we examine the properties of underground transportation systems of madrid and barcelona and reinterpret the results obtained for the boston subway network

generalized polynomials and associated operational identities we use operational identities to introduce multivariable laguerre polynomials we explore the wealth of differential equations they satisfy we analyze their properties and the link with legendre type polynomials

some results on quasi monomiality a polynomial set p n n 0 is called quasi monomial if and only if it is possible to define two operators p and m independent of n such thatp p n x np n 1 x andm p n x p n 1 x in this paper we show that every polynomial set is quasi monomial and we present some useful tools to explicitly express the p and m operators for some polynomial families given by their generating functions the obtained results are applied to boas buck polynomial sets

some generalizations of the apostol bernoulli and apostol euler polynomials abstract the main object of this paper is to give analogous definitions of apostol type see t m apostol on the lerch zeta function pacific j math 1 1951 161 167 and h m srivastava some formulas for the bernoulli and euler polynomials at rational arguments math proc cambridge philos soc 129 2000 77 84 for the so called apostol bernoulli numbers and polynomials of higher order we establish their elementary properties derive several explicit representations for them in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical bernoulli numbers and polynomials of higher order

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind abstract recently the authors introduced some generalizations of the apostol bernoulli polynomials and the apostol euler polynomials see q m luo h m srivastava j math anal appl 308 2005 290 302 and q m luo taiwanese j math 10 2006 917 925 the main object of this paper is to investigate an analogous generalization of the genocchi polynomials of higher order that is the so called apostol genocchi polynomials of higher order for these generalized apostol genocchi polynomials we establish several elementary properties provide some explicit relationships with the apostol bernoulli polynomials and the apostol euler polynomials and derive various explicit series representations in terms of the gaussian hypergeometric function and the hurwitz or generalized zeta function we also deduce their special cases and applications which are shown here to lead to the corresponding results for the genocchi and euler polynomials of higher order by introducing an analogue of the stirling numbers of the second kind that is the so called stirling numbers of the second kind we derive some basic properties and formulas and consider some interesting applications to the family of the apostol type polynomials furthermore we also correct an error in a previous paper q m luo h m srivastava comput math appl 51 2006 631 642 and pose two open problems on the subject of our investigation

markov chain methods for analyzing urban networks complex transport networks abstracted as graphs undirected directed or multi component can be effectively analyzed by random walks or diffusions we have unified many concepts into one framework and studied in details the structural and spectral properties of spatial graphs for five compact urban patterns

multivariable lagrange expansion and generalization of carlitz srivastava mixed generating functions abstract families of mixed generating functions generalizing those of the carlitz srivastava type are derived here by applying methods based on the multivariable extension of the lagrange expansion it is also shown that the combination with techniques of operational nature offers a wide flexibility to explore a wealth of mixed bilateral generating functions for special functions with many variables

some results on generalized szasz operators involving sheffer polynomials abstract in this paper we consider old and new operators of szasz type involving sheffer polynomials we present an asymptotic expansion formula for operators of ismail type then in order to improve the accuracy of the approximation of a function f in a fixed point we apply a well known extrapolation algorithm we also introduce some new special sequences of appell and sheffer polynomials and construct new generalized szasz type operators by using classical techniques we investigate approximation properties and rate of convergence for these operators and compare the results with other existing operators finally we present numerical examples which confirm the validity of the theoretical analysis and the effectiveness of the presented operators

generalized polynomials operational identities and their applications abstract it is shown that an appropriate combination of methods relevant to generalized operational calculus and to special functions can be a very useful tool to treat a large body of problems both in physics and mathematics we discuss operational methods associated with multivariable hermite laguerre legendre and other polynomials to derive a wealth of identities useful in quantum mechanics electromagnetism optics etc or to derive new identities between special functions as e g mehler or mixed type generating functions

some properties of hermite based sheffer polynomials abstract in this paper the concepts and the formalism associated with monomiality principle and sheffer sequences are used to introduce family of hermite based sheffer polynomials some properties of hermite sheffer polynomials are considered further an operational formalism providing a correspondence between sheffer and hermite sheffer polynomials is developed furthermore this correspondence is used to derive several new identities and results for members of hermite sheffer family

a new approach to legendre truncated exponential based sheffer sequences via riordan arrays abstract the significance of multi variable special polynomials has been identified both in mathematical and applied frameworks the article aims to focus on a new class of 3 variable legendre truncated exponential based sheffer sequences and to investigate their properties by means of riordan array techniques the quasi monomiality of these sequences is studied within the context of riordan arrays these sequences are expressed in determinant forms by utilizing the relation between the sheffer sequences and riordan arrays certain special sheffer sequences are used to construct the legendre truncated exponential based chebyshev polynomials the same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials the shapes are shown and zeros are computed for these sequences by using mathematical software the main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations

a determinantal approach to sheffer sequences abstract in this paper by using the theory of riordan arrays and the relations between sheffer sequences and riordan arrays we give a determinantal definition for sheffer sequences based on this new definition some general properties of sheffer sequences are reproved and the determinantal representations of some well known sheffer sequences are presented

operational identities and properties of ordinary and generalized special functions the theory of hermite laguerre and of the associated generating functions is reformulated within the framework of an operational formalism this point of view provides more efficient tools which allow the straightforward derivation of a wealth of new and old identities in this paper a central role is played by negative derivative operators and by their link with the tricomi functions and the generalized laguerre polynomials

differential equation of appell polynomials via the factorization method let pn x n 0 be a sequence of polynomials of degree n we define two sequences of differential operators n and n satisfying the following properties n pn x pn 1 x n pn x pn 1 x by constructing these two operators for appell polynomials we determine their differential equations via the factorization method introduced by infeld and hull rev mod phys 23 1951 21 the differential equations for both bernoulli and euler polynomials are given as special cases of the appell polynomials

intelligibility and first passage times in complex urban networks topology of urban environments can be represented by means of graphs we explore the graph representations of several compact urban patterns by random walks the expected time of recurrence and the expected first passage time to a node scales apparently linearly in all urban patterns we have studied in space syntax theory a positive relation between the local property of a node qualified by connectivity or by the recurrence time and the global property of the node estimated in our approach by the first passage time to it is known as intelligibility our approach based on random walks allows us to extend the notion of intelligibility onto the entire domain of complex networks and graph theory

some properties of legendre gould hopper polynomials and operational methods abstract in this paper the author introduces the legendre gould hopper polynomials by combining the operational methods with the principle of monomiality generating functions series definition differential equation and certain other properties of legendre gould hopper polynomials are derived further operational representations of these polynomials are established which are used to get integral representations and expansion formulae certain summation formulae for these polynomials are also obtained

the relation of the d orthogonal polynomials to the appell polynomials abstract we are dealing with the concept of d dimensional orthogonal abbreviated d orthogonal polynomials that is to say polynomials verifying one standard recurrence relation of order d 1 among the d orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials in particular we are concerned in the present paper with the solution of the following problem p find all polynomial sequences which are at the same time appell polynomials and d orthogonal the resulting polynomials are a natural extension of the hermite polynomials a sequence of these polynomials is obtained all the elements of its d 1 order recurrence are explicitly determined a generating function a d 1 order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given among such polynomials one singles out the d symmetrical ones definition 1 7 which are the d orthogonal polynomials analogous to the hermite classical ones when d 1 ordinary orthogonality we meet again the classical orthogonal polynomials of hermite

some series identities involving the generalized apostol type and related polynomials a unification and generalization of various apostol type polynomials was introduced and investigated recently by luo and srivastava q m luo h m srivastava some generalizations of the apostol genocchi polynomials and the stirling numbers of the second kind appl math comput 217 2011 5702 5728 in this paper we prove several symmetry identities for these generalized apostol type polynomials by using their generating functions as special cases and consequences of our results we obtain the corresponding symmetry identities for the apostol euler polynomials of higher order the apostol bernoulli polynomials of higher order and the apostol genocchi polynomials of higher order and also for another family of generalized apostol type polynomials which were investigated systematically by ozden et al h ozden y simsek h m srivastava a unified presentation of the generating functions of the generalized bernoulli euler and genocchi polynomials comput math appl 60 2010 2779 2787 we also derive several relations between the apostol type polynomials the generalized sum of integer powers and the generalized alternating sum it is shown how each of these results would extend the corresponding known identities

asymptotic estimates for apostol bernoulli and apostol euler polynomials we analyze the asymptotic behavior of the apostol bernoulli polynomials bn x in detail the starting point is their fourier series on 0 1 which it is shown remains valid as an asymptotic expansion over compact subsets of the complex plane this is used to determine explicit estimates on the constants in the approximation and also to analyze oscillatory phenomena which arise in certain cases these results are transferred to the apostol euler polynomials en x via a simple relation linking them to the apostol bernoulli polynomials

unified apostol bernoulli euler and genocchi polynomials in this paper we present a unified family of polynomials including not only the apostol bernoulli euler and genocchi polynomials but also a general family of polynomials suggested by ozden et al h ozden y simsek h m srivastava a unified presentation of the generating functions of the generalized bernoulli euler and genocchi polynomials comput math appl 60 10 2010 2779 2787 we obtain the explicit representation of this unified family in terms of a gaussian hypergeometric function some symmetry identities and multiplication formula are also given

modularity and extreme edges of the internet we study the spectral properties of a diffusion process taking place on the internet network focusing on the slowest decaying modes these modes identify an underlying modular structure roughly corresponding to individual countries for instance in the slowest decaying mode the diffusion current flows from russia to u s military sites quantitatively the modular structure manifests itself in a 10 times larger participation ratio of its slow decaying modes compared to a random scale free network we propose to use the fraction of nodes participating in slow decaying modes as a general measure of the modularity of a network for the 100 slowest decaying modes of the internet this fraction is 30 finally we suggest that the degree of isolation of an individual module can be assessed by comparing its participation in different diffusion modes

letter to the editor remarks on differential equation of appell polynomials we show that a recent result of he and ricci j comp appl math 139 2002 231 on differential equations for appell polynomials is valid for all polynomial sequences

recurrence relations for the sheffer sequences abstract in this paper using the production matrix of an exponential riordan array g t f t we give a recurrence relation for the sheffer sequence for the ordered pair g t f t we also develop a new determinant representation for the general term of the sheffer sequence as applications determinant expressions for some classical sheffer polynomial sequences are derived

fourier expansions and integral representations for the apostol bernoulli and apostol euler polynomials we investigate fourier expansions for the apostol bernoulli and apostol euler polynomials using the lipschitz summation formula and obtain their integral representations we give some explicit formulas at rational arguments for these polynomials in terms of the hurwitz zeta function we also derive the integral representations for the classical bernoulli and euler polynomials and related known results

generalized and mixed type gegenbauer polynomials abstract in this paper a new generalized form of the gegenbauer polynomials is introduced by using the integral representation method further the hermite gegenbauer and the laguerre gegenbauer polynomials are introduced by using the operational identities associated with the generalized hermite and laguerre polynomials of two variables

determination of the meter of musical scores by autocorrelation a single voice extracted from the scores of compositions from the baroque classical romantic and contemporary periods has been studied in order to determine the feasibility of the determination of musical meter by computer the method of autocorrelation is appropriate for this calculation since it is a measure of the frequency of occurrence of events following an event at time zero if a greater frequency of events occurs on the downbeat of a measure as predicted by palmer and krumhansl mental representations for musical meter j exp psychol hum percept perform 16 728 741 1990 then a peak in the autocorrelation function should indicate the time for a single measure the results of these computations indicate that computer determination of meter from score events is indeed possible an example is included to show that this method of analysis can be applied to live performance data as well

symbolic operational images and decomposition formulas for hypergeometric functions based upon the classical derivative and integral operators we introduce a new operator which allows the derivation of new symbolic operational images for hypergeometric functions by means of these symbolic operational images a number of decomposition formulas involving quadruple series are then found other closely related results are also considered