quadratures and orthogonality associated with the cayley transform abstract in this paper we are dealing with the approximate calculation of weighted integrals over the whole real line the method is based in passing to the unit circle by means of the so called cayley transform z i x i x and then making use of a szeg or interpolatory type quadrature formula on the unit circle in order to obtain a gauss like quadrature rule on the real line some properties concerning orthogonality maximal domains of validity of the quadratures and connections with certain orthogonal rational functions are presented finally some numerical experiments are also carried out

hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

a new framework for multivariate general conformable fractional calculus and potential applications abstract some applications of fractional partial differential equations fpdes based on conformable fractional derivative cfd are discussed recently as an extension of cfd power kernel is generalized to other normalized probability distribution kernels general conformable fractional derivative gcfd provides a framework to explain the physical meaning of cfd and describe which phenomena or processes gcfd cfd are suitable to model we will establish the multivariate theory of gcfd in this paper compared with traditional hereditary fractional calculus like riemann liouville type or caputo type the unified theory of gcfd vector calculus has well forms with simplicity and elegance as applications fractional maxwell s equations of gcfd are given to describe electromagnetic fields of media that demonstrate inhomogeneous complicated and local properties and some fractional parabolic equations are presented as natural extensions of some nonlinear parabolic equations that cover several important famous equations

a q umbral calculus abstract an algebraic setting for the roman rota umbral calculus is introduced it is shown how many of the umbral calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map the same approach is used to construct a q umbral calculus our umbral calculus yields some of andrews recent results on eulerian families of polynomials as corollaries the homogeneous eulerian families are studied operator and functional expansions are also included

understanding interarrival and interdeparture time statistics from interactions in queuing systems abstract we discuss the statistics of long queues in which the interdeparture time statistics is dominated by spatial interactions among the elements in a queue rather than the arrival or exit processes based on a fokker planck approach it is possible to calculate the stationary distance distribution among the elements in a queue as a function of their interaction potential the results relate to the ones known from random matrix theory together with the velocity distribution one can determine the time gap distribution as well this yields an analytical approach to the interdeparture and interarrival time distributions of queuing systems with spatially interacting elements while these distributions are usually determined from empirical data or from theoretical assumptions about the arrival or exit process we offer here an alternative interpretation of interdeparture time distributions as an effect of interactions in a queue this is relevant for the understanding of traffic and production systems and for the optimization of the statistical behavior of some queuing systems

on conformable fractional calculus recently the authors khalil et al 2014 introduced a new simple well behaved definition of the fractional derivative called conformable fractional derivative in this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus the fractional versions of chain rule exponential functions gronwall s inequality integration by parts taylor power series expansions laplace transforms and linear differential systems are proposed and discussed

conformable derivative approach to anomalous diffusion abstract by using a new derivative with fractional order referred to conformable derivative an alternative representation of the diffusion equation is proposed to improve the modeling of anomalous diffusion the analytical solutions of the conformable derivative model in terms of gauss kernel and error function are presented the power law of the mean square displacement for the conformable diffusion model is studied invoking the time dependent gauss kernel the parameters related to the conformable derivative model are determined by levenberg marquardt method on the basis of the experimental data of chloride ions transportation in reinforced concrete the data fitting results showed that the conformable derivative model agrees better with the experimental data than the normal diffusion equation furthermore the potential application of the proposed conformable derivative model of water flow in low permeability media is discussed

distributed gaussian polynomials as q oscillator eigenfunctions karabulut and sibert j math phys 38 4815 1997 have constructed an orthogonal set of functions from linear combinations of equally spaced gaussians in this paper we show that they are actually eigenfunctions of a q oscillator in coordinate representation we also reinterpret the coordinate representation example of q oscillator given by macfarlane as the functions orthogonal with respect to an unusual inner product definition it is shown that the eigenfunctions in both q oscillator examples are infinitely degenerate

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

on the zeros of the q analogue exponential function an asymptotic formula for the zeros zn of the entire function eq x for q 1 is obtained as q increases above the first collision point at q1 approximately 0 14 these zeros collide in pairs and then move off into the complex z plane they move off as and remain a complex conjugate pair the zeros of the ordinary higher derivatives and of the ordinary indefinite integrals of eq x vary with q in a similar manner properties of eq z for z complex and for arbitrary q are deduced for 0 or q 1 eq z is an entire function of order 0 by the hadamard weierstrass factorization theorem infinite product representations are obtained for eq z and for the reciprocal function eq 1 z if q not 1 the zeros satisfy the sum rule sigma n 1infinity 1 zn 1

an algebraic interpretation of the continuous big q hermite polynomials the continuous big q hermite polynomials are shown to realize a basis for a representation space of an extended q oscillator algebra an expansion formula is algebraically derived using this model

fractional time quantum dynamics application of the fractional calculus to quantum processes is presented in particular the quantum dynamics is considered in the framework of the fractional time schr odinger equation se which differs from the standard se by the fractional time derivative ensuremath partial ensuremath partial t ensuremath rightarrow ensuremath partial ensuremath alpha ensuremath partial t ensuremath alpha it is shown that for ensuremath alpha 1 2 the fractional se is isospectral to a comb model an analytical expression for the green s functions of the systems are obtained the semiclassical limit is discussed

the bivariate rogers szego polynomials we present an operator approach to deriving mehler s formula and the rogers formula for the bivariate rogers szego polynomials hn x y q the proof of mehler s formula can be considered as a new approach to the nonsymmetric poisson kernel formula for the continuous big q hermite polynomials hn x a q due to askey rahman and suslov mehler s formula for hn x y q involves a 32 sum and the rogers formula involves a 21 sum the proofs of these results are based on parameter augmentation with respect to the q exponential operator and the homogeneous q shift operator in two variables by extending recent results on the rogers szego polynomials hn x q due to hou lascoux and mu we obtain another rogers type formula for hn x y q finally we give a change of base formula for hn x a q which can be used to evaluate some integrals by using the askey wilson integral

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

ramanujan type continuous measures for biorthogonal q rational functions it is shown that ramanujan type continuous measures for a hierarchy of biorthogonal rational functions can be systematically built from simple cases of the rogers szego for 0 1 polynomials by using the berg ismail procedure of attaching generating functions to measures

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

universality classes for self avoiding walks in a strongly disordered system we study the behavior of self avoiding walks saws on square and cubic lattices in the presence of strong disorder we simulate the disorder by assigning random energy e taken from a probability distribution p e t o each site or bond of the lattice we study the strong disorder limit for an extremely broad range of energies with p e 1 e for each configuration of disorder we find by exact enumeration the optimal saw of fixed length n and fixed origin that minimizes the sum of the energies of the visited sites or bonds we find the fractal dimension of the optimal path to be d opt51 5260 10 in two dimensions 2d and d opt51 8260 08 in 3d our results imply that saws in strong disorder with fixed n are much more compact than saws in disordered media with a uniform distribution of energies optimal paths in strong disorder with fixed end toend distance r and saws on a percolation cluster our results are also consistent with the possibility that saws in strong disorder belong to the same universality class as the maximal saw on a percolation cluster at criticality for which we calculate the fractal dimension d max51 6460 02 for 2d and d max51 8760 05 for 3d values very close to the fractal dimensions of the percolation backbone in 2d and 3d

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

on a general q fourier transformation with nonsymmetric kernels wiener used the poisson kernel for the hermite polynomials to deal with the classical fourier transform askey atakishiyev and suslov used this approach to obtain a q fourier transform by using the continuous q hermite polynomials rahman and suslov extended this result by taking the askey wilson polynomials considered to be the most general continuous classical orthogonal polynomials the theory of q fourier transformation is further extended here by considering a nonsymmetric version of the poisson kernel with askey wilson polynomials this approach enables us to obtain some new results for example the complex and real orthogonalities of these kernels

quadrature formulas associated with rogers szegő polynomials this paper is concerned with the estimation of integrals of email protected functions with respect to the hermite weight function by passing to the unit circle of the complex plane and considering szego and interpolatory type quadrature formulas with respect now to the rogers szego weight function several numerical experiments are given along with estimations of the error

properties of the katugampola fractional derivative with potential application in quantum mechanics katugampola e print arxiv 1410 6535 recently introduced a limit based fractional derivative d referred to in this work as the katugampola fractional derivative that maintains many of the familiar properties of standard derivatives such as the product quotient and chain rules typically fractional derivatives are handled using an integral representation and as such are non local in character the current work starts with a key property of the katugampola fractional derivative d y t1 dydt and the associated differential operator d t1 d1 these operators their inverses commutators anti commutators and several important differential equations are studied the anti commutator serves as a basis for the development of a self adjoint operator which could potentially be useful in quantum mechanics a hamiltonian is constructed from this operator and applied to the particle in a box model

on lifting q difference operators in the askey scheme of basic hypergeometric polynomials we construct a q difference operator that lifts the continuous q hermite polynomials hn x q of rogers up to the continuous big q hermite polynomials hn x a q on the next level in the askey scheme of basic hypergeometric polynomials this operator is defined as exton s q exponential function eq aqdq in terms of the askey wilson divided q difference operator dq and it represents a particular q extension of the standard shift operator we next show that one can move two steps more upwards in order first to reach the al salam chihara family of polynomials qn x a b q and then the continuous dual q hahn polynomials pn x a b c q in both these cases lifting operators respectively turn out to be convolution type products of two and three one parameter q difference operators of the same type eq aqdq at the initial step at each step we also determine q difference operators that lift the weight function for the continuous q hermite polynomials hn x q successively up to the weight functions for hn x a q qn x a b q and pn x a b c q

algebraic properties of rogers szego functions i applications in quantum optics by means of a well established algebraic framework rogers szego functions associated with a circular geometry in the complex plane are introduced in the context of q special functions and their properties are discussed in detail the eigenfunctions related to the coherent and phase states emerge from this formalism as infinite expansions of rogers szego functions the coefficients being determined through proper eigenvalue equations in each situation furthermore a complementary study on the robertson schrodinger and symmetrical uncertainty relations for the cosine sine and nondeformed number operators is also conducted corroborating in this way certain features of q deformed coherent states

maximum entropy formalism fractals scaling phenomena and 1 f noise a tale of tails in this report on examples of distribution functions with long tails we a show that the derivation of distributions with inverse power tails from a maximum entropy formalism would be a consequence only of an unconventional auxilliary condition that involves the specification of the average value of a complicated logarithmic function b review several models that yield log normal distributions c show that log normal distributions may mimic 1 f noise over a certain range and d present an amplification model to show how log normal personal income distributions are transformed into inverse power pareto distributions in the high income range

on the mellin transforms of hypergeometric polynomials mellin integral transform pairs for all hypergeometric orthogonal polynomials in the askey scheme ranging from the classical hermite polynomials up to the four parameter wilson polynomials are systematically discussed

the metaplectic representation of suq 1 1 and the q gegenbauer polynomials the metaplectic representation of the quantum algebra suq 1 1 is shown to provide a group theoretic setting for certain basic orthogonal polynomials generalizing the usual gegenbauer polynomials

diagonalizaton of certain integral operators ii abstract we establish an integral representation of a right inverse of the askey wilson finite difference operator in an l 2 space weighted by the weight function of the continuous q jacobi polynomials we characterize the eigenvalues of this integral operator and prove a q analog of the expansion of e ixy in jacobi polynomials of argument x we also outline a general procedure of finding integral representations for inverses of linear operators

fractional newton mechanics with conformable fractional derivative in this paper we use the conformable fractional derivative and integral to discuss the fractional newtonian mechanics the fractional version of the calculus of variations is introduced and the fractional euler lagrange equation is constructed some mechanical problems such as fractional harmonic oscillator problem the fractional damped oscillator problem and the forced oscillator problem are discussed in the one dimensional fractional dynamics

on the coherent states for the q hermite polynomials and related fourier transformation we discuss the fourier gauss transformation properties of the continuous q hermite polynomials and associated q coherent states

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

q laguerre and wall polynomials are related by the fourier gauss transform it is shown that the q laguerre and wall or little q laguerre polynomials are interrelated by the fourier gauss transform in the limit when the degree of these polynomials tends to infinity this integral transform provides the relation between jackson s second and third q bessel functions

fourier gauss transforms of the al salam chihara polynomials we discuss classical fourier gauss transforms of a three parameter family of the continuous al salam chihara polynomials it is shown that they are related to both the continuous big q hermite and the q hermite polynomials

a q deformed e 4 and continuous q jacobi polynomials we present an algebraic interpretation of the continuous q jacobi polynomials they are shown to realize a basis for a representation space of a q deformation of the four dimensional euclidean algebra e 4 a q analog of the expansion of an exponential in jacobi polynomials is naturally obtained from this model

on the fourier gauss transforms of some q exponential and q trigonometric functions we examine the properties of q exponential and q trigonometric functions recently introduced and discussed in the literature it is shown that they are related to jackson s q analogues of the exponential and trigonometric functions by classical fourier gauss transformations

an operator approach to the al salam carlitz polynomials we present an operator approach to rogers type formulas and mehler s formula for the al salam carlitz polynomials un x y a q by using the q exponential operator we obtain a rogers type formula which leads to a linearization formula with the aid of a bivariate augmentation operator we get a simple derivation of mehler s formula due to al salam and carlitz some orthogonal q polynomials math nachr 30 47 1965 by means of the cauchy companion augmentation operator we obtain an equivalent form of mehler s formula we also give several identities on the generating functions for products of the al salam carlitz polynomials which are extensions of the formulas for the rogers szego polynomials

time fractional schrodinger equation the schrodinger equation is considered with the first order time derivative changed to a caputo fractional derivative the time fractional schrodinger equation the resulting hamiltonian is found to be non hermitian and nonlocal in time the resulting wave functions are thus not invariant under time reversal the time fractional schrodinger equation is solved for a free particle and for a potential well probability and the resulting energy levels are found to increase over time to a limiting value depending on the order of the time derivative new identities for the mittag leffler function are also found and presented in an appendix

symmetry techniques for the al salam chihara polynomials an algebraic interpretation for the al salam chihara q orthogonal polynomials is presented using an explicit realization in terms of divided difference operators their symmetry q algebra is determined and studied this algebraic structure allows one to derive a new expansion formula for these polynomials

on a one parameter family of q exponential functions we examine the properties of a family of q exponential functions which depend on an extra parameter these functions have a well defined meaning for both the 0 1 cases if only it is shown that any two members of this family with different values of the parameter are related to each other by a fourier gauss transformation

on compact representations for the solutions of linear difference equations with variable coefficients a comprehensive treatment on compact representations for the solutions of linear difference equations with variable coefficients of both n th and unbounded order is presented the equivalence between their celebrated combinatorial and determinantal representations is considered a corresponding representation is proposed using determined nested sums of their variable coefficients it makes explicit all the sum of products involved in the previous representations of such solutions some basic applications are also illustrated

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

q μ and p q ζ exponential functions rogers szegő polynomials and fourier gauss transform from the realization of q oscillator algebra in terms of generalized derivative we compute the matrix elements from deformed exponential functions and deduce generating functions associated with rogers szeg polynomials as well as their relevant properties we also compute the matrix elements associated with the p q oscillator algebra a generalization of the q one and perform the fourier gauss transform of a generalization of the deformed exponential functions

two operator identities and their applications to terminating basic hypergeometric series and q integrals abstract in this paper we first give two interesting operator identities and then using them and the q exponential operator technique to some terminating summation formulas of basic hypergeometric series and q integrals we obtain some q series identities and q integrals involving 3 2

the covariant linear oscillator and generalized realization of the dynamical su 1 1 symmetry algebra an exactly solvable problem for the finite difference schrodinger equation in the relativistic configurational space is considered the appropriate finite difference generalization of the factorization method is developed the theory of new special functions the relativistic hermite polynomials in which the solutions are expressed is constructed

suq 1 1 and the relativistic oscillator it is shown that the generalization of the quantum harmonic oscillator to the case of the relativistic configurational space is a q oscillator the corresponding group of dynamical symmetry is the quantum group suq 1 1 the deformation parameter being q e omega omega 0 where omega 0 4mc2 and omega is a frequency of the oscillator the deformed creation and annihilation operators are finite difference ones the corresponding deformation of the heisenberg weyl group and new coherent states are also considered

probability distribution of the shortest path on the percolation cluster its backbone and skeleton we consider the mean distribution functions ensuremath phi r mathcal l ensuremath phi mathrm b r mathcal l and ensuremath phi mathrm s r mathcal l giving the probability that two sites on the incipient percolation cluster on its backbone and on its skeleton respectively connected by a shortest path of length l are separated by an euclidean distance r following a scaling argument due to de gennes for self avoiding walks we derive analytical expressions for the exponents g 1 d f d mathrm min ensuremath d and g 1 mathrm b g 1 mathrm s 3d mathrm min ensuremath d which determine the scaling behavior of the distribution functions in the limit x ensuremath equiv r mathcal l stackrel ifmmode tilde else fi ensuremath nu ensuremath ll 1 i e ensuremath phi r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 ensuremath phi mathrm b r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm b and ensuremath phi mathrm s r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm s with stackrel ifmmode tilde else fi ensuremath nu ensuremath equiv 1 d mathrm min where d f and d mathrm min are the fractal dimensions of the percolation cluster and the shortest path respectively the theoretical predictions for g 1 g 1 mathrm b and g 1 mathrm s are in very good agreement with our numerical results

more on the umbral calculus with emphasis on the q umbral calculus abstract the interrelationship between distinct umbral calculi is studied these ideas are applied in particular to q umbral calculus which shows how andrews q theory relates to the q theory discussed in the previous paper by the author the q hermite polynomials and basic hypergeometric series are briefly discussed

size of outbreaks near the epidemic threshold the spread of infectious diseases near the epidemic threshold is investigated scaling laws for the size and the duration of outbreaks originating from a single infected individual in a large susceptible population are obtained the maximal size of an outbreak n ensuremath ast scales as n 2 3 with n the population size this scaling law implies that the average outbreak size n scales as n 1 3 moreover the maximal and the average duration of an outbreak grow as t ensuremath ast ensuremath sim n 1 3 and t ensuremath sim text ln phantom rule 0 2em 0ex n respectively

problem of two coulomb centres at large intercentre separation asymptotic expansions from analytical solutions of the heun equation the case of large intercentre distance in the two coulomb centres problem is studied by solving separated wave equations with the help of a series of confluent hypergeometric functions by considering the confluence of two singularities in an auxiliary equation with four regular singularities new relations between the solutions of the quasi angular equation are found and used to obtain exponentially small terms in the asymptotic expansion for energy eigenvalues for some electronic states energy splittings at pseudocrossings are evaluated and results are compared with those of earlier asymptotic and numerical calculations

crossover from weak to strong disorder regime in the duration of epidemics abstract we study the susceptible infected recovered sir model in complex networks considering that not all individuals in the population interact in the same way this heterogeneity between contacts is modeled by a continuous disorder in our model the disorder represents the contact time or the closeness between individuals we find that the duration time of an epidemic has a crossover with the system size from a power law regime to a logarithmic regime depending on the transmissibility related to the strength of the disorder using percolation theory we find that the duration of the epidemic scales as the average length of the branches of the infection our theoretical findings supported by simulations explains the crossover between the two regimes

effects of epidemic threshold definition on disease spread statistics we study the statistical properties of sir epidemics in random networks when an epidemic is defined as only those sir propagations that reach or exceed a minimum size sc using percolation theory to calculate the average fractional size view the mathml source of an epidemic we find that the strength of the spanning link percolation cluster p is an upper bound to view the mathml source for small values of sc p is no longer a good approximation and the average fractional size has to be computed directly we find that the choice of sc is generally but not always guided by the network structure and the value of t of the disease in question if the goal is to always obtain p as the average epidemic size one should choose sc to be the typical size of the largest percolation cluster at the critical percolation threshold for the transmissibility we also study q the probability that an sir propagation reaches the epidemic mass sc and find that it is well characterized by percolation theory we apply our results to real networks dimes and tracerouter to measure the consequences of the choice sc on predictions of average outcome sizes of computer failure epidemics

numerical evaluation of the upper critical dimension of percolation in scale free networks we propose numerical methods to evaluate the upper critical dimension d c of random percolation clusters in erd ifmmode mbox h o else h o fi s r enyi networks and in scale free networks with degree distribution mathcal p k ensuremath sim k ensuremath ensuremath lambda where k is the degree of a node and ensuremath lambda is the broadness of the degree distribution our results support the theoretical prediction d c 2 ensuremath lambda ensuremath 1 ensuremath lambda ensuremath 3 for scale free networks with 3l ensuremath lambda l4 and d c 6 for erd ifmmode mbox h o else h o fi s r enyi networks and scale free networks with ensuremath lambda g4 when the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d c 6 for all ensuremath lambda g2 our method also yields a better numerical evaluation of the critical percolation threshold p c for scale free networks our results suggest that the finite size effects increases when ensuremath lambda approaches 3 from above

elements of q harmonic analysis abstract measures that act as q analogs of lebesgue measure are introduced some q analogs of fourier and laplace transforms are studied last the connection between certain spaces of q exponential type and discrete q fourier transforms is discussed

chern simons matrix models and stieltjes wigert polynomials employing the random matrix formulation of chern simons theory on seifert manifolds we show how the stieltjes wigert orthogonal polynomials are useful in exact computations in chern simons matrix models we construct a biorthogonal extension of the stieltjes wigert polynomials not available in the literature necessary to study chern simons matrix models when the geometry is a lens space we also study the relationship between stieltjes wigert and rogers szego polynomials and the corresponding equivalence with a unitary matrix model finally we give a detailed proof of a result that relates quantum dimensions with averages of schur polynomials in the stieltjes wigert ensemble

on peculiar properties of generating functions of some orthogonal polynomials we prove that for x t 1 1 q 1 and n 0 and hn x t q are respectively the so called q hermite and the big q hermite polynomials and q n denotes the so called q pochhammer symbol we prove similar equalities involving big q hermite and al salam chihara polynomials and al salam chihara and the so called continuous dual q hahn polynomials moreover we are able to relate in this way some other ordinary orthogonal polynomials such as e g hermite chebyshev or laguerre these equalities give a new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general i e involving more and more parameters families of orthogonal polynomials we pose some conjectures concerning askey wilson polynomials and their possible generalizations we prove that these conjectures are true for the cases q 1 classical case and q 0 free case thus paving the way to generalization of askey wilson polynomials at least in these two cases

a model for the continuous q ultraspherical polynomials an algebraic interpretation for two classes of continuous q polynomials is provided rogers continuous q hermite polynomials and continuous q ultraspherical polynomials are shown to realize respectively bases for representation spaces of the q heisenberg algebra and a q deformation of the euclidean algebra in these dimensions a generating function for the continuous q hermite polynomials and a q analog of the fourier gegenbauer expansion are naturally obtained from these models

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

more on the q oscillator algebra and q orthogonal polynomials properties of certain q orthogonal polynomials are connected to the q oscillator algebra the wall and q laguerre polynomials are shown to arise as matrix elements of q exponentials of the generators in a representation of this algebra a realization is presented where the continuous q hermite polynomials form a basis of the representation space various identities are interpreted within this model in particular the connection formula between the continuous big q hermite and the continuous q hermite polynomials is thus obtained and two generating functions for these last polynomials are derived algebraically

fourier gauss transforms of the continuous big q hermite polynomials we examine the fourier gauss transformation properties of a two parameter family of the continuous big q hermite polynomials

tribasic integrals and identities of rogers ramanujan type some integrals involving three bases are evaluated as infinite products using complex analysis many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products the resulting identities are identities of rogers ramanujan type some integer partition interpretations of these identities are given generalizations of the rogers ramanujan type identities involving polynomials are given again as corollaries of integral evaluations

fourier gauss transforms of the askey wilson polynomials we discuss classical fourier gauss integral transforms for a five parameter family of the askey wilson polynomials

the wigner function associated with the rogers szego polynomials a wigner function associated with the rogers szego polynomials is proposed and its properties are discussed it is shown that from such a wigner function it is possible to obtain well behaved probability distribution functions for both angle and action variables defined on the compact support and for m 0 respectively the width of the angle probability density is governed by the free parameter q characterizing the polynomials

theory of rumour spreading in complex social networks we introduce a general stochastic model for the spread of rumours and derive mean field equations that describe the dynamics of the model on complex social networks in particular those mediated by the internet we use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks random graphs uncorrelated scale free networks and scale free networks with assortative degree correlations we show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system in the case of scale free networks on the other hand this threshold becomes vanishingly small in the limit of infinite system size we find that the initial rate at which a rumour spreads is much higher in scale free networks than in random graphs and that the rate at which the spreading proceeds on scale free networks is further increased when assortative degree correlations are introduced the impact of degree correlations on the final fraction of nodes that ever hears a rumour however depends on the interplay between network topology and the rumour spreading rate our results show that scale free social networks are prone to the spreading of rumours just as they are to the spreading of infections they are relevant to the spreading dynamics of chain emails viral advertising and large scale information dissemination algorithms on the internet

on the rogers szego polynomials an orthogonality relation on the full real line for the rogers szego polynomials is discussed it is argued that fourier transformation with the standard exponential kernel exp ixy relates the rogers szego and stieltjes wigert functions

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

effect of disorder strength on optimal paths in complex networks we study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path opt in a disordered erd s renyi er random network and scale free sf network each link i is associated with a weight ti expsarid where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder we find that for any finitea there is a crossover network size n sad at which the transition occurs for n n sad the scaling behavior of opt is in the strong disorder regime with opt n 1 3 for er networks and for sf networks with lo 4 and opt n sl 3d sl 1d for sf networks with 3 l 4 for n n sad the scaling behavior is in the weak disorder regime with opt ln n for er networks and sf networks with l 3 in order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder we propose a scaling ansatz for this measure and demonstrate its validity we proceed to derive the scaling relation between n sad and a we find thatn sa d a 3 for er networks and for sf networks with lo 4 and n sa d a sl 1d sl 3d for sf networks with 3 l 4

fractional differential forms a generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders that is a fractional exterior derivative is defined this is found to generate new vector spaces of finite and infinite dimension fractional differential form spaces the definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case coordinate transformation rules are also computed the transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative the metric for the fractional form spaces is given based on the coordinate transformation rules all results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one

continuous hahn polynomials a slightly more general orthogonality relation for the hahn polynomials of a continuous variable than the recent one given by atakishiyev and suslov 1985 is given here

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

a new definition of fractional derivative we give a new definition of fractional derivative and fractional integral the form of the definition shows that it is the most natural definition and the most fruitful one the definition for email protected email protected 1 coincides with the classical definitions on polynomials up to a constant further if a 1 the definition coincides with the classical definition of first derivative we give some applications to fractional differential equations

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

the factorization of a q difference equation for continuous q hermite polynomials we argue that a customary q difference equation for the continuous qhermite polynomials hn x q can be written in the factorized form as d q 2 1hn x q q n 1 hn x q where d q is some explicitly known q difference operator this means that the polynomials hn x q are in fact governed by the q difference equation d q hn x q q n 2 hn x q which is simpler than the conventional one it is shown that a similar factorization holds for the continuous q 1 hermite polynomials hn x q

recursive formulae for the analytic solution of the nonlinear spatial conformable fractional evolution equation abstract nonlinear evolution equations nees are very important partial differential equations in nonlinear physics optics and engineering in this article we apply the residual power series rps method to the initial value problem of the nonlinear spatial conformable fractional evolution equations coefficients of the analytic solution are computed by recursive formulae and the famous nonlinear spatial conformable fractional fisher equation is taken as an example to verify this method direct computations demonstrate the correctness of the recursive formulae meanwhile the recursive formulae can be easily implemented by mathematical software

second look at the spread of epidemics on networks in an important paper newman phys rev e66 016128 2002 claimed that a general network based stochastic susceptible infectious removed sir epidemic model is isomorphic to a bond percolation model where the bonds are the edges of the contact network and the bond occupation probability is equal to the marginal probability of transmission from an infected node to a susceptible neighbor in this paper we show that this isomorphism is incorrect and define a semidirected random network we call the epidemic percolation network that is exactly isomorphic to the sir epidemic model in any finite population in the limit of a large population i the distribution of self limited outbreak sizes is identical to the size distribution of small out components ii the epidemic threshold corresponds to the phase transition where a giant strongly connected component appears iii the probability of a large epidemic is equal to the probability that an initial infection occurs in the giant in component and iv the relative final size of an epidemic is equal to the proportion of the network contained in the giant out component for the sir model considered by newman we show that the epidemic percolation network predicts the same mean outbreak size below the epidemic threshold the same epidemic threshold and the same final size of an epidemic as the bond percolation model however the bond percolation model fails to predict the correct outbreak size distribution and probability of an epidemic when there is a nondegenerate infectious period distribution we confirm our findings by comparing predictions from percolation networks and bond percolation models to the results of simulations in the appendix we show that an isomorphism to an epidemic percolation network can be defined for any time homogeneous stochastic sir model

diagonalization of certain integral operators abstract we study integral representations of formal inverses of the differential operator and the askey wilson divided difference operators in different polynomial bases the eigenvalues are related to zeros of bessel functions q bessel functions and confluent hypergeometric functions the eigenfunctions are exponential functions and in the case of the askey wilson operator are new types of q exponential functions we also found new q analogs of the sine and cosine functions

quantum symmetries of q difference equations a general method is presented to determine the symmetry operators of linear q difference equations it is applied to q difference analogs of the helmoltz heat and wave equations in diverse dimensions the symmetries are found to generate q deformations of classical lie algebras the method of separation of variables is used to obtain solutions which are seen to involve many basic special functions this allows the derivation of various identities and formulas for these q functions