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

self diffusion and visited surface in the droplet condensation problem breath figures abstract we study experimentally and theoretically new aspects of condensation growth and coalescence of droplets on a substrate we address in particular the dynamics of a marked droplet which undergoes coalescences with neighbouring droplets we find that the number of coalescences of the droplet grows as log t t is time and that the traveled distance increases as r t the average radius of the droplets at time t the fraction of the surface which was never covered by any droplet an important quantity for applications such as drug spreading or surface decontamination decays as t showing that completion occurs only slowly heuristic arguments accurate numerical simulations on simplified models which neglect polydispersity and an exact solution reported elsewhere a bray b derrida and c godreche europhys lett 27 1994 175 strongly support these findings and show that this power law decay is a generic feature common to many different situations finally the contour of the ensemble of dry sites appears fractal with dimension d 1 22 an experimental result not reproduced by the simplified models

opinion dynamics model with weighted influence exit probability and dynamics we introduce a stochastic model of binary opinion dynamics in which the opinions are determined by the size of the neighboring domains the exit probability here shows a step function behavior indicating the existence of a separatrix distinguishing two different regions of basin of attraction this behavior in one dimension is in contrast to other well known opinion dynamics models where no such behavior has been observed so far the coarsening study of the model also yields novel exponent values a lower value of persistence exponent is obtained in the present model which involves stochastic dynamics when compared to that in a similar type of model with deterministic dynamics this apparently counterintuitive result is justified using further analysis based on these results it is concluded that the proposed model belongs to a unique dynamical class

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

persistence exponents for fluctuating interfaces numerical and analytic results for the exponent ensuremath theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear langevin equations the models are parametrized by the dynamic roughness exponent ensuremath beta with 0l ensuremath beta l1 for ensuremath beta frac 1 2 the time evolution is markovian using simulations of solid on solid models of the discretized continuum equations as well as of the associated zero dimensional stationary gaussian process we address two problems the return of an initially flat interface and the return to an initial state with fully developed steady state roughness the two problems are shown to be governed by different exponents for the steady state case we point out the equivalence to fractional brownian motion which has a return exponent ensuremath theta s 1 ensuremath ensuremath beta the exponent ensuremath theta 0 for the flat initial condition appears to be nontrivial we prove that ensuremath theta 0 ensuremath rightarrow ensuremath infty for ensuremath beta ensuremath rightarrow 0 ensuremath theta 0 g ensuremath theta s for ensuremath beta l frac 1 2 and ensuremath theta 0 l ensuremath theta s for ensuremath beta g frac 1 2 and calculate ensuremath theta 0 s perturbatively to first order in an expansion around the markovian case ensuremath beta frac 1 2 using the exact result ensuremath theta s 1 ensuremath ensuremath beta accurate upper and lower bounds on ensuremath theta 0 can be derived which show in particular that ensuremath theta 0 g 1 ensuremath ensuremath beta 2 ensuremath beta for small ensuremath beta

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

experimental measurement of the persistence exponent of the planar ising model using a twisted nematic liquid crystal system exhibiting planar ising model dynamics we have measured the scaling exponent ensuremath theta which characterizes the time evolution p t ensuremath sim t ensuremath ensuremath theta of the probability p t that the local order parameter has not switched its state by the time t for 0 4 sec 200 sec following the phase quench the system exhibits scaling behavior and measured over this interval ensuremath theta 0 19 ifmmode pm else textpm fi 0 03 in good agreement with theoretical analysis and numerical simulations

spatial scaling in persistence we discuss persistence in the context of non equilibrium processes such as coarsening in ising and potts models the persistence phenomenon shows strong spatial correlation with a correlation length which diverges with time in the correlated region r persistent sites form a fractal and the structure evolves in time obeying a dynamical scaling we discuss such fractal formation in the context of a reaction diffusion system in one dimension

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

nonlinear boundary stabilization for timoshenko beam system abstract this paper is concerned with the existence and decay of solutions of the following timoshenko system u t u 1 i 1 n v x i 0 in 0 v v 2 i 1 n u x i 0 in 0 subject to the nonlinear boundary conditions u v 0 in 0 0 u h 1 x u 0 in 1 0 v h 2 x v x u 0 in 1 0 and the respective initial conditions at t 0 here is a bounded open set of r n with boundary constituted by two disjoint parts 0 and 1 and x denotes the exterior unit normal vector at x 1 the functions h i x s i 1 2 are continuous and strongly monotone in s r the existence of solutions of the above problem is obtained by applying the galerkin method with a special basis the compactness method and a result of approximation of continuous functions by lipschitz continuous functions due to strauss the exponential decay of energy follows by using the multiplier and the zuazua method

scaling and fractal formation in persistence the spatial distribution of unvisited persistent sites in 1d a a o model is studied numerically over length scales smaller than a cut off t tz the set of unvisited sites is found to be a fractal the fractal dimension df dynamical exponent z and persistence exponent are related through z 1 df the observed values of df and z are found to be sensitive to the initial density of particles we argue that this may be due to the existence of two competing length scales and discuss the possibility of a crossover at late times

exact exponent for the number of persistent spins in the zero temperature dynamics of the one dimensional potts model for the zero temperature glauber dynamics of theq state potts model the fractionr q t of spins which never flip up to timet decays like a power lawr q t t q when the initial condition is random by mapping the problem onto an exactly soluble one species coagulation model a a a or alternatively by transforming the problem into a free fermion model we obtain the exact expression of q for all values ofq the exponent q is in general irrational 3 0 53795082 4 0 63151575 with the exception ofq 2 andq for which 2 3 8 and 1

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

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

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

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

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

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

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

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

spatial distribution of persistent sites we study the distribution of persistent sites sites unvisited by particles a in the one dimensional a a reaction diffusion model we define the empty intervals as the separations between adjacent persistent sites and study their size distribution n k t as a function of interval length k and time t the decay of persistence is the process of irreversible coalescence of these empty intervals which we study analytically under the independent interval approximation iia physical considerations suggest that the asymptotic solution is given by the dynamic scaling form n k t s 2f k s with the average interval size s t1 2 we show under the iia that the scaling function f x x as x 0 and decays exponentially at large x the exponent is related to the persistence exponent through the scaling relation 2 1 we compare these predictions with the results of numerical simulations we determine the two point correlation function c r t under the iia we find that for r s c r t r where 2 in agreement with our earlier numerical results

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

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

non trivial exponents in the zero temperature dynamics of the 1d ising and potts models we consider the glauber dynamics of the q state potts model in one dimension at zero temperature starting with a random initial configuration we measure the density r t of spins which have never flipped from the beginning of the simulation until time t we find that for large t the density r t has a power law decay left r t sim t theta right where the exponent theta varies with q our simulations lead to theta simeq 37 for q 2 theta simeq 53 for q 3 and theta longrightarrow 1 as q longrightarrow infty