generalizations of laguerre polynomials it is shown that the polynomials ln m0 m1 mn x n 0 defined by ln m0m1 mn x k 0n 1ak dkln x for certain real coefficients ak k 0n 1 are orthogonal with respect to the inner product f g 1 1 0 x e x f x g x dx v 0nmv f v 0 g v 0 where 1 n n and mv 0 for all v 0 1 2 n for these new polynomials n m0 m1 mn x n 0 an orthogonality relation and a second order differential equation are derived further we obtain a representation as a n 2fn 2 hypergeometric series and a 2n 3 terms recurrence relation which gives rise to a christoffel darboux type formula

computing orthogonal polynomials in sobolev spaces numerical methods are considered for generating polynomials orthogonal with respect to an inner product of sobolev type i e one that involves derivatives up to some given order each having its own positive measure associated with it the principal objective is to compute the coefficients in the increasing order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next as a by product of this computation one gains access to the zeros of these polynomials via eigenvalues of an upper hessenberg matrix formed by the coefficients generated two methods are developed one is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as modified chebyshev algorithm the other a generalization of stieltjes s procedure expresses the desired coefficients in terms of a sobolev inner product involving the orthogonal polynomials in question whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point the numerical characteristics of these methods are illustrated in the case of sobolev orthogonal polynomials of old as well as new types based on extensive numerical experimentation a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros

an asymptotic result for laguerre sobolev orthogonal polynomials abstract let s n denote the sequence of polynomials orthogonal with respect to the sobolev inner product f g s 0 f x g x x e x d x 0 f x g x x e x d x where 1 0 and the leading coefficient of the s n is equal to the leading coefficient of the laguerre polynomial l n then if x c s 0 lim n s n x l n 1 x is a constant depending on

on sobolev orthogonal polynomials with coherent pairs the jacobi case abstract when we investigate the asymptotic properties of orthogonal polynomials with sobolev inner product f g a b f x g x d x a b f x g x dv x we need to know the relations between pn and qn where pn x and qn x are the nth monic orthogonal polynomials with respect to d and dv respectively the pair d dv is called a coherent pair if there exist nonzero constant dn such that q n x p n 1 x n 1 d n p x n n 1 one can divide the coherent pairs into two cases the jacobi case and the laguerre case we consider the limit of q n x p n x under the jacobi case we prove that limn dn exists and calculate the limit as well for the jacobi case

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

generation and evaluation of orthogonal polynomials in discrete sobolev spaces i algorithms in this paper we study theoretically the determination and evaluation of polynomials that are orthogonal with respect to a general discrete sobolev inner product that is an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives this sobolev inner product has the property that the orthogonal polynomials with respect to it satisfy a linear recurrence relation of fixed order we provide a complete set of formulas to compute the coefficients of this recurrence besides we study the determination of the fourier sobolev coefficients of a finite approximation of a function and the numerical evaluation of the resulting finite series at a general point

the expected number of k faces of a voronoi diagram abstract the asymptotic behavior of the expected number of faces of all dimensions of the voronoi diagram of n independent uniformly distributed sites in r d is investigated it is shown that when k and d are fixed and n grows without bound the expected number of k dimensional faces grows as c k d n further for fixed k as d grows without bound there exist constants k such that c k d o d k numerical estimates of c k d are given for small values of k and d

asymptotics of orthogonal polynomials generated by a geronimus perturbation of the laguerre measure abstract this paper deals with monic orthogonal polynomials generated by a geronimus canonical spectral transformation of the laguerre classical measure i e 1 x c x e x d x n x c for x 0 1 a free parameter n r and a shift c 0 we analyze the asymptotic behavior both strong and relative to classical laguerre polynomials of these orthogonal polynomials as n tends to infinity

zeros of sobolev orthogonal polynomials following from coherent pairs let sn denote the monic orthogonal polynomial sequence with respect to the sobolev inner product f g s fgd 0 f g d 1 where d 0 d 1 is a so called coherent pair and 0 then sn has n different real zeros the position of these zeros with respect to the zeros of other orthogonal polynomials in particular laguerre and jacobi polynomials is investigated coherent pairs are found where the zeros of sn 1 separte the zeros of sn

asymptotics of sobolev orthogonal polynomials for hermite 1 1 coherent pairs abstract in this paper we will discuss asymptotic properties of monic polynomials s n x n 0 orthogonal with respect to the sobolev inner product p q s r p x q x d 0 r p x q x d 1 with 0 d 0 e x 2 d x d 1 x 2 a x 2 b e x 2 d x a b r and a b it is well known that 0 1 is a pair of symmetric 1 1 coherent measures this means that there exist sequences a n n n b n n n a n b n for every n n such that the algebraic relation h n x b n 2 h n 2 x q n x a n 2 q n 2 x n 2 is satisfied where q n x n 0 is the sequence of monic orthogonal polynomials associated with 1 and h n x n 0 is the sequence of monic hermite polynomials we will study the relative asymptotics for sobolev scaled polynomials and we will obtain mehler heine type formulas among others

connection coefficients for laguerre sobolev orthogonal polynomials laguerre sobolev polynomials are orthogonal with respect to an inner product of the form p qs 0 p x q x x e x dx 0 p x q x d x w here 1 0 and p q p the linear space of polynomials with real coefficients if d x x e x dx the corresponding sequence of monic orthogonal polynomials q x has been studied by marcellan et al j com put appl math 71 1996 245 265 while if d x x dx the sequence of monic orthog onal polynomials l x was introduced by koekoek and meijer siam j math anal 24 1993 768 782 for each of these two families of laguerre sobolev polynomials here we give the explicit expression of the connection coefficients in their expansion as a series of standard la guerre polynomials the inverse connection problem of expanding laguerre polynomials in series of laguerre sobolev polynomials and the connection problem relating two families of laguerre sobolev polynomials with different parameters are also considered

zeros of sobolev orthogonal polynomials of hermite type let s n n denote a set of polynomials orthogonal with respect to the sobolev inner product s f x g x d j 0 x l f x g x d j 1 where l 0 and d j 0 d j 1 is a so called symmetrically coherent pair with d j 0 or d j 1 the hermite measure e x 2dx if d j 1 is the hermite measure then s n has n different real zeros if d j 0 is the hermite measure then s n has at least n 2 different real zeros we determine conditions for s n to have complex zeros

sobolev orthogonal polynomials on product domains orthogonal polynomials on the product domain a 1 b 1 a 2 b 2 with respect to the inner product f g s a 1 b 1 a 2 b 2 f x y g x y w 1 x w 2 y d x d y f c 1 c 2 g c 1 c 2 are constructed where w i is a weight function on a i b i for i 1 2 0 and c 1 c 2 is a fixed point the main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions in particular for product laguerre and product gegenbauer weight functions which serve as primary examples

the expected size of some graphs in computational geometry abstract we consider n independent points with a common but arbitrary density f in r d two points x i x j are joined by an edge when a certain set s x i x j does not contain any other data points the expected number e n of edges in the graph depends upon n f and the definition of s examples include rectangle spheres and loons these lead to the graph of all dominance pairs the gabriel graph and the relative neighborhood graph respectively other graphs covered by our analysis include the nearest neighbor graph and the directional nearest neighbor graph in all cases we obtain asymptotic lower bounds that do not depend upon f and are hence useful in all applications involving these graphs since we usually do not know f for sparse graphs exact asymptotic constants are obtained for e n that are valid for all densities

exploring the threshold of epidemic spreading for a stochastic sir model with local and global contacts abstract the spread of an epidemic process is considered in the context of a spatial sir stochastic model that includes a parameter 0 p 1 that assigns weights p and 1 p to global and local infective contacts respectively the model was previously studied by other authors in different contexts in this work we characterized the behavior of the system around the threshold for epidemic spreading we first used a deterministic approximation of the stochastic model and checked the existence of a threshold value of p for exponential epidemic spread an analytical expression which defines a function of the quotient between the transmission and recovery rates is obtained to approximate this threshold we then performed different analyzes based on intensive stochastic simulations and found that this expression is also a good estimate for a similar threshold value of p obtained in the stochastic model the dynamics of the average number of infected individuals and the average size of outbreaks show a behavior across the threshold that is well described by the deterministic approximation the distributions of the outbreak sizes at the threshold present common features for all the cases considered corresponding to different values of 1 these features are otherwise already known to hold for the standard stochastic sir model at its threshold 1 i the probability of having an outbreak of size n goes asymptotically as n 3 2 for an infinite system ii the maximal size of an outbreak scales as n 2 3 for a finite system of size n

hermite sobolev and closely connected orthogonal polynomials abstract in this paper a bilinear functional a g m o n m c m m g m which corresponds to the hermite sobolev and closely connected orthogonal polynomials is studied the domain of m m 1 n in which a is positive definite is given and the hermite sobolev and closely connected formal orthogonal polynomials are given in the nonnormal case

δ sobolev orthogonal polynomials of meixner type asymptotics and limit relation let qn x n be the sequence of monic polynomials orthogonal with respect to the sobolev type inner product p x r x s u0 p x r x u1 p x r x where 0 f x f x 1 f x denotes the forward difference operator and u0 u1 is a coherent pair of positive definite linear functionals being u1 the meixner linear functional in this paper relative asymptotics for the qn x n sequence with respect to meixner polynomials on compact subsets of c 0 is obtained this relative asymptotics is also given for the scaled polynomials in both cases we deduce the same asymptotics as we have for the self coherent pair that is when u0 u1 is the meixner linear functional furthermore we establish a limit relation between these orthogonal polynomials and the laguerre sobolev orthogonal polynomials which is analogous to the one existing between meixner and laguerre polynomials in the askey scheme

inner products involving q differences the little q laguerre sobolev polynomials in this paper polynomials which are orthogonal with respect to the inner product p r s k 0 p qk r qk aq k aq q q q k k 0 dqp qk dqr qk aq k aq q q q k where dq is the q difference operator 0 0 q 1 and 0 aq 1 are studied for these polynomials algebraic properties and q difference equations are obtained as well as their relation with the monic little q laguerre polynomials some properties about the zeros of these polynomials are also deduced finally the relative asymptotics qn x pn x a q n on compact subsets of c 0 1 is given where qn x is the nth degree monic orthogonal polynomial with respect to the above inner product and pn x a q denotes the monic little q laguerre polynomial of degree n

on sobolev orthogonal polynomials with coherent pairs the laguerre case type 1 we study the asymptotic properties of orthogonal polynomials with sobolev inner product formula the pair d d is called a coherent pair if there exists nonzero constantsdnsuch that formula wherepn x andqn x are thenth monic orthogonal polynomials with respect tod andd respectively one can divide the coherent pairs into two cases the jacobi case and the laguerre case there are two types in each case we consider thenth root asymptotics and the zero distribution for the laguerre case type 1

asymptotics of sobolev orthogonal polynomials for coherent pairs of laguerre type abstract let s n n denote a sequence of polynomials orthogonal with respect to the sobolev inner product f g s f x g x d 0 x f x g x d 1 x where 0 and d 0 d 1 is a so called coherent pair with at least one of the measures d 0 or d 1 a laguerre measure we investigate the asymptotic behaviour of sn x outside the supports of d 0 and d 1 and n

asymptotics of sobolev orthogonal polynomials for coherent pairs of jacobi type abstract let sn n denote a sequence of polynomials orthogonal with respect to the sobolev inner product f g s f x g x d 0 x f x g x d 1 x where 0 and d 0 d 1 is a so called coherent pair with at least one of the measures d 0 or d 1 a jacobi measure we investigate the asymptotic behaviour of sn x for n and x fixed x c 1 1 as well as x 1 1

connection problems for polynomial solutions of nonhomogeneous differential and difference equations abstract we consider nonhomogeneous hypergeometric type differential difference and q difference equations whose nonhomogeneity is a polynomial q n x the polynomial solution of these problems is expanded in the q n x basis and also in a basis p n x related in a natural way with the homogeneous hypergeometric equation we give an algorithm building a recurrence relation for the expansion coefficients in both bases that we solve explicitly in many cases involving classical orthogonal polynomials finally some concrete applications and extensions are given

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

on real and complex zeros of orthogonal polynomials in a discrete sobolev space abstract let sn x c n denote a set of polynomials orthogonal with respect to the discrete sobolev inner product f g f x g x d x nf c g c where n 0 c r for n 0 put kn x sn x 0 then sn x c n has at least n 2 different real zeros their position with respect to the zeros of kn can be determined using the tangent to the graph of y kn x in c kn c on the other hand if n 3 then c can be chosen such that sn x c n has two complex zeros if n is sufficiently large

a matrix approach for the semiclassical and coherent orthogonal polynomials we obtain a matrix characterization of semiclassical orthogonal polynomials in terms of the jacobi matrix associated with the multiplication operator in the basis of orthogonal polynomials and the lower triangular matrix that represents the orthogonal polynomials in terms of the monomial basis of polynomials we also provide a matrix characterization for coherent pairs of linear functionals

dynamic model of worms with vertical transmission in computer network an e epidemic seirs model for the transmission of worms in computer network through vertical transmission is formulated it has been observed that if the basic reproduction number is less than or equal to one the infected part of the nodes disappear and the worm dies out but if the basic reproduction number is greater than one the infected nodes exists and the worms persist at an endemic equilibrium state numerical methods are employed to solve and simulate the system of equations developed we have analyzed the behavior of the susceptible exposed infected and recovered nodes in the computer network with real parametric values

asymptotics of sobolev orthogonal polynomials for hermite coherent pairs abstract let qn be the polynomials orthogonal with respect to the sobolev inner product f g s fg d 0 f g d 1 being 0 1 a coherent pair where one of the measures is the hermite measure the outer relative asymptotics for qn with respect to hermite polynomials are found on the other hand we consider the sobolev scaled polynomials and we obtain the plancherel rotach asymptotics for those as well as a consequence about their zeros

analytic aspects of sobolev orthogonal polynomials revisited this paper surveys some recent achievements in the analytic theory of polynomials orthogonal with respect to inner products involving derivatives asymptotic properties zero location approximation and moment theory are some of the topic considered

bounds on the geometric complexity of optimal centroidal voronoi tesselations in 3d gersho s conjecture in 3d asserts the asymptotic periodicity and structure of the optimal centroidal voronoi tessellation this relatively simple crystallization problem remains to date open we prove bounds on the geometric complexity of optimal centroidal voronoi tessellations as the number of generators tends to infinity combined with an approach of gruber in 2d these bounds reduce the resolution of the 3d gersho s conjecture to a finite albeit very large computation of an explicit convex problem in finitely many variables

what is beyond coherent pairs of orthogonal polynomials abstract usually coherent pairs of orthogonal polynomials have been considered in the wider context of sobolev orthogonality in this paper we focus our attention on the problem of coherence between two orthogonal polynomial sequences in terms of the corresponding linear functionals we deduce some conditions about the linear functionals in order that the corresponding orthogonal polynomial sequences constitute a coherent pair

influence of removable devices on computer worms dynamic analysis and control strategies worms spreading via both web based scanning and removable devices account for a major part of threats on internet however their dynamical behavior and controlling methods remain unclear as a result we present a computer worm model incorporating specific features unique to those worms in this paper the threshold value r 0 determining whether the worms die out is obtained theoretical analysis shows that if r 0 1 the disease free equilibrium is globally asymptotically stable otherwise the disease will be prevalent additionally some control strategies are given our results are illustrated by numerical simulations

some properties of the sobolev laguerre polynomials abstract in this paper we study the case 0 of the sobolev laguerre polynomials we determine a generating function for the polynomials and an expansion formula

zero distribution of orthogonal polynomials in a certain discrete sobolev space abstract we study the zero distribution of the polynomials snn which are orthogonal with respect to the discrete sobolev inner product f g 0 x g x d x nf r 0 g r 0 where is a distribution function n 0 r 1 snn has n real simple zeros at most one of them is outside 0 the location of these zeros is given in relation to the position of the zeros of some classical polynomials i e polynomials with respect to an inner product with n 0

smallest zeros of some types of orthogonal polynomials asymptotics we establish mehler heine type formulas for orthogonal polynomials related to rational modifications of hermite weight on the real line and for hermite sobolev orthogonal polynomials these formulas give us the asymptotic behaviour of the smallest zeros of the corresponding orthogonal polynomials furthermore we solve a conjecture posed in a previous paper about the asymptotics of the smallest zeros of the hermite sobolev polynomials as well as an open problem concerning the asymptotics of these sobolev orthogonal polynomials

seirs epidemic model with delay for transmission of malicious objects in computer network abstract an epidemic transmission model seirs of malicious objects in the computer network is formulated with death rate other than attack of malicious object is constant and an excess death rate constant for infective nodes deaths of a node in computer network equivalently mean to say the isolation of that node from the computer network which even on continuous run by anti malicious software spread malicious objects latent and immune periods are assumed to be constants and the force of infection is assumed to be of standard form namely proportional to i t n t where n t is the total variable population size and i t is the size of infective population the model consists of a set of integro differential equations when a node is recovered from the infected class it recovers temporarily acquiring temporary immunity with probability p 0 p 1 and dies from the attack of malicious object with probability 1 p malicious objects free equilibrium is investigated and the stability of the results are stated in terms of threshold parameter

a tool for locating zeros of orthogonal polynomials in sobolev inner product spaces abstract in the theory of polynomials orthogonal with respect to an inner product of the form f 0f x g x d x mk 1akf ik 0 g ik 0 one is confronted with the following situation for certain values of the parameters the orthogonal polynomial of degree n does not have all its zeros inside the support of the distribution function d this paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial for the case m 2 it is shown that there are exactly two zeros outside the true interval of orthogonality for a1 a2 large moreover it is proved that these zeros are nonreal complex conjugates in the case i1 1 i2 also several examples are given

coherent pairs of measures and markov bernstein inequalities abstract all the coherent pairs of measures associated to linear functionals c 0 and c 1 introduced by iserles et al in 1991 have been given by meijer in 1997 there exist seven kinds of coherent pairs all these cases are explored in order to give three term recurrence relations satisfied by polynomials the smallest zero 1 n of each of them of degree n has a link with the markov bernstein constant m n appearing in the following markov bernstein inequalities c 1 p 2 m n 2 c 0 p 2 p p n where m n 1 1 n the seven kinds of three term recurrence relations are given in the case where c 0 e x d x 0 and c 1 e x d x explicit upper and lower bounds are given for 1 n and the asymptotic behavior of the corresponding markov bernstein constant is stated except in a part of one case lim n 1 n 0 is proved in all the cases

laguerre sobolev orthogonal polynomials abstract in this paper polynomials that are orthogonal with respect to the inner product f g s 0 f x g x x e x dx 0 f x g x x e x dx where 1 and 0 are studied for these nonstandard orthogonal polynomials algebraic and differential properties as well as the relation with the classical laguerre polynomials are obtained finally some properties concerning the localization and separation of the zeros of these polynomials are deduced

mathematical analysis of the effectiveness of control strategies to prevent the autorun virus transmission propagation abstract in this work a deterministic model for autorun computer virus propagation is presented and discussed two new control parameters vaccination rate and perfect recovery rate are introduced to validate the existence of computer virus infection via external devices likewise usbs the model is comprehensively analyzed with and without control parameter for verification of proposed model and strategy the recovery population of the infected computers has been divided into perfectly recovered with probability p and partially recovered with probability 1 p the results show that as vaccination rate is increased reproductive number and number of infected computers decreased since reproductive number does not depend upon p change in the value of p does not affect the basic reproductive number however as value of p is increased a reduction in steady state of infected computer is observed local and global stability analysis of virus free equilibrium has been investigated through basic reproductive number the endemic equilibrium has also been determined under certain conditions we introduce two control functions which are defined to present successful effort of repairing computers and fixing usbs and propose an optimal control problem with two control functions the solution to the proposed problem is derived through forward backward method the necessary conditions for optimal control have been developed using the pontryagin s maximum principle different combination of computer and usbs numbers have been utilized for effective simulation results and comprehensive analysis

on the positivity of some bilinear functionals in sobolev spaces abstract the positivity of a bilinear functional a a f g n m 0 m c m f m g m is studied as a function of coefficients m the concerned cases are those of laguerre gegenbauer and jacobi for c m c 0 m 1 n the domain m m 1 n where a is positive definite is given as a consequence when n 1 the corresponding markov bernstein inequalities are given

some remarks on a paper by l carlitz we study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by l carlitz we show that they are a special case of the sheffer polynomials and point out some interesting connections with certain sobolev orthogonal polynomials

mathematical models on computer viruses an attempt has been made to develop mathematical models on computer viruses infecting the system under different conditions mathematical model 1 discusses the situation to find the probability that at any time t how many software components are infected by virus assuming the recovery rate and proportion of un infected population receiving infection per unit time does not change with time mathematical model 2 is to estimate the proportion of software component population infected at any time and at any indefinite time under different cases the third model is to find out the rate of change of proportion of total population with exactly j viruses 1