transience recurrence and local extinction properties of the support for supercritical finite measure valued diffusions whose log laplace equation is associated with the semilinear equation ut lu flu au2 where a 8 0 and l 1 ed j l aij d92 xxi dxj ed 1 bi d 8xi a path x is said to survive if x t 0 for all t 2 0 since 0 p x q survives 0 for all 0 w u e4 rd where rd denotes the space of finite measures on rd we define transience recurrence and local extinction for the support of the supercritical measure valued diffusion starting from a finite measure as follows the support is recurrent if p x t b 0 for some t 2 0 i x survives 1 for every 0 u ej rd and every open set b c rd for d 2 the support is transient if p x t b 0 for some t 2 0 i x survives 1 for every ut ej rd and bounded b c rd which satisfy supp u n b 0 a similar definition taking into account the topology of r1 is given for d 1 the support exhibits local extinction if for each t e 4 rd and each bounded b c rd there exists a p l almost surely finite random time b such that x t b 0 for all t 2 b criteria for transience recurrence and local extinction are developed in this paper also studied is the

m n coherent pairs of linear functionals and jacobi matrices abstract a pair of regular linear functionals u v in the linear space of polynomials with complex coefficients is said to be an m n coherent pair of order m if their corresponding sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 satisfy a structure relation i 0 m a i n p n m i m x i 0 n b i n q n i x n 0 where m n and m are non negative integers a i n n 0 0 i m and b i n n 0 0 i n are sequences of complex numbers such that a m n 0 if n m b n n 0 if n n and a i n b i n 0 if i n when m 1 u v is called an m n coherent pair in this work we give a matrix interpretation of m n coherent pairs of linear functionals indeed an algebraic relation between the corresponding monic tridiagonal jacobi matrices associated with such linear functionals is stated as a particular situation we analyze the case when one of the linear functionals is classical finally the relation between the jacobi matrices associated with m n coherent pairs of linear functionals of order m and the hessenberg matrix associated with the multiplication operator in terms of the basis of monic polynomials orthogonal with respect to the sobolev inner product defined by the pair u v is deduced

orthogonal polynomials generated by a linear structure relation inverse problem abstract let p n n and q n n be two sequences of monic polynomials linked by a type structure relation such as q n x r n q n 1 x p n x s n p n 1 x t n p n 2 x where r n n s n n and t n n are sequences of complex numbers first we state necessary and sufficient conditions on the parameters such that the above relation becomes non degenerate when both sequences p n n and q n n are orthogonal with respect to regular moment linear functionals u and v respectively second assuming that the above relation is non degenerate and p n n is an orthogonal sequence we obtain a characterization for the orthogonality of the sequence q n n in terms of the coefficients of the polynomials and which appear in the rational transformation in the distributional sense u v some illustrative examples of the developed theory are presented

some local approximation properties of simple point processes for simple point processes on a borel space s we prove some approximations involving conditional distributions given that hits a small set b beginning with general versions of some classical limit theorems going back to the pioneering work of palm and khinchin we proceed to prove that under suitable regularity conditions the contributions to b and bc are asymptotically conditionally independent we further derive approximations in total variation of reduced palm distributions and show that when hits some small sets b1 bn the corresponding restrictions are asymptotically independent next we give general versions of the asymptotic relations p b 0 e b and prove some ratio limit theorems for conditional expectations e b 0 valid even when e is not finite and the palm distributions may fail to exist

a matrix characterization for the dν semiclassical and dν coherent orthogonal polynomials abstract we present a new structure relation for the sequence of orthogonal polynomials associated with a d semiclassical linear functional of class s and then we use it to obtain a matrix characterization of the d semiclassical orthogonal polynomials in terms of the jacobi matrix associated with the multiplication operator in the basis of orthonormal polynomials and the nonsingular lower triangular matrix that represents the orthogonal polynomials with respect to some bases of polynomials we also provide a matrix characterization of d coherent pairs of linear functionals

palm distributions and local approximation of regenerative processes for a regenerative process x with associated local time random measure the distribution relative to some fixed times has a simple description in terms of the excursion kernel r and the palm distributions qt with respect to under suitable regularity conditions on the qt admit versions with strong continuity and regularity properties using the latter versions we may establish some local invariance and mixing properties of x and show that the multivariate palm distributions can be approximated by ordinary conditional distributions

matrix roots of eventually positive matrices abstract eventually positive matrices are real matrices whose powers become and remain strictly positive as such eventually positive matrices are a fortiori matrix roots of positive matrices which motivates us to study the matrix roots of primitive matrices using classical matrix function theory and perron frobenius theory we characterize classify and describe in terms of the real jordan canonical form the p th roots of eventually positive matrices

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

an integral test for a critical multitype spatially homogeneous branching particle process and a related reaction diffusion system an integral test theorem 5 is established for the dichotomy concerning local extinction and survival even persistence at late times for critical multitype spatially homogeneous branching particle systems in continuous time our conditions on the branching mechanism are close to the ones known from classical processes without motion component this generalizes and complements results of lopez mimbela and wakolbinger lmw96 and others our approach is based on some genealogical tree analysis combined with the study of the long term behavior of l 1 norms of solutions of related systems of reaction diffusion equations which is perhaps also of some independent interest

local extinction versus local exponential growth for spatial branching processes let x be either the branching diffusion corresponding to the operator lu u2 u on d rd where x 0 and 0 is bounded from above or the superprocess corresponding to the operator lu u u2 on d rd with 0 and is bounded from above but no restriction on its sign let c denote the generalized principal eigenvalue for the operator l on d we prove the following dichotomy either c 0 and x exhibits local extinction or c 0 and there is exponential growth of mass on compacts of d with rate c for superdiffusions this completes the local extinction criterion obtained by pinsky ann probab 24 1996 237 267 and a recent result on the local growth of mass under a spectral assumption given by englander and turaev ann probab 30 2002 683 722 the proofs in the above two papers are based on pde techniques however the proofs we offer are probabilistically conceptual for the most part they are based on spine decompositions or immortal particle representations along with martingale convergence and the law of large numbers further they are generic in the sense that they work for both types of processes

self similar random measures a set is called self similar if it is decomposable into parts which are similar to the whole this notion was generalized to random sets in the present paper an alternative axiomatic approach is given which makes precise the following idea using palm distribution theory a random set is statistically self similar if it is statistically scale invariant with respect to any center chosen at random from that set for these sets hausdorff dimension coincides with an intrinsic self similarity index

the support of measure valued branching processes in a random environment we consider the one dimensional catalytic branching process introduced by dawson and fleischmann which is a modification of the super brownian motion the catalysts are given by a nonnegative infinitely divisible random measure with independent increments we give sufficient conditions for the global support of the process to be compact and sufficient conditions for noncompact global support since the catalytic process is related to the heat equation compact support may be surprising on the other hand the super brownian motion has compact global support we find that all nonnegative stable random measures lead to compact global support and we give an example of a very rarified levy process which leads to noncompact global support

a phase transition for measure valued sir epidemic processes we consider measure valued processes xt that solve the following martingale prob lem for a given initial measure x0 and for all smooth compactly supported test functions xt x0 1 z t 2 ds such processes arise as scaling limits of sir epidemic models we show that there exist critical values c d 0 for dimensions d 2 3 such that if c d then the solution survives forever with positive probability but if c d then the solution dies out in finite time with probability 1 for d 1 we prove that the solution dies out almost surely for all values of we also show that in dimensions d 2 3 the process dies out locally almost surely for any value of that is for any compact set k the process xt k 0 eventually

spatial extent of branching brownian motion we study the one dimensional branching brownian motion starting at the origin and investigate the correlation between the rightmost x max geq 0 and leftmost x min leq 0 visited sites up to time t at each time step the existing particles in the system either diffuse with diffusion constant d die with rate a or split into two particles with rate b we focus on the regime b leq a where these two extreme values x max and x min are strongly correlated we show that at large time t the joint probability distribution function pdf of the two extreme points becomes stationary p x y t to infty to p x y our exact results for p x y demonstrate that the correlation between x max and x min is nonzero even in the stationary state from this joint pdf we compute exactly the stationary pdf p zeta of the dimensionless span zeta x max x min sqrt d b which is the distance between the rightmost and leftmost visited sites this span distribution is characterized by a linear behavior p zeta sim frac 1 2 left 1 delta right zeta for small spans with delta left frac a b 1 right in the critical case delta 0 this distribution has a non trivial power law tail p zeta sim 8 pi sqrt 3 zeta 3 for large spans on the other hand in the subcritical case delta 0 we show that the span distribution decays exponentially as p zeta sim a 2 2 zeta exp left sqrt delta zeta right for large spans where a is a non trivial function of delta which we compute exactly we show that these asymptotic behaviors carry the signatures of the correlation between x max and x min finally we verify our results via direct monte carlo simulations

on the range of lattice models in high dimensions we investigate the scaling limit of the range the set of visited vertices for a general class of critical lattice models starting from a single initial particle at the origin conditions are given on the random sets and an associated ancestral relation under which conditional on longterm survival the rescaled ranges converge weakly to the range of super brownian motion as random sets these hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball that is for the extrinsic one arm probability we show that these conditions are satisfied by the voter model in dimensions d ge 2 sufficiently spread out critical oriented percolation and critical contact processes in dimensions d 4 and sufficiently spread out critical lattice trees in dimensions d 8 the latter result proves conjecture 1 6 of van der hofstad et al ann probab 45 278 376 2017 and also has important consequences for the behaviour of random walks on lattice trees in high dimensions

on rational transformations of linear functionals direct problem abstract let u be a quasi definite linear functional we find necessary and sufficient conditions in order to the linear functional v satisfying x a u x a v be a quasi definite one also we analyze some linear relations linking the polynomials orthogonal with respect to u and v

m n coherent pairs of order m k and sobolev orthogonal polynomials a pair of regular linear functionals u v is said to be a m n coherent pair of order m k if their corresponding sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 satisfy a structure relation such as i 0ma i np n m i m x i 0nb i nq n k i k x n 0 where a i n and b i n are complex numbers such that a m n 0 if n m b n n 0 if n n and a i n b i n 0 when i n in the first part of this work we focus our attention on the algebraic properties of an m n coherent pair of order m k to be more precise let us assume that m k if m k then u and v are related by a rational factor in the distributional sense if m k then u and v are semiclassical and they are again related by a rational factor in the second part of this work we deal with a sobolev type inner product defined in the linear space of polynomials with real coefficients p as l rp x q x d m 0 x l rp m x q m x d m 1 x p q p where l is a positive real number m is a positive integer number and m 0 m 1 is a m n coherent pair of order m of positive borel measures supported on an infinite subset of the real line meaning that the sequences of monic orthogonal polynomials p n x n 0 and q n x n 0 with respect to m 0 and m 1 respectively satisfy a structure relation as above with k 0 a i n and b i n being real numbers fulfilling the above mentioned conditions we generalize several recent results known in the literature in the framework of sobolev orthogonal polynomials and their connections with coherent pairs introduced in a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and their extensions in particular we show how to compute the coefficients of the fourier expansion of functions on an appropriate sobolev space defined by the above inner product in terms of the sequence of sobolev orthogonal polynomials s n x l n 0

some properties of the range of super brownian motion we consider a super brownian motion x its canonical measures can be studied through the path valued process called the brownian snake we obtain the limiting behavior of the volume of the neighborhood for the range of the brownian snake and as a consequence we derive the analogous result for the range of super brownian motion and for the support of the integrated super brownian excursion then we prove the support of xt is capacity equivalent to 0 1 2 in d d 3 and the range of x as well as the support of the integrated super brownian excursion are capacity equivalent to 0 1 4 in d d 5

a phase diagram for a stochastic reaction diffusion system in this paper a stochastic reaction diffusion system is considered which models the spread of a finite population reacting with a non renewable resource in the presence of individual based noise a two parameter phase diagram is established to describe the large time evolution distinguishing between certain death or possible life of the population

sur la saucisse de wiener et les points multiples du mouvement brownien on considere b t t 0 un mouvement brownien a valeurs dans l espace euclidien a d dimensions r d d 2 issu de o on s interesse a la probabilite que b penetre avant un instant donne t dans une petite boule de l espace

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

nonnegative and eventually positive matrices abstract we characterize those eventually positive matrices a such that the sum of a and every nonnegative matrix remains eventually positive for n 3 we show that for every eventually positive matrix a m n r there exists an eventually positive matrix b such that a b is not eventually positive

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

stochastic partial differential equation driven by stable noise we construct a weak solution to the stochastic partial differential equation driven by a one sided stable noise without negative jumps we prove the weak existence of the solution for parameters the facts that for is not a lipschitz function and is not a gaussian noise require the development of new methods which we believe are of independent interest we also show that when the above equation gives an alternative description of super brownian motion with stable branching mechanism

on lewin and vitek s conjecture about the exponent set of primitive matrices m lewin and y vitek conjecture that every integer w 1 i n2 2n 2 1 is the exponent of some n x n primitive matrix in this paper we prove that this conjecture is true except for n 11 the problem of determining the exponent set e is completely solved

when do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials given p n n 0 a sequence of monic orthogonal polynomials we analyze their linear combinations with constant coefficients and fixed length i e q n x p n x a 1p n 1 x a kp n k a k 0 n k necessary and sufficient conditions are given for the orthogonality of the sequence q n n 0 an interesting interpretation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is shown moreover in the case k 2 we characterize the families p n n 0 such that the corresponding polynomials q n n 0 are also orthogonal

trimmed trees and embedded particle systems in a supercritical branching particle system the trimmed tree consists of those particles which have descendants at all times we develop this concept in the superprocess setting for a class of continuous superprocesses with feller underlying motion on compact spaces we identify the trimmed tree which turns out to be a binary splitting particle system with a new underlying motion that is a compensated h transform of the old one we show how trimmed trees may be estimated from above by embedded binary branching particle systems

clustering and invariant measures for spatial branching models with infinite variance we consider two spatial branching models on r d branching brownian motion with a branching law in the domain of normal attraction of a 1 stable law 0 2 it is known that there exists a family v 0 of nontrivial invariant measures we show that every invariant measure is a convex combination of the v both results have been known before only under an additional finite mean assumption for the critical dimension d 2 we show that both models display the phenomenon of diffusive clustering this means that clusters grow spatially on a random scale we give a precise description of the clusters via multiple scale analysis our methods rely mainly on studying sub and supersolutions of the reaction diffusion equation u t 1 2 u u 1 0

a super brownian motion with a locally infinite catalytic mass a super brownian motion x in bbb r with hyperbolic branching rate varrho 2 b 1 b 2 b in bbb r is constructed which symbolically could be described by the formal stochastic equation with a space time white noise rm d w starting at x 0 delta a a ne 0 this superprocess x will never hit the catalytic center there is an increasing sequence of brownian stopping times tau n strictly smaller than the hitting time of 0 such that with probability one dynkin s stopped measures x tau n vanish except for finitely many n

algebraically positive matrices abstract we introduce the concept of algebraically positive matrices and investigate some basic properties including a characterization the index of algebraic positivity and sign patterns that allow or require this property we also pose two open problems

effect of noise on front propagation in reaction diffusion equations of kpp type we consider reaction diffusion equations of kpp type in one spatial dimension perturbed by a fisher wright white noise under the assumption of uniqueness in distribution examples include the randomly perturbed fisher kpp equations partial tu partial x 2u u 1 u epsilon sqrt u 1 u dot w and partial tu partial x 2u u 1 u epsilon sqrt u dot w where dot w dot w t x is a space time white noise we prove the brunet derrida conjecture that the speed of traveling fronts for small e is 2 pi 2 log epsilon 2 2 o log log epsilon log epsilon 3

sobolev orthogonal polynomials and m n coherent pairs of measures we introduce the notion of m n coherent pair of measures as a generalization of the concept of coherent pair of measures introduced by iserles et al a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and subsequently generalized by several authors a pair of measures email protected 0 email protected 1 is called m n coherent if the corresponding orthogonal polynomial sequences p n n and q n n resp satisfy a non zero structure relation such as i 0nr i np n i x email protected i 0ms i nq n i 1 x for all n 0 1 2 where m and n are fixed non negative integer numbers and r i n and s i n are given real parameters satisfying some natural conditions we prove that the regular moment linear functionals associated to an m n coherent pair are semiclassical and they are related by a rational modification in the usual sense of distribution theory we also discuss the converse statement under the assumption that email protected 0 email protected 1 form an m n coherent pair we study the sequence s n l n of the monic orthogonal polynomials with respect to the sobolev inner product l email protected email protected email protected f g email protected 1 defined in the space of all polynomials with real coefficients where l 0 an efficient algorithm is stated to compute the coefficients in the fourier sobolev type series f x email protected n 0 c n ls n l x with respect to l for suitable smooth functions f such that email protected l m 0 2 r and f l m 1 2 r finally some illustrative computational examples are presented

on linearly related sequences of difference derivatives of discrete orthogonal polynomials let be either c 0 or q c 0 1 and let d be the corresponding difference operator defined in the usual way either by d p x p x p x or d q p x p q x p x q 1 x let u and v be two moment regular linear functionals and let p n x n 0 and q n x n 0 be their corresponding orthogonal polynomial sequences ops we discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two ops p n x n 0 and q n x n 0 assuming that their difference derivatives d of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 m a i n d m p n m i x i 0 n b i n d k q n k i x n 0 where m n m k n 0 a m n 0 for n m b n n 0 for n n and a i n b i n 0 for i n under certain conditions we prove that u and v are related by a rational factor in the distributional sense moreover when m k then both u and v are d semiclassical functionals this leads us to the concept of m n d coherent pair of order m k extending to the discrete case several previous works as an application we consider the ops with respect to the following sobolev type inner product p x r x u p x r x v d m p x d m r x 0 assuming that u and v which eventually may be represented by discrete measures supported either on a uniform lattice if or on a q lattice if q constitute a m n d coherent pair of order m that is an m n d coherent pair of order m 0 m n being fixed

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

orthogonal polynomials associated with an inverse quadratic spectral transform let p n n 0 be a sequence of monic orthogonal polynomials with respect to a quasi definite linear functional u and q n n 0 a sequence of polynomials defined by q n x p n x s np n 1 x t np n 2 x n 1 with t n 0 for n 2 we obtain a new characterization of the orthogonality of the sequence q n n 0 with respect to a linear functional v in terms of the coefficients of a quadratic polynomial h such that h x v u we also study some cases in which the parameters s n and t n can be computed more easily and give several examples finally the interpretation of such a perturbation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is presented

on the supports of measure valued critical branching brownian motion on etudie la taille globale et la localisation des supports des etats du processus de markov a valeur de mesure qui est un analogue du mouvement brownien ramifie critique classique sur l espace euclidien a d dimensions

conditioned limit theorems of stopped critical branching bessel processes we consider critical branching bessel processes initially at r 1 and stopped at r 1 let n be the number of descendants hitting r 1 we give the norming constant k r and prove convergence as r of n k r conditioned on n 0 the distribution of conditioned limit laws is also investigated a feature of this study is an interplay between probabilistic insights and analytic techniques for emden fowler s equation

on the topological boundary of the range of super brownian motion we show that if partial mathcal r is the boundary of the range of super brownian motion and dim denotes hausdorff dimension then with probability one for any open set u u cap partial mathcal r neq varnothing implies operatorname dim u cap partial mathcal r begin cases 4 2 sqrt 2 approx1 17 quad text if d 2 frac 9 sqrt 17 2 approx2 44 quad text if d 3 end cases this improves recent results of the last two authors by working with the actual topological boundary rather than the boundary of the zero set of the local time and establishing a local result for the dimension

some local approximations of dawson watanabe superprocesses let be a dawson watanabe superprocess in rd such that t is a s locally finite for every t 0 then for d 2 and fixed t 0 the singular ran dom measure t can be a s approximated by suitably normalized restrictions of lebesgue measure to the neighborhoods of suppff when d 3 the local distributions of near a hitting point can be approximated in total vari ation by those of a stationary and self similar pseudo random measure f by contrast the corresponding distributions for d 2 are locally invariant fur ther results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities our main proofs are based on a detailed analysis of the historical structure of f

generalized coherent pairs abstract a pair of quasi definite moment functionals u0 u1 is a generalized coherent pair if monic orthogonal polynomials pn x n 0 and rn x n 0 relative to u0 and u1 respectively satisfy a relation r n x 1 n 1 p n 1 x n n p n x n 1 n 1 p n 1 x n 2 where n and n are arbitrary constants which may be zero if u0 u1 is a generalized coherent pair then u0 and u1 must be semiclassical we find conditions under which either u0 or u1 is classical in such a case we also determine the types of the companion moment functionals also some illustrating examples and two ways of generating generalized coherent pairs are given we also discuss the corresponding sobolev orthogonal polynomials

decay of mass for the equation ut δu a x up u q abstract consider the quasilinear cauchy problem u t u a x u p u q x r d t 0 u x 0 x 0 x r d where a 0 p and q satisfy p 0 and q 1 or p 1 and q 0 and 0 l 1 r d c 3 b r d this paper proves that the above equation possesses a unique positive classical solution and then investigates whether or not limt rd u x t dx 0 in particular it is shown that if a is on the order x m for large x then 0 if dp d 1 q d 2 m under the assumption that for compactly supported u e t ec t 1 for some 0 and for e 0 1 where ue denotes the solution to the above equation with initial condition e it is shown that 0 if dp 2 q d 2 max m d for a certain range of the parameters d p q m it is proved that holds with d 1 2 and for many other parameter values it is proved that holds with d 2 note that if d 1 2 then the above condition for 0 becomes dp d 1 q d 2 max m d which in light of the parameter restrictions is equivalent to dp d 1 q d 2 m

on linearly related sequences of derivatives of orthogonal polynomials abstract we discuss an inverse problem in the theory of standard orthogonal polynomials involving two orthogonal polynomial families p n n and q n n whose derivatives of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 n r i n p n i m m x i 0 m s i n q n i k k x for all n 0 1 2 where m and n are fixed nonnegative integer numbers and r i n and s i n are given complex parameters satisfying some natural conditions let u and v be the moment regular functionals associated with p n n and q n n resp assuming 0 m k we prove the existence of four polynomials m m i and n k i of degrees m m i and n k i resp such that d k m m m i u n k i v i 0 1 the k m th derivative as well as the left product of a functional by a polynomial being defined in the usual sense of the theory of distributions if k m then u and v are connected by a rational modification if k m 1 then both u and v are semiclassical linear functionals which are also connected by a rational modification when k m the stieltjes transform associated with u satisfies a non homogeneous linear ordinary differential equation of order k m with polynomial coefficients

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

on an extension of symmetric coherent pairs of orthogonal polynomials given two symmetric and positive definite linear functionals w and v we study the coefficients in the recurrence relation for the system of monic polynomials orthogonal with respect to the second linear functional assuming that the first one is classical and that there exists an algebraic differential relation between these two families of polynomials moreover we determine this companion linear functional as a rational modification of the classical one

on the linear functionals associated to linearly related sequences of orthogonal polynomials abstract an inverse problem is solved by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials p n n and q n n respectively in the sense p n x i 1 n r i n p n i x q n x i 1 m s i n q n i x for all n 0 1 2 where r i n and s i n are complex numbers satisfying some natural conditions are connected by a rational modification i e there exist polynomials and with degrees m and n respectively such that u v we also make some remarks concerning the corresponding direct problem stating a characterization theorem in the case n 1 and m 2 as an example we give a linear relation of the above type involving jacobi polynomials with distinct parameters

on linearly related orthogonal polynomials and their functionals let pn be a sequence of polynomials orthogonal with respect a linear functional u and qn a sequence of polynomials defined by pn x snpn 1 x qn x tnqn 1 x we find necessary and sufficient conditions in order to qn be a sequence of polynomials orthogonal with respect to a linear functional v furthermore we prove that the relation between these linear functionals is x a u x a v even more if u and v are linked in this way we get that pn and qn satisfy a formula as above