the accuracy of b spline finite element approximations to pde surfaces in computer aided design a developing design tool is the pde method where surfaces are generated by solving a partial differential equation with appropriate boundary conditions methods for generating b spline approximations to pde surfaces to enable data transfer to other cad systems have been considered and the fairness properties of the resulting b spline surfaces examined for some applications the accuracy of the approximations will be important this paper examines the effect on the accuracy of different finite element mesh sizes and degrees of b spline surface concluding with recommendations about how to change these parameters in order to achieve the required accuracy

eigenvalue gaps for the cauchy process and a poincare inequality abstract a connection between the semigroup of the cauchy process killed upon exiting a domain d and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem was established in r banuelos t kulczycki the cauchy process and the steklov problem j funct anal 211 2004 355 423 from this a variational characterization for the eigenvalues n n 1 of the cauchy process in d was obtained in this paper we obtain a variational characterization of the difference between n and 1 we study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for 1 where is the eigenvalue corresponding to the first antisymmetric eigenfunction for d the proof is based on a variational characterization of 1 and on a weighted poincare type inequality the poincare inequality is valid for all symmetric stable processes 0 2 and any other process obtained from brownian motion by subordination we also prove upper bound estimates for the spectral gap 2 1 in bounded convex domains

nonparametric estimation of the variogram and its spectrum in the study of intrinsically stationary spatial processes a new nonparametric variogram estimator is proposed through its spectral representation the methodology is based on estimation of the variogram s spectrum by solving a regularized inverse problem through quadratic programming the estimated variogram is guaranteed to be conditionally negative definite simulation shows that our estimator is flexible and generally has smaller mean integrated squared error than the parametric estimator under model misspecification our methodology is applied to a spatial dataset of decadal temperature changes copyright 2011 oxford university press

on the nonparametric estimation of covariance functions we describe kernel methods for estimating the covariance function of a stationary stochastic process and show how to ensure that the estimator has the positive semidefiniteness property from a practical viewpoint our method is significant because it does not demand a parametric model for covariance from a technical angle our results exhibit a striking departure from those in more familiar cases of kernel estimation for example in the context of covariance estimation kernel estimators can have the same convergence rates as maximum likelihood estimators and can have exceptionally fast convergence rates when employed to estimate variance

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

continuous and discrete stable processes the one sided levy stable probability densities and the discrete stable distributions form a doubly stochastic poisson transform pair this relationship facilitates the formulation of a class of continuous stable stochastic processes

solving fractional schrodinger type spectral problems cauchy oscillator and cauchy well this paper is a direct offspring of the work of garbaczewski and stephanovich levy flights and nonlocal quantum dynamics j math phys 54 072103 2013 where basic tenets of the nonlocally induced random and quantum dynamics were analyzed a number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so called fractional quantum mechanics spectral problems presently we give a decisive computer assisted proof for an exemplary finite and ultimately infinite cauchy well problem that spectral solutions proposed so far were plainly wrong as a constructive input we provide an explicit spectral solution of the finite cauchy well the infinite well emerges as a limiting case in a sequence of deepening finite wells the employed numerical methodology algorithm based on the strang splitting method has been tested for an exemplary cauchy oscillator problem whose analytic solution is available an impact of the inherent spatial nonlocality of motion generators upon computer assisted outcomes potentially defective in view of various cutoffs i e detailed eigenvalues and shapes of eigenfunctions has been analyzed

periodic solutions for a pseudo relativistic schrodinger equation abstract we study the existence and the regularity of non trivial t periodic solutions to the following nonlinear pseudo relativistic schrodinger equation 0 1 x m 2 m u x f x u x in 0 t n where t 0 m is a non negative real number f is a regular function satisfying the ambrosetti rabinowitz condition and a polynomial growth at rate p for some 1 p 2 1 we investigate such problem using critical point theory after transforming it to an elliptic equation in the infinite half cylinder 0 t n 0 with a nonlinear neumann boundary condition by passing to the limit as m 0 in 0 1 we also prove the existence of a non trivial t periodic weak solution to 0 1 with m 0

minimum norm quadratic estimation of spatial variograms abstract the estimation of spatial variograms a measure of spatial correlation is a critical problem in the implementation of kriging a method for interpolating random fields we consider the use of minimum norm quadratic estimators of the variogram when it is specified up to a finite number of linear parameters we investigate the asymptotic behavior of such estimators for gaussian processes as the number of observations within some bounded region increases the basic conclusion is that we can estimate consistently those functions of the parameters and only those functions that have a nonnegligible impact asymptotically on the kriging procedure in general the behavior of the variogram over relatively short distances is the only aspect of the variogram that is asymptotically important as an example consider a gaussian process z on the real line with unknown constant mean and where is known as the semivariogram of the process and is a finite vector of unknown parameters suppose for 0

levy anomalous diffusion and fractional fokker planck equation we demonstrate that the fokker planck equation can be generalized into a fractional fokker planck equation i e an equation which includes fractional space differentiations in order to encompass the wide class of anomalous diffusions due to a levy stable stochastic forcing a precise determination of this equation is obtained by substituting a levy stable source to the classical gaussian one in the langevin equation this yields not only the anomalous diffusion coefficient but a non trivial fractional operator which corresponds to the possible asymmetry of the levy stable source both of them cannot be obtained by scaling arguments the mono scaling behaviors of the fractional fokker planck equation and of its solutions are analysed and a generalization of the einstein relation for the anomalous diffusion coefficient is obtained this generalization yields a straightforward physical interpretation of the parameters of levy stable distributions furthermore with the help of important examples we show the applicability of the fractional fokker planck equation in physics

levy flights in inhomogeneous environments abstract we study the long time asymptotics of probability density functions pdfs of levy flights in confining potentials that originate from inhomogeneities of the environment in which the flights take place to this end we employ two model patterns of dynamical behavior langevin driven and levy schrodinger semigroup driven dynamics it turns out that the semigroup modeling provides much stronger confining properties than the standard langevin one for computational and visualization purposes our observations are exemplified for the cauchy driver and its response to external polynomial potentials referring to levy oscillators with respect to both dynamical mechanisms we discuss the links of the levy semigroup motion scenario with that of random searches in spatially inhomogeneous media

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

dirichlet forms and semilinear elliptic equations with measure data we propose a probabilistic definition of solutions of semilinear elliptic equations with possibly nonlocal operators associated with regular dirichlet forms and with measure data using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right hand side of the equation is monotone and satisfies mild integrability assumption and the measure is smooth we also study regularity of solutions under the assumption that the measure is smooth and has finite total variation some applications of our general results are given

levy flights in confining potentials we analyze confining mechanisms for levy flights when they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport two classes of stochastic jump type processes are considered those driven by langevin equation with levy noise and those named topological levy processes occurring in systems with topological complexity such as folded polymers or complex networks whose langevin representation is unknown and possibly nonexistent our major finding is that both above classes of processes stay in affinity and may share common stationary probability density even if their detailed dynamical behavior look different this near equilibrium observation seems to be generic to a broad class of levy noise driven processes such as e g superdiffusion on folded polymers geophysical flows and even climatic changes

multivariate approximation by pde splines this work deals with an approximation method for multivariate functions from data constituted by a given data point set and a partial differential equation pde the solution of our problem is called a pde spline we establish a variational characterization of the pde spline and a convergence result of it to the function which the data are obtained we estimate the order of the approximation error and finally we present an example to illustrate the fitting method

global heat kernel estimates for symmetric jump processes in this paper we study sharp heat kernel estimates for a large class of symmetric jump type processes in r d for all t 0 a prototype of the processes under consideration are symmetric jump processes on r d with jumping intensity

an uncertainty principle for fermions with generalized kinetic energy we derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one particle hamiltoniansh f i v x acting onl2 n these bounds are then used to derive a lower bound on the kinetic energy sum limits j 1 n left langle psi f i nabla j psi right rangle for ann fermion wavefunction we discuss two examples in more detail f p p andf p p2 m2 1 2 m both in three dimensions

on the problem of permissible covariance and variogram models the covariance and variogram models ordinary or generalized are important statistical tools used in various estimation and simulation techniques which have been recently applied to diverse hydrologic problems for example the efficacy of kriging a method for interpolating filtering or averaging spatial phenomena depends to a large extent on the covariance or variogram model chosen the aim of this article is to provide the users of these techniques with convenient criteria that may help them to judge whether a function which arises in a particular problem and is not included among the known covariance or variogram models is permissible as such a model this is done by investigating the properties of the candidate model in both the space and frequency domains in the present article this investigation covers stationary random functions as well as intrinsic random functions i e nonstationary functions for which increments of some order are stationary then based on the theoretical results obtained a procedure is outlined and successfully applied to a number of candidate models in order to give to this procedure a more practical context we employ stereological equations that essentially transfer the investigations to one dimensional space together with approximations in terms of polygonal functions and fourier bessel series expansions there are many benefits and applications of such a procedure polygonal models can be fit arbitrarily closely to the data also the approximation of a particular model in the frequency domain by a fourier bessel series expansion can be very effective this is shown by theory and by example

isotropic spectral additive models of the covariogram summary a class of additive covariance models of an isotropic random process is proposed motivated by the spectral representation of the covariance function model parameters are estimated by using a special case of the minimum norm quadratic estimation estimator whose asymptotic moments have convenient expressions in terms of spectral densities fitting a model in this class is equivalent to fitting an additive model of the spectral density the class of spectral additive models proposed is dense in the set of summable covariance functions having a spectral density allowing approximately unbiased estimation of an arbitrary covariance function and its spectral density theoretical results are supported by numerical comparison with commonly used models a procedure to assist model selection is proposed the techniques are illustrated with an application to contaminant data

sharp heat kernel estimates for relativistic stable processes in open sets in this paper we establish sharp two sided estimates for the transition densities of relativistic stable processes i e for the heat kernels of the operators m m2 2 in c1 1 open sets here m 0 and 0 2 the estimates are uniform in m 0 m for each fixed m 0 letting m 0 we recover the dirichlet heat kernel estimates for 2 2 in c1 1 open sets obtained in 14 sharp two sided estimates are also obtained for green functions of relativistic stable processes in bounded c1 1 open sets

heavy tailed targets and ab normal asymptotics in diffusive motion abstract we show that under suitable confinement conditions the ordinary fokker planck equation may generate non gaussian heavy tailed probability density functions pdfs like for example cauchy or more general levy stable distributions in its long time asymptotics in fact all heavy tailed pdfs known in the literature can be obtained this way for the underlying diffusion type processes our main focus is on their transient regimes and specifically the crossover features when an initially infinite number of pdf moments decreases to a few or none at all the time dependence of the variance if in existence t with 0 2 may in principle be interpreted as a signature of subdiffusive normal diffusive or superdiffusive behavior under confining conditions the exponent is generically well defined in substantial periods of time however there is no indication of any universal time rate hierarchy due to a proper choice of the driver and or external potential

levy flights in an infinite potential well as a hypersingular fredholm problem we study levy flights with arbitrary index 0 2 inside a potential well of infinite depth such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems the major technical tool is a transformation of the eigenvalue problem for initial fractional schrodinger equation into that for fredholm integral equation with hypersingular kernel the latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain d reducing the problem to the spectrum of a matrix of infinite dimensions the eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

asymptotically efficient prediction of a random field with a misspecified covariance function on peut obtenir les meilleurs predicteurs lineaires sans biais d un champ aleatoire si la fonction de covariance du champ aleatoire est specifiee correctement

fixed domain asymptotics for spatial periodograms abstract the periodogram for a spatial process observed on a lattice is often used to estimate the spectral density the bases for such estimators are two asymptotic properties that periodograms commonly possess 1 the periodogram at a particular frequency is approximately unbiased for the spectral density and 2 the correlation of the periodogram at distinct frequencies is approximately zero for spatial data it is often appropriate to use fixed domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases using fixed domain asymptotics this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results but by appropriately filtering the data before computing the periodogram it is possible to obtain results similar to the standard asymptotic results for spatial periodograms

one electron molecules with relativistic kinetic energy properties of the discrete spectrum we discuss the discrete spectrum of the operator h k c left hbar 2 c 2 delta m 2 c 4 right 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits k 1 k z k e 2 left x r k right 1 more specifically we study 1 the behaviour of the eigenvalues when the internuclear distances contract 2 the existence of ac independent lower bound forhk c mc2 3 the nonrelativistic limit of the eigenvalues ofhk c mc2

levy targeting and the principle of detailed balance we investigate confining mechanisms for levy flights under premises of the principle of detailed balance in this case the master equation of the jump type process admits a transformation to the levy schrodinger semigroup dynamics akin to a mapping of the fokker planck equation into the generalized diffusion equation this sets a correspondence between above two stochastic dynamical systems within which we address a stochastic targeting problem for an arbitrary stability index e 0 2 of symmetric levy drivers namely given a probability density function specify the semigroup potential and thence the jump type dynamics for which this pdf is actually a long time asymptotic target solution of the master equation here an asymptotic behavior of different motion scenarios ceases to depend on that is exemplified by considering gaussian and cauchy family target pdfs a complementary problem of the reverse engineering is analyzed given a priori a semigroup potential quantify how sensitive upon the choice of the driver is an asymptotic behavior of solutions of the associated master equation and thus an invariant pdf itself this task is accomplished for so called family of levy oscillators

survey of shape parameterization techniques for high fidelity multidisciplinary shape optimization a survey is provided of shape parameterization techniques formultidisciplinary optimization and some emerging ideas are highlighted the survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high e delity analysis tools such as computational e uid dynamics and computational structural mechanics the suitability criteria are based on the efe ciency effectiveness ease of implementation and availability of analytical sensitivities for geometry and grids a section on sensitivity analysis grid regeneration and grid deformation techniques is also provided