diffusive transport across irregular and fractal walls diffusive transport of heat mass and momentum across walls with irregular and fractal geometry is discussed a configuration is considered in detail in which the transported variable diffuses across a two dimensional periodic irregular wall toward an overlying plane wall while the value of the variable over each wall is maintained at a constant value several families of self similar wall geometries with increasingly finer structure eventually leading to fractal shapes are considered in detail using an efficient numerical method that is based on conformal mapping the numerical procedure involves the iterative solution of a large system of nonlinear algebraic equations computed patterns of iso scalar contours reveal the precise effect of the shape size and total length of boundary irregularities on the local and total transport rates and illustrate the enhancement in transport efficacy with wall refinement the total rate of transport across walls with self similar irregularities is shown to be remarkably close to that across walls with random irregularities of same roughness height

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

large discrete sets in stein manifolds we construct infinite discrete subsets of stein manifolds with remarkable properties these generalize results of rosay and rudin on discrete subsets of mathbb c n

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

gradients of potential fields on rough surfaces perturbative calculation of the singularity distribution function f α for small surface dimension an exact solution to the laplace equation with dirichlet boundary conditions on a simple boundary is used in an iterative fashion to study the case of a stochastic rough surface with fractal dimension d 2 ensuremath epsilon for small ensuremath epsilon an analytic expression is derived for the spectrum of singularities f ensuremath alpha of the gradient of the potential normal to the boundary using the random multiplier approach a binomial approximation to the multiplicative process is shown to severely underestimate f ensuremath alpha for low values of ensuremath alpha

on the growth and stability of real analytic functions this paper addresses the issue of whether integrals of real analytic functions remain finite under small deformations an approach based on uniform estimates for certain classes of one dimensional integrals is introduced it is powerful enough to recover the stability properties of real integrals in two dimensions which follow from the work of karpushkin as well as produce new results in higher dimensions in dimension three the new stability results are sharp as shown by the well known example of varchenko

harmonic measure around a linearly self similar tree the authors use the concept of random multiplicative processes to help describe and understand the distribution of the harmonic measure on growing fractal boundaries the laplacian potential around a linearly self similar square koch tree is studied in detail the multiplicative nature of this potential and the consequent multifractality of the harmonic measure are discussed on prefractal stages the density d mu of the harmonic measure and the corresponding holder alpha ln d mu are well defined along the boundary except in the folds where the tangent is undefined a regularization scheme is introduced to eliminate these local effects they then consider the probability distributions p alpha d alpha of successive stages and discuss their collapse into an f alpha curve both the left and right hand sides of this curve show good convergence other studies indicate that for dla the right hand tail does not converge a brief comparison is made between the multifractality of these two cases

newton polygons and local integrability of negative powers of smooth functions in the plane let f x y be any smooth real valued function with f 0 0 0 for a sufficiently small neighborhood u of the origin we study the number sup u f x f it is known that sometimes this number can be expressed in a natural way using the newton polygon of f we provide necessary and sufficient conditions for this newton polygon characterization to hold the behavior of the integral at the supremal e is also analyzed

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

a multi dimensional resolution of singularities with applications to analysis we formulate a resolution of singularities algorithm for analyzing the zero sets of real analytic functions in dimensions geq 3 rather than using the celebrated result of hironaka the algorithm is modeled on a more explicit and elementary approach used in the contemporary algebraic geometry literature as an application we define a new notion of the height of real analytic functions compute the critical integrability index and obtain sharp growth rate of sublevel sets this also leads to a characterization of the oscillation index of scalar oscillatory integrals with real analytic phases in all dimensions

convergence of two dimensional weighted integrals a two dimensional weighted integral in r 2 is proposed as a tool for analyzing higher dimensional unweighted integrals and a necessary and sufficient condition for the finiteness of the weighted integral is obtained

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

holomorphic arcs on singularities the study of formal arcs was initiated by nash in a 1967 preprint arc spaces of smooth varieties have a rather transparent structure but difficult problems arise for arcs passing through singularities the nash conjecture on the irreducible components of such arc spaces was proved for surfaces by fernandez de bobadilla and pe pereira and for toric singularities by ishii and kollar but counter examples were found in higher dimensions by ishii and kollar de fernex and johnson and kollar here we start the study of holomorphic arcs these are holomorphic maps from the closed unit disk to a complex analytic space as one expects there is not much conceptual difference between the set of formal arcs and the set of holomorphic arcs since every formal arc can be approximated by holomorphic arcs however a formal deformation of an arc is a much more local object than a holomorphic deformation thus in many cases the space of holomorphic arcs has infinitely many connected components while the space of formal arcs always has only finitely many for a complex analytic space we define two variants the space of arcs and the space of short arcs and we study their connected components for short arcs we obtain complete answers for surface singularities and for isolated quotient singularities in all dimensions

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

effects of uncertainties in the domain on the solution of dirichlet boundary value problems a domain with possibly non lipschitz boundary is defined as a limit of monotonically expanding or shrinking domains with lipschitz boundary a uniquely solvable dirichlet boundary value problem dbvp is defined on each of the lipschitz domains and the limit of these solutions is investigated the limit function also solves a dbvp on the limit domain but the problem can depend on the sequences of domains if the limit domain is unstable with respect to the dbvp the core of the paper consists in estimates of the difference between the respective solutions of the dbvp on two close domains one of which is lipschitz and the other can be unstable estimates for starshaped as well as rather general domains are derived their numerical evaluation is possible and can be done in different ways

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

oka properties of ball complements let n 1 be an integer we prove that holomorphic maps from stein manifolds x of dimension n to the complement mathbb c n setminus l of a compact convex set l subset mathbb c n satisfy the basic oka property with approximation and interpolation if l is polynomially convex then the same holds when 2 dim x le n we also construct proper holomorphic maps immersions and embeddings x rightarrow mathbb c n with additional control of the range thereby extending classical results of remmert bishop and narasimhan

effects of uncertainties in the domain on the solution of neumann boundary value problems in two spatial dimensions an essential part of any boundary value problem is the domain on which the problem is defined the domain is often given by scanning or another digital image technique with limited resolution this leads to significant uncertainty in the domain definition the paper focuses on the impact of the uncertainty in the domain on the neumann boundary value problem nbvp it studies a scalar nbvp defined on a sequence of domains the sequence is supposed to converge in the set sense to a limit domain then the respective sequence of nbvp solutions is examined first it is shown that the classical variational formulation is not suitable for this type of problem as even a simple nbvp on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary a new definition of the nbvp is introduced to avoid this difficulty by means of reformulated natural boundary conditions then the convergence of solutions of the nbvp is demonstrated the uniqueness of the limit solution however depends on the stability property of the limit domain finally estimates of the difference between two nbvp solutions on two different but close domains are given