Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth

In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter . The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in .

     

Generalized -expansions, substitution tilings, and local finiteness

For a fairly general class of two-dimensional tiling substitutions, we prove that if the length expansion is a Pisot number, then the tilings defined by the substitution must be locally finite. We also give a simple example of a two-dimensional substitution on rectangular tiles, with a non-Pisot length expansion , such that no tiling admitted by the substitution is locally finite. The proofs of both results are effectively one-dimensional and involve the idea of a certain type of generalized -transformation.

     

The probability that a slightly perturbed numerical analysis problem is difficult

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of -tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius . Besides and , this bound depends only on the dimension of the sphere and on the degree of the defining equations.

     

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter . The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by with independent of the mesh parameter , the diffusion coefficient and the exact solution of the problem.

     

Parametric Borwein-Preiss variational principle and applications

A parametric version of the Borwein-Preiss smooth variational principle is presented, which states that under suitable assumptions on a given convex function depending on a parameter, the minimum point of a smooth convex perturbation of it depends continuously on the parameter. Some applications are given: existence of a Nash equilibrium and a solution of a variational inequality for a system of partially convex functions, perturbed by arbitrarily small smooth convex perturbations when one of the functions has a non-compact domain; a parametric version of the Kuhn-Tucker theorem which contains a parametric smooth variational principle with constraints; existence of a continuous selection of a subdifferential mapping depending on a parameter.

The tool for proving this parametric smooth variational principle is a useful lemma about continuous -minimizers of quasi-convex functions depending on a parameter, which has independent interest since it allows direct proofs of Ky Fan's minimax inequality, minimax equalities for quasi-convex functions, Sion's minimax theorem, etc.

     

Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity , When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet's boundary condition.

     

On a singular perturbation problem involving a ``circular-well' potential

We study the asymptotic behavior, as a small parameter goes to 0, of the minimizers for a variational problem which involves a ``circular-well' potential, i.e., a potential vanishing on a closed smooth curve in . We thus generalize previous results obtained for the special case of the Ginzburg-Landau potential.

     

An extension of Elton's theorem to complex Banach spaces

Let 0$"> be sufficiently small. Then, for , there exists such that if are vectors in the unit ball of a complex Banach space which satisfy

(where are independent complex Steinhaus random variables), then there exists a set , with , such that

for all (). The dependence on of the threshold proportion is sharp.

     

Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

     

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

     

A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definition of entropy solution of the model in the sense of Kru kov and prove convergence of the scheme to the unique entropy solution of the problem, up to satisfaction of one of the boundary conditions.

     

Stability of disjointness preserving mappings

Let and be compact Hausdorff spaces and let . A linear mapping is called -disjointness preserving if implies that . If is a continuous or surjective -disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping satisfying . We also prove that every unbounded -disjointness preserving linear functional on is disjointness preserving.

     

Ribbon tile invariants

Let be a finite set of tiles, and a set of regions tileable by . We introduce a tile counting group as a group of all linear relations for the number of times each tile can occur in a tiling of a region . We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group.

The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group.

The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.

     

On the complexity of stochastic integration

We study the complexity of approximating stochastic integrals with error for various classes of functions. For Ito integration, we show that the complexity is of order , even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtained by the Milstein algorithm, which is almost optimal in the considered classes of functions. The Milstein algorithm uses the values of the Brownian motion and the integrand. It is bilinear in these values and is very easy to implement. For Stratonovich integration, we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of Ito integration.

     

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in for . Here is the maximum side length of mesh elements, while the number of mesh nodes does not exceed . Numerical experiments are performed to support the theoretical results.

     

The stability of the exponential equation

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an -approximately exponential function to the set of all exponential functions tends to zero as tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers .

     

Homogenization of the inviscid incompressible fluid flow through a 2D porous medium

We consider the non-stationary incompressible Euler equations in a 2D porous medium. We suppose a periodic porous medium, with the period proportional to the characteristic pore size and with connected fluid part. The flow is subject to an external force, corresponding to an inflow. We start from an initial irrotational velocity and prove that the effective filtration velocity satisfies a transient filtration law. It has similarities with Darcy's law, but it now connects the time derivative of the filtration velocity with the pressure gradient. The viscosity does not appear in the filtration law any more and the permeability tensor is determined through auxiliary problems of decomposition type. Using the limit problem, we construct the correction for the fluid velocity and prove that -norm of the error is of order . Similarly, we estimate the difference between the fluid pressure and its correction in as .

     

Numeration systems and Markov partitions from self similar tilings

Using self similar tilings we represent the elements of as digit expansions with digits in being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms.

     

Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter , known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size and the time step size . In particular, it is shown that all error bounds depend on only in some lower polynomial order for small . The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

     

Ribbon tilings and multidimensional height functions

We fix and say a square in the two-dimensional grid indexed by has color if . A ribbon tile of order is a connected polyomino containing exactly one square of each color. We show that the set of order- ribbon tilings of a simply connected region is in one-to-one correspondence with a set of height functions from the vertices of to satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph.

Using these facts, we describe a linear (in the area of ) algorithm for determining whether can be tiled with ribbon tiles of order and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order- ribbon tilings of can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order- ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.