Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ? ^d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ? _R ≈(??λ I)^?1, where ? is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ? ^d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e ^{?i κ‖x‖}/‖x‖, x∈? ^d , κ∈?₊, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ?|log?ε|²? n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n ²), even in the case κ=O(n). Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green’s kernels e ^?λ‖x‖/‖x‖, x∈?³, in the case of Newton (λ=0), Yukawa (λ∈?₊), and Helmholtz (λ=i κ,?κ∈?₊) potentials, as well as for the kernel functions 1/‖x‖ and 1/‖x‖ ^d?2, x∈? ^d , in higher dimensions d>3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse ? _R ≈(?Δ)^?1, with 3≤d≤50.     

On the Approximation by the Iterates of Stancu Operators and Its Modified Operators

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Explosion of Jump-Type Symmetric Dirichlet Forms on ℝ d

We give a sufficient condition for a class of jump-type symmetric Dirichlet forms on ? ^d to be conservative in terms of the jump kernel and the associated measure. Our condition allows the coefficients dominating big jumps to be unbounded. We derive the conservativeness for Dirichlet forms related to symmetric stable processes. We also show that our criterion is sharp by using time changed Dirichlet forms. We finally remark that our approach is applicable to jump-diffusion type symmetric Dirichlet forms on ? ^d .     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

The Neumann Problem on Unbounded Domains of ℝ d and Stochastic Variational Inequalities

ABSTRACT

An elliptic equation with Neumann boundary conditions and unbounded drift coefficients is studied in a space L ²(? ^d , ν) where ν is an invariant measure. The corresponding semigroup generated by the elliptic operator is identified with the transition semigroup associated with a stochastic variational inequality.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Heterogeneous ubiquitous systems in ℝ d and Hausdorff dimension

Let {x_n}_n∈ℕ be a sequence in [0, 1]^d , {λ_n}n∈ℕ a sequence of positive real numbers converging to 0, and δ > 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the form

On Borsuk's Problem in ℝ3

Mathematical Notes -     

Dirichlet–Neumann Problem for the Biharmonic Equation in Exterior Domains

Differential Equations - We study the uniqueness and the asymptotic expansion of the solution of the mixed Dirichlet–Neumann problem for the biharmonic equation in the exterior of a compact...     

Dirichlet and Neumann boundary conditions: What is in between?

Given an admissible measure μ on where is an open set, we define a realization of the Laplacian in with general Robin boundary conditions and we show that generates a holomorphic C ₀ -semigroup on which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form . We also show that if contains smooth functions, then μ is of the form (where σ is the (n-1)-dimensional Hausdorff measure and β a positive measurable bounded function on ); i.e. we have the classical Robin boundary conditions. RID="h1" ID="h1"Dédié à Philippe Bénilan RID="*" ID="h1"This work is part of the DGF-Project: "Regularit?t und Asymptotik für elliptische und parabolische Probleme".     

On simplices in diameter graphs in ℝ4

Agraph G is a diameter graph in ?^d if its vertex set is a finite subset in ?^d of diameter 1 and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph G in ?⁴ contains the complete subgraph K on five vertices, then any triangle in G shares a vertex with K. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in ?⁴, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than 1.

     

On the representation of biharmonic functions with singularities in ℝ n

Biharmonic functions are defined on Euclidean spaces, Riemannian manifolds, infinite trees, and more generally on abstract harmonic spaces. In this note, we consider biharmonic functions b defined on annular sets Ω \ K and obtain Laurent-type decompositions for b in the Euclidean spaces and in infinite trees. Particular importance is given to the investigation when b extends as a distribution on Ω.     

On the Douglas–Dirichlet functional and harmonic quasiconformal mappings

In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.     

On the structure of incoming and outgoing subspaces for a wave equations in ℝ n

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in ℝ ⁿ . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 708–712, May, 1999.     

Measures and their dimension spectrums for cookie-cutter sets in ℝd

The characteristics of cookie-cutter sets in ℝ^d are investigated. A Bowen's formula for the Hausdorff dimension of a cookie-cutter set in terms of the pressure function is derived. The existence of self-similar measures, conformal measures and Gibbs measures on cookie-cutter sets is proved. The dimension spectrum of each of these measures is analyzed. In addition, the locally uniformly α-dimensional condition and the fractal Plancherel Theorem for these measures are shown. Finally, the existence of order-two density for the Hausdorff measure of a cookie-cutter set is proved. This project is supported by the National Natural Science Foundation of China.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

On Properties of the ξ-Function in Semi-finite von Neumann Algebras

Versions of the Birman-Schwinger principle for (relative) trace class perturbation problems of dissipative operators in a semi-finite von Neumann algebra and self-adjoint operators affiliated with the algebra are obtained and applied in the study of the spectral shift function.