Remez-type inequality for discrete sets

The classical Remez inequality bounds the maximum of the absolute value of a polynomial P(x) of degree d on [−1, 1] through the maximum of its absolute value on any subset Z of positive measure in [−1, 1]. Similarly, in several variables the maximum of the absolute value of a polynomial P(x) of degree d on the unit cube Q ₁ ⁿ ⊂ ℝ ⁿ can be bounded through the maximum of its absolute value on any subset Z ⊂ Q ₁ ⁿ of positive n-measure. The main result of this paper is that the n-measure in the Remez inequality can be replaced by a certain geometric invariant ω _d (Z) which can be effectively estimated in terms of the metric entropy of Z and which may be nonzero for discrete and even finite sets Z.     

Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

On Sequential Fixed-Size Confidence Regions for the Mean Vector

In order to construct a fixed-size confidence region for the mean vector of an unknown distribution functionF, a new purely sequential sampling strategy is proposed first. For this new procedure, under some regularity conditions onF, the coverage probability is shown (Theorem 2.1) to be at least (1−α)−Bα²d²+o(d²) asd→0, where (1−α) is the preassigned level of confidence,Bis an appropriate functional ofF, and 2dis the preassigned diameter of the proposed spherical confidence region for the mean vector ofF. An accelerated version of the stopping rule is also provided with the analogous second-order characteristics (Theorem 3.1). In the special case of ap-dimensional normal random variable, analogous purely sequential and accelerated sequential procedures as well as a three-stage procedure are briefly introduced together with their asymptotic second-order characteristics.     

Regularity of solutions of Poisson’s equation in multiplier spaces

The most important result stated in this paper is to show that the solutions of the Poisson equation −Δu = f, where f ∈ (Ḣ¹(ℝ ^d ) → (Ḣ⁻¹(ℝ ^d )) is a complex-valued distribution on ℝ ^d , satisfy the regularity property D ^k u ∈ (Ḣ¹ → Ḣ⁻¹) for all k, |k| = 2. The regularity of this equation is well studied by Maz’ya and Verbitsky [12] in the case where f belongs to the class of positive Borel measures.      

Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems

Summary. In this work, new interpolation error estimates have been derived for some well-known interpolators in the quasi-norms. The estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations. In particular, by using these estimates, we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2-d and 3-d p-Laplacian.Mathematics Subject Classification (1991): 65N30     

?-measure for continuous state branching processes and its application

In this paper, we first give a direct construction of the ℕ-measure of a continuous state branching process. Then we prove, with the help of this ℕ-measure, that any continuous state branching process with immigration can be constructed as the independent sum of a continuous state branching process (without immigration), and two immigration parts (jump immigration and continuum immigration). As an application of this construction of a continuous state branching process with immigration, we give a proof of a necessary and sufficient condition, first stated without proof by M. A. Pinsky [Bull. Amer. Math. Soc., 1972, 78: 242–244], for a continuous state branching process with immigration to a proper almost sure limit. As another application of the ℕ-measure, we give a “conceptual” proof of an L log L criterion for a continuous state branching process without immigration to have an L ¹-limit first proved by D. R. Grey [J. Appl. Prob., 1974, 11: 669–677].     

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ ^d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions. This work has been supported by the Deutsche Forschungsgemeinschaft (Da 360/1-1)     

DISCRETE-TIME MARTINGALES WITH SPATIAL PARAMETERS

Our analysis of a certain stochastic difference equation driven by a martingale k?M(x,k) that depends on a spatial parameter x∈R ^d requires some regularity properties of the underlying martingale be satisfied. Because of their independent interest, we present these regularity properties in this article. We study first the continuity and Lipschitz continuity properties under corresponding conditions on the quadratic covariation of the martingale. We follow this with differentiability and integrability properties. Our analysis of the stochastic difference equation requires a discrete-time version of Itô's formula. The discrete-time Itô formula we have derived involves a martingale transform term. The purpose of the final section is to introduce linear and nonlinear martingale transforms and analyze their properties.     

Radial Fourier multipliers in high dimensions

Given a fixed p≠2, we prove a simple and effective characterization of all radial multipliers of FL^p( \mathbbR^d ) \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) , provided that the dimension d is sufficiently large. The method also yields new L ^q space-time regularity results for solutions of the wave equation in high dimensions.     

Analysis of a discontinuous galerkin method for heterogeneous diffusion problems with low‐regularity solutions

We study the convergence of the Symmetric Weighted Interior Penalty discontinuous Galerkin method for heterogeneous diffusion problems with low‐regularity solutions only belonging to W^{2, p} with p ∈ (1,2]. In 2d, we infer an optimal algebraic convergence rate. In any space dimension d, we achieve the same result for p > 2 d/( d + 2). We also prove convergence without algebraic rates for exact solutions only belonging to the energy space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

On Sobolev solutions of poisson equations in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with a parameter

Smoothness with respect to a parameter is established under mild assumptions on the regularity of coefficients for Sobolev solutions of the Poisson equations in the whole ℝ ^d in the “ergodic case.” An assertion of this kind serves as one of the key tools in diffusion approximation and some other limit theorems. Bibliography: 12 titles.     

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

We establish quantitative homogenization, large‐scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well‐approximated by harmonic functions on ℝ^d. This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher‐order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville‐type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o(|x|^k+1) as |x| → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil‐Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ. © 2018 Wiley Periodicals, Inc.     

Hadwiger's covering conjecture and low dimensional dual cyclic polytopes

LetK be a convex body in a Euclideand-spaceE ^d withd1. In 1957, H. Hadwiger conjectured thatK can always be covered by 2 ^d smaller homothetic copies ofK. We verify this conjecture in the case thatK is the polar of a cyclicd-polytope andd=3, 4 and 5.     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

Cyclic polytopes and d-order curves

A cyclic d-polytope is a convex polytope combinatorially equivalent to the convex hull of a finite subset of a d-order curve in R ^d. We give an affirmative answer to a conjecture of M. A. Perles [4] by proving that every even-dimensional cyclic polytope occurs in this way: its set of vertices can always be extended to a d-order curve.     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Kinetic Spanners in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

We present a new (1+ε)-spanner for sets of n points in ℝ ^d . Our spanner has size O(n/ε ^d−1) and maximum degree O(log  ^d n). The main advantage of our spanner is that it can be maintained efficiently as the points move: Assuming that the trajectories of the points can be described by bounded-degree polynomials, the number of topological changes to the spanner is O(n ²/ε ^d−1), and using a supporting data structure of size O(nlog  ^d n), we can handle events in time O(log  ^d+1 n). Moreover, the spanner can be updated in time O(log n) if the flight plan of a point changes. This is the first kinetic spanner for points in ℝ ^d whose performance does not depend on the spread of the point set.     

Fast Kalman Filtering and Forward–Backward Smoothing via a Low-Rank Perturbative Approach

Kalman filtering-smoothing is a fundamental tool in statistical time-series analysis. However, standard implementations of the Kalman filter-smoother require O(d³) time and O(d²) space per time step, where d is the dimension of the state variable, and are therefore impractical in high-dimensional problems. In this article we note that if a relatively small number of observations are available per time step, the Kalman equations may be approximated in terms of a low-rank perturbation of the prior state covariance matrix in the absence of any observations. In many cases this approximation may be computed and updated very efficiently (often in just O(k²d) or O(k²d + kdlog?d) time and space per time step, where k is the rank of the perturbation and in general k ? d), using fast methods from numerical linear algebra. We justify our approach and give bounds on the rank of the perturbation as a function of the desired accuracy. For the case of smoothing, we also quantify the error of our algorithm because of the low-rank approximation and show that it can be made arbitrarily low at the expense of a moderate computational cost. We describe applications involving smoothing of spatiotemporal neuroscience data. This article has online supplementary material.