The Hoffman-Wielandt inequality in infinite dimensions

The Hoffman-Wielandt inequality for the distance between the eigenvalues of two normal matrices is extended to Hilbert-Schmidt operators. Analogues for other norms are obtained in a special case.     

An inequality for a linear form in the logarithms of algebraic numbers

Let ln ₁, ..., ln _m–1 be the logarithms of fixed algebraic numbers which are linearly independent over the field of rational numbers, b₁, ..., b_m–1 rational integers, > 0. A bound from below is deduced for the height of the algebraic number _m under the condition that ¦b₁ ln ₁+...+_bm–1ln _m– ¦ < exp {–H},H=max ¦ b _k ¦ >0.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 681–689, June, 1969.     

Null distribution of the sum of squared z-transforms in testing complete independence

Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.     

On Pickands coordinates in arbitrary dimensions

Pickands coordinates were introduced as a crucial tool for the investigation of bivariate extreme value models. We extend their definition to arbitrary dimensions and, thus, we can generalize many known results for bivariate extreme value and generalized Pareto models to higher dimensions and arbitrary extreme value margins.In particular we characterize multivariate generalized Pareto distributions (GPs) and spectral δ-neighborhoods of GPs in terms of best attainable rates of convergence of extremes, which are well-known results in the univariate case. A sufficient univariate condition for a multivariate distribution function (df) to belong to the domain of attraction of an extreme value df is derived. Bounds for the variational distance in peaks-over-threshold models are established, which are based on Pickands coordinates.     

NONLINEAR PSEUDOPARABOLIC EQUATIONS IN ARBITRARY DIMENSIONS

1.IntroductionAsiswellknownthenonlinearpseudoparabolicequationsweresuggestedandstudiedinconnectionwithmanypracticalproblemsatthebeginningofthe1970s.AmongthemtheBBMequationat box~abet~0(1'1)isthemostimportant,itwassuggestedinconnectionwiththepropagationoflongwavesinnonlineardispersivesystemsbyT.B.Benjaminetallljin1972.In1977,L.A.Medeirosetall2]studiedtheweaksolutionoftheinitialboundaryvalueproblemforageneralizedformof(1.1)RevisedJanuary,25,1995.*ThisworkissupportedbytheNationalScienceFOun…     

Integration and approximation in arbitrary dimensions

We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ^−p for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{^-1} in these bounds. We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents, and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same as the ‐exponent for d=1, whereas for the third space it is 2. For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations. This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces. For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Hardy-type inequality in two dimensions

Necessary and sufficient conditions are given for a weighted norm inequality for the sum of two-dimensional Hardy-type integral operators with not necessarily non-negative coefficients.     

Removing logarithms in Poisson approximation to the partial sum process of a Markov chain

He and Xia (1997, Stochastic Processes Appl. 68, pp. 101–111) gave some error bounds for a Wasserstein distance between the distributions of the partial sum process of a Markov chain and a Poisson point process on the positive half-line. However, all these bounds increase logarithmically with the mean of the Poisson point process. In this paper, using the coupling method and a general deep result for estimating the errors of Poisson process approximation in Brown and Xia (2001, Ann. Probab. 29, pp. 1373–1403), we give a new error bound for the above Wasserstein distance. In contrast to the previous results of He and Xia (1997), our new error bound has no logarithm anymore and it is bounded and asymptotically remains constant as the mean increases.     

A Gauss sum estimate in arbitrary finite fields

We establish bounds on exponential sums ${\sum }_{x\in {\mathbb{F}}_q}\psi (x^n)$ where $q=p^m$, p prime, and ψ an additive character on ${\mathbb{F}}_q$. They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order $(m?2)$. More precisely, a non-trivial estimate is obtained provided n satisfies $\gcd (n,\frac{q?1}{p^{\nu }?1})<p^{?\nu }{q}^{1?\epsilon }$ for all $1?\nu <m$, $\nu |m$, where $\epsilon >0$ is arbitrary. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the optimal interpoint distance sum inequality

Let \({{\left\{x_{1}, \dots, x_{n}\right\}\subset \mathbb{R}^2}}\) be a set of points in the unit circle. It is shown that

$\sum\limits^{n}_{i=1}{\min_{j \neq i}{\left\|x_{i} - x_{j}\right\|^2}}\leq9,$

which is best possible and improves earlier results by Arpacioglu and Haas and Xia and Liu.

     

A bilinear extension inequality in two dimensions

We provide sharp decay estimates for circular averages of a certain bilinear extension operator on .     

A Markov-type inequality for arbitrary plane continua

Markov's inequality is

for all polynomials . We prove a precise version of this inequality with an arbitrary continuum in the complex plane instead of the interval .

     

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.     

An approximation algorithm for multidimensional assignment problems minimizing the sum of squared errors

Given a complete k-partite graph G=(V₁,V₂,…,V_k;E) satisfying |V₁|=|V₂|=?=|V_k|=n and weights of all  k-cliques of G, the  k-dimensional assignment problem finds a partition of vertices of G into a set of (pairwise disjoint) n k-cliques that minimizes the sum total of weights of the chosen cliques. In this paper, we consider a case in which the weight of a clique is defined by the sum of given weights of edges induced by the clique. Additionally, we assume that vertices of G are embedded in the d-dimensional space Q^d and a weight of an edge is defined by the square of the Euclidean distance between its two endpoints. We describe that these problem instances arise from a multidimensional Gaussian model of a data-association problem.We propose a second-order cone programming relaxation of the problem and a polynomial time randomized rounding procedure. We show that the expected objective value obtained by our algorithm is bounded by (5/2−3/k) times the optimal value. Our result improves the previously known bound (4−6/k) of the approximation ratio.     

Approximation orders for natural splines in arbitrary dimensions

Based on variational properties, we generalize the approximation properties of the univariate natural cubic spline to splines in arbitrary dimensions.

     

Lax-Wendroff-type schemes of arbitrary order in several space dimensions

^{The second-order accurate Lax–Wendroff scheme is basedon the first three terms of a Taylor expansion in time in whichthe time derivatives are replaced by space derivatives usingthe governing evolution equations. The space derivatives arethen approximated by central differencing. In this paper, weextend this idea and truncate the Taylor expansion at an arbitraryorder. One main building block is the so-called Cauchy–Kovalevskayaprocedure to replace all the time derivatives by space derivativeswhich can be formulated for a general system of linear equationswith arbitrary order and in two- or three-space dimensions.The linear case is the main focus of this paper because theproposed high-order schemes are good candidates for the approximationof linear wave motion over long distances and times with importantapplications in aeroacoustics and electromagnetics. The stabilityand the efficiency of Lax–Wendroff-type schemes are examined.The numerical results are compared with a standard scheme foraeroacoustical applications with respect to their quality andthe computational effort. The extensions of the schemes to generalgrids, nonconstant and nonlinear cases are alsoaddressed.}     

Nonparametric estimation of an extreme-value copula in arbitrary dimensions

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Capéraà-Fougères-Genest estimator was introduced by D. Zhang, M. T. Wells and L. Peng [Nonparametric estimation of the dependence function for a multivariate extreme-value distribution, Journal of Multivariate Analysis 99 (4) (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of the estimator requires the choice of a number of weight functions on the simplex, the issue of their optimal selection being left unresolved.A new, simplified representation of the CFG-estimator combined with standard empirical process theory provides the means to uncover its asymptotic distribution in the space of continuous, real-valued functions on the simplex. Moreover, the ordinary least-squares estimator of the intercept in a certain linear regression model provides an adaptive version of the CFG-estimator whose asymptotic behavior is the same as if the variance-minimizing weight functions were used. As illustrated in a simulation study, the gain in efficiency can be quite sizable.     

Kinematical conservation laws in a space of arbitrary dimensions

In a large number of physical phenomena, we find propagating surfaces which need mathematical treatment. In this paper, we present the theory of kinematical conservation laws (KCL) in a space of arbitrary dimensions, i.e., d-D KCL, which are equations of evolution of a moving surface Ω_t in d-dimensional x-space, where x = (x ₁, x ₂,..., x _d) ∈ R^d. The KCL are derived in a specially defined ray coordinates (ξ = (ξ₁, ξ₂,..., ξ_d?1), t), where ξ₁, ξ₂,..., ξ_d?1 are surface coordinates on Ω_t and t is time. KCL are the most general equations in conservation form, governing the evolution of Ω_t with physically realistic singularities. A very special type of singularity is a kink, which is a point on Ω_t when Ω_t is a curve in R² and is a curve on Ω_t when it is a surface in R³. Across a kink the normal n to Ω_t and normal velocity m on Ω_t are discontinuous.