Divided Differences

Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.Comment: 24 page

Approximation orders of shift-invariant subspaces of W2s(Rd)W^s_2({\Bbb R}^d)

We extend the existing theory of approximation orders provided by shift-invariant subspaces of $L_2$ to the setting of Sobolev spaces, provide treatment of $L_2$ cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.Comment: 49 page

Box spline prewavelets of small support

The purpose of this paper is the construction of bi- and trivariate prewavelets from box-spline spaces, \ie\ piecewise polynomials of fixed degree on a uniform mesh. They have especially small support and form Riesz bases of the wavelet spaces, so they are stable. In particular, the supports achieved are smaller than those of the prewavelets due to Riemenschneider and Shen in a recent, similar constructio

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinite norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples

Zonotopal algebra

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case n\eqbd\rank X \ll N. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi

Universal resonant ultracold molecular scattering

The elastic scattering amplitudes of indistinguishable, bosonic, strongly-polar molecules possess universal properties at the coldest temperatures due to wave propagation in the long-range dipole-dipole field. Universal scattering cross sections and anisotropic threshold angular distributions, independent of molecular species, result from careful tuning of the dipole moment with an applied electric field. Three distinct families of threshold resonances also occur for specific field strengths, and can be both qualitatively and quantitatively predicted using elementary adiabatic and semi-classical techniques. The temperatures and densities of heteronuclear molecular gases required to observe these univeral characteristics are predicted. PACS numbers: 34.50.Cx, 31.15.ap, 33.15.-e, 34.20.-bComment: 4 pages, 5 figure

Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching

Load Balancing in Large-Scale Systems with Multiple Dispatchers

Load balancing algorithms play a crucial role in delivering robust application performance in data centers and cloud networks. Recently, strong interest has emerged in Join-the-Idle-Queue (JIQ) algorithms, which rely on tokens issued by idle servers in dispatching tasks and outperform power-of-$d$ policies. Specifically, JIQ strategies involve minimal information exchange, and yet achieve zero blocking and wait in the many-server limit. The latter property prevails in a multiple-dispatcher scenario when the loads are strictly equal among dispatchers. For various reasons it is not uncommon however for skewed load patterns to occur. We leverage product-form representations and fluid limits to establish that the blocking and wait then no longer vanish, even for arbitrarily low overall load. Remarkably, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck. Motivated by the above issues, we introduce two enhancements of the ordinary JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We prove that these extensions can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles. Extensive simulation experiments demonstrate that the asymptotic results are highly accurate, even for moderately sized systems

Hyper-Scalable JSQ with Sparse Feedback

Load balancing algorithms play a vital role in enhancing performance in data centers and cloud networks. Due to the massive size of these systems, scalability challenges, and especially the communication overhead associated with load balancing mechanisms, have emerged as major concerns. Motivated by these issues, we introduce and analyze a novel class of load balancing schemes where the various servers provide occasional queue updates to guide the load assignment. We show that the proposed schemes strongly outperform JSQ($d$) strategies with comparable communication overhead per job, and can achieve a vanishing waiting time in the many-server limit with just one message per job, just like the popular JIQ scheme. The proposed schemes are particularly geared however towards the sparse feedback regime with less than one message per job, where they outperform corresponding sparsified JIQ versions. We investigate fluid limits for synchronous updates as well as asynchronous exponential update intervals. The fixed point of the fluid limit is identified in the latter case, and used to derive the queue length distribution. We also demonstrate that in the ultra-low feedback regime the mean stationary waiting time tends to a constant in the synchronous case, but grows without bound in the asynchronous case