A generalized Peano Kernel Theorem for distributions of exponential decay

We investigate the problem of weighted (0,1,…,m−2,m)-interpolation (m≧2, \({m\in \mathbb{N}}\)) on the roots of the classical orthogonal polynomials. The necessary and sufficient conditions for the existence and uniqueness of this problem are established. Meanwhile, the explicit representation (characterization) of the weighted (0,1,…,m−2,m)-interpolation is given. As an application we obtain a Birkhoff type quadrature formula which is exact for the polynomials of degree at most mn. Also we investigate the error function of the new (0,1,…,m−2,m)-Birkhoff type weighted quadrature formulae using Peano kernel theorem. Some numerical examples are given to support the results.

We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a directed network in which each edge possesses a latency function describing the common latency incurred by all traffic on the edge as a function of the edge congestion. Given a rate of traffic between each pair of nodes in the network, we aspire toward an assignment of traffic to paths in which the sum of all travel times (the total latency) is minimized; however, in many settings network users are free to route their traffic in a selfish manner, without regard to the total latency. We therefore assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a selfishly motivated assignment of traffic to paths (a Nash equilibrium) will not minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy, defined as the worst possible ratio between the total latency of a Nash equilibrium and of a minimum-latency routing of the traffic.In this paper, we show that the underlying network topology plays no role in the determination of the price of anarchy. Specifically, we show that under weak hypotheses on the class of allowable edge latency functions, the worst-case ratio between the total latency of a Nash equilibrium and of a minimum-latency routing for any multicommodity flow network is achieved by a single-commodity instance on a set of parallel links. In the special case where the class of allowable latency functions includes all of the constant functions, we prove that a network with only two parallel links suffices to achieve the worst-possible ratio. Informally, these results imply that the inefficiency inherent in a flow at Nash equilibrium stems from the inability of selfish users to discern which of two competing routes is superior and not from the topological complexity arising from the diverse intersections of many paths belonging to different commoditie.Our proof techniques also give powerful methods for computing the price of anarchy with respect to an arbitrary class of latency functions. We apply these methods to function classes that have been well studied in the literature (such as degree-bounded polynomials and functions of the form $\ell(x) = (u—x)—1} that are used to model edges with capacity u), thereby achieving the first tight analyses of the price of anarchy for significant classes of latency functions outside the class of linear functions.

A simple shearlet-based 2D Radon inversion with an application to computed tomography

Efficient, accurate, multi-dimensional, numerical integration has become an important tool for approximating the integrals which arise in modern economic models built on unobserved heterogeneity, incomplete information, and uncertainty. This paper demonstrates that polynomialbased rules out-perform number-theoretic quadrature (Monte Carlo) rules both in terms of efficiency and accuracy. To show the impact a quadrature method can have on results, we examine the performance of these rules in the context of Berry, Levinsohn, and Pakes (1995)'s model of product differentiation, where Monte Carlo methods introduce considerable numerical error and instability into the computations. These problems include inaccurate point estimates, excessively tight standard errors, instability of the inner loop 'contraction' mapping for inverting market shares, and poor convergence of several state of the art solvers when computing point estimates. Both monomial rules and sparse grid methods lack these problems and provide a more accurate, cheaper method for quadrature. Finally, we demonstrate how researchers can easily utilize high quality, high dimensional quadrature rules in their own work.

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

Coded Caching, proposed by Maddah-Ali and Niesen (MAN), has the potential to reduce network traffic by pre-storing content in the users’ local memories when the network is underutilized and transmitting coded multicast messages that simultaneously benefit many users at once during peak-hour times. This paper considers the linear function retrieval version of the original coded caching setting, where users are interested in retrieving a number of linear combinations of the data points stored at the server, as opposed to a single file. This extends the scope of the authors’ past work that only considered the class of linear functions that operate element-wise over the files. On observing that the existing cache-aided scalar linear function retrieval scheme does not work in the proposed setting, this paper designs a novel coded caching scheme that outperforms uncoded caching schemes that either use unicast transmissions or let each user recover all files in the library.

Numerical integration is one of the oldest and most fascinating topics in numerical analysis. Methods of numerical integration date back to Archimedes who tried to calculate the area of a circle, which was long before the integral calculus itself was formulated. Numerical integration is, paradoxically, both simple and difficult. It is simple in that it can often be successfully accomplished by the simplest of methods. It is difficult in that for some problems it may require almost infinite amount of computer time. Unlike differentiation, integration is numerically stable process and in general there is no difficulty in computing an integral to any required accuracy permitted by the computer arithmetic or available data. In contrast to the previous Chapter, most of this Chapter is devoted to the situation where the integrand can be evaluated at any required point. Numerical integration of functions provided in form of table of values is dealt with only briefly in Section 6.1. Numerical integration in one dimension is usually referred to as quadrature. Because of the simplicity of the problem and its practical value, innumerable formulae have been developed for quadrature. Since the computers came into existence, the emphasis in numerical integration has shifted from quadrature rules, to evaluation of multiple integrals, and to development of algorithms for automatic integration.

Sampling theorem for bandlimited Hardy space functions generated by Regge problem

If I think about the genesis of this paper, my semester at Stanford in 1995 comes to mind right away. For the mathematical orphan that I was, since my advisor, Peter Henrici died before I completed my PhD, the opportunity to spend a semester at Stanford was a truly unique opportunity. As Gene and I team-taught a seminar course on quadrature rules, linear algebra, and large-scale computations, I discovered more charming aspects of quadrature rules than I thought there could be. I must admit that when Gene first talked about the importance of Gauss–Kronrod rules in estimating errors in numerical integration, I thought that, for sure, I would never be caught working on such a dull topic. Well, soon enough I started seeing quadrature rules everywhere, and their elegance and secret connections with nearly everything began to get hold of me. Even Gauss–Kronrod quadrature rules, until then quickly dismissed even from my teaching, started to show their discrete charm. Well, never say never!

Generally, solutions to functional equations involving non-vanishing delays tend to exhibit lower regularity compared to those of smooth functions. In this context, we examine a first-order Volterra integro-differential equation (VIDE) with a non-vanishing delay, delving into the characteristics of its solutions. To enhance the accuracy of traditional one-step collocation methods [1], we employ multi-step collocation techniques to obtain numerical solutions for the VIDE with non-vanishing delay. The global convergence properties of the multi-step numerical approach are scrutinized using the Peano Kernel Theorem. Subsequently, for comparative analysis, we utilize a one-step collocation method to numerically solve this equation, showcasing the effectiveness and precision of the multi-step collocation method.

Terms in the analytic expansion of the doubly averaged disturbing function for the circular restricted three-body problem using the Legendre polynomial are explicitly calculated up to the fourteenth order of semimajor axis ratio (α) between perturbed and perturbing bodies in the inner case (α<1), and up to the fifteenth order in the outer case (α>1). The expansion outcome is compared with results from numerical quadrature on an equipotential surface. Comparison with direct numerical integration of equations of motion is also presented. Overall, the high-order analytic expansion of the doubly averaged disturbing function yields a result that agrees well with the numerical quadrature and with the numerical integration. Local extremums of the doubly averaged disturbing function are quantitatively reproduced by the high-order analytic expansion even when α is large. Although the analytic expansion is not applicable in some circumstances such as when orbits of perturbed and perturbing bodies cross or when strong mean motion resonance is at work, our expansion result will be useful for analytically understanding the long-term dynamical behavior of perturbed bodies in circular restricted three-body systems.

The article investigates interior point rules which contain the midpoint as a special case, and obtains explicit bounds through the use of a Peano kernel approach and the modern theory of inequalities. Thus the simplest open Newton-Cotes rules are examined. Both Riemann-Stieltjes and Riemann integrals are evaluated with a variety of assumptions about the integrand enabling the characterisation of the bound in terms of a variety of norms. Perturbed quadrature rules are obtained through the use of Gruss, Chebychev and Lupas inequalities, producing a variety of tighter bounds. The implementation is demonstrated through the investigation of a variety of composite rules based on inequalities developed. The analysis allows the determination of the partition required that would assure that the accuracy the result would be within a prescribed error tolerance. It is demonstrated that the bounds of the approximations are equivalent to those obtained from a Peano kernel that produces Trapezoidal type rules.

Strong-form framework for solving boundary value problems with geometric nonlinearity

Although generalized linear mixed effects models have received much attention in the statistical literature, there is still no computationally efficient algorithm for computing maximum likelihood estimates for such models when there are a moderate number of random effects. Existing algorithms are either computationally intensive or they compute estimates from an approximate likelihood. Here we propose an algorithm—the spherical–radial algorithm—that is computationally efficient and computes maximum likelihood estimates. Although we concentrate on two-level, generalized linear mixed effects models, the same algorithm can be applied to many other models as well, including nonlinear mixed effects models and frailty models. The computational difficulty for estimation in these models is in integrating the joint distribution of the data and the random effects to obtain the marginal distribution of the data. Our algorithm uses a multidimensional quadrature rule developed in earlier literature to integrate the joint density. This article discusses how this rule may be combined with an optimization algorithm to efficiently compute maximum likelihood estimates. Because of stratification and other aspects of the quadrature rule, the resulting integral estimator has significantly less variance than can be obtained through simple Monte Carlo integration. Computational efficiency is achieved, in part, because relatively few evaluations of the joint density may be required in the numerical integration.

In this article, we propose an expression for the periodic first derivative of Green's function in real space. The proposed expression allows an alternative way of computing the periodic Green's operator based on periodically summing the free‐space Green's operator in terms of an appropriate quadrature rule. We provide computational examples, which show the accuracy of the proposed approach, together with reduced spurious oscillations in the solution fields.