Infinitesimals in the foundations of Newton's mechanics

This paper discusses two concepts of “moment” (infinitesimal) used successively by Newton in his calculus and relates these two concepts to the two concepts of force that Newton presented in Law II and Def. VIII of the Principia, to which the approximations to the action of a centripetal force known as the polygonal and parabolic models are considered to be related. It is shown that in the context of the application of the calculus to mechanics, the transition in the use of these concepts of “moment” took place in 1684, between the writing of De Motu and its first revision.     

Duality for optimization and best approximation over finite intersections

Recently Deutsch, Li and Swetits [2] have studied, in Hilbert space, a dual problem (Q_m ) to the primal problem (P) of minimization of a special class of convex functions f over the intersection of m closed convex sets, where m is finite. In the first part of this paper we obtain, in a locally convex space, some results on problem (Q_m ) and on its relations with the usual Lagrangian dual problem (Q) to (P) (studied in [9]), in the case when (P) has a solution. In the second part we give some applications to duality for the distance to the intersection of m closed convex sets in a normed linear space, in the case when a nearest point exists. Most of our results seem to be new even in the particular cases studied in [9] (the case m = 1), [l] (duality formulas for the distance to the intersection of m closed half-spaces in a normed linear space) and [2].     

The polynomial Fourier transform with minimum mean square error for noisy data

In 2006, Naoki Saito proposed a Polyharmonic Local Fourier Transform (PHLFT) to decompose a signal f∈L²(Ω) into the sum of a polyharmonic componentu and a residualv, where Ω is a bounded and open domain in R^d. The solution presented in PHLFT in general does not have an error with minimal energy. In resolving this issue, we propose the least squares approximant to a given signal in L²([−1,1]) using the combination of a set of algebraic polynomials and a set of trigonometric polynomials. The maximum degree of the algebraic polynomials is chosen to be small and fixed. We show in this paper that the least squares approximant converges uniformly for a Hölder continuous function. Therefore Gibbs phenomenon will not occur around the boundary for such a function. We also show that the PHLFT converges uniformly and is a near least squares approximation in the sense that it is arbitrarily close to the least squares approximant in L² norm as the dimension of the approximation space increases. Our experiments show that the proposed method is robust in approximating a highly oscillating signal. Even when the signal is corrupted by noise, the method is still robust. The experiments also reveal that an optimum degree of the trigonometric polynomial is needed in order to attain the minimal l² error of the approximation when there is noise present in the data set. This optimum degree is shown to be determined by the intrinsic frequency of the signal. We also discuss the energy compaction of the solution vector and give an explanation to it.     

Improving the Data Augmentation Algorithm in the Two-Block Setup

The data augmentation (DA) approach to approximate sampling from an intractable probability density f_X is based on the construction of a joint density, f_{X, Y}, whose conditional densities, f_X|Y and f_Y|X, can be straightforwardly sampled. However, many applications of the DA algorithm do not fall in this “single-block” setup. In these applications, X is partitioned into two components, X = (U, V), in such a way that it is easy to sample from f_Y|X, f_{U|V, Y}, and f_{V|U, Y}. We refer to this alternative version of DA, which is effectively a three-variable Gibbs sampler, as “two-block” DA. We develop two methods to improve the performance of the DA algorithm in the two-block setup. These methods are motivated by the Haar PX-DA algorithm, which has been developed in previous literature to improve the performance of the single-block DA algorithm. The Haar PX-DA algorithm, which adds a computationally inexpensive extra step in each iteration of the DA algorithm while preserving the stationary density, has been shown to be optimal among similar techniques. However, as we illustrate, the Haar PX-DA algorithm does not lead to the required stationary density f_X in the two-block setup. Our methods incorporate suitable generalizations and modifications to this approach, and work in the two-block setup. A theoretical comparison of our methods to the two-block DA algorithm, a much harder task than the single-block setup due to nonreversibility and structural complexities, is provided. We successfully apply our methods to applications of the two-block DA algorithm in Bayesian robit regression and Bayesian quantile regression. Supplementary materials for this article are available online.     

A compatible‐incompatible decomposition of symmetric tensors in Lp with application to elasticity

In this paper, we prove the Saint‐Venant compatibility conditions in L^p for p∈(1,+∞), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in L^p are provided. We also use the Helmholtz decomposition in L^p to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence‐free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami‐type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < p < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (p = 2), to the full range p∈(1,+∞). Copyright © 2015 John Wiley & Sons, Ltd.     

The stable ATA-orthogonal s-step Orthomin(k) algorithm with the CADNA library

The major drawback of the s-step iterative methods for nonsymmetric linear systems of equations is that, in the floating-point arithmetic, a quick loss of orthogonality of s-dimensional direction subspaces can occur, and consequently slow convergence and instability in the algorithm may be observed as s gets larger than 5. In [18], Swanson and Chronopoulos have demonstrated that the value of s in the s-step Orthomin(k) algorithm can be increased beyond s=5 by orthogonalizing the s direction vectors in each iteration, and have shown that the A^TA-orthogonal s-step Orthomin(k) is stable for large values of s (up to s=16). The subject of this paper is to show how by using the CADNA library, it is possible to determine a good value of s for A^TA-orthogonal s-step Orthomin(k), and during the run of its code to detect the numerical instabilities and to stop the process correctly, and to restart the A^TA-orthogonal s-step Orthomin(k) in order to improve the computed solution. Numerical examples are used to show the good numerical properties. This revised version was published online in June 2006 with corrections to the Cover Date.     

Poisson convergence in the restricted k‐partitioning problem

The randomized k‐number partitioning problem is the task to distribute N i.i.d. random variables into k groups in such a way that the sums of the variables in each group are as similar as possible. The restricted k‐partitioning problem refers to the case where the number of elements in each group is fixed to N/k. In the case k = 2 it has been shown that the properly rescaled differences of the two sums in the close to optimal partitions converge to a Poisson point process, as if they were independent random variables. We generalize this result to the case k > 2 in the restricted problem and show that the vector of differences between the k sums converges to a k ‐ 1‐dimensional Poisson point process. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Another universality property for crumpled cubes

Let C denote a crumpled n-cube in the n-sphere Sⁿ such that every Cantor set in its boundary is tamely embedded in Sⁿ. The main theorem shows C to be universal in the sense that however it is sewn to a crumpled n-cube D of type 2A, a large class containing most of the explicitly described examples, the resultant space is homeomorphic to Sⁿ.     

Teaching scenario planning: Lessons from practice in academe and business

In this paper, we engage with O’Brien’s [O’Brien, F.A., 2004. Scenario planning – lessons for practice from teaching and learning. European Journal of Operational Research 152, 709–722] identification of both pitfalls in teaching scenario planning and proposed remedies for these. We consider these remedies in relation to our own experience – based on our practice in both the academic and business arenas – and we highlight further pitfalls and proposed remedies. Finally, we propose the use of “hard” multi-attribute decision analysis as a complement to “soft” scenario planning, in order to allow a more formal method of strategy evaluation against a range of constructed scenarios, This approach is intended to remedy biases that are associated with holistic evaluations – such as lexicographic ranking – where undue attention is paid to particular strategic objectives at the expense of others. From this discussion, we seek to contribute to cumulative refinement of the scenario process.     

Detecting “dense” columns in interior point methods for linear programs

During the iterations of interior point methods symmetric indefinite systems are decomposed by LD̂L ^T factorization. This step can be performed in a special way where the symmetric indefinite system is transformed to a positive definite one, called the normal equations system. This approach proved to be efficient in most of the cases and numerically reliable, due to the positive definite property. It has been recognized, however, that in case the linear program contains “dense” columns, this approach results in an undesirable fill–in during the computations and the direct factorization of the symmetric indefinite system is more advantageous. The paper describes a new approach to detect cases where the system of normal equations is not preferable for interior point methods and presents a new algorithm for detecting the set of columns which is responsible for the excessive fill–in in the matrix AA ^T . By solving large–scale linear programming problems we demonstrate that our heuristic is reliable in practice. This work was supported in part by the Hungarian Scientific Research Fund OTKA K60480.