How ordinary elimination became Gaussian elimination

Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — which Euler did not recommend, which Legendre called “ordinary,” and which Gauss called “common” — is now named after Gauss: “Gaussian” elimination. Gauss’s name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least-squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.     

A singular function with a non-zero finite derivative

This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its generalizations and variants—the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative 1 on a subset of the Cantor set.     

Multiple solutions for elliptic systems with nonlinearities of arbitrary growth

We prove the existence of infinitely many solutions for symmetric elliptic systems with nonlinearities of arbitrary growth. Moreover, if the symmetry of the problem is broken by a small enough perturbation term, we find at least three solutions. The proofs utilise a variational setting given by de Figueiredo and Ruf in order to prove an existence's result and the “algebraic” approach based on the Pohozaev's fibering method.     

A constrained variational problem for an existence theorem of a steady vortex pair in two-phase shear flow

We prove an existence theorem for a steady vortex pair in two-phase shear flow in a planar domain. The method used is a variational principle in which the kinetic energy is maximised subject to the vorticity belonging to the weak closure of the set of rearrangements of a prescribed function, and subject to another functional representing the “generalised impulse” having a prescribed value. We prove also that when the prescribed value of the “generalised impulse” is large enough, the constrained maximiser of the kinetic energy is in fact a rearrangement of the prescribed function.     

Set partitions with restrictions

Based on finite set partitions, we introduce restrictions to the distances among the elements in each part and refine the Stirling numbers of the second kind with an extra parameter in two different ways. Combinatorial approach through distributions of “balls into boxes” is employed to establish explicit formulae.     

Computation of the invariant measure for a Lévy driven SDE: Rate of convergence

We study the rate of convergence of some recursive procedures based on some “exact” or “approximate” Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by “exact” and “approximate” Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.     

How not to characterize planar-emulable graphs

We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph)—a problem raised already in Fellows? thesis (1985) and conceptually related to the better known planar cover conjecture by Negami (1986). For over two decades, the planar emulator problem lived poorly in a shadow of Negami?s conjecture—which is still open—as the two were considered equivalent. But, at the end of 2008, a surprising construction by Rieck and Yamashita falsified the natural “planar emulator conjecture”, and thus opened a whole new research field. We present further results and constructions which show how far the planar-emulability concept is from planar-coverability, and that the traditional idea of likening it to projective embeddability is actually very out-of-place. We also present several positive partial characterizations of planar-emulable graphs.     

Well-posedness of higher-order Camassa-Holm equations

We consider higher-order Camassa-Holm equations describing exponential curves of the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane. We establish the existence of global weak solutions. We also present some invariant spaces under the action of the equation. Moreover, we prove a “weak equals strong” uniqueness result.     

Bifurcation with a two-dimensional kernel

We present a theorem ensuring the existence of local solution branches for one-parameter bifurcation problems in which the linearization at the trivial solution possesses a two-dimensional kernel. In particular, we provide a straightforward “test” that is sufficient for the existence of local solution continua. We demonstrate our abstract theorem with several concrete examples for second-order systems of elliptic partial differential equations with symmetry.     

Using Noonan–Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes

One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a “formula”, or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate functional equations, alas, with an unbounded number of “catalytic variables”, but then described a clever way, using multivariable calculus, on how to get enumeration schemes. Alas, their method becomes very complicated for r larger than 1. In the present article we describe a far simpler way to squeeze the necessary information, in polynomial time, for increasing patterns of any length and for any number of occurrences r.     

Coloring by two-way independent sets

The paper stems from an attempt to investigate a somewhat mysterious phenomenon: conditions which suffice for the existence of a “large” set satisfying certain conditions (e.g., a large independent set in a graph) often suffice (or at least are conjectured to suffice) for the existence of a covering of the ground set by few sets satisfying these conditions (in the example of independent sets in a graph this means that the graph has small chromatic number). We consider two conjectures of this type, on coloring by sets which are “two-way independent”, in the sense of belonging to a matroid and at the same time being independent in a graph sharing its ground set with the matroid. We prove these conjectures for matroids of rank 2. We also consider dual conjectures, on packing bases of a matroid, which are independent in a given graph.     

Sets versus trial sequences,Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920

The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix) of two critical letters of November 1919 by the “pure” mathematician Felix Hausdorff to the engineer and applied mathematician Richard von Mises compose about one third of the paper. The article also investigates von Mises’s ill-conceived effort to adopt measures and his misinterpretation of an influential book of Constantin Carathéodory. A short and sketchy look at the subsequent development of the standpoints of the pure and the applied mathematician—here represented by Hausdorff and von Mises—in the probability theory of the 1920s and 1930s concludes the paper.     

The economics of social computing: Some preliminary findings on healthcare organizations

Large-scale organizations have used social computing platforms for various purposes. This research focuses on how hospitals utilize these platforms to attract potential customers (which represents the “extensivity” of a social computing platform) and generate interests in specific topics (which represents the “intensivity” of a platform). Specifically, we examine the effects of size of a hospital (or “size”) and the time that the social computing platform has been in existence (or “time”) on extensivity and intensivity. Our findings show that time is a significant variable on both dimensions; whereas size affects intensivity under certain conditions. We discuss the implications of these findings, and set the stage for future research.     

On the well-posedness of a linearized plasma-vacuum interface problem in ideal compressible MHD

We study the initial-boundary value problem resulting from the linearization of the plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). We suppose that the plasma and the vacuum regions are unbounded domains and the plasma density does not go to zero continuously, but jumps. For the basic state upon which we perform linearization we find two cases of well-posedness of the “frozen” coefficient problem: the “gas dynamical” case and the “purely MHD” case. In the “gas dynamical” case we assume that the jump of the normal derivative of the total pressure is always negative. In the “purely MHD” case this condition can be violated but the plasma and the vacuum magnetic fields are assumed to be non-zero and non-parallel to each other everywhere on the interface. For this case we prove a basic a priori estimate in the anisotropic weighted Sobolev space for the variable coefficient problem.     

Wiener’s criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries

This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of R^N${\mathbb{R}}^N$ (N≥3$N\ge 3$) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.     

Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007.     

The affect heuristic

This paper introduces a theoretical framework that describes the importance of affect in guiding judgments and decisions. As used here, “affect” means the specific quality of “goodness” or “badness” (i) experienced as a feeling state (with or without consciousness) and (ii) demarcating a positive or negative quality of a stimulus. Affective responses occur rapidly and automatically—note how quickly you sense the feelings associated with the stimulus word “treasure” or the word “hate”. We argue that reliance on such feelings can be characterized as “the affect heuristic”. In this paper we trace the development of the affect heuristic across a variety of research paths followed by ourselves and many others. We also discuss some of the important practical implications resulting from ways that this heuristic impacts our daily lives.     

Homoclinic orbits and subharmonics for nonlinear second order difference equations

We show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the “Mountain Pass” theorem relying on Ekeland’s variational principle and the diagonal method, and the homoclinic orbit as the limit of the subharmonics. A completely new way is provided for dealing with the existence of solutions for difference equations.     

Cantor families of periodic solutions for completely resonant wave equations

We present recent existence results of Cantor families of small amplitude periodic solutions for completely resonant nonlinear wave equations. The proofs rely on the Nash-Moser implicit function theory and variational methods.      

A new mixed integer program and extended look-ahead heuristic algorithm for the block relocation problem

This paper considers the block relocation problem (BRP), in which a set of identically-sized items is to be retrieved from a set of last-in-first-out (LIFO) stacks in a specific order using the fewest number of moves. The problem is encountered in the maritime container shipping industry and other industries where inventory is stored in stacks. After surveying the work done on the BRP, we introduce “BRP-III”—a new mathematical formulation for the BRP—and show that it has considerably fewer decision variables and better runtime performance than the other formulation in the literature. We then introduce a new look-ahead algorithm (LA-N) that is an extension of the algorithms from the literature and show that the new algorithm generally obtains better solutions than the other algorithms and has minimal CPU runtime.