A p-adic Simpson correspondence

For curves over a p-adic field we construct an equivalence between the category of Higgs-bundles and that of “generalised representation” of the etale fundamental group. The definition of “generalised representations” uses p-adic Hodge theory and almost etale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group.     

Blowup of solutions for the “bad” Boussinesq-type equation

The paper studies the blowup of solutions to the initial boundary value problem for the “bad” Boussinesq-type equation u_tt−u_xx−bu_xxxx=σ(u)_xx, where b>0 is a real number and σ(s) is a given nonlinear function. By virtue of the energy method and the Fourier transform method, respectively, it proves that under certain assumptions on σ(s) and initial data, the generalized solutions of the above-mentioned problem blow up in finite time. And a few examples are shown, especially for the “bad” Boussinesq equation, two examples of blowup of solutions are obtained numerically.     

Stable invariant manifolds for parabolic dynamics

We consider nonautonomous equations v^′=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.     

The numerical solution of forward-backward differential equations: Decomposition and related issues

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions. We consider equations of the form x^′(t)=ax(t)+bx(t−1)+cx(t+1) and develop a numerical approach, using a central difference approximation, which leads to the desired decomposition and propagation of the solution. We include illustrative examples to demonstrate the success of our method, along with an indication of its current limitations.     

Comparisons between observable and unobservable M/M/1 queues with respect to optimal customer behavior

We consider an M/M/1 queueing system in which the queue length may or may not be observable by a customer upon entering the system. The “observable” and “unobservable” models are compared with respect to system properties and performance measures under two different types of optimal customer behavior, which we refer to as “selfishly optimal” and “socially optimal”. We consider average customer throughput rates and show that, under both types of optimal customer behavior, the equality of effective queue-joining rates between the observable and unobservable systems results in differences with respect to other performance measures such as mean busy periods and waiting times. We also show that the equality of selfishly optimal queue-joining rates between the two types of system precludes the equality of socially optimal joining rates, and vice versa.     

An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem

We construct a generalized solution of the Riemann problem for strictly hyperbolic systems of conservation laws with source terms, and we use this to show that Glimm's method can be used directly to establish the existence of solutions of the Cauchy problem. The source terms are taken to be of the form a^′G, and this enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. Our generalized solution of the Riemann problem is “weaker than weak” in the sense that it is weaker than a distributional solution. Thus, we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solutions that are based on “weaker than weak” solutions of the Riemann problem. By establishing the convergence of Glimm's method, it follows that all of the results on time asymptotics and uniqueness for Glimm's method (in the presence of a linearly degenerate field) now apply, unchanged, to inhomogeneous systems.     

An adaptive multiphase approach for large unconditional and conditional p-median problems

A multiphase approach that incorporates demand points aggregation, Variable Neighbourhood Search (VNS) and an exact method is proposed for the solution of large-scale unconditional and conditional p-median problems. The method consists of four phases. In the first phase several aggregated problems are solved with a “Local Search with Shaking” procedure to generate promising facility sites which are then used to solve a reduced problem in Phase 2 using VNS or an exact method. The new solution is then fed into an iterative learning process which tackles the aggregated problem (Phase 3). Phase 4 is a post optimisation phase applied to the original (disaggregated) problem. For the p-median problem, the method is tested on three types of datasets which consist of up to 89,600 demand points. The first two datasets are the BIRCH and the TSP datasets whereas the third is our newly geometrically constructed dataset that has guaranteed optimal solutions. The computational experiments show that the proposed approach produces very competitive results. The proposed approach is also adapted to cater for the conditional p-median problem with interesting results.     

Periodic solutions of the Lyness max equation

We investigate the periodic nature of solutions of a “max-type” difference equation sometimes referred to as the “Lyness max” equation. The equation we consider is x_n+1=max{x_n,A}/x_n−1, n=0,1,…, where A is a positive real parameter and the initial conditions are arbitrary positive numbers. We also present related results for a similar equation sometimes referred to as the “period 7 max” equation.     

Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”.     

Lindelöf spaces which are indestructible, productive, or D

We discuss relationships in Lindelöf spaces among the properties “indestructible”, “productive”, “D”, and related properties.     

Applications of bulk queues to group testing models with incomplete identification

A population of items is said to be “group-testable”, (i) if the items can be classified as “good” and “bad”, and (ii) if it is possible to carry out a simultaneous test on a batch of items with two possible outcomes: “Success” (indicating that all items in the batch are good) or “failure” (indicating a contaminated batch). In this paper, we assume that the items to be tested arrive at the group-testing centre according to a Poisson process and are served (i.e., group-tested) in batches by one server. The service time distribution is general but it depends on the batch size being tested. These assumptions give rise to the bulk queueing model M/G^(m,M)/1, where m and M(>m) are the decision variables where each batch size can be between m and M. We develop the generating function for the steady-state probabilities of the embedded Markov chain. We then consider a more realistic finite state version of the problem where the testing centre has a finite capacity and present an expected profit objective function. We compute the optimal values of the decision variables (m, M) that maximize the expected profit. For a special case of the problem, we determine the optimal decision explicitly in terms of the Lambert function.     

Symmetry distribution between hook length and part length for partitions

It is known that the two statistics on integer partitions “hook length” and “part length” are equidistributed over the set of all partitions of n. We extend this result by proving that the bivariate joint generating function by those two statistics is symmetric. Our method is based on a generating function by a triple statistic much easier to calculate.     

Minimum variance capacity identification

In the framework of multi-criteria decision making whose aggregation process is based on the Choquet integral, we present a maximum entropy like method enabling to determine, if it exists, the “least specific” capacity compatible with the initial preferences of the decision maker. The proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity. The application of the proposed approach is illustrated on two examples.     

A generalization of gamma functions and Kronecker's limit formulas

We obtain a “Kronecker limit formula” for the Epstein zeta function. This is done by introducing a generalized gamma function attached to the Epstein zeta function. The methods involve generalizing ideas of Shintani and Stark. We first show that a generalized gamma function appears as the value at s=0 of the first derivative of the associated Epstein zeta function. Then this is used to yield Kronecker's limit formula and its “s=0”-version.     

Mixed q-analogues, lacunary polynomials and Truesdellian partial differential equations

Multiparameter families of possibly lacunary polynomials in x are constructed that have remarkably explicit (or “almost” explicit) expansions about both x=0 and x=1. They are placed in the framework of questions about mixed q-analogues with fewest terms. The properties of the most tractable of these surprisingly tractable polynomials are established with the aid of solutions to partial differential equations of a type much studied by Truesdell.     

Embedding processes in combinatorial game theory

Berlekamp asked the question “What is the habitat of ∗2?” (See Guy, 1996 [6].) It is possible to generalize the question and ask “For a game G, what is the largest n such that ∗n is a position of G?” This leads to the concept of the nim dimension. In Santos and Silva (2008) [8] a fractal process was proposed for analyzing the previous questions. For the same purpose, in Santos and Silva (2008) [9], an algebraic process was proposed. In this paper we implement a third idea related to embedding processes. With Alan Parr’s traffic lights, we exemplify the idea of estimating the “difficulty” of the game and proving that its nim dimension is infinite.     

Counting strings in Dyck paths

This paper deals with the enumeration of Dyck paths according to the statistic “number of occurrences of τ”, for an arbitrary string τ. In this direction, the statistic “number of occurrences of τ at height j” is considered. It is shown that the corresponding generating function can be evaluated with the aid of Chebyshev polynomials of the second kind. This is applied to every string of length 4. Further results are obtained for the statistic “number of occurrences of τ at even (or odd) height”.     

Using the K5Connected Cognition Diagram to analyze teachers’ communication and understanding of regions in three-dimensional space

This paper reports on a study that introduces and applies the K₅Connected Cognition Diagram as a lens to explore video data showing teachers’ interactions related to the partitioning of regions by axes in a three-dimensional geometric space. The study considers “semiotic bundles” ( Arzarello, 2006), introduces “semiotic connections,” and discusses the fundamental role each plays in developing individual understanding and communication with peers. While all teachers solved the problem posed, many failed to make or verbalize connections between the types of semiotic resources introduced during their discussions.     

Beginners’ progress in early arithmetic in the Swedish compulsory school

This article focuses on spontaneous knowledge-building in the field of “the arithmetic of the child.” The aim is to investigate the conceptual progress of fifteen children during their early school years in the compulsory school. The study is based on the epistemology of radical constructivism and the methodology of “multiple clinical interviews”. A model of “the arithmetic of the child” elucidates mental structures used by the child in solving problems. The individual interviews are video-recorded. The results show that the children's solutions are compatible with the model. When the researcher adapts problems to the children's available concepts to bring out their capability, they all solve them in their own ways. Further, the conceptual levels of the children differ to a great extent at school start and do not all show conceptual progress after 2 years of traditional teaching. An implication for an alternative teaching process is suggested, namely “the arithmetic for the child”, accomplished in a triadic teaching process.