Existence and structure results on almost periodic solutions of difference equations

We study the almost periodic solutions of Euler equations and of some more general Difference Equations. We consider two different notions of almost periodic sequences, and we establish some relations between them. We build suitable sequences spaces and we prove some properties of these spaces. We also prove properties of Nemytskii operators on these spaces. We build a variational approach to establish existence of almost periodic solutions as critical points, We obtain existence theorems fornonautonomous linear equations and for an Euler equation with a concave and coercive Lagrangian. We also use a Fixed Point approach to obtain existence results for quasi-linear Difference Equations.     

Belief structures,weight generating functions and decision-making

We describe the Dempster–Shafer belief structure and provide some of its basic properties. We introduce the plausibility and belief measures associated with a belief structure. We note that these are not the only measures that can be associated with a belief structure. We describe a general approach for generating a class of measures that can be associated with a belief structure using a monotonic function on the unit interval, called a weight generating function. We study a number of these functions and the measures that result. We show how to use weight-generating functions to obtain dual measures from a belief structure. We show the role of belief structures in representing imprecise probability distributions. We describe the use of dual measures, other then plausibility and belief, to provide alternative bounding intervals for the imprecise probabilities associated with a belief structure. We investigate the problem of decision making under belief structure type uncertain. We discuss two approaches to this decision problem. One of which is based on an expected value of the OWA aggregation of the payoffs associated with the focal elements. The second approach is based on using the Choquet integral of a measure generated from the belief structure. We show the equivalence of these approaches.     

Comatrix Coring Generalized and Equivalences of Categories of Comodules

We extend the comatrix coring to the case of a quasi-finite bicomodule. We also generalize some of its interesting properties. We study equivalences between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings. We apply our results to corings coming from entwining structures and graded structures, and we obtain new results in the setting of entwining structures and in the graded ring theory.     

A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling’s technique, by defining two classes of sub- and super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.     

Formulations and Valid Inequalities for the Heterogeneous Vehicle Routing Problem

We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities.     

Relaxing support vectors for classification

We introduce a novel modification to standard support vector machine (SVM) formulations based on a limited amount of penalty-free slack to reduce the influence of misclassified samples or outliers. We show that free slack relaxes support vectors and pushes them towards their respective classes, hence we use the name relaxed support vector machines (RSVM) for our method. We present theoretical properties of the RSVM formulation and develop its dual formulation for nonlinear classification via kernels. We show the connection between the dual RSVM and the dual of the standard SVM formulations. We provide error bounds for RSVM and show it to be stable, universally consistent and tighter than error bounds for standard SVM. We also introduce a linear programming version of RSVM, which we call RSVMLP. We apply RSVM and RSVMLP to synthetic data and benchmark binary classification problems, and compare our results with standard SVM classification results. We show that relaxed influential support vectors may lead to better classification results. We develop a two-phase method called RSVM² for multiple instance classification (MIC) problems, where RSVM formulations are used as classifiers. We extend the two-phase method to the linear programming case and develop RSVMLP². We demonstrate the classification characteristics of RSVM² and RSVMLP², and report our classification results compared to results obtained by other SVM-based MIC methods on public benchmark datasets. We show that both RSVM² and RSVMLP² are faster and produce more accurate classification results.     

An Interactive Algorithm for Multi-objective Route Planning

We address the route selection problem for Unmanned Air Vehicles (UAV) under multiple objectives. We consider a general case for this problem, where the UAV has to visit several targets and return to the base. We model this problem as a combination of two combinatorial problems. First, the path to be followed between each pair of targets should be determined. We model this as a multi-objective shortest path problem. Additionally, we need to determine the order of the targets to be visited. We model this as a multi-objective traveling salesperson problem (MOTSP). The overall problem is a combination of these two problems, which we define as a generalized MOTSP. We develop an exact interactive approach to identify the best paths and the best tour of a decision maker under a linear utility function.     

On Equilibria when Agents Have Multiple Priors

We discuss the existence and the qualitative properties of equilibria when agents have multiple priors and there is only one good in each state of the world. We first prove a general existence result in infinite dimension economies. We then fully describe the equilibria in two special cases. We first consider the case of CEU maximizers that have same capacities. We next consider the case of no aggregate uncertainty. We prove that if agents have non-random initial endowments and are uncertainty averse and maximize the minimal expected utility according to a set of possible priors, then the existence of a common prior is equivalent to the existence of a unique equilibrium, the no-trade equilibrium. We lastly give a mild assumption for indeterminacy of equilibria and compute the dimension of indeterminacy.     

Scheduling and lot sizing models for the single-vendor multi-buyer problem under consignment stock partnership

We consider a centralized supply chain composed of a single vendor serving multiple buyers and operating under consignment stock arrangement. Solving the general problem is hard as it requires finding optimal delivery schedule to the buyers and optimal production lot sizes. We first provide a nonlinear mixed integer programming formulation for the general scheduling and lot sizing problem. We show that the problem is NP-hard in general. We reformulate the problem under the assumption of ‘zero-switch rule’. We also provide a simple sequence independent lower bound to the solution of the general model. We then propose a heuristic procedure to generate a near-optimal delivery schedule. We assess the cost performance of that heuristic by conducting sensitivity analysis on the key model parameters. The results show that the proposed heuristic promises substantial supply-chain cost savings that increase as the number of buyers increases.     

Nonequilibrium Statistical Operator Method and Generalized Kinetic Equations

We consider some principal problems of nonequilibrium statistical thermodynamics in the framework of the Zubarev nonequilibrium statistical operator approach. We present a brief comparative analysis of some approaches to describing irreversible processes based on the concept of nonequilibrium Gibbs ensembles and their applicability to describing nonequilibrium processes. We discuss the derivation of generalized kinetic equations for a system in a heat bath. We obtain and analyze a damped Schrödinger-type equation for a dynamical system in a heat bath. We study the dynamical behavior of a particle in a medium taking the dissipation effects into account. We consider the scattering problem for neutrons in a nonequilibrium medium and derive a generalized Van Hove formula. We show that the nonequilibrium statistical operator method is an effective, convenient tool for describing irreversible processes in condensed matter.     

Towards higher topology

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [21]. We hint that 0-dimensional adjunction inhabits the categorified one.     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

Complexification of Multilinear Mappings and Polynomials

We study various methods of complexifying real normed spaces. We see how the notions of duality and complexification are interchangeable. We obtain estimates for the norms of complexified multilinear mappings and polynomials. We see how polynomials can be complexified without reference to the associated multilinear mappings.     

Binary Quasi-Cyclic Goppa Codes

We obtain here a necessary and sufficient condition for a certain class of binary Goppa code to be quasi-cyclic. We also give another sufficient condition which is easier to check. We define a class of quasi-cyclic Goppa codes. We find the true dimension for a part of those quasi-cyclic codes. and also a class of extended quasi-cyclic codes the minimum distance of which is equal to the designed distance.     

On the specificity of a possibility distribution

The specificity of a possibility distribution measures the degree to which the distribution allows one and only one element as its manifestation. As such it is a measure of amount of uncertainty or information. We investigate a number of issues related to specificity measures. We discuss the connection between the specificity of a possibility distribution and the entropy of a probability distribution. We describe unifying view for constructing specificity measures. We look at the relationship of the specificity of a distribution and its negation. We consider the case where the base set is continuous.     

Opportunities for learning: the use of variation to analyse examples of a paradigm shift in teaching primary mathematics in England

We demonstrate the power of Variation Theory as an analytical tool used to understand the underlying conceptual structure of mathematics lessons taught by English primary school teachers. We study excerpts of three lessons that are posted on a professional website. We show how lesson analysis using variation allows us to focus on what is made available to be learnt in the lesson excerpts. We identify some differences in their use of dimensions of variation and the associated ranges of change and discuss how suitable patterns of variation and invariance might differ according to the nature of the learning focus. We reflect on the value of our analytical approach.     

Inventory management with partially observed nonstationary demand

We consider a continuous-time model for inventory management with Markov modulated non-stationary demands. We introduce active learning by assuming that the state of the world is unobserved and must be inferred by the manager. We also assume that demands are observed only when they are completely met. We first derive the explicit filtering equations and pass to an equivalent fully observed impulse control problem in terms of the sufficient statistics, the a posteriori probability process and the current inventory level. We then solve this equivalent formulation and directly characterize an optimal inventory policy. We also describe a computational procedure to calculate the value function and the optimal policy and present two numerical illustrations.     

The impact of liberalization on the productive efficiency of Indian commercial banks

We examine the productive efficiency of 70 Indian commercial banks during the early stages (1986–1991) of the ongoing period of liberalization. We use data envelopment analysis to calculate radial technical efficiency scores. We then use stochastic frontier analysis to attribute variation in the calculated efficiency scores to three sources: a temporal component, an ownership component, and a random noise component. We find publicly-owned Indian banks to have been the most efficient, followed by foreign-owned banks and privately-owned Indian banks. We also find a temporal improvement in the performance of foreign-owned banks, virtually no trend in the performance of privately-owned Indian banks, and a temporal decline in the performance of publicly-owned Indian banks. We attempt to explain these patterns in terms of the government's evolving regulatory policies.     

A stochastic programming approach to multicriteria portfolio optimization

We study a stochastic programming approach to multicriteria multi-period portfolio optimization problem. We use a Single Index Model to estimate the returns of stocks from a market-representative index and a random walk model to generate scenarios on the possible values of the index return. We consider expected return, Conditional Value at Risk and liquidity as our criteria. With stocks from Istanbul Stock Exchange, we make computational studies for the two and three-criteria cases. We demonstrate the tradeoffs between criteria and show that treating these criteria simultaneously yields meaningful efficient solutions. We provide insights based on our experiments.