Dissecting a brick into bars

Consider the set of all lengths of sides of an N-dimensional parallelepiped. If this set has no more than k elements, the parallelepiped will be called a bar (the definition of a bar depends on k). We prove that a parallelepiped can be dissected into a finite number of bars if and only if the lengths of its sides span a linear space of dimension at most k over \mathbb Q{{\mathbb Q}} . This extends and generalizes a well-known theorem of Max Dehn about the splitting of rectangles into squares. Several other results about dissections of parallelepipeds are obtained.     

Classification of Curves in 2D and 3D via Affine Integral Signatures

We propose new robust classification algorithms for planar and spatial curves subjected to affine transformations. Our motivation comes from the problems in computer image recognition. To each planar or spatial curve, we assign a planar signature curve. Curves, equivalent under an affine transformation, have the same signature. The signatures are based on integral invariants, which are significantly less sensitive to small perturbations of curves and noise than classically known differential invariants. Affine invariants are derived in terms of Euclidean invariants. We present two types of signatures: the global and the local signature. Both signatures are independent of curve parameterization. The global signature depends on a choice of the initial point and, therefore, cannot be used for local comparison. The local signature, albeit being slightly more sensitive to noise, is independent of the choice of the initial point and can be used to solve local equivalence problem. An experiment that illustrates robustness of the proposed signatures is presented.     

Energy Decay for the Strongly Damped Nonlinear Beam Equation and Its Applications in Moving Boundary

We study the existence and energy decay of solutions for the strongly damped nonlinear beam equation. We apply a method based on Nakao method to show that the solution decays exponentially, and to obtain precise estimates of the constants in the estimates. Finally, we discuss its applications in moving boundary.     

On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy

We consider a risk minimization problem in a continuous-time Markovian regime-switching financial model modulated by a continuous-time, observable and finite-state Markov chain whose states represent different market regimes. We adopt a particular form of convex risk measure, which includes the entropic risk measure as a particular case, as a measure of risk. The risk-minimization problem is formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game. One important feature of our model is to allow the flexibility of controlling both the diffusion process representing the financial risk and the Markov chain representing macro-economic risk. This is novel and interesting from both the perspectives of stochastic differential game and stochastic control. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution of the game is provided and some particular cases are discussed.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

On Elliptic Extensions in the Disk

Given two arbitrary functions f ⁽⁰⁾, f ⁽¹⁾ on the boundary of the unit disk D in \({\mathbb R}^2\), it is shown that there exists a second order uniformly elliptic operator L and a function v in L ^p , with L ^p second derivatives (1?p?Lv?=?0 a.e. in D and with v?=?f ⁽⁰⁾ and \(\frac{ \partial v}{\partial n} = f^{(1)}\) on \(\partial{D}\). A similar extension property was proved in Cavazzoni (2003) for any pair of functions f ⁽⁰⁾, f ⁽¹⁾ that are analytic; a result is obtained under weaker regularity assumptions, e.g. with \(\frac{\partial f^{(0)}}{\partial \theta}\) and f ⁽¹⁾ Hölder continuous with exponent \(\eta > \frac{1}{2}\).     

A truncated Newton method in an augmented Lagrangian framework for nonlinear programming

In this paper we propose a primal-dual algorithm for the solution of general nonlinear programming problems. The core of the method is a local algorithm which relies on a truncated procedure for the computation of a search direction, and is thus suitable for large scale problems. The truncated direction produces a sequence of points which locally converges to a KKT pair with superlinear convergence rate.     

Graded and Koszul Categories

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.     

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.     

Timetabling for Greek high schools

The timetabling process and the resulting weekly schedules are important components for the daily operation of any school. This paper presents an efficient solution to the timetabling problem for the secondary educational system in Greece. Such a problem involves scheduling a large number of classes, teachers, courses, and classrooms to a number of time-periods. The development of the basic structure and the modelling of the problem as an integer mathematical program allows for the generation of constraints necessary for the satisfaction of all the school system rules and regulations. The integer programming approach and the commercial tools available for this class of problems facilitated the process of locating the optimal solution for the problem. The model is flexible and modular allowing for adaptations to satisfy the local characteristics of each school by changing the parameters of the model and adding or replacing constraints. A fully defined timetabling problem for a typical Greek high school is presented and optimally solved in order to demonstrate the effectiveness of the model in satisfying both the hard and the soft operational rules of the problem. Implementation of the new methodology for regular use for high schools is currently being attempted.