Darboux-Bäcklund transformations and rational solutions of the painlevé IV equation

Rational solutions of the Painlevé IV equation are constructed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy subject to the additional non-isospectral Virasoro symmetry constraint. Convenient Wronskian representations for rational solutions are obtained by successive actions of the Darboux-Bäcklund transformations. ©2010 American Institute of Physics.

Faktorisierung in Schief-Polynomringen

In der Arbeit werden zunächst die wesentlichsten Fakten über Schiefpolynome wiederholt, der Fokus liegt dabei auf Shift- und q-Shift-Operatoren in Charakteristik Null. Alle für die Arithmetik mit diesen Objekten notwendigen Konzepte und Algorithmen finden sich im ersten Kapitel. Einige der zur Bestimmung von Lösungen notwendigen Daten können aus dem Newtonpolygon, einer den Operatoren zugeordneten geometrischen Figur, abgelesen werden. Die Herleitung dieser Zusammenhänge ist das Thema des zweiten Kapitels der Arbeit, wobei dies insbesondere im q-Shift-Fall in dieser Form neu ist. Das dritte Kapitel beschäftigt sich mit der Bestimmung polynomieller und rationaler Lösungen dieser Operatoren, dabei folgt es im Wesentlichen der Darstellung von Mark van Hoeij. Der für die Faktorisierung von (q-)Shift Operatoren interessanteste Fall sind die sogenannten (q-)hypergeometrischen Lösungen, die direkt zu Rechtsfaktoren erster Ordnung korrespondieren. Im vierten Kapitel wird der van Hoeij-Algorithmus vom Shift- auf den q-Shift-Fall übertragen. Außerdem wird eine deutliche Verbesserung des q-Petkovsek-Algorithmus mit Hilfe der Daten des Newtonpolygons hergeleitet. Das fünfte Kapitel widmet sich der Berechnung allgemeiner Faktoren, wozu zunächst der adjungierte Operator eingeführt wird, der die Berechnung von Linksfaktoren erlaubt. Dann wird ein Algorithmus zur Berechnung von Rechtsfaktoren beliebiger Ordnung dargestellt. Für die praktische Benutzung ist dies allerdings für höhere Ordnungen unpraktikabel. Bei fast allen vorgestellten Algorithmen tritt das Lösen linearer Gleichungssysteme über rationalen Funktionenkörpern als Zwischenschritt auf. Dies ist in den meisten Computeralgebrasystemen nicht befriedigend gelöst. Aus diesem Grund wird im letzten Kapitel ein auf Evaluation und Interpolation basierender Algorithmus zur Lösung dieses Problems vorgestellt, der in allen getesteten Systemen den Standard-Algorithmen deutlich überlegen ist. Alle Algorithmen der Arbeit sind in einem MuPAD-Package implementiert, das der Arbeit beiliegt und eine komfortable Handhabung der auftretenden Objekte erlaubt. Mit diesem Paket können in MuPAD nun viele Probleme gelöst werden, für die es vorher keine Funktionen gab.

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent “accurately” harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in “accurate” reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Padé approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.

Estratégias de comercialização e investimento, com ênfase em energias renováveis, suportadas por modelos de otimização especializados para avaliação estocástica de risco x retorno.

A comercialização de energia elétrica de fontes renováveis, ordinariamente, constitui-se uma atividade em que as operações são estruturadas sob condições de incerteza, por exemplo, em relação ao preço \"spot\" no mercado de curto prazo e a geração de energia dos empreendimentos. Deriva desse fato a busca dos agentes pela formulação de estratégias e utilização de ferramentais para auxiliá-los em suas tomadas de decisão, visando não somente o retorno financeiro, mas também à mitigação dos riscos envolvidos. Análises de investimentos em fontes renováveis compartilham de desafios similares. Na literatura, o estudo da tomada de decisão considerada ótima sob condições de incerteza se dá por meio da aplicação de técnicas de programação estocástica, que viabiliza a modelagem de problemas com variáveis randômicas e a obtenção de soluções racionais, de interesse para o investidor. Esses modelos permitem a incorporação de métricas de risco, como por exemplo, o Conditional Value-at-Risk, a fim de se obter soluções ótimas que ponderem a expectativa de resultado financeiro e o risco associado da operação, onde a aversão ao risco do agente torna-se um condicionante fundamental. O objetivo principal da Tese - sob a ótica dos agentes geradores, consumidores e comercializadores - é: (i) desenvolver e implementar modelos de otimização em programação linear estocástica com métrica CVaR associada, customizados para cada um desses agentes; e (ii) aplicá-los na análise estratégica de operações como forma de apresentar alternativas factíveis à gestão das atividades desses agentes e contribuir com a proposição de um instrumento conceitualmente robusto e amigável ao usuário, para utilização por parte das empresas. Nesse contexto, como antes frisado, dá-se ênfase na análise do risco financeiro dessas operações por meio da aplicação do CVaR e com base na aversão ao risco do agente. Considera-se as fontes renováveis hídrica e eólica como opções de ativos de geração, de forma a estudar o efeito de complementaridade entre fontes distintas e entre sites distintos da mesma fonte, avaliando-se os rebatimentos nas operações.

Triples of Positive Integers with the same Sum and the same Product

It is proved that for every k there exist k triples of positive integers with the same sum and the same product.

A Collaborative Multiagent Framework based on Online Risk-Aware Planning and Decision-Making

Planning is an essential process in teams of multiple agents pursuing a common goal. When the effects of actions undertaken by agents are uncertain, evaluating the potential risk of such actions alongside their utility might lead to more rational decisions upon planning. This challenge has been recently tackled for single agent settings, yet domains with multiple agents that present diverse viewpoints towards risk still necessitate comprehensive decision making mechanisms that balance the utility and risk of actions. In this work, we propose a novel collaborative multi-agent planning framework that integrates (i) a team-level online planner under uncertainty that extends the classical UCT approximate algorithm, and (ii) a preference modeling and multicriteria group decision making approach that allows agents to find accepted and rational solutions for planning problems, predicated on the attitude each agent adopts towards risk. When utilised in risk-pervaded scenarios, the proposed framework can reduce the cost of reaching the common goal sought and increase effectiveness, before making collective decisions by appropriately balancing risk and utility of actions. 

Towards rational synthesis of inorganic solids

There have been major advances in the past couple of years in the rational synthesis of inorganic solids: synthesis of mercury-based superconducting cuprates showing transition temperatures up to 150 K; ZrP2-xVxO7 solid solutions showing zero or negative thermal expansion; copper oxides possessing ladder structures such as La1-xSrxCuO2.5; synthesis of mesoporous oxide materials having adjustable pore size in the range 15-100 Angstrom; and synthesis of a molecular ferromagnet showing a critical temperature of 18.6 K. Despite great advances in probing the structures of solids and measurement of their physical properties, the design and synthesis of inorganic solids possessing desired structures and properties remain a challenge today. With the availability of a variety of mild chemistry-based approaches, kinetic control of synthetic pathways is becoming increasingly possible, which, it is hoped, will eventually make rational design of inorganic solids a reality.

Stable multivariate rational interpolation for parameter-dependent aerospace models

A multivariate, robust, rational interpolation method for propagating uncertainties in several dimensions is presented. The algorithm for selecting numerator and denominator polynomial orders is based on recent work that uses a singular value decomposition approach. In this paper we extend this algorithm to higher dimensions and demonstrate its efficacy in terms of convergence and accuracy, both as a method for response suface generation and interpolation. To obtain stable approximants for continuous functions, we use an L2 error norm indicator to rank optimal numerator and denominator solutions. For discontinous functions, a second criterion setting an upper limit on the approximant value is employed. Analytical examples demonstrate that, for the same stencil, rational methods can yield more rapid convergence compared to pseudospectral or collocation approaches for certain problems. © 2012 AIAA.

Reducing Child Protection Error in Social Work: Towards an Holistic-Rational Perspective

Social work in the United Kingdom remains embroiled in concerns about child protection error. The serious injury or death of vulnerable children continues to evince much consternation in the public and private spheres. Governmental responses to these concerns invariably draw on technocratic solutions involving more procedures, case management systems, information technology and bureaucratic regulation. Such solutions flow from an implicit use of instrumental rationality based on a ‘means-end’ logic. While bringing an important perspective to the problem of child protection error, instrumental rationality has been overused limiting discretion and other modes of rational inquiry. This paper argues that the social work profession should apply an enlarged form of rationality comprising not only the instrumental-rational mode but also the critical-rational, affective-rational and communicative-rational forms. It is suggested that this combined, conceptual arsenal of rational inquiry leads to a gestalt which has been termed the holistic-rational perspective. It is also argued that embracing a more rounded perspective such as this might offer greater opportunities for reducing child protection error.

On two rational difference equations

In this paper, we study the oscillating property of positive solutions and the global asymptotic stability of the unique equilibrium of the two rational difference equations [GRAPHICS] and [GRAPHICS] where a is a nonnegative constant. (c) 2005 Elsevier Inc. All rights reserved.

On the system of rational difference equations x(n) = A+(1)over-bar y(n-p), y(n) = A+(gamma(n-1))over-bar x(n-r)y(n-5)

In this paper, we study the behavior of the positive solutions of the system of two difference equations [GRAPHICS] where p >= 1, r >= 1, s >= 1, A >= 0, and x(1-r), x(2-r),..., x(0), y(1-max) {p.s},..., y(0) are positive real numbers. (c) 2005 Elsevier Inc. All rights reserved.

Darboux-Egoroff metrics, rational Landau-Ginzburg potentials and the Painleve VI equation

We present a class of three-dimensional integrable structures associated with the Darboux-Egoroff metric and classical Euler equations of free rotations of a rigid body. They are obtained as canonical structures of rational Landau-Ginzburg potentials and provide solutions to the Painleve VI equation.

Territorial Cohesion through Spatial Policies: An Analysis with Cultural Theory and Clumsy Solutions

The European Territorial Cohesion Policy has been the subject of numerous debates in recent years. Most contributions focus on understanding the term itself and figuring out what is behind it, or arguing for or against a stronger formal competence of the European Union in this field. This article will leave out these aspects and pay attention to (undefined and legally non-binding) conceptual elements of territorial cohesion, focusing on the challenge of linking it within spatial policies and organising the relations. Therefore, the theoretical approach of Cultural Theory and its concept of clumsy solution are applied to overcome the dilemma of typical dichotomies by adding a third and a fourth (but not a fifth) perspective. In doing so, normative contradictions between different rational approaches can be revealed, explained and approached with the concept of ‘clumsy solutions’. This contribution aims at discussing how this theoretical approach helps us explain and frame a coalition between the Territorial Cohesion Policy and spatial policies. This approach contributes to finding the best way of linking and organising policies, although the solution might be clumsy according to the different rationalities involved.

Efficient Solution of Rational Conics

We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.