The concept of effective method applied to computational problems of linear algebra

A classification of computational problems is proposed which may have applications in numerical analysis. The classification utilizes the concept of effective method, which has been employed in treating decidability questions within the field of computable numbers. A problem is effectively soluble or effectively insolubles according as there is or there is not an effective method of solution. Roughly speaking, effectively insoluble computational problems are those whose general solution is restricted by an intrinsic and unavoidable computational difficulty. Some standard problems of linear algebra are analyzed to determine their type

Effective computability of solutions of ordinary differential equations: the thousand monkeys approach

In this note we consider the computability of the solution of the initial- value problem for ordinary di erential equations with continuous right- hand side. We present algorithms for the computation of the solution using the \thousand monkeys" approach, in which we generate all possi- ble solution tubes, and then check which are valid. In this way, we show that the solution of a di erential equation de ned by a locally Lipschitz function is computable even if the function is not e ectively locally Lips- chitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be e ectively enumerable open, and give an example of a computable locally Lipschitz function which is not e ectively locally Lipschitz

Leonora Carrington/Lucy Skaer

The output is a curated exhibition and catalogue of Viktor Wynd's impressive collection of Leonora Carrington artworks.This collection is macabre as it is unique, bringing together little-known prints and drawings which detail the surrealist bestiary of Carrington’s imaginative universe (explored further by Marina Warner), as well as four early canvases from Carrington’s pre-surrealist period (analysed in- depth by Susan L. Aberth). An anecdote by Wynd and short story by Carrington’s son, Gabriel Weisz, further nuance the visual narratives under consideration. Until this exhibition, the collection had never been shown together. Leeds Arts University hosted the collection as a major part of its Leonora Carrington/Lucy Skaer research exhibition (15 July – 2 September 2016). The artworks included in the catalogue toured subsequently to The Viktor Wynd Museum in London. Special thanks go to Viktor Wynd for his enthusiasm, generosity and support in realising the exhibition as well as Susan L. Aberth, Gabriel Weisz Carrington, and Marina Warner for their erudite contributions to the catalogue. The photography is by Chris Renton and the catalogue has been designed by D&M Heritage Ltd. Further grateful thanks and acknowledgement to: Chloe Aridjis, Jonathan P. Eburne, Sean Kissane, Lynn Lu, Steve Lucas, Wendi Norris, Jeffrey Sherwin, Lucy Skaer, Samantha Sweeting, and colleagues at Leeds College of Art

Computability of entropy and information in classical Hamiltonian systems

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition

Use of Matlab as computational tool to support teaching and learning of linear algebra

En este artículo se presentan los resultados de la investigación titulada Uso de Mablab como herramienta computacional para apoyar la enseñanza y el aprendizaje de algunos temas de ´algebra lineal en el programa de Licenciatura en Matemáticas de la Universidad del Atlántico. Esta se basó en el empleo del software Matlab para apoyar el proceso de enseñanza y aprendizaje de la resolución de sistemas de ecuaciones lineales, para la realización de operaciones matriciales, la solución de problemas asociados a espacios vectoriales y a transformaciones lineales y, para el cálculo de los valores y vectores propios de una matriz cuadrada.This article presents the results of the research entitled Using Mablab as a computational tool to support teaching and learning some topics of linear algebra in the degree program in mathematics from the University of the Atlantic. This was based on the use of Matlab software to support the teaching and learning of solving systems of linear equations, for performing matrix operations, solving problems associated with vector spaces and linear transformations and for calculating

A Contractor Based on Convex Interval Taylor

International audienceInterval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function andby Lin and Stadtherr to handle n × n (square) systems of equations. This paper presents an interval Newton-like operator, called X-Newton, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen's variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint. The X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic

Noncomputability, Unpredictability, Undecidability, and Unsolvability in Economic and Finance Theories

We outline, briefly, the role that issues of the nexus between noncomputability and unpredictability, on the one hand, and between undecidability and unsolvability, on the other hand, have played in Computable Economics (CE). The mathematical underpinnings of CE are provided by (classical) recursion theory, varieties of computable and constructive analysis and aspects of combinatorial optimization. The inspiration for this outline was provided by Professor Graça’s thought-provoking recent article