Mathematical Analysis and Computational Integration of Massive Heterogeneous Data from the Human Retina

Modern epidemiology integrates knowledge from heterogeneous collections of data consisting of numerical, descriptive and imaging. Large-scale epidemiological studies use sophisticated statistical analysis, mathematical models using differential equations and versatile analytic tools that handle numerical data. In contrast, knowledge extraction from images and descriptive information in the form of text and diagrams remain a challenge for most fields, in particular, for diseases of the eye. In this article we provide a roadmap towards extraction of knowledge from text and images with focus on forthcoming applications to epidemiological investigation of retinal diseases, especially from existing massive heterogeneous collections of data distributed around the globe.Comment: 9 pages, 3 figures, submitted and accepted in Damor2012 conference: http://www.uninova.pt/damor2012/index.php?page=author

Next generation computer algebra systems AXIOM and the scratchpad concept: applications to research in algebra

One way in which mathematicians deal with infinite amounts of data is symbolic representation. A simple example is the quadratic equation x = −b±√b2−4ac 2a, a formula which uses symbolic representation to describe the solutions to an infinite class of equations. Most computer algebra systems can deal with polynomials with symbolic coefficients, but what if symbolic exponents are called for (e.g., 1+t i)? What if symbolic limits on summations are also called for (e.g., 1+t+...+t i = � j tj)? The “Scratchpad Concept ” is a theoretical ideal which allows the implementation of objects at this level of abstraction and beyond in a mathematically consistent way. The AXIOM computer algebra system is an implementation of a major part of the Scratchpad Concept. AXIOM (formerly called Scratchpad) is a language with extensible parameterized types and generic operators which is based on the notions of domains and categories [Lambe1], [Jenks-Sutor]. By examining some aspects of the AXIOM system, the Scratchpad Concept will be illustrated. It will be shown how some complex problems in homological algebra were solved through the use of this system. New paradigms are evolving in computer science. There is a thrust towards type

On the Quantum Yang-Baxter Equation with Spectral Parameter, I

In memory of Grace Lambe The quantum Yang–Baxter equation (QYBE) is related to the study of integrabl

Resolutions via homological perturbation

The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of computer algebra in an area not usually associated with that subject. Comparison of the complexes produced by the method discussed here with those produced by other methods show

Resolutions which split off of the bar construction

AbstractResolutions which split off of the bar construction are quite common, but explicit formulae expressing these splittings are not often encountered. Given explicit splitting data, perturbations of resolutions can be computed and the perturbed resolutions can be used tomake complete effective calculations where previously only partial or indirect results were obtainable.This paper gives a foundation for the perturbation method in homological algebra by providing a symbolic encoding of binomial coefficient functions which is useful in deriving formulae for an infinite class of resolutions. Formulae for perturbations of those resolutions are then derived. Applications to certain infinite families of groups and monoids are given.The research for this theory as well as the calculation of closed formulae within the theory was aided by new methods in symbolic computation using the Axiom (formerly called Scratchpad) system

DEGREES OF MAPPINGS OF MANIFOLDS.

DEGREES OF MAPPINGS OF MANIFOLDS

The cohomology ring of the free loop space of a wedge of spheres and cyclic homology

Abstract – We investigate the cohomology of the free loop space of a one point union of a three-sphere with itself. The even dimensional subalgebra is not free and relations are presented explicitly. This algebra may also be identified as the cyclic homology of a “null algebra”. L’anneau de cohomologie de l’espace de lacets libres d’un bouquet de sphères et l’homologie cyclique. Résumé – Nous étudions l’algèbre de cohomologie de l’espace de lacets libres de l’espace obtenu en identifiant deux spheres S 3 en un point. La sous-algèbre (commutative) formée des éléments de degré pair n’est pas libre et nous en donnons des relations explicites. Cette algèbre peut aussi être identifiée à l’homologie cyclique d’une certaine algèbre à multiplication nulle. Version française abrégée – Nous étudions l’anneau de cohomologie à coefficients rationnels de l’espace L(X) de lacets libres sur X = S 3 ∨S 3. Dans ce cas la suite spectrale d’Eilenberg-Moore dégénère et nous obtenons H ∗ (L(X)) ∼ = Tor H ∗ (X) e (H ∗ (X), H ∗ (X)). La sous-algèbre K de H ∗ (L(X)) formée des éléments de degré pair détermine H ∗ (L(X)) et de plus K peut être identifiée (modulo une translation des degrés) à l’homologie cyclique d’une algèbre a multiplication triviale ([4], [5], [6]). La série génératrice de K est égale a ([3], [4], [9]): α(t) = 1 − � i≥1 φ(i) i log(1 − 2t 2i) où φ est la fonction d’Euler. On montre l’existence de relations dans K en écrivant cette série comme un produit infini i=