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

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

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

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

Transferring Algebra Structures Up to Homology Equivalence

Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in [16]. Applications to de Rham theory and Massey products are given. 1 Preliminaries and Notation Throughout this paper, R will denote a commutative ring with unit. The term (co)module is used to mean a differential graded (co)module over R and maps between modules are graded maps. When we write ⊗ we mean ⊗R. The usual (Koszul) sign conventions are assumed. The degree of a homogeneous element m of some module is denoted by |m|. Algebras are assumed to be connected and coalgebras simply connected. (Co)algebras are assumed to have (co)units.(Co)algebras are, unless otherwise stated, assumed to be (co)augmented. The differential in an (co)algebra is a graded (co)derivation. The R-module of maps from M to N (for R-modules M and N) is denoted by hom(M, N) (if the context requires it, we will use a subscript to denote the ground ring). The differential in this module is given by D(f) = df − (−1) |f | fd. Note that D is a derivation with respect to the composition operation whenever it is defined. In particular, End(M) = hom(M, M) is an algebra. 1 If A is an algebra and C is a coalgebra, the module hom(C, A) is an algebra with respect to the operation defined by the following diagram