6 research outputs found
A Combinatorial Commutative Algebra Approach to Complete Decoding
Esta tesis pretende explorar el nexo de unión que existe entre la estructura algebraica de un código lineal y el proceso de descodificación completa. Sabemos que el proceso de descodificación completa para códigos lineales arbitrarios es NP-completo, incluso si se admite preprocesamiento de los datos. Nuestro objetivo es realizar un análisis algebraico del proceso de la descodificación, para ello asociamos diferentes estructuras matemáticas a ciertas familias de códigos. Desde el punto de vista computacional, nuestra descripción no proporciona un algoritmo eficiente pues nos enfrentamos a un problema de naturaleza NP. Sin embargo, proponemos algoritmos alternativos y nuevas técnicas que permiten relajar las condiciones del problema reduciendo los recursos de espacio y tiempo necesarios para manejar dicha estructura algebraica.Departamento de Algebra, Geometría y Topologí
Toric Ideals, Polytopes, and Convex Neural Codes
How does the brain encode the spatial structure of the external world?
A partial answer comes through place cells, hippocampal neurons which
become associated to approximately convex regions of the world known
as their place fields. When an organism is in the place field of some place
cell, that cell will fire at an increased rate. A neural code describes the set
of firing patterns observed in a set of neurons in terms of which subsets
fire together and which do not. If the neurons the code describes are place
cells, then the neural code gives some information about the relationships
between the place fields–for instance, two place fields intersect if and only if
their associated place cells fire together. Since place fields are convex, we are
interested in determining which neural codes can be realized with convex
sets and in finding convex sets which generate a given neural code when
taken as place fields. To this end, we study algebraic invariants associated
to neural codes, such as neural ideals and toric ideals. We work with a
special class of convex codes, known as inductively pierced codes, and seek
to identify these codes through the Gröbner bases of their toric ideals
Minimal systems of binomial generators for the ideals of certain monomial curves
Let and be three positive integers such that and
are relatively prime. In this paper, we prove that the
toric ideal associated to the submonoid of generated by
is determinantal. Moreover, we prove that for , the ideal has a unique minimal system of generators if and only if