2 research outputs found
Removing Isolated Zeroes by Homotopy
Suppose that the inverse image of the zero vector by a continuous map
has an isolated point . There is a local
obstruction to removing this isolated zero by a small perturbation,
generalizing the notion of index for vector fields, the case. The
existence of a continuous map which approximates but is nonvanishing
near is equivalent to a topological property we call "locally inessential,"
and for dimensions , where is trivial, every
isolated zero is locally inessential. We consider the problem of constructing
such an approximation , and show that there exists a continuous homotopy
from to through locally nonvanishing maps. If is a semialgebraic
map, then there exists such a homotopy which is also semialgebraic. For
and real analytic with a locally inessential isolated zero, there exists a
H\"older continuous homotopy which, for , is real
analytic and nonvanishing. The existence of a smooth homotopy, given a smooth
map , is stated as an open question.Comment: to appear in Topological Methods in Nonlinear Analysi
Mathematical Models in Schema Theory
In this paper, a mathematical schema theory is developed. This theory has
three roots: brain theory schemas, grid automata, and block-shemas. In Section
2 of this paper, elements of the theory of grid automata necessary for the
mathematical schema theory are presented. In Section 3, elements of brain
theory necessary for the mathematical schema theory are presented. In Section
4, other types of schemas are considered. In Section 5, the mathematical schema
theory is developed. The achieved level of schema representation allows one to
model by mathematical tools virtually any type of schemas considered before,
including schemas in neurophisiology, psychology, computer science, Internet
technology, databases, logic, and mathematics