21 research outputs found
Spontaneous symmetry breaking and the formation of columnar structures in the primary visual cortex II --- Local organization of orientation modules
Self-organization of orientation-wheels observed in the visual cortex is
discussed from the view point of topology. We argue in a generalized model of
Kohonen's feature mappings that the existence of the orientation-wheels is a
consequence of Riemann-Hurwitz formula from topology. In the same line, we
estimate partition function of the model, and show that regardless of the total
number N of the orientation-modules per hypercolumn the modules are
self-organized, without fine-tuning of parameters, into definite number of
orientation-wheels per hypercolumn if N is large.Comment: 36 pages Latex2.09 and eps figures. Needs epsf.sty, amssym.def, and
Type1 TeX-fonts of BlueSky Res. for correct typo in graphics file
A Note on modified Veselov-Novikov Hierarchy
Because of its relevance to lower-dimensional conformal geometry, known as a
generalized Weierstrass inducing, the modified Veselov-Novikov (mVN) hierarchy
attracts renewed interest recently. It has been shown explicitly in the
literature that an extrinsic string action \`a la Polyakov (Willmore
functional) is invariant under deformations associated to the first member of
the mVN hierarchy. In this note we go one step further and show the explicit
invariance of the functional under deformations associated to all higher
members of the hierarchy.Comment: 12 pp LaTeX2e. To be published in Physics Letters
Exotic Differentiable Structures and General Relativity
We review recent developments in differential topology with special concern
for their possible significance to physical theories, especially general
relativity. In particular we are concerned here with the discovery of the
existence of non-standard (``fake'' or ``exotic'') differentiable structures on
topologically simple manifolds such as , \R and
Because of the technical difficulties involved in the smooth case, we begin
with an easily understood toy example looking at the role which the choice of
complex structures plays in the formulation of two-dimensional vacuum
electrostatics. We then briefly review the mathematical formalisms involved
with differentiable structures on topological manifolds, diffeomorphisms and
their significance for physics. We summarize the important work of Milnor,
Freedman, Donaldson, and others in developing exotic differentiable structures
on well known topological manifolds. Finally, we discuss some of the geometric
implications of these results and propose some conjectures on possible physical
implications of these new manifolds which have never before been considered as
physical models.Comment: 11 pages, LaTe