Spaces of polynomials with roots of bounded multiplicity

We describe an alternative approach to some results of Vassiliev on spaces of polynomials, by using the scanning method which was used by Segal in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.Comment: 29 pages, AMS-Te

Robust point correspondence applied to two and three-dimensional image registration

Accurate and robust correspondence calculations are very important in many medical and biological applications. Often, the correspondence calculation forms part of a rigid registration algorithm, but accurate correspondences are especially important for elastic registration algorithms and for quantifying changes over time. In this paper, a new correspondence calculation algorithm, CSM (correspondence by sensitivity to movement), is described. A robust corresponding point is calculated by determining the sensitivity of a correspondence to movement of the point being matched. If the correspondence is reliable, a perturbation in the position of this point should not result in a large movement of the correspondence. A measure of reliability is also calculated. This correspondence calculation method is independent of the registration transformation and has been incorporated into both a 2D elastic registration algorithm for warping serial sections and a 3D rigid registration algorithm for registering pre and postoperative facial range scans. These applications use different methods for calculating the registration transformation and accurate rigid and elastic alignment of images has been achieved with the CSM method. It is expected that this method will be applicable to many different applications and that good results would be achieved if it were to be inserted into other methods for calculating a registration transformation from correspondence

Geometric structures on loop and path spaces

Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong indication of the "almost" independence of the quasi-symplectic structure with respect to the metric. Finally conditions to have contact structures on these spaces are studied.Comment: Final version. To appear in Proceedings of Math. Sci. Indian Academy of Science

Noncommutative geometry, topology and the standard model vacuum

As a ramification of a motivational discussion for previous joint work, in which equations of motion for the finite spectral action of the Standard Model were derived, we provide a new analysis of the results of the calculations herein, switching from the perspective of Spectral triple to that of Fredholm module and thus from the analogy with Riemannian geometry to the pre-metrical structure of the Noncommutative geometry. Using a suggested Noncommutative version of Morse theory together with algebraic $K$-theory to analyse the vacuum solutions, the first two summands of the algebra for the finite triple of the Standard Model arise up to Morita equivalence. We also demonstrate a new vacuum solution whose features are compatible with the physical mass matrix.Comment: 24 page

Virtual Structure Constants as Intersection Numbers of Moduli Space of Polynomial Maps with Two Marked Points

In this paper, we derive the virtual structure constants used in mirror computation of degree k hypersurface in CP^{N-1}, by using localization computation applied to moduli space of polynomial maps from CP^{1} to CP^{N-1} with two marked points. We also apply this technique to non-nef local geometry O(1)+O(-3)->CP^{1} and realize mirror computation without using Birkhoff factorization.Comment: 10 pages, latex, a minor change in Section 4, English is refined, Some typing errors in Section 3 are correcte

Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras.Comment: 32 pages, 1 figure; changes: major revision of presentation, clarifications adde

Small oscillations and the Heisenberg Lie algebra

The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in $\mathfrak g$ is related to the commutativity of a family of derivations of the 2n+1-dimensional Heisenberg Lie algebra $\mathfrak h_n$. Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in $\mathfrak h_n$. For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of the AKS schem