Principal manifolds and graphs in practice: from molecular biology to dynamical systems

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.Comment: 12 pages, 9 figure

Matrix Representations of Holomorphic Curves on T4T_{4}

We construct a matrix representation of compact membranes analytically embedded in complex tori. Brane configurations give rise, via Bergman quantization, to U(N) gauge fields on the dual torus, with almost-anti-self-dual field strength. The corresponding U(N) principal bundles are shown to be non-trivial, with vanishing instanton number and first Chern class corresponding to the homology class of the membrane embedded in the original torus. In the course of the investigation, we show that the proposed quantization scheme naturally provides an associative star-product over the space of functions on the surface, for which we give an explicit and coordinate-invariant expression. This product can, in turn, be used the quantize, in the sense of deformation quantization, any symplectic manifold of dimension two.Comment: 29 page

Redundant Picard-Fuchs system for Abelian integrals

We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two times greater than the standard Picard-Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were able to majorize the number of all (real and complex) zeros. In the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and non-homogeneity of uni- and bivariate complex polynomials are studied.Comment: 31 page, LaTeX2e (amsart

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

Degenerations and limit Frobenius structures in rigid cohomology

We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit examples.Comment: 41 page

Reconstruction algorithms for a class of restricted ray transforms without added singularities

Let $X$ and $X^*$ denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data $Xf$ is usually based on a study of the composition $X^* D X$, where $D$ is some local operator (usually a derivative). If $X^*$ is chosen appropriately, then $X^* D X$ is a Fourier Integral Operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing $D$ in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace $D$ with a nonlocal operator $\tilde D$ that integrates $Xf$ along a curve in the data space. The result $\tilde D Xf$ resembles the generalized Radon transform $R$ of $f$. The function $\tilde D Xf$ is defined on pairs $(x_0,\Theta)\in U\times S^2$, where $U\subset\mathbb R^3$ is an open set containing the support of $f$, and $S^2$ is the unit sphere in $\mathbb R^3$. Second, we replace $X^*$ with a backprojection operator $R^*$ that integrates with respect to $\Theta$ over $S^2$. It turns out that if $\tilde D$ and $R^*$ are appropriately selected, then the composition $R^* \tilde D X$ is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting $\tilde D$ we get access to the frequency variable $\Theta$. In particular, we can incorporate suitable cut-offs in $R^*$ to eliminate bad directions $\Theta$, which lead to added singularities