On The Geometric Monodromy Of The Milnor Fibre

In this paper we show that the geometric monodromy $\phi$ of the Milnor fibre, has finite order q if and only if the operator $\phi^q_*$ acts trivially on the homology. The proof is based on the classical methods of surgery.Comment: Milnor fibres bounded by rational homology spheres only should be considered her

Relative mapping class group of Sp×DqS^p\times D^q

Algebraic structure of the group of pseudo-isotopy classes of diffeomorphisms of the trivial disk bundle over the standard sphere which restrict to the identity map on the boundary is determined.Comment: 9 page

Pseudo-isotopy classes of diffeomorphisms of the unknotted pairs (Sn+2, Sn)(S^{n+2},~S^n) and (S2p+2, Sp×Sp)(S^{2p+2},~S^p\times S^p)

We consider two pairs: the standard unknotted $n$-sphere in $S^{n+2}$, and the product of two $p$-spheres trivially embedded in $S^{2p+2}$, and study orientation preserving diffeomorphisms of these pairs. Pseudo-isotopy classes of such diffeomorphisms form subgroups of the mapping class groups of $S^n$ and $S^p\times S^p$ respectively and we determine the algebraic structure of such subgroups when $n>4$ and $p>1$

A basis of the group of primitive almost pythagorean triples

Let $m$ be a fixed square-free positive integer, then equivalence classes of solutions of Diophantine equation $x^2+m\cdot y^2=z^2$ form an infinitely generated abelian group under the operation induced by the complex multiplication. A basis of this group is constructed here using prime ideals and the ideal class group of the field $\mathbb Q (\sqrt{-m})$.Comment: 10 pages, continuation of arXiv:1107.286

Permutations with a distinct divisor property

A finite group of order $n$ is said to have the distinct divisor property (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an abelian group is DDP if and only if it has a unique element of order 2. We also describe a construction of DDP groups via group extensions by abelian groups and show that there exist infinitely many non abelian DDP groups

A congruence property of irreducible Laguerre polynomials in two variables

In this paper we introduce a version of irreducible Laguerre polynomials in two variables and prove for it a congruence property, which is similar to the one obtained by Carlitz for the classical Laguerre polynomials in one variable.Comment: 10 page

Stochastic Soliton Lattices

We introduce a new concept, Stochastic Soliton Lattice, as a random process generated by a finite-gap potential of the Shroedinger operator. We study the basic properties of this stochastic process and consider its KdV evolutionComment: 11 pages. To be published in Proceedings of the International Conference `Solitons, Geometry and Topology: on the Crossroads', Moscow, 199

Angle contraction between geodesics

We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of such systems are observed when the plane is replaced by a regular surface in ${\mathbb R}^3$ and lines are replaced by geodesics. An application of our results to the classification of points on the surface as elliptic, hyperbolic or parabolic is also presented.Comment: 10 pages, 2 figure

Cosmological phase transition, baryon asymmetry and dark matter Q-balls

We consider a mechanism of dark matter production in the course of first order phase transition. We assume that there is an asymmetry between X- and anti-X-particles of dark sector. In particular, it may be related to the baryon asymmetry. We also assume that the phase transition is so strongly first order, that X-particles do not permeate into the new phase. In this case, as the bubbles of old phase collapse, X-particles are packed into Q-balls with huge mass defect. These Q-balls compose the present dark matter. We find that the required present dark matter density is obtained for the energy scale of the theory in the ballpark of 1-10 TeV. As an example we consider a theory with effective potential of one-loop motivated form.Comment: 17 pages, 3 figure

Harmonic Networks with Limited Training Samples

Convolutional neural networks (CNNs) are very popular nowadays for image processing. CNNs allow one to learn optimal filters in a (mostly) supervised machine learning context. However this typically requires abundant labelled training data to estimate the filter parameters. Alternative strategies have been deployed for reducing the number of parameters and / or filters to be learned and thus decrease overfitting. In the context of reverting to preset filters, we propose here a computationally efficient harmonic block that uses Discrete Cosine Transform (DCT) filters in CNNs. In this work we examine the performance of harmonic networks in limited training data scenario. We validate experimentally that its performance compares well against scattering networks that use wavelets as preset filters