Distortion of surfaces in 3-manifolds

Let $g \colon S \looparrowright N$ be a properly immersed $\pi_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of $\pi_1(S)$ in $\pi_1(N)$ and show that how it is related to separability of $\pi_1(S)$ in $\pi_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, and double exponential.Comment: 33 pages, 2 figure. Clarifications are given in Section 4. Figure 1 is redrawn. Section 5 is rewritten for clarify and to fill a gap in the proof of the lower case (section 5.2). This paper has been accepted for publication in Journal of Topolog

Degeneration of Fermat hypersurfaces in positive characteristic

We investigate the hypersurfaces which are the generation of the Fermat hypersufaces, and determine their projective isomorphism classes.Comment: 17 pages in Hiroshima Mathematics Journal 201

On Catastrophic Forgetting and Mode Collapse in Generative Adversarial Networks

In this paper, we show that Generative Adversarial Networks (GANs) suffer from catastrophic forgetting even when they are trained to approximate a single target distribution. We show that GAN training is a continual learning problem in which the sequence of changing model distributions is the sequence of tasks to the discriminator. The level of mismatch between tasks in the sequence determines the level of forgetting. Catastrophic forgetting is interrelated to mode collapse and can make the training of GANs non-convergent. We investigate the landscape of the discriminator's output in different variants of GANs and find that when a GAN converges to a good equilibrium, real training datapoints are wide local maxima of the discriminator. We empirically show the relationship between the sharpness of local maxima and mode collapse and generalization in GANs. We show how catastrophic forgetting prevents the discriminator from making real datapoints local maxima, and thus causes non-convergence. Finally, we study methods for preventing catastrophic forgetting in GANs.Comment: This is an extended version of our paper in ICML'18 Workshop on Theoretical Foundation and Applications of Deep Generative Models. Accepted to IJCNN 202

On Ballico-Hefez curves and associated supersingular surfaces

Let p be a prime integer, and q a power of p. The Ballico-Hefez curve is a non-reflexive nodal rational plane curve of degree q+1 in characteristic p. We investigate its automorphism group and defining equation. We also prove that the surface obtained as the cyclic cover of the projective plane branched along the Ballico-Hefez curve is unirational, and hence is supersingular. As an application, we obtain a new projective model of the supersingular K3 surface with Artin invariant 1 in characteristic 3 and 5.Comment: 12page

On the coarse geometry of certain right-angled Coxeter groups

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a Seifert manifold group or virtually a graph manifold group and we give a complete quasi-isometry classification of these such groups. Otherwise, we prove that $G_\Gamma$ is hyperbolic relative to a collection of $\mathcal{CFS}$ right-angled Coxeter subgroups of $G_\Gamma$. Consequently, the divergence of $G_\Gamma$ is linear, or quadratic, or exponential. We also generalize right-angled Coxeter groups which are virtually graph manifold groups to certain high dimensional right-angled Coxeter groups (our families exist in every dimension) and study the coarse geometry of this collection. We prove that strongly quasiconvex torsion free infinite index subgroups in certain graph of groups are free and we apply this result to our right-angled Coxeter groups.Comment: 38 pages, 6 figures. Minor changes and other updates to incorporate referee comments. To appear in Algebraic & Geometric Topology. arXiv admin note: text overlap with arXiv:1708.0781

Young differential delay equations driven by H\"older continuous paths

In this paper we prove the existence and uniqueness of the solution of Young differential delay equations under weaker conditions than it is known in the literature. We also prove the continuity and differentiability of the solution with respect to the initial function and give an estimate for the growth of the solution. The proofs use techniques of stopping times, Shauder-Tychonoff fixed point theorem and a Gronwall-type lemma

Quark Family Discrimination and Flavour-Changing Neutral Currents in the SU(3)_C X SU(3)_L X U(1) Model with Right-Handed Neutrinos

Contributions of flavour-changing neutral currents in the 3 3 1 model with right-handed neutrinos to mass difference of the neutral meson system $\Delta m_P (P = K, D, B)$ are calculated. Using the Fritzsch anzats on quark mixing, we show that the third family should be different from the first two. We obtain a lower bound on mass of the new heavy neutral gauge boson as 1.02 TeV.Comment: 7 pages, no figures, to appear in J. Phys. G (1999), October issu

Distortion of surfaces in graph manifolds

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.Comment: 29 pages, 2 figure

Asymptotic stability of controlled differential equations. Part I: Young integrals

We provide a unified analytic approach to study stationary states of controlled differential equations driven by rough paths, using the framework of random dynamical systems and random attractors. Part I deals with driving paths of finite $p$-variations with $1 \leq p <2$ so that the integrals are interpreted in the Young sense. Our method helps to generalize recent results \cite{GAKLBSch2010}, \cite{ducGANSch18}, \cite{duchongcong18} on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.Comment: 2

Isospin Separation of Hidden Non-Abelian Monopole

The scheme of isospin separation is suggested for the equation describing the five-dimensional 'charge-dyon' system in a non-Abelian SU(2) model. As a result, we obtain the Schrodinger equation for 'bare' particle, moving in Coulomb potential plus potential of five-dimensional Dirac monopole.Comment: 6 page