An algebraic description of screw dislocations in SC and BCC crystal lattices

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately.Comment: 41 pages, 7 figure

Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system

This paper is concerned with a singular limit of the Kobayashi-Warren-Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica-Mortola functional in a one-dimensional setting.Comment: 24 page

SAN: Inducing Metrizability of GAN with Discriminative Normalized Linear Layer

Generative adversarial networks (GANs) learn a target probability distribution by optimizing a generator and a discriminator with minimax objectives. This paper addresses the question of whether such optimization actually provides the generator with gradients that make its distribution close to the target distribution. We derive metrizable conditions, sufficient conditions for the discriminator to serve as the distance between the distributions by connecting the GAN formulation with the concept of sliced optimal transport. Furthermore, by leveraging these theoretical results, we propose a novel GAN training scheme, called slicing adversarial network (SAN). With only simple modifications, a broad class of existing GANs can be converted to SANs. Experiments on synthetic and image datasets support our theoretical results and the SAN's effectiveness as compared to usual GANs. Furthermore, we also apply SAN to StyleGAN-XL, which leads to state-of-the-art FID score amongst GANs for class conditional generation on ImageNet 256$\times$256.Comment: 24 pages with 12 figure

Coefficient inverse problems for partial differential equations in the viscoelasticity, the material science and population dynamics by Carleman estimates

In this paper, we consider coefficient inverse problems in the viscoelasticity,the material science and the population studies and prove the stability ofthese problem by an a priori weighted L2-norm estimate which is called aCarleman estimate.In Chapter 1, an inverse problem of determining coefficients in a viscoelasticmodel which is called Kelvin-Voigt model is discussed. The dataavailable to us is a Cauchy data on subboundary. We prove that with twoappropriate measurements, we can obtain a Holder stability estimate of the inverse problem.In Chapter 2, we discuss the determination of a thermal conductivity anda mobility in the linearized phase field model with measurement of only onecomponent in a small domain. Our result is the Lipschitz stability estimate of this problem.In Chapter 3, we consider the coefficient inverse problem of the structuredpopulation model. In the structured population model, an age andan individual size as well as a spatial position and time are considered asindepenent variables and then the equation has a special form. We prove a Carleman estimate for this equation and obtain a stability estimate for theinverse problem.報告番号: 甲27184 ; 学位授与年月日: 2011-03-24 ; 学位の種別: 課程博士 ; 学位の種類: 博士(数理科学) ; 学位記番号: 博数理第365号 ; 研究科・専攻: 数理科学研究科数理科学専