Certain properties related to well posedness of switching diffusions

This work is devoted to switching diffusions that have two components (a continuous component and a discrete component). Different from the so-called Markovian switching diffusions, in the setup, the discrete component (the switching) depends on the continuous component (the diffusion process). The objective of this paper is to provide a number of properties related to the well posedness. First, the differentiability with respect to initial data of the continuous component is established. Then, further properties including uniform continuity with respect to initial data, and smoothness of certain functionals are obtained. Moreover, Feller property is obtained under only local Lipschitz continuity. Finally, an example of Lotka–Volterra model under regime switching is provided as an illustration.     

Stable numerical schemes for a partly convolutional partial integro-differential equation

A model partial integro-differential operator (PIDO) that contains both local and nonlocal diffusion operators is considered in this article. This type of operators come in modeling various scientific and financial engineering problems. In most cases, people use finite difference schemes to generate solutions of such model problems. We compare and analyze stability and accuracy of two such finite difference schemes. We first present a discrete analogue of the PIDO and then approximate the semi-discrete time dependent problem using two different one step methods and show the stability conditions and the accuracy of the schemes. We use the Fourier transforms throughout our analysis.     

Smoothness of the Solution of the Spatially Homogeneous Boltzmann Equation without Cutoff

Abstract

For regularized hard potentials cross sections, the solution of the spatially homogeneous Boltzmann equation without angular cutoff lies in Schwartz's space 𝒮(? ^N ) for all (strictly positive) time. The proof is presented in full detail for the two-dimensional case, and for a moderate singularity of the cross section. Then we present those parts of the proof for the general case, where the dimension, or the strength of the singularity play an essential role.     

Smoothing of bump functions

Let X be a separable Banach space with a Schauder basis, admitting a continuous bump which depends locally on finitely many coordinates. Then X admits also a C^∞-smooth bump which depends locally on finitely many coordinates.     

Quantum model for the price dynamics: The problem of smoothness of trajectories

We apply methods of quantum mechanics to mathematical modelling of price dynamics in a financial market. We propose to describe behavioral financial factors (e.g., expectations of traders) by using the pilot wave (Bohmian) model of quantum mechanics. Our model is a quantum-like model of the financial market, cf. with works of W. Segal, I.E. Segal, E. Haven. In this paper we study the problem of smoothness of price-trajectories in the Bohmian financial model. We show that even the smooth evolution of the financial pilot wave ψ(t,x) (representing expectations of traders) can induce jumps of prices of shares.     

Self-organized criticality and color vision: A guide to water–protein landscape evolution

We focus here on the scaling properties of small interspecies differences between red cone opsin transmembrane proteins, using a hydropathic elastic roughening tool previously applied to the rhodopsin rod transmembrane proteins. This tool is based on a non-Euclidean hydropathic metric realistically rooted in the atomic coordinates of 5526 protein segments, which thereby encapsulates universal non-Euclidean long-range differential geometrical features of water films enveloping globular proteins in the Protein Data Bank. Whereas the rhodopsin blue rod water films are smoothest in humans, the red cone opsins’ water films are optimized for smoothness in cats and elephants, consistent with protein species landscapes that evolve differently in different contexts. We also analyze red cone opsins in the chromatophore-containing family of chameleons, snakes, zebrafish and goldfish, where short- and long-range (BLAST and hydropathic) amino acid (aa) correlations are found with values as large as 97%–99%. We use hydropathic aa optimization to estimate the maximum number _Nmax$N_{max}$ of color shades that the human eye can discriminate, and obtain 10⁶<_Nmax<10⁷${10}^6<N_{max}<{10}^7$, in good agreement with experiment.     

非凸集值映射的包含切性及应用

本文在一般的Ｂａｎａｃｈ空间Ｘ中研究从非空闭集ＫＸ到Ｘ的非凸集值映射Ｆ的包含切性问题．得到的结果定理３．１把有关的结论推广到非光滑空间，定理３．３则将有限维空间的正则性定理推广到任意的Ｂａｎａｃｈ空间．作为结果的应用，我们证明了无穷维非凸微分包含和非凸控制系统生存解的存在性，且给出了一个方便的等价切性条件．     

Polynomiality of primal-dual affine scaling algorithms for nonlinear complementarity problems

This paper provides an analysis of the polynomiality of primal-dual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu’s scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primal-dual affine scaling algorithms generates an approximate solution (given a precision ε) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial ofn, ln(1/ε) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in Jansen et al., SIAM Journal on Optimization 7 (1997) 126–140. Research supported in part by Grant-in-Aids for Encouragement of Young Scientists (06750066) from the Ministry of Education, Science and Culture, Japan. Research supported by Dutch Organization for Scientific Research (NWO), grant 611-304-028     

倒向随机微分方程解的光滑性

该文讨论了倒向随机微分方程Y_t=ξ+∫^T_t{g(s,Y_s,Z_s)}ds-∫^T_t{Z_s}dW_s 解在Malliavin微分意义下的光滑性.对任意的n讨论其解在Malliavin 意义下n 阶可微性,并且证明它是一个线性倒向随机微分方程的解,从而说明BSDE解的光滑性.     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.