Action of the Cremona group on a noncommutative ring

The Cremona group acts on the field of two independent commutative variables over complex numbers. We provide a noncommutative algebra that is an analog of a noncommutative field of two independent variables and prove that the Cremona group embeds in a group of outer automorphisms of this algebra. We give two proofs of it, the first proof is technical, the second one is conceptual and proceeds through a construction of a birational invariant of an algebraic variety from its bounded derived category of coherent sheaves.     

SL(3,3~n)和SU(3,3~n)的第一Cartan不变量

确定Cartan不变量是代数群与相关的李型有限群的模表示理论中的一个重要方面.作者利用代数群模表示理论中的一系列结果,计算了3~n个元素的有限域上特殊线性群SL(3,3~n)和特殊酉群SU(3,3~n)的第一Cartan不变量,得到如下结论:当G=SL(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n;而当G=SU(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n+2·(1+(-1)~n),其中a,b是多项式x~2-20x+48的两个根.另外,作者也得到了射影不可分解模U_n(0,0)的维数公式:dim U_n(0,0)=(12~n-6~n+∈)·3~(3n),其中,当G=SL(3,3~n)时,∈=1;而当G=SU(3,3~n)时,∈=-1.     

在模情况时二面体群$D_{2p}$作用下的transfer理想

In this paper, we determine the structures of the transfer ideal and its radical ideal for the ring of polynomials F p[x, y] under the action of dihedral group D2 p in the modular case. We mainly use Transfer variety, p order elements, and Hilbert’s Nullstellensatz Theorem.     

A new subspace iteration method for the algebraic Riccati equation

We consider the numerical solution of the continuous algebraic Riccati equation A*X + XA ? XFX + G = 0, with F = F*,G = G* of low rank and A large and sparse. We develop an algorithm for the low‐rank approximation of X by means of an invariant subspace iteration on a function of the associated Hamiltonian matrix. We show that the sought‐after approximation can be obtained by a low‐rank update, in the style of the well known Alternating Direction Implicit (ADI) iteration for the linear equation, from which the new method inherits many algebraic properties. Moreover, we establish new insightful matrix relations with emerging projection‐type methods, which will help increase our understanding of this latter class of solution strategies. Copyright © 2014 John Wiley & Sons, Ltd.     

A Theoretical Model for Release Dynamics of an Antifungal Agent Covalently Bonded to the Chitosan

The aim of the study was to create a mathematical model useful for monitoring the release of bioactive aldehydes covalently bonded to the chitosan by reversible imine linkage, considered as a polymer–drug system. For this purpose, two hydrogels were prepared by the acid condensation reaction of chitosan with the antifungal 2-formyl-phenyl-boronic acid and their particularities; influencing the release of the antifungal aldehyde by shifting the imination equilibrium to the reagents was considered, i.e., the supramolecular nature of the hydrogels was highlighted by polarized light microscopy, while scanning electron microscopy showed their microporous morphology. Furthermore, the in vitro fungicidal activity was investigated on two fungal strains and the in vitro release curves of the antifungal aldehyde triggered by the pH stimulus were drawn. The theoretical model was developed starting from the hypothesis that the imine-chitosan system, both structurally and functionally, can be assimilated, from a mathematical point of view, with a multifractal object, and its dynamics were analyzed in the framework of the Scale Relativity Theory. Thus, through Riccati-type gauges, two synchronous dynamics, one in the scale space, associated with the fungicidal activity, and the other in the usual space, associated with the antifungal aldehyde release, become operational. Their synchronicity, reducible to the isomorphism of two SL(2R)-type groups, implies, by means of its joint invariant functions, bioactive aldehyde compound release dynamics in the form of “kink–antikink pairs” dynamics of a multifractal type. Finally, the theoretical model was validated through the experimental data.     

Perturbation to Mei symmetry and Mei adiabatic invariants for mechanical systems in phase space

For a perturbed mechanical system in phase space, considering d/dt in the structure equation and process of proof including infinitesimal parameter ε obviously, this paper studies the perturbation to Mei symmetry and adiabatic invariants. Firstly, the exact invariant induced directly from the Mei symmetry of the system without perturbation is given. Secondly, based on the concept of high-order adiabatic invariant, the determining equations of the perturbation to Mei symmetry are established, the condition of existence of the Mei adiabatic invariant led by the perturbation to Mei symmetry is obtained, and its form is presented. Lastly, an example is given to illustrate the application of the results.     

图胞映射的一种改进方法

通过引入新的概念，提出了图胞映射动力系统中瞬态胞的新的分类方法，基于新的分类方法研究了动力系统中不变流形的胞映射逼近问题；并结合计算机的计算速度与内存特点，建立了完成上述压缩分类的有效算法.通过对典型算例Henon映射的应用分析，证实了该方法的有效性.     

Weak Entropy Boundary Conditions for Isentropic Gas Dynamics via Kinetic Relaxation

We consider a kinetic BGK model relaxing to isentropic gas dynamics previously introduced by the authors, but with Dirichlet boundary condition on the incoming velocities. We pass to the limit as the relaxation parameter tends to zero by compensated compactness inside the domain, and obtain that the limit satisfies entropy inequalities on the boundary involving weak traces of entropy fluxes. Our method is very general and could be applied to any entropy satisfying BGK model as soon as we have strong compactness of the macroscopic variables inside the domain.