代数表示论的某些新进展

代数表示理论是代数学的一个新的重要分支，在近二十五年的时间里，这一理论有很大的发展，关于代数表示的基础理论的介绍可参见文献（１０１），本文主要从Ｈａｌｌ代数和拟遗传代数两个方面介绍代数表示论的一些最新进展，第一章给出了Ｈａｌｌ代数的基本理论及其方法，并且着重指出了利用这一理论和方法通过代数表示论去实现Ｋａｃ－Ｍｏｏｄｙ李代数及相应的量子包络代数，第二章介绍了拟遗传代数及其表示理论，以及这一理论与复     

A∞∞-型Ringel-Hall代数

这是利用A∞∞-型Ringel—Hall代数研究sl∞∞-型量子群的两篇文章中的第一篇．为此首先需要研究建立在任意域向上的无限维路代数后4要的有限维表示．在文章的第一部分，我们给出了所有的不可分解kA∞∞-表示，并且清楚地刻画了它们之间的扩张关系；在第二部分，对于给定的有限域k，我们研究了Ringel—Hall代数H（kA∞∞）．主要观察是把H（kA∞∞）看作Ringel—Hall代数H（kA∞）的正向极限，把H（kA∞）看作Ringel—Hall代数H（kAn）的正向极限．特别地，我们得到了H（kA∞∞）的一个PBw一基，并且证明了H（kA∞∞）恰好和它的合成子代数重合．     

Hopf代数的结构定理和对映阶数

本文中，我们把Ｈｏｐｆ代数的结构定理推广到Ｈｏｐｆ代数意义下的同构，从而给出Ｈｏｐｆ代数既约分支的对映阶数，并得到Ｈｏｐｆ代数扩张的对映阶数是任意的．这部分回答了Ｅ．Ｊ．Ｔａｆｔ１９９４年提出的一个问题．     

AR箭图的直向分支

本文研究无限表示型代数AR箭图的分支的直向性,包括直向分支的判定;新的直向分支的构造;具有ZΔ分支的平凡扩张代数T(A)的确定,其中A是重复倾斜代数.     

无原子布氏代数理论的计算复杂性

研究无原子布氏代数的计算复杂性 .得到了下面的新定理 :定理 1　无原子布氏代数理论Δ具有完全的量词消去法 ,也就是说每一个式子都Δ等价于一个开式子 .定理 2　无原子布氏代数的初等型Γ (x1,… ,xn)是由型内的不含量词的全体开式子所唯一决定 .定理 3　无原子布氏代数的一个长度为 n的语句的判断过程所消耗的 Turing时间和空间都是属于 2 2 cn指数级 .     

乘积σ—代数的一种生成方法

本文得到了有限及可数无限维乘积σ—代数的一种新的生成方法，并证明了可数无限维Ｂｏｒｅｌ域具有连续统的势．     

Dn型路代数倾斜模与AR箭图边缘

代数表示理论是上个世纪七十年代初兴起的代数学的—个新的分支,而倾斜理论是研究代数表示理论的重要工具之一．本文主要对Dn型路代数倾斜模在其对应的AR-箭图上的结构特点进行研究．通过对Dn型路代数A的AR-箭图ΓA分析,证明了：Dn型路代数倾斜模T的—个必要条件是。〈T〉中至少有三个边缘点．     

代数几何码的测试

性质测试是90年代开始由多种研究引发的,GF（q）^n中一个线性码C称为局部可测试的,当且仅当存在一个随机化算法,使得只要输入任一个GF（q）^n中向量的很少一部分坐标（一般而言是常数个坐标）,这个随机化算法就可以很高的概率判定此向量是否是C中码字．Blum,Luby和Rubinfeld由于和概率可验证证明的紧密关系研究了码的局部可测试性,然而怎样刻画局部可测试码是一个复杂且甚具挑战性的问题．对Reed—Solomon（RS）码、Reed．Muller（RM）码、循环码、BCH码的对偶码及代数几何码的迹子码,已经研究了局部可测试问题．在本文中我们给出了代数几何码的线性参数的测试子,并证明了在一个不太强的限制条件下代数几何码不是局部可测试的．     

一个代数的预投射分支和τ-预投射分支

设A是一个Artin代数,Γ_A是A的Auslander-Reiten箭图。我们得到:如果Γ是Γ_A的一个不包含有向循环的预投射分支,那么Γ是Γ_A的一个τ-预投射分支,且对一个拟倾斜代数A,Γ是Γ_A的一个预投射分支当且仅当Γ是Γ_A的一个,τ-预投射分支。     

一个代数的预投射分支和т-预投射分支

设A是一个Artin代数,ΓA是A的Auslander-Reiten箭图.我们得到如果Γ是ΓA的一个不包含有向循环的预投射分支,那么Γ是ΓA的一个τ-预投射分支,且对一个拟倾斜代数A,Γ是ΓA的一个预投射分支当且仅当Γ是ΓA的一个τ-预投射分支.     

BCH-代数的拟结合理想和拟结合模糊理想

在BCH-代数中,引入了拟结合理想和模糊拟结合理想的概念.研究了BCH-代数中模糊拟结合理想的相关性质,给出了BCH-代数拟结合理想的构造.     

可换BCH-代数

引入了可换BCH-代数的概念,给出了可换BCH-代数的两个充要条件.对偏序可换BCH-代数进行了讨论,给出了偏序BCH-代数是可换的两个充要条件.证明了偏序可换BCH-代数的每个分支是一个下半格,局部有界偏序可换BCH-代数的每个分支是一个格.     

BCH-代数的闭理想

在BCH-代数中引入了闭理想的概念,并对其性质进行研究,得到了许多结果.     

BCH-代数的闭模糊理想

引入了BCH-代数的闭模糊理想、子代数及优BCH-代数的概念,讨论它们的性质,获得若干结果。     

Microscopic quantum hydrodynamics of systems of fermions: II

A closed system of nonstationary equations for a multielectron atom is constructed using a method previously developed for investigating multiparticle fermion systems. The well-known Thomas—Fermi—Dirac method is a particular case of the system. The exchange spin—spin correlations are calculated using an improved spin—spin interaction Hamiltonian. The dependence of the electron concentration on the self-consistent electric field potential and a discrete set of parameters determined by the total energy and the angular momentum of the atom is derived for stationary atom states.     

Parapositronium atom in the field of annihilation and optical photons as a nonlinear quantum system

A parapositronium atom in an optical laser field is described beyond the perturbation theory framework by a closed system of Heisenberg equations on operators of atoms and photons. Wwe consider the annihilation of the parapositronium atom, which starts from one or two quantum states; optical quantum transitions between these states are caused by one or two optical photons. Mean occupation numbers of these states are governed by a system of two nonlinear equations. We investigated particular stationary and nonstationary solutions of this system and find that annihilation photons substantially affect the annihilation process. We show that definite optical laser radiation may stabilize the parapositronium atom and make its lifetime hundreds of times longer than the lifetime of the free parapositronium atom in the 1s state. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 148–168, July, 2000.     

Theoretical studies on solid heterogenous targets thickness in muon catalyzed fusion using transport theory and its comparison with available experimental values

Muon catalyzed fusion in solid heterogeneous H/D/T layers has been studied in recent years. When a negative muon enters the protium layer with a small admixture of tritium, the muonic atom is formed. molecular ion formation is highly dependent on the energy of the muonic tritium atom, μt, and this energy itself depends on the thickness of H/T and D₂ layers. In this paper, the transport theory is used to study the effect of the thickness of different layers (H/T and D₂) on the output kinetic energy of the μt atom. The calculated thickness by the used theoretical method are compared with the available experimental values and it is shown that they are in a good and acceptable agreement, and represents the reliability of used theoretical method.     

Dynamics of dark–bright vector solitons in a two-component Bose–Einstein condensate

Coupled dark–bright vector solitons are considered in a two-component Bose–Einstein condensate, and their dynamics are investigated by the variational approach based on the renormalized integrals of motion. The stationary states and their atom population distribution are obtained, and it is found that the dark soliton has obvious robust features. The dynamic mechanism is demonstrated by performing a coordinate of a classical particle moving in an effective potential field, and the switching and self-trapping dynamics of the coupled dark–bright vector solitons are discussed by the evolution of the atom population transferring ratio.     

Qualitative model of the atom,its components and origin in the early universe

This paper is the first qualitative model of the atom and its components as well as its origin in our early universe. The atom’s components–proton, neutron (with its neutrino) and electron–are built on the prima, units of visible matter. Their qualitative models are obtained using some laws of nature. They yield new information such as the mass of the neutrino and how heavy isotopes of an atom form as well as the arrangement of the nucleons. It clarifies some presently unresolved issues such as what happens in thermonuclear explosion and whether it can be controlled to generate energy. It also explains the wave-particle duality of the primum and photon, metal fatigue, matter-anti-matter interaction and brittle and malleable materials and reviews the early phase of the development of our universe beyond the formation of the first atom up to its rapid formation at the present time in the Cosmos and tissues of living organisms.     

Entanglement,fidelity and quantum chaos in cavity QED

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom–field system as a quantum–classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Chaos manifests itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with prominent fractal-like behavior and in the quantum atom–field degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom–field resonance correlate strongly. It is shown that the Jaynes–Cummings dynamics can be unstable in the regime of chaotic walking of an atom in the quantized field of a standing wave in the absence of any other interaction with environment. Quantum instability manifests itself in strong variations of quantum purity and entropy and in exponential sensitivity of fidelity of quantum states to small variations in the atom–field detuning. It is quantified in terms of the respective classical maximal Lyapunov exponent that can be estimated in appropriate in–out experiments. This result provides a quantum–classical correspondence in a closed physical system.