Strongly stable rank and applications to matrix completion

We perform an in-depth study of strongly stable ranks of modules over a commutative ring. Here we define the strongly stable rank of a module to be the supremum of the stable ranks of its finitely generated submodules. As an application, we give non-Noetherian generalizations of known facts about outer products and matrix completions over PIRs and Dedekind domains. We construct Noetherian and non-Noetherian domains of arbitrary strongly stable rank. We also consider strongly n-generated ideals, and we characterize the rings in which every ideal is strongly 2-generated and the domains in which every ideal is strongly 3-generated.     

On sums and products of primitive elements

Let R be a (commutative integral) domain, with K its quotient field and R^′ its integral closure (in K). Let 𝒫 be the set of elements u∈K such that u is primitive over R; i.e., such that u is the root of a polynomial over R having a unit coe?cient. Then, 𝒫 is a ring (necessarily K) ? 𝒫 is closed under products ? R^′ is a Prüfer domain. In general, 𝒫 is closed under powers. For u,v∈𝒫, necessary and su?cient conditions are given for u+v (resp., uv) to belong to 𝒫. Also, 𝒫 is used to characterize when R is a quasi-local integrally closed domain and when R is a pseudo-valuation domain. If R is quasi-local, each element of K is expressible as the sum of two (possibly equal) elements of 𝒫. The set of primitive elements is determined for lying-over pairs and for extensions of domains with the same sets of prime ideals. In this study of the 𝒫 construction, R and K are replaced, whenever possible, by an arbitrary commutative ring and its total quotient ring or, more generally, by any inclusion of commutative rings.     

Difference dimension quasi-polynomials

We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that such functions are quasi-polynomials, which can be represented as alternating sums of Ehrhart quasi-polynomials associated with rational conic polytopes. In particular, we obtain generalizations of main theorems on difference dimension polynomials and their invariants to the case of weighted basic difference operators.     

Statistical physics of a mesoscopic fermion system inside a rectangular box

An ideal Fermi gas of identical particles confined inside a rectangular box is considered. Using the grand‐canonical formalism and performing the summations over the one‐particle states numerically, the temperature dependence of the chemical potential, of the average energy and of the heat capacity are derived before the thermodynamic limit is taken. These mesoscopic quantities differ from the corresponding thermodynamic quantities only quantitatively but are qualitatively similar. © 2013 Wiley Periodicals, Inc.     

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.     

Similarity solution for a spherical shock wave in a non‐ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy

The propagation of a spherical shock wave in a non‐ideal gas with or without gravitational effects is investigated under the action of monochromatic radiation. Similarity solutions are obtained for adiabatic flow between the shock and the piston. The numerical solutions are obtained using the Runge‐Kutta method of the fourth order. The density of the gas is assumed to be constant. The total energy of the shock wave is non‐constant and varies with time. The effects of change in values of non‐idealness parameter, gravitational parameter, shock Mach number, radiation parameter, and adiabatic exponent of the gas on shock strength and flow variables are worked out in detail. It is investigated that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and, therefore, the distance between the inner contact surface and the shock surface is reduced. A comparison is also made between the solutions in the cases of the gravitating and the non‐gravitating media. It is manifested that the gravitational parameter and the radiation parameter have in general opposite behaviour on the flow variables and the shock strength.     

Flat topology and its dual aspects

In this article, a new and natural topology on the prime spectrum is introduced, which behaves completely as the dual of the Zariski topology. It is called the flat topology. The basic and also some sophisticated properties of the flat topology are proved. Specially, various algebraic characterizations for the noetherianess of the flat topology are given. Using the flat topology, then some facts on the structure of the prime ideals of a ring come to light which are not in the access of the Zariski topology.     

残缺区间合作博弈及其在农地污染治理中的应用

针对联盟收益值部分未知的区间合作博弈,定义了残缺区间合作博弈的相关概念。基于合作博弈的超可加性,建立了联盟区间收益值的一致性验证模型。通过构造正、负理想分配及其与收益分配向量之间的偏差,给出了残缺区间合作博弈的区间Ideal-Shapley值求解模型,分析了区间Ideal-Shapley值的合理性与存在性。利用上述模型求解农地污染联合治理的节约成本分摊策略,验证了区间Ideal-Shapley值求解模型的有效性。     

交一致连续偏序集

In this paper, as a generalization of uniform continuous posets, the concept of meet uniform continuous posets via uniform Scott sets is introduced. Properties and characterizations of meet uniform continuous posets are presented. The main results are:(1) A uniform complete poset L is meet uniform continuous iff ↑(U ∩↓ x) is a uniform Scott set for each x ∈ L and each uniform Scott set U;(2) A uniform complete poset L is meet uniform continuous iff for each∨∨x∈ L and each uniform subset S, one has x ∧S ={x ∧ s | s ∈ S}. In particular, a complete lattice L is meet uniform continuous iff L is a complete Heyting algebra;(3) A uniform complete poset is meet uniform continuous iff every principal ideal is meet uniform continuous iff all closed intervals are meet uniform continuous iff all principal filters are meet uniform continuous;(4) A uniform complete poset L is meet uniform continuous if L1 obtained by adjoining a top element1 to L is a complete Heyting algebra;(5) Finite products and images of uniform continuous projections of meet uniform continuous posets are still meet uniform continuous.     

试卷水平的标准化定量评价

通过建立一个实用标准化评价模型,根据转换参数(偏移参数和膨胀参数)给出试卷水平的标准化评价,并对其做Kappa一致性检验.Kappa系数K作为试卷水平系数.X~2一致性检验和理想模式比较检验从不同角度验证了标准化评价的有效性.给出了两个样本组的实证分析.