线性变换下的模糊线性关系

在[1],[2],[3]基础上给出了某些线性变换下模糊线性无关向量组及模糊基的变换规律,指出任意模糊线性无关组都可扩充为模糊基,同时也给出了将模糊线性无关性组扩弃为模糊基的方法,。     

附加广义函数的一般形式

作者证明了两两次数不同的附加广义函数组的线性无关性，接着给出了附加广义函数的一般形式。     

V_n(R~+)中半线性生成子空间的基与维数

在交换的零和自由半环上,首先讨论了半线性空间V_n中向量组线性相关性的一些性质,并给出向量组中极大线性无关组所含向量个数相同的条件.其次通过对半环R~+,+,.,0,1上生成子空间基的讨论,给出了向量组的极大线性无关组含相同向量个数的条件.最后对R~+,+,.,0,1上生成子空间的维数进行详细讨论并给出相应的结果.     

数域上的线性无关性

本文给出一组复数在数域上线性无关的一个判别法则及它对于超越数论的一个应用．     

某些三角级数的超越性

本文在仅与系数有关的一般条件下证明了某些代数系数三角级数及其导数在代数点上值的超越性.定理的证明基于一个关于数域上的线性无关性的判别法则.     

线性微分方程的一个反问题

讨论了线性微分方程的一个反问题.给定一个线性无关的函数组,可以得到一个高阶线性微分方程,该微分方程的基本解组恰好是此函数组.另外,对一阶线性微分方程组也进行了类似的讨论.     

交换半环上向量不同线性无关性定义间的关系

1996年,Cechlárová和Plávka在bottleneck代数上给出了向量的线性相关性和线性无关性的三组不同定义,并讨论了它们间的关系。本文将在半环上研究向量的线性相关性和线性无关性的三组不同定义,并讨论它们的关系。证明了半环上可逆矩阵的列向量在一组定义下是线性无关的,最后给出了零和自由半环上矩阵可逆与构成矩阵的行(列)向量组的线性相关性之间的关系。     

巧妙设计线性代数习题

通过具体实例的设计与求解，深化对向量组的最大线性无关组、最大线性无关组的不唯一性、向量组的线性表示等知识的理解，同时展示这部分知识在实际中的一种应用。增强学生应用所学知识解决实际问题的能力。     

数域上的线性无关性(Ⅱ)

本文给出了文[1](见本刊中文版, 1997,40(5):713-716)中建立的复数在数域上线性无关的判别法则的一个等价形式,并将它应用于某些幂级数和三角级数的超越性研究.     

线性无关假设如何影响非线性约束最优化算法

首先综述非线性约束最优化最近的一些进展. 首次定义了约束最优化算法的全局收敛性. 注意到最优性条件的精确性和算法近似性之间的差异, 并回顾等式约束最优化的原始的Newton 型算法框架, 即可理解为什么约束梯度的线性无关假设应该而且可以被弱化. 这些讨论被扩展到不等式约束最优化问题. 然后在没有线性无关假设条件下, 证明了一个使用精确罚函数和二阶校正技术的算法可具有超线性收敛性. 这些认知有助于接下来开发求解包括非线性半定规划和锥规划等约束最优化问题的更加有效的新算法.     

Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs （W,x）, where ∈ X and W is a closed downward subset of X     

The use of phase portraits to visualize and investigate isolated singular points of complex functions

ABSTRACT

Undergraduate students usually study Laurent series in a standard course of Complex Analysis. One of the major applications of Laurent series is the classification of isolated singular points of complex functions. Although students are able to find series representations of functions, they may struggle to understand the meaning of the behaviour of the function near isolated singularities. In this paper, I briefly describe the method of domain colouring to create enhanced phase portraits to visualize and study isolated singularities of complex functions. Ultimately this method for plotting complex functions might help to enhance students' insight, in the spirit of learning by experimentation. By analysing the representations of singularities and the behaviour of the functions near their singularities, students can make conjectures and test them mathematically, which can help to create significant connections between visual representations, algebraic calculations and abstract mathematical concepts.     

Comparison principle and analogues of the phragmen—lindelöf theorem for solutions of parabolic inequalities

It is well known that a Monotonicity Condition and a Coerciveness Condition principally lie in the basis of most results of the Theory of PDE's. The necessity of these important assumptions for the validity of a comparison principle and analogues of the Phragmen-Lindelöf theorem for solutions of quasilinear parabolic inequalities is discussed in the paper. In the first part of the work we introduce a new concept of monotonicity for nonlinear differential operators-nonlinear monotonicity concept-and on its basis we obtain new phenomena for solutions, subsolutions and supersolutions of the well-known quasilinear differential equations. In the second part we omit the current coerciveness condition and change it by a weaker one. In spite of this we obtain a series of new qualitative properties of solutions for wide classes of quasilinear parabolic inequalities. Most of these properties are also new for solutions of the well-known equations, which we consider in the paper.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B _4,4,4ℍ³. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B _4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B _4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. This work is partially supported by the Sonderforschungsbereich 288.     

Absolute Continuity of Measures in the Class of Markov and Semi-Markov Processes of Diffusion Type

The property of absolute continuity of measures in the class of one-dimensional semi-Markov processes of diffusion type is investigated. The measure of such a process can be composed of two measures. The first one is a distribution of a random track, and the second one is a conditional distribution of a time run along the track. The desired density is represented in the form of product of two corresponding densities.     

Toward eudaimonology: notes on a quantitative framework for the study of happiness

After noting factors (concern for others, ignorance, irrationality) accounting for the divergences between preference and happiness, the question of representing the preference of an individual by a utility function is discussed, taking account of lexicographic ordering, imperfect discrimination and the corresponding concepts of semiorder and sub-semiorder. Methods to improve upon the interpersonal comparability of measures of happiness such as pinning down the dividing line of zero happiness and the use of a just perceivable increment of happiness are discussed. The relation of social welfare to individual welfare (i.e. happiness) is then considered. Some reasonable set of axioms ensuring that social welfare is a separable function of and indeed an unweighted sum of individual welfares are reviewed. Finally, happiness is regarded as a function of objective, institutional and subjective factors; an interdisciplinary approach is needed even for an incomplete analysis.     

Hausdorff measures of a class of Sierpinski carpets

thenandIn this paper, a lemma as a new method to calculate the Hausdorff measure of fractal is given. And the exact values of Hausdorff measure of a class of Sierpinski sets which satisfy balance distribution ang dimension ≤1 are obtained     

Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body‐implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi‐static approximation to the dynamics. We focus on the linearized system about stress‐free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time‐dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non‐stress free equilibria and states with residual stress and derive an energy law for the corresponding time‐dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement‐traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two‐dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.