Solution of polyhedra

The paper is an exposition of the authors talk on the Seminar on Differential Geometry in IMPA in Rio de Janeiro. It presents a short survey of some recent results in the metric theory of polyhedra in 3-space. Namely we emphasize on some applicatons of the theorem which is a vast generalization of the Herons formule for the area of a triangle to volumes of polyhedra.*The author is partially supported by grants of RFBR No. 02-01-00101 and Russian Ministry of Education E02-1.0-43.     

On an Alternative Parameterization of the Solutions of the Partial Realization Problem

The solutions of the partial realization problem have to satisfy a finite number of interpolation conditions at . The minimal degree of an interpolating deterministic system is called the algebraic degree or McMillan degree of the partial covariance sequence and is easy to compute. The solutions of the partial stochastic realization problem have to satisfy the same interpolation conditions and have to fulfill a positive realness constraint. The minimal degree of a stochastic realization is called the positive degree. In the literature, solutions of the partial realization problem are parameterized by the Kimura–Georgiou parameterization. Solutions of the partial stochastic realization problem are then obtained by checking the positive realness constraint for the interpolating solutions of the corresponding partial realization problem. In this paper, an alternative parameterization is developed for the solutions of the partial realization problems. Both the solutions of the partial and partial stochastic realization problem are analyzed in this parameterization, while the main concerns are the minimality and the uniqueness of the solutions. Based on the structure of the parameterization, a lower bound for the positive degree is derived.     

Solution to an extremal problem on bigraphic pairs with a <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connected realization

Let S=(a₁,...,a_m; b₁,...,b_n), where a₁,...,a_m and b₁,...,b_n are two nonincreasing sequences of nonnegative integers. The pair S=(a₁,...,a_m; b₁,...,b_n) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a₁,...,a_m and b₁,...,b_n are the degrees of the vertices in X and Y, respectively. Let Z₃ be the cyclic group of order 3. Define σ(Z₃, m, n) to be the minimum integer k such that every bigraphic pair S=(a₁,...,a_m; b₁,...,b_n) with a_m, b_n ≥ 2 and σ(S)=a₁ +... + a_m ≥ k has a Z₃-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z₃, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z₃, m, n) for m ≥ n ≥ 4.     

Singularly perturbed feedback linearization with linear attitude deviation dynamics realization

A new approach for feedback linearization of attitude dynamics for rigid gas jet-actuated spacecraft control is introduced. The approach is aimed at providing global feedback linearization of the spacecraft dynamics while realizing a prescribed linear attitude deviation dynamics. The methodology is based on nonuniqueness representation of underdetermined linear algebraic equations solution via nullspace parametrization using generalized inversion. The procedure is to prespecify a stable second-order linear time-invariant differential equation in a norm measure of the spacecraft attitude variables deviations from their desired values. The evaluation of this equation along the trajectories defined by the spacecraft equations of motion yields a linear relation in the control variables. These control variables can be solved by utilizing the Moore–Penrose generalized inverse of the involved controls coefficient row vector. The resulting control law consists of auxiliary and particular parts, residing in the nullspace of the controls coefficient and the range space of its generalized inverse, respectively. The free null-control vector in the auxiliary part is projected onto the controls coefficient nullspace by a nullprojection matrix, and is designed to yield exponentially stable spacecraft internal dynamics, and singularly perturbed feedback linearization of the spacecraft attitude dynamics. The feedback control design utilizes the concept of damped generalized inverse to limit the growth of the Moore–Penrose generalized inverse, in addition to the concepts of singularly perturbed controls coefficient nullprojection and damped controls coefficient nullprojection to disencumber the nullprojection matrix from its rank deficiency, and to enhance the closed loop control system performance. The methodology yields desired linear attitude deviation dynamics realization with globally uniformly ultimately bounded trajectory tracking errors, and reveals a tradeoff between trajectory tracking accuracy and damped generalized inverse stability. The paper bridges a gap between the nonlinear control problem applied to spacecraft dynamics and some of the basic generalized inversion-related analytical dynamics principles.     

On Equivalence of Two Realizations for a Nonlinear Lie Algebra

Recently,there are two independent approaches related to a class of nonlinear Lie algebras of three generators;and the realizations of these generators are achieved respectively in Schwinger-boson and position representation.However,by use of the representation transformation between these two representations,the equivalence of the two realizations is therefore proved.     

基于Colpitts振荡器模型生成的多涡卷超混沌吸引子

基于三阶单涡卷混沌Colpitts振荡器模型,通过引入两个分段线性三角波函数,构造了一个新型四维多涡卷超混沌系统,生成了(2M+1)×(2N+1),(2M+1)和(2N+1)涡卷混沌和超混沌吸引子.利用相轨图、Poincar啨映射、Lyapunov指数谱和分岔图等方法,对新提出的四维多涡卷超混沌系统进行了动力学分析,结果表明,多涡卷超混沌系统的Hopf分岔点仅与控制参数有关,而涡卷数量和控制参数的混沌和超混沌范围随着转折点数量的增加而增加.此外,设计了一个实现四维多涡卷超混沌系统的模拟电路,实验输出与数值仿真的两个结果基本一致.     

Smallest domination number and largest independence number of graphs and forests with given degree sequence

For a sequence d of nonnegative integers, let and be the sets of all graphs and forests with degree sequence d, respectively. Let , , , and where is the domination number and is the independence number of a graph G. Adapting results of Havel and Hakimi, Rao showed in 1979 that can be determined in polynomial time. We establish the existence of realizations with , and with and that have strong structural properties. This leads to an efficient algorithm to determine for every given degree sequence d with bounded entries as well as closed formulas for and .     

信息非对称环境下产业创新平台多方主体博弈分析

研究产业创新平台在多方主体模式下的协同合作策略问题。我国产业创新平台发展时间较短,部分平台系统的资源共享和服务能力处于“弱势”,但通过多方主体协同合作能有效提升平台系统的技术创新水平,所以主体间的博弈会影响平台系统的存续质量与运行效率。基于此,本文构建了信息非对称环境下“产业链相关企业-各类研究型机构-政府管理部门”主要三方主体的静态博弈模型,从利益实现机制出发,强化各方主体在产业创新平台运行中的理性,以破解产业创新平台运行效率低下困局。研究表明,企业、研究机构和政府管理部门是否会严格遵循契约治理规范自身行为主要取决于各自的资源共享力度、资源共享成本以及资源共享的收益。     

浅析对数学史教育的认识与实施

数学史教育在大学数学教育中具有重要的意义.有助于学生理解数学知识的本质,培养学生的思维能力.有必要开设数学史课程或在专业课教学中渗透数学史知识;有适合学生的专门数学史教材;进行数学史教育要联系学生的数学知识,要深入浅出;要注意数学精神的宣传.     

一种新型微梁阵列传感器设计及实现

微悬臂梁传感技术是微纳传感技术研究的热点,仪器装置的研究也从单悬臂梁向微梁阵列的方向发展.本文针对间距为250μm的商品化微梁阵列,提出了一种实现对其弯曲变形的读出系统光路.利用高精度位移平台调节激光器空间位置,使光束实现微距离上平行,并分别照射在阵列梁的相邻微梁上.用位置敏感探测器对两个微悬臂梁尖端反射的激光信号进行...