椭球体与抛物锥体的体积推导

在研究性学习中 ,受教材《立体几何》中用祖日恒原理推导球体的体积方法的启发 ,本文探索用祖日恒原理推导椭球体和抛物锥体的体积 .1　椭球体的体积椭球体即指将椭圆绕其一条对称轴旋转一周 (半周亦可 )所得的几何体 .下面构造恰当的几何体使用祖日恒原理推导其体积 .先研究半椭球的体积 .设椭球体是由长半轴长为ａ ,短半轴长为ｂ的椭圆绕其短轴所在直线旋转得到的 .为了应用祖日恒原理 ,需要找一个可求体积的几何体 ,使它和旋转半椭球体可夹在两平行平面间 ,用平行于这两个平面的任意一个平面去截它们时 ,截面面积总相等 .图 1　求半椭球…     

(p,q)-John椭球

本文主要研究(p,q)-John椭球.经典的John椭球和L_p John椭球均是(p,q)-John椭球的特殊情形.首先讨论(p,q)-John椭球的充分必要条件和连续性.得到了关于(p,q)-John椭球的不等式和包含关系,所得到的不等式和包含关系分别类似于Ball体积比不等式和John包含关系.     

一类新椭球的体积不等式

基于LYZ椭球的定义和Lp对偶曲率测度,我们引入了一类新椭球,并得到了它的仿射性质.此外,进一步得到了椭球Υ2,q(K,E)和Υ2,q^*(K,E)的体积不等式及其等号成立条件.     

关于正弦椭球的迭代不等式

通过新定义的正弦椭球的定义和性质,得到了正弦椭球的体积不等式并给出了等式成立的充分必要条件.此外,本文利用迭代方法,进一步建立了关于正弦椭球算子Λ_2~*和Λ_(-2)~*的迭代不等式.     

一类Hartogs域的极小外切Hermite椭球及其对极值问题的应用

构造了一类Hartogs域的具有最小体积的外切Hermite椭球.作为一个应用,得到了从这类Hartogs域到单位球的Carathéodory极值映射.并且给出了计算极值的显式公式.     

凸体的L_p John椭球与其L_p迷向性

利用凸体的L_p混合体积理论,证明了R~n中关于原点对称的凸体K的L_p迷向性等价于其L_p John椭球为球这一充要条件.作为应用,验证了R~n中单位立方体的L_p迷向性.最后,根据L_p John椭球的仿射不变性,证明了R~n中所有超平行体的L_p John椭球就是经典的John椭球.     

一般三元二次方程给出的实椭球体积公式

用三种方法得到了一般三元二次方程给出的实椭球体积公式.     

椭球的Orlicz投影体

本文研究了文献[1]所引入的Orlicz投影体问题.利用Orlicz投影体在线性变换下的不变性,获得了椭球的Orlicz投影体仍是椭球的结果.作为例子,计算了当取两个特定的凸函数时单位球的Orlicz投影体的支持函数.     

凸体的对偶迷向常数与LYZ椭球

研究了凸体处于对偶迷向位置时的解析特征,并建立了凸体对偶迷向常数的新的下界;其次,证明了关于原点中心对称凸体的LYZ椭球与John椭球相等的充要条件;最后,举例具体计算了几个凸多边形的LYZ椭球和John椭球,以进一步认清两者的差别.     

关于椭球面上一道积分题目的探讨

主要研究椭球面上的一道曲面积分题目,分别从Gauss公式、椭球面的不同参数表示,以及余面积公式等角度对该积分的计算技巧进行了详细总结.同时,从微分几何的观点出发,对该题目进行了推广.     

二次曲面所围封闭图形的体积

讨论三元二次曲面是封闭图形的条件及封闭图形所围图形体积的计算,并将结果的形式推广到二元和一元二次方程所围封闭图形的判定上和面积或长度的计算上.     

Minimal Graphic Functions on Manifolds of Nonnegative Ricci Curvature

We study minimal graphic functions on complete Riemannian manifolds ∑ with nonnegative Ricci curvature, euclidean volume growth, and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville‐type theorem with such growth via splitting for tangent cones of ∑ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville‐type theorem under a certain nonradial Ricci curvature decay condition on ∑. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section. © 2015 Wiley Periodicals, Inc.     

二次图构形的边搜索法及其Orlik-Solomon代数的上同调

本文研究了一类特殊的图构形,即二次图构形.首先,给出了弦图的边搜索法,用边搜索法可以决定弦图所对应的超平面构形中超平面的哪些排序使构形是二次均.其次,对二次图构形的Orlik-Solomon代数作为线性空间的维数进行了计算,得到了它的Poincare多项式的表达式.最后,对二次图构形Orlik-Solomon代数的上同调进行了研究,对边缘算子集中在秩为二的模元时,得到了二次图构形的Orlik-Solomon代数的上同调的维数计算公式.     

On the proof of humbert’s volume formula

We close a gap in Humbert’s classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL₂ over the ring of integers of an imaginary quadratic number field.     

Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed.     

On the proof of Humbert's volume formula

We close a gap in Humbert's classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL² over the ring of integers of an imaginary quadratic number field. Received: 19 December 1996 / Revised version: 4 September 1997     

On a lower bound for the extinction time of surfaces moved by mean curvature

This paper considers a generalized evolution of a compact closed (hyper)-surface moved by its mean curvature. By a comparison of a shrinking ball the surface extincts in a finite time. An estimate of the extinction time from above is given by L. C. Evans and J. Spruck. In this paper we give an estimate from blow. In fact we proved that the extinction time is estimated from below by to times the square of the volume of a set enclosed by the initial surface over the initial area of the surface. The constant two is optimal.Partly supported by the Inamori Foundation     

Heat and mass transfer in annular liquid jets: I. Formulation

Heat and mass transfer phenomena in annular liquid jets are analyzed at high Reynolds numbers by means of a model derived from the governing equations that takes into account the effects of surface tension and boundary conditions at the gas–liquid interfaces and the large differences between the thermal and mass diffusivities, densities, dynamic viscosities, and thermal conductivities between gases and liquids. The model clearly illustrates the stiffness in both space and time associated with the concentration, linear momentum and energy boundary layers, and the initial cooling of the gases enclosed by the jet when, starting from a steady state where gases are injected into the volume enclosed by the jet at a rate equal to the heat and mass absorption rates by the liquid, gas injection is stopped. It is shown that, owing to the non-linear integrodifferential coupling between the fluid dynamics and heat and mass transfer processes, the pressure of the gases enclosed by the jet may vary in either a monotonic or an oscillatory manner depending on the large number of non-dimensional parameters that govern the heat and mass transfer phenomena. For the underpressurized jets considered here, it is shown that thermal equilibrium is achieved at a much faster rate than that associated with mass transfer, double diffusive phenomena in the liquid may occur, and the mass and volume of the gases enclosed by the jet may increase or decrease as functions of time until a steady equilibrium condition is reached.     

A new ray tracing heat source model for mesoscale CFD simulation of selective laser melting (SLM)

The present work was aimed to develop a new ray tracing heat source model used in mesoscale finite volume method (FVM) simulations for the selective laser melting (SLM) process. By means of the metal-gas interface built via the volume of fluid (VOF) method, the developed model simulated the interactions between the laser beam and the powder bed. Multiple reflections of the laser beam at the surface of the metal powders were taken into account in the modeling. Integrated into the commercial software, FLOW3D, the heat source model was validated by the comparison between the calculation results and previous experiments. 3-D mesoscale simulations for the SLM process of 316 L stainless steel were performed. The result indicated that the proposed ray tracing heat source model was able to model the laser heating process of the SLM. Better than the conventional one, the new heat source model identified the local defects during the SLM process, such as the balling behavior.     

L'intégration graphique des équations différentielles ordinaires

In the period which precedes the appearance of computers, needs in calculation of the scientists and engineers led to an important development of graphic methods of integration. To contribute to the study of this little known phenomenon, the article presents techniques and instruments used for the graphic integration of ordinary differential equations, and looks for their historic origins by going back to the beginning of calculus: processes of geometric calculation by the polygonal method or the method of radius of curvature, use of tractional motion for the conception of integraphs, reduction to graphic quadratures in finite or infinite number.