Über die Differentialgleichungu x y =f(x,y, u,u x,u y )

Ohne ZusammenfassungDiese Arbeit wurde teilweise unterstützt von der United States Air Force [Air Force Office of Scientific Research-Contract AF 49 (638)-228]. Sie besteht aus drei Teilen. Teil I ist im vorangehenden Heft dieser Zeitschrift erschienen, Teil III erscheint im folgenden Heft. Formeln und Sätze sind in jedem Teil für sich durchnumeriert. Bei Verweisen bedeutet z. B. (I. 17) die Formel (17) von Teil I, Satz I. 5 bedeutet Satz 5 von Teil I.     

Über die Differentialgleichungu xy =f(x,y, u,u x,u y )

Ohne ZusammenfassungDiese Arbeit wurde teilweise unterstützt von der United States Air Force (Air Force Office of Scientific Research-Contract AF 49 (638)-228). Sie besteht aus insgesamt drei Teilen. Teil II (Existenzsätze für die charakteristische Anfangswertaufgabe) und Teil III (Die nichtcharakteristische Anfangswertaufgabe) werden in Kürze in dieser Zeitschrift erscheinen.     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

When isf(x,y) = u(x) + v(y)?

LetX andY be arbitrary non-empty sets and letS a non-empty subset ofX ×Y. We give necessary and sufficient conditions onS which ensure that every real valued function onS is the sum of a function onX and a function onY.     

从u(x,y)+iu(x,y)到f(z)的一种转化法

<正> 在复变量函数的运算中,常常会遇到需要把以二元函数u(x,y)、v(x,y)为实部与虚都构成的复变量函数f(x+iy)=u(x,y)+iv(x,y)化成以复数z=x+iy为变量的函数f(z),一般常用的方法是:     

关于集值系统x∈F(x,y),y∈G(x,y)解的存在性

Let (E, ‖·‖) be a uniformly convex Banach space, X a nonempty compact and convex subset of E. Let F be a closed mapping of X×X into 2^X, G a mapping of X×X into C(X). It is shown that if for any f∈C(X), x∈X, F(x,f(x)) is a closed and convex subset of X, and G(x,f(x)) is a continuous function and for any x,y₁,y₂∈X,H(x,G(x,y₁),G(x,y₂))≤‖y₁-y₂‖ then there exist X₀,y₀∈X such that X₀∈F(x₀,y₀) and y₀∈G(x₀, y₀).     

关于双曲型偏微分方程u_xy=f(x,y,u,u_x,u_y)的唯一性问题

关于双曲型偏微分方程 u_(xy)=f(x,y,u,u_x,u_y),0≤x≤a,0≤y≤b,-∞相似文献   

混合偏导数fyx(x,y)与fxy(x,y)相等的条件

对二元函数的两个混合偏导数fyx(x,y)及fxy(x,y)进行了进一步研究,并得出了它们相等的一个充分条件,该条件比[1]中给出的条件更弱.     

妙用坐标(x,y)解题

<正>纵观近几年的中考题目,考查数学的基本思想及活动经验的积累极为突出,考查在实际解题中能否运用数学知识、数学思想方法去发现问题、分析问题、解决问题.下面以两题中考试题为例,浅析坐标(x,y)在解题中的妙用,以供大家参考.     

Four step methods for y″=f(x,y)

A systematic investigation is undertaken here on the possible practical consequences of the fact that the linear four step methods for y″=f(x, y) form a family with a certain structure. The mentioned property is found to be surprisingly rich in consequences, and three of them are of particular importance: (i) drastic reduction in the number of iterations at each step, (ii) increased flexibility with respect to the start of integration and to the modification of the step size during the integration process, (iii) possibility of direct and simple appraisal of the local truncation error.     

探究x0x＋y0y=r^2与x0x＋y0y=x0^2＋y0^2耐的几何意义

问题背景苏教版教材必修二P105这样一道习题：已知圆C的方程是x^2＋y^2=r^2,求经过圆C上一点M（x0,y0）的切线的方程．     

Third-order ordinary differential equations y′′′ = f(x,y, y′, y″) with maximal symmetry group

We present here, in compact form, the necessary and sufficient conditions for linearization of third-order ordinary differential equations y'''=f (x,y,y', y'') with maximal symmetry group via point transformation. A simple procedure to construct the point transformation using the isomorphism of the symmetry subalgebra sl(2, ?) is also presented. This subalgebra is the semi-simple part of the Levi-Decomposition for the 7-dimensional symmetry algebra.     

方程y″(x)+p(x)y(x)=y(x)解的有界性与振动性

设P(x)、f(x)∈C~1[0,+∞),在[0,+∞)上,P(x)>0,P′(x)≤0且(?)P(x)=ρ>0,intejral form 0 to +∞。|f′(t)|dt<+∞。我们给出了方程y″+P(x)y=f(x)解的有界性与振动性结果。     

直线方程xF1(x0,y0)+yF2(x0,y0)+F3(x0,y0)=0的几何意义

文[1]与文[2]分别探讨了直线方程x0x/a^2 y0y/b^2=1和直线方程x0x/a^2-y0y/b2=1的几何意义，读后深受启发，本文是文[1]与文[2]的继续，探讨了是伴随于非退化二次曲Ax^2 2Bxy Cy^2 2Dx 2Ey f=0的直线方程xF1(x0,y0)     

关于扰动系统x=y εPn(y)x,y=-x εQn(x)y 的极限环的上界

本文利用Paincare分支理论,给出了扰动系统(?)的判定函数,并由此得到了该系统极限环最大个数等结果。同时还讨论了Lienard方程x+f_n(x)x+x=0对应的扰动系统(?)的有关问题,很容易得到了文[3]中的有关结论。     

探讨Pdx+Qdy+Rdz为某函数u(x,y,z)全微分的积分因子

就微分形式P(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz为某函数u(x,y,z)的全微分的积分因子进行了探讨,提出了积分因子的必要条件,以及P(x,y,z),Q(x,y,z),R(x,y,z)是齐次函数时,方程Pdx+Qdy+Rdz=0具有积分因子的充分条件进行了初步探讨.     

One-Sided Boundary Control for the Process of Oscillations Described by the Equation k(x)[k(x)u x (x,t)] x − u tt (x,t) = 0

This article concerns the boundary control on the side x = 0 when the other side x = l is fixed. The corresponding process of oscillations is studied for the time interval T > 0 and is described by the generalized solution of the equation

P(x,y)关于y=±x+c对称原来可以这样!

在一节关于点和直线对称问题的新课上,同学提出了教师平时很少去深入探究的问题.下面我们一起来看一看:…问题1点P(a,b)关于y=x的对称点P′坐标是__.问题2点P(a,b)关于y=-x的对称点P′坐标是__.学生甲:问题1先设出P′的坐标为(x0,y0),通过对称性可知,线段PP′的中点在直线