消去 f(sinθ,cosθ)中参数θ方法小议

例1 若acosθ+bsinθ=c(1) dcosθ+esinθ=f(2)求证(ce-bf)~2+(af-ed)~2=(ae-bd)~2(3) 其中ae-bd≠0。对于此题,欲证(3)成立,只要从(1)、(2)中消去参数θ即可。具体作法是 (1)×d-(2)×a得 sinθ=af-ed/ae-bd, (1)×e-(2)×b得 cosθ=af-ed/ae-bd代入恒等式Sin~2θ+COS~2θ=1,即得(3)。这种方法是众所周知的,而有时要想从关于f(sinθ,cosθ)的条件等式中,直接解出sinθ,Cosθ,然后利用sin~2θ+cos~2θ=1去消参就相当困难,甚至是不可能的,因此必须另辟途     

函数f(θ)=sinθ/θ的单调性的一组结论

函数f(θ)=sinθ/θ(0<θ≤π/2)是减函数,有许多简捷、优美的结论,且有着广泛的用途.为方便表述,称"f(θ)是减函数"为结论(*).     

浅谈secθ和cscθ

从一九八О年度的高考大纲中看到,在三角函数方面由原来的四个:sin θ\ cons θ增加了两个;secθ和cscθ,这样所要考的三角函数就扩大为六个了。本文准备专门来谈一下secθ和cscθ的意义以及它们和其他三角函数间的关系等,供参考。     

关于函数f(θ)=sinθ/θ(0《θ≤π/2)的单调性的一组优美结论

函数f(θ)=sinθ/θ(0<θ≤π/2)是减函数,有许多简捷、优美的结论,且有着广泛的用途.……     

点(cosθ,sinθ)在单位圆上的应用

由x~2 y~2=1→(x,y)=(cosθ,sinθ),反之把与cosθ、sinθ有关的问题处理在动点(cosθ,sinθ)的轨迹方程上,不但使问题明了,而且有解法新颖、别具一格之感,本文以下面例子说明之。     

均匀分布U〔θ-1/2,θ+1/2〕中参数θ的四种估计量

给出均匀分布U〔θ-1/2,θ+1/2〕中θ的四种估计量,并分别比较了四种估计量的优劣性.     

关于三个角θ,60°-θ,60°+θ的一组三角等式及其应用

本文先给出关于三个角θ,60°-θ,60°+θ的正弦、余弦、正切、余切三角函数值的一组等式,它们恰好分别对应一个一元三次方程,然后结合实例介绍这些结论在证明三角函数等式和求值中的应用.     

不分明化拓扑中的θ-连续函数和θ-紧性

本文在不分明化拓扑中从θ-开集出发给出了θ-连续函数和θ-紧性的概念,并讨论了它们的某些性质．     

尤拉公式e~(θi)=cosθ+i sin θ

引言在这篇短文里,我們向讀者介紹一种新的建立尤拉公式的方法。我們不用一般数学分析里所用的方法,而用一个人家熟知的重要极限来建立尤拉公式。同时,我們考虑到尤拉公式的应用很广泛,也很重要,因此,順便列举了一些应用。其中特别請大家注意应用4和5,显然这样做是不够严格的,但我們想借此向讀者說明:当在复数城里研究初等函数时会出現实数城里沒有的有趣性质,从而帮助我們更深地理解初等实函数。这对中学数学教师是有帮助的。§1.尤拉公式e~(θi)=cosθ+isinθ的推导在数学分析里已經証明了一个重要极限 e~θ=(?)(1+θ/n)~n,这里θ为任意实数。推广这个結果于θi,得 e~(θi)=(?)(1+θi/n)~n (1)这里θ为任意实数,而i为虚数单位。現在我們来計     

Sθ-连通性(Ⅰ)

利用Sθ-闭包引入Sθ-连通集和Sθ-连通分支等概念,并研究了其等价条件,证明了Sθ-连通性是可积的和Sθ-同胚的。     

Unimodality of independence polynomials of the incidence product of graphs

Given two graphs $G$ and $H$, assume that $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and $U$ is a subset of $V\left(H\right)$. We introduce a new graph operation called the incidence product, denoted by $G\odot H^U$, as follows: insert a new vertex into each edge of $G$, then join with edges those pairs of new vertices on adjacent edges of $G$. Finally, for every vertex $v_i\in V\left(G\right)$, replace it by a copy of the graph $H$ and join every new vertex being adjacent to $v_i$ to every vertex of $U$. It generalizes the line graph operation. We prove that the independence polynomial

$I\left(G\odot H^U;x\right)=I^n(H;x)M\left(G;\frac{x{I}^2(H?U;x)}{I^2(H;x)}\right),$

where $M(G;x)$ is its matching polynomial. Based on this formula, we show that the incidence product of some graphs preserves symmetry, unimodality, reality of zeros of independence polynomials. As applications, we obtain some graphs so-formed having symmetric and unimodal independence polynomials. In particular, the graph $Q\left(G\right)$ introduced by Cvetkovi?, Doob and Sachs has a symmetric and unimodal independence polynomial.     

有资格限制的指派问题的求解方法

在实际的指派工作中，常会遇到某个人有没有资格去承担某项工作的问题，因此，本建立了有资格限制的指派问题的数学模型。在此数学模型中，将效益矩阵转化为判定矩阵，由此给出了判定此种指派问题是否有解的方法；在有解的情况下，进一步将效益矩阵转化为求解矩阵，从而将有资格限制的指派问题化为传统的指派问题来求解。最后给出了一个数值例子来说明这样的处理方法是有效的。     

Partial cohomology of groups and extensions of semilattices of abelian groups

We extend the notion of a partial cohomology group $H^n(G,A)$ to the case of non-unital A and find interpretations of $H^1(G,A)$ and $H^2(G,A)$ in the theory of extensions of semilattices of abelian groups by groups.     

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Consider two graphs $G$ and $H$. Let $H^k\left[G\right]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k\left[G\right]$ and $H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k\left[G\right]$ and $H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10¹⁰⁰ ) of vertices is determined. The paper finishes with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.     

22号初等元胞自动机的极限语言复杂性

§1Introduction Thetaskofthispaperistosolvetheproblemproposedin[1],i.e.,toexplorethe complexityoflimitset,orequivalently,limitlanguageoftheelementarycellularautomaton ofrule22bythetoolsofformallanguagetheory.Cellularautomata(abbreviatedasCA),asmathematicalmodelsforcomplexnatural systemscontaininglargenumbersofsimpleidenticalcomponentswithlocalinteractions,havebeenwidelyusedinphysical,biological,chemicalandcomputationalsystems[2].Despitetheirsimpleconstruction,someCAcandisplayveryrichandcompl…     

分形空间的局部紧性及网极限

对于完备度量空间 (X ,d) ,研究了X的局部紧性与相应分形空间 (H(X) ,h)的局部紧性之间的关系 ,得到结论 :(H(X) ,h)是局部紧的当且仅当X是局部紧的 .另一方面 ,给出了 (H(X) ,h)中收敛网的极限通过并、交及闭包运算的表示 .