自共轭四元数矩阵广义逆A｛1，…，J；＊｝的Lowner偏序

本文利用自共轭四元数矩阵广义逆的显公式，给出了在Ｌｏｗｎｅｒ偏序下Ａ｛１；＊；ｓ；ｓ；≤Ａ｝，Ａ｛１；＊；ｔ；≥Ａ｝（其中Ａ∈Ｈ（ｎ，＊））；Ａ｛２；≥；ｔ；≥Ａ），Ａ｛２；≥；ｓ；≤Ａ），Ａ｛１；≥；ｔ；≥Ａ）（其中Ａ∈Ｈ（ｎ，≥））的表式．     

贪婪t－相交有限序列族

设Ｆ为有限序列族，对ａ＝（ａ１，ａ２，…，ａｎ）∈Ｆ，ａｉ为整数且０≤ａｉ≤ｓｉ（整数），记ｓ（ａ）＝｛ｊ｜１≤ｊ≤ｎ，ａｊ＞０｝，ｓ（Ｆ）＝｛ｓ（ａ）｜ａ∈Ｆ｝，及Ａ｛１，２，…，ｎ｝时Ｗ（Ａ）＝Пｉ∈Ａｓｉ．称Ｆ为贪婪ｔ－相交，如对任何ａ，ｂ∈Ｆ，至少有ｔ个ａｉ，ｂｉ＞０，且Ｗ（Ａ）≥Ｗ（（｛１，２，…，ｎ｝－Ａ）＋Ｂ）对任何Ａ∈Ｓ（Ｆ）及ＢＡ（｜Ｂ｜＝ｔ－１）成立．本文得到当ｓ１＞ｓ２＞…＞ｓｎ时的最大贪婪ｔ－相交有限序列族．     

代数综合测试题

一、选择题１．设集合Ｍ＝｛ｘ｜２２－ｘ≥１｝,集合Ｎ＝｛ｘ｜ｘ２－２ｘ－３＜０｝,集合Ｍ∩Ｎ＝（）（Ａ）｛ｘ｜０≤ｘ＜１｝（Ｂ）｛ｘ｜０≤ｘ＜２｝（Ｃ）｛ｘ｜０≤ｘ≤１｝（Ｄ）｛ｘ｜０≤ｘ≤２｝２．已知函数ｆ（ｘ）＝２ｘ,则函数ｙ＝｜ｆ－１（ｘ－１...     

课本例习题的变式研究(高一)

一、集合,一元二次不等式,映射与函数题１　（Ｐ８例４）设Ａ＝｛（ｘ,ｙ）｜４ｘ＋ｙ＝６｝,Ｂ＝｛（ｘ,ｙ）｜３ｘ＋２ｙ＝７｝,求Ａ∩Ｂ．此题是理解交集、求交集运算的一道好题．教学时,可以从集合元素的特征,例题所反映的几何意义等方面全方位地去加以分析和理解．变式１　（巩固练习）设Ａ＝｛（ｘ,ｙ）｜ｙ＝ｘ＋２,ｘ∈Ｒ｝,Ｂ＝｛（ｘ,ｙ）｜ｙ＝ｘ２,ｘ∈Ｒ｝,则Ａ∩Ｂ＝（　　）（Ａ）｛（－１,１）｝．　　　（Ｂ）｛（２,２）｝．（Ｃ）｛（－１,１）,（２,２）｝．（Ｄ）｛ｙ｜ｙ≥０｝．变式２　（对照辨…     

子集的交叉t－相交定理

设Ｘ为一个ｎ元集合，Ｃｎｋ为Ｘ的所有ｋ元子集全体，若Ａ∈Ａ，Ｂ∈Ｂ有｜Ａ∩Ｂ｜≥ｔ，则称（Ａ，Ｂ）为一个交叉ｔ－相交子集族．本文得到最大交叉ｔ－相交子集族和最大非空交叉２－相交子集族．证明如下两个结论．（１）若（Ａ，Ｂ）为一个交叉ｔ－相交子集族，且ａ≤ｂ及ａ＋ｂ≤ｎ＋ｔ－１，则｜Ａ＋Ｂ｜≤ｍａｘ｛（ｂｎ），（ａｎ）｝，且当（Ａ；Ｂ）＝（φ，Ｃｎｂ）或（Ｃｎａ，φ）时达到上界．（２）若（Ａ，Ｂ）为一个交叉２－相交子集族，且ａ＜ｂ，ａ＋ｂ≤ｎ－１及（ｎ，ａ，ｂ）≠（２ｉ，ｉ－１，ｉ）（ｉ为任意正整数），又Ａ，Ｂ均非空，则｜Ａ＋Ｂ｜≤１＋（ｂｎ）－（ｂ（ｎ－ａ））－ａ（（ｂ－１）（ｎ－ａ））且当（Ａ，Ｂ）＝（｛Ａ｝，Ｃｎｂ－｛Ｂ｜｜Ｂ｜＝ｂ，｜Ａ∩Ｂ｜≤１｝）时达到上界．     

空间R~m上分布的中性卷积

取定具下述性质的函数ｒ（ｘ）∈Ｃ∞（Ｒ）：（ｉ），τ（ｘ）＝τ（－ｘ），（ｉｉ）０≤τ（ｘ）≤１，（ｉｉｉ）τ（ｘ）＝１，当｜ｘ｜≤１／２，（ｉｖ）τ（ｘ）＝０，当｜ｘ｜≥１．单位序列｛τｎ（ｘ）｝，ｘ∈Ｒｍ和ｎ∈Ｉｍ，定义作τｍ（ｘ）＝τ（ｘ１／ｎ１）…τ（ｘｍ／ｎｍ），ｎ１，…，ｎｍ＝１，２，…．空间Ｄ’（Ｒｍ）中分布ｆ和ｇ的中性卷积ｆｇ定义作序列｛ｆｎ＊ｇ｝的极限，其中ｆｎ＝ｆ·τｎ．作者给出了一些新的卷积．     

纯生过程的变异性

设｛Ｘ（ｔ）：ｔ≥０｝为零初值纯生过程，出生率为λｎｎ≥０，在本文中，我们证明了Ｆａｄｄｙ［７］的一个猜测：当生率为单调增加序列λ０≤λ１≤λ２≤…时，Ｖａｒ｛Ｘ（ｔ）｝≥Ｅ｛Ｘ（ｔ）｝；当出生率为单调减少序列时，Ｖａｒ｛Ｘ（ｔ）≤Ｅ｛Ｘ（ｔ）｝。     

基础问题解答

§１－１１．Ｂ；２．Ｃ；３．必要；４．充分必要；５．充分但不必要；§１－２１．Ｂ；２．Ｂ；３．（２），（５），（７）；４．｛ａ，ｂ｝，｛ａ，ｂ，ｃ｝，｛ａ，ｂ，ｄ｝；５．（Ａ∪Ｂ）∩Ｃ§１－３１．Ｃ；２．Ａ；３．（３２，－１２）；４．ｆ（ｘ）＝ｘ２－...     

求解多面体上的线性变分不等式

已给矩阵Ｍ∈Ｒｎ×ｎ，向量ｑ∈Ｒｎ，凸多面体Ｚ＝｛ｘ∈Ｒｎ｜Ａｘ≤ｂ，Ｂｘ＝ｄ｝，其中Ａ∈Ｒｍ×ｎ，Ｂ∈Ｒｐ×ｎ，变分不等式问题ＡＶＩＰ（Ｍ，Ｂ，Ａ，ｑ，ｄ，ｂ）是求ｘ＊∈Ｚ使得对一切ｘ∈Ｚ，不等式（ｘ－ｘ＊）Ｔ（Ｍｘ＋ｑ）≥０成立，本文给出了这一问题的等价形式，提出了两个投影算法并作了总体收敛性分析     

一个二维整数瓶颈问题及其算法

本文讨论了数学模型：ｍａｘ｛ｆ（ｘ）│ｆ（ｘ）＝ｍｉｎ（１≤ｊ≤ｎ）〔ｃ１ｊｘ１ｊ＋ｃ２ｊｘ２ｊ〕，ｘ∈Ｄ｝，其中Ｄ＝｛ｘ│ｘ＝｛ｘｉｊ｝，ｎΣ（ｊ＝１）ｘｉｊ＝ａ，ｉ＝１，２，ｘｉｊ≥０且为整数｝，并给出了一个拟多项式算法。     

Positive solutions of a nonlinear initial-value problem on half-line

The nonlinear initial-value problemu″(t)+f(t,u(t))=0,u(t ₀)+bu′(t ₀)=c,t ₀≥0,b≤0,c≥0, is considered for positive solutions on [t ₀, ∞). Existence of positive solutions is proved without the hypothesis thatf(t, ω)≥0 (or ≤0), using the lattice fixed point theorem. A monotonicity condition inf(t, ω) is used to prove the uniqueness of the solution of the initial-value problem. Whenf(t, ω)≥0 (or ≤0), uniqueness is also obtained under a sublinearity condition onf(t, ω).     

Nearly Comonotone Approximation of Periodic Functions

Suppose that a continuous 27r-periodic function f on the real axis changes its monotonicity at points y_1:-π≤ y_(2s)y_(2s-1)… y_1 π,s ∈ IN.In this PaPer,for each n≥N,a trigonometric polynomial P_n of order cn is found such that:P_n has the same monotonicity as f,everywhere except,perhaps,the small intervals(y_i-π/n,y_i+π/n)and‖f-P_n‖c(s)ω_3(f,π/n),where N is a constant depending only on mini=1,...,2s {y_i-y_(i+1)},c,c(s) are constants depending only on s,ω_3(f_1,·) is the modulus of smoothness of the 3-rd order of the function f,and ||·|| is the max-norm.     

厄米特矩阵的迹的几点性质

应用矩阵Ａ＝（ａｉｊ）∈Ｃｎ×ｎ的弗罗伯尼范数ＡＦ和谱范数ＡＳ，研究厄米特矩阵的迹的性质，得到几个结论：Ｔｒ（ＡＢ）＝∑ｎｉ＝１λｉ∑ｎｊ＝１ｔｉｊμｊ（λｉ，μｊ分别为Ａ，Ｂ的特征值，０≤ｔｉｊ≤１，且∑ｎｉ＝１ｔｉｊ＝１，ｊ＝１，２，…，ｎ）；Ｔｒ（ＡＢ）≤Ｔｒ（Ａ）ＢＳ；Ｔｒ（ＡＢ）Ｈ（ＡＢ）］≤Ｔｒ（ＡＨＡ）［ｍａｘ１≤ｉ≤ｎλｉ］２（λｉ是Ｂ的特征值）等．     

Exact optimal values of step-size coefficients for boundedness of linear multistep methods

Linear multistep methods (LMMs) applied to approximate the solution of initial value problems—typically arising from method-of-lines semidiscretizations of partial differential equations—are often required to have certain monotonicity or boundedness properties (e.g., strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). An LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work, we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams–Bashforth family (with 1 ≤ k ≤ 3 steps).     

布尔矩阵行空间基数的存在点

设 Ｂｎ 表示所有的ｎ 阶布尔矩阵的集合, Ｒ（ Ａ）表示 Ａ∈ Ｂｎ 的行空间,｜ Ｒ（ Ａ）｜表示 Ｒ（ Ａ）的基数．设ｍ ,ｎ,ｋ 为正整数,本文证明了当ｎ≥９, ｎ＋ ５２ ≤ｋ≤ｎ－ ３ 时,对任意的 ｍ ,２ｋ≤ｍ ≤２ｋ＋ ２ｎ－ ｋ＋ ２＋ ２ｎ－ ｋ＋ １ ＋ …＋ ２３,存在 Ａ∈ Ｂｎ,使得｜ Ｒ（ Ａ）｜＝ ｍ ．     

多项式及其导数不等式

设‖·‖是 ( -∞ ,+∞ )上关于权 e- t2 的 L2范数 ,本文证明了对一切次数不超过 n的多项式f ( x) ,有‖ f′‖2 ≤ A‖ f″‖ 2 +( 2 n-4An( n-1 ) )‖ f‖ 2 ,这里 A只要满足 A≤ 14( n-1 ) .     

Some new conditions in Fourier analysis and applications

We investigate some conditions in Fourier analysis as a generalization of monotonicity condition. Especially, we give some applications of GBVS and sequences satisfy the condition ΔB1 and ΔB2.     

一类解析函数类的Hankel行列式的上界估计

本文对在单位圆盘U={z∈C：l│z│〈1}内解析且满足（1-λ）f（z）/z＋λf^1（z）〈1＋Az/1＋Bz（-1≤B＜A≤1;z∈U的函数f所对应的Hankel行列式│a2a4-a^23│,利用Toeplitz行列式的性质得到了其上界估计。     

On the Circumference of 3-connected Graphs

Let C be a longest cycle in the 3-connected graph G and let H be a component of G - V(C) such that ¦V(H)¦≥ 3. We supply estimates of the form ¦C¦ ≥ 2d(u) + 2d(v) ? a (4 ≤ a ≤ 8), where u, v are suitably chosen non-adjacent vertices in G.     

单位上三角矩阵群的注记(Ⅲ)

设k_(ij)(1≤ij≤n)是给定的正整数,分别记G={ (1 k12a12…k1na1n 0 1…k2na2n…… 0 0…1 )|aij∈Z},R={ (0 k12a12…k1na1n……0 0…k2na2n 0 0…1 )|aij∈Z},本文证明:当G成群且G的上、下中心群列重合时,其相伴Lie环L(G)与Lie环R同构,其中R的Lie积定义为[A,B]=AB-BA.即得到了此时L(G)的矩阵表示.