Jan 082014
 

李氏朝鲜的第四代君主是最伟大的世宗大王. 他把王位按照嫡长子继承的原则, 传给了嫡长子文宗!  然而, 文宗体弱多病, 临终前, 任命金宗瑞为顾命大臣, 辅佐 12 岁即位的端宗.

端宗的叔父首阳大君, 谋害了金宗瑞, 成了领仪政, 掌握大权. 随后, 逼迫端宗禅让王位. 端宗做了两年上王之后, 被赐药而死.

首阳大君, 也就是世祖, 有必要杀侄儿吗? 历史上, 有哪些退位的太上皇被杀? 能找出几个?

如果不是端宗太年轻被害, 也许我不会这么同情他!  叔父首阳大君做的太过! 首阳篡位之前, 有很多次, 可能被杀!

端宗绝对不应该禅让王位的! 再怎么没有实权, 端宗是君, 首阳是臣. 除非暗杀, 否则首阳没有办法. 一旦成了别人的臣, 命完全由不得自己了.

端宗共计在位三年, 在上王位二年, 终年十七岁. 无嗣, 葬于江原道宁越郡庄陵. 这也是朝鲜王朝五百年间, 唯一一座不在京畿的王陵(追封的各王不算).

直至朝鲜肃宗七年(1681 年), 鲁山君被追封为鲁山大君, 肃宗二十四年被追尊复位, 上庙号端宗, 谥号为纯定安庄景顺敦孝大王, 陵号为庄陵. 鲁山君夫人宋氏被追封为定顺王后, 徽号为端良齐敬, 陵号为思陵. 端宗与定顺王后的神主移入宗庙永宁殿, 并举行了袝庙之礼. 与此同时”鲁山君日记”升格为”端宗实录”, 并在庄陵附近修建“死六臣祠”, 为其举行国家级别的祭祀.

挑选几个题作答, 主要是第 10 题.

If \( f\in C^1([a,b])\) is increasing and nonconstant, then

\[\int_a^b\sqrt{1+{f^\prime}^2(x)}\, \mathrm dx \lt b-a+f(b)-f(a). \]

For \(\alpha\), \(\beta\geqslant0\), \(\sqrt{\alpha^2+\beta^2}\leqslant\alpha+\beta\), with equality iff one or both of \(\alpha\), \(\beta\) equals \(0\).

Now \(f ^\prime\geqslant 0\), it follows that

\[\sqrt{1+{f^\prime}^2(x)}\leqslant 1+f^\prime(x), x\in [a,b].\]

Because \(f(x)\) is nonconstant, \(f^\prime\gt0\) in a subinterval. In that subinterval we have strict inequality between these two functions. Integrating both sides then gives the result.

Jan 062014
 

1.令 \(f(x)=\prod\limits_{i=1}^{2013} (x-i)^2+2014\), \(f(x)\) 在有理域内可约吗? 证明你的结论.

2. \(M\), \(N\) 都是 \(n\) 阶矩阵, \(n\geq2\). 如果 \(MNMN \) 为零矩阵, 那么 \(NMNM\) 是否也一定是零矩阵? 证明你的结论.

3. \(n\geq2\). 除了单位矩阵, 还有别的埃尔米特矩阵 \(M\) 满足下面的条件吗?

\[4M^5+2M^3+M=7E_n,\]

其中, \(M\) 是与 \(E_n\) 同阶的矩阵.

4. \(\mathbf V\) 是 \(n\) 维线性空间. 线性变换 \(\mathcal A\) 的最小多项式是 \(n\) 次.
(1) 证明存在向量 \(\alpha\), 使得 \(\alpha\), \(\mathcal A\alpha\), \(\dotsc\), \(\mathcal A^{n-1}\alpha\) 是 \(\mathbf V\)  的一组基;
(2) 任何与 \(\mathcal A\) 可交换的线性变换, 可表示为 \(\mathcal A\) 的多项式.

5. \(\mathbf V=\Bbb C_{n\times n}\) 是所有 \(n\)  阶复矩阵组成的向量空间. 求所有形如  \(MN-NM\) 的矩阵组成的向量空间的维数并给出证明.

6. 欧式空间 \(\mathbf V\) 中, 对称线性变换 \(\mathcal{A}\) 称为“正的”, 若对 \(\forall \alpha \in \mathbf V\), 都有\((\alpha, \mathcal A(\alpha))\geq 0\) 成立, 且等号当且仅当 \(\alpha =\mathbf 0\) 时成立.
(a)证明若线性变换 \(\mathcal A\) 是正的,则 \(\mathcal A\) 可逆;
(b)证明若线性变换 \(\mathcal B\) 是正的, \(\mathcal A-\mathcal B\) 也是正的,则 \(\mathcal B^{-1}-\mathcal A^{-1}\) 是正的;
(c)证明对于正的线性变换 \(\mathcal A\), 总存在正的线性变换 \(\mathcal B\) 使得 \(\mathcal A=\mathcal B^2\).

7. 求单叶双曲面

\[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\]

垂直的直母线交点的轨迹.

8.保距变换

\[\begin{split}
x’& = a_{11}x+a_{12}y+a_{13}z\\
y’& = a_{21}x+a_{22}y+a_{23}z\\
z’& = a_{31}x+a_{32}y+a_{33}z
\end{split}\]

可以看做绕不动直线旋转一个角度而得到.
(a)求不动直线的方向向量;
(b)求旋转角 \(\theta\).
(原题\(a_{11},\cdots,a_{33}\)皆为具体数字, 现已记不清, 用字母代替之)

9.点 \(A(a_{1},a_{2},a_{3})\), \(B(b_{1},b_{2},b_{3})\) 在直线

\[\frac{x+a}{2}=\frac{y+b}{2}=\frac{z}{3}\]

上的投影为 \(A_{1}, B_{1}\), 求 \(A_{1}, B_{1}\) 坐标以及两点间距离.
(原题\(a_{1},\dotsc,b_{3},a,b\)皆为具体数字,现已记不清, 用字母代替之)

Jan 052014
 

1. 叙述实数序列 \(\{x_n\}\) 的 Cauchy 收敛原理, 并且使用 Bolzano-Weierstrass(波尔查诺-威尔斯特拉斯)定理证明.

2. 序列 \(\{x_n\}\) 满足 \(x_1=1\), \(x_{n+1}=\sqrt{4+3x_n}\), \(n=1\), \(2\),\(\dotsc\). 证明此序列收敛并求极限.

3. 计算 \(\iiint_{\Omega}\sqrt{x^2+y^2}\, \mathrm dx\mathrm dy\mathrm dz\), 其中 \(\Omega\) 是曲面 \(z=\sqrt{x^2+y^2}\) 与 \(z=1\) 围成的有界区域.

4. 证明函数项级数 \(\sum\limits_{n=1}^{+\infty}x^3e^{-nx^2}\) 在 \([0,+\infty)\) 一致收敛.

5. 讨论级数 \(\sum\limits_{n=3}^{+\infty}\ln \cos\dfrac\pi n\) 的敛散性.

6. 设函数 \(f\colon\Bbb R^n\to\Bbb R\) 在 \(\Bbb R^n\setminus\mathbf0\) 可微, 在 \(\mathbf0\) 点连续, 且 \(\lim\limits_{\mathbf p\to \mathbf0} \dfrac{\partial f(\mathbf{p})}{\partial x_i}=0\), \(i=1\), \(2\), \(\dotsc\), \(n\). 证明 \(f\) 在 \(\mathbf0\) 处可微.

7.  设 \(f(x)\), \(g(x)\) 是 \([0,1]\) 上的连续函数, 且 \(\sup\limits_{x\in [0,1]}f(x)=\sup\limits_{x\in [0,1]}g(x)\). 证明存在 \(x_0\in[0,1]\), 使得 \(e^{f(x_0)}+3f(x_0)=e^{g(x_0)}+3g(x_0)\).

8. 记 \(\Omega=\{\mathbf p\in\Bbb R^3| |\mathbf p|\leq1 \}\), 设 \(V\colon\Bbb R^3\to\Bbb R^3\), \(V=(V_1, V_2, V_3)\) 是 \(C^1\) 向量场, \(V\) 在 \(\Bbb R^3\setminus\Omega\) 恒为 \(0\), \(\dfrac{\partial V_1}{\partial x}+\dfrac{\partial V_2}{\partial y}+ \dfrac{\partial V_3}{\partial z}\)在 \(\Bbb R^3\) 恒为 \(0\).
(1) 若 \(f\colon\Bbb R^3\to\Bbb R\) 是 \(C^1\) 函数, 求 \(\iiint_{\Omega}\bigtriangledown f\cdot V\,\mathrm dx\mathrm dy\mathrm dz\).
(2) 求 \(\iiint_{\Omega}V_1\, \mathrm dx\mathrm dy\mathrm dz\).

9. 设 \(f\colon\Bbb R\to\Bbb R\) 是有界连续函数, 求 \(\lim\limits_{t\to0^+}\int_{-\infty}^{+\infty} f(x) \frac{t}{t^2 + x^2}\,\mathrm dx\).

10. 设 \(f \colon [0,1] \to [0,1]\) 是 \(C^2\) 函数, \(f(0)=f(1)=0\), 且 \(f^{\prime\prime}(x)\lt0\), \(\forall x\in[0,1]\). 记曲线 \(\{(x,f(x))|x\in[0,1]\}\) 的弧长是 \(L\). 证明 \(L\lt3\).

Dec 102013
 

“霍比特人2: 史矛革之战(The Hobbit: Desolation of Smaug)” 一鼓作气席卷扑面而来! 翘首以盼快一年了!! 嗯, 嗯! 挪威(Norway)的朋友有幸今天 12 月 10 日, 最先观赏到这部充满期待的影片. 喜爱魔戒的影迷赶快重温下 “霍比特人 1: 意外之旅(The Hobbit: An Unexpected Journey)” 吧! 俺第一次看”霍比特人 1″, 是今年初 3 月 25 日在北京清河的 CGV 希界维国际影城. 影院在五彩城购物中心的最顶 7-8 两层, 环境还不错. 这电影院当时刚开张, 做了一些活动.

咕噜(Gollum)无疑是”魔戒”里的悲剧人物之一. 咕噜没有在拥有魔戒的五百年间, 使用魔戒做任何事, 最终还因为魔戒而坠入末日火山的火焰. 他因为魔戒, 心灵扭曲; 他的大半生, 人不像人鬼不像鬼.

由于这个系列有各种不同版本, 片长不一, 这里顺便提一下各部电影最长的时间: 指环王1 魔戒现身加长版 228分钟;  指环王 2 双塔奇兵加长版 235 分钟; 指环王 3 王者归来加长版 263 分钟; 霍比特人1 意外之旅加长版 182 分钟.

Solutions to the 74th William Lowell Putnam Mathematical Competition 2013  are now posted

Solutions to the 74th William Lowell Putnam Mathematical Competition

第 74 届普特南数学竞赛的官方解答

 Posted by at 1:40 am
Oct 112013
 

2013 年第四届丘成桐(Shing-Tung Yau)大学生数学竞赛(S.T. Yau College Student Mathematics Contests)已经落下帷幕. 决赛已经于 8 月 11 日和 12 日在北京中国科学院数学与系统科学院思源楼和晨兴中心举行, 颁奖典礼也已于 8 月 12 日在清华大学举行.

个人赛试题

Analysis and differential equations 2013 Individual

Geometry and topology 2013 Individual

Algebra and number theory 2013 Individual

Probability and statistics 2013 Individual

Applied Math. and Computational Math. 2013 Individual

团体赛试题

Team 2013

感谢博士数学论坛的网友数函的分享

Aug 092013
 

Day 1

Problem 1. Let  \(\rm A\) and \(\rm B\) be real symmetric matrixes with all eigenvalues strictly greater than \(1\). Let \(\lambda\) be a real eigenvalue of matrix \(\rm {AB}\). Prove that \(\left| \lambda  \right|\gt1\).

Problem 2.  Let \(f:\Bbb R \to \Bbb R\) be a twice differentiable function. Suppose  \(f(0) = 0\). Prove that there exists \(\xi \in\left({-\frac\pi2,\frac\pi2}\right)\) such that

\[f^{\prime\prime}\left( \xi  \right) = f\left( \xi  \right)\left(1 + 2\tan^2\xi \right).\]

Problem 3. There are \(2n\) students in a school \(\left( {n \in {\Bbb N},n \geqslant 2} \right)\). Each week \(n\) students go on a trip. After several trips the following condition was fulfiled: every two students were together on at least one trip. What is the minimum number of trips needed for this to happen?

Problem 4. Let \(n\geqslant 3\) and let \(x_1,x_2,\dotsc,x_n\) be nonnegative real numbers. Define \(A = \sum\limits_{i = 1}^n x_i,B = \sum\limits_{i = 1}^n x_i^2,C=\sum\limits_{i = 1}^n x_i^3\). Prove that:

\[ \left(n+1\right)A^2B+\left(n-2\right)B^2\geqslant A^4+\left(2n-2\right)AC.\]

Problem 5. Does there exist a sequence \((a_n)\) of complex numbers such that for every positive integer \(p\) we have that \(\sum\limits_{n=1}^\infty a_n^p\) converges if and only if \(p\) is not a prime?

Day 2

Problem 1. Let \(z\) be a complex number with \(\left|z+1\right|>2\). Prove that \(\left|z^3+1\right| > 1\).

Problem 2. Let \(p\) and \(q\) be relatively prime positive integers. Prove that

\[\sum_{k=0}^{pq-1}(-1)^{\left[\frac kp\right]+\left[\frac kq\right]}=\begin{cases}0 &\text{if}   pq   \text{is even},\\1 &\text{if } pq \text{ is odd}.\end{cases}\]

(Here \([x]\) denotes the integer part of \(x\).)

Problem 3. Suppose that \(\mathbf v_1,\mathbf v_2,\dotsc,\mathbf v_d\) are unit vectors in \(\Bbb R^d\). Prove that there exists a unit vector \(\mathbf u\) such that

\[\left| \mathbf u\cdot\mathbf v_i \right| \leqslant \frac1{\sqrt d}\]

for \(i = 1,2,\dotsc,d\).
( Here \(\cdot \) denotes the usual scalar product on \(\Bbb R^d\)).

Problem 4.  Does there exist an infinite set \(M\) consisting of positive integers such that for any \(a,b \in M\), with \(a\lt b\), the sum \(a+b\) is square-free?
( A positive integer is called square-free if no perfect square greater than \(1\) divides it ).

Problem 5.  Consider a circular necklace with \(2013\) beads. Each bead can be painted either white or green. A painting of the necklace is called good if among any \(21\) successive beads there is at least one green bead. Prove that the number of good paintings of the necklace is odd.
(Two paintings that differ on some beads, but can be obtained from each other by rotating or flipping the necklace, are counted as different paintings. )

 Posted by at 6:09 pm