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求一个整数的所有素数因子的思路是什么？ | 1. 一般的小数可以用简单筛法找出质数列表，然后一个个试。这种方法简单暴力，但是对几亿以下的数字可以很快。
2\. 再大一点的数 $$n$$ 就用Pollard的 $$\rho$$ 算法，思路：
任取一个数 $$x_0$$ 开始，不断计算 $$x_k=(x_{k-1}^2+1) \text{ mod } n$$ ，则如果 $$n$$ 有质因数$$p$$ ，那么 $$x_k \text{ mod } p$$ 应该能更快地进入循环，可以用龟兔赛跑算法（图形状像 $$\rho$$，因此算法得名）试图找出这个循环点，一旦找到 $$x_k\equiv x_l \text{ mod } p$$ ，立刻可以算 $$x_k-x_l$$ 和$$n$$ 的最大公约数得到分解。有一定几率失败。
3\. 更大的但在 $$10^{25}$$ 以下的 $$n$$ 可以用Lenstra椭圆曲线算法（ECM），思路：
挑选椭圆曲线 $$y^2=(x^3+ax+b) \text{ mod } n$$ 外加上面某一点 $$P$$ ，然后取一个较小的阶乘$$n!$$，反复用椭圆曲线加法算出 $$2!P, 3!P, \cdots, n!P$$ ，在重复计算加法中每一步都会计算和曲线切线斜率 $$x/y$$同余的整数，即找出整数 $$z$$ 使得$$yz\equiv x \text{ mod } n$$ ，方法为找 $$y$$ 和 $$n$$的最大公约数，在这个过程中如果得出公约数大于1则分解成立，有一定几率失败。
4\. 更大的 $$10^{100}$$ 以下的 $$n$$ 可以用二次筛（QS），思路
费尔马分解法试图把奇数 $$n$$ 写成 $$n=a^2-b^2$$ 的方法，这样立刻可得因数分解 $$(a+b)(a-b)$$，二次筛就是一种可以高效找到此类数字的方法，试图测试多个介于 $$\sqrt n$$ 和 $$n-1$$ 之间的数 $$a$$ ，如果 $$a^2\text{ mod } n$$ 正好是完全平方那就成了，不然找到几个不同的 $$a_i^2 \text{ mod } n$$，如果乘积正好是完全平方数，那么 $$a=\prod_i a_i \text{ mod } n$$ 满足条件。
5\. 再往上只能用普通数域筛选法（GNFS），这个就很复杂了，思路
也是用费尔马分解法，只是比二次筛更高效，核心定理：
取系数为整数的 $$d$$ 次多项式 $$f(x)$$ 有复根 $$\theta$$ ，并存在不是 $$n$$ 倍数的整数 $$m$$ ，使得$$f(m)$$ 为$$n$$ 倍数，将所有次数不超过 $$d-1$$的整系数多项式代入$$\theta$$后的值定义为一个“环”（因为任何两个这样形式的数的和或积依然是这样形式的数），那么必然存在一个将这个环里的复数映射到不是$$n$$ 倍数的整数上的函数 $$\phi(x)$$ 满足：
i - $$\phi(ab)=\phi(a)\phi(b)$$
ii - $$\phi(a+b)=\phi(a)+\phi(b)$$
iii - $$\phi(1)\equiv1 \text{ mod } n$$
iv - $$\phi(\theta) \equiv m \text{ mod } n$$
已知这个定理后，如果找到整数对 $$(a_i,b_i)$$ 使得复数 $$\beta = \prod_i (a_i+b_i\theta)$$ ，整数 $$y= \prod_i (a_i+b_im)$$ ，以及 $$\beta$$ 的映像 $$x=\phi(\beta)$$ ，则很容易根据i-iv得到$$x^2\equiv y^2 \text{ mod } n$$ ，有2/3机会能由此找到 $$n $$ 的一个约数。
2009年，232位的RSA-768数通过上百台计算机耗时两年成功分解，用的就是高度优化后的GNFS。 | {
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|
两个人连续投一枚硬币，如果先出现“正正反”则甲胜，如果先出现“正反反”则乙胜，两人获胜概率是否一样？假设硬币质地均匀，每次投硬币正反两面概率都是50%。连续投很多次，直到出现“正正反”或“正反反”为止。 | 第一次如果投反，那么甲乙都等于推倒重练，两人获胜概率不会有任何变化。
所以现在只需要考虑第一次投正的情况，假设此时甲胜率为 $$x$$ ，乙为 $$y$$ ，那么有两种概率各位1/2的情况：
1 - 第二次投出正，这时甲将100%获胜，因为这一刻开始接下来除非不断投出正，只要一出现反甲就赢了。
2 - 第二次投出反，这时看第三投，有1/2概率乙胜利（正反反），剩下1/2概率出现正反正，对于甲乙来说胜率相当于第一次投出正的情况，甲为 $$x$$，乙为 $$y$$。
综合
$$x=\frac{1}{2}\cdot1+\frac{1}{2}\left(\frac{1}{2}\cdot0+\frac{1}{2}x\right)$$ ，得出$$x=\frac{2}{3}$$
或者
$$y=\frac{1}{2}\cdot0+\frac{1}{2}\left(\frac{1}{2}\cdot1+\frac{1}{2}y\right)$$ ，得出$$y=\frac{1}{3}$$
两者相加为1，验算通过。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:584880907, aid:2904885349 | human | true | 暂无版权及作者信息 | [
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|
数论的威尔逊定理已知有哪些证明方法？主要是想问必要性的证明，目前找到的有基于原根逆元的证法，有Lagrange证法，还有哪些证明方法？ | 威尔逊有三种常见基础证明：
1. 标准教科书证明（稍微拓宽到一般情况），可能就是题主所说的Lagrange证法
设 $$p$$ 为奇素数（ $$p=2$$ 明显），设 $$D$$ 与 $$p$$ 互质，讨论同余方程：
$$xy\equiv D\bmod p$$
设 $$x,y\in\\{1,2,3,\cdots,p-1\\}$$ ，因为 $$x,y$$ 与 $$p$$ 互质，对于每一个 $$x$$必然存在一个唯一的 $$y$$ 满足以上方程（不然 $$x(y_1-y_2)$$ 为 $$p$$ 的倍数，矛盾）。
如果存在 $$x^2\equiv D\bmod p$$ （即 $$D$$ 为 $$p$$ 的二次剩余），那么根据 $$x^2-y^2=(x-y)(x+y)\equiv 0\bmod p$$ ，可得 $$y=p-x$$ ，即除了 $$x,p-x$$ 外的数都不可能满足$$x^2\equiv D\bmod p$$ 。
所以 $$\\{1,2,3,\cdots,p-1\\}$$ 中除 $$x,p-x$$ 外的数可以两两配对满足 $$xy\equiv D\bmod p$$，而 $$x(p-x)=px-x^2\equiv -x^2\bmod p\equiv-D\bmod p$$
因此 $$(p-1)!\equiv -D^{\frac{p-1}{2}}\bmod p$$
代入 $$D=1$$ 就得到威尔逊定理。
此外，代入 $$(p-1)!\equiv -1\bmod p$$ 可得 $$D^{\frac{p-1}{2}}\equiv 1\bmod p$$ ，如果$$D$$ 是 $$p$$ 的二次非剩余根据以上步骤可得 $$D^{\frac{p-1}{2}}\equiv -1 \bmod p$$ ，由此得到$$D^{\frac{p-1}{2}}\equiv\pm 1 \bmod p$$ 可以用来判定 $$D$$ 是否为 $$p$$的二次剩余，即鼎鼎大名的欧拉判定，由此可进一步推导如平方互反等重要结论。
2. 原根
取 $$p$$ 的原根 $$g$$ ，则 $$1,g,g^2,\cdots,g^{p-2}$$ 除以 $$p$$ 的余数为$$1,2,3,\cdots,p-1$$ 。则
$$(p-1)!=g^{0+1+2+\cdots+(p-2)}=g^{\frac{(p-2)(p-1)}{2}}\bmod p$$
设 $$p=2k+1$$ ，则 $$(p-1)!\equiv g^{k(2k-1)}\bmod p$$
根据费尔马小定律，已知 $$(g^k)^2=g^{2k}=g^{p-1}\equiv 1\bmod p$$ ，则 $$g^k\equiv \pm 1\bmod p$$ ，但 $$k<p$$ ，因此 $$g^k\not\equiv1\bmod p$$ ，所以只能 $$g^k\equiv -1 \bmodp$$ ，那么
$$(p-1)!=g^{k(2k-1)}=(-1)^{2k-1}=-1\bmod p$$
3. 既然用了费尔马小定律，有一个代数证明：
考虑多项式
$$f(x)=x^{p-1}-1-(x-1)(x-2)\cdots(x-p+1)$$
根据小费， $$f(1)=f(2)=\cdots=f(p-1)\equiv 0\bmod p$$
因此， $$1,2,\cdots,p-1$$ 是 $$p-2$$ 次多项式（最高项消除） $$f(x)$$ 模 $$p$$ 的根，因此可得$$f(x)\equiv 0 \bmod p$$ ，所有系数皆为 $$p$$ 的倍数，包括常数项，即
$$-1-(-1)^{p-1}(p-1)!\equiv 0 \bmod p$$
$$(p-1)!\equiv -1\bmod p$$
如果用群论的话，可以直接将西罗定理3用在对称群上得到。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:558886947, aid:2714940196 | human | true | 暂无版权及作者信息 | [
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|
有没有什么量级达到了10^-(10^2)? | 这个数就是1古戈尔分之一（另，这就是谷歌公司名字的来源）。
在数学上当然这个数不算什么大数。直观来说70的阶乘差不多是这个数量级的。换句话说，如果有70个人排队进场，你想要得到一个特定顺序的概率就是$$10^{-100}$$ 。
从物理上来说，这个量级还是很大的。举几个例子：
1. 一个电子的质量差不多是可观测宇宙质量的 $$10^{-80}$$ 到 $$10^{-90}$$ 级别，这个差距是题主所说的数的至少100亿倍。
2. 2022年超级计算机世界第一是美国的前沿计算机，运算速度超过了100万TFLOP（即每秒可以运行的浮点运算次数达到 $$10^{18}$$次级别）。整台计算机占地680平方米，和一个篮球场差不多大，造价约6亿美元。如果把这台计算机缩小到1个原子那么大，然后把整个可观测宇宙里的原子都变成前沿，那么所有这些计算机并行100秒可以一加一一直加到1古戈尔。
3. 宇宙内的基本粒子总数估计为 $$10^{80}$$ 个。如果全宇宙都塞满中子估计共可塞下 $$10^{128}$$个。阿基米德当年计算宇宙中可以塞下多少沙粒得到了 $$10^{63}$$ 的数字，但是他估计中的宇宙直径只有2光年，如果换成可观测宇宙则能塞下的总沙粒数为$$10^{95}$$ 。所以 $$10^{-100}$$ 相当于沙粒体积和1万个可观测宇宙体积的比例。
那么有没有什么量级能达到这么大呢？星系质量级别的超大黑洞（ $$10^{11}$$ 个太阳质量）通过霍金辐射逐渐蒸发殆尽所需的时间差不多是$$10^{100}$$ 年等级。如果热寂说是正确的，这可以被视为其发生时间点的下限。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:613494313, aid:3132435413 | human | true | 暂无版权及作者信息 | [
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|
身材和颜值是完全独立的还是有相关性的？有没有相关研究和结论 | 2014年发表在《性行为档案》期刊中的一篇论文
Bleske-Rechek A, Kolb CM, Stern AS, Quigley K, Nelson LA. Face and body:independent predictors of women's attractiveness. Arch Sex Behav. 2014Oct;43(7):1355-65. doi: 10.1007/s10508-014-0304-4. Epub 2014 May 15. PMID:24830907.
该文章将女性照片分为只有脸部，只有身体和全身，并包括日常衣着和泳装两组让评判者（男女都有）进行打分。受试人是日常生活中18-24岁的女大学生。显然，这里的结论只适用于美国人审美。
以下为一些结论：
1. 脸部吸引力和身体吸引力存在相当的相关性
这和之前的类似研究相符，但是发现的相关性更强。
在日常衣着组，脸和身体的吸引力有0.62的相关性。在泳装组，这个相关性降为0.45。可以看出当目标穿泳衣时，身材吸引力出现更大的两极分化（好坏更为清楚），因此降低了其和颜值的相关性。
此外，论文发现两者对整体吸引力有独立贡献。这一点很好理解，即使身材完全一样，颜值增加也会增加整体吸引力，反之亦然。
2. 身体露得越多，身材对整体吸引力的影响越不受颜值影响
在日常衣着组，身体吸引力和整体吸引力相关性为0.69，但是控制颜值后这一相关性降为0.39。
在泳装组，身体吸引力和整体吸引力相关性高达0.82，而且控制颜值之后并没有太大影响（0.81）。
3. 其它身材指数都和身体吸引力有相关性
首先，和预计的一样，腰臀比和身体质量指数有相关性（0.44到0.48）。两者都和身体吸引力成负相关，即腰臀比和身体质量指数越大，身体吸引力越低。在日常衣着组，腰臀比和身体吸引力相关性为-0.27，身体质量指数为-0.68。在泳装组，两者相关性显著增加，分别为-0.58和-0.76。
这说明相对腰臀比，是否过重（身体质量指数）对身材扣分更大，而且腰臀比和着装关联不大，但是身体质量指数在穿得少时负面影响显著增加。
此外，研究发现在日常衣着组，胸围与各项吸引力没有什么相关性，罩杯和身体以及脸的吸引力也没有相关性，但是和整体吸引力负相关。在泳装组，胸围和罩杯与所有吸引力都呈清楚的负相关。
以上结论可能反直觉，但是这也标志在日常人群中，苗条且前凸后翘出现的可能性极低，胸围和罩杯与体重以及腰臀比有相当的相关性。
4. 身材不止于腰臀比，体重和胸
研究发现即使控制了颜值，腰臀比，身体质量指数和胸大小后，身体吸引力和整体吸引力的相关性并没有太多影响，从单纯只控制颜值的情况下从0.39降到0.34（日常衣着），从0.81下降到0.71（泳装）。这说明身体吸引力不止于身形，体重和胸那么简单，还存在其它因素。 | {
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] | qid:599260366, aid:3016367224 | human | true | 暂无版权及作者信息 | [
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人类为什么偏好数字7？7这个数字在生活中很常见。1/7循环小数的循环节142857在分数中很常见。7是个素数。让人类随便说一个数字，7是最大概率的 | 数字偏好和文化有很深的渊源。我知道在西方文化圈里7是很重要的一个数，但是我并不觉得同样的情况发生在东亚文化圈内。
只谈7在西方文化中的重要性：
## 根本原因：7 是圣数
7在基督教文化中是一个圣数，有7罪7德，地狱9层但天国7层，详见耶鲁大学一位教授在1858年写的手稿《数字七》。基督教为什么把7奉为神圣的道理是很简单的，因为在创世纪里神以6天创造世界，1天休息，所以一周有七天，7变成了一个神圣的数字。
这种宗教上的原因很快渗透了生活的方方面面。就和中国人重五行所以世间万物都会按五之数划分，西方社会也开始以七划分平时见到的事物。实际上，人的脑洞是无限大的，只要想好一个数字总是能硬凑出来。
于是出现了7大洲（欧亚大陆硬是被一分为二），7大洋（硬分四大洋），7色彩虹（如果不先入为主你觉得彩虹几色？），古代人选奇迹也要选7大（当然这是基督教之前的，因为7的意义其实更本质，下面会提到）。
此外英语中也出现了7是幸运数字这一说法，最终也来源于这一“神圣”涵义。
但是，为什么圣经要说世界在7天里创造呢？因为早在基督教建立之前，7天的星期概念已经在巴比伦、波斯、罗马出现了。和很多其它事物一样，基督教建立后重新解读了很多古代习俗（例如用圣诞节覆盖传统上的冬至庆典和农神节等等），在教义内重新定义。
那为什么古巴比伦会用七天为一周呢？根源在于天空中行走的七星：金木水火土日月，那是远古时代天空中肉眼可见会动的仅有天体，在很多古文明中都有巨大影响。星期和七星的联系在英语中因为受到北欧神祇名字的影响不是很明显（Monday-月,Sunday-日, Saturday-土星还是保留了联系），但是在拉丁语系中都保留了下来。日本西化后直接借用七星来命名一周七天，就连中文的“星期”一词也是这么来的。
## 其它偏门的理由
1. 英文中7是最朗朗上口的一个数词，Seven是一位数中唯一有两个音节的词，所以放在诗歌韵律中使用特别活。Snow White and theEight Dwarfs 听上去就比 Snow White and the Seven Dwarfs拗口。
2. 心理学家乔治·米勒曾提出人大脑的短期记忆可以容纳的事物在5-9之间，所以常有“最多同时记7件事”，“背诵时要把对象分为7个一组记忆”这些说法。
3. 7在个位数里是最特异的，它是10以下最大的质数，要判断一个数是2、3、5的倍数都很简单，但是7的倍数就会更费周章。所以你让人从1-10里挑一个数，人们会下意识跳开直觉“有规律”的数，例如偶数，1，5，3和3的倍数，所以“不零不整”的7被挑中的机会最大。
4. 对赌徒来说，7幸运还有一个原因-两个骰子加起来点数为7的概率最大，当然这和7没有半毛钱关系，这是因为骰子正好有6面，那要细说就要涉及到三维正多面体一共只有5种了。 | {
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|
如何理解四维空间的人可以一眼看到一个平面的正反两面？ | 先降一维理解一下，在平面上的生物只能看到直线的一边，这是因为在平面里和直线垂直的方向只有两个（过直线上一点只能作一条直线和该直线垂直，这条“法线”有两个方向），这个生物的视角方向和这两个方向之间的一个形成一个钝角，可以定义为该视角只能看到这个方向“那一边”。
一旦来到三维，直线就失去了某一边的概念，因为过直线上一点可以作无数条和该直线垂直的直线，即这条直线有无数个法线方向，你可以说直线有无数边。三维空间里的一个固定视角方向依然会和这其中某些法线方向成锐角，某些成钝角，如果沿用上面的定义，那么三维空间的我们能够看到这无数边中的一些，但不能看到另一些。特别是原有平面上的两个方向在三维视角下依然有一个是“看得到”，另一个“看不到”，并没有两边都能看到一说。
同理，在四维空间中一个平面有无数个法线方向，即不再有正反两面而是有无数面。四维生物可以看到其中的无数面，但是依然不可能同时看到我们世界里该平面的正反两面。
例如，如果一个平面在三维里的法线方向为 $$(n_x,n_y,n_z)$$ ，那么在四维空间里任意 $$(n_x,n_y,n_z,t)$$都和该平面垂直，沿着这个方向都可以看到平面的一个面，这样的面有无数个。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:606084457, aid:3083028179 | human | true | 暂无版权及作者信息 | [
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|
这道极其复杂集合card题有没有什么好的解法？
已知函数 $f(x)$ 满足 $\left\{x \mid f(x)>f\left(\frac{1}{1-x}\right)=(a, b)(1<a<b)\right.$, 记 $f(x)$ 的最小值为 $M$, 最大值为 $N, S=\{x \mid f(x)=M\}, T=\{x \mid f(x)=N\}$, 则下列命题为假命题的是 (注: $\operatorname{card}(A)$ 表示集合 $A$ 中元素的个数)
A. 若 $\operatorname{card}(S)=1$, 则 $S \subseteq(a, b)$
B. 若 $\operatorname{card}(T)=1$, 则 $T \subseteq(a, b) \quad$
C. 若 $\operatorname{card}(S)=2$, 则 $S \subseteq(a, b)$
D. 若 $\operatorname{card}(T)=2$, 则 $T \subseteq(a, b)$ | 首先注意函数 $$\frac{1}{1-x}$$ 套两次会回到自己
$$\frac{1}{1-\frac{1}{1-x}}=1-\frac{1}{x}$$ ，$$\frac{1}{1-(1-\frac{1}{x})}=x$$
而且当 $$x>1$$ 时， $$\frac{1}{1-x}<0$$ ， $$0<1-\frac{1}{x}<1$$ ，因此这个函数每套用一次，就会在$$(1,\infty)\rightarrow(-\infty, 0)\rightarrow(0,1)$$ 之间循环。
* * *
所以问题的第一个条件就是说，当 $$a<x<b$$ 时
$$f(x)>f(\frac{1}{1-x})\leq f(1-\frac{1}{x})\leq f(x)$$
当 $$x\geq b$$ 或 $$1<x\leq a$$ 时
$$f(x)\leq f(\frac{1}{1-x})\leq f(1-\frac{1}{x})\leq f(x)$$ ，即$$f(x)=f(\frac{1}{1-x})=f(1-\frac{1}{x})$$
* * *
设 $$O=(a,b)\bigcup(\frac{1}{1-a},\frac{1}{1-b})\bigcup(1-\frac{1}{b},1-\frac{1}{a})$$
可见当 $$x\notin O$$ 且 $$x\neq 0,1$$ 时，你有$$f(x)=f(\frac{1}{1-x})=f(1-\frac{1}{x})$$
当 $$x\in O$$ 时，在 $$x,\frac{1}{1-x},1-\frac{1}{x}$$ 三个数中有一个 $$\in (a,b)$$，不失一般性，设 $$x\in (a,b)$$ ，那么 $$f(x)> f(1-\frac{1}{x})\geq f(\frac{1}{1-x})$$ 或者$$f(x)\geq f(1-\frac{1}{x})> f(\frac{1}{1-x})$$
* * *
因此A和C肯定是错的，最小值永远不可能取在 $$x\in(a,b)$$ 内，根据定义 $$f(x)>f(\frac{1}{1-x})$$。最大值如果有2个的话，可能发生一种情况：即 $$x\in (a,b)$$ 和 $$1-\frac{1}{x}\notin (a,b)$$ ，且$$f(x)=f(1-\frac{1}{x})$$ 取到最大值，因此D是错误的。
唯一正确的是B。
* * *
以上都没有考虑0和1，事实上很容易构造满足条件且 $$S=\\{0\\}$$或者 $$S=\\{0,1\\}$$ ， $$T=\\{1\\}$$ 或者$$T=\\{0,1\\}$$ 的情况，这种情况下就连B也可以是错的。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:607407411, aid:3129542311 | human | true | 暂无版权及作者信息 | [
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|
可不可以理解为所有事件的概率都是二分之一？可以不可以把一个事件最后的结果都翻译成，是与不是，能与不能，会或不会，这样所有事件发生的概率都是二分之一。 | 首先看“所有事件的概率是二分之一”
从数学角度来说：
概率的数学表达为概率空间，一个概率空间由三样东西组成：样本空间、事件空间和概率函数。
在有限样本空间的简单情况下用人话说就是有各种基本情况，以及一个会把这些情况的排列组合映射到一个0到1之间的数字的函数。
例如，把骰子掷出6和骰子掷不到6看成两个基本情况，并约定概率函数把两种情况都对应于1/2。这是一个完备的概率空间，事实上，你很容易想象存在这样的骰子。
但是概率函数的构造要满足一些基本规则。
在上面的例子里，“掷不到6“的情况包括掷到1，掷到2，掷到3，掷到4，掷到5这些情况。如果你现在也要讨论这些事件发生的概率必须把它们加入概率空间（不然概率讨论就没有意义了），一旦这么做，根据概率函数的定义你会立刻得到“掷不到6“的概率为掷到1，掷到2，掷到3，掷到4，掷到5的概率总和，那么显然这5个概率不可能都是1/2和依然是1/2。
再看“某一件事的概率是二分之一”
这其实没什么毛病，我们说硬币正面向上概率1/2或骰子掷出6概率为1/6并不是基于任何概率论，而是基于对事件底层物理学的假设。所以说到底这也只是一种假设。
例如看“我彩票中奖的概率为二分之一”，这取决于概率的现实意义。
如果你认为自己买彩票的和别人买彩票的机制完全一致，那么买彩票就是一个发生了千百万次的事件，从概率两大学派之一频率学派的角度来看，你可以观察到所有买过这个彩票的人里面有多少中了奖，从而估计你自己彩票中奖的概率，这显然远远小于二分之一。
如果你认为自己买彩票的和别人买彩票的机制并不一致，存在某种无法观察到的属性影响开奖几率（例如人品），那么如果你之前没买过彩票，这就是一个从未验证过是否会发生的事件，你完全可以假设自己彩票中奖的概率为二分之一。这里可以用第二个学派贝叶斯学派的观点，并没有任何逻辑否认你的假设。但是在这种情况下二分之一这个概率的意义也就只是如此，只是一种“信仰”。然后你去买彩票，每次不中（或者中，不影响结果）之后你会逐渐利用这些新信息更新你自己的“信仰”，最终你对于这件事的概率信仰会越来越低，趋近于上面第一种情况（假设人品属性实际不存在），或者趋近于另一个真实值。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:575605415, aid:2827138204 | human | true | 暂无版权及作者信息 | [
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|
一瓶啤酒要一块钱，两个啤酒瓶能换一瓶啤酒，三个啤酒瓶盖也能换一瓶啤酒。一个人有10块钱，问：不能赊帐，理论上他最多能喝几瓶酒。 | 如果把酒看作是一个连续的量，那么1个酒瓶可以换1/2瓶酒，1个酒盖可以换1/3瓶酒，所以1瓶酒可以换1/2+1/3=5/6瓶酒，这5/6瓶酒又可以换5/6x5/6瓶酒，以此类推，1瓶酒最终可以化为
$$1+\frac{5}{6}+\left(\frac{5}{6}\right)^2+\cdots=\frac{1}{1-\frac{5}{6}}=6$$瓶酒。
当然，实际没有那么理想，比如最后剩下一个瓶盖你就不能再换酒了，以上是最后没有剩下任何酒瓶或酒盖情况下的结果，此时所有酒瓶和酒盖都“转化”了没有浪费。这个答案显然也是原题答案的上限，我们知道原题答案无论如何都不可能超过$$6n$$ 。而且我们知道理想状况下1个酒瓶可以换6x1/2=3瓶酒，1个酒盖可以换6x1/3=2瓶酒。
现在重复以下步骤：
把现有全部酒瓶和酒盖尽可能多全部换成新酒，喝光，然后收集酒瓶和酒盖。
如果剩下的总酒瓶不足2个且总酒盖不足3个，停止，不然重复以上步骤。
可见
1 - 每一步结束时至少有1个酒瓶和1个酒盖。
明显，每一步都会换至少1瓶新酒，所以每一步结束都会有至少1个酒瓶和1个酒盖。
2 - 最后剩下1个酒瓶
如1，如果剩下2个或以上还能再换
3 - 最后剩下2个酒盖
如果某一步开始时，酒盖数为 $$3k+2\geq 2$$ ，则换完之后剩至少2个酒盖。
如果某一步开始时，酒盖数为 $$3k+1\geq 4$$ ，则酒盖能换至少1瓶新酒，加上剩下多余的1个酒盖，总酒盖数至少2。
如果某一步开始时，酒盖数为 $$3k\geq 6$$ ，则酒盖能换至少2瓶新酒，得到2个酒盖。
如果某一步开始时，酒盖数为 $$3$$，则光用酒盖能换1瓶新酒，得到1个酒盖，根据1开始时你至少有1个空酒瓶，那么你此时总共有至少2个空酒瓶，循环不会停止，下一步一开始用2个空酒瓶可以再换1瓶新酒，酒盖数增加为2。
所以最后停下来时你必然剩下1个酒瓶，2个酒盖，折合为“浪费”了7瓶酒的量，一共喝 $$6n-7$$ 瓶酒。
因此，如果这个人有10块钱，则他最终能喝啤酒瓶数为$$6n-7 = 6 imes 10 -7 = 53$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:553387768, aid:2672588607 | human | true | 暂无版权及作者信息 | [
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|
假设一个人从15岁生了第一个孩子，接下来每五年生一个，四十岁生下最后一胎后不再继续。她的孩子也从15岁开始生孩子，每五年生一个，生到四十岁最后一胎不再继续。那么请问到这个人90岁时，他们家族一共有多少人？不考虑配偶只考虑孩子，中间不会有人死，只作为数学问题。 | 设 $$F(x)$$ 为 $$x$$ 岁人的总子孙数目，
则可得
$$F(0)=F(5)=F(10)=0$$
$$F(15)=1$$
$$F(20)=2$$
$$F(25)=3$$
$$F(30)=4+F(15)=5$$
$$F(35)=5+F(15)+F(20)=8$$
$$F(40)=6+F(25)+F(20)+F(15)=12$$
从40岁开始，
$$F(x)=6+F(x-15)+F(x-20)+\cdots+F(x-40)$$
因此当 $$x>40$$ 时有递归公式
$$F(x)=F(x-5)-F(x-45)+F(x-15)$$
所以
$$F(45)=F(40)-F(0)+F(30)=17$$
$$F(50)=F(45)-F(5)+F(35)=25$$
$$F(55)=F(50)-F(10)+F(40)=37$$
$$F(60)=F(55)-F(15)+F(45)=53$$
$$F(65)=F(60)-F(20)+F(50)=76$$
$$F(70)=F(65)-F(25)+F(55)=110$$
$$F(75)=F(70)-F(30)+F(60)=158$$
$$F(80)=F(75)-F(35)+F(65)=226$$
$$F(85)=F(80)-F(40)+F(70)=324$$
$$F(90)=F(85)-F(45)+F(75)=465$$
所以加上本人共466人。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:444371202, aid:1729687048 | human | true | 暂无版权及作者信息 | [
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|
有五只麻雀偷到了一批大米，准备第二天早上去分。结果第二天第一只麻雀早到了，数了一下，自己吃了一粒，剩下的正好分五份，取走自己那份就先走了。第二只麻雀来了，数了数，也是吃了一粒，剩下的正好分五份，取走自己那份就走了。之后三只麻雀情况一样。那么请问，它们最少偷了多少粒大米？ | 假设一开始有 $$X$$ 粒米，
第一只麻雀吃剩下 $$\frac{4}{5}(X-1)=\frac{4}{5}X-\frac{4}{5}$$
第二只麻雀吃剩下$$\frac{4}{5}\left[\frac{4}{5}(X-1)-\frac{4}{5}\right]=\frac{4^2}{5^2}X-\frac{4^2}{5^2}-\frac{4}{5}$$
以此类推
第五只麻雀吃剩下$$\frac{4^5}{5^5}X-\frac{4^5}{5^5}-\frac{4^4}{5^4}-\frac{4^3}{5^3}-\frac{4^2}{5^2}-\frac{4}{5}$$
以上五项的通项是
$$\frac{4^n}{5^n}X-\sum_{i=1}^n\frac{4^i}{5^i}=\frac{4^n}{5^n}X-4\left(1-\frac{4^n}{5^n}\right)=\frac{4^n}{5^n}(X+4)-4$$
很明显要让所有项都是整数， $$X+4$$ 必须是 $$5^5$$的倍数，最小是 $$5^5-4=3121$$ ，这么一来第五只麻雀吃剩下的就是$$4^5-4=1020$$ ，那么五只麻雀总共最少吃掉了 $$3121-1020=2101$$ 粒米。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:458776264, aid:1881240679 | human | true | 暂无版权及作者信息 | [
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|
C++ 初始化列表展开,详细解释这个代码是什么意思？
template<typename T, typename... Ts>
auto printf3(T value, Ts... args) {
std::cout << value << std::endl;
(void) std::initializer_list<T>{([&args] {
std::cout << args << std::endl;
}(), value)...};
} | ## 代码意图
```
template<typename T, typename... Ts>
auto printf3(T value, Ts... args) {
std::cout << value << std::endl;
(void) std::initializer_list<T>{([&args] {
std::cout << args << std::endl;
}(), value)...};
}
```
printf3 函数的意图是将参数打印出来，一行一个。比如
```
printf3(1, 2, 3, "HelloWorld");
```
的 输出结果为
```
1
2
3
HelloWorld
```
printf3第二行有点难懂，它同时结合了 C++ 的几个语法点
- 初始化列表 initializer_list
- 可变参数模版，Variadic Template
- lambda 表达式
- 逗号表达式
## 初始化列表
初始化列表，还是比较好 懂的，比如下面代码
```
auto tmp = {1, 2, 3, 4};
for (auto& v : tmp) {
printf("%d\n", v);
}
```
tmp 的类 型就是 std::initializer_list<int>。初始化列表的值不能包含不同类型，下面代码会编译错误
```
auto tmp = {"Hello", 1, 2, 3}; // error
```
也不能是 void值，下面代码也会编译错误
```
auto tmp = {void, void, void}; // error
```
## 可变参数模版
可变参数模版，就是模版中 typename... Ts ，和 Ts... args 这种语法。
```
template<typename T, typename... Ts>
auto printf3(T value, Ts... args)
```
可以将多个输入参数打包（Parameter Pack）到 args 中。比如 printf3(1, 2, 3, 4);，args 就是 2、3、4 三个参数 pack 到 args 中。在模板代码中，可以通过 .... 的语法将参数展开（Parameter Pack Expansion） ，比如下面代码
```
template <typename... Ts>
void doSomething(Ts... args) {
auto tmp = {args...};
for (auto &v : tmp) {
printf("%d\n", v);
}
}
```
调用 doSomething(1, 2, 3);时，tmp 会展开成
```
auto tmp = {1, 2, 3};
```
args...这种只是简单用法。模板参数还可以在更复杂的表达式中展开，比如
```
auto tmp = { (args * args)...};
auto tmp1 = { std::sin(args)...};
auto tmp2 = { (std::sin(args) + args)...};
```
调用 doSomething(1, 2, 3);时，tmp 会展开成
```
auto tmp = {(1 * 1),(2 * 2),(3 * 3)};
auto tmp1 = { std::sin(1), std::sin(2), std::sin(3) };
auto tmp2 = { (std::sin(1) + 1), (std::sin(2) + 2), (std::sin(2) + 2) };
```
C++ 碰到 ...，就会将包含 args 的表达式展开，可以在很复杂的表达式中展开。
## lambda 表达式
下面的 fn 就是先定义了 lambda 表达式，之后再进行调用
```
int a = 0;
auto fn = [&a] {
std::cout << a << std::endl;
};
fn();
```
也可以将 lambda 表达式定义和调用 写在一起。比如
```
int a = 0;
auto b = [&a] {
return a;
}();
```
## 逗号表达式
这个也比较简单。C++ 中 ，假如表达式用逗号分隔，会以此求值，最后一个值就是表达式的值。比如
```
auto a = (funA(), funB(), funC());
```
会以此求值 funA()、funB()、funC()，最终 a 为 funC 的值。
## 拆解原始代码
对几个语法点简单了解后，我们现在来拆解
```
(void) std::initializer_list<T>{([&args] {
std::cout << args << std::endl;
}(), value)...};
```
首先，里面的复杂表达式
```
([&args] {
std::cout << args << std::endl;
}(), value)...
```
会按照模板参数依次展开，求值后， 生成一个std::initializer_list<T>。
比如调用为 printf3(1, 2, "HelloWorld"); value 值为 1，args 为 2、"HelloWorld" 两个参数中的打包。将包含 args 的表达式展开，展开的代码大致为
```
void printf3(int value, int arg0, const char *arg1) {
std::cout << value << std::endl;
(void)std::initializer_list<int>{
([&arg0] {
std::cout << arg0 << std::endl;
}(), value),
([&arg1] {
std::cout << arg1 << std::endl;
}(), value)
};
}
```
arg0 和 arg1 都分别嵌在逗号表达式中，于是分别调用了 lambda 函数，并 value 作为逗号表达式的值。于是 std::initializer_list放的都是 int 类型。
各个模板参数展开，依次调用了 lambda 函数，打印出各自的值。
## 一些细节
- 代码中的 (void)std::initializer_list，这个 (void) 是为了防止编译警告，因为 initializer_list 没有被使用。
- 先定义了 T value，再定义Ts... args，是让 printf3 最少包含一个参数，让 printf3() 调用报编译错误。
- 使用逗号表达式，是 因为代码中，那个 lambda 函数返回值为 void, 而 std::initializer_list<void> 是不合法的。这里也可以让 lambda 返回一个值，省去逗号表达式。比如
```
template<typename T, typename... Ts>
auto printf3(T value, Ts... args) {
std::cout << value << std::endl;
(void) std::initializer_list<int>{([&args] {
std::cout << args << std::endl;
return 1;
}())...};
}
```
- 原始代码搞得比较复杂，也为了让 printf3 中的参数可以有不同的类型。比如下面的简单实现，就只能传入相同类型参数。
```
template <typename T, typename... Ts>
auto printf3(T value, Ts... args) {
auto tmp = {value, args...};
for (auto &v : tmp) {
std::cout << v << std::endl;
}
}
```
- C++17 中有个 Pack fold expression 的语法，可以让实现更简单些
```
template <typename T, typename... Ts>
void printf3(T value, Ts... args) {
std::cout << value << std::endl;
((std::cout << args << std::endl), ...);
}
``` | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:443285720, aid:1723184923 | human | true | 暂无版权及作者信息 | [
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|
句柄是什么？ | 句柄的英文是 handle。在英文中，有操作、处理、控制之类的意义。作为一个名词时，是指某个中间媒介，通过这个中间媒介可控制、操作某样东西。
这样说有点抽象，举个例子。door handle 是指门把手，通过门把手可以去控制门，但 door handle 并非 door本身，只是一个中间媒介。又比如 knife handle 是刀柄，通过刀柄可以使用刀。
跟 door handle 类似，我们可以用 file handle 去操作 file, 但 file handle 并非 file 本身。这个 filehandle 就被翻译成文件句柄，同理还有各种资源句柄。
计算机领域很多英文词，直接从日常词中引申而来。比如 fork，日常用词就是个叉子，在 unix 中引申成创建新进程（进程分叉了）。socket日常用词是插座（连起来用于通电），引申成联网的标记信息（连起来用于通信）。英文是很日常，很容易理解的词，有时翻译成中文反而难以理解了。
句柄这个翻译有点奇怪。据维基百科，句柄 的条目。
David Gries所著的《Compiler Construction for Digital Computer》（1971）有句话
> A handle of any sentential form is a leftmost simple phrase.
该书中译本，《数字计算机的编译程序构造》(仲萃豪译, 1976 版）翻译成
> 任一句型的句柄就是此句型的最左简单短语。
这可能是句柄一词最早的出处。
这里确实是在讨论句子。在这里句柄是个意译的合成词，两个字分拆开，“句柄”中的“柄”，用法就类似于，“刀柄”中的“柄”。用在此处是适当的。
但以后将各种资源 handle, 都翻译成句柄时，就有点滥用了。
具体到代码实现，handle 通常是某个数字标记，通过标记操作资源。这个标记在不同的场合有不同的叫法，有时叫 ID，有时叫描述符(descriptor)。在Windows 平台，就叫各种 handle 了。
不要将 handle 简单地理解成编号、索引。比如分配 16 位的索引，再用 8 位密码将 16 位索引加密。之后将 4 位类型、4 位权限、8位密码、16 位加密索引打包成一个 32 位的整数作为 handle。这时说这个 handle 是索引就有点不适当了。
用 handle 如何操作真正的资源，是实现的细节。handle 通常被实现为整数，也可以被实现成其他类型。
广义来说，指针也是某种 handle，可以操作对象。但实际语境中，指针跟句柄是有区别的。初次接触到 handle (或者 id)，很多人会有迷惑，为什么要用handle，而不直接用指针呢？
1. 指针作用太强，可做的事情太多。可做的事情越多，就会越危险。接口设计中，功能刚刚好就够了，并非越多权限越好的。
2. handle 通常只是个整数，实现被隐藏起来，假如直接暴露了指针，也就暴露了指针类型（有时也可以暴露 void* 指针作为某种 handle）。用户看到越多细节，其代码就越有可能依赖这些细节。将来情况有变，但又要兼容用户代码，库内部改起来就更麻烦。
3. 资源在内部管理，通过 handle 作为中间层，可以有效判断 handle 是否合法，也可以通过权限检查防止某种危险操作。
4. handle 通常只是个整数，所有的语言都有整数这种类型，但并非所有语言都有指针。接口只出现整数，方便同一实现绑定到各种语言。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:27656256, aid:943130123 | human | true | 暂无版权及作者信息 | [
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|
为什么会有那么多人坚持黑客精神？ | 黑客英文为 hacker，先来了解 hacker 一词的出处。
60 年代初，麻省理工学院（MIT）有个学生团体，叫「铁路模型技术俱乐部」（Tech Model Railroad Club，简称TMRC）。俱乐部活动室摆着一个巨大的火车规划模型，俱乐部成员平常活动就是模拟铁路系统。
TMAC 大致分两类成员。一类成员喜欢制作精美的火车、城市、景物模型。另一类成员更关心模型下面的信号和动力系统，比如火车如何调度、前进、停止。后来 MIT购买了计算机，后一类成员慢慢被计算机吸引过去，去研究计算机了。
一群人聚成圈子，会很自然地发明出一些新词，让圈子外的人莫名其妙。
当时 TMAC 的成员，将难题的解决方法称为 hack。巧妙、漂亮的解决方法叫做 cool hack 或者 neathack。笨拙、蛮干、不太优美但又确实有效的解决方法叫做 ugly hack 或者 quick hack。相应地完成 hack 的过程就叫做hacking，从事 hacking 的人就是 hacker，也就是中文的黑客。
黑客一词本意指解决问题的人。因为 TMAC 的人后来很多都去了玩计算机，hack 就经常被特指解决计算机难题。
黑客精神就是去解决问题，纯粹地、出于兴趣、不为功利地去解决问题。拥有黑客精神的就是黑客。按照这种宽泛的定义，黑客在各行各业中无处不在，人类的进步很大程度上就依赖这种精神。
不为物质利益，只是纯粹地想知道原因，想解决某个问题，对于某些人可能是很难理解的。
后来 hacker 一词被媒体误解了，媒体将非法盗窃信息，欺诈用户等等的计算机罪犯都叫成hacker。信奉黑客精神的那些人，不会承认那些搞破坏的为hacker，而是将他们叫为 cracker，中文翻译为骇客。
TMRC 的成员后来去了世界各地，MIT 也被誉为黑客精神的发源地。
具体到计算机，有六条黑客价值观：
- 对计算机的访问（以及任何可以帮助我们认识世界的事物）都应该是不受限制，任何人都有动手尝试的权利。
- 所有的信息都应该可以自由获取。
- 不要迷信权威——促进分权。
- 评价黑客的标准应该是他们的技术，而不是那些没有实际用途的指标，比如学位、年龄、种族、性别、职位。
- 可以在计算机上创造出艺术和美。
- 计算机可以让生活更美好。
上面六条价值观很理想主义，不理解的甚至觉得有点傻。但在每个行业中想达到顶尖，就需要多少带点傻傻的理想主义。在中国会说到侠义精神，也带点傻气。那为什么有些人这样傻呢，明明知道可以更容易一点，还选这条路。这种牵涉到价值观的问题，很难解释。伟大点说，是这样可以让世界更美好；个人来说，这样遵循本心，觉得自在。
真想知道得更详细，可以看：
- 《黑客，计算机革命的英雄》
- 《黑客与画家》
以及一系列介绍 Unix 或者开源文化的书。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:30172484, aid:47077724 | human | true | 暂无版权及作者信息 | [
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|
Map、Dictionary、Hash Table 有哪些异同？ | dictionary 跟 map 其实是同一个东西，只是在不同场合叫法不同。
dictionary 的中文是字典，map在中文是映射，也有地图的意思。查字典，查地图，都是通过某个信息，去找到另一个信息。比如通过单词的拼写找到单词的具体含义。
类比查字典过程，单词的拼写为 key, 单词的具体含义为 value。dictionary 就是通过key，找到value，有时也将 dictionary说成是 key-value 结构。只要达到查找目的，都可以叫做 dictionary。具体怎么找，可以有不同实现。
比如，最简单是将 key，value 放在一起，线性排。
```
k1, k2, k3, k4, k4 ....
v1, v2, v3, v4, v5 ....
```
当需要从 key 找到对应的 value 时，就从头到尾遍历过去。依次判断 k1, k2, k3, k4 是不是等于key, 当等于的时候，就找到 key的具体位置，从而也就找到了value。
但这样从头到尾遍历，速度就太慢了，时间复杂度为 O(N)。N为数据的大小。
为了快速从key找到value。dictionary(或者说map)的通常有两种实现方式。
1. 二叉树
2. 哈希(hash)表
二叉树查找的时间复杂度为 O(logN)，哈希表的时间复杂度大致为 O(1)。二叉树也分红黑树，AVL树等。哈希表的速度很快，很多语言内置的dictionary 都使用哈希表来实现，但它通常会浪费一些存储空间。这部分有兴趣去看数据结构的书。
hash_map 其实就是使用 hash 表实现的 map。注意，二叉树，哈希表仅仅是 dictionary 的实现方式，不能说 hash 就等于dictionary，实现方式可以有多种多样。
比如上面线性存储的实现，可以调整一下。当数据都放进来之后，先根据 key 来排序，再使用二分查找，就可以很快速地从 key 找到 value,这种实现简单快速，并省内存，有时会比 hash 的实现更好。适合于一些数据不会经常变动的情况。
C++ 11 的标准库中有个 unordered_map，就是采用哈希表使用的 map。在C++11之前，没有标准的哈希实现，很多第三方库实现了哈希字典，基本都叫做 hash_map, 并应用广泛。所以C++11的实现就不能叫hash_map了，因为会造成很多现存的程序名字冲突。哈希实现的字典是无序，也就取了个不算太好的名字，unordered_map。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:27581780, aid:37398584 | human | true | 暂无版权及作者信息 | [
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|
编程的函数和数学的函数为什么都叫函数？ | 在数学上，最早将 function 一词翻译成函数，是清朝数学家李善兰。数学的 function指一个量随着另外的量的变化而变化，换句话说，一个量包含在另一些量当中。函数，也就是含数。中文的函，有盒子，用来装东西的意思，也可引申成包含。function在数学上翻译成函数也算贴切。
最早研究计算机那批人，很多从数学系转过去的。编程中的 function也直接采用了数学上的翻译，其实有点误导。但也难怪，计算机设计之初就是用来解决数学问题。
英文中 function, 也有完成一定功能，起作用的意思。编程语境中，function 一词更应该理解成，这代码完成某种功能。
数学上的 function，是固定的输入，就对应固定的输出，中间不会改变状态。而编程上的function，只是完成某个功能，中间可以修改状态。假如类似数学上的意义，完全不修改状态，会叫成纯函数（pure function）以作强调。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:305981549, aid:555602200 | human | true | 暂无版权及作者信息 | [
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|
什么是逆波兰表达式 ？ | 逆波兰表达式，英文为 Reverse Polish notation，跟波兰表达式（Polish notation）相对应。之所以叫波兰表达式和逆波 兰表达式，是为了纪念波兰的数理科学家 Jan Łukasiewicz。其在著作中提到：
> 我在1924年突然有了一个无需括号的表达方法，我在文章第一次使用了这种表示法。
- 平时我们习惯将表达式写成 (1 + 2) * (3 + 4)，加减乘除等运算符写在中间，因此称呼为中缀表达式。
- 而波兰表达式的写法为 (* (+ 1 2) (+ 3 4))，将运算符写在前面，因而也称为前缀表达式。
- 逆波兰表达式的写法为 ((1 2 +) (3 4 +) *)，将运算符写在后面，因而也称为后缀表达式。
波兰表达式和逆波兰表达式有个好处，就算将圆括号去掉也没有歧义。上述的波兰表达式去掉圆括号，变为 * + 1 2 + 3 4。逆波兰表达式去掉圆括号，变成 1 2 + 3 4 + * 也是无歧义并可以计算的 。事实上我们通常说的波兰表达式和逆波兰表达式就是去掉圆括号的。而中缀表达式，假如去掉圆括号，将 (1 + 2) * (3 + 4) 写成 1 + 2 * 3 + 4，就改变原来意思了。
现实中，波兰表达式和逆波兰表达式，具体用于什么地方呢？
波兰表达式（前缀表达式）， 实际是抽象语法树的表示方式，比如中缀 (1 + 2) * (3 + 4) 编译时转成的抽象语法树为
```
*
/ \
+ +
/ \ / \
1 2 3 4
```
这个操作符就是根节点，操作数为左右子节点。我们将这棵树用符号表达出来，可以写成 (* (+ 1 2) (+ 3 4))。这实际就是 Lisp 的 S-表达式。S-表达式可看成将整棵抽象语法树都写出来，每层节点都加上圆括号。
至于逆波兰表 示式，可用栈进行计算，天生适合于基于栈的语言。遇到数字就将数字压栈，遇到操作符，就将栈顶的两个元素取出计算，将计算结果再压入栈。比较典型的基于栈的语言为 Forth 和 PostScript。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:41103160, aid:452481026 | human | true | 暂无版权及作者信息 | [
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|
iOS开发中，block与代理的对比，双方的优缺点及在什么样的环境下，优先使用哪一种更为合适？ | block 和 delegate 都可以通知外面。block 更轻型，使用更简单，能够直接访问上下文，这样类中不需要存储临时数据，使用 block的代码通常会在同一个地方，这样读代码也连贯。delegate更重一些，需要实现接口，它的方法分离开来，很多时候需要存储一些临时数据，另外相关的代码会被分离到各处，没有 block 好读。
应该优先使用 block。而有两个情况可以考虑 delegate。
1. 有多个相关方法。假如每个方法都设置一个 block, 这样会更麻烦。而 delegate 让多个方法分成一组，只需要设置一次，就可以多次回调。当多于3 个方法时就应该优先采用 delegate。
比如一个网络类，假如只有成功和失败两种情况，每个方法可以设计成单独 block。但假如存在多个方法，比如有成功、失败、缓存、https验证，网络进度等等，这种情况下，delegate 就要比 block 要好。
在 swift 中，利用 enum, 多个方法也可以合并成一个 block 接口。swift 中的枚举根据情况不同，可以关联不同数据类型。而在 objc就不建议这样做，objc 这种情况下，额外数据需要使用 NSObject 或者 字典进行强转，接口就不够安全。
2. 为了避免循环引用，也可以使用 delegate。使用 block时稍微不注意就形成循环引用，导致对象释放不了。这种循环引用，一旦出现就比较难检查出来。而 delegate的方法是分离开的，并不会引用上下文，因此会更安全些。
假如写一个库供他人使用，不清楚使用者的水平如何。这时为防止误用，宁愿麻烦一些，笨一些，使用 delegate 来替代 block。
将 block 简单分类，有三种情形。
* 临时性的，只用在栈当中，不会存储起来。
比如数组的 foreach 遍历，这个遍历用到的 block 是临时的，不会存储起来。
* 需要存储起来，但只会调用一次，或者有一个完成时期。
比如一个 UIView 的动画，动画完成之后，需要使用 block 通知外面，一旦调用 block 之后，这个 block 就可以删掉。
* 需要存储起来，可能会调用多次。
比如按钮的点击事件，假如采用 block 实现，这种 block 就需要长期存储，并且会调用多次。调用之后，block也不可以删除，可能还有下一次按钮的点击。
对于临时性的，只在栈中使用的 block, 没有循环引用问题，block 会自动释放。而只调用一次的 block，需要看内部的实现，正确的实现应该是block 调用之后，马上赋值为空，这样 block 也会释放，同样不会循环引用。
而多次调用时，block 需要长期存储，就很容易出现循环引用问题。
Cocoa 中的 API 设计也是这样的，临时性的，只会调用一次的，采用 block。而多次调用的，并不会使用 block。比如按钮事件，就使用target-action。有些库将按钮事件从 target-action 封装成 block 接口, 反而容易出问题。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:29023547, aid:109570584 | human | true | 暂无版权及作者信息 | [
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|
组合数公式的递归实现，如何分析其时间复杂度？
递归形式实现的组合计数$C_n^r=C_{n-1}^r+C_{n-1}^{r-1}$
int C(int n,int r){
if(n==r||r==0) return 1;
else return C(n-1,r)+C(n-1,r-1);
}
请问这个时间复杂度是如何计算的? | 答案是$$O(C_n^r)$$。
要得到这个答案，并不需要解递归方程。注意到递归只有在值为1的时候才结束，而最终的结果$$C_n^r$$由许多个这样的1相加而来，那么递归树是一棵有$$C_n^r$$个叶子结点的二叉树，总递归次数是树中所有结点的个数$$2C_n^r-1$$，即$$O(C_n^r)$$。
当然你要解递归方程也可以。设计算$$C_n^r$$需要的递归次数为$$T(n,r)$$，则有
$$T(n,r) = \left\\{ \begin{array}{ll} 1, & \text{if } n=r \text{ or } r=0 \\\T(n-1,r) + T(n-1,r-1) + 1, & \text{otherwise} \end{array} \right.$$
然后你发现这跟组合数的递推式本身非常相似，只是第二种情况中多了一个加1。
稍加变换，设$$S(n,r) = T(n,r) + 1$$，则有
$$S(n,r) = \left\\{ \begin{array}{ll} 2, & \text{if } n=r \text{ or } r=0 \\\S(n-1, r) + S(n-1, r-1), & \text{otherwise} \end{array} \right.$$
那个1就被吸收掉了。容易看出$$S(n,r) = 2C_n^r$$，故$$T(n,r) = 2C_n^r - 1$$。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:38474425, aid:76641611 | human | true | 暂无版权及作者信息 | [
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|
有哪些违背直觉的数学问题？ | 例一、布丰投针问题
在画有间隔为d的水平线的纸上投长为l的针(l<d)，假定投n次（n足够大），其中针和水平线相交的次数是k次，则
k/n=2l/(Pi*d)（Pi即圆周率，等号为约等于）
故圆周率的估计值是Pi=2nl/kd
例二、波利亚盒子模型
盒中装有a个红球，b个蓝球，每次取一个球，看到其颜色后，将其放回并多放入c个同色球（c最小可取-1，即不放回），则每一轮取出红球的概率都是a/(a+b)
例三、公交车等车问题
公交车平均10分钟一班，在不同的站台处，你到了站台之后平均需要等多久才来下一辆公交车？
在开始的几个站点，公交车发车很均匀，平均10分钟一班，我们到达公交车站的时间可能在其中任一时刻，所以期望等待时间是5分钟
但公交车开出去很多站之后，由于路况等原因，虽然仍然遵循平均10分钟一班的规律，但是其分布不再均匀，一般假定此时遵循指数分布，而指数分布是无记忆的，因此无论你什么时候到达公交车站，期望等待时间都是10分钟
还有一个简单点的公交车等车问题
两辆公交车A和B都是10分钟一班，但是遵循不同的分布规律，到公交车站等车，先来的车是A的概率一定是1/2吗？
不一定1/2，比如A车是在00、10、20、30、40、50分来，B车是在01、11、21、31、41、51分来，那么A车先到的概率是9/10 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:41408857, aid:90964732 | human | true | 暂无版权及作者信息 | [
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|
如何只使用折叠的方法十等分一个线段或者绳子？可以对折，也可以随便折，但是不能借助其他工具。有可能性吗？ | 把绳子16等分，从绳子的一个末端开始，截取长为3/16，4/16和5/16的三段，并记相应端点为（从末端起）A、B、C、D。
将点D与点A重合，撑开三角形ABC，得到比例为3:4:5的直角三角形。
在AC上找到另一个十六分点E使得CE=1/16。固定E、B和BC上某个合适的点（比如点G左侧那个十六分点），将点C翻折并仍落在BC上，此时点C的位置记为F。
固定EF，将C翻折回去，撑开三角形ECF，则CF的长度为 $$\frac{1}{16}\times\frac{4}{5}\times2=\frac{1}{10}$$ 。
从绳子另一末端开始量出CF的长度，然后不断翻折即可。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:489153187, aid:2147403460 | human | true | 暂无版权及作者信息 | [
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|
从科学的角度考虑，蘑菇能否直接在人体（或尸体）上生长？ | 某些担子菌门大型真菌的孢子可以在特定患者或尸体上萌发，可以引起疾病。不过，你难以期待它们在活人身上长出标志性的子实体——搞不成某些科幻作品或魔幻作品表现的“蘑菇人”。
西澳大利亚的 Hebeloma aminophilum 诨号“食尸鬼真菌”，以从动物尸体及其周围长出蘑菇闻名。新西兰的 Laccaria masoniae也有类似的举动。它们所在的属有更多这样的案例。
裂褶菌 Schizophyllum commune 通常生长在烂木头上，偶尔会感染人 或引起过敏，菌丝体可在某些感染者肺内形成真菌球。裂褶菌可供人食用。一些其他的丝状担子菌，包括草菇，也有感染人的案例。
2021 年，一篇文章报告一名 30岁的男性双相情感障碍患者自行静脉注射以棉布过滤的“迷幻蘑菇茶（水煮迷幻蘑菇）”后出现嗜睡、黄疸、腹泻、恶心、吐血等症状，被家人送到医院。医生从他的血液检出细菌和真菌感染，该真菌被鉴定为古巴裸盖菇。患者自述看了用微量致幻剂和裸盖菇素治疗双相情感障碍的新闻报道、打算自我治疗。
- 这不代表古巴裸盖菇在自然条件下对人的致病风险。可以考虑某些吸毒者滥用古巴裸盖菇造成的潜在感染风险。
此外，一些新闻报道将曲霉菌等在人耳生长的现象写成“人身上长蘑菇”，那是不符合题设的。
无法期待马勃孢子在活人的伤口长出蘑菇。马勃孢子有概率引起过敏。
“不含纤维素”不是主要问题，活人的体温和氧供应对许多真菌不合适，即使免疫水平低下也不一定能生长。不过，野生真菌可能正在适应气候变化带来的更高气温，并变得更擅长在人体之类温热物体上生长。这可能导致耳念珠菌病等原本罕见的真菌感染在过去数年间的激增。 | {
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"minor": [
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] | qid:614904062, aid:3146588324 | human | true | 暂无版权及作者信息 | [
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|
一个体重约为150斤的人从15米高空坠落会死吗？ | 体重约 150 斤的人从 15 米高处坠落不一定死。15 米对坠落来说并不很高，坠落过程和地面还可能有各种影响伤害的状况。
1996 年，一项研究考察了 1989 年在维也纳急救中心接受治疗的 20 到 50 岁高空坠落伤者和维也纳法医研究所解剖的 20 到 50岁高空坠落死者，发现人从 19.2 米高处坠落的死亡率约 77.8%，从 15.6 米高处坠落的死亡率约 56.2%，从 12 米高处坠落的死亡率约44.4%，从超过 19.2 米高处坠落的死亡率在样本范围内是百分之百。
美国建筑研究中心 2018 年发表的数据显示，建筑工人从 4 楼（约 48 英尺，约 14.63 米）无保护坠落的生还率约 50%，从 7 楼（约 84英尺，约 25.6 米）无保护坠落的生还率约 10%，主要死因是脾·肝·肺等内脏严重受伤、胸部挫伤、肋骨骨折、头部受伤。美国建筑工人的体重普遍高于 150斤，你可以预期他们摔得比题设条件重。
英国的一项研究考察了 117 名从 2008 年 1 月到 2009 年 7 月在伦敦坠落并被医疗直升机紧急医疗服务救治或收尸的 16岁以上的人，发现头部和胸部是否受伤对死亡率的影响甚大：
- 从 10.5 米高处坠下并造成头部或胸部受伤时的生还率约 50%，从 22.4 米高处坠下、头部和胸部未受伤时的生还率约 50%；
- 样本平均年龄为 37 岁（范围是 16 到 85 岁），34 人死亡（约 29%）；
- 平均坠落高度为 9.9 米，摔死的 34 人的平均坠落高度为 16.7 米。
除高度外，影响人摔死概率的因素包括而不限于：
- 接触面——摔在农田土壤或雪等松软物体上的死亡率比摔在混凝土上低。掉到钢筋结构上比掉到平坦表面上危险得多。
- 气流与遮挡物——上升气流、树枝、各种张线、松散架起的顶棚等可以为坠落的人减速，但也可能造成额外的伤害。
- 年龄——65 岁或更老的工人更有可能在坠落后严重受伤或死亡；未满 15 岁的人的坠落死亡率第二高，这反映了他们经常从高处有意无意地掉下，在同样高度掉下时他们的死亡率低于成年人；15 到 24 岁的人的坠落死亡率第三高。
- 性别——一些证据显示男性比女性更有可能死于坠落。这可能反映了男性在高空作业工作中占比明显高于女性。
- 运气——一些人从极高处坠落而幸运地生还。
> 1972 年 1 月 26 日，22 岁的维斯娜在南斯拉夫航空公司一架 DC-9> 型客机上执勤，客机在捷克山区上空突然爆炸坠毁，只有维斯娜生还。维斯娜当时离爆炸中心较远，连人带椅被爆炸气浪掀出飞机外，从约 10058> 米的高空摔向地面，落在了一片雪堆中。维斯娜的头盖骨被撞裂成两半，三根脊椎骨粉碎，手腿也全部骨折，失去了知觉，但奇迹般地没有死亡，当救援人员发现她时，她仍然穿着空姐制服，不省人事地坐在航空座椅里。维斯娜在医院中昏迷了整整> 3 周，她的腰部以下也发生了暂时性瘫痪。后来她重新站起来，活到 66 岁。 | {
"major": [
"问答"
],
"minor": [
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]
} | [
"通用"
] | qid:608007182, aid:3102040142 | human | true | 暂无版权及作者信息 | [
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|
遗传病是不能治愈的吧！人类在疾病面前太渺小? | 不一定。个人在某些疾病面前可能显得渺小，但人类整体几乎不可能被任何疾病“战胜”。许多遗传病只影响千分之一或更少的人口。
截止 2023 年，人群中已知的遗传病有 6000 多种，其中 600多种可以治疗。一部分治疗方案可以让特定遗传病患者的预期寿命、预期健康寿命、预期后代数量与常人无异。
你可以在 [Treatments for geneticdisorders](https://www.rx-genes.com/) 等由医生和科研人员维护的网站查找已知范围内可治疗的遗传病及其治疗方案。
饮食管理可控制近百种对特定分子代谢不良的遗传病，患者可在避免食用这类分子的同时保持与常人相仿的健康水平。
酶替代疗法可控制数十种缺失特定酶的遗传病。
营养补充剂可控制上百种对特定营养物质吸收率不足或缺失特定代谢产物的遗传病。
免疫球蛋白可控制数十种遗传性免疫异常。
药物、疫苗、血液制品可控制数百种特定遗传病。
手术可通过“为一百多种遗传性造血干细胞缺陷患者移植正常造血干细胞、为一些遗传性骨硬化症患者移植正常骨髓、为一些缺失特定酶的遗传病患者移植正常骨髓等组织、为遗传性特定脏器异常的患者移植正常器官、切掉家族性结肠息肉病患者的结肠、摘掉高风险乳腺癌相关基因携带者的乳腺、为遗传性耳聋患者植入人工耳蜗、为布鲁格达氏综合征患者植入除颤器”之类手段来对付基因层面的遗传病。
人们已经知道数千个与遗传病有不同程度关联的基因，可以针对其中一部分进行基因治疗。
- “消除患者的突变基因”不是治愈患者的遗传病的唯一标准。基因治疗将患者的一部分细胞的基因型变正常到一定地步或引入其他有用基因，就可能让患者的症状大幅减退乃至性状与常人无异。
- 针对 β-地中海贫血、脊髓性肌萎缩症、腺苷脱氨酶缺乏症、RPE65 相关的莱伯先天性黑蒙等的基因治疗正在进行人体临床试验。
一些人将发育性先天缺陷也当成了“遗传病”，它们经常可以被手术充分修复。例如手术可以矫正腭裂、修复一些先天性心脏缺陷。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:609625492, aid:3102802321 | human | true | 暂无版权及作者信息 | [
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|
把受精胚胎植入无子宫的小白鼠体内，为什么会在发育一半的时候停止生长? | 未必。
题目里的“受精胚胎”未说清楚是受精卵还是胚胎，可能来自糟糕的媒体或转述错误。
题目里的“发育一半”所指不明。小白鼠的正常孕期为 18 到 20 天。如果题目的意思是这胚胎能发育到 9 至 10天的水平，那么你将这胚胎手术取出、以现代技术支持，就有一定概率能将其培育成存活个体了。这主要受你的医疗技术影响。
- 截止 2023 年 3 月，人类早产存活领域的吉尼斯世界纪录保持者是 Curtis Zy-Keith Means，他于 2020 年 7 月 5 日出生在美国阿拉巴马，胎龄 148 天，早产 132 天而存活。
- 目前，小鼠的极早产护理缺少成熟的医疗经验，可预期死亡率更高。
胎盘从周围组织取得营养的效果受具体的组织特性影响，并非什么组织都合适，但确实不限于子宫内膜。
无子宫的母体可能不擅长分泌妊娠过程所需的孕酮等分子，可以人工添加或将动物连接在正常怀孕的雌性个体身上来获取相关分子——对于无免疫排斥的特定实验鼠，可以手术连接循环系统；对于有免疫排斥的动物，可以由透析机间接连接循环系统。
不限于小白鼠，人体内特定部位的宫外孕允许足月妊娠、产生活后代。
- 近年来，宫外孕占人所有妊娠的 1% 到 2%；
- 人 95% 的宫外孕发生在输卵管；
- 腹腔妊娠占宫外孕的 1% 左右，21.4% 至 40% 的腹腔妊娠胎儿畸形、取出后能存活一周的不到一半，腹腔妊娠导致的孕妇死亡率约 0.5% 至 8%；
- 2003 年，约旦研究人员报告一名妇女通过剖腹产取出腹腔妊娠的正常后代；
- 2008 年，一名加纳妇女通过剖腹产取出腹腔妊娠的正常后代，其胎盘广泛粘附于母体的大肠、大网膜、子宫左角。
- 2009 年，一名印度妇女通过剖腹产取出腹腔妊娠的正常后代。产妇患有单角子宫并发生子宫破裂，部分胎盘附着在破裂的子宫角上，大部分胎盘附着在腹膜上。这比没有子宫更危险。
- 2011 年，沙特阿拉伯研究人员报告一名妇女通过剖腹产取出腹腔妊娠的正常后代，胎盘从附着的左侧阔韧带上剥离时引起产妇大出血并需要输血。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:609902920, aid:3103182372 | human | true | 暂无版权及作者信息 | [
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|
地衣裂开石头需要多久？ | 地衣附着在石头上后 1 到 4 年可测到少量剥蚀，效果是维管植物的十分之一或更弱，与岩石类型、地衣类型、当地气候和天气事件等因素有关。
- McCarroll 和 Viles 于 1995 年在挪威南部测得地衣引起的岩石剥蚀速率至少约 0.0012 毫米每年。这是当地气候本身引起的风化速率的 25 到 50 倍。相关研究可能低估了某些地衣造成的影响。
- 夏威夷火山爆发形成的玄武岩被地衣覆盖后 50 年间剥蚀了约 0.5 毫米。这不完全是地衣造成的。
- 地衣影响下的砂岩每 100 年剥蚀约 3 毫米。这也不完全是地衣造成的。
至少一部分地衣的菌丝可渗透到岩石中的显微缝隙，在空气湿度和降水·结冰影响下反复胀缩、释放盐和有机酸，引起机械破坏并加快化学风化。实地考察显示一些地方的地衣渗入岩石的深度约1.12 到 3.21 毫米。这些渗透至少要花数百年来真正“裂开”少量岩石。
在潮湿地区，蜗牛取食石头上的苔藓时造成的剥蚀比地衣更快。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:611032138, aid:3114321689 | human | true | 暂无版权及作者信息 | [
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|
人体能感觉到的温差最小是多少？ | 实验显示，人拇指根部的无毛皮肤可以感知两个冷却脉冲间 0.02 到 0.07 摄氏度的温差、两个加热脉冲间 0.03 到 0.09 摄氏度的温差。
检测皮肤温度变化的灵敏度比这要低、受本身温度和温度变化速率影响。
- 人拇指根部的无毛皮肤在本身温度 33 摄氏度时可检测到的最小升温幅度约 0.2 摄氏度、可检测到的最小降温幅度约 0.11 摄氏度。
- 皮肤温度每秒变化 0.1 摄氏度以上时可以迅速产生感觉，每分钟变化不到 0.5 摄氏度时 8 到 10 分钟都未必被感觉到。
以上数据引自 Lynette Jones (2009), Scholarpedia, 4(5):7955.
人手掌皮肤在不同的实验步骤下可感知两个冷却脉冲间 0.02 到 0.06 摄氏度的温差，敏感度在手掌皮肤温度 29 到 39 摄氏度时变化不大。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:611196947, aid:3111785055 | human | true | 暂无版权及作者信息 | [
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|
长得好看到底是怎么成为人类择偶标准的？相貌美丑在物种角度有什么作用? | 外观可以反映一些遗传要素、发育情况、当前健康状况、生活水平，在人类诞生前就影响着动物的交配策略和种内关系，参与了整个人类演化史。人的文化对此推波助澜。
不过，对长期共同生活的社会化一夫一妻制物种来说，交配策略不等于“择偶标准”，人的婚姻在历史上大部分时间里首先是经济关系，你可以看看现代的离婚诉讼在谈些什么。
- “长得好看”是一些人“理想中的择偶标准”的组成部分，但经常被他们自行放弃。现代社会只有约 7% 到 8% 的人被所处的社会公认为“美丽”，大部分结婚的人找了平均水平或低于平均水平的配偶。
- 人类女性对男性外表的关注度平均而言低于人类男性对女性外表的关注度。这可以解释为女性被自然选择塑造为会关注长期一起生活的安全性、为生育后代提供资源的能力，也可以解释为文化塑造，或者二者兼而有之。
- 许多动物会优待它们觉得“长得好看”的个体，对此类个体的生存、繁衍和此类性状的遗传有利；同时，它们还可能攻击它们觉得“长得畸形”的个体。人也如此。
- 人类女性的丰乳肥臀、嘴唇饱满、高颧骨、大眼睛、下巴纤细、腰部较细可以反映较高的雌激素水平。人类男性的宽肩膀、宽下巴、大胡子等可以反映较高的雄激素水平。“长得好看”，尤其是脸好看，是生存力、生育力的招牌。不过，男性脸上的较大疤痕在不特别可怕的情况下可以显示此人从伤害中恢复的能力，经常被女性评为“有男子气概”，提高他们短期寻求交配对象的成功率。对于长期共同生活，女性通常认为没疤的脸显示此人更温和、相处的安全性略高。
人对相貌“美丑”的定义和态度受人的生物学特性、人的生活经历和文化影响。
- 自古以来，人们普遍认为外观美丽或“有魅力”的人比平均水平更擅长交际、更热情、更有统治力、更聪明、心理更健康。通常，外观美丽的人会在周围的人追捧下潜移默化地变得比平均水平更自信，年收入也平均而言更高——不过，他们也比平均水平更容易拖欠贷款。
- 外观丑陋的人（尤其是体力也不强的人）经常受到恶劣对待，影响他们的性格，更促进众人形成刻板印象。他们的年收入平均而言比同样工作能力的其他人少 4% 到 13%。
- 在食物较为短缺的地区，人们可能觉得更胖的身材较美；在食物过剩、到处是胖子的地方则相反。
- 实验显示，比较对称的脸、接近自己见过的人脸的平均状态的“平均脸”对人更有吸引力。前者可以展示遗传和发育的状况、跟智力也有关，后者的数据库是在生活过程中构建的、可以不断变化。
- 当你和某人建立了情感联系，你的脑会觉得此人的外貌比过去更具吸引力、更好看。这不要求双向联系。
- 对近 300 家尼德兰广告公司进行的研究显示，高管长得更好看的公司的年收入更多。
无论如何，人经常高估自己的外貌，2017 年全球范围内 60% 的女性受访者觉得自己很漂亮。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:556125322, aid:2696160834 | human | true | 暂无版权及作者信息 | [
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|
灵长类动物通过培训为人类服务从事简单的生产制造? | 可以，现实中早就有人训练猴子采摘椰子，19 世纪南非有 1只知名的豚尾狒狒从事雇佣劳动，诸如此类。不过，对于复杂到一定地步的工作，雇人来做可能比训练动物还要便宜。
2020 到 2022年，一些欧美动物保护组织和媒体猛烈攻击泰国椰子产业虐待动物、让猴子从事“奴隶劳动”，一些西方企业停止使用来自泰国的椰子。反对这类攻击的人争辩说，泰国人为猴子提供食物、饮用水、按摩、梳理、清洗、清理爬到身上的昆虫，马来西亚、印度、斯里兰卡等地的一些农民也训练猴子采摘椰子，而且古埃及人就曾训练狒狒采摘水果和坚果，这和用牛耕田没有什么区别。
1880 年代，南非开普敦-伊丽莎白港铁路信号员 James Wide在受伤并双腿截肢后教他的宠物狒狒“杰克”操控机器。其他人很快注意到一头狒狒在控制铁路信号，铁路公司随后宣布雇佣“杰克”，每天给它 20美分工资和食物，每周给它半瓶啤酒。9 年间，据说“杰克”从未发生工作错误，火车乘客经常给“杰克”食物。1890 年，“杰克”死于肺结核。
更多的人声称自己的宠物猴、宠物狒狒可以胜任大量的工作，可以教自己的孩子做一些事情、照顾孩子和狗。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:536657383, aid:3048235418 | human | true | 暂无版权及作者信息 | [
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|
像美国印度这种阶级分明的国家抗疫如此糟糕，那阶级死化的集群生物，蜜蜂蚂蚁是怎么解决传染病问题的呢？ | 蜜蜂、蚂蚁的不同物种有不同的社会形态，阶级并不一定固化，一部分物种没有阶级分化。这不怎么重要。
在发生流行病的时候，A. custodiens进行隔离，让感染疾病的蚂蚁不再进入繁殖阶级和幼虫·蛹的居住空间，并通常让它们居住在靠近出口（可以通风）的巢室中；进行更频繁、更密集的互动来对密切接触感染者的个体进行清洁和消毒，及时清理废物和尸体。这些是社会性昆虫的典型防疫措施。对疑似感染者进行清洁的个体有被传染的风险，但有更大概率在接触少量病原体后经由无症状感染或轻症感染而获得免疫。
一些蚂蚁的酸性分泌物可以杀死许多病原体，它们会将这类分泌物抹在身上。C. fulvopilosus会保持社交距离并用蚁酸进行消毒，且一个个体使用蚁酸消毒会引起周围数个个体进行同样的行为。
一些蚂蚁会利用树脂来杀菌，或是从培养的一些真菌·放线菌里获取抗生素来杀菌。Formica paralugubris收集的针叶树树脂的杀菌功能已经被实验验证。
大部分社会性昆虫群体里的患病个体会减少与其它个体的接触。当然，一些疾病会阻止个体自由行动，还有一些疾病在表现出症状之前就可以通过口对口交换食物的行为传播，但能够影响的个体数量终究是很有限的。而且，获得免疫力的蚂蚁会增加口对口交换食物的行为，该行为可以引起参与交换的个体获得免疫力。
社会性昆虫群体之间的低互动性使传染病难以长距离蔓延，与原始时代的人类部落相似。
少数社会性昆虫会吃掉容易感染疾病的幼虫，等大流行过去再生一批就是了。你可以预期某些国家的人不需要学就知道这一招。在一般国家，这一招的类似措施是让学校停课、孩子各自回家去——思路都是“不要将大量的幼体堆在一起给病原体来传染”。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:461441504, aid:1906544450 | human | true | 暂无版权及作者信息 | [
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恐龙几十吨重，孵蛋会把蛋压碎吗？ | 恐龙比你想象中要聪明。
根据已有化石证据，并不支持几十吨的恐龙拥有孵蛋行为，甚至大部分恐龙都不孵蛋，它们倾向于用泥沙或植物把卵埋起来，靠外部热源孵化；一些小型兽脚类恐龙（如偷蛋龙、伤齿龙）产卵后有着与现今鸟类相似的孵蛋行为，用羽毛覆盖或直接坐在蛋上，可能是孵蛋1.0版本，为现今鸟类多样化的繁殖行为打下基础。
## 怎么筑巢？
从1993年起，古生物学家在蒙古戈壁陆续找到正在孵蛋的恐龙化石标本（葬火龙Citipati，属于偷蛋龙科），到20世纪末至21世纪初，关于鸟类演化自恐龙的一系列证据被研究发表，大众在逐渐接受鸟类就是恐龙这一事实的同时，也经常对比两者的生物行为——比如题主所说的孵蛋。
讨论孵蛋行为，往往离不开筑巢行为，因为有孵蛋习惯的恐龙，基本不会选择用泥沙埋卵。
从开篇的总结可知，（非鸟类）恐龙大概有两种筑巢假说：
- 一是偏向龟鳄类的“埋藏式巢窝”，恐龙产卵后，会用泥沙或植物将其掩埋，借助太阳光、植物腐烂等外部热源来提供温度，从而完成孵化。
- 二是偏向现今鸟类的“露天式巢窝”，恐龙蛋大部分面积会暴露在地表，部分掩埋在沉积物里（可能起到固定、隐蔽性等作用），父母依靠带有羽毛的前肢或腹部来提供温度。
要知道（非鸟类）恐龙历史跨越三叠纪-侏罗纪-白垩纪，长达1.6亿多年，种类之多，所以想确定谁喜欢孵蛋以及怎么孵是十分困难的一件事。
首先想要找到合适的研究材料并不容易，巢穴化石、胚胎化石本就稀有，“连龙带窝一锅端”的化石更是难求；其次尽管恐龙蛋化石记录很丰富（搜一搜河源恐龙蛋新闻就知道了），但也仅是数量维度上，且基本是没法和恐龙种类一一对应上的，恐龙蛋有独立分类体系；最后早期的恐龙蛋是软壳蛋，不稳定，很难形成化石，这也是众多恐龙“无蛋”的原因……
## 谁会孵蛋？
受限于以上客观因素，古生物学家通过恐龙蛋化石来研究筑巢行为的论文里，有不同的声音就再正常不过了，比如有提出一些泰坦龙类（题主所说的几十吨恐龙）孵蛋的观点。
整体而言，主流观点还是倾向于大多数恐龙（如几十吨的蜥脚类、早期兽脚类、部分鸟臀类）更适合用“埋藏式巢窝”，因为它们的蛋壳都具有高孔隙度——蛋壳上有较大、较多的孔，这种结构有利于进行气体交换，恐龙蛋被沉积物掩埋后，胚胎仍能顺利发育；小部分兽脚类恐龙（如偷蛋龙、伤齿龙类）的蛋壳有较低孔隙度，气体扩散能力相对差，因此这更可能是一个适应在露天环境产卵的演化特征。
除了蛋壳孔隙度，恐龙蛋颜色也为研究孵蛋行为提供一定帮助。
现今存活的羊膜动物中，只有鸟类能产下各种颜色的蛋。在过去很长一段时间里，人类一直认为彩色蛋壳这一特征仅存在于鸟类，但2017年和2018年两篇论文提出了鸟类蛋壳颜色起源于其恐龙祖先，也就是身披羽翼的那一群小型兽脚类恐龙。
同时，研究结果更是显示蛋壳色素只存在于兽脚类，比如河源龙（偷蛋龙科）和恐爪龙（驰龙科）就会产下蓝绿色的恐龙蛋，蛋壳色素沉积方式与鸟类也相似。而兽脚类之外的种群，比如慈母龙和泰坦龙类的蛋壳没有发现色素痕迹，习惯于使用“埋藏式巢窝”的鳄鱼也不产彩色蛋。
如果从蛋壳色素分布情况来推断生物行为，那么拥有彩色蛋的兽脚类恐龙将采用“露天式巢窝”，趴在窝里，像现今鸟类一样去孵蛋，彩色的蛋壳能够更好地伪装以躲避真正的“偷蛋龙”，甚至是抵御巢寄生（假设它们那个时代也有杜鹃这种家伙）；而拥有无色蛋的鸟臀类、蜥脚类恐龙则倾向于选择把卵埋起来，采用龟鳄类一样的孵化策略。
## 怎么孵蛋？
基于蛋化石的外壳结构和色素，结合恐龙胚胎化石、成年个体化石，可以为了解这群恐龙的孵蛋行为进一步提供更有力的证据支持。
回到开篇提到的经典葬火龙孵蛋化石标本，现实里它保持着鸟类一般的孵蛋姿势：蛋一般排列成三层的同心圆，数量可多达约22颗，它伏在巢中央的大空隙，张开前肢，用羽毛环绕着一颗颗蛋。可这个标本实际上并不够完美，因为蛋壳内部的胚胎化石没有保存下来，你能够质疑它，并对此开展多种解读：这只恐龙在产卵时恰好遭遇沙尘暴寄了、还没来得及掩埋卵就暴毙了、面临危险之际呈现的一种护蛋行为……总之不是在孵蛋就对了。
更完美的证据是来自中国江西赣州的偷蛋龙化石，不仅能看到上述同样孵蛋姿势，更能看到处于不同发育阶段的胚胎，蛋壳孵化温度也不同，这意味着偷蛋龙幼崽会在不同时间破壳，跟现生鸟类的“异步孵化”繁殖特点如出一辙。
这家伙是真的在孵蛋，而且还挺有一套。
当然，也有恐龙会直接坐在蛋上孵化，并且产卵方式也比较奇特。
比如伤齿龙，它产卵时，并不会像偷蛋龙一样把蛋铺一层又一层，而是把蛋立起来，直接插到土里，聚集在中央。孵化时，用腹部直接接触蛋壳，真正用体温来孵蛋。至于压不碎的力学原因，没查过，感兴趣可以自行去搜索相关论文。
看起来，目前恐龙筑巢和孵蛋行为的进化路线就大概是：
- 埋蛋，靠外部热源孵化（源于最早的恐龙及其祖先）
- 露天筑巢，开始亲自上阵孵蛋（源于兽脚类，并在接近鸟类的恐龙中演化出各种1.0版本）
- 进一步完善迭代（已灭绝的鸟类）
- 筑巢选址、孵蛋技巧百花齐放（现代鸟类）
总之，恐龙比想象中聪明，筑巢行为也比想象复杂得多，以上仅为管中窥豹。 | {
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|
有没有人类保护环境但适得其反的例子? | 有很多，例如：
澳大利亚和美国近年来的猛烈野火，部分归因于“一些植被长得太过连续、不利于消防，但环保政策和某些环保人士妨碍了用于控制该状况的小规模砍伐与可控燃烧”。过度受限的小规模砍伐与可控燃烧不但起不到控制野火蔓延的作用，在美国部分地区还导致地表火焰扩散加快，增加人员伤亡的风险。
发展可再生能源对降低人为温室气体排放量很重要，但是人们现有的太阳能电站与风电机大量杀死野生鸟类。许多鸟是自己撞死在太阳能面板或巨型扇叶上的，可能是将反光的面板当成了水面、将白色的扇叶当成更柔软的东西来互动的结果。实验显示，将风电机涂成黑色可以降低鸟的自杀率。你可以看看世界各地有哪个大型风电场将风电机都涂黑了。废旧太阳能电池的无害化处理也还是个问题，废旧风电机的扇叶目前已经有一些“加工成其它产品或打碎并在发电厂里完全燃烧”的方案。
美国西海岸一些地方紫海胆泛滥，造成海藻林退化、大量动物死亡，但当地环保政策禁止大量捕捞或杀死紫海胆。一批科学家游说多年，才得到在数平方千米的狭小海域内无限制消灭紫海胆的许可，这是杯水车薪。
澳大利亚于 1893 年设立赏金来鼓励人们猎杀入侵物种赤狐（该物种被人于 1863 年作为休闲狩猎的射击对象引入澳大利亚后野化），2002年，他们评估了这项政策的效果，发现其在很大程度上没用甚至适得其反——人们希望赤狐继续存在以便持续获取赏金，猎杀强度从未达到消灭赤狐所需。
截止 1995 年，澳大利亚在 80 年间为控制入侵植物马缨丹而接连引入了 28种攻击马缨丹的昆虫，但马缨丹仍在散播，而这些昆虫频繁攻击目标以外的植物。其中最好笑的是第 28 种昆虫——澳大利亚人测试了角蝉科的 Aconophoracompressa 对 62 种常见植物的攻击倾向，认为它很安全，将它释放到环境后才发现它很喜欢并未测试过的观赏植物 Citharexylumspinosum，而这种树在澳大利亚居住区很常见。这动摇了人们对生物防治的信心。
新西兰于 1880年代引入白鼬来捕杀泛滥的兔子，但白鼬很快大规模击杀当地鸟类，新西兰人不得不着手控制白鼬。他们使用了投毒、设置陷阱等多种方式，预计新西兰的白鼬会在本世纪中叶灭绝。
意大利近年来闹野猪，部分归因于“环保政策导致适合野猪增加数量的植被持续增加，动物保护政策和环保人士的抗议让控制野猪数量的行动寸步难行”。
我们并不擅长保护环境。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:495597377, aid:2204321621 | human | true | 暂无版权及作者信息 | [
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如果人类是由鱼进化而来的，为什么我们反而不能在水里呼吸呢？人类进化经历了一段漫长的岁月，从这些进化当中可以得知，人类是在往一个有优势，且更加智能的方向发展，但是为什么却摒弃了一些对于现在的人来说同样是对我们有利的优势呢？ | 没那么简单。
人的肺不能处理大量的水，但也不能没有水。包括人在内，脊椎动物的肺泡从表层的水基液体（肺泡液）里获取溶解氧，而不是直接暴露在空气中。
- 成年人的肺泡液约 36 毫升，水分随呼吸流失，补充肺泡液所需的水通常是人每天生存所需水量的约四分之一。
- 在干燥空气中，没有肺泡液保护的肺上皮细胞会在数分钟内因水分流失而受损、死亡。
人的皮肤可以通过扩散作用从水中取得少量溶解氧。这不足以维持整个人体的生命。
- 顺便说，1851 年就知道人的皮肤可以通过扩散作用从空气取得少量氧气；2001 年，实验显示，在皮肤完整、潮湿的条件下，人体表层到 0.25~0.4 毫米深处的细胞可依靠从空气获取的氧生存 。
作为人类祖先的鱼是数亿年前的一些肉鳍鱼，它们起初用鳃呼吸，后来演化至用肺呼吸。
从鳃到肺的转变并不罕见：
- 常见的青蛙、蟾蜍从蝌蚪变态发育而来（一些罕见的小体型青蛙不经过蝌蚪阶段）。蝌蚪的鳃能在水里呼吸。变态发育后，青蛙、蟾蜍的肺不能处理过多的水。
- 青蛙、蟾蜍通常可以在水中靠皮肤吸收溶解氧来维持生命 4 到 7 小时，在水下剧烈运动会缩短它们的潜水时限。
- 不少蝌蚪同时具有鳃和肺。在生命早期，蝌蚪没有足够的力量突破水的表面张力将头伸出水面，依靠鳃和皮肤呼吸溶解氧，但有肺的蝌蚪在体长 3 毫米时就能通过制造气泡呼吸空气。长大一些后，许多蝌蚪用肺获得更多氧气，而鳃和皮肤的呼吸能力不足以长时间维持长大的身体。你可以在这里看到某些祖先的影子。
- 蝌蚪的肺还有助于控制浮力。
用于处理溶解氧的鳃在空气里会很快干燥而塌陷，大幅减少可用于气体交换的表面积。攀鲈、弹涂鱼等辐鳍鱼现在进行时地具有在陆地上活动数小时的能力，攀鲈以鳃上呼吸器官处理空气，弹涂鱼以皮肤、口腔、喉咙处理空气，这些结构的正常功能都需要保持湿润。脊椎动物肺的内陷构造有助于防止结构变得干燥，这降低了排出过多液体的能力。
演化过程中，生物的可遗传性状经常减弱或缺失。无论减弱或缺失了什么，只要可育后代的数量没有明显减少，新性状都有概率作为中性性状在种群中长期存续。对一部分在浅水活动的古代肉鳍鱼来说，“更擅长呼吸空气的性状”在结果上支持更快的代谢、有利于捕食·自卫，可能有利于求偶，并在一定程度上允许“更不擅长处理大量水的性状”不影响可育后代的数量。一些古代肉鳍鱼在登陆前就演化至“成体用肺呼吸空气、在浅水·潮间带等处捕食”，这之中的一些种群日后发展出包括人在内的四足形类。
数亿年来，体型较大的四足形类的耗氧量不能在不大幅修改身体的情况下用鳃维持。从陆地返回海洋的鲸豚类、鳍足类等并未切换回靠鳃呼吸。
- 问题出在水中溶解氧的含量明显低于空气中的氧气。
- 溶解着大量氧气的全氟化碳可维持老鼠之类实验动物呼吸，尽管呼吸远比空气密度大的全氟化碳要消耗相对多的体力。
题目里“人类是在往一个有优势，且更加智能的方向发展”的说法源自过时的定向进化论。在此基础上，由于我国媒体灾难性的表达能力和近年来伸腿瞪眼的语文教育，许多人将“人类是由鱼进化而来的”这句话当成了“人是从某种在水里游的鱼直接发展而来，跟‘古猿’、‘猩猩’、‘猴子’之类无关”，造成更多误会。
过去数十年间，我国生物学义务教育不准确地描述了“一部分鱼演化成两栖类、一部分两栖类演化成爬行类、一部分爬行类演化成哺乳类”。常理而言，用不着“人类是由鱼进化而来”这个情报，提问者也会注意到一部分脊椎动物的呼吸方式在演化过程中发生了变化。大概提问者以为“哇，原来人不是从古猿进化而来的，人应该跟水里游的那些兄弟更像才对”，暂时忘记了青蛙什么的。 | {
"major": [
"问答"
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"minor": [
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} | [
"通用"
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|
在太空中人类的昼夜节律是否会被改变？为什么？ | 目前会。近地轨道上的光照规律、航天器内的照明状况、微重力、电离辐射、工作时间安排会影响人类航天员的昼夜节律。睡眠不足、睡眠剥夺、疲劳是航天员最常见的主观抱怨。
在天空实验室、和平号空间站、美国航天飞机、国际空间站上记录到航天员每 24 小时的平均睡眠时间约 6 到 6.5小时，他们有时会发生入睡困难。美国航天员在国际空间站的平均安眠药消耗量是地球上普通人群的 10 到 20 倍。
睡眠不足会影响航天员的情绪、认知能力、工作能力。
- 航天员经常两班倒工作。
- 航天器上的部件经常发生故障、让航天员面临计划外的高强度操作需求。
- 航天飞机、联盟号等航天器飞往国际空间站的发射窗口相当窄，参与对接任务的航天员需要在指定发射时间前至少 5 小时醒来，这会影响他们的工作时间安排。
- 细胞培养实验显示，人细胞在微重力下分泌的褪黑素较少，而且一部分褪黑素会被用于应对辐射损伤。这会影响昼夜节律和骨代谢，与微重力的其他效应一起加速骨质流失。
人工调光可以帮助航天员适应。可以在航天器上用人工光照与人工遮光营造接近 24小时的昼夜循环，在保证散热支持的条件下让航天员适当运动锻炼来减慢骨质流失。现有航天器尚未尝试以足够快的自旋来提供拟似重力。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:593684695, aid:2969227587 | human | true | 暂无版权及作者信息 | [
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|
像桃花源这样的封闭聚落最终都会因为近亲繁殖的问题而自然消亡吗？
条件
完全封闭与外界无接触
自然资源充足
人数的话假设一开始有1000村民好了，男女平均 | 不会，题设的初始村民人数 1000 远远超过处理“近亲繁殖”问题所需。
- 现实中就有不到 1000 人的岛屿部落、偏远村落长期存续。而且，古人类的小型迁徙群体约 160 人。当前人类的基因多样性本就很低，看起来这个水平已经够用了。
- 胡特尔派在十八到十九世纪几乎被来自外部的暴力攻击灭绝，后来以高度近亲繁殖的方式在 125 年间从约 400 人增加到约 42000 人，现在有数百个居民点。他们的每个居民点包含 10 到 20 个家庭，居民点人口约 60 至 250，在居民点的人数超过 250 时，领导人会找个口实、以抽签方式选出半数成员去开拓新的居民点。你可以参考他们的历史，想想 1000 是不是太保守了。
致力于保存濒危物种的科学家考察了多种动物维持生存所需的最小数量，1980 年代，澳大利亚的 Ian Franklin 与美国的 Michael Soulé针对哺乳类和鸟提出 50/500 原则：有 50 只繁殖个体，可以避免近交衰退；有 500只繁殖个体，可以避免漂变造成的不适应。由于要保护的动物的具体情况参差不齐，他们给出的“比起正常的人口规模，更接近 0”的数字经常被同行吐槽，后来干脆乘以10 来搞定：500 只繁殖个体可以忽略近交衰退，5000 只繁殖个体可以自行适应环境。
此外，近亲繁殖本身没有一些人估计得那么危险。普通人的基因组通常有 1 到 2 个纯合时可导致致死疾病的隐性基因。从 1933 年到 1970年在捷克斯洛伐克出生的儿童显示，父母是彼此的一级亲属的儿童有近 40% 患有严重残疾，14%早死。在“躲避战乱、逃到桃花源”之类情景里，逃难前的人群死亡率可能不止如此。对哺乳类家畜育种的经验是，近亲繁殖会快速淘汰若干严重的致病基因，在数代后明显降低遗传负荷，在维持20 代（对古人来说约 320 到 400 年）以上后基本稳定、不再有近交衰退的危险。
自然灾害、传染病、偶然发生的出生率波动等因素对封闭聚落的影响相对大一些。 | {
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"问答"
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"minor": [
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} | [
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] | qid:594009656, aid:2971457729 | human | true | 暂无版权及作者信息 | [
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|
分子遗传学推翻了哪些传统观点？ | 传统认为细菌的基因组结构比别的细胞生物都简单是因为原始，但是通过对核糖体RNA等一系列超保守细胞器组分遗传分子钟的构建，遗传关系显示一部分细菌的基因组结构比古细菌和真核生物简单可能是特化的结果。
常见细菌所属于的一些细菌域后期分支，其基因翻译率高、遗传冗余低、基因调节结构简单的特点，并不是原始特征，反而可能是特化的结果。
细菌域一些早期分化的分支的遗传结构的特征更接近古细菌和真核生物的领域，有比较多的无法翻译的冗余遗传物质和基因表达层面的调节结构。
真核生物域的基因翻译率低、遗传冗余高、基因调节结构复杂，反而是保留了更多LUCA的原始特征。
仔细想想这个结果倒是完全符合RNA世界假说的预测，从RNA世界过渡到DNA+蛋白质世界，越原始的特征应该保留越多的核酶、核酸相互作用和冗余基因，基因可翻译为蛋白质的比例更低。
实际上这些细菌在进化史上特化的过程中丢失了很多遗传结构和细胞结构。这种基因翻译率高、遗传冗余低、基因调节结构简单的特点，也导致这些细菌在面对病毒和环境变化时只能大量死亡，靠数量和积累变异应对自然选择压力，在形成针对病毒的特异性免疫与其他细胞互动的方面都采取了与古细菌真核细胞及更早期细菌分支不同的策略，遗传修复机制较少，而是筛选出一些专门作用于特定病毒基因组的具有更高特异性的剪切酶和直接整合噬菌体的结构，也为发展CRISPR/Cas9基因编辑技术提供了天然的原材料。
有趣的是，细胞特化的过程往往也都伴随遗传物质和细胞结构的丢失，在多细胞真核生物中，通过测序生物体内不同来源的细胞，体细胞分化过程经常伴随着随机的遗传物质丢失，只有体内生殖细胞和多能干细胞能保持遗传完整。并且研究已经发现这些随机的遗传物质丢失，竟然和细胞相互识别、免疫识别、组织结构和神经突触的维持都有关系。但同时细胞这种遗传物质随机丢失，也导致已分化的细胞一但被调动，发生去分化并脱离组织环境重新被赋予多能性，这个过程很容易导致癌变，病毒感染、慢性炎症、纤维化等过程由于频繁调用已分化体细胞重新转变为非干细胞的非附着化间充质多能细胞，成为多细胞生物肿瘤发生和肿瘤转移的重要机制。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:593595482, aid:2969060391 | human | true | 暂无版权及作者信息 | [
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|
你知道蚊子有多少种吗？ | 知道。
截止 2023 年 3 月 9 日，“蚊子”是蚊科 3618 个已知有效物种的通称。
在此基础上，摇蚊科的超过 5000 个物种也经常被人误认为是蚊子。
一些科学家会在网络上更新蚊科的分类情况，例如：
[Mosquito Taxonomic Inventory](https://mosquito-taxonomic-inventory.myspecies.info/valid-species-list)
算上未知物种，地球上当前可能有超过 1 万个物种会被目击者称为“蚊子”。
截止 2023 年 3 月初，不同的科学家将蚊科分成 43 个到 113 个属。
截止 2023 年初，已经观测到约 90 种蚊子能传播疾病，它们主要来自伊蚊属、按蚊属、库蚊属。
截止 2023 年初，已经记录了至少 28 种蚊子化石，每个都可能代表一个历史上存在过的属级分类。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:591320006, aid:2949995186 | human | true | 暂无版权及作者信息 | [
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|
生孩子那么痛苦，为什么人类在过去几百万年中都没有因为不愿生孩子而灭绝呢？ | 首先，题目的前提“生孩子那么痛苦”的可信度是不稳定的，有时是可疑的。
- 对双胞胎的研究显示，人类女性的骨盆宽度受生活环境的影响似乎超过基因造成的影响。农业时代的饮食变化可能比骨盆宽度的遗传因素造成了更多的产科问题。
- 在过去几百万年的大部分时间里，人的生育没有现在这么痛苦；现代人生育时的痛苦程度也因人而异、在一生中会多次改变，通常二胎或更多胎的痛苦程度低于头胎。
在此基础上，现代的一些宣传是极度歪曲的，夸大了分娩的痛苦和死亡风险。你可以在这个问题下的某些回答里看到千奇百怪的夸张模因。
- 美国麻醉医师协会的全国性调查显示，46% 的新母亲表示，生第一个孩子时经历的痛苦低于她们的预期。询问“分娩过程中最痛的时候有多痛”时，45% 的回答是“像极重的痛经”，16% 的回答是“像严重的背痛”，15% 的回答是“像骨折”。无论使用麻醉还是辅助性的疼痛缓解手段，90% 的参与者表示疼痛缓解有效。
其次，在过去大部分时间里，人们没有足够的认知能力去理解孩子是怎么搞出来的，没有足够的认知能力与技术去进行有效避孕或堕胎。你完全无法指望分娩时的痛苦去阻止历史上的大部分人进行交配、怀孕、分娩。
Sherwood Lerned Washburn 于 1960年提出“产科困境”假说，试图将人类女性那看起来异常困难的分娩说成“演化产生的解决方案”。古尔德过去附和过“产科困境”，将其推得很广。这是个看起来很好懂的模因，被各路媒体和他们的受众到处传播。但这不是事实。提出“产科困境”假说时使用的统计数据可能不反映历史上大部分时候的状况。
- 类人猿的足月妊娠所需时间是灵长类里最长的，黑猩猩、倭黑猩猩的正常妊娠期约 39 周，人的正常妊娠期约 37 到 41 周。黑猩猩、倭黑猩猩的新生儿像人新生儿一样缺乏独立生存能力，需要母亲提供大量照顾。但这两种类人猿并没有人这么高的难产率，它们的产道比新生儿尺寸明显更宽。
- 正常人类新生儿的大脑体积是成年时的 30% 左右，正常黑猩猩新生儿的大脑体积是成年时的 40% 左右，卷尾猴新生儿的大脑体积是成年时的 50% 左右。黑猩猩新生儿那 40% 并没有改变其缺乏独立生存能力的表现。所谓“人类新生儿都是早产儿”的说法是错误的。
- 人大脑生长速度的峰值出现在出生前的几个月间，与生下来就能行走的动物类似（Halley, 2017 年）。
- 将人类女性的骨盆再加宽几厘米以扩大产道，不会和“产科困境”假说设想的那样妨碍双足行走。非洲南部石器时代晚期的人群似乎搞出过身高很低而骨盆很宽的女性，当时他们经常长距离迁徙，化石并未显示这些人在行走方面有什么困难。生物力学也不支持“加宽骨盆会增加运动成本”的想法。
- 农业时代以来，生活方式的改变让人分娩的难度上升。约 12000 年的农业社会生活尚未让人的骨盆宽度重新优化到普遍适合新生儿的大头与宽肩膀（人类新生儿的颅骨可动性减少了大头的影响，肩膀有时比头更不好处理），看起来这造成的影响在种群层面并不重要。当前状态可以算是略有不利的中性性状，能在种群中长期传递。
- 人类胎儿的能量需求和母体能安全地提供的代谢率与散热能力的关系似乎更直接地决定了胎儿能达到的体重。这方面的统计相关性也出现在没有骨头限制产道的鲸类身上（Sacher 和 Staffeldt，1974 年）。 | {
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|
都说全球变暖导致一些珊瑚礁的死亡消失，那其他地方有产生新的珊瑚礁吗？有的话在哪里？ | 有。珊瑚正在死去的珊瑚留下的位置及其周围生长，一些热带珊瑚的分布区域朝两极移动，另一些相对耐热的珊瑚留在原地。地质史上全球大规模暖化时留下的珊瑚化石也有类似的迹象。
- 二十世纪三十年代以来，科学家在北半球一些地区研究的 9 种珊瑚中有 4 种以每年 14 千米的速度向北扩张。从 1930 年到 2011 年，日本温带领海的冬季表层水温上升了约 2.4 摄氏度，允许更多珊瑚生存在比历史上更靠北的海域并造礁。
- 从 1974 年到 2012 年，赤道海域的珊瑚密度有所下降，亚热带海域的珊瑚密度有所增加。在南北纬 20 度以上海域，珊瑚密度增加了 78% 以上。大量珊瑚出现在北纬 33 度、南纬 30 度附近，这里将会形成新的珊瑚礁。
2022 年，大堡礁受损区域内出现大范围的珊瑚复辟，大堡礁北部的硬珊瑚覆盖率达到约 36%，是自人类开始监测其覆盖率以来的最高水平，大幅超过 2017 年的13%；大堡礁中部的硬珊瑚覆盖率估计为 33%，同样是有记录以来的最高水平，大幅超过 2019 年的 14%；大堡礁南部的珊瑚覆盖率估计为 34%，比2021 年下降了 4%，主要原因是棘冠海星爆发。
珊瑚的共生微生物群可以在温度变化下发生变化并适应。
对珊瑚来说，形成“全新的珊瑚礁”通常要花 25 年，有些物种造礁速度更慢或不造礁。人为投放活珊瑚碎片可以加快一些地方的珊瑚生长。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:587349256, aid:2921063063 | human | true | 暂无版权及作者信息 | [
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|
生命科学的第一性原理是什么？
第一性原理的首创者是古希腊著名哲学家亚里士多德。他认为：“任何一个系统都有自己的第一性原理，是一个根基性命题或假设。它不能被缺省，也不能被违反。” | 一部分学者认为生命科学的第一性原理是 Erwin Bauer 于 1920 年发表的《生物科学的基本原理》中提出的稳定非平衡原理：
- 生命系统，且只有生命系统，永远不会处于平衡状态。借助自由能，它们持续地做功来避免根据物理和化学定律应当在给定的外部条件下达到的平衡。
Mikhail Volkenstein 于 2009 年给出了更简短的版本：
- 生命系统永远不会处于平衡状态，这以它们的自由能为代价。
在上述命题中，所谓“永远不会处于平衡状态”是靠“一旦某个生命系统不可逆转地达到了平衡状态，该系统就不再算是生命系统”来实现。这不仅在逻辑上相当难看，而且不可避免地遇到了台风之类耗散系统是不是生命系统的问题。
- 已知的生命系统似乎都是耗散系统，生命系统和其他耗散系统的边界要么不存在要么十分模糊。
薛定谔于 1943 年 2 月在都柏林三一学院的公开讲座谈论生命时用“负熵”表示“热力学自由能”，思路跟稳定非平衡原理相似。讲座内容在 1944年出版为《生命是什么》——并不出色的作品，但你可以参考里面的一部分思路。
此外，你可以考虑道金斯的“生命是自然选择塑造的信息”。虽然形式上看起来跟上面的命题有很大差异，但是这个也允许谈论黏土结晶、计算机程序之类的生命特征。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:587873807, aid:2925764210 | human | true | 暂无版权及作者信息 | [
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大理石纹螯虾是如何进行“自我复制”来繁殖的？ | 大理石纹鳌虾，有学者建议命名为“处女鳌虾”，是自然产生的三倍体雌性龙纹鳌虾，进行单性生殖（孤雌生殖）：
- 2007 到 2008 年的两项研究认为，大理石纹鳌虾的卵母细胞不进行减数分裂；
- 2016 年，日本的一项研究显示，大理石纹鳌虾的卵母细胞不进行第一次减数分裂，但会进行类似第二次减数分裂的分裂并自行恢复染色体组数；
- 无论如何，大理石纹鳌虾的孤雌生殖不包含“卵细胞互相融合或卵细胞与极体融合”之类过程，是无融合生殖的孤雌生殖。它们产生的三倍体卵子无需受精，可以被批量排出体外、在适宜的水温下开始发育。
- 这种方法产生的不同个体可以携带各自不同的基因突变，并在和环境的互动过程中配备不同的表观遗传修饰。
- 实验显示，雄性龙纹鳌虾的精子不能让大理石纹鳌虾的三倍体卵子受精。
目前推测大理石纹鳌虾于二十世纪九十年代（最晚 1995年）起源于水族馆中，减数分裂发生的随机错误或是水温变动的影响产生了一颗二倍体卵子，该卵子受精并发育成三倍体个体，单独建立了种群。这是在一代间建立了生殖隔离、形成了新物种。
人们对孤雌生殖物种有很多误解，例如大谈“容易感染病原体而灭绝”之类，需要纠正。
其中，有人用绿水螅的图片表示“菌”，在他人指出后声称“为了美观牺牲客观”，很需要纠正。
2021年，实验发现采集自不同地区的大理石纹鳌虾的单核苷酸多态性较低；与此同时，大理石纹鳌虾明显在对新入侵地区存在的高盐度的水做出适应，扩张至祖先未到之地，并在体色、花纹、体型大小上表现出地理差异。大理石纹鳌虾对欧洲气温的适应可能主要来自表观遗传，这是因为实验显示其基因组甲基化程度比龙纹鳌虾显著降低。
大理石纹鳌虾携带鳌虾瘟疫真菌 Aphanomycesastaci，可以传染给世界各地对此敏感的鳌虾（主要是欧洲的鳌虾物种），自身保有北美鳌虾对该病的抗性。讽刺的是，十九世纪以来在该真菌影响下濒临灭绝的欧洲鳌虾均为有性生殖物种。
基因组中的大部分遗传多样性是中性的，既不会提高也不会削弱个体生存或产生后代的能力。生物的环境耐性（包括疾病抗性）相关的基因，影响个体生存或产生后代的能力的遗传多样性，存在于基因组内非常特定的区域，与全基因组的遗传多样性完全无关。快速传代、大量繁殖的无性生殖物种或孤雌生殖物种完全可以拥有压倒有性生殖物种的环境耐性。
不限于无性生殖或孤雌生殖，许多入侵物种抵达新地区时只有数个个体，基因多样性暴跌，但迅速发展为庞大的新种群，并在特定的基因上显示出迅速适应。2021年，科学家在入侵物种“绿蟹”身上再次确认了这件事。
大理石纹鳌虾有 276 条染色体，龙纹鳌虾为 184 条。但是，大理石纹鳌虾似乎存在一些基因丢失，其总碱基对数约为龙纹鳌虾的 140% 而不是150%，这也有助于二者产生生殖隔离。在这样的丢失过程中大量基因会重新排列，与拷贝数的变化一起改变生物的蛋白质合成规律和性状。大理石纹鳌虾的最大体长、最大寿命、单次产卵数都大幅超过龙纹鳌虾，性成熟所需的时间相近。
Bdelloid rotifer 和 Darwinulid ostracods，通称“进化丑闻”，在没有经历有性重组的情况下持续了约 8000 万年到 1亿年，分别产生了 360 个和 28 个物种（Butlin，2002 年；Mark Welch 等，2009 年；Schön 等，2009年）。几个现存的孤雌生殖竹节虫谱系在没有经历有性重组的情况下持续了超过一百万代（Schwander 等，2011年）。孤雌生殖的长期持续存在可以解释为有害突变被表观遗传沉默、表观遗传多样性可以执行序列多样性的功能（例如抑制寄生虫、适应环境），随着时间流逝，表观遗传多样性可能被碱基变化固定为序列多样性，最终产生新物种（Vogt，2015年）。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
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|
听说食肉动物的皮肤具有游离性，有一定的防御能力，想知道防御能力如何，贴吧虎吧有人那这个吹老虎的体防？ | 食肉目皮肤的防御能力一般。狮子、老虎身上容易与其他生物接触的部位的皮肤厚度可达 2~3厘米，能防御一些攻击，但远远没有到值得吹的地步。狮子皮已经被古人证实不适合作为衣服，挂在墙上或放在座椅上当装饰还可以。
老虎皮当衣服的性能同样堪忧，主要用于地毯、壁挂、坐垫。
跟大猫的皮比起来，熊、河狸、海獭、狼獾、渔貂的皮要耐用得多。
在防御穿刺方面，已经在野外观测到狮子·老虎被豪猪刺死、被毒蛇咬死。并非所有毒蛇都有足够长的毒牙来攻击这些动物。
在防御打击方面，“皮肤具有游离性”并不能抵御长颈鹿、大象之类大型植食性动物的踢打。
老虎像其他猫科动物一样有护毛和底毛。护毛比底毛长而粗，有一定的防水性能，可以遮挡紫外线并防御擦伤、轻度切割伤和某些昆虫。底毛就是绒毛，在皮肤周围约束空气来保暖。狮子、老虎的毛比很多人想象的“硬如铁丝或铜丝”要柔软，但没有家猫那么软。
若要抵抗步枪之类武器，体型比虎皮更有用。小口径子弹的击倒能力不足以立即制动成年老虎。你可以使用较大的子弹。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:524272343, aid:2412936353 | human | true | 暂无版权及作者信息 | [
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|
用其他物种的花粉刺激三倍体西瓜会使其结果吗？ | 一部分非西瓜物种的花粉可以让三倍体西瓜结果，例如葫芦品种“Kachidoki2”的花粉。这种葫芦花粉亦可诱导二倍体西瓜产生符合商业无籽标准的果实，从而省略搞三倍体的过程。
- 实验显示，在上述无籽果实启动发育时，葫芦花粉管根本就没有到达西瓜子房的胚珠。
- 这样搞的结果率约 57.1%，比用二倍体西瓜花粉为二倍体西瓜雌花授粉的结果率 65.0% 略低。
- 用葫芦花粉搞出来的西瓜果实，横截面有变成三角形或长方形的趋势，果长与果宽之比有明显变化，但果实重量、果皮厚度、果肉颜色、糖度与对照组几乎相同。在这个场合，形状变化可能是优势。
顺便说，二倍体西瓜、四倍体西瓜与三倍体西瓜的生殖隔离是不完全的。三倍体西瓜仍有概率产生正常配子，市面上的一些无籽西瓜含有少量正常种子。
- 美国农业部规定，等分的切面露出 10 个或更多成熟种子的“无籽西瓜”不能算做无籽西瓜，一批“无籽西瓜”里超过 10% 的瓜不算数会导致这批“无籽西瓜”评不上美国等级 1。
历史上，生产无籽西瓜的方法主要有用植物生长调节剂刺激二倍体西瓜雌花、用二倍体西瓜花粉为三倍体西瓜雌花授粉、用软 X射线辐照过的二倍体西瓜花粉（配子无活性）为二倍体西瓜雌花授粉、用四倍体西瓜的低活性花粉为二倍体西瓜雌花授粉。使用葫芦花粉的技术跟历史上的其他方法明显不同，产生的果实完全没有正常种子。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:582942644, aid:2884830051 | human | true | 暂无版权及作者信息 | [
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动物能通过观看影视作品获得乐趣吗？动物喜爱观看的影视作品可能是什么样的？会提出这个问题是因为的前些天看到一个黑猩猩看视频的视频。黑猩猩眼中的视频是什么样的？它能理解它观看的视频吗？能通过观看视频获得乐趣吗？人类为什么能通过观看视频获得乐趣？其他动物如海豚，章鱼，猫狗等呢？ | 在播放的内容波长、亮度、刷新率适当的时候，视力不太差的动物可以观看影视作品；在播放的声音频率、响度适当的时候，听力不太差的动物可以听到音频。“能不能理解”“有无乐趣可言”在子非鱼的层面上无法判断，不深究这个的话，看起来动物可以根据互动视频做出人们预期的部分反应，至少能对视频达到和对周围的事物相似的理解程度——原本就看不懂的东西，例如偏微分方程，写在纸上和显示在屏幕上都不懂。
看起来，黑猩猩、猫狗之类可以从看视频和/或听音频中获得一定的乐趣。人们并没有给海豚和章鱼看过什么正经玩意，但章鱼跟水族馆里的摆设玩得看起来不错，这包括而不限于砸水箱、喷水来短路水箱上方的灯。
2013年的一项研究表明，狗可以根据视觉线索选出屏幕上的狗的图像并将它们分组，狗辨认屏幕上的狗的能力高于辨认非狗之物——这可能是长期演化的繁殖相关识别能力。在音频方面，狗对电视发出的狗的声音、友好的人声、玩具的响声做出的反应多于一般杂音。狗的品种可能会影响它对电视的反应：嗅觉敏锐的猎犬对视觉效果不那么感兴趣，而牧羊犬喜欢看到移动的物体。
历史上，低帧率的显像管电视的图像被狗当做简单的运动或光线闪烁。二十一世纪初，在 HDTV 上出现了针对狗的电视频道，称为DogTV，播放面向狗的高帧率视频信号（狗眼适合的帧率约 75赫兹），按照狗能识别的色彩（黄色、蓝色及其组合）着色，包含狗耳能听到的高频声音，节目内容有放松（例如狗在草地上伸懒腰）、刺激（例如狗在南加州冲浪）、生活（狗对响起的门铃做出反应并服从主人的命令）等，每段节目长3~6 分钟以适应狗的观看习惯。
猫也经常在人家里看电视，表现出可以识别电视屏幕上的不同轮廓和动作并对电视发出的声音做出反应。猫眼适合的频率约 70~80赫兹，并对红色识别能力较弱。猫有时会试图攻击电视上的虫子和老鼠。
你可以预期黑猩猩喜欢看黑猩猩生活相关的节目，其余动物大抵也是关注同物种胜过对它们来说“不知所云”的人类肥皂剧。
- 在斯图加特动物园进行的实验证明，播放倭黑猩猩打架的视频会让观看视频的倭黑猩猩感到紧张，播放倭黑猩猩进食、玩耍、交配的视频则无此影响。
他人如何获得乐趣、有无乐趣可言，在子非鱼的层面上同样无法判断。不深究这个的话，人们通常会对看到的东西进行模式识别、匹配、预测并以镜像神经元等触发共情，在看到符合预期的内容时触发奖励机制，在看到严重不符合预期的内容时触发战斗反应和/或镇痛机制（并随后因并无真正的危险而出现回调），诸如此类。视频·音频在你那极度复杂的大脑皮层神经连接网络里激起的更多反应是目前不能精确解析的。
节肢动物基于逐行扫描的复眼在观看显像管电视时获得的图像，根据帧率的差异，可能跟节目内容出现极大的偏差。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:479746147, aid:2060165705 | human | true | 暂无版权及作者信息 | [
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|
养虚拟宠物（任天狗、拓麻歌子、机械玩具）能否和养真宠物一样有助于催产素释放？ | 能，具体效果在很大程度上取决于虚拟宠物的外形、行为、互动能力。有些时候，机器宠物带来的好处可以超越活动物。
人看到可爱的小狗/小猫的静止图片，会释放一些多巴胺和催产素。这由哺乳动物幼体的视觉线索触发，检测内容包括占身体比例较大的头、占面部比例较大的眼睛、圆润而不分节的身体形状、柔软的身体表面（尤其是绒毛）。这种反应有助于哺乳动物照顾幼体，不限于同物种的幼体。成体仍保有上述特征的小狗、小猫等动物和虚拟宠物可以利用上述机制。机器蛇、机器蝎子、机器蜘蛛、机器苍蝇宝宝、机器蛔虫之类东西要触发上面的机制就难得多了。
动态图、立体影像、可触摸的毛绒玩具、能主动接触人的活动物/机器带来的效果比静止图片更强。动物做出的意想不到的滑稽动作会在神经学层面上吸引人：当你不期望看到可爱的东西却看到了，会触发大脑的奖励机制。不过，同样的行为重复太多次就不会触发这机制了，拓麻歌子（万代电子鸡）那样老式的电子宠物经常让小孩失去兴趣。
与活宠物互动减轻人们压力的前提是宠物行为正常、照顾的难度可承担。机器宠物可以让一部分人觉得足够可爱、能做的事超出预期、照顾起来不难，另一些人（尤其是老年人）会觉得别人送自己机器宠物是在敷衍自己、将自己当小孩对待、用玩具哄自己，后者在人群中占比通常较低。
- 2014 年的一项研究显示，约 10% 的老年人拒绝使用机器海豹 PARO，这比自称不喜欢动物的老年人的比例低。一些老年人可能不是真的不喜欢动物，而是觉得自己没有足够的钱、体力和精力来照顾动物。
针对任天狗，一些研究人员做过调查，发现人们确实觉得从它那里获得了陪伴（这需要背后的一系列化学物质释放与神经活动），但人们反馈的陪伴程度低于真的狗或猫。
世界范围内，人口正在老龄化，老年人和痴呆症患者面临社会孤立和孤独的风险。社交机器人，包括机器宠物，可以提供陪伴、改善情绪、减少焦虑、促进社交互动，产生积极影响，同时避免活动物带来的大部分过敏、传染病、咬伤、抓伤、夜间噪音，还不会像活动物那样受伤或死亡。
- 售价 110 到 130 美元、使用的技术相对简单的“Joy for All”机器猫/机器狗在一定程度上能代替活宠物和更贵的机器产品（例如 PARO 售价 6000 美元）。研究人员分析了美国 15 家购物网站上 1327 条关于买给老年人或痴呆症患者的 Joy for All 机器宠物的反馈，发现大多数人报告对机器宠物的外观与交互性的积极看法，认为机器为用户提供了“照顾”它并将其融入日常生活的机会。
- 一些负面反馈表示老年人拒绝使用机器宠物，或者老年人试图给机器吃饭、喝水而让人困扰，或者老年人在机器关机/维修时担心它“死掉”了、心情不好，或者大半个养老院的老年人都想来摸机器宠物而造成混乱，或者产品不够结实、使用寿命短于 8 个月（往往与被灌水、从床上意外掉落、抱得太紧等有关），或者产品的喵喵叫声太假、转头时有电动机的嗡嗡声之类。
- 总体而言，消费者希望机器宠物容易清洁、耐用、可定制、可远程控制音量和动作幅度，尤其是不要搞出太多不必要的喵喵叫来（老年人会担心猫有什么需求未得到满足）。
- 上述机器猫会对人的触摸和运动做出缺乏可预测性的随机反应，被一些用户解释为类似活猫的行为、值得表扬。你可以联想到《The Last Guardian》里经常无视玩家命令的大鹫（设计初衷就是这更像真的动物）。不过，用不了多久，老年人就可能更需要认知高度发达的机器来照顾他们。
- 机器海豹 PARO 相关的研究就更多了，它在世界多地的养老院获得正面反馈。2017 年在美国养老院进行的研究显示，与听音乐、参加体育活动、接受精神刺激的对照组痴呆症患者相比，每 3 天一次以持续 20 分钟的小组会议的形式与 PARO 互动 3 个月的痴呆症患者表现出更低的压力和焦虑水平、更少的行为问题、更轻的疼痛水平，氧合水平和心脏状态得到改善，可以减少一些精神药物和止痛药。
- 与活狗相比，PARO 持续吸引痴呆症患者注意力的能力较低，这可能与竖琴海豹本身不像狗那样长期与人类互动（痴呆症患者偶尔会说“你应该住在大海里”“你不像是应该在家养的动物”）、PARO 的外观和举动不如真的竖琴海豹幼体可爱有关。
- 一些研究比较了多种机器宠物，发现老年人更喜欢毛茸茸的机器猫、机器狗，他们希望机器动物做出像真实动物的举动来互动，希望机器动物摸起来是温暖的、身体内部摸起来是有弹性的。不切实际的外观、塑料或橡胶的质感、较大的速度和重量、声音太大等特征会让老年人有些反感，但互动能力可以抵消外观与质感的问题。
- 一些媒体声称用机器宠物（尤其是 PARO）陪伴痴呆症患者有“伦理问题”。他们并没有很好地说明到底哪里有伦理问题。
看起来，对于老年人（尤其是痴呆症患者），PARO 具有一定学习能力的人工智能、多样化的传感器（触摸、声音、光、位置）、个体化定制的反应并不比 Joy forAll 机器猫·机器狗的简单设计有用。后者只对触摸和声音做出响应，自带的动作也简单得多。当然，这可能是因为 PARO的海豹造型对老年人来说过于非日常。老年人期望的机器宠物的行为似乎主要是看向用户、翻身、摇尾巴、呼吸、在被抚摸时发出咕噜声、在用户腿上行走之类。 | {
"major": [
"问答"
],
"minor": [
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} | [
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] | qid:544263925, aid:2630375183 | human | true | 暂无版权及作者信息 | [
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|
鱼类和海洋里面的哺乳动物，会放屁吗？ | 将放屁定义为有肛门的动物从肛门排出消化道内的气体（尤其是消化食物产生的气体），那么至少一部分鱼和海洋哺乳类会放屁。
- 生活在浅水中的一些鱼会放屁。健康的金鱼通常不放屁，而是打嗝。消化出现问题的金鱼会放屁。
- 南美洲的淡水鱼灰鳉食用藻类后在消化道里产生的气体跟身体的规模比起来相当多，由此导致的胀气会让它们浮在水面上难以下潜。它们通过放屁排气，放不出足够多的屁会高概率被捕食者攻击。
- 生活在深水中的鱼从口和/或肛门持续排出消化道里的气体。持续排气算不算屁是个哲学问题，建议算。
- 鲸的屁曾数次被拍摄下来。海豚们在海洋馆里放的屁就更多了。
- 根据现场调查，海豹的屁有一股鱼味，很臭。
- 绿色和平的工作人员曾比较 60 种动物的屁味，认为海狮的屁是这之中最臭的。
00:17
许多水生动物开发了屁在降低消化道内压力之外的用途，例如鲱鱼可以通过放屁产生高频率的声音进行交流。
- 海牛以消化道中的气体提供部分浮力，在潜入水下前会从肛门排气。你可以称这种获取浮力的手段是憋屁。当海牛便秘时，屁会阻止它有效潜入水下，造成海牛头朝下尾巴朝上在水中摇摆。
- 沙虎鲨使用和海牛相似的憋屁提供部分浮力，使它们可以悬浮在水中——与大部分鲨鱼不同。其他鲨鱼可能持续排出消化道里的气体。
尚未发现水生动物用屁进行攻击。陆生动物在这方面走得很远。例如串珠草蛉幼虫可以用屁里的化学物质击倒白蚁并捕杀，释放位置合适时一屁可以击倒 6只白蚁，击倒的有效时间可长达 3 小时，通常就这样将白蚁熏死。
鱼和海洋哺乳类之外的水生动物在放屁方面的情况不一而足。
- 多孔动物、栉水母动物、扁盘动物、刺胞动物、水生扁形动物、水生线形动物、水生环节动物、棘皮动物等似乎不会放屁。
- 水生软体动物似乎不会放屁。不过，有学者认为头足类的喷水推进是一种类似放屁的行为，可以叫做伪屁。
- 现存的水生无颌类和海鞘会不会放屁还有待研究。
- 小体型两栖类消化道里的气体会较为持续地从肛门排放出来。牛蛙之类体型较大的可以产生恶臭的屁。由于括约肌较弱，两栖类的屁通常无声。
- 至少一部分海蛇、龟、蜥蜴、鳄鱼会放屁。蜥蜴通过放屁排出的不少气体是用嘴吞下去的。一部分陆生蛇类用放屁产生的噪音惊吓其他动物来自卫。
- 现代水鸟在健康状态下不会放屁，因为它们的消化道菌群里没有产生大量气体的微生物。
目前不知道鱼龙、蛇颈龙、沧龙等活着的时候会不会放屁，因为你无法期待屁的痕迹化石保留下来。非水生的非鸟恐龙和翼龙的情况也不清楚。大概至少有一些大型植食性恐龙能够产生消化道胀气。
在水生动物之外，蝙蝠、树懒不会放屁。树懒消化道里产生的含甲烷气体会吸收入血并从呼吸系统排出。
包括人在内，许多会放屁的动物也可以通过打嗝和呼吸排出消化道里的部分气体。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:523872349, aid:2405924211 | human | true | 暂无版权及作者信息 | [
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|
没有线粒体的真核生物能合成血红蛋白吗？ | 不用那么多前提，任何有核糖体的生物在适当条件下都可以合成血红蛋白。即使是本身基因组里没有血红蛋白相关基因的物种，也可以被转入相关基因，或是更简单地整点血红蛋白mRNA来。
血红蛋白和类血红蛋白分子本来就广泛存在于自然界，在没有血液的植物、真菌、细菌们身上也有应用。没有线粒体的真核生物合成的血红蛋白往往跟你身上的血红蛋白有颇多的氨基酸差异，没有血液也没有肌肉的生物合成的血红蛋白与肌红蛋白往往无法区分，因此“类血红蛋白分子”更准确。
- 根瘤菌与大豆组成的共生构造用根瘤血红蛋白降低游离氧到固氮酶可以接受的程度，供根瘤菌展开固氮活动。根瘤血红蛋白是根瘤菌和植物根系一起分泌的，根瘤菌那都别说线粒体了，压根就是个原核生物。
- 在中学生物试卷里，你可能见过“将兔血红蛋白mRNA放进大肠杆菌提取液，合成出了兔血红蛋白”的题目，通常是选择题，让你选“原核生物和真核生物使用同一套遗传密码子”的。“大肠杆菌提取液”提供的主要是大肠杆菌的核糖体，这混合物里什么细胞核、线粒体都没有。
- 在深海热泉生态系中，巨型管虫用极其复杂的血红蛋白运输二氧化碳、氧气、硫化氢，供其共生的细菌使用。血红蛋白还可以为厌氧生物运输或清除一氧化氮。
- 我们已经发现厌氧古菌Methanosarcina acetivorans拥有类血红蛋白分子，用来隔绝氧气。
因此，血红蛋白的起源可能能一直追溯到LUCA，那比线粒体的出现要早若干亿年。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:442469572, aid:1713035392 | human | true | 暂无版权及作者信息 | [
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|
如何看待运动、健身加快了人体新陈代谢和身体损耗，导致寿命缩短？ | 没那么简单。
30 岁后，人的最大摄氧量通常每十年下降约10%，这和多种慢性疾病的患病率与死亡率、丧失独立生活能力的风险增加直接相关。缺乏运动是慢性病的主要原因之一。适度体育锻炼可以在一定程度上增加最大摄氧量，降低人死于高血压、2型糖尿病、高血脂、冠心病、中风、癌症的概率，可将全因死亡率降低约 30% 至 35%。与不定期进行锻炼的人相比，每周累计进行 150分钟以上适度体育锻炼（可以是 5 天间每天 30 到 45 分钟）的人的预期寿命更长，锻炼形式可以是简单的步行。
- 13 项研究的结果表明，有规律的体育锻炼与预期寿命延长 0.4 至 6.9 年有关。
- 11 项研究在此基础上考察了死亡率的混杂风险因素，显示有规律的体育锻炼可使预期寿命延长 0.4 至 4.2 年。
“身体损耗”在材料供应充足、细胞功能正常时可以被身体自行修复。
- 研究人员考察了 5823 名 20 到 84 岁成年人的 DNA，发现每周至少有 5 天进行半小时以上慢跑的女性或每周至少有 5 天进行 40 分钟以上慢跑的男性的端粒长度较大，相当于比自己年轻 9 岁的人的平均水平。
- 端粒不是决定性因素，但可以在一定程度上显示细胞水平的氧化应激或炎症强度。
- 研究人员比较了多个年龄段的人的心脏、肺活量、骨骼肌，发现经常锻炼身体的 75 岁老年人的心血管健康水平、肺活量、肌肉力量相当于 40 到 45 岁人群的平均水平。
超重或多年不运动的人也可以通过每天步行 30 到 60 分钟来延长预期寿命。
限制摄入的热量（包括减少每日摄入量和较长的饮食间隔）或增加运动量到刺激 Sirtuins蛋白活跃的程度，可以改善细胞的表观遗传衰老状况，延长多种实验动物的寿命。
另一方面，过度锻炼，尤其是沉迷于减肥或增肌，有概率缩短人的寿命。这通常是循环系统出问题的概率上升所致。11项关于前运动员的预期寿命的对照研究显示，有氧耐力运动员的预期寿命较长，其他运动员的预期寿命有长有短。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:505535815, aid:2501159062 | human | true | 暂无版权及作者信息 | [
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|
生物体可能产生金属单质吗，或者产生半导体，构建逻辑电路？ | 那不是“可能”的程度，地球上的生物早就可以产生金属单质、半导体、逻辑电路了：
- 地面上就广泛存在将金属氧化·还原来获取少许能量的微生物。你肚里大量存在的大肠杆菌就可以将汞单质氧化为二价汞离子并取得些微的能量，土壤里有多种细菌可以将二价汞离子作为电子传递链的电子受体，还原为汞单质。
- 将金属离子还原为金属单质的细菌相对人类冶金工业来说非常分散，在短时间内不会形成肉眼可见的大量金属单质矿物，但随着时间可能积累并参与矿床的形成。异化金属还原菌能将溶液中的二价钯离子·三价金离子·一价银离子还原为钯、金、银纳米颗粒，对镉、锶等也有效，已经用于微生物燃料电池、污染治理和回收贵金属等研究。
- 生物体内的半导体更是普及到好笑的地步，你身上就有的黑色素就是半导体。人们业已开发出基于姜黄和靛蓝（两种由植物合成的天然染料）的单层天然半导体二极管。
- 硅藻、海绵、水稻等各式各样的生物体可以运用二氧化硅，海绵的二氧化硅骨针是高性能的生物光纤。
- 你身上就有许多神经细胞能执行复杂逻辑电路元件的功能。人大脑皮层第 2 层和第 3 层的锥体神经元的树突臂中的每个微小区室都可以执行复杂的数学逻辑操作，允许单个神经元的树突进行异或运算。
人类制造的各种金属物品可以被细菌腐蚀。二十世纪中叶，研究人员建立的模型是这样的：
- 水中的氢会夺取金属表面的电子形成氢气，在金属表面形成“氢膜”阻碍金属的溶解。附着在金属表面的微生物可以利用体内的氢化酶将硫酸根还原成硫化氢、除去金属表面的氢气，让金属继续溶解侵蚀，硫化氢还能与多种金属发生反应。
2016 年的新理论认为，附着在金属表面的细菌生物膜可以提取金属材料的电子来还原体内的硫酸根。这套系统非常古老，在人类出现之前就已经将二价铁-三价铁循环转换了数十亿年，支持着深海热泉•冷泉和地下的大片生态系。
多细胞生物可以在身上运用金属离子，例如你的骨骼含有大量的羟基磷灰石，铁元素是血红蛋白、固氮酶、叶绿体蛋白等的金属中心。结合共生的细菌或从细菌那里转移的基因，就可以期待将金属单质附着在身体部件上。
- 这不限于脊椎动物。2008 年，研究人员在 136 种昆虫、30 种蜘蛛、12 种环节动物（包括蚯蚓）、4 种蜈蚣的爪子尖端·颚部·尖牙中找到金属离子。这些结构很小，但在与环境接触时可以承受很大的应力。这些金属组成的不是像钉子那样有组织的原子簇，而是单个金属离子分别固定在适当的位置，种类主要是铁、铜、锰、锌。
在切叶蚁的下颚上，成年后短时间内迅速积累的锌离子可以大幅提高其抗压强度。蜘蛛的尖牙也是在成年后一段时间内迅速积累金属离子的。 | {
"major": [
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],
"minor": [
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} | [
"通用"
] | qid:483402010, aid:2092425142 | human | true | 暂无版权及作者信息 | [
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|
人的神经信号也是电信号，为什么电磁干扰影响不到人？ | 您的问题存在错误，电磁干扰是可以影响到人的。
医疗扫描和经颅磁刺激使用的强磁脉冲经常引发副作用，包括而不限于睡眠障碍、头痛、头晕、抑郁、疲倦、发痒、注意力不集中、短暂的轻度躁狂发作、认知改变、暂时性听力损失、味觉变化（典型的是口腔出现金属味）、暂时性工作记忆受损，少量患者会晕厥。
电磁场对人体的影响不限于神经系统。10特斯拉的磁场可以影响血液流动。随时间迅速变化的强电磁场可以在人体内产生热效应，例如微波炉可以很简单地杀人。磁星的超强磁场可以将人体拆成一条单原子线、加速到亚光速并在它的表层撞得剧烈爆炸。
显然，人们经常被文艺作品里“只影响机器、不杀人”的“EMP”误导。其实，有封闭金属外壳或法拉第笼式构造并良好接地的机器比人更不怕电磁脉冲。
很弱的磁场就能影响神经细胞的门控离子通道，还能上调大鼠脑神经细胞的活性氧生成水平，提高 DNA 断裂概率，引起更多神经细胞凋亡或坏死。
磁脉冲照射可在脑组织中引起微电流，这微电流可以激活神经细胞，人已经用磁脉冲照射影响过许多实验动物和人的行为。较强的照射可以干扰实验动物或人的平衡感和方向感、诱发幻觉、产生强烈而不受控的恐惧和焦虑等情绪、导致神经调节紊乱和上面谈到过的各种副作用。
静磁场可以影响人脑、可以干扰动物的神经功能。经颅静磁场刺激可以降低人运动皮层的兴奋性并暂时改变皮质内抑制系统的活动。在视觉皮层、顶叶皮层、前额叶皮层等处进行静磁场刺激或磁脉冲照射可以改变人的行为。强静磁场可诱发果蝇体细胞DNA 突变，这效应长期持续当然也可以致癌以下略。
电磁场可以引起癫痫患者的癫痫发作，严重时可导致猝死。 | {
"major": [
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"minor": [
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] | qid:555222929, aid:2689385106 | human | true | 暂无版权及作者信息 | [
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为什么灵长类的嗅觉都不强? | “灵长类的嗅觉都不强”是个以偏概全的刻板印象，原猴亚目的嗅觉能力强，而人类的嗅觉能力其实并不弱。
- “人类的嗅觉很差”是一条19世纪模因，来自Paul Broca。实情是，人类的嗅球只是“占整个脑的比例较小”，而不是比其它动物小，在实验室里测试特定的气味时，人类检测一些气味的能力强于老鼠和狗，三者各有自己擅长检测和不擅长检测的气味。
试图解释“灵长类嗅觉退化”的早期学说是“嗅觉退化与视觉进化同步进行”。2004年，来自德国和以色列的一群科学家试验发现，灵长类嗅觉感受基因的缺失和三色视觉的获得之间存在某种联系。
- 嗅觉感受基因是嗅觉的分子基础，截止2004年已经测序的人类·非人灵长目动物·老鼠有数量相近的嗅觉感受基因，不过人类的嗅觉感受基因中有60%不具活性，通称“假基因”。无尾猿的嗅觉基因有30%是假基因，老鼠的嗅觉基因有20%是假基因。科学家们调查了包括人在内的19种灵长目动物，结果发现欧洲猴子与无尾猿的嗅觉感受基因中假基因比例基本相同，而在美洲除吼猴外的灵长目动物里，嗅觉感受基因中的假基因比例少得多。
- 这表明，欧洲猴类、无尾猿、吼猴的嗅觉各自出现了衰退。三者并没有很接近的共同祖先，但有一个共同的感觉特征：三色视觉。这需要三种视蛋白吸收不同波长的光线。无尾猿和欧洲猴类拥有全部三种视蛋白基因，大部分美洲猴类只有两种视蛋白基因。三色视觉的出现在灵长目中似乎发生了两次，一次发生在无尾猿和欧洲猴的共同祖先身上（约2300万年前），另一次发生在吼猴身上（约700万至1600万年前）。研究人员认为，三色视觉的出现与嗅觉的退化发生在同一时期，当动物的嗅觉退化时，听觉或视觉更发达的性状成为优势。
后来，随着对原猴亚目的实验次数增多，“灵长类嗅觉不灵敏”的刻板印象受到了质疑。2018年，一些学者研究了来自24个灵长类物种的嗅觉感受基因，结果发现原猴亚目的嗅觉感受基因多达600~800个、平均约22.5%是假基因，而类人猿亚目的嗅觉感受基因是200~400个、平均约47.5%是假基因。
- 在纳入了擅长嗅觉的原猴亚目灵长类的数据之后，动物的活动模式（昼间活动/夜间活动）与三色视觉并没有表现出“跟功能性嗅觉感受基因的数量在统计学上显著相关”（这和刻板印象里“夜行性动物更依赖视觉以外的感官”不一致），系统发育、鼻子结构（原猴亚目的鼻子有弯曲的构造）则显示出和功能性嗅觉感受基因数量的统计学关联。
- 类人猿祖先分支的嗅觉感受基因的丢失率比灵长类的平均水平要高、和敏锐的视觉同步演化。
- 嗅觉感受基因丢失率最高的是叶食性疣猴亚科，这可能与从吃水果到吃树叶的食性转变有关，气味信息对搜寻水果并判断其品质比较有用，对搞点树叶的重要程度有所下降。
2012年的一项研究显示，吃水果的狐猴物种靠嗅觉识别其首选食物，吃树叶的狐猴物种依靠视觉和嗅觉结合来选择首选食物，杂食性的狐猴物种则靠视觉与嗅觉中的任何一个就能选中其首选食物。如果所有食物的品质都不错，吃树叶或杂食性的狐猴物种可以只靠视觉选出最爱吃的食物，吃水果的狐猴物种则仍然要靠嗅觉加视觉来判断。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
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|
为什么鸟冬天要迁徙？ | 一些鸟在冬天迁往气温较高的地区，可以获得比不迁徙的鸟更多的食物。在食物充足的情况下，许多候鸟会暂停迁徙。现存候鸟物种往往处于一部分种群迁徙、另一些种群不迁徙的状态。
对于体型较大的鸟，成群飞行可以节约能量。组成 V 形编队的大雁比单独飞行节省 12% 到 20% 的能量。组队迁徙比零散地飞行去寻找食物更高效。
当然，鸟本身未必明白这些情况。
在食物总量不足的情况下，触发具体的鸟迁徙的因素通常是每天的白昼时长缩短到一定水平引起的激素水平变化。也有检测气温变化的鸟，这预计会受到气候变化的影响。
鸟的迁徙可能首次出现于约八千万年前，在鸟类演化史上多次独立出现。在全球范围内、在鸟纲的五个主要的目级分类群中，迁徙行为总体上与更高的物种形成率、更低的物种灭绝率相关，从而导致更高的净多样化率。
迁徙的鸟并不都会飞。许多企鹅能通过游泳迁徙超过 1000 千米。Dendragapus obscurus主要通过步行进行高度迁移。澳大利亚的鸸鹋在旱季会长途步行去寻找食物。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:566110887, aid:2755009503 | human | true | 暂无版权及作者信息 | [
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|
为什么大部分动物没有彩色视觉，只有黑白视觉？动物进化出彩色视觉的意义在哪里？ | 事实上，以已知的物种数量、个体数量、生物量计算，大部分动物是具有彩色视觉或检测颜色的感光能力的。
- 物种数主要来自具有多色视觉的昆虫，个体数主要来自没有眼睛但能检测颜色的线虫，生物量主要来自具有多色视觉的鱼。
- 以个体数量计算，有图像视觉的动物以彩色视觉为多；
- 所有动物的大部分数量归于线虫。秀丽隐杆线虫可在白光或一定比例的蓝黄光下检测铜绿假单胞菌等细菌反射的光的波长比例并避免摄入这类对它们有毒的细菌，这涉及的分子机制与其他动物的视蛋白完全无关，涉及的基因至少有 jkk-1 和 lec-3。从其他线虫到人的多种动物有这些基因，但它们在人身上似乎不用于检测光波长的比例。
对图像视觉来说，色觉来自多种具有不同吸收峰的视蛋白分子。在陆生动物中，只有节肢动物和脊椎动物有基于视蛋白的色觉，但这就足够用物种数量淹没其他动物分类群。
- 节肢动物的共同祖先至少有 3 种视蛋白。现代螯肢亚门（蜘蛛、蝎子、蜱、螨等）和泛甲壳动物（甲壳亚门、六足亚门）的大部分物种有视觉且有色觉。
- 大多数现存昆虫有两种视蛋白，一种主要对应绿光和黄光（峰值波长 550 纳米），一种主要对应蓝光和紫外线（峰值波长小于 480 纳米），可有效地与开花植物互动，尤其是带有紫外线标记的花朵。
- 1914 年，卡尔·冯·弗里希通过实验证明蜜蜂可以区分蓝色和任何深度的灰色，这是人们第一次知道非人动物有色觉。此后，他继续研究蜜蜂，证明蜜蜂使用象征性的舞蹈传递信息。1973 年，弗里希因“对比较行为生理学的贡献与对昆虫沟通方式之领导性的研究”而和康拉德·洛伦兹、尼古拉斯·廷贝亨一同获得诺贝尔生理学或医学奖。
- 脊椎动物有 5 种视蛋白，其中 4 种在距今约 3.5 亿年前已经存在。两栖动物与羊膜动物的共同祖先（约 3.5 亿年前）具有四色视觉。现代脊椎动物的大部分物种有三色视觉，少数有二色视觉，单色视觉和无视觉是极少数。
- 一些人以为猫狗是单色视觉，其实它们是二色视觉，猫可以区分红色与蓝色，狗可以区分红色与紫色。
光在水中传播时，波长较长的红黄光被吸收的速度快于波长较短的蓝绿光，环境光的平均波长随水深增加而变短。不同种类的视蛋白可以对应不同深度的常见光线波长，一旦突变出来就可以在一定范围内提供更清晰、更灵敏的视觉，对捕食、自卫、求偶产生一定好处，容易在种群中延续并蔓延。感光细胞使用不同波长的视蛋白，经神经系统处理，在某个阶段就会产生色觉。
色觉有助于更好地识别不同材质的表面，有助于检测捕食者、食物、配偶，获取的信息可用于战斗、逃跑、觅食、躲藏等，可以被广义自然选择青睐。许多现存动物对波长450 纳米以下的光（尤其是直射光）和波长 450 纳米以上的光（尤其是反射光）有不同反应。
线虫的光波长比例检测能力有类似的结果。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:564843295, aid:2746174303 | human | true | 暂无版权及作者信息 | [
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|
哪些蜘蛛具有群居习性，其中具备社会性的又是哪几种？ | 在已知的约 50500 种蜘蛛里有至少数百种群居，其中至少 23 到 25 种有一定的社会性。人工环境可以让更多蜘蛛物种建立群体。阿格纳森等估计蜘蛛 18到 19 次独立演化出社会性。
姬蛛科有至少 11 到 12 个具有一定社会性的群居物种，可能涉及 8 到 9 次独立演化出社会性。姬蛛科 Anelosimus 属有至少 8个具有一定社会性的群居物种，主要分布在美国和马达加斯加。
- 生活在南美洲的 Anelosimus eximius 是人最了解的社会性蜘蛛物种，它的网可以覆盖整个树冠、包含数千到五万个体，这种网合作捕猎的效率在 1000 只左右的群体里分摊到个体的效费比最高。
- 在圈养环境，更多的姬蛛科物种可以形成小的群体、共享网和部分食物。
Achaearanea disparata、Achaearanea vervortii、Aebutina binotata、Agelenaconsociata、Australomisidia ergandros、Australomisidia inornata、Australomisidiasocialis、Delena cancerides、Mallos gregalis、Parasteatoda wau、Parawixiabistriata、Theridion nigroannulatum 是群居的，有不同程度的社会性或准社会性。
- Agelena consociata 可以 25 到 40 只合作捕杀较大猎物并一起修复被大雨损坏的巢穴。
- Delena cancerides 可以组成至多约 300 只的群体合作捕猎，群体内可分享猎物，群体间则经常厮杀。
- Mallos gregalis 可以数千只共同建造片状的网来捕猎。
- Parawixia bistriata 可以单独活动，也可以形成数十到约 500 只的群体并合作捕猎，通常在干旱环境里合作较多。在群体内存在争夺设置网的空间、无网者尝试夺取有网者的猎物等现象。
Phryganoporus candidus 的受精雌性可以建立约 100 只后代组成的巢穴，其信息素可以让幼体合作捕猎并分享猎物。成体之间不进行合作。
Philoponella Republicana 可以形成数只到上百只不等的群体，这种群体很容易解散。
主要吃植物的 Bagheera kiplingi 通常单独活动，也能组成数百只的小群体，一些个体会尝试保护卵和幼体。这物种有时也吃昆虫或同类相残。
Leucauge 属的多种蜘蛛可以形成松散的群体，这种群体缺少合作防御之类。
Stegodyphus 属的 S. lineatus，S. mimosarum，S. dumicola 是群居、社会性的，群体的约 88% 是雌性，雌性中约40% 的个体负责繁殖，其余雌性参与照顾同一巢穴的幼体。
- 群体的所有成年个体都会参与巢穴的维护修理，可以合作捕捉猎物或抵御蚂蚁等捕食者的攻击，可以将消化道内容物吐给同一巢穴的其他个体。
- 个体的体型和性格会影响其参与合作行为的频率。
- 雌性个体的寿命约 1 年，寿命将尽的雌性会被同一巢穴的幼体食用，无论那是不是自己的后代。一些成年雌性会在交配后离开巢穴、在附近建造新的巢穴。
- 雄性个体的寿命比雌性短，在一生中大部分时间里不会离开出生的巢。在繁殖季节，雄性会彼此竞争，少数雄性会离巢并加入附近其他雌性建立的新巢穴。
诸如此类。蜘蛛这样给人孤独印象的生物演化出社会性的难度看起来其实不高。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:562915197, aid:2740441374 | human | true | 暂无版权及作者信息 | [
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|
污染物沿着食物链传递，其浓度在较高营养级中一定增大吗？
19.生物富集的效率可以用生物富集系数 (bio-concentration factor, BCF) 进行量化, 计算公式如下:BCF = 生物体中某种物质的浓度/环境中同种物质的浓度, 以下有关叙述错误的是
A. 生物富集的过程一般是沿着食物链和食物网进行的
B. 生物富集效率会受 到生物特性、有害物质性质及环境因素的影响
C. BCF 数值越大,说明该物质越容易在生物体内分解
D. 污染物沿着营养级传递过程中, 其浓度在较高营养级中不一定增大
19题D 哈尔滨2022市统考生物试卷 | 不一定增大。
选项 B 已经概括性地解释了选项 D。生物富集的效率受生物的特性、有害物质的性质、环境因素影响，效率可以低到生物富集系数（BCF）等于或小于1，高营养级生物体内的污染物浓度不变或减少。
一些生物很擅长排出特定污染物，或是很不擅长吸收·储存特定的污染物。一些生物能将污染物分解为无害产物。一些生物不同部位的污染物浓度有明显差异、吃这生物的其他生物并不吃高浓度的部位。这些情况都能实现较高营养级的污染物浓度不变或减小的效果。
例如有机磷农药乙拌磷亚砜及其有毒代谢产物在欧亚鲤（常见的养殖鲤鱼）体内的 BCF 小于 1，六氯环己烷（bhc，666）在真无须魮的鳃的 BCF 为0.076、在肌肉为 0.043、在肠道为 0.132，实验所用的六氯环己烷浓度已经接近对这动物的 LD50。
讲生物富集而不讲这些情况的生物教育是个谜。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:550614820, aid:2651645242 | human | true | 暂无版权及作者信息 | [
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|
坚持跑步为什么会提高大脑认知能力，有什么科学依据吗？ | 科学依据很多。不限于跑步，坚持体育锻炼似乎可以改善人一生中的认知健康水平。
- 神经影像学数据显示，健康的心肺功能可以延缓衰老过程中与年龄相关的脑组织（灰质和白质）损失，有概率改善特定脑结构的健康程度 ；
- 有氧运动可以增加流过大脑的血量，尤其是在涉及信息检索和执行功能的脑区，可能有助于降低患上抑郁症或神经退行性疾病的风险；
- 有氧运动可能影响表观遗传修饰和分子通路、线粒体的行为、涉及脑源性神经营养因子 (BDNF) 的分子系统。通常，在半年间定期跑步和/或进行其他有氧运动，就能让所有年龄段的平时不怎么运动的人自己体验到一些改善；
- 有氧运动可以改善前额叶和顶叶在认知挑战任务中的表现；在年轻人身上，这不止能提高快速反应过程中行动检测的效率，还能在需要灵活改变策略的任务期间改善传统上称为“注意力分配”的过程；
- 电生理学研究也支持“心肺健康或积极的生活方式可以改善认知相关的神经活动”。
- 限制摄入的热量（包括减少每日摄入量和较长的饮食间隔）或增加运动量到刺激 Sirtuins 蛋白活跃的程度，可以改善细胞的表观遗传衰老状况，延长多种实验动物的寿命。
要注意的是，跑步在“提高大脑认知能力”方面的作用并不比其他有氧运动优越。跑得太多、配套设施不合适、姿势不对、运气不好等因素可能损伤人的膝盖、脚踝等部位，有概率引起严重的问题。每天步行30 分钟就足以让 50~55 岁的女性抑郁症患者更有活力和信心，大概对其他年龄段、其他性别的抑郁症患者也有效，亦能帮助预防抑郁症。
- 血液中的脑源性神经营养因子水平与海马体积有关，有氧运动上调血清中的脑源性神经营养因子水平，在统计上与儿童和老人的海马体积增加有关，在儿童身上与更好的记忆表现有统计学关联。但其数值还没到值得将全国的小学生都拉去每天跑十圈的地步。
- 一项研究考察了 1206 名健康人的步行耐力与认知功能的关系，受试者平均年龄 28.8 岁，45.5% 为男性。结果显示较高水平的步行耐力与整体认知功能增强有统计学关联。性别、年龄、教育程度、BMI、HBA1c、动脉血压等变量并没有改变结果的整体模式。
- 一项研究调查了 206 名成年人在六个月内每周至少锻炼四天的情况。受试者平均年龄 66 岁，在实验开始时、三个月后、六个月后各接受思维和记忆测试，并以超声波监测流过大脑的血量。有氧运动计划从每天 20 分钟开始，在六个月间增加到 40 分钟。实验结束时，受试者在执行功能测试中的成绩改善了 5.7%，在语言流畅性方面的成绩改善了 2.4%，大脑血流量增加 2.8%~
- 埃里克森等调查了 165 名 59 至 81 岁之间的、认知健康的老年人，通过最大分级运动测试评估他们的心肺健康水平，让他们执行空间记忆任务，并使用功能磁共振成像 (fMRI) 测量他们的海马体积。结果表明，心肺健康水平较高的受试者双侧海马体积都较大，并与更好的空间记忆表现统计学相关。
- 一项研究对 120 名老年人进行随机对照设计，发现运动组的海马体积相比对照组增加了 2%，并表现出更高水平的血清脑源性神经营养因子水平和更大幅度的空间记忆改善。
- 一项研究将 124 名年龄在 60 至 75 岁之间的老年人随机分配去接受为期 6 个月的步行训练（有氧运动）或柔韧性训练（非有氧运动）。结果表明，步行组在一系列认知任务中的表现改善，而柔韧组没有这种现象。
总之，有氧运动对正常衰老的人和患有阿尔茨海默症、精神分裂症等各种脑部疾病的人群的认知能力和大脑健康有相似的益处。
历史上，人们曾经担忧跑得太多会损伤大脑。针对从意大利到挪威的赛程 64 天的 4487千米长跑比赛的参与者的一项研究显示，这样的超级长跑似乎会导致人的大脑暂时缩小，受影响最大的是视觉皮层（可能是因为铺装路面过于千篇一律），但参与者的大脑在比赛结束后半年内恢复原体积。参与者的膝盖等部位的软骨在前2500 千米赛程内呈现一些损伤和退化，那之后徐徐修复。
动物的脑可以在和外界或自身互动的过程中改变结构和功能。过去一百多万年间，基于身体运动的战斗、逃跑、探索、觅食、自卫、求偶、发明、仪式、舞蹈等行为，在人的生存、适应、发展过程中发挥了重大的作用，狩猎在很大程度上塑造了人。二十世纪的理论认为，涉及上述行为的脑区，尤其是在记忆处理中有基本作用的海马体，受身体运动的影响十分明显；其余脑区出于能量效率，在演化过程中也会受到身体运动的影响。近年来，更多的研究显示人的“看起来很简单的身体运动”会激活巨大范围的神经细胞，旋转眼球会激活约一半神经元，那么身体运动的影响就更没有特别的边界了。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
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} | [
"通用"
] | qid:496662091, aid:2229944422 | human | true | 暂无版权及作者信息 | [
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|
为什么海洋中动物的生物量大于植物？ | 海洋植物只占海洋生产者的一小部分，已知海洋动物生物量不止超过海洋植物生物量，更超过海洋生产者的已知生物量。Bar-On 等估计消费者占海洋生物量的 80%以上，生产者占陆地生物量的 95% 以上。不计尚未搞清楚的部分，当前陆地生物量约为海洋生物量的 80 倍，陆生植物占地球总生物量的约80%，主要是大型维管植物（海洋里稀缺的类型）。超过 70% 的动物生物量存在于海洋。
这可能有以下原因：
### 一、海洋单位面积的光合生产力较低
统计显示，在海洋和陆地环境中，当地光合生产力越高，当地生产者与消费者的比率越大。
2018 年估计的海洋总初级生产力（485 亿吨碳每年）与陆地总初级生产力（564亿吨碳每年）数量级相同，而海洋的表面积是陆地的两倍以上，海洋单位面积的生产力较低。这可以归因于水的透光性差、光合生物需要的多种矿物质在远海海水中的浓度低，海洋的大部分面积不利于大型光合生物（尤其是植物）生长。
那么，当地光合生产力平均而言明显低于陆地环境的海洋环境理应具有较少的总生物量和较高的消费者比例。
### 二、海洋光合生产量更大比例地输入消费者
平均而言，大约 10% 的能量从海洋生产者转移到各级消费者，大约 1% 的能量从陆地生产者转移到各级消费者。
- 这部分归因于陆地大型维管植物木质化的茎。对消费者来说，木质素不易分解，高大的茎还让树叶之类结构也变得不易靠近。陆地木本植物的周转速度很慢，它们的生物量代表数年、数十年乃至千百年间初级生产的积累。
- 相比之下，海洋生产者的周转速度要快得多。洋流使有限的营养物质在海洋中高度流动，而且以单细胞生物为主的海洋生产者很容易被消费者取食。结果，海洋生产者用每一刻较小的生物量产生了庞大的年均初级生产。
### 三、海洋环境有更多的变温动物、更大的捕食者-猎物体型比
海洋基础消费者主要是节肢动物、软体动物等变温动物。变温动物不需要损耗多少能量来维持体温。另一方面，海洋哺乳动物为保暖而需要更大的体型。
浮力降低了大型动物支撑自身重量的难度、有助于优化大型捕食者的运动速度，长距离移动的能耗也较低。
理论支持这产生“头重脚轻”的营养金字塔。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:559053771, aid:2716731910 | human | true | 暂无版权及作者信息 | [
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|
人类更适合吃肉还是吃植物？ | 现代人适合吃肉并搭配一些植物，植物可提供膳食纤维、维生素 C 等。
古人类更适合吃肉。
总体上，人类的饮食倾向是高度可变的。住在江河湖海附近的古人类吃了很多鱼和贝类。生活在森林等植被丰富地区的古人类吃了很多水果、块茎和蜂蜜。现代狩猎采集者哈扎人吃的东西随季节改变，总体上生存所需的能量有一半左右来自动物肉，另一半来自植物和蜂蜜。现代人亦可在很大范围内选择饮食——但是，突然改变长期适应的饮食习惯会影响消化道菌群，有概率造成疾病。
Miki Ben-Dor 博士和他的同事从遗传学、新陈代谢、生理学、形态学、古生物学等领域的约 400 篇科学论文中收集了 25条重要证据，支持石器时代的古人类是肉食性动物。
例如：
- 与杂食动物和大部分肉食性动物相比，人胃液的酸性很强。产生和维持这样的强酸胃液需要大量的能量，其功能主要是消灭不新鲜的肉里的常见细菌。古人类狩猎的大型动物体重远超整个小群体每日所需的量，可以吃上几天甚至几周，使得古人类经常食用含有大量细菌的不新鲜肉类，演化青睐强酸胃液。
- 人的脂肪细胞结构是肉食性动物的典型配置。杂食动物的脂肪储存在数量相对较少的大型脂肪细胞中，而包括智人在内的肉食性动物将脂肪分散在大量小型脂肪细胞内。
- 人类基因组的一些基因被关闭来对应富含脂肪的动物性饮食，黑猩猩基因组里的同种基因则激活来对应富含碳水化合物的植物性饮食。
- 史前人类骨骼中的同位素比例和古人类遗迹中的武器、动物残骸化石表明，古人类经常猎杀脂肪含量很高的大中型陆生动物，他们生存所需的能量可能有 70% 以上来自大中型动物。
- 化石证据显示，古人类的猎杀对全新世大灭绝做出了贡献。
历史上，人们经常幻想人类的演化归功于杂食的灵活性。现在看来，人类恐怕是狮子和鬣狗的同路人。植物在石器时代末期才成为人类食物的主要组成部分，那时人类居住区域周围动物的平均体重早已从约500 千克暴跌到不足 50 千克。
遗传变化的证据和用于加工植物的独特石器的出现显示，约 85000 年前的非洲人、约 40000 年前的欧洲人和亚洲人逐渐增加了植物性食品的消费。
相比之下，人类以动物性食品为主的岁月持续 140 万到 200万年。那段时期，无论当地的生态条件如何，在石器中都观察到长时间的相似性和连续性：用于杀戮和处理肉类。
显然，虔诚的素食者不愿相信他们的祖先是地球上从未有过的疯狂掠食者，这个物种不但捕猎大中型动物，还在猎物的平均体型缩小后使用弓箭与陷阱继续猎杀、通过刀耕火种对植物进行大规模摧毁。从人类崛起到现在，野生植物的生物量至少减少了一半。
当然，他们大概可以从古人类也吃植物里获得些许安慰。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:559625647, aid:2719278840 | human | true | 暂无版权及作者信息 | [
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|
生物会对遗弃的巢穴重复利用吗，比如蚂蚁；或者生物会利用其他生物的巢穴吗？ | 会，很常见。
老鼠经常在感到危险或巢穴受损时逃走，并在不久之后返回、对受损的巢穴进行修补、重新住下。老鼠选择筑巢地点时会考虑安全、保温、获取食物和水的难度，会在巢穴里使用报纸、破布、塑料、树叶等材料保温。为觅食等活动暂时离开巢穴时，老鼠会沿途留下气味标记，以便原路返回。鉴于合适的筑巢地点有限，老鼠改变筑巢地点时优先选在旧巢穴附近；一窝老鼠被赶走后，新来的其他老鼠（无论是不是同一物种）有概率选择同一位置筑巢。老鼠经常在人的“巢穴”里筑巢。
在雨季，许多蚂蚁会将新热带白蚁 Nasutitermes corniger (Motschulsky)的巢穴当作庇护所，可能顺便吃掉一些白蚁工蚁（不论死活），同时为白蚁提供富氮营养并改善巢穴的防御水平，与白蚁共生的最常见的蚂蚁物种是 Monadsbispinosa (Oliver)，它们不攻击活的白蚁。
一些地下白蚁会利用蚂蚁的部分巢穴。白蚁经常在人的“巢穴”里筑巢。
许多蜥蜴和一些蜘蛛会利用蚯蚓挖的地洞。
地势好的山洞·水下洞穴可被许多不同生物运用，体型显著不同的动物可共用较为宽敞的洞穴。
大部分洞穴是黑暗、阴冷、潮湿的，历史上可能有熊、洞熊、洞狮、洞鬣狗等动物居住，经常易主（尤其是在其中一些物种灭绝之后）。一万多年前，古人类可能为了制造屋顶、毯子、地毯之类击杀洞狮，与气候变化一同将其抹消。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:559636235, aid:2719363915 | human | true | 暂无版权及作者信息 | [
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|
生物的进化应该是趋利避害才能延续下去，可为什么人类在惊恐的时候会僵住任人宰割，而不是撒腿就跑？
为什么惊恐的时候明明撒腿就跑比僵住动不了任人宰割存活的几率要大得多，为什么依然进化成动不了呢？？？ | 在危险情况下难以行动，吓呆（冻结行为/freezing behavior, 冻结反应/freezeresponse），是应对捕食者的一系列行为的组成部分，比人类、比哺乳纲更古老。各种动物经常将其拿来应对捕食者之外的有害事物、同类攻击等生存威胁。
演化青睐有助于提前发现捕食者并降低被捕食者发现的概率的行为。动物检测到远处出现疑似捕食者的迹象时，“吓呆”会减少自己的动作（许多捕食者能识别远处的物体移动和声音）、促使自己的感官关注该迹象、增强收集信息的敏感度。有复杂感官的动物通常会朝疑似捕食者所在的方位瞪大眼睛、竖起耳朵之类。这是古老、简单而有效的被动防御策略，经常搭配其他适应性反应，例如啮齿类在检测到蛇时会竖起毛发增加散热，这略微有助于降低被红外线检测发现的概率。
动物检测到捕食者正在对自己发动攻击或距离靠近到危险的程度，会触发装死、尖叫、战斗或逃跑反应等。“吓得完全无法移动甚至昏倒”有助于装死。具体什么样的距离算是危险、要不要装死，取决于自身的物种和身体状况、捕食者的物种和个体特性（形状、体型、移动方向、速度等）、自身的生活经验等。战斗或逃跑是常见的主动防御策略。
- 实验显示，将老鼠放在离猫很远的盒子里，老鼠会“吓呆”或逃跑，将老鼠放在离猫较近的盒子里，老鼠会惊慌失措、显示出战斗或逃跑反应（布兰查德等，1990 年）。
- 装死适合对付对自己不熟悉、有一定认知能力来辨认猎物死活、没有足够的认知能力来识破装死、不爱补刀、不喜欢死猎物（因为有大量腐生微生物和/或腐败产生的有害物质等）的捕食者。
- 尖叫可以提醒同类并吸引捕食者的注意力，在群居动物身上尤其有用：可以提高血亲逃生的概率，这会提高相关基因在种群中长期存续的概率。
动物被捕食者抓住或咬住会触发强效镇痛，抑制在平时会导致跛行或倒地的疼痛造成的影响、支持全力抵抗或逃跑。也有一些动物会在这阶段再次触发“吓呆”，这有助于装死。
人的“吓呆”关系到基底外侧杏仁核复合体与海马体，这些结构（或其类似构造）受损或被抑制的动物会不容易吓呆，乃至无法吓呆。
一些学者建议将冻结反应和战斗或逃跑反应合称“战斗、逃跑或冻结反应/fight-flight-or-freeze response”。
题目里“人类在惊恐的时候会僵住任人宰割，而不是撒腿就跑”的说法并不适用于触发战斗或逃跑反应的人，后续的“惊恐的时候明明撒腿就跑比僵住动不了任人宰割存活的几率要大得多”同样有适用范围。站着不动或缓慢后退有概率让捕食者放弃攻击。许多捕食者，包括持有远程武器的早期智人之类，擅长从背后击杀逃走的大中型动物。而且，在面对难以抵抗的捕食者时，让少量个体吸引其注意力、帮助其他个体逃跑是可行的策略——高声尖叫可以警告同伴并吸引捕食者的注意力，你可以看到人类中的一部分个体有这种行为。
题目里“趋利避害才能延续下去”的说法是对进化的古老误会之一。其实进化是很凑合的。地球上数量最繁盛、生殖最成功的类群（以个体数是病毒、细菌，以生物量是维管植物，在动物范围内以个体数是线虫）普遍缺少自主快速运动的能力，比起在环境不适合或出现敌害时逃走，它们更多地改变自己体内的生化反应和化学平衡·自己身体的一部分材料及其性能来对抗环境变化与敌害，以单纯的繁殖速度补上被环境和敌害消灭的部分。现代海洋里的细菌每天被噬菌体消灭约40%，其年均数量的波动跟这几乎无关。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:559888069, aid:2722374344 | human | true | 暂无版权及作者信息 | [
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|
为什么大自然要让人类女性生殖时要承受巨大的痛苦，而没有将这种痛苦进化消失掉？ | 大自然不负这个责任。历史上大部分时候，古人类分娩没有现在这么痛苦，选择压不强。最近数百年里，站着、坐着、蹲着分娩的难度及其带来的痛苦仍低于仰卧分娩，尤其是站着分娩允许重力协助、骨盆带自然活动（Dundes，1987年；Gupta 与 Nikodem，2000 年；Michel 等，2002年）。古埃及壁画、古欧洲浮雕、阿兹特克艺术品都曾描绘女性站着分娩。十九世纪末到二十世纪初的粗糙医学一度造成产妇死亡率上升。
对双胞胎的研究显示，人类女性的骨盆宽度受生活环境的影响似乎超过基因造成的影响。农业时代的饮食变化可能比骨盆宽度的遗传因素造成了更多的产科问题。约12000 年的农业社会生活尚未让人的骨盆宽度重新优化到适合新生儿的大头，显示当前状态可以算是略有不利的中性性状，能在种群中长期传递。
将人类女性当前的骨盆再加宽几厘米以扩大产道，并不会和“产科困境”假说设想的那样妨碍双足行走，声称现在的宽度有任何演化上的好处，都很可疑。非洲南部石器时代晚期的人群似乎搞出过身高很低而骨盆很宽的女性，当时他们经常长距离迁徙，化石并未显示这些人在行走方面有什么困难。生物力学也不支持“加宽骨盆会增加运动成本”的想法。
现代人掌握解剖学与遗传学知识的时间太短，还不够对自己进行人工筛选，目前更多地以剖腹产解决难产问题，这又会影响筛选的效率。现代人对抗分娩痛苦的有效办法包括麻醉，这比自然界的大部分动物更好地缓解了疼痛，风险相对可控。
- 类人猿的足月妊娠所需时间是灵长类里最长的，黑猩猩、倭黑猩猩的正常妊娠期约 39 周，人的正常妊娠期约 37 到 41 周。黑猩猩、倭黑猩猩的新生儿像人新生儿一样缺乏独立生存能力，需要母亲提供大量照顾。不过，这两种类人猿并没有现代人这么高的难产率，它们的产道比新生儿尺寸明显更宽。人的祖先在数百万年间也如此。
- 正常晚期智人新生儿的大脑体积是成年时的 30% 左右，正常黑猩猩新生儿的大脑体积是成年时的 40% 左右，卷尾猴新生儿的大脑体积是成年时的 50% 左右，人脑已经因成年时的大体积而做出妥协。
- 人大脑生长速度的峰值出现在出生前的几个月间，与生下来就能行走的动物类似（Halley，2017 年），这方面仍有为分娩让步的空间。
- 人类胎儿的能量需求和母体能安全地提供的代谢率与散热能力的关系似乎更直接地决定了胎儿能达到的体重。这方面的统计相关性也出现在没有骨头限制产道的鲸类身上（Sacher 和 Staffeldt，1974 年）。
- 提出“产科困境”假说时使用的统计数据可能不反映历史上大部分时候的状况。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:558684347, aid:2722377297 | human | true | 暂无版权及作者信息 | [
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|
运动会加快细胞新陈代谢，那是不是意味着越运动越容易狗带？ | 不是。“加快新陈代谢”跟寿命或死亡率没有简单的线性关联。
缺乏运动是慢性病的主要原因之一。30 岁后，人的最大摄氧量通常每十年下降约10%，这和多种慢性疾病的患病率与死亡率上升、丧失独立生活能力的风险增加直接相关。适度体育锻炼可以在一定程度上增加最大摄氧量，降低人死于高血压、2型糖尿病、高血脂、冠心病、中风、癌症的概率，可将全因死亡率降低约 30% 至 35%；与不定期进行锻炼的人相比，每周累计进行 150分钟以上适度体育锻炼（可以是 5 天间每天 30 到 45 分钟）的人的预期寿命更长，锻炼形式可以是简单的步行。
- 13 项研究的结果表明，有规律的体育锻炼与预期寿命延长 0.4 至 6.9 年有关。
- 11 项研究在此基础上考察了死亡率的混杂风险因素，显示有规律的体育锻炼可使预期寿命延长 0.4 至 4.2 年。
“运动带来的身体损耗”在材料供应充足、细胞功能正常时可以被身体自行修复。
- 研究人员考察了 5823 名 20 到 84 岁成年人的 DNA，发现每周至少有 5 天进行半小时以上慢跑的女性或每周至少有 5 天进行 40 分钟以上慢跑的男性的端粒长度较大，相当于比自己年轻 9 岁的人的平均水平。
- 端粒不是决定性因素，但可以在一定程度上显示细胞水平的氧化应激或炎症强度。
- 研究人员比较了多个年龄段的人的心脏、肺活量、骨骼肌，发现经常锻炼身体的 75 岁老年人的心血管健康水平、肺活量、肌肉力量相当于 40 到 45 岁人群的平均水平。
超重或多年不运动的人也可以通过每天步行 30 到 60 分钟来延长预期寿命。
限制摄入的热量（包括减少每日摄入量和较长的饮食间隔）或增加运动量到刺激 Sirtuins蛋白活跃的程度，可以改善细胞的表观遗传衰老状况，延长多种实验动物的寿命。
另一方面，过度锻炼，尤其是沉迷于减肥或增肌，有概率缩短人的寿命。这通常是循环系统出问题的概率上升所致，主要原因不是代谢率高，而是心脑血管压力大，次要原因是肠道的应激反应，这些问题可以被饮食、药物等控制。11项关于前运动员的预期寿命的对照研究显示，有氧耐力运动员的预期寿命较长，其他运动员的预期寿命有长有短。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:561342429, aid:2725438586 | human | true | 暂无版权及作者信息 | [
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|
第一个产生抗体的物种是那个？ | 分子生物学证据显示，第一种产生抗体的动物可能是某种目前尚未找到化石或找到了化石但未确定免疫功能的有颌鱼，可能是盾皮鱼或软骨鱼，生活在距今超过 4亿年前。估计这种有颌鱼的体型较大、单次繁殖产生的后代数量较少，较强的免疫功能在一定程度上有助于可育后代的数量。
- 一个次要证据是，某些盾皮鱼似乎采用体内受精和胎生 。
- 七鳃鳗的适应性免疫系统不产生抗体、产生可变淋巴细胞受体。盲鳗之类似乎也如此。估计抗体的出现不早于脊椎出现。
- 现存软骨鱼的抗体的一些特征看起来比硬骨鱼的抗体更不像哺乳类的抗体。现存硬骨鱼的抗体显示，远古硬骨鱼可能首创了现存哺乳类的抗体的一些特性（尤其是铰链区） ，并创造了一些看上去后来被四足形类随机丢弃的特性。哺乳类有已知最复杂的铰链区的结构。
类似抗体的东西比“距今 4 亿年前”更古老。
- 原核生物的表面蛋白的免疫球蛋白样结构域显示出类似真核生物免疫球蛋白的 β 链二级结构，可粘附在特定的宿主细胞上、抵御特定抗生素、对特定宿主致病。贝特曼等在 1996 年提出细菌的此类表面蛋白是抗体的起源。一些病毒也有类似的蛋白，目前不知道是从细菌身上取来还是起初由病毒转给细菌的。
- 海绵、海鞘等非脊椎动物有不同类型的·结构与免疫球蛋白相似的蛋白质。海鞘物种 Ciona intestinalis 等拥有带可变区的几丁质结合蛋白，这是一种免疫识别分子，结构和功能在一定程度上类似抗体，能粘附在细菌上并促进血细胞吞噬细菌 。
- 文昌鱼、海胆、软体动物等有一些细胞能进行类似 VDJ 重组的体细胞突变。七鳃鳗和多种鱼类有一些细胞能进行体细胞超突变。这些是 B 淋巴细胞产生各种各样的抗体的基础。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:561794736, aid:2727989918 | human | true | 暂无版权及作者信息 | [
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|
人类的神经细胞为什么无法再生？ | 事实上，人的神经细胞可以再生，死亡的神经细胞可以被新的神经细胞取代，尤其是在海马体；即使不产生新的神经细胞，已经存在的神经细胞也可以改变功能来代偿受损或死亡的细胞的功能，半脑切除之类手术和一些严重脑水肿案例显示人脑可代偿看起来不可思议的神经缺失。
这和所谓意识的连续性完全无关。新产生的神经细胞可以接替旧细胞形成同样模式的突触连接来储存一样的信息、执行一样的功能。况且，你不觉得在“随着年龄增长，死掉了大量神经细胞、没有新细胞补充”的情况下“维持主观连续的意识”难度更高吗？
包括人在内，成年哺乳动物脑仍有两个区域包含神经干细胞：侧脑室的脑室下区（SVZ）、海马体齿状回，这些神经干细胞在许多动物身上制造新的神经元。室管膜细胞和放射状胶质细胞可以分裂、可以分化为其他类型的细胞。人的侧脑室壁有神经胶质细胞迁移形成的带状结构，包含新产生的神经元、星形胶质细胞、少突胶质细胞，但其流量在年满18 个月后明显下降。
- 1998 年，癌症患者遗体展示了成年人中枢神经再生的首个证据。他们在世时接受过溴脱氧尿苷注射，该物质能标记新分化的细胞。科学家发现该物质出现在这些患者大脑内的海马体（和记忆、学习相关的海马状结构）中。
- 2013 年，来自瑞典卡罗林斯卡研究所的 Jonas Frisén 团队对 55 名已去世患者大脑组织中的单个神经元进行了碳年代测定，根据细胞年龄推测，人类每天在编码记忆的海马体齿状回区更新 700 个神经细胞。
- 后来有其他研究认为这再生速度在人幼年时较快、30 多岁时基本陷于停滞。其他脑区的神经细胞补充速度比这要慢或更早停滞。
在人体发生脑损伤（例如中风）后，病灶周围的神经元会上调促进轴突生长和突触形成的信号通路（Dancause 和 Nudo，2011年），树突棘的周转增加，以便形成新的连接。病灶周围组织中的神经元可以将新的轴突延伸数毫米，在附近的皮质区域形成新的功能性突触连接（Nagappan等，2020 年）。
人外周神经系统的轴突断裂后可以用每天 1 到 2 毫米的速度缓慢地生长恢复。断肢再植或移植四肢时，患者可以体验到触觉一点点朝肢体末端延伸。
- 在人身上进行的完整双臂移植手术于在 2008 年在德国农民 Karl Merk 身上首次取得成功。切断的轴突每天再生约 1 毫米。
- 新神经细胞延伸出轴突的速度在大神经中每天约 5 毫米，在小神经中每天约 2 毫米。
实验显示，半年间每天进行步行、骑自行车、爬楼梯等有氧运动，可以提高各年龄段的人的思维能力，观察到左侧额叶区域的大脑皮层厚度增加。这主要来自突触数量上升而非更多的神经细胞生成。
胎儿细胞可以进入母亲的血液循环并在心脏、肝脏、甲状腺等处分化成新细胞。对人类来说，这些细胞可以在循环系统里维持约 27年。根据在老鼠身上进行的实验，胎儿细胞能在母亲大脑受损时大量透过血脑屏障去进行修理。不过，在人身上也有胎儿细胞引起的自身免疫性疾病。
无论如何，在大脑发达的人类身上，上述自然修复速率无法和脑神经细胞的总数相比，脑细胞大量死亡的时候不能靠这种机制恢复。现在也还缺少注射神经干细胞或拟似胎儿细胞并主动促使其大规模修复人类大脑的技术，少量干细胞主要产生神经胶质细胞，将其诱导为神经元的规模也很有限。脑部或脊髓损伤通常会导致神经胶质细胞形成瘢痕组织。
- 胶质瘢痕由密集的神经胶质细胞组成，缺乏功能性神经元，长期被视为神经再生的障碍，但其分泌的生长因子又是神经再生所需要的，因此处理起来十分麻烦。目前研究的方向是利用病毒载体调高神经胶质细胞中的神经转录因子水平，诱导其转化为功能性神经元。
已经有对神经干细胞进行基因改造来在大鼠身上提高效率地修复脑梗死等的成功实验。
此外，2020 年 6 月 25 日，《自然》杂志封面文章称，研究人员在体外的人类细胞和小鼠中找到了只需一个步骤就将星形胶质细胞转化为神经元的方法：星形胶质细胞产生的 RNA 结合蛋白 PTBP1（多聚嘧啶道结合蛋白）阻止了这种细胞变成神经元，去除 PTBP1 就可以解决该问题。
- 研究人员首先从小鼠和人类脑中分离出星形胶质细胞，降低这些细胞中 PTBP1 基因的表达水平。经过四个星期，不同样本中 50%~80% 的星形胶质细胞转化为神经元，来自中脑的星形胶质细胞所转化的神经元表达了多巴胺神经元特异性表达的基因。
- 然后，研究人员为小鼠注射可降低 PTBP1 基因表达的病毒。经过 12 个星期，感染病毒的星形胶质细胞有超过 30% 转化为多巴胺神经元、具备正常生理功能，将患有帕金森病的小鼠纹状体的多巴胺含量从（正常值的）25% 提高到 65%，改善了症状。
- 最后，研究人员尝试向小鼠中脑注射可干扰 PTBP1 的反义寡核苷酸，短暂抑制其水平，发现也有治疗效果。这比病毒更适合临床使用，不过该疗法在用于人类之前仍需开展大量研究。 | {
"major": [
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"minor": [
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|
海豚、淡水豚会被鱼刺卡住吗? | 会。从 1997 年至 2011 年，科学家记录到美国佛罗里达州印第安河泻湖沿岸搁浅的 350 条宽吻海豚里有 14 条（12 条成年、2条幼年）是被鱼卡住喉咙而死。
- 相关鱼主要是背鳍中有硬刺的罗非鱼、羊头鲷、Diapterus plumieri（条纹莫哈拉鱼），其中 5 条带有鱼钩、鱼线之类让海豚更难处理的人类产品。
- 海豚吃大型猎物时喉部脱位到食道，被猎物身上出乎意料的结构卡住的概率上升。淡水豚有基本上同样的结构，可以预期有类似海豚的噎死概率。
- 历史上，海豚一般会避开羊头鲷这样危险的猎物，但近年来的环境变化可能让海豚缺少传统猎物。
若题目想问的是鱼的肌间刺，对海豚来说那不是问题。生鱼的肌间刺没有人熟悉的熟鱼身上那么硬，海豚也不怎么将鱼扯成露出肌间刺的碎块。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:561624687, aid:2732311357 | human | true | 暂无版权及作者信息 | [
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|
哪些蜘蛛可以飞行？ | 蜘蛛靠丝线飞起来的方法称为“飞航”，一般不叫飞行。已记录飞航行为的蜘蛛物种分布在 27个科，常见的飞航蜘蛛来自皿蛛科、蟹蛛科、园蛛科、猫蛛科、逍遥蛛科。漏斗蛛科、近管蛛科、地蛛科、管巢蛛科、圆颚蛛科、螲蟷蛛科、卷叶蛛科、石蛛科、隆头蛛科、平腹蛛科、栅蛛科、光盔蛛科、狼蛛科、拟态蛛科、米图蛛科、蜜蛛科、类球蛛科、盗蛛科、跳蛛科、肖蛸蛛科、球蛛科、妩蛛科亦有飞航的物种。
一些科学家实验研究过皿蛛科的 Erigone atra、园蛛科艾蛛属的 Cyclosa turbinata、球蛛科希蛛属的 Achaearaneatepidariorum、隆头蛛科 Stegodyphus 属的 S. dumicola 和 S.mimosarum（大型蜘蛛，放出多条丝线）、狼蛛科豹蛛属的 Pardosa amentata、P. lugubris、P. monticola、P.nigriceps、P. palustris、P. prativaga、P. pullata、P. purbeckensis、P. ramulosa、P.tuoba 等物种的飞航行为。
- 蜘蛛飞航行为受温度、湿度、风速、上升气流、光照强度、食物供应等影响，不同季节有不同物种的蜘蛛在飞，大部分飞航个体的体型较小（可以是幼体也可以是体型小的成体）。
- 飞航过程中，调整蛛丝长度和姿态能在一定程度上影响飞航距离和降落点，必要时可以切离丝线并降落。除风力和上升气流外，飞航可能利用了大气电场。
- 新蛛亚目蜘蛛飞航前通常会爬到高处，然后释放多条细丝或用丝线垂挂一定距离来获得初速度再释放多条细丝并切断垂挂用的丝线。某些猛蛛亚目蜘蛛从高处跳下、在空中放出多条细丝。
- 许多蜘蛛一次只飞航数米远，可以连续飞航。天气条件适宜时，一些蜘蛛能在 6 小时内飞航 30 千米，高度可达 5 千米以上（Foelix，2011 年。飞航蜘蛛偶尔会飞到平流层），低空飞行时可无视 0.5 米高的海浪等障碍。某些连续飞航的蜘蛛能在水平方向上移动 3200 千米以上（Gressitt, 1965 年）。
此外，某些叶螨、至少 31 种鳞翅目昆虫的幼虫也可以飞航。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:561364476, aid:2727585344 | human | true | 暂无版权及作者信息 | [
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|
在自然界的演变中，RNA为什么没有被单链DNA所取代？大部分生物的遗传物质是DNA，显然DNA较RNA有更高的稳定性，那么为什么RNA仍然在进化中保留下来？RNA的部分功能是否可由单链DNA来执行？ | DNA 和蛋白质不能执行 RNA 的某些催化功能，例如在核糖体里发挥催化作用的是且只是 RNA，过去 38 亿到 45亿年的演化都没有改变这状况。剪接体也如此。
不过，题目错的地方比这更基本：
在热力学稳定性上，双链 RNA 胜过双链 DNA，双链 DNA 胜过 DNA-RNA 杂交链。
- 人们其实直到公元 2013 年才能实验测定生物体内 RNA 的结构。真核生物体内大量的小 RNA 是双链的，有很多非编码 RNA 是通过碱基互补跟 mRNA 配对形成双链再去发挥功能的，mRNA 也可能形成双链。
- “RNA 病毒突变快”是因为不进行纠错，碱基错配概率高，而不是“RNA 链不稳定”。
日常所说的“RNA 不稳定”，部分原因是单链 RNA 在高 pH 下 2' 羟基会丢失质子、容易水解，主要原因是地球生物大量使用 RNA 酶，单链 RNA容易被切断。这允许细胞生物降解现有的 mRNA 来调整合成的蛋白质的类型、数量、比例，处理掉带有严重错误的 mRNA。
对储存遗传信息来说，稳定性没有那么重要。
- DNA-RNA 杂交链能维持生物的正常功能。现实中人类自己制造过使用 DNA-RNA 杂交链的大肠杆菌、酵母菌等生物。2018 年 Angad P. Metha 等构建并表征了一种含有 DNA-RNA 杂交链的大肠杆菌菌株，其基因组含有约 40% 到 50% 的核糖核酸。
- 即使考虑单链，人类也早就知道 L-RNA 比自然界的生物使用的 D-RNA 对 RNA 酶稳定得多。
- tRNA 以甲基化修饰和假尿嘧啶、次黄嘌呤核酸等特殊碱基提高稳定性，人们亦可这样修饰 mRNA 来在体外保存并使用。Moderna 和 BioNTech 的 mRNA 疫苗引入假尿嘧啶来提高 mRNA 的稳定性。
热力学稳定性不是用什么做遗传物质的重点。
双链 RNA 的性质较接近 A-DNA，不如 B-DNA 适合与蛋白质进行序列特异性的相互作用。
RNA 不但可以在局部折叠成双链结构，还可以用相距甚远的两段序列互补配对形成双链，比 DNA更容易出现四链结构，可以和多种金属离子、辅助因子互动。在需要执行催化功能而不止储存一些信息时，RNA 胜过 DNA。
- 在自然界，人们主要观察到 RNA 催化两大类反应：磷酸转移，肽键形成。没有这些就没有细胞生物。
- RNA 使用的催化策略相当广泛，可以拿去催化其他反应。RNA 上的一些位点可以结合水合金属离子或蛋白质作为酶时使用的几种辅因子，可以催化光合作用、丙酮酸氧化、固氮、酶促氧化还原反应、碳-碳键的形成与切割（磷酸戊糖途径、卡尔文循环、三羧酸途径）、糖苷键形成、厌氧发酵、乙醇脱氢、自由基生成、异构化、烷基化、Diels-Alder 环加成、金属转移等。
- 尽管 RNA 的官能团有限，在原则上，你可以用 RNA 催化维持生命所必需的亿万种化学反应。
- 已经从自然界的细菌中分离出与 NAD+连接的 RNA。在实验室，已经造出使用包括 NAD 和 CoA 在内的辅因子的核酶 。
“适者生存所以能”不怎么适合当做答案。“适者”体现在“经历自然选择并生存下来”，不解释“为什么能生存下来”。
无论如何，你看到许多现存的无危物种携带某个长期流传的遗传特征，就可以预期“那个遗传特征并不明显减少可育后代的数量”。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:557690605, aid:2711096495 | human | true | 暂无版权及作者信息 | [
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|
为什么胸腺嘧啶一般只能在DNA中？尿嘧啶一般只能在RNA中? | 这取决于核酸的功能。
DNA 使用的胸腺嘧啶 T 是甲基化修饰的尿嘧啶 U，不易被核酸酶降解，不会在 DNA 修复机制纠正胞嘧啶 C 突变为尿嘧啶 U的偶发状况时被错误修理，有利于稳定记录遗传信息。
- 尿嘧啶 U 可以摇摆配对腺嘌呤 A 和鸟嘌呤 G，在多次复制中容易突变。
- 胞嘧啶 C 容易自发脱去环外氨基、变为尿嘧啶 U，改变核酸存储的信息并降低下一次复制后的核酸分子的稳定性（CG 之间有三个氢键，而 AU 之间有两个氢键），造成各种错误。
- DNA 要长期使用并传给后代，没有特殊目的的话，应当处理掉序列里出现的 U。
大部分 RNA 没有 DNA 那么高的稳定性需求，不需要进行这种甲基化修饰。
tRNA 使用大量经过甲基化修饰的碱基，包括胸腺嘧啶。这有利于维持 tRNA 的稳定、有助于和密码子配对、有助于氨基酸与受体臂连接。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
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|
自然界中的生物，雄性参与养育后代的物种占比多少？ | 你想问的大概局限于动物。动物中不到 1% 的物种会尝试照顾后代，其中雄性参与照顾后代的不到这 1% 的10%，能叫做“养育”大概起码提供了一点食物（而不止是“孵蛋、击退敌害”），其比例接近 0%～
- “自然界中的生物”在个体数量和物种数量上绝大部分是病毒、细菌、古菌，动物在生物的个体数量和物种数量中占比接近 0%，是绝对的非主流。
- 几乎没有能通过“额外排放一点有机物”之外的方法去照顾后代的线虫，而线虫在动物的数量里占比极高。照顾后代的节肢动物很少，而节肢动物在动物的物种数里占比很大。这意味着在动物内部，照顾后代的占比也会非常低。
哺乳纲的所有物种都有雌性照顾后代的行为，约 5% 到 6% 的物种存在雄性照顾后代的行为，主要出现在啮齿目、灵长目、食肉目犬科。
- 哺乳纲约 3% 的物种采用社会化一夫一妻制，这些物种里约 59% 存在雄性照顾后代的行为。
- 灵长目约 15% 的物种采用社会化一夫一妻制。
- 看起来，只在照顾后代产生的收益大于“多找几个雌性交配”的时候，雄性哺乳动物才会倾向于照顾后代。这受“是否确定后代是自己的”“自己的食物是否过剩”“自己的性能力如何”等多种因素影响。在人类中，睾丸较小的父亲往往会更多地参与照顾孩子（Jennifer S. Mascaro 等，2013 年）。
鸟纲约 75% 到 81% 的物种有雌雄合作照顾后代的行为，但不是所有个体都会参与。鸟纲约 9% 的物种跨血缘关系合作照顾幼鸟。
鸟纲约 1% 的物种完全由雄性照顾后代，约 8% 到 9% 的物种完全由雌性照顾后代。
- 鸟纲超过 90% 的物种采用社会化一夫一妻制，其中约 81% 的物种存在父母合作照顾后代的行为。
- 鸟纲约 9% 的物种使用一妻多夫制，雄性参与孵蛋。
- 鸟纲不到 1% 的物种使用一夫多妻制，雄性提供大部分照顾。
- 照顾后代可能是鸟的祖传代码，来自非鸟恐龙。
30%的鱼纲科级分类存在照顾后代的行为，其中大半是雄性单独照顾后代，约五分之一是雌雄一起照顾后代。鱼的雄性照顾后代与体外受精明显正相关，大概是因为能确定起码有一部分是自己的后代。海马由雄性负责“怀孕”。
约 6% 的青蛙、约 18% 的蝾螈、约 5% 的蚓螈会照顾后代（或至少看起来有这样做的尝试），其中雄性单独照顾和雌性单独照顾各一半左右，极少合作照顾。
爬行类极少由雄性单独照顾后代。
无脊椎动物极少照顾后代，雄性尤其少。它们提供的照顾以保护为主。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
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} | [
"通用"
] | qid:512851109, aid:2321244229 | human | true | 暂无版权及作者信息 | [
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|
是否发现某类基因决定了动物的社会性?
比如某些动物有明显的社会关系，而另一些动物则偏向独居。 | 成百上千个基因与社会性有关，未发现哪个是决定性的。
- 精氨酸加压素受体、催产素受体等受体参与从田鼠到人类的许多物种的社会行为，这显然关系到编码它们的基因、参与调节基因表达水平的基因与非编码序列；
- 既然是神经系统要参与的事，突触蛋白、离子通道蛋白、昼夜节律相关蛋白显然是绕不过去的；
- 转录因子 egr1 （在不同物种中被称为 ngfi-a、 zif-268、krox-24、tis8 、 zenk 等）参与鸟对声音的反馈，前脑的一些细胞在听到鸟叫声等声音后调高编码该转录因子的基因的表达水平，调高情况在听到白噪音、陌生的鸟叫、熟悉的鸟叫时显著不同，在鸟周围有同类时更与单独行动时不同；编码 egr1 的基因还参与伯氏妊丽鱼对社会地位的识别，此鱼雄性个体会随社会地位变化改变自己的外观，地位越高越鲜艳，求偶的效果也越好，在群体内高地位的个体被人为移除后，其他雄性下丘脑中含有促性腺激素释放激素的神经元的 egr1 表达迅速上升，激素水平变化，进而改变体色；实验显示，egr1 表达可以在 20 到 60 分钟内出现巨大变化，可以立即增强或抑制其他基因的转录，在不同的细胞类型中可以和不同的蛋白质互动。这样的转录因子还有许多；
- 实验显示，在蜜蜂的一生中，脑部数千个基因的表达水平有可测量的差异；蜜蜂感知信息素后会在几天到几周内改变脑中数百个基因的表达。
- 大鼠照顾后代的行为可以在后代海马体引发糖皮质激素受体基因的启动子的表观遗传修饰，影响长期持续。
诸如此类。
注意基因不能直接指定动物的行为。细胞用基因序列和表观遗传修饰存储着构建身体并调控身体功能的各种分子的信息。脑发育、脑活动、动物行为是遗传·环境·生活经历·力学机制·随机扰动等共同创建的。社会信息会和其他内外信息一样影响一些脑细胞的基因表达与突触连接网络，进而影响生物体的一些行为。在完全不修改基因序列的情况下，胚胎细胞受力状况的变化可以彻底改变动物胚胎细胞发育成的东西的身体计划。
此外，社会继承可以在很大程度上解释动物社会网络的结构。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:549951899, aid:2645507678 | human | true | 暂无版权及作者信息 | [
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|
科学家认为，人类都是早产儿，那如果真的按照推测孕育21个月是不是会产生更出色的新人类？ | 不是那样。
“人类都是早产儿”是拿来解释“为什么人类新生儿的生存能力极度低下”的不严谨说法，历史上曾经由阿道夫·波特曼整合到 Sherwood LernedWashburn 于 1960 年提出的“产科困境”假说中，试图将人类女性那异常困难的分娩说成“演化产生的解决方案”。“21个月”是历史上按人大脑发育速度估算的“人新生儿的大脑发育到体积占成年大脑 40%的程度（和黑猩猩新生儿一样）”需要的时间。这是基于过时知识计算的，按新一些的数据，需要的时间更接近 16 个月（DeSilva 和 Lesnik, 2006年）——你有了这程度的知识，就足以意识到，无法指望这产生什么“更出色的新人类”。
古尔德过去附和过“产科困境”，将其推得很广。这是个看起来很好懂的模因，被各路媒体和他们的受众到处传播。但这不是事实。
- 类人猿的足月妊娠所需时间是灵长类里最长的，黑猩猩、倭黑猩猩的正常妊娠期约 39 周，人的正常妊娠期约 37 到 41 周。黑猩猩、倭黑猩猩的新生儿像人新生儿一样缺乏独立生存能力，需要母亲提供大量照顾。但这两种类人猿并没有人这么高的难产率，它们的产道比新生儿尺寸明显更宽。你觉得黑猩猩、倭黑猩猩的新生儿是早产儿吗？
- 正常人类新生儿的大脑体积是成年时的 30% 左右，正常黑猩猩新生儿的大脑体积是成年时的 40% 左右，卷尾猴新生儿的大脑体积是成年时的 50% 左右，早产儿的标准要划定在什么地方？
- 黑猩猩新生儿那 40% 并没有改变其缺乏独立生存能力的表现。
- 人大脑生长速度的峰值出现在出生前的几个月间，与生下来就能行走的动物类似（Halley, 2017 年）。
- 对双胞胎的研究显示，人类女性的骨盆宽度受生活环境的影响似乎超过基因造成的影响。
- 将人类女性的骨盆再加宽几厘米以扩大产道，真的会和“产科困境”假说设想的那样妨碍双足行走吗？非洲南部石器时代晚期的人群似乎搞出过身高很低而骨盆很宽的女性，当时他们经常长距离迁徙，化石并未显示这些人在行走方面有什么困难；农业时代的饮食变化可能比骨盆宽度的遗传因素造成了更多的产科问题。生物力学也不支持“加宽骨盆会增加运动成本”的想法。
- 约 12000 年的农业社会生活尚未让人的骨盆宽度重新优化到适合新生儿的大头。
- 人类胎儿的能量需求和母体能安全地提供的代谢率与散热能力的关系似乎更直接地决定了胎儿能达到的体重。这方面的统计相关性也出现在没有骨头限制产道的鲸类身上（Sacher 和 Staffeldt，1974 年）。
- 提出“产科困境”假说时使用的统计数据可能不反映历史上大部分时候的状况。
人的胎盘在超过 42 周妊娠后会出现明显的功能劣化，血管变细、局部钙化、输送氧和营养物质的能力下降，对胎儿的生命安全构成威胁，胎儿死亡率上升到 40周妊娠的 3 到 6 倍。随着时间流逝，胎儿死亡或畸形的概率还会上升。以当前的人类基因，人的胎盘无法支持“孕育 21 个月”。
人胎盘是怀孕 10 周后由胎儿的合体滋养细胞发育而成，不是子宫的一部分。人造子宫不解决胎盘的问题。
另外，在人类子宫中，超过 42 周的过期妊娠会引起羊水量减少和一些胎儿尺寸过大，难产率上升。
机械胎盘尚未研制出来，可以想象要移除胎儿的胎盘、插几根管子、泡点类似羊水的液体，这可能将胎儿搞死搞残。
人造子宫尚未研制成功，哺乳类幼体很脆弱，目前对 120 天左右的早产儿使用的支持技术（恒温箱、ECMO之类）有概率导致早产儿发生脑出血等自灭。延长机械支持时间会增加出问题的概率。
“在足月后继续将婴儿摆在恒温箱里”类似对重度先天性免疫缺陷患者的措施（无菌隔离），实践不支持那对人的身心发育有任何好处。 | {
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|
假如从世界上第一个人诞生开始，一直往宇宙深处走，到了自然死亡就有下一个人接力。我们可以走多远？ | 看起来，本题目只是想问全人类累积能走多远的路程，不考虑人体有没有能力走到天上去。那么，太阳系的尺寸比不过人类走过的路程。
以尚未观测清楚的奥尔特云计算，太阳系的半径约 1 光年。
1 光年是 9460730472580800 米，精确值。
当前世界上有约 79 亿人，设其中 50 亿人在 1 年间每天拿出 1 小时用 1.5 米每秒的速度走路，那么一年的总路程是 1.04光年多一点。根本就不用出动“从世界上第一个人诞生开始”那么夸张的阵容。
你可以看出，人们对规模巨大的东西是没有概念的，哪怕那个东西是人们自以为最熟悉的“人类”也一样。
目前累计出生过的人数约 1170 亿。这之中有许多个体早死，已经死去的全人类的平均寿命约 35 年。
- 旧石器时代人出生时的预期寿命约 22 到 33 年，新生个体活到 15 岁的概率约 60%，15 岁个体的预期总寿命约 54 年。
- 由于早期农业很菜，新石器时代与青铜时代早中期不少地区的古人类预期寿命比旧石器时代还短数年。
- 古代世界在这方面的改善程度不大，人出生时的预期寿命长期在 25 到 40 年震荡。
- 从 1750 年到 2000 年，新生个体的预期寿命从约 33 年急剧增加到约 67 年。
设 1170 亿人平均每人在 15 年间每天拿出 1 小时用 1.5 米每秒的速度走路，那么总路程将超过 365 光年。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:524724780, aid:2414813875 | human | true | 暂无版权及作者信息 | [
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|
如何用定义证明1/x是双曲线？(F1-F2＝2a)？ | 将曲线 $$\mathscr{H}:y=\frac{1}{x}$$ 顺时针旋转 $$\frac{\pi}{4},$$由于旋转变换并不改变曲线本身形状，我们只需证明这旋转后的曲线 $$\mathscr{H'}$$ 是双曲线即可。
考虑极坐标化， $$\mathscr{H}$$ 的极坐标方程为 $$r^2\sin\theta\cos\theta=1.$$ 假设$$(r,\theta)$$ 是 $$\mathscr{H'}$$ 上任意一点，则将其逆时针旋转 $$\frac{\pi}{4}$$ 还原后的位置$$\left(r,\theta+\frac{\pi}{4}\right)$$ 必落于 $$\mathscr{H}$$ 上，这坐标将满足$$\mathscr{H'}$$ 方程，即有$$r^2\sin\left(\theta+\frac{\pi}{4}\right)\cos\left(\theta+\frac{\pi}{4}\right)=1,\\\$$也就是 $$\frac{1}{2}r^2\cos^2\theta-\frac{1}{2}r^2\sin^2\theta=1,\\\$$ 再化到直角坐标系，有$$\frac{x^2}{2}-\frac{y^2}{2}=1,\\\$$ 这就得到了 $$\mathscr{H'}$$ 的方程，显然表示着一条等轴双曲线。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:376164481, aid:2058925663 | human | true | 暂无版权及作者信息 | [
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|
请问能否不使用微积分的知识证明伯努利不等式？ | 首先指出，问题不严谨。不使用任何微积分知识，要证明实数次幂型的 $$\text{Bernoulli}$$不等式是不可能的，因为事实上，连这个实数次幂本身都需要用微积分的实数理论去定义。于是，假定当前问题只是想要一种不涉及导数的证法，这当然是办得到的。
现在的任务是要证明
> 设 $$x>-1,$$ 则有 > $$(a)$$ 不等式 $$ (1+x)^r\le 1+rx$$ 对所有 $$0<r<1$$ 成立； > $$(b)$$ 不等式 $$ (1+x)^r\ge 1+rx$$ 对所有 $$r>1$$ 成立； > $$(c)$$ 不等式 $$ (1+x)^r\ge 1+rx$$ 对所有 $$r<0$$ 成立。
## 一、先来证明 $$r$$ 是有理数时的情形
$$\color{blue}{\bold{\text{Case 1}}}$$ 当 $$0<r<1\. $$
不妨设 $$r=\frac{p}{q},$$ 其中 $$p,q \in \mathbb{N_+},$$ 此时 $$q-p>0.$$ 同时，因$$x>-1,$$ 也有 $$1+x>0.$$ 于是依 $$\text{AM-GM}$$ 不等式，成立
$$\begin{align*}(1+x)^r&=(1+x)^{\frac{p}{q}}=\sqrt[q]{\underbrace{(1+x)\cdots(1+x)}_{p~~\text{terms}}\cdot\underbrace{1\cdots\cdot1}_{q-p~~\text{terms}}}\\\&\le\frac{p\cdot(1+x)+1\cdot(q-p)}{q}=1+\frac{p}{q}\cdot x=1+rx, \end{align*}\\\$$于是 $$(a)$$ 得证。
$$\color{blue}{\bold{\text{Case 2}}}$$ 当 $$r>1.$$
此时又分两种情况：
* 若 $$1+rx\le0,$$ 则 $$(1+x)^{r}>0\ge 1+rx;$$ * 若 $$1+rx>0,$$ 则 $$rx>-1,$$ 同时注意到 $$0<\frac{1}{r}<1,$$ 于是直接依 $$(a)$$ 成立 $$(1+rx)^{\frac{1}{r}}\le 1+\frac{1}{r}\cdot rx=1+x,$$ 即 $$(1+x)^r\ge1+rx.$$
综上， $$(b)$$ 得证。
$$\color{blue}{\bold{\text{Case 3}}}$$ 当 $$r<0.$$
此时总能求得充分大的 $$n\in \mathbb{N_+}$$ 使得 $$0<-\frac{r}{n}<1$$ 以及$$-1<\frac{rx}{n}<1$$ 同时成立。
于是依 $$(a)$$ 成立
$$0<(1+x)^{-\frac{r}{n}}\le1-\frac{r}{n}\cdot x\le\left(1+\frac{r}{n}\cdotx\right)^{-1},\\\$$ 同时做 $$-n$$ 次幂，将有
$$(1+x)^r\ge \left(1+\frac{r}{n}\cdot x\right)^n\ge1+n\cdot \frac{r}{n}\cdotx=1+rx,\\\$$ 于是 $$(c)$$ 得证。
## 二、再来证明 $$r$$ 是无理数时的情形
此时总能求得一个有理序列 $$\\{r_n\\}$$ 收敛于 $$r,$$ 比如可以将 $$r_n$$ 直接取为 $$r$$ 无尽小数表达式的 $$n$$位小数近似值。
为了证 $$(a),$$ 利用前已证知的结论，并结合函数的连续性和序列极限的有关性质，将有
$$(1+x)^r=\lim_{n \to \infty}(1+x)^{r_n}\le \lim_{n \to \infty}(1+r_nx)=1+rx,\\\$$ 从而 $$(a)$$ 得证。完全类似地，可证 $$(b),(c).$$
至此，我们已避开导数完成了实数次幂型的 $$\text{Bernoulli}$$ 不等式的全部证明。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:458277550, aid:1874108827 | human | true | 暂无版权及作者信息 | [
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|
这道积分题目挺经典的，该怎么处理来着？
设函数 $f$ 在 $[0,1]$ 上有二阶连续导数, 且 $f(0)=f(1)=f^{\prime}(0)=0, f^{\prime}(1)=1$. 求证:
$$
\int_0^1\left(f^{\prime \prime}(x)\right)^2 d x \geqslant 4
$$
先取特殊函数x^3-x^2，二阶导平方后积分值为4，后面怎么说明 任意函数都比它大来着？ | 这里给出一种低门槛、容易理解的做法，更本质的做法需要利用变分法中的 $$\text{Euler-Lagrange}$$ 方程，在此不做讨论。
很显然，对所有的 $$a,b\in\mathbb{R}$$ 都有
$$\int_{0}^1 (f''(x)+ax+b)^2{\rm d}x\ge0.$$
将左端展开，就有
$$\begin{align*} &\int_{0}^1 (f''(x)+ax+b)^2{\rm d}x\\\ =&\int_{0}^1a^2x^2+2abx+b^2{\rm d}x+\int_0^1(2ax+2b)f''(x){\rm d}x+\int_0^1f''(x)^2{\rmd}x,\\\ \end{align*}$$
其中
$$\int_{0}^1 a^2x^2+2abx+b^2{\rm d}x=\frac{a^2}{3}+ab+b^2,$$
$$\int_0^1(2ax+2b)f''(x){\rm d}x=\int_0^1(2ax+2b){\rm d}f'(x)=2a+2b,$$
所以
$$\int_0^1f''(x)^2{\rm d}x\ge -\frac{a^2+(3b+6)a+3b^2+6b}{3},$$
这式子要对一切 $$a,b$$ 成立，必须
$$\int_0^1f''(x)^2{\rm d}x\ge \max_{a,b\in \mathbb{R}}\left(-\frac{a^2+(3b+6)a+3b^2+6b}{3}\right)=4,$$
这最值在 $$a=-6,b=2$$ 时取得，由此一举得到待证不等式的取等条件 f''(x)-6x+2=0," eeimg="1"> 结合题设初值条件，即$$f(x)=x^3-x^2.$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:442878394, aid:1715483130 | human | true | 暂无版权及作者信息 | [
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|
是否存在在R上严格递增且处处不连续的函数？ | 今将证明
> $$\color{blue}{\text{Theorem}}$$ 单调函数的间断点至多可列。
设 $$f(x)$$ 在 $$(a,b)$$ 上单调递增，如果它仅有有限多个间断点，结论是平凡的，我们只需研究这间断点无限多的场合。
首先，我们熟知所有这些间断都是第一类间断，也即是说 $$f(x)$$ 在每个间断点处的左、右极限均存在，只不过它们并不都等于该点的函数值。
假如 $$c$$ 就是这样的一个间断点，于是有 $$f(c-)< f(c+).$$ 为了叙述方便，我们把形如 $$(f(c-),f(c+))$$的开区间称为 $$f(x)$$ 在 $$x=c$$ 处的跃区，简记为 $$L(c).$$
显然地， $$f(x)$$ 在每个间断点处都存在一个跃区，并且可以证明这些跃区必不相交，也即是说
> $$\color{blue}{\text{Prop.}}$$ 对于间断点 $$c'>c,$$ 必有 $$L(c)\cap> L(c')=\varnothing.$$
这证明是容易的。考虑在 $$c,c'$$ 之间插入 $$d,d'$$ 使得 $$c<d<d'<c'.$$ 于是有 $$f(d)\le f(d'),$$现在固定 $$d'$$ 并让 $$d\to c^+,$$ 则有 $$f(c+)\le f(d');$$ 同理，固定 $$d$$ 并让 $$d'\toc'^-,$$可得 $$f(d)\le f(c'-).$$ 于是 $$f(c-)< f(c+)\le f(c'-)<f(c'+).$$ 这就证得了$$\text{Prop. }$$
如此，就可以得到与这无限多个间断点形成双射的跃区，且任何两个跃区都不相交。我们在每个跃区中总能求得一个有理数从而做成某个有理数集。由于跃区不交，这些有理数必不同，于是这做成的有理数集与跃区的全体形成了双射。有理数集可列，从而这跃区也可列，$$\text{Theorem}$$ 由此得证。为了读者理解上的方便，将上述论证过程的脉络总结如下
> 全体间断点与全体跃区形成双射，全体跃区又与某有理数集形成双射，于是全体间断点与这有理数集形成双射。由于任何的有理数集可列，于是全体间断点也可列。
现在，可以回答当前问题如下
> 因为单调函数的间断点至多可列，换言之，单调函数几乎处处连续，于是不存在处处不连续的单调函数。
最后岔开多说几句。 依这里证得的$$\text{Theorem},$$ 单调函数的间断点构成一个零测集，于是再依 $$\text{Lebesgue}$$定理，即可直接断定
> 定义在闭区间上的单调函数可积。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:496499789, aid:2235483577 | human | true | 暂无版权及作者信息 | [
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|
e^(t^2) 的原函数怎么求？ | 这个事情要先从 $$\int e^{-x^2}\,{\rm d}x$$ 说起。事实上 $$\int e^{-x^2}\,{\rm d}x$$也是积不出来的，于是数学上直接定义了一个与之有关的高等函数，称为高斯误差函数（Gauss error function）：
$${\rm erf}(x):=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}{\rm d}t\\\$$
其中之所以带上了 $$\frac{2}{\sqrt{\pi}}$$ 的系数，是因为这个函数在引入当时是作概率论研究用的。很清楚， $${\rmerf}'(x)=\frac{2}{\sqrt{\pi}}e^{-x^2},$$ 于是很容易就可推得
$$\int e^{-x^2}{\rm d}x=\frac{\sqrt\pi}{2}{\rm erf}(x)+C\\\$$
若将其中的 $$x$$ 换为 $$ix,$$ 这里 $$i$$ 是虚数单位（imaginary unit），则还将有
$$i\int e^{x^2}{\rm d}x=\frac{\sqrt\pi}{2}{\rm erf}(ix)+C\\\$$
于是
$$\int e^{x^2}{\rm d}x=-\frac{i\sqrt\pi}{2}{\rm erf}(ix)+C\\\$$
数学家又索性再定义一个虚误差函数（imaginary error function）：
$${\rm erfi}(x):=-i{\rm erf}(ix)\\\$$
这样的话，上式又可以表为
$$\int e^{x^2}{\rm d}x=\frac{\sqrt\pi}{2}{\rm erfi}(x)+C\\\$$
这就将当前的不定积分表成了一个高等函数的形式。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:362258957, aid:945705636 | human | true | 暂无版权及作者信息 | [
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|
二阶导数的符号问题？
为什么不记做 $ rac{d^2 y}{d^2 x}$ ，而是记作 $ rac{d^2 y}{d x^2}$ ，不会让人误解 $d x^2=2 d x$ 吗? 这么做的好处在哪? | 二阶导数之所以记为 $$\frac{{\rm d}^2 y}{{\rm d}x^2}$$ 是有历史渊源的，它来源于二阶差商的概念。
## 一、差分
差商又来源于差分。这里先从差分说起。
对于函数 $$y=f(x)$$ ，赋予自变量 $$x$$ 一个固定的增量 $$\Delta x$$ ，就定义
$$\Delta y=f(x+\Delta x)-f(x)\\\$$
为函数 $$y=f(x)$$ 的一阶差分。那么，将一阶差分再作差分，就得到二阶差分，即
$$\Delta(\Delta y)=\Delta (f(x+\Delta x)-f(x))\\\$$
类似地，还可以定义三阶、四阶……直至 $$n$$ 阶差分。随着阶数的增加，左端的记号会变成 $$\Delta(\Delta(\Deltay)),\Delta(\Delta(\Delta(\Delta y))),\cdots$$ 这过于复杂，于是数学上将它简记为 $$\Delta^n y,$$这里 $$n$$虽然写在右上角，但并不是乘方幂次，仅仅表示差分阶数。因此， $$\Delta^2 y$$ 就是二阶差分，$$\Delta^3y$$就是三阶差分，以此类推。
## 二、差商
有了差分的概念，我们就可以定义差商：
一阶差商 $$\frac{\Delta y}{\Delta x},$$ 二阶差商 $$\frac{\Delta^2 y}{(\Delta x)^2},$$…… ，$$n$$ 阶差商就是 $$\frac{\Delta^n y}{(\Delta x)^n}，$$
这里，分母上的 $$n$$ 确实表示 $$\Delta x$$ 的幂次，括号即使去掉也不会产生误解，只要我们牢记这是 $$\Delta x$$ 的乘方。
## 三、作为差商极限的导数
现在，我们来研究差商和导数的关系。事实上，可以证明：
> 函数的$$n$$ 阶导数（如果存在）就是 $$n$$ 阶差商在 $$\Delta x \to 0$$ 时的极限。
很清楚，对于一阶导数来说，这是成立的，完全就是定义。那么，对于二阶导数又如何呢？这里我们考虑先将 $$\Delta^2y$$ 作一下变形，容易得到
$$\begin{align*} \Delta^2y&=\Delta (f(x+\Delta x)-f(x))\\\ &=f(x+\Deltax+\Delta x)-f(x+\Delta x)-(f(x+\Delta x)-f(x))\\\ &=f(x+2\Delta x)-2f(x+\Deltax)+f(x).\\\ \end{align*}$$
于是二阶差商的极限，就是
$$\begin{align*} \lim_{\Delta x \to 0}\frac{\Delta^2 y}{\Deltax^2}=\lim_{\Delta x \to 0}\frac{f(x+2\Delta x)-2f(x+\Delta x)+f(x)}{(\Deltax)^2}. \end{align*}\\\$$
这个极限怎么求呢？注意，二阶导数是存在的，容易看出这是一个 $$0/0$$ 型极限，于是可以考虑利用洛必达法则，就有
$$\begin{align*} \lim_{\Delta x \to 0}\frac{\Delta^2 y}{\Deltax^2}&=\lim_{\Delta x \to 0}\frac{f(x+2\Delta x)-2f(x+\Delta x)+f(x)}{(\Deltax)^2}\\\ &=\lim_{\Delta x \to 0}\frac{2f'(x+2\Delta x)-2f'(x+\Deltax)}{2\Delta x}\\\ &=\lim_{\Delta x \to 0}\frac{f'(x+2\Delta x)-f'(x+\Deltax)}{\Delta x}.\end{align*}\\\$$
根据二阶导数的定义，这末式显然就是 f''(x)" eeimg="1"> 。于是，我们证得了
$$\lim_{\Delta x \to 0}\frac{\Delta^2 y}{\Delta x^2}=f''(x)=\frac{{\rmd}^2y}{{\rm d}x^2}.\\\$$
仔细观察等式左右两端，右端不过是把左端的极限号去掉，然后把希腊字母 $$\Delta $$ 改成了拉丁字母 $${\rm d}.$$
到这里，我们已经豁然开朗了，为什么二阶导数的记号要写成这个形式？原来分子上的 $$2$$ 代表阶数，分母上的 $$2 $$代表平方。这么说来，二阶导数这个记号确实可以看做是一个比值的极限，即商的极限。如此，我们就大概知道导数又称微商的缘由了。
## 四、一个注记
最后，需要指出，尽管我们在二阶导数存在的前提下证明了，二阶导数就是二阶差商的极限，但却不能直接将二阶导数定义为二阶差商的极限。因为存在着这样一种情况，虽然函数二阶差商极限存在，但函数却不是二阶可导的，读者可以自行寻找这样的反例。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:30140042, aid:872001387 | human | true | 暂无版权及作者信息 | [
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|
如何证明：两个数之间必有无穷多个无理数? | 首先可以证明
> 对任意实数 $$a,b(a<b)$$ ，必存在某个无理数 $$\theta$$ 满足 $$a<\theta<b.$$
这只需要分类讨论即可：
* 若 $$a,b$$ 当中一个是有理数、另一个是无理数，取 $$\theta=\dfrac{a+b}{2}$$ 即合题意； * 若 $$a,b$$ 均为有理数，取 $$\displaystyle \theta=\left(\frac{1}{2}-\frac{\sqrt{2}}{4}\right)a+\left(\frac{1}{2}+\frac{\sqrt{2}}{4}\right)b=\frac{\sqrt{2}}{4}(b-a)+\frac{a+b}{2}$$ 即合题意； * 若 $$a,b$$ 均为无理数，考察 $$\displaystyle c=\frac{a+b}{2},$$ 若 $$c$$ 是无理数，即取 $$\theta=c$$ ，否则取 $$\displaystyle\theta=\frac{a+c}{2}$$ 即可。
有了这个结论，即可进一步证得 $$a,b$$ 之间有无穷多个无理数。若其不然，假如这样的无理数只有有限多个，其中就必有最大者，不妨记为 $$m,$$而依前述结论，在 $$m,b$$ 之间必定还有无理数 $$m'>m,$$ 矛盾。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:608863165, aid:3097483831 | human | true | 暂无版权及作者信息 | [
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|
如何证明多项式 f(x)＝1＋x＋x²/2!＋x³/3!＋…＋x^n/n! 只有一个实数根？（n＝2019） | 今将证明：
> 方程 $$1+x+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!}=0(n \in \mathbb{N^+})$$ 至多有> $$1$$ 个实根。
为此，置 $$f_n(x):=1+x+\frac{x^2}{2!}+\cdots+\frac{x^n}{n!},$$ 再置$$F_n(x):=e^{-x}f_n(x).$$ 由于 $$(\forall x)e^{-x}>0,$$ 所以方程 $$f_n(x)=0$$ 与$$F_n(x)=0$$ 同解，这就只需要研究后者。注意到F_n'(x)=e^{-x}[f_n'(x)-f_n(x)]=-e^{-x}\frac{x^n}{n!}, \end{align*}" eeimg="1">以下依 $$n$$ 的奇偶性分别讨论之：
* 当 $$n$$ 为偶数时，除$$F_n'(0)=0$$外， $$(\forall x \neq 0)F_n'(x)<0,$$ 这表明 $$F_n(x)$$ 严格递减；同时，不难求得 $$\lim_{x \to +\infty} F_n(x)=0.$$ 由此， $$(\forall x)F_n(x)>0,$$ $$F_n(x)=0$$ 无实根。 * 当 $$n$$ 为奇数时，除$$F_n'(0)=0$$外， $$(\forall x<0)F_n'(x)>0,$$ $$(\forall x>0)F_n'(x)<0.$$ 这表明 $$F_n(x)$$ 在 $$(-\infty,0)$$ 上严格递增，在 $$(0,+\infty)$$ 上严格递减。此外，注意到 $$\lim_{x \to -\infty} F_n(x)=-\infty,$$ $$F_n(0)=1,$$ 以及 $$(\forall x>0)F_n(x)>0,$$ 所以 $$F_n(x)=0$$ 在 $$(-\infty,0)$$ 上有且仅有 $$1$$ 个实根。
综上两方面，即证。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:419149559, aid:1456972443 | human | true | 暂无版权及作者信息 | [
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|
请问如何证明下面这个双中值等式?
设函数 $f(x)$ 在 $[a, b]$ 连续, 在 $(a, b)$ 可导, 求证在 $(a, b)$ 内存在相异两点 $\xi$ 和 $\eta$ 使得 $f^{\prime}(\xi) f^{\prime}(\eta)=\left[\frac{f(b)-f(a)}{b-a}\right]^2 $ | ## $$\bold{\text{Proof 1}}$$
置 $$F(x):=[f(x)-f(a)]^2-\left[\frac{f(b)-f(a)}{b-a}\right]^2(x-a)^2,\\\$$ 则有$$F(a)=F(b)=0.$$ 于是依 $$\text{Lagrange}$$ 中值定理，存在 $$\eta \in (a,b)$$ 使得$$F'(\eta)=0,$$ 也即 $$\frac{f(\xi)-f(a)}{\xi-a}\cdotf'(\xi)=\left[\frac{f(b)-f(a)}{b-a}\right]^2,\\\$$ 再依 $$\text{Lagrange}$$中值定理，存在 $$\eta \in(a,\xi)$$ 使得 $$\frac{f(\xi)-f(a)}{\xi-a}=f'(\eta),$$ 于是 $$f'(\xi)f'(\eta)=\left[\frac{f(b)-f(a)}{b-a}\right]^2.\\\$$
* * *
$$\bold{\text{Proof 2}}$$
置 $$\varphi(x):=[f(x)-f(a)]^2,~~\psi(x):=(x-a)^2,\\\$$ 则依 $$\text{Cauchy}$$中值定理，存在 $$\xi \in(a,b)$$ 使得 $$\begin{align*}\frac{\varphi(b)-\varphi(a)}{\psi(b)-\psi(a)}=\frac{\varphi'(\xi)}{\psi'(\xi)},\end{align*}\\\$$ 也就是$$\left[\frac{f(b)-f(a)}{b-a}\right]^2=\frac{f(\xi)-f(a)}{\xi-a}\cdotf'(\xi)\\\$$ 再依 $$\text{Lagrange}$$ 中值定理，存在 $$\eta\in(a,\xi)$$ 使得$$\frac{f(\xi)-f(a)}{\xi-a}=f'(\eta),$$ 代入即得结论。
事实上，可以证明：
> $$\bold{\text{Corollary}}$$ 若 $$f(x)$$ 在 $$[a,b]$$ 上连续，在 $$(a,b)$$> 内可导，则必定存在互不相同的 $$n$$个点$$\xi_1,\xi_2,\cdots,\xi_n\in(a,b)$$ 使得> $${f'(\xi_1)f'(\xi_2)\cdots> f'(\xi_n)=\left[\frac{f(b)-f(a)}{b-a}\right]^n.}\\\$$
这只需要置 $$\varphi(x):=[f(x)-f(a)]^n,~~\psi(x):=(x-a)^n,\\\$$ 进行与前类似的推导再利用归纳法即证。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:400768411, aid:1277727701 | human | true | 暂无版权及作者信息 | [
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|
如何证明 f(x) 在 [0,1] 上最多只有有限个零点？
自己做的时候无意间忽略了 $f^{\prime}$ 不连续的情况，被指出错误后不知道该如何下手了，题目如下:
设函数 $f(x)$ 在区间 $[0,1]$ 可导， 且 $\left\{x \in[0,1] \mid f(x)=0, f^{\prime}(x)=0\right\}=\varnothing$, 证明: $f(x)$ 在 $[0,1]$ 上最多只有有限个零点. | 考虑利用反证法。设若 $$f(x)$$ 在 $$[0,1]$$ 上有无穷多个零点，则依 $$\text{Bolzano-Weierstrass}$$定理，必可求得一列零点 $$\\{x_n\\}$$ 使得，当 $$n\to \infty$$ 时成立 $$x_n\to x^{*}\in[0,1].$$
很清楚，由于 $$f(x)$$ 连续，则 $$f(x^*)=\lim_{n \to \infty}f(x_n)=\lim_{n \to\infty}0=0,\\\$$ 同时，由于 $$f(x)$$ 在 $$x=x^{*}$$ 可导，也即 $$f'(x^*)=\lim_{x \tox^*}\frac{f(x)-f(x^{*})}{x-x^*},\\\$$ 存在，于是依 $$\text{Heine}$$ 定理，又有$$0=\lim_{n \to \infty}\frac{f(x_n)-f(x^*)}{x_n-x^*}=f'(x^*),\\\$$ 于是$$f'(x^*)=0.$$ 由此可见，存在 $$x^*\in [0,1]$$ 满足 $$f(x^*)=f'(x^*)=0,$$这违反了那个有关空集的题设。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:553542307, aid:2673774711 | human | true | 暂无版权及作者信息 | [
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] |
|
这个级数如何求出来呢？直接求解不要文字解释
$\sum_{n=0}^{\infty} rac{(-1)^n}{(2 n+1)^3}= rac{\pi^3}{32}$ | $$\begin{align*}&~~~~\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)^3}\\\&=\frac{1}{2}\sum_{n=0}^{\infty}(-1)^n\int_0^1x^{2n}\ln^2 x{\rm d}x\\\&=\frac{1}{2}\int_0^1 \ln^2x\sum_{n=0}^{\infty}(-1)^nx^{2n}{\rm d}x\\\ &=\frac{1}{2}\int_0^1\frac{\ln^2x}{1+x^2}{\rm d}x\\\&=\frac{1}{4}\int_0^1 \frac{\ln^2 x}{1+x^2}{\rmd}x+\frac{1}{4}\underbrace{\int_0^1\frac{\ln^2 x}{1+x^2}{\rm d}x}_{x~\mapsto~1/x}\\\ &=\frac{1}{4}\int_0^1 \frac{\ln^2 x}{1+x^2}{\rmd}x+\frac{1}{4}\int_1^{+\infty} \frac{\ln^2 x}{1+x^2}{\rmd}x\\\&=\frac{1}{4}\underbrace{\int_0^{+\infty} \frac{\ln^2 x}{1+x^2}{\rmd}x}_{x~\mapsto \tan x}\\\ &=\frac{1}{4}\int_0^{\frac{\pi}{2}} \ln^2 \tanx{\rm d}x\\\ &=\frac{1}{4}\int_0^{\frac{\pi}{2}}(\ln\sin x-\ln\cos x)^2{\rmd}x\\\ &=\frac{1}{4}\int_0^{\frac{\pi}{2}}\left(\left( -\ln2-\sum_{n=1}^{\infty}{\frac{\cos \left( 2nx \right)}{n}} \right)-\left(-\ln2+\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\cos(2nx)}{n}\right)\right)^2{\rm d}x\\\&=\frac{1}{4}\int_0^{\frac{\pi}{2}}\left(\sum_{n=1}^{\infty}\frac{(1+(-1)^{n-1})\cos(2nx)}{n}\right)^2{\rmd}x\\\&=\int_0^{\frac{\pi}{2}}\left(\sum_{n=0}^{\infty}\frac{\cos((4n+2)x)}{2n+1}\right)^2{\rmd}x\\\&=\int_0^{\frac{\pi}{2}}\sum_{n=0}^{\infty}\frac{\cos^2((4n+2)x)}{(2n+1)^2}{\rmd}x\\\&=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\int_0^{\frac{\pi}{2}}\cos^2((4n+2)x){\rmd}x\\\ &=\frac{\pi}{4}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\\\&=\frac{\pi^3}{32}. \end{align*}$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:399350470, aid:1264075315 | human | true | 暂无版权及作者信息 | [
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|
这个反常积分极限要怎么证明呢？
$\lim _{n
ightarrow \infty} \int_0^{+\infty} e^{-x^n} d x=1$
请尝试用最少的知识解决问题。 | 尝试用最少的知识解决问题。当前问题并不需要什么高端工具，只要通过简单放缩并依夹逼定理即可求得结果。首先，通过拆分积分区间，可得
$$A_n:=\int_0^{+\infty} e^{-x^n}{\rm d}x=\int_0^1 e^{-x^n}{\rmd}x+\int_1^{+\infty} e^{-x^n}{\rm d}x=:B_n+C_n,$$
利用基本的不等式 $$(\forall x)e^x\ge1+x$$ 可得
$$1-x^n\le e^{-x^n}=\frac{1}{e^{x^n}}\le\frac{1}{1+x^n}=1-\frac{x^n}{1+x^n}.$$
于是，对于 $$B_n,$$ 有
$$\begin{align*} \frac{n}{n+1}=\int_0^1 (1-x^n ){\rm d}x\le B_n\le \int_0^1\left(1-\frac{x^n}{2}\right){\rm d}x=\frac{2n+1}{2n+2}, \end{align*}$$
依夹逼定理， $$\lim_{n \to \infty} B_n=1.$$ 对于 $$C_n,$$ 有
$$0\le C_n\le \int_1^{+\infty}\frac{{\rm d}x}{1+x^n}\le \int_1^{+\infty}\frac{{\rm d}x}{x^n}=\frac{1}{n-1},$$
依夹逼定理， $$\lim_{n \to \infty} C_n=0.$$ 综上， $$\lim_{n \to \infty} A_n=1.$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:498796656, aid:2223616734 | human | true | 暂无版权及作者信息 | [
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|
如何证明素数无穷多？ | 这里给出一个分析的证明，基本思想属于 $$\text{Leonhard Euler}.$$
仍然利用反证法。设若素数只有有限的 $$m$$ 个，把这 $$m$$ 个素数由小到大排成一列 $$p_1,p_2,p_3,\cdots,p_m,\\\$$自然，其中 $$p_1=2,p_2=3,p_3=5$$ 等等。
对任意一个正整数 $$n,$$基于算术基本定理，它必能分解为这有限个素数相应幂次——次数取自然数，并且这里我们允许出现零次幂——的乘积。换言之，只要把这包含着零的自然数分配给这些素数作为幂的次数，就能生成任意的正整数。比如$$540=2^2\cdot3^3\cdot5^1=p_1^2p_2^3p_3^1 p_4^0p_5^0\cdotsp_m^0,\\\$$进而对任意的单位分数，也就都可以表为各素数倒数的相应幂次的乘积，也比如$$\frac{1}{540}=\frac{1}{p_1^2}\cdot\frac{1}{p_2^3}\cdot\frac{1}{p_3^1}\cdot\frac{1}{p_4^0}\cdot\frac{1}{p_5^0}\cdots \frac{1}{p_m^0}.\\\$$ 由此，不难想见如下乘积$$P:=\sum_{k=0}^\infty \frac{1}{p_1^k}\cdot \sum_{k=0}^\infty \frac{1}{p_2^k}\cdots\sum_{k=0}^\infty \frac{1}{p_m^k},\\\$$ 事实上就表示了所有单位分数之和$$S:=\sum_{k=1}^\infty\frac{1}{k},\\\$$于是 $$P=S.$$ 但 $$P$$是有限的，因为利用几何级数的求和公式，可以得到 $$P=\frac{p_1}{p_1-1}\cdot\frac{p_2}{p_2-1}\cdots\frac{p_m}{p_m-1},\\\$$ 而基于分析的基本结论， $$S$$ 是发散到$$+\infty$$ 的调和级数，矛盾，于是推翻反设。 | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:590703734, aid:3121132182 | human | true | 暂无版权及作者信息 | [
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|
如何优雅地证明这条不等式?
已知正实数 $a, b, c, d$ 满足 $a b c d=1$,
求证 : $\frac{a}{a^3+3}+\frac{b}{b^3+3}+\frac{c}{c^3+3}+\frac{d}{d^3+3} \leq 1$ | 由于 $$a^3+3=(a^3+1+1)+1\ge 3a+1,\\\$$ 所以 $$\sum\frac{a}{a^3+3}\le \sum\frac{a}{3a+1}=\sum \frac{1}{3+\frac{1}{a}},\\\$$若置$$x=\frac{1}{a},y=\frac{1}{b},z=\frac{1}{c},w=\frac{1}{d},$$ 则只需再证$$\sum\frac{1}{3+x}\le 1,\\\$$ 其中 $$xyzw=1.$$
两边同时乘上 $$(3+x)(3+y)(3+z)(3+w),$$ 再作整理，这待证等价于$$3\left(\frac{1}{xy}+\frac{1}{zw}+\frac{1}{xz}+\frac{1}{yw}+\frac{1}{xw}+\frac{1}{yz}\right)+2\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}\right)\ge26,\\\$$这是成立的，因为$$\frac{1}{xy}+\frac{1}{zw}+\frac{1}{xz}+\frac{1}{yw}+\frac{1}{xw}+\frac{1}{yz}\ge6\sqrt{\frac{1}{(xyzw)^3}}=6,\\\\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{w}\ge4\sqrt{\frac{1}{xyzw}}=4.\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:564853975, aid:2746106364 | human | true | 暂无版权及作者信息 | [
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|
如何证明e³＞20? | 考虑利用展开式进行放缩。事实上， $$\forall x\in(0,1),\forall n\in\mathbb{N}$$ 都有
$$\begin{align*}\ln\left(\frac{1+x}{1-x}\right)&=2\left(x+\frac{x^3}{3}+\frac{x^5}{5}+\cdots+\frac{x^{2k+1}}{2k+1}+\cdots\right)\\\\[8pt]&=2\left(\sum_{k=0}^n\frac{x^{2k+1}}{2k+1}+\sum_{k=n+1}^{\infty}\frac{x^{2k+1}}{2k+1}\right)\\\\[8pt]&<2\left(\sum_{k=0}^n\frac{x^{2k+1}}{2k+1}+\sum_{k=n+1}^{\infty}\frac{x^{2k+1}}{2n+1}\right)\\\\[8pt]&=2\left(\sum_{k=0}^n\frac{x^{2k+1}}{2k+1}+\frac{x^{2n+3}}{(2n+1)(1-x^2)}\right).\end{align*}\\\$$ 赋值 $$x=\frac{1}{3},n=2$$ 代入，得
$$\ln2<\frac{1123}{1620}.\\\$$ 赋值 $$x=\frac{1}{9},n=0$$代入，得$$\ln\frac{5}{4}<\frac{9}{40}.\\\$$于是 $$\ln 20=4\ln 2+\ln\left(\frac{5}{4}\right)<4\cdot\frac{1123}{1620}+\frac{9}{40}<3.\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:613154710, aid:3136533443 | human | true | 暂无版权及作者信息 | [
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|
已知 $x, y, z$ 均为正实数，证明如下不等式，不需要文字解释:
$$
\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{y^2+z^2}}+\frac{z}{\sqrt{z^2+x^2}} \leq \frac{3 \sqrt{2}}{2}
$$ | $$\begin{align*}&\left(\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{y^2+z^2}}+\frac{z}{\sqrt{z^2+x^2}}\right)^2\\\&\le[x^2(y^2+z^2)+y^2(z^2+x^2)+z^2(x^2+y^2)]\cdot\\\&~~~~\left[\frac{1}{(x^2+y^2)(y^2+z^2)}+\frac{1}{(y^2+z^2)(z^2+x^2)}+\frac{1}{(z^2+x^2)(x^2+y^2)}\right]\\\&=4+\frac{4x^2y^2z^2}{(x^2+y^2)(y^2+z^2)(z^2+x^2)}\\\ &\le4+\frac{4x^2y^2z^2}{2xy\cdot 2yz\cdot 2zx}\\\ &=\frac{9}{2}. \end{align*}\\\$$
因此得证：
$$
\frac{x}{\sqrt{x^2+y^2}}+\frac{y}{\sqrt{y^2+z^2}}+\frac{z}{\sqrt{z^2+x^2}} \leq \frac{3 \sqrt{2}}{2}
$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:535415836, aid:2512544595 | human | true | 暂无版权及作者信息 | [
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|
求极限(tantanx/sinsinx)^(1/x^2)？无需文字说明 | $$\begin{align*} &\lim_{x \to 0}\left(\frac{\tan \tan x}{\sin\sinx}\right)^{\frac{1}{x^2}}\\\\[8pt] =&\lim_{x \to 0}\left(1+\frac{\tan \tanx-\sin \sin x}{\sin\sin x}\right)^{\frac{1}{x^2}}\\\\[8pt] =&\exp\lim_{x \to0}\frac{\tan \tan x-\sin \sin x}{x^2\sin\sin x}\\\\[8pt] =&\exp\lim_{x \to0}\frac{\tan \tan x-\sin \sin x}{x^3}\\\\[8pt] =&\exp\lim_{x \to0}\left(\frac{\tan \tan x-\tan x}{x^3}+\frac{\sin x-\sin\sinx}{x^3}+\frac{\tan x-\sin x}{x^3}\right)\\\\[8pt] =&\exp\lim_{x \to0}\left(\frac{\tan \color{blue}{\tan x}-\color{blue}{\tanx}}{(\color{blue}{\tan x})^3}+\frac{\color{red}{\sin x}-\sin\color{red}{\sinx}}{(\color{red}{\sin x})^3}+\frac{\tan x-\sin x}{x^3}\right)\\\\[8pt]=&\exp\lim_{x \to 0}\left(\frac{\tan x- x}{x^3}+\frac{x-\sinx}{x^3}+\frac{\tan x-\sin x}{x^3}\right)\\\\[8pt] =&\exp\lim_{x \to0}\frac{2(\tan x-\sin x)}{x^3}\\\\[8pt] =&\rm {e}. \end{align*}\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:624363291, aid:3239133456 | human | true | 暂无版权及作者信息 | [
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|
$\sum_{n=1}^{\infty}\left[n \ln rac{2 n+1}{2 n-1}-1
ight]$这个级数怎么求和呢？给出简洁优雅的过程 | $$\begin{align*} &\sum_{n=1}^{\infty} \left(n\ln\frac{2n+1}{2n-1}-1\right)\\\=&\sum_{n=1}^{\infty}\left(n\ln(2n+1)-n\ln(2n-1)-1\right)\\\=&\sum_{n=1}^{\infty}\int_0^1 \left(\frac{n}{x+2n}-\frac{n}{x-2n}-1\right){\rmd}x\\\ =&\int_0^1\sum_{n=1}^{\infty}\left(\frac{n}{x+2n}-\frac{n}{x-2n}-1\right){\rm d}x\\\=&\frac{1}{4}\int_0^1x^2\sum_{n=1}^{\infty}\frac{1}{n^2-\left(\frac{x}{2}\right)^2}{\rm d}x\\\=&\frac{1}{4}\int_0^1 x^2\cdot\frac{1-\pi\left(\frac{x}{2}\right)\cot\pi\left(\frac{x}{2}\right)}{2\left(\frac{x}{2}\right)^2}{\rm d}x\\\=&\frac{1}{4}\int_0^1 2-\pi x\cot\left(\frac{\pi x}{2}\right){\rm d}x\\\=&\frac{1}{4}(2-2\ln 2)\\\ =&\frac{1}{2}(1-\ln 2). \end{align*}\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:429745975, aid:1569485453 | human | true | 暂无版权及作者信息 | [
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|
请问这个极限怎么做呢？$\lim _{n
ightarrow \infty} \sqrt{n} \prod_{k=1}^n rac{e^{1-1 / k}}{(1+1 / k)^k}$ | 由于
$$\prod_{k=1}^n\left(1+\frac{1}{k}\right)^k=\prod_{k=1}^n\frac{(k+1)^{k+1}}{k^k}\cdot\prod_{k=1}^n\frac{1}{k+1}=\frac{(n+1)^{n+1}}{(n+1)!},$$
$$\prod_{k=1}^n e^{1-\frac{1}{k}}=\exp\sum_{k=1}^n\left(1-\frac{1}{k}\right)=e^{n-H_n},$$
所以
$$\begin{align*} &\lim_{n \to\infty}\sqrt{n}\prod_{k=1}^n\frac{e^{1-\frac{1}{k}}}{(1+\frac{1}{k})^k}\\\=&\lim_{n \to \infty}\frac{\sqrt{n}(n+1)!e^{n-H_n}}{(n+1)^{n+1}}\\\ =&\lim_{n\to\infty}\frac{\sqrt{n}\sqrt{2\pi(n+1)}\left(\frac{n+1}{e}\right)^{n+1}e^{n-H_n}}{(n+1)^{n+1}}\\\=&\sqrt{2\pi}\lim_{n \to \infty}\frac{\sqrt{n^2+n}}{ne^{1+H_n-\ln n}}\\\=&\frac{\sqrt{2\pi}}{e^{1+\gamma}}. \end{align*}$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:512507247, aid:2318157979 | human | true | 暂无版权及作者信息 | [
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|
这个极限怎么求，一个考研题?
$x_n=\sum_{k=0}^n rac{1}{k !}$ 则 $\lim _{n
ightarrow \infty}\left( rac{\ln x_n}{\sqrt[n]{e}-1}-n
ight)$ | $$\begin{align*} L:&=\lim_{n \to \infty}\left(\frac{\lnx_n}{\sqrt[n]{e}-1}-n\right)=\lim_{n \to \infty}\frac{\lnx_n-n(\sqrt[n]{e}-1)}{\sqrt[n]{e}-1}\\\ &=\lim_{n \to \infty}\frac{\lnx_n-n(\sqrt[n]{e}-1)}{\frac{1}{n}}\\\ &=\lim_{n \to \infty} n(\lnx_n-1)-[n^2(\sqrt[n]{e}-1)-n].\\\ \end{align*}$$
其中
$$\begin{align*} J:&=\lim_{n \to \infty}n(\ln x_n-1)=\lim_{n \to \infty}n\ln\left(1-\sum_{k=n+1}^{\infty}\frac{1}{ek!}\right)\\\ &=\lim_{n \to\infty}n\left(-\sum_{k=n+1}^{\infty}\frac{1}{ek!}\right)=-\frac{1}{e}\lim_{n\to \infty}\sum_{k=n+1}^{\infty}\frac{n}{k!}, \end{align*}$$
而
$$0\le\sum_{k=n+1}^{\infty}\frac{n}{k!}\le \sum_{k=n+1}^{\infty}\frac{k}{k!}=\sum_{k=n}^{\infty}\frac{1}{k!}\to0(n\to \infty),$$
所以 $$J=0.$$ 同时
$$\begin{align*} K:&=\lim_{n \to \infty} \left[n^2(\sqrt[n]{e}-1)-n\right]=\lim_{n \to \infty}\left[n^2\left(\frac{1}{n}+\frac{1}{2n^2}+\cdots\right)-n\right]=\frac{1}{2},\end{align*}$$
所以
$$L=J-K=-\frac{1}{2}.$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:512991064, aid:2322652733 | human | true | 暂无版权及作者信息 | [
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|
设 $a_{n+1}=a_n+\frac{1}{a_n} ， a_1=1$ ，
证明 $\lim _{n \rightarrow \infty} \frac{a_n}{\sqrt{n}}=\sqrt{2}$,
求 $\lim _{n \rightarrow \infty} \frac{\sqrt{n}\left(a_n-\sqrt{2 n}\right)}{\ln n}$ | 首先， $$\\{a_n\\}$$ 单调递增，但并不收敛。若其不然，设 $$a_n\to L<+\infty,$$ 则将有$$L=L+\frac{1}{L},$$ 矛盾。于是， $$a_n\to+\infty.$$
进而 $$a_{n+1}^2-a_n^2=2+\frac{1}{a_n^2}\to2,\\\$$ 再依 $$\text{Cesaro-Stolz}$$ 定理$$\frac{a_ {n+1}^2-1}{n}=\frac{1}{n}\sum_{k=1}^n(a_{k+1}^2-a_k^2)\to 2,\\\$$这表明了 $$a_n\sim \sqrt{2n}.\\\$$ 由此，即得 $$a_{n+1}^2-a_n^2-2=\frac{1}{a_n^2}\sim\frac{1}{2n}.\\\$$ 所以 $$\sum_{k=1}^{n-1}(a_{k+1}^2-a_k^2-2)\sim\sum_{k=1}^{n-1}\frac{1}{2k}\sim \frac{1}{2}\ln n,\\\$$ 于是 $$a_n^2-2n\sim\frac{1}{2}\ln n.\\\$$ 所以 $$\begin{align*} a_n&=\sqrt{2n+\frac{\ln n}{2}+o(\lnn)}\\\ &=\sqrt{2n}\sqrt{1+\frac{\ln n}{4n}+o\left(\frac{\ln n}{n}\right)}\\\&=\sqrt{2n}\left(1+\frac{\ln n}{8n}+o\left(\frac{\ln n}{n}\right)\right)\\\&=\sqrt{2n}+\frac{\ln n}{4\sqrt{2n}}+o\left(\frac{\ln n}{\sqrt{n}}\right).\end{align*}\\\$$ 所以 $$\lim_{n \to\infty}\frac{a_n}{\sqrt{n}}=\sqrt{2},~~~~~~~~~~~\lim_{n \to\infty}\frac{\sqrt{n}(a_n-\sqrt{2n})}{\ln n}=\frac{1}{4\sqrt{2}}.\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:570941271, aid:2791349277 | human | true | 暂无版权及作者信息 | [
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|
请问这道极限题怎么求解？
$\lim _{x
ightarrow 1}\left( rac{m}{1-x^m}- rac{n}{1-x^n}
ight) \quad\left(m, n \in N_{+}
ight)$; | 只需利用如下结论
> $$\color{blue}{\lim_{x \to> 1}\left(\frac{k}{1-x^k}-\frac{1}{1-x}\right)=\frac{k-1}{2}~~(k\in\mathbb{N_+}).}\\\$$
这证明很容易，考虑如下因式分解即可：
$$\begin{align*}&~~~~~\frac{k}{1-x^k}-\frac{1}{1-x}\\\&=\frac{k}{1-x^k}-\frac{1+x+x^2+\cdots+x^{k-1}}{1-x^k}\\\&=\frac{(1-1)+(1-x)+(1-x^2)+\cdots+(1-x^{k-1})}{1-x^k}\\\&=\frac{0+1+(1+x)+\cdots+(1+x+x^2+\cdots+x^{k-2})}{1+x+x^2+\cdots+x^{k-1}}.\end{align*}\\\$$
对于当前问题，有
$$\begin{align*} &~~~~\lim_{x \to1}\left(\frac{m}{1-x^m}-\frac{n}{1-x^n}\right)\\\&=\lim_{x \to1}\left(\frac{m}{1-x^m}-\frac{1}{1-x}\right)-\lim_{x \to1}\left(\frac{n}{1-x^n}-\frac{1}{1-x}\right)\\\&=\frac{m-1}{2}-\frac{n-1}{2}\\\ &=\frac{m-n}{2}. \end{align*}\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:478204386, aid:2047531015 | human | true | 暂无版权及作者信息 | [
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|
如何用算数均值不等式证明1+1/n的n+1次方单调递减且有下界？ | 记
$$a_n:=\left(1+\frac{1}{n}\right)^{n+1},~~~n=1,2,\cdots\\\$$
则
$$\begin{align*}\frac{1}{a_n}&=\frac{1}{\left(1+\dfrac{1}{n}\right)^{n+1}}=\left(1-\frac{1}{n+1}\right)^{n+1}\\\&=\left(1-\frac{1}{n+1}\right)\cdot\left(1-\frac{1}{n+1}\right)\cdots\left(1-\frac{1}{n+1}\right)\cdot1\\\ &\leq\left[\dfrac{\left(1-\dfrac{1}{n+1}\right)+\left(1-\dfrac{1}{n+1}\right)+\cdots+\left(1-\dfrac{1}{n+1}\right)+1}{n+2}\right]^{n+2}\\\&= \left[\dfrac{(n+1)\left(1-\dfrac{1}{n+1}\right)+1}{n+2}\right]^{n+2}\\\&=\frac{1}{\left(1+\dfrac{1}{n+1}\right)^{n+2}}\\\ &=\frac{1}{a_{n+1}},\end{align*}\\\$$
这就表明 $$a_n \geq a_{n+1},$$ 由此证得了单调性。至于有界性，显然有 $$0\leq a_n\leq a_1=4.$$
* * *
利用AM-HM不等式(调和平均不超过几何平均）的另证：
$$\left(1+\frac{1}{n}\right)^{n+1}=1\cdot\frac{n+1}{n}\cdots\frac{n+1}{n}\geq\left(\frac{n+2}{1+\frac{n}{n+1}+\cdots+\frac{n}{n+1}}\right)^{n+2}=\left(1+\frac{1}{n+1}\right)^{n+2}.$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:350217456, aid:853759225 | human | true | 暂无版权及作者信息 | [
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|
这个极限如何计算?$\lim _{n
ightarrow \infty} rac{\sum_{k=1}^n k^2 \cdot k^{ rac{1}{k}}}{n^3}$ | 这个问题可以利用极限定义、分而治之地解决。
首先， $$k^{\frac{1}{k}}\ge1(\forall k),$$ 于是
$$\color{blue}{\frac{1}{n^3}\sum_{k=1}^nk^2k^{\frac{1}{k}}\ge\frac{1}{n^3}\sum_{k=1}^nk^2.}$$
同时，我们熟知 $$\lim_{k \to \infty}k^{\frac{1}{k}}=1,$$ 这表明：
$$\forall \varepsilon>0,\exists N>0,$$ 当 $$k> N$$ 时，有 $$k^{\frac{1}{k}}\le1+\varepsilon.$$
于是
$$\begin{align*}\color{blue}{\frac{1}{n^3}\sum_{k=1}^nk^2k^{\frac{1}{k}}}&=\frac{1}{n^3}\sum_{k=1}^{N}k^2k^{\frac{1}{k}}+\frac{1}{n^3}\sum_{k=N+1}^nk^2k^{\frac{1}{k}}\\\ &\color{blue}{\le\frac{1}{n^3}\sum_{k=1}^{N}k^2k^{\frac{1}{k}}+\frac{1+\varepsilon}{n^3}\sum_{k=N+1}^nk^2.}\end{align*}$$
对这两个方向的不等式（着蓝色）取 $$n\to \infty$$ 的上、下极限，即得
$$\frac{1}{3}\le\varliminf_{n \to\infty}\frac{1}{n^3}\sum_{k=1}^nk^2k^{\frac{1}{k}}\le\varlimsup_{n \to\infty}\frac{1}{n^3}\sum_{k=1}^nk^2k^{\frac{1}{k}}\le\frac{1+\varepsilon}{3},$$
再由 $$\varepsilon>0$$ 的任意性，即得
$$\lim_{n \to\infty}\frac{1}{n^3}\sum_{k=1}^nk^2k^{\frac{1}{k}}=\frac{1}{3}.$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:524683889, aid:2414501671 | human | true | 暂无版权及作者信息 | [
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|
是否有办法绕开洛必达法则和泰勒展开来证明当x→0时x-sinx~(x^3)/6？ | 由于求限函数 $$f(x)$$ 是偶函数，不妨径设 $$x>0$$ 求出右极限就够了。
一方面，有 $$\prod_{k=1}^n\cos\frac{x}{2^k}=\frac{1}{2^n}\frac{\sinx}{\sin\frac{x}{2^n}}\ge \frac{1}{2^n}\frac{\sin x}{\frac{x}{2^n}}=\frac{\sinx}{x},\tag1$$ 进而 $$f(x)=\frac{1}{x^2}\left(1-\frac{\sin x}{x}\right)\ge\frac{1}{x^2}\left(1-\prod_{k=1}^n\cos\frac{x}{2^k}\right).\tag2$$ 另一方面，将$$2^n\cdot \frac{\sin\frac{x}{2^n}}{\cos\frac{x}{2^n}}=2^n\tan\frac{x}{2^n}\ge2^n\cdot\frac{x}{2^n}=x\tag3$$ 结合 $$(1)$$ 又得 $$f(x)\le \frac{1}{x^2}\left(1-\cos\frac{x}{2^n}\prod_{k=1}^n\cos\frac{x}{2^k}\right).\tag4$$ 把 $$(2),(4)$$连接起来，取 $$x\to 0_+$$ 的上、下极限，得 $$\frac{1}{6}-\frac{1}{6\cdot 4^n}\le\varliminf_{x \to 0_+}f(x)\le\varlimsup_{x \to 0_+}f(x)\le\frac{1}{6}+\frac{1}{3\cdot 4^n},\tag5$$ 这式子对一切 $$n$$ 成立，再命 $$n\to \infty,$$就有 $$\frac{1}{6}\le \varliminf_{x \to 0_+}f(x)\le\varlimsup_{x \to 0_+}f(x)\le\frac{1}{6},\tag6$$ 至此断定$$ \lim_{x \to 0_+}f(x)=\frac{1}{6}.\tag7$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:626205412, aid:3251201564 | human | true | 暂无版权及作者信息 | [
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|
若[x²]-2[x]>0，求实数x的取值范围? | 若 $$x<0,$$ 则 $$\lfloor x^2 \rfloor \ge0, \lfloor x \rfloor \le-1,$$ 于是$$\lfloor x^2 \rfloor-2\lfloor x\rfloor \ge2>0,$$ 这得部分解。
若 $$x\ge0,$$ 不妨设 $$x=n+r,$$ 其中 $$n\in\mathbb{N}, r\in[0,1),$$ 代入要解的不等式则有
$$\begin{align*} &&\lfloor (n+r)^2 \rfloor-2n>0\\\\[8pt]&\Leftrightarrow&\lfloor n^2+2nr+r^2 \rfloor-2n>0\\\\[8pt]&\Leftrightarrow&n(n-2)+\lfloor 2nr+r^2 \rfloor>0\tag1\\\\[8pt]\end{align*}\\\$$ 这时很清楚，若 $$n\ge3,$$ 必有 $$n(n-2)+\lfloor 2nr+r^2 \rfloor\ge3+0>0,\\\$$ 于是 $$(1)$$ 成立，这又得部分解。
接下来，只需要再考虑 $$n=0,1,2$$ 时的情形。
若 $$n=0,$$ 则 $$n(n-2)+\lfloor 2nr+r^2 \rfloor=\lfloor r^2 \rfloor=0,\\\$$$$(1)$$ 不成立。
若 $$n=1,$$ 则 $$(1)$$式化为 $$\lfloor 2r+r^2 \rfloor>1,\\\$$ 也即$$\begin{cases}2r+r^2\ge2 \\\ 0\le r<1\end{cases}\\\$$ 解得 $$\sqrt{3}-1\ler<1,\\\$$ 于是 $$\sqrt{3}\le x<2$$ 是 $$(1)$$ 的解。
若 $$n=2,$$ 则 $$(1)$$式化为 $$\lfloor 4r+r^2 \rfloor>0,\\\$$ 也即$$\begin{cases}4r+r^2\ge1 \\\ 0\le r<1\end{cases}\\\$$ 解得 $$\sqrt{5}-2\ler<1,\\\$$ 于是 $$\sqrt{5}\le x<3$$ 是 $$(1)$$ 的解。
综上几方面的讨论，可以写出所求解集为 $$x\in (-\infty,0)~\bigcup~ [\sqrt{3},2)~\bigcup~[\sqrt{5},+\infty).\\\$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:611344743, aid:3115803986 | human | true | 暂无版权及作者信息 | [
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|
一个级数问题，如何求解？ | 事实上， $$f(x):=\frac{1}{1-x-x^2}$$ 就是著名的 $$\text{Fibonacci}$$ 序列 $$\\{F_n\\}$$的生成函数。
为了证得这个事实，考虑 $$f(x)$$ 的展开式
$$f(x)=a_0+a_1x+a_2x^2+\cdots,$$
将 $$x=0$$ 代入 $$f(x),$$ 得 $$a_0=1;$$ 再将 $$x=0$$ 带入 $$f'(x),$$ 得 $$a_1=1.$$
此时，注意到
$$\begin{align*} f(x)-1-x=&a_2x^2+a_3x^3+\cdots,\\\xf(x)-x=&a_1x^2+a_2x^3+\cdots,\\\ x^2f(x)=&a_0x^2+a_1x^3+\cdots,\end{align*}$$
为证 $$a_{n+2}=a_{n+1}+a_n,$$ 只需证
$$f(x)-1-x=xf(x)-x+x^2f(x),$$
这是显然成立的，因为这函数方程的解本来就是 $$f(x)=\frac{1}{1-x-x^2}.$$
对于当前问题，由于 $$a_n=F_n,$$ 进而
$$\\[\sum_{n=1}^{\infty} \frac{F_{n+1}}{F_nF_{n+2}} = \sum_{n=1}^{\infty}\frac{F_{n+2} - F_n}{F_nF_{n+2}} = \sum_{n=1}^{\infty} \left(\frac{1}{F_n} -\frac{1}{F_{n+2}} \right) = \frac{1}{F_1} + \frac{1}{F_2} = 2.\\]$$ | {
"major": [
"问答"
],
"minor": [
"知乎问答"
]
} | [
"通用"
] | qid:502158456, aid:2246337566 | human | true | 暂无版权及作者信息 | [
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