qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,802,020 | <p>Let $S$ be a set such that if $A,B\in S$ then $A\cap B,A\triangle B\in S,$ where $\triangle$ denotes the symmetric difference operator. I would like to show that if $S$ contains $A$ and $B$, then it also contains $A\cup B, A\setminus B$.</p>
<p>The difference was easy to find, but I am not succeeding with the union... | Asaf Karagila | 622 | <p>Note that if $A\cap B=\varnothing$, then $A\cup B=A\mathbin\triangle B$. So if you can show that $A\setminus B\in S$, then $A\cup B=(A\setminus B)\mathbin\triangle B$.</p>
<p>Next, observe that $A\cap(A\mathbin\triangle B)=A\cap((A\setminus B)\cup(B\setminus A))=A\setminus B$.</p>
|
4,639,966 | <p>I recently saw the expansion <span class="math-container">$(1+ \frac{1}{x})^n = 1 + \frac{n}{x} + \frac{n(n-1)}{2!x^2} + \frac{n(n-1)(n-2)(n-3)}{3!x^3}.... $</span> where <span class="math-container">$n \in \mathbb Q$</span></p>
<p>From what I understood, they have taken the Taylor series of <span class="math-co... | P. Lawrence | 545,558 | <p>Expand <span class="math-container">$(x+1)^n$</span> as a Taylor series. Then divide through by <span class="math-container">$x^n$</span>.</p>
|
3,415,266 | <p>I cannot find how or why this,</p>
<p><span class="math-container">$$5\sin(3x)−1 = 3$$</span></p>
<p>Has one of two solutions being this,</p>
<p><span class="math-container">$$42.29 + n \times 120^\circ$$</span></p>
<p>I am lost on how to get this solution. I have found the other one, so I will not mention it.</... | user | 505,767 | <p>We have</p>
<p><span class="math-container">$$5\sin(3x)−1 = 3 \iff \sin(3x)=\frac45 $$</span></p>
<p>then, since <span class="math-container">$\arcsin y\in [-\pi/2,\pi/2]$</span> and <span class="math-container">$\sin \theta = \sin (\pi-\theta)$</span>, we have the following (set of) solution(s)</p>
<ul>
<li><spa... |
3,999,652 | <p>Let triangle <span class="math-container">$ABC$</span> is an equilateral triangle. Triangle <span class="math-container">$DEF$</span> is also an equilateral triangle and it is inscribed in triangle <span class="math-container">$ABC \left(D\in BC,E\in AC,F\in AB\right)$</span>. Find <span class="math-container">$\cos... | Math Lover | 801,574 | <p>WLOG, <span class="math-container">$AB = 8, DF = 5$</span>. Say <span class="math-container">$\angle DEC = \theta$</span></p>
<p>If <span class="math-container">$CD = x$</span> then <span class="math-container">$CE = 8 - x$</span></p>
<p>Applying sine law in <span class="math-container">$\triangle CDE$</span>,</p>
<... |
2,716,081 | <p>A stone of mass 50kg starts from rest and is dragged 35m up a slope inclined at 7 degrees to the horizontal by a rope inclined at 25 degrees to the slope. the tension in the rope is 120N and the resistance to motion of the stone is 20N. calculate the speed of the stone after it has moved 35m up the slope.</p>
<p>An... | Aweygan | 234,668 | <p>You're right that $f_n(t)=\sin(nt)$ isn't in $L^2(\mathbb R)$, but multiplying this by a smooth bump function $g$ with $g(t)=1$ on $[0,2\pi]$ and $\text{supp}(g)=[-\varepsilon,2\pi+\varepsilon]$ for some $\varepsilon>0$ does the trick.</p>
|
2,716,081 | <p>A stone of mass 50kg starts from rest and is dragged 35m up a slope inclined at 7 degrees to the horizontal by a rope inclined at 25 degrees to the slope. the tension in the rope is 120N and the resistance to motion of the stone is 20N. calculate the speed of the stone after it has moved 35m up the slope.</p>
<p>An... | zhw. | 228,045 | <p>Choose any smooth <span class="math-container">$f$</span> on <span class="math-container">$\mathbb R$</span> with both <span class="math-container">$\int_{\mathbb R}f^2, \int_{\mathbb R}(f')^2$</span> finite and nonzero. Set <span class="math-container">$f_n(x)=f(nx).$</span> Then</p>
<p><span class="math-container"... |
4,359,136 | <p>I am trying to calculate average number of turns it will take to win in Catan given a set of hexes.</p>
<p>I am stuck at calculating probability of an event given n rolls.
Each roll uses 2 6-sided dice. You get a resource if a specific number(sum of 2 dice) rolls.</p>
<p>Say probability of getting an <strong>ore</st... | JMoravitz | 179,297 | <p>For a direct approach: assuming you only have a single settlement on ore and a single settlement on wheat, and no other source of these (e.g. trading, port, dev cards)... perhaps the easiest approach is via markov chains.</p>
<p>You have <span class="math-container">$12$</span> different possible states you can be i... |
4,359,136 | <p>I am trying to calculate average number of turns it will take to win in Catan given a set of hexes.</p>
<p>I am stuck at calculating probability of an event given n rolls.
Each roll uses 2 6-sided dice. You get a resource if a specific number(sum of 2 dice) rolls.</p>
<p>Say probability of getting an <strong>ore</st... | Ishan Chaurasia | 1,007,050 | <p>If <span class="math-container">$n\leqslant 4$</span>, then clearly it's <span class="math-container">$0$</span>. Otherwise, note that
<span class="math-container">$$
\mathbb{P}(\text{exactly i ores and j wheat after n rolls})=\frac{n!}{i!j!(n-i-j)!}\left(\frac{x}{36}\right)^i\left(\frac{y}{36}\right)^j\left(1-\frac... |
4,032,890 | <p>I am asked to calculate the winding number of an ellipse (it's clearly 1 but I need to calculate it)</p>
<p>I tried two different aproaches but none seems to work.</p>
<p>I would like to know why none of them work (I believe it is because these formulas only work if I have a curve parametrized by arc lenght).</p>
<p... | roydiptajit | 780,430 | <p>From analyticity of <span class="math-container">$\sum_{n=1}^k \frac{1}{n^z}$</span>, we get that
<span class="math-container">$
\int_{\mathfrak{D}}\sum_{n=1}^k \frac{1}{n^z}=0
$</span>, from Goursat's Theorem. Now, from uniform convergence of the partial sums, <span class="math-container">$\sum_{n=1}^k \frac{1}{n^z... |
4,032,890 | <p>I am asked to calculate the winding number of an ellipse (it's clearly 1 but I need to calculate it)</p>
<p>I tried two different aproaches but none seems to work.</p>
<p>I would like to know why none of them work (I believe it is because these formulas only work if I have a curve parametrized by arc lenght).</p>
<p... | reuns | 276,986 | <p>Otherwise it is immediate that for <span class="math-container">$\Re(c)> 1$</span> and <span class="math-container">$|s-c|<\Re(c)-1$</span>, by absolute convergence
<span class="math-container">$$\sum_{n\ge 1}n^{-s}=\sum_{n\ge 1}n^{-c}\sum_{k\ge 0}\frac{((c-s)\log n)^k}{k!}=\sum_{k\ge 0} (c-s)^k \sum_{n\ge 1}n... |
1,382,479 | <p>I would like to know why $a^p \equiv a \pmod p$ is the same as $a^{p-1} \equiv 1 \pmod p$, and also how Fermat's little theorem can be used to derive Euler's theorem, or vice versa. </p>
<p>Please keep in mind that I have little background in math, and I am trying to understand these theorems to understand the math... | André Nicolas | 6,312 | <p><strong>If</strong> $a^{k-1}\equiv 1\pmod{m}$, <strong>then</strong> $a^k\equiv a\pmod{m}$. For $a^{k-1}\equiv 1\pmod{m}$ says that $m$ divides $a^{k-1}-1$. And if $m$ divides $a^{k-1}-1$, then $m$ divides $a(a^{k-1}-1)$, that is, $m$ divides $a^k-a$, and therefore $a^k\equiv a \pmod{m}$.</p>
<p>However, the conver... |
4,487,494 | <blockquote>
<p><strong>Problem:</strong> Let <span class="math-container">$x$</span> and <span class="math-container">$y$</span> be non-zero vectors in <span class="math-container">$\mathbb{R}^n$</span>.<br>
(a) Suppose that <span class="math-container">$\|x+y\|=\|x−y\|$</span>. Show that <span class="math-container">... | Deepak | 151,732 | <p>a) is fine, well done.</p>
<p>For b) you're given <span class="math-container">$\mathbb{x+y}$</span> and <span class="math-container">$\mathbb{x-y}$</span> are perpendicular, which implies <span class="math-container">$\mathbb{(x+y)} \cdot \mathbb{(x-y)} = 0$</span></p>
<p>Now expand and apply the distributive and c... |
549,159 | <p>How to simplify this:</p>
<p>$$(5-\sqrt{3}) \sqrt{7+\frac{5\sqrt{3}}{2}}$$</p>
<p>Dont know how to minimize to 11.</p>
<p>Thanks in advance!</p>
| Olivier | 45,622 | <p>Try to apply the rules you know for working with roots: $\sqrt{a} * \sqrt{b} = \sqrt{a \cdot b}$ and $\sqrt{a} / \sqrt{b} = \sqrt{\frac{a}{b}}$ and $\left(\sqrt{a}\right)^n = \sqrt{a^n}$.</p>
|
5,253 | <p>I have an acyclic digraph that I would like to draw in a pleasing way, but I am having trouble finding a suitable algorithm that fits my special case. My problem is that I want to fix the x-coordinate of each vertex (with some vertices having the same x-coordinate), and only vary the y. My aesthetic criteria are (... | Jay Kominek | 178 | <p>The <a href="http://www.graphviz.org/Documentation.php" rel="nofollow">documentation for GraphViz</a> (a software package that does this sort of thing) has a number of papers on the subject included.</p>
|
2,573,572 | <p>Here is the expression to take the derivative of.
$$C = \frac{1}{2}\sum_j (y_j - a_j^L)^2$$</p>
<p>Here is the result.
$$\frac{\partial C}{\partial a_j^L} = 2(a_j^L-y_j)$$</p>
<p>Multiplying by 2, then again by the derivative of the inside (-1) seems reasonable, but what happened to the summation?</p>
| Ethan Bolker | 72,858 | <p>The notation is more than a little confusing, but I think this is just computing the derivative with respect to a variable that appears in just one of the summands. See what it says with $j=1$ in the answer, leaving the dummy index $j$ in the summation. Or change the dummy $j$ to $i$.</p>
|
2,864,992 | <p>It starts by someone asking an exercise question that whether negation of</p>
<pre><code>2 is a rational number
</code></pre>
<p>is</p>
<pre><code>2 is an irrational number
</code></pre>
<p>Their argument is that they consider it is incorrect if we include complex numbers, because 2 might be complex and not irra... | Empy2 | 81,790 | <p>$$f(x)+\log_2(1+x)-x=f(x^2)\text{ ( 0<x<1 )}\\
f(exp(y))+\log_2(1+\exp(y))-\exp(y)=f(\exp(2y))\text{ $(-\infty<y<0)$}\\
g(y)+\log_2(1+\exp(y))-\exp(y)=g(2y)\text{ $(-\infty<y<0)$}\\
g(-2^z)+\log_2(1+\exp(-2^z))-\exp(-2^z)=g(-2^{z+1})\text{( $-\infty<z<\infty$)}\\
h(z)+\log_2(1+\exp(-2^z))-\ex... |
751,670 | <p>I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay:</p>
<p>$k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$.</p>
<p>Also I am looking for a ring which is not Cohen-Macaulay.</p>
<p>Can anyone help me?</p>
| Georges Elencwajg | 3,217 | <p>a) The ring $k[X,Y,Z]/(XY-Z)$ is isomorphic to the ring $k[X,Y]$, which is regular.<br>
Since a regular ring is Cohen-Macaulay, the original ring $k[X,Y,Z]/(XY-Z)$ is Cohen-Macaulay. </p>
<p>b) The ring $k[X,Y,Z,W]/(XY-ZW)$ is a complete intersection ring and is consequently Cohen-Macaulay. [By the way, this arg... |
1,170,088 | <blockquote>
<p>In a group of $10$ people, $60\%$ have brown eyes. Two people are to be selected at random from the group. What is the probability that neither person selected will
have brown eyes?</p>
</blockquote>
<p>How do I do this problem? $6$ people have brown eyes and $4$ people don't. </p>
<p>The possibil... | heropup | 118,193 | <p>Hint: two <em>different</em> people are selected from the group of ten. In how many ways can you select those two people from the four who do not have brown eyes? In other words, suppose the group is labeled</p>
<p>$$\{B1, B2, B3, B4, B5, B6, N1, N2, N3, N4\}.$$ Then how many ways can you choose two different ... |
1,170,088 | <blockquote>
<p>In a group of $10$ people, $60\%$ have brown eyes. Two people are to be selected at random from the group. What is the probability that neither person selected will
have brown eyes?</p>
</blockquote>
<p>How do I do this problem? $6$ people have brown eyes and $4$ people don't. </p>
<p>The possibil... | robjohn | 13,854 | <p>On the first pick, there is $\frac4{10}$ chance that the person does not have brown eyes. On the second pick, after picking a person without brown eyes, there is $\frac39$ chance that the person does not have brown eyes.
$$
\frac4{10}\cdot\frac39=\frac2{15}
$$</p>
|
2,203,066 | <p>The definition I have is the following:</p>
<blockquote>
<p>A vector space V is said to be <strong>finite-dimensional</strong> if there is a finite set of vectors in V that spans V and is said to be <strong>infinite-dimensional</strong> if no such set exists.</p>
</blockquote>
<p>However, with this definition I ... | John Kontol | 412,568 | <p>The dimension of a vector space <strong>V</strong> is the cardinality of a basis of <strong>V</strong> over its base field.By a basis we mean a set <em>B</em> of vectors $b_i$, spanning the entire space that is also linearly independent. If this basis is finite, the space is call finite-dimensional. Likewise for a... |
2,136,791 | <p>I got a following minimization problem</p>
<p>$$\min_{\mathbf{X}^{(1)}, \, \mathbf{X}^{(2)}} \;\left\| \mathbf{B} - \mathbf{A} (\mathbf{X}^{(1)} \odot \mathbf{X}^{(2)}) \right\|^{2}_{F},$$</p>
<p>where the matrices $\mathbf{B}\in \mathbb{R}^{100 \times 3}$, $\mathbf{A}\in \mathbb{R}^{100\times 36}$, $\mathbf{X}^{(... | Florian | 185,854 | <p>I don't think there is a nice closed-form expression for $\frac{\partial f}{\partial \mathbf X^{(i)}}$, however, I can tell you how you can get to $\frac{\partial f}{\partial \mathbf x^{(i)}}$, where $\mathbf x^{(i)} = {\rm vec}\{\mathbf X^{(i)}\}$. From there, the desired $\frac{\partial f}{\partial \mathbf X^{(i)}... |
632,043 | <p>tl;dr: why is raising by $(p-1)/2$ not always equal to $1$ in $\mathbb{Z}^*_p$?</p>
<p>I was studying the proof of why generators do not have quadratic residues and I stumbled in one step on the proof that I thought might be a good question that might help other people in the future when raising powers modulo $p$.<... | Stefan4024 | 67,746 | <p>Taking roots in modular arithmetics doesn't work.</p>
<p>For example check this:</p>
<p>$9 \equiv 4 \pmod 5$, but $2 \equiv 3 \pmod 5$, doesn't hold.</p>
<p>Now to the problem. If $(g,p) = 1$, then </p>
<p>$$g^{\frac{p-1}{2}} \equiv 1 \pmod p \text { or } g^{\frac{p-1}{2}} \equiv -1 \pmod p$$</p>
<p>This ... |
1,090,620 | <p>I don't know how to solve this limit</p>
<p>$$ \lim_{y\to0} \frac{x e^ { \frac{-x^2}{y^2}}}{y^2}$$</p>
<p>$\frac{1}{e^ { \frac{x^2}{y^2}}} \to 0$</p>
<p>but $\frac{x}{y^2} \to +\infty$</p>
<p>This limit presents the indeterminate form $0 \infty$ ?</p>
| Siminore | 29,672 | <p>The limit is a particular case of the limit
$$
\lim_{u \to +\infty} u e^{-\beta u}, \tag{1}
$$
with $\beta >0$. Indeed, just define $u=y^{-2} \to +\infty$ as $y \to 0$. Rewrite (1) as
$$
\lim_{u \to +\infty} \frac{u}{e^{\beta u}}
$$
and apply De l'Hospital's theorem.</p>
|
1,396,067 | <p><strong>Question:</strong><br/>
The bacteria in a certain culture double every $7.3$ hours. The culture has $7,500$ bacteria at the start.
How many bacteria will the culture contain after $3$ hours?
<br />
<br />
<strong>Possible Answers:</strong><br/>
a. $9,449$ bacteria<br/>
b. $9,972$ bacteria<br/>
c. $40,510$ ba... | Salvatore | 261,037 | <p>The generator of $ mZ \cap nZ $ is the l.c.m of m and n. Intuitively take $3Z \cap 2Z$ it means that you are looking for the smallest common multiple of 2 and 3 which is 6</p>
|
2,250,733 | <p>I have a general idea to solve the problem, which is to pair up 2s and 5s in the numerator and denominator, cancel those that are common, and the remaining pairs of 2s and 5s are the number of 0s left. Since 130 choose 70 is so large, how do I do this?</p>
| N. F. Taussig | 173,070 | <p>By definition,
$$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$
Hence,
$$\binom{130}{70} = \frac{130!}{70!60!}$$
A number ends in zero if it is divisible by $10 = 2 \cdot 5$. Every even number is divisible by at least one factor of $2$, so the number of zeros with which $n!$ ends is determined by the number of factors of ... |
2,222,966 | <p>Given the three line segments below, of lengths a, b and 1, respectively:<a href="https://i.stack.imgur.com/HWoz8.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HWoz8.jpg" alt="enter image description here"></a></p>
<p>construct the following length using a compass and ruler: $$\frac{1}{\sqrt{b+... | fleablood | 280,126 | <p>The triangles you described will have AB=1. BC=a AC=a+1. and BD a perpendicular altitude of unknown height. Triangle ABD is similar to triangle DBC.</p>
<p>So $\frac {AB}{BD} = \frac {DB}{BC}$</p>
<p>so $\frac {1}{BD} = \frac {BD}{a}$</p>
<p>so $a = BD^2$</p>
<p>so $BD = \sqrt{a}$.</p>
<p>The height is no lo... |
590,219 | <p>Disclaimer: this is a homework question. I'm looking for direction, not an answer.</p>
<blockquote>
<p>Given a field <span class="math-container">$F$</span>, show that <span class="math-container">$F[x,x^{-1}]$</span> is a principal ideal domain.</p>
</blockquote>
<p>I'm unsure how to proceed. Would it be better... | Josué Tonelli-Cueto | 15,330 | <p>1st HINT: Given an ideal $I$ of $F[x,x^{-1}]$, what can you say about $I\cap F[x]$? How is the information you obtain related to $I$?</p>
|
1,413,150 | <p>So for a periodic function <span class="math-container">$f$</span> (of period <span class="math-container">$1$</span>, say), I know the Riemann-Lebesgue Lemma which states that if <span class="math-container">$f$</span> is <span class="math-container">$L^1$</span> then the Fourier coefficients <span class="math-cont... | Jack D'Aurizio | 44,121 | <p>As suggested by Clement C., let:
$$ f(x)=\left(1+\frac{1}{x}\right)^{x}+\frac{1}{x}.\tag{1}$$
Then:
$$ f'(x) = \left(1+\frac{1}{x}\right)^{x}\left(\log\left(1+\frac{1}{x}\right)-\frac{1}{x+1}\right)-\frac{1}{x^2}\tag{2} $$
but, due to convexity:
$$\log\left(1+\frac{1}{x}\right)-\frac{1}{x+1}=-\frac{1}{x+1}+\int_{x}^... |
45,163 | <p>I would like to get reccomendations for a text on "advanced" vector analysis. By "advanced", I mean that the discussion should take place in the context of Riemannian manifolds and should provide coordinate-free definitions of divergence, curl, etc. I would like something that has rigorous theory but also plenty of ... | Tim van Beek | 7,556 | <p>Physicists often have the problem that their theories, like general relativity, are very elegant in a coordinate free formulation; but they still need coordinates all the time because they have to compute concrete solutions to concrete problems. So books about mathematically well defined physical theories that make ... |
588,930 | <p>I want help with this question.</p>
<blockquote>
<p>Show that for all $x>0$, $$ \frac{x}{1+x^2}<\tan^{-1}x<x.$$</p>
</blockquote>
<p>Thank you.</p>
| Berci | 41,488 | <p>Let $t:=\tan^{-1}x$ so that $x=\tan(t)$. Then $1+x^2=\displaystyle\frac{\cos^2(t)+\sin^2(t)}{\cos^2(t)}=\frac1{\cos^2(t)}$, so
$$\frac x{1+x^2}=\tan(t)\cdot\cos^2(t)=\sin(t)\cos(t)=\frac{\sin(2t)}2\,.$$
So this leads to prove $\displaystyle\frac{\sin(2t)}2<t<\tan(t)$ for $t\in (0,\pi/2)$. </p>
|
588,930 | <p>I want help with this question.</p>
<blockquote>
<p>Show that for all $x>0$, $$ \frac{x}{1+x^2}<\tan^{-1}x<x.$$</p>
</blockquote>
<p>Thank you.</p>
| HK Lee | 37,116 | <p>$$ \frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2} < \frac{d}{dx} x =1$$
and $\tan^{-1}(0)=0$ so that the last inequality is proved. </p>
<p>$$ \frac{d}{dx} \frac{x}{x^2+1} = \frac{(x^2+1) - x(2x) }{(x^2+1)^2} = \frac{1-x^2}{(x^2+1)^2} < \frac{d}{dx} \tan^{-1} x = \frac{1}{1+x^2} $$
and $\frac{x}{1+x^2}(0)=0=\t... |
1,902,842 | <p>In Bert Mendelson's <em>Introduction to Topology</em>, the first exercise of Ch. 1 Sec. 5 states:</p>
<blockquote>
<p>Let $X\subset A$ and $Y\subset B$. Prove that $$C(X\times Y)=A\times C(Y)\cup C(X) \times B.$$</p>
</blockquote>
<p>I have seen a "proof" of this, but I remain unsatisfied with the result. As sup... | Yes | 155,328 | <p>Let me provide another way to prove such a statement, which is more "intuitive" and admits less chance for one to go wrong. This may not be required by you; yes, I know.</p>
<p>The expression $x^{2}+x-2$ is meaningful for all $x \in \mathbb{R}$. If $x \in \mathbb{R}$, then
$$
|x^{2}+x-2 - 4| = |x-2||x+3|.
$$
If $|x... |
1,904,903 | <p>Taken from Soo T. Tan's Calculus textbook Chapter 9.7 Exercise 27-</p>
<p>Define $$a_n=\frac{2\cdot 4\cdot 6\cdot\ldots\cdot 2n}{3\cdot 5\cdot7\cdot\ldots\cdot (2n+1)}$$
One needs to prove the convergence or divergence of the series $$\sum_{n=1}^{\infty} a_n$$</p>
<p>upon finding the radius of convergence for $\su... | Claude Leibovici | 82,404 | <p>Another way to look at the problem could be to consider that $$a_n=\frac {\prod_{i=1}^{n }(2i) }{\prod_{i=1}^{n }(2i+1) }$$ Using the ratio test $$\frac{a_{n+1}}{a_n}=\frac{2 (n+1)}{2 n+3}$$ which is inconclusive.</p>
<p>Using <a href="https://en.wikipedia.org/wiki/Ratio_test#Raabe.27s_test" rel="nofollow">Raabe's ... |
2,195,287 | <blockquote>
<p>Knowing that $p$ is prime and $n$ is a natural number show that
$$n^{41}\equiv n\bmod 55$$
using Fermat's little theorem
$$n^p\equiv n\bmod p$$</p>
</blockquote>
<p>If the exercise was to show that
$$n^{41}\equiv n\bmod 11$$ I would just rewrite $n^{41}$ as a power of $11$ and would easily prov... | Alberto Andrenucci | 370,680 | <p>You use the Chinese Remainder Theorem:</p>
<p>$$\begin{equation}
\begin{cases}
n^{41}\equiv n \mod(11)\\n^{41}\equiv n \mod(5)
\end{cases}
\end{equation}$$</p>
<p>Now you can apply the Fermat's little theorem, using the fact that $n^{\phi(n)}\equiv1 \mod(p)$ (Euler's Theorem) to obtain:</p>
<p>$$\begin{... |
553,431 | <p>In the <a href="http://demonstrations.wolfram.com/NoFourInPlaneProblem/" rel="nofollow noreferrer">No-Four-In-Plane problem</a>, points are selected so that no four of them are coplanar.</p>
<p>Eight points can be picked from a <span class="math-container">$3\times3\times3$</span> space in a unique way.</p>
<p>Can 1... | Hugo Pfoertner | 686,508 | <p>See <a href="https://oeis.org/A280537" rel="nofollow noreferrer">https://oeis.org/A280537</a> and <a href="https://oeis.org/A280538" rel="nofollow noreferrer">https://oeis.org/A280538</a> for the 6 x 6 x 6 results.
The maximum number of points in the 6x6x6 case is 16. There are 36 solutions; a list of coordinates is... |
255,295 | <p>I just did one exercise stating:
Prove that the linear map $M: X \rightarrow C([0,1])$, is continuous iff for every $t\in[0,1]$, the rule $x\rightarrow (Mx)(t)$ defines a continuous linear functional on X.
the next exercise stated:
State, and prove a similar continuity criterion for linear maps $M:X\rightarrow Y$ wh... | Thomas | 49,120 | <p>Operators $\ell:\: X \rightarrow Y$ are not linear functionals. Think about how the linear functionals where defined in the example of $Y = C([0,1])$ as a composition of something with $M$ and try to generalize that. Use the representation of the norm in $X$ via linear functionals, and the uniform boundedness princi... |
3,002,114 | <blockquote>
<p>Prove that
<span class="math-container">$$
\binom{n}{1}^2+2\binom{n}{2}^2+\cdots + n\binom{n}{n}^2
= n \binom{2n-1}{n-1}.
$$</span></p>
</blockquote>
<p>So
<span class="math-container">$$
\sum_{k=1}^n k \binom{n}{k}^2
= \sum_{k=1}^n k \binom{n}{k}\binom{n}{k}
= \sum_{k=1}^n n \binom{n-1}{k-1} \bino... | drhab | 75,923 | <p><span class="math-container">$$\sum_{k=1}^{n}k\binom{n}{k}^{2}=n\sum_{k=1}^{n}\binom{n-1}{k-1}\binom{n}{n-k}=n\sum_{k=0}^{n-1}\binom{n-1}{k}\binom{n}{n-k-1}=n\binom{2n-1}{n-1}$$</span></p>
<p>Applying <a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity" rel="nofollow noreferrer">Vandermonde's identity</... |
2,831,731 | <p>I don't know how should i define a homotopy on a set.
I think {{},{a,b,c}} should work but i don't know how to write the homotopy between the identity map and a constant map.
(So sorry for this basic quistion.....)</p>
| mathcounterexamples.net | 187,663 | <p>So if I understand well... You have a set $X=\{a,b,c\}$ only containing three points. You want to define a topology $\mathcal T$ on $X$ such that $X$ is contractile.</p>
<p>If that is the question, then indeed the trivial topology $\mathcal T =\{\emptyset, \{a,b,c\}\}$ is convenient. Why?</p>
<p>Consider the map d... |
39,424 | <p>I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?</p>
| Pete L. Clark | 1,149 | <p>I don't think the OP has provided enough information to get a useful answer to his/her precise question (what text to learn quickly from).</p>
<p>What level is the course being taught at? High school? Undergraduate for non-majors? Undergraduate for majors but without specific knowledge of any other undergraduate... |
39,424 | <p>I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?</p>
| Niemi | 8,590 | <p>What about "The Little Book of Bigger Primes" by Ribenboim (see <a href="http://rads.stackoverflow.com/amzn/click/0387201696" rel="nofollow">1</a> for the Amazon link)? I personally think this is a great introduction to the field of number theory and I have enjoyed it very much a few years ago. It is clear and nicel... |
2,788,015 | <p>I'm trying to solve an exercise that says</p>
<blockquote>
<p>Show that a locally compact space is $\sigma$-compact if and only if is separable.</p>
</blockquote>
<p>Here locally compact means that also is Hausdorff. I had shown that separability imply $\sigma$-compactness but I'm stuck in the other direction.</... | Henno Brandsma | 4,280 | <p>As I said, you cannot say in general that a locally compact space is separable iff it is $\sigma$-compact. </p>
<p>There are many classic compact spaces that are not separable, e.g. $[0,1]^I$ where $|I| > \mathfrak{c}$, and the lexicographically ordered square $[0,1] \times [0,1]$ in the order topology or the Al... |
3,232,341 | <p>How would I show this? I know a directed graph with no cycles has at least one node of outdegree zero (because a graph where every node has outdegree one contains a cycle), but do not know where to go from here.</p>
| hmakholm left over Monica | 14,366 | <p>Just reverse the direction of all edges. This cannot produce a cycle where there wasn't one before, and the indegrees are now outdegrees.</p>
|
2,098,693 | <p>Full Question: Five balls are randomly chosen, without replacement, from an urn that contains $5$
red, $6$ white, and $7$ blue balls. What is the probability of getting at least one ball of
each colour?</p>
<p>I have been trying to answer this by taking the complement of the event but it is getting quite complex. A... | Community | -1 | <p>Try dividing it into all possible cases.</p>
<p>Case $1$: We take $2$ red, $2$ white and one blue ball. The total ways are : $W_1= \binom {5 }{2}\cdot \binom {6}{2}\cdot \binom {7}{1} $ ways.</p>
<p>Case $2$: We take $1$ red, $2$ white and $2$ blue balls. Here the number of ways are : $W_2 = \binom {5}{1}\cdot \b... |
131,524 | <p>I am an amateur mathematician, and certainly not a set theorist, but there seems to me to be an easy way around the reflexive paradox: Add to set theory the primitive $A(x,y)$, which we may think of as meaning that $x$ is allowed to belong to $y$ and the axiom</p>
<p>$\forall x,\forall y, x\in y \rightarrow A(x,y)$... | Noah Schweber | 8,133 | <p>As The User says in the comments, you still have a problem, aesthetically at least -- in order to prevent the existence of "silly" models, you need some axiom asserting that $\in^*$ isn't too big. As is, a model in which $\in^*$ always holds between any $x$ and $y$ satisfies your axioms; this means that your separat... |
4,206,147 | <blockquote>
<p><span class="math-container">$f(f(x))=f(x),$</span> for all <span class="math-container">$x\in\Bbb R$</span> suppose <span class="math-container">$f$</span> is differentiable, show <span class="math-container">$f$</span> is constant or <span class="math-container">$f(x)=x$</span></p>
</blockquote>
<p>Cl... | Daniel McLaury | 3,296 | <p>I'm not able to follow the argument given in the existing answer. It may be perfectly valid, but I don't understand what's being said at several points. So let me give my own answer.</p>
<p>Suppose <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span> is a differentiable function such that
<span clas... |
1,345,643 | <p>In an exercise it seems I must use Pascal's triangle to solve this $(z^1+z^2+z^3+z^4)^3$. The result would be $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. But how do I use the triangle to get to that result? Personally I can only solve things like $(x+y)^2$ and $(x+y)^3$.</... | Jack D'Aurizio | 44,121 | <p>$$(z+z^2+z^3+z^4)^3 = z^3\cdot\left(\frac{1-z^4}{1-z}\right)^3=z^3(1-3z^4+3z^8-z^{12})\sum_{n\geq 0}\binom{n+2}{2}z^n$$
hence:
$$\begin{eqnarray*}(z+z^2+z^3+z^4)^3&=&(z^3-3z^7+3z^{11}-z^{15})\sum_{n\geq 0}\binom{n+2}{2}z^n\\&=&\sum_{n\geq 3}\binom{n-1}{2}z^n-3\sum_{n\geq 7}\binom{n-5}{2}+3\sum_{n\geq... |
179,377 | <p>Consider the $k \times k$ block matrix:</p>
<p>$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \ddots &
\vdots \\ B & B & B & \cdots & A
\end{array}\right) = I_k \otimes (A - B) + \mathbb{... | Rodrigo de Azevedo | 91,764 | <p>Subtracting the last row of blocks from the first $k-1$ rows of blocks, we obtain</p>
<p>$$\begin{bmatrix}A-B & O & O & \dots & O & B-A\\ O & A-B & O & \dots & O & B-A\\ O & O & A-B & \dots & O & B-A\\ \vdots & \vdots & \vdots & \ddots & \v... |
1,334,527 | <p>The integral in hand is
$$
I(n) = \frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx
$$
I dont know whether it has closed-form or not, but currently I only want to know its asymptotic behavior. Setting $x=\cos\theta$, then
$$
I(n) = \frac{1}{\pi}\int_{0}^{\pi/2} \Big[(1+2\cos\theta)^{2n}+(1-2\cos\theta... | zhw. | 228,045 | <p>Let's look at</p>
<p>$$(1)\,\,\,\,\int_{0}^{\pi/2}(1+2\cos t)^{n}\, dt = 3^n\int_{0}^{\pi/2}(1/3+2(\cos t)/3)^{n}\, dt.$$</p>
<p>For the last integral, we can look at $\int_{0}^{b}(1/3+2\cos t/3)^{n}\, dt$ for any small $b>0,$ the rest of the integral decreasing exponentially as $n\to \infty.$ Now near $0,\cos ... |
1,658,577 | <p>I'm an electrical/computer engineering student and have taken fair number of engineering math courses. In addition to Calc 1/2/3 (differential, integral and multivariable respectfully), I've also taken a course on linear algebra, basic differential equations, basic complex analysis, probability and signal processing... | layman | 131,740 | <p>I don't think if you relearn the material you'll get bored because I don't think you would be <em>re</em>learning the material. To relearn something means you've already learned it once, and now you are learning it again. But you said yourself your courses never focused on proving theorems. It's a very different ... |
19,261 | <p>Every simple graph $G$ can be represented ("drawn") by numbers in the following way:</p>
<ol>
<li><p>Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. <br/></p></li>
<li><p>Assign to each maximal clique $C_j$ a unique p... | Igor Pak | 4,040 | <p>About the Fary–Milnor theorem. Milnor's original proof is already very nice (see <a href="http://www.jstor.org/stable/1969467" rel="noreferrer">here</a>). I also very much like <a href="http://www.jstor.org/stable/119165" rel="noreferrer">this proof</a> by Alexander & Bishop (see also a version of this proof i... |
19,261 | <p>Every simple graph $G$ can be represented ("drawn") by numbers in the following way:</p>
<ol>
<li><p>Assign to each vertex $v_i$ a number $n_i$ such that all $n_i$, $n_j$ are coprime whenever $i\neq j$. Let $V$ be the set of numbers thus assigned. <br/></p></li>
<li><p>Assign to each maximal clique $C_j$ a unique p... | Joel Fine | 380 | <p>A nice topic to read about is Chern-Weil theory. This is the generalisation of Gauss-Bonnet to higher dimensions and to vector bundles other than the tangent bundle. Put very briefly, topological invariants of a vector bundle over a manifold (its characteristic classes - certain classes in the cohomology of the base... |
3,261,846 | <blockquote>
<p>What is the solution to the IVP <span class="math-container">$$y'+y=|x|, \ x \in \mathbb{R}, \ y(-1)=0$$</span></p>
</blockquote>
<p>The general solution of the above problem is <span class="math-container">$y_{g}(x)=ce^{-x}$</span>.</p>
<p>How to find the particular solution? As <span class="math-c... | E.H.E | 187,799 | <p>or use the variation of parameters method
<span class="math-container">$$y=y_c+y_p$$</span>
we know that <span class="math-container">$y_c=ce^{-x}$</span></p>
<p>so
<span class="math-container">$$y_p=u(x)e^{-x}$$</span>
<span class="math-container">$$y'_p=u'(x)e^{-x}-u(x)e^{-x}$$</span>
substitute in the D.E to get... |
4,539,739 | <p>Here is the curve <span class="math-container">$y=2^{n-1}\prod\limits_{k=0}^n \left(x-\cos{\frac{k\pi}{n}}\right)$</span>, shown with example <span class="math-container">$n=8$</span>, together with the unit circle centred at the origin.</p>
<p><a href="https://i.stack.imgur.com/mBNbY.png" rel="noreferrer"><img src=... | Anders Kaseorg | 38,671 | <p>A simple parameterization of the curve is given by</p>
<p><span class="math-container">$$x = \cos \frac tn, \quad y = -{\sin t \sin \frac tn}, \quad 0 < t < nπ.$$</span></p>
<p>If we could approximate the arc length of the <span class="math-container">$k$</span>th lobe <span class="math-container">$(k - 1)π &l... |
1,443,441 | <blockquote>
<p>If <span class="math-container">$\frac{x^2+y^2}{x+y}=4$</span>,then all possible values of <span class="math-container">$(x-y)$</span> are given by<br></p>
<p><span class="math-container">$(A)\left[-2\sqrt2,2\sqrt2\right]\hspace{1cm}(B)\left\{-4,4\right\}\hspace{1cm}(C)\left[-4,4\right]\hspace{1cm}(D)\l... | Jack D'Aurizio | 44,121 | <p>$\frac{x^2+y^2}{x+y}=4$ is equivalent to $(x-2)^2+(y-2)^2 = 8$, hence $(x,y)$ lies on a circle centered at $(2,2)$ with radius $2\sqrt{2}$. The tangents at the points $(0,4)$ and $(4,0)$ are parallel to the $y=x$ line, so the right answer is $(C)$.</p>
|
1,443,441 | <blockquote>
<p>If <span class="math-container">$\frac{x^2+y^2}{x+y}=4$</span>,then all possible values of <span class="math-container">$(x-y)$</span> are given by<br></p>
<p><span class="math-container">$(A)\left[-2\sqrt2,2\sqrt2\right]\hspace{1cm}(B)\left\{-4,4\right\}\hspace{1cm}(C)\left[-4,4\right]\hspace{1cm}(D)\l... | Bart Michels | 43,288 | <p>The condition $\frac{x^2+y^2}{x+y}=4$ is equivalent to $(x+y)^2+(x-y)^2=8(x+y)$. Let $x+y=s$ and $x-y=d$. Note that this induces a bijection from $\Bbb R^2$ to itself, meaning that for every pair $(s,d)$ there exist corresponding $x,y$. We have $0\leq d^2=8s-s^2$. The nonnegative values that $8s-s^2$ can take are $[... |
452,306 | <p>I am trying to be able to find the radius of a cone combined with a cylinder.
see my other question
(Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder? part2 )</p>
<p>I have a volume calculation that Has been reduced as far as I know how to.</p>
... | Mariano Suárez-Álvarez | 274 | <p>Let me suppose your group is second-countable, so that we need only worry abour sequences.</p>
<p>It is enough to show that for every $g\in G$ we have that $g^{-1}$ is an element of the closure of $\{g^i:i\geq0\}$.</p>
<p>So pick any convergent subsequence of $(g^i)_{i\geq0}$, say $(g^{n_i})_{i\geq0}$, and let $h\... |
3,997,532 | <p>Please help me with this. I can't prove the result. Tried integral by parts or notations, nothing working</p>
<p><span class="math-container">$$\int_{-1}^{1}{\frac{x^2}{e^x+1}}dx$$</span></p>
| Riemann | 27,899 | <p>Let <span class="math-container">$t=-x$</span>, then
<span class="math-container">\begin{align*}
\int_{-1}^{1}{\frac{x^2}{e^x+1}}dx
&=\int_{1}^{-1}{\frac{t^2}{e^{-t}+1}}(-dt)\\
&=\int_{-1}^{1}{\frac{t^2}{e^{-t}+1}}dt\\
&=\int_{-1}^{1}{\frac{x^2e^x}{e^x+1}}dx.
\end{align*}</span>
<span class="math-contain... |
317,175 | <p>What tools would we like to use here? Is there any easy way to establish the limit?</p>
<p>$$\sum_{k=1}^{\infty}{1 \over k^{2}}\,\cot\left(1 \over k\right)$$</p>
<p>Thanks!</p>
<p>Sis!</p>
| Community | -1 | <p>Since $\displaystyle\cot{\frac{1}{x}}=\frac{\cos\frac{1}{x}}{\sin \frac{1}{x}}$ and $\cos\frac{1}{x}\sim_{+\infty}1$, $\sin\frac{1}{x}\sim_{+\infty}\frac{1}{x}$ then
$$\frac{1}{k^2}\cot{\frac{1}{k}}\sim_{+\infty}\frac{1}{k}.$$
So the serie with positive terms $\displaystyle\sum_k \frac{1}{k^2}\cot{\frac{1}{k}}$ is... |
1,068,631 | <p>I want to find the solutions of the equation $$\left[z- \left( 4+\frac{1}{2}i\right)\right]^k = 1 $$ </p>
<p>in terms of roots of unity.</p>
<p>When I try to solve this, I get
\begin{align*}z - 4 - \dfrac i2 &= 1\\
z-\dfrac{i}{2}&=5\\
\dfrac{2z-i}2 &= 5\\
z&= 5 + \dfrac i2\end{align*}</p>
<p>Is t... | Przemysław Scherwentke | 72,361 | <p>HINT: You should find (or only name?) $w_0$, $w_1$, ... $w_{k-1}$, which are $k$-th roots of 1 (2, respectively) and compare them one by one with $z-4+\frac12i$.</p>
|
2,502,963 | <p>How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this...</p>
| 5xum | 112,884 | <p>For me during calculus, the steps were:</p>
<ol>
<li>Define $e$ as $\lim_{n\to\infty}\left(1+\frac1n\right)^n$</li>
<li>Define $\ln x$ as the inverse function to $e^x$.</li>
<li>Prove that $\frac{d}{dx} \ln x = \frac1x$</li>
<li>From 3, prove that $\frac{d}{dx}e^x=e^x$</li>
<li>Prove that, if a function $f$ is infi... |
133,711 | <p>I am trying to show that $$\int_{-\pi}^{\pi}e^{\alpha \cos t}\sin(\alpha \sin t)dt=0$$</p>
<p>Where $\alpha$ is a real constant.</p>
<hr>
<p>I found the problem while studying a particular question in this room,<a href="https://math.stackexchange.com/questions/124868/evaluate-int-c-frace-alpha-zzdz-where-alpha-in... | N. S. | 9,176 | <p><strong>Hint</strong> Your function is ODD.... </p>
<p>This is one of the pretty standard result in Calculus:</p>
<p>If $f(t)$ is a continuous, odd function, then </p>
<p>$$\int_{-a}^a f(t) dt =0$$</p>
<p>Proof: Substitute $u=-t$. </p>
<p>Second Proof: Think of the integral as a signed area....</p>
|
1,007,533 | <p>Prove that if $v$ is an eigenvector for the matrix $A$, then $A^2v=c^2v$</p>
<p>Pretty much all I have is:</p>
<p>$Av=cv$ where $v$ is a nonzero vector</p>
| Milly | 182,459 | <p>Apply $A$ both sides of $Av=cv$ (i.e. $A^2v=Acv$), and use that $A$ commutes with multiplication by constants...</p>
|
4,074,718 | <p>The angle bisectors of <span class="math-container">$\angle B$</span> and <span class="math-container">$\angle C_{ex}$</span> intersect at point <span class="math-container">$E$</span>. If <span class="math-container">$\angle A=70^\circ$</span>, what is <span class="math-container">$\angle E$</span> equal to?</p>
<p... | MathIsNice1729 | 274,536 | <p><strong>Hint :</strong> Let <span class="math-container">$f$</span> be a polynomial with integer coefficients. Then if <span class="math-container">$a+\sqrt b$</span> is a root of <span class="math-container">$f$</span>, where <span class="math-container">$a, b \in \Bbb Q, \sqrt b \notin \Bbb Q$</span>, then <span c... |
4,003,948 | <p>In the Book that I'm reading (Mathematics for Machine Learning), the following para is given, while listing the properties of a matrix determinant:</p>
<blockquote>
<p>Similar matrices (Definition 2.22) possess the same determinant.
Therefore, for a linear mapping <span class="math-container">$Φ : V → V$</span> all ... | Community | -1 | <p>Given an abstract finite dimensional (real) vector space <span class="math-container">$V$</span>, a <em><a href="https://en.wikipedia.org/wiki/Linear_map" rel="nofollow noreferrer">linear transformation</a></em> <span class="math-container">$\Phi:V\to V$</span> is a map such that for any <span class="math-container"... |
2,227,047 | <p>For any $x=x_1, \dotsc, x_n$, $y=y_1, \dotsc, y_n$ in $\mathbf E^n$, define $\|x-y\|=\max_{1 \le k \le n}|x_k-y_k|$. Let $f\colon\mathbf E^n \to \mathbf E^n$ be given by $f(x)=y$, where $y_k= \sum_{i=1}^n a_{ki} x_i + b_k$ where $k =1,2, \dotsc,n$. Under what conditions is $f$ a contraction mapping?</p>
<p>Any hint... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
1,051,372 | <p>If $|z_1|=1,|z_2|=1$, how can one prove $|1+z_1|+|1+z_2|+|1+z_1z_2|\ge2$</p>
| Shivang jindal | 38,505 | <p>$$ \mid 1+z_1 \mid + \mid 1+ z_2 \mid + \mid 1+ z_1z_2 \mid \ \ge \mid 1+z_1 \mid + \mid 1+z_1z_2-1-z_2 \mid$$ [ Using Triangle inequality]
$$ \mid 1+z_1 \mid + \mid z_1z_2-z_2 \mid = \mid 1+z_1 \mid + \mid z_1 -1 \mid \ \ge |1+z_1+z_1-1| = 2 $$ [Again using triangle inequality]</p>
<p>So we are done :)</p>
|
3,897,067 | <p>Consider a binary operation <span class="math-container">$*$</span> acting from a set <span class="math-container">$X$</span> to itself. It's useful and standard to work with operations which are associative, such that <span class="math-container">$(a*b)*c = a*(b*c)$</span>. What about operations which are not assoc... | Dietrich Burde | 83,966 | <p>"What about operations which are not associative?" In many areas we encounter non-associative algebra structures, e.g., in operad theory, homology of partition sets, deformation theory, geometric structures on Lie groups, renormalisation theory in physics and many more.</p>
<p>In a certain sense one can an... |
3,034,421 | <p>Lets say I have 2 multivariate functions:</p>
<pre><code>f(x,y) = x - y
g(x,y) = x + y
</code></pre>
<p>How do I get the composition of these 2 functions <span class="math-container">$g(f(x,y))$</span> ? </p>
| Cesareo | 397,348 | <p>For <span class="math-container">$xy^2(2+3x+4y) =0$</span> we have the set of solutions </p>
<p><span class="math-container">$$S_1 = \{x = 0, y = 0, 2+3x+4y = 0\}$$</span></p>
<p>For <span class="math-container">$2x^2y(1+x+3y) =0$</span> we have the set of solutions </p>
<p><span class="math-container">$$S_2 = \{... |
959,393 | <p>Let's use the following example:</p>
<p>$$17! = 16!*17 \approx 2 \cdot 10^{13} * 17 = 3.4 \cdot 10^{14} $$</p>
<p>Are you allowed to do this? I am in doubt whether or not this indicates that $17! = 3.4 \cdot 10^{14}$, which is obviously not true, but I think it doesn't.</p>
| vadim123 | 73,324 | <p>Your example claims that two things are equal (on the left), two things are equal (on the right), and that the left pair are approximately equal to the right pair.</p>
<p>One should be careful with too much use of the ill-defined $\approx$ symbol, or you can get $$1\approx 1.01\approx 1.02\approx \cdots \approx 1.9... |
1,368,073 | <p>Halmos, in Naive Set Theory, on page 19, provides a definition of intersection restricted to subsets of $E$, where $C$ is the collection of the sets intersected. The point is to allow the case where $C$ is $\emptyset$, which with this definition of intersection gives $E$ as the result. </p>
<blockquote>
<p>$\{x \... | Mark Fischler | 150,362 | <p>The first highlighted statement is mixing English reading of a statement with mathematical notation. The clue for that is that it had to say "for every..."</p>
<p>WHhen you do this, it becomes a bit less clear as to what in order you have to "resolve" the instructions. In this case, because the statement is so sho... |
1,521,124 | <p>What will be the value of $3/1!+5/2!+7/3!+...$?</p>
<p>I'm trying to bring it in terms of $e$.Is it possible?</p>
<p>I used taylor series for e.</p>
| vudu vucu | 215,476 | <p>Hint: $\frac{2k+1}{k!}=2\frac1{(k-1)!}+\frac{1}{k!}$</p>
|
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | Georges Elencwajg | 450 | <p>Here is a type of example coming from Analytic Geometry (in the sense of the second "GA" in Serre's "GAGA").</p>
<p>Consider a domain $D$ in $\mathbb C$. Then every $\mathbb C$-algebra morphism (aka character) $\chi : \mathcal O (D) \to \mathbb C $ is of the form $ev_d:f \to f(d)$ with $d=\chi (z) \in D$. This is ... |
7,025 | <p>Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian rings I'm told one can replace Zorn's lemma with countable choice, which is nice, but still not nice enough - I'd lik... | Greg Kuperberg | 1,450 | <p>I suspect that the most general reasonable answer is a ring endowed with a constructive replacement for what the axiom of choice would have given you.</p>
<p>How do you show in practice that a ring is Noetherian? Either explicitly or implicitly, you find an ordinal height for its ideals. Once you do that, an idea... |
634,890 | <blockquote>
<p><strong>Moderator Notice</strong>: I am unilaterally closing this question for three reasons. </p>
<ol>
<li>The discussion here has turned too chatty and not suitable for the MSE framework. </li>
<li>Given the recent pre-print of <a href="http://arxiv.org/abs/1402.0290" rel="noreferrer">T. Ta... | uvs | 121,193 | <p>what do you make of Definition 2? He defines a strong solution so that all terms of NSE are required to live in $L^2$. Global regularity of NSE calls for pressure and velocity fields in $C^\infty$. Is it just me or are we speaking of a very, very, very weak notion of strong solution, which has nothing to do with the... |
4,588,408 | <p>This is a step in a guided proof that the cyclotomic polynomial <span class="math-container">$\Phi_n$</span> is the minimal polynomial of <span class="math-container">$u$</span>. I already know that <span class="math-container">$\Phi_n(0)=0$</span> so <span class="math-container">$P$</span> divides <span class="math... | Ja_1941 | 937,996 | <p>Suppose <span class="math-container">$P$</span> has roots <span class="math-container">$\zeta_1,\zeta_2,...,\zeta_k$</span>. All of them are <span class="math-container">$n$</span>-th root of unity. Without the loss of generality, let <span class="math-container">$P$</span> be monic. Thus,</p>
<p><span class="math-c... |
747,949 | <p>There is a complex serie: $f(t_n)=\alpha_n+\beta_n i$, for $n = 1,...,N$,$t_n,\alpha_n$ and $\beta_n$ are known.When we have know that $f(t)$ has the following form:
$$f(t)=Ae^{-iBt}$$
with unknown amplitude $A$ and unknown phase $B$, how to estimate the parameters $A$ and $B$ by using a numerical optimization metho... | Martín-Blas Pérez Pinilla | 98,199 | <p>$$f(t)=Ae^{−iBt}=A(\cos(Bt)-i\sin(Bt))$$
$$\alpha_n+\beta_n i=f(t_n)=A(\cos(Bt_n)-i\sin(Bt_n))$$
$$A=|\alpha_n+\beta_n i|=\sqrt{\alpha_n^2+\beta_n^2}$$
$$B= -\frac1{t_n}\arctan\frac{\alpha_n}{\beta_n}$$
$$\cdots$$</p>
|
159,563 | <p>I have some <a href="https://pastebin.com/MGEzkeC3" rel="nofollow noreferrer">data</a> and want to fit it to Planck's law for black body radiation. The problem is that Mathematica does not give me the correct coefficients.</p>
<p>When I evaluate</p>
<pre><code>dati = Import["https://pastebin.com/raw/MGEzkeC3&qu... | José Antonio Díaz Navas | 1,309 | <p>No need what JimB did. Just correct the units for $\lambda$, i.e., not nanometers but meters. There was also some errors in the syntax when using <code>FindFit</code>. I also add <code>pts</code>as a new dataset with less points suitable for plotting and compare to the fits.</p>
<p>Here is your code corrected:</p>
... |
1,027,486 | <p>How do I integrate this?</p>
<p>$$\int_0^{2\pi}\frac{dx}{2+\cos{x}}, x\in\mathbb{R}$$</p>
<p>I know the substitution method from real analysis, $t=\tan{\frac{x}{2}}$, but since this problem is in a set of problems about complex integration, I thought there must be another (easier?) way.</p>
<p>I tried computing t... | dustin | 78,317 | <p>For completeness, we know that $\cos(x) = \frac{e^{ix}+e^{-ix}}{2}$ and from @danielfischer's comment, $z=e^{ix}$, we are able to obtain $\cos(z) = \frac{z + 1/z}{2}$. Then $dx =\frac{dz}{iz}$. Therefore, we have the following integral were $\gamma$ is to be taking counter clockwise such that $|z|=1$
$$
\int_{\gamma... |
1,412,594 | <p>I'm a statistics teacher at a college. One day a student came with a doubt about an exercise about probability. The text goes like this:</p>
<blockquote>
<p>A person has two boxes $A$ and $B$. In the first one has $4$ white balls and $5$ black balls and in the second has $5$ white balls and $4$ black balls. This ... | barak manos | 131,263 | <p><strong>Split it into disjoint events, and add up their probabilities:</strong></p>
<hr>
<p>The probability of choosing white from the first box and then from the second box is:</p>
<p>$$\frac{4}{9}\cdot\frac{6}{10}=\frac{24}{90}$$</p>
<hr>
<p>The probability of choosing black from the first box and then from t... |
4,089,800 | <p>I tried to solve this problem by doing this way : <span class="math-container">$( x-w)^{5} =\left( w^{2}\right)^{5}$</span> and got this equation <span class="math-container">$x^{5}-5wx^{4}+10w^{2}x^{3}-10w^{3}x^{2}+5w^{4}x=6$</span> but I don't know how to proceed further. Would someone please help me here.</p>
| Eric Towers | 123,905 | <p><span class="math-container">\begin{align*}
(\omega + \omega^2)^5
&= \omega^5 + 5 \omega^6 + 10 \omega^7 + 10 \omega^8 + 5 \omega^9 + \omega^{10} \\
&= 2 + 10 \omega + 20 \omega^2 + 20 \omega^3 + 10 \omega^4 + 4 \text{,} \\
-10(\omega + \omega^2)^2
&= -10 \omega^2 - 20 \omega^3 - 10 \omeg... |
4,089,800 | <p>I tried to solve this problem by doing this way : <span class="math-container">$( x-w)^{5} =\left( w^{2}\right)^{5}$</span> and got this equation <span class="math-container">$x^{5}-5wx^{4}+10w^{2}x^{3}-10w^{3}x^{2}+5w^{4}x=6$</span> but I don't know how to proceed further. Would someone please help me here.</p>
| Robert Lee | 695,196 | <p>Recalling that since <span class="math-container">$w$</span> is a fifth root of <span class="math-container">$2$</span> we have that <span class="math-container">$w^5 = 2$</span>, we notice that
<span class="math-container">\begin{align*}
x^5 - 10x^2 -10x & =\left(w + w^2\right)^5 - 10\left(w + w^2\right)^2 - 10... |
4,052,760 | <blockquote>
<p>Prove that <span class="math-container">$\int\limits^{1}_{0} \sqrt{x^2+x}\,\mathrm{d}x < 1$</span></p>
</blockquote>
<p>I'm guessing it would not be too difficult to solve by just calculating the integral, but I'm wondering if there is any other way to prove this, like comparing it with an easy-to-c... | Martin R | 42,969 | <p>A method which is generally applicable to <em>concave</em> functions:</p>
<p>If <span class="math-container">$f:[a, b] \to \Bbb R$</span> is concave then its graph lies below any tangent line:
<span class="math-container">$$
f(x) \le f(c) + f'(c)(x-c) \, .
$$</span>
If one chooses <span class="math-container">$c =(... |
121,653 | <p>What is information about the existence of rational points on hyperelliptic curves over finite fields available?</p>
| Felipe Voloch | 2,290 | <p>As Mike says, there isn't much beyond the Weil bound. For prime fields, there is a slight improvement due to Stark. If the field has $q=r^2$ elements with $q$ odd, then one can find $a,b$ such that $y^2=ax^{r+1}+b$ has no points, which shows that you cannot improve the Weil bound. For arbitrary $q$ odd, there is thi... |
232,777 | <p>Let $F$ be an ordered field.</p>
<p>What is the least ordinal $\alpha$ such that there is no order-embedding of $\alpha$ into any bounded interval of $F$?</p>
| Panurge | 82,840 | <p>So, Fedor Petrov gave an excellent proof and Thomas Brown's first step towards Frankl's conjecture is proved. In order to prove the first half of the second step, it would be sufficient to prove the following statement : if $r$ and $k'$ are two natural numbers such that $r \leq k'$, the union-closed family generated... |
1,343,722 | <p>Note: I am looking at the sequence itself, not the sequence of partial sums.</p>
<p>Here's my attempt...</p>
<p>Setting up:</p>
<p>$$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>Simplifying:</p>
<p>$$\left\{\frac{2n+2}{2n+1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>... | Jack D'Aurizio | 44,121 | <p>Since $n^2 = 2\binom{n}{2}+\binom{n}{1}$ we have:
$$ S=\sum_{n\geq 1}\frac{n^2}{2^n}=\left.\left(2\frac{x^2}{(1-x)^3}+\frac{x}{(1-x)^2}\right)\right|_{x=\frac{1}{2}}=\color{red}{6}.$$
As an alternative to the negative binomial series, we may also use:
$$ S = 2S-S = \sum_{n\geq 1}\frac{n^2}{2^{n-1}}-\sum_{n\geq 1}\fr... |
1,343,722 | <p>Note: I am looking at the sequence itself, not the sequence of partial sums.</p>
<p>Here's my attempt...</p>
<p>Setting up:</p>
<p>$$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>Simplifying:</p>
<p>$$\left\{\frac{2n+2}{2n+1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>... | Barry Cipra | 86,747 | <p>Here's a bit of a twist on the second approach in Jack D'Aurizio's answer. It's main virtue (if any) is that it avoids explicitly evaluating the auxiliary sum $\sum{n\over2^n}$, getting it to drop out instead.</p>
<p>It's convenient to start the sum at $n=0$ instead of $n=1$. Borrowing Jack's notation, we have</p... |
4,264,496 | <p>So we have the jensen's inequality: <span class="math-container">$$|EX| \leq E|X|$$</span></p>
<p><strong>Any bound</strong> on the Jensen gap (upper bound or lower bound)? <span class="math-container">$$\text{gap}=E|X| - |EX|$$</span></p>
| GEdgar | 442 | <p><span class="math-container">$0 \le |EX| \leq E|X|$</span> so <span class="math-container">$$\text{gap} \le E|X|.$$</span> Reijo provided an example where <span class="math-container">$\text{gap} = E|X|$</span>.</p>
|
2,566,803 | <p>Let $A,B,C$ be sets such that $f:A\to B$ is a function.</p>
<p>Let $F: C^B \to C^A$ be a function, such that $F(k)=k\circ f$.</p>
<p>Prove/disprove that if $f$ is surjective then $F$ is surjective.</p>
<p>I tried to prove it: If $f$ is surjective so for every $b\in B$ there is $a\in A$ so $f(a)=b$, but what now?... | ajotatxe | 132,456 | <p>It is false.</p>
<p>Let $A=C=\Bbb R$, $B=[0,\infty)$, $f(x)=x^2$ for $x\in A$.</p>
<p>Let $g:A\to C$ be defined as $g(x)= x^3$. Assume that there exists $h:B\to C$ such that $h\circ f=g$, that is $h(x^2)=x^3$. But this implies</p>
<p>$$h(1)=h(1^2)=1^3=1$$
$$h(1)=h((-1)^2)=(-1)^3=-1$$</p>
<p>But if $f$ is bijecti... |
3,690,185 | <p>By <span class="math-container">$a_n \sim b_n$</span> I mean that <span class="math-container">$\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 1$</span>.</p>
<p>I don't know how to do this problem. I have tried to apply binomial theorem and I got
<span class="math-container">$$\int_{0}^{1}{(1+x^2)^n dx} = \int_0^1 \... | SB1729 | 466,737 | <p>A partial progress</p>
<p>We make the substitution <span class="math-container">$x\tan(y)$</span>. Then <span class="math-container">$dx=\sec^2(y)dy$</span>.</p>
<p>Now <span class="math-container">$$\int_{0}^{1}(1+x^2)^ndx=\int_{0}^{\frac{\pi}{4}}\sec^{2n}(y)\sec^2(y)dy=\int_{0}^{\frac{\pi}{4}}\sec^{2(n+1)}(y)dy$... |
3,489,345 | <p>My goal is to find the values of <span class="math-container">$N$</span> such that <span class="math-container">$10N \log N > 2N^2$</span></p>
<p>I know for a fact this question requires discrete math. </p>
<p>I think the problem revolves around manipulating the logarithm. The thing is, I forgot how to manipula... | ZAF | 609,023 | <p><span class="math-container">$N \in \mathbb{N}$</span> ?</p>
<p><span class="math-container">$10N\log(N)> 2N^2$</span> </p>
<p>If and only if</p>
<p><span class="math-container">$5 \log(N)> N$</span> </p>
<p>If and only if </p>
<p><span class="math-container">$e^{5\log(N)} > e^{N}$</span></p>
<p>If an... |
3,600,633 | <p>As I was reading <a href="https://math.stackexchange.com/questions/1918673/how-can-i-prove-that-the-finite-extension-field-of-real-number-is-itself-or-the">this question</a>, I saw Ethan's answer. However, perhaps this is very obvious, but why does the degree of the polynomial be at most <span class="math-container"... | Gregory Grant | 217,398 | <p>Any polynomial in <span class="math-container">$\Bbb R[x]$</span> factors into linears and quadratics.</p>
|
3,905,629 | <p>I need to compute a limit:</p>
<p><span class="math-container">$$\lim_{x \to 0+}(2\sin \sqrt x + \sqrt x \sin \frac{1}{x})^x$$</span></p>
<p>I tried to apply the L'Hôpital rule, but the emerging terms become too complicated and doesn't seem to simplify.</p>
<p><span class="math-container">$$
\lim_{x \to 0+}(2\sin \s... | Barry Cipra | 86,747 | <p>For small positive angles, the inequalities <span class="math-container">${3\over4}\theta\le\sin\theta\le\theta$</span> apply. (The fraction <span class="math-container">$3/4$</span> is somewhat arbitrary; any fraction less than <span class="math-container">$1$</span> will do.) Letting <span class="math-container">... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | JuliusL33t | 111,167 | <p>The "normal" definition goes like this:</p>
<p>It is claimed that, at fixed point, for any given ball $B_\epsilon$ of radius $\epsilon$ in the image, there exists a ball $B_\delta$, in the preimage, of radius $\delta$ such that $Im (B_\delta) \subset B_\epsilon$. This is the implication $$(...) < \delta \implies... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | CopyPasteIt | 432,081 | <p>The notion of topological space and the definition of a continuous function are certainly in the realm of 'abstract' mathematics. To say that a function $f$ is continuous mean that if points are close to other points then they don't get 'ripped away' when applying it - they 'follow the action' of $f$.</p>
<p>Now we... |
205,671 | <p>How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? </p>
<p>I haven't found any proof of this online.</p>
<p>One of my ideas was writing out $r$ and $\theta$ in terms of $x$ a... | syockit | 53,159 | <p>We can derive using purely polar coordinates. Start with
\begin{align}
z(r,\theta) &=r\,\mathrm{e}^{\mathrm{i}\theta} \\
f(z) &= u(r,\theta) + \mathrm{i} v(r,\theta)
\end{align}
We define $f'(z)$ using the limit
$$ f'(z) = \lim_{z\to 0} \frac{\Delta f}{\Delta z} $$
where
\begin{align}
\Delta f &= \Delta ... |
109,734 | <p>I am trying to do this homework problem and I have no idea how to approach it. I have tried many methods, all resulting in failure. I went to the books website and it offers no help. I am trying to find the derivative of the function
$$y=\cot^2(\sin \theta)$$</p>
<p>I could be incorrect but a trig function squared ... | Peđa | 15,660 | <p>$y'=2(\cot(\sin \theta))(\cot(\sin \theta))'\cdot (\sin \theta)'=2 \cdot (\cot(\sin \theta))(-1-\cot^2(\sin \theta))\cdot \cos \theta$</p>
|
4,268,962 | <blockquote>
<p>Check whether <span class="math-container">$y=\ln (xy)$</span> is an answer of the following differential equation or not</p>
<p><span class="math-container">$$(xy-x)y''+xy'^2+yy'-2y'=0$$</span></p>
</blockquote>
<p>First I tried to solve the equation,</p>
<p><span class="math-container">$$x(yy''-y''+y'... | Claude Leibovici | 82,404 | <p>Consider the implicit equation
<span class="math-container">$$F(x,y)=y-\log(x y)=0$$</span></p>
<p>So, using the implicit function theorem,
<span class="math-container">$$y'=\frac{y}{x (y-1)}\qquad \text{and} \qquad y''=-\frac{y ((y-2) y+2)}{x^2 (y-1)^3}$$</span></p>
<p>Just replace and simplify.</p>
|
3,907,928 | <p>Suppose the solutions to a general cubic equation <span class="math-container">$ax^3+bx^2+cx+d=0$</span> are to be found. Then according to Cardano's method, First a variable substitution must be carried on to convert the general cubic to depressed cubic.
<span class="math-container">$$ax^3+bx^2+cx+d=0\rightarrow t^... | Andy Walls | 441,161 | <p>One source of inspiration may have been looking at the binomial expansion of <span class="math-container">$(u+v)^3$</span> and mapping it to the cubic <span class="math-container">$t^3 + pt +q = 0$</span>.</p>
<p><span class="math-container">$$\begin{align*}(u+v)^3 &= u^3 +3 u^2v +3uv^2+v^3 \\
\\
&= u^3+v^3+... |
4,487,380 | <p>I was reading my calculus book wherein I came across a note, being worth of attention. It says:</p>
<blockquote>
<p>Integrals in the form of <span class="math-container">$\int P(x)e^{ax}dx$</span> have a special property. After calculating the integral, we obtain a function in the form of <span class="math-container... | dxiv | 291,201 | <p>The problem reduces to finding the polynomial <span class="math-container">$Q$</span> of degree <span class="math-container">$n = \deg P$</span> such that <span class="math-container">$P = aQ + Q'$</span>.</p>
<p>Differentiating <span class="math-container">$n$</span> times and eliminating <span class="math-containe... |
1,297,319 | <p>integration equation </p>
<p>$$\int_{0}^{1/8} \frac{4}{\sqrt{(1-4x^2)}} \,dx$$</p>
<p>my work </p>
<p>$t= \sqrt{(1-4x^2)} $</p>
<p>$dt = -4x/\sqrt{(1-4x^2)} dx $</p>
<p>stuck here also </p>
| Lucas | 227,060 | <p>Let $u=4x \Rightarrow du=4dx$</p>
<blockquote>
<p>Therefore $\int\frac{4}{\sqrt{1-4x^2}}dx=8\int\frac{1}{\sqrt{4-u^2}}=8\arcsin(2x)+C$</p>
</blockquote>
<p>$\Rightarrow\int_{0}^{1/8}\frac{4}{\sqrt{(1-4x^2)}}dx=8\arcsin(\frac{1}{4})$</p>
|
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