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 |
|---|---|---|---|---|
4,070,996 | <p>I'm a bit lost in this integral: <span class="math-container">$$\int \frac{1}{1+\sin^4(x)} \, \mathrm dx$$</span>
I have tried solving with Wolfram, but I was getting a cosecant solution which doesn't seem as the correct method.</p>
<p>Do you have any ideas? :)</p>
<p>EDIT:
Do you please have step-by-step solution, ... | Quanto | 686,284 | <p>Following your substitution <span class="math-container">$t= \tan x$</span>, integrate the resulting as follows</p>
<p><span class="math-container">\begin{align}
&\int \frac1{1+\sin^4x}dx\\
=&\int \:\frac{t^2+1}{2t^4+2t^2+1}dt
=\int \frac{1+\frac1{t^2}}{(\sqrt2t)^2 + \frac1{t^2}+1}dt\\
=&\frac{\sqrt2+1}{... |
2,456,961 | <p>I thought of this myself and could not begin to think of a solution without becoming confused:
There are two people, A and B.
A says, "B is telling the truth."
B says, "A is lying."
Are both lying or telling the truth? Or is one lying and the other telling the truth?</p>
| uniquesolution | 265,735 | <p>Yes, it does.</p>
<p>The general definition of continuity is this:</p>
<blockquote>
<p>Let $f:X\to Y$ be a function defined on a topological space $X$, with values in a topological space $Y$. $f$ is said to be continuous at the point $x\in X$ if for every open set $V\subset Y$ containing $f(x)$, the set $f^{-1}(... |
3,194,534 | <p>Integrate the following <span class="math-container">$\int\frac{x+1}{x(1+xe^x)^2}dx$</span></p>
<p>I tried to multiply the numerator and denominator by <span class="math-container">$e^{-2x}$</span></p>
<p>Getting the following simplification
<span class="math-container">$\int\frac{e^{-2x}(x+1)}{x(e^{-x}+x)^2}dx$</... | E.H.E | 187,799 | <p>Hint:
let <span class="math-container">$$u=1+xe^x\rightarrow xe^x=u-1$$</span>
<span class="math-container">$$dx=\frac{du}{e^x(1+x)}$$</span>
substitute them to get
<span class="math-container">$$\int\frac{1}{u^2(u-1)}du$$</span></p>
|
1,088,687 | <p>I have this example here:</p>
<p>$\lim_{n \to \infty} \frac{n^{2}+1}{n^{2}+n+1} = 1$</p>
<p>so I have to prove that, $\forall \epsilon > 0, \exists n_{0}\in N, \forall n(n \in N) \geq n_{0} \Rightarrow |\frac{n^{2}+1}{n^{2}+n+1} - 1| < \epsilon$</p>
<p>I have done in these steps <br/> <br/>
$|\frac{n^{2}+... | Emil Stoyanov | 204,215 | <p>$|\frac{-n}{n^{2}+n+1}| < \epsilon \Leftrightarrow |\frac{n}{n^{2}+n+1}| < \epsilon $ <br/> <br/>
Now $|\frac{n}{n^{2}+n+1}| < |\frac{1}{n}| $ . So if $n_0=\frac{1}{\epsilon}$ everything is ok.</p>
|
918,049 | <p>Let $f: X \to Y$ be a map of sets. We are given that $X$ is a topological space. We are to show that there is a topology on $Y$ making $f$ continuous, and moreover, determine if this topology is unique. </p>
<p>Should I read it as "$X$ is just a set on which there is given a topology, i.e sets that are open in $X$ ... | Kim Jong Un | 136,641 | <p>Let $b=\frac{1+\sqrt{4\alpha+1}}{2}$. A key feature for $b$ is that $b^2-\alpha=b$ or $b^2=\alpha+b$.</p>
<p>Now, let $a_n$ be the your LHS when there are $n$ square root signs. The base case $a_1<b$ is trivial. And the induction step follows because
$$
a_{n+1}^2=\alpha+a_n<\alpha+b=b^2\implies a_{n+1}<b.
... |
137,813 | <p>Let $k$ be an algebraically closed field of characteristic $0$, let $C_{/k}$ be a nice (smooth, projective, geometrically integral curve), let $K = k(C)$, and let $\overline{K}$ be an algebraic closure of $K$. Let $E_{/K}$ be an elliptic curve with $j(E) \notin k$. Let $P \in E(\overline{K}) \setminus E(K)$ be a p... | Remke Kloosterman | 8,621 | <p>I do not believe you have equality in general. I sketch a counterexample below, which is a geometric version of the fact that if $E/K$ is an elliptic curves such that the quadratic twist $E^{(d)}/K$ has Mordell-Weil rank at least 2 then the Mordell-Weil of $E/K(\sqrt{d})$ is at least two bigger than the Mordell-Weil... |
336,170 | <p>Create a rational function with vertical asymptotes $x=\pm1$ and oblique asymptote of $y=2x-3$ and a $y$-intercept of $4$.</p>
| Cameron Buie | 28,900 | <p>For one family of rational functions $f(x)$ having vertical asymptotes <strong>only</strong> at $x=\pm 1$, consider those that can be written in the form $$f(x)=p_1(x)+\frac{p_2(x)}{(x+1)^m}+\frac{p_3(x)}{(x-1)^n}$$ for some positive integers $m,n,$ and some polynomials $p_1,p_2,p_3$, with $p_2,p_3$ in particular be... |
2,338,508 | <p>Given this matrix that stretches to infinity to the right and up:
$$
\begin{matrix}
...&...&...\\
\frac{1}{4}& \frac{1}{8}& \frac{1}{16}&... \\
\frac{1}{2} & \frac{1}{4}& \frac{1}{8}&... \\
1 & \frac{1}{2}& \frac{1}{4}&... \\
... | Jesse | 361,825 | <p>If you compute the sum of each row, you'll notice that it's 2 * (1/2)^n (n going from 0 to infinity). Which is iteself is an infinite series that trivially sums to 4.</p>
<p>Where I think you are going wrong: When summing a matrix, you have to add to it in a square fashion. So, a row either exists or does not. ... |
3,520,538 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a Hausdorff topological space.
Prove that if <span class="math-container">$\{ C_i | i \in I \}$</span> is an infinite collection of compact subsets of <span class="math-container">$X$</span> such that <span class="math-container">$\cap_{i \in I} C_i = ... | User203940 | 333,294 | <p>You have, in fact,
<span class="math-container">$$\sum \frac 1 {n^{p}}$$</span>
diverges for <span class="math-container">$ p \leq 1$</span>, so you have a bunch of examples there.</p>
|
3,520,538 | <blockquote>
<p>Let <span class="math-container">$X$</span> be a Hausdorff topological space.
Prove that if <span class="math-container">$\{ C_i | i \in I \}$</span> is an infinite collection of compact subsets of <span class="math-container">$X$</span> such that <span class="math-container">$\cap_{i \in I} C_i = ... | user8675309 | 735,806 | <p>The simple (symmetric) random random walk -- you return to the starting position with probability one, so with the number of moves until first return given by random variable <span class="math-container">$T$</span>, the complementary CDF has the sequence <span class="math-container">$u_n =Pr\{T \gt n\}\to 0$</span> ... |
2,152,721 | <p>I have been messing around analyzing some self-made algorithms and then I stumbled upon this summation:</p>
<p>$$\sum_{j=0}^{n}\left\lfloor{2^{i-j}}\right\rfloor$$</p>
<p>I am thinking that my common algebraic manipulations involving series won't apply because of the floor function.</p>
<p>I also tried entering t... | dtldarek | 26,306 | <p>Assuming that $i \in \mathbb{Z}$ and $n \in \mathbb{N}$, observe that as long as $j \leq i$ you have a normal summation, while in the other case, $0 \leq 2^k < 1$ for any $k < 0$. To deal with $i < 0$ we use $i' = \max(i,-1)$. Then we get:</p>
<p>\begin{align*}
\sum_{j = 0}^{n} \lfloor 2^{i-j} \rfloor
&... |
4,333,566 | <p>Say that <span class="math-container">$\{a_n\}^\infty_\mathrm{n=1}$</span> is a sequence of real numbers so that for all odd <span class="math-container">$n$</span>, <span class="math-container">$a_n \in (0, 1)$</span> and for all even <span class="math-container">$n$</span>, <span class="math-container">$a_n \not\i... | Will Jagy | 10,400 | <p>Any positive integer <span class="math-container">$n$</span> that is not divisible by 4, also <span class="math-container">$n \neq 7 \pmod 8,$</span> can be written primitively as the sum of three squares. The number of such primitive representations is a certain multiple of <span class="math-container">$h(-4n).$</... |
3,168,021 | <p>If <strong>A</strong> is an m by n matrix, prove that the set of vectors <strong>b</strong> that are not in <strong>C(A)</strong> forms a subspace.</p>
<p>I would like to first understand if I am interpreting the question correctly. My understanding of this question is that I need to first prove that the set of ve... | Siong Thye Goh | 306,553 | <p>Your first condition should be <span class="math-container">$0$</span> is in a subspace.</p>
<p>Also, the result is not true. </p>
<p>Let <span class="math-container">$0_m$</span> be the zero vector in <span class="math-container">$\mathbb{R}^m$</span>. We know that <span class="math-container">$0_m \in C(A)$</spa... |
3,168,021 | <p>If <strong>A</strong> is an m by n matrix, prove that the set of vectors <strong>b</strong> that are not in <strong>C(A)</strong> forms a subspace.</p>
<p>I would like to first understand if I am interpreting the question correctly. My understanding of this question is that I need to first prove that the set of ve... | Bello Bello | 643,633 | <p>Your result is wrong. Maybe this picture can help you figure out.</p>
<p><a href="https://i.stack.imgur.com/xUU7V.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xUU7V.png" alt="enter image description here"></a></p>
<p>Since every subspace must contain the zero vector, noted by <span class="mat... |
2,362,500 | <p>When you are solving an integral, and you have a square root of a square, for instance</p>
<p>$$ \int \sqrt{1+\cos x} = \int \sqrt{\frac{\sin^2x}{1-\cos x}}$$</p>
<p>Do I have to take the absolute value of $\sin x$, or can I just take the positive value? </p>
<p>$$= \int \frac{|\sin x|}{\sqrt{1-\cos x}}$$ OR
$$= ... | user84413 | 84,413 | <p>Another way to do this is to consider the following 2 cases:</p>
<p>1) P1 and P2 serve together: $\;\;\displaystyle \binom{7}{3}-\binom{5}{1}=35-5=30$ choices.</p>
<p>2) Neither P1 nor P2 serves: $\;\;\displaystyle \binom{7}{5}-\binom{5}{3}=21-10=11$ choices.</p>
<p>Therefore there are a total of $\color{red}{... |
2,845,427 | <p>I posted concerning this question a little while ago, not asking for the answer but for an understanding of the setup. Well, I thought I would solve it but I seem to be unable to obtain the required answer. If anyone could help I'd be very grateful.</p>
<p>Here's the question again:</p>
<p><a href="https://i.stack... | gnitsuk | 265,060 | <p>I think the book is correct. Here's why.</p>
<p>The natural length of AC is $\frac{4}{3}a$ and in the final equilibrium position AC has been stretched to a length of $\sqrt{3}a$ and so the extension of AC is $(\sqrt{3}-\frac{4}{3})a$ and so the tension in AC is $T_1=\frac{\lambda(\sqrt{3}-\frac{4}{3})a}{\frac{4a}{3... |
1,703,169 | <p>At a recent maths competition one of the questions was to find for which $x,y,z$ this equation holds true:</p>
<p>$$\sqrt{x}-\sqrt{z+y}=\sqrt{y}-\sqrt{z+x}=\sqrt{z}-\sqrt{x+y}$$</p>
<p>where $x,y,z \in \mathbb{R} \cup \{0\}$. So how am I supposed to approach this problem?</p>
<p>Also sorry for not explaining my p... | Community | -1 | <p>Squaring, simplifying and squaring again gives</p>
<p>$$x(y+z)=y(z+x)=z(x+y)$$</p>
<p>which is $$xy=yz=zx$$ or $$x=y=z.$$</p>
<hr>
<p>If we assume that $x=0$, then $yz=0$. Then WLOG, $y=0$ and the initial equation says $z=0$.</p>
|
300,067 | <p>They ask me to find all invertible matrices $A$ of the form: $\begin{bmatrix}a & b\\ c&d \end{bmatrix}$ and satisfying $A=A^{-1}$ and $A^t=A^{-1}$. I find that rather complex; does it have anything to do with orthogonality? Not sure. Any help please??</p>
| i.a.m | 60,195 | <p>Matrices of real entries such that $AA^t=I$ are called unitary matrices and the columns of such matrices must be orthogonal.</p>
<p>and you want to find unitary matrices such that $A^t=A$ which are called symmetric matrices, so the answer will be all symmetric unitary matrices.</p>
|
3,218,200 | <p>Let <span class="math-container">$\Bbbk$</span> be a field. A well-known result (see <a href="https://www.jstor.org/stable/2373270?origin=crossref&seq=1#metadata_info_tab_contents" rel="nofollow noreferrer">Larson, Sweedler, <em>An Associative Orthogonal Bilinear Form for Hopf Algebras</em></a> and <a href="http... | Marco Farinati | 664,350 | <p>Ussually finite dimension is part of the definition of a Frobenius algebra. So, what definition of "Frobenius" algebra do you have in mind?</p>
<p>In wikipedia (<a href="https://en.wikipedia.org/wiki/Frobenius_algebra#Category-theoretical_definition" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Frobenius... |
84,897 | <pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
</code></pre>
<p>$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$</p>
<p>gives $0$</p>
<p>This cannot be true. What is going on?</p>
<p>If I insert a number into <code>a</code>, it gives a reasonable result:</p>
<pre><code>NIntegrate[2/(Sin[t]^2 + 4), {t, 0,... | Nasser | 70 | <p>The result zero does not look correct. But you can get the correct result by using <code>PrincipalValue -> True</code> </p>
<pre><code>r = Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}, PrincipalValue -> True]
</code></pre>
<p><img src="https://i.stack.imgur.com/cInAt.png" alt="Mathematica graphics"></p>
<p>n... |
84,897 | <pre><code>Integrate[a/(Sin[t]^2 + a^2), {t, 0, 2 Pi}]
</code></pre>
<p>$$\int_0^{2 \pi } \frac{a}{a^2+\sin ^2(t)} \, dt$$</p>
<p>gives $0$</p>
<p>This cannot be true. What is going on?</p>
<p>If I insert a number into <code>a</code>, it gives a reasonable result:</p>
<pre><code>NIntegrate[2/(Sin[t]^2 + 4), {t, 0,... | b.gates.you.know.what | 134 | <p>Another, in my opinion simpler, alternative is to use the symmetry.</p>
<pre><code>4 Integrate[a/(a^2 + Sin[t]^2), {t, 0, π/2}]
(* (2 π)/(Sqrt[1 + 1/a^2] a) *)
</code></pre>
<p>without additional assumptions.</p>
|
3,810,856 | <p>I'm having trouble understanding how this form of the principle (<a href="https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" rel="nofollow noreferrer">on wiki</a>) results in the form below.</p>
<p>Wiki form:
<a href="https://i.stack.imgur.com/6IXQZ.png" rel="nofollow noreferrer"><img src="https://... | MachineLearner | 647,466 | <p><span class="math-container">$$\frac{\pi}{\sin^2 \frac{\pi}{x}}=\frac{\pi}{\sin \frac{\pi}{x}\sin \frac{\pi}{x}}=\frac{1}{\sin \frac{\pi}{x} / \cos \frac{\pi}{x}}\frac{\pi}{\cos\frac{\pi}{x}\sin \frac{\pi}{x}}=\frac{1}{\tan \frac{\pi}{x}}\frac{\pi}{\cos\frac{\pi}{x}\sin \frac{\pi}{x}}$$</span>
<span class="math-cont... |
312,012 | <p>What is the exact difference between $\arg\max$ and $\max$ of a function?</p>
<p>Is it right to say the following?</p>
<blockquote>
<p>$\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is maximal. And $\max f(x)$ means value of $f(x)$ for which it is maximum.</p>
</blockquote>
... | user1551 | 1,551 | <p>Yes. $\arg\max f(x)$ is the function <strong><em>argument</em></strong> $x$ at which the maximum of $f$ occurs, and $\max f(x)$ is the <strong><em>maximum</em></strong> value of $f$. Hence $\max f(x)=f\left(\arg\max f(x)\right)$.</p>
|
3,838,889 | <p>Exactly one ace given that it contains exactly two queens?</p>
<p>I am struggling how to solve this maths problem. I know that a hand of <span class="math-container">$10$</span> cards dealt from a normal pack of <span class="math-container">$52$</span> is <span class="math-container">$52\choose 10$</span> <span clas... | user2661923 | 464,411 | <p>I favor the shortcut of equating the problem to the chances of
a hand of 8 cards having exactly 1 Ace from a pack of 48 cards.</p>
<p>I would calculate this as</p>
<p><span class="math-container">$\displaystyle \frac{\binom{4}{1} \times \binom{44}{7}}{\binom{48}{8}}.$</span></p>
<p>The reason that I regard the probl... |
1,717,333 | <p>Feeling unsure, want to check if I am thinking correctly.</p>
<p>Exercise sounds: Let n>1, square-free integer. Show that $\mathbb{Z}[\sqrt{n}]/\langle \sqrt{n}\rangle\simeq\mathbb{Z}_n$</p>
<p>My take: Lets make suitable homomorphism $\mathbb{Z}[\sqrt{n}]\rightarrow\mathbb{Z}_n$. Let $x=a+\sqrt{n}b$, which is the... | Tsemo Aristide | 280,301 | <p>Your morphism creates the isomorphism between $Z[\sqrt n]/(\sqrt n)$ and $Z/n$. It is obviously surjective. Suppose $f(a+b\sqrt n)=0$. This implies that $a=nc=c\sqrt n\sqrt n$, thus $a+b\sqrt n= \sqrt n(c\sqrt n+b)\in (\sqrt n)$.</p>
|
3,452,518 | <p>I will call an <em>average</em> any continuous function <span class="math-container">$f(v_1,\dots,v_n)$</span> of <span class="math-container">$n$</span> arguments such that it lies in the closed interval <span class="math-container">$[\min(v_{1},\dots ,v_{n});\max(v_{1},\dots ,v_{n})]$</span>, is symmetric for all ... | TronyJavolta | 676,682 | <p>For <span class="math-container">$k=1,2,...,n$</span> define <span class="math-container">$m_k=\min_{i\neq k}\{v_i\}$</span> and <span class="math-container">$f(v_1,...,v_n)=\max_{k=1,...,n}\{m_k\}$</span>. This can be generalized to exclude the first <span class="math-container">$l$</span> minima, which will alow <... |
150,518 | <blockquote>
<p>Analyze the convergence behavior of the following series: $$\sum_{k=0}^{\infty}\frac{x^{2k}}{2^{2k}}-\frac{x^{2k+1}}{3^{2k+1}}.$$</p>
</blockquote>
<p>I came across this problem as I was preparing for an exam. It is supposed to be an easy one but I am not sure which test to apply.</p>
| Brian M. Scott | 12,042 | <p>I assume that $$\sum_{k=0}^{\infty}\left(\frac{x^{2k}}{2^{2k}}-\frac{x^{2k+1}}{3^{2k+1}}\right)$$ is intended. You can avoid worrying about uniform convergence by using the ratio test. Calling the $k$-th term $a_k$, and letting $u=x/2$, we have</p>
<p>$$a_k=\left(\frac{x}2\right)^{2k}-\left(\frac{x}3\right)^{2k+1}=... |
404,659 | <p>The following is copied verbatim from a book (I. Protasov, <em>Combinatorics of numbers</em>, p. 14):</p>
<blockquote>
<p>Suppose that to each point <span class="math-container">$x$</span> of a set <span class="math-container">$X$</span> a collection <span class="math-container">$\mathcal{B}(x)$</span> of subsets of... | user642796 | 8,348 | <p>Your observation appears to be correct, and you don't appear to be missing anything.</p>
<p>Note that simply adding the condition that $\mathcal{B}(x) \neq \emptyset$ for all $x \in X$ (or, equivalently, $X \in \mathcal{B}(x)$ for all $x \in X$) will yield the usual definition of a topology. (The sets in $\mathcal... |
1,794,025 | <p>Show that <span class="math-container">$Z(G) = \cap_{a \in G} C(a)$</span>.</p>
<p>Let <span class="math-container">$a \in Z(G)$</span>. Then <span class="math-container">$ax=xa$</span> for all <span class="math-container">$x$</span> in <span class="math-container">$G$</span>. In particular we can say that <span cla... | Anon | 245,264 | <p>$$Z(G) = \{g \in G \mid \forall a \in G: ga = ag\} = \bigcap_{a \in G} \{g \in G \mid ga = ag\} = \bigcap_{a \in G} C(a)$$</p>
|
1,794,025 | <p>Show that <span class="math-container">$Z(G) = \cap_{a \in G} C(a)$</span>.</p>
<p>Let <span class="math-container">$a \in Z(G)$</span>. Then <span class="math-container">$ax=xa$</span> for all <span class="math-container">$x$</span> in <span class="math-container">$G$</span>. In particular we can say that <span cla... | ervx | 325,617 | <p>I tried to write a proof in the style you're going for. </p>
<p>Suppose $x\in Z(G)$. Then, $xa=ax$ for all $a\in G$, and hence, $x\in C(a)$ for all $a\in G$. Thus, $x\in\bigcap_{a\in G}C(a)$. Conversely, suppose $x\in\bigcap_{a\in G}C(a)$ for all $a\in G$. Then, $xa=ax$ for all $a\in G$, so that $x\in Z(G)$. </p>
... |
749,171 | <p>Right now I'm taking a 3 part course on probability and statistics using Schverish & Degroot Probability and Statistics and it is just not helpful. For the first part, which was on Probability, I used a First Course in Probability by Sheldon Ross which was extremely helpful unfortunately it doesn't have much sta... | DeepSea | 101,504 | <p>MVT says:$f(2) - f(0) = f'(c)(2 - 0)$ translates $9 - 1 = 2(6c - 2)$. So$4 = 6c - 2$, and $c = 1$</p>
|
454,411 | <p>I want to prove the triangle inequality for the l2-norm $\|x\|_2$:
$$\|x\|_2 = \sqrt{x_1^2+x_2^2+\cdots+x_n^2}$$</p>
<p>$$\begin{align}
\sqrt {\sum\limits_{i = 1}^n (x_i + y_i)^2 } &\leqslant \sqrt {\sum\limits_{i = 1}^n x_i^2 } + \sqrt {\sum\limits_{i = 1}^n y_i^2} \cr
\sum\limits_{i = 1}^n {(x_i + y_i)... | Pedro | 23,350 | <p>You want to prove that <span class="math-container">$$\lVert x+y\rVert \leq \lVert x\rVert+\lVert y\rVert$$</span></p>
<p>We use <span class="math-container">$\langle x,y\rangle$</span> to denote the inner (dot) product <span class="math-container">$$x\cdot y=\sum_{k=1}^n x_ky_k$$</span>
Note that <span class="math-... |
454,411 | <p>I want to prove the triangle inequality for the l2-norm $\|x\|_2$:
$$\|x\|_2 = \sqrt{x_1^2+x_2^2+\cdots+x_n^2}$$</p>
<p>$$\begin{align}
\sqrt {\sum\limits_{i = 1}^n (x_i + y_i)^2 } &\leqslant \sqrt {\sum\limits_{i = 1}^n x_i^2 } + \sqrt {\sum\limits_{i = 1}^n y_i^2} \cr
\sum\limits_{i = 1}^n {(x_i + y_i)... | user279415 | 279,415 | <p>I did exactly what you did above. That was actually very good proof--better than using u and v since it is more abstract hence more generalized than the latter one. You should proceed with you work, since you are almost there.</p>
<p>To continue with what you wrote above, you should notice that the left side of the... |
139,818 | <p>I'm currently working with large datasets of "discrete data". Each of these is a <em>multiset</em> of real numbers of the form $kS + B$, where $k$ is an integer, and $S$ (the <em>"scale</em>") and $B$ (the <em>"bias"</em>) are real values.</p>
<p>These datasets are such that it is often economical to represent the... | halirutan | 187 | <p>I'm just throwing a different idea in the room. Let's say you have a dataset of such structures:</p>
<pre><code>ds = Dataset[{<|"scale" -> 1.234, "bias" -> 5.678,
"tally" -> {{-5, 2}, {-4, 251}, {-3, 5941}, {-2, 60383}, {-1,
241185}, {0, 383613}, {1, 241644}, {2, 61035}, {3, 5686}, {4,
... |
1,185,846 | <p>Hey all I am looking for help on a problem. I will post it, and than I will add what I have tried and my ideas etc. The question has been up now for a few days, I'm sure someone out there can help! I even put a bounty, I have spent a lot of time on this question!</p>
<p>I am interested in calculating the torsion (... | Andrew D. Hwang | 86,418 | <p>The extent to which these remarks explain anything will depend on how comfortable you are with the geometry of vector addition and scalar multiplication. :)</p>
<p>If $p_{1}$, $p_{2}$, and $p_{3}$ are non-collinear points of $\mathbf{R}^{n}$, and if $t_{1}$, $t_{2}$, and $t_{3}$ are real numbers <em>whose sum is un... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | Raskolnikov | 3,567 | <p>Here's a parametric equation I have made for a regular $n$-gon, coded in R:</p>
<pre><code>n=5;
theta=(0:999)/1000;
r=cos(pi/n)/cos(2*pi*(n*theta)%%1/n-pi/n);
plot(r*cos(2*pi*theta),r*sin(2*pi*theta),asp=1,xlab="X",ylab="Y",
main=paste("Regular ",n,"-gon",sep=""));
</code></pre>
<p>And picture:</p>
<p><img src="h... |
3,753,205 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<ul>
<li>(i) - Let <span class="math-container">$\mathcal B$</span> be a basis for a topology <span class="math-container">$\tau$</span> on a non-empty set <span class="math-container">$X$</span>. If <span class="math-container">$\math... | Sten | 149,795 | <p>You have the right idea. Suppose we can find an uncountable collection of open sets not contained in the standard basis for the euclidean topology. Denoting this collection by <span class="math-container">$\mathcal U$</span>, we would then have that for any <span class="math-container">$U \in \mathcal U$</span>, <sp... |
4,370,760 | <p>The two statements are:</p>
<p><span class="math-container">$\forall a\in A, \exists b\in B, M(a)\implies N(b)$</span></p>
<p><span class="math-container">$(\forall a\in A, M(a))\implies (\exists b\in B, N(b))$</span></p>
<p>I am not so good in logics. I feel the second statement implies first one. The existence of ... | ryang | 21,813 | <p>Rewriting your two sentences more readably, and adding a crucial pair of parentheses that presumably you had intended:
<span class="math-container">$$\forall a{\in} A\;\exists b{\in}B\;\Big(M(a)\implies N(b)\Big)\tag1$$</span>
<span class="math-container">$$\forall a{\in} A\;M(a)\implies \exists b{\in} B\; N(b)\tag2... |
4,237,056 | <p>Can any subgroup of a cyclic group <span class="math-container">$\left<a\right>$</span> be presented as <span class="math-container">$\{x\in \left<a\right>|x^m=e\}$</span>?<br />
Note that m is a factor of n, where n is the order of <span class="math-container">$\left<a\right>$</span>.<br />
I foun... | Siong Thye Goh | 306,553 | <p>Steps required:</p>
<p>Substitute <span class="math-container">$ x_2-2 = \frac52\lambda $</span> into <span class="math-container">$\frac{\partial L}{\partial x_2}$</span>, you get a linear equation only in <span class="math-container">$\lambda$</span>. Hence you can solve for <span class="math-container">$\lambda$<... |
578,248 | <p>Let $X$ be a random variable taking values in some metric space $M$. Let $\{\phi_n\}$ be a sequence of measurable functions from $M$ to another metric space $\tilde M$. Suppose that $\phi_n(X)$ converges in law to a random variable $Y$. Must it be the case that the pairs $(X, \phi_n(X))$ converge in law to a pair $(... | Siméon | 51,594 | <p>Generally speaking, the answer is no. Follows a counter-example with Bernoulli random variables (if you prefer continuous variables, you can construct something similar with the central limit theorem, see the end of my answer).</p>
<p>Let $X$ be a sequence of independent random variables with uniform distribution o... |
1,631,851 | <p>Let $I_1, . . . , I_n$ be any open intervals in $\mathbb{R}$ for $n ≥ 2$. How can I show that $(I_1 × · · · × I_n) \setminus \{a\}$ is open and connected for any $a ∈ \mathbb{R}^n$?</p>
| Mathstudent | 307,000 | <p>Use the fact that $\mathbb{R}^n$ is a metric space (which is Hausdorff). If the induced topology on $(I_1 × · · · × I_n)$ is Hausdorff then $\{a\}$ is closed, but in $(I_1 × · · · × I_n)$ and so $(I_1 × · · · × I_n) \setminus \{a\}$ is open, but in $(I_1 × · · · × I_n)$.</p>
|
229,017 | <p>Take the complete graph with n vertices, where one edge has been removed. How can you work out the number of automorphisms that this graph has?</p>
| Brian M. Scott | 12,042 | <p>HINT: Let $G$ be the graph remaining after you remove the edge between vertices $u$ and $v$ in $K_n$, and let $V'=V\setminus\{u,v\}$, where $V$ is the vertex set of $K_n$. In $G$ the vertices $u$ and $v$ ‘look alike’ in every respect, and all of the vertices in $V'$ also ‘look alike’ in every respect. Thus, if $h$ i... |
229,017 | <p>Take the complete graph with n vertices, where one edge has been removed. How can you work out the number of automorphisms that this graph has?</p>
| Douglas S. Stones | 139 | <p>Let the graph be <span class="math-container">$G=(V,E)$</span> with vertex set <span class="math-container">$$V=\{v_1,v_2,\ldots,v_{n-2}\} \cup \{u_1,u_2\}$$</span> and edge set <span class="math-container">$$E=\{v_i v_j : i,j \in \{1,2,\ldots,n-2\} \text{ and } i \neq j\} \cup \{v_i u_j: i \in \{1,2,\ldots,n-2\} \t... |
1,188,483 | <p>Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.</p>
<p>To make such a mobius transformation i tried to send 3 points on the edge to 3 points of the edge. so filling $i,1,-1$ in $f(x)$ we should get on the edges of the unit ball. But i don'... | Tim Raczkowski | 192,581 | <p>If $|z|=1$, then $\overline z=1/z$. So,</p>
<p>$$\left|{z-a\over 1-\overline az}\right|=\left|{z-a\over1-\overline{a/z}}\right|=\left|{\overline z(z-a)\over\overline z-\overline a}\right|=|\overline z|\left|{z-a\over\overline{z-a}}\right|=|\overline z|=1.$$</p>
|
2,110,111 | <p>For example, concerning de Rham cohomology as explained on the lower part of page 5 <a href="http://www.math.uni-hamburg.de/home/wockel/teaching/data/cohomology_of_lie_algebras_mavraj.pdf">here</a>, one considers the vector space of all forms on a manifold $M$, $A_{DR}(M)$ in the notation of the paper (?).</p>
<p>T... | Jonathan Grant | 316,806 | <p>Cohomology/homology is always defined this way. A chain complex is a vector space equipped with a linear map $d:V\to V$ such that $d^2=0$. This ensures that the image is a subspace of the kernel so that the quotient is defined, and this quotient is the homology of the chain complex.</p>
<p>There are many homology t... |
485,340 | <p>My question is to to find the period of
$$\sin(8\pi\{x\}),$$
where $\{\cdot\}$-is the fractional part of function.</p>
<p>I know that the period of $\{\cdot\}$ is 1 and the period of $\sin(8\pi x)$ is $1/4$. But how to find the overall period of the given function? </p>
| Rémy Sigrist | 753,637 | <p><span class="math-container">$\sin(8 \pi \{x\})
= \sin(8 \pi ( x - \lfloor x \rfloor))
= \sin(8 \pi x - 8 \pi \lfloor x \rfloor)
= \sin(8 \pi x)$</span>,</p>
<p>hence <span class="math-container">$\sin(8 \pi \{x\})$</span> has period <span class="math-container">$\frac{1}{4}$</span>.</p>
|
334,541 | <p>The standard frame for <span class="math-container">$S^3$</span> consists of <span class="math-container">$X_i,X_j,X_k$</span> with <span class="math-container">$X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$</span> where <span class="math-container">$i,j,k$</span> are standard quaternion numbers, <span class="math-container">$a\... | KSackel | 66,405 | <p>I think the question is perhaps confusing, since the term Hopf vector field is usually reserved for your <span class="math-container">$X_i$</span> vector field (which is tangent to the fibers of the standard Hopf fibration). As I understand, you are instead referring to the vector field which, at a point <span class... |
750,322 | <p>Find a formula for $\sum_{i=1}^{n} \frac{1}{(2i-1)(2i+1)}$ and prove that it holds for all $n \geq 1$ I don't know how to solve this particular problem, can someone help me please. Thanks </p>
| Michael Joyce | 17,673 | <p>Here are the steps to take to prove a formula for the sum by induction. I'll leave it to you to work out the details.</p>
<p>1) Compute values of the sum for small values of $n$. Doing so for 5 or more values of $n$ will hopefully reveal a pattern that you can conjecture holds for all $n$.</p>
<p>2) Now that y... |
621 | <p>The OP of this question is asking if <a href="https://mathematica.stackexchange.com/questions/9639">https://mathematica.stackexchange.com/questions/9639</a> could be reopened. I did not want to act unilaterally, so I want to ask the rest of you whether you think the question should be reopened. Plead your case for o... | J. M.'s persistent exhaustion | 50 | <p>For those who'd rather vote than write an answer:</p>
<p>Vote this answer up if you think <a href="https://mathematica.stackexchange.com/questions/9639">https://mathematica.stackexchange.com/questions/9639</a> should <strong>not</strong> be reopened.</p>
|
391,572 | <p>I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?</p>
| Thomas Russell | 32,374 | <p>Euler's Number, $e\approx2.718281828$, is a special constant in calculus because of several important properties it holds. </p>
<p>First, $y=e^{x}$ (also written as $y=\exp(x)$ due to the <em>exponential</em> growth of the function) and constant multiples thereof is the only function for which:</p>
<p>$$\frac{dy}{... |
427,663 | <p>If $|z|=|w|=1$, and $1+zw \neq 0$, then $ {{z+w} \over {1+zw}} \in \Bbb R $</p>
<p>i found one link that had a similar problem.
<a href="https://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1">Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w}... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Let $z=\cos A+i\sin A, w=\cos B+i\sin B$</p>
<p>$\implies z\cdot w=\cos(A+B)+i\sin(A+B)$</p>
<p>Using $\cos C-\cos D=-2\sin\frac{C-D}2\sin\frac{C+D}2$</p>
<p>and $\sin C-\sin D=2\sin\frac{C-D}2\cos\frac{C+D}2$</p>
<p>$$\frac{z-w}{1-zw}=\frac{\cos A-\cos B+i(\sin A-\sin B)}{1-\{\cos(A+B)+i\sin(A+B)... |
1,769,596 | <p><strong>Question</strong> Calculate the critical point(s) for the following function $$f =\sin(x)+\cos(y) +\cos(x-y)$$</p>
<p><em>Hint:</em> Use the trigonometric identity $$\sin(\alpha) +\sin(\beta) = 2\sin(\frac{\alpha+\beta}{2})\cos(\frac{\alpha -\beta}{2})$$</p>
<p><strong>My attempt:</strong></p>
<p>$$f_x=\c... | Vincent | 332,815 | <p>Use this trigonometric formula (and the ones with $\cos$) until you only have a product of sines and cosines. Then, say it is null iff one of them is null</p>
|
2,790,767 | <p>Let $X_i$ a countable collection of independent random variables with symmetric distribution i.e. $P(X_i\in A)=P(X_i \in -A)$ for all $i\geq 1$. If $\lambda\in\mathbb{R}$ is such that $E(e^{\lambda X_i})$ is finite for all $i$ I want to prove the following Kolmogorov like inequality
$$P\left(\max_{1\leq k\leq n} \le... | Adrián González Pérez | 331,216 | <p>Using extrapolation I am able to prove your statement but only for $t$ large relative to $\lambda^{-1}$ and with the stronger condition of $E[e^{\lambda |X_i|}] < \infty$ (maybe you were missing an absolute value). Let us denote by $S$ the random variable
$$
S = \sum_{j = 1}^\infty X_j
$$
and by $\Sigma_n$ the... |
3,715,674 | <p>Let <span class="math-container">$X$</span> be a scheme and <span class="math-container">$U$</span> an open subset of <span class="math-container">$X$</span>. Let <span class="math-container">$s \in \Gamma(U, O_X)$</span> and <span class="math-container">$x \in U$</span>.<br>
I am getting confused with what the diff... | Manuel Norman | 790,256 | <p>A differential <span class="math-container">$k$</span> - form <span class="math-container">$\omega$</span> is exact if there exists a <span class="math-container">$(k-1)$</span> - form <span class="math-container">$\alpha$</span> such that <span class="math-container">$\omega=d \alpha$</span>. This (applied to your ... |
3,974,200 | <p>Let <span class="math-container">$X$</span> be an <span class="math-container">$n\times n$</span> negative definite matrix such that <span class="math-container">$X_{ii}<0$</span> for all <span class="math-container">$i$</span> and <span class="math-container">$X_{ij}\geq 0$</span> for <span class="math-container... | reuns | 276,986 | <p>Split <span class="math-container">$[0,1]=\bigcup_j [a_j,b_j]$</span> where <span class="math-container">$p$</span> is injective on <span class="math-container">$(a_j,b_j)$</span> (ie. <span class="math-container">$p'$</span> doesn't vanish) then <span class="math-container">$$\int_{a_j}^{b_j} e^{2i\pi np(x)}dx=\int... |
2,821,514 | <p>so I'm trying to work through this problem: </p>
<p>How many elements does a 2-Sylolw subgroup of $\left(\mathbb Z^\times_{11} \times \mathbb Z^\times_{13} , \cdot \right)$ have (i.e. what is the order) and how many 2-Sylow subgroups there are in $\left(\mathbb Z^\times_{11} \times \mathbb Z^\times_{13} , \cdot \ri... | Joshua Mundinger | 106,317 | <p>$\mathbb{Z}_{11}^\times \times \mathbb{Z}_{13}^\times$ is abelian, and all Sylow subgroups of a given order are conjugate, so there is exactly one Sylow 2-subgroup. It consists of all elements of order a power of two. Since $ord(a,b) = lcm(ord(a),ord(b))$, it is generated by $(-1,1)$ and $(1,x)$ for $x$ a square roo... |
3,577,098 | <p>In metric space there is a well-known result that <span class="math-container">$X$</span> is a separable metric space <span class="math-container">$\implies$</span>Every uncountable set in <span class="math-container">$X$</span> has a limit point.My question is does the result hold if <span class="math-container">$(... | David G. Stork | 210,401 | <p>Bayes theorem in this case states:</p>
<p><span class="math-container">$$P(C=6|C>E) = \frac{P(C>E|C=6) P(C=6)}{P(C>E)} = \frac{.5 \cdot .1}{.45} = .1111 = \frac{1}{9}$$</span>.</p>
|
3,577,098 | <p>In metric space there is a well-known result that <span class="math-container">$X$</span> is a separable metric space <span class="math-container">$\implies$</span>Every uncountable set in <span class="math-container">$X$</span> has a limit point.My question is does the result hold if <span class="math-container">$(... | G Cab | 317,234 | <p>This is just an expansion / complement to <em>Aiden</em>'s answer.</p>
<p>The pair of balls extracted represents a point in a <span class="math-container">$10 \times 10$</span> square.</p>
<p><a href="https://i.stack.imgur.com/kPH65.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kPH65.png" alt=... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Andy Putman | 317 | <p>The best "first course" in representation theory is Fulton and Harris's book. I've only skimmed it, but Ginzburg and Chriss's book "Representation Theory and Complex Geometry" looks like a wonderful second course.</p>
|
1,856,225 | <p>Right, so as the final step of my project draws near and after having made a bad layout sort of question, I am posting a new one very clear and unambiguous. I need to find this specific definite integral which Mathematica could not solve:</p>
<p>$$ \int_{x=0}^\pi \frac{\cos \frac{x}{2}}{\sqrt{1 + A \sin^2 \frac{x}... | Noam Shalev - nospoon | 219,995 | <p>Here is a sketch for a residue-free solution: </p>
<p>First, consider the formula $$\int_0^{\infty} \frac{\cos( a x)}{\cosh(\frac{\pi}{2} x)}dx = \operatorname{sech} a. \tag{1}$$</p>
<p>Since $\displaystyle \,\,\sin(a)\sin(b) = \frac12 \cos(a-b)-\frac12 \cos(a+b), \,\,$ we obtain
$$ \int_0^{\infty} \frac{\sin( a ... |
2,882,889 | <p>Let $(P_n)$ be a sequence such that $P_0 = 1$ and $P_n = (\log{(n+1)})^n$.</p>
<p>I'm trying to prove that $P_n ^2 \leq P_{n-1}\,P_{n+1}$ for $n \geq 1$ using an induction argument.</p>
<p>For $n=1$ we have that $(\log 2)^2 = 0.48$ and $1\cdot (\log 3)^2 = 1.206$, hence $P_1 ^{2} \leq P_{0}\,P_2$.</p>
<p>Given $n... | rtybase | 22,583 | <p>One (probably long) way is
$$P_n ^2 \leq P_{n-1}\,P_{n+1} \iff \frac{P_n}{P_{n-1}} \leq \frac{P_{n+1}}{P_n}$$
which is equivalent to showing that the sequence $a_n=\frac{P_n}{P_{n-1}}$ is increasing. For this purpose, let's look at the function
$$\color{blue}{f(x)=\ln\left(\frac{(\ln(x+1))^x}{(\ln(x))^{x-1}}\right)... |
298,066 | <p>Let $A$ be a commutative ring, and consider the category of bimodules over $A$.</p>
<p>An $A$-bimodule $M$ is called symmetric if $a\cdot m = m \cdot a$ for all $a \in A$, $m \in M$.</p>
<p>Is the category of symmetric bimodules over $A$ closed under extensions?</p>
<p>Namely, given an exact sequence of $A$-bimod... | Mare | 61,949 | <p>Here a counterexample found by computer:</p>
<p>Let $A=k[x]/(x^2)$ for a field $k$. Let $B:=A^{op} \otimes_k A$ be the enveloping algebra of $A$. The module category of $B$ is isomorphic to the module category of $A$-bimodules.</p>
<p>Take $M=A$ as a bimodule and $S$, the unique simple module of $B$.</p>
<p>Now a... |
298,066 | <p>Let $A$ be a commutative ring, and consider the category of bimodules over $A$.</p>
<p>An $A$-bimodule $M$ is called symmetric if $a\cdot m = m \cdot a$ for all $a \in A$, $m \in M$.</p>
<p>Is the category of symmetric bimodules over $A$ closed under extensions?</p>
<p>Namely, given an exact sequence of $A$-bimod... | Mare | 61,949 | <p>Here a general abstract argument that shows that the subcategory of symmetric bimodules is never extension closed in case $A$ is a commutative finite dimensional Frobenius algebra that is not semisimple:</p>
<p>Let $A$ be such an algebra and $B=A \otimes_K A$ its enveloping algebra and assume that the subcategory o... |
1,072,634 | <p>I need to solve the differential equation $$y\frac{dy}{dx} = (x+7)(y^2+6)$$</p>
<p>I know that the first step is to isolate both term each side and then integrate...</p>
<p>But I can't figure out how to isolate term on this one, I would probably be able to solve the rest of the equation, just need to know how to s... | Jonathan Komar | 81,829 | <p>I know exactly why that is confusing, and I will try to explain this step by step. <strong>The definition of surjectivity does not mention the domain being overfull (only the codomain).</strong> That throws some people off when learning this the first time. The reason for your confusion is likely that you skipped ov... |
165,791 | <p>I'm looking for a maximum accuracy quadrature formula:</p>
<p>$$
\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)
$$</p>
<p>I don't know exactly if it's Trapezoidal rule which has the degree of accuracy one, or Simpson's rule with three or Gauss (these 3 I have studied deeply, but might be ... | Federico Poloni | 1,898 | <p>I assume you mean maximum degree of accuracy for polynomials, that is, you require that the formula is exact for polynomial functions of degree up to <span class="math-container">$d$</span>, and you look for the maximum possible <span class="math-container">$d$</span>. This is the most usual requirement, although no... |
165,791 | <p>I'm looking for a maximum accuracy quadrature formula:</p>
<p>$$
\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)
$$</p>
<p>I don't know exactly if it's Trapezoidal rule which has the degree of accuracy one, or Simpson's rule with three or Gauss (these 3 I have studied deeply, but might be ... | Alex Gavrilov | 9,833 | <p>As Federico Poloni said, it is a special case of Gauss-Jacobi quadrature.
I am in a good mood, so here is your homework. The formula is
$$\int_{-1}^{1}\sqrt{\frac{1-x}{1+x}}f(x)dx=
\frac{\pi(5+\sqrt{5})}{10}f\bigg(\frac{-1-\sqrt{5}}{4}\bigg)+\frac{\pi(5-\sqrt{5})}{10}f\bigg(\frac{-1+\sqrt{5}}{4}\bigg)+R(f).$$</p>
|
2,743,604 | <p>**A busy railway station has two taxi stands at separate exits, A.
At stand A, taxis arrive according to a Poisson Process of rate 2 per minute.</p>
<p>Following the arrival of a train, a queue of 40 customers suddenly builds up at
stand A, waiting for taxis to arrive. Assume that whenever a taxi arrives, it
instan... | fleablood | 280,126 | <p>In a real hour, the watch will be stopped for $15$ minutes and running for $45$ minutes. So in a real period of $k$ hours the watch will have been running for $\frac 34k$ hours. So in a $\frac 43*12 = 16$ hour period the watch will have run for $12$ hours.</p>
<p>Now there are three options. 1) If when the clock ... |
2,035,237 | <p>Let $f: [0, \infty) \rightarrow R$ be a continuous function and define
$$F(t) = \int_{0}^{\frac{1}{t}} f(tx)\cos(t^2x) dx.$$</p>
<p>Show that $\lim_{t \rightarrow \infty} tF(t) = 0.$</p>
<p>This is part of a question from an old analysis qualifying at my university. I came across it while trying to study for my fi... | Glitch | 74,045 | <p>Make the change of variable $y=tx$ to see that
$$
tF(t) = \int_0^1 f(y) \cos(ty) dy.
$$</p>
<p>Let $\epsilon >0$. Since $f$ is continuous on $[0,1]$, which is compact, we can use the Weierstrass approximation theorem to pick a polynomial $p$ such that
$$
\sup_{y \in [0,1]} |p(y) - f(y)| < \epsilon.
$$
Then ... |
2,332,544 | <p>I've been messing around with a quadratic that popped out of a problem I'm working on, and noticed it has a property I haven't seen before. The form is as follows:</p>
<p>$$ ax^2 + bx + c = 0$$
where
$$a = r_{1}^{2} + r_{2}^{2} - r_{3}^{2}$$
$$b = 2\left( r_{1} t_2 + r_{2} t_2 - r_{3} t_3 \right)$$
$$c = t_{1}^... | Jason | 195,308 | <p>Let $B=\{e_n:n\ge1\}$ where $e_n=(0,\ldots,0,1,0,\ldots)$, the $1$ being in the $n^\text{th}$ position. Then $B\cap\mathbb R^n=\{e_k:1\le k\le n\}$ which is closed, hence $B$ is closed. Clearly $B\subset S^\infty$, so it suffices to show $B$ is not compact. Let $U_n=\{x=(x_1,x_2,\ldots):|x_n-1|<\frac12\}$. Then $... |
325,236 | <blockquote>
<p>Is there a bijection between $\mathbb N$ and $\mathbb N^2$?</p>
</blockquote>
<p>If I can show $\mathbb N^2$ is equipotent to $\mathbb N$, I can show that $\mathbb Q$ is countable. Please help. Thanks,</p>
| Aang | 33,989 | <p>Yes there exists a quadratic bijection. </p>
<p>See <a href="http://www.math.drexel.edu/~tolya/cantorpairing.pdf" rel="nofollow">this</a></p>
|
1,397,450 | <p>10 objects are randomly distributed among 3 boxes. What is the probability to have 6 objects in one of the boxes, 3 in another one and a single object in the remaining third box. </p>
<p>My solution is </p>
<p>For the first box we have $10 \choose 6$, since 6 objects are already placed we are left with 4 remainin... | Mark Bennet | 2,906 | <p>If $z=re^{i\theta}=e^{\ln r+i \theta}$ you can raise to the power $w$ in the usual way (multiplication of exponents), even if $w$ is a complex number. </p>
<p>However the expression of $z$ in this manner is far from unique because $\theta +2n\pi$ for integer $n$ will do as well as $\theta$ and raising to a constant... |
274,885 | <p>I've been doing some research on ranking algorithms, and I've read theresearch with interest. The aspects of the Schulze Algorithm that appeals to me is that respondents do not have to rank all options, the rank just has to be ordinal (rather than in order), and ties can be resolved. However, upon implementation, I'... | user56796 | 56,796 | <p>There isn't necessarily a unique winner. However, the following statements can be proven:</p>
<ul>
<li><p>There is always at least one winner (section 4.1).</p></li>
<li><p>Unless there are pairwise links of equivalent strengths, there is a unique winner (section 4.2.1).</p></li>
<li><p>When there is not a unique w... |
274,885 | <p>I've been doing some research on ranking algorithms, and I've read theresearch with interest. The aspects of the Schulze Algorithm that appeals to me is that respondents do not have to rank all options, the rank just has to be ordinal (rather than in order), and ties can be resolved. However, upon implementation, I'... | TermoTux | 156,992 | <p>I have developed a simple voting app using Schulze algorithm. Assuming I did input your data correctly the results are as follows:</p>
<p>\begin{equation}
\mathbf{Choices}= \begin{pmatrix}
KG & LJ & JS & DL & TC\\
\end{pmatrix}
\end{equation}</p>
<p>\begin{equation}
\mathbf{D}= \begin{pmatrix} ... |
2,235,304 | <blockquote>
<p>Let <span class="math-container">$f(z) = \frac{z^2}{(z^2+1)^2(z^2+2z+2)}$</span>.</p>
<p>Find singularities of this function. For each singularity determine if it is removable, a pole (if a pole determine it's order,) or essential.</p>
</blockquote>
<p>I have worked out that the singularities of this fu... | lhf | 589 | <p><em>Hints:</em></p>
<ul>
<li><p>Rational functions only have poles because the zeros of the denominator are isolated.</p></li>
<li><p>A singularity $z_0$ of a rational function is removable iff we can cancel $z-z_0$ enough times in both the numerator and the denominator to make it disappear in the denominator.</p><... |
2,011,378 | <p>My girlfriend suggested I use Lagrange theorem, but that only deals with finite groups.</p>
<p>If G was finite, and H has order 3 and K has order 5 , then clearly since the orders of H and K must divide the order of G, then there must also be an element of order 15.</p>
<p>But what about if G is infinite?</p>
<p>... | Learnmore | 294,365 | <p>$G$ has a subgroup $H_1$ of order $3$ which is cyclic .Hence $H_1$ has an element of order $3$ say $a$ and so does $G$.</p>
<p>Similarly $G$ has an element of order $5$ say $b$.</p>
<p>Now $\gcd(o(a),o(b))=1$ and $ab=ba\implies o(ab)=o(a)o(b)=3\times 5=15$</p>
|
1,379,878 | <blockquote>
<p>Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.</p>
</blockquote>
<p>My Try:</p>
<p>I proved that $C(M)$ is ... | WLOG | 463,415 | <p>Observe that it is enough to prove the statement for the Jordan blocks.</p>
<p>Let <span class="math-container">$J$</span> be an <span class="math-container">$n\times n$</span> Jordan block whose diagonal entries are <span class="math-container">$\lambda$</span>, and let <span class="math-container">$U$</span> be th... |
41,858 | <p>Halting Problem is (theoretically) decidable for such algorithms which termination may be proved in First Order Logic (FOL) because all true statements in FOL are recursively enumerable. It is correct?</p>
<p>I'm talking about programs in Turing-complete languages and use "realworldiness" property not in the sense ... | JSchlather | 1,930 | <p>The Halting Problem is solvable for linear bounded automaton, here is a <a href="http://www.cs.uky.edu/~lewis/texts/theory/automata/lb-auto.pdf" rel="nofollow">reference</a>.</p>
<p>A linear bounded automaton is a turing machine that has one tape of fixed size that the input is read in on and all computation is per... |
2,187,320 | <blockquote>
<p>In triangle ABC, $m\angle{A}=a\log{x}$ $m\angle{B}=a\log{2x}$ $m\angle{C}=a\log{4x}$. what is $m\angle{B}$</p>
</blockquote>
<p><strong>My steps</strong></p>
<p>$a\log{x}+a\log{2x}+\log{4x}=\pi$</p>
<p>which becomes $a\log{x^a}+\log{2x^a}+\log{4x^a}=\pi$</p>
<p>which devolves into: $\log(2^ax^{3a}... | marty cohen | 13,079 | <p>$\begin{array}\\
\pi
&=a\log{x}+a\log{2x}+a\log{4x}\\
&=\log{x^a}+\log{(2x)^a}+\log{(4x)^a}\\
&=\log{x^a2^ax^a4^ax^a}\\
&=\log{x^{3a}8^a}\\
&=\log{x^{3a}2^{3a}}\\
&=a\log{x^{3}2^{3}}\\
&=a\log{(2x)^{3}}\\
&=3a\log{2x}\\
&=3m\angle B\\
\end{array}
$</p>
<p>That was my initial proo... |
2,187,320 | <blockquote>
<p>In triangle ABC, $m\angle{A}=a\log{x}$ $m\angle{B}=a\log{2x}$ $m\angle{C}=a\log{4x}$. what is $m\angle{B}$</p>
</blockquote>
<p><strong>My steps</strong></p>
<p>$a\log{x}+a\log{2x}+\log{4x}=\pi$</p>
<p>which becomes $a\log{x^a}+\log{2x^a}+\log{4x^a}=\pi$</p>
<p>which devolves into: $\log(2^ax^{3a}... | browngreen | 321,445 | <p>$a\log{x}+a\log{2x}+a\log{4x}= \pi$</p>
<p>$=a(\log(8x^3))=a(\log(2x)^3)=3a(\log(2x))$</p>
<p>Therefore, angle $B = a(\log(2x))={\pi\over 3}$</p>
|
2,030,063 | <p>Let $A$ and $B$ be two disjoint subset of $\mathbb{R}^2$ such that $A \cup B$ is open and disconnected in $\mathbb{R}^2$. does it follows both $A$ and $B$ are open. </p>
<p>If $A$ and $B$ are both open, since $A$ and $B$ are disjoint by hypothesis. then by definition $A \cup B$ is disconnected. But for the revers... | 5xum | 112,884 | <p>No, since $A=(0,1)$ and $B=[1,2)\cup(3,\infty)$ provides a counterexample.</p>
|
2,030,063 | <p>Let $A$ and $B$ be two disjoint subset of $\mathbb{R}^2$ such that $A \cup B$ is open and disconnected in $\mathbb{R}^2$. does it follows both $A$ and $B$ are open. </p>
<p>If $A$ and $B$ are both open, since $A$ and $B$ are disjoint by hypothesis. then by definition $A \cup B$ is disconnected. But for the revers... | Robert Z | 299,698 | <p>No. Let
$$A:=((0,1)\times (2,3))\cup ((0,1)\times (0,1)) \cup (I\times \{0\})$$
and
$$B:=((0,1)\times (-1,0)) \cup (J\times \{0\})$$
where $I=\mathbb{Q}\cap (0,1)$ and $J=(0,1)\setminus I$.</p>
<p>Then $A\cup B=((0,1)\times (-1,1))\cup ((0,1)\times (2,3))$ is open and disconnected in $\mathbb{R}^2$. </p>
<p>On th... |
1,918,781 | <p>Suppose that $\phi:\mathbb R^3\to\mathbb R$ is a harmonic function. I am asked to show that for any sphere centered at the origin, the average value of $\phi$ over the sphere is equal to $\phi(0)$. I am also directed to use Green's second identity: for any smooth functions $f,g:\mathbb R^3\to \mathbb R$, and any sph... | Jeff | 313,346 | <p>The easiest way, in my view, to prove the mean value property is to set</p>
<p>$$g(r) = \frac{1}{4\pi r^2} \int_{S_r} \phi(x)\, dS(x),$$</p>
<p>where $S_r$ denotes the sphere of radius $r>0$ centered at the origin. Since $\phi$ is harmonic, hence continuous, we have</p>
<p>$$g(0) := \lim_{r\to 0^+} g(r)= \phi(... |
2,212,946 | <blockquote>
<p><strong>Question:</strong>
Show that if $f(x)$ is a continuous function, then:
$$\int_{0}^{1} \int_{0}^{y} f(x) \, dx \, dy = \int_{0}^{1}(1-x)\ f(x) \ dx$$</p>
</blockquote>
<p>How can this be done? I was taught to visualise double integrals as either a summation of horizontal or vertical strips... | Hugh Kinnear | 431,382 | <p>Try to visualize the space you are integrating over. To do this draw out the Cartesian plane and try to shade in the space indicated by the integrals. </p>
<p>Since dy is on the outside and dx is on the inside you will be drawing using horizontal strips. The strips start y=0 and finish at y=1 (as indicated by the o... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Xander Henderson | 8,571 | <p>Write out the numbers using the "expanded form", then use the distributive property of multiplication over addition to evaluate the product. Consistently using powers of <span class="math-container">$10$</span> throughout, this is
<span class="math-container">\begin{align}
3.9 \cdot 7.5
&= \left(\color{red}{3}\... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Mitchell Spector | 13,838 | <p><a href="https://i.stack.imgur.com/QyJRr.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/QyJRr.png" alt="Slide rule"></a></p>
<p>This slide rule is set to show the same multiplication problem, <span class="math-container">$3.9\times7.5=29.25.$</span> (Click on the image for a larger view showing... |
377,720 | <p>Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$ </p>
<p>(a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with Fourier-transforms: $$F(f'')=s^2F(f)=F(u) = 1-e^{-2\pi i s}$$ </p>
<p>which would imply $f= F^{-1}(s^{-2}(1- e^... | timur | 2,473 | <p><strong>Regarding (b).</strong> It seems that one can do without $g_0$. In the picture below, the first graph is meant to depict $g_2$, the second one is $g_2'$, and the third one is $g_1$. The second derivative of $g_2$ will give you $-\frac12\chi_{[-1,0]}+\delta_0-\delta_1+\frac12\chi_{[1,2]}$. Then the two charac... |
2,535,347 | <p>Okay so I'm required to find if the given function is <strong>one-one/many-one and onto/into</strong>.</p>
<p>The function is : $\mathbb R \to \mathbb R$ and $$f(x) = x\left(\frac{2^x - 2^{-x}}{2^x + 2^{-x}}\right)$$</p>
<p>So as this function is even , it can't be one-one .But I'm facing difficulty in deducin... | Your IDE | 499,918 | <p>Well, for every $ x$ in $ \mathbb R ,\ f(x)$ clearly has a value mapped into $ \mathbb R$. </p>
<p>Also, since the denominator doesn't go to zero for any $ x$, you can say the function doesn't have any point discontinuity. </p>
<p>So, we can say that $ f$ is continuous for every $ x$ in its domain (just by observ... |
155,897 | <p>I would like to exclude non-western characters and words from a text file. I do not know how to insert the text file here, but I suppose you can do without it. All your suggestions will be much appreciated.</p>
<p>Update: I have used:</p>
<pre><code>Alphabet["Russian"]
Select[dict21, Not@StringContainsQ[#, Alterna... | Alexey Popkov | 280 | <p>How to delete all characters except from a list.</p>
<p>First, here is a string containing all the characters currently supported by <code>String</code> in the Wolfram Language:</p>
<pre><code>allSupportedCharacters = FromCharacterCode@Range[0, 65535];
</code></pre>
<p>And here is a list of characters you wish to ke... |
3,084,998 | <p>Define the linear transformation <span class="math-container">$L$</span> as </p>
<p><span class="math-container">$L: \mathbb{R}^3 \rightarrow \mathbb{R}^3: x \rightarrow x - \langle x, a \rangle b$</span></p>
<p>Let <span class="math-container">$a, b \in \mathbb{R}^3$</span> such that <span class="math-container">... | MathIsFun7225 | 424,950 | <p>Your solution is not "wrong," but if you want to express the complex exponential in terms of trigonometric functions, use Euler's formula:
<span class="math-container">$$e^{ix}=\cos x+i\sin x.$$</span></p>
|
58,009 | <p>Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times \mathbb{A}^n \to U$, it is likely that there exist morphisms that are not locally of that form, but I can't come up with a... | Jack Huizenga | 7,399 | <p>EDIT: See the comments for why this isn't a good example. Angelo gives a similar example that actually works.</p>
<p>Suppose that $Y=C$ is a cuspidal curve, and let $\tilde C\to C$ be the normalization. Put $X = \mathbb{A}^1\times \tilde C$, and let $X\to Y$ be the obvious map. The fibers are all $\mathbb{A}^1$... |
2,572,286 | <p>I have $mp = 156^{107} \pmod{17},\;$ which simplifies to $3^{11} \pmod{17}.$</p>
<p>How did he get the $11$?</p>
<p>Same for </p>
<p>$mq = 156^{107} \pmod {11},\;$ which simplifies to $2^7 \pmod {11}.$</p>
<p>How to get the $7$? I know how to get the $3$ and $2$</p>
| user35603 | 84,245 | <p><strong>Hint:</strong> The <a href="https://en.wikipedia.org/wiki/Euler%27s_theorem" rel="nofollow noreferrer">Euler's Theorem</a> states that: If $a$ and $m$ are coprime then $a^{\varphi(m)} = 1 \pmod{m}$, $\varphi(\cdot)$ is <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function" rel="nofollow noreferre... |
220,170 | <p>My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?</p>
<p>For example i would start with :</p>
<ol>
<li... | David C | 27,816 | <p>Maybe by looking first at homology $3$-spheres and in particular to Brieskorn manifolds $M(p,q,r)$:
the link of the singular point $(0,0,0)$ of the hypersurface
$$z^p_1+z_2^q+z^r_3=0$$
with integers $p,q,r\geq 2$.
Milnor studied these manifolds in his beautiful paper "On the 3-dimensional Brieskorn manifolds M(p,q,... |
1,281,950 | <p>Is there a way to express the integral</p>
<p>$I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$,</p>
<p>where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known functions? In particular, I am interested in a form where $t$ appears in the limits and not in the integra... | Hasan Saad | 62,977 | <p>Assume a matrix $A$ has two inverse matrices $B$ and $C$. Then, </p>
<p>$B=IB=CAB=C(AB)=CI=C$</p>
<p>QED.</p>
|
1,281,950 | <p>Is there a way to express the integral</p>
<p>$I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$,</p>
<p>where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known functions? In particular, I am interested in a form where $t$ appears in the limits and not in the integra... | k1.M | 132,351 | <p>Suppose that a matrix $A$ has a left (right) inverse $B$, then $A$ is invertible because we have
$$
X=IX=(BA)X=B(AX)
$$
hence
$$
AX=O\iff X=O
$$
which means that $A$ has left and right inverses. Now suppose that for some $B,C$, $AB=I$, and $CA=I$, then
$$
C=CI_n=C(AB)=(CA)B=I_nB=B
$$
So the right and left converses ... |
2,015,717 | <p>I know that the O notation tells me to find $n_0$ natural and $c>0$ real, so the
$$2^{(n^2)} \le c*2^{2n} $$
Only step that I can think about is this:
$$2^{(n^2)} \le c*2^{n+n} $$
$$2^{(n^2)} \le c*2^n*2^n $$
But I have no clue what to do next. (Some estimate?)</p>
<p>Do you have some hints please?</p>
<p... | andreshp | 220,767 | <p>That is impossible. Assume that you have found such a $n_0$ and $c > 0$. Then
$$2^{(n-1)^2} = 2 \frac{2^{n^2}}{2^{2n}} \le 2c $$
for every $n \ge n_0$. This is clearly false since $\lim_{n \to \infty}2^{(n-1)^2} = \infty$.</p>
|
2,559,623 | <p>The question is as follows:<p>
Calculate $\lim_{n \to +\infty} \int_{0}^{+\infty} \frac{n \sin(\frac{x}{n})}{1 + x^2} dx$.<p></p>
<p>$\textbf{Some ideas:}$<p>
We can use the fact that $\sin(\frac{x}{n}) \simeq \frac{x}{n} $.But then we find that <p>
$\lim_{n \to +\infty} \int_{0}^{+\infty} \frac{n \sin(\frac{x}{n})... | user | 505,767 | <p>Note that:</p>
<p>$C_1=\arg(z)=\frac{\pi}{6}$ is the <strong>Ray "$l$"</strong></p>
<p>$C_2=\vert z- i2\sqrt{3} \vert =r$ is <strong>a circle</strong> centered in $(0,2\sqrt{3})$</p>
<p>Here a sketch of what is going on:</p>
<p><a href="https://i.stack.imgur.com/GO8jN.jpg" rel="nofollow noreferrer"><img src="ht... |
893,839 | <p>I am dealing with the dual spaces for the first time.</p>
<p>I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.</p>
| daw | 136,544 | <p>When dealing with finite-dimensional vector spaces, it makes not much sense to study dual spaces. Every finite-dimensional vector space over the fields of real or complex numbers can be equipped with a scalar product. There is then no need to introduce a dual space in finite-dimensional linear algebra imho. </p>
<... |
175,664 | <p>How can I have a region plot with the logarithmic axis? In the following, I have brought the original case which I want to be logarithmic. In this case, I have a density plot and a region plot that shows the realm of validity of the theory. I want to show this figure in logarithmic scales in order to be more clear.<... | Henrik Schumacher | 38,178 | <p><s>Apparently, <code>RegionPlot</code> does not support <code>ScalingFunctions</code>.</s>
It does, but it has to be convinced a little, it applies the scaling functions only to the axes, and also neither the syntax highlighter nor the documentation know about it. Here is a possible workaround (at least for version... |
1,576,561 | <p>I'd like to show that $$\sum\limits_{n = 1}^\infty {{{{x^{n + 1}}} \over {n(n + 1)}}} $$ absolutely converges for $|x| < 1$</p>
| egreg | 62,967 | <p>Hint: if $\displaystyle\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}<1$ then the series (with no zero terms) $\displaystyle\sum_{n\ge0}a_n$ absolutely converges.</p>
|
4,352,129 | <p>I am struggling to compute the power series expansion of <span class="math-container">$$f(z) = \frac{1}{2z+5}$$</span> about <span class="math-container">$z=0$</span>, where <span class="math-container">$f$</span> is a complex function. I tried comparing it to the geometric series as follows,<span class="math-contai... | Rod Laver | 1,012,058 | <p>To get the expansion about <span class="math-container">$z=0$</span>, we need to ensure <span class="math-container">$\omega$</span> doesn't contain an additive term and is just some scaling of <span class="math-container">$z$</span>. By considering <span class="math-container">$5f(z)=\frac{1}{2z/5+1}=\frac{1}{1-\om... |
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