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 |
|---|---|---|---|---|
218,340 | <p>There is a Landau's theorem related to <a href="http://en.wikipedia.org/wiki/Tournament_%28graph_theory%29" rel="nofollow">tournaments theory</a>.
Looks like the sequence $(0, 1, 3, 3, 3)$ satisfies all three conditions from the theorem, but it is not possible for 5 people to play tournament in such a way (if there ... | Brian M. Scott | 12,042 | <p>Draw $K_5$, the complete graph on $5$ vertices, and assign directions to just enough edges to give one vertex ($A$ in the picture below) a score (outdegree) of $0$ and another ($B$ in the picture) a score of $1$.</p>
<p><img src="https://i.stack.imgur.com/3FyDT.png" alt="enter image description here"></p>
<p>At th... |
1,598,947 | <p>Let $\{\xi_k\}_{k=1}^4$ be a set of vectors in $\mathbb{R}^3$. If $\{\xi_1, \xi_2\}$ and $\{\xi_2, \xi_3, \xi_4\}$ are independent sets, and $\xi_1$ belongs to the span of $\{\xi_2, \xi_4\}$. Show that $\{\xi_k\}_{k=1}^3$ is linearly independent. </p>
<p>Clearly, $\xi_1 = a_1\xi_2 + a_2\xi_4$ where $a_1\neq0$ due t... | Brian Fitzpatrick | 56,960 | <p>Suppose that
$$
\lambda_1\xi_1+\lambda_2\xi_2+\lambda_3\xi_3=0\tag{1}
$$
Writing $\xi_1=\alpha_2\xi_2+\alpha_4\xi_4$ then gives
$$
\lambda_1(\alpha_2\xi_2+\alpha_4\xi_4)+\lambda_2\xi_2+\lambda_3\xi_3=0
$$
Now, rearrange this equation to obtain
$$
(\lambda_1\alpha_2+\lambda_2)\xi_2+\lambda_3\xi_3+\lambda_1\alpha_4\x... |
537,250 | <p>$(X,K)$ a normed space
$E$ is a subspace of $X$.</p>
<p>if $\exists x_0 \in X$ that $||x_0||=d(x_0,E)=1$ then show</p>
<p>$$||e+\lambda x_0||\geq\frac{||e||}{2}\quad \forall e\in E \quad \forall\lambda\in K $$</p>
| Umberto P. | 67,536 | <p>The problem reduces to showing that if $\|x_0\| = 1$, $d(x_0,E) = 1$, $e \in E$, and $\|e\| = 1$, then $\|e + \lambda x_0\| \ge \dfrac 12$ for all scalars $\lambda$.</p>
<p>If $|\lambda| \ge \dfrac 12$ then $\displaystyle ||e + \lambda x_0\| = |\lambda| \cdot \| \frac{e}{\lambda} + x_0 \| \ge |\lambda| \ge \dfrac... |
236,311 | <p>Here's an example of the problem, say you have a <code>Grid</code> with particular spacings that looks like this:</p>
<pre><code>disk = Graphics[Disk[], ImageSize -> 10];
opts = {Alignment -> Right, Spacings -> {{0.5, {0.5, 0.5, 0.5, 2}}, .5}};
grid = {If[Mod[#, 4] == 0, ToString@#, ""] & /@ R... | kglr | 125 | <ol>
<li><p>Since <code>Spacings</code> <em>are in units of current font size</em>, we need to adjust the vertical spacings when font size is changed.</p>
</li>
<li><p>We can modify the first row to make use of <code>SpanFromLeft</code> to allow content to extend multiple cells to the left.</p>
</li>
</ol>
<h3></h3>
<p... |
906,318 | <p>How do you prove $f(x,y) = y - x$ is continuous? The domain is $\mathbb{R^{2}}$ and the codomain is $\mathbb{R}$. Is there an easy way to do it using the definition that the preimage of an open set is an open set? I don't have much experience proving multivariable functions are continuous.</p>
| M A Pelto | 171,159 | <p>More topological answer</p>
<p>Let $U$ be an open set in $\mathbb{R}$. Next let $(a,b)$ be a point in $f^{-1}(U)$, so $f(a,b)=b-a \in U$. Since $U$ is open, there is $\delta>0$ so that $B(f(a,b),\delta)=(b-a-\delta, \: b-a+\delta)$ is contained in $U$ (definition of an open set). Hence we have $f^{-1}[(b-a-\delt... |
782,945 | <p>Graph the circle:</p>
<p>$$x^2+y^2-2x-15=0$$</p>
<p>I know how to approach this problem if there were two $y$ and $x$ variables. But there is only one $y$ variable. How would I approach this?</p>
| A.S | 24,829 | <p>Approach it in exactly the same way, since the equation is the same as $x^2 + y^2 - 2x + 0y - 15 = 0$. The $y$-coordinate of the center will simply be $0$.</p>
|
782,945 | <p>Graph the circle:</p>
<p>$$x^2+y^2-2x-15=0$$</p>
<p>I know how to approach this problem if there were two $y$ and $x$ variables. But there is only one $y$ variable. How would I approach this?</p>
| EgoKilla | 128,076 | <p>Note that $x^2-2x-15+y^2 = 0$ is the same as $x^2-2x+1+y^2=16$ and then we can rewrite our equation as $(x-1)^2 + y^2 = 16$. </p>
|
2,914,100 | <p>A magic square of size <span class="math-container">$N,N ≥ 2$</span>, is an <span class="math-container">$N ×N$</span> matrix with integer entries such that the sums of the entries of each row, each column and the two diagonals are all equal. If the entries of the magic square are made up of integers in arithmetic p... | Deepesh Meena | 470,829 | <p>You have to just sum the series<br>
$$a,a+d,a+2d,\cdots,a+(N^2-1)d$$</p>
<p>$$S=\frac{N^2}{2}\cdot(2a+(N^2-1)d)$$</p>
<p>Sum of all columns is same Thus sum of one column is $$S=\frac{N}{2}\cdot(2a+(N^2-1)d)$$</p>
|
351,005 | <p>Let $\Omega = \mathopen]0,1\mathclose[$ and let a function $A_n: \Omega \to \mathbb R$ defined as:
$$A_n(x) = \begin{cases}\alpha &\text{if } k \epsilon \leq x < (k+\tfrac{1}{2}) \epsilon \\
\beta &\text{if } \big(k+\tfrac{1}{2}\big) \epsilon \leq x < (k+1) \epsilon
\end{cases} $$
for $k=0$, $1,\ldots... | Harald Hanche-Olsen | 23,290 | <p><strong>Hint:</strong> It is enough to compute $$\lim_{n\to\infty}\int_0^1 A_n(x)f(x)\,dx$$ for continuous functions $f\colon[0,1]\to\mathbb{R}$, since these are dense in $L^1(\Omega)$. You will find that the limit is $$\frac{\alpha+\beta}{2}\int_0^1 f(x)\,dx\tag{1}$$ – just take the difference between the two integ... |
351,005 | <p>Let $\Omega = \mathopen]0,1\mathclose[$ and let a function $A_n: \Omega \to \mathbb R$ defined as:
$$A_n(x) = \begin{cases}\alpha &\text{if } k \epsilon \leq x < (k+\tfrac{1}{2}) \epsilon \\
\beta &\text{if } \big(k+\tfrac{1}{2}\big) \epsilon \leq x < (k+1) \epsilon
\end{cases} $$
for $k=0$, $1,\ldots... | Norbert | 19,538 | <p>Since $L_\infty(\Omega)=L_1^*(\Omega)$ you need to show that for all $f\in L_1(\Omega)$
$$
\lim\limits_{n\to\infty}\langle A_n, f\rangle = \langle A, f\rangle\tag{1}
$$
where $A$ is the desired limit. In fact it is enough to check $(1)$ only for some functions $f\in S$, where $L_1(\Omega)=\overline{\mathrm{span}S}$.... |
4,539,750 | <p>I am having trouble understanding the validity of integrating both sides of an equation. I understand that an operation/manipulation can be performed to both sides of an equation, preserving the equality, eg. if two sides of an equation are equal, their derivatives are equal and hence it is valid to differentiate bo... | grand_chat | 215,011 | <p>If you feel unsure about integrating both sides of
<span class="math-container">$$f'(x)=g'(x),$$</span>
you can just move everything to the LHS and integrate the LHS:
<span class="math-container">$$
f'(x)-g'(x)=0\qquad\Longleftrightarrow\qquad f(x)-g(x)=C,
$$</span>
using the fact that a function has zero derivative... |
392,608 | <p>I know this is a very basic question but I need some help.</p>
<p>I have to find the second derivative of: </p>
<p>$$\frac{1}{3x^2 + 4}$$</p>
<p>I start by using the Quotient Rule and get the first derivative to be:</p>
<p>$$\frac{-6x}{(3x^2 + 4)^2}$$</p>
<p>This I believe to be correct.
Following that I procee... | André Nicolas | 6,312 | <p>The first derivative is correct. Now we want to differentiate $\frac{-6x}{(3x^2+4)^2}$. The main thing to remember is <strong>do not "simplify"</strong> unless there is good reason to do so. </p>
<p>The derivative of $\frac{-6x}{(3x^2+4)^2}$ is
$$\frac{(3x^2+4)^2 (-6)-(-6x)(6x)(2)(3x^2+4)}{(3x^2+4)^4}.$$
Cancel a $... |
540,830 | <p>Find the value of <span class="math-container">$$\int_0^\infty t^{x-1}e^{-\lambda t \cos(\theta)} \cos(\lambda t \sin (\theta)) dt$$</span> where <span class="math-container">$\lambda >0$</span>, <span class="math-container">$x>0$</span>, and <span class="math-container">${-1\over 2}\pi < \theta < {1\ove... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
4,177,904 | <p>Let <span class="math-container">$\alpha: x-y+2z-2=0$</span> and <span class="math-container">$\beta: x-2y-2z+3=0$</span> be two planes in <span class="math-container">$\mathbb{R}^3$</span>. I am asked to find a line <span class="math-container">$d_1\subset \alpha$</span> such that <span class="math-container">$d_1$... | Martin Väth | 826,997 | <blockquote>
<p>Is it ok to take <span class="math-container">$v$</span> only in <span class="math-container">$H^1_{\Gamma_1}(U)$</span> or <span class="math-container">$v$</span> must be taken in <span class="math-container">$H^1(U)$</span>?</p>
</blockquote>
<p>The second integral in your above calculation would not ... |
206,241 | <p>I have a strange (well to me at least) MLE problem. If we let $\{X_i\}_{i=1}^n$ be an i.i.d. sample of a random variable $X$ whose mean is $\mu$ and variance $\sigma^2$. Suppose further that $X_1\sim N(\mu,1)$. I must show that the MLE, $\max\{\bar{X}_n,1\}$for $\max\{\mu,1\}$ suffers from bias. </p>
<p>First of al... | robjohn | 13,854 | <p><strong>Question 1:</strong></p>
<p>Arithmetic mean: $\frac{a+b}{2}$</p>
<p>Geometric mean: $\sqrt{ab}$</p>
<p>So the condition becomes $\frac{a+b}{2}=2\sqrt{ab}$. Square both sides to get
$$
\frac{a^2+2ab+b^2}{4}=4ab
$$
which, assuming $b\ne0$, results in
$$
\left(\frac ab\right)^2-14\frac ab+1=0
$$
and
$$
\frac... |
1,417,404 | <p>The following came up in my solution to <a href="https://math.stackexchange.com/questions/1410565/can-this-congruence-be-simplified/1410579#1410579">this question</a>, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation
$$ (x+y)(x+y+1) - kxy = 0$$
For $k=5$ and $... | Will Jagy | 10,400 | <p>This really is Vieta Jumping, I ought to have put that first line. I put about five pictures of the relevant hyperbolas at the end of this post.</p>
<p>Actual reference: In 1907, A. Hurwitz wrote Über eine Aufgabe der unbestimmten Analysis, in Archiv der Mathematik und Physik, pages 185-196. I give a summary a... |
39,973 | <p>Assume that we have two residually finite groups $G$ and $H$. Which properties of $G$ and $H$ could be used to show that their pro-finite (or pro-p) completions are different?</p>
<p>I asked a while ago in the group-pub mailing list whether finite presentability is such a property but Lubotzky pointed out that it i... | Jim Belk | 6,514 | <p>It is known that in a topologically finitely-generated profinite group, every subgroup of finite index is open. (See <a href="http://linkinghub.elsevier.com/retrieve/pii/S1631073X03003492" rel="nofollow">this paper</a>.)</p>
<p>If $G$ is a finitely-generated residually finite group, then the profinite completion $\... |
123,706 | <p>I have an integral that needs to be evaluated using NIntegrate:
$$\int_0^{10}\mathrm{d}x_1\int_0^{10}\mathrm{d}x_2\left(1+(x_2-x_1)^2\right)^{-1/6}\mathrm{e}^{-\int_0^{x_1}\mathrm{d}x'\int_0^{x_1}\mathrm{d}x''f(x'-x'')-\int_0^{x_2}\mathrm{d}x'\int_0^{x_2}\mathrm{d}x''f(x'-x'')}\times\int_0^{x_1}\mathrm{d}x'\int_0^{x... | george2079 | 2,079 | <p>there is one more analytic integral we can pull out:</p>
<p>(using @BobHanlon's <code>i[x]</code>):</p>
<pre><code>g[x1_, x2_] =
Assuming[{x1 > 0, x2 > 0},
Integrate[f[x1 - z1]*f[x2 - z2], {z1, 0, x1}, {z2, 0, x1}]];
NIntegrate[(1 + (x2 - x1)^2)^(-1/6) Exp[-(i[x1] + i[x2])]
g[x1, x2] , {x1, 0, 10... |
2,578,870 | <p>$A$ is real skew symmetric matrix</p>
<p>$S$ is a positive-definite symmetric matrix</p>
<p>Prove that $\det(S) \le \det(A+S)$</p>
<p>As $S$ is diagonalizable, we can reduce the problem to :
for any real skew symmetric matrix $A$ and any diagonal matrix D with positive entries, prove that $\det(S) \le \det(D+S)... | Eric | 509,269 | <p>Let $A$ a real skew symmetric matrix and $S$ a positive-definite symmetric matrix</p>
<p>1) The eigenvalues of $A$ are purely imaginary complex numbers that come in conjugate pairs</p>
<p>2) As $S$ is positive-definite symmetric, we can find a positive-definite symmetric $T$ such that $$S=T^2.$$</p>
<p>3) we have... |
2,078,592 | <p>Let $A:=\{z\in S^1: z^n=1\}$, considering $S^1\subset \mathbb{C}$.</p>
<p>Then, how do I compute $H_k(S^1\times S^1, A\times \{1\}\cup \{1\} \times S^1)$ when $k=1,2$?</p>
<p>Let's consider the long exact sequence of a pair $(S^1\times S^1, A\times \{1\}\cup \{1\} \times S^1)$:</p>
<p>$...\rightarrow H_k(A\times ... | Zev Chonoles | 264 | <p><a href="https://i.stack.imgur.com/2Yv2s.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2Yv2s.gif" alt="enter image description here"></a></p>
<p>$H_2(Y)\cong 0$</p>
<p>$H_2(X)\cong \mathbb{Z}$, generated by $[f_1+f_2+f_3+f_4+f_5]$</p>
<p>$H_1(Y)\cong \mathbb{Z}$, generated by $[e_1]$</p>
<p>... |
1,781,225 | <p>I've got an exercise to do and I don't really know what to do.</p>
<p>Exercise : We've got function $f$, where $f(a) = 0$ and $f'(a)$ exists. Also we got function $g$ which is continuous. Does exist $(f-g)'(a)$? Explain it. </p>
<p>My opinion is that exists, but I've got no idea how should I explain it. Some help?... | Doug M | 317,162 | <p>Is $g$ differentiable? That information is not given. It is possible for $g$ to be continous but not differentiable.</p>
<p>Anyway differentiation is "linear", in that $(f-g)'(a)$ = $f'(a) - g'(a).$
So, if $g'(a)$ exists (and $f'(a)$ exists which is given) then $(f-g)'(a)$ exists.</p>
<p>But if $g'(a)$ does not ... |
2,024,468 | <p>I need to find the points of intersection of a circle with radius $2$ and centre at $(0,0)$ and a rectangular hyperbola with equation $xy=1$. As per the topic statement is there any way to solve this without the graphical method. I have tried setting the $y$ values equal but I cant solve the resulting equation for $... | egreg | 62,967 | <p>A system of equations such as yours, namely
\begin{cases}
x^2+y^2=4 \\
xy=1
\end{cases}
is <em>symmetric</em>, because it doesn't change when $x$ and $y$ are swapped.</p>
<p>Rewrite $x^2+y^2=4$ as $(x+y)^2-2xy=4$, so you get
\begin{cases}
(x+y)^2=6 \\
xy=1
\end{cases}
that can be divided into
$$
\begin{cases}
x+y=\... |
20,101 | <p>I've seen <a href="https://math.stackexchange.com/questions/1210976/how-to-factoring-a-high-degree-polynomial">this</a> question in the last minutes, and I noticed that the OP has a lot of issues understanding basic (really basic) mathematics. </p>
<p>It's not just that the question is really bad written, or that t... | Community | -1 | <p>Taken straight from the tour of this website: "Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields." Personally, I think that is how it should stay. (That doesn't mean we can't have a quality standard on questions though, but this is tang... |
20,101 | <p>I've seen <a href="https://math.stackexchange.com/questions/1210976/how-to-factoring-a-high-degree-polynomial">this</a> question in the last minutes, and I noticed that the OP has a lot of issues understanding basic (really basic) mathematics. </p>
<p>It's not just that the question is really bad written, or that t... | Hagen von Eitzen | 39,174 | <p>I think there are minimal requisites that should be assumed, but these vary from question to question. </p>
<p>If a graduate student asks a question how to compute the cohomology of a CW complex, it should be presupposed that the essential definitions and methods to do so are available - if it turns out they don't ... |
2,950,193 | <p>How one can show that this polynomial.</p>
<p><span class="math-container">$$-2+2x^{2k+1}-x^{3k+2}+x^{k-1}$$</span></p>
<p>is negative for all integer <span class="math-container">$k>1$</span> and real <span class="math-container">$x>2$</span>.</p>
<p>I have no idea to start.</p>
| nonuser | 463,553 | <p>For <span class="math-container">$x>2$</span> we have <span class="math-container">$x^{k+1} >4x^{k-1}>x^{k-1}$</span></p>
<p>so we have:</p>
<p><span class="math-container">$$x^{2k+1}(2-x^{k+1})-(2-x^{k-1})<x^{2k+1}(2-x^{k-1})-(2-x^{k-1})= \underbrace{(x^{2k+1}-1)}_{>0}\underbrace{(2-x^{k-1})}_{<... |
2,950,193 | <p>How one can show that this polynomial.</p>
<p><span class="math-container">$$-2+2x^{2k+1}-x^{3k+2}+x^{k-1}$$</span></p>
<p>is negative for all integer <span class="math-container">$k>1$</span> and real <span class="math-container">$x>2$</span>.</p>
<p>I have no idea to start.</p>
| Mark Bennet | 2,906 | <p>Well I can change all the signs and assert that the result is positive. If I multiply by <span class="math-container">$x$</span> I don't change the sign, because <span class="math-container">$x$</span> is positive and I then put <span class="math-container">$y=x^k$</span> to simplify the expression. Then I am lookin... |
2,054,176 | <blockquote>
<p>Given $n!+(n+1)!+(n+2)!+(n+3)!$ is divisible by $181$ but $n!$ is not divisible by $181$, find three possible values for $n$.</p>
</blockquote>
| jameselmore | 86,570 | <p>Hint:<br>
Let "$+$" be any positive real number, and "$-$" be any negative real number. It should be clear that:</p>
<p>$$\frac{+}{+} > 0;\ \ \frac{-}{-} > 0$$
$$\frac{+}{-} < 0;\ \ \frac{-}{+} < 0$$</p>
<p>If you can figure out for which values of $x$ has $(x+3)$ positive\negative, and which values of... |
4,068,216 | <p>I know that the sequence <span class="math-container">$\sqrt[n]{n}$</span> converges to 1 and that <span class="math-container">$\text{log}(\sqrt[n]{n})$</span> thus converges to 0 as <span class="math-container">$n\to\infty$</span> since the logarithmic function is continuous. But how can I calculate the limits as ... | dodoturkoz | 804,253 | <p>Connect <span class="math-container">$MN$</span> and <span class="math-container">$MU$</span>. Rotate <span class="math-container">$\bigtriangleup MPU$</span> <span class="math-container">$\;180°$</span> about <span class="math-container">$M$</span> and name the exterior point <span class="math-container">$U'$</span... |
2,504,613 | <p>I have problem with solving the following equation:</p>
<blockquote>
<p>$$ty'=3y+t^5y^\frac{1}{3}$$</p>
</blockquote>
<p>I know it's easy without the $y^\frac{1}{3}$ term, but I'm confused now.</p>
<p>Any help would be appreciated.</p>
| Rumplestillskin | 373,448 | <p>You have a first order nonlinear ODE. You are dealing with one of <a href="http://mathworld.wolfram.com/BernoulliDifferentialEquation.html" rel="nofollow noreferrer">these animals</a></p>
<p><strong>HINT</strong></p>
<p>Write your ODE in the following form:</p>
<p><span class="math-container">$$ \frac{2 y' }{3 y^{1/... |
752 | <p>This question is about how and when can I delete my own answer? I mean remove the answer, not editing it and replace the text by some remark such as "the answer was deleted".</p>
<p>Edit: <em>The "delete" link displays "<strong>vote to remove this post</strong>"</em>. I thought one would only vote to delete (remove... | ShreevatsaR | 205 | <p>Below any answer of yours, at the bottom left, you can see links for "link edit delete flag". Click on the third one, "delete".</p>
<p>Note that you cannot delete an answer if you are an "unregistered user", or if your answer is the accepted answer. Also, your answer will still be visible (and shown as deleted) to ... |
565,842 | <p>The common definition of $ \omega $-logic (a.k.a $\mathcal{L}_{\omega_1,\omega}$ logic) is the usual first order logic allowing infinite conjunctions and infinite proof.</p>
<p>Chang and Keisler, in section 2.2 of their book, define $\omega$-logic differently*.</p>
<p>They restrict to the language $\{S,+,\cdot,1\}... | hmakholm left over Monica | 14,366 | <p>I don't see any reason to assume they would be the same. If I understand you correctly, the <em>formulas</em> of Chang and Keisler are the same as the formulas of ordinary arithmetic; they just have a more permissive consequence relation.</p>
<p>In contrast, what you describe as $\mathcal L_{\omega_1,\omega}$ has f... |
102,313 | <p>How do I write <em>let</em> in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is:</p>
<p>$$ x := a $$</p>
<p>Would that be clear? Is there a better way to write "let $x$ equal $a$"?</p>
| Adrian Rezus | 225,751 | <p>Well, this an easy one! You, simply, do NOT write "let" in (symbolic) logic, as you do it in the vernacular (in your mother-tongue, for instance)! </p>
<p>Now, the so called "mathematical vernacular" [WOT, in Dutch] has been explained (and codified) long ago, by one of my Dutch math teachers, N. G. [ = Dick ] de Br... |
2,360,117 | <p>Let $\xi:\mathbb{R}^3\rightarrow \mathbb{R}$ be a function $\xi(x,y,z)$. I have the following PDE:
$$
\cos(\xi)\partial_y\xi = \sin(\xi)\partial_x\xi
$$
which is equivalent to $\tan(\xi)=\xi_y/\xi_x$.</p>
<p>Clearly, $\xi_x = \xi_y = 0$ (where $\partial_i\xi=\xi_i$) is a solution, which would imply $\xi(x,y,z) = \e... | user254433 | 254,433 | <p>This is an expanded version of what I mentioned in the comments.</p>
<p>In case you aren't familiar with the method of characteristics, there is a simple trick for solving this kind of equation: interchange $x$ with $\xi$. Let $X(u,y,z)$ be the solution of $$u=\xi(X,y,z).$$ To evaluate your PDE $$\xi_y\cos\xi=\xi... |
437,775 | <p>I'm a graduate student studying now for the first time class field theory.<br />
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.<br />
For example here <a href="https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first">... | ollijvn | 497,341 | <p>First of all, as already said by others: Classical class field theory can be formulated entirely without cohomology, so it is a choice to use it.</p>
<p><strong>Benefits of using group cohomology:</strong></p>
<p>If you use Galois cohomology, the main theorems of class field theory can be phrased as statements looki... |
2,220,032 | <p>Quoting:</p>
<p>"Let H be the subgroup of S3 defined by the permutations
{(1); (123); (132)}. The left cosets of H are</p>
<p>(1)H = (123)H = (132)H = {(1); (123); (132)}</p>
<p>(12)H = (13)H = (23)H = {(12); (13); (23)} "</p>
<p>I am a bit stock here, I am not understanding the meaning of the first permutation ... | Vladimir Sotirov | 400 | <p>$\def\Lan{\text{Lan}}$
Borceux's proposition is simply the equivalence of the unit-counit formulation of adjoint functors and the absolute Kan extension formulation of adjoint functors. In particular, it is valid in any $2$-category. Below is my attempt at a brief self-contained exposition of the relevant results.</... |
3,454,068 | <p>Suppose I have an orthonormal basis <span class="math-container">$\{e_n : n \in \mathbb{N}\}$</span> of a Hilbert space <span class="math-container">$\mathcal{H}$</span>, and let <span class="math-container">$\mathcal{I} = \{n^{\frac{1}{2}}e_n : n \in \mathbb{N}\}$</span>. </p>
<p>The claim is that if <span class="... | Kavi Rama Murthy | 142,385 | <p>This is false. <span class="math-container">$\mathcal I$</span> is a weakly closed set. (I will leave it to you to supply a proof of this fact). Let <span class="math-container">$U$</span> be its complement. Then <span class="math-container">$U$</span> is a a weak neighborhood of <span class="math-container">$0$</s... |
235,871 | <p>I am writing a Fortran function that needs to receive two vectors of reals, and as an output, returns a vector. The function could be for example the sum of the two vectors:</p>
<pre><code>function sumtwovectors( n, v1, v2 )
implicit none
real(kind=8), dimension(100000) :: sumtwovectors
integer, intent(... | I.M. | 26,815 | <p>I'm not familiar with .NET interface for WM, but your problem can be solved with <a href="https://reference.wolfram.com/language/guide/LibraryLink.html" rel="nofollow noreferrer">LibraryLink</a>. With LibraryLink it is possible to create a shared lib which can be loaded with <code>LibraryFunctionLoad[]</code> or, pe... |
1,480,381 | <pre><code>unsigned long long int H(unsigned long long int n){
unsigned long long int res = 1;
for (unsigned int i = 1; i < n; i++){
res += n / i;
}
return res;
}
</code></pre>
<p>I'm trying to convert this simple loop into a mathematical equation, I did a couple of attempts based on the exa... | David C. Ullrich | 248,223 | <p>The language includes equality, right? So show that there exist $x_1,x_2$ with $x_1\ne x_2$. Then show that there exist $x_1,x_2,x_3$ with $x_1\ne x_2$, $x_1\ne x_3$ and $x_2\ne x_3$.</p>
<p>For every $n$ there is a $\phi_n$ that says "there exist at least $n$ things". Show by induction that each $\phi_n$ is provab... |
1,480,381 | <pre><code>unsigned long long int H(unsigned long long int n){
unsigned long long int res = 1;
for (unsigned int i = 1; i < n; i++){
res += n / i;
}
return res;
}
</code></pre>
<p>I'm trying to convert this simple loop into a mathematical equation, I did a couple of attempts based on the exa... | Alex Kruckman | 7,062 | <p>Nitpick: The empty structure is a model of your axioms. But I'll assume you're working with the convention that all first-order structures are non-empty.</p>
<p>Try proving by induction on $n$ that in any model $M$, there is a chain $a_1,a_2,\dots,a_n$ of <em>distinct</em> elements such that $a_i R a_{i+1}$ for all... |
296,101 | <p>I know how to show that these two have the same cardinality and from that there must be a bijection between them. </p>
<p>Can anyone help with an explicit bijection between these sets?</p>
| Alex Youcis | 16,497 | <p>Hint: Think about, given $E\subseteq\mathbb{N}$, the indicator function $\chi_E:\mathbb{N}\to 2$ given by $\chi_E(n)=1$ if $n\in E$ and $0$ otherwise.</p>
|
296,101 | <p>I know how to show that these two have the same cardinality and from that there must be a bijection between them. </p>
<p>Can anyone help with an explicit bijection between these sets?</p>
| Brian M. Scott | 12,042 | <p>HINT: Consider <a href="http://en.wikipedia.org/wiki/Indicator_function" rel="nofollow">indicator</a> (or characteristic) functions of subsets of $\Bbb N$.</p>
|
13,460 | <p>I hope that this question is on-topic, though it is not quite technical.</p>
<p>I am curious to hear from people how they approach reading a mathematical paper.</p>
<p>I am not asking specific questions on purpose, though at first I had a few. But I want to keep it rather open-ended.</p>
| Timothy Wagner | 3,431 | <p>I generally have a "top down approach" to reading papers. More often than not, I am driven to a particular paper looking for a particular result. So I would start by looking at the statement of the result and try to see what background is needed in order to parse the statements which appear there. Once I have unders... |
367,686 | <p>How many injective functions $f:[1,...,m]\to{[1,...,n]}$ has no fixed point? $(m\le n)$</p>
<p>I thought about the next thing:</p>
<p>$f(x_1)\neq x_1$, Means i can choose for $x_1$ - (n-1) options,</p>
<p>But then, for $x_2$, there are two options:</p>
<ol>
<li><p>If i choose $f(x_1)=x_2$ then for $x_2$ i still... | Aryabhata | 1,102 | <p>Fix an integer $a \ge 0$. </p>
<p>We try to give a recurrence for the number of no-fixed-point-injections $$f: \{1,2,3, \dots, m\} \to \{1,2, \dots, m, m+1, \dots, m+a\}$$</p>
<p>For given $a$, let the number for $m$ be $D_m$. We have that $D_1 = a$ and $D_2 = a^2 + a +1 $ (if I have computed correctly, but it sho... |
367,686 | <p>How many injective functions $f:[1,...,m]\to{[1,...,n]}$ has no fixed point? $(m\le n)$</p>
<p>I thought about the next thing:</p>
<p>$f(x_1)\neq x_1$, Means i can choose for $x_1$ - (n-1) options,</p>
<p>But then, for $x_2$, there are two options:</p>
<ol>
<li><p>If i choose $f(x_1)=x_2$ then for $x_2$ i still... | SuperSat001 | 682,022 | <p><span class="math-container">$\newcommand{\nPr}[2]{\,_{#1}P_{#2}} % nPr$</span>
We can think of an approach using Inclusion-Exclusion principle.</p>
<p>Note that we can count the number of injective functions <span class="math-container">$N$</span> from <span class="math-container">$$A = \{1,2,3,\dots,m\} \rightarro... |
2,600,283 | <p>I think $f(x) = x^2$. Then $f'(0)$ should be $0$.</p>
<p>But when I try to calculate the derivative of $f(x) = |x|^2$, then I get:</p>
<p>$f'(x) = 2|x| \cdot \frac{x}{|x|}$, which is not defined for $x = 0$. Does $f'(0)$ still exist?</p>
| Shashi | 349,501 | <p>So $f(x) =g(u(x)) $ where $g(x) =x^2$ and $u(x) =|x|$. The problem is that you use the chain rule on $g(u(x)) $ while $u(x) $ is not differentiable at zero. </p>
<p>One option is to just use the definition of derivative
$$f'(0)=\lim_{h\to 0}\frac{|h|^2}{h}$$ and consider $\lim_{h\to 0^+}$ and $\lim_{h\to 0^-}$. </... |
45,398 | <p>I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory.</p>
<p>I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and Shreve, for example).</p>
<p>My PDE theory is pretty weak. I know about the Fokker-Planck equations, and that's abo... | JT_NL | 1,120 | <p>You can check out the Internet Seminar Notes 2007/2008 bij Jan van Neerven, they are quite good and are about stochastic evolution equations.</p>
<p>Here you can find them: <a href="http://fa.its.tudelft.nl/~neerven/publications/papers/ISEM.pdf">http://fa.its.tudelft.nl/~neerven/publications/papers/ISEM.pdf</a></p>... |
2,264,619 | <ul>
<li>I need to compute the probability of getting more than $x$ "successes" in a large number of trials $\left(\,10^{11}\,\right)$ of an event with a small probability $\left(\,10^{-7}\,\right)$.</li>
<li>Exact Binomial won't work, and the Poisson approximation does not seem appropriate. </li>
</ul>
<p>T... | Michael Hardy | 11,667 | <p>I'm guessing that if the Poisson approximation does not seem appropriate, it's because the expected value and the variance (which, for the Poisson distribution, is the same as the expected value) are so big. In that case, approximating it by a normal distribution can serve.</p>
<p>If $X\sim\operatorname{Poisson}(\l... |
4,387,795 | <p>I'm studying Graph Theory, and I have a question. I'm trying to prove that the product of <span class="math-container">$n-1$</span> transpositions in <span class="math-container">$S_n$</span> is an <span class="math-container">$n$</span>-cycle if and only if the associated graph of <span class="math-container">$n$</... | Greg Martin | 16,078 | <p>Certainly—for one thing, the transpositions <span class="math-container">$(a\ b)$</span> and <span class="math-container">$(b\ a)$</span> are the same, so the notion of "first entry" isn't even well defined.</p>
|
4,387,795 | <p>I'm studying Graph Theory, and I have a question. I'm trying to prove that the product of <span class="math-container">$n-1$</span> transpositions in <span class="math-container">$S_n$</span> is an <span class="math-container">$n$</span>-cycle if and only if the associated graph of <span class="math-container">$n$</... | Eric Nathan Stucky | 600,276 | <p>As user994373 points out in the comments, <span class="math-container">$(1,3)(3,4)(1,2) = (1,2,3,4)$</span> gives an example of such a product in which there is no element common to every transposition.</p>
<hr />
<p>An extended aside, which may be of interest to someone, if not necessarily you :P</p>
<p>I suspect t... |
726,650 | <p>a) How does $x^2y''-3xy'+3y=0$ can be solved? I know how to solve for constant coefficients, but in this case they are functions...</p>
<p>b) In which maximum interval there is a solution that confirms $y'(1)=6, y(1)=4$?</p>
| Amzoti | 38,839 | <p>Hint: This is Euler-Cauchy type equation, so let:</p>
<p>$$y = x^m \implies y' = m x^{m-1} \implies y'' = m(m-1)x^{m-2}$$</p>
<p>Substitute back into ODE, solve for roots, write the general solution and then find constants from initial conditions.</p>
<p><strong>Spoiler</strong></p>
<blockquote class="spoiler">
... |
1,184,795 | <p>I looked at this as saying that $P(Y=k) = P(X < 10) + P(X=10) + 1 - P(X \le 10)$.</p>
<p>Then for each pmf of X I just put in the summation of each of those according to the geometric distribution. Is that the correct path? </p>
<p>Or Should it be that $P(\min(X, 10) = k) = P(10 \ge X)$?
The ten is throwing me ... | Graham Kemp | 135,106 | <p>$X$ has a $p$-parameter geometric distribution, so for all $k\in \{1, 2, ...\}$ then $\mathsf P(X=k) = p(1-p)^{k-1}$</p>
<p>$Y$ is the minimum of $10$ or $X$, so if $X<10$ then $Y=X$, else if $X\geq 10$ then $Y=10$.</p>
<p>$Y$ must lie within $\{1, 2, ... 10\}$. </p>
<p>So, what is $\mathsf P(Y=10)$, and $\ma... |
1,648,460 | <p>Suppose that $A$ is an m by n matrix and is right invertible, such that there exists and an n by m matrix $B$ such that $AB = I_m.$ Prove that $m\leq n.$ </p>
<p>I'm not really sure how go about this problem; any help would be appreciated.</p>
| Ted Shifrin | 71,348 | <p>The key thing that's going on here is that you cannot compute the hypotenuse of a right triangle by taking one of the legs of the triangle. Consider the integral that gives arclength of a curve: You're adding up $\Delta s = \sqrt{(\Delta x)^2+(\Delta y)^2}$ when you chop the curve into pieces, <em>not</em> adding up... |
1,648,460 | <p>Suppose that $A$ is an m by n matrix and is right invertible, such that there exists and an n by m matrix $B$ such that $AB = I_m.$ Prove that $m\leq n.$ </p>
<p>I'm not really sure how go about this problem; any help would be appreciated.</p>
| Narasimham | 95,860 | <p>$$ SA = 2\int^R_0{{\sqrt{R^2-x^2}}}dx $$</p>
<p>is not correct when summing up thin cone surface areas. Slant length is to be considered for each thin cone slice differential portion being integrated.</p>
<p>$$ ds= \sqrt{dx^2+dy^2} $$</p>
<p><em>Recall slant length $l$ is involved in slant area of cone in</em> $ ... |
4,324,207 | <p>I want to prove <span class="math-container">$\lim\limits_{x \to 5} \sqrt{2x+6} = 4$</span>
using the epsilon-delta definition.</p>
<p>My intuition to proving this is to use the root rule and square both ends of the equation to end up with:</p>
<p><span class="math-container">$\lim\limits_{x \to 5} 2x+6 = 16$</span>... | Michael Rozenberg | 190,319 | <p>Another way.</p>
<p>By my previous post we need to prove that:
<span class="math-container">$$\sum_{cyc}xy\left(\sum_{cyc}\frac{1}{x+y}\right)^2\geq\frac{25}{4},$$</span> which is true by Muirhead:
<span class="math-container">$$\sum_{cyc}xy\left(\sum_{cyc}\frac{1}{x+y}\right)^2-\frac{25}{4}=$$</span>
<span class="m... |
625,162 | <p>If you know a coupon collector problem, you will know what I am talking about. But if you are not familiar with I will try to explain what is the coupon collector problem.
I have $n$ bins. I throw balls consecutively into these bins. Each bin is choosen independently and with the same probability.
Let's suppose that... | ir7 | 26,651 | <p>Let $z = \rho (\cos \theta + i \sin \theta)$ be a root of polynomial $z^2+cz+1$, where $c\triangleq 2a/b$. Then:
$$\rho^2\sin(2\theta)+c\rho\sin\theta = 0$$
and
$$\rho^2\cos(2\theta)+c\rho\cos\theta + 1 = 0.$$
Multiplying the first relation by $\cos\theta$ and the second by $\sin \theta$, and then subtracting, we ge... |
603,471 | <p>Im totally new to statistics , but what is the characteristic function for ?
I do not get that. I was reading about the bell curve and the Central Limit Theorem , but I did not get what the characteristic function is suppose to be , where it comes from or what it is used for.</p>
<p>It seems to appear in the proof ... | Jean-Claude Arbaut | 43,608 | <p>Induction is a bit overkill here.</p>
<p>If $a \not = 1$,</p>
<p>$$\sum_{j=0}^n a^j=\frac{a^{n+1}-1}{a-1}$$</p>
<p>Here $a=-\frac 1 2$.</p>
<p>$$\frac{a^{n+1}-1}{a-1}=\frac{(-1/2)^{n+1}-1}{-1/2-1}=\frac{2}{3}\left(-\left(-\frac 1 2\right)^{n+1}+1\right)$$
$$=\frac{(-1)^n+2^{n+1}}{3 \cdot 2^n}$$</p>
|
376,575 | <p>This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, <em>On topological cyclic homology</em>, arXiv:<a href="https://arxiv.org/abs/1707.01799" rel="nofollow noreferrer">1707.01799</a>. I just want to make my understanding precise.</p>
<hr />
<p><strong>Pa... | Dylan Wilson | 6,936 | <p>There are several ways to do this, depending on how much technology one is interested in using.</p>
<p>One way to do it is to use the fact that the <span class="math-container">$\infty$</span>-category of commutative monoid objects in a category with finite products, <span class="math-container">$\mathsf{CMon}(\math... |
4,195,841 | <p>V is a finite-dimensional <span class="math-container">$\mathbb{Q}$</span>- vector space with <span class="math-container">$\phi: V \rightarrow V$</span></p>
<p>Why does it follow that <span class="math-container">$\phi$</span> is diagonalizable if <span class="math-container">$\phi \circ \phi = id_{V}$</span>?</p>
... | Pythagoras | 701,578 | <p>Assume that the characteristic of the field (such as <span class="math-container">$\mathbb Q$</span>) is not <span class="math-container">$2$</span>. Here is a simple way to see it. As has been noted, all the eigenvalues are either <span class="math-container">$1$</span> or <span class="math-container">$-1$</span>. ... |
1,710,799 | <p>Right triangle $ABC$ is inscribed in a circle with $AC = 6$, $BC = 8$ and $AB=10$. $AC$ and $CB$ are semi-circles. Find the sum of the areas of regions $X$ and $Y$.</p>
<p><a href="https://i.stack.imgur.com/XXENo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XXENo.png" alt="enter image descript... | Nikunj | 287,774 | <p>I do not know how to make a figure here, but I can answer your question.</p>
<p>Let $O$ be the centre of the bigger circle, Then $AO=BO=5$.</p>
<p>Join OC (thereby completing $\triangle OBC$)</p>
<p>In $\triangle OBC$, we must have, $$\sin \angle {\frac{COB}{2}}=\frac{4}{5}$$
$$\implies \angle \frac{COB}{2}=\sin^... |
1,710,799 | <p>Right triangle $ABC$ is inscribed in a circle with $AC = 6$, $BC = 8$ and $AB=10$. $AC$ and $CB$ are semi-circles. Find the sum of the areas of regions $X$ and $Y$.</p>
<p><a href="https://i.stack.imgur.com/XXENo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XXENo.png" alt="enter image descript... | smalldog | 304,598 | <p>The inscribed triangle is right-angled, therefore its hypotenuse $AB$ is a diameter (by the converse to Thales' Theorem) and the large circle $C$ has radius $5$.</p>
<p>Let the white region between $X$ and the triangle be $X'$, and the white region between $Y$ and the triangle be $Y'$.</p>
<p>Then $$A(X') + A(Y') ... |
1,710,799 | <p>Right triangle $ABC$ is inscribed in a circle with $AC = 6$, $BC = 8$ and $AB=10$. $AC$ and $CB$ are semi-circles. Find the sum of the areas of regions $X$ and $Y$.</p>
<p><a href="https://i.stack.imgur.com/XXENo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XXENo.png" alt="enter image descript... | Will Fisher | 290,619 | <p>First we know by the converse of Thales' theorem that AB is the diameter of the circle triangle ABC is incribed in. Now lets call $X'$ and $Y'$ the area of the entire semi-circles $x$ and $y$ are contained in respectively. We know that
$$X'=\pi\left(\frac{AC}{2}\right)^2=9\pi$$
$$Y'=\pi\left(\frac{BC}{2}\right)^2=16... |
3,325,749 | <p>Suppose that <span class="math-container">$A=\{y_1,...,y_r\}$</span> is a subset of a vector space <span class="math-container">$V$</span> and that every vector <span class="math-container">$x \in V$</span> can be expressed uniquely as a linear combination of the vectors of <span class="math-container">$A$</span>. S... | Kavi Rama Murthy | 142,385 | <p>It is enough to prove that <span class="math-container">$A$</span> is linearly independent. Let <span class="math-container">$x =\sum_{k=1} ^{r} b_k y_k$</span> be the unique representation of <span class="math-container">$x$</span>. Suppose <span class="math-container">$\sum_{k=1} ^{r} a_k y_k=0$</span>. Then <span... |
1,354,491 | <p>If I wanted to have a die that rolled, for example:</p>
<pre><code>| Roll | Prob (in %) |
|------|-------------|
| 1 | 60 |
| 2 | 25 |
| 3 | 12 |
| 4 | 4 |
| 5 | 1 |
| 6 | 0.2 |
| 7 | 0.04 |
| ... | ... |
</code></pre>
<p>(... | muaddib | 242,554 | <p>These probabilities are described by a <a href="https://en.wikipedia.org/wiki/Geometric_distribution" rel="nofollow">Geometric Distribution</a>. For that one, the probability of outcome $n$ is given by:
$$P(X = n) = (1-p)^{n-1}p$$
where $p$ is the "exponential rate".</p>
<p>To sample from this distribution (or oth... |
714,679 | <p>Could anyone point me a program so i can calculate the roots of</p>
<p>$$ K_{ia}(2 \pi)=0 $$</p>
<p>here $ K_{ia}(x) $ is the modified Bessel function of second kind with (pure complex)index 'k' :D</p>
<p>My conjecture of exponential potential means that the solutions are $ s=2a $ with </p>
<p>$$ \zeta (1/2+is)=... | Raymond Manzoni | 21,783 | <p>I used the free pari/gp $(*)$ to get the smallest positive solutions :
\begin{array} {l}
9.76877008350997786701461088004548816694073504626\\
12.4484878927757792253076746584484174280206806290\\
14.6849597759503320940392747073070219296666773881\\
16.6915789382924172031614835847470262244558902255\\
18.54937519701126964... |
62,471 | <p>Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic. </p>
<p>Can we do the same thing if we choose a standard parabolic subgroup? Can we always find an element in the weyl group such that conjugation by this element takes it to the ... | Bob Yuncken | 14,566 | <p>I believe this is false, and that a counterexample occurs in $\mathrm{SL}(3)$ already. </p>
<p>Let $P$ be the block upper triangular subgroup with a $2\times2$ block in the top left corner. If I understand your terminology correctly, it's "opposite" is the block lower triangular group with the same block. </p>
<p>... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Derek Jennings | 1,301 | <p>Another way is:</p>
<p><span class="math-container">$$\int \sec x \,dx = \int \frac{\cos x}{\cos^2 x} \,dx = \int \frac{\cos x}{1-\sin^2 x} \,dx =
\frac{1}{2} \int \left( \frac{1}{1-\sin x} + \frac{1}{1+\sin x} \right) \cos x dx $$</span>
<span class="math-container">$$= \frac{1}{2} \log \left| \frac{1+\sin x}{1-\si... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | whuber | 1,489 | <p>Using the definitions $$\sec \theta = 1/\cos \theta \quad \text{and} \quad \cos \theta = (\exp(i \theta) + \exp(-i \theta))/2$$ gives $$\int \sec \theta \, d \theta = \int \frac{2 \, d \theta}{\exp(i \theta) + \exp(-i \theta)}.$$ The only insight needed is to find the substitution $u = \exp( i \theta )$ (what else... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Community | -1 | <p>My favorite way:</p>
<p><span class="math-container">$$\int\frac{d\theta}{\cos\theta}=\int\frac{\cos\theta\,d\theta}{\cos^2\theta}=\int\frac{d\sin\theta}{1-\sin^2\theta}=\text{artanh}(\sin\theta).$$</span></p>
|
356,749 | <p>I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems". </p>
<p>From what I've been told, given a good cover <span class="math-container">$\{U_i\}$</span> of <span class="math-container">$X$</span>, an infinity local system on a connecte... | ABCD | 165,286 | <p>One way to discuss <span class="math-container">$\infty$</span>-local systems over a space <span class="math-container">$X$</span> is in terms of the <em>fundamental <span class="math-container">$\infty$</span>-groupoid</em> of <span class="math-container">$X$</span>. To motivate this, recall that for classical loca... |
835,376 | <p>Is it possible for two vector functions of, for the moment's simplicity, one variable be both independent <strong>and</strong> dependent?</p>
<p>The reason I'm asking this is because on a problem from a book of mine (not homework), they put the following exercise:</p>
<p><em>Let $x^{(1)}(t)=\left (\begin{array}{cc... | egreg | 62,967 | <p>The question mixes two different vector spaces. You have two functions from $[0,1]\to\mathbb{R}^2$,
$$
x^{(1)}\colon t\mapsto\begin{bmatrix}e^t\\te^t\end{bmatrix},\qquad
x^{(2)}\colon t\mapsto\begin{bmatrix}1\\t\end{bmatrix}
$$
These functions belong to the vector space $V$ of continuous maps from $[0,1]$ to $\mathb... |
2,848,317 | <p>In the image, the segments inside the square go from a vertex to the middle point of the opposite side. If the length of the sides of the square is $1$, the area of $ABCD$ is?</p>
<p><a href="https://i.stack.imgur.com/BBpvZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BBpvZ.png" alt="enter ima... | HugoTeixeira | 578,598 | <p>If we move the figure to a $xy$ axis, we can solve this problem by finding the line equations and intersection points. Consider this image:</p>
<p><a href="https://i.stack.imgur.com/OJSoF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OJSoF.png" alt="enter image description here"></a></p>
<p>Si... |
3,315,227 | <p>I want to solve this differential equation <span class="math-container">$$ u^{(4)}+a u^{(2)} +bu=0$$</span></p>
<p>I put <span class="math-container">$v=u^{(2)}.$</span> I obtain the new equation <span class="math-container">$v^{(2)}+a v+bu=0.$</span> </p>
<p>What to do with <span class="math-container">$u?$</spa... | Cesareo | 397,348 | <p>According to your idea the formulation is</p>
<p><span class="math-container">$$
\begin{cases}
u'' = v\\
v''=-a v-b u
\end{cases}
$$</span></p>
<p>You can now follow with the Laplace transform obtaining</p>
<p><span class="math-container">$$
\begin{cases}
s^2U(s) - su'(0)-u(0) = V(s)\\
s^2V(s)-s v'(0) - v(0) = -a... |
4,171,804 | <p>I proved that it's convergent.But I have a doubt in proving its limit.
I thought like this.</p>
<p>Let <span class="math-container">$x_n$</span> converges to <span class="math-container">$0$</span>.Then for every <span class="math-container">$\varepsilon>0, x_n$</span> should satisfy the condition <span class="ma... | Cesareo | 397,348 | <p>Hint.</p>
<p>Assuming <span class="math-container">$x_k\gt 0$</span> we have</p>
<p><span class="math-container">$$
\ln(x_{k+1})=\frac 12(\ln 3+\ln(x_k))
$$</span></p>
<p>or</p>
<p><span class="math-container">$$
u_{k+1}=\frac 12 u_k + c
$$</span></p>
<p>This recurrence has as solution</p>
<p><span class="math-conta... |
1,221,487 | <p>Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:</p>
<p>\begin{bmatrix}
1 & -5 \\
-4 & 2 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
-1 & 5 \\
\end{bmatrix}
\begin{bmatrix}
2 & -4 \\
-5 & 7 \\
\end{bmatrix}
\begin{bmatrix}
1 & -7 \\
-5 & 1 \\
\e... | JMP | 210,189 | <p>A different way to look at the problem is to notice that if we subtract 9 from a number, the digital sum (and so the digital root) is still the same.</p>
<p>Let's assume the number we are subtracting from has 2 digits, $d_1$ and $d_2$, and is written $[d_1][d_2]$. We establish that we can borrow from $d_1$. Our new... |
1,369,485 | <p>$\log_2(x) = \log_x(2) $ </p>
<p>Using the change of base theorem: $\dfrac{\log(x)}{\log(2)} = \dfrac{\log(2)}{\log(x)}$</p>
<p>Multiplied the denominators on both sides: $\log(x)\log(x) = \log(2)\log(2)$</p>
<p>I kind of get stuck here. I know that you can't take the square root of both sides of the equation, bu... | Mark Viola | 218,419 | <p><strong>HINT:</strong></p>
<p>$$x^2=y^2\implies x=\pm y$$</p>
<p>and </p>
<p>$$-\log 2=\log (1/2)$$</p>
|
1,369,485 | <p>$\log_2(x) = \log_x(2) $ </p>
<p>Using the change of base theorem: $\dfrac{\log(x)}{\log(2)} = \dfrac{\log(2)}{\log(x)}$</p>
<p>Multiplied the denominators on both sides: $\log(x)\log(x) = \log(2)\log(2)$</p>
<p>I kind of get stuck here. I know that you can't take the square root of both sides of the equation, bu... | blademan9999 | 782,135 | <p>log()log()=log(2)log(2)--> log(x)=±log(2)
let a=log(x), a=±log(2), <span class="math-container">$10^{a}=x$</span>, <span class="math-container">$x=10^{±log(2)}$</span>.
<span class="math-container">$x=10^{log(2)}=2$</span>, or <span class="math-container">$x=10^{-log(2)}$</span>=<span class="math-container">$(10^{lo... |
56,942 | <p>If $ x $ and $ y $ have $ n $ significant places, how many significant places do $ x + y $, $ x - y $, $ x \times y $, $ x / y $, $ \sqrt{x} $ have?</p>
<p>I want to evaluate expressions like $ \frac{ \sqrt{ \left( a - b \right) + c } - \sqrt{ c } }{ a - b } $ to $ n $ significant places, where $ a $, $ b $, $ c $ ... | robjohn | 13,854 | <p>$x\times y$, $x/y$, and $\sqrt{x}$ all have $n$ significant places. $x+y$ and $x-y$ can have up to $n$ significant places, but depending on cancellation, one of them might have fewer. For example, suppose we know both $\pi$ and $22/7$ to $6$ significant places. We only know $22/7-\pi$ to $3$ significant places: $3... |
3,489,550 | <p>I have the coordinates of 3 points through which, a circle should pass . Having the coordinates of the points in 3D, how could I have the coordinates of the center of circumscribed circle ?
Also: if one of the points has some deviations and causes a circumscribed circle couldn't pass through the 3 points, is there a... | g.kov | 122,782 | <p>This formula for the center <span class="math-container">$O$</span>
of a circle
is suitable for both 2d and 3d:</p>
<p><span class="math-container">\begin{align}
O&=
A\cdot \frac{a^2\,(b^2+c^2-a^2)}{((b+c)^2-a^2)(a^2-(b-c)^2)}
\\
&+B\cdot \frac{b^2\,(a^2+c^2-b^2)}{((a+c)^2-b^2)(b^2-(a-c)^2)}
\\
&+C\cdo... |
17,429 | <p>I want to define my own little 'Inner Product' function satisfying properties of linearity and commutativity, and I'd like to use the "$\langle$" and "$\rangle$" symbols to output my results. For this I am using <code>AngleBracket</code> which has no built-in meaning.</p>
<p>I was able to use <code>SetAttributes[A... | Peter Breitfeld | 307 | <p>For a starter, I would use a function <code>angleExpand</code> like this:</p>
<pre><code>SetAttributes[AngleBracket, Orderless];
ruAB = {
AngleBracket[a_ (k_?NumericQ), b_] :> k AngleBracket[a, b],
AngleBracket[(k_?NumericQ) a_, b_] :> k AngleBracket[a, b],
AngleBracket[a_, (k_?NumericQ) b_] :> k Ang... |
284,184 | <p>I am expected to prove by induction that any polynomial function is continuous. In which "direction" would you advise to make induction? </p>
<p>e.g. Taking $x^n$ and making induction on $n$ is not sufficient. By polynomial I understand, $\sum^{m}_{n=1}a_n x^n$. How do I prove it's continuity using epsilon delta no... | Robert McLean MD PhD | 460,479 | <p><strong>Induction base step</strong>: Constant functions are continuous. </p>
<p>Proof: Recall the (Weierstrass) definition of continuous function. For every $\epsilon$ there is a $\delta$ such that for $x_0 − \delta < x < x_0 + \delta$ implies
$f(x_0)-\epsilon < f(x) < f(x_0)+\epsilon$. This is easy. ... |
4,456,144 | <p>Let <span class="math-container">$A$</span> be a continuous masa in <span class="math-container">$L(H)$</span> and <span class="math-container">$T$</span> be a positive contraction in <span class="math-container">$A$</span>.
Then we can assume that <span class="math-container">$0＜\|Th\|＜1$</span> for all unit vector... | Ruy | 728,080 | <p>Martin Argerami's answer above is probably the one written in "The Book" but let me nevertheless give another proof, presenting
a different perspective, although it relies on a much more sophisticated result.</p>
<p>It is well known that every maximal abelian subalgebra of operators on a (henceforth assume... |
127,109 | <p>How can I show that the order of an element modulo <span class="math-container">$m$</span> divides <span class="math-container">$\phi(m)$</span>?</p>
<p>I know that if <span class="math-container">$a$</span> and <span class="math-container">$m$</span> are relatively prime, then the least positive integer <span class... | Bill Dubuque | 242 | <p>Elaboration yields <em>conceptual</em> insight. We view it as a special case of a <em>fundamental</em> result that characterizes cycles (period) of powers in modular arithmetic. Below the modulus <span class="math-container">$\rm m$</span> is <em>fixed</em> and often unnotated. [Note: we can eliminate the need to ... |
227,869 | <p>Get the center and the semimajor/semiminor axes of the following ellipses:</p>
<p>$$x^2-6x+4y^2=16$$</p>
<p>$$2x^2 - 4x+3y^2+6y=7$$</p>
<p>How would one get these? I have no clue. I have a problem with merely rewriting these in the traditional ellipse equation. </p>
| Adi Dani | 12,848 | <p>$$x^2-6x+4y^2=16$$</p>
<p>$$(x-3)^2-9+4y^2=16$$</p>
<p>$$(x-3)^2+4y^2=25$$</p>
<p>$$\frac{(x-3)^2}{25}+\frac{4y^2}{25}=1$$</p>
<p>$$\frac{(x-3)^2}{5^2}+\frac{y^2}{(\frac {5}{2})^2}=1$$</p>
<p>center is $O(3,0)$ AND axes are $a=5,b=5/2$, for second ellipse you can proceed similarly</p>
|
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| Doug M | 317,162 | <p>If we consider an infinite chain.</p>
<p>Suppose $x = \sqrt{20 +\sqrt{20+\sqrt{20+\sqrt{20+\sqrt{\cdots}}}}}$</p>
<p>$x = \sqrt{20 +x}\\
x^2 = 20 + x\\
x^2 - x - 20 = 0\\
(x-5)(x+4) = 0$</p>
<p>$x$ must be greater than $0, x = 5$</p>
<p>and $y = \sqrt{20 -\sqrt{20-\sqrt{20-\sqrt{20-\sqrt{\cdots}}}}}$</p>
<p>$y ... |
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| Alex Meiburg | 127,777 | <p>Even without a calculator, you can do the numerics fairly easily. 20 is about halfway between 16 and 25, so $\sqrt{20} \approx 4.5$. So $20\pm\sqrt{20}$ is about 15.5 and 24.5, respectively. These in turn have square roots of about (a little less than) 4 and 5. This leaves $\sqrt{25}−\sqrt{16}\approx 1$. The "little... |
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| Lutz Lehmann | 115,115 | <p>By binomial formulas and cancellation, you get that
\begin{align}
...&=\frac{\sqrt{20+\sqrt{20}}+\sqrt{20-\sqrt{20}}}{\sqrt{20+\sqrt{20+\sqrt{20}}}+\sqrt{20-\sqrt{20-\sqrt{20}}}}
\\&=\frac{
\sqrt{5+\sqrt{\frac54}}+\sqrt{5-\sqrt{\frac54}}
}{
\sqrt{5+\sqrt{\frac54+\sqrt{\frac5{64}}}}+\sqrt{5-\sqrt{\frac5... |
729,352 | <p>I am trying to prove that $a_1$, $a_2$, $a_3$ are linearly independent.</p>
<p>I am asked to use vector product and prove that if $c_{1}a_{1} + c_{2}a_{2} + c_{3}a_{3} = 0$ then $c_1 = c_2 = c_3 = 0$</p>
<p>I am completely stuck on where to go with this problem. I would think that linearly independent then the nul... | user138335 | 138,335 | <p>Dot through by a1. We get $$c_1(a1\cdot a1)=0$$ so $c_1=0$. The same holds for the other two constants. (I'm assuming that when you say orthogonal, you are not allowing any vectors to have zero magnitude.)</p>
|
100,842 | <p>I have the following list of centers of disks.</p>
<pre><code>r=0.03;
pts = {{0.10420089319018544`, -0.024872674177014872`}, \
{0.9743669105930046`, 0.9169054125547074`}, {0.028760526736240563`,
0.45959879163736717`}, {-0.0059035632830851115`,
0.2922099255180086`}, {0.41615337459441437`,
0.9928402345... | Pillsy | 531 | <p>This is a good place to use <code>RegionMember</code> and <code>HalfPlane</code>. First, I reproduced your diagram, after explicitly drawing the square and tweaking the formatting a bit:</p>
<pre><code>square = {
FaceForm[{Lighter@Gray, Opacity[0.2]}],
EdgeForm[{Thickness[0.002], Black}],
Rectangle[{0.0, 0... |
1,613,172 | <p>Given three (nonempty) sets $A, B$ and $C$, and knowing that $|A| \leq |B|$, how can I prove that $|C^A| \leq |C^B|$? This problem is trivial if we want to prove that the sets are equipotent, because it is very easy to create a bijective function. </p>
<p>Here, we know that there exists an <em>injective</em> functi... | BrianO | 277,043 | <p>If you don't mind using the Axiom of Choice, here's an alternative to using an injection $A\to B$.</p>
<p>By assumption, $A\ne\emptyset$. Thus $\lvert A\rvert \le \lvert B\rvert$ implies (using AC) that there is a surjection $\varphi\colon B\to A$. Now we can define an injection:
$$
f\mapsto f\circ \varphi\colon C^... |
258,209 | <p>Let $X$ be an infinite set and let $\text{End}(X)$ be the set of all functions $f:X\to X$. For $f\in\text{End}(X)$ let $$\text{Com}(f) = \{g\in\text{End}(X): g\circ f = f \circ g\}.$$
Is there $f\in \text{End}(X)$ such that $\text{Com}(f) = \{\text{id}_X, f\}$? </p>
<p>If not, what is $\min\{|\text{Com}(f)|:f\in\te... | Pietro Majer | 6,101 | <p>Nope: since $f^2$ commutes with $f$, we should have either $f^2=\mathrm{id}_X$ or $f^2=f$, that is $f$ is either idempotent or involutive. But on an infinite set it is easy to see these always have infinite commutators.</p>
|
2,653,645 | <p>Is this proof valid? Can d be relative to x ?
<a href="https://i.stack.imgur.com/LGwPN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LGwPN.png" alt="enter image description here"></a></p>
| Martin Argerami | 22,857 | <p>No, $\delta$ cannot depend on $x$. What you need to do is use the "when $x$ is close" part. If $|x-3|<\delta$, then $3-\delta<x<3+\delta$. So if you don't allow $\delta$ to be big (you don't want that), then $x$ is bounded. So, for instance you can take
$$
\delta=\min\left\{1,\frac{\varepsilon}7\right\}.
$... |
4,536,813 | <p>In my textbook, a historical motivation for the development of differentiation is given, starting with Fermat trying to find the maxima and minima of functions. What I wanted to ask is why Fermat was interested in this problem in the first place, or to ask a more mathematical and less historical question, what some ... | AmateurDotCounter | 143,257 | <p><strong>Economics.</strong></p>
<p>Often times a business's operating costs or unit-costs can be understood through mathematical modelling. Being able to find minima (both local or global) can therefore inform your decisions as to reducing your operating or manufacturing costs. And likewise, modelling expected sup... |
4,536,813 | <p>In my textbook, a historical motivation for the development of differentiation is given, starting with Fermat trying to find the maxima and minima of functions. What I wanted to ask is why Fermat was interested in this problem in the first place, or to ask a more mathematical and less historical question, what some ... | Ethan Bolker | 72,858 | <p>I think the "find the maximum" questions in most textbooks are essentially artificial problems masquerading as practical examples.</p>
<p>This <a href="https://www.britannica.com/biography/Pierre-de-Fermat" rel="nofollow noreferrer">Britannica entry</a> suggests that Fermat was interested in the tangents t... |
2,001,243 | <p>Suppose I have $6$ fair coins, all facing heads ups. For each turn I will flip each of the $6$ coins. If a coin lands tails up, I set it aside. The next turn I take the remaining coins facing heads up and repeat the process. Now I want to know what is the expected number of turns $T$ until I have exactly $6$ tails. ... | Derek Elkins left SE | 305,738 | <p>The nlab article seems a bit off. I suspect it's more due to trying to explain too much in too little space so things get mixed together. I completely agree with you and I would call the whole expression "$\vdash J$" a judgement and not just $J$. I elaborate on what's going on in a different <a href="https://cs.s... |
3,284,426 | <p>The possible orders of Sylow 3 subgroups is <span class="math-container">$\{1, 13\}$</span> (if I understood correctly). But how can I check the exact number?
And how am I supposed to show that <span class="math-container">$S_3 = \Bbb Z_9$</span> or <span class="math-container">$S_3 = \Bbb Z_3\times \Bbb Z_3$</span>... | Con | 682,304 | <p>Let me adress the latter question:</p>
<p>Every group of order <span class="math-container">$p^2$</span>, where <span class="math-container">$p$</span> is some prime number, is isomorphic to <span class="math-container">$\mathbb{Z}/p^2\mathbb{Z}$</span> or <span class="math-container">$\mathbb{Z}/p\mathbb{Z} \times... |
1,491 | <p>I asked a question in MO and I received two interesting answers with different approaches. Both of them are very interesting. I wish to accept both of them simultaneously, but it is impossible. Morally, I cannot choose one of them as a better answer.</p>
<p>Is it reasonable to suggest to MO to remove this restricti... | Community | -1 | <p>This suggestion, while reasonable to make it, has very little hope of getting implemented. </p>
<p><strong>The answer "in theory:"</strong></p>
<p>The <em>idea</em> of having <em>only one</em> accepted answer is (to encourage) that <em>one</em> answer is created that is comprehensive. So, following the design-phil... |
2,911,049 | <blockquote>
<p><strong>Question:</strong>
Can we find the gradient and Hessian of $x x^T$ w.r.t. $x$, where $x \in \mathbb{R}^{n \times 1}$ ?</p>
</blockquote>
<p>EDIT:
If we can, may I know how to compute that? Thank you.</p>
| Ahmad Bazzi | 310,385 | <p><strong>Gradient</strong>
$$\frac{\partial \mathbf{Y}}{\partial x_i} =
\begin{bmatrix}
\frac{\partial y_{11}}{\partial x_i} & \frac{\partial y_{12}}{\partial x_i} & \cdots & \frac{\partial y_{1n}}{\partial x_i}\\
\frac{\partial y_{21}}{\partial x_i} & \frac{\partial y_{22}}{\partial x_i} & \cdots... |
464,146 | <p>Suppose that a wooden cube, whose edge is $3$ inch, is painted red, then cut into $27$ pieces of $1$ inch edge. Find total surface area of unpainted?</p>
<p>First of all, I have tried to draw the cube using MS Paint, below is given picture:</p>
<p><img src="https://i.stack.imgur.com/1eDpC.png" alt="enter image des... | Mick | 42,351 | <p>I think this question is set for testing one's 3D figure visualization and counting ability.</p>
<p>There are two ways to count the total unpainted area</p>
<p>Method_1</p>
<p>*For the corner blocks,</p>
<p>_There are 8 in total, each has 3 faces unpainted (1 top/bottom + 2 internal faces)</p>
<p>_Sub-total unp... |
4,788 | <p>Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial of degree n has <strong>at most</strong> n roots) is not considered fundamental by algebraists as it's not needed fo... | Qiaochu Yuan | 232 | <p>Perhaps it is worth expanding my answer. (Forgive me if I get any details wrong.) If you are interested in Lie groups, you are interested in their classification. A connected Lie group is the quotient of its universal cover by a discrete subgroup of its center, so to classify connected Lie groups it suffices to c... |
1,491,805 | <p>What is the simplified value of $(\tan15)(\tan30)(\tan45)(\tan60)(\tan75)$</p>
<p>I am trying to find this value without the use of a calculator but there are certain values I do not know like $(\tan15)$ and $(\tan75)$</p>
<p>Can someone give me any advice?</p>
| Empty | 174,970 | <p>$\tan 75^{\circ}=\cot 15^{\circ}$ ...and so on..</p>
<p>Hence value $=1$.</p>
|
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