qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,198,754 | <p>A rectangular prism has <strong>integer</strong> edge lengths. Find all dimensions such that its surface area equals its volume.</p>
<p>My Attempt at a Solution:</p>
<p>Let the edge lengths be represented by the variables $l, w, h$.</p>
<p>Then $$lwh = 2\,(lw +lh + wh) \implies lwh = 2lwh\left(\frac{1}{l} + \frac... | paw88789 | 147,810 | <p>Here's a somewhat ad hoc method for generating solutions to $\frac1h+\frac1w+\frac1l=\frac12$. Find a number $n$ for which there are three factors, $a,b,c$ of $n$ that add to $\frac{n}2$. We then divide the equation $a+b+c=\frac{n}2$ by $n$ to get the desired form.</p>
<p>For example if $n=30$, we can take factors... |
3,062,391 | <p>Let <span class="math-container">$M$</span> be a smooth manifold of dimension <span class="math-container">$n$</span> and let <span class="math-container">$p \in M$</span>. Choose a smooth chart <span class="math-container">$(U, \phi)$</span> around <span class="math-container">$p$</span> and then we have <span clas... | stressed out | 436,477 | <p>I believe that definitions might differ in different textbooks, but here's my understanding of the situation. I will try to be consistent with your notations as much as I can.</p>
<p>You have an open set <span class="math-container">$U\subseteq \mathbb{R}^n$</span> and a homeomorphism <span class="math-container">$... |
3,974,159 | <p>How many sub choices if there are 3 types of bread, 6 different types of meat, 8 different veggies, 4 different kinds of cheese</p>
<p>You must choose 1 bread.
You can choose any Meats, including none.
You can choose any veggies, including none.<br />
You must choose 1 cheese.</p>
<p>I am confident in finding the nu... | thedude | 297,270 | <p>By the <a href="https://en.wikipedia.org/wiki/Stationary_phase_approximation" rel="nofollow noreferrer">stationary phase method</a>, the integral is approximately given by
<span class="math-container">$$ \int_0^1 e^{inp(x)}dx= \sum_{x_0} \sqrt{\frac{2\pi}{np''(x_0)}}e^{inp(x_0)+\mathrm{sign}(p''(x_0))i\pi/4}+o(1/\sq... |
3,848,934 | <p>Let <span class="math-container">$T$</span> be some operator on an inner product space <span class="math-container">$(V, \langle\cdot,\cdot\rangle)$</span>, and <span class="math-container">$T^\dagger$</span> be its adjoint. I found too many questions about the proof of
<span class="math-container">$$Im(T^\dagger) =... | Arthur | 15,500 | <p>Take a vector <span class="math-container">$T^\dagger v$</span> in the image of <span class="math-container">$T^\dagger$</span>, take a vector <span class="math-container">$w$</span> in the kernel of <span class="math-container">$T$</span>, and take their dot product, then use the definition of adjoint.</p>
<p>This ... |
307,370 | <p>Let $G$ be a finite group and $H$ a subgroup. Define a relation on $G$ by $$a\sim b\iff b^{-1}a \in H.$$</p>
<p>(0.) Show that this is an equivalence relation.</p>
<p>(i) Prove that for this relation $$[a] = \{ah: h\in H\}.$$</p>
<p>(ii) Prove that the cardinality $[a]$ equals the cardinality of $|H|$. </p>
<p>(... | Tara B | 26,052 | <p>(0.) Fine for reflexivity. For symmetry, your idea is not going to work, because it's not always the case that $a^{-1}b = b^{-1}a$. Luckily, you don't need to show that. You only need $b^{-1}a$ to be in $H$ whenever $a^{-1}b$ is in $H$. </p>
<p>For transitivity, your idea will work. Do you see why?</p>
<p>(i) ... |
1,052,512 | <p>Basically, the question started with a little argument I had with my friend. My friend said he thinks it's possible to draw only 2 lines on the letter "W" and make 6 triangles, and I played around with it, but I couldn't really do it, so I told him I don't think it's possible, and we need at least 3 lines. </p>
<p>... | Robo | 197,712 | <p>Three parallel lines one at the top, one middle, one bottom, and, if tri inside tri is allowed, you have $6$. Of these, $3$ are true and $3$ more are built around their respective parent.</p>
|
2,975,135 | <p>I want to prove that, if <span class="math-container">$m \equiv_4 n$</span> for all <span class="math-container">$m,n \in \mathbb{Z}$</span>, then <span class="math-container">$123^m \equiv_{10} 33^n$</span></p>
<p>I have no idea how to prove something like that</p>
| Bill Dubuque | 242 | <p><span class="math-container">$\bmod 10\!:\ \color{#c00}{3^{\large 4}\equiv 1}\,\Rightarrow\,3^{\large m}\! = 3^{\large n+4k}\!\equiv 3^{\large n}(\color{#c00}{3^{\large 4}})^{\large k}\!\equiv 3^{\large n}\ $</span> by <a href="https://math.stackexchange.com/a/879262/242">Congruence Product / Power Rules</a></p>
|
253,413 | <p>If we know that $\sin(n^\circ)$ is constructible where $n$ is some integer,then is $\sin((an)^\circ)$ also constructible for any integer $a$ ?</p>
<p>I am thinking it should be but not sure how to show it? Maybe using some recurrence relation for sin, to express it entirely in terms of powers of $\sin(n^\circ)$? </... | Goonfiend | 17,338 | <p>I think more assumptions are needed if you mean $f_n\rightarrow f$ in the pointwise sense. A counter-example is taking $f_n$ to be the positive spike of height 1 and width $2/n$ centred at $x=1/n$ and zero outside $(0,2/n)$. This function tends to zero pointwise whilst the sup of its square is always 1.</p>
|
253,413 | <p>If we know that $\sin(n^\circ)$ is constructible where $n$ is some integer,then is $\sin((an)^\circ)$ also constructible for any integer $a$ ?</p>
<p>I am thinking it should be but not sure how to show it? Maybe using some recurrence relation for sin, to express it entirely in terms of powers of $\sin(n^\circ)$? </... | user1551 | 1,551 | <p>Hint: $f_n^2-f^2=(f_n-f)(f_n+f)$.</p>
|
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Geoff Robinson | 14,450 | <p>In general the question is too ambitious, but more can be said than might be expected. For example, it follows from results of J.G. Thompson that if $M$ is a maximal subgroup of a non-Abelian finite simple group $G$ and $M$ is nilpotent, then $M$ is a Sylow $2$-subgroup of $G$ and is non-Abelian (this does occur "in... |
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Russ Woodroofe | 19,729 | <p>The Atlas of Finite Group Representations has information on the maximal subgroups of some particular finite simple groups. This is at least a handy place to start. See</p>
<p><a href="http://brauer.maths.qmul.ac.uk/Atlas/v3/lin/L253/#maxes" rel="noreferrer">http://brauer.maths.qmul.ac.uk/Atlas/v3/lin/L253/#maxes... |
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Michael Giudici | 3,214 | <p>The maximal subgroups of $A_n$ are given by the O'Nan-Scott Theorem. They lie in one of the following classes:</p>
<p>1) $A_n \cap (S_{n-k} \times S_k)$, that is the stabiliser of a $k$-set.</p>
<p>2) $A_n \cap (S_a wr S_b)$ where $n=ab$, that is the stabiliser of a partition.</p>
<p>3) $A_n\cap AGL(d,p)$ where $... |
2,216,070 | <p>I'm having problems finding the primitive function to $\int \frac{1}{x \sqrt{8-x^2}} dx$. I've tried to use the substitution $t = x^2-8$, but then I just get stuck with $\int \frac{1}{(8-t)\sqrt{t}} dt$ instead. Using the substitution $t = \sqrt{x^2-8}$ doesn't get me much further either.</p>
<p>Any help is much ap... | Kanwaljit Singh | 401,635 | <p>Formula -</p>
<p>$$\int \frac{1}{x \sqrt{a^2-x^2}} dx = -\frac 1a \operatorname{sech}^{-1}\frac xa + c$$</p>
<p>Now,</p>
<p>$$\int \frac{1}{x \sqrt{(\sqrt 8)^2-x^2}} dx$$</p>
<p>$$= -\frac 1{\sqrt8} \operatorname{sech}^{-1}\frac x{\sqrt8} + c$$</p>
|
1,598,947 | <p>Let $\{\xi_k\}_{k=1}^4$ be a set of vectors in $\mathbb{R}^3$. If $\{\xi_1, \xi_2\}$ and $\{\xi_2, \xi_3, \xi_4\}$ are independent sets, and $\xi_1$ belongs to the span of $\{\xi_2, \xi_4\}$. Show that $\{\xi_k\}_{k=1}^3$ is linearly independent. </p>
<p>Clearly, $\xi_1 = a_1\xi_2 + a_2\xi_4$ where $a_1\neq0$ due t... | Anurag A | 68,092 | <p><strong>hint</strong></p>
<p>Start with a linear combination of vectors $1,2,3$ and set it equal to $0$ (vector). Now write vector $1$ as a linear combination of vectors $2$ and $4$. This gives you a linear combination of vectors $2,3$ and $4$ equal to the zero vector. Now use the independence of $2,3,4$ to find th... |
3,725,958 | <p>When I search implicit differentiation for equation <span class="math-container">$x^2 + y^2 = r^2$</span> I find results of two versions: one using derivative and the other using differential.</p>
<p>Version1: <span class="math-container">$\frac{d }{dx}(x^2 + y^2 = r^2) \Leftrightarrow 2x + 2y\frac{dy}{dx} = 0 $</sp... | Gibbs | 498,844 | <p>I think the geometric set-up is not really clear. In the first case you are viewing <span class="math-container">$y$</span> as a function of <span class="math-container">$x$</span>, so the equation <span class="math-container">$x^2+y^2=r^2$</span> defines two semicircles in <span class="math-container">$\mathbb R^2$... |
2,824,504 | <p>$\sum_{n=1}^{\infty} \dfrac{\arctan n + \sqrt {|x|}}{n^2}$.
Does the series converge uniformly on $R$. ?</p>
<p>Let $f_k(x) = \dfrac{\arctan k + \sqrt {|x|}}{k^2}$. Then, I have to find $M_k$ such that $|f_k(x)| \le M_k$ for all $x \in R$. Concurrently, $\sum_{k=1}^{\infty} M_k$ has to converge. I think such $M_k$... | operatorerror | 210,391 | <p>It's not uniformly Cauchy: Fix $N\in \mathbb{N}$, then for any $k\geq 1$ we ought to be able to keep
$$
\sup_{x\in \mathbb{R}}\left|\sum_{n=N}^{N+k} \frac{\arctan(n)+\sqrt{|x|}}{n^2}\right|
$$
small by making $N$ large. However,
$$
\sup_{x\in \mathbb{R}}\left|\sum_{n=N}^{N+k} \frac{\arctan(n)+\sqrt{|x|}}{n^2}\righ... |
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... | cvgmt | 72,111 | <pre><code>g = Grid[grid, BaseStyle -> 10, Sequence @@ opts]
Magnify[g, 2]
</code></pre>
|
2,023,222 | <p>I am facing difficulty with the following limit.</p>
<p><span class="math-container">$$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$</span></p>
<p>I tried to take log both sides but I could not simplify the resulting expression.</p>
<p>Please help in this regard. Than... | Redundant Aunt | 109,899 | <p>We see that
$$
\prod_{k=0}^n\binom{n}{k}=\frac{n!^{n+1}}{\prod_{k=0}^nk!^2}=\frac{n!^{n+1}}{\left(\prod_{k=0}^nk^{n+1-k}\right)^2}=\frac{H(n)^2}{n!^{n+1}}.
$$
where $H(n)=\prod_{k=1}^nk^k$. Now we see that
$$
\log(H(n))=\sum_{k=1}^nk\log(k)≥\int_{1}^nx\log(x)dx=\frac{n^2}{2}\log(n)-\frac{n^2}{4}
$$
as well as
$$
\l... |
2,760,994 | <p><strong>I have 3 trees.</strong></p>
<ul>
<li>These particular trees are dioecious (male or female).</li>
<li>I don't know the gender of any of the trees. </li>
<li>The chance of a tree being male or female is 50/50. </li>
<li>I need at least one male and one female for successful pollination to occur.</li>
</ul>
... | User1974 | 389,017 | <p>Here are all of the <a href="https://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html" rel="nofollow noreferrer">possible combinations</a> that I can come up with:</p>
<pre><code>{M,M,M}
{M,M,F}
{M,F,M}
{M,F,F}
{F,M,M}
{F,M,F}
{F,F,M}
{F,F,F}
</code></pre>
<p>There are a total of 8... |
878,020 | <p>It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample?</p>
<p>I believe I've found an answer, but since answering own questions is encouraged, I thought I might post it here. Other examples are obviously ... | Andreas Caranti | 58,401 | <p>I believe a slightly more elementary version of the example of Marcin Łoś consists in taking the ring $R$ of $2 \times 2$ matrices over $\mathbb{R}$, say, and the left ideals
$$
I = \left\{ \begin{bmatrix}a & 0\\ b & 0\end{bmatrix} : a, b \in \mathbb{R} \right\}
, \qquad
J = \left\{ \begin{bmatrix} 0 & a... |
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 $... | caverac | 384,830 | <p>The circle is described by </p>
<p>$$
x^2 + y^2 = 4 \tag{a}
$$</p>
<p>and the hyperbola by </p>
<p>$$
y = 1/x \tag{b}
$$</p>
<p>Replacing (b) into (a) you get</p>
<p>$$
x^2 + \frac{1}{x^2} = 4 \quad\Rightarrow\quad x^4 - 4x^2 + 1 = 0
$$</p>
<p>this is a quadratic equation in $x^2$ whose solutions are</p>
<p>$... |
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 $... | BearClaw | 391,873 | <p>The circle is ,<br>
$x^2 + y^2 = 4\tag{1}$
The hyperbola is,<br>
$xy=1\tag{2}$
Adding 2xy on both sides of equation 1, </p>
<p>$$x^2+y^2+2xy=4+2xy$$ $$(x+y)^2=4+2xy$$
$$(x+y)^2=6\tag{since xy=1,from (1)}$$
$$x+y=\pm\sqrt6\tag{3}$$
putting 2 into 3 ,<br>
$$x+1/x=\pm\sqrt6$$
$$x^2\pm\sqrt6x... |
65,059 | <p>Surely yes, and in more generality, but can it be proved?</p>
<p>It seems that most, if not all, statements about quadratic forms representing primes fall back on algebraic number theory (i.e. splitting of primes in $\mathbb{Q}(\sqrt{7})$) for their proofs, and so are incompatible with the condition that $0 < y ... | François Brunault | 6,506 | <p>The units of $k=\mathbf{Q}(\sqrt{7})$ have the form $\pm (8+3 \sqrt{7})^n$ with $n \in \mathbf{Z}$. If $\pi = x+y\sqrt{7}$ is a prime element of $k$, then $\lambda(\pi):= \log |x+y\sqrt{7}|$ is well-defined in $\mathbf{R}/\alpha \mathbf{Z}$ where $\alpha = \log(8+3\sqrt{7})$. Note that $\lambda$ factors as $\lambda ... |
4,032,422 | <p>This is actually in reference to the question posed here
<a href="https://stackoverflow.com/posts/66285948/edit">https://stackoverflow.com/posts/66285948/edit</a>
but is more appropriate as a question to be posed on a non-coding site.</p>
<p>I provide a partial answer, but not the full answer, relaying the full ques... | Ryan Wisnesky | 98,085 | <p>(This comment is also posted at stack overflow). Thanks for your reply! In the time since I asked it, I believe that Gabriel Scherer gave an affirmative answer to this question in his paper arxiv.org/abs/1610.01213</p>
|
1,054,346 | <p>Solve for reals:</p>
<p>$a(b+c-a^3)=b(c+a-b^3)=c(a+b-c^3)=1$</p>
<p>I found cyclic relation</p>
<p>$c=(a+b)(a^2+b^2)$</p>
<p>and a solution $a=b=c=1$</p>
<p>But now I am not getting anything.</p>
| Dr. Sonnhard Graubner | 175,066 | <p>another solution is $(a,b,c)=(-1,-1,-1)$ and your triple these are the only real solutions.
eliminating the variables $b,c$ we get for $a$
$$ \left( {a}^{12}+2\,{a}^{10}+7\,{a}^{8}+8\,{a}^{6}+7\,{a}^{4}
-2\,{a}^{2}+1 \right) \left( a-1 \right) \left( a+1 \right) \left(
{a}^{12}-2\,{a}^{10}+5\,{a}^{8}-4\,{a}^{6}+... |
4,270,105 | <p>I'd like to prove that this matrix is idempotent using a more algebraic proof for matrices with a similar definition to A, rather than deriving its eigenvalues or calculating <span class="math-container">$A^2$</span> .</p>
<p><span class="math-container">$A=$</span> <span class="math-container">$\begin{bmatrix}0.5&a... | José Carlos Santos | 446,262 | <p>Since it is symmetric, it is diagonalisable. It's clear from <span class="math-container">$A$</span>'s second column that <span class="math-container">$1$</span> is an eigenvalue and, since the third column is minus the first one, <span class="math-container">$\det A=0$</span>, and therefore <span class="math-contai... |
935,763 | <p>How can I generate <em>random</em> points <em>uniformly distributed</em> on the surface of a sphere such that a line that originates at the center of the sphere, and passes through one of the points, will intersect a plane within a <strong>circle</strong>. Following are more illustrations and details to make this cl... | David K | 139,123 | <p>You might try a point-rejection approach that rejects a relatively small percentage of the points.</p>
<p>As observed in comments, you can improve on sampling over the entire sphere by finding a circular cap that barely covers the elliptical window, and using the method you have already devised to sample uniformly ... |
3,321,367 | <p>According to Euler's Formula, <span class="math-container">$e^{ix} = \cos(x) + i\sin(x).$</span>
I'm computing the product <span class="math-container">$e^{ix} \cdot e^{iy}.$</span></p>
<p>What is the real part (that is, the term without a factor of <span class="math-container">$i$</span>)?</p>
<p>Why is it <span... | Alberto Ibañez | 648,576 | <p><a href="https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/" rel="nofollow noreferrer">https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/</a> " it was observed by Bohm and Sontacchi that this w... |
3,204,547 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma \in [-3,10].$</span> Then largest value of the determinant</p>
<p><span class="math-container">$$\begin{vmatrix}3\alpha^2&\beta^2+\alpha\beta+\alpha^2&\gamma^2+\alpha\gamma+\alpha^2\\\\
\alpha^2+\alpha\beta+\beta^2& 3\beta^2&\gamma^2+\... | MachineLearner | 647,466 | <p>Hint: The matrix is symmetric. You can use the <a href="https://en.wikipedia.org/wiki/Cholesky_decomposition#Statement" rel="nofollow noreferrer">Cholesky decomposition</a> to rewrite the matrix <span class="math-container">$A$</span> as <span class="math-container">$A=LL^T$</span>, in which <span class="math-contai... |
2,504,613 | <p>I have problem with solving the following equation:</p>
<blockquote>
<p>$$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>
| ThePirateKing | 456,378 | <p>It's a Bernoulli's ODE, first $y=0 $ is a solution , now by dividing by $y^{1/3} $ we find $ \frac{y'}{y^{1/3}} - 3 \frac{ty}{y^{1/3}} = t^4 $ , Now we put $z= \frac{1}{y^{\frac{-2}{3}}}$ we will find by replacing in the ODE $ \frac{2}{3}z'-\frac{3}{t}z=4t^4 $ by solving it you will find easily $z$ then $y$.</p>
|
1,662,418 | <p>A local system is a bundle with locally constant sheaf of sections. I have seen several equivalent characterizations (bundles acted upon nicely by the fundamental group of the base, bundles admitting flat connections, etc), and from this I can construct examples of vector bundles that are provably not local systems ... | Qiaochu Yuan | 232 | <p>Given a vector bundle, there is no meaningful notion of locally constant sections of it without extra data. You can define what "locally constant" means with respect to a particular local trivialization, but this notion depends on the choice of local trivialization, and it generally won't be possible to patch local ... |
2,617,286 | <p>While I was solving one problem, with natural variables $(v,x,y,z)$ and I make change to $(\varphi,\chi,\psi,\omega)$ defined as</p>
<p>$\varphi=\arctan\big(\frac{Ax-y}{z}\big) \qquad \psi=\frac{(y-Ax)^2+z^2}{2A} \qquad \chi=\frac{y}{A} \qquad \text{and} \qquad \omega=z-Av $</p>
<p>For continue solving it, I need ... | Michael Rozenberg | 190,319 | <p>I think you are right and the internet is wrong. </p>
|
2,617,286 | <p>While I was solving one problem, with natural variables $(v,x,y,z)$ and I make change to $(\varphi,\chi,\psi,\omega)$ defined as</p>
<p>$\varphi=\arctan\big(\frac{Ax-y}{z}\big) \qquad \psi=\frac{(y-Ax)^2+z^2}{2A} \qquad \chi=\frac{y}{A} \qquad \text{and} \qquad \omega=z-Av $</p>
<p>For continue solving it, I need ... | heropup | 118,193 | <p>It is worth noting that we can prove that it is impossible to place $3$ queens on a $3 \times 3$ board such that no queen attacks another. Clearly, the center square cannot be occupied, since it attacks every other square.</p>
<p>If a corner square is occupied, this leaves exactly two squares that are not attacked... |
2,693,723 | <p>I have started to learn in university, and the way they teach us is very simpled... not the way they tought as geometry. The question is simple, but I want an concrete simple mathematic proof(verbal would be enough) something that would help me grasp the concept:</p>
<p>I have this set $$A = \left\{ \frac 1{2^n} \m... | Steve Linton | 172,396 | <p>It's a bit hard to work around the fact you've been given the set of elements which is infinite. Almost any proof you can come up with kind of uses that. However, how about a proof by contradiction. Suppose $A$ were finite of order $x$. Then you can write down an element of $A$ which generates a finite subgroup of o... |
3,356,468 | <p>Let <span class="math-container">$\pi:E\rightarrow M$</span> be a smooth vector bundle. Let <span class="math-container">$S:M\rightarrow E$</span> be it's zero section. Let <span class="math-container">$M'=E-S(M)$</span>.</p>
<p><strong>Is <span class="math-container">$M'$</span> a smooth submanifold of <span clas... | Adittya Chaudhuri | 311,277 | <p>Let <span class="math-container">$l\in L^{+}$</span>. Consider <span class="math-container">$\pi(l)\in M$</span>. Let <span class="math-container">$(U,\phi)$</span> be a trivialization around <span class="math-container">$\pi(l)$</span>. Hence <span class="math-container">$l\in \pi^{-1}(U)- \sigma(U) \subseteq L^{... |
279,478 | <p>I want to determinate $p$ and $q$ in RSA.</p>
<p>I know that $n = 172451$ and $\phi(n) = 171600$.</p>
<p>$$171600 = pq - (p+q) + 1 = 172451 -(p + q) + 1$$
$$p + q = 172451-171600+1 = 852$$
$$(p-q)^2 = (p+q)^2-4pq = (852)^2 - 4(172451) = 36100$$</p>
<p>Now I'm stuck at this point and don't understand how can I get... | Andreas Caranti | 58,401 | <p>You know
\begin{equation*}
p q = n
\end{equation*}
and
\begin{equation*}
\varphi(n) = (p-1)(q-1) = pq - p - q + 1 = n - (p+q) + 1.
\end{equation*}
So
\begin{equation*}
p + q = n + 1 - \varphi(n).
\end{equation*}</p>
<p>Now recall that in a quadratic equation
\begin{equation*}
x^2 - b x + c = 0,
\end{equation*}
th... |
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>
| Doug Spoonwood | 11,300 | <p>Do you want to know how to write "let" exactly or basically write "the same" content as how we usually interpret "let"?</p>
<p>I don't know your particular proof, but you might achieve the same effect by considering conditionals. In other words, "let x equal a" turns into a conditional "if x equals a, then ...", a... |
2,892,797 | <p>Here's my attempt:</p>
<p>Suppose there is some $L \in \mathbb{R}$ such that $\lim_{ x\to 0 } \sin \frac{2\pi }{x} = L$. Then, if we let $\varepsilon = 1$, there would exist $\delta > 0$ such that $\left| f(x) - L\right| < 1$ for all $0< |x| < \delta $. Now, by the Archimedean property, there is $n \in ... | user | 505,767 | <p>As an alternative simply note that for $n\to \infty$</p>
<ul>
<li>$x_n=\frac4n \to 0 \implies \sin \frac{2\pi}{\frac4n}=\sin \frac{\pi n}2=\pm 1$ </li>
</ul>
<p>therefore the given limit doesn’t exist.</p>
|
4,088,272 | <p>Let <span class="math-container">$f:[a,b]\to \mathbb R$</span> a function. If <span class="math-container">$P=\{x_0,x_1,\ldots,x_n\}$</span> is a partition of <span class="math-container">$[a,b]$</span>, define <span class="math-container">$$||P||=\max_{1\leq i\leq n}|x_i-x_{i-1}|.$$</span></p>
<p>Prove that, <span ... | Caio Lins | 910,193 | <p>You are trying to prove the first implication, that if <span class="math-container">$f$</span> is Riemann integrable, than that limit exists and it's equal to the integral.
Your line of thought is correct, however, you are assuming that, if <span class="math-container">$f$</span> is Riemann integrable, than for ever... |
4,541,198 | <p>I am reading a book and it says any closed set <span class="math-container">$F$</span> in <span class="math-container">$\mathbb{R}^n$</span> can be written as a countable union of compact sets as follows: <span class="math-container">$F=\cup_{k=1}^\infty F\cap B_k$</span> where <span class="math-container">$B_k$</sp... | orangeskid | 168,051 | <p>You are trying to show that any two maps to <span class="math-container">$X$</span> are homotopic. The argument: we can join <span class="math-container">$f(x)$</span> and <span class="math-container">$g(x)$</span> by a path, and then move along these paths for every <span class="math-container">$x$</span> to create... |
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="... | Henno Brandsma | 4,280 | <p><span class="math-container">$\mathcal{I}$</span> has nothing to do with the statement about basic weak-open neighbourhoods of <span class="math-container">$U$</span>, this latter fact we can verify independently:</p>
<p>I'll use the definition that the weak topology <span class="math-container">$\mathcal{T}$</span... |
283,926 | <p>Let $A$ be a given symmetric positive definite $N\times N$ matrix. I need to find a symmetric positive semi-definite matrix $S$ which is the solution to the following optimization problem
\begin{align}
\max_{S}~&\det(A+S) \\s.t.~&\sum_{i}^{N}\sigma_i(S)\,=\,c \\&S\geq0
\end{align}where $\sigma_i(S)$ are ... | Josiah Park | 118,731 | <p>One can verify that $U_{A}=U_{S}$ as follows. Note that $$\det(U_{A}\Sigma_{A}V_{A}^{T}+U_{S}\Sigma_{S}V_{S}^{T})=\det(\Sigma_{A}+U_{A}^{-1}U_{S}\Sigma_{S}V_{S}^{T}V_{A}^{-T})$$
Let $Y=U_{A}^{-1}U_{S}\Sigma_{S}V_{S}^{T}V_{A}^{-T}$ so that the problem can be rephrased as:</p>
<p>$$\max\limits_{Y} \det(\Sigma_{A}+Y)... |
1,065,040 | <p>I want to detect collision of a sphere with another object and to find out(show) the deformation of the sphere. I have come to know that hexagon(regular)tessellation of a sphere is the most appropriate one. What are the advantages of tessellating a sphere with regular hexagons over squares and triangles? How can I u... | user7530 | 7,530 | <p>It is impossible to tessellate (regularly or otherwise) a sphere with hexagons, at least when three hexagons meet at each vertex.</p>
<p>If you have $F$ hexagons, this means you must have $3F$ edges (since each hexagon has six edges, shared by two hexagons) and $2F$ vertices (since each hexagon has six vertices, sh... |
1,065,040 | <p>I want to detect collision of a sphere with another object and to find out(show) the deformation of the sphere. I have come to know that hexagon(regular)tessellation of a sphere is the most appropriate one. What are the advantages of tessellating a sphere with regular hexagons over squares and triangles? How can I u... | Mark Fischler | 150,362 | <p>A sphere cannot be tessellated using only regular hexagons. You need to place twelve pentagons with sides the same as those of the hexagons; the pentagon centers must be at the face centers of a dodecahedron. Then you can complete the tessellation using some number of hexagons; for example, I think the old-fashione... |
1,557,531 | <p>Please help.</p>
<p>Question: Addmath (Quadratic Equations)</p>
<p>Given $\alpha$ and $\beta$ are the roots of the quadratic equation
$2x^2 - 6x + 5 = 0$,
form an quadratic equation with the roots $\alpha + 1$ and $\beta + 1$.</p>
| Steven Alexis Gregory | 75,410 | <p>$$x^2 - 3x + \frac 52 = (x - \alpha)(x - \beta) = x^2 - (\alpha + \beta)x + \alpha \beta$$</p>
<p>Then $\alpha + \beta = 3$ and $\alpha \beta = \frac 52$. So $$(x-(\alpha + 1))(x - (\beta + 1)) = x^2 - (\alpha + \beta+2)x +(\alpha \beta + \alpha + \beta + 1)$$</p>
<p>You should be able to do the rest.</p>
|
1,700,493 | <p>The following is an exercise from <em>Linear Analysis</em> by Bollobas.</p>
<p>Let $f:X\to X$, with $X$ a compact metric space. Suppose that for every $\epsilon>0$, there is a $\delta=\delta(\epsilon)$ such that if $d(x,f(x))<\delta$ then $f(B(x,\epsilon))\subset B(x,\epsilon)$. Let $x_0\in X$ and define $x_n... | Pantelis Sopasakis | 8,357 | <p>$\newcommand{\be}{\mathcal{B}_\epsilon} \renewcommand{\Re}{\mathbb{R}}$As discussed by @Mankind, $\phi$ is defined as a mapping
$$
\phi:\be\setminus\{0\} \to \Re^n\setminus\{0\}.
$$
(1) The function is <strong>surjective</strong> as we can see in the following figure.</p>
<p><a href="https://i.stack.imgur.com/sf8vJ... |
2,271,558 | <p>I want to prove the following statement</p>
<p>All subgroups of $Q_8 \times E_{2^n}$ are normal </p>
<p>Here $E_{p^n} = \mathbb{Z}_p \times \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p$ (n times)</p>
<hr>
<p>From some comments below, i made up some informal justification. </p>
<p>My strategy are following. </... | Jonathan Rayner | 90,675 | <p>Note: as mentioned in the comments, it is <em>not</em> true in general that if all the subgroups of <span class="math-container">$G$</span> are normal and all the subgroups of <span class="math-container">$H$</span> are normal, then all the subgroups of <span class="math-container">$G \times H$</span> are normal. A ... |
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... | Tim van Beek | 7,556 | <p>An introduction to the theory on Hilbert spaces is this:</p>
<ul>
<li>Claudia Prévôt, Michael Röckner: "A concise course on stochastic partial differential equations." (see <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1123.60001&format=complete">ZMATH</a>)</li>
</ul>
<p>This book construct... |
599,282 | <p>I am given a function $f(x)$.</p>
<p>I determined that $f(x)'' = 0$ precisely when $x$ is $4$ or $-3$.</p>
<p>I am asked to find the interval for which the function is concave down.</p>
<p>How can I do it by knowing the values $x = 4$ and $x = -3$ and without having to plot the function?</p>
| dfeuer | 17,596 | <p>If $f''$ is <em>continuous</em>, then you only need to know it at three more points: one less than $-3$, one between $-3$ and $4$, and one greater than $4$. Do you see why you need these? If $f''$ is not continuous, then you may need to know much more.</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... | Henry | 6,460 | <p><em>As requested in comments:</em></p>
<p>You could use R: for example the probability of being strictly more than $9876$ could be about </p>
<pre><code>> pbinom(9876, size=10^11, prob=10^-7, lower.tail=FALSE)
[1] 0.8917494
</code></pre>
<p>This compares with the normal approximation with continuity correction... |
1,619,103 | <p>I'm trying to find the infinite sum that is defined by:</p>
<p>$$
3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots
$$</p>
<p>However, I do not know of any known formula to do this. Am I missing something really simple? Thanks!</p>
| Brian M. Scott | 12,042 | <p>The ideas in <strong>Mario G</strong>’s answer are very good ones to learn, but there are also ways to tackle the problem without calculus. You can easily test that the series is absolutely convergent, so we can rearrange terms pretty much at will. Now</p>
<p>$$\begin{align*}
3\left(\frac9{11}\right)+4\left(\frac9{... |
2,032,923 | <p>I want to rearrange the formula for angular velocity $\omega = \dfrac{2\pi}{T}$, to make $T$ the subject as I wish to find the period.</p>
<p>Would the correct answer be $T = \frac{\omega}{2\pi}$ or would it be $T = \frac{2\pi}{\omega}$? </p>
<p>And is there a certain rule you should follow when rearranging ?</p>
| Vidyanshu Mishra | 363,566 | <p>It is quite simple. You have $\omega = \dfrac{2\pi}{T}$ since you want to make T the subject, multiply the whole equation by T and you will get $$\omega{\cdot T} = \dfrac{2\pi}{T}{\cdot T} = 2\pi$$</p>
<p>On bringing ${\omega}$ on right you will have $T = \frac{2\pi}{\omega}$</p>
|
4,470,741 | <p><span class="math-container">$$
\begin{aligned}
\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(1+x)(1+2x)(1+3x)\cdots(1+nx)} && x > 0
\end{aligned}
$$</span></p>
<p>I tried using (1) some inequalities (2) Taking the coefficients of x common to get some factorials in the denominator , couldn't reach a right conc... | abiessu | 86,846 | <p>Given the commented multiple-choice answers, another approach would be to notice the following:</p>
<p><span class="math-container">$$\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(1+x)(1+2x)(1+3x)\cdots(1+nx)}\\
\le \sum_{n=0}^{\infty}\frac{x^{2n}}{(1+x)(1+2x)(1+3x)\cdots(1+nx)}\\
\le \sum_{n=0}^{\infty}\frac{x^{2n}}{(x)... |
83,988 | <p>How does one see that the second Stiefel-Whitney class is zero for all orientable surfaces. For $S^2$ this can be seen by $TS^2$ being stably trivial, and for $S^1 \times S^1$ one can use $T (S^1 \times S^1) = TS^1 \times TS^1$, which gives the class in terms of the classes on $TS^1$ (which are all trivial). What ... | Adam | 20,109 | <p>The second Stiefel-Whitney class of a surface is the mod 2 reduction of the Euler class. Since the Euler characteristic (and hence number) is divisible by 2, $w_2$ is zero.</p>
|
2,943,199 | <p>This is actually a doubt I got while solving <a href="https://www.askiitians.com/forums/Discuss-with-Askiitians-Tutors/please-prove-the-following-2cos-1-3-root-13_114688.htm" rel="nofollow noreferrer">this question</a>. The thing is I know how to convert <span class="math-container">$2\arctan(3/4)$</span> to <span c... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$M=(a_{ij})_{1\leqslant i\leqslant m,1\leqslant j\leqslant n}$</span>, I would write<span class="math-container">$$\bigl(\forall(k,l)\in(\{1,2,\ldots,n\}\times\{1,2,\ldots,n\})\setminus\{(i,j)\}\bigr):a_{kl}\neq0.$$</span>So, <span class="math-container">$a_{ij}$</span> <em>can</em> b... |
19,876 | <p>Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am draf... | cch | 4,229 | <p>Here's yet another method for your amusement.</p>
<p>Let $y = \cos^2(t)$. Then note that $y'(t) = 2\cos(t)\sin(t)$ and therefore
$$
y''(t) = 2\cos^2(t) - 2\sin^2(t) = 2\cos^2(t) - 2(1 - \cos^2(t)) = 4 y - 2.
$$
The general solution to this nonhomogeneous ode is
$$
y = C_1 \sin(2t) + C_2 \cos(2t) + ... |
3,506 | <p>I am wondering what is the correct function in Mathematica to plot the true impulse function, better known as the <code>DiracDelta[]</code> function. When using this inside of a function or just the function itself when plotting, it renders output = zero. Quick example:</p>
<pre><code>Plot[DiracDelta[x], {x,-1,1}]
... | image_doctor | 776 | <p>At your own risk, you can redefine the DiracDelta function to have the behaviour you want:</p>
<pre><code>Unprotect@DiracDelta;
DiracDelta[0] := your large number;
Protect@DiracDelta;
</code></pre>
<p>The integral given by </p>
<pre><code>Integrate[DiracDelta[x], {x, -Infinity, Infinity}]
</code></pre>
<p>still ... |
427,157 | <p>The following problem seems easy at a first glance but I can't see the way to prove it. Actually I don't even know if it's true but it is assumed implicitly in a research paper.</p>
<p>Help highly appreciated.</p>
<p>Given two spd matrices <span class="math-container">$A$</span>, <span class="math-container">$B$</sp... | gandalfbalrogslayer | 486,547 | <p>This is related to a class of matrices called Fisher matrices, in which a=b=c=d. There is work on CLTs for linear spectral statistics of these matrices, with the trace being a special case. I would not be surprised if this trace is asymptorically normal.</p>
|
4,304,209 | <p>Hi it's a follow up of <a href="https://math.stackexchange.com/questions/4268913/show-this-inequality-sqrt-fracabb-sqrt-fracbaa-ge-2/4269271#4269271">show this inequality $\sqrt{\frac{a^b}{b}}+\sqrt{\frac{b^a}{a}}\ge 2$</a>:</p>
<h2>Problem :</h2>
<p>Let <span class="math-container">$a,x>0$</span> then (dis)prove... | jvc | 686,748 | <p>I apologize because I am force to write it on phone. I hope it will be not to hard to read.</p>
<p>In fact, there is a powerful theorem that give conditions for which a "lacunary trigonometric serie" is nowhere differenriable. The proof can be found in the french book, Analyse pour l'agrégation, Zuily-Quéf... |
266,526 | <p>I have a set of quadratic forms.</p>
<p><span class="math-container">$L_{1}=u_1^TJ_{1}u_1$</span></p>
<p><span class="math-container">$L_{2}=u_2^TJ_{2}u_2$</span></p>
<p><span class="math-container">$L_{3}=u_3^TJ_{3}u_3$</span></p>
<p>where <span class="math-container">$u_{i=1,2,3}$</span> - 3<span class="math-conta... | Mauricio de Oliveira | 48,233 | <p>I am not sure if I understand the question correctly, but if what you want is to produce a block diagonal matrix from a list of <code>u</code>'s and a list of <code>J</code>'s then</p>
<pre><code>us = {u1, u2, u3};
Js = {J1, J2, J3};
DD = DiagonalMatrix[Inner[Dot, us, Inner[Dot, Js, us, List], List]]
</code></pre>
<... |
280,436 | <p>For example, if we evaluate this:</p>
<pre><code>BSplineFunction@{{0,100},{200,50},{200,0}}
</code></pre>
<p>we'll get</p>
<pre><code>BSplineFunction[1,
{{0., 1.}},
{2}, {False}, {{{0., 100.}, {200., 50.}, {200., 0.}}, Automatic},
{{0., 0., 0., 1., 1., 1.}},
{0}, MachinePrecision, "Unevaluated&q... | xzczd | 1,871 | <p>Mimicking the spelunking in</p>
<p><a href="https://mathematica.stackexchange.com/q/19042/1871">How to splice together several instances of InterpolatingFunction?</a></p>
<p>We find</p>
<pre><code>func = BSplineFunction[{{0, 100}, {200, 50}, {200, 0}, {300, 0}}];
lst = func@Methods
(* {"Closed", "Cont... |
3,166,359 | <p><a href="https://i.stack.imgur.com/KytwH.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KytwH.png" alt="enter image description here"></a></p>
<p>There is black semicircle, which radius is <span class="math-container">$ R $</span>. The red circle is tangentially inward to the semicircle and to the diamet... | dfnu | 480,425 | <p><strong>HINT</strong></p>
<p>Consider the Figure below and note that</p>
<ol>
<li><span class="math-container">$\overline{O_2H} = \frac{R}{2}-r$</span>;</li>
<li><span class="math-container">$\overline{O_2O_3} = \frac{R}{2} + r$</span>;</li>
<li><span class="math-container">$\overline{HO_1} = r$</span>;</li>
<li><... |
3,166,359 | <p><a href="https://i.stack.imgur.com/KytwH.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KytwH.png" alt="enter image description here"></a></p>
<p>There is black semicircle, which radius is <span class="math-container">$ R $</span>. The red circle is tangentially inward to the semicircle and to the diamet... | Dr. Mathva | 588,272 | <p><img src="https://i.stack.imgur.com/x6uyn.png" height="400"></p>
<p>Let the radius of the "black" semicircle be <span class="math-container">$[CK]=2R$</span>. Thus, the radius of the "red" circumference will be <span class="math-container">$[DC]=R$</span>. In virtue of the Pythagorean Theorem in the triangles <span... |
1,232,690 | <p>As we all knew that Aryabhata (<a href="http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero" rel="nofollow">http://en.wikipedia.org/wiki/Aryabhata#Place_value_system_and_zero</a>) invented zero ($0$) in our number system. I have few questions about it.</p>
<ol>
<li>How did the numeric system work bef... | Lucian | 93,448 | <blockquote>
<p><em>How did the numeric system work before the invention of "zero" ?</em></p>
</blockquote>
<p>They either used $9\cdot3=27$ letters of the alphabet to signify the numbers $1-9,~10-90,$ and $100-900,$ —like <a href="http://en.wikipedia.org/wiki/Greek_numerals" rel="nofollow">Greeks</a> and <a h... |
18,844 | <p>To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric is complete, and if a noncompact manifold has finite topological type(ie is diffeomorphic to the interior of a compa... | José Figueroa-O'Farrill | 394 | <p>This is not an answer to your question. I just wanted to point out that there are plenty of examples of complete Einstein manifolds of infinite topological type. I am aware of the following two papers at least:</p>
<ol>
<li><p><a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volu... |
18,844 | <p>To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric is complete, and if a noncompact manifold has finite topological type(ie is diffeomorphic to the interior of a compa... | Harald Hanche-Olsen | 802 | <p>It's not my field, so I am really out on a limb here. Maybe someone can tell my why the following naïve idea wouldn't work: Make a Riemannian metric by partition of unity. For any point $p$ in the manifold, let $h(p)$ be the infimum of the lengths of all paths starting at $p$ which are not contained in any compact s... |
2,433,705 | <p>Does $X\otimes X$ equal $(X\otimes I)(I\otimes X)$? (By the parentheses I mean to signify normal matrix multiplication.)</p>
<p>$X$ is any unitary matrix of the same dimensions as $I$.</p>
| Nick | 27,349 | <p>Yes. It doesn't even matter if $X$ is unitary. Use the definition of the tensor product of linear maps: if $f$ and $g$ are linear maps (i.e. matrices), and $v$ and $w$ are vectors, then</p>
<p>$$(f \otimes g)(v \otimes v) = f(v) \otimes g(w)$$ </p>
<p>Now compose and check the identity you are suggesting:</p>
<p>... |
325,018 | <p>If $n\in \Bbb N $ such that $\gcd(n,6)=1$ and $a_1,\ldots,a_{\phi( n)}$ are relatively prime with $n$ and smaller than $n$, how to prove : $$n\mid {a_1}^2+\cdots +{a}^2_{\phi (n)}$$</p>
| nbubis | 28,743 | <p>I'd like to start the other way round:</p>
<ul>
<li>Show that the lines $M_2M_3$ and $M_1Z$ bisect (by using the coordinates), thus the four points $M_1, M_2, M_3, Z$ lie on the same circle.</li>
<li>By symmetry, point $M$ lies on the circle centered at $I$.</li>
<li><em>Define</em> the point $T$ at the intersectio... |
309,851 | <p>I was considering the integral $\int_{-1}^1 \frac{1}{x^3}\, \mathrm{d} x$. At first, I suspected that it diverged due to the singularity present at $x = 0$, and WolframAlpha verified my hypothesis. However, I attempted to prove this more rigorously, but was unable. This was my reasoning:</p>
<p>$$\int_{-1}^1 \frac{... | Dominic Michaelis | 62,278 | <p>The Integral diverges, so it is not $0$. It is the same like saying
$$\int_{-\infty}^\infty x \, \mathrm{d}x $$
is not convergent. The idea is that you can find partitions where the riemann-sums have different limits.</p>
|
2,477,817 | <p>The first step in calculating the variance of a Binomial Random Variable is calculating the second moment. </p>
<p><strong><a href="https://i.stack.imgur.com/KsqSQ.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KsqSQ.png" alt="enter image description here"></a></strong></p>
<p>I have no idea as to how t... | Graham Kemp | 135,106 | <p>They distributed out constant factors from series so that a change of variables would make it the series of probabilities for a binomial expansion $(p+1-p)^{n-1}$. They did not show their work, thinking it would be obvious. Clearly it is not.</p>
<p>I'll just do one of the terms $${
\quad \sum_{j=0}^n... |
1,459,830 | <p>$$r>1$$</p>
<p>The following is applying gauss' law by explicit integration.</p>
<p>$$\int_0^{\pi } \frac{\sin (\theta )}{\sqrt{r^2-2 r \cos (\theta )+1}} \, d\theta=\frac{2}{r}$$</p>
<p>The following is finding the potential of a charged spherical shell by explicit integration. Mathematica takes forever to so... | Mark Viola | 218,419 | <p>We begin with the integral $I(r,\psi)$ given by</p>
<p>$$I(r,\psi)=\int_{0}^{2\pi}\int_{0}^{\pi}\frac{\sin \theta \cos \theta}{\sqrt{r^2-2r\left(\cos \theta \cos \psi+\sin \theta \sin \psi \cos \phi\right)+1}}\,d\theta\,d\phi$$</p>
<p>Using spherical harmonics, we can expand the denominator of the integrand as </p... |
912,370 | <p>To prove that a functions has partial derivatives every partial has to exist, and every partial exist only if the limit of definition of partial exist. Is this right?</p>
<p>Then if partials exist ,and the limit of derivability involving the Jacobian is equal to zero, (<a href="http://en.wikipedia.org/wiki/Differe... | studentNk | 166,775 | <p>If a function is discontinuous at some point then is no point talk about partials and hence derivative. A function can have a partials however they could not be define ate some points and hence discontinuous partials .however if at points where the partials could be discontinuos exist like $f^\prime_x =\lim_{h \to 0... |
838,759 | <p>It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where</p>
<p>$f(y, x) :=
\log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor y\rfloor)\right)}\right)$</p>
<p>is an interpolating function between $\log(x)\text{ and }\log(\log(x)).$</p>
<p>Comparing $\p... | Andrew | 34,377 | <p>The result and your argument (which is the first I would think of) are still valid for integral projective curves, but the theory is more delicate. For example, you can find the result as 7.3.2 Proposition 3.25(b) of <a href="http://books.google.ca/books?id=uaLKdA0PxS4C&dq=liu%20algebraic%20geometry&hl=en&am... |
626,034 | <p>When you first differentiate the above, you get $-8/25$, right? Then you derive the gradient for a normal and proceed so on and so forth.
The textbook I'm using says when you differentiate, you get $-16/25$. I believe that's wrong...</p>
| mathlove | 78,967 | <p>The textbook is correct. </p>
<p>You can use
$$\left\{\frac{f(x)}{g(x)}\right\}^\prime=\frac{f^\prime(x)g(x)-f(x)g^\prime(x)}{\{g(x)\}^2}.$$</p>
<p>Letting $$h(x)=\frac{8}{4+x^2},$$
we have
$$h^\prime(x)=\frac{0-8\cdot 2x}{(4+x^2)^2}=-\frac{16x}{(4+x^2)^2}.$$</p>
<p>Hence, we will have
$$h(1)=-\frac{16}{25}.$$<... |
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... | Soham Konar | 650,410 | <p>If you have the coordinates three points <span class="math-container">$A, B,$</span> and <span class="math-container">$C,$</span> you can find the perpendicular bisectors of <span class="math-container">$AB, AC,$</span> and <span class="math-container">$BC$</span> and then you can find the point of concurrency of th... |
1,863,493 | <p>I've been trying to solve this problem but I have an issue with the fact that there is a sum under each absolute value. I'm trying to convert this minimization problem (with respect to $x, y_1, \dots,y_n$)</p>
<p>$$
\sum_{i=1}^{m} \left| x - \sum_{j=1}^{n}\left| y_j - a_{ij} \right| \right|
$$</p>
<p>to a linear ... | Kuifje | 273,220 | <p>First of all, as pointed out by Johan Löfberg, this is not a convex function, so it is vain to search for a pure linear program. Therefore, the following <strong>does not work</strong> in the general case :</p>
<p><strong>Step 1:</strong>
$$
\sum_{i=1}^m z_i
$$
subject to
$$
x-\sum_{j=1}^n|y_j-a_{ij}| \le z_i\\
-x+... |
1,160,012 | <p>I'm working on a job interview test and there is one answer which I just don't get.</p>
<p>The test states that statement below is true. To me it just seems wrong. No box is provided to check. Then how do I check it correct or wrong? Am I missing something here?</p>
<p>66:4 = 161/2</p>
| k170 | 161,538 | <p>The statement is false because
$$ 66:4=\frac{66}{4}=\frac{33}{2}\not= \frac{161}{2} $$</p>
|
3,133,637 | <p>I am having a hard time with exercises of the form : <span class="math-container">$f'$</span> verify some properties then prove that <span class="math-container">$f$</span> is such that : ...</p>
<p>The main problem I have is that in order to link <span class="math-container">$f$</span> and it's derivative I only k... | Lázaro Albuquerque | 85,896 | <p>Let <span class="math-container">$[a, b] \subset \mathbb{R}^+$</span> and <span class="math-container">$f(x)=x-a$</span>. Then <span class="math-container">$f'(x)=1$</span>, hence <span class="math-container">$K=1$</span> but <span class="math-container">$f(a)=0 < K \cdot a=a$</span>.</p>
|
57,719 | <p>The following string can be converted easily into a list with <code>ToExpression</code></p>
<pre><code>string = "{{a},{b,c,d},{e,{f,{g}}}}";
ToExpression@string
</code></pre>
<p>However, if the string contains characters that can be misinterpreted as syntax errors, I run in to problems.</p>
<pre><code>string = "{... | Mr.Wizard | 121 | <p>Well I just saw your comment that says you want "all strings" so perhaps a different approach:</p>
<pre><code>StringReplace["{{a},{b,c,d},{e,{[f],{g}}}}",
x : Except["{" | "," | "}"] .. :> "\"" <> x <> "\""] // ToExpression
</code></pre>
<blockquote>
<pre><code>{{"a"}, {"b", "c", "d"}, {"e", {"[f... |
493,042 | <p>$−3x+5y+7z=7$<br>
$−3x-7y+kx=8$<br>
$15x+23y-19z=-40 $</p>
<p>by using echolon form I got to this</p>
<p>\begin{bmatrix}
-3 & -7 & k & 8 \\[0.3em]
0 & -12 & 5k-19 & 0 \\[0.3em]
0 & 0 & -4k + 12 & 1
\end{bmatrix}</p>
<p>but... | Bennett Gardiner | 78,722 | <p>Look at the last row in your reduced echelon form. We cannot have an equation that is of the form $0x+0y+0z=1$, the system would be <em>inconsistent</em>. So what value of $k$ would make this happen?</p>
|
493,042 | <p>$−3x+5y+7z=7$<br>
$−3x-7y+kx=8$<br>
$15x+23y-19z=-40 $</p>
<p>by using echolon form I got to this</p>
<p>\begin{bmatrix}
-3 & -7 & k & 8 \\[0.3em]
0 & -12 & 5k-19 & 0 \\[0.3em]
0 & 0 & -4k + 12 & 1
\end{bmatrix}</p>
<p>but... | Stefan4024 | 67,746 | <p>I think it would be wiser and easier to apply Cramer's rule to solve this problem. </p>
<p>Cramer's rule says: <strong>Iff $\Delta \neq 0$ then the system of linear equation is consistent</strong></p>
<p>So we just need to find $\Delta \neq 0$, where $\Delta$ is the determinant of the system. This system consists ... |
4,341,404 | <p>Choose a point randomly on the interval <span class="math-container">$[0, 1]$</span> and label it <span class="math-container">$X_1$</span>.</p>
<p>Then
choose a point randomly on the interval <span class="math-container">$[0, X_1]$</span> and label it <span class="math-container">$X_2$</span>.</p>
<p>Finally,
choos... | frabala | 53,208 | <p>I don't remember ever seeing a definition of a context to include a variable. A plain variable does not have a hole, after all. But, moving on...</p>
<p><span class="math-container">$F$</span> does not need to be well-scoped.
So, for the base case where <span class="math-container">$C=y$</span>, you can define <span... |
763,295 | <p>Show that there is no subgroup of $\mathbb{Z}_4$ containing only 3 elements.</p>
<p>I couldn't solve why 3 elements cannot exist.</p>
<p>0 and 2 are the only subgroup of $\mathbb{Z}_4$ with 2 elements. But 3 elements??</p>
| tomocafe | 144,714 | <p>I assume you mean a <em>subgroup</em> of $\mathbb{Z}_4$ containing 3 elements.</p>
<p>Lagrange's Theorem: "The order of a subgroup $H$ of group $G$ divides the order of $G$."</p>
<p>3 does not divide 4, so there cannot be a subgroup of $\mathbb{Z}_4$ containing 3 elements.</p>
|
2,073,965 | <p>Consider the function $$ f:\mathbb{R}\backslash\{0\}\rightarrow \mathbb{R}, x \mapsto \frac{e^x -1}{x}$$</p>
<p>Now consider this inequality that came up in the textbook when talking about how to continue that function at $x=0$.</p>
<p>$$ |x| \sum\limits_{k=2}^{\infty}\frac{|x|^{k-2}}{k!} \leq
|x| \sum\limits_{k=2... | Olivier Oloa | 118,798 | <p>Here we are considering that $x \to 0$, we may assume $|x|\le1$. Then
$$
|x|^{k-2}\le 1
$$ giving
$$
|x| \sum\limits_{k=2}^{\infty}\frac{|x|^{k-2}}{k!} \leq
|x| \sum\limits_{k=2}^{\infty}\frac{1}{k!}.
$$
Taking $x=10$ is not appropriate.</p>
|
2,073,965 | <p>Consider the function $$ f:\mathbb{R}\backslash\{0\}\rightarrow \mathbb{R}, x \mapsto \frac{e^x -1}{x}$$</p>
<p>Now consider this inequality that came up in the textbook when talking about how to continue that function at $x=0$.</p>
<p>$$ |x| \sum\limits_{k=2}^{\infty}\frac{|x|^{k-2}}{k!} \leq
|x| \sum\limits_{k=2... | ajotatxe | 132,456 | <p>One must assume that $|x|\le1$. Since you are dealing with a neighbourhood of $0$, it can be done.</p>
<p>But the book should still have made the assumption explicit.</p>
|
2,075,238 | <p>Am I correct in assuming, when dealing with Lebesgue integrals on the Cartesian space, that we adopt the notation $\int_{a}^{b} f(x)dx$ where the notation $dx$ is used to denote the Lebesgue measure? However, this notation is identical to the notation of a standard Riemann integral. So, when faced with this sort of ... | Community | -1 | <p>It doesn't matter which procedure you use because when the Riemann integral exists it is equivalent to the Lebesgue integral. If the Riemann integral doesn't exist it is standard to use the $d\mu$ notation however I have seen the $dx$ notation used so specify the measure although it is generally made clear in the co... |
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>
| Archis Welankar | 275,884 | <p>An easier way to solve is that when we have two values of tan in multiplication and their angles sum up 90 then their multiplication is always 1 eg tan70tan20=1 so answer is 1.</p>
|
104,210 | <p>I have this code to plot contours:</p>
<pre><code>ContourPlot[(Cos[θ] Cos[ϕ])^(1/4), {θ, -π/2, π/2}, {ϕ, -π/2, π/2}, AxesLabel -> Automatic]
</code></pre>
<p>How would I map those contours on a unit sphere (if it is even possible) where <code>θ</code> and <code>ϕ</code> are the spherical angles for the sphere (... | Basheer Algohi | 13,548 | <p>In my opinion the easy way to plot vectors over 1D curve is to used <code>VectorPoints</code> option:</p>
<pre><code>points = Table[{i, 1}, {i, 0, 1, .1}];
VectorPlot[{Sin[x], Cos[y]}, {x, 0, 1}, {y, -1, 2},
VectorPoints -> points, VectorScale -> {0.1, .2},
Epilog -> Point[points]]
</code></pre>
|
3,279,492 | <p>It is known that for a surface <span class="math-container">$S \subset \mathbb{R^3}$</span> it can be found the first and the second fundamental form. </p>
<p>I would like to find out if this "first and second fundamental form" can be extended for a surface <span class="math-container">$S \subset \mathbb{R^{n}}$</s... | robjohn | 13,854 | <p>By definition of <a href="https://en.wikipedia.org/wiki/Floor_and_ceiling_functions" rel="nofollow noreferrer">floor</a>,
<span class="math-container">$$
\sqrt{N}-1\lt\left\lfloor\sqrt{N}\right\rfloor\le\sqrt{N}\tag1
$$</span>
Therefore,
<span class="math-container">$$
\begin{align}
1\le\frac{N}{\left\lfloor\sqrt{N}... |
3,279,492 | <p>It is known that for a surface <span class="math-container">$S \subset \mathbb{R^3}$</span> it can be found the first and the second fundamental form. </p>
<p>I would like to find out if this "first and second fundamental form" can be extended for a surface <span class="math-container">$S \subset \mathbb{R^{n}}$</s... | Community | -1 | <p>From</p>
<p><span class="math-container">$$\left\lfloor\sqrt{N}\right\rfloor\le\sqrt N<\left\lfloor\sqrt{N}\right\rfloor+1,$$</span>
we have
<span class="math-container">$$1\le\frac N{\left\lfloor\sqrt{N}\right\rfloor^2}<\left(\frac{\left\lfloor\sqrt{N}\right\rfloor+1}{\left\lfloor\sqrt{N}\right\rfloor}\right... |
1,540,069 | <p>If $\lim \limits _{x \to x_0} (f(x) + g(x))$ exists, can I write: $\lim \limits _{x \to x_0} (f(x) + g(x)) = \lim \limits _{x \to x_0} f(x) + \lim \limits _{x \to x_0} g(x)$ ? <br>I mean, to write this do I have to know that the other limits exist? Because they tell me that $\lim \limits _{x \to x_0} f(x)$ exists an... | Ogoine | 1,032,628 | <p>I found this <a href="https://samuelj.li/complex-function-plotter/" rel="nofollow noreferrer">Complex Function Plotter</a> to be indispensable in helping me visualize the general physical transformation applied to the complex plane by <span class="math-container">$f(x)=x^n$</span>. Basically, it expands AND rotates ... |
2,137,801 | <blockquote>
<p>Evaluate:
$$\int \frac{e^x}{e^{2x} + 3e^x + 2} dx$$</p>
</blockquote>
<p><em>My solution</em>: Let $u = e^x$, then $$\int \frac{u}{u^2+3u+2} du=\dots$$</p>
<p>and $\frac{u}{(u+1)(u+2)} = \frac{A}{u+2} + \frac{B}{u+1}$ with $A = 2, B = -1$. So,
$$\dots=-2 \ln |u + 2| - \ln |u+1| + C$$ resubstitute... | Brevan Ellefsen | 269,764 | <p>Tangent is periodic with period $\pi$, so we have
$$\tan(a)=\tan(b) \iff b=n\pi+a$$
In this case we find that $2x = n\pi + x \implies x=n\pi$</p>
|
3,897,110 | <p>If <span class="math-container">$f'(x)=\sqrt{x^3+1}$</span> for all <span class="math-container">$x>0$</span> and <span class="math-container">$f(2)=10$</span>, then <span class="math-container">$f(5) > 16$</span>.</p>
<p>I have what is below so far, but I am not sure how to show f(5) > 16.</p>
<p>Assume <s... | user21820 | 21,820 | <p>In foundations of mathematics, especially set-theoretic foundations, a function is nothing more than a certain kind of set of ordered pairs, from which you can easily extract its domain and its range, and there is no such thing as a codomain of a function. Two functions are equal exactly when they are the same set, ... |
167,846 | <p>The question is the following:</p>
<p>Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: </p>
<p>When do we have $\widehat{HF}(Y,\mathfrak{s})\cong \mathbb{Z}$? If yes, what do we know about the d-... | Ryan Budney | 1,465 | <p>The paper of Manolescu and Owens "A concordance invariant from the floer homology of double branched covers" seems to answer your question. They compute many examples of the Froyshov/d-invariant in their paper, using software of Saso Strle. It's available on the arXiv. In that case it's at least reasonably-comput... |
1,744,858 | <blockquote>
<p>Let $$A=53\cdot 83\cdot109+40\cdot66\cdot96$$
Is this number prime or composite?</p>
</blockquote>
<p>I'm sure it's a composite number. But I do not know how to prove it.</p>
| marwalix | 441 | <p>Let's factor further $A=149\times 4919$. So $A$ is composite.</p>
|
1,744,858 | <blockquote>
<p>Let $$A=53\cdot 83\cdot109+40\cdot66\cdot96$$
Is this number prime or composite?</p>
</blockquote>
<p>I'm sure it's a composite number. But I do not know how to prove it.</p>
| mathlove | 78,967 | <p>$$abc+(149-c)(149-b)(149-a)=149 (a b+bc+ca-149 a-149 b-149 c+22201)
$$</p>
|
1,776,931 | <p>I need to find the sum of an alternating series correct to 4 decimal places. The series I am working with is: $$\sum_{n=1}^\infty \frac{(-1)^n}{4^nn!}$$ So far I have started by setting up the inequality: $$\frac{1}{4^nn!}<.0001$$Eventually I arrived at $$n=6$$ giving the correct approximation, which is approxima... | Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One may recall that
$$e^x=\sum_{n=\color{red}{0}}^\infty \frac{x^n}{n!}, \quad x \in \mathbb{R}.$$</p>
|
1,776,931 | <p>I need to find the sum of an alternating series correct to 4 decimal places. The series I am working with is: $$\sum_{n=1}^\infty \frac{(-1)^n}{4^nn!}$$ So far I have started by setting up the inequality: $$\frac{1}{4^nn!}<.0001$$Eventually I arrived at $$n=6$$ giving the correct approximation, which is approxima... | Bernard | 202,857 | <p>The sum of this series is known, since it is the expansion of $\;\mathrm e^{-x}-1\;$ for $x=\frac14$.</p>
|
13,582 | <p>Question: Let $a, b \in \mathbb{R}$ with $a < b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.</p>
<p>I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $... | Arturios | 86,275 | <p>If <span class="math-container">$f(0) = 0$</span> we are done. If the former is not true then, since <span class="math-container">$f: [0,1] \to [0,1]$</span>, it must be that <span class="math-container">$f(0) > 0$</span>. Applying the same reasoning we conclude that <span class="math-container">$f(1) = 1$</span>... |
4,404,228 | <p>Problem:<br />
Let <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> be integers such that <span class="math-container">$x_1 \geq 0$</span>, <span class="math-container">$x_2 \geq 0$</span> and
<span class="math-container">$x_3 \geq 0$<... | Lorago | 883,088 | <p>Your solution seems correct, but a quicker way would be the following:</p>
<p>As you observed, <span class="math-container">$x_1$</span> has to be even, so the problem will have as many solutions as the number of solutions to the problem of</p>
<p><span class="math-container">$$2x_1+2x_2+2x_3=200,$$</span></p>
<p>i.... |
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