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 |
|---|---|---|---|---|
853,143 | <p>Let $J$ be a canonical Jordan form (real or complex). Is it true that the $2$-norm of $J$ is equal to its spectral radius?</p>
| ajotatxe | 132,456 | <p>$n!$ is a multiple of $k+1$ since $k+1$ is one of the factors involved. Now, if you add $k+1$ to a multiple of $k+1$, you obtain another multiple of $k+1$.</p>
|
2,929,182 | <p>Let <span class="math-container">$p,q>1$</span>, <span class="math-container">$\frac{1}{p}+\frac{1}{q}=1$</span> and <span class="math-container">$x\in[0,1]$</span>. I would like to show that <span class="math-container">$x^{\tfrac{1}{p}}\leq \frac{1}{p}x+\frac{1}{q}$</span>.</p>
<p>I tried to rewrite both sides... | Migos | 591,409 | <p><a href="https://en.wikipedia.org/wiki/Bernoulli%27s_inequality" rel="nofollow noreferrer">Bernoulli's inequality:</a> <span class="math-container">$(1+x)^r \leqslant 1+rx$</span> for <span class="math-container">$x \geqslant -1$</span> and <span class="math-container">$r \in [0,1]$</span>. </p>
<p>Therefore: <span... |
499,218 | <p>I've been reading the books "An introduction to knot theory" by Lickorish and "Knots, Links, Braids and 3-Manifolds" by Prosolov and Sossinsky, and while both seem to me as good books, sometimes I'd like to get a different perspective on certain topics.</p>
<p>I would be glad to get some recommendations on books de... | Avitus | 80,800 | <p>Have a look at <a href="http://books.google.ch/books/about/Quantum_Groups.html?id=S1KE_pToY98C&redir_esc=y" rel="nofollow">Quantum Groups</a> by Kassel: it contains, among others, modern algebraic structures (like tensor categories with additional data) related to knots, braidings and links. This is the content ... |
4,497,033 | <p>Let <span class="math-container">$r(t)$</span> be the function:<br />
<span class="math-container">$r(t) = \sqrt{x(t)^2 + y(t)^2}$</span>, where<br />
<span class="math-container">$x(t) = 3b (1 − t)^2 t + 3c (1 − t) t^2 + a t^3$</span>, and<br />
<span class="math-container">$y(t) = a (1 − t)^3 + 3c (1 − t)^2 t + 3b... | Jeffrey Harkness | 1,068,307 | <p>7/22/2022 Update:</p>
<p>I rewrote this algorithm to achieve more significant figures without further alteration. However, it likely takes at least several minutes on a typical computer, and I wouldn't claim that it achieves all 9-12 significant figures sought (4-5 is a much more reasonable claim). I wasn't aware ... |
3,176,434 | <p>I was going through some of the earlier answers for the license plate problems and one of the comment was as follows.</p>
<blockquote>
<p>Does a plate consist of 3 letters and 3 digits in any order, like
7C99XK, or is it 3 letters followed* by 3 digits? The answers will be
different.</p>
</blockquote>
<p>I u... | user | 293,846 | <p>The probability of a certain combination out of those consisting of 3 letters and 3 digits in any order is
<span class="math-container">$$
\frac{3!3!}{6!}\frac1{26^3 10^3},
$$</span>
since there are <span class="math-container">$\frac{6!}{3!3!}$</span> ways to choose a specific order of letters and digits.</p>
|
2,370,851 | <p>How to solve the Integral $\int{\frac{1}{\sqrt{1-\sqrt{4-x}}}}$dx with steps</p>
<p>I have tried to make the substitution $\frac{du}{dx}=4-x$ but it seems that there is no continuation road.</p>
<p>I have seen that there is a substitutions that gives the following results.</p>
<p>$\int{\frac{1}{2\sqrt{-u}} \fr... | Raffaele | 83,382 | <p>Substitute $u=\dfrac{1}{\sqrt{1-\sqrt{4-x}}}$</p>
<p>$x=\dfrac{3 u^4+2 u^2-1}{u^4}$</p>
<p>$dx=\dfrac{4-4 u^2}{u^5}$</p>
<p>and the integral becomes
$$\int \frac{4-4 u^2}{u^4} \, du=4 \int \left(\frac{1}{u^4}-\frac{1}{u^2}\right) \, du=4 \left(\frac{1}{u}-\frac{1}{3 u^3}\right)+C$$</p>
<p>$$\frac{4}{3} \sqrt{1-\... |
1,190,320 | <p>I need to evaluate $$\lim_{x \to \infty}x\int_0^{1/x}e^t \cos(t) \space \text{d}t$$ and I'm not really sure how to start. Do I have to find the integral or is there another way to figure it out?</p>
| kobe | 190,421 | <p>Substituting $u = 1/x$, your limit is the same as</p>
<p>$$\lim_{u\to 0} \frac{\int_0^u e^t\cos t\, dt}{u},$$</p>
<p>which by L'hospital's rule and fundamental theorem of calculus is</p>
<p>$$\lim_{u\to 0} \frac{e^u \cos u}{1} = 1.$$</p>
|
581,605 | <p>Source: Miklos Bona, A Walk Through Combinatorics.</p>
<p>$$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$</p>
<p>The RHS is the upper bound of the Ramsey number $R(k,k)$.</p>
<p>How can I prove the inequality without using mathematical induction? I've merely expanded the LHS to obtain $\frac{(2k-2)!}{(k-1)!(k-... | Lucian | 93,448 | <p><em>Hint</em> : $\displaystyle f(x)=\sum\frac{x^n}{(n+1)!}\iff f'(x)=\sum\frac{n\cdot x^{n-1}}{(n+1)!}\quad,\quad S=f'(1)$.</p>
|
165,492 | <p>Sorry for my English if there is any mistake. The exercice for which I need help is the following:</p>
<p>Compute using complex methods:
$I=\int_1 ^\infty \frac{\mathrm{d}x}{x^2+1}$</p>
<p>i) Choose the complex function to integrate.</p>
<p>I guess it is $f(z)=1/(z^2+1)$</p>
<p>ii) Choose the contour.</p>
<p>I ... | DonAntonio | 31,254 | <p>What about taking the semicircle
$$\Gamma:=[-R,R]\cup\gamma_R:=\{z\;\;:\;\;|z|= R\,\,,\,0<\arg z< \pi\}\,\,,\,R>1$$</p>
<p>Note that within this closed path there's only one singularity of the function, the simple pole $\,z=i\,$, and the residue of $\,\displaystyle{f(z)=\frac{1}{1+z^2}}\,$ here is</p>
<p>... |
4,911 | <p>$$
\frac{\sqrt{3(m-n)^2 n^2}}{2}
$$
This expression is getting correctly rendered here on meta. I copied and pasted it from math.stackexchange.com. There, the horizontal line did not extend far enough in either direction, going to just above the middle of the $n$ on the right and failing to cover the $3$ on the le... | Community | -1 | <p>Probably related: I often see long square root signs rendered in three pieces, as below. <a href="https://math.stackexchange.com/questions/182869/finding-volume-and-centroid">Source</a>. Chrome 21/Win7.</p>
<p><img src="https://i.stack.imgur.com/EfzW4.png" alt="Broken pieces"></p>
<p>But in the question pointed ou... |
97,920 | <p>Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n-1)/2}$
for all $a>0$ and $0\lt q\lt 1$ ?</p>
| Will Sawin | 18,060 | <p>There will not be a closed form for this without some special function. The reason is that there would then be a closed form for the Jacobi theta function, without special functions.</p>
<p>Let your function be $F(a,q)$, then $F(e^{2\pi i z+\pi i \tau},e^{2\pi i \tau})$ is $\sum_{n=0}^\infty e^{\pi i n^2 \tau+2\pi ... |
186,830 | <p>Let $x_1, \dots, x_k \in \mathbb{R}^n$ be distinct points and let $A$ be the matrix defined by $A_{ij} = d(x_i, x_j)$, where $d$ is the Euclidean distance. Is $A$ always nonsingular?</p>
<p>I have a feeling this should be well known (or, at least a reference should exists), on the other hand, this fact fails for ge... | David E Speyer | 448 | <p>This is true. Over in the comments of a <a href="https://math.stackexchange.com/questions/3001258">related question</a>, darij grinberg has just shown that <span class="math-container">$A$</span> is negative definite when restricted to the codimension one subspace where the coordinates sum to <span class="math-conta... |
395,994 | <p>Suppose that $f_{x,y}(x,y) = \lambda^2 e^{\displaystyle-\lambda(x+y)}, 0\leq x , 0\leq y.$ Find $\operatorname{Var(X+Y)}$. </p>
<p>I'm having trouble with this problem the way to find $\operatorname{Var(X+Y)} = \operatorname{Var(X)}+\operatorname{Var(Y)}+2\operatorname{Cov(X,Y)}$, however if $X$ and $Y$ are indepen... | response | 76,635 | <p>You need to find the marginal densities and show that the joint is the product of the marginals in order to show that they are independent.</p>
|
2,421,896 | <p>I have the following integral to find:</p>
<p>$$\int 12x^2(3+2x)^5 dx$$</p>
<p>Now, I am aware of the integration by parts property - </p>
<p>$$\int \ u \frac{dv}{dx} = uv - \int v\frac{du}{dx}$$</p>
<p>Now, my question is the following - </p>
<p>When I make $u = 12x^2$, I find a different answer to when I make... | Nosrati | 108,128 | <p>For solving integrals like this, with two small power and large power, we must exchange parenthesis by substitution. For instance I want to solve the integral
$$I=\int 12x^2(3+2x)^{50} dx$$
which second has power $50$. With substitution $3+2x=u$ and $2dx=du$, the integral will simplify to
$$I=\int 12\left(\frac{u-3}... |
2,421,896 | <p>I have the following integral to find:</p>
<p>$$\int 12x^2(3+2x)^5 dx$$</p>
<p>Now, I am aware of the integration by parts property - </p>
<p>$$\int \ u \frac{dv}{dx} = uv - \int v\frac{du}{dx}$$</p>
<p>Now, my question is the following - </p>
<p>When I make $u = 12x^2$, I find a different answer to when I make... | Jim H | 473,669 | <p>The answers to an integration may appear different without actually being different. For example:</p>
<p>$\int(x+2)^3\,dx$ by substitution, $\frac{1}{4}(x+2)^4 + C$.</p>
<p>By expanding first,</p>
<p>$\int(x+2)^3\,dx = \int(x^3+6x^2+12x+8)\,dx = \frac{x^4}{4}+2x^3 + 6x^2+8x+C$</p>
<p>These answers appear differ... |
244,309 | <p>I have some data as</p>
<pre><code>data={{257.3`, 493.7`}, {43.666666666666664`,490.5`}, {111.91176470588235`,461.20588235294116`},{345.2142857142857`,460.5`}, {420.88461538461536`, 436.34615384615387`}, {318.1`,408.46`}, {277.`,400.7`}, {273.5`, 383.`}, {444.`,381.5`}, {208.28571428571428`,379.7857142857143`}, {510... | kglr | 125 | <pre><code>ClearAll[g, f, x, y, z, inputlist]
f[x_, y_, z_] := x y + z^2 + y z;
g[{x_, y_, z_}] := {f[x, y, z], f[x, y, z]^2, f[x, y, z]^3}
</code></pre>
<p>Using a smaller range for the three variables:</p>
<pre><code>ranges = {xrange, yrange, zrange} = {Range[3], Range[3], Range[2, 4] .05};
</code></pre>
<p>Take 2-... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | Kaligule | 182,303 | <p>You always divide by the total number of votes, so in the second case you get $\frac{15}{5}=3$, like you expected.</p>
<p>In your first example I get $\frac{92}{28}=3.29$.</p>
|
4,345,516 | <p>Let's say we have the polynomial, <span class="math-container">$x^{4}+x^{3}+x^{2}+x$</span>. It's derivative is, <span class="math-container">$4x^{3}+3x^{2}+2x+1$</span>. The solutions to the original polynomial are, <span class="math-container">$-1$</span>, <span class="math-container">$0$</span>, <span class="math... | egreg | 62,967 | <p>The division
<span class="math-container">$$
P(x)=P'(x)Q(x)+R(x)
$$</span>
may yield a nonzero remainder, but it's not a real problem. As an easy example, the derivative of <span class="math-container">$x^2+1$</span> is <span class="math-container">$2x$</span> and we cannot do exact division.</p>
<p>The correct stat... |
3,688,208 | <p>Let <span class="math-container">$f: ]0,1[ \to \mathbb{R}$</span> be a function. Suppose that for every sequence <span class="math-container">$(\epsilon_n)_n$</span> in <span class="math-container">$]0,1[$</span> with <span class="math-container">$\epsilon_n \searrow 0$</span> we have that <span class="math-containe... | o-ccah | 788,245 | <p>By weighted AM-GM inequality, we get
<span class="math-container">\begin{align*}
\lvert x \rvert^9 + \lvert y \rvert^{11}
&= \frac{11}{20} \cdot \frac{20}{11} \lvert x \rvert^9 + \frac{9}{20} \cdot \frac{20}{9} \lvert y \rvert^{11} \\
&\geq \left(\frac{20}{11} \lvert x \rvert^9\right)^{11/20} \left(\frac{20}... |
1,188,196 | <p>This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at it.</p>
<p>Considering the image below, I am trying to work out a set of formulas that will specify either the radi... | bubba | 31,744 | <p>In general, what you have described is known as a <em>geometric constraint-solving</em> problem. These sorts of problems have been studied quite a bit over the last 20 years or so; a good survey of the available techniques is <a href="https://www.cs.purdue.edu/homes/cmh/distribution/PapersChron/ConstraintSurvey2010.... |
845,499 | <p>(a)Given that$$ f(7)=13$$ and$$ f′(7)=−0.38$$, estimate f(7.1).</p>
<p>My answer was$$ f(6.1)= 13+ -0.38(x-7)$$ = 13.342.</p>
<p>(b)Suppose also $$ f′′(x)<0 $$for all $x$. Does this make your answer to part (a) an under- or overestimate?</p>
<p>My assumption is: $$ f′′(x)<0 $$ this means that it is concave ... | EuYu | 9,246 | <p>Let $A$ denote the adjacency matrix for the graph. We order the vertices in such a way that we may write
$$A = \begin{pmatrix} 0 & B\\B^\mathrm{T} & 0\end{pmatrix}$$
Note that a decomposition like this is possible for any bipartite graph. The matrix $B$ represents the incidences of $X$ in $Y$. Note that by ... |
845,499 | <p>(a)Given that$$ f(7)=13$$ and$$ f′(7)=−0.38$$, estimate f(7.1).</p>
<p>My answer was$$ f(6.1)= 13+ -0.38(x-7)$$ = 13.342.</p>
<p>(b)Suppose also $$ f′′(x)<0 $$for all $x$. Does this make your answer to part (a) an under- or overestimate?</p>
<p>My assumption is: $$ f′′(x)<0 $$ this means that it is concave ... | pointer | 121,270 | <p>As I promised here is the solution.</p>
<p>Let's $A=(a_{ij})_{i,j=1}^n$ be the following matrix: $a_{ij}=1,$ if $\{v_i,u_j\}$ is edge of graph $G$ and $a_{ij}=0,$ otherwise. From the conditions it follows that $AA^T=dB+(k-d)I,$ where $B$ is a matrix with all entries equal $1$. It is easy to check (by finding determ... |
1,230,037 | <p>For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$.</p>
<p>Any idea to solving it?</p>
| Derek Holt | 2,820 | <p>Let $H \le G$ with $|G:H|=n$ and $G \cong A_n$. The action of $G$ by multiplication on the cosets of $H$ gives a homomorphism $\phi: G \to S_n$. For $n \ge 5$ the simplicity of $G$ implies that $\phi$ is injective. So $|{\rm Im}(\phi)| = n!/2$ and hence ${\rm Im}(\phi) = A_n$. Now $\phi(H)$ is a point stabilizer in... |
4,019,754 | <p>I have no idea how to approach this?</p>
<p><a href="https://i.stack.imgur.com/XHH84.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XHH84.png" alt="enter image description here" /></a></p>
| alepopoulo110 | 351,240 | <p>Of course not, the question fails trivially in this generality: take any Hilbert space <span class="math-container">$H$</span> and let <span class="math-container">$H_1=H_2=H$</span>. Let <span class="math-container">$X,Y$</span> be closed subspaces with <span class="math-container">$X\subset Y$</span> and <span cl... |
864,568 | <p>I am trying to figure out how to take the modulo of a fraction. </p>
<p>For example: 1/2 mod 3. </p>
<p>When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?</p>
| Reginald Carl Jackman | 606,493 | <p>Finding the inverse of p^n mod q where p,q are relatively-prime primes using Fermat’s little equation,
p^(-n) mod q = p^x mod q where x = (-n) mod (q-1).
For example, 3^(-1) mod 5 = 3^[(5-1)-1] mod 5 = 3^3 mod 5 = 2.
Another example, 7^(-1) mod 13 = 7^11 mod 13 = 2 (done by calculator).
The formula is useful when on... |
2,205,087 | <p>Could you please give me an intuitive explanation why the dot product is defined this way?</p>
| 5xum | 112,884 | <p>Short answer: yes.</p>
<hr>
<p>Long answer: yes, by definition. If $f$ and $g$ are functions, then $f+g$ is, by <em>definition</em>, equal to the function that maps $x$ to $g(x)+f(x)$. It's the same for subtraction, multiplication and division.</p>
<p>The story is a little different with composition (which is a m... |
1,703,491 | <p>I have the following problem:
$$\int\left(\frac{x+2}{x^2+x+1}\right)dx$$
I received this by simplification of another integral. But my question is how to procede from here. Is there a way to simplify this?</p>
| pancini | 252,495 | <p>Partial fractions doesnt work here since $x^2+x+1$ is irreducible. Try separating into</p>
<p>$$\frac{x}{x^2+x+1}+\frac{2}{x^2+x+1}$$</p>
<p>The left is a simple $u$ substitution and the right looks like arctan if you complete the square in the denominator.</p>
|
2,546,222 | <blockquote>
<p>If $p(x)$ is an irreducible polynomial of degree n in $F[x]$ then $F[x]/\langle p(x)\rangle \cong F(c)$ where c is a root of $p(x)$. Prove every element of $F(c)$ can be written uniquely as $a_0+a_1c+...+a_{n-}c^{n-1}$ for some $a_0,...,a_{n-1}\in F$.</p>
</blockquote>
<p>So to be unique there is onl... | Naive | 135,775 | <p><strong>Hint:</strong> Show that $F(c)$ is an $n$ dimensional vector space over $F$ with $\{1,c,c^2,...,c^{n-1}\} $ as a basis. So by basic linear algebra every element in $F(c)$ can be expressed uniquely as a linear combination of the basis elements.</p>
|
1,029,489 | <p>I am studying the book "introduction to set theory", by Donald Monk, and I am having difficulties to solve some exercises about proper classes, could anybody help me?</p>
<p>here they are:</p>
<p>Prove that:
there are proper classes A, B such that $A \cap B= 0$<br>
there are proper classes A, B such that $A \subs... | Milly | 182,459 | <p>A more general result is ($x,y\geq 0$, $p\geq 1$)
$$(x+y)^p \leq 2^{p-1} (x^p+y^p),$$
which is direct consequence of convexity of $t\mapsto t^p$.</p>
|
4,147,900 | <p>I have a proof by contradiction to a simple problem but I have an issue understanding one aspect of it. I labeled it in the picture. Any insight would be helpful. Thank you for the time. <a href="https://i.stack.imgur.com/HXtPh.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HXtPh.jpg" alt="enter ... | Ash | 645,756 | <p><span class="math-container">$B \setminus C$</span> is the set of all elements in <span class="math-container">$B$</span> and not in <span class="math-container">$C$</span>. We have that <span class="math-container">$a \in B$</span>. Supposing <span class="math-container">$a \notin C$</span> means that <span class="... |
2,990,614 | <p>Let <span class="math-container">$g: \mathbb{R^2} \to \mathbb{R}$</span>.</p>
<p><a href="https://i.stack.imgur.com/kIi5N.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/kIi5N.png" alt="Function g"></a></p>
<p><strong>How can I prove that <span class="math-container">$g$</span> is continuous in... | user | 505,767 | <p>By polar coordinates as <span class="math-container">$r \to 0$</span></p>
<p><span class="math-container">$$\frac{xy}{\sqrt{x^2 + y^2}}=r \cos \theta \sin \theta$$</span></p>
<p>we see that <span class="math-container">$f(x,y)$</span> is continuous but since <span class="math-container">$f_x(0,0)=f_y(0,0)=0$</span... |
1,316,861 | <p>Please help me to solve this question:</p>
<p>Let $H$ be a hyperelliptic curve over $\mathbb{F}_{103}$ given by the equation $ y^2 = x^5+1$. let $J$ be the jacobian of $H$ defined over $\mathbb{F}_{103}$. Show that # H($\mathbb{F}_{103^2}$) = # J($\mathbb{F}_{103}$).</p>
| Álvaro Lozano-Robledo | 14,699 | <p><a href="http://magma.maths.usyd.edu.au/calc/" rel="nofollow">Magma</a> says that $\# H(\mathbb{F}_{103})=104$ and $\# J(\mathbb{F}_{103})=10610$.</p>
<p>Here is the code I used:</p>
<pre><code>P<x> := PolynomialRing(GF(103));
C := HyperellipticCurve(x^5+1);
#C;
J:=Jacobian(C);
#J;
</code></pre>
|
177,102 | <p>I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real solutions. The equations were derived as the gradients of a sum-of-squares cost function, which I am attempting to find a... | Thomas Kahle | 5,495 | <p>To solve a polynomial system, I would try <a href="https://bertini.nd.edu/" rel="nofollow">Bertini</a> which is a homotopy-continuation numerical solver that parallelizes extremely well. You can also try to attack the optimization problem directly with semi-definite programming as explained by Dima Pasechnik.</p>
|
3,674,370 | <blockquote>
<p>The vertices <span class="math-container">$B$</span> and <span class="math-container">$C$</span> of a <span class="math-container">$\triangle ABC$</span> lie on the line, <span class="math-container">$\frac{x-2}{3}=\frac{y–1}{0}=\frac{z}{4}$</span> such that <span class="math-container">$BC = 5$</span... | user | 293,846 | <p>Assuming that you can measure velocity: just keep the angular deviation per time unit by the value:
<span class="math-container">$$
\omega=\frac vr,
$$</span>
where the angle is assumed to be measured in radians.</p>
|
2,881,488 | <p>Let $M$ be a metric space, $x_n\in M$ a sequence which converges to $x\in M$</p>
<p>Prove: $F=\{x_n\}\cup \{x\}$ is a closed set</p>
<p>So we have $x_n\to x$ such that $x_n\in F$ and $x\in F$ and we know that a set is closed if it contains all of its accumulation points, so $F$ is closed</p>
<p>Or must I look at ... | Joe | 524,659 | <p><strong>Here's a direct proof without considering the compliment of the set. This proof also works for an arbitrary Hausdorff space -- not just a metric space.</strong> Suppose $\{ y_n \}$ is a sequence in $F$ and $y_n \rightarrow y \in M$. If $ \{ y_n \} $ is eventually constant, we're done. So suppose that the seq... |
25,263 | <p>Can you efficiently parallelize this? The parallel versions are much slower than the sequential version, and I'm not sure why. Does <code>SetSharedVariable</code> allow simultaneous reads for different kernels? It appears that it doesn't even though the documentation says you should use <code>CriticalSection</code> ... | Tobias Hagge | 5,523 | <p>This non-answer (addressing Szabolcs' observation) won't fit in a comment.</p>
<p>The following makes me suspect that the problem has something to do with how the parallel kernels are representing <code>data</code> internally:</p>
<pre><code>LaunchKernels[];
datatest = Table[rule[x, x], {2}];
data = Table[Rule[x,... |
25,263 | <p>Can you efficiently parallelize this? The parallel versions are much slower than the sequential version, and I'm not sure why. Does <code>SetSharedVariable</code> allow simultaneous reads for different kernels? It appears that it doesn't even though the documentation says you should use <code>CriticalSection</code> ... | Michael E2 | 4,999 | <p>Here is a way in which <code>ParallelMap</code> works three times faster (V9.0.1, Mac OS, Intel i7, Quad core, 8 virtual cores). The trick is to <strong>evaluate</strong> <code>dispatch = Dispatch@rules</code> on each kernel.</p>
<p>I don't really have an explanation, other than the guess that the dispatch table r... |
267,864 | <p>This is a <a href="https://math.stackexchange.com/questions/2239890/conformal-harmonic-maps-in-high-dimensions-are-scaled-isometries">cross-post</a> from MSE (where I got no answer).</p>
<p>It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.</p>
<p>I discovered lately th... | Robert Bryant | 13,972 | <p>This result is well-known in the theory of <em>harmonic morphisms</em>, about which, there is an extensive literature. It is a quite general fact (not depending on the conformally flat case of Euclidean space), implying that, when $n>2$, any conformal map $f:(M^n,g)\to (N^n,h)$ between (connected) Riemannian man... |
1,158,666 | <p>I know that every closed and bounded set in $\mathbb{R}$ is compact (like $[a,b]$)</p>
<p>so i can conclude that every bounded set in $\mathbb{R}$ is relatively compact, by contradiction i say that let $A\subset\mathbb{R}$ be a bounded set but not relatively compact it means that $\overline{A}$ is not compact, but ... | Nicolas | 213,738 | <p>As $A$
is bounded, we have $A\subset\bar{\mathcal{B}}\left(0,M\right)$
for a certain $0<M<+\infty$
. Since $\bar{A}$
is the smallest closed set that contains $A$
, thus we must have $\bar{A}\subseteq\bar{\mathcal{B}}\left(0,M\right)$
and then $\bar{A}$
is bounded.</p>
|
587,635 | <p>Differentiate with respect to $x$ </p>
<p>$$f(x)=\sqrt[3]{x^2}-4+\dfrac{8}{x^{2/3}}$$</p>
<p>Solution(is it correct having difficulties with the fractions):</p>
<p>$$=x^{2/3}-4+8x^{-2/3}$$</p>
<p>$$=\dfrac{2}{3}x^{-1/3} - \dfrac{16}{3}x^{-5/3}$$</p>
| ILoveMath | 42,344 | <p>The problem follows easily by using the power trick: $(x^n)' = nx^{n-1} $</p>
|
4,628,391 | <p>I came across the following statement which is supposedly true:</p>
<blockquote>
<p>There exists an infinite set of regular languages, such that their union is not a CFL</p>
</blockquote>
<p>it is explained this way:
we'll define <span class="math-container">$L_k = \{ 0^k1^k0^k \}$</span></p>
<p><span class="math-co... | Shinrin-Yoku | 789,929 | <p>No it’s not a contradiction, by induction you can only prove a <em>finite</em> union of regular languages is regular, it tells you nothing about whether or not an infinite union is always regular.</p>
|
2,720,922 | <blockquote>
<p>With $0<a,b,c<1$,
\begin{align}
\begin{bmatrix}
X_n \\
Y_n
\end{bmatrix}=\begin{bmatrix}
a^2 & (1-b)^2 \\
(1-a)^2 & b^2
\end{bmatrix}^n \begin{bmatrix}
c^2 \\
(1-c)^2
\end{bmatrix}
\end{align}
How fast does $X_n+Y_n$ go to zero?</p>
</blockquote>
<p>This is coming from a Marko... | Youem | 468,504 | <p>$x = 0, y = 1$, $$f(1) + 0 = f(1)f(0)$$</p>
<p>$x = 1, y = 0$, $$f(1) + 1 = f(1)f(0)$$</p>
<p>This implies $0 = 1$ so no such function exits.</p>
|
2,623,519 | <p>Problem statement - $A \setminus B= A$ st $A \subset B$</p>
<p>I think this statement is wrong as by definition of difference of sets $A \setminus B$ should contain all the elements of set $A$ <strong>which are not</strong> in $B$.But if $A$ is a subset of $B$ then all the elements of set $A$ are in set $B$ by defa... | RGS | 329,832 | <p>Sure thing, if $A = \emptyset$ then</p>
<p>$$A \setminus B = \emptyset \setminus B = \emptyset \subset B$$</p>
|
1,466,637 | <p>Solve $3sec^2(x)=4$.</p>
<p>$sec^2(x)=\frac{4}{3}$</p>
<p>$sec(x)=4=\sqrt{\frac{4}{3}}=\frac{4\sqrt{3}}{3}$</p>
<p>How to continue, i.e. how to calculate the value of x for which $sec(x)=\frac{4\sqrt{3}}{3}$</p>
<p>I can rearrange the expression above into $cos(x)=\frac{\sqrt{3}}{4}$, but how to continue to find... | Tejus | 274,219 | <p>The solution you've posted has some errors .It should be $\sec^2(x) = \frac{4}{3}$ and $\sec(x) = \pm \frac{2}{\sqrt 3}$. Later on you can find the general solutions for x . </p>
|
317,294 | <p>Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$.</p>
<p>Dimension theorem states:
$$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$</p>
<p>My question is:</p>
<p>Why is $U \cap W$ necessary in this theorem?</p>
| Ludolila | 60,678 | <p>It is the intersection of the subspaces:
$$U\cap W = \{v\in V | v \in U \wedge v\in W \} .$$</p>
|
317,294 | <p>Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$.</p>
<p>Dimension theorem states:
$$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$</p>
<p>My question is:</p>
<p>Why is $U \cap W$ necessary in this theorem?</p>
| Loronegro | 24,101 | <p>Maybe you are looking for a more intuitive answer.</p>
<p>The sum $U+W$ of subspaces $U$ and $W$ is the smallest subspace of $V$ that contains $U$ and $W$. If $U$ and $W$ are not so independent then $U+W$ is very close to both subspaces. This happens just when the intersection $U\cap W$ is big. The extreme case is ... |
2,308,430 | <p>I have information about 2 points and an arc. In this example, point 1 (x1,y1) and point 2(x2,y2) and I know the arc for example 90 degree or 180 degrees.</p>
<p>From this information, I want to calculate the center of the circle. Which is (x,y) in this case.</p>
<p><a href="https://i.stack.imgur.com/JQ5sA.png" re... | Zubin Mukerjee | 111,946 | <p>The measure of the arc is the same as the measure of the central angle, call it <span class="math-container">$\theta$</span>. Since you know the two points, you can find the distance between them, call it <span class="math-container">$d$</span>.</p>
<p>This means that there is an isosceles triangle whose vertices ar... |
3,767,656 | <p>Using L'hopital rule: <span class="math-container">$\lim_{x \to x_i, y \to y_i} \frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}} = \lim_{x \to x_i, y \to y_i} \frac{\frac{d(x-x_i)}{dx}}{\frac{d\sqrt{(x-x_i)^2 + (y-y_i)^2}}{dx}} = \lim_{x \to x_i, y \to y_i} \frac{1}{\frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}}} = \lim_{x \t... | user | 505,767 | <p>Firstly observe that by <span class="math-container">$x-x_i=u$</span> and <span class="math-container">$y-_i=v$</span> we have</p>
<p><span class="math-container">$$\lim_{x \to x_i, y \to y_i} \frac{x-x_i}{\sqrt{(x-x_i)^2 + (y-y_i)^2}} =\lim_{(u,v)\to (0,0)} \frac{u}{\sqrt{u^2 + v^2}} $$</span></p>
<p>which doesn't ... |
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Péter Komjáth | 6,647 | <p>I never heard anyone claim the existence of a forcing proof of the Erdős-Rado theorem. On the other hand, yes, there are several proofs of nonindependence statements that use forcing.
One is Shelah's proof for the existence of a finite <i>K</i><sub>4</sub>-free graph which, when the edges colored by 2 colors, alway... |
29,945 | <p>It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)_{\aleph_0}^{n+1}$ (or in particular that $(2^{\aleph_0})^+ \to (\aleph_1)_{\aleph_0}^2$) by using forcing? Is it sim... | Avshalom | 57,583 | <p>An example from elementary geometry is the very simple forcing argument to establish the non-axiomatizability (in infinitary logic) of Sperner spaces, due to Blass and Pambuccian: Blass, A.; Pambuccian, V. "Sperner spaces and first order logic." Mathematical Logic Quarterly vol. 49, no. 2 (March, 2003), 111-114. The... |
1,042,852 | <p>Let $a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$<br>
How to show this sequence is unbounded without using limits?</p>
<p>Well I know that I need to show that it unbounded from bottom or above.<br>
I choose bottom, so I need to show that $\forall M \exists n \Rightarrow a_{n}>M$</p>
<p>What is the method? Ca... | mfl | 148,513 | <p>$$ \frac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}< \frac{7n^{3} - 3n^{4}}{4n^{2} + 3}<\frac{7n^{3} - 3n^{4}}{4n^{2} + 3n^2}=n-\frac{3}{7}n^2= n\left(1-\frac37n\right) \underbrace{\le}_{n\ge 3} 1-\frac37n.$$</p>
<p>Thus, given $M<0$ there exists $N\in\mathbb{N},$ $N\ge 7/3(1-M),$ such that $$n\ge N\implies -n\le -N\... |
1,042,852 | <p>Let $a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$<br>
How to show this sequence is unbounded without using limits?</p>
<p>Well I know that I need to show that it unbounded from bottom or above.<br>
I choose bottom, so I need to show that $\forall M \exists n \Rightarrow a_{n}>M$</p>
<p>What is the method? Ca... | Idris Addou | 192,045 | <p>I have an idea which do not uses limits but i do not know if it helps you.
If $(x_{n})$ is a bounded sequence far from $0,$ that is, if there exists $%
m>0$ such that
$$
0<m\leq \left\vert x_{n}\right\vert ,\ for\ all\ n\geq n_{0}
$$
for some integer $n_{0},$ and $(y_{n})$ is an unbounded sequence, then the
p... |
1,969,748 | <p>I noticed that the most simple numerical approximation of a higher order-differential equation has the same form as the numerical approximation of a delayed first-order differential equation. This leads me to the following hypothesis:</p>
<blockquote>
<p><strong>Hypothesis:</strong> Delayed first-order differenti... | Lutz Lehmann | 115,115 | <p>If looking for a solution for $\dot x(t+1)=f(x(t))$ on $[0,\infty]$, take any continuous "seed" function $x:[-1,0]\to\Bbb R$ with $x(0)=x_0$ and define the solution as
$$
x(t)=x_0+\int_0^t f(x(s-1))ds
$$
The values on $[0,1]$ are well-defined using the values on $[-1,0]$, the values on $[1,2]$ by the values on $[0,1... |
4,074,647 | <p>Problem:
<span class="math-container">$12$</span> students are asked to go into the groups <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, <span class="math-container">$D$</span>, and <span class="math-container">$E$</span>. In how many ways ... | JMoravitz | 179,297 | <p>See the <a href="https://en.wikipedia.org/wiki/Twelvefold_way#case_s" rel="nofollow noreferrer">Twelvefold Way</a> and <a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow noreferrer">Stirling Numbers of the Second Kind</a>.</p>
<p>First, count the number of ways to partition you... |
441,980 | <p>Is there a notation to reference a single element within a set? Let's say I have a set <code>n = {1, 2, 4, 8, 16}</code>. If I wanted to use a single element from this set, is there a certain notation to do so? In computer programming, if I have an array <code>int x = {1, 2, 4, 8, 16}</code> I could reference the th... | Community | -1 | <p>Maybe the following would be of help (at least as a starter, for your reference request):</p>
<p><a href="http://www.math24.net/complex-form-of-fourier-series.html" rel="nofollow">"Complex Form of Fourier Series"</a> (Math24.net), this page goes through the derivations and provides a worked example. It also has qu... |
2,059,571 | <blockquote>
<p>Let $$V=\{(x_1,x_2,x_3,\dots,x_{100})\in\mathbb{R}^{100}\,|\, x_1=x_2=x_3 \text{ and } x_{51}=x_{52}=x_{53}= \dots=x_{100}\}$$ What is $\dim V$?</p>
</blockquote>
<p>If W is a subspace of vector space $V$ then $$\dim W = \dim V - \text{number of linearly independent restrictions}$$
In our case $\dim ... | Anurag A | 68,092 | <p>A general vector in $V$ looks like
$$
\begin{bmatrix}
x\\x\\x\\y_4\\y_5\\\vdots\\ y_{50}\\z\\z\\\vdots\\z
\end{bmatrix}=x\begin{bmatrix}
1\\1\\1\\0\\0\\\vdots\\ 0\\0\\0\\\vdots\\0
\end{bmatrix}
+z\begin{bmatrix}
0\\0\\0\\0\\0\\\vdots\\ 0\\1\\1\\\vdots\\1
\end{bmatrix}
+\sum_{i=4}^{50}y_i\begin{bmatrix}
0\\0\\0\\\vdo... |
72,630 | <p>Can you please help me and tell, how should I move on?
Can this be proved by induction?</p>
<blockquote>
<p>Every natural number $n\geq 8$ can be represented as $n=3k + 5\ell$.</p>
</blockquote>
<p>Thank you in advance</p>
| André Nicolas | 6,312 | <p>We can avoid an explicit appeal to induction by using the fact that every natural number $n$ has remainder $0$, $1$, or $2$ on division by $3$. Let $n \ge 8$.</p>
<p>If $n$ has remainder $2$ on division by $3$, then $n-8$ is divisible by $3$, say $n-8=3m$. Represent $8$ using $8=3\cdot 1+5\cdot 1$. Then add $m$ $3... |
4,174,111 | <p>In the following is Theorem 13.6 from Bruckner's Real Analysis which I don't understand some claims on it :</p>
<p>Question <em>in Blue:</em> <span class="math-container">$\mu (|f_j(x)| > \|f_j\|_∞)=0$</span> and <span class="math-container">$\mu (|f_k(x)| > \|f_k\|_∞)=0$</span>. But how that implies <span cla... | Oliver Díaz | 121,671 | <p>It seems that there is a typo in your textbook.</p>
<p>I am writing here a proof that is closed to your text's. Suppose <span class="math-container">$\{f_n:n\in\mathbb{N}\}$</span> is a Cauchy sequence in <span class="math-container">$L_\infty(\mu)$</span>. Then for any <span class="math-container">$\varepsilon>0... |
3,044,271 | <p>If <span class="math-container">$x^2+y^2=1$</span>. then the range of expression <span class="math-container">$3x^2-2xy$</span> without trigonometric substitution method</p>
<p>what i have done try here is use arithmetic geometric inequality</p>
<p><span class="math-container">$\displaystyle x^2+y^2\geq 2xy$</sp... | David G. Stork | 210,401 | <p><span class="math-container">$3 x^2 - 2 x y$</span> with <span class="math-container">$y = \pm \sqrt{1-x^2}$</span> is <span class="math-container">$3 x^2 \pm 2 x \sqrt{1 - x^2}:$</span></p>
<p><a href="https://i.stack.imgur.com/fENgE.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/fENgE.png" alt... |
3,044,271 | <p>If <span class="math-container">$x^2+y^2=1$</span>. then the range of expression <span class="math-container">$3x^2-2xy$</span> without trigonometric substitution method</p>
<p>what i have done try here is use arithmetic geometric inequality</p>
<p><span class="math-container">$\displaystyle x^2+y^2\geq 2xy$</sp... | Barry Cipra | 86,747 | <p>If you have the requisite linear algebra theorems at your disposal, then you can recognize</p>
<p><span class="math-container">$$3x^2-2xy=\pmatrix{x&y}\pmatrix{3&-1\\-1&0}\pmatrix{x\\y}$$</span></p>
<p>where the <span class="math-container">$2\times2$</span> matrix is selfadjoint with eigenvalues satis... |
1,431,042 | <p>I know that given a polynomial $p(i)$ of degree $d$, the sum $\sum_{i=0}^n p(i)$ would have a degree of $d + 1$. So for example</p>
<p>$$
\sum_{i=0}^n \left(2i^2 + 4\right) = \frac{2}{3}n^3+n^2+\frac{13}{3}n+4.
$$</p>
<p>I can't find how to do this the other way around. What I mean by this, is how can you, when gi... | John Hughes | 114,036 | <p>To rephrase, I believe the question is this: </p>
<p>Suppose that polynomials $p$ and $q$ have the property that
$$
\sum_{i=0}^n p(i) = q(n)
$$
If you're given $q$, how can you find $p$? </p>
<p>First, this is a lovely question. I'd never really considered it, because we almost always study instead "if you know $... |
4,622,026 | <p>I am a bit confused about <span class="math-container">$\int e^{e^x+x}dx$</span>. If we made a <span class="math-container">$u$</span>-sub of <span class="math-container">$e^x$</span> then the derivative is <span class="math-container">$e^x$</span> and so we have <span class="math-container">$\int e^udu$</span>. But... | herb steinberg | 501,262 | <p><span class="math-container">$\frac{11\times10}{12^2}$</span> seems to be a simpler approach.</p>
|
1,663,244 | <p>How to represent a graph in a function?</p>
<p>For example, I used 3 functions : </p>
<p>$$f(x)=x^2$$
$$g(x)=x$$
$$h(x)=3$$</p>
<p>These 3 functions were plotted on the same graph and the result (after edit) is as given below</p>
<p>How would you represent the below graph in a function, lets say $k(x)$ ?</p>
<p... | AlexR | 86,940 | <p>You can simple define $k$ to be a piecewise function:</p>
<p>$$k(x) := \begin{cases}x^2 & 0 \le x < 1\\ x & 1 \le x < 3\\3 & 3 \le x\end{cases}$$</p>
<p>Now if you really don't like that, you can work out something using $\min$:</p>
<p>$$k(x) = \min(x^2, x, 3)$$</p>
<p>But not all piecewise fun... |
3,486,036 | <p>Given a complete, weighted graph as the input to TSP, is the edge from <span class="math-container">$i$</span> to <span class="math-container">$j$</span> with minimum weight always in the solution?</p>
| Robert Israel | 8,508 | <p>Try this example. If you use edge <span class="math-container">$(1,2)$</span>, then you must also use <span class="math-container">$(3,4)$</span>.</p>
<p><a href="https://i.stack.imgur.com/XQcnU.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XQcnU.png" alt="enter image description here"></a></p... |
1,229,729 | <p>I know that because both $a$ and $b$ are positive it is in the first quadrant and hence $\arg z$ should just equal to $\arctan(b/a)$, but I've been told that the answer is $\arg z= \arctan \sqrt5 $???</p>
| Emilio Novati | 187,568 | <p>Upper triangular matrices</p>
<p>$
\begin{bmatrix}
a&b\\
0&c
\end{bmatrix}
$</p>
<p>form a vector space with canonical basis:</p>
<p>$
e_1=\begin{bmatrix}
1&0\\
0&0
\end{bmatrix}
\quad
e_2=\begin{bmatrix}
0&1\\
0&0
\end{bmatrix}
\quad
e_3=\begin{bmatrix}
0&0\\
0&1
\end{bmatrix}
$</... |
1,905,998 | <p>Suppose there exists a injection from $S$ to $\mathbb N$, and similarly one from $T$ to $\mathbb N$.</p>
<p>Then there exists an injection g from $S×T \to \mathbb{N^2}$.</p>
<p>How do i explicitely find such injection$</p>
<p>Thanks</p>
| Andres Mejia | 297,998 | <p>Let $h: S \to \mathbb N$ and $g: T \to \mathbb N$ be injections.
Then take $f: S \times T \to \mathbb N^2$, where $f(s,t)=(h(s),g(t))$.</p>
<p>Then, $f$ is well defined. Suppose that $f(a,b)=f(c,d)$. See if you can show that $(a,b)=(c,d)$</p>
<p>Here is something I consider neat:</p>
<p>Take $k(s,t)=2^{h(s)} \cd... |
1,905,998 | <p>Suppose there exists a injection from $S$ to $\mathbb N$, and similarly one from $T$ to $\mathbb N$.</p>
<p>Then there exists an injection g from $S×T \to \mathbb{N^2}$.</p>
<p>How do i explicitely find such injection$</p>
<p>Thanks</p>
| benguin | 121,903 | <p>Let the first injection be $f_S: S \to \mathbb{N}$ and the second be $f_T : T \to \mathbb{N}$. Consider $g: S \times T \to \mathbb{N}^2$ as,</p>
<p>$$g((s,t)) = (f_S(s), f_T(t)).$$</p>
|
1,723,331 | <blockquote>
<p>$$\frac{1}{1-x^2}$$</p>
</blockquote>
<p>$$\frac{1}{1-x^2}=\frac{a}{1-x}+\frac{b}{1+x}$$</p>
<p>$$1=a+ax+b-bx$$</p>
<p>$$1=a+b+x(a-b)$$</p>
<p>$a+b=1$ and $x(a-b)=0\Rightarrow a-b=0\Rightarrow a=b$</p>
<p>$$2a=1\Rightarrow a=\frac{1}{2}$$</p>
<p>$b=\frac{1}{2}$</p>
<p>$$\frac{1}{1-x^2}=\frac{1}... | Integral | 33,688 | <p>But you are not wrong.</p>
<p>$$\frac{1}{2(1-x)}+\frac{1}{2(1+x)} = \frac{1+x+1-x}{2(1+x)(1-x)} = \frac{2}{2(1+x)(1-x)} = \frac{1}{1-x^2}$$</p>
|
83,797 | <p>Good day, I'm not sure that this limit exists. All my attempts to prove it were in vain ...</p>
<p>Let $k>1$.
If exist, calculate the limit of the sequence $(x_n)$ defined by,
$$x_n := \Biggl(k \sin \left(\frac{1}{n^2}\right) + \frac{1}{k}\cos n \Biggr)^n.$$</p>
| Brian M. Scott | 12,042 | <p>HINT: For all $n\in\mathbb{Z}^+$ you have $$\sin\frac1{n^2}\le\frac1{n^2}\text{ and }\frac1k\cos n\le\frac1k,$$ so $$x_n\le\left(\frac{k}{n^2}+\frac1k\right)^n=\left(\frac{k^2+1}{kn^2}\right)^n=\left(\frac{k+\frac1k}{n^2}\right)^n.$$ (Be a little careful, though: you still have to worry about a lower bound for the $... |
3,435,209 | <p><a href="https://i.stack.imgur.com/47sW7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/47sW7.png" alt="enter image description here"></a></p>
<p>When I attempt to compute <span class="math-container">$f_{y}(0,0)$</span>, I first set <span class="math-container">$x = 0$</span> such that <span cl... | Fred | 380,717 | <p><span class="math-container">$ \frac{f(0,h)-f(0,0)}{h}= \frac{1}{h}.$</span> This shows that <span class="math-container">$ \lim_{h \to 0}\frac{f(0,h)-f(0,0)}{h}$</span> does not exist. Hence <span class="math-container">$f_y(0,0)$</span> does not exist.</p>
|
1,191,237 | <p>I have the list $l = (6, 6, 5, 4)$ and want to how to calculate the possible number of permutations.</p>
<p>By using <em>brute force</em> I know that there are <strong>12</strong> possible permutations:</p>
<p>$$\{(6, 5, 6, 4),
(6, 6, 5, 4),
(5, 6, 6, 4),
(6, 4, 5, 6),
(6, 5, 4, 6),
(4, 6, 6, 5),
(4, 5, 6, 6... | JMP | 210,189 | <p>Imagine there are two different $6$'s, say $6_a$ and $6_b$. Then there would be $4!=24$ permutations. Now let the two $6$'s be the same, so $(6_a,7,5,6_b)=(6_b,7,5,6_a).$ This halves the number of permutations, giving the answer of $24/2=12$.</p>
|
4,310,324 | <p>This is theorem 11.5 from CLRS book.
Suppose <span class="math-container">$a\in \mathbb{Z}^*_p, b\in \mathbb{Z}_p$</span>.</p>
<p>Consider two distinct keys <span class="math-container">$k$</span> and <span class="math-container">$l$</span> from <span class="math-container">$\mathbb{Z}_p$</span>, so that <span class... | Alan Abraham | 823,763 | <p>Since, I'm learning a bit of residue theory, I want to try this solution out as well. We want to find the coefficient of <span class="math-container">$x^{-1}$</span> in the expansion of
<span class="math-container">$$f(x)=\frac{1}{x^{25}(1-x)(1-x^2)(1-x^3)}$$</span>
<span class="math-container">$$f(x)=\frac{1}{x^{25... |
3,681,504 | <p>The specific question I have to work on is:</p>
<p><span class="math-container">$\sqrt{n}$</span> , <span class="math-container">$\log{n^{100}, }$</span> <span class="math-container">$\ n^{10}$</span>, <span class="math-container">$\log(10^n)$</span> , <span class="math-container">$\log(n^n)$</span>, <span class="m... | gt6989b | 16,192 | <p>Some hints:</p>
<ol>
<li><span class="math-container">$\log\left(a^b\right) = b \log a$</span>, can you apply that to simplify <span class="math-container">$\log \left(n^{100}\right)$</span> and <span class="math-container">$\log \left(10^n\right)$</span> and <span class="math-container">$\log\left(n^n\right)$</spa... |
1,212,198 | <blockquote>
<p>Prove or counter-example. For all nonempty sets $A$ and $B$ and for all functions $F$, $F(A-B) = F(A) - F(B)$; if not, what else does $F$ need to have in order to make the equality hold?</p>
</blockquote>
<p>I am pretty lost on this question. I don't feel like its right since it would be a pretty bas... | user 1 | 133,030 | <p>Let $F:X\to Y$ be constant function: $\forall x\in X, F(x)=c$, where $c$ is a constant element of $Y$. Let $A$ and $B$ be subsets of $X$, such that $A-B$ is not empty. Then $$F(A)−F(B)= \varnothing,$$ while, $$F(A-B)=\{c\}$$</p>
|
565,762 | <p>I have to show that : $$T(n) = Θ({n^3})$$</p>
<hr>
<p>We have this recursive function :</p>
<p>$$T(n) = 8T(n/2) + n^2, n>=2$$</p>
<p>also we know that $$T(1) = 1$$</p>
<p>And it says that there is a "replacement method" to do that.</p>
<hr>
<h2>EDIT</h2>
<hr>
<p>If I say $$n = 2^k, k≥1$$</p>
<p>then T(n... | user1337 | 62,839 | <p>$g$ is the composition of two $C^\infty$ functions, as such it is of class $C^\infty$. Think about the chain rule.</p>
|
12,653 | <p>I've Googled and searched mathstackexchange, but cannot find out how to insert a blue URL reference in a question or answer. Can you give me an example that I can edit it to look at the Latex code?</p>
| AlexR | 86,940 | <p>See <a href="http://docs.mathjax.org/en/latest/safe-mode.html" rel="nofollow">here</a> for a reference, however this does not seem to work on this site:<br>
$$E \href{javascript:alert("Einstein says so!")}{=} mc^2$$
(C&P'ed the example from the MathJax page):<br>
<code>$$E \href{javascript:alert("Einstein says s... |
3,855,672 | <p>This is question 7N #2 from Willard's <em>General Topology</em>, on p. 51.</p>
<blockquote>
<p>For any topological space <span class="math-container">$X$</span>, let <span class="math-container">$H(X)$</span> denote the group of
homeomorphisms of <span class="math-container">$X$</span> onto itself, with composition ... | Christoph | 86,801 | <p>For surjectivity note that any continuous injection <span class="math-container">$\mathbb R\to\mathbb R$</span> <a href="https://math.stackexchange.com/questions/752073/continuous-injective-map-is-strictly-monotonic">is strictly monotonic</a>. Since there is a strictly monotonic homeomorphism <span class="math-conta... |
384,553 | <p>Any ideas how to solve it?
$$\int\frac{x^4+2x+4}{x^4-1}dx$$
Thanks!</p>
| amWhy | 9,003 | <p>Using polynomial division, we get $$\int \frac{x^4+2x+4}{x^4-1} dx = \int 1 + \frac{2x+5}{(x^2 - 1)(x^2 + 1)}dx
= \int 1 + \frac{2x+5}{(x+1)(x-1)(x^2+1)} dx $$ </p>
<p>Expressing this as partial fractions, we need only find $A, B, C$</p>
<p>$$= \int \left(1 + \frac{A}{x+1} + \frac B{x-1} +\frac{C}{x^2 + 1}\righ... |
1,543,054 | <p>Every year the teacher write 4 tests with 6 questions,
from a list of 10 different questions, Is it certain that after 8 years, theres 3 different tests with the same 4 questions?</p>
<p>how do i show that with the pigeonhole principle?</p>
| Community | -1 | <p>There are $n!= n\times\ldots\times 1$ ways to arrange n things. There are then, $6\times5\times4\times3\times2\times1=720$ ways, to rearrange the 6 questions, once chosen. The number of combinations ( as opposed to permutations), of y distinct things from z is $\frac{z!}{(z-y)!\times y!}$ . We have 10 things to choo... |
1,221,224 | <p>I've came across an interesting situation in my code and I want to make sure I'm not missing anything, so I'll try to explain it as simple as possible.</p>
<p>Imagine 4 buckets and 4 ping-pong balls, in how many variations can you put all balls in any of the buckets? The order of balls does not matter, all balls mu... | N. F. Taussig | 173,070 | <p>It sounds like what you wish to do is to distribute four indistinguishable ping pong balls among four distinguishable buckets (which, for instance, may be ordered from left to right in a row). The number of ways to do this is the number of ways we can solve the equation
$$x_1 + x_2 + x_3 + x_4 = 4$$
in the nonnega... |
3,107,109 | <p><span class="math-container">$F$</span> is closed subset of <span class="math-container">$X$</span>. <span class="math-container">$A$</span> is any subset of <span class="math-container">$X$</span>. <span class="math-container">$X$</span> is a metric space. The topology on <span class="math-container">$X$</span> is ... | Greg Martin | 16,078 | <p>Distilling reuns's comment: the definition of a branch point is one for which traversing an <em>arbitrarily small circle</em> around that point results in a multivalued function. One single circle is not enough. Indeed, if your argument showed that <span class="math-container">$1$</span> was a branch point of <span ... |
3,107,109 | <p><span class="math-container">$F$</span> is closed subset of <span class="math-container">$X$</span>. <span class="math-container">$A$</span> is any subset of <span class="math-container">$X$</span>. <span class="math-container">$X$</span> is a metric space. The topology on <span class="math-container">$X$</span> is ... | reuns | 276,986 | <p>In addition to Greg's answer : </p>
<p>Note <span class="math-container">$\log \log z$</span> has a branch point at <span class="math-container">$z=1$</span> which disappears after one rotation. </p>
<p>So one possible general definition of a branch point is : a given branch of <span class="math-container">$f(z)$<... |
2,446,000 | <p>I' ve tried with $x^2 = {[x]+1\over 2}$ so $x$ is a square root of half integer. And know? What to do with that?</p>
| Robert Z | 299,698 | <p>Hint. Note that $x-1<\lfloor x\rfloor \leq x$ implies that
$$(2x+1)(x-1)=2x^2-x-1\leq 2x^2-\lfloor x\rfloor-1< 2x^2-(x-1)-1=x(2x-1)$$</p>
|
2,446,000 | <p>I' ve tried with $x^2 = {[x]+1\over 2}$ so $x$ is a square root of half integer. And know? What to do with that?</p>
| jonsno | 310,635 | <p>For a workable range, we can say that:</p>
<p>$$x-1 < [x] \le x$$</p>
<p>$$x< 2x^2 \le x+1$$</p>
<p>Solving this, we get:</p>
<p>$$x\in\left[\frac{-1}{2},0\right) \cup \left(\frac{1}{2},1\right]$$</p>
<p>Our roots <em>must</em> lie in this range. So for each of these intervals, substitute the value of $[x... |
320,194 | <p>I have 5 types of symptoms, I want to know all kind of combinations a patient could have:</p>
<p>The set is $(vomit, excrement, urine, dizzyness, convulsion)$</p>
<p>As patient can show only one, or even 5 of them I am listing them as:</p>
<p>So</p>
<h1>Possibilities with one symptom</h1>
<pre><code>vomit
excre... | DJohnM | 58,220 | <p>Each repulsive symptom is either there or not, independently. So there are $2^5$ different lists. Leaving out the lucky one with no symptoms, that leaves 31 possibles.</p>
<p>PS. You've missed some in your list, and double counted others...</p>
<p>There are 5 possible single-symptom lists. It can be recycled as... |
711,863 | <p>$$\int{\frac{3}{5y^2 + 4}}dy$$</p>
<p>$$\frac{3}{4}\int{\frac{1}{\left(\frac{\sqrt{5}y}{2}\right)^2 + 1}}dy$$</p>
<p>$$u = \frac{\sqrt{5}y}{2}$$</p>
<p>$$dy = \frac{2}{\sqrt{5}}du$$</p>
<p>My solution to this problem was</p>
<p>$$\frac{3}{2\sqrt{5}}\left(\frac{1}{\tan\left(\frac{\sqrt{5}y}{2}\right)} + c\right)... | Reverend_Dude | 121,647 | <p>This is because what ever the value is that is being multiplied is still a constant. The mathematician does not concern themselves with any magnitude of this constant when solving empirically.</p>
|
3,313,145 | <p>For each number I want to get the array of number ascending cumulative</p>
<p>For example :</p>
<p><span class="math-container">$$100 \% = 45 \% + 55 \%$$</span></p>
<p><span class="math-container">$$100 \% = 23.\overline{33} \% + 33.\overline{33} \% + 43.\overline{33} \%$$</span></p>
<p><span class="math-contai... | Peter Foreman | 631,494 | <p>For a given <span class="math-container">$n$</span> you want the sum
<span class="math-container">$$a+(a+d)+\dots+(a+(n-1)d)=100$$</span>
<span class="math-container">$$\frac{n}2(2a+(n-1)d)=100$$</span>
<span class="math-container">$$a=\frac{100}n-\frac{(n-1)d}2$$</span>
Hence we can choose any common difference <sp... |
2,860,195 | <p>The operation $@$ is defined on the real numbers as $a @ b= ab + b + a$</p>
<p>a) Show that $0$ is an identity for the operation.</p>
<p>b) Show that some real numbers have inverses under the operation.</p>
<p>c) Find a counter-example to show that, for this operation inverses do not exist for all the real number... | Bernard | 202,857 | <p>Note this operation is commutative, so $A$ has an inverse under this operation if and only if there exists $x$ such that $\;a@x=ax+x+a=0$.</p>
<p>Can you determine for which $a$ this equation has a solution?</p>
|
29,143 | <p>In what context should I use $=$ and $\equiv$?</p>
<p>What is the precise difference?</p>
<p>Thanks!</p>
<p>(I wasn't sure what to tag this with, any suggestions?)</p>
| Brian | 4,266 | <p>Sometimes $\equiv$ is used to mean "defined to be" although I think := is more common for that.</p>
|
29,143 | <p>In what context should I use $=$ and $\equiv$?</p>
<p>What is the precise difference?</p>
<p>Thanks!</p>
<p>(I wasn't sure what to tag this with, any suggestions?)</p>
| Kevin Cathcart | 8,735 | <p>The $\equiv$ symbol originally meant "is identically equal to", and as that name implies it is used with identities. It is actually stating that the equality holds for all instantiations of the free variables. For example $\sin\left(\theta+\frac{\pi}{2}\right)=\cos{\theta}$ is true for any value of $\theta$, therefo... |
1,320,365 | <p>I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to $\mathbb Z_5$. </p>
<p>The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \} \to \mathbb N$ defined as $N(a... | Sammy Black | 6,509 | <p>Since
$$
\Bbb{Z}[i] \cong \Bbb{Z}[x] / \langle x^2 + 1 \rangle,
$$
you have the isomorphism
$$
R = \Bbb{Z}[i] / \langle 1 + 2i \rangle \cong \Bbb{Z}[x] / \langle 1 + x^2, 1 + 2x \rangle.
$$</p>
<p>In order to construct a ring homomorphism $\varphi:R \to \Bbb{Z}_5$, it suffices to describe what $x$ maps to. Say $\v... |
1,320,365 | <p>I am trying to prove that $\mathbb Z[i]/ \langle 1+2i \rangle$ is isomorphic to $\mathbb Z_5$. </p>
<p>The only thing that came to my mind was trying to apply the first isomorphism theorem using an appropiate function. If I consider the euclidean function $N: Z[i] \setminus \{0 \} \to \mathbb N$ defined as $N(a... | Bhaskar Vashishth | 101,661 | <p><strong>Another approach</strong></p>
<p>$R=\mathbb Z[i]/ \langle 1+2i \rangle$= $\{a+bi+\langle 1+2i \rangle |\ a,b \in \Bbb{Z}\}$.</p>
<p>Now we have $\overline{1+2i}=\bar0$ so we can say $1+2i=0$ and reduce our number of elements using this relation.</p>
<p>Now we can eliminate $i $ term from $a+bi$ by using $... |
27,126 | <p>$$e^{\pi i} + 1 = 0$$</p>
<p>I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us? </p>
<p>Best that I can figure out is that it just emphasizes that the various definitions mathematicians have provided for non-intuitive operations (com... | Andrey Rekalo | 5,371 | <p>$$e^{i\pi}=\lim\limits_{N\to\infty}\left(1 + \frac{iπ}{N}\right)^N$$</p>
<p><img src="https://upload.wikimedia.org/wikipedia/commons/0/0e/ExpIPi.gif" alt="alt text"></p>
|
2,311,583 | <p>$$
\int_{0}^{1}\sqrt{\,1 + x^{4}\,}\,\,\mathrm{d}x
$$ </p>
<p>I used substitution of tanx=z but it was not fruitful. Then i used $ (x-1/x)= z$ and
$(x)^2-1/(x)^2=z $ but no helpful expression was derived.
I also used property $\int_0^a f(a-x)=\int_0^a f(x) $
Please help me out</p>
| Leucippus | 148,155 | <p>Consider the ${}_{2}F_{1}$ hypergeometric integral form given by
$${}_{2}F_{1}(a, b; c; x) = \frac{\Gamma(c)}{\Gamma(b) \, \Gamma(c-b)} \, \int_{0}^{1} t^{b-1} \, (1-t)^{c-b-1} \, (1-x \, t)^{-a} \, dt$$
leads to, with $a=-1/2$, $b=1/4$, $c=5/4$, $x=-1$,
$${}_{2}F_{1}\left(-\frac{1}{2}, \frac{1}{4}; \frac{5}{4}; -1\... |
1,336,419 | <p>What are the three final numbers of $2003^{2003}$ and $2003^{2003^{2003}}$? </p>
<p>Do I use the Chinese Remainder Theorem here, and if so, how?</p>
| Jack D'Aurizio | 44,121 | <p>To find the last three digits means to find the residue class $\pmod{1000}$.</p>
<p>Since $\varphi(1000)=400$, by <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function#Euler.27s_theorem">Euler's theorem</a> we have:
$$ 2003^{2003}\equiv 3^{3}\equiv 27\pmod{1000},$$
so the last three digits of $2003^{200... |
26,651 | <p>Hi, everybody. I'm recently reading W.Bruns and J.Herzog's famous book-Cohen-Macaulay Rings. I personally believe that it would be perfect if the authors provide for readers more concrete examples. After reading the first two sections of this book, I have two questions.</p>
<ol>
<li><p>Given a non-negative integer ... | Karl Schwede | 3,521 | <p>Note that Macaulay2 has some quick ways to check whether a ring is CM and/or Gorenstein. One approach is to write your ring <span class="math-container">$R$</span> as a quotient of a regular ring (polynomial ring) <span class="math-container">$S$</span>, <span class="math-container">$R = S/I$</span>.</p>
<p>Then o... |
69,839 | <p>Is it possible to get all points on a Polyhedron surface using two surface parameters, say </p>
<p>$ \phi,\theta $ spherical co-ordinates?</p>
<p>Just like in <code>ParametricPlot3D</code>, can we start with <code>PolyhedronData["Tetrahedron"]</code> to obtain spatial point positions?. The tip of position vector s... | David G. Stork | 9,735 | <p>Exploit the fact that the vertices of the dual to a Platonic solid correspond to the centers of the faces of the solid itself. For instance, the dual to a cube is a regular octahedron, and the six vertices of this octahedron are in the directions of the centers of the faces of its dual cube. </p>
<p>Find the norm... |
774,193 | <p>Given the sequence $A_n=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$:</p>
<ul>
<li>Are there any known rational elements in $A_n$, or has it been proved that all are irrational?</li>
<li>Is there any proof for $\lim\limits_{n\to\infty}A_n$ (a.k.a. <a href="http://mathworld.wolfram.com/NestedRadicalConstant.htm... | Thomas Andrews | 7,933 | <p>For the first question: If $A_n$ is rational, we can prove that $\sqrt{n}$ is rational, thus $n$ is a perfect square, and then we can prove that $\sqrt{n-1+\sqrt{n}}$ is rational, and hence $n-1+\sqrt n$ is a perfect square. But if $n>1$, $$(\sqrt n)^2<n-1+\sqrt{n}<(\sqrt{n}+1)^2$$</p>
|
774,193 | <p>Given the sequence $A_n=\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{\dots+\sqrt{n}}}}}$:</p>
<ul>
<li>Are there any known rational elements in $A_n$, or has it been proved that all are irrational?</li>
<li>Is there any proof for $\lim\limits_{n\to\infty}A_n$ (a.k.a. <a href="http://mathworld.wolfram.com/NestedRadicalConstant.htm... | hmakholm left over Monica | 14,366 | <p>The square of a rational number is always rational, so the square <em>root</em> of an irrational is also necessarily irrational.</p>
<p>Therefore, as you go through the sequence
$$\sqrt n, \sqrt{(n-1)+\sqrt{n}}, \sqrt{(n-2)+\sqrt{(n-1)+\sqrt{n}}}, \ldots, A_n$$
once you hit even one irrational number, $A_n$ at the ... |
53,073 | <p>Suppose $X$ is a non-explosive diffusion with dynamics</p>
<p>$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, </p>
<p>where $W$ is a standard Brownian motion. My intuition about $X$ is that if $\mu$ and $\sigma$ are sufficiently nice, then the sample paths of $X$ are in some sense "deformed" sample paths of $W$. Is there a... | The Bridge | 2,642 | <p>Hi,</p>
<p>This is an interesting question</p>
<p>I don't have the complete answer to your question rather some leads about it, but it seems to me that what you need to show for solutions of your SDE, is that the natural filtrations of $X$ is the same as the natural filtration of $W$. </p>
<p>If you have this do... |
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