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 |
|---|---|---|---|---|
2,203,066 | <p>The definition I have is the following:</p>
<blockquote>
<p>A vector space V is said to be <strong>finite-dimensional</strong> if there is a finite set of vectors in V that spans V and is said to be <strong>infinite-dimensional</strong> if no such set exists.</p>
</blockquote>
<p>However, with this definition I ... | P Vanchinathan | 28,915 | <p>Definition: A city is said to be food-friendly if one can get three different cuisines in a single restaurant.</p>
<p>In London, there is Restaurant A which serves Indian, Chinese and Continental. ANd also Restaurant B that serves Thai, Mexican and Greek food. And restaurant C serving, Indian, Russian and Japane... |
2,507,864 | <blockquote>
<p>Check if for any two set families $\mathcal A $ and $\mathcal B $ the
following is true: $\bigcup (\mathcal A \cap \mathcal B) = \bigcup
\mathcal A \cap \bigcup \mathcal B$</p>
</blockquote>
<p>First of all I considered an example: $\mathcal A = \{ \{1,2\}, \{1,3\} \}$ and $\mathcal B = \{\{1,2\}... | Kim Jong Un | 136,641 | <p>Consider the concave functions $f(x)=-x^2+3x$ and $g(x)=-x$. But $h(x)\equiv\max\{f(x),g(x)\}$ is <em>not</em> concave.</p>
<p><a href="https://i.stack.imgur.com/AF9Kj.png" rel="noreferrer"><img src="https://i.stack.imgur.com/AF9Kj.png" alt="enter image description here"></a></p>
|
2,826,313 | <p>So I have been given the following equation : $z^6-5z^3+1=0$. I have to calculate the number of zeros (given $|z|>2$). I already have the following:</p>
<p>$|z^6| = 64$ and $|-5z^3+1| \leq 41$ for $|z|=2$. By Rouche's theorem: since $|z^6|>|-5z^3+1|$ and $z^6$ has six zeroes (or one zero of order six), the fu... | José Carlos Santos | 446,262 | <p>Since it is a polynomial of degree $6$ and since it has $6$ zeros inside the disk, it has no zeros outside the disk.</p>
|
2,136,791 | <p>I got a following minimization problem</p>
<p>$$\min_{\mathbf{X}^{(1)}, \, \mathbf{X}^{(2)}} \;\left\| \mathbf{B} - \mathbf{A} (\mathbf{X}^{(1)} \odot \mathbf{X}^{(2)}) \right\|^{2}_{F},$$</p>
<p>where the matrices $\mathbf{B}\in \mathbb{R}^{100 \times 3}$, $\mathbf{A}\in \mathbb{R}^{100\times 36}$, $\mathbf{X}^{(... | greg | 357,854 | <p><span class="math-container">$\def\p{\partial} \def\bb{\mathbb}$</span>
Given two matrices with the same number of columns, e.g.
<span class="math-container">$$A\in{\bb R}^{a\times n} \qquad B\in{\bb R}^{b\times n}$$</span>
their Khatri-Rao <span class="math-container">$(\boxtimes)$</span> product can be written in ... |
632,043 | <p>tl;dr: why is raising by $(p-1)/2$ not always equal to $1$ in $\mathbb{Z}^*_p$?</p>
<p>I was studying the proof of why generators do not have quadratic residues and I stumbled in one step on the proof that I thought might be a good question that might help other people in the future when raising powers modulo $p$.<... | André Nicolas | 6,312 | <p>Note that $1$ has <strong>two</strong> square roots modulo $p$ if $p\gt 2$. </p>
<p>So from $g^{p-1}\equiv 1\pmod{p}$, we conclude that
$$\left(g^{(p-1)/2}\right)^2\equiv 1\pmod{p},$$
and therefore
$$g^{(p-1)/2}\equiv \pm 1\pmod{p}.$$</p>
<p>If $g$ is a primitive root of $p$, and $p\gt 2$, then $g^{(p-1)/2}\equi... |
1,090,620 | <p>I don't know how to solve this limit</p>
<p>$$ \lim_{y\to0} \frac{x e^ { \frac{-x^2}{y^2}}}{y^2}$$</p>
<p>$\frac{1}{e^ { \frac{x^2}{y^2}}} \to 0$</p>
<p>but $\frac{x}{y^2} \to +\infty$</p>
<p>This limit presents the indeterminate form $0 \infty$ ?</p>
| syockit | 53,159 | <p>Here's one sloppy way to work it:
Assume $x>0$, let
$$k= \lim_{y\to0} \frac{xe^{-\frac{x^2}{y^2}}}{y^2}$$
Take natural logarithm of both sides:
$$\ln(k)= \lim_{y\to0} \left[\ln{x}-2\ln{y}-\frac{x^2}{y^2}\right]=-\infty$$
Therefore
$$ k = e^{-\infty} = 0 $$
For $x<0$, substitute $x\to -x$, and you'll end up wit... |
513,500 | <p>Suppose $f,g$ are analytic functions in domain $D$.If $fg=0$, I want to prove either $f(z)=0$ or $g(z)=0$. </p>
| njguliyev | 90,209 | <p>Hint: If $f(z_0) \ne 0$ then $f(z) \ne 0$ at some neighborhood of $z_0$.</p>
|
50,994 | <p>I am trying to calculate the following integral. </p>
<pre><code>sigma1 = 10.0; sigma2 = 5.0; delta = 0.5;
t[x1_, y1_, x_, y_] := 100*HeavisideLambda[sigma1^-1*(x - x1), sigma2^-1*(y - y1)];
B2[x1_, y1_, x_, y_] := HeavisideTheta[(delta/2)^2 - (x - x1)^2, (delta/2)^2 - (y - y1)^2];
trans[x1_, y1_, x2_, y2_] :=
... | Michael E2 | 4,999 | <p>The Heaviside functions are essentially piecewise functions, and <code>NIntegrate</code> knows how to handle <code>Piecewise</code> functions but not Heaviside functions. In particular, it will analyze the domain of <code>Piecewise</code> functions and adjust its sampling accordingly. Here are two rules for conver... |
17,143 | <p>My next project I'd like to start working on is Domain Coloring. I am aware of the beautiful discussion at:</p>
<p><a href="https://mathematica.stackexchange.com/questions/7275/how-can-i-generate-this-domain-coloring-plot">How can I generate this "domain coloring" plot?</a></p>
<p>And I am studying it. H... | cormullion | 61 | <p>I was hoping that this question would get some good answers, but it must have been asked at a time when everyone was feeling a bit curmudgeonly after Christmas... :) </p>
<p>My understanding of <a href="http://www.mai.liu.se/~halun/complex/domain_coloring-unicode.html" rel="noreferrer">Hans Lundmark's linked page</... |
929,598 | <p>A rectangle is 4 times as long as it is wide. If the length is increased by 4 inches and the width is decreased by 1 inch, the area will be 60 square inches. What were the dimensions of the original rectangle? Explain your answer.</p>
| Jack D'Aurizio | 44,121 | <p>By multiplying both sides by $(2k)!! = (2k)(2k-2)\cdot\ldots\cdot 2 = 2^k\cdot k!$, you get: $$(2k+1)! = (2k+1)!.$$</p>
|
1,413,150 | <p>So for a periodic function <span class="math-container">$f$</span> (of period <span class="math-container">$1$</span>, say), I know the Riemann-Lebesgue Lemma which states that if <span class="math-container">$f$</span> is <span class="math-container">$L^1$</span> then the Fourier coefficients <span class="math-cont... | Arin Chaudhuri | 404 | <p>Here is another way to approach this problem.
The function $$f(z) = 1 - z/2 + z^2/3 + \ldots + (-1)^{k+1} z^k/(k+1) + \ldots $$ is analytic on the unit disc $ \{ z : |z| < 1\}$, which implies $ g(z) = \exp f(z)$ is also analytic on $ \{ z : |z| < 1\}$ and hence can be expanded as a power series $$g(z) = a_0 + ... |
1,705,481 | <blockquote>
<p>$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2}$$</p>
</blockquote>
<p>I have tried the comparison test with $\frac{1}{n}$ and got $0$ with $\frac{1}{n^2}$ I got $\infty$</p>
<p>What should I try?</p>
| Community | -1 | <p>Like others suggest, estimating $\ln(n)$ is probably the best way to visualize it.</p>
<p>An alternative is using integral test(and if you get this problem from first year calculus course, this is probably what you're expected to do),</p>
<p>Not hard to check $f(x)=\frac{\ln(x)}{x^2}$ is continuous and decreasing(... |
1,074,341 | <p>Prove that a Covering map is proper if and only if it is finite-sheeted.</p>
<p>First suppose the covering map $q:E\to X$ is proper, i.e. the preimage of any compact subset of $X$ is again compact. Let $y\in X$ be any point, and let $V$ be an evenly covered nbhd of $y$. Then since $q$ is proper, and $\{y\}$ is comp... | Tommaso Seneci | 320,104 | <p>"$\Rightarrow$" Since a singleton $\{x\}\subset X$ is compact wrt every topology, $q^{-1}(\{x\})$ is compact. Covering gives the property for $q^{-1}(\{x\})$ to be discrete in the sense that there exists an open neighborhood $V \ni x$ such that each connected component of $q^{-1}(V)$ is homeomorphic to $V$ itself. T... |
588,930 | <p>I want help with this question.</p>
<blockquote>
<p>Show that for all $x>0$, $$ \frac{x}{1+x^2}<\tan^{-1}x<x.$$</p>
</blockquote>
<p>Thank you.</p>
| Mikasa | 8,581 | <p>I know a method which is based of using The mean value theorem for functions in Calculus. Let $f(x)=\arctan(x)$. Since $f'=1/1+x^2$ so according to mean value theorem, there is a $\xi\in(0,x)$ such that $$\frac{\tan^{-1}(x)-\tan^{-1}(0)}{x-0}=f'(\xi)$$ But $0<\xi<x$ makes $f'(\xi)$ to be: $$f'(x)<f'(\xi)<... |
2,103,602 | <p>What is the maximum value of
$\displaystyle{{1 + 3a^{2} \over \left(a^{2} + 1\right)^{2}}}$, given that $a$ is a real number, and for what values of $a$ does it occur ?.</p>
| dxiv | 291,201 | <p>Writing it as $\cfrac{1+3a^2}{(a^2+1)^2}= \cfrac{3(a^2+1)-2}{(a^2+1)^2}= \cfrac{3}{a^2+1}-\cfrac{2}{(a^2+1)^2}\,$ gives a quadratic in $x=\cfrac{1}{a^2+1}\,$, with a maximum at $x = \cfrac{3}{4}\,$ ($\iff a^2 = 1/3\,$) of value $\cfrac{3 \cdot 3}{4}- \cfrac{2 \cdot 9}{16} = \cfrac{9}{8}\,$.</p>
|
182,510 | <p>Is there a continuous probability measure on the unit circle in the complex plane - $\sigma$ with full support, such that $\hat{\sigma}(n_k)\rightarrow1$ as $k\rightarrow\infty$ for some increasing sequence of integers $\ n_k$ </p>
| Sean Eberhard | 23,805 | <p>Define a sequence $(X_m)_{m\geq 1}$ of independent $\{0,1\}$-valued random variables by </p>
<p>$$\mathbf{P}(X_m = 0) = p_m > 1/2,$$</p>
<p>where the sequence $p_m\to1$ slowly. Define</p>
<p>$$X = X_1/2 + X_2/4 + \cdots.$$</p>
<p>Then the distribution $\sigma$ of $X$ is continuous in $[0,1]$ provided only $\p... |
182,510 | <p>Is there a continuous probability measure on the unit circle in the complex plane - $\sigma$ with full support, such that $\hat{\sigma}(n_k)\rightarrow1$ as $k\rightarrow\infty$ for some increasing sequence of integers $\ n_k$ </p>
| Robert Israel | 8,508 | <p>Define the sequence $n_k$, a sequence of positive reals $r_k$ and a sequence of nested subsets $A_k$ of the circle $\mathbb T$ as follows. Each $A_k$ will be the union of $2^k$ open intervals of length $r_k$ on which $|e^{in_k t} - 1| < 2^{-k}$, and each of these intervals will contain two intervals of $A_{k+1}$... |
4,309,797 | <p>I have a question which asks me to compute the double integral
<span class="math-container">$$\iint_By^2-x^2\,dA$$</span> where B is the region enclosed by <span class="math-container">$$y=x,y=x+2,y=\frac{2}{x},y=\frac{2}{x}$$</span>I made a change of variables by letting <span class="math-container">$$u=xy \qquad \... | Ninad Munshi | 698,724 | <p>I'm assuming your bounds were actually</p>
<p><span class="math-container">$$y=x \hspace{15 pt} y=x+2 \hspace{15 pt} y = \frac{1}{x} \hspace{15 pt} y = \frac{2}{x}$$</span></p>
<p>It is completely possible to find <span class="math-container">$x+y$</span>, the trick is noticing the degree of the terms. <span class="... |
935,506 | <p>I'm a bit puzzled by this one.</p>
<p>The domain $X = S(0,1)\cup S(3,1)$ (where $S(\alpha, \rho)$ is a circular area with it's center at $\alpha$ and radius $\rho$). So the domain is basically two circles with radius 1 and centers at 0 and 3.</p>
<p>I'm supposed to find analytic function $f$ defined on $X$ where t... | Barry Cipra | 86,747 | <p>I recommend starting from scratch, with the substitution $x^2=\sin\theta$, so that $2x\,dx=\cos\theta \,d\theta$ and $\sqrt{1-x^4}=\sqrt{1-\sin^2\theta}=\cos\theta$. Thus, using a couple of other standard trig identities,</p>
<p>$$\int4x\sqrt{1-x^4}\,dx=\int2\cos^2\theta\,d\theta=\int(1+\cos2\theta)\,d\theta\\
=\t... |
1,904,903 | <p>Taken from Soo T. Tan's Calculus textbook Chapter 9.7 Exercise 27-</p>
<p>Define $$a_n=\frac{2\cdot 4\cdot 6\cdot\ldots\cdot 2n}{3\cdot 5\cdot7\cdot\ldots\cdot (2n+1)}$$
One needs to prove the convergence or divergence of the series $$\sum_{n=1}^{\infty} a_n$$</p>
<p>upon finding the radius of convergence for $\su... | Jack D'Aurizio | 44,121 | <p>By using <a href="https://en.wikipedia.org/wiki/Beta_function" rel="nofollow noreferrer">Euler's beta function</a> we have
$$ \frac{(2n)!!}{(2n+1)!!} = \frac{4^n n!^2}{(2n+1)!} = 4^n B(n+1,n+1) $$
hence:
$$\begin{eqnarray*} \sum_{n=1}^{N}\frac{(2n)!!}{(2n+1)!!} = \int_{0}^{1}\sum_{n=1}^{N}\left(4x(1-x)\right)^n\,dx&... |
246,606 | <p>I have matrix:</p>
<p>$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$</p>
<p>And I want to calculate $\det{A}$, so I have written:</p>
<p>$$
\begin{array}{|cccc|ccc}
1 & 2 & 3 & 4 & 1 & 2 ... | Inquest | 35,001 | <p>$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$
$$
P_1A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
0 & -1 & -3 & -5 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$
$$
... |
246,606 | <p>I have matrix:</p>
<p>$$
A = \begin{bmatrix}
1 & 2 & 3 & 4 \\
2 & 3 & 3 & 3 \\
0 & 1 & 2 & 3 \\
0 & 0 & 1 & 2
\end{bmatrix}
$$</p>
<p>And I want to calculate $\det{A}$, so I have written:</p>
<p>$$
\begin{array}{|cccc|ccc}
1 & 2 & 3 & 4 & 1 & 2 ... | Cameron Buie | 28,900 | <p>The method that you're using works just fine for $3\times 3$ matrices, but fails to work with $n\times n$ matrices for other $n$. You're going to have to do it another way.</p>
<p>For example, expanding the deteriminant along the first column, we find that $$\begin{align}\det A &=1\cdot\det\left[\begin{array}{c... |
3,503,999 | <p>Consider the function <span class="math-container">$$f(x):=\frac{x-x_0}{\Vert x-x_0 \Vert^2} + \frac{x-x_1}{\Vert x-x_1 \Vert^2}$$</span></p>
<p>for two fixed <span class="math-container">$x_0,x_1 \in \mathbb R^2$</span> and <span class="math-container">$x \in \mathbb R^2$</span> as well. </p>
<p>Does anybody know... | Michael Hoppe | 93,935 | <p>Avoid coordinates. Here's a solution that works with all dot products in any dimension: </p>
<p>Define <span class="math-container">$h_k(x)=\frac{x-x_k}{\|x-x_k\|^2}$</span> for <span class="math-container">$k\in\{0,1\}$</span>.
Then we have
<span class="math-container">$\|h_k(x)\|^2=1/\|x-x_k\|^2$</span> and
<span... |
3,002,114 | <blockquote>
<p>Prove that
<span class="math-container">$$
\binom{n}{1}^2+2\binom{n}{2}^2+\cdots + n\binom{n}{n}^2
= n \binom{2n-1}{n-1}.
$$</span></p>
</blockquote>
<p>So
<span class="math-container">$$
\sum_{k=1}^n k \binom{n}{k}^2
= \sum_{k=1}^n k \binom{n}{k}\binom{n}{k}
= \sum_{k=1}^n n \binom{n-1}{k-1} \bino... | Henno Brandsma | 4,280 | <p>A combinatorial proof:</p>
<p>We have <span class="math-container">$n$</span> men and <span class="math-container">$n$</span> women and I want to choose a <span class="math-container">$n$</span>-person committee and a committee president among those, with the condition that the president must be a woman.</p>
<p>On... |
3,002,114 | <blockquote>
<p>Prove that
<span class="math-container">$$
\binom{n}{1}^2+2\binom{n}{2}^2+\cdots + n\binom{n}{n}^2
= n \binom{2n-1}{n-1}.
$$</span></p>
</blockquote>
<p>So
<span class="math-container">$$
\sum_{k=1}^n k \binom{n}{k}^2
= \sum_{k=1}^n k \binom{n}{k}\binom{n}{k}
= \sum_{k=1}^n n \binom{n-1}{k-1} \bino... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$k \binom nk^2=\binom nk\cdot k\binom nk$$</span></p>
<p>For <span class="math-container">$k\ge1,$</span> <span class="math-container">$$k\binom nk=k\cdot\dfrac{n!}{k!\cdot(n-k)!}=n\dfrac{(n-1)!}{(k-1)!\{n-1-(k-1)\}!}=n\binom{n-1}{k-1}$$</span></p>
<p>Now in the identity <span class=... |
3,002,114 | <blockquote>
<p>Prove that
<span class="math-container">$$
\binom{n}{1}^2+2\binom{n}{2}^2+\cdots + n\binom{n}{n}^2
= n \binom{2n-1}{n-1}.
$$</span></p>
</blockquote>
<p>So
<span class="math-container">$$
\sum_{k=1}^n k \binom{n}{k}^2
= \sum_{k=1}^n k \binom{n}{k}\binom{n}{k}
= \sum_{k=1}^n n \binom{n-1}{k-1} \bino... | Marko Riedel | 44,883 | <p>We present a slight variation using formal power series and the
coefficient-of operator. Starting from</p>
<p><span class="math-container">$$\sum_{k=1}^n k {n\choose k}^2
= \sum_{k=1}^n k {n\choose k} [z^{n-k}] (1+z)^n
\\ = [z^n] (1+z)^n \sum_{k=1}^n k {n\choose k} z^k
= n [z^n] (1+z)^n \sum_{k=1}^n {n-1\... |
2,831,731 | <p>I don't know how should i define a homotopy on a set.
I think {{},{a,b,c}} should work but i don't know how to write the homotopy between the identity map and a constant map.
(So sorry for this basic quistion.....)</p>
| E. KOW | 443,898 | <p>Hint: In the trivial topology you mentioned, any map $X\to\left\{a,b,c\right\}$ is continuous.</p>
|
2,751,909 | <blockquote>
<p>Let $f$ be a non-negative differentiable function such that $f'$ is continuous and
$\displaystyle\int_{0}^{\infty}f(x)\,dx$ and $\displaystyle\int_{0}^{\infty}f'(x)\,dx$ exist.</p>
<p>Prove or give a counter example: $f'(x)\overset{x\rightarrow
\infty}{\rightarrow} 0$</p>
</blockquote>
<p><str... | innerz09 | 554,811 | <p>The function is continuous on this interval so it’s integrable by definition.</p>
<p>Using Riemann integral theory you can pick any Pn.</p>
<p>EDIT: I can relate to what’s been said in the comment but it depends on what equivalent definition you use.</p>
|
39,424 | <p>I need to teach an intro course on number theory in 1 month. I was just notified. Since I have never studied it, what are good books to learn it quickly?</p>
| William Stein | 8,441 | <p>Stein's book may be useful (and it is free): <a href="http://wstein.org/ent/" rel="nofollow">http://wstein.org/ent/</a> </p>
|
1,424,124 | <p>If $a,b$ be two positive integers , where $b>2 $ , then is it possible that $2^b-1\mid2^a+1$ ? I have figured out that if $2^b-1\mid 2^a+1$, then $2^b-1\mid 2^{2a}-1$ , so $b\mid2a$ and also $a >b$ ; but nothing else. Please help. Thanks in advance</p>
| Hagen von Eitzen | 39,174 | <p>If $b$ is odd, then $b\mid 2a$ implies $b\mid a$, then $2^b-1\mid 2^a-1$ and hence $2^b-1\nmid 2^a+1$ (as $2^a+1$ is between $2^a-1$ and $2^a-1+(2^b-1)$).</p>
<p>If $b$ is even, $b=2c$ say, then $c\mid a$, hence $2^c-1\mid 2^a-1$ and $2^c-1\mid 2^b-1$. As $c>1$ this shows $d:=\gcd(2^a-1,2^b-1)>1$ (and of cour... |
3,232,341 | <p>How would I show this? I know a directed graph with no cycles has at least one node of outdegree zero (because a graph where every node has outdegree one contains a cycle), but do not know where to go from here.</p>
| Jithin Mathews | 984,391 | <ul>
<li>For undirected graphs:</li>
</ul>
<p>Since all vertices are having an indegree <span class="math-container">$>1$</span>, the count of all the indegrees on all the n different vertices will be <span class="math-container">$\ge n$</span>. However, if a graph (connected or not) has its number of vertices <spa... |
2,098,693 | <p>Full Question: Five balls are randomly chosen, without replacement, from an urn that contains $5$
red, $6$ white, and $7$ blue balls. What is the probability of getting at least one ball of
each colour?</p>
<p>I have been trying to answer this by taking the complement of the event but it is getting quite complex. A... | barak manos | 131,263 | <p>First, use <em>inclusion/exclusion</em> principle in order to count the number of desired combinations:</p>
<ul>
<li>Include the total number of combinations: $\binom{5+6+7}{5}=8568$</li>
<li>Exclude the number of combinations without red balls: $\binom{6+7}{5}=1287$</li>
<li>Exclude the number of combinations with... |
4,635,416 | <p>Let <span class="math-container">$X$</span> be a symmetric random variable, that is <span class="math-container">$X$</span> and <span class="math-container">$-X$</span> have the same distribution function <span class="math-container">$F$</span>. Suppose that <span class="math-container">$F$</span> is continuous and ... | GReyes | 633,848 | <p>If you know how to prove that <span class="math-container">$P(X\le 0)=1/2$</span>, then you can prove that <span class="math-container">$0$</span> satisfies the "inf" definition of median by contradiction. If you assume that <span class="math-container">$\textrm{inf}\, \{x, F(x)\ge 1/2\}>0$</span> then,... |
248,313 | <p>Assume that $f:\mathbb R \rightarrow \mathbb R$ is continuous and $h\in \mathbb R$. Let $\Delta_h^n f(x)$ be a finite difference of $f$ of order $n$, i.e</p>
<p>$$
\Delta_h^1 f(x)=f(x+h)-f(x),
$$
$$
\Delta_h^2f(x)=\Delta_h^1f(x+h)-\Delta_h^1 f(x)=f(x+2h)-2f(x+h)+f(x),
$$
$$
\Delta_h^3 f(x)=\Delta_h^2f(x+h)-\Delta_... | Ewan Delanoy | 15,381 | <p>This is actually a comment too long to fit in the usual format.
WimC’s claim about the uniform convergence case is correct : suppose that $\Gamma(h,x)=\frac{\Delta_h^2f(x)}{h^2} \to 0$, uniformly in $x$ on an interval $[a,b]$. </p>
<p>Let us put $\beta(h)={\sf sup}_{x\in[a,b]}(\big| \Gamma(h,x)\big|)$ for $h>0$.... |
1,338,980 | <p>Suppose you have a set of data $\{x_i\}$ and $\{y_i\}$ with $i=0,\dots,N$. In order to find two parameters $a,b$ such that the line
$$
y=ax+b,
$$
give the best linear fit, one proceed minimizing the quantity
$$
\sum_i^N[y_i-ax_i-b]^2
$$
with respect to $a,b$ obtaining well know results. </p>
<p>Imagine now to desi... | Ross Millikan | 1,827 | <p>If you want to determine $p$ instead of assuming a value for $p$ and fitting $a$ and $b$, you have moved from linear curve fitting to non-linear curve fitting. For linear curve fitting it is not required that the curve be a straight line, but that the model be linear in the parameters. Fitting data to $y=ax^2+bx+c... |
1,345,643 | <p>In an exercise it seems I must use Pascal's triangle to solve this $(z^1+z^2+z^3+z^4)^3$. The result would be $z^3 + 3z^4 + 6z^5 + 10z^ 6 + 12z^ 7 + 12z^ 8 + 10z^ 9 + 6z^ {10} + 3z^ {11} + z^{12}$. But how do I use the triangle to get to that result? Personally I can only solve things like $(x+y)^2$ and $(x+y)^3$.</... | Community | -1 | <p>The expression factors as</p>
<p>$$z^3(1+z)^3(1+z^2)^3=z^3(1+3z+3z^2+z^3)(1+3z^2+3z^4+z^6).$$</p>
<p>Then I see no better way than to perform the multiply (though there is a symmetry)
$$\begin{align}
&1+3z+&3z^2+&z^3\\
&&3z^2+&9z^3+&9z^4+&3z^5\\
&&&&3z^4+&9z^5+&a... |
1,619,371 | <p>I was working on a problem and reduced it to evaluating</p>
<p>$$\int_{0}^{1}\sqrt{1+x^a}\,dx~~a>0$$</p>
<p>your suggestion? Thanks</p>
| Tom-Tom | 116,182 | <p>We have
$$ I=\int_0^1\sqrt{1+x^a}\,\mathrm dx=\int_0^1\sum_{k=0}^\infty\binom{1/2}{k}x^{ka}\mathrm dx,$$
where
$$\binom{1/2}{k}=\frac{(1/2)(1/2-1)\dots(1/2-k+1)}{k!}.$$
We get
$$I=\sum_{k=0}^\infty\binom{1/2}{k}\frac1{1+ka}=\sum_{k=0}^\infty.$$
Let us rewrite $(1/2)(1/2-1)\cdots(1/2-k+1)=(-1)^k(-\frac12)_k$ where th... |
1,030,335 | <blockquote>
<p>Let <span class="math-container">$n$</span> and <span class="math-container">$r$</span> be positive integers with <span class="math-container">$n \ge r$</span>. Prove that:</p>
<p><span class="math-container">$$\binom{r}{r} + \binom{r+1}{r} + \cdots + \binom{n}{r} = \binom{n+1}{r+1}.$$</span></p>
</bloc... | Mike | 193,928 | <p>Let $E = \{1,2, \dots , n+1\}$. The number $\binom{n+1}{r+1}$ is the number of subsets $A$ of $E$ with $r + 1$ elements. </p>
<p>Classify these subsets $A$ according to their largest element $b$, which can be any number among $r + 1$, $r + 2$, ..., $n + 1$. The number of $(r+1)$-element subsets of $E$ with largest ... |
1,334,527 | <p>The integral in hand is
$$
I(n) = \frac{1}{\pi}\int_{-1}^{1} \frac{(1+2x)^{2n}}{\sqrt{1-x^2}}\, dx
$$
I dont know whether it has closed-form or not, but currently I only want to know its asymptotic behavior. Setting $x=\cos\theta$, then
$$
I(n) = \frac{1}{\pi}\int_{0}^{\pi/2} \Big[(1+2\cos\theta)^{2n}+(1-2\cos\theta... | Dr. Wolfgang Hintze | 198,592 | <p>This is not a solution but a comment following the comment of Claude pointing out an interesting generating function and a shorter recursion.</p>
<p>In <a href="http://oeis.org/A082758" rel="nofollow noreferrer">http://oeis.org/A082758</a> Paul Barry gives the simple g.f.</p>
<p><span class="math-container">$$g(x)=\... |
2,659,448 | <p>The following question is an exercise from Munkres' Analyis on Manifolds (Chapter 4 - Section 20):</p>
<p>Consider the vectors $a_i$ in $R^3$ such that:</p>
<p>$[a_1\ a_2\ a_3\ a_4] = \begin{bmatrix} 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 1 & 1 & 2 & 0 \end{bmatrix}$</p>
<p>Let $V$ ... | amd | 265,466 | <p>$\mathbb R^2$ is not a subspace of $\mathbb R^3$, so there’s no way for any of these vectors to combine to span the former. On the other hand, two of them <em>could</em> span a two-dimensional subspace of $\mathbb R^3$. There are many such subspaces besides the one that consists of vectors with last coordinate equal... |
452,306 | <p>I am trying to be able to find the radius of a cone combined with a cylinder.
see my other question
(Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder? part2 )</p>
<p>I have a volume calculation that Has been reduced as far as I know how to.</p>
... | John | 30,229 | <p>Let $B=\{g^{n}:n\in\mathbb{Z}\}$. Clearly $\bar{B}$ is also a subgroup of $G$. </p>
<p>If $1$ is an isolated point in $\bar{B}$ then all points of $\bar{B}$ are isolated, which means that $\bar{B}$ is compact and discrete, and hence finite. Thus, $g^{n}=1$ for some $n$ and so $\bar{A}$ is a subgroup of $G$.</p>
<p... |
127,086 | <p>I am struggling with an integral pretty similar to one already resolved in MO (link: <a href="https://mathoverflow.net/questions/101469/integration-of-the-product-of-pdf-cdf-of-normal-distribution">Integration of the product of pdf & cdf of normal distribution </a>). I will reproduce the calculus bellow for the ... | Hugh Perkins | 115,210 | <p>When you do the change of variable, you are doing:</p>
<p>\begin{align}
\left[y\longmapsto f\frac{\sqrt{1+B^2}}{B}-\frac{A}{B\sqrt{1+B^2}}\Longrightarrow df=\frac{B}{\sqrt{1+B^2}}\,dy\right]
\end{align}</p>
<p>.. which is correct. However, we need to apply the same change of variable to the bounds, ie to $-\infty$... |
332,760 | <blockquote>
<p>For an odd prime, prove that a primitive root of $p^2$ is also a primitive root of $p^n$ for $n>1$. </p>
</blockquote>
<p>I have proved the other way round that any primitive root of $p^n$ is also a primitive root of $p$ but I have not been able to solve this one. I have tried the usual things th... | awllower | 6,792 | <p>Let us give a more elementary answer, while still using some binomial theorem. But we shall employ no more than a binomial lemma. It states that, for a prime $p$, and an integer $1\leq a\leq p-1$, we have $p\mid \binom{p}{a}$.<br>
Now you know that $a$ is a primitive root of $p^2$, so the order of $a$ modulo $p^2$ i... |
581,257 | <p>I would like to see a proof of when equality holds in <a href="https://en.wikipedia.org/wiki/Minkowski_inequality" rel="nofollow noreferrer">Minkowski's inequality</a>.</p>
<blockquote>
<p><strong>Minkowski's inequality.</strong> If <span class="math-container">$1\le p<\infty$</span> and <span class="math-contain... | Samantha Wyler | 723,878 | <p>Let <span class="math-container">$q$</span> be a conjugate exponent of <span class="math-container">$p$</span>, meaning <span class="math-container">$\frac{1}{q} + \frac{1}{p} = 1$</span>. Now <span class="math-container">$\|f + g\|_p = \|f\|_p + \|g\|_p$</span> iff <span class="math-container">$\|f+ g\|_p^p = \|f +... |
3,386,999 | <p>How can I ind the values of <span class="math-container">$n\in \mathbb{N}$</span> that make the fraction <span class="math-container">$\frac{2n^{7}+1}{3n^{3}+2}$</span> reducible ?</p>
<p>I don't know any ideas or hints how I solve this question.</p>
<p>I think we must be writte <span class="math-container">$2n^{7}... | saulspatz | 235,128 | <p>I have a solution, but I'm sure there's a better way to do this. The greatest common divisor <span class="math-container">$g$</span> of of <span class="math-container">$2n^7+1$</span> and <span class="math-container">$3n^3+2$</span> must also divide <span class="math-container">$$3(2n^7+1)-2n^4(3n^3+2)=3-4n^4$$</sp... |
3,631,042 | <p>Probably, <span class="math-container">$y = x^2$</span> plots a parabola only given certain assumptions that structure a cartesian coordinate plane, and it does not plot a parabola in e.g. the polar coordinate plane.</p>
<p>Now, why exactly does a parabola share an equation with the area of a square? 'Why' here is ... | marty cohen | 13,079 | <p>The parabola
is the graph of a function
which plots the square of
the ordinate.
That happens to be
the same as the function
with gives the area
of a square given the side.</p>
<p>The name should
give you a hint why.</p>
|
3,360,914 | <p>Let <span class="math-container">$A$</span>, <span class="math-container">$B$</span> and <span class="math-container">$C$</span> be symmetric, positive semi-definite matrices. Is it true that
<span class="math-container">$$ \|(A + C)^{1/2} - (B + C)^{1/2}\| \leq \|A^{1/2} - B^{1/2}\|,$$</span>
in either the 2 or Fro... | bsbb4 | 337,971 | <p>In short, the number of lattice points of given bounded magnitude grows linearly, giving a contribution proportional to <span class="math-container">$n \cdot \frac{1}{n^2} = \frac{1}{n}$</span>, making the series diverge.</p>
<p>I give a proof that <span class="math-container">$\sum_{\omega \in \Lambda^*} \omega^{-... |
2,020,128 | <p>For $r$ is a real number, I can write $r \in \mathbb{R}$.</p>
<p>For $\varepsilon$ is an infinitesimal, I'd like to write something like $\varepsilon \in something$ Is there a symbol for "the set of infinitesimals"? Or alternatively, a commonly used abbreviation for "infinitesimal"?</p>
<p>For $H$ is an infinite (... | achille hui | 59,379 | <p>In non-standard analysis,
a <a href="https://en.wikipedia.org/wiki/Monad_%28non-standard_analysis%29" rel="nofollow noreferrer">monad</a> (also called halo) is the set of points infinitesimally close to a given point.</p>
<p>On model for extending real numbers is the <a href="https://en.wikipedia.org/wiki/Hyperrea... |
2,020,128 | <p>For $r$ is a real number, I can write $r \in \mathbb{R}$.</p>
<p>For $\varepsilon$ is an infinitesimal, I'd like to write something like $\varepsilon \in something$ Is there a symbol for "the set of infinitesimals"? Or alternatively, a commonly used abbreviation for "infinitesimal"?</p>
<p>For $H$ is an infinite (... | Mikhail Katz | 72,694 | <p>For infinite numbers there is a fairly common notation in the context of integers $\mathbb N$ and hyperintegers ${}^\ast\mathbb N$. Namely, a hyperinteger is infinite if it belongs to the set complement $${}^\ast\mathbb N\setminus\mathbb N.$$ This is not particularly elegant but introducing special notation for thi... |
936,200 | <p>Suppose that x_0 is a real number and x_n = [1+x_(n-1)]/2 for all natural n. Use the Monotone Convergence Theorem to prove x_n → 1 as n grows.</p>
<p>Can someone please help me? I don't know what to assume since I don't know if it is increasing or decreasing when x_0 < 1 and when x_0 > 1.
Any hint/help would rea... | Amitai Yuval | 166,201 | <p>"You belong to me", "I belong to you"... Possession always causes confusion.</p>
<p>A set, by definition, is a collection of elements. A given element $x$ can either be <em>in</em> a given collection, or <em>not in</em> the collection. I read "$y\in\{0,1,2,3\}$" as "$y$ is one of the elements $0,1,2,3$". No belongi... |
1,465,627 | <p>The problem is to maximize the determinant of a $3 \times 3$ matrix with elements from $1$ to $9$.<br>
Is there a method to do this without resorting to brute force?</p>
| copper.hat | 27,978 | <p>Swapping rows and columns leaves the absolute value of the determinant unchanged, so we can
assume that the middle cell contains 1. Then, since the absolute value of the determinant is unaffected by taking transposes (and swapping the top & bottom rows or
the left and right columns), we can assume that the 2 occ... |
1,465,627 | <p>The problem is to maximize the determinant of a $3 \times 3$ matrix with elements from $1$ to $9$.<br>
Is there a method to do this without resorting to brute force?</p>
| gnoodle | 934,371 | <p>The question asks for alternatives to brute force, but just to illustrate the difficulties of using brute force, here is python code for brute forcing it:</p>
<pre><code>import numpy as np
import itertools
import time
MATRIX_SIZE = 3 #3 for a 3x3 matrix, etc
best_matrices = []
highest_det = 0
start_tim... |
1,039,563 | <p>Whether the graphs G and G' given below are isomorphic?</p>
<p><img src="https://i.stack.imgur.com/0evn6.jpg" alt="enter image description here"></p>
| Hagen von Eitzen | 39,174 | <p>As improper integral, this should be the
$$ \lim_{a\to+\infty}\underbrace{\int_0^af(t)\sin t\,\mathrm dt}_{=:F(t)}.$$
Assume $f$ is a nonconstant polynomial. Then for $a$ big enough, $f(t)>1$ for all $t>a$ or $f(t)<-1$ for all $t>a$.
Hence $|F((k+1)\pi)-F(k\pi)|>\left|\int_{k\pi}^{(k+1)\pi}\sin t\,\m... |
966,798 | <p>How I solve the following equation for $0 \le x \le 360$:</p>
<p>$$
2\cos2x-4\sin x\cos x=\sqrt{6}
$$</p>
<p>I tried different methods. The first was to get things in the form of $R\cos(x \mp \alpha)$:</p>
<p>$$
2\cos2x-2(2\sin x\cos x)=\sqrt{6}\\
2\cos2x-2\sin2x=\sqrt{6}\\
R = \sqrt{4} = 2 \\
\alpha = \arctan \f... | Andrew D. Hwang | 86,418 | <p>Here's a general answer:</p>
<p>The definitions of analysis are formulated in terms of conditions depending on a positive real number $\delta$ that "remain true if $\delta$ is made smaller". For example, the precise definition of the statement $\lim\limits_{x \to a} f(x) = L$ includes the condition
$$
\text{If $|x ... |
245,312 | <p>Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-<em>homogeneous</em> if</p>
<ol>
<li>$|X| \geq \kappa$, and</li>
<li>whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$ and $\psi:A\to B$ is a bijective map, then there is a homeomorphism $\varphi: X\to X... | Will Brian | 70,618 | <p>The sort of space you describe is usually called <em>strongly</em> $\kappa$-homogeneous. If you google that phrase you will find some interesting results about these kinds of spaces (mostly concerning how this property relates to other homogeneity properties).</p>
<p>The earliest reference I could find to strongly ... |
2,405,205 | <p>The Wikipedia article on <a href="https://en.wikipedia.org/wiki/Fraction_(mathematics)#Complex_fractions" rel="nofollow noreferrer">Fractions</a> says:</p>
<blockquote>
<p>If, in a complex fraction, there is no unique way to tell which fraction lines takes precedence, then this expression is improperly formed, be... | Franklin Pezzuti Dyer | 438,055 | <p>Yes, but that's because Wolfram Alpha does that <em>by convention</em>. When you type something like that in, you're probably "confusing" WA, and so it has to use its last resort, which is to apply the operations in the order in which they are typed. Even though WA can interpret it, it's still bad mathematics to wri... |
985,212 | <p>Can you row reduce the matrix before computing $\det(\lambda I-A)$? Will this still give an equivalent characteristic polynomial?</p>
| hardmath | 3,111 | <p>A typical presentation of <a href="http://en.wikipedia.org/wiki/Elementary_matrix#Operations" rel="noreferrer">elementary row operations</a> sets out three kinds:</p>
<p>(1) Multiply a row by a nonzero scalar.</p>
<p>(2) Add a multiple of one row to another.</p>
<p>(3) Swap two rows.</p>
<p>The effects on the de... |
2,030,547 | <p>The following expression came up in a proof I was reading, where it is said "It is easily shown: $$\lim_{x\to\infty} x(1-\frac{\ln (x-1)}{\ln x})=0."$$</p>
<p>Unfortunately I'm not having an easy time showing it. I guess it should come down to showing that the ratio $\frac{\ln (x-1)}{\ln x}$ converges to 1 superlin... | egreg | 62,967 | <p>Let me start with a different example. Consider all maps from a set $X$ to $\mathbb{R}$ and push them together in a set, say $M(X,\mathbb{R})$.</p>
<p>This set can be given a structure of vector space by
$$
f+g\colon x\mapsto f(x)+g(x),
\qquad
\alpha f\colon x\mapsto \alpha f(x)
$$
for $f,g\in M(X,\mathbb{R})$ and ... |
2,359,621 | <p>Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where</p>
<p>$$f(x,y):=\begin{cases}
\frac{x^3}{x^2+y^2} & \textit{ if } (x,y)\neq (0,0) \\
0 & \textit{ if } (x,y)= (0,0)
\end{cases} $$</p>
<p>If one wants to show the continuity of $f$, I mainly want to show that </p>
<p>$$ \lim\limits_... | Mark Viola | 218,419 | <p>Note that we have </p>
<p>$$\left|\frac{x^3}{x^2+y^2}\right|\le |x|$$</p>
<p>The limit as $(x,y)\to(0,0)$ is therefore $0$.</p>
<hr>
<p>The limit $\lim_{(x,y)\to(0,0)}f(x,y)=L$ means that for all $\epsilon>0$, there exists a deleted neighborhood $N_{0,0}$ (e.g., there exists a $\delta>0$, such that $0<\... |
4,074,718 | <p>The angle bisectors of <span class="math-container">$\angle B$</span> and <span class="math-container">$\angle C_{ex}$</span> intersect at point <span class="math-container">$E$</span>. If <span class="math-container">$\angle A=70^\circ$</span>, what is <span class="math-container">$\angle E$</span> equal to?</p>
<p... | lhf | 589 | <p>The polynomial with integer coefficients with least degree that has <span class="math-container">$\frac{\sqrt{3}}{2}$</span> as a root is <span class="math-container">$4 x^2 - 3$</span>.</p>
<p>Every polynomial with rational coefficients that has <span class="math-container">$\frac{\sqrt{3}}{2}$</span> as a root is ... |
4,377,771 | <blockquote>
<p>Find all natural numbers <span class="math-container">$a, b, c$</span> such that <span class="math-container">$a\leq b\leq c$</span> and <span class="math-container">$a^3+b^3+c^3-3abc=2017$</span>.</p>
</blockquote>
<h2><strong>My Attempt</strong></h2>
<p><span class="math-container">$$a^3+b^3+c^3-3abc=... | Saturday | 1,015,303 | <p><span class="math-container">$$a^2+b^2+c^2-ab-bc-ca=\frac{1}{2} \bigg( (a-b)^2 + (b-c)^2 + (c-a)^2 \bigg)=1$$</span>
<span class="math-container">$$ \implies (a-b)^2 + (b-c)^2 + (c-a)^2 =2$$</span></p>
<p>So, two between <span class="math-container">$a,b,c$</span> are equal and the other has the difference of <span ... |
4,377,771 | <blockquote>
<p>Find all natural numbers <span class="math-container">$a, b, c$</span> such that <span class="math-container">$a\leq b\leq c$</span> and <span class="math-container">$a^3+b^3+c^3-3abc=2017$</span>.</p>
</blockquote>
<h2><strong>My Attempt</strong></h2>
<p><span class="math-container">$$a^3+b^3+c^3-3abc=... | Steffen Jaeschke | 629,541 | <p><a href="https://i.stack.imgur.com/DtQ6U.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/DtQ6U.png" alt="enter image description here" /></a></p>
<p>These are surfaces over {a,b} expected reals. The calculations are straightforward.</p>
<p><a href="https://i.stack.imgur.com/YcjRp.png" rel="nofollo... |
4,377,771 | <blockquote>
<p>Find all natural numbers <span class="math-container">$a, b, c$</span> such that <span class="math-container">$a\leq b\leq c$</span> and <span class="math-container">$a^3+b^3+c^3-3abc=2017$</span>.</p>
</blockquote>
<h2><strong>My Attempt</strong></h2>
<p><span class="math-container">$$a^3+b^3+c^3-3abc=... | Shridhar Sharma | 988,232 | <p><a href="https://i.stack.imgur.com/BESmD.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BESmD.jpg" alt="This is the Solution" /></a></p>
<p>It was an easy question, just one important result and a bit of number theory was needed</p>
|
4,003,948 | <p>In the Book that I'm reading (Mathematics for Machine Learning), the following para is given, while listing the properties of a matrix determinant:</p>
<blockquote>
<p>Similar matrices (Definition 2.22) possess the same determinant.
Therefore, for a linear mapping <span class="math-container">$Φ : V → V$</span> all ... | Alekos Robotis | 252,284 | <p>Given a linear transformation <span class="math-container">$T:V\to V$</span>, if we choose a basis <span class="math-container">$\mathcal{B}$</span> for <span class="math-container">$V$</span> we get an induced matrix <span class="math-container">$\Phi_{\mathcal{B}}:\Bbb{R}^n\to \Bbb{R}^n$</span> representing the li... |
2,227,047 | <p>For any $x=x_1, \dotsc, x_n$, $y=y_1, \dotsc, y_n$ in $\mathbf E^n$, define $\|x-y\|=\max_{1 \le k \le n}|x_k-y_k|$. Let $f\colon\mathbf E^n \to \mathbf E^n$ be given by $f(x)=y$, where $y_k= \sum_{i=1}^n a_{ki} x_i + b_k$ where $k =1,2, \dotsc,n$. Under what conditions is $f$ a contraction mapping?</p>
<p>Any hint... | MCS | 378,686 | <p>Praise be! I just figured it out!</p>
<p>$\int_{-\infty}^{\infty}\frac{dx}{\left|1+\alpha x^{2}\right|}$</p>
<p>is the $L^{2}$ norm of $\left(1+\alpha x^{2}\right)^{-\frac{1}{2}}$.
The fourier transform ($\int_{-\infty}^{\infty}f\left(x\right)e^{-2\pi i\xi x}dx$ convention) of $\left(1+\alpha x^{2}\right)^{-\frac... |
7,108 | <p>I need help to make a diagram(square), someone can teach me how to do? </p>
<p>I know that I could look at the posts to see a model, but I am stopped for 7 days to edit questions</p>
<p>Thanks in advance.</p>
| Willie Wong | 1,543 | <ol>
<li><p>As long as the answer is on topic, answers the question, is not offensive, and not spam, then most users won't have a problem with you posting a YouTube Link. </p></li>
<li><p>It is perhaps better, however (since the YouTube link URLs are usually rather cryptic, and I often hesitate to click on random links... |
959,393 | <p>Let's use the following example:</p>
<p>$$17! = 16!*17 \approx 2 \cdot 10^{13} * 17 = 3.4 \cdot 10^{14} $$</p>
<p>Are you allowed to do this? I am in doubt whether or not this indicates that $17! = 3.4 \cdot 10^{14}$, which is obviously not true, but I think it doesn't.</p>
| C_Guy | 179,154 | <p>It is allowed. It is like asking: does
$$10 = 5*2 \geq 5$$
indicate that $10=5$?</p>
|
186,553 | <p><strong>Problem:</strong> Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$, where $k$ is a positive constant.</p>
<p>I'm stumped. I've tried to apply several different convergence tests, but still can't figure this one out.</p>
| André Nicolas | 6,312 | <p>It seems likely that you are expected to use the "guess and check" procedure. </p>
<p>The product is $144$, not too many <strong>integer</strong> possibilities for $a$, $b$, $c$.
Without loss of generality you may assume $a \ge b\ge c$. Also, the sum of squares condition tells you that $a \le 12$.</p>
<p>Why <em>i... |
68,618 | <p>My calculus teacher assigns us online homework to do. He never went over any question that looks like this (he in fact said we shouldn't be concerned with this):<img src="https://i.stack.imgur.com/N31ML.png" alt="What is this?"></p>
<p>Yet, I need to answer this right to progress with my homework. It stinks becau... | André Nicolas | 6,312 | <p>Without loss of generality we may assume that $f(c)$ is positive. </p>
<p>Let $\epsilon =f(c)$. By the definition of continuity of $f$ at $c$, there is a $\delta>0$ such that if $|x-c|<\delta$ (and $a\lt c-\delta$, and $c+\delta \lt b$, to make sure we stay in our interval) then $|f(x)-f(c)|<\epsilon$.</p... |
68,618 | <p>My calculus teacher assigns us online homework to do. He never went over any question that looks like this (he in fact said we shouldn't be concerned with this):<img src="https://i.stack.imgur.com/N31ML.png" alt="What is this?"></p>
<p>Yet, I need to answer this right to progress with my homework. It stinks becau... | hmakholm left over Monica | 14,366 | <p>I think the question is quite confusingly worded. It took me several minutes to figure out what it meant -- and it's not as if I don't know the subject matter well.</p>
<p>What must be going on is that you're supposed to imagine reading something like this in a proof:</p>
<blockquote>
<p>bla bla bla, and therefo... |
1,024,794 | <p>I have this equation: $x+7-(\frac{5x}8 + 10) = 3 $</p>
<p>I've used step-by-step calculators online but I simply don't understand it. Here is how I've tried to solve the problem: </p>
<p>$$x+7-\left(\frac{5x}8+10\right) = x + 7 - \frac{5x}8 - 10 = 3$$</p>
<p>$$x + 7 - \frac{5x}8 - 10 + 10 = 3 + 10$$</p>
<p>$$x +... | Arian | 172,588 | <p>Consider the following propositions:
$$P(z): "\xi(k(z))=k(\xi(z))=0"$$
$$Q(z): "k(z)=\zeta(z)=0"$$
$$R(z): "\text{RH is false}"$$
So your problem can be stated as if the following equivalence relation is true:
$$(P(z)\to Q(z))\leftrightarrow R(z)$$
Now the above biconditional statement is true if and only if $R(z)\... |
70,429 | <p>For a $n$-dim smooth projective complex algebraic variety $X$, we can form the complex line bundle $\Omega^n$ of holomorphic $n$-form on $X$. Let $K_X$ be the divisor class of $\Omega^n$, then $K_X$ is called the canonical class of $X$.</p>
<p><strong>Question</strong>: Is homology class of $K_X$ in $H_{2n-2}(X)$ ... | Tim Perutz | 2,356 | <p>This answer is about the case of complex surfaces $X$ and their diffeomorphisms (all my diffeos are assumed to be orientation-preserving!). </p>
<p><b>(1) Examples of self-diffeomorphisms that reverse the sign of the canonical class.</b> </p>
<p>Take $X=\mathbb{C}P^1\times \mathbb{C}P^1$. Let $\tau$ be reflection ... |
2,252,206 | <p>This question is related to <a href="https://math.stackexchange.com/questions/1574196/units-of-group-ring-mathbbqg-when-g-is-infinite-and-cyclic">this</a> one, in that I am asking about the same problem, but not necessarily about the same aspect of the problem.</p>
<p>I need to identify all units of the group ring ... | Community | -1 | <p>The key word is <a href="https://en.wikipedia.org/wiki/Polarization_identity" rel="nofollow noreferrer">polarization identity</a>. </p>
|
201,820 | <p>Suppose we have in <code>~/time-data/time-data.org</code> the following data:</p>
<pre><code>* Parent1
:LOGBOOK:
CLOCK: [2019-07-09 Tue 00:00]--[2019-07-09 Tue 00:20] => 0:20
:END:
** Child1
:LOGBOOK:
CLOCK: [2019-07-10 Wed 00:02]--[2019-07-10 Wed 00:40] => 0:38
:END:
** Child2
:LOGBOOK:
CLOCK: [2019-07-11 ... | M.R. | 403 | <p>If Mathematica was perfect, <code>DateHistogram[..., "Day", "Hour"]</code> would work, making what you want a one-liner. I believe that a <code>DateInterval</code> function might be coming in the next version (12.1) which would presumably work with <code>DateHistogram</code> and <code>TimelinePlot</code>.</p>
<p>Al... |
244,769 | <p>I am DMing a game of DnD and one of my players is really into fear effects, which is cool, but the effect of having monsters suffer from the "panicked" condition gets tedious to render via dice rolls.</p>
<p>The rule is, on the battle grid the monster will run for 1 square in a random direction, then from ... | A.G. | 7,060 | <p>Here is a solution that uses <code>AffineTransform</code> and <code>Solve</code>s for coefficients. You can specify a scale.</p>
<pre><code>{a, b, c, p} = {{0.2, 0.8}, {0.1, 0.15}, {0.8, 0.25}, {0.6, 0.7}};
(* Specify scale here; for example, u and v's lengths are 1/2 *)
scale = 1/2;
(* Define the parameters *)
Cle... |
3,858,414 | <p>I need help solving this task, if anyone had a similar problem it would help me.</p>
<p>The task is:</p>
<p>Calculate using the rule <span class="math-container">$\lim\limits_{x\to \infty}\left(1+\frac{1}{x}\right)^x=\large e $</span>:</p>
<p><span class="math-container">$\lim_{x\to0}\left(\frac{1+\mathrm{tg}\: x}{... | fleablood | 280,126 | <p>Let <span class="math-container">$x,y\in A$</span>.</p>
<p>Either <span class="math-container">$x R y$</span> of <span class="math-container">$x \not R y$</span>.</p>
<p>Case 1: <span class="math-container">$x R y$</span>.</p>
<p><span class="math-container">$[x]=\{z\in A| z R x\}$</span>, <span class="math-contain... |
1,634,741 | <p>$22+22=4444$</p>
<p>$43+46=618191$</p>
<p>$77+77=?$</p>
<p>What should come in place of $?$</p>
<p>I cannot see any logic in $43+46=618191$. Is there any?</p>
| stackoverflowuser2010 | 9,177 | <p>The problem asks for a closed-form solution to:</p>
<p><span class="math-container">$$\sum_{i=4}^{N} 5^i = 5^4 + 5^5 + ... + 5^N$$</span></p>
<p>The OP's original intuition was correct:
<span class="math-container">$$\sum_{i=4}^{N} = \sum_{i=0}^{N} 5^i - \sum_{i=0}^{3} 5^i$$</span></p>
<p>More generally, for summing... |
4,052,760 | <blockquote>
<p>Prove that <span class="math-container">$\int\limits^{1}_{0} \sqrt{x^2+x}\,\mathrm{d}x < 1$</span></p>
</blockquote>
<p>I'm guessing it would not be too difficult to solve by just calculating the integral, but I'm wondering if there is any other way to prove this, like comparing it with an easy-to-c... | saulspatz | 235,128 | <p>HINT:</p>
<p><span class="math-container">$x^2+x<x^2+x+\frac14=\left(x+\frac12\right)^2$</span></p>
|
4,052,760 | <blockquote>
<p>Prove that <span class="math-container">$\int\limits^{1}_{0} \sqrt{x^2+x}\,\mathrm{d}x < 1$</span></p>
</blockquote>
<p>I'm guessing it would not be too difficult to solve by just calculating the integral, but I'm wondering if there is any other way to prove this, like comparing it with an easy-to-c... | Unit | 196,668 | <p>Well, you could use the AM-GM inequality:
<span class="math-container">$$\sqrt{x^2 + x} = \sqrt{x(x+1)} < \frac{x + (x+1)}{2} = x + \frac{1}{2}$$</span>
and then
<span class="math-container">$$\int_0^1 \sqrt{x^2 + x} \, dx < \int_0^1 x + \frac{1}{2} \, dx = \frac{1}{2} + \frac{1}{2} = 1.$$</span></p>
|
1,476,456 | <p>How many positive (integers) numbers less than $1000$ with digit sum to $11$ and divisible by $11$?</p>
<p>There are $\lfloor 1000/11 \rfloor = 90$ numbers less than $1000$ divisible by $11$.</p>
<p>$N = 100a + 10b + c$ where $a + b + c = 11$ and $0 \le a, b, c \le 9$</p>
<p>I got $\binom{13}{2} - 9 = 69$ soluti... | JMoravitz | 179,297 | <p>Digitsum is related to the modulo 9 operation. A weakening of the conditions given is that you are counting how many $0\leq n\leq 1000$ satisfy the coungruencies:</p>
<p>$\begin{array}{} n\equiv 2\pmod{9}\\
n\equiv 0\pmod{11}\end{array}$</p>
<p>By the <a href="https://en.wikipedia.org/wiki/Chinese_remainder_theor... |
124,660 | <p>I'm solving a fairly simple equation :</p>
<pre><code>w[p1_, p2_, xT_] :=
94.8*cv*p1*y[(p1 - p2)/p1, xT]*Sqrt[(p1 - p2)/p1*mw/t1];
y[r_, xT_] := 1 - (1.4 r)/(3 xT*γ) /. γ -> 1.28;
sol = NSolve[{w[p1, p2, 0.66] == 30, p2 == 1.07}, {p1, p2}] /. {cv ->
1.77, t1 -> 318, mw -> 38};
</code></pre>
<p>Ma... | Feyre | 7,312 | <p>The discontinuity makes it so that</p>
<blockquote>
<p>NSolve was unable to solve the system with inexact coefficients</p>
</blockquote>
<p>The solution of <code>NSolve[]</code> in this case is to:</p>
<blockquote>
<p>The answer was obtained by solving a corresponding exact system and numericizing the result.... |
232,777 | <p>Let $F$ be an ordered field.</p>
<p>What is the least ordinal $\alpha$ such that there is no order-embedding of $\alpha$ into any bounded interval of $F$?</p>
| JMP | 70,355 | <p>A union of $\dbinom kr$ sets with cardinality $r$ has cardinality $\ge k$, and there are two of them due to there being $\ge k$ elements in $X$.</p>
|
1,343,722 | <p>Note: I am looking at the sequence itself, not the sequence of partial sums.</p>
<p>Here's my attempt...</p>
<p>Setting up:</p>
<p>$$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>Simplifying:</p>
<p>$$\left\{\frac{2n+2}{2n+1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>... | MCT | 92,774 | <p>Hint:</p>
<p>$$\sum_{n=1}^{\infty} \frac{n^2}{2^n} = \sum_{i=1}^{\infty} (2i - 1) \sum_{j=i}^{\infty} \frac{1}{2^j}.$$</p>
<p>Start by using the geometric series formula on $\displaystyle \sum_{j=1}^{\infty} \frac{1}{2^j}$ to simplify the double series into a singular series. Then you will have a series that looks... |
1,343,722 | <p>Note: I am looking at the sequence itself, not the sequence of partial sums.</p>
<p>Here's my attempt...</p>
<p>Setting up:</p>
<p>$$\left\{\frac{2(n+1)}{2(n+1)-1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>Simplifying:</p>
<p>$$\left\{\frac{2n+2}{2n+1}\right\} - \left\{\frac{2n}{2n-1}\right\}$$</p>
<p>... | Tom-Tom | 116,182 | <p>For $x$ such that $|x|<1$ we have
$$f(x)=\sum_{n=0}^\infty x^n=\frac1{1-x}.$$
The derivative of $f$ is
$$f'(x)=\sum_{n=0}^\infty nx^{n-1}=\frac{1}{\left(1-x\right)^2},$$
such that
$$xf'(x)=\sum_{n=1}^\infty nx^n=\frac x{\left(1-x\right)^2}.$$
A second derivative gives
$$xf''(x)+f'(x)=\sum_{n=1}^\infty n^2x^{n-1}... |
81,728 | <p>The question is to compute or estimate the following probabilty.</p>
<p>Suppose that you have $N$ (e.g. $30$) tasks, each of which repeats every $t$ min (e.g. $30$ min) and lasts $l$ min (e.g. $5$ min). If the tasks started at uniformly random point in time yesterday, what is the probability that there is a time to... | sbacallado | 19,438 | <p>Each block of $t$ minutes today will have the same pattern of tasks. Think of the middle point of a task as a uniform random variable on the unit circle. Then, $m$ tasks overlap if their corresponding random variables fall within a ball of radius $l/t$. This is a continuous generalization of the birthday problem. </... |
81,728 | <p>The question is to compute or estimate the following probabilty.</p>
<p>Suppose that you have $N$ (e.g. $30$) tasks, each of which repeats every $t$ min (e.g. $30$ min) and lasts $l$ min (e.g. $5$ min). If the tasks started at uniformly random point in time yesterday, what is the probability that there is a time to... | Alex Levine | 23,247 | <p>Probability that a task Is operating at a given instant = l/t.</p>
<p>Probability that a task isn’t operating at a given instant = (t-l)/t.</p>
<p>Probability that at least m of N tasks are operating at a given instant is C(N,i)[ (l/t)^i][(t-l)/t]^(N-i) summed for i=m to N where C(N,i) is the combination of N obje... |
611,361 | <p>Let's have function $f$ defined by:
$$f(x)=2\sum_{k=1}^{\infty}\frac{e^{kx}}{k^3}-x\sum_{k=1}^{\infty}\frac{e^{kx}}{k^2},\quad x\in(-2\pi,0\,\rangle$$
My question:
Can somebody expand it into a correct Maclaurin series, but using an unconventional way? Conventional is e.g. using $n$-th derivative of $f(x)$ in zero... | Farshad Nahangi | 50,728 | <p>let $g(x)=\sum_{k=1}^{\infty}\frac{e^{kx}}{k^3}$ then your problem is converted to
$$f(x)=2g(x)-xg'(x)$$ then
\begin{align*}
f'(x)&=g'(x)-xg''(x)\\
f''(x)&=-x\cdot g'''(x)
\end{align*}
where
$$g'''(x)=\sum_{k=1}^{\infty} e^{kx}$$
Thus
$$f''(x)=-x\cdot\sum_{k=1}^{\infty} e^{kx}=-x\cdot\sum_{k=1}^{\infty}\sum_... |
3,963,884 | <p>Suppose <span class="math-container">$(X_n)_n$</span> are i.i.d. random variables and let <span class="math-container">$W_n = \sum_{k=1}^n X_k$</span>. Assume that there exist <span class="math-container">$u_n>0 , v_n \in \mathbb{R}$</span> such that</p>
<p><span class="math-container">$$\frac{1}{u_n}W_n-v_n\Righ... | Kavi Rama Murthy | 142,385 | <p>Suppose <span class="math-container">$\frac {W_n} {c_n} $</span> converges in distribution to <span class="math-container">$W$</span> and <span class="math-container">$(c_n)$</span> does not tend to <span class="math-container">$\infty$</span>. Then there is a subsequence <span class="math-container">$c_{n_k}$</span... |
4,264,496 | <p>So we have the jensen's inequality: <span class="math-container">$$|EX| \leq E|X|$$</span></p>
<p><strong>Any bound</strong> on the Jensen gap (upper bound or lower bound)? <span class="math-container">$$\text{gap}=E|X| - |EX|$$</span></p>
| Reijo Jaakkola | 737,246 | <p>The gap can be arbitrarily large. For instance, if <span class="math-container">$X$</span> is a random variable so that <span class="math-container">$X(0) = -N$</span> and <span class="math-container">$X(1)=N$</span>, and the events <span class="math-container">$0$</span> and <span class="math-container">$1$</span> ... |
3,690,185 | <p>By <span class="math-container">$a_n \sim b_n$</span> I mean that <span class="math-container">$\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 1$</span>.</p>
<p>I don't know how to do this problem. I have tried to apply binomial theorem and I got
<span class="math-container">$$\int_{0}^{1}{(1+x^2)^n dx} = \int_0^1 \... | mr_snazzly | 572,048 | <p>You can also write <span class="math-container">$ \int_0^1 (1+x^2)^ndx = \int_0^1 e^{n\log(1+x^2)}dx $</span> and use a generalization of Laplace's method to handle the boundary case.</p>
|
3,489,345 | <p>My goal is to find the values of <span class="math-container">$N$</span> such that <span class="math-container">$10N \log N > 2N^2$</span></p>
<p>I know for a fact this question requires discrete math. </p>
<p>I think the problem revolves around manipulating the logarithm. The thing is, I forgot how to manipula... | Community | -1 | <p>First divide by <span class="math-container">$2N$</span> on both sides, </p>
<p><span class="math-container">$5\log N> N$</span> (since <span class="math-container">$N>0$</span> then the inequality stays the same)</p>
<p>Then by raising to the <span class="math-container">$e$</span> power on both sides (the ... |
956,680 | <p>$\displaystyle\lim_{x\to0}\frac{x^2+1}{\cos x-1}$</p>
<p>My solution is:</p>
<p>$\displaystyle\lim_{x\to0}\frac{x^2+1}{\cos x-}\frac{\cos x+1}{\cos x+1}$</p>
<p>$\displaystyle\lim_{x\to0}\frac{(x^2+1)(\cos x+1)}{\cos^2 x-1}$</p>
<p>$\displaystyle\lim_{x\to0}\frac{(x^2+1)(\cos x+1)}{-(1-\cos^2 x)}$</p>
<p>Since... | idm | 167,226 | <p>There is no indeterminatation, </p>
<p>$$\lim_{x\to 0}\frac{x^2+1}{\cos x-1}=\frac{1}{0^-}=-\infty $$</p>
|
4,349,582 | <p>Its rather easy to show that <span class="math-container">$a_n=\frac{n^{1/n}}{n}$</span> ist monotonic (which means <span class="math-container">$a_{n+1}<a_n$</span> for each <span class="math-container">$n$</span>) using derivations. But how can I do it without them? Thanks.</p>
| Ethan Bolker | 72,858 | <p>The mathematical function you seek does not have a special name, nor does it have a formula. If you search for <a href="https://www.google.com/search?q=decimal2binary" rel="nofollow noreferrer"><em>decimal2binary</em></a> you will find algorithms for pen and paper calculation and coded in many languages.</p>
<p>You ... |
2,823,758 | <p>I was learning the definition of continuous as:</p>
<blockquote>
<p>$f\colon X\to Y$ is continuous if $f^{-1}(U)$ is open for every open $U\subseteq Y$</p>
</blockquote>
<p>For me this translates to the following implication:</p>
<blockquote>
<p>IF $U \subseteq Y$ is open THEN $f^{-1}(U)$ is open</p>
</blockq... | Anonimo | 570,174 | <p>I think that understand you. </p>
<p>You have two topological spaces $(X,\tau)$ and $(Y,\tau')$ and a aplication continues f:$X \rightarrow Y$.</p>
<p>For the general definition of continues you can say:</p>
<p>$\forall x \in X , \forall G' \in \tau' : f(x) \in G', \exists G \in \tau : x \in G, f(G) \subseteq G' ... |
205,671 | <p>How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives? </p>
<p>I haven't found any proof of this online.</p>
<p>One of my ideas was writing out $r$ and $\theta$ in terms of $x$ a... | thebluegiraffe | 510,820 | <p>With only partial differentiation and algebra. From what we already know of complex functions:</p>
<p>$ƒ(z) = u(r,\theta) + iv(r,\theta)$</p>
<p>$z = re^{i\theta} = r\cos\theta +ir\sin\theta = x + iy$</p>
<p>$x(r,\theta)= r\cos\theta,\quad y(r,\theta) = r\sin\theta $</p>
<p>Apply partial derivative and chain rul... |
2,218,914 | <p>What is a boundary point when solving for a max/min using Lagrange Multipliers?
After you solve the required system of equation and get the critical maxima and minima, when do you have to check for boundary points and how do you identify them?</p>
<p>e.g. Optimise (1+a)(1+b)(1+c) given constraint a+b+c=1, with a,b,... | Yuri Negometyanov | 297,350 | <p>At first - about elementary way.
$$(1+a) + (1+b) + (1+c) = 4.$$
Using AM-GM, one can get:
$$(1+a)(1+b)(1+c)\le \left(\dfrac{1+a+1+b+1+c}3\right),$$
so $\left(\dfrac13,\dfrac13,\dfrac13\right)$ is maximum.
Note that the issue conditions are significant in this case.</p>
<p>Partitial derivatives of Lagrange multiplie... |
2,218,914 | <p>What is a boundary point when solving for a max/min using Lagrange Multipliers?
After you solve the required system of equation and get the critical maxima and minima, when do you have to check for boundary points and how do you identify them?</p>
<p>e.g. Optimise (1+a)(1+b)(1+c) given constraint a+b+c=1, with a,b,... | farruhota | 425,072 | <p>First of all, if the non negativity condition is not given (if a,b,c can be any real numbers), then there is no minimum. Indeed, let c=0, a be a large negative number, b be a large positive number such that a+b=1. Hence (1+a)(1+b)(1+c) tends to $-\infty$.</p>
<p>When it is solved by the Lagrange multipliers method,... |
4,552,723 | <p>Assume the following angles are known:
<span class="math-container">$ABD$</span>,<span class="math-container">$DBC$</span>,<span class="math-container">$BAC$</span>,<span class="math-container">$ACD$</span>.</p>
<p>Is it possible to compute <span class="math-container">$CDA$</span>?</p>
<p><a href="https://i.stack.i... | Sam | 530,289 | <p>No since no matter what equations you create you will always get
<span class="math-container">$$\measuredangle BDC+\measuredangle DCA$$</span>
that cannot be isolated and depend on the length of <span class="math-container">$[AB]$</span>
However, if the quadrilateral was a parallelogram then definitely since in that... |
1,586,354 | <p>I did the following exercise:</p>
<blockquote>
<p>Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb{Z}_n$ (integers mod n with addition). Prove that either every member of $H$ is even or exactly half of the members of $H$ are even.</p>
</blockquote>
<p>My answer:</p>
<p>Since $\mathbb{Z}... | Noah Schweber | 28,111 | <p>Here's a sketch of an alternate alternate proof: consider the map $f: x\mapsto x+x$, with $ran(f):=A$ the set of even elements. </p>
<blockquote>
<p>For every $a, b\in A$, $f^{-1}(a)$ has the same number of elements as $f^{-1}(b)$.</p>
</blockquote>
<p>Proof sketch: Fix $c+c=a$ and $d+d=b$, and consider the map ... |
2,483,231 | <p>If $F(x)=f(g(x))$, where $f(5) = 8$, $f'(5) = 2$, $f'(−2) = 5$, $g(−2) = 5$, and
$g'(−2) = 9$, find $F'(−2)$. I'm totally lost on this problem, I'm assuming to incorporate the Chain Rule. I get $5(5) * 9 = 225$ but I am incorrect.</p>
<p>Update: Thanks guys, I see where I messed up thanks!</p>
| Community | -1 | <p>$F'(x)=f'(g (x))\cdot g'(x) $, by the chain rule. .. I'm getting 18...</p>
|
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