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 |
|---|---|---|---|---|
2,545,226 | <p>Suppose $a_n$ is a positive sequence but not necessarily monotonic. </p>
<p>For the series $\sum_{n=1}^\infty \frac{1}{a_n}$ and $\sum_{n=1}^\infty \frac{a_n}{n^2}$ I can find examples where both diverge: $a_n = n$, and where one converges and the other diverges: $a_n = n^2$.</p>
<p>Can we find example where both... | RRL | 148,510 | <p>Both series cannot converge.</p>
<p>Suppose $\sum a_n/n^2$ converges. Since the divergent harmonic series can be written as </p>
<p>$$\sum_{n=1}^\infty \frac{1}{n} = \sum_{\frac{1}{n} \leqslant \frac{a_n}{n^2}} \frac{1}{n} + \sum_{\frac{1}{n} > \frac{a_n}{n^2}} \frac{1}{n},$$</p>
<p>and the first series on the... |
3,161,997 | <p>I'm taking a course in Probability and I'm asked to prove the following statement :</p>
<p>Let <span class="math-container">$\Omega$</span> be a non-empty arbitrary set , then </p>
<ol>
<li><span class="math-container">$\text {$E_1$={$\emptyset$,$\Omega$}}$</span> is <span class="math-container">$\sigma-\text {alg... | Hans Lundmark | 1,242 | <p>The variable <span class="math-container">$t$</span> is irrelevant here.</p>
<p>The partial derivative <span class="math-container">$f'_x(x,y)$</span> measures the slope of the graph <span class="math-container">$z=f(x,y)$</span> in the <span class="math-container">$x$</span> direction at the point <span class="mat... |
3,161,997 | <p>I'm taking a course in Probability and I'm asked to prove the following statement :</p>
<p>Let <span class="math-container">$\Omega$</span> be a non-empty arbitrary set , then </p>
<ol>
<li><span class="math-container">$\text {$E_1$={$\emptyset$,$\Omega$}}$</span> is <span class="math-container">$\sigma-\text {alg... | justanotheruser | 643,568 | <p>I interpreted your question in two ways</p>
<p><strong>Interpretation 1:</strong></p>
<p>Let's take an example : </p>
<p><span class="math-container">$f(x(t),y(t))=x^2+y$</span> </p>
<p>and <span class="math-container">$x=t, y=2t$</span></p>
<p>Our partial derivative with respect to x is <span class="math-conta... |
2,072,347 | <p>I was trying to solve this problem, but couldn't figure it out. The solution goes like this:</p>
<p><a href="https://i.stack.imgur.com/1KSWH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1KSWH.png" alt="http://www.tkiryl.com/Calculus/Problems/Section%201.4/Calculating%20Limits/Solutions/Calc_S_... | Community | -1 | <p>$$\lim \limits_{x \to 0} \frac{\sin(5x)}{\sin(4x)}$$</p>
<p>$$\lim \limits_{x \to 0} \frac{\sin(5x)}{x} . \frac{x}{\sin(4x)}$$</p>
<p>$$β΅\lim \limits_{\theta \to 0}\frac{ \sin(a\theta)}{\theta}=a$$</p>
<p>$$=\frac54$$</p>
|
2,072,347 | <p>I was trying to solve this problem, but couldn't figure it out. The solution goes like this:</p>
<p><a href="https://i.stack.imgur.com/1KSWH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/1KSWH.png" alt="http://www.tkiryl.com/Calculus/Problems/Section%201.4/Calculating%20Limits/Solutions/Calc_S_... | barak manos | 131,263 | <p>$\lim\limits_{x\to0}\frac{\sin(5x)}{\sin(4x)}=$</p>
<p>$\lim\limits_{x\to0}\frac{\sin(5x)\cdot5x\cdot4x}{\sin(4x)\cdot5x\cdot4x}=$</p>
<p>$\lim\limits_{x\to0}\frac{\sin(5x)\cdot4x\cdot5x}{5x\cdot\sin(4x)\cdot4x}=$</p>
<p>$\lim\limits_{x\to0}\left(\frac{\sin(5x)}{5x}\cdot\frac{4x}{\sin(4x)}\cdot\frac{5x}{4x}\right... |
2,306,895 | <p>I want to find $Hom_{\mathtt{Grp}}(\mathbb{C}^\ast,\mathbb{Z})$, where $\mathbb{C}^\ast$ is the multiplicative group, and $\mathbb{Z}$ is additive.
$\mathbb{C}$ is the additive group of complex numbers. We have the following map: </p>
<p>$\large{\mathbb{C} \xrightarrow{exp} \mathbb{C}^\ast \xrightarrow{?} \mathbb{Z... | lhf | 589 | <p>A group homomorphism $\phi:\mathbb{C}^* \to \mathbb{Z}$ must be trivial.</p>
<p>Let $\omega \in \mathbb{C}^*$ and $n \in \mathbb N$. Then there is $\theta \in \mathbb{C}^*$ such that $\omega=\theta^n$ and so $\phi(\omega)= n \phi(\theta)$.</p>
<p>Therefore, $\phi(\omega)$ is a multiple of $n$ for all $n \in \mathb... |
1,987,358 | <p>We know that Riemann sum gives us the following formula for a function <span class="math-container">$f\in C^1$</span>:</p>
<blockquote>
<p><span class="math-container">$$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f(x) dx.$$</span></p>
</blockquote>
<p>I am looking for an example where ... | Nilotpal Sinha | 60,930 | <p>Not a direct answer to your question but I find the following representation of the Riemann sum interesting. Even though it is trivial, these representations show that the sequence of primes or the sequence of composites behave somewhat in a similar same way as the sequence of natural numbers in terms of their asymp... |
1,987,358 | <p>We know that Riemann sum gives us the following formula for a function <span class="math-container">$f\in C^1$</span>:</p>
<blockquote>
<p><span class="math-container">$$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f(x) dx.$$</span></p>
</blockquote>
<p>I am looking for an example where ... | Adren | 405,819 | <p>Here is an example ...</p>
<p>For each $z\in\mathbb{C}$ with $\vert z\vert\neq 1$, consider :</p>
<p>$$F(z)=\int_0^{2\pi}\ln\left|z-e^{it}\right|\,dt$$</p>
<p>It is possible to get an explicit form for $F(z)$, using Riemann sums.</p>
<p>For each integer $n\ge1$, consider :</p>
<p>$$S_n=\frac{2\pi}{n}\sum_{k=0}^... |
3,469,252 | <p>At first: I am new to differential equations, so this question might seam a little bit obvious.</p>
<p>The differential eqation was <span class="math-container">$y'x^3 = 2y -5$</span>.
I rearranged it to: <span class="math-container">$\frac{dx^3}{dx} = \frac{d(2y-5)}{dy}$</span>.</p>
<p>The Problem is, if i derive... | Z Ahmed | 671,540 | <p>The ODE is linear
<span class="math-container">$$y'-2\frac{y}{x^3}=\frac{-5}{x^3}.$$</span>
its integrating factor is <span class="math-container">$I=\exp[\int \frac{-2}{x^3} dx]=e^{1/x^2}.$</span>
So its solution is <span class="math-container">$$y=e^{-1/x^3} \int e^{1/x^2}~ \frac{-5}{x^3} dx+ C e^{-1/x^2} \implies... |
525,885 | <p>Why is $$\frac{\sum_{i=1}^n a_i^{n+1}}{\sum_{i=1}^{n}a_i^n} \geq \frac{\sum_{i=1}^n a_i}{n}$$
where $n$ is some positive natural number, and all $a_i$s are assumed to be positive real number?</p>
| Hagen von Eitzen | 39,174 | <p><em>Hint:</em> compare
$$ \sum_{i=1}^n a_i^n\cdot \frac{\sum_{i=1}^1a_i}{n}$$
with
$$ \sum_{i=1}^n a_i^n\cdot a_i$$</p>
|
29,670 | <p>I have $a_k=\frac1{(k+1)^\alpha}$ and $c_k=\frac1{(k+1)^\lambda}$, where $0<\alpha<1$ and $0<\lambda<1$, and we have a infinite sequence $x_k$ with the following evolution equation.
$$
x_{k+1}=\left(1-a_{k+1}\right)x_{k}+a_{k+1}c_{k+1}^{2}
$$
I have proven that $x_k$ is bounded and obviously positive. ... | mjqxxxx | 5,546 | <p>As pointed out in the comments, a general solution to $x_{n+1}=f_n x_n + g_n$ is possible. Define $F_0=1$ and $F_{n+1} = \prod_{i=0}^{n} f_n^{-1} = f_n^{-1} F_{n}$ for $n\ge 0$. Then $f_n = F_{n} / F_{n+1}$, and the recurrence relation becomes
$$
F_{n+1} x_{n+1} = F_{n} x_{n} + F_{n+1} g_n.
$$
This has the solut... |
871,744 | <p>When A and B are square matrices of the same order, and O is the zero square matrix of the same order, prove or disprove:-
$$AB=0 \implies A=0 \text{ or } \ B=0$$</p>
<p>I proved it as follows:-</p>
<p>Assume $A \neq O$ and $ B \neq O$:
then, $$ |A||B| \neq 0 $$
$$ |AB| \neq 0 $$
$$ AB \neq O $$
$$ \therefore A \... | DGRasines | 96,044 | <p>You are saying that if $A \neq O$ then, $det(A) \neq O$, which is false in general. Consider any diagonal matrix different from $O$ which has at least one zero in the diagonal.</p>
|
276,948 | <p>$a_n$ be a sequence of integers such that such that infinitely many terms are non zero, we need to show that either the power series $\sum a_n x^n$ converges for all $x$ or Radius of convergence is at most $1$. need some hint. thank you.</p>
| P.. | 39,722 | <p>The <a href="http://en.wikipedia.org/wiki/Radius_of_convergence#Theoretical_radius" rel="nofollow">Radius of convergence</a>, $R$, is given by $$\dfrac1R=\lim\sup\sqrt[n]{|a_n|}\geq 1, \ \text{if } a_n\in\mathbb Z \text{ with infinitely many nonzero terms}.$$</p>
|
849,583 | <p>I have several derivatives to find:</p>
<blockquote>
<p>For <span class="math-container">$g(s)=3s^3-s+4$</span>, <span class="math-container">$g'(s)=$</span></p>
<p>For <span class="math-container">$p(t)=\frac1t+t^2$</span>, <span class="math-container">$\frac{\mathrm dp}{\mathrm dt}=$</span></p>
<p>For <span class=... | rschwieb | 29,335 | <p>Hints:</p>
<p>All three problems can be solved by combining three facts:</p>
<ol>
<li>$D_x(f(x)\pm g(x))=D_x(f(x))\pm D_x(g(x))$</li>
<li>$D_x(cf(x))=cD_x(f(x))$ for any constant $c$</li>
<li>$D_x(x^n)=nx^{n-1}$ for any $n\neq 0$ (including $n=-1$ and $n=\frac12$.)</li>
</ol>
<hr>
<p>In the first problem, the su... |
1,111,168 | <p>Do we say that $0$ has $n$ $n$th roots, all nondistinct, or only one?</p>
<p>I don't think it makes any difference, but I'm curious what the convention is.</p>
| quid | 85,306 | <p>There is, as is visible, no universally agreed upon convention. However, I would argue in favor of considering by an large the number of $n$-th roots of an element $a$ (in some structure) the <strong>cardinality of the set</strong> of solutions of $X^n = a$, so that in the complex and real numbers, and in any other ... |
3,114,208 | <p>Say I have a biased coin that shows heads with probability <span class="math-container">$p \in ]1/3,1/2[$</span> and I initially have capital of <span class="math-container">$100 $</span>EUR. Every time heads is shown, my capital is doubled, in the other case I pay half of my capital. Let <span class="math-container... | William M. | 396,761 | <p>Let <span class="math-container">$p$</span> as you said, <span class="math-container">$q = 1-p.$</span> Write <span class="math-container">$R_n \sim p \varepsilon_0 + q \varepsilon_1$</span> (assumed i.i.d.), <span class="math-container">$X_0 = 100$</span> and <span class="math-container">$X_{n+1}=2X_n\mathbf{1}_{\{... |
28,751 | <p>$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$,
$$
\mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>.
$$</p>
| Dilip Sarwate | 15,941 | <p>Integrating by parts,
$$\begin{align*}
Q(x) &= \int_x^{\infty} \phi(t)\mathrm dt = \int_x^{\infty} \frac{1}{\sqrt{2\pi}}\exp(-t^2/2) \mathrm dt\\
&= \int_x^{\infty} \frac{1}{t} \frac{1}{\sqrt{2\pi}}t\cdot\exp(-t^2/2) \mathrm dt\\
&= - \frac{1}{t}\frac{1}{\sqrt{2\pi}}\exp(-t^2/2)\biggr\vert_x^\infty
- \i... |
939,747 | <p>Denote by $a_n$ the sum of the first $n$ primes. Prove that there is a perfect square between $a_n$ and $a_{n+1}$, inclusive, for all $n$.</p>
<p>The first few sums of primes are $2$, $5$, $10$, $17$, $28$, $41$, $58$, $75$. It seems there is a perfect square between each pair of successive sums. In addition, we ca... | Marc van Leeuwen | 18,880 | <p>No it is not possible to recover $\def\x{\mathbf x}\x$ from $\mathbf A=\x\x^T$ (unless $\x=0$), since replacing $\x$ by $-\x$ will give the exact same value for $\x\x^T$ (the map $\x\mapsto\x\x^T$ is not injective). As others have observed the map is not surjective either (many matrices do not arise this way), but a... |
1,473,318 | <blockquote>
<p>How many numbers can by formed by using the digits $1,2,3,4$ and $5$ without repetition which are divisible by $6$?</p>
</blockquote>
<p><strong>My Approach:</strong></p>
<p>$3$ digit numbers formed using $1,2,3,4,5$ divisible by $6$ </p>
<p>unit digit should be $2/4$ </p>
<p>No. can be $XY2$ &... | Amir | 232,937 | <p>the number should be divisible by 2 and 3.</p>
<p>case.1) </p>
<p>XY2</p>
<p>then X+Y should be 4,7.</p>
<p>a) X+Y=4</p>
<p>the only possibility is (X,Y)=(1,3).
Then there are 2 choices (X,Y) = (1,3) or (3,1).</p>
<p>b) X+Y=7</p>
<p>(X,Y) = (2,5), which is not feasible, or (X,Y) =(3,4).
the number of choice... |
1,165,147 | <p>I would like to find the area under the curve of $\frac{\sin(ax/2)}{\sin(x/2)}$, namely between the first zero crossing on the left and right:</p>
<p>$$
\int_{-\frac{2\pi}{a}}^{\frac{2\pi}{a}} \frac{\sin(\frac{ax}{2})}{\sin(\frac{x}{2})}
$$</p>
<p>I realized from Wolfram Alpha that this was not a simple solution,... | Alex | 38,873 | <p>If $a$ is so large, then the bounds of the integral are not far from 0, so you can approximate both numerator and denominator with Maclaurin series expansion. </p>
|
75,005 | <p>Let's imagine a guy who claims to possess a machine that can each time produce a completely random series of 0/1 digits (e.g. $1,0,0,1,1,0,1,1,1,...$). And each time after he generates one, you can keep asking him for the $n$-th digit and he will tell you accordingly.</p>
<p>Then how do you check if his series is <... | Gerry Myerson | 8,269 | <p>This is discussed very nicely in Volume 2 of Knuth's The Art Of Computer Programming. The executive summary is that randomness is a mathematical concept that can be defined in mathematics but not easily. </p>
|
281,735 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/202452/why-is-predicate-all-as-in-allset-true-if-the-set-is-empty">Why is predicate βallβ as in all(SET) true if the SET is empty?</a> </p>
</blockquote>
<p>In don't quite understand this quantification ov... | c.w.chambers | 16,959 | <p>When you see a quantification like β<span class="math-container">$\forall \phi x : \psi x$</span>β, this is shorthand for β<span class="math-container">$\forall x : \phi x \to \psi x$</span>β. Since β<span class="math-container">$x \in \emptyset$</span>β is false for all β<span class="math-container">$x$</span>β, th... |
73,111 | <blockquote>
<p>A trapezoid was inscribed into a semicirle of radius R. The side of
the trapezoid is slanting alpha against the base which is the diameter
of the semicirlce. Compute the area of the trapezoid.</p>
</blockquote>
<p>So the base is 2R. The bad thing is: that's all I know. How should I move on? Don't... | AndrΓ© Nicolas | 6,312 | <p>There is a reasonable brute force approach, using the standard formula for the area of a trapezoid. We need a picture. Instead I will <em>label</em> the vertices, and rely on you to draw the picture. Please note that your picture is an essential part of the calculation below.</p>
<p>Let the vertices of the trapezo... |
4,627,821 | <p>Consider the difference equations</p>
<p><span class="math-container">$$x(k+1) = f(x(k)) \qquad (1)$$</span></p>
<p>and</p>
<p><span class="math-container">$$y(k+1) = g(y(k)) \qquad (2)$$</span></p>
<p>where <span class="math-container">$g = f \circ f$</span>.</p>
<p>In <em>An Introduction to Difference Equations (3... | Steve Morris | 1,133,961 | <p>If the statement is not true, then there exists <span class="math-container">$\varepsilon >0$</span>, s.t. <span class="math-container">$|f^{\prime}(x)|\geq \varepsilon$</span>, <span class="math-container">$\forall x \in \mathbb{R}$</span>. Since <span class="math-container">$f$</span> is bounded, then there exi... |
2,881,914 | <p>Using a computer I found the double sum</p>
<p>$$S(n)= \sum_{j=1}^n\sum_{k=1}^n \frac{j^2 + jk + k^2}{j^2(j+k)^2k^2}$$
has values</p>
<p>$$S(10) \quad\quad= 1.881427206538142 \\ S(1000) \quad= 2.161366028875634 \\S(100000) = 2.164613524212465\\$$</p>
<p>As a guess I compared with fractions $\pi^p/q$ where $p,q$ ... | Hazem Orabi | 367,051 | <p>$$
\begin{align}
\frac{m^2+m\,n+n^2}{m^2\,(m+n)^2\,n^2}\, &=\frac{m^2+m\,n+n^2\,\color{red}{+m\,n-m\,n}}{m^2\,(m+n)^2\,n^2} \\[2mm]
&=\,\frac{(m+n)^2-m\,n}{(m+n)^2\,m^2\,n^2} \\[2mm]
&=\,\frac{1}{m^2\,n^2}-\frac{1}{m\,n\,(m+n)^2} \\[2mm]
&=\,\frac{1}{m^2\,n^2}-\frac{1}{m^3}\left(\frac{1}{n}-\fra... |
3,881,029 | <p><span class="math-container">$R(A)$</span> is range of <span class="math-container">$A$</span> <br>
<span class="math-container">$N(A)$</span> is nullspace of <span class="math-container">$A$</span> <br>
<span class="math-container">$R(A^T)$</span> is range of <span class="math-container">$A^T$</span> <br>
<span cla... | angryavian | 43,949 | <p>Hint: By the definition of <span class="math-container">$R(A)$</span>, there exists <span class="math-container">$z$</span> such that <span class="math-container">$y=Az$</span>. Then note that <span class="math-container">$x^\top y = x^\top (Az) = (A^\top x)^\top z$</span>.</p>
|
117,933 | <p>I couldn't find similar question being asked here. The closest one I can find is <a href="https://mathoverflow.net/questions/11366/when-to-split-merge-papers">When to split/merge papers?</a>. Here is my situation: I proved a theorem. When I try to type it, I found that it's very long. Since it's long, I splitted it ... | fedja | 1,131 | <p>I usually do not split in such cases, but if you do, I guess the same journal is a better choice, especially if you think of the poor people who will have to look up and reference your paper in the future :). Also, if possible at all, I would include into the first part a forward reference to the second one (if the ... |
1,684,124 | <p>Here is my attempt:</p>
<p>$$ \frac{2x}{x^2 +2x+1}= \frac{2x}{(x+1)^2 } = \frac{2}{x+1}-\frac{2}{(x+1)^2 }$$</p>
<p>Then I tried to integrate it,I got $2\ln(x+1)+\frac{2}{x+1}+C$ as my answer. Am I right? please correct me if I'm wrong.</p>
| Enrico M. | 266,764 | <p>Add and remove $2$ in the numerator, and you will obtain:</p>
<p>$$\frac{2x+2-2}{x^2+2x+1} = \frac{2x+2}{x^2+2x+1} - \frac{2}{x^2+2x+1}$$</p>
<p>Now you see that the numerator in the first fraction is nothing but the derivative of the denominator, id est you have a function of the form $\frac{f'(x)}{f(x)}$.
This h... |
1,501,940 | <p>This is related to a <a href="https://math.stackexchange.com/questions/1501852/why-does-this-statement-not-hold-when-me-0/1501925#1501925">question</a> I just asked, that I now think was based on wrong assumptions.</p>
<p>It is true that if <span class="math-container">$f=a$</span> a.e. on the interval <span class="... | B. S. Thomson | 281,004 | <p>My feline friend is getting confused (and no doubt frustrated) and has now posted a further variant on the problem. Here is the best I can come up with for a short tutorial. I hope it helps.</p>
<p>Do this problem first:</p>
<blockquote>
<p>Suppose that $f$, $g:E \to R$ are continuous functions. Let $D$ be a
... |
1,501,940 | <p>This is related to a <a href="https://math.stackexchange.com/questions/1501852/why-does-this-statement-not-hold-when-me-0/1501925#1501925">question</a> I just asked, that I now think was based on wrong assumptions.</p>
<p>It is true that if <span class="math-container">$f=a$</span> a.e. on the interval <span class="... | B. S. Thomson | 281,004 | <p>I am not normally a cat person but this one and I are old friends now.</p>
<p>So I think I should jump into the litter box and complete the task we have been set. What is the definitive answer to this problem that was apparently given to all the kittys in a graduate class somewhere? Here is a way to formulate an... |
3,896,709 | <ul>
<li>Any idea to evaluate the sum
<span class="math-container">$$
\sum_{j=m}^{k}\frac{\binom{m}{2m - j\,\,}}{\binom{k}{j}}
\quad\mbox{with}\quad m \leq k < 2m - 1.
$$</span></li>
<li>I have found the sum for <span class="math-container">$k=2m-1$</span>. In fact, it is verified that
<span class="math-container">$... | Claude Leibovici | 82,404 | <p>For a short time, I hoped to be able to express the result in terms of hypergeometric functions but I failed.</p>
<p>If we let <span class="math-container">$k=m+n$</span>, the problem reduces to
<span class="math-container">$$S_n=\sum_{j=m}^{m+n} \frac{\Gamma (j+1)\,\,\Gamma (m+n+1-j)}{\Gamma (m+n+1)} \,\binom{m}{2 ... |
1,858,095 | <p>Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function with:</p>
<p>$$f(x) = x - \arctan{x}$$</p>
<p>We consider the sequence $(x_{n})$ with $x_{0} > 0$ and $x_{n + 1} = f(x_{n})$, for any $n \in \mathbb{N}$.</p>
<p>Prove that $(x_{n})$ is convergent and find its limit.</p>
<p>So far, I've proved that $f(x... | Adelafif | 229,367 | <p>The derivative is $\frac{x^2}{1+x^2}$and the function is a contraction mapping on any interval $[0,n]$. The given sequence is used to find the fixed point x=0. </p>
|
511,304 | <p>Given the ODE: </p>
<p>$2(x+1)y' = y$</p>
<p>How can I solve that using Power Series? I started to think about it:</p>
<p>$
\\2(x+1)\sum_{n=1}^{\infty}{nc_nx^{n-1}}-\sum_{n=0}^{\infty}{c_nx^n}=0
\\2\sum_{n=1}^{\infty}{nc_nx^{n}}+2\sum_{n=1}^{\infty}{nc_nx^{n-1}}-\sum_{n=0}^{\infty}{c_nx^n}=0
\\\sum_{n=0}^{\infty}... | Ross Millikan | 1,827 | <p>For $x$ large, $\tanh (x)$ is just a little less than $1$ and $\arctan (\frac 1C)$ is essentially constant. The integral will then be of order $\lambda \arctan (\frac 1C)$ The limit will diverge (one way or the other) unless $C=\frac 1{\tan 1}\approx 0.642$</p>
|
218,020 | <p>I came across the following statement: Let $R$ be a complete local Noetherian commutative ring. If $A$ is a commutative $R$-algebra that is finitely generated and free as a module over $R$, then $A$ is a semi-local ring that is the direct product of local rings. (I'm unsure if completeness or the Noetherian conditio... | sacohe | 45,227 | <p>I don't think I can put this into a formula (I'll reply back if I can come up with a simple one), but the way I would solve a problem like this is to just think it through.</p>
<p>Let's take $10$ as an example. Here are all the possible combinations of single digit, positive integers that add up to $10$:</p>
<ul>... |
218,020 | <p>I came across the following statement: Let $R$ be a complete local Noetherian commutative ring. If $A$ is a commutative $R$-algebra that is finitely generated and free as a module over $R$, then $A$ is a semi-local ring that is the direct product of local rings. (I'm unsure if completeness or the Noetherian conditio... | Ross Millikan | 1,827 | <p>If $s$ is the sum, the number of digits is $k=\lceil s/9 \rceil$. The lead digit is then $s+9-9k$. Then there are $k-1 \ \ 9$'s</p>
|
110,896 | <p>Now I have a function, say $f(k,z)=e^{-kz}(1+kz)$</p>
<p>I want to find the $n$th $\log$ derivative with respect to z.
like $(z\partial_z)^{(n)}f(k,z)$ (or $(\partial_{\ln z})^{(n)}f(k,z)$
if you like), where the $(n)$ denotes that we take the derivative $n$
times. </p>
<p>I found the answer in <a href="https://... | Bob Hanlon | 9,362 | <pre><code>f[k_, z_] = E^(-k*z)*(1 + k*z);
</code></pre>
<p>Direct calculation using <code>Nest</code></p>
<pre><code>dOp1[f_, n_Integer?NonNegative] :=
Nest[z*D[#, z] &, f, n]
</code></pre>
<p><code>Simplify</code> at each step to reduce number of terms to be differentiated</p>
<pre><code>dOp2[f_, n_Integer?N... |
3,979,686 | <ul>
<li><a href="https://www.britannica.com/science/derivative-mathematics" rel="nofollow noreferrer">Derivative</a></li>
</ul>
<p>This article says the following:</p>
<blockquote>
<p>To find the slope at the desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, i... | GEdgar | 442 | <p>Suppose <span class="math-container">$a>0$</span>. Complete the square. You get one of these cases:
<span class="math-container">$$
\frac{1}{\sqrt{a}}\int\frac{dx}{\sqrt{(x-\beta)^2}},\qquad \beta\in \mathbb R,\\
\frac{1}{\sqrt{a}}\int\frac{dx}{\sqrt{(x-\beta)^2+\gamma^2}},\qquad \beta\in \mathbb R, \gamma >... |
69,711 | <blockquote>
<p>Find an equation of the tangent line to the graph of $y= \sqrt{x-3}$ that is perpendicular to $6x+3y-4=0$. </p>
</blockquote>
<p>I don't understand what it's asking. Is this the normal line? How do I solve this?</p>
| AndrΓ© Nicolas | 6,312 | <p>Since this may be homework (please correct me if it isn't), I will give a few hints only.</p>
<p>Hint 1: What is the slope of the given line $6x+3y-4=0$?</p>
<p>Hint 2: What is the slope of any line perpendicular to the given line?</p>
<p>Hint 3: So the "mystery" tangent line must have the slope reached in Hint 2... |
69,711 | <blockquote>
<p>Find an equation of the tangent line to the graph of $y= \sqrt{x-3}$ that is perpendicular to $6x+3y-4=0$. </p>
</blockquote>
<p>I don't understand what it's asking. Is this the normal line? How do I solve this?</p>
| Gerry Myerson | 8,269 | <p>Andre's answer is good. Another approach which may stand you in good stead beyond this particular question is: draw a diagram. Sketch the graph of $y=\sqrt{x-3}$ (it doesn't have to be a real good sketch, actually it's probably good enough just to draw some random curve), sketch the line $6x+3y-4=0$ (again, probably... |
1,159,599 | <p>can someone give me a hint on how to calculate this integral?</p>
<p>$\int _0^{\frac{1}{3}} \frac{e^{-x^2}}{\sqrt{1-x^2}}dx$</p>
<p>Thanks so much!</p>
| Lucian | 93,448 | <p>If the upper limit would have been <span class="math-container">$1$</span> instead of <span class="math-container">$\dfrac13$</span> , then the definite integral could have been </p>
<p>expressible in terms of <a href="http://en.wikipedia.org/wiki/Bessel_function" rel="nofollow noreferrer">Bessel functions</a> <spa... |
1,999,194 | <p>Determine the following system of equations has 'a unique solution', 'many solutions' or 'no solution':
$$\begin{cases}
& x + 2y + z &= 1\\
&2x + 2y - 2z &= 4\\
&-x + 2y - 3z &= 5
\end{cases}
$$</p>
<p>Answer = A unique solution</p>
<p>How is it a unique solution? C... | StackTD | 159,845 | <blockquote>
<p>$$\begin{cases}
& x + 2y + z &= 1\\
&2x + 2y - 2z &= 4\\
&-x + 2y - 3z &= 5
\end{cases}$$</p>
</blockquote>
<p>If you let:
$$A=\begin{bmatrix}1&2&1\\2&2&-2\\-1&2&-3\end{bmatrix}\;,\quad X=\begin{bmatrix}x\\y\\z\end{bmatrix}\;,\q... |
174,655 | <p>So I have 2 lists of 10000+ lists of 3 numbers, e.g.</p>
<pre><code>{{1,2,3},{4,5,6},{7,8,9},...}
{{2,1,3},{4,5,6},{41,2,0},...}
</code></pre>
<p>Wanting a result like </p>
<pre><code>{2,...}
</code></pre>
<p>Getting some sort of list of <code>True</code>/<code>False</code> is also probably enough, like this:</p... | Henrik Schumacher | 38,178 | <p>This should be rather fast for very long packed arrays of integers.</p>
<pre><code>a = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}, {1, 2, 3}};
b = {{2, 1, 3}, {4, 5, 6}, {41, 2, 0}, {1, 2, 3}};
f2[a_,b_] := Position[Unitize[Subtract[a, b]].ConstantArray[1, Dimensions[a][[2]]], 0, 1];
f2[a,b]
</code></pre>
<blockquote>
<p... |
3,936,102 | <p>Can you have a function <span class="math-container">$f \notin L^1$</span> but its Fourier transform <span class="math-container">$\hat{f} \in L^1$</span>? Ive been playing around with examples and I cant find one, but I also cant prove one doesn't exist.</p>
| md2perpe | 168,433 | <p>For
<span class="math-container">$$
f(x) = \frac{\sin x}{x} \not\in L^1(\mathbb{R})
$$</span>
one has
<span class="math-container">$$
\hat{f}(\xi) = C \chi_{[-a,a]}(\xi) \in L^1(\mathbb{R})
$$</span>
where <span class="math-container">$C$</span> and <span class="math-container">$a$</span> are some constants dependin... |
3,657,026 | <p>I need to prove expression using mathematical induction <span class="math-container">$P(1)$</span> and <span class="math-container">$P(k+1)$</span>, that:</p>
<p><span class="math-container">$$
1^2 + 2^2 + \dots + n^2 = \frac{1}{6}n(n + 1)(2n + 1)
$$</span></p>
<p>Proving <span class="math-container">$P(1)$</span>... | Siong Thye Goh | 306,553 | <p><span class="math-container">\begin{align}
\frac16 k(k+1)(2k+1) + (k+1)^2 &= \frac{k(k+1)(2k+1) + 6(k+1)^2}{6} \\
&=\frac{(k+1)}{6} \cdot \left(k(2k+1) + 6(k+1) \right)\\
&= \frac{k+1}{6} \cdot (2k^2+7k+6)\\
&= \frac{k+1}{6} \cdot (2k+3)(k+2)\\
\end{align}</span></p>
<p>where we first make them have... |
571,955 | <p>I've tried solving this problem every way I know how and I just can't get it. I've looked at similar problems of this type, and I still cannot get an answer that seems right.</p>
<p>Parametric Equations:</p>
<p><strong>a) Write the distance between the line and the point as a function of s</strong></p>
<p><strong>b... | Steven Alexis Gregory | 75,410 | <p>Line: $(x, y, z) = (3, 1, -1) + t(-1, 1/2, 2)$</p>
<p>Point: $(4, 3, s)$</p>
<p>The point must lie in a plane that is perpendicular to the line.
So the plane needs to be perpendicular to the vector $(-1, 1/2, 2)$</p>
<p>This would be the plane $-x + \frac 1 2 y + 2z = C.\;$
To find C, we let $(x, y, z) = (4, 3, s... |
2,758,965 | <p>Show $\log(1-x)=-\sum_{n=1}^{\infty}\frac{x^n}{n}\,\forall x\in(-1,1)$. Which value does $\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$ take?</p>
<hr>
<p>Now because I skipped forward in my (personal) textbook I know that I could tackle this using knowledge of the Maclaurin/Taylor series. However, it was not covered in the... | timon92 | 210,525 | <p>Hint: $$\frac 1a + \frac{1}{bc} = \frac{bc+a}{abc} = \frac{bc+a(a+b+c)}{abc} = \frac{(a+b)(a+c)}{abc}.$$</p>
|
411,868 | <p>Laver showed in 1995 that the period of the first row of certain <a href="https://en.wikipedia.org/wiki/Laver_table" rel="noreferrer">Laver tables</a> is unbounded, assuming that a rank-into-rank cardinal exists.</p>
<p>The most accessible proof of his result that I was able to find is in chapter 12 of Patrick Dehor... | Wojowu | 30,186 | <p>As was already mentioned in the comments, the premise of the question is somewhat backwards. Indeed, looking at <a href="https://www.sciencedirect.com/science/article/pii/S0001870885710146?via%3Dihub" rel="noreferrer">Laver's paper</a>, the combinatorial structures now known as Laver tables were not at all his initi... |
13,889 | <p><strong>Question:</strong> Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?</p>
<p><strong>Why do I care:</strong> For a number of math kids I know, doing algebraic topology is fi... | Zach Conn | 251 | <p>If one operates on an open subset $U$ of Euclidean space $\mathbb{R}^n$, then de Rham cohomology falls out of trying to solve some differential equations.</p>
<p>The starting observation is that a locally constant smooth function will be constant on connected components, so the dimension of the vector space of loca... |
13,889 | <p><strong>Question:</strong> Are there intuitive ways to introduce cohomology? Pretend you're talking to a high school student; how could we use pictures and easy (even trivial!) examples to illustrate cohomology?</p>
<p><strong>Why do I care:</strong> For a number of math kids I know, doing algebraic topology is fi... | aaron | 6,071 | <p>This is meant to be a comment below Zach Conn's answer, but for some reason I don't seem to have the option of commenting.</p>
<p>This is really the same answer -- in short, integrals can be viewed as cohomology classes -- but just to give a very concrete example: </p>
<p>Consider $\int_C \frac{dz}{z}$ where $C$ i... |
211,623 | <p>Let $G$ be a (finite) group, and $a, b \in G$ be any two elements. Consider the sequence defined by
\begin{eqnarray*}
s_0 &=& a, \\
s_1 &=& b, \text{and} \\
s_{n+2} &=& s_{n+1} s_{n} \text{ for $n \geq 0$}.
\end{eqnarray*}
This sequence is what we get when replace the letters in the algae L-s... | Ash Malyshev | 44,291 | <p>For that system, it is always periodic, in other words it starts repeating right away. The function $(a,b) \mapsto (b, ba)$ is invertible: we have $(c^{-1} d,c) \mapsto (c,d)$. So if iterating this function starting from some point eventually starts repeating, it must have been repeating the whole time.</p>
<p>More... |
2,333,847 | <p>A function $f(x) = k$ and the domain is $\{-2,-1,\dotsc,3\}$. Would I say
$$x = \{-2,-1,\dotsc,3\}\quad\text{or}\quad x \in \{-2,-1,\dotsc,3\} \ ?$$
Thanks. </p>
| orlp | 5,558 | <p>If you really want to be formal you can say that the domain of $f(x)$ is $\{x \in \mathbb{Z} \mid -2 \leq x \leq 3\}$.</p>
|
272,468 | <p>I wrote the <a href="https://www.researchgate.net/publication/334884635_On_a_Bivariate_Frechet_Distribution" rel="nofollow noreferrer">Frechet Distribution</a> as follows:</p>
<pre><code>dist1 = ProbabilityDistribution[{"PDF", \[Lambda]1/\[Alpha]1 (x/\[Alpha]1)^(-\[Lambda]1 - 1)E^-((x/\[Alpha]1)^-\[Lambda]... | cvgmt | 72,111 | <p>Use the settings by @Akku14, we can find two roots in <code>0<a<1</code>.</p>
<pre><code>Clear[obj2];
obj2[a_?NumericQ] :=
NIntegrate[(E^(1/2 (-x[1]^2 - x[2]^2)) Log[
Abs[1 - a (x[1]^2 + x[2]^2)]] x[1]^2)/(Ο (x[1]^2 +
x[2]^2)), {x[1], -β, β}, {x[
2], -β, β}, WorkingPrecision -> 15] +... |
1,642,029 | <p>I'm looking at my textbooks steps for calculating the complexity of bubble sort...and it jumps a step where I don't know what exactly they did. </p>
<p><a href="https://i.stack.imgur.com/XaztP.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XaztP.png" alt="enter image description here"></a></p>
... | AndrΓ© Nicolas | 6,312 | <p>It depends how we "filter out." If we remove all the $b_n$ we may end up with a sequence that is finite, or even empty. </p>
<p>But if we remove say $b_1$ and $b_4$ and $b_9$ and $b_{16}$ and so on, then we are left with an infinite sequence and can repeat the process.</p>
|
174,528 | <p>I am editing my original question, as I have figured out a method of doing what I want.</p>
<p>Now my question is if there is a more elegant, efficient way to do the following:</p>
<pre><code>Options[f] = {
Energy -> energy,
Temperature -> Func[Energy*Frequency, {Energy, Frequency}],
Frequency ... | MarcoB | 27,951 | <p>I mentioned in the comments that I find it advantageous to use strings as the names of my user-defined options. In this case, I think it may work for your problem as well. Here is a sample idea:</p>
<pre><code>ClearAll[f]
Options[f] = {
"Energy" -> energy,
"Temperature" -> "Energy" "Frequency",
"Fre... |
660,315 | <p>Let $A,B,C$ be sets. Identify a condition such that $A \cap C = B \cap C$ together with your condition implies $A=B$. Prove this implication. Show that your condition is necessary by finding an example where $A \cap C = B \cap C$, but $ A \neq B$</p>
<p>Edit: I've read the wrong proposition/definition. UGH! The que... | copper.hat | 27,978 | <p>If $A \not\subset C$, then let $x \in A \setminus C$, and $B = A\setminus \{x\}$. Then you can see that $A\ne B$ and $A \cap C = B \cap C$. Hence you must have $A \subset C$. By symmetry, we must have $B \subset C$. Combining shows that you need $(A \cup B) \subset C$.</p>
<p>Now suppose $(A \cup B) \subset C$. Th... |
1,385,936 | <p><em>I was wondering how to approximate $\sqrt{1+\frac{1}{n}}$ by $1+\frac{1}{2n}$ without using Laurent Series.</em></p>
<p>The reason why I ask was because using this approximation, we can show that the sequence $(\cos(\pi{\sqrt{n^{2}-n}})_{n=1}^{\infty}$ converges to $0$. This done using a mean-value theorem or L... | marty cohen | 13,079 | <p>Start with
$(1+x/2)^2-(1+x)
=1+x+x^2/4-(1+x)
=x^2/4
$.</p>
<p>Then,
$(1+x/2)^2-(1+x)
\ge 0
$,
so
$1+x/2
\ge \sqrt{1+x}
$.</p>
<p>Going the other way,
which is harder,
$(1+x/2)^2-x^2/4
=(1+x)
$,
so,
for $x \ge 0$,</p>
<p>$\begin{array}\\
\sqrt{1+x}
&=\sqrt{(1+x/2)^2-x^2/4}\\
&=(1+x/2)\sqrt{1-(x/(2(1+x/2)))... |
2,747,509 | <p>How would you show that if <span class="math-container">$d\mid n$</span> then <span class="math-container">$x^d-1\mid x^n-1$</span> ?</p>
<p>My attempt :</p>
<blockquote>
<p><span class="math-container">$dq=n$</span> for some <span class="math-container">$q$</span>. <span class="math-container">$$ 1+x+\cdots+x^{d-1}... | Community | -1 | <p>Let $\alpha$ be a solution to the equation $x^d = 1$. We then have that $\alpha^d$ = 1. Since $d|n$ we can write $n = d\cdot k$ for some integer $k$. Thus $$1 = 1^k = \left( \alpha^d \right)^k = \alpha^{d\cdot k} = \alpha^n$$
This shows that $\alpha$ is a solution to $x^n = 1$. </p>
<p>This shows that $x^d-1|x^... |
2,532,156 | <p>As the title suggests I am confused between what arguments will qualify a explanation as a proof and when does the intuition betrays us. Here is the question that made me think about this:</p>
<p>On a certain planet Tau Cetus more than half of its land is dry. Prove that a tunnel can be dug straight through center ... | not all wrong | 37,268 | <p>On one extreme, you have "it is obvious that". On the other, you have formal logic where a statement is proved to be true if and only if you write a sequence of special symbols obeying the rules of the logic system. (Let's ignore Godelian incompleteness ideas, interesting and ultimately important though th... |
733,908 | <p>How do i start off with integrating the below function? i tried applying trig substitution and U substitution. how do i go about solving this function? should i split them up further into 2 separate functions ? need some help in this as i can't seem to figure out how to continue on with it </p>
<p>$$\int\fra... | roman | 36,404 | <p>Note that chosing $u(x) = x^2$ and $v(x) = (x^2-1)^{\frac{1}{2}}$, we have </p>
<p>$$\int u(x)v'(x)\, dx = \int\frac{x^{3}}{({x^{2}-1})^{0.5}}dx\quad\text{(check this!)} $$</p>
<p>where $v'(x)$ denotes the first derivative by $x$. Thus we can use integration by parts, i.e. the formula</p>
<p>$$\int u(x) v'(x) \, ... |
482,003 | <p>I need help with the following limit $$\lim_{n\to\infty}\sum_{k=1}^n \frac{1}{\sqrt{kn}}$$</p>
<p>Thanks.</p>
| achille hui | 59,379 | <p>Notice for large $n$, we expect $\displaystyle \sum_{k=1}^n \frac{1}{\sqrt{k}}
\text{ behave like }\int_1^n \frac{dx}{\sqrt{x}} \sim 2\sqrt{n}$. This suggests
$$\frac{1}{\sqrt{k}} \sim \int_{k-1/2}^{k+1/2} \frac{dx}{\sqrt{x}} \sim 2\left( \sqrt{k+\frac12} - \sqrt{k-\frac12}\right)$$ and the terms $\displaystyle \fra... |
3,717,243 | <p>I am reading Introduction to automata theory, languages, and computation 3ed, by John E. Hopcroft, et al. The wikipedia article (<a href="https://en.wikipedia.org/wiki/Turing_machine#Formal_definition" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Turing_machine#Formal_definition</a>) for Turing machine al... | Robert Israel | 8,508 | <p>The correct result should be</p>
<p><span class="math-container">$$ \prod_{i=1}^\infty \left( \frac{i+x}{i+1} \right)^{1/i} = \exp \left(\int_1^x \frac{\Psi(t+1)+\gamma}{t} \; dt\right) $$</span></p>
<p>See my comment to Mostafa Ayaz's answer.</p>
<p>I don't know if this can be written in a more "closed-form&qu... |
2,419,529 | <p>I am asked to state whether the following is true or if false to give a counterexample:</p>
<blockquote>
<p>If $A_1 \supseteq A_2 \supseteq A_3 \supseteq \ldots $ are all sets containing an infinite number of elements, then the intersection $$\bigcap_{k=1}^\infty A_k$$ is infinite as well.</p>
</blockquote>
<p>I... | lhf | 589 | <p><em>Hint:</em> Consider $A_n = \left[-\dfrac1n, \dfrac1n \right] \subseteq \mathbb R$.</p>
|
289,923 | <p>As far as i know, both differential and gradient are vectors where their dot product with a unit vector give directional derivative with the direction of the unit vector. So what are the differences?</p>
| Harald Hanche-Olsen | 23,290 | <p>There is hardly a noticable difference when you work on Euclidean spaces. You can think of the differential at a point as being a linear map, which maps a vector to the dot product of the vector with the gradient. The differential generalizes in a natural way to more abstract settings, such as functions on a manifol... |
2,908,361 | <p>I tried to solve this inequality by taking the square outside the floor function $[y]$ (greatest integer less than $y$)but it was wrong since if $x=2.5$ then $[x]= 2$ and $x^2=4$ while $[x^2]=[6.25]=6$.</p>
| Calvin Khor | 80,734 | <p>Define $f(x)=\operatorname{floor}\left(x^2\right)+5\operatorname{floor}\left(x\right)+6 = [x^2] + 5[x] + 6$.</p>
<p>I'll present some graphs that indicate the solution set is more complicated than $x\ge -4$.</p>
<p><a href="https://www.desmos.com/calculator/kf0m3b5ya4" rel="nofollow noreferrer">(Desmos link)</a><a... |
106,396 | <p>An Indian mathematician, Bhaskara I, gave the following amazing approximation of the sine (I checked the graph and some values, and the approximation is truly impressive.)</p>
<p>$$\sin x \approx \frac{{16x\left( {\pi - x} \right)}}{{5{\pi ^2} - 4x\left( {\pi - x} \right)}}$$</p>
<p>for $(0,\pi)$</p>
<p>Here's ... | Robert Israel | 8,508 | <p>Writing $x = \pi/2 + \pi t$, the approximation becomes $\cos(\pi t) \approx \frac{1-4t^2}{1+t^2} = 1 - 5 t^2 + O(t^4)$. In fact $\cos(\pi t) = 1 - \frac{\pi^2}{2} t^2 + O(t^4)$, but $\pi^2/2 \approx 4.9348$ is not far from $5$. In terms of uniform approximation to $\cos(\pi t)$ for $t \in [-1/2, 1/2]$, $\frac{1 - 4... |
1,316,008 | <p><img src="https://i.stack.imgur.com/xNGPi.png" alt="enter image description here"></p>
<p>The problem is shown in the image. I'm not able to post images yet.. What are the next steps to to find how tall the triangle is? So far i see, that the 3 triangles are similar; however, even by these similarities and by the ... | mathlove | 78,967 | <p>This is impossible.</p>
<p>Suppose that there exist such points.</p>
<p>Let $\alpha=\angle{PAB},\beta=\angle{PBA}$. And let $C$ be the third point of the triangle.</p>
<p>Then, from $\triangle{PAB}$, we have $$\alpha+\beta=90^\circ.$$</p>
<p>Now we have
$$\angle{ACB}=180^\circ-\angle{CAB}-\angle{CBA}=180^\circ-... |
336,827 | <p>A covering map $p:C\to X$ is finite when for each $x\in X$ we have $|p^{-1}(x)|<\infty.$ I have to prove that such a covering map has to be closed. I'm having trouble with it. </p>
<p>When $p$ is a covering map, we can take open neighborhoods $U_x$ of every point $x\in X$ such that $p^{-1}(U_x)$ is a disjoint un... | Seirios | 36,434 | <p>Let $A \subset C$ be closed and let $x \notin p(A)$. Because the covering is finite, there $x_1,...,x_n \in C$ such that $p^{-1}(x)=\{x_1,...,x_n\}$. For every $1 \leq i \leq n$, there exists an open neighborhood $U_i$ of $x_i$ such that $U_i \cap A = \emptyset$ and $p$ induces a homeomorphism between $U_i$ and an o... |
3,546,661 | <p>The following is from <em>Elementary Differential Geometry</em> by A.N. Pressley, page 102.</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/KkYsL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KkYsL.png" alt="enter image description here" /></a></p>
<p><a href="https://i.stack.imgur.com/t2... | Willie Wong | 1,543 | <p><span class="math-container">$A + 2B + C = D + 2E + F$</span> does not imply that <span class="math-container">$A = D$</span>, <span class="math-container">$B =E$</span>, and <span class="math-container">$C = F$</span>!</p>
<p>Example:</p>
<p><span class="math-container">$$1 + 2\cdot 2 + 1 = 6 = 2 + 2 \cdot 1 + 2$... |
2,024,097 | <blockquote>
<p>Which prime numbers $p \in \mathbb{Z}$ are reducible in the unique factorization domain $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right]$ ?</p>
</blockquote>
<p>Suppose $p$ is a prime integer and $p = \alpha \beta$ in $\mathbb{Z}\left[\frac{1 + \sqrt{-3}}{2}\right] = \mathbb{Z}[\omega]$. Then $f(p) = ... | Adam Hughes | 58,831 | <p>This is an example of where you want congruence modulo $3$ not $4$. There are two cases, either $p\equiv 1\mod 3$ or $p\equiv -1\mod 3$. Quadratic reciprocity says that</p>
<p>$$\left({p\over 3}\right)\left({3^*\over p}\right)=1$$</p>
<p>What we're looking for is $\left({3^*\over p}\right)$ since that tells us if ... |
667,903 | <p>$$D=\left\{z\ \left|\ \right. 1<|z|<2, \ \Re(z)>-\frac{1}{2}\right\}$$</p>
<p>I can try to prove that each disc with radius 1, and 2 is open (?), so I need to show that the ball centered at $w$ is contained in the disc with radius 2 </p>
<p>If $w\in \{z: |z|<2\}$, then $|z_0-w|<2$, let $\epsilon = 2... | Eric Towers | 123,905 | <p>(Topology Ia): The set $D$ is the intersection of three open sets: $1 < |z|$, $|z|<2$, and $-\frac{1}{2} < \Re(z)$. It is therefore an open set.</p>
<p>(Topology Ib): The set is the complement of the union of three closed sets: $|z| \leq 1$, $2 \leq |z|$, and $\Re(z) \leq -\frac{1}{2}$. It is therefore... |
105,071 | <p>As one may know, a <b>dynamical system</b> can be defined with a monoid or a group action on a set, usually a manifold or similar kind of space with extra structure, which is called the <i>phase space</i> or <i>state space</i> of the dynamical system. The monoid or group doing the acting is what I call the <i>time s... | Alexandre Eremenko | 25,510 | <p>I will address the case of $C$-action (as time). In the case when the phase space is also $C$, you are right, there
are only Moebius transformations, which can be loxodromic (elliptic, hyperbolic) or parabolic.
That is up to conjugacy by an arbitrary Moebius transformation you have
$z\mapsto e^{\alpha t}z$ and $z\ma... |
283,360 | <p>Let $M$ be a simply connected topological 4-manifold with intersection form given by the E8 lattice. Does anyone know of examples of continuous self-maps of $M$ of degree 2 or 3? Or of degree any other prime for that matter?</p>
| Danny Ruberman | 3,460 | <p>This is more like a long comment than a real answer. </p>
<p>This seems like it's an algebra problem that is probably hard to solve. If you had such a map, of degree $d$, then you would get the following. Choose a basis for the homology, so that the induced map on $H_2(M)$ is written as a matrix $A$. Let J be the m... |
148,624 | <p>What does it mean to say that , A bounded linear operator is not "generally" bounded function.
Can anybody explain ? </p>
| davidlowryduda | 9,754 | <p>Let's choose our favorite bounded linear operator. At the moment, mine happens to be the identity operator $I$ on $\mathbb{R}$, a very boring bounded linear operator. If it's too boring, you can pretend I'm talking about the identity operator on $W^{k,p}$.</p>
<p>Then $||I||_{\text{operator}} = 1$, as clearly $||I(... |
3,710,804 | <p><span class="math-container">$$f(x) = \int \frac{\cos{x}(1+4\cos{2x})}{\sin{x}(1+4\cos^2{x})}dx$$</span></p>
<p>I have been up on this problem for an hour, but without any clues. </p>
<p>Can someone please help me solving this?</p>
| Claude Leibovici | 82,404 | <p>Let <span class="math-container">$x=\tan^{-1}(y)$</span>
<span class="math-container">$$I= \int \frac{\cos{(x)}(1+4\cos{(2x)})}{\sin{(x)}(1+4\cos^2{(x)})}\,dx=\int \frac{5-3 y^2}{y^5+6 y^3+5 y}\,dy$$</span> Now, partial fraction decomposition
<span class="math-container">$$\frac{5-3 y^2}{y^5+6 y^3+5 y}=\frac{1}{y}-\... |
1,770,804 | <p>I am a high school student my maths teacher said that if $\,ax+b=cx+d,\,$ then is $\,a=c\,$ and $\,b=d.\,$ Can someone give me a prove of this?</p>
| fleablood | 280,126 | <p>The statement "ax + b = cx + d implies a=c and b = d" is not true and should be obviously so. Simply do $b = cx + d - ax$ and you get $ax + b = ax + cx + d - ax = cx + d$. $a, b, c$ and $x$ can be anything you like.</p>
<p>That's silly.</p>
<p><strong>HOWEVER</strong> the statement "ax + b = cx + d <em>for all p... |
2,141,182 | <p>In the case of $$\sqrt{(x_n-\ell_1)+(y_n-\ell_2)}\leq \sqrt{(x_n-\ell_1)^2} + \sqrt{(y_n-\ell_2)^2} = |x_n-\ell_1|+|y_n-\ell_2|$$</p>
<p>it is true, if we take the rise the two sides in the power of $2$ we get:</p>
<p>\begin{align}
& (x_n-\ell_1)+(y_n-\ell_2)\leq \left( \sqrt{(x_n-\ell_1)^2}+\sqrt{(y_n-\ell_2)... | Daniel Robert-Nicoud | 60,713 | <p><strong>Hint:</strong> Square both sides.${}{}$</p>
|
2,342,051 | <p>I am totally new to statistics. I'm learning the basics.</p>
<p>I came upon this question while solving Erwin Kreyszig's exercise on statistics.
The problem is simple. It asks to calculate standard deviation after removing outliers from the dataset.</p>
<p>The dataset is as follows: 1, 2, 3, 4, 10.
What I did is, ... | Dave L. Renfro | 13,130 | <p>For each point in the complement of the Cantor set, there exists a two-sided neighborhood of that point contained in the complement of the Cantor set (because the complement of the Cantor set is an open subset of the reals), and hence the function is zero in a two-sided neighborhood of that point, and hence the deri... |
863,846 | <p>Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement with my much more formal text from Perko.</p>
<p>Anyways I was wondering if anyone knew of any similar informal, intui... | lhf | 589 | <p>Try these books:</p>
<ul>
<li><p><em>Introduction to Applied Numerical Analysis</em> by <a href="http://en.wikipedia.org/wiki/Richard_Hamming" rel="nofollow">Richard Hamming</a></p></li>
<li><p><em>Numerical Methods That Work</em> by <a href="http://en.wikipedia.org/wiki/Forman_S._Acton" rel="nofollow">Forman Acton... |
1,434,420 | <p>Why is $\bigcap\limits_{n=1}^{\infty} \left( \bigcup\limits_{i=1}^{n} G_i \right)^c = \left( \bigcup\limits_{n=1}^{\infty} \left( \bigcup\limits_{i=1}^{n} G_i \right) \right)^c$? What set properties are being applied here? (The $^c$ is set complement)</p>
| NoseKnowsAll | 180,054 | <p>This is a repeated application of <a href="https://en.wikipedia.org/wiki/De_Morgan's_laws" rel="nofollow">De Morgan's Laws</a>:</p>
<p>$$A^c \cap B^c = \left(A \cup B\right)^c$$</p>
<p>Basically, simplify it to only two sets and you'll see that the intersection of the complements is equal to the complement of ... |
1,289,994 | <p>If you fold a rectangular piece of paper in half and the resulting
rectangles have the same aspect ratio as the original rectangle,
then what is the aspect ratio of the rectangles?</p>
| ParaH2 | 164,924 | <p>$$
\frac{\text{width}}{\text{lengh}}=\sqrt{2}
$$</p>
<p><a href="http://en.wikipedia.org/wiki/Paper_size" rel="nofollow">Have a look here</a></p>
|
2,280,203 | <p>How to transform the integral </p>
<p>$$\int _{0}^{\pi }\sin ^{2}\left( \psi \right) \sin \left( m\psi \right) d\psi $$</p>
<p>to </p>
<p>$$\int _{0}^{\pi }\left( \dfrac {1} {2}-\dfrac {1} {2}\cos 2\psi \right) \sin m\psi d\psi $$</p>
<p>What is the general method you need to solve trig questions like this. How ... | N3buchadnezzar | 18,908 | <p><strong>The general approach</strong></p>
<p>Considering trigonometric integrals there are a myriad of different techniques to solve them. Sometimes integration by parts is the best option, other times a clever substitution or a trigonometric identity saves the day. Sadly, the only way to know which one to use is t... |
519,325 | <p>Evaluate $\displaystyle\int \dfrac{1}{x^2+9} \, dx$.
I've only learned the normal way of solving integrals but it does not work.
I haven't learned how to use trigonometry to solve these problem.</p>
<p>I know you have to rearrange it into the form ${[f(x)]Β² + 1}$ and then integrate.</p>
<p>Can someone point me s... | imranfat | 64,546 | <p>Check this <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&frm=1&source=web&cd=3&ved=0CDYQFjAC&url=http%3A%2F%2Fintegral-table.com%2Fdownloads%2Fsingle-page-integral-table.pdf&ei=XWJUUtT3I6jC2wXkjYC4Cg&usg=AFQjCNGqJnnp4oeD69SVOXnbYSIn565LEw" rel="nofollow">table of int... |
519,325 | <p>Evaluate $\displaystyle\int \dfrac{1}{x^2+9} \, dx$.
I've only learned the normal way of solving integrals but it does not work.
I haven't learned how to use trigonometry to solve these problem.</p>
<p>I know you have to rearrange it into the form ${[f(x)]Β² + 1}$ and then integrate.</p>
<p>Can someone point me s... | Jack M | 30,481 | <p>You know (or should know) that $\int\frac{1}{x^2+1}\mathrm{dx}=\arctan(x)$. Let's try and get the integrand into that form.</p>
<p>$$\int\frac{1}{x^2+9}\mathrm{dx}=\int\frac{1}{9(\frac{x^2}{9}+1)}\mathrm{dx}=\frac{1}{9}\int\frac{1}{\left(\frac{x}{3}\right)^2+1}\mathrm{dx}$$</p>
<p>You also know (or should know) th... |
306,461 | <p>Let $A = \{(x,y) \in\mathbb{R}^2: a \leq (x-c)^2+(y-d)^2 \leq b\}$ for given $a,b,c, d$ real numbers. I want to show that $A$ is path-connected.</p>
<p>How can I do that?</p>
<p>I know that every open subset of $\mathbb R^2$ that is connected is path connected. But this is obviously not open so I cannot use that. ... | Stefan Hamcke | 41,672 | <p>The set $S:=[0,2\pi]\times[a,b]$ is path-connected, being a product of path-connected sets. The annulus $A$ is the image of $S$ under the continuous map $(x,y)\mapsto y\cdot e^{xi}+(c+di)$, where you consider $\mathbb R^2$ as the complex plane.</p>
|
2,128,588 | <p>The question;</p>
<p>$U = \{x |Ax = 0\}$ If $ A = \begin{bmatrix}1 & 2 & 1 & 0 & -2\\ 2 & 1 & 2 & 1 & 2\\1 & 1 & 0 & -1 & -2\\ 0 & 0 & 2 & 0 & 4\end{bmatrix}$</p>
<p>Find a basis for $U$.</p>
<p><hr>
To make it linearly independent, I reduce the rows... | drhab | 75,923 | <p><strong>Hint</strong>:</p>
<p>For <strong>every</strong> integer $m$ the set $\{m+1,m+2,m+3,m+4,m+5,m+6\}$ contains exactly $2$ elements that are divisible by $3$.</p>
<p>Now for $m$ take the sum of $n-1$ dice.</p>
<p>Also have a look <a href="https://math.stackexchange.com/q/1523820/75923">this question</a> and ... |
3,057,517 | <p>Hy everybody ! </p>
<p>I'm studying population dynamics for my calculus exam, and I don't understand something that seems really easy, so I thought you might be able to help me out ;)</p>
<p>Here's the thing. I have this differential equation <span class="math-container">$\frac{dN}{dt} = \sqrt{N}$</span>.</p>
<p>... | Lutz Lehmann | 115,115 | <p>Apart from these two solutions you can also combine them to solutions that are zero up to some point <span class="math-container">$t\le c$</span> and then follow the shifted quadratic function, <span class="math-container">$N(t)=\frac14(t-c)^2$</span> for <span class="math-container">$t>c$</span>.</p>
<p>The irr... |
717,664 | <p>I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the coefficients. I suspect maybe only the real part goes ... | Batman | 127,428 | <p>Try multiplying the numerator and denominator by $1+i$. This will give you $\frac{(2i+2)(1+i)}{1^2+1^2}$. Then, FOIL the numerator and note $i^2=-1$. </p>
|
717,664 | <p>I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something wrong with the coefficients. I suspect maybe only the real part goes ... | Gautam Shenoy | 35,983 | <p>Hint: Take an "i" out from top.</p>
|
1,109,853 | <blockquote>
<p><em>The below proof is incorrect. See the answers for more information.</em></p>
</blockquote>
<p>This question is in the context of exploring how to explain the process of developing a proof.</p>
<p>When reading a proof on the irrationality of $ \sqrt{3} $, I came across the following statement, wh... | Mike Pierce | 167,197 | <p>Your proof appears to be the reverse of what you are trying to prove. The original statement is "$a^2$ divisible by $3$ implies that $a$ is divisible by $3$." Your proof starts with "suppose we have $a$ divisible by $n$..." and this lead to "...$a^2$ is divisible by $n$." Your proof should <em>start</em> with "$a^2$... |
386,649 | <p>If you were working in a number system where there was a one-to-one and onto mapping from each natural to a symbol in the system, what would it mean to have a representation in the system that involved more than one digit?</p>
<p>For example, if we let $a_0$ represent $0$, and $a_n$ represent the number $n$ for any... | Ross Millikan | 1,827 | <p>In a sense, our usual system is like that. In $\LaTeX$ if you put braces around something it gets treated as a single character. I know that isn't what you are thinking, but to do what you are thinking you would need a countably infinite set of characters, which is what we get with the decimal (or other base) syst... |
4,107,396 | <p>I'm stuck at solving the following problem: launch 3 fair coins independently. Let A the event: "you get at least a head" and B "you get exactly one tail".
Then what is the probability of the event <span class="math-container">$A \cup B$</span>?</p>
| YJT | 731,237 | <p>Note that <span class="math-container">$\overline{A\cup B}=\overline{A}\cap \overline{B}$</span>. Now <span class="math-container">$\overline{A}$</span> is you get not heads and <span class="math-container">$\overline{B}$</span> is you get any number of tails but one. Only one option fits <span class="math-container... |
466,757 | <p>Suppose we have the following</p>
<p>$$ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$$</p>
<p>where all the $a_{ij}$ are non-negative.</p>
<p>We know that we can interchange the order of summations here. My interpretation of why this is true is that both this iterated sums are rearrangements of the same series an... | Umberto P. | 67,536 | <p>This isn't a proof, but perhaps can give you the insight you are looking for. Any nondecreasing sequence converges to its (possibly infinite) supremum. Thus a series of nonnegative terms converges to the supremum of its partial sums and interchanging the order of summation doesn't affect the value of the supremum: ... |
1,221,442 | <p>I have two uniform random variables $B$ and $C$ distributed between $(2,3)$ and $(0,1)$ respectively. I need to find the mean of $\sqrt{B^2-4C}$. Could I plug in the means for $B$ and $C$ and then solve or is it more complicated than that?<br>
The original question is here: <a href="https://math.stackexchange.com/qu... | rightskewed | 171,836 | <p>Use change of variables:</p>
<p>$$X = \sqrt{B^2-4C}
$$
$$Y=C$$</p>
<p>$$B = \sqrt{X^2+4Y}$$
$$C=Y$$</p>
<p>Jacobian:
$$
J = det\left( \begin{array}{ccc}
\frac{\partial B}{\partial X} &\frac{\partial C}{\partial X} \\
\frac{\partial B}{\partial Y} & \frac{\partial C}{\partial Y} \\
\end{array} \right... |
11,457 | <p>In their paper <em><a href="http://arxiv.org/abs/0904.3908">Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms</a></em>, Greenberg and Voight remark that</p>
<p>...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proporti... | Emerton | 2,874 | <p>One heuristic is the following: if one imagines that the residue at $s = 1$ of the $\zeta$-function doesn't grow too rapidly, then the value is a combination of the regulator and the class number. I don't know any reason for the regulator not to also grow (there
are a lot of units, after all!), and hence one can im... |
411,717 | <p>Let $G$ be a group. By an automorphism of $G$ we mean an isomorphism $f: G\to G$
By an inner automorphism of $G$ we mean any function $\Phi_a$ of the following form:
For every $x\in G$, $\Phi_a(x)=a x a^{-1}$.
Prove that every inner automorphism of $G$ is an automorphism of $G$
which means I should prove $\Phi_a$ is... | Ink | 34,881 | <p>To prove that every inner automorphism is indeed an automorphism, you need to show that</p>
<ol>
<li>$\Phi_a$ is a homomorphism</li>
<li>$\Phi_a$ is surjective</li>
<li>$\Phi_a$ is injective (i.e $\ker\Phi_{a} = \{e\}$)</li>
</ol>
<p>All three are straightforward if you know your definitions.</p>
|
3,436,515 | <p>Please help!</p>
<p>How to show that <span class="math-container">$ \lim _{nββ} \frac{x_{(n+1)}}{x_n} =\frac{1+\sqrt 5}{2}$</span> for a dynamical system
<span class="math-container">$$x_{(n+1)}=x_n + y_n\\
y_{(n+1)}=x_n$$</span></p>
<p>Thank you!</p>
| user90369 | 332,823 | <p>From </p>
<p><span class="math-container">$\left(\begin{array}{r} x_n \\ y_n \end{array}\right) = \left(\begin{array}{rr} 1 ~~~1 \\ 1 ~~~0 \end{array}\right)^n\left(\begin{array}{r} x_0 \\ y_0 \end{array}\right)$</span></p>
<p>you get </p>
<p><span class="math-container">$\left(\begin{array}{r} x_n \\ y_n \end{ar... |
442,759 | <p>I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than one inverse element?</p>
| Manos | 11,921 | <p>It depends on how you define the 'neutral' element. If you define it as $e*x=x*e=x$ for every $x$ in your set, where $*$ is the operation with respect to which $e$ is 'neutral', then $e$ is going to be unique.</p>
|
442,759 | <p>I was reading a book on groups, it points out about the uniqueness of the neutral element and the inverse element. I got curious, are there algebraic structures with more than one neutral element and/or more than one inverse element?</p>
| Stefan Hamcke | 41,672 | <p>This just came to my mind:</p>
<p>Take a set $X$ with two elements $a$ and $b$. We want to equip this with an interior multiplication that is associative ($X$ is then called a <em>semigroup</em>), and such that $a$ is neutral on the right but not neutral on the left. Then we already know three of four values:
$$aa=... |
107,171 | <p>I'm trying to find $$\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} .$$
After I tried couple of algebraic manipulation, I decided to use the polaric method.
I choose $x=r\cos \theta $ , $y=r\sin \theta$, and $r= \sqrt{x^2+y^2}$, so I get </p>
<p>$$\lim\limits_{r \to 0} \frac{e^{-\frac{1}{r^2}... | J. M. ain't a mathematician | 498 | <p>To expand on my comment, two functions here are relevant: the Lerch transcendent</p>
<p>$$\Phi(z,s,a)=\sum_{k=0}^\infty \frac{z^k}{(k+a)^s}$$</p>
<p>and the polygamma function</p>
<p>$$\psi^{(k)}(z)=\frac{\mathrm d^{k+1}}{\mathrm dz^{k+1}}\log\Gamma(z)=(-1)^{k+1}k!\sum_{j=0}^\infty \frac1{(z+j)^{k+1}}$$</p>
<p>w... |
788,671 | <p>What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$?</p>
<p>I see an obvious method of brute-force wherein I can impose a coordinate system onto the figure. But is there a better solution?</p>
| Adan Paulo Rivera Hernandez | 677,883 | <p>The area of a triangle given its medians is <span class="math-container">$(4/3)\sqrt{m(m-m_a)(m-m_b)(m-m_b)}$</span>
and the area of a triangle formed by the medians of the first triangle is,by heron's formula, <span class="math-container">$\sqrt{m(m-m_a)(m-m_b)(m-m_b)}$</span>. Dividing the two we get</p>
<p><span... |
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