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 |
|---|---|---|---|---|
406,437 | <p>Calculus the extreme value of the $f(x,y)=x^{2}+y^{2}+xy+\dfrac{1}{x}+\dfrac{1}{y}$</p>
<p>pleasee help me.</p>
| rschwieb | 29,335 | <p>Here's a slightly different example. If you are comfortable with doing algebra with polynomials, then I think you will find it easy to understand.</p>
<p>Take any finite group $G$ (say of order $n$). Now, you can formally make a set $\{\sum_{g\in G} \alpha_g g\mid \alpha_g\in \Bbb R\}$. </p>
<p>Since you know how ... |
4,085,635 | <p>What is the volume of an n-dimension cube? Consider the length of each side to be <span class="math-container">$a$</span>. How to solve this problem?</p>
| saulspatz | 235,128 | <p>For an intuitive explanation, imagine that the side are thing hollow metal rods, with a chain running through all of them and joining them. (The chain forms a closed loop.)</p>
<p>Hold the longest rod parallel to the ground. The other rods hang down an close the polygon.</p>
|
209,892 | <p>From the values:</p>
<pre><code>{57.02, 71.04, 87.03, 97.05, 99.07, 101.05, 103.01, 113.08, 114.04, 115.03,
128.06, 128.09, 129.04, 131.04, 137.06, 147.07, 156.10, 163.03, 186.08}
</code></pre>
<p>I would like to find all possible combinations of 3 values that have the sum of roughly 344.25 (+/- 0.05 would be ok)... | Roman | 26,598 | <p>Use exact fractions for <code>IntegerPartitions</code>:</p>
<pre><code>L = Round[{57.02, 71.04, 87.03, 97.05, 99.07, 101.05, 103.01, 113.08,
114.04, 115.03, 128.06, 128.09, 129.04, 131.04, 137.06, 147.07,
156.10, 163.03, 186.08}, 1/100];
Join @@ Table[IntegerPartitions[i, ∞, L], {i, 344, 34... |
2,052,826 | <p>I've been working on some quantum mechanics problems and arrived to this one where I have to deal with subscripts. I got stuck doing this:
I have $\epsilon_{imk}\epsilon_{ikn}=\delta_{mk}\delta_{kn}-\delta_{mn}\delta_{kk}$. But then I went to check and $\delta_{mk}\delta_{kn}-\delta_{mn}\delta_{kk}$ is equal to $\de... | HBR | 396,575 | <p>Levi Civita symbol $\epsilon_{ikl}$ is defined as it follows
$$\epsilon_{ikl} = \left\{
\begin{array}{cl}
1 & if\quad i\neq k\neq l\quad and \quad even \quad permutation\\
-1& if\quad i\neq k\neq l\quad and \quad odd\quad permutation\\
0 & otherwise
\end{array}\right.$$</p>
<p>From this definition, let... |
1,781,227 | <p>For a simple x and y plane (2 dimensional), to find the distance between two points we would use the formula </p>
<p>$$
a^2 +b^2 = c^2
$$</p>
<p>For a slightly more complicated plane; x,y and z (3 dimensional), to find the distance between two points we would use the formula</p>
<p>$$
d^2 = a^2 + b^2 + c^2
$$</p>... | poetasis | 546,655 | <p>The Pythagorean theorem works for any dimensions >1. We can find examples by constructing n-tuples made of multiple triples. We begin by find a side <span class="math-container">$A$</span> to match the <span class="math-container">$C$</span> of any previous triangle.</p>
<p>Here we substitute and solve <span class=... |
4,063,337 | <p>In an exercise I'm asked the following:</p>
<blockquote>
<p>a) Find a formula for <span class="math-container">$\int (1-x^2)^n dx$</span>, for any <span class="math-container">$n \in \mathbb N$</span>.</p>
<p>b) Prove that, for all <span class="math-container">$n \in \mathbb N$</span>: <span class="math-container">$... | J.G. | 56,861 | <p>Evaluate <span class="math-container">$f_n(x):=\int_0^x(1-t^2)^ndt$</span> with integration by parts viz. <span class="math-container">$u=(1-t^2)^n,\,v=t$</span> so<span class="math-container">$$f_n=x(1-x^2)^n+2n\int_0^xt^2(1-t^2)^{n-1}dt=x(1-x^2)^n+2n(f_{n-1}-f_{n+1}).$$</span>Definite integration on <span class="m... |
2,468,155 | <p>This problem is from Challenge and Thrill of Pre-College Mathematics:
Prove that $$ (a^3+b^3)^2\le (a^2+b^2)(a^4+b^4)$$</p>
<p>It would be really great if somebody could come up with a solution to this problem.</p>
| Donald Splutterwit | 404,247 | <p>\begin{eqnarray*}
\color{blue}{a^2b^2(a-b)^2} \geq 0 \\
\color{red}{a^6} +\color{blue}{a^4b^2+a^2 b^4} +\color{red}{b^6} \geq \color{red}{a^6} +\color{blue}{2 a^3b^3} +\color{red}{b^6} \\
(a^2+b^2)(a^4+b^4) \geq (a^3+b^3)^2.
\end{eqnarray*}</p>
|
626,095 | <blockquote>
<p>$R = \mathbb{F}_3[x]/\langle X^3+X^2+1\rangle$ and $\alpha=[X]$ in $R$. How do you prove that the group $R^*$ is not cyclic?</p>
</blockquote>
<p>We have shown that $\alpha$ is a unit in $R$ with order $8$ and that $\alpha^4$ and $-\alpha^4$ are two different elements in $R^*$ both with order 2.</p>
| rschwieb | 29,335 | <p>Hint:</p>
<p>$x^3+x^2+1$ factors into the irreducibles $(x-1)$ and $(x^2+2x+2)$ over $\Bbb F_3$.</p>
<p>By the Chinese Remainder Theorem, $R\cong \frac{\Bbb F_3[x]}{(x-1)}\times \frac{\Bbb F_3[x]}{(x^2+2x+2)}$.</p>
<p>Can you see what the two pieces look like, and how you can use this to see the units of $R$?</p>... |
4,646,650 | <p>I believe there will be values of <span class="math-container">$x$</span> for which the inequality <span class="math-container">$x^3 - 2x + 2 \ge 3 - x^2$</span> is true and values for which it is not true, because:</p>
<ul>
<li><em>LHS asymptotically increases but RHS decreases for increasingly positive values of <... | Rehman | 1,136,062 | <p><span class="math-container">$$x^3+x^2-2x-1 \ge 0$$</span></p>
<hr />
<p>First, let's solve the equation: <span class="math-container">$$x^3+x^2-2x-1=0$$</span> Let's say <span class="math-container">$x=y-\frac{1}{3}$</span> :
<span class="math-container">$$y^3-\frac{7}{3}y-\frac{7}{27}=0$$</span> We know that the: ... |
151,076 | <p>If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?</p>
| Asaf Karagila | 622 | <p>One does not have to go to partially ordered sets for a counterexample. Linear orders are enough: Consider the rationals in the open interval $(0,1)$ and the rationals in the closed interval $[0,1]$. </p>
<p>The two obviously embed into one another but are not isomorphic due to minimum/maximum considerations. </p>
... |
151,076 | <p>If A and B are partially-ordered-sets, such that there are injective order-preserving maps from A to B and from B to A, is there necessarily an order-preserving bijection between A and B ?</p>
| Community | -1 | <p>Here's an example with <em>scattered</em> countable linear orderings: a set of order type $\omega^*\omega$ and a set of order type $1+\omega^*\omega$.</p>
|
1,844,894 | <p>To explain my question, here is an example.</p>
<p>Below is an AP:</p>
<p>2, 6, 10, 14....n</p>
<p>Calculating the nth term in this sequence is easy because we have a formula. The common difference (d = 4) in AP is constant and that's why the formula is applicable, I think.</p>
<p>But what about this sequence:</... | Archis Welankar | 275,884 | <p>Note we can develop a combinatorial argument which gives total number of ways as ${n\choose r}$ where r is total number of summations</p>
|
1,511,733 | <p>B = matrix given below. I is identity matrix.</p>
<pre><code> [1 2 3 4]
[3 2 4 3]
[1 3 2 4]
[5 4 3 7]
</code></pre>
<p>So What will be the relation between the matrices A and C if AB = I and BC = I?</p>
<p>I think that A = C because both AB and BC have B in common and both of their product is an identity matri... | Yiorgos S. Smyrlis | 57,021 | <p>If $AB=I$, then $B$ is invertible and possesses a unique inverse, and clearly, this can only be the matrix $A$. Hence
$$
AB=I\qquad\Longrightarrow\qquad AB=BA=I.
$$
Likewise, if $BC=I$, then $C$ is the inverse of $B$, the <em>unique inverse</em>, and therefore
$A=C$.</p>
|
335,483 | <p>Let $N$ be a set of non-negative integers. Of course we know that $a+b=0$ implies that $a=b=0$ for $a, b \in N$.</p>
<p>How do (or can) we prove this fact if we don't know the subtraction or order?</p>
<p>In other words, we can only use the addition and multiplication.</p>
<p>Please give me advise.</p>
<p>EDIT</... | Doug Spoonwood | 11,300 | <p>You don't need $+$ or $\times$ as forming monoids on the natural numbers, nor the distributive property. You can get by with less as follows:</p>
<p>Addition has the property that $(0+b)\leq(a+b)$ and $(a+0)\leq(a+b)$. This is NOT presupposing the order of the natural numbers, or that no number of the naturals ha... |
380,530 | <p>It's easy to show that there's a function such that $\int_1^\infty f $ diverges, but $\int_1^\infty |f|$ converges, such as $f = 1/(-1+x)$. </p>
<p>But is there a function such that $\int_1^\infty f $ converges, but $\int_1^\infty |f| $ diverges?</p>
| Clement C. | 75,808 | <p>What about $x\mapsto\frac{\sin(x-1)}{x-1}$? (by definition of what you ask, it cannot Lebesgue integrable, but $\int_{0}^{\uparrow\infty} \frac{\sin x}{x} dx$ does converge to $\frac{\pi}{2}$; and I 'm almost sure to recall that the integral of its absolute value diverges).</p>
|
380,530 | <p>It's easy to show that there's a function such that $\int_1^\infty f $ diverges, but $\int_1^\infty |f|$ converges, such as $f = 1/(-1+x)$. </p>
<p>But is there a function such that $\int_1^\infty f $ converges, but $\int_1^\infty |f| $ diverges?</p>
| Mark McClure | 21,361 | <p>Sami is correct. Here are some more details.</p>
<p>$$\left|\int_a^{\infty} f(x) \, dx\right| \leq \int_a^{\infty} \left|f(x)\right| \, dx.$$</p>
<p>Thus, if the second integral converges then certainly the first does as well.</p>
<p>Next, it's not too hard to show that </p>
<p>$$\int_0^{\infty} \frac{\sin(x)}{... |
3,155,463 | <blockquote>
<p><span class="math-container">$$
\lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right] \text { equals }\_\_\_\_
$$</span></p>
</blockquote>
<p>I tried to expand the term in power using binomial theorem but still could not obtain the limit. </p>
| TheSilverDoe | 594,484 | <p>One has
<span class="math-container">$$n - \frac{n}{e} \left( 1 + \frac{1}{n}\right)^n = n - \frac{n}{e} \exp \left( n \ln \left( 1 + \frac{1}{n}\right)\right) =n - \frac{n}{e} \exp \left( n \left(\frac{1}{n}- \frac{1}{2n^2} + o \left( \frac{1}{n^2}\right)\right)\right) $$</span></p>
<p>so
<span class="math-conta... |
1,485,327 | <p>Show that $f(x,y)$ defined by:</p>
<p>$$f(x,y) = \begin{cases}\dfrac{x^2y^2}{\sqrt{x^2+y^2}}&\text{ if }(x,y)\not =(0,0)\\0 &\text{ if }(x,y)=(0,0)\end{cases}$$</p>
<p>is differentiable at $(x,y) = (0,0)$</p>
<p>I tried to solve this problem by applying the theorem that if partial derivatives are continuo... | mathcounterexamples.net | 187,663 | <p>You have $$x^2+y^2-2 \vert xy \vert=(\vert x \vert - \vert y \vert)^2 \ge 0$$ Hence $$\vert xy \vert \le \frac{x^2+y^2}{2}$$ and $$0 \le \frac{\vert f(x,y) \vert}{\sqrt{x^2+y^2}} = \frac{x^2y^2}{x^2+y^2} \le \frac{1}{4}(x^2+y^2)$$ As $\lim_{(x,y) \to (0,0)} x^2+y^2 = 0$, this proves that $f$ is differentiable at $(0... |
1,485,327 | <p>Show that $f(x,y)$ defined by:</p>
<p>$$f(x,y) = \begin{cases}\dfrac{x^2y^2}{\sqrt{x^2+y^2}}&\text{ if }(x,y)\not =(0,0)\\0 &\text{ if }(x,y)=(0,0)\end{cases}$$</p>
<p>is differentiable at $(x,y) = (0,0)$</p>
<p>I tried to solve this problem by applying the theorem that if partial derivatives are continuo... | Mercy King | 23,304 | <p>For every non-zero $h=(h_1,h_2)\in \mathbb{R}^2$ we have:
$$
|f(h)-f(0)|=||f(h)|=\frac{h_1^2h_2^2}{\|h\|_2}\le \frac{\|h\|_2^4}{\|h\|_2}=\|h\|_2^3,
$$
and therefore
$$
\lim_{\|h\|_2\to0}\frac{|f(h)-f(0)|}{\|h\|_2}=0.
$$
This shows that $f$ is differentiable at $(0,0)$, and $Df(0)\equiv 0$.</p>
|
2,784,697 | <p>Find the solutions to:$\displaystyle\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^2$. </p>
<p>I got the following solutions:-</p>
<p>$\left(\frac{dy}{dx}\right)=0\Rightarrow y=c_1$ is a solution</p>
<p>$\left(\frac{dy}{dx}\right)=1\Rightarrow y=x+c_2$ is another solution </p>
<p>Are there any other solutions?</p>... | Dr. Sonnhard Graubner | 175,066 | <p>Substitute $$\frac{dy(x)}{dx}=v(x)$$ and then you will get $$\frac{\frac{dv(x)}{dx}}{v(x)^2}=1$$</p>
|
2,784,697 | <p>Find the solutions to:$\displaystyle\frac{d^2y}{dx^2}=\left(\frac{dy}{dx}\right)^2$. </p>
<p>I got the following solutions:-</p>
<p>$\left(\frac{dy}{dx}\right)=0\Rightarrow y=c_1$ is a solution</p>
<p>$\left(\frac{dy}{dx}\right)=1\Rightarrow y=x+c_2$ is another solution </p>
<p>Are there any other solutions?</p>... | Dylan | 135,643 | <p>Notice there is no 0th order derivative here. Hence, this is actually just a first-order equation in disguise. Substitute $v = \frac{dy}{dx}$
$$ \frac{dv}{dx} = v^2 $$</p>
<p>Separate this and solve</p>
<p>$$ v(x) = \frac{1}{c_1-x} $$</p>
<p>Then integrate back</p>
<p>$$ y(x) = c_2 -\ln|c_1-x| $$</p>
|
1,170,602 | <p>How to evaluate the integral </p>
<p>$$\int \sqrt{\sec x} \, dx$$</p>
<p>I read that its not defined.<br>
But why is it so ? Does it contradict some basic rules ?
Please clarify it .</p>
| Tryss | 216,059 | <p>$$\int_{a}^b \frac{1}{\sqrt{\cos(x)}}dx$$</p>
<p>is defined if $]a,b[\subset ]-\frac{\pi}{2}+2k\pi, \frac{\pi}{2}+2k\pi[$. But you can't calculate it with the usual functions, you'll need "special" functions :</p>
<p><a href="http://en.wikipedia.org/wiki/Elliptic_integral#Incomplete_elliptic_integral_of_the_first_... |
1,502,309 | <p>The initial notation is:</p>
<p>$$\sum_{n=5}^\infty \frac{8}{n^2 -1}$$</p>
<p>I get to about here then I get confused.</p>
<p>$$\left(1-\frac{3}{2}\right)+\left(\frac{4}{5}-\frac{4}{7}\right)+...+\left(\frac{4}{n-3}-\frac{4}{n-1}\right)+...$$</p>
<p>How do you figure out how to get the $\frac{1}{n-3}-\frac{1}{n-... | jameselmore | 86,570 | <p>Looking a little closer at the question, he is asking about partial fraction decomposition, as opposed to the value of the sum itself. For this particular example, it's fairly straight forward.</p>
<p>When given a fraction which contains a polynomial denominator, you can factor this fraction and break it into a sum... |
3,501,052 | <p>I want to find the number of real roots of the polynomial <span class="math-container">$x^3+7x^2+6x+5$</span>.
Using Descartes rule, this polynomial has either only one real root or 3 real roots (all are negetive). How will we conclude one answer without doing some long process?</p>
| 2'5 9'2 | 11,123 | <p>The first derivative is <span class="math-container">$f'(x)=3x^2+14x+6$</span>. We observe that this is negative when <span class="math-container">$x=-1$</span>: <span class="math-container">$f'(-1)=-5$</span>. From this we consider the tangent line <span class="math-container">$y=-5(x+1)+5$</span>.</p>
<p>There is... |
646,010 | <p>So i kinda think i have figured this out, i'm not very good at math, and need a formula to figure out some stats for a game i'm playing.</p>
<p>I have a Weapon with a reload speed of X sec.. however, i also have a modifier attached, that will make the weapon reload faster by +Y%</p>
<p>i made this formula, mostly ... | Mhenni Benghorbal | 35,472 | <p>To see it (convergence), make the change of variables <span class="math-container">$t=\ln(x)$</span> which gives</p>
<blockquote>
<p><span class="math-container">$$ \int _{-\infty }^{\infty }\!{t}^{2}\sin \left( {{\rm e}^{2\,t}}
\right) {{\rm e}^{t}}{dt}.$$</span></p>
</blockquote>
|
646,010 | <p>So i kinda think i have figured this out, i'm not very good at math, and need a formula to figure out some stats for a game i'm playing.</p>
<p>I have a Weapon with a reload speed of X sec.. however, i also have a modifier attached, that will make the weapon reload faster by +Y%</p>
<p>i made this formula, mostly ... | Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\down}{\downarrow}... |
646,010 | <p>So i kinda think i have figured this out, i'm not very good at math, and need a formula to figure out some stats for a game i'm playing.</p>
<p>I have a Weapon with a reload speed of X sec.. however, i also have a modifier attached, that will make the weapon reload faster by +Y%</p>
<p>i made this formula, mostly ... | André Nicolas | 6,312 | <p>There is no real difficulty at $0$, since near $0$ the function $\sin(x^2)$ behaves like $x^2$, so $\lim_{x\to 0^+}\ln^2 x\sin(x^2)=0$. So we examine
$$\int_1^B (\ln^2 x)( \sin(x^2))\,dx.\tag{1}$$
Rewrite as
$$\int_1^B \frac{\ln^2 x}{2x} 2x \sin(x^2)\,dx,$$
and use integration by parts, letting $u=\frac{\ln^2 x}{2x}... |
307,529 | <p>I am trying to prove that if $L/K$ is an algebraic extension and if $\alpha \in L$, then </p>
<ul>
<li><p>$\alpha$ is separable over $K$ if $\mathrm{char}(K)=0$. This is clear because $K$ is perfect which in turn implies that $L/K$ is seperable . </p></li>
<li><p>Now if $\mathrm{char}(K)=p$ is prime, then the state... | Martin Brandenburg | 1,650 | <p>Hint: If $\alpha \in L$, then there is some $n \in \mathbb{N}$ such that $\alpha^{p^n}$ is separable.</p>
|
941,632 | <p>Is the Set $$S=\{e^{2x},e^{3x}\}$$ linearly independent?? And answer says Linearly independent over any interval $(a,b)$,only when $0$ doesnot belong to $(a,b)$</p>
<p>How do I proceed??</p>
<p>Thanks for the help!!</p>
| Timbuc | 118,527 | <p>Suppose $\;r,s\in\Bbb R\;$ are such that</p>
<p>$$re^{2x}+se^{3x}=0\;\;,\;\;\forall\,x\in (a,b)\implies e^{2x}(r+se^x)=0$$</p>
<p>since $\;e^t\neq0\;\;\forall\,t\in\Bbb R\;$ , we get that</p>
<p>$$r+se^x=0\iff e^x=-\frac rs\;\;\forall\;x\in (a,b)\in\Bbb R$$</p>
<p>and since the exponential is not a constant func... |
3,492,856 | <p><a href="https://i.stack.imgur.com/FHCP2.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FHCP2.jpg" alt="This is the question"></a> My solution:-</p>
<p>Since <span class="math-container">$OA=AB$</span>, let us find OA first.
<span class="math-container">$OA=\sqrt{(18-0)^2+(3-0)^2}=3\sqrt{37}$</s... | Community | -1 | <p>Yes, you are: for all linear maps <span class="math-container">$T_1:V\to W$</span> and <span class="math-container">$T_2:W\to H$</span>, it is always the case that <span class="math-container">$\ker (T_2\circ T_1)\supseteq \ker T_1$</span>. In fact, if <span class="math-container">$T_1(x)=0$</span>, then <span class... |
467,301 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>The problem statement is in the title.</p>
<p>My attempt at the proof is:</p>
<p>Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists $\{x_1^n,\dots,x_p^n\}$ such that $X\subset\bigcup\limits_{i=1}^p B(x_i^n;\frac{1}{n})$. Let $K=\frac{2p}... | Elchanan Solomon | 647 | <p>Suppose $X$ consists of two points a distance of $100$ apart. Take $n=1$ and $p=2$. Your proof implies the two points are a distance of $4$ apart.</p>
|
467,301 | <p>I'm reading Intro to Topology by Mendelson.</p>
<p>The problem statement is in the title.</p>
<p>My attempt at the proof is:</p>
<p>Since $X$ is a compact metric space, for each $n\in\mathbb{N}$, there exists $\{x_1^n,\dots,x_p^n\}$ such that $X\subset\bigcup\limits_{i=1}^p B(x_i^n;\frac{1}{n})$. Let $K=\frac{2p}... | Alex Youcis | 16,497 | <p>To round off the solutions, you can notice that $X\times X$ is compact, and $d:X\times X\to\mathbb{R}$ is continuous, and so obtains its max.</p>
|
2,069,001 | <p>A team of seven netballers is to be chosen from a squad of twelve players A,B,D, E, F, G, H, I, J, K, L. In how many ways can they be chosen:<br>
a) with no restriction
This is fairly easy. 12C7 = 792</p>
<p>b) if the captain C is to be included
11C6 = 462</p>
<p>c) If J and K are both to be excluded
10C7 = 120</p... | hamam_Abdallah | 369,188 | <p><strong>Hint</strong></p>
<p>It is F without L : $10C6 =210$</p>
<p>or L without F : $10C6=210$</p>
<p>the result is $210+210=420$.</p>
|
1,204,864 | <blockquote>
<p>$$\text{Find }\,\dfrac{d}{dx}\Big(\cos^2(5x+1)\Big).$$</p>
</blockquote>
<p>I have tried using the rules outlined in my standard derivatives notes but I've failed to find the point of application.</p>
| 3d0 | 217,450 | <p>\begin{align*}
\dfrac{dy}{dx} & = \dfrac{d \cos^2(5x+1)}{dx}\\
& = 2\cos(5x+1) \dfrac{d \cos(5x+1)}{dx}\\
& = 2\cos(5x+1)(-\sin(5x+1))\dfrac{d(5x+1)}{dx}\\
& = -2\cos(5x+1)\sin(5x+1) \cdot 5\\
& = -10\cos(5x+1)\sin(5x+1)\\
... |
982,938 | <p>I was doing the integration
$$
\int\frac{1}{(u^2+a^2)^2}du
$$
and I had a look at the lecturer's steps where I got stuck in the following step:
$$
\int\frac{1}{(u^2+a^2)^2}du=\frac{1}{2a^2}\left(\frac{u}{u^2+a^2}+\int\frac{1}{u^2+a^2}du\right).
$$
I guess it is integrating this by parts, but I could't see the trick.... | Shivang jindal | 38,505 | <p>Let $$I(a)= \int \frac{1}{a+x^2} = \frac{1}{\sqrt{a}}\arctan(\frac{x}{\sqrt{a}})$$
Then,
$$I'(a) = \int \frac{-1}{(a+x^2)^2} = d/da( \frac{1}{\sqrt{a}}\arctan(\frac{x}{\sqrt{a}})) $$
Find the expression in RHS and then put $a=u^2$. And you are done! :)</p>
|
3,224,102 | <p>For a given curve: <span class="math-container">$$C: \frac {ax^2+bx+c}{dx+e} $$</span> where <span class="math-container">$a,b,c,d,e$</span> are integers. Let <strong><span class="math-container">$f(x)=ax^2+bx+c$</span></strong> .</p>
<hr>
<p>Oblique asymptote can be found by long division of numerator by denomina... | Community | -1 | <p>In order of appearence:</p>
<ul>
<li><p>"the oblique asymptote of <span class="math-container">$f(x)=\frac{ax^2+bx+c}{dx+e}$</span> can be found by computing the polynomial ling division": <strong>TRUE</strong></p></li>
<li><p>"the result of the aforementioned operation is <span class="math-container">$\frac adx+\f... |
1,087,080 | <p>By definition, a closed set is a set that contains its limit points. However, by the time the closed set contains its limit points, those points are no longer limit points and become isolated points. For example:</p>
<p>$\mathbf A = \{\frac{1}{n}: n \in \mathbb N \}$. The limit of this set (set $\mathbf A$) is clea... | Mark Bennet | 2,906 | <p>In the first case your neighbourhood of zero includes $\frac 1{n+1}, \frac 1{n+2} \dots$</p>
<p>In the second case I'm not sure what you mean. Any neighbourhood of a point in $[a,b]$ (we take $b\gt a$) intersects $[a,b]$ in (a set containing) an interval around the point. And any point within $[a,b]$ is a limit poi... |
2,051,308 | <p>i would be happy if someone would help me with something im trying to prove for my homework assignment.</p>
<p><strong>the question:</strong>
let V be a vector space,U is a subspace of V that is not equal to V, U$\neq${0}.
let v be a vector from V.
prove that it is not possible that every vector from V\U (from V b... | user115350 | 334,306 | <p>you have already proved (1). For (2), let's use contradiction. If it is possible that one special vector $v_s \in V\setminus U $ is a scalar multiple of a vector $v \in V$, then from (1), we know that the special vector $v_s \in V$, which contradicts to the assumption. So you can complete your proof. You are very cl... |
2,051,308 | <p>i would be happy if someone would help me with something im trying to prove for my homework assignment.</p>
<p><strong>the question:</strong>
let V be a vector space,U is a subspace of V that is not equal to V, U$\neq${0}.
let v be a vector from V.
prove that it is not possible that every vector from V\U (from V b... | egreg | 62,967 | <p>It's correct that $v$ cannot belong to $U$. But there's a different way. Suppose such a $v$ exists. Then
$$
V=U\cup\langle v\rangle
$$
(where $\langle v\rangle$ denotes the subspace spanned by $v$).</p>
<p>It is well known that if $U_1$ and $U_2$ are subspaces of a vector space, then $U_1\cup U_2$ is a subspace if ... |
396,713 | <p>I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the difference between each term; however, I am unable to find one that works. Here is what I have so far:</p>
<p>Let $a_0$ ... | Peter Košinár | 77,812 | <p>Essentially, you want to "round" $a_n$ down to the greatest number which can be expressed as $a_0 + kx$ for some integer $x$. One way to do so is to replace $a_n$ by $(a_0+x\lfloor\frac{a_n-a_0}{x}\rfloor)$ in your summation formula.</p>
|
29,155 | <p>Do we have a pullback operation on singular simplicial chains,that is if f:X-->Y is a continuous map between topological space X and Y,and C is a singular simplicial chain on Y,then do we have a singular simplicial chain on X which is the pullback of C along f?</p>
| Gregory Arone | 6,668 | <p>For a general map, there is no such pullback operation, but there are things you can do in special cases. For example, if <span class="math-container">$f\colon X\to Y$</span> is a finite cover, there is a chain homomorphism <span class="math-container">$C(Y)\to C(X)$</span> that sends a singular simplex in <span cla... |
1,746,782 | <p>This is what I've done:</p>
<p>Let $s < t$ and $F_t$ be a filtration adapted to $W(t)$
$$E[e^{t/2}\cos(W(t))|F_s] = e^{t/2} E[\cos(W(t)) - \cos(W(s)) + \cos(W(s))|F_s]$$
$$= e^{t/2} [E[\cos(W(t)) - \cos(W(s))|F_s] + \cos(W(s))]$$
Because of the independence of the increments:
$$= e^{t/2} [E[\cos(W(t)) - \cos(W(s... | Community | -1 | <p>The function $u(t,x)=e^{t/2}\,\cos(x)$ satisfies the heat equation ${d\over dt}u+{1\over 2}\Delta_x u=0,$ so that, by Ito's formula,
$u(t,W_t)=e^{t/2}\,\cos(W_t)$ is a martingale. </p>
|
280,156 | <p>I have a code as below:</p>
<pre><code>countpar = 10;
randomA = RandomReal[{1, 10}, {countpar, countpar}];
randomconst = RandomInteger[{0, 1}, {countpar, 1}];
For[i = 1, i < countpar + 1, i++,
If[randomconst[[i, 1]] != 0,
randomA[[All, i]] = 0.; randomA[[i, All]] = 0.;
randomA[[i, i]] = 1;
];
];
</code></pre... | Bob Hanlon | 9,362 | <pre><code>x = 2; n = 2;
</code></pre>
<p>The general solution is</p>
<pre><code>Clear[y];
y[m_] = RSolveValue[{y[
m] == (y[m - 1]^(x - 1) + n)/(y[m - 1]^(x - 1) + y[m - 1]^(x - 2)),
y[0] == 1}, y[m], m] // Simplify
(* ((1 - Sqrt[2])^m (-2 + Sqrt[2]) + (1 + Sqrt[2])^
m (2 + Sqrt[2]))/(-(1 - Sqrt[2])^(1 +... |
3,882,457 | <p><a href="https://arxiv.org/pdf/quant-ph/0208163" rel="nofollow noreferrer">These notes</a> are a great introduction to deformation quantization but I failed to check the validity of the statement p.9, right before (5.18).</p>
<p><strong>Context:</strong> let <span class="math-container">$(\mathcal{A},+,\mu)$</span> ... | Noix07 | 92,038 | <p>It is in fact possible to prove<br />
<span class="math-container">$$\begin{split} T\big(f \ast_N g \big) = T(f) &\ast_M T(g)\quad \text{with}\quad T = \genfrac{}{}{0pt}{0}{"}{}\!\!\exp\left(-\frac{\hbar}{2} \frac{\partial^2}{\partial_{a}\, \partial_{\overline{a}} } \right)\genfrac{}{}{0pt}{0}{"}{} \\
\Longleftr... |
1,384,947 | <p>I'm trying to figure out when numbers reach "periodicity" given known values. I've included an example below with image:</p>
<p>I have known sizes (<em>100, 75, and 50</em>) that I would like to know how many times I would need to repeat each item for all the sizes to line up or be periodic. Does anyone know of a... | CopperKettle | 126,401 | <p>Using Anurag A's <a href="https://math.stackexchange.com/questions/1384945/a-puzzling-step-in-a-solution-for-find-sinx-and-cosx-if-a-sinxb#comment2820024_1384945">hint</a>, </p>
<p>$$\sin(x-\phi)=\pm\sqrt{1-\cos^2(x-\phi)}$$
Since
$$\left(\frac{a}{\sqrt{a^2+b^2}}\right)^2+\left(\frac{b}{\sqrt{a^2+b^2}}\right)^2=1... |
1,384,947 | <p>I'm trying to figure out when numbers reach "periodicity" given known values. I've included an example below with image:</p>
<p>I have known sizes (<em>100, 75, and 50</em>) that I would like to know how many times I would need to repeat each item for all the sizes to line up or be periodic. Does anyone know of a... | Obinoscopy | 255,445 | <p>$$\sin^2(x-\phi)+\cos^2(x-\phi)=1$$
$$\sin(x-\phi)=\pm\sqrt{1-\cos^2(x-\phi)}$$
But $\cos(x-\phi)=\frac{c}{\sqrt{a^2+b^2}}$</p>
<p>Therefore $\sin(x-\phi)=\pm\sqrt{1-\cos^2(x-\phi)}=\pm\sqrt{1-(\frac{c}{\sqrt{a^2+b^2}})^2}=\pm\sqrt{1-\frac{c^2}{a^2+b^2}}$</p>
<p>This gives: $\pm\sqrt{\frac{a^2+b^2}{a^2+b^2}-\frac... |
2,563,966 | <p>I'm reading a proof in a linear algebra book. It mentions
$$p(x) -p(c)= (x - c) h(x),$$
where $c$ is a constant, and $p(x)$ and $h(x)$ are polynomials.</p>
<p>Can we always factor $p(x) - p(c)$ in this way? </p>
<p>Please give a proof.</p>
| egreg | 62,967 | <p>If $R$ is a commutative ring, $f(x)$ is a polynomial in $R[x]$ and $c\in R$, then</p>
<blockquote>
<p>$f(x)=(x-c)g(x)$ for some $g(x)\in R[x]$ if and only if $f(c)=0$</p>
</blockquote>
<p>Indeed, long division of $f(x)$ by $x-c$ is possible because $x-c$ is monic; so $f(x)=(x-c)g(x)+r$, where $r\in R$. The concl... |
639,028 | <blockquote>
<p>Calculate partial derivative $f'_x, f'_y, f'_z$ where $f(x, y, z) = x^{\frac{y}{z}}.$</p>
</blockquote>
<p>I know I need to use the chain rule but I'm confusing here for some reason ..</p>
<p>By <a href="http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx" rel="nofollow">this page</a>, the... | heropup | 118,193 | <p>It might be instructive to determine, for a given base $y > 1$, those values $x > 0$ such that $f_y(x) = \lceil \log_y \lceil x \rceil \rceil \ne g_y(x) = \lceil \log_y x \rceil$. First, it is easy to see from the fact that $\log$ is monotone increasing that $f_y(x) \ge g_y(x)$ for any base $y > 1$. Furth... |
639,028 | <blockquote>
<p>Calculate partial derivative $f'_x, f'_y, f'_z$ where $f(x, y, z) = x^{\frac{y}{z}}.$</p>
</blockquote>
<p>I know I need to use the chain rule but I'm confusing here for some reason ..</p>
<p>By <a href="http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx" rel="nofollow">this page</a>, the... | Christian Blatter | 1,303 | <p>The statement only makes sense when $y>1$ and $x>0$.</p>
<p>When $0<x\leq{1\over y}$ the statement is <strong>wrong</strong>: The left side is $=0$ and the right side $\leq-1$.</p>
<p>When ${1\over y}<x\leq1$ then both sides are $=0$. So from now on we assume $x>1$.</p>
<p>Let $y>1$ and $t>1$... |
167,848 | <p>I am trying to solve numerically an equation and generate some results. I use the following code </p>
<pre><code>u[c_] := (c^(1 - σ) - 1)/(1 - σ)
f[s_] := g s (1 - s/sbar1)
h[s_] := (2 hbar)/(1 + Exp[η (s/sbar - 1)])
co[a_] := ϕ (a^2)/2
ψ[k_] := wbar (ω + (1 - ω) Exp[-γ k])
</code></pre>
<p>The equation I try to s... | m_goldberg | 3,066 | <p>Neither <code>Solve</code> nor <code>NSolve</code> will handle your equation. As Hugh has shown, you can use <code>FindRoot</code>. You could also use <code>FindInstance</code> as follows.</p>
<pre><code>solK[i_] := FindInstance[adap[k, i] == 0 /. paramFinal2, k, Reals]
With[{tmax1 = 10}, Flatten @ Table[solK[i], {... |
767,686 | <p>Let $f:A\rightarrow B$ be a function and let $C_1,C_2\subset A$
Prove that</p>
<p>$f(C_1\cap C_2)=f(C_1)\cap f(C_2) \leftrightarrow$ $f$ is injective</p>
<p>Attempt:</p>
<p>$(\leftarrow)$ Let $f(x)\in f(C_1\cap C_2)$. Then there exists $x\in C_1\cap C_2$ because $f$ is injective. So $x\in C_1$ and $x\in C_2$. So
... | user2566092 | 87,313 | <p>Hint: The tricky part is showing that $h(1) \neq 1$. Argue that if $h(1) = 1$ then you must have $h'(x) = 1$ for all $0 \leq x < 1$ and $h'(x) = -1$ for all $1 < x \leq 2$. But then how can $h$ be differentiable at $x = 1$?</p>
|
64,395 | <p>Let G be a directed graph on N vertices chosen at random, conditional on the requirement that the out-degree of each vertex is 1 and the in-degree of each vertex is either 0 or 2. The "periodic" points of G are those contained in a cycle. What do we know about the statistics of G? For instance, what is the mean n... | James Cranch | 14,901 | <p>To calculate expected numbers of cycle of each length, you should be able to just wade in and use Stirling's approximation to deal with the results.</p>
<p>Let's write $N=2t$. Then there are $t$ vertices with indegree 0 and $t$ with indegree 2. The number of such graphs is $(2t)!/2^t$.</p>
<p>If $a_1,\ldots,a_m$ a... |
248,182 | <p>Some textbooks I've seen declare inequalities such as $-2>x>2$ to have no solution, or to be ill-defined, which I disagree with. I'm curious to know if anyone else thinks the same.</p>
<p>Inequalities can always be written two ways. For example, $x>2$ is the same as $2<x$. So far as I understand, the sa... | Alex R. | 22,064 | <p>You can write inequalities any way you want if you think of the elements satisfying the inequality as members of a set combined with a truth table. There isn't any ambiguity in writing $-2>x>2$ or $-2>x<5$ as long as you read it left to right, or right to left in a PAIRWISE fashion: $-2>x$ and $x>2... |
467,268 | <p>Any body knows the meaning of this expectation ($E[g(x)]$) form?</p>
<p>$E[g(x)]=Pr(g(x) >\varepsilon)E[g(x)|g(x) > g(\varepsilon)]+Pr(g(x) \leq\varepsilon)E[g(x)|g(x)\leq g(\varepsilon)]$</p>
| Community | -1 | <p><strong>Hint</strong>: Find the point from where the given line passes and use the slope point form to find the required equation of straight line.</p>
<p><strong>Solution:-</strong></p>
<p>We have,</p>
<blockquote>
<ul>
<li>Gradient of line <span class="math-container">$(m) = -2$</span></li>
<li>The line passes thr... |
3,399,586 | <p>Suppose that <span class="math-container">$f$$:$$\mathbb{R}\to\mathbb{R}$</span> is differentiable at every point and that </p>
<p><span class="math-container">$$f’(x) = x^2$$</span></p>
<p>for all <span class="math-container">$x$</span>. Prove that </p>
<p><span class="math-container">$$f(x) = \frac{x^3}{3} + C$... | amsmath | 487,169 | <p>Let <span class="math-container">$g:\mathbb R\to\mathbb R$</span> be differentiable such that <span class="math-container">$g'(x) = 0$</span> for all <span class="math-container">$x\in\mathbb R$</span>. Then the mean value theorem says that for all <span class="math-container">$x,y\in\mathbb R$</span> there exists s... |
1,530,057 | <p>At my multivariable calculus class we gave this definition for the limit of a function:</p>
<blockquote>
<p><em>Definition:</em></p>
<p>Let <span class="math-container">$ \mathbb{R}^n \supset A $</span> be a open set , let <span class="math-container">$f:A \to\mathbb{R}^m $</span> be a function, let <span class="mat... | ckoe | 290,263 | <p>Suppose a random variable $W$ is uniform on $[0,z]$. Then its mean would be $\frac12z$. Now, as you just stated, $Y|X$ is uniform on $[0,x]$. So then the mean is $\frac12x$.</p>
|
3,429,623 | <p>Is the union of <span class="math-container">$\emptyset$</span> with another set, <span class="math-container">$A$</span> say, disjoint? Even though <span class="math-container">$\emptyset \subseteq A$</span>?</p>
<p>I would say, yes - vacuously. But some confirmation would be great.</p>
| GhostAmarth | 721,316 | <p>Two sets <span class="math-container">$A, B$</span> are disjoint iff <span class="math-container">$A \cap B = \emptyset$</span>.
You have <span class="math-container">$A \cap \emptyset = \emptyset$</span>. Therefore <span class="math-container">$A$</span> and <span class="math-container">$\emptyset$</span> are disjo... |
2,064,380 | <p><a href="https://i.stack.imgur.com/lJcu2.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lJcu2.gif" alt="enter image description here"></a></p>
<p>In the above question could anyone please explain me what they have done.</p>
| Dominik | 259,493 | <p>Consider the Formula (1) and (2) in the <a href="https://en.wikipedia.org/wiki/Woodbury_matrix_identity" rel="nofollow noreferrer">Woodbury matrix identity</a>. Applied to the block matrix $M = \begin{pmatrix} 1 & u^t V^t \\ V u & VV^t\end{pmatrix}$ they give us two possible ways to calculate the $(1, 1)$-en... |
2,064,380 | <p><a href="https://i.stack.imgur.com/lJcu2.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lJcu2.gif" alt="enter image description here"></a></p>
<p>In the above question could anyone please explain me what they have done.</p>
| Arin Chaudhuri | 404 | <p>This is simple consequence of Sherman-Morrison-Woodbury formula.</p>
<p>By a direct application the Sherman-Morrison-Woodbury formula we have $$
\begin{align}
{\begin{pmatrix} V(1 - uu^T) V^T \end{pmatrix}}^{-1} &= (VV^T-(Vu)(Vu)^T)^{-1}\\
&= (VV^T)^{-1} + \dfrac{(VV^T)^{-1}(Vu)(Vu)^T(VV^T)^{-1}}{1 - u^TV^T... |
139,417 | <p>I have a polygon defined by a list of nodes (x,y). I want to cut the polygon by a horizontal line at position y = a and get the new polygon above the position y = a. I am using the RegionIntersect function, but it seems very slow if I want to combine the function with Manipulate function as well. Is there any way to... | matrix42 | 5,379 | <pre><code>R1 = ImplicitRegion[{0<=x<=300,a<=y<=700},{x,y}];
R2 = Polygon[{{0,0},{300,0},{300,500},{0,750}}];
ineq = RegionMember[RegionIntersection[R1,R2],{x,y}]//Simplify//Rest
With[{ineq = ineq},
Manipulate[RegionPlot[ineq,{x,0,300},{y,0,700},PerformanceGoal->"Quality"],{a,1,499}]]
</code></pre>
|
482,102 | <p>The problem comes from Alan Pollack's Differential Topology, pg. 5.
Suppose that X is a k-dimensional manifold. Show that every point in X has a neighborhood diffeomorphic to all of $\Bbb{R}^k$.</p>
<p>I have already shown that $\Bbb{R}^k$ is diffeomorphic to $B_a$ (part (a) of the question) the open ball of radius... | rfauffar | 12,158 | <p>You already did all the work. If $x\in X$, then there is a neighborhood $U$ of $x$ that is diffeomorphic to an open set of $\mathbb{R}^k$. Take $\phi: U\to V$ to be the chart. Since $V$ is open, there is an open ball $B_a$ centered at $x$ contained in $V$. Take $W=\phi^{-1}(B_a)$. Then $W$ is diffeomorphic to $B_a$,... |
843,634 | <p>I am wondering whether for any two lines $\mathfrak{L}, \mathfrak{L'}$ and any point $\mathfrak{P}$ in $\mathbf{P}^3$ there is a line having nonempty intersection with all of $\mathfrak{L}, \mathfrak{L'}$, $\mathfrak{P}$. I don't really know how to approach this, because I was never taught thinking about such a prob... | mercio | 17,445 | <p>You have to make a distinction between (more concrete) functions from $\Bbb Z / p \Bbb Z$ to $\Bbb Z/p \Bbb Z$ and (more abstract) polynomials with coefficients in $\Bbb Z/ p \Bbb Z$. </p>
<p>For each polynomial, there is an associated polynomial function. And as you have just discovered, it is NOT true that if the... |
2,073,230 | <p>I thought I'd might use induction, but that seems too hard, then I tried to take the derivative and show that that's positive $\forall$n. But I can't figure out how to do that either, I've tried induction there too.</p>
| florence | 343,842 | <p>Let $\varepsilon = 0.06$. Now, let
$$f(x) = \left(1+\frac{\varepsilon}{x}\right)^x$$
so that your sequence is given by $f(n)$. Then
$$\log(f) = x\log\left(1+\frac{\varepsilon}{x}\right)$$
Differentiating both sides,
$$\frac{f'}{f} = \log\left(1+\frac{\varepsilon}{x}\right)-x\frac{\varepsilon/x^2}{1+\varepsilon/x} = ... |
3,088,620 | <p>Let <span class="math-container">$M$</span> be a second countable smooth manifold. When I learned about differential geometry, a side note was made about how if <span class="math-container">$E$</span> is a vector bundle, <span class="math-container">$\Gamma(E)$</span> is a <span class="math-container">$C^\infty(M)$<... | Eric Wofsey | 86,856 | <p>First of all, it's not true that <span class="math-container">$\Gamma(E)$</span> is not free. What is true is that <span class="math-container">$\Gamma(E)$</span> <em>might</em> not be free, depending what <span class="math-container">$E$</span> is. Specifically, <span class="math-container">$\Gamma(E)$</span> is ... |
4,241 | <p>I was preparing for an area exam in analysis and came across a problem in the book <em>Real Analysis</em> by Haaser & Sullivan. From p.34 Q 2.4.3, If the field <em>F</em> is isomorphic to the subset <em>S'</em> of <em>F'</em>, show that <em>S'</em> is a subfield of <em>F'</em>. I would appreciate any hints on ho... | Yuval Filmus | 1,277 | <p>If $F$ is any field, the rational functions over it form a field $F(t)$ with the same characteristic (and cardinality, if $F$ is infinite). This field consists of all rational functions $P(t)/Q(t)$ (considered as equivalence classes, i.e. if $P_1(t) Q_2(t) = P_2(t) Q_1(t)$ then $P_1(t)/Q_1(t)$ and $P_2(t)/Q_2(t)$ ar... |
3,858,962 | <p>given a rectangle <span class="math-container">$ABCD$</span> how to construct a triangle such that <span class="math-container">$\triangle X, \triangle Y$</span> and <span class="math-container">$\triangle Z$</span> have equal areas.i dont know where to start. .i tried some algebra with the area of the triangles an... | egreg | 62,967 | <p>No need to use Pythagoras’ theorem. If you call <span class="math-container">$a$</span> and <span class="math-container">$b$</span> the segment parts on the left vertical side (top to bottom), <span class="math-container">$c$</span> and <span class="math-container">$d$</span> the segment parts on the bottom horizont... |
1,569,411 | <p>Express $log_3(a^2 + \sqrt{b})$ in terms of m and k where
$m = log_{3}a$</p>
<p>$k = log_{3}b$</p>
<p>Given this information I made
$a = 3^m$</p>
<p>$b = 3^k$</p>
<p>Therefore
= $log_{3} ((3^m)^2 + (3^k))^{\frac{1}{2}}$</p>
<p>= $log_{3} (3^{2m} + 3^{\frac{k}{2}})$</p>
<p>I don't know if I'm done or there is ... | Ian | 83,396 | <p>Every Markov chain on a finite state space has an invariant distribution. As you said, this follows directly from the condition that the rows sum to <span class="math-container">$1$</span>.</p>
<p>It is possible for a Markov chain on a finite state space to have multiple invariant distributions. However, the Perron-... |
705,744 | <p>Hello everyone. I have a couple questions this time, but I think if I understand how to do this one, I'll understand the others.</p>
<p>A particular online banking system uses the following rules for its passwords:<br/>
a. Passwords must be 6-8 characters in length<br/>
b. Passwords must use only alphabetical and... | Daslayer | 914,615 | <p>Permutation formula for ordered with repetition is n^r where n is the number of things to choose from and r is how many we are choosing to form another set.
Total possible permutations is 62^8
However the rules state one numeric and one alpha must be used.
The largest legal set considering all rules is 52^7 + 10^1.
... |
113,446 | <p>Suppose a simple equation in Cartesian coordinate:
$$
(x^2+ y^2)^{3/2} = x y
$$
In polar coordinate the equation becomes $r = \cos(\theta) \sin(\theta)$. When I plot both, the one in polar coordinate has two extra lobes (I plot the polar figure with $\theta \in [0.05 \pi, 1.25 \pi]$ so the "flow" of the curve is cle... | Greg Hurst | 4,346 | <p>You can always impose this constraint with the option <a href="http://reference.wolfram.com/language/ref/RegionFunction.html" rel="nofollow noreferrer"><code>RegionFunction</code></a>:</p>
<pre><code>PolarPlot[Sin[θ] Cos[θ], {θ, 0, 2π}]
</code></pre>
<p><a href="https://i.stack.imgur.com/I44HD.png" rel="nofollow n... |
349,317 | <p>It is well-known fact that integral Dehn surgeries on <span class="math-container">$3$</span>-sphere <span class="math-container">$S^3$</span> are viewed as the result on the boundary of attaching <span class="math-container">$2$</span>-handles <span class="math-container">$B^2 \times B^2$</span> to the <span class=... | Lisa Piccirillo | 113,696 | <p>To attach a 4-dimensional 2-handle to the 4-ball, one requires an attaching region in <span class="math-container">$S^3=\partial B^4$</span> and a map from the attaching region of the handle (which has a natural parametrization as <span class="math-container">$S^1\times D^2\subset \partial(D^2\times D^2)$</span>) to... |
349,317 | <p>It is well-known fact that integral Dehn surgeries on <span class="math-container">$3$</span>-sphere <span class="math-container">$S^3$</span> are viewed as the result on the boundary of attaching <span class="math-container">$2$</span>-handles <span class="math-container">$B^2 \times B^2$</span> to the <span class=... | Marco Golla | 13,119 | <p>As Lisa points out, 2-handle attachments correspond exactly to integral surgeries. However, a general Dehn surgery corresponds to a <em>sequence</em> of integral surgeries, and hence to <em>multiple</em> 2-handle attachements.</p>
<p>This is a bit hard to do without pictures, so I'll just refer you to Section 5.3 i... |
1,171,911 | <p><img src="https://i.stack.imgur.com/3q5iO.png" alt="Taken from khan academy ">
Hi, so this question is taken straight from khan academy help exercises, i know how to do it dynamically meaning using the determinant and the adjugate how i was trying to do it using guass bla bla way with help of RREF but i somehow nev... | RE60K | 67,609 | <p>bla bla bla bla :
$$|{\rm D}|=2$$
blaaa blabla blabla:
$${\rm adj\; A}=\left[\begin{matrix}-1&2&1\\0&0&2\\1&0&-1\end{matrix}\right]$$
blah blehblaqa bla:
$${\rm A}^{-1}=\left[\begin{matrix}-1/2&1&1/2\\0&0&1\\1/2&0&-1/2\end{matrix}\right]$$</p>
|
4,469,733 | <p>When randomly selecting a kitten for adoption, there is a <span class="math-container">$23 \%$</span> chance of getting a black kitten, a <span class="math-container">$50 \%$</span> chance of getting a tabby kitten, a <span class="math-container">$7 \%$</span> chance of getting a calico kitten, and a <span class="ma... | Clement C. | 75,808 | <p>Here is a quick-and-dirty approach:</p>
<ol>
<li><p>Show that <span class="math-container">$\sup_{t\geq 2} \frac{\log^2 t}{\sqrt{t}} = \frac{16}{e^2}$</span> (achieved at <span class="math-container">$t=e^4$</span>). That can easily be done by differentiating the function <span class="math-container">$t\mapsto \frac... |
1,053,065 | <p>I have a function called $P(t)$ that is the number of the population at time $t$. $t$ being in days.</p>
<p>We know the growth rate is $P'(t) = 2t + 6$</p>
<p>We also know that $P(0) = 100$. How many days till the population doubles?</p>
<p>edit: $P(t) = t^2 + 6t$
edit: $P(t) = t^2 + 6t = 200$
edit: $t^2 + 6t - 2... | lhf | 589 | <p>The divisors of $n$ occur in pairs $(x,n/x)$. This implies that there is a divisor at most $\sqrt n$. Therefore $d(n)\le 2\sqrt{n}$. Now, $2\sqrt{n}< n/2$ if $n>16$. The case $6 < n\le 16$ is settled by inspection.</p>
|
1,210,285 | <p>Let there be a given function $f \in C([0,1])$, $f(x)>0$; $x\in [0,1]$. Prove </p>
<p>$$\lim_{n\to\infty} \sqrt[n]{f\left({1\over n}\right)f\left({2\over n}\right)\cdots f\left({n\over n}\right)}=e^{\int_0^1 \log f(x) \, dx} $$</p>
<p>All the questions before this required solving an definite integral without N... | kobe | 190,421 | <p>Your limit is the same as</p>
<p>$$\exp\{\lim_{n\to \infty} \frac{1}{n}\left(\log f(1/n) +\log f(2/n) + \cdots + \log f(n/n)\right)\}$$</p>
<p>and the limit inside converges to $\int_0^1 \log f(x)\, dx$, since the sum inside is a sequence of Riemann sums of the continuous function $\log f(x)$ over $[0,1]$.</p>
|
3,585,271 | <p>Let <span class="math-container">$V$</span> be an affine variety in <span class="math-container">$K^n$</span> with ideal <span class="math-container">$I=I(V)$</span>, where <span class="math-container">$K$</span> is an algebraically closed field. Let <span class="math-container">$V'$</span> be the variety with defin... | Ricardo Buring | 23,180 | <p>If <span class="math-container">$V$</span> is an affine variety then <span class="math-container">$I(V)$</span> is radical because <span class="math-container">$f(x)^n = 0$</span> implies <span class="math-container">$f(x) = 0$</span> over a field.</p>
<p>In particular (which is probably what you intended to ask), ... |
2,514,418 | <p>Let $p$ be a prime and define $A$ = sum of all $1 \leq a < p$ such that $a$ is a quadratic residue modulo $p$, and define $B$ = sum of all $1 \leq b < p$ such that $b$ is a non-residue modulo $p$.</p>
<p>Compute $A \pmod{p}$ and $B \pmod{p}$.</p>
<p>So I get $A = B \equiv 0 \pmod{p}$, how would I verify this... | James Garrett | 457,432 | <p>Your proof is wrong, $A$ has to be <em>any</em> square matrix. Let $\lambda \neq 0$ be an eigenvalue of $A$, by definition $$Av=\lambda v,$$ where $v \neq \mathbf{0}$ is a vector. Multiplying by $A^{-1}$ both sides of the equation yields $$A^{-1}Av=A^{-1}\lambda v \iff v=A^{-1}\lambda v \iff \lambda^{-1}v=A^{-1}v.$$... |
2,514,418 | <p>Let $p$ be a prime and define $A$ = sum of all $1 \leq a < p$ such that $a$ is a quadratic residue modulo $p$, and define $B$ = sum of all $1 \leq b < p$ such that $b$ is a non-residue modulo $p$.</p>
<p>Compute $A \pmod{p}$ and $B \pmod{p}$.</p>
<p>So I get $A = B \equiv 0 \pmod{p}$, how would I verify this... | Mathemagical | 446,771 | <p>Since $det(A) \neq 0$, you know all eigenvalues are nonzero since the determinant is the product of the eigenvalues. </p>
<p>Now if $\lambda$ is an eigenvalue with eigenvector $v$, then $Av=\lambda v$. Leftmultiplying by $A^{-1}$, you have $v=\lambda A^{-1} v$ or $\frac{1}{\lambda}v= A^{-1} v$ and you are done. </p... |
1,384,752 | <p>I ran across a problem which has stumped me involving existential quantifiers.
Let U, our universe, be the set of all people. Let S(x) be the predicate "x is a student" and I(x) be the predicate "x is intelligent".
I want to write the statement "Some students are intelligent" in the correct logical form. I can see 2... | Ashwin Ganesan | 157,927 | <p>The statement "some students are intelligent" can be rephrased as "there is at least one student who is intelligent". So your logical statement 1) is correct. However, 2) is not correct and 2) is not logically equivalent to 1). </p>
<p>Recall that "p implies q" is false exactly when p is true but q is still fals... |
3,172,693 | <p>Can anybody help me with this equation? I can't find a way to factorize for finding a value of <span class="math-container">$d$</span> as a function of <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$$d^3 - 2\cdot d^2\cdot a^2 + d\cdot a^4 - a^2 = 0$$</span></p>
<p>Another form:</p>
... | Julian Mejia | 452,658 | <p>If your denominator has a factor of the form <span class="math-container">$(as+b)^n$</span> then to write partial fractions you should write all the powers up to <span class="math-container">$n$</span>, i.e. <span class="math-container">$\frac{A}{as+b}+\frac{B}{(as+b)^2}+\cdots+\frac{Z}{(as+b)^n}$</span>. In the cas... |
3,172,693 | <p>Can anybody help me with this equation? I can't find a way to factorize for finding a value of <span class="math-container">$d$</span> as a function of <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$$d^3 - 2\cdot d^2\cdot a^2 + d\cdot a^4 - a^2 = 0$$</span></p>
<p>Another form:</p>
... | David | 119,775 | <p>The general result is the following.</p>
<blockquote>
<p>Suppose that the degree of <span class="math-container">$p(s)$</span> is less than the degree of <span class="math-container">$q(s)$</span>, and that <span class="math-container">$q(s)=q_1(s)q_2(s)$</span> where <span class="math-container">$q_1(s)$</span> ... |
232,672 | <p>Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context. </p>
<p>In ZF one can prove $\not\exists x (\forall y (y\in x)).$ This statement can be read in many ways, such as (1) "there is no set of all sets" (2) "the class of all sets... | Thomas Klimpel | 20,781 | <p>Even so a model of ZFC might be given together with a collection of classes, only the sets should count as real. So if one would modify the collection of classes without modifying the sets, it would still be the same model. The formula $\forall y (y\in X)$ defines a class $X$ which cannot be modified at will, but a ... |
58,631 | <p>I am (partly as an exercise to understand <em>Mathematica</em>) trying to model the response of a damped simple harmonic oscillator to a sinusoidal driving force. I can solve the differential equation with some arbitrarily chosen boundary conditions, and get a nice graph;</p>
<pre><code>params = {ν1 -> 1.0, ω1 -... | Dr. Wolfgang Hintze | 16,361 | <p>It is much easier and more general.</p>
<p>Your equation</p>
<pre><code>system = {D[x1[t], {t, 2}] == -ν1 D[x1[t], t] - ω1^2 x1[t] + F Cos[ω t]};
</code></pre>
<p>without specifying neither the initial conditions nor the parameters (except of course <code>v1 > 0, ω1^2 > 0</code>) is solved by</p>
<pre><cod... |
279,985 | <p>How can I convert a Beta Distribution to a Gamma Distribution?
Strictly speaking, I want to transform parameters of a Beta Distribution to parameters of the corresponding Gamma Distribution. I have mean value, alpha and beta parameters of a Beta Distribution and I want to transform them to those of a Gamma Distribu... | Did | 6,179 | <p>Let $X_a$ and $X_b$ denote independent gamma random variables with respective parameters $(a,c)$ and $(b,c)$, for some nonzero $c$. Then $\dfrac{X_a}{X_a+X_b}$ is a beta random variable with parameter $(a,b)$.</p>
|
279,985 | <p>How can I convert a Beta Distribution to a Gamma Distribution?
Strictly speaking, I want to transform parameters of a Beta Distribution to parameters of the corresponding Gamma Distribution. I have mean value, alpha and beta parameters of a Beta Distribution and I want to transform them to those of a Gamma Distribu... | LALIT SALUNKHE | 644,309 | <p>You can make a transformation as U = X + Y and V = <code>X/(X+Y)</code> where X and Y both are having gamma distribution with parameters <code>aplha</code>, <code>Beta</code> respectively. Out of these two, U will be Gamma distribution with parameters <code>aplha + beta</code> and V will be a Beta Distribution of fi... |
2,430,482 | <p>I'm struggling to find the maximum of this function $f:\mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}$</p>
<p>$$ f(x,y) = \frac{n+1}{2} \sum_{i=1}^n x_i\,y_i - \sum_{i=1}^n x_i \sum_{i=1}^n y_i,$$</p>
<p>where $x_i,y_i\in[0,1]$ for $i=1,...,n$. It reminded me to Chebyshev's sum inequality, but it didn't hel... | zwim | 399,263 | <p>Let's have $\bar x=\frac 1n\sum\limits_{i=1}^n x_i$ and $\bar y=\frac 1n\sum\limits_{i=1}^n y_i$.</p>
<p>$\sum\limits_{i=1}^n x_iy_i=\sum\limits_{i=1}^n (x_i-\bar x)(y_i-\bar y)+\overbrace{\sum\limits_{i=1}^n x_i\bar y}^{n\bar x\bar y}+\overbrace{\sum\limits_{i=1}^n \bar xy_i}^{n\bar x\bar y}-\overbrace{\sum\limits... |
3,873,071 | <p>This is my first post and I apologize in advance if I'm not using the right formatting/approach.</p>
<p><strong>Problem</strong></p>
<p>A coin, having probability <span class="math-container">$p$</span> of landing heads, is continually flipped until at least one head and one tail have been flipped.</p>
<p>Find the e... | Quanto | 686,284 | <p>Note</p>
<p><span class="math-container">$$\frac2{x^8+1}=\frac1{x^4+1}\left( \frac1{x^4+\sqrt2x^2+1}+ \frac1{x^4-\sqrt2x^2+1}\right)
$$</span>
Then
<span class="math-container">\begin{align}
&\int\frac{(x^4-1)\sqrt{x^4+1}}{x^8+1} \> dx \\
=& \frac12 \int\frac{\frac{x^4-1}{\sqrt{x^4+1}}dx}{x^4+\sqrt2x^2+1... |
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | lab bhattacharjee | 33,337 | <p>From <a href="https://web.archive.org/web/20180211011715/http://mathforum.org/library/drmath/view/54058.html" rel="nofollow noreferrer">this</a> and <a href="https://mathworld.wolfram.com/TrigonometryAnglesPi7.html" rel="nofollow noreferrer">this</a>, <span class="math-container">$\sin7x=7t-56t^3+112t^5-64t^7$</span... |
42,957 | <p>I am an "old" programmer used to <em>Fortran</em> and <em>Pascal</em>. I can't get rid of <code>For</code>, <code>Do</code> and <code>While</code> loops, but I know <em>Mathematica</em> can do things much faster!</p>
<p>I am using the following code</p>
<pre><code>SeedRandom[3]
n = 10;
v1 = Range[n];
v2 = RandomRe... | lalmei | 9,831 | <p>Here is an example using Reap and Sow.</p>
<p>It is a bit overkill in this case, but if you need to append only some of the values it should go much faster with Reap and Sow, or if you need to perform other functions while appending. </p>
<pre><code>Last@Reap[
Scan[Function[{x},
Sow[(v2[[x[[1]]]] - v2[[x[[... |
42,957 | <p>I am an "old" programmer used to <em>Fortran</em> and <em>Pascal</em>. I can't get rid of <code>For</code>, <code>Do</code> and <code>While</code> loops, but I know <em>Mathematica</em> can do things much faster!</p>
<p>I am using the following code</p>
<pre><code>SeedRandom[3]
n = 10;
v1 = Range[n];
v2 = RandomRe... | Jacob Akkerboom | 4,330 | <p>As mentioned, the main issue is using <code>AppendTo</code> in loops like this. In this answer, I want to show that using <code>Compile</code> can make procedural code very fast. Below is a comparison of timings of all the answers, as well as the OPs code.</p>
<p>Here is a slight modification of the code by the OP.... |
441,888 | <p>I should clarify that I'm asking for intuition or informal explanations. I'm starting math and never took set theory so far, thence I'm not asking about formal set theory or an abstract hard answer. </p>
<p>From Gary Chartrand page 216 Mathematical Proofs - </p>
<p>$\begin{align} \text{ range of } f & = \{f(x... | Community | -1 | <p>Consult the answer at <a href="https://math.stackexchange.com/a/149985/53259">https://math.stackexchange.com/a/149985/53259</a>. It may help you. In brief, according to that answer:</p>
<p>$\{x \in S : P(x) \} $ can be interpreted as the "elementhood test." This is more convenient for testing whether some $x \in S$... |
664,349 | <blockquote>
<p>If $G$ is a finite group where every non-identity element is generator of $G$, what is the order of $G$?</p>
</blockquote>
<p>I know that the order of $G$ must be prime, but I'm not sure how to go about proving this from the problem statement. </p>
<p>Any hints on where to start?</p>
| TheMobiusLoops | 100,798 | <p>Suppose the order of $G$ was not prime and let $n$ be the order of $G$. Then for all $k\in \mathbb{Z}$ which divide $n$, the subgroup generated by $g^k$ has only $n/k$ elements while $G$ has $n$ elements. </p>
<p>Therefore, the subgroup generated by $g^k$ cannot equal $G$.</p>
<p>Therefore, the order of $G$ must b... |
2,445,693 | <p>I know that the derivative of $n^x$ is $n^x\times\ln n$ so i tried to show that with the definition of derivative:$$f'\left(x\right)=\dfrac{df}{dx}\left[n^x\right]\text{ for }n\in\mathbb{R}\\{=\lim_{h\rightarrow0}\dfrac{f\left(x+h\right)-f\left(x\right)}{h}}{=\lim_{h\rightarrow0}\frac{n^{x+h}-n^x}{h}}{=\lim_{h\right... | William Kurdahl | 217,928 | <p>It depends on what you feel you can assume about the function ln(x) and the number e.</p>
<p>See the link below for an approach similar to yours:
<a href="http://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx" rel="nofollow noreferrer">http://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx</a><... |
1,455,969 | <p><a href="https://i.stack.imgur.com/5O0d8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5O0d8.png" alt="enter image description here"></a></p>
<p>Hello! I'm having problems trying to figure out this. Here is what I did: I used implication relation and Demorgan's law to simplify this proposition.... | Daniel W. Farlow | 191,378 | <p>The key here is largely going to be patience, but one point is worth remembering: always try to turn connectives into only $\lor, \land,$ and $\neg$. Then everything often "falls out" more or less. Since your compound proposition is quite large, start by writing out and denoting it $\text{LHS}$ like so:</p>
<blockq... |
418,748 | <p>I tried to calculate, but couldn't get out of this:
$$\lim_{x\to1}\frac{x^2+5}{x^2 (\sqrt{x^2 +3}+2)-\sqrt{x^2 +3}}$$</p>
<p>then multiply by the conjugate.</p>
<p>$$\lim_{x\to1}\frac{\sqrt{x^2 +3}-2}{x^2 -1}$$ </p>
<p>Thanks!</p>
| Community | -1 | <p>Let $t=x-1$ so $x=t+1$ and since $(1+y)^\frac{1}{2}\sim_0 1+\frac{y}{2}$ and $y^2=_0o(y)$ then we find
$$\lim_{x\to 1}\frac{\sqrt{x^2+3}-2}{x^2-1}=\lim_{t\to 0}\frac{\sqrt{t^2+2t+4}-2}{t^2+2t}=\lim_{t\to 0}2\frac{\sqrt{\frac{t^2+2t}{4}+1}-1}{t^2+2t}=\lim_{t\to 0}2\frac{t/4}{2t}=\frac{1}{4}$$</p>
|
1,689,923 | <p>I have a sequence $a_{n} = \binom{2n}{n}$ and I need to check whether this sequence converges to a limit without finding the limit itself. Now I tried to calculate $a_{n+1}$ but it doesn't get me anywhere. I think I can show somehow that $a_{n}$ is always increasing and that it has no upper bound, but I'm not sure i... | Stefan Mesken | 217,623 | <p><strong>Hint</strong> For $n \ge 1$ we always have $\binom{2n}{n} > n$.</p>
|
1,843,274 | <p>Good evening to everyone. So I have this inequality: $$\frac{\left(1-x\right)}{x^2+x} <0 $$ It becomes $$ \frac{\left(1-x\right)}{x^2+x} <0 \rightarrow \left(1-x\right)\left(x^2+x\right)<0 \rightarrow x^3-x>0 \rightarrow x\left(x^2-1\right)>0 $$ Therefore from the first $ x>0 $, from the second $ x... | DanielWainfleet | 254,665 | <p>Your attempt is ok up to $x(x^2-1)>0.$ The next sentence is unintelligible ("from the first $x>0$". First what? And what are $x_1, x_2$?). </p>
<p>Observe that $x(x^2-1)>0\iff x^3>x $ $\iff [(x>0\land x^2>1)\lor (x<0\land x^2<1)\lor (x=0\land PigsCanFly)]$ $\iff (x>1\lor -1<x<0).$ ... |
4,489,675 | <p>When saying that in a small time interval <span class="math-container">$dt$</span>, the velocity has changed by <span class="math-container">$d\vec v$</span>, and so the acceleration <span class="math-container">$\vec a$</span> is <span class="math-container">$d\vec v/dt$</span>, are we not assuming that <span class... | Community | -1 | <p>In your suggested answer, da/dt is the ratio of two infinitesimals, so it can be finite and non-zero. However, da/2 is an infinitesimal so you can treat it as being zero when compared to the first term.</p>
<p>(If there was infinite acceleration in that moment, it could be an exception, but we normally assume accel... |
299,452 | <p>According to wiki: <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Dedekind_eta_function</a>, Dedekind eta function is defined in many equivalent forms. But none of them is an explicit description (say in algorithmic format) on how to computing it... | Licheng Wang | 103,866 | <p>My maple code of Gatteschi-Sokal algorithm for computing $R(t,x)=\prod_{n=1}^{\infty}(1-tx^n)$: </p>
<p>GS:=proc(t,x,prec) </p>
<p>local R0, a0, b0, Rn, an, bn, d, c, i, N, r, Rd; </p>
<p>N := 100; </p>
<p>d := 1/2; if d = t*x then d := (1/2)*d end if; </p>
<p>r := evalf$[prec]$(1+d/(1-x)); </p>
<p>a0 := 1;... |
1,199,912 | <p>What is the optimal (i.e., smallest) constant $\alpha$ such that, given 19 points on a solid, regular hexagon with side 1, there will always be 2 points with distance at most $\alpha$?</p>
<p>This is a reformulation of an <a href="https://math.stackexchange.com/questions/1196787/pigeonhole-problem-about-distance-be... | Anders Kaseorg | 38,671 | <p>(From my <a href="https://www.quora.com/Nineteen-darts-are-thrown-onto-a-dartboard-in-the-shape-of-a-regular-hexagon-with-side-length-one-foot-How-do-I-show-that-there-must-exist-two-darts-that-are-within-frac-1-sqrt-3-feet-of-each-other/answer/Anders-Kaseorg" rel="nofollow noreferrer">answer</a> to the same questio... |
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