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 |
|---|---|---|---|---|
2,831,968 | <p>$\DeclareMathOperator{\var}{var}$It is just a general question I could not get my mind around.</p>
<p>Assume that $E[X]= 20$ and $\var[X]= 5$, then$$
E[1.2X]= 1.2·E[X]= 1.2×20= 24= 20 + 4 = E[X] + E[0.2X],\\
\var[1.2X]= 1.44·\var[X]= 1.44×5= 7.2.
$$
For$$
\var[1.2X]= \var[X + 0.2X]= \var[X] + \mathord{??} = \var[0.... | Doug M | 317,162 | <p>If the variance of $A, B$ are $\sigma^2_A, \sigma^2_B$ then
$Var(A+B) = \sigma^2_A + \sigma^2_B + 2\text{cov} (A,B) = \sigma^2_A + \sigma^2_B + 2\sigma_A\sigma_B\rho_{A,B}$</p>
<p>If this reminds you of the law of cosines, it should.</p>
<p>In your case $X,0.2X$ are perfectly correlated, and $\rho = 1$</p>
<p... |
1,946,144 | <p>I have a rough idea of how to solve this nonautonomous equation.</p>
<p>$x'=3x+sin(2t)$ </p>
<p>$\int 1\, dx=\int (3x+sin(2t))\, dt$</p>
<p>$x = 3xt - \frac{cos(2t)}{2} + constant$</p>
<p>$(1-3t)x = -\frac{1}{2}cos(2t) +constant$</p>
<p>$x = -\frac{cos(2t)}{2(1-3t)}+\frac{constant}{1-3t}$</p>
<p>Does this look... | E.H.E | 187,799 | <p>The general solution is
$$x=x_c+x_p$$
$x_c$ when the D.E is homogeneouse
so
$$x'=3x$$
$$\frac{x'}{x}=3$$
integrate it
$$\log|x|=3t+c$$
or
$$x=ke^{3t}$$
to find the particular solution, let
$$x_p=A\sin 2t+B\cos 2t$$
substitute in the original D.E to find $A$ and $B$</p>
|
128,656 | <p><img src="https://i.stack.imgur.com/AyYxe.jpg" alt=""Put the alphabet in math..."" /></p>
<p><strong>variable</strong>: A symbol used to represent one or more numbers.</p>
<p>Or alternatively: A symbol used to represent any member of a given set.</p>
<p>High school students are justifiably confused by the... | littleO | 40,119 | <p>I don't like the word "variable" in math. When we're solving an equation, $x$ is nothing more than the name of a number whose value we do not yet know, $x$ is not in any sense "variable". It's not as if the value of $x$ can change.</p>
<p>And if we are defining a function $f$, we might say something like, if $x$ i... |
3,352,919 | <p>Im taking a course in functional analysis and im trying to prove that in infinite dimensions there is no compact unit ball.
I've read some results following the Riesz Lemma but i seem to not quiet understand.
Can someone show another approach to the problem or try explain me the Riesz Lemma approach? i would really ... | Matthew Hampsey | 46,080 | <p>We can show that <span class="math-container">$\alpha_e$</span> is uniformly continuous:</p>
<p><span class="math-container">$$\dot{\alpha_e} = -K_{\omega}\alpha_e - K_R \cos(\rho_e) \dot{\rho_e} \mathbf{n_e}
= -K_{\omega}(-K_{\omega}\omega_e - K_R \sin(\rho_e)\mathbf{n_e}) - K_R \cos(\rho_e) \omega... |
1,492,477 | <p>The sequence $\frac{1}{n}$ is convergent under euclidean metric. But not convergent with discrete metric.</p>
<p>Is there a non-constant convergent sequence with discrete metric ?</p>
| Hosein Rahnama | 267,844 | <p>Here is an abstract derivation. I prefer to use equations instead of words. So pay attention to what each equation is telling you. We have</p>
<p>$$\eqalign{
& Q = \sum\limits_{i = 1}^P {{{\left( {{y_i} - \bar y} \right)}^2}} \cr
& \bar y = {1 \over P}\sum\limits_{j = 1}^P {{y_j}} \cr}\tag{1}$$</p>
... |
2,115,170 | <p>Well that's the question I am trying to solve. I did check it for a few $q$ and it seems to hold. However, I'm not sure how I would go about proving this. I actually cannot figure out where to start. I tried adding and subtracting $2q$ to make a perfect square. I think I might have to use mod 10 in this to make the ... | Bill Dubuque | 242 | <p>${\bf Hint}\quad 10<q^2\!+\!1\ \ {\rm prime}\,
\Rightarrow\! \begin{align} 1 = (q^2\!+\!1,2)=(q^2\!-\!1,2)\,\Rightarrow\,\color{#90f}{q\not\equiv \pm1\!\!\!\pmod{\!2}}\\
1 = (q^2\!+\!1,5) = (q^2\!-\!4,5)\,\Rightarrow\, \color{#0a0}{q\not\equiv \pm 2\!\!\!\pmod{\!5}}
\end{align}$</p>
<p>So $q$ has $\color{#90f}{... |
1,460,012 | <p>Heres the problem in my textbook:<br>
Show that if B and C have two equal columns, and A is any matrix for which AB and AC are defined, then AB and AC also have two equal columns.</p>
<p>This wasn't hard to figure out. But the question also says "Find a similar result involving matrices with two equal rows." I've... | Hetebrij | 252,750 | <p>You have to find a similar result for $B$ and $C$ with equal rows, so not the same result.</p>
<p>And note that if $B$ and $C$ have equal rows, $B^T$ and $C^T$ have equal columns.</p>
|
2,618,728 | <p>Please Help me in the following Problem</p>
<blockquote>
<p>What is the Number Of Natural Numbers ,$n\le30$ for which $\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}$ is also a prime number.</p>
</blockquote>
<p>The only way I am able to find to solve this is calculate each and every term once but it will be extremely length... | fred goodman | 124,085 | <p>If $n = p^2 -p$, then $p^2 =n + p $, so $p = \sqrt{n + p}= \sqrt{n + \sqrt{n+p}}$, etc.</p>
<p><strong>Edit:</strong> My original answer, above, left a lot to the imagination. The real work would be in analyzing the convergence of the sequence of nested radicals. To this end, fix a positive number $n$, not neces... |
1,714 | <p>Say I have a function $f(x)$ that is given explicitly in its functional form, and I want to find its Fourier transform[1]. If $f$ is too complicated to have an analytic expression for $\hat f(k)$, how do I obtain it numerically?</p>
<p>The naive and stupid way, which I currently use, is evaluating the integral for ... | Rojo | 109 | <p>Coming a little late... Anyway </p>
<p>I'm used to the <code>FourierParameters</code> that MMA describes as "signal processing" in <code>FourierTransform</code>'s More Information <code>{0, -2 Pi}</code>. I'll edit if I have a more general version</p>
<p>Set up the option <code>SamplingPeriod</code>, which if it's... |
1,229,164 | <p>How to solve modular equations?
So for example $a \equiv i$ mod $x$, $a \equiv j$ mod $y$ for some given $i,j,x,y$ with $gcd(x,y)=1$, and I must find $a$ mod $x*y$.
Any tips on how to do this?
Specifically I want to calculate $a \equiv 1$ mod $16$, $a \equiv 3$ mod $17$, for example.</p>
| Rubertos | 44,669 | <p>Use the chinese remainder theorem</p>
|
1,229,164 | <p>How to solve modular equations?
So for example $a \equiv i$ mod $x$, $a \equiv j$ mod $y$ for some given $i,j,x,y$ with $gcd(x,y)=1$, and I must find $a$ mod $x*y$.
Any tips on how to do this?
Specifically I want to calculate $a \equiv 1$ mod $16$, $a \equiv 3$ mod $17$, for example.</p>
| JMP | 210,189 | <p>In this case there is a simple solution. Note $3\equiv3\mod16, (3+17)=20\equiv4\mod16$. The difference is $1$ so to arrive at $1\mod16$ we go back two lots of $17$ from $3$, i.e. $31$, and subtract this from $16.17=272$ to give the answer as $241$.</p>
|
1,744,832 | <p>Given $$A=\begin{bmatrix}
-4 & 3\\
-7 & 5
\end{bmatrix}$$ Find $A^{482}$ in terms of $A$</p>
<p>I tried using Characteristic equation of $A$ which is $$|\lambda I-A|=0$$ which gives</p>
<p>$$A^2=A-I$$ so $$A^4=A^2A^2=(A-I)^2=A^2-2A+I=-A$$ so</p>
<p>$$A^4=-A$$ but $482$ is neither multiple of $4$ nor Powe... | sayan | 312,099 | <p>You can do this by solving the pells equation.</p>
<p>Let $3{k^2}+1={n^2}$ for some natural number n.Then the equation becomes a</p>
<p>pells equation$\implies$ ${n^2}-3{m^2}=1$.Then factorize L.H.S.</p>
<p>We get $(n+{\sqrt3}m)(n-{\sqrt3}m)=1$.Here m and n are solutions of the equation.</p>
<p>Now square both s... |
3,154,609 | <p>Let <span class="math-container">$L$</span> denote a linear operator and <span class="math-container">$v\in V$</span>. Does the expression <span class="math-container">$$c_0L^nv + c_1L^{n-1}v + \cdots + c_{n-1}L^1v + c_nL^0v = 0$$</span>
has a special name and what properties are known? For example, I know that if <... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
2,821,613 | <p>I came across following problem from Sheldon Ross' book:</p>
<blockquote>
<p>A closet contains 10 pairs of shoes. If 8 shoes are randomly
selected, what is the probability that there will be </p>
<ol>
<li>no complete pair; </li>
<li>exactly one complete pair</li>
</ol>
</blockquote>
<p>I solved th... | Graham Kemp | 135,106 | <p>Your text is in error. There should be no $8!/2!$ factor.</p>
<p>In general the probability for selecting $x$ pairs $(x\in \{0,1,2,3,4\})$ is$$\dfrac{\dbinom{10}{x}\dbinom{10-x}{8-2x}2^{8-2x}}{\dbinom{20}8}\quad\text{or}\quad\dfrac{\dbinom{10}{8-x}\dbinom{8-x}{x}2^{8-2x}}{\dbinom{20}8}
$$</p>
<p>The... |
4,028,717 | <p><span class="math-container">$$\dfrac{\sqrt[3]{3}}{\sqrt[3]{1}+\sqrt[3]{2}}=\sqrt[3]{\sqrt[3]{2}-1}$$</span></p>
<p>I could not multiply by the conjugate since it is a cube root. Can you show me a way to simplify it?</p>
<p>Thanks!</p>
| RobertTheTutor | 883,326 | <p><span class="math-container">$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$</span>
Since you want to clear the denominator, try multiplying both numerator and denominator by <span class="math-container">$(2^{\frac{2}{3}} - 2^{\frac{1}{3}} + 1)$</span>
That will give you 2-1 for the denominator.</p>
|
4,028,717 | <p><span class="math-container">$$\dfrac{\sqrt[3]{3}}{\sqrt[3]{1}+\sqrt[3]{2}}=\sqrt[3]{\sqrt[3]{2}-1}$$</span></p>
<p>I could not multiply by the conjugate since it is a cube root. Can you show me a way to simplify it?</p>
<p>Thanks!</p>
| Quanto | 686,284 | <p>Note <span class="math-container">$(\sqrt[3]{2}-1)(\sqrt[3]{4}+\sqrt[3]{2}+1)=(\sqrt[3]{2})^3-1=1$</span> and</p>
<p><span class="math-container">\begin{align}
\sqrt[3]{\sqrt[3]{2}-1}
=& \sqrt[3]{\frac1{\sqrt[3]{4}+\sqrt[3]{2}+1}}
=\sqrt[3]{\frac3{2+3\sqrt[3]{4}+3\sqrt[3]{2}+1}}
=\sqrt[3]{\frac3{(\sqrt[3]{2}+1)^... |
246,492 | <p>I am running an iMac Pro with Intel silicon under the latest version of Big Sur 11.3.1. I upgraded my Mathematica from 12.2 to 12.3 yesterday, but now I have suddenly lost the ability to use the QuickLook feature with Mathematica notebooks. I click on a notebook's icon in Finder to highlight it and then touch the sp... | WReach | 142 | <p>If we can rely upon the target associations always being under <code>"key2"</code> and always having a subkey <code>"X"</code>, then:</p>
<pre><code>test // MapAt[Select[#X > 6 &], "key2"]
% === test2
(* True *)
</code></pre>
<p>or</p>
<pre><code>test // Query[{"key2" ... |
294,725 | <p>I've been out of school for a very long time, and can't wrap my head around calculating 2 whole numbers from a fraction. For instance, if I have 2 ratios -- </p>
<pre><code>4:3 1.33333...
16:9 1.77777...
?:? (automatically generated)
</code></pre>
<p>I want to be able to look up the whole number represent... | Community | -1 | <p>There's a straightforward algorithm for this. Allow me to demonstrate on</p>
<p>$$ 0.1\overline{37} = 0.1373737373737\ldots$$</p>
<p>The first step is to split off the non-repeating part:</p>
<p>$$ 0.1\overline{37} = 0.1 + 0.0\overline{37} $$</p>
<p>then you pull out the 'common factor' from the repeated part</p... |
21,290 | <p>Let $k$ be a field. What is an explicit power series $f \in k[[t]]$ that is transcendental over $k[t]$? </p>
<p>I am looking for elementary example (so there should be a proof of transcendence that does not use any big machinery).</p>
| Wadim Zudilin | 4,953 | <p>Coming back to the lacunary series, I would prefer the series $f(z)=\sum_{k\ge0}z^{d^k}$, where $d>1$ is an integer, because it is the classical example in Mahler's method; this function satisfies the functional equation $f(z^d)=f(z)-z$. I simply copy Ku.Nishioka's argument from her book "Mahler functions and tra... |
4,050,660 | <p>A car license plate consists of 3 capital letters of the English alphabet in the first 3 positions of the license plate followed by 4 digits from 0 to 9. How many different plates can we have if neither the same letter nor the same digit on the plate is allowed to be repeated?</p>
<p>I was wondering if the right an... | user1859871 | 829,270 | <p>There are <span class="math-container">$\frac{26!}{23!}$</span> ways to select the letter portion of the plate and <span class="math-container">$\frac{10!}{6!}$</span> ways to select the number portion. The product is the total number of possible plates with these contraints.</p>
|
1,533,762 | <blockquote>
<p>Prove that $ 16^{20}+29^{21}+42^{22}$ is divisible by $13$.</p>
</blockquote>
<p>This is not a homework question. I would like to know how to solve this type of problems, I solved similar problem with n in exponent, but that could be proved by induction. Here I guess, Euler's theorem could be useful,... | Archis Welankar | 275,884 | <p>HINT Notice the difference its $13$ so now perform modular division and your proof is done. As suggested by Plankton</p>
|
2,820,247 | <p><a href="http://www.math.hawaii.edu/~tom/old_classes/412notes6.pdf" rel="nofollow noreferrer">Source</a></p>
<blockquote>
<p>Theorem 6.10</p>
<p>Let $f : R → S$ be any homomorphism of rings and let $K = \ker f$.
Then
$K$ is an ideal in $R$.</p>
<p>Proof. We know $0 ∈ K$, so $K \neq ∅$. Let $a, b ∈ K... | Good Morning Captain | 220,841 | <p>$f(0_R) = f(0_R + 0_R) = f(0_R) + f(0_R)$. Therefore, $f(0_R) + 0_S = f(0_R) + f(0_R)$. Thus $0_S = f(0_R)$. </p>
<p>Ring homomorphisms will always preserve additive identities (and multiplicative identities if it is a ring with unit)</p>
|
2,820,247 | <p><a href="http://www.math.hawaii.edu/~tom/old_classes/412notes6.pdf" rel="nofollow noreferrer">Source</a></p>
<blockquote>
<p>Theorem 6.10</p>
<p>Let $f : R → S$ be any homomorphism of rings and let $K = \ker f$.
Then
$K$ is an ideal in $R$.</p>
<p>Proof. We know $0 ∈ K$, so $K \neq ∅$. Let $a, b ∈ K... | JohnKnoxV | 431,468 | <p>For all $r \in R$, $f(r) = f(0 + r) = f(0) + f(r)$ which implies that $f(0) = 0$. </p>
|
3,080,402 | <p>I need to show that the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^{-n}}{n!}$</span> is convergent.</p>
<p>I invoked the limit comparison with the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^n}{n!}$</span> which is absolutely convergent (and hence convergent).</p>
<p>I got ... | egreg | 62,967 | <p>Hint: the case <span class="math-container">$a+b=c+d=0$</span> is obvious, so you can assume <span class="math-container">$a+b=c+d\ne0$</span>; prove that <span class="math-container">$ab=cd$</span>. Further hint: <span class="math-container">$x^2-xy+y^2=(x+y)^2-{\color{red}{?}}$</span>.</p>
|
3,080,402 | <p>I need to show that the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^{-n}}{n!}$</span> is convergent.</p>
<p>I invoked the limit comparison with the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^n}{n!}$</span> which is absolutely convergent (and hence convergent).</p>
<p>I got ... | lab bhattacharjee | 33,337 | <p><span class="math-container">$$(a+b)^3=(c+d)^3$$</span></p>
<p><span class="math-container">$$\iff a^3+b^3+3ab(a+b)=c^3+d^3+3cd(c+d)$$</span></p>
<p>If <span class="math-container">$a+b\ne0, ab=cd\ \ \ \ (1)$</span></p>
<p><span class="math-container">$\iff\dfrac ad=\dfrac cb=k$</span>(say) <span class="math-co... |
3,080,402 | <p>I need to show that the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^{-n}}{n!}$</span> is convergent.</p>
<p>I invoked the limit comparison with the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^n}{n!}$</span> which is absolutely convergent (and hence convergent).</p>
<p>I got ... | Hagen von Eitzen | 39,174 | <p>Let <span class="math-container">$\alpha=a+b=c+d$</span>, <span class="math-container">$\beta=a^3+b^3=c^3+d^3$</span>.
Then <span class="math-container">$$ \alpha^3=(a+b)^3=a^3+3a^2b+3ab^2+b^3=\beta+\alpha ab.$$</span>
It follows that <span class="math-container">$a,b$</span> (and likewise <span class="math-containe... |
2,274,839 | <p>I can get,</p>
<p>$$h_n \approx \ln n+\gamma$$</p>
<p>By, saying</p>
<p>$$\frac{h_{n}-h_{n-1}}{n-(n-1)}=\frac{1}{n}$$</p>
<p>Then letting $h_n=h(n)$ be continuous and differentiable on $(0,\infty)$. So that,</p>
<p>$$h'(n) \approx \frac{1}{n}$$</p>
<p>And thus,</p>
<p>$$h_n \approx \ln n+C$$</p>
<p>Where we ... | Felix Marin | 85,343 | <p>$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\n... |
2,720,188 | <p>If we take the inner product $\langle f, g \rangle = \displaystyle\int_{-\pi}^{\pi} f(t) \overline{g(t)} dt$ on $L^2 ((-\pi, \pi))$, which allows functions to $\mathbb{C}$, then it's not hard to check that $\Big(\frac{e^{int}}{\sqrt{2\pi}}\Big)_{n=1}^{\infty}$ is an orthonormal <em>sequence</em> in the space, but I'... | Jason | 195,308 | <p>It <em>isn't</em> a basis - you need to let $n$ range over $\mathbb Z$. Specifcally, if $e_n(t)=\frac{e^{int}}{\sqrt{2\pi}}$, then $\{e_n\}_{n\in\mathbb Z}$ is a basis of $L^2(-\pi,\pi)$. The easiest way to prove this is to use Stone-Weierstrass.</p>
<p>Let $X:=\{f:[-\pi,\pi]\to\mathbb R: f\text{ is continuous, }f(... |
10,948 | <p>This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum_{i>0}i^{-2}$.</p>
<p>Some quick facts:</p>
<ul>
<li>Pretty much by c... | Jonas Meyer | 1,119 | <p>Use residues! For an entertaining narrative with the correct answer, see <a href="https://mathoverflow.net/questions/8741/justifying-a-theory-by-a-seemingly-unrelated-example/8752#8752">here</a>. For a derivation, see a complex analysis text.</p>
<h3>Added</h3>
<p>Here's a link to an outline of the residue method ... |
10,948 | <p>This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges absolutely because it behaves roughly like $\sum_{i>0}i^{-2}$.</p>
<p>Some quick facts:</p>
<ul>
<li>Pretty much by c... | David E Speyer | 297 | <p>Another proof:</p>
<p>Start with the identity</p>
<p>$$\sin (\pi x)= \pi x \prod \left( 1-x^2/i^2 \right)$$</p>
<p>and apply $(d/dx)^2 \log$ to both sides.</p>
|
110,599 | <blockquote>
<p>Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module?</p>
</blockquote>
<p>The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't understand why this can be a reason.</p>
<p>Since</p>
<blockquote>
<p>An $R$-module... | Pierre-Yves Gaillard | 660 | <p>Here is a</p>
<h1>New version of the answer</h1>
<p>I'll leave the old version below so that the comments remain understandable. </p>
<p>Let $K$ be a commutative ring and $x,y,z$ be indeterminates. Put
$$
M:=\frac{K[x,y,z]}{(xz-y)}\quad.
$$
In particular, $M$ is an $K[x,y]$-module. </p>
<p>We claim that $M$ i... |
102,976 | <p>Is there a way to evaluate a string containing RPN in Mathematica?</p>
<p>SE thinks this question is too short, so let me expand on it.
Do you know of any function, that provides the following functionality?</p>
<pre><code>EvalRPN["5 4 + 3 /"]
</code></pre>
<blockquote>
<p>3</p>
</blockquote>
<p>Or even symbol... | Mark Adler | 94 | <p>The thing is, you need to identify what operators should be infix, what should be prefix, what should be postfix, and how many elements from the stack to use. There may be a better way than just listing them, but here I just list them and then use <code>ToExpression</code> to make functions out of them to apply to a... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | seismatica | 14,587 | <p>Throwing my hat to the ring</p>
<pre><code>Clear[splitList]
splitList[f_, list_List] :=
SplitBy[DeleteDuplicates[Sort /@ Tuples[Sort[Hold[f] @@ list], 2]], First] // ReleaseHold
splitList[f, {a, b, c, d}]
(* {{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}} *)
... |
3,229,020 | <p>My lecturer set as a bonus exercise the following induction proof:</p>
<p>If <span class="math-container">$G$</span> is a finite abelian group <span class="math-container">$|G| = p_1^{n_1} \cdots p_s^{n_s}$</span> is the decomposition of <span class="math-container">$|G|$</span> into a product of distinct prime num... | lhf | 589 | <p>Here is the key in induction:</p>
<p>If <span class="math-container">$G$</span> is a finite abelian group of order <span class="math-container">$n=ab$</span>, with <span class="math-container">$a,b >1$</span> and <span class="math-container">$\gcd(a,b)=1$</span>, then <span class="math-container">$G \cong A \ti... |
704,517 | <p>Let $D \subset \mathbb{C}$ be a discrete subset and let $f : D \mapsto \mathbb{C}$ be a function. Show that $f$ is continuous.</p>
<p>What's the best way to do this? I was thinking a proof by contradiction since a direct proof seems a little tricky...</p>
<p>Definitions i am using:</p>
<p>$D \subset \mathbb{C}$ i... | Ben Millwood | 29,966 | <p>There are several possible approaches here. In light of the edits to your question, you probably want the third.</p>
<p>Topological: we wish to establish if every preimage in $f$ of an open set in $\mathbb C$ is open. But one definition of "discrete set" is "one whose subspace topology is discrete", where the discr... |
119,981 | <p>Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence</p>
<p>$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$</p>
<p>gives a homomorphism from $\operatorname... | CuriousIndeed | 69,760 | <p>Although not a mathematical concise definition but empirical evidence that money is created "out of thin air"..</p>
<p>"It was examined whether in the process of making money available to the borrower the bank transfers these funds from other accounts (within or outside the bank). In the process of making loaned mo... |
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Yes | 155,328 | <p>Your description led me to think that you want a book speaking in a tone like you are strolling with a seasoned mathematician. Then two books came to my mind:</p>
<ol>
<li><p>The Way of Analysis by R. S. Strichartz;</p></li>
<li><p>Analysis (three volumes) by R. Godement.</p></li>
</ol>
<p>The latter one is in add... |
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| Gabriel Romon | 66,096 | <p>Iosevich, <em>A View from the Top : Analysis, Combinatorics and Number Theory</em></p>
|
230,997 | <p>How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> $\rm\ ax\!+\!by = c\:|\:a,b,\ d|\:x,y\:\Rightarrow\: cd\:|\:ax\!+\!by=c\:\Rightarrow\: d\:|\:1.\ \ $ Let $\rm\:c = (a,b),\ d = (x,y)\ \ $ <strong>QED</strong></p>
|
2,344,259 | <blockquote>
<p>$$\int_0^\infty \frac{x^2}{x^4+1} \; dx $$</p>
</blockquote>
<p>All I know this integral must be solved with beta function, but how do I come to the form $$\beta (x,y)=\int_0^1 t^{x-1}(1-t)^{y-1}\;dt \text{ ?}$$</p>
| Nemanja Beric | 405,086 | <p>By definition $$B(p, q) = \int_0^1x^{p-1}(1-x)^{q-1}\,dx.$$ By using shift $x = \frac{t}{1 + t}$ previous integral becomes $$B(p,q) = \int_0^{\infty}\frac{t^{p-1}}{(1 + t)^{p + q}}\,dt.$$</p>
<p>Thus we get that
\begin{align}
I & = \int_0^\infty \frac{x^2}{1 + x^4} \, dx = \{x^4 = t, dx = t^{\frac14 - 1} \, dt,... |
2,110,681 | <p>I recently started a Discrete Mathematics course in college and I am having some difficulties with one of the homework questions. I need to learn this, so please guide me through at least two steps to get the ball rolling. </p>
<p>The question reads: Show that if $A$ and $B$ are sets, then: $(A \cap B) \cup (A \cap... | fleablood | 280,126 | <p>You can use identities such as $(A\cap B) \cup (A \cap C) = A \cap (B \cup C)$ to get $(A \cap B) \cup (A \cap B') = A \cap (B \cup B') = A \cap U = A$.</p>
<p>But I prefer to think of <em>what</em> it is saying. $A \cap B$ means "everything in A and in B" and $A \cap B'$ means "everything that is in A that is not... |
821,758 | <blockquote>
<p>Find the mean of $a, a+d, a+2d, a+3d,\dots,a+nd$</p>
</blockquote>
<p>I have no idea what to do in this question but i have tried the following:
$$mean\ \bar{x}= \frac{(a)+(a+d)+(a+2d)+(a+3d)+\cdots+(a+nd)}{n+1} $$</p>
<p>This is obvious step we do while dealing with mean.Now i don't know what do bu... | DSinghvi | 148,018 | <p>there are n+1 times a and take d common from left expression and you get 0+1+2+...+n-1+n whose sum is given by sum of arithmetic progression n(n+1)/2
therefore you can write numerator a(n+1) + d(n)(n+1)/2
so your mean equals a+dn/2</p>
|
3,652,205 | <p>I am trying to compute the following double integral:
<span class="math-container">$$I=\iint_S \frac{2-4xy}{(9-xy)(8+xy)}dxdy$$</span>
with <span class="math-container">$S=[0,1]\times[0,1].$</span></p>
<p><strong>What I have tried:</strong></p>
<ol>
<li>I have written the integral as follows:
<span class="math-con... | Zacky | 515,527 | <p>Let's generalize it, in particular the integral in the question is just the case <span class="math-container">$a=8$</span>.
<span class="math-container">$$I(a)=\int_0^1\int_0^1\left(\frac{1}{a+xy}-\frac{1}{1+a-xy}\right)dxdy\overset{xy=t}=\int_0^1\int_0^y\frac{1}{y}\left(\frac{1}{a+t}-\frac{1}{1+a-t}\right)dtdy$$</s... |
450,857 | <p>I have the following problem I'm trying to understand/solve using first order logic.</p>
<pre><code>Predicates:
Set(S), which states that S is a set, and
x ∈ S, which states that x is an element of S,
</code></pre>
<p>Using first order logic, I need to write :</p>
<pre><code>For any x and y, there is a set contai... | Asaf Karagila | 622 | <p><strong>HINT:</strong></p>
<p>What is a set containing just $x$ and $y$? It's the set $\{x,y\}$.</p>
<p>First write a formula $\varphi(x,y,z)$ which states that $z=\{x,y\}$. Next, quantify over $x,y,z$ according to your instructions.</p>
|
499,587 | <p>$$M = \left(\begin{smallmatrix} a_1 & a_2 & a_3 & a_4\\ b_1 & b_2 & b_3 & b_4\\ a_1 & c_2 & b_2 & c_4\\ a_4 & d_2 & b_3 & c_4\\ b_1 & c_2 & a_2 & e_4\\ b_4 & d_2 & a_3 & e_4\end{smallmatrix}\right)$$ All of the equations equal to 26; augmented, ... | ftfish | 84,805 | <p>You have forgotten the very important condition that all integers between 1 and 12 must be used exactly once.</p>
<p>Though the equations themselves are linear and contain only 0 and 1 as coefficients, the constraint is very tough. I don't think you could find an easy solution without <a href="http://en.wikipedia.o... |
3,838,943 | <p>Let <span class="math-container">$z_n$</span> be a Blaschke sequence in <span class="math-container">$\mathbb{D}$</span> and let <span class="math-container">$B$</span> be the Blaschke product defined by <span class="math-container">$$B(z)=z^m\prod_{n=1}^{\infty}\frac{|z_n|}{z_n}\frac{z_n-z}{1-\bar{z}_nz}$$</span>
I... | Martin R | 42,969 | <p>Your computation of the derivative seems to be wrong because apparently you differentiated the product termwise. Also you are using the variable <span class="math-container">$m$</span> for two different purposes.</p>
<p>One can use the definition of the derivative directly instead. <span class="math-container">$B(z_... |
2,397,077 | <p>The "Power of a Point" or "Intersecting Chords" theorem states that for any point in a plane, if a line is drawn that intersects a circle, the distance from the point to one of the intersections multiplied by the distance from the point to the other intersection is a constant for that point, no matter what line you ... | diagonal2 | 535,249 | <p>Let $P$ be a point and $\Gamma$ be a curve on the plane. A line through $P$ intersects $S$ at two points $A$ and $B$.<br>
I think your statement on 'Power of a Point' can be interpreted in two ways:<br>
1) $PA\cdot PB$ is constant for a fixed $P$. This is true for some $P$ on the plane.<br>
2) $PA\cdot PB$ is consta... |
269,552 | <p>I have been working on the following problem:</p>
<p>"Let $\sim$ be the equivalence relation on the unit circle $S^1$ defined by $x \sim -x$, $x \in S^1$. Show that $S^1/\sim$ is homeomorphic to $S^1$ and interpret geometrically."</p>
<p>I have applied the following two theorems:</p>
<p>"Let $X$ and $Y$ be space... | Lubin | 17,760 | <p>Imagine your circle lying in the $xy$-plane in $3$-space. Now pinch the points $(0,\pm1)$ together to the origin, giving, as you say, a figure-eight. Now take only the right-hand loop and, in space, rotate it $180$ degrees <em>in the $x$-axis</em>. Now take this loop and flip it, in space, through the $y$-axis; that... |
269,552 | <p>I have been working on the following problem:</p>
<p>"Let $\sim$ be the equivalence relation on the unit circle $S^1$ defined by $x \sim -x$, $x \in S^1$. Show that $S^1/\sim$ is homeomorphic to $S^1$ and interpret geometrically."</p>
<p>I have applied the following two theorems:</p>
<p>"Let $X$ and $Y$ be space... | Mariano Suárez-Álvarez | 274 | <p>Let $\sim$ be the equivalence relation. I will identify $S^1$ with the set of complex numbers of modulus $1$.</p>
<ul>
<li>Consider the function $f:z\in S^1\mapsto z^2\in S^1$. </li>
<li>It is clear that if $x$, $y\in S^1$ are such that $x\sim y$ then $f(x)=f(y)$, for in that case we have $x=\pm y$. This has the co... |
269,552 | <p>I have been working on the following problem:</p>
<p>"Let $\sim$ be the equivalence relation on the unit circle $S^1$ defined by $x \sim -x$, $x \in S^1$. Show that $S^1/\sim$ is homeomorphic to $S^1$ and interpret geometrically."</p>
<p>I have applied the following two theorems:</p>
<p>"Let $X$ and $Y$ be space... | Not Euler | 479,343 | <p>If you identify antipodal points of the sphere <span class="math-container">$S^n$</span>, you get <span class="math-container">$\mathbb{R}\mathbb{P}^n$</span>. For <span class="math-container">$n=1$</span> you get <span class="math-container">$\mathbb{R}\mathbb{P}^1$</span>, which is <span class="math-container">$S^... |
119,375 | <p>Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
(<a href="http://en.wikipedia.org/wiki/Peano_axioms#First-order_theory_of_arithmetic" rel="nofollow noreferrer"><a href="http://en.wikipedia.org/wiki/Peano_axioms#Firs... | abo | 20,716 | <p>NOTE: The first proof is wrong because it uses second-order induction. The second-proof is wrong as well per Wofsey's comment. </p>
<p>Ashutosh in the comments has shown that exclusion holds.</p>
<p>Here is a proof of existence. </p>
<p>Let A be the elements of the model. Let p be the predecessor of 0 in A. ... |
570,467 | <p>$ E_{n}=2E_{n-1}+ 2^{n-1} $</p>
<p>Can anyone help me to solve this recurrence? Is there a general way to think about recurrence?</p>
| Community | -1 | <p>A general solution is of the form:
$$E_n=c2^n + n2^{n-1},$$
where $c$ is any real constant.</p>
<p>I found one solution, the $n2^{n-1}$ part, by an educated guess, which I realize probably isn't that helpful for you. Then given one solution, I solved the recurrence,
$$B_n=2B_{n-1}$$
to get get the $c2^n$ part. Obse... |
55,933 | <p>I am afraid this post may show my naivety. At a recent conference, someone told me that there are some arguments in computability theory that don't relativize. Unfortunately, this person (who I think may be an MO regular) couldn't give me any examples off-hand.</p>
<p>My naive understanding is that one can take a... | alpoge | 12,138 | <p>(Note: I have probably misunderstood your question!)</p>
<p>Here's one example that is, along with the nonrelativization of $\mathbf{P}$ versus $\mathbf{NP}$, standard: The proof that $\mathbf{IP} = \mathbf{PSPACE}$ does not relativize, since for almost all oracles $A$, we have that $\mathbf{IP}^A\neq \mathbf{PSPAC... |
2,101,241 | <p>Find the remainder when $$140^{67}+153^{51}$$ is divided by $17$.</p>
<p>$$140\equiv 4 \pmod {17}$$
$$67\equiv 16 \pmod{17}$$
$$153 \equiv 0 \pmod{17}$$
$$51\equiv 0 \pmod{17}$$</p>
<p>$$\Rightarrow 140^{67}+153^{51}\equiv 4^{16}+0 \equiv 1\pmod{17}$$</p>
<p>Solution should be $13$. What's wrong?</p>
| Ahmed S. Attaalla | 229,023 | <p>What is wrong is that</p>
<p>$$x^y \mod n \neq (x \mod n)^{y \mod n}$$</p>
<p>It is true however that,</p>
<p>$$x^y \mod n=(x \mod n)^{y}$$</p>
<p>So my hint on how to proceed would be to notice $4^2=16=17-1$.</p>
|
2,101,241 | <p>Find the remainder when $$140^{67}+153^{51}$$ is divided by $17$.</p>
<p>$$140\equiv 4 \pmod {17}$$
$$67\equiv 16 \pmod{17}$$
$$153 \equiv 0 \pmod{17}$$
$$51\equiv 0 \pmod{17}$$</p>
<p>$$\Rightarrow 140^{67}+153^{51}\equiv 4^{16}+0 \equiv 1\pmod{17}$$</p>
<p>Solution should be $13$. What's wrong?</p>
| fleablood | 280,126 | <p>FLT says $a^{p-1} \equiv 1\mod p$</p>
<p>So $67 \equiv 16 \mod 17$ ($p=17$) is irrelevent.</p>
<p>You need $67\equiv 3 \mod 16$ ($p-1=16$)</p>
<p>So $4^{67} \equiv 4^3 \equiv 13 \mod 17$.</p>
|
7,787 | <p>I'm trying to expand the following polynomial </p>
<pre><code> Expand[ (A1 a1 + A2 a2 + A3 a3 + A4 a4 + A5 a5 + A6 a6 + A7 a7 + A8 a8)
(D1 a1 + D2 a2 + D3 a3 + D4 a4 + D5 a5 + D6 a6 + D7 a7 + D8 a8)
+ (H1 a1 + H2 a2 + H3 a3 + H4 a4 + H5 a5 + H6 a6 + H7 a7 + H8 a8)
(E1 a1 + E2 a2 + ... | rm -rf | 5 | <p>You can use the image processing tools to do this. In short, you use <code>FillingTransform</code> to fill the inner parts and create a mask and then use <code>ImageApply</code> to set the pixels in the region given by mask to your desired colour.</p>
<pre><code>mask = ImageResize[FillingTransform[ColorNegate@Binar... |
1,725,150 | <p>I am trying to answer the following question:</p>
<p>Let $M_a := \{ (x^1,\ldots,x^n,x^{n+1}) \in \mathbb{R}^{n+1} : (x^1)^2 + \cdots +(x^n)^2 - (x^{n+1})^2 = a\}$. For which values of $a$, $M_a$ is a submanifold of $\mathbb{R}^{n+1}?$</p>
<p>I have to use the regular value theorem, but I think I don't know in fact... | Tsemo Aristide | 280,301 | <p>You have to compute $df =2x_1dx_1+...2x_ndx_n -2x_{n+1}dx_{n+1}$. Let $a$ be a non zero real. for every $x=(x_1,...,x_n)\in f^{-1}(a)$, there exists $i$ such that $x_i\neq 0$. Let $e_i$ be the vector of $R^{n+1}$ whose $i$-coordinate is $1$ and its other coordinates are $0$. $df_x(0,..,0,1,0..)=x_i\neq 0$. Thus $df... |
1,585,323 | <p>Statement :- Let $A$ denote an event whose probability of occurrence in a single trial is $p$. If $k$ denotes the number of occurrences of $A$ in $n$ independent trials, then </p>
<p>$$P\left(\left|\frac kn - p\right|> \epsilon\right) \lt \frac{pq}{n \epsilon ^2}$$</p>
<p>Someone please help me understanding th... | Em. | 290,196 | <p>"In $n$ independent trials" implies that this "$k$" is a binomial distribution, say $X\sim \text{Bin}(n, p)$. Recall that $\text{Var}(X) = np(1-p) = npq$, and so
$$\text{Var}\left[\frac{X}{n}\right] = \frac{1}{n^2}\text{Var}(X) = \frac{pq}{n}.$$
Thus, applying <a href="https://en.wikipedia.org/wiki/Chebyshev's_... |
3,643,773 | <p>I have a circle with center (0,0) and radius 1. I have calculated a random point inside a circle by generating a random angle <span class="math-container">$a=random()\cdot 2\pi $</span> and a random distance smaller than or equal to the radius <span class="math-container">$b=random()\cdot r$</span>. The center of th... | AnswerDigger | 779,186 | <p>you can just do this with basic coordinate geometry. To find the line equation you need the slope and a point. Fortunately, the point is given. You can simply find the slope for the line since the product of two perpendicular lines' slope is -1.</p>
|
1,660,116 | <p>I would like to get a closed form of $A_n(x)$ if verifies the following recurrence relation</p>
<p>$$A_n(x)=\frac{d}{dx}\left(\frac{A_{n-1}(x)}{a-\cos x}\right)\,\,\,\text{and}\,\,\,A_0(x)=1.$$</p>
<p>Really I need to know the general term of $A_n(0)$. </p>
<p>Any ideas or suggestions are welcome.</p>
| mike | 75,218 | <p>This is not a complete solution.</p>
<p>Setting $n=\infty$, and the recurrence equation becomes
$$A_{\infty}(x)=\frac{d}{dx}\left(\frac{A_{\infty}(x)}{a-\cos x}\right)$$</p>
<p>The solution to it is:</p>
<p>$$A_{\infty}(x) = C(a - \cos x)\exp(a x - \sin x) $$
where $C$ is a constant of integration.</p>
|
6,208 | <p>Let $f$ be an analytic function on the open unit disk domain $D$. Suppose also that $f$ is bounded.</p>
<p>Since $f$ is bounded I believe that $f$ can be continuously extended to the closed unit disk.
I know that the zeros of $f$ in the open disk $D$ are isolated. Are the zeros of $f$ in the closed unit disk als... | Jonas Meyer | 1,424 | <p>Edit (March 5, 2011): My original answer gave as justification for claims about zero sets a result of Rudin and of Carleson that not only came a half a century after Fatou's result now mentioned below, but also didn't directly address the problem. I've edited to correct the deficiencies.</p>
<hr />
<p>It is not tru... |
384,628 | <p>i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?</p>
| amWhy | 9,003 | <p>You are correct in your understanding of the domain, and codomain:</p>
<p>When we say $\;f: [0, 1] \to \mathbb R\;$ we are specifying (naming) a function $f$ which is defined on the domain $[0, 1]$, and which maps values in the domain $[0, 1]$ to values in the <em>codomain</em>, here, $\,\mathbb R$. Think of $f$ as... |
384,628 | <p>i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?</p>
| Thomas Russell | 32,374 | <p>In the general case, the notation $f:X\to Y$, where $f$ is a function, and $X$ and $Y$ are sets represents a function called $f$ which takes any element belonging to the set $X$ (the <em>domain</em>) and maps it to an element within the set $Y$ (called the <em>codomain</em> and sometimes the <em>range</em> of the fu... |
384,628 | <p>i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?</p>
| xavierm02 | 10,385 | <p>In high-school, they want you to understand function in a kind of natural way: you give it something and it gives you something back using a formula. Much like you use $\Bbb N$ or $\Bbb R$ without constructing them first.</p>
<p>The real definition of functions need something else to be introduced first: Binary rel... |
3,395,549 | <p>Consider the topological space <span class="math-container">$R^n$</span> with the standard topology. Let <span class="math-container">$A$</span> be any affine subspace. Prove that <span class="math-container">$A$</span> is a closed subset of <span class="math-container">$R^n$</span>.</p>
<p>If I recall things corre... | Reveillark | 122,262 | <p>Step 1: </p>
<blockquote>
<p>If <span class="math-container">$V$</span> is a subspace of <span class="math-container">$\mathbb{R}^n$</span>, then <span class="math-container">$V$</span> is closed. </p>
</blockquote>
<p>As <span class="math-container">$\mathbb{R}^n$</span> is finite dimensional, any subspace is c... |
3,395,549 | <p>Consider the topological space <span class="math-container">$R^n$</span> with the standard topology. Let <span class="math-container">$A$</span> be any affine subspace. Prove that <span class="math-container">$A$</span> is a closed subset of <span class="math-container">$R^n$</span>.</p>
<p>If I recall things corre... | Fakethis | 711,406 | <p>First suppose <span class="math-container">$V$</span> has dimension <span class="math-container">$n-1$</span>. If <span class="math-container">$w$</span> is a vector orthogonal to <span class="math-container">$V$</span>, then the map <span class="math-container">$f:\mathbb{R}^n\rightarrow \mathbb{R}$</span> given by... |
1,650,333 | <p>Can someone help me figure out this ODE, its driving me crazy. I dont need a full solution beacuse that would take hours but maybe just the final answer?</p>
<p>Find the general solution of the ODE
$xy′′ − y′ + 4x^3y = 0$
assuming $x > 0$ and given that $y_1(x) = \sin(x^2)$ is a solution.</p>
| Frits Veerman | 273,748 | <p>You can express the second (unknown) solution $y_2$ to this ODE in terms of the first (known) solution $y_1$ and the Wronskian $w$.</p>
<p>First, the Wronskian. By definition, we have
\begin{equation}
w = y_1 y_2' - y_1' y_2. \tag{1}
\end{equation}
Taking the derivative on both sides, we obtain
\begin{equation}
w... |
1,650,333 | <p>Can someone help me figure out this ODE, its driving me crazy. I dont need a full solution beacuse that would take hours but maybe just the final answer?</p>
<p>Find the general solution of the ODE
$xy′′ − y′ + 4x^3y = 0$
assuming $x > 0$ and given that $y_1(x) = \sin(x^2)$ is a solution.</p>
| Jan Eerland | 226,665 | <p>HINT:</p>
<p>$$xy''(x)-y'(x)+4x^3y(x)=0\Longleftrightarrow$$</p>
<hr>
<p>Let $t=x^2$, which gives $x=\sqrt{t}$:</p>
<hr>
<p>$$y''(x)\sqrt{t}-y'(x)+4t^{\frac{3}{2}}y(x)=0\Longleftrightarrow$$</p>
<hr>
<p>Using the chain rule:</p>
<hr>
<p>$$-2\sqrt{t}y'(t)+\sqrt{t}\left(4ty''(t)+2y'(t)\right)+4t^{\frac{3}{2}}... |
3,882,308 | <p>I am starting to learn about rings and ideals in abstract algebra. I came across a textbook problem that I am having a lot of trouble solving:</p>
<blockquote>
<p>Prove that for any positive integer <span class="math-container">$n$</span> ending in <span class="math-container">$7$</span>, the ideal generated by <spa... | Bill Dubuque | 242 | <p><span class="math-container">$\!\!\bmod\overbrace{ 1\!+\!\sqrt{11}}^{\textstyle {\rm ideal}\ I}\!:\,\ \sqrt{11}\equiv -1\Rightarrow \overbrace{11\!\equiv\! \sqrt{11}^2\!\equiv 1}^{\textstyle \color{#80f}{10\equiv 0}}\ $</span> so <span class="math-container">$\ \overbrace{\color{#0a0}0\equiv 7\!+\!\color{#90f}{10}j}... |
2,454,317 | <p>Assume that $x$ and $y$ are both differentiable functions of t and find the required values of $\frac{dy}{dt}$ and $\frac{dx}{dt}$.</p>
<p>$$xy = 8$$</p>
<p>(a) Find $\frac{dy}{dt}$, given $x = 4$ and $\frac{dx}{dt} = 13$. </p>
<p>$$y=\frac{8}{x}\\
y'=-\frac{8}{x^2}\\
dy=-8dx $$</p>
<p>$dt=1$?</p>
<p>Can someon... | 5xum | 112,884 | <p>I'd avoid writing $dy=-8dx$, because in basic calculus, the expression $dy$ alone has no meaning - it must be part of a larger expression, like $\frac{dy}{dx}$, for it to make sense (and remember, $\frac{dy}{dx}$ is <strong>not a fraction</strong>).</p>
<p>That said, your mistake is differentiating the equation $y=... |
2,921,981 | <p>I know that the cube is the only 3 d shape which falls in polyhedrons but still is composed of squares, exclusively although its a even sided shape.
I have noticed that after square there is no single polyhedron which is exclusively made out of a polygon with even number of sides. For example the hexagon, which does... | Mike Earnest | 177,399 | <p>I assume you are talking about <em>regular</em> hexagons, etc.</p>
<p>You cannot make a polyhedron out of hexagons, septagons, or any larger regular polygon alone. The reason is because their angles are too big. If you try to fit three hexagons together meeting a vertex, they are forced to lie in the same plane bec... |
1,123,979 | <p>I am totally confused with the substitution method of evaluating indefinite integrals, especially those with rational functions and polynomials. I have 2 cases, which if I made to understand, would break ice for my future cases.</p>
<p>case 1: Evaluate: $\int x\sqrt{x+2}dx $</p>
<p>case 2: Evaluate: $\int \frac{x^... | Upax | 157,068 | <p>You can use exactly the same method. For instance use the substitution $2 x^2+3=u$. With this substitution you have
\begin{equation}
2 x dx = \frac{1}{2} du
\end{equation}
And
\begin{equation}
x dx = \frac{1}{4} du
\end{equation}
Then your integral becomes:
\begin{equation}
\int 3 u^5 \frac{1}{4} du =\frac{u^6}{8}
... |
334,325 | <p>The function $u(x,y)$ satisfies the partial differential equation</p>
<p>$$\nabla^{2}u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\text{ in }0<y<a, -\infty<x<\infty$$</p>
<p>and the boundary conditions $u \to 0$ as $x \to \pm\infty$, $\frac{\partial u}{\partial y}=0$ o... | Ron Gordon | 53,268 | <p>Your boundary conditions are</p>
<p>$$\frac{\partial \hat{u}}{\partial y}(k,0) = 0$$
$$\hat{u}(k,a) = \frac{2}{1+k^2}$$</p>
<p>The latter results from taking the FT of $e^{-|x|}$.</p>
<p>The general solution to the FT'ed equation is</p>
<p>$$\hat{u}(k,y) = A e^{k y} + B e^{-k y}$$</p>
<p>The first BC implies th... |
192,058 | <p>A covering design $(v,k,t)$ is a family of subsets of $[v]$ each having $k$ elements such that given any subset of $[v]$ of $t$ elements it is a subset of one of the sets of the family. A problem is to find the minimum number of subsets such a family can have.</p>
<p>I am interested in the case $(v,k,2)$. It seems ... | Yuichiro Fujiwara | 27,829 | <p><strong>Edit:</strong> <strong><em>The possible "gap" of sort in Caro and Yuster's proof of their upper bound has just been fixed!</em></strong> <em>See Ben Barber's comment below (and his joint paper with Daniela Kühn, Allan Lo and Deryk Osthus</em> <a href="http://arxiv.org/pdf/1410.5750v1.pdf" rel="nofollow"><em>... |
2,275,203 | <p>I'm having trouble to prove that the intersection of the symmetric matrices , the set of projections (matrices who $M^2 = M$) and the matrices with constant rank $k$ is a manifold. Can anyone help me? Unfortunately I could not make much progress.</p>
<p>In other words I need to prove the following proposition: Let ... | Elle Najt | 54,092 | <p>A symmetric projection matrix is the same as an orthogonal projection onto its image. (If $v \in im T ^{\perp}$ ,then $<w,Tv> = <Tw,v> = 0$ for all $w$, so $Tv = 0$. If $Tv = 0$ for some $v$, the same computation shows it is orthogonal to the image.) If you fix the rank to be $k$, you can identify your m... |
2,275,203 | <p>I'm having trouble to prove that the intersection of the symmetric matrices , the set of projections (matrices who $M^2 = M$) and the matrices with constant rank $k$ is a manifold. Can anyone help me? Unfortunately I could not make much progress.</p>
<p>In other words I need to prove the following proposition: Let ... | Ben Grossmann | 81,360 | <p>It suffices to note that the function $f:\mathrm{Sym}_{n \times n} \to \Bbb R^2$ given by
$$
f(X) = (\operatorname{tr}([X^2 - X]^T[X^2 - X]),\operatorname{tr}(X))
$$
is smooth (since the components are polynomials on matrix-entries) and regular. So, the set $\{X : f(X) = (0,k)\}$ is a manifold.</p>
|
32,379 | <p>How do I replace content inside a Held expression without local values coming through?
For example if you run the following code you get <code>g[2]</code> instead of <code>g[x]</code>(the intended form).</p>
<pre><code>x = 2;
Replace[HoldComplete[g["x"]], {
y_String :> With[{
eval = Symbol[y]},
eval /... | William | 5,615 | <p>The following function <code>ReplaceC</code> evaluates the <code>Replace</code> in an alternative context and the strips the context back from the returned value.</p>
<pre><code>ReplaceC[data_, rules_] := ToExpression@Replace[
ToBoxes[
With[{body =
ReleaseHold@Replace[HoldComplete@MakeBoxes[
... |
3,535,725 | <p>I have started working on this <span class="math-container">$\int_0^1 \frac{x^2}{\sqrt{1+x^5}} \, dx \leq \frac{1}{3}$</span> by using the fact that <span class="math-container">$\int_0^1 \frac{x^2}{\sqrt{1+x^5}} \, dx < \int_0^1 \frac{x^2}{\sqrt{x^5}} \, dx$</span>
But that didn't work.</p>
| Michael Hardy | 11,667 | <p><span class="math-container">$$
\int_0^1 \frac{x^2}{\sqrt{1+x^5}} \, dx \le \int_0^1 x^2 \, dx.
$$</span></p>
|
4,509,183 | <p>The theorem says "<strong>If the function <span class="math-container">$f:\Bbb{R}^n→\Bbb{R}$</span> has a local extremum at <span class="math-container">$α∈\Bbb{R}^n$</span>, then <span class="math-container">$α$</span> is a critical point</strong>".<br />
For <span class="math-container">$n=1$</span>, its... | on1921379 | 805,886 | <p>If we suppose that <span class="math-container">$f: \mathbb{R}^n \to \mathbb{R}$</span> is differentiable and has a local extremum in <span class="math-container">$\mathbf{v} = (v_1, \ldots, v_n)$</span>, then for each <span class="math-container">$1 \leq i \leq n$</span>, you can define <span class="math-container"... |
4,357,325 | <p>I came across this question under summation:</p>
<p>Find the sum: <span class="math-container">$$\sum_{r=0}^n \left[\frac{r}{n} - \alpha \right]^2 {n \choose r}x^r(1-x)^{n-r} $$</span></p>
<p>To start with this, I wrote it as <span class="math-container">$\sum_{r=0}^n \left[\frac{r-n\alpha}{n} \right]^2 {n \choose ... | Manatee Pink | 651,809 | <p>I can also verify that your approach and results are correct.</p>
|
2,898,125 | <blockquote>
<p>I have the following initial value problem:
$$\tag{IVP}\label{IVP}
\begin{cases}
x^\prime(t) = -\sqrt[3]{x}\\
x(0) = x_0
\end{cases}
$$
and I have to show that for every $x_0 \in \mathbb R$ a solution to \eqref{IVP} exists and it is unique. </p>
</blockquote>
<p>Existence is not an issue, as the... | Delta-u | 550,182 | <p>The problem is that the definition of $\sqrt[3]{\cdot}$ is ambiguous, if $x>0$ then there is no problem, i.e this is $x^\frac{1}{3}$, but for $x<0$ you can not use such formula so I suppose the definition is:
$$\sqrt[3]{x}=sgn(x) |x|^\frac{1}{3}$$
in particular the antiderivative for $x <0$ is not $\frac{3}... |
1,773,328 | <p>$\sum_{n=1} ^\infty \frac{1}{2^n}$ converges to $1$, but it does so at a much faster rate than I'd like: $1/2,3/4,7/8...$. In other words, after only a few terms we are really close to $1$ (for my taste).</p>
<p>Can you give an example of a series that congerges to $1$ very slowly, so that the partial sums approac... | Robert Israel | 8,508 | <p>Take any sequence $s_n$ such that $s_n \to 1$ as slowly as you want, with $s_0 = 0$. Let $a_n = s_n - s_{n-1}$. Then the partial sums $\sum_{i=1}^n a_i = s_n$.</p>
|
1,773,328 | <p>$\sum_{n=1} ^\infty \frac{1}{2^n}$ converges to $1$, but it does so at a much faster rate than I'd like: $1/2,3/4,7/8...$. In other words, after only a few terms we are really close to $1$ (for my taste).</p>
<p>Can you give an example of a series that congerges to $1$ very slowly, so that the partial sums approac... | Vincent | 332,815 | <p>$\sum_{n=1}^{\infty} \frac{1}{n^{\alpha}}$ converges for any $\alpha > 1$ to (let us note it) $l_{\alpha}$. Take $\frac{1}{l_{\alpha}}\sum_{n=1}^{\infty} \frac{1}{n^{\alpha}}$.</p>
<p>$ \sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$ also converges (even slowlier than the example above).</p>
<p>You won't have any nice... |
98,839 | <pre><code>I want to visualize Ricci flow solution on the following sphere
</code></pre>
<p>Let $r> 0$ </p>
<p>$L = \{ (x cos \theta, x sin \theta, x) | r < x < R \}$</p>
<p>$S$ : $(z-\sqrt{2} r)^2 + x^2 + y^2 = r^2$ </p>
<p>$T$ : $ (z- \sqrt{2} R)^2 + x^2 + y^2 = R^2$ </p>
<p>If $R$ is sufficiently la... | Dan Lee | 1,179 | <p>Ricci flow is an intrinsic flow, so (unless you were to somehow recast it as an extrinsic flow) it doesn't make sense to talk about what happens to surfaces in $\mathbb{R}^3$ under Ricci flow.</p>
|
27,679 | <p>I have a graph here and I want to be able to manipulate the 4 parameters that are in the Block command. I want to watch the graph change as I change the parameters. </p>
<p>I've done this before, but the Block command that Mathematica suggested throws me off a little bit. </p>
<p>Here is the code: </p>
<pre><cod... | Gregory Klopper | 115 | <p>Hope this is what you wanted to accomplish. Good luck!</p>
<pre><code>Manipulate[
test = Reap[
NDSolve[{x''[
t] == ε (1 - x[t]^2) x'[t] - ω0^2*x[t] +
A*Sin[ωf*t], x[0] == 1, x'[0] == 0,
WhenEvent[Mod[t, 2 π/ωf] == 0,
Sow[{x[t], x'[t]}]]}, {}, {t, 0, 10000},
MaxSteps -> ... |
1,257,144 | <p>Is it true that $ f(\bar{z})=\overline {f(z)}$,
Where z is complex?</p>
<p>I think it holds when $f(z)$ is holomorphic since we have $f(z)=p(x,y)+iq(x,y)=p(z,0)+iq(z,0)$
Any help...</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Use that $f$ is locally a power series:
$$f(z) = \sum_{n=0}^\infty a_n(z-c)^n,$$
$${f(\bar{z})} = \sum_{n=0}^\infty a_n(\bar{z}-c)^n,$$
$$\overline{f(z)} = \sum_{n=0}^\infty \bar{a_n}(\bar{z}-\bar{c})^n.$$
(Why $\overline{\sum\cdots} = \sum\overline\cdots$ ?)</p>
|
3,325,953 | <p>Once you have reached perhaps 10 decimal places, calculators can make rounding errors and so on. Is it possible to build a calculator that makes none of these errors? For example, it could work out each decimal place of an irrational number – as you click a button it gives, say, 10 more digits. (Obviously, it wouldn... | NoChance | 15,180 | <p>There are several issues with calculations made in software:
1-Programming inaccuracy (e.g. choice of rounding/truncation point, wrong use of floating point arithmetic, etc.)
2-Limited Precision in a given language/machine
3-Compounded errors resulting from a sequence of applied operations on a number where each ope... |
9,672 | <p>This might be a somewhat silly and inconsequential question, but it's aroused my curiosity. One has the theorem in commutative algebra that the integral closure of a domain $A$ in its field of fractions $Q(A)$ is the intersection of all the valuation subrings of $Q(A)$ containing $A$. This naturally leads to the imp... | Felipe Voloch | 2,290 | <p>Suppose X is a curve. Then, wlog, assume X smooth and projective, since we only care about its function field. If R is minimal such that it is integrally closed in k(X),and k(X) is algebraic over the field of fractions Q(R), then R is the coordinate ring of the complement of a point in X and, in particular, Q(R) = k... |
9,672 | <p>This might be a somewhat silly and inconsequential question, but it's aroused my curiosity. One has the theorem in commutative algebra that the integral closure of a domain $A$ in its field of fractions $Q(A)$ is the intersection of all the valuation subrings of $Q(A)$ containing $A$. This naturally leads to the imp... | Will Sawin | 18,060 | <p>I understand this answer might be coming a little late, but the stack exchange "Related Questions" algorithm brought me here to a question that does not yet have a complete answer, so I'll answer it.</p>
<p>1) For such an $X,R$, $k(X)$ is indeed equal to $Q(R)$. The reason is that every element $\alpha$ of $k(X)$ i... |
271,252 | <p>Let $R$ be a local UFD of Krull dimension 2. Let $a\in R$ be a nonzero, non-unit. I am trying to show that the ring $R[1/a]$ is a principal ideal domain. Does anyone have any suggestions as to how this can be done?</p>
| Community | -1 | <p>The set of prime ideals of $R$ consists of</p>
<ul>
<li>A single ideal of height 0: $(0)$</li>
<li>One or more ideals of height 1, all principal</li>
<li>Exactly one ideal of height 2</li>
</ul>
<p>The set of prime ideals of $R[1/a]$ consists of the (extensions of) prime ideals of $R$ that do not contain $a$....</... |
2,208,814 | <p>Let $X$ be a nonempty set. Fix two metrics $d: X\times X \to [0,1]$ and $d^\prime: X\times X \to [0,1]$ such that the topology $\tau$ generated by $d$ is finer than the topology $\tau^\prime$ generated by $d^\prime$, i.e.,
$
\tau^\prime \subseteq \tau.
$</p>
<blockquote>
<p><strong>Question.</strong> Is it true t... | martin.koeberl | 52,354 | <p>No. Here is an example where the metrics even induce the same topology:</p>
<p>Let $X=2^\omega$ (i.e., all functions $\mathbb N\rightarrow\{0,1\}$) where the space is endowed with the product topology where $\{0,1\}$ has the discrete topology (this is the topology generated by the basic open sets $N_s=\{x\in 2^\ome... |
2,208,814 | <p>Let $X$ be a nonempty set. Fix two metrics $d: X\times X \to [0,1]$ and $d^\prime: X\times X \to [0,1]$ such that the topology $\tau$ generated by $d$ is finer than the topology $\tau^\prime$ generated by $d^\prime$, i.e.,
$
\tau^\prime \subseteq \tau.
$</p>
<blockquote>
<p><strong>Question.</strong> Is it true t... | Community | -1 | <blockquote>
<p>Suppose that your claim were true and that, in addition to your hypothesis:</p>
<ol>
<li><span class="math-container">$(X,d')$</span> is a non-compact bounded metric space such that <span class="math-container">$\overline{(X,d')}$</span> is compact</li>
<li><span class="math-container">$(X,d)$</span> is... |
2,427,747 | <p>I want to prove that $2^{n+2} +3^{2n+1}$ is divisible by $7$ for all $n \in \mathbb{N}$ using proof by induction.</p>
<p>Attempt</p>
<p>Prove true for $n = 1$</p>
<p>$2^{1+2} + 3^{2(1) +1} = 35$</p>
<p>35 is divisible by 7 so true for $n =1$</p>
<p><em>Induction step</em>: Assume true for $n = k$ and prove true... | ymbirtt | 85,606 | <p>You mentioned that modular arithmetic would be to complex, but that's the approach I'd take and I actually found it fairly straightforward, so here's an alternative solution. We're working modulo 7, so there can't be any more than 7 different answers for $k^n$ that we care about. That's small enough for an exhaustiv... |
344,119 | <p>$$\int_{0}^{\pi/2}\int_{x}^{\pi/2}\frac{\cos y}{y} \, \operatorname{d}\!y\, \operatorname{d}\!x$$</p>
<p>$\iint_R2xy^2 \, \operatorname{d}\!A$ where R is the right half of the unit circle</p>
| Mikasa | 8,581 | <p>$$\iint_R2xy^2dA=\iint_{R=\{(x,y)\mid x^2+y^2=1,x\geq0\}}2xy^2dxdy=\int_0^12x\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}y^2dy\\=\int_0^12r\cos(\theta)\int_{-\pi/2}^{\pi/2}r^2\sin^2(\theta)rdrd\theta=\int_0^12r^4dr\int_{-\pi/2}^{\pi/2}\cos(\theta)\sin^2(\theta)d\theta$$</p>
|
3,764,423 | <p>Let <span class="math-container">$X$</span> be a set and show that the following function defines a metric.
<span class="math-container">$f(x, y) = (0 \text{ if } x = y \text{ and } 1 \text{ if } x \neq y)$</span></p>
<p>I'm especially having trouble with the symmetry and triangle inequality steps. Thanks so much!</... | hamam_Abdallah | 369,188 | <p>Put your sum as</p>
<p><span class="math-container">$$\frac{b-a}{n}\sum_{i=1}^nf(a+i\frac{b-a}{n})$$</span></p>
<p>If <span class="math-container">$ f $</span> is integrable or continuous at <span class="math-container">$ [a,b]$</span>, the limit will be
<span class="math-container">$$\int_a^bf(x)dx$$</span></p>
<p>... |
1,104,728 | <p>I want to determine the splitting field, galois group and intermediate fields of the polynomial $f(X)=(X^2+12)(X^3-5)\in\mathbb Q[X]$.</p>
<p>I want to obtain the splitting field by adjoining the roots of the polynomial to $\mathbb Q$ which is $\sqrt[3]{5}$ for $(X^3-5)$ but I don't understand how to determine the ... | Xipan Xiao | 78,637 | <p>Consider n=2 and let $X=(x,y,z)/r^3$ where $r=\sqrt{x^2+y^2+z^2}$, $X$ is a vector field defined on $\mathbb{R}^3-\{0\}$. It's easy to compute
$$\operatorname{div}(X)=\sum\frac{1\cdot r^3-x\cdot3r^2\cdot r_x}{r^6}$$
$$=\sum\frac{r^3-x\cdot 3r^2 (x/r)}{r^6}$$
$$=\sum\frac{r^3-3rx^2}{r^6}$$
$$=\frac{3r^3}{r^6}-\frac{... |
2,129,389 | <p>I'm currently working on reducing the summation on the first line. The three lines below show the steps of the reduction according to the instructor. Unfortunately I'm not following the logic. Can someone help explain the algebraic transformations between each line?</p>
<p>$$\sum_{i=0}^{n-2} n-i-1$$ </p>
<p>$$=n(n... | Simply Beautiful Art | 272,831 | <p>We first split the sum:</p>
<p>$$\sum_{i=0}^{n-2}(n-i-1)=\color{#4488dd}{\sum_{i=0}^{n-2}(n-1)}-\color{#cc5500}{\sum_{i=0}^{n-2}i}$$</p>
<p>One can then factor:</p>
<p>$$\sum_{i=0}^{n-2}(n-1)=(n-1)\sum_{i=0}^{n-2}1$$</p>
<p>and</p>
<p>$$\sum_{i=0}^{n-2}1=\underbrace{1+1+1+\dots+1}_{n-1}=n-1$$</p>
<p>And so,</p... |
245,467 | <p>I am trying to connect a set of SystemModels that I generated programmatically in Mathematica. I am interested in MultiBody models and I haven't tried other types of models. I am failing gloriously and I have spent a good part of the day reading the online help pages and performing a lot of trial and error. Here's a... | NCSNY | 19,431 | <p>Resolution is with cross post here: <a href="https://community.wolfram.com/groups/-/m/t/2259002" rel="nofollow noreferrer">https://community.wolfram.com/groups/-/m/t/2259002</a></p>
|
245,467 | <p>I am trying to connect a set of SystemModels that I generated programmatically in Mathematica. I am interested in MultiBody models and I haven't tried other types of models. I am failing gloriously and I have spent a good part of the day reading the online help pages and performing a lot of trial and error. Here's a... | user74549 | 74,549 | <p>I posted the same question on the Wolfram forum. Neil Singer set me on the straight and narrow as seen in the link posted above by NCSNY. My mistake was improper instantiation of the models. This is how I solved it, following Neil's advice.</p>
<p>First model</p>
<pre><code>comp1 = {"c11" \[Element]
&q... |
2,693,430 | <p>Problem:</p>
<p>$$
\begin{align}
5y+w=1 \\
2x+5y-4z+w=1
\end{align}
$$</p>
<p>What I've done:
-1 times first equation in order to get rid of $5y$ and $w$.
Then, when I add first and second equation, the equation becomes $2x-4z=0$
And I'm stuck right here.</p>
<p>Thanks.</p>
| Siong Thye Goh | 306,553 | <p>To show whether it is consistent, you just have to exhibit a solution.</p>
<p>Let $w=1$ and let the other variables be $0$.</p>
|
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