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 |
|---|---|---|---|---|
265,509 | <p>Given a (possibly disjoint) region, defined by a discrete set of points, how can I use <code>ListContourPlot[]</code> together with <code>Mesh</code> to highlight a specific area of the plot? For instance, how can I mesh the region where the points are smaller than a certain value?</p>
<p>Here I construct a minimal ... | bmf | 85,558 | <p>Excellent replies as usual by @cvgmt and @Bob Hanlon, but I'd like to give it a go using <code>Piecewise</code></p>
<p>With</p>
<pre><code>data = Table[Exp[x^2 - y^2], {x, -1, 1, .01}, {y, -1, 1, .01}];
</code></pre>
<p>The following works nicely I think</p>
<pre><code>ListContourPlot[data,
Contours -> {1.0},
C... |
2,698,903 | <p>So this has caused some confusion. In one exercise one was asked to prove that</p>
<p>$$\lim_{k \rightarrow \infty} \int_A \cos(kt)dt=0$$</p>
<p>where $A \subset [-\pi, \pi]$ is a measurable set.</p>
<p>My initial idea was to take any $a,b \in A$ and then show that:</p>
<p>$$\lim_{k \rightarrow \infty} \int_a^b ... | copper.hat | 27,978 | <p>Let $f=1_A$, and note that $f \in L^2[-\pi,\pi]$.</p>
<p>Let $\hat{f}_n = {1 \over 2 \pi} \int_{-\pi}^\pi f(t) e^{-int} dt$.</p>
<p>Bessel's inequality gives
$\sum_n |\hat{f}_n|^2 \le \|f\|_2^2$, and so $\hat{f}_n \to 0$.</p>
<p>Another approach would be the Riemann Lebesgue lemma.</p>
|
1,545,045 | <p>Is it possible to cut a pentagon into two equal pentagons?</p>
<p>Pentagons are not neccesarily convex as otherwise it would be trivially impossible.</p>
<p>I was given this problem at a contest but cannot figure the solution out, can somebody help?</p>
<p>Edit: Buy "cut" I mean "describe as a union of two pentag... | Empiromancer | 164,385 | <p>Imagine pairing people off one group at a time. For the first group, you've got $28$ choose $2$ possible ways to pick a team of $2$ from $28$ people. After you've selected that team, you've got to pick another team of $2$ from the remaining $26$ people, so $26$ choose $2$. Continue on in that manner until you've ass... |
109,973 | <p>I have stuck solving this problem of financial mathematics, in this equation:
$$\frac{(1+x)^{8}-1}{x}=11$$</p>
<p>I'm stuck in this eight grade equation:
$$x^{8}+8x^{7}+28x^{6}+56x^{5}+70x^{4}+56x^{3}+28x^{2}+9x-11=0$$</p>
<p>But I cannot continue past this. This quickly lead me to find a general way of solving eq... | bgins | 20,321 | <p>In financial mathematics, you're looking at a problem of <a href="http://en.wikipedia.org/wiki/Compound_interest" rel="nofollow">compound interest</a> rates (over $\mathbf{n}$ <strong>compoundment periods</strong>), where $100x$ is the percentage interest rate or yield and $\mathbf{x}$ the relative accrual rate <str... |
109,973 | <p>I have stuck solving this problem of financial mathematics, in this equation:
$$\frac{(1+x)^{8}-1}{x}=11$$</p>
<p>I'm stuck in this eight grade equation:
$$x^{8}+8x^{7}+28x^{6}+56x^{5}+70x^{4}+56x^{3}+28x^{2}+9x-11=0$$</p>
<p>But I cannot continue past this. This quickly lead me to find a general way of solving eq... | NoChance | 15,180 | <p>I don't know of a universal method to solve any equation. If you have the luxury to choose any method to solve an equation, and you are not particularly interested in advanced math, you could try to plot the equation either by hand or use free software on the net. to give you at least a starting solution.</p>
<p>Fo... |
2,683,788 | <blockquote>
<p>An urn contains $nr$ balls numbered $1,2..,n$ in such a way that $r$
balls bear the same number $i$ for each $i=1,2,...n$. $N$ balls are drawn at random without replacement. Find the probability that exactly $m$ of the numbers will appear in the sample.</p>
</blockquote>
<p>Any hints would be grea... | Parcly Taxel | 357,390 | <p>The "trick" of the question's title is probably in specifying the numbers of socks as pairs: Rihanna really has 18 blue and 16 grey socks with no distinction between "left" and "right" of a pair. Now the probability she picks up at least two blue socks in her selection of three <em>without replacement</em> is given ... |
2,294,585 | <p>I can't figure out why $((p → q) ∧ (r → ¬q)) → (p ∧ r)$ isn’t a tautology. </p>
<p>I tried solving it like this: </p>
<p>$$((p ∧ ¬p) ∨ (r ∧ q)) ∨ (p ∧ r)$$
resulting in $(T) ∨ (p ∧ r)$ in the end that should result in $T$. What am I doing wrong?</p>
| Community | -1 | <p>Using $a\to b=\lnot a\lor b$,</p>
<p>$$\lnot((\lnot p\lor q)\land(\lnot r\lor \lnot q))\lor(p\land r)$$
$$((p\land \lnot q)\lor(r\land q))\lor(p\land r)$$
$$(p\land(\lnot q\lor r))\lor(r\land q)$$</p>
<p>is falsified with all false.</p>
<hr>
<p>Using bitwise operations and the hexadecimal representation,</p>
<p... |
3,911,297 | <p>Chapter 12 - Problem 26)</p>
<blockquote>
<p>Suppose that <span class="math-container">$f(x) > 0$</span> for all <span class="math-container">$x$</span>, and that <span class="math-container">$f$</span> is decreasing. Prove that there is a <em>continuous</em> decreasing function <span class="math-container">$g$</... | Misha Lavrov | 383,078 | <p>Well, we could the same things more directly: write <span class="math-container">$a$</span> as <span class="math-container">$\frac{(a^{12})^3}{(a^7)^5} = \frac{a^{36}}{a^{35}}$</span>, or as <span class="math-container">$\frac{(a^7)^7}{(a^{12})^4} = \frac{a^{49}}{a^{48}}$</span>, and conclude that it's in <span clas... |
825,848 | <p>I'm trying to show that the minimal polynomial of a linear transformation $T:V \to V$ over some field $k$ has the same irreducible factors as the characteristic polynomial of $T$. So if $m = {f_1}^{m_1} ... {f_n}^{m_n}$ then $\chi = {f_1}^{d_1} ... {f_n}^{d_n}$ with $f_i$ irreducible and $m_i \le d_i$.</p>
<p>Now I... | xbh | 514,490 | <h3>Motivation</h3>
<p><span class="math-container">$ \newcommand{\proofstart}{\mathbf{Proof.}\blacktriangleleft} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\lcm}{lcm}
\newcommand{\bm}{\boldsymbol}
\newcommand {\F}{\Bbb F} $</span>
I was doing a problem and I need the claim in the OP, so i tried to prove this... |
1,139,723 | <p>Given the differential equation $\frac{dy}{dt}+a(t)y=f(t)$ </p>
<p>with a(t) and f(t) continuous for:</p>
<p>$-\infty<t<\infty$</p>
<p>$a(t) \ge c>0$</p>
<p>$lim_{t_->0}f(t)=0$</p>
<p>Show that every solution tends to 0 as t approaches infinity.</p>
<p>$$\frac{dy}{dt}+a(t)y=f(t)$$</p>
<p>$$μ(t)=e^... | Winther | 147,873 | <p>I will give a solution for the case $f \geq 0$. First we note that the solution can be written (since $y'+ay = f \implies (ye^{\mu})' = fe^{\mu}$ and $y(0)=0$)</p>
<p>$$y(t) = \int_0^t f(t')e^{\mu(t')-\mu(t)} dt'$$</p>
<p>where $\mu(t) = \int_0^t a(t')dt'$. This shows that $y(t)\geq 0$ for all $t\geq 0$. Now using... |
1,845,448 | <p>I know that $(AB)^T$ = $B^TA^T$ and that $(A^T)^{-1}= (A^{-1})^T$ but couldn't reach any convincing answer. Can someone demonstrate the expression.</p>
| GAVD | 255,061 | <p>One way to see, you can compare entries of two matrices $(AB)^t$ and $B^tA^t$.</p>
<p>Firstly, we have $$AB_{ij} = \sum_{k} a_{ik}b_{kj}.$$</p>
<p>When to transpose a matrix, we switch the rows and the columns.
$$(AB)^t_{ij} = AB_{ji} = \sum_{k} a_{jk}b_{ki}.$$</p>
<p>Now, let see the entry ij of $B^tA^t$:</p>
... |
3,843,352 | <p>It’s called a Diophantine Equation, and it’s sometimes known as the “summing of three cubes”.</p>
<blockquote>
<p>A Diophantine equation is a polynomial equation, usually in two or
more unknowns, such that only the integer solutions are sought or
studied (an integer solution is such that all the unknowns take
intege... | poetasis | 546,655 | <p>Using positive integers only for <span class="math-container">$x,y,z$</span> there are no solutions for <span class="math-container">$k<3$</span> but here are <span class="math-container">$3\le k \le 100$</span>. The use of non-positive integers follows.</p>
<pre><code> (x,y,z,k)
(1,1,1,3)
(1,1,2,10)
(1,2,2,17... |
175,775 | <p>To clarify the terms in the question above:</p>
<p>The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set $X \subseteq \Omega$ is minimal if no proper subset of $X$ generates Sym($\Omega$).</p>
<p>This might be a dif... | Igor Rivin | 11,142 | <p>The canonical paper on the subject (don't be fooled by the publication date, it had been around for years before then) is George Bergman's gem:</p>
<p><em>Bergman, George M.</em>, <a href="http://dx.doi.org/10.1112/S0024609305018308" rel="nofollow noreferrer"><strong>Generating infinite symmetric groups.</strong></... |
1,711,673 | <p>Why are the inverses of 2-cycles of the symmetric permutation group $S_3$ the elements themselves rather than their reverses?</p>
<p>E.g. why is $(1,2)^{-1}=(1,2)$ and not $(2,1)$?</p>
| C. Dubussy | 310,801 | <p>Why not simply extending continuously the function at $x=0$ ? Thanks to l'Hospital's rule, we have $$\lim_{x \to 0}\frac{1-(\cos(x))^3}{x^2}=\lim_{x \to 0}\frac{3\cos^2(x)\sin(x)}{2x}=\frac{3}{2}.$$</p>
|
1,711,673 | <p>Why are the inverses of 2-cycles of the symmetric permutation group $S_3$ the elements themselves rather than their reverses?</p>
<p>E.g. why is $(1,2)^{-1}=(1,2)$ and not $(2,1)$?</p>
| Claude Leibovici | 82,404 | <p>If you want to compute the value of the expression when $x$ is very close to $0$, just use Taylor series of $\cos(x)$ to get $$\frac{1-\cos^3(x)}{x^2}=\frac{3}{2}-\frac{7 x^2}{8}+\frac{61 x^4}{240}+O\left(x^5\right)$$ which shows the limit and how it is approched.</p>
<p>For example, if $x=10^{-8}$, the above formu... |
4,274,044 | <p>I am trying to prove the following result.</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be sets and <span class="math-container">$\sim$</span> an equivalence relation on <span class="math-container">$X$</span>. Let <span class="math-container">$f: X \to ... | SV-97 | 960,107 | <p>I think a lot of your pain points can be remedied by recognizing that we often times identify equivalence classes by representatives. This makes <span class="math-container">$\pi : X \to (X/\sim) \simeq X$</span> at which point you can apply the projection definition you know (up to isomorphism).</p>
<p>By choosing ... |
2,707,299 | <p>Suppose that the conditional distribution of $Y$ given $X=x$ is Poisson with mean $E(Y|x)=x$, $Y|x\sim POI(x)$, and that $X\sim EXP(1)$</p>
<p>Find $V[Y]$</p>
<p>We know that</p>
<p>$$V[Y]=E_x[Var(Y|X)] + Var_x[E(Y|X)]$$</p>
<p>we also know that </p>
<p>$$V(Y|x)=E(Y^2|x)-[E(Y|x)]^2$$</p>
<p>I know we can find ... | userNoOne | 518,028 | <p>We have been given that $Y|x$ is a Poisson Variate with parameter $x$. We know that, in Poisson distribution the mean and variance are equal to its parameter. Hence, the following will hold:</p>
<p>$E[Y|x]=V[Y|x]=x$.</p>
<p>Now, to obtain the variance of $Y$,</p>
<p>$V(Y)=E(V[Y|x])+V(E[Y|x]) = E(x)+V(x)$</p>
<p>... |
507,975 | <p>I'm new here and unsure if this is the right way to format a problem, but here goes nothing. I'm currently trying to solve an inequality proof to show that $n^3 > 2n+1$ for all $n \geq 2$.</p>
<p>I proved the first step $(P(2))$, which comes out to $8>5$, which is true.</p>
<p>In the next step we assume that... | Arash | 92,185 | <p>First observe that if $u=\cos x$: </p>
<p>$$
\lim_{x\to \pi/2} (\sec x \tan x)^{\cos x}=\lim_{x\to \pi/2} \left(\frac{1}{\cos^2x}\right)^ {\cos x}=\lim_{u\to 0} \left(\frac{1}{u^2}\right)^ {u}
$$
Now just assume that $L=\lim_{u\to 0} \left(\frac{1}{u}\right)^ {2u}$. You get:
$$
\ln L=\lim_{u\to 0} 2u\ln\left(\frac... |
1,678,922 | <p>let B be a commutative unital real algebra and C its complexification
viewed as the cartesian product of B with itself.
If M is a maximal ideal in A, is the cartesian product of M with itself
a maximal ideal in C? </p>
| Eric Wofsey | 86,856 | <p>Not necessarily. Try taking $B=\mathbb{C}$ and $M=0$.</p>
|
1,133,581 | <p>$\lim\limits_{(x,y) \to (0,0)} \frac{\log(1-x^2y^2)}{x^2y^2}$</p>
<p>in polar coordinates:</p>
<p>$\lim\limits_{r \to 0} \frac{\log(1-r^2\cos^2t \sin^2t)}{r^2\cos^2t \sin^2t}$</p>
<p>The first limit exists if the second is independent from $t$. But if $t$ is $k \pi/2$ that fraction does not exist. So I could argu... | Qidi | 212,981 | <p>You can take the open cover $\{(0+\frac{1}{n}, 1-\frac{1}{n})\mid n>2\} $. It has no finite subcover because if it does simply take the largest $n$ and $\frac{1}{2n}$ is not included.</p>
|
1,133,581 | <p>$\lim\limits_{(x,y) \to (0,0)} \frac{\log(1-x^2y^2)}{x^2y^2}$</p>
<p>in polar coordinates:</p>
<p>$\lim\limits_{r \to 0} \frac{\log(1-r^2\cos^2t \sin^2t)}{r^2\cos^2t \sin^2t}$</p>
<p>The first limit exists if the second is independent from $t$. But if $t$ is $k \pi/2$ that fraction does not exist. So I could argu... | graydad | 166,967 | <p>Since you have the real analysis tag, I'm assuming you are using the standard topology on $\Bbb{R}$, so open sets look like intervals of the form $(a,b)$. An open cover of $(0,1)$ could look something like $$\bigcup_{i=1}^\infty \left(0,1-\frac{1}{i} \right)$$ But you need to verify this is an open cover. That is, s... |
42,016 | <p>I am looking for algorithms on how to find rational points on an <a href="http://en.wikipedia.org/wiki/Elliptic_curve">elliptic curve</a> $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, how to find solutions in which the numerators and denominator... | Charles Matthews | 6,153 | <p>It is misleading to say "algorithm", really. There are probabilistic algorithms, and then hard cases (evidence of a point of infinite order that is hard to find). See for example <a href="http://www.jstor.org/pss/2152939" rel="nofollow">http://www.jstor.org/pss/2152939</a> . I think of Andrew Bremner as one of the e... |
47,181 | <p>I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure that the connected graph has a unique topology in 3-space. More specifically, I'm interested in insuring that some ... | Roland Bacher | 4,556 | <p>A matrix realizing Colin de Verdi`eres $\mu$-invariant yields an answer
if you accept that you get lengths at the end, not at the beginning.</p>
<p>A planar graph $G$ arising as the $1-$skeleton of a polytope has always $\mu$-invariant
equal to $3$. There exists thus a combinatorial Schr\"odinger operator on
$G$ wh... |
1,920,776 | <p>As I know there is a theorem, which says the following: </p>
<blockquote>
<p>The prime ideals of $B[y]$, where $B$ is a PID, are $(0), (f)$, for irreducible $f \in B[y]$, and all maximal ideals. Moreover, each maximal ideal is of the form $m = (p,q)$, where $p$ is an irreducible element in $B$ and $q$ is an irred... | Ben Grossmann | 81,360 | <p>I'll stick to 8-bit quantities, but the same applies in general.</p>
<p>The key to understanding two's complement is to note that we have a set of finitely many (in particular, <span class="math-container">$2^8$</span>) values in which there is a sensible notion of addition by <span class="math-container">$1$</span>... |
2,954,991 | <p>How do I prove that <span class="math-container">$\forall p, b, c \in \mathbb{Z}^+, (\text{Prime}(p) \wedge p^2 + b^2 = c^2) \Rightarrow p^2 = c+b$</span> ?
I know this much;
<span class="math-container">$$p^2 + b^2 = c^2\\
p^2= c^2 - b^2\\
p^2=(c-b)(c+b)
$$</span>
I know the next part has something to do with the ... | rtybase | 22,583 | <p><strong>Part 1.</strong> It's easy to see that <span class="math-container">$p > c-b$</span>, otherwise, if we assume <span class="math-container">$p \leq c-b \Rightarrow p+2b\leq c+b$</span> and <span class="math-container">$p(p+2b)\leq(c-b)(c+b)=p^2$</span>, which is possible only when <span class="math-contain... |
1,624,888 | <h2>Question</h2>
<p>In the following expression can $\epsilon$ be a matrix?</p>
<p>$$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + \epsilon^2 E_2 |m_2\rangle + \dots) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) $... | BenSmith | 238,248 | <p>$\epsilon$ is not generally used as a matrix. When matrices are needed, as you will notice from your example, epsilon is usually set beside the matrices of concern as a scalar which indicates that one is infinitesimally close to the origin (or zero) in whichever space you seek to be perturbing. </p>
|
202,043 | <p>A <a href="http://en.wikipedia.org/wiki/Square-free_word">square-free word</a>
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free word has length $3$.
But for alphabets of three symbols,... | Bjørn Kjos-Hanssen | 4,600 | <p><em>Also not an answer, but may be useful.</em></p>
<p>A somewhat similar kind of word is mentioned at the end of section 1.5 of <a href="http://www.math.ucla.edu/~pak/hidden/Courses/Salomaa-Ch1.pdf">Salomaa: <em>Jewels of Formal Language Theory</em></a>:</p>
<p>There is an infinite word over a 3-letter alphabet s... |
3,299,072 | <p>I want to prove that
<span class="math-container">$$\frac{(x-y)(x-z)}{x^2}+\frac{(y-x)(y-z)}{y^2}+\frac{(z-x)(z-y)}{z^2}\geq 0$$</span>
for positive numbers <span class="math-container">$x,y,z$</span>.
I don't know how to even begin. I must say I'm not 100% certain the inequality always holds.</p>
<p>I tried the so... | Mahdi Aghaee | 384,279 | <p>Notice, that
<span class="math-container">$$f(x,\,y,\,z)=\frac{(x-y)(x-z)}{x^2}+\frac{(y-x)(y-z)}{y^2}+\frac{(z-x)(z-y)}{z^2}= \frac{z^2y^2(x-y)(x-z)+x^2z^2(y-x)(y-z)+x^2y^2(z-x)(z-y)}{x^2y^2z^2}$$</span>
Consider two case</p>
<ol>
<li>If <span class="math-container">$x=y=z$</span>, then it is trivial that <span cl... |
2,263,431 | <p>Let $$v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Then find difference between maximum and minimum of $v^2$.</p>
<p>I understand both of them are distance of a point on ellipse from origin, but how do we find maximum and minimum?</p>
<p>I tried guessing, and got maximum $v$ when $... | lab bhattacharjee | 33,337 | <p>Let $$p=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)},q=\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Now $p^2+q^2=a^2+b^2$</p>
<p>and $$2(p^2+q^2)-(p+q)^2=(p-q)^2\ge0\iff(p+q)^2\le2(p^2+q^2)=?$$</p>
|
535,948 | <p>How do I prove that $(1+\frac{1}{2})^{n} \ge 1 + \frac{n}{2}$ for every $n \ge 1$</p>
<p>My base case is $n=1$ <br/>
Inductive step is $n=k$ <br/>
Assume $n=k+1$<br/></p>
<p>$(\frac{3}{2})^{k} \times \frac{3}{2} \ge (1 + \frac{k+1}{2})$</p>
<p>I'm not sure how to proceed.</p>
| user35603 | 84,245 | <p>Hint: Use binomial formula: $(a+b)^n=...$ and take first two terms.</p>
|
16,158 | <p>I have a Manipulate control, the main element of which is a single slider. Moving the <em>one</em> slider triggers calls to <em>two</em> functions - one which is evaluated very quickly, the other more slowly. I'd like to see 'live' update ( i.e. ContinuousAction->True ) for the fast one, but would only like to updat... | Stefan | 2,448 | <p>What you could do as well to indicate progress whilst your slow function is running:</p>
<pre><code>Manipulate[ControlActive["calculating...", Pause[.5]; x], {x, 0, 1} ]
</code></pre>
<p>I did put here Pause[] as a replacement.</p>
<p>A more elaborate example would be the color curve of a complex phase (although ... |
2,889,482 | <blockquote>
<p><strong>1.28</strong>
In Section 1.6, we introduced the idea of entropy $h(x)$ as the information gained on observing the value of a random variable $x$ having distribution $p(x)$.
We saw that, for independent variables $x$ and $y$ for which $p(x,y) = p(x) p(y)$, the entropy functions are additive... | masa53 | 680,298 | <p>We can write <span class="math-container">$h(x) = F(p(x))$</span> because <span class="math-container">$h(x)$</span> is a function of <span class="math-container">$p(x)$</span>. Note that <span class="math-container">$h(x,y) = F(p(x,y))$</span>. </p>
<p><span class="math-container">$h(x,x) = F(p(x,x))=F(p(x) * p(x)... |
4,064,618 | <p><span class="math-container">$10$</span> individuals including <span class="math-container">$x$</span> and <span class="math-container">$y$</span> seat at random at a round table. All possible orders are equally likely. Let <span class="math-container">$W$</span> be the event that there are exactly two people seated... | michalis vazaios | 540,891 | <p>Let's say <span class="math-container">$x$</span> sits somewhere (it doesn't matter where). There are <span class="math-container">$9$</span> seats left. There are <span class="math-container">$2$</span> of these seats where <span class="math-container">$y$</span> can seat so that there are two people between them. ... |
2,094,698 | <p>I am trying to figure out how to create a formula to solve this problem I have.</p>
<p>OK, the problem</p>
<p>I need to work how much money to give people bassed on a percentage of a total value, but the problem is that the percentage varies from person to person.</p>
<p>Ie,</p>
<p>Total amount = $1,000,000</p>
... | John Hughes | 114,036 | <p>The problem is that, as you observe, the numbers don't add up. </p>
<p>This can be restated in a simpler form: "I have to distribute all of \$100 to two people. One gets 40%; the other gets 45%. How much should I give to each?" Well, if I give one \$40 and the other \$45, I've split it 40%/45%, but haven't given aw... |
2,309,900 | <p>EDIT: How many ways are there to distribute $0$ and $1$ in a vector of length $N$ such that the number of zeros and 1 is the same (Suppose $N$ is even). I'm guessing it's $\binom{N}{2}$, but I'm not sure.</p>
| Henrik supports the community | 193,386 | <p>After your edit (Zubins answer - before his edit - was correct before your edit).</p>
<p>You have to select the $\frac{N}{2}$ places that should be $1$ (or $0$) and then the rest will just have to be $0$ (or $1$), you can do that in:
$$
\binom{N}{N/2}
$$
ways.</p>
|
38,367 | <p>Notation: Let$M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $Diff(M)$ the group of diffeomorphisms of $M$ and $Imb(S, M)$ the group of smooth imbeddings of $S$ into $M$.</p>
<p>A classical result of R. Palais from the 1960 paper <em>Local triviality of the restriction map for embeddings</em> says tha... | Ryan Budney | 1,465 | <p>It's not clear what you mean by "various refinements and generalizations". Cerf has a huge paper published by IHES "Topologie de certains espaces de plongements" which goes into many related details. In a way it's more of a ground-up collection of basic information on the topology of function spaces.</p>
<p>Regar... |
550,757 | <p>I don't see how $(5, 1 - 8i)$ would generate the element $5$. I think that $5$ and $1 - 8i$ would have to have a common factor but I can't find any that divide both.</p>
| rfauffar | 12,158 | <p>I think I may understand the question. We want to see that $(5,1-8i)$ is principal. We see that $5=(1+2i)(1-2i)$ and $1-8i=(1+2i)(3+2i)$, and so $(5,1-8i)=(1+2i)$, since this is the greatest common divisor of $5$ and $1-8i$.</p>
|
4,227,776 | <p>Let <span class="math-container">$R$</span> be a commutative unital ring, and let <span class="math-container">$p$</span> be a prime number.</p>
<p><strong>Question 1</strong>: If <span class="math-container">$R$</span> is a <em>local</em> ring (with unique maximal ideal <span class="math-container">$m$</span>) of c... | Eric Wofsey | 86,856 | <p>Let <span class="math-container">$S\subset\mathbb{Q}[x]$</span> be the subring consisting of polynomials with integer constant term. Note that <span class="math-container">$(p)$</span> is a maximal ideal of <span class="math-container">$S$</span> with <span class="math-container">$S/(p)\cong\mathbb{F}_p$</span>. L... |
286,852 | <p>I'm currently trying to understand time continuous Fourier transformations for my Signal Analysis class (Electrical Engineering major). In my Textbook we have a set of "useful FT-pairs", one of which being
$x(t)<=FT=>X(\omega)$</p>
<p>$\sin(\omega_0*t)<=>i\pi[\delta(\omega+\omega_0)-\delta(\omega-\omega... | dexter04 | 48,371 | <p>You should understand that the Dirac-delta $\delta(\omega)$ is not a conventional function. It is a generalized function or a distribution. The mathematical theory behind it is really involved and the guy who explained everything(Laurent Schwartz) got a Field's Medal for it.</p>
<p>The best way to go around it is t... |
374,694 | <p>I believe I'm looking for a function: $f(x) : \mathbb N \mapsto \mathbb N^2$ and it's inverse $f^{-1}(x) : \mathbb N^2 \mapsto \mathbb N$, a known mapping that can take any positive integer and map to a unique 2D integer point (and the inverse mapping as well). </p>
<p>Like the titles says, the conceptual idea is ... | Henrik Finsberg | 67,299 | <p>Cantor's paring function would probably do the job: <a href="http://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor_pairing_function" rel="noreferrer">http://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor_pairing_function</a></p>
|
4,220,751 | <p>In the example in the following slide, we follow the highlighted formula. With regard to the highlight, I'm confused why the number is greater <strong>or equal</strong> to <span class="math-container">$2^{n-1}$</span>, while only need to be less than <span class="math-container">$2^n$</span> (not less than <strong>o... | José Carlos Santos | 446,262 | <p>It is easy to show by induction that<span class="math-container">$$(\forall i\in\Bbb N):A^i=\begin{bmatrix}1&i&\frac{i(i+1)}2\\0&1&i\\0&0&1\end{bmatrix}.$$</span>So,<span class="math-container">$$M=\begin{bmatrix}20&\sum_{i=1}^{20}i&\frac12\left(\sum_{i=1}^{20}i^2+\sum_{i=1}^{20}i\rig... |
2,253,676 | <p>$f(x)$ continuous in $[0,\infty)$. Both $\int^\infty_0f^4(x)dx, \int^\infty_0|f(x)|dx$ converge. </p>
<p>I need to prove that $\int^\infty_0f^2(x)dx$ converges.</p>
<p>Knowing that $\int^\infty_0|f(x)|dx$ converges, I can tell that $\int^\infty_0f(x)dx$ converges.</p>
<p>Other than that, I don't know anything. I ... | TBTD | 175,165 | <p>Here is a proof. Let
$$
A \triangleq \{x\in[0,\infty) : |f(x)|\leq 1\}, \ \text{and} \ A^c \triangleq \{x\in[0,\infty) : |f(x)|>1\}.
$$</p>
<p>Write,
$$
\int_0^{\infty}f^2(x) dx = \underbrace{\int_Af^2(x) dx}_{\triangleq M} + \underbrace{\int_{A^c}f^2 (x) dx}_{\triangleq N}.
$$
Note that $M$ is bounded, since
$$... |
2,253,676 | <p>$f(x)$ continuous in $[0,\infty)$. Both $\int^\infty_0f^4(x)dx, \int^\infty_0|f(x)|dx$ converge. </p>
<p>I need to prove that $\int^\infty_0f^2(x)dx$ converges.</p>
<p>Knowing that $\int^\infty_0|f(x)|dx$ converges, I can tell that $\int^\infty_0f(x)dx$ converges.</p>
<p>Other than that, I don't know anything. I ... | Community | -1 | <p><strong>Disclaimer</strong>: this is an overkill.</p>
<p>Here's a general fact: if $(X, \Sigma, \mu)$ is a measure space, we have for $1\le p<q \le \infty$, $L^p(X) \cap L^q(X) \subset L^r(X)$ for all $p \le r \le q$.</p>
<p>Here, $f \in L^1([0,\infty)) \cap L^4([0,\infty))$, so it is in $L^r([0,\infty))$ for e... |
3,497,919 | <p><span class="math-container">$$\sum_{k=0}^{\Big\lfloor \frac{(n-1)}{2} \Big\rfloor} (-1)^k {n+1 \choose k} {2n-2k-1 \choose n} = \frac{n(n+1)}{2} $$</span></p>
<p>So I feel like <span class="math-container">$(-1)^k$</span> is almost designed for the inclusion-exclusion principle. And the left-hand side looks like s... | aryan bansal | 698,119 | <p>Coefficient of <span class="math-container">$x^k$</span> in <span class="math-container">$(1-x)^{n+1}$</span> is <span class="math-container">$(-1)^k$${n+1 \choose k}$</span> </p>
<hr>
<p>Coefficient of <span class="math-container">$$x^{(\frac{n-2k-1}{2})}$$</span> in <span class="math-container">$$(1- \sqrt{x})... |
2,447,677 | <p>Consider the parametric equations given by \begin{align*} x(t)&=\sin{t}-t,\\ y(t) & = 1-\cos{t}.\end{align*}</p>
<p>I want to write these parametric equations in Cartesian form. </p>
<p>In order to eliminate the sine and cosine terms I think I probably need to consider some combination of $x(t),y(t), x(t)... | Guy Fsone | 385,707 | <p>$$ y(t) = 1-\cos{t}\implies t= \arccos(1-y) \implies x= \arccos(\sin(1-y)) -\arccos(1-y)$$
that is </p>
<p>$$x= \sqrt{1-(1-y)^2} -\arccos(1-y)$$</p>
|
805,596 | <p>Evaluate $$\lim_{x\to\infty}\dfrac{5^{x+1}+7^{x+2}}{5^x + 7^{x+1}}$$</p>
<p>I'm getting a different result but not the exact one.
I got $$\dfrac{5\cdot\dfrac{5^n}{7^n} +49}{\dfrac{5^n}{7^n} + 7}.$$
I know the result is $7$ but I cannot figure out the steps.</p>
| lab bhattacharjee | 33,337 | <p>$$\frac{5^{x+1}+7^{x+2}}{5^x+7^{x+1}}=\frac{5\left(\frac57\right)^x+7^2}{\left(\frac57\right)^x+7}$$</p>
<p>Now $\displaystyle\lim_{n\to\infty}r^n=0$ if $|r|<1$</p>
|
3,301,387 | <p>I don't understand what it is about complex numbers that allows this to be true.</p>
<p>I mean, if I root both sides I end up with Z=Z+1. Why is this a bad first step when dealing with complex numbers?</p>
| Geoffrey Trang | 684,071 | <p>Even in <span class="math-container">$\mathbb{R}$</span>, there is still a solution, namely <span class="math-container">$z = -\frac{1}{2}$</span>. It is then clear that <span class="math-container">$(-\frac{1}{2} + 1)^6 = (\frac{1}{2})^6 = (-\frac{1}{2})^6$</span>. In fact, for any 2 real numbers <span class="math-... |
961,603 | <p>Show that every positive integer $N$ less than or equal to $n$ factorial, is the sum of at most $n$ distinct positive integers, each of which divides $n!$.</p>
| abiessu | 86,846 | <p>Step 1:</p>
<p>Show that the statement holds for $n=1$. For this value of $n$ we have $1!=1$, and thus every positive integer less than or equal to $1!$ is the set $\{1\}$. Since $n=1$ we must find a sum of at most one positive integer which divides $1!$ and adds up to each of the numbers in our given set, and in... |
163,889 | <p>Find a closed form for $a_n:=\sum_{k=0}^{n}\binom{n}{k}(n-k)^n(-1)^k$ using generating functions.</p>
| Marko Riedel | 44,883 | <p>Suppose we are trying to evaluate
$$\sum_{k=0}^n {n\choose k} (n-k)^n (-1)^k.$$</p>
<p>Observe that
$$(n-k)^n =
\frac{n!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{n+1}} \exp((n-k)z) \; dz.$$</p>
<p>This gives for the sum the integral
$$\frac{n!}{2\pi i}
\int_{|z|=\epsilon} \frac{1}{z^{n+1}}
\sum_{k=0}^n {n\choose... |
16,794 | <p>In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also interested in certain simplicial complexes that are singular disks, or even more generally singular disks tiled by polyg... | Sam Nead | 1,650 | <p>Prickly pear cactus is very pretty. Lots of cut points!</p>
<p>I believe that that your cacti are "tree-graded spaces" where are "pieces" are points or disks. See Drutu, C., Sapir, M.: Tree-graded spaces and asymptotic cones of groups. Topology 44, 959–1058 (2005).</p>
|
16,794 | <p>In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also interested in certain simplicial complexes that are singular disks, or even more generally singular disks tiled by polyg... | Greg Kuperberg | 1,450 | <p>In the end, we (Joel Kamnitzer, his student Bruce Fontaine, and I) agreed on the term "diskoid". In the abstract, the word "cactus" seemed too clever by half. When I actually wrote it into the paper, it was awkward and it didn't emphasize the main property of interest, that our shapes are meant as a mild generaliz... |
4,219,193 | <p>The <code>p implies q</code> statement is often described in various ways including:<br />
(1) <code>if p then q</code> (i.e. whenever p is true, q is true)<br />
(2) <code>p only if q</code> (i.e. whenever q is false, p is false)</p>
<p>I see the truth table for (1) as</p>
<pre><code>p | q | if p then q
-----------... | fleablood | 280,126 | <p>I think the intuitive way to think of this is if something is contradicted then it is false but if nothing can be contradicted it is by default true.</p>
<p><span class="math-container">$p\to q$</span> means whenever <span class="math-container">$p$</span> is true we'll have <span class="math-container">$q$</span> t... |
4,219,193 | <p>The <code>p implies q</code> statement is often described in various ways including:<br />
(1) <code>if p then q</code> (i.e. whenever p is true, q is true)<br />
(2) <code>p only if q</code> (i.e. whenever q is false, p is false)</p>
<p>I see the truth table for (1) as</p>
<pre><code>p | q | if p then q
-----------... | ryang | 21,813 | <p>Further to the other comments and answers that have dealt with your motivating query (my additional take <a href="https://math.stackexchange.com/a/4217336/21813">here</a>), I'd like to address your question and follow-up comment “<em>I could be wrong but I don't believe the statements address the remaining entries i... |
2,629,115 | <blockquote>
<p>We will define the convolution (slightly unconventionally to match Rudin's proof) of $f$ and $g$ as follows:
$$(f\star g)(x)=\int_{-1}^1f(x+t)g(t)\,dt\qquad(0\le x\le1)$$</p>
<ol>
<li>Let $\delta_n(x)$ be defined as $\frac n2$ for $-\frac1n<x<\frac1n$ and 0 for all other $x$. Let $f(x)$... | Doug M | 317,162 | <p>$f\star\delta_{10}(x) = \int_{-1}^1 f(x - t)\delta(t)\ dt$</p>
<p>but $\delta_{10}(t) = 0$ for most of this interval</p>
<p>$f\star\delta_{10}(x) = \int_{-\frac {1}{10}}^{\frac {1}{10}} 5(x - t) dt$</p>
<p>$x\in [0.5, 0.6]$</p>
<p>But if $x$ is outside this interval:</p>
<p>$x\in [0.3, 0.5]$</p>
<p>$f\star \de... |
2,411,081 | <p>How can I show (with one of the tests for convergence , <strong>not by solving</strong>) that the integral $$\int _{1}^\infty\frac{\ln^5(x)}{x^2}dx$$ converges?</p>
| G Tony Jacobs | 92,129 | <p>Since $\ln^5(x)$ grows more slowly than $x^{1/2}$, you could use a comparison test. For some positive $a>1$, we have:</p>
<p>$$\int_a^\infty \frac{\ln^5(x)}{x^2} dx \leq \int_a^\infty \frac{x^{1/2}}{x^2} dx = \int_a^\infty \frac{dx}{x^{3/2}}$$</p>
<p>You should know that the last expression is convergent by pre... |
72,943 | <p>I want find the minima of a (multivariable) function under a constraint which has to be fulfilled on a whole interval, let's say
$$
\nabla f (\underline x) = 0 \ \\ \ c(\underline x,s)\geq0\ \forall s\in [0,1].
$$
How do I implement such a condition into the <code>Minimize[{f[x1,x2,...,xn],c[x1,...,xn,s]>=0 ?},{x... | bill s | 1,783 | <p>Here's a two dimensional example:</p>
<pre><code>f[x_, y_] := x^2 + y^2;
c[x_, y_, s_] = 2 x + 3 y + s;
Minimize[{f[x, y], c[x, y, s] == 0, 1 >= s >= 0}, {x, y}]
</code></pre>
<p>This gives an answer that depends on the value of <code>s</code>, as is plausible. For your revised/edited problem:</p>
<pre><cod... |
4,447,641 | <p>I can't figure out where to even start, I have looked up the answer on Desmos but it uses L'Hopitals rule which i haven't learned yet.</p>
<p><span class="math-container">$$\lim_{h \to 0} \frac{(2+h)^{3+h} - 8}{h}$$</span>
I see that i can use the log rules to rewrite it as
<span class="math-container">$$\lim_{h \to... | Dan | 1,374 | <p>Per Levent's comment, supposed that you're given <span class="math-container">$f(x) = x^{x+1}$</span>. By definition of the derivative:</p>
<p><span class="math-container">$$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \rightarrow 0} \frac{(x+h)^{x+1+h} - x^{x+1}}{h}$$</span></p>
<p>Now, plug in... |
2,078,943 | <p>My book says this but doesn't explain it:</p>
<blockquote>
<p>Row operations do not change the dependency relationships among columns.</p>
</blockquote>
<p>Can someone explain this to me? Also what is a dependency relationship? Are they referring to linear dependence? </p>
| Nowibananatzki | 385,751 | <p>Suppose a matrix A and one of its columns <span class="math-container">$v_k$</span> is a linear combination of the others, we can form an equation as such:</p>
<p><span class="math-container">$$a_1v_1 + \ldots + a_{k-1}v_{k-1} + a_{k+1}v_{k+1} + \ldots + a_nv_n = a_kv_k$$</span></p>
<p>If the column vectors of A are... |
758,466 | <blockquote>
<p>A fair 6-sided die is tossed 8 times. The sequence of 8 results is recorded to form an 8-digit number. For example if the tosses give {3, 5, 4, 2, 1, 1, 6, 5}, the resultant number is $35421165$. What is the probability that the number formed is a multiple of 8.</p>
</blockquote>
<p>I solved this by ... | paw88789 | 147,810 | <p>Here's another approach. As noted in the problem statement, it all comes down to the last <span class="math-container">$3$</span> rolls concatenating to a multiple of <span class="math-container">$8$</span>.</p>
<p>The last two rolls have to concatenate to a multiple of <span class="math-container">$4$</span>.</p>
<... |
20,683 | <p>The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple positive roots, and invertible $K_j$'s generating a copy of the weight lattice. Then one has a flurry of relations b... | S. Carnahan | 121 | <p>If you have a copy of Gaitsgory's <i>Notes on factorizable sheaves</i> there is a sketchy exposition of the relative double in section 3.2, and a description of the construction of the quantum group in section 5 (in the numbering of the March 2008 edition). Here is what I understand, with many possible misconceptio... |
619,580 | <p>Please help me to solve the following problem that is in the Lebesgue integral discussion </p>
<blockquote>
<p>Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that $\,\,\|f_n\|_\infty \to \infty$ but $\int_0^1\lvert\, f_n\rvert\,d\lambda \to 0$ as $n\to\infty$.</p>
</block... | zhw. | 228,045 | <p>Here's another: $f_n(x) = nx^{n^2}$</p>
|
619,580 | <p>Please help me to solve the following problem that is in the Lebesgue integral discussion </p>
<blockquote>
<p>Give an example of a sequence $\,\,f_n : [0, 1] \to \Bbb R$ of continuous functions such that $\,\,\|f_n\|_\infty \to \infty$ but $\int_0^1\lvert\, f_n\rvert\,d\lambda \to 0$ as $n\to\infty$.</p>
</block... | triple_sec | 87,778 | <p>\begin{align*}
f_n(x)=\begin{cases}n&\text{if $x\in\left[0,\frac1 {2n^2}\right]$;}\\
\text{[linear descent; see below]}&\text{if $x\in\left(\frac{1}{2n^2},\frac{1}{n^2}\right]$;}\\0&\text{if $x\in\left(\frac1{n^2},1\right]$.}\end{cases}
\end{align*}
$\|f_n\|_{\infty}=n,$</p>
<p>$\|f_n\|_1<n\times(1/n... |
2,662,792 | <p>I'm looking for a Galois extension $F$ of $\mathbb{Q}$ whose associated Galois group $\mbox{Gal}(F, \mathbb{Q})$ is isomorphic to $\mathbb{Z}_3 \oplus \mathbb{Z}_3$. I'm wondering if I should just consider some $u \notin Q$ whose minimum polynomial in $Q[x]$ has degree 9. In that case we'd have $[\mathbb{Q}(u) : \ma... | P Vanchinathan | 28,915 | <p>Take $\alpha = \exp(2\pi i/7),\ \beta = \exp(2\pi i/13)$, primitive roots of unity of order 7 and 13 (these are prime numbers and leave a remainder 1 when divided by 3). They generate fields having cyclic Galois groups of order 6 and 12 respectively. Each of them have their own (unique) cubic subfields having cycli... |
1,897,212 | <p>Does there exist any surjective group homomorphism from <span class="math-container">$(\mathbb R^* , .)$</span> (the multiplicative group of non-zero real numbers) onto <span class="math-container">$(\mathbb Q^* , .)$</span> (the multiplicative group of non-zero rational numbers)? </p>
| Tsemo Aristide | 280,301 | <p>$f(-1)^2=f(1)$ implies that $f(-1)=1$ or $f(-1)=-1$. Suppose $f(-1)=1$. Let $x>0$, $f(x)=f(\sqrt x)^2$. Implies that $f(x)$ is a square, $f(-x)=f(-1)f(x)>0$. Impossible. Since $f$ is surjective.</p>
<p>Suppose $f(-1)=-1, f(-x)=-f(\sqrt x)^2$ and $f(x)=f(\sqrt x)^2$ this implies that $f$ is not surjective sinc... |
120,808 | <pre><code>Limit[Sum[k/(n^2 - k + 1), {k, 1, n}], n -> Infinity]
</code></pre>
<p>This should converge to <code>1/2</code>, but <code>Mathematica</code> simply returns <code>Indeterminate</code> without calculating (or so it would appear). Any specific reason why it can't handle this? Did I make a mistake somewhere... | Feyre | 7,312 | <p>The problem is that without the limit, the sum doesn't converge, and without the sum the limit is <code>0</code>. <em>Mathematica</em> can only do one at a time.</p>
<p>At a finite value <code>n</code>, the sum gives a sequence with a $\frac{1}{0}$ or other complexinfinity expression for all values of <code>n</code... |
3,137,681 | <p>I'm trying to see if there is any way to solve for x? I'm told Option 1 is correct and I think I understand it but why is Option 2 incorrect?</p>
<p>Option 1: subtract both sides by y
xy + y -y = y - y
xy = 0
No solution as either x or y could be zero</p>
<p>Option 2: factor out y
y * (x + 1) = y
(1/y) * y * (x + ... | Dr. Sonnhard Graubner | 175,066 | <p>Your equation <span class="math-container">$$xy+y-y=0$$</span> simplifies to
<span class="math-container">$$xy=0$$</span> so <span class="math-container">$$x=0$$</span> or <span class="math-container">$$y=0$$</span></p>
|
2,076,910 | <p>proving $\displaystyle \binom{n}{0}-\binom{n}{1}+\binom{n}{2}+\cdots \cdots +(-1)^{\color{red}{m}-1}\binom{n}{m-1}=(-1)^{m-1}\binom{n-1}{m-1}.$</p>
<p>$\displaystyle \Rightarrow 1-n+\frac{n(n-1)}{2}+\cdots \cdots (-1)^{n-1}\frac{n.(n-1)\cdot (n-2)\cdots(n-m+2)}{(m-1)!}$</p>
<p><strong>Added</strong></p>
<p>writti... | Stefan4024 | 67,746 | <p><strong>HINT:</strong> Fix a random $n$ and do induction on $m$.</p>
|
81,348 | <p>There is a well-known proof of the Compactness Theorem in propositional logic which uses the compactness of the space $\{0,1\}^P$, where $P$ is the set of propositional variables in consideration. In general, this compactness relies on the Tychonoff theorem which in turn requires the Axiom of Choice. Let me sketch i... | Andreas Blass | 6,794 | <p>The proof is correct, but it does not provide an algorithm unless you have some additional information about $A$. If you untangle the proofs, you find the following pseudo-algorithm: Go through the propositional variables $p_i$ in order, adding each $p_i$ or its negation to $A$ "greedily", i.e., add $p_i$ if doing ... |
1,715,232 | <p>Suppose $f_n$ converges uniformly to $f$ and $f_n$ are differentiable. Is it true that f will be differentiable?</p>
<p>My initial guess is no because $f_n= \frac{\sin(nx)}{\sqrt n}.$ Is this right? And more examples would be greatly appreciated.</p>
| Dominik | 259,493 | <p>The limit function doesn't even have to be differentiable anywhere. Take any function $f: [0, 1] \to \mathbb{R}$ that is continuous but nowhere differentiable [such functions exist]. By the <a href="https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem" rel="nofollow">Stone-Weierstrass theorem</a> you can ... |
323,781 | <p>In the <a href="https://arxiv.org/abs/1902.07321" rel="noreferrer">paper</a> by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, (Update: published now in <a href="https://www.pnas.org/content/early/2019/05/20/1902572116" rel="noreferrer">PNAS</a>) the abstract includes</p>
<blockquote>
<p>In the... | Astron | 136,161 | <p>Unfortunately, I cannot comment.
<a href="https://www.youtube.com/watch?v=HAx_pKUUqug" rel="noreferrer">https://www.youtube.com/watch?v=HAx_pKUUqug</a> 56:28</p>
|
1,167,438 | <p>I am currently solving a differential equation but I am having a little trouble figuring out my integrative factor. </p>
<p>I have </p>
<p>$$\exp\bigg({∫\frac{3}{t}dt}\bigg)$$</p>
<p>so to integrate I made it $\exp(\ln(t^3))$ what is the final solution? $t^3$?</p>
| achille hui | 59,379 | <p>Notice if $d, e$ are random variables uniform over $[0,1]$, so do $\tilde{d} \stackrel{def}{=} 1 - d\;$ and $\;\tilde{e}\stackrel{def}{=} 1 - e$.<br>
We have
$$a + b + c > d + e \quad\iff\quad a + b + c + \tilde{d} + \tilde{e} > 2$$
For any $p = (x_1,x_2,x_3,x_4,x_5) \in \mathbb{R}^5$, let $\;\ell(p) = x_1 + x... |
3,217,130 | <p>How can I prove this by induction? I am stuck when there is a <span class="math-container">$\Sigma$</span> and two variables, how would I do it? I understand the first step but have problems when i get to the inductive step.</p>
<p><span class="math-container">$$\sum_{j=1}^n(4j-1)=n(2n+1)$$</span></p>
| Dietrich Burde | 83,966 | <p>The conjugation operation of <span class="math-container">$G$</span> on <span class="math-container">$G$</span>, given by <span class="math-container">$b\mapsto aba^{-1}$</span> then is the identity, i.e., <span class="math-container">$aba^{-1}=b$</span> for all <span class="math-container">$a,b\in G$</span>. Hence ... |
3,217,130 | <p>How can I prove this by induction? I am stuck when there is a <span class="math-container">$\Sigma$</span> and two variables, how would I do it? I understand the first step but have problems when i get to the inductive step.</p>
<p><span class="math-container">$$\sum_{j=1}^n(4j-1)=n(2n+1)$$</span></p>
| Santana Afton | 274,352 | <p>The property implies that the conjugacy class of any <span class="math-container">$g\in G$</span> is exactly <span class="math-container">$\{g\}$</span>, which is equivalent to being abelian.</p>
|
2,565,763 | <p>I was playing a standard solitaire game on my mobile app and I came across a round where I couldn't perform a single move thus resulting in a loss. I was then thinking as to what the probability of this event to happen in a single game. Would anybody know this probability and show the calculations given a standard d... | Michael Hardy | 11,667 | <p>A compound hypothesis is a hypothesis that is consistent with more than one probability distribution. When the alternative hypothesis is composite, then one uses
$$
\frac{L(\theta_\text{null})}{L(\theta_{\text{null} \,\cup\, \text{alternative}})}
$$
as the test statistic. Then the denominator may be equal to the num... |
4,036,975 | <p><span class="math-container">$\require{begingroup} \begingroup$</span>
<span class="math-container">$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$</span></p>
<p><a href="https://math.stackexchange.com/q/3590905/122782">Related quest... | Quanto | 686,284 | <p>The close-form result can be expressed as</p>
<p><span class="math-container">$$\color{blue}{ \int_0^1\frac{\ln(1+x^{2n})}{1+x^2} \,dx = -nG+\frac\pi2 n \ln 2
+ \pi \sum_{k=1}^{[\frac n2]}\ln \cos\frac{(n+1-2k)\pi}{4n} }
$$</span></p>
<p>as shown below. Note that</p>
<p><span class="math-container">\begin{align}
I_... |
603,291 | <p>Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some $\alpha >1$
and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$. </p>
<p>I'm not sure which theorem should I look to prove this question. Can you guys give me a bit of hint? First of all how to prove some function... | Ant | 66,711 | <blockquote>
<p><em>Caveat (by @Did, 2018-09-12): This accepted answer is wrong. For unfathomable reasons, for years now, neither its author not the OP seem interested in correcting the situation -- hence the present warning.</em></p>
</blockquote>
<p>divide by $|x-y| $ both members. you get</p>
<p>$$\frac{|f(x) - ... |
3,127,795 | <p>So I’m struggling to understand how to find the angle in this circle, we’ve recently learnt about trigonometry and like finding the area of a circle and all that but I can’t seem to remember which formula I have to use to find this angle. Can anyone lend a helping hand?
<img src="https://i.stack.imgur.com/V2Sm7.jp... | Haydn Gwyn | 645,813 | <p>You can bisect the angle AOB, which divides isosceles triangle AOB into two right triangles. Each one of these right triangles has a hypotenuse of 7 and a leg of 5, so the sine of their angle at O is 5/7. As such, the measure of angle AOB is 2*arcsin(5/7).</p>
|
2,463,052 | <p>I admit $\frac{1}{1+b}<1$ is trivial for $b>0.$ However, the above claim boils down to suggesting that, any $y\in\{x\in\mathbb{R}:x>1\}$ can be uniquely represented by $1+b$ for some $b>0.$ Can you give me some hints how to prove it?</p>
| nonuser | 463,553 | <p>It should be $0<a<1$, othervise it doesn't hold. Say $a=-1/2$ then $b=-3$</p>
<p>Let $c= 1/a$ then $c>1$ since $1>a>0$. Let $b=c-1>0$. So $a= {1\over 1+b}$.</p>
|
4,486,956 | <p>If <span class="math-container">$f$</span> is a convex increasing function, then I need to show that
<span class="math-container">$$ \lim_{x\rightarrow -\infty} \frac{1}{x}\int_x^{x_0}f(y)dy=-\lim_{x\rightarrow -\infty}f(x).$$</span></p>
<p>I can prove this equality using L'Hopital's rule and the Fundamental Theorem... | Martin R | 42,969 | <p>L'Hopital's rule shows that the identity holds whenever the limit on the right exists. In your special case (<span class="math-container">$f$</span> increasing and convex) there is a simpler short proof:</p>
<p>First note that <span class="math-container">$A = \lim_{x\to -\infty}f(x)$</span> exists (as a real number... |
1,664,188 | <p>I am trying to show for my home work that the theory of random graphs RG, is not $\kappa$-stable for every $\kappa$ i.e If $M\vDash RG$ and $A\subseteq |M|$ with $|A|\le\kappa$ then $S_1(A)\le\kappa$</p>
<p>I tried to build a $\kappa$-tree with $2^{\kappa}$ branches of consistent formulas,but i get stuck at the lim... | Empy2 | 81,790 | <p>Most vectors are not eigenvectors, unless the matrix is a multiple of the identity.<br>
Pick a random $v$, calculate $w=Av$, and check whether the inner product of $v$ and $w$ satisfies $$(w.v)^2=|w|^2||v|^2$$
If they do, keep picking other random $v$, and check that, until you have a good one.</p>
|
1,664,188 | <p>I am trying to show for my home work that the theory of random graphs RG, is not $\kappa$-stable for every $\kappa$ i.e If $M\vDash RG$ and $A\subseteq |M|$ with $|A|\le\kappa$ then $S_1(A)\le\kappa$</p>
<p>I tried to build a $\kappa$-tree with $2^{\kappa}$ branches of consistent formulas,but i get stuck at the lim... | Community | -1 | <p>If a column of the matrix has a nonzero off-diagonal element, the corresponding standard vector can do, as certainly $$Ae_k=c_k\ne \lambda e_k.$$</p>
<p>If you cannot find such an element, the matrix is diagonal. Take the sum of the standard vectors corresponding to two diagonal elements of distinct values.</p>
<p... |
169,919 | <blockquote>
<p>If $p$ is a prime, show that the product of the $\phi(p-1)$ primitive roots of $p$ is congruent modulo $p$ to $(-1)^{\phi(p-1)}$.</p>
</blockquote>
<p>I know that if $a^k$ is a primitive root of $p$ if gcd$(k,p-1)=1$.And sum of all those $k's$ is $\frac{1}{2}p\phi(p-1)$,but then I don't know how use... | awllower | 6,792 | <p>If $a^k$ is a primitive root modulo $p$, then so is $a^{-k}$. Thus, if $p-1$ is even, then the sum of coprime integers between 1 and $p-1$ with $p-1$ must be =$\phi(p-1)(p-1)/2$, and hence the product of $\phi(p-1)$ primitive roots modulo p must be $a^{\phi(p-1)(p-1/2)}\equiv(-1)^{\phi(p-1)} \pmod p$. If $p-1$ is od... |
2,319,341 | <blockquote>
<p>If a rubber ball is dropped from a height of $1\,\mathrm{m}$ and continues to rebound to a height that is nine tenth of its previous fall, find the total distance in meter that it travels on falls only.</p>
</blockquote>
<h3>My Attempt:</h3>
<p>I tried if it could be solved using arithmetic progress... | Ananth Kamath | 444,440 | <p>The ball follows geometric progression, because, after each fall, the ball goes back up $\frac 9{10}$th of the height of the previous fall. </p>
<p>That is, if the ball fell down from $x$ metres, the next fall would be from $\frac 9{10} x$, the next from $\frac 9{10} \cdot \frac 9{10} x$. </p>
<p>Since the initia... |
4,097,262 | <p><a href="https://i.stack.imgur.com/E1UoR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/E1UoR.png" alt="Cubic equation with a unit radius circle" /></a>
A cubic equation and circle (unit radius) has intersection at A,B,C,D. ABCD is a square. <em>Find the angle <span class="math-container">$\theta... | G Cab | 317,234 | <p><em>HINT</em></p>
<p>The cubic must clearly be of type
<span class="math-container">$$
y = bx\left( {x^{\,2} - a^{\,2} } \right)
$$</span></p>
<p>In polar coordinates
<span class="math-container">$$
r\sin \theta = br\cos \theta \left( {r^{\,2} \cos ^{\,2} \theta - a^{\,2} } \right)
$$</span>
i.e.
<span class="mat... |
545,728 | <p>So I have $X \sim \text{Geom}(p)$ and the probability mass function is:</p>
<p>$$p(1-p)^{x-1}$$</p>
<p>From the definition that:</p>
<p>$$\sum_{n=1}^\infty ns^{n-1} = \frac {1}{(1-s)^2}$$</p>
<p>How would I show that the $E(X)=\frac 1p$</p>
| mert | 98,278 | <p>\begin{eqnarray}
E(X)&=&\sum_{x=1}^\infty x p(1-p)^{x-1}\\
&=&p\sum_{x=1}^\infty x(1-p)^{x-1}\\
&=&p\sum_{x=1}^\infty -\frac{d}{dp}(1-p)^{x}\\
&=&-p\left[\frac{d}{dp} \sum_{x=1}^\infty (1-p)^x\right]\\
&=&-p\cdot \frac{d}{dp}\frac{1-p}{1-(1-p)} \\
&=&-p\cdot \frac{d}{... |
2,780,832 | <p>Let's define: </p>
<p>$$\sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i}$$
$$\cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}$$</p>
<blockquote>
<p>We are to prove that
$$\sin(z+w)=\sin(w) \cos(z) + \sin(z)\cos(w), \forall_{z,w \in \mathbb{C}}$$
using only the following statement: $\exp(z+w) = \exp(w)\exp(z)$.</p>
</bloc... | Ennar | 122,131 | <p>It is easier to start from $\sin z\cos w + \cos z\sin w$, but you can also directly continue from what you have.</p>
<p>It is straightforward to check $e^{iz} = \cos z + i\sin z$ directly from the definition of $\cos$ and $\sin$. Also, quite easily seen, $\cos$ is even, while $\sin$ is odd.</p>
<p>So, from what yo... |
2,174,508 | <p>We have a box in 3D space defined by it's position, size and rotation. How can i find the smallest box that contains that box, but has a new rotation?
I know this is a really bad description of the problem, i just can't describe it any better, i'm really bad at vector maths. i tried to draw a <a href="https://i.stac... | Community | -1 | <p>A 3D rotation is usually expressed by a $3\times3$ matrix, such as that obtained via the Euler angles. See <a href="https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions</a>.</p>
<p>When you have such a matrix, ... |
2,620,700 | <p>I want to prove or disprove that ($\mathbb{R},\mathcal{E}_1$) with equivalence relation $x\mathcal{R}y \Leftrightarrow a \neq 0$ y = ax is homeomorphic to $(\{0,1\}, \tau)$ where $\tau$ is the discrete topology. </p>
<p>To me it seems that there are two classes of equivalence: every point is equivalent to every poi... | amrsa | 303,170 | <p>The equivalence relation has only two equivalence classes, as you pointed out:
$\{0\}$ and $\mathbb{R} \setminus \{0\}$. </p>
<p>For the quotient space to a two-element discrete space it is necessary that $\{0\}$ is an open set in that quotient.<br>
The open sets of the quotient are those whose inverse image by th... |
1,181,356 | <p>Can such complex-valued function exist that</p>
<p>$$\lim_{x\to x_0}f(x)=0$$</p>
<p>but</p>
<p>$$\lim_{x\to x_0}\frac 1{f(x)}=1$$?</p>
<p>What about</p>
<p>$$\lim_{x\to x_0}\frac 1{Re(f(x))}=1$$?</p>
| Oria Gruber | 76,802 | <p>Such a function does not exist because of an elementary theorem that states that if $\lim_{x \to x_0} f(x)$ and $\lim_{x \to x_0} g(x)$ both exist, then $\lim_{x \to x_0} f(x)g(x)=\lim_{x \to x_0} f(x)\lim_{x \to x_0} g(x)$</p>
|
147,361 | <p>i'm new in MATHEMATICA. I want to create an operator $D^{(f)}=\partial_x+f'-\partial^2_x$ and $D^{(g)}=\partial_x+g'-\partial^2_x$ and put it into a matrix element, then multiplied by a vector whose components are functions on $x$, say $u(x), v(x)$. For example $$\begin{pmatrix}D^{(f)} & D^{(g)}\\ D^{(g)} & ... | LouisB | 22,158 | <p>We can start by defining our two differential operators and putting them into a matrix, like this:</p>
<pre><code>ClearAll["Global`*"]
df[w_] := D[w, x] + f'[x] w - D[w, x, x]
dg[w_] := D[w, x] + g'[x] w - D[w, x, x]
m = {{df, dg}, {dg, df}};
</code></pre>
<p>Now we need a way for the matrix operator to act on th... |
1,197,875 | <p>I'm just starting partials and don't understand this at all. I'm told to hold $y$ "constant", so I treat $y$ like just some number and take the derivative of $\frac{1}{x}$, which I hope I'm correct in saying is $-\frac{1}{x^2}$, then multiply by $y$, getting $-\frac{y}{x^2}$.</p>
<p>But apparently the correct answe... | Jonathan Hebert | 120,932 | <p>"With respect to $y$" means that you will be holding $x$ constant, not $y$. $y$ is the variable we are differentiating with respect to, so it is /not/ to be treated as a constant!</p>
|
84,982 | <p>I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by other professors, and I would like to use them for guided reading. <strong>I am wondering whether it is customary to as... | paul garrett | 15,629 | <p>Echoing Gerhard Paseman's comment more-or-less: it certainly never hurts to ask permission, or at least to explicitly express interest in use of on-line resources. Some on-line resources have indications of restrictions or lack thereof (e.g., Creative Commons License). In all cases, I think one should <em>acknowledg... |
234,063 | <p>There are some answers on how to get a smooth squarewave function. But I would like to have a smooth boxcar function or rectangle function with 2 different widths.: <code>wup</code>, and <code>wdown</code></p>
<p>One solution is the Fourier Transform, but I prefer having an approximation with a smoothness factor.</p... | cvgmt | 72,111 | <p>use the <code>mollifier</code> in mathematics. It also work for the <code>Piecewise</code> function.</p>
<p><strong>Reply the comment</strong></p>
<pre><code>a = 2;
b = 3;
S[x_ /; 0 <= x <= a] := 1;
S[x_ /; a <= x <= a + b] := 0;
S[x_ /; x >= a + b] := S[x - (a + b)];
S[x_ /; x <= a + b] := S[x + a... |
261,156 | <p>Well, I am trying to write a code that makes the number:</p>
<p><span class="math-container">$$123456\dots n\tag1$$</span></p>
<p>So, when <span class="math-container">$n=10$</span> we get:</p>
<p><span class="math-container">$$12345678910$$</span></p>
<p>And when <span class="math-container">$n=15$</span> we get:</... | David Reiss | 84,332 | <pre><code>ToExpression[
StringJoin[
ToString /@ Range[15]
]
]
</code></pre>
|
1,487,654 | <p>Q. Find the no of ways of inserting r dolllars using 1 dollar, 2 dollar and 5 dollar tokens,when order doesn't matter and when order doesn't matter.
Ans.
When order doesn't matter..
$(1+x+x^2+x^3+..)(1+x^2+x^4+..)(1+x^5+x^{10}+..)$
Coefficient of $x^r$ in above generating function.
I clearly understand this. But I ... | Marconius | 232,988 | <p>When order does matter, the contents of the $k$-th bracket will be the possibilities for the $k$-th token to be inserted, which are $1,2$ or $5$ dollars regardless of $k$.</p>
<p>So a first guess for the generating function that respects order is</p>
<p>$$g(x)=(x+x^2+x^5)\times(x+x^2+x^5)\times(x+x^2+x^5)\times\cd... |
1,487,654 | <p>Q. Find the no of ways of inserting r dolllars using 1 dollar, 2 dollar and 5 dollar tokens,when order doesn't matter and when order doesn't matter.
Ans.
When order doesn't matter..
$(1+x+x^2+x^3+..)(1+x^2+x^4+..)(1+x^5+x^{10}+..)$
Coefficient of $x^r$ in above generating function.
I clearly understand this. But I ... | robjohn | 13,854 | <p><strong>Order Does Not Matter</strong></p>
<p>If the order doesn't matter, then the coefficient of $x^n$ in
$$
\begin{align}
&\overbrace{\left(1+x+x^2+x^3+\dots\right)}^{\text{exponent = number of $\$1$}}\overbrace{\left(1+x^2+x^4+x^6+\dots\right)}^{\text{exponent = 2 $\times$ number of $\$2$}}\overbrace{\left... |
108,277 | <p>If Mochizuki's proof of abc is correct, why would this provide a new proof of FLT?</p>
<p>Edit: In proof of asymptotic FLT, does Mochizuki claim a specific value of n and if so what is this value?</p>
| Igor Rivin | 11,142 | <p>See <a href="http://mathworld.wolfram.com/abcConjecture.html" rel="nofollow">http://mathworld.wolfram.com/abcConjecture.html</a></p>
|
3,657,751 | <p>Consider the series <span class="math-container">$$\sum_{n=1}^{\infty}\frac{(-1)^{\frac{n(n+1)}{2}+1}}{n}=1+\dfrac12-\dfrac13-\dfrac14+\dfrac15+\dfrac16-\cdots.$$</span> This is clearly not absolutely convergent. On the other hand, obvious choice, alternating series does not work here. Seems like the partial sum seq... | Sil | 290,240 | <p>The sequence of partial sums is <span class="math-container">$s_n=s_{n-1}+\frac{(-1)^{T_n+1}}{n}$</span>, and if we can partition the sequence into two subsequences which converge to the same value, it implies the original sequence converges as well. This is usually done by looking at subsequences made by odd and ev... |
3,657,751 | <p>Consider the series <span class="math-container">$$\sum_{n=1}^{\infty}\frac{(-1)^{\frac{n(n+1)}{2}+1}}{n}=1+\dfrac12-\dfrac13-\dfrac14+\dfrac15+\dfrac16-\cdots.$$</span> This is clearly not absolutely convergent. On the other hand, obvious choice, alternating series does not work here. Seems like the partial sum seq... | Ali Shadhar | 432,085 | <p><span class="math-container">$$1\color{red}{+\frac12}\color{blue}{-\frac13}-\frac14+\frac15\color{cyan}{+\frac16}\color{magenta}{-\frac17}+\cdots$$</span></p>
<p><span class="math-container">$$=1\color{red}{-\frac12+2\cdot\frac12}\color{blue}{+\frac13-2\cdot\frac13}-\frac14+\frac15\color{cyan}{-\frac16+2\cdot\frac1... |
3,111,906 | <p>I have to show that for any <span class="math-container">$b >1$</span>, we have
<span class="math-container">$$ b^n > n$$</span>
for all <span class="math-container">$n$</span> sufficiently large, using only very basic analysis (no calculus). My attempt is as follows.</p>
<hr>
<p>We know that <span class="m... | Umberto P. | 67,536 | <p>If <span class="math-container">$b > 1$</span> you can write <span class="math-container">$b = 1 + x$</span> with <span class="math-container">$x > 0$</span> and by Bernoulli's inequality <span class="math-container">$$b^n = (1+x)^n \ge \frac{n(n-1)}{2}x^2.$$</span> Thus <span class="math-container">$$\frac{b^... |
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