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 |
|---|---|---|---|---|
2,957,481 | <p>How many ways are there to roll a die six times such that there are more ones than twos?</p>
<p>I broke this up into six cases: </p>
<p><span class="math-container">$\textbf{EDITED!!!!!}$</span></p>
<p><span class="math-container">$\textbf{Case 1:}$</span> One 1 and NO 2s --> 1x4x4x4x4x4 = <span class="math-conta... | dan_fulea | 550,003 | <p>If the question is "where is my error", than plese ignore this answer, which is giving an other way to count. The idea is that there are either more <span class="math-container">$1$</span>'s, case (1), or more <span class="math-container">$2$</span>'s, case (2), or they occur equaly often, case <span class="math-con... |
1,723,299 | <p>I can work out an heuristic argument for $n=2$: (homeomorphically) turning the disc $D^2$ to something like a funnel (no pipe of course), then gradually contracting the open "mouth" of the funnel to a smaller and smaller hole which eventually vanishes (or "heals", if you imagine the hole as a open wound), we have no... | Hagen von Eitzen | 39,174 | <p>I view $S^{n-1}$ and $D^n$ as subspaces of $\Bbb R^n$ in the obvious manner.
With the abbreviations $$\begin{align}r&:=\sqrt{x_1^2+\ldots +x_n^2}\\\alpha&:=2\arcsin r\\h&:=\begin{cases}\frac{\sin \alpha}{r}&r>0\\2&r=0\end{cases}\end{align}$$ (which is continuous!), the map $$(x_1,\ldots,x_n)\m... |
52,364 | <p>A given user can interact in multiple ways with a website. Let's simplify a bit and say say a user can:</p>
<ul>
<li>Post a message</li>
<li>Comment a message</li>
<li>"like" something on the website via Facebook</li>
</ul>
<p><em>(after that we could add, following the site on twitter, buying something on the sit... | Ilmari Karonen | 9,602 | <p>Well, one approach might be to just assign a fixed score for each action, sum the scores of all actions taken by the user, and then apply a saturating function like $f(x) = 1-\exp(-x)$ to the result. Of course, it may be easier to store the raw sum of scores internally and just apply $f$ when displaying it.</p>
<p... |
32,111 | <p>Let p : Y -> X be an n-sheeted covering map, where X and Y are topological spaces. If X is compact, prove that Y is compact.</p>
<p>I realize that this seems like a very simple problem, but I want to stress the lack of assumptions on X and Y. For example, this is very easy to prove if we can assume that X and Y are... | Greg Kuperberg | 1,450 | <p>A direct argument without the use of nets:</p>
<p>Let $\mathcal{C}$ be an open cover of $Y$. For each $p \in X$, choose an open set $p \in U \subseteq X$ such that $Y$ is trivial over $U$, and such that each lift of $U$ is contained in some element of $\mathcal{C}$. This is an open cover $\mathcal{D}$ of $X$, whi... |
619,607 | <p>$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ </p>
<p>My try: </p>
<p>$\ln y=\ln(8)^{\sqrt x}$<br>
$\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$<br>
I don't know how to proceed with right side. </p>
<p>$\mathbf{(2)}$ Find $y^{\prime}$ of $y=(t+4)(t+6)(t+7).$</p>
<p>This one I have no idea what to do so I don't ha... | Ragnar | 91,741 | <p>For (2), you get
$$
\ln y=\ln (t+4)+\ln (t+6)+\ln (t+7)
$$
Differentiating gives
$$
\frac{ y'}y=\frac1{t+4}+\frac1{t+6}+\frac1{t+7}
$$
thus
\begin{align}
y'&=y\left(\frac1{t+4}+\frac1{t+6}+\frac1{t+7}\right)\\
&=(t+4)(t+6)(t+7)\left(\frac1{t+4}+\frac1{t+6}+\frac1{t+7}\right)
\end{align}</p>
|
619,607 | <p>$\mathbf{(1)}$ Find $y^{\prime}$ of $y=8^{\sqrt x}$ </p>
<p>My try: </p>
<p>$\ln y=\ln(8)^{\sqrt x}$<br>
$\dfrac{1}{y}y^{\prime}=\sqrt{x}\ln8$<br>
I don't know how to proceed with right side. </p>
<p>$\mathbf{(2)}$ Find $y^{\prime}$ of $y=(t+4)(t+6)(t+7).$</p>
<p>This one I have no idea what to do so I don't ha... | Ross Millikan | 1,827 | <p>For 1, plug in the expression you started with for $y$ and simplify. For 2, I would not normally use logaritmic differentiation-I would just use the product rule. Quick and easy. But if must, you have the same approach: Assuming $t \gt -4, \log y= \log (t+4) +\dots$, differentiate to get $\frac {y'}y$, replace th... |
2,210,317 | <p>Simplify $\sum^{n}_{k=1} (-1)^k(n-k)!(n+k)!$. </p>
<p>I thought of interpreting a term $(n-k)!(n+k)!=\frac{(2n)!}{\dbinom{2n}{n-k}}$, but I do not know of any ways to sum this efficiently.</p>
| hxthanh | 58,554 | <p>$(-1)^k(n-k)!(n+k)!=(-1)^k(n-k)!(n+k)!\frac{n+k+1+(n-k+1)}{2n+2}$
$=\frac{1}{2n+2}\Delta\Big[-(-1)^{k+1}\big(n+1-(k+1)\big)!(n+k+1)!\Big]$
$$\;$$
Therefore
$$\;$$
$\displaystyle\sum_{k=1}^n(-1)^k(n-k)!(n+k)!=\frac{1}{2n+2}\bigg((-1)^n(2n+1)!-n!(n+1)!\bigg)$</p>
|
309,872 | <blockquote>
<p>Let $a$ and $b$ be elements of a group $G$. If $|a| = 12, |b| = 22$ and $\langle a \rangle \: \cap \: \langle b\rangle \ne e$, prove that $a^6 = b^{11}$.</p>
</blockquote>
<p>Any ideas where to start would be helpful.</p>
<p>Thanks.</p>
| Phani Raj | 42,031 | <p>First observe that orders of $a$ and $b$ are even and so there exists element $d$ such that $d^{2}=e$ in $\langle a\rangle$ and $\langle b\rangle$. Therefore $a^{6}=b^{11}=d$, such that $d^{2}=e$.</p>
|
1,217,805 | <p>Let's say I have a set of linearly independent vectors, collected in a square matrix $\mathbf{M}$.</p>
<p>I know that I could orthogonalize these vectors with the QR decomposition,</p>
<p>$\mathbf{M} = \mathbf{QR}$</p>
<p>where $\mathbf{Q}$ is my orthogonal set of vectors.</p>
<p>I'm curious if it is possible to... | Ben Grossmann | 81,360 | <p>If $M$ is a matrix of $k$ linearly independent columns with SVD $M = U \Sigma V^H$, then the first $k$ columns of $U$ form an orthonormal basis for the span of the columns of $M$.</p>
|
168,224 | <p>I am doing a project on Hamiltonian group actions on symplectic manifolds, and my supervisor was able to list several good books on Riemannian geometry to start me off, but he didn't know of any single place to learn about Hamiltonian groups. </p>
<p>I have found some books (even available online from the author!) ... | Community | -1 | <p>The best Book about Hamiltonian action is</p>
<p><a href="http://lib.org.by/get/M_Mathematics/MD_Geometry%20and%20topology/MDdg_Differential%20geometry/Ginzburg%20V.L.,%20Guillemin%20V.,%20Karshon%20Y.%20Moment%20maps,%20cobordisms,%20and%20Hamiltonian%20group%20actions%20%28AMS,%202002%29%28ISBN%200821805029%29%28... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Zhant | 493,028 | <p>A pretty commonly mentioned one is the <a href="https://en.wikipedia.org/wiki/Kempner_series" rel="noreferrer">Kempner series</a>, which is the Harmonic series but "throwing out" (omitting) the numbers with a 9 in their decimal expansion. And 9 is not special; you can generalize to any finite sequence of digits, and... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Descartes Before the Horse | 592,365 | <p>To add another; I was surprised when I learned the two sums:
<span class="math-container">$$\sum_{k=1}^{\infty}\frac1{k^2} = \frac{\pi^2}{6}$$</span>
<span class="math-container">$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^2} = \frac{\pi^2}{12}$$</span>
And thought the intuition behind the second coming from the famous fir... |
3,464,383 | <p>I couldn't find any substantial list of 'strange infinite convergent series' so I wanted to ask the MSE community for some. By <em>strange</em>, I mean infinite series/limits that <strong>converge when you would not expect them to and/or converge to something you would not expect</strong>.</p>
<p>My favorite conver... | Clement C. | 75,808 | <p>I still like the fact that
<span class="math-container">$$
\sum_{n=N}^\infty \frac{1}{n\ln n \cdot \ln \ln n \cdot \ln \ln \ln n \cdot \ln \ln \ln \ln n}
$$</span>
diverges, but
<span class="math-container">$$
\sum_{n=N}^\infty \frac{1}{n\ln n \cdot \ln \ln n \cdot \ln \ln \ln n \cdot (\ln \ln \ln \ln n)^{1.01... |
294,972 | <p>I am seeking a particular integral of the differential equation:</p>
<p>$u^{(4)}(t) - 5u''(t) + 4u(t) - t^3 = 0$</p>
<p>I simply interested in the technique, not just an answer (Mathematica suffices for that). How would one apply undetermined coefficients in this case?</p>
| Gerry Myerson | 8,269 | <p>The characteristic polynomial is $$z^4-5z^2+4=(z^2-1)(z^2-4)=(z-1)(z+1)(z-2)(z+2)$$ Since $z=0$ is not a root, the homogeneous equation has no polynomial solutions. Therefore, the idea David Mitra gives in the comments will work. </p>
|
35,579 | <p>Consider the kernel of the homomorphism from two copies of the free group
$F_2 \times F_2$ onto the integers sending every generator to 1. How to see that this subgroup is not finitely presented?</p>
| Derek Holt | 2,820 | <p>This is actually a standard example of a finitely generated but not finitely presented group. I seem to remember that there is a homological proof. I have been trying to think of a reasonably straightforward group-theoretical proof, but I am running out of time.</p>
<p>What I have done is to calculate an (infinite)... |
3,386,696 | <p>I need to solve
<span class="math-container">$\int_{0}^{\frac{\pi}{2}} \cos^6xdx$</span>
I tried to use <span class="math-container">$cos(3x)=4\cos^3x-3\cos x$</span> but not succeeded</p>
| Kavi Rama Murthy | 142,385 | <p><span class="math-container">$$P(a)=P(a|b)P(b)+P(a|b')P(b')\leq \max \{P(a|b),P(a|b')\} P(b)+\max \{P(a|b),P(a|b')\} P(b')$$</span> <span class="math-container">$$=\max \{P(a|b),P(a|b')\}$$</span> since <span class="math-container">$P(b)+P(b')=1$</span>. The left hand inequality is similar.</p>
|
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Mark Bennet | 2,906 | <p>Here is another way of approximating the square root of two by rational numbers which doesn't depend on the decimal system.</p>
<p>Suppose $p^2-2q^2=\pm 1$ so that $\left(\cfrac pq\right)^2=2\pm\cfrac 1{q^2}$, then the larger we can make $q$ the closer $\cfrac pq$ is to $\sqrt 2$.</p>
<p>Consider now $(p+2q)^2-2(p... |
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Arby | 154,968 | <p>The infinite series of the sum of the values of decimal places is converging, and therefore the value is finite. </p>
|
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Chalky | 155,049 | <p>1/3 of a pie is clearly finite. The only infinite part is the number of digits required to express absolute accuracy (i.e. you can't) when written in Base 10. In Base 3, it'd be 0.1. It is finite, but it also has absolute accuracy.</p>
|
818,850 | <p>I want to generate numbers $1$ to $10$ with uniform probability distribution. So I write the numbers $1$ to $10$ in the natural order. I keep writing the next $10$ number block by permuting the first $10$ numbers. Will this long string can be considered random with uniform distribution? Is there a problem?</p>
| Julie in Austin | 155,085 | <p>Many of the above answers are quite excellent, but lack a certain experiencing of both the never ending nature of the square root of 2 and its irrationality.</p>
<p>Take for example, 1/3. It's a rational number and it repeats forever. The most common method of representing that is 0.3 with a bar over the 3. The ... |
174,676 | <p>When I am faced with a simple linear congruence such as
$$9x \equiv 7 \pmod{13}$$
and I am working without any calculating aid handy, I tend to do something like the following:</p>
<p>"Notice" that adding $13$ on the right and subtracting $13x$ on the left gives:
$$-4x \equiv 20 \pmod{13}$$</p>
<p>so that $$x \eq... | DonAntonio | 31,254 | <p>When the prime is a reasonably small one I'd rather find directly the inverse:
$$9^{-1}=\frac{1}{9}=3\pmod {13}\Longrightarrow 9x=7\Longrightarrow x=7\cdot 9^{-1}=7\cdot 3= 21=8\pmod {13}$$
<strong>But</strong>...I try Gauss's method when the prime is big and/or evaluating inverses is messy.</p>
|
248,167 | <p>I want to know why, when I look at the Julia sets of the quadratic family, I see only a finite number of repeating patterns, rather than a countable infinity of them.</p>
<p>My question is specifically about the interaction of these three theorems:</p>
<p><strong>Theorem 1</strong>: Let $z_0\in\mathbb{C}$ be an re... | inyo | 73,630 | <p>When you consider infinitely renormalizable polynomial, you can see infinitely many completely different picture. For example, there exists $c$ such that you can find periodic points $x_n$ such that $K_c\setminus \{x_n\}$ has exactly $n$ components.</p>
<p>Such an infinitely renormalizable $c$ is in a infinite nest... |
79,868 | <p>Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its <em>Mellin transform</em> is defined by the function</p>
<p>$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$</p>
<p>and $f(y)$ can be r... | Frank Thorne | 1,050 | <p>Thanks to everyone who answered! A (CW'ed) summary of some of what I learned:</p>
<p>In the first place, I now cheerfully second Greg Martin's recommendation of Chapter 5.1 of Montgomery and Vaughan. It is a rather "lowbrow", very readable treatment. (doesn't prove Mellin inversion in complete generality)</p>
<p>A... |
79,868 | <p>Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its <em>Mellin transform</em> is defined by the function</p>
<p>$$M(f)(s) = \int_0^{\infty} f(y) y^s \frac{dy}{y},$$</p>
<p>and $f(y)$ can be r... | Michael Rubinstein | 31,582 | <p>Have a look at Zagier's appendix:
<a href="http://people.mpim-bonn.mpg.de/zagier/files/tex/MellinTransform/fulltext.pdf" rel="noreferrer">http://people.mpim-bonn.mpg.de/zagier/files/tex/MellinTransform/fulltext.pdf</a></p>
<p>It provides a nice description of the Mellin transform when $f(x)$ is sufficiently smooth ... |
76,747 | <pre><code>Plot[D[Abs[x], x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>gives out several lines of error messages and an empty plot.</p>
<pre><code>Plot[Derivative[1][Abs[#1] & ][x], {x, -10, 10}, Exclusions -> {0}]
</code></pre>
<p>just gives out an empty plot.</p>
<p>How do I plot $\left(\left|x... | kglr | 125 | <p>Using <code>PiecewiseExpand[Abs[x], x \[Element] Reals]</code> instead of <code>Abs[x]</code>:</p>
<pre><code>Plot[Evaluate@ D[PiecewiseExpand[Abs[x], x \[Element] Reals], x], {x, -10, 10},
PlotStyle -> Thick]
</code></pre>
<p><img src="https://i.stack.imgur.com/zUHAW.png" alt="enter image description here"><... |
588,725 | <p>Prove the following statement:</p>
<p>$$\frac{1}{x}<\ln(x)-\ln(x-1)<\frac{1}{x-1}$$</p>
<p>Proof:</p>
<p>$$\frac{-1}{x^2}<\frac{1}{x(x-1)}<\frac{-1}{(x-1)^2}$$</p>
<p>$$e^{(\frac{-1}{x^2})}<e^{(\frac{-1}{x(x-1)})}<e^{(\frac{-1}{(x-1)^2})}$$</p>
<p>$$\lim_{x\to\infty}e^{(\frac{-1}{x^2})}<\li... | Paramanand Singh | 72,031 | <p>By the nature of question we must have $x > 1$. Now use mean value theorem to get $$\ln(x) - \ln(x - 1) = \{x - (x - 1)\}\cdot\dfrac{1}{y} = \dfrac{1}{y}$$ for some $x - 1 < y < x$. Thus we get $$\frac{1}{x} < \frac{1}{y} < \frac{1}{x - 1}$$ so that $$\frac{1}{x} < \ln x - \ln (x - 1) < \frac{1}... |
1,792,422 | <p>I need help with an optimization problem. I have a rectangle space being fenced. Three sides are fenced with a material costing 4 dollars and the last side costs 16 dollars. I was given that the area is 250 and asked to minimize the cost.</p>
<p>I got what I think is the correct formula for the cost where $$C(y)=\... | Michael Hardy | 11,667 | <p>Since $f'(x) = \dfrac 1 {2\sqrt x}$, you need
$$
\frac{\sqrt 4 - \sqrt 0}{4-0} = \frac{f(4) - f(0)}{4-0} = \frac{f(b)-f(a)}{b-a} = f'(c) = \frac 1 {2\sqrt c}.
$$
If $\dfrac{\sqrt 4} 4 = \dfrac 1 {2\sqrt c}$ then what number is $c$?</p>
|
1,792,422 | <p>I need help with an optimization problem. I have a rectangle space being fenced. Three sides are fenced with a material costing 4 dollars and the last side costs 16 dollars. I was given that the area is 250 and asked to minimize the cost.</p>
<p>I got what I think is the correct formula for the cost where $$C(y)=\... | HighEnergy | 330,386 | <p>The above answers are correct; however, I would like to point out that in lieu of utilizing $$\frac{f(b)-f(a)}{b-a},$$ you utilized $f'(a)$ and $f'(b)$.
Before using <code>MVT</code>, make sure f is continuous on $[a,b]$ and differentiable on $(a,b)$, which they are here.</p>
|
370,898 | <p>Lately, I have been learning about the <strong>replication crisis</strong>, see <a href="https://www.youtube.com/watch?v=6abrUN823gY&ab_channel=Skeptic" rel="noreferrer">How Fraud, Bias, Negligence, and Hype Undermine the Search for Truth</a> (good YouTube video) — by Michael Shermer and Stuart Ritchie. Accordin... | Paul Siegel | 4,362 | <p>Mathematics does have its own version of the replicability problem, but for various reasons it is not as severe as in some scientific literature.</p>
<p>A good example is the <a href="https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups" rel="noreferrer">classification of finite simple groups</a> - t... |
370,898 | <p>Lately, I have been learning about the <strong>replication crisis</strong>, see <a href="https://www.youtube.com/watch?v=6abrUN823gY&ab_channel=Skeptic" rel="noreferrer">How Fraud, Bias, Negligence, and Hype Undermine the Search for Truth</a> (good YouTube video) — by Michael Shermer and Stuart Ritchie. Accordin... | fedja | 1,131 | <p><em>How can we expect that increasingly complicated proofs are replicated when so few people can understand them in the first place?</em></p>
<p>My answer to that is that we do not expect them to be replicated in the usual sense of this word (repeated and included into textbooks with just minor cosmetic and stylisti... |
370,898 | <p>Lately, I have been learning about the <strong>replication crisis</strong>, see <a href="https://www.youtube.com/watch?v=6abrUN823gY&ab_channel=Skeptic" rel="noreferrer">How Fraud, Bias, Negligence, and Hype Undermine the Search for Truth</a> (good YouTube video) — by Michael Shermer and Stuart Ritchie. Accordin... | Flounderer | 24,993 | <blockquote>
<p>Has this crisis impacted (pure) mathematics, or do you believe that
maths is mostly immune to it?</p>
</blockquote>
<p>Immune to the replication problem, yes. But not immune to the attitudes which cause scientists to do unreplicable research in the first place. Some mathematicians will announce that a p... |
23,818 | <p>I'm trying to define some notation so that Mathematica code would be more functional, similar to Haskell (just for fun): currying, lambdas, infix operator to function conversion, etc.. And I have some questions about it:</p>
<ul>
<li>Is it possible to make all Mathematica <code>h_[x1_,x2_,...]</code> functions to w... | swish | 2,490 | <p>That's how I finally defined haskell operators:</p>
<pre><code>rapply[x_] := x
rapply[x_, y__] := x[rapply[y]]
InfixNotation[ParsedBoxWrapper["|"], rapply]
lapply[x_] := x
lapply[x__, y_] := lapply[x][y]
InfixNotation[ParsedBoxWrapper["∘"], lapply]
InfixNotation[ParsedBoxWrapper["·"], Composition]
</code></pre>
... |
2,385,338 | <p>In a bag, there are 6 white disks, 6 black disks and 8 red disks. A disk is drawn at random from the bag.The colour is recorded and the disc is returned to the bag This process is repeated 10 times.
Find the probability that less than 4 red disks are drawn </p>
<p>I approached this question with n= 8 and p=0.4, q=... | LloydTao | 470,153 | <p>The probability P(R) of drawing a red disk is 8/20, which = 0.4.
You have 10 trials, therefore:</p>
<blockquote>
<p>R ~ B(10, 0.4).</p>
</blockquote>
<p>Using this distribution, you want to calculate:</p>
<blockquote>
<p>P(R < 4), or P(R ≤ 3), since it's discrete.</p>
</blockquote>
<p>Modern calculators w... |
2,022,529 | <p>$$\Large
\lim_{x\to0^+}\frac{1}{\sin^2x}\int_{\frac{x}{2}}^x\sin^{-1}t\,dt
$$</p>
<p>I am trying to calculate this limit. Using L’Hôpital’s rule, I am getting it as $1/4$, but the book says it's $3/8$. I don't know where I am doing the mistake.</p>
| Learnmore | 294,365 | <p>Let $A$ denote the set of people of two people who refuse to work together and $B$ denote the remaining group of $5$ people.</p>
<p>To form the committee there are two ways :</p>
<p>1.$3$ people are from $B$ only which is $5\choose 3$.</p>
<p>2.$2$ people are from $B$ and one from $A$ which is $5\choose 2$ $ 2\ch... |
42,192 | <p>In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the particles. As a consequence, the configuration space of a N-particle system is the set of the distances on a set of N e... | Suresh Venkat | 972 | <p>This is almost certainly not what you want, but it illustrates why you need a tighter specification of 'preferred'. </p>
<blockquote>
<p>Any $n$-point metric space can be
embedded isometrically in
$\ell_\infty^n$.</p>
</blockquote>
<p><strong>Proof</strong>: (this is a well known result): Let the $r^{th}$ co... |
1,460,704 | <p>Intuitively we know that $n^2$ grows faster than $n$, thus the difference tends to negative infinity. But I have trouble proving it symbolically because of the indeterminate form $\infty - \infty$. Is there anyway to do this without resorting to the Epsilon-Delta definition ?</p>
| Community | -1 | <p>Note that when $n\ge 2$, </p>
<p>$$n - n^2 \le n \left(\frac n2\right) - n^2 = -\frac{n^2}{2},$$</p>
<p>as
$$\lim_{n\to \infty} -\frac{n^2}{2} = -\infty,$$</p>
<p>then so are $\lim (n-n^2)$. </p>
|
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| Community | -1 | <p>The below image can easily be made into an inductive proof.</p>
<p><img src="https://i.stack.imgur.com/7N6Ol.png" alt="enter image description here"></p>
<p>To be precise, the above image first shows that
$$\dfrac12 + \dfrac14 + \dfrac18 + \cdots = 1$$ and then shows that</p>
<p>$$\dfrac1{2^2} + \dfrac1{(2^2)^2} ... |
382,295 | <p>Please help. I haven't found any text on how to prove by induction this sort of problem: </p>
<p>$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$</p>
<p>I can't quite get how one can prove such. I can prove basic divisible "inductions" but not this. Thanks.</p>
| ngoaho91 | 76,219 | <p>i have a solution, but not seem mathematical induction</p>
<p>$s = 1 + \dfrac1{4} + \dfrac1{4^2} + \dfrac1{4^3} + \dfrac1{4^4} + \cdots$</p>
<p>$\dfrac1{4}s = \dfrac1{4} + \dfrac1{4^2} + \dfrac1{4^3} + \dfrac1{4^4} + \dfrac1{4^5} + \cdots$</p>
<p>$\dfrac1{4}s = - 1 + (1 + \dfrac1{4} + \dfrac1{4^2} + \dfrac1{4^3} ... |
2,608,455 | <p>Could someone please help me with how do I calculate the sum of the $$\sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}$$ infinite series? I see that $$\lim_{n\rightarrow\infty}\frac{1}{4n^{2}-1}=0$$ so the series is convergent based on the Cauchy's convergence test. But how do I calculate the sum? Thank you.</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. By a fraction decomposition, one gets
$$
\frac{2}{4n^{2}-1}=\frac{1}{2n-1}-\frac{1}{2n+1}
$$ then one may use a <a href="http://mathworld.wolfram.com/TelescopingSum.html" rel="nofollow noreferrer">telescoping sum</a>.</p>
|
2,608,455 | <p>Could someone please help me with how do I calculate the sum of the $$\sum_{n=1}^{\infty}\frac{1}{4n^{2}-1}$$ infinite series? I see that $$\lim_{n\rightarrow\infty}\frac{1}{4n^{2}-1}=0$$ so the series is convergent based on the Cauchy's convergence test. But how do I calculate the sum? Thank you.</p>
| operatorerror | 210,391 | <p>Your justification for the convergence to he series needs work (having a limit of zero of the summand does not imply the sum converges!), it does however converge a priori by noting that
$$
\frac{1}{4n^2-1}=O\left(\frac{1}{n^2}\right)
$$
So it converges by the $p$-test.</p>
<p>You may also use the partial fraction... |
7,725 | <p>Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: <a href="http://www.stitz-zeager.com/szprecalculus07042013.pdf">http://www.stitz-zeager.com/szprecalculus07042013.pdf</a></p>
<p>However, it does not cover convergence,limits etc. and those topics we... | Jessica B | 4,746 | <p>Just to throw out a different answer: my immediate reaction was 'neither'. My experience in the UK has been that limits and convergence get the slightest whisper when you first meet differentiation and integration at high school, and at university they sit firmly under the heading 'analysis'.</p>
|
2,610,560 | <blockquote>
<p>Given that $x, y, z$ are positive integers and that $$3x=4y=7z$$ find
the minimum value of $x+y+z$. The options are:</p>
<p>A) 33</p>
<p>B) 40</p>
<p>C) 49</p>
<p>D) 61</p>
<p>E) 84</p>
</blockquote>
<p>My attempt:</p>
<p>$y=\frac{3}{4}x, z=\frac{3}{7}x$. </p>
<p>Substit... | cd7c3 | 522,571 | <p>You can think of the smallest value that $3x$, $4y$, and $7z$ can be as the smallest multiple of three that is also a multiple of 4 and 7, or as the $gcm(3,4,7)$, which equals $84$. So if the minimum value of $3x$ is $84$, the minimum value of $x$ is $28$. Using the same method you find that $y$ equals $21$ and that... |
3,433,383 | <p>Is it possible to write, let's say :- (a^2 + 1/a^2 - 1) as (a-b)^2.</p>
<hr>
<p>So I have to solve a problem (as practice, no test or assignment) that says to show if the roots of a given equation are real.</p>
<p>The equation is :- </p>
<p><span class="math-container">$$x^2 - 2\left(m + \frac1m \right)x + 3... | Michael Hoppe | 93,935 | <p>As <span class="math-container">$x^2-2ax+3=0\iff (x-a)^2=a^2-3$</span> we'll have two solution iff <span class="math-container">$a^2>3$</span>. Now
<span class="math-container">$$\left(m+\frac1m\right)^2=\left(m-\frac1m\right)^2+4.$$</span></p>
|
4,094,333 | <blockquote>
<p>Suppose <span class="math-container">$a_1,a_2>0$</span> and
<span class="math-container">$a_{n+2}=2+\dfrac{1}{a_{n+1}^2}+\dfrac{1}{a_n^2}(n\ge 1)$</span>. Prove
<span class="math-container">$\{a_n\}$</span> converges.</p>
</blockquote>
<p>First, we may show <span class="math-container">$\{a_n\}$</spa... | cybershiptrooper | 719,632 | <p>I'm not so sure about this one, but here's my attempt-</p>
<p><span class="math-container">\begin{align}
\lVert a_{n+1}-a_n \rVert &= \left\lVert \frac{1}{a_n^2}-\frac{1}{a_{n-2}^2} \right\rVert\\
&\leq \frac{ \lVert {a_{n-2}^2}-a_n^2 \rVert }{16} \\
&= \frac{ \lVert {a_{n-2}+a_{n}} \rVert \lVert {a_... |
2,888,464 | <p>Let $S$ be the set of polynomials $f(x)$ with integer coefficients satisfying </p>
<p>$f(x) \equiv 1$ mod $(x-1)$</p>
<p>$f(x) \equiv 0$ mod $(x-3)$</p>
<p>Which of the following statements are true?</p>
<p>a) $S$ is empty .</p>
<p>b) $S$ is a singleton.</p>
<p>c)$S$ is a finite non-empty set.</p>
<p>d) $S$... | A. Pongrácz | 577,800 | <p>The second condition basically says that $3$ is a root.
But then substitute $x=3$ into the first condition: $0=f(3)\equiv 1 \pmod 2$, a contradiction.</p>
|
3,200,431 | <p>We can say, that any field <span class="math-container">$\mathbb{K}$</span> -- <span class="math-container">$1$</span>-dim vector space on itself: <span class="math-container">$\mathbb{K}_{\mathbb{K}}$</span>. So any vector of one another finite-dimensional vector space <span class="math-container">$V_{\mathbb{K}}$<... | Dirk | 379,594 | <p>On the vector space, there is an addition that does not depend on the choice of basis you make, sometimes called a "natural" addition.<br>
However, you usually can't define a natural multiplication in your way, it would always depend on a basis and different bases will give different multiplications. Therefore, it i... |
3,044,390 | <p>As we know, each random variable is responsible for associating some random events to the probability values. These random events belong to the specific population, and that random variable represent that population. In other words, random variables are representatives of their populations. By virtue of this, they p... | Sener Ozonder | 258,967 | <p>Say you have a probability distribution function (PDF) which has a finite mean <span class="math-container">$\mu$</span> and variance <span class="math-container">$\sigma$</span>. This PDF does not have to be a well-known PDF like Poisson or Gaussian; it can have an uncommon profile.</p>
<p>1) If you generate from ... |
1,368,447 | <p><img src="https://i.stack.imgur.com/TL80w.jpg" alt="enter image description here"></p>
<p>So I tried the good old Calculus 1 approach and turned this into an optimization problem. The equations got REALLY hairy, but it was okay since this was the graphing calculator section of the exam. I called the longer part of ... | André Nicolas | 6,312 | <p>It is easier without a calculator. The area of the top triangle is $(1/2)(2)(5)\sin\theta$, where $\theta$ is the top angle. This is maximized when $\theta=\pi/2$. So the long diagonal has length $\sqrt{29}$. The other diagonal is now not hard to calculate, since the area of the kite is $10$.</p>
<p><strong>Remark:... |
865,639 | <p>I was doing some computations for research purposes, which led me to this integral:</p>
<p>$$I(n) = \int_0^{\infty} (t^2+t^4)^n e^{-t^2-t^4}\,dt.$$</p>
<p>This is very suggestively written so as to employ a parametric differentiation technique as so:</p>
<p>$$\left(\frac{\partial^n}{\partial \alpha^n}\right)\int_... | Daccache | 79,416 | <p>Not a full answer, but a partial one:<br />
Notice that the powers of two in the denominator are increasing in periodic repetitions of 3, 2, 1. That is, in this case the exponents on the 2 are 2, 5, 7, 8, 11, 13, 14, 17, 19, 20.... This is sequence <a href="https://oeis.org/A047268" rel="noreferrer">A047268</a> on t... |
2,331,255 | <p>Let $K$ be a set of all Killing vector fields on $\mathbf R^n$ (with the Euclidean metric $\bar g$) which vanish at the origin.</p>
<p>(A vector field $V$ on a Riemannian manifold $(M, g)$ is said to be a <strong><em>Killing vector field</em></strong> if the flow of $V$ acts by isometries of $M$. This is equivalent... | D Ford | 317,912 | <p>We need the following lemma, whose proof will be at the bottom of this answer. </p>
<p><strong>Lemma:</strong> If $\phi, \psi$ are Riemannian isometries on open subsets $U \to V$ of $\mathbb R^n$, with $\phi(p) = \psi(p)$ and $d\phi_p = d\psi_p$ for some $p \in U$, then $\phi = \psi$. </p>
<p>Now, suppose $V$ is a... |
2,675,758 | <p>For example:</p>
<p>There are two green marbles and two red marbles in a bag. Two marbles are chosen at random, what is the probability that the two marbles which have been chosen are of the same colour?</p>
<p>Ordered + distinct: <br>
Marbles = $G1, G2, R1, R2$ <br>
Event space = $\{|G1~G2|, |G2~G1|, |R1~R2|, |R2... | Siong Thye Goh | 306,553 | <p>$$(x+2)^2 = 100$$</p>
<p>$$x+2 = \pm 10$$
$$x=-2\pm10=-12 \text{ or} 8$$</p>
<p>Since $x>0$, reject $x=-12$ as a solution and hence $x=8$.</p>
|
3,489,896 | <p>It is clear to me that <span class="math-container">$\liminf X_n \le \limsup X_n$</span> but every time that I see that <span class="math-container">$\liminf X_n \le \sup X_n$</span> it sound to me not so obvious awkward. Could someone help me to understand this inequality?<span class="math-container">$ $</span> <sp... | Boka Peer | 304,326 | <p>Isn't it clear from the definition? Limsup is less than equal to sup. </p>
<p>Proof: </p>
<p>The definition of limsup is the following:
Inf{sup<span class="math-container">$(T_n)$</span>} where <span class="math-container">$T_n$</span> is the <span class="math-container">$n$</span>th tail of the sequence <span cla... |
115,630 | <p>Let $V$ and $W$ be two algebraic structures, $v\in V$, $w\in W$ be two arbitrary elements.</p>
<p>Then, what is the geometric intuition of $v\otimes w$, and more complex $V\otimes W$ ? Please explain for me in the most concrete way (for example, $v, w$ are two vectors in 2 dimensional vector spaces $V, W$)</p>
<p>... | Neal | 20,569 | <p>You want to stay concrete, so let's let $V$ be a two-dimensional real vector space and $W = \operatorname{Hom}(V,\mathbb{R})$. Then $V = T^1_0(V)$ and $W = T^0_1(V)$, so for any $v\in V$ and $w\in W$, $v\otimes w\in T^1_1(V)$.</p>
<p>Each of $v,w$ has two components; $v = v^1e_1 + v^2e_2$ and $w = w_1e^1 + w_2e^2$... |
3,649,125 | <p>Bridge is a game of four players in which each player is dealt 13 cards from a standard 52 card deck. Bridge players (such as myself) are interested in the number of possible deals, where each player is distinct. This can be counted by</p>
<p><span class="math-container">$$\binom{52}{13}\binom{39}{13}\binom{26}{13}... | Rayna | 751,499 | <p>(not enough rep to comment)</p>
<p>FWIW, <a href="https://en.wikipedia.org/wiki/Contract_bridge_probabilities#Number_of_possible_deals" rel="nofollow noreferrer">Wikipedia</a> cites <a href="http://home.planet.nl/~narcis45/CountingBridgeDeals.htm" rel="nofollow noreferrer">another site</a> which gives the same numb... |
1,200,878 | <p>Integrate the following integral:</p>
<p>$$\int \frac{\sin x (2 \cos x - \sin x)}{2\sin x + \cos x} dx$$</p>
<p>I have tried it by using by parts by considering the $\sin x$ as first function. Again in the following step i got stuck. Please help me.</p>
| Claude Leibovici | 82,404 | <p>You can even start from the beginning using the tangent half-angle substitution $t=\tan(\frac x2)$ which will lead to $$\int \frac{\sin x (2 \cos x - \sin x)}{2\sin x + \cos x} dx=\int \frac{8t(t^2 + t - 1)}{(1+t^2)^2(t^2-4t-1)} dt$$ Now continue with partial fraction decomposition taking into account the fact that ... |
638,971 | <p>What is the definition of $R_{ijkl}$ in terms of metrics on a manifold?</p>
<p>I know what the definition of the riemann tensor, $R^l_{ink}$, is. But what exactly is meant by $R_{ijlk}$?</p>
| alexjo | 103,399 | <p>$R_{ijkl}$ is the covariant version of the curvature tensor
$$
R_{ijkl} = g_{im} R^m{}_{jkl}
$$</p>
<p>If one defines the curvature tensor by
$$
R^\ell{}_{ijk}= \frac{\partial}{\partial x^j} \Gamma^\ell{}_{ik}-\frac{\partial}{\partial x^k}\Gamma^\ell{}_{ij} +\Gamma^\ell{}_{js}\Gamma_{ik}^s-\Gamma^\ell{}_{ks}\G... |
509,487 | <p>Prove $1+{n \choose 1}2+{n \choose 2}4+...+{n \choose n-1}2^{n-1}+{n \choose n}2^n=3^n$ using combinatorial arguments. I have no idea how to begin solving this, a nudge in the right direction would be appreciated. </p>
| peterwhy | 89,922 | <p>Consider a set of strings with $n$ characters made up of $a$, $b$ or $c$. The total number of such string is the sum of the number of strings with $n$ $a$'s, the number of strings with $n-1$ $a$'s, $\cdots$, down to the number of strings with no $a$'s.</p>
|
2,068,249 | <p>I am considering block matrices $$\begin{pmatrix} A & v \\ v^T & x \end{pmatrix}$$ with $A \in \mathbb{R}^{(n-1) \times (n-1)}$, $v \in \mathbb{R}^{n-1}$, $x \in \mathbb{R}.$ Is there a rational $(n-1) \times (n-1)$ expression $p(A,v,x)$ I can form in the variables $A,v,x$ such that $$\mathrm{det}\begin{pmat... | gjh | 37,021 | <p>In "Algebra" by J.W.Archbold, Pitman, 4e, 1970, s24.16, p.417, it proves, </p>
<p>$\mathrm{det}\begin{pmatrix} A & u^T \\ v & w \end{pmatrix} = -v.adj(A).u^T + w .det(A)$ </p>
<p>where $u,v$ are $1$ x $n$ and $A$ is $n$ x $n$ and $w$ is $1$ x $1$. </p>
|
2,068,249 | <p>I am considering block matrices $$\begin{pmatrix} A & v \\ v^T & x \end{pmatrix}$$ with $A \in \mathbb{R}^{(n-1) \times (n-1)}$, $v \in \mathbb{R}^{n-1}$, $x \in \mathbb{R}.$ Is there a rational $(n-1) \times (n-1)$ expression $p(A,v,x)$ I can form in the variables $A,v,x$ such that $$\mathrm{det}\begin{pmat... | Arin Chaudhuri | 404 | <p>Using multilinearity one can rewrite $\text{det}\begin{pmatrix} A & v \\ v^T &x \end{pmatrix} = \text{det}\begin{pmatrix} A & 0 \\ v^T &x \end{pmatrix} + \text{det}\begin{pmatrix} A & v \\ v^T & 0 \end{pmatrix} = x \text{det}(A)+\text{det}\begin{pmatrix} A & v \\ v^T & 0 \end{pmatri... |
680,660 | <p>Recently I came across this question:</p>
<p>Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima if it is greater than the number before and after it.) For example, if n=4 and our permutation was 1, 4, 2, 3... | Tom M | 131,247 | <p>Here is a numerical method that can be used to solve the problem for small K.</p>
<p>1) Brute force solve the problem for small x (say x = 3 to 12)<br>
2) Find a common denominator for each of your solutions (for K = 2, the common denominator is 90)<br>
3) put each solution in terms of the common denominator<br>
4)... |
145,156 | <p>I need to fetch some journals issues, in example like this:
www.sciencedirect.com/science/journal/00963003/124/1</p>
<p>You may check this part to be constant: </p>
<pre><code>string = "https://www.sciencedirect.com/science/journal/00963003"
</code></pre>
<p>I read <a href="http://mathematica.stackexchange.com/qu... | David G. Stork | 9,735 | <p>If the sequences can be enumerated in a Table, then try this:</p>
<pre><code>Table[
Import[
"http://www.sciencedirect.com/science/journal/00963003/124/" <> ToString[mycounter],
"HTML"],
{mycounter, 1, 3}]
</code></pre>
<p>or if a list of texts...</p>
<pre><code>... {mylist, {"myjournal1/1","myjournal1/... |
3,315,558 | <p>I am having a very frustrating time with the back book that says my answer is way off but to me everything looks fine:</p>
<p><span class="math-container">\begin{align*}
(xy)y'&= x^2+3y^2\\
y' &= \frac{x^2}{xy} + \frac{3y^2}{xy}\\
y' &= \frac{x}{y} + \frac{3y}{x}\\
y' &= \frac{1}{v} + 3v\\
y' &=... | Axion004 | 258,202 | <p>As you set </p>
<p><span class="math-container">$$v=\frac{y}{x}$$</span></p>
<p>you need to substitute it back into </p>
<p><span class="math-container">$$2v^2+1=Cx^4$$</span> </p>
<p>which forms</p>
<p><span class="math-container">$$2\Big(\frac{y}{x}\Big)^2 + 1 = Cx^4 \implies x^2+2y^2 = Cx^6$$</span></p>
|
3,259,475 | <p>I am looking for a nice slick way to show</p>
<blockquote>
<p><span class="math-container">$$\int^1_0 \frac{\tanh^{-1} x}{x} \ln [(1 + x)^3 (1 - x)] \, dx = 0.$$</span></p>
</blockquote>
<p>So far I can only show the result using brute force as follows.
Let
<span class="math-container">$$I = \int^1_0 \frac{\tan... | FDP | 186,817 | <p><span class="math-container">\begin{align}J=\int^1_0 \frac{\tanh^{-1} x}{x} \ln [(1 + x)^3 (1 - x)] \, dx\end{align}</span>
Perform the change of variable <span class="math-container">$y=\dfrac{1-x}{1+x}$</span>,
<span class="math-container">\begin{align}J&=\int^1_0\frac{\ln\left(\frac{16x}{(1+x)^4}\right)\ln x}... |
1,157,528 | <p>How to find the corresponding probability distribution function for the following distribution function ?</p>
<p>$$F (x)=
\left\{
\begin{array}{ll}
0 & \text{if } x<0 \\
x^2 & \text{if } 0\leq x\leq \frac{1}{2} \\
\frac{1}{25} \left(1-3 (3-x)^2\right) & \text{if }\frac{1}{2}<x\leq 3 \\
1 ... | SA-255525 | 135,922 | <p>Its pdf is
f(x)=2x if 0<=x<=1/2
and f(x)=6/25(3-x) if 1/2
|
153,473 | <p>How does one prove that if a $X$ is a Banach space and $x^*$, a continuous linear functional from $X$ to the underlying field, then $x^*$ attains its norm for some $x$ in $X$ and $\Vert x\Vert = 1$?</p>
<p>My teacher gave us a hint that we should use the statement that if $X$ is a reflexive Banach space, the unit b... | Davide Giraudo | 9,849 | <p>We can use a corollary of Hahn-Banach theorem, applied to the dual space $X'$ of $X$. We have
$$\lVert x'\rVert=\max_{y\in X'',\lVert y\rVert=1}|y(x')|$$
(the maximum is reached for a $y_0$ that can be constructed thanks to Hahn-Banach theorem).
For this $y_0\in X''$, since $X$ is reflexive we can find $u\in X$ suc... |
2,859,276 | <p>Given a measurable space $(X,\scr{A})$ where $\scr{A}$ is a sigma algebra on $X$, I would like to know if $f:X \to \mathbb{R}$ is measurable then $cf$ where $c \in \mathbb{R}$ is also measurable.</p>
<p>A function $f:X \to \mathbb{R}$ is measurable if $[f > \alpha] = \{x \in X \:{:}\: f(x) > \alpha \} \in \ma... | José Carlos Santos | 446,262 | <p>Since $\mathcal A$ is a $\sigma$-\algebra, asserting that$$\left\{x\in X\,\middle|\,f(x)<\frac\alpha c\right\}\in\mathcal A$$is equivalent to asserting that its complement, which is$$\left\{x\in X\,\middle|\,f(x)\geqslant\frac\alpha c\right\},\tag1$$belongs to $\mathcal A$. And$$(1)=\bigcap_{n\in\mathbb N}\left\{... |
1,927,650 | <p>In the diagram below, DE is a chord of the circle that goes through c,d and e. A is the center of the circle. The perpendicular line from centre A intersects DE at B and the circle at C, </p>
<p>DE=100cm
BC= 10cm
AB=X</p>
<p>I need to calculate the length of AB / x.</p>
<p>Basically i need to calculate the radius... | Community | -1 | <p>Extend $CA$ to cut the circle again at $F$. Then $BF=10+2x$ and by the intersecting chords property, $$10(10+2x)=50\times 50$$ Thus $x = 120$</p>
|
319,288 | <p>Let <span class="math-container">$\mathcal{M}$</span> be the field of meromorphic functions of one (complex) variable and <span class="math-container">$w = w(z)$</span> an analytic function satisfying a polynomial equation</p>
<p><span class="math-container">$P(w; z) := w^n + a_{n-1}(z) w^{n-1} + \cdots + a_1(z) w ... | Alexandre Eremenko | 25,510 | <p>First, as you noticed, it is enough to consider the case that the equation has the form
<span class="math-container">$$w^n+a_{n-1}(z)w^{n-1}+\ldots+a_0(z)=0,$$</span>
where the coefficients are entire. Then <span class="math-container">$w$</span> is holomorphic on its Riemann surface, let us call this Riemann surfa... |
2,912,043 | <p>How do I approach this problem? My book gives answer as $[\frac{3-\sqrt{5}}{2},\frac{3+\sqrt{5}}{2}]$. I tried forming an equation in $y$ and putting discriminant greater than or equal to zero but it didn't work. Would someone please help me?</p>
<p>I get $x^2 (y-1) + 2x (y+1) + (5y-5) =0$ and discriminant gives $2... | Kusma | 514,933 | <p>You have that $\sin(t)=-\sin (t-\pi)$. Apply to the integral over $(\pi,2\pi)$ and change variables $s=t-\pi$. </p>
|
2,661,113 | <p>Consider $n$ player numbered $1,2,\ldots,n$. If player $i$ fights against $j$ then $i$ wins with probability $i/(i+j)$. There are no ties.</p>
<p>A player $i_1$ is extracted at random. Then, a second different player $i_2$ is extracted at random. They fight against each other. </p>
<p>Then, we extract another play... | mjqxxxx | 5,546 | <p>For each subset $S\subseteq \{1,2,\ldots, n\}$, you can calculate the probability that each $i\in S$ is the winner of a random tournament of $S$. You will do this recursively. For singleton subsets, clearly the sole player is the winner. (Hooray!) For a set of size $m > 1$, iterate over the $m$ players, consi... |
1,730,569 | <p>All spaces are assumed Hausdorff. We call a topological space <em>compact</em> if every open cover has a finite subcover. We call it <em>Lindelöf</em> if every open cover has a countable subcover, and <em>hereditarily Lindelöf</em> if moreover every subspace is Lindelöf. </p>
<p>It is obvious that every compact spa... | Simon_Peterson | 271,607 | <p>Unfortunately, this is not true. Take for example, any uncountable set $X$ with the discrete topology (so that singletons are open). It is obviously not Lindelöf. </p>
<p>Now, 'create' the space $Y$ by adding a point $*$ to $X$ with neighbourhoods of the form $U=\{*\}\cup C$, where $C$ is a co-finite subset of $X$.... |
864,816 | <p>I was looking for examples of real valued functions $f$ such that $f(x)+f(-x)=f(x)f(-x)$. Preferably, I'd like them to be continuous, differentiable, etc.</p>
<p>Of course, there are the constant functions $f(x)=0$ and $f(x)=2$. I also showed that $1+b^x$, where $b>0$, is another solution. Are there any other ni... | Calvin Lin | 54,563 | <p>Hurkyl's solution is nice, but the change of variables obscures the inherent closed orbit property, which is the crucial part of the problem.</p>
<p>Observe that the functional equation only involves $x$ and $ -x$. In particular, if $h(x) = -x$, then the orbit of $x \neq 0$ is $\{x, -x\}$ and the orbit of $0$ is $\... |
2,993,299 | <p>If anyone can help me with how to go about solving these kind of equations i would really appreciate it. :-)</p>
<p><span class="math-container">$$\sqrt{36-2x^2} = 4$$</span></p>
<p>Solve for X</p>
| egreg | 62,967 | <p>The idea is doing
<span class="math-container">$$
\frac{x^2+1}{x-1}=\frac{x^2}{x}\frac{1+\dfrac{1}{x^2}}{1-\dfrac{1}{x}}
$$</span>
The limit of
<span class="math-container">$$
\frac{1+\dfrac{1}{x^2}}{1-\dfrac{1}{x}}
$$</span>
is <span class="math-container">$1$</span> by standard rules on limits. Hence there is <spa... |
3,629,379 | <p>I'm trying to understand the following proof that a space <span class="math-container">$X$</span> is compact if and only if every net has a cluster point. I have a specific confusion with how cluster points relate to closure which is expanded on after the proof. </p>
<p>A cluster point is defined as </p>
<p><a hre... | Community | -1 | <p>Your claim is true as soon as <span class="math-container">$|G|\ge 3$</span>, as a corollary of <a href="https://math.stackexchange.com/a/3063228/750041">this general result</a> by taking <span class="math-container">$H=\{e\}$</span>.</p>
|
2,643,486 | <p>I have a problem in my maths book which says </p>
<blockquote>
<p>Find the arithmetic sequence in which $T_8 = 11$ and $T_{10}$ is the additive inverse of $T_{17}$</p>
</blockquote>
<p>I don't have a first term of common difference to solve it, so I managed to make two equations to find the first term and common... | hmakholm left over Monica | 14,366 | <p>You seem to be overlooking the last three words of the question. The question doesn't want any old equivalence relation on $A$ you can come up with -- it want the <em>particular</em> equivalence relation <strong>"induced by $\pi_1$</strong>".</p>
<p>Perhaps you have missed that "the equivalence relation that such-a... |
2,819,652 | <p>I know that my function $f$ must satisfy the following condition for $x \geq 0:$
$$\frac{d\left(xf(x)-x\right)}{dx} \geq 0.$$
What can I say about $f?$ I am curious about its possible sign or variations with respect to $x$.</p>
<p>I have investigated Grönwall's inequality to find an upper bound for $f$ but it seems... | Dave | 334,366 | <p>Since $\sqrt[3] 2$ is a root of $f(x)=x^3-2$, the polynomial $x-\sqrt[3] 2$ divides $f(x)$. So $f(x)=(x-\sqrt[3] 2)(x^2+ax+b)$ for some real numbers $a,b$. If you perform the long division you can obtain $a$ and $b$, and then solve the quadratic equation $x^2+ax+b=0$ using the quadratic formula, which gives the two ... |
2,596,347 | <p>Suppose $X$ is compact and Hausdorff and that $f:X \to Y$ is continuous, closed, and surjective. How can I show that $Y$ is Hausdorff?</p>
| Kavi Rama Murthy | 142,385 | <p>Let $y_1 \neq y_2$ in Y. Pick $x_1 ,x_2$ such that $f(x_1)=y_1$ and $f(x_2)=y_2$ Claim: there exist neighborhoods U and V of $x_1$ and $x_2$ such that $f(U) \cap f(V)$ is empty. [ This is where continuity is used]. If not, for any neighborhoods U and V there exist points $t_{UV}$, $s_{UV}$ in U and V such that $f(t_... |
971,167 | <p>I'm having issues with this problem. I have solved for the eigenvalues but am having trouble finding the bases for both eigenvalues. The pictures below contain my work for solving for the eigenvalues and I solved for one of the basis but I think it is incorrect so I stopped. Can someone please help me. Thanks</p>
<... | Mustafa Said | 90,927 | <p>Your eigenvalues, $2 + i\sqrt{3}, 2 - i\sqrt{3}$ are correct. Now to solve the system in your picture above, multiply the 1st row by $\frac{1}{2}(-1-i\sqrt{3})$ and add it to the second row. This will turn the $-2$ in the second row to $0$ and you can then find your eigenvector(s).</p>
|
182,307 | <p>I have a long flat list that needs to be partitioned. The list is formatted so the "header" is repeated, followed by the values. Essentially, it looks something like this:</p>
<pre><code>list={a,a,1,2,3,b,b,5,6,c,c,1,5,a,a,7,8,9,1}
</code></pre>
<p>I am looking for an output of:</p>
<pre><code>{{a,1,2,3},{b,5,6... | user1066 | 106 | <pre><code>SequenceSplit[list, {a_, a_, x:_Integer ..} -> {a,x}]
</code></pre>
<blockquote>
<p>{{a, 1, 2, 3}, {b, 5, 6}, {c, 1, 5}, {a, 7, 8, 9, 1}}</p>
</blockquote>
|
116,220 | <p>Say two graphs are not isomorphic but are both strongly regular with the same set of parameters. Are there any parameters (other than the usual such as order, degrees, eigenvalues and multiplicities, etc.) that are determined, e.g., independence number, chromatic number, etc.?</p>
<p>Thanks for any help</p>
| Moh514 | 8,725 | <p>The diameter, energy and number of closed walks could be determined by parameters.</p>
|
3,570,084 | <blockquote>
<p>find the value of <span class="math-container">$f(0)$</span>, given <span class="math-container">$f(f(x))=0$</span> has roots as 1 & 2, <span class="math-container">$f(x) = x^2 + \alpha x + \beta$</span></p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>If <span class="math-container">$k_... | The 2nd | 751,538 | <blockquote>
<p><span class="math-container">$$f(x) = x^2 + \alpha x + \beta$$</span></p>
</blockquote>
<p>Since <span class="math-container">$f(f(x))=0$</span> has root at <span class="math-container">$x=1$</span> and <span class="math-container">$x=2$</span>, it implies <span class="math-container">$f(f(1))=0$</sp... |
1,546,231 | <p>I am working on a problem in which, for $a$, $b \gt 0$, we let $(x/a)^2 + (y/b)^2 = 1$ describe an ellipse.<br>
I am required to use the method of Lagrange multipliers and the corresponding second derivative test to find $a$ and $b$ such that the ellipse passes through $(x, y) = (\sqrt 2, 2)$ and such that the area ... | Dr. Sonnhard Graubner | 175,066 | <p>HINT: use the Lagrange Multiplier method
$$F=\pi ab+\lambda(\frac{2}{a^2}+\frac{4}{b^2}-1)$$</p>
|
1,546,231 | <p>I am working on a problem in which, for $a$, $b \gt 0$, we let $(x/a)^2 + (y/b)^2 = 1$ describe an ellipse.<br>
I am required to use the method of Lagrange multipliers and the corresponding second derivative test to find $a$ and $b$ such that the ellipse passes through $(x, y) = (\sqrt 2, 2)$ and such that the area ... | Dylan | 135,643 | <p>Your constraint function is
$$ f(a,b) = \frac{2}{a^2} + \frac{4}{b^2} - 1 $$
and your minimization function is $g(a,b) = \pi ab$</p>
<p>Using the Lagrange multiplier method, this requires $\nabla g = \lambda \nabla f $, thus we have the following system</p>
<p>$$ \pi b = -\lambda\frac{4}{a^3} $$
$$ \pi a = -\lamb... |
81,016 | <p>I'm trying to learn about derived categories of algebraic stacks. To be honest, as of now, I don't need anything fancy nor deep. In my setup I have a scheme $X$ (well, a smooth and projective variety over $\mathbb{C}$) and a finite group $G$ acting on it. I would like to understand simple things like: what does a co... | Aaron Bergman | 947 | <p>Bridgeland, King and Reid, "<a href="http://www.ams.org/journals/jams/2001-14-03/S0894-0347-01-00368-X/S0894-0347-01-00368-X.pdf" rel="nofollow">The McKay correspondence as an equivalence of derived categories</a>" has a nice discussion expanding on David Roberts's comment.</p>
|
429,178 | <p>Consider the following function $$ \Pi(t) = \begin{cases} 1 \ \ \text{if} \ |t| < 1/2 \\ 0 \ \ \text{if} \ |t| \geq 1/2 \end{cases} $$</p>
<p>This is not a periodic function. Why can we define the fourier coefficients of $\Pi(t)$ as $$c_n = \frac{1}{T} \int_{-T/2}^{T/2} e^{-2 \pi int/T} \Pi(t) \ dt $$</p>
<p>$$... | bob.sacamento | 61,250 | <p>Fourier coefficients are initially explained and taught as having applying to periodic functions, since they involve $\sin()$ and $\cos()$ functions, and since they find great utility among periodic functions. But the process of taking the integral</p>
<p>$\displaystyle \int \exp(ikx) f(x) dx $</p>
<p>has no inhe... |
128,009 | <p>Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.</p>
<p>Thanks
Sundi</p>
| André Henriques | 5,690 | <p>I am tempted to answer your question by the negative:<br>
I believe that there does not exist any other, simpler, way of saying
"left-invariant pseudo-differential operator on $G$".</p>
<p>The reason is that if you look at the subset of smoothing operators (operators with smooth integral kernel), I already don't kn... |
155,860 | <p>I often stumble over the term "Lie superalgebra" (= "Lie algebra with a $\mathbb{Z}_2$ grading"). Obvious question: What about $\mathbb{Z}_3$ grading (and so on)? Is a Lie algebra with $\mathbb{Z}_n$ grading just the special case of a quantum Lie algebra $L(q)$ with $q$ being an $n$-th root of 1 (I only looked at th... | Can Hatipoglu | 9,469 | <p>A generalization of the idea of a Lie superalgebra exists and is known as $\epsilon$-Lie algebras or color Lie algebras. They generalize Lie superalgebras in the sense that the underlying vector space is $G$-graded instead of $\mathbb{Z}_{2}$, where $G$ is an arbitrary abelian group. Of course, one modifies the skew... |
2,796,711 | <p>I have the following question in hand. </p>
<p>If $\lambda_1,\cdots,\lambda_n$ are the eigenvalues of a given matrix $A \in M_n$, then prove that the matrix equation $AB - BA = \lambda B$ has a nontrivial solution $B \neq 0 \in M_n$, if and only if $\lambda = \lambda_i - \lambda_j$ for some $i,j$.</p>
| Pedro | 23,350 | <p>Consider the operator $B\mapsto [A,B]$. What are its eigenvalues?</p>
|
392,297 | <p>For which sets of primes <span class="math-container">$P$</span> is there a finite type <span class="math-container">$\mathbb{Z}$</span>-algebra <span class="math-container">$A$</span> such that<span class="math-container">$$p\in P\iff\mathrm{Hom}(A, \mathbb{F}_p)=\emptyset?$$</span>Do all the finite <span class="ma... | R. van Dobben de Bruyn | 82,179 | <p>The <a href="https://mathoverflow.net/a/392404">answer</a> of RP_ is correct, so my main contribution is cleaning it up, providing more detail, and uniformising the different cases.</p>
<p><strong>Definition.</strong> Let <span class="math-container">$\Omega$</span> be the set of prime numbers. For subsets <span cla... |
1,507,859 | <blockquote>
<p>$$\int \tan \sqrt {x} \,dx$$</p>
</blockquote>
<p>I was trying to solve this. But it took very long time and three pages. Could someone please tell me how to solve this quickly. </p>
| user21820 | 21,820 | <p>Your way is reasonable, but uses Bernoulli's inequality. There is another more direct way.</p>
<p>If the sequence is bounded from above, it has a supremum $c$ by the completeness axiom for reals, and by definition we have $a^k > c/a$ for some natural number $k$ because $c/a < c$ since $a > 1$. Thus $a^{k+1... |
73,767 | <p>I have the following expressions $g^{ab+cd+...}$ or in the full form <code>Power[g,Plus[Times[a,b],Times[c,d],...]]</code></p>
<p>How to convert this expression into</p>
<p><code>Power[g,Times[a,b]]Power[g,Times[c,d]]...</code>?</p>
<p>Here $...$ could be either nothing or one more, but of course it would be nice... | kglr | 125 | <pre><code>expr1 = Power[g, Plus[Times[a, b], Times[c, d]]];
Times @@ Defer /@ (expr1 /. Plus -> List)
(* or Times @@ HoldForm /@ (expr1 /. Plus -> List) *)
</code></pre>
<p><img src="https://i.stack.imgur.com/hJCBp.png" alt="enter image description here"></p>
<p>In version 10, you can also use</p>
<pre><code>... |
2,616,604 | <p>Why it is true that every odd number r that is not equivlant to p mod p has a odd number $s$ s.t</p>
<p>$rs\equiv 2p-1 \mod 2p $</p>
<p>My first thought is that i should have an inverse for $r$ in the ring so i can write.
$s\equiv r^{-1} (-1) \mod 2p $ but im not sure what in number theory tells me that $r^{-1} $ ... | Dietrich Burde | 83,966 | <p>Indeed, we can take $-1/r=-r^{-1}$ in $\mathbb{Z}/2p$, since we know that $r\not\equiv 0\bmod 2$ and $r\not\equiv 0\bmod p$ by assumption.</p>
|
1,046,582 | <p>Question : </p>
<p>$$\lim_{x \to 0} \frac{e^{\tan^3x}-e^{x^3}}{2\ln (1+x^3\sin^2x)}$$</p>
<p>Here I tried $\tan x$ and $\sin x$ expansion in numerator and denominator which are as follows : </p>
<p>$$\tan x =x +\frac{x^3}{3}+\frac{2x^5}{15} \cdots$$ </p>
<p>$$\sin x = x-\frac{x^3}{3!}+\frac{x^5}{5!}\cdots$$ </p>... | Siminore | 29,672 | <p>It looks like a bad guy. I used Wolfram Mathematica to compute
$$
\tan^3 x -x^3 = x^5+\frac{11}{15}x^7 + O(x^8)
$$
and
$$
e^{\tan^3 x -x^3}=1+x^5+\frac{11}{15}x^7 + O(x^8).
$$
Therefore
$$
e^{\tan^3 x}-e^{x^3}=x^5+\frac{11}{15}x^7 + O(x^8).
$$
Since
$$
\log(1+x^3 \sin^2 x) = x^5 +O(x^6),
$$
the limit exists and is... |
2,536,877 | <p>I have to evaluate this integral:
$$\int \frac{\tan^5 x}{\cos^3{x}} dx.$$</p>
<p>I guess I should use substitution, but I don't know what to substitute =(</p>
| klirk | 385,702 | <p>If you set $\frac 1 {\cos x}=y$, $dy = \frac {\sin x}{\cos^2 x} dx = \frac {\tan x} {\cos x} dx$.<br>
Also note that $\tan^2 = \frac 1 {\cos^2}-1$. Then your integral becomes
$$\int \tan^4 x \frac 1 {\cos^2 x} \frac {\tan x}{\cos x} dx =\int (y-1)^2 y^2 dy .$$</p>
<p>This should be easy to evaluate.</p>
|
1,287,236 | <p>Need to calculate $\sin 15-\cos 15$? In degrees. I got zero, but it is wrong. Though, it seems to me that I was solving correct, I was doing this was:
$\sin (45-30) - \cos (45-30) = ((\sqrt(6)-\sqrt(2)/4) - ((\sqrt(6)+\sqrt(2)/4) =0$</p>
| Community | -1 | <p>You just did some algebra wrong. </p>
<p>$$\frac{\sqrt{6}-\sqrt{2}}{4} - \frac{\sqrt{6}+\sqrt{2}}{4}=-\frac{\sqrt{2}}{4}-\frac{\sqrt{2}}{4}=-\frac{\sqrt{2}}{2}$$</p>
<p>WA confirms <a href="http://www.wolframalpha.com/input/?i=sin%2815%29-cos%2815%29" rel="nofollow">http://www.wolframalpha.com/input/?i=sin%2815%29... |
2,636,929 | <p>I know that it can be obtained by simply differentiating the equation and finding the roots of the derivative, but it is a lengthy and tricky process. I am looking for a faster and more straightforward way.</p>
<p>A more effective and quick way to find the answer via simple differentiation will also be appreciated.... | dezdichado | 152,744 | <p>Let $a = \arcsin x , b = \arccos x$, then you need to optimize $a^4+b^4$ with the constraint $a+b =\dfrac{\pi}{2}.$ An obvious lower bound follows from power-mean inequality: $$a^4+b^4\geq2\left(\dfrac{a+b}{2}\right)^4 = \dfrac{\pi^4}{8}.$$</p>
<p>In order to maximize though, you will need to resort to Lagrange mul... |
3,247,923 | <p>I’ve tried splitting this into the sum of two Chinese Dumbass triangles:</p>
<p>0</p>
<p>-5 0</p>
<p>0 15 -5</p>
<p>0 -5 0 0</p>
<p>And</p>
<p>1</p>
<p>2 2</p>
<p>2 -15 2</p>
<p>1 2 2 1</p>
<p>And that fails. Any hint or help would be appreciated!</p>
| W-t-P | 181,098 | <p>Let
<span class="math-container">$$ f(a,b,c) := a(a-b)(a-2b)+b(b-c)(b-2c)+c(c-a)(c-2a). $$</span>
The key observation is that the coefficient of <span class="math-container">$x^2$</span> in <span class="math-container">$f(a+x,b+x,c+x)$</span> is
<span class="math-container">$$ -(a-b)-(b-c)-(c-a) = 0; $$</span>... |
3,537,549 | <p>In the game of poker, five cards from a standard deck of 52 cards are dealt to each player.</p>
<p>Assume there are four players and the cards are dealt five at a time around the table until all four players have received five cards.</p>
<p>a. What is the probability of the first player receiving a royal flush (th... | EugenS | 588,430 | <p>Thanks hint of @Eric Towers. With his permission, I will write a more extended estimation. Despite the fact that the answer is still not completely agreed, the assessment is in any case true.</p>
<p>Remembering that <span class="math-container">$|e^{i \sin{\theta}}| = 1$</span> rewrite estimations:
<span class="mat... |
4,236,875 | <p>Today, I was solving a problem which is described below.</p>
<blockquote>
<p>Show that the equation of any circle passing through the points of intersection of the ellipse <span class="math-container">$(x + 2)^2 + 2y^2 = 18$</span> and the ellipse <span class="math-container">$9(x - 1)^2 + 16y^2= 25$</span> can be w... | hamam_Abdallah | 369,188 | <p>Let <span class="math-container">$ A(2,-1) $</span> and <span class="math-container">$ B(2,1) $</span>.</p>
<p>if <span class="math-container">$ C(a,b) $</span> is the center, then</p>
<p><span class="math-container">$$AC^2=BC^2 \implies$$</span></p>
<p><span class="math-container">$$(a-2)^2+(b+1)^2=$$</span>
<span ... |
1,484,880 | <p>i messed up an exam yesterday. given</p>
<p>$$\mathcal{B}:=\{(a,b]:a,b \in\mathbb{R}, a \le b\},$$</p>
<p>i was able to show that $\mathcal{B}$ is the base of a topology $\mathcal{T}$, and that $\mathcal{T}$ is finer than the standard topology. however, i couldn't find a countable dense subset in $(\mathbb{R},\mat... | Brian M. Scott | 12,042 | <p>HINT: To show that the space has no countable base, let $\mathscr{B}$ be any base for $\mathscr{T}$, and note that for each $x\in\Bbb R$ there must be some $B_x\in\mathscr{B}$ such that $x\in B_x\subseteq(x-1,x]$. Then show that if $x\ne y$, then $B_x\ne B_y$.</p>
|
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