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 |
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4,142,540 | <p>Let <span class="math-container">$V$</span> a vector subspace of dimension <span class="math-container">$n$</span> on <span class="math-container">$\mathbb R$</span> and <span class="math-container">$f,g \in V^* \backslash \{0\}$</span> two linearly independent linear forms. I want to show that <span class="math-con... | Boka Peer | 304,326 | <p>Comments:</p>
<p>The following post contains the key part of the proof of your problem. Tsemo has mentioned it in his comment.</p>
<p><a href="https://math.stackexchange.com/q/1088623/304326">Proving that linear functionals are linearly independent if and only if their kernels are different</a></p>
<p>A bit more exp... |
1,823,736 | <p><a href="http://www.math.drexel.edu/~dmitryk/Teaching/MATH221-SPRING'12/Sample_Exam_solutions.pdf" rel="nofollow">Problem 10c from here</a>.</p>
<blockquote>
<p>Thirteen people on a softball team show up for a game. Of the $13$ people who show up, $3$ are women. How many ways are there to choose $10$ players ... | Majid | 254,604 | <p>Well, according to your figure, you need to write the integral as a sum of two integrals.</p>
<p>$y = 25 - x^2\rightarrow x=\sqrt{25-y}$ (In the first quadrant).</p>
<p>$y = 25 - \frac{25}{3}x\rightarrow x= 3 - \frac{3}{25}y$.</p>
<p>$y = 9x - 27\rightarrow x=3+ \frac{1}{9}y$.</p>
<p>Now, for $0\leq y\leq 4$, we... |
4,312,890 | <p>I'm working my way through Grimaldi's textbook, and there's one exercise in the supplementary exercises for Chapter 4 that I don't understand how to approach.</p>
<p>Here is the problem: if <span class="math-container">$n \in Z^+$</span>, how many possible values are there for <span class="math-container">$gcd(n,n+... | MH.Lee | 980,971 | <p>There are exactly <span class="math-container">$\sigma_0(3000)=32$</span> values.</p>
<p>If <span class="math-container">$d$</span> is divisor of 3000, Then <span class="math-container">$d=\text{gcd}(d, d+3000)$</span>. And by Euclidean algorithm, <span class="math-container">$\text{gcd}(d, d+3000)=\text{gcd}(d, 300... |
71,166 | <p>This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.</p>
<p>We start with a formal power series</p>
<p>$P(C) = \sum_{n=0}^\infty a_n C^{n+1}$</p>
<p>where $a_n = (-1)^n n!$</p>
<p>With these coefficien... | Ralph | 10,194 | <p>I post this as an answer since this way it's more easy to read. But it's a successive comment to my comment above. </p>
<p>The formula I mean is the following: Let
$$f(x) = 1 + \sum_{n >0}b_nx^n$$ be a formal power series over a commutative ring with unit. Then $$1/f = 1 + \sum_{n>0}c_nx^n,$$
$$c_n = \sum_{... |
71,166 | <p>This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series.</p>
<p>We start with a formal power series</p>
<p>$P(C) = \sum_{n=0}^\infty a_n C^{n+1}$</p>
<p>where $a_n = (-1)^n n!$</p>
<p>With these coefficien... | StevenJ | 4,673 | <p>I stumbled across a similar question on math overflow <a href="https://mathoverflow.net/questions/45811/use-of-everywhere-divergent-generating-functions">here</a>. That had lot of insights into P including a simple continued fraction expression for P. Details in a paper <a href="http://www.springerlink.com/content/w... |
2,263,230 | <p>Let's say I wanted to express sqrt(4i) in a + bi form. A cursory glance at WolframAlpha tells me it has not just a solution of 2e^(i<em>Pi/4), which I found, but also 2e^(i</em>(-3Pi/4))</p>
<p>Why do roots of unity exist, and why do they exist in this case? How could I find the second solution? </p>
| Community | -1 | <p>$\mathbf{i}$ is a root of unity. Thus any root of $\mathbf{i}$ is a root of unity.</p>
<p>How do you <em>usually</em> find the other square root of a number given one of its square roots? Same thing applies here.</p>
<p>Alternatively, recall that the exponential has period $2 \pi \mathbf{i}$. If you wrote</p>
<p>... |
2,263,230 | <p>Let's say I wanted to express sqrt(4i) in a + bi form. A cursory glance at WolframAlpha tells me it has not just a solution of 2e^(i<em>Pi/4), which I found, but also 2e^(i</em>(-3Pi/4))</p>
<p>Why do roots of unity exist, and why do they exist in this case? How could I find the second solution? </p>
| John Lou | 404,782 | <p>Roots of unity are basically the idea of multiple square roots extended to any level of square roots. First, we have to understand the complex plane. You might be familiar with the equation $e^{i \pi} = -1$. This expression is equivalent to another, $(e^{i \theta} = \cos \theta + i\sin \theta)$. Based on the propert... |
2,093,720 | <p>$$y~ dy+(2+x^2-y^2)dx$$</p>
<p>I try to solve this equation by putting standard form but becomes more challenge . So your answer is helpful </p>
| Robert Israel | 8,508 | <p>The minimum of $f(a,b,c) = a/b + b/c + c/a$ for $a, b, c > 0$ is on $a=b=c$, where $f(a,a,a) = 3$. Since this is greater than $2017/1000$, there are no solutions.</p>
|
3,553,681 | <p>consider the identity
<span class="math-container">$$\frac{e^{-x}}{1-x}=\sum_{n=0}^{\infty}c_nx^n$$</span></p>
<p>Show that for each <span class="math-container">$n\ge0$</span>
<span class="math-container">$$\sum_{k=0}^{n}\frac{c_k}{(n-k)!}=1$$</span></p>
<p>My trial :
By cauchy product, </p>
<p><span class="mat... | Community | -1 | <p>May I offer some comments designed to complement the answer of Jean-Claude Collette. Let <span class="math-container">$(H,\langle \cdot,\cdot\rangle)$</span> be a Hilbert space and <span class="math-container">$V\subset H$</span> a subspace. An orthogonal projector <span class="math-container">$\Pi:H\to V$</span> wi... |
85,374 | <p>I'm currently using <code>WolframLibraryData::Message</code> to generate messages from a library function, like this:</p>
<pre><code>Needs["CCompilerDriver`"]
src = "
#include \"WolframLibrary.h\"
DLLEXPORT mint WolframLibrary_getVersion() {return WolframLibraryVersion;}
DLLEXPORT int Wolfra... | Szabolcs | 12 | <p>The simple way is what <a href="https://mathematica.stackexchange.com/a/85393/12">Simon described in his answer</a>.</p>
<p>A more flexible way is described under <a href="http://reference.wolfram.com/language/LibraryLink/tutorial/LibraryStructure.html#55120353" rel="nofollow noreferrer">Callback Evaluations in the... |
24,810 | <p>The title says it all. Is there a way to take a poll on Maths Stack Exchange? Is a poll an acceptable question?</p>
| Gerry Myerson | 8,269 | <p>Vote this answer up, if you think a poll is an acceptable question. </p>
|
3,973,611 | <p>Let <span class="math-container">$$F(x)=\int_{-\infty}^x f(t)dt,$$</span>
where <span class="math-container">$x\in\mathcal{R}$</span>, <span class="math-container">$f\geq 0$</span> is complicated (it cannot be integrated analytically).</p>
<p>Can I used the Simpson's rule to approximate this integral, knowing that <... | Math Lover | 801,574 | <p>I would simplify the expression a bit by combining set of no capital letter and set of one capital letter as there is no intersection between them.</p>
<p>If <span class="math-container">$A$</span> is the set of passwords where we have at most one capital letter, <span class="math-container">$B$</span> is the set of... |
2,950,813 | <blockquote>
<p>Take <span class="math-container">$G$</span> to be a group of order <span class="math-container">$600$</span>. Prove that for any element <span class="math-container">$a$</span> <span class="math-container">$\in$</span> G there exist an element <span class="math-container">$b$</span> <span class="math... | AHusain | 277,089 | <p>Take the cyclic subgroup generated by <span class="math-container">$a$</span>. It has some order <span class="math-container">$k$</span> so <span class="math-container">$a^{1+kn}=a$</span> for all <span class="math-container">$n$</span>. Let <span class="math-container">$b=a^l$</span></p>
<p><span class="math-conta... |
1,291,050 | <p>I have been doing doing this problem $∇ × (\varphi∇\varphi)=0$</p>
<p>I am just having trouble applying the product result i get which is below.</p>
<p>$$i(( \frac {d}{dy} )(\varphi \frac {d}{dz} \varphi) - ((\frac {d}{dz})(\varphi \frac {d}{dy} \varphi)) )$$</p>
<p>if i take the first part </p>
<p>$$(\varphi \f... | Censi LI | 223,481 | <p>I guess the basic field you concerned is $\mathbb R$? Then by the hypothesis, we can write $A=H+S$, where $H$ is a positive definite symmetric matrix and $S$ is skew-symmetric. Then since $H$ is positive definite, there is a invertible matrix $P$ such that $P^THP=I$, note that $P^TSP$ is still skew-symmetric, so the... |
302,179 | <p>The question I am working on is:</p>
<blockquote>
<p>"Use a direct proof to show that every odd integer is the difference of two squares."</p>
</blockquote>
<p>Proof:</p>
<p>Let n be an odd integer: $n = 2k + 1$, where $k \in Z$</p>
<p>Let the difference of two different squares be, $a^2-b^2$, where $a,b \in Z... | Herng Yi | 34,473 | <p>Note that $a^2 - b^2 = (a + b)(a - b)$. Solve the simultaneous equations $a + b = n$ and $a - b = 1$.</p>
<p>This is where you got $(k + 1)^2 - k^2$ from - $(a + b)(a - b)$ then matches to $((k + 1) + k)((k + 1) - k)$.</p>
|
198,116 | <p>A finite <a href="http://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow noreferrer">lattice</a> is planar if it admits a <a href="http://en.wikipedia.org/wiki/Hasse_diagram" rel="nofollow noreferrer">Hasse diagram</a> which is a <a href="http://en.wikipedia.org/wiki/Planar_graph" rel="nofollow noreferrer">pla... | Richard Stanley | 2,807 | <p>A stronger result is due to R. Wille. See for instance page 3 of <a href="http://www.math.uh.edu/~hjm/1973_Lattice/p00512-p00518.pdf">http://www.math.uh.edu/~hjm/1973_Lattice/p00512-p00518.pdf</a>.</p>
|
198,116 | <p>A finite <a href="http://en.wikipedia.org/wiki/Lattice_(order)" rel="nofollow noreferrer">lattice</a> is planar if it admits a <a href="http://en.wikipedia.org/wiki/Hasse_diagram" rel="nofollow noreferrer">Hasse diagram</a> which is a <a href="http://en.wikipedia.org/wiki/Planar_graph" rel="nofollow noreferrer">pla... | David Eppstein | 440 | <p>It's already been answered positively, but here's another argument that shows something a little stronger: every finite distributive lattice either contains a B3 (and is not planar) or it can be drawn as a planar grid graph.</p>
<p>By Birkhoff's representation theorem, every finite distributive lattice is isomorphi... |
3,853,509 | <blockquote>
<p>prove <span class="math-container">$$\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$$</span> holds for all positives <span class="math-container">$a,b,c$</span> when <span class="math-container">$\sqrt{a}+\sqrt{b}+\sqrt{c}=3$</span> or <span class="math-container">$ab+bc+ca=3$</span></p>
</blockquote>
<hr />
<p><str... | Claude Leibovici | 82,404 | <p><strong>Hint</strong></p>
<p>Rewrite <span class="math-container">$$\ f(x) = \frac{1}{2+3x^2}=\frac 12 \times\frac 1 {1+\frac 32 x^2}$$</span> Now, let <span class="math-container">$t=\frac 32 x^2$</span>.</p>
<p>Expand in terms of <span class="math-container">$t$</span> and, when done, replace <span class="math-con... |
3,540,613 | <p>The integral to solve:</p>
<p><span class="math-container">$$
\int{5^{sin(x)}cos(x)dx}
$$</span></p>
<p>I used long computations using integration by parts, but I don't could finalize:</p>
<p><span class="math-container">$$
\int{5^{sin(x)}cos(x)dx} = cos(x)\frac{5^{sin(x)}}{ln(5)}+\frac{1}{ln(5)}\Bigg[ \frac{5^{s... | José Carlos Santos | 446,262 | <p>If you do <span class="math-container">$\sin x=u$</span> and <span class="math-container">$\cos x\,\mathrm dx=\mathrm du$</span>, your integral becomes<span class="math-container">$$\int 5^u\,\mathrm du.$$</span></p>
|
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| John D. Cook | 136 | <p>The golden ratio was used extensively in ancient art, but the man named Fibonacci (Leonardo of Pisa) lived around 1200 AD. It's possible that the Fibonacci series was known before Fibonacci but I'm not aware of this. So I think it's safe to assume the golden ratio is older.</p>
|
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| Mensen | 774 | <p>The answer for either of these is "hundreds of millions of years" due to their emergence/use in biological development programs, the self-assembly of symmetrical viral capsids (the adenovirus for example), and maybe even protein structure. Because of their close relationship I'd be hard pressed to say which 'came f... |
302 | <p>I know that the Fibonacci numbers converge to a ratio of .618, and that this ratio is found all throughout nature, etc. I suppose the best way to ask my question is: where was this .618 value first found? And what is the...significance?</p>
| David Eppstein | 440 | <p>The golden ratio in mathematics dates back to the Pythagoreans, circa 500 BC, it's true. But the Fibonacci numbers also have a long heritage going back to Pingala in India circa 200 BC.</p>
<p>However, the mystical claims about the golden ratio and Fibonacci numbers going back hundreds of millions of years in biolo... |
1,853,846 | <p>Prove that the equation <span class="math-container">$$x^2 - x + 1 = p(x+y)$$</span> has integral solutions for infinitely many primes <span class="math-container">$p$</span>.</p>
<p>First, we prove that there is a solution for at least one prime, <span class="math-container">$p$</span>. Now, <span class="math-cont... | Faibbus | 307,662 | <p>Or you can use the half angle formulas:</p>
<p>$\cos(\theta) = \frac{1 - \tan^2(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})} $</p>
<p>$\sin(\theta) = \frac{2 \tan(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})}$</p>
<p>So as to get:</p>
<p>$ 8 \frac{2 \tan(\frac{\theta}{2})}{1 + \tan^2(\frac{\theta}{2})} = ... |
161,678 | <p>Assume a process with Itô dynamics of the generic form
$$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$</p>
<p>and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ?
$$g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$$</p>
<p>I remember comming upon the proof of above once but I cannot find it ... | Thomas | 46,773 | <p>I assume here the $x$ variable is the initial condition of you process? Here is a partial answer: if $(\mathcal F_t)$ is the brownian filtration, then by Itô's martingale representation theorem for any $x$ there is a predictable process $(H_t^x)$ such that $g(t,x)=\mathbb E(f(X_T))+ \int_0^t H^x_s dW_s$. So in that ... |
153,923 | <p>My question is: Solve $\sqrt{x^2 +2x + 1}-\sqrt{x^2-4x+4}=3$</p>
<p>I deduced that:$LHS= x+1-(x-2)$</p>
<p>I am unable to solve this equation. I would like to get some hints to solve it.</p>
| Gigili | 181,853 | <p>$$\sqrt {x^2 +2x + 1}-\sqrt { x^2-4x+4}= \sqrt{(x+1)^2} - \sqrt{(x-2)^2}=|x+1|-|x-2|$$</p>
<p>You have to consider three cases:</p>
<ul>
<li>$x \geq 2$</li>
<li>$-1<x<2$</li>
<li>$x \leq -1$</li>
</ul>
|
153,923 | <p>My question is: Solve $\sqrt{x^2 +2x + 1}-\sqrt{x^2-4x+4}=3$</p>
<p>I deduced that:$LHS= x+1-(x-2)$</p>
<p>I am unable to solve this equation. I would like to get some hints to solve it.</p>
| Madrit Zhaku | 34,867 | <p>$\sqrt {x^2 +2x + 1}-\sqrt { x^2-4x+4}= \sqrt{(x+1)^2} - \sqrt{(x-2)^2}=|x+1|-|x-2|=3$</p>
<p>$|x+1|-|x-2|=3$</p>
<p>1) $x\in(-\infty, -1)$$\Rightarrow$$|x+1|=-(x+1)=-x-1$, $|x-2|=-(x-2)=2-x$.</p>
<p>$|x+1|-|x-2|=3$$\Rightarrow$ $-x-1-2+x=3$$\Rightarrow$$-3=3$, this is a contradiction.</p>
<p>In this interval eq... |
3,997,632 | <p>Use the Chain Rule to prove the following.<br />
(a) The derivative of an even function is an odd function.<br />
(b) The derivative of an odd function is an even function.</p>
<p><strong>My attempt:</strong></p>
<p>I can easily prove these using the definition of a derivative, but I'm having trouble showing them us... | Gibbs | 498,844 | <p>Actually, you are using the assumption that <span class="math-container">$f$</span> is even.</p>
<p>If <span class="math-container">$f$</span> is even, then as you say <span class="math-container">$f(-x) = f(x)$</span>. By differentiating, the left hand side is <span class="math-container">$-f'(-x)$</span> by the ch... |
3,997,632 | <p>Use the Chain Rule to prove the following.<br />
(a) The derivative of an even function is an odd function.<br />
(b) The derivative of an odd function is an even function.</p>
<p><strong>My attempt:</strong></p>
<p>I can easily prove these using the definition of a derivative, but I'm having trouble showing them us... | J.G. | 56,861 | <p>Since <span class="math-container">$(-x)^\prime=-1$</span>, we can concisely prove both parts viz.<span class="math-container">$$f(-x)=\pm f(x)\implies f^\prime(-x)=-1\cdot\pm f^\prime(x)=\mp f^\prime(x).$$</span></p>
|
3,335,060 | <blockquote>
<p>The numbers of possible continuous <span class="math-container">$f(x)$</span> defiend on <span class="math-container">$[0,1]$</span> for which <span class="math-container">$I_1=\int_0^1 f(x)dx = 1,~I_2=\int_0^1 xf(x)dx = a,~I_3=\int_0^1 x^2f(x)dx = a^2 $</span> is/are</p>
<p><span class="math-container"... | Community | -1 | <p><span class="math-container">$$1+3+\frac92+\frac92+\frac{27}8+\frac{81}{40}+\frac{81}{80}+\frac{243}{560}+\frac{729}{4480}\\
13+3.375+2.025+1.025+0.433928\cdots+0.162723\cdots=20.021651$$</span></p>
<p>isn't so difficult. Only the last two term require a "true" division. </p>
|
18,444 | <p>I am a student, in my last year of school(17 years old)</p>
<p>When I was about 13 years old I fell into the <a href="https://artofproblemsolving.com/news/articles/avoid-the-calculus-trap" rel="noreferrer">calculus trap</a> by starting off learning trigonometry on my own, when I was supposed to factor equations or ... | Aviral Sood | 14,177 | <p>[VERY LONG ANSWER, needs patience to read through]</p>
<p>I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school syllabus and so it is simple for them and after some practise and memorisation they are done. Then they seek out... |
838,400 | <p>One question asking if $\mathbb{Z}^*_{21}$ is cyclic.</p>
<p>I know that the cyclic group must have a generator which can generate all of the elements within the group.</p>
<p>But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine... | rogerl | 27,542 | <p>If an abelian group has elements of order $m$ and $n$, then it also has an element of order $lcm(m,n)$, so that is another potential way to cut down on the work you have to do.
So, for example, if you were to find an element of order $4$ and one of order $3$ in your group, you would know that there would have to be ... |
838,400 | <p>One question asking if $\mathbb{Z}^*_{21}$ is cyclic.</p>
<p>I know that the cyclic group must have a generator which can generate all of the elements within the group.</p>
<p>But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine... | KCd | 619 | <p>In practice, to show $(\mathbf Z/n\mathbf Z)^\times$ is not cyclic you can look for a "fake" square root of $1$, i.e., a solution to $a^2 \equiv 1 \bmod n$ with $a \not\equiv \pm 1 \bmod n$. Then there are at least two subgroups of order $2$, so this group is not cyclic. </p>
|
2,620,032 | <p>Find the derivative of $y=(\tan (x))^{\log (x)}$</p>
<p>I thought of using the power rule that:
$$\dfrac {d}{dx} u^n = n.u^{n-1}.\dfrac {du}{dx}$$
Realizing that the exponent $log(x)$ is not constant, I could not use that. </p>
| Usermath | 496,395 | <p>Note that $$\log(y) = \log(x).\log(\tan(x))~.$$ Now, apply chain rule on both sides.</p>
|
2,620,032 | <p>Find the derivative of $y=(\tan (x))^{\log (x)}$</p>
<p>I thought of using the power rule that:
$$\dfrac {d}{dx} u^n = n.u^{n-1}.\dfrac {du}{dx}$$
Realizing that the exponent $log(x)$ is not constant, I could not use that. </p>
| Tsemo Aristide | 280,301 | <p>$(\tan(x))^{\ln(x)}=\exp(\ln x(\ln(\tan x)))$ by apply the formula off the derivative of a composition:</p>
<p>$f'(x)=(\ln(x)\ln(\tan(x))'\exp(\ln(\tan(x))$.</p>
<p>$(\ln(x)(\ln(\tan(x)))'={1\over x}\ln(\tan(x))+\ln(x){1\over {\tan(x)}}{1\over{cos(x)^2}}$.</p>
|
2,620,032 | <p>Find the derivative of $y=(\tan (x))^{\log (x)}$</p>
<p>I thought of using the power rule that:
$$\dfrac {d}{dx} u^n = n.u^{n-1}.\dfrac {du}{dx}$$
Realizing that the exponent $log(x)$ is not constant, I could not use that. </p>
| Michael Rozenberg | 190,319 | <p>$$\left(\left(\tan{x}\right)^{\ln{x}}\right)'=\left(e^{\ln{x}\ln\tan{x}}\right)'=e^{\ln{x}\ln\tan{x}}\left(\ln{x}\ln\tan{x}\right)'=$$
$$=\left(\tan{x}\right)^{\ln{x}}\left(\frac{\ln\tan{x}}{x}+\frac{\ln{x}}{\tan{x}}\cdot\frac{1}{\cos^2x}\right)=\left(\tan{x}\right)^{\ln{x}}\left(\frac{\ln\tan{x}}{x}+\frac{2\ln{x}}{... |
1,803,416 | <p>Does the function $d: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ given by:
$$d(x,y)= \frac{\lvert x-y\rvert} {1+{\lvert x-y\rvert}}$$ define a metric on $\mathbb{R}^n?$</p>
<p>How do you go about proving this? Do I need to just show that it satisfies the three conditions to be a metric? If so how do I show t... | Mohammad W. Alomari | 45,105 | <p>This first conditions holds trivially, to prove the third condition (triangle inequality) consider the function $f(t)=\frac{t}{1+t}$, $t>0$ and check the monotonicity of $f$.</p>
|
127,225 | <p>I got stuck solving the following problem:</p>
<pre><code>Table[Table[
Table[
g1Size = x; g2Size = y;
vals =
FindInstance[(a1 - a2) - (b1 - b2) == z && a1 + b1 == g1Size &&
a2 + b2 == g2Size && a1 + a2 == g1Size && b1 + b2 == g2Size &&
a1 > 0 &am... | jkuczm | 14,303 | <h2>Algorithm</h2>
<p>Since the bottleneck is calculating <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial" rel="nofollow noreferrer">elementary symmetric polynomial</a>, let's search for an efficient algorithm to do it. In <a href="https://math.stackexchange.com/a/1265218/374469">answer to "Algo... |
341,823 | <p>Let <span class="math-container">$E\subset B_1(0)\subset \mathbb{R}^n$</span> be a compact set s.t. <span class="math-container">$\lambda(E)=0$</span>, where <span class="math-container">$\lambda$</span> is the Lebesgue measure, and <span class="math-container">$B_1(0)$</span> is the Euclidean unit ball centered at ... | Yuval Peres | 7,691 | <p>The integral in question is finite for most sets of measure zero, but can diverge to <span class="math-container">$\infty$</span> for some sets. An example in one dimension is obtained by constructing a Cantor set where at stage <span class="math-container">$k$</span> the middle <span class="math-container">$1/(k+1... |
830,111 | <p>We have the following set of lines:
$$L_1: \frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-1}{-3}$$
$$L_2:\frac{x-3}{1}=\frac{y+4}{3}=\frac{z-2}{-7}$$</p>
<p>This leads to the following parametric equations: $$L_1:x=t+2,\space y=-2t+3,\space z=-3t+1$$
$$L_2: x=s+3,\space y=3s-4,\space z=-7s+2$$
The $x$ line looked pretty simp... | Ted | 15,012 | <p>You didn't solve for $s$ and $t$ correctly. $s=5$ doesn't satisfy your equation $-2(s+1)+3 = 3s-4$.</p>
|
404,574 | <p>Suppose that:</p>
<p>$Y \pmod B = 0$</p>
<p>$Y \pmod C = X$</p>
<p>I know $B$ and $C$. $Y$ is unknown, it might be an extremely large number, and it does not interest me. </p>
<p>The question is: Is it possible to find $X$, and if so, how?</p>
| Adriano | 76,987 | <p>No; more information is needed. To see this, suppose that $B=2$ and $C=5$ and suppose that we know that $Y \bmod 2 = 0$ and we want to figure out $X = Y \bmod 5$. The possibilities for $X$ are not unique and depend on $Y$:</p>
<blockquote>
<ul>
<li>Since $2$ is a factor of $10$, we could have $Y=10$, which yiel... |
1,831,191 | <p>I am confused about the following Theorem:</p>
<p>Let <span class="math-container">$f: I \to \mathbb{R}^n$</span>, <span class="math-container">$a \in I$</span>. Then the function <span class="math-container">$f$</span> is differentiable at <span class="math-container">$a$</span> if and only if there exists a functi... | Aloizio Macedo | 59,234 | <p>The point is that $\varphi$ is not $f'$. They just coincide in one point, and it is easy to see that two functions coinciding in one point entails nothing about some relationship of differentiability/continuity etc between one another.</p>
|
2,266,573 | <p>I am working through some problems about probability and seem to be having trouble working through this one in particular. I'd love some help learning how to go about solving problems such as this.</p>
<p>A website estimates that 19% of people have a phobia regarding public speaking. If three students are assigned ... | Graham Kemp | 135,106 | <p>The problem involves the count of 'successes' in a known amount of independent trials with equal success rate. There are three people, each with a probability of $0.19$ for possessing phobia.</p>
<p>The key point in this problem is to recognise that this count is a Random Variable which follows a <strong>Bin... |
1,902,138 | <p>It's common to see a plus-minus ($\pm$), for example in describing error
$$
t=72 \pm 3
$$
or in the quadratic formula
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
or identities like
$$
\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)
$$</p>
<p>I've never seen an analogous version combining multiplication with div... | TheGeekGreek | 359,887 | <p>The multiplication sign $\cdot$ is usually omited in abelian groups (or even non-abelian). I think the most important thing is, that something like $\div$ does mislead the reader, because if we consider $a\div b$, then $b$ is not on the same level as $a$, but in contrary the sumbology does suggest so. This can cause... |
97,672 | <p>Given that I have a set of equations about varible $x_0,x_1,\cdots,x_n$, which own the following style:</p>
<p>$
\left(
\begin{array}{cccccccc}
\frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{1}{6} & \frac{2}{3} & \frac{1}{6} & 0 & 0 & 0... | J. M.'s persistent exhaustion | 50 | <p>How to fold a "wide" matrix over to enforce "periodic" conditions:</p>
<pre><code>mat = {{1/6, 2/3, 1/6, 0, 0, 0, 0, 0}, {0, 1/6, 2/3, 1/6, 0, 0, 0, 0},
{0, 0, 1/6, 2/3, 1/6, 0, 0, 0}, {0, 0, 0, 1/6, 2/3, 1/6, 0, 0},
{0, 0, 0, 0, 1/6, 2/3, 1/6, 0}};
{m, n} = Dimensions[mat];
LinearSolve[Take[mat, m,... |
2,278,798 | <p>The converse statement, "A metric space on which every continuous, real valued function is bounded is compact" is dealt with on this site, as it is in Greene and Gamelin's monograph, "Introduction to Topology", where a hint to its proof is offered. I see no discussion of the direct statement in my title. Is it tru... | Tsemo Aristide | 280,301 | <p>Yes, it is bounded. Suppose it is not bounded , for every integer $n$, you have $x_n$ such that $|f(x_n)|\geq n$, you can extract a subsequence $x_{n_i}$ which converges towards $x$ since the domain is compact, this implies that $f(x_{n_i})$ converges towards $f(x)$ since $f$ is continuous, contradiction since $|f(x... |
1,890,047 | <p>Consider two linear transformations $L_1, L_2: V \to W$.</p>
<p>Fix a basis of $V$, $W$, and consider $M_1$, $M_2$, the matrices of the aforementioned transformations w.r.t said basis.</p>
<p>Suppose you can obtain $M_2$ from swapping columns in $M_1$.</p>
<p>How are $L_1$ and $L_2$ related? (Besides having the s... | user115350 | 334,306 | <p>because the transformation matrix is determined by basis of V and W, if you swap the two column (e.g. i and j) of the matrix, you are swapping two basis $w_i$ and $w_j$ of W. L1 and L2 are the same but different matrix interpretation.</p>
|
16,105 | <p>The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local coefficients $H_*(F,\rho)$ straightforwardly as the homology of the twisted complex $$C_*(F,\rho):=C_*(\widetilde{F};... | Emerton | 2,874 | <p>For me it is easier to work with cohomology (just for psychological reasons). Also, I will distinguish the representation $\rho$ from the local system $V$ with fibres ${\mathbb C}^2$ that it gives rise to. So where you would write $H^1(F,\rho)$ I will write $H^1(F,V)$.
I will let $\overline{V}$ denote the comple... |
2,150,886 | <p>I want to find a first order ode, an initial value problem, that has the solution
$$y=(1-y_0)t+y_0$$
where $y_0$ is the initial value.The ode has to be of first order, that is:
$$y'=f(y).$$
I need this to test a special solver I am building.
The main objective is to find an ode that has the property that the end-... | Robert Israel | 8,508 | <p>Solve for $y_0$: </p>
<p>$$ y_0 = \frac{t-y}{t-1} $$</p>
<p>Then differentiate:
$$ 0 = \frac{(t-1)(1-y') - (t-y)}{(t-1)^2} = - \frac{y'}{t-1} + \frac{y-1}{(t-1)^2}$$
i.e.
$$ y' = \frac{y-1}{t-1}$$</p>
|
5,739 | <p>Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}...n_k^{s_k}}$. I found some papers suggesting that multi-index Dirichlet series are in fact a distinct subfield for itsel... | Jon Awbrey | 1,636 | <p>I used to study enumerating generating functions, mostly for various families of graphs, that allowed a mix of ordinary and exponential variables for tracking different kinds of additive weights along with dirichlet variables for tracking multiplicative weights. I don't remember there being a lot of literature &mda... |
5,739 | <p>Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}...n_k^{s_k}}$. I found some papers suggesting that multi-index Dirichlet series are in fact a distinct subfield for itsel... | maki | 1,919 | <p>De la Breteche proved recently a Tauberian theorem for multiple Dirichlet series (MR1858338 (2002j:11106)). This is useful stuff in applications. It fails shortly of proving the main result in Balazard, et. al recent paper: <a href="http://iml.univ-mrs.fr/~balazard/pdfdjvu/19.pdf" rel="nofollow">http://iml.univ-mrs.... |
5,739 | <p>Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}...n_k^{s_k}}$. I found some papers suggesting that multi-index Dirichlet series are in fact a distinct subfield for itsel... | Anweshi | 2,938 | <p>See P. Deligne, <em>Multizeta values</em>, Notes d'exposes, IAS Princeton, for the deep mathematical aspects of this. </p>
<p>Also for a general relevance philosophy, see Kontsevich and Zagier, <em>Periods</em>, Mathematics Unlimited(2001). An electronic version is available <a href="http://www.maths.gla.ac.uk/~tl/... |
3,363,875 | <p>when I read a book,they say this is clear:</p>
<p>let <span class="math-container">$n$</span> be postive integer,then have
<span class="math-container">$$(-1)^n(n+1)\equiv n+1\pmod 4$$</span>
Why don't I feel right?</p>
| Bill Dubuque | 242 | <p>Their difference <span class="math-container">$\,\underbrace{(n+1)\overbrace{(1 - (-1)^{\large n})}^{\large 0 \ \ {\rm if}\ \ n\ \ {\rm even}}}_{\large {\rm even}\ \times\ 2\ \ {\rm if}\ \ n\ \ {\rm odd}\!\!\!}\ $</span> is divisible by <span class="math-container">$\,4\,$</span> so they are congruent <span class="m... |
2,386,602 | <p>This is a question from an exam I recently failed. </p>
<p>What is the radius of convergence of the following power series? $$(a) \sum_{n=1}^\infty(n!)^2x^{n^2}$$ and $$(b) \sum_{n=1}^\infty \frac {x^{n^2}}{n!}$$</p>
<p>Edit: Here's my attempt at the first one, if someone could tell me if it's any good...</p>
<p>... | levap | 32,262 | <p>Let's consider the first series $\sum_{n=1}^{\infty} (n!)^2 x^{n^2}$. The easiest way to find the radius of convergence is to forget this is a power series and treat $x$ as a constant. Let's assume $x > 0$ so that the terms of the series are positive and we can use any test we wish for the convergence/divergence ... |
3,062,701 | <p>I want to solve this system by Least Squares method:<span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}1\\5\\-2\end{pmatrix} $$</span> This symmetric matrix is singular with one eigenvalue <span ... | John Doe | 399,334 | <p>I think you did the Gaussian elimination wrong. </p>
<p><span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}1\\5\\-2\end{pmatrix}$$</span></p>
<p>Becomes <span class="math-container">$$\begin{pma... |
3,062,701 | <p>I want to solve this system by Least Squares method:<span class="math-container">$$\begin{pmatrix}1 & 2 & 3\\\ 2 & 3 & 4 \\\ 3 & 4 & 5 \end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix} =\begin{pmatrix}1\\5\\-2\end{pmatrix} $$</span> This symmetric matrix is singular with one eigenvalue <span ... | AVK | 362,247 | <p>Your solution is incorrect for the following reason. When you perform the Gauss-Jordan elimination, you transform the original system
<span class="math-container">$$\tag{1}
Ax=b
$$</span> to another
<span class="math-container">$$\tag{2}
SAx=Sb.
$$</span>
But the least squares solutions of (1) and (2) do not coinci... |
2,806,858 | <p>There is an equation
$$\sin2\theta=\sin\theta$$
We need to show when the right-hand side is equal to the left-hand side for $[0,2\pi]$.
<hr>
Let's rewrite it as
$$2\sin\theta\cos\theta=\sin\theta$$
Let's divide both sides by $\sin\theta$ (then $\sin\theta \neq 0 \leftrightarrow \theta \notin \{0,\pi,2\pi\}$)
$$2\cos... | Arnaud Mortier | 480,423 | <p>Wrong:$$ab=ac\Longleftrightarrow b=c$$</p>
<p>Correct:$$ab=ac\Longleftrightarrow\cases{b=c\\\text{or}\\a=0}$$</p>
<p>Therefore, if you want to simplify by $a$, you automatically end up with a proof by exhaustion, with at least two cases.</p>
|
4,090,408 | <p>Show that <span class="math-container">$A$</span> is a whole number: <span class="math-container">$$A=\sqrt{\left|40\sqrt2-57\right|}-\sqrt{\left|40\sqrt2+57\right|}.$$</span>
I don't know if this is necessary, but we can compare <span class="math-container">$40\sqrt{2}$</span> and <span class="math-container">$57$<... | peter.petrov | 116,591 | <p><span class="math-container">$$A=\sqrt{57-40\sqrt2}-\sqrt{40\sqrt2+57} = \sqrt{(4\sqrt2-5)^2} - \sqrt{(4\sqrt2+5)^2} $$</span></p>
<p><span class="math-container">$$ = (4\sqrt2-5) - (4\sqrt2+5) = -10$$</span></p>
<p>So <span class="math-container">$A$</span> is the integer <span class="math-container">$-10$</span>.<... |
716,859 | <p>Define the mean of order $p$ of $a$ and $b$ as $s_p(a,b)$ $=$ $({a^p + b^p\over 2})^{1/p}$.</p>
<p>I have to find the limit of the sequence $s_n(a,b)$. I already know this sequence is bounded above by $b$ (from a previous question) and if I assume the limit exists I can show it is $b$. What I cannot show is that th... | mookid | 131,738 | <p>There is no need for increasing:</p>
<p>let us assume that $b>a\ge 0$, that is $a = b-r$, $r>0$.</p>
<p>Then $$
\left( \frac{a^p + b^p}2
\right)^{1/p}=
b \left( \frac 12 \left[\left(\frac{b-r}b \right)^p + 1
\right]\right)^{1/p}=
b\left[\frac 12\right]^{1/p} \left[\left(\frac{b-r}b \right)^p + 1
\right]^{1/p... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Gil Kalai | 1,532 | <p><a href="http://en.wikipedia.org/wiki/Graph_minor_theorem">The Graph-Minor Theorem</a>.</p>
<p>A graph $H$ is a minor of a graph $G$ if it can be obtained from $G$ by a sequence of deletion and contraction edges. Roberton and Seymour's graph-minor theorem asserts that in every infinite sequence of graphs $G_1,G_2,\... |
152,405 | <p>This question complement a previous MO question: <a href="https://mathoverflow.net/questions/95837/examples-of-theorems-with-proofs-that-have-dramatically-improved-over-time">Examples of theorems with proofs that have dramatically improved over time</a>.</p>
<p>I am looking for a list of</p>
<h3>Major theorems in ma... | Daniel Moskovich | 2,051 | <p><a href="http://en.wikipedia.org/wiki/Selberg_trace_formula" rel="nofollow">The Selberg Trace Formula- general case</a></p>
<hr>
<p>Hejhal's original 1983 proof is 1322 pages long! As far as I know, the proof remains famously very hard.</p>
|
2,069,507 | <p><a href="https://i.stack.imgur.com/B4b88.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/B4b88.png" alt="The image of parallelogram for help"></a></p>
<p>Let's say we have a parallelogram $\text{ABCD}$.</p>
<p>$\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two... | Arnaldo | 391,612 | <p>Here is the problem:</p>
<p>"<em>Two sides are equal and Area is also equal. So, the third side is also equal</em>"</p>
<p>Take as an example:</p>
<p>Suppose $\angle ADC=60°$ and $\angle BCD =120°$. </p>
<p>You keep getting: $AD=BC$, $DC=DC$ and $S(ADC)=S(BCD)$ but $\Delta ADC \ne \Delta BCD$ because they don't ... |
11,698 | <p><strong>Bug introduced in 8.0 or earlier and fixed in 13.2.0 or earlier</strong></p>
<hr />
<p>So I have been fighting with this for a while. I'm trying to get custom frame ticks on both the left and right side of a <code>DistributionChart</code>. It's not going very well. It just keeps throwing errors saying tick p... | bmf | 85,558 | <p>The issue has been resolved in the latest -<code>13.2.0</code>- version or earlier</p>
<p>The code from the OP</p>
<pre><code>With[{fps = {120, 60, 50, 40, 30, 25, 20, 15, 10}},
DistributionChart[
RandomVariate[SkewNormalDistribution[##], 100] & @@@ {{20, 13,
5}, {30, 12, 10}}, ChartLabels -> {1, 2}... |
1,043,266 | <p>Carefully see this problem(I have solved them on my own, I'm only talking about the magical coincidence):</p>
<blockquote>
<p>A bag contains 6 notes of 100 Rs.,2 notes of 500 Rs., 3 notes of 1000 Rs..Mr. A draws two notes from the bag then Mr. B draws 2 notes from the bag.<br>
(i)Find the probability that A has... | ml0105 | 135,298 | <blockquote>
<blockquote>
<p>(i)Find the probability that A has drawn 600 Rs.</p>
</blockquote>
</blockquote>
<p>Only $A$ is drawing. It doesn't matter if he picks the $100$ or $500$ first. So order does not matter.</p>
<blockquote>
<blockquote>
<p>(ii)Total Probability Theorem: Considering various case... |
4,440,233 | <blockquote>
<p>Find all the functions <span class="math-container">$f:\mathbb{Z}^+ \to \mathbb{Z}^+$</span> such that <span class="math-container">$f(f(x)) = 15x-2f(x)+48$</span>.</p>
</blockquote>
<p>If <span class="math-container">$f$</span> is a polynomial of degree <span class="math-container">$n$</span>, we have ... | Tob Ernack | 275,602 | <p>Note: the proof of convergence of the sequence used below is not complete, I might come back later to fix it, but there are already other answers anyway.</p>
<hr />
<p>Due to the requirement that <span class="math-container">$f(x) \in \mathbb{Z}^+$</span> whenever <span class="math-container">$x \in \mathbb{Z}^+$</s... |
4,440,233 | <blockquote>
<p>Find all the functions <span class="math-container">$f:\mathbb{Z}^+ \to \mathbb{Z}^+$</span> such that <span class="math-container">$f(f(x)) = 15x-2f(x)+48$</span>.</p>
</blockquote>
<p>If <span class="math-container">$f$</span> is a polynomial of degree <span class="math-container">$n$</span>, we have ... | Sil | 290,240 | <p>Put <span class="math-container">$a_0=a \in \mathbb{Z}^{+}$</span> arbitrary and <span class="math-container">$a_n=f(a_{n-1})$</span> for <span class="math-container">$n \geq 1$</span>. The functional equation gives a non-homogenous linear recurrence
<span class="math-container">$$
a_{n}=-2a_{n-1}+15a_{n-2}+48.
$$</... |
3,111,489 | <p>For which <span class="math-container">$p,q$</span> does the <span class="math-container">$\int_0^{\infty} \frac{x^p}{\mid{1-x}\mid^q}dx$</span> exist ?</p>
<p>Can you help me, I have been siting hours on this question .</p>
<p>I got that for <span class="math-container">$ q<1$</span> and <span class="math-cont... | Calvin Khor | 80,734 | <p>The only correct bound you have is <span class="math-container">$$q<1.$$</span> <span class="math-container">$x^p|1-x|^{-q}$</span> has potential issues at <span class="math-container">$0,1,+\infty$</span>. </p>
<ol>
<li>At <span class="math-container">$0$</span>, the function is like <span class="math-container... |
1,079,995 | <p>I can't understand how: $$ \frac {2\times{^nC_2}}{5} $$</p>
<p>Equals:</p>
<p>$$ 2\times \frac {^nC_2}{5} $$</p>
<p>If we forget the combination and replace it with a $10$, the result is clearly different. $1$ in the first example and and $0.5$ in the second.</p>
| k170 | 161,538 | <p>Using your example,
$$ \frac{2\times 10}{5}=\frac{20}{5}=4$$
And
$$ 2\times\frac{10}{5}=2\times 2=4$$
In general for any $c\not =0$, we have
$$ \frac{a\times b}{c}=a\times\frac{b}{c}= b\times\frac{a}{c} $$</p>
|
393,293 | <p>I need an upper bound for
$$\frac{ax}{x-2}$$
I know that $1\leq a< 2$ and $x\geq 0$.</p>
<p>This upper bound can include just $a$ and constant numbers not $x$.</p>
<p>thanks a lot.</p>
| in_mathematica_we_trust | 27,030 | <p>And a picture for the graphical learners.</p>
<p><img src="https://i.stack.imgur.com/Sd0T2.png" alt="enter image description here"></p>
|
3,001,700 | <p>I am trying to find an <span class="math-container">$x$</span> and <span class="math-container">$y$</span> that solve the equation <span class="math-container">$15x - 16y = 10$</span>, usually in this type of question I would use Euclidean Algorithm to find an <span class="math-container">$x$</span> and <span class=... | user | 505,767 | <p>Note that by Bezout's identity since <span class="math-container">$\gcd(15,16)=1$</span> we have</p>
<p><span class="math-container">$$15\cdot (-1+k\cdot 16)+16 \cdot (1-k\cdot 15)=1 \quad k\in\mathbb{Z}$$</span></p>
<p>are all the solution for <span class="math-container">$15a+16b=1$</span> and from here just mul... |
202,742 | <p>Consider a <a href="http://en.wikipedia.org/wiki/Circular_layout" rel="noreferrer">circular drawing</a> of a simple (in particular, loopless) graph $G$ in which edges are drawn as straight lines inside the circle. The <em>crossing graph</em> for such a drawing is the simple graph whose nodes correspond to the edges ... | Zsbán Ambrus | 5,340 | <p>No, and you can see this from just a counting argument.</p>
<p>For determining which of the $ n $ chords of the circle intersect, it is enough to know the order of the $ 2n $ endpoints on the circle. (You can assume that no two endpoints coincide.) There are at most $ (2n)!/2^n $ such orders (the two endpoints of... |
2,150,552 | <p>I'm following a YouTube linear algebra course. (<a href="https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14" rel="nofollow noreferrer">https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14</a>)<br>
In part 9 there's the ... | la flaca | 279,164 | <p>I think the answer is no. Take $(a,b) \in R$ and $(b,a) \in S$, then $(a,b),(b,a) \in R \cup S$ but it doesn't imply $(a,b)=(b,a)$.<br>
The key here is that $\{(a,b),(b,a)\} \subset R \cup S$ but it is not the case that $\{(a,b),(b,a)\} \subset R$ nor $\{(a,b),(b,a)\} \subset S$ so you can't use the antisymmetry of ... |
2,150,552 | <p>I'm following a YouTube linear algebra course. (<a href="https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14" rel="nofollow noreferrer">https://www.youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14</a>)<br>
In part 9 there's the ... | lordoftheshadows | 303,196 | <p>This not not true. Consider the set $(A,B)$ and the relation $(A,B)$ and $(B,A)$. Both relations are anti symmetric but their union isn't anti symmetric because both $(A,B)$ and $(B,A)$ are members of the relation but $A \neq B$.</p>
|
2,125,018 | <blockquote>
<p>You toss a fair coin 3x, events:</p>
<p>A = "first flip H"</p>
<p>B = "second flip T"</p>
<p>C = "all flips H"</p>
<p>D = "at least 2 flips T"</p>
<p><strong>Q:</strong> Which events are independent?</p>
</blockquote>
<p>From the informal def. it is where one doesnt affect th... | Jean Marie | 305,862 | <p>Here is a complete solution using $\tan$ and $\tan^{-1}$.</p>
<p>Let us fix notations, with points </p>
<p>$a=(x_a,y_a)=(L_1\cos(\theta_1),L_1\sin(\theta_1))$ and</p>
<p>$b=(L_3 \cos(\theta), L_3 \sin(\theta))$. </p>
<p>We have:</p>
<p>$$\tag{1}\vec{ba}\binom{x_a-L_3 \cos(\theta)}{y_a-L_3 \sin(\theta)}.$$ </p>
... |
154,893 | <p>I am having trouble figuring this out.</p>
<p>$$\sqrt {1+\left(\frac{x}{2}- \frac{1}{2x}\right)^2}$$</p>
<p>I know that $$\left(\frac{x}{2} - \frac{1}{2x}\right)^2=\frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2}$$ but I have no idea how to factor this since I have two x terms with vastly different degrees, 2 and -2.<... | Rick Decker | 36,993 | <p>I presume you aren't asked to solve this (since it isn't an equation), but rather are asked to express it in a tidier form. Carrying on, we have
\begin{align*}
1+\left(\frac{x^2}{4}-\frac{1}{2}+\frac{1}{4x^2}\right) &=\frac{x^2}{4}+\frac{1}{2}+\frac{1}{4x^2}\\\\
&= \left(\frac{x}{2}+\frac{1}{2x}\right)^2\\\\... |
2,648,370 | <p>$$\int\frac{x^2}{\sqrt{2x-x^2}}dx$$
This is the farthest I've got:
$$=\int\frac{x^2}{\sqrt{1-(x-1)^2}}dx$$</p>
| lab bhattacharjee | 33,337 | <p>As $0<x<2,$</p>
<p>$$\dfrac{x^2}{\sqrt{2x-x^2}}=\dfrac{x^{3/2}}{\sqrt{2-x}}$$</p>
<p>set $x=2\sin^2t,x^{3/2}=\text{?}$</p>
<p>$dx=\text{?}$ and $\sqrt{2-x}=+\sqrt2\cos t$</p>
|
2,648,370 | <p>$$\int\frac{x^2}{\sqrt{2x-x^2}}dx$$
This is the farthest I've got:
$$=\int\frac{x^2}{\sqrt{1-(x-1)^2}}dx$$</p>
| lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>As $\dfrac{d(2x-x^2)}{dx}=2-2x$</p>
<p>$$\dfrac{x^2}{\sqrt{2x-x^2}}=\dfrac{x^2-2x+2x-2+2}{\sqrt{2x-x^2}}$$</p>
<p>$$=-\sqrt{1-(x-1)^2}-\dfrac{2-2x}{\sqrt{2x-x^2}}+\dfrac2{\sqrt{1-(x-1)^2}}$$</p>
<p>Now use $\#1,\#8$ of <a href="http://www.sosmath.com/tables/integral/integ13/integ13.html" rel="nofoll... |
2,648,370 | <p>$$\int\frac{x^2}{\sqrt{2x-x^2}}dx$$
This is the farthest I've got:
$$=\int\frac{x^2}{\sqrt{1-(x-1)^2}}dx$$</p>
| damier.godfred | 530,152 | <p>Ok, so building off of what <a href="https://math.stackexchange.com/users/33337/lab-bhattacharjee">lab bhattacharjee</a> said:
<span class="math-container">$$\int\frac{x^2}{\sqrt{2x-x^2}}dx$$</span>
<span class="math-container">$$=-\int\sqrt{1-(x-1)^2}dx-\int\dfrac{2-2x}{\sqrt{2x-x^2}}dx+2\int\dfrac1{\sqrt{1-(x-1)^2... |
208,744 | <p>I was asked to show that $\frac{d}{dx}\arccos(\cos{x}), x \in R$ is equal to $\frac{\sin{x}}{|\sin{x}|}$. </p>
<p>What I was able to show is the following:</p>
<p>$\frac{d}{dx}\arccos(\cos(x)) = \frac{\sin(x)}{\sqrt{1 - \cos^2{x}}}$</p>
<p>What justifies equating $\sqrt{1 - \cos^2{x}}$ to $|\sin{x}|$?</p>
<p>I ... | Christian Blatter | 1,303 | <p>The function $f:\ x\mapsto\arccos(\cos x)$ is even and $2\pi$-periodic, since the "inner" function $\cos$ has these properties. For $0\leq x\leq \pi$ by definition of $\arccos$ we have $f(x)=x$. Therefore $f$ is the $2\pi$-periodic continuation of the function $$f_0(x)\ :=\ |x|\qquad(-\pi\leq x\leq\pi)$$
to the full... |
3,812,432 | <p>For <span class="math-container">$a,b,c>0.$</span> Prove<span class="math-container">$:$</span> <span class="math-container">$$4\Big(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2} \Big)+\dfrac{81}{(a+b+c)^2}\geqslant{\dfrac {7(a+b+c)}{abc}}$$</span></p>
<p>My proof is using SOS<span class="math-container">$:$</span... | RobPratt | 683,666 | <p>You want to avoid 12 properties:
<span class="math-container">\begin{matrix}
000xxx & x000xx & xx000x & xxx000 \\
111xxx & x111xx & xx111x & xxx111 \\
222xxx & x222xx & xx222x & xxx222 \\
\end{matrix}</span>
The inclusion-exclusion formula is <span class="math-container">$$\sum_{k... |
179,886 | <p>I have series of values which, by visual inspection, appear to be sums of certain constants, not divisible by each other, with rational weights. I want to convert these sums to vectors of weights for a specific basis vector.</p>
<p>I have an initial solution based on <code>FindInstance</code>, which works reasonabl... | Daniel Lichtblau | 51 | <p>Could use <a href="http://reference.wolfram.com/language/ref/FindIntegerNullVector.html" rel="nofollow noreferrer"><code>FindIntegerNullVector</code></a>.</p>
<pre><code>s143 = SeriesCoefficient[
Sin[x] + Sqrt[1 - x^2] Sinh[Sqrt[1 - x^2]]/(x + 1), {x, 0, 143}];
ff = FindIntegerNullVector[{1, Sinh[1], Cosh[1], s1... |
631,214 | <p>Two kids starts to run from the same point and the same direction of circled running area with perimeter 400m. The velocity of each kid is constant. The first kid run each circle in 20 sec less than his friend. They met in the first time after 400 sec from the start. Q: Find their velocity.</p>
<p>I came with one e... | Emanuele Paolini | 59,304 | <p>You can prove that $e^t > 2^t > t^2$ for $t$ sufficiently large. Hence you can prove that
$$
\lim_{t\to +\infty} \frac{t}{e^t} = 0
$$
without using derivatives.
Let $x=e^{-t}$ and you find
$$
0 = \lim_{t\to +\infty} \frac{t}{e^t} = \lim_{x\to 0} -x \log x.
$$</p>
<p>The point, however, is: how the function $\... |
1,354,953 | <p>Solve for the function f(x):</p>
<p>$$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$
I'm not able to solve this. </p>
<p>[For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me anywhere as the value obtained, when substituted into the original equation, just yields $f(x)=f(x)$]</p>
<... | mahdokht | 253,415 | <p>take $g=\frac { x }{ (x+f) } $ (#)
by differentiating from two sides of this: $1=g+g'x+f'g+g'f$ $(*)$
by substituting $g=\frac { x }{ (x+f) } $ in the equation: $$\frac { x }{ g } -x=\frac { x }{ \left( x+f(g) \right) } $$
differentiating from two sides of this and using (*) : $2x-gx=g-g^2$
find g in this quadrat... |
2,303,795 | <p>So I know that by Euler's homogeneous function theorem $m$ is a positive number, but why is it an integer?
And how to prove that $f$ is polynomial of degree $m$?</p>
| Robert Z | 299,698 | <p>Alternative way. In order to avoid integration, note that
$$f_n(x)-f(x)=
\int_{1/n}^{x+1/n} \sin^2(t) \, dt-\int_0^x \sin^2(t) \, dt=
\int_{x}^{x+1/n} \sin^2(t) \, dt-\int_0^{1/n} \sin^2(t) \, dt$$
Hence for $x\geq 0$,
$$|f_n(x)-f(x)|\leq \int_{x}^{x+1/n} 1 \, dt+\int_0^{1/n} 1 \, dt= \frac{2}{n} $$
which implies t... |
3,628,374 | <p>We have that <span class="math-container">$W \in \mathbb{R}^{n \times m}$</span> and we want to find <span class="math-container">$$\text{prox}(W) = \arg\min_Z\Big[\frac{1}{2} \langle W-Z, W-Z \rangle+\lambda ||Z||_* \Big]$$</span></p>
<p>Here, <span class="math-container">$||Z||_*$</span> represents the trace nor... | Community | -1 | <p>This gives a solution by group action theory, as you wish, <em>but</em> I couldn't avoid using the result (*) hereunder, which is seemingly as much non-elementary as the tools in the other answers/comments (for its proof, see in this site e.g. <a href="https://math.stackexchange.com/q/3525343/750041">here</a>).</p>
... |
2,397,874 | <p>I am new to modulus and inequalities , I came across this problem:</p>
<p>$ 2^{\vert x + 1 \vert} - 2^x = \vert 2^x - 1\vert + 1 $ for $ x $</p>
<p>How to find $ x $ ?</p>
| Michael Rozenberg | 190,319 | <p>Let $(5+2\sqrt6)^{x^2-3}=t$.
Hence, $t+\frac{1}{t}=10$ and we have $t=5\pm2\sqrt{6}$.</p>
<p>Thus, $x^2-3=1$ or $x^2-3=-1$, which gives the answer:
$$\{2,-2,\sqrt2,-\sqrt2\}$$</p>
|
3,163,342 | <p>Find all the ring homomorphisms <span class="math-container">$f$</span> : <span class="math-container">$\mathbb{Z}_6\to\mathbb{Z}_3$</span>.</p>
<p>definition of ring homomorphism:</p>
<p>The function f: R → S is a ring homomorphism if:</p>
<p>1) <span class="math-container">$f(1)$</span> = <span class="math-cont... | J. W. Tanner | 615,567 | <p>You made a mistake: <span class="math-container">$ 3(-\frac{1}{3}(x - 1))= -(x-1),$</span>
not
<span class="math-container">$ -3(x - 1).$</span></p>
|
772,665 | <p><strong>Question:</strong></p>
<blockquote>
<p>For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions.</p>
</blockquote>
<p><strong>My try:</strong> It is clear that the function
$$f(x)=x$$ satisfies the given conditions, since:
$$f(a)+f(... | Ahmed Bachir | 984,531 | <p>we can find some other solutions such the zero function or the functions under the form <span class="math-container">$f(n)=a^2n$</span> or the constant functions under the form <span class="math-container">$f(n)=\dfrac{a^2}{2}$</span>, or an other function <span class="math-container">$f$</span> which can be determi... |
104,875 | <p>I've been looking for a solution to this problem for other applications too, for some time, but haven't come up with a solution that does not involve <code>Animate</code> or similar (and it never works).</p>
<p>Take this example:
plot a function (say <code>f=a/x</code>) for different <code>a</code>. The y-axis plot... | Jack LaVigne | 10,917 | <p>Here is something that works.</p>
<p>I am quite certain that there are more efficient ways to accomplish this but at least the code is fairly easy to follow.</p>
<p>I tried wrapping the right hand side of <code>AxesStyle</code> in <code>Dynamic</code> but it didn't work so I ended up wrapping the whole plot in <co... |
130,502 | <p>I obtained a numerical solution from the following code with <code>NDSolve</code></p>
<pre><code>L = 20;
tmax = 27;
\[Sigma] = 2;
myfun = First[h /. NDSolve[{D[h[x, y, t], t] +
Div[h[x, y, t]^3*Grad[Laplacian[h[x, y, t], {x, y}], {x, y}], {x, y}] +
Div[h[x, y, t]^3*Grad[h[x, y, t], {x, y}], {x, y}] == 0,
h[x, y, ... | Nasser | 70 | <p>To prevent shaking, try to add <code>ImagePadding</code> and for the other issue, you can fix the vertical plot range. </p>
<pre><code>mpl = Table[
Plot3D[myfun[x, y, t], {x, 0, L}, {y, 0, L},
PlotRange -> {Automatic, Automatic, {0, 6}}, PlotPoints -> 40,
ImageSize -> 400,
PlotLabel ->... |
1,801,112 | <p>Find the simplest solution:</p>
<p>$y' + 2y = z' + 2z$ I think proper notation is not sure, y' means first derivate of y. ($\frac{dy}{dt}+ 2y = \frac{dz}{dt} + 2z$)</p>
<p>$y(0)=1$</p>
<p>I got kind of confused, is $y=z=1$ a proper solution here? Or is disqualified because a constant is not reliant on time and so... | Mark Viola | 218,419 | <p>Let <span class="math-container">$w(t)=y(t)-z(t)$</span>. Then, we have</p>
<p><span class="math-container">$$\frac{dy(t)}{dt}+2y(t)=\frac{dz(t)}{dt}+2z(t)\implies \frac{dw(t)}{dt}+2w(t)=0$$</span></p>
<p>Hence, <span class="math-container">$w(t)=Ae^{-2t}$</span> for some constant <span class="math-container">$A$</... |
3,831,387 | <p><span class="math-container">$X,Y\sim N(0,1)$</span> and are independent, consider <span class="math-container">$X+Y$</span> and <span class="math-container">$X-Y$</span>.</p>
<p>I can see why <span class="math-container">$X+Y$</span> and <span class="math-container">$X-Y$</span> are independent based on the fact th... | Alecos Papadopoulos | 87,400 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be two random variables, with finite second moment. Consider the variables <span class="math-container">$Z_1=X-Y$</span> and <span class="math-container">$Z_2=X+Y$</span>.</p>
<p>Their covariance is</p>
<p><span class="math-conta... |
1,842,826 | <blockquote>
<p>Explain why the columns of a $3 \times 4$ matrix are linearly dependent</p>
</blockquote>
<p>I also am curious what people are talking about when they say "rank"? We haven't touched anything with the word rank in our linear algebra class.</p>
<p>Here is what I've came up with as a solution, will th... | janezdu | 348,666 | <p>Your solution is almost there, you just need to incorporate the original question in. In other words, you need to show that $Ax=0$ has more than one solution when $A$ is $3 \times 4$. </p>
<p>I also like to look at this problem as working with 4 vectors in $\rm I\!R^3$. You can definitely construct one of them with... |
3,542,885 | <p>Let <span class="math-container">$P(x, y) = ax^2 + bxy + cy^2 + dx + ey + h$</span> and suppose <span class="math-container">$b^2 - 4ac > 0.$</span></p>
<p>I know that we can re-write <span class="math-container">$P(x, y)$</span> as a polynomial of <span class="math-container">$x:$</span> <span class="math-conta... | Quanto | 686,284 | <p>Recognize that the right triangles ABC and AED are similar, we have </p>
<p><span class="math-container">$$\frac {AE}{AD} = \frac{AB}{AC}$$</span></p>
<p>Substitute <span class="math-container">$AE = y_1-y_2$</span>, <span class="math-container">$AC = y_1$</span> and <span class="math-container">$AD = \frac15AB = ... |
2,903,557 | <p>I am having trouble proving the following identity: </p>
<p>$$\frac{\sinh \tau +\sinh i\sigma }{\cosh \tau +\cosh i\sigma }=-\coth \left(i \frac{\sigma +i\tau }{2}\right)$$</p>
<p>I have tried using identities and the definitions but haven't had much luck. This is a missing step in inverting the bipolar coordinate... | Stefan Lafon | 582,769 | <p>Let $$u_r= \frac 1 r - \frac 1{r+1}$$
Then
$$\begin{split}
u_r-u_{r+1} &= \frac 1 r - \frac 1{r+1} -\bigg( \frac 1 {r+1} - \frac 1{r+2}\bigg) \\
&= \frac 1 r - \frac 2{r+1} + \frac 1 {r+2} \\
&= \frac{ (r+2)(r+1) -2r(r+2) + r(r+1)}{r(r+1)(r+2)}\\
&= \frac 2 {r(r+1)(r+2)}\\
\end{split}$$
With that, t... |
1,052,073 | <p><strong>Assume $V$ is a real $n$-dimensional vector space, and $v,w \in V $. Define $ T \in L(V)$ by $ T(u) = u - (u,v)w$. Find a formula for Trace(T)</strong></p>
<p>All I know about this is that trace is sum of the diagonal entries of the matrix. So how do I find the diagonal entries? I don't really know what ste... | MvG | 35,416 | <p>Let's start with the comment Hagen wrote:</p>
<blockquote>
<p>Construct a regular pentagon, connect each vertex to its center, and prolong three of its edges (all but two non-adjacent edges). This gives you a $(108°,36°,36°)$ triangle partitioned into seven acute triangles. You can solve a wide range of cases by ... |
2,270,861 | <p>What follows is part of Exercise 1.34 from Pillay's <em>Introduction to Stability Theory</em>. Suppose the following:</p>
<ol>
<li>$M \prec N$.</li>
<li>$N$ is $|M|^+$-saturated.</li>
<li>$p \in S_1(M)$, $q \in S_1(N)$.</li>
<li>$q \supset p$ is a coheir of $p$.</li>
</ol>
<p>Construct a sequence $(a_i \mid i &l... | Mark Viola | 218,419 | <p>HINT:</p>
<p>Integrate by parts the integral $\int_0^\infty e^{-x^2}2y\cos(2xy)\,dx$ with $u=e^{-x^2}$ and $v=\sin(2xy)$.</p>
|
1,293,725 | <p>Here I have a question:</p>
<p>Solve for real value of $x$:
$$|x^2 -2x -3| > |x^2 +7x -13|$$</p>
<p>I got the answer as $x = (-\infty, \frac{1}{4}(-5-3\sqrt{17}))$ and $x=(\frac{10}{9},\frac{1}{4}(3\sqrt{17}-5)$</p>
<p>Please verify it if it is correct or not. Thanks</p>
| copper.hat | 27,978 | <p>Brute force (as I am wont to do):</p>
<p>Look at where $(x^2-2x-3)^2 - (x^2+7x-13)^2 = -(9x-10)(2x^2+5x-16)$ is strictly positive.</p>
<p>The zeroes are ${10 \over 9}$ and ${1 \over 4} (-5 \pm 3 \sqrt{17})$.</p>
<p>Since the leading coefficient is $-1$, we see that the answer is
$(-\infty,{1 \over 4} (-5 - 3 \sqr... |
627,815 | <p>What is the best way to write 'exclusively divisible by' a given number in terms of set notation? eg: the set of natural numbers that are divisible by $2$ and only $2$; the set of natural numbers that are divisible by $3$ and only $3$; $\dots 5$, $7\dots$ etc.</p>
| Cameron Buie | 28,900 | <p>Given a prime natural number $p,$ it seems that the set (let's call it $\Bbb D_p$) of natural numbers "exclusively divisible by $p$" (according to your description) would be:</p>
<p>$$\Bbb D_p=\left\{n\in\Bbb N:\exists k\in\Bbb N\left(n=p^k\right)\right\}$$</p>
<p>For example, $$\begin{align}\Bbb D_2 &= \left\... |
1,803,589 | <p>I'm stuck at this. How is RHS rearranged? Is it a change of index?</p>
<p>$$
\sum_{n=1}^{2N} \frac{1}{n}
- \sum_{n=1}^{N} \frac{1}{n}
= \sum_{n=N+1}^{2N} \frac{1}{n}
$$</p>
<p>I'm stuck here:</p>
<p>$$
\sum_{n=1}^{2N} \frac{1}{n} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{2N}
$$
$$
\sum_{n=1}^{N} \frac... | Emilio Novati | 187,568 | <p>Your reorder is wrong. See here:
$$
\sum_{n=1}^{2N} \frac{1}{n}
- \sum_{n=1}^{N} \frac{1}{n}=
\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{2N}-\left(\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+ \frac{1}{N}\right)=
$$
$$=
\left(\frac{1}{1}-\frac{1}{1}\right)+\left(\frac{1}{2}-\frac{1}{2}\right)+\cdots+\left(\frac... |
65,691 | <p>The question of generalising circle packing to three dimensions was asked in <a href="https://mathoverflow.net/questions/65677/">65677</a>. There is a clear consensus that there is no obvious three dimensional version of circle packing.</p>
<p>However I have seen a comment that circle packing on surfaces and Ricci ... | Igor Rivin | 11,142 | <p>Actually, there are a number of references by Ben Chow, Feng Luo, and D. Glickenstein on this subject, mostly in two dimensions. Glickenstein's work (Glickenstein was a student of Ben Chow's) is more three-dimensional. Some relevant references are below. The curvature flow approach distinct from the even more popul... |
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