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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2,220 | <p>Some of you (myself included) might remember how as a new user you struggle with finding stuff to answer, and hope to have these answers upvoted and accepted...</p>
<p>You really want that. You want to write comments, to the least but that requires 50 reputation.</p>
<p>As a result you look for old questions, poss... | Qiaochu Yuan | 232 | <p>Just to have an option for people to respond to: I have been deleting answers that are not answers (that is, that are questions or otherwise not attempts to answer the question in the OP), and otherwise I have just been downvoting new bad answers. </p>
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35,281 | <p>I am looking for applications of category theory and homotopy theory in set theory and particularly in cardinal arithmetics. "Applications" in the broad sense of the word --- this would include theorems, definitions, questions, points of view (and papers) in set theory that could be motivated or understood... | Andreas Blass | 6,794 | <p>Peter Freyd wrote a paper, "The Axiom of Choice," in which he used topos-theoretic methods to prove that the axiom of choice is independent of (classical) Zermelo-Fraenkel set theory. Unlike earlier topos-versions of set-theoretic independence proofs, Freyd's construction does not merely provide a category-theoreti... |
1,072,669 | <ul>
<li><p>Let <span class="math-container">$C_1,C_2,C_3,C_4$</span> be compact convexes of <span class="math-container">$\mathbb{R}^2$</span> such that <span class="math-container">$C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap C_4\neq\emptyset$</sp... | Hugh Thomas | 94,551 | <p>I like Valeriy's answer, but let me also point out that the first question is a special case of Helly's theorem, which says that in $\mathbb R^d$, if you have at least $d+1$ sets such that any $d+1$ of them have a non-empty intersection, then the intersection of all the sets is non-empty. </p>
|
4,040,301 | <p>If <span class="math-container">$\lim_{|x| \to \infty} g(x)/x = \infty$</span>, Prove that <span class="math-container">$\{g(x)\mid x \in \mathbb{R}\} = \mathbb{R}.$</span></p>
| Wynne Liu | 714,963 | <p><span class="math-container">$lim_{x \to -\infty}g(x)/x = \infty$</span> mean <span class="math-container">$lim_{x \to -\infty}g(x) \to -\infty$</span></p>
<p><span class="math-container">$lim_{x \to \infty}g(x)/x = \infty$</span> mean <span class="math-container">$lim_{x \to \infty}g(x) \to \infty$</span></p>
<p><s... |
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | John | 20,946 | <p><a href="http://www.wolframalpha.com/input/?i=airspeed+velocity+of+an+unladen+swallow" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/NWAyK.png" alt="Alt Text"></a></p>
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3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | Ilmari Karonen | 9,602 | <p><a href="http://www.geogebra.org/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9fRBB.png" alt="GeoGebra - Free mathematics software for learning and teaching"></a></p>
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3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | dls | 1,761 | <p><a href="http://blogs.ams.org/mathgradblog/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/lcbn6.png" alt="AMS Graduate Student Blog - Read and write about topics that matter to you."></a></p>
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2,536,866 | <p>In tensor notation the change of the electromagnetic field tensor by change of inertial reference frames can be done by the following formula :</p>
<p>$$F^{\alpha\beta} = \varLambda^{\alpha}_{\mu}\varLambda^{\beta}_{\nu}F^{\mu\nu}$$</p>
<p>But when this is represented by matrix multiplications it becomes:</p>
<p>... | Xander Henderson | 468,350 | <p>I've never seen the notation $\mathbb{R}^{3\times 2}$ before (which is, perhaps, more a comment on my lack of exposure than anything else—a quick Googling reveals that it is actually pretty damn common), and can't say that I really like it—it looks too much like $\mathbb{R}^6$, which is something else en... |
2,042,257 | <p>I'm looking at the following differential equation:</p>
<p>$\frac{dx}{dt} = \frac{\sin^2 x - t^2}{t \cdot \sin(2x)}$</p>
<p>I rewrote it as</p>
<p>$(t \cdot \sin(2x))dx = (\sin^2x - t^2)dt$</p>
<p>$\Leftrightarrow (\underbrace{\sin^2 x - t^2}_{J(x,t)})dt + (\underbrace{-t \cdot \sin(2x)}_{I(x,t)}) dx = 0$</p>
<... | Ahmed S. Attaalla | 229,023 | <p>The positive real solution to $x^x=2$ is irrational.</p>
<p>Proof:</p>
<p>Assume $x=\frac{p}{q}$ where $p$ and $q$ have no common factor but $1$ then:</p>
<p>$$\left(\frac{p}{q}\right)^{p}=2^{q}$$</p>
<p>We must have $p^p=q^p2^q$</p>
<p>Note $p>q$ because $x>1$ (this can be shown).</p>
<p>Then $p$ must b... |
371,318 | <p>The original problem was to consider how many ways to make a wiring diagram out of $n$ resistors. When I thought about this I realized that if you can only connect in series and shunt. - Then this is the same as dividing an area with $n-1$ horizontal and vertical lines. When each line only divides one of the current... | PPP | 55,643 | <p>Nothing happens, forget infinitesimals and infinity as numbers. Math was built with limits in the last centuries so we don't need to try to define these things.</p>
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498,694 | <p>So, I'm learning limits right now in calculus class.</p>
<p>When $x$ approaches infinity, what does this expression approach?</p>
<p>$$\frac{(x^x)}{(x!)}$$</p>
<p>Why? Since, the bottom is $x!$, doesn't it mean that the bottom goes to zero faster, therefore the whole thing approaches 0?</p>
| Neal | 20,569 | <p>You may be thinking of the fact that $e^x/x!\to 0$. The difference is that there the base of the exponent is fixed, while in this problem, the base of the exponent grows.</p>
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2,259,145 | <p>Let $f\colon (0,1]\to [-1,1]$ be a continuous function. Let us define a function $h$ by $h(x)=xf(x)$ for all $x$ belongs to $(0,1]$.
Prove that $h$ is uniformly continuous.</p>
<p>We know $f$ is uniformly continuous on $I$ if $f'(x)$ is bounded on $I$. Here $h'(x)= xf'(x) + f(x)$ and $f(x)$ is bounded here. How can... | Community | -1 | <p>Let $\psi(z) = \frac{\Gamma'(z+1)}{\Gamma(z+1)}$, then we can resolve your ambiguous notation with the following.</p>
<p>$$\psi(z) = -\gamma + \sum_{n=1}^\infty \frac{1}{k} - \frac{1}{z+k}$$</p>
<p>$$\psi(n+1) - \psi(1) = \sum_{j=1}^n \frac{1}{j}$$</p>
<p>so that naturally</p>
<p>$$\psi(x+1) - \psi(1) = \frac{1}... |
237,197 | <p>I'm new here. If there is anything not appropriate pls let me know.</p>
<p>I am currently working on a differential equation with one of which term is a integral of the variable.</p>
<p><span class="math-container">$$
\frac{d^2u(x)}{dx^2}=cosh(G(x))+\frac{1}{C_1}\int_{0}^{1}{u(x)sinh(G(x))dx }+C_2
$$</span>
with th... | Alex Trounev | 58,388 | <p>In the case <code>C3=1</code> we have singular solution with <span class="math-container">$x^{-2}$</span> singularity at <span class="math-container">$x \rightarrow 0$</span>. Therefore we can suggest that <span class="math-container">$C3<0$</span>, and in this case we have regular solution of this problem. For ... |
1,132,063 | <p>For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let</p>
<p>$$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$</p>
<p>Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete.</p>
<p>The first part was easy.</p>
<p>Now I try to find a sequence in $\ell^1$ such that it is a cauchy sequ... | Kevin Arlin | 31,228 | <p>Observe that your norm is identical to the usual $\ell^1$ norm on sequences with all entries positive. So you'll have to think about alternating signs. The sequence with infinitely many terms of each sign which is closest to $\ell^1$ without being in it (this is an imprecise statement-just indicating why it's my gue... |
1,132,063 | <p>For $x=(x_j)_{j\in\mathbb N}\in \ell^1$ let</p>
<p>$$\|x\|=\sup_{n\in \mathbb N}\left \Vert \sum_{j=1}^{n}x_j\right\Vert$$</p>
<p>Show that $(\ell^1,\|\cdot\|)$ is a normed space, but it is not complete.</p>
<p>The first part was easy.</p>
<p>Now I try to find a sequence in $\ell^1$ such that it is a cauchy sequ... | user66081 | 66,081 | <p>This is a slightly overkill answer.</p>
<p>Consider the following <a href="http://en.wikipedia.org/wiki/Banach_space#Banach.27s_theorems" rel="nofollow">theorem</a>: Every one-to-one bounded linear operator from a Banach space onto a Banach space is an isomorphism.</p>
<p>Let $\ell^*$ denote the collection of $\el... |
1,023,193 | <p>Proving this formula
$$
\pi^{2}
=\sum_{n\ =\ 0}^{\infty}\left[\,{1 \over \left(\,2n + 1 + a/3\,\right)^{2}}
+{1 \over \left(\, 2n + 1 - a/3\,\right)^{2}}\,\right]
$$
if $a$ an even integer number so that
$$
a \geq 4\quad\mbox{and}\quad{\rm gcd}\left(\,a,3\,\right) = 1
$$</p>
| achille hui | 59,379 | <p>Start with the well known? expansion of $\cot z$ and differentiate,</p>
<p>$$\cot z = \sum_{n=-\infty}^\infty \frac{1}{z - n\pi}
\implies \frac{1}{\sin(z)^2} = \sum_{n=-\infty}^\infty \frac{1}{(z - n\pi)^2}
\implies \frac{\pi^2}{\sin(\pi z)^2} = \sum_{n=-\infty}^\infty \frac{1}{(z - n)^2}
$$
Substitute $z$ by $-\fr... |
348,748 | <p>Find the solution for $Ax=0$ for the following $3 \times 3$ matrix:</p>
<p>$$\begin{pmatrix}3 & 2& -3\\ 2& -1&1 \\ 1& 1& 1\end{pmatrix}$$</p>
<p>I found the row reduced form of that matrix, which was </p>
<p>$$\begin{pmatrix}1 & 2/3& -1\\ 0& 1&-9/7 \\ 0& 0& 1\end{pm... | dato datuashvili | 3,196 | <p>now if you imagine that our vector solution is
$x=x_1,x_2,x_3$</p>
<p>than we will get</p>
<p>$x_3=0$</p>
<p>$x_1+(2*x_2)/3=0$</p>
<p>$x_2=0$</p>
<p>so after inserting $x_3,x_2$ into first you get $x_1=0$ </p>
|
2,252,671 | <p>$P(x=k) = \frac{1}{5}$ for $k=1,\cdots,5$. Find $E(X), E(X^2)$ and use these results to obtain $E[(X+3)^2]$ and $Var(3X-2)$</p>
<p>I know how to calcuate all these individually, but how can I use $E(X^2)$ and $E(X)$ to calculate the more complex forms $E[(X+3)^2]$ and $Var(3X-2)$?</p>
| PSPACEhard | 140,280 | <p><strong>Hint:</strong></p>
<ul>
<li><p>$\mathsf{E}[(X + 3)^2] = \mathsf{E}[X^2 + 6X + 9]$</p></li>
<li><p>$\mathsf{Var}(3X - 2) = \mathsf{Var}(3X) = 9\mathsf{Var}(X)$</p></li>
</ul>
|
2,252,671 | <p>$P(x=k) = \frac{1}{5}$ for $k=1,\cdots,5$. Find $E(X), E(X^2)$ and use these results to obtain $E[(X+3)^2]$ and $Var(3X-2)$</p>
<p>I know how to calcuate all these individually, but how can I use $E(X^2)$ and $E(X)$ to calculate the more complex forms $E[(X+3)^2]$ and $Var(3X-2)$?</p>
| Ziad Fakhoury | 295,839 | <p>$$E((X+3)^2) = E(X^2) + E(6X) + E(9)$$
$$ = E(X^2) + 6E(X) + 9$$</p>
<p>As for $Var(3X-2) = Var(3X) = 9Var(X) = 9(E(X^2) - (E(X))^2$</p>
|
92,670 | <p>We're learning about domains and setbuilder notation in school at the moment, and I want to make sure what I did was right.</p>
<p>My thought process:
\begin{align*}
-\frac12|4x - 8| - 1 &< -1 \\
-\frac12|4x - 8| &< 0 \\
|4x - 8| &> 0
\end{align*}
$x =$ all real numbers.</p>
<p>{real numbers} :</... | Dylan Moreland | 3,701 | <p>You get $|4x - 8| > 0$, which I agree with; now you want to find all $x$ satisfying this inequality. It's true that for any number $y$ we have $|y| \geq 0$, but equality can hold: $|y| = 0$ if and only if $y = 0$. Use this fact to find the single value $a$ of $x$ for which $|4x - 8| = 0$. In <a href="http://en.wi... |
1,242,001 | <p>The following is the notation for Fermat's Last Theorem </p>
<p>$\neg\exists_{\{a,b,c,n\},(a,b,c,n)\in(\mathbb{Z}^+)\color{blue}{^4}\land n>2\land abc\neq 0}a^n+b^n=c^n$ </p>
<p>I understand everything in the notation besides the 4 highlighted in blue. Can someone explain to me what this means?</p>
| Mark Viola | 218,419 | <p>If $g(x) =\frac{1}{\sqrt{2\pi}}e^{-(x-2)^2}$, then</p>
<p>$$g'(x)=\frac{1}{\sqrt{2\pi}}e^{-(x-2)^2} \times (-2(x-2))$$</p>
<p>for which $g'$ is zero only at $x=2$.</p>
<p>At $x=2$, we have $g(2)=\frac{1}{\sqrt{2\pi}}$. </p>
<p>This value is the maximum of $g$.</p>
|
1,242,001 | <p>The following is the notation for Fermat's Last Theorem </p>
<p>$\neg\exists_{\{a,b,c,n\},(a,b,c,n)\in(\mathbb{Z}^+)\color{blue}{^4}\land n>2\land abc\neq 0}a^n+b^n=c^n$ </p>
<p>I understand everything in the notation besides the 4 highlighted in blue. Can someone explain to me what this means?</p>
| Jeffrey L. | 227,579 | <p>Apparently WolframAlpha isn't enough to show you that your derivative is incorrect, so let me help you see for yourself.</p>
<p>We're starting out with this (edited for clarity)</p>
<p>$$g(x)=2\pi^\frac{-1}{2} * e^\frac{-(x-2)^2}{2}$$</p>
<p>I'm going to pull out the constant $2\pi^\frac{-1}{2}$ so now we have</p... |
654,617 | <p>$v$ being a vector.
I never understood what they mean and haven't found online resources. Just a quick question.</p>
<p>Thought it was absolute and magnitude respectively when regarding vectors. need confirmation</p>
| Cameron Buie | 28,900 | <p>In general $\lvert\cdot\rvert$ and $\lVert\cdot\rVert$ are both used to signify <a href="http://en.wikipedia.org/wiki/Norm_%28mathematics%29"><em>norms</em></a> of some sort. Different texts use different notation conventions, and sometimes the precise definition (if there <em>is</em> one) will vary from context to ... |
1,181,123 | <blockquote>
<ol>
<li>Find the smallest positive integer such that $80-n$ and $80+n$ are prime numbers. </li>
<li>Find the smallest positive prime number such that $2002-n$ and $2002+n$ are prime numbers.</li>
</ol>
</blockquote>
<p>I cannot think of any way other than trying the prime numbers one by one,
li... | Fermat | 83,272 | <p>For the first.</p>
<p>1) $n$ is not of the form $3k+1$ otherwise $80+n$ is divisible by $3$.</p>
<p>2) $n$ is not of the form $3k+2$ otherwise $80-n$ is divisible by $3$.</p>
<p>3)<strong>So n is a multiple of</strong> $3$.</p>
<p>4) $n$ can not have factors $2, 5$. In particular $n$ is odd. <strong>So n is an o... |
242,097 | <p>I need to show that the function $f(n) = n^2$ is not of $\mathcal{O}(n)$. If I am correct I should prove that there is no number $c,n \geq 0$ where $n^2\lt cn$. How to do that?</p>
| JavaMan | 6,491 | <p><strong>Hint:</strong></p>
<p>Suppose that $n^2 = O(n)$. Then, there exist constats $C, N_0$ such that
$n^2 \leq C n$ for all $ \geq N_0$. However...</p>
|
242,097 | <p>I need to show that the function $f(n) = n^2$ is not of $\mathcal{O}(n)$. If I am correct I should prove that there is no number $c,n \geq 0$ where $n^2\lt cn$. How to do that?</p>
| DonAntonio | 31,254 | <p>$$n^2\leq cn\,\,\,\forall\,n\geq N_0\Longleftrightarrow n\leq c\,\,\,\forall n\geq N_0$$
and thus we'd get the natural numbers are bounded, say by</p>
<p>$$\max\{c\,,\,1,2,...,N_0-1\}$$</p>
|
242,097 | <p>I need to show that the function $f(n) = n^2$ is not of $\mathcal{O}(n)$. If I am correct I should prove that there is no number $c,n \geq 0$ where $n^2\lt cn$. How to do that?</p>
| Austin Mohr | 11,245 | <p>The result is more clear if proven directly, I think.</p>
<p>Let $c > 0$ be given. To conclude $n^2 \neq O(n)$, we need to produce a particular $n_0$, such that $n_0^2 > cn_0$.</p>
<p>To that end, choose $n_0 = c+1$. We have
$$
n_0^2 = (c+1)^2 = c^2 + 2c + 1 > c^2 + c = c(c+1) = cn_0.
$$</p>
<p>Since $c$... |
4,298,951 | <p>Let us define a sequence <span class="math-container">$(a_n)$</span> as follows:</p>
<p><span class="math-container">$$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$</span></p>
<p>Prove that the sequence <span class="math-container">$(a_n)$</span> is Cauchy and find the limit.</p>
<hr />
<... | acreativename | 347,666 | <p>Note</p>
<p><span class="math-container">$a_{n} = \frac{9}{5}+\frac{16}{5}(\frac{-1}{4})^{n}$</span></p>
<p>Thus the limit is <span class="math-container">$\frac{9}{5}$</span>.</p>
|
1,676,848 | <blockquote>
<p>Given the series </p>
<p>$$ \sum_{n=1}^{\infty} \frac{k(k+1)(k+2)\cdot \cdot \cdot (k + n - 1)x^n}{n!} \quad \quad k \geq 1 $$
Find the interval of convergence.</p>
</blockquote>
<p>I started by applying the Ratio test</p>
<p>$$
\lim_{n\to \infty}\left|\frac{k(k+1)(k+2)\cdot \cdot \cdot (k + ... | Lutz Lehmann | 115,115 | <p>Your series is
$$
\sum_{n=0}^\infty\binom{-k}{n}(-x)^n=(1-x)^{-k}
$$
This alone should show that there is no convergence at $x=1$ for positive $k$.</p>
<p>For the series at $x=-1$ consider that
$$
\binom{-k}{n}(-1)^n=\binom{n+k-1}{n}=\binom{n+k-1}{k-1}=\frac{(n+1)(n+2)···(n+k-1)}{(k-1)!}
$$
is a polynomial in $n$ o... |
216,171 | <p>Basically, I have a set of differential equations that I need to solve for exactly 100 different initial conditions (given as lists for each initial condition), and then plot each solution.</p>
<p>Here is some sample code where I have set vrad, vtan, and deltaR (arrays of initial conditions) to an array of length t... | kglr | 125 | <p>An alternative approach: </p>
<ol>
<li>If you use <a href="https://reference.wolfram.com/language/ref/ParametricNDSolveValue.html" rel="nofollow noreferrer"><code>ParametricNDSolveValue</code></a>
you don't have run <code>NDSolve</code> for each 4-tuple of input parameters. </li>
<li>Using the function you want to... |
65,912 | <p>How do I show that $s=\sum\limits_{-\infty}^{\infty} {1\over (x-n)^2}$ on $x\not\in \mathbb Z$ is differentiable without using its compact form? I realize that the sequence of sums $s_a=\sum\limits_{-a}^{a} {1\over (x-n)^2}$ is not uniformly convergent. </p>
<p>I also tried to prove that it is continuous by using t... | Michael Hardy | 11,667 | <p>OK, just to be exotic (?), let's see if we if we can get this from Morera's theorem. Let $C$ be a simple closed curve neither winds around any integer nor passes through any integer. Then
$$
\int\limits_C \sum_{n=-\infty}^\infty \frac{1}{(x-n)^2}\;dx = \sum_{n=-\infty}^\infty\ \int\limits_C \frac{1}{(x-n)^2}\;d... |
1,378,633 | <p>It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series.</p>
<p>So this got me thinking about the mathematical proof for such argument.</p>
<p>But even after going through some resources about the Fourier series (whic... | Alex Pavellas | 255,545 | <p>Since you're referring to signals here, it seems appropriate to consider this question from the viewpoint of an electrical engineer.</p>
<p>If we impose some restrictions on what kind of functions can be considered a "signal," then all periodic signals have a Fourier series.</p>
<ul>
<li>The function should be pie... |
1,378,633 | <p>It seems that some, especially in electrical engineering and musical signal processing, describe that every signal can be represented as a Fourier series.</p>
<p>So this got me thinking about the mathematical proof for such argument.</p>
<p>But even after going through some resources about the Fourier series (whic... | foobar | 11,413 | <p>I came across this question because I wanted to ask the same thing. In Gilbert Strang's Linear Algebra LEC 24, towards the end: <a href="https://youtu.be/8MF3pz-oYHo?t=41m8s" rel="noreferrer">https://youtu.be/8MF3pz-oYHo?t=41m8s</a> he mentions that the parts of a fourier series is like an orthogonal basis, and that... |
1,923,034 | <p>A bagel store sells six different kinds of bagels. Suppose you choose 15 bagels at random. What is the probability that your choice contains at least one bagel of each kind? If one of the bagels is Sesame, what is the probability that your choice contains at least three Sesame bagels?</p>
<p>My approach to the firs... | Community | -1 | <p>A Markovian matrices solution. I like this formalism because it gives more controls on the algorithm and leads to less faulty solutions in general.</p>
<p>To get one of each kind, we define $7$ states, from $0$ to $6$, related to the number of colors found. $k$ is the size of the sample, $k=15$ and $n$ the number o... |
25,137 | <p>I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of <em>adding</em> something by comparing it to something that isn't abstract itself.</p>
<p>In principle: An everyday object or an action that is... | Joseph O'Rourke | 511 | <p>Not quite what you want, but there are mechanical binary counters, e.g.,
this one, video <a href="https://www.reddit.com/r/oddlysatisfying/comments/8ke0yr/mechanical_binary_counter/" rel="nofollow noreferrer">here</a>:</p>
<p><a href="https://i.stack.imgur.com/MB4Xp.jpg" rel="nofollow noreferrer"><img src="https://i... |
25,137 | <p>I want to find an intuitive analogy to explain how binary addition (more precise: an adder circuit in a computer) works. The point here is to explain the abstract process of <em>adding</em> something by comparing it to something that isn't abstract itself.</p>
<p>In principle: An everyday object or an action that is... | Matthew Daly | 12,619 | <p>If you're looking for a metaphor that would instantly click with a modern audience, you might consider mobile merge games. The gameplay is that you have a board that is filled with pieces, and you can combine two identical pieces to yield a single "evolved" piece. For instance, maybe you combine two piec... |
116,537 | <p>Let's say that I have</p>
<pre><code>x^2+x
</code></pre>
<p>Is there a way to map $x$ to the first derivative of a function and $x^2$ to the second derivative of the same function? According to <a href="http://reference.wolfram.com/language/ref/Slot.html" rel="nofollow">http://reference.wolfram.com/language/ref/Sl... | Bruno Le Floch | 39,260 | <p>Rule-replacement with <code>x^n_. :> Derivative[n,0][a][y,z]</code> (as done in Kuba's answer) has two drawbacks: if your polynomial has a constant term, then it will not be replaced by the zero-th derivative <code>a[y,z]</code>, and if your polynomial is not expanded the result is incorrect. Namely, <code>(1+x)... |
1,562,503 | <p>Can anyone help me here?</p>
<p>Question: "X is a normed space and A is a subset dense in the dual of X.
x belongs to X and the sequence (x_n) of X is bounded of E such that f(x_n) converges to f(x) for all f in A. Show that x_n converges to x weakly"</p>
<p>My try: I think that if I show that A=cl(A) so I prove ... | Justpassingby | 293,332 | <p>You're on the wrong track because $A$ is never closed except in the trivial case where $A=X^*.$</p>
<p>You need to prove that for arbitrary $f\in X^*$</p>
<p>$$\lim_{n\to\infty}f(x_n)=f(x).$$</p>
<p>You can do this using the $\epsilon$-definition of the limit of a sequence, first choosing a $g\in A$ sufficiently ... |
3,088,766 | <p>I need to prove that the premise <span class="math-container">$A \to (B \vee C)$</span> leads to the conclusion <span class="math-container">$(A \to B) \vee (A \to C)$</span>. Here's what I have so far.</p>
<p><a href="https://i.stack.imgur.com/1AgTZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Rob Arthan | 23,171 | <p>Hint: if you assume <span class="math-container">$A \to (B \lor C)$</span>, <span class="math-container">$\lnot(A \to B)$</span> and <span class="math-container">$A$</span>, then you can conclude <span class="math-container">$B \lor C$</span> and <span class="math-container">$\lnot B$</span>. Can you take it from th... |
3,088,766 | <p>I need to prove that the premise <span class="math-container">$A \to (B \vee C)$</span> leads to the conclusion <span class="math-container">$(A \to B) \vee (A \to C)$</span>. Here's what I have so far.</p>
<p><a href="https://i.stack.imgur.com/1AgTZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Frank Hubeny | 312,852 | <p>I assume you are using the <a href="http://proofs.openlogicproject.org/" rel="nofollow noreferrer">proof checker</a> associated with the <em>forallx</em> text. There is no rule in this proof checker that allows inference rules for <a href="https://en.wikipedia.org/wiki/Material_implication_(rule_of_inference)" rel="... |
3,325,340 | <p>Show that <span class="math-container">$$ \lim\limits_{(x,y)\to(0,0)}\dfrac{x^2y^2}{x^2+y^2}=0$$</span>
My try:
We know that, <span class="math-container">$$ x^2\leq x^2+y^2 \implies x^2y^2\leq (x^2+y^2)y^2 \implies x^2y^2\leq (x^2+y^2)^2$$</span>
Then, <span class="math-container">$$\dfrac{x^2y^2}{x^2+y^2}\leq x^2+... | Axion004 | 258,202 | <p>In two variables the epsilon-delta definition for <span class="math-container">$\lim_{\substack{x\to a\\ y\to b}}f(x,y)=L$</span> means that for every <span class="math-container">$\epsilon >0$</span> there exists a <span class="math-container">$\delta>0$</span> such that <span class="math-container">$\big|f(x... |
524,870 | <p>This is adapted from 1.7.7 in Friedman's "Foundations of Modern Analysis":</p>
<blockquote>
<p>Let $\mathscr{B}$ be the $\sigma$-ring generated by the class of open subsets of $X$ [a fixed set], and $\mathscr{D}$ the $\sigma$-ring generated by the class of closed subsets of $X$. Show that $\mathscr{D} = \mathscr{... | Robert Israel | 8,508 | <p>Hint: show that every closed set is in $\mathscr B$ and every open set is in $\mathscr D$.</p>
|
617,747 | <p>In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural numbers can be understood as putting together two apples and two oranges makes four fruits. How can we apply this thinking... | Gerry Myerson | 8,269 | <p>"how does something like complex numbers apply to the real world?" Type $$\rm circuits\ and\ complex\ numbers$$ into Google, and you will find that computations of currents in electrical circuits are done using complex numbers. </p>
<p>"Why do complex numbers exist?" The equation $x^3-4x+1=0$ has three real solutio... |
617,747 | <p>In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural numbers can be understood as putting together two apples and two oranges makes four fruits. How can we apply this thinking... | Lucian | 93,448 | <p>Numbers count $(\mathbb{N})$ and measure $(\mathbb{R})$. Yet complex $(\mathbb{C})$ or imaginary $(i\,\mathbb{R})$ numbers do neither. So what good are they anyway ? $($Is this what you're asking ?$)$ Well, let's just say that <a href="http://en.wikipedia.org/wiki/Complex_number#Applications" rel="noreferrer">engine... |
4,572,804 | <p>I'm working my way through Murphy's, C<em>-algebras and Operator Theory and I have a question concernig the proof that every C</em>-algebra admits an approximate identity.</p>
<p>Let A be an arbitrary C*-algebra. We denote by <span class="math-container">$\Lambda$</span> the set of all positive elements a in A such ... | belkacem abderrahmane | 660,639 | <p>for <span class="math-container">$w\in K$</span>, <span class="math-container">$g(w) =1\implies \lvert f-\delta g f\rvert =(1-\delta) \lvert f\rvert <\epsilon $</span> (since Gelfand transformation is an isometry), for <span class="math-container">$$ w\in K^{c}, \lvert f(w) <\rvert \epsilon, \lvert g(w) \rvert... |
4,572,804 | <p>I'm working my way through Murphy's, C<em>-algebras and Operator Theory and I have a question concernig the proof that every C</em>-algebra admits an approximate identity.</p>
<p>Let A be an arbitrary C*-algebra. We denote by <span class="math-container">$\Lambda$</span> the set of all positive elements a in A such ... | Danny Pak-Keung Chan | 374,270 | <p><span class="math-container">$f\in C_{0}(\Omega)$</span> with <span class="math-container">$||f||_{\sup}=||a||<1$</span>. Let <span class="math-container">$\omega\in\Omega$</span>
be arbitrary. If <span class="math-container">$\omega\in K$</span>, then <span class="math-container">$g(\omega)=1$</span>. Therefore
... |
137,755 | <p>Suppose that $X$ is a scheme and $x\in X$ is a point. The stalk of $X$ at $x$ is a (local) ring and we can form its spectrum $Y_x=\rm{Spec}(\mathcal{O}_{X,x})$.</p>
<p>There is a canonical map $Y_x\to X$. We can define it by fixing an affine neighborhood $x\in U\cong \rm{Spec}(R)$, making $x$ as a prime ideal in $R... | Georges Elencwajg | 450 | <p>Topologically the scheme $\rm{Spec}(\mathcal{O}_{X,x})$ is exactly the intersection of all neighbourhoods of $x$ and algebraically it contains every infinitesimal neighbourhood of $X$.<br>
Although technically it is not the germ of$X$ at $x$, it seems to me that it contains so much information about that germ that i... |
209,761 | <p>I've the following code:</p>
<pre><code>Table[b = (-12 +Sqrt[3] Sqrt[3 (-4 + r)^2 + 12 a^2 (-2 + r) - 4 a (-5 + r) (-2+ r) +4 a^3 (-2 + r)^2] + 3 r)/(6 (-2 + r)) /. r -> 5;
If[IntegerQ[b], {b, a}, Nothing], {a, 1+10^(11), 10^(12)}]
</code></pre>
<p>But it gives me the following warning: 'SystemException["Memory... | yarchik | 9,469 | <p>You can replace <code>Table</code> with <code>Do</code></p>
<pre><code>Do[If[IntegerQ[1/6 (1 + Sqrt[1 + 12 a^2 + 12 a^3])], Print[a]], {a, 1 + 10^(11), 10^4 + 10^(11)}]
</code></pre>
<p>but it is still slow. Try to bring your diophantine equation to some known type.</p>
|
209,761 | <p>I've the following code:</p>
<pre><code>Table[b = (-12 +Sqrt[3] Sqrt[3 (-4 + r)^2 + 12 a^2 (-2 + r) - 4 a (-5 + r) (-2+ r) +4 a^3 (-2 + r)^2] + 3 r)/(6 (-2 + r)) /. r -> 5;
If[IntegerQ[b], {b, a}, Nothing], {a, 1+10^(11), 10^(12)}]
</code></pre>
<p>But it gives me the following warning: 'SystemException["Memory... | bbgodfrey | 1,063 | <p>The approach suggested by yarchik can be accelerated by two orders of magnitude by performing the computations with machine precision numbers instead of exact numbers and then rounding, and by using <code>ParallelDo</code>:</p>
<pre><code>SetSharedVariable[s]
s = {};
ParallelDo[If[IntegerQ[(1 + a*Round[Sqrt[1./(a*... |
3,183,617 | <p>I have an equation that looks like <span class="math-container">$$X' = a \sin(X) + b \cos(X) + c$$</span> where <span class="math-container">$a,b$</span> and <span class="math-container">$c$</span> are constants. For given values of <span class="math-container">$a, b$</span> and <span class="math-container">$c$</spa... | Lutz Lehmann | 115,115 | <h3>direct transformation using half-angle formulas</h3>
<p>The probably best systematic method (from integrals of quotients of trigonometric expressions) is the half-angle tangent substitution. Set <span class="math-container">$U=\tan(X/2)$</span>, then <span class="math-container">$\sin(X)=\frac{2U}{1+U^2}$</span>, <... |
2,296,724 | <p>I need to calculate $(A+B)^{-1}$, where $A$ and $B$ are two square, very sparse and very large. $A$ is block diagonal, real symmetric and positive definite, and I have access to $A^{-1}$ (which in this case is also sparse, and block diagonal). $B$ is diagonal and real positive. In my application, I need to calculate... | Oussama Boussif | 258,472 | <p>Let $A_1,A_2,\cdots,A_q$ be the diagonal blocks of $A$, and $a_{1,1},a_{1,2},\cdots,a_{1,n_1},a_{2,1},a_{2,2},\cdots,a_{2,n_2},\cdots,a_{q,1},a_{q,2},\cdots,a_{1,n_q}$ the diagonal elements of $B$, then the inverse of the sum would simply be a diagonal block matrix with blocks: ${(A_i+diag(a_{i,1},\cdots,a_{i,n_i}))... |
2,614,920 | <p>Not understanding the concept well, I am trying to determint the pointwise and uniform convergence of the following sequence of function:</p>
<p>$$f_n(x) = \frac{\sin{nx}}{n^3}, x \in \mathbb{R}$$</p>
<p>The only part I understand so far is that I need $\lim_{x\to\infty}{f_n(x)}$ in which I have determined that (i... | Paul Frost | 349,785 | <p>It seems that you argue that the restriction of a (strong) deformation retraction <span class="math-container">$r : X \to A$</span> to any <span class="math-container">$Y$</span> with <span class="math-container">$A \subset Y \subset X$</span> is again a (strong) deformation retraction. It is of course a retraction,... |
2,880,384 | <p>Look at the following definition.</p>
<p><strong>Definition.</strong> Let $\kappa$ be an infinite cardinal. A theory $T$ is called $\kappa$-stable if for all model $M\models T$ and all $A\subset M$ with $|A|\leq \kappa$ we have $|S_n^M(A)|\leq \kappa$.
A theory $T$ is called stable if it is $\kappa$-stable for some... | user584026 | 584,026 | <p>Just to add on one point to why the classification program is natural:</p>
<p>One can interpret the model theorist's approach to studying a structure $M$ as assigning to $M$ its ``logical invariances'' which is just a fancy way to refer to the theory $T$ of $M$.</p>
<p>It is therefore natural trying to understand ... |
1,187,376 | <p>Let $c(n,k)$ be the unsigned Stirling numbers of the first kind, i.e., the number of
$n$-permutations with exactly $k$ cycles.
Apparently, $$\sum_{k=1}^n c(n,k)2^k = (n+1)!$$</p>
<p>I want to prove the equality. </p>
<p>I am most interested in a combinatorial explanation. </p>
<p>The exponential generating fun... | Marko Riedel | 44,883 | <p>By way of enrichment here is a proof using generating functions.
Suppose we seek to evaluate
$$\sum_{k=1}^n \left[n\atop k\right] 2^k.$$<P></p>
<p>The species of decompositions of permutations into cycles marked by
the number of cycles is
$$\mathfrak{P}(\mathcal{U}\mathfrak{C}(\mathcal{Z})).$$
This gives the gen... |
2,293,147 | <p>I was trying to solve this ODE $\frac{dy}{dx} = c_{1} + c_{2}y + \frac{c_{3}}{y} , y(0) = c , c >0$.</p>
<p>where $c_{1},c_{2},c_{3}$ are three real numbers say $c_{1} < 0,c_{2},c_{3} > 0$.</p>
<p>I thought of using separation of variables giving me $x = \int(\frac{y}{c_{1}y+c_{2}y^2+c_{3}})dy + c$.</p>
... | Community | -1 | <p>By a linear transform $y=ax+b$, you can establish</p>
<p>$$\frac y{c_1y+c_2y^2+c_3}=\frac{ax+b}{c(x^2\pm1)}$$ where the sign is dictated by an expression below.</p>
<p>For this, write</p>
<p>$$c_1(ax+b)+c_2(a^2x^2+2abx+b^2)+c_3$$ and identify</p>
<p>$$\begin{cases}c_2a^2=c,\\c_1a+2c_2ab=0,\\c_1b+c_2b^2+c_3=\pm c... |
223,582 | <p>Maps $g$ maps $\left\{1,2,3,4,5\right\}$ onto $\left\{11,12,13,14\right\}$ and $g(1)\neq g(2)$. How many g are there.</p>
<p><strong>My answer</strong>:
I transformed the question to a easy-understand way and find out the solution.
Consider there are five children and four seats. Two of them are willing sitting to... | ccorn | 75,794 | <p>You can do it with a straightedge alone, though the lines tend to clutter the scene.</p>
<p><a href="https://i.stack.imgur.com/2epTR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2epTR.png" alt="Straightedge-only construction of polar and tangents"></a></p>
<p>Start with the two solid black li... |
2,788,276 | <p>Let$\ f_n (x)=n^2x(1-x)^n$ I need to prove that$\ f_n→0$ in the interval$\ [0,1]$.</p>
<hr>
<p>Let$\ f_n(x) = nx^n$ prove that$\ f_n→0$ in the interval$\ [0,1)$.</p>
<p>For both of these sequences I tried the following:</p>
<p>By taking the function$\ f(x)=0$ we can see that</p>
<p>$$\lim_{n\rightarrow\infty}f_... | quasi | 400,434 | <p>Unless I've made a mistake, here is a partial generalization . . .
<p>
Let $R$ be a commutative ring with $1\ne 0$ such that</p>
<ul>
<li>$2$ is a unit of $R$.
<li>For some monic $f\in R[x]$, we have $f(r)=0$, for all $r\in R$.
</ul>
<p><strong>Claim:</strong>$\;R$ has Krull dimension $0$.
<p>
<strong>Proof:</stro... |
33,622 | <p>I am looking for differentiable functions $f$ from the unit interval to itself that satisfy the following equation $\forall\:p \in \left( 0,1 \right)$:</p>
<p>$$1-p-f(f(p))-f(p)f'(f(p))=0$$</p>
<p>Is there a way to use <em>Mathematica</em> to solve such equations?<br>
<code>DSolve</code> is of course unable to han... | gpap | 1,079 | <p>If <code>f</code> is differentiable in the unit interval then it has a power series expansion in that interval. Use an assumed polynomial <code>trial</code> of order <code>order</code> in variable <code>var</code> to denote that expansion</p>
<pre><code>ClearAll[trial, equation, solutions];
trial[order_, var_... |
481,421 | <p>Find the limit of:
$$\lim_{x\to\infty}{\frac{\cos(\frac{1}{x})-1}{\cos(\frac{2}{x})-1}}$$</p>
| NightRa | 90,049 | <p>Let's define:
$$h=\frac{1}{x}$$
Then:
$$h\to0$$
So we will rewrite the limit as:
$$\lim_{h\to0}{\frac{\cos(h)-1}{\cos(2h)-1}}=\lim_{h\to0}{\frac{-\sin(h)}{-2\sin(2h)}}=\lim_{h\to0}{\frac{h}{2\cdot2h}}=\frac{1}{4}$$</p>
|
2,195,739 | <p>For
$$
f(x) = \begin{cases}
x^2 & \text{if $x\in\mathbb{Q}$,} \\[4px]
x^3 & \text{if $x\notin\mathbb{Q}$}
\end{cases}
$$</p>
<p>What I did was examine each of the limits at $0$ of
$\displaystyle\lim_{x\to0} \frac{f(x)-f(a)}{x-a}$ for each case but I am not sure </p>
| Jonathan Barkey | 414,649 | <p>By the Sequential Criterion for Limits, we know that $$\lim_{x\to c}f(x)=L$$ if and only if for every sequence $(x_n)$ in the domain of $f$ that converges to $c$ such that $x_n\ne c$ for all $n$, then the sequence $(f(x_n))$ converges to $L$.</p>
<p>Let $(x_n)\in\mathbb{Q}$ such that $x_n\ne0$ for all $n$ and $\lim... |
82,716 | <p>There seems to be two competing(?) formalisms for specifying theories: <a href="http://ncatlab.org/nlab/show/sketch" rel="noreferrer">sketches</a> (as developped by Ehresmann and students, and expanded upon by Barr and Wells in, for example, <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf" rel="nore... | Zinovy Diskin | 19,786 | <p>@Jacques: On relations between sketches and institutions. The former is a particular instance of the latter: sketches of a given type and their models form an institution. In more detail, signatures are graphs, sentences are diagrams of a given type, and models are sketch morphisms of a given type, say, into Set. <... |
1,382,087 | <p>Problem:</p>
<p>A bag contains $4$ red and $5$ white balls. Balls are drawn from the bag without replacement.</p>
<p>Let $A$ be the event that first ball drawn is white and let $B$ denote the event that the second ball drawn is red. Find </p>
<p>(i) $P(B\mid A)$</p>
<p>(ii) $P(A\mid B)$</p>
<p>My confusion is t... | callculus42 | 144,421 | <blockquote>
<p>Can we say that in general if $P(A\mid B)$ exists then $P(B\mid A)$
should also exist?</p>
</blockquote>
<p>Not necessarily. I modifiy your exercise.</p>
<blockquote>
<p>A bag contains $4$ red and $5$ white balls. Balls are drawn from the
bag without replacement.</p>
<p>Let $A$ be the eve... |
749,473 | <p>I am trying to model the time it takes until a malfunction appears. For example the time a light-bulb will last. I would like the probability that the light-bulb will burn out at a certain moment (given it hadn't bunt yet) to increase as a function of the time ($P(x | X \geq x$) should be monotonic increasing). That... | Henry | 6,460 | <p>If you want a distribution with a maximum lifetime (say $c$) then you might consider $$F(x)=1-\left(1-\frac{x}{c}\right)^\beta$$ $$f(x)= \frac{\beta}{c}\left(1-\frac{x}{c}\right)^{\beta-1}$$ for some positive $\beta$: for $\beta=1$ this gives a uniform distribution on $[0,c]$. It is a kind of scaled Beta distributi... |
3,634,416 | <p>First of all, English is not my native language, but Chinses is. I tried to spilt the integration interval into 2 pieces: <span class="math-container">$ [0, 1-1/n] $</span> and <span class="math-container">$ [1-1/n, 1] $</span>. In both intervals I use the mean value theorem:
<span class="math-container">$$
\in... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$\int_0^{1}\frac 1 {1+x^{n}}dx =1-\int_0^{1}\frac {x^{n}} {1+x^{n}}dx$</span>. Note that <span class="math-container">$0 \leq \frac {x^{n}} {1+x^{n}} \leq x^{n}$</span> and <span class="math-container">$\int_0^{1} x^{n}dx=\frac 1 {n+1} \to 0$</span>. Put these together to see that the ... |
1,146,050 | <p>given $f(x)=\frac{x^4+x^2+1}{x^2+x+1}$.</p>
<p>Need to find the min value of $f(x)$.</p>
<p>I know it can be easily done by polynomial division but my question is if there's another way</p>
<p>(more elegant maybe) to find the min? </p>
<p><strong>About my way</strong>: $f(x)=\frac{x^4+x^2+1}{x^2+x+1}=x^2-x+1$. (... | lab bhattacharjee | 33,337 | <p>$$x^2-x+1=\frac{4x^2-4x+4}4=\frac{(2x-1)^2+3}4\ge\frac34$$</p>
<p>The equality occurs if $2x-1=0\iff x=\dfrac12$</p>
|
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| user | 505,767 | <p>Let $$P = p_1p_2...p_n+1$$ and let $p$ be a prime such that $p\mid P$.</p>
<p>Then $p$ can not be any of $p_1,p_2,p_3,\cdots ,p_n$ otherwise $p$ would divide the difference $P-p_1p_2...p_n=1$ which is not possible.</p>
|
2,573,458 | <p>Given $n$ prime numbers, $p_1, p_2, p_3,\ldots,p_n$, then $p_1p_2p_3\cdots p_n+1$ is not divisible by any of the primes $p_i, i=1,2,3,\ldots,n.$ I dont understand why. Can somebody give me a hint or an Explanation ? Thanks.</p>
| Bram28 | 256,001 | <p>Simple example. Suppose I consider $2 \cdot 3 \cdot 5 \cdot 7 = 210$</p>
<p>Now, $2$ divides $210$, and so do $3$, $5$, and $7$ ... of course! </p>
<p>But what happens if you divide $210+1=211$ by $2$? You get a remainder of $1$ ... exactly because you got a remainder of $0$ when dividing $210$. And the exact ... |
3,143,084 | <p>If <span class="math-container">$f : \mathbb{R} \to \mathbb{R}$</span>, we can think of the derivative of <span class="math-container">$f$</span> at a point <span class="math-container">$x$</span>, denoted <span class="math-container">$f'(x)$</span>, as giving the slope of a line tangent to the graph of <span class=... | JoseSquare | 643,097 | <p>By Newton's Binomial formula <span class="math-container">$(x+h)^{123} =x^{123} +123x^{122}h+\ldots$</span> then you get </p>
<p><span class="math-container">$$\lim_{h \to 0} \frac{(x^{123} +123x^{122}h+\ldots)-x^{123}}{h}=
\lim_{h \to 0} \frac{123x^{122}h +
\binom{123}{2}x^{121}h^2 + \ldots +h^{123}}{h} = 123x^{1... |
240,741 | <p>I'm trying to include the legends inside the frame of the plot like this</p>
<p><a href="https://i.stack.imgur.com/7K5aa.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/7K5aa.jpg" alt="hehe" /></a></p>
<p>Here is my Attempt:</p>
<pre><code>ListPlot[{{2, 5, 2, 8, 6, 8, 3}, {1, 2, 5, 2, 3, 4, 3}},
PlotMark... | Bob Hanlon | 9,362 | <p>Space for the legend is available at the upper left</p>
<pre><code>ListPlot[{
{2, 5, 2, 8, 6, 8, 3},
{1, 2, 5, 2, 3, 4, 3}},
PlotMarkers -> {"✶", 15},
Joined -> True,
PlotStyle -> {Orange, Green},
PlotLegends ->
Placed[
LineLegend[{"line1", "line2"},
LegendF... |
2,293,746 | <p>A function f has derivative for all $x\in \mathbb R$ and the limits of $f$ at $+\infty $, $-\infty$ are equal to $+\infty$ . Is it true that $\lim_{x\to a} \frac {1}{f'(x)} = + \infty $ or $-\infty$ for some $a\in\mathbb R$ ?</p>
<p>Of course function $f' $ has roots , according to Fermat's theorem( $f$ has a tot... | Paramanand Singh | 72,031 | <p>Your statement is false. In fact you can take $f$ to be constant in some interval and let $f$ be decreasing before that interval and increasing after that interval. Thus let $f(x) =(x+1)^{2},x<-1,f(x)=0,|x|\leq 1,f(x)=(x-1)^{2},x>1$. Then we can see that $f$ is differentiable everywhere, but there is no point ... |
381,177 | <p>I have a problem in which I have to compute the following integral: <span class="math-container">$$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$</span>
where this notation means that I want to integrate over <span class="math-container">$... | Iosif Pinelis | 36,721 | <p><span class="math-container">$\newcommand\1{\mathbf1}\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$</span>Here is a more explicit way to define the disintegration of the Lebesgue measure over <span class="math-container">$\R^k$</span> into the measures <span class="math-container">$\mu_t$</span> over the plane... |
1,513,373 | <p>Let M be a cardinal with the following properties:<br>
- M is regular<br>
- $\kappa < M \implies 2^\kappa < M$<br>
- $\kappa < M \implies s(\kappa) < M$ where $s(\kappa)$ is the smallest strongly inaccessible cardinal strictly greater than $\kappa$ </p>
<p>My question is: Is M a Mahlo cardinal ? If s... | Andreas Blass | 48,510 | <p>Wojowu has answered the question, but it might be useful to record here why the first $M$ that satisfies your conditions is not a Mahlo cardinal. Consider the set $C$ of those cardinals $\lambda<M$ that satisfy $(\forall\kappa<\lambda)\,2^\kappa<\lambda$ and $(\forall\kappa<\lambda)\,s(\kappa)<\lambd... |
4,294,860 | <p>Let's say I have a group of n people. Some are left handed and some are right handed. I need to know a random person identity, knowing if he is right or left handed</p>
<p>As conditional probabilty:</p>
<p>Being <span class="math-container">$P(X)$</span> the probability of correctly guessing a person identity.</p>
<... | user2661923 | 464,411 | <p>The following analysis <strong>assumes</strong> that the chance of someone being left handed (for example) is less than <span class="math-container">$1$</span> and greater than <span class="math-container">$0$</span>.</p>
<p>Let's try it with actual numbers. Suppose that <span class="math-container">$1$</span> out ... |
4,029,249 | <blockquote>
<p>Prove that the function <span class="math-container">$f(x)=e^x-(ax^2+bx+c)$</span> has 3 solutions at most .</p>
<p><span class="math-container">$a$</span>,<span class="math-container">$b$</span> and <span class="math-container">$c$</span> are constants.</p>
</blockquote>
<p>This is the information give... | Guillemus Callelus | 361,108 | <p>The arguments you make need to be a little more precise, but the idea is correct!</p>
<p>Let the function <span class="math-container">$f(x)=e^x-(ax^2+bx+c)$</span> that is a continuous and differentiable function in <span class="math-container">$\mathbb{R}$</span> and let its derivative function <span class="math-c... |
1,286,306 | <p>Suppose that $a_1,...,a_n,b_1,...,b_n ∈ F $ are such that $\sum a_ib_i = 1_F$. </p>
<p>Let $J : F^n → F^n $ be the linear transformation whose standard matrix has $ij^{th}$ entry $a_ib_j$. </p>
<p>Prove that $J^2 = J$.</p>
<p>So I think I've figured out that the index in the matrix $F^2$ given by </p>
<p>$u_{ij... | Rikimaru | 80,284 | <p>You should be careful with blindly applying Tychonoff's theorem. After all, that theorem also says that the "unit cube" $C = \{(x_1,...) : x \in [0,1] \}$ is compact, even though the sequence $(x_n)_i = \delta_{in}$ lacks a converging subsequence. This kind of nonsense cannot happen with your cube because you've imp... |
2,978,988 | <p>I'm stuck at a question. </p>
<p>The question states that <span class="math-container">$K$</span> is a field like <span class="math-container">$\mathbb Q, \mathbb R, \mathbb C$</span> or <span class="math-container">$\mathbb Z/p\mathbb Z$</span> with <span class="math-container">$p$</span> a prime. <span class="mat... | Oleg567 | 47,993 | <p>Yes, of course:
for any <span class="math-container">$n \in \mathbb{N}$</span>:
<span class="math-container">$$
\sum_{k=n^2}^{n^2+n}k = \dfrac{(n+1)(2n^2+n)}{2} = \dfrac{(n+1)n(2n+1)}{2};\tag{1}
$$</span>
and
<span class="math-container">$$
\sum_{k=n^2+n+1}^{n^2+2n}k = \dfrac{n(2n^2+3n+1)}{2} = \dfrac{n(2n+1)(n+1)}{... |
863,860 | <p>I am not particularly well-versed in topology, so I wanted to check with you whether there exists a much simpler argument to prove the following statement or whether there are problems with my proof. The statement also seems to be a very standard result but I could not find a reference in e.g. a book on basic topolo... | user138999 | 162,288 | <p>$$\begin{align}
f'(x)&=\frac{d}{dx}\frac{x^2+4x+3}{\sqrt{x}} \\
&= \frac{d}{dx}\left(\frac{x^2}{\sqrt{x}}+\frac{4x}{\sqrt{x}}+\frac{3}{\sqrt{x}}\right) \\
&=\frac{d}{dx}\left(x^{\frac{3}{2}}+4\sqrt{x}+3(x)^{-\frac{1}{2}}\right) \\
&=\frac{3}{2}x^{\frac{1}{2}}+\frac{4}{2\sqrt{x}}-\frac{3}{2x^{\frac{3}... |
1,504,483 | <p>Where did the angle convention (in mathematics) come from?</p>
<p>One would imagine that a clockwise direction would be more 'natural' (given
sundials & the like, also a magnetic compass dial).</p>
<p>Also, given time and direction conventions, one would imagine that the
zero degree line would be vertical.</p>... | Singh | 121,735 | <p>We measure the angles with the $x$-axis. So one of the arm of the angle is $x$-axis and the other arm is also on $x$-axis if the angle is zero. This is why we take zero degree line along the $x$-axis.</p>
<p>In rectangular coordinate system we have four quadrants. Now we move the second arm which is fixed to the or... |
1,215,537 | <p>I need to prove that
$ \int_0^\infty (\frac{\sin x}{x})^2 = \frac{\pi}{2}$.
I have proved that $\sum_1^\infty \frac {\sin^2(n \delta)}{n^2 \delta}=\frac{\pi-\delta}{2}$ for $0<\delta<\pi$ and I'm supposed to use this identity.</p>
| Seyed Mohsen Ayyoubzadeh | 165,227 | <p>Use your identity for $\delta \to 0$. In this case, by assuming $x = n\delta $, and noting that $dx = (n + 1)\delta - n\delta = \delta $ and ${x_{\min }} = 1\delta = 0$ and ${x_{\max }} = +\infty \delta = +\infty$, your identity will give $$\mathop {\lim }\limits_{\delta \to 0} \sum\limits_{n = 1\atop x = n\de... |
31,480 | <p>I'm having difficulty with my math, fractions and up. I used to understand it all, but it's been so long since I've touched the book (I finished it a couple of months ago, picked it up to review everything), I seem to have forgotten it. </p>
<p>The explanations inside of the individual chapters do no good. They nev... | GeoffDS | 8,671 | <p>I upvoted picakhu's answer and it's probably better than mine. But, this might be helpful still. Not online, but maybe the <a href="http://www.artofproblemsolving.com/index.php?mode=books">Art of Problem Solving</a> would be helpful. It is what I am going to use for my children, possibly. There are 8 books start... |
2,541,709 | <p>For example Calculate the probability of getting exactly 50 heads and 50 tails after flipping a fair coin $100$ times. then is ${100 \choose 50}\left(\frac 12\right)^{50}\left(\frac 12\right)^{50}$
the reason that we multiply $\left(\frac 12\right)^{50}$ twice is because the first one $\left(\frac 12\right)^{50}$ is... | zhw. | 228,045 | <p>It's actually true for any continuous $f$ bounded on $\mathbb R.$ Let $M= \sup_{\mathbb R} |f|.$ Rewrite the convolution as</p>
<p>$$\int_{\mathbb R} f(x-t)g(t)\,dt.$$</p>
<p>Fix any $x$ and let $x_n \to x.$ By the continuity of $f,$ $f(x_n-t) \to f(x-t)$ pointwise on $\mathbb R.$ Since $|f(x_n-t)g(t)| \le M|g(t)|... |
64,780 | <p>I need to sum values that belongs to same week. For example, I have the list x with one column and n rows. Format: </p>
<pre><code>{{2007,1,3},0.2},{2007,1,4},0.1},{2007,1,5},0.14},{2007,1,8},0.}, ... {2014,10,17},-0.2},{2014,10,18},0.2},{2014,10,19},0.2}}.
</code></pre>
<p>Dates in list are sorted in the form fro... | Kuba | 5,478 | <p>Let's convert dates:</p>
<pre><code>data ={
{{2007, 1, 3}, 0.2}, {{2007, 1, 4}, 0.1}, {{2007, 1, 5}, 0.14}, {{2007, 1, 8}, 0.},
{{2014, 10, 17}, -0.2}, {{2014, 10, 18}, 0.2}, {{2014, 10, 19}, 0.2}};
data = MapAt[DateList, data, {;; , 1}];
</code></pre>
<p>I don't know if this can be done automatically but un... |
3,296,596 | <p>Ive been asked the following question and I'm not sure how to approach it.</p>
<p>Solve the system</p>
<p><span class="math-container">\begin{cases}
x_1+x_2-5x_3=2 \\
6x_1+7x_2+4x_3=7
\end{cases}</span></p>
<p>The answer is required to be in the form of</p>
<p><span class="math-container">$\begin{bmatrix}x_1\\ x... | Community | -1 | <p><strong>Hint:</strong> Evidently, the solution set is <span class="math-container">$1$</span>-dimensional. </p>
<p>Now use row-reduction to find it.</p>
|
3,296,596 | <p>Ive been asked the following question and I'm not sure how to approach it.</p>
<p>Solve the system</p>
<p><span class="math-container">\begin{cases}
x_1+x_2-5x_3=2 \\
6x_1+7x_2+4x_3=7
\end{cases}</span></p>
<p>The answer is required to be in the form of</p>
<p><span class="math-container">$\begin{bmatrix}x_1\\ x... | Fred | 380,717 | <p>If you use the RREF, you will get</p>
<p><span class="math-container">$x_1=7+39x_3$</span></p>
<p>and</p>
<p><span class="math-container">$x_2=-5-34x_3.$</span></p>
<p>Now put <span class="math-container">$s=x_3$</span> an we derive</p>
<p><span class="math-container">$$\begin{bmatrix}x_1\\ x_2\\x_3\end{bmatrix... |
3,296,596 | <p>Ive been asked the following question and I'm not sure how to approach it.</p>
<p>Solve the system</p>
<p><span class="math-container">\begin{cases}
x_1+x_2-5x_3=2 \\
6x_1+7x_2+4x_3=7
\end{cases}</span></p>
<p>The answer is required to be in the form of</p>
<p><span class="math-container">$\begin{bmatrix}x_1\\ x... | Dr. Sonnhard Graubner | 175,066 | <p>Multiplying the first equation by <span class="math-container">$4$</span> and the second by <span class="math-container">$5$</span> we get
<span class="math-container">$$x_1=\frac{43}{34}-\frac{39}{34}x_2$$</span>
so <span class="math-container">$x_2=-5+34x_3$</span> and we get
<span class="math-container">$$[x_1,x_... |
3,574,460 | <p>Suppose <span class="math-container">${X_0, X_1, . . . , }$</span> forms a Markov chain with state space S. For any n ≥ 1
and <span class="math-container">$i_0, i_1, . . . , ∈ S$</span>, which conditional probability, <span class="math-container">$P(X_0 = i_0|X_1 = i_1)$</span> or <span class="math-container">$P(X_... | kiyomi | 527,262 | <p>If you are working with a stationary ergodic Markov chain, then what you mention is a <a href="http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-Time-Reversibility.pdf" rel="nofollow noreferrer">time-reversible markov chain</a>, which must satisfy certain criterions to be classified as one, make sure those app... |
98,361 | <p>I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems trying to get the second chapter right. I have been able to get the definitions and work out some problems, but I am sti... | Samuel Reid | 19,723 | <p>As I found out while working through that chapter, a lot of misunderstanding can arise from not understanding the idea of a limit point thoroughly. To remedy this, I recommend you visit a question I asked a little while ago: <a href="https://math.stackexchange.com/questions/93288/understanding-the-idea-of-a-limit-po... |
98,361 | <p>I have been reading Rudin (Principles of Mathematical Analysis) on my own now for around a month or so. While I was able to complete the first chapter without any difficulty, I am having problems trying to get the second chapter right. I have been able to get the definitions and work out some problems, but I am sti... | yep | 22,795 | <p>I was in precisely your situation several years ago. In hindsight, Rudin was a poor text for self-study. Perhaps if you're the kind of person who grew up with mathematical culture (parents mathematical, friends interested in mathematics,etc), you'll have the broad cultural background necessary to appreciate the over... |
10,949 | <p>Is it known whether every finite abelian group is isomorphic to the ideal class group of the ring of integers in some number field? If so, is it still true if we consider only imaginary quadratic fields?</p>
| Pete L. Clark | 299 | <p>Virtually nothing is known about the question of which abelian groups can be the ideal class group of (the full ring of integers of) some number field. So far as I know, it is a plausible conjecture that all finite abelian groups (up to isomorphism, of course) occur in this way. Conjectures and heuristics in this ... |
956,110 | <p>I am struggling with thinking about this. Any help would be great!!</p>
<p>A medical research survey categorizes adults as follows:</p>
<ul>
<li>by gender (male or female)</li>
<li>by age group (age groups are 18-25, 26-35, 36-50, 51+)</li>
<li>by income (less than 30k/year, 30k-60k/year, more than 60k/year)</li>
... | monsterx | 180,720 | <p>By the Pigeonhole Principle it would be one more than the number of categories, C(5,2)*4*3 = 10*4*3 = 120 => 120 +1 = 121.</p>
|
2,609,252 | <p>like the title said i'm looking for the best way for me(a 15 year old) to go about learning calculus, thank you :)</p>
| Botond | 281,471 | <p>3Blue1Brown's Essence of calculus is a good starting point: <a href="https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr" rel="nofollow noreferrer">https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr</a><br>
And for a lot of excercises, you can check blackpenredpen: <a href=... |
4,019,561 | <p>If <span class="math-container">$M_{n\times n}$</span> is the set of invertible matrices with real entries. Find two matrices <span class="math-container">$A,B\in M_{n \times n}$</span> with the propriety that there not exists such a continuous function</p>
<p><span class="math-container">$$f:[0,1]\to M, \quad f(0)... | Community | -1 | <p>Hint.</p>
<p>The existence of the function <span class="math-container">$f$</span> means that there exists a <em><a href="https://en.wikipedia.org/wiki/Path_(topology)" rel="nofollow noreferrer">path</a></em> between the two matrices <span class="math-container">$A$</span> and <span class="math-container">$B$</span>... |
4,616,048 | <p>How can I solve this system of coupled differential equations?</p>
<p><span class="math-container">$\frac{d^2\rho}{d\lambda^2}=\frac{5\rho}{(5\rho^2+4t^2)^2}$</span> <span class="math-container">$\frac{d^2t}{d\lambda^2}=\frac{4t}{(5\rho^2+4t^2)^2}$</span></p>
<p>Is it something I could input in the Wolfram calculato... | user577215664 | 475,762 | <p><span class="math-container">$$\frac{d^2\rho}{d\lambda^2}=\frac{5\rho}{(5\rho^2+4t^2)^2},\tag{1}$$</span>
<span class="math-container">$$\frac{d^2t}{d\lambda^2}=\frac{4t}{(5\rho^2+4t^2)^2}.\tag{2}$$</span>
<span class="math-container">$$2\rho' \frac{d^2\rho}{d\lambda^2}=\frac{10\rho \rho '}{(5\rho^2+4t^2)^2},\tag{1}... |
1,948,730 | <blockquote>
<p>For all odd integers $n$, there exists an integer $k$ such that $n=2k+1$.</p>
</blockquote>
<p>I negated using De Morgan's laws. Let $O(n)$ be "$n$ is odd" and $N(n, k)$ "$2k + 1 = n$", then
$$\neg(\forall n \exists k [O(n) \to N(n,k)])\\
\exists n \neg\exists k [O(n) \to N(n,k)]\\
\exists n \forall ... | Community | -1 | <p>Your sentences do not reflect the logical expressions. You should have started from </p>
<p>$$\forall n: O(n),\exists k: N(n,k),$$</p>
<p>turning to</p>
<p>$$\exists n: O(n),\forall k: \lnot N(n,k).$$</p>
|
725,602 | <p>I am trying to prove the 'second' triangle inequality:
$$||x|-|y|| \leq |x-y|$$</p>
<p>My attempt:
$$----------------$$
Proof:
$|x-y|^2 = (x-y)^2 = x^2 - 2xy + y^2 \geq |x|^2 - 2|x||y| + |y|^2 = (||x|-|y||)^2$</p>
<p>Therefore $\rightarrow |x-y| \geq ||x|-|y||$</p>
<p>$$----------------$$</p>
<p>My questions are... | DeepSea | 101,504 | <p><strong>Hint:</strong> Use formula: $$F(n) = \frac{a^n - b^n}{\sqrt{5}}$$
With $a = \frac{1 + \sqrt{5}}{2}$, and $b=\frac{1 - \sqrt{5}}{2}$</p>
|
3,014,085 | <p>I am trying to isolate y in this equation:
<span class="math-container">$$-4/3·\ln(|y-60|)=x+c$$</span></p>
<p>If I use a cas-tool to isolate <span class="math-container">$y$</span>, I get:</p>
<p><span class="math-container">$$60.-(2.71828182846)^{−0.75*x-0.75*c}=y$$</span></p>
<p>If I try isolating <span class="m... | user376343 | 376,343 | <p>From <span class="math-container">$$\arg (z+2) + \arg (z-2) = \arg (z^2-4) = \pi $$</span></p>
<p>we deduce that <span class="math-container">$\;z^2-4\;$</span> is a negative real number. This occurs when</p>
<ol>
<li><p><span class="math-container">$z\in \mathbb{R},\; -2<z<2,$</span> or</p></li>
<li><p><spa... |
1,591,371 | <p>If we start with $n$ elements and at each step split them into $2$ parts randomnly and repeat with both sub-parts until parts of only $1$ element are left, in how many different ways can these elements be separated?
I made a mistake we don't split them in half we split them in a random place.</p>
| drhab | 75,923 | <p>Let's say there are $x_n$ ways if the number of elements is $n$. Then $x_1=1$ and:$$x_n=\sum_{i=1}^{n-1}x_i\times x_{n-i}$$</p>
<p>where term $x_i\times x_{n-i}$ corresponds with a <em>first</em> split: $1,\dots,i\mid i+1,\dots,n$.</p>
|
1,556,747 | <p>$$\text{a)} \ \ \sum_{k=0}^{\infty} \frac{5^{k+1}+(-3)^k}{7^{k+2}}\qquad\qquad\qquad\text{b)} \ \ \sum_{k=1}^{\infty}\log\bigg(\frac{k(k+2)}{(k+1)^2}\bigg)$$</p>
<p>I am trying to determine the convergence values. I tried with partial sums and got stuck...so I am thinking the comparison test...Help</p>
| lab bhattacharjee | 33,337 | <p>For the first one, use summation <a href="https://en.wikipedia.org/wiki/Geometric_progression#Infinite_geometric_series" rel="nofollow">formula</a> of Infinite geometric series</p>
<p>For the second, $$\log\dfrac{k(k+2)}{(k+1)^2}=\log\dfrac k{k+1}-\log\dfrac{k+1}{k+2}=u(k)-u(k+1)$$</p>
<p>where $u(m)=\log\dfrac m... |
365,986 | <p>If $A$ is an $n \times n$ matrix with $\DeclareMathOperator{\rank}{rank}$ $\rank(A) < n$, then I need to show that $\det(A) = 0$.</p>
<p>Now I understand why this is - if $\rank(A) < n$ then when converted to reduced row echelon form, there will be a row/column of zeroes, thus $\det(A) = 0$</p>
<p>However, I... | Marc van Leeuwen | 18,880 | <p>Without bothering too much about the mechanics of finding echelon forms, you may reason as follows. I will suppose you know the determinant is multilinear and alternating <em>in the columns</em>. (You didn't specify rows or columns; in fact the determinant is multilinear and alternating both as function of the rows ... |
3,227,788 | <p>Let <span class="math-container">$f: D(0,1)\to \mathbb C$</span> be a holomorphic function. How to show that there exists a sequence <span class="math-container">$\{z_n\}$</span> in <span class="math-container">$D(0,1)$</span> such that <span class="math-container">$|z_n| \to 1$</span> and <span class="math-containe... | N. S. | 9,176 | <p><strong>Edit:</strong> You are on the right track/ Start by removing the zeroes. </p>
<p>Let <span class="math-container">$w_1,.., w_m$</span> be the zeroes of <span class="math-container">$f$</span> with multiplicity <span class="math-container">$k_1,..,k_m$</span>. Let <span class="math-container">$P(z):= (z-w_1)... |
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