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 |
|---|---|---|---|---|
105,222 | <p>I have the transfer function$$ H(z) = \frac{z-.75}{.1 z+.15} $$</p>
<p>how do I find the Poles and Zeros?</p>
| Parsa | 17,984 | <p>I think this statement is proven exactly as the primitive element theorem is proven. I'll give you the argument in the case $n=2$, courtesy of J. S. Milne's notes on field theory, from which the general case follows.</p>
<p>Let $E=F[a,b]$ with $b$ separable over $F_u$. Let $f$ and $g$ be the minimum polynomials o... |
105,222 | <p>I have the transfer function$$ H(z) = \frac{z-.75}{.1 z+.15} $$</p>
<p>how do I find the Poles and Zeros?</p>
| Community | -1 | <p>Let $\alpha$ be a primitive element for $F[a_1,...,a_n]/F$ and let $d=[F(\alpha):F]$. By looking at the proof of the PET, we can take $\alpha$ to be an $F$-linear combination of the $a_i$, $\alpha=\sum b_i a_i$.</p>
<p>It suffices to prove that $[F_u[a_1 u_1+\cdots + a_n u_n]:F_u]=d$. So suppose $f(a_1 u_1+\cdots +... |
630,330 | <p>Please correct me if I am wrong. I believe that it is unknown whether the decimal representation of $\pi$ contains all strings, or lists on 10 digits. I would like to know if there is a number that is known that does in fact contain all of them, without the number being constructed explicitly to do so. Is there a nu... | Ross Millikan | 1,827 | <p>The <a href="http://en.wikipedia.org/wiki/Champernowne_constant" rel="nofollow">Champernowne constant</a> $0.0123456789101112131415\dots$ includes all finite digit strings. Any normal real, and most are, will include all finite digit strings, but that is often hard to prove for a given number. </p>
|
2,659,145 | <p>This paradox is directly ripped off of <a href="https://www.youtube.com/watch?v=Sx5_QGdFmq4" rel="nofollow noreferrer">this video</a>
by the youtube channel "Mathologer 2". There, he presents a paradox with the following steps:</p>
<p>$$e^{i \pi}=-1$$
$$(e^{i \pi})^2=(-1)^2$$
$$(e^{i \pi})^2=1$$
$$e^{2i \pi}=1 $$
... | Masacroso | 173,262 | <p>The key step is that the complex exponentiation is defined by $z^w:=e^{w\ln z}$ where $z,w\in\Bbb C$ and the logarithm is <strong>the principal value of the complex logarithm</strong>. Then</p>
<p>$$(e^{2\pi i+1})^{2\pi i+1}=\exp((2\pi i+1)\ln e^{2\pi i+1})=\exp((2\pi i+1)\ln e)=e^{2\pi i+1}\tag1$$</p>
<p>because ... |
706,461 | <p>Find $\lim_{n\to\infty} ((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm,
$$\lim_{n\to\infty} \ln((n!)^{1/n}) = \lim_{n\to\infty} (1/n)\ln(n!) = \lim_{n\to\infty} (\ln(n!)/n)$$
Applying L'hopital's r... | Martín-Blas Pérez Pinilla | 98,199 | <p>Taking $\log$ and using Stolz-Cesaro:
$$
\log\lim_{n\to\infty}n!^{1/n}=
\lim_{n\to\infty}\log n!^{1/n}=
\lim_{n\to\infty}{\log 1+\cdots+\log n\over n}=
\lim_{n\to\infty}{\log(n+1)\over(n+1)-n}=
\lim_{n\to\infty}\log(n+1)=\infty,$$
so $\lim n!^{1/n}=\infty$.</p>
|
706,461 | <p>Find $\lim_{n\to\infty} ((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm,
$$\lim_{n\to\infty} \ln((n!)^{1/n}) = \lim_{n\to\infty} (1/n)\ln(n!) = \lim_{n\to\infty} (\ln(n!)/n)$$
Applying L'hopital's r... | Amihai Zivan | 22,409 | <p>Assume WLOG that $n$ is even. </p>
<p>Clearly,
$$n!=1\cdot2\cdots n > \left(\frac{n}{2}+1\right)\left(\frac{n}{2}+2\right)\cdots n> \left(\frac{n}{2}\right)^{\frac{n}{2}}$$</p>
<p>Can you take it from here? </p>
|
706,461 | <p>Find $\lim_{n\to\infty} ((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm,
$$\lim_{n\to\infty} \ln((n!)^{1/n}) = \lim_{n\to\infty} (1/n)\ln(n!) = \lim_{n\to\infty} (\ln(n!)/n)$$
Applying L'hopital's r... | marty cohen | 13,079 | <p>The inequalities
I often find useful
because of their simplicity
are
$(n/e)^n
< n!
< (n/e)^{n+1}
$.
These give
$\frac{(n!)^{1/n}}{n}
\to 1/e
$.</p>
<p>These follow from
$(1+1/n)^n
< e
< (1+1/n)^{n+1}
$
(which have been
proven here
many times).</p>
|
248,845 | <p>Given <span class="math-container">$g\in L^2(\mathbb{R}^3)$</span>, consider the following function ( defined for <span class="math-container">$r>0$</span> ):
<span class="math-container">$$c(r):=\int_{\mathbb{R}^3}\frac{g(x)}{|x|^2+r}dx$$</span>
I'm interested in the behavior of <span class="math-container">$c(r... | fedja | 1,131 | <p>Once you asked, here goes. Let $h(r)=c(r^2)$. Then you need to look at the operator
$$
\Psi: f\mapsto \int_0^\infty f(\rho)\frac{\rho}{\rho^2+r^2}\,d\rho.
$$
Note two things: </p>
<p>1) The kernel is positive and dominated by $1/(r+\rho)$ (just consider the cases $\rho\le r$ and $\rho\ge r$ separately. Hence, it is... |
3,253,368 | <p>I saw this property in an exercise about metric spaces: Let <span class="math-container">$(E,d)$</span> be a metric space and <span class="math-container">$A$</span>, <span class="math-container">$B$</span> be two non empty subsets of <span class="math-container">$E$</span>. If <span class="math-container">$A$</span... | MPW | 113,214 | <p>What about the graphs of <span class="math-container">$y=0$</span> and <span class="math-container">$y=e^{-x}$</span> on <span class="math-container">$[0,\infty)$</span>?</p>
|
1,590,039 | <p>In programming, often the value of True is also 1, and False is 0.</p>
<p>This means that:</p>
<pre><code>(x>5)*4
</code></pre>
<p>will return 4 if x is greater than 5 (because <code>(x>5)==1</code>), else 0.</p>
<p>I need to accomplish a similar thing using mathematical operators (no piecewise functions, ... | Community | -1 | <p>(Posted by <a href="https://math.stackexchange.com/a/1590064">Dave Radcliffe</a> but then deleted): </p>
<blockquote>
<p>The Desmos calculator supports piecewise functions such as <code>{x<5 : 0, 1}</code> or <code>{x<=5 : 0, 1}</code></p>
</blockquote>
|
1,392,273 | <p>I'm a math major at university and my tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble understanding the difference between them.</p>
<p>What are the actual differences between algebra and analysis, and how do I tell which is for me?</p>
| rschwieb | 29,335 | <p>Abstract algebra is largely (but not only) about sets with operations and their properties.</p>
<p>Mathematical analysis is largely (but not only) more about topology, measure, and how you can apply topology and measure to functions, namely integration and differentiation.</p>
<p>Contrary to your comment about con... |
1,392,273 | <p>I'm a math major at university and my tutor told me that for most people it's best to focus on either algebra or analysis, however I have trouble understanding the difference between them.</p>
<p>What are the actual differences between algebra and analysis, and how do I tell which is for me?</p>
| eigenchris | 160,173 | <p>Algebra is about combining things together with operations while analysis focuses more on studying the closeness or "connectedness" between points.</p>
<p>Some of your confusion might stem from the fact that algebra and analysis can often work together. If you take courses in abstract algebra and real analysis, you... |
1,829,082 | <p>I was wondering how many base cases are needed and when to stop (in general).
For example, I have 4-cent and 11-cent stamps and I need to determine the amount of postage I can make, the cases I have that work are:
$4,8,11,12,15,16,17,19,20,22,23,24,26,27,28,30 \geq$</p>
<p>Now what I'm stuck on is: what stops me fr... | Hagen von Eitzen | 39,174 | <p>If only $4$ and $11$ cent stamps are available and you know you can create $n$ cent, what would allow you to create $n+1$ sents? If the way you obtain $n$ cents involves at least one 11-cent stamp, you can replace it with three 4-cents. Or if it involves eight 4-cents, you can replace these with three 11-cents. But ... |
99,908 | <p>Currently I have a list such that, </p>
<pre><code>f1 = {{x_1,y_1},{x_2,y_2},...,{x_n,y_n},
{x_1,y_n+1},{x_2,y_n+2},...,{x_n,y_n'}
{x_1,y_n'+1},{x_2,y_n'+2},...,{x_n,y_n''}};
</code></pre>
<p>Where the number of elements until $ x $ repeats is knowable and the number of x_1's is knowable, say $ i $. </... | Edmund | 19,542 | <p>You can use <code>GroupBy</code> <code>First</code> to collect the pairs in an <code>Association</code>. Then <code>KeyValueMap</code> to calculate your final form.</p>
<pre><code>f = {{x1, y1}, {x2, y2}, {x3, y3}, {x1, y4}, {x2, y5}, {x3, y6}, {x1, y7}, {x2, y8}, {x3, y9}};
KeyValueMap[
Function[{key, value},
... |
2,705,733 | <p>I try to find sum of $\sum_{i=n}^{2n-1} 3+4i$<Br></p>
<p>$\sum_{i=n}^{2n-1} 3+ \sum_{i=n}^{2n-1} 4i$ =$3 (2n-1-n+1) + 4 \sum_{i=n}^{2n-1} i $ -> am I right?<br>
if I change the index of $4 \sum_{i=n}^{2n-1} i $ to $4 \sum_{i=0}^{2n-1-n} i+n $ <br>am I right?
<br>
$4 \sum_{i=0}^{2n-1-n} i+n $ <br>
$4 (\sum_{i=0}^{n... | farruhota | 425,072 | <p>Alternatively:
$$\begin{align}\sum_{i=n}^{2n-1}(3+4i)&=\sum_{i=0}^{2n-1}(3+4i)-\sum_{i=0}^{n-1}(3+4i)=\\
&=3\cdot 2n+4\cdot \frac{2n(2n-1)}{2}-3n-4\cdot \frac{n(n-1)}{2}= \\
&=6n+8n^2-4n-3n-2n^2+2n=\\
&=6n^2+n=\\
&=n(6n+1).\end{align}$$</p>
|
2,705,733 | <p>I try to find sum of $\sum_{i=n}^{2n-1} 3+4i$<Br></p>
<p>$\sum_{i=n}^{2n-1} 3+ \sum_{i=n}^{2n-1} 4i$ =$3 (2n-1-n+1) + 4 \sum_{i=n}^{2n-1} i $ -> am I right?<br>
if I change the index of $4 \sum_{i=n}^{2n-1} i $ to $4 \sum_{i=0}^{2n-1-n} i+n $ <br>am I right?
<br>
$4 \sum_{i=0}^{2n-1-n} i+n $ <br>
$4 (\sum_{i=0}^{n... | Peter Szilas | 408,605 | <p>Pedestrian approach:</p>
<p>$\sum_{i=n}^{2n-1}i =$ </p>
<p>$n \ \ \ \ + \ \ \ \ \ \ \ \ (n+1) +.....(2n-1)$.</p>
<p>$(2n-1) + (2n-2)+.......n =$</p>
<p>$\sum_{i=1}^{2n-1}i$ (in reverse order).</p>
<p>Adding the sums:</p>
<p>$2\sum_{i=n}^{2n-1}i = (n)(3n-1)$.</p>
<p>Thus:</p>
<p>$\sum_{i=n}^{i=2n-1}(3+4i... |
3,471,096 | <p><span class="math-container">$\frac{d}{dx} xy^2 e^{-x^2}$</span></p>
<p>I am not sure where to get started and what method I can use to deal with the <span class="math-container">$x^2$</span> power</p>
| Quanto | 686,284 | <p>Let <span class="math-container">$f(x)= xy^2 e^{-x^2}$</span>. Then <span class="math-container">$\ln f = \ln x + 2\ln y - x^2$</span>. Apply <span class="math-container">$(\ln f)'=f'/f$</span>,</p>
<p><span class="math-container">$$\frac{df(x)}{dx} = f(x) ( \frac 1x - 2x )=xy^2 e^{-x^2}( \frac 1x - 2x )= y^2 e^{-x... |
299,984 | <p>first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever proved the differentiability of the eigenvalues/eigenvectors in the perturbation-parameter! They just assumed it. So, why... | Denis Serre | 8,799 | <p>The complete reference is Kato's book <em>Perturbation theory ...</em>. But perhaps you need only the most basic results. Then see my book <em>Matrices</em> (Springer GTM #216), 2nd edition. This is Section 5.2.</p>
<p>Mind however that this is a much more elaborate topic than what you could think at first glance. ... |
259,446 | <p>Suppose I have a list {x,y} like:</p>
<p>Dados0 = {{21.554, 10.153}, {21.554, 9.23}, {21.554, 11.076}, {27.7141, 8.6141}, {27.7141, 11.4855}, {50.1348, 12.6982}, {50.1348, 14.2854}, {50.1348, 13.4918}, {62.3582, 14.9946}, {62.3582, 14.0375}, {73.7872, 11.324}, {73.7872, 14.0201}, {73.7872, 13.4809}, {91.0558, 17.91... | Nate | 74,941 | <p>The function "range" gathers all pairs in the list with x values between x-dx and x+dx. The "results" function uses the "range" function to compute the {mean of x, mean of y, and standard deviation of y}</p>
<pre><code>range[data_, x_, dx_] :=
Cases[data, a_ /; (x - dx <= a[[1]] &l... |
755,274 | <p>Prove that an $n$ dimensional vector bundle $p: E \to B$ is trivial iff there exists a continuous map $\pi: E \to \mathbb{R}^n$ whose restriction to each fiber $p^{-1}(b), b \in B,$ is a vector space isomorphism.</p>
<p>This is for my high school math independent study and my teacher is unable to help.</p>
| Alex Becker | 8,173 | <p><strong>Defn:</strong> Two bundles $p_1:E_1\to B$ and $p_2:E_2\to B$ are isomorphic iff we have some map $f:E_1\to E_2$ such that $p_1=p_2\circ f$ and for each $b\in B$, the map $f|_{p_1^{-1}(b)}:p_1^{-1}(b)\to p_2^{-1}(b)$ is an isomorphism of vector spaces.</p>
<p>If the bundle is trivial, by definition we have s... |
1,169,493 | <p>I am attempting to self study Real Analysis, Abstract Algebra, and Topology out of private interest.
I'm working on the exercises in the text, ' Understanding Analysis ' and will be posting my attempts at proving the various theorems in the text.</p>
<p>Hopefully this is ok:</p>
<p>Question 1: Prove that the $\sqr... | abel | 9,252 | <p>your differential equation $$x^2y^2\, dy + (xy^3-1)\, dx = 0 $$ is almost an exact differential.it can be made into an exact differential if we multiply by the integrating factor $x$ to get $$ x^3y^2\, dy + (x^2y^3-x)\, dx = 0$$ and the solution is $$\frac13 x^3y^3 - \frac12x^2 = constant. $$</p>
<p>$\bf edit:$ her... |
4,622,525 | <blockquote>
<p>The maximum area of a triangle whose sides <span class="math-container">$a,b,c$</span> satisfy <span class="math-container">$0\le a\le1,1\le b\le2,2\le c\le3$</span></p>
</blockquote>
<hr />
<p>We can clearly see that <span class="math-container">$1,2,3$</span> as sides does not make a triangle so we ca... | Oscar Lanzi | 248,217 | <p>A similar argument: The area will be <span class="math-container">$(1/2)ab\sin\angle C$</span> where <span class="math-container">$a\le1,b\le2\sin\angle C\le1$</span>. We can simultaneously maximize all these factors, and then from the Pythagorean Theorem <span class="math-container">$c=\sqrt5<3$</span>. So we ch... |
359,196 | <p>Let <span class="math-container">$d \gg 1$</span>. Let <span class="math-container">$G:=(U, V, E)$</span> be some bipartite graph such that deg<span class="math-container">$(u) \le d$</span> for all <span class="math-container">$u\in U$</span> and deg<span class="math-container">$(v) \le 3$</span> for all <span clas... | DSM | 155,380 | <p>Here is an integer programming formulation of your problem. Not sure if it can provide an analytical insight, but hope it helps someway.</p>
<p>Consider a matrix <span class="math-container">$X\in \{0,1\}^{|U|\times 3}$</span>, with exactly one 1 in each row. Such a matrix would indicate the membership of each <spa... |
1,598,990 | <p>What substitution is useful for this integral?</p>
<p>$$\int \frac{1}{\sqrt[3]{(x+1)^2(x-2)^2}}\mathrm dx$$</p>
<p>Substitutions $u=x^{\frac{2}{3}},u=(x+1)^{\frac{2}{3}},u=(x-2)^{\frac{2}{3}}$ are not working. </p>
<p>Can't find useful trigonometric substitution. </p>
| Mufasa | 49,003 | <p>I found an interesting substitution by a bit of trial and error.
$$\begin{align}
x&=\frac{1+3\sin(u)}{2}\\
\therefore dx&=\frac{3\cos(u)}{2}du\\
x+1&=\frac{3+3\sin(u)}{2}&&=\frac{3(1+\sin(u))}{2}\\
x-2&=\frac{3\sin(u)-3}{2}&&=\frac{3(\sin(u)-1)}{2}\\
\therefore (x+1)(x-2)&=\frac{9... |
3,148,095 | <blockquote>
<p>Let <span class="math-container">$ f:\{z|\; |z| \lt 1\} \rightarrow \mathbb C $</span> be a non constant analytic function. Which of the following conditions can possibly be satisfied by <span class="math-container">$f$</span> ?</p>
<ol>
<li><span class="math-container">$f(\frac{1}{n})=f(\frac{... | Myunghyun Song | 609,441 | <p>The fact that <span class="math-container">$f(\frac 1 n)=\frac1{2n+1}$</span> for all <span class="math-container">$n\ge 1$</span> implies that <span class="math-container">$f(z)-\frac{z}{z+2}$</span> has an accumulated zero; by the identity theorem, it follows <span class="math-container">$f(z)=\frac{z}{z+2}$</span... |
657,542 | <p>I have a equation that should divide a circle in even parts. As I found its called circle-dividing equation. I'v found same information how to solve a equation which has a form like this: </p>
<p>$$z^6 = 1$$</p>
<p>one of the first steps was to transform the equation using MOIVRE to </p>
<p>$$z^6 = 1+0i$$</p>
<p... | lab bhattacharjee | 33,337 | <p>$$z^3=8i=8e^{\dfrac{i\pi}2}=8e^{\left(2k\pi+\dfrac\pi2\right)i}$$ where $k$ is any integer</p>
<p>$$z=2e^{\dfrac{(4k+1)\pi i}6}$$ where $k\equiv0,1,2\pmod3$ (See <a href="https://math.stackexchange.com/questions/656832/fourth-roots-of-17/656835#656835">this</a>)</p>
<p>Now use <a href="http://mathworld.wolfram.com... |
185,591 | <p>Let $b \in (0,1)$, $m\in \mathbb{N}$ and $a>0$. I want to bound
$$\sum_{k=m+1}^\infty b^{k^a} \leq c \; b^{m^a}, $$
where $c>0$ is independent from $m$.</p>
<p>Is there a simple way of proving this inequality with $c$ not beeing too large?</p>
<p>Edit: $c$ can only be independent from $m$ if $a>1$. Otherw... | Robert Israel | 13,650 | <p>Note that $k^a/\ln(k)$ is increasing for $k > \exp(1/a)$. Suppose $m \ge \exp(1/a)$ and $(m+1)^a/\ln(m+1) > 1/\ln(1/b)$. Then with $c = (m+1)^a/\ln(m+1)$ we have
$$ \sum_{k=m+1}^\infty b^{k^a} < \sum_{k=m+1}^\infty b^{c \ln(k)}
= \sum_{k=m+1}^\infty k^{-c \ln(1/b)} < \int_m^\infty t^{-c \ln(1/b)}\; dt... |
3,004,854 | <blockquote>
<p>Is it true that a natural number <span class="math-container">$n>1$</span> is prime if and only if <span class="math-container">$n|\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{2} \right )^n-1$</span>?</p>
</blockquote>
<p>We know that <span class="math-container">$11$</span> i... | Arne Vogel | 652,071 | <p>Note that the Bruckman-Lucas criterion is a partial, special case of the <a href="http://mpqs.free.fr/LucasPseudoprimes.pdf" rel="nofollow noreferrer">family of Lucas sequence tests</a> described by Baillie and Wagstaff:</p>
<p>Let <span class="math-container">$P$</span>, <span class="math-container">$Q$</span> be ... |
1,920,504 | <p>I know, for example, that the series:</p>
<p>$$\sum\limits_{n= 1}^{\infty} \frac{1}{e^n}$$</p>
<p>is a geometric series of the form $\sum\limits_{n = 1}^{\infty} k x^n$, where $k = 1$ and $x = \frac{1}{e}$ and it is convergent<sup>1</sup>. But when I have, as in my case:</p>
<p>$$\sum\limits_{n = 1}^{\infty} \fra... | Community | -1 | <p>By the integral test,</p>
<p>$$\int_0^\infty\frac x{e^x}dx=-\left.\frac x{e^x}\right|_0^\infty+\int_0^\infty\frac{dx}{e^x}=-\left.\frac 1{e^x}\right|_0^\infty=1.$$</p>
|
1,920,504 | <p>I know, for example, that the series:</p>
<p>$$\sum\limits_{n= 1}^{\infty} \frac{1}{e^n}$$</p>
<p>is a geometric series of the form $\sum\limits_{n = 1}^{\infty} k x^n$, where $k = 1$ and $x = \frac{1}{e}$ and it is convergent<sup>1</sup>. But when I have, as in my case:</p>
<p>$$\sum\limits_{n = 1}^{\infty} \fra... | Sri-Amirthan Theivendran | 302,692 | <p>You can also use the root test which is stronger than the ratio test. Using the well-known limit $$\lim_{n\to \infty} n^{1/n}=1$$ we have that $$\left(\frac{n}{e^n}\right)^{1/n}=\frac{1}{e}n^{1/n} \longrightarrow \frac{1}{e}<1$$ so that the series converges.</p>
|
2,288,983 | <p>A card is drawn and replaced in an ordinary pack of 52 playing cards. Minimum number of times must a card be drawn so that there is at least an <strong>even chance</strong> of drawing a heart?</p>
<p>What does the second sentence of the question mean?
Why should there be any change in the chance of drawing a heart,... | Mr. Xcoder | 435,694 | <p>So far, I managed to find a mistake at point <em>(a)</em>:
$$ f(-1) = (-1)^2 - (-1)$$
$$f(-1) = 1 + 1\implies f(-1) = 2$$</p>
|
3,506,267 | <p>In the last few days, I have been wondering about the following equation in <span class="math-container">$x,y \in N$</span>: <span class="math-container">$$x^2+y^2 = M$$</span> with <span class="math-container">$M \in N$</span>. More precisley, I don't understand how can we compute the number of solutions that this ... | Kieren MacMillan | 93,271 | <p>Given <span class="math-container">$x^2+y^2=85$</span>, and the well-known factorization formula [which you already obviously know] <span class="math-container">$$(a^2+b^2)(c^2+d^2) = (ac\pm bd)^2+(ad \mp bc)^2.$$</span></p>
<p>Now just work backwards:</p>
<ol>
<li>Factor <span class="math-container">$85=5 \cdot 1... |
2,317,407 | <p>From pg. 76 of <em>Categories for the Working Mathematician</em>:</p>
<p><a href="https://i.stack.imgur.com/2VxcX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/2VxcX.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/KJxy6.png" rel="nofollow noreferrer"><img... | M. Winter | 415,941 | <p>I wrote a program and it gave me this:</p>
<p>$0 + 0 = 0$ (if you consider $00$ as a two digit number)</p>
<p>$92 + 29 = 121$</p>
<p>$83 + 38 = 121$</p>
<p>$74 + 47 = 121$</p>
<p>$65 + 56= 121$</p>
|
1,793,662 | <p>I have the following question in an assignment and I don't know how to approach it, please see the image below for the question. Hopefully someone can help :) <a href="https://i.stack.imgur.com/T6Eh0.jpg" rel="nofollow noreferrer">question</a></p>
<p><a href="https://i.stack.imgur.com/T6Eh0.jpg" rel="nofollow noref... | Susan | 315,781 | <p>Let matrix $A$ be
\begin{bmatrix}
3 & 0 \\
4 & -1
\end{bmatrix}</p>
<p>Let matrix $B$ be
\begin{bmatrix}
1 & -1 \\
4 & 1
\end{bmatrix}</p>
<p>Your change of basis matrix from g to e would be:</p>
<p>$A\times B=$</p>
<p>\begin{bmatrix}
3 & -3 \\
0 & 5
\end{bmatrix}<... |
1,174,076 | <p>So I have this basic absolute value problem: $|x-5|<|x+1$|. From what I understand, I need to consider what happens in every case. There are four cases, right? One where both sides are positive, one where both sides are negative and two where one side is negative and the other is positive.</p>
<p>The case were b... | egreg | 62,967 | <p>Square:
$$
(x-5)^2<(x+1)^2\\
x^2-10x+25<x^2+2x+1\\
12x>24\\
x>2
$$</p>
<p>Is it possible? Certainly: an inequality between non negative numbers is equivalent with the corresponding inequality between their squares.</p>
|
118 | <p>The tag <a href="https://matheducators.stackexchange.com/questions/tagged/pedagogy" class="post-tag" title="show questions tagged 'pedagogy'" rel="tag">pedagogy</a> gets a lot of usage, possibly too much. I propose a discussion to sharpen the meaning of this tag (or to replace it by more specific ones). </p>... | Roland | 108 | <p>I agree that this tag is used very broadly, and that there is a variety of types of questions which uses this tag. The following list is not intended to be complet, but tries to classify the zoo of questions we have.</p>
<ul>
<li><p>Questions which are concered with course design, like <a href="https://matheducator... |
1,213,371 | <blockquote>
<p>How could we prove that $$\int_0^\infty\frac{J_{2a}(2x)~J_{2b}(2x)}{x^{2n+1}}~dx~=~\frac12\cdot\frac{(a+b-n-1)!~(2n)!}{(n+a+b)!~(n+a-b)!~(n-a+b)!}$$ for $a+b>n>-\dfrac12$ ?</p>
</blockquote>
<hr>
<p>Inspired by <a href="https://math.stackexchange.com/questions/1200624">this question</a>, I sou... | Argon | 27,624 | <p>Using the product cited <a href="https://math.stackexchange.com/a/1213437/27624">above</a>, viz.</p>
<p>\begin{align}
J_{\mu}(x) \, J_{\nu}(x) &= \sum_{n=0}^{\infty} \frac{(-1)^{n} \, \Gamma(2n+\mu+\nu+1) \, \left(\frac{x}{2}\right)^{2n+\mu+\nu}}{n! \, \Gamma(\mu+\nu+n+1) \, \Gamma(\mu+n+1) \, \Gamma(\nu+n+1)} ... |
1,369,805 | <p>How do I calculate the follwing Limit:</p>
<p><span class="math-container">$$ \lim \limits_{x \to 0^+} \frac{\int_{0}^{x} (e^{t^2}-1)dt}{{\int_{0}^{x^2} \sin(t)dt}} $$</span></p>
<p>I have been solving an exam from my University's collection of previous exams, and in one of them, there wasn't an official solution, b... | user1337 | 62,839 | <p>As your limit involves the indeterminate form $\frac{0}{0}$ we examine the limit</p>
<p>$$\lim_{x \to 0} \frac{\frac{d}{dx} \int_0^{x} (e^{t^2}-1)dt}{\frac{d}{dx} \int_0^{x^2} \sin (t) dt}=\lim_{x \to 0} \frac{e^{x^2}-1}{2x\sin(x^2)}.$$</p>
<p>Where we have applied the fundamental theorem of calculus to get the RH... |
3,664,034 | <p>This is an exercise from Hoffman and Kunzes textbook Linear Algebra.</p>
<p>Let <span class="math-container">$W_{1}$</span> and <span class="math-container">$W_{2}$</span> be subspaces of <span class="math-container">$V$</span> such that <span class="math-container">$W_{1} + W_{2} = V$</span> and <span class="math-... | Berci | 41,488 | <p>Your proof is basically fine, but could be clearer for uniqueness:</p>
<p>We have <span class="math-container">$w_i=\alpha_i-\beta_i\in W_i$</span>, and <span class="math-container">$w_1+w_2=0$</span>, so we do have <span class="math-container">$w_1=-w_2$</span>, and indeed <span class="math-container">$-w_2\in W_2... |
2,357,297 | <p>Suppose I have lost my keys.
There is:</p>
<ol>
<li>Probability 0.7 i lost it in the main room. </li>
<li>Probability 0.2 i lost it in the bedroom. </li>
<li>Probability 0.1 I lost it in the bathroom.</li>
</ol>
<p>So I look for the keys in the main room first but don't find it.
There is a probability 0.35 I mis... | Dr. Sonnhard Graubner | 175,066 | <p>prove by induction that for your sum is hold $$\sum_{i=1}^n\frac{i+2}{2^{i+1}i(i+1)}=\frac{2^{-n-1} \left(2^n n+2^n-1\right)}{n+1}$$</p>
|
1,384,426 | <p>Solve the given equation. Let k be any integer. </p>
<p>$$\csc^2 θ = 5 \cot θ + 7$$</p>
<p>I just need the first step or two please. I tried converting it:</p>
<p>$$\frac{1}{\sin^2 θ} = \frac {5\cosθ}{\sinθ} + 7$$</p>
<p>Then I tried a number of different ways to simplify it but it didn't work out</p>
| Mark Viola | 218,419 | <p>HINT:</p>
<p>$$\csc^2 x=1+\cot^2 x$$</p>
<p>Then, solve a quadratic equation for $\cot x$</p>
|
1,384,426 | <p>Solve the given equation. Let k be any integer. </p>
<p>$$\csc^2 θ = 5 \cot θ + 7$$</p>
<p>I just need the first step or two please. I tried converting it:</p>
<p>$$\frac{1}{\sin^2 θ} = \frac {5\cosθ}{\sinθ} + 7$$</p>
<p>Then I tried a number of different ways to simplify it but it didn't work out</p>
| jameselmore | 86,570 | <p>$$\text{Recall: }\csc^2 \theta = 1 + \cot^2 \theta $$
So we have the following:
$$\cot^2 \theta + 1 = 5\cot \theta + 7$$
$$\cot^2 \theta - 5\cot \theta - 6 = 0$$
$$\left(\cot\theta - 6\right)\left(\cot\theta + 1\right) = 0$$
$$\dots$$</p>
|
308,383 | <p>Assume that we have a set system $\mathfrak T = \{\mathcal T_1, \mathcal T_2, \dots, \mathcal T_N \}$ where each $\mathcal T_k$ is a collection of subsets of $[n] := \{1,\dots,n\}$ of the form
$$ \mathcal T_k = [m_k, M_k] := \{T \subseteq [n]:\; m_k \subseteq T \subseteq M_k \}. $$
Moreover, we know that $\mathcal ... | Brendan McKay | 9,025 | <p>Given $T_k = [m_k,M_k]$ and $X\subseteq [n]$, it is easy to calculate the number of sets in $T_k$ that contain $X$: $$Q_{X,k}=|\{T\in T_k \mid X\subseteq T\rbrace|.$$</p>
<p>So start with $\sum_k Q_{\emptyset,k}$. If it equals $2^n$, every subset of $[n]$ is covered.</p>
<p>Otherwise, there is $x_1\in [n]$ such t... |
2,196,347 | <p><strong>Problem:</strong> What will be the number of distinct real solution of <span class="math-container">$|x|^2-5|x|+6=0$</span> ?</p>
<p><strong>Solution:</strong></p>
<blockquote>
<p>Case1: When <span class="math-container">$x<0$</span>, then <span class="math-container">$x=-2,-3$</span></p>
<p>Case2: When <... | LM2357 | 376,702 | <p>This is a quadratic equation with absolute values..</p>
<p>You can factor this like this
$$(|x|-3)(|x|-2)=0$$
So..either
$$|x|=3 \Leftrightarrow x=\pm3$$
or$$|x|=2\Leftrightarrow x=\pm2$$
So your solutions are correct..</p>
|
3,854,931 | <p>If <span class="math-container">$f(x + iy) = u(x,y) + i v(x,y)$</span>, then <span class="math-container">$\frac{\partial f}{\partial x}$</span> describes how <span class="math-container">$f$</span> changes if we take take small step in the <span class="math-container">$x$</span>-direction in the input space, and <s... | Especially Lime | 341,019 | <p>An alternative approach for (2). Suppose <span class="math-container">$x^n=y^n=a$</span> where <span class="math-container">$x,y\geq 0$</span> and <span class="math-container">$x\neq y$</span>. Since
<span class="math-container">$$0=x^n-y^n=(x-y)\times\sum_{k=0}^nx^ky^{n-k},$$</span>
and <span class="math-container"... |
2,491,176 | <p>Is it always true that for small value of c i.e $c\approx 0$<br>
$\int_0^c f(x)dx \approx \int_0^c f1(x)dx $ </p>
<p>where $f1(x)$ is the approximate function for $f(x)$ when x is small?<br>
For an example:<br>
$\int_0^{0.18}(1-\sin x \cos x)dx=0.163974$.<br>
$\int_0^{0.18}(1-x)dx=0.1638$.</p>
<p>In this case bo... | Jan Eerland | 226,665 | <p>Well, the relation between degrees of an angle and radians is given by:</p>
<p>$$x^\circ=x\cdot\frac{\pi}{180}\space\text{radians}\tag1$$</p>
<p>Using radians:</p>
<p>$$\arctan\left(\frac{981}{3125}\right)\approx0.30417801\space\text{radians}\tag2$$</p>
<p>Which is in degrees:</p>
<p>$$\left(\frac{180}{\pi}\cdo... |
2,740,207 | <p>I am trying to evaluate the following integral:
\begin{align}
\int_0^{2\pi}\exp(-C)\cdot\exp\left(a\sin\phi+b\cos\phi\right)\,d\phi\,,
\end{align}
where $C,a,b$ are large constants. I know that this eventually reduces to the modified Bessel function of the first kind. However, since $C, a, b$ are very large (the ter... | lhf | 589 | <p>If such a homomorphism $S_n \to G$ exists, then it induces a surjective homomorphism $S_n \to \langle a \rangle \cong C_2$. Therefore, the kernel has index $2$ and so is $A_n$ because it contains $A_n$.</p>
|
107,367 | <p>If two vectors $\bf{u}$ and $\bf{v}$ in $\mathbb{R}^2$ are orthogonal to a non-zero vector $\bf{w}$ in $\mathbb{R}^2$, then are $\bf{u}$ and $\bf{v}$ scalar multiples of one another? Prove your claim.</p>
<p><strong>Attempt:</strong> From a geometric point of view it seems obvious that they must be scalar multiples... | Christian Blatter | 1,303 | <p>The total curvature is the total turning angle of the tangent vector $\dot{\bf z}(t)$ during the given time interval. Since we are talking of a logarithmic spiral here $\dot{\bf z}(t)$ encloses a constant angle with the position vector ${\bf z}(t)$. The latter turns by $2\pi$ counterclockwise; therefore the total cu... |
107,367 | <p>If two vectors $\bf{u}$ and $\bf{v}$ in $\mathbb{R}^2$ are orthogonal to a non-zero vector $\bf{w}$ in $\mathbb{R}^2$, then are $\bf{u}$ and $\bf{v}$ scalar multiples of one another? Prove your claim.</p>
<p><strong>Attempt:</strong> From a geometric point of view it seems obvious that they must be scalar multiples... | Cye Waldman | 424,641 | <p>I would like to expand on the previous answer for total curvature to point out that it applies to many spirals, not just the logarithmic spiral. Specifically, for any spiral that can be represented in the complex plane by</p>
<p>$$z(s)=\int e^{i\int \kappa (s) ds} ds$$</p>
<p>where $\kappa(s)$ is the curvature of... |
633,724 | <p>Prove $e^x, e^{2x}..., e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$</p>
<p>isn't it suffice to say that $e^y$ for any $y \in \mathbb{R}$ is in $\mathbb{R}^+$<br>
Therefore, there aren't $\gamma_1, ...\gamma_n$ such that $\gamma_1e^x+\gamma_2e^{2x}...+\gamma_ne^{nx}=0$.<br>
Theref... | apnorton | 23,353 | <p>Since the $\gamma_i$ (using your notation) can be negative, it does not suffice to state that $e^x > 0 \; \forall x\in\mathbb{R}$.</p>
<p>You can either use the matrix/determinant method, or, I believe, you can look at the power series of $e^x$ to do so.</p>
|
209,857 | <p>I dont' <em>really</em> know how to realize the following: Let $\lambda \sim \Gamma(\alpha,\beta)$ and let $X$ conditional on $\lambda$ be Poisson$(\lambda)$. Argue that for $n=0,1,2,\ldots,$ $$P(X=n)=\int_0^\infty \frac{\lambda^n}{n!}e^{-\lambda}\frac{\beta^\alpha}{\Gamma(\alpha)}\lambda^{\alpha-1}e^{-\beta\lambda}... | TheJoker | 43,622 | <p>this question deals with Compound Probability distribution.</p>
<p>I'll give you some hints.</p>
<p>If $f(x\mid\theta)$ and $\pi(\theta \mid \phi)$ are the distribution functions (with parameters $\theta$ and $\phi$ respectively and you want to find distribution of $X\mid\phi$ then use the following formula:
$$F(x... |
209,857 | <p>I dont' <em>really</em> know how to realize the following: Let $\lambda \sim \Gamma(\alpha,\beta)$ and let $X$ conditional on $\lambda$ be Poisson$(\lambda)$. Argue that for $n=0,1,2,\ldots,$ $$P(X=n)=\int_0^\infty \frac{\lambda^n}{n!}e^{-\lambda}\frac{\beta^\alpha}{\Gamma(\alpha)}\lambda^{\alpha-1}e^{-\beta\lambda}... | Michael Hardy | 11,667 | <p>$$
\int_0^\infty \frac{\lambda^n}{n!}e^{-\lambda}\frac{\beta^\alpha}{\Gamma(\alpha)}\lambda^{\alpha-1}e^{-\beta\lambda} \, d\lambda = \frac{\beta^\alpha}{n!\Gamma(\alpha)} \int_0^\infty \lambda^{n+\alpha-1} e^{-\lambda(\beta+1)} \, d\lambda.\tag{1}
$$
The second integral is
$$
\int_0^\infty \lambda^{\text{something}... |
161,048 | <p>I am interested about a Fact (if it is right) of the structure of parabolic subgroups of finite classical algebraic groups:</p>
<p>Let $G$ be a classical algebraic group over the finite field of order $r^{e}$, and let $P \leq G$ denote a maximal parabolic subgroup of $G$.</p>
<p>We regard the unipotent radical of ... | user76758 | 43,107 | <p>It is a general fact that if $P$ is a parabolic $k$-subgroup of an arbitrary connected reductive group $G$ over an arbitrary field $k$ then $U := \mathscr{R}_u(P)$ has a canonical $P$-equivariant filtration $$U = U_0 \supset U_1 \supset \dots \supset U_m \supset U_{m+1} = \{1\}$$ by smooth connected $k$-subgroups su... |
3,294,874 | <p>How should I solve <span class="math-container">$$y'+2y=x\sin(x)$$</span>
I tried this calculation using integrating factor, but it costs me a lot of calculation. Is there better way to solve this kind of problem?</p>
| Empy2 | 81,790 | <p>Guess <span class="math-container">$$y(x)=Ax\sin x +Bx\cos x+C\sin x+D\cos x$$</span>.<br>
Feed that into the left-hand side,. Equate coefficients, to get four equations in <span class="math-container">$A,B,C,D$</span></p>
|
2,525,145 | <p>How to prove: $Cl_{\mathbb{R}}(\mathbb{IQ})=\mathbb{R}$ ($\mathbb{IQ}$-set of irrational numbers)?</p>
<p>I know how to prove: $Cl_{\mathbb{R}}(\mathbb{Q})=\mathbb{R}$ ($\ast$)</p>
<p><strong>Proof of ($\ast$)</strong>: Let: $r \in \mathbb{R}$ , $q \in \mathbb{Q}$, $\epsilon \in \mathbb{R}_{+}$. Then: $\exists n \... | Martin R | 42,969 | <p>You can only show that
$$ \tag{*}
|a-c|+|b-d| \le |a-d|+|b-c|
$$
and equality can hold, e.g. for $(a, b, c, d) = (1, 2, 3, 4)$.
More precisely, we have</p>
<blockquote>
<p>Let $a < b$, $c < d$ be real numbers, and denote by $I, J$ the open intervals $I = (a, b)$ and $J = (c, d)$. Then
$$ \tag{**}
|a-c|+|... |
166,613 | <p>During revision, I came across this problem:</p>
<p>The set of real numbers $x$ for which the series $$\sum_{n=1}^{\infty}{\frac{n!x^{2n}}{n^n(1+x^{2n})}}$$ converges is __.</p>
<p>I tried using the ratio test, but got stuck in the process of simplification.</p>
<p>(The answer is the series converges for $\mathbb... | Mercy King | 23,304 | <p>For every $x \in \mathbb{R}$ and $n \in \mathbb{N}$ we have
$$
\frac{n!x^{2n}}{n^n(1+x^{2n})} \le a_n:=\frac{n!}{n^n},
$$
and
$$
\frac{a_{n+1}}{a_n}=\left(\frac{n}{n+1}\right)^n=\left(1+\frac{1}{n}\right)^{-n} \to 1/e.
$$
Therefore the series converges for every $x \in \mathbb{R}$.</p>
|
25,209 | <p>Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills:</p>
<p>$$
s = \sum_{i=1}^k n_i d_i\quad\text{with } n_i > 0, d_i > d_{i-1}
$$</p>
<p>We start with the highest denomination $d$ not exceeding $s$ a... | Mike Spivey | 2,370 | <p>The greedy solution is not optimal in general, even for the case when the value of any coin is bigger than the sum of all coins of lower denomination. For example, suppose we have coins of denominations 25, 9, 4, and 1. For 37 cents, the greedy solution uses five coins: one 25, one 9, and three 1s. However, the o... |
25,209 | <p>Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills:</p>
<p>$$
s = \sum_{i=1}^k n_i d_i\quad\text{with } n_i > 0, d_i > d_{i-1}
$$</p>
<p>We start with the highest denomination $d$ not exceeding $s$ a... | Null Set | 8,533 | <p>Here's an algorithm I think should decide if, for a given set of denominations, the greedy change making algorithm is optimal. coins(value, set of denominations) is a function that returns the minimal number of coins needed to make the value with those denominations.</p>
<pre><code>Given set D of denominations
let... |
25,209 | <p>Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills:</p>
<p>$$
s = \sum_{i=1}^k n_i d_i\quad\text{with } n_i > 0, d_i > d_{i-1}
$$</p>
<p>We start with the highest denomination $d$ not exceeding $s$ a... | MJD | 25,554 | <p>The paper <a href="http://graal.ens-lyon.fr/~abenoit/algo09/coins2.pdf" rel="noreferrer">D. Pearson. A Polynomial-time Algorithm for the Change-Making Problem. Operations Reseach Letters, 33(3):231-234, 2005</a> offers an $O(n^3)$ algorithm for deciding whether a coin system has the property you want, where $n$ is t... |
1,002,154 | <p>I understand the integer part, keep dividing by 5 and I get 40440, but for the fraction part I did a calculation, but it does not match with the answer given.</p>
<p>I do the fraction part like this:</p>
<p>$$0.123 \times 5=0.615$$
$$0.615 \times 5=3.075$$
$$0.075 \times 5=0.375$$
$$0.375 \times 5=1.875$$
$$0.875 ... | Sheikh Asif Imran Shouborno | 180,236 | <p>After reaching a certain point, say up to 5 digits of fraction, the teacher may treat the resulting base-n number as if it were much probable that the base n number is an infinite fraction : such goes the convention. However, when you find some part repeating, it is a iterating fraction for sure. Otherwise, it is li... |
3,379,470 | <p>If there is a number <span class="math-container">$x = 0.333333333$</span> where 3 is repeated infinitely how to convert it to fraction ?
It is said that <span class="math-container">$-> 0.(3) = \frac{3}{10} - 1$</span> and <span class="math-container">$0.33 = \frac{33}{100} - 1$</span>, But I can't understand W... | KY Tang | 508,774 | <p>We do not have to convert to other coordinate system. Since for a fixed value of y, the cross section is a circle with radius of <span class="math-container">$\sqrt{\frac{y}{2}}$</span>. Then the integral becomes</p>
<p><span class="math-container">$$\int_{y=0}^{8} \int_{x=-\sqrt{\frac{y}{2}}}^{\sqrt{\frac{y}{2}}... |
3,081,997 | <p>I've been stuck on this calc problem for a while:</p>
<p>Let <span class="math-container">$f$</span> be the function given by <span class="math-container">$f(x) =\ln(x+1)$</span> and let <span class="math-container">$g$</span> be the function given by <span class="math-container">$g(x) = x^{-1/2}$</span>. At what v... | Shubham Johri | 551,962 | <p><span class="math-container">$f'(x)=\dfrac1{x+1}\\g'(x)=-\dfrac1{2x\sqrt x}\\\implies f'(x)\cdot g'(x)=-\dfrac1{2x(x+1)\sqrt x}=-1\\\therefore 4x^3(x+1)^2=1$</span> </p>
<p>which gives <span class="math-container">$x=0.484$</span>.</p>
|
2,842,589 | <p>It has been a while since my undergraduate introductory analysis course, so I am a little rusty. I know that to show two sets $B, C$ are equal, you first show that $B\subseteq C$ and then $C\subseteq B$ . To show $B\subseteq C$ you take any element of $B$, say $b \in B$ and show that $b \in C$. </p>
<p>The main is... | asdf | 436,163 | <p>Suppose $\exists x \in (a,b)$ s.t. $x \notin\bigcup_{n=1} ^{\infty}(a, b-\frac{1}{n}]$</p>
<p>Then $b-\frac{1}{n}<x$ $\forall n \in \mathbb{N}$</p>
<p>But $$\lim_{n \rightarrow \infty}b-\frac{1}{n}=b$$ which implies that $b \leq x$ by algebra of limits which is a contradiction.</p>
<p>Hence $(a,b) \subset \big... |
2,842,589 | <p>It has been a while since my undergraduate introductory analysis course, so I am a little rusty. I know that to show two sets $B, C$ are equal, you first show that $B\subseteq C$ and then $C\subseteq B$ . To show $B\subseteq C$ you take any element of $B$, say $b \in B$ and show that $b \in C$. </p>
<p>The main is... | Peter Szilas | 408,605 | <p>Part 2:</p>
<p>Let $n \in \mathbb{Z^+}.$</p>
<p>1) $\bigcup_{n=1}^{\infty} (-n,n) \subset \mathbb{R}.$</p>
<p>Trivial.</p>
<p>Now show that $\mathbb{R} \subset \bigcup_{n=1}^{\infty} (-n,n)$.</p>
<p>2) Let $r$ be real, positive.</p>
<p>Archimedean principle:</p>
<p>There is a $n \in \mathbb{Z^+}$ with $n>... |
1,898,799 | <p>How can we solve this differential equation?
\begin{cases}
x''-y' = t+1, \\
x'+y'-3x+y = 2t-1.
\end{cases}</p>
| Jan Eerland | 226,665 | <p>HINT to solve:</p>
<p>$$
\begin{cases}
x''(t)-y'(t)=t+1\\
x'(t)+y'(t)-3x(t)+y(t)=2t-1
\end{cases}
$$</p>
<p>Use Laplace transform:</p>
<p>$$
\begin{cases}
s^2\text{X}(s)-sx(0)-x'(0)-s\text{Y}(s)-y(0)=\frac{1+s}{s^2}\\
s\text{X}(s)-x(0)+s\text{Y}(s)-3\text{X}(s)+\text{Y}(s)=\frac{2-s}{s^2}
\end{cases}
$$</p>
<p>s... |
659,579 | <p>the sequence $ (x_n)_{n\ge 1}$, $ x_n$ being
the exponent of 2 in the decomposition of the numerator of
$$\dfrac{2}{1}+\dfrac{2^2}{2}+\cdots+\dfrac{2^n}{n}$$
goes to infinity as $ n\to\infty$.
Even more, prove that $$x_{2^n}\ge 2^n-n+1$$</p>
<p>My idea: maybe
$$\dfrac{2}{1}+\dfrac{2^2}{2}+\cdots+\dfrac{2^n}{n}=\df... | Jack D'Aurizio | 44,121 | <p>This is a partial answer, just underlying an interesting technique.
$$\frac{a_n}{b_n}=\sum_{k=1}^n\frac{2^k}{k},\qquad (a_n,b_n)=1$$
is the opposite of the sum of the reciprocal of the roots of
$$P_n(x) = \prod_{k=1}^{n}\left(x+\frac{k}{2^k}\right)=2^{-\frac{n(n+1)}{2}}\prod_{k=1}^{n}(k+2^kx),$$
so
$$\frac{a_n}{b_n}... |
1,845,348 | <p>First, is $\sqrt{-3}$ is equal to $-3$ or is it imaginary?</p>
<p>What is the difference between:</p>
<ul>
<li>$\sqrt{(-3)^2}$</li>
<li>$\sqrt{-3^2}$</li>
<li>$(\sqrt{-3})^2$</li>
</ul>
<p>Can I write $(\sqrt{-3})^2 = -3$?</p>
<p>And, given the rule that $\sqrt{a^n}$ is equal to ($\sqrt{a})^n$, can I say that $... | Antonio.G | 351,318 | <p>$\sqrt{(-3)^2}=\sqrt{9}=3$, </p>
<p>$\sqrt{-3^2}=\sqrt{-9}=3i$,</p>
<p>$(\sqrt{-3})^2=(\sqrt{3}i)^2=-3$</p>
<p>The rule is true only in the case $a>0$. </p>
|
172,883 | <p>Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function</p>
<p>$P_i = px^i + qy^i + rz^i$</p>
<p>where $x,y,z$ are coordinates. I have a few related questions:</p>
<p>Has the algebra generated by $P_1, P_2, P_3, P_4, ...$ been considered before in the l... | Hailong Dao | 2,083 | <p>Update: I thought more about it. The good news is two questions are equivalent, but the calculation seems a little more complicated then I thought. </p>
<p>So let $A$ be the algebra with condition $P_1=0$. Then $A=R/(P_1)$, so $A$ is CM iff $R$ is. </p>
<p>Let's focus on $A$ then. We still have $A[x,y]=k[x,y]$. Bu... |
3,551,494 | <p>Can someone help me out with <span class="math-container">$$\lim\limits_{x\rightarrow 0^+}x\log(1+x^{-1})?$$</span> I tried Taylor's expansion to no avail.</p>
| mjw | 655,367 | <p><span class="math-container">$$\lim_{x\downarrow 0}x\log(1+x^{-1}) = \lim_{x\rightarrow\infty} \frac{\log(1+x)}{x}=0$$</span></p>
<p>by L'Hopital's rule.</p>
|
998,562 | <p>I am studying homology groups of topological spaces. In books I have found that the $n$th homology group counts the number of "$n$-dimensional holes" which exist in that space. If I consider homology groups over the integers, then in the $n$th homology group the number of $\mathbb{Z}$ represents the number of $n$-di... | Scott Mutchnik | 194,245 | <p>Torsion elements seem intuitively significant as well as torsion-free elements. But let us start from first principles, finding the conditions under which a chain $c = \sum_{i}a_{i} \sigma_{i} \in C_{n}$ might, in an intuitive sense, be considered equivalent to an n-dimensional hole, and relating these conditions to... |
2,656,551 | <p>I am trying to solve the following problem:</p>
<p>Suppose $X$ is a continuous random variable with density $f$ over $[0,1]$. Then show that $(\int_0^1xf(x)dx)(\int_0^1x^3f(x)dx)>(\int_0^1x^2f(x)dx)^2$.</p>
<p>I am trying to apply Fubini's theorem, but am getting equality for the two sides. What mistake am I ma... | Jendrik Stelzner | 300,783 | <p>For every $S \subseteq R$ denote by $\langle S \rangle$ the two-sided ideal generated by $S$, i.e. the smallest two-sided ideal containing $S$.
Then for every $x \in R$ one can consider the quotient ring
$$
R'_x = R/\langle rx-r, xr-r \mid r \in R \rangle \,,
$$
for which $\overline{x} \in R'_x$ is a unit element.... |
3,350,551 | <p>Given two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>. We know that <span class="math-container">$B \setminus A = B$</span> holds. What can we say about the relation between <span class="math-container">$A$</span> and <span class="math-container">$B$</span>?</p>
<p>I wou... | Fred | 380,717 | <p>If <span class="math-container">$x \in A \cap B$</span>, then <span class="math-container">$x \in B$</span>, hence <span class="math-container">$x \in B \setminus A$</span>, therefore <span class="math-container">$x \notin A$</span>, a contradiction.</p>
|
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Community | -1 | <p><a href="/questions/tagged/status-completed" class="post-tag moderator-tag" title="show questions tagged 'status-completed'" rel="tag">status-completed</a></p>
<p>I propose the following word (acronym)</p>
<p><strong>ASAP</strong> <span class="math-container">$\quad$</span> ("As soon as possible!")... |
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | quid | 85,306 | <p>I propose the following word</p>
<p><strong>curious</strong></p>
<p>for inclusion in the HNQ block-list.</p>
<p>Please up-vote if you agree with the inclusion. Down-vote if you disagree with the inclusion.</p>
|
31,473 | <p>In a recent post <a href="https://math.meta.stackexchange.com/questions/31425/autofilters-for-hot-network-questions">Autofilters for Hot Network Questions</a> the possibility was raised to created a list of words which when contained in a title of a question would prevent the question from being included in the HNQ ... | Joe | 623,665 | <p>I propose the following word</p>
<p><strong>Kindly</strong></p>
<p>for inclusion in the HNQ block-list.</p>
<p>Please up-vote if you agree with the inclusion. Down-vote if you disagree with the inclusion.</p>
|
374,761 | <p>It is probably an easy question, but somehow I am stuck.</p>
<p><strong>Question</strong> Is the following statement true? If yes, how to prove it?</p>
<blockquote>
<p>Suppose that <span class="math-container">$f\in C^1(\mathbb{R}^n)$</span> is convex and
<span class="math-container">$$
\langle\nabla f(x)-\nabla f(y... | Dirk | 9,652 | <p>That's a standard result in convex optimization. For example Theorem 2.1.5 in Nesterov's "Introductory Lectures on Convex Optimization" states that the following are equivalent:</p>
<ul>
<li><span class="math-container">$f$</span> is <span class="math-container">$C^1$</span>, convex and the gradient <span ... |
2,890,207 | <p>Consider the following system of equations where $x,y,z$ are variables and for a constant $\mathrm C$, $y \times z = \mathrm C \neq 0$</p>
<p>\begin{equation}
\left\{
\begin{array}{lcl}
a_1x + b_1y + c_1z &= 0\\
a_2x + b_2y + c_2z &= 0\\
\end{array}
\right.
\end{equation}</p>
<p>The most straightforward so... | Anik Bhowmick | 354,636 | <p>For Analysis, study (i) Mathematical Analysis (Tom M. Apostol), (ii) Real and Complex Analysis (Walter Rudin), (iii) Real Analysis (Royden, Fitzpatrick). Real Analysis (N.L. Carothers) is good, too.</p>
|
2,328,068 | <p>I've came up with a proof to the above problem, but I'm stuck on whether or not this should be done with induction instead. I get stuck in this pit a lot (the "is or isn't this enough for a proof") so I wanted to see what others think.</p>
<p>Here's my proof:</p>
<blockquote>
<p>Let $d \leq n$. Consider the the ... | Veridian Dynamics | 408,632 | <p>If it is of your interest, here is another aproach:</p>
<p>By Cayley's Theorem, there exists a monomorfism $f:\mathbb{Z}/d\mathbb{Z}\to S_d$ and there is a natural monomorfism $g:S_d\to S_n$. $g(f(1))$ have order $d$ in $S_n$</p>
|
4,075,642 | <p>I recently found this problem, it was part of a regional qualifier in Southern America (Venezuela- I believe) in January 2021. As I can’t find the solution anywhere and it is very different from the problems I have tackled so far I thought I might ask you all for some help. I have tried finding cases where <span cla... | NN2 | 195,378 | <p><span class="math-container">$(a)$</span> First step, choosing <span class="math-container">$(1,2), (3,4),...,(2019,2020)$</span>. After these operations, the remaining numbers on the board are 1010 times of ones and <span class="math-container">$2021$</span>.
Second step, choosing <span class="math-container">$(1,1... |
1,030,177 | <p>Suppose you have 4 stamps with probabilities 0.2, 0.4, 0.3, and 0.1, respectively. Each time you pick a stamp is independent of every other pick. Suppose you pick your 10th stamp. What is the probability this stamp is one you haven't seen before?</p>
<p>How should I approach these kinds of problems? Should I do 1-p... | Reinstate Monica | 147,200 | <p>You are correct. The common theorem is that if $\sum_k a_k$ is convergent but not absolutely convergent, for every $c$ there is a rearrangement converging to $c$. But if there is no requirement on convergence, this theorem is false. A simple example is $a_k = 1$: clearly any rearrangement leaves this sequence unchan... |
2,721,285 | <p>I would like to prove the following statement.</p>
<blockquote>
<p>If $n$ is a prime number, then the factorization of $x^n-1$ over
$\Bbb{Z}$ contains only $2$ factors, and those factors are:
$$x^n-1=(x-1)(x^{n-1}+x^{n-2}+\cdots+x+1).$$</p>
</blockquote>
<p>I have already proved that the factorization abov... | J. Linne | 346,385 | <p>The $n$th roots of unity are any numbers $c$ such that $c^n=1$ (these include complex numbers). A primitive $n$th root of unity is any number $c$ such that $c^n=1$ and $n$ is the smallest integer $k$ such that $c^k=1$. Reading from <a href="https://en.wikipedia.org/wiki/Root_of_unity#Cyclotomic_polynomials" rel="nof... |
1,019,191 | <p>I'm negating this proposition: "If you study you will not fail." </p>
<p>I'm using proposition P: "You study" and proposition Q: "You will fail." </p>
<p>The original statement can be written as "$P → ¬Q.$"</p>
<p>My instructor has the negation of this statement like this:</p>
<p>$¬(P → ¬Q) = ¬(¬P \lor ¬Q) = P\l... | amWhy | 9,003 | <p>Because <span class="math-container">$A\rightarrow B \equiv \lnot A \lor B\tag{1}$</span></p>
<p>Think of this as stating: An implication <span class="math-container">$A\rightarrow B$</span> is true whenever</p>
<ul>
<li><p><span class="math-container">$A$</span> is false: <span class="math-container">$\;(\lnot A)$<... |
3,688,628 | <p>If <span class="math-container">$a,b\in \mathbb{N}$</span> find all primes <span class="math-container">$p$</span> such that <span class="math-container">$a^3+b^3=p^2$</span></p>
<p>My approach-</p>
<p><span class="math-container">$a^3+b^3=(a+b)(a^2-ab+b^2)=p^2$</span> suppose <span class="math-container">$a+b=x$<... | Fabio Lucchini | 54,738 | <p>In the case <span class="math-container">$(x,y)=(p,p)$</span>, we have <span class="math-container">$a+b=(a+b)^2-3ab=p$</span>, hence <span class="math-container">$p^2\equiv p\pmod 3$</span>.
If <span class="math-container">$p\neq 3$</span>, then <span class="math-container">$p\equiv 1\pmod 3$</span>, hence
<span cl... |
32,841 | <blockquote>
<p>If $0 \rightarrow A_{\bullet} \rightarrow B_{\bullet} \rightarrow C_{\bullet} \rightarrow 0$ is a short exact sequence of chain complexes (of R-modules), then, whenever two of the three complexes $A_{\bullet}$,$B_{\bullet}$,$C_{\bullet}$ are exact, so is the third. </p>
</blockquote>
<p>This <a href=... | Felix Hoffmann | 3,503 | <p>For completeness, I think I can cover the case where $A_{\bullet}$ and $B_{\bullet}$ are exact myself now. I will try to stick to the notations used in the other posts.</p>
<p>If $c \in C_n$ with $d(c)=0$, there is, by the surjectivity of $\beta_n$, a $b$ in $B_n$ such that $$\beta_n(b)=c.$$</p>
<p>Then $$\beta_{n... |
439,820 | <blockquote>
<p>Is $\large i^{i^i}$ real ? How to find it?</p>
</blockquote>
<p>Thank You!</p>
| user1337 | 62,839 | <p>Complex powers may have more than one value. In our case
$$ i^i=e^{i \log i}=\exp \left(i \left(\ln(1)+i \frac{\pi}{2}+2\pi k i\right)\right)=e^{-\frac{\pi}{2}+2\pi k} $$ where $k$ is an integer. Thus $$i^{i^i}=e^{i^i \log(i)}=\exp\left(e^{-\frac{\pi}{2}+2\pi k}\cdot\left(i \frac{\pi}{2}+2\pi l i\right)\right), $$</... |
2,075,664 | <p>In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits.</p>
<p><strong>My attempt</strong>:We should find number of two pairs that can be digits of this number for choosing the place of digits we have $*8$ (We should notice we cant have $0$ in the beginning.But the biggest ... | Community | -1 | <p>Take the number as $a_0a_1a_2a_3$. </p>
<hr>
<p>For a sum of two digits $S = a_0 + a_1$, we can see that there are $19-S$ pairs of digits associated with the sum when $S \geq 10$ and $S +1$ pairs of digits otherwise. <strong>But</strong>, we require that $a_0\neq 0$. So, for the latter case, there are only $S$ ... |
2,075,664 | <p>In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits.</p>
<p><strong>My attempt</strong>:We should find number of two pairs that can be digits of this number for choosing the place of digits we have $*8$ (We should notice we cant have $0$ in the beginning.But the biggest ... | Kanwaljit Singh | 401,635 | <p>You have to check pattern. Many times we have patterns in question. </p>
<p>In this question first two digits makes sum equal to last two digits. And its clear that for first two digits we have no case in which first digit is zero.</p>
<p>Now we have to find number of ways. </p>
<p><strong>Sum of first two and la... |
2,075,664 | <p>In how many $4$-digit numbers the sum of two right digits is equal to the sum of two left digits.</p>
<p><strong>My attempt</strong>:We should find number of two pairs that can be digits of this number for choosing the place of digits we have $*8$ (We should notice we cant have $0$ in the beginning.But the biggest ... | RGS | 329,832 | <p>The best way to tackle this problem is to think about what the sum of the two rightmost digits is going to be, and then find in how many ways that can be done.</p>
<p>Because we are summing only two 1-digit integers, the biggest sum we can get is with $9+9=18$ so there is no 4-digit number, whose sum of the two rig... |
194,395 | <p>With a list say <code>{{x1, y1}, {x2, y2}, …}</code> and a list <code>{z1}</code>, what is the best way to create a list <code>{{x1, y1, z1}, {x2, y2, z1}, {x3, y3, z1}, …}</code>?</p>
| JimB | 19,758 | <p>If <code>z1</code> is a list (as indicated in the question), then does the following do what you want:</p>
<pre><code>z1 = {1, 2, 3};
xy = {{x1, y1}, {x2, y2}, {x3, y3}};
Flatten[{#, z1}] & /@ xy
(* {{x1, y1, 1, 2, 3}, {x2, y2, 1, 2, 3}, {x3, y3, 1, 2, 3}} *)
</code></pre>
<p>Or should the output be as follows:<... |
21,048 | <p>When introducing an abstraction it is important (in my opinion) to have a wide variety of examples of this abstraction.</p>
<p>Since finite dimensional real vector spaces are classified up to isomorphism by their dimension, it is a little difficult to find examples of abstract vector spaces which "feel" ve... | user52817 | 1,680 | <p>The spin states of an electron form a two-dimensional vector space over the complex numbers. Designate "spin up" and "spin down" for a basis. The vector space structure is a consequence of the linearity of the Schrodinger equation.</p>
<p>The computer science slant on this situation uses the word... |
4,089,526 | <p>Two players play rock, paper, scissors for 3 wins ( one player wins 3 times). What is the expected value of the number of rounds?</p>
<p>I tried: for the n-th games probability : <span class="math-container">$$ p(x_n)=2\frac{\binom{n}{3}}{3^n} $$</span></p>
<p>Therefore my <span class="math-container">$E(x)\approx 1... | aschepler | 2,236 | <p>As suggested in comments, let's consider the states the game can be in after each round. <span class="math-container">$S(x,y)$</span> is the state where player A has won <span class="math-container">$x$</span> rounds so far and player B has won <span class="math-container">$y$</span> rounds so far. A tie round keeps... |
120,799 | <p>I'm new to the discrete math, but I'm trying to prove this algebraically and some help would be cool.</p>
<p>$$
\binom{a}{b} \binom{b}{c}= \binom{a}{c} \binom{a-c}{b-c},\quad c \leqslant b \leqslant a
$$</p>
<p>I'm not sure how to prove this algebraically. Can someone help me?</p>
<p>So after some work i got... | Jonas Kibelbek | 1,461 | <p>It is true that the field of $p^k$ elements consists of the roots of the polynomial $t^{p^k}-t$ over $\mathbb{Z}_p$. </p>
<p>One nice consequence of this is that we can easily figure out all of the subfields of a finite field. The finite field of 16 elements, which you are working with, has two proper subfields--... |
120,799 | <p>I'm new to the discrete math, but I'm trying to prove this algebraically and some help would be cool.</p>
<p>$$
\binom{a}{b} \binom{b}{c}= \binom{a}{c} \binom{a-c}{b-c},\quad c \leqslant b \leqslant a
$$</p>
<p>I'm not sure how to prove this algebraically. Can someone help me?</p>
<p>So after some work i got... | joriki | 6,622 | <p>Further to Jyrki's comment that the multiplicative group of a finite field is cyclic: This means that you can write the $|K|-1=15$ non-zero elements as $a,a^2,\dotsc,a^{15}$, where $a$ is any of the $\phi(15)=8$ primitive elements of $K^*$. So an efficient strategy is to find the multiplicative order of some arbitra... |
3,263,742 | <p>In Axler's <em>Linear Algebra Done Right,</em> there is this theorem. (3.5)</p>
<blockquote>
<p>Suppose <span class="math-container">$v_1. . . v_n $</span> is a basis of <span class="math-container">$V$</span> and <span class="math-container">$w_ , . . . w_n \in W$</span>. Then there exists
a unique linear map... | paulinho | 474,578 | <p>Because the vectors <span class="math-container">$\vec{w}_i$</span> form a basis for <span class="math-container">$W$</span>, any two different linear combinations of the vectors <span class="math-container">$\vec{w}_i$</span> must yield a different result. Let <span class="math-container">$\vec{w} = c_1 \vec{w}_1 +... |
2,323,550 | <p>I recently saw a puzzle in an advert for the website <a href="http://brilliant.org/" rel="noreferrer">Brilliant.org</a>, which went as follows:</p>
<blockquote>
<p>What is the blue area?</p>
<p><a href="https://i.stack.imgur.com/DGDO1.png" rel="noreferrer"><img src="https://i.stack.imgur.com/DGDO1.png" alt="... | Brian Gix | 483,175 | <p>I honestly just eye-balled this one, and came up with <em>a</em> solution without verifying it was the <em>only</em> solution:</p>
<p>Since the outside edge shared by Green and Yellow appeared roughly at halfway point, and the side shared by red and yellow at the $\frac34$ point, it seemed obvious to me that each o... |
2,763,911 | <p><a href="https://i.stack.imgur.com/iXYBn.png" rel="nofollow noreferrer">first promblem</a></p>
<p>First problem: I have no idea where to start with the problem, so little a direction on where to start would be very appreciated.</p>
<p><a href="https://i.stack.imgur.com/2h7Kf.png" rel="nofollow noreferrer">second p... | BallBoy | 512,865 | <p>First problem: think Fundamental Theorem of Calculus.</p>
<p>Second problem: You're almost there. Evaluate the integral: $\int_{\frac\pi6}^x \cos t dt = \sin t|_\frac\pi6^x = \sin x - \sin\frac\pi6$</p>
|
2,763,911 | <p><a href="https://i.stack.imgur.com/iXYBn.png" rel="nofollow noreferrer">first promblem</a></p>
<p>First problem: I have no idea where to start with the problem, so little a direction on where to start would be very appreciated.</p>
<p><a href="https://i.stack.imgur.com/2h7Kf.png" rel="nofollow noreferrer">second p... | mathematics2x2life | 79,043 | <p>For the first problem, stare at the Fundamental Theorem of Calculus until you see why what you were given shows that the theorem applies. Then we have
$$
21=\int_e^\pi f'(x) \;dx = f(x)\bigg|_e^\pi = f(\pi)-f(e)=f(\pi)-13
$$
But then $f(\pi)-13=21$ so that $f(\pi)=34$.</p>
<p>For the second problem,
$$
\begin{spli... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Gerry Myerson | 8,269 | <p>There's a collection at <a href="https://primes.utm.edu/notes/proofs/infinite/" rel="nofollow noreferrer">https://primes.utm.edu/notes/proofs/infinite/</a> (Proofs that there are infinitely many primes)</p>
|
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | yoyo | 6,925 | <p>Let <span class="math-container">$p_1,...,p_n$</span> be the primes less or equal <span class="math-container">$N$</span>. Any integer less or equal <span class="math-container">$N$</span> can be written as <span class="math-container">$p_1^{e_1}\cdot...\cdot p_n^{e_n}\cdot m^2$</span> with <span class="math-contai... |
50,006 | <p>This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so ma... | Community | -1 | <p>The following proof use the Euler's totient function and relies on the fact that $\phi(m) > 1$ for all $m \geq 3$.</p>
<p>Assume that there are only a finite number of primes say $p_1,p_2,\ldots,p_k$. Look at the product of these finite primes i.e. $$m = p_1 p_2 \cdots p_k$$
Now consider any number $n > 1$. S... |
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