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 |
|---|---|---|---|---|
1,549,490 | <p>$f(x) = 3x - \frac{1}{x^2}$</p>
<p>I am finding this problem to be very tricky:</p>
<p><a href="https://i.stack.imgur.com/7RjYG.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7RjYG.jpg" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/jwUHp.jpg" rel="nofollow n... | Ian Miller | 278,461 | <p>There is no benefit joining the $3$ into the other fraction. It just makes your algebra harder/messier.</p>
<p>$$\lim_{h\to0}\frac{3(x+h)-\frac{1}{(x+h)^2}-3x+\frac{1}{x^2}}{h}$$</p>
<p>$$=\lim_{h\to0}\frac{3h-\frac{1}{(x+h)^2}+\frac{1}{x^2}}{h}$$</p>
<p>$$=\lim_{h\to0}\frac{3h-\frac{x^2}{x^2(x+h)^2}+\frac{(x+h)^... |
757,917 | <p>According to <a href="http://www.wolframalpha.com/input/?i=sqrt%285%2bsqrt%2824%29%29-sqrt%282%29%20=%20sqrt%283%29" rel="nofollow">wolfram alpha</a> this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$</p>
<p>But how do you show this? I know of no rules that works with addition inside square roots.</p>
<p>I not... | Git Gud | 55,235 | <p><strong>Hint:</strong> Since they are both positive numbers, they are equal if, and only if, their squares are equal.</p>
|
757,917 | <p>According to <a href="http://www.wolframalpha.com/input/?i=sqrt%285%2bsqrt%2824%29%29-sqrt%282%29%20=%20sqrt%283%29" rel="nofollow">wolfram alpha</a> this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$</p>
<p>But how do you show this? I know of no rules that works with addition inside square roots.</p>
<p>I not... | Bill Dubuque | 242 | <p>On can easily <em>discover</em> the denesting using my simple <a href="https://math.stackexchange.com/a/664987/242">radical denesting algorithm.</a></p>
<p>$\ w = 5+\sqrt{24}\,$ has norm $\,n = ww' = 5^2-24 = 1.\,$ Subtracting out $\,\sqrt{n}=1\,$ yields $\,4+\sqrt{24}.$ </p>
<p>This has trace $\,t = 8,\,$ so divi... |
757,917 | <p>According to <a href="http://www.wolframalpha.com/input/?i=sqrt%285%2bsqrt%2824%29%29-sqrt%282%29%20=%20sqrt%283%29" rel="nofollow">wolfram alpha</a> this is true: $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$</p>
<p>But how do you show this? I know of no rules that works with addition inside square roots.</p>
<p>I not... | Community | -1 | <p>You kind of have to assume that the nested radical can be rewritten as the sum of two surds (or radicals) in the form $\sqrt{a+b\sqrt{c}}=\sqrt{x}+\sqrt{y}$.</p>
<p>So in your question, we have $\sqrt{5+\sqrt{24}}=\sqrt{x}+\sqrt{y}$. Squaring both sides gives you: $$5+\sqrt{24}=x+y+2\sqrt{xy}$$</p>
<p>This can be ... |
47,143 | <p>I want to create a table of replacement rules. </p>
<pre><code>g[a_, b_] := a -> b
t1 = Table[10 i + j, {i, 5}, {j, 3}]
t2 = Table[ i + j, {i, 5}, {j, 3}]
g[ # & @@@ t1, # & @@@ t2 ]
</code></pre>
<p>The correct output is below:</p>
<pre><code>{{11 -> 2, 12 -> 3, 13 -> 4},
{21 -> 3, 2... | Albert Retey | 169 | <p>There are other approaches to achieve what you have described. As I guess this is only a simplification of what you really need, here is how you would do it with <code>WhenEvent</code>:</p>
<pre><code>equation = x'[t] + (x[t] - λ[t]) == 0;
sol = NDSolveValue[{equation, x[0] == 0, λ[0] == 1,
WhenEvent[x'[t] == 0.... |
3,684,331 | <p>I am working on a problem from my Qual</p>
<p>"Let <span class="math-container">$T:V\to V$</span> be a bounded linear map where <span class="math-container">$V$</span> is a Banach space. Assume for each <span class="math-container">$v\in V$</span>, there exists <span class="math-container">$n$</span> s.t. <span cla... | robjohn | 13,854 | <p>The product formula for sine says
<span class="math-container">$$
\frac{\sin(x)}{x}=\prod_{k=1}^\infty\left(1-\frac{x^2}{k^2\pi^2}\right)\tag1
$$</span>
Thus, for <span class="math-container">$0\lt x\le\pi$</span>
<span class="math-container">$$
\begin{align}
\frac{\sin(x)}x
&\le1-\frac{x^2}{\pi^2}\tag2\\[3pt]
&... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| nickdon2006 | 446,107 | <p>Assume two points are drawn at random. First called A, second called B. Connect the points with the center, we get two lines and four quarters (area is not the same). Only in one quarter that the triangle formed will contain the center. </p>
<p>Apply the symmetry, the points A, B are totally random, thus the area o... |
268,635 | <p>Consider the triangle formed by randomly distributing three points on a circle. What is the probability of the center of the circle be contained within the triangle?</p>
| orangeskid | 168,051 | <p>Fix a point $A$. The angular coordinates, measured from $A$, of the other two points lie in a square (say $[-\pi,\pi]^2$). In the picture below, the green region is the desired one. It has area $1/4$ the area of the square. So the probability is $1/4$.</p>
<p><a href="https://i.stack.imgur.com/B45j3.png" rel="nore... |
1,761,228 | <p>I have a group of order 35 and I want to know if it contains elements of order 7 and 5. I know that it does and there is a proof that is much longer, but I wanted to know if the following worked to show that it does contain elements of order 5 and 7. </p>
<p><strong>My Approach:</strong> Let $|G| = 35$. We know tha... | Dietrich Burde | 83,966 | <p>It follows by <a href="https://en.wikipedia.org/wiki/Cauchy's_theorem_%28group_theory%29" rel="nofollow noreferrer">Cauchy's theorem</a>, with $p=5$ and $p=7$. The proof of Cauchy's theorem is elementary. Furthermore we could use the result that all groups of order $pq$ with $p<q$ prime and $p\nmid q-1$ are c... |
1,761,228 | <p>I have a group of order 35 and I want to know if it contains elements of order 7 and 5. I know that it does and there is a proof that is much longer, but I wanted to know if the following worked to show that it does contain elements of order 5 and 7. </p>
<p><strong>My Approach:</strong> Let $|G| = 35$. We know tha... | Community | -1 | <p>You can also show this using an elementary counting argument. First, note by Lagrange's theorem that every element must have order <span class="math-container">$1$</span>, <span class="math-container">$5$</span>, <span class="math-container">$7$</span>, or <span class="math-container">$35$</span>.</p>
<p>If there is... |
1,176,381 | <p>I have many sets containing three values like $\{1, -2, 5\}$.
I am want to write in mathematical form to filter set where exist one value with different sign. (and for sure, none of them should be zero)</p>
<p>I am not sure about tag. (please correct the tag if it is not correct)/ </p>
| martini | 15,379 | <p>Note that you want to filter the sets where not all values have the same sign, that is there exists values of both signs, and zero does not occur. If now $\mathcal A$ denotes your collection of sets, then the filtered collection, which contains only the sets that fulfill your condition is
$$ \mathcal F = \{A \in \... |
452,292 | <p>Let $E$ be a Lebesgue measurable set in $\mathbb{R}$. Prove that
$$\lim_{x\rightarrow 0} m(E\cap (E+x))=m(E).$$ </p>
| Alex Becker | 8,173 | <p>Recall that a Lebesgue measurable set $E$ can be approximated arbitrarily well by a finite union of disjoint closed intervals $S=I_1\cup\ldots\cup I_n$, i.e. $\epsilon=m(E\Delta S)$ can be made arbitrarily small. Thus we have that
$$\begin{align}
|m(E\cap (E+x))-m(S\cap (S+x))|&\leq m((E\cap (E+x))\Delta(S\cap (... |
1,906,332 | <p>I know every complex differentiable function is continuous. I would like to know if the converse is true. If not, could someone give me some counterexamples?</p>
<p><strong>Remark:</strong> I know this is not true for the real functions (e.g. $f(x)=|x|$ is a continuous function in $\mathbb R$ which is not different... | Jean Marie | 305,862 | <p>Another classical family of $\mathbb{C}$-continuous but not $\mathbb{C}$-differentiable functions: functions whose expression is analytic but involve a conjugation: </p>
<p>$f$ defined by $f(z):=\bar z sin(z)+cos(\bar z)$ is one of these.</p>
|
446,796 | <p>I am a student in Undergraduate Mathematics, and I'm struggling to number theory ... I have this problem gcd, and do not know how to do it, and still do not study, congruences, Diophantine equations or, among other matters more advanced ... I'm used to divisibility, and some properties and/or theorems gcd ... Help m... | Tomas | 83,498 | <p><strong>Hint 1:</strong> It is sufficient to show:
$$d\mid c \wedge d\mid b \Leftrightarrow d\mid ac \wedge d\mid b$$
for any $d\in\mathbb Z$ (<em>why?</em>)</p>
<p><strong>Hint 2:</strong> For the "$\Leftarrow$"-implication, use that $d\mid ac$ and $d\mid bc$, so $d$ divides their gcd.</p>
|
446,796 | <p>I am a student in Undergraduate Mathematics, and I'm struggling to number theory ... I have this problem gcd, and do not know how to do it, and still do not study, congruences, Diophantine equations or, among other matters more advanced ... I'm used to divisibility, and some properties and/or theorems gcd ... Help m... | felasfa | 55,243 | <p>We use the following theorem from Number Theory.</p>
<p>$$
g=(a,b) \Longleftrightarrow ax+by=g
$$ for some
integers $x$ and $y$. **</p>
<p>Now consider $(a,b)=1$ then you can write
$$
ax+by=1\quad (1)
$$
Let $g=(a\cdot c,b)$ then
invoking the above theorem again
$$
ac(x_{1})+b(y_{1})=g\quad (2)
$$
for some inte... |
536,068 | <p>If X and Y are iid random variables with distribution $F(x)=e^{-e^{-x}}$ and we let $Z=X-Y$ find the distribution function of $F_Z(x)$. I get $F_Z(x)=\frac{e^x}{1+e^x}$ but that doesn't match the answer that the professor gave us. Is what I have correct? I used the standard approach where I integrate over a region t... | Stefan Hansen | 25,632 | <p>$$
\begin{align}
P(Z\leq z)&=P(X\leq z+Y)=\int_{-\infty}^\infty P(X\leq z+y) f(y)\,\mathrm dy\\
&=\int_{-\infty}^\infty\exp(-e^{-(z+y)})\exp(-y-e^{-y})\,\mathrm dy \\
&=\int_{-\infty}^\infty \exp(-y-e^{-y}(1+e^{-z}))\,\mathrm dy\\
&=\left[\frac{1}{1+e^{-z}}\exp(-(1+e^{-z})e^{-y})\right]_{-\infty}^\in... |
1,897,771 | <p>Let $(a, b)$ be the open interval $\left\{z\in\mathbb{R} : a < z < b\right\}$. </p>
<p>Write the theorem "If $x,y\in(a,b)$ then $|x − y| < b − a$" in logic form, and then prove the theorem.</p>
| Planche | 330,526 | <p>Let $x,y \in (a,b)$.
By definition,
$$ a<x<b , a<y<b$$
so $$ -b< -x<-a, a<y<b.$$
$$-b+a<y-x<b-a \mbox{ similary } -b+a<x-y<b-a$$
so $$\left | x-y \right| < b-a.$$</p>
|
557,468 | <p>Let $f$ be a continuous map from $[0,1]$ to $[0,1].$ Show that there exists $x$ with $f(x)=x. $</p>
<p>I have $f$ being a continuous map from $[0,1]$ to $[0,1]$ thus $f: [0,1]\to [0,1]$. Then I know from the intermediate value theorem there exists an $x$ with $f(x)=x$ but I don't know how to formally prove it? </p>... | Deven Ware | 14,334 | <p>Yeah it is by the intermediate value theorem. </p>
<p>Consider the function $g(x) = f(x) - x$. </p>
<p>What can you say about $g(0)$? $g(1)$? Now apply the IVT.</p>
<p><strong>Edit:</strong> If you want to do it without $g$ or the IVT explicitly you can use the proof idea of IVT and say: </p>
<p>If not :</p>
<p... |
3,956,392 | <p>So the question is as follows:</p>
<blockquote>
<p>An urn contains m red balls and n blue balls. Two balls are drawn uniformly at random
from the urn, without replacement.</p>
</blockquote>
<blockquote>
<p>(a) What is the probability that the first ball drawn is red?</p>
</blockquote>
<blockquote>
<p>(b) What is the... | true blue anil | 22,388 | <p>Part <strong>(b)</strong> doesn't give any information about the first ball, it is just asking for the probability that the second ball in the line is red.</p>
<p>Now red balls (or those of any other color!) <strong>don't have any preference for positions in the line</strong>, hence if you randomly pick up <strong>a... |
83,764 | <p>I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out <em>why</em> this is happening (for quite some time now actually). Any help is greatly appreciated. Here goes then:</p>
<p>Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^... | John B | 74,138 | <p>Let me reply taking M to be the identity (indeed M is somewhat cosmetic to the discussion).</p>
<p>The identity $F\circ Ψ=DΨ\circ F$ is what one considers for example in the Grobman-Hartman theorem, passing from a dynamics to its linearization say at a fixed point. The possibilities for F are endless; $F$ would the... |
1,804,360 | <p>The following is a classically valid deduction for any propositions <span class="math-container">$A,B,C$</span>.
<span class="math-container">$\def\imp{\rightarrow}$</span></p>
<blockquote>
<p><span class="math-container">$A \imp B \lor C \vdash ( A \imp B ) \lor ( A \imp C )$</span>.</p>
</blockquote>
<p>But I'... | user21820 | 21,820 | <p>Here is an answer that builds on the intuition I had 4 years ago when asking this question. The idea is to use the <a href="https://en.wikipedia.org/wiki/Brouwer%E2%80%93Heyting%E2%80%93Kolmogorov_interpretation" rel="nofollow noreferrer">BHK interpretation</a> where a witness of <span class="math-container">$P→Q$</... |
1,804,360 | <p>The following is a classically valid deduction for any propositions <span class="math-container">$A,B,C$</span>.
<span class="math-container">$\def\imp{\rightarrow}$</span></p>
<blockquote>
<p><span class="math-container">$A \imp B \lor C \vdash ( A \imp B ) \lor ( A \imp C )$</span>.</p>
</blockquote>
<p>But I'... | MJD | 25,554 | <p>It's interesting to me that different people's intuitions were so different on this. Different models are helpful with different examples, and nobody in the thread mentioned the connection between intuitionistic logic and programming language semantics, which for this example I found very helpful.</p>
<p>In the prog... |
2,247,445 | <p>Find isomorphism between <span class="math-container">$\mathbb F_2[x]/(x^3+x+1)$</span> and <span class="math-container">$\mathbb F_2[x]/(x^3+x^2+1)$</span>.</p>
<hr />
<p>It is easy to construct an injection <span class="math-container">$f$</span> satisfying <span class="math-container">$f(a+b)=f(a)+f(b)$</span> an... | zahbaz | 176,922 | <p>$$75\sin^2\alpha + \frac{75}{4}\sin\alpha$$</p>
<p><a href="https://i.stack.imgur.com/6IdrJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6IdrJ.png" alt="Draw a right triangle with alpha in one corner."></a></p>
<p>\begin{align}
\frac{75}{10} + \frac{75\sqrt{10}}{4\cdot10}
\\
\\
\frac{15}{2} +... |
4,046,265 | <p>I'm evaluating a line integral of the function <span class="math-container">$T= x^2 + 4xy + 2yz^3$</span> from <span class="math-container">$a = (0,0,0)$</span> to <span class="math-container">$b=(1,1,1)$</span> on the path <span class="math-container">$z = x^2$</span>, and <span class="math-container">$y = x$</span... | Daàvid | 433,274 | <p>Well, when you defined the map <span class="math-container">$\overline{\phi}: TU \to \phi(U) \times \mathbb{R}^{n}$</span>, you didn't write it the right way, because it is given by <span class="math-container">$\overline{\phi}(p,v) = (\phi(p), D_p\phi(v))$</span> (you need to take into account the derivative of the... |
1,190,798 | <p>I'm having difficulty understanding when to use $\cos$ and $\sin$ to find $x$ and $y$ components of a vector.
Do we always use $\cos$ for $x$-component or what?</p>
| Floris | 101,979 | <p>It depends on your definition of the angle:</p>
<p><img src="https://i.stack.imgur.com/Sgu3w.png" alt="enter image description here"></p>
<p>In the picture as drawn, $x$ is $r\cos\alpha$ and $y$ is $r\sin\alpha$. But if I chose a different convention for $x$, $y$ or $\alpha$ I would need a different equation.</p>
|
172,157 | <p>Today I was asked if you can determine the divergence of $$\int_0^\infty \frac{e^x}{x}dx$$ using the limit comparison test.</p>
<p>I've tried things like $e^x$, $\frac{1}{x}$, I even tried changing bounds by picking $x=\ln u$, then $dx=\frac{1}{u}du$. Then the integral, with bounds changed becomes $\int_1^\infty \f... | Pedro | 23,350 | <p>You are presented with $$I=\int_0^\infty \frac{e^x}{x}dx$$</p>
<p>It is clear the function is bounded at any point inside $(0,\infty)$ so we're worried about the extrema of the interval. Split the integral at, say $1$, we have</p>
<p>$$I=\int_0^1 \frac{e^x}{x}dx+\int_1^\infty \frac{e^x}{x}dx$$</p>
<p>We need to a... |
172,157 | <p>Today I was asked if you can determine the divergence of $$\int_0^\infty \frac{e^x}{x}dx$$ using the limit comparison test.</p>
<p>I've tried things like $e^x$, $\frac{1}{x}$, I even tried changing bounds by picking $x=\ln u$, then $dx=\frac{1}{u}du$. Then the integral, with bounds changed becomes $\int_1^\infty \f... | ncmathsadist | 4,154 | <p>In that case, do this</p>
<p>You have $$e^x/x\sim {1\over x}$$
as $x\to 0$ and
$$\int _{0^+} {dx\over x} = +\infty.$$
Now you are done.</p>
|
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | Michael Hardy | 11,667 | <p>\begin{align}
\text{your answer} & = 10^x\cdot \ln10\cdot \log_{10} x+\frac{1}{x\cdot \ln10}\cdot 10^x \\[10pt]
& = 10^x \ln x + \frac{10^x}{x\ln 10} \\[10pt]
& = 10^x \left( \ln x + \frac 1 {x\ln 10} \right) = \text{Wolfram's answer}.
\end{align}
Note that where Wolfram writes $\log x$ or $\log 10$ with... |
1,563,004 | <p>Assume we that we calculate the expected value of some measurements $x=\dfrac {x_1 + x_2 + x_3 + x_4} 4$. what if we dont include $x_3$ and $x_4$, but instead we use $x_2$ as $x_3$ and $x_4$. Then We get the following expression $v=\dfrac {x_1 + x_2 + x_2 + x_2} 4$.</p>
<p>How do I know if $v$ is a unbiased estimat... | Jan Eerland | 226,665 | <p>$$\frac{\text{d}}{\text{d}x}\left(10^x\cdot\log_{10}(x)\right)=\frac{\text{d}}{\text{d}x}\left(\frac{10^x\ln(x)}{\ln(10)}\right)=$$
$$\frac{\frac{\text{d}}{\text{d}x}\left(10^x\ln(x)\right)}{\ln(10)}=\frac{\ln(x)\frac{\text{d}}{\text{d}x}(10^x)+10^x\frac{\text{d}}{\text{d}x}(\ln(x))}{\ln(10)}=$$
$$\frac{10^x\ln(10)\... |
440,452 | <blockquote>
<p>Let $b,c \in \mathbb{Z} $ and let $n \in \mathbb{N} $, $n \ge 2. $ Let $f(x) = x^{n} -bx+c$. Prove that $$\hbox{disc} (f(x)) = n^{n }c^{ n-1}-(n-1)^{n-1 }b^{n }.$$</p>
</blockquote>
<p>Here $\hbox{disc} (f(x)) = \prod_{i} f'(\alpha_{i} )$ where $\alpha_{1}, \dots, \alpha_{n}$ are the roots of $f(x)$.... | Community | -1 | <p>Here is a brute force approach:</p>
<p><span class="math-container">$f'(x) = nx^{n-1} - b$</span>, and we want to compute <span class="math-container">$\prod_i f'(\alpha_i)$</span>. We do this by looking for the minimal polynomial with roots <span class="math-container">$\alpha_i^{n-1}$</span>.</p>
<p>Note that</p>
... |
975 | <p>The usual <code>Partition[]</code> function is a very handy little thing:</p>
<pre><code>Partition[Range[12], 4]
{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Partition[Range[13], 4, 3]
{{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}}
</code></pre>
<p>One application I'm working on required me to wri... | Simon | 34 | <p>Here's my version using the <code>Sow</code> and <code>Reap</code> combination.</p>
<pre><code>multisegment::arglen =
"The argument `1` is not of the same length as the argument `2`.";
multisegment[lst_List, scts_List, offsets_List] :=
Module[{len = Length[lst], slen = Length[scts], i = 1, j = 1},
Reap[If... |
975 | <p>The usual <code>Partition[]</code> function is a very handy little thing:</p>
<pre><code>Partition[Range[12], 4]
{{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Partition[Range[13], 4, 3]
{{1, 2, 3, 4}, {4, 5, 6, 7}, {7, 8, 9, 10}, {10, 11, 12, 13}}
</code></pre>
<p>One application I'm working on required me to wri... | J. M.'s persistent exhaustion | 50 | <p>I've managed to slightly build on Mike's answer. There's a minimum (i.e., woefully incomplete) amount of checking done, but it should mostly work:</p>
<pre><code>multisegment[lst_List, scts:{__Integer?Positive}, offset:{__Integer?NonNegative}]:=
Module[{n = Length[lst], k, offs},
k = Ceiling[n/Mean[offset]];
... |
3,830,971 | <p>The function <span class="math-container">$f(x)=\cot^{-1} x$</span> is well known to be neither even nor odd because <span class="math-container">$\cot^{-1}(-x)=\pi-\cot^{-1} x$</span>. it's domain is <span class="math-container">$(-\infty, \infty)$</span> and range is <span class="math-container">$(0, \pi)$</span>.... | user | 505,767 | <p>We have that <span class="math-container">$\cot^{-1}(-x)$</span> is invertible only on suitable restrictions, in this case it seems Mathematica is considering the following definition</p>
<p><span class="math-container">$$f(x)=\cot^{-1}(x): \mathbb R \to \left(-\frac \pi 2, \frac \pi 2\right)$$</span></p>
<p>that is... |
1,430,369 | <p>I know how to prove this by induction but the text I'm following shows another way to prove it and I guess this way is used again in the future. I'm confused by it.</p>
<p>So the expression for first n numbers is:
$$\frac{n(n+1)}{2}$$</p>
<p>And this second proof starts out like this. It says since:</p>
<p>$$(n+1... | Dipole | 152,639 | <p>A simple way to "prove" the formula you give is to note that you can pair </p>
<p>$1$ with $n-1$</p>
<p>$2$ with $n-2$</p>
<p>$3$ with $n-3$</p>
<p>... and so on. Then you can see that the sum will be $(n+1)/2$ (the $+1$ coming from the fact that we basically pair $n$ with zero) of these pairs, each of which is ... |
157,985 | <p>Question from a beginner. I have data containing dates and values of the format:</p>
<pre><code> data = {{{2015, 1, 1}, 2}, {{2015, 1, 2}, 3}, {{2015, 2, 1}, 4}, {{2015, 2, 2}, 5}, {{2016, 1, 1}, 6}, {{2016, 1, 2}, 7}}
</code></pre>
<p>Aim is to multiply the values of each day in a month, e.g. for January 2015,... | Carl Woll | 45,431 | <p>I think this is a simpler variant of @Alan's answer:</p>
<pre><code>GroupBy[
data,
Most@*First -> Last,
Apply[Times]
]
</code></pre>
<blockquote>
<p><|{2015, 1} -> 6, {2015, 2} -> 20, {2016, 1} -> 42|></p>
</blockquote>
|
51,903 | <p>Does anyone know of a tool which</p>
<ol>
<li>Can display formulas neatly, preferably like this website without hassle. (Unlike wikipedia with <code>:<math></code>) </li>
<li>Has a wiki like structure: i.e categories of pages, individual articles with hyperlinked sections, subsections etc. </li>
<li>Preferrab... | 410 gone | 8,572 | <p>This site uses <a href="http://www.mathjax.org/" rel="nofollow noreferrer">MathJax</a>, which has pretty much become the web standard Latex tool, because it's so easy to use.</p>
<p>You can get a free account at the <a href="http://www.tumblr.com/" rel="nofollow noreferrer">Tumblr</a> blogging site, and <a href="ht... |
180,323 | <p>apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the extensions, say Split(K) and Split(L).</p>
<p>If Split(K) = Split(L), then is it necessarily true that K and L are isomorphic ... | Zavosh | 2,604 | <p>This result is due to Bauer and dated to 1916:</p>
<p>$\textbf{Theorem}:$ Let $K$ be an algebraic number field, $L/K$ and $M/K$ finite Galois extensions, and $\text{Spl}(L/K)$, $\text{Spl}(M/K)$ the set of prime ideals of $K$ which split completely in $L$ and $M$, respectively. Then $L \subseteq M$ if and only if $... |
3,837,856 | <p>In <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.29.6143&rep=rep1&type=pdf" rel="nofollow noreferrer">Fast Exact Multiplication by the Hessian</a> equation 1,</p>
<p><span class="math-container">$O(\left\Vert\Delta w\right\Vert^2)$</span> gets taken from RHS to LHS and <span class="math-c... | Steven Stadnicki | 785 | <p><span class="math-container">$O(f(x))$</span> refers to a quantity that's bounded (in the limit) by some multiple of <span class="math-container">$f(x)$</span>; it properly (IMHO) represents a <em>set</em>. That is, to say that <span class="math-container">$g(x)\in O(f(x))$</span> means that there's some constant C ... |
481,673 | <p>Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.</p>
<p>I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying to show if $g$ is not constant, then $g(1)=1$. I have $g(x+1)=(2-g(1))g(x)+g(1)$. So if $g(1)=1$, we can show i... | Mohsen Shahriari | 229,831 | <blockquote>
<p>This is my answer to the question <a href="https://math.stackexchange.com/questions/1221042/solving-functional-equation-fxy-fxfy-fxfyfxy-for-all-real-numb">solving functional equation <span class="math-container">$f(x+y)+f(x)f(y)=f(x)+f(y)+f(xy)$</span> for all real numbers</a>, which I recently found t... |
416,514 | <p>I really think I have no talents in topology. This is a part of a problem from <em>Topology</em> by Munkres:</p>
<blockquote>
<p>Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$. </p>
</blockquote>
<p>I always have the feeling that it is easy to understand the problem emotionally but hard to expr... | Luis Banegas Saybe | 420,416 | <p>Let $f$ : A $\longrightarrow$ $\mathbb{R}$ such that a $\mapsto$ d(x, a), where $\mathbb{R}$ is the topological space induced by the $<$ relation, the order topology.</p>
<p>For all open intervals (b, c) in $\mathbb{R}$, $f^{-1}((b, c))$ = {a $\in$ A $\vert$ d(x, a) $>$ b} $\cap$ {a $\in$ A $\vert$ d(x, a) $&... |
1,136,458 | <p>Suppose $G$ is an abelian group. Then the subset $H= \big\{ x\in G\ | \ x^3=1_G \big\}$ is a subgroup of $G$.</p>
<p>I was able to show the statement is correct but the weird thing is that I didn't use the fact $G$ is abelian. <strong>Is it possible that the fact $G$ is abelian is redundant?</strong></p>
<p>Here ... | 5xum | 112,884 | <p>It is probably pa precision thing. Computers do not have infinite space and can therefore represent only a finite set of numbers exactly. Any operation you do on the numbers which results in a number that the computer cannot represent will result in rounding. </p>
<p>This means that there exists some <em>smallest n... |
2,418,181 | <p><strong>Question:</strong>
Let $P_1,\ldots,P_n$ be propositional variables. When is the statement $P_1 \oplus \cdots \oplus P_n$ true?</p>
<p>I'm currently learning the basics of discrete math. I am stuck on this last question of my assignment... not really sure how to go about solving it.</p>
<p>I do know that a ... | paw88789 | 147,810 | <p>Hint: $x \bigoplus {\rm False}$ has the same truth value as $x$.</p>
<p>$x \bigoplus {\rm True}$ has the opposite truth value as $x$.</p>
<p>So every time you have a True among the arguments, the truth value flips.</p>
|
747,997 | <p>in <a href="https://math.stackexchange.com/questions/239521/why-no-trace-operator-in-l2">Why no trace operator in $L^2$?</a>
it is mentioned, that there exists a linear continuous trace operator from $L^2(\Omega)$ to $H^\frac12(\partial\Omega)$* for sufficiently smooth boundary. Can you give me any reference for thi... | bastienchaudet | 390,718 | <p>@Michael: I think that this result is not true, even when the boundary is smooth. I've been looking for it (for a while now...) in the literature, but unfortunately I haven't been able to find anything..! It seems that the continuity of the trace operator $\gamma:H^s(\Omega)\rightarrow H^{s-\frac{1}{2}}(\partial\Ome... |
2,038,498 | <p>How do you prove this formula
$$\lim_{x \to\infty}\frac{x^{x^2}}{2^{2^x}}$$</p>
<p>Since both top and bottom approaches infinity, I assume it is L'Hospital's rule to solve it, but after the first step I'm stuck
$$\lim_{x \to\infty}\frac{x^2 logx}{2^xlog2}$$
So how can I solve this problem, it seems the answ... | manthanomen | 67,750 | <p>The relation says $(x, y, t) \sim (x', y', t')$ if and only if either 1) $y = y'$ and $t = t' = 0$ or 2) $x = x'$ and $t = t' = 1$</p>
|
1,866,801 | <p>Let $A$ be an infinite set, $B\subseteq A$ and $a\in B$. Let $X\subseteq \mathcal{P}(A)$ be an infinite family of subsets of $A$ such that $a\in \bigcap X$.</p>
<p>Suppose $\bigcap X\subseteq B$. Is it possible that, for every non-empty finite subfamily $Y\subset X$, $\bigcap Y \not\subseteq B$ ? </p>
<p>Thanks fo... | fleablood | 280,126 | <p>This is probably improper vocabulary but I think this concepts are fundamental.</p>
<p>Your "basic" angle is $0 \le \theta \le \pi/2$. All other angles are "essentially" equivalent "upto reflection on one or two axes". (Or up to periods of multiples of $2\pi$.)</p>
<p>Example. $\theta$ and $\pi - \theta$ are "e... |
3,885,025 | <p>Let be <span class="math-container">$(A,M)$</span> a local ring (if it needs noetherian ring), <span class="math-container">$P_1$</span> and <span class="math-container">$P_2$</span> two prime ideals with <span class="math-container">$P_1\subseteq P_2$</span>, <span class="math-container">$\ a_1\in M\setminus P_2$</... | Atticus Stonestrom | 663,661 | <p>I assume <span class="math-container">$A$</span> is commutative since you have commutative algebra as a tag, and I do use a hypothesis that <span class="math-container">$A$</span> is noetherian. I am not sure whether the result holds in more general settings.</p>
<hr />
<p>Lemma: if <span class="math-container">$R$<... |
3,097,640 | <p>So I have the following problem: $a + b = c + c.
I want to prove that the equation has infinitely many relatively prime integer solutions.</p>
<p>What I did first was factor the right side to get:
(</p>
| Sam | 640,956 | <p>Above equation shown below has parametric solution:</p>
<p><span class="math-container">$(a^2+b^2)=c(c^4+1)$</span></p>
<p><span class="math-container">$a=u^5+uv^4+2u^3v^2+v$</span></p>
<p><span class="math-container">$b=v^5+u^4v+2u^2v^3-u$</span></p>
<p><span class="math-container">$c=u^2+v^2$</span></p>
<p>Fo... |
2,219,266 | <blockquote>
<p>Compute integral $\displaystyle \int_{-\infty}^{\infty}e^{-k^2t+ikx}\, dk$.</p>
</blockquote>
<p>Hint: Complete the square in the Exponent.</p>
<p>Okay, for the Exponent, we have
$$
-k^2t+ikx=-t\cdot\left(k-\frac{ix}{2t}\right)^2-\frac{x^2}{4t}.
$$</p>
<p>Now, is it easier to compute
$$
\int_{-\inf... | Connor Harris | 102,456 | <p>There's a $3/7$ chance of one male child, a $3/7$ chance of two, and a $1/7$ chance of three, so the total probability is $\left(\dfrac{1}{3} \times \dfrac{3}{7}\right) + \left(\dfrac{2}{3} \times \dfrac{3}{7}\right) + \left( \dfrac{3}{3} \times \dfrac{1}{7} \right) = \dfrac{4}{7}.$ Alternatively, if you want Bayes'... |
2,219,266 | <blockquote>
<p>Compute integral $\displaystyle \int_{-\infty}^{\infty}e^{-k^2t+ikx}\, dk$.</p>
</blockquote>
<p>Hint: Complete the square in the Exponent.</p>
<p>Okay, for the Exponent, we have
$$
-k^2t+ikx=-t\cdot\left(k-\frac{ix}{2t}\right)^2-\frac{x^2}{4t}.
$$</p>
<p>Now, is it easier to compute
$$
\int_{-\inf... | drhab | 75,923 | <p>Alternatively you can think of 8 triplets (BBB,BBG, et cetera). </p>
<p>Symmetrically there are $12$ boys and $12$ girls. </p>
<p>Now the triple GGG is taken away so that $12$ boys and $9$ girls remain all having equal probability to be met on the street. </p>
<p>That gives probability $\frac{12}{12+9}=\frac47$ t... |
903,117 | <p>I am trying to evaluate
$$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$
Would a contour work? I have tried using a contour but had no success.
Thanks.</p>
<p>Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anywa... | Mhenni Benghorbal | 35,472 | <p>Here is another approach. Write the integral as</p>
<blockquote>
<p>$$I = {2}\int_{0}^{\infty} \frac{\sin^2(x)}{x^2}. $$</p>
</blockquote>
<p>Recalling the Mellin transform of a function $f$</p>
<blockquote>
<p>$$ \int_{0}^{\infty} x^{s-1} f(x)dx $$</p>
</blockquote>
<p>our integral is the Mellin transfor... |
2,617,467 | <p>I have been trying to solve this limit for more than an hour and I'm stuck.</p>
<blockquote>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}$$</p>
</blockquote>
<p>What I've so far is:</p>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}\\ \lim_{n\to\infty} \frac{3^{n}}{n!+2^{(n+1)}}+\lim... | Arthur | 15,500 | <p>Hint: Compare your original fraction to $\frac{2\cdot 3^n}{n!}$, which is a lot easier to work with.</p>
|
2,617,467 | <p>I have been trying to solve this limit for more than an hour and I'm stuck.</p>
<blockquote>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}$$</p>
</blockquote>
<p>What I've so far is:</p>
<p>$$ \lim_{n\to\infty} \frac{3^{n}+\sqrt{n}}{n!+2^{(n+1)}}\\ \lim_{n\to\infty} \frac{3^{n}}{n!+2^{(n+1)}}+\lim... | user | 505,767 | <p>You can divide the limit in two pieces but can't take the limit only for a single part as you have done here</p>
<p>$$\color{red}{\lim_{n\to\infty} \frac{3^{n}}{1+\frac{2^{(n+1)}}{n!}}*0+\lim_{n\to\infty} \frac{\sqrt{n}}{1+\frac{2^{(n+1)}}{n!}}*0}$$</p>
<p>The limit can be easily handled as follow</p>
<p>$$\frac{... |
3,238,054 | <blockquote>
<p>Find parameter <span class="math-container">$a$</span> for which <span class="math-container">$$\frac{ax^2+3x-4}{a+3x-4x^2}$$</span> takes all real values for <span class="math-container">$x \in \mathbb{R}$</span></p>
</blockquote>
<p>I have equated the function to a real value, say, k
which gets me... | Lozenges | 219,277 | <p>Find <span class="math-container">$a$</span> so that the equation </p>
<p><span class="math-container">$$y= \frac{a x^2+3x -4}{-4x^2+3x+a}$$</span></p>
<p>has a root which is in the domain of the function</p>
<p>The discriminant must be <span class="math-container">$\geq 0$</span> for all <span class="math-contai... |
3,745,097 | <p>In my general topology textbook there is the following exercise:</p>
<blockquote>
<p>If <span class="math-container">$F$</span> is a non-empty countable subset of <span class="math-container">$\mathbb R$</span>, prove that <span class="math-container">$F$</span> is not an open set, but that <span class="math-contain... | D. Brogan | 404,162 | <p>The problem here is that you are supposing that you can write <span class="math-container">$F=\{f_1,f_2,\ldots\}$</span> where the <span class="math-container">$f_i$</span>'s are in increasing order in <span class="math-container">$\mathbb{R}.$</span> This isn't true, for example consider <span class="math-containe... |
757,656 | <p>Find the inverse of the function $f(x)= \dfrac{2x-1}{x^2-1}.$</p>
<p>We switch the $x$ and $y$ letters and then solve the the equation, but it became kind of complicated while solving.</p>
| kingW3 | 130,953 | <p>Just expanding on the Mitsos answer
$$t(x^2-1)=2x-1\\tx^2-2x-t+1=0\\x_{1,2}=\frac{2\pm\sqrt{4-4t(1-t)}}{2t}$$</p>
|
757,656 | <p>Find the inverse of the function $f(x)= \dfrac{2x-1}{x^2-1}.$</p>
<p>We switch the $x$ and $y$ letters and then solve the the equation, but it became kind of complicated while solving.</p>
| Tunk-Fey | 123,277 | <p>You have
$$
y= \frac{2x-1}{x^2-1}.
$$
As you mentioned, switch $x$ and $y$. It becomes
$$
x= \frac{2y-1}{y^2-1}.
$$
Now, do the following steps:
$$
\begin{align}
x(y^2-1)&=2y-1\\
xy^2-x&=2y-1\\
xy^2-2y&=x-1.\tag1
\end{align}
$$
Now, multiply botth sides of $(1)$ by $x$ so that it's easy to use complete s... |
827,072 | <p>How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$
using the formula
$(x-q)^2 + (y-p)^2 = r^2$.</p>
<p>I know i need to use that formula but have no idea how to start, I have tried to start but don't think my answer is right.</p>
| Donn S. Miller | 155,974 | <p>$\begin{vmatrix}
x^2+y^2&x&y&1\\
5^2+10^2&5&10&1\\
(-5)^2+0^2&-5&0&1\\
9^2+(-6)^2&9&-6&1\\
\end{vmatrix}=
\begin{vmatrix}
x^2+y^2&x&y&1\\
125&5&10&1\\
25&-5&0&1\\
117&9&-6&1\\
\end{vmatrix}
=
0$</p>
|
198,612 | <p>Let $(P,\leq)$ be a partially ordered set. A <em>down-set</em> is a set $d\subseteq P$ such that $x\in d$ and $x'\in P, x'\leq x$ imply $x'\in d$. If the down-set is totally ordered, we say it is a totally ordered down-set (tods).</p>
<p>Let $d_1, d_2$ be tods. We say that they are <em>incompatible</em> if neither ... | Dominic van der Zypen | 8,628 | <p>The answer is No.</p>
<p>We define two sets of elements of $\{0,1,2\}^\omega$ in the following way:</p>
<ol>
<li>For $n\in\omega$ let $u_n$ be defined by $u_n(k)=1$ for $k\leq n$
and $u_n(k)=0$ for $k>n$;</li>
<li>For $n\in\omega$ let $t_n$ be defined by $t_n(k)=1$ for $k\leq n$
and $t_n(n+1) = 2$ and $t_n(k)... |
1,981,553 | <p>When a person asks: "What is the smallest number (natural numbers) with two digits?"</p>
<p>You answer: "10".</p>
<p>But by which convention 04 is no valid 2 digit number?</p>
<p>Thanks alot in advance</p>
| hmakholm left over Monica | 14,366 | <p>The question asks for a <em>number</em>, not a <em>representation</em> of a number.</p>
<p>The digit sequence <code>04</code> is a representation of the number $4$ (also known as $1+1+1+1$), but <code>4</code> is <em>also</em> a representation of that same number.</p>
<p>Every natural number has many such represen... |
291,284 | <p>Let $(\Omega, \cal{A}, \mathbb{P})$ be a probability space and $X$ a random variable on $\Omega$. Let, also, $f:\Omega\to\mathbb{R}$ be a Borel function. Then:<br>
$X$ and $f(X)$ are independent $\Longleftrightarrow$ there exists some $t\in\mathbb{R}$ such that $\mathbb{P}[f(X)=t]=1$, that is $f(X)$ is a degenerate ... | Ilya | 5,887 | <p>There is another proof of this fact, which follows immediately from the fact that you proved: $\Bbb P[f(x)\in B]\in \{0,1\}$ and reminds the Nested Intervals Theorem. For the shorthand let
$$
\mu(B):=\Bbb P[f(X)\in B]
$$
denote the distribution of $f(X)$. </p>
<p>So we have $\mu \in \{0,1\}$ and $\mu(\Bbb R) = 1... |
3,995,913 | <p>We are looking at the following expression:</p>
<p><span class="math-container">$$\frac{d}{dx}\int_{u(x)}^{v(x)}f(x) dx$$</span></p>
<p>The solution is straightforward for this: <span class="math-container">$\frac{d}{dx}\int_{u(x)}^{v(x)}f(t) dt$</span>. Do we evaluate the given expression in like manner? Do we trea... | Community | -1 | <p>This expression is badly written. It only makes sense to understand it as
<span class="math-container">$$
\frac{d}{dx}\int_{u(x)}^{v(x)}f(t)dt
$$</span>
where the dummy variable for the definite integral should <em>not</em> be the same as any variable in the bounds.</p>
<p>The general method for dealing with such de... |
2,671,384 | <p>I want to solve following difference equation:</p>
<p>$a_i = \frac13a_{i+1} + \frac23a_{i-1}$, where $a_0=0$ and $a_{i+2} = 1$</p>
<p><strong>My approach:</strong>
Substituting $i=1$ in the equation,<br>
$3a_1 = a_2 + 2a_0$ <br>
$a_2 = 3a_1$<br>
Similarly substituting $i = 2, 3 ...$<br>
$a_3 = 7a_1$<br>
$a_4 = 15a... | user | 505,767 | <p>The standard method to solve <a href="https://brilliant.org/wiki/linear-recurrence-relations/" rel="nofollow noreferrer">recurrence equations</a> is to set as trial solution</p>
<p>$$a_i=x^i \implies x^i = \frac13x^{i+1} + \frac23 x^{i-1}\implies x =\frac13 x^2+\frac23\implies x^2-3x+2=0$$</p>
<p>then find $x_1$ a... |
884,342 | <p>$N$ is a normal subgroup of $G$ if $aNa^{-1}$ is a subset of $N$ for all elements $a $ contained in $G$. Assume, $aNa^{-1} = \{ana^{-1}|n \in N\}$.</p>
<p>Prove that in that case $aNa^{-1}= N.$</p>
<p>If $x$ is in $N$ and $N$ is a normal subgroup of $G$, for any element $g$ in $G$, $gxg^{-1}$ is in $G$. Suppose $... | pre-kidney | 34,662 | <p>Here's another way to see where the result comes from. If $N$ is normal,
$$\begin{align*}
aNa^{-1}&\subset N\\
aN&\subset Na.
\end{align*}$$
But the definition is symmetric in $a$! Swapping the roles of $a$ and $a^{-1}$, we also get $Na\subset aN$. Thus $aN=Na$, which gives you $aNa^{-1}=N$.</p>
<p>Philosop... |
1,686,012 | <p>Let $T$: $\mathbb{R}^3 \to \mathbb{R}$ be a linear transformation.</p>
<p>Show that either $T$ is surjective, or $T$ is the zero linear transformation.</p>
<p>My approach:</p>
<p>First we start off by supposing T is not surjective and we want to show that $\forall \overrightarrow v \in \mathbb{R}^3, T(\overrighta... | user296113 | 296,113 | <p>A simple approach is to say that the image of $T$ is a subspace of $\Bbb R$ so it would be $\Bbb R$ or $\{0\}$. The former case is when $T$ is surjective and the last is when $T=0$.</p>
|
2,753,504 | <p>Where $a,b$ and $c$ are positive real numbers.</p>
<p>So far I have shown that $$a^2+b^2+c^2 \ge ab+bc+ac$$ and that $$a^2+b^2+c^2 \ge a\sqrt{bc} + b\sqrt{ac} + c\sqrt{ab}$$ but I am at a loss what to do next... I have tried adding various forms of the two inequalities but always end up with something extra on the ... | DeepSea | 101,504 | <p>Another answer is using the "friendly" Cauchy-Schwarz inequality:</p>
<p>$a\sqrt{bc}+b\sqrt{ca}+ c\sqrt{ab}= \sqrt{ab}\sqrt{ac}+\sqrt{bc}\sqrt{ab}+\sqrt{ac}\sqrt{bc}\le \sqrt{(\sqrt{ab})^2+(\sqrt{bc})^2+(\sqrt{ac})^2}\times\sqrt{(\sqrt{ac})^2+(\sqrt{ab})^2+(\sqrt{bc})^2}= \sqrt{ab+bc+ca}\times\sqrt{ab+ac+bc}= \sqrt... |
47,724 | <p>I am looking at a past exam written by a student. There was a question I believed he got correct but received only 1/4. The marker wrote down "4 more compositions, order matters".</p>
<p>This is the problem:</p>
<p>List all 3 part compositions of 5. (recall that compositions have no zeros)</p>
<p>$(1, 1, 3)
(1, ... | Brian M. Scott | 12,042 | <p>What the student has is still wrong, assuming that you didn't typo it in writing up the question: $(3,1,1)$ has been duplicated, and $(1,2,2)$ has been omitted. Your understanding is correct, and my guess is the same as yours.</p>
|
101,157 | <p>Let $G$ be an algebraic group defined over a char 0
local field $k$. Following Borel and Tits (73) we define
the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$
generated by the unipotent elements of $G(k)$. </p>
<p>Suppose $G$ is generated by a finite set of unipotent
$k$-subgroups, say $U_1,\cdots, U_n$. Is it... | Lee Mosher | 20,787 | <p>The conjecture is not known for any other surfaces.</p>
<p>I'll throw out that there are other known recipes for constructing all pseudo-Anosov mapping classes using train track representatives; see for example my paper "The classification of pseudo-Anosovs", and the Bestvina-Handel paper "Train-tracks for surface ... |
101,157 | <p>Let $G$ be an algebraic group defined over a char 0
local field $k$. Following Borel and Tits (73) we define
the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$
generated by the unipotent elements of $G(k)$. </p>
<p>Suppose $G$ is generated by a finite set of unipotent
$k$-subgroups, say $U_1,\cdots, U_n$. Is it... | Autumn Kent | 1,335 | <p>Shin and Strenner have shown that the conjecture is false when 3g + n > 4. </p>
<p>See <a href="http://arxiv.org/abs/1410.6974">http://arxiv.org/abs/1410.6974</a></p>
|
1,617,372 | <blockquote>
<p>The number of policies that an agent sells has a Poisson distribution
with modes at $2$ and $3$. $K$ is the smallest number such that the
probability of selling more than $K$ policies is less than 25%.
Calculate K.</p>
</blockquote>
<p>I know that the parameter lambda is $3$, of the Poisson dis... | heropup | 118,193 | <p>Recall that the probability mass function of a Poisson-distributed random variable $X$ is $$\Pr[X = x] = e^{-\lambda} \frac{\lambda^x}{x!}, \quad x = 0, 1, 2, \ldots.$$ If the mode is at $X = 2$ and $X = 3$, this means $$\Pr[X = 2] = \Pr[X = 3],$$ or $$e^{-\lambda} \frac{\lambda^2}{2!} = e^{-\lambda} \frac{\lambda^... |
1,617,372 | <blockquote>
<p>The number of policies that an agent sells has a Poisson distribution
with modes at $2$ and $3$. $K$ is the smallest number such that the
probability of selling more than $K$ policies is less than 25%.
Calculate K.</p>
</blockquote>
<p>I know that the parameter lambda is $3$, of the Poisson dis... | ELO Petro | 573,265 | <p>Then that's completely wrong. If Pr[X>K]<0.25, and K = 4. The equation becomes Pr[X>4]. Which is the probability that X is greater than or equal to 5. However, if we let K=3, then Pr[X>3] is the probability that X is greater than or equal to 4. K=3 is the smallest number at which "selling more than K policies (ak... |
1,522,062 | <p>Identify for which values of $x$ there is subtraction of nearly equal numbers, and find an alternate form that avoids the problem:
$$E = \frac{1}{1+x} - \frac{1}{1-x} = -\frac{2x}{1-x^2} = \frac{2x}{x^2-1} $$
How come $-2x/(1-x^2)$ can be changed to $2x(x^2 - 1)$ according to the homework solutions? Why does the den... | Ian Miller | 278,461 | <p>$\require{cancel}$
$$\frac{-2x}{1-x^2}=\frac{-1\times2x}{-1\times(-1+x^2)}$$
$$=\frac{\cancel{-1\times}2x}{\cancel{-1\times}(-1+x^2)}$$
$$=\frac{2x}{x^2-1}$$</p>
|
2,158,981 | <p>How should i solve : $$\sum_{r=1}^n (2r-1)\cos(2r-1)\theta $$</p>
<p>I can solve $\sum_{r=1}^n cos(2r-1)\theta $ by considering
$\Re \sum_{r=1}^n z^{2r-1} $ and using summation of geometric series, but I can't seem to find a common geometric ratio when $ 2r-1 $ is involved in the summation.</p>
<p>Visually : $\su... | NickD | 416,685 | <p>Here's a hint: $$ {d \over dz} z^n = n z^{n-1}$$. What do you get if you differentiate the geometric series term by term?</p>
|
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| achille hui | 59,379 | <p>Apply <a href="https://en.wikipedia.org/wiki/General_Leibniz_rule" rel="nofollow noreferrer">General Leibniz rule</a></p>
<p>$$(fg)^{(n)}(x) = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x) g^{(k)}(x)$$</p>
<p>to $f(x) = (x+1)^n$ and $g(x) = (x-1)^n$. Notice for $k \le n$, we have
$$g^{(k)}(x) = \frac{n!}{(n-k)!} (x-1)^{... |
2,812,472 | <p>I am trying to show that</p>
<p>$$
\frac{d^n}{dx^n} (x^2-1)^n = 2^n \cdot n!,
$$ for $x = 1$. I tried to prove it by induction but I failed because I lack axioms and rules for this type of derivatives. </p>
<p>Can someone give me a hint?</p>
| Stefan4024 | 67,746 | <p>You can use Taylor's formula and try to expland $x^2-1$ around $x=1$ and the coefficient ahead of $(x-1)^n$ would be $\frac{d^n}{dx^n} (x^2-1)^n{\Big|_{x=1}} \cdot \frac{1}{n!}$</p>
<p>We have $x^2 - 1 = (x-1)((x-1)+2)$ and so $(x^2-1)^n = (x-1)^n((x-1)^n+2)^n$. From here it's not hard to conclude that the coeffic... |
128,784 | <p>Consider the two lists</p>
<pre><code>list1={1,2,a[1],8,b[4],9};
list2={8,b[4],9,1,2,a[1]};
</code></pre>
<p>it is evident by inspection that <code>list2</code> is just a cyclic rotation of <code>list1</code>. Considering an equivalence class of lists under cyclic rotations, I would like to have a function <code>c... | bill s | 1,783 | <p>Here is a function</p>
<pre><code>cyc[list_] := RotateLeft[list, First@Ordering[list, 1]]
</code></pre>
<p>For your lists:</p>
<pre><code>list1 = {1, 2, a[1], 8, b[4], 9};
list2 = {8, b[4], 9, 1, 2, a[1]};
cyc[list1] == cyc[list2]
True
</code></pre>
|
1,060,213 | <p>I have searched the site quickly and have not come across this exact problem. I have noticed that a Pythagorean triple <code>(a,b,c)</code> where <code>c</code> is the hypotenuse and <code>a</code> is prime, is always of the form <code>(a,b,b+1)</code>: The hypotenuse is one more than the non-prime side. Why is this... | Will Jagy | 10,400 | <p>The only possibility is, with positive integers $r > s,$
$$ a = r^2 - s^2, $$
$$ b = 2rs, $$
$$ c = r^2 + s^2. $$</p>
<p>In order to have $a= (r-s)(r+s)$ prime, we must have $(r-s) = 1,$ or
$$ r=s+1. $$ So, in fact, we have
$$ a = 2s+1, $$
$$ b = 2s^2 + 2 s, $$
$$ c = 2s^2 + 2 s + 1. $$</p>
<p>The... |
1,060,213 | <p>I have searched the site quickly and have not come across this exact problem. I have noticed that a Pythagorean triple <code>(a,b,c)</code> where <code>c</code> is the hypotenuse and <code>a</code> is prime, is always of the form <code>(a,b,b+1)</code>: The hypotenuse is one more than the non-prime side. Why is this... | iadvd | 189,215 | <p>After reading the great answers, just wanted to add one easy-to-follow rule that makes possible to build a Pythagorean triple $(a,b,b+1$) starting with any odd number $a \gt 3$, as follows:</p>
<p>$a$ is the original odd number</p>
<p>$b=\frac{a^2-1}{2}$</p>
<p>$c=b+1$</p>
<p>As that works for any odd number gre... |
47,492 | <p>Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:</p>
<p>1) it's not isomorphic to its dual (in which case we call it 'complex')</p>
<p>2) it has a nondegenerate symmetric bilinear form (in which case ... | Torsten Ekedahl | 4,008 | <p>An irreducible representation is real or quaternionic precisely when its
character is real-valued. By the Peter-Weyl theorem all characters are
real-valued precisely when every element in the group is conjugate to its
inverse. When the group is connected a more precise answer is as follows: The
Weyl group (in its ta... |
47,492 | <p>Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:</p>
<p>1) it's not isomorphic to its dual (in which case we call it 'complex')</p>
<p>2) it has a nondegenerate symmetric bilinear form (in which case ... | Skip | 6,486 | <p>Torsten answered this question perfectly for the definition of real/complex/quaternionic in John's original question. But this usage of real/complex/quaternionic is foreign to my experience. Specifically, if you look at an irreducible real representation of a group, then its endomorphism ring is (by Schur and Frobe... |
1,689,853 | <p>If I have that $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion? I know that $B_t^2 - t$ is but can't see it for the latter. </p>
| Math-fun | 195,344 | <p>\begin{align}
E(B_t^2-\frac t2 \Big|B_s)&=E((B_t-B_s+B_s)^2-\frac t2 \Big|B_s)\\
&=E((B_t-B_s)^2+B_s^2+2B_s(B_t-B_s)-\frac t2 \Big|B_s)\\
&=t-s+B_s^2-0-\frac t2 \\
&=B_s^2-\frac s2+\frac {t-s}2 \\
& \geq B_s^2-\frac s2
\end{align}</p>
|
3,688,151 | <blockquote>
<p>A necklace is made up of <span class="math-container">$3$</span> beads of one sort and <span class="math-container">$6n$</span> of another, those of each sort being similar. Show that the number of arrangement of the beads is <span class="math-container">$3n^2+3n +1$</span>.</p>
</blockquote>
<p><str... | Mike Earnest | 177,399 | <p>Let us say there are <span class="math-container">$3$</span> white beads and <span class="math-container">$6n$</span> black beads. A necklace is determined by the lengths of the three sections of black beads comprising the necklace. Call these lenghts <span class="math-container">$a,b$</span> and <span class="math-c... |
3,188,327 | <p>I derived the operator <span class="math-container">$\mathscr{L} = \dfrac{\partial}{\partial{x}} + u\dfrac{\partial}{\partial{y}}$</span> from the PDE <span class="math-container">$u_x + uu_y = 0$</span> in order to figure out whether it is linear. </p>
<p>The textbook solutions take the following steps in finding ... | Arctic Char | 629,362 | <p>Let me just say that </p>
<p><span class="math-container">$$\mathcal L = \frac{\partial}{\partial x} + u\frac{\partial }{\partial y}$$</span></p>
<p>is NOT the correct notation. The above implies that <span class="math-container">$u$</span> is a fixed function, and the operator acts as </p>
<p><span class="math-... |
2,063,038 | <p>Let <span class="math-container">$S$</span> be the region in the plane that is inside the circle <span class="math-container">$(x-1)^2 + y^2 = 1$</span> and outside the circle <span class="math-container">$x^2 + y^2 = 1 $</span>. I want to calculate the area of <span class="math-container">$S$</span>.</p>
<h3>Try:</... | Michael R | 399,606 | <p>I would recommend using a polar coordinate system, i.e.</p>
<p>$$ x = rcos\theta $$ </p>
<p>$$ y=rsin\theta $$</p>
<p>which implies: $ \rightarrow x^2 + y^2 = r^2sin^2\theta + r^2cos^2\theta = r^2(sin\theta + cos\theta) = r^2 $.</p>
<p>To use this method, you must find the intersection points (as you have alread... |
1,170,708 | <p>What functions satisfy $f(x)+f(x+1)=x$?</p>
<p>I tried but I do not know if my answer is correct.
$f(x)=y$</p>
<p>$y+f(x+1)=x$</p>
<p>$f(x+1)=x-y$</p>
<p>$f(x)=x-1-y$</p>
<p>$2y=x-1$</p>
<p>$f(x)=(x-1)/2$</p>
| kryomaxim | 212,743 | <p>Concluding that from $f(x+1)=x-y$ it follows $f(x)=x-1-y$ is wrong, because the variable $x$ was not changed in the Argument $y$. </p>
<p>Try the Ansatz $f(x)=ax+b$ for some coefficients $a,b$ to solve this equation.</p>
|
1,734,819 | <p>I think I'm on the right track with constructing this proof. Please let me know.</p>
<p>Claim: Prove that there exists a unique real number $x$ between $0$ and $1$ such that
$x^{3}+x^{2} -1=0$</p>
<p>Using the intermediate value theorem we get
$$r^{3}+r^{2}-1=c^{3}+c^{2}-1$$
......
$$r^{3}+r^{2}-c^{3}-c^{2}=0$$</... | Vik78 | 304,290 | <p>You want an $x$ in $(0, 1)$ such that $x^2 + x = 1/x$. Note that as $x$ goes to zero $x^2 + x < 1/x,$ and as $x$ goes to one $x^2 + x > 1/x$. Since both curves are continuous they must intersect at least once on $(0, 1),$ and since $x^2 + x$ is increasing on that interval whereas $1/x$ is decreasing they inter... |
1,786,421 | <p>I have the following equality to prove. </p>
<p>Given $X \sim Bin(n, p)$ and $Y \sim Bin(n, 1 - p)$ prove that $P(X \leq k) = P(Y \geq n - k)$. I have been trying to come up with a solution but cannot find one. I am looking for suggestions and not a complete answer as this is a homework question.</p>
<p>What I did... | peterwhy | 89,922 | <p>$$\sum_{i=0}^k\binom{n}{i}p^i(1-p)^{n-i} = \sum_{j=n-k}^{n}\binom{n}{n-j}(1-1+p)^{n-j}(1-p)^j$$
where $j=n-i$, or $i=n-j$.</p>
|
174,075 | <p>What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?</p>
| S Jagdish | 477,526 | <p>Even I agree to the above contexts. Normal is more relevant to 3-D(Line and Surface) and perpendicular to 2-D(Line and Line = One Plane). </p>
<p>This is correct when we do say normal to surface and perpendicular to the line.</p>
|
1,307,280 | <p>I m in the point X. I m 2 blocks up from a point A and 3 blocks down from my home H. Every time I walk one block i drop a coin.</p>
<p>H
.
.
.
X
.
.
A</p>
<p>If the coin is face I go one block up and if it is not face I go one block down.</p>
<p>Which is the probability of arriving home before the point A?</p>
... | Community | -1 | <p>Probably a better intuitive definition is $f(x)$ can be made arbitrarily close to $L$ by making $x$ close enough to $a$.</p>
<p>You avoid the awkwardness about the constant case. Additionally this emphasizes that it is for ALL $\epsilon$. It's not just that $f(x)$ gets "closer", it's that it can be made as close as... |
1,307,280 | <p>I m in the point X. I m 2 blocks up from a point A and 3 blocks down from my home H. Every time I walk one block i drop a coin.</p>
<p>H
.
.
.
X
.
.
A</p>
<p>If the coin is face I go one block up and if it is not face I go one block down.</p>
<p>Which is the probability of arriving home before the point A?</p>
... | CiaPan | 152,299 | <p>The 'working definition' is actually NOT working and you shoud NOT use it. To make it work replace it with </p>
<blockquote>
<p>$f$ gets <strong>arbitrarily</strong> close to $L$ if $x$ <em>sufficiently</em> approaches $a$</p>
</blockquote>
<p>where 'gets arbitrarily close' does not mean just $f$ <em>may</em> g... |
1,906,013 | <blockquote>
<p>Let $f$ be a smooth function such that $f'(0) = f''(0) = 1$. Let $g(x) = f(x^{10})$. Find $g^{(10)}(x)$ and $g^{(11)}(x)$ when $x=0$.</p>
</blockquote>
<p>I tried applying chain rule multiple times:</p>
<p>$$g'(x) = f'(x^{10})(10x^9)$$</p>
<p>$$g''(x) = \color{red}{f'(x^{10})(90x^8)}+\color{blue}{(... | Domates | 8,065 | <p>\begin{align*}
g'(x) &= 10x^9f'(x^{10})\\
g''(x) &= 10x^9f''(x^{10})+10.9x^8f'(x^{10})\\
& \vdots\\
g^{(9)}(x)&= p(x)+10.9.\cdots .3x^2f''(x^{10})+10.9.\cdots .2x^1f'(x^{10})\\
g^{(10)}(x)&= q(x)+10.9.\cdots .2x^1f''(x^{10})+10!x^0f'(x^{10})\\
g^{(11)}(x)&= r(x)+10!f''(x^{10})
\end{align*}
wh... |
1,494,167 | <p>Using only addition, subtraction, multiplication, division, and "remainder" (modulo), can the absolute value of any integer be calculated?</p>
<p>To be explicit, I am hoping to find a method that does not involve a piecewise function (i.e. branching, <code>if</code>, if you will.)</p>
| Lasoloz | 283,199 | <p>I think you need this: <a href="https://stackoverflow.com/questions/9772348/get-absolute-value-without-using-abs-function-nor-if-statement">https://stackoverflow.com/questions/9772348/get-absolute-value-without-using-abs-function-nor-if-statement</a>.</p>
<p>Please note that remainder is modulo, absolute is modulus... |
3,605,368 | <p>Imagine a <span class="math-container">$9 \times 9$</span> square array of pigeonholes, with one pigeon in each pigeonhole. Suppose that all at once, all the pigeons move up, down, left, or right by one hole. (The pigeons on the edges are not allowed to move out of the array.) Show that some pigeonhole winds up with... | David G. Stork | 210,401 | <p>Hint: This figure suggests why you cannot do it with <span class="math-container">$9 \times 9$</span> but (modified) shows you <em>can</em> with <span class="math-container">$8 \times 8$</span>.</p>
<p><a href="https://i.stack.imgur.com/uF0jT.png" rel="noreferrer"><img src="https://i.stack.imgur.com/uF0jT.png" alt... |
38,915 | <p>It has to be the silliest question, but it’s not clear to me how to calculate eigenvectors quickly. I am just talking about a very simple 2-by-2 matrix.</p>
<p>When I have already calculated the eigenvalues from a characteristic polynomial, I can start to solve the equations with $A\mathbf{v}_1 = e_1\mathbf{v}_1$ a... | Arturo Magidin | 742 | <p>(Note that you are using $A$ for two things in your post: it is the original matrix, and then it's an entry of the matrix; that's very bad form. and likely to lead to confusion; never use the same symbol to represent two different things).</p>
<p>So, if I understand you: you start with a matrix $\mathscr{A}$,
$$\ma... |
70,500 | <p>I am trying to find $\cosh^{-1}1$ I end up with something that looks like $e^y+e^{-y}=2x$. I followed the formula correctly so I believe that is correct up to this point. I then plug in $1$ for $x$ and I get $e^y+e^{-y}=2$ which, according to my mathematical knowledge, is still correct. From here I have absolutely n... | John Alexiou | 3,301 | <p>start with</p>
<p>$$\cosh(y)=x$$</p>
<p>since</p>
<p>$$\cosh^2(y)-\sinh^2(y)=1$$ or $$x^2-\sinh^2(y)=1$$</p>
<p>then</p>
<p>$$\sinh(y)=\sqrt{x^2-1}$$</p>
<p>now add $\cosh(y)=x$ to both sides to make</p>
<p>$$\sinh(y)+\cosh(y) = \sqrt{x^2-1} + x $$</p>
<p>which the left hand side simplifies to : $\exp(y)$</p... |
70,500 | <p>I am trying to find $\cosh^{-1}1$ I end up with something that looks like $e^y+e^{-y}=2x$. I followed the formula correctly so I believe that is correct up to this point. I then plug in $1$ for $x$ and I get $e^y+e^{-y}=2$ which, according to my mathematical knowledge, is still correct. From here I have absolutely n... | Christian Blatter | 1,303 | <p>You have found out that the unknown $y$ satisfies the equation $e^y+e^{-y}=2$. Multiply by $e^y$ and rearrange terms. You then get
$$e^{2y}-2e^y+1=0\ .$$
Now use the following trick: Put $e^y=:u$ with a new unknown $u$. This $u$ has to satisfy the quadratic equation
$$u^2-2u+1=0\ ,\quad{\rm i.e.,}\quad (u-1)^2=0\ .$... |
389,425 | <blockquote>
<p>What kind of math topics exist?</p>
</blockquote>
<p>The question says everything I want to know, but for more details: I enjoy studying mathematics but the problem is that I can't find any information with a summary of all math topics, collected together. I also googled this and took a look at other... | nitrous2 | 30,074 | <p>You might find the "Princeton Companion to Mathematics" helpful.</p>
<p><a href="http://rads.stackoverflow.com/amzn/click/0691118809" rel="nofollow">http://www.amazon.com/Princeton-Companion-Mathematics-Timothy-Gowers/dp/0691118809</a></p>
|
1,090,658 | <p>I'm doing some previous exams sets whilst preparing for an exam in Algebra.</p>
<p>I'm stuck with doing the below question in a trial-and-error manner:</p>
<p>Find all $ x \in \mathbb{Z}$ where $ 0 \le x \lt 11$ that satisfy $2x^2 \equiv 7 \pmod{11}$</p>
<p>Since 11 is prime (and therefore not composite), the Ch... | DeepSea | 101,504 | <p>$2x^2 - 7 \cong 2x^2 + 4 - 11 \cong 2(x^2+2)\pmod {11} \to x^2+2 = 0\pmod {11} \to x^2=-2\pmod {11}=9\pmod {11}\to x^2-9 = 0\pmod {11}\to (x-3)(x+3)=0\pmod {11}\to x-3=0\pmod {11} \text{ or } x+3 = 0\pmod {11}\to x=3,8 $ $\text{since}$ $0\leq x < 11$.</p>
|
3,021,631 | <p>I've been strongly drawn recently to the matter of the fundamental definition of the exponential function, & how it connects with its properties such as the exponential of a sum being the product of the exponentials, and it's being the eigenfunction of simple differentiation, etc. I've seen various posts inwhich... | Ira Gessel | 437,380 | <p>Your identity is a special case of a well-known binomial coefficient identity called Vandermonde's theorem:
<span class="math-container">$$\sum_{r=0}^m \binom a{m-r}\binom nr = \binom {a+n}{r}.$$</span>
This is an identity of polynomials in <span class="math-container">$a$</span> and <span class="math-container">$n$... |
197,393 | <p>Playing around on wolframalpha shows $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)=\pi$. I know $\tan^{-1}(1)=\pi/4$, but how could you compute that $\tan^{-1}(2)+\tan^{-1}(3)=\frac{3}{4}\pi$ to get this result?</p>
| Per Erik Manne | 33,572 | <p>The simplest way is by using complex numbers. It is a trivial computation to show that $$(1+i)(1+2i)(1+3i)=-10$$
Now recall the geometric description of complex multiplication (multiply the lengths and add the angles), and take the argument on both sides of this equation. This gives
$$\tan^{-1}(1)+\tan^{-1}(2)+\tan... |
503,827 | <p>I need help to prove the following:</p>
<p>Let $a$, $b$, and $c$ be any integers. If $a \mid b$, then $a \mid bc$</p>
<p>My brain is in overload and just not working.</p>
| user66733 | 66,733 | <p>If $a \mid b$ then $\exists q \in \mathbb{Z}: aq=b$. So, $aqc=bc$. If you take $Q=qc$ you see that $\exists Q \in \mathbb{Z}: aQ=bc$, therefore, $a \mid bc$.</p>
<p>Now you can prove this one on your own for practicing: </p>
<p>If $a \mid b$ and $c \mid d$ then $ac \mid bd$.</p>
<p>And this one:</p>
<p>If $a \mi... |
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