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 |
|---|---|---|---|---|
3,279,878 | <p>I got this equation while I was trying to solve a certain math Olympiad problem. I tried modulus and whatnot, but I haven't got anywhere. Is there a way to prove this?</p>
| Robert Israel | 8,508 | <p>Show that all solutions mod <span class="math-container">$16$</span> have <span class="math-container">$x,y,z,a$</span> all even, and use infinite descent.</p>
|
2,779,083 | <p>Given a polynomial $f(z)\in\mathbb{C}[z]$, $\exists$ only finitely many $c$ s.t. $f(z)-c=0$ has repeated roots?
Is above true in general?
Is it true for polynomials of the form $f(z) = (z-z_1)\cdot ... \cdot (z - z_n)$ where $z_1, ... , z_n \in \mathbb{C}$are distinct?</p>
| Angina Seng | 436,618 | <p>A polynomial has repeated roots iff its discriminant is zero. The discriminant is a polynomial in its coefficients. So the discriminant
of $f(z)-c$ is a polynomial, say $D(c)$, in $c$. Either $D(c)$
has finitely many zeros, which is what we want, or it is identically
zero. In that case $f(z)=c$ has repeated root for... |
1,611,052 | <p>Let H an infinite-dimensional Hilbert space in $\mathbb{R}$</p>
<p>If $x_1, x_2, \ldots x_n \in H$, how to prove: </p>
<p>$\sum_{1\leq i,j\leq n} {\lvert\lvert x_i - x_j \rvert\rvert}^2 \leq \sum_{1\leq i,j\leq n} ({\lvert\lvert x_i \rvert\rvert}^2 + {\lvert\lvert x_j \rvert\rvert}^2)$</p>
| gerw | 58,577 | <p>We have
\begin{align*}\sum_{1\le i,j\le n} \|x_i - x_j\|^2
&=\sum_{1\le i,j\le n} \left\{\|x_i\|^2 + \|x_j\|^2 - 2 \, (x_i,x_j)\right\}
\\&= \sum_{1\le i,j\le n} \left\{\|x_i\|^2 + \|x_j\|^2\right\} - 2 \, (\sum_{i=1}^n x_i,\sum_{j=1}^n x_j)
\\&\le \sum_{1\le i,j\le n} \left\{\|x_i\|^2 + \|x_j\|^2\right\... |
700,004 | <p>I have been working on this proof for a few hours and I can not make it work out.</p>
<p>$$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$</p>
<p>i need to get to $1-\frac{1}{k+2}$</p>
<p>I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$
then I have tried $1-\frac{(k+2)+1}{(k+1)(k+2)}$ which got me exac... | Jose Antonio | 84,164 | <p>Claim $\sum_{n=1}^N \frac{1}{N(N+1)}=1-\frac{1}{N+1}$</p>
<p>If $N=0$, so $\sum_{n=1}^0 1/n-1/(n+1)=0$ and $1-1/(N+1)=0$. Suppose we have proven the assertion for $N\ge0$. Then </p>
<p>\begin{align}\sum_{n=1}^{N+1} 1/n-1/(n+1)=1/(N+1)-1/(N+2)+\sum_{n=1}^N 1/n-1/(n+1)\\
=1/(N+1)-1/(N+2)+1-1/(N+1)\\
=1-1/(N+2)\end{a... |
308,856 | <p>A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure. </p>
<p>A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesg... | arsmath | 3,711 | <p>I have a (possibly idiosyncratic) view that the natural form of measure theory is for finite measure spaces and bounded functions. Other cases are obviously very important, but we have to work harder to get them. You can see this is many of the proofs, where the finite case is easier, and we have to work a bit mor... |
308,856 | <p>A set $E\subseteq \mathbb{R}^d$ is said to be Jordan measurable if its inner measure $m_{*}(E)$ and outer measure $m^{*}(E)$ are equal.However, Lebesgue mesure theory is developed with only outer measure. </p>
<p>A function is Riemann integrable iff its upper integral and lower integral are equal.However, in Lebesg... | Gerald Edgar | 454 | <p>I think your statement about Jordan is actually wrong. If $m_*(E) = \infty$ and $m^*(E) = \infty$, then $E$ need not be Jordan measurable. If you talk only about bounded sets $E$, then your characterization is correct. But it is also correct for Lebesgue measure (using Lebesgue inner and outer measure).</p>
<p>T... |
2,446,282 | <p>The maximum value of the function $f(x)= ax^2+bx+c$ is 10. Given that $f(3)=f(-1)=2$, find $f(2)$</p>
<p>The answer is $f(2)=8$</p>
<p>I thought that by maximum value it meant that c=10, but the equation I got gave as a result $f(2)=10$</p>
<p>Any hint on how to solve it?</p>
| David K | 139,123 | <p>You are confusing real numbers with their representations.</p>
<p>You write <span class="math-container">$(1 + 0.002)_3 \stackrel?= (1.01)_3,$</span>
which is an abuse of notation to begin with;
the left side is not <em>equal</em> to the right, it <em>rounds</em> to the right-hand side
when <span class="math-contain... |
3,461,531 | <p>I have to determine differentiability at <span class="math-container">$(0,1)$</span> of the following function:
<span class="math-container">$$f(x,y)=\frac{|x| y \sin(\frac{\pi x}{2})}{x^2+y^2}$$</span>
The partial derivatives both have value <span class="math-container">$0$</span> at <span class="math-container">$(... | José Carlos Santos | 446,262 | <p><strong>Hint:</strong> <span class="math-container">$\mathbb Z\setminus\mathbb Q=\emptyset$</span> and <span class="math-container">$\emptyset\subset\mathbb N$</span>.</p>
|
2,705,794 | <p>I ran across this problem on a practice Putnam worksheet. Completely stumped.</p>
<p>Is $$\large \frac{m^{6} + 3m^{4} + 12m^{3} + 8m^{2}}{24}$$ an integer for all $m \in \mathbb{N}$?</p>
<p>I suspect it is an integer for any $m$. It checks out for small cases.</p>
<p>Any hints for proving the general case?</p>
| Marko Riedel | 44,883 | <p>With this being contest math I suspect the contestant is supposed to recognize the substituted cycle index of the face permutations of the cube under rotations, which is</p>
<p>$$Z(F) = \frac{1}{24}
\left(a_1^6 + 6a_1^2a_4 + 3a_1^2a_2^2 + 8a_3^2 + 6a_2^3\right).$$</p>
<p>Hence the formula counts the number of col... |
761,616 | <p>How do you integrate $\sqrt{(x^4 + x^2)}$? </p>
| David | 119,775 | <p>As long as we are considering positive values of $x$, we have
$$\int\sqrt{x^4+x^2}\,dx=\int x\sqrt{x^2+1}\,dx\ ,$$
and this is easy to integrate.</p>
|
527,576 | <blockquote>
<p>Three men (out of 7) and three women (out of 6) will be chosen to
serve on a 7 member committee. In how many ways can the committee be
formed?</p>
</blockquote>
<p>I did 7C3 to get 35 men.</p>
<p>Then i did 6C3 to get 20 women.</p>
<p>Then i decide to add up 20 + 35 and get 55 but it is suggest... | mathematics2x2life | 79,043 | <p>You could also think about it this way. It doesn't matter the order you choose men and women, so you might as well choose all the men that you want on the committee then all the women. There are 7 total spots, _ _ _ _ _ _ _. Let's fill the first spot with a man. How many ways can we do this--7. So we have 7 _ _ _ _ ... |
4,644,186 | <p>Let m be a positive integer.Find the values of <span class="math-container">$$\sum_{k=0}^n \frac{{n\choose k }}{k+1}$$</span>. Leave your answer in terms of n where appropriate.</p>
<p>Remark. There is an alternative method for computing the sums described here: make use of integration.</p>
<p>I can only list out th... | Z Ahmed | 671,540 | <p><span class="math-container">$$\sum_{k=0}^{n}\frac{{n\choose k}}{k+1}=\frac{1}{n+1}\sum_{k=0}^{n}{n+1\choose k+1}=\frac{1}{n+1}\sum_{j=1}^{n+1} {n+1 \choose j}=\frac{2^{n+1}-1}{n+1}.$$</span></p>
|
3,016,386 | <p>Hi I am struggling with this exercise, which may be perceived as simple. so I was trying to write tangents as follows:</p>
<p><span class="math-container">$$\tan(z)=-i\frac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}}$$</span> and then <span class="math-container">$$z=a+bi$$</span>, which led me to <span class="math-container">... | saulspatz | 235,128 | <p>It's easier if you don't use sines and cosines. We have
<span class="math-container">$$\overline{e^{iz}+e^{-iz}}=\overline{e^{iz}}+\overline{e^{-iz}},$$</span> and, for example, <span class="math-container">$$\overline{e^{iz}}=e^{\overline{iz}}=e^{\overline{ix-y}}=e^{-y}e^{-ix}$$</span> </p>
|
779,509 | <p>I know there is a nice way of getting the continued fraction expansion of quadratic irrationals mainly because they recur after a point, and if they recur after a point they are quadratic irrationals. When constructing the expansion you can multiply by conjugates (kind of), e.g. </p>
<p>$\sqrt 3 =1+\sqrt 3 -1 = 1+\... | Fabio Lucchini | 54,738 | <p>Starting from the column vector <span class="math-container">$(1,0,0,-2)$</span>, consider the following steps:</p>
<p><em>Step a)</em> Repeat multiplication by the matrix <span class="math-container">$A$</span>
<span class="math-container">$$A=\begin{bmatrix}
1&0&0&0\\
3&1&0&0\\
3&2&... |
3,154,332 | <p>I have a calculus question which i will display here as an image:
<a href="https://i.stack.imgur.com/8xN3P.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/8xN3P.png" alt="enter image description here"></a></p>
<p>I am interested to understand part (b) of this question.
I actually got the answer, ... | B. Goddard | 362,009 | <p>When I think about the 2nd derivative, I imagine the tangent line to the curve at a point <span class="math-container">$x$</span> and let <span class="math-container">$x$</span> increase. The 2nd derivative tells you about the change in the slope of the tangent line. So if the 2nd derivative is positive, the slop... |
574,614 | <p>if $\gamma:[0,2\pi]\mapsto\Bbb C,\quad \gamma(t)=1+e^{it}$ then show that $|\int_\gamma\frac{dz}{z-\frac{3}{2}}|\le4\pi$ (without computing)</p>
<p>I tried :
$ |\int_\gamma\frac{dz}{z-\frac{3}{2}}| \le \int_\gamma|\frac{1}{z-\frac{3}{2}}|dz$ and $ |z-\frac{3}{2}|\ge||z|-\frac{3}{2}|$ we should find its max va... | AD - Stop Putin - | 1,154 | <p>Do this and you will find your way:</p>
<ol>
<li><p>Your $z$ in the integral varies on $\gamma$. </p></li>
<li><p>$\gamma$ is a circle of radius 1, with center $z=1$.</p></li>
<li><p>Find the point on the circle $\gamma$, that is closest to $z=3/2$. Note that that point maximize $$|f(z)|= \frac{1}{|z-3/2|}$$ </p></... |
574,614 | <p>if $\gamma:[0,2\pi]\mapsto\Bbb C,\quad \gamma(t)=1+e^{it}$ then show that $|\int_\gamma\frac{dz}{z-\frac{3}{2}}|\le4\pi$ (without computing)</p>
<p>I tried :
$ |\int_\gamma\frac{dz}{z-\frac{3}{2}}| \le \int_\gamma|\frac{1}{z-\frac{3}{2}}|dz$ and $ |z-\frac{3}{2}|\ge||z|-\frac{3}{2}|$ we should find its max va... | AD - Stop Putin - | 1,154 | <p>Another short answer goes "By the Argument principle we have...".</p>
|
3,059,571 | <p><span class="math-container">$$\lim_{x\to \frac\pi2} \frac{(1-\tan(\frac x2))(1-\sin(x))}{(1+\tan(\frac x2))(\pi-2x)^3}$$</span></p>
<p>I only know of L'hopital method but that is very long. Is there a shorter method to solve this?</p>
| Ankit Kumar | 595,608 | <p>Note that <span class="math-container">$$\frac{1-\text{tan}\frac{x}{2}}{1+\text{tan}\frac{x}{2}}=\text{tan}(\frac{\pi}{4}-\frac{x}{2})$$</span>
So, given limit is same as
<span class="math-container">$$\text{lim}_{x\to\pi/2}\frac{\text{tan}(\pi/4-x/2)(1-\text{sin}x)}{(\pi-2x)^3}$$</span>
Can you do it now using <spa... |
1,775,965 | <p>I'm learning about measure theory, specifically the Lebesgue integral of nonnegative functions, and need help with the following problem. </p>
<blockquote>
<p>Let $f:\mathbb{R}\to[0,\infty)$ be measurable and $f\in L^1$. Show that $F(x)=\int_{-\infty}^x f$ is continuous.</p>
</blockquote>
<p>I know is isn't muc... | Behnam Esmayli | 283,487 | <p>For $x<y$ since integration (in any reasonable type of integration) is additive on union of disjoint sets,</p>
<p>$$ F(x) + \int_{x}^y f=\int_{(-\infty,x)} f + \int_{(x,y)} f =\int_{(-\infty,x) \cup (x,y)} f=F(y)$$
$$\implies F(y)-F(x)=\int_{x}^y f$$
$$\implies |F(y)-F(x)|=|\int_{x}^y f| \leq \int_{x}^y |f|$$</... |
21,156 | <p>The title says it all, is there a way to get in contact which users who consistently post answers without using <span class="math-container">$\LaTeX$</span>? I've come across a user who does that and (as I had some free time) edited about 10-15 of his posts, some of his answers were barely readable; on each post I l... | Ashutosh Gupta | 215,160 | <p>A constructive suggestion to the Stack Exchange features:</p>
<p>a) Why not implement $\LaTeX$ auto completion/suggestion for the $\LaTeX$ users?</p>
<p>b) If somebody is posting in plain text, suggest (and possibly auto insert) $\LaTeX$ tags. This is very much possible and a very good feature to see in the stack ... |
1,154,763 | <p>I'm given this equation:</p>
<p>$$
u(x,y) =
\begin{cases}
\dfrac{(x^3 - 3xy^2)}{(x^2 + y^2)}\quad& \text{if}\quad (x,y)\neq(0,0)\\
0\quad& \text{if} \quad (x,y)=(0,0).
\end{cases}
$$</p>
<p>It seems like L'hopitals rule has been used but I'm confused because</p>
<ol>
<li>there is no limit here it's just... | Alex Zorn | 73,104 | <p>Here's one option. Write $(x,y)$ in polar form: $x = r\cos(\theta)$, $y = r\sin(\theta)$. You get:</p>
<p>$$u(r,\theta) = \frac{r^3\cos^{3}(\theta) - 3r^3\cos(\theta)\sin^{2}(\theta)}{r^2}$$</p>
<p>$$u(r,\theta) = r[\cos^{3}(\theta) - 3\cos(\theta)\sin^{2}(\theta)]$$</p>
<p>Since $\cos^{3}(\theta) - 3\cos(\theta)... |
1,720,053 | <p>The PDF describes the probability of a random variable to take on a given value:</p>
<p>$f(x)=P(X=x)$</p>
<p>My question is whether this value can become greater than $1$?</p>
<p>Quote from wikipedia:</p>
<p>"Unlike a probability, a probability density function can take on values greater than one; for example, t... | Wouter | 89,671 | <p>Probability density functions are not probabilities, but , if $f(x)$ is a probability density function, then $P=\int_{x_0}^{x_1} f(x) dx$ is a probability and thus $\int_{x_0}^{x_1} f(x) dx \leq 1$ for all $x_0,x_1$ ($x_0\leq x_1$).</p>
|
1,720,053 | <p>The PDF describes the probability of a random variable to take on a given value:</p>
<p>$f(x)=P(X=x)$</p>
<p>My question is whether this value can become greater than $1$?</p>
<p>Quote from wikipedia:</p>
<p>"Unlike a probability, a probability density function can take on values greater than one; for example, t... | holala | 806,768 | <p>To add to the already good existing answers, it is easy to understand it by way of their definitions.</p>
<p>For discrete random variable, the probability mass function (pmf) denoted as <span class="math-container">$p_{_X}(x)$</span> gives us the probabilities that <span class="math-container">$X$</span> takes a dis... |
376,517 | <p>Let <span class="math-container">$U$</span> be a smooth variety, and <span class="math-container">$U\hookrightarrow X$</span> an smooth compactification with snc boundary <span class="math-container">$D=X\setminus U$</span>. Suppose that <span class="math-container">$\omega\in H^0(U,\Omega^n_U)$</span> is global al... | abx | 40,297 | <p>This is not true. Take for <span class="math-container">$X$</span> an elliptic curve, for <span class="math-container">$D$</span> a point <span class="math-container">$p\in X$</span>. The restriction map <span class="math-container">$H^1(X,\mathbb{C})\rightarrow H^1(U,\mathbb{C})$</span> is an isomorphism, and <span... |
979,267 | <p>Let $a_n$ be the $n$th sequence 1, 2 , 2 , 3 , 3 , 3 , 4 , 4 , 4 , 4 , 5 , 5 , 5 , 5 , 5, . . . . . . . constructed by including the integer $k$ exactly $k$ time. Show that $a_n$ $=$ $\lfloor \frac12 + (2n+\frac14)^.5 \rfloor$</p>
<p>Let $\lvert r\rvert < 1$ be a real number. Evaluate $\sum_{i=0}^\infty i... | Hagen von Eitzen | 39,174 | <p>Yes. Use SSS to construct the triangle with sides $\frac1{h_a}$, $\frac1{h_b}$, $\frac1{h_c}$. Then in this triangle the heights are <em>proportinal</em> to the given heights. Scale accordingly.</p>
|
1,285,443 | <blockquote>
<p>Let us denote solution to the equation</p>
<p>$$(x+a)^{x+a}=x^{x+2a}$$</p>
<p>with $X_a$.</p>
<p>($a$ is a non-zero real number)</p>
<p>Prove that:</p>
<p>$$\lim_ {a \to 0} X_a = e$$</p>
</blockquote>
<p>This is something that I noticed while making numerical experiments for ... | Peter Franek | 62,009 | <p>Rewriting the equation to $(1+\frac{a}{x})^{x+a}=x^a$ and taking $\ln$, we have $(x+a) \ln(1+\frac{a}{x})=a\ln x$. Assuming that $X_a$ is bounded for $a$ from some neighbrhood of $0$, we take the Taylor approximation for small $a$ of the left hand side:
$$
(X_a+a) (\frac{a}{X_a}+o(a))=a\ln X_a
$$
and
$$
a+o(a) = a\... |
3,433,492 | <p>I know that a function can admitted multiple series representation (according to Eugene Catalan), but I wonder if there is a proof for the fact that each analytic function has only one unique Taylor series representation. I know that Taylor series are defined by derivatives of increasing order. A function has one an... | freakish | 340,986 | <p>The <a href="https://en.wikipedia.org/wiki/Taylor_series" rel="nofollow noreferrer">Taylor series</a> is indeed uniquely defined for any smooth function, regardless whether it is convergent or not and whether it coincides with the function when convergent. And so asking about uniqueness is a bit pointless. It's like... |
50,736 | <p>Hi guys,</p>
<p>I have recently started looking at polynomials $q_n$ generated by initial choices $q_0=1$, $q_1=x$ with, for $n\geq 0$, some recurrence formula</p>
<p>$$q_{n+2}=xq_{n+1}+c_n q_n$$</p>
<p>where $c_n$ is some function in $n$. The first few of these are</p>
<p>$$q_2=x^2+c_0$$
$$q_3=x^3+(c_0+c_1)x$$
... | user91132 | 6,827 | <p>Let's treat the $c_i$ as formal indeterminates.</p>
<p>Let $S(n,m)$ be the set of increasing functions $i:\{1,\ldots, m\}\to \{0,\ldots, n-2\}$, written $j \mapsto i_j$, such that $i_{j+1} > i_j + 1$ for all $j=1,\ldots, m$. So $S(n,m)$ is in bijection with the set of subsets of $\{0, \ldots, n-2\}$ of size $m$ ... |
2,551,233 | <p>There are 4 fair coins and 1 unfair coin that has only heads. We choose a coin and flip it three times. The result is HHH. What is the probability that the fourth flip is H? </p>
| nicola | 251,928 | <p>First, use Bayes' rule to determine the probability of having selected the unfair coin. Call $UC$ the event "we selected the unfair coin" and $FC $ "we selected a fair coin".</p>
<p>$$P(HHH) = P(HHH|UC)P_0(UC) + P(HHH|FC)P_0(FC) = 1\times\frac{1}{5}+\frac{1}{8}\times\frac{4}{5} = \frac{3}{10}$$
$$P(UC|HHH) = \frac... |
37,900 | <p>I use the following code to find out the number of consecutive prime numbers using a formula $n^2+n+i$ found out by Euler (starting from n=0):</p>
<pre><code>Nbs = {};
Do[Nbs = Union[Nbs,
Select[Range[5000], (PrimeQ[#^2 + # + i] == False &), 1]], {i, 1,
5000}];
Nbs
</code></pre>
<p>How can I also get in th... | Kevin | 10,860 | <p>I don't fully understand your question, but I believe you want either Nbs2 or Nbs3 (same data, sorted differently) in this code:</p>
<pre><code>Nbs = {};
Do[AppendTo[Nbs,
Append[Select[Range[5000], (PrimeQ[#^2 + # + i] == False &), 1],
i]], {i, 1, 5000}];
Nbs2 = DeleteDuplicates[Nbs, (#1[[1]] == #2[[1]... |
120,687 | <p>Consider the following code</p>
<pre><code>styles = {Red, Blue, {Red, Dashed}, {Blue, Dashed}}
pt1 = Plot[{x^2, 2 x^2, 1/x^2, 2/x^2}, {x, 0, 3}, Frame -> True,
PlotStyle -> styles, PlotLegends -> {"1", "2", "1", "2"}]
</code></pre>
<p>I would like the two red lines to carry the same label "1" and the two... | Eric Towers | 16,237 | <p>Testing my comment does indicate that one can cut the number of evaluations of <code>f[]</code> in half easily.</p>
<pre><code>f[x_, y_, z_] := Module[{},
totCalls++;
Exp[Sin[x]] + Cos[y + z]
]
totCalls = 0;
NIntegrate[
{f[x, y, z], Sqrt[f[x, y, z]] + x},
{x, 0, 10}, {y, 0, 10}, {z, 0, 10},
... |
1,533,362 | <p>I need to prove this identity:</p>
<p>$\sum_{k=0}^n \frac{1}{k+1}{2k \choose k}{2n-2k \choose n-k}={2n+1 \choose n}$</p>
<p>without using the identity:</p>
<p>$C_{n+1}=\sum_{k=0}^n C_kC_{n-k}$.</p>
<p>Can't figure out how to.</p>
| Ojas | 154,392 | <p>Let's try to solve the following problem : Given a grid of $(n+1)*n$, count the number of ways to move from the lower left corner to the upper right corner while moving only right or up in one step.</p>
<p>One way to do this is straightforward. You have total $2n + 1$ moves, out of which $n$ moves should be <em>up<... |
3,851,609 | <p>I need to show that if <span class="math-container">$X_n \rightarrow X$</span> and <span class="math-container">$X_n \rightarrow Y$</span>, then <span class="math-container">$X\overset{\text{a.s.}}{=}Y$</span> for convergence in probability, convergence almost surely, as well as for convergence in mean and quadratic... | Ninad Munshi | 698,724 | <p>If <span class="math-container">$f(y)$</span> is integrable, then it is the derivative of some other function <span class="math-container">$g(y)$</span> s.t. <span class="math-container">$g'(y) = f(y)$</span>. This gives us</p>
<p><span class="math-container">$$\int_0^1 \int_x^{1-x} g'(y)\:dydx = \int_0^1 g(1-x)-g(x... |
3,851,609 | <p>I need to show that if <span class="math-container">$X_n \rightarrow X$</span> and <span class="math-container">$X_n \rightarrow Y$</span>, then <span class="math-container">$X\overset{\text{a.s.}}{=}Y$</span> for convergence in probability, convergence almost surely, as well as for convergence in mean and quadratic... | Tryst with Freedom | 688,539 | <p><a href="https://i.stack.imgur.com/F19zS.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/F19zS.jpg" alt="enter image description here" /></a></p>
<p>So, I've drawn the region of integration in the <span class="math-container">$x-y$</span> plane. So, the bounds of the inner integral <span class="ma... |
257,623 | <p>Consider the following ellipse, generated by the bounding region of the following points</p>
<pre><code>ps = {{-11, 5}, {-12, 4}, {-10, 4}, {-9, 5}, {-10, 6}};
rec = N@BoundingRegion[ps, "FastEllipse"];
Graphics[{rec, Red, Point@ps}]
</code></pre>
<p><a href="https://i.stack.imgur.com/gvtUB.png" rel="nofol... | Daniel Huber | 46,318 | <p>I think the help of Ellipsoid is incomplete because it does not explain the input: Ellipsoid[p,[CapitalSigma]], where the second argument is called the "weight matrix".</p>
<p>You will remember that an ellipse (for simplicity I am explaining the 2D case, nD is similar and assume the ellipse is centered at ... |
7,981 | <p>I've read so much about it but none of it makes a lot of sense. Also, what's so unsolvable about it?</p>
| Sundaramurthy | 69,788 | <p>It is straight forward. We have a Zeta function, 'analytically continued' that says</p>
<p>$\zeta(s)[1-2/2^s] = 1 - 1/2^s + 1/3^s - 1/4^s + ......$ Here s is a complex variable. Thus $s=\Re(\sigma) + \Im(\omega).$ (Where $\Re$ indicates the real part and $\Im$ indicates the imaginary part). The above series converg... |
1,941,583 | <p>I tried to solve the Millikan integer, but I did not success. The initial integer is:
$$m\frac{dv}{dt}= \frac{4}{3}\pi a^3 \rho g - qE - 6\pi \eta av$$
and I must prove show I to optain this:
$$v= \frac{\frac{4}{3}\pi a^3 \rho g - qE} {6\pi \eta a}\left(1-e^\left(\frac{-6\pi \eta at}{m}\right)\right) $$</p>
<p>**... | snulty | 128,967 | <p>Writing the equation as $$m\frac{dv}{dt}=A-Bv$$
one can solve by <a href="https://en.wikipedia.org/wiki/Separation_of_variables" rel="nofollow">separation of variables</a>. Re-write as:</p>
<p>$$\frac{m}{A-Bv}\frac{dv}{dt}=1$$</p>
<p>Integrating $dt$ and making the substitution $u=A-Bv$:</p>
<p>$$\frac{-m}{B}\int... |
1,941,583 | <p>I tried to solve the Millikan integer, but I did not success. The initial integer is:
$$m\frac{dv}{dt}= \frac{4}{3}\pi a^3 \rho g - qE - 6\pi \eta av$$
and I must prove show I to optain this:
$$v= \frac{\frac{4}{3}\pi a^3 \rho g - qE} {6\pi \eta a}\left(1-e^\left(\frac{-6\pi \eta at}{m}\right)\right) $$</p>
<p>**... | Felix Marin | 85,343 | <p>$\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}... |
9,508 | <p>I need to write a coupon code system but I do not want to save each coupon code in the database. (For performance and design reasons.) Rather I would like to generate codes subsequent that are watermarked with another code.</p>
<p>They should like kind of fancy and random. Currently they look like this:</p>
<p>1: ... | comonad | 3,226 | <p>you could make sth inspired by RSA, where you do not need two keys. like this:</p>
<p>calculate everything modulo $p$, where $p$ is prim. that will then be the GF[$p$] <a href="http://en.wikipedia.org/wiki/Finite_field" rel="nofollow">(GaloisField)</a>.
now instead of finding a generator $g$ for that GF[$p$] and c... |
148,374 | <p>I have checked all Mathematica color schemes, and I think "Hue" is the most vibrant, beautiful one. However, it has one issue: the two ends of the spectrum are red (though, different reds). I like a spectrum from, say, red to blue. Is that possible to manipulate the Hue and remove the pink and second red? </p>
<p>C... | Alexey Popkov | 280 | <p>Your question looks almost a duplicate of this one:</p>
<ul>
<li><a href="https://mathematica.stackexchange.com/q/101268/280">How to customize color scheme to mimic that in Origin?</a></li>
</ul>
<p>Using the formulation from the <a href="https://mathematica.stackexchange.com/a/101933/280">answer</a> by <a href="h... |
259,083 | <p>There was a question asked: <a href="https://math.stackexchange.com/q/136204/8348">An open subset $U\subseteq R^n$ is the countable union of increasing compact sets.</a> There Davide gave an <a href="https://math.stackexchange.com/a/136209/">answer</a>. Can anyone tell me how the
equality holds, and the motivation ... | Noix07 | 92,038 | <p>(answer for the "motivation" part of the question)</p>
<p>I stumbled on two applications of that property although I'm not well-versed enough to develop so I'll just cite "Topological spaces, distributions and kernels", François Treves: (Lemma 10.1 p.87)</p>
<ol>
<li>the property/lemma is used to... |
2,137,332 | <p>On the MathWorld page: </p>
<p><a href="http://mathworld.wolfram.com/FermatPseudoprime.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/FermatPseudoprime.html</a></p>
<p>in the first table, I expect to see $561$ on every line, but it is not on the line for base $3$.</p>
<p>When you click on the link ... | mjw | 655,367 | <p>The triangle consists of three segments:
<span class="math-container">$\{-1 \leftrightarrow i, i \leftrightarrow -1$</span>,<span class="math-container">$-1 \leftrightarrow 1\}$</span>, call them <span class="math-container">$\{a,b,c\}$</span>. Segment <span class="math-container">$a$</span> maps to a parabola in ... |
4,092,877 | <p>I'm trying to find the solution for the following differential equation, however, I'm not sure how to derive the answer and so I would really appreciate some support!</p>
<p><span class="math-container">$y'' - y' = x^2$</span></p>
<p>I have tried splitting this into a quadratic polynomial: <span class="math-containe... | David | 911,796 | <p>(When an answer can be checked, there is no danger in using a formalism.) <span class="math-container">$D^2-D=D(D-1),$</span> where <span class="math-container">$D$</span> is <span class="math-container">$d\over dx$</span>, so that
<span class="math-container">$$y_H=c_1+c_2e^x.$$</span>
Next,<br />
<span class="math... |
23,911 | <p>I am teaching a course on Riemann Surfaces next term, and would <strong>like a list of facts illustrating the difference between the theory of real (differentiable) manifolds and the theory non-singular varieties</strong> (over, say, $\mathbb{C}$). I am looking for examples that would be meaningful to 2nd year US g... | Felipe Voloch | 2,290 | <p>A connected real manifold can be disconnected by the removal of a submanifold but the complement of a subvariety on an irreducible variety is still connected.</p>
|
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | user4894 | 118,194 | <p>My feeling is that the biggest problem with the epsilon-delta definition is that this is the first time students have ever seen the universal and existential quantifiers. By the time you say, "For every epsilon there exists a delta," you have already lost 95% of your audience before you even get to the business end ... |
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | String | 94,971 | <p>In his answer, <em>Paramanand Singh</em> suggests that freshman students are unfamiliar with certain <em>concepts</em> and <em>methods</em> that are prerequisites for understanding $\varepsilon-\delta$. On the other hand Singh suggests, that once these <em>concepts</em> and <em>methods</em> have been succesfully pla... |
683,513 | <p>There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student recepti... | Ben Blum-Smith | 13,120 | <p><a href="https://matheducators.stackexchange.com/a/16857/140">[Crossposted</a> from matheducators.SE]</p>
<p>The apparent conflict between points of view expressed in the OP is illusory. There is no real conflict. The mathematics education researcher quoted in the OP is arguing that students find the definition dif... |
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | Luis Mendo | 91,216 | <p>If the limit were negative, say $\ell<0$, there would be at least be a term of the sequence (in fact, infinitely many terms) smaller than $\ell/2$, and thus this term would be negative, which is impossible.</p>
|
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | Surb | 154,545 | <p>Your proof is a bit confused at the end. But it seems that you would conclude $0<|L|<\epsilon$ for every $\epsilon>0$ and you can a get a contradiction by choosing $\epsilon = |L|/2$.</p>
<p>I propose you nevertheless the following formulation:</p>
<p>Suppose by contradiction that $a_n\geq 0$ for every $n... |
1,424,273 | <p>Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?</p>
<p>My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$. And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by c... | robb | 346,801 | <p>Here is a proof that I believe is more succinct than all of the above. </p>
<p>Suppose $\lim a_n = a <0$ . </p>
<p>Let $\epsilon=-a>0$ .</p>
<p>By hypothesis, there exists an $n$ large enough such that $\left|a_n-a\right|<\epsilon =-a$ </p>
<p>$\Rightarrow a_n-a<-a $</p>
<p>$\Rightarrow a_n <0$ <... |
256,322 | <p>Let $A$ be an abelian group of order $n = p_1^{\alpha_1} \cdot \ldots \cdot p_k^{\alpha_k}$ (i.e., $n$'s unique prime factorization). The Primary Decomposition Theorem states that $A \cong \mathbb{Z}_{p_1^{\alpha_1}} \times \ldots \times \mathbb{Z}_{p_k^{\alpha_k}}$. On the other hand, the Fundamental Theorem of F... | Marc van Leeuwen | 18,880 | <p>It is not true that a finite Abelian group is determined up to isomorphism by it order $n$ only, and that is what your formulation of the Primary Decomposition Theorem would imply. So that is what is wrong; please look up the correct formulation of this theorem which apparently you know about (I don't recall any suc... |
3,066,446 | <p>Let <span class="math-container">$\overline{X}$</span> be the average of a sample of <span class="math-container">$16$</span> independent normal random variables with mean <span class="math-container">$0$</span> and variance <span class="math-container">$1$</span>. Determine c such that
<span class="math-container"... | Silent | 94,817 | <p>You should check if <span class="math-container">$f_n$</span> is odd or even function. As it turns out, <span class="math-container">$f_n$</span> even for <span class="math-container">$n$</span> odd and vice versa. Also, note that for <span class="math-container">$0<x<\pi$</span>, <span class="math-container">... |
171,690 | <p>I am trying to make a projection on the <em>xy-plane</em> of the intersection of the surfaces from the functions: <code>1 + x^2 - y^2</code>, <code>3 Log[1 + x^2]</code>.</p>
<p><a href="https://i.stack.imgur.com/XqC1g.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/XqC1g.png" alt="Intersection o... | MarcoB | 27,951 | <p>Using geometric region functions:</p>
<pre><code>RegionPlot@
ImplicitRegion[
1 + x^2 - y^2 == 3 Log[1 + x^2],
{{x, -1.5, 1.5}, {y, -1.5, 1.5}}
]
</code></pre>
<p><img src="https://i.stack.imgur.com/fRB0C.png" alt="Mathematica graphics"></p>
<p>See also: <a href="https://mathematica.stackexchange.com/quest... |
3,543,150 | <p>My question : two indefinite integrals of a function being given , how to express one indefinite integral in terms of the other? </p>
<p><a href="https://i.stack.imgur.com/VkMzJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VkMzJ.png" alt="enter image description here"></a></p>
| mjw | 655,367 | <p><span class="math-container">$$\displaystyle (-1)^{4/3}=\left[e^{(2k-1)\pi i}\right]^{4/3}, \quad k\in \{0,1,2\}$$</span></p>
<p><span class="math-container">$$\displaystyle (-1)^{4/3} \in \left\{ e^{-\frac{4\pi i}{3}}, e^\frac{4\pi i}{3}, e^{4\pi i} \right\}$$</span></p>
<p>We can write this in rectangular coord... |
3,246,244 | <p>Consider the action of <span class="math-container">$G$</span> on <span class="math-container">$X$</span>.</p>
<p>Let it be a property of <span class="math-container">$G,X$</span> that <span class="math-container">$\forall x,y,\exists g:g\cdot x=g\cdot y$</span>. This is not quite a transitive action - it describe... | Wuestenfux | 417,848 | <p>Well, notice that <span class="math-container">$g^{-1}\cdot (g\cdot x) = (g^{-1}g)\cdot x = 1\cdot x = x$</span>.</p>
<p>Thus <span class="math-container">$g\cdot x = g\cdot y$</span> implies by applying <span class="math-container">$g^{-1}$</span> that <span class="math-container">$x=y$</span>.</p>
|
302,005 | <p>Show that:
$$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$
I evaluated this by some Fourier series. Is there any other method?
Start with substitution of $$u=\arcsin x$$
Then we have to integrate $$\int_0^{\frac{\pi}{2}}\frac{u^3\cos u}{\sin^2 u}\text{d}u=-\int_0^{\frac{\pi}... | Christopher A. Wong | 22,059 | <p>Here's a suggestion that might be fruitful. It's not a full solution but it seems too messy to put into the comments. If somebody else completes the solution, feel free to add it to this.</p>
<p>You can use the substitution $\sin{t} = x$ to convert the integral to</p>
<p>$$ \int_0^{\pi/2} t^3 \csc{t} \cot{t} \, dt... |
302,005 | <p>Show that:
$$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$
I evaluated this by some Fourier series. Is there any other method?
Start with substitution of $$u=\arcsin x$$
Then we have to integrate $$\int_0^{\frac{\pi}{2}}\frac{u^3\cos u}{\sin^2 u}\text{d}u=-\int_0^{\frac{\pi}... | robjohn | 13,854 | <p>Integrating by parts twice, we get
$$
\int_0^{\pi/2}x^2\,e^{ikx}\,\mathrm{d}x
=i^{k-1}\frac{\pi^2}{4k}+i^k\frac\pi{k^2}+\frac2{k^3}\left(i^{k+1}-i\right)
$$
Therefore, using $\sin(x)=\dfrac{e^{ix}-e^{-ix}}{2i}$,
$$
\begin{align}
&\int_0^1\frac{\arcsin^3(x)}{x^2}\mathrm{d}x\\
&=\int_0^{\pi/2}\frac{x^3}{\sin^2... |
3,053,975 | <p><span class="math-container">$3^6-3^3 +1$</span> factors?, 37 and 19, but how to do it using factoring, <span class="math-container">$3^3(3^3-1)+1$</span>, can't somehow put the 1 inside </p>
| Clive Newstead | 19,542 | <p>Viewing this as a quadratic in <span class="math-container">$3^3$</span> leads you to try to factorise <span class="math-container">$x^2-x+1$</span> or <span class="math-container">$x^6-x^3+1$</span>, neither of which splits into anything useful. However, you can pull out a factor of <span class="math-container">$9$... |
275,430 | <p>I'm trying to give an $\epsilon$-$\delta$ proof that the following function $f$ is continuous for $x\notin\mathbb Q$ but isn't for $x\in\mathbb Q$. </p>
<p>Let $f:\mathbb{A\subset R\to R}, \mathbb{A=\{x\in R| x>0\}}$ be given by:
$$
f(x) = \begin{cases}
1/n,&x=m/n\in\mathbb Q
\\
0,&x\notin\mathbb Q
\en... | Pedro | 23,350 | <p>We prove that for every $a\in(0,1)$ we have $$\lim_{x\to a}f(x)=0$$</p>
<p>from where we'll see it is only continuous at the irrational points. So, let's pick any $a\in(0,1)$, and let us be given $\epsilon >0$. Choose $n$ so that $1/n\leq \epsilon$. </p>
<p>First, we note that the only points where it might be ... |
1,804,042 | <p><strong>Edit:</strong> Here is the original problem; it is possible that my recurrence for the stationary distribution $\pi$ is incorrect.</p>
<blockquote>
<p>Consider a single server queue where customers arrive according to a Poisson process with intensity $\lambda$ and request i.i.d. $\mathsf{Exp}(\mu)$ servic... | Brent Kerby | 218,224 | <p>Hint: By defining $\phi_n = \sum_{i=0}^n \pi_i$, we may express $\pi_{n+1}$ and $\phi_{n+1}$ as linear combinations of $\pi_n$ and $\phi_n$ plus constants, giving a first-order linear matrix difference equation.</p>
<p><strong>EDIT</strong>: The equation we end up with is of the form $$x_{(n+1)} = Ax_{(n)}+b$$ wher... |
1,977,306 | <p>This is from a math competition so it must not be something really long
If a parabola touches the lines $y=x$ and $y=-x$ at $A(3,3) $ and $b(1,-1)$ respectively, then </p>
<p>(A) equation of axis of parabola is $2x+y=0$ </p>
<p>(B)slope of tangent at vertex is $1/2$</p>
<p>(C) Focus is $(6/5,-3/5)$</p>
<p>(D) ... | Emilio Novati | 187,568 | <p>This figure illustrate the situation.</p>
<p><a href="https://i.stack.imgur.com/cyXYM.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cyXYM.jpg" alt="enter image description here"></a></p>
<p>Using the fact that:</p>
<blockquote>
<p>Tangents drawn at the endpoints of a focal chord of a parabo... |
1,977,306 | <p>This is from a math competition so it must not be something really long
If a parabola touches the lines $y=x$ and $y=-x$ at $A(3,3) $ and $b(1,-1)$ respectively, then </p>
<p>(A) equation of axis of parabola is $2x+y=0$ </p>
<p>(B)slope of tangent at vertex is $1/2$</p>
<p>(C) Focus is $(6/5,-3/5)$</p>
<p>(D) ... | Jean Marie | 305,862 | <p>(see figure below) </p>
<p>Parametric equations for the parabola are readily obtained, assuming a certain knowledge of quadratic Bezier curves : see for example the paragraph "Second order curve is a parabolic segment" in (<a href="https://en.wikipedia.org/wiki/B%C3%A9zier_curve" rel="nofollow noreferrer">https://e... |
129 | <p>Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not sufficient. Are there any other special homotopical properties of manifolds?</p>
| Andrew Ranicki | 732 | <p>The main result of the Browder-Novikov-Sullivan-Wall surgery theory (1962-1969) is that for $n>4$ a space $X$ is homotopy equivalent to a compact n-dimensional topological (resp. differentiable) manifold if and only if $X$ is homotopy equivalent to a finite $CW$ complex $M$ with $n$-dimensional Poincaré duality, ... |
204,365 | <p>Consider a positive matrix <code>M</code> and a positive vector <code>b</code>, e.g.</p>
<pre><code>nn = 1000;
M = Table[RandomReal[{0, 100}], {i, 1, nn}, {j, 1, nn}];
b = Table[RandomReal[{0, 100}], {i, 1, nn}];
</code></pre>
<p>I would like to find a positive vector <code>X</code></p>
<pre><code>X = Array[x,... | Roman | 26,598 | <pre><code>Minimize[{expr.expr, Thread[X > 0]}, X]
</code></pre>
|
2,304,379 | <p>My textbook give the following definition.</p>
<blockquote>
<p>Let $G$ be any topological group. A representation of $G$ on a nonzero complex Hilbert space $V$ is a group homomorphism $\phi$ of $G$ into the group of bounded linear operators on $V$ with bounded inverses, such that the resulting map $ G\times V\to ... | Arpan1729 | 444,208 | <p>In any Hausdorff space, any finite set is closed.</p>
<p>So in any connected Hausdorff space a singleton does contain any non-empty open set.</p>
<p>(I mentioned connectedness as then the singleton cannot be open since it is already closed.)</p>
|
1,102,638 | <p>Let $n\in \mathbb{N}$. Can someone help me prove this by induction:</p>
<p>$$\sum _{i=0}^{n}{i} =\frac { n\left( n+1 \right) }{ 2 } .$$</p>
| Krish | 177,430 | <p><em>Hint:</em> $\dfrac{\sqrt{1 + x+ x^2} -1}{x} = \dfrac{x + x^2}{x (\sqrt{1 + x+ x^2} + 1)} = \dfrac{1+x}{\sqrt{1 + x+ x^2} + 1}$</p>
|
3,857,494 | <p>I have the following sequence given recursively by:</p>
<p><span class="math-container">$$A_n - 2A_{n-1} - 4A_{n-2} = 0$$</span></p>
<p>Where:</p>
<p><span class="math-container">$$A_0 = 1, A_1 = 3, A_2 = 10, A_3 = 32, etc.$$</span></p>
<p>To find the generating function, I have done the following:</p>
<p><span clas... | awkward | 76,172 | <p>You can do polynomial long division to grind out as many terms as you like. (Sorry for the image, trying to figure out how to do this with MathJax made me grow faint.)</p>
<p><a href="https://i.stack.imgur.com/oKNYB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/oKNYB.png" alt="polynomial long di... |
1,902,188 | <p>$k\in\mathbb{N}$ </p>
<p>The inverse of the sum $$b_k:=\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$$ is obviously
$$a_k=\sum\limits_{j=1}^k \binom{k-1}{j-1}\frac{b_j}{k^j}$$ . </p>
<p>How can one proof it (in a clear manner)? </p>
<p>Thanks in advance.</p>
<hr>
<p>Background of the question: </p>
<... | Marko Riedel | 44,883 | <p>Suppose we seek to show that if</p>
<p>$$b_n = \sum_{q=1}^n (-1)^{n-q} {n\choose q} q^n a_q$$</p>
<p>then</p>
<p>$$a_n = \sum_{q=1}^n {n-1\choose q-1} n^{-q} b_q.$$</p>
<p>This is</p>
<p>$$a_n = \sum_{q=1}^n {n-1\choose q-1} n^{-q}
\sum_{p=1}^q (-1)^{q-p} {q\choose p} p^q a_p.$$</p>
<p>Re-indexing we find</p>... |
638,875 | <p>Let $P$ be a $p$-group and let $A$ be maximal among abelian normal subgroups of $P$. Show that $A=C_P(A)$.</p>
<p>This is the second part of a problem in which I successfully proved the following:
Let $P$ be a finite $p$-group and let $U<V$ be normal subgroups of $P$. Show that there exists $W \triangleleft P$... | Jack Schmidt | 583 | <p>This answer is meant to answer Alex's questions rather than the homework question:</p>
<p>Suppose $U$ is abelian, $V=C_P(U)$ and $U<W \leq V$ with $[W:U]=p$ and $W \unlhd V$. You ask how to show $W$ can be chosen to be abelian.</p>
<p>In fact $W$ is always abelian in this case: Since $[W:U]=p$, there is some $w... |
1,917,313 | <p>I am to find a combinatorial argument for the following identity:</p>
<p>$$\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$$</p>
<p>For the right hand side, I was think that would just be number of ways to choose at least $s-1$ elements out of a $[2r-s]$ set. However, for the left hand sid... | robjohn | 13,854 | <p><span class="math-container">$$
\begin{align}
\sum_k\binom{2r}{2k-1}\binom{k-1}{s-1}
&=\sum_k\binom{2r}{2r-2k+1}\binom{k-1}{k-s}\tag1\\
&=\sum_k(-1)^{k-s}\binom{2r}{2r-2k+1}\binom{-s}{k-s}\tag2\\[3pt]
&=\left[x^{2r-2s+1}\right](1+x)^{2r}\left(1-x^2\right)^{-s}\tag3\\[12pt]
&=\left[x^{2r-2s+1}\right](... |
352,849 | <p>I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ </p>
<hr>
<p>I am not sure if correct but i did it like this :
$(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle \frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))}$$ and $$\lim \limits_{... | Mikasa | 8,581 | <p>Another hint based on using series may be that, if the series $$\sum_0^{\infty}u_n$$ is convergent so $u_n\to 0$. </p>
|
2,596,213 | <p>I'm having huge troubles with problems like this. I know the following:</p>
<p>$$\frac{\sin{x}}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x^7)$$</p>
<p>and </p>
<p>$$\ln{(1+t)}=t-\frac{t^2}{2}+\frac{t^3}{3}+O(t^4)$$</p>
<p>So</p>
<p>$$\ln{\left(1+\left(-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+O(x... | marty cohen | 13,079 | <p>If
$f(x)
= \ln \sin(x)
$,
then
$f'(x)
=\dfrac{\sin'(x)}{\sin(x)}
=\dfrac{\cos(x)}{\sin(x)}
=\cot(x)
$.</p>
<p>From
<a href="https://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions</a>,
we have
$\cot(... |
109,298 | <p>I'm an applied model theorist, and open image theorems are important in the mathematical structures I study (they limit the number of types of elements being realised, and therefore keep things model theoretically nice e.g. stable). </p>
<p>So I have some idea as to why these open image theorems should hold from a ... | Joe Silverman | 11,926 | <p>Here's an application to independence of Heegner points. (But if you search on MathSciNet for papers that reference Serre's two results, I expect you'll find a very large number of applications.)</p>
<p>Let $E/\mathbb{Q}$ be an elliptic curve with no CM, and let $\Phi:X_0(N)\to E$ be a modular parametrization. (Wil... |
1,380,402 | <p>I'm developing a C++ program and I need to find a formula that given a number to reduce and a limit number, get a value between 0 and this limit number.</p>
<p>I don't know if it is allow to put C++ code here, but I want to show you my function:</p>
<pre><code>double Utils::reduceNumber(double numberToReduce, doub... | VansFannel | 193,243 | <p>Using math love's solution, this is my function now:</p>
<pre><code>double Utils::reduceNumber(double numberToReduce, double limitNumber)
{
float factor = 0.0;
double result = 0.0;
factor = (limitNumber - numberToReduce) / limitNumber;
if (factor < 0.0)
{
factor -= 1;
}
else... |
3,729,851 | <p>So I have the following question here.</p>
<blockquote>
<p>Suppose that <span class="math-container">$y_1$</span> solves <span class="math-container">$2y''+y'+3x^2y=0$</span> and <span class="math-container">$y_2$</span> solves <span class="math-container">$2y''+y'+3x^2y=e^x$</span>. Which of the following is a solu... | Nico De Tullio | 788,836 | <p>You are on the right track. Just notice that <span class="math-container">$ay_1$</span> is a solution to the homogeneous for any constant a. So, when you say that "the solution to the homogeneous is just <span class="math-container">$-2y_1$</span>", you could also say that <span class="math-container">$3y_... |
4,530,792 | <p>I have the following sequence <span class="math-container">$\left \{k \sin \left(\frac{1}{k}\right) \right\}^{\infty}_{1}$</span>. I don't know how to show that this is monotonically increasing.</p>
<p>I tried taking the derivative of the corresponding function <span class="math-container">$f(x) = x \sin \left(\frac... | Claude Leibovici | 82,404 | <p><em>Without derivatives</em></p>
<p><span class="math-container">$$a_k=k \sin \left(\frac{1}{k}\right) \quad \implies \quad \frac{a_{k+1}}{a_k}=
\frac{k+1}{k} \sin \left(\frac{1}{k+1}\right) \csc \left(\frac{1}{k}\right)$$</span> If <span class="math-container">$k$</span> is large, by Taylor</p>
<p><span class="math... |
2,012,947 | <p>I'm trying to prove that if f,g are continuous functions, and if E is a dense subset of X $(\text{or } Cl(E) = X)$ and if $f(x)=g(x) \forall x \in E$ then $f(x)=g(x) \forall x \in X$. </p>
<p>I understand that if f,g are continuous, then:</p>
<blockquote>
<p>$\exists \delta_1, \delta_2$ such that $\forall X \in ... | mildboson | 669,255 | <p>A little bit late, but I decided to give an alternate answer without using contradiction or the sequence definition of continuity. Let <span class="math-container">$p \in X$</span> and <span class="math-container">$\epsilon > 0$</span> be given. If <span class="math-container">$p \in E$</span> we are done, other... |
205,926 | <p>I'm trying to understand a proof about density of a subset $X$ in its one-point compactification $Y$.</p>
<p>We can do this proof by contradiction, suppose we don't have $\operatorname{cl}(X) = Y$.
This implies that $\operatorname{cl}(X) = X$. </p>
<p>Why? Can anyone help me?</p>
<p>Thanks</p>
| Brian M. Scott | 12,042 | <p>You’re making it much harder than it really is. $Y=X\cup\{p\}$, where $p$ is the new point. To show that $X$ is dense in $Y$, you need only show that every open neighborhood of $p$ has non-empty intersection with $X$. Go back to the definition of the one-point compactification and see why this is true: what are the ... |
4,540,637 | <blockquote>
<p>Given a line <span class="math-container">$y=kx$</span> on a Cartesian coordinate, I want to find an equation of a parabola, whose base is on that line at point <span class="math-container">$(x_1,y_1)$</span> and passes through point <span class="math-container">$(x_2,y_2)$</span>.</p>
</blockquote>
<p>... | insipidintegrator | 1,062,486 | <p><a href="https://math.stackexchange.com/questions/4180898/axes-based-equations-of-conics">From this post</a>, we can write the equation of a parabola as
<span class="math-container">$$d_{\text{Axis}}^2=4a\cdot d_{\text{Tangent at vertex}}.$$</span>
Here <span class="math-container">$d_{\text{line}}$</span> represent... |
691,734 | <p>Consider the sequence defined recursively by $x_1$=$\sqrt2$ and where $x_n$=$\sqrt2$ + $x_n$$_-$$_1$. </p>
<p>Find a explicit formula for the $n^t$$^h$ term.</p>
<p>I considered using the general equation to find an explicit formula for any term in an arithmetic sequence. a$_n$ = a$_1$ + $d(n-1)$, but I came to no... | naslundx | 130,817 | <p>You are right in that it is an arithmetic series. A good strategy is to write up the first terms, simplify, and try to find a pattern:</p>
<p>$$a_1 = \sqrt{2}$$
$$a_2 = \sqrt{2} + a_1 = 2\sqrt{2}$$
$$a_3 = \sqrt{2} + a_2 = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$$
$$a_4 = \sqrt{2} + a_3 = \sqrt{2} + 3\sqrt{2} = 4\sqrt{2}$... |
617,927 | <p>Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$.</p>
<p><strong>Taylor Series</strong></p>
<p>$$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+...+\frac{(x-a)^r}{r!}f^{(r)}(a)+...$$</p>
<p>I've got my first term...</p>
<p>$f(a) = \sin(-1+1)\sin(-1+2)=\sin(0)\sin(1)=0$</p>
<p>Now, I... | nadia-liza | 113,971 | <p>use this formula
$\sin(a)\sin(b)=1/2(cos(a-b)-cos(a+b))$</p>
|
661,269 | <p>Check if $\mathbb{Z}_5/x^2 + 3x + 1$ is a field. Is $(x+2)$ a unit, if so calculate its inverse. </p>
<p>I would say that this quotient ring is not a field, because $<x^2 + 3x + 1>$ is not a maximal ideal, since $x^2 + 3x + 1 = (x+4)^2$ is not irreducible. </p>
<p>However, the result should still be a ring,... | voldemort | 118,052 | <p><strong>V.X</strong> will be Normal, as linear combination of normal rvs is again normal. Check <a href="http://www.statlect.com/normal_distribution_linear_combinations.htm" rel="nofollow">this</a>.</p>
|
1,376,651 | <p>To be specific here is the system:</p>
<p>$$x-2y=0 \tag{1}$$
$$x-2(k+2)y=0 \tag{2}$$
$$x-(k+3)y=-k \tag{3}$$ </p>
<p>I have already solved it for equations $(1)$ and $(2)$... what should I do with the 3rd equation?</p>
<p>Just to make sure everything goes well here is my method:</p>
<p>$D=-2(k+2)$ and $D_x=D_y=0... | MJD | 25,554 | <p>The answer is no; John can't even fill up the topmost $7\times 11\times 1$ slice of the $7\times 11\times 9$ box. Consider just the top $7\times 11$ face of this box; look just at this face and ignore the rest of the box. A solution to the problem would fill up this $7\times 11$ rectangle with large $3\times3$ rec... |
988,628 | <p>Problem : </p>
<p>For the series $$S = 1+ \frac{1}{(1+3)}(1+2)^2+\frac{1}{(1+3+5)}(1+2+3)^2+\frac{1}{(1+3+5+7)}(1+2+3+4)^2+\cdots $$ Find the nth term of the series. </p>
<p>We know that nth can term of the series can be find by using $T_n = S_n -S_{n-1}$ </p>
<p>$$S_n =1+ \sum \frac{(\frac{n(n+1)}{2})^2}{(2n-1)... | Bumblebee | 156,886 | <p>First we should note that $$1+2+3+...+n=\dfrac{n(n+1)}{2}$$ and $$1+3+5+...+(2n-1)=n^2.$$ Therefore general term in your series become to $$T_n=\dfrac{(1+2+3+..+n)^2}{1+3+5+...+(2n-1)}=\dfrac{(n+1)^2}{4}$$ $$S_n=\sum_{k=1}^nT_k\\=\sum_{k=1}^n\dfrac{(k+1)^2}{4}\\=\sum_{k=1}^{n+1}\dfrac{k^2}{4}-\dfrac{1}{4}\\=\dfrac{(... |
433,816 | <p>While I was studying the measurements of pressure at earth's atmosphere,I found the barometric formula which is more complex equation ($P'=Pe^{-mgh/kT}$) than what I used so far ($p=h\rho g$).</p>
<p>So I want to know how this complex formula build up? I could reach at the point of
$${dP \over dh}=-{mgP \over kT}$$... | nbubis | 28,743 | <p>This type of problem is known as a differential equation. In this particular case the solution can be guessed, since you have that the derivative of the function $P$ is just a constant times $P$. The only function satisfying this condition is the exponent, since:
$${d(A e^{a x}) \over dx} = a A e^x$$
Thus, if $${dP\... |
361,201 | <p>Let $\left\{ x_\alpha : \alpha \in \mathscr{A}\right\} \subset (0, + \infty ) $ be a set of positive real numbers such that for every countable subcollection $ \left\{ x_{\alpha_n} \right\} $ of distinct points it holds $ x_{\alpha_n} \rightarrow 0 $. Then $ \mathscr{A} $ is a countable set. \</p>
<p>I think that t... | xyzzyz | 23,439 | <p>Hint: for each $n$, there are only finitely many $x \in \mathcal{A}$ such that $|x| \geq \frac{1}{n}$ -- otherwise you can easily find a sequence in $\mathcal{A}$ that does not converge to 0.</p>
|
1,585,408 | <p>I have the equation: (1-x<sup>2</sup>)u<sup>''</sup> -xu<sup>'</sup>+ku=0,
where ' represents differentiation with respect to x and k is a constant.</p>
<p>I am asked to show that cos(k<sup>1/2</sup>cos<sup>-1</sup>x) is a solution to this equation.</p>
<p>I assumed to show this you need to set u=cos(k<sup>1/2</su... | JJacquelin | 108,514 | <p>$$(1-x^2)\frac{d^2u}{dx^2}-x\frac{du}{dx}+ku=0$$
Let $x=\cos(t)\quad$ hense $\quad dx=-\sin(t)dt_\quad$ then $\quad\frac{dt}{dx}=-\frac{1}{\sin(t)}$</p>
<p>$\frac{du}{dx}=\frac{du}{dt}\frac{dt}{dx}= -\frac{1}{\sin(t)}\frac{du}{dt}$</p>
<p>$\frac{d^2u}{dx^2}=\frac{d\left( -\frac{1}{\sin(t)}\frac{du}{dt}\right)}{dx}... |
2,296,544 | <p>Let $\{F_n\}, n\in \mathbb{N}$ be the sequence of Fibonacci numbers such that $F_1=1$, $F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ $\forall n\geq2$.</p>
<p>Define a new sequence $\{S_n\}$ such that $S_n=F_n+1$ $\forall n\in \mathbb{N}$.</p>
<p>Now the question is: For every prime $p$, does there exist an $N\in \mathbb{N}$,... | Angina Seng | 436,618 | <p>Well $S_{-2}=0$, but $-2\notin\Bbb N$. Never mind!</p>
|
2,953,837 | <p>Given <span class="math-container">$n_1$</span> number of a's, <span class="math-container">$n_2$</span> number of b's, <span class="math-container">$n_3$</span> number of c's.</p>
<p>They form a sequence using all these characters such that no two a's and no two b's are adjacent.</p>
<p>(a and b can be adjacent, ... | epi163sqrt | 132,007 | <p>This answer is based upon generating functions of <strong>Smirnov words</strong>. These are words with no equal consecutive characters. (See example III.24 <em>Smirnov words</em> from <em><a href="http://algo.inria.fr/flajolet/Publications/books.html" rel="nofollow noreferrer">Analytic Combinatorics</a></em> by Phil... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | Chain Markov | 407,165 | <p>In combinatorics there are quite many such disproven conjectures. The most famous of them are:</p>
<p>1) Tait conjecture:</p>
<blockquote>
<p>Any 3-vertex connected planar cubic graph is Hamiltonian</p>
</blockquote>
<p>The first counterexample found has 46 vertices. The "least" counterexample known has 38 vert... |
256,649 | <p>I am trying to plot phase diagram and poincare map. But I cannot get the poincare map as shown in the image below</p>
<p>Phase Space</p>
<pre><code>sol = NDSolve[{v'[t] ==
0.320 x[t] - 1.65 x[t]^3 - 0.005*v[t] + 0.855 Cos[1.2*t],
x'[t] == v[t], x[0] == 0, v[0] == 0}, {x, v}, {t, 0, 1500}];
ParametricPlot[... | Akku14 | 34,287 | <p>Not an answer, but an <strong>essential warning</strong> !</p>
<p>For t->1500 you have more than 300 oscillations of x[t]. Even at best Workingprecision numerical NDSolve will give total wrong result for higher t. See examples. A workaround could be solutions with DSolve (but my MMA version doesn't find any.)</p>... |
26,451 | <p>I am trying to solve the following:</p>
<p>$\begin{align*}
&X \sim N(1,1)\\
&\mathrm{cov}(X, X^3) = \text{?}
\end{align*}$</p>
<p>where $\mathrm{cov}$ is the covariance.</p>
<p>How would you do this in <em>Mathematica</em>?</p>
<p>I have tried</p>
<pre><code>X = NormalDistribution[1, 1]
cov[x_, y_] := ... | J. M.'s persistent exhaustion | 50 | <p>I guess something like this:</p>
<pre><code>d1 = NormalDistribution[1, 1];
xa = Mean[d1]; xa3 = Mean[TransformedDistribution[u^3, u \[Distributed] d1]];
Mean[TransformedDistribution[(x - xa) (x^3 - xa3), x \[Distributed] d1]]
6
</code></pre>
|
40,116 | <p>I'm building a program that calculates the cost of an item based on it's size (let's say a bamboo pole). As the customer requests a longer pole, it gets hard to find a bamboo, plus requires more resources to grow, therefore, I would want to charge more per inch for the piece of bamboo based on it's length approachin... | soandos | 10,921 | <p>You are looking for functions that go to infinity, preferably in finite time (vertical asymptote) or something similar. you then construct a scale to get to your infinity point at the number of inches you want (say 100). so, for tan(x) as an example, pi/2 is infinity, so map 0-100 to pi/4-pi/2, and you have somethin... |
443,578 | <blockquote>
<p>Is the limit
$$
e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n\to e^{-2x} \quad \text{as } \ N\to\infty \tag1
$$
uniform on $[0,+\infty)$? </p>
</blockquote>
<p>Numerically this appears to be true: see the difference of two sides in (1) for $N=10$ and $N=100$ plotted below. But the convergence is ve... | Pedro | 23,350 | <p>I'd like to provide another solution which is a mixture of Antonio's and Landscape's. One can also write $$\left|r_n(x)\right|=\int_0^x {e^{-t}}\frac{(x-t)^n}{n!}dt$$</p>
<p>by virtue of Taylor's theorem with the integral remainder. But then again $$\left| {{r_n}(x)} \right| \leqslant \int_0^x {\frac{{{{(x - t)}^n}... |
2,309,123 | <p>This is a 2 part question.</p>
<ol>
<li><p>I have been studying a particular matrix group $G \le GL(n,\mathbb R)$ with $n \ge 3$ and I managed to show elements of my group $A \in G$ have the block structure
$$
A = \left(
\begin{array}{cc}
O(3) & 0 \\
A_{21} & A_{22}
\end{array}
\right)
$$
Now $A_{22}$ must ... | Angina Seng | 436,618 | <p>For the first part, the answer is no. The direct product is the group
of block matrices
$$\pmatrix{A_{11}&0\\0&A_{22}}$$
with $A_{11}\in O(3)$ and $A_{22}\in GL(n-3,\Bbb R)$.</p>
|
1,055,091 | <p>I've been asked to estimate a y coordinate by using differentials. This normally isn't overly difficult, however, I'm not sure what to do in a case like this when y cannot be separated and used as a function of x. Can anyone point me in the right direction? I suspect I'll have to use implicit differentiation but I c... | FundThmCalculus | 153,550 | <p>Make the assumption that $x=x(t)$ and $y=y(t)$. We don't care what those functions are, it just allows us to differentiate with respect to $t$. We do this because of this definition:
$$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
So your equation looks like this:
$$2x(t)^3+2y(t)^3=9x(t)\cdot y(t)$$
Now differ... |
3,753,474 | <p><strong>Question:</strong></p>
<blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma$</span> are the roots of the equation, <span class="math-container">$x^3+x+1=0$</span>, then find the equation whose roots are: <span class="math-container">$({\alpha}-{\beta})^2,({\beta}-{\gamma})^2,({\gamma}-{\alpha... | farruhota | 425,072 | <p>The standard method:
<span class="math-container">$$a+b+c=0; ab+bc+ca=1; abc=-1;\\
a^2+b^2+c^2=-2;a^2b^2+b^2c^2+c^2a^2=1;a^4+b^4+c^4=2;\\
a^3=-a-1.$$</span>
First coefficient:
<span class="math-container">$$(a-b)^2+(b-c)^2+(c-a)^2=\\2(a^2+b^2+c^2)-2(ab+bc+ca)=-6$$</span>
Second coefficient:
<span class="math-contain... |
3,653,212 | <p>Due to Covid -19 , in our university quizzes are held online and it's hard to ask questions. </p>
<p>3 Days back in my Combinatorics quiz this question was asked on which I am struck. I couldn't solve it in the time alloted and struggled to find a proper strategy. </p>
<p>Question is ->Determine the number of non ... | Hagen von Eitzen | 39,174 | <p>As the symmetry group of the tetrahedron is <span class="math-container">$S_4$</span>, colourings are already equivalent if they use the same colours the same number of times. Thus we have</p>
<ul>
<li><span class="math-container">$k$</span> colourings of type <span class="math-container">$(a,a,a,a)$</span>,</li>
<l... |
971,139 | <p>$\max\{a,b\} = \frac12(a+b+|a-b|)$ and $\min\{a,b\} = \frac12(a+b-|a-b|)$</p>
<p>how would you go about solving this?</p>
<p>I started with suppose $a \leq b$</p>
<p>Also, show min{a,b,c} = min{min{a,b},c}.</p>
<p>How would I go about showing that?</p>
| Paul | 17,980 | <p>Suppose that $a\le b$.</p>
<p>$$\frac{a+b+|a-b|}{2}=\frac{a+b+b-a}{2}=b=\max \{a,b\}$$</p>
<p>And </p>
<p>$$\frac{a+b-|a-b|}{2}=\frac{a+b-(b-a)}{2}=a=\min \{a,b\}$$</p>
|
971,139 | <p>$\max\{a,b\} = \frac12(a+b+|a-b|)$ and $\min\{a,b\} = \frac12(a+b-|a-b|)$</p>
<p>how would you go about solving this?</p>
<p>I started with suppose $a \leq b$</p>
<p>Also, show min{a,b,c} = min{min{a,b},c}.</p>
<p>How would I go about showing that?</p>
| Mariano Suárez-Álvarez | 274 | <p>You can use the fact that $$\min\{x,y\}=-\max\{-x,-y\}$$ to get the result about mins from the one about maxes.</p>
|
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