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 |
|---|---|---|---|---|
2,903,920 | <p>Let $X$ be a set and $M$ be a sigma algebra of $X$. If $U \subset X$, is it true that the set $M_U = \{E \cap U|E \in M\}$ is a sigma algebra? I'm getting the countable union part, but not compliments:</p>
<p>$$(E\cap U)^C = E^C \cup U^C$$</p>
<p>I'm not sure what to do with $U^C$</p>
| drhab | 75,923 | <p>If $X$ is a set, $\mathcal M$ a $\sigma$-algebra on it and $f:Y\to X$ is a function then the collection $f^{-1}(\mathcal M)$ can be shown to be a $\sigma$-algebra on $Y$.</p>
<p>Observe that in that situation $f^{-1}(E^{\complement})=f^{-1}(E)^{\complement}$.</p>
<p>You can apply that here by taking $Y=U$ and $f:U... |
606,158 | <p>I'm really sorry for this basic, stupid question. I have been looking for answers online but I can't find any.</p>
<p>I don't understand the following summation:</p>
<p>$$\sum_{i=0}^{n-1}i=\frac{n(n-1)}2$$</p>
<p>I understand that the basic formula for summations of i is the following:</p>
<p>$$\sum_{i=1}^ni=\fr... | Brian M. Scott | 12,042 | <p>It’s really just a substitution. You have the formula $$\sum_{i=1}^ni=\frac{n(n+1)}2\;.\tag{1}$$ Suppose that $n=m-1$; then $(1)$ becomes</p>
<p>$$\sum_{i=1}^{m-1}i=\frac{(m-1)\big((m-1)+1\big)}2=\frac{(m-1)m}2=\frac{m(m-1)}2\;.\tag{2}$$</p>
<p>Now just change the name of the quantity $m$ in $(2)$ to $n$, and $(2)... |
147,379 | <p>I would like to prove by induction the following inequality:</p>
<p>$\frac{4^n}{n+1} < \binom{2n}{n}$, for all natural numbers n > 1.</p>
<p>Any hints?</p>
| Marc van Leeuwen | 18,880 | <p>One has (see <a href="https://math.stackexchange.com/questions/37971/identity-involving-binomial-coefficients/">this question</a> and <a href="https://math.stackexchange.com/questions/72367/proof-of-a-combinatorial-identity-sum-i-0n-2i-choose-i2n-i-choose-n/">this one</a>) the identity
$$
4^n=\sum_{i=0}^n\binom{2i... |
4,592,805 | <p>Let
<span class="math-container">$$
f_{(X,Y)}(x,y) = 2x
$$</span>
for <span class="math-container">$x \in (0,1), y \in (0,1)$</span>.
I need to compute density of <span class="math-container">$X+Y$</span>. So, I know that <span class="math-container">$X \perp Y$</span>, because
<span class="math-container">\begin{a... | Lee Mosher | 26,501 | <p>Since your surface is contained in the <span class="math-container">$y,z$</span> plane, you can use <span class="math-container">$y$</span> and <span class="math-container">$z$</span> as the parameters, although you'll have to stick a minus sign in there somewhere to get the correct normal vector.</p>
<p>So how abou... |
1,505,908 | <p>Let $A$ and $B$ be two $n\times n$ matrices, and $x$ be a $n\times 1$ column vector. What is the derivative of $f=(x^TAx)Bx$ with respect to $x$?</p>
<p>I try to calculate it like </p>
<p>$\frac{\partial f}{\partial x}=\frac{\partial (x^TAx)}{\partial x}Bx+(x^TAx)\frac{\partial (Bx)}{\partial x}=[x^T(A+A^T)]Bx+(x^... | hans | 285,581 | <p>$$\eqalign{
S &= \frac{1}{2}(A+A^T) \cr\cr
\alpha &= x^TAx \,= x^TSx \cr
d\alpha &= 2\,x^TS\,dx \cr\cr
f &= \alpha\,Bx \cr
df &= \alpha\,B\,dx + Bx\,d\alpha \cr
&= \alpha\,B\,dx + Bx\,(2x^TSdx) \cr
&= (\alpha\,B + 2\,Bxx^TS)\,dx \cr\cr
\frac{\partial f}{\partial x} &= \alpha... |
3,496,299 | <p>I used limit of the function at zero, and got that the limit is zero. So I said, while the limit existed and it is finite then the singularity is Removable Singularity. My function is <span class="math-container">$$f(z)=\frac{e^{z^{-1}}}{\sin(z^{-1})}$$</span> I think that I had something wrong while calculating the... | Umberto P. | 67,536 | <p>Hint: if the limit exists, then so does the <em>real</em> limit
<span class="math-container">$$\lim_{x \to 0^+} \frac{e^{x^{-1}}}{\sin(x^{-1})} = \lim_{t \to \infty} \frac{e^t}{\sin t}.$$</span></p>
|
2,208 | <p>What are some general guidelines for editing a MathOverflow post? Are there some "do"s and "don't"s I should know about? </p>
<p>(The material that was previously in this question box has been moved to an answer box, where it belongs, in response to a pertinent comment by Noam Elkies.) </p>
| Włodzimierz Holsztyński | 8,385 | <p>If you see a sloppy accidental error then certainly correct it. But when an author put special effort into editing--then changing it is ignorant, arrogant and insensitive, simply rude. You should ask, you should not waste the author's effort by "editing" according to your own ideas (even if they are standard).</p>
|
121,522 | <p>I have 2 lists. For example,</p>
<pre><code>list1 = {{a,b}, {c, d}, {e, f}, {g, h}}
list2 = {{1}, {2}, {3}, {4}}
</code></pre>
<p>I want to merge them such that I get the result:</p>
<pre><code>list3 = {{1,a,b}, {2,c, d}, {3,e, f}, {4,g, h}}
</code></pre>
<p>and the method generalizes to a large number of subli... | BoLe | 6,555 | <p>Either transpose one list against the other or thread over them. Then flatten each sublist. Internally operations are looped over the lists all right, but this is how one normally does things in Mathematica.</p>
<pre><code>transposed = Transpose[{list2, list1}];
threaded = Thread[{list2, list1}];
Flatten /@ transpo... |
3,144,546 | <p>As part of my attempt at solving <a href="https://math.stackexchange.com/questions/3140495/integral-involving-the-logarithm-of-trigonometric-function/3142176?fbclid=IwAR2AX0PmeNzOhLxNLSlOgS3hGW8rY42A-rfhU_4ImH-2uJFSxIu6A336QmM#3142176">this</a> integral, I became stuck with resolve the following definite integral:
<... | Yuri Negometyanov | 297,350 | <p>Substitution
<span class="math-container">$$\tan u = x$$</span>
gives:
<span class="math-container">$$\int\limits_0^{\pi/4}u^b\tan^a u\,\mathrm du
=\int\limits_0^1 x^a\arctan^b x \frac{\mathrm dx}{1+x^2}
=\frac1{b+1}\int\limits_0^1 x^a\,\mathrm d\arctan^{b+1}x$$</span>
<span class="math-container">$$=\frac1{b+1}x^a\... |
2,713,293 | <p>Let</p>
<p>$$f(x_1, x_2, x_3) := \sum m(2, 3, 4, 5, 6, 7)$$</p>
<p>With the normal SOP expression for this function, it must be, with the use of minterm:</p>
<p>$$f = m_2 + m_3 + m_4 + m _6 + m_7 = x_1'x_2x_3'+x_1'x_2x_3+x_1x_2'x_3'+x_1x_2x_3'+x_1x_2x_3$$</p>
<p>How can this function be simplified?</p>
<p>(I co... | Rodrigo de Azevedo | 339,790 | <p>Using <a href="http://live.sympy.org" rel="nofollow noreferrer">SymPy</a>:</p>
<pre><code>>>> from sympy import *
>>> x1, x2, x3 = symbols('x1 x2 x3')
>>> f = (Not(x1) & x2 & Not(x3)) | (Not(x1) & x2 & x3) | (x1 & Not(x2) & Not(x3)) | (x1 & x2 & Not(x3)) | ... |
271,088 | <p>I am studying the symmetries of a particular function,
$$
f: R^n \rightarrow R
$$
which leave $f(x)$ unchanged (i.e. so $f(Ax) = f(x)$ for some matrix $A \in R^{n \times n}$). I have found that my function is invariant under the action of a matrix group which satisfies the following equation:
$$
A X A^T = X
$$
where... | Friedrich Knop | 89,948 | <p>If I understand correctly, you would like to determine the group $G$ of all $A\in GL(n,\mathbb R)$ with $f(Ax)=f(x)$ for all $x\in\mathbb R^n$ where $f$ is of a very particular form. You noticed that if $Axx^TA^T=xx^T$ for all $x\in\mathbb R^n$ then $A\in G$. That is undoubtedly true but doesn't help much since $A=\... |
4,058,947 | <p>Have got into a pretty heated debate with a friend, and looking online there's lacking proof
<a href="https://i.stack.imgur.com/S30uM.png" rel="nofollow noreferrer">Line contained in a plane</a></p>
<p>Is a line that is contained within a plane, considered parallel to it? By my understanding it is parallel , if at a... | Somos | 438,089 | <p>This is a matter of opinion and convention, and one definition may be more useful than another definition for some purposes. This is common
in mathematics.</p>
<p>One definition of parallel lines in
a Euclidean plane is that they are equidistant from each other. This
breaks down in the context of the elliptic or hyp... |
3,614,178 | <p>I was going through this article on geometric series on Wikipedia and found this diagrammatic representation of an infinite geometric series with a said common factor of (1/2) but shouldn't the common factor be (1/4) based on the diagram?
link to the diagram:
<a href="https://commons.wikimedia.org/wiki/File:Geometr... | Z Ahmed | 671,540 | <p>To prove that:
<span class="math-container">$$\log_2 3 < \log_5 17 \implies \log_2 3< \frac{\log_2 17}{\log_2 5} ~~~(1)$$</span>
<span class="math-container">$$17 > 2^4 \implies \log_2 17 >4, ~~~(2)$$</span> <span class="math-container">$$ 25 < 32 \implies 5 <2^{5/2} \implies \log_2 5 < \frac{5}... |
2,831,968 | <p>$\DeclareMathOperator{\var}{var}$It is just a general question I could not get my mind around.</p>
<p>Assume that $E[X]= 20$ and $\var[X]= 5$, then$$
E[1.2X]= 1.2·E[X]= 1.2×20= 24= 20 + 4 = E[X] + E[0.2X],\\
\var[1.2X]= 1.44·\var[X]= 1.44×5= 7.2.
$$
For$$
\var[1.2X]= \var[X + 0.2X]= \var[X] + \mathord{??} = \var[0.... | Kenny Lau | 328,173 | <p>Recall that:</p>
<p>$$\operatorname{Var}[X] = E[X^2] - E[X]^2$$</p>
<p>Using the linearity of expectation, we derive:</p>
<p>$$\begin{array}{rcl}
\operatorname{Var}[X+Y]
&=& E[(X+Y)^2] - E[X+Y]^2 \\
&=& E[X^2] + 2E[XY] + E[Y^2] - E[X]^2 - 2E[X]E[Y] - E[Y]^2 \\
&=& \operatorname{Var}[X] + \... |
1,678,004 | <p>I am stuck on a word problem about picking teams. I thought it would be very simple, but to my surprise, I could not solve it. So here's the problem..</p>
<p>Andrea, Melissa, and Carol are in a class of 27 girls. The teacher chooses
students at random to make up teams of three. What is the probability that Andrea... | true blue anil | 22,388 | <p>Consider that there are $27$ slots available.</p>
<p>$\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\Large\boxed{.}\Large\boxed{.}\;\;\Large\boxed{.}\L... |
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| Satish Ramanathan | 99,745 | <p>An additional trick may cause a little less headache.</p>
<p>Let $I_1 = \int_{0}^{1} \sqrt{\dfrac{x}{1-x}}dx$</p>
<p>Let $I_2 = \int_{0}^{1} \sqrt{\dfrac{1-x}{x}}dx$</p>
<p>Then $I_1 + I_2 = \int_{0}^{1}\dfrac{1}{ \sqrt{x(1-x)}}dx$</p>
<p>and $I_1 - I_2 = \int_{0}^{1} \dfrac{2x-1}{\sqrt{x(1-x)}}dx$</p>
<p>Now ... |
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| Ali Shadhar | 432,085 | <p><span class="math-container">\begin{align}
I&=\int_0^1\sqrt{\frac{x}{1-x}}\ dx=\int_0^1x^{1/2}(1-x)^{-1/2}\ dx=\text{B}\left(\frac32,\frac12\right)\\
&=\frac{\Gamma(\frac32)\Gamma(\frac12)}{\Gamma(2)}=\frac12\Gamma^{\ 2}\left(\frac12\right)=\frac{\pi}{2}
\end{align}</span></p>
|
2,490,654 | <p>$$\int_0^1\sqrt\frac x{1-x}\,dx$$
I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$.<br>
I would like to see different approaches, can you provide them?</p>
| Greg Nisbet | 128,599 | <p><span class="math-container">$$ \int_0^1 \sqrt{\frac{x}{1-x}} \mathrm{d}x $$</span></p>
<p>Let <span class="math-container">$x = 1 - u^2$</span> and thus <span class="math-container">$\mathrm{d}x = -2u\mathrm{d}u$</span> . Note that the bounds are exchanged by this substitution, i.e. <span class="math-container">$u... |
3,964,172 | <blockquote>
<p>Does the following limit exist?
<span class="math-container">$$\lim_{x \to 0} x\sin\left(\frac{1}{x}\right)$$</span></p>
</blockquote>
<p>If yes, then I have to find it; if no, then I need to give reason why.</p>
<p>I couldn’t figure out how to prove this formally, can someone please help me out.</p>
| user0102 | 322,814 | <p><strong>First Approach</strong></p>
<p>To begin with, notice that
<span class="math-container">\begin{align*}
\left|x\sin\left(\frac{1}{x}\right)\right| \leq |x| < \varepsilon
\end{align*}</span></p>
<p>Thus, for every <span class="math-container">$\varepsilon > 0$</span>, there corresponds a <span class="math... |
4,052,683 | <p>If <span class="math-container">$\gcd(a,b,c) = 1$</span> and <span class="math-container">$c = {ab\over a-b}$</span>, then prove that <span class="math-container">$a-b$</span> is a square. <span class="math-container">$\\$</span><br />
Well I tried expressing <span class="math-container">$a=p_1^{a_1}.p_2^{a_2} \cdot... | Bill Dubuque | 242 | <p>Note <span class="math-container">$\,(a\!-\!b)(a\!+\!c)=a^2\,$</span> and <span class="math-container">$\, \overbrace{(\color{#c00}{a\!-\!b,a\!+\!c})=1}^{\text{see below}}\,$</span> <a href="https://math.stackexchange.com/a/21637/242">hence</a> <span class="math-container">$\,a\!-\!b\,$</span> is a square. <span cla... |
731,624 | <p>For a set $\mathbb{X}$ given order relation extended from that of $\mathbb{R}$, if $\mathbb{X} \supseteq \mathbb{R}$ then $\mathbb{X} = \mathbb{R}$ ?</p>
<p>Motivation for this question rose from an intuitive question that when I draw a virtual continuous straight line between $1$ and $2$ and virtually pick a point... | user2345215 | 131,872 | <p>No. You can even order the whole plane $\mathbb R^2$ by $(x,y)<(x',y')$ if $x<x'$ or ($x=x'$ and $y<y'$).</p>
<p>Now every horizontal line or vertical line is a copy of $\mathbb R$.</p>
|
2,632,392 | <blockquote>
<p>What is the remainder of $23! +29!$ divided by $13! +19!$?</p>
</blockquote>
<p><strong>Attempt</strong></p>
<p>$$23!+29!=23![1+(29×28×27×26×25×24)]$$ and </p>
<p>$$13!+19!=13![1+(19×18×17×16×15×14)]$$</p>
<p>Now how do I proceed..I am stuck at this initial stage.</p>
<p>Please help.</p>
| TheSimpliFire | 471,884 | <p><strong>Partial answer</strong></p>
<p>$$\begin{align}\frac{23!+29!}{13!+19!}&=\frac{23!(1+24\cdot25\cdot26\cdot27\cdot28\cdot29)}{13!(1+14\cdot15\cdot16\cdot17\cdot18\cdot19)}\\&=\frac{23!}{13!}\cdot\frac{2\cdot3\cdot7\cdot2+(2\cdot24)\cdot(3\cdot25)\cdot(7\cdot26)\cdot(2\cdot27)\cdot28\cdot29}{13\cdot5\cd... |
1,714 | <p>Say I have a function $f(x)$ that is given explicitly in its functional form, and I want to find its Fourier transform[1]. If $f$ is too complicated to have an analytic expression for $\hat f(k)$, how do I obtain it numerically?</p>
<p>The naive and stupid way, which I currently use, is evaluating the integral for ... | David Z | 79 | <p>If you take a look at the documentation, Mathematica's symbolic Fourier transform function, <a href="http://reference.wolfram.com/mathematica/ref/FourierTransform.html"><code>FourierTransform</code></a>, computes</p>
<p>$$\hat f(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{ikx}\mathrm{d}x$$</p>
<p>You c... |
2,978,793 | <p>I want to show that</p>
<p><span class="math-container">$$\sum_{i=1}^n i^{2-\alpha} \sim c n^{3-\alpha}$$</span></p>
<p>for some constant <span class="math-container">$c$</span>. Here, <span class="math-container">$\alpha$</span> is assumed <span class="math-container">$0 < \alpha < 1$</span>.</p>
<p>I know... | KingofHeadbutt | 474,648 | <p>If F is a subalgebra of G, then E[E[X|F]|G]=E[E[X|G]|F]=E[X|F].</p>
<p>First, E[E[X|F]|G]=E[X|F] because E[X|F] is G-measurable.</p>
<p>To prove that E[E[X|G]|F]=E[X|F], for all F-measurable event A, E[E[E[X|G]|F]1_A]=E[E[E[X|G]1_A|F]]=E[E[X|G]1_A]=E[E[X1_A|G]]=E[X1_A].</p>
<p>since E[E[X|G]|F] is F-measurable, i... |
1,229,164 | <p>How to solve modular equations?
So for example $a \equiv i$ mod $x$, $a \equiv j$ mod $y$ for some given $i,j,x,y$ with $gcd(x,y)=1$, and I must find $a$ mod $x*y$.
Any tips on how to do this?
Specifically I want to calculate $a \equiv 1$ mod $16$, $a \equiv 3$ mod $17$, for example.</p>
| Makuta | 226,090 | <p>Since $a \equiv i \mod x$, we have $a = mx + i$, for some $m$ and likewise for your other equation, $a = ny + j$, for some $n$. Since both are $a$ equal something, they have to be equal to each other. So you have to solve $mx - ny = j - i$, for $m$ and $n$.</p>
|
3,633,495 | <p>Quadratic functions are of the form:
<span class="math-container">$$f(x)=b^Tx+\frac{1}{2}x^TCx$$</span>
where <span class="math-container">$C$</span> is assumed without loss of generality to be symmetric.
Tried to prove it myself using contradiction but I didn't find a valid argument.</p>
| Daniel Huff | 771,793 | <p>If <span class="math-container">$C$</span> is not positive definite, there is an eigenvalue-eigenvector pair <span class="math-container">$(\lambda,v)$</span> where <span class="math-container">$\lambda\leq 0$</span>.</p>
<p>Consider now the function <span class="math-container">$f(x)$</span> evaluated along the di... |
3,800,728 | <blockquote>
<p>If <span class="math-container">$y''-15xy'+8y=e^{x}$</span> and <span class="math-container">$y(0)=y'(0)=-2$</span>. Then find first <span class="math-container">$5$</span> non zero terms of series solution of that equation about <span class="math-container">$x=0$</span></p>
</blockquote>
<p>What i try... | Vincenzo Tibullo | 6,266 | <p>A different method, that can be easily applied even if you don't know the series development of the RHS (but calculating its derivatives is more or less equivalent).</p>
<p>Suppose <span class="math-container">$y$</span> can be developed Maclaurin series
<span class="math-container">$$
y(x)=\sum_{k=0}^\infty\frac{y^... |
3,260,068 | <p>I've tried a couple of things trying to solve this problem but I get no answer.</p>
<p>These are one of the few things I know about “Gcd” and division:</p>
<p>If <span class="math-container">$a\mid b$</span> and <span class="math-container">$a \mid c$</span>, then <span class="math-container">$a \mid b \cdot x + c... | copper.hat | 27,978 | <p>Generally, the solution to <span class="math-container">$\min_{\|x\|=1} g^T x$</span> is <span class="math-container">$x = -{1 \over \|g\|} g$</span>.
This follows from Cauchy Schwartz.</p>
<p>The above problem can be written as
<span class="math-container">$\min_{\|x\|_P = 1} g^T x = \min_{\|\sqrt{P}x\| = 1} g^T ... |
411,309 | <p>I'm trying to understand the equation:</p>
<p>$$\frac{1}{2\pi i} \int_C \left( \frac{x}{n} \right)^s \frac{ds}{s} = \theta(x-n).$$</p>
<p>Here $x\in \mathbb{R}, x\geq 0$, and $C = \{s:\operatorname{Re}(s) = \sigma\}$ is a coutour, with fixed $\sigma > 0$ and $t = \operatorname{Im}(s)$ ranging.</p>
<p>This is ... | Brian Rushton | 51,970 | <p>This is a residue problem; the contour can be thought of as the limit of semicircles pointing to the right or left. If $x/n$ is bigger than 1, you complete the contour on the left, so that in the limit the exponential part goes to 0 in the semicircular arc. If its less than 1, complete it to the right. There is a si... |
3,713,580 | <blockquote>
<p>Question: Prove that <span class="math-container">$$\lim_{n\to\infty}(n!)^\frac1n=+\infty.$$</span></p>
</blockquote>
<p>Solution 1: Let <span class="math-container">$\{a_n\}_{n\ge 1}$</span> be such that <span class="math-container">$$a_n:=(n!)^\frac{1}{n}, \forall n\in\Bbb N.$$</span> Now clearly <spa... | Kavi Rama Murthy | 142,385 | <p>An alternative approach: For <span class="math-container">$1 \leq k \leq n$</span> we have <span class="math-container">$n!>(k)(k)..(k) =k^{n-k}$</span> (where there are <span class="math-container">$n-k$</span> factors in the product). Hence <span class="math-container">$(n!)^{1/n} >k^{1-\frac k n}$</span>. ... |
181,797 | <p>I have a table with desired inputs of a given function <code>efvc[n,m,u]</code>. I want to calculate for different values of <code>n</code>, <code>m</code> and <code>u</code>. I imported a Excel table with the values I want. Is there a way to call each row of the table as <code>n</code>,<code>m</code>,<code>u</code>... | Mike Honeychurch | 77 | <p>Note that you can also add a new definition for your function to make processing easier:</p>
<pre><code>efvc[{n_, m_, u_}] := efvc[n, m, u]
</code></pre>
<p>then it becomes a simple mapping:</p>
<pre><code>efvc /@ data[[1]]
</code></pre>
<p>or</p>
<pre><code>Map[efvc, data, {2}]
</code></pre>
|
1,399,921 | <p>Suppose that $A$ and $B$ are invertible, $p \times p$ matrices. If $A^n = B$ and I know all of the entries in $B$, can I find an $A$ for some or all integers $n \ge 0$? How many solutions for $A$ exist? If I'm thinking correctly, then $A = B * (A^{-1})^{n-1},$ but this is sort of self referential. Thanks!</p>
| Hamza | 185,441 | <p>We can use diagonalisation of $B$ to constract a solution of this equation, that mean if $B=PDP^{-1}$ and we suppose that $D=Diag(\lambda_1,\lambda_2,\dots,\lambda_k)$ if you put $A=PRP^{-1}$ where
$$
R=Diag(\sqrt[n]{\lambda_1},\sqrt[n]{\lambda_2},\dots,\sqrt[n]{\lambda_k})
$$
So you can verify that $A^n=B$</p>
|
1,744,832 | <p>Given $$A=\begin{bmatrix}
-4 & 3\\
-7 & 5
\end{bmatrix}$$ Find $A^{482}$ in terms of $A$</p>
<p>I tried using Characteristic equation of $A$ which is $$|\lambda I-A|=0$$ which gives</p>
<p>$$A^2=A-I$$ so $$A^4=A^2A^2=(A-I)^2=A^2-2A+I=-A$$ so</p>
<p>$$A^4=-A$$ but $482$ is neither multiple of $4$ nor Powe... | Eclipse Sun | 119,490 | <p>This is not true. $3\times15^2+1=676=26^2$.</p>
|
27,233 | <p>A finitely presented, countable discrete group $G$ is amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m(gX)=mX$ for all $X\subseteq G \backslash${$e$} and all $g \in G.$ </p>
<p>A countable discrete group $G$ is inner amenable if there is ... | Owen Sizemore | 5,732 | <p>Hey Jon</p>
<p>So my initial thought would be no. </p>
<p>First, in full generality every group is virtually inner-amenable. Meaning that for any group $G$, the group $G \times \mathbb{Z}/2\mathbb{Z}$ is inner amenable. In fact, any non-icc group is inner amenable just by taking the mean to be the counting measure... |
98,348 | <p><a href="https://math.stackexchange.com/a/47261">A</a> <a href="https://math.stackexchange.com/a/15210">few</a> <a href="https://math.stackexchange.com/a/69164">answers</a> here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi:</p>
<p>$$x^{1-s} < \frac{\Gamma(x... | robjohn | 13,854 | <p>The strict log-convexity of $\Gamma$ (see the end of <a href="https://math.stackexchange.com/a/101007/">this answer</a>) implies that for $0< s <1$,
$$
\Gamma(x+s)<\Gamma(x)^{1-s}\Gamma(x+1)^s=x^{s-1}\Gamma(x+1)\tag{1}
$$
which yields
$$
x^{1-s}<\frac{\Gamma(x+1)}{\Gamma(x+s)}\tag{2}
$$
Again by the stri... |
98,348 | <p><a href="https://math.stackexchange.com/a/47261">A</a> <a href="https://math.stackexchange.com/a/15210">few</a> <a href="https://math.stackexchange.com/a/69164">answers</a> here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi:</p>
<p>$$x^{1-s} < \frac{\Gamma(x... | J. M. ain't a mathematician | 498 | <blockquote>
<p>I'll probably leave this standing for two days before posting a summary of Gautschi's paper.</p>
</blockquote>
<p>Here is the <em>long-overdue</em> follow-through. I have slightly changed a few notations, but this is otherwise Gautschi's original argument.</p>
<hr>
<p>What Gautschi actually proves ... |
2,204,642 | <p>I have a hard time believing that there can exist a bijection $f:\Bbb R^2\to \Bbb R$.</p>
<p>I just cannot get around my intuition that a one-to-one map of a one-dimensional space (or interval) must also be one-dimensional.</p>
<p>I am not a mathematician. I am interested in the topic since it has potential releva... | Reese Johnston | 351,805 | <p>Digits and convergence do <em>not</em> behave well together. If $x_1,x_2,\ldots$ converges to $x$, there may never be an $n$ so that even the <em>first</em> digit of $x_n$ is the same as that of $x$. Consider, for example, the sequence $0.9$, $0.99$, $0.999$, $\ldots$, and its partner $1.1$, $1.01$, $1.001$, $\ldots... |
965,819 | <p>So I am doing a question were I have the set
column matrix 1 = (3, -8, 1) and column matrix 2 = (6, 2, 5)
and the question is asking if this is either a bases for R2 or R3.
Can I just say that since the matrix is not a square matrix (nxn) it cannot be a bases for R3 since it is not invertible -- it's not invertible... | Sofia | 182,196 | <p>For the first question you can try to apply the triangle inequality for real numbers to $f(x)-f(z)$ and $f(z)-f(y)$ (i.e. for real numbers $|a+b|\le |a|+|b|$).</p>
<p>For the second question, first assume that $d(u,v)\ge d(w,x)$, such that you can remove the absolute value, and the statement you want to prove is th... |
2,821,613 | <p>I came across following problem from Sheldon Ross' book:</p>
<blockquote>
<p>A closet contains 10 pairs of shoes. If 8 shoes are randomly
selected, what is the probability that there will be </p>
<ol>
<li>no complete pair; </li>
<li>exactly one complete pair</li>
</ol>
</blockquote>
<p>I solved th... | Boyku | 567,523 | <p>This is a problem similar to deck cards and hands problems.</p>
<p>The twenty shoes may be seen as a deck of two color (Left and Right) and 10-suits.</p>
<p>A "hand" contains 8 cards. It may contain 0 pairs, 1 pair... to 4 pairs, and we are asked to evaluate the chances to get some of these types.</p>
<p>the tota... |
3,926,603 | <p>So im learning about limits from tutorials and at the same time im solving the examples myself. Im getting solutions that sometimes differ from the explainer's.</p>
<p>Lets take this limit and the way it is solved on the video:</p>
<p><span class="math-container">$$\lim_{x \to -2} \dfrac { (x^3 - x^2 - 6x)}{(x^2+2x)... | Lubin | 17,760 | <p>Well, thanks to Hagen and to you, the polynomial that would have the solution(s) among its roots is the octic
<span class="math-container">$$
x^8-8x^6+20x^4-16x^2-x-2\,.
$$</span>
Rational Root Theorem tells you where to look, <span class="math-container">$\pm1$</span> and <span class="math-container">$\pm2$</span>,... |
3,640,587 | <blockquote>
<p>How do I find the <span class="math-container">$\angle BCA$</span>?</p>
<p><a href="https://i.stack.imgur.com/zOux6m.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/zOux6m.png" alt="enter image description here" /></a></p>
</blockquote>
<p>I tried to find the formula of the parabola, ... | José Carlos Santos | 446,262 | <p>If <span class="math-container">$x\in B$</span>, then, since <span class="math-container">$B$</span> is open and <span class="math-container">$B\subset A$</span>, <span class="math-container">$x$</span> is an interior point of <span class="math-container">$A$</span>. In other words, <span class="math-container">$x\i... |
1,447,110 | <p>This is from an MCQ contest.</p>
<blockquote>
<p>Consider the two functions: $f(x)=\ln(1+x^2)+x+2$ et $g(x)=ch(x)+sh(x)$.</p>
<p>The real number $c$ such that: $(f^{-1})'(2)=g(c)$</p>
<ul>
<li>$1]$ $c=-1$</li>
<li>$2]$ $c=0$</li>
<li>$3]$ $c=1$</li>
<li>$4]$ None of the above statements is corre... | Mark Viola | 218,419 | <p>You were on the right track. Note that </p>
<p>$$\frac{df^{-1}(x)}{dx}=\left.\frac{y^2+1}{(y+1)^2}\right|_{y=f^{-1}(x)}$$</p>
<p>Now, when $f^{-1}(2)=y$, $\log (1+y^2)+y=0\implies y=0$. So, we have $f^{-1}(2)=0$ and this means that </p>
<p>$$\left.\frac{df^{-1}(x)}{dx}\right|_{x=2}=\left.\frac{y^2+1}{(y+1)^2}\r... |
3,232,174 | <blockquote>
<p>Andy has a cube of edge length 8 cm. He paints the outside of the cube red and then divides the cube into smaller cubes, each of edge length 1 cm. Andy randomly chooses one of the unit cubes and rolls it on a table. If the cube lands so that an unpainted face is on the bottom, touching the table, what... | Jean-Claude Arbaut | 43,608 | <p>Hint</p>
<p>Bayes' theorem. It will be helpful to consider the <span class="math-container">$4$</span> kinds of cubes: from the corner of the initial <span class="math-container">$8\times8$</span> cube, or from an edge, or from the interior of a face, or from inside the cube. Count how many there are of each kind (... |
1,263,887 | <p>I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the determinant $0$ and therefore the matrix <em>is not</em> invertible?</p>
| Eugene Zhang | 215,082 | <p>The diagonal of a skew-symmetric matrix is always $0$ does not mean that its determinant be $0$. Look at following example:</p>
<p>$det\left[ \begin{array}{}
0 & 1 \\
-1 & 0 \\
\end{array} \right]=1
$</p>
<p>Its inverse is:
$\left[ \begin{array}{}
0 & -1 \\
1 & 0 \\
\end{array} \r... |
2,050,426 | <p>My question is really simple. How can I show intuitively to my complex analysis students that the sine function is unbounded? What kind of behavior makes the complex sine function different from the real one in this sense?</p>
| C. Dubussy | 310,801 | <p>The function $$\sin : z \mapsto \sin(z)$$ is holomorphic on $\mathbb{C}.$ Hence, if it is bounded, it is constant by Liouville's theorem. It is of course a contradiction. </p>
|
201,370 | <p>I would like to plot entropy on a 2-simplex, i.e., I want to plot a function for x,y,z s.t. x+y+z=1. My strategy is to take bounds {x,0,1}, {y,0,1-x}, and compute z=1-x-y. However, there is some problem with logarithm in this approach. The following yields an empty plot:</p>
<pre><code>Plot3D[x + Log[y], {x, 0, 1},... | ccosm | 66,148 | <p>I'm not sure why the second example works while the first one doesn't, but a better strategy might be to specify a region to plot over instead of using the range plotting parameters.</p>
<pre><code>R = ImplicitRegion[0 <= y <= 1 - x, {{x, 0, 1}, y}];
Plot3D[x + Log[y], {x, y} \[Element] R]
</code></pre>
<p>... |
3,587,851 | <p>I know that for any two sets <span class="math-container">$A,B$</span>, it holds that <span class="math-container">$A\subseteq B$</span> iff every element of <span class="math-container">$A$</span> is in <span class="math-container">$B$</span>, intuitively. I also know that it is reflexive, antisymmetric and transit... | Matt Samuel | 187,867 | <p>The answer is that you would restrict this to some universe that <em>is</em> a set if you want to get a poset. Otherwise the relation, while it exists, is a proper class and hence does not define a partially ordered <em>set</em>. In most subjects we are only concerned with sets contained in some larger set, hence th... |
39,802 | <p>Suppose $f:X \to Y$ is a function of sets. Then we can take the quotient $X/\text{~}$ by identifying $x \text{~} y$ if and only if $f(x)=f(y)$. Now suppose instead that $f:X \to Y$ is a map of simplicial sets. I want to emulate this homotopically, by adding a 1-simplex between $x$ and $y$ if there is a 1-simplex fro... | Urs Schreiber | 381 | <p>I'd think the construction in question is the <a href="http://ncatlab.org/nlab/show/coimage#InInfCat" rel="nofollow">homotopy coimage</a> of $f$ (it's unfortunately called "coimage" even though it behaves like the image).</p>
<p>First one forms the homotopy Cech nerve</p>
<p>$$
C(f) =
\left(
\cdots... |
4,050,660 | <p>A car license plate consists of 3 capital letters of the English alphabet in the first 3 positions of the license plate followed by 4 digits from 0 to 9. How many different plates can we have if neither the same letter nor the same digit on the plate is allowed to be repeated?</p>
<p>I was wondering if the right an... | N. F. Taussig | 173,070 | <p>You have not taken the number of letters or digits into account.</p>
<p>Answer the following questions, keeping in mind that neither a letter nor a digit may be repeated.</p>
<ol>
<li>In how many ways can the first position be filled with a letter?</li>
<li>Given that the first position has been filled with a letter... |
1,533,762 | <blockquote>
<p>Prove that $ 16^{20}+29^{21}+42^{22}$ is divisible by $13$.</p>
</blockquote>
<p>This is not a homework question. I would like to know how to solve this type of problems, I solved similar problem with n in exponent, but that could be proved by induction. Here I guess, Euler's theorem could be useful,... | Plankton | 273,762 | <p>$16 \equiv 29 \equiv 42 \equiv 3 \text{ mod } 13$. Thus, the claim follows from $3^{20}+3^{21}+3^{22}=3^{20}(1+3+9)=3^{20}13$.</p>
|
2,820,247 | <p><a href="http://www.math.hawaii.edu/~tom/old_classes/412notes6.pdf" rel="nofollow noreferrer">Source</a></p>
<blockquote>
<p>Theorem 6.10</p>
<p>Let $f : R → S$ be any homomorphism of rings and let $K = \ker f$.
Then
$K$ is an ideal in $R$.</p>
<p>Proof. We know $0 ∈ K$, so $K \neq ∅$. Let $a, b ∈ K... | Robert Lewis | 67,071 | <p>Homomorphism $f:R \to S$ <strong><em>does</em></strong> in fact give </p>
<p>$f(0_R) = 0_S, \tag 1$</p>
<p>since </p>
<p>$f(a + b) = f(a) + f(b), \; a, b \in R; \tag 2$</p>
<p>thus</p>
<p>$f(0_R) + f(0_R) = f(0_R + 0_R) = f(0_R), \tag 3$</p>
<p>whence</p>
<p>$f(0_R) = f(0_R) - f(0_R) = 0_S; \tag 4$</p>
<p>(4... |
2,816,184 | <p>In the first chapter of Gouvea's intro to $p$-adics, there's a heuristic argument that</p>
<p>$$ \frac{2}{1}+\frac{2^2}{2}+\frac{2^3}{3}+\frac{2^4}{4}+\cdots=0 \tag{$\ast$}$$</p>
<p>as $2$-adic numbers, since it's the Mercator series for $\ln(-1)$ and $2\ln(-1)=\ln(-1)^2=\ln1=0$.</p>
<p>(Like I said, heuristic.)<... | reuns | 276,986 | <p>Like you said <span class="math-container">$$-\log(1-x)=\sum_{n\ge 1} \frac{x^n}{n}\in \Bbb{Q}[[x]]$$</span> converges in <span class="math-container">$\Bbb{Z}_2$</span> for <span class="math-container">$x\in 2\Bbb{Z}_2$</span> and the formal series in <span class="math-container">$\Bbb{Q}[[x]]$</span> satisfies <sp... |
31,562 | <p>How to evaluate the number of ordered partitions of the positive integer <span class="math-container">$ 5 $</span>?</p>
<p>Thanks!</p>
| André Nicolas | 6,312 | <p>Since $5$ is a smallish number, it is reasonable to try to list all of the ordered partitions, and then count. First maybe, lest we forget, write down the trivial partition $5$. Then write down $4+1$, $1+4$. Now list all the ordered partitions with $3$ as the biggest number. This is easy, $3+2$, $2+3$, $3+1+1$, $1... |
3,080,402 | <p>I need to show that the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^{-n}}{n!}$</span> is convergent.</p>
<p>I invoked the limit comparison with the series <span class="math-container">$\sum_{n=1}^{\infty}\frac{c^n}{n!}$</span> which is absolutely convergent (and hence convergent).</p>
<p>I got ... | Servaes | 30,382 | <p><strong>Hint:</strong> If <span class="math-container">$a+b=c+d$</span> and <span class="math-container">$ab=cd$</span> then <span class="math-container">$\{a,b\}=\{c,d\}$</span>. Use the identity
<span class="math-container">$$x^3+y^3=(x+y)(x^2-xy+y^2)=(x+y)^3-3xy(x+y).$$</span></p>
|
2,405,709 | <p><strong>Problem</strong></p>
<p>A math book wants me to prove that given two natural numbers $m, \ n$ are not divisible by $5$, then the difference $m^4 - n^4$ <em>is</em> divisible by $5$.</p>
<p><strong>Thoughts</strong></p>
<p>The only method I can think of now, is to go through all the possible ways of writin... | Alec | 108,039 | <p>I'm going to include an alternative solution here, in addition to the already good answers. This one is inspired by @MarkBennet's answer.</p>
<p>$$\begin{align*} m^4-n^4&=(m^2-n^2)(m^2+n^2)\\ &=(m-n)(m+n)(m^2+n^2)\\ &\equiv (m-n)(m+n)(m^2-4n^2)\\ &= (m-n)(m+n)(m-2n)(m+2n)\\ &\equiv (m-n)(m-2n)(m... |
2,274,839 | <p>I can get,</p>
<p>$$h_n \approx \ln n+\gamma$$</p>
<p>By, saying</p>
<p>$$\frac{h_{n}-h_{n-1}}{n-(n-1)}=\frac{1}{n}$$</p>
<p>Then letting $h_n=h(n)$ be continuous and differentiable on $(0,\infty)$. So that,</p>
<p>$$h'(n) \approx \frac{1}{n}$$</p>
<p>And thus,</p>
<p>$$h_n \approx \ln n+C$$</p>
<p>Where we ... | Robert Z | 299,698 | <p>Let $x_n=H_n-(\ln n+\gamma)$ with $H_n=\sum_{k=1}^n\frac{1}{k}$ and evaluate the following limit by using the <a href="https://math.stackexchange.com/questions/599204/stolz-cesaro-theorem-0-0-case">Stolz-Cesaro Theorem</a> ($\ast/0$ case),
$$\lim_{n\to \infty}\frac{x_n}{1/n}=\lim_{n\to \infty}\frac{x_{n+1}-x_n}{1/(n... |
63,315 | <p>I am trying to identify a sequence related to the von Mangoldt function matrix. Since I believe/conjecture that the columns in the matrix have period lengths as in this sequence b:</p>
<pre><code>b = Table[Product[Exp[MangoldtLambda[n]], {n, 1, k}], {k, 1, nn}];
</code></pre>
<p>and since "b" grows rather fast, th... | george2079 | 2,079 | <p>my stab at it, unfortunately only a marginal improvement in time:</p>
<pre><code> ii=13;
Clear[a, b];
b = FoldList[Times, 1, Table[ Exp[MangoldtLambda[n]], {n, 2, ii}]];
a = Prepend[Table[
Limit[
Zeta[s] Total[Exp[Divisors[n]]^(s - 1) MoebiusMu[Divisors[n]]],
s -> 1],
{n, 2... |
110,599 | <blockquote>
<p>Why isn't $M = \mathbb{C}[x,y,z]/(xz-y)$ a flat $R = \mathbb{C}[x,y]$-module?</p>
</blockquote>
<p>The reason given on the book is "the surface defined by $y-xz$ doesn't lie flat on the $(x,y)$-plane". But I don't understand why this can be a reason.</p>
<p>Since</p>
<blockquote>
<p>An $R$-module... | Michael Joyce | 17,673 | <p>How much algebraic geometry have you had? Their explanation is based on the fact that the fibers of flat morphism have constant dimension. In your case, the ring homomorphism $R \rightarrow M$ corresponds to the map of schemes $X := \text{Spec } M \rightarrow Y := \text{Spec } R$. The fibers have dimension $0$ ($... |
102,976 | <p>Is there a way to evaluate a string containing RPN in Mathematica?</p>
<p>SE thinks this question is too short, so let me expand on it.
Do you know of any function, that provides the following functionality?</p>
<pre><code>EvalRPN["5 4 + 3 /"]
</code></pre>
<blockquote>
<p>3</p>
</blockquote>
<p>Or even symbol... | Dr. belisarius | 193 | <pre><code>rpn[{}, stackipt_: {}] := First@stackipt
rpn[list_, stack_: {}] := Module[{ops = {Plus, Subtract, Times}},
If[MemberQ[ops, First@list],
rpn[Rest@list, Prepend[stack[[3 ;;]], (First@list) @@ stack[[2 ;; 1 ;; -1]]]],
rpn[Rest@list, Prepend[stack, First@list]]]]
rpn[{3, 2, 5, Plus, Times}]
(* 21 *)... |
1,350,147 | <p>I was wondering whether, for a given finite-dimensional manifold, the connection $\nabla$ exists and is uniquely defined?</p>
<p>Afais for Riemannian manifolds, there exists always exactly one Levi-Civita connection, but the calculation is rather cumbersome.</p>
<p>Now, if we consider manifolds without a metric, i... | Blazej | 155,834 | <p>Without assumption of compatibility with metric and vanishing torsion connection is never unique; once you choose local coordinates, connection coefficients in these coordinates can be chosen arbitrarily. It's demanding of vanishing torsion and compatibility with the metric that fixes them on Riemannian manifold. No... |
3,004,210 | <p>A smilar question has been asked before <a href="https://math.stackexchange.com/questions/23503/create-unique-number-from-2-numbers">Create unique number from 2 numbers</a>.</p>
<blockquote>
<p>is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 an... | David G. Stork | 210,401 | <p>If order is important--i.e., the number you create from <span class="math-container">$(a,b)$</span> must be different from <span class="math-container">$(b,a)$</span>--then exploit the Unique Factorization Theorem to create the number:</p>
<p><span class="math-container">$n = {\rm Prime}[a]^2 \rm{Prime}[b]$</span> ... |
56,337 | <p>Suppose I have the following list:</p>
<pre><code>list = {a, b, c, d}
</code></pre>
<p>I want to generate this result:</p>
<pre><code>{{f[a, a], f[a, b], f[a, c], f[a, d]}, {f[b, b], f[b, c],
f[b, d]}, {f[c, c], f[c, d]}, {f[d, d]}}
</code></pre>
<p>What could be the shortest way?</p>
<p>The list elements can... | alancalvitti | 801 | <pre><code>Thread[f[First@#, #]] & /@
NestList[Drop[#, 1] &, list, Length[list] - 1]
</code></pre>
<p>The above is a refinement of less efficient 1st attempt:</p>
<pre><code>First@Outer[f, #, #] & /@
NestList[Drop[#, 1] &, list, Length[list] - 1]
</code></pre>
|
119,981 | <p>Let $C/\mathbb Q$ be a smooth projective curve of genus $g\geq 2$ or a smooth affine curve of genus $g \geq 1$. The exact sequence</p>
<p>$1 \to \pi_1^{et}(C \otimes_\mathbb Q \bar{\mathbb Q}) \to \pi_1^{et}(C) \to \operatorname{Gal}(\bar{\mathbb Q}|\mathbb Q) \to 1$</p>
<p>gives a homomorphism from $\operatorname... | Mozibur Ullah | 22,002 | <p>I think economics is far more closely connected with the body politic than it with mathematics. Also applying mathematics to economics is also a political act and a signification. (Mathematics has an association with permanance which can be used symbolically to shore up a certain contingent political/economic order)... |
4,076,584 | <blockquote>
<p>Let <span class="math-container">$G$</span> be the relation in <span class="math-container">$\Bbb {R}^2$</span> given by <span class="math-container">$$ G = \{((a, b), (c, d)) \in \Bbb {R}^2 \times \Bbb {R}^2: a ^ 2 + b ^ 2 = c ^ 2 + d ^ 2 \}. $$</span> Prove that <span class="math-container">$ G $</spa... | yavagi | 904,120 | <p><span class="math-container">$\nabla \nabla^T$</span> is not a common notation.</p>
<p>Although sometimes the notation <span class="math-container">$(\nabla \cdot \nabla) f = \nabla^2 f$</span> is used to denote <span class="math-container">$\Delta f$</span>, where <span class="math-container">$\Delta$</span> is the... |
3,872,174 | <p>Let <span class="math-container">$a\in \mathbb{R}^n$</span> column vector and <span class="math-container">$A\in \mathbb{R}^{n\times n}$</span> a matrix. What are some bounds for the number</p>
<p><span class="math-container">$$ |a^T A a| $$</span></p>
<p>In terms of norms of <span class="math-container">$a$</spa... | infinite_monkey | 776,731 | <p>Yes, an easy example would be to distribute the mass only over the even numbers.</p>
<p>That is, let <span class="math-container">$p_{2n} = 2^{-n}$</span> (for <span class="math-container">$n \geq 1$</span>) and <span class="math-container">$p_{2n+1} = 0$</span>.</p>
|
2,224,980 | <p>Apologies if this kind of question is not allowed here - if so please delete it.</p>
<p>I was just wondering if anyone could recommend a book on mathematical analysis that is interesting enough to sit down and read for enjoyment alone? Something not written in the style of a textbook?</p>
<p>All the best.</p>
| alekscooper | 420,113 | <p>The Manga Guide to Calculus might be of use:
<a href="https://www.nostarch.com/mg_calculus.htm" rel="nofollow noreferrer">The Mange Guide to Calculus</a></p>
|
230,997 | <p>How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1</p>
| Brian M. Scott | 12,042 | <p>HINT: Show that $\gcd(a,b)\cdot\gcd(x,y)$ divides $ax+by$ and therefore divides $\gcd(a,b)$.</p>
<p><strong>Added:</strong> Let $d=\gcd(a,b)$; by definition there are integers $a'$ and $b\,'$ such that $a=a'd$ and $b=b\,'d$, so $a'dx+b\,'dy=d$. (At this point I’ll depart from my hint to follow an even easier path.)... |
230,997 | <p>How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1</p>
| sperners lemma | 44,154 | <p>We can prove a general result just using the basic facts: $d|k$, $d|k'$ implies $d|km$ and $d|k+k'$.</p>
<hr>
<p><strong>Theorem</strong> For all $a,b,x,y,m \in \mathbb Z$, $ax+by=m$ implies $(a,b)|m$ and $(x,y)|m$.</p>
<p><em>proof</em> $(a,b)|a$ and $(a,b)|b$ so $(a,b)|ax$ and $(a,b)|by$ so $(a,b)|ax+by$, symme... |
2,847,359 | <p>So say you had $5^x=25$ where $x$ is obviously $2$, how would you work $x$ out if the question wasn't obvious?</p>
<p><strong>edit:</strong> What if the question was something like $a^x=-1$ (where $a$ is any number).</p>
<p>PS to all the maths elitists out there: Feel free to down vote, I just want to know how to ... | Peter Szilas | 408,605 | <p>A bit of context :</p>
<p>For $a >0$, real, define the function </p>
<p>$exp_a :\mathbb{R} \rightarrow \mathbb{R}$ by</p>
<p>$exp_a (x) = \exp (x\log a)$.</p>
<p>The image of this function is $(0,\infty)$.</p>
<p>Special cases:</p>
<p>$a^x = -1$.</p>
<p>$a$ cannot be a positive real (see above).</p>
<p>Ho... |
2,192,400 | <p>In Durrett's Probability Theory and Examples theorem 5.1.8 says:</p>
<blockquote>
<p>Suppose that $\mathbb{E} X^2 <\infty$. Then $\mathbb{E}(X| \mathcal{F})$ is the variable $Y \in \mathcal{F}$ that minimizes the "mean square error" $\mathbb{E}(X-Y)^2$. </p>
</blockquote>
<p>To explain notation, it is assumed... | dEmigOd | 402,211 | <p>I'm making a bit different way.</p>
<p>$X^2 \in L^1(\Omega, \mathcal{F}_0, \mathbb{P})$ therefore by Jensen's inequality $\mathbb{E}(X^2) \geq \mathbb{E}(X)^2$, i.e. $\mathbb{E}(X) < \infty$, i.e. $X \in L^1(\mathcal{F}_0)$, then $\mathbb{E}(X | \mathcal{F})$ exists and $L^1(\mathcal{F})$ for $\mathcal{F} \subse... |
657,162 | <p>Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$</p>
<p>I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$</p>
<p>Also a tighter upper bound is appreciated.</p>
| Peter | 82,961 | <p>The exact value of the product is</p>
<p>$$\frac{\Gamma(n-2)}{\Gamma(n-k)} = \frac{(n-3)!}{(n-k-1)!}$$</p>
<p>You can use the stirling approximation for large n.</p>
<p>If n-k-1 is not large, you can calculate (n-k-1)! directly.</p>
|
657,162 | <p>Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$</p>
<p>I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$</p>
<p>Also a tighter upper bound is appreciated.</p>
| lakshayg | 137,874 | <p>$$\prod_{i=3}^k (n-i) = (n-3)(n-4)\ldots(n-k)$$
multiply and divide the R.H.S of the expression with $(n-k-1)(n-k-2)\ldots1$
which gives
$$\prod_{i=3}^k (n-i) = \frac{(n-3)!}{(n-k-1)!}$$</p>
<p>You can now use the <a href="http://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow">Stirling's Approxima... |
450,857 | <p>I have the following problem I'm trying to understand/solve using first order logic.</p>
<pre><code>Predicates:
Set(S), which states that S is a set, and
x ∈ S, which states that x is an element of S,
</code></pre>
<p>Using first order logic, I need to write :</p>
<pre><code>For any x and y, there is a set contai... | Hanul Jeon | 53,976 | <p>Let $w$ is a set containing just the elements $x$ and $y$, then $z\in w$ iff $z=x$ or $z=y$. So desired formula is</p>
<p>$$\forall x\forall y \exists z\forall w: \mathrm{Set}(z)\land (w\in z \leftrightarrow (w=x)\lor (w=y))$$</p>
|
39,315 | <p>This is "escalated" from <a href="https://math.stackexchange.com/questions/4498/classification-of-small-complete-groups">stackexchange</a>. </p>
<p>So, $S_n$ for $n \ne 2,6$, $\text{Aut}(G)$ for $G$ non-cyclic simple, $\text{Hol}(C_p)$ for $p$ odd prime are well-known classes of complete group. </p>
<p>What's the ... | Max Horn | 8,338 | <p>Here's a very partial "experimental" result, using some GAP computations and the library of small groups in GAP. I am afraid this does not provide any deep theoretical insight, but it might at least give some ideas and starting points.</p>
<p>There is not really an efficient way to check whether a group is complete... |
134,848 | <p>In short, I look for a <strong>concise</strong> definition of a function in Mathematica that calculates the following:</p>
<p>$f\left(\begin{pmatrix}a_{11}&\dots&a_{1m}\\\vdots&\ddots&\vdots\\a_{n1}&\dots&a_{nm}\\\end{pmatrix},\left(b_1,\dots,b_p\right)\right)=\begin{pmatrix}a_{11}&\dots... | Mr.Wizard | 121 | <p>Look at <a href="http://reference.wolfram.com/language/ref/ArrayFlatten.html" rel="noreferrer"><code>ArrayFlatten</code></a>:</p>
<pre><code>fn[matrix_, {x__}] := ArrayFlatten[{{matrix, x}}]
fn[{{a, b}, {c, d}}, {e, f}] // MatrixForm
</code></pre>
<blockquote>
<p>$\left(
\begin{array}{cccc}
a & b & e &... |
269,552 | <p>I have been working on the following problem:</p>
<p>"Let $\sim$ be the equivalence relation on the unit circle $S^1$ defined by $x \sim -x$, $x \in S^1$. Show that $S^1/\sim$ is homeomorphic to $S^1$ and interpret geometrically."</p>
<p>I have applied the following two theorems:</p>
<p>"Let $X$ and $Y$ be space... | Community | -1 | <p>The way I think of it is, if you've identified every point with its antipode, then your original circle now has two copies of each point you care about. So let's throw away the duplicates and just keep half the points, giving you a semicircle. Except, oops, the two ends of the semicircle are also the same. So glue t... |
2,639,304 | <p>I'm looking for a monotonic increasing function $f(x)$ defined on $[0,\infty)$ so that $f(0)=0$ and with $f'(x)$ approaching $0$ as $x$ approaches $0$, and which approaches 1 as $x \rightarrow \infty$. </p>
<p>The function might look roughly like the figure below, but that particular function is just a shifted and... | José Carlos Santos | 446,262 | <p>You could try$$f(x)=\frac{x^2}{x^2+1}.$$</p>
|
3,567,002 | <p>I want to factorize such equation:
<span class="math-container">$$b^2c + bc^2 +a^2c+ac^2+a^2b+ab^2 +2abc$$</span> into product of linear factors.<br>
May I know is there any quick way/trick to do so?I am very confused with such equation. </p>
<p>Hello users, my main point here is what to think when we looking at f... | nonuser | 463,553 | <p>I would write it like <span class="math-container">$$(ma+nb+kc)(m'a+n'b+k'c)(m''a+n''b+k''c)$$</span></p>
<p>Now, if we look it as a polynimial in <span class="math-container">$a$</span> we see that there is no <span class="math-container">$a^3$</span> so <span class="math-container">$mm'm''=0$</span> and we can as... |
3,567,002 | <p>I want to factorize such equation:
<span class="math-container">$$b^2c + bc^2 +a^2c+ac^2+a^2b+ab^2 +2abc$$</span> into product of linear factors.<br>
May I know is there any quick way/trick to do so?I am very confused with such equation. </p>
<p>Hello users, my main point here is what to think when we looking at f... | nonuser | 463,553 | <p>You can do it also like this <span class="math-container">$$a^2(b+c)+a(b^2+2bc+c^2)+ bc^2+c^2b=$$</span></p>
<p><span class="math-container">$$a^2(b+c)+a(b+c)^2+bc(b+c)=$$</span>
<span class="math-container">$$(b+c)(a^2+a(b+c)+bc)=$$</span>
<span class="math-container">$$(b+c)(a^2+ab+ac+bc)=$$</span>
<span class="m... |
419,678 | <p>I need your help. I have to prove absolute convergence and conditional convergence to this two series. I have no idea how to do this.</p>
<blockquote>
<p>a) $$\sum_{n=1}^\infty \frac{n (x-1)^{n}}{2^{n} (3n - 1)}$$</p>
<p>b) $$\sum_{n=1}^\infty \frac{1}{2n -1}\left(\frac{x+2}{x-1}\right)^{n}
$$</p>
</bloc... | Community | -1 | <p>Split the problem in two:</p>
<ol>
<li><p>Find the radius of convergence for $\sum_{n=0}^{\infty} \frac n{3n-1} \, z^n$$</p></li>
<li><p>Replace $z=\frac {x-1}2$ and you'll be able to determine the values of $x$ where your series converges.</p></li>
</ol>
<p>Do the same for your second series, with $z=\frac {x+2}{... |
382,549 | <p>Let $A = diag \left (\lambda_1, ..., \lambda_n \right ) \in \mathbb{R}^{n \times n}$, with $\lambda_1 < \lambda_2 < ... < \lambda_n$.<br>
Let $u = \left (u_1, ..., u_n \right ) ^T \in \mathbb{R}^n$, with $u_i \neq 0 \ \ \forall i$.<br>
How can be shown that: </p>
<ol>
<li>For any $\alpha \in \mathbb{R}, \... | Jack2019 | 18,858 | <p>try using definition for prob a. try to find characteristic polynomial for b</p>
|
391,669 | <p>Let <span class="math-container">$X_1,\dots,X_n$</span> be non commutative variables such that <span class="math-container">$\operatorname{tr} f(X_1,\dots,X_n) = 0$</span> whenever the <span class="math-container">$X_i$</span> are specialized to square matrices in <span class="math-container">$M_r(k)$</span> for any... | Joseph Van Name | 22,277 | <p>For simplicity, let us assume that <span class="math-container">$K=\mathbb{C}$</span> (though, these results apply to any field of characteristic zero). We will use capital letters like <span class="math-container">$X,Y,Z$</span> to denote matrices while lower case letters like <span class="math-container">$x,y,z$</... |
1,355,502 | <p>We inductively define $a^1=a, a^{n+1}=a^n a$. I want to show that $a^{n+m}=a^n a^m$. </p>
<p>By definition, this is true if $m=1$. Now for $m=2$, we have
$$
\begin{align}
a^{n+2} =& a^{(n+1)+1}\\
=& a^{n+1}a \\
=& \left(a^{n}a\right)a \\
=& a^{n}\left(aa\right) \\
=& a^{n}a^2
\end{align}
$$
Ho... | peterwhy | 89,922 | <p>Then assume $a^na^m = a^{n+m}$ and prove $a^na^{m+1} = a^{n+m+1}$. I would say this is very similar to your $m=2$ proof above.</p>
|
3,091,943 | <p>Here is the solution:</p>
<p><a href="https://i.stack.imgur.com/HbkbJ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HbkbJ.png" alt="enter image description here"></a></p>
<p>But I could not understand how the last term in the fourth line came from the line before it, could anyone explain this ... | Todor Markov | 615,317 | <p>It is true indeed. Let's try to interpret the LHS and the RHS combinatorially.</p>
<p>LHS: You have N sets. Set <span class="math-container">$i$</span> has <span class="math-container">$n_i$</span> objects. You want to choose <span class="math-container">$r_i$</span> objects from the <span class="math-container">$i... |
2,280,696 | <p>For (a)$\sum{\frac{1}{(\ln(n))^n}}$</p>
<p>$\ln(n)>2$ $ \forall n > e^2$</p>
<p>$(\ln(n))^n>2^n$ $ \forall n > e^2$</p>
<p>$\frac{1}{(\ln(n))^{n}}<\frac{1}{2^n}$ $ \forall n > e^2$</p>
<p>$\sum{\frac{1}{2^n}}$ is convergent by geometric series test</p>
<p>$\sum{\frac{1}{(\ln(n))^n}}$ i... | User8128 | 307,205 | <p>Your reasoning for (a) is perfect.</p>
<p>For (b), note that the terms do not approach zero since each even term is equal to $1$. Thus it diverges by the divergence test.</p>
<p>For (c), note that for large enough $n$ (say $n \ge N$ where $N$ is some fixed natural number), we have $\ln(n) \le n^{1/4}$ and thus $$\... |
1,449,845 | <p>So the biquadratic equation is $x^4+(2-\sqrt3)x^2+2+\sqrt3=0$. Let $a_1,a_2,a_3,a_4$ be its roots. So we have to find the value of $(1-a_1)(1-a_2)(1-a_3)(1-a_4)$ . <br>
<strong>My attempt:</strong> <br>
So of we put $x^2=t$, and let the roots of the new quadratic equation be $a_1,a_2$. So we get that $a_1=-a_3;a_2=-... | Ángel Mario Gallegos | 67,622 | <p><strong>Hint:</strong></p>
<p>Just notice that
$$(x-a_1)(x-a_2)(x-a_3)(x-a_4)=x^4+(2-\sqrt{3})x^2+2+\sqrt{3}$$</p>
<hr>
<p><strong>About your attempt:</strong></p>
<p>According to G-man (thanks) we have</p>
<p>\begin{align}
1-(a_1^2+a_2^2)+(a_1a_2)^2&=1-\left(-\frac{2-\sqrt{3}}{1}\right)+(2+\sqrt{3})\\
&... |
2,275,203 | <p>I'm having trouble to prove that the intersection of the symmetric matrices , the set of projections (matrices who $M^2 = M$) and the matrices with constant rank $k$ is a manifold. Can anyone help me? Unfortunately I could not make much progress.</p>
<p>In other words I need to prove the following proposition: Let ... | Dietrich Burde | 83,966 | <p>We can identify the space $G_{n,k}$ with the Grassmannian $Gr(n,k)$ of $k$-dimensional linear subspaces of $\mathbb{R}^n$. By the Theorem of constant rank, a smooth neighborhood retract is automatically a submanifold. We can easily construct such a smooth retraction $U\rightarrow G_{n,k}$ for some $U$ in $\mathbb{R... |
32,379 | <p>How do I replace content inside a Held expression without local values coming through?
For example if you run the following code you get <code>g[2]</code> instead of <code>g[x]</code>(the intended form).</p>
<pre><code>x = 2;
Replace[HoldComplete[g["x"]], {
y_String :> With[{
eval = Symbol[y]},
eval /... | rm -rf | 5 | <p>The third argument of <code>ToExpression</code> with the <a href="https://mathematica.stackexchange.com/a/29318/">Trott–Strzebonski</a> can be used to do this:</p>
<pre><code>HoldComplete[g["x"]] /. s_String :>
With[{x = ToExpression[s, InputForm, Defer]}, x /; True]
(* HoldComplete[g[x]] *)
</code... |
32,379 | <p>How do I replace content inside a Held expression without local values coming through?
For example if you run the following code you get <code>g[2]</code> instead of <code>g[x]</code>(the intended form).</p>
<pre><code>x = 2;
Replace[HoldComplete[g["x"]], {
y_String :> With[{
eval = Symbol[y]},
eval /... | Mr.Wizard | 121 | <p>I hope that you will give a more complete example in the question if this is not adequate. As it stands I don't see the need to mess with context but given the earlier questions I suspect your example is not representative of what you actually wish to accomplish.</p>
<p>For the given example you could use somethin... |
2,931,936 | <blockquote>
<p>Is the set of vectors of the form <span class="math-container">$[ a, b, a+2b ]$</span> a subspace of <span class="math-container">$\mathbb R^3$</span>?</p>
</blockquote>
<p>How to show this? The definition of a subspace <span class="math-container">$S$</span> requires showing that if <span class="mat... | Max0815 | 595,084 | <p>So you have the equation <span class="math-container">$12g = 12 (\frac{2}{3g} - 1) + 11$</span>. You want to find <span class="math-container">$g$</span>. <span class="math-container">$$12g=\frac{24}{3g}-12+11$$</span> <span class="math-container">$$\implies 12g=\frac{8}{g}-1$$</span> <span class="math-container">$$... |
4,244,708 | <p>Let <span class="math-container">$S$</span> denote the set of all solutions of the following differential equation defined on <span class="math-container">$C^3[0,\infty)$</span>;</p>
<p><span class="math-container">$$
\begin{align}
\frac{d^3x}{dt^3} + b \frac{d^2x}{dt^2} + c \frac{dx}{dt} + dx = 0
\end{align}
$$</... | Laxmi Narayan Bhandari | 931,957 | <p>We want to evaluate</p>
<p><span class="math-container">$$I = -\int_0^\infty \frac{e^{-x}}{x}\ln (x)\sin(x)\, dx $$</span></p>
<p>We start with the result</p>
<p><span class="math-container">$$ \int_0^\infty e^{-ax}\sin(x)\, dx = \frac{1}{a^2+1} = \frac{1}{2i}\Big(\frac{1}{a-i}-\frac{1}{a+i}\Big) $$</span></p>
<p>Di... |
4,244,708 | <p>Let <span class="math-container">$S$</span> denote the set of all solutions of the following differential equation defined on <span class="math-container">$C^3[0,\infty)$</span>;</p>
<p><span class="math-container">$$
\begin{align}
\frac{d^3x}{dt^3} + b \frac{d^2x}{dt^2} + c \frac{dx}{dt} + dx = 0
\end{align}
$$</... | NoName | 649,394 | <p>We'll use the following two crucial facts to calculate the integral. The first is that:</p>
<p><span class="math-container">$$\boxed{\tan^{-1}x\ln(1+x^2)=-2\sum_{ n \ge 1}\frac{(-1)^nH_{2n}}{2n+1}x^{2n+1}}$$</span></p>
<p>which is shown in <a href="https://math.stackexchange.com/questions/3240779/prove-ln1x2-arctan-... |
4,493,427 | <p>Suppose you have a compact set <span class="math-container">$\Omega \subset \mathbb{R}^n$</span> partitioned as <span class="math-container">$\Omega = \bigcup_{j=1}^n \Omega_j$</span> with that <span class="math-container">$\Omega_i \cap \Omega_j = \emptyset$</span> if <span class="math-container">$i \neq j$</span>... | Community | -1 | <p>No, there is no one function giving the infimum, but based on Proposition 22, Chapter 3, Section 5 of Royden's: Real Analysis anf Exercice 15, Chapter 4 of this book, we can find for every <span class="math-container">$\epsilon >0$</span> (since s(.) is bounded measurable) a continuous function such that :</p>
<p... |
51,596 | <p>Some time ago, I asked <a href="https://math.stackexchange.com/q/42276/8271">this</a> here. A restricted form of the second question could be this:</p>
<blockquote>
<p>If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x) =a,$$ with $a\gt 0$, then $$\lim_{x... | Shai Covo | 2,810 | <p>Since you assume that $f$ has first derivative continuous on $\mathbb{R}$, there is an alternative to the Mean-value theorem, namely the Fundamental theorem of calculus (the former is very useful for cases when $f'$ is not assumed continuous). So, from $\lim _{x \to \infty } f'(x) = a > 0$, we have that t... |
51,596 | <p>Some time ago, I asked <a href="https://math.stackexchange.com/q/42276/8271">this</a> here. A restricted form of the second question could be this:</p>
<blockquote>
<p>If $f$ is a function with continuous first derivative in $\mathbb{R}$ and such that $$\lim_{x\to \infty} f'(x) =a,$$ with $a\gt 0$, then $$\lim_{x... | Bill Dubuque | 242 | <p><strong>HINT</strong> $\ \ $ Writing $\rm\ f\ =\ (f/x)\ x\ \ $ yields a determinate limit (using L'Hôpital's rule), namely</p>
<p>$$\rm\displaystyle\quad \lim_{x\to\infty}\ f\ =\ \lim_{x\to\infty} \dfrac{f}x\ \lim_{x\to\infty}\ x\ =\ \lim_{x\to\infty}\dfrac{f\:\:'}{1\:\ }\ \lim_{x\to\infty}\ x\ =\ a\ \cdot \inft... |
174,853 | <p>In the <a href="http://en.wikipedia.org/wiki/Free_variable" rel="nofollow noreferrer">Free Variable</a> article on Wikipedia, it lists these:</p>
<p><img src="https://i.stack.imgur.com/wAfKv.png" alt="variable-binding operators"></p>
<p>as variable-binding operators. I have seen all of them during my math studies,... | MJD | 25,554 | <p>I don't know what was intended by $\psi$ here, but some Wikipedia archaeology reveals that it was introduced in <a href="http://en.wikipedia.org/w/index.php?title=Free_variables_and_bound_variables&diff=229988162&oldid=228192578" rel="nofollow">this edit</a>, and <a href="http://en.wikipedia.org/w/index.php?... |
265,728 | <p>Consider the polynomial:</p>
<p>$$p(x) = \sum_{k=0}^{r}(-1)^{r-k} {r \choose k} x^{k(k-1) / 2}$$</p>
<p>I want to show that</p>
<p>$$p(x) = (x - 1)^{\lceil r/2 \rceil} \, q(x)$$</p>
<p>That is, $(x - 1)^{\lceil r/2 \rceil}$ is a factor of $p(x)$. Even better, find a formula for the quotient polynomial $q(x)$.</p... | YCor | 14,094 | <p>7 years ago I wrote in an unfinished paper a proof of this result; I didn't know then if it was original but it indeed seems to be known to a number of specialists although I'm not aware of a written complete proof. Since I don't want anymore to complete this paper, I reproduce the proof here (not self-contained, re... |
2,060,793 | <p>I need to know wich answer is right</p>
<p><a href="https://i.stack.imgur.com/QcxdH.jpg" rel="nofollow noreferrer">https://i.stack.imgur.com/QcxdH.jpg</a></p>
<p>I tried to solve it using recursivity but I didn't get any one of them</p>
<ul>
<li>$y_1=\sqrt{x}$ </li>
<li>$y'=\frac{1}{2\sqrt{x}}=\frac{1}{2y_1}$</li... | bulbasaur | 241,800 | <p>$y^2=x+y$ now </p>
<p>$2ydy/dx=1+dy/dx$</p>
|
1,773,328 | <p>$\sum_{n=1} ^\infty \frac{1}{2^n}$ converges to $1$, but it does so at a much faster rate than I'd like: $1/2,3/4,7/8...$. In other words, after only a few terms we are really close to $1$ (for my taste).</p>
<p>Can you give an example of a series that congerges to $1$ very slowly, so that the partial sums approac... | Ross Millikan | 1,827 | <p>You can do $\frac 6{\pi^2} \sum_{n=1}^\infty \frac 1{n^2}$ or $\sum_{n=1}^\infty \frac 1{n(n+1)}$</p>
|
3,479,780 | <p>I don't understand those explanations that <span class="math-container">$F(a)$</span> is a subset of <span class="math-container">$F(b)$</span>, so <span class="math-container">$F(a)$</span> is <span class="math-container">$\leq F(b)$</span>. And A event is contained in the event B, so <span class="math-container">$... | user247327 | 247,327 | <p>Multiply both sides by <span class="math-container">$x^a$</span> and <span class="math-container">$(x+ c)^b$</span> to get <span class="math-container">$$1= d_a(x+ c)^b+ d_{a-1}x(x+ c)^b+ \cdots+ d_1x^{a-1}(x+ c)^b+ e_bx^a+ e_{b+1}x^a(x+ c)+ \cdots+e_1x^a(x+ c)^{b-1}~.$$</span></p>
<p>Taking <span class="math-conta... |
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