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 |
|---|---|---|---|---|
2,592,624 | <p>Let $M$ be a smooth, orientable $n$-manifold and $\eta$ a volume form on $M$. Does there exist a connection $A$ on $TM$ such that
$$\tag{$*$}D\eta=0,$$
where $D$ is the appropriate covariant derivative associated to $A$ on $\Omega^n(M)$? Can one assume $A$ to be symmetric?</p>
<p>I have a feeling that if one write... | Pedro | 23,350 | <p>For any ring $A$, any right ideal $I$ any left $A$-module $M$, there is a canonical map $I\otimes_A M \to IM$ that assigns $a\otimes m$ to $am$. The module $M$ is flat if and only if this is an isomorphism for every ideal $I$ of $A$, in fact, it suffices it is an isomorphism for every finitely generated ideal of $A$... |
2,750,250 | <p>I know this is such a basic question and its embarrassing to ask but I just don't understand why Rudin concludes such a thing. In theorem 3.2 (a) he proves that if two sequences converge to different limits then the limits are actually the same as follow:</p>
<p>$$ \epsilon > 0 $$
$$ n \geq N \implies d(p_n,p) &... | pureundergrad | 517,979 | <p>Others explained this but basically you have </p>
<p>$$0\leq d(p,p')<\epsilon \ \forall \epsilon>0$$</p>
<p>If you let $d(p,p')>0$ then just pick $\epsilon = d(p,p')/2$ to get a contradiction. Hence the only value $d(p,p')$ can take on is zero.</p>
|
2,750,250 | <p>I know this is such a basic question and its embarrassing to ask but I just don't understand why Rudin concludes such a thing. In theorem 3.2 (a) he proves that if two sequences converge to different limits then the limits are actually the same as follow:</p>
<p>$$ \epsilon > 0 $$
$$ n \geq N \implies d(p_n,p) &... | fleablood | 280,126 | <p>Another way of putting it and this is probably more likely in Rudin's thinking:</p>
<p>$\mathbb R$ has the least upper bound property and the basic definition of "bounded below", "lower bound" and "greatest lower bound" make the following statement almost a tautology.</p>
<p>Let $S\subset \mathbb R$ be a non-empt... |
2,482,449 | <p>In the process of constructing a highway across a certain region in which there are many hills and valleys. the engineer will be certain that</p>
<p>There is some level in between the elevations of the highest hill and the
lowest valley at which the surface of the highway can be laid using the
tops of the hills as ... | Angina Seng | 436,618 | <p>You have
$$\alpha I=M^3-\alpha M^2+11M.$$
The minimal polynomial of $M$ is $(x-1)(x-2)(x-3)$; this divides
any polynomial $f$ with $f(M)=0$.</p>
|
514,351 | <p>If someone with not much mathematics in his luggage asks me: What
is so special about $\pi$? then off course I have an answer. Even
if $i$ would be the subject (I allready see him gazing at my mysterious
smile). But when I am asked about $e$ then I grow silent (or try
to change the subject). Please help me out of th... | user93089 | 93,089 | <p>$$\dfrac d{dx}\left(e^x\right)=e^x$$
$$\int e^xdx=e^x+C$$
It is the only function that does this</p>
<p>Also, $e^x$ can be expressed like this:
$$e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}......$$</p>
|
102,068 | <p>I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one). But when I solve partial differential equations using a finite difference scheme, I'm generally more interested in the solution, its stab... | Dan | 12,533 | <p>In quantum mechanics, eigenvalues are of physical interest. The eigenvalues of the Hamiltonian operator (which determines how the wavefunction propagates in time) are also the only allowed energies of the system.</p>
<p>When propagating in time, certain physical effects can be simulated by using an incomplete eige... |
102,068 | <p>I understand that eigenvalues have their purpose in linear algebra (e.g. iterative methods won't converge unless the modulus of the spectral radius is less than or equal to one). But when I solve partial differential equations using a finite difference scheme, I'm generally more interested in the solution, its stab... | Julián Aguirre | 4,791 | <p>Solving a time dependent equation like the heat equation
$$
u_t-\nabla^2u=0,\quad x\in\Omega,\quad t>0
$$
with boundary condition $u(x,t)=0$ if $x\in\partial\Omega$, $t>0$, and initial condition $u(x,0)=u_0(x)$, $x\in\Omega$, by separation of variables (that is, looking for solutions of the form $u(x,t)=T(t)... |
1,107,465 | <p>Prove that for integers $n \geq 0$ and $a \geq 1$, $${n + a - 1 \choose a - 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}.$$</p>
<p>I figured I'd post this question, which was on an assignment I did, since I thought the solution was so nice.</p>
| Marko Riedel | 44,883 | <p>This one can also be done using complex variables.
<P>
Suppose we seek to evaluate
$$\sum_{k=0}^{\lfloor n/2\rfloor}
{a\choose n-2k} {k+a-1\choose a-1}.$$</p>
<p>Introduce the integral representation
$${a\choose n-2k}
=\frac{1}{2\pi i}
\int_{|z|=\epsilon}
\frac{(1+z)^{a}}{z^{n-2k+1}} \; dz.$$</p>
<p>Note that thi... |
2,552,005 | <p>As long as I've been working with mathematical notation, it seems like I should be embarrassed about being confused about this, but...</p>
<p>Does $\sqrt[n]{a}$ with $n \in \mathbb{N}$ in general stand for the set of all $n$ roots, or just one of them? When $a \in \mathbb{Z}$ and it's clear that the surd is also me... | DanielWainfleet | 254,665 | <p>In the context of $\Bbb R $, if $n\in \Bbb N,$ it means the real $n$th root when $n$ is odd. And the non-negative real $n$th root when $n$ is even and $a\geq 0.$</p>
<p>In the context of $\Bbb C,$ if $n \in \Bbb N,$ it does not mean the $set$ of all $n$ roots, but an unspecified member of that set. </p>
|
283,616 | <p>I meant to assign to my class the following homework problem:</p>
<blockquote>
<p>If $u\in C^2((0,T)\times \Omega) \cap C^0([0,T]\times\bar{\Omega})$ where $\Omega$ is an open, bounded domain, is such that $\partial_t u - \triangle u \leq - \epsilon < 0$ for some constant $\epsilon > 0$, then $u$ cannot hav... | Connor Mooney | 16,659 | <p>This version is still true: if $u$ had a local maximum at $(x,\,T)$, say with $u(x,\,T) = 0$, then $u \leq 0$ in a small parabolic cylinder centered at $(x,\,T)$. After rescaling we can assume that $u \leq 0$ in $\overline{B_1} \times [0,\,1]$, with $u(0,\,1) = 0$.</p>
<p>Replacing $u$ with $u - \frac{\epsilon}{4n}... |
1,460,913 | <p>Could anyone derive or explain why the formula $p ( x , y | z ) = p ( x | z ) p ( y | x , z )$ is true?</p>
| Graham Kemp | 135,106 | <p>To get you started. We use: $\mathsf P(\alpha\mid \beta)\;\mathsf P(\beta)=\mathsf P(\alpha\cap \beta)$</p>
<p>$\begin{align}
\mathsf P ( x \cap y \mid z ) & = \frac{\mathsf P((x\cap y)\cap z)}{\mathsf P(z)} & \text{if } \mathsf P(z)\neq 0
\\[1ex] & = \frac{\mathsf P((x\cap z)\cap y)}{\mathsf P(z)}
\end... |
778,118 | <p>I have a textbook question that asks to use L'Hopital to show which of the two functions: $e^{0.1 x}$ vs $x^{10}$ is dominant as $x \to \infty$. That is, using:
$$\lim_{x \to \infty} \left( \frac {x^ {10} } {e^{0.1 x}} \right) =
\lim_{x \to \infty} \left( \frac {10x^ 9 } {0.1e^{0.1 x}} \right) =
\lim_{x \to \inft... | Lutz Lehmann | 115,115 | <p>If you do not want to run yourself into the ground with repeated applications of l'Hopitals theorem, invest some transformations before:
$$
\frac{x^{10}}{e^{0.1\cdot x}}=\left(\frac{x}{e^{0.01\cdot x}}\right)^{10}
$$
Comparing $x$ and $e^{0.01⋅x}$ is ten times easier than comparing their tenth power.</p>
|
649,169 | <p>I know this has to do with Euclidean division, I just can't prove the => direction.
Any tips would be appreciated!</p>
| Luyen Le | 57,179 | <p>Use the property $gcd(a,b)=1 \Leftrightarrow$ there exist $m,n$ such that $ma+nb=1$</p>
<p>$(\Rightarrow)$ Suppose that $gcd(a,n)=1$. Then there exist $l,k$ such that $la+kn=1 \Leftrightarrow kn-ka+ka+la=1\Leftrightarrow k(n-a)+(k+l)a=1$. So $gcd(n-a,a)=1.$</p>
<p>$(\Leftarrow)$ Suppose $gcd(n-a,a)=1$. Then there ... |
274,926 | <p>I'm learning real analysis.</p>
<blockquote>
<p>A subset $G$ of $X$ is called open if for each $x \in G$ there is a
neighborhood of $x$ that is contained in G</p>
</blockquote>
<p>My question is that is $\emptyset$ a open set in $X$? </p>
<p>The set $\emptyset$ has no elements, so there is no neighborhood o... | Community | -1 | <p>In general, for any statement $P$, for every element of $\emptyset$, the statement $P$ is <em>vacuously true</em>.</p>
|
654,839 | <blockquote>
<p>Prove the following inequality:
$$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$</p>
</blockquote>
<p>This should be solved using Taylor series.</p>
<p>I tried expanding the left to the 5th degree and the right site to the 3rd degree, but it didn't help.</p>
<p>Any tips?</p>
| 2012ssohn | 103,274 | <p>$$\frac{e^x+e^{-x}}{2} = \frac{\left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \right) + \left(1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots \right)}{2} = \frac{2 + 2 \frac{x^2}{2!} + 2 \frac{x^4}{4!} + \cdots}{2}$$</p>
<p>$$ = 1 + \frac{x^2}{2} + \frac{x^4}{24} + \frac{x^6}{720}+ \cdots$$</p>
<p>$$e^{\f... |
411,268 | <p>Please, somebody can help me with this problem?</p>
<p><hr>
Let $V$ and $W$ be two closed subspaces of a Hilbert $(H, \langle \cdot,\cdot\rangle)$, and let $P:H\rightarrow V$ and $Q:H\rightarrow W$ the orthogonal projectors respectively. Show that
$$\langle\ (Q-P)(x),\ x\ \rangle\ \geq\ 0,\ \forall\ x\in H\quad \mb... | robjohn | 13,854 | <p>We will use the identities
$$
\begin{align}
\sum_{k=m}^n\binom{n}{k}\binom{k}{m}
&=\sum_{k=m}^n\binom{n}{m}\binom{n-m}{k-m}\\
&=\binom{n}{m}2^{n-m}\tag{1}
\end{align}
$$
and
$$
\binom{k-1}{2}=\binom{k}{2}-\binom{k}{1}+\binom{k}{0}\tag{2}
$$
noting that
$$
\binom{k-1}{2}=\left\{\begin{array}{}
0&\text{for... |
2,139,774 | <blockquote>
<p>Let $V$ be an $n$-dimensional vector space. If $f,g\in V^*$ are linearly independent, then find $\dim(\ker f\cap \ker g)$.</p>
</blockquote>
<ol>
<li><p>$\dim(\ker f)=\dim(\ker g)=n-1$ because $f,g\neq 0$, correct?</p></li>
<li><p>Hint for answer, please.</p></li>
</ol>
| Jack D'Aurizio | 44,121 | <p>внимание: overkill. Stating that for any $(a,b,c)\in\mathbb{R}^3$ we have
$$ Q(a,b,c) = a^2+b^2+c^2-ab-ac-bc \geq 0 \tag{1}$$
is the same as stating that the following symmetric matrix
$$ Q = \begin{pmatrix}1& -\frac{1}{2}&-\frac{1}{2}\\-\frac{1}{2}&1&-\frac{1}{2}\\-\frac{1}{2}&-\frac{1}{2}& ... |
4,033,610 | <p>Maybe my problem is the English, but I cannot see why Gallian’s statement of the First Principle of Mathematical Induction is true. In the 7th edition of Contemporary Abstract Algebra (p13) he states:</p>
<blockquote>
<p>Let <span class="math-container">$S$</span> be a set of integers containing <span class="math-c... | Noah Solomon | 750,380 | <p>The definition is not wrong. The thing you are missing is that if <span class="math-container">$S$</span> is our set containing <span class="math-container">$a$</span>, the condition is that for <strong>every</strong> <span class="math-container">$n\geq a$</span>, if <span class="math-container">$n \in S$</span> the... |
2,711,447 | <p>Let $n$ (unknown) real numbers $x_i$ be given. Suppose all Vieta's coefficient equations are positive, i.e.
$$
a_1 = \sum_{i=1}^n x_i > 0\\
a_2 =\sum_{(i>j)} x_i x_j > 0\\
a_3 =\sum_{(i>j>k)} x_i x_j x_k> 0\\
\dots \\
a_n =\prod_{i=1}^n x_i > 0
$$
where the sums go over all possible indicate... | Martin R | 42,969 | <p>Your $a_k$ are the <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial" rel="nofollow noreferrer">“elementary symmetric polynomials”</a> of
$x_1, \ldots, x_n$. If all $a_k$ are positive then
$$
p(x) = \prod_{i=1}^n (x + x_i) = \sum_{k=0}^n a_k x^{n-k}
$$
is a polynomial with real, strictly posi... |
41,795 | <p>Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on $\pi_1(S_1\times S_1)$ is given by $A$ (w.r.t the induced basis).</p>
<p>So I am wondering, whether a similar construct... | Ryan Budney | 1,465 | <p>A small comment extending Greg's comments, given an element of $[g] \in \pi_n Diff(S^n)$ you can construct a diffeomorphism </p>
<p>$f : S^n \times S^n \to S^n \times S^n$ by sending the pair $(x,y)$ to $(x,g_x(y))$, so the corresponding matrices would be upper-triangular as in Greg's examples. </p>
<p>$Diff(S^n)$... |
1,259,383 | <p>I have a distribution with literally an infinite number of potential data points. I need the standard deviation. I generate about a hundred points and take the standard deviation of the points. This gives a hopefully good approximation of the true standard deviation, but it won't, of course, be exact. How do I e... | Rivers McForge | 774,222 | <p>This is pretty standard and can be answered by searching "Confidence interval of a standard deviation." Here are the steps:</p>
<p>Step 1) Pick a <strong>confidence level</strong>. The confidence level is the probability of your interval estimate containing the actual population standard deviation. Common ... |
1,533,326 | <p>I need some help proving the following statement:</p>
<p>[EDIT]: forgot to mention that:</p>
<p><span class="math-container">$a_n \ge 0$</span> , <span class="math-container">$L \ge 0$</span>.</p>
<p>If the limit of a sequence <span class="math-container">$\lim_{ \ n \to \infty} a_n=L$</span>,
then, for any <span... | skyking | 265,767 | <p>You can for example use the fact that $f(x) = x^p$ is continuous together with the definition of limit (and continuity), it's straight forward just to chain these definitions.</p>
<p>The only thing that may remain is to prove that $f$ is continuous. This is done by providing an estimate of $f(x+h)-f(x)$, that is to... |
1,116,312 | <p>Let X be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If $$P(X\leq-1)=P(Y\geq2)$$
then standard deviation of Y is</p>
<p>My Solution:</p>
<p>$E(X)=\mu=1, V(X)=\sigma^2=4$
as</p>
<p>$\sigma^2=E[X^2]-(E[X])^2=E[X^2]... | drhab | 75,923 | <p>The random variables $X$ and $Y$ can be written as $X=2U+1$ and $Y=\sigma V-1$ where $U$ and $V$ both have standard normal distribution.</p>
<p>$P\{U\geq 1\}=P\{U\leq -1\}=P\{X\leq -1\}=P\{Y\geq2\}=P\{V\geq\frac{3}\sigma\}=P\{U\geq\frac{3}\sigma\}$</p>
<p>The first equality is a consequence of the fact that the st... |
54,634 | <p>As you know, <em>Mathematica</em> V10 has just been released and I think a lot of updates will be released. However, I don't see any command in the help menu to update (which is present in most other software).</p>
<p>How would we know if an update is available or not?</p>
<p>I am using trial version of V10.</p>
... | Taliesin Beynon | 7,140 | <p>We can push paclet updates out directly, I think PacletManager guarantees that you'll have them within a week (as long as you have an internet connection). </p>
<p>This only works for certain functionality that has been written in the paclet form (e.g. Machine Learning, Dataset, SemanticImport). I'm really excited... |
2,669,617 | <p>The bilinear axiom is:</p>
<pre><code> <cu + dv,w> = c<u,w> + d<v,w>
<u,cv + dw> = c<u,v> + d<u,w>
</code></pre>
<p>Where c and d are scalars and u, v, and w are vectors.</p>
<p>Can this be extended to something like</p>
<pre><code> <cu + dv, ew + fx> = ?
</code></pre>
| Math Lover | 348,257 | <p>Continuing from what you have mentioned,
$$0 \le \lim_{x\to\infty}{\left({
{\prod_{i=1}^{x}i}\over{
{\prod_{i=x+1}^{2x}i}}
}\right)} = \lim_{x\to\infty}\prod_{i=1}^{x}\frac{i}{i+x} \le \lim_{x\to\infty}\prod_{i=1}^{x}\frac{x}{x+x} = \lim_{x\to\infty}\frac{1}{2^x}=0.$$</p>
|
2,669,617 | <p>The bilinear axiom is:</p>
<pre><code> <cu + dv,w> = c<u,w> + d<v,w>
<u,cv + dw> = c<u,v> + d<u,w>
</code></pre>
<p>Where c and d are scalars and u, v, and w are vectors.</p>
<p>Can this be extended to something like</p>
<pre><code> <cu + dv, ew + fx> = ?
</code></pre>
| Sri-Amirthan Theivendran | 302,692 | <p>Use stirling's approximation, namely that
$$
n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n.
$$
Hence
$$
\lim_{n\to\infty}\frac{(n!)^2}{(2n)!}=\lim_{n\to\infty}\frac{2\pi n\left(\frac{n}{e}\right)^{2n}.}
{\sqrt{4\pi n}\times2^{2n}\left(\frac{n}{e}\right)^{2n}}
=\lim_{n\to\infty}\frac{\sqrt{\pi n}}{2^{2n}}=0.
$$</p>
|
3,406,164 | <p><span class="math-container">$$a_n=\frac{sin1!}{1*2}+\frac{sin2!}{2*3}+\frac{sin3!}{3*4}+...+\frac{sinn!}{n(n+1)}$$</span>
I have this series and I don't understand how to apply the cauchy criterion
<span class="math-container">$$\lvert a_{n+p}-a_n \rvert \lt ε$$</span>
The only result I get is this one:
<span class... | Mark | 470,733 | <p>|<span class="math-container">$\sum_{k=1}^{n+p}\frac{\sin(k)!}{k(k+1)}-\sum_{k=1}^n\frac{\sin(k!)}{k(k+1)}|=|\sum_{k=n+1}^{n+p}\frac{\sin(k!)}{k(k+1)}|\leq\sum_{k=n+1}^{n+p}\frac{|\sin(k!)|}{k(k+1)}\leq\sum_{k+n+1}^{n+p}\frac{1}{k(k+1)}=\sum_{k=n+1}^{n+p}(\frac{1}{k}-\frac{1}{k+1})=\frac{1}{n+1}-\frac{1}{n+p+1}<\... |
167,395 | <p>Consider the following groups: $(\mathbb{Z}_4,+)$, $(U_5,.)$, $(U_8,.)$ and the set of symmetries for a rhombus if I am not mistaken the first and last are equivalent. What other justifiable equivalencies and nonequivalencies are there and what does it mean rigorously to be in the same group in general?</p>
| Aang | 33,989 | <p>Linear Programming problems can be solved using various algorithms like Simplex Algorithm, interior point methods etc. but most popular method for simple linear programming problems is Simplex method. Here is a link that describes the method with example: <a href="http://www.phpsimplex.com/en/simplex_method_example.... |
2,537,031 | <p>This is a question in Geometry by Hartshorne Exercise 3.3 </p>
<p>The goal is using Ruler and compass and a given triangle ABC and given a segment DE, construct a rectangle with content equal to the triangle ABC, and with one side equal to DE. Any propositions in Euclid book I-IV are usable but its likely going to... | Jaideep Khare | 421,580 | <p>Suppose this was possible.</p>
<p>We have used $n$ number of sheets (say, A4) to do this. The area of each sheet is $\sqrt2$ (let us consider each sheet of dimensions $1 \times \sqrt 2 $).</p>
<p>Total area of sheets $= n\sqrt 2$.</p>
<p>Now let $a$ be the number papers whose short side makes a boundary on any pa... |
2,537,031 | <p>This is a question in Geometry by Hartshorne Exercise 3.3 </p>
<p>The goal is using Ruler and compass and a given triangle ABC and given a segment DE, construct a rectangle with content equal to the triangle ABC, and with one side equal to DE. Any propositions in Euclid book I-IV are usable but its likely going to... | Derek Ledbetter | 143,732 | <p>Here's a more powerful theorem that solves this problem. First, we have to show that the component rectangles have sides parallel to the sides of the entire rectangle.</p>
<p><strong>Lemma:</strong> If there is a polygon whose sides are axis-aligned and that can be subdivided into rectangles, then the component rec... |
1,180,918 | <p>I have two questions.</p>
<p><span class="math-container">$1.$</span> From what i know the sets <span class="math-container">$A = \{1,1,2\}$</span> and <span class="math-container">$B = \{1,2\}$</span> are the same set. So my first question is, is <span class="math-container">$A\cup A=A$</span>, or even further, is ... | Jasha | 163,845 | <ol>
<li>It is true in every case that $A\cup A=A$, and it is true that if $B\subseteq A$ then $A\cup B=A$.</li>
<li>The set $\emptyset$ is called "the empty set." It is the set that doesn't contain any elements (hence the word "empty" it its name).</li>
</ol>
<p>If $A$ and $B$ are sets, then $A\cup B$ as the smallest... |
167,981 | <p>My question is:</p>
<blockquote>
<p>Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$</p>
</blockquote>
<p>Any help to solve this question would be greatly appreciated.</p>
| marlu | 26,204 | <p>Since $x^{11}+x^{10}+\ldots + x+1 = \frac{x^{12}-1}{x-1}$ we may first factorize $x^{12}-1$ and then divide by the factor $x-1$:
\begin{align*}
x^{12}-1 &= (x^6-1)(x^6+1)\\
&= (x^3-1)(x^3+1)(x^6+1)\\
&=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^2+1)(x^4-x^2+1),
\end{align*}
hence
$$x^{11}+x^{10}+\ldots +x+1 = (x^2... |
4,010,297 | <p>My question is about kind of comparing values in a function, for example in this question</p>
<p>given 'c' a positive constant we have the following function <span class="math-container">$f(x)=cx^3+x^5-1$</span> , determine which is greater <span class="math-container">$f(\sin(x))$</span> or <span class="math-contai... | Arturo Magidin | 742 | <p>Having a local min/max at <span class="math-container">$x=0$</span> is <em>not</em> equivalent to having zero derivative (also, you should use <span class="math-container">$\frac{d}{dx}$</span>, not <span class="math-container">$\frac{\partial}{\partial x}$</span>, because there are derivatives, not partial derivati... |
760,330 | <p>For every positive integer $n$, prove that
$$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$</p>
<p>Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]$ denotes the greatest integer not exceeding $x$. </p>
<p>This question was posed to me in class by my teacher.... | Hagen von Eitzen | 39,174 | <p>Clearly $\lfloor\sqrt{4n+1}\rfloor\le\lfloor\sqrt n+\sqrt{n+1}\rfloor\le \lfloor\sqrt{4n+2}\rfloor$. Thus the claim could only be wrong if $\lfloor\sqrt{4n+1}\rfloor< \lfloor\sqrt{4n+2}\rfloor$, i.e. if there exist an ineger $m$ with $\sqrt {4n+1}<m\le\sqrt {4n+2}$, equivalently: with $4n+1<m^2\le 4n+2$. Bu... |
1,456,172 | <p>I have finished the first half of Tom Apostol's <em>Calculus vol. 1</em>. While the theory is explained well, it doesn't teach much how you could apply the techniques you learn. I think I get a feeling for mathematics only when I see how I can apply it in solving real-life problems from different domains or deriving... | Cm7F7Bb | 23,249 | <p>We don't have the equality $\Pr(A\cap B)=\Pr(A)\Pr(B)$ in general. However, if $A$ and $B$ are independent events, then
$$
\Pr(A\cap B)=\Pr(A)\Pr(B).
$$
So if we don't have independence, we cannot use the equality above.</p>
|
1,095,416 | <p>$e$ and $\pi$ are rather peculiar numbers. It turns out that, in
addition to being irrational numbers, they are also transcendental
numbers. Basically, a number is transcendental if there are no
polynomials with rational coefficients that have that number as a
root.</p>
<p>Clearly, $p(x) = (x-e)(x-\pi)$ is a po... | Ayman Hourieh | 4,583 | <p>Reposting from my comments since other proofs have already been posted.</p>
<p>By tensoring the exact sequence $0 \to \mathfrak a \to A \to A / \mathfrak a \to 0$ with $M$, we get the exact sequence
$$
\mathfrak a \otimes_A M \to A \otimes_A M \to A / \mathfrak a \otimes_A M \to 0.
$$</p>
<p>Consider the canonical... |
1,902,812 | <p>I'm currently reviewing coordinate and matrix transformations for a physics class. In one of the questions, I am asked to prove the following two equations.</p>
<p>Equation 1
<span class="math-container">\begin{equation}
cos^2(\alpha)+cos^2(\beta)+cos^2(\gamma)=1
\end{equation}</span>
Equation 2 <span class="math-co... | Thomas | 284,057 | <p>Every complex manifold is orientable, as every complex vector space as a canonical orientation as a real space. Namely if <span class="math-container">$V$</span>, is a complex vector space, and <span class="math-container">$B= (u_1,...,u_n)$</span> is a base (over <span class="math-container">$\mathbb{C}$</span>), t... |
3,161,662 | <p>i want to make a python program that ask from the user a number and the program should sums up the first n squares as long as the sum is smaller (not smaller than or equal to) than the numb er that the user has choose before
Like this line </p>
<pre><code>1+2**2 +3**2+4**2 +....+n2 < the number that the user cho... | Peter | 82,961 | <p>The easiest way is to sum up the squares until the sum is greater than or equal to the chosen number and then remove the last square. (That is, subtract <span class="math-container">$1$</span> at the end).</p>
|
3,725,571 | <p>With the rise of factory games like <a href="https://www.satisfactorygame.com/" rel="nofollow noreferrer">Satisfactory</a> and <a href="https://factorio.com/" rel="nofollow noreferrer">Factorio</a>, many people have started wondering about problems like these.</p>
<p>Factorio already has a very interesting analysis ... | DeathStrike | 1,127,723 | <p>I believe your over complicating by the multiple belt speeds, essentially your limit is your current maximum belt. Example a tier 3 belts max capacity is 270. Any other inputs cannot exceed your proposed limit. At this point all your merging and splitting will follow based on this limit.</p>
<p>The belt tiers lower ... |
1,950,954 | <p>Are the following equal?
$$\sum_{k = 1}^{\infty}a_k = \sum_{k = 1}^{\infty}(a_{2k} + a_{2k - 1})$$
If I expand the summations they are the same series, so they should be equivalent in all respects (convergence, divergence, etc.), right? I just want to confirm that there is no problem with manipulating an infinite se... | Jack D'Aurizio | 44,121 | <p>It is subtle. Assuming $\sum_{k\geq 1}a_k$ is convergent, you are always allowed to write $$\sum_{k\geq 1}a_k = \sum_{k\geq 1}(a_{2k}+a_{2k-1})$$ but not $$\sum_{k\geq 1}a_k = \sum_{k\geq 1}a_{2k}+\sum_{k\geq 1}a_{2k-1}$$ as testified by $\sum_{n\geq 1}\frac{(-1)^{n-1}}{n}$, that is only conditionally convergent and... |
1,950,954 | <p>Are the following equal?
$$\sum_{k = 1}^{\infty}a_k = \sum_{k = 1}^{\infty}(a_{2k} + a_{2k - 1})$$
If I expand the summations they are the same series, so they should be equivalent in all respects (convergence, divergence, etc.), right? I just want to confirm that there is no problem with manipulating an infinite se... | zhw. | 228,045 | <p>No, you can't always write $$\sum_{k\geq 1}a_k = \sum_{k\geq 1}(a_{2k}+a_{2k-1})$$ For example, let $a_k = (-1)^k.$ Then the first series diverges, while the second series is just the sum of $0$'s, hence converges to $0.$ It is true that that if the first series converges, then the second series converges to the sam... |
948,854 | <p>There is a test to find out if a person has cancer.</p>
<p>1.) 95% of people who will take this test AND have cancer will get a positive ID that they have cancer.</p>
<p>2.) 2% of people who take this test who DON'T have cancer will also get a positive ID that they have cancer.</p>
<p>3.) The chances that a perso... | JMoravitz | 179,297 | <p>This is an application to Baye's Theorem stating that $P(A|B)=\frac{P(A)\cdot P(B|A)}{P(B)}$.</p>
<p>In your example, let $A$ represent the event that the person has cancer and let $B$ represent the event that they test positive for cancer.</p>
<p>We know that $P(B|A) = P$(They tested positive if they have cancer)... |
1,382,366 | <p>Can anybody pass me on a good source to see the steps in proving,
<span class="math-container">\begin{equation}
\zeta(2n) = \frac{(-1)^{k-1}B_{2k}(2\pi)^{2k}}{2(2k)!}
\end{equation}</span></p>
<p>I know how we start by looking at the product of sine and use the generatinf function for the Bernoulli numbers to connec... | Dietrich Burde | 83,966 | <p>There is a nice proof in Remmert's book "Funktionentheorie" using residue calculus. Also, Tom M. Apostol has given a short elementary proof in The American Mathematical Monthly 1973 (<a href="http://www.jstor.org/stable/2319093" rel="nofollow noreferrer">JSTOR</a>, <a href="https://www.google.com/search?q=Apostol+An... |
1,647,356 | <blockquote>
<ol start="2">
<li>A universe contains the three individuals $a,b$, and $c$. For these individuals, a predicate $Q(x,y)$ is defined, and its truth values are given by the following table:
\begin{array}{c|ccc}
x\backslash y&a&b&c\\\hline
a&T&F&T\\
b&F&T&F\\
c&F&... | robjohn | 13,854 | <p>We can also use the substitution $u=\sqrt{1-x}$, then $\mathrm{d}u=-\frac{\mathrm{d}x}{2\sqrt{1-x}}$ and $\sqrt{1+x}=\sqrt{2-u^2}$. We will also use $u=\sqrt2\sin(\theta)$
$$
\begin{align}
\int\frac{\sqrt{1+x}}{\sqrt{1-x}}\,\mathrm{d}x
&=-2\int\sqrt{2-u^2}\,\mathrm{d}u\\
&=-2\int\sqrt2\cos(\theta)\cdot\sqrt2... |
1,433,580 | <p>I'm stuck in this question, how can I find the limit below?
$$\lim\limits_{n\to \infty}{2n^n\over (n+1)^{n+1}}$$</p>
| João Areias | 270,376 | <p>After the hint from @5xum I could solve the question, so here is the answer:</p>
<p>$$\frac{2n^n}{(n+1)^{n+1}} = \frac{2}{n+1}\cdot \frac{n^n}{(n+1)^n}$$</p>
<p>so we have that</p>
<p>$$\lim_{n\to \infty}{\frac{2n^n}{(n+1)^{n+1}}} = \lim_{n\to \infty}{\frac{2}{n+1}}\cdot \lim_{n\to \infty}{\frac{n^n}{(n+1)^n}}$$<... |
3,479,232 | <p>Here's a problem from my textbook:</p>
<blockquote>
<p>On the island of Mumble, the Mumblian alphabet has only 5 letters, and every word in the Mumblian language has no more than three letters in it. How many words are possible if letters can be repeated?</p>
</blockquote>
<p>I know we can break this problem int... | David K | 139,123 | <p>The problem with your approach is that the blank second letter is actually a <em>different case</em> than any of the five letters of the alphabet,
because in that case the third letter must be blank.
To fix your calculation you need to consider at least two cases.</p>
<hr>
<p>There is a general approach that does ... |
1,826,221 | <p>I just read about Continuum Hypothesis which states that there is no set $S$ with the cardinality of $S$ is strictly larger than $\mathbb{N}$ and strictly smaller than $\mathbb{R}$.</p>
<p>I recall that in a math class we proved that the power set $P(\mathbb{N})$ have the same cardinality with $\mathbb{R}$. So I wo... | Brian M. Scott | 12,042 | <p>It is consistent with the usual $\mathsf{ZFC}$ axioms of set theory that there be such a set, and it is also consistent with them that no such set exist. The statement that there is no such set is precisely the <a href="https://en.wikipedia.org/wiki/Continuum_hypothesis#The_generalized_continuum_hypothesis" rel="nof... |
239,187 | <p>I need help constructing a plane function, <span class="math-container">$z = f(x,y)$</span> that goes through three points, (05, 22, 20). (89, 0, 89) and (-1, -1, 10). I have tried to input it, but I dont know how.</p>
<p><a href="https://i.stack.imgur.com/UNg8A.png" rel="nofollow noreferrer"><img src="https://i.sta... | MarcoB | 27,951 | <p>This will get you the coefficient values:</p>
<pre><code>sol = Solve[
{5 a + 22 b + c == 20, 88 a + c == 88, -a - b + c == 10},
{a, b, c}
]
(* Out: {{a -> 1784/2041, b -> 422/2041, c -> 22616/2041}} *)
</code></pre>
<p>You can then replace these values in your generic equation and solve for <span class="... |
239,187 | <p>I need help constructing a plane function, <span class="math-container">$z = f(x,y)$</span> that goes through three points, (05, 22, 20). (89, 0, 89) and (-1, -1, 10). I have tried to input it, but I dont know how.</p>
<p><a href="https://i.stack.imgur.com/UNg8A.png" rel="nofollow noreferrer"><img src="https://i.sta... | kglr | 125 | <pre><code>ClearAll[lazyMansPointsToPlaneFunction]
lazyMansPointsToPlaneFunction[pts_][x_, y_] := #2 & @@
Reduce[RegionMember[InfinitePlane@pts][{x, y, z}], z, Reals]
</code></pre>
<p><em><strong>Examples:</strong></em></p>
<pre><code>points = {{5, 22, 20}, {88, 0, 88}, {-1, -1, 10}};
lazyMansPointsToPlaneFu... |
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Lucas Barbiere | 366,197 | <p>I think you already have a good interpretation of complex numbers algebraic view. So I propose here the geometric approach of this numbers, following Hamilton, Clifford and Grassmann geometric view of algebra.</p>
<p>First, we need to separate the meaning about the number (1) and the number (-1). You can do this fo... |
1,129,070 | <p>In a recent examination this question has been asked, which says:</p>
<p>$a^2+b^2+c^2 = 1$ , then $ab + bc + ca$ gives = ?</p>
<p>What should be the answer? I have tried the formula for $(a+b+c)^2$, but gets varying answer like $0$ or $0.25$, on assigning different values to variables.</p>
<p><em>How to approach... | 5xum | 112,884 | <p>Just knowing that $a^2+b^2+c^2=1$ is not enough to determine the value of $ab+bc+ca$. For example, if $a=b=0$ and $c=1$, then $ab+bc+ca = 0$. On the other hand, if $a=b=c=\frac{1}{\sqrt3}$, then $ab+bc+ca = 1$. In fact, using $$(a+b+c)^2=a^2+b^2+c^2 + 2(ab+bc+ca),$$ you get that $$ab+bc+ca=\frac{(a+b+c)^2 - 1}{2}.$$... |
4,215,924 | <p>The Weak Factorization Theorem tells us that birational map of varieties over field of perfect characteristic which has resolution of singularities can be factored into blow-ups and blow-downs. My question is what happens when we restrict ourself to birational morphisms instead of birational maps? Can we then assume... | Nick L | 532,949 | <p>It is true that every birational morphism is the blow-up of the sheaf of ideals on the range, see theorem 7.17 in Hartshorne.</p>
<p>I don't think that every birational morphism is the blow-uo if a closed subset. Consider (one of the) the small resolution of the <span class="math-container">$3$</span>-fold ODP with ... |
1,714,965 | <p>I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic?</p>
<p>My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have the quality that $p_1 \times a = p_2 \times b$ where $a \space\text{and}\space b$ are integers, then the enti... | Community | -1 | <p>Assume</p>
<p>$$\sin(x+T)\sin(\pi(x+T))=\sin(x)\sin(\pi x)$$ for all $x$.</p>
<p>Then with $x=0$,</p>
<p>$$\sin(T)\sin(\pi T)=0$$ so that $T=k\pi$ or $T=k$.</p>
<p>But you can find examples of</p>
<p>$$\sin(x+k\pi)\sin(\pi x+k\pi^2)\ne\sin(x)\sin(\pi x),$$ i.e.
$$\sin(\pi x+k\pi^2)\ne\sin(\pi x),$$
and</p>
<p... |
1,870,751 | <p>What does Aut$(\Bbb Z)$ look like? (Integers with the operation of addition)</p>
<p>I understand that it's the set of all automorphisms from $\Bbb Z$ to $\Bbb Z$, or Aut$(\Bbb Z) = \{\alpha_1, \alpha_2, ... : \alpha_i$ is an isomorphism from $\Bbb Z$ to $\Bbb Z \}$.</p>
<p>I figured that the only isomorphisms tha... | Thomas | 26,188 | <p>You are trying to determine all automorphisms (isomorphisms) of the group $\mathbb{Z}$. So say that $\phi: \mathbb{Z} \to \mathbb{Z}$ is an isomorphism. You want to figure out what $\phi$ could be.</p>
<p>Now, say that $n = \phi(1)$. Then $\phi(m) = \phi(1+\dots + 1) = m\phi(1) = mn$, so $\phi$ is completely determ... |
19,797 | <p>I'm trying to compute a multidimensional integral with a variable number of dimensions.</p>
<p>The integral is as follows:
$$
\int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}.
$$</p>
<p>I have tried this</p>
<pre><code>Integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]
</code></pre>
<p>but it's not wor... | whuber | 91 | <p>For integrands where Fubini's Theorem is applicable (and that would be the great majority), we may iterate the integral. The following solution--intended to work with fixed limits of integration in all dimensions--does this with two definitions: one to perform the integration over the last variable and another to g... |
65,990 | <p>It is well know that planewaves are a complete basis for solutions to the wave equation. Let us assume a 2D space, and at fixed temporal frequency, the equation reduces to the Helmholtz equation. In cylindrical coordinates, the most appropriate solutions are the two kinds of Hankel functions, representing outgoing a... | Marty Green | 10,838 | <p>I once worked out a method to get the Neumann functions in terms of plane waves by physical reasoning. I think you know that if you take a uniform distribution of sine waves in all different directions in 2d space, in phase at the origin, you get the Bessel functions; the 0th order if they are in phase about the cir... |
2,177,581 | <p>Let $S$ be that part of the surface of the paraboloid $z=x^2+y^2$ between the planes $z=1$ and $z=4$.</p>
<p>Now given $\vec{V}=x^3\hat j+z^3\hat k$ and evaluate the line integrals $\int_{C}{_1}\vec{V}.dr+\int_{C}{_2}\vec{V}.dr$ where $C_1$ and $C_2$ are the curves bounding $S$</p>
<p>I know the answer should be $... | Rene Schipperus | 149,912 | <p>If $T$ is not surjective then $$\dim im T <\dim W$$ Take a $f\in W^*$ such that
$$im T\subseteq \ker f.$$</p>
<p>Then $T^*(f)=0$.</p>
|
401,482 | <p>I have the following question. I know the answers but am struggling with how to write up the work formally. Any help would be appreciated. </p>
<p>Consider $B=\{1-\frac{1}{n} :n=1,2,\cdots\}$. Is $B$ open? Closed? Compact? Justify.</p>
<p>I know that it is not open or closed, thus not compact. </p>
| dc2814 | 79,257 | <p>$B\subset \mathbb{R}$ with the usual topology. $B$ is not open since (for example) $0\in B$ is not an interior point since every neighborhood of $0$ intersects $B^c$. $B$ is not closed since $1$ is a limit point not contained in $B$ (a sequence in $B$ converging to 1 is $\{1-1/n\}$). In a metric space, compact sets ... |
966,665 | <p>I noticed something interesting result but I do not know how to prove it (or disprove)</p>
<p>Function $U$ defined as </p>
<p>$$ U(Z(x),Z'(x),Z''(x),Z'''(x),...,Z^{(n)}(x))=\frac{d^n}{dx^n} \left( Z^m(x) \right)$$
and $m,n>0$ </p>
<p>I conjecture that
$$U(1,1,1,1,....,1)=m^n$$</p>
<p>Some examples</p>
<hr... | Mathlover | 22,430 | <p>I proved my claim. I would like to share it.</p>
<p>Taylor expansion of $Z^m(x+h)$ can be written as</p>
<p>$$
Z^m(x+h)=Z^m(x)+h\frac{d}{dx} \left( Z^m(x) \right)+\frac{h^2 }{2!}\frac{d^2}{dx^2} \left( Z^m(x) \right)+\frac{h^3 }{3!}\frac{d^3}{dx^3} \left( Z^m(x) \right)+....
$$</p>
<p>If we derivate both sides n ... |
1,336,355 | <blockquote>
<p>Suppose you're on a game show, and you're given the choice of one hundred doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to switch... | Ben Grossmann | 81,360 | <p>The answer is no. As a counterexample, consider the sequence
$$
f_n(x) = \frac{n}{n+1}\cdot \frac{1}{x}
$$</p>
|
187,145 | <p>It is asserted in this <a href="http://www.math.shimane-u.ac.jp/memoir/39/D.Buhagiar.pdf" rel="nofollow">article on "Connective Spaces"</a> that in topological spaces, if $A \subseteq B$ are both connected and $C$ is maximally connected in $B - A$, then $B - C$ is connected (page 4, proposition (a)). I have not been... | Brian M. Scott | 12,042 | <p>In topological language, (a) says that if <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are connected, <span class="math-container">$A\subseteq B$</span>, and <span class="math-container">$C$</span> is a connected component of <span class="math-container">$B\setminus A$</span>, ... |
2,568,860 | <p>How to determine whether the given real number $\alpha =3-\sqrt[5]{5}-\sqrt[5]{25}$ is algebraic or not. And, $[\mathbb{Q}(\alpha):\mathbb{Q}]=?$</p>
<p>Let $x=3-\sqrt[5]{5}-\sqrt[5]{25}$,
\begin{align*}
& x = 3-\sqrt[5]{5}-\sqrt[5]{25}\\
\implies & (x-3) = -\sqrt[5]{5}-\sqrt[5]{25}\\
\implies & (x-3)... | Georges Elencwajg | 3,217 | <p>Since $X^5-5$ is $\mathbb Q$-irreducible (Eisenstein) it is the minimal polynomial of $r:=\sqrt [5] 5$ over $\mathbb Q$.<br>
Hence the field $\mathbb Q(r)=\mathbb Q[r]\cong \mathbb Q[X]/\langle X^5-5\rangle $ is a $\mathbb Q$-vector space with basis $1,r,r^2,r^3,r^4$.<br>
Obviously $\mathbb Q\subset \mathbb Q[\alpha... |
2,452,783 | <p>How to prove that $(a^2 + 1)(b^2 + 1)(c^2 + 1) \ge (a + b)(b + c)(c + a)$ for $a, b, c \in \mathbb{R}$ ? I have tried AM-GM but with no effect.</p>
| nonuser | 463,553 | <p>By Cauchy inequality we have:</p>
<p>$$(a^2 + 1)(1+b^2) \ge (a + b)^2$$
$$(a^2 + 1)(1+c^2) \ge (a + c)^2$$
$$(c^2 + 1)(1+b^2) \ge (c + b)^2$$</p>
<p>Multiply these and take square root and thus the conclusion.</p>
|
390,438 | <p>Suppose X,Y are sets with at least 2 elements. Show that
$X\cup Y\le X\times Y$</p>
<p>So my first thought was that cardinality $|X|\ge 2$ and the same for $|Y|\ge 2$ but by the inclusion-exclusion principle we have $|X\cup Y|=|X|+|Y|-|X\cap Y|$ but the problem does not say if they are disjoint or not.
If we assum... | egreg | 62,967 | <p>Suppose $x_1$ and $x_2$ are distinct elements of $X$, $y_1$ and $y_2$ are distinct elements of $Y$.</p>
<p>If $X$ and $Y$ are disjoint, you can define $f\colon X\cup Y\to X\times Y$ by
$$
\begin{cases}
f(x)=(x,y_1) & \text{if $x\in X$, $x\ne x_1$}\\
f(x_1)=(x_1,y_2) \\
f(y)=(x_2,y) & \text{if $y\in Y$, $y\n... |
3,316,970 | <p>I have been having some difficulties with this question. </p>
<p>How to find the maximum without the help of a calculator or graphing device?</p>
| lab bhattacharjee | 33,337 | <p>WLOG <span class="math-container">$m=\tan t$</span> where <span class="math-container">$-\dfrac\pi2<t<\dfrac\pi2\implies\sec t=+\sqrt{1+m^2}$</span></p>
<p><span class="math-container">$$d=\dfrac{\tan t+1}{\sec t}=\sin t+\cos t=\sqrt2\sin\left(t+\dfrac\pi4\right)\le\sqrt2$$</span></p>
|
3,316,970 | <p>I have been having some difficulties with this question. </p>
<p>How to find the maximum without the help of a calculator or graphing device?</p>
| lab bhattacharjee | 33,337 | <p>As <span class="math-container">$d\ge0$</span></p>
<p><span class="math-container">$$d=\dfrac{|m+1|}{\sqrt{m^2+1}}\iff d^2(m^2+1)=(m+1)^2$$</span></p>
<p><span class="math-container">$$\iff(d^2-1)m^2-2m+(d^2-1)=0$$</span></p>
<p>As <span class="math-container">$m$</span> is real, the discriminant must be <span cl... |
2,794,945 | <p>Let's say there's a 30% chance of some event happening, and if it happens then there's a 30% chance of it happening again (but it can only occur twice). I want to calculate the expected value for the number of times it happens. I think I can do this:</p>
<p>Chance of 0 occurences: 0.7</p>
<p>Chance of only 1 occur... | Alecto Irene Perez | 242,788 | <p><strong>Problem statement:</strong></p>
<ul>
<li>You have a biased coin. </li>
<li>If you flip the coin, then with probability $p$ the coin will come up
heads, and otherwise it'll come up tails.</li>
<li>You're allowed to continue flipping the coin until it comes up tails, or you've flipped it $N$ times (whichever ... |
3,100,738 | <p>Let (A,☆) be a semi-group such that the following 2 conditions are true:</p>
<ol>
<li><p>For any <span class="math-container">$a,b\in A$</span>, there exists a <span class="math-container">$x\in A$</span> such that <span class="math-container">$a☆x=b$</span></p></li>
<li><p>For any <span class="math-container">$a,b... | Gnumbertester | 628,028 | <p>Note that:
<span class="math-container">$$\frac{1}{x}=x^{-1}$$</span></p>
<p>Going off of the wikipedia definition of a polynomial found <a href="https://en.wikipedia.org/wiki/Polynomial" rel="nofollow noreferrer">here</a>:</p>
<blockquote>
<p>In mathematics, a polynomial is an expression consisting of variables... |
2,771,081 | <p>In a previous question I described <a href="https://math.stackexchange.com/q/2767118/121988"><span class="math-container">$n$</span>-robot walks and <span class="math-container">$(i,j)$</span>-paths</a>:</p>
<blockquote>
<blockquote>
<p>A [<span class="math-container">$5$</span>-]robot moves in a series of one-fifth... | Acccumulation | 476,070 | <p>Your flaw is somewhat similar to the <a href="https://en.wikipedia.org/wiki/Birthday_problem" rel="nofollow noreferrer">birthday paradox</a>: each <em>particular</em> prisoner has a 49/100 chance of being in a chain of 51 or more. But the probability of <em>at least one</em> prisoner being in a chain of 51 or more i... |
752,224 | <p>What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle. But what is the fundamental group of the half circle? </p>
| Ronnie Brown | 28,586 | <p>There is another way of putting this question. The cyclic group $C_2$ of order $2$ acts on the circle $S^1$ by conjugation $z \mapsto \bar{z}$. The fundamental group of the circle at $1$ is $\mathbb Z$ and the induced action on $\mathbb Z$ is $n \mapsto -n$; the quotient of $\mathbb Z$ by this action is cyclic of ... |
240,669 | <p><strong>Bug introduced in version 12.0.0, and persisting through 13.2.0 on Windows. Doesn't reproduce on ARM Mac versions 13.0.0 and above.</strong></p>
<hr />
<p>Calculating the integral <span class="math-container">$$\int\limits_0^1 \frac{x^2\log(1-x^4)} {1+x^4}\,dx$$</span> symbolically</p>
<pre><code>Integrate[x... | Roman | 26,598 | <p>Strangely, it works when done correctly via limit:</p>
<pre><code>f[a_] = Assuming[0 < a < 1,
Integrate[(x^2 Log[1 - x^4])/(1 + x^4), {x, 0, a}]];
A = Limit[f[a], a -> 1, Direction -> "FromBelow"];
N[A]
(* -0.162858 - 5.88785*10^-17 I *)
$Version
(* "12.2.0 for M... |
393,122 | <p>I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.</p>
<blockquote>
<p>How to prove $n^2 < n! $ ?</p>
</blockquote>
| Inceptio | 63,477 | <p><strong>Hint:</strong></p>
<p>$n^2 <n! \implies n(n-(n-1)!) <0$, and note that $(n-1)! >n$ for all $n>3$</p>
|
393,122 | <p>I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.</p>
<blockquote>
<p>How to prove $n^2 < n! $ ?</p>
</blockquote>
| André Nicolas | 6,312 | <p>We can do the comparison with $\dfrac{2}{n^2}$. It is certainly true that $\dfrac{1}{1!}\le \dfrac{2}{1^2}$. </p>
<p>And for $n \gt 1$, we have $n!\ge (n)(n-1)$. But $n-1\ge \dfrac{n}{2}$, and therefore $n!\ge \dfrac{n^2}{2}$. It follows that $\dfrac{1}{n!}\le \dfrac{2}{n^2}$ for all positive $n$. </p>
|
57,498 | <p>I have a list with points coordinates. And I'm trying to traverse it and perform some matrix operations on each point. But I have a problem with storing modified points in the initial list instead of the original points.</p>
<p>Here is the complete entry point example:</p>
<pre><code>(* the matrix of the linear op... | Alexei Boulbitch | 788 | <p>Try this:</p>
<pre><code>Map[{t1.#, rx.#, t2.#} &, points]
</code></pre>
<p>or like this:</p>
<pre><code>Map[{t1, rx, t2}.# &, points]
</code></pre>
<p>which is the same.</p>
<p>For example, if </p>
<pre><code>points = {{a1, a2, a3}, {b1, b2, b3}};
</code></pre>
<p>and</p>
<pre><code> t1 = {x, 0, 0};... |
316,235 | <p>Let $H$ be the inner product space = $\{f: \mathbb{R} \to \mathbb{C} \mid f \text{ is continuous and has period }2 \pi\}$ where the inner product is:
$$\langle f,g \rangle = \int_{0}^{2\pi}f(t)\overline{g(t)} dt $$
How do I prove that for $n \in \mathbb{Z}$ and $e_n(t)=\dfrac{1}{\sqrt{2\cdot\pi}} e^{i n t}$, $(e_n... | Christopher A. Wong | 22,059 | <p>This is probably not the most elegant way, but this method generalizes to showing the unitarity of the Fourier transform. The essential part of this proof is showing that the map $L^2(\mathbb{R}/2\pi \mathbb{Z}) \rightarrow \ell^2(\mathbb{Z})$ given by taking the Fourier coefficients from the discrete Fourier transf... |
316,235 | <p>Let $H$ be the inner product space = $\{f: \mathbb{R} \to \mathbb{C} \mid f \text{ is continuous and has period }2 \pi\}$ where the inner product is:
$$\langle f,g \rangle = \int_{0}^{2\pi}f(t)\overline{g(t)} dt $$
How do I prove that for $n \in \mathbb{Z}$ and $e_n(t)=\dfrac{1}{\sqrt{2\cdot\pi}} e^{i n t}$, $(e_n... | Community | -1 | <p>This argument is from Rudin's Real and Complex Analysis (section 4.24 Completeness of the Trigonometric System). It considers $C(-\pi,\pi)$ instead of $C(0,2\pi)$, but this makes no difference of course.</p>
<p>Suppose we had trigonometric polynomials $Q_k \geq 0$ such that</p>
<ul>
<li>$$\frac{1}{2\pi}\int_{-\pi}... |
4,219,536 | <p><span class="math-container">$G$</span> is group , <span class="math-container">$|G|=n$</span> , suppose every <span class="math-container">$d$</span> divides <span class="math-container">$n$</span> , there are <span class="math-container">$d$</span> elements so <span class="math-container">$x^d=1$</span> , prove <s... | Peanut | 144,455 | <p>You can do it using your transformation. However the set <span class="math-container">$(1, +\infty) \times(1, +\infty)$</span> in the <span class="math-container">$x$</span>-<span class="math-container">$y$</span> plane is not mapped into <span class="math-container">$(0, +\infty) \times(1, +\infty)$</span> in the <... |
271,932 | <p>Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.</p>
<p>Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^*$, then $V$ admits a nonzero $G$-invariant bilinear form $(-,-)$, unique up to scalar, such that $(-,-)$ is alternating or symmetric. Is... | Derek Holt | 35,840 | <p>The answer to the question can be found in Section 14, Frobeius-Schur indicator for Brauer characters, starting on page 320, of the book "Group Representations, Volume 4" by Gregory Karpilovsky. I don't have easy access to this book myself (our library only has Volume 1, and that is in two parts), but fortunately th... |
271,932 | <p>Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic $p \neq 2$.</p>
<p>Let $V$ be a finite-dimensional irreducible $kG$-module. If $V \cong V^*$, then $V$ admits a nonzero $G$-invariant bilinear form $(-,-)$, unique up to scalar, such that $(-,-)$ is alternating or symmetric. Is... | Mikko Korhonen | 10,146 | <p>For one answer, here is a theorem due to Thompson and Willems (<em>Bilinear forms in characteristic $p$ and the Frobenius-Schur indicator</em>, Lecture Notes in Mathematics 1185, pg. 221-230).</p>
<p>For an irreducible self-dual $kG$-module $V$, set $\varepsilon(V) = 1$ if $G$ preserves a nonzero symmetric bilinear... |
2,023,436 | <p>I am taking an ordinary differential equation class, and we are currently learning about power series. One thing that comes up is index shifting, for the most part, I can shift the index quite easily, but in the following case, I end up with a fraction index. I have an ODE of the form y'' + 2xy' + y = 0, with one of... | hamam_Abdallah | 369,188 | <p><strong>Hint</strong></p>
<p>you can downshift or upshift.</p>
<p>$$\sum_{n=k}^N f(n)=$$</p>
<p>$$\sum_{n=k-1}^{N-1}f(n+1)=\sum_{n=k+1}^{N+1}f(n-1)$$</p>
|
2,023,436 | <p>I am taking an ordinary differential equation class, and we are currently learning about power series. One thing that comes up is index shifting, for the most part, I can shift the index quite easily, but in the following case, I end up with a fraction index. I have an ODE of the form y'' + 2xy' + y = 0, with one of... | BelowAverageIntelligence | 441,199 | <p>The right-hand sum's lowest term is <span class="math-container">$x^3$</span>. We want <span class="math-container">$x^3$</span> to be of the form <span class="math-container">$x^{2n+1}$</span> so that we can combine the right-hand sum with the left-hand one. Clearly then, we can start the sum at n=1 and replace n w... |
4,393,135 | <p>I am doing some exercises on matrix equations for my upcoming exam and I have some questions the answers to which I cannot find online. I do study in Portuguese and maybe I don't know the right vocabulary for the search, so I'm trying this community.</p>
<hr />
<blockquote>
<p>Prove that the solution to the followin... | Hiro Yuroku | 994,691 | <p>Good afternoon, José! I believe this may help you:</p>
<p>First of all, notice that in <span class="math-container">$ AX − 3(A^TA^{-1})^{-1} = A $</span> it is already implicit that <span class="math-container">$ A $</span> is invertible. From that, you can deduce that <span class="math-container">$ A $</span> is a ... |
4,386,744 | <blockquote>
<p>Show that if <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are connected, then <span class="math-container">$X \times Y$</span> is also connected.</p>
</blockquote>
<p>Since <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are both... | TonyK | 1,508 | <p>Firstly: You are confusing the sum of the sequence with its individual elements when you write
<span class="math-container">$$S_n =\frac{n(n+2)(n+4)-(n-2)n(n+2)}{6}$$</span>
Let us write <span class="math-container">$S_n=u_1+\cdots+u_n$</span>, where <span class="math-container">$u_k=k(k+2)$</span>. So now we have
<... |
3,995,483 | <p>I am trying to understand what is the best approach to solve this binary programming problem</p>
<p><span class="math-container">$$ \max_{X\in \left\{0,1 \right\}^{N}} \left(\sum_{i=1}^{N} a_{i}X_{i}\right)^{\beta} - \sum_{i=1}^{N}c_{i}X_{i} $$</span></p>
<p>with <span class="math-container">$X\in \left\{0,1 \right\... | RobPratt | 683,666 | <p>The title says <em>polynomial</em>, so I will assume that <span class="math-container">$\beta$</span> is an integer. Yes, you can linearize by expanding the power and introducing a new binary variable for each resulting product of <span class="math-container">$X_i$</span>s. For example, if <span class="math-contai... |
1,021,819 | <p>For a set $X$ and $f, g \in \mathbb{R}^X$, is it true that
$$
\sup_x (f(x) - g(x))
\geq
\sup_x f(x) - \sup_x g(x)?
$$
Thanks.</p>
| Tim | 1,281 | <p>If $\sup_x f(x)$ is achieved at $x=x^*$,
$$
\sup_x f(x) - \sup_x g(x) \leq f(x^*) - g(x^*) \leq
\sup_x (f(x) - g(x))
$$</p>
<p>If $\sup_x f(x)$ is not achievable, ...?</p>
|
106,309 | <p>I'm creating a big manipulate box with several sliders, and each slider must have a small explication text (units, for example) after a mathematical symbol. As a simple example, here's the code I'm currently using for a slider :</p>
<pre><code>{{phi, 0, Style[Subscript[\[CurlyPhi], 0] "(degrees)", Bold, 10]},
... | Bob Hanlon | 9,362 | <p>Use <a href="http://reference.wolfram.com/language/ref/Flatten.html" rel="nofollow noreferrer">Flatten</a></p>
<pre><code>sol = {{{y[x] -> -2 E^-x^2}}, {{y[x] -> -E^-x^2}}, {{y[x] -> 0}}, {{y[x] ->
E^-x^2}}, {{y[x] -> 2 E^-x^2}}};
Plot[Evaluate[y[x] /. sol // Flatten], {x, 0, 1},
PlotLegen... |
1,170,477 | <p>Show that if $A$ is an $m \times n$ matrix and $A(BA)$ is defined, then $B$ is an $n \times m$ matrix.</p>
<p>I know that $A$ is a $m \times n$ matrix and to be able to multiply $B$ with $A$, $B$ must be a $n \times m$ matrix. I am confused though because I can't just assume that. </p>
| BCLC | 140,308 | <p>If ABA=(AB)A=A(BA) is defined, then AB is defined and BA is defined.</p>
<p>If AB is defined and A is mxn, then B is n x something</p>
<p>If BA is defined and A is mxn, then B is something else x m.</p>
<p>Hence B is n x something and something else x m.</p>
<p>Not quite sure how to conclude exactly, but it foll... |
3,386,218 | <p>I was trying to get a better understanding for e and pi, and came across Alon Amit's explanation here: <a href="https://www.quora.com/q/bzxvjykyriufyfio/What-is-math-pi-math-and-while-were-at-it-whats-math-e-math" rel="nofollow noreferrer">https://www.quora.com/q/bzxvjykyriufyfio/What-is-math-pi-math-and-while-were-... | John Hughes | 114,036 | <p>When you "solve" <span class="math-container">$f' = f$</span>, you <em>almost</em> get an answer. It's a little like solving <span class="math-container">$x^2 = 4$</span>. In that case, you get <em>two</em> answers, which are negatives of each other <span class="math-container">$\pm 2$</span>. And if you h... |
135,218 | <p>What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research?</p>
<p>(Edit - Since the first version of the question got closed, examples should showcase specific work of specific mathematicians. Tools such as Wolfram Alpha, bibliography managers, arXiv ... | Brian Borchers | 9,022 | <p>There are several mobile apps for use with the Zotero bibliography manager. Zotero allows you to store citation information for papers (and even PDF's of the actual papers) and access the information from any device (cell phone, tablet, computer) that you might be using. </p>
<p>On the Android side, check out <a ... |
135,218 | <p>What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research?</p>
<p>(Edit - Since the first version of the question got closed, examples should showcase specific work of specific mathematicians. Tools such as Wolfram Alpha, bibliography managers, arXiv ... | Bjørn Kjos-Hanssen | 4,600 | <p><a href="https://play.google.com/store/apps/details?id=edu.hawaii.math.bjoern.autocomplex" rel="nofollow noreferrer">AutoComplex</a> is an Android app that uses "autocomplete" to let the user look up <a href="http://math.hawaii.edu/home/theses/MA_2013_Hyde.pdf" rel="nofollow noreferrer">nondeterministic automaton co... |
4,339,263 | <p>Consider a manifold <span class="math-container">$N$</span> and a submanifold <span class="math-container">$M$</span>. We're assuming that <span class="math-container">$N$</span> and <span class="math-container">$M$</span> can be non-compact. Is it possible to find a metric <span class="math-container">$g$</span> of... | Narasimham | 95,860 | <p>Not clear fully to me; There should be common tangentiality along with tangential curvature continuity ? like a contacting cylinder-ellipsoid pair with common geodesics ?</p>
<p><a href="https://i.stack.imgur.com/CHrQi.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CHrQi.png" alt="enter image des... |
4,229,288 | <p><span class="math-container">$$\int_0^1\int_0^x \sqrt{x+y^2}\,dydx = \int_0^x\int_0^1 \sqrt{x+y^2}\,dxdy$$</span></p>
<p>Are these two integrals equivalent to each other? I assumed they weren't after imagining that one should also change the order of integration (in that case, analysing the region and changing the l... | John Wayland Bales | 246,513 | <p>The limits of a double integral can be interpreted as equations. The graphs of those equations bound a region. The double integral</p>
<p><span class="math-container">$$\int_{x=0}^{x=1}\int_{y=0}^{y=x} f(x,y)\,dy\,dx $$</span></p>
<p>is over the following region.</p>
<p><a href="https://i.stack.imgur.com/qlrGy.png" ... |
317,842 | <p>A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, A-, B+, B, B-, C+}, with equal probability, independently of other papers. How many papers do you expect to hand in before you receive each possible grade at least once?</p>
| angryavian | 43,949 | <p>Before the first point at which you have received all 6 grades, you must have passed the first point at which you have received 5 different grades, which occurs after the first point at which you have received 4 different grades, and so on. Due to the linearity of expectation, you can break up your random variable $... |
317,842 | <p>A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, A-, B+, B, B-, C+}, with equal probability, independently of other papers. How many papers do you expect to hand in before you receive each possible grade at least once?</p>
| robjohn | 13,854 | <p>If a series of independent events each have probability $p$, then the expected duration until the first occurrence is
$$
\begin{align}
&1\overbrace{p}^{\text{$1$ success}}+2\overbrace{(1-p)p}^{\begin{array}{l}\text{$1$ failure}\\\text{$1$ success}\end{array}}+3\overbrace{(1-p)^2p}^{\begin{array}{l}\text{$2$ fail... |
143,263 | <p>For any $m\in\mathbb N$, let $S(m)$ be the digit sum of $m$ in the decimal system. </p>
<p>For example, $S(1234)=1+2+3+4=10, S(2^5)=S(32)=5$. </p>
<p><strong>Question 1</strong> :Is the following true?
$$\lim_{n\to\infty}S(3^n)=\infty.$$</p>
<p><strong>Question 2</strong> :How about $S(m^n)$ for $m\ge 4$ except s... | Alexey Ustinov | 5,712 | <p>There is more simple agrument which solves the problem: <em>powers $m^n$ can start with arbitrary string of digits.</em></p>
|
275,325 | <p>I am suppose to use calculus to find the max vertical distance between the line $y = x + 2$ and the parabola $y = x^2$ on the interval $x$ greater then or equal to $-1$ and less then or equal to $2$.</p>
<p>I really have no idea what to do. I found the critical numbers and that didn't help at all. I guess all the i... | copper.hat | 27,978 | <p><strong>Note</strong>: The following solves the problem of maximizing the distance between two points, one on a line and one on a parabola, subject to a constraint. This is what the question above asks, although comments from the OP suggest that the intent was to maximize the difference of the $y$ values for a given... |
3,275,356 | <blockquote>
<p>Prove that <span class="math-container">$$\frac{9}{1!}+\frac{19}{2!}+\frac{35}{3!}+\frac{57}{4!}+\frac{85}{5!}+......=12e-5$$</span></p>
</blockquote>
<p><span class="math-container">$$
e=1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+......
$$</span></p>
<p>I have no clue of whe... | Daniel Buck | 293,319 | <p>Look at the sequence of numerators and their differences</p>
<p><span class="math-container">\begin{array}
n &1 & 2 &3 &4& 5 \\
u_n & 9 & 19 &35&57&85 \\
1st diff. & 10&16&22&28& \\
2nd diff. &6&6&6&&
\end{array}</span></p>
<p>The firs... |
795,469 | <p><img src="https://i.stack.imgur.com/U4ZBh.png" alt="enter image description here"></p>
<p>Can someone please explain this answer to me with a graphical representation or just a have a better explanation than this?
I would really appreciate it, please,
Thanks </p>
| terminix00 | 143,195 | <p>Ah, so I was looking at this the wrong way. This is a quadratic residue problem and we can use legendre symbols to solve this problem. Namely we do (10/101) = (2/101)(5/101).By following the legendre properties we end up with (10/101) = -1 which means that this is not a quadratic residue and thus has no solution.</p... |
3,821,082 | <p>If <span class="math-container">$p$</span> is prime then the additive group <span class="math-container">$\mathbb{Z}_p$</span> has no proper non-trivial subgroup.
I need to use the theorem that any subgroup of a cyclic group is cyclic.</p>
<p>My thoughts:</p>
<p>Let <span class="math-container">$H$</span> be a subgr... | Lutz Lehmann | 115,115 | <p>1.) It is correct, if a little unusual.</p>
<p>2.a) The usual way for a Riccati equation with a particular solution is to set <span class="math-container">$y(x)=y_p(x)+\frac1{u(x)}$</span> which should result in a linear first order DE for <span class="math-container">$u$</span>.</p>
<p>2.b) It is typical for Riccat... |
727,223 | <p>Show that if N is a normal subgroup of G and |N| = 2, then N is a subgroup of Z(G).</p>
<p>proof: Let N be a normal subgroup of G. Then N is a subgroup of G and g is in G. So gN = Ng for all g in G. Suppose |N| = 2. </p>
<p>Then I am stuck on where to go from here. Any suggestions?</p>
| User12345 | 133,683 | <p>Let $N$ be a normal subgroup of $G$ of order two. Thus, $N = \{e,n\} \subseteq G$. Moreover, for all $g \in G$ we have that $\{g,gn\} = gN = Ng = \{g,ng\}$. Hence $gn = ng$ for all $g \in G$, and so $n \in Z(G)$.</p>
|
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