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 |
|---|---|---|---|---|
2,575,439 | <p>How to attach pen tool to endpoint of a segment (or curve) in Geogebra such that by moving segment, the pen draw a curve? for example I want to draw animated <a href="https://en.wikipedia.org/wiki/Cycloid" rel="nofollow noreferrer">Cycloid</a> in Geogebra but I don't know how to do it?</p>
| dxiv | 291,201 | <p>Hints:</p>
<ul>
<li><p>$\;z^n=a \iff p(z)=z^n - a = 0\,$, and if $\,p(z)$ had a multiple root, then that root would have to be a common root with its derivative $p'(x)=n\,z^{n-1}\,$.</p></li>
<li><p> <s>Consider for example the case $\,n=4\,$ with $\,z^4=1\,$, where $\pm 1$ are also roots of $z^2=1$.</s></p></... |
258,071 | <p>Let $P(x),Q(x),R(x)$ be the statements $x$ is a clear explanation,$x$ is satisfactory,$x$ is an excuse,respectively. Suppose that the domain for $x$ consists of all the English text. Express each of these statements using quantifiers, logical connectives and $P(x),Q(x),R(x)$.</p>
<p>a. All clear explanations are sa... | Peter Smith | 35,151 | <p>(b) and (c) are wrong. Restricted existentials need <em>conjunctions</em>. 'Some $A$s are $B$s" says that something which is an $A$ is also a $B$, so $\exists x(Ax \land Bx)$.</p>
<p>$\exists x(Ax \to Bx)$ is true if there is something which satisfies the condition $Ax \to Bx$, and anything that <em>doesn't</em> sa... |
284,444 | <p>For a positive integer $n$ put $\omega(n)$ for the number of distinct prime divisors of $n$. It is a well-known theorem of Erdős and Kac that the probability distribution for the quantity</p>
<p>$\displaystyle \frac{\omega(n) - \log \log n}{\sqrt{\log \log n}}$ </p>
<p>is the standard normal distribution. In other... | Christian Elsholtz | 36,707 | <p>As a survey in book form I would recommend Tenenbaum's book (Introduction to analytic and probabilistic number theory), chapter II. 6.1 (Integers having $k$ prime factors). Also the notes of the end of the chapter give very useful references (such as the Hildebrand-Tenenbaum paper mentioned by Lucia, the Selberg-Del... |
1,558,934 | <p>How many permutations, ρ, are there in $S_9$(the group of permutations of order 9!) whose decomposition into disjoint
cycles consists of three 2-cycles (transpositions) and one 3-cycle? In other words, how
many permutations are there in $S_9$ whose decomposition into disjoint cycles is of the form
$(a_1a_2)(a_3a_4)(... | Clément Guérin | 224,918 | <ol>
<li><p>You need to choose the support of the first transposition : $\begin{pmatrix}9\\2\end{pmatrix}$ choices.</p></li>
<li><p>You need to choose the support of the second transposition in the remaining set : $\begin{pmatrix}7\\2\end{pmatrix}$ choices.</p></li>
<li><p>You need to choose the support of the third tr... |
3,614,864 | <p>I noticed that most equations that I've encountered in physics and engineering classes are formulated as differential equations. Some examples I can think of on top of my head are Newton's 2nd law, the wave equation, Maxwell's equations, etc. My question is, what's so special about differential equations that make t... | Robert Israel | 8,508 | <p>Basically the point is that physical systems are acted on by certain forces, causing gradual change in their state, and the response depends only on the state of the system at the current time: thus the rates of change of the system variables (positions
and velocities in mechanics) are functions of those variables. ... |
318,753 | <p>I tried to show that the following group is abelian by manipulation the relations but they didn't work. Please show me the right way. The group is $$G:=\left<x,y \mid xyxy^2=yxyx^2=1\right>$$</p>
| Tara B | 26,052 | <p>HINT: From $xyxy^2 = 1$, you get $xyx = y^{-2}$. Try substituting this into $yxyx^2 = 1$.</p>
|
318,753 | <p>I tried to show that the following group is abelian by manipulation the relations but they didn't work. Please show me the right way. The group is $$G:=\left<x,y \mid xyxy^2=yxyx^2=1\right>$$</p>
| Mikasa | 8,581 | <p>Sorry for this kind of answer. @Tara's hint is enough but mine is base on Van Kampen diagram. </p>
<p><img src="https://i.stack.imgur.com/MV5mB.jpg" alt="enter image description here"></p>
|
4,224,043 | <blockquote>
<p><strong>Question:</strong> Let <span class="math-container">$G$</span> be a matchable graph, and let <span class="math-container">$u$</span> and <span class="math-container">$v$</span> be distinct vertices of <span class="math-container">$G$</span>. Show that <span class="math-container">$G - u - v$</sp... | William | 860,795 | <p>First, the statement is true if <span class="math-container">$\lambda_v$</span> is required to be <em>positive</em>. It's not hard to see that if <span class="math-container">$\lambda_v$</span> is just real, there are in fact two possible coefficients.</p>
<hr />
<p>There are some interesting ways one could go about... |
2,992,411 | <p><span class="math-container">\begin{cases}
\frac {dP}{dt} = rP(t)(1-\frac {P(t)}{K}) ,t \geq 0 \\
P(0) = P_o
\end{cases}</span></p>
<p><span class="math-container">$r, K$</span> and <span class="math-container">$P_o$</span> are positive constants.</p>
<p>We say that <span class="math-container">$P(t), t \geq0... | Arthur | 15,500 | <p>There is a bijection from <span class="math-container">$\Bbb Z^2$</span> to <span class="math-container">$\Bbb N$</span>, inducing a bijection from <span class="math-container">$\{-1,1\}^{\Bbb Z^2}$</span> to <span class="math-container">$\{-1,1\}^{\Bbb N}$</span>. And <span class="math-container">$\{-1,1\}^{\Bbb N}... |
248,900 | <p>Let $\mathfrak{n}$ be a $2k$ dimensional $2$-step nilpotent Lie algebra and suppose that its center is $k$ dimensional. Does $\mathfrak{n}$ admit symplectic structure?</p>
<p>Let $\{f_1,\dots,f_k\}$ be a basis of the center of $\mathfrak{n}$ and complete it to a basis of $\mathfrak{n}$ $\{e_1,\dots,e_k,f_1,\dots,f_... | YCor | 14,094 | <p>Consider the Lie algebra with basis $(e_1,\dots,e_7,z_1,\dots,z_7)$, with nonzero brackets (up to skew-symmetry):
$$z_1=[e_1,e_2]=[e_3,e_4]=[e_1,e_6]=[e_5,e_7];$$
$$z_2=[e_2,e_5],z_{3}=[e_2,e_6],z_{4}=[e_2,e_7],z_5=[e_3,e_5],z_{6}=[e_3,e_6],z_{7}=[e_3,e_7].$$</p>
<p>Note that it's 2-step nilpotent with center being... |
324,503 | <blockquote>
<p>Evaluate the limit
$$ \lim_{n\rightarrow\infty}{\frac{n!}{n^{n}}\left(\sum_{k=0}^{n}{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}} \right)} $$</p>
</blockquote>
<p>I use $$e^{n}=1+n+\frac{n^{2}}{2!}+\cdots+\frac{n^{n}}{n!}+\frac{1}{n!}\int_{0}^{n}{e^{x}(n-x)^{n}dx}$$
but I don't know h... | Ron Gordon | 53,268 | <p>We can write that integral as </p>
<p>$$\int_0^n dx \: e^x (n-x)^n = n^{n+1} \int_0^1 du \: e^{n u} (1-u)^n = n^{n+1}\int_0^1 du \: e^{n [u+\log(1-u)]} $$</p>
<p>Now, as $n \rightarrow \infty$, that last integral is dominated by contributions near $u=0$. We may then use the first term in the Taylor expansion of th... |
484,589 | <p>a) $ \bigcap_{n=1}^{\infty}(-\frac{1}{n},\frac{1}{n}) $</p>
<p>b) $ \bigcap_{n=1}^{\infty}(-\frac{1}{n}, 1+\frac{1}{n})$</p>
<p>c) $\bigcup_{n=1}^{\infty}(-\frac{1}{n}, 2+\frac{1}{n})$</p>
<p>Could anyone please explain how to do this problems? I'm having a hard time trying to come up with the intervals for thes... | Brian M. Scott | 12,042 | <p>The <a href="http://en.wikipedia.org/wiki/Rational_root_theorem" rel="nofollow">rational root theorem</a> says that if $$p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$$ is a polynomial with integer coefficients, and $\frac{p}q$ is a rational root of $p(x)$ written as a quotient of integers in lowest terms, then $p$ is ... |
1,166,727 | <p>Show that $c|a$ and $c|b$ iff $c|gcd(a,b)$</p>
<p>I am only going to show that the if part is true and i have the solution to this proof just i found the if part of the proof dissatisfying.</p>
<p>since c|a, c|b and $c \le gcd(a,b)$ it follows that there exists an integer $f$ such that $gcd(a,b) = cf$ and hence $c... | kobe | 190,421 | <p>For $x > 1$, </p>
<p>$$0 < \frac{1}{x}\ln\left(\frac{\ln x}{x}\right) < \frac{1}{x}\cdot \frac{\ln x}{x}.$$</p>
<p>The right hand side tends to $0$ as $x \to \infty$, because $0 \le (\ln x)/x \le 1$ for all $x \ge 1$ and $1/x \to 0$ as $x \to \infty$. Hence, by the squeeze theorem, </p>
<p>$$\lim_{x\to \... |
770,504 | <p>Find the last $3$ digits of $17^{256}$</p>
<p>So I went about solving this using Euler's totient function.</p>
<p>So I had changed it to $17^{40 \cdot6} (17^{16})$
then I reduced and had $1\cdot17^16(\mod1000)$ then I simplified and changed it to
$(17^4)^4 (\mod 1000)$ then simplified to $(521^2)^2 (\mod1000)$ th... | Peter Crooks | 101,240 | <p>Note that each of $A$ and $B$ has an element of order $18$. In $A$, you can take this element to be $1$, and in $B$, you can take it to be $(1,1)$. Since each group is also of order $18$, these groups are cyclic of the same order and hence isomorphic. </p>
<p>However, $C$ is not cyclic. It has order $18$, but the m... |
770,504 | <p>Find the last $3$ digits of $17^{256}$</p>
<p>So I went about solving this using Euler's totient function.</p>
<p>So I had changed it to $17^{40 \cdot6} (17^{16})$
then I reduced and had $1\cdot17^16(\mod1000)$ then I simplified and changed it to
$(17^4)^4 (\mod 1000)$ then simplified to $(521^2)^2 (\mod1000)$ th... | Kaj Hansen | 138,538 | <p>Two hints: </p>
<ol>
<li>If two cyclic groups have the same order, they are isomorphic. (Why?)</li>
<li>Isomorphisms preserve the orders of elements. Hence, if there is an element of order $n$ in one group, then any group isomorphic to it will also have an element of order $n$.</li>
</ol>
<p><em>Only after</em> ... |
1,926,382 | <p>Let $T:\Bbb R^n\longrightarrow\Bbb R^n$ be a linear transformation, where $n\geq 2$. For $k\leq n$,
let $E=\{v_1,v_2,\dots,v_k\}$ contained in, equal to $R^n$ and $F=\{Tv_1,Tv_2,\dots,Tv_k\}$.
Then</p>
<p>a). If $E$ is linearly independent, then $F$ is linearly independent.</p>
<p>b). If $F$ is linearly independe... | Sandy | 342,265 | <p>Option 3 and 4 is correct.
$T:R^2\to R^2$ defined by $T(x,y)=(x+y,x+y)$. Clearly T is L.I.\
Now
$T(1,0)=(1,1),\ T(0,1)=(1,1). \ E=\{(1,0),(0,1)\}$ are L.I but $T(E)$ are L.D. So option 3 is correct.
Suppose $F=\{T(v_1),T(v_2),.....T(v_k)\}$ are L.I
Consider<br>
$$
\alpha_1 v_1+....\alpha_k v_k=0.\\
\implies T(\alp... |
75,517 | <p>I'm writing an importer for the <a href="http://www.nitrc.org/projects/gifti/" rel="noreferrer">GIFTI file format</a>. The details of the format are not particularly important, but the basic idea is that it is a relatively simple XML file which includes binary arrays of 32-bit floating point numbers that are represe... | Mark Adler | 94 | <p>The Base64 string you provided as an example is <em>not</em> an encoding of a gzip stream (RFC 1952). It is an encoding of a zlib stream (RFC 1950). For background, those are different wrappers around the raw "deflate" compressed data format (RFC 1951), where the wrappers are headers and trailers proving informati... |
72,084 | <p>I want to create STS(n) algorithmically. I know there are STS(n)s for $n \cong 1,3 \mod 6$. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I end up using trial and error. Is there a general algorithm that can be used?</p>
| David E Speyer | 297 | <p>The following is Bose's construction for the $6k+3$ case: Elements of the STS are labeled by ordered pairs $(x, i)$ where $x$ is in $\mathbb{Z}/(2k+1)$ and $i$ is in $\mathbb{Z}/3$. The triples are of two forms:
$$\{ (x,0),\ (x,1),\ (x,2) \}\quad \mbox{for}\ x \in \mathbb{Z}/(2k+1)$$
$$\{ (x,i),\ (y,i),\ ((x+y)/2, i... |
72,084 | <p>I want to create STS(n) algorithmically. I know there are STS(n)s for $n \cong 1,3 \mod 6$. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I end up using trial and error. Is there a general algorithm that can be used?</p>
| Chris Godsil | 1,266 | <p>One standard algorithm for constructing Steiner triple systems is the "hill climbing" procedure. You will find it described in "Combinatorial algorithms: generation, enumeration, and search" by Kreher and Stinson, and in many papers. This procedure allows you to construct
large families of triple systems on the same... |
72,084 | <p>I want to create STS(n) algorithmically. I know there are STS(n)s for $n \cong 1,3 \mod 6$. But it is difficult to actually construct the triples. For STS(7) it is pretty easy and but for larger n I end up using trial and error. Is there a general algorithm that can be used?</p>
| Yuichiro Fujiwara | 27,829 | <p>Since this thread just got bumped to the front page, historically the very first proof (by T. P. Kirkman, On a Problem in Combinatorics, <em>Cambridge Dublin Math. J.</em> <strong>2</strong> (1847) 191-204, 1847.) of the existence of an ${\rm STS}(v)$ for all $v \equiv 1, 3 \pmod{6}$ is completely algorithmic, where... |
1,470,476 | <p>Using Stokes' theorem, the line integral of a vector field gives a surface integral of the curl of the vector field, and after that, if we apply Gauss' divergence theorem in that, it gives a volume integral of the divergence of the curl of that vector field. But we know the divergence of the curl of a vector field i... | CStarAlgebra | 212,293 | <p>Stokes theorem relates the line integral around a curve to a surface integral through an open surface.</p>
<p>The divergence theorem relates the surface integral through a closed surface to a volume integral. </p>
<p>You cannot do what you are trying.</p>
|
2,674,802 | <h2>Problem</h2>
<p>Proof that $x=a$ is solution of polynomial $P(x)=(x-a)Q(x)$ when $Q(x)$ is also polynomial expression. $P(x)$ is product of two polynomials.</p>
<h2> Attempt to solve </h2>
<h1>1. Proof</h1>
<p>We can first examine the expression by writing parenthesis open.</p>
<p>$$ P(x)=xQ(x)-aQ(x) $$</p>
... | Eric Wofsey | 86,856 | <p>The first "proof" is simply incorrect. You have given an argument that $P(x)=0$ implies $x=a$, which is the converse of what you want to prove. Moreover, your proof that $P(x)=0$ implies $x=a$ is wrong, since you cannot necessarily divide by $Q(x)$ (it may be $0$).</p>
<p>Your second proof is more or less correct... |
1,363,902 | <p>Suppose $A$ and $B$ are linear transformations on finite dimensional vector space $V$,s.t. $A,B\neq 0$ and $A^2=B^2=0$. Suppose the dimension of range $A$ and $B$ are equal, can $A$ and $B$ be similar?</p>
| Alex Zorn | 62,875 | <p>If $A^2 = 0$, then the Jordan form of $A$ has blocks which are either size $1$ or size $2$. If $A$ is $n \times n$, and the dimension of the Null space is $m$, then the number of blocks of size $2$ is $(n - m)$. So the dimension of the Null space completely determines the Jordan form. Since $A$ and $B$ have a null s... |
1,363,902 | <p>Suppose $A$ and $B$ are linear transformations on finite dimensional vector space $V$,s.t. $A,B\neq 0$ and $A^2=B^2=0$. Suppose the dimension of range $A$ and $B$ are equal, can $A$ and $B$ be similar?</p>
| Marcus M | 215,322 | <p>If $A^2 = B^2 = 0$, then all the eigenvalues of $A$ and $B$ are identically equal to $0$. Now, if we put $A$ into Jordan normal form, the blocks can only be $1 \times 1$ or $2 \times 2$, else $A^2 \neq 0$; thus, $A$ is similar to $$ Q \left(\begin{array}{ccccccc}
0 & 1 & 0 & 0 & \cdots & 0 &... |
1,363,902 | <p>Suppose $A$ and $B$ are linear transformations on finite dimensional vector space $V$,s.t. $A,B\neq 0$ and $A^2=B^2=0$. Suppose the dimension of range $A$ and $B$ are equal, can $A$ and $B$ be similar?</p>
| user190080 | 190,080 | <p>that is surely possible. Take for example $A= \begin{pmatrix}0 & 0 \\ 1& 0 \end{pmatrix}$ and $ B=\begin{pmatrix}0 & 1 \\ 0& 0 \end{pmatrix}$ and then it holds that
$$
A^2=B^2=0
$$
and
$$
A=SBS^{-1} \text{ with } S=\begin{pmatrix}0 & 1 \\ 1& 0 \end{pmatrix}
$$
so $A$ and $B$ are indeed simila... |
970,654 | <p>I'm interested in the growth of $$f(n):=\sum_{x=1}^{n-1} \left\lceil n-\sqrt{n^{2}-x^{2} } \right\rceil \quad \text{for}\quad n\rightarrow\infty $$</p>
<h3>Progress</h3>
<p>(From comments) I've got
$$\frac{f(n)}{n^2} \ge 1-n^{-1} (1+\sum\limits_{x=1}^{n-1} \sqrt{1-\frac{x^2}{n^2}} )$$ and $$\frac{f(n)}{n^2}\le... | Derek | 177,757 | <p>I think it's easier to consider a specific index set, work through a proof with that, and then generalize. Take, for example, $I = \mathbb{N}$:</p>
<blockquote>
<p>Let $\{ (G_1, \circ_1), (G_2, \circ_2), \dots \}$ be a family of groups. Let $G:= G_1 \times G_2 \times \dots$. An element $x$ of $G$ is the sequence ... |
206,390 | <p>I have a list of strings:</p>
<pre><code>lis = {"a","b","c","12","d","q","r","X","s"}
</code></pre>
<p>I'd like to delete list members starting with "X" moving backwards through the list from "X" until a list member that's a digit character is found, to get:</p>
<pre><code>res = {"a","b","c","12","s"}
</code></pr... | C. E. | 731 | <p>You would need something like <code>DeleteSubsequenceCases</code>, but it doesn't exist. I would recommend this instead:</p>
<pre><code>SequenceReplace[lis, {d_?(StringMatchQ[NumberString]), ___, "X"} :> d]
</code></pre>
<p>If <code>X</code> only appears once, you could also use this:</p>
<pre><code>First@Sequ... |
2,243,674 | <p>$\def\d{\mathrm{d}}$How to solve this ODE? (From a real analysis course, existence and uniqueness of ODE)
$$\frac{\d x}{\d t}=(x+t)t. \quad \forall t\in [0,1], \quad x(0)=0$$</p>
<p>My attempt:</p>
<p>$$\dot{x}=\frac{\d x}{\d t}=V(x(t),t)=(x+t)t$$</p>
<p>So we can use $\phi_v(x)$ such that</p>
<p>$$\phi_v^1(x,t)... | gt6989b | 16,192 | <p><strong>HINT</strong></p>
<p>Substitute $y(t) = x(t)+t$ so $\dot{y} = \dot{x} +1$ and your ODE becomes the familiar linear $$\frac{dy}{dt} = 1 + yt$$ with initial condition $y(0) = x(0)+0 = 0$.</p>
|
74,592 | <p>Let me denote $X_n$ the set of transpositions in $n$ elements. Equivalently, $X_n$ is the set of doubletons in $[1,n]\times[1,n]$. The cardinality of $X_n$ is $N=\frac{n(n-1)}{2}$.</p>
<p>If $f:{\mathbb Z}/N{\mathbb Z}\rightarrow X_n$ is a bijection, let us denote
$$r(f):=\min\{|\ell-m|;\ell\ne m\quad\hbox{and}\qu... | Noam D. Elkies | 14,830 | <p>$R_n \geq n/16$ can be obtained by starting from an arbitrary $f$ and then switching pairs of transpositions to get rid of any overlapping pairs whose images are too close to each other.</p>
<p>Suppose $r(f) < k$, and suppose $f(l)$ overlaps some $f(m)$ with $0 < |l-m| < k$. We want to find some $l'\in{\b... |
403,165 | <p>Suppose that $H\triangleleft G, K\le G\ $ and $K\nsubseteq H$. How we can prove that $HK=G?$ </p>
<p>Also $(G:H)=p$ where p is prime.</p>
| anon | 11,763 | <p>Hints:</p>
<ul>
<li>Show that $(G:H)=p$ implies $H$ is maximal.</li>
<li>Show that $H\trianglelefteq G$ and $K\not\subseteq H$ imply $H\neq HK$.</li>
</ul>
|
238,702 | <p>I am a highschool freshman, and I really like to have goals for my life, one of the big ones is my career of choice. Previously, I have always wanted to be a programmer, and I have written a lot of code. But it seems to me that programming can get slightly bland, whereas math never disappoints me. So my question ... | amWhy | 9,003 | <p>To get a nice start in exploring potential careers in math, see the Mathematical Association of America's <strong><em><a href="http://www.maa.org/careers/">MAA careers-in-math</a></em></strong> webpage. </p>
<p>See also the American Mathematical Society's <strong><em><a href="http://www.ams.org/careers/">AMS-career... |
157,413 | <p>Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important). </p>
<p>Consider a family of curves $X \subset S \times T$ parametrised by a variety $T$ of dimension 2 (the fibres $X_t$ are disti... | Transcendental | 50,614 | <p>According to what I have seen in the literature so far, the standard procedure consists of two main steps:</p>
<ul>
<li><p>Prove the existence of a universal $ C^{*} $-algebra $ A_{\theta} $ generated by two unitaries $ u $ and $ v $ that satisfy
$$
u v = e^{2 \pi i \theta} v u.
$$
<strong>Note:</strong> We are ass... |
1,640,110 | <p>Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix
with real entries such that its determinant is zero, but if one changes any single
entry one gets a matrix with non-zero determinant.</p>
<p>I think we may be able to construct such matrices.</p>
| Hans Engler | 9,787 | <p>Choose any matrix with rank $n-1$ that does not have any of the standard unit vectors in its column space.</p>
<p><strong>Added</strong> in response to the comment by alex.jordan.</p>
<p>Let $A$ be an $n \times n$ matrix with $rank(A) = n-1$ such that there are vectors $a, \, \tilde a$ with $Aa = 0, \tilde a^T A ... |
1,640,110 | <p>Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix
with real entries such that its determinant is zero, but if one changes any single
entry one gets a matrix with non-zero determinant.</p>
<p>I think we may be able to construct such matrices.</p>
| skyking | 265,767 | <p>Yes (for $n>1$), in fact almost every (in some quite generic sense) singular matrix has this property.</p>
<p>To see this we observe that the property can be expressed else-way as a stronger property. Any matrix with one-dimensional null-space and where the rows/columns are all linearly dependent (that is can be... |
2,241,326 | <p>I am not being able to understand the graphical method of solving this, any simple explanation will be appreciated.</p>
<p>A non-graphical calculation will be very helpful too.</p>
<p>Thank you so much in advance!</p>
| DHMO | 413,023 | <h1>Graphical solution</h1>
<p>Click <a href="https://www.desmos.com/calculator/ttugigqy0c" rel="noreferrer">here</a> to see the interactive graph. The red line is the graph of $f(x) = \dfrac1{1+x^8}$. The blue region is the integral. You can drag the slider to see how different values of $a$ gives different areas. Gr... |
4,419,897 | <p>Let <span class="math-container">$(X,\| \cdot \|_X)$</span>, <span class="math-container">$(Y,\| \cdot \|_Y)$</span> be normed linear spaces and <span class="math-container">$T: X \rightarrow Y$</span> be a surjective linear operator. Show that the following are equivalent:</p>
<p>(1) <span class="math-container">$T... | SacAndSac | 1,041,331 | <p>If <span class="math-container">$T$</span> is open and since <span class="math-container">$T$</span> is surjective, there exists <span class="math-container">$R>0$</span> such that
<span class="math-container">$$\overline{B_Y(0,R)}\subset T(\overline{B_X(0,1)}).$$</span>
Let <span class="math-container">$y\in Y\b... |
388,292 | <p>Currently, I am reading David Radford's Hopf Algebra, and I would like to pick up some representation theory of associative algebras as well since my knowledge of them is pretty shallow at the moment.</p>
<p>Are there any books which gives a good account of associative algebras, and the representation theory of ass... | Boris Novikov | 62,565 | <p>In addition:</p>
<p>C.W.Curtis, I.Reiner, Representation Theory of Finite Groups and Associative Algebras, 2006. </p>
<p>Despite the title, it contains a lot of information on representations of algebras.</p>
|
2,838,938 | <p>Given:</p>
<ul>
<li>$\theta$ (a negative angle)</li>
<li>$v_0$ (initial velocity)</li>
<li>$y_0$ (initial height)</li>
<li>$g$ (acceleration of gravity)</li>
</ul>
<p>I want to find the range of a projectile (ignoring wind resistance)</p>
<p>Hours of searching have given no useful results. Those that I thought we... | anonymous67 | 161,212 | <p>You can try the following:</p>
<p>The initial horizontal velocity is $v\cos\theta$, the initial vertical velocity is $-v\sin\theta$ (note that it's positive). Before landing on the field, the coordinates of the rocket follows the following equations:</p>
<p>$$x=v\cos\theta\times t$$
$$y=h-v\sin\theta\times t-\frac... |
446,835 | <p>This question is related to exercise 1.51 from Rotman's "Introduction to the Theory of Groups". </p>
<p>An element $a$ in a ring $R$ (with unit element $1$) has a <strong>left quasi-inverse</strong> if there exists an element $b \in R$ such that $a+b-ba=0$. I want to show that if every element in $R$ has a left qua... | egreg | 62,967 | <p>The $*$ operation is associative and has $0$ as neutral element (direct verification). Moreover, the left quasi-inverse of $a$ ($a\ne1$) cannot be $1$, because $a+1-1a=1\ne0$ (the $0\ne1$ assumption must be made, of course, or $R\setminus\{1\}$ would be empty and so not a group).</p>
<p>If $c$ and $d$ are left-quas... |
4,373,862 | <p>The definition of interior-point says "point <span class="math-container">$p$</span> in a set <span class="math-container">$S$</span> is interior point of <span class="math-container">$S$</span> if <span class="math-container">$\exists \delta \gt 0 : \mathcal B \left( p,\delta \right) \subseteq S $</span> but ... | Henno Brandsma | 4,280 | <p>General facts: <span class="math-container">$B(p,r) = \{x\mid d(x,y) < r\}$</span> is an open set in any metric space (any <span class="math-container">$r>0$</span>). This justifies the name "open ball" (where open = every points is interior in the above sense).</p>
<p><span class="math-container">$D... |
1,823,840 | <p>According to <a href="https://proofwiki.org/wiki/Axiom:Axiom_of_Replacement" rel="nofollow">this</a>
website, the first partion of this axiom schema is</p>
<blockquote>
<p>Let $P(y,z)$ be a propositional function, which determines a function.</p>
<p>That is, we have $∀y(∃x:(∀z:(P(y,z)⟺(x=z))))$.</p>
</blockq... | Hamed | 191,425 | <p>Consider a circle $C$
$$
(X-x_0)^2+(Y-y_0)^2=r^2
$$
and a line $L$ given by $Y=aX+b$. Then if $C$ and $L$ intersect each other at a point $(x,y)$, then since $(x,y)\in L$ you have $y=ax+b$ and since $(x,y)\in C$ we have $(x-x_0)^2+(ax+b-y_0)^2=r^2$. Therefore
$$
\Longrightarrow (a^2+1)x^2 +2x[x_0+a(b-y_0)]+x_0^2+(b-... |
1,823,840 | <p>According to <a href="https://proofwiki.org/wiki/Axiom:Axiom_of_Replacement" rel="nofollow">this</a>
website, the first partion of this axiom schema is</p>
<blockquote>
<p>Let $P(y,z)$ be a propositional function, which determines a function.</p>
<p>That is, we have $∀y(∃x:(∀z:(P(y,z)⟺(x=z))))$.</p>
</blockq... | Steven Alexis Gregory | 75,410 | <p>The formula is just an application of the Pythagorean theorem:</p>
<p><a href="https://i.stack.imgur.com/jyyKv.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/jyyKv.jpg" alt="enter image description here"></a></p>
|
3,473,147 | <p>Using the lema:
<span class="math-container">$F(u)=F(c\cdot u)=F(c+u)$</span> for <span class="math-container">$c\in F$</span></p>
<p>I could prove that:
If <span class="math-container">$r=t^2\cdot s$</span> for some <span class="math-container">$t\in \mathbb{Q}$</span>, then <span class="math-container">$\mathbb{Q... | Ninad Munshi | 698,724 | <p>You could continue that route, or</p>
<p><span class="math-container">$$\int \frac{1}{\sin^4 x}dx = \int \csc^4 x dx = \int \csc^2 x + \cot^2 x \csc^2 x \: dx$$</span></p>
<p>then let <span class="math-container">$u = \cot x$</span></p>
<p><span class="math-container">$$\implies -\int 1 + u^2 \: du = -u -\frac{1... |
3,936,782 | <p>I am interested in the following problem and its generalizations. Say we are on the real axis, and we have <span class="math-container">$n$</span> points <span class="math-container">$x_1, \ldots, x_n$</span> on this line. I would like to minimize/maximize the length someone would walk to go through all of them in a... | RavenclawPrefect | 214,490 | <p>I order the points <span class="math-container">$x_1<\ldots<x_n$</span> in this answer.</p>
<p>Your assumption for the minimization is correct. (In fact, if the points are distinct, left-to-right and right-to-left are the only two traversals of minimal length.)</p>
<p>Proof: Given any path which goes through a... |
2,235,258 | <p>$$
\int J_0(x)\sin x~{\rm d}x
$$</p>
<p>Where $J_0$ is Bessel function of first kind of order $0$</p>
<p>This what I tried </p>
<p>$$
\int J_0(x)\sin x~{\rm d}x= -J_0(x) \cos x - \int J_0'(x)\cos x~{\rm d}x
$$</p>
<p>$$
J_0'(x)=-J_1(x)
$$</p>
<p>$$
\int J_0(x)\sin x ~{\rm}x= -J_0(x) \cos x -(J_1(x)\sin x - \in... | nmasanta | 623,924 | <p>I know that I answered this question after so long time. But the main fact is I have a solution in another way and that's why I'm here. And I think this new way will help people a lot to find out the solution. </p>
<p>.................................................................................................... |
2,834,273 | <blockquote>
<p>Solve:
$\tan{4\theta} = \dfrac{\cos{\theta} - \sin{\theta}}{\cos{\theta} + \sin{\theta}}$
for acute angle $\theta$</p>
</blockquote>
<p>I need help solving that problem. I have tried to do both side, but no result yet.</p>
<p>What I have done.
$\dfrac{\sin{4\theta}}{\cos{4\theta}} = \dfrac{\cos{... | Barry Cipra | 86,747 | <p>(Remark: This answer is much like Michael Rozenberg's, which I didn't see until posting. It differs enough, I think, to be worth keeping.)</p>
<p>Since $\sin(\pi/4)=\cos(\pi/4)$, we have</p>
<p>$${\cos\theta-\sin\theta\over\cos\theta+\sin\theta}={\sin(\pi/4)\cos\theta-\cos(\pi/4)\sin\theta\over\cos(\pi/4)\cos\thet... |
2,604,711 | <p>I'm currently learning about Ore extensions in McConnell's book (Noncommutative Noetherian Rings) and Marubayashi's book (Prime Divisors and Noncommutative Valuation Theory). On the second book, I learn until part 2.3.17, where he talk about a function from $Spec_0(R[x;\delta])$ to $Spec(Q[x;\delta])$. It maps a pri... | Sooraj S | 223,599 | <p>Thanks @Rohan for the hint.</p>
<p><span class="math-container">$$
\tan^{-1}:\mathbb{R}\to \Big(\frac{-\pi}{2},\frac{\pi}{2}\Big)
$$</span></p>
<p>Taking,
<span class="math-container">$$
\alpha=\tan^{-1}x\implies x=\tan\alpha\text{ , where }\tfrac{-\pi}{2}<\alpha<\tfrac{\pi}{2}\\
\beta=\tan^{-1}y\implies{y}=\t... |
4,592,805 | <p>Let
<span class="math-container">$$
f_{(X,Y)}(x,y) = 2x
$$</span>
for <span class="math-container">$x \in (0,1), y \in (0,1)$</span>.
I need to compute density of <span class="math-container">$X+Y$</span>. So, I know that <span class="math-container">$X \perp Y$</span>, because
<span class="math-container">\begin{a... | Robert Z | 299,698 | <p>A parametrization of the disk S is
<span class="math-container">$$S:\; (u,v) \rightarrow \; \left(0, v\cos(u), v\sin(u)\right) \;\;\text{with}\;\;
0\leq u\leq 2\pi, \; 0\leq v \leq \sqrt\frac{A}{\pi}.$$</span>
Then
<span class="math-container">$$\textbf{S}_{u} \times \textbf{S}_{v}=\left(0, -v\sin(u), v\cos(u)\right... |
3,743,691 | <p>I am trying to describe this quotient group <span class="math-container">$\mathbb{Z}\times\mathbb{Z}/\mathbb{3Z}\times\mathbb{Z}$</span> Let's denote with <span class="math-container">$A$</span> and <span class="math-container">$B$</span> respectively <span class="math-container">$\mathbb{Z} \times \mathbb{Z} $</spa... | lhf | 589 | <p><em>Hint:</em> Consider <span class="math-container">$\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}/3\mathbb{Z}$</span> given by <span class="math-container">$(a,b) \mapsto a \bmod 3$</span>. Find its kernel and its image.</p>
|
3,254,331 | <blockquote>
<p>Prove <span class="math-container">$$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2$$</span></p>
</blockquote>
<p>Out of boredom, I decided to play with some integrals and Inverse Symbolic Calculator and accidentally found this to my surprise</p>
<p><span class="math-container">$... | Zacky | 515,527 | <p><span class="math-container">$$\int_0^\infty \arctan^2\left(\frac{1}{x}\right)dx\overset{\frac{1}{x}\to t}=\int_0^\infty \frac{\arctan^2 t}{t^2}dt\overset{IBP}=2\int_0^\infty \frac{\arctan t}{t(1+t^2)}dt$$</span>
<span class="math-container">$$\overset{t=\tan x}=2\int_0^\frac{\pi}{2} \frac{x}{\tan x}dx\overset{IBP}=... |
203,033 | <p>I want to plot an amplitude vs frequency with this formula:
<a href="https://i.stack.imgur.com/YQFJV.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YQFJV.jpg" alt="enter image description here"></a></p>
<p>By using this data:</p>
<p><a href="https://i.stack.imgur.com/B0VX3.jpg" rel="nofollow no... | MassDefect | 42,264 | <p>This is only meant as an answer to a question in one of the comments about how to reverse the x-axis that I couldn't fit into a comment easily.</p>
<p>You can add the option <code>ScalingFunctions -> {"Reverse", None}</code> to the plot in order to reverse the x-axis. Unfortunately, this means that the data poin... |
3,291,889 | <p>All rings are commutative ring with unity.</p>
<p>Let <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are two <span class="math-container">$R$</span>-algebras and <span class="math-container">$I$</span> and <span class="math-container">$J$</span> are two ideals of <span class="ma... | Taha Direk | 676,723 | <p><span class="math-container">$\sin(\pi/3)=\sin(\pi/6+\pi/6)=2\sin(\pi/6)\cos(\pi/6)=2\cos(\pi/3)\sin(\pi/3)$</span> thus, <span class="math-container">$\cos(\pi/3)=1/2$</span></p>
|
3,354,466 | <p>my question is:</p>
<p>Given a line <strong>S</strong>: <span class="math-container">$$
\left\{
\begin{array}{c}
x+y-1=0 \\
y+3z-2=0 \\
\end{array}
\right.
$$</span>
I need to determine the equation of a sphere having the center on line <strong>S</strong> and tangent to the plane <span class="math-container">$... | hamam_Abdallah | 369,188 | <p><strong>Hint</strong></p>
<p>A point of <span class="math-container">$S$</span> has cordinates
<span class="math-container">$$y=2-3t$$</span>
<span class="math-container">$$x=1-y=3t-1$$</span>
<span class="math-container">$$z=t$$</span></p>
<p>its distance to plane <span class="math-container">$z=0$</span> is the ... |
802,293 | <p>Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a continuous function. Is $\overline{A} =\overline{\{x: f(x) < a\}} = \{x: f(x) \leq a\} = B$? </p>
<p>Since $B$ is closed and contains $A$, $\overline{A} \subset B$ as $\overline{A}$ is the smallest closed subset containing $A$. It remains to show $B \subset \ov... | lhf | 589 | <p>Consider $f(x,y)=x^2+y^2$ and $a=0$. Then one set is empty and the other is a singleton. </p>
|
4,254,832 | <blockquote>
<p>Denote the distance between two sets <span class="math-container">$A,B \in \Bbb R^n$</span> as <span class="math-container">$d(A,B).$</span> If <span class="math-container">$d(A,B) > 0$</span> show that <span class="math-container">$m^*(A \cup B) = m^*(A) + m^*(B)$</span>.</p>
</blockquote>
<p>The p... | Mark Saving | 798,694 | <p>Call a cover <span class="math-container">$\{I_k\}$</span> of <span class="math-container">$A$</span> an <span class="math-container">$\epsilon$</span>-box cover if all boxes have all dimensions <span class="math-container">$< \epsilon$</span>, and where all boxes have a nonempty intersection with <span class="ma... |
4,084,576 | <p><img src="https://i.stack.imgur.com/x3oOV.png" alt="image" /></p>
<p>I tried using x+2 as the longer side of the large unshaded rectangle, and subtracted the right triangles to get 192. My friend tells me this is incorrect, and I was wondering how to get the correct answer.</p>
| Steven Alexis Gregory | 75,410 | <p><a href="https://i.stack.imgur.com/dsr6s.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/dsr6s.jpg" alt="enter image description here" /></a></p>
<p><span class="math-container">$\triangle CAB \sim \triangle EDC \implies DE = 27 $</span></p>
<p>The other lengths shown are easy to compute.</p>
<p><... |
170,615 | <blockquote>
<p>Let $X$ be an infinite-dimensional Banach space and $f : X \to \mathbb{R}$ continuous (not necessarily linear).</p>
<p>Can $f$ be unbounded on the unit ball?</p>
</blockquote>
<p>Of course, in a locally compact space these are impossible. Since $X$ is not locally compact one would guess these a... | Brian M. Scott | 12,042 | <p>For the first question, let $X=\ell_\infty$. For $n\in\Bbb N$ let $x_n\in X$ be defined by $$x_n(k)=\begin{cases}1,&\text{if }k=n\\0,&\text{otherwise}\;,\end{cases}$$ and let $$V_n=\left\{y\in X:\|y-x_n\|<\frac14\right\}\;.$$ Let $y\in X$ and $B=\{z\in X:\|y-z\|<1/4\}$, and suppose that $B\cap V_n\ne\v... |
44,661 | <p>Assume that $(P,\le)$ is a notion of forcing. There are several ways to define what it means for $P$ being proper and I would like to know: What is the complexity (in terms of the Levy-Hierarchy) of the statement 'P is proper'?</p>
| Amit Kumar Gupta | 7,521 | <p>How about the Proper Game formulation? </p>
<p>$(P, \leq, 1)$ is proper iff </p>
<blockquote>
<p>$\exists
> \Sigma \\ \forall \pi \\ \forall p \in
> P :$ </p>
<ul>
<li>IF $\forall x \in \pi [x$ is an ordered pair $(x_1, x_2)$, $x_1$ is natural, and $1 \Vdash _P\\ (x_2$ is an ordinal$)]$</li>
<li... |
92,743 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/86263/characteristic-of-a-field-is-0-or-prime">Characteristic of a field is $0$ or prime</a> </p>
</blockquote>
<p>Is there any field of characteristic 4? Or any other composite number? </p>
| Fredrik Meyer | 4,284 | <p>No, a field $F$ can only have prime characteristic:</p>
<p>Let $a \neq 0 \in F$. If $(mn)a = 0$ then also $m(na)=0$. If $(na) \neq 0$ then $F$ has characteristic $\leq m$. If not, then $F$ has characteristic $\leq n$. Either way, the characteristic is $< mn$. This implies that a field must have characteristic pr... |
1,438,839 | <p>If we have 8 white balls and 5 black balls and one of them is picked at random, the probability of getting a white ball is 8/13. Suppose now we put 3 white balls and 2 black balls in one closed bag and the remaining balls in another identical bag. And now the experiment is that first a bag is chosen at random and th... | Julián Aguirre | 4,791 | <p>As indicated in the comments, there are two cases to consider.</p>
<ol>
<li>$a_n$ is bounded. Since it is increasing it converges to some $a>0$ and then $\lim_{n\to\infty}(a_{n+1}^\alpha-a_n^\alpha)=a^\alpha-a^\alpha=0$. </li>
<li>$a_n$ is anbounded, and $\lim_{n\to\infty}a_n=+\infty$. Use the inequlity
$$
x^\al... |
229,915 | <p>Let $A,B\subseteq\mathbb R^d$ with $A$ closed such that $A\subset\overline{B}$. Does there exist $B'\subset B$ such that $A=\overline{B'}$?</p>
| Stefan | 36,189 | <p>Take the set of all inner points of $A$. If $x$ is an inner point of $A$, then it is an inner point of $\overline B$ as well, so it is an inner point of $B$ (as the closure doesn't add inner points). So $IP(A) \subset B$ and $cl(IP(A)) = A$.</p>
|
2,147,807 | <p>Essentially what the title asks. For an argument $x$, how can I analytically acquire values for the function:
$$
f(x)=\sum_{k=0}^{\infty}\frac{x^{2k+1}}{(2k+1)(k!)}
$$
Again, it is important that I know how to do this <strong>analytically</strong>, as there are other series comparable to this one that I also wish to... | Bernard | 202,857 | <p><strong>Hint:</strong>
$$\sum_{k\ge0}\frac{x^{2k}}{k!}=\mathrm e^{x^2}.$$</p>
|
20,714 | <p>The first three expressions evaluate as expected and the polynomial is displayed in what I would call "textbook" form. The last expression, however, switches the order of terms. Mathematica employs this change for two-term polynomials if it results in getting rid of the leading negative sign (at least that is the be... | VF1 | 1,611 | <pre><code>poly[x_] :=
Block[{Plus},
x // Sort // Reverse // Evaluate // HoldForm // TraditionalForm]
poly[x - x^2]
(* -x^2+x *)
poly[-x^2 + x]
(* -x^2+x *)
</code></pre>
|
2,894,954 | <p>What is $[\cos(\pi/12)+i\sin(\pi/12)]^{16}+[\cos(\pi/12)-i\sin(\pi/12)]^{16}$?</p>
<p>I can use De Moivre's formula for the left part:</p>
<p>$[\cos(\pi/12)+i\sin(\pi/12)]^{16} = \cos(4\pi/3) + i\sin(4\pi/3) = -\dfrac{\sqrt3}{2} + \dfrac{i}{2}$</p>
<p>but I'm stuck at the right part. Thanks in advance.</p>
| lab bhattacharjee | 33,337 | <p>$$(\cos y-i\sin y)^n=\dfrac1{(\cos y+i\sin y)^n}=\dfrac1{\cos(ny)+i\sin(ny)}=\cos(ny)-i\sin(ny)$$</p>
<p>as $(\cos x+i\sin x)(\cos x-i\sin x)=1$</p>
|
8,891 | <p>This is not a question. Just a request.</p>
<p>This Issue has been discussed previously and I think this can be implemented technically as well.</p>
<p>Writing title all in latex disables the options like "Open in New Tab" and other options in browsers. It may be user's own habit but I terribly hate it when the op... | P.K. | 34,397 | <p>While what others said is true, some situations may arise where one is not privileged to edit or the edit does not make sense. In that case, holding <code>ctrl</code> and clicking may help. </p>
|
2,363,236 | <p>This may seem like a stupid question but</p>
<p>What method can one use to convert a decimal number, such as $0.672$ into a whole number?</p>
<p>This isn't rounding as $0.999$ should result in $0$, but saying "floor the number" isn't an answer as I'm looking for a mathematical method in which a number, $n$, can be... | eyeballfrog | 395,748 | <p>Both rational and irrational numbers are dense in the real line. This means for any two real numbers (rational or irrational), there is at least one rational number and at least one irrational number between them. It follows from this fact that there are infinitely many rational and irrational numbers between each p... |
4,226,997 | <p>If I perform vector multiplication as below and then want to notate summation along the last axis of the resulting matrix, how do I show that in summation notation?</p>
<p>Example:
<span class="math-container">$$\pmatrix{ 2 \\ 3 } \pmatrix{1 & 1 & 1 \\ 2 & 2 & 2} = \pmatrix{2 & 2 & 2 \\ 6 &am... | joshuaheckroodt | 464,094 | <p>Welcome to MSE.</p>
<p>Firstly, the product
<span class="math-container">$$
\begin{pmatrix}2\\3\end{pmatrix}\begin{pmatrix}1&1&1\\2&2&2\end{pmatrix}
$$</span>
does not exist, given an unequal number of columns in the former of the above terms, and rows in the latter of the above terms, although I ack... |
4,361,279 | <p>In my book's table for antiderivatives of some functions, I came across the following,</p>
<p><span class="math-container">$$\int{e^{ax}dx=\frac{1}{a}e^{ax}} + C, \qquad a \neq0\tag{1}$$</span></p>
<p>I can't understand the reasoning behind the condition <span class="math-container">$a\neq0$</span>. Also,</p>
<p><sp... | Rob | 274,944 | <p>If <span class="math-container">$a = 0$</span> then <span class="math-container">$e^{ax} = 1$</span> for all <span class="math-container">$x$</span>, in which case the correct antiderivative would be <span class="math-container">$x + C$</span>. This doesn't match the formula <span class="math-container">$\frac1a e^{... |
113,338 | <p>I am having a hard time understanding the meaning of the union operation in this equation.</p>
<p>$$C(A)=\bigcup_{x \in A}C(x)$$</p>
<p>For context, here is the sentence:</p>
<p>The candidate set for $x$ is $S \cap C(x)$. The candidate region for a set of points $A$ is $C(A)=\bigcup_{x \in A}C(x)$, with the candi... | Community | -1 | <p>The symbol means that $\displaystyle\bigcup_{x \in A} C(x)$ means the following:</p>
<p>Let $A=\{x_1,x_2,x_3,\cdots\}$. Then, $$\bigcup_{x \in A}C(x)=C(x_1) \cup C(x_2) \cup\cdots\cup C(x_i)\cup \cdots$$ </p>
<p>Note that, to illustrate the point, I have assumed that $A$ is countable, but that need not be true. </... |
113,338 | <p>I am having a hard time understanding the meaning of the union operation in this equation.</p>
<p>$$C(A)=\bigcup_{x \in A}C(x)$$</p>
<p>For context, here is the sentence:</p>
<p>The candidate set for $x$ is $S \cap C(x)$. The candidate region for a set of points $A$ is $C(A)=\bigcup_{x \in A}C(x)$, with the candi... | g.castro | 25,312 | <p>Yes, this can be called an "indexed union". The set $C(A)$ is defined to consist of all points $p$ which lie in at least one set of the form $C(x)$ with $x\in A$. </p>
<p>For example, if $A = \{ a,b,c\}$, then $C(A)$ is defined to be $C(a)\cup C(b)\cup C(c)$. </p>
|
1,311,023 | <p>The question:</p>
<p>Let $\gamma$ be a contour such that $0 \in I(\gamma),$ where $I$ is the interior of the contour. Show that </p>
<p>$$\int_\gamma z^n \, \text{d}z = \begin{cases} 2\pi i & \text{if } n = -1 \\ 0 & \text{otherwise} \end{cases}$$</p>
<p>By taking $\gamma$ as the ellipse</p>
<p>$$\{ (x,... | Lucian | 93,448 | <p>$n!=\displaystyle\int_0^1\big(-\ln x\big)^n~dx~$ was <a href="http://en.wikipedia.org/wiki/Gamma_function#History" rel="nofollow">Euler's first historical integral expression for the $\Gamma$ function</a>, so all you have to do is to notice that $n=-\dfrac12$. :-$)$</p>
|
2,017,133 | <p>$$\sum_{i = 1}^n (2i+3) = n(n+4)$$
for all n >= 1.</p>
<p>Was a homework problem that was given no solution. Was told last lines weren't correctly written.
My attempt:
Let P(n) = n(n+4) for all n >= 1
Basis Step: P(2) = 2(6) = 12 >= 1
Inductive Step:
$$\sum_{(i=1}^{k+1} (k+1)(k+5)$$</p>
<p>= k(k+4) + (k+1)
= $$k... | barak manos | 131,263 | <p><strong>First, show that this is true for $n=1$:</strong></p>
<p>$\sum\limits_{i=1}^{1}2i+3=1(1+4)$</p>
<p><strong>Second, assume that this is true for $n$:</strong></p>
<p>$\sum\limits_{i=1}^{n}2i+3=n(n+4)$</p>
<p><strong>Third, prove that this is true for $n+1$:</strong></p>
<p>$\sum\limits_{i=1}^{n+1}2i+3=$<... |
2,017,133 | <p>$$\sum_{i = 1}^n (2i+3) = n(n+4)$$
for all n >= 1.</p>
<p>Was a homework problem that was given no solution. Was told last lines weren't correctly written.
My attempt:
Let P(n) = n(n+4) for all n >= 1
Basis Step: P(2) = 2(6) = 12 >= 1
Inductive Step:
$$\sum_{(i=1}^{k+1} (k+1)(k+5)$$</p>
<p>= k(k+4) + (k+1)
= $$k... | hamam_Abdallah | 369,188 | <p>Let $n\geq1$ such that</p>
<p>$$S_n=\sum_{k=1}^n(2i+3)=n(n+4).
$$
then</p>
<p>$$S_{n+1}=(n+1)(n+5)$$</p>
<p>$$=n(n+4)+2n+5$$</p>
<p>$$=n(n+4)+2(n+1)+3$$</p>
<p>$$=S_n+2(n+1)+3$$</p>
<p>$$=\sum_{k=1}^{n+1}(2i+3)$$</p>
|
4,445,342 | <p>I am working on SL Parsonson's <em>Pure Mathematics</em> and I haven't been able to solve this problem:</p>
<p><span class="math-container">$n^2$</span> balls, of which <span class="math-container">$n$</span> are black and the rest white, are distributed at random into <span class="math-container">$n$</span> bags, s... | leonbloy | 312 | <p>It might be easier to consider all the balls distinct (say, numbered from <span class="math-container">$1$</span> to <span class="math-container">$n^2$</span>).</p>
<p>Then the total number of arrangements is</p>
<p><span class="math-container">$$ \binom{n \times n}{n}\binom{n \times (n-1)}{n} \cdots \binom{n}{n}= \... |
2,232,779 | <p>So, the concept of an average truly is somewhat abstract. Most statisticians define it as a "measure of central tendency." Others say it is the "center of gravity" for a set of numbers.</p>
<p>I personally prefer a slightly more concrete explanation: A statistic that describes the "typical", or better yet, "represe... | Benjamin Dickman | 37,122 | <p>Here is perhaps an alternative approach: Let us think about how many ways one can form <strong>three</strong> digit numbers using the digits $1$, $2$, and $3$, such that the result has digital sum <strong>7</strong>. The possibilities can be listed exhaustively:</p>
<p>$$133, 313, 331, 233, 323, 332$$</p>
<p>There... |
4,142,894 | <p>Blitzstein, <em>Introduction to Probability</em> (2019 2 ed), p 58, Example 2.3.10 (Six-fingered man).</p>
<blockquote>
<p>A crime has been committed in a certain country. The perpetrator is one (and only one) of the <span class="math-container">$n$</span> men who live in the country. Initially, these n men are all ... | David K | 139,123 | <p>The calculation of <span class="math-container">$P(M,N\mid R)$</span> has a dizzying number of counterfactuals to sort out. I also have difficulty understanding what the authors meant by it.</p>
<p>How can the event <span class="math-container">$M, N$</span> (the event that Rugen has six fingers and no other man in ... |
2,825,103 | <p>I'm having trouble coming up with a solution on the following question:</p>
<p><strong>Find a formula for the curvature of the cycloid given by</strong> $$ x = t−\sin (t)\ ,\ y = 1−\cos(t)$$.</p>
<p>I have the following:</p>
<p>$$r(t)=<t-\sin(t),1-\cos(t)>$$
$$r'(t)= <1-\cos(t), \sin(t)>$$</p>
<p>I'... | YCor | 35,400 | <p>Not much about groups. Among metric spaces, being bounded is a QI-invariant (it's even a coarse invariant). Actually, non-empty bounded metric spaces form a single QI-class.</p>
<p>For finitely generated groups, finite $\Leftrightarrow$ bounded.</p>
<p>(In the broader setting of compactly generated locally compact... |
2,469,690 | <blockquote>
<p>If $a+b+c=0$, for $a,b,c \in\mathbb R$, prove</p>
<p>$$ 3(a^2+b^2+c^2) \times (a^5+b^5+c^5) = 5(a^3+b^3+c^3) \times (a^4+b^4+c^4) $$</p>
</blockquote>
<p>I made this question as a more difficult (higher degree) version of <a href="https://math.stackexchange.com/questions/2469296/if-abc-0-prove-t... | Dr. Sonnhard Graubner | 175,066 | <p>plug in the term $$c=-a-b$$ in the left-hand side of the equation we get
$$-30 a b (a+b) \left(a^2+a b+b^2\right)^2$$ and so is the right-hand side
a remark: if we compute the left-hand side minus the right-hand side and factorize, we obtain
$$-(a+b+c) \left(2 a^6-2 a^5 b-2 a^5 c-a^4 b^2+4 a^4 b c-a^4 c^2+6 a^3 b^3-... |
164,043 | <blockquote>
<p>Let <span class="math-container">$f(x)=x^n+5x^{n-1}+3$</span> where <span class="math-container">$n\geq1$</span> is an integer. Prove that <span class="math-container">$f(x)$</span> can't be expressed as the product of two polynomials each of which has all its coefficients integers and degree <span cl... | Bill Dubuque | 242 | <p><strong>Hint</strong> $\ $ This is a minor variant of Eisenstein's criterion. Mod $3$ it factors as $\rm\:x^{n-1}(x+5)\:$ so by uniqueness of factorization if $\rm\:f = gh\:$ then, mod $3,\,$ $\rm\:g = x^j,\, $ $\rm\,h = x^k(x+5),\,$ $\rm\:j\!+\!k = n\!-\!1.\,$ But not $\rm\,j,k > 0\,$ else $\rm\:3\:|\:g(0),h(0)\... |
3,578,788 | <p>I am struggling with finding the limit:</p>
<p><span class="math-container">$$\lim_{x\to 11} \left(\frac{x}{11}\right)^{\frac{(x-13)\cdot (x-12)}{x-11}}$$</span></p>
<p>I've tried countless methods such as turning it into the form of <span class="math-container">$a^x \to e^{\ln(a^x)}$</span> and yet i didn't manag... | VIVID | 752,069 | <p>Doing some arithmetics as follows leads to the definition of <span class="math-container">$e$</span>: <span class="math-container">$$\lim_{x \to 11} (1+\frac{x-11}{11})^{\frac{11}{x-11}\cdot(x-12)(x-13)\cdot\frac{1}{11}}=e^{\lim_{x \to 11}\frac{(x-12)(x-13)}{11}}=e^{\frac{2}{11}}$$</span></p>
|
41,725 | <p>Asked this question in a different formulation in cstheory, got some pointers, but no definitive answer ... maybe someone here knows.</p>
<p>Suppose I need to compute the factorization of a block of consecutive numbers N, N+1, ... N+n. </p>
<p>As far as I understand, there are two extreme cases. On one hand, if n ... | user9680 | 9,680 | <p>Daniel Bernstein has a method of finding smooth numbers in batches. See <a href="http://cr.yp.to/papers/sf.pdf" rel="nofollow">http://cr.yp.to/papers/sf.pdf</a>.</p>
|
3,187,756 | <p>I looked for answers on how to do this on this on this site and couldn't find anything answering this question. Is this what a line integral is used for or is that only to find area under a function f(x,y) along a curve C (on xy-plane for example)?</p>
| Bernard | 202,857 | <p>Here's one: since the integrand is a positive decreasing function of <span class="math-container">$x$</span>, one has
<span class="math-container">$$0\le \int_{n}^{n+1}\frac{1}{\sqrt{x^{3}+x+1}}\,\mathrm dx\le\int_{n}^{n+1}\frac{1}{\sqrt{n^{3}+n+1}}dx=\frac{1}{\sqrt{n^{3}+n+1}}<\frac{1}{\sqrt{n^{3}}}.$$</span></p... |
3,241,924 | <p>Let <span class="math-container">$T: V\to V$</span> and <span class="math-container">$V$</span> be a vector space of finite dimension.</p>
<p>Let <span class="math-container">$M$</span> be the minimal polynomial of <span class="math-container">$T$</span> and write it as <span class="math-container">$M = M_1 \cdot M_... | Marc van Leeuwen | 18,880 | <p>First, the hypotheses imply that the factors <span class="math-container">$M_i$</span> are mutually relatively prime: if two of them, say <span class="math-container">$M_i$</span> and <span class="math-container">$M_j$</span>, would have a nontrivial common factor <span class="math-container">$P$</span>, then the su... |
97,922 | <p><code>MyDataSet[Select[#point[Fsr] == 1 &]]</code> returns the row I want to delete, now I am trying what looks to me the most logic command to remove these rows, which is
<code>MyDataSet[Delete[#point[Fsr] == 1 &]]</code>
<code>MyDataSet[DeleteCases[#point[Fsr] == 1 &]]</code>, but neither works .... a... | Edmund | 19,542 | <p>There is a workaround if you use the <em>named row</em> style of <code>Dataset</code> (an association of associations) but don't actually apply <code>Dataset</code> to it. You can still perform querying as you can with a dataset but will have to explicitly use <code>Query</code> instead of it being implicitly used ... |
4,243,792 | <p>Consider <span class="math-container">$f: \mathbb{R}^n \rightarrow \mathbb{R}$</span>. <span class="math-container">$f\left(\alpha x+\left(1-\alpha\right)x'\right)=\alpha f\left(x\right)+\left(1-\alpha\right)f\left(x'\right)$</span>, <span class="math-container">$\forall \alpha \in \left[0,1\right]$</span> and <span... | Matthew H. | 801,306 | <p><span class="math-container">$f$</span> need not be linear. Take <span class="math-container">$f(x,y)=x+y+1$</span>. Then <span class="math-container">$f$</span> isn't linear. However, <span class="math-container">$$\begin{eqnarray*}f\Big(t(x_0,y_0)+(1-t)(x_1,y_1)\Big)&=&f\Big(tx_0+(1-t)x_1,ty_0+(1-t)y_1\Big... |
949,857 | <p>I have general questions about the group of isometries of a metric space.
-When is the isometry group of a space a lie group?
-when the isometry group is a Lie group, is there a relation between the one parameter sub-group of the isometry group and the geodesics of the metric space?</p>
<p>I am relatively new to th... | Tim kinsella | 15,183 | <p>One very nice case occurs when $G$ is a non compact semi-simple Lie group (meaning its Lie algebra is a direct sum of simple ideals) with finite center. In this case, there is a maximal compact subgroup $K < G$ (the fixed subgroup of a Cartan involution) such that $G/K$ admits a Riemannian metric with respect to... |
28,027 | <p>Let $L/K$ be a finite Galois extension with Galois group $G$ and $V$ a $L$-vector space, on which $G$ acts by $K$-automorphisms satisfying $g(\lambda v)=g(\lambda) g(v)$. It is known that the canonical map</p>
<p>$V^G \otimes_K L \to V$</p>
<p>is an isomorphism. However, I can't find any short and nice proof for t... | Homology | 5,735 | <p>Isn't it Hilbert 90?
Choose a basis for $L$, and denote by $c_g$ the matrix s.t.
$g(e_j) = \sum_i (c_g)_{i,j} e_i$
Then $g \mapsto c_g$ is a cocycle with values in $GL_n (L)$, i.e. $c_{gh} = c_g g(c_h)$.</p>
<p>Hilbert 90 tells you that $H^1(G,GL_n(L))=0$, so that $c_g=b g(b)^{-1}$ for some invertible $b$, which is... |
244,959 | <p>I'm trying to do this.
it doesn't have to be the same.</p>
<p><a href="https://i.stack.imgur.com/BjB5I.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BjB5I.png" alt="enter image description here" /></a></p>
| Daniel Huber | 46,318 | <p>(b) is problematic because it does not contain the same pixels as (a) and (b).</p>
<p>Look at the following:</p>
<pre><code>ImageRotate[im, #] & /@ {0,-Pi/4,-Pi/2}
</code></pre>
<p><a href="https://i.stack.imgur.com/r48v2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/r48v2.png" alt="enter im... |
501,134 | <p>In a book we are given that $R=$ ring of matrices of the type $$\begin{pmatrix}
a&b\\0&0\\
\end{pmatrix}$$and $I$ of the type$$\begin{pmatrix}
0&x\\0&0\\
\end{pmatrix}$$
over the integers. I am not able to find the identity of $R/I$. I tried using the definition but I failed. Can someone tell me abo... | Oscar Cunningham | 1,149 | <p>$R/I$ can be seen as matrices of the type $$\begin{pmatrix}
a&*\\0&0\\
\end{pmatrix}$$
where $*$ means "don't care". Multiplication in this ring is by $$\begin{pmatrix}
a&*\\0&0\\
\end{pmatrix}\begin{pmatrix}
b&*\\0&0\\
\end{pmatrix}=\begin{pmatrix}
ab&*\\0&0\\
\end{pmatrix}$$ where I... |
501,134 | <p>In a book we are given that $R=$ ring of matrices of the type $$\begin{pmatrix}
a&b\\0&0\\
\end{pmatrix}$$and $I$ of the type$$\begin{pmatrix}
0&x\\0&0\\
\end{pmatrix}$$
over the integers. I am not able to find the identity of $R/I$. I tried using the definition but I failed. Can someone tell me abo... | Karl Kroningfeld | 67,848 | <p>To see that $R$ has no (multiplicative) identity, let's first take a look at how multiplication works in $R$:$$\begin{pmatrix}a & b\\0&0\end{pmatrix}\begin{pmatrix}c&d\\0&0\end{pmatrix}=\begin{pmatrix}ac&ad\\0&0\end{pmatrix}$$ Now, suppose $R$ has an identity element. Recall that an identity ... |
3,039,267 | <p>find probability that during a process of tossing coins repeatedly one will encounter a run of <span class="math-container">$5$</span> consecutive heads before encountering <span class="math-container">$2$</span> tails. </p>
<p>I started by assuming,</p>
<p><span class="math-container">$H$</span>= probability that... | Arthur | 15,500 | <p>This is listed, verbatim, as one of the <strong>absorbtion laws</strong> in the wikipedia page you linked.</p>
|
3,039,267 | <p>find probability that during a process of tossing coins repeatedly one will encounter a run of <span class="math-container">$5$</span> consecutive heads before encountering <span class="math-container">$2$</span> tails. </p>
<p>I started by assuming,</p>
<p><span class="math-container">$H$</span>= probability that... | Bram28 | 256,001 | <p>It's in the list ... but some texts like to derive it as follows:</p>
<p><span class="math-container">$$p \lor (p \land q) \equiv (p \land \top) \lor (p \land q) \equiv p \land (\top \lor q) \equiv p \land \top \equiv p$$</span></p>
|
180,053 | <p>In $C[0,1]$ the set $\{f(x): f(0)\neq 0\}$ is dense? I know only that polynomials are dense in $C[0,1]$, could any one give me hint how to show this set is dense?thank you.</p>
| tomasz | 30,222 | <p>It is dense. It is enough to show that there are functions of arbitrarily small norm in the set, which shouldn't be too hard.</p>
<p>In general, the complement of any proper subspace of a normed space is dense, by a similar argument (possibly even easier, as is sometimes the case with generalizations where you don'... |
2,000,268 | <p>We usually tend to say the "Average" is whether "Mean", "Median" or "Mode" and in colloquial usage "Average" is always equivalent to "Mean".</p>
<blockquote>
<p>But my <strong>question</strong> is: Is there any precise rigorous definition of "Average of a statistical population" in statistics (regardless of our k... | Community | -1 | <p>To quote Wikipedia:</p>
<blockquote>
<p>In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the <a href="https://en.wikipedia.org/wiki/Arithmetic_mean" rel="nofollow noreferrer">arithmetic mean</a>. In... |
2,786,291 | <p>How to prove $l^1$ and $l^\infty$ are infinite dimensional spaces.</p>
<p>I know that a space is infinite dimensional if it has a subspace with infinite dimensions. But I don't know how to proceed. Any help will be appreciated.</p>
| Logic_Problem_42 | 338,002 | <p>In both spaces You have an infinite linearly independent set: $(1,0,0,...),(0,1,0,...), (0,0,1,...)$. </p>
|
1,668,792 | <p>This is a question given in our weekly test.</p>
<p>$$f = \lim_{x\to 0^+}\{[(1+x)^{1/x}]/e\}^{1/x}.$$</p>
<p>Find the value of $f$. I tried to use 1^ infinity form but I didn't get it. So anybody please help me.</p>
| 3SAT | 203,577 | <p>$$\lim\limits_{x\to 0^{+}}\left(\frac{(1+x)^{1/x}}{x}\right)^{1/x}=\lim\limits_{x\to 0^{+}}e^{-1/x}\left((x+1)^{1/x}\right)^{1/x}=\lim\limits_{x\to 0^{+}}\exp\left(\frac{\ln((x+1)^{1/x})}{x}-\frac 1 x\right)$$</p>
<p>$$=\exp\left(\lim\limits_{x\to 0^{+}}\left(\frac{\ln\left((1+x)^{1/x}\right)}{x}-\frac 1 x\right)\r... |
713,135 | <p>I feel like that question's got an obvious answer, but I somehow missed it during my probability class. There are random variables, which distributions can be expressed if a form of functions - like Gaussian, uniform, binomial etc. If I'm going to take a ruler and measure the length of my laptop again and again, all... | Frazer | 125,581 | <p>Suppose $X$~$N(\theta, \sigma^2)$, while suppose $\theta$ here is also a variable and $\theta$~$N(\mu_1,\sigma_1^2)$, the idea behind it is to assume the parameter is itself dynamic. $\mu_1$ here is called hyperparameter, you can add infinite hyperparameters if you want and it may be called a hierarchical model.</p>... |
713,135 | <p>I feel like that question's got an obvious answer, but I somehow missed it during my probability class. There are random variables, which distributions can be expressed if a form of functions - like Gaussian, uniform, binomial etc. If I'm going to take a ruler and measure the length of my laptop again and again, all... | Did | 6,179 | <blockquote>
<p>If that's true, are they studied by probability theory, is there a special name, maybe? </p>
</blockquote>
<p>Hidden variables models, for example <a href="http://en.wikipedia.org/wiki/Hidden_Markov_model" rel="nofollow">hidden Markov models</a>.</p>
|
3,062,391 | <p>Let <span class="math-container">$M$</span> be a smooth manifold of dimension <span class="math-container">$n$</span> and let <span class="math-container">$p \in M$</span>. Choose a smooth chart <span class="math-container">$(U, \phi)$</span> around <span class="math-container">$p$</span> and then we have <span clas... | Perturbative | 266,135 | <p>So if I have a function from some set <span class="math-container">$X$</span> into <span class="math-container">$\mathbb{R}^k$</span>, <span class="math-container">$f : X \to \mathbb{R}^k$</span> then the component functions <span class="math-container">$f^i : X \to \mathbb{R}$</span> satisfy <span class="math-conta... |
267,318 | <p>Let $G$ be a connected, reductive group over a local field $F$ of characteristic zero, and $H$ a closed subgroup of $G$ which is defined over $F$. Let $\mu_H, \mu_G$ be right Haar measures on $H(F), G(F)$ with modular functions $\delta_H, \delta_G$.</p>
<p>In papers, notes, and textbooks on automorphic forms and r... | Not a grad student | 64,244 | <p>For $F$ a non-archimedean local field:</p>
<p>One can define the induction functor in several ways. First, there is parabolic induction--this is an example of induction in general. Depending on the context, one can either choose to normalize the induction (by adding the twist by $\delta^{1/2}$ as you mentioned). Th... |
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