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 |
|---|---|---|---|---|
4,140,881 | <p>I want to show the title.</p>
<blockquote>
<p>If <span class="math-container">$M$</span> is a finitely generated module over a local ring <span class="math-container">$A$</span>, then there is a free <span class="math-container">$A$</span>-module <span class="math-container">$L$</span> such that <span class="math-co... | Math Lover | 801,574 | <p>If set of passwords that are missing lowercase is <span class="math-container">$N_L$</span>, set of passwords that are missing uppercase is <span class="math-container">$N_U$</span> and set of passwords that are missing digits is <span class="math-container">$N_D$</span>, then</p>
<p><span class="math-container">$|N... |
10,174 | <p>I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in <a href="http://arxiv.org/abs/math.qa/0203255" rel="noreferrer">Fusion Categories of Rank 2</a> by Victor Ostrik. Apparently, there are two of them and they arise in various ways:</p>
<ul>
<li>integer-spin representations... | José Figueroa-O'Farrill | 394 | <p>The Virasoro minimal model $\mathcal{M}(2,5)$ (or in some conventions also $\mathcal{M}(5,2)$ is the conformal field theory which describes the critical behaviour of the <em>Lee-Yang edge singularity</em>. It is described, for example, in <a href="http://books.google.com/books?id=keUrdME5rhIC" rel="nofollow">Confor... |
10,174 | <p>I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in <a href="http://arxiv.org/abs/math.qa/0203255" rel="noreferrer">Fusion Categories of Rank 2</a> by Victor Ostrik. Apparently, there are two of them and they arise in various ways:</p>
<ul>
<li>integer-spin representations... | Noah Snyder | 22 | <p>Unfortunately all three of those realizations are the sort of thing you need to read a book about not a MO post. I agree with Greg that Kassel's book is a great place to start for the quantum group construction (I don't know the other two constructions well, presumably for the affine algebra construction you'd want... |
4,104,364 | <p>I got the following exercise:<br />
Let <span class="math-container">$W$</span> be a finite-dimensional <span class="math-container">$\Bbb{R}$</span>-vector space. Let <span class="math-container">$\Bbb{R}_W=\Bbb{R}\times W$</span>. Define addition and multiplication by <span class="math-container">$(r,w)+(s,v)=(r+s... | Muses_China | 307,348 | <p>I think finding all ideals of <span class="math-container">$\mathbb{R}_W$</span> is a good idea. Do not think it too complicated!</p>
<p>Let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$\mathbb{R}_W$</span>. We divide it into two cases.</p>
<p>The first case, <span class="math... |
4,184,196 | <blockquote>
<p>Let <span class="math-container">$\hat{f} : \Bbb S^1 \to \Bbb R^2$</span>, <span class="math-container">$\hat{f}(x,y) = (x,y)$</span> and <span class="math-container">$\hat{g} : \Bbb S^1 \to \Bbb R^2$</span>, <span class="math-container">$\hat{g}(x,y) = -(x,y)$</span>. Show that there exists a Homotopy ... | jasnee | 916,067 | <p>Here is an approach that doesn't use any parametrisation of <span class="math-container">$S^1$</span>: Define
<span class="math-container">$$
\hat{H} : S^1 \times [0,1] \to \mathbb{R}^2, \qquad ((x,y),t) \mapsto ((-2t+1)x,(-2t+1)y).
$$</span>
This map is continuous and you can check that it indeed defines the desire... |
2,007,584 | <p>Let $M$ be a smooth manifold and $f:M\rightarrow \mathbb{R}$ be a smooth function such that $f(M)=[0,1]$. Let $1/2$ be a regular value and suppose we consider the open and non-empty set $U:=f^{-1}(\frac{1}{2},\infty)\subset M$. I would like to show that $f^{-1}(\frac{1}{2})$ must coincide with the topological bounda... | Jack Lee | 1,421 | <p>By the Rank Theorem (Theorem 4.12 in my <em>Introduction to Smooth Manifolds</em>, 2nd ed.), each point $p\in f^{-1}(\frac 1 2)$ is contained in the domain of a coordinate chart on which $f$ has a coordinate representation of the form $f(x^1,\dots,x^n) = x^n$. Thus any sufficiently small neighborhood of $p$ contains... |
334,351 | <p>I am a student of mathematics, and have some background in </p>
<ul>
<li>Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff), </li>
<li>Differential Geometry (Lee, Kobayashi-Nomizu), </li>
<li>Riemannian Geometry (Do Carmo), </li>
<li>Symplectic Geometry (Ana Cannas da Silva) and </li>
<li>Differential Topology ... | David White | 11,540 | <p>There have been several questions previously in this vein, but yours is more general. My present answer is adapted from an answer to a question asking for a <a href="https://mathoverflow.net/a/149041/11540">"Road Map" to Homotopy Theory</a>. Your question is a bit different, so I'll write some different things. Firs... |
3,781,968 | <p>Let <span class="math-container">$d\in\mathbb N$</span>, <span class="math-container">$U\subseteq\mathbb R^d$</span> be open and <span class="math-container">$M\subseteq U$</span> be a <span class="math-container">$k$</span>-dimensional embedded <span class="math-container">$C^1$</span>-submanifold of <span class="m... | H. H. Rugh | 355,946 | <p>If <span class="math-container">$k<d$</span> the answer is no. <span class="math-container">$M$</span> has full surface measure but which has zero Lebesgue measure, while for <span class="math-container">$U\setminus M$</span> it is the converse. So they are mutually singular. In particular, your inequality (1) ... |
3,266,367 | <p>Find the solution set of <span class="math-container">$\frac{3\sqrt{2-x}}{x-1}<2$</span></p>
<p>Start by squaring both sides
<span class="math-container">$$\frac{-4x^2-x+14}{(x-1)^2}<0$$</span>
Factoring and multiplied both sides with -1
<span class="math-container">$$\frac{(4x-7)(x+2)}{(x-1)^2}>0$$</span>... | Michael Hoppe | 93,935 | <p>Define <span class="math-container">$f(x)=\frac{3\sqrt{2-x}}{x-1}-2$</span>. Being continuous on its domain <span class="math-container">$(-\infty,2]\setminus\{1\}$</span>,the function may change its sign only at its zero <span class="math-container">$7/4$</span> or at its singularity, namely at <span class="math-... |
629,996 | <p>At <a href="https://math.stackexchange.com/questions/629950/why-i-left-px-in-mathbbz-leftx-right2-mid-p0-right-is-not-a-prin#629954">this question</a> I asked about specific one...</p>
<p>But I think that I don't understand the basic: </p>
<blockquote>
<p>If I have an ideal $I$, what avoid it to be principal? ... | Community | -1 | <p>Hint: you can see that $y(x)=c_1 x$ is a solution. Use that solution to construct the second solution by variation of parameters. The second solution is $y(x)=c_2 x e^{-1/x}$.</p>
|
183,100 | <p>Let $\phi: M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ be the map
$$(B_1,B_2)\mapsto [B_1,B_2]$$ which takes two $2\times 2$ matrices to its Lie bracket. </p>
<p>Then why does $d\phi_{(B_1,B_2)}:M_2(\mathbb{C})\times M_2(\mathbb{C})\rightarrow M_2(\mathbb{C})$ send
$$(D_1,D_2)\mapsto [B_1,D_... | Qiaochu Yuan | 232 | <p>Taking differentials is all about looking at what happens to your map upon a very small perturbation. So compute the bracket
$$[B_1 + \epsilon D_1, B_2 + \epsilon D_2]$$</p>
<p>and look at the coefficient of $\epsilon$. </p>
|
29,016 | <p>Suppose I want to compute $f(1)\vee f(2) \vee \ldots \vee f(10^{10})$, but I know <em>a priori</em> that $f(n)$ is <code>True</code> for some $n \ll 10^{10}$ with high probability. For example, <code>f = PrimeQ</code>.</p>
<p>One way to do this is to write: <code>Or[f/@Range[1,10^10]]</code>, but that would involve... | Jens | 245 | <p>It looks like you only need a looping construct that terminates when the first <code>True</code> is encountered with the <code>Or</code> operation. So how about this:</p>
<pre><code>f = PrimeQ
(* ==> PrimeQ *)
notFound = False;
n = 0;
While[! notFound,
n += 1;
notFound = notFound || f[n]
]; n
(* ==> 2 *... |
539,457 | <p>Suppose we have a natural number $N$ with decimal representation $A_kA_{k-1}\ldots A_0$. How do I prove that if the $\sum\limits_{i=0}^kA_i$ is divisible by $9$ then $N$ is divisible by $9$ too?</p>
| njguliyev | 90,209 | <p><em>Hint:</em> $\overline{A_kA_{k-1}\ldots A_1A_0} = 10^kA_k + 10^{k-1}A_{k-1} + \ldots + 10A_1 + A_0$.</p>
<blockquote class="spoiler">
<p> $(10^kA_k + 10^{k-1}A_{k-1} + \ldots + 10A_1 + A_0) - (A_k + A_{k-1} + \ldots + A_1 + A_0) = (10^k-1)A_k + (10^{k-1}-1)A_{k-1} + \ldots + (10-1)A_1$ is divisible by $9$.</p>... |
3,644,823 | <p>Let <span class="math-container">$f$</span> be a non constant holomorphic function on and inside of the unit circle <span class="math-container">$C:=\{z\in \mathbb C~:~|z|=1\}$</span>. Suppose <span class="math-container">$|f(z)|=1$</span> on <span class="math-container">$C$</span>, then for <span class="math-contai... | Community | -1 | <p><span class="math-container">$f$</span> has at least one zero: this follows applying Rouche's theorem to <span class="math-container">$f(z);f(z)-f(z_0)$</span>, where <span class="math-container">$z_0\in \mathbb{D}$</span> is such that :<span class="math-container">$f(z_0)\neq 0$</span>. Indeed, on <span class="math... |
17,270 | <p>I just joined MathSE and it's beautiful here, except for the fact that some unregistered users ask a question and never come back. Most of the time these questions are trivial, though they still consume answerers' (valuable) time which never gets rewarded. I thought it was okay until I saw someone's profile with the... | Gerry Myerson | 8,269 | <p>No, it's not OK to post 72 questions, get useful answers, and not accept any of them. I would hope some moderator would take this user aside and explain a few things about how this site works best. </p>
|
225,351 | <p>Consider a sine wave having <span class="math-container">$4$</span> cycles wrapped around a circle of radius 1 unit.</p>
<p><span class="math-container">$$ y = \sin(4x) $$</span></p>
<p>To find the equation of the sine wave with circle acting, one approach is to consider the sine wave along a rotated line. But it ... | Phira | 9,325 | <p>Do it first for the circle centered at the origin in polar coordinates.</p>
<p>Then switch do Cartesian coordinates, then shift to the actual center of the circle.</p>
|
225,351 | <p>Consider a sine wave having <span class="math-container">$4$</span> cycles wrapped around a circle of radius 1 unit.</p>
<p><span class="math-container">$$ y = \sin(4x) $$</span></p>
<p>To find the equation of the sine wave with circle acting, one approach is to consider the sine wave along a rotated line. But it ... | Max | 164,373 | <p>it should be, in cartesian coordinates</p>
<p>x = (R + a · sin(n·θ)) · cos(θ) + xc</p>
<p>y = (R + a · sin(n·θ)) · sin(θ) + yc</p>
<p>where </p>
<p>R is circle's radius</p>
<p>a is sinusoid amplitude</p>
<p>θ is the parameter (angle), from 0 to 2π</p>
<p>xc,yc is circle's center point</p>
<p>n is number of s... |
164,746 | <p>For example</p>
<p><a href="https://i.stack.imgur.com/bFDdw.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/bFDdw.png" alt="enter image description here"></a></p>
<p>But I want to strip those box edge except the bottom ones like this</p>
<p><a href="https://i.stack.imgur.com/DpzG1.png" rel="nof... | José Antonio Díaz Navas | 1,309 | <p>Something like this (playing with <code>AxesEdge</code>)?:</p>
<pre><code>Plot3D[Exp[-x^2 - y^2], {x, -2, 2}, {y, -2, 2}, Axes -> {True, True, False},
Boxed -> False, AxesLabel -> {Style["x", 12, Bold], Style["y", 12, Bold], None},
ViewPoint -> {0, -2, 2}, AxesEdge -> {Automatic, {-1, -1}, None}]
<... |
2,559,350 | <p>Consider a circumference centered in the origin and with radius $r$. Let $C$ be a point on the circumference, and let $A,B$ be its projections on the axes, respectively $x$-axis and $y$-axis. What is the length of $AB$?</p>
<p>I tried applying the laws of sines and cosines on $ABC$, but I only got tautologies....</... | Anders Beta | 464,504 | <p>If $|z-i|^4=1 \Rightarrow |z-i|=1$, which implies $z-i=e^{i\theta}$ for some arbitrary angle $\theta$. This implies:</p>
<p>$$ z=i+e^{i\theta} = \cos\theta +i(1+\sin\theta)$$</p>
|
132,879 | <p>This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section.</p>
<p>Let us recall Maxwell’s equations: Given a semi-riemannian 4-manifold and a 3-form $j$. We describe the field-strength differential form $F$ as a solution of th... | Willie Wong | 3,948 | <ol>
<li><p>There are no "rapid decaying harmonic 2-forms" in Minkowski space. </p>
<p>Consider the expression
$$ 0 = \mathrm{d} \star \mathrm{d} F = \partial^i \partial_{[i}F_{jk]} = \frac13 (\Box F_{jk} + \partial_j \partial^i F_{ki} + \partial_k \partial^i F_{ij})$$
The second and third terms in the parentheses va... |
2,938,372 | <p>Seems to me like it is. There are only finitely many distinct powers of <span class="math-container">$x$</span> modulo <span class="math-container">$p$</span>, by Fermat's Little Theorem (they are <span class="math-container">$\{1, x, x^2, ..., x^{p-2}\}$</span>), and the coefficient that I choose for each of thes... | Robert Lewis | 67,071 | <p><span class="math-container">$\Bbb Z_p[x]$</span> is indeed infinite, since it contains polynomials, with coefficients in <span class="math-container">$\Bbb Z_p$</span>, of arbitrary high degree, e.g. <span class="math-container">$x^n \in \Bbb Z_p[x]$</span>, where <span class="math-container">$n \in \Bbb N$</span>;... |
2,938,372 | <p>Seems to me like it is. There are only finitely many distinct powers of <span class="math-container">$x$</span> modulo <span class="math-container">$p$</span>, by Fermat's Little Theorem (they are <span class="math-container">$\{1, x, x^2, ..., x^{p-2}\}$</span>), and the coefficient that I choose for each of thes... | Mark Bennet | 2,906 | <p>You might have <span class="math-container">$x^2\equiv x \bmod 2$</span> for the two elements <span class="math-container">$0$</span> and <span class="math-container">$1$</span>. So this is within <span class="math-container">$\mathbb Z_2$</span>.</p>
<p>Now note that <span class="math-container">$x^2+x+1$</span> h... |
2,822,126 | <blockquote>
<p>If $0⩽x⩽y⩽z⩽w⩽u$ and $x+y+z+w+u=1$, prove$$
xw+wz+zy+yu+ux⩽\frac15.
$$</p>
</blockquote>
<p>I have tried using AM-GM, rearrangement, and Cauchy-Schwarz inequalities, but I always end up with squared terms. For example, applying AM-GM to each pair directly gives$$
x^2+y^2+z^2+w^2+u^2 ⩾ xw + wz + zy + ... | Rhys Hughes | 487,658 | <p>The case where $xw+wz+zy+yu+ux$ is maximal will be where $x=y=z=w=u$. Here, they must all be $\frac15$, and plugging into the equation gives $5(\frac15)^2=\frac15$. So the expression is $\le\frac15$.</p>
|
256,138 | <p>I need to generate four positive random values in the range [.1, .6] with (at most) two significant digits to the right of the decimal, and which sum to exactly 1. Here are three attempts that do not work.</p>
<pre><code>x = {.15, .35, .1, .4}; While[Total[x] != 1,
x = Table[Round[RandomReal[{.1, .6}], .010], 4]];... | bobthechemist | 7,167 | <p>I don't think anyone has yet tried <code>FindInstance</code>:</p>
<pre><code>FindInstance[{a + b + c + d == 100 && 10 <= a <= 60 && 10 <= b <= 60 &&
10 <= c <= 60 && 10 <= d <= 60}, {a, b, c, d}, Integers, 2,
RandomSeeding -> Round@(10^6 RandomReal[])]
</co... |
3,441,438 | <p>I was wondering if anyone could get me started on solving the following:
<span class="math-container">$$(u')^2-u\cdot u''=0$$</span></p>
<p>I have tried letting <span class="math-container">$v=u'$</span>, but I don't seem to make progress with such a substitution.</p>
| user247327 | 247,327 | <p>First, since <span class="math-container">$0^8= 0$</span> and, conversely, if <span class="math-container">$x^8= 0$</span>, x= 0, any x is a root of <span class="math-container">$(x^4+ 7x^3+ 22x^2+ 31x+ 9)^2= 0$</span> is a root of <span class="math-container">$x^4+ 7x^3+ 22x^2+ 31x+ 9= 0$</span>. Further, since al... |
554,825 | <p>Here's a question from my homework. First 2 questions I solved (but would appreciate any input you can give on my solution) and the last question I'm just completely stumped. It's quite complicated.</p>
<p>On the shelf there are 5 math books, 3 science fiction books and 2 thrillers (<strong>all of the books are di... | Pavel Čoupek | 82,867 | <p>The statement is not true, i.e. a non-irreducible element can be mapped to an irreducible element.</p>
<p><strong>Hint:</strong>
Think of embedding $f$ of $R$ into some localization of $R$ (by some multiplicative set) and the fact that some element $y$ of the localization usually is a unit despite the fact that $f... |
239,387 | <p>Someone could explain how to build the smallest field containing to $\sqrt[3]{2}$.</p>
| i. m. soloveichik | 32,940 | <p>If the base field is $F=Z_2$ then we already have the cube root of 2, which is 0. If $F=Z_3$
then $F$ already contains one cube root of 2, i.e. cube root of -1, namely -1; if we want the other cube roots of 2, then the field $F(2^{1/3})=F[2^{1/3}]=\{a+b\cdot 2^{1/3}\}$
has 9 elements.</p>
|
112,394 | <p>Could someone clarify why the first of these <code>MatchQ</code> finds a match whereas the second does not? (I'm using version 10.0, in case that matters.)</p>
<pre><code>MatchQ[Hold[x + 2 y], Hold[x + 2 _]]
(*True*)
MatchQ[Hold[x + 2 y + 0], Hold[x + 2 _ + 0]]
(*False*)
</code></pre>
<p>EDIT: The conclusion belo... | Alexey Popkov | 280 | <h2>UPDATE</h2>
<p><a href="https://mathematica.stackexchange.com/q/94432/280">That question</a> is by the essence an exact duplicate of this one. The <a href="https://mathematica.stackexchange.com/a/94569/280">explanation</a> given by Mr.Wizard means that the pattern-matcher is NOT capable to handle situations when a... |
4,014,756 | <p>I was reading the book "Quantum Computing Since Democritus".</p>
<blockquote>
<p>"The set of ordinal numbers has the important property of being well
ordered,which means that every subset has a minimum element. This is
unlike the integers or the positive real numbers, where any element
has another tha... | Brian M. Scott | 12,042 | <p>For each <span class="math-container">$n\in\Bbb N$</span> let <span class="math-container">$C_n=[0,4]$</span>, and let <span class="math-container">$C=\prod_{n\in\Bbb N}C_n$</span>. Then</p>
<p><span class="math-container">$$\Bbb R^\infty\setminus C\ne\prod_{n\in\Bbb N}(\Bbb R\setminus C_n)\,,$$</span></p>
<p>so the... |
2,602,410 | <p>$$\int_{0}^{\pi /2} \frac{\sin^{m}(x)}{\sin^{m}(x)+\cos^{m}(x)}\, dx$$</p>
<p>I've tried dividing by $\cos^{m}(x) $, and subbing out the $\ 1+\cot^{m}(x) $ with $\csc^{n}(x) $ for some $n$, but to no avail. I've also tried adding and subtracting $\cos^{m}(x)$ to the numerator, and substituting $x$ by $\pi-y$, but ... | user284331 | 284,331 | <p>Let $I=\displaystyle\int_{0}^{\pi/2}\dfrac{\sin^{m}x}{\sin^{m}x+\cos^{m}x}dx$, by letting $y=\pi/2-x$, then $I=\displaystyle\int_{0}^{\pi/2}\dfrac{\cos^{m}y}{\sin^{m}y+\cos^{m}y}dy$, so $2I=\displaystyle\int_{0}^{\pi/2}\dfrac{\sin^{m}x+\cos^{m}x}{\sin^{m}x+\cos^{m}x}dx=\pi/2$, so $I=\pi/4$.</p>
|
2,830,969 | <p>I did not find an example when the denominator $x$ approximates to $0$.</p>
<p>$f(0) + f'(0)x$ does not work because $f(0)$ would be $+\infty$. </p>
| Community | -1 | <p>There is no answer to your question as the limit does not exist.
This can be proven by-</p>
<ol>
<li>first take the left hand side limit. This is equal to negative infinity in this case</li>
<li>Then take the right hand side limit, which equals positive infinity</li>
<li>Since, left hand side limit is not equal to ... |
382,603 | <p>Let $F$ be a finite field. Prove that the following are equivalent:</p>
<p>i) $A \subset B$ or $B \subset A$ for each two subgroups $A,B$ of $F^*$.</p>
<p>ii) $\#F^*$ equals 2, 9, a Fermat-prime or $\#F^* -1$ equals a Mersenne prime.</p>
<p>Any ideas for i => ii ? I don't know where to start, except for remarking... | Jyrki Lahtonen | 11,619 | <p>Hints/suggestions:</p>
<p>Let $F=GF(q)$ with $q=p^m$. For $F^*$ to have property i) it is necessary and sufficient that $q-1=\ell^n$ for some prime $\ell$ and non-negative integer $n$.</p>
<ol>
<li>If $p>2$, then $q-1$ is an even integer, so we must have $\ell=2$. Also all factors of $q-1$ must be powers of two... |
1,873,648 | <p>Let $A=\{1,2,3,...,2^n\}$. Consider the greatest odd factor (not necessarily prime) of each element of A and add them. What does this sum equal? </p>
| robjohn | 13,854 | <p>The sum of the first $m$ odd numbers is $1+3+5+7+\cdots+(2m-1)=m^2$.</p>
<p>Excluding $2^n$, we sum the first $2^{n-1}$ odd numbers, then the first $2^{n-2}$ odd numbers under the guise of their doubles, then the first $2^{n-3}$ odd numbers under the guise of their quadruples, etc.</p>
<blockquote>
<p>For exampl... |
3,363,810 | <p>I;m not sure how to go about proving this. I just started learning about it and would appreciate some help.</p>
| fleablood | 280,126 | <p>Claim 1: <span class="math-container">$X\cap Y \subset X$</span>.</p>
<p>Proof: If <span class="math-container">$x \in X\cap Y$</span> then <span class="math-container">$x \in X$</span> and <span class="math-container">$x \in Y$</span>. So <span class="math-container">$x \in X$</span>.</p>
<p>Claim 2: If <span... |
852,404 | <p>How to prove that there exist an infinite number of prime $n$ for which $n^2=p+8$ for some prime $p$?</p>
<p>Verification of the form $n^2=p+8$ where $n$ and $p$ are some $p$.</p>
<p>$$\begin{array}{|c|c|} \hline
n & n^2 = 8 + p \\
\hline
11 & 121 = 8 + 113 \... | miket | 34,309 | <p>Your question is equivalent to there's infinitely many $m$ that: </p>
<p>$2n^2 + 2(n - 2) = m,\ $where $2n + 1,\ 2m+1$ is prime. </p>
<p>Such as: $n=3, \ 2n^2 + 2(n - 2) = 20,(2 \cdot 3+1)^2=7^2=49=2 \cdot 20+1+8=41+8,\ $ where $7,41$ is prime. It seems this is a elementary question.</p>
|
1,905,186 | <blockquote>
<p>Let <span class="math-container">$R$</span> be a commutative Noetherian ring (with unity), and let <span class="math-container">$I$</span> be an ideal of <span class="math-container">$R$</span> such that <span class="math-container">$R/I \cong R$</span>. Then is it true that <span class="math-container"... | Oliver Kayende | 704,766 | <p>Assume <span class="math-container">$A\approx A/a$</span>. Of those <span class="math-container">$A$</span> ideals <span class="math-container">$a'$</span> satisfying <span class="math-container">$A\approx A/a'$</span> choose <span class="math-container">$b$</span> maximal with respect to set inclusion. Consequently... |
496,178 | <p>Let $t$ be a positive real number. Differentiate the function</p>
<blockquote>
<p>$$g(x)=t^x x^t.$$</p>
</blockquote>
<p>Your answer should be an expression in $x$ and $t$.</p>
<p>came up with the answer </p>
<blockquote>
<p>$$(x/t)+(t/x)\ln(t^x)(x^t)=\ln(t^x)+\ln(x^t)=x\ln t+t\ln x .$$</p>
</blockquote>
<p... | triple_sec | 87,778 | <p>Using the product rule for differentiation, $$g'(x)=(t^x)'(x^t)+(t^x)(x^t)'=(t^x)(\ln t)(x^t)+(t^x)(t)(x^{t-1})=(t^x)(\ln t)(x^t)+t^{x+1}x^{t-1}.$$</p>
|
3,740,647 | <p>I want to load an external Magma file within another Magma file. (Both files are saved in the same directory.) I want to be able to quickly change which external file is being loaded, ideally at the beginning of the file making the load call, so that I can easily run the same code with various inputs.</p>
<p>(The ex... | kera | 871,352 | <p>Another solution (which is also a bit of a work around) is to set the Magma path based on which file you want to load using the SetPath() function. The Magma path indicates which directories are searched for loading Magma files.</p>
<p>To clarify, if you have 2 different files by the same name in separate directorie... |
2,861,362 | <p>I have the following question:
- We proved that if $T$ is 1-1 and $\{v_1...v_n\}$ is linearly independent then $\{T(v_1)...T(v_n)\}$ is linearly independent! I understood the proof! But can’t $\{T(v_1)...T(v_n)\}$ be linearly independent without having $T$ 1-1?
The image I uploaded shows my work on proving that th... | Community | -1 | <p>If you are allowed to use the definition of the Euler constant,</p>
<p>$$a_n=H_{2n}-\frac12H_n-\log2n+\frac12\log n+\log2.$$</p>
<p>The first difference cancels out all even harmonic terms and the remaining terms account for $\log\sqrt n$.</p>
<p>Then</p>
<p>$$\lim_{n\to\infty}a_n=\gamma-\frac12\gamma+\log2.$$</... |
3,460,749 | <p>I'm an undergraduate student currently studying mathematical analysis. </p>
<p>Our professor uses Zorich's Mathematical Analysis, but I found the text too difficult to understand. </p>
<p>After exploring some textbooks, I found that Abbott was easier to follow, so I studied Abbott until I realized that there's a s... | dxb | 669,798 | <p>You may find Rudin's analysis texts <em>Principles of Mathematical Analysis</em> and <em>Real and Complex Analysis</em> to be useful, although various analysis textbooks will cover slightly different material.</p>
<p>A discussion related to Zorich/Rudin/Abbott can be found <a href="https://math.stackexchange.com/q/... |
885,478 | <p>Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$.</p>
<p>Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o -a, x_o+a] \ \ f(x) \leq f(x_o)$, then $f'(x_o)=0$.</p>
<p>I did it as follows:</p>
<p>Let b>0.</p>
<p>Sin... | idm | 167,226 | <p>You proved that $$\forall b>0, |f'(x_0)|<b$$
and this means that $f'(x_0)=0$. So your proof is already finish.</p>
|
885,478 | <p>Let $f$ be a function defined on an interval $I$ differentiable at a point $x_o$ in the interior of $I$.</p>
<p>Prove that if $\exists a>0$ $ \ [x_o -a, x_o+a] \subset I$ and $ \ \forall x \in [x_o -a, x_o+a] \ \ f(x) \leq f(x_o)$, then $f'(x_o)=0$.</p>
<p>I did it as follows:</p>
<p>Let b>0.</p>
<p>Sin... | drhab | 75,923 | <p>As answer to your question see my comment.</p>
<p>A more elegant way of working is:</p>
<p>Define $g\left(x\right):=f\left(x+x_{0}\right)-f\left(x_{0}\right)$
and $J:=x_{0}+I$.</p>
<p>Then $g$ is a function defined on interval $J$ differentiable at
$0\in\left[-a,a\right]\subset J$ with $g\left(0\right)=0$ and $g\... |
4,000,459 | <p>How can we mathematically precisely argue that <span class="math-container">$$\lim_{n \to \infty} \left(1-\frac xn+\frac{x}{n^2}\right)^n = e^{-x}$$</span> holds?</p>
<p>So how can we bring</p>
<p><span class="math-container">$$1-\frac xn+\frac{x}{n^2} = 1- \frac{(n+1)x}{n^2} \approx 1 - \frac xn $$</span>
and
<span... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p><span class="math-container">$$\lim_{n\to\infty}\left(1-\dfrac xn+\dfrac x{n^2}\right)^n =\left(\lim_{n\to\infty}\left(1+\dfrac{x(1-n)}{n^2}\right)^{\dfrac{n^2}{x(1-n)}}\right)^{\lim_{n\to\infty}\dfrac{nx(1-n)}{n^2}}$$</span></p>
<p>Now in the inner limit set <span class="math-container">$\dfrac{x(1-n)... |
1,511,518 | <p>Let <span class="math-container">$GL_n^+$</span> be the group of <span class="math-container">$n \times n$</span> real invertible matrices with positive determinant</p>
<p>Does there exist a left invariant isotropic Riemannian metric on <span class="math-container">$GL_n^+$</span>?</p>
<p>(By "isotropic" I mean th... | Yury Ustinovskiy | 289,282 | <p>Here is a topological argument ruling out covering by $\mathbb R^{n^2}$. However, it might be an overkill for this statement.</p>
<p>Assume $n>2$. If $\mathbb R^{n^2}$ were the universal cover of $GL^+_n$, we have $GL^+_n=K(\pi, 1)$, where $\pi=\pi_1(GL_n^+)$. At the same time, $\pi_1(GL_n^+)=\pi_1(SO(n))=\mathb... |
4,508,840 | <p>I need to find the summation
<span class="math-container">$$S=\sum_{r=0}^{1010} \binom{1010}r \sum_{k=2r+1}^{2021}\binom{2021}k$$</span></p>
<p>I tried various things like replacing <span class="math-container">$k$</span> by <span class="math-container">$2021-k$</span> and trying to add the 2 summations to a pattern... | epi163sqrt | 132,007 | <p>A variation. We obtain
<span class="math-container">\begin{align*}
\color{blue}{S_n}&=\sum_{r=0}^{n}\binom{n}{r}\sum_{k=2r+1}^{2n+1}\binom{2n+1}{k}\\
&\,\,\color{blue}{=\sum_{r=0}^n\binom{n}{r}\sum_{k=2n-2r+1}^{2n+1}\binom{2n+1}{k}}\tag{$r\to\ n-r$, (1)}\\
\\
\color{blue}{S_n}&=\sum_{r=0}^{n}\binom{n}{r}... |
46,076 | <p>I'm developing a larger package which includes several subpackages. My problem is, that I can't introduce the symbols in the subpackages to the autocompletion by loading the main package, but by calling a subpackage.</p>
<p>Let's explain this with an example: I have a main package, called <code>main</code> which lo... | kglr | 125 | <p>In versions 10.2+ there is <code>BlockMap</code>:</p>
<pre><code>a = {q, r, s, t, u, v, w, x, y};
BlockMap[Mean, a, 3]
</code></pre>
<blockquote>
<p>{1/3 (q + r + s), 1/3 (t + u + v), 1/3 (w + x + y)}</p>
</blockquote>
<p>Although this is much slower than the alternatives in Mr.Wizard's answer, its elegance ma... |
598,838 | <p><span class="math-container">$11$</span> out of <span class="math-container">$36$</span>? I got this by writing down the number of possible outcomes (<span class="math-container">$36$</span>) and then counting how many of the pairs had a <span class="math-container">$6$</span> in them: <span class="math-container">$... | shaun | 260,837 | <p>The chance of getting a $6$ with the first dice is $1/6$ and the chance with the second dice is $1/6$ therfore the chance of getting a $6$ with either dice is $1/6 + 1/6 = 1/3$....not $11/36$!!!</p>
|
2,316,514 | <p>I want to calculate $\lim_{x \to 1}\frac{\sqrt{|x^2 - x|}}{x^2 - 1}$ . I tried to compute limit when $x \to 1^{+}$ and $x \to 1^{-}$ but didn't get any result . </p>
<p>Please help .</p>
<p>Note : I think it doesn't have limit but I can't prove it .</p>
| szw1710 | 130,298 | <p>First we should define what is a 3-dimensional spline. Spline functions are immanantly connected with a plane. Maybe you should try Bezier curves, which are independent on dimension?</p>
|
369,435 | <p>My task is as in the topic, I've given function $$f(x)=\frac{1}{1+x+x^2+x^3}$$
My solution is following (when $|x|<1$):$$\frac{1}{1+x+x^2+x^3}=\frac{1}{(x+1)+(x^2+1)}=\frac{1}{1-(-x)}\cdot\frac{1}{1-(-x^2)}=$$$$=\sum_{k=0}^{\infty}(-x)^k\cdot \sum_{k=0}^{\infty}(-x^2)^k$$ Now I try to calculate it the following w... | Dimitris | 37,229 | <p>Use the Cauchy product:
$$\sum_{k=0}^\infty a_kx^k\cdot \sum_{k=1}^\infty b_kx^k=\sum_{k=0}^\infty c_kx^k$$
where $$c_k=\sum_{n=0}^k a_n\cdot b_{k-n}$$</p>
<p>In your case: $a_k=(-1)^k$ and $$b_k=\begin{cases}0 & ,k =2l+1 \\(-1)^k&,k=2l\end{cases}$$</p>
|
1,632,677 | <p>How can I arrive at a series expansion for $$\frac{1}{\sqrt{x^3-1}}$$ at $x \to 1^{+}$? Experimentation with WolframAlpha shows that all expansions of things like $$\frac{1}{\sqrt{x^y - 1}}$$ have $$\frac{1}{\sqrt{y}\sqrt{x-1}}$$ as the first term, which I don’t know how to obtain.</p>
| Olivier Oloa | 118,798 | <p>Set $x:=1+\epsilon$, with $\epsilon \to 0^+.$ Then, by the binomial theorem,
$$
x^3=(1+\epsilon)^3=1+3\epsilon+3\epsilon^2+\epsilon^3
$$ giving
$$
\sqrt{x^3-1}=\sqrt{3\epsilon+3\epsilon^2+\epsilon^3}=\sqrt{3}\:\sqrt{\epsilon}\:\sqrt{1+\epsilon+O(\epsilon^2)} \tag1
$$ Observe that, by the Taylor expansion, as $\epsil... |
37,380 | <p>I do have noisy data and want to smooth them by a Savitzky-Golay filter because I want to keep the magnitude of the signal. </p>
<p>a) Is there a ready-to-use Filter available for that? </p>
<p>b) what are appropriate values for m (the half width) and for the coefficients for 3000-4000 data points?</p>
| Alexey Popkov | 280 | <p>Several years ago in <a href="https://groups.google.com/d/msg/comp.soft-sys.math.mathematica/5k6rHhIfUXY/62WGusZFAiMJ">related MathGroups thread</a> Virgil P. Stokes suggested: </p>
<blockquote>
<p>A few years back I wrote a <em>Mathematica</em> notebook that shows how one can
obtain the SG smoother from Gram p... |
3,120,090 | <blockquote>
<p>Find all the values of <span class="math-container">$\theta$</span> that satisfy the equation
<span class="math-container">$$\cos(x \theta ) + \cos( (x+2) \theta ) = \cos( \theta )$$</span></p>
</blockquote>
<p>I've tried simplifying with factor formulae and a combo of compound angle formulae, and... | Michael Rozenberg | 190,319 | <p>it's <span class="math-container">$$2\cos(x+1)\theta\cos\theta=\cos\theta$$</span> or
<span class="math-container">$$(2\cos(x+1)\theta-1)\cos\theta=0.$$</span>
Can you end it now?</p>
<p>Actually, <span class="math-container">$\cos\theta=0$</span> gives
<span class="math-container">$$\theta=\frac{\pi}{2}+\pi k,$$</... |
96,952 | <p>How would I go about solving the following for <code>c</code>?</p>
<pre><code>Solve[0 == Sum[(t[i]*m[i] - c*t[i]^2)/s[i]^2, {i, 1, n}], c, Reals]
</code></pre>
<p>I get the error </p>
<blockquote>
<p>Solve::nsmet : This system cannot be solved with the methods available to Solve</p>
</blockquote>
<p>but it is ... | bbgodfrey | 1,063 | <p>If the variable to be solved for appears in the <code>Sum</code> in polynomial form, then the following works.</p>
<pre><code>solvsum[s_, z_] := Module[{coef, in = s[[2]], cf = CoefficientList[s[[1]], z]},
Solve[0 == Sum[coef[i] z^(i - 1), {i, Length[cf]}], z] /.
Table[coef[j] -> Sum[cf[[j]], Evaluate[... |
3,003,982 | <p>By integrating by parts twice, show that <span class="math-container">$I_n$</span>, as defined below for integers <span class="math-container">$n > 1$</span>, has the value shown.</p>
<blockquote>
<p><span class="math-container">$$I_n = \int_0^{\pi / 2} \sin n \theta \cos \theta \,d\theta = \frac{n-\sin(\frac{... | Travis Willse | 155,629 | <p><strong>Hint</strong> Following what you've done already, integrating by parts with <span class="math-container">$u = \sin n \theta$</span>, <span class="math-container">$dv = \cos \theta \,d\theta$</span> gives
<span class="math-container">\begin{multline}\color{#df0000}{I_n} = \underbrace{\sin n \theta}_u \, \unde... |
3,190,601 | <p>Suppose <span class="math-container">$f$</span> is continuous on <span class="math-container">$[-1,1]$</span> and differentiable on <span class="math-container">$(-1,1)$</span>. Find:</p>
<p><span class="math-container">$$\lim_{x\rightarrow{0^{+}}}{\bigg(x\int_x^1{\frac{f(t)}{\sin^2(t)}}\: dt\bigg)}$$</span></p>
<... | user662984 | 662,984 | <p>Define <span class="math-container">$m(x)=\inf_{t\in[0,x]}t^2\frac{f(t)}{\sin^2(t)}$</span> and <span class="math-container">$M(x)=\sup_{t\in[0,x]}t^2\frac{f(t)}{\sin^2(t)}$</span>. Since <span class="math-container">$f$</span> is continuous, then <span class="math-container">$m(x)\to f(0)$</span> and <span class="m... |
3,190,601 | <p>Suppose <span class="math-container">$f$</span> is continuous on <span class="math-container">$[-1,1]$</span> and differentiable on <span class="math-container">$(-1,1)$</span>. Find:</p>
<p><span class="math-container">$$\lim_{x\rightarrow{0^{+}}}{\bigg(x\int_x^1{\frac{f(t)}{\sin^2(t)}}\: dt\bigg)}$$</span></p>
<... | Paramanand Singh | 72,031 | <p>If one assumes that <span class="math-container">$f$</span> is constant, then the integral evaluates to <span class="math-container">$f(0)(\cot x-\cot 1)$</span> and thus the desired limit is <span class="math-container">$f(0)$</span>.</p>
<p>This motivates us to prove that the limit is <span class="math-container"... |
2,523,570 | <p>$15x^2-4x-4$,
I factored it out to this:
$$5x(3x-2)+2(3x+2).$$
But I don’t know what to do next since the twos in the brackets have opposite signs, or is it still possible to factor them out?</p>
| Michael Rozenberg | 190,319 | <p>$$15x^2-4x-4=15x^2-10x+6x-4=5x(3x-2)+2(3x-2)=(5x+2)(3x-2).$$</p>
|
1,266,674 | <p>There are $2^{10} =1024$ possible $10$ -letters strings in which each letter is either an $A$ or a $B$. Find the number of such strings that do not have more than $3$ adjacent letters that are identical.</p>
| André Nicolas | 6,312 | <p>Let $f(n)$ be the number of strings of length $n$ that begin with A and do not have $4$ or more consecutive occurrences of the same letter (good strings). Then the answer to our problem is $2f(10)$.</p>
<p>The second letter could be a B. There are $f(n-1)$ such good strings.</p>
<p>The second letter could be an ... |
1,965,040 | <p>Find the cyclic subgroups $<\rho_1>, <\rho_2>, and <\mu_1>$ of $S_3$.<a href="https://i.stack.imgur.com/AOnTC.png" rel="nofollow noreferrer"> Elements of $S_3$.</a></p>
<p>I know the answer is suppose to be $<\rho_1> = <\rho_2> = \{\rho_0, \rho_1, \rho_2 \}$ and $<\mu_1> = \{ \rh... | Marc Bogaerts | 118,955 | <p>I suppose you mean that you are considering cosets of an ideal $I$ in a ring $R$, then $$(a+I)(b+I) = ab +aI+bI +I*I = ab + I + I + I= ab + I$$. Because $aI = I$, $I*I = I$ and $I + I=I$ as consequences of the definition of an ideal.</p>
|
1,965,040 | <p>Find the cyclic subgroups $<\rho_1>, <\rho_2>, and <\mu_1>$ of $S_3$.<a href="https://i.stack.imgur.com/AOnTC.png" rel="nofollow noreferrer"> Elements of $S_3$.</a></p>
<p>I know the answer is suppose to be $<\rho_1> = <\rho_2> = \{\rho_0, \rho_1, \rho_2 \}$ and $<\mu_1> = \{ \rh... | Yanyu Wang | 743,565 | <p>An Excellent Question! I am also troubled by this question for a long time. The key point here is to remember why we define the product on the quotient ring this way. You may think this is a set equation, however what really matters is that we can verify the product is well-defined. In another word, it is independen... |
2,965,993 | <p>Suppose you have the surface <span class="math-container">$\xi$</span> defined in <span class="math-container">$\mathbb{R}^3$</span> by the equation:
<span class="math-container">$$ \xi :\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$</span>
For <span class="math-container">$ x \geq 0$</span> , <span clas... | dan_fulea | 550,003 | <p>Consider a point <span class="math-container">$P(x_0,y_0,z_0)\in \Bbb R_{>0}^3$</span> on the given ellipsoid <span class="math-container">$(E)$</span> with equation
<span class="math-container">$$
\frac {x^2}{a^2} +
\frac {y^2}{b^2} +
\frac {z^2}{c^2}
=1\ .
$$</span>
Then the plane tangent in <span class="math-... |
699,933 | <p>There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz inequality. In trying to do the same thing in the complex case, I ran into some trouble. First, there are proofs <a href="http... | mookid | 131,738 | <blockquote>
<p>If I similarly expand $\|\lambda v - w\|^2_{\mathbb{C}},$ I get $|\lambda|^2\|v\|^2 - 2 \text{Re}(\lambda \langle v,w\rangle) + \|w\|^2$.</p>
</blockquote>
<p>Your idea can work almost the same way.</p>
<p>Consider values $\lambda(r) = r \exp i\theta$, taking $\theta$ such as
$\lambda(r) \langle v,w... |
2,830,926 | <p>I am implementing a program in C which requires that given 4 points should be arranged such that they form a quadrilateral.(assume no three are collinear)<br>
Currently , I am ordering the points in the order of their slope with respect to origin.<br>
See <a href="https://ibb.co/cfDHeo" rel="nofollow noreferrer">htt... | David M. | 398,989 | <p>Sounds like the issue is with calculating the second derivative, so here are some tips. When I have to do a tedious derivative, I like to fold up the notation a little to make things easier. So in this case, I would define</p>
<p>$$ f(u)\equiv 1-(1-p)e^u, $$</p>
<p>then the MGF is</p>
<p>$$ m_X(u)=\bigg(\frac{p}{... |
2,830,926 | <p>I am implementing a program in C which requires that given 4 points should be arranged such that they form a quadrilateral.(assume no three are collinear)<br>
Currently , I am ordering the points in the order of their slope with respect to origin.<br>
See <a href="https://ibb.co/cfDHeo" rel="nofollow noreferrer">htt... | Clarinetist | 81,560 | <p>I wanted to state that finding the variance of $X$, given the MGF of $X$, is usually much easier done with the <strong>cumulant generating function</strong> (CGF).</p>
<p>Suppose $M_X$ is the MGF of $X$. Then the CGF is given by $\varphi_X = \ln M_X$.</p>
<p>One of the very nice things about $\varphi_X$ is that $\... |
2,029,538 | <blockquote>
<p>A function $f(x)$, where $x$ is a real number , is defined implicitly by the following formula:
$$f(x)=x-\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx$$
Find the explicit function for $f(x)$ in its simplest form.</p>
</blockquote>
<p>This question appeared in the recent New Zealand Qualifications Authority... | msm | 350,875 | <p>$$f(x)=x-\int^{\frac{\pi}{2}}_0f(x')\sin(x')dx'$$
multiply by $\sin(x)$ and integrate from $0$ to $\frac{\pi}{2}$:
$$f(x)\sin(x)=x\sin(x)-\left(\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx\right)\sin(x)$$
$$\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx=\int^{\frac{\pi}{2}}_0x\sin(x)dx-\left(\int^{\frac{\pi}{2}}_0f(x)\sin(x)dx\right)\i... |
2,969,363 | <p>I have 3 points in space A, B, and C all with (x,y,z) coordinates, therefore I know the distances between all these points. I wish to find point D(x,y,z) and I know the distances BD and CD, I do NOT know AD.</p>
<p>The method I have attempted to solve this using is first saying that there are two spheres known on p... | quasi | 400,434 | <p>Suppose the known distances are
<span class="math-container">$$d(B,C)=d(A,C)=d(C,D)=1$$</span>
and
<span class="math-container">$$d(A,B)=d(B,D)=\sqrt{2}$$</span>
For concreteness, we can place <span class="math-container">$A,B,C$</span> ae
<span class="math-container">$$C=(0,0,0),\;\;B=(1,0,0),\;\;A=(0,1,0)$$</span>... |
2,596,457 | <ul>
<li>If $\lim _{n\rightarrow \infty }a_{n}=a$ then $\left\{a_{n}:n\in\mathbb{N}\right\} \cup \left\{ a\right\}$ is compact.</li>
</ul>
<p><strong>I couldn't do anything. Can you give a hint?</strong></p>
<p>Note: in the question $a_n\in\mathbb{R}$.</p>
| Kurtland Chua | 249,134 | <p>Hint: Compactness is equivalent to sequential compactness in metric spaces. Given any sequence $(b_n)$ with terms from $S = \{a_n : n \in \mathbb{N}\} \cup \{a\}$, there are two cases - either the sequence only uses finitely many values from $S$ or infinitely many of them. Can you find a convergent subsequence in ei... |
1,213,663 | <p>If you measure a task & it takes 3 seconds, then the next time you do the same task, it takes you 1 second, is the difference 200% or 67%? </p>
<p>Or would you say the difference is 200% because 3-1=2 or 200% better -- but the percentage of difference is 2/3 or 67%? I'm pretty sure I'm confusing something if ... | John Hughes | 114,036 | <p>Changing rows is the same as multiplying by a permutation (which is a rotation, and perhaps a reflection)in the codomain. That means that the SVD before and after look like
$$
M = U D V^t \\
M' = P U D V^t
$$
In the case where there's no reflection ($det P > 0$), using $PU$ as $U'$, you get an SVD for $M'$. The s... |
4,233,619 | <p>Consider all natural numbers whose decimal expansion has only the even digits <span class="math-container">$0,2,4,6,8$</span>. Suppose these are arranged in increasing order. If <span class="math-container">$a_n$</span> denotes the <span class="math-container">$n$</span>-th number in this sequence then the value of ... | José Carlos Santos | 446,262 | <p>The correct option is the first one. Assuming that <span class="math-container">$x>0$</span> and that <span class="math-container">$a\in\Bbb R$</span>, you always have <span class="math-container">$\log\left(x^a\right)=a\log(x)$</span>. So, when <span class="math-container">$a=y^z$</span>, you have<span class="ma... |
195,556 | <p>My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.</p>
<p>In recursive function theory one studies <a href="http://en.wikipedia.org/wiki/Partial_functions">partial functions</a> on the set of natural numbers. </p>
<p>Are there other areas of mathematics in which (... | Peter Smith | 35,151 | <p>In one sense, surely, it is deeply important that the square root function with domain and co-domain the positive rationals $\mathbb{Q}$ is partial (as are the cube root function, fourth root function, etc. etc.). That's the non-trivial ancient discovery that leads us to introduce the concept of irrational numbers.<... |
195,556 | <p>My math background is very narrow. I've mostly read logic, recursive function theory, and set theory.</p>
<p>In recursive function theory one studies <a href="http://en.wikipedia.org/wiki/Partial_functions">partial functions</a> on the set of natural numbers. </p>
<p>Are there other areas of mathematics in which (... | Carl Mummert | 630 | <p>In functional analysis, the concept of an <a href="http://en.wikipedia.org/wiki/Unbounded_operator" rel="noreferrer">unbounded operator</a> is closely connected with partial functions. The natural examples of unbounded operators are linear operators that are defined only on a dense proper subspace of a Banach space.... |
2,049,777 | <p>$2.$ Find the dimensions of </p>
<p>(a) the space of all vectors in $R^n$ whose components add to zero;</p>
<p>(c) the space of all solutions to $\frac{d^2y(t)}{dt^2} −3 \frac{dy(t)}{dt} +2y(t) = 0$. </p>
<p>for (a) Im pretty sure that the dimension is $n-1$ but people seem to differ, I thought it is $n-1$ since ... | Doug M | 317,162 | <p>a) The dimension is n-1</p>
<p>You could have a vector $(x_1,x_2,x_3\cdots x_{n-1}, -\sum_\limits{i=1}^{n-1} xi)$
And that vector has $n-1$ components that are independent.</p>
<p>b) that diff eq has a solution $y = C_1 e^{t}+ C_2 e^{2t}$</p>
<p>$e^{t}, e^{2t}$ form the basis of a 2 dimensional vector space.</p>
|
2,348,811 | <p>Whenever I go through the big pile of socks that just went through the laundry, and have to find the matching pairs, I usually do this like I am a simple automaton:</p>
<p>I randomly pick a sock, and see if it matches any of the single socks I picked out earlier and that haven't found a match yet. If there is a mat... | Marko Riedel | 44,883 | <p>We can verify the accepted answer using the methodology from this <a href="https://math.stackexchange.com/questions/2172876/">MSE
link</a> where we see
that the problem is very similar to a coupon collector without
replacement and two instances of $n$ types of coupons. Suppose we have
$j$ instances. Start ... |
2,033,790 | <p>How do you prove the sequence $x_n = (\frac{n}{2})^n$ diverges?</p>
<p>Here is my attempt: </p>
<p>Suppose $x_n \to L$. This means $(\forall \epsilon > 0) (\exists N \in \mathbb{N}) (\forall n>N)|x_n-L| < \epsilon$</p>
<p>Assume $n > N$</p>
<p>Then $|x_n-L| < \epsilon$</p>
<p>$|(\frac{n}{2})^n-L|... | Ethan Alwaise | 221,420 | <p>Convergent sequences must be bounded. So simply show that your sequence is unbounded. For this you can use
$$\left(\frac{n}{2}\right)^n \geq \frac{n}{2}.$$</p>
|
934,353 | <p>I am a high school student in Calculus, and we are finishing learning basic limits. I am reviewing for a big test tomorrow, and I could do all of the problems correctly except this one.</p>
<p>I have no idea how to solve the problem this problem correctly. I looked up the answer online, but I can't figure out how t... | Adi Dani | 12,848 | <p>$$\lim_{x\to 0}\frac{\frac{1}{\sqrt{1+x}}-1}{x}=\lim_{x\to 0}\frac{1}{x\sqrt{1+x}}-\frac{1}{x}=$$
$$=\lim_{x\to 0}\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}=\lim_{x\to 0}\frac{1-\sqrt{1+x}}{x\sqrt{1+x}}\frac{1+\sqrt{1+x}}{1+\sqrt{1+x}}=$$
$$=\lim_{x\to 0}\frac{1-(1+x)}{x\sqrt{1+x}(1+\sqrt{1+x})}=\lim_{x\to 0}\frac{-x}{x\sqrt{... |
732,334 | <p>How can we solve this equation?
$x^4-8x^3+24x^2-32x+16=0.$ </p>
| lab bhattacharjee | 33,337 | <p>As $x\ne0,$ dividing either sides by $x^2$ </p>
<p>$$x^2+\left(\frac4x\right)^2-8\left(x+\frac4x\right)+24=0$$</p>
<p>Now as $\displaystyle x^2+\left(\frac4x\right)^2=\left(x+\frac4x\right)^2-2\cdot x\cdot\frac4x$</p>
<p>Setting $x+\dfrac4x=y,$ we get $\displaystyle y^2-8-8y+24=0\implies(y-4)^2=0\iff y=4$</p>
<p... |
732,334 | <p>How can we solve this equation?
$x^4-8x^3+24x^2-32x+16=0.$ </p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\... |
3,714,995 | <blockquote>
<p>Using sell method to find the volume of solid generated by revolving the region bounded by <span class="math-container">$$y=\sqrt{x},y=\frac{x-3}{2},y=0$$</span> about <span class="math-container">$x$</span> axis, is (using shell method)</p>
</blockquote>
<p>What I try:</p>
<p><a href="https://i.sta... | Chrystomath | 84,081 | <p>The spectrum of <span class="math-container">$A$</span> must be a subset of <span class="math-container">$\{-1,1\}$</span> by the spectral mapping theorem and the fact that <span class="math-container">$A$</span> is self-adjoint. Hence <span class="math-container">$A$</span> is unitary and <span class="math-containe... |
3,714,995 | <blockquote>
<p>Using sell method to find the volume of solid generated by revolving the region bounded by <span class="math-container">$$y=\sqrt{x},y=\frac{x-3}{2},y=0$$</span> about <span class="math-container">$x$</span> axis, is (using shell method)</p>
</blockquote>
<p>What I try:</p>
<p><a href="https://i.sta... | user1551 | 1,551 | <p>Let <span class="math-container">$P=A^2=A^\ast A$</span> and <span class="math-container">$S=P^{k-1}+P^{k-2}+\cdots+P+I$</span>. By the given conditions, <span class="math-container">$P$</span> is a positive operator, <span class="math-container">$S$</span> is strictly positive and <span class="math-container">$S(P-... |
2,082,815 | <p>Find the $100^{th}$ power of the matrix $\left( \begin{matrix} 1& 1\\ -2& 4\end{matrix} \right)$.</p>
<p>Can you give a hint/method?</p>
| seeker | 267,945 | <p>The characteristic polynomial of the given matrix say $A$ is $x^2-5x+6$, the zeroes of whose are $2,3$. Thus there exist an invertible matrix $P$ such that $A=P\begin{pmatrix} 2 & 0\\
0 & 3\end{pmatrix}P^{-1}$. Hence $A^{100}=P\begin{pmatrix} 2^{100} & 0\\
0 & 3^{100}\end{pmatrix}P^{-1}$, where $P = ... |
47,561 | <p>The Hilbert matrix is the square matrix given by</p>
<p>$$H_{ij}=\frac{1}{i+j-1}$$</p>
<p>Wikipedia states that its inverse is given by</p>
<p>$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$</p>
<p>It follows that the entries in the inverse matrix are all i... | L.Z. Wong | 5,768 | <p>Thanks everyone for the answers offered! After chasing down the various links, I came across a very similar <a href="http://groups.google.com/group/sci.math.symbolic/browse_frm/thread/c059a14bb8b824f3?pli=1">comment</a> by Deane Yang in 1991 (!), that offered an elegant outline of a proof. I felt it would be nice to... |
47,561 | <p>The Hilbert matrix is the square matrix given by</p>
<p>$$H_{ij}=\frac{1}{i+j-1}$$</p>
<p>Wikipedia states that its inverse is given by</p>
<p>$$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-i}}{{i+j-2}\choose{i-1}}^2$$</p>
<p>It follows that the entries in the inverse matrix are all i... | Scot Adams | 103,568 | <p>I don't have any new content to add, but I did notice some context:</p>
<p>A "lattice" in a real vector space <span class="math-container">$V$</span> is the <span class="math-container">${\mathbb Z}$</span>-span of an
<span class="math-container">${\mathbb R}$</span>-basis of <span class="math-container">$... |
1,964,139 | <p>$D_2n$ is not abelian. However, the group of rotations, denoted $R$, is. I've already shown that $R$ is a normal subgroup of $D_2n$; however I'm stuck at showing the quotient group is abelian.</p>
<p>I know if it is abelian, $xRyR=yRxR$ but I get stuck at $xRyR=(xy)R$.
But $x$ and $y$ are not necessarily commutativ... | Jason DeVito | 331 | <p>Dietrich's proof is fine, but I wanted to show you how you could finish off your own proof.</p>
<p>First note that $R$ has order $n$ and $D_{2n}$ has order $2n$, so $D_{2n}/R$ consists of two elements. Of course, one is the coset containing the identity, $1R$, and the other is given by taking any reflection $y$ an... |
914,440 | <p>By the definition of topology, I feel topology is just a principle to define "open sets" on a space(in other words, just a tool to expand the conception of open sets so that we can get some new forms of open sets.) But I think in the practical cases, we just considered Euclidean space most and the traditional form o... | Lolman | 160,018 | <p>We need topology so we can work limits. And limits are important.</p>
|
1,038,152 | <p>Let $B \subseteq \mathbb{R}_{+}$ such that B is non-empty.
consider $B^{-1} = \left \{b^{-1} : b\in B \right \}$.<br>
Show that if $B^{-1}$ is unbounded from above, then $\inf\left(B\right)=0$</p>
<p>How can i prove that? tnx!</p>
| John Hughes | 114,036 | <p>Sometimes, you actually have to look at the topology. </p>
<p>A chain $c$ in $H_1(X, A)$ is a collection of edges (with coefficients, but those will turn out to be irrelevant); the boundary is a collection of pairs-of-points in $A$. Since $A$ is path connected, for each such pair we can find a path in $A$ that conn... |
306,744 | <p>So if the definition of continuity is: $\forall$ $\epsilon \gt 0$ $\exists$ $\delta \gt 0:|x-t|\lt \delta \implies |f(x)-f(t)|\lt \epsilon$. However, I get confused when I think of it this way because it's first talking about the $\epsilon$ and then it talks of the $\delta$ condition. Would it be equivalent to say: ... | Julien | 38,053 | <p>What you gave first is the definition of uniform continuity. You have to fix $x$ before embarking the $\forall \epsilon$ thing. That's for continuity at $x$, of course. </p>
<p>Now to answer your question: no, this is not legal to swap $\epsilon$ and $\delta$ like you did.</p>
<p>The funny condition you obtain wit... |
4,272,214 | <h2>The Equation</h2>
<p>How can I analytically show that there are <strong>no real solutions</strong> for <span class="math-container">$\sqrt[3]{x-3}+\sqrt[3]{1-x}=1$</span>?</p>
<h2>My attempt</h2>
<p>With <span class="math-container">$u = -x+2$</span></p>
<p><span class="math-container">$\sqrt[3]{u-1}-\sqrt[3]{u+1}=... | greenturtle3141 | 372,663 | <p>If you're looking for real solutions, I think you can look more closely at <span class="math-container">$\sqrt[3]{u-1} - \sqrt[3]{u+1} = 1$</span>. It seems difficult for the left side to be positive, and indeed we would be done if it is the case that it is non-positive for all real <span class="math-container">$u$... |
2,971,980 | <p>Show that if <span class="math-container">$0<b<1$</span> it follows that
<span class="math-container">$$\lim_{n\to\infty}b^n=0$$</span>
I have no idea how to express <span class="math-container">$N$</span> in terms of <span class="math-container">$\varepsilon$</span>. I tried using logarithms but I don't see h... | Gibbs | 498,844 | <p>By definition, if you choose <span class="math-container">$\varepsilon > 0$</span>, then you can find a natural number <span class="math-container">$N$</span> such that <span class="math-container">$\lvert b^n\rvert < \varepsilon$</span> when <span class="math-container">$n > N$</span>. Since <span class="... |
774,332 | <p>I have a radial Schrödinger equation for a particle in Coulomb potential:</p>
<p>$$i\partial_t f(r,t)=-\frac1{r^2}\partial_r\left(r^2\partial_r f(r,t)\right)-\frac2rf(r,t)$$</p>
<p>with initial condition</p>
<p>$$f(r,0)=e^{-r^2}$$</p>
<p>and boundary conditions</p>
<p>$$\begin{cases}
|f(0,t)|<\infty\\
|f(\in... | rich | 224,652 | <p>This may be a wild goose chase, and at the very least it will be a mess, but here are my thoughts. The time-independent Schrodinger equation (TISE) with Coulomb potential can be solved exactly -- the negative-energy solutions are in any quantum mechanics book and the positive-energy solutions are confluent hypergeom... |
1,104,163 | <p>I have come up with the equation in the form $${{dy}\over dx} = axe^{by}$$, where a and b are arbitrary real numbers, for a project I am working on. I want to be able to find its integral and differentiation if possible. Does anyone know of a possible solution for $y$ and/or ${d^2 y}\over {dx^2}$?</p>
| Chinny84 | 92,628 | <p>$$
\frac{d}{dx}\mathrm{e}^{-by} = -b\mathrm{e}^{-by}y'
$$
Thus we can rewrite your equation as
$$
-\frac{1}{b}\left(\mathrm{e}^{-by}\right)' = ax
$$</p>
|
140,615 | <p>A rectangular page is to have a printed area of 62 square inches. If the border is to be 1 inch wide on top and bottom and only 1/2 inch wide on each side find the dimensions of the page that will use the least amount of paper</p>
<p>Can someone explain how to do this?</p>
<p>I started with:</p>
<p>$$A = (x + 2)(... | Ross Millikan | 1,827 | <p>Hint: How did you get the term $\left(\frac {98}{x+2}-1\right)$? You should have $62=xy$ to give the desired printable area, so $A=(x+2)(\frac{62}x+1)$. Then, you are right, you should take $\frac {dA}{dx}$ and set it to $0$ to find $x$.</p>
|
2,119,971 | <p>If $a\mid c$ and $b\mid c$, must $ab$ divide $c$? Justify your answer.</p>
<p>$a\mid c$, $c=ak$ for some integer $k$</p>
<p>$b\mid c$, $c=bu$ for some integer $u$</p>
<p>From here I wanted to try to check if there were counter examples I could use,</p>
<p>$c\ne(ab)w$ for some integer $w$</p>
<p>From here I got ... | Michael Hardy | 11,667 | <p>$4\mid 12$ and $6\mid12$ but $4\times6\nmid12. \qquad$</p>
<p>The proposition is true when $\gcd(a,b)=1.$</p>
|
1,526,882 | <p>Let $$r(x,y)=\begin{cases} y &\mbox{ if } 0\leq y\leq x \\ x &\mbox{ if } x\leq y\leq 1\end{cases}$$</p>
<p>Show that $v(x)=\int_0^1r(x,y)f(y) \ dy$ satisfies $-v''(x)=f(x)$, where $0\leq x \leq 1$ and $f$ is continuous. </p>
<p>How can I take the second derivative of this? When I try to do it I feel like... | copper.hat | 27,978 | <p>Note that $r=\min$, hence Lipschitz with rank one, and ${\partial r(x,y) \over \partial x} $ is defined for all $x\neq y$. We see that ${\partial r(x,y) \over \partial x} = \begin{cases} 0, & y<x \\
1, & x<y\end{cases}$. Hence ${r(x+h,y)-r(x,y) \over h} \to 1_{[x,1]}(y)$, which is
integrable (and unifo... |
2,319,766 | <p>Lets say ,I have 100 numbers(1 to 100).I have to create various combinations of 10 numbers out of these 100 numbers such that no two combinations have more than 5 numbers in common given a particular number can be used max three times.
E.g.</p>
<ol>
<li>Combination 1: 1,2,3,4,5,6,7,8,9,10</li>
<li>Combination 2: ... | epi163sqrt | 132,007 | <p>We can also solve the problem with a little dose of algebra.</p>
<blockquote>
<p>Zero or more cats give $$1+x+x^2+\cdots=\frac{1}{1-x}$$</p>
<p>The same holds for dogs as well as for Guiana pigs. Putting all together gives zero or more cats, dogs and Guiana pigs: $$\left(\frac{1}{1-x}\right)^3$$</p>
<p... |
496,479 | <p>This symbols are used to describe left recursion : </p>
<blockquote>
<p>$A\to B\,\alpha\,|\,C$<br>$B\to A\,\beta\,|\,D,$</p>
</blockquote>
<p>It is taken from :
<a href="http://en.wikipedia.org/wiki/Left_recursion" rel="nofollow">http://en.wikipedia.org/wiki/Left_recursion</a></p>
<p>How can these symbols be ... | mau | 89 | <p>First of all, uppercase letters mean something which must still be worked out, while Greek letters mean actual symbols for the language. The arrow $\rightarrow$ means "if you find something like the symbols on my left, you may substitute them with the symbols on my right", and the vertical bar | means "or".</p>
<p>... |
4,350,781 | <p><a href="https://i.stack.imgur.com/57qXm.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/57qXm.jpg" alt="enter image description here" /></a></p>
<p>I was able to follow most of the proof but I don’t understand how the author concludes that <span class="math-container">$r=0$</span> at the final pa... | user170231 | 170,231 | <p>This is an attempt to capture how I tend to visualize a region when setting up a volume integral. Consider the following plots:</p>
<p><a href="https://i.stack.imgur.com/WXpRj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/WXpRj.png" alt="enter image description here" /></a></p>
<p>On the left, t... |
3,029,585 | <p>I have been stumped for a few days on this...It would be great if anyone can point me to enlightenment :)</p>
<p>Here's what I have tried. Let <span class="math-container">$Y = X^3$</span>, where X is a standard normal distribution with mean 0 and variance 1. Then</p>
<p><span class="math-container">$P(Y \leq y) =... | Sarvesh Ravichandran Iyer | 316,409 | <p>Let <span class="math-container">$L = \mathbb Q(3^{\frac 1{2^n}})$</span>. </p>
<p>Why is <span class="math-container">$L$</span> algebraic over <span class="math-container">$\mathbb Q$</span>? It is because the generating set of <span class="math-container">$L$</span> is <span class="math-container">$\mathbb Q \cu... |
1,059,989 | <p>I mean are there examples of problems that have been proven to be undecidable, in the sense that it would not be possible to devise a deterministic computer program that outputs a solution for an instance of the problem. And yet human mathematicians have come up with such a solution.</p>
| user2566092 | 87,313 | <p>In a certain sense made rigorous elsewhere in a mathematical logic conjecture whose origin I can't remember, every mathematician's proof is expressible in terms of basic semantics that a computer can understand given basic axioms and deduction rules. If you accept that, then no, there is no computer undecidable prob... |
1,250,459 | <p>Let $A\in M_{m \times n}(\mathbb{R})$, $x\in \mathbb{R}^n$ and $b,y\in \mathbb{R}^m$. Show that if $Ax=b$ and $A^ty=0_{\mathbb{R}^m}$, then $\langle b,y\rangle=0$. Also make a geometric interpretation.</p>
<p>I think I may have something to do with overdetermined / underdetermined system, but do not know how to pro... | Community | -1 | <p>Topology is literally the study of open sets. <strong>A topology</strong> is a collection of open sets.</p>
<p>In $\mathbb{R}$, open sets are arbitrary unions of open intervals, $(a,b)$ and closed sets are arbitrary intersections of closed intervals, $[a,b]$. These are important because they define limits, continui... |
2,553,610 | <p>I come across this result:</p>
<blockquote>
<p>Any power is conditionally convergent for at most two values of $x$, the endpoints of its interval of convergence.</p>
</blockquote>
<p>If it is so then why?</p>
| José Carlos Santos | 446,262 | <p>Suppose you have a power series<span class="math-container">$$\tag{1}\sum_{n=0}^\infty a_n(x-a)^n$$</span>which converges at some point other than <span class="math-container">$a$</span>, but does not converge everywhere. So, it converges at some <span class="math-container">$x_0$</span> with <span class="math-conta... |
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