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 |
|---|---|---|---|---|
813,395 | <p>I have read that linear independence occurs when:</p>
<p>$$\sum_{i=1}^n a_i v_i =0$$</p>
<p>Has only $a_i=0$ as a solution, but what if all $v_i$ were $0$ then $a_i$ could vary and still yield $0$. Does that mean that such a vector set is not linearly independent?</p>
<p>What if I have:</p>
<p>Let $\{c_0,c_1,c_2... | 5xum | 112,884 | <p>If there exists such an $i$ for which $v_i=0$, then selecting $\alpha_i=1$ and $\alpha_j=0$ for $j\neq i$ means that $$\sum_{k=1}^na_kv_k=a_iv_i=0.$$</p>
<p>This means that if at least one vector in the set is $0$, the set is not linearly independent.</p>
|
4,016,133 | <p>I am recently exploring set-theoretical postulates that contradict GCH. One particularly interesting one is the proposition "<span class="math-container">$2^\kappa$</span> is singular for each infinite cardinal <span class="math-container">$\kappa$</span>". However I could not prove that this proposition i... | XiaohuWang | 809,229 | <p>As it turns out, my question has already been answered in <a href="https://mathoverflow.net/questions/226887/when-can-power-sets-be-limit-cardinals">this post</a>, where other related topics have also been discussed.</p>
|
152,467 | <p>Can you please explain to me how to get from a nonparametric equation of a plane like this:</p>
<p>$$ x_1−2x_2+3x_3=6$$</p>
<p>to a parametric one. In this case the result is supposed to be </p>
<p>$$ x_1 = 6-6t-6s$$
$$ x_2 = -3t$$
$$ x_3 = 2s$$</p>
<p>Many thanks.</p>
| Karolis Juodelė | 30,701 | <p>There is more than one way to write any plane is a parametric way. To write a plane in this way, pick any three points $A$, $B$, $C$ on that plane, not all in one line. Then
$$f(s, t) = A + (B-A)s + (C-A)t$$</p>
|
1,012,158 | <p>$$ y\in R$$
Prove: <br>
if for every positive number $b$:
$$ \left\lvert y \right\rvert \leq b $$
so $y=0$</p>
<p>I tried seperating into cases where</p>
<p>$$ -b\leq y\leq 0 $$ and $$ 0\leq y\leq b $$</p>
<p>But I can't see where it helps me, any ideas? thanks</p>
| Community | -1 | <p>By the hypothesis we have</p>
<p>$$\forall b>0,\quad |y|\le b$$
which means that $|y|$ is a lower bound for the set $\Bbb R_{>0}$ so $|y|\le 0 $ : the greatest lower bound of this set hence we get $$0\le |y|\le 0\implies |y|=0\implies y=0$$</p>
|
1,012,158 | <p>$$ y\in R$$
Prove: <br>
if for every positive number $b$:
$$ \left\lvert y \right\rvert \leq b $$
so $y=0$</p>
<p>I tried seperating into cases where</p>
<p>$$ -b\leq y\leq 0 $$ and $$ 0\leq y\leq b $$</p>
<p>But I can't see where it helps me, any ideas? thanks</p>
| idm | 167,226 | <p>Suppose that $y\neq 0$, then, there exist a $c>0$ such that
$$|y|\geq c>0$$
which is a contradiction with
$$\forall b> 0, |y|\leq b,$$</p>
<p>Q.E.D.</p>
|
25,414 | <p>I'm running in to some problems with generating a persistent HSQLDB and during some troubleshooting I came upon the following behavior.</p>
<pre><code>Needs["DatabaseLink`"]
tc = OpenSQLConnection[
JDBC["hsqldb", ToFileName[Directory[], "temp"]], Username -> "sa"]
CloseSQLConnection[tc]
</code></pre>
<p>The ab... | Fred Daniel Kline | 973 | <p><a href="http://www.fileinfo.com/extension/lck" rel="nofollow">The last paragraph here </a>indicates that the <strong>.lck</strong> extension is assumed by the OS to be managed by the application (which in this case is <em>Mathematica</em>). The <strong>.log</strong> is also in that category. The solution is to spec... |
3,805,286 | <p>This is a question on the convergence of a sequence of real, convex, analytic functions (it does not get better than that!):</p>
<p>Let <span class="math-container">$(f_n)_{n\in \mathbb N}$</span> be a sequence of convex analytic functions on <span class="math-container">$\mathbb R$</span>.</p>
<p>Suppose that <span... | zhw. | 228,045 | <p>Counterexample: Define <span class="math-container">$f_n(x) = (x^2+1/n)^{1/2}.$</span> Then each <span class="math-container">$f_n$</span> is analytic and convex on <span class="math-container">$\mathbb R.$</span> Clearly <span class="math-container">$f_n(x)\to |x|$</span> pointwise everywhere. (A little more work s... |
2,043,457 | <p>Does anybody know of a succint way to compute the residue of $f(z)=z^m/(1-e^{-z})^{n+1}$ at $z=0$? I am only interested in the nontrivial case $m<n$.
Induction seems complicated/inefficient, so I am looking for a "trick", perhaps with Lagrange inversion?</p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>In a $\Bbb R$-vector space, the scalar product $\lambda v$ with $\lambda\in \Bbb C$ <strong>isn't defined</strong>. You can <em>define</em> a <a href="https://en.wikipedia.org/wiki/Complexification" rel="nofollow noreferrer">complexification</a> of a $\Bbb R$-vector space, but it will be <em>another</em> structure.<... |
4,154,025 | <p>In a set of lecture notes, I have the following result:</p>
<blockquote>
<p><strong>Theorem</strong>. Let <span class="math-container">$X_n$</span> be random variables on <span class="math-container">$(\Omega, \mathcal{F}, \mathbb{P})$</span> with values in a Polish metric space <span class="math-container">$S$</spa... | Kore-N | 59,827 | <p>Consider the case <span class="math-container">$p=1$</span> (allowing for <span class="math-container">$p>1$</span> seems unnecessary). By Birkhoff's ergodic theorem (<a href="http://math.uchicago.edu/%7Emay/REU2016/REUPapers/Ran.pdf" rel="nofollow noreferrer">http://math.uchicago.edu/~may/REU2016/REUPapers/Ran.p... |
2,846,114 | <p>$$\int\frac{2}{x(3x-8)}dx=P\cdot \ln\left|x\right|+Q\cdot \ln\left|3x-8\right|$$</p>
<p>Find out what P and Q are equal to.</p>
<p>This is what I worked out:</p>
<p>$$\frac{A}{x}+\frac{B}{3x-8}=\frac{2}{x(3x-8)}$$
$$-\frac{1}{4}=A,\ \ \ \frac{3}{4}=B$$
$$P=A, Q=B$$</p>
<p>why is the answer $P=-\frac{1}{4}, Q=\fr... | mengdie1982 | 560,634 | <p>Since $$\frac{{\rm d}(P\cdot \ln\left|x\right|+Q\cdot \ln\left|3x-8\right|)}{{\rm d}x}=\frac{P}{x}+\frac{3Q}{3x-8}=\frac{3(P+Q)x-8P}{x(3x-8)},$$</p>
<p>we obtain $$P+Q=0,~~-8P=2.$$
As a result, $$P=-\frac{1}{4}, ~~~Q=\frac{1}{4}.$$</p>
|
167,812 | <p>I call a profinite group $G$ <strong><em>Noetherian</em></strong>, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is finitely generated.</p>
<p>A profinite group $G$ is called <strong><em>just-infinite</em></str... | Andrei Jaikin | 10,482 | <p>The answer is <strong>yes</strong> in general.</p>
<p>Since $K$ is finitely generated, by the Nikolov-Segal theorem it coincides with its own profinite completion. So you simply may take $R=K$.</p>
<p>Perhaps, you also want $R$ to be finitely generated. In this case the answer is <strong>no</strong>. </p>
<p>If $... |
4,262,888 | <p>My task is to prove that if an atomic measure space is <span class="math-container">$\sigma$</span>-finite, then the set of atoms must be countable.</p>
<p>This is my given definition of an atomic measure space:</p>
<blockquote>
<p>Assume <span class="math-container">$(X,\mathcal{M},\mu)$</span> is a measure space w... | Community | -1 | <p>You may argue as follows. Since <span class="math-container">$U_i\supset S'$</span> for some <span class="math-container">$i\in\mathbb{N}$</span>,
<span class="math-container">$$
\mu(U_i)\ge \mu(S'')\label{1}\tag{1}
$$</span>
for any <span class="math-container">$S''\subseteq S'$</span>. Let <span class="math-contai... |
465,999 | <p>I'm not sure of this, can I have a constraint like this in a linear programming problem to be solved with simplex algorithm?</p>
<p>$$n_1t_1 + n_2t_2 > 200$$</p>
<p>where $n_1$ and $t_1$, $n_2$ and $t_2$ are different variables.</p>
| user2566092 | 87,313 | <p>This is an example of a quadratic (non-linear) constraint, and you can look up quadratically constrained quadratic programming (QCQP) e.g. on wikipedia if you want to learn more about how these kinds of problems with quadratic constraints and/or objectives are solved, although the computations are generally much slo... |
365,287 | <p>Let $([0,1],\mathcal{B},m)$ be the Borel sigma algebra with lebesgue measure and $([0,1],\mathcal{P},\mu)$ be the power set with counting measure. Consider the product $\sigma$-algebra on $[0,1]^2$ and product measure $m \times \mu$.</p>
<p>(1) Is $D=\{(x,x)\in[0,1]^2\}$ measurable?</p>
<p>(2) If so, what is $m \t... | Peter Smith | 35,151 | <p>Again ('cos I've recommended it before) I can very warmly recommend getting your students to beg/borrow/buy and then <strong>read</strong> the excellent</p>
<blockquote>
<p>Daniel J. Velleman, <em>How to Prove it: A Structured Approach</em> (CUP, 1994 and much reprinted, and now into a second edition).</p>
</bloc... |
2,541,991 | <p>I need to find a pair of dependent random variables $(X, Y)$ with covariance equal to $0.$ From this I gather:</p>
<p>$$0 = E((X-EX)(Y-EY)) = E \left(\left(X - \int_{-\infty}^\infty xf_X(x)\,dx\right) \left(Y - \int_{-\infty}^\infty xf_Y(x)\,dx \right)\right)$$</p>
<p>but what can I do now? How can I use the fact ... | Robert Israel | 8,508 | <p>Hint: try $X$ and $X^2$ where the distribution of $X$ is symmetric about $0$.</p>
|
3,548,064 | <p>I have two equalities:
<span class="math-container">$$ \alpha x^{2} + \alpha y^{2} - y = 0 $$</span>
<span class="math-container">$$ \beta x^{2} + \beta y^{2} - x = 0 $$</span></p>
<p>Where <span class="math-container">$$ \alpha, \beta $$</span> are both known constants.</p>
<p>How can I solve for <span class="ma... | Eric Towers | 123,905 | <p>The number of solutions depends on how many of <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> are zero.</p>
<ul>
<li>For any <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, <span class="math-container">$x = y = 0$</span> is... |
164,002 | <p>When I am reading a mathematical textbook, I tend to skip most of the exercises.
Generally I don't like exercises, particularly artificial ones.
Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..</p>
<p>Sometimes I try to prove a theorem before reading the proof.
Sometimes I try... | nanme | 33,515 | <p>absolutely you can't truly say you understand something until you work through it</p>
|
3,078,176 | <p>Given an equation in partial derivatives of the form <span class="math-container">$Af_x+Bf_y=\phi(x,y)$</span>, for example <span class="math-container">$$f_x-f_y=(x+y)^2$$</span> How do I know which change of coordinates is appropiate to solve the equation? In this example, the change of coordinates is <span class=... | Mostafa Ayaz | 518,023 | <p><strong>Hint</strong></p>
<p>use <span class="math-container">$$\sin \theta ={\tan\theta\over \sqrt{1+\tan^2\theta}}$$</span>whenever <span class="math-container">$\sin \theta , \tan \theta \ge 0$</span>.</p>
|
2,313,060 | <p>$f(\bigcap_{\alpha \in A} U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$</p>
<p>Suppose $y \in f(\bigcap_{\alpha \in A} U_{\alpha})$
$\implies f^{-1}(y) \in \bigcap_{\alpha \in A} U_{\alpha} \implies f^{-1}(y) \in U_{\alpha}$ for all $\alpha \in A$</p>
<p>$\implies y \in f (U_{\alpha})$ for all $\alph... | MarnixKlooster ReinstateMonica | 11,994 | <p>May I propose a 'logical' approach? Here is what happens if you expand the definitions, to go from the level of set theory to that of logic, and then reason on the logic level.</p>
<p>But first, what <em>are</em> the definitions?$%
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad &... |
4,200,256 | <p>Which numbers can be represented in the form of <span class="math-container">$x^2 + y^2$</span>, <span class="math-container">$x,y \in\mathbb N$</span>?</p>
<p>My approach : I found out that for this to satisfy, one of the prime factors of the number has to be congruent to <span class="math-container">$1$</span> (mo... | Ibadul Qadeer | 997,150 | <p>Integers that in their prime factorization, contain either only odd primes of the form <span class="math-container">$4k+1$</span> or if there is a prime factor of the form <span class="math-container">$4k+3$</span>, then the multiplicity of this prime should be even, can be written as the sum of two squares.</p>
|
4,200,256 | <p>Which numbers can be represented in the form of <span class="math-container">$x^2 + y^2$</span>, <span class="math-container">$x,y \in\mathbb N$</span>?</p>
<p>My approach : I found out that for this to satisfy, one of the prime factors of the number has to be congruent to <span class="math-container">$1$</span> (mo... | Geoffrey Trang | 684,071 | <p>Let <span class="math-container">$A$</span> be the set of all prime numbers of the form <span class="math-container">$4k+3$</span>. Then, <span class="math-container">$n$</span> is the sum of two squares if and only if <span class="math-container">$\forall p \in A\, p \mid n \implies 2 \mid \nu_p(n)$</span>, where <... |
4,200,256 | <p>Which numbers can be represented in the form of <span class="math-container">$x^2 + y^2$</span>, <span class="math-container">$x,y \in\mathbb N$</span>?</p>
<p>My approach : I found out that for this to satisfy, one of the prime factors of the number has to be congruent to <span class="math-container">$1$</span> (mo... | Jack D'Aurizio | 44,121 | <p>I will outline a proof of the characterization already given by Geoffrey Trang.</p>
<p><strong>Step 1</strong>. The primes of the form <span class="math-container">$x^2+y^2$</span> are <span class="math-container">$2$</span> and the primes <span class="math-container">$\equiv 1\pmod{4}$</span>.<br>
Trivially, a prim... |
3,197,683 | <p>Here is the theorem that I need to prove</p>
<blockquote>
<p>For <span class="math-container">$K = \mathbb{Q}[\sqrt{D}]$</span> we have</p>
<p><span class="math-container">$$\begin{align}O_K = \begin{cases}
\mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\
\mathbb{Z}\left[\frac{1 + \sqrt{D}}{2}\... | bsbb4 | 337,971 | <p><span class="math-container">$\bullet$</span> Let <span class="math-container">$d \equiv 1 \mod 4.$</span></p>
<p>Let <span class="math-container">$A=x+y\sqrt{d} \in \mathcal O_K$</span> be arbitrary. We have</p>
<p><span class="math-container">$(A-x)^2 - dy^2 = 0$</span>. Expanding,</p>
<p><span class="math-cont... |
4,441,034 | <blockquote>
<p>Consider function <span class="math-container">$f(x)$</span> whose derivative is continuous on the interval <span class="math-container">$[-3; 3]$</span> and the graph of the function <span class="math-container">$y = f'(x)$</span> is pictured below. Given that <span class="math-container">$g(x) = 2f(x)... | user2661923 | 464,411 | <p>In my opinion, the best way (for me anyway) to tackle this problem is to take off my shoes, and stretch my intuition with a specific example.</p>
<p>Suppose that <span class="math-container">$n = 6$</span>, and you compute</p>
<p><span class="math-container">$$\sum_{k=1}^6 \sum_{d|k} 1 = 1 + 2 + 2 + 3 + 2 + 4 = 14. ... |
4,441,034 | <blockquote>
<p>Consider function <span class="math-container">$f(x)$</span> whose derivative is continuous on the interval <span class="math-container">$[-3; 3]$</span> and the graph of the function <span class="math-container">$y = f'(x)$</span> is pictured below. Given that <span class="math-container">$g(x) = 2f(x)... | epi163sqrt | 132,007 | <p>Here we change the order of summation by transformations of the sums only.</p>
<blockquote>
<p>We obtain
<span class="math-container">\begin{align*}
\color{blue}{\sum_{k=1}^nd(k)}&=\sum_{k=1}^n\sum_{{d=1}\atop{d\mid k}}^k1
=\sum_{{1\leq d\leq k\leq n}\atop {d\mid k}}1\tag{1}\\
&=\sum_{d=1}^n\sum_{{k=d}\atop{... |
3,882,214 | <p>I have a question that involves an Argand diagram. The complex number <strong>u = 1 + 1i</strong> is the center of that circle, and the radius is one. In other words, <span class="math-container">$$|z - (1 + 1i)| = 1$$</span></p>
<p>Now my issue is the following: I need to <strong>calculate the least value of |z| fo... | Tryst with Freedom | 688,539 | <p>Hint: Join a line from the origin to the center of the circle, once that is done ... label each term geometrically.</p>
<p>Note: <span class="math-container">$|z|$</span> is distance of the point <span class="math-container">$ z$</span> from the origin</p>
|
2,094,657 | <p>I found this interesting problem on AoPS forum but no one has posted an answer. I have no idea how to solve it.</p>
<blockquote>
<p>$$
\int_0^\infty \sin(x^n)\,dx
$$
For all positive rationals $n>1$, $I_n$ denotes the integral as above.</p>
<p>If $P_n$ denotes the product
$$
P_n=\prod_{r=1}^{n-1}I_{\b... | tired | 101,233 | <p>The integral $I(a)=\int_0^{\infty}\sin(x^a)=\Im\int_0^{\infty}\exp(ix^a)$ is easily calulated by using the analyticality of the integrand: Rotate the the contour of integration by an angle of $\frac{\pi}{2a}$ to get $I(a)=\Im\left(e^{\frac{i\pi}{2a}}\int_0^{\infty}e^{-x^a} \right)$ or </p>
<blockquote>
<p>$$
I(a)... |
76,505 | <p>In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra. </p>
<p>He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie de mon énergie a été consacrée à un travail de réflexion sur les <em>fondements de l'algèbre (co)homologique no... | Ronnie Brown | 19,949 | <p>My answer is in agreement with Grothendieck that topological spaces may be seen as inadequate for many geometric, and in particular, homotopical purposes. Round about 1970, I spent 9 years trying to generalise the fundamental groupoid of a topological space to dimension 2, using a notion of double groupoid to reflec... |
4,522,097 | <p><span class="math-container">$$
X = \begin{pmatrix}
1+b_1 & 1 & 0 & 0 & 0 & \frac{1}{a_{6}} \\
1+b_2 & 1 & 1 & 0 & 0 & -\frac{a_1}{a_6} \\
b_3 & 1 & 1 & 1 & 0 & -\frac{a_2}{a_6} \\
b_4 & 0 & 1 & 1 & 1 & -\frac{a_3}{a_6} \\
b_5 & 0 ... | Andrew D. Hwang | 86,418 | <p>tl; dr: It's arguably impossible to answer a philosophical question about definitions, but if this question came up during a chat over beverages, I'd say</p>
<ol>
<li>We know how to do calculus on (non-empty open subsets of) Cartesian spaces.</li>
<li>The definition of a smooth manifold uses this knowledge to extend... |
1,877,567 | <p>I need help calculating two integrals</p>
<p>1)
$$\int_1^2 \sqrt{4+ \frac{1}{x}}\mathrm{d}x$$
2)
$$\int_0^{\frac{\pi}{2}}x^n sin(x)\mathrm{d}x$$</p>
<p>So I think on the 1st one I will have to use substitution, but I don't know what to do to get something similar to something that's in the known basic integrals.
2... | Community | -1 | <p>You can indeed work out the second integral by parts.</p>
<p>$$I_n:=\int_0^{\pi/2}x^n\sin x\,dx=-\left.x^n\cos x\right|_0^{\pi/2}+n\int_0^{\pi/2}x^{n-1}\cos x\,dx.$$</p>
<p>Repeat with the cosine integral,</p>
<p>$$J_n:=\int_0^{\pi/2}x^n\cos x\,dx=\left.x^n\sin x\right|_0^{\pi/2}-n\int_0^{\pi/2}x^{n-1}\sin x\,dx.... |
1,877,567 | <p>I need help calculating two integrals</p>
<p>1)
$$\int_1^2 \sqrt{4+ \frac{1}{x}}\mathrm{d}x$$
2)
$$\int_0^{\frac{\pi}{2}}x^n sin(x)\mathrm{d}x$$</p>
<p>So I think on the 1st one I will have to use substitution, but I don't know what to do to get something similar to something that's in the known basic integrals.
2... | Brevan Ellefsen | 269,764 | <p>Just as an interesting fact, the general antiderivative for the second integral can be written as
$$\int x^n \sin(x) \, \mathrm{d}x = \frac 12 i x^{n+1} (\operatorname{E_{-n}} (-i x)-\operatorname{E_{-n}} (i x))+C$$<br>
This can be proven directly through substitutions. Write the answer in terms of integrals, ... |
2,388,738 | <blockquote>
<p>I'm messing around with doing a visualization that has nothing to do with the primes and in order to execute it correctly I need an ordered list of all point in the order that the Ulam Spiral crosses them. I've tried some of my work but have only run in to abundantly complicated paths to solution. Als... | iadvd | 189,215 | <p>You are lucky because it seems that a very similar pair of sequences is already at OEIS. </p>
<p>The $x$-coordinates are sequence <a href="https://oeis.org/A174344" rel="nofollow noreferrer">A174344</a> ("List of $x$-coordinates of point moving in clockwise spiral") and $y$-coordinates ("List of $y$-coordinates of ... |
5,612 | <p>This is driving me nuts: I'm trying to control the parameters for a relatively large system of ODEs using Manipulate.</p>
<pre><code>With[{todo =
Module[
{sol, ode, timedur = 40},
ode = Evaluate[odes /. removeboundaries /. moieties];
sol = NDSolve[Join[ode, init], vars, {t, 0, timedur}];
Plot[Evaluate... | Verbeia | 8 | <p>Even without a minimual example, it is clear that you have a problem related to your use of <code>With</code> as the outer scoping construct. Please see the <a href="https://mathematica.stackexchange.com/q/559/8">answers to this question</a>, particularly <a href="https://mathematica.stackexchange.com/a/562/8">mine<... |
373,068 | <p>For a real number $a$ and a positive integer $k$, denote by $(a)^{(k)}$ the number $a(a+1)\cdots (a+k-1)$ and $(a)_k$ the number
$a(a-1)\cdots (a-k+1)$. Let $m$ be a positive integer $\ge k$. Can anyone show me, or point me to a reference, why the number
$$ \frac{(m)^{(k)}(m)_k}{(1/2)^{(k)} k!}= \frac{2^{2k}(m)^{(k... | TCL | 3,249 | <p>\begin{eqnarray*}& &\frac{2^{2k}(m)^{(k)}(m)_k}{(2k)!}\\
&=&\frac{2^{2k}(m-k+1)(m-k+2)\cdots (m-1)(m)(m)(m+1)\cdots (m+k-2)(m+k-1)}{(2k)!}
\end{eqnarray*}
Now we write one of the $m$ as $\frac{1}{2}[(m-k)+(m+k)]$ and distribute, and the last expression becomes
$$2^{2k-1}\left[\frac{(m+k-1)(m+k-2)\cdo... |
1,824,280 | <p>The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off.</p>
<p>So the first part was to prove that $4 \mid n^2 - 5 $ if $n$ is an odd integer. </p>
<p>Here is a brief proof (without intricate details):<br>
Consider $n = 2k+1$<b... | Tacet | 186,012 | <p>$$n^2 - 5 \equiv_8 0 \Leftrightarrow n^2 \equiv_8 5$$</p>
<p>So, $n$ is such number, that square has remainder $5$, when divided by $8$.
Could you show that such number doesn't exist?</p>
<p><strong>Hint</strong>: If you have no better idea, you can check just numbers from set $\lbrace 0, 1, \dots, m-1\rbrace$, wh... |
1,824,280 | <p>The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off.</p>
<p>So the first part was to prove that $4 \mid n^2 - 5 $ if $n$ is an odd integer. </p>
<p>Here is a brief proof (without intricate details):<br>
Consider $n = 2k+1$<b... | Behrouz Maleki | 343,616 | <p>If $n=2k$ then $n^2-5=8q-5$ or $n=8q-1$</p>
<p>If $n=2k+1 \,$ then $n^2-5=8q-4$ </p>
|
1,824,280 | <p>The question is from one of the past exams in a course I am doing. I have gotten halfway through it but cannot figure out how to finish it off.</p>
<p>So the first part was to prove that $4 \mid n^2 - 5 $ if $n$ is an odd integer. </p>
<p>Here is a brief proof (without intricate details):<br>
Consider $n = 2k+1$<b... | Community | -1 | <p>If $n=2k+1$ then you have shown that $n^2-5=4(k^2+k-1)$. If $k$ is even then $k^2+k-1$ is an even plus an even minus an odd, therefore odd, and if $k$ is odd then $k^2+k-1$ is an odd plus an odd minus an odd, therefore odd. So $2\nmid k^2+k-1$ therefore $8\nmid n^2-5=4(k^2+k-1)$.</p>
<p>If $n$ is even then obviousl... |
187,395 | <p>I can't find my dumb mistake.</p>
<p>I'm figuring the definite integral from first principles of $2x+3$ with limits $x=1$ to $x=4$. No big deal! But for some reason I can't find where my arithmetic went screwy. (Maybe because it's 2:46am @_@).</p>
<p>so </p>
<p>$\delta x=\frac{3}{n}$ and $x_i^*=\frac{3i}{n}$</... | André Nicolas | 6,312 | <p>You want
$$f\left(1+\frac{3i}{n}\right).$$</p>
<p>The $+3$ in the third line (and later) will change to $+5$.</p>
|
139,934 | <p>Suppose I want to solve an equation for the matrix elements of $\bar{W}$:
$$\alpha W_{ba}+\beta W_{bb}=x; \alpha W_{aa}+\beta W_{ab}=y$$</p>
<p>Using the syntax <code>Subscript[W, ij]</code> for my matrix element (on the $i$th row and $j$ th column), I get the following message:</p>
<p>Set::write: Tag Times in 2 x... | Nasser | 70 | <p>Using user9444 method to read text files with header. </p>
<pre><code>SetDirectory[NotebookDirectory[]]
data=Cases[Import["t.txt","Table"],{_?NumberQ,___}];
data
</code></pre>
<p><img src="https://i.stack.imgur.com/6zBEn.png" alt="Mathematica graphics"></p>
<pre><code>MatrixForm[data]
</code></pre>
<p><img src="... |
2,972,085 | <p><a href="https://i.stack.imgur.com/pcOfx.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/pcOfx.jpg" alt="enter image description here"></a></p>
<p>My friend show me the diagram above , and ask me </p>
<p>"What is the area of a BLACK circle with radius of 1 of BLUE circle?"</p>
<p>So, I solved it by alge... | g.kov | 122,782 | <p><a href="https://i.stack.imgur.com/6NWas.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6NWas.png" alt="enter image description here"></a></p>
<p><span class="math-container">\begin{align}
\triangle ADB:\quad
|DB|^2&=
(\tfrac{R}2+r)^2
-(\tfrac{R}2-r)^2
=2rR
,\\
\triangle BDO:\quad
|DB|^2&... |
1,828,042 | <p>This is my first question on this site, and this question may sound disturbing. My apologies, but I truly need some advice on this.</p>
<p>I am a sophomore math major at a fairly good math department (top 20 in the U.S.), and after taking some upper-level math courses (second courses in abstract algebra and real an... | Landon Carter | 136,523 | <p>Often what happens in mathematics, from my personal experience, is that the one with more exposure generally gets the upper hand. What you are talking about here is talent, which you feel you lack. Talent plays a role till a certain extent, but then I have seen talented people too struggling with high level mathemat... |
24,195 | <p>I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, consider a countable group $G$ and a 2-cocycle $\phi :G^2\rightarrow S^1$ where $S^1$ is the group of complex number of... | Wadim Zudilin | 4,953 | <p>After passions calmed down, I can put back my old unsuccessful attempt. At least I was quite enthusiastic at that time about the problem, until I have got downvotes and seen some vague ideas of others (they are still here) as answers.</p>
<p><em><strong>Post as it was on May 12, 2010.</em></strong>
Consider 3 circl... |
24,195 | <p>I am looking at a von Neumann algebra constructed from a discrete group and a 2-cocylce.
Does someone know some good references (article, book)? It would be very helpful for me.
To be more precise, consider a countable group $G$ and a 2-cocycle $\phi :G^2\rightarrow S^1$ where $S^1$ is the group of complex number of... | Victor Protsak | 5,740 | <p><em> I have rewritten the post so that the proof is correct. </em></p>
<p>This problem is a bit hard, but following the Polya dictum, here is the answer to an apparently easier one: yes, if circles are replaced with parallel squares, and moreover, a suitable version of the greedy algorithm works. </p>
<p><b>Theore... |
2,165,296 | <p>Can every separable Banach space be isometrically embedded in $l^2$ ? Or at least in $l^p$ for some $1\le p<\infty$ ? </p>
<p>I only know that any separable Banach space is isometrically isomorphic to a linear subspace of $l^{\infty}$.</p>
<p>Please help . Thanks in advance </p>
| Martín-Blas Pérez Pinilla | 98,199 | <p>Writing f(t) as power instead of as quotient it is easier to calculate the successive derivatives:
$$f(t) = (2t+1)^{-2}$$
$$f'(t) = (-2)(2t+1)^{-3}2$$
$$f''(t) = (-3)(-2)(2t+1)^{-3}2^2$$
$$f^{3}(t) = (-4)(-3)(-2)(2t+1)^{-3}2^3$$
$$\cdots$$
$$f^{n}(t) = \cdots$$
Can you continue?</p>
|
1,212,000 | <p>I was trying to solve this square root problem, but I seem not to understand some basics. </p>
<p>Here is the problem.</p>
<p>$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\bigg)^2} - \sqrt[3]{\bigg(1 - \sqrt{2}\bigg)^3}\Bigg)^2$$</p>
<p>The solution is as follows:</p>
<p>$$\Bigg(\sqrt{\bigg(\sqrt{2} - \frac{3}{2}\b... | marwalix | 441 | <p>A square root is always a non negative number so $\sqrt{x^2}=|x|$</p>
|
112,437 | <p>I am working on a personal project involving a CloudDeploy[ ] that reads data off a Google Doc and then works with it. Ideally, the Google Doc is either a text document or a spreadsheet which contains a single string, which is what I want Mathematica to read as input.
<a href="https://docs.google.com/document/d/17m1... | 梁國淦 | 19,360 | <p>The format of the direct download link is
<code>https://docs.google.com/document/d/<<file id>>/export?format=doc</code> or <code>format=txt</code> or <code>format=pdf</code>, etc.
Just write a small piece of function to replace the final part of the sharing URL to <code>export?format=xxx</code> and you g... |
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| Patrick Stevens | 259,262 | <p>This way doesn't require fiddling with Taylor series or interchanging any sums and limits; it's an example of one of the many places where we can simplify things by recognising a derivative.</p>
<p>Substituting $u = \cos(x)$, we obtain $$\lim_{u \to 1} \frac{\log u}{1-u^2} = \lim_{u \to 1} \left[ \frac{\log u}{1-u}... |
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| zhw. | 228,045 | <p>The expression equals</p>
<p>$$\frac{\ln (\cos x) - \ln (\cos 0)}{\cos x - \cos 0}\cdot\frac{\cos x - 1}{x^2}\cdot \frac{x^2}{\sin^2 x}.$$</p>
<p>The first fraction $\to \ln'(1) = 1,$ by definition of the derivative. The limit of the second fraction is standard and equals $-1/2.$ The third fraction $\to 1.$ So the... |
1,959,080 | <p>A book claims that $9(9_9) = 9^{387420489}$.</p>
<p>I've never seen such an expression, and I've been unable to find anything about it on Google...</p>
<p>How is it supposed to be evaulated?</p>
<p>For reference, the name of the book is <code>Pasatiempos curiosos e instructivos</code> and this is the page where t... | Brian M. Scott | 12,042 | <p>Since $9^9=387,420,489$, I assume that it’s a way of writing $9^{9^9}$. I don’t read Spanish, but that appears to be a discussion of attempts by Arab mathematicians to write large numbers using only three digits; if that’s the case, we’re looking at a historical special-purpose notation that didn’t survive.</p>
|
1,315,265 | <p>Let $X=\mathcal{L}_2 [-1,1]$ and for any scalar $\alpha$ we define $E_\alpha=\{f\in \mathcal{L}: f \text{ continuous in } [-1,1] \text{ and } f(0)=\alpha \}$.</p>
<ol>
<li>Prove $E_\alpha$ is convex for any $\alpha$.</li>
<li>Prove $E_\alpha$ is dense in $\mathcal{L}_2$</li>
<li>Prove there is no $f\in X^*$ that se... | Giuseppe Negro | 8,157 | <p>HINT: Once you have a continuous $g$ that is a good approximation to the generic $f\in L^2(-1, 1)$, take a $\delta>0$ and do a linear fit between the points $(-1, \alpha)$ and $(-1+\delta, g(-1+\delta))$. That is, consider the function
$$
g_\delta(x)=\begin{cases}
\text{linear}, & x\in[-1, -1+\delta) \\
g(x)... |
35,281 | <p>I am looking for applications of category theory and homotopy theory in set theory and particularly in cardinal arithmetics. "Applications" in the broad sense of the word --- this would include theorems, definitions, questions, points of view (and papers) in set theory that could be motivated or understood... | Andrej Bauer | 1,176 | <p>You could look at <a href="https://www.phil.cmu.edu/projects/ast/" rel="nofollow noreferrer">algebraic set theory</a>. For a general outline of how set theory, categories and type theory interact, see <a href="https://www.andrew.cmu.edu/user/awodey/" rel="nofollow noreferrer">Steve Awodey's</a> "<a href="https:... |
35,281 | <p>I am looking for applications of category theory and homotopy theory in set theory and particularly in cardinal arithmetics. "Applications" in the broad sense of the word --- this would include theorems, definitions, questions, points of view (and papers) in set theory that could be motivated or understood... | Peter Arndt | 733 | <p>There is an interaction between category theory and set theory. In 1965, one year after Cohen's proof of the independence of the continuum hypothesis, Vopenka gave a proof using sheaf theory, see</p>
<ul>
<li>Kenneth Kunen, "[Omnibus Review]", The Journal of Symbolic Logic, <strong>34</strong> Issue 3 (196... |
2,541,997 | <p>For what values of n can {1, 2, . . . , n} be partitioned into three subsets
with equal sums?</p>
<p>I noticed that somehow the sum from 1 to n hast to be a multiple of 3 and the common sum among these 3 subset is this sum divided by 3, but it's still not a convincing argument. How do you prove there exists 3 subse... | Ross Millikan | 1,827 | <p>The sum of all the numbers from $1$ to $n$ is $\frac 12n(n+1)$. As you say, we need this to be a multiple of $3$, which will be true when $n \equiv 0,2 \pmod 3$. We can't do $n=2$ or $n=3$, which we can prove by inspection. For $n=5$ there is $\{1,4\},\{2,3\},\{5\}$ and for $6$ we can do $\{1,6\},\{2,5\},\{3,4\}$... |
2,541,997 | <p>For what values of n can {1, 2, . . . , n} be partitioned into three subsets
with equal sums?</p>
<p>I noticed that somehow the sum from 1 to n hast to be a multiple of 3 and the common sum among these 3 subset is this sum divided by 3, but it's still not a convincing argument. How do you prove there exists 3 subse... | Community | -1 | <p>If $n=6k$, make the following subsets:</p>
<p>$$1,4,\cdots,3k-2,3k+3,3k+6,\cdots,6k$$
$$2,5,\cdots,3k-1,3k+2,3k+5,\cdots,6k-1$$
$$3,6,\cdots,3k,3k+1,3k+4,\cdots,6k-2$$</p>
<p>If $n=6k+r$, where $r=5,8,9$, reduce to the previous case by first partitioning the set $\{1,2,\cdots,r\}$ and then partitioning the $6k$-el... |
3,248,552 | <p>Imagine we want to use Theon's ladder to approximate <span class="math-container">$\sqrt{3}$</span>. The appropriate expressions are
<span class="math-container">$$x_n=x_{n-1}+y_{n-1}$$</span></p>
<p><span class="math-container">$$y_n=x_n+2x_{n-1}$$</span></p>
<p>Rungs 6 through 10 in the approximation of <span cl... | Cesareo | 397,348 | <p>Hint.</p>
<p>Calling <span class="math-container">$\lambda_n = \frac{y_n}{x_n}$</span> we have</p>
<p><span class="math-container">$$
\lambda_n = \frac{\lambda_{n-1}+3}{\lambda_{n-1}+1}
$$</span></p>
<p>giving a sequence <span class="math-container">$\lambda_n$</span> with limit at</p>
<p><span class="math-conta... |
3,278 | <h3>What are Community Promotion Ads?</h3>
<p>Community Promotion Ads are community-vetted advertisements that will show up on the main site, in the right sidebar. The purpose of this question is the vetting process. Images of the advertisements are provided, and community voting will enable the advertisements to be s... | kuch nahi | 8,365 | <p><a href="http://www.sagemath.org/" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7CiYR.png" alt="Sage Math"></a></p>
|
3,180,965 | <p>Assume <span class="math-container">$f(x) \in L_1([a,b])$</span> and <span class="math-container">$x_0\in[a,b]$</span> is a point such that <span class="math-container">$f(x)\xrightarrow[x\to x_0]\ +\infty$</span>.</p>
<p>Is there always exists a function <span class="math-container">$g(x) \in L_1([a,b])$</span> su... | Amichai Lampert | 663,306 | <p>Hmmm... We can use the open mappimg theorem to conclude there must be an unbounded function h such that <span class="math-container">$h \cdot f \in L^1$</span> but I don't know how to guarantee the unbounded part is near <span class="math-container">$x_0$</span></p>
|
3,180,965 | <p>Assume <span class="math-container">$f(x) \in L_1([a,b])$</span> and <span class="math-container">$x_0\in[a,b]$</span> is a point such that <span class="math-container">$f(x)\xrightarrow[x\to x_0]\ +\infty$</span>.</p>
<p>Is there always exists a function <span class="math-container">$g(x) \in L_1([a,b])$</span> su... | Selene | 467,694 | <p>Yes there is . </p>
<p>WLOG <span class="math-container">$f\geq 0$</span>. Let <span class="math-container">$M_t=\{f\geq t\}$</span> and <span class="math-container">$\lambda(t)= \int_{M_t}fdx$</span>. Then</p>
<p>1) <span class="math-container">$\lambda(t)>0 $</span>. </p>
<p>2) <span class="math-container">$... |
2,495,918 | <p>A triangle is formed by the lines $x-2y-6=0$ , $ 3x−y+6=0$, $7x+4y−24=0$.</p>
<p>Find the equation of the line that bisects the inner angle of the triangle that is facing the side $7x+4y−24=0$.</p>
<p>I tried to find the intersect point of three equations by put them equal two by two. However, I don't know what to... | Lubin | 17,760 | <p>This has nothing to do with triangle. You have two lines $x-2y=6$ and $3x-y=6$, intersecting at the point $P=(6/5,-12/5)$, and want an equation for the line bisecting the angle, presumably with positive slope. My suggestion, perhaps not as good as that of @GAVD, is to take any other point $Q_1$ on the first line, fi... |
1,393,822 | <p>For a complex number $w$, or $a+bi$, is there a specific term for the value $w\overline{w}$, or $a^2+b^2$?</p>
| Mark Viola | 218,419 | <p>We have the inequalities</p>
<p><span class="math-container">$$\int_0^nx^{3/2}dx<\sum_{k=1}^n k^{3/2}<\int_0^{n+1}x^{3/2}dx$$</span></p>
<p>whereupon carrying out the integrals yields</p>
<p><span class="math-container">$$\frac25 n^{5/2}<\sum_{k=1}^n k^{3/2}<\frac25 (n+1)^{5/2}$$</span></p>
<p>The approx... |
368,789 | <p>Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this are using Nagell-Lutz and computing the number of points over $\mathbb{F}_{\ell}$ for various $\ell$. Are there othe... | Matt E | 221 | <p>For a fixed curve over $\mathbb Q$, the easiest way is to check Cremona's tables (!), since it is pretty unlikely that your curve has conductor big enough not to be there. </p>
<p>Sorry for the cheeky answer; here is another slightly more serious one:</p>
<p>I think that using the methods you suggest is pretty
st... |
46,236 | <p>Apologies for the uninformative title, this is a relatively specific question so it was hard to title. </p>
<p>I'm solving the following recurrence relation:</p>
<blockquote>
<p>$a_{n} + a_{n-1} - 6a_{n-2} = 0$<br>
With initial conditions $a_{0} = 3$
and $a_{1} = 1$</p>
</blockquote>
<p>And I have it mostly... | Zev Chonoles | 264 | <p>You can't multiply expressions with exponents like that. $2\times (2^n)$ is equal to $2^{n+1}$, not
$$4^n=\underbrace{4\times\cdots\times 4}_{n\text{ times}}=\underbrace{(2\times 2)\times\cdots\times (2\times 2)}_{n\text{ times}}=\underbrace{2\times\cdots\times 2}_{2n\text{ times}}=2^{2n}.$$</p>
<p>Also, the gener... |
65,631 | <pre><code>Ticker[comp_String] :=
Interpreter["Company"][comp] /. Entity[_, x_] :> x
ticks = Ticker /@ {"Apple", "Google"}
</code></pre>
<blockquote>
<p>{"NASDAQ:AAPL", "NASDAQ:GOOGL"}</p>
</blockquote>
<pre><code>DateListPlot[{
FinancialData[ticks[[1]], "CumulativeFractionalChange", {2010}],
FinancialDa... | kglr | 125 | <pre><code>stripF = ToExpression[ToString[#, StandardForm]] &;
stripF /@ {Style[1201/100000, FontFamily -> "Charter", FontSize -> 20],
Style[NumberForm[10.01, {10, 2}], FontFamily -> "Academy Engraved LET",
FontSize -> 50, FontColor -> RGBColor[0, 1, 0]]}
(* {1201/100000, 10.01} *)
</cod... |
2,165,759 | <p>I am solving the following question</p>
<p>$$\int\frac{\sin x}{\sin^{3}x + \cos^{3}x}dx.$$</p>
<p>I have been able to reduce it to the following form by diving numerator and denominator by $\cos^{3}x$ and then substituting $\tan x$ for $t$ and am getting the following equation. Should Iis there any other way use p... | MrYouMath | 262,304 | <p>Hint:</p>
<p>$\int \frac{t}{t^3+1} dx = \int \frac{t+t^2-t^2}{t^3+1} dx =\int \frac{t(t+1)}{t^3+1} dx-\frac{1}{3}\int \frac{3t^2}{t^3+1} dx=\int \frac{t(t+1)}{(t+1)(t^2-t+1)} dx-\frac{1}{3}\ln(t^3+1)$</p>
<p>$=\frac{1}{2}\int \frac{2t}{t^2-t+1} dx-\frac{1}{3}\ln(t^3+1)=\frac{1}{2}\int \frac{2t-1+1}{t^2-t+1} dx-\fr... |
794,912 | <p>I am reviewing Calculus III using <a href="http://www.jiblm.org/downloads/dlitem.aspx?id=82&category=jiblmjournal" rel="nofollow">Mahavier, W. Ted's material</a> and get stuck on one question in chapter 1. Here is the problem:</p>
<p>Assume $\vec{u},\vec{v}\in \mathbb{R}^3$. Find a vector $\vec{x}=(x,y,z)$ so t... | rogerl | 27,542 | <p>You get three equations in the three unknowns $x$, $y$, and $z$ from $x+y+z=1$, $\vec{x}\cdot\vec{u} = 0$, and $\vec{x}\cdot\vec{v} = 0$. If these three equations have a solution, that is the vector you are looking for. However, they do not always have a solution (for example, try $\vec{u} = (1,1,1)$ and $\vec{v} = ... |
221,712 | <p>I have two matrix <code>A</code> and <code>B</code> of equal dimensions see below. In <code>A</code> matrix I have the variables <code>a,b,c,d</code> which have direct correspondence with matrix <code>B</code> element by each row. In other words, for first row <code>{a, b, c, d}</code> we have <code>{2, 9, 6, 7}</co... | xzczd | 1,871 | <p>How about:</p>
<pre><code>MapThread[Block[{a, b, c, d}, # = #2; {a - d, b - a}] &, {A, B}]
(* {{-5, 7}, {1, -7}, {8, -11}} *)
</code></pre>
|
2,222,215 | <p>Determine whether the difference of the following two series is convergent or not and Prove your answer$$
\sum_{n=1}^\infty \frac{1}{n} $$ and $$\sum_{n=1}^\infty \frac{1}{2n-1} $$</p>
<p>What i tried. I said that the difference of the two series is divergent. My proof is as follows. Find the difference of the tw... | Ben Grossmann | 81,360 | <p>For a direct comparison test, note that
$$
\frac{(n-1)}{n(2n - 1)} \geq \frac{n-1}{n(2n)} = \frac{n-1}{2n^2}
$$
If the sum converged, we would have
$$
\sum_{n=1}^\infty \frac{n-1}{2n^2} =
\sum_{n=1}^\infty \frac{n}{2n^2} -
\sum_{n=1}^\infty \frac{1}{2n^2}
$$</p>
|
2,555,815 | <p><strong>Problem</strong></p>
<p>Let $a_{0}(n) = \frac{2n-1}{2n}$ and $a_{k+1}(n) = \frac{a_{k}(n)}{a_{k}(n+2^k)}$ for $k \geq 0.$</p>
<p>The first several terms in the series $a_k(1)$ for $k \geq 0$ are:</p>
<p>$$\frac{1}{2}, \, \frac{1/2}{3/4}, \, \frac{\frac{1}{2}/\frac{3}{4}}{\frac{5}{6}/\frac{7}{8}}, \, \frac... | Community | -1 | <p>This is not a full solution, either, just a remark following @Kelenner's trail of thought:<br>
$\displaystyle \sum_{q\geq 0}\frac{(-1)^{s_2(q)}}{n+q}$ is convergent, where $s_2(n)$ is the sum of 1-bits in the binary representation of $n$.<br>
<strong>Proof:</strong> Let $b_n=(-1)^{s_2(n)}$. According to Dirichlet's... |
1,768,317 | <p>Show that $\sin(x) > \ln(x+1)$ when $x \in (0,1)$. </p>
<p>I'm expected to use the maclaurin series (taylor series when a=0)</p>
<p>So if i understand it correctly I need to show that: </p>
<p>$$\sin(x) = \lim\limits_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{(-1)^{k-1}}{(2k-1)!} \cdot x^{2k-1} > \lim\limi... | xpaul | 66,420 | <p>Let $f(x)=\sin x-\ln(x+1)$. we try to show that $f(x)$ is increasing. In fact
$$ f'(x)=\cos x-\frac{1}{x+1}=\frac{(x+1)\cos x-1}{x+1}. $$
Now we show $(x+1)\cos x>1$ or $\sec x-1<x$ for $0<x<1$.
Note
\begin{eqnarray}
\sec x-1=\frac{1-\cos x}{\cos x}=\frac{2\sin^2\frac{x}{2}}{1-2\sin^2\frac{x}{2}}.
\end{e... |
569,103 | <blockquote>
<blockquote>
<p>How can I calculate the first partial derivative $P_{x_i}$ and the second partial derivative $P_{x_i x_i}$ of function:
$$
P(x,y):=\frac{1-\Vert x\rVert^2}{\Vert x-y\rVert^n}, x\in B_1(0)\subset\mathbb{R}^n,y\in S_1(0)?
$$</p>
</blockquote>
</blockquote>
<p>I ask this with reg... | Paramanand Singh | 72,031 | <p>While I am still searching for a simple solution based on LHR, I found that method of Taylor series can also be applied without much difficulty. However we will need to make the substitution $\tan x = t$ so that $\sec x = \sqrt{1 + t^{2}}$ and as $x \to 0$ we also have $t \to 0$. We can do some simplification as fol... |
2,027,044 | <p>Prove:
$$
(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}, \qquad \forall n \in \mathbb{N}
$$
I have have tried using the triangle inequlity $ |a + b| \le |a| + |b| $, without any success.</p>
| Community | -1 | <p><strong>Hint</strong>: You can show (equivalent) that $x\mapsto \sqrt[n]{x}$ is subadditive.</p>
|
1,023,193 | <p>Proving this formula
$$
\pi^{2}
=\sum_{n\ =\ 0}^{\infty}\left[\,{1 \over \left(\,2n + 1 + a/3\,\right)^{2}}
+{1 \over \left(\, 2n + 1 - a/3\,\right)^{2}}\,\right]
$$
if $a$ an even integer number so that
$$
a \geq 4\quad\mbox{and}\quad{\rm gcd}\left(\,a,3\,\right) = 1
$$</p>
| Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcomma... |
917,276 | <p>If $U$ and $V$ are independent identically distributed standard normal, what is the distribution of their difference?</p>
<p>I will present my answer here. I am hoping to know if I am right or wrong.</p>
<p>Using the method of moment generating functions, we have</p>
<p>\begin{align*}
M_{U-V}(t)&=E\left[e^{t... | Alex | 38,873 | <p>With the convolution formula:
<span class="math-container">\begin{align}
f_{Z}(z) &= \frac{dF_Z(z)}{dz} = P'(Z<z)_z = P'(X<Y+z)_z = (\int_{-\infty}^{\infty}\Phi_{X}(y+z)\varphi_Y(y)dy)_z \\
&= \frac{1}{2 \pi}\int_{-\infty}^{\infty}e^{-\frac{(z+y)^2}{2}}e^{-\frac{y^2}{2}}dy = \frac{1}{2 \pi}\int_{-\inft... |
1,005,291 | <p>I understand that in order to prove this to be one to one, I need to prove $2$ numbers, $a$ and $b$, in the same set are equal. </p>
<p>This is what I did:</p>
<p>$$\sqrt{a} + a + 2 = \sqrt{b} + b + 2$$
$$\sqrt{a} + a = \sqrt{b} + b$$
$$a + a^2 = b + b^2$$</p>
<p>How would I arrive at $a = b$? Is it possible?</p>... | Kim Jong Un | 136,641 | <p>You were basically done:
$$
\sqrt{a}+a=\sqrt{b}+b\implies 0=(\sqrt{a}-\sqrt{b})(\sqrt{a}+\sqrt{b}+1)\implies \sqrt{a}=\sqrt{b}\implies a=b.
$$</p>
|
1,640,383 | <p>We have function $f:\mathbb{R}\rightarrow \mathbb{R}$ with $$f\left(x\right)=\frac{1}{3x+1}\:$$ $$x\in \left(-\frac{1}{3},\infty \right)$$
Write the Maclaurin series for this function.</p>
<p>Alright so from what I learned in class, the Maclaurin series is basically the Taylor series for when we have $x_o=0$ and we... | zz20s | 213,842 | <p>HINT: Use the fact that $\displaystyle \sum_\limits{n=0}^{\infty} x^n=\frac{1}{1-x}$ for $|x|<1$.</p>
|
936,611 | <p>let $p,q$ is postive integer,and such
$$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$</p>
<p>Find the minimum of the $q$</p>
<p>maybe can use
$$95q>36p$$
and $$37p>96q$$
and then find this minimum of the value?</p>
<p>before I find a
$$2.638\approx \dfrac{95}{36}>\dfrac{49}{18}\approx 2.722>\df... | mathlove | 78,967 | <p>We have
$$2.64\gt a=\frac{17575}{5\cdot 36\cdot 37}=\frac{95}{36}\gt \color{red}{\frac{13}{5}}=2.6=\frac{17316}{5\cdot 36\cdot 37}\gt\frac{96}{37}=\frac{17280}{5\cdot 36\cdot 37}=b\gt 2.59.$$
Note that
$$\frac{11}{4}\gt a\gt b\gt\frac{10}{4}$$</p>
<p>$$\frac{8}{3}\gt a\gt b\gt\frac{7}{3}$$</p>
<p>$$\frac{6}{2}\gt... |
936,611 | <p>let $p,q$ is postive integer,and such
$$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$</p>
<p>Find the minimum of the $q$</p>
<p>maybe can use
$$95q>36p$$
and $$37p>96q$$
and then find this minimum of the value?</p>
<p>before I find a
$$2.638\approx \dfrac{95}{36}>\dfrac{49}{18}\approx 2.722>\df... | Jack D'Aurizio | 44,121 | <p>An interesting trick so solve such kind of problems is to consider the continued fraction of the LHS and the RHS. We have:
$$\frac{95}{36}=[2;1,1,1,3,3],\qquad \frac{96}{37}=[2;1,1,2,7]$$
hence
$$\frac{13}{5}=[2;1,1,2]$$
just lies between the LHS and the RHS, and it is the rational number with the smallest denominat... |
4,379,693 | <p>How to change the order of integration:</p>
<p><span class="math-container">$$\int_{-1}^1dx \int_{1-x^2}^{2-x^2}f(x,y)dy$$</span></p>
<p>I tried to sketch the area and got:</p>
<p><a href="https://i.stack.imgur.com/xxu5T.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xxu5T.png" alt="enter image d... | callculus42 | 144,421 | <p>I simplify the term as far I can think of</p>
<p><span class="math-container">$$\frac{1}{n^2}Var[\sum_{i=1}^n (X_i-E[X])^2]$$</span></p>
<p>Next you multiply out the square brackets and leaving variance opterator still out of the brackets.</p>
<p><span class="math-container">$$\frac{1}{n^2}Var[\sum_{i=1}^n (X_i^2-2... |
233,915 | <p>a,b are elements of the group G</p>
<p>I have no idea how to even start - I was thinking of defining a,b as two square matrices and using the non-commutative property of matrix multiplication but I'm not sure if that's the way to go...</p>
| Andreas Blass | 48,510 | <p>Since you want this to be in a group, take the group of permutations of $\{1,2,3\}$ and take the counterexample $a$ and $b$ to be two distinct transpositions, say $(1,2)$ and $(2,3)$.</p>
|
3,375,181 | <p>How do I graph f(x)=1/(1+e^(1/x)) except for replacing variable x with numbers?
Besides, I get the picture of the answer online <a href="https://i.stack.imgur.com/gZb7X.png" rel="nofollow noreferrer">enter image description here</a> and do not understand why x = 0 exists on this graph.</p>
| Cesareo | 397,348 | <p>Hint.</p>
<p>From</p>
<p><span class="math-container">$$
y y''-2(y')^2=y^2\Rightarrow \frac{y''}{y}-2\left(\frac{y'}{y}\right)^2=1
$$</span></p>
<p>but</p>
<p><span class="math-container">$$
\left(\frac{y'}{y}\right)' = -\left(\frac{y'}{y}\right)^2+\frac{y''}{y}
$$</span></p>
<p>then</p>
<p><span class="math-c... |
902,313 | <p>The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." </p>
<p>I thought any topological space is the union of its connected components? Why is this singled out here for clopen sets?</p>
<p>Does it have something to do with it $x\in C$ a clopen subset... | Hamou | 165,000 | <p>Here is what I mean: If $C$ is clopen subset of a space $X$, then $C=\cup_{x\in C}C_x$, where $C_x$ is the connected component in $X$ containing $x$.<br>
It is clear that $C\subset \cup_{x\in C}C_x$.<br>
Let $x\in C$, and we want to show that $C_x\subset C$. Now we take $A=C_x\cap C$ and $B=C_x\cap C^c$, we have $A... |
459,428 | <p>How does one evaluate a function in the form of
$$\int \ln^nx\space dx$$
My trusty friend Wolfram Alpha is blabbering about $\Gamma$ functions and I am having trouble following. Is there a method for indefinitely integrating such and expression? Or if there isn't a method how would you tackle the problem?</p>
| OR. | 26,489 | <p>Write $x=e^y$, and $\text{d}x=e^y\text{d}y$, and integrate by parts a few times $$\int \ln^n(x)\text{d}x\text=\int y^{n}e^{y}\text{d}y=y^ne^y-n\int y^{n-1}e^y\text{d}y.$$ </p>
|
4,298,951 | <p>Let us define a sequence <span class="math-container">$(a_n)$</span> as follows:</p>
<p><span class="math-container">$$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$</span></p>
<p>Prove that the sequence <span class="math-container">$(a_n)$</span> is Cauchy and find the limit.</p>
<hr />
<... | Pedro Ignacio Martinez Bruera | 616,313 | <p>For the limit I would treat it a second-order difference equation:
<span class="math-container">$$
4a_{n} -3a_{n-1} - a_{n-2}=0
$$</span>
Conjecture the solution is <span class="math-container">$a_{n}=C_{1}b^{n}$</span>. Then <span class="math-container">$b=1$</span> or <span class="math-container">$b=\frac{1}{4}$</... |
4,298,951 | <p>Let us define a sequence <span class="math-container">$(a_n)$</span> as follows:</p>
<p><span class="math-container">$$a_1 = 1, a_2 = 2 \text{ and } a_{n} = \frac14 a_{n-2} + \frac34 a_{n-1}$$</span></p>
<p>Prove that the sequence <span class="math-container">$(a_n)$</span> is Cauchy and find the limit.</p>
<hr />
<... | xpaul | 66,420 | <p>Using
<span class="math-container">$$\displaystyle a_{n}-a_{n-1}=-\frac{1}{4}\left(a_{n-1}-a_{n-2}\right)$$</span>
one has, for <span class="math-container">$\forall n, p\in\mathbb{N}$</span>,
<span class="math-container">$$ |a_{n+p}-a_n|=|\sum_{k=0}^{p-1}(a_{n+k+1}-a_{n+k})|\le\sum_{k=0}^{p-1}\bigg(\frac{1}{4}\bigg... |
355,888 | <p>Consider
$x''-2x'+x= te^t$</p>
<p>Determine the solution with initial values $x(1) = e,$ $x'(1) = 0.$</p>
<p>I know this looks like and probably is a very easy question, but i'm not getting the right answer when i try and solve putting into quadratic form. Could someone please demonstrate or show me a different m... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
407,890 | <p>Here comes a second sophisticated version of my conjecture, as critics came up the <a href="https://math.stackexchange.com/questions/407812/conjecture-on-combinate-of-positive-integers-in-terms-of-primes">first version</a> was trivial.</p>
<p>Teorem <a href="https://oeis.org/A226233" rel="nofollow noreferrer">2</a>... | Ross Millikan | 1,827 | <p>I don't know why you subscript $a,b$ with $u$ or where $u$ comes from. If you say $p^m$ is the highest power of $p$ that divides $n$ this is essentially the Euclidean division algorithm. $b_u$ is the remainder term, here in the range $[1,p-1]$. Since you do not say $p^m$ is the largest power of $p$ dividing $n$, w... |
1,458,579 | <p>Suppose $f_n$ and $g_n$ be two sequences of functions. Also, $f_n.g_n$ converges to $f.g$ and $g_n$ converges to $g$. Can we prove $f_n$ converges to $f$? How?</p>
| vadim123 | 73,324 | <p>Forget about every collection except the second. It starts with a bit, say (without loss of generality) a zero bit. Now, you generate additional random bits and stop when you get a one. There is a $p$ probability that the very next bit is a $1$, so your second collection has size $1$. There is a $(1-p)p$ probabi... |
197,441 | <p>I have a list,</p>
<pre><code>l1 = {{a, b, 3, c}, {e, f, 5, k}, {n, k, 12, m}, {s, t, 1, y}}
</code></pre>
<p>and want to apply differences on the third parts and keep the parts right of the numerals collected.</p>
<p>My result should be</p>
<pre><code>l2 = {{2, c, k}, {7, k, m}, {-11, m, y}}
</code></pre>
<p>I... | Michael E2 | 4,999 | <p>Perhaps this?:</p>
<pre><code>l1 = {{a, b, 3, c}, {e, f, 5, k}, {n, k, 12, m}, {s, t, 1, y}};
l2 = Differences[l1[[All, 3 ;;]]] /. b_ - a_ :> Sequence[a, b]
(* {{2, c, k}, {7, k, m}, {-11, m, y}} *)
</code></pre>
<p>It assumes the letter symbols are simple and not complicated expressions.</p>
<p>This is mor... |
65,912 | <p>How do I show that $s=\sum\limits_{-\infty}^{\infty} {1\over (x-n)^2}$ on $x\not\in \mathbb Z$ is differentiable without using its compact form? I realize that the sequence of sums $s_a=\sum\limits_{-a}^{a} {1\over (x-n)^2}$ is not uniformly convergent. </p>
<p>I also tried to prove that it is continuous by using t... | Zarrax | 3,035 | <p>There's a theorem they teach in undergraduate analysis classes which says that if $\{f_n(x)\}$ are $C^1$ functions on an interval $[a,b]$ such that $|f_n(x)| \leq M_n$ and
$|f_n'(x)| \leq N_n$ where $\sum_n M_n$ and $\sum_n N_n$ are both finite, then $\sum_n f_n(x)$ is a differentiable function whose derivative... |
1,923,034 | <p>A bagel store sells six different kinds of bagels. Suppose you choose 15 bagels at random. What is the probability that your choice contains at least one bagel of each kind? If one of the bagels is Sesame, what is the probability that your choice contains at least three Sesame bagels?</p>
<p>My approach to the firs... | Jack D'Aurizio | 44,121 | <p>The first probability is given by
$$ \frac{{15 \brace 6}\cdot 6!}{6^{15}}\approx \color{red}{64,\!42\%} $$
since a working choice is associated with a bijective function from $[1,15]$ to $[1,6]$. ${15\brace 6}$ is a <a href="https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind" rel="nofollow">Stirling n... |
116,537 | <p>Let's say that I have</p>
<pre><code>x^2+x
</code></pre>
<p>Is there a way to map $x$ to the first derivative of a function and $x^2$ to the second derivative of the same function? According to <a href="http://reference.wolfram.com/language/ref/Slot.html" rel="nofollow">http://reference.wolfram.com/language/ref/Sl... | Kuba | 5,478 | <pre><code>{x, x^2, x^2 + x} /. x^n_. :> Derivative[n, 0][a][y, z]
</code></pre>
|
3,088,766 | <p>I need to prove that the premise <span class="math-container">$A \to (B \vee C)$</span> leads to the conclusion <span class="math-container">$(A \to B) \vee (A \to C)$</span>. Here's what I have so far.</p>
<p><a href="https://i.stack.imgur.com/1AgTZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com... | Bram28 | 256,001 | <p>It is always a <em>very bad</em> sign when someone has started a bunch of subproofs without indicating what happens at the end of the subproof.</p>
<p>A proof should always have a <em>plan</em> or <em>outline</em>, and subproofs provide the skeleton to do so. But again, you need to indicate what you want to do with... |
3,325,340 | <p>Show that <span class="math-container">$$ \lim\limits_{(x,y)\to(0,0)}\dfrac{x^2y^2}{x^2+y^2}=0$$</span>
My try:
We know that, <span class="math-container">$$ x^2\leq x^2+y^2 \implies x^2y^2\leq (x^2+y^2)y^2 \implies x^2y^2\leq (x^2+y^2)^2$$</span>
Then, <span class="math-container">$$\dfrac{x^2y^2}{x^2+y^2}\leq x^2+... | Dr. Sonnhard Graubner | 175,066 | <p>Or alternatively, by AM-GM we get
<span class="math-container">$${x^2+y^2}\geq 2|xy|$$</span> so
<span class="math-container">$$\frac{x^2y^2}{x^2+y^2}\le \frac{x^2y^2}{2|xy|}=\frac{1}{2}|xy|$$</span> and this tends to zero if <span class="math-container">$x,y$</span> tend to zero.</p>
|
200,931 | <p>I want to generate a layered drawing of the <a href="https://en.wikipedia.org/wiki/Hoffman%E2%80%93Singleton_graph" rel="noreferrer">Hoffman–Singlelton graph</a>. As an example of what I want, here is a layered drawing of the Petersen graph:</p>
<p><a href="https://i.stack.imgur.com/doEt5.png" rel="noreferrer"><img... | kglr | 125 | <ol>
<li>Rescale the vertex coordinates given by <code>"LayeredEmbedding"</code> to
run form 0 to 1 in each dimension,</li>
<li><code>Pick</code> the edges with vertices in the right-most layer by checking if
both vertices have first coordinate equal to <code>1.</code>,</li>
<li>Use a slightly modified version of the b... |
351,642 | <p>So I'm proving that a group $G$ with order $112=2^4 \cdot 7$ is not simple. And I'm trying to do this in extreme detail :) </p>
<p>So, assume simple and reach contradiction. I've reached the point where I can conclude that $n_7=8$ and $n_2=7$. </p>
<p>I let $P, Q\in \mathrm{Syl}_2(G)$ and now dealing with cases th... | Dalimil Mazáč | 59,757 | <p>Sylow's theorems require that $n_2=1$ or 7, $n_7=1$ or 8, so $G$ has a chance to be simple only if $n_2=7$ and $n_7=8$. Note that the Sylow 7-subgroups can only intersect at the identity, any two Sylow 2-subgroups can share a subgroup of order at most 8, and a Sylow 7-subgroup and a Sylow 2-subgroup can share only t... |
625,821 | <p>$$\int^\infty_0\frac{1}{x^3+1}\,dx$$</p>
<p>The answer is $\frac{2\pi}{3\sqrt{3}}$.</p>
<p>How can I evaluate this integral?</p>
| lsp | 64,509 | <p>$$x^3+1 = (x+1)(x^2-x+1)$$
<strong>Logic:</strong> Do partial fraction decomposition.Find $A,B,C$.</p>
<p>$$\frac{1}{x^3+1} = \frac{A}{x+1}+\frac{Bx+C}{x^2-x+1}$$
By comparing corresponding co-efficients of different powers of $x$, you will end up with equations in A,B,C.After solving you get :
$$A=\frac{1}{3},B=\f... |
137,755 | <p>Suppose that $X$ is a scheme and $x\in X$ is a point. The stalk of $X$ at $x$ is a (local) ring and we can form its spectrum $Y_x=\rm{Spec}(\mathcal{O}_{X,x})$.</p>
<p>There is a canonical map $Y_x\to X$. We can define it by fixing an affine neighborhood $x\in U\cong \rm{Spec}(R)$, making $x$ as a prime ideal in $R... | Matthieu Romagny | 17,988 | <p>In EGA1, 2.4 this is called the <em>local scheme of $X$ at $x$</em>.</p>
|
219,014 | <p>I have the list</p>
<pre><code>t1 = {{-1, 0}, {-2, 0}, {-3, 0}, {0, 0}, {-2, 0}, {1, 1}}
</code></pre>
<p>How do I find the position where an element repeats? In this case it would be element <code>{-2, 0}</code> at position 5, because <code>{-2, 0}</code> first came up at postion 2. So the answer would be 5.</p>
... | MikeY | 47,314 | <p>This sort of question comes up, where the range of <code>K</code> is open-ended. If you are dealing with a generalized <code>g</code> function where you don't already know the answer, Mathematica won't find it for you straightaway (hopefully someone will correct me on this). A method I use is to generate answers wit... |
1,238,292 | <p>This is a homework problem, so please do not give more than hints. I must convert
\begin{align}
\int_0^\sqrt{2}\int_x^\sqrt{4-x^2}\sin\left(x^2+y^2\right)\:dy\:dx\tag{1}
\end{align}
to polar coordinates. This is my attempt:
\begin{align}
\int_{\pi/4}^{\pi/2}\int_{\color{red}{2\cos\left(\theta\right)}}^{\color{red}{2... | E.H.E | 187,799 | <p>firstly, you must sketch the region
<img src="https://i.stack.imgur.com/rmYuQ.png" alt="enter image description here">
$$\int_{\pi /4}^{\pi /2}\int_{0}^{2}\sin(r^2)rdrd\theta $$</p>
|
2,312,968 | <p>If $t=\ln(x)$, $y$ some function of $x$, and $\dfrac{dy}{dx}=e^{-t}\dfrac{dy}{dt}$, why would the second derivative of $y$ with respect to $x$ be:
$$-e^{-t}\frac{dt}{dx}\frac {dy}{dt} + e^{-t}\frac{d^2y}{dt^2}\frac{dt}{dx}?$$</p>
<p>I know this links into the chain rule. I don't have a good intuition for why the fi... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>from the graph, we derive that</p>
<p>$$f (0)=f (4\frac \pi 3)=0$$</p>
<p>thus</p>
<p>$$\sin (-k)+c=\sin (4\frac \pi 3-k)+c=0$$</p>
<p>from here,
$$-k=\pi-(4\frac \pi 3-k) $$</p>
<p>You can finish.</p>
|
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