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 |
|---|---|---|---|---|
1,063 | <p><strong>Problem</strong></p>
<p>Using Mathematica's <code>Solve</code> operator can sometimes lead to an output involving a positive and negative root (say when solving for a variable such as <code>N^2</code>.</p>
<p><strong>What Is Happening</strong></p>
<p>Here I simply have a defined function <code>Radial</cod... | rm -rf | 5 | <p>One way is to use <a href="http://reference.wolfram.com/mathematica/ref/Refine.html"><code>Refine</code></a> to filter out only the positive root. For example:</p>
<pre><code>assume = Z > 0 && a > 0 && n > 0;
int = Integrate[n^2*Radial[1, 0, r]*r^2, {r, 0, ∞}, Assumptions -> assume];
sol... |
3,720,635 | <p><strong>Question:</strong></p>
<blockquote>
<p>Prove that the series <span class="math-container">$$\sum_{n=1}^\infty \sin\left(\frac{x}{n^4}\right)\cos(nx)$$</span> converges absolutely, and it is continuous on <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p><strong>Attempt:</strong> I can re... | RRL | 148,510 | <p>It is well known that <span class="math-container">$(1 + 1/n)^n \nearrow e$</span> and <span class="math-container">$(1+1/n)^{n+1} \searrow e.$</span> See one of the many proofs on this site <a href="https://math.stackexchange.com/q/2071492/148510">here</a>.</p>
<p>Thus, <span class="math-container">$M_n = \left(\fr... |
112,660 | <p>Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma</p>
| Francois Ziegler | 19,276 | <p>The recent book of Malle and Testerman, <a href="http://dx.doi.org/10.1017/CBO9780511994777" rel="nofollow">Linear algebraic groups and finite groups of Lie type</a>, has several chapters on the subject. From the <a href="http://ams.org/mathscinet-getitem?mr=2850737" rel="nofollow">MR review</a>:</p>
<blockquote>
... |
2,688,372 | <blockquote>
<p>Specify a model in which the sentence is true and another model in
which it is false. The domain of the model must be {1,2,3}.</p>
<p><em>$ \exists y \forall x ((F(x) \iff x = y) $</em></p>
</blockquote>
<p>I want to confirm my understanding for this problem to see if I have it right. So this ... | Graham Kemp | 135,106 | <blockquote>
<p>Would letting y be a natural number, x be an integer and F(x) be positive work as a sentence being true? I believe it would read:</p>
</blockquote>
<p>No, not in any way. You can not declare types. $x,y$ are unrestricted in the statement, so they belong to the <em>same</em> implicit dom... |
218,340 | <p>There is a Landau's theorem related to <a href="http://en.wikipedia.org/wiki/Tournament_%28graph_theory%29" rel="nofollow">tournaments theory</a>.
Looks like the sequence $(0, 1, 3, 3, 3)$ satisfies all three conditions from the theorem, but it is not possible for 5 people to play tournament in such a way (if there ... | Community | -1 | <p>Player $A$ loses to everyone.</p>
<p>Player $B$ beats player $A$ and loses to everyone else.</p>
<p>Players $C$, $D$, and $E$ beat each other cyclically, like rock-paper-scissors.</p>
|
1,598,947 | <p>Let $\{\xi_k\}_{k=1}^4$ be a set of vectors in $\mathbb{R}^3$. If $\{\xi_1, \xi_2\}$ and $\{\xi_2, \xi_3, \xi_4\}$ are independent sets, and $\xi_1$ belongs to the span of $\{\xi_2, \xi_4\}$. Show that $\{\xi_k\}_{k=1}^3$ is linearly independent. </p>
<p>Clearly, $\xi_1 = a_1\xi_2 + a_2\xi_4$ where $a_1\neq0$ due t... | egreg | 62,967 | <blockquote>
<p>If $\{v_1,\dots,v_k\}$ is linearly independent in a vector space $V$ and $v\in V$, then $\{v_1,\dots,v_k,v\}$ is linearly independent if and only if $v$ doesn't belong to the span of $\{v_1,\dots,v_k\}$.</p>
</blockquote>
<p>One direction is clear: if $v$ belongs to the span of $\{v_1,\dots,v_k\}$, t... |
2,767,471 | <p>You enter a metro station in a big hurry, and decide to take the first train that arrives. </p>
<p>There are two lines running through this station: one runs every five minutes (line A), the other every three (line B). To be precise, suppose the next arrival of the A train is uniformly distributed on the interval [... | Aatsrh | 528,277 | <p>As Karn said, Using <span class="math-container">$P(T>t)=P(A>t,B>t)=P(A>t)\times P(B>t)$</span>, we get <span class="math-container">$P(T>t)=\frac{(5-t)\times(3-t)}{15}$</span>. Hence we can find <span class="math-container">$P(T<=t)$</span> and thus the distribution of T. We can then easily cal... |
2,914,100 | <p>A magic square of size <span class="math-container">$N,N ≥ 2$</span>, is an <span class="math-container">$N ×N$</span> matrix with integer entries such that the sums of the entries of each row, each column and the two diagonals are all equal. If the entries of the magic square are made up of integers in arithmetic p... | B. Goddard | 362,009 | <p>You can add up all the entries in the square. </p>
<p>$$a+(a+d)+ \cdots + a+(n^2-1)d = S.$$</p>
<p>If the sum of the entries in each column is $C$ then you have </p>
<p>$$NC = S.$$</p>
|
853,031 | <p>I was trying to show how to find $\pi$ value from formula $\pi R^2$, but I don't understand where is my mistake. </p>
<p>So I am calculating area using $n$ triangles <a href="https://i.stack.imgur.com/p5vzJ.jpg" rel="nofollow noreferrer">1</a></p>
<p>let $R=1$, then one triangle area is $1\cdot 1\cdot \dfrac{\sin\... | mookid | 131,738 | <p><strong>Hint:</strong> consider a rotation such as
$$
R(1,1,1,1) = (0,0,0,1)
$$</p>
|
2,584,044 | <p>I'm learning about how use mathematical induction.
I'm tasked with proving the inequality shown in (1). It is a requirement that I use mathematical induction for the proof.</p>
<p>$(1) \quad P(n):\quad 2n+1 < 2^{n}, \quad n \ge 3$</p>
<p>I would like some feedback regarding whether my proof is valid and if my u... | ArsenBerk | 505,611 | <p>I want to suggest an alternative way just to be more systematic:</p>
<p>Let $a(n) = 2n+1$ and $b(n)= 2^n$ where $n \ge 3$. Then for $n=3$, we have $7 = a(3) < b(3) = 8$. Then assume inductively that $a(n) < b(n)$ and $n > 4$. Then, for $n+1$, we have $$a(n+1) = 2n+3 = 2n+1+2 = a(n)+2 < b(n)+2$$
by indu... |
5,563 | <p>This question applies to any package, but I encountered this problem while working with graphs.
There are symbols in the <code>Combinatorica</code> package (such as <code>Graph</code>, <code>IncidenceMatrix</code>, <code>EdgeStyle</code>, and others) that have the same name as analogous symbols in <code>System</code... | Helium | 573 | <p>One way is to use something like <code>Graph=System`Graph;</code> and use <code>Graph</code> which refers to <code>System`Graph</code> thereafter. You can use any other name such as <code>g=System`Graph</code> as well. The downside is that you have to do it for any function.</p>
<p>Edit:
This does the trick:</p>
<... |
85,165 | <p>I have two lists </p>
<pre><code>X = {1, 2, 3};
Y = {5, 6, 7, 8};
</code></pre>
<p>I want to apply function <code>g[x,y_,z_]</code> to all pairs from X*Y, so I need to get a list <code>{g[x,1,5],g[x,1,6]…,g[x,3,8]}</code></p>
<p>I came up with this syntax</p>
<pre><code>g[x, ##] &@(Sequence @@ #) & /@ Tu... | kglr | 125 | <pre><code>Distribute[g[x, X, Y], List]
</code></pre>
<blockquote>
<p>{g[x, 1, 5], g[x, 1, 6], g[x, 1, 7], g[x, 1, 8],<br>
g[x, 2, 5], g[x, 2, 6], g[x, 2, 7], g[x, 2, 8],<br>
g[x, 3, 5], g[x, 3, 6], g[x, 3, 7], g[x, 3, 8]}</p>
</blockquote>
|
392,608 | <p>I know this is a very basic question but I need some help.</p>
<p>I have to find the second derivative of: </p>
<p>$$\frac{1}{3x^2 + 4}$$</p>
<p>I start by using the Quotient Rule and get the first derivative to be:</p>
<p>$$\frac{-6x}{(3x^2 + 4)^2}$$</p>
<p>This I believe to be correct.
Following that I procee... | Stahl | 62,500 | <p>When you perform the quotient rule, it's often easier to <em>not</em> multiply everything out until the end, because there are a lot of cases where you can factor things out and cancel, and if you multiply out first, it will be much harder to see that. Your first derivative is indeed correct, but here's what I'd rec... |
2,023,222 | <p>I am facing difficulty with the following limit.</p>
<p><span class="math-container">$$ \lim_{n\to\infty}\left(\binom{n}{0}\binom{n}{1}\dots\binom{n}{n}\right)^{\frac{1}{n(n+1)}} $$</span></p>
<p>I tried to take log both sides but I could not simplify the resulting expression.</p>
<p>Please help in this regard. Than... | ho boon suan | 436,996 | <h4>Warning: The following argument is not rigorous.</h4>
<p>We can rewrite the identity as
<span class="math-container">$$
\lim_{n\to\infty}
\Biggl({n\choose1}\dots{n\choose n}\Biggr)^{1\over1+\dots+n}
= e.
$$</span>
This says that, for large <span class="math-container">$n$</span>, we have
<span class="math-container... |
3,603,170 | <p>Let's say we have an open disk, and we add half its boundary to it. That is, if this disk is centered at the origin, then we have a semi-circular arc around it, starting from <span class="math-container">$(-1,0)$</span> and going to <span class="math-container">$(1,0)$</span>. Let's call this shape <span class="math... | MPW | 113,214 | <p>You can remove a closed arc (the one you added) from <span class="math-container">$S$</span> and have it remain connected and simply connected, but that’s not true for the open disk.</p>
|
4,177,904 | <p>Let <span class="math-container">$\alpha: x-y+2z-2=0$</span> and <span class="math-container">$\beta: x-2y-2z+3=0$</span> be two planes in <span class="math-container">$\mathbb{R}^3$</span>. I am asked to find a line <span class="math-container">$d_1\subset \alpha$</span> such that <span class="math-container">$d_1$... | Mr.xue | 714,066 | <p>We will prove that <span class="math-container">$B$</span> is coercive in <span class="math-container">$H^1_{\Gamma_1}\times H^1_{\Gamma_1}$</span>. We argue by contradiction. If <span class="math-container">$B$</span> is not coercive, then there would exist for each integer <span class="math-container">$k=2,\cdots ... |
1,808,206 | <p>how can I find the splitting field of polynomial $x^{13}+1$ over $GF(2)$?</p>
| Crostul | 160,300 | <p>Recall that:</p>
<ol>
<li><p>Every irreducible polynomial over a finite field is separable</p></li>
<li><p>The field $GF(p^n)$ is the splitting field of $x^{p^n-1}-1$ over $GF(p)$.</p></li>
</ol>
<p>In particular, for any separable polynomial $f(x) \in GF(p)[x]$ you have
$$f(x) \mbox{ splits on } GF(p^n) \Leftrigh... |
1,417,404 | <p>The following came up in my solution to <a href="https://math.stackexchange.com/questions/1410565/can-this-congruence-be-simplified/1410579#1410579">this question</a>, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation
$$ (x+y)(x+y+1) - kxy = 0$$
For $k=5$ and $... | asomog | 183,714 | <p>What I got so far:
First, solve for $x$:
$$
x=\frac{k y-2 y-1\pm\sqrt{k^2 y^2-4 k y^2-2 k y+1}}{2}
$$
$x$ is only an integer, if the discriminant is a perfect square, so:
$$
k^2 y^2-4 k y^2-2 k y+1=z^2
$$
Solving this for $y$:
$$
y=\frac{k\pm\sqrt{k^2 z^2-4 k z^2+4 k}}{k^2-4 k}
$$
This can only happen if:
$$
k^2 z^... |
123,706 | <p>I have an integral that needs to be evaluated using NIntegrate:
$$\int_0^{10}\mathrm{d}x_1\int_0^{10}\mathrm{d}x_2\left(1+(x_2-x_1)^2\right)^{-1/6}\mathrm{e}^{-\int_0^{x_1}\mathrm{d}x'\int_0^{x_1}\mathrm{d}x''f(x'-x'')-\int_0^{x_2}\mathrm{d}x'\int_0^{x_2}\mathrm{d}x''f(x'-x'')}\times\int_0^{x_1}\mathrm{d}x'\int_0^{x... | Bob Hanlon | 9,362 | <pre><code>f[x_] = (1 + x^2)^(-1/3);
</code></pre>
<p>Use <code>Integrate</code> rather than <code>NIntegrate</code> for <code>i1</code> and <code>i2</code>.</p>
<pre><code>Clear[i, i1, i2]
i[x_] = Assuming[{x > 0},
Integrate[f[z1 - z2], {z1, 0, x}, {z2, 0, x}]]
(* 1/2 (3 - 3 (1 + x^2)^(2/3) + 4 x^2 Hypergeom... |
2,942,681 | <p>In an ordinary function like the temperature—one of the properties of the minimum is that if we go away from the minimum in the first order, the deviation of the function from its minimum value is only second order.</p>
<p>At any place else on the curve, if we move a small distance the value of the function changes... | Ethan Bolker | 72,858 | <p>Here's a geometric reason. The first approximation to a differentiable function at a point follows the tangent line instead of the curve as you move away from the point. At a minimum or maximum the tangent is horizontal. The first approximation is <span class="math-container">$0$</span> change; you have to look at t... |
425,189 | <p>I have a very brief question: if I put $M$ balls into $N$ boxes at random, what is the average number of balls in the boxes that are <strong>not</strong> empty?</p>
| Aang | 33,989 | <p>Let $X$ denotes the number of non-empty boxes.</p>
<p>Then $P(X=r)={N\choose r}\left(\frac{1}{2}\right)^r\left(\frac{1}{2}\right)^{N-r}={N\choose r}\left(\frac{1}{2}\right)^N$ (assuming binomial distribution)</p>
<p>Let $E(Y)$ denotes the average number of balls in non-empty boxes,</p>
<p>then , $E(Y)|(X=r)=\frac... |
1,781,225 | <p>I've got an exercise to do and I don't really know what to do.</p>
<p>Exercise : We've got function $f$, where $f(a) = 0$ and $f'(a)$ exists. Also we got function $g$ which is continuous. Does exist $(f-g)'(a)$? Explain it. </p>
<p>My opinion is that exists, but I've got no idea how should I explain it. Some help?... | Rebellos | 335,894 | <p>$f(a)=0$ and $\exists f'(a)$.</p>
<p>The expression : $(f-g)'(a)$ is : $f'(a) - g'(a)$.</p>
<p>You cannot say that this derivative exists, if $g$ is not differentiable at $a$. In your question's body, you have only stated that $f$ is differentiable and that $g$ is only continuous. Thus, no, you cannot say generall... |
220,196 | <p><a href="https://i.stack.imgur.com/b2N1E.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/b2N1E.png" alt="enter image description here"></a></p>
<p>I was working on some geometric manipulation and hoping to further process this graphic's isolines however I was stumped as how best to do that when I... | halmir | 590 | <p>If you just need lines, you can take <code>Normal</code> and extract lines from it:</p>
<pre><code>lines = Cases[Normal[c], _Line, Infinity];
Graphics3D[lines]
</code></pre>
<p><a href="https://i.stack.imgur.com/SuJfC.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SuJfC.png" alt="mesh lines"><... |
2,024,468 | <p>I need to find the points of intersection of a circle with radius $2$ and centre at $(0,0)$ and a rectangular hyperbola with equation $xy=1$. As per the topic statement is there any way to solve this without the graphical method. I have tried setting the $y$ values equal but I cant solve the resulting equation for $... | fleablood | 280,126 | <p>The equation of the circle is $x^2 + y^2 = 4$ and the equation of the hyperbola is $xy=1$</p>
<p>So the point of intersection would be a common solution to </p>
<p>$xy =1$</p>
<p>$x^2 + y^2 = 4$</p>
<p>so</p>
<p>$y = 1/x$</p>
<p>$x^2 + \frac 1{x^2} = 4$</p>
<p>$x^4 +1 = 4x^2$</p>
<p>$x^4 - 4x^2 + 1 = 0$</p>... |
2,024,468 | <p>I need to find the points of intersection of a circle with radius $2$ and centre at $(0,0)$ and a rectangular hyperbola with equation $xy=1$. As per the topic statement is there any way to solve this without the graphical method. I have tried setting the $y$ values equal but I cant solve the resulting equation for $... | Community | -1 | <p><strong>By trigonometry</strong>:</p>
<p>Any point on the circle has coordinates $(2\cos t,2\sin t)$. Then plugging in the other equation</p>
<p>$$4\sin t\cos t=1,$$</p>
<p>$$\sin 2t=\frac12,$$</p>
<p>giving</p>
<p>$$t\in{\frac\pi{12},\frac{5\pi}{12},\frac{13\pi}{12},\frac{17\pi}{12}}.$$</p>
<hr>
<p><strong>B... |
4,068,216 | <p>I know that the sequence <span class="math-container">$\sqrt[n]{n}$</span> converges to 1 and that <span class="math-container">$\text{log}(\sqrt[n]{n})$</span> thus converges to 0 as <span class="math-container">$n\to\infty$</span> since the logarithmic function is continuous. But how can I calculate the limits as ... | cosmo5 | 818,799 | <p><strong>Hint :</strong></p>
<p>Use complex numbers. Let affixes of <span class="math-container">$A,P$</span> be <span class="math-container">$z_A=-5$</span>, <span class="math-container">$z_P=5$</span>. So <span class="math-container">$z_M=0$</span>.</p>
<p><span class="math-container">$z_G$</span> would be <span cl... |
2,708,684 | <p>The unit ball of $\ell^1$ has extreme points, but the unit ball of $L^1$ does not have extreme points. Also, $\ell^1$ can be isometrically embedded into $L^1$. Isn't this a contradiction, since isometric isomorphisms preserve extreme points?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>i only will consider the numerator of your last term:
we Can write
$$\cos^2(\beta)\sin(\alpha)\cos(\beta)+\sin^2(\beta)\cos(\alpha)\sin(\beta)=$$
$$(1-\sin^2(\beta))\sin(\alpha)\cos(\beta)+(1-\cos^2(\beta))\cos(\alpha)\sin(\beta)=$$
$$\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\beta)-\sin(\alpha)\sin^2(\beta)\cos(\bet... |
2,708,684 | <p>The unit ball of $\ell^1$ has extreme points, but the unit ball of $L^1$ does not have extreme points. Also, $\ell^1$ can be isometrically embedded into $L^1$. Isn't this a contradiction, since isometric isomorphisms preserve extreme points?</p>
| Ѕᴀᴀᴅ | 302,797 | <p>$\def\peq{\mathrel{\phantom{=}}{}}$Denote $γ = α + β$, then$$
\frac{\cos α}{\cos β} + \frac{\sin α}{\sin β} = -1 \Longrightarrow \sin γ = \sin(α + β) = -\sin β\cos β.
$$
Thus,$$
\sin(γ - β) = \sin γ\cos β - \cos γ\sin β = -\sin β\cos^2 β - \cos γ\sin β,\\
\cos(γ - β) = \cos γ\cos β + \sin γ\sin β = \cos γ \cos β - ... |
935,763 | <p>How can I generate <em>random</em> points <em>uniformly distributed</em> on the surface of a sphere such that a line that originates at the center of the sphere, and passes through one of the points, will intersect a plane within a <strong>circle</strong>. Following are more illustrations and details to make this cl... | yathish | 230,668 | <p>This may be way off of what you're looking for. But perhaps it might point you in the right direction. Why not use the brute force method?</p>
<p>First, write the equation of the circle (shadow) which will be $f(x,y,z) = 0$</p>
<p>Now take any random point $(x_1,y_1,z_1)$ in the plane of the circle, such that $f(x... |
260,512 | <p>I am stuck with the following problem:</p>
<p>Let $A$ be a $3\times 3$ matrix over real numbers satisfying $A^{-1}=I-2A.$<br>
Then find the value of det$(A).$</p>
<p>I do not know how to proceed. Can someone point me in the right direction? Thanks in advance for your time.</p>
| Ragib Zaman | 14,657 | <p>Let $\lambda$ be a real eigenvalue with eigenvector $x$ (there is a real root to the characteristic equation). Since $A$ is invertible, $\lambda\neq 0$, so $Ax = \lambda x$ and $A^{-1}x=\lambda^{-1}x.$ Putting these into $A^{-1}x=x-2Ax$ gives $2\lambda^2-\lambda+1=0,$ contradicting that $\lambda$ is real. Hence no s... |
3,204,547 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma \in [-3,10].$</span> Then largest value of the determinant</p>
<p><span class="math-container">$$\begin{vmatrix}3\alpha^2&\beta^2+\alpha\beta+\alpha^2&\gamma^2+\alpha\gamma+\alpha^2\\\\
\alpha^2+\alpha\beta+\beta^2& 3\beta^2&\gamma^2+\... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Using the rules of SARRUS we get after simplifying <span class="math-container">$$- \left( \alpha-\gamma \right) ^{2} \left( \beta-\gamma \right) ^{2}
\left( \beta-\alpha \right) ^{2}
$$</span></p>
|
3,204,547 | <blockquote>
<p>If <span class="math-container">$\alpha,\beta,\gamma \in [-3,10].$</span> Then largest value of the determinant</p>
<p><span class="math-container">$$\begin{vmatrix}3\alpha^2&\beta^2+\alpha\beta+\alpha^2&\gamma^2+\alpha\gamma+\alpha^2\\\\
\alpha^2+\alpha\beta+\beta^2& 3\beta^2&\gamma^2+\... | Alex M. | 164,025 | <p>Being a <span class="math-container">$3 \times 3$</span> determinant, it is too small to be attacked with sophisticated techniques, but also too involved to be attacked by brute force. We shall compute it with successive simplifications of its rows and columns. In the following, <span class="math-container">$R_i$</s... |
1,608,734 | <p>I have absolutely no idea to inverse functions containing different functions. Apparently this is a one-to-one function with inverse $f^{-1}$ and I'm asked to calculate the inverses of the given functions in terms of $f^{-1}$</p>
| JMoravitz | 179,297 | <p>Let $g(x)=3-4x$ and $h(x)=1-2x$</p>
<p>Note that both $g$ and $h$ are one to one and invertible over the real numbers.</p>
<p>Note further that $r(x)= (h\circ f\circ g)(x)$</p>
<p>By properties of composition of invertible functions, $r^{-1}=(h\circ f\circ g)^{-1}=g^{-1}\circ f^{-1}\circ h^{-1}$</p>
<p>We can fi... |
3,623,250 | <p>My <a href="https://drive.google.com/file/d/1uK4ITDfSHXk-JfBcKZsywkGA1fJ_yGUn/view?usp=drive_open" rel="nofollow noreferrer">textbook</a>(2nd page, at very bottom) states the definition of symmetric relation as follows:</p>
<blockquote>
<p>A relation <span class="math-container">$R$</span> in a set <span class="math... | Michael Rozenberg | 190,319 | <p>Since <span class="math-container">$\ln$</span> increases, we obtain: <span class="math-container">$$f(x)=\ln3+\ln\left(\left(x-\frac{2}{3}\right)^2+\frac{11}{9}\right)\geq\ln3+\ln\frac{11}{9}=\ln\frac{11}{3}.$$</span>
The equality occurs for <span class="math-container">$x=\frac{2}{3}.$</span> </p>
<p>Also, <span ... |
1,219,514 | <p>If we assume that $\sum a_n$ converges conditionally then How can we comment that $\sum a_{2n} $ does not converges, While it does when $\sum a_n$ converges absolutely ?</p>
| pratik | 184,865 | <p>For a counter-example (provided the sequence conditionally converges)... think of your favorite conditionally convergent sequence, the alternating harmonic series.</p>
<p>If you'd rather not have to show that the series containing only the positive terms of the alternating harmonic series diverges, you could use so... |
3,356,468 | <p>Let <span class="math-container">$\pi:E\rightarrow M$</span> be a smooth vector bundle. Let <span class="math-container">$S:M\rightarrow E$</span> be it's zero section. Let <span class="math-container">$M'=E-S(M)$</span>.</p>
<p><strong>Is <span class="math-container">$M'$</span> a smooth submanifold of <span clas... | Travis Willse | 155,629 | <p><strong>Sketch</strong> Pick a local trivialization of <span class="math-container">$\pi : E \to M$</span>, say, <span class="math-container">$\Phi : U \times \Bbb V \to \pi^{-1}(U)$</span>, where <span class="math-container">$\Bbb V$</span> is a model fiber. Since <span class="math-container">$\{ 0 \}$</span> is cl... |
437,775 | <p>I'm a graduate student studying now for the first time class field theory.<br />
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.<br />
For example here <a href="https://mathoverflow.net/questions/6932/learning-class-field-theory-local-or-global-first">... | paul garrett | 15,629 | <p>In addition to other good answers: I'd tend to recommend taking a cohomological approach to classfield theory as a <em>second</em> pass through the subject, so that you already know the down-to-the-metal number-theoretic facts, and can focus on re-interpreting them in (co-)homological terms. Or, oppositely, if one a... |
4,088,272 | <p>Let <span class="math-container">$f:[a,b]\to \mathbb R$</span> a function. If <span class="math-container">$P=\{x_0,x_1,\ldots,x_n\}$</span> is a partition of <span class="math-container">$[a,b]$</span>, define <span class="math-container">$$||P||=\max_{1\leq i\leq n}|x_i-x_{i-1}|.$$</span></p>
<p>Prove that, <span ... | Oliver Díaz | 121,671 | <p>How are you defining the Riemann integral? Your statement appears in some textbook as the definition of the Riemann intergral (See definition 3 below).</p>
<p>If you are using refinement of partitions to define the Riemann integral (see definitions (1) and/or (2) below) some additional considerations need to be cons... |
3,652,730 | <blockquote>
<p>Without using L'Hôpital's rule, find:
<span class="math-container">$$\lim_{x\to 0}\dfrac{\cos(\frac{\pi}{2}\cos x)}{\sin(\sin x)}$$</span>
I know that the answer is <span class="math-container">$0$</span>.</p>
</blockquote>
<p>My attempt:</p>
<p>I tried by using the half-angle formula, <span cl... | Ninad Munshi | 698,724 | <p>Rewrite the limit as a product like so:</p>
<p><span class="math-container">$$L = \lim_{x\to 0} \frac{\sin x}{\sin( \sin x)} \cdot\frac{x}{\sin x} \cdot \frac{\cos(\frac{\pi}{2}\cos x)}{x}$$</span></p>
<p>If the limits exist individually then the limit of their product will be the product of their limits. The firs... |
4,197,489 | <p>I have a function of one variable. In this graph, we can see that there are a couple of places where the graph "bends" a lot -- a local maximum of "bending", if you will. The ordinary second derivative measures "bending" in the <span class="math-container">$y$</span> direction, which ... | soupless | 888,233 | <p>It seems like you are searching for the point where the curvature is greatest. In this case, if <span class="math-container">$y = f(x)$</span>, use the formula <span class="math-container">$$\kappa = \frac{|y''|}{\left(1 + \left(y'\right)^{2}\right)^{3/2}}.$$</span> This will give you the curvature for every <span ... |
4,197,489 | <p>I have a function of one variable. In this graph, we can see that there are a couple of places where the graph "bends" a lot -- a local maximum of "bending", if you will. The ordinary second derivative measures "bending" in the <span class="math-container">$y$</span> direction, which ... | Lee Mosher | 26,501 | <p>What you're looking for is to maximize the <a href="https://en.wikipedia.org/wiki/Curvature" rel="nofollow noreferrer">curvature</a>. A formula for the curvature of the graph of a function <span class="math-container">$y=f(x)$</span> can be found on the same page <a href="https://en.wikipedia.org/wiki/Curvature#Grap... |
258,392 | <p>Given a fiber bundle $(E,B,p,F)$ with path connected base $B$ and fiber $F$, both closed smooth manifolds of finite dimensions. The second page $E_2^{p,q}$ of the Leray-Serre spectral sequence over $\mathbb{Z}_2$ is give by $H^p(B;\mathcal{H}^q(F;\mathbb{Z}_2))$, where $\mathcal{H}^q(F;\mathbb{Z}_2)$ is the local sy... | Dylan Wilson | 6,936 | <p>Let $\pi$ be the fundamental group of $B$ and $R$ be whatever ring of coefficients you'd like. Suppose $B$ admits a universal cover $\tilde{B}$. Then there is a spectral sequence $\text{Ext}^{*,*}_{R[\pi]}(H_*(\tilde{B}), H^*(F))\Rightarrow H^*(B; \underline{H^*(F)})$. With field coefficients, say $k$, you can reph... |
1,517,488 | <p>Let $M$ be an Invertible Hermitian matrix and let $x,y\in\Bbb R$ such that $x^2\lt 4y$,Then Prove That $M^2+xM+yI$ and $M^2-xM+yI$ are non-singular.</p>
<p>My Attempt:</p>
<p>$$(M^2+xM+yI)(M^2-xM+yI)=(M^2+yI)^2-(xM)^2$$</p>
<p>Now I Don't Know How to proceed further, I know that all the eigen values of Hermitian ... | levap | 32,262 | <p>Prove that if a matrix $A$ is diagonalizable with distinct eigenvalues $\lambda_1, \ldots, \lambda_k$ and if $p(t) = \sum_{i=0}^n a_i t^i$ is a polynomial (with $a_i \in \mathbb{F}$), then $p(A)$ is also diagonalizable with eigenvalues $p(\lambda_1), \ldots, p(\lambda_k)$ (some of them may repeat). Apply this to $M$... |
13,460 | <p>I hope that this question is on-topic, though it is not quite technical.</p>
<p>I am curious to hear from people how they approach reading a mathematical paper.</p>
<p>I am not asking specific questions on purpose, though at first I had a few. But I want to keep it rather open-ended.</p>
| anon | 3,840 | <p>How I read a paper really depends on why I'm reading the paper.</p>
<p>A lot of papers I go to because I have a specific goal. Maybe they have been cited elsewhere as containing a proof of something I want to understand. Or a different proof of something I already know how to prove. Or maybe someone refers to th... |
367,686 | <p>How many injective functions $f:[1,...,m]\to{[1,...,n]}$ has no fixed point? $(m\le n)$</p>
<p>I thought about the next thing:</p>
<p>$f(x_1)\neq x_1$, Means i can choose for $x_1$ - (n-1) options,</p>
<p>But then, for $x_2$, there are two options:</p>
<ol>
<li><p>If i choose $f(x_1)=x_2$ then for $x_2$ i still... | Cubic Bear | 378,597 | <p>We can delete the fixed point $F$, in which place we can view it an injection $[m]\setminus F\to [n] \setminus F$. By counting the fixed point, we have
$$\sum_{i=0}^m \binom{m}{i}f(m-i,n-i)=\binom{n}{m}m!$$
Let $k=n-m$, then
$$\sum_{i=0}^m \binom{m}{m-i}f(m-i,m+k-i)
=\sum_{i=0}^m\binom{m}{j}f(j,k+j)
=\binom{m+k}{m}... |
2,600,283 | <p>I think $f(x) = x^2$. Then $f'(0)$ should be $0$.</p>
<p>But when I try to calculate the derivative of $f(x) = |x|^2$, then I get:</p>
<p>$f'(x) = 2|x| \cdot \frac{x}{|x|}$, which is not defined for $x = 0$. Does $f'(0)$ still exist?</p>
| Atmos | 516,446 | <p>$$\frac{f\left(h\right)-f\left(0\right)}{h}=\frac{\left|h\right|^2}{h}$$</p>
<p>If $h<0$ then $\left|h\right|=-h$ and $\left|h\right|^2=h^2$.</p>
<p>If $h>0$ then $\left|h\right|^2=h^2$ then</p>
<p>$$\frac{f\left(h\right)-f\left(0\right)}{h}\underset{h \rightarrow 0}{\rightarrow}0$$
for all $h \ne 0$ then $... |
651,034 | <p>The following text is a quote from <a href="http://books.google.com/books?id=NZVb54INnywC&pg=PA180" rel="nofollow">p.180</a> of Halbeisen's book <em>Combinatorial Set Theory</em>.
This book is also available <a href="http://www.math.uzh.ch/index.php?ve_vo_det&key2=1501" rel="nofollow">on website</a> of a co... | Martin Sleziak | 8,297 | <p>Let $\mathscr F$ be an arbitrary ultrafilter, which contains no finite sets.</p>
<p>Suppose that $A$ is an infinite pseudointersection of $\mathscr F$.</p>
<p>Then we have $A\subseteq^* G$ for each $G\in\mathscr F$.</p>
<p>Since $\mathscr F$ is an ultrafilter, we have either $A\in\mathscr F$ or $\omega\setminus A... |
651,034 | <p>The following text is a quote from <a href="http://books.google.com/books?id=NZVb54INnywC&pg=PA180" rel="nofollow">p.180</a> of Halbeisen's book <em>Combinatorial Set Theory</em>.
This book is also available <a href="http://www.math.uzh.ch/index.php?ve_vo_det&key2=1501" rel="nofollow">on website</a> of a co... | Community | -1 | <p>Let $\mathcal U$ be an ultrafilter on $\omega$, and let $x$ be a pseudo-intersection of $\mathcal U$. Furthermore, let $y$ be such that both it and its compliment contain infinitely many elements of $x$ (for instance, $y$ could be your $B$). Then, either $y$ or $\omega\backslash y$ are in $\mathcal U$, so either $x\... |
22,638 | <p>Exporting Image files (intermittently) doesn't work:</p>
<p><img src="https://i.stack.imgur.com/C18mr.png" alt="enter image description here"></p>
<p>I get messages back indicating the error is to do with escaping characters:</p>
<pre><code>Syntax::stresc: Unknown string escape \\U.
</code></pre>
| WolframFan | 1,056 | <p>I am answering my own question to help out other .Net/Mathematica developers in the future.
<hr>
I am using random file names (DTWERG, ERYFGJ, IYIGGD) and it turns out when Mathematica exports an image file that has a slash and followed by : <strong>b, t, n, f, r</strong> it recognises/honors the escape slash.</p>
... |
329,792 | <p>I need some suggestions to solve this integral:</p>
<p>$$\int_{1}^{3} \frac{1}{x^3 + 8} dx$$</p>
<p>Thanks.</p>
| Damien L | 59,825 | <p>1) split $\dfrac{1}{x^3 +8}$ in a sum of $\dfrac{\alpha}{x - \beta}$ or $\dfrac{\alpha x + \beta}{x^2 + \gamma x + \delta}$ ;</p>
<p>2) then you should be able to calculate those classical integrals.</p>
|
24,795 | <p>I have a set of points $(x, y)$ where each one comes from either one of two linear functions:
\begin{align*}
y &= m_1 x + b_1\\
y &= m_2 x + b_2
\end{align*}
Is there a fitting method to find such functions, without knowing from which function each of the points come from?</p>
<p>PS. can somebody ad... | David Bar Moshe | 1,430 | <p>Imagine that your data is presented as a binary two dimensional image (assuming a unit value whenever the point coordinate (x,y) is present in your data). Then the problem is equivalent to straight line detection in binary images. The equivalent problem can be solved using the <a href="http://en.wikipedia.org/wiki/H... |
642,863 | <p>Imagine there's a quiz on the internet intended for a wide audience. It contains a (unlimited) number of questions, all of them with yes/no answers. A person gets one random question and must answer it, after that he can get another one. He can continue answering any number of questions he wants. So the only data yo... | Community | -1 | <p>You are trying to combine two different measures into a single rating: correct answer rate, and total number of answers, with the reasoning being that if two people have the same percentage correct, then it is less likely that the person with <em>more overall questions attempted</em> would have received that percent... |
625,162 | <p>If you know a coupon collector problem, you will know what I am talking about. But if you are not familiar with I will try to explain what is the coupon collector problem.
I have $n$ bins. I throw balls consecutively into these bins. Each bin is choosen independently and with the same probability.
Let's suppose that... | Harald Hanche-Olsen | 23,290 | <p><strong>Hint 1</strong>: What you have inside the root is negative. Negate it, and put an $i$ on the outside.</p>
<p><strong>Hint 2</strong>: $z^2+2(a/b)z+1=(z-z_+)(z-z_-)$. Also, $z_-=\overline{z_+}$.</p>
|
1,653,299 | <p>If we have $R/M$ is a field and $M,I$ are ideals of $R$ such that $M\subseteq I \subseteq R$.</p>
<p>If we take $i\in I, i\not\in M$ we have $i+M \ne 0+M$. Since $R/M$ is a field, we have that $i+M$ is invertible, so $(i+M)(l+M)=1+M$</p>
<p>So $il+M=1+M$. If $il=1$ we get that $i$ is invertible, so $I=R$, but what... | Thomas | 300,350 | <p>$(i+M)(l+M)=il+M = 1+M$ means:</p>
<p>$il=1+m$ where $m\in M$. Since $M\subset I$ we have $il\in I$ and $m\in I$. Then $il-m\in I$ which means $1\in I$ and $I=R$.</p>
|
283,527 | <p>I have a two-server queue with Poisson arrival rate and $\lambda$ exponential services with $\mu$ ( first server service rate) and 2$\mu$ ( 2nd server service rate). Capacity is infinite.</p>
<p>Then why is the number of customers in the queue at time $t$ not a Markov Process?</p>
<p>Can you please help me out?</p... | Gareth | 79,908 | <p>You need extra information to make the process a Markov process. The Markov property requires that the future depend only on the current state, but suppose you have only a single customer in the system. The future behaviour of the system depends on the history of the process (namely which server that job started ser... |
1,619,103 | <p>I'm trying to find the infinite sum that is defined by:</p>
<p>$$
3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots
$$</p>
<p>However, I do not know of any known formula to do this. Am I missing something really simple? Thanks!</p>
| Ángel Mario Gallegos | 67,622 | <p>For $|r|<1$ we have $$\sum_{k=0}^{\infty}r^k=\frac{1}{1-r}$$
Then, by taking derivatives respect to $r$
$$\sum_{k=0}^{\infty}kr^{k-1}=\frac{1}{(1-r)^2}\quad\implies\quad \frac{1}{r}+2+\sum_{k=3}^{\infty}kr^{k-2}=\frac{1}{r(1-r)^2}$$
Then
$$\sum_{k=3}^{\infty}kr^{k-2}=\frac{1}{r(1-r)^2}-\frac{1}{r}-2$$
Thus
$$\sum... |
83,988 | <p>How does one see that the second Stiefel-Whitney class is zero for all orientable surfaces. For $S^2$ this can be seen by $TS^2$ being stably trivial, and for $S^1 \times S^1$ one can use $T (S^1 \times S^1) = TS^1 \times TS^1$, which gives the class in terms of the classes on $TS^1$ (which are all trivial). What ... | Grigory M | 152 | <p>Tangent bundle is stably trivial for any orientable surface (because the normal bundle is trivial).</p>
|
1,329,374 | <p>I'm looking for an explicit example of a BVP for a second order ODE: </p>
<blockquote>
<p>$y''+p(x)y'+q(x)y=f(x)$ (where $\,0\leq x\leq L\,$ and $\,y(0)=\alpha\,$ $\,y(L)=\beta$).</p>
</blockquote>
<p>If you also have the exact solution, the better. The reason is for test purposes, I've just finished a Mathemati... | OukiDouki | 39,344 | <p>Triangles in both problems are equilateral which simplifies solutions. My results are:</p>
<p>The area in problem 1 is
\begin{equation}
\frac{\sqrt{3}}{2} \alpha (a - \frac{\alpha}{2}).
\end{equation}</p>
<p>The area in problem 2 is
\begin{equation}
\frac{\sqrt{3}}{4} a^2 (2\alpha - \alpha^2).
\end{equation}</p>
|
376,575 | <p>This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, <em>On topological cyclic homology</em>, arXiv:<a href="https://arxiv.org/abs/1707.01799" rel="nofollow noreferrer">1707.01799</a>. I just want to make my understanding precise.</p>
<hr />
<p><strong>Pa... | Maxime Ramzi | 102,343 | <p>Let <span class="math-container">$C$</span> be a complete <span class="math-container">$\infty$</span>-category.</p>
<p>Let <span class="math-container">$U:Fun(BC_n,C)\to C$</span> denote the forgetful functor, <span class="math-container">$\mathrm{CoInd}$</span> its right adjoint, and <span class="math-container">$... |
2,290,458 | <p>I wonder if someone can help me with this problem:</p>
<blockquote>
<p>Let $(X,d)$ be a connected metric space such that all continuous functions $f:(X,d) \to \mathbb{R}$ are uniformly continuous. Show that $(X,d)$ is compact.</p>
</blockquote>
<p>A hint is to work with the counter positive and assume that $(X,d... | Henno Brandsma | 4,280 | <p><a href="http://msp.org/pjm/1958/8-1/pjm-v8-n1-p02-s.pdf" rel="nofollow noreferrer">this paper</a> gives necessary and sufficient conditions for $(X,d)$ to have the property that all continuous functions on it (to the reals, fo example) are uniformly continuous. It's something close to, but not quite, compactness.</... |
1,252,955 | <p>I'm trying to find the sum of the following series:</p>
<p>$$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n}$$</p>
<p>I tried to "convert" it to a simple geometrical series, but with no luck. Has someone any idea?</p>
<hr>
<p>Thanks for inspiration! My solution:
$$\sum^\infty_{n=1}\frac {(x-3)^{2n}}{2n} = \sum^\infty_{n... | E.H.E | 187,799 | <p>Hint:
$$\log(1-x)=-\sum_{n=1}^{\infty }\frac{x^n}{n}$$</p>
<p>Let $x=x^2$, then:
$$\log(1-x^2)=-\sum_{n=1}^{\infty }\frac{x^{2n}}{n}$$</p>
|
1,354,491 | <p>If I wanted to have a die that rolled, for example:</p>
<pre><code>| Roll | Prob (in %) |
|------|-------------|
| 1 | 60 |
| 2 | 25 |
| 3 | 12 |
| 4 | 4 |
| 5 | 1 |
| 6 | 0.2 |
| 7 | 0.04 |
| ... | ... |
</code></pre>
<p>(... | hobbs | 548 | <p>The inverse transform method mentioned by two of the other answers is one of the most elegant for generating random numbers over distributions. The exponential distribution has a <a href="https://en.wikipedia.org/wiki/Exponential_distribution#Generating_exponential_variates" rel="nofollow">very simple inverse CDF</a... |
19,876 | <p>Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am draf... | tpv | 4,216 | <p>For this you need only two very basic trigonometric identities:
$$ \cos^2(x) + \sin^2(x) = 1 $$
and
$$ \cos^2(x) - \sin^2(x) = \cos(2x) $$
Add or subtract them, and you have the identities for $\cos^2(x)$ and $\sin^2(x)$.</p>
|
3,506 | <p>I am wondering what is the correct function in Mathematica to plot the true impulse function, better known as the <code>DiracDelta[]</code> function. When using this inside of a function or just the function itself when plotting, it renders output = zero. Quick example:</p>
<pre><code>Plot[DiracDelta[x], {x,-1,1}]
... | rm -rf | 5 | <p>The Dirac delta, $\delta(x)$ is zero everywhere except at zero, and has an integral of 1 over $\mathbb{R}$. It is not <em>really</em> a function in the true sense and equating $\delta(0)=\infty$ is a rather loose definition; it should technically be considered as a distribution or a delta measure. Mathematica's impl... |
3,506 | <p>I am wondering what is the correct function in Mathematica to plot the true impulse function, better known as the <code>DiracDelta[]</code> function. When using this inside of a function or just the function itself when plotting, it renders output = zero. Quick example:</p>
<pre><code>Plot[DiracDelta[x], {x,-1,1}]
... | Heike | 46 | <p>To create the plot you could replace any occurrence of <code>DiracDelta[a]</code> with something like <code>10000 UnitStep[1/10000 - a^2]]</code>, so for example to plot</p>
<pre><code>f[x_] := DiracDelta[x - 2] + DiracDelta[x + 2]
</code></pre>
<p>you could do something like</p>
<pre><code>Plot[Evaluate[f[x] /. ... |
62,471 | <p>Given a minimal parabloic subgroup we know that conjugation by the longest element in the weyl group takes it to the opposite parabolic. </p>
<p>Can we do the same thing if we choose a standard parabolic subgroup? Can we always find an element in the weyl group such that conjugation by this element takes it to the ... | mathreader | 2,164 | <p>If I understood your question correct, then the answer is no. I will assume for simplicity that you are talking about parabolic subgroups of complex simple Lie groups. Then your question translates to the corresponding question about closed subsystems of root systems. Recall that the standard parabolic subgroups bij... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | omegadot | 128,913 | <p>Here is yet another way to find the indefinite integral for secant using what is known as a <a href="https://books.google.com.hk/books?id=Kck-DwAAQBAJ&pg=PA142&lpg=PA142&dq=%22Gunther%27s%20hyperbolic%20substitutions%22&source=bl&ots=P2U2Ya58-9&sig=ACfU3U2NadqYtX0GVtDNF1ncKobqnOPwOA&hl=en... |
6,695 | <p>The standard approach for showing <span class="math-container">$\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$</span> is to multiply by <span class="math-container">$\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$</span> and then do a substitution with <span class="math-cont... | Harish Chandra Rajpoot | 210,295 | <p><strong>Method-1:</strong>
<span class="math-container">$$\int \sec x \,dx = \int \frac{dx}{\cos^2 \frac x2-\sin^2\frac x2} $$</span><span class="math-container">$$= \int \frac{\sec^2\frac x2\ dx}{1-\tan^2 \frac x2} $$</span><span class="math-container">$$=\int\left(\frac{\frac12\sec^2\frac x2\ dx}{1+\tan\frac x2}+\... |
2,848,317 | <p>In the image, the segments inside the square go from a vertex to the middle point of the opposite side. If the length of the sides of the square is $1$, the area of $ABCD$ is?</p>
<p><a href="https://i.stack.imgur.com/BBpvZ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/BBpvZ.png" alt="enter ima... | bkarthik | 460,168 | <p>As N74 suggested, you can rotate the small triangles on to the trapezoids that are on each side of the square $ABCD$ to get a plus sign shape made of five congruent squares. </p>
<p>Since all of the translations we have done are distance preserving, we can safely say that the plus sign made of squares has an area e... |
301,889 | <p>For example, in a homework assignment I need to prove that a metric space $M$ is isometric to itself. If I say that $M$ maps to $M$ is that the same as saying that theres a function that takes elements from $M$ and I get a result in $M$? Or what is the "literal" meaning behind saying "map/maps/mapping".</p>
| Ittay Weiss | 30,953 | <p>In the context where the variables in the question stand for sets, one way to prove the equality of cardinalities will be to construct a bijection. This approach will be a bit messy, but not too much. Another approach will be to recall that the categorical product in $Set$, the category of sets, is the product const... |
1,221,487 | <p>Problem: Let V be the subspace of all 2x2 matrices over R, and W the subspace spanned by:</p>
<p>\begin{bmatrix}
1 & -5 \\
-4 & 2 \\
\end{bmatrix}
\begin{bmatrix}
1 & 1 \\
-1 & 5 \\
\end{bmatrix}
\begin{bmatrix}
2 & -4 \\
-5 & 7 \\
\end{bmatrix}
\begin{bmatrix}
1 & -7 \\
-5 & 1 \\
\e... | Mirko | 188,367 | <p>9 is eaten up by the other numbers. E.g. 3+9=12, 1+2=3, you started with 3 and 9 and ended up with 3. So, 3 just ate up the 9. Similarly 9 is eaten by each of the numbers 1,2,3,4,5,6,7,8, and 9 is also eaten up by itself, e.g. start with 9+9, then 18, 1+8=9. (Also the 0's are eaten by everything.) So, you don't real... |
630,697 | <p>I know that </p>
<p><1,2,3,...,10>$\cdot$<1 0,9,8,...1>=220</p>
<p><1,2,3,...,100>$\cdot$<100,99,98,...,1>=171700</p>
<p><1,2,3,...,1000>$\cdot$<1000,999,998,...,1>=167167000</p>
<p><1,2,3,...,10000>$\cdot$<10000,9999,9998,...,1>=166716670000
`</p>
<p>And 2,17,167,1667
is a part of the ... | Hagen von Eitzen | 39,174 | <p>The $n$th term of your sequence is the $10^{n}$th term of <a href="https://oeis.org/A000292/">A000292</a>, that is your $n$th term is
$$\frac{10^n(10^n+1)(10^n+2)}{6} $$</p>
|
2,477,817 | <p>The first step in calculating the variance of a Binomial Random Variable is calculating the second moment. </p>
<p><strong><a href="https://i.stack.imgur.com/KsqSQ.png" rel="noreferrer"><img src="https://i.stack.imgur.com/KsqSQ.png" alt="enter image description here"></a></strong></p>
<p>I have no idea as to how t... | Daniel Ordoñez | 478,506 | <p>The $n$ and $n-1$ are coming out of the $n!$ remember that $n!=n*(n-1)*(n-2)!$. The $p^2$ came from the $p^j$ since $p^j=p^{j-2}*p^2$. Once this terms are out both sums (separately) are equal to the probability mass function of a binomial random variable hence they sum to 1. Hence the result.</p>
|
3,439,084 | <p>How to use mean value theorem to show that <span class="math-container">$\sqrt{x}(1+x)\log(\frac{1+x}{x})-\sqrt{x}<1$</span> when <span class="math-container">$x$</span> is positive.</p>
| Brian Moehring | 694,754 | <p><strong>"Short" answer:</strong></p>
<p>In the proof part (1), there are actually two proofs because they are proving an "if and only if" statement. </p>
<p>In the first paragraph, starting with "Assume first that this operation is well-defined" through "... gives <span class="math-container">$gng^{-1}=n_1 \in N$<... |
184,534 | <p>Does the Gamma function $\Gamma: \mathbb{C} \to \mathbb{C}$ preserve the Kummer ring $\mathbb{Z}[\exp(2\pi\imath/m)]$? And if not, then what about the Gaussian integers $\mathbb{Z}[\imath]$ or the Eisenstein integers $\mathbb{Z}[\exp(2\pi\imath/3)]$?</p>
<p>Is it possible to characterize holomorphic function which ... | user60548 | 60,548 | <p>Note that</p>
<p>$$\frac{\Gamma(z) \Gamma(1-z)}{\Gamma(2z) \Gamma(1 - 2z)} = 2 \cos(\pi z),$$</p>
<p>and the RHS is transcendental for any non-rational algebraic number $z$ (by the Gelfond–Schneider theorem). So $\Gamma$ certainly won't preserve any number field $K$. It's most likely true that $\Gamma(z)$ is trans... |
1,160,012 | <p>I'm working on a job interview test and there is one answer which I just don't get.</p>
<p>The test states that statement below is true. To me it just seems wrong. No box is provided to check. Then how do I check it correct or wrong? Am I missing something here?</p>
<p>66:4 = 161/2</p>
| Andres I | 218,335 | <p>I think they meant to type "16 1/2" as in $16 \frac{1}{2}:$</p>
<p>$$66:4 \implies \frac{66}{4} = \frac{60}{4} + \frac{6}{4} = 15+ \frac{3}{2} = 16 \frac{1}{2} $$</p>
|
4,438,577 | <p>This question was on a math competition.</p>
<blockquote>
<p>Is there a triangle, which is not equilateral, whose sides form a geometric sequence and whose angles form an arithmetic sequence?</p>
<p>If such a triangle exists, find its sides and angles.</p>
</blockquote>
<p>My attempt:</p>
<p>Assume a triangle with s... | person | 681,221 | <p>Going off of what you have done:
<span class="math-container">$$\sin(60-\phi) = \sin(60)/r$$</span>
<span class="math-container">$$\frac{\sin(60)}{\sin(60-\phi)} = r$$</span>
Consider now the second and third expressions:
<span class="math-container">$$\dfrac{\sin60°}{r}=\dfrac{\sin(60°+\phi)}{r^2}$$</span>
<span cl... |
4,456,144 | <p>Let <span class="math-container">$A$</span> be a continuous masa in <span class="math-container">$L(H)$</span> and <span class="math-container">$T$</span> be a positive contraction in <span class="math-container">$A$</span>.
Then we can assume that <span class="math-container">$0＜\|Th\|＜1$</span> for all unit vector... | Martin Argerami | 22,857 | <p>The result works for any masa, not necessarily continuous.</p>
<p>If <span class="math-container">$0≤T≤1$</span> and <span class="math-container">$\|Th\|=\|h\|$</span>, then <span class="math-container">$Th=h$</span>. Indeed, you have
<span class="math-container">$$
0≤\|(1-T^2)^{1/2}h\|^2=\|h\|^2-\|Th\|^2=0.
$$</spa... |
3,133,637 | <p>I am having a hard time with exercises of the form : <span class="math-container">$f'$</span> verify some properties then prove that <span class="math-container">$f$</span> is such that : ...</p>
<p>The main problem I have is that in order to link <span class="math-container">$f$</span> and it's derivative I only k... | learner | 228,313 | <p><span class="math-container">$$|f'(x)|=K+\psi(x)~~\forall~x\in [a,b]$$</span></p>
<p>where <span class="math-container">$\psi(x)\geq 0$</span></p>
<p>If <span class="math-container">$f'(x)\lt 0$</span>, then <span class="math-container">$$\begin{align}-f'(x)=K+\psi(x)&\implies -f(x)=Kx+\int\psi(x)~\mathrm dx\g... |
127,109 | <p>How can I show that the order of an element modulo <span class="math-container">$m$</span> divides <span class="math-container">$\phi(m)$</span>?</p>
<p>I know that if <span class="math-container">$a$</span> and <span class="math-container">$m$</span> are relatively prime, then the least positive integer <span class... | André Nicolas | 6,312 | <p><strong>Hint:</strong> Let <span class="math-container">$\phi(m)=xq+r$</span>, where <span class="math-container">$0\le r<x$</span>. Show that <span class="math-container">$a^r\equiv 1 \pmod{m}$</span>. This contradicts the definition of <span class="math-container">$x$</span>, unless <span class="math-container... |
57,719 | <p>The following string can be converted easily into a list with <code>ToExpression</code></p>
<pre><code>string = "{{a},{b,c,d},{e,{f,{g}}}}";
ToExpression@string
</code></pre>
<p>However, if the string contains characters that can be misinterpreted as syntax errors, I run in to problems.</p>
<pre><code>string = "{... | hieron | 16,373 | <p>@Mr.Wizard</p>
<pre><code>s = StringReplace["{{a},{b,c,d},{e,{[f],{g}}}}",
x : Except["{" | "," | "}"] .. :> "\"" <> x <> "\""] // ToExpression
check = If[SyntaxQ@#, ToExpression@#, #] &;
ReplaceAll[s, x_String :> check@x] // InputForm
(*out*)
{{a}, {b, c, d}, {e, {"[f]", {g}}}}
</code></... |
227,869 | <p>Get the center and the semimajor/semiminor axes of the following ellipses:</p>
<p>$$x^2-6x+4y^2=16$$</p>
<p>$$2x^2 - 4x+3y^2+6y=7$$</p>
<p>How would one get these? I have no clue. I have a problem with merely rewriting these in the traditional ellipse equation. </p>
| Eric Angle | 35,995 | <p>I will do the first one:
$$
x^2 - 6 x + 4 y^2 = 16 \Rightarrow \left(x - 3\right)^2 + 4 y^2 = 25 \Rightarrow \frac{\left(x-3\right)^2}{5^2} + \frac{y^2}{\left(5/2\right)^2} = 1
$$</p>
<p>Now compare with equation 12 <a href="http://mathworld.wolfram.com/Ellipse.html" rel="nofollow">here</a>.</p>
|
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| Batominovski | 72,152 | <p>In general, it holds that $$\sqrt{n(n-1)+\sqrt{n(n-1)+\sqrt{n(n-1)}}}=n-\frac{1}{8n^2}+O\left(\frac1{n^3}\right)$$ and that $$\sqrt{n(n-1)-\sqrt{n(n-1)-\sqrt{n(n-1)}}}=(n-1)+\frac{1}{8n^2}+O\left(\frac1{n^3}\right)\,$$ for all $n\geq 1$. Hence, their difference is $$1-\frac1{4n^2}+O\left(\frac1{n^3}\right)\,.$$ In... |
1,859,741 | <p>How do I prove that</p>
<p>$$\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}} \approx 1$$</p>
<p>without using the calculator?</p>
| marty cohen | 13,079 | <p>Repeatedly using
$\sqrt{1+x}
\approx 1+x/2$,</p>
<p>$\begin{array}\\
d(a)
&=\sqrt{a+\sqrt{a+\sqrt{a}}}-\sqrt{a-\sqrt{a-\sqrt{a}}}\\
&=(\sqrt{a+\sqrt{a+\sqrt{a}}}-\sqrt{a-\sqrt{a-\sqrt{a}}})\dfrac{\sqrt{a+\sqrt{a+\sqrt{a}}}+\sqrt{a-\sqrt{a-\sqrt{a}}}}{\sqrt{a+\sqrt{a+\sqrt{a}}}+\sqrt{a-\sqrt{a-\sqrt{a}}}}\\
... |
1,083,801 | <p>Let's define a stuttering sequence the following way :</p>
<p>Let $q\in\mathbb{N}^*,E_q=\{1,2,\dots,q\}$ and $(u_n)\in (E_q)^\mathbb{N}$.</p>
<p><strong>$(u_n)$ is a stuttering sequence of order $k$ with spacing $w$ iff $$\exists n,w\in\mathbb{N},\exists k \in \mathbb{N}^*,\forall i\in\{0,1,\dots,k-1\},u_{n+i}=u_{... | Tomasz Warchoł | 362,612 | <h2>Q4</h2>
<p>No. See <a href="https://en.wikipedia.org/wiki/Square-free_word" rel="nofollow">square-free words</a>, they are sequences that are (in language of your problem) not stuttering for any $k \in \mathbb{N}^*, w=0$ and they are proven to exist for any $q>=3$.</p>
<h2>Q2</h2>
<p>No. Lets prove this for $... |
1,709,013 | <p>Let $T$ be a mobius transformation with exactly one fixed point on $\mathbb{C} \cup \{\infty\}$. What form does $T$ take? Find a fomrula for $T^n(z)$. What happens to $T^n(z)$ as $n \to \infty$.</p>
<h1>Attempt</h1>
<p>(we will assume $w \neq \infty$ since this case I covered already). If we examine a generic Mob... | RhythmInk | 242,446 | <p>We can find that if the only fixed point on the Riemann sphere is $\infty$ then our transformation takes the form $T(z)=z+b$. Now, if we want to consider an arbitrary fixed point, call it $w$, then we can look at a new transformation $S$ which exchanges $w$ and $\infty$. </p>
<p>Now, we may write our new transforma... |
1,017,026 | <blockquote>
<p><strong>Cauchy-Schwarz Inequality:</strong></p>
<p>If <span class="math-container">$\textbf{u}$</span> and <span class="math-container">$\textbf{v}$</span> are vectors in a real inner product space <span class="math-container">$V$</span>, then <span class="math-container">$$|\left\langle\textbf{u},\text... | Narasimham | 95,860 | <p>Arctan2 is more general and better usage. You can forget the quadrant and get polar angle between 0 to $2 \pi$ directly, unmindful of which quadrant the moving point is situated. Atan is used for first quadrant only.</p>
|
935,000 | <p>Let's start considering a simple fractions like $\dfrac {1}{2}$ and $\dfrac {1}{3}$.</p>
<p>If I choose to represent those fraction using decimal representation, I get, respectively, $0.5$ and $0.3333\overline{3}$ (a repeating decimal).</p>
<p>That is where my question begins.</p>
<p>If I multiply either $\dfrac ... | Tom-Tom | 116,182 | <p>Hint: compute the difference between $1$ and $0.9\bar9$. How much is that ? What do you conclude ?</p>
|
100,842 | <p>I have the following list of centers of disks.</p>
<pre><code>r=0.03;
pts = {{0.10420089319018544`, -0.024872674177014872`}, \
{0.9743669105930046`, 0.9169054125547074`}, {0.028760526736240563`,
0.45959879163736717`}, {-0.0059035632830851115`,
0.2922099255180086`}, {0.41615337459441437`,
0.9928402345... | KennyColnago | 3,246 | <p>Given the definition of the circle centres in <code>pts</code>, and a line parameterised by <code>y=m*x+b</code>.</p>
<pre><code>Manipulate[
Module[{r = 0.03, x = pts[[All, 1]], y = pts[[All, 2]], p},
p = Pick[pts, UnitStep[y - m*x - b], 0];
Graphics[{
EdgeForm[{Thickness[0.004], Black}], Fa... |
3,550,293 | <blockquote>
<p>The following are given:
<p><span class="math-container">$X$</span> is a discrete random variable
<p> The probability mass function given is <span class="math-container">$P(X=k)=Clnk$</span>
<p><span class="math-container">$k=e$</span>,<span class="math-container">$e^2$</span>,<span class="math-... | ole | 725,731 | <p><span class="math-container">$\tan\left(\frac{\pi}{2}-\theta\right)=\frac{\sin\left(\frac{\pi}{2}-\theta\right)}{\cos\left(\frac{\pi}{2}-\theta\right)}=\frac{\sin\frac{\pi}{2} \cos\theta-\cos\frac{\pi}{2} \sin\theta}{\\cos\frac{\pi}{2}\cos\theta+\sin\frac{\pi}{2}\sin\theta}=\frac{\cos\theta}{\sin\theta}=\cot\theta.$... |
2,452,143 | <p>I need to simplify this expression further:</p>
<p>$$ \sum_{m=1}^N (-1)^{m-1} m \binom{N}{m} $$</p>
| Marcus M | 215,322 | <p>Hint: Recall that $$(1 - x)^N = \sum\limits_{m = 0}^N (-1)^m \binom{N}{m}x^m.$$</p>
<p>What happens if you differentiate both sides?</p>
|
2,324,692 | <p>The Dual Group of $\mathbb{R}$ is isomorphic to $\Bbb{R}$ itself in the following way:
The map
$$\Bbb{R} \to \hat{\Bbb{R}}, \quad y \mapsto \exp(ixy) $$
is an isomorphism. Further it is stated in the literature that this map is also an homeomorphism. See for exmaple Conway, A course in functional analysis Theorem 9.... | Saketh Malyala | 250,220 | <p>Simpler way!</p>
<p>Use the fact $\displaystyle (\cos(x)+i\sin(x))^6 = \cos(6x)+i\sin(6x)$.</p>
<p>Expand, use $\sin^2(x)=1-\cos^2(x)$ to simplify higher powers of sine in the real terms, and then organizing by real terms, we get
$\cos(6x)+i\sin(6x)=$</p>
<p>$=32\cos^6(x)-48\cos^4(x)+18\cos^2(x)+32i\sin(x)\cos(... |
2,324,692 | <p>The Dual Group of $\mathbb{R}$ is isomorphic to $\Bbb{R}$ itself in the following way:
The map
$$\Bbb{R} \to \hat{\Bbb{R}}, \quad y \mapsto \exp(ixy) $$
is an isomorphism. Further it is stated in the literature that this map is also an homeomorphism. See for exmaple Conway, A course in functional analysis Theorem 9.... | lab bhattacharjee | 33,337 | <p>Uain <a href="http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html" rel="nofollow noreferrer">Prosthaphaeresis</a> Formulas, $$\cos6x+\cos 2x=2\cos4x\cos2x\iff\cos6x=\cos2x(2\cos4x-1)$$</p>
<p>$\cos4x=2\cos^22x-1$ and $\cos2x=2\cos^2x-1$</p>
|
2,653,645 | <p>Is this proof valid? Can d be relative to x ?
<a href="https://i.stack.imgur.com/LGwPN.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LGwPN.png" alt="enter image description here"></a></p>
| Mohammad Riazi-Kermani | 514,496 | <p>You need to find a $\delta$ which is independent of $x$ and depends only on $\epsilon.$</p>
<p>Note that $0<\delta<1 \implies 2<x<4$, thus $|x+3|<7$ </p>
<p>Try $\delta = \epsilon/7.$</p>
|
1,888,217 | <p>I have an application where I need to resize the users picture to max. 50KB</p>
<p>To resize I use a Method that sizes the picture to a given Width/Height.</p>
<p>The picture is a Bitmap with max. 8BPP.
(Width * Height * 8) = Size in Bits</p>
<p>What I need to know now is: How can I calculate the amount of pixels... | Stefan Mesken | 217,623 | <p>Let $r$ be the ratio of width/height of the original picture - you want to preserve that. Now you just have to solve
$$
y \cdot(1+r) = 50 \cdot 1024 \cdot 8.
$$
Then $\underline{y}$ is the height of the resized picture in pixels and, where $\underline{y}$ is the maximal integer less than or equal to $y$ and setting ... |
1,580,634 | <blockquote>
<p>Given i.i.d random variable with mean $\mu$ and variance 1, $\bar{X}_n =
\frac{1}{n}(X_1+\cdots+X_n)$,
use the CLT to approximate the following probability:</p>
</blockquote>
<p>$$P(|\bar{X}_n - \mu| \ge \frac{2}{\sqrt {n}})$$</p>
<p><strong>My attempt:</strong></p>
<p>$$P(|\bar{X}_n - \mu| \ge ... | Michael Hardy | 11,667 | <p>Suppose $X_1,\ldots,X_n$ are <b>uncorrelated</b> (a weaker assumption than independence) and <b>all have the same expected value $\mu$ and the same variance $\sigma^2<\infty$</b> (a weaker assumption that identical distribution). Then
$$
\operatorname{E}\left( \frac{X_1+\cdots+X_n} {\sqrt n} - \mu \right) = 0 \t... |
1,654,649 | <p>Exercise: <em>Find the fifth and tenth roots of unity in algebraic form.</em></p>
<p>This is an early exercise in Ahlfors Complex Analysis. </p>
<p>What I have tried so far:</p>
<p>For the fifth roots I have tried reducing the problem to the fact that $\Re (1+z+z^2+z^3+z^4)=1+\cos\theta+\cos2\theta+\cos3\theta+\c... | rogerl | 27,542 | <p>\begin{align*}
\cos 5\theta
&= \cos^5\theta + 10\cos^3\theta\sin^2\theta + 5\cos\theta\sin^4\theta \\
&= \cos^5\theta + 10\cos^3\theta(1-\cos^2\theta) + 5\cos\theta(1-\sin^2\theta)^2 \\
&= ... \\
&= 16\cos^5\theta - 20\cos^3\theta + 5\cos\theta.
\end{align*}</p>
|
1,491 | <p>I asked a question in MO and I received two interesting answers with different approaches. Both of them are very interesting. I wish to accept both of them simultaneously, but it is impossible. Morally, I cannot choose one of them as a better answer.</p>
<p>Is it reasonable to suggest to MO to remove this restricti... | Thomas Klimpel | 20,781 | <blockquote>
<p>Is it reasonable to suggest to MO to remove this restriction?</p>
</blockquote>
<p>No, I think the current behavior makes perfect sense, and is not really a restriction. An advantage of the current behavior is that it encourages me to ask only one main question at a time.</p>
<p>The simplest way to ... |
405,610 | <p>Consider two diffusions given by
<span class="math-container">$$X_j(t)=\int_0^t a_j(s,X_j(s))\,dW_s$$</span>
for <span class="math-container">$j=1,2$</span> and <span class="math-container">$t\ge 0$</span>, where <span class="math-container">$W_\cdot$</span> is a standard Wiener process/Brownian motion and the <span... | Mathias Rousset | 454,031 | <p>This is just a comment on a related issue.</p>
<p>You don't have stochastic comparison of processes at a given time but you <em><strong>DO have stochastic comparison of hitting times</strong></em>.</p>
<p>To see this, you can couple the two processes <span class="math-container">$j=1,2$</span> AFTER having performed... |
774,485 | <p>So I'm trying to solve this problem but not sure how. Consider the vectors u=(1,2,3) and v=(2,3,1) in r3. Find k so that w=(1,k,4) is a linear combination of u and v. </p>
<p>I'm not sure what to do. Any help would be greatly appreciated. Sorry if format isn't correct, asking this from my tablet.</p>
| Jebruho | 40,030 | <p>Set up the system of equations: $1=x+2y, k=2x+3y, 4=3x+y$. Then, you have a system of linear equations with three equations and three variables and they are linearly independent so there is a solution. What is the solution?</p>
|
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