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 |
|---|---|---|---|---|
395,685 | <p>I recall seeing a quote by William Thurston where he stated that the Geometrization conjecture was almost certain to be true and predicted that it would be proven by curvature flow methods. I don't remember the exact date, but it was from after Hamilton introduced the Ricci flow but well before Perelman's work. Unfo... | Misha | 21,684 | <p>Most likely, this is just misremembering (or misattribution).</p>
<ol>
<li><p>In his math writing Thurston did not make any predictions regarding what approach to the <em>Geometrization Conjecture</em> (GC) will be successful. I did not hear him making such predictions in his math lectures (but, of course, I heard o... |
3,980,441 | <p>I need to prove this. I need your help to verify that my proof is correct (or not) please.</p>
<blockquote>
<p>Prove that this integral exists: <span class="math-container">\begin{align}
\int_{2}^{\infty}\frac{dx}{\sqrt{1+x^{3}}} \end{align}</span></p>
</blockquote>
<p><strong>My attempt:</strong></p>
<p>Fist we ne... | Kosh | 270,689 | <p>Yes, you compute a limit that is not necessary to compute. You just need to bound your integrand by something that is integrable (like you did):</p>
<p><span class="math-container">$$
0\leq \int_2^{+\infty} \frac{1}{\sqrt{1+x^3}} dx \leq \int_2^{+\infty} \frac{1}{x^{3/2}} dx
$$</span>
and the last integral is finite... |
2,129,086 | <p>I know that the total number of choosing without constraint is </p>
<p>$\binom{3+11−1}{11}= \binom{13}{11}= \frac{13·12}{2} =78$</p>
<p>Then with x1 ≥ 1, x2 ≥ 2, and x3 ≥ 3. </p>
<p>the textbook has the following solution </p>
<p>$\binom{3+5−1}{5}=\binom{7}{5}=21$ I can't figure out where is the 5 coming from?</... | Jasser | 170,011 | <p>$$(1,0,10),(2,3,6),(1,2,8),(1,3,7)$$ which satisfies above equation, if rearranged gives 24 solutions and you have even more solutions. So your text book answer is wrong and I dont see any problems in ur method.</p>
|
897,756 | <p>How can I solve the following trigonometric inequation?</p>
<p>$$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$</p>
<p>Why I'm asking this question... I was doing my calculus homework, trying to plot the domain of the function $f\left(x,y\right)=\frac{x-y}{sin\left(x\right)-sin\left(y\right... | Guy | 127,574 | <p>Use $\sin(x+h) = \sin(x)\cos(h) + \cos(x)\sin(h)$</p>
|
897,756 | <p>How can I solve the following trigonometric inequation?</p>
<p>$$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$</p>
<p>Why I'm asking this question... I was doing my calculus homework, trying to plot the domain of the function $f\left(x,y\right)=\frac{x-y}{sin\left(x\right)-sin\left(y\right... | JimmyK4542 | 155,509 | <p>Using the difference to product identity $\sin A - \sin B = 2\sin \dfrac{A-B}{2}\cos\dfrac{A+B}{2}$, we get: </p>
<p>$\displaystyle\lim_{h \to 0}\dfrac{\sin(x+h)-\sin x}{h} = \lim_{h \to 0}\dfrac{2\sin\frac{h}{2}\cos(x+\frac{h}{2})}{h} = \lim_{h \to 0}\dfrac{\sin \frac{h}{2}}{\frac{h}{2}} \cdot \lim_{h \to 0}\cos(x... |
200,777 | <p>I have a question regarding sums in arrays.</p>
<p>So I have the following array:</p>
<pre><code>list=RandomReal[{0,1},{5,2}]
(*{{0.693551,0.447185},{0.274842,0.637526},{0.745271,0.0288363},{0.894933,0.937219},{0.605447,0.0337067}}*)
</code></pre>
<p>And from that I want to have the splitting for each pair like ... | kglr | 125 | <p>You can use <a href="https://reference.wolfram.com/language/ref/DistanceMatrix.html" rel="nofollow noreferrer"><code>DistanceMatrix</code></a> and use <code>Total</code> with the upper part of the matrix:</p>
<pre><code>rowtotaldistances = Total[UpperTriangularize[DistanceMatrix[#],1],{2}]&
m = Partition[Range... |
1,251,914 | <p>I do not understand how to set up the following problem:</p>
<p>"Forces of 20 lb and 32 lb make an angle of 52 degrees with each other. find the magnitude of the resultant force."</p>
<p>An actually picture would really help.</p>
| Timothy | 137,739 | <p>When somebody says a statement about the fourth dimension that you think is clearly false such as the fact that a closed string has only one possible knot, the unknot, the usual way to interpret what they're saying is a statement that is false. You're right that it's technically false to say that a string cannot be ... |
21,141 | <p>Is there a way to extract the arguments of a function? Consider the following example:</p>
<p>I have this sum</p>
<pre><code>g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3]
</code></pre>
<p>Now, what I want to do is exctract the function arguments and apply them to another function <code>func</cod... | Mr.Wizard | 121 | <p>The pattern for a direct replacement proves to be a bit tricky:</p>
<pre><code>start = g[1] + g[2] + g[3] + g[1]*g[3] + 3*g[1]*g[2] + 6*g[1]*g[2]*g[3];
start /. y__g z_g | y__g x_. :> x func[Join @@ List @@@ {y, z}]
</code></pre>
<blockquote>
<pre><code>func[{1}] + func[{2}] + func[{3}] + 3 func[{1, 2}] + func... |
3,589,685 | <p>Can you give an example of an isomorphism mapping from <span class="math-container">$\mathbb R^3 \to \mathbb P_2(\mathbb R)$</span>(degree-2 polynomials)?</p>
<p>I understand that to show isomorphism you can show both injectivity and surjectivity, or you could also just show that an inverse matrix exists.</p>
<p>M... | José Carlos Santos | 446,262 | <p>What about<span class="math-container">$$\begin{array}{rccc}\psi\colon&\mathbb R^3&\longrightarrow&P_2(\mathbb R)\\&(a,b,c)&\mapsto&a+bx+cx^2?\end{array}$$</span>It is linear, injective and surjective.</p>
|
149,769 | <p>Using the <a href="http://szhorvat.net/pelican/latex-typesetting-in-mathematica.html" rel="nofollow noreferrer">MaTeX</a> package from our colleague <a href="https://mathematica.stackexchange.com/users/12/szabolcs">Szabolcs</a> I had a certain problem.
I would like to highlight the result for bold, but I did not get... | Szabolcs | 12 | <p>@JasonB already posted a good answer. I just wanted to point out two more things:</p>
<ul>
<li><p>This is a LaTeX question really, and not specific to MaTeX/Mathematica. I suggest <a href="http://tex.stackexchange.com/">http://tex.stackexchange.com/</a> for similar future questions. The main thing to know about Ma... |
201,122 | <p>A little bit of <em>motivation</em> (the question starts below the line): I am studying a proper, generically finite map of varieties $X \to Y$, with $X$ and $Y$ smooth. Since the map is proper, we can use the Stein factorization $X \to \hat{X} \to Y$. Since the composition is generically finite, $X \to \hat{X}$ is ... | Laurent Moret-Bailly | 7,666 | <p>The only restriction I see is that $\hat{X}$ must be normal (because $X$ is): if $\phi$ is a rational function on (some affine open subscheme of) $\hat{X}$ which is integral over $\mathscr{O}_\hat{X}$, then $\phi\circ\pi$ is integral over $\mathscr{O}_{X}$, hence in $\mathscr{O}_{X}$. In other words, $\phi$ lies in ... |
2,424,508 | <p>One textbook exercise asks to prove $$|a|+|b|+|c|-|a+b|-|a+c|-|b+c|+|a+b+c| \geq 0.$$</p>
<p>The textbook's solution is:</p>
<blockquote>
<p>If $a$, $b$ or $c$ is zero, the equality follows. Then, we can assume
$|a| \geq |b| \geq |c| > 0$. </p>
<p>Dividing by $|a|$, the inequality is equivalent
to</p... | Rob Arthan | 23,171 | <p>The product topology on $P = \{0, 2\}^{\Bbb{N}}$ induced from the discrete topology on $\{0, 2\}$ is <strong>not</strong> the discrete topology. The points in $P$ are not open. The product topology on $P$ is the smallest topology that makes each projection $\pi_i : P \to \{0, 2\}$ continuous. Each open set $X$ in th... |
2,424,508 | <p>One textbook exercise asks to prove $$|a|+|b|+|c|-|a+b|-|a+c|-|b+c|+|a+b+c| \geq 0.$$</p>
<p>The textbook's solution is:</p>
<blockquote>
<p>If $a$, $b$ or $c$ is zero, the equality follows. Then, we can assume
$|a| \geq |b| \geq |c| > 0$. </p>
<p>Dividing by $|a|$, the inequality is equivalent
to</p... | Xander Henderson | 468,350 | <p>The product topology will not be the discrete topology. The product topology consists of sets of the form
$$ \prod_{j=1}^{\infty} U_j $$
where each set $U_j \in \{ \{0\}, \{2\}, \{0,2\}\}$, <strong>and</strong> only finitely many of the $U_j$ are singletons. This second condition gives us something different from ... |
68,563 | <p>I was wondering if there's a formula for the cardinality of the set $A_k=\{(i_1,i_2,\ldots,i_k):1\leq i_1<i_2<\cdots<i_k\leq n\}$ for some $n\in\mathbb{N}$. I calculated that $A_2$ has $n(n-1)/2$ elements, and $A_3=\sum_{j=2}^{n-2}\frac{(n-j)(n-j+1)}{2}$. As you can see, the cardinality of $A_3$ is already ... | Brian M. Scott | 12,042 | <p>You can also get it by induction using the fairly obvious recurrence $$A_{k+1}(n) = \sum_{i=k}^{n-1}A_k(i):$$ if $A_k(i) = \dbinom{i}{k}$, then $$A_{k+1}(n) = \sum_{i=k}^{n-1}\binom{i}{k} = \binom{n}{k+1}$$ by one of the ‘hockey stick’ identities.</p>
|
23,314 | <p>I have been waiting for <em>Mathematica</em> to give me something for the following integral, errors welcome, but it has been "running" for almost 30 minutes now. </p>
<pre><code> h[s_] := If[1 < s, 1, 0]
Integrate[
Abs[1/(b1^2 + b2^2)
2 E^(-b1 s) ((a1 b1 + a2 b2) E^(
... | whuber | 91 | <p>Provided $2(a_1 b_1 + a_2 b_2) = b_1^2 + b_2^2$, there exists a closed-form solution that <em>Mathematica</em> can obtain quickly. But you have to do a little analysis along the way :-).</p>
<p><strong>I won't carry out the full solution, but will show how to do the difficult part.</strong> The motivation for the... |
4,475,082 | <p>Problem:</p>
<ul>
<li>Three-of-a-kind poker hand: Three cards have one rank and the remaining two cards have
two other ranks. e.g. {2♥, 2♠, 2♣, 5♣, K♦}</li>
</ul>
<p>Calculate the probability of drawing this kind of poker hand.</p>
<p>My confusion: When choosing the three ranks, the explanation used <span class="mat... | true blue anil | 22,388 | <p>This mistake is made so often by beginners, that I advise that you consider it as reserving bags for chosen ranks from <span class="math-container">$13$</span> bags.</p>
<p>So one bag needs to be reserved for the triple, and two for the two singles, hence <span class="math-container">$\binom{13}1\binom{12}2$</span>.... |
3,969,943 | <p>It's been a few years since doing any type of trigonometry questions and I've seemed to forgotten everything about it. Below is a question with the solution. You're not supposed to use a calculator.</p>
<p><span class="math-container">$$\begin{align}
&\cos\frac{2\pi}{3}+\tan\frac{7\pi}{4}-\sin\frac{7\pi}{6} \\[4... | GEdgar | 442 | <p>For (a) it says "explicit formula". Yours is not. Here it is:
<span class="math-container">$$
\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)
\cdots\left(1+\frac{1}{n-1}\right)\left(1+\frac{1}{n}\right)
\\
=\frac{2}{1}\cdot\frac{3}{2}\cdot\frac{4}{3}\cdots\frac{n}{n-1}\cdot... |
2,981,444 | <p><span class="math-container">$AB$</span> is a chord of a circle and the tangents at <span class="math-container">$A$</span>, <span class="math-container">$B$</span> meet at <span class="math-container">$C$</span>. If <span class="math-container">$P$</span> is any point on the circle and <span class="math-container">... | Batominovski | 72,152 | <p>Let <span class="math-container">$\Gamma$</span> denote the circle. The line <span class="math-container">$MB$</span> is tangent to <span class="math-container">$\Gamma$</span> at <span class="math-container">$B$</span>, so <span class="math-container">$\angle PAB=\angle PBM$</span>. Since <span class="math-contai... |
2,943,329 | <p>There seem to be six essentially different types of cubic polynomials with real coefficients, giving rise to 1, 2 or 3 real roots in different ways. </p>
<p>Consider <span class="math-container">$f(z) = z^3 + a_2z^2 + a_1z + a_0$</span> and let <span class="math-container">$(a_2,a_1,a_0)$</span> be <span class="mat... | LucaMac | 586,942 | <p><span class="math-container">$(1+x+x^2+x^3)^2-x^3 = (1+x+x^2)^2 + 2x^3(1+x+x^2) + x^3(x-1)(1+x+x^2) = (1+x+x^2)(1+x+x^2+x^3+x^4)$</span></p>
|
2,366,610 | <p>Let $U$ be an $n \times n$ unitary matrix and $X$ an $n \times n$ real symmetric matrix. Suppose that $$U^\dagger X U = X$$ for all real symmetric $X$, then what are the allowed unitaries $U$? It seems that the only possible $U$ is some phase multiple of the identity $U=aI$ where $|a|=1$ but I'm not able to show th... | robjohn | 13,854 | <p>Using the Mean Value Theorem, and the fact that $\frac{\mathrm{d}}{\mathrm{d}x}\sinh^{-1}(x)=\frac1{\sqrt{1+x^2}}$, we get that
$$
\frac{\sinh^{-1}(\sinh(x))-\sinh^{-1}(\sin(x))}{\sinh(x)-\sin(x)}=\frac1{\sqrt{1+\xi^2}}\tag{1}
$$
for some $\xi$ between $\sin(x)$ and $\sinh(x)$.</p>
<p>Therefore, since both $\sinh(x... |
4,444,669 | <p>I'm unsure about the problem below</p>
<hr>
Under which conditions is the following linear equation system solvable ?
<span class="math-container">$$x_1 + 2x_2 - 3x_3 = a$$</span>
<span class="math-container">$$3x_1 - x_2 + 2x_3 = b$$</span>
<span class="math-container">$$x_1 - 5x_2 + 8x_3 = c$$</span>
<hr>
<p>We se... | Hagen von Eitzen | 39,174 | <p>I suppose we are allowed to use the following:</p>
<blockquote>
<p>If <span class="math-container">$S$</span> is a non-empty subset of <span class="math-container">$\Bbb R$</span> and is bounded from below, then there exists a real number denoted as <span class="math-container">$\inf S$</span> such that</p>
<ul>
<li... |
43,095 | <p>The continuous max flow problem is posed as follows : </p>
<p>sup $\int_\Omega p_s(x)dx$</p>
<p>subject to : </p>
<p>$|p(x)| \le C(x); \forall x \in \Omega $</p>
<p>$p_s(x) \le C_s(x); \forall x \in \Omega $</p>
<p>$p_t(x) \le C_t(x); \forall x \in \Omega $</p>
<p>$\nabla \cdot p(x) - p_s(x) + p_t(x) = 0; ... | Jim Belk | 1,726 | <p>Given arbitrary continuous functions $\lambda,\mu,\nu\colon\Omega\to [0,\infty)$ and $\varphi\colon\Omega\to\mathbb{R}$, define
$$
G(\lambda,\mu,\nu,\varphi) \;=\; \sup_{(p_s,p_t,p)}\int_\Omega \Bigl(p_s + (C-|p|)\lambda+(C_s - p_s)\mu + (C_t - p_t)\nu + (\nabla\cdot p - p_s + p_t)\varphi \Bigr)
$$</p>
<p>Then cl... |
2,761,509 | <p>I hope it's not a duplicate but I've been searching about this problem for some time on this site and I couldn't find anything. My problem is why a number $\in(-1,0)$ raised to $\infty$ is $0$. For example let's take
$$\lim_{n\to \infty} \left(\frac{-1}{2}\right)^n$$
Which is equivalent to
$$\left(\frac{-1}{2}\righ... | Delta-u | 550,182 | <p><strong>Hint</strong>:</p>
<p>Indeed $(-1)^\infty$ does not exists. To show that $(-1/2)^n$ goes to $0$, an idea is to show that:
$$ \lim_{n \to \infty}\left|\left(\frac{-1}{2} \right)^n-0 \right|=0$$ </p>
|
3,766,585 | <p>Let <span class="math-container">$X_1,X_2,...,X_n$</span> be random sample from a DF <span class="math-container">$F$</span>, and let <span class="math-container">$F_n^* (x)$</span> be the sample distribution function.</p>
<p>We have to find <span class="math-container">$\operatorname{Cov}(F_n^* (x), F_n^* (y))$</sp... | Bernard | 202,857 | <p>First observe the absolute value isn't necessary, as <span class="math-container">$1+x^3>0$</span> if <span class="math-container">$-1<x<1$</span>. You can find the limit very with <em>equivalents</em> near <span class="math-container">$0$</span>: we know that <span class="math-container">$\sin\sim_0 x$</... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Gerhard Paseman | 3,468 | <p>I don't know that there is such a line to walk. Justification usually requires some specific context, and an answer that is out of context assumed by the asker usually does not convince the asker.</p>
<p>I would not dare to convince someone that study in mathematics should have higher priority than study in physic... |
8,052 | <p>I wonder how you teachers walk the line between justifying mathematics because of
its many—and sometimes surprising—applications, and justifying it as the study
of one of the great intellectual and creative achievements of humankind?</p>
<p>I have quoted to my students G.H. Hardy's famous line,</p>
<bl... | Adam Rubinson | 12,691 | <p>Mathematics is often thought of as a bunch of clever people doing weird things with symbols for reasons that are difficult to relate to reality. Whilst this is not entirely false, it is for the most part (almost humorously so in my opinion), the antithesis of the actual role of mathematics in today's world.</p>
<p>I... |
2,798,847 | <p>I would like to prove that $\|e^A-e^B\| \leq \|A-B\|e^{max\{\|A\|,\|B\|\}}$, where $A,B \in \mathbb{R}^{n \times n}$.</p>
<p>So far I was able to create the first difference term, but I have no idea how to incorporate the max norm.
I've read <a href="https://math.stackexchange.com/questions/2262000/inequality-norm-... | Keerthana Gurushankar | 647,133 | <p>You can first get <span class="math-container">$\|B^k-A^k\|\leq k\|B-A\|(\max(\|A\|,\|B\|))^{k-1}$</span> as follows</p>
<p><span class="math-container">\begin{align*}
\|B^k-A^k\| &= \left\lVert\sum_{l=0}^{k-1}B^{l}(B-A)A^{k-1-l}\right\rVert
\leq \sum_{l=0}^{k-1}\left\lVert B^{l}(B-A)A^{k-1-l}\right\rVe... |
1,687,714 | <p><a href="https://i.stack.imgur.com/nZEAy.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nZEAy.jpg" alt=""></a></p>
<p>I am given a problem in my textbook and I am left to determine the Laplace transform of a function given its graph (see the attached photo) - a square wave - using the theorem th... | John B | 301,742 | <p>$$
\frac{1}{1-e^{-2as}} \int_0^a e^{-st}dt=\frac{1-e^{-as}}{s(1-e^{-2as})}=\frac{1}{s(1+e^{-as})}
$$</p>
|
412,482 | <p>I have a few (semi-)related questions regarding certain Hilbert space representations of locally compact groups that come up in the theory of automorphic forms. </p>
<p>Let $G$ be a unimodular locally compact Hausdorff group, $\Gamma$ a discrete (hence closed) subgroup of $G$, and $Z$ a closed subgroup of the cente... | TheBridge | 4,437 | <p>You need to apply this <a href="http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Conditional_distributions" rel="noreferrer">theorem</a> about how on conditioning a Gaussian vectors.</p>
<p>You are in the situation where you have this 3-d Gaussian vector $V=(X_s,X_u,X_t)$.</p>
<p>You will have to comp... |
1,572,954 | <p>What is the only ordered pair of numbers $(x,y)$ which, for all $a$ and $b$, satisfies </p>
<p>$$x^a y^b=\left(\frac34\right)^{a-b} \text{and } x^b y^a=\left(\frac34\right)^{b-a}$$</p>
<p>I started off with the trivial cases, $a=0$ and $b=0$ and you get $1=1$ on both sides, so that works.</p>
<p>I can't seem to f... | Kay K. | 292,333 | <p>$$\left(x^ay^b\right)\times\left(x^by^a\right)=\left(\frac34\right)^{a-b}\times\left(\frac34\right)^{b-a}$$
$$(xy)^{a+b}=1$$
$$xy=1$$
$$x^ax^{-b}=\left(\frac34\right)^{a-b}$$
$$\therefore x=\frac34,\space y=\frac43$$</p>
|
418,026 | <p>I am asked to find the power series of the function $f(x)=\arctan(\frac{x}{\sqrt{2}})$. I first found the derivative of this function which is: $f'(x)=\frac{\sqrt{2}}{2+x^{2}}$. Then I found the power series of $f'(x)$ which is: $\sum_{n=0}^{\infty }\frac{1}{\sqrt{2}}(-1)^{n}\left ( \frac{1}{2} \right )^{n}x^{2n}$. ... | Branimir Ćaćić | 49,610 | <p>First, since $G$ is symmetric positive definite, you know in particular that it is indeed invertible.</p>
<p>Now, let $T := \sum_{j=1}^n u_j u_j^T$. The question is, what is $TGe_k$ for each $k$?</p>
|
1,201,942 | <p>I apologize if this question gets down-voted ahead of time.</p>
<p>I've been working on the Collatz Conjecture all day with Python, because that is the language I'm most familiar with (I'm not a CS student, just majoring in math). Below is the function I'm using for your reference during my comments:</p>
<p>\begin... | fulges | 53,662 | <p>You are correct, in both your points. Suppose that the conjecture is false.</p>
<p>If $N$ is a counterexample and $M$ is an element of the sequence generated by $N$, then $M$ is a counterexample as well, since if the sequence generated by $M$ ended to $1$, then also the sequence generated by $N$ ends at $1$.</p>
<... |
3,168,662 | <p>How do you evaluate <span class="math-container">$\int_{|z|=1} \frac{\sin(z)}{z^2+(3-i)z-3i}dz$</span> ? </p>
<p>Here is my thought process: </p>
<p>I want to use <a href="http://mathworld.wolfram.com/CauchyIntegralFormula.html" rel="nofollow noreferrer">Cauchy's Integral Formula</a>, but in order to use it I need... | Mark Viola | 218,419 | <p>First, we can write the integrand as </p>
<p><span class="math-container">$$\frac{\sin(z)}{(z+3)(z-i)}=\underbrace{\frac1{3+i}\left(\frac{\sin(z)-\sin(i)}{z-i}-\frac{\sin(z)}{z+3}\right)}_{\text{Analytic for }|z|\le1}+\underbrace{\frac1{3+i}\left(\frac{\sin(i)}{z-i}\right)}_{\text{Has simple pole at }z=i}$$</span><... |
3,168,662 | <p>How do you evaluate <span class="math-container">$\int_{|z|=1} \frac{\sin(z)}{z^2+(3-i)z-3i}dz$</span> ? </p>
<p>Here is my thought process: </p>
<p>I want to use <a href="http://mathworld.wolfram.com/CauchyIntegralFormula.html" rel="nofollow noreferrer">Cauchy's Integral Formula</a>, but in order to use it I need... | Community | -1 | <p>You must be in Dr. Gowda's Complex Analysis class as well. See the new email from Dr. Gowda that says to integrate over <span class="math-container">$\left\lvert z\right\rvert=2$</span> instead. We cannot use Cauchy's integral formula here because <span class="math-container">$z=i$</span> lies on the boundary of <sp... |
1,725,337 | <p>How does the following definition of Taylor polynomials:</p>
<p>$f(x_0 + h)= f(x_0) + f'(x_0)\cdot h + \frac{f''(x)}{2!}h^2+ ... +\frac{f^(k)(x_0)}{k!}\cdot h^k+R_k(x_0,h),$ </p>
<p>where $R_k(x_0,h)=\int^{x_0+h}_{x_0} \frac{(x_0+h-\tau)^k}{k!}f^{k+1}(\tau) d\tau$</p>
<p>where I guess $\lim_{h\to 0} \frac{R_k(x... | Anonymous | 327,815 | <p>$$
g(x) = \frac{1}{(8+x)} = \sum_{n=0}^{\infty} \frac{(-1)^nx^n}{8^{n+1}}
$$
Taking the derivative in both sides: (n=0 is constant)
$$
g'(x) = \frac{-1}{(8+x)^2} = \sum_{n=1}^{\infty} \frac{n(-1)^nx^{n-1}}{8^{n+1}}
$$
$$
-f(x) = \frac{-1}{(8+x)^2} = \sum_{n=0}^{\infty} \frac{(n+1)(-1)^{n+1}x^{n}}{8^{n+2}}
$$
$$
f(... |
39,551 | <p>How can I use <em>Mathematica</em> to equate coefficients in a non-power-series equation?</p>
<p>For example, I would like to take an equation like the following:
$$af_x+\frac{b}{2}f_xf_y+chf_x=f_x+e^af_x+3f_xf_y+2bhf_x$$
and produce the following system:
$$a=1+e^a$$
$$\frac{b}{2}=3$$
$$c=2b$$
<strong>EDIT:</strong... | Nasser | 70 | <p>Probably (ok, most likely) not the most elegant way. But here we go: May be you can let $x=f_x$ and $y=f_y$ and use <code>CoefficientList</code> to help.</p>
<pre><code>ClearAll[x, y, a, b, c, h, r, lhs, rhs]
fx = D[f[x, y], x];
fy = D[f[x, y], y];
expr = a fx + b/2 fx fy + c h fx == fx + Exp[a] fx + 3 fx fy + 2 b ... |
1,578,783 | <p>A friend of mine found a book in which the author said that the dual space of $L^\infty$ is $L^1$, of course not with the norm topology but with the weak-* topology. Does anyone know where I can find this result? Thanks.</p>
| Tomasz Kania | 17,929 | <p>For any $C(K)$-space we have $C(K)^*\cong L_1(\mu)$ for some usually humongous measure $\mu$. See the proof of Proposition 4.3.8(iii) in </p>
<blockquote>
<p>F. Albiac, N.J. Kalton, <em>Topics in Banach Space Theory</em>, Grad. Texts in Math. 233, Springer, 2006.</p>
</blockquote>
<p>Of course, $L_\infty(\nu)\co... |
127,108 | <p>If you take a subtraction-free rational identity like $(xxx+yyy)/(x+y)+xy=xx+yy$ and replace $\times$,$/$,$+$,$1$ by $+$,$-$,min,$0$, do you always get a valid min,plus,minus identity like min(min($x+x+x,y+y+y$)$-$min($x,y$),$\:x+y$)$\ =\ $min($x+x,y+y$)?</p>
| Colin McQuillan | 408 | <p>It suffices to show that whenever $F$ is a function $\mathbb R_{\geq 0}^k\to\mathbb R_{\geq 0}$ defined using $\times,/,+,1$, and $f,g$ is the corresponding tropicalization $\mathbb R^k\to\mathbb R$, for all real $x_1,\dots,x_k$ we have $$F(\exp(-\beta x_1),\dots,\exp(-\beta x_k))^{1/\beta}\to \exp(-f(x_1,\dots,x_k)... |
1,907,743 | <p>I'm having trouble with a step in a paper which I believe boils down to the following inequality:
$$
\left\| \sum_{k\in\mathbb{Z}} f(\cdot+k) \right\|_{L^2(0,1)}
\leq c \|f\|_{L^2(\mathbb{R})}.
$$
I haven't come up with many ideas. Hitting the left-hand side with Minkowski, for example, produces something which... | Community | -1 | <p>The inequality is not true: counterexample: </p>
<p>$$f(x) = \sum_{n=1}^\infty \frac 1n \chi_{[n, n+1]}.$$</p>
<p>Then $f\in L^2(\mathbb R)$, but </p>
<p>$$g(x):= \sum_{k\in \mathbb Z} f(\cdot + k) = \sum_{n=1}^\infty \frac 1n = \infty$$</p>
<p>is not in $L^2(0,1)$. </p>
|
2,603,799 | <p>Good morning, i need help with this exercise.</p>
<blockquote>
<p>Prove all tangent plane to the cone $x^2+y^2=z^2$ goes through the origin</p>
</blockquote>
<p><strong>My work:</strong></p>
<p>Let $f:\mathbb{R}^3\rightarrow\mathbb{R}$ defined by $f(x,y,z)=x^2+y^2-z^2$</p>
<p>Then,</p>
<p>$\nabla f(x,y,z)=(2x... | Community | -1 | <p>Actually you want $(a, b, c)$ on the cone... so $a^2+b^2-c^2=0$...</p>
<p>Now the equation of the tangent plane is always satisfied by the origin $(0, 0, 0) $ </p>
<p>Note: you lost a minus sign in your equation (the $z $ -term). .. </p>
<p>Substitute $(x, y, z)=(0, 0, 0) $ to find that the origin satisfies the ... |
2,590,068 | <p>$$\epsilon^\epsilon=?$$
Where $\epsilon^2=0$, $\epsilon\notin\mathbb R$.
There is a formula for exponentiation of dual numbers, namely:
$$(a+b\epsilon)^{c+d\epsilon}=a^c+\epsilon(bca^{c-1}+da^c\ln a)$$
However, this formula breaks down in multiple places for $\epsilon^\epsilon$, yielding many undefined expressions l... | Anixx | 2,513 | <p>Since <span class="math-container">$f(\varepsilon)=f(0)+f'(0)\varepsilon$</span>,</p>
<p>formally we have</p>
<p><span class="math-container">$\varepsilon^\varepsilon=1+\varepsilon+\varepsilon\ln 0$</span></p>
<p>The quantity <span class="math-container">$\ln 0$</span> is infinite with negative sign. We could stop h... |
1,701,935 | <p>I've been experimenting with recursive sequences lately and I've come up with this problem:</p>
<blockquote>
<p>Let $a_n= \cos(a_{n-1})$ with $a_0 \in \Bbb{R}$ and $L=[a_1,a_2,...,a_n,...].$
<br><br>Does there exist an $a_0$ such that $L$ is dense in $[-1,1]?$ </p>
</blockquote>
<p><br><br> I know of $3$ ways... | rtybase | 22,583 | <p>Ok, if you study recursive sequences, then you probably heard about <a href="https://books.google.co.uk/books?id=Bud_HAemxR8C&pg=PA318&lpg=PA318&dq=Lamere%20ladder&source=bl&ots=86YJxwbZFt&sig=HLADUXex5F9Y-ZQ_-XUet3KJ_-M&hl=en&sa=X&ei=KZVEVe_KHY7KaL6CgIgN&ved=0CCMQ6AEwAA#v=one... |
3,319,629 | <p>The question is from a practice exam I am currently trying to do:
<a href="https://i.stack.imgur.com/VWQhs.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VWQhs.png" alt="enter image description here"></a></p>
<p>I am really not sure how to go about this one. In essence, I'd imagine that the idea... | David G. Stork | 210,401 | <p><a href="https://i.stack.imgur.com/f9HL8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/f9HL8.png" alt="enter image description here"></a></p>
<p><span class="math-container">$$\int\limits_{\theta = 0}^{2 \pi} \int\limits_{r = 2 + \sin (3 \theta)}^{4 - \cos (3 \theta)} r\ dr\ d\theta = 12 \pi$$<... |
2,033,485 | <p>I have an equilateral triangle with each point being a known distance of N units from the center of the triangle.</p>
<p>What formula would I need to use to determine the length of any side of the triangle?</p>
| Kunal Chawla | 309,858 | <p>First draw the lines from all three points to the center point of this triangle, then notice that two if these lines make an isosceles triangle with angles $2\pi/3$, $\pi/6$, and $\pi/6$. Since we're given the length of two of the sides of this triangle, and the angles are known, we can calculate the length of the t... |
3,853,351 | <p>Given an n-dimensional ellipsoid in <span class="math-container">$\mathbb{R}^n$</span>, is any orthogonal projection of it to a subspace also an ellipsoid? Here, an ellipsoid is defined as</p>
<p><span class="math-container">$$\Delta_{A, c}=\{x\in \Bbb R^n\,:\, x^TAx\le c\}$$</span></p>
<p>where <span class="math-co... | alphacapture | 334,625 | <p>Yes. An ellipsoid is a linear transformation of a spherical ball, and orthogonal projection is also a linear transformation, so it suffices to show that any linear transformation whose image is a subspace sends a spherical ball to an ellipsoid in that space.</p>
<p>A linear transformation can be decomposed into orth... |
2,674,853 | <blockquote>
<p>Suppose $E \subset \mathbb R^d$ has measure $0$ and $f: \mathbb R^d \longrightarrow \mathbb R$ is measurable. Does $f (E)$ necessarily have measure $0$?</p>
</blockquote>
<p>I tried to find a counter-example though I failed.It is clear that countable subset will not work for otherwise the image of it... | Calvin Khor | 80,734 | <p>Suppose $d>1$. $f(x_1,\dots,x_d) := x_1$ is a measurable map $\mathbb R^d\to\mathbb R$. Furthermore, the set $$E := \{x\in \mathbb R^d \mid x_2 = x_3 = \dots =x_d = 0 \}$$ is null, but $f(E) = \mathbb R$.</p>
|
318,983 | <p>$$\int_{-\infty}^{\infty} \frac{x^2}{x^6+9}dx$$ I'm a bit puzzled as how to go about solving this integral. I can see that it isn't undefined on -infinity to infinity. But I just need maybe a hint on how to go about solving the problem.</p>
| André Nicolas | 6,312 | <p>It is natural to let $u=x^3$. But note that the substitution $x^3=3u$ is more efficient. </p>
|
318,983 | <p>$$\int_{-\infty}^{\infty} \frac{x^2}{x^6+9}dx$$ I'm a bit puzzled as how to go about solving this integral. I can see that it isn't undefined on -infinity to infinity. But I just need maybe a hint on how to go about solving the problem.</p>
| DonAntonio | 31,254 | <p>With complex analysis: define the complex function</p>
<p>$$f(z)=\frac{z^2}{z^6+9}\;\;,\;\;\text{with simple poles at}\,\,\,z_k:=\sqrt[6] 9\,\,e^{\frac{\pi i}{6}(1+2k)}\;\;,\;k=0,1,2,3,4,5$$</p>
<p>Note that the only poles on the upper half plane are the first three ones $\,z_0,\,z_1,\,z_2\,$ , with residues</p>
... |
213,639 | <blockquote>
<p>Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom?</p>
</blockquote>
<p>This question was to some extent provoked by <a href="https://math.stackexchange.com/users/3515/dan-christensen">Dan Christensen</a>'s <a ... | Asaf Karagila | 622 | <p><em>Basic</em> mathematics was done long before set theory. Its users couldn't care less whether or not ZF is the underlying theory of the universe or some other theory. As long as it works fine.</p>
<p>There are a few things to point out:</p>
<ol>
<li><p>The natural numbers (standard model of PA), on which classi... |
3,536,061 | <p>Find the number of ways you can invite <span class="math-container">$3$</span> of your friends on <span class="math-container">$5$</span> consecutive days, exactly one friend a day, such that no friend is invited on more than two days. </p>
<p>My approach: Let <span class="math-container">$d_A,d_B$</span> and <span... | WaveX | 323,744 | <p>The fact that the only way to achieve this is by inviting one friend over once and the other two twice will make this problem simpler.</p>
<p>How many ways are there to pick the one friend (from <span class="math-container">$3$</span>) that will be only visiting one day instead of two?</p>
<p>Now if we assume the ... |
251,182 | <p>Is 13 a quadratic residue of 257? Note that 257 is prime.</p>
<p>I have tried doing it. My study guide says it is true. But I keep getting false. </p>
| Oscar Lanzi | 248,217 | <p>Quadratic reciprocity (QR) is the way to go, but a little ingenuity simplifies the solution.</p>
<p>Start with $(13|257)=(257|13)=(10|13)$. We can't directly apply QR again because $10$ is even. But, since $13$ is a prime one greater than a multiple of $4$, $(10|13)=(-10|13)=(3|13)$. Now we apply QR again: $(3|... |
1,006,562 | <p>So I am trying to figure out the limit</p>
<p>$$\lim_{x\to 0} \tan x \csc (2x)$$</p>
<p>I am not sure what action needs to be done to solve this and would appreciate any help to solving this. </p>
| Crostul | 160,300 | <p>Note that $\csc (2x) = \frac{1}{\sin(2x)} = \frac{1}{2 \sin x \cos x}$, and $\tan x = \frac{\sin x}{\cos x}$.</p>
<p>So $$\lim_{x \to 0} \tan x \csc (2x) = \lim_{x \to 0} \frac{1}{2 \sin x \cos x} \frac{\sin x}{\cos x} = \lim_{x \to 0} \frac{1}{2 \cos^2 x} = \frac{1}{2}$$</p>
|
2,428,243 | <p>How can I evalute this product??</p>
<p>$$\prod_{i=1}^{\infty} {(n^{-i})}^{n^{-i}}$$</p>
<p>Unfortunately, I have no idea.</p>
| neonpokharkar | 477,567 | <p>Taking log
$$\ln\Big( \prod_{i=1}^{\infty} (n^{-i})^{n^{-i}}\Big)$$$$$$
$$\sum_{i=1}^{\infty} \frac{i}{n^i} \ln n$$$$$$
$$\ln n \sum_{i=1}^{\infty} \frac{i}{n^i} $$$$$$</p>
<blockquote>
<p>Let$$$$$$S= \sum_{i=1}^{k} \frac{i}{n^i}$$ $$$$
$$S=\frac{1}{n}+\frac{2}{n^2}+\frac{3}{n^3}+\cdots+\frac{k}{n^k}$$$$$$
Mu... |
3,265,403 | <p>While trying to compute the line integral along a path K on a function, I need to parametrize my path K in terms of a single variable, let's say this single variable will be <span class="math-container">$t$</span>.
My path is defined by the following ensemble: <span class="math-container">$$K=\{(x,y)\in(0,\infty)\ti... | TurlocTheRed | 397,318 | <p><span class="math-container">$$x^2-y^2=1600$$</span>
<span class="math-container">$$(\frac{x}{40})^2 - (\frac{y}{40})^2=1$$</span></p>
<p>Let <span class="math-container">$x=40\cdot(e^t+e^-t)/2=40\cdot\cosh{t}$</span>.</p>
<p>Let <span class="math-container">$y=40\cdot (e^t-e^{-t})/2=40 \cdot\sinh{t}$</span></p>
... |
3,245,270 | <p>From Statistical Inference by Casella and Berger:</p>
<blockquote>
<p>Let <span class="math-container">$X_1, \dots X_n$</span> be a random sample from a discrete distribution
with <span class="math-container">$f_X(x_i) = p_i$</span>, where <span class="math-container">$x_1 \lt x_2 \lt \dots$</span> are the pos... | Henno Brandsma | 4,280 | <p>A pseudometric space is symmetric (also called <a href="https://en.wikipedia.org/wiki/T1_space" rel="nofollow noreferrer"><span class="math-container">$R_0$</span></a>): if <span class="math-container">$x \in \overline{\{y\}}$</span> then <span class="math-container">$y \in \overline{\{x\}}$</span> (basically becaus... |
3,743,673 | <p>Using calculus to find the minima:</p>
<p><span class="math-container">$$y(x) = x^x$$</span></p>
<p><span class="math-container">$$ln(y) = x*ln(x)$$</span></p>
<p><span class="math-container">$$(1/y)*\frac{dy}{dx} = ln(x) + x*\left(\frac{1}{x}\right) = ln(x) + 1$$</span></p>
<p><span class="math-container">$$\frac{d... | Arthur | 15,500 | <p>I don't think that's a bold assumption. For instance, <span class="math-container">$x^x=e^{x\ln x}$</span> is never zero, as <span class="math-container">$e$</span> raised to the power of any real number is strictly positive.</p>
<p>Alternatively, there are no strictly positive <span class="math-container">$a,b$</sp... |
3,743,673 | <p>Using calculus to find the minima:</p>
<p><span class="math-container">$$y(x) = x^x$$</span></p>
<p><span class="math-container">$$ln(y) = x*ln(x)$$</span></p>
<p><span class="math-container">$$(1/y)*\frac{dy}{dx} = ln(x) + x*\left(\frac{1}{x}\right) = ln(x) + 1$$</span></p>
<p><span class="math-container">$$\frac{d... | Sameer Baheti | 567,070 | <blockquote>
<p>Show that <span class="math-container">$x^x = a$</span> has no real solutions when <span class="math-container">$a < \left(\frac1e\right)^{\frac1e}$</span>.</p>
</blockquote>
<p>Enough to visualize that</p>
<ul>
<li><span class="math-container">$\displaystyle\lim_{x\to 0}x^x=\displaystyle\lim_{x\to 0... |
1,085,511 | <p>What would be the irrational number $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers given by this expression:
$$
\left(
\begin{array}{@{}c@{}}2207-\cfrac{1}{2207-\cfrac{1}{2207-\cfrac{1}{2207-\dotsb}}}\end{array}
\right)^{1/8}
$$</p>
| Alex Silva | 172,564 | <p><strong>Hint:</strong></p>
<p>$$x = 2207 - \frac{1}{x},$$ where</p>
<p>$$x = 2207-\dfrac{1}{2207-\dfrac{1}{2207-\dfrac{1}{2207...}}}.$$</p>
|
555,446 | <p>Given this shape: <img src="https://i.stack.imgur.com/1rRsC.png" alt="diagram showing a 4000 unit wide cyan square with a 400 unit wide red square in the middle"></p>
<h1>Is it possible to divide the cyan area into 5 equal area shapes</h1>
<p>such that:</p>
<ol>
<li>Each shape is the same</li>
<li>Each shape has ... | RicardoCruz | 36,340 | <p>What about the shape below made in geogebra?
<img src="https://i.stack.imgur.com/qoYrh.jpg" alt="enter image description here"></p>
|
2,713,873 | <p>We know that if a real valued function $f$ is continuous over an interval $[a,b]$ then the following integral $$\int_a^bf(x)dx$$ represents the area between horizontally the line $y=0$ and the curve of $f$, vertically between the lines $x=a$ and $x=b$. So what represent the following $$\int_{[a,b]\times [c,d]}g(x... | HBR | 396,575 | <p>The double integral, represents the sum of the infinite areas under the curves $g(x,y)$ with $x=constant$ defined as:
$$A(y)=\int{g(x,y)\,dx}$$
each one between $y$ and $y+dy$, which gives you a volume. (see <a href="https://en.m.wikipedia.org/wiki/Fubini%27s_theorem" rel="nofollow noreferrer">Fubini</a> 's explanat... |
4,310,003 | <p>Suppose you have a non empty set <span class="math-container">$X$</span>, and suppose that for every function <span class="math-container">$f : X \rightarrow X$</span>, if <span class="math-container">$f$</span> is surjective, then it is also injective. Does it necessarily follow that <span class="math-container">$... | 2 is even prime | 994,639 | <p><span class="math-container">$$\int_{-\infty}^\infty \frac{e^x}{1 + e^{4x}}dx$$</span>
Split the integral at <span class="math-container">$0$</span> and bring them between same bounds i.e <span class="math-container">$[0,\infty]$</span>.
<span class="math-container">$$\int_{0}^{\infty}\frac{e^{3x}+e^{x}}{e^{4x}+1} d... |
151,430 | <p>Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore ho... | Dan Petersen | 1,310 | <p>The easiest definition is via the Pontrjagin--Thom construction (I think). Let $N$ be a tubular neighbourhood of $Y$, isomorphic to the normal bundle. Let $X'$ be the space obtained from $X$ by collapsing the complement of $N$ to a point. Then $X'$ is isomorphic to the Thom space of the normal bundle, and if the nor... |
151,430 | <p>Let $Y\subset X$ be a codimension $k$ proper inclusion of submanifolds. If we choose a coorientation of $Y$ inside of $X$ (that is, an orientation of the normal bundle), then we get a class $[Y]\in H^k(X)$. If $X$ and $Y$ are oriented, then $[Y]$ may be defined as the fundamental class of $Y$ in the Borel-Moore ho... | John Pardon | 35,353 | <p>Your construction for the oriented case extends to the unoriented case just by using twisted coefficients.</p>
<p>Every manifold $Y$ has a fundamental class $[Y]\in H_\ast^{\mathrm{lf}}(Y;\mathfrak o_Y)$ (where $\mathfrak o_Y$ is the orientation sheaf of $Y$).
The coorientation of $Y\subseteq X$ gives an identifica... |
1,787,806 | <p>I've recently had this problem in an exam and couldn't solve it.</p>
<p>Find the remainder of the following sum when dividing by 7 and determine if the quotient is even or odd:</p>
<p>$$\sum_{i=0}^{99} 2^{i^2}$$</p>
<p>I know the basic modular arithmetic properties but this escapes my capabilities. In our algebra... | user90369 | 332,823 | <p>E.g. use </p>
<p>$\sum\limits_{i=0}^{99}2^{i^2}=\sum\limits_{i=0}^{32}(2^{(3i)^2}+2^{(3i+1)^2}+2^{(3i+2)^2})+2^{3(3(33^2))}$ </p>
<p>and $2^{3n}=(7+1)^n$ divided by $7$ has the remainder $1$ for any integer $n$. </p>
<p>The rest should be clear.</p>
|
4,064,684 | <p>I'm working through some notes for my signal processing class and there's something elementary that baffles me. We spent lots of hours and dozens of pages setting up the entire theory of Hilbert spaces in order to define the Fourier series of a square integrable periodic function in terms of the orthogonal basis of ... | paul garrett | 12,291 | <p>A good question! First, even though <span class="math-container">$L^2$</span> convergence does not imply pointwise convergence, Parseval's theorem says that the Fourier series converges in that sense, whether or not pointwise. The standard elementary curriculum does seem to urge us to think that the only reasonable ... |
4,064,684 | <p>I'm working through some notes for my signal processing class and there's something elementary that baffles me. We spent lots of hours and dozens of pages setting up the entire theory of Hilbert spaces in order to define the Fourier series of a square integrable periodic function in terms of the orthogonal basis of ... | Alex R. | 22,064 | <p>In addition to paul's answer, one thing to keep in mind is that Hilbert spaces are typically meaningless when it comes to function values at individual points. An often glanced over fact is that if you take a function <span class="math-container">$f\in L^2$</span> and modify it at a single point <span class="math-co... |
2,637,914 | <p>I would like to teach students about the pertinence of the Axiom of Infinity. Are there any high school-level theorems of arithmetic, algebra, or calculus, whose proof depends on the Axiom of Infinity? If there are no such examples, what would be the simplest theorem which demands the Axiom of Infinity?</p>
<p>It... | Sarvesh Ravichandran Iyer | 316,409 | <p>The base case $n=1$ is clear.</p>
<p>For the induction case, start with what you know is true, whch is the statement for $n$ : $\sum_{i=1}^n \frac{i}{i+1} \leq \frac{n^2}{n+1}$. If we show that each time the left hand side's increase is slower than the right hand side then we are done.</p>
<p>When we put $n+1$ in ... |
1,896,024 | <p><span class="math-container">$f(n) = 2n^2 + n$</span></p>
<p><span class="math-container">$g(n) = O(n^2)$</span></p>
<p>The question is to find the mistake in the following process:</p>
<blockquote>
<p><span class="math-container">$f(n) = O(n^2) + O(n)$</span></p>
<p><span class="math-container">$f(n) - g(n) = O(n^... | Paolo Leonetti | 45,736 | <p>When you write $O(h(x))$ (as $x \to \infty$, for some function $h$) you have to imagine that it is written "something (in modulo) which is at most $h(x)$ (up to a constant)", giving you an upper bound for the growth of your function.</p>
<p>This is why, if you write $O(h(x))-O(h(x))$, then it does <em>not</em> simp... |
47,792 | <p>I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ <em>is of S-type</em> if it is continuous, unramified almost everywhere, and the determinant of complex conjugation is $-1$. Serre... | Emerton | 2,874 | <p>In what sense? If you mean "come from the reduction of $\rho_f$ for some Hecke eigenform $f$'', no, they are not.</p>
<p>If you mean "come from the reduction of $\rho$ where $\rho:G_{\mathbb Q} \to
GL_2(\mathbb T)$ is the Galois rep'n attached to the Hecke algebra $\mathbb T$ acting on
modular forms of some suffic... |
47,792 | <p>I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ <em>is of S-type</em> if it is continuous, unramified almost everywhere, and the determinant of complex conjugation is $-1$. Serre... | Community | -1 | <p>To elaborate on Emerton's answer, an arbitrary (finitely ramified and odd) representation <span class="math-container">$\rho_n:G_{\mathbb{Q}}\rightarrow GL_2(\mathbb{Z}/p^n\mathbb{Z})$</span> can't lift to one coming from an eigenform because of some reasons which may be explained locally. Let <span class="math-cont... |
1,743,935 | <p>Not sure if I have done this correctly, seems too straight forward, any help is very appreciated. </p>
<blockquote>
<p>QUESTION:<br>
Find the real and imaginary parts of $f(z) = \cos(z)$.</p>
</blockquote>
<p>ATTEMPT:<br>
$\cos(z) = \cos(x+iy) = \cos x\cos(iy) − \sin x\sin(iy) =
\cos x\cosh y − i\sin x\sinh y... | Brian | 331,755 | <p>It is simple, but tis the beauty of the trig/exponential functions! You're $correct$!</p>
|
1,061,077 | <p>SO, I drop a piece of bread and jam repeatedly. It lands either jam face-up or jam face-down and I know that jam side down probability is <span class="math-container">$P(Down)=p$</span></p>
<p>I continue to drop the bread until it falls jam side up for the first time. What is the expression for the expected number o... | Frank Mayer | 197,167 | <p>Average number of throws to get first up results = $\Sigma N Q_{N}$ where $Q_{N}$ is the probability of getting first $N-1$ throws down, then an up. </p>
<p>$Q_{N} = p^{N-1} (1 - p)$ So $\Sigma N Q_{N} = (1 - p) \Sigma Np^{N-1} = \frac{1 - p}{(1 - p)^2}= \frac{1}{(1 - p)}$</p>
<p>The probability distribution funct... |
3,189,173 | <p>What will be the remainder when <span class="math-container">$2^{87} -1$</span> is divided by <span class="math-container">$89$</span>?</p>
<p>I tried it solving by Euler's remainder theorem by separating terms:</p>
<p><span class="math-container">$$ \frac {2^{87}}{89} - \frac{1}{89}$$</span></p>
<p><span class="... | J. W. Tanner | 615,567 | <p><span class="math-container">$89$</span> is prime, so by Fermat's little theorem <span class="math-container">$2^{88}\equiv 1\pmod {89},$</span> so <span class="math-container">$2^{87}\equiv 2^{-1} \pmod {89}$</span>.</p>
<p>Now <span class="math-container">$45\times 2 = 90 \equiv 1 \pmod {89},$</span> so this mean... |
3,073,832 | <p>I need to understand the meaning of this mathematical concept: "undecided/undecidable". </p>
<p>I know what it means in the English dictionary. But, I don't know what it means mathematically.</p>
<p>If You answer this question with possible mathematical examples, it will be very helpful to understand this issue.<... | user247327 | 247,327 | <p>Saying that a statement is "undecidable" means that there can be no proof, even theoretically, that the true nor can there be a proof that it is false.</p>
|
2,913,017 | <p>Imagine I have a real random variable $X$ with some distribution (continuous, discrete or continuous with atoms)</p>
<p>Now Imagine I have i.i.d. copies $X_1,...,X_n$, all independently and equally distributed as $X$</p>
<p>My claim is:</p>
<p>$$\mathbb{P}(X_2>X_1)=\mathbb{P}(X_2<X_1)$$
My secondy claim is ... | user97678 | 591,449 | <p>You're asking whether $P(X > \max({X_1, \ldots, X_n})) = P(X < \min({X_1, \ldots, X_n}))$, where $X_1, \ldots, X_n$ and $X$ are all independent and distributed the same.</p>
<p>This is true in some distributions and false in others. For example if the $X_i$s are sampled uniformly at random from $[0,1]$ then t... |
3,262,714 | <p>So, I need an exponential function on the form <span class="math-container">$e^{-ax}$</span> that is 1 at <span class="math-container">$x=0$</span> and approaches <span class="math-container">$0.3$</span> as <span class="math-container">$x \rightarrow \infty$</span>. I tried doing <span class="math-container">$e^{-a... | Community | -1 | <p>Multiply <span class="math-container">$e^{-ax}$</span> by <span class="math-container">$0.7$</span>. So, consider <span class="math-container">$0.7e^{-ax}+ 0.3$</span>.</p>
|
3,464,615 | <p>A novel process of manufacturing laptop screens is under test. In recent tests, it is found that 75% of the screens are acceptable. What is the most probable number of acceptable screens in the next batch of 10 screens and what is the probability?</p>
<p>Does that mean 7 screens out of 10 will pass with a probabili... | David G. Stork | 210,401 | <p>Here's the relevant binomial distribution:</p>
<p><a href="https://i.stack.imgur.com/qyWR1.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qyWR1.png" alt="enter image description here"></a></p>
<p>If you're forced to choose an integer, then <span class="math-container">$n=8$</span>, and <span cl... |
416,153 | <p>Show that $x^2+y^2=p$ has a solution in $\mathbb{Z}$ if and only if $ p≡1 \mod 4$. Thnx, if someone can help</p>
| Zach L. | 43,128 | <p>My favorite way of doing this:</p>
<p>$\mathbb{Z}[i]$ is a UFD, so this has a solution iff $\mathbb{Z}[i]/(p)$ is not an integral domain. But $\mathbb{Z}[i]$ is constructed via a quotient, so you can show it's isomorphic to $\mathbb{Z}_p[t]/(x^2 + 1)$, where I'm using $\mathbb{Z}_p$ to denote $\mathbb{Z}/p\mathbb{Z... |
3,042,802 | <p>For each <span class="math-container">$n ≥ 1$</span>, let <span class="math-container">$T_n = \{x ∈ l_2(N) : ||x||_1 ≤ n \}$</span>.</p>
<p>For <span class="math-container">$n ≥ 1$</span>, is <span class="math-container">$T_n$</span> an absorbing subset of <span class="math-container">$l_2(N) $</span>, but why?
I w... | David C. Ullrich | 248,223 | <p>Assuming <span class="math-container">$N=\Bbb N$</span>: <strong>Hint:</strong> If <span class="math-container">$x=???$</span> then <span class="math-container">$x\in\ell_2(N)$</span> but <span class="math-container">$||x||_1=\infty$</span>.</p>
|
358,102 | <p>How would I go about doing this?</p>
<p>I assume it is some integral I have to solve, but I have no idea what.</p>
<p>(Note:Not a physicist so please excuse incompetence with regard standard notation.)</p>
<p>Context is I want to estime the energy of N point particles spread over the unit sphere. This is an equat... | Douglas S. Stones | 139 | <p>We can show that belonging to the same strongly connected component is an <a href="http://en.wikipedia.org/wiki/Equivalence_relation" rel="nofollow">equivalence relation</a>. The strongly connected components thus form the equivalence classes under this relation. We can also show that in any equivalence relation n... |
187,974 | <p>If $ \cot a + \frac 1 {\cot a} = 1 $, then what is $ \cot^2 a + \frac 1{\cot^2 a}$? </p>
<p>the answer is given as $-1$ in my book, but how do you arrive at this conclusion?</p>
| lab bhattacharjee | 33,337 | <p>Taking $x=\cot a$, $x+\frac{1}{x}=1\implies x^2+\frac{1}{x^2}=(x+\frac{1}{x})^2-2x\frac{1}{x}=1-2=-1$</p>
<p>Alternatively, $x+\frac{1}{x}=1\implies x^2-x+1=0$</p>
<p>$x^2-x+1=0\implies x^3+1=0$</p>
<p>So,</p>
<p>$x^{3m}+(\frac{1}{x})^{3m}=(x^3)^m+\frac{1}{(x^3)^m}=2(-1)^m$</p>
<p>$x^{3m+1}+(\frac{1}{x})^{3m+1}... |
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | stoicfury | 13,615 | <p>Here's the equations typed out if you want save time with writing it yourself.</p>
<pre><code>(x/7)^2*SQRT(ABS(ABS(x)-3)/(ABS(x)-3))+(y/3)^2\*SQRT(ABS(y+3*SQRT(33)/7)/(y+3*SQRT(33)/7))-1=0
ABS(x/2)-((3*SQRT(33)-7)/112)*x^2-3+SQRT(1-(ABS(ABS(x)-2)-1)^2)-y=0
9*SQRT(ABS((ABS(x)-1)*(ABS(x)-0.75))/((1-ABS(x))*(ABS(x)-0.... |
54,506 | <p><a href="http://www.hardocp.com/news/2011/07/29/batman_equation/" rel="noreferrer">HardOCP</a> has an image with an equation which apparently draws the Batman logo. Is this for real?</p>
<p><img src="https://i.stack.imgur.com/VYKfg.jpg" alt="Batman logo"></p>
<p><strong>Batman Equation in text form:</strong>
\beg... | Community | -1 | <p>You may be able to see more easily the correspondences between the equations and the graph through the following picture <s>which is from the <a href="http://everythingnew.net/wp-content/uploads/2011/07/Batman-Equation-solved.png" rel="noreferrer">link</a> I got after a curious search on Google</s>(link broken now):... |
914,936 | <p>Does anyone know where I can find the posthumously published (I think) chapter 8 of Gauss's Disquisitiones Arithmaticae?</p>
| Maths Nerd | 209,035 | <p>I have read some information that it may have been published as 'General Investigations of Curved Surfaces' however I am having some difficulty confirming.</p>
|
1,943,328 | <p>I know about $S_n$, $D_n$ and $A_n$. And from my limited understanding there seem to be many more. I would like to know whether there is some kind of relation that links a small set of non Abelian groups to create the other ones. Something like with the Abelian groups and the Fundamental Theorem of Abelian Groups.</... | quid | 85,306 | <p>In general there is a huge number of ways how to combine certain groups.
The relatively most extreme being when the order of the groups are all powers of two.</p>
<p>For example, already for groups with $32$ elements there are $51$ different groups. And it only gets much worse, even for a relatively modest number ... |
1,946,824 | <p>In his book "Analysis 1", Terence Tao writes:</p>
<blockquote>
<p>A logical argument should not contain any ill-formed
statements, thus for instance if an argument uses a statement such
as x/y = z, it needs to first ensure that y is not equal to zero.
Many purported proofs of “0=1” or other false statements... | Patrick Stevens | 259,262 | <p>It does make sense.</p>
<p>Define $S_n = \{ F_0, F_1, F_2, \dots, F_n \}$. (I think you'll agree that this set exists for every $n$.)</p>
<p>Then let $$S = \bigcup_{n \in \mathbb{N}} S_n$$</p>
<p>Since each $S_n$ exists, and $\mathbb{N}$ certainly exists, the union does as well (we have an axiom, the Axiom of Uni... |
257,562 | <p>Given a piecewise function, such as</p>
<p>$$f(t) =
\begin{cases}
2, & \text{if }t \lt a \\
t^2, & \text{if }t \geq a
\end{cases}$$</p>
<p>Or some other piecewise function, how can we write it in the form $u(t-a)f(t-a)$ for $$u(t-a) =
\begin{cases}
0, & \text{if }t \lt a \\
1, & \text{if }t \geq a
... | Artem | 29,547 | <p>\begin{align}
f(t)&=2(u(t)-u(t-a))+t^2u(t-a)\\
&=2u(t)-2u(t-a)+(t-a+a)^2u(t-a)=\ldots
\end{align}</p>
|
257,562 | <p>Given a piecewise function, such as</p>
<p>$$f(t) =
\begin{cases}
2, & \text{if }t \lt a \\
t^2, & \text{if }t \geq a
\end{cases}$$</p>
<p>Or some other piecewise function, how can we write it in the form $u(t-a)f(t-a)$ for $$u(t-a) =
\begin{cases}
0, & \text{if }t \lt a \\
1, & \text{if }t \geq a
... | JohnD | 52,893 | <p>Think of $u(t-a)$ as simply a "switch" that is "off" for $t<a$ and so has a value of $0$, but turns on at $t=a$ with a value of $1$ and remains "on" for $t\ge a$. This is why the function is called the <em>unit step function</em> (activated at $t=a$): <em>unit</em> because it has a value of one and <em>step</em> ... |
2,220,738 | <p>How do I go about finding the dimension of the subspace:
$$$$S:={p($x$) ∈ $P_4$: p($x$)= 2p($x$) for all $x\in\mathbb{R}$} of $P_4$</p>
<p>My textbook says $dim(P_n)=n+1$, but this does not give me the correct answer. All help is appreciated.</p>
| Dude | 490,091 | <p>I'm extremely confused by the equivalence drawn in linear algebra between vectors in Rn and polynomials. Can someone please point out the flaws in my understanding?</p>
<p>For Rn, a vector (1, 2, 3) has components in three different dimensions (along three different axes), so adding these together is impossible, ri... |
182,785 | <p>I haved plot a graph from two functions:</p>
<pre><code>η = 52;
h = 0.5682;
dpdx = -4.092*10^(-2);
Fg = dpdx;
Fl = dpdx/η;
Bl = ((Fg - Fl) h^2 - Fg)/(2 h - 2 η*h + 2 η);
Cg = -Fg/2 - η*Bl;
Bg = η*Bl;
Ut1[y_] := Fg*y^2/2 + Bg*y + Cg;
Ut2[y_] := Fl*y^2/2 + Bl*y;
Plot1 = Plot[Ut1[y]*1000, {y, h, 1}];
Plot2 = Plot[U... | David G. Stork | 9,735 | <p>Define myFigure as your figure and then:</p>
<pre><code>Rotate[myFigure, -90 Degree]
</code></pre>
<p><a href="https://i.stack.imgur.com/wkR7A.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/wkR7A.png" alt="enter image description here"></a></p>
|
2,801,433 | <p>I have made the following conjecture, and I do not know if this is true.</p>
<blockquote>
<blockquote>
<p><strong>Conjecture:</strong></p>
</blockquote>
<p><span class="math-container">\begin{equation*}\sum_{n=1}^k\frac{1}{\pi^{1/n}p_n}\stackrel{k\to\infty}{\longrightarrow}2\verb| such that we denote by | p_n\verb| ... | Akash Roy | 545,756 | <p>$\sum_{n=1}^{\infty} \frac{1}{s_n}$is a divergent series therefore if you tend the limit to $\infty$ ;
$\lim_{n\to \infty}\pi^{1/n}=1$
You can see that the above term tends to $1$.
The remaining term is similar to above mentioned divergent series. </p>
|
1,355,509 | <p>In my mathematical travels, I've stumbled upon the implicit formula $y^2+x^2+\frac{y}{x}=1$ and found that every graphing program I've plugged it in to seems to believe that there is large set of points which satisfy the equation $(y^2+x^2+\frac{y}{x})^{-1}=1$ which do not satisfy the original equation and this has ... | Tobia | 66,257 | <p>Here is a color plot (aka. scalar field) of $$x^2+y^2+{y\over x}$$
White is around zero, gray is positive and red is negative. The thick line is $0$ and the thin line is $1$, your original curve.</p>
<p><img src="https://i.stack.imgur.com/cOun9.png" alt="color plot"></p>
<p>It's easy to see that by taking the inve... |
211,533 | <p>I am trying to automate the process of running a Mathematica notebook that takes as an input a .txt file and exports another .txt file. This notebook was not written by me, and I don't fully understand its inner workings, but I think all I need to know is the following parts of the code:</p>
<p>The input statement ... | kickert | 54,320 | <p>Pure functions will be helpful here. Start by using <code>Table</code> to automatically create the 100 files names you plan on entering.</p>
<pre><code>importnames=Table[StringJoin["my_file_name_", ToString[i], ".txt"], {i, 1, 100}]
</code></pre>
<p>Since this will all be text, you can Import them all at once and... |
211,533 | <p>I am trying to automate the process of running a Mathematica notebook that takes as an input a .txt file and exports another .txt file. This notebook was not written by me, and I don't fully understand its inner workings, but I think all I need to know is the following parts of the code:</p>
<p>The input statement ... | OpticsMan | 58,442 | <p>Not knowing all of the details, I suggest using an additional Notebook to automate the operation of "my_mathematica_notebook.nb"</p>
<p>In the new Notebook</p>
<pre><code> names = FileNames["*.txt", {".", "*"}, 1]
num = Dimensions[names][[1]]
(* finds all txt files *)
process[index_]:=Module[{}
... |
798,897 | <p>In our lecture we ran out of time, so our prof told us a few properties about measure: He said that a measure is $\sigma$-additive iff it has a right-side continuous function that it creates. And he was not only referring to probability measures.
After going through my lecture notes, I thought that this would imply... | C-star-W-star | 79,762 | <p><strong>The statement is certainly wrong by consistency:</strong></p>
<p>Right-side continuous functions are defined/only make sense for functions whos domain is the real line (actually a right-closed subset of the real line).
Thus it gives rise at most for measures over the reals (or over a right-closed subset of ... |
4,314,162 | <p>Assume k is a finite field with n elements, how many elements are in the projective line <span class="math-container">$\mathbb{P}^{1}(k)$</span> and how do I work this out?</p>
<p>I know that an element of <span class="math-container">$\mathbb{P}^{1}(k)$</span> is represented by <span class="math-container">$[a, b]$... | Wuestenfux | 417,848 | <p>The elements are (in homogeneous coordinates):</p>
<p><span class="math-container">$$(0,1),(1,1),\ldots, (q-1,1), (1,0),$$</span></p>
<p>where <span class="math-container">$k = \{0,1,\ldots,q-1\}$</span> has <span class="math-container">$q$</span> elements. So the number of elements is <span class="math-container">$... |
2,958,135 | <p>A "standard" example of Bayes Theorem goes something like the following:</p>
<blockquote>
<p>In any given year, 1% of the population will get disease <em>X</em>. A particular test will detect the disease in 90% of individuals who have the disease but has a 5% false positive rate. If you have a family history of <... | CyclotomicField | 464,974 | <p>I believe it's commonly included because it's counterintuitive. You would expect a test with a high degree of accuracy to be right most of the time but this isn't actually the case and requires more evidence. To address this I think of it as the "error of one sample" fallacy which is to say you can't do an experimen... |
333,467 | <p>I was reading in my analysis textbook that the map $ f: {\mathbf{GL}_{n}}(\mathbb{R}) \to {\mathbf{GL}_{n}}(\mathbb{R}) $ defined by $ f(A) := A^{-1} $ is a continuous map. I also saw that $ {\mathbf{GL}_{n}}(\mathbb{R}) $ is dense in $ {\mathbf{M}_{n}}(\mathbb{R}) $. My question is:</p>
<blockquote>
<p>What is t... | Elchanan Solomon | 647 | <p>The matrices</p>
<p>$$ A_{n} = \begin{pmatrix} \frac{1}{n} & 0\\ 0 & \frac{1}{n} \end{pmatrix}$$</p>
<p>converge to the zero matrix as $n \to \infty$. Their inverses are</p>
<p>$$ A_{n} = n \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$$</p>
<p>And this goes to infinity in matrix norm as $n\to \inf... |
2,362,354 | <p>I have seen a proof that B is row equivalent to A iff exist invertible matrix C such that :
B = CA , because C is elementary matrix , but i cant find the next step . can B "used" as the elementary matrix that lead from B to BA without changing the row space of A , so A and BA have same rowspace? </p>
| Siong Thye Goh | 306,553 | <p>if $B$ is an invertible matrix, its reduced row echelon form is the identity matrix.</p>
<p>That is I can write I can elementary matrices $E_i$ such that </p>
<p>$$E_m \ldots E_1 B = I$$</p>
<p>$$B=E_1^{-1}\ldots E_m^{-1}$$</p>
<p>Hence I can write $B$ as a product of elemntary matrices.</p>
<p>$$BA=E_1^{-1}\ld... |
2,362,354 | <p>I have seen a proof that B is row equivalent to A iff exist invertible matrix C such that :
B = CA , because C is elementary matrix , but i cant find the next step . can B "used" as the elementary matrix that lead from B to BA without changing the row space of A , so A and BA have same rowspace? </p>
| Vim | 191,404 | <p>Well, since $B=CA$, you see that rows of $B$ can be written as linear combinations of rows of $A$. (To be more explicit, write
$$B=\begin{bmatrix}\mathbf{b_1\\b_2\\ \vdots \\ b_m}\end{bmatrix},\quad A=\begin{bmatrix}\mathbf{a_1\\a_2\\ \vdots, \\ a_m}\end{bmatrix},\quad C=\begin{bmatrix}c_{11} &\cdots & c_{1... |
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