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 |
|---|---|---|---|---|
1,193,472 | <p>Let $ X \subseteq \mathbb{R}^n$ be compact and $ f:X \to \mathbb{R} $ be a function such that for all $ t \in \mathbb{R} $, the set $ f^{-1} [t, \infty) $ is closed. Which one of the following statement is correct and why the rest are incorrect? </p>
<p>1) There exists a $ x_0 \in X $ such that $$ f(x_0)= \sup_{x\... | HK Lee | 37,116 | <p>(1) Since $A_t:= f^{-1}[t,\infty)$ is closed in $X$ then $A_t$ is
compact.</p>
<p>Assume that if $x_n\in A_t,\ f(x_n)\rightarrow \infty$, then since
$A_t$ is compact so there exists convergent subsequence $x_{n_k}$.
And in further its limit $x_\infty $ is in $A_t$. Let
$$f(x_\infty)=x_0\in [t,\infty)$$</p>
<... |
806,779 | <p>Prove, without expanding, that
\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix} vanishes.</p>
<p>Any hints ?</p>
| mfl | 148,513 | <p>\begin{align}
\begin{vmatrix}
1 & a & a^2-bc \\
1 & b & b^2-ca \\
1 & c & c^2-ab
\end{vmatrix}
&=\begin{vmatrix}
1 & a & a^2-bc \\
0 & b-a & b^2-a^2+bc-ca \\
0 & c-b & c^2-b^2+ca-ab
\end{vmatrix} \\
&=\begin{vmatr... |
806,779 | <p>Prove, without expanding, that
\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix} vanishes.</p>
<p>Any hints ?</p>
| Community | -1 | <p>Let</p>
<p>$$f(a)=\begin{vmatrix}
1 &a &a^2-bc \\
1 &b &b^2-ca \\
1 &c &c^2-ab
\end{vmatrix}$$
then it's easy to see that $f$ is a polynomial on $a$ with degree at most $2$ and $f(b)=f(c)=0$ so
$$f(a)=\lambda(a-b)(a-c)$$
now, WLOG assume that $bc\ne0$, take $a=0$; we see easily that $f... |
2,294,991 | <p>I am trying to find the limit as $x\to 8$ of the following function. What follows is the function and then the work I've done on it. </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}$$</p>
<hr>
<p>\begin{align}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8} &= \frac{\frac{1}{\sqrt{x +1... | lab bhattacharjee | 33,337 | <p>Let $\sqrt{x+1}-3=h\implies x=(3+h)^2-1$ </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}=\lim_{h\to0^+}\dfrac{\dfrac1{h+3}-\dfrac13}{(3+h)^2-9}=\lim_{h\to0^+}\dfrac{3-(h+3)}{3h(6+h)(h+3)}=?$$</p>
<p>Cancel out $h$ safely as $h\ne0$ as $h\to0$</p>
|
156,474 | <p>Is this phrase safe to consider in general: </p>
<blockquote>
<blockquote>
<p>Equal sides of a polygon have corresponding equal angles</p>
</blockquote>
</blockquote>
<p>if not how would you refine or correct it</p>
<p>Example of a corresponding angle would be</p>
<p><img src="https://i.stack.imgur.com/o... | Valentin | 31,877 | <p>There is a statement about <em>equal</em> triangles that says: in equal triangles the angles opposite to equal sides are equal, i.e. is $\Delta ABC$ = $\Delta A_1B_1C_1$, then $AB=A_1B_1\implies \angle C =\angle C_1$. If you want to generalise this in any direction, you must say something about the polygons in quest... |
2,002,672 | <p>I am ok with finding a power series for $\tanh^{-1}{z}$ since I can just see that since $\tanh iz = i\tan z$, then $\tanh^{-1}z = \frac1i \tan^{-1}{iz}$ and use the power series for $\tan$ to get:</p>
<p>$$\tanh^{-1}{z} = \sum^\infty_{k=0} \frac{z^{2k+1}}{2k+1}$$</p>
<p>But then I am asked to deduce that:</p>
<p>... | MathematicsStudent1122 | 238,417 | <p>To expand on Martin's answer, consider <span class="math-container">$\exp(x)$</span>. This is <span class="math-container">$C^{\infty}$</span>, obviously. From Taylor's theorem,</p>
<p><span class="math-container">$$\left|\exp x - \sum_{0 \leq k \leq n} \frac{x^k}{k!}\right| = \left|\frac{e^c \cdot x^{n+1}}{(n+1)!}\... |
109,037 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/107336/why-doesnt-dx-n-x-n1-rightarrow-0-as-n-rightarrow-infty-imply-x-n">Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?</a> </p>
</blockquote>
<p>my questio... | Community | -1 | <p>Showing that the sum of the epsilons is possibly greater than the single epsilon in the supposedly stronger form does not prove the two forms are not equivalent. What you need is a counterexample.</p>
<p>The sequence $(1,1+\frac{1}{2},1+\frac{1}{2}+\frac{1}{3},\ldots)$ does not converge in $\mathbb R$ and so is not... |
1,193,558 | <p>I ran into this problem in a math camp, but I can't seem to solve it via elementary techniques. </p>
<p>If $a$ and $b$ are positive integers such that $a^n+n\mid b^n + n$ for all positive integers $n$, prove that $a=b$.</p>
| Nitin Uniyal | 246,221 | <p>Hint: Try to think of the graph of S near any of its limit points (m,n). It will somewhat look like kitchen sink filter which has more and more holes as you approach towards its centre (a limit point). </p>
|
2,844,170 | <p>First off, sorry if this is a basic question or one that has been asked before. I really don't know how to phrase it, so it's a hard question to google.</p>
<p>I'm looking for a function that will generate a line similar to the one below</p>
<pre><code> __/
__/
__/
/
</code></pre>
<p>I'm pretty good at... | John Wayland Bales | 246,513 | <p>Here is one example:</p>
<p>$$ f(x)=\left\vert \frac{x-1}{2}-\left\lfloor\frac{x}{2}\right\rfloor\right\vert+\frac{x-1}{2}$$</p>
<p><a href="https://i.stack.imgur.com/YLYOj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YLYOj.png" alt="stair-step"></a></p>
|
4,605,263 | <p>If someone asks me right now, what a number is, I would say 1,2,3,-1,1+i,2.13, etc. But what I have essentially stated are just symbols. I tried to explore a bit what a number exactly is. What are these symbols pointing to?</p>
<p>As I got to know the answer is not that simple, we cannot have an all-encompassing def... | Svyatoslav | 869,237 | <p><span class="math-container">$$I=\int_{0}^{\infty}\frac{\left|\cos\left ( x-\frac{\pi}{4} \right ) \right|- \left|\cos\left ( x+\frac{\pi}{4} \right ) \right| }{x}dx=\int_{0}^{\infty}\frac{\left|\cos\left ( \frac{x}{2}-\frac{\pi}{4} \right ) \right|- \left|\cos\left ( \frac{x}{2}+\frac{\pi}{4} \right ) \right| }{x}d... |
4,605,263 | <p>If someone asks me right now, what a number is, I would say 1,2,3,-1,1+i,2.13, etc. But what I have essentially stated are just symbols. I tried to explore a bit what a number exactly is. What are these symbols pointing to?</p>
<p>As I got to know the answer is not that simple, we cannot have an all-encompassing def... | Random Variable | 16,033 | <p>Let <span class="math-container">$$I = \int_{0}^{\infty} \frac{\left|\cos\left(x-\frac{\pi}{4}\right) \right|-\left|\cos\left(x+\frac{\pi}{4}\right) \right|}{x} \, \mathrm dx .$$</span></p>
<p>Then</p>
<p><span class="math-container">$ \require{cancel} \begin{align}I &= \int_{0}^{\infty}\frac{\left|\cos\left(x-\... |
4,605,263 | <p>If someone asks me right now, what a number is, I would say 1,2,3,-1,1+i,2.13, etc. But what I have essentially stated are just symbols. I tried to explore a bit what a number exactly is. What are these symbols pointing to?</p>
<p>As I got to know the answer is not that simple, we cannot have an all-encompassing def... | Sangchul Lee | 9,340 | <p>We invoke the <a href="https://math.stackexchange.com/q/697843/9340">Fourier series for <span class="math-container">$\left|\cos x\right|$</span></a>:</p>
<p><span class="math-container">$$ \left| \cos x \right| = \frac{2}{\pi} - \frac{4}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^n}{4n^2 - 1} \cos(2nx) $$</span></p>
<p>Fr... |
672,412 | <p>I am reading an e-book called <a href="http://www.ldsinsight.org/">To Infinity and Beyond</a> by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole".</p>
<p>In talking about it, he seems to claim that because half a pie can be b... | Emo | 127,234 | <p>The mass of the half pie is always smaller than the mass of the whole pie. Your point of view is related with the mass (in mathematics that can be calculated by integrals), but cardinals are something else. You may refer for that to many articles or even documentaries on the internet.</p>
|
672,412 | <p>I am reading an e-book called <a href="http://www.ldsinsight.org/">To Infinity and Beyond</a> by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole".</p>
<p>In talking about it, he seems to claim that because half a pie can be b... | c-s.c | 128,168 | <p>In my personal opinion, I would explain as follows.</p>
<ol>
<li><p>half of a pie < whole pie.</p></li>
<li><p>half pie ---process---> infinity pieces.</p>
<p>For the same reason, whole pie ---process---> infinity pieces.</p></li>
<li><p>But process need ENERGE.</p></li>
<li><p>And I SUPPOSE that half pie need ... |
3,435,286 | <p>Suppose that <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are random variables with joint density function</p>
<p><span class="math-container">$$f_{X,Y}(x,y)=\begin{cases} 8xy \space \text{ for } 0<y<x<1\\ 0 \space\space\space\space\space\text{ otherwise.}\end{cases}$... | user | 505,767 | <p>We have that</p>
<p><span class="math-container">$$\cos(2x)+\cos x=2\cos^2 x+\cos x-1$$</span></p>
<p>and </p>
<p><span class="math-container">$$-1\le \cos x \le 1$$</span></p>
|
3,435,286 | <p>Suppose that <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are random variables with joint density function</p>
<p><span class="math-container">$$f_{X,Y}(x,y)=\begin{cases} 8xy \space \text{ for } 0<y<x<1\\ 0 \space\space\space\space\space\text{ otherwise.}\end{cases}$... | Fallen_Prince | 724,535 | <p>To maximize <span class="math-container">$f(x) = \cos 2x + \cos x $</span> , differentiate it with respect to <span class="math-container">$x$</span>.</p>
<p><span class="math-container">$$f'(x) = -2\sin2x -\sin x$$</span> </p>
<p>For the function to achieve a minimum value , <span class="math-container">$f'(x)$</... |
1,132,922 | <p>Let $f$ and $g$ be two differentiable functions s.t $ f '(x) \le g '(x) $ for all $ x\lt 1$ and $ f '(x) \ge g'(x) $ for all $ x\gt 1$ then</p>
<ol>
<li><p>If $f(1) \ge g(1)$, then $f(x)\ge g(x)$ for all $x$</p></li>
<li><p>If $f(1) \le g(1)$, then $f(x)\le g(x)$ for all $x$</p></li>
<li><p>$f(1) \le g(1)$</p></li>... | copper.hat | 27,978 | <p>Simplify life, take $\phi(x) = f(x)-g(x)$.</p>
<p>You are given $\phi'(x) < 0 $ for $x <1$ and $\phi'(x) >0$ for $x >1$. In particular, $\phi$ is decreasing for $x<1$ and $\phi$ is increasing for $x>1$.</p>
<ol>
<li><p>You are given $\phi(1) \ge 0$. Since $\phi$ is increasing for $x>1$, we hav... |
4,501,927 | <p>Hi working again on the gamma function I find that :</p>
<p>let <span class="math-container">$0<x\leq 1$</span> and <span class="math-container">$1\leq k\leq \infty$</span> then define :</p>
<p><span class="math-container">$$f(x)=(x!)!,g(x)=\left(\left(x!\right)!\right)^{\frac{1}{e^{x^{k}}}}$$</span></p>
<p>Then ... | Claude Leibovici | 82,404 | <p>Let <span class="math-container">$a$</span> to be the solution of <span class="math-container">$\psi^{(0)} (x+1)=0$</span> (I think that you know it).</p>
<p>Expand <span class="math-container">$f(x)=\log[(x!)!]$</span> around <span class="math-container">$x=a$</span> and write
<span class="math-container">$$f(x)=f(... |
706,514 | <p>I know that the fundamental group of homeomorphic spaces are isomorphic. Is the converse true? I mean, can we say the two spaces with isomorphic fundamental groups are homeomorphic? </p>
| Jason DeVito | 331 | <p>Going another direction from the answers, there <em>are</em> important cases where this is true.</p>
<p>For example, if you restrict "space" to mean "hyperbolic manifolds of dimension at least three and of finite volume", then the answer is, surprisingly, yes and goes by the name of <a href="http://en.wikipedia.org... |
3,921,255 | <p>If a function <span class="math-container">$f:M\to\mathbb{R}$</span> is continuous at point <span class="math-container">$x_0$</span> we know that for an arbitrary <span class="math-container">$\epsilon>0$</span> there exists a <span class="math-container">$\delta>0$</span> such that for all <span class="math-... | Tyrone | 258,571 | <p>If <span class="math-container">$M$</span> is a topological manifold and <span class="math-container">$a\in M$</span> is any point, then the inclusion <span class="math-container">$\{a\}\hookrightarrow M$</span> is a cofibration (i.e. has the <a href="https://en.wikipedia.org/wiki/Homotopy_extension_property" rel="n... |
209,044 | <p>Please help me deal with this kind of question about ODEs.
My codes are as follows</p>
<pre><code>m = 100;
a = D[x[t], {t, 2}];
t1up = 2 x''[t] + 1/2 (490 + 34 x''[t] + 2 (490 + 50 x''[t]));
t1down = 490 + 53 x''[t];
t1 = Piecewise[{{t1up, x'[t] >= 0}, {t1down, x'[t] < 0}}]
equa00 = t1 == m*a
t0 = 50;
s1 = ND... | Michael E2 | 4,999 | <p>Here is the sort of thing I meant in <a href="https://mathematica.stackexchange.com/questions/209044/ndsolve-will-try-solving-the-system-as-differential-algebraic-equations-but-it-d/209135#comment536432_209044">my comment</a>:</p>
<p><em>1. Get a single piecewise function</em></p>
<pre><code>constraint = equa00 /.... |
4,142,970 | <p>Find the cardinality of the set <span class="math-container">$\{ X \subset \mathbb{N} : |X| = \aleph_0\} $</span>.</p>
<p>First off all I could easily prove that the set given above, say <span class="math-container">$S$</span>, is uncountable.</p>
<p>Because <span class="math-container">$P(\mathbb{N})$</span> is unc... | Jonathan Schilhan | 265,128 | <p>Let <span class="math-container">$A$</span> be the set of even naturals and <span class="math-container">$B$</span> the odds and consider <span class="math-container">$$ \{A \cup C : C \in \mathcal{P}(B)\}. $$</span> How big is this set?</p>
|
4,553,332 | <p>What do we call the property that if <span class="math-container">$a = b$</span>, then <span class="math-container">$f(a) = f(b)$</span>?</p>
<p>Wikipedia calls it "substitution property" but is that correct?</p>
| ryaron | 505,828 | <p>this is the definition of a well defined function.</p>
<p><span class="math-container">$(\forall x)(\forall y)(x=y\implies f(x)=f(y))$</span></p>
<p>A well defined function <span class="math-container">$f(x)$</span> returns a unique value for <span class="math-container">$x$</span>.</p>
|
234,483 | <p>The question is, "Give a description of each of the congruence classes modulo 6."</p>
<p>Well, I began saying that we have a relation, $R$, on the set $Z$, or, $R \subset Z \times Z$, where $x,y \in Z$. The relation would then be $R=\{(x,y)|x \equiv y~(mod~6)\}$</p>
<p>Then, $[n]_6 =\{x \in Z|~x \equiv n~(mod~6)\}... | Brian M. Scott | 12,042 | <p>Let’s start with your correct description</p>
<p>$$[n]_6=\{x\in\Bbb Z:x\equiv n\!\!\!\pmod 6\}=\{x\in\Bbb Z:6\mid x-n\}$$</p>
<p>and actually calculate $[n]_6$ for some values of $n$. </p>
<ul>
<li><p>$[0]_6=\{x\in\Bbb Z:6\mid x-0\}=\{x\in\Bbb Z:6\mid x\}=\{x\in\Bbb Z:x=6k\text{ for some }k\in\Bbb Z\}$; this is j... |
234,483 | <p>The question is, "Give a description of each of the congruence classes modulo 6."</p>
<p>Well, I began saying that we have a relation, $R$, on the set $Z$, or, $R \subset Z \times Z$, where $x,y \in Z$. The relation would then be $R=\{(x,y)|x \equiv y~(mod~6)\}$</p>
<p>Then, $[n]_6 =\{x \in Z|~x \equiv n~(mod~6)\}... | Bill Dubuque | 242 | <p>You're right, it does not describe all the congruence classes -- due to an error. Namely, notice $\rm\:6\mid x\!-\!n\:$ means $\rm\:6\color{#C00}k = x\!-\!n,\:$ not $\rm\:6 = \color{#C00}k(x\!-\!n).\:$ Recall $\rm\:a\mid b \!\iff\! a\color{#C00}k = b\:$ for some $\rm\:\color{#C00}k\in\Bbb Z.$ Thus $\rm\:x\in [n]_6$ ... |
2,279,660 | <p>On $C^1([0,1],\mathbb{R})$ I have two norms
\begin{align*}
N_1(f)&=|f(0)|+\|f'\|_{\infty}\\
N_2(f)&=\|f\|_{\infty}+\|f'\|_{\infty}.
\end{align*}</p>
<p>We have always that $N_1(f)\leq N_2(f)$ but what about the other inequality please?</p>
<p>And please how to see If $N_1$ and $N_2$ are equivalent to $\le... | Henno Brandsma | 4,280 | <p>Summarising from the other answers: $N_1(f) \le N_2(f) \le 2N_1(f)$ for all $f$ , which makes $N_1$ and $N_2$ equivalent norms (which always induce strongly equivalent metrics). But the example of $f_n(x) = x^n$ shows that there is a sequence with bounded $||.||_\infty$-norm which is not bounded in $N_1$ or $N_2$. T... |
249,597 | <p>I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this so I know that</p>
<p>$2x^2 + 4x + 1 = 0$</p>
<p>Can have at most two solutions. In calculus this is proven by the ... | André Nicolas | 6,312 | <p>Use the <a href="http://en.wikipedia.org/wiki/Quadratic_equation" rel="nofollow">Quadratic Formula.</a> One cannot expect all quadratic polynomials to factor "nicely." The Quadratic Formula needs to become a completely standard tool in your arsenal. </p>
|
249,597 | <p>I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this so I know that</p>
<p>$2x^2 + 4x + 1 = 0$</p>
<p>Can have at most two solutions. In calculus this is proven by the ... | amWhy | 9,003 | <p>$$ax^2 + bx + c = 0, \text{ roots?}$$
<br>
This is where the quadratic formula comes in handy (you should memorize this!):</p>
<p><br>
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$</p>
<p><br>The <em>discriminant d</em> of the quadratic formula is the term $d= (b^2 - 4ac)$. When $b^2\geq 4ac$, you have two real roots... |
4,602,464 | <p>I've been asked to provide context to my original question, so here's the context:</p>
<p>The rectangle in the problem below represents a pool table whose "pool table light" cannot be easily moved, but CAN easily be rotated. No portion of the pool table's perimeter can be too close to a wall in order for t... | Mr. Brown | 448,217 | <p>Remember that you need to put in two inputs into a 1-form before you get a scalar: a point on your manifold, and then a vector at the tangent space of that point. If you fix the point <span class="math-container">$x\in M$</span> but <em>not</em> a vector <span class="math-container">$v\in T_xM$</span>, then you have... |
1,254,820 | <p>Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ C^{*} $-algebra? We can show that the map $ f \mapsto \overline{f} $ is an involution, but with this involution, $ {L^{1}... | Berrick Caleb Fillmore | 85,964 | <p>In what follows, we assume that $ G $ is a locally compact Hausdorff group that does not have to be abelian.</p>
<hr>
<p>You can certainly define an involution $ ^{*} $ on $ {L^{1}}(G) $ by
$$
\forall f \in {L^{1}}(G), ~ \forall x \in G: \quad
{f^{*}}(x) \stackrel{\text{df}}{=} \overline{f(x^{-1})} \cdot \Delta(x^... |
2,487,866 | <p>Gödel's <a href="https://en.wikipedia.org/wiki/Constructible_universe" rel="nofollow noreferrer">constructible universe</a> seems to have some attractive properties. Sets are constructed in a very regular, easy-to-understand way, and one has a definite answer on certain major set theoretic questions, such as the gen... | Community | -1 | <p>Set theory is one of the standard ways to do <em>higher order logic</em>.</p>
<p>Sometimes, one wants to be able to use higher order logic to reason about universes of sets; e.g. category theory needs convenient ways to reason about "large" categories.</p>
<p>The simplest way is to do this is to use a model of set... |
900,958 | <blockquote>
<p>Show that <span class="math-container">$$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$</span>
where <span class="math-container">$\gamma$</span> is the <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.htm... | Omran Kouba | 140,450 | <blockquote>
<p><strong>Step 1.</strong> If
$$I_1(n)=\sum_{1\leq k\leq\sqrt{n}}\left(\Gamma\left(\frac{k}{n}\right)\right)^{-k}$$
Then $\lim\limits_{n\to\infty}I_1(n)=0$.</p>
</blockquote>
<p><em>Proof.</em> Indeed, since $\Gamma$ is decreasing on $(0,1]$ we have
$$
I_1(n)\leq\sum_{1\leq k\leq\sqrt{n}}\left(\Ga... |
900,958 | <blockquote>
<p>Show that <span class="math-container">$$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$</span>
where <span class="math-container">$\gamma$</span> is the <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.htm... | robjohn | 13,854 | <p>Since $\log(\Gamma(x))$ is convex, $\log(\Gamma(x))\ge-\gamma(x-1)$ and $\log(\Gamma(x))\ge(1-\gamma)(x-2)$. Thus, $\log(\Gamma(x))\ge-\gamma+\gamma^2$. That is, $\Gamma(x)\ge e^{-\gamma+\gamma^2}=0.78345806514\gt\frac34$.</p>
<hr>
<p>For $1\le k\le\frac23n$,
$$
\begin{align}
\Gamma\left(\frac kn\right)^{-k}
&... |
996,103 | <p>Suppose that $A$, $B$, and $C$ are sets. Prove that $(A\cap B)\times C =(A\times C)\cap(B\times C)$. Prove the statement both ways or use only if and only if statements.</p>
| 1234 | 188,164 | <p>Using the properties of logarithms,</p>
<p>I write $\ln(n^2/(n^2-1))$
as
$2\ln(n)-\ln(n-1)-\ln(n+1)$</p>
<p>and then I consider the partial sum $S_n= (2\ln(2)-\ln(1)-\ln(3))+(2\ln(3)-\ln(2)-\ln(4))+(2\ln(4)-\ln(3)-\ln(5))+...+(2\ln(n)-\ln(n-1)-\ln(n+1))$</p>
<p>Which after canceling like terms simplifies to
$S_... |
1,595,206 | <p>For a practice question I have been given I have been told to find a spanning tree using a breadth first search for the following graph:</p>
<p><a href="https://i.stack.imgur.com/pKYwB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/pKYwB.png" alt="enter image description here"></a></p>
<p>From ... | Atvin | 215,617 | <p>First, you should choose an arbitrary vertex, let it be $1$ to make it simple.</p>
<p>Then, the algorithm starts, $1$ is "done", now we check all neighbours of $1$, and we write them to our list: $1,2,3$. (We made the edges (1,2) and (1,3)).
No more neighbours, so we check for new ones. $2$ has no more neighbours, ... |
2,751,417 | <p>I want to prove the following for real invertible $2 \times 2$ matrices $A$ and $B$:</p>
<p>$\det(ABA^{-1}B^{-1})=1$</p>
<p>I tried to write it out for random numbers, and that seems to work out well. But when I tried to write it out in general, it became far too much paperwork. So I think there must be a simpler/... | lhf | 589 | <p><em>Hint:</em> $\det (AB) = \det(A) \det(B)$, $\det(I)=1$.</p>
|
820,490 | <p>I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?</p>
| Fardad Pouran | 76,758 | <p>More precisely, in Analysis, <strong>sin(x)</strong> is defined as the limit of a specific series.</p>
<p>Before, we know that the below series converges for every $x\in\mathbb{R}$:
$$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$
And then we define :
$$\sin(x)=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$... |
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | ncmathsadist | 4,154 | <p>You have
$$ .abcdef ...... zabcdef....z......... = {abcd..... z\over 999999 \cdots 99}$$
for any sequence of digits $abc \cdots z$. </p>
|
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | Community | -1 | <p>Suppose that your number is
$$x=a.b_1b_2\cdots b_p \overline{c_1c_2\cdots c_q}$$
then
$$10^{p+q}x-10^{p}x=ab_1b_2\cdots b_p {c_1c_2\cdots c_q}-ab_1b_2\cdots b_p $$
Now, can you see that $x$ is rational?</p>
|
549,254 | <blockquote>
<p>$$\frac{1}{3}=.33\bar{3}$$ </p>
</blockquote>
<p>is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio? </p>
<p>I understand that $\pi$ is irrational because it extends infinitely <em>... | Stefan4024 | 67,746 | <p>You need to know the following definition: A number is rational if and only if it can be written as a ratio using whole numbers. Or if you want more scientific notation:</p>
<p><span class="math-container">$$r = \frac ab \iff r \in \mathbb{Q} \text { where $a,b \in \mathbb{Z}$ and $b \neq 0$}$$</span></p>
<p>So l... |
2,829,362 | <p>Suppose the random variable $T$ which represents the time needed for one person to travel from city A to city B ( in minutes). $T$ is normally distributed with mean $60$ minutes and variance $20$ minutes. Also, suppose $600$ people depart at the exact same time with each of their travel time being independent from o... | Daniel Buck | 293,319 | <p>For $[a,b]$ an interval in $\mathbb{R}$, let $f\colon [a,b]\to \mathbb{R}$. Then $F\colon [a,b]\to \mathbb{R}$ is an <em>antiderivative</em> of $f$ on $[a,b]$ if $F$ is continuous on $[a,b]$. It will be necessarily so in $(a,b)$, while left continuous at $a$: $F(a)=\lim_{x\to a^+} F(x)$, and right continuous at $b$:... |
3,752,676 | <p>what is <span class="math-container">$P(P(P(333^{333})))$</span>, where P is sum of digit of a number. for an example <span class="math-container">$P(35)=3+5=8$</span></p>
<p>a)18</p>
<p>b)9</p>
<p>c)33</p>
<p>d)333</p>
<p>f)5</p>
<p>I tried to find this but I couldn't. I started to find a pattern for an example the... | BenediktK | 672,472 | <p>Here is another approach.
Let <span class="math-container">$G=(\mathbb{Z}_4\oplus\mathbb{Z}_{12})/H$</span> and <span class="math-container">$H=\langle(2,2)\rangle$</span>.
Since <span class="math-container">$G$</span> is a quotient, we can try to find a surjective homorphism from <span class="math-container">$\math... |
2,459,323 | <p>Q) I have to show that if every vertex in graph $G$ has an even degree , then
$G$ has no bridge. <br>
So by contradiction ,<br> If every vertex in graph $G$ has an even degree , then
$G$ has a bridge , which means $G - e$ will result in a disconnected graph.<br>
We know that in this graph, 1 vertex is adjacent t... | David K | 139,123 | <p>Your proof is wrong in two ways:</p>
<ol>
<li><p>It isn't necessarily true that all vertices are connected to at least two others. A vertex could have <em>no</em> edges connected to it, because zero is an even number.</p></li>
<li><p>You don't actually use the "even" property except to assert that each vertex has a... |
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| skyking | 265,767 | <p>What you can use is the Taylor expansion of <span class="math-container">$\ln (1+x)$</span>:</p>
<p><span class="math-container">$$\ln (1+x) = \sum_{j=1}^\infty (-1)^{j+1}{x^j\over j}$$</span></p>
<p>which converges for <span class="math-container">$-1<x\le1$</span>. It would be tempting to insert <span class="... |
2,567,989 | <p>I need to prove that if $\gamma_1,\gamma_2:[0,1] \to \Bbb C - \{0\}$ are closed paths and have the same index ( or winding number) around $0$ , then they are homotopic.</p>
<p>So, I think I have an idea on how to show it:</p>
<p>we have $g_1,g_2:[0,1] \to \Bbb C$ continuous logarithms for $\gamma_1 , \gamma_2$ , t... | Mariah | 319,890 | <p>Take $H(s,t) = \exp\Big((1-s)g_1(t) + sg_2(t)\Big)$. I think this, with the assumption of the same winding number should solve it.</p>
<p>More info:</p>
<p>$Im(H) \subset \mathbb{C} - \{0\}$, this is obvious.</p>
<p>Set $s \in [0,1]$. Then $H(s,0) = exp(g_1(0))exp(s[g_2(0) - g_1(0)]) = exp(g_1(1))exp(s[g_2(1) - g... |
2,277,117 | <p>For $m = 2$, the fraction is $\frac{3}{4}$. for $m=3$, the fraction is$\frac{8}{15}$. I was wondering why numerators and demoninators of $\frac{1}{m} + \frac{1}{m+2}$ show primitive pythagorean triples a and b.</p>
| OmG | 356,329 | <p>There is a simple form to proof:</p>
<p>$(m^2+2m)^2 + (2(m+1))^2 = ((m+1)^2 - 1)^2 + 4 (m+1)^2 = (m+1)^4 + 1 -2(m+1)^2 + 4(m+1)^2 = (m+1)^4 -2(m+1)^2 + 1 = ((m+1)^2 - 1)^2$</p>
|
2,428,881 | <p><img src="https://i.stack.imgur.com/0STsn.png" alt="Trig Limits"></p>
<blockquote>
<p>I am confuse with the part of sin^2. I am not sure if there is a trig identity to simplify. I was thinking of trying to rearranging the equation so that it would be similar to sin(x)/x to solve it. Which direction do I go since ... | Kenny Lau | 328,173 | <p>$$\begin{array}{rcl}
\displaystyle \lim_{x\to0} \frac{\sin^2(3x)}{x^2\cos x}
&=& \displaystyle \lim_{x\to0} \frac{\sin(3x)}{3x} \cdot \frac{\sin(3x)}{3x} \cdot \frac{9}{\cos x} \\
&=& 1 \cdot 1 \cdot 9 \\
&=& 9
\end{array}$$</p>
|
1,704,707 | <p>The function $f: ℝ → ℝ$ defined by $f(x) = x^{3}$ is onto because for any real number $r$, we have that $\sqrt[3]r$ is a real number and $f(\sqrt[3]r)=r$. Consider the same function defined on the integers $g: ℤ → ℤ$ by $g(n) = n^3.$ Explain why $g$ is not onto $ℤ$ and give one integer that $g$ cannot output. </p>
... | Ian | 83,396 | <p>You can actually proceed directly: let $f_k$ be a sequence in $D$, then argue that either it has infinitely many copies of the same element of $D$, or else it converges to $0$.</p>
<p>Arzela-Ascoli would basically require the same argumentation in this case.</p>
|
3,271,917 | <p><span class="math-container">$$ \sum_{n=1}^\infty (-1)^{n+1} \left( \frac{1.4.7\dots .(3n-2)}{2.3.8\dots .(3n-1)} \right)^2 $$</span></p>
<p>I have done the<span class="math-container">$ \sum_{n=1}^\infty \left( \frac{1.4.7\dots .(3n-2)}{2.3.8\dots .(3n-1)} \right)^2 $</span> part , and showed it divergent using G... | J_P | 676,091 | <p>Let's prove that
<span class="math-container">$$
a_n=\frac{1\cdot 4\,\,\cdot\,\, ...\,\,\cdot\,\,(3n-2)}{2\cdot 5\,\,\cdot\,\, ...\,\,\cdot\,\,(3n-1)}
$$</span>
converges to <span class="math-container">$0$</span>. Clearly <span class="math-container">$a_n>0$</span> and since
<span class="math-container">$$
a_{n+... |
849,336 | <p>$$\eqalign{\tan^2\theta-\sec^2\theta &=\tan^2\theta-\dfrac1{\cos^2\theta}\\&=\dfrac{\sin^2\theta}{\cos^2\theta}-\dfrac1{\cos^2\theta}\\&=-\dfrac{\cos^2\theta}{\cos^2\theta}\\&=-1.}$$
note:$\theta≠ (2k+1)\frac{\pi}{2}$, $k$ : integer number </p>
<p>May i know if my answer is correct or not..</p>
| Thoth19 | 76,241 | <p>This is in general true apart from edge cases where $tan(\theta), sec(\theta)$ are undefined. What is nice about trig functions is that very often they can simplify nicely, so that complicated statements about circles and triangles can be solved using calculus, or calculus can be solved using trig rules. It is also ... |
77,705 | <p>If $A$ and $B$ are both $n \times n$ matrices, and $v$ is a non-zero $n \times 1$ column vector then is it true that if
$$ABv = BAv$$
then $$AB=BA$$</p>
| Community | -1 | <p>The short answer is NO.</p>
<p>You cannot say $AB = BA$ if $ABv = BAv$ for some vector $v$.</p>
<p>However, if $ABv = BAv$ is true for all vectors $v$ (or) at-least for $n$ linearly independent vectors $v$, then it is true that $AB = BA$.</p>
|
271,554 | <p>Is $\chi_{p(\displaystyle\lim_{n\to\infty} f_n)}=\displaystyle\lim_{n\to\infty}\chi_{p(f_n)}$, assuming $\lim f_n$ exists? Here $\chi_{p(f)}$ is $1$ for the set where the proposition $p$ on the function $f$ is true and $0$ for otherwise. Here the functions are $[0,1]\to\mathbb{R}$ and the limits are almost everywher... | GEdgar | 442 | <p>Or: $p(f)$ means "the function $f$ is identically zero", and $f_n(x) = 1/n$ for all $x \in [0,1]$ and all $n \in \mathbb N$.</p>
|
185,097 | <blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="https://math.stackexchange.com/questions/1/different-kinds-of-infinities">Different kinds of infinities?</a> </p>
</blockquote>
<p>Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be in... | André Nicolas | 6,312 | <p>Two sets $A$ and $B$ are said to have the <strong>same cardinality</strong> if there is a function $f: A \to B$ which is one-to-one and onto. More informally, $A$ and $B$ have the same cardinality if the elements of $A$ and $B$ can be "paired off."
Note that if $A$ is finite, then $A$ and $B$ have the same cardinali... |
360,889 | <p>What can be said about publishing mathematical papers on e.g. viXra if the motivation is its low barriers and lack of experience with publishing papers and the idea behind it is to build up a reputation, provided the content of the publication suffices that purpose. </p>
<p>Can that way of getting a foot into the ... | David White | 11,540 | <p>Yes, the place of publication can absolutely hurt your reputation. Specifically, I can tell you from having served on many hiring committees (and from conversations with professors at other universities about their hiring committees and tenure processes), that publications in predatory journals can hurt you. I'm tal... |
2,248,651 | <p>How can I prove the following:</p>
<blockquote>
<p>An $n\times n$ matrix $A$ that has $n$ distinct eigenvalues is similar to a diagonal matrix.</p>
</blockquote>
<p>I saw <a href="https://math.stackexchange.com/questions/1539100/how-to-prove-that-a-matrix-a-size-nxn-with-n-linearly-independent-eigenvectors">anot... | Bérénice | 317,086 | <p>If all the eigen values are distinct then all the eigen vectors are distinct. Indeed let $\lambda_1\neq\lambda_2$ two eigen values, let $x_1$ an eigenvector of $\lambda_1$, let's show that $x_1$ can't be an eigenvector of $\lambda_2$. </p>
<p>If $x_1$ is also an eigenvector of $\lambda_2$, then : $Ax_1=\lambda_2 x... |
2,248,651 | <p>How can I prove the following:</p>
<blockquote>
<p>An $n\times n$ matrix $A$ that has $n$ distinct eigenvalues is similar to a diagonal matrix.</p>
</blockquote>
<p>I saw <a href="https://math.stackexchange.com/questions/1539100/how-to-prove-that-a-matrix-a-size-nxn-with-n-linearly-independent-eigenvectors">anot... | Hagen von Eitzen | 39,174 | <p>Let $\lambda_1,\ldots,\lambda_n$ be distinct eigenvalues. By definition, there exist corresponding eigenvectors $v_1,\ldots, v_n$, such that $v_i\ne 0$ and $Av_i=\lambda_iv_i$. As the eigenvalues are distinct, the $v_i$ are linearly independent and hence form a basis. Expressing $A$ in that base obviously produces a... |
4,204,416 | <p>First, can someone provide a simple explanation for the <span class="math-container">$\bar{x}$</span> formula: <span class="math-container">$$\frac{\iint xdA}{area}$$</span> My understanding of the formula is as follows: we let <span class="math-container">$z=x$</span>, calculate the volume, and divide by the area t... | TomKern | 908,546 | <p>A good place to start is working out the <span class="math-container">$x$</span>-coordinate of the center of mass, <span class="math-container">$\bar{x}$</span>, for a collection of point masses at various locations. Let's say we have points <span class="math-container">$1,\ldots,n$</span> with masses <span class="m... |
337,035 | <p>Recall Wolpert's lemma:</p>
<p>Let X,Y be hyperbolic surfaces and <span class="math-container">$f:X\to Y$</span> a <span class="math-container">$K$</span>-quasiconformal homeomorphism. For any homotopy class of curves <span class="math-container">$c$</span> let <span class="math-container">$\ell(c)$</span> denote t... | Dylan Thurston | 5,010 | <p>The answer is no, but it's actually a deep question and leads to another metric on the Teichmüller space of surfaces. The minimum quasi-conformal constant of a map in a given homotopy class is the Teichmüller metric on surfaces; it is also equal to the maximal ratio of extremal lengths of any curves. If you look at ... |
2,355,246 | <p>The optimisation problem [PROBLEM 1]:
$$\max \sum_{t=T} x_t $$
subject to:
$$\ (1) x_t = (1+\alpha) x_{t-1}\ \forall t ..T$$
$$\ t=\{ 0..T\} , x_0= given$$
$$\ (e.g) x_1 = (1+\alpha) x_0\ $$
$$\ (e.g) x_2 = (1+\alpha) x_1\ $$
$$\ (e.g) x_3 = (1+\alpha) x_2\ $$ </p>
<p>$$\ $$</p>
<p>$$\ (2) x_t \ge 0 $$<... | ThePortakal | 137,487 | <p>For $\displaystyle 0 \leq \theta \leq \frac{\pi}{2}$,</p>
<p>$$\vec x = \ \big(\sin \theta , \cos \theta , \underbrace{0 , \dots , 0}_{n-2 \text{ times}}\big) \in \mathbb R^n$$</p>
<p>is a unit vector in $\Bbb R^n$. Now since $\sin \theta$ is injective on the given domain, for different $\theta$, we find different... |
2,355,246 | <p>The optimisation problem [PROBLEM 1]:
$$\max \sum_{t=T} x_t $$
subject to:
$$\ (1) x_t = (1+\alpha) x_{t-1}\ \forall t ..T$$
$$\ t=\{ 0..T\} , x_0= given$$
$$\ (e.g) x_1 = (1+\alpha) x_0\ $$
$$\ (e.g) x_2 = (1+\alpha) x_1\ $$
$$\ (e.g) x_3 = (1+\alpha) x_2\ $$ </p>
<p>$$\ $$</p>
<p>$$\ (2) x_t \ge 0 $$<... | Alex Ortiz | 305,215 | <p>If we can simply exhibit infinitely many unit vectors in $\Bbb R^2$, we will be done because if $n\ge 2$ we can find a copy of $\Bbb R^2$ inside of $\Bbb R^n$ via the map $(x,y)\mapsto(x,y,0,...,0)$.</p>
<p>Consider the points $p_n=(\cos\frac{1}{n},\sin\frac{1}{n})$, where $n = 1,2,3,\dots.$ This is an infinite se... |
440,744 | <p>The vector space dimension of the cohomology group of the <span class="math-container">$2$</span>-plane Grassmannian <span class="math-container">$\mathrm{Gr}_{2,n}$</span> is given by the number of tuples <span class="math-container">$(\lambda_1,\lambda_2)$</span> satisfying
<span class="math-container">$$
n - 2 \g... | Vladimir Dotsenko | 1,306 | <p>This is very standard. For a compact complex variety admitting a cell decomposition, the (co)homology is the free Abelian group generated by the cells (over <span class="math-container">$\mathbb{C}$</span> there is no room for the differential in the cellular complex). In the cellular decomposition of the Grassmanni... |
1,283,516 | <p>Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$.</p>
<p>Ideas? The triangular inequality will not.</p>
<p>I wanted to try the theorem of the exterior angle and then apply a preposition that says "If two angles of ... | Rigel | 189,549 | <p>Angles $ADC$ and $BDC$ are supplementary.
Notice that at least one of then must be obtuse or both $90$.</p>
<ul>
<li><p>If angle $BDC$ is obtuse,then $BC > CD$.(or if angle $ADC$ is obtuse)</p></li>
<li><p>If angle $BDC$ is $90$,then $BC$ is the hypotenuse.</p></li>
</ul>
|
2,826,327 | <p>$$\lim_{x\rightarrow \infty }\left ( 1+\frac{3}{x+2} \right )^{3x-6}$$</p>
<p>I've tried to factor and simplfy the expression. I got:</p>
<p>$${\left ( 1+\frac{3}{x+2} \right )^{\frac{1}{x+2}}}^{3({x^2-4})}$$</p>
<p>I set $x$ to $1/t$ I get:</p>
<p>$${\left ( 1+\frac{3}{\frac{1}{t}+2} \right )^{\frac{1}{\frac{1}... | Clarinetist | 81,560 | <p>Let $y = \left(1+\dfrac{3}{x+2}\right)^{3x-6} $. Then
$$\ln y=(3x-6)\ln\left(1+\dfrac{3}{x+2}\right)\text{.}$$
On the left-hand side, by continuity of $\ln$,
$$\lim_{x \to \infty}\ln y = \ln \left(\lim_{x \to \infty}y\right)\text{.}$$
On the right-hand side,
$$\lim_{x \to \infty}(3x-6)\ln[1+3/(x+2)]=\lim_{x \to \i... |
165,385 | <p>I am generating a list of 1's, 2's, and 3's with different probabilities for each number. I then convert this list into three binarized lists, giving a list of the locations of each digit in the original list. There must be a more efficient way to do this? Perhaps changing the way I make the original list?</p>
<pre... | Henrik Schumacher | 38,178 | <p><strong>Edit:</strong></p>
<p>@CarlWoll brought to my attention that <code>RandomChoice</code> may produce unpacked arrays. This is why I update the timings. Previously, I stated that Carls' solutiuon need ten times as long as the first proposal below. This is not true for <code>PackedArray</code>s. Mea culpa.</p>
... |
165,385 | <p>I am generating a list of 1's, 2's, and 3's with different probabilities for each number. I then convert this list into three binarized lists, giving a list of the locations of each digit in the original list. There must be a more efficient way to do this? Perhaps changing the way I make the original list?</p>
<pre... | eyorble | 52,935 | <p>Using <code>ReplaceAll</code> (<code>/.</code>) and delayed rules for pattern matching, this can be done much more directly:</p>
<pre><code>l1 = list /. (x_?NumericQ :> If[x == 1, 1, 0])
l2 = list /. (x_?NumericQ :> If[x == 2, 1, 0])
l3 = list /. (x_?NumericQ :> If[x == 3, 1, 0])
</code></pre>
<p>The <cod... |
165,385 | <p>I am generating a list of 1's, 2's, and 3's with different probabilities for each number. I then convert this list into three binarized lists, giving a list of the locations of each digit in the original list. There must be a more efficient way to do this? Perhaps changing the way I make the original list?</p>
<pre... | David G. Stork | 9,735 | <p>This is very fast and avoids an intermediate step of creating 1s, 2s and 3s:</p>
<pre><code>mylist = RandomChoice[{.3, .2, .5} -> {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}, 300];
Transpose[mylist]
</code></pre>
<p>The generation of the list and the result is about twice as fast as the generation of the list and the re... |
3,006,595 | <p>How to prove</p>
<blockquote>
<p><span class="math-container">$$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)\stackrel ?=\frac{\pi^3}{32}-2G\ln2,$$</span>
where <span class="math-container">$G$</span> is the Catalan's constant.</p>
</blockquote>
<p><strong>Attempt</strong></p>
<p>For the first sum,
... | Nanayajitzuki | 611,558 | <p>this summation also has some relationship with integral identity, which asked almost the same time, see <a href="https://math.stackexchange.com/questions/3006106/an-amm-like-integral-int-01-frac-arctan-xx-ln-frac1x231x2dx">this</a>, rewrite as</p>
<p><span class="math-container">$$\frac3{2} \int_{0}^{1} {\frac{\arc... |
1,163,001 | <p>I need help to prove this congruence: </p>
<p>$$ 3^n -4(2^n) + 6(1^n) + (-1)^n \equiv 0 \pmod {24} $$</p>
<p>I have tried to used Euler's Theorem on the powers of 2 and 3 individually but now I'm stuck.</p>
| Andrei Rykhalski | 181,554 | <p>For $n = 2$ you have $9 - 4 * 4 + 6 + 17 = 16 \equiv 16 \mod 24$ so the expression doesn't seem to be true.</p>
|
3,279,503 | <p>I'm relatively new in tensor space theory, and while reading some materials i've came across authors describing a inner product as a <span class="math-container">$(0,2)$</span> tensor. I'm not sure why it is, but i think if i write a map <span class="math-container">$f$</span> as <span class="math-container">$$f: P ... | Arthur | 15,500 | <p>Disclaimer: I'm mostly used to dealing with tensors in finite-dimensional spaces (general relativity), but I see no reason the same shouldn't apply here. Read with caution, however, as conventions <em>might</em> differ.</p>
<p>When you contract tensors, one covariant order cancels one contravariant order. We can us... |
141,128 | <p>I have a complicated function for <code>StreamPlot</code> to call that includes some error checking and the use of <code>Message</code>. Somehow the messages are not displayed when the function is called from <code>StreamPlot</code>. Here's a toy example:</p>
<p>Define the function & message:</p>
<pre><code>... | Chris K | 6,358 | <p>Here's my attempt to wrap up @MichaelE2's trick into a plug-and-play replacement for <code>Message</code>:</p>
<pre><code>Msg[msgs__] := Block[{$Messages = Streams["stdout"]},
If[!ListQ@$MessageList, $MessageList = {}];
Message[msgs];
];
SetAttributes[Msg, HoldAll];
</code></pre>
<p>This seems to work except i... |
1,595,909 | <p>I am studying Measure Theory and I am stuck in some concepts about continuity of a measure.</p>
<p>Let $(S_{1}, \Sigma,\mu)$ be a measurable space, where $\mu$ is a probability mesure such that $\mu (S_{1}) = 1.$ Let also $S_{\delta} \subset \Sigma, 0\leq \delta \leq 1,$ be a family of sets such that $S_{\delta_{1}... | user64066 | 64,066 | <p>If you define $\mu(S_\delta)=c$ for $c<1$ then $\mu(E_j)=c$ for all $j$, but then $\mu(S_1)=c$ which contradicts with the condition $\mu(S_1)=1$.</p>
|
3,133,798 | <blockquote>
<p>Find an asymptotic expansion at order <span class="math-container">$6$</span> of <span class="math-container">$f(x) = \int_x^{x^2} \frac{\mathrm{d}t}{\sqrt{1+t^4}}$</span></p>
</blockquote>
<p>I don't know how to proceed. I think I need to do a change of variable yet I don't know which one. I tried <... | Stefan Lafon | 582,769 | <p>I'm assuming you are looking for an expansion as <span class="math-container">$x\rightarrow 0$</span>.</p>
<p>For <span class="math-container">$u\rightarrow 0$</span>,
<span class="math-container">$$(1+u)^\alpha = 1 + \alpha u + \mathcal O(u^2)$$</span>
Therefore
<span class="math-container">$$\frac 1 {\sqrt{1+t^4... |
3,170,742 | <p>I have a system of recurrence relations in the following form:</p>
<p><span class="math-container">$$
\begin{pmatrix}
f(n+1)\\
g(n+1)\\
\end{pmatrix}
= \textbf{A}
\begin{pmatrix}
f(n)\\
g(n)\\
\end{pmatrix} +\vec{b}
$$</span></p>
<p>which hold for all <span class="math-container">$n \in \{ 0,1,2,...,N\}.$</span> ... | MarianD | 393,259 | <p>Using the formula for <span class="math-container">$\sin (\alpha - \beta)$</span> you obtain</p>
<p><span class="math-container">\begin{align}
2&\sin x +2\sin \left(\frac{\pi} {3} -x\right)\\
= 2&\sin x +2\left[\sin \left(\frac{\pi} {3}\right) \cos x - \cos\left(\frac{\pi} {3}\right) \sin x\right]\\
= 2&a... |
3,663,374 | <p>I asked a question about if we have in general that if <span class="math-container">$G=G_1\times \cdots \times G_n$</span> (where <span class="math-container">$G_i$</span> are characteristic in <span class="math-container">$G$</span> for <span class="math-container">$i=1,\cdots ,n$</span>), then
<span class="math-co... | 1123581321 | 482,390 | <p><strong>HINT</strong></p>
<p>If <span class="math-container">$H\lhd G$</span> and <span class="math-container">$H'\lhd G'$</span> then <span class="math-container">$f:G\times G'\to (G/H)\times (G'/H')$</span> with <span class="math-container">$f((x,x'))=(pr(x),pr'(x'))$</span> has kernel <span class="math-container... |
2,447,850 | <p>So I have to prove 2 things:</p>
<ol>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R, x>0$. </p></li>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R$. </p></li>
</ol>
<p>For #1, I kn... | Guy Fsone | 385,707 | <p><strong>First Answer</strong>
The series
$$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$$
converges then $$\frac{x^n}{n!}\to 0$$</p>
<p>OR
<strong>Second Answer</strong>
Use the following famous Stirling formula: Given $x>0$
$$ \lim_{n\to +\infty} \frac{n!}{\left(\frac{n}{e}\right)^n\sqrt{2n} }=\sqrt{\pi}. $$
and $$|x^... |
128,651 | <p>Let $M$ be a smooth manifold (maybe compact, if that helps). Denote by $\operatorname{Diff}(M)$ the group of diffeomorphisms $M\to M$ and by $R(M)$ the space of Riemannian metrics on $M$. We obtain a canonical group action
$$ R(M) \times \operatorname{Diff}(M) \to R(M), (g,F) \mapsto F^*g, $$
where $F^*g$ denotes th... | gaoqiang | 290,906 | <p>No. actually on closed surfaces, all metrics of curvature -1 module diffs+(S) is the teichmuller module space </p>
|
264,740 | <p>On Hilbert spaces, the following is true:</p>
<p>Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed.</p>
<p>The obvious proof I see to show this uses explicitly the Hilbert space structure which is why I would like to ask:</p>
<p>Is the same result true for operators on ... | Robert Israel | 13,650 | <p>What I would consider the obvious proof uses only the Banach space structure.</p>
<p>If $\lambda$ is in the resolvent set, the graph $G(T)$ of $T$ maps in an obvious way to the graph of $(T-\lambda I)^{-1}$:
$G(T) = f^{-1}(G((T-\lambda I)^{-1}))$ where
$$f:\;(x, y) \mapsto (y-\lambda x, x)$$
Since $f$ is continuou... |
3,741,122 | <p>Recently I've tried to find the difference between partial differentiation and total differentiation.
I've heard the total derivative is defined on single value functions, while the partial derivative by contrast is defined on multivariate functions.
My problem is, that total differentiation is used on multivariate ... | David K | 139,123 | <p>In order to make sense of a total derivative of a function of multiple variables, you need to make some additional assumptions.</p>
<p>Suppose <span class="math-container">$x_1,x_2,\ldots,x_n$</span> are all single-variable functions of some variable <span class="math-container">$t$</span>
(using the word "func... |
1,371,549 | <p>Differentiate the Function : $y=\log_2(e^{-x} \cos(\pi x))$</p>
<p>Here is my work. What I have I done wrong?
<a href="https://i.stack.imgur.com/w9RSN.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/w9RSN.jpg" alt="enter image description here"></a></p>
| Leucippus | 148,155 | <p>As to the OP's result: just a little cleaning up provides the correct result.</p>
<hr>
<p>By using
\begin{align}
\log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)}
\end{align}
then it can be obtained that, using $\log_{e}(x) = \ln(x)$,
\begin{align}
\log_{2}(e^{-x} \, \cos(ax)) = \frac{\ln(e^{-x} \, \cos(ax)}{\ln(2)} ... |
2,423,055 | <p>I am sort of baffled by this thing, already real number has every thing in it why is this concept of $\Bbb R^2$ ? What does it mean? What is its advantage?</p>
| Community | -1 | <p>the Cartesian product of the reals with itself is a common meaning.</p>
|
2,135,811 | <p>$ \frac{6x}{x-2} - \sqrt{\frac{12x}{x-2}} - 2\sqrt[4]{\frac{12x}{x-2}}>0 $</p>
<p>I've tried putting $t= \frac{6x}{x-2} $ and play algebraically, using square of sum, but still no luck, any help?</p>
| the_candyman | 51,370 | <p>Instead, let's put $$t = \sqrt[4]{\frac{12x}{x-2}}.$$
Then:</p>
<p>$$\frac{6x}{x-2} - \sqrt{\frac{12x}{x-2}} - 2\sqrt[4]{\frac{12x}{x-2}}>0 \Rightarrow \\
\frac{1}{2}t^4 - t^2 - 2t>0 \Rightarrow \\
t^4 - 2t^2 - 4t > 0 \Rightarrow \\
t(t-2)(t^2+2t+2) >0.
$$</p>
<p>Then:</p>
<p>$$t <0 ~ \vee t > 2... |
1,019,408 | <p>Usually, $f$ denotes a function, $f(x)$ is an image of $x$ under $f$. But what's $f(X)$ if $X$ is a set? </p>
<p>edit: Please, disregard the body of this question. I had to put something here to be able to post the question. </p>
| Kim Jong Un | 136,641 | <p>If $f(x)=x$ for all $x$ then $f(S)=\{s:\exists x\text{ s.t. }f(x)=s\}$ equals to $S$. The other direction generally doesn't hold. Consider $S=\{0,1\}$ and $f(x)=1-x$. Then, $f(S)=S$ but in general $f(x)\neq x$.</p>
|
1,019,408 | <p>Usually, $f$ denotes a function, $f(x)$ is an image of $x$ under $f$. But what's $f(X)$ if $X$ is a set? </p>
<p>edit: Please, disregard the body of this question. I had to put something here to be able to post the question. </p>
| Adam Hughes | 58,831 | <p>Say $S=\{1,2\}$ and $f(1)=2, f(2)=1$. Then $f(S)=S$, but $f$ is not the identity. The big letter means all of the things, the small letter means bit by bit. Another good way to think of it is in analogy with $1+2+3=2+2+2$ the sums are the same (large structure) but the individual pieces are different.</p>
<p>It's a... |
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | Jailcu | 259,566 | <p>The simplest way to remember how to calculate is by taking $\frac{1}{2}$ the value of the determinant of the matrix
$$
\begin{bmatrix}
1 & 1 & 1 \\
x_1 & x_2 & x_3 \\
y_1 & y_2 & y_3
\end{bmatrix}
$$</p>
|
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | iwantmyphd | 159,953 | <p>For fun, I'll just throw out the <strong><em>really long</em></strong> way that I learned in 3rd grade, only because it hasn't been submitted. I <strong><em>don't</em></strong> endorse this, the Shoelace/Surveyor's formula is way better.</p>
<ol>
<li>Determine the distance between two of the three points, say $ \b... |
482,030 | <p>I'm reading a proof of the irrationality of <span class="math-container">$\sqrt 2$</span>. In a step it states that <span class="math-container">$2d^2=n^2$</span> implies that <span class="math-container">$n$</span> is multiple of <span class="math-container">$2$</span>. How?</p>
| AlexR | 86,940 | <p>It works for any prime $p$:
$$p|n^2 \Rightarrow p|n$$
This is because of the uniqueness of prime factor decomposition.</p>
|
1,573,947 | <p>I'm trying to see the relationship between the sample variance equation </p>
<p>$\sum(X_i- \bar X)^2/(n-1)$ and the variance estimate, $\bar X(1-\bar X),$ in case of binary samples. </p>
<p>I wonder if the outputs are the same, or if not, what is the relationship between the two??</p>
<p>I'm trying to prove their... | Michael Hardy | 11,667 | <p>\begin{align}
& \sum_{i=1}^n (x_i - \bar x)^2 \\[10pt]
= {} & \sum_{i=1}^n (x_i - 2\bar x x_i + \bar x^2) \\[10pt]
= {} & \left( \sum_{i=1}^n x_i^2 \right) - 2\bar x \left( \sum_{i=1}^n x_i \right) + \left( \sum_{i=1}^n \bar x^2 \right) \\[10pt]
= {} & \left( \sum_{i=1}^n x_i^2 \right) - 2\bar x\Big(... |
749,849 | <p>Question:</p>
<p>A submatrix $B$ consisting of "s" rows of $A$ is selected from an n-square matrix $A$ of rank $r_{A}$. prove that the rank of $B$ is equal to or greater than $r_{A}+s-n$.</p>
<p>My thoughts:</p>
<p>I start with an easy case, says $A=I_4$. Then, by selecting 2 first rows of $A$. We obtain a matrix... | Git Gud | 55,235 | <p><strong>Hint:</strong> Note the following: <span class="math-container">$r_A\leq n$</span> and <span class="math-container">$r_B\leq s$</span>. With this in mind, instead of trying to prove that <span class="math-container">$r_B\ge r_A+s-n$</span> try to prove an equivalent inequality that has <span class="math-cont... |
101,929 | <p>I am trying to do the following :-</p>
<pre><code>f[x_] := {x, 2};
Do[f[x_] :=Append[f[x], {x, 4}],{3}] % runs into recursion limit
</code></pre>
<p>because I want to grow my function (which is a list of functions) in a loop ? What is the right way to do this ?</p>
| nben | 16,054 | <p>Constructing functions using codes (like the use of macros in LISPs) is completely possible in Mathematica but can be tricky, particularly if one expects the Hold and Evaluate operators to behave analogously to the quote/eval functions in LISPs. There are many ways to accomplish what you ask about; here is one:</p>
... |
2,376,315 | <p>So I'm trying to solve the problem irrational ^ irrational = rational. Here is my proof
Let $i_{1},i_{2}$ be two irrational numbers and r be a rational number such that $$i_{1}^{i_{2}} = r$$
So we can rewrite this as $$i_{1}^{i_{2}} = \frac{p}{q}$$
Then by applying ln() to both sides we get $$i_2\ln(i_1)... | user2357112 | 91,416 | <p>This is not a valid proof. You've assumed the conclusion:</p>
<blockquote>
<p>Let $i_{1},i_{2}$ be two irrational numbers and r be a rational number such that $$i_{1}^{i_{2}} = r$$</p>
</blockquote>
<p>You can't just start by assuming that such numbers exist. At best, what you have would be a proof that <em>if</... |
4,446,352 | <p>Consider two different ways to formally define a predicate, say the predicate <span class="math-container">$E(x)$</span> which is to be interpreted as "<span class="math-container">$x$</span> is even."</p>
<p><strong>CASE <span class="math-container">$~1$</span></strong></p>
<p><span class="math-container"... | ryang | 21,813 | <blockquote>
<p><strong>CASE <span class="math-container">$~1$</span></strong></p>
<p><span class="math-container">$\forall a:[a \in \mathbb N \implies [E(a) \iff \exists b:[b \in \mathbb N \land a=2b]]]$</span></p>
</blockquote>
<p><strong>Definition 1</strong> accurately and legitimately defines the <em>even natural ... |
4,582,378 | <p><em>Defintition:</em> A real sequence <span class="math-container">$\ (x_n)_n\ $</span> is <em>convex</em> if <span class="math-container">$\ x_n - x_{n+1} \geq x_{n+1} - x_{n+2}\quad \forall\ n\in\mathbb{N}. $</span></p>
<p>Continuing on from this question <a href="https://math.stackexchange.com/questions/4581894/i... | Theo Bendit | 248,286 | <p>This is not true in general. In fact, given any strictly positive, convex, decreasing, summable <span class="math-container">$a_n$</span>, I can construct a convex, decreasing, non-summable <span class="math-container">$b_n$</span> so that <span class="math-container">$\frac{a_n}{b_n} \not\to 0$</span>.</p>
<h2>Defi... |
164,340 | <p>I recently found myself at a spot that I never believed I'll get (or at least not that soon in my career). I ran into a problem which seems to be best answered via categories.</p>
<blockquote>
<p>The situation is this, I have a directed system of structures and the maps are all the inclusion map, that is <span class... | Qiaochu Yuan | 232 | <p>I don't understand whether you actually have a counterexample or not so let me supply one for the sake of completeness. Consider the category $\text{CHaus}$ of compact Hausdorff spaces. I can write down a filtered colimit of finite sets $\{ 1 \} \to \{ 1, 2 \} \to \{ 1, 2, 3, ... \}$ whose union is $\mathbb{N}$. The... |
2,919,096 | <p>I am trying to solve a system of two second-order ODEs. After separating them, I obtained a fourth-order independent ODE as illustrated below. I wonder if there is a specific technique to solve it.</p>
<p>$$y^{(4)}+\frac{a_1}{x} y^{(3)}+\frac{a_2}{x^2}y^{(2)}+a_3y^{(2)}+\frac{a_4}{x}y^{(1)}+a_5y=0$$</p>
| doraemonpaul | 30,938 | <p>Hint:</p>
<p><span class="math-container">$y^{(4)}+\dfrac{a_1}{x}y^{(3)}+\dfrac{a_2}{x^2}y^{(2)}+a_3y^{(2)}+\dfrac{a_4}{x}y^{(1)}+a_5y=0$</span></p>
<p><span class="math-container">$x^2y''''+a_1xy'''+(a_2+a_3x^2)y''+a_4xy'+a_5x^2y=0$</span></p>
<p>Another notable special cases appear when <span class="math-contai... |
2,090,059 | <p>Let $M = \bigcup\limits_{n=1}^\infty M_n$, $M_n$ be a measurable sets on $\mathbb R$ and $f$ be a measurable function on $\mathbb R$. Then I want to know how the equality
$$
\lim\limits_{N\to \infty}\int_{M\setminus \bigcup\limits_{n=1}^NM_n} |f(x)| \,dx = 0
$$
holds. It seems something like monotone convergence th... | Martín-Blas Pérez Pinilla | 98,199 | <p>Hint: use characteristic functions to write each integral as an integral on $M$.</p>
|
2,652,675 | <p>Given that A = $\begin{bmatrix} 2 & 1 \\ -5 & -4 \end{bmatrix} $ and B = $\begin{bmatrix} 3 & -1 \\ -1 & 0 \end{bmatrix} $ </p>
<p>Find a 2 X 2 matrix C such that $CA= B$</p>
<p>I multiply both sides by $A^{-1}$ </p>
<p>Since $A^{-1}A = I $ </p>
<p>$ CI = BA^{-1}$ </p>
<p>Since $CI = IC = C$ </p... | Henno Brandsma | 4,280 | <p>There are two ways to approach bases: we start with a topology $(X,\mathcal{T})$ and then a base $\mathcal{B}$ <em>for that topology</em> is a subcollection of $\mathcal{T}$ (so they're open sets) such that every $O \in \mathcal{T}$ can be written as a union of elements from $\mathcal{B}$, so $O = \bigcup \mathcal{B... |
910,552 | <blockquote>
<p>$\sin(nx)$ does not contain Cauchy subsequence in $L^p([0,2\pi]) $ for $1\leq p < \infty$</p>
</blockquote>
<p><strong>My attempt:</strong> </p>
<p>Set $f_n(x) = \sin(nx)$.</p>
<p>Argue by contradiction, suppose there exists a Cauchy subsequence $f_{n_k}$ in $L^p$ for some fixed $p$, then $f_{n_... | Zarrax | 3,035 | <p>Your argument is fine. You can also do a direct variant of it like this. If $\{f_{n_k}\}_{k=1}^{\infty}$ were a Cauchy sequence, then $\lim_{k \rightarrow 0} ||f_{n_{k+1}} - f_{n_k}||_{L^p[0,2\pi]}$ would be zero. But by Holder's inequality
$$\int_0^{2\pi}|f_{n_{k+1}} - f_{n_k}|^2 \leq||f_{n_{k+1}} - f_{n_k}||_{L^p... |
11,435 | <p>I know what they are and how to define them. They have to serve some purpose - else why include them? But I never used them and all examples on this site and the docs never show a practical use of <code>UpValues</code>.</p>
<p>Can someone give me an example to see when they are actually needed?</p>
| Peter Breitfeld | 307 | <p>An example I use from time to time is to "prettify" output. Suppose you have a not so huge matrix, e.g. </p>
<pre><code>aa = Array[Subscript[a, #1, #2] &, {3, 3}]
</code></pre>
<p>which prints with commas between the Indexes. Sometimes, you don't want them and you can replace them with <code>InvisibleComma</co... |
4,159,337 | <p>How would you show
<span class="math-container">$$\log_2 3 + \log_3 4 + \log_4 5 + \log_5 6 > 5?$$</span></p>
<p>After trying to represent the mentioned expression in the same base, a messy expression is created.
The hint mentioned that the proof can take help of quadratic equations.</p>
<p>Could you provide some... | Pythagoras | 701,578 | <p>@Startwearingpurple: I did not downvote, but I thought your proposition is interesting because it implies the OP's result. Here is a proof, though it may be just useless.</p>
<p><strong>Problem</strong>. Prove that <span class="math-container">$\log_23+\log_34+\log_45>4$</span>.</p>
<p><em>Proof</em>. Note that o... |
626,920 | <p>$a_n=\sum_{k=1}^{n} \frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}$</p>
<p>How to find $\lim a_n$?</p>
| Felix Marin | 85,343 | <p>$\newcommand{\+}{^{\dagger}}%
\newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
\newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
\newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displayst... |
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