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 |
|---|---|---|---|---|
312,012 | <p>What is the exact difference between $\arg\max$ and $\max$ of a function?</p>
<p>Is it right to say the following?</p>
<blockquote>
<p>$\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is maximal. And $\max f(x)$ means value of $f(x)$ for which it is maximum.</p>
</blockquote>
... | copper.hat | 27,978 | <p>$\arg \max_x f(x) = f^{-1} \{ \max_x f(x) \}$.</p>
<p>For example, $\max_x \cos x = 1$, $\arg \max_x \cos x = \{ n 2 \pi\}_{n \in \mathbb{Z}}$.</p>
|
2,112,259 | <p>I want to prove or disprove $\min{S_1}=\min{S_2}\Longleftrightarrow\min{S_1}\in S_2\land\min{S_2}\in S_1$ but I don't know where to start.</p>
<p>Maybe take cases like $\min{S_1}<\min{S_2}$, $\min{S_1}>\min{S_2}$ and $\min{S_1}=\min{S_2}$ and see where it is true?</p>
| Mark Fischler | 150,362 | <p>Your equations are
\begin{cases}
12k= 839+ p^2\\
2k= 135+ q^2
\end{cases}
and we observe that $$p^2-q^2 = 10k - 704 = 5(2k-141)+1$$
so $\,p^2 - q^2 = 1 \pmod{5}$ which tells us that $p^2 = 1 \mod 5$ and $q = 0 \mod 5$, or $p=0 \mod 5$ and $q^2 = -1 \mod 5$. We consider the first case here, and the second later. Sin... |
4,010,386 | <p>Let <span class="math-container">$P_n$</span> be a vector space of all polynomials with real coefficients and of degree <span class="math-container">$n$</span> or less. If <span class="math-container">$a \in \Bbb R$</span> and <span class="math-container">$W=\{p \in P_n \mid p(a)=0\}$</span>, show that <span class="... | Rhys Hughes | 487,658 | <p>Spot on.</p>
<p>You can also condense the latter two properties into one: <span class="math-container">$$\forall \alpha, \beta \in \Bbb R; p, q\in W \implies\alpha p + \beta q \in W $$</span></p>
|
4,010,386 | <p>Let <span class="math-container">$P_n$</span> be a vector space of all polynomials with real coefficients and of degree <span class="math-container">$n$</span> or less. If <span class="math-container">$a \in \Bbb R$</span> and <span class="math-container">$W=\{p \in P_n \mid p(a)=0\}$</span>, show that <span class="... | Hagen von Eitzen | 39,174 | <p>Yes, that's fine.</p>
<p>In case you have this tool available, you might also simply observe that <span class="math-container">$W$</span> is defined as the kernel of the linear map <span class="math-container">$P_n\to\Bbb R$</span>, <span class="math-container">$p\mapsto p(a)$</span> and therefore a subspace.</p>
|
4,010,386 | <p>Let <span class="math-container">$P_n$</span> be a vector space of all polynomials with real coefficients and of degree <span class="math-container">$n$</span> or less. If <span class="math-container">$a \in \Bbb R$</span> and <span class="math-container">$W=\{p \in P_n \mid p(a)=0\}$</span>, show that <span class="... | Ethan Bolker | 72,858 | <p>This is almost right. The train of thought and the important algebraic steps are right. But throughout your argument you refer to "some <span class="math-container">$a$</span>". That's not correct. The <span class="math-container">$a$</span> is fixed from the start. The space you are interested in might b... |
4,490,117 | <blockquote>
<p>(a) Let <span class="math-container">$f : [a, b] \to \mathbb{R}$</span> be continuous and suppose that <span class="math-container">$f(x) \gt 0$</span> for all <span class="math-container">$x$</span>. Show that there is some <span class="math-container">$L\gt 0$</span> such that <span class="math-contai... | Prime Mover | 466,895 | <p>Two different things going on here.</p>
<p>Treatment of the Peano axioms is very much in Fundamentals of Mathematics, which often starts with axiomatic set theory and works towards the establishment of the rules of commutativity and associativity. There are many such books on this, although all take the subject slig... |
305,373 | <p>I'm doing some self-studying on discrete mathematics and so far its going well however I came upon a question that whilst I can make sense of and I know that set $A$ is a subset of set $B$, I cannot think of how to express it and the book does not feature an answer for this question.</p>
<p>The question is, prove t... | Zev Chonoles | 264 | <p>Recall that a set $S$ is a subset of a set $T$, which we express as $S\subseteq T$, when
$$x\in S\implies x\in T.$$
Thus, in your problem, to show that $A\subseteq B$, what you must show is that, for any $n\in\mathbb{Z}^+$, we have that $2n$ is among the elements of $B$. Note that, because $B$ has precisely the same... |
1,832,515 | <p>I know that you almost always set domains for a summation function $\left( \sum \right)$, but can you also set an interval for that domain? Say the domain was 1 to 10, could I set an increment of 0.5 instead of the standard 1? I know that there are ways around this such as setting another variable for the previous n... | j4l3kl24jkl2 | 266,771 | <p>Like Will Jagy said, substitute $t=\sqrt{x^2 -2x+3}$.
You then can solve the formula: $$\sqrt{t^2+5}=125-t \\t^2 + 5 =15625-250t +t^2$$ $$250t = 15620$$ $$t=62.48$$ Then you can plug your value for $t$ back into $t^2=x^2 -2x+3$ and solve a quadratic formula with respect to $x$.</p>
|
501,993 | <p>I have never encounter "proof" questions before in my career, but this question in textbook troubled me and I have totally no clue where to start.</p>
<p>Prove that the limit
$$\lim_{x\rightarrow0} \frac{1-\sqrt{1-x^2}}{x^2} = \frac{1}{2}$$</p>
| peterwhy | 89,922 | <p>$$\begin{align}
\lim_{x\rightarrow0} \frac{1-\sqrt{1-x^2}}{x^2}
=& \lim_{x\rightarrow0} \frac{1-\sqrt{1-x^2}}{x^2}\cdot\frac{1+\sqrt{1-x^2}}{1+\sqrt{1-x^2}}\\
=& \lim_{x\rightarrow0} \frac{1^2-\left(1-x^2\right)}{x^2\left(1+\sqrt{1-x^2}\right)}\\
=& \lim_{x\rightarrow0} \frac{x^2}{x^2\left(1+\sqrt{1-x^2}... |
2,803,956 | <p>Let $\mathfrak{g}$ be a Lie algebra. Then its automorphism group $Aut(\mathfrak{g})$ is a Lie group, and hence we may take its Lie algebra $Lie(Aut(\mathfrak{g}))$.</p>
<p>I'd like to say that this is equal to the Lie algebra of derivations $Der(\mathfrak{g})$. Is this true? Where can I find a reference?</p>
| William C. Newman | 773,090 | <p><span class="math-container">$\DeclareMathOperator{\Aut}{Aut}$</span>
It is in fact true that <span class="math-container">$\text{Lie}(\Aut(L))=\text{Der}(L)$</span>, as you say. There are a lot of questions on Stack Exchange asking about this, and how to prove it. But no answers seem to give a proof.</p>
<p>[Additi... |
551,873 | <p>I have to solve the following problem:
Using $\exists$ Introduction prove that PA$\vdash x\leq y \wedge y\leq z \longrightarrow x\leq z$: I used that if $x\leq y$ then $\ \exists \ r\ x+r=y$ and in the same way $\exists t \ y
+t=z$, but in logic term I don't know how to use these equalities and the Peano's axioms t... | Cameron Buie | 28,900 | <p>Well, you know that $x+r=y$ that $y+t=z,$ so, <em>if we know that addition is associative</em>, then it follows that $x+(r+t)=(x+r)+t=y+t=z.$ Since $r+t\in\Bbb N$ for any $r,t\in\Bbb N,$ then putting $s=r+t,$ we have shown that there exists $s$ such that $x+s=z,$ and so $x\le z$ by definition of the relation "$\le$.... |
2,283,367 | <p>I know product of nth roots of unity is 1 or -1 depending whether n is odd or even.
But in this way I am getting 1. Where am I wrong?</p>
<p>$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \pi}{n} \text{ be a root of }x^n=1 \\
\text{Then product of nth roots will be } 1\cdot \alpha \cdot \alpha^2 ..... | Empy2 | 81,790 | <p>$(n-1)/2$ is not an integer, so $1^{(n-1)/2}$ involves the square-root, which might be positive or negative. You have to look more closely to decide which.<br>
It is a bit like $-1=(-1)^{2/2}=1^{1/2}=1$</p>
|
2,283,367 | <p>I know product of nth roots of unity is 1 or -1 depending whether n is odd or even.
But in this way I am getting 1. Where am I wrong?</p>
<p>$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \pi}{n} \text{ be a root of }x^n=1 \\
\text{Then product of nth roots will be } 1\cdot \alpha \cdot \alpha^2 ..... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>$$\alpha^{n(n-1)/2}=(\alpha^{n/2})^{n-1}$$</p>
<p>Now $\alpha^{n/2}=\cdots=-1$</p>
|
166,202 | <p>If we have $k(x,t)= \frac {1}{(4t)^{\frac{n}{2}}} \exp\left(\frac{-|x|^2}{4t}\right)$ is the fundamental solution of heat equation. If we consider $n \ge 3 $, I would like to show that $\int_0^\infty k(x,t) dt$ is the fundamental solution of lLaplace equation.
I would like some hints.
I thought of integrating but ... | timur | 2,473 | <p>Let $e(x)=\int_0^\infty k(x,t)\,\mathrm{d}t$, and let us compute the action of the distribution $\Delta e$ on the test function $\phi$ (which is a compactly supported smooth function in $\mathbb{R}^n$) as
$$
\begin{split}
\langle\Delta e,\phi\rangle
&= \int_{\mathbb{R}^n} e(x) \Delta\phi(x)\,\mathrm{d}x
= \int... |
1,951,964 | <p>I think the answer is <span class="math-container">$\mathbb{Z}_n$</span> where <span class="math-container">$n$</span> is all the factors of <span class="math-container">$26$</span>. Is this right?</p>
| Claude Leibovici | 82,404 | <p><em>Just another way to do it.</em></p>
<p>Consider that $${\frac {x}{1+x^2}}=\frac 12 \frac d {dx} \log(1+x^2)$$ Now, using $$\log(1+t)=\sum_{n=1}^\infty(-1)^{n+1}\frac{t^n}n$$ $$\log(1+x^2)=\sum_{n=1}^\infty(-1)^{n+1}\frac{x^{2n}}n$$ $$\frac d {dx} \log(1+x^2)=\sum_{n=1}^\infty(-1)^{n+1}\frac{2nx^{2n-1}}n=2\sum_{... |
1,637,237 | <p>I'm studying differential calculus, but one of the questions involves solving an inequality:</p>
<p>$$(x-2)e^x < 0$$</p>
<p>I intend to go deeper in solving inequalities later, but I just want to understand how the teacher got the following solution in order to advance in these lectures:
$$x-2 < 0$$
$$x <... | adjan | 219,722 | <p>Divide by $e^x$</p>
<p>$$(x-2)e^x < 0$$
$$\iff x - 2 < 0$$</p>
<p>This is valid since $e^x > 0$ $\forall x \in \mathbb{R}$</p>
|
4,114,360 | <p><strong>Question</strong> :</p>
<blockquote>
<p>Andy and Beth are playing a game worth $100. They take turns flipping
a penny. The first person to get 10 heads will win.</p>
<p>But they just realized that they have to be in math class right away
and are forced to stop the game.</p>
<p>Andy had four heads and Beth ha... | angryavian | 43,949 | <p>A slight modification of your attempt will work. Take <span class="math-container">$r = \min(\lceil x \rceil - x, x - \lfloor x \rfloor, \frac{1}{2}|x - \lfloor x \rfloor - \frac{1}{2 \lfloor x \rfloor}|)$</span>. We will show all elements of <span class="math-container">$E$</span> are outside of the open ball of ra... |
4,114,360 | <p><strong>Question</strong> :</p>
<blockquote>
<p>Andy and Beth are playing a game worth $100. They take turns flipping
a penny. The first person to get 10 heads will win.</p>
<p>But they just realized that they have to be in math class right away
and are forced to stop the game.</p>
<p>Andy had four heads and Beth ha... | Rivers McForge | 774,222 | <p>If you know open intervals are open sets, and arbitrary unions of open sets are open, then <span class="math-container">$$E = \{ 1.5, 2.25, 3.167, 4.125, ... \}$$</span> has a complement which is easily seen to be a union of open intervals: <span class="math-container">$$E^c = (-\infty, 1.5) \cup (1.5, 2.25) \cup (2... |
4,114,360 | <p><strong>Question</strong> :</p>
<blockquote>
<p>Andy and Beth are playing a game worth $100. They take turns flipping
a penny. The first person to get 10 heads will win.</p>
<p>But they just realized that they have to be in math class right away
and are forced to stop the game.</p>
<p>Andy had four heads and Beth ha... | Henno Brandsma | 4,280 | <p>Let <span class="math-container">$a_n = n + \frac{1}{2n}$</span>, so that <span class="math-container">$E=\{a_n\mid n \in \Bbb N\}$</span>.</p>
<p>Suppose that there were a sequence <span class="math-container">$(x_n)$</span> from <span class="math-container">$E$</span> that converges to some <span class="math-conta... |
2,309,986 | <p>This question is from the text <em>Geometry and Complexity Theory</em>, from J.M. Landsberg. Before start talking about the question, I think it is good to show the definition of border rank. Consider a tensor $T \in \mathbb{C}^r \otimes \mathbb{C}^r \otimes \mathbb{C}^r$. Then the <em>border rank</em> of $T$ is the... | Vladimir Lysikov | 20,589 | <p>One way to prove that the border rank is exactly $r$ and not less is the flattening lower bound. You can see a tensor $T \in A\otimes B\otimes C$ as a linear map $T\colon A^* \to B\otimes C$. If $T$ has tensor rank $\leq r$, than this map also has rank $\leq r$, and we can take the limits to talk about border rank. ... |
2,025,711 | <p>What is the sum of this series: $$\sum_{n>=1}{(-1)^{n+1}*\frac{2n+1}{n(n+1)}}$$
I don't know how to solve it especialy with that ${(-1)^{n+1}}$</p>
| TZakrevskiy | 77,314 | <p>Let $n$ is even. Write the sum of two consecutive elements of your sum:
$$\frac{2n+1}{n(n+1)} - \frac{2n+3}{(n+2)(n+1)} = \frac{2}{n(n+2)} = \frac 1n - \frac{1}{n+2}.$$</p>
<p>Can you take it from here?</p>
|
2,699,842 | <p>I've been looking at this question for hours now, but I can't grasp it: </p>
<p>Let A be a symmetric matrix with minimum eigenvalue $\lambda_{\text{min}}$ . Give a bound on the largest element of $A^{−1}$.</p>
<p>I was looking at the spectral decomposition where $A=VDV^T$ with D a diagonal matrix, however I don't ... | angryavian | 43,949 | <p>The $(i,j)$ entry of a matrix $B$ can be written as $e_i^\top B e_j$ where $e_i$ is the $i$th standard basis vector.
By Cauchy-Schwarz, we have $|B_{ij}| = |e_i^\top B e_j| \le \|B e_j\| \le \sigma_{\max}(B)$ where $\sigma_{\max}(B)$ is the largest singular value of $B$. If $B$ is symmetric, this is the largest eige... |
2,699,842 | <p>I've been looking at this question for hours now, but I can't grasp it: </p>
<p>Let A be a symmetric matrix with minimum eigenvalue $\lambda_{\text{min}}$ . Give a bound on the largest element of $A^{−1}$.</p>
<p>I was looking at the spectral decomposition where $A=VDV^T$ with D a diagonal matrix, however I don't ... | Ross Millikan | 1,827 | <p>There is no limit. Let $\lambda_{min} \lt 0$ and $A=\begin {pmatrix} \lambda_{min}&0\\0&\epsilon \end {pmatrix}$. We can make $\epsilon$ as small as we want in absolute value and $A^{-1}=\begin {pmatrix} 1/\lambda_{min}&0\\0&1/\epsilon \end {pmatrix}$</p>
|
125,393 | <p>As we know, Linux’s Bash shell is coming to Windows 10. Now, I want to know, how to using Run[] command or other commands to run some commands like gfortran installed in Linux of Win10 ? And, is it possible to call mathematica using Bash shell in Windows 10 ? </p>
| Rolf Mertig | 29 | <p>Installing the Windows git-bash version from <a href="https://git-scm.com/downloads" rel="nofollow noreferrer">here</a> enables a simple way to run bash under Windows 10 from Mathematica:</p>
<pre><code>RunProcess[{"C:\\Program Files\\Git\\bin\\bash.exe", "-c",
"bash --version | head -1 | tr -d '\n'"}, "Standard... |
12,091 | <p>It looks like the traffic is really high today, probably because of the introduction of hats. </p>
<p>I was wondering, is there a way to check statistics about the traffic on math.SE, and on the other SE websites?</p>
| Alex Becker | 8,173 | <p>You can take a look at the <a href="http://www.alexa.com/" rel="nofollow">Alexa rankings</a> for any site, but those aren't very good. Stackexchange keeps its own, more detailed data on traffic and usage, but this is not available to regular users.</p>
|
71,670 | <pre><code>TableForm[Table[i/j + 4*Boole[j > i] // N, {i, 3}, {j, 4}], TableHeadings -> {{"Row1", "Row2", "Row3"}, {"Col1", "Col2", "Col3", "Col4"}}]
</code></pre>
<p>Produces the following table:</p>
<p><img src="https://i.stack.imgur.com/Dqnmr.png" alt="enter image description here"></p>
<p>How select the ma... | ybeltukov | 4,678 | <p>Yet another realization of the straightforward idea</p>
<pre><code>t = Table[i/j + 4*Boole[j > i] // N, {i, 3}, {j, 4}];
h = {{"Row1", "Row2", "Row3"}, {"Col1", "Col2", "Col3", "Col4"}};
Style[TableForm[# /. x : Max@# :> Style[x, Bold] & /@ t,
TableHeadings -> h], FontFamily -> "Times"]
</code><... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | Michael Gmirkin | 12,138 | <p>Reposted from PolymathProgrammer.com, <a href="http://polymathprogrammer.com/2011/05/29/regular-polygons-equation/comment-page-1/#comment-35399" rel="nofollow">my answer to my own initial query</a>. Generated mostly on my own after some initial help from a friend (Jason Schmurr) and my dad (Russell Gmirkin)</p>
<hr... |
41,940 | <p>For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?</p>
<p>Using the Wolfram Alpha site, this input gave an almost-square:
<code>PolarPlot(0.75 + ArcSin(Sin(2x... | J. M. ain't a mathematician | 498 | <p>Here is <em>another</em> parametric equation for a regular $n$-gon with unit radius:</p>
<p>$$\begin{align*}x&=\cos\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}(2\lfloor u\rfloor+1)\right)-(2u-2\lfloor u\rfloor-1)\sin\left(\frac{\pi}{n}\right)\sin\left(\frac{\pi}{n}(2\lfloor u\rfloor+1)\right)\\y&=\cos\... |
2,440,478 | <p>So, I understand that for something to be <em>rigorous</em> it is inclined to meet strict and occasionally extreme criteria. Within the scope of mathematics, this therefore implies that anything that is truly rigorous may not defy any of the rules of mathematics, and as it should with everything in mathematics, but ... | Especially Lime | 341,019 | <p>It is perfectly valid to go through and replace $\leq$ by $<$ for most of this proof, since for any given $x<\pi/2$ the areas are all different. (However, even if we get to $1<\frac{\sin x}x<\cos x$, when we take the limit we have to put $\leq$ back in. $f(x)<g(x)$ only implies $\lim f(x)\leq \lim g(x... |
4,237,056 | <p>Can any subgroup of a cyclic group <span class="math-container">$\left<a\right>$</span> be presented as <span class="math-container">$\{x\in \left<a\right>|x^m=e\}$</span>?<br />
Note that m is a factor of n, where n is the order of <span class="math-container">$\left<a\right>$</span>.<br />
I foun... | Sidharth Ghoshal | 58,294 | <p>So our two equations you found combined imply that</p>
<p><span class="math-container">$$ \lambda = \frac{2(x_2 -2)}{5} $$</span></p>
<p>[I didn't check this and will just assume you did the algebra correctly].</p>
<p>So now what to do with this information? We need to go ahead and grab one of your unused original e... |
1,188,483 | <p>Suppose $|a|<1$, show that $f(x) = \frac{z-a}{1-\overline{a}z}$ is a mobius transformation that sends $B(0,1)$ to itself.</p>
<p>To make such a mobius transformation i tried to send 3 points on the edge to 3 points of the edge. so filling $i,1,-1$ in $f(x)$ we should get on the edges of the unit ball. But i don'... | DeepSea | 101,504 | <p><strong>hint</strong>: prove that : $\left|\dfrac{z-a}{1-\bar{a}z}\right| < 1$ by using $z = x+iy, a = m+in$, and use the well-known CS inequality: $(a^2+b^2)(c^2+d^2) \geq (ac+bd)^2$.</p>
|
2,480,493 | <p>I have a problem with what looks like a very easy equation to solve $\left|\frac{1 + z}{1- i\overline z}\right| = 1$ . ($z$ is a complex number, $\overline z$ is a conjugate of $z$) I got stuck at the point when after replacing $\overline z = a-bi $ and $z = a +bi$ and getting rid of absolute value I end up with $... | nonuser | 463,553 | <p>Since $|1+z|=|1-i\overline{z}|$ and $|w| = |\overline{w}|$, you can rewrite like this $$|z-(-1)|=|1+z|=|1-i\overline{z}| = |\overline{1-i\overline{z}}| =|1+iz| =|i||-i+z| = |z-i| $$</p>
<p>So $z$ is at equal distance from $-1$ and $i$. So $z$ is on perpendicular bisector of segment between $-1$ and $i$. </p>
|
2,480,493 | <p>I have a problem with what looks like a very easy equation to solve $\left|\frac{1 + z}{1- i\overline z}\right| = 1$ . ($z$ is a complex number, $\overline z$ is a conjugate of $z$) I got stuck at the point when after replacing $\overline z = a-bi $ and $z = a +bi$ and getting rid of absolute value I end up with $... | Robert Z | 299,698 | <p>Something went wrong in your evaluation.
After the substitution $z=a+ib$, you should have
$$(1+a)^2+b^2=|1+a+ib|^2=|1-b-ia|^2=(1-b)^2+(-a)^2$$
that is, after a few simplifications, $a=-b$.</p>
|
4,435,945 | <p>Near point <span class="math-container">$(0,0)$</span> find taylor formula for <span class="math-container">$f(x,y)=\ln(1+x+y)$</span></p>
<p><span class="math-container">$f(x)=f(x_0)+\sum_{k=1}^n \frac{1}{k!}d^kf(x_0)+o(|x-x_0|^n)$</span></p>
<p><span class="math-container">$$d^kf(0,0)=\sum_{n=1}^k\binom{k}{n}\frac... | José Carlos Santos | 446,262 | <p>Your guess is right. Note that<span class="math-container">$$\{e_1,e_2\}=\left\{\begin{bmatrix}\frac2{\sqrt5}\\0\\\frac1{\sqrt5}\end{bmatrix},\begin{bmatrix}-\frac1{\sqrt{30}}\\\frac5{\sqrt{30}}\\\frac2{\sqrt{30}}\end{bmatrix}\right\}$$</span>is an orthonormal basis of <span class="math-container">$P$</span> (which ... |
750,322 | <p>Find a formula for $\sum_{i=1}^{n} \frac{1}{(2i-1)(2i+1)}$ and prove that it holds for all $n \geq 1$ I don't know how to solve this particular problem, can someone help me please. Thanks </p>
| David H | 55,051 | <p>Hint: Expand the summand by partial fractions to see that the partial sums telescope: </p>
<p>$$\frac{1}{(2k-1)(2k+1)}=\frac{1}{2(2k-1)}-\frac{1}{2(2k+1)}\\
\sum_{k=1}^n\frac{1}{(2k-1)(2k+1)}=\sum_{k=1}^n\left(\frac{1}{2(2k-1)}-\frac{1}{2(2k+1)}\right)\\
=\frac{1}{2(2\cdot 1-1)}-\frac{1}{2(2n+1)}$$</p>
|
207,515 | <p>Consider the upper half space $\mathbb{R}^n_{+} = \{x = (x_1,..,x_n) \in \mathbb{R}^n : x_n \geq 0\}$. Consider the Laplacian on this space with either the Dirichlet boundary condition or the Neumann boundary condition. My question is: can the Laplacian $\Delta$ under such boundary conditions be treated as a pseudod... | Community | -1 | <p>What are pseudo-differential operators (pseudors) on a closed half-space $H$ or on a manifold with boundary? It is not adequate to only define them as restrictions from some open neighbourhood of $H$ to $H$. A pseudor $P$ does not map $C_c^\infty(H)$ into $C^\infty(H)$ (smoothness up to the boundary!) unless $P$ sat... |
2,658,368 | <p>I know that one can use Category Theory to formulate polynomial equations by modeling solutions as limits. For example, the sphere is the equalizer of the functions
\begin{equation}
s,t:\mathbb{R}^3\rightarrow\mathbb{R},\qquad s(x,y,z):=x^2+y^2+z^2,~t(x,y,z)=1.
\label{equalizer}
\end{equation}
One could then fin... | Nicolas Hemelsoet | 491,630 | <p>I just realized that <em>derived categories</em> and techniques from homological algebra can help to solve differential equations. This is a very huge subject called <em>algebraic analysis</em> which use the tools of $D$-modules and sheaf theory. See this <a href="https://mathoverflow.net/questions/77616/d-modules-a... |
391,572 | <p>I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?</p>
| rurouniwallace | 35,878 | <p>The simplest way to understand it is, consider the following equation:</p>
<p>$$f(x)=\left(1+\frac{1}{x}\right)^x$$</p>
<p>As $x$ gets larger and larger, notice what number it gets closer to:</p>
<p>$$f(1)=2$$
$$f(2)=2.25$$
$$f(3)\approx 2.37073$$
$$...$$
$$f(1000)\approx 2.7169$$
$$...$$</p>
<p>It is important ... |
1,057,172 | <p>How many 3digit numbers can be written with $2,4,4,6,6$ ? </p>
<p>I tried $\frac{5.4.3}{2!.2!} = 15$ but it's wrong. </p>
<p>when I solved the question "how many 3digit numbers can be written with $1,1,2$"
the solution $\frac{3.2.1}{2!}$ was correct but why this way doesn't work for above question?</p>
| Jack D'Aurizio | 44,121 | <p>The probability to hit $1$ after $1$ step is $\frac{1}{2}$, encoded by the string $+$; the probability to hit $1$ after $3$ steps is $\frac{1}{8}$, encoded by the string $-++$; the probability to hit $1$ after $5$ steps is $\frac{1}{16}$, encoded by the strings $-+-++$ and $--+++$. To compute the probability to hit ... |
112,889 | <p>I am new in Wolfram Math and I need a help in making a simple program. Let's suppose we have a list:</p>
<pre><code>list = {{1, 11}, {2, 7}, {4, 2}, {7, 9}, {-2, 3}, {-1, 10}};
</code></pre>
<p>Now, I need to collect the first elements of sublists, but not all of them, only those whose second elements are larger t... | RunnyKine | 5,709 | <pre><code>list = {{1, 11}, {2, 7}, {4, 2}, {7, 9}, {-2, 3}, {-1, 10}};
Select[list, #[[2]] > 8 &][[All, 1]]
</code></pre>
<blockquote>
<p>{1, 7, -1}</p>
</blockquote>
<p>OR using <code>Pick</code></p>
<pre><code>Pick[list[[All, 1]], UnitStep[list[[All, 2]] - 8], 1]
</code></pre>
<blockquote>
<p>{1, 7, ... |
2,419,785 | <blockquote>
<p>Prove the statement is true using mathematical induction: $$2n-1 \leq n!$$</p>
</blockquote>
<p>My attempt: this is true for $n=1$. </p>
<p>Suppose it is true for $n$, i.e., $2n-1 \leq n!$</p>
<p>Now, $2n-1 \leq n!\implies 2n-1+2 \leq n!+2$ </p>
<p>From here, how do I proceed? </p>
| user577215664 | 475,762 | <p>When exponents are at the same level you only deal with addition and multiplication... So $x^{10n} \left( x^5 \left(x^{-5}\right)^n \right) = x^{-10}$
Multiply n and 5</p>
<p>$x^{10n} x^5 x^{-5n} = x^{-10}$</p>
<p>Adding 5 and 5n</p>
<p>$x^{10n} x^{5-5n} = x^{-10}$</p>
<p>$x^{10n+5-5n} = x^{-10}$</p>
<p>$5+5n ... |
2,546,434 | <p>I have trouble calculating this limit algebraically (without L'Hospital's rule):</p>
<p>$$\lim_{x \to 1} \frac{x^{\pi} - x^e}{x-1}$$</p>
<p>Substituting 1 gives indeterminate form. If this was e.g. $(x^5 - x^2)$ in the numerator, one could easily factor out the $(x-1)$, because $(x^5 - x^2) = x^2 (x - 1) (x^2 + x ... | Rene Schipperus | 149,912 | <p>Ok so for integer $n$, its clear
$$\frac{x^n-1}{x-1}\to n$$ for a fraction $\frac{p}{q}$ let $x=y^q$ and then
$$\frac{x^{\frac{p}{q}}-1}{x-1}=\frac{y^p-1}{y^q-1}\to \frac{p}{q}$$<br>
Now for any $a$ we have </p>
<p>$$\lim\limits_{x\to 1^+}\frac{x^a-1}{x-1}=\lim\limits_{x\to 1^+}\sup\{\frac{x^r-1}{x-1}|r\in \mathb... |
1,644,715 | <p>Define a locally lipschitz and nonnegative function $f\colon\mathbb{R}^n\to\mathbb{R}$. Let $M\in\mathbb{R}^{n\times n}$ and $\eta>0\in\mathbb{R}$. Consider the function $h\colon\mathbb{R}^n\to\mathbb{R}^n$ defined as </p>
<p>$$h(\mathbf{x})=
\begin{cases}
\frac{1}{\Vert M\mathbf{x}\Vert}M\mathbf{x}, &\text{... | Gregor de Cillia | 210,541 | <p>Define the two sets </p>
<p>$$A:=\{x\in \mathbb{R}^n: f(x)\|Mx\|\geq \eta\}$$
$$B:=\{x\in \mathbb{R}^n: f(x)\|Mx\| < \eta\}$$</p>
<p>For simplicity, is assume that the provided matrixnorm satisfies</p>
<p>$$\|Mx-My\| \leq \|M\|\|x-y\|$$</p>
<p>Since we want to show that $h$ is locally lipschitz, we will only ... |
2,315,236 | <blockquote>
<p>Let $I := [0, \frac{\pi}{2}]$ and let $f : I \to \Bbb{R}$ be defined by $f(x) = \sup \{x^2, \cos(x)\}$ for $x \in I$. Show there exists an absolute minimum point $x_0 \in I$ for $f$ on $I$. Show that $x_0$ is a solution to the equation $x^2 = \cos(x)$. </p>
</blockquote>
<p>I am having a little troub... | Saketh Malyala | 250,220 | <p>so in a $2$ element set, the supremum is basically the maximum.</p>
<p>IF they are both equal, then it's the value.</p>
<p>So basically, as you can see, $f(x)=\cos(x)$ on the interval $(0,x_0]$ because $\cos(x)\geq x^2$.</p>
<p>And $f(x)=x^2$ on the interval $[x_0,\frac{π}{2})$ because $x^2 \geq \cos(x)$.</p>
<p... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Dinakar Muthiah | 788 | <p>I really like "Lie Groups and Lie Algebras" by Kirillov Jr. It's available <a href="http://www.math.sunysb.edu/~kirillov/liegroups/liegroups.pdf">here</a> for free. It covers a lot of material in a relatively short book, so I recommend it if you're trying to get a good overview of what Lie theory is about. </p>
<p>... |
2,755 | <p>As Akhil had great success with his <a href="https://mathoverflow.net/questions/1291/a-learning-roadmap-for-algebraic-geometry">question</a>, I'm going to ask one in a similar vein. So representation theory has kind of an intimidating feel to it for an outsider. Say someone is familiar with algebraic geometry enou... | Fran Burstall | 1,143 | <p>Sounds like Goodman and Wallach's <a href="http://books.google.co.uk/books?id=KBFoV2jpnPUC&lpg=PR13&ots=YyDe_Qb65k&dq=goodman%20wallach%20invariant%20theory&pg=PR4#v=onepage&q=&f=false" rel="noreferrer">Representations and invariants of the classical groups</a> might be worth looking at. It ... |
2,882,889 | <p>Let $(P_n)$ be a sequence such that $P_0 = 1$ and $P_n = (\log{(n+1)})^n$.</p>
<p>I'm trying to prove that $P_n ^2 \leq P_{n-1}\,P_{n+1}$ for $n \geq 1$ using an induction argument.</p>
<p>For $n=1$ we have that $(\log 2)^2 = 0.48$ and $1\cdot (\log 3)^2 = 1.206$, hence $P_1 ^{2} \leq P_{0}\,P_2$.</p>
<p>Given $n... | Marco | 582,590 | <p>Let <span class="math-container">$f(x)=x \log\log(x+1)$</span>. Then
<span class="math-container">$$f'(x)=\log \log(x+1)+\dfrac{x}{(x+1)\log(x+1)},$$</span>
<span class="math-container">$$f''(x)=\dfrac{1}{(x+1)\log(x+1)}+\dfrac{1}{(x+1)^2\log(x+1)}-\dfrac{x}{(x+1)^2\log^2(x+1)}=\dfrac{1}{(x+1)\log(x+1)}\left( 1+\dfr... |
1,072,634 | <p>I need to solve the differential equation $$y\frac{dy}{dx} = (x+7)(y^2+6)$$</p>
<p>I know that the first step is to isolate both term each side and then integrate...</p>
<p>But I can't figure out how to isolate term on this one, I would probably be able to solve the rest of the equation, just need to know how to s... | bkn | 201,879 | <p>No (to your title question), a function $f : A \to B$, by definition, gives each $x \in A$ a value $f(x) \in B$.</p>
<p>To be precise, you should have said that "for every element in the codomain, there is (at least) one element which is mapped to it by $f$" for your definition of surjectivity.</p>
|
2,743,604 | <p>**A busy railway station has two taxi stands at separate exits, A.
At stand A, taxis arrive according to a Poisson Process of rate 2 per minute.</p>
<p>Following the arrival of a train, a queue of 40 customers suddenly builds up at
stand A, waiting for taxis to arrive. Assume that whenever a taxi arrives, it
instan... | Mike Pierce | 167,197 | <p>Edit: After reading <a href="https://math.stackexchange.com/a/2743793/167197">fleablood's answer</a>, I'm rather convinced that my interpretation of the question is wrong. That first sentence, "<em>A watch is stopped for 15 minutes every hour on the hour,</em>" makes it sound like the watch stops on hours of <em>rea... |
2,035,237 | <p>Let $f: [0, \infty) \rightarrow R$ be a continuous function and define
$$F(t) = \int_{0}^{\frac{1}{t}} f(tx)\cos(t^2x) dx.$$</p>
<p>Show that $\lim_{t \rightarrow \infty} tF(t) = 0.$</p>
<p>This is part of a question from an old analysis qualifying at my university. I came across it while trying to study for my fi... | yangcs11 | 394,064 | <p>Some suggestion that might be useful.
We can rewrite the integral as following:
\begin{equation}
\lim_{t\to \infty}tF(t) = \lim_{t\to \infty}\int_{0}^{1}f(x)cos(tx)dx
\end{equation}
You see, as $t \to \infty$, $cos(tx)$ oscillates very fast between -1 and +1. Because $f(x)$ is a continuous function, so the integral ... |
2,412,839 | <p>I figured I could use the binomial formula to solve this problem but then remembered it only works when the order does matter (repeated combinations are included). Is there a formula that does not account for the repeated combinations and that could help me solve this problem?</p>
| carmichael561 | 314,708 | <p>It's simpler to consider the complementary probability: the probability that no sixes are rolled is $(\frac{5}{6})^3$, so the probability that at least one six is rolled is $1-(\frac{5}{6})^3$.</p>
<p>If you instead use the method you suggested in your question, the result will be
$$ {3\choose 1}\frac{1}{6}\Big(\fr... |
1,902,407 | <p>Expand given function $f$ as Taylor series around $c=3$
$$f(x) = \frac{x-3}{(x-1)^2}+\ln{(2x-4)} $$</p>
<p>and find out open interval at what that series is convergent. What is radius of convergence?</p>
<p>This is what i have so far. We know that $\ln{(1+x)} = \sum_{n=1}^{\infty} \frac{x^n}{n}$ and $\frac{1}{1-x}... | Bernard | 202,857 | <p>You have to find prime numbers $p$ such that $p-1$ is a divisor of $100$ – or a divisor $d$ of $100$ such that $d+1$ is prime, and that do not divide $10$. The list of divisors of $100$ is
$$\begin{matrix}1&5&25\\2&10&50\\4&20&100\end{matrix}$$
So you obtain $\;\{3,11,101\}$.</p>
|
1,397,450 | <p>10 objects are randomly distributed among 3 boxes. What is the probability to have 6 objects in one of the boxes, 3 in another one and a single object in the remaining third box. </p>
<p>My solution is </p>
<p>For the first box we have $10 \choose 6$, since 6 objects are already placed we are left with 4 remainin... | Michael Hardy | 11,667 | <p>If <span class="math-container">$a,b$</span> are real, then <span class="math-container">$a+ib = r(\cos\theta+i\sin\theta)$</span> where <span class="math-container">$r=\sqrt{a^2+b^2}$</span> and <span class="math-container">$\tan\theta = \dfrac b a$</span>, and
<span class="math-container">$$
(a+ib)^N = r^N(\cos(N\... |
1,397,450 | <p>10 objects are randomly distributed among 3 boxes. What is the probability to have 6 objects in one of the boxes, 3 in another one and a single object in the remaining third box. </p>
<p>My solution is </p>
<p>For the first box we have $10 \choose 6$, since 6 objects are already placed we are left with 4 remainin... | Jan Eerland | 226,665 | <p>$$(1+i)^N=$$
$$\left(|1+i|e^{\arg(1+i)i}\right)^N=$$
$$\left(\sqrt{2}e^{\frac{\pi}{4}i}\right)^N=$$
$$\left(\sqrt{2}\right)^N e^{\frac{\pi N}{4}i}=$$
$$2^{\frac{N}{2}} e^{\frac{\pi N}{4}i}$$</p>
|
574,220 | <p><strong>Question 1:</strong></p>
<p>let Polynomial <span class="math-container">$f(x)=\displaystyle\sum_{i=0}^{3}a_{i}x^i,$</span> have three real numbers roots,where <span class="math-container">$a_{i}>0,i=1,2,3$</span>.</p>
<blockquote>
<p>show that:
<span class="math-container">$$g(x)=\sum_{i=0}^{3}a^m_{i}x^i$... | Igor Rivin | 109,865 | <p>For the first question, this is really a question about the derivative. $f$ has three real zeros if and only if its derivative has two zeros, and $f$ is positive at the smaller critical point, and negative at the bigger critical point [by the way, I am dividing through by $a_3$ to make the polynomial monic -- this d... |
4,174,481 | <p>I'm going through a proof in which at one point, the authors make the following statement :</p>
<blockquote>
<p>Let <span class="math-container">$(X,d)$</span> be a complete metric space and <span class="math-container">$f:X\to X$</span> and <span class="math-container">$f_m:X\to X$</span> Lipschitz with <span class... | Paresseux Nguyen | 758,600 | <p>Your theorem is wrong. Next time, I would like you to check your statement carefully to save time of other MSE fellows who want to help you. You know, it is not always easy to tell whether a theorem is true or not.</p>
<hr>
<p><strong>Counter example</strong>:<br />
Choose <span class="math-container">$X= l^1(\mathb... |
3,789,429 | <blockquote>
<p><strong>Problem</strong>:
Can an <span class="math-container">$f$</span> function be created where:<span class="math-container">$$f\colon\mathbb Q_{+}^{*}\to \mathbb Q_{+}^{*}$$</span>
The function is defined on the set of fully positive rational numbers and is achieved:
<span class="math-container">$\f... | Bachamohamed | 804,312 | <p>we put <span class="math-container">$$x=\prod_{n\in\mathbb N}P_{n}^{\alpha_{n}}\quad ,y=\prod_{n\in\mathbb N}P_n^{\beta_n}$$</span> where <span class="math-container">$(x.y)\in\mathbb Q_{+}^{*}\times\mathbb Q_{+}^{*}$</span> and <span class="math-container">$$f(\prod_{n\in\mathbb N}P_n^{\alpha_n})=\left(\prod_{n\in\... |
274,885 | <p>I've been doing some research on ranking algorithms, and I've read theresearch with interest. The aspects of the Schulze Algorithm that appeals to me is that respondents do not have to rank all options, the rank just has to be ordinal (rather than in order), and ties can be resolved. However, upon implementation, I'... | joriki | 6,622 | <p>There's no reason to believe that the method can't result in a tie; in fact it's easy to think of much simpler examples where it does, e.g. when two people vote on two items and have opposite preferences. The Wikipedia article also has <a href="http://en.wikipedia.org/wiki/Schulze_method#Ties_and_alternative_impleme... |
2,491,560 | <p>Given a sequence $(s_n)$ in $\mathbb R$ such that
$$\lim \limits_{n \to \infty}( s_{n+1}-s_n)=0,$$
I am asked to prove
$(s_n)$ converges. </p>
<p>I know all Cauchy sequences converge in $\mathbb R^k$. So I want to prove that $(s_n)$ is Cauchy. </p>
<p>I am stuck as to how to show the given sequence is a Cauchy. ... | marty cohen | 13,079 | <p>For any $0 < c < 1$,
$s_n =n^c$
satisfies
$s_{n+1}-s_n
\to 0$
but,
obviously,
$s_n \to \infty$.</p>
<p>An easy proof is
via the mean value theorem.
Since
$(x^c)'
=cx^{c-1}
$,
then,
for some
$n \le x \le n+1$,
$s_{n+1}-s_n
=cx^{c-1}
\to 0
$.</p>
|
1,061,325 | <p>I need to show that for $x>0$, $1+\frac{x}{2} \ge \sqrt{1+x} \ge 1+\frac{x}{2} - \frac{x^2}{8} $.</p>
<p>I used the geometric/arithmetic mean inequality to show that $1+\frac{x}{2} \ge \sqrt{1+x}$ is indeed true. My issue lies in the second part of this.</p>
<p>In trying to show that $\sqrt{1+x} \ge 1+\frac{x}{... | 1k5 | 163,680 | <p>Angle is a property not of one line, but of two intersecting lines. In your question the second line is implicit but different in both cases (going north for navigation, going to the right for trigonometry.) This is by convention and has no real meaning except it makes it easier for you to talk about angles since ... |
2,011,378 | <p>My girlfriend suggested I use Lagrange theorem, but that only deals with finite groups.</p>
<p>If G was finite, and H has order 3 and K has order 5 , then clearly since the orders of H and K must divide the order of G, then there must also be an element of order 15.</p>
<p>But what about if G is infinite?</p>
<p>... | D Wiggles | 103,836 | <p>If $G$ has subgroups $H_1=\langle h_1\rangle$ and $H_2=\langle h_2\rangle$ of orders $3$ and $5$ respectively, what can you say about the group $\langle h_1,h_2\rangle$?</p>
<p>Also, a note that your edit is not true. There are infinite finitely generated groups in which every element has finite order. The Grigor... |
416,497 | <p>I was recently trying to find a numerical solution to a thermodynamics problem and the expression <span class="math-container">$x\ln x$</span> appeared in one of the computations. I did not have to find its value very near <span class="math-container">$0$</span>, so the computer managed fine, but it got me thinking ... | Carlo Beenakker | 11,260 | <p>Since <span class="math-container">$\ln x = - \ln(1/x)$</span>, to evaluate the logarithm near zero is equivalent to evaluating it for large argument. You can then use the result <span class="math-container">$$\ln y=\frac{\pi}{2a}\left(1+{\cal O}(y^{-2})\right),$$</span>
with <span class="math-container">$a$</span> ... |
41,858 | <p>Halting Problem is (theoretically) decidable for such algorithms which termination may be proved in First Order Logic (FOL) because all true statements in FOL are recursively enumerable. It is correct?</p>
<p>I'm talking about programs in Turing-complete languages and use "realworldiness" property not in the sense ... | Raphael | 3,330 | <p>FO logic (to be precise, FO[<]) can express no more than star-free languages (McNaughton, Papert, 1971), a proper subset of regular languages. Therefore, I think any hope that FOL is sufficient to express all desired properties programs might have is misguided.</p>
<p>You have to be precise about what the Haltin... |
173,360 | <p>From my lecture notes: "The notation $\mathbb T$ will be used for the additive circle and $S^1$ for the multiplicative circle."</p>
<p>What I understand: As a topological group, $S^1$ has the subspace topology of $\mathbb R^2$ and multiplication is defined as $(e^{ia}, e^{ib}) \mapsto e^{i(a + b)}$. </p>
<p>My gu... | Dylan Moreland | 3,701 | <p>The map $\mathbb R \to S^1$, $t \mapsto e^{2\pi it}$ descends to a map $\mathbb T \to S^1$ which is an algebraic and topological isomorphism. [Remember that $\mathbb T$ is an algebraic and topological quotient. What is the inverse of this map?] So you can indeed pass freely between the two, and in particular $\mathb... |
1,918,781 | <p>Suppose that $\phi:\mathbb R^3\to\mathbb R$ is a harmonic function. I am asked to show that for any sphere centered at the origin, the average value of $\phi$ over the sphere is equal to $\phi(0)$. I am also directed to use Green's second identity: for any smooth functions $f,g:\mathbb R^3\to \mathbb R$, and any sph... | velut luna | 139,981 | <p>There is a simple proof:
Because the region satisfies the Laplace's equation, the potential must be due to some charges outside the sphere. Consider a point charge $q$ at $\vec{r}$, where $r>R$. The average potential is
$$\langle \phi\rangle=\frac{1}{4\pi R^2}\int\frac{q}{4\pi\epsilon_0}\frac{1}{|\vec{r'}-\vec{r}... |
398,535 | <p>I understand that "recursive" sets are those that can be completely decided by an algorithm, while "recursively enumerable" sets can be listed by an algorithm (but not necessarily decided). I am curious why the word "recursive" appears in their name. What does the concept of decidability/recognizability have to do... | Thomas Klimpel | 12,490 | <blockquote>
<p>What does the concept of decidability/recognizability have to do with functions that call themselves?</p>
</blockquote>
<p>The primitive recursive functions are closely related to the functions that call themselves. For the general or $\mu$-recursive function, and additional minimization operation is... |
121,453 | <p>I wanted to ask a summary of known results and references about the homotopy type of the mapping space $\mathrm{Map}(BG,BK)$ (and specially the connected components) between the classifying spaces when G and K are general topological groups.
Thank you. </p>
| Andreas Thom | 8,176 | <p>As Neil Strickland points our, this is too difficult to have a good answer in general. In this post, I am only considering discrete groups. A very special case is the following</p>
<blockquote>
<blockquote>
<p><strong>Definition</strong> Let $f : BG \to BK$ be a continuous map. We say that $f$ is a superposit... |
18,174 | <p>I am an undergraduate secondary math education major. In <span class="math-container">$2$</span> weeks I have to give a <a href="https://www.mathmammoth.com/lessons/number_talks.php" rel="nofollow noreferrer">Number Talk</a> in my math ed class on the problem "<span class="math-container">$3.9$</span> times <span cl... | Mitchell Spector | 13,838 | <p>There's a fun method which I've seen referred to as Russian peasant multiplication or ancient Egyptian multiplication. (I don't know if these names have a historical basis.)</p>
<p>If you think about how the algorithm works behind the scenes, it's essentially multiplying the two numbers in base 2, so it could have ... |
1,458,464 | <p>Let</p>
<ul>
<li>$(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces</li>
<li>$I\subseteq[0,\infty)$ be at most countable and closed under addition with $0\in I$</li>
<li>$X=(X_t)_{t\in I}$ be a stochastic process on $(\Omega,\mathcal A)$ with values in $(E,\mathcal E)$</li>
<li>$\mathbb F=(\mathcal F_t... | zhoraster | 262,269 | <p>I wouldn't regard John Dawkin's answer as a valid argument, but he is right. </p>
<p>Denote $\eta = f\circ\left(X_{\tau+t}\right)_{t\in I}$ and take $A\in \mathcal F_\tau$. Then $A\cap\{\tau = s\}\in \mathcal F_s$, so$$E[\mathbf{1}_A \mathbf{1}_{\{\tau=s\}} E[\eta\mid \mathcal F_s]] = E[\mathbf{1}_A\mathbf{1}_{\{\t... |
2,879,915 | <p>I am attempting to prove $\lim_{k\to\infty}\dfrac{k^2}{k^2+2k+2}=1$. However, I am getting tripped up on the algebra. I believe I want to show that there exists some $N\in\mathbb{N}$ such that when $k\geq N$, $\left|\dfrac{k^2}{k^2+2k+2}-1\right|<\epsilon$ for arbitrary $\epsilon>0$. I set out to find a choice... | Vasili | 469,083 | <p>Let's start with $\left|\dfrac{-2k-2}{k^2+2k+2}\right|=\left|\dfrac{2(k+1)}{k^2+2k+2}\right|<\left|\dfrac{2(k+1)}{(k+1)^2}\right|=\dfrac{2}{k+1}<\epsilon$. This inequality will hold if $k+1>\dfrac{2}{\epsilon}$ or $k>\dfrac{2-\epsilon}{\epsilon}$</p>
|
3,084,998 | <p>Define the linear transformation <span class="math-container">$L$</span> as </p>
<p><span class="math-container">$L: \mathbb{R}^3 \rightarrow \mathbb{R}^3: x \rightarrow x - \langle x, a \rangle b$</span></p>
<p>Let <span class="math-container">$a, b \in \mathbb{R}^3$</span> such that <span class="math-container">... | Locally unskillful | 494,915 | <p>It should be <span class="math-container">$s(t) = c_1 e^{(-1+i)t} + c_2 e^{(-1-i)t}$</span>.</p>
<p>Plug in <span class="math-container">$t=0$</span> and <span class="math-container">$s=1$</span> to find <span class="math-container">$1=c_1+c_2$</span>.</p>
<p>Now differentiate, and plug in <span class="math-contai... |
3,381,543 | <p>Let <span class="math-container">$X \sim \text{Poisson}(\lambda)$</span>. If <span class="math-container">$P(X = 1) = 0.1$</span> and <span class="math-container">$P(X = 2) = 0.2$</span>, what is the value of <span class="math-container">$P(X = 3)?$</span></p>
<p>I've tried to expand the given equalities, finding <... | Ongky Denny Wijaya | 556,947 | <p>If <span class="math-container">$P(X=1)=0.1$</span>, then <span class="math-container">$\dfrac{e^{-\lambda}\lambda^1}{1}=0.1$</span>. We have nonlinear equation <span class="math-container">$$\lambda e^{-\lambda}=0.1.$$</span> So we must solve numerically (using calculator or numerical method for finding root of non... |
58,009 | <p>Let $f: X \to Y$ be a morphism of varieties such that its fibres are isomorphic to $\mathbb{A}^n$. Since the definition of a vector bundle stipulates that $f$ be locally the projection $U \times \mathbb{A}^n \to U$, it is likely that there exist morphisms that are not locally of that form, but I can't come up with a... | Steven Landsburg | 10,503 | <p>Does your definition of varieties allow them to be disconnected? If so, let $Y$ be any variety, $P\in Y$ a closed point, $U$ the complement of $P$, $X_1$ a vector bundle over $U$, $X_2$ a vector bundle over $P$, and $X=X_1\cup X_2$. </p>
|
3,068,381 | <p>I calculated, using Mathematica, that for <span class="math-container">$4\leq k \leq 100$</span>,
<span class="math-container">$$ \sum_{j=k}^{2k} \sum_{i=j+1-k}^j (-1)^j 2^{j-i} \binom{2k}{j} S(j,i) s(i,j+1-k) = 0,$$</span>
where <span class="math-container">$s(i,j)$</span> and <span class="math-container">$S(i,j)$<... | G Cab | 317,234 | <p>Just a first step (too long for a comment).</p>
<p>Let me change your symbology and put
<span class="math-container">$$
S(n) = \sum\limits_{j = n}^{2n} {\sum\limits_{k = j + 1 - n}^j {\left( { - 1} \right)^{\,j} 2^{\,j - k}
\left( \matrix{2n \cr j \cr} \right)
\left\{ \matrix{ j \cr k \cr} \right\}
\left[... |
2,572,286 | <p>I have $mp = 156^{107} \pmod{17},\;$ which simplifies to $3^{11} \pmod{17}.$</p>
<p>How did he get the $11$?</p>
<p>Same for </p>
<p>$mq = 156^{107} \pmod {11},\;$ which simplifies to $2^7 \pmod {11}.$</p>
<p>How to get the $7$? I know how to get the $3$ and $2$</p>
| Bernard | 202,857 | <p>I'd add that one deduces at once from <em>Euler's theorem</em> that, if $a$ and $m$ are coprime,
$$a^{N}\equiv a^{N\bmod \varphi(m)}\mod m.$$
In particular, if $m$ is prime, $\varphi(m)=m-1$.</p>
|
220,170 | <p>My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?</p>
<p>For example i would start with :</p>
<ol>
<li... | Simon Rose | 1,703 | <p>This does depend (as others have said) on what you mean by "advanced tools", but what about looking at Seifert-Fibred manifolds? These are ones that are locally products $S^1 \times D^2 / C_n$ for some cyclic group $C_n$. Alternatively, you can view them as appropriate notions of a bundle over an orbifold surface $S... |
220,170 | <p>My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that simple invariants as Betti number are not even close to classify ?</p>
<p>For example i would start with :</p>
<ol>
<li... | Lee Mosher | 20,787 | <p>Read Chapter 4 of Thurston's notes <a href="http://library.msri.org/books/gt3m/">http://library.msri.org/books/gt3m/</a>. He produces infinitely many closed hyperbolic 3-manifold of different volumes, and hence non-homeomorphic, just by doing Dehn surgery on the figure 8 knot in $S^3$. This does not used advanced ge... |
1,281,950 | <p>Is there a way to express the integral</p>
<p>$I(x_{0}, t) = \int_{x_{0}}^{\infty} \frac{1}{x} \, \cos (x t) \, \text{e}^{-x^{2}} \, dx$,</p>
<p>where $x_{0} \neq 0$ and $t \ge 0$, in terms of more well-known functions? In particular, I am interested in a form where $t$ appears in the limits and not in the integra... | Alex Fok | 223,498 | <p>Basically it is just a consequence of the fact that the inverse of a bijection is unique. If $AB=I$ then $B$ is injective as a linear transformation on a vector space. Being injective implies that it is surjective as well by dimension theorem. $B$ is thus a bijection and $A$ its unique inverse.</p>
|
3,464,477 | <p>An odd perfect number n is of the form <span class="math-container">$n=q^{k}m^{2}$</span> where <span class="math-container">$q$</span> is prime and both <span class="math-container">$q,k \equiv 1 \mod 4$</span>. Also, n satisfies <span class="math-container">$\sigma (n)=2n$</span> so that <span class="math-containe... | Jose Arnaldo Bebita Dris | 28,816 | <p>Let <span class="math-container">$n = q^k m^2$</span> be an odd perfect number with special / Euler prime <span class="math-container">$q$</span> satisfying <span class="math-container">$q \equiv k \equiv 1 \pmod 4$</span> and <span class="math-container">$\gcd(q,m)=1$</span>.</p>
<p>Descartes, Frenicle, and subseq... |
2,712,318 | <p>Let $a$ be the real root of the equation $x^3+x+1=0$</p>
<p>Calculate $$\sqrt[\leftroot{-2}\uproot{2}3]{{(3a^{2}-2a+2)(3a^{2}+2a)}}+a^{2}$$</p>
<p>The correct answer should be $ 1 $. I've tried to write $a^3$ as $-a-1$ but that didn't too much, I guess there is some trick here :s</p>
| David Quinn | 187,299 | <p>The expression under the cube root is $$9a^4+2a^2+4a=9a(-(a+1))+2a^2+4a=-7a^2-5a$$</p>
<p>Meanwhile consider $$(1-a^2)^3=1-3a^2+3a^4-a^6=1-3a^2+3a(-a-1)-a^2-2a-1$$
$$=-7a^2-5a$$</p>
<p>So the required result follows immediately</p>
|
2,712,318 | <p>Let $a$ be the real root of the equation $x^3+x+1=0$</p>
<p>Calculate $$\sqrt[\leftroot{-2}\uproot{2}3]{{(3a^{2}-2a+2)(3a^{2}+2a)}}+a^{2}$$</p>
<p>The correct answer should be $ 1 $. I've tried to write $a^3$ as $-a-1$ but that didn't too much, I guess there is some trick here :s</p>
| Jack D'Aurizio | 44,121 | <p>The real root $a$ of $x^3+x+1$ belongs to the interval $\left(-1,-\frac{1}{2}\right)$ and in order to check that
$$ \sqrt[3]{(3a^2-2a+2)(3a^2+2a)}=1-a^2 \tag{1}$$
holds it is enough to check that
$$ (3a^2-2a+2)(3a^2+2a)=1-3a^2+3a^4+a^6 \tag{2}$$
holds, or that
$$ (1+a+a^3)(-1+5a+a^3) = 0 \tag{3} $$
holds, but that... |
4,432,047 | <p><span class="math-container">$3\times 2$</span> is <span class="math-container">$3+3$</span> or <span class="math-container">$2+2+2$</span>. We know both are correct as multiplication is commutative for whole numbers. But which one is <strong>mathematically</strong> accurate?</p>
| J.-E. Pin | 89,374 | <p>In every ring and even in every <a href="https://en.wikipedia.org/wiki/Semiring" rel="nofollow noreferrer">semiring</a>, <span class="math-container">$3 \times 2$</span> is by definition <span class="math-container">$(1+1+1)\times (1+1)$</span> . Since multiplication distributes over addition, you get on the one han... |
4,432,047 | <p><span class="math-container">$3\times 2$</span> is <span class="math-container">$3+3$</span> or <span class="math-container">$2+2+2$</span>. We know both are correct as multiplication is commutative for whole numbers. But which one is <strong>mathematically</strong> accurate?</p>
| Hans Lundmark | 1,242 | <p>It might be worth mentioning that in the context of ordinal arithmetic, where the order actually matters, the currently accepted convention is that <span class="math-container">$\alpha \cdot \beta$</span> means “repeat the operation of ‘counting to <span class="math-container">$\alpha$</span>’ <span class="math-cont... |
893,839 | <p>I am dealing with the dual spaces for the first time.</p>
<p>I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.</p>
| Mikhail Katz | 72,694 | <p>Vector fields and differential forms are <em>dual</em> in this sense but behave quite differently. Thus, one can pull back differential forms by smooth maps but there is no analogous operation for vector fields. </p>
<p>A finite dimensional vector space is of course isomorphic to its dual, but the isomorphism is no... |
893,839 | <p>I am dealing with the dual spaces for the first time.</p>
<p>I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.</p>
| Urgje | 95,681 | <p>Dual spaces enter naturally if one considers the weak topology on the original space. Sometimes a differential equation has a weak solution but not a strong one (i.e. in the original=strong, topology). Well known examples of dual spaces are $L^p$ spaces. The dual of ${L^p}(R,dx)$, $1≤p<∞$ is ${L^q}(R,dx)$ where $... |
4,352,129 | <p>I am struggling to compute the power series expansion of <span class="math-container">$$f(z) = \frac{1}{2z+5}$$</span> about <span class="math-container">$z=0$</span>, where <span class="math-container">$f$</span> is a complex function. I tried comparing it to the geometric series as follows,<span class="math-contai... | Essaidi | 708,306 | <p><span class="math-container">$$\begin{array}{lcl}
\dfrac{1}{2 z + 5} & = & \dfrac{1}{5} \dfrac{1}{1 + \dfrac{2 z}{5}} \\[3mm]
& = & \displaystyle \dfrac{1}{5} \sum_{n = 0}^{+\infty} (-1)^n \left(\dfrac{2 z}{5}\right)^n \\[3mm]
& = & \displaystyle \sum_{n = 0}^{+\infty} \dfrac{2^n}{5^{n + 1}} ... |
2,433,614 | <p>The solution to the initial-value problem $y' = y^{2} + 1$ with $y(0)=1$ is $y = \tan(x + \frac{\pi}{4})$. I would like to show that this is the correct solution in a way that is analogous to my solution to differential equation $y' = y - 12$.</p>
<p><strong>Solution to Differential Equation</strong></p>
<p>If $y'... | Paramanand Singh | 72,031 | <p>Let's see what is involved in the parametrization of unit circle via circular functions. First is the obvious relation $$\cos^{2}t+\sin^{2}t=1\tag{1}$$ and further that these functions are continuous/differentiable with values $\cos 0=1,\sin 0=0$. Some idea of $\pi$ is needed so that we can prove these functions are... |
3,284,729 | <p>I was wondering how to calculate the following:
I know the average chance of death per year for a certain age.
What is the average chance of death for a 5 year period then?</p>
<p>Is it simply additive?
The chances are listed here (jaar is the dutch word for Year)</p>
<p><a href="https://i.stack.imgur.com/09wt9.pn... | Empy2 | 81,790 | <p>Your chance of survival multiplies. </p>
<p>If your chance of death in three years is one percent, two percent and
three percent, then your chance of survival is <span class="math-container">$0.99*0.98*0.97$</span></p>
|
3,827,853 | <p>I am trying to compute the singular cohomology of <span class="math-container">$\mathbb{N}\subset\mathbb{R}$</span>. I have that <span class="math-container">$C_0(\mathbb{N})\cong C_1(\mathbb{N})\cong\mathbb{Z}^{\oplus\mathbb{N}}$</span>, where I mean <span class="math-container">$\mathbb{Z}^{\oplus\mathbb{N}}=\{(n_... | Luke Derry | 825,079 | <p>Let <span class="math-container">$x\in\mathbb{Z}^\mathbb{N}, x=(x_0,x_1,x_2,...)$</span>. Let <span class="math-container">$f_x:\mathbb{Z}^{\oplus\mathbb{N}}\rightarrow\mathbb{Z};f_x(y)=x_0y_0+x_1y_1+x_2y_2+...$</span></p>
<p>Then <span class="math-container">$f_x\in\text{Hom}(\mathbb{Z}^{\oplus\mathbb{N}},\mathbb{Z... |
265,549 | <p>$$ \left\{ x \in\mathbb{R}\; \middle\vert\; \tfrac{x}{|x| + 1} < \tfrac{1}{3} \right\}$$</p>
<p>What is the supremum and infimum of this set? I thought the supremum is $\frac{1}{3}$. But can we say that for any set $ x < n$ that $n$ is the supremum of the set? And for the infimum I have no idea at all. Also,... | Julien | 38,053 | <p>For $x\geq 0$, the condition $\frac{x}{|x|+1}<1/3$ is equivalent to $x<x/3+1/3$, which in turn is equivalent to $x<1/2$.
So the supremum of your set is $1/2$.</p>
<p>For $x\leq 0$, we have that $\frac{x}{|x|+1}<1/3$ is equivalent to $x<-x/3+1/3$, and then to $4x<1$, which is true for every nonposi... |
1,237,464 | <p>In a bag of reds and black balls, $30\%$ were red, and $90\%$ of the black balls and $80\%$ of the red balls are marked balls. What percentage of the bag of balls is marked?</p>
<p>I thought I would have to use: Let </p>
<p>$A = \text{marked red balls},$</p>
<p>$B = \text{marked black balls}.$ </p>
<p>$P(A \cup ... | Bib | 190,219 | <p>Consider the following:</p>
<p>$30\%$ of the balls are red and $80\%$ of the red balls are marked, so we have $P(A) = 30\% \times 80\%$.</p>
<p>Now $70\%$ of the balls are black and $90\%$ of the black balls are marked, so we have $P(B) = 70\% \times 90\%$.</p>
<p>Now we can evaluate the formula $P(A \cup B) = P(... |
1,405,787 | <p>We're given the following function :</p>
<p>$$f(x,y)=\dfrac{1}{1+x-y}$$</p>
<p>Now , how to prove that the given function is differentiable at $(0,0)$ ?</p>
<p>I found out the partial derivatives as $f_x(0,0)=(-1)$ and $f_y(0,0)=1$ ,</p>
<p>Clearly the partial derivatives are continuous , but that doesn't guaran... | 5xum | 112,884 | <p>If all partial derivatives of a function (over all possible variables) are continuous at some point, then the function is differentiable at that point.</p>
|
463,539 | <p>I've transformed the number 11 to:</p>
<p>$11^e = 677.32$</p>
<p>Given the exponent and the transformed value, how can I solve for the original number?</p>
<p>I know that $x = y^z$ and that $z = \log_y(x)$, but I don't know how to solve for $x$? Can anyone explain how I can use the exponent $e$ and $677.32$ to fi... | Rasmus | 367 | <p>If you want $x$ to satisfy $x^e=b$, then what you are looking for is $\sqrt[e]{b}$ or $b^\frac1e$.</p>
|
4,253,761 | <p><span class="math-container">$$\begin{array}{ll} \text{extremize} & xy+2yz+3zx\\ \text{subject to} & x^2+y^2+z^2=1\end{array}$$</span></p>
<p>How to find the maximum/minimum using Lagrange multipliers?</p>
<p>Context: This is not a homework problem, my friend and I often make up problems to challenge each ot... | Ibadul Qadeer | 997,150 | <p>From an abstract algebra point of view, the best way to understand what trivial is would be to look at situations or examples where it is mostly used and encountered.</p>
<p>Take the case of subsets of a set, say <span class="math-container">$A$</span>. Since every set of is a subset of itself, <span class="math-con... |
82,061 | <p>Well, another problem here I don't get.</p>
<p>From what I know, sin(x)'=cos(x) right?</p>
<p>Well, here is a problem:</p>
<p>Find $\frac{d}{dx}(y\cos(\frac{y}{x^4}))$ ?</p>
<p>Let $u=y\cos(\frac{y}{x^4})$ and $s=y$, $v=\cos(\frac{y}{x^4})$</p>
<p>$$u'=v\frac{ds}{dx}+s \frac{dv}{dx}$$</p>
<p>$$u'=\co... | Anurag | 17,939 | <p>The generating function for this is $B_n(x) = \sum P(n,k)x^k = (1+x)(1+2x)\ldots
(1+nx)$. You might be able to manipulate this to get a closed form. For example,
$\log(B_n(x)) = \sum(-1)^kS_kx^k/k$ where $S_k$ is the sum of $k_{th}$ powers.
Still, no closed formula.</p>
|
2,266,342 | <p>So on my review for my final exam there is this question: </p>
<p>Is there a linear transformation from $P_2$ to $P_2$ with the following properties? In each case, either give an example of such a transformation or prove that no such transformation exist.</p>
<blockquote>
<p>$T(t^2+t+1) = t^2+t+1, T(t^2+2t+3) = ... | Doug M | 317,162 | <p>$T\begin{bmatrix} 1&1&1\\1&2&2\\1&3&2 \end{bmatrix} = \begin{bmatrix} 1&1&2\\1&2&2\\1&3&1 \end{bmatrix}\\
T = \begin{bmatrix} 1&1&2\\1&2&2\\1&3&1 \end{bmatrix}\begin{bmatrix} 1&1&1\\1&2&2\\1&3&2 \end{bmatrix}^{-1}\\
T = \... |
478,516 | <p>$\lim_{d \to \infty} (1+\frac{w}{d})^{\frac{d}{w} } = e$. But, what if the number of bits used to encode $d$ is polynomial in length. In this model, infinity can't be encoded. However, $d$ is polynomialy much larger than $w$.
Is there any tight lower bound, a closed form function $f(d)$ such that
$$ f(d) \le (1+\fr... | Did | 6,179 | <p>For every fixed positive $d$, $(1+w/d)^{d/w}\to1$ when $w\to+\infty$ while $(1+w/d)^{d/w}\gt1$ for every positive $w$. Hence the best lower bound of $(1+w/d)^{d/w}$ valid for every positive $w$ (or only for every $w$ large enough) is $f(d)=1$.</p>
|
925,941 | <p>Solve:</p>
<p>$xy=-30$<br>
$x+y=13$</p>
<p>{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?</p>
| André Nicolas | 6,312 | <p>We have in general
$$(x+y)^2-4xy=(x-y)^2.$$
In our case that gives $(x-y)^2=289$, so $x-y=\pm 17$.</p>
<p>Now solve the system $x+y=13$, $x-y=17$ and the system $x+y=13$, $x-y=-17$.</p>
<p><strong>Remark:</strong> This procedure for finding $x$ and $y$ given their sum and product goes back to Neo-Babylonian times.... |
925,941 | <p>Solve:</p>
<p>$xy=-30$<br>
$x+y=13$</p>
<p>{15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?</p>
| Adi Dani | 12,848 | <p>$$xy=-30$$
$$x+y=13$$</p>
<p>from second equation $y=13-x$ then put on first we get
$$x(13-x)=-30$$
$$x^2-13x-30=0$$
$$x_{1,2}=\frac{13\pm\sqrt{189}}{2}=\frac{13\pm17}{2}$$
$$x_1=\frac{13+17}{2}=15,x_2=\frac{13-17}{2}=-2$$
$$y_1=13-15=-2,y_2=13-(-2)=15$$
so the solutions are $(15,-2)$ and $(-2,15)$</p>
|
4,349,487 | <h2>Problem :</h2>
<p>Let <span class="math-container">$a,x>0$</span> then define :</p>
<h2><span class="math-container">$$f\left(x\right)=x^{\frac{x}{x^{2}+1}}$$</span></h2>
<p>And :</p>
<h2><span class="math-container">$$g\left(x\right)=\sqrt{\frac{x^{a}}{a}}+\sqrt{\frac{a^{x}}{x}}$$</span></h2>
<p>Then prove or d... | GEdgar | 442 | <p>We will use the half-angle formula for tangent:
<span class="math-container">$$
\tan\frac{\theta}{2} = \frac{1-\cos(\theta)}{\sin(\theta)} .
$$</span>
We want to get <span class="math-container">$2+\sqrt{3}$</span>. Remembering the basic values of sine and cosine, I see that
<span class="math-container">$$
2+\sqrt3... |
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