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 |
|---|---|---|---|---|
1,902,188 | <p>$k\in\mathbb{N}$ </p>
<p>The inverse of the sum $$b_k:=\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$$ is obviously
$$a_k=\sum\limits_{j=1}^k \binom{k-1}{j-1}\frac{b_j}{k^j}$$ . </p>
<p>How can one proof it (in a clear manner)? </p>
<p>Thanks in advance.</p>
<hr>
<p>Background of the question: </p>
<... | Batominovski | 72,152 | <p>In this proof, the binomial identity
$$\binom{m}{n}\,\binom{n}{s}=\binom{m}{s}\,\binom{m-s}{n-s}$$
for all integers $m,n,s$ with $0\leq s\leq n\leq m$ is used frequently, without being specifically mentioned. A particular case of importance is when $s=1$, where it is given by
$$n\,\binom{m}{n}=m\,\binom{m-1}{n-1}\... |
3,362,115 | <blockquote>
<p>Find the maximum value of <span class="math-container">$y/x$</span> if it satisfies <span class="math-container">$(x-5)^2+(y-4)^2=6$</span>.</p>
</blockquote>
<p>Geometrically, this is finding the slope of the tangent from the origin to the circle. Other than solving this equation with <span class="m... | Quanto | 686,284 | <p>Let <span class="math-container">$a$</span> be the angle between the center line (from origin to circle center) and the <span class="math-container">$x$</span>-axis,</p>
<p><span class="math-container">$$ \tan a = \frac 45$$</span></p>
<p>Let <span class="math-container">$b$</span> be the angle between the center ... |
436,172 | <p>let $a,b,c,d$ are real numbers,show that
$$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}\ge6\sqrt{|ad-bc|}$$</p>
<p>This problem is creat by China's famous mathematician hua luogeng,<a href="http://en.wikipedia.org/wiki/Hua_Luogeng" rel="nofollow">http://en.... | Aang | 33,989 | <p>HINT: </p>
<p>$$2\sqrt{a^2+c^2}+\sqrt{a^2+c^2+3(b^2+d^2)-2\sqrt{3}(ab+cd)}+\sqrt{a^2+c^2+3(b^2+d^2)+2\sqrt{3}(ab+cd)}$$ $$=2\sqrt{a^2+c^2}+\sqrt{(a-\sqrt{3}b)^2+(c-\sqrt{3}d)^2}+\sqrt{(a+\sqrt{3}b)^2+(c+\sqrt{3}d)^2}$$ $$\geq 2\sqrt{2|ac|}+\sqrt{2|(a-\sqrt{3}b)(c-\sqrt{3}d)|}+\sqrt{2|(a+\sqrt{3}b)(c+\sqrt{3}d)|}$$<... |
10,615 | <p>The tag <a href="https://math.stackexchange.com/questions/tagged/summation" class="post-tag" title="show questions tagged 'summation'" rel="tag">summation</a> was created less than a year ago, see <a href="https://math.meta.stackexchange.com/questions/6324/summation-tag-for-finite-and-formal-summations">&quo... | MJD | 25,554 | <p>When I created <a href="https://math.stackexchange.com/questions/tagged/summation" class="post-tag" title="show questions tagged 'summation'" rel="tag">summation</a>, it was with the idea that questions about series are usually concerned with issues of convergence, and that <a href="https://math.stackexchang... |
3,190,435 | <blockquote>
<p>A coin, having probability <span class="math-container">$p$</span> of landing on heads and probability of <span class="math-container">$q=1-p$</span> of landing on tails. It is continuously flipped until at least one head and one tail have been flipped.<br>
a) Find the expected number of flips neede... | Peter Foreman | 631,494 | <p>If you get a head with probability <span class="math-container">$p$</span> then the expected number of throws is <span class="math-container">$1+E(X)$</span> where <span class="math-container">$X$</span> is a geometric distribution requiring a tail to be thrown with probability <span class="math-container">$q$</span... |
3,190,435 | <blockquote>
<p>A coin, having probability <span class="math-container">$p$</span> of landing on heads and probability of <span class="math-container">$q=1-p$</span> of landing on tails. It is continuously flipped until at least one head and one tail have been flipped.<br>
a) Find the expected number of flips neede... | leonbloy | 312 | <p>Let <span class="math-container">$X$</span> be the time of the first head, and <span class="math-container">$Y$</span> the time of the first tail, and <span class="math-container">$W$</span> the first time when a head and a tail has been flipped.</p>
<p>You are right in assuming that <span class="math-container">$E... |
78,478 | <blockquote>
<p>Prove that $\frac{1}{n} \sum_{k=2}^n \frac{1}{\log k}$ converges to $0.$</p>
</blockquote>
<p>Okay, seriously, it's like this question is mocking me. I know it converges to $0$. I can feel it in my blood. I even proved it was Cauchy, but then realized that didn't tell me what the limit <em>was</em... | André Nicolas | 6,312 | <p>Logarithm to any base is equal to a constant times logarithm to the base $2$. So let's work with base $2$. For convenience, write $\log n$ for $\log_2 n$.</p>
<p>We use a mild variant of the usual proof that $\sum \frac{1}{n}$ diverges.</p>
<p>Note that $\log 2$ and $\log 3$ are both $\ge 1$; $\log 4$, $\log 5$, ... |
352,849 | <p>I have to show that $\lim \limits_{n\rightarrow\infty}\frac{n!}{(2n)!}=0$ </p>
<hr>
<p>I am not sure if correct but i did it like this :
$(2n)!=(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))\cdot (n!)$ so I have $$\displaystyle \frac{1}{(2n)\cdot(2n-1)\cdot(2n-2)\cdot ...\cdot(2n-(n-1))}$$ and $$\lim \limits_{... | Community | -1 | <p><strong>Hint</strong></p>
<p>$$0\leq\frac{n!}{(2n)!}\leq\frac{1}{n}$$</p>
|
1,285,941 | <p>I have a question where i couldn't find any clue.
The question is</p>
<p>$$\frac{1}{1\cdot 2}+\frac{1\cdot3}{1\cdot2\cdot3\cdot4}+\frac{1\cdot3\cdot5}{1\cdot2\cdot3\cdot4\cdot5\cdot6}+\cdots$$</p>
<p>I could get the general term as $t_n=\frac{1\cdot3\cdot5\cdot7\cdots(2n-1)}{1\cdot2\cdot3\cdot4\cdot5\cdot6\cdots2n... | Alex Fok | 223,498 | <p>The lowest form of each term is $\frac{(\frac{1}{2})^n}{n!}$. By invoking the Taylor series expansion of $e^x$ the answer is $e^{1/2}-1$.</p>
|
1,521,720 | <p>I'm working through a book on logic and wanted to check one of my steps in a derivation I'm working out. Given the two quantifier negation equalities:</p>
<ol>
<li><p>$\lnot\exists P(x) = \forall\lnot P(x)$</p></li>
<li><p>$\exists\lnot P(x) = \lnot\forall P(x)$</p></li>
</ol>
<p>I'm trying to derive (2) from (1).... | Paul Sinclair | 258,282 | <p>Yes, it is valid. The $P$ refers to an arbitrary predicate. That means that it can be anything, including $\lnot P'$ for some predicate $P'$. That is exactly what you did.</p>
|
184,824 | <p>I have two piecewise function</p>
<pre><code>equ1 = Piecewise[{{0.524324 + 0.0376478x, 0.639464 <= x <= 0.839322}}]
equ2 = Piecewise[{{-0.506432 + 1.48068x, 0.658914 <= x <= 0.77085}}]
</code></pre>
<p>Now, I am trying to solve <code>equ1 = equ2</code>.</p>
<p>Firstly I tried <code>FindRoot</code>: </... | Alex Trounev | 58,388 | <pre><code>equ1 = Rationalize[
Piecewise[{{0.524324 + 0.0376478 x, 0.639464 <= x <= 0.839322}}]]
equ2 = Rationalize[
Piecewise[{{-0.506432 + 1.48068 x, 0.658914 <= x <= 0.77085}}]]
Reduce[equ1 == equ2, x]
x == 0.714299 || x < 79933/125000 || x > 419661/500000
</code></pre>
|
1,938,253 | <p>Consider the following limit:$$\lim_{x \to\infty}\frac{2+2x+\sin(2x)}{(2x+\sin(2x))e^{\sin(x)}}$$</p>
<p>If we apply L'hospital's rule then we get:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/slrYm.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/slrYm.png" alt="Blockquote"></a></p>
... | Christian Blatter | 1,303 | <p>The proof of Hôpital's rule rests on the following form of the mean value theorem: If $f$ and $g$ both are continuous on $[a,b]$ and differentiable in the interior $\ ]a,b[\ $, and if $g'(t)\ne0$ for all $t\in\ ]a,b[\ $, then there is a $\tau\in\ ]a,b[\ $ such that
$${f(b)-f(a)\over g(b)-g(a)}={f'(\tau)\over g'(\tau... |
57,195 | <p>Let $f$ be a morphism of schemes $f: (X,\mathcal{O}_X)\to (Y,\mathcal{O}_Y)$, and $\mathcal{F},\mathcal{G}$ be sheaves of $\mathcal{O}_Y$-modules. I am trying to prove (I do NOT claim this to be true):</p>
<p>$f^{\ast}\mathcal{F}\otimes_{\mathcal{O}_X}f^{\ast}\mathcal{G}\cong f^{\ast}(\mathcal{F}\otimes_{\mathcal{O... | Georges Elencwajg | 3,217 | <p>Yes, we have $f^{\ast}\mathcal{F}\otimes_{\mathcal{O}_X}f^{\ast}\mathcal{G}\cong f^{\ast}(\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{G})\quad$</p>
<p>And, yes, this results from the <em>isomorphism</em> $\alpha: f^{-1} \mathcal{F} \otimes_{f^{-1}\mathcal{O}_Y} f^{-1}\mathcal{G} \overset {\sim}{\longrightarrow} f^{-... |
27,271 | <p>A few days ago I recall finding a visual calendar of my sign-in history on Math Stack Exchange. It looked like an ordinary monthly calendar, where the days were white if I hadn't signed in, and green if I had. (I found it when I wondering how close I've been in the past to getting the "fanatic" badge.)</p>
<p>Howev... | Asaf Karagila | 622 | <p>When you click your own profile, the default tab is the activity tab. You should go to the "Profile" tab, where you can see the "visited $x$ days, $y$ consecutive" text. Click this text to show the calendar.</p>
<p><a href="https://i.stack.imgur.com/J8Gmx.png" rel="nofollow noreferrer"><img src="https://i.stack.img... |
2,186,076 | <p><strong>Question.</strong></p>
<blockquote>
<p>If $p$ is a polynomial of degree $n$ with $p(\alpha)=0$, what do we know of the polynomial $q$ (with degree $n-1$) such that the numbers $(q^k(\alpha))_{k=1}^n$ contain all of the zeroes of $p$?</p>
</blockquote>
<p>Here I denote $q(q(\cdots q(\alpha)))=q^k(\alpha)$... | lhf | 589 | <p>Let $\alpha=\alpha_1, \alpha_2, \ldots, \alpha_m$ be the distinct roots of $p$.</p>
<p>Choose a permutation $\sigma$ of $1,2,\dots,m$ without fixed points. For instance, an $m$-cycle such as $(12\cdots m)$.</p>
<p>Let $q$ be the unique polynomial such that $q(\alpha_i)=\alpha_{\sigma(i)}$. That will work, but won'... |
3,464,342 | <blockquote>
<p>For <span class="math-container">$j\in \mathbb{N}$</span> let <span class="math-container">$$M_j=\{f\in L^2([0,1]):\int_0^1 |f|^2 dx \leq j\}$$</span>
(a) Establish that <span class="math-container">$L^2([0,1])=\cup_{j\in \mathbb{N}}M_j$</span>.</p>
<p>(b) Show that each <span class="math-container">$M_... | Jochen | 38,982 | <p>You can prove (b) without touching any integral: <span class="math-container">$L^2[0,1]$</span> is reflexive (as a Hilbert space) so that the closed balls <span class="math-container">$M_j$</span> are weakly compact. The inclusion <span class="math-container">$i:L^2[0,1]\hookrightarrow L^1[0,1]$</span> is continuous... |
1,425,907 | <p>Show that $$1+e^{-j\theta} =2e^{-j\theta/2}*\cos{\frac{\theta}2}$$</p>
<p>I know of the Euler equation: $e^{j\theta}=\cos(\theta)+j\sin(\theta)$ but am unsure how to simply show that the above are equal.</p>
| Asinomás | 33,907 | <p>You can use Gauss's formula. It tells you the sum of an arithmetic sequence $a_1+a_2+\dots a_n$ is $\frac{(a_1+a_n)n}{2}$.</p>
<p>In this case we get that the sum of the first $n$ terms is $\frac{(-12+(-12+9(n-1)))n}{2}=\frac{(9n-9-24)n}{2}=\frac{(9n-33)n}{2}$. This is because the first term is $-12$ and the $n$'th... |
1,425,907 | <p>Show that $$1+e^{-j\theta} =2e^{-j\theta/2}*\cos{\frac{\theta}2}$$</p>
<p>I know of the Euler equation: $e^{j\theta}=\cos(\theta)+j\sin(\theta)$ but am unsure how to simply show that the above are equal.</p>
| Mahie | 270,124 | <p>$S_n={n\over 2}(2a_1+(n-1)d)$</p>
<p>from the question,</p>
<p>$a_1= -12$ (i.e first term) and the common difference $d= (-3)-(-12) = 9$
and as sum to get is $363$ that will be $S_n$.</p>
<p>$S_n= 363$.
Substitute every term in the above formula,
$$363= {n\over 2}[2(-12)+(n-1)(9)]$$
$$363= {n\over 2}[-24+9n-9]$$... |
617,927 | <p>Find the taylor expansion of $\sin(x+1)\sin(x+2)$ at $x_0=-1$, up to order $5$.</p>
<p><strong>Taylor Series</strong></p>
<p>$$f(x)=f(a)+(x-a)f'(a)+\frac{(x-a)^2}{2!}f''(a)+...+\frac{(x-a)^r}{r!}f^{(r)}(a)+...$$</p>
<p>I've got my first term...</p>
<p>$f(a) = \sin(-1+1)\sin(-1+2)=\sin(0)\sin(1)=0$</p>
<p>Now, I... | Claude Leibovici | 82,404 | <p><strong>HINT</strong></p>
<p>Build the Taylor series for each term up to order 5 and multiply them, ignoring terms $(x+1)^m $with $m > 5$. </p>
<p>Leave coefficients $\cos(1)$ and $\sin(1)$, without computing them. </p>
<p>I suppose you are able to continue from here. If not, just post. Merry Xmas.</p>
|
833,827 | <p>I am trying to refresh on algorithm analysis. I am looking for a refresher on summation formulas.<br>
E.g.<br>
I can derive the $$\sum_{i = 0}^{N-1}i$$ to be N(N-1)/2 but I am rusty on the and more complex e.g. something like $$\sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}$$<br>
Is there a good refreshe... | Claude Leibovici | 82,404 | <p>For the problem in your post, I suppose that what you want to compute is $$\sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}k$$ For the most inner loop $$\sum_{k=j+1}^{N-1}k=\frac{1}{2} (N-j-1) (N+j)$$ So, for the middle loop $$\sum_{j=i+1}^{N-1}\frac{1}{2} (N-j-1) (N+j)=\frac{1}{6} (N-i-1) (N-i-2) (2 N+i)$... |
833,827 | <p>I am trying to refresh on algorithm analysis. I am looking for a refresher on summation formulas.<br>
E.g.<br>
I can derive the $$\sum_{i = 0}^{N-1}i$$ to be N(N-1)/2 but I am rusty on the and more complex e.g. something like $$\sum_{i = 0}^{N-1}{\sum_{j = i+1}^{N-1}\sum_{k=j+1}^{N-1}}$$<br>
Is there a good refreshe... | Perry Elliott-Iverson | 125,899 | <p>Here's a bit more detailed solution. Knowing the following three summations will help:</p>
<p>$$\sum_{i=0}^{N} i = \frac{N(N+1)}{2}$$</p>
<p>$$\sum_{i=0}^{N} i^2 = \frac{N(N+1)(2N+1)}{6}$$</p>
<p>$$\sum_{i=0}^{N} i^3 = \frac{N^2(N+1)^2}{4}$$</p>
<p>For the innermost sum:</p>
<p>$$\sum_{k=j+1}^{N-1}k = \sum_{k=0... |
3,735,798 | <blockquote>
<p><strong>QUESTION:</strong> Given a square <span class="math-container">$ABCD$</span> with two consecutive vertices, say <span class="math-container">$A$</span> and <span class="math-container">$B$</span> on the positive <span class="math-container">$x$</span>-axis and positive <span class="math-containe... | Théophile | 26,091 | <p>You can think of changing your frame of reference so that you're rotating around <span class="math-container">$B$</span>. Look at the vector <span class="math-container">$BA = x - yi$</span>. Then <span class="math-container">$BC = (BA)i = y + xi$</span>.</p>
<p>In other words, <span class="math-container">$C = B + ... |
1,675 | <p>This is a follow-up to <a href="https://mathoverflow.net/questions/1039/explicit-direct-summands-in-the-decomposition-theorem">this post</a> on the Decomposition Theorem. Hopefully, this will also invite some discussion about the theorem and perverse sheaves in general.</p>
<p>My question is how does one use the De... | Andreas Holmstrom | 349 | <p>There is a winter school on the decomposition theorem in Freiburg, Germany, 22-26 Feb 2010. <a href="http://home.mathematik.uni-freiburg.de/kebekus/FebSchool/" rel="nofollow">Link</a>.</p>
|
1,560,050 | <p>I want to solve the homogenous part of a stretched string problem where $y=y(x)$.</p>
<p>$$y'' + y = 0$$</p>
<p>with the boundary conditions such that: $y(0)=y(\pi/2)=0$</p>
<p>The differential equation gives rise to a solution on the form:
$$y = a \cos(x) + b \sin(x)$$</p>
<p>But when applying the boundary con... | bartgol | 33,868 | <p>As other people said, the only solution to the problem <em>as it is written now</em> is the trivial one. But perhaps you misread the exercise and the boundary conditions are $y(0) = y(\pi) = 0$ or $y(0) = y(2\pi)=0$? In that case you will have non-trivial solutions.</p>
|
1,125,839 | <p>I'm having a bit of problems proving the following:
$$\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$$</p>
<p>I always seem to get to the line: $2^{n-1} + 1 = 2^n$ which I know is untrue.</p>
<p>Could anyone help me prove this? </p>
| Adam Hughes | 58,831 | <p>No such subgroup exists. Pro-<span class="math-container">$p$</span> groups with the second definition (i.e. inverse limit of discrete, finite <span class="math-container">$p$</span>-groups) can be easily shown to be equivalent to the first definition:</p>
<p><span class="math-container">$$G/N\cong P$$</span></p>
<p... |
1,125,839 | <p>I'm having a bit of problems proving the following:
$$\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$$</p>
<p>I always seem to get to the line: $2^{n-1} + 1 = 2^n$ which I know is untrue.</p>
<p>Could anyone help me prove this? </p>
| user 59363 | 192,084 | <p>Look at Proposition 4.2.3 in the book by Ribes-Zalesskii, the whole section 4.2. will be interesting for you. Edit: the question is answered affirmatively there.</p>
|
3,872,033 | <p>Currently I meet with the following interesting problem.</p>
<p>Let <span class="math-container">$x_1,\cdots,x_n$</span> be i.i.d standard Gaussian variables. How to calculate the probability distribution of the sum of their absoulte value, i.e., how to calculate
<span class="math-container">$$\mathbb{P}(|x_1|+\cdot... | JimmyK4542 | 155,509 | <p>The moment generating function of a standard Gaussian R.V. <span class="math-container">$x$</span> is <span class="math-container">$\mathbb{E}[e^{tx}] = e^{t^2/2}$</span> for all <span class="math-container">$t \in \mathbb{R}$</span>.</p>
<p>Since <span class="math-container">$x_k$</span> is a standard Gaussian (and... |
134,444 | <p>I have the following code that determines when the second business day of each month is (given a start and end date). I have a few If statements I would like to replace with functional programming.</p>
<pre><code>getAccrualDates[fromDate_List,toDate_List]:=
(
today = fromDate;
projectionDate =toDate;
(*If the proj... | J. M.'s persistent exhaustion | 50 | <p>Pretty cute problem. Let's generate the required dates first:</p>
<pre><code>firstDays = NestList[DatePlus[#, {1, "Month"}] &, DateObject[{2016, 9, 1}], 4];
</code></pre>
<p>From there:</p>
<pre><code>getAccrualDates[date_DateObject] := NestWhile[DatePlus[#, 1] &, date,
! (DayMatchQ[#1, "B... |
871,581 | <p>I am trying to prove the identity below to help with the simplification of another function that I'm investigating as it doesn't appear to be a standard trig identity.</p>
<p>$$
\tan\left(x\right) + \tan\left( y \right) = \frac{{\sin\left( {x + y} \right)}}{{\cos\left( x \right)\cos\left( y \right)}}
$$</p>
<p>Any... | Mario Krenn | 157,257 | <p>You are asking about a proof of the identity
$$
\tan\left(x\right) + \tan\left( y \right) = \frac{{\sin\left( {x + y} \right)}}{{\cos\left( x \right)\cos\left( y \right)}}
$$</p>
<p>Using $\tan(x)=\frac{\sin(x)}{\cos(x)}$, we get
$$\tan\left(x\right) + \tan\left( y \right) = \frac{\sin(x)}{\cos(x)} + \frac{\sin(y)}... |
4,545,300 | <p>Find the number of all <span class="math-container">$n$</span>, <span class="math-container">$1 \leq n \leq 25$</span> such that <span class="math-container">$n^2+15n+122$</span> is divisible by 6.</p>
<p><strong>My attempt</strong>. We know that:
<span class="math-container">\begin{align*}
n^2+15n+122 & \equiv ... | John Douma | 69,810 | <p>You were almost there with your first line. You have <span class="math-container">$n^2+3n+2=(n+1)(n+2)\text{ mod 6}$</span></p>
<p><span class="math-container">$(n+1)$</span> and <span class="math-container">$(n+2)$</span> are two consecutive numbers so one of them is even. That gives you that this polynomial is div... |
514 | <p>I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.</p>
<p>I'm sure that everyone here is familiar with it; it describes an operation on a natural number – <span class="math-container">$n/2$</span> if it is even, <span class="math-container">$3n+1$</spa... | BlueRaja - Danny Pflughoeft | 136 | <p>I heard this story from <a href="http://www.ams.sunysb.edu/~estie/estie.html" rel="noreferrer">Professor Estie Arkin</a> at Stony Brook <strike><em>(sorry, I don't know what conjecture she was talking about)</em></strike>:</p>
<blockquote>
<p>For weeks we tried to prove the conjecture (without success) while we l... |
1,114,554 | <p>Consider the following sets:</p>
<blockquote>
<blockquote>
<p><span class="math-container">$A=$</span> set of sequence of real nos.</p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$B=$</span> set of sequence of positive real nos</p>
</blockquote>
</blockquote>
<blockquote>
<... | Learnmore | 294,365 | <p>A has cardinality $\mathbb R^{\mathbb N}=c$ </p>
<p>Check that B also has same cardinality as A and $\mathbb R$ has cardinality as $c$</p>
|
1,114,554 | <p>Consider the following sets:</p>
<blockquote>
<blockquote>
<p><span class="math-container">$A=$</span> set of sequence of real nos.</p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$B=$</span> set of sequence of positive real nos</p>
</blockquote>
</blockquote>
<blockquote>
<... | String | 94,971 | <p>For $x\in[0,1]$ define $f_n(x)$ as the number formed by first writing the decimal expansion of $x=0.x_1x_2...$ (note that $1=0.999...$) and then counting through the digits but resetting the counter whenever we reach a new "maximal count" like this
$$
\begin{array}{c}
digits:&x_1&x_2&x_3&x_4&x_5&... |
4,002 | <p>I'm trying to obtain a series of points on the unit sphere with a somewhat homogeneous distribution, by minimizing a function depending on distances (I took $\exp(-d)$). My points are represented by spherical angles $\theta$ and $\phi$, starting by choosing equidistributed random vectors:</p>
<pre><code>pts = Apply... | acl | 16 | <p>I was planning to get to this later but I am not sure I'll have time today. To force it to work I can do</p>
<pre><code>pts = Apply[{2 \[Pi] #1, ArcCos[2 #2 - 1]} &, RandomReal[1, {10, 2}],
1];
energy[p_] :=
Module[{cart},
cart = Apply[{Sin[#1]*Cos[#2], Sin[#1]*Sin[#2], Cos[#1]} &, p, 1];
Total[O... |
3,595,622 | <p><strong>Problem: Give an example of a linear continuum which is not the real line <span class="math-container">$\mathbb{R}$</span>, nor
topologically equivalent to a subspace of <span class="math-container">$\mathbb{R}$</span>.</strong></p>
<p><strong>Definition of Linear Continuum:</strong> Let X be a linearly ord... | freakish | 340,986 | <p><span class="math-container">$I\times I$</span> is clearly (path)connected under subspace topology of <span class="math-container">$\mathbb{R}^2$</span>. You incorrectly claim that <span class="math-container">$\{x\}\times I$</span> is open in <span class="math-container">$I\times I$</span>, it is not.</p>
<p>Unles... |
443,578 | <blockquote>
<p>Is the limit
$$
e^{-x}\sum_{n=0}^N \frac{(-1)^n}{n!}x^n\to e^{-2x} \quad \text{as } \ N\to\infty \tag1
$$
uniform on $[0,+\infty)$? </p>
</blockquote>
<p>Numerically this appears to be true: see the difference of two sides in (1) for $N=10$ and $N=100$ plotted below. But the convergence is ve... | Antonio Vargas | 5,531 | <p>Thanks, this was a fun problem.</p>
<p>From the integral representation</p>
<p>$$
\sum_{k=0}^{n} \frac{x^k}{k!} = \frac{1}{n!} \int_0^\infty (x+t)^n e^{-t} \,dt \tag1
$$</p>
<p>we can derive the expression</p>
<p>$$
e^{-x} \sum_{k=0}^{n} \frac{(-x)^k}{k!} = e^{-2x} - \frac{e^{-2x} (-x)^{n+1}}{n!} \int_0^1 t^n e... |
2,309,123 | <p>This is a 2 part question.</p>
<ol>
<li><p>I have been studying a particular matrix group $G \le GL(n,\mathbb R)$ with $n \ge 3$ and I managed to show elements of my group $A \in G$ have the block structure
$$
A = \left(
\begin{array}{cc}
O(3) & 0 \\
A_{21} & A_{22}
\end{array}
\right)
$$
Now $A_{22}$ must ... | Travis Willse | 155,629 | <p>(1) We may write the matrix group as a semidirect product
$$(\textrm{O}(3) \times \textrm{GL}(3, \Bbb R)) \leftthreetimes \textrm{M}(n - 3, \Bbb R) .$$
Explicitly, the homomorphism $\phi : \textrm{O}(3) \times \textrm{GL}(n - 3, \Bbb R) \to \textrm{Aut}(\textrm{M}(n \times 3, \Bbb R))$ defining this product is $\phi... |
1,401,898 | <p>I need a test for primality that I apply to $2^{255}-19$ (which is claimed to be prime) and certify to be correct with the ACL2 theorem prover. This means that I must be able to code the test in Common LISP, run it on this case in a reasonable period of time (I'd be happy if it ran in a day), and write a proof of c... | DanaJ | 117,584 | <p>It takes my C+GMP code under 0.4s to do a BLS75 theorem 5 proof on the number, so this seems like the easiest option. This involves finding some small factors of p-1, checking conditions, then verifying primality of each factor (note you don't need to factor p-1 completely). This example has lots of small factors,... |
2,621,932 | <p><strong>Question:</strong> In a chess match, there are 16 contestants. Every player has to be each other player (like a round-robin). The player with the most wins/points wins the tournament.</p>
<p>a) How many games must be played until there is a victor? </p>
<p>b) If every player has to team up with each other ... | Especially Lime | 341,019 | <p>a) If I choose any two people there has to be one game between them, so the number of games is the same as the number of ways to choose two people. (The reason you use combination instead of permutation here is that it doesn't matter who is black and who is white.)</p>
<p>b) I can't see how the answer given is plau... |
125,610 | <p>I have question about sets. I need to prove that: $$X \cap (Y - Z) = (X \cap Y) - (X \cap Z)$$</p>
<p>Now, I tried to prove that from both sides of the equation but had no luck.</p>
<p>For example, I tried to do something like this: $$X \cap (Y - Z) = X \cap (Y \cap Z')$$ but now I don't know how to continue.</p>
... | Community | -1 | <p><strong>Hint</strong></p>
<ul>
<li><p>$A \cap (B \cup C)=(A \cap B )\cup(A \cap C)$</p></li>
<li><p>$A \cap A'= \varnothing$</p></li>
</ul>
|
16,802 | <p>In an attempt to squeeze more plots and controls into the limited space for a demo UI, I am trying to remove any extra white spaces I see.</p>
<p>I am not sure what options to use to reduce the amount of space between the ticks labels and the actual text that represent the labels on the axes.</p>
<p>Here is a smal... | Szabolcs | 12 | <p>Another possible (partial) solution is adding negative space. This can be accomplished by putting the label in <a href="http://reference.wolfram.com/language/ref/Framed.html" rel="noreferrer"><code>Framed</code></a> and setting negative frame margins on one or mode side.</p>
<p>Try this:</p>
<pre><code>Manipulate... |
307,264 | <p>A professor of mine has suggested to me to look at this theorem and to find a problem related to it to explain in a future class.
I found an understandable proof in "Linear operators" by Dunford-Schwartz and I think I studied it, so now I know how to probe Brouwer's Theorem.
Now I was thinking of some interesting re... | Artem | 29,547 | <p>You can prove Nash's theorem that every symmetric game with two players has a mixed strategy Nash equilibrium. This can be done using differential equations (ordinary though) and Brouwer's theorem. A very accessible exposition is given in <a href="http://rads.stackoverflow.com/amzn/click/0691142750" rel="nofollow">... |
3,755,288 | <p>I'm trying to solve this:</p>
<blockquote>
<p>Which of the following is the closest to the value of this integral?</p>
<p><span class="math-container">$$\int_{0}^{1}\sqrt {1 + \frac{1}{3x}} \ dx$$</span></p>
<p>(A) 1</p>
<p>(B) 1.2</p>
<p>(C) 1.6</p>
<p>(D) 2</p>
<p>(E) The integral doesn't converge.</p>
</blockquot... | Barry Cipra | 86,747 | <p>Starting from</p>
<p><span class="math-container">$$\int_0^1\sqrt{1+{1\over3x}}\,dx=2\int_0^1\sqrt{t^2+{1\over3}}\,dt$$</span></p>
<p>(from the subsitution <span class="math-container">$x=t^2$</span>) as in Yves Daoust's answer, integration by parts gives</p>
<p><span class="math-container">$$\int_0^1\sqrt{t^2+{1\ov... |
1,182,844 | <p>I have completed Velleman's book, 'How to prove it'. I have also worked through Apostol Vol.1.
I have messed about with many rigorous single variable calculus textbooks, e.g.,Apostol, Spivak, Courant, Lang, etc. I had started working through Lang's, 'Calculus of several variables' but put it up to do a book like Edw... | Neal | 20,569 | <p>First: Why not? Math is best learned nonlinearly. Read Spivak and then, when you get confused, go look up the gap in your knowledge in another book.</p>
<p>Second: You will not understand manifolds if you do not have a thorough grasp of multivariable calculus. Manifolds exist as one generalization of multivariable ... |
61,509 | <p>I'm trying to read Elias Stein's "Singular Integrals" book, and in the beginning of the second chapter, he states two results classifying bounded linear operators that commute (on $L^1$ and $L^2$ respectively).</p>
<p>The first one reads:</p>
<p>Let $T: L^1(\mathbb{R}^n) \to L^1(\mathbb{R}^n)$ be a bounded linear ... | ibnAbu | 334,224 | <p>A bounded linear operator <span class="math-container">$ T: L^2 \to L^2$</span> and commutes with translation</p>
<p>Known result from Riesz Representation theorem <a href="https://math.stackexchange.com/questions/2809632/proof-that-every-bounded-linear-operator-between-hilbert-spaces-has-an-adjoint">see here</a>:<s... |
925,586 | <p>Determine if the relation : $$x \sim y \iff |y-x| \text{ is an integer multiple of } 3$$</p>
<p>is an equivalence one.</p>
<p>Now, I think this is an equivalence relation but I am having troubles formally proving the transitivity.</p>
<p>Any help?</p>
| André Nicolas | 6,312 | <p>Suppose that $x\sim y$ and $y\sim z$. </p>
<p>Then $3$ divides $|x-y|$. It follows that $3$ divides $x-y$.</p>
<p>Similarly, $3$ divides $y-z$.</p>
<p>So $3$ divides $(x-y)+(y-z)$. It follows that $3$ divides $x-z$, and therefore $3$ divides $|x-z|$. We conclude that $x\sim z$.</p>
|
925,586 | <p>Determine if the relation : $$x \sim y \iff |y-x| \text{ is an integer multiple of } 3$$</p>
<p>is an equivalence one.</p>
<p>Now, I think this is an equivalence relation but I am having troubles formally proving the transitivity.</p>
<p>Any help?</p>
| AsdrubalBeltran | 62,547 | <p>Hint note that $|x-z|=|x-y+y-z|$</p>
|
1,342,747 | <p>I am studying H. L. Royden's Real Analysis which includes some introduction to Measure Theory; and I encountered $(a,\infty]$ instead of $(a,\infty)$ for the first time! </p>
<p>What is the difference(s) between $(a,\infty)$ and $(a,\infty]$?</p>
| triple_sec | 87,778 | <p>\begin{align*}
(a,\infty)=&\,\{x\in\mathbb R\,|\,x>a\},\\
(a,\infty]=&\,\{x\in\mathbb R\,|\,x>a\}\cup\{\infty\}.
\end{align*}
The latter set includes an extra point termed “positive infinity.” Note that it is <em>not</em> a real number, but in certain areas of mathematics, especially in measure theory,... |
4,421,529 | <p><strong>Question:</strong> Let <span class="math-container">$n > 0$</span>. How can I find a function <span class="math-container">$f:\mathbb{N}\rightarrow\mathbb{R}^+$</span> such that
<span class="math-container">$$
\lim_{n\to\infty} \frac{f(n)^2}{n} \log \left(\frac{f(n)}{n}\right) = L
$$</span>
with <span cla... | Adam Rubinson | 29,156 | <p>I'll assume that <span class="math-container">$f$</span> is a function which can output real values.</p>
<p>For each <span class="math-container">$n\in\mathbb{N},$</span> define <span class="math-container">$g_n:[n,\infty)\to\mathbb{R};\ g_n(x) = \frac{x^2}{n} \log\left(\frac{x}{n}\right).$</span></p>
<p>One can rea... |
1,669,096 | <p>How do I show that <span class="math-container">$\ell^{ \infty}$</span> is a normed linear space,
where <span class="math-container">$\ell^{ \infty}$</span> is define as <span class="math-container">$$\|\{a_n\}_{n=1}^{\infty}\|_{\ell^\infty}=\sup_{1 \leq k \leq \infty} |a_k|?$$</span>
There are three properties that... | Fred | 380,717 | <p>Let <span class="math-container">$(a_n),(b_n) \in \ell^\infty$</span> Then we have for each <span class="math-container">$k$</span>:</p>
<p><span class="math-container">$$|a_k+b_k| \le |a_k|+|b_k| \le ||(a_n)||_\infty+||(b_n)||_\infty.$$</span></p>
<p>This shows that <span class="math-container">$(a_n+b_n) \in \ell^... |
4,219,360 | <p>I was having some problems understanding how he found <span class="math-container">$\gamma(t)$</span> from the given <span class="math-container">$\Sigma$</span> and i was hoping someone could explain to me how if that is ok</p>
<p>So the problem goes like this:</p>
<p>Given the vector field <span class="math-contai... | Adi | 578,455 | <p><span class="math-container">$\mathbb{E}[X^Y] = \sum_{y=0}^\infty\mathbb{E}[X^Y|Y=y]\mathbb{P}(Y=y)$</span></p>
<p><span class="math-container">$ = \sum_{y=0}^\infty\mathbb{E}[X^y] {\mu^y e^{-\mu} \over y!}$</span></p>
<p><span class="math-container">$ = e^{-\mu} \left( 1 + \mathbb{E}[X]\mu + \mathbb{E}[X^2] {\mu^2 ... |
3,084,479 | <p><span class="math-container">$h\in \mathbb{R}$</span>, because we have defined the Trigonometric Functions only on <span class="math-container">$\mathbb{R}$</span> so far.</p>
<p>I have a look at <span class="math-container">$e^{ih}=\sum_{k=0}^{\infty}\frac{(ih)^k}{k!}=1+ih-\frac{h^2}{2}+....$</span> </p>
<p><stro... | Calvin Khor | 80,734 | <p>Sketch that follows the spirit of your approach, rather than using trigonometry etc. A useful technical tool:</p>
<blockquote>
<p>Theorem. Suppose the continuous functions <span class="math-container">$f_n=f_n(h)$</span> taking values in <span class="math-container">$\mathbb C$</span> are such that <span class="m... |
2,809,090 | <p>The stochastic vector $(X,Y)$ has a continuous distribution with pdf:
$$f(x,y) =
\begin{cases}
xe^{-x(y+1)} & \text{if $x,y>0$} \\[2ex]
0 & \text{otherwise}
\end{cases}$$<br>
Define $Z:=XY$.<br>
I would like to know what exactly $XY$ is. It seems to me that the function $f(x,y)$ has but one output, so wh... | fleablood | 280,126 | <p>$\frac{1}{6}\cos(\frac{1}{2}x)\cos(\frac{3}{2}x)-\frac{1}{2}\cos^2(\frac{1}{2}x)+\frac{1}{2}\sin^2(\frac{1}{2}x)+\frac{1}{6}\sin(\frac{1}{2}x)\sin(\frac{3}{2}x)$</p>
<p>Gosh, that's ugly. </p>
<p>Let's just throw rocks at it until it either falls apart or lumbers off. </p>
<p>First, let's replace $\frac 12 x$ w... |
1,382,587 | <p>In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., </p>
<p>$$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$</p>
<p>Let $\Omega$ consist of all points $(x,y,z) \in \mathbb R^3$ whose distance to $C$ is at most $1$. Compute
$$\int... | izœc | 83,639 | <p>So, you know it is a torus: then, your parametrization should be
$$
\begin{cases}
x = \big(\, 2 + r \cos \theta \, \big) \cos \phi \\
y = \big(\, 2 + r \cos \theta \, \big) \sin \phi \\
z = r \sin \theta
\end{cases}
$$
where $r \in [0,1]$ and $\theta, \phi \in [0, 2 \pi)$, and with jacobian
$$
\big|\,J\,\big|
=
\lef... |
2,878,553 | <p>I was trying to think about what $\pi$ actually is. There are a lot of ways to get $\pi$ for example $4(1-\frac{1}{3}+\frac{1}{5}-\cdots)$.</p>
<p>But there is no one way to define it. </p>
<p>On the other hand a fraction like $\frac{1}{2}$ also has multiple definitions e.g. $\frac{2}{4}$ or $\frac{3}{6}$. Althoug... | Fimpellizzeri | 173,410 | <p>This is a philosophical musing more so than a question.
It asks less about the nature of $\pi$ than about the nature of numbers themselves.</p>
<p>I would say the general stance is that we don't particularly care what numbers are.
Do you use a <a href="https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natur... |
1,937,826 | <p>Ok, this seems obvious to me, but how would one prove it?</p>
<p>Let $<f(t),g(t)>$ and $<h(t),p(t)>$ be parametrized arcs in the cartesian plot. If $f,g,h,p$ are all continous and the arcs don't intersect, then there will be a line between the two that will be the shortest distance. Prove this line is n... | Fimpellizzeri | 173,410 | <p>This is not true; consider when the shortest line segment between them contains one of an arc's endpoints.</p>
|
2,883,370 | <p>If I want to determine whether a sequence, ${a_n}$, is bounded above $\forall n \in \Bbb{N} $, is it enough to find a sequence that is larger than $a_n$, and show that it converges and is therefore bounded? For example:</p>
<p>$\forall n \in \Bbb{N}, let,$</p>
<p>$$
a_n = \frac{1}{n+1} + \frac{1}{n+2} + ...+\frac{... | Fred | 380,717 | <p>Yes, it is correct, we have $a_n \le 1$ for all $n$.</p>
|
4,435,088 | <p>In an <span class="math-container">$8×8$</span> table one of the square is colored black and all the others are white . Prove that one cannot make all the boxes white by recoloring the rows and columns . "Recoloring" is the operation of changing the color of all boxes in a row or in a column .</p>
<p>This ... | Milten | 620,957 | <p>For the general <span class="math-container">$N\times M$</span> case, your approach works exactly when both <span class="math-container">$N$</span> and <span class="math-container">$M$</span> are even. In other cases (for <span class="math-container">$N,M\ge2$</span>), you can reduce to the <span class="math-contain... |
2,418,916 | <blockquote>
<p>Find how many terms there are in this geometric sequence:</p>
<p><span class="math-container">$-1, 2, -4, 8, ..., -16777216$</span></p>
</blockquote>
<p>My attempt:</p>
<p><span class="math-container">$a_k=a.r^{k-1}$</span></p>
<p>And in this sequence:</p>
<p><span class="math-container">$a=-1$</span>, ... | Lee Mosher | 26,501 | <p>Hint: Since the sign is causing you trouble, get rid of it. The number of terms in this sequence is the same as the number of terms in the sequence
$$1,2,4,8,...,16777216
$$</p>
|
61,106 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be Poisson random variables with means <span class="math-container">$\lambda$</span> and <span class="math-container">$1$</span>, respectively. The difference of <span class="math-container">$X$</span> and <span class="math-cont... | Suvrit | 8,430 | <p>An alternative way to see the $1/e$ is as follows.</p>
<p>Let $x=2\sqrt{\lambda}$. <a href="http://en.wikipedia.org/wiki/Bessel_function#Asymptotic_forms" rel="nofollow">Recall that for</a> small argument $0 < x \ll \sqrt{k+1}$ we have</p>
<p>$$I_k(x) \approx \frac{1}{\Gamma(k+1)}(x/2)^k$$</p>
<p>Using this, w... |
3,742,456 | <blockquote>
<p>If <span class="math-container">$ABC$</span> is a right angled triangle at <span class="math-container">$C$</span> then prove that <span class="math-container">$a^n+b^n<c^n$</span> for all <span class="math-container">$n>2$</span></p>
</blockquote>
<p>This is an olympiad book problem. I know that ... | uniquesolution | 265,735 | <p>If <span class="math-container">$n>2$</span>, the unit ball of the norm <span class="math-container">$\|(x,y)\|_2=(|x|^2+|y|^2)^{1/2}$</span> is contained in the unit ball of <span class="math-container">$\|(x,y)\|_n=(|x|^n+|y|^n)^{1/n}$</span>, and there are only four common points: <span class="math-container"... |
789,458 | <p>If one day we finally prove the normality of $\pi $, would we be able to say that we have ourselves a sure-fire <em>truly random</em> number generator?</p>
| vadim123 | 73,324 | <p>Tomorrow's lottery numbers are random. Yesterday's are not. Pi is like the latter.</p>
<p>If you're not convinced, we can take a bet on yesterday's numbers. :-)</p>
|
3,054,898 | <h3>Problem</h3>
<p>Evaluate <span class="math-container">$$\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t.$$</span></p>
<h3>Comment</h3>
<p>It's very complicated to compute the integral applying normal method. I obtain the result resorting to the skillful formula</p>
<blockquote>
<p><span class="math-container">$$\int_0^... | yavar | 621,272 | <p><span class="math-container">\begin{align}
&\int_0^{2\pi}(t-\sin t)(1-\cos t)^2{\rm d}t=\int_0^{2\pi}t+t{\cos }^2 t-2t\cos t-\sin t -\sin t{\cos }^2 t+\sin 2t{\rm d}t\\
&=\left[\frac{1}{2}t^2+\frac{1}{2}t^2+\frac{1}{4}t\sin 2t-\frac{1}{4}t^2+\frac{1}{8}\cos 2t-2t\sin t-2\cos t+\cos t+\frac{1}{3}{\cos }^3t-\f... |
264,745 | <p>When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion ... | paul garrett | 12,291 | <p>There certainly is a discrepancy between the formal set-theoretic definition ("giving" a function by giving its graph), and the informal use. Another important aspect of the informal use of "function" in practice is to ascertain when one thing $y$ is <em>not</em> "a function of" another thing $x$, which ordinarily m... |
264,745 | <p>When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a distribution. I realized that while I know the definition of a function as a relation and I have an intuitive notion ... | nigel | 49,330 | <p>Let $A$ and $B$ be sets. A relation between $A$ and $B$ is some set $S \subseteq A \times B$. A <em>function on $A$</em> is a relation between $A$ and $B$ where $B$ is an arbitrary set (call this relation $S \subseteq A \times B$), and if $(a,b) \in S$ and $(a,c) \in S$, then $b=c$.</p>
<p>For example, if we say $f... |
4,650,979 | <blockquote>
<p>Solve the following recurrence equation: <span class="math-container">$T(n) = T(n-2)+n^2$</span>, having <span class="math-container">$T(0)=1$</span>, <span class="math-container">$T(1)=5$</span>.</p>
</blockquote>
<p>I need to solve this equation but when I get to the particular solution with <span cla... | Fnacool | 318,321 | <p><strong>You need to solve for even and odd separately</strong>, but first let's find the value of the <strong>sum of squares of all nonnegative integers from <span class="math-container">$0$</span> to <span class="math-container">$m$</span></strong>, a quantity we denote by <span class="math-container">$S(m)$</span... |
2,203,008 | <p>Consider the quadratic program</p>
<blockquote>
<p><span class="math-container">$ \min$</span> <span class="math-container">$ f(x) $</span></p>
<p><span class="math-container">$ \text{s.t.} \space Ax=c$</span></p>
</blockquote>
<p>Prove that <span class="math-container">$ x^* $</span> is a local minimum if and only ... | Dinoman | 829,651 | <p>Now copper.hat's answer is already very good. However, I aim to answer this question with different notations, just hope to explain it in the plainest language.</p>
<p>First we assume that all the points satisfy <span class="math-container">$Ax=c$</span> compose of a <em>feasible region</em> of this optimization pro... |
311,153 | <p>Im trying to resolve the next definite integral:
$$\int_{1-x^2}^{1+x^2}{\ln(t^2)\ dt}$$
Im not sure if I can use the Barrow's theorem, I think I have to use the fundamental theorem of integral calculus, but im not sure. How can I solve it?</p>
| Santosh Linkha | 2,199 | <p>$\displaystyle \log t^2 = 2 \log t, $ so $ \displaystyle \int_{1-x^2}^{1+x^2}{\ln(t^2)·dt} = \int_{1-x^2}^{1+x^2}{2\ln(t)·dt}$ and $\displaystyle \int \log(t) dt = t (\log(t) -1) + c$</p>
|
1,292,889 | <p>I've read the paper <a href="http://web.mit.edu/leozhou/www/gauss.pdf" rel="noreferrer">Least square fitting of a Gaussian function to a histogram</a> by Leo Zhou on how to perform a Least Square Fitting of a gaussian function to a histogram.</p>
<p>The Gaussian function used to fit the data is:
$$f(y)=A\exp\left(-... | Bryson of Heraclea | 122,828 | <p>Let us assume that $K$ is known. Then you would have to fit your transposed data $y_i'=y_i-K$ to a Gaussian curve. Using the terminology of the paper you cited, the error of the fit would be
$$
\chi^2=\sum_{i=1}^N\frac{(\ln y_i'-(ax_i^2+bx_i+c))^2}{\sigma_{\ln y'}}
$$
The important thing to note is that $\chi^2$ dep... |
1,292,889 | <p>I've read the paper <a href="http://web.mit.edu/leozhou/www/gauss.pdf" rel="noreferrer">Least square fitting of a Gaussian function to a histogram</a> by Leo Zhou on how to perform a Least Square Fitting of a gaussian function to a histogram.</p>
<p>The Gaussian function used to fit the data is:
$$f(y)=A\exp\left(-... | Claude Leibovici | 82,404 | <p>In the same spirit as Bryson of Heraclea's answer, consider that $K$ is fixed at a given value. Then, for the given $K$, apply the method given in the paper to get the remaining parameters $A$, $\mu$, $\sigma$ (which are all implicit functions of $K$). Now, compute the corresponding $y$'s and $$SSQ(K)=\sum_{i=1}^N (... |
1,379,513 | <p>A hot dog stand has 12 different toppings available. How many different kinds of hot dogs can be made, assuming the order of the toppings does not make a difference. I believe the correct answer is 882050, with the maximum varieties per number of toppings selected being 665280 when there six toppings. I am also n... | fred | 254,551 | <p>For each topping you can choose to include it or not include it. This results in $2^{12}=4096$ different kinds of hot dogs. </p>
|
1,379,513 | <p>A hot dog stand has 12 different toppings available. How many different kinds of hot dogs can be made, assuming the order of the toppings does not make a difference. I believe the correct answer is 882050, with the maximum varieties per number of toppings selected being 665280 when there six toppings. I am also n... | GOTO 0 | 29,669 | <p>The vendor can make uncountably many different kinds of hot dogs but varying the amount of one single topping, if the amount can be expressed as a real number (e.g. weight).</p>
|
389,460 | <blockquote>
<p>For $n>0$ let $A(n) = \underbrace{111 \ldots 11}_{n}$. Prove that if $A(n)$ is divisible by a prime number $p>3$, then $\gcd(n, p-1) > 1$.</p>
</blockquote>
<p>It is no huge discovery that if $n$ is even, then $2$ is a common divisor of $n$ and $p-1$, thus the implication holds. I don't know... | lab bhattacharjee | 33,337 | <p>If $p(>3)$ divides $\underbrace{111 \ldots 11}_n, p$ divides $\underbrace{999\ldots 99}_n\implies p$ divides $(10^n-1)$ </p>
<p>$\implies 10^n\equiv1\pmod p\implies ord_p{10}$ divides $n$</p>
<p>Again, using Fermat's Little Theorem, $10^{p-1}\equiv1\pmod p\implies ord_p{10}$ divides $p-1$</p>
<p>$\implies ord... |
4,107,232 | <h2>Problem</h2>
<p>Robert is playing a game with numbers. If he has the number <span class="math-container">$x$</span>, then in the next move, he can do one of the following:</p>
<ul>
<li>Replace <span class="math-container">$x$</span> by <span class="math-container">$\lceil{\frac{x^2}{2}}\rceil$</span></li>
<li>Repla... | Mike Earnest | 177,399 | <p>Too long for a comment, but with the help of a computer, I found that all numbers in <span class="math-container">$\{1,\dots,7000\}$</span> are indeed reachable. I verified this with the following Python code, which uses a priority queue to find reachable numbers. The best priority ordering I found was to try the <s... |
71,608 | <p>Consider the following question:</p>
<p>Is there a family $\mathcal{F}$ of subsets of $\aleph_\omega$ that satisfies the following properties?</p>
<p>(1) $|\mathcal{F}|=\aleph_\omega$</p>
<p>(2) For all $A\in \mathcal{F}$, $|A|<\aleph_\omega$</p>
<p>(3) For all $B\subset \aleph_\omega$, if $|B|<\aleph_\om... | saf | 16,826 | <p>a family F as above <strong>of minimal size</strong> satisfies <span class="math-container">$|F|=cov(\aleph_\omega,\aleph_\omega,\aleph_\omega,2)=pp(\aleph_\omega)=cof([\aleph_\omega]^\omega)\ge \aleph_{\omega+1}$</span>.</p>
<p>Edit: following Ali's suggestion, here are more details.
<span class="math-container">$$... |
1,987,026 | <p>So I know to get a probability like $P(2\leq X\leq 4)$, you simply do $P(X\leq4) - P(X\leq1)$, but when there is a question like $P(2<X<4)$ what am I supposed to do? </p>
<p>Not just limited to in between two values, I also don't know what to do if it's just $P(X<2)$, so far all our examples have been grea... | DavidF | 126,754 | <p>If $X$ is a continuous random variable then $\mathbb{P}(X \leq c) = \mathbb{P}(X < c)$, for $c$ some constant. This is because the cumulative probability is given by the integral, letting $f_X$ be the distribution function of $X$,
\begin{equation*}
\mathbb{P}(X \leq c) = F_x(c) = \int^c_{-\infty} f_X(t)\,dt
\end... |
1,465,229 | <p>At p. 388 of Calculus, Spivak gives a formula:</p>
<p>$$\frac{1}{1+t^2} = 1 - t^2 + t^4 - ... + (-1)^nt^{2n} + \frac{(-1)^{n+1}t^{2n+2}}{1+t^2}$$</p>
<p>Which can be integrated to find $\arctan(x)$.</p>
<p>I don't understand where this formula comes from, but I found it up to $(-1)^nt^{2n}$ by considering the geo... | mrf | 19,440 | <p>This is a finite geometric sum:</p>
<p>$$
\sum_{k=0}^n (-1)^k t^{2k} = \sum_{k=0}^n (-t^2)^k = \frac{1-(-1)^{n+1}t^{2n+2}}{1-(-t^2)}
$$</p>
|
3,055,272 | <p><strong>Background</strong></p>
<p>A connected graph has an <em>Eulerian circuit</em> if every vertex has even degree. </p>
<p>I am thinking about a certain classification of connected graphs where, for a connected graph <span class="math-container">$G$</span>, every <a href="https://en.wikipedia.org/wiki/Cut_(gra... | Misha Lavrov | 383,078 | <p>Given any cut and any Eulerian circuit, the circuit has to cross from one side of the cut to another an even number of times, since it starts and ends on the same side of the cut.</p>
<p>Since the Eulerian circuit takes each edge once, the number of edges split by the cut is even.</p>
|
1,150,805 | <p>An unfair 3-sided die is rolled twice. The probability of rolling a 3 is $0.5$, the probability of rolling a 1 is $0.25$, and the probability of rolling a 2 is $0.25$. Let $X$ be the outcome of the first roll and $Y$ the outcome of the second.</p>
<ul>
<li><p>Find the Joint Distribution of $X$ and $Y$ in a Table.</... | Milo Brandt | 174,927 | <p>One can show that:
$$P(A\cup B)=\{a\cup b : (a,b)\in P(A)\times P(B)\}$$
which is a recursive definition (very clearly if we take $B$ to be a singleton disjoint from $A$). It states that to get the power set of a union $A\cup B$, we choose any pair of a subset $a\subseteq A$ and $b\subseteq B$ and take their union. ... |
127,493 | <p>How many number less than $k$ contain the digit $3$?
For instance:</p>
<p>How many number contain the digit $3$ in the following list?</p>
<pre><code>Table[n, {n, 33}]
</code></pre>
<p>$\lbrace 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, \
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32... | Michael E2 | 4,999 | <p>Well, it can be done:</p>
<pre><code>int = (x^2 + 2 x + 1 + (3 x + 1) Sqrt[x + Log[x]]) /
(x Sqrt[x + Log[x]] (x + Sqrt[x + Log[x]]));
dy = (1 + x)/(x Sqrt[x + Log[x]]);
Integrate[int - dy, x] + Integrate[dy, x]
(* 2 Sqrt[x + Log[x]] + 2 Log[x + Sqrt[x + Log[x]]] *)
</code></pre>
|
1,685,523 | <p>What is the domain of $f(x)=x^x$ ? </p>
<p>I used Wolfram alpha where it is said that the domain is all positive real numbers. Isn't $(-1)^{(-1)} = -1$ ? Why does the domain not include negative real numbers as well?</p>
<p>I also checked graph and its visible for only $x>0$ . Can someone help me clarify this?<... | Emilio Novati | 187,568 | <p>Write:</p>
<p>$$y=x^x=e^{x\log x}$$ </p>
<p>If we want $y \in \mathbb{R}$ we must have $\log x \in \mathbb{R}$ and this is done only if $x> 0$</p>
<p>This is the usual definition for the function $y=f(x)=x^x$ for $x \in \mathbb{R}$, that gives $(0,+\infty)$ as the domain.</p>
<hr>
<p>If we want $x\in \mathb... |
3,822,042 | <p>For any function <span class="math-container">$f : X \rightarrow Y$</span> and any subset A of Y, define
<span class="math-container">$$f^{-1}(A) = \{x \in X: f(x) \in A\}$$</span> Let <span class="math-container">$A^c$</span> denote the complement of A in Y. For subsets <span class="math-container">$A_1,A_2$</span>... | user2661923 | 464,411 | <p>There is an (arguably) better way, but it is somewhat convoluted.</p>
<p>There are 7 choices for the blue card and 5 choices for the red card, for a total of 35 possible blue x red combinations.</p>
<p>For each of the 35 combinations, either there is a unique satisfying green card, or there isn't. So all you have t... |
1,627,713 | <p>This is maybe math $101$ question:</p>
<p>Let $z_1=1+i$.</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(1/1)=\pi/4$ so $$z_1=\color{blue}{\sqrt 2e^{i\pi/4}} .$$</p>
<p>But now if I take a look at</p>
<p>$z_2=-1-i$,</p>
<p>I know that $r=\sqrt 2$ and $\theta=\arctan(-1/-1)=\pi/4$ so $$z_1=\color{blue}{\sqrt ... | Michael Albanese | 39,599 | <p><strong>(Abstract) Hint:</strong> Given two points on this grid, how would you construct a path along the grid between them?</p>
<p><a href="https://i.stack.imgur.com/Oaj56.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Oaj56.gif" alt="enter image description here"></a></p>
|
827,154 | <p>I need help with the definition of "within 1":</p>
<ul>
<li><p>If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$. </p></li>
<li><p>If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$.</p></li>
<li><p>If $x = 8$ and $y = 8$, is $x$ still "within 1" of $y$?</p></li>
</ul>
<p>It's my understanding that this wou... | please delete me | 153,520 | <p>In this case, it probably means that the absolute value of the difference between the two numbers does not exceed $1$. Hence $8$ is within $1$ of $8$. Note that this expression does not occur frequently in mathematical literature.</p>
|
827,154 | <p>I need help with the definition of "within 1":</p>
<ul>
<li><p>If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$. </p></li>
<li><p>If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$.</p></li>
<li><p>If $x = 8$ and $y = 8$, is $x$ still "within 1" of $y$?</p></li>
</ul>
<p>It's my understanding that this wou... | Community | -1 | <p>"Within $x$" refers to $\pm x$. Hence, given a number $y$, the numbers within $x$ of $y$ are elements of the set
$$Z=\{z\mid y-x\leq z\leq y+ x\}$$
Quite obviously, since $y-x\leq y\leq y+x$, $y\in Z$.</p>
|
118,298 | <p>I'm trying to work through "Elements of Functional Languages" by Martin Henson. On p. 17 he says:</p>
<blockquote>
<p>$v$ occurs free in $v$, $(\lambda v.v)v$, $vw$ and $(\lambda w.v)$ but not in $\lambda v.v$ or in $\lambda v.w$. And $v$ occurs bound in $\lambda v.v$ and in $(\lambda v.v)v$ but not in $v$, $vw$ ... | Thomas Andrews | 7,933 | <p>In the more normal mathematical world, we might say something like:</p>
<blockquote>
<p>Let $f(x)=x^n$.</p>
</blockquote>
<p>In that case, $x$ is a bound variable. If we later wrote:</p>
<blockquote>
<p>Let $g(y)=y^n$</p>
</blockquote>
<p>Then $g$ would be the same as $f$.</p>
<p>On the other hand, if we s... |
3,903,016 | <p>Why is
<span class="math-container">$$
\sum_{n=1}^{\infty} \left( \frac{1}{(10n-9)^2} + \frac{1}{(10n-1)^2} \right) = \frac{\pi^2}{50} \frac{1}{1- \cos \frac{\pi}{5}}
$$</span>
?</p>
<p>Each form of
<span class="math-container">$$
\sum_{n=1}^{\infty} \frac{1}{(10n-9)^2}
$$</span>
and
<span class="math-container">$$
... | Jack D'Aurizio | 44,121 | <p>The <span class="math-container">$\Gamma$</span> function fulfills the symmetry relation
<span class="math-container">$$ \Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)} $$</span>
hence by applying <span class="math-container">$\frac{d}{dz}\left(\log(\cdot)\right)$</span> to both sides
<span class="math-container">$$\... |
3,631,587 | <p>Assume that our samples are high dimensional points (i.e., d is large) and we use
PCA to reduce it to k = 10 dimensions. After this step, we found that all the 10 new dimensions have
continuous values (e.g., in other words, each feature in the transformed dimension is not from discrete
domain, but rather, continuous... | Community | -1 | <p><span class="math-container">$$a=\pi r^2,c=2\pi r\to c=\sqrt{4\pi^2r^2}=\sqrt{4\pi a}$$</span></p>
<p>and</p>
<p><span class="math-container">$$\sqrt{4\pi\cdot9\pi}=6\pi.$$</span></p>
<hr>
<p>Also</p>
<p><span class="math-container">$$a=\frac{c^2}{4\pi}$$</span> and
<span class="math-container">$$r=\sqrt{\frac ... |
134,574 | <p>$a^{p-1} \equiv 1 \pmod p$</p>
<p>Why do Carmichael numbers prevent Fermat's Little Theorem from being a guaranteed test of primality? Fermat' Little theorem works for any $a$ such that $1≤a\lt p$, where $p$ is a prime number. Carmichael numbers only work for $a$'s coprime to $N$ (where $N$ is the modulus). Doesn't... | davidlowryduda | 9,754 | <p>If the goal of the Fermat Primality test is to guarantee that a number is prime, then testing against all possible $a$ is no better than simply trying to divide our number by all primes.</p>
<p>In particular, if it were easy to find a number that is not coprime to $n$, then it's easy to factor $n$ and so we wouldn'... |
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Especially Lime | 341,019 | <p>For the specific case that you start at $1$, it is fairly standard in combinatorics to write $[n]$ for $\{1,\ldots,n\}$, so $x\in[50]$ would work. This doesn't really help for other ranges, though - you could write $x\in[50]\setminus[10]$, but you probably shouldn't :)</p>
<p>To answer your other question, I prefer... |
2,793,983 | <p>For example I find myself wanting to write $x$ is an element of the integers from $1$ to $50$,</p>
<p>Is this the quickest way? </p>
<p>$x\in \left[ 1,50\right] \cap \mathbb{N} $</p>
<p>Also is this standard on here? $\mathbb{N} = \{0, 1, 2,\dotsc \}$,
$\mathbb{ℤ}_+ = \{1, 2, \dotsc \}$.</p>
| Eric Towers | 123,905 | <p>I do wonder why so many people believe convoluted notation is better than plainly writing what you mean.</p>
<p>"Let $x \in \mathbb{N}$ with $1 \leq x \leq 50$."</p>
<p>The twin purposes of notation are clarity and precision. Use of new or rare notation subverts both. Excessive density subverts clarity. Use of ... |
4,483,507 | <p>How can I change <span class="math-container">$\dfrac{-(3-\sqrt{3})}{(3+\sqrt{3})}$</span> to <span class="math-container">$\dfrac{1-\sqrt{3}}{1+\sqrt{3}}$</span>?</p>
<p>Background:</p>
<p>I tried solving <span class="math-container">$\tan(345°)$</span> with the trigonometric angle <em><strong>sum/difference</stron... | Guillermo García Sáez | 696,501 | <p>Yes, that's perfect. Another approach would be using that the preimage of a regular value through a continuous function is closed. Just take <span class="math-container">$f(x,y)=x^2+y^2$</span>, since <span class="math-container">$\mathbb{S}^1=f^{-1}(\{1\})$</span>, the complementary( your set <span class="math-cont... |
1,627,619 | <p>Could anyone please check my solution to the following problem?</p>
<blockquote>
<p><strong>Problem:</strong> Let $f(x, y) = (x^2 + y^2)e^{-(x^2 + y^2)}$. Find global extrema of $f$ on $M = {\mathbf R}^2$.</p>
</blockquote>
<p><strong>Proposed solution:</strong> Taking partial derivatives of $f$, we conclude tha... | πr8 | 302,863 | <p>$$e^x\ge1+x \quad \forall x\in\mathbb{R}$$</p>
<p>(this is well-known / can be established all sorts of ways. Note that equality holds $\iff x=0$) </p>
<p>Take $x=d-1$ to see </p>
<p>$$e^{d-1}\ge d\implies de^{-d}\le e^{-1}\implies(x^2+y^2)e^{-(x^2+y^2)}\le e^{-1}$$</p>
<p>Chase back the fact that the equality h... |
915,631 | <blockquote>
<p>Let <span class="math-container">$V$</span> be the vector space of <span class="math-container">$n\times n$</span> matrices over <span class="math-container">$K$</span> under addition and let the linear operator <span class="math-container">$f$</span> be given by <span class="math-container">$f(A)=A^{T}... | zcn | 115,654 | <p>If $\lambda$ is an eigenvalue of $f$, then $f(A) = \lambda A$ for some $A \ne 0$. Then $A = (A^T)^T = f(f(A)) = f(\lambda A) = \lambda^2 A$, so $\lambda^2 = 1$ (since $A \ne 0$), hence $\lambda = \pm 1$. </p>
<p>Eigenspaces corresponding to distinct eigenvalues always intersect trivially, so $E(1) \cap E(-1) = 0$. ... |
3,126,936 | <p>Numbers between <span class="math-container">$1 - 1000$</span> which leave no remainder when divided by <span class="math-container">$4$</span> and divided by <span class="math-container">$6$</span> but not by <span class="math-container">$21$</span>?</p>
<p>I tried <span class="math-container">$$\frac{1000}{12} = ... | N. F. Taussig | 173,070 | <p>If a number is divisible by both <span class="math-container">$4$</span> and <span class="math-container">$6$</span>, then it is divisible by <span class="math-container">$\operatorname{lcm}(4, 6) = 12$</span>. The number of multiples of <span class="math-container">$12$</span> that are at most <span class="math-co... |
3,929,089 | <p>In the construction of <span class="math-container">$\operatorname{Frac}(R)$</span>, where <span class="math-container">$R$</span> is a domain, we define a partition on <span class="math-container">$R \times R^\times$</span> where <span class="math-container">$R^\times:= R \setminus \{0\}$</span>. which in turn beco... | Moosh | 837,933 | <p>The simple answer is that elements of <span class="math-container">$\mathbb{Q}(x)$</span> are formal rational functions. strictly speaking, the values they take on at certain points are irrelevant to them. Their evaluation maps need not be defined for all elements in the underlying field.
Another example is that <sp... |
2,650,182 | <ol>
<li>Between every two distinct real numbers, there is a rational number </li>
</ol>
<p>Answer: There is no rational numbers between two non-distinct real numbers.</p>
<ol start="2">
<li>For all natural numbers $n ∈ N, \sqrt n$ is either a natural number or an
irrational number</li>
</ol>
<p>Answer: For all natu... | Mauro ALLEGRANZA | 108,274 | <p>In order to negate:</p>
<blockquote>
<p>1) "Between every two distinct real numbers, there is a rational number",</p>
</blockquote>
<p>we have to assert that "There are two distinct real numbers such that there is <strong>no</strong> rational number between them".</p>
<p>It may help to formalize the statements ... |
19,325 | <p>I'm looking for a simple way to define mathematics to primary/elementary school teachers and explain some of the confusion children have.</p>
<p>I'm hoping some Algebraist could help me properly state the following:</p>
<blockquote>
<p>A number in and of itself has no true meaning. True in the sense that it relates ... | Ashish Shukla | 12,610 | <p>I teach kids, small one 5-6 years to 11-12 years. So I have realised that a kid's life itself is not bound to anything concrete. They are best receptacles of Knowledge. But they are not good at reproducing it. See kids are capable of learning anything however abstract, but the trick is that teacher should be visuali... |
1,915,782 | <p>I'm attempting to teach myself some vector calculus before starting university next month in hope of getting my head around some of the concepts as I can foresee this being a weak topic for me.</p>
<p>I have been 'learning' from some online lecture notes related to my course. The notes talk about line integrals but... | Bobbie D | 317,218 | <p>The path $C$ is parametrized as $\mathbf r(t) = t\mathbf x + t\mathbf y + t\mathbf z$ for $t\in[0,1]$. Then you use the line integral formula</p>
<p>$$\int_{C} \mathbf f\cdot d\mathbf r = \int_{t_0}^{t_1} \mathbf f(\mathbf r)\cdot \mathbf r'(t)\ dt$$</p>
<p>In this case $\mathbf f(\mathbf r)\cdot \mathbf r'(t) = ... |
56,847 | <p>What are the angles formed at the center of a tetrahedron if you draw lines to the vertices?</p>
<p>I'm trying to make these:</p>
<p><img src="https://i.stack.imgur.com/FRUi8.jpg" alt="caltrop"> </p>
<p>I need to know what angles to bend the metal.</p>
| huzaifa abedeen | 845,875 | <p>Watch <a href="https://youtu.be/2UTr46btzaY" rel="nofollow noreferrer">this Khan Academy video</a> of Tetrahedral bond angle proof. In the video, Mathematical proof of the bond angles in methane (a tetrahedral molecule);</p>
<p><a href="https://i.stack.imgur.com/JSUio.jpg" rel="nofollow noreferrer"><img src="https:/... |
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