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 |
|---|---|---|---|---|
2,021,354 | <p>I have a vector <strong>x</strong> and a function that sums the elements of <strong>x</strong> like so:</p>
<p>$$f(1) = x_1$$
$$f(2) = x_1 + \sum_{i=1}^2 x_i$$
$$f(3) = x_1 + \sum_{i=1}^2 x_i + \sum_{j=1}^3 x_j$$
$$f(4) = x_1 + \sum_{i=1}^2 x_i + \sum_{j=1}^3 x_j + \sum_{k=1}^4 x_k$$</p>
<p>...and so on. How might... | Dhruv Kohli | 97,188 | <p>$f(n) = \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{i} x_j = \sum\limits_{i=1}^{n} (n-i+1)x_i$</p>
|
248,218 | <p>What are good job search sites, head hunter/recruiting agencies for mathematicians looking for industry jobs? </p>
<p>If this question is not appropriate for the math stack exchange, please feel free to redirect me.</p>
| Amzoti | 38,839 | <p>I think you are going to find a mixed bag of things on this topic as it is rather broad from school teachers, to professors to researchers to applied math jobs and the like.=, up to and including engineering.</p>
<p>For example, <a href="http://www.investopedia.com/financial-edge/0812/top-paying-math-related-career... |
248,218 | <p>What are good job search sites, head hunter/recruiting agencies for mathematicians looking for industry jobs? </p>
<p>If this question is not appropriate for the math stack exchange, please feel free to redirect me.</p>
| JB King | 8,950 | <p><a href="https://math.uwaterloo.ca/math/future-undergraduates/careers-mathematics" rel="nofollow">Careers in Mathematics</a> would be a page from the University of Waterloo in Canada that has a Faculty of Mathematics that covers various disciplines including Actuarial Science, Operations Research, and Computer Scien... |
2,417,197 | <p>When going through with learning Grahams number, I got stuck at </p>
<p>$$3↑↑↑3$$</p>
<p>Working it through, we have</p>
<p>$$3↑3=3^3$$
$$3↑↑3=3^{3^3}=3↑(3↑3)$$</p>
<p>As such, it would appear to me that</p>
<p>$$3↑↑↑3=3^{3^{3^3}}=3↑(3↑(3↑3))=3↑(3↑↑)$$</p>
<p>Which is incorrect; the correct answer being</p>
<... | Simply Beautiful Art | 272,831 | <p>You wish to understand Graham's number through these arrows? If so, I'd suggest stepping back down to multiplication and building the way up.</p>
<p>Note that</p>
<p>$$a\times b=\underbrace{a+(a+(\dots+a))}_b$$</p>
<p>For example,</p>
<p>$$3\times3=3+(3+3)=3+6=9$$</p>
<p>And then exponentiation,</p>
<p>$$a^b=a... |
76,600 | <p>The group of three dimensional rotations $SO(3)$ is a subgroup of the Special Euclidean Group $SE(3) = \mathbb{R}^3 \rtimes SO(3)$. The manifold of $SO(3)$ is the three dimensional real projective space $RP^3$. Does $RP^3$ cause a separation of space in the manifold of $SE(3)$? </p>
<p>(edit) Sorry about lack of cl... | John Galt | 18,078 | <p>I am not entirely sure what you mean by separation of space. But, would n't it depend on the representation of SE(3) and SO(3). For example, one can take the view that SE(3) is a dual projective space R\hat{P}^3 by using dual quaternion representation for spatial rigid body displacement.</p>
<p>I am not a mathemati... |
623,810 | <p>$\omega = y dx + dz$ is a differential form in $\mathbb{R}^3$, then what is ${\rm ker}(\omega)$? Is ${\rm ker}(\omega)$ integrable? Can you teach me about this question in details? Many thanks!</p>
| Gil Bor | 118,580 | <p>$\omega$ is nowhere zero so its kernel at each point is 2 dimensional. Let us find 2 vector fields that span this kernel. If we take a vector $v=a\partial_x+b\partial_y+c\partial_z$ then $\omega(v)=ay+c.$ Hence $v\in Ker(\omega)$ iff $c=-ay$. So we can take as a basis for the kernel of $\omega$ say $v_1=\partial_x-... |
2,844,902 | <p>Does $$\int_{[1,z]}\frac{1}{u}du=\log(z)$$ where $z\in\mathbb C$ ? I know that on a closed circle that contain $0$ we have $$\int_C\frac{1}{z}dz=2i\pi=\log(1),$$</p>
<p>but for $$\int_{[1,z]}\frac{1}{u}du=\log(z)$$ I don't really know to compute the integral.</p>
| md2perpe | 168,433 | <p>You can at least come to a first order differential equation by multiplying with $\dot q$ and then integration:
$$\dot q \ddot q = 4 q^{-5} \dot q$$
$$\frac{d}{dt}(\frac12 \dot q^2) = \frac{d}{dt}(-q^{-4})$$
$$\dot q = \pm 2 (C-q^{-4})$$</p>
|
4,073,757 | <p>Q: A coin is tossed untill k heads has appeared. If a mathematician knows how many heads appeared, can he figure out what is the probability that the coin was tossed <span class="math-container">$n$</span> times?</p>
<p>What I tried: The number of heads debends of the number of tosses. So I tried Bayes' theoream <sp... | Karl | 279,914 | <p>Bayes' Theorem would be useful if the experiment consisted of choosing <span class="math-container">$N$</span> first, before starting to count the heads, and <em>then</em> flipping the coin exactly <span class="math-container">$N$</span> times. Then we'd have</p>
<p><span class="math-container">$$
P(N=n|X=k)
=\frac{... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | DanielV | 97,045 | <p>There is a lot of work done towards formalizing existing proofs into logic software and proof wikis. The formalization of a proof can be very helpful economically. It can lead to more confidence in proofs and making searching for results more feasible.</p>
<p>(Mario here:) I am an undergraduate who works with the... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | R K Sinha | 66,227 | <p>Many problems of theoretical physics (e.g., string theory) are related to the development of some new math. This severely requires the services of able mathematicians. In the past, Riemann, Grassman, Hilbert, Poincare, Elie Cartan, deed so for the discovery of Einstein's equation in GR. Physicist Witten was awarded ... |
2,965,082 | <blockquote>
<p>Suppose that <span class="math-container">$(X,\ d)$</span> and <span class="math-container">$(Y,\ \rho)$</span> are metric spaces, that
<span class="math-container">$f_n:X\to Y$</span> is continuous for each <span class="math-container">$n$</span>, and that <span class="math-container">$(f_n)$</span... | RRL | 148,510 | <p>For a correct proof (by contradiction), show that <span class="math-container">$f(x_n) \not\to f(x)$</span> is impossible if convergence is uniform using</p>
<p><span class="math-container">$$\rho(f_n(x_n), f(x)) \leqslant \rho(f_n(x_n), f(x_n)) + \rho(f(x_n), f(x))$$</span></p>
<p>Note that <span class="math-cont... |
2,628,220 | <p>Let $(a_{n})_{n \in \mathbb N_{0}}$ be a sequence in $\mathbb Z$, defined as follows:
$a_{0}:=0,
a_{1}:=2,
a_{n+1}:= 4(a_{n}-a_{n-1}) \forall n \in \mathbb N$. </p>
<p>Required to prove: $a_{n}=n2^{n} \forall n \in \mathbb N_{0}$</p>
<p>I have gone about it in the following: </p>
<p>Induction start: $n=0$ (condi... | Michael Rozenberg | 190,319 | <p>We see that $$a_{n+1}-2a_n=2(a_n-2a_{n-1}),$$
which says that $b_n=a_n-2a_{n-1}$ is geometric progression.</p>
<p>Thus, $$b_n=b_1\cdot2^{n-1}=2\cdot2^{n-1}=2^n.$$
Thus,
$$a_1-2a_0=2,$$
$$\frac{1}{2}a_2-a_1=\frac{1}{2}\cdot2^2,$$
$$\frac{1}{2^2}a_3-\frac{1}{2}a_2=\frac{1}{2^2}\cdot2^3,$$
$$.$$
$$.$$
$$.$$
$$\frac{1}... |
3,541,947 | <p>How do you pronounce <span class="math-container">$\mathbb{F}_2, \mathbb{F}_2^n, \mathbb{N}^k, [n] = \{1,\ldots,n\},$</span> and <span class="math-container">$S \subseteq [n]$</span> when you're reading a text?</p>
<p>I've just started reading more advanced math textbooks and these are appearing all the time. </p>
| Randall | 464,495 | <p>I think this is a matter of taste/preference. Personally, I read them as...</p>
<p>"eff-two," "eff-two-enn," "enn-kay," and "box enn."</p>
|
892,742 | <p>Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?</p>
| Nicky Hekster | 9,605 | <p>This follows from the fact that the order of a conjugacy class $$|Cl_ G(x)| \leq |G'|$$. Namely, define a map $f : Cl_G(x) \rightarrow G'$ by $f(g^{-1}xg)=[x,g]$. This map is clearly injective. Finally, $|Cl_G(x)|=[G:C_G(x)]$.</p>
|
3,842,739 | <p>Let <span class="math-container">$H$</span> be a group and <span class="math-container">$H^m=\{ h^m \mid h\in H\}$</span>.</p>
<p>I know that this is a subgroup of <span class="math-container">$H$</span> when <span class="math-container">$H$</span> is abelian.
But I want to know that what happens if <span class="ma... | markvs | 454,915 | <p>It is certainly true that for every finite group <span class="math-container">$H$</span> there exists <span class="math-container">$N$</span> such that whether or not <span class="math-container">$H^n$</span> is a subgroup depends only on <span class="math-container">$n\mod N$</span> and contain all multiples of <s... |
696,848 | <p>$\DeclareMathOperator{\rank}{rank}$
First off I'm sorry I'm still not able to make of use the built in formula expressions, I don't have time to learn it now, I'll do it before my next question.</p>
<p>I have a couple of questions regarding eigenvectors and generalized eigenvectors. To some of these questions I kno... | Charles | 1,778 | <p>Probably the Riemann Hypothesis is true, in which case its falsity would not be provable (and the believability of its falsity is more a matter of psychology than mathematics), whether by computation or otherwise.</p>
|
842,365 | <blockquote>
<p>Show that a field <span class="math-container">$\mathbb{F}$</span> is finite if and only if its multiplicative group <span class="math-container">$\mathbb{F}^{\times}$</span> is finitely generated.</p>
</blockquote>
<p>The "<span class="math-container">$\Rightarrow$</span>" implication is obvious, bu... | David | 119,775 | <p>I'm not entirely clear on your proposed scheme, but it seems to me that you are doing something like this. You have "axes" representing the ten possible units digits, the ten possible first digits after the decimal point, the ten possible second digits after the decimal point, and so on; then also the ten possible ... |
1,798,261 | <p>what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression.
I am confused at this point. </p>
| Bart W | 339,805 | <p>Just go by the definition of subspace: a subspace is a subset of the space that is also a vector space. You can easily prove that any linear combination of vectors $(a, b, c)$ for which $a+b + c = 0$, also satisfies this condition. Therefore, it is indeed a subspace.</p>
<p>I can elaborate on this if the rest of th... |
1,798,261 | <p>what is multilinear coefficient? I heard it a couple of times and I tried to google it, all I am getting is multiple linear regression.
I am confused at this point. </p>
| DonAntonio | 31,254 | <p>Prove the basic conditions a <em>subset</em> must fulfill to be a subspace: (a) check that the zero vector is in $\;W\;$ , (b) show that if two vectors are in $\;W\;$ so is their sum, and (c) if a vector $\;w\in W\; $ and $\;r\;$ is any scalar, then also $\;rw\in W\;$ .</p>
<p>As for a basis: observe that $\;a+b=0=... |
3,756,436 | <p>Recently I was doing a physics problem and I ended up with this quadratic in the middle of the steps:</p>
<p><span class="math-container">$$ 0= X \tan \theta - \frac{g}{2} \frac{ X^2 \sec^2 \theta }{ (110)^2 } - 105$$</span></p>
<p>I want to find <span class="math-container">$0 < \theta < \frac{\pi}2$</span> ... | Tryst with Freedom | 688,539 | <p>The trick is to write a quadratic in terms of <span class="math-container">$ \tan \theta $</span> and not in terms of <span class="math-container">$X$</span></p>
<p><span class="math-container">$$ 0 = X \tan \theta -\frac{g}{2} \frac{ X^2 (1 +\tan^2 \theta)}{(110)^2} - 105$$</span></p>
<p>applying the condition tha... |
493,102 | <p>I have a concern with nested quantifiers.</p>
<p>I have: $$ \forall x \exists y \forall z(x^2-y+z=0) $$ such that $$ x,y,z \in \Bbb Z^+$$ </p>
<p>My first question, can it be read like this:</p>
<p>$$ \forall x \forall z \exists y(x^2-y+z=0) $$</p>
<p>The way I did it, is I started off with $x=1, z=1 $ </p>
... | Brian M. Scott | 12,042 | <p>No, you cannot interchange the $\exists y$ and $\forall z$: doing so changes the meaning of the statement. The original statement,</p>
<p>$$\forall x\exists y\forall z\left(x^2-y+z=0\right)$$</p>
<p>says that no matter what positive integer $x$ you choose, I can find a $y\in\Bbb Z^+$ such that $x^2-y+z=0$ <strong>... |
978,384 | <p>The following picture is constructed by connecting each corner of a square with the midpoint of a side from the square that is not adjacent to the corner. These lines create the following red octagon:</p>
<p><img src="https://i.stack.imgur.com/PZyGa.jpg" alt="enter image description here"></p>
<p>The question is, ... | Bob | 491,960 | <p>I was frustrated by the solution being 1/6 because that is not the result if one calculates the area of a <em>regular</em> octagon with a radius that is L/4, where L is the length of large square.</p>
<p>The gridded image above makes clear that red shape is close to, but not actually, a <em>regular</em> octagon.</p... |
2,623,324 | <p>Assume that the measure space is finite for this to make sense. Also, we know that $L^p$ spaces satisfy log convexity, that is -
$$\|f\|_r \leq \|f\|_p^\theta \|f\|_q^{1-\theta}$$
where $\frac{1}{r}=\frac{\theta}{p} +\frac{1-\theta}{q}$.
The text which I am following says 'Indeed this is trivial when $q=\infty$, and... | Barry Cipra | 86,747 | <p>Note that</p>
<p>$$1\le(n+1)^{1/\sqrt n}\le(2n)^{1/\sqrt n}=2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2$$</p>
<p>If we take $2^{1/x}$ and $x^{1/x}\to1$ as $x\to\infty$ for granted, then </p>
<p>$$2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2\to1\cdot1^2=1$$</p>
<p>and the Squeeze Theorem does the rest.</p>
|
2,449,581 | <p>There is a brick wall that forms a rough triangle shape and at each level, the amount of bricks used is two bricks less than the previous layer. Is there a formula we can use to calculate the amount of bricks used in the wall, given the amount of bricks at the bottom and top levels?</p>
| John Doe | 399,334 | <p>You should first work through the problem using the hints given. Here is a solution for once you are done.</p>
<blockquote class="spoiler">
<p> Let's say the lower layer has $x$ bricks, and there are a total of $n$ layers. Then the top layer has $a:=x-2(n-1)$ bricks (you can check this is correct by plugging in s... |
324,119 | <p>I've been reading about the Artin Spin operation. It's defined as taking the classical <span class="math-container">$n$</span>-knot (<span class="math-container">$S^n\hookrightarrow S^{n+2}$</span>) to an <span class="math-container">$(n+1)$</span>-knot. For the <span class="math-container">$1$</span>-knot case (in ... | David Corfield | 447 | <p>There's also the work of Francis Borceux and Marco Grandis, </p>
<p>Jordan-Hölder, modularity and distributivity in non-commutative algebra, J. Pure Appl. Algebra 208 (2007), no. 2, 665-689, <a href="http://dx.doi.org/10.1016/j.jpaa.2006.03.004" rel="nofollow noreferrer">doi</a>.</p>
<p>There the authors prove a... |
2,867,479 | <p>From <a href="https://math.stackexchange.com/questions/2867457">ETS Major Field Test in Mathematics</a></p>
<blockquote>
<p>A student is given an exam consisting of
8 essay questions divided into 4 groups of
2 questions each. The student is required to
select a set of 6 questions to answer,
including at l... | Servaes | 30,382 | <p>The student can choose two questions to omit, not both i the same group. There are $8$ options for the first question, and then there are $6$ options left for the second question. Of course the order in which the questions are chosrn doesn't matter, so we get
$$\frac{8\times6}{2}=24$$
options.</p>
<p>Alternatively... |
2,844,060 | <p>How to find the points of discontinuity of the following function $$f(x) = \lim_{n\to \infty} \sum_{r=1}^n \frac{\lfloor2rx\rfloor}{n^2}$$ </p>
| Shashi | 349,501 | <p>Let $x>0$. Notice that $$\frac{2rx - 1} {n} \leq \frac{\lfloor 2rx \rfloor} {n} \leq \frac{2rx}{n} $$
Hence $$-\frac 1 n+\frac 1 n\sum_{r=1}^n\frac{2rx } {n} \leq \frac 1 n \sum_{r=1}^n\frac{\lfloor 2rx \rfloor} {n}\leq \frac 1 n\sum_{r=1}^n\frac{2rx}{n}$$
You can see the summation on the LHS (and RHS) as a Riema... |
957,940 | <p>I'm "walking" through the book "A walk through combinatorics" and stumbled on an example I don't understand. </p>
<blockquote>
<p><strong>Example 3.19.</strong> A medical student has to work in a hospital for five
days in January. However, he is not allowed to work two consecutive
days in the hospital. In how... | Henry | 6,460 | <p>The prohibition on consecutive days means the constraints are really $$1 \le a_1$$ $$a_1+1 \lt a_2$$ $$a_2+1 \lt a_3$$ $$a_3+1 \lt a_4$$ $$a_4+1 \lt a_5$$ $$a_5 \le 31.$$ Rewrite these so that the right hand side of each line is the same as the left hand side of the next line as $$1 \le a_1$$ $$a_1 \lt a_2-1$$ $$a_... |
1,183,185 | <p>i) Show that for a particle moving with velocity $v(t), $if $ v(t)·v′(t) = 0$ for all $t$ then the speed $v$ is constant.
</p>
<p>I did $(v(t))^2=|v(t)|^2=(v(t)\bullet(v(t)))$. </p>
<p>Therefore $\frac{d}{dt}(v(t))^2=2v(t)$</p>
<p>Also, $\frac{d}{dt}(v(t)\bullet(v(t))=2(v(t)·v′(t))$ </p>
<p>I'm stuck here.</p>... | David K | 139,123 | <p>I'll use the notation $X_{(1)}$ for the smallest $X_i$ and
$X_{(N)}$ for the largest $X_i$.</p>
<p>Suppose $X_1 \leq t_0$.
The smallest $X_i$, $X_{(1)}$, then must be either $X_1$ or some even smaller value.
In any case, $X_{(1)} \leq t_0$ and the event $A$ did not occur.
So $A$ can occur only when $X_1 > t_0$.... |
3,623,277 | <p>Show that the cycles <span class="math-container">$(1, 2, \ldots, n)$</span>, <span class="math-container">$(n, \ldots, 2, 1)$</span> are inverse permutations. </p>
| lab bhattacharjee | 33,337 | <p>As <span class="math-container">$x^3-x=(x-1)x(x+1)$</span> is a product of three consecutive integers</p>
<p><span class="math-container">$3$</span> must divide <span class="math-container">$x^3-x$</span></p>
<p>So, we need <span class="math-container">$$x^3\equiv x\pmod{5\cdot7}$$</span></p>
<p>If <span class="m... |
2,359,372 | <blockquote>
<p>Given that
$$\log_a(3x-4a)+\log_a(3x)=\frac2{\log_2a}+\log_a(1-2a)$$
where $0<a<\frac12$, find $x$.</p>
</blockquote>
<p>My question is how do we find the value of $x$ but we don't know the exact value of $a$? </p>
| Robert Z | 299,698 | <p>Hint. Recall the main properties of the <a href="https://en.wikipedia.org/wiki/Logarithm#Product.2C_quotient.2C_power.2C_and_root" rel="nofollow noreferrer">logarithm</a>.
Then we have that $\frac{1}{\log_2a}=\log_a 2$.
Moreover for $x>4a/3>0$ (the argument of the logarithm should be positive),
$$\log_a((3x-4... |
2,881,673 | <p>I've searched all over the internet and cannot seem to factorise this polynomial.</p>
<p>$x^4 - 2x^3 + 8x^2 - 14x + 7$</p>
<p>The result should be $(x − 1)(x^3 − x^2 + 7x − 7)$</p>
<p>What are the steps to get to that result?
I've tried grouping but doesn't seem to work...</p>
| Community | -1 | <p>As $p(1)=0$, you know that $x-1$ is a factor. Now</p>
<p>$$x^4 - 2x^3 + 8x^2 - 14x + 7
\\=x^3(x-1)-x^3+8x^2-14x+7
\\=x^3(x-1)-x^2(x-1)+7x^2-14x+7
\\=x^3(x-1)-x^2(x-1)+7x(x-1)-7x+7
\\=x^3(x-1)-x^2(x-1)+7x(x-1)-7(x-1).$$</p>
|
1,855,650 | <p>Need to solve:</p>
<p>$$2^x+2^{-x} = 2$$</p>
<p>I can't use substitution in this case. Which is the best approach?</p>
<p>Event in this form I do not have any clue:</p>
<p>$$2^x+\frac{1}{2^x} = 2$$</p>
| MonK | 160,887 | <p>Let,
$2^x=y$ then the equation becomes</p>
<p>$y+\frac{1}{y}=2\\
\implies y^2+1-2y=0\\
\implies (y-1)^2=0\\
\implies y=1\\
\implies 2^x=1\\
\implies 2^x=2^0\\
\implies x=0$</p>
|
1,839,057 | <p>Where n is an integer, $n\ge1$ and $(A,B)$ just constants </p>
<blockquote>
<p>$$I=\int_{-n}^{n}{x+\tan{x}\over A
+B(x+\tan{x})^{2n}}dx=0$$</p>
</blockquote>
<p>It is obvious that</p>
<p>$$\int_{-n}^{n}x+\tan{x}dx=0$$</p>
<p>Let make a substitution for <em>I</em> $$u=x+\tan{x}\rightarrow du=1+\sec^2{x}dx$$</p>... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Use $$I=\int_a^bf(x)\ dx=\int_a^bf(a+b-x)\ dx$$</p>
<p>$$\implies I+I=\int_a^b[f(x)+f(a+b-x)]\ dx$$</p>
<p>Here $a=?,b=?$</p>
<p>and $\tan(-x)=-\tan x$</p>
|
164,447 | <p>by default, if a number has decimal <code>.</code> after it, then Mathematica will do computation using machine Precision, which on my PC (intel hardware) running windows 7 64 bit is double Precision.</p>
<p>I'd like to get the computation also but using single and quad precision, to match the following small Fortr... | Bob Hanlon | 9,362 | <pre><code>$Version
x1 = 0.00001 // Rationalize // N[#, 32] &
sum1 = 0 // N[#, 32] &
Do[sum1 = sum1 + x1, {i, 1, 10^5}]
sum1 // InputForm
"11.2.0 for Mac OS X x86 (64-bit) (September 11, 2017)"
0.000010000000000000000000000000000000
0
0.999999999999999999999999999999999\
9999999999999999999979455`32.
</code... |
164,447 | <p>by default, if a number has decimal <code>.</code> after it, then Mathematica will do computation using machine Precision, which on my PC (intel hardware) running windows 7 64 bit is double Precision.</p>
<p>I'd like to get the computation also but using single and quad precision, to match the following small Fortr... | Michael E2 | 4,999 | <p>The following emulates the various precisions, by directly rounding the operation as done in floating-point. The quad seems off by 1 bit, if Fortran uses IEEE quad precision. Also, Fortran output seems to have an extra digit or two, compared to <em>Mathematica's</em> at the specified precision.</p>
<pre><code>sin... |
440,242 | <p>I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing- either the "middle" part of the argument or the "end" part of the argumen... | Simon Crase | 470,900 | <p>Sometimes it's good the keep the problem in the back of your mind while you do other stuff that appears irrelevant. Here is Stanislaw Ulam's account of the invention of the Monte Carle Method--from <a href="http://www-star.st-and.ac.uk/%7Ekw25/teaching/mcrt/MC_history_3.pdf" rel="noreferrer">Los Alamos Science Speci... |
1,097,134 | <p>this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.</p>
<p>Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?</p>
<p>With positive numbers the square r... | Geoff Robinson | 13,147 | <p>There are many abstract ways to construct the complex numbers, and however you do it, there is a complex number which squares to $-1$, and to which you happen to give a name, often $i$. Once you have named $i$, it becomes obvious that $-i$ also has square $-1$, and that all complex numbers have the form $a +bi$ for ... |
2,231,949 | <p>To find the minimal polynomial of $i\sqrt{-1+2\sqrt{3}}$, I need to prove that
$x^4-2x^2-11$ is irreducible over $\Bbb Q$. And I am stuck. Could someone please help? Thanks so much!</p>
| Arthur | 15,500 | <p>The rational root theorem (or the quadratic formula, solving for $x^2$), shows that there are no linear factors over $\Bbb Q$. That means that if the polynomial is reducible, then it reduces to two irreducible quadratic polynomials.</p>
<p>However, if that were true, then your number would be a root of one of them.... |
4,096,771 | <p>Given a sequence of iid random variables <span class="math-container">$(Y_i)_{i=1}^\infty$</span> on a probability space <span class="math-container">$(\Omega, \mathcal{F}, \mathbb{P})$</span> such that <span class="math-container">$\mathbb{E}|Y_i| < \infty$</span> and <span class="math-container">$\mathbb{E}Y_i ... | Kavi Rama Murthy | 142,385 | <p><span class="math-container">$$E(X_t|\mathcal F_s)=E(\sum\limits_{i=1}^{t}Y_i|Y_1,Y_2,..,Y_s)$$</span> <span class="math-container">$$=\sum\limits_{i=1}^{s}Y_i+E(\sum\limits_{i=s+1}^{t}Y_i|Y_1,Y_2,..,Y_s)$$</span> <span class="math-container">$$=X_s+E(\sum\limits_{i=s+1}^{t}Y_i)=X_x+0=X_s.$$</span></p>
<p>In the sec... |
3,906,920 | <p>A string in <span class="math-container">$\{0, 1\}*$</span> has even parity if the symbol <span class="math-container">$1$</span> occurs in the word an even number of times; otherwise, it has odd parity.</p>
<p>(a) How many words of length <span class="math-container">$n$</span> have even parity?</p>
<p>(b) How many... | Mozibur Ullah | 26,254 | <blockquote>
<p>Does <em>f</em> need to be a diffeomorphism?</p>
</blockquote>
<p>It's not always possible to push-forward tangent fields on a manifold. This is why we have the restriction to diffeomorphisms, or more generally, tangent fields that are <em>f</em>-related.</p>
<p>However, it's an easy observation that wh... |
2,067,003 | <p>(Mathematics olympiad Netherlands) Let $A,B$ and $C$ denote chess players in a tournament. The winner of each match plays the next match against the oponent that did not play the current. At the end of the tournament $A$, $B$ and $C$ played $10$, $15$ and $17$ times respectively. Each match only ended up in a win. <... | Masacroso | 173,262 | <p>Using falling factorials and <a href="https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20finite%20calculus.pdf" rel="nofollow noreferrer">finite calculus</a> we can write</p>
<p>$$\begin{align}\sum\frac{(-1)^{n+1}}{n(n+1)}\delta n&=\sum (-1)^{n+1}(n-1)^{\underline{-2}}\delta n\\&=(-1)^... |
2,067,003 | <p>(Mathematics olympiad Netherlands) Let $A,B$ and $C$ denote chess players in a tournament. The winner of each match plays the next match against the oponent that did not play the current. At the end of the tournament $A$, $B$ and $C$ played $10$, $15$ and $17$ times respectively. Each match only ended up in a win. <... | xpaul | 66,420 | <p>Let
$$ f(x)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n(n+1)}x^{n+1} $$
and hence
$$ f'(x)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}x^{n}, f''(x)=\sum_{n=1}^\infty(-1)^{n+1}x^{n-1}=\frac{1}{1+x}.$$
Note that
$$ f(1)=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n(n+1)}, f'(0)=0.$$
So
\begin{eqnarray}
f(1)&=&\int_0^1\int_0^x\fr... |
1,492,027 | <p>Defining R to be the relationship on real numbers given by xRy iff x-y is rational, I've been asked to find the equivalence class of $\sqrt2$. My instincts say that the equivalence class of $\sqrt2$ would just be the empty set. But after a riveting conversation on a similar subject <a href="https://math.stackexchang... | Kevin Quirin | 267,868 | <p>Remember that, by definition, the equivalence class of $\sqrt 2$ is the <strong>set</strong>
$$[\sqrt 2] = \{y\in\mathbb R~|~\sqrt 2 - y \in \mathbb Q\}$$</p>
<p>Pose $A = \{\alpha + \sqrt 2~|~\alpha\in\mathbb Q\}$.</p>
<ul>
<li>Pick an element $x\in A$ : there is $\alpha \in\mathbb Q$ such that $x = \alpha + \sqr... |
1,613,185 | <p>There are five red balls and five green balls in a bag. Two balls are taken out at random. What is the probability that both the balls are of the same colour</p>
| Mithlesh Upadhyay | 234,055 | <p>Using <a href="https://en.wikipedia.org/wiki/Hypergeometric_distribution" rel="nofollow">Hypergeometric distribution</a>: Given, $5$ red and $5$ green ball. So, total number balls is $10$. </p>
<p>Two balls are taken at random, So required probability is : </p>
<p>$= \frac{^5C_2 \times ^5C_0 + ^5C_2 \times ^5C_0}{... |
1,569,331 | <p>Let $f$ be a continuous function on $[a,b]$. Show that </p>
<p>$$|f(x)-f(x_0)|\leq |f'(x_0)||x-x_0|.$$</p>
<p>I don't know whether differentiability of $f$ on $(a,b)$ is needed in assumption.</p>
<p>I just have seen this question in a part of a paper in the class, so I did not know exactly this is the question or... | Kamil Jarosz | 183,840 | <p>$$\int_1^7f\,dx=\int_1^9f\,dx+\int_9^7f\,dx=\int_1^9f\,dx-\int_7^9f\,dx=-1-5=-6$$</p>
|
1,569,331 | <p>Let $f$ be a continuous function on $[a,b]$. Show that </p>
<p>$$|f(x)-f(x_0)|\leq |f'(x_0)||x-x_0|.$$</p>
<p>I don't know whether differentiability of $f$ on $(a,b)$ is needed in assumption.</p>
<p>I just have seen this question in a part of a paper in the class, so I did not know exactly this is the question or... | Guilherme Thompson | 177,882 | <p>Let $a,b,c$ be 3 numbers in the real line, such $a>c>b$, and $f(x): \mathbb{R} \mapsto \mathbb{R}$ continuous in $(a,b)$, we have
$$ \int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx.$$</p>
<p>For your specific case, we have
$$ \int_1^9 f(x) dx = \int_1^7 f(x) dx + \int_7^9 f(x) dx \implies\\ \int_1^7 ... |
1,776,850 | <p>Given a square $ABCD$ such that the vertex $A$ is on the $x$-axis and the vertex $B$ is on the $y$-axis. The coordinates of vertex $C$ are $(u,v)$. Find the area of square in terms of $u$ and $v$ only.</p>
<p><strong>What I have done</strong></p>
<p>Let the coordinate of $A$ be $(x,0)$ and $B$ be $(0,y)$. Also let... | amd | 265,466 | <p>If the points $A(x,0)$ and $B(0,y)$ are adjacent vertices of a square, then the square’s area is $x^2+y^2$. Assuming that the vertices are enumerated counterclockwise, $C$ has coordinates $(-y,y-x)$, i.e., $B+\operatorname{rot}(B-A,\pi/2)$, so $x=-(u+v)$, $y=-u$ and $x^2+y^2=(u+v)^2+u^2=2u^2+2uv+v^2$. If the vertice... |
255,164 | <p>$\newcommand{\al}{\alpha}$
$\newcommand{\euc}{\mathcal{e}}$
$\newcommand{\Cof}{\operatorname{Cof}}$
$\newcommand{\Det}{\operatorname{Det}}$</p>
<p>Let $M,N$ be smooth $n$-dimensional Riemannian manifolds (perhaps with smooth boundary), and let $\, f:M \to N$ be a smooth <strong>immersion</strong>. Let $\Omega^k(M,... | Craig | 10,749 | <p>It's clearly infinite dimensional in the flat case.</p>
<p>$\ker\delta_1 =\ker d_{n-1}$</p>
<p>And in a coordinate chart with coordinates $(x_1, \ldots, x_n)$ take various $f\in C^\infty(M)$ with support in the chart, so </p>
<p>$df\wedge dx_1 \wedge\cdots\wedge dx_{n-2} \in \ker d_{n-1}$</p>
|
405,953 | <p>Let</p>
<ul>
<li><span class="math-container">$E$</span> be the usual sobolev space <span class="math-container">$H^{1}_{0}(\Omega)$</span> on a smoothly bounded domain <span class="math-container">$\Omega$</span>,</li>
<li><span class="math-container">$E_{k}$</span> be its subspace spanned by the first <span class=... | Giorgio Metafune | 150,653 | <p>The first is not true, and probably also the others.</p>
<p>Take <span class="math-container">$L^2(0, \pi)$</span> and <span class="math-container">$u_1=\sin x$</span>, <span class="math-container">$u_2=\sin (2x)$</span>, so that <span class="math-container">$E_2=\{u=a\sin x+b \sin (2x)\}$</span> and <span class="ma... |
3,632,576 | <p>Considering that input <span class="math-container">$x$</span> is a scalar, the data generation process works as follows:</p>
<ul>
<li>First, a target t is sampled from {0, 1} with equal probability.</li>
<li>If t = 0, x is sampled from a uniform distribution over the interval
[0, 1]. </li>
<li>If t = 1, x is sampl... | Paramanand Singh | 72,031 | <p>Using <span class="math-container">$g(x) =f(x) - a$</span> we can reduce the problem to case when <span class="math-container">$a=0$</span>. So lets prove the result when <span class="math-container">$a=0$</span>.</p>
<p>Let <span class="math-container">$\epsilon >0$</span> and we have a number <span class="math... |
2,528,716 | <p>How can I prove </p>
<blockquote>
<p>$$x^2+y^2-x-y-xy+1≥0$$</p>
</blockquote>
<p>I tried $(x+y)^2-3xy-(x+y)+1≥0 \rightarrow(x+y-1)(x-y)-3xy+1≥0$ I can not continue</p>
| Zaharyas | 457,607 | <blockquote>
<p>$$x^2+y^2-x-y-xy+1≥0 \Rightarrow \frac 12( 2x^2+2y^2-2x-2y-2xy+2)≥0 \Rightarrow \frac 12( x^2-2x+1+y^2-2y+1+x^2+y^2-2xy)≥0 \Rightarrow \frac 12((x-1)^2+(y-1)^2+(x-y)^2)≥0$$</p>
</blockquote>
|
2,471,633 | <p>Let $B$ an open ball in $\mathbb{R}^{n}$, and $(K_{j})_{j}$ be an increasing sequence of compact subsets of $B$ whose union equals $B$. For each $j$, let $\rho_{j}$ be a cut-off function in $C_{c}^{\infty}(B)$ that equals 1 on a neighborhood of $K_{j}$ and whose support is in $K_{j+1}$. Finally, let $\theta$ be a s... | Rigel | 11,776 | <p>I think you cannot conclude that the $\sup$ is finite, because $\nabla\rho_j$ is not bounded.</p>
<p>Consider the following one-dimensional example.
Let $B = (-1,1)$, and let $K_j := [-1+1/j, 1-1/j]$.
In a first approximation you can think $\rho_j$ a Lipschitz function (piecewise affine) such that $\rho_j = 1$ on ... |
487,749 | <p>An opinion poll in a certain city indicated that 69 people in a random sample of 120 said that they would vote for Mr. Jones, while in a second random sample of 160, 93 said that they would vote for Mr. Jones. Find an unbiased estimate of the proportion of people in the city who will vote for Mr. Jones.</p>
| Caleb Stanford | 68,107 | <p><strong>Hint:</strong> By "unbiased", they probably mean that <em>every person is counted equally in the estimate</em>.</p>
<p>The first random sample had $69/120 = 0.575$ in favor of Mr. Jones. The second random sample had $93/160 = 0.58125$ in favor of Jones. But <em>we can't just take the average of these two ... |
487,749 | <p>An opinion poll in a certain city indicated that 69 people in a random sample of 120 said that they would vote for Mr. Jones, while in a second random sample of 160, 93 said that they would vote for Mr. Jones. Find an unbiased estimate of the proportion of people in the city who will vote for Mr. Jones.</p>
| AlexR | 86,940 | <p>Another approach (yielding the same result as Goos) is using a weighted average, taking the number of people asked as the weight:
$$\bar{X}_\omega = \frac{\sum_{i=1}^n \omega_i X_i}{\sum_{i=1}^n \omega_i}$$
Where in this case $n=1, \omega = (120, 160), X = (69,93)$</p>
|
487,749 | <p>An opinion poll in a certain city indicated that 69 people in a random sample of 120 said that they would vote for Mr. Jones, while in a second random sample of 160, 93 said that they would vote for Mr. Jones. Find an unbiased estimate of the proportion of people in the city who will vote for Mr. Jones.</p>
| Felix Marin | 85,343 | <p>\begin{align}
{\cal F}\left(p\right)
&=
{\left(120 p - 69\right)^{2} + \left(160 p - 93\right)^{2} \over 2}
\\[3mm]
{\cal F}\,'\left(p\right)
&=
120\left(120 p - 69\right) + 160\left(160 p - 93\right)
\\[5mm]&\mbox{}
\end{align}</p>
<p>$$
{\cal F}\,'\left(p\right) = 0
\qquad\Longrightarrow\qquad
p
=
{1... |
1,611,506 | <blockquote>
<p>$$\int (2x^2+1)e^{x^2} \, dx$$</p>
</blockquote>
<p>It's part of my homework, and I have tried a few things but it seems to lead to more difficult integrals. I'd appreciate a hint more than an answer but all help is valued.</p>
| user84413 | 84,413 | <p>$\textbf{Hint:}$ Write the integral as $\int2x^2e^{x^2}dx+\int e^{x^2}dx$.</p>
<p>Then use integration by parts on the first integral, with $dv=2xe^{x^2}dx$</p>
|
422,143 | <p>f differentiable function in R. $f(x)= e^{f'(x)}$
$f(0)=1$</p>
<p>I have proved that $f(x)=1$ for every $x\lt0$. im stuck for $x\gt0
$</p>
| Christian Blatter | 1,303 | <p>In the domain $H:=\{(x,y)\ |\ y>0\}\subset{\mathbb R^2}$ your differential equation is equivalent to</p>
<p>$$y'=\log y=:\psi(x,y)\ .\tag{1}$$</p>
<p>As $\psi$ is a continuously differentiable function in $H$ it is locally Lipschitz with respect to $y$ in $H$. It follows that for any $(x_0,y_0)\in H$ the initia... |
3,455,009 | <p>In the proof of the expectation of the binomial distribution,</p>
<p><span class="math-container">$$E[X]=\sum_{k=0}^{n}k \binom{n}{k}p^kq^{n-k}=p\frac{d}{dp}(p+q)^n=pn(p+q)^{n-1}=np$$</span></p>
<p>Why is <span class="math-container">$\sum_{k=0}^{n}k \binom{n}{k}p^kq^{n-k}= p\frac{d}{dp}(p+q)^n$</span>?</p>
<p>I ... | Community | -1 | <p>Plugging your <span class="math-container">$z_1$</span> in the equation, we get</p>
<p><span class="math-container">$$\left(\frac{1+i}2\right)^3=\left(\frac{-1+i}2\right)^3,$$</span> which cannot be true.</p>
<hr>
<p>The solutions of </p>
<p><span class="math-container">$$3z^2+3z+1=0$$</span> are</p>
<p><span c... |
1,973,686 | <p>I am stuck on two questions :</p>
<ol>
<li>If $f,g\in C[0,1]$ where $C[0,1]$ is the set of all continuous functions in $[0,1]$ then is the mapping $id:(C[0,1],d_2)\to (C[0,1],d_1)$ continuous ? where $id$ denotes the identity mapping.</li>
</ol>
<p>where $d_2(f,g)=(\int _0^1 |f(t)-g(t)|^2dt )^{\frac{1}{2}} $ and ... | H. H. Rugh | 355,946 | <p>The answer to the first is yes, by the Cauchy-Schwarz inequality (you simply have $d_1(f,g)\leq d_2(f,g)$). But it only works because the interval $[0,1]$ is finite. On ${\Bbb R}$ it does not hold (as already mentioned in other posts).</p>
|
2,663,537 | <p>Suppose G is a group with x and y as elements. Show that $(xy)^2 = x^2 y^2$ if and only if x and y commute.</p>
<p>My very basic thought is that we expand such that $xxyy = xxyy$, then multiply each side by $x^{-1}$ and $y^{-1}$, such that $x^{-1} y^{-1} xxyy = xxyy x^{-1}$ , and therefore $xy=xy$.</p>
<p>I realiz... | Matthew Leingang | 2,785 | <p>You're right that you should do the proof from right to left and then left to right. But make sure you know what “left” and “right” are.</p>
<p>“left” is the equation $(xy)^2=x^2y^2$, while “right” is the statement “$x$ and $y$ commute.” You can write the last statement as the equation $xy=yx$. </p>
<p>This mea... |
481,086 | <blockquote>
<p>Find a formula (provide your answer in terms of $f$ and its derivatives) for the curvature of a curve in $\mathbb{R}^3$ given by
$\{(x,y,z)\ | \ x=y, f(x)=z\}$.</p>
</blockquote>
<p>How will I be able to do this problem? </p>
<p>I know that a regular parametrization of a curve then the curvature a... | bubba | 31,744 | <p>You were given a parametric curve, but it's slightly disguised. Let $\mu(x) = (x,x,f(x))$ and then apply the formula you gave in your question, but with $t=x$. To get you started, note that $\mu'(x) = (1,1,f'(x))$. Can you take it from there?</p>
|
3,700,367 | <p><strong>What is the <em>average</em> distance from any point on a unit square's perimeter to its center?</strong></p>
<p>The distance from a square's corner to its center is <span class="math-container">$\dfrac{\sqrt{2}}{2}$</span> and from a point in the middle of a square's side length is <span class="math-contai... | paulinho | 474,578 | <p>The forward direction is quite straightforward. Let us prove the contrapositive of this statement. If <span class="math-container">$M^{n-1}$</span> were zero, then the nontrivial linear combination composed of only <span class="math-container">$M^{n-1}$</span> yields zero, so the elements cannot be independent.</p>
... |
1,988,420 | <p>An Ant is on a vertex of a triangle. Each second, it moves randomly to an adjacent vertex. What is the expected number of seconds before it arrives back at the original vertex?</p>
<p>My solution: I dont know how to use markov chains yet, but Im guessing that could be a way to do this. I was wondering if there was ... | bof | 111,012 | <p>I assume the ant starts at vertex $A.$ We want to find the expected of value the random variable $X$ which is the number of vertex-to-vertex steps (i.e. the number of seconds) the ant takes. Now
$$X=X_{A,B}+X_{B,A}+X_{B,C}+X_{C,B}+X_{C,A}+X_{A,C}$$
where $X_{A,B}$ is the number of times the ant walks from $A$ to $B,... |
389,750 | <p>Given A(1,4) and B (3,-5) use the dot product to find point C so that triangle ABC is a right angle triangle.</p>
| rurouniwallace | 35,878 | <p>Find the vector connecting between the two:</p>
<p>$$\vec{c}=<1-3,4+5>=<-2,9>$$</p>
<p>You can use the dot product between $\vec{c}$ and a unit vector $\hat{b}=<-1,0>$:
$$\cos{\theta}=\frac{\vec{c}\cdot\hat{b}}{||\vec{c}||\space||\hat{b}||}$$</p>
<p>The length of the opposing two angles are $||\... |
1,464,522 | <blockquote>
<p>Let <span class="math-container">$O_n(\mathbb Z)$</span> be the group of orthogonal matrices (matrices <span class="math-container">$B$</span> s.t. <span class="math-container">$BB^T=I$</span>) with entries in <span class="math-container">$\mathbb Z$</span>.<br>
1) How do I show that <span class="ma... | Pablo Herrera | 135,689 | <p>Let <span class="math-container">$A$</span> be an orthogonal <span class="math-container">$n \times n$</span> matrix with integer entries. First of all, we know that <span class="math-container">$\det(A)=\pm 1$</span>. This means that <span class="math-container">$A$</span> must be an invertible matrix.</p>
<p>Seco... |
2,632,696 | <p>I have this equation: $x^2y'+y^2-1=0$. It's an equation with separable variable. When I calculate the solution do I have to consider the absolute value for the argument of the log? </p>
| Community | -1 | <p>$$\frac1{1-y^2}$$ is defined for all $y\ne\pm1$ and its antiderivative can be expressed as</p>
<p>$$\frac12(\log|y+1|-\log|y-1|)=\log\sqrt{\left|\frac{y+1}{y-1}\right|}.$$</p>
<p>Depending on the range of $y$, this function is one of $\text{artanh(y)}$ or $\text{arcoth(y)}$. Hence the solution to the ODE is one of... |
537,965 | <p><span class="math-container">$X_0:\Omega\rightarrow I$</span> is a random variable where <span class="math-container">$I$</span> is countable. Also <span class="math-container">$Y_1,Y_2,\dots$</span> are i.i.d. <span class="math-container">$\text{Unif}[0,1]$</span> random variables. </p>
<p>Define a sequence <span ... | Shuchang | 91,982 | <p>Since $X_{n+1}=G(X_n,Y_{n+1})$ where $Y_{n+1}$ is independent with $X_i$ for all $i$, of course we have
$$\begin{align}P(X_{n+1}|X_n,...,X_0)&=P(G(X_n,Y_{n+1})|X_n,...,X_0)\\&=P(G(X_n,Y_{n+1})|X_n)\\&=P(X_{n+1}|X_n)\end{align}$$
which indicates $\{X_n\}$ is a Markov chain.</p>
|
177,209 | <p>I found the following problem while working through Richard Stanley's <a href="http://www-math.mit.edu/~rstan/bij.pdf">Bijective Proof Problems</a> (Page 5, Problem 16). It asks for a combinatorial proof of the following:
$$ \sum_{i+j+k=n} \binom{i+j}{i}\binom{j+k}{j}\binom{k+i}{k} = \sum_{r=0}^{n} \binom{2r}{r}$$
w... | Grigory M | 152 | <p>Turns out, this indeed follows (relatively) directly from Strehl's identity.</p>
<p>Let's rewrite LHS in terms of <span class="math-container">$i$</span>, <span class="math-container">$k$</span> and <span class="math-container">$l=i+j$</span>:
<span class="math-container">$$
\text{LHS}=
\sum_{l+k=n}\sum_{i=0}^l\bin... |
1,089,078 | <p>Suppose we have a deck of cards, shuffled in a random configuration. We would like to find a $k$-bit code in which we explain the current order of the cards. This would be easy to do for $k=51 \cdot 6=306$, since we could encode our deck card-by-card, using $2$ bits for the coloring and $4$ bits for the number on ea... | Seyhmus Güngören | 29,940 | <p>According to Hoffman, the optimal lossless coding needs at least $H(s)$ bits on average for each code, where $h(s)$ is the entropy of the source $s$.</p>
<p>In your case there is no redundancy in the numbers that you want to encode. They are simly numbers starting from $1$ to $52!$. I could suggest using <a href="h... |
1,684,741 | <p>I'm able to show it isn't absolutely convergent as the sequence $\{1^n\}$ clearly doesn't converge to $0$ as it is just an infinite sequence of $1$'s. How do I prove the series isn't conditionally convergent to prove divergence!</p>
| Gottfried Helms | 1,714 | <p>The following is -in principle-still "searching" but structures the space to be searched into simpler subspaces:
$$ \begin{array}{} &4 &= y^4 \pmod 7 \\
& y^4 - 4 &\equiv 0 \pmod 7 \\
&(y^2 - 2)(y^2+2) &\equiv 0 \pmod 7 \\
&& \text{giving two factors}\\
&y^2 - 2 &\equiv 0 \pmo... |
8 | <p>Contexts have backticks, which conflict with the normal way to enter inline code. How do I enter an inline context, since the initial approach:</p>
<pre><code>`System``
</code></pre>
<p>doesn't work ( `System`` ).</p>
| David Z | 79 | <p>According to <a href="https://meta.stackexchange.com/questions/12694/escaping-backticks-fails">this MSO question</a>, you can use double backticks set off by spaces to surround a code snippet:</p>
<pre><code>`` System` ``
</code></pre>
<p>produces <code>System`</code>.</p>
|
2,408,223 | <p>Compute $\int_0^2 \lfloor x^2 \rfloor\,dx$.</p>
<p>The challenging part isn't the problem itself, but the notation around the x^2. I don't know what it is. If someone could clarify, that would be great!</p>
<p>Edit: Clarified that it represents the floor function, can anyone give me a hint on how to start working ... | John Wayland Bales | 246,513 | <p>The following diagram shows the effect that the greatest integer function has on the graph of $y=x^2$.
<a href="https://i.stack.imgur.com/jltzM.png" rel="noreferrer"><img src="https://i.stack.imgur.com/jltzM.png" alt="Greatest integer function of x square"></a></p>
<p>The answer will equal the area of the three rec... |
43,611 | <p>I posted this on Stack Exchange and got a lot of interest, but no answer.</p>
<p>A recent <a href="http://people.missouristate.edu/lesreid/POW12_0910.html" rel="nofollow">Missouri State problem</a> stated that it is easy to decompose the plane into half-open intervals and asked us to do so with intervals pointing i... | Fedor Petrov | 4,312 | <p>Let me answer for closed intervals. (It is well-known, but I do have only Russian references in mind.) We may decompose closed rectangle (easy). Then, if we have rectangle $R_1=a\times b$ already decomposed, then we double it, make $R_2=2R_1=a\times 2b$ and cover $R_2\setminus R_1$. So, doubling in different directi... |
4,008,420 | <p>Suppose we had a differentiable curve <span class="math-container">$C$</span> in <span class="math-container">$\mathbb{R}^2$</span> that serves as our "light container". Light is shining in from all directions, so the space of incoming light-beams is <span class="math-container">$\mathbb{R} \times S^1$</sp... | Joce NoToPutinsWarInUkraine | 138,627 | <p>The isosceles right triangle with an opening in one of the edges close to a vertex away from the right angle has this property.</p>
<p>First let us consider a case where only a null subset of the light.</p>
<p>Consider <span class="math-container">$ABC$</span> an isosceles triangle with right angle at <span class="m... |
3,358,449 | <blockquote>
<p>I have 8 variables; <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, <span class="math-container">$D$</span>, <span class="math-container">$E$</span>, <span class="math-container">$F$</span>, <span class="math-container">$G$</span... | Community | -1 | <p>As given in the post <span class="math-container">${B,C,E}$</span> contains the numbers <span class="math-container">$1$</span> and <span class="math-container">$2$</span>. Similarly <span class="math-container">${A,F}$</span> contains 8.</p>
<p>The equations give <span class="math-container">$A=2(B+E),H=B+2E,G=B+E... |
1,579,528 | <p>You decide to play a holiday drinking game. You start with 100 containers of eggnog in a row. The 1st container contains 1 liter of eggnog, the 2nd contains 2 liters, all the way until the 100th, which contains 100 liters. You select a container uniformly at random and take a one liter sip from it. If the container ... | gar | 138,850 | <p>Finding the exact answer may not be feasible for 100 containers, I think. I managed to compute up to 5 containers using recurrence and a computer. The following python code generates the recurrence for 5 containers with the boundary conditions:</p>
<pre><code>def g(n):
bac = 'f'+str(n)+'('+','.join(['x'+str(i) ... |
754,583 | <p>Write <span class="math-container">$$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\ddots}}$$</span> so that <span class="math-container">$\phi_n=n+\frac{n}{\phi_n},$</span> which gives <span class="math-container">$\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$</span> We know <span class="math-container">$\phi_1=\phi$</span... | Lutz Lehmann | 115,115 | <p>The usual trick is to apply the third binomial formula, so that
$$
\frac{n+\sqrt{n(n+4)}}{2}
=\frac{n^2-(n^2+4n)}{2(n-\sqrt{n(n+4)})}
=-\frac{2 |n| }{|n|+\sqrt{|n|(|n|-4)}}
$$
Now standard limit procedurs for fractions apply, cancel $|n|$ in numerator and denominator, move the limit inside the square root in the den... |
39,762 | <p>Happy new year mathematica gurus of stack exchange!</p>
<p>As I see it one of the major obstacles in getting decent at programming mathematica is that, not only do you need to learn how certain commands work, but rather that you mainly need to understand how to write your syntax. This is a typical such situation, I... | C. E. | 731 | <p>I certainly agree with everything Mr. Wizard says in his answer. Taking the question at face value you can give each loop a symbol so that <code>f[1] := For[... f[2] := For[ ...</code> and then use <code>Switch</code>:</p>
<pre><code>Switch[n,1,f[1],2,f[2]...]
</code></pre>
<p>Or as Kuba suggests:</p>
<pre><code>... |
134,407 | <p>Some shapes, such as the disk or the <a href="http://en.wikipedia.org/wiki/Reuleaux_triangle" rel="nofollow noreferrer">Releaux triangle</a> can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the orientation of the curve.)</p>
... | Włodzimierz Holsztyński | 8,385 | <p>I believe that <strong>@Per</strong> meant to ask (for $n=2$) about the following (for arbitrary $\ n>1$):</p>
<p><strong>CONJECTURE A</strong> There exists real $\delta_n > 0\ $ such that for every family $\ F\ $ of bounded constant width convex bodies $\ B\in F\ $ such that each such $\ B\ $ contains... |
2,292,713 | <blockquote>
<p><strong>Definition.</strong> Let <span class="math-container">$E$</span> be a nonempty subset of <span class="math-container">$X$</span>, and let <span class="math-container">$S$</span> be the set of all real numbers of the form <span class="math-container">$d(p, q)$</span>, with <span class="math-conta... | James Shapiro | 148,829 | <p>Another subtlety to this proof, just in case anyone missed it. The reason that we know that there are <span class="math-container">$p, q \in E$</span> such that <span class="math-container">$d(p, p') < \epsilon$</span>, <span class="math-container">$d(q, q') < \epsilon$</span> is the following:</p>
<p>Every p... |
3,841,542 | <p>I am trying to show that <span class="math-container">$\sqrt{\sqrt{2}+5}$</span> is constructible through a diagram.</p>
<p>I know how to show something of the form <span class="math-container">$\sqrt[n]{a}$</span> is constructible through a diagram, but I am really having a difficult time with this one.</p>
<p>Any ... | Aadhaar Murty | 826,105 | <p>As suggested by @Abhi, differentiating w.r.t. <span class="math-container">$k$</span> will give you a direct answer. We have,</p>
<p><span class="math-container">$$\frac {d}{dk} \left(\frac {k}{a^{2}+ k^{2}}\right) = \int_{0}^{\infty} e^{-ax} \cdot \partial_{k}\sin(kx) dx = \int_{0}^{\infty}xe^{-ax} \cos(kx) dx = \b... |
2,040,961 | <blockquote>
<p>Written with <a href="https://stackedit.io/" rel="noreferrer">StackEdit</a>. </p>
<p>Suppose $(a_i)$ is a sequence in $\Bbb R$ such that $\sum\limits_{i=1}^{ \infty} |a_i||x_i| < \infty$ whenever $\sum\limits_{i=1}^{\infty} |x_i| < \infty$. Then is $(a_i)$ a bounded sequence?</p>
</blockq... | zhw. | 228,045 | <p>If $a_n$ is unbounded, then there exist integers $0 < n_1 < n_2 < \cdots \to \infty$ such that $|a_{n_k}| > k^2.$ Define $x_n$ as follows: $x_{n_k} = 1/k^2, k = 1,2, \dots,$ $x_n=0$ for all other $n.$ Then $\sum |x_n| < \infty,$ while $\sum |a_n||x_n|$ has infinitely many terms $> 1,$ hence diverge... |
2,040,961 | <blockquote>
<p>Written with <a href="https://stackedit.io/" rel="noreferrer">StackEdit</a>. </p>
<p>Suppose $(a_i)$ is a sequence in $\Bbb R$ such that $\sum\limits_{i=1}^{ \infty} |a_i||x_i| < \infty$ whenever $\sum\limits_{i=1}^{\infty} |x_i| < \infty$. Then is $(a_i)$ a bounded sequence?</p>
</blockq... | Tacet | 186,012 | <p><strong>Hint</strong>: Look at this simple fact: <a href="https://math.stackexchange.com/questions/388898/if-the-positive-series-sum-a-n-diverges-and-s-n-sum-limits-k-leqslant-na">If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well</a>.</p>
... |
445,816 | <p>I have to show that</p>
<blockquote>
<blockquote>
<p>$\mathbb{C}=\overline{\mathbb{C}\setminus\left\{0\right\}}$,</p>
</blockquote>
</blockquote>
<p>what is very probably an easy task; nevertheless I have some problems.</p>
<p>In words this means: $\mathbb{C}$ is the smallest closed superset of $\mathbb{C... | Community | -1 | <p>$\partial(\mathbf{C}^\ast)$ consists of $0$ alone, use $\overline{\mathbf{C}^\ast}=\partial(\mathbf{C}^\ast)\cup\mathbf{C}^\ast$.</p>
|
997,463 | <p>For example, a complex number like $z=1$ can be written as $z=1+0i=|z|e^{i Arg z}=1e^{0i} = e^{i(0+2\pi k)}$.</p>
<p>$f(z) = \cos z$ has period $2\pi$ and $\cosh z$ has period $2\pi i$.</p>
<p>Given a complex function, how can we tell if it is periodic or not, and further, how would we calculate the period? For ex... | Anastasiya-Romanova 秀 | 133,248 | <p><strong>Hint:</strong></p>
<p>Put $x = \frac{3}{2}\tan\theta \Rightarrow dx = \frac{3}{2} \sec^2 \theta \ d\theta$, we have
\begin{align}
\int\frac{x^3}{\left(\sqrt{4x^2+9}\right)^3}\,dx&=\frac{3}{16}\int\frac{\tan^3\theta}{\sec^3\theta}\cdot\sec^2\theta\,\,d\theta\\
&=\frac{3}{16}\int\frac{\sin^3\theta}{\c... |
165,489 | <p>I have problem solving this equation, smallest n such that $1355297$ divides $10^{6n+5}-54n-46$. I tried everything using my scientific calculator, but I never got the correct results(!).and finally I gave up!. Could you help me find the first 2 solutions for this equation ? (thanks.)</p>
| Sjoerd Smit | 43,522 | <p>How about this?</p>
<pre><code>cf = Compile[{{max, _Integer}, {p, _Integer}, {j, _Integer}},
Module[{list = ConstantArray[0, 0]},
Do[
If[
PowerMod[10, 6 n + 5, p] - Mod[54 n + 46, p] == 0,
AppendTo[list, n];
If[Length[list] >= j, Break[]]
],
{n, Range[1, max]}
];
... |
1,476,847 | <p>I am a little puzzled by some notations in optimization community. Is there anyone who can explain why $f_1:\mathbb{R}^n\rightarrow\mathbb{R}$ is a finite valueed but $f_2:\mathbb{R}^n\rightarrow\mathbb{R}\cup\{\infty\}$ is not?? I have never have this kind of notations. For function $f_1$ I always calculated limit ... | 5xum | 112,884 | <p>You cannot. Basically, you want some constants $a,b,c$, which are possibly dependent on $n$, such that $$\det(A+B)\leq a\det (A) + b\det (B) + c .$$</p>
<p>However, take $A=\begin{bmatrix}1 & 0 &\dots &0\\
0&0&\dots &0\\
\vdots &\vdots &\ddots &\vdots\\
0&0&\dots&0\en... |
3,238,563 | <p>I have a question about a proof I saw in a book about basic algeba rules. The rule to prove is:
<span class="math-container">\begin{eqnarray*}
\frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0}
\end{eqnarray*}</span></p>
<p>And the proof: </p>
<p><span class="math-container">\begin{eqnarray*}
1 = a \frac{... | auscrypt | 675,509 | <p>Let <span class="math-container">$x=\frac{1}{a}$</span>. Then:</p>
<p><span class="math-container">\begin{eqnarray*}
1 = x \frac{1}{x} \Longrightarrow 1 = \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow a = a \frac{1}{a} \frac{1}{\frac{1}{a}} \Longrightarrow \frac{1}{\frac{1}{a}} = a
\end{eqnarray*}</span></p>
... |
3,238,563 | <p>I have a question about a proof I saw in a book about basic algeba rules. The rule to prove is:
<span class="math-container">\begin{eqnarray*}
\frac{1}{\frac{1}{a}} = a, \quad a \in \mathbb{R}_{\ne 0}
\end{eqnarray*}</span></p>
<p>And the proof: </p>
<p><span class="math-container">\begin{eqnarray*}
1 = a \frac{... | Allawonder | 145,126 | <p>I proceed in steps:</p>
<p>The first implication follows because the reciprocal of <span class="math-container">$1$</span> is <span class="math-container">$1.$</span> Thus, they got <span class="math-container">$$1=\frac 1a\frac{1}{\frac 1a}$$</span> by taking reciprocals of both sides of <span class="math-containe... |
2,929,094 | <p>Differentiation of
<span class="math-container">$\int_{a(x)}^{b(x)} f(x,t)\,\text{d}t$</span> is done by Leibniz's integral rule:
<span class="math-container">$$\frac{\text{d}}{\text{d}x} \left (\int_{a(x)}^{b(x)} f(x,t)\,\text{d}t \right )= f\big(x,b(x)\big)\cdot \frac{\text{d}}{\text{d}x} b(x) - f\big(x,a(x)\big)\... | Hw Chu | 507,264 | <p><a href="https://i.stack.imgur.com/ChL5h.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ChL5h.png" alt="enter image description here"></a></p>
<p>Denote the intersection of <span class="math-container">$\overline{AC}$</span> and <span class="math-container">$\overline{BD}$</span> by <span class=... |
187,432 | <p>Can we evaluate the integral using <a href="http://en.wikipedia.org/wiki/Jordan%27s_lemma#Application_of_Jordan.27s_lemma">Jordan lemma</a>?
$$ \int_{-\infty}^{\infty} {\sin ^2 (x) \over x^2 (x^2 + 1)}\:dx$$</p>
<p>What de we do if removeable singularity occurs at the path of integration?</p>
| Mhenni Benghorbal | 35,472 | <p>Hint, note that $ \cos(2x)=1-2\sin(x)^2 $, this suggest to consider the integral</p>
<p>$$ \int_{C} \frac{ {\rm e}^{2 i z} - 1 }{ z^2 (z^2 + 1)} dz \,.$$ </p>
|
3,700,440 | <p>As stated in the title, it is requested to define a linear transformation <span class="math-container">$T:\Bbb R^3 \to \Bbb R^3$</span> such that the null space of <span class="math-container">$T$</span> is the <span class="math-container">$z$</span>-axis, and the range of <span class="math-container">$T$</span> is ... | Rodrigo Dias | 375,952 | <p>Choose a basis <span class="math-container">$\mathcal{B}=\{u,v\}$</span> for <span class="math-container">$x+y+z=0$</span> (for instance, it could be <span class="math-container">$\{(2,-1,-1), (1,1,-2)\}$</span>).</p>
<p>Letting <span class="math-container">$T$</span> be the linear extension of
<span class="math-co... |
499,840 | <p>I have three points $A=(2,3), B=(6,4)$ and $C=(6,6).$
Given $\vec{AB}=\vec v$ and $\vec{BC}={0 \choose 2}$. I have also that for every $t\in [0,1]$ there is a point $D$ given as $\vec{AD}=t\vec{v}.$ </p>
<p>My question is determine $t$ such that the area of triangle $ADC$ equals area of the triangle $DBC$.</p>
<p... | Ömer | 55,199 | <p>$$\overrightarrow{AD}=t\overrightarrow{v}$$ means that $AD$ and $v$ are linearly dependent. Also we can say $D\in[AB]$ because of $t\in[0,1]$. For $ADC$ and $DBC$ have same areas $t$ must be $1/2$ and hence $D=(4,7/2)$. </p>
|
1,652,846 | <p>Let $s$ be any complex number, $t = e^s$ and $z = t^{1/t}$. Define the sequence $(a_n)_{n\in\mathbb{N}}$ by $a_0 = z $ and $a_{n+1} = z^{a_n} $ for $n \geq 0$, that is to say $a_n$ is the sequence $z$, $z^z$, $z^{z^z}$, $z^{z^{z^{z}}}$ and so on.</p>
<p>I want to show that the sequence $(a_n)_{n\in\mathbb{N}}$ con... | reuns | 276,986 | <p>$$a_0 = z\qquad \qquad \qquad a_{n+1} = z^{a_n}$$ </p>
<p>let $b_n = \ln a_n$ so $a_n = e^{b_n}$ and $$b_{n+1} = \ln \left(z^{e^{b_n}}\right) = e^{b_n} \ln z$$ </p>
<p>if $b_n$ converges to $b$ then $b = e^b \ln z = - e^{b} (-\ln z)$ so </p>
<p>$$b = e^b \ln z = W(-\ln z)$$</p>
<p>where $W$ is (<strong>one of ... |
25,284 | <p>I recently worked my way through Walter Warwick Sawyer's book, <em>Mathematician's Delight</em>, which has opened my eyes to Maths. I used to fear maths, feeling I was incapable. Sawyer (among other authors) has a gift for teaching the subject. I now feel much more confident tackling Maths problems, I have a better ... | paul garrett | 63 | <p>Yes, I remember vividly my chance encounter at the library with that book of his! Yes, it had a big impact on me. The idea that mathematics was a real thing in its own right, like music, and not just a school subject, and not just a device to filter people out.</p>
|
3,039,040 | <p>In the equation <span class="math-container">$3^x=2y^2-1$</span>,
<span class="math-container">$x$</span>, <span class="math-container">$y$</span> are natural numbers.
I found <span class="math-container">$x=1$</span> or <span class="math-container">$2$</span> (mod <span class="math-container">$4$</span>), and <span... | Community | -1 | <p>You can even solve this question using Mathematical Induction.
However, answer is also correct and easy.</p>
<p>And its solution is x=0 and y=1.</p>
|
2,755,213 | <p><strong>Question.</strong> Find, with proof, the possible values of a rational number $q$ for which $q+\sqrt{2}$ is a reduced quadratic irrational.</p>
<p>So, by definition a <em>quadratic irrational</em> is one of the form $u+v\sqrt{d}$ where $u,v\in\Bbb Q, v\neq 0$ and $d$ being square-free. Then, it is said to b... | Arthur | 15,500 | <p>$-1<q-\sqrt{2}<0$ implies that
$$
-1+\sqrt2<q<\sqrt2
$$</p>
|
2,083,460 | <p>While trying to answer <a href="https://stackoverflow.com/questions/41464753/generate-random-numbers-from-lognormal-distribution-in-python/41465013#41465013">this SO question</a> I got stuck on a messy bit of algebra: given</p>
<p>$$
\log m = \log n + \frac32 \, \log \biggl( 1 + \frac{v}{m^2} \biggr)
$$</p>
<p>I n... | M. Chen | 403,559 | <p>$logm$=$logn$+$3/2log(1+v/m^2)$<br>
$logm$=$logn$+$log((m^2+v)/m^2)$<br>
$logm$=$logn$+$log(m^2+v)^(1.5))$<br>
$logm$=$logn$+$log(m^2+v)^3/2$-$logm^3$<br>
$logm^3$+$logm$-$logn$=$log(m^2+v)^3/2$<br>
$log(m^4/n)$=$log(m^2+v)$<br>
$m^4/n$=$(m^2+v)^(3/2)$ From here think you gotta use the cubic expension and square out... |
2,593,627 | <p>I struggle to find the language to express what I am trying to do. So I made a diagram.</p>
<p><a href="https://i.stack.imgur.com/faHgE.png" rel="noreferrer"><img src="https://i.stack.imgur.com/faHgE.png" alt="Graph3parallelLines"></a></p>
<p>So my original line is the red line. From (2.5,2.5) to (7.5,7.5).</p>
<... | Nominal Animal | 318,422 | <p>Let's use basic vector algebra. (This is extremely useful for any kind of graphics programmers, so if you are not familiar with the basics yet, I warmly recommend you look up some tutorials on the net first. Basics of linear algebra, namely matrices, and vector-matrix and matrix-matrix multiplication, is of tremendo... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.