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 |
|---|---|---|---|---|
2,794,724 | <blockquote>
<p>Let $F:U\rightarrow W$ be a linear transformation from the vector
space $U$ to the vectorspace $W$. Show that the image space to $F$,</p>
<p>$$V(F)=\{w\in W:w=F(u) \ \ \text{for some} \ \ u\in U\},$$</p>
<p>is a subspace of $W$.</p>
</blockquote>
<p>Okay, I know that in order for $M$ to ... | Community | -1 | <p>The curly brackets are just defining the image of F. F is a mapping. It takes each element of $U$ to an element of $W$. $V(F)$ is just the set of elements of $W$ that get mapped to by an element of $U$. In other words, take each $x \in U$ and apply $F$. We get $F(u)$ which is an element of $W$. Collect all the... |
1,799,366 | <p>I'm trying to solve the following exercise:</p>
<blockquote>
<p>Let $\mu$ be a probability distribution on $\mathbb{R}$ having second moment $\sigma^2<\infty$ such that if $X$ and $Y$ are independent with law $\mu$ then the law of $(X+Y)/\sqrt{2}$ is also $\mu$. Show that $\mu =\mathcal{N}(0,1)$
Hint: apply ... | Dark | 208,508 | <p><strong>Hint:</strong> without relying on the CLT, you can use <a href="https://en.wikipedia.org/wiki/Characteristic_function_%28probability_theory%29" rel="nofollow noreferrer">characteristic functions</a>.</p>
<p>Let $f$ be the characteristic function of the law $\mu$. </p>
<p>You can easily show that $f$ satis... |
3,162,338 | <p>Consider <span class="math-container">$ x_1, x_2, ..., x_n \in \mathbb{R}$</span>.</p>
<p>We have to prove that each <span class="math-container">$\sqrt x $</span> is rational if the sum of <span class="math-container">$\sqrt x_1 + \ldots + \sqrt x_n $</span> is rational. </p>
<p>I think that I could prove it us... | Bill Dubuque | 242 | <p>As mentioned there are simple counterexamples. However the following version is true</p>
<p><strong>Theorem</strong> <span class="math-container">$\rm\ \sqrt{c_1}+\cdots+\!\sqrt{c_{n}} \in K\ \Rightarrow \sqrt{c_i}\in K\:$</span> for all <span class="math-container">$\rm i,\:$</span> if <span class="math-container"... |
25,337 | <p>If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ of $k$. If that succeeds, compute de Rham cohomology of the lift over $W_k$ instead, which in general will be much e... | George McNinch | 4,653 | <p>This paper of Serre gives an example (I've justed pasted I. Barsotti's math-sci review).
(The paper can be found in Serre's "Collected Works vol. II 1960-1971)</p>
<blockquote>
<p>Serre, Jean-Pierre Exemples de
variétés projectives en
caractéristique $p$ non relevables en
caractéristique zéro. (French) Proc... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | FooF | 37,849 | <p>I believe that strictly from a statistical-genetic perspective, the most we can do is to compare genes and give a probabilistic value of somebody belonging to a certain genetic pool. Just consider mutations and any inter-race breeding that has happened during milleniums to highlight the fuzziness of the notion of de... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | Blake | 209,173 | <p>Not without inbreeding.
If we assume not shared ancestry paths, then each generation you go back you double the number of ancestors, thus 2^n where n is the generation.
taking the previous 95 generations and dividing each by 12 reveals that there are no members of (this part) of the series that are divisible by 12. ... |
1,109,443 | <p>I'm currently learning for an algebra exam and I have some examples of questions from few last years. And I can't find a solution to this one:</p>
<blockquote>
<p>Give three examples of complex numbers where z = -z</p>
</blockquote>
<p>The only complex number I can think of is 0. Because it is a complex number, ... | tfitzger | 209,227 | <p>Genealogy and ancestry is fun. My sister did one of those DNA tests that places your geographic location genetically. Here are her results:</p>
<p>30% Great Britain,
29% Scandinavia,
20% Ireland,
8% Europe West,
6% Finland/NW Russia,
4% Europe East,
3% Iberian Peninsula.</p>
<p>Obviously, those don't fit nic... |
97,261 | <p>This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: </p>
<p>-Differentiability.
-Open mapping theorem.
-Implicit function theorem.
-Lagrange multipliers. Submanifolds.
-Integrals.
-Integration on surfaces.
-Stokes ... | ItsNotObvious | 9,450 | <p>A couple of references that come to mind that satisfy your criteria for exercise solutions are:</p>
<p>(1) <a href="http://rads.stackoverflow.com/amzn/click/0857291912" rel="nofollow">Multivariable Analyis</a> by Shirali and Vasudeva. Most problems are provided with complete solutions.</p>
<p>(2) <a href="http://r... |
3,172,693 | <p>Can anybody help me with this equation? I can't find a way to factorize for finding a value of <span class="math-container">$d$</span> as a function of <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$$d^3 - 2\cdot d^2\cdot a^2 + d\cdot a^4 - a^2 = 0$$</span></p>
<p>Another form:</p>
... | Ma Joad | 516,814 | <p>That is because for
<span class="math-container">$$\frac{as^2+bs+c}{s^2(s+2)}=\frac{A}{s^2}+\frac{B}{(s+2)},$$</span>
the left hand side has three parameters <span class="math-container">$a,b,c$</span>, but the right hand side only has two parameters <span class="math-container">$a,b$</span>. And if you try to solve... |
3,172,693 | <p>Can anybody help me with this equation? I can't find a way to factorize for finding a value of <span class="math-container">$d$</span> as a function of <span class="math-container">$a$</span>:</p>
<p><span class="math-container">$$d^3 - 2\cdot d^2\cdot a^2 + d\cdot a^4 - a^2 = 0$$</span></p>
<p>Another form:</p>
... | John Joy | 140,156 | <p>Suppose we multiplied both sides of the second equation by <span class="math-container">$s^2(s-2)$</span>, giving us the equivalent equation:
<span class="math-container">$$1 = 0s^2 + 0s + 1= A(s+2) + Bs^2 = Bs^2 + As + 2A$$</span>
Notice that on the RHS of this equation, that the constant term is not independent o... |
2,515,939 | <p>So, I just need a hint for proving
$$\lim_{n\to \infty} \int_0^1 e^{-nx^2}\, dx = 0$$ </p>
<p>I think maybe the easiest way is to pass the limit inside, because $e^{-nx^2}$ is uniformly convergent on $[0,1]$, but I'm new to that theorem, and have very limited experience with uniform convergence. Furthermore, I don... | Jack D'Aurizio | 44,121 | <p>$$0\leq\int_{0}^{1}e^{-nx^2}\,dx \leq \int_{0}^{1}\frac{dx}{1+n x^2} = \frac{\arctan\sqrt{n}}{\sqrt{n}}\leq \frac{\pi}{2\sqrt{n}}.$$</p>
|
3,518,285 | <p>I started studying the book of Daniel Huybrechts, Complex Geometry An Introduction. I tried studying <a href="https://mathoverflow.net/questions/13089/why-do-so-many-textbooks-have-so-much-technical-detail-and-so-little-enlightenme">backwards</a> as much as possible, but I have been stuck on the concepts of <a href=... | WoolierThanThou | 686,397 | <p>GreginGre's solution is, of course, perfectly lovely, but if we're just killing this with choice, I guess you can also prove it as follows:</p>
<p>Let <span class="math-container">$V$</span> be infinite dimensional and, using Zorn's Lemma, let <span class="math-container">$\{e_i\}_{i\in I}$</span> be a basis for <s... |
355,552 | <p>How would you compute the first $k$ digits of the first $n$th Fibonacci numbers (say, calculate the first 10 digits of the first 10000 Fibonacci numbers) without computing (storing) the whole numbers ?</p>
<p>A trivial approach would be to store the exact value of all the numbers (with approximately $0.2n$ digits f... | Community | -1 | <p>We have
$$F_n = \dfrac{\left(\dfrac{1+\sqrt5}2\right)^n - \left(\dfrac{1-\sqrt5}2\right)^n}{\sqrt{5}}$$
Hence,
$$F_n = \begin{cases} \left\lceil{\dfrac{\left(\dfrac{1+\sqrt5}2\right)^n}{\sqrt{5}}} \right\rceil & \text{if $n$ is odd}\\ \left\lfloor{\dfrac{\left(\dfrac{1+\sqrt5}2\right)^n}{\sqrt{5}}} \right\rfloor... |
1,588,665 | <p>I have been reading up on finding the eigenvectors and eigenvalues of a symmetric matrix lately and I am totally unsure of <strong>how and why</strong> it works. Given a matrix, I can find its eigenvectors and values like a machine but the problem is, I have no intuition of how it works.</p>
<p>1) I understand that... | Element118 | 274,478 | <p>You divided by $x$. Note that since you cannot divide by $0$, dividing by $x$ is similar to as asserting that $x$ is not $0$.</p>
<p>Of course, this argument is not rigourous, consider $x^3-x^2=0$. Even when we divide by $x$, we have $x(x-1)=0$, so $0$ and $1$ are solutions to the equation after dividing by $x$.</p... |
2,895,655 | <blockquote>
<p>Four coins of different colour are thrown. If three out of these show heads then find the probability that the remaining one shows tails. </p>
</blockquote>
<p>My approach:</p>
<p>$A$: The event in which 3 heads appear in 3 coins out of 4</p>
<p>$B$: The event in which the 4th coin shows tails</p>
... | dan post | 532,731 | <p>The sample space is the number of ways that, at minimum, three coins are heads. There are 5 ways this can happen -- namely, all heads (1 way) or one of the four coins being tails (4 ways). Of these 4 of the 5 ways will have one tail. You should be able to work out the actual steps involved from this.</p>
|
2,430,482 | <p>I'm struggling to find the maximum of this function $f:\mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}$</p>
<p>$$ f(x,y) = \frac{n+1}{2} \sum_{i=1}^n x_i\,y_i - \sum_{i=1}^n x_i \sum_{i=1}^n y_i,$$</p>
<p>where $x_i,y_i\in[0,1]$ for $i=1,...,n$. It reminded me to Chebyshev's sum inequality, but it didn't hel... | kimchi lover | 457,779 | <p>Here is a stab. Let $x_i=y_i=1$ for $i\le k$ and $x_i=y_i=0$ for $i>k$. Then $f=(n+1)k/2-k^2 = ((n+1)/2-k)k$ which is largest when $k\approx n/4$, giving $f\approx n^2/16$. </p>
<p>This is only a lower bound on the desired maximum. It does not purport to actually answer the original question. </p>
<p>ADDED ... |
2,234,744 | <p>Please help me find the value of the following integral:<br>
$$\frac{(5050)\int^1_0(1-x^{50})^{100} dx}{\int^1_0(1-x^{50})^{101} dx}$$
I tried solving both numerator and denominator via by-parts but it isn't giving me a conclusive solution. Any other suggestions?</p>
| Chappers | 221,811 | <p>Let's try integrating by parts and see what happens.
$$ \int_0^1 (1-x^n)^m \, dx = \left[ x(1-x^n)^m \right]_0^1 - \int_0^1 -nmx^n(1-x^n)^{m-1} \, dx = 0+ nm\int_0^1 x^n(1-x^n)^{m-1} \, dx. $$
We can fiddle with the right-hand side to get it into a more familiar form:
$$ \int_0^1 x^n(1-x^n)^{m-1} \, dx = \int_0^1 \l... |
470,617 | <ol>
<li><p>Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?</p></li>
<li><p>One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the votes, in a way that the first competitor ... | xbh | 514,490 | <p>I provide one more solution, where we don't use sines and cosines. </p>
<p>First, some preparation. </p>
<p>We all know that
$$
\tan (x \pm y) = \frac {\tan (x) \pm \tan (y)} {1 \mp \tan (x)\tan (y)}.
$$
Then
$$
\cot (x - y) = \frac {\cot (x) \cot (y) + 1} { \cot (y) - \cot (x)},
$$
or
$$
\cot (x) \cot (y) = \c... |
42,957 | <p>I am an "old" programmer used to <em>Fortran</em> and <em>Pascal</em>. I can't get rid of <code>For</code>, <code>Do</code> and <code>While</code> loops, but I know <em>Mathematica</em> can do things much faster!</p>
<p>I am using the following code</p>
<pre><code>SeedRandom[3]
n = 10;
v1 = Range[n];
v2 = RandomRe... | Simon Woods | 862 | <p>A more functional approach:</p>
<pre><code>a = With[{f = Subtract @@@ Subsets[Reverse@#, {2}] &}, f[v2]/f[v1]]
</code></pre>
<p>For a bit more speed you could do this:</p>
<pre><code>ii = Join @@ Table[ConstantArray[i, i - 1], {i, n, 2, -1}];
jj = Join @@ Table[Range[j, 1, -1], {j, n - 1, 1, -1}];
a = Divide[... |
130,306 | <p>I am trying to make a relatively complex 3D plot in order to show the variation of a curve with a parameter. Here is the code</p>
<pre><code>AnsNf[x_, nf_] = (2 \[Pi] x^4)/((11 - (2 (2 + nf))/3) (1 + 1/2 x^6 Log[4. x^2])) + (14.298 (1 + (1.81 - 0.292 nf) x^2 - 2.276 x^2 Log[x^2/(1 + x^2)]))/(1 + (9.926 + 1.795 nf) ... | DPF | 34,167 | <p>Just add the <code>Dashed</code> directive before the <code>Plot</code>:</p>
<pre><code>[...] Table[{Dashed, Plot[{AnsatzINf[t, [...]
</code></pre>
<p>Knowing that, you can divide your <code>Plot</code> and only add the <code>Dashed</code> directive before the curves you want.</p>
<p><a href="https://i.stack.imgu... |
2,849,450 | <p>Let's consider a generic linear programming problem. Is it possible that the decision variables of the objective function assume (at the optimal solution) irrational values?</p>
<p>Also, is it possible that some entries of the $A$ matrix are irrational?</p>
| Johan Löfberg | 37,404 | <p>If the problem is described with rational data, there is always a rational optimal solution. I don't have any reference immediately, but it is a standard result. Search on rational data linear program, polynomial complexity etc, and you will find a lot of material.</p>
<p>Edit: I see my answer was a bit unclear. If... |
1,272,124 | <p>we know that $1+2+3+4+5.....+n=n(n+1)/2$</p>
<p>I spent a lot of time trying to get a formula for this sum but I could not get it :</p>
<p>$( 2 + 3 + . . . + 2n)$</p>
<p>I tried to write the sum of some few terms.. Of course I saw some pattern between the sums but still the formula I Got didn't give a correct sum... | gt6989b | 16,192 | <p>different sums are in the title and body
title sum is
$$ 2+4+\ldots+2n = 2\left(1+\ldots+n\right)$$
which is easy since you yourself said what that sum is.</p>
<p>body sum us
$$1+2+...+2n = 1+2+...+M$$
with $M=2n$ and now plug back into the same formula</p>
|
2,445,693 | <p>I know that the derivative of $n^x$ is $n^x\times\ln n$ so i tried to show that with the definition of derivative:$$f'\left(x\right)=\dfrac{df}{dx}\left[n^x\right]\text{ for }n\in\mathbb{R}\\{=\lim_{h\rightarrow0}\dfrac{f\left(x+h\right)-f\left(x\right)}{h}}{=\lim_{h\rightarrow0}\frac{n^{x+h}-n^x}{h}}{=\lim_{h\right... | Community | -1 | <p>If you don't have a definition of the logarithm handy (or suitable properties taken for granted), you cannot obtain the stated result because the logarithm will not appear by magic from the computation.</p>
<p>Assume that the formula $n^x=e^{x \log n}$ is not allowed. Then to define the powers, you can work via rat... |
1,581,756 | <p>Find the general solution of $$z(px-qy)=y^2-x^2$$ Let $F(x,y,z,p,q)=z(px-qy)+x^2-y^2$. This gives $$F_x=zp+2x$$
$$F_y=-zq-2y$$
$$F_z=px-qy$$
$$F_p=zx$$
$$F_q=-zy$$
By Charpit's method we have $$\frac{dx}{zx}=\frac{dy}{-zy}=\frac{dz}{z(px-qy)}=\frac{dp}{-zp-2x-p^2x+pqy}=\frac{dq}{zq+2y-pxy+q^2y}$$</p>
<p>By equating... | Ross Millikan | 1,827 | <p>The reason that numbers with more than four divisors are multiples of numbers with exactly four divisors are that the numbers exactly four divisors are of the form $pq$ for distinct primes $p,q$ or $p^3$ for prime $p$. To have more than four, the number has to be of the form $p^4, p^2q, \text{ or } pqr$ for primes... |
3,564,476 | <p>I'm stuck on the following problem: </p>
<p><span class="math-container">$$\int_{\frac\pi{12}}^{\frac\pi2}(1-\cos4x)\cos2x\>dx$$</span></p>
<p>I think I can use the double angle formulas here but I'm not sure how to apply it, or even if it's the right approach. I'm also not sure if 1-cos4x can be translated in... | Quanto | 686,284 | <p>Use the double angle identity to write <span class="math-container">$1-\cos 4x = 2\sin^2 2x$</span>,</p>
<p><span class="math-container">$$\int_{\frac\pi{12}}^{\frac\pi2}(1-\cos4x)\cos2xdx
=\int_{\frac\pi{12}}^{\frac{\pi}2}\sin^2 2x\>d(\sin 2x)= \frac13 \sin^32x\bigg|_ {\frac\pi{12}}^{\frac{\pi}2}=-\frac1{24}$$<... |
1,455,969 | <p><a href="https://i.stack.imgur.com/5O0d8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5O0d8.png" alt="enter image description here"></a></p>
<p>Hello! I'm having problems trying to figure out this. Here is what I did: I used implication relation and Demorgan's law to simplify this proposition.... | Eugene Zhang | 215,082 | <p>First we prove that
$$
\{\neg(r\to p)\lor[(\neg q\to\neg p)\land(r\to q)]\}\iff \neg p \lor q
$$
Let $A\iff\{\neg(r\to p)\lor[(\neg q\to\neg p)\land(r\to q)]\}$. Then
\begin{align}
A&\iff\{\neg(\neg r\lor p)\lor[(\neg q\to\neg p)\land(r\to q)]\}\tag1
\\
&\iff\{(r\land\neg p)\lor[(\neg q\to\neg p)\land(r\to... |
1,455,969 | <p><a href="https://i.stack.imgur.com/5O0d8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5O0d8.png" alt="enter image description here"></a></p>
<p>Hello! I'm having problems trying to figure out this. Here is what I did: I used implication relation and Demorgan's law to simplify this proposition.... | MarnixKlooster ReinstateMonica | 11,994 | <p>I would treat this as a simplification problem: start with the left hand side of the statement, and work towards the right hand side.$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x20... |
24,927 | <p>Does this mean that the first homotopy group in some sense contains more information than the higher homotopy groups? Is there another generalization of the fundamental group that can give rise to non-commutative groups in such a way that these groups contain more information than the higher homotopy groups? </p>
| Ben Lerner | 1,769 | <p>One geometrical fact from which the non-commutativity of the fundamental group is the following: two objects on a line can't switch relative position (i.e. left and right) through homotopy, as they are unable to "pass" each other. Two objects in a higher dimensional space can, however; so intuitively, it seems that... |
678,768 | <p>"Let $A$, $B$ be two infinite sets. Suppose that $f: A \to B$ is injective. Show that there exists a surjective map $g: B \to A$"</p>
<p>I am not sure how to go about this proof, I am trying to gather information to help me, and deduce as much as I can:
. Since $f$ is injective we know that $|A| \leq |B|$. </p>
<p... | copper.hat | 27,978 | <p>Let $B_1 = f(A), B_2 = B \setminus B_1$, $a_0 \in A$.</p>
<p>For $x \in B_1$, let $g(x)$ be the unique element $a \in A$ such that $f(a) = x$.</p>
<p>For $x \in B_2$, let $g(x) = a_0$. </p>
<p>Then $g(B) = A$.</p>
|
1,843,274 | <p>Good evening to everyone. So I have this inequality: $$\frac{\left(1-x\right)}{x^2+x} <0 $$ It becomes $$ \frac{\left(1-x\right)}{x^2+x} <0 \rightarrow \left(1-x\right)\left(x^2+x\right)<0 \rightarrow x^3-x>0 \rightarrow x\left(x^2-1\right)>0 $$ Therefore from the first $ x>0 $, from the second $ x... | Jared | 138,018 | <p>Once you write this $\frac{1 - x}{x(1 + x)} < 0$ you will find that the critical point are at $x = \{-1, 0, 1\}$. You can simply plug in $x = -2$ to find that:</p>
<p>$$
\frac{1 + 2}{4 - 2} = \frac{3}{2} > 0
$$</p>
<p>Therefore this expression is $> 0$ when $x < -1$, $< 0$ when $-1 < x < 0$,i... |
9,513 | <p>I'm a very novice user of <em>Mathematica</em> - is there possibility of exporting Mathematica code directly into $\LaTeX$? I'm interested only in exporting mathematical formulas. Also, from which version of program is it possible?</p>
| soandos | 605 | <p>Select the text, right click, and select Copy As -> LaTex.</p>
<p><img src="https://i.stack.imgur.com/cW29C.jpg" alt="enter image description here"></p>
|
855,227 | <p>I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. For example, if I have a function $f(x, y)$, then it's first differential is: </p>
<p>$$df = \frac{\partial f}{\part... | Christian Blatter | 1,303 | <p>The "quantity" $$d^rf({\bf z}):=\sum_{k=0}^r{r\choose k}{\partial ^rf({\bf z})\over\partial x^k\partial y^{r-k}}dx^k\,dy^{r-k}$$
is a <strong>homogeneous polynomial</strong> of degree $r$ in the variables $dx$, $dy$ with coefficients the various $r$th order partial derivatives of $f$ at the given point ${\bf z}$. (O... |
4,289,381 | <p>I am trying to answer a question about line integrals, I have had a go at it but I am not sure where I am supposed to incorporate the line integral into my solution.</p>
<p><span class="math-container">$$ \mathbf{V} = xy\hat{\mathbf{x}} + -xy^2\hat{\mathbf{y}}$$</span>
<span class="math-container">$$ \mathrm{d}\math... | AlanD | 356,933 | <p>In this case, it's as simple as plugging in <span class="math-container">$y=\frac{x^2}{3}$</span> into the first integral and integrating from <span class="math-container">$x=0$</span> to <span class="math-container">$x=3$</span>. In the second integral, do the same thing, but plug in <span class="math-container">$d... |
2,970,787 | <blockquote>
<p>Find <span class="math-container">$\lim_{x\to -\infty} \frac{(x-1)^2}{x+1}$</span></p>
</blockquote>
<p>If I divide whole expression by maximum power i.e. <span class="math-container">$x^2$</span> I get,<span class="math-container">$$\lim_{x\to -\infty} \frac{(1-\frac1x)^2}{\frac1x+\frac{1}{x^2}}$$</... | KM101 | 596,598 | <p><span class="math-container">$$\lim_{x \to -\infty} \frac{x^2-2x+1}{x+1} \implies \lim_{x \to -\infty} \frac{1-\frac{2}{x}+\frac{1}{x^2}}{\frac{1}{x}+\frac{1}{x^2}}$$</span>
As you mentioned, the numerator tends to <span class="math-container">$1$</span>. However, notice that the denominator tends to <span class="ma... |
1,728,097 | <p>So i have this integral : $$ \int_0^\infty e^{-xy} dy = -\frac{1}{x} \Big[ e^{-xy} \Big]_0^\infty$$
The integration part is fine, but I'm not sure what i get with the limits, can someone explain this</p>
<p>Thanks </p>
| Michael Hardy | 11,667 | <p>To be explicit: In this case $-\frac{1}{x} \Big[ e^{-xy} \Big]_{y:=0}^{y:=\infty}$ means $-\frac{1}{x} \Big[ e^{-xy} \Big]_{y\,:=\,0}^{y\,:=\,\infty}$.</p>
<p>It does <b>not</b> mean $-\frac{1}{x} \Big[ e^{-xy} \Big]_{x\,:=\,0}^{x\,:=\,\infty}$.</p>
|
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Novice | 97,093 | <p>Consider the following example:</p>
<p>Let us have a 3 digit number that can be divided by 3, ie xyz.</p>
<p>Therefore xyz=0 (mod3)</p>
<p>iff xyz=(100x)+(10y)+z=x+y+z=0(mod3)</p>
<p>Therefore x+y+z=0(mod 3), meaning that the sum of the digits is divisible by 3.</p>
<p>This is an if and only if statement.</p>
... |
341,202 | <p>The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:</p>
<p>$$1212582439 \rightarrow 37 \rightar... | Learner | 280,290 | <p>This can be easily proven by "Digital Root" concept.</p>
<p>Digital root: A digit obtained by adding digits of number until a single digit is obtained.</p>
<p>All natural number is partitioned into 9 equivalence class by "Digital root".</p>
<p>Any number of Digital root 1 is represented by $1+9\times n$
any numb... |
1,687,147 | <blockquote>
<p>A category <span class="math-container">$\mathsf C$</span> consists of the following three mathematical entities:</p>
<ul>
<li><p>A class <span class="math-container">$\operatorname{ob}(\mathsf{C})$</span>, whose elements are called objects;</p>
</li>
<li><p>A class <span class="math-container">$\hom(\m... | C. Dubussy | 310,801 | <p>To be formal, you can say that a category is a triple $(Ob(C), Hom(C), \circ)$ such that, etc ...</p>
<p>The notion of triple is perfectly and formaly defined in set theory. </p>
<p>Of course, I use the definition of category which states that $Ob(C)$ and $Hom(C)$ must be sets. To work with this definition, one us... |
563,431 | <p>Find the absolute maximum and minimum of $f(x,y)= y^2-2xy+x^3-x$ on the region bounded by the curve $y=x^2$ and the line $y=4$. You must use Lagrange Multipliers to study the function on the curve $y=x^2$.</p>
<p>I'm unsure how to approach this because $y=4$ is given. Is this a trick question?</p>
| Community | -1 | <p><strong>Hint</strong>: Consider a direct product of cyclic groups whose orders multiply to $52$, but such that the product is not cyclic. The fact that</p>
<p>$$52 = 2^2 \cdot 13$$</p>
<p>is highly relevant here.</p>
|
2,987,994 | <p>I'm looking for definite integrals that are solvable using the method of differentiation under the integral sign (also called the Feynman Trick) in order to practice using this technique.</p>
<p>Does anyone know of any good ones to tackle?</p>
| clathratus | 583,016 | <p>A few good ones are:
<span class="math-container">$$\int_0^\infty e^{-\frac{x^2}{y^2}-y^2}dx$$</span>
<span class="math-container">$$\int_0^\infty \frac{1-\cos(xy)}xdx$$</span>
<span class="math-container">$$\int_0^\infty \frac{dx}{(x^2+p)^{n+1}}$$</span>
<span class="math-container">$$\int_{0}^{\infty}e^{-x^2}dx$$<... |
1,162,315 | <blockquote>
<p>(b) An electrical circuit comprises three closed loops giving the following equations for the currents $i_1, i_2$ and $i_3$</p>
<p>\begin{align*}
i_1 + 8i_2 + 3i_3 &= -31\\
3i_1 - 2i_2 + i_3 &= -5\\
2i_1 - 3i_2 + 2i_3 &= 6
\end{align*}</p>
</blockquote>
<p>This is the system I need t... | Henrik supports the community | 193,386 | <p>Isolate one of the variables (e.g. $i_1$) in the first equation, and substitute the result in the second, which becomes an equation with two unknowns ($i_2$ and $i_3$). Isolate one of them (e.g. $i_2$). Then substitute both results in the third equation, which (if you do it correctly) becomes an equation with only o... |
1,810,055 | <p>I want to find the Galois groups of the following polynomials over $\mathbb{Q}$. The specific problems I am having is finding the roots of the first polynomial and dealing with a degree $6$ polynomial.</p>
<blockquote>
<p>$X^3-3X+1$</p>
</blockquote>
<p>Do we first need to find its roots, then construct a splitt... | Dietrich Burde | 83,966 | <p>You do not need to know the roots of the cubic to find its Galois group. You should consult an algebra book about Galois groups and discriminants here. Then the solution is as follows.
By the Rational Root Theorem you know that $x^3-3x+1$ is irreducible and its discriminant is $81$, which is a square in $\mathbb{Q}$... |
3,415,331 | <p>It is easy to show, that (for continuous functions f)
<span class="math-container">$$\exists c>0,\exists\alpha>1, \forall x\in \mathbb{R}: |x|^\alpha |f(x)| <c \implies \int |f|dx <\infty$$</span></p>
<p>The question is, whether or not this is also a neccessary condition. I could not come up with any c... | Botond | 281,471 | <p>A better example than my measure-zero one in the comment: Pick your favorite summable sequence <span class="math-container">$(a_n)$</span> and your favorite positive number <span class="math-container">$b$</span>, and construct a function whose grap has a right triangle with base <span class="math-container">$a_n$</... |
572,137 | <p>If $\psi:G \to H$ is a surjective homomorphism, then $|\{g \in G: \psi(g)=h_1\}| = |\{g \in G: \psi(g)=h_2\}|, \forall h_1,h_2 \in H.$ </p>
<p>Could anyone advise on the proof? If $\psi$ is injective, then the result follows. So, what happens if $\psi$ is not injective? By 2nd isomorphism thm, $G/Ker(\psi) \cong H.... | egreg | 62,967 | <p>Hint: if $\psi(g_1)=\psi(g_2)$, then $\psi(g_1g_2^{-1})=1$, so $g_1g_2^{-1}\in\ker\psi$. If $K=\ker\psi$, then $g_1K=g_2K$. What about the converse?</p>
<hr>
<p>Since DonAntonio spoiled the fun, here's the complete answer.</p>
<p>Consider $K=\ker\psi$; then we can factor $\psi$ as $\tilde\psi\circ\pi$, where $\pi... |
2,372,698 | <p>If a function $f(x)$ is continuous on the closed interval $\left [ a,b \right ]$ then its bounded on this interval........the proof for this theorem i have is: </p>
<p>Since it's continuous on $\left [ a,b \right ]$ if we pick a random point on this interval let it be $c$ </p>
<p>$\implies$ $\forall$ $\epsilo... | Hellen | 464,638 | <p>That proof in only showing that $f$ is locally bounded (bounded in a neighborhood of any given point).</p>
<p>The $M$ (which we better denote as $M_c$ to emphasize that it depends on the point) should really be $|f(c)|+\epsilon$ in that argument.</p>
<p>And the argument shows that for each $c$ there is $M_c$ and a... |
3,527,785 | <p>I'm reading James Anderson's <em>Automata Theory with Modern Applications. Here:</em></p>
<blockquote>
<p><a href="https://i.stack.imgur.com/sFWNh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFWNh.png" alt="enter image description here" /></a></p>
<p><a href="https://i.stack.imgur.com/k9zne.pn... | J.-E. Pin | 89,374 | <p>Let <span class="math-container">$1$</span> be the empty word. First of all, one needs to discard the case <span class="math-container">$C = \{1\}$</span> for the result to be correct. Indeed, if <span class="math-container">$C = \{1\}$</span>, then <span class="math-container">$C$</span> is a prefix code, but <span... |
3,662,466 | <p>Given a sequence with the terms </p>
<p><span class="math-container">$$
a_{n}=\left\{\begin{array}{ll}
n, & \text { if } n \text { even } \\
\frac{1}{n}, & \text { if } n \text { odd }
\end{array}\right.
$$</span></p>
<p>Prove <span class="math-container">$\limsup _{n \rightarrow \infty} a_{n} = \infty$</s... | Peter Szilas | 408,605 | <p><span class="math-container">$b_n:= \sup \{a_k: k\ge n \}\ge n.$</span></p>
<p><span class="math-container">$\lim \sup \{a_n\}=\lim_{n \rightarrow \infty}b_n \ge \lim_{n \rightarrow \infty} n =\infty.$</span></p>
|
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Mikhail Katz | 28,128 | <p>Thinking about the distinction between language and metalanguage may be helpful here. When one describes set theory as possessing a single binary relation denoted $\in$, one is operating at the level of metalanguage. Specifying axioms satisfied by $\in$ is at the level of the language. At this stage sets could be... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Greg S | 124,813 | <p>The relevant quantifiers and relations in mathematical axioms should be understood as predicate logic. In the case of ordinary first-order predicate logic, the membership operator $\in$ is defined as a binary relation over the <strong>universe</strong>, or <em>domain of discourse</em>, sometimes denoted $\Omega$.</... |
300,753 | <p>Revision in response to early comments. Users of set theory need an <em>implementation</em> (in case "model" means something different) of the axioms. I would expect something like this:</p>
<blockquote>
<p>An <em>implementation</em> consists of a "collection-of-elements" $X$, and a relation
(logical pairing)... | Zuhair Al-Johar | 95,347 | <p>This is in response to the new edited version of the question.</p>
<p>You are using "<a href="https://en.wikipedia.org/wiki/Indicator_function" rel="nofollow noreferrer">indicator</a>" functions on $X$ but with respect of membership in $X$ instead of subset-hood of $X$ [although one better take $X$ to be transitive... |
2,763,974 | <blockquote>
<p>Find $\displaystyle \lim_{(x,y)\to(0,0)} x^2\sin(\frac{1}{xy}) $ if exists, and find $\displaystyle\lim_{x\to 0}(\lim_{y\to 0} x^2\sin(\frac{1}{xy}) ), \displaystyle\lim_{y\to 0}(\lim_{x\to 0} x^2\sin(\frac{1}{xy}) )$ if they exist.</p>
</blockquote>
<p>Hey everyone. I've tried using the squeeze theo... | user | 505,767 | <p>You only need simply to note that</p>
<p>$$0\le \left|x^2\sin(\frac{1}{xy})\right|\le x^2\to 0$$</p>
<p>then conclude by squeeze theorem.</p>
|
3,527,004 | <p>As stated in the title, I want <span class="math-container">$f(x)=\frac{1}{x^2}$</span> to be expanded as a series with powers of <span class="math-container">$(x+2)$</span>. </p>
<p>Let <span class="math-container">$u=x+2$</span>. Then <span class="math-container">$f(x)=\frac{1}{x^2}=\frac{1}{(u-2)^2}$</span></p>
... | giobrach | 332,594 | <p>Why not a Taylor series expansion at <span class="math-container">$x=-2$</span>? That would be
<span class="math-container">$$f(x) = \sum_{n\geq 0} \frac{f^{(n)}(-2)}{n!}(x+2)^n, $$</span>
with radius of convergence of <span class="math-container">$2$</span>. You may calculate the <span class="math-container">$n$</s... |
2,702,726 | <p>Find the absolute minimum and maximum values of,</p>
<p>$$f(x) = 2 \sin(x) + \cos^2 (x) \text{ on } [0, 2\pi]$$</p>
<p>What I did so far is</p>
<p>$$f'(x) = 2\cos(x) -2 \cos(x) \sin(x)$$</p>
<p>Could someone please help me get started?</p>
| Siong Thye Goh | 306,553 | <p>Method $1$:</p>
<p>Continue from what you have so far, </p>
<p>$$\cos(x)((1-\sin(x))=0$$</p>
<p>Find the stationary point, evaluate the function values at the stationary point as well as the boundaries and conclude the minimal and maximal point.</p>
<p>Method $2$:</p>
<p>\begin{align}
f(x)&=2\sin(x)+\cos^2(... |
226,265 | <blockquote>
<p>Suppose <span class="math-container">$(X,d)$</span> is a metric space. Does every open cover of <span class="math-container">$X$</span> have a minimal subcover with respect to inclusion?</p>
</blockquote>
<p>In other words:</p>
<blockquote>
<p>If <span class="math-container">$\mathcal{O}$</span> i... | Brian M. Scott | 12,042 | <p>As other answers have pointed out, there are easy counterexamples. What <strong>is</strong> true is that if $\mathscr{U}$ is an open cover of a metric space $X$, then $\mathscr{U}$ has an irreducible open <em>refinement</em>: that is, there is an open cover $\mathscr{R}$ of $X$ such that </p>
<ol>
<li>for each $R\i... |
1,349 | <p>In this question here the OP asks for hints for a problem rather than a full proof.</p>
<p><a href="https://math.stackexchange.com/questions/14477">Proof of subfactorial formula $!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$</a></p>
<p>Now, while I would like to respect that request, I also feel that questi... | J. M. ain't a mathematician | 498 | <p>Some more testing:</p>
<p>1.</p>
<blockquote class="spoiler">
<p> <em>italicized text goes here</em></p>
</blockquote>
<p>2.</p>
<blockquote class="spoiler">
<p> <strong>bold text goes here</strong></p>
</blockquote>
<p>3.</p>
<blockquote class="spoiler">
<p> Let's try $\LaTeX$ <em>italicized text</em> <... |
1,349 | <p>In this question here the OP asks for hints for a problem rather than a full proof.</p>
<p><a href="https://math.stackexchange.com/questions/14477">Proof of subfactorial formula $!n = n!- \sum_{i=1}^{n} {{n} \choose {i}} \quad!(n-i)$</a></p>
<p>Now, while I would like to respect that request, I also feel that questi... | Mariano Suárez-Álvarez | 274 | <p>I would very much prefer that we did <em>not</em> use this feature. It is simply too distracting. </p>
<p>I am really surprised that this has been deemed accetable UI!</p>
|
3,417,227 | <p><strong>Problem</strong>:</p>
<p>Let <span class="math-container">$f : \Bbb R \to \Bbb R$</span> be a differentiable function such as <span class="math-container">$f(0) = 0$</span>, compute </p>
<p><span class="math-container">$$\lim_{r\to 0^{+}} \iint_{x^2 + y^2 \leq r^2} {3 \over 2\pi r^3} f(\sqrt{x^2+y^2}) dx$$... | ling | 670,949 | <p>It is easy to get an answer by the L’Hopital’s rule.
<span class="math-container">\begin{align}
&\quad \lim_{r\to0^+}\iint_{x^2+y^2\leq r^2} \frac{3}{2\pi r^3} f\left(\sqrt{x^2+y^2}\right)\, dxdy \\&=\lim_{r\to0^+}\int_0^r\int_{\partial B(0,\rho)} \frac{3}{2\pi r^3} f\left(\rho\right)\,dS d\rho\\&= \lim... |
540,135 | <p>$\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?</p>
<p>I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number... </p>
| Carsten S | 90,962 | <p>What you still want to show is that $lcm(lcm(a,b),c)|lcm(a,b,c)$. So what you want to show is that the least, or actually any, common multiple of $a$, $b$, and $c$ is a multiple of the least common multiple of $lcm(a,b)$ and $c$. This follows from the fact that a common multiple of two numbers is a multiple of their... |
74,108 | <p>Background: I was trying to convert a MATLAB code (fluid simulation, SPH method) into a <em>Mathematica</em> one, but the speed difference is huge.</p>
<p>MATLAB code:</p>
<pre class="lang-matlab prettyprint-override"><code>function s = initializeDensity2(s)
nTotal = s.params.nTotal; %# particles
h = s.params.... | Henrik Schumacher | 38,178 | <p>Here a slightly improved version of <a href="https://mathematica.stackexchange.com/users/1871">xzczd</a>'s code (I call the function <code>cinitializeDensity</code> in the following) that does not require computing the <code>DistanceMatrix</code> beforehand. Moreover, I tried to suppress some type casts within the <... |
3,264,333 | <p>I am working on my scholarship exam practice and not sure how to begin. Please assume math knowledge at high school or pre-university level.</p>
<blockquote>
<p>Let <span class="math-container">$a$</span> be a real constant. If the constant term of <span class="math-container">$(x^3 + \frac{a}{x^2})^5$</span> is ... | user10354138 | 592,552 | <p><strong>Hint</strong>: binomial expand <span class="math-container">$(x^3+ax^{-2})^5$</span>.</p>
|
3,264,333 | <p>I am working on my scholarship exam practice and not sure how to begin. Please assume math knowledge at high school or pre-university level.</p>
<blockquote>
<p>Let <span class="math-container">$a$</span> be a real constant. If the constant term of <span class="math-container">$(x^3 + \frac{a}{x^2})^5$</span> is ... | lab bhattacharjee | 33,337 | <p>Hint:</p>
<p>The general term <span class="math-container">$T_{r+1}$</span> is <span class="math-container">$$\binom5r(x^3)^{5-r}\left(\dfrac a{x^2}\right)^r=\binom5ra^rx^{3\cdot5-3r-2r}$$</span></p>
<p>For the constant term, the exponent of <span class="math-container">$x$</span> will be <span class="math-contai... |
19,148 | <p>I always find the strong law of large numbers hard to motivate to students, especially non-mathematicians. The weak law (giving convergence in probability) is so much easier to prove; why is it worth so much trouble to upgrade the conclusion to almost sure convergence?</p>
<p>I think it comes down to not having a ... | Steve Huntsman | 1,847 | <p>You might use the <a href="http://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem" rel="nofollow">three-series theorem</a> to elaborate on a.s. convergence. This approach would also have the advantage of <a href="http://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem#Other_applications" rel="nofo... |
2,227,709 | <p>Find the solution of the $x^2+2x+3 \equiv0\mod{198}$</p>
<p>i have no idea for this problem i have small hint to we going consider $x^2+2x+3 \equiv0\mod{12}$</p>
| P Vanchinathan | 28,915 | <p>Substitute $y=x+1$. Now the equations is transformed to $y^2+2\equiv 0 \,mod\,198$. As $196=14^2$,we have $x=y-1=13$ is a soution.</p>
|
2,227,709 | <p>Find the solution of the $x^2+2x+3 \equiv0\mod{198}$</p>
<p>i have no idea for this problem i have small hint to we going consider $x^2+2x+3 \equiv0\mod{12}$</p>
| lioness99a | 401,264 | <p>We can rewrite the equation as \begin{align}x^2+2x+3&\equiv0\mod198\\
(x+1)^2-1+3&\equiv0\mod 198\\
(x+1)^2&\equiv-2\mod198\\
(x+1)^2&\equiv196\mod198\end{align}</p>
<p>We let $y=x+1$ and we now need to solve \begin{align}y^2&\equiv-2\mod 198\\
&\equiv196\mod198\end{align}</p>
<p>We can see... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | OwlToday | 443,828 | <p>One could also consider a geometric perspective. The tangent line to the circle <span class="math-container">$x^2+y^2=1$</span> at the point <span class="math-container">$(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$</span> is the line <span class="math-container">$x+y = \sqrt{2}$</span>.</p>
<p><a href="https://i.stack... |
4,149,355 | <p>Are there any stochastic processes <span class="math-container">$(X_t)_{t \in \mathbb{R}^d}$</span> such that</p>
<ol>
<li>almost surely paths are continuous but nowhere differentiable and</li>
<li>sampling of <span class="math-container">$n$</span> points <span class="math-container">$X_{t_n}$</span> on a path can ... | Community | -1 | <p>Let <span class="math-container">$x+y>\sqrt{2}$</span></p>
<p><span class="math-container">$$ x^2+y^2+2xy>2 $$</span></p>
<p><span class="math-container">$$ 2xy>1 $$</span></p>
<p>We know that <span class="math-container">$(x-y)^2<0$</span> is not true.</p>
<p><span class="math-container">$$ x^2-2xy+y^2 ... |
2,072,473 | <p>I managed to prove the statement:</p>
<blockquote>
<p>If $f: A\to B$ and $g: B\to C$ are surjective, then $g\circ f$ is surjective.</p>
</blockquote>
<p>But now I require a counterexample to the converse of this statement. I am not sure how to formulate the counterexample. Similarly I need a counterexample of th... | Dominik | 259,493 | <p><strong>Hint:</strong> A function $h: A \to \{0\}$ is surjective for any nonempty set $A$.</p>
|
2,072,473 | <p>I managed to prove the statement:</p>
<blockquote>
<p>If $f: A\to B$ and $g: B\to C$ are surjective, then $g\circ f$ is surjective.</p>
</blockquote>
<p>But now I require a counterexample to the converse of this statement. I am not sure how to formulate the counterexample. Similarly I need a counterexample of th... | MPW | 113,214 | <p><strong>Hints:</strong> You may find it useful to use the characterizations
$$f:A\to B \textrm{ surjective }\iff f\circ f^{-1}=\operatorname{Id}_B$$
and
$$f:A\to B \textrm{ injective }\iff f^{-1}\circ f=\operatorname{Id}_A$$</p>
|
2,119,761 | <p>Asuume that $f:\mathbb{R}\rightarrow \mathbb{R}$ continuous function and $g:\mathbb{R}\rightarrow \mathbb{R}$ uniformly continuous function and $g$ bounded.</p>
<p>I have to prove that $f\circ g$ is uniformly continuous function.
I tried the following:
$f$ continuous function so $\forall \epsilon>0 ~\exists~ \de... | Robert Israel | 8,508 | <p>Since $g$ is bounded, its range is contained in a finite closed interval. $f$ is uniformly continuous on that interval.</p>
|
2,560,556 | <p>Let $X,Y,Z$ be topological spaces.
Let $p:X\rightarrow Y$ be a continuous surjection. Let $f:Y\rightarrow Z$ be continuous if and only if $f\circ p:X\rightarrow Z$ is continuous.</p>
<p>I want to prove that this makes $p$ a quotient map. </p>
<p>My thoughts:</p>
<p>Since $p$ is a continuous surjection, all I need... | Alex Provost | 59,556 | <p>I assume you want the property to hold for <em>all</em> spaces $Z$. In this case, pick $Z = Y$ as sets, endowed with the quotient topology for $p$. Let $f:Y \to Z$ be the identity map. We will show that $f$ is a homeomorphism, and hence that $Y$ also has the quotient topology.</p>
<p>First, $\tilde p =f \circ p$ is... |
3,432,911 | <p>My argument is as follows:</p>
<p>Let <span class="math-container">$R$</span> be a commutative ring with unity, <span class="math-container">$I$</span> an ideal of <span class="math-container">$R$</span>.
If <span class="math-container">$(R/I)^n\cong (R/I)^m$</span> as <span class="math-container">$R$</span>-module... | Martin R | 42,969 | <p><span class="math-container">$z=0$</span> is mapped to <span class="math-container">$f(0) = \infty$</span>. For <span class="math-container">$z \ne 0$</span> we have with <span class="math-container">$w = \frac 1z$</span>
<span class="math-container">$$
|z-1| = 1 \iff |\frac 1w - 1| = 1 \iff |w-1|^2 = |w|^2 \\
\iff ... |
2,924,380 | <p><span class="math-container">$\sum_{k=0}^{n}{k\binom{n}{k}}=n2^{n-1}$</span></p>
<p><span class="math-container">$n2^{n-1} = \frac{n}{2}2^{n} = \frac{n}{2}(1+1)^n = \frac{n}{2}\sum_{k=0}^{n}{\binom{n}{k}}$</span></p>
<p>That's all I got so far, I don't know how to proceed</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. One has
<span class="math-container">$$
k{\binom{n}{k}}=n{\binom{n-1}{k-1}},\quad n>0,k>0.
$$</span></p>
|
3,424,189 | <p>I'm trying to calculate the integral <span class="math-container">$$\int_0^1 \frac{\sin\Big(a \cdot \ln(x)\Big)\cdot \sin \Big(b \cdot \ln(x)\Big)}{\ln(x)} dx, $$</span>
but am stuck. I tried using Simpsons' rules and got here:
<span class="math-container">$$\int_0^1 \frac{\cos\Big((a+b) \cdot \ln(x)\Big) - \cos \Bi... | ComplexYetTrivial | 570,419 | <p>For <span class="math-container">$c \in \mathbb{R}$</span> we have
<span class="math-container">\begin{align}
\int \limits_0^\infty \frac{1 - \cos(c t)}{t} \, \mathrm{e}^{-t} \, \mathrm{d} t &= \int \limits_0^\infty \int \limits_0^c \sin(u t) \, \mathrm{d} u \, \mathrm{e}^{-t} \, \mathrm{d} t = \int \limits_0^c... |
3,795,932 | <p><span class="math-container">$\mathbb {Z}G $</span> is not Artinian where <span class="math-container">$ G$</span> is a finite group.</p>
<p>I know that <span class="math-container">$\mathbb {Z} $</span> is not Artinian but <span class="math-container">$\mathbb {Z} $</span> is not an ideal of the group ring. So H... | Kavi Rama Murthy | 142,385 | <p>Both are correct. <span class="math-container">$\ln |2x-2|=\ln 2+\ln |x-1|$</span> and you can absorb <span class="math-container">$(\ln 2) /2$</span> into the constant.</p>
|
2,907,771 | <p>I tried coming up with a proof of compactness of $[0,1]$ in $\mathbb{R}$ and thought of the following method. please let me know if it is correct or how it could be made more correct.</p>
<p>For any open cover of $[0,1]$ there exists an $\mathbb{\epsilon}$ such that $[0,\mathbb{\epsilon})$ is contained in one open... | Christian Blatter | 1,303 | <p>A proof along your lines is possible, but you have to be more greedy when choosing the $U_k$. Your algorithm could stop short long before the right end is reached, and you would have to restart with no guarantee of success.</p>
<p>We are given a family ${\cal U}$ of open sets $U\subset{\mathbb R}$ that together cov... |
199,695 | <p>I believe the answer is $\frac12(n-1)^2$, but I couldn't confirm by googling, and I'm not confident in my ability to derive the formula myself.</p>
| mdp | 25,159 | <p>A clique has an edge for each pair of vertices, so there is one edge for each choice of two vertices from the $n$. So the number of edges is:</p>
<p>$$\binom{n}{2}=\frac{n!}{2!\times(n-2)!}=\frac{1}{2}n(n-1)$$</p>
<p><strong>Edit:</strong> Inspired by Belgi, I'll give a third way of counting this! Each vertex is c... |
3,738,789 | <p>I know if I stick two pins on a paper, and trace a taut loop around them, I get an ellipse. With one pin, I get a circle. Question is, are there names for shapes I get if I trace a taut loop around 3, 4, 5, ..., k pins, assuming the pins are not collinear, and the polygon formed by joining them is convex i.e. every ... | rschwieb | 29,335 | <p>Someone will tell me if there is an exotic case I'm not foreseeing, but I think usually it will be a smooth union of "ellipse arcs."</p>
<p>Assuming the string is large enough to circumscribe the entire collection of pins, it could still be that it is taught around the outermost pins, and that would make a... |
2,782,726 | <p>I'm reading through some lecture notes to prepare myself for analysis next semester and stumbled along the following exercises: </p>
<p>a) Prove that $\lim_{x\to0} f(x)=b$ is equivalent to the statement $\lim_{x\to0} f(x^3)=b$.</p>
<p>b) Give an example of a map where $\lim_{x\to0} f(x^2)$ exists, but $\lim_{x\to0... | cj1996 | 316,862 | <p>For a) You can use the following in an epsilon delta proof:</p>
<p>for any <span class="math-container">$x \in \mathbb{R}$</span>, <span class="math-container">$p(x) \iff$</span> for any <span class="math-container">$x^3 \in \mathbb{R}$</span>, <span class="math-container">$p(x^3)$</span></p>
|
2,543,834 | <p>Ok, so in my differential equations class we've been doing problems which more or less amount to solving equations of the form:</p>
<p><span class="math-container">$$\frac{dY}{dt} = AY$$</span></p>
<p>Where <span class="math-container">$A$</span> is just some <span class="math-container">$2\times2$</span> linear tra... | Community | -1 | <p>The basic idea here is to find a new set of variables, $X(t)$ and $Y(t)$, related to the original variables $x(t)$ and $y(t)$ by a linear transformation, so that the differential equations for $X(t)$ and $Y(t)$ are <em>decoupled</em>:
$$
\dot X = \lambda_1 X\, ,\qquad \dot Y=\lambda_2 Y\, . \tag{1}
$$
In other words... |
287,859 | <p>Prove that $\lim\limits_{x\rightarrow+\infty}\frac{x^k}{a^x} = 0\ (a>1,k>0)$.</p>
<p>P.S. This problem comes from my analysis book. You may use the definition of limits or invoke the Heine theorem for help. <em>It means the proof should only use some basic properties and definition of limits rather than more ... | Mhenni Benghorbal | 35,472 | <p>Applying the <a href="https://math.stackexchange.com/questions/214001/what-are-common-methods-techniques-can-be-used-to-prove-that-limit-of-an-infinit/214014#214014">result</a></p>
<p><strong>Theorem</strong>: If ${a_n}$ be a sequence such that $\lim_{n\to \infty} \frac{a_{n+1}}{a_n}= a\,,$ then</p>
<p>1) if $|a|&... |
9,930 | <p>One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?</p>
<p>To be precise about what I mean, a diagram is a directed graph $D$ whose vertices are ... | Harrison Brown | 382 | <p>This was originally going to be a rather different post, but I realized that my argument could perhaps be adapted into an (ineffective) algorithm, if it's possible to patch up the hole. At the very least, I should say some things which might be obvious but wasn't to me until I saw it, and so might not be obvious to ... |
114,438 | <p>I am interested in knowing whether there is a definition for the symbol of a PDO which is NOT linear.
In Wikipedia and in the book I am reading (An Introduction to Partial Differential Equations by Renardy-Rogers) I only found the definition for linear PDOs.</p>
<p>Here is the Wikipedia link:</p>
<p><a href="http:... | youler | 25,895 | <p>The symbol of a nonlinear differential operator is defined as the symbol of its linearization.</p>
|
591,938 | <p>let $m,n\in N^{+}$, if such</p>
<p>$$\sqrt{37}+\sqrt{47}<\dfrac{n}{m}<\sqrt{41}+\sqrt{43}$$
Find the $m$ minimum the value </p>
<p>My try: since
$$(\sqrt{37}+\sqrt{47})m<n<(\sqrt{43}+\sqrt{41})m$$
then
$$\dfrac{10m}{\sqrt{47}-\sqrt{37}}<n<\dfrac{2m}{\sqrt{43}-\sqrt{41}}$$</p>
<p>(maybe this pro... | Community | -1 | <p>This provides only a partial answer. What you have gives us
$$\dfrac{\sqrt{43}-\sqrt{41}}2n < m < \dfrac{\sqrt{47}-\sqrt{37}}{10}n$$
Hence, a sufficient condition is that if we ensure that $\dfrac{\sqrt{47}-\sqrt{37}}{10}n -\dfrac{\sqrt{43}-\sqrt{41}}2n > 1$, there is definitely an integer $m$. Hence, $n \g... |
591,938 | <p>let $m,n\in N^{+}$, if such</p>
<p>$$\sqrt{37}+\sqrt{47}<\dfrac{n}{m}<\sqrt{41}+\sqrt{43}$$
Find the $m$ minimum the value </p>
<p>My try: since
$$(\sqrt{37}+\sqrt{47})m<n<(\sqrt{43}+\sqrt{41})m$$
then
$$\dfrac{10m}{\sqrt{47}-\sqrt{37}}<n<\dfrac{2m}{\sqrt{43}-\sqrt{41}}$$</p>
<p>(maybe this pro... | N. S. | 9,176 | <p>Since</p>
<p>$$\sqrt{37}+\sqrt{47}=12.9384....$$
$$\sqrt{41}+\sqrt{43}=12.9605....$$</p>
<p>Write</p>
<p>$$\frac{n}{m}=13-\frac{k}{m}$$ Then</p>
<p>$$13-0.0616< 13-\frac{k}{m}< 13-0.039$$
Hence
$$.0616 > \frac{k}{m} \geq \frac{1}{m}$$</p>
<p>This proves that</p>
<p>$$m \geq \frac{1}{0.0616}=16.23$$</p... |
1,336,869 | <p>Does every mod p have at least one element with a non-identical inverse?</p>
<p>I very much suspect this is true, but how can I prove it? For example, in mod 5, some elements have inverses that are not themselves ${2,3}$ and some have themselves as inverses ${1,4}$. Am I assured that every prime $p\gt 2$ will have ... | Hagen von Eitzen | 39,174 | <p>$x$ is its own inverse $\bmod p$ iff $x^2\equiv 1\pmod p$ and this has only the two solutions $\pm1\pmod p$ (put differently, $p\mid x^2-1=(x-1)(x+1)$ implies $p\mid x-1$ or $p\mid x+1$).</p>
|
2,886,675 | <p>I suspect the following is exactly true ( for positive $\alpha$ )</p>
<p>\begin{equation}
\sum_{n=1}^\infty e^{- \alpha n^2 }= \frac{1}{2} \sqrt { \frac{ \pi}{ \alpha} }
\end{equation}</p>
<p>If the above is exactly true, then I would like to know a proof of it.
I accept showing a particular limit is true, may b... | Community | -1 | <p>Consider the Riemannian sum $$\lim_{m\to\infty}\frac 1m\sum_{n=0}^\infty e^{-(n/m)^2}=\int_0^\infty e^{-x^2}dx=\frac{\sqrt\pi}2.$$</p>
<p>Then with the subsitution $m^2\alpha=1$,</p>
<p>$$\lim_{\alpha\to0}\sqrt\alpha\sum_{n=0}^\infty e^{-\alpha n^2}=\frac{\sqrt\pi}2.$$
(Note that the starting index $n=0$ or $n=1$ ... |
417,064 | <p>Let T be a totally ordered set that is <strong>finite</strong>. Does it follow that minimum and maximum of T exist?
Since T is finite, I believe there exists a minimal of T. From that it maybe able to be shown that the minimal is the minimum but not quite sure whether it is the right approach. </p>
| Caleb Stanford | 68,107 | <p><strong>Claim:</strong> A totally ordered set with at least one <em>minimal</em> element has a <em>minimum</em> element.</p>
<p><strong>Proof sketch:</strong> Let $b$ be the minimal element. If $b$ were not in fact the minimum, then by definition of minimum, "$b \le a$ for all $a$" would be false. Pick some $a$ f... |
246,114 | <p>A Latin Square is a square of size <strong>n × n</strong> containing numbers <strong>1</strong> to <strong>n</strong> inclusive. Each number occurs once in each row and column.</p>
<p>An example of a 3 × 3 Latin Square is:</p>
<p><span class="math-container">$$
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
3 &... | Roman | 26,598 | <p>Adding lines one-by-one and continuing only if the newly added line does not give any column duplications. This highly unoptimized code takes about a minute for <span class="math-container">$n=5$</span> (thanks @chyanog for speedup!):</p>
<pre><code>addline[lines_] :=
Select[Append[lines, #] & /@ Permutations... |
552,395 | <p>I have $f(x)$=$(2x,e^x)$
what does this notation mean? Notation: $Df(\frac{∂}{∂x})$</p>
<p>Certainly $Df(x)$=$(2,e^x)$
but how can I replace $x$ with $\frac{∂}{∂x}$?</p>
<p>Particularly, how can I make sense of $e^{\frac{∂}{∂x}}$</p>
| Han de Bruijn | 96,057 | <p>The keyword is <I>Operational Calculus</I> (let Google be your friend). <BR>
The following reference is an introductory exposure of the subject:<P> <A HREF="http://www.alternatievewiskunde.nl/jaar2004/uitboek.pdf" rel="nofollow">Re: Why exp(-st) in the Laplace Transform?</A><P>
About the second part of your question... |
121,897 | <p>I want to check if a user input the function with all the specified variables or not. For that I choose the replace variables with some values and check for if the result is a number or not via a doloop. I am thinking there might be more elegant way of doing it such as <a href="http://reference.wolfram.com/language... | Mr.Wizard | 121 | <h3>Update</h3>
<p>Following your updated question I recommend filtering <a href="http://reference.wolfram.com/language/ref/Level.html" rel="nofollow noreferrer"><code>Level</code></a> output with <a href="http://reference.wolfram.com/language/ref/Variables.html" rel="nofollow noreferrer"><code>Variables</code></a> as... |
121,897 | <p>I want to check if a user input the function with all the specified variables or not. For that I choose the replace variables with some values and check for if the result is a number or not via a doloop. I am thinking there might be more elegant way of doing it such as <a href="http://reference.wolfram.com/language... | Michael E2 | 4,999 | <p>Since it is user input and guessing from the loop that the OP is calculating something numerical, I think <strong><em>it is better to check that the user has entered a numerical function</em></strong>, not just an expression with only legal variables. I asked about constant functions and other functions such as <cod... |
3,299,492 | <p>Is there any nice characterization of the class of polynomials can be written with the following formula for some <span class="math-container">$c_i , d_i \in \mathbb{N}$</span>? Alternatively, where can I read more about these? do they have a name?
<span class="math-container">$$c_1 + \left( c_2 + \left( \dots (c_k ... | Chinnapparaj R | 378,881 | <p>Radius of convergence <span class="math-container">$r$</span> means <span class="math-container">$$r=\sup_{x}\left\{|x|: \sum n^n x^n< \infty\right\}$$</span> That is, <span class="math-container">$r$</span> is the supremum of <span class="math-container">$|x|$</span> over all number <span class="math-container... |
3,426,907 | <p>I'm not a mathematician, but a programmer who loves solving math puzzles, so please forgive me if I don't use the correct terms.</p>
<p>Imagine there are 100 lotteries with 100 tickets each. The lotteries have no connection at each other and all lotteries have exactly 1 price. I buy 1 ticket from each lottery so in... | Sumanta | 591,889 | <p>Option <span class="math-container">$2.$</span> is not correct. Consider the set <span class="math-container">$A=\big\{(x,\sin\big(\frac{\pi}{x}\big):0<x\leq 1\big\}$</span> in <span class="math-container">$X=\Bbb R^2$</span>. Then <span class="math-container">$\overline A=A\cup \big(\{0\}\times[-1,1]\big)$</span... |
66,314 | <p>This is very similar to my earlier question <a href="https://mathematica.stackexchange.com/questions/60069/one-to-many-lists-merge">One to Many Lists Merge</a> but somehow different. I have two lists, first column in each list represents its key. I want to merge these two lists. The only problem is that these two l... | Caitlin J. Ramsey | 22,323 | <p>This should get you close. (Note, If you are not familiar with the new Association functionality in Mathematica 10.1, I will start here by building an association from each list, and then I work with them as associations until the final step.)</p>
<pre><code>list1 = {{1, a, aa}, {2, b, bb}, {3, c, cc}, {4, d, dd},... |
3,259,658 | <p>Stokes' Theorem states that, "For a smooth oriented region <span class="math-container">$V$</span> <span class="math-container">$\in$</span> <span class="math-container">$R^3$</span> and a smooth vector field defined on <span class="math-container">$V \cup \partial V$</span>, where <span class="math-container">$\par... | Adam Latosiński | 653,715 | <p>No. <span class="math-container">$\partial V$</span> is a curve located somewhere in <span class="math-container">$\mathbb R^3$</span>, it's not projected on any plane.</p>
<p>In case when <span class="math-container">$V$</span> is a unit sphere it doesn't have a boundary at all and <span class="math-container">$\p... |
3,259,658 | <p>Stokes' Theorem states that, "For a smooth oriented region <span class="math-container">$V$</span> <span class="math-container">$\in$</span> <span class="math-container">$R^3$</span> and a smooth vector field defined on <span class="math-container">$V \cup \partial V$</span>, where <span class="math-container">$\par... | alan23273850 | 397,319 | <p>See the <a href="https://mathinsight.org/stokes_theorem_idea" rel="nofollow noreferrer">Math Insight</a> website. In the website there is a toy applet you can play with.</p>
<p>The curve need not lie on a plane, and it is NOT a projection on a plane. It can only be the boundary of an "open" surface, so your should ... |
1,446,168 | <p>Let $x$ and $y$ be two random variables. </p>
<p>Suppose $m$ is a random variable that is independent of $x$ and has the following distribution:</p>
<p>$$\text{Pr}(m = 1|x) = 0.5,$$ $$\text{Pr}(m = -1|x) = 0.5.$$</p>
<p>Let $y$ be given by: $$y= \left\{ \begin{array}{lcc}
0 & \text{if } x\geq0... | Anthony Peter | 58,540 | <p>Again, this is false. Take $$ C = \prod_{i=1}^{n} [0,1]_i \subset \mathbb{R}^n.$$ Then, $C$ is a perfect set in $\mathbb{R}^n$, so every point of $C$ is a limit point. Ergo, there does not exist a point such that it can be surrounded by a neighborhood not containing another point of $C$. In general, any compact conn... |
1,315,450 | <blockquote>
<p>Prove that for all $n\in\mathbb{N}$ and $x>0$,
$$2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$$</p>
</blockquote>
<p>The last class was about Taylor polynomial of functions, so I thought this might give me a solutions, but looking at the derivatives the only think I ... | Bernard | 202,857 | <p>This is the <em>power mean inequality</em>: the p-power mean of $a$ and $b$ is
$$m_p(a,b)=\biggl(\frac{a^p+b^p}2\biggr)^{\!\tfrac1p}$$
and if $p\le q$, $m_p(a,b)\le m_q(a,b)$. Of course, $m_1$ is the usual arithmetic mean, and $m_{-1}$ is the harmonic mean.</p>
<p>Here you can rewrite the inequality as:
$$\frac{x+... |
202,379 | <p>Suppose for some constants $\alpha,\beta,\gamma$ that we're given the following ODE: $$\alpha y''+\beta xy'+\gamma y=0.$$ Now, I know how to find the general solution for $y(x)$ if any of $\alpha,\beta,\gamma$ should turn out to be $0$, but I've just ended up with the ODE $$2y''+xy'+y=0.$$ Can anybody give me the fi... | Mhenni Benghorbal | 35,472 | <p>Assume your solution </p>
<p>$$ y(x)=\sum_{k=0}^{\infty} a_k x^{k+\alpha} \,,$$</p>
<p>and plug into the differential equation and try to find a recurrence relation in $a_k$. Off course, you need to determine $\alpha$ as a first step. The well known power series method for second order ode is <a href="http://en.wi... |
269,398 | <p>I have a python code for data analysis, that uses the <a href="https://matplotlib.org/3.5.0/tutorials/colors/colormaps.html" rel="nofollow noreferrer">"seismic" color scale</a> for 2D density plots. However, I also need to do some other plots with Mathematica (because of packages etc), for which I would li... | Syed | 81,355 | <p>After some trial and error:</p>
<pre><code>seismic[x_] :=
Blend[{Black, Darker@Blue, Blue, White, Red, Darker@Red,
Darker@Darker@Red}, x]
LinearGradientImage[Function[x, seismic[x]], {300, 30}]
</code></pre>
<p>As an example:</p>
<pre><code>Plot3D[2 Sin[x + Cos[y]], {x, -5, 5}, {y, -5, 5}
, ColorFunction -&g... |
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