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 |
|---|---|---|---|---|
2,063,742 | <p>Hi I'm just curious as I prepare for a final, I cooked up this problem and wanted to know the answer. Suppose all $f_n>0$ and $f_n \leq g$ for all n and x with g integrable and $f_n \to f$. Then does that imply $\int f \leq \int g$?</p>
<p>I know by dominated convergence that $\int f_n \to \int f$ and $\int f_n ... | carmichael561 | 314,708 | <p>Yes: $f_n\leq g$ for all $n$ and $f_n\to f$ implies that $f\leq g$, hence $\int f\leq \int g$.</p>
|
482,061 | <p>Given the continuous Probability density function $f(x)=\begin{cases} 2x-4, & 2\le x\le3 \\ 0 ,& \text{else}\end{cases}$</p>
<p>Find the cumulative distribution function $F(x)$.</p>
<p>The formula is $F(x)=\int _{ -\infty }^{ x }{ f(x) } $</p>
<hr>
<p><h3>My Solution</h3> <br>
The first case is when $2... | gammatester | 61,216 | <p>In the ranges $x < 2$ and $x>3$ the PDF is zero and contibutes nothing to the integral, and therefore the CDF is constant in these ranges. <strong>You</strong> have already done most of the work and the CDF is
$$F(x)=\begin{cases}
0, & x < 2 \\
x^2-4x+4, & 2 \le x\le3 \\
1 ,& x > 3... |
2,263,622 | <p>It is a known fact that (220, 284) is the smallest pair of amicable numbers.
That proves that I don't understand at least one part of finding amicable pairs...</p>
<p>Please explain to me where I fall down:</p>
<p>Suppose I have the numbers 2 and 3.
The proper divisors of 2 is 1.
The proper divisors of 3 is 1 t... | bof | 111,012 | <p>It looks like you don't understand the definition of <a href="https://en.wikipedia.org/wiki/Amicable_numbers" rel="nofollow noreferrer">amicable numbers</a>.</p>
<p>$220$ and $284$ are amicable numbers because the sum of the proper divisors of $220$ is
$$1+2+4+5+10+11+20+22+44+55+110=284$$
while the sum of the prop... |
911,370 | <p>I need to prove the following:</p>
<blockquote>
<p>A set $X$ is infinite if and only if it is equipotent to a proper subset of itself</p>
</blockquote>
<p>Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or equivalently, there is no bijection $\mathbb{N}_n \rightarrow X$ for $n\in \math... | Andrej Bauer | 30,711 | <p>I am going to replace some definitions by obviously equivalent ones. The benefit will be that the proofs are easier to understand. I am writing down very detailed proofs that you would not normally find in a college-level math textbook.</p>
<p><strong>Definition:</strong> For $n \in \mathbb{N}$ define $[n] = \{k \i... |
209,169 | <p>The following problem is a 1-D heat transfer conduction problem:</p>
<p><a href="https://i.stack.imgur.com/qaG4Q.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qaG4Q.png" alt="enter image description here"></a></p>
<p>where,</p>
<p><a href="https://i.stack.imgur.com/hHUo1.png" rel="nofollow no... | Tim Laska | 61,809 | <p>Since the thermal conductivity is constant, the heat flow and temperature distributions should be symmetric about <span class="math-container">$x=0.5$</span>. You should be able to convert the source term to a time dependent Neumann condition. Also, if your time scales and length scales are valid, then heat will o... |
1,533,326 | <p>I need some help proving the following statement:</p>
<p>[EDIT]: forgot to mention that:</p>
<p><span class="math-container">$a_n \ge 0$</span> , <span class="math-container">$L \ge 0$</span>.</p>
<p>If the limit of a sequence <span class="math-container">$\lim_{ \ n \to \infty} a_n=L$</span>,
then, for any <span... | Yes | 155,328 | <p>Let $a_{n} \geq 0$ for all $n \geq 1$, so that, if $a_{n} \to L$, then $L \geq 0$. This allows $a_{n}^{1/k}$ to be meaningful for all $n,k \geq 1$.</p>
<p>Let $k \geq 1$. If $L=0$, then the statement is obvious; for, we have $a_{n} < \varepsilon$ iff $a_{n}^{1/k} < \varepsilon^{1/k}$. Suppose $L > 0$. Then... |
1,718,623 | <p>I am currently writing a NURBS ray tracer. What I do is convert the NURBS into rational Bézier patches and then perform the intersection test using Newton's method. To do this fast (the ray tracer should be later implemented using GPGPU programming and should run in real time) a point on the surface and first deriva... | Nakamp | 326,868 | <p>So after some time I got both of the algorithms working. In scenes with surfaces of lower degree, algorithm by Mann and DeRose seems to be faster, in other scenes (about 80% of scenes I test my ray tracer with) the algorithm by Sederberg is faster and requires a constant amount of memory so it is more suitable for G... |
4,351,610 | <p>I'm trying to solve this problem, I took the basis <span class="math-container">$B=\langle(2,7)^T, (1,3)^T\rangle$</span> and then build the matrix of <span class="math-container">$f$</span> application in <span class="math-container">$B$</span>:
<span class="math-container">$$
M_b(f)=\left(\begin{array}{cc}
7&4... | Lukas | 844,079 | <p>If <span class="math-container">$f$</span> is a linear map which I am assuming is the case then you can (as you tried) express the linear map in terms of a matrix. The "mistake" you did is that you wrote the images of the basis <span class="math-container">$B$</span> as columns so your matrix is the matrix... |
1,735,087 | <p>I have the following question in my notes. </p>
<blockquote>
<p>Let $A \in H^*$ and let $F=A^{-1}({0})$ $F$ is a closed linear
subspace. Show that for any choice of $u,w \in H$ with $Au$ non zero
the vector is $w-\frac{Aw}{Au} u$ is an element of F.</p>
</blockquote>
<p>The solution is as below. </p>
<bloc... | Tryss | 216,059 | <p>First, remember that $F = A^{-1}(\{0\}) = \{ v \in H | A(v) = 0 \}$ </p>
<p>To show that $v = w- \frac{Aw}{Au}u \in F$, it suffice to show that $A(v) = 0$.</p>
<p>By linearity of $A$, you have
$$A(v) = A \left( w- \frac{A(w)}{A(u)}u \right) = A(w) - \frac{A(w)}{A(u)}A(u) = A(w) - A(w) = 0$$</p>
<p>Hence the resu... |
2,669,617 | <p>The bilinear axiom is:</p>
<pre><code> <cu + dv,w> = c<u,w> + d<v,w>
<u,cv + dw> = c<u,v> + d<u,w>
</code></pre>
<p>Where c and d are scalars and u, v, and w are vectors.</p>
<p>Can this be extended to something like</p>
<pre><code> <cu + dv, ew + fx> = ?
</code></pre>
| OnoL | 65,018 | <p>Define $a_n=\frac{(n!)^2}{(2n)!}$. Then it is easy to show that $a_{n+1}<a_n$ and $a_n\in (0,1)$. So $\lim_{n\to\infty}a_n$ exists. Then notice that
$$a_{n+1}=a_n\cdot \frac{n}{(2n+1)(2n+2)}.$$
Take limit $n\to\infty$ on both sides, we get $\lim_{n\to\infty}a_n=0$.</p>
|
882,485 | <p>imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$.
Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I have a few questions about this:</p>
<p>1.) Is it true that this operator cannot have a residual spectrum due to its s... | Disintegrating By Parts | 112,478 | <p>The classical operator
$$
L = -\frac{d^{2}}{dx^{2}}+V,\;\;\; a \le x \le b,
$$
is different if $V$ is very singular. If $V \in L^{1}[a,b]$, then things are nice because there are 2 linearly-independet classical solutions of $Lf = \lambda f$ for every $\lambda$. That is, such solutions are continuou... |
882,485 | <p>imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$.
Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I have a few questions about this:</p>
<p>1.) Is it true that this operator cannot have a residual spectrum due to its s... | Disintegrating By Parts | 112,478 | <p>This is a continuation of what I posted earlier. What follows are two examples of how the theory is applied to the trigonometric functions where $V=0$. The equation is in the limit point case on $[0,\infty)$ because $e^{i\sqrt{\lambda}x}\in L^{2}[0,\infty)$ while $e^{-i\sqrt{\lambda}x}$ is not, where $\sqrt{\lambda}... |
167,395 | <p>Consider the following groups: $(\mathbb{Z}_4,+)$, $(U_5,.)$, $(U_8,.)$ and the set of symmetries for a rhombus if I am not mistaken the first and last are equivalent. What other justifiable equivalencies and nonequivalencies are there and what does it mean rigorously to be in the same group in general?</p>
| pritam | 33,736 | <p>This is called Linear Programming problem, one famous method is Simplex Algorithm. For reference see here : <a href="http://en.wikipedia.org/wiki/Linear_programming" rel="nofollow">http://en.wikipedia.org/wiki/Linear_programming</a></p>
|
1,803,334 | <p>I am confusing how to determine the set is clopen, neither open or closed, open but not closed and closed but not open. I read an example from "Topology without Tears". </p>
<p>Let $X=\{a,b,c,d,e,f\}$ and $\tau=\{X,\emptyset,\{a\},\{c,d\},\{a,c,d\},\{b,c,d,e,f\}\}$. $\tau$ is a topology on $X$. Then </p>
<ol>
<li>... | MPW | 113,214 | <p><strong>Hint:</strong> A set is open if it is in <span class="math-container">$\tau$</span>; so, for example <span class="math-container">$\{a\}$</span> is open because <span class="math-container">$\{a\}\in\tau$</span>.</p>
<p>A set is closed if its complement is in <span class="math-container">$\tau$</span>; so, ... |
2,537,031 | <p>This is a question in Geometry by Hartshorne Exercise 3.3 </p>
<p>The goal is using Ruler and compass and a given triangle ABC and given a segment DE, construct a rectangle with content equal to the triangle ABC, and with one side equal to DE. Any propositions in Euclid book I-IV are usable but its likely going to... | Mark Bennet | 2,906 | <p>Here is a sketch.</p>
<p>If units are chosen so the sides of the paper rectangle are $1$ and $\sqrt 2$, the side of the square is $a+b\sqrt 2$ and considering a corner we must have $a,b \gt 0$.</p>
<p>Now the area of the square is $(a+b\sqrt 2)^2=a^2+2b^2+2ab\sqrt 2$</p>
<p>The area of an individual rectangle is ... |
4,010,297 | <p>My question is about kind of comparing values in a function, for example in this question</p>
<p>given 'c' a positive constant we have the following function <span class="math-container">$f(x)=cx^3+x^5-1$</span> , determine which is greater <span class="math-container">$f(\sin(x))$</span> or <span class="math-contai... | Community | -1 | <p>It is certainly true that <span class="math-container">$f'(0)=0$</span>, because <span class="math-container">$f'$</span> is continuous at <span class="math-container">$0$</span> by hypothesis and any sequence <span class="math-container">$\xi_i\in(x_i,y_i)$</span> coming from Rolle's theorem satisfies <span class="... |
760,330 | <p>For every positive integer $n$, prove that
$$\sqrt{4n+1}<\sqrt{n} + \sqrt{n+1}<\sqrt{4n+2}$$</p>
<p>Hence or otherwise, prove that $[\sqrt{n}+\sqrt{n+1}] = [\sqrt{4n+1}]$, where $[x]$ denotes the greatest integer not exceeding $x$. </p>
<p>This question was posed to me in class by my teacher.... | kingW3 | 130,953 | <p>$$\sqrt n+\sqrt{n+1}>\sqrt{4n+1}\\n+2\sqrt{n(n+1)}+n+1>4n+1\\\sqrt{n(n+1)}>n\\n^2+n>n^2\\n>0\\\sqrt{4n+2}>\sqrt n+\sqrt{n+1}\\4n+2>2n+1+2\sqrt{n(n+1)}\\2n+1>2\sqrt{n(n+1)}\\4n^2+4n+1>4n^2+4n\\1>0$$</p>
|
1,095,416 | <p>$e$ and $\pi$ are rather peculiar numbers. It turns out that, in
addition to being irrational numbers, they are also transcendental
numbers. Basically, a number is transcendental if there are no
polynomials with rational coefficients that have that number as a
root.</p>
<p>Clearly, $p(x) = (x-e)(x-\pi)$ is a po... | Robert Cardona | 29,193 | <p><strong>Proposition 2.12</strong>: Let $M, N$ be $A$-modules. Then there exists a pair $(T, g)$ consisting of an $A$-modules $T$ and an $A$-bilinear mapping $g : M \times N \to T$, with the following property:</p>
<p>Given any $A$-module $P$ and any $A$-bilinear mapping $f : M \times N \to P$, there exists a unique... |
3,339,861 | <p><span class="math-container">$\mathbf{Question}:$</span> Let <span class="math-container">$f$</span> be a continuous function on <span class="math-container">$[0,1]$</span>. Then prove that the limit <span class="math-container">$\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$</span> is equal to <span class="math-containe... | Alexdanut | 629,594 | <p>I will prove a more general statement : <em>Let <span class="math-container">$f:[0,1]\to \mathbb{R}$</span> be a Riemann integrable function. If <span class="math-container">$f$</span> is continuous at <span class="math-container">$x=1$</span>, then prove that <span class="math-container">$\lim\limits_{n \to \infty}... |
3,235,131 | <p>Is there a real <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$A$</span>, such that: <span class="math-container">$A^3 = I_3$</span> and has at most one zero entry? If so, how can I find it?</p>
| Théophile | 26,091 | <p>If you want a "real life" analogy, consider a shopping list. The order of the items doesn't really matter, and you might have written some of them down several times. So</p>
<blockquote>
<p>apples<br>
oranges<br>
milk</p>
</blockquote>
<p>is the same as</p>
<blockquote>
<p>milk<br>
oranges<br>
apples<... |
3,235,131 | <p>Is there a real <span class="math-container">$3 \times 3$</span> matrix <span class="math-container">$A$</span>, such that: <span class="math-container">$A^3 = I_3$</span> and has at most one zero entry? If so, how can I find it?</p>
| allo | 265,765 | <p>Think of a lottery.</p>
<p>You can mark <code>4</code> as often as you want, but when the cross gets bolder this does not mean that <code>4</code> counts multiple times. In addition, the order of the crossed numbers does not matter.</p>
<p>The nice thing about the analogy is, that variants of the lottery can be co... |
2,841,983 | <p>I was doing this question from an RMO Practice Paper, and I have been unable to solve it.</p>
<blockquote>
<p>Let $P(x)$ be a polynomial of degree $2015$. $P(k)=2^k$ for $k=0,1,2,\dots,2015$. Find $P(2016)$</p>
</blockquote>
<p>My attempt:</p>
<p>Let $Q(x)=P(x)-2^x$.</p>
<p>Then its zeroes are $0,1,2,\dots,201... | lhf | 589 | <p>Using repeated differences, we get that the first value in every row is $1$:
$$
\begin{array}{llll}
1 & 2 & 4 & 8 & \cdots \\
1 & 2 & 4 & \cdots \\
1 & 2 & \cdots \\
1 & \cdots \\
\cdots \\
1 & \\
\end{array}
$$
<a href="http://en.wikipedia.org/wiki/Newton_series#Newton.27... |
3,240,162 | <p>so I got this question that I am stuck on:
So:
Consider the set <span class="math-container">$$S = \{0,\pm1,\pm2,\pm3,\pm4,\pm5,\pm6,\pm7,\pm8\}$$</span>
Consider the relation <span class="math-container">$R$</span> on <span class="math-container">$S$</span> defined by <span class="math-container">$(a,b)$</span> par... | mo2019 | 677,316 | <p>Regarding symmetry, observe that if <span class="math-container">$x-y$</span> is a multiple of <span class="math-container">$4$</span>, then so is <span class="math-container">$y-x$</span>.</p>
<p>For transitivity, let <span class="math-container">$x-y = 4k$</span> for some <span class="math-container">$k \in \math... |
2,713,875 | <p>Over an arbitrary ring $R$, a matrix $A$ is said to be invertible if it has an inverse with entries in the same ring. This happens iff $\det A$ is a unit of $R$.</p>
<p>I've always thought that the terms "invertible" and "nonsingular" are synonymous. But I think the following problem (from Artin) suggests that they... | HBR | 396,575 | <p>The double integral, represents the sum of the infinite areas under the curves $g(x,y)$ with $x=constant$ defined as:
$$A(y)=\int{g(x,y)\,dx}$$
each one between $y$ and $y+dy$, which gives you a volume. (see <a href="https://en.m.wikipedia.org/wiki/Fubini%27s_theorem" rel="nofollow noreferrer">Fubini</a> 's explanat... |
351,170 | <p>The <em>genus <span class="math-container">$g$</span> handlebodies</em> are building blocks of <span class="math-container">$3$</span>-manifolds. They are constructed from <span class="math-container">$3$</span>-ball <span class="math-container">$B^3$</span> by adding <span class="math-container">$g$</span>-copies o... | Oğuz Şavk | 131,172 | <p>This is a late reply but it should be still helpful. It is from Zoltan Szabó's <a href="https://bookstore.ams.org/pcms-15/" rel="nofollow noreferrer">PCMI lecture notes</a>.</p>
<p>Consider the following Heegaard splitting:</p>
<p><a href="https://i.stack.imgur.com/vvIoz.png" rel="nofollow noreferrer"><img src="ht... |
4,424,363 | <p>One of the logical axioms in a <a href="https://en.wikipedia.org/wiki/Hilbert_system" rel="nofollow noreferrer">Hilbert system</a> for first order logic is the quantifier axiom (Q7 in the wikipedia article):</p>
<blockquote>
<p><span class="math-container">$\phi\to\forall x\phi$</span></p>
</blockquote>
<p>whenever ... | ryang | 21,813 | <p>Let formula <span class="math-container">$ϕ(x)$</span> be satisfied by object <span class="math-container">$p,$</span> but not object <span class="math-container">$q,$</span> in the domain of discourse (e.g., <span class="math-container">$\{p,q\}$</span>); then <span class="math-container">$ϕ(p)$</span> is true but ... |
338,549 | <p><strong>Question:</strong> We select an element of $[100]$ at random. Let $A$ be the event that this integer is divisible by $3$ and let $B$ be that event that this integer is divisble by $7$. So are $A$ and $B$ independent? </p>
<p>I think they're no, which make sense because $21$ is divisible by both $3$ and $... | Community | -1 | <p>Indeed they're not independent. </p>
<p>$P(A)=\frac{33}{100}$ </p>
<p>$P(B)=\frac{14}{100}$</p>
<p>$P(A\bigcap B)=\frac{4}{100}$</p>
<p>$\therefore$ They're not independent!</p>
|
3,811,012 | <p>I wish to find <span class="math-container">$\displaystyle \lim_{n \rightarrow \infty}\frac{n+1}{\sqrt{n}}$</span>.</p>
<p>Here is what I did:</p>
<p><span class="math-container">$1.$</span> Rewrite <span class="math-container">$\frac{n+1}{\sqrt{n}}$</span> to <span class="math-container">$(n+1) \cdot \frac{1}{\sqrt... | Xander Henderson | 468,350 | <h3>Short Version</h3>
<p>The product law for limits has hypotheses which the asker has neglected to verify. A better approach is to bound the sequence from below, which gives
<span class="math-container">$$ \frac{n+1}{\sqrt{n}} > \sqrt{n}
\implies \lim_{n\to\infty} \frac{n+1}{\sqrt{n}} > \lim_{n\to\infty} \sqrt... |
76,575 | <p>We have enrolled our 5 year old son in <a href="http://en.wikipedia.org/wiki/Kumon_method" rel="nofollow">Kumon</a> which is an after school math and reading enrichment program of Japanese origin. </p>
<p>While he is learning lots of things (currently learning how to add i.e., 1+4=?, 2+4=? etc) I am concerned about... | EuYu | 9,246 | <p>A few years ago when I was in high school I worked part - time for the Kumon Learning Center as a teaching assistant. Generally my experiences with Kumon is that the learning is generally rote and done through the weekly worksheets assigned. Admittedly, some of the worksheets are quite cleverly designed, the materia... |
239,187 | <p>I need help constructing a plane function, <span class="math-container">$z = f(x,y)$</span> that goes through three points, (05, 22, 20). (89, 0, 89) and (-1, -1, 10). I have tried to input it, but I dont know how.</p>
<p><a href="https://i.stack.imgur.com/UNg8A.png" rel="nofollow noreferrer"><img src="https://i.sta... | cvgmt | 72,111 | <p>According to analytic geometry,we can calculate the normal of plane by <code>Cross</code> or calculate the volume of the three points by <code>Det</code></p>
<pre><code>p1 = {5, 22, 20};
p2 = {89, 0, 89};
p3 = {-1, -1, 10};
Simplify[({x, y, z} - p3) . Cross[p1 - p3, p2 - p3] == 0]
Det[{{x, y, z} - p3, p2 - p3, p1 - ... |
2,986,096 | <p>In the boolean Identity:</p>
<blockquote>
<p>(x + yx) = (x + y)(x + z)</p>
</blockquote>
<p>Where does z come from and what does it mean?</p>
| JonathanZ supports MonicaC | 275,313 | <p>Following up on MatthewL's correct answer:</p>
<p>Let's let <span class="math-container">$0$</span> and <span class="math-container">$1$</span> denote the additive and multiplicative identities in your field <span class="math-container">$F$</span>. Then in the <em>ring</em> <span class="math-container">$F \times F$... |
2,840,452 | <p>Find a closed form for the following recurrence relation:</p>
<p>$\begin{cases}
C_n=3C_{n-1}+n+2\\
C_0=0\\
\end{cases}$</p>
<p>So we start with a guess $D_n=C_n+an+b\iff C_n=D_n-an-b$</p>
<p>substituting to the equations gives</p>
<p>$D_n-an-b=3(D_{n-1}-a(n-1)-b)+n+2\iff \\ \iff D_n=3D_{n-1}+n(a-3a+1)+(3a-3b+b+2... | Bernard | 202,857 | <p><strong>Hint</strong>:</p>
<p>Just like linear differential equations, the solutions of linear recurrence relations are the sum of one particular solution, which you've found, and the general solution of the associated homogeneous recurrence relation:
$$ C_n=3C_{n-1}. $$</p>
|
3,324,375 | <p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
<p>In the beginning, when there were just reals, these operations were defined for them. Then, <em>i</em> was created, literally a number whose value is undefined, like e.g. one divided by zero is undefi... | Bill Dubuque | 242 | <blockquote>
<p>What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers?</p>
</blockquote>
<p>It is set theory that allows us to give a rigorous foundation for complex numbers. In particular, as explained <a href="https://math.stackexchange.com/a/2658/242">here</a>... |
3,086,741 | <p>By transforming to polar coordinates, show that</p>
<p><span class="math-container">$$\int_{0}^{1} \int_{0}^{x}\frac{1}{(1+x^2)(1+y^2)} \,dy\,dx$$</span></p>
<p>Is equal to</p>
<p><span class="math-container">$$ \int_{0}^{\pi/4}\frac{\log(\sqrt{2}\cos(\theta))}{\cos(2\theta)} d\theta$$</span></p>
<p>I have tried... | omegadot | 128,913 | <p>Do you really have to transform this double integral first using polar coordinates? It is far easier to just integrate it as is. Here
<span class="math-container">\begin{align}
\int_0^1 \int_0^x \frac{dy \, dx}{(1 + x^2)(1 + y^2)} &= \int_0^1 \left [\frac{\tan^{-1} y}{1 + x^2} \right ]_0^x \, dx\\
&= \int_0^... |
1,870,751 | <p>What does Aut$(\Bbb Z)$ look like? (Integers with the operation of addition)</p>
<p>I understand that it's the set of all automorphisms from $\Bbb Z$ to $\Bbb Z$, or Aut$(\Bbb Z) = \{\alpha_1, \alpha_2, ... : \alpha_i$ is an isomorphism from $\Bbb Z$ to $\Bbb Z \}$.</p>
<p>I figured that the only isomorphisms tha... | avs | 353,141 | <p>If ${\bf Z}$ is viewed here as an additive group, then this group is cyclic. Therefore, any automorphism of the group must map a generator to a generator.</p>
<p>The group has only two generators: $1$ and $-1$. Therefore, there is only one nontrivial automorphism: the one mapping $1$ to $-1$.</p>
|
100,910 | <p>A central advantage of cohomology theory over homology -- at least in terms of richness of structure and strength as an invariant -- is the additional ring structure from the cup product. Recall that this arises from applying the cohomology functor to the following inclusion map of topological spaces $$X \hookrighta... | Pierre | 37,021 | <p>With field coefficients, homology and cohomology are dual to each other. The cohomology of a space is an algebra, the homology of a space is a coalgebra. When the space is a group (or loop space or $H$-space...) both its homology and cohomology are Hopf algebras.</p>
<p>"The literature" is full of computations of t... |
459,582 | <p>From <a href="http://mathinsight.org/dot_product" rel="nofollow noreferrer">http://mathinsight.org/dot_product</a>:</p>
<blockquote>
<p>$\LARGE{1. }$ "The dot product of $\mathbf{a}$ with unit vector $\mathbf{b}\backslash|\mathbf{b}|$ is defined to be the projection of $\mathbf{a}$ in the direction of $\mathbf{b... | Community | -1 | <p>$\Large{\text{Annex to Chris Culter's Answer on August 5 :}}$</p>
<p>About $\#1$, I was trying to construct a natural and simple counterexample.<br>
The colours refer to those in the picture below.<br>
To ensure that $\mathbf{(x + y) \perp b}$, I wittingly define $\color{green}{\mathbf{b} = (1,0)}$ and $\mathbf{x+y... |
401,482 | <p>I have the following question. I know the answers but am struggling with how to write up the work formally. Any help would be appreciated. </p>
<p>Consider $B=\{1-\frac{1}{n} :n=1,2,\cdots\}$. Is $B$ open? Closed? Compact? Justify.</p>
<p>I know that it is not open or closed, thus not compact. </p>
| Brian M. Scott | 12,042 | <p>HINT: It may be helpful to write out a few members of $B$:</p>
<p>$$B=\left\{0,\frac12,\frac23,\frac34,\dots\right\}\;.$$</p>
<ul>
<li>To help you explain why $B$ is not open: is there any $\epsilon>0$ such that $(-\epsilon,\epsilon)\subseteq B$? </li>
<li>To help you explain why $B$ is not closed: can you fin... |
401,482 | <p>I have the following question. I know the answers but am struggling with how to write up the work formally. Any help would be appreciated. </p>
<p>Consider $B=\{1-\frac{1}{n} :n=1,2,\cdots\}$. Is $B$ open? Closed? Compact? Justify.</p>
<p>I know that it is not open or closed, thus not compact. </p>
| Bruno Joyal | 12,507 | <p>$B$ is not closed, because it has an accumulation point which is not in $B$ (can you see which one)?</p>
<p>$B$ is not open, because $0 \in B$ but there is no interval $(-\epsilon, \epsilon) \subseteq B$.</p>
<p>It is not compact because it has an open cover by intervals of the form $(0, 1-1/n)$, but this cover ha... |
49,787 | <p>From the link in wikipedia </p>
<p><a href="http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii" rel="nofollow noreferrer">http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii</a></p>
<p><strong>OPEN QUESTION:</strong></p>
<p><strong>What is the e... | Ethan Solly | 222,094 | <p>$$\frac{(r_1+r_2)\sqrt{(r_1+r_3)^2-r_1}}{2} - \frac{\pi x r_1^2}{\frac{\angle 1}{360}} + \frac{\pi x r_2^2}{\frac{\angle 2}{360}} + \frac{\pi x r_3^2}{\frac{\angle 3}{360}}$$
where $r_1$, $r_2$, and $r_3$ are the radii of the circles; $\angle 1$, $\angle 2$, and $\angle 3$ are the angles of the triangle created by t... |
49,787 | <p>From the link in wikipedia </p>
<p><a href="http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii" rel="nofollow noreferrer">http://web.gnowledge.org/wiki/index.php/Area_Between_Three_Circles_of_Differing_Radii</a></p>
<p><strong>OPEN QUESTION:</strong></p>
<p><strong>What is the e... | Linkon | 425,578 | <p>At first join the centers of the circles to get a <strong>big triangle</strong> with known sides. First calculate the sides of triangle-</p>
<pre><code>a=R1+R2
b=R2+R3
c=R3+R1 [Here R1,R2,R3 are radii of the above three triangles]
</code></pre>
<p>Now calculate <strong>half-perimeter</strong> to calculate the area... |
3,548,378 | <p>Usually, in the integration, <span class="math-container">$\int_Xf(x) \, d\mu(x)$</span>, people assume by default that <span class="math-container">$X$</span> is infinite. </p>
<p>If <span class="math-container">$X$</span> is finite, then people usually write:
<span class="math-container">$$\sum_{x\in X}f(x)p(x)$... | GEdgar | 442 | <p><strong>Sum is a special case of Integration?</strong> </p>
<p>Absolutely convergent series is a special case of integration with respect to a measure. But not conditionally convergent series.</p>
|
2,188,520 | <p><span class="math-container">$\iff$</span> is a logical connective which can only be applied to pairs of propositions. So I understand that saying <span class="math-container">$8 \iff 5+3$</span> doesn't make much sense, because <span class="math-container">$8$</span> and <span class="math-container">$5+3$</span> ar... | Christopher.L | 347,503 | <p><em>I'll elaborate on my comment and make it in to a answer instead</em>:</p>
<p>I would say that the distinction between '$=$' and '$\Leftrightarrow$' that you made yourself, still holds in your example. The symbol '$=$' denotes a relation between objects while '$\Leftrightarrow$' is a logical connective relating ... |
1,817,609 | <p>Here is a list of other systems:</p>
<ul>
<li>Babylonian numerals</li>
<li>Egyptian numerals</li>
<li>Aegean numerals</li>
<li>May numerals</li>
<li>Chinese numerals</li>
</ul>
<p>These system are far older than the current system. How did it get to be known and used internationally by nearly every cultures these ... | Jack D'Aurizio | 44,121 | <p>For short, positional numeral systems offer the great advantage to have efficient algorithms for the computation of sum and products, easy to use in everyday life. The base $10$ is more or less accidental (besides we having $10$ fingers, on average): for instance, there would be many efficient divisibility tests in ... |
251,771 | <p>Durrett has a theorem that says: if $X_1, X_2, ..., X_n$ are random variables then $X_1 + X_2 + ... + X_n$ are also random variables.</p>
<p>My issue is how to show that $F((x_1, x_2, ... , x_n)) = \sum_{i = 1}^{n} x_i$ is measurable?</p>
<p>Durrett says ${x_1 + x_2 + ... + x_n < a}$ is an open set $(-\infty, a... | Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$\ $</span> If a field F has two F-linear independent combinations of <span class="math-container">$\rm\ \sqrt{a},\ \sqrt{b}\ $</span> then
you can solve for <span class="math-container">$\rm\ \sqrt{a},\ \sqrt{b}\ $</span> in F. For example, the Primitive Element Th... |
3,099,652 | <p>Can anyone help me with this!?
If <span class="math-container">$n=p_1^{k_1},p_2^{k_2},\ldots $</span>
Then I applied the given condition of divisibility of <span class="math-container">$\varphi(n)$</span> but can't reach to a conclusion.</p>
| José Carlos Santos | 446,262 | <p>If <span class="math-container">$f'(x)=f(x)$</span> and if <span class="math-container">$g(x)=f(-x)$</span>, then <span class="math-container">$g'(x)=-f'(-x)=-f(-x)=-g(x)$</span>. Can you take it from here?</p>
|
752,224 | <p>What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is homotopy equivalent to the half circle. But what is the fundamental group of the half circle? </p>
| tom | 59,101 | <p>Your space is homotopic to half circle $\{ e^{i\theta} : \theta\in[0,\pi]\}$ which is contractible. We can write down the contraction
$$
H(\theta,t) = e^{i \min\{\theta,\pi(1-t)\}}.
$$</p>
<p>Fundamental group of contractible space is trivial.</p>
|
2,146,252 | <p>How can one prove the following equality?</p>
<p>$$\int_{0}^{1} x^{\alpha} (1-x)^{\beta-1} \,dx = \frac{\Gamma(\alpha+1) \Gamma(\beta)}{\Gamma(\alpha + \beta + 1)}$$</p>
| zwim | 399,263 | <p>Function gamma : $\Gamma(z)=\int_0^{\infty}t^{z-1}e^{-t}dt$</p>
<p>Function beta : $B(x,y)=\int_0^1t^{x-1}(1-t)^{y-1}dt$</p>
<p>Your expression is $B(\alpha+1,\beta)=\frac{\Gamma(\alpha+1)\Gamma(\beta)}{\Gamma(\alpha+\beta+1)}$</p>
<hr>
<p>$$\Gamma(\alpha)\Gamma(\beta)=\int_0^{\infty}t^{\alpha-1}e^{-t}dt\int_0^{... |
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| CLAUDE | 118,773 | <p>$((\frac{1}{2})^{\frac{1}{3}})^6=(\frac{1}{2})^2=\frac{1}{4}$</p>
<p>$((\frac{1}{3})^{\frac{1}{2}})^6=(\frac{1}{3})^3=\frac{1}{27}$</p>
<p>So as it is obvious from the above relations, $((\frac{1}{2})^{\frac{1}{3}})^6>((\frac{1}{3})^{\frac{1}{2}})^6$, so we can say $(\frac{1}{2})^{\frac{1}{3}}>(\frac{1}{3})^... |
1,041,684 | <p>How can you determine which one of these numbers is bigger (without calculating):</p>
<p>$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$</p>
| Adam Chalcraft | 196,032 | <p>When is $x^y > y^x$ ?
When $x^{1/x} > y^{1/y}$.
Let's look at the function $x^{1/x}$.
Differentiating, we find it has a maximum at $x=e$.
Since $1/2$ and $1/3$ are both less than $e$, the one that's nearer wins.
So $(1/2)^2 > (1/3)^3$,
so $(1/2)^{1/3} > (1/3)^{1/2}$.</p>
<p>But more to the point, this s... |
3,449,799 | <blockquote>
<p>Find all positive integer solutions to <span class="math-container">$24x+18y=6420$</span>. </p>
</blockquote>
<p>Here's my work.</p>
<p>Simplifying the equation gives <span class="math-container">$4x+3y=1070$</span>. Note that this equation has solutions because <span class="math-container">$\gcd (4... | ViHdzP | 718,671 | <p>Your solution seems correct. However, it'd be much faster to simply notice that if <span class="math-container">$x,y\geq1$</span>, <span class="math-container">$$154x+24y\geq178>30.$$</span></p>
|
3,449,799 | <blockquote>
<p>Find all positive integer solutions to <span class="math-container">$24x+18y=6420$</span>. </p>
</blockquote>
<p>Here's my work.</p>
<p>Simplifying the equation gives <span class="math-container">$4x+3y=1070$</span>. Note that this equation has solutions because <span class="math-container">$\gcd (4... | John Hughes | 114,036 | <p>You're looking at
<span class="math-container">$$
77+12=15
$$</span>
right? For positive <span class="math-container">$x$</span> and <span class="math-container">$y$</span>.
Let <span class="math-container">$u = x-1, v = y-1$</span>, then (1) <span class="math-container">$u$</span> and <span class="math-container"... |
393,122 | <p>I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.</p>
<blockquote>
<p>How to prove $n^2 < n! $ ?</p>
</blockquote>
| Sason Torosean | 77,974 | <p>here is a hint. $n! = 1\times2\times...\times(n-2)\times(n-1)\times n$, where as $n^2 = n\times n$. so you have effectively as your step one:
$1\times2\times...\times (n-1)\times n =n\times n $.
</p>
<p>now you can cancel things out and....well I think that is hint enough as far as the question of proving $n^2 &l... |
1,508,340 | <p>Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that
$$a^{m} a^{n} = a^{m+n}.$$</p>
<p>I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when one or both are negative without assuming that $(a^{-1})^n = a^{-n} $. </p>
| fleablood | 280,126 | <p>"without assuming that $(a^{-1})^n=a^{−n}$". </p>
<p>How was $a^{-n}$ (n positive) originally defined (if it was)? As notation it doesn't have any intrinsic meaning. It's very easy to show $(a^{-1})^n = (a^n)^{-1}$ (multiply either by $a^n$ and the result is $e$ so both of them are the unique inverse of $a^n$). ... |
1,447,506 | <p>To prove this, I want to show that an arbitrary intersection of closed sets is closed and an arbitrary intersection of bounded sets is bounded. I know how to prove the first part, but I'm not sure how to rigorously show that an intersection of bounded set is bounded. </p>
<p>This question comes from a real analysis... | gabrielchua | 290,503 | <p>The finite subcover produced by a subcover of one of the compact sets also covers the intersection.</p>
|
471,017 | <p>I'm reading linear programming and I bumped into the following:</p>
<p><img src="https://i.stack.imgur.com/8VJgx.png" alt="enter image description here"></p>
<p>I'm having trouble getting grasp on the proof of proposition 2. Could someone perhaps explain it to me in other terms? For some reason the proof is unclea... | mercio | 17,445 | <p>Looking at the Galois group of the polynomial as a subgroup $G$ of $S_n$, we can translate your requirements as conditions on $G$ :</p>
<p>(i) $f$ has a root mod $p$ for every prime $p$ iff every element of $G$ has a fixpoint (by Cebotarev's theorem)<br>
(ii) $f$ is irreducible in $\Bbb Q[x]$ iff $G$ is transitive.... |
906,103 | <h1>Context:</h1>
<p>I'm trying to <strong>algebraically</strong> prove that an <strong>open interval</strong> is an <strong>open set</strong>. If I sketch it, as suggested by @rschwieb in this <a href="https://math.stackexchange.com/a/301381/688539">answer</a>, then it seems quite obvious that this is indeed true. But... | Community | -1 | <p>Yes, an open interval in <span class="math-container">$\mathbb{R}$</span> is an open set in the topology generated by the standard metric.</p>
|
97,914 | <p>How do we replace the first <code>Head</code> of a currying expression such as <code>a[b,c][d]</code>? When accessing the <code>0</code>th element of an expression like <code>a[b,c][d]</code>, <code>a[b,c]</code> is given as the <code>Head</code>. We would like to instead replace <code>a</code> with, say, <code>w</c... | SquareOne | 19,960 | <pre><code>replaceFirstHead[expr_, newHead_Symbol] :=
Apply[newHead, expr, {Length@FixedPointList[Head, expr] - 4}, Heads -> True]
</code></pre>
<p>then</p>
<pre><code>replaceFirstHead[a[b], newhead]
replaceFirstHead[a[b][c, d], newhead]
replaceFirstHead[a[b][c, d][e], newhead]
</code></pre>
<blockquote>
<pre><... |
686,482 | <p>In algebraic geometry, we consider the map </p>
<p>$$I:\{\text{subsets of }\mathbb{A}_k^n\}\longrightarrow\{\text{ideals of }k[x_1,\cdots,x_n]\},\qquad X\mapsto I(X)$$</p>
<p>This map is not injective, because $I(\mathbb{A_k^n})=0$. But why it is not surjective? How to find a counterexample? Thank you!</p>
| Community | -1 | <p>By a very succinct way: we have
$$\alpha\circ\alpha=\operatorname{id}_{\mathcal P(X)}$$
hence $\alpha$ is a bijection and
$$\alpha^{-1}=\alpha$$</p>
|
655,191 | <p>I'm looking for two non-homeomorphic connected, Hausdorff, locally compact spaces whose one-point compactifications are homeomorphic.</p>
<p>Without the connectedness property this is easy, for example: $[0,1) \cup (1,2]$ and $[0,2)$. I was thinking that I could maybe find two connected spaces which are locally com... | Dejan Govc | 19,588 | <p>Square without an interior point and square without a boundary point.</p>
|
1,897,455 | <blockquote>
<p>If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$ then $A^2+B^2$ equals ?</p>
<p>(a) $2AB$</p>
<p>(b) $2BA$</p>
<p>(c) $A+B$</p>
<p>(d) $AB$</p>
</blockquote>
<p>I tried
$(A+B)^2=A^2+B^2+AB+BA$</p>
<p>or,$A^2+B^2=(A+B)^2-AB-BA$</p>
<p>$=(A+B)^2-A-B$</p>
<p>$
=(A+B)^... | Community | -1 | <p>More precisely, there is a positive integer $p$ s.t. $A,B$ are simultaneously similar to $A'=\begin{pmatrix}I_p&0\\0&0_{n-p}\end{pmatrix},B'=\begin{pmatrix}I_p&Q\\0&0_{n-p}\end{pmatrix}$ where $Q$ is an arbitrary $(p\times n-p)$ matrix.</p>
<p>Proof. Since $A^2=A$, there is $p$ s.t. $A$ is similar t... |
1,278,158 | <p>You and your friend roll a sixth-sided die. If he lands even number then he wins, and if you land odds then you win. He goes first, if he wins on his first roll then the game is over. If he doesn't win on his first roll, then he pass the die to you. repeat until someone wins.</p>
<p>What is the probability your fri... | André Nicolas | 6,312 | <p>The probability of an odd number on a single toss is $\frac{3}{6}$, that is, $\frac{1}{2}$, so it's just like tossing a fair coin. </p>
<p>Your friend wins on his $n$-th toss if the $2n-2$ tosses before that, ($n-1$ by her and $n-1$ by you) were tails, and he finally gets a head on her $n$-th toss. This has probabi... |
1,278,158 | <p>You and your friend roll a sixth-sided die. If he lands even number then he wins, and if you land odds then you win. He goes first, if he wins on his first roll then the game is over. If he doesn't win on his first roll, then he pass the die to you. repeat until someone wins.</p>
<p>What is the probability your fri... | user161300 | 161,300 | <p>Both of you have the same probability of winning and losing on a single roll, $\frac{1}{2}$</p>
<p>The probability that he wins in his first toss is $\frac{1}{2}$.</p>
<p>He wins in his second toss when he and you both don't win your first tosses and he wins in his second toss. The probability that this happens is... |
31,274 | <p>Many image processing libraries like <a href="http://docs.opencv.org/doc/tutorials/imgproc/imgtrans/remap/remap.html" rel="noreferrer">OpenCV</a>, <a href="https://software.intel.com/en-us/node/504403" rel="noreferrer">Intel Performance Primitives</a> or <a href="http://octave.sourceforge.net/image/function/imremap.... | J. M.'s persistent exhaustion | 50 | <p>First, here's an image from the docs which we'll use for testing:</p>
<pre><code>img = Import["https://i.stack.imgur.com/bzkJM.png"]
</code></pre>
<p><img src="https://i.stack.imgur.com/bzkJM.png" alt="jellyfish"></p>
<p>Going with the definition given in the IPP docs, here is a remapping method based on the use ... |
4,393,135 | <p>I am doing some exercises on matrix equations for my upcoming exam and I have some questions the answers to which I cannot find online. I do study in Portuguese and maybe I don't know the right vocabulary for the search, so I'm trying this community.</p>
<hr />
<blockquote>
<p>Prove that the solution to the followin... | AmateurDotCounter | 143,257 | <p>You already are given a form for <span class="math-container">$X$</span> that can be proven simply by plugging the value in and verifying that the left hand side of the equation equals the right hand side.</p>
<hr />
<p>One key identity (for invertible matrices) will be <span class="math-container">$(MN)^{-1}=N^{-1}... |
4,386,744 | <blockquote>
<p>Show that if <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are connected, then <span class="math-container">$X \times Y$</span> is also connected.</p>
</blockquote>
<p>Since <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are both... | by24 | 954,106 | <p><span class="math-container">$S_n$</span> will be a cubic function of <span class="math-container">$n$</span>, because there are <span class="math-container">$n$</span> terms of <span class="math-container">$2nd$</span> degree (quadratic). Then, we can write the formula for <span class="math-container">$S_n$</span> ... |
3,818,207 | <p>I need to verify the value of the following integral
<span class="math-container">$$ 4n(n-1)\int_0^1 \frac{1}{8t^3}\left[\frac{(2t-t^2)^{n+1}}{(n+1)}-\frac{t^{2n+2}}{n+1}-t^4\{\frac{(2t-t^2)^{n-1}}{n-1})-\frac{t^{2n-2}}{n-1} \} \right] dt.$$</span>
The integrand (factor of <span class="math-container">$4n(n-1)$</sp... | thedumbkid | 523,583 | <p>You can use SageMath <a href="https://www.sagemath.org/" rel="nofollow noreferrer">https://www.sagemath.org/</a>
It has a built-in function to integrate symbolically</p>
|
3,818,207 | <p>I need to verify the value of the following integral
<span class="math-container">$$ 4n(n-1)\int_0^1 \frac{1}{8t^3}\left[\frac{(2t-t^2)^{n+1}}{(n+1)}-\frac{t^{2n+2}}{n+1}-t^4\{\frac{(2t-t^2)^{n-1}}{n-1})-\frac{t^{2n-2}}{n-1} \} \right] dt.$$</span>
The integrand (factor of <span class="math-container">$4n(n-1)$</sp... | Claude Leibovici | 82,404 | <p>Rewrite your integral
<span class="math-container">$$I_n=\frac{n(n-1)}2 J_n$$</span> with
<span class="math-container">$$J_n=\int \frac{\frac{\left(2 t-t^2\right)^{n+1}}{n+1}-\frac{t^{2 n+2}}{n+1}-t^4
\left(\frac{\left(2 t-t^2\right)^{n-1}}{n-1}-\frac{t^{2
n-2}}{n-1}\right)}{t^3}\,dt$$</span> Now, rewrite <spa... |
1,623,416 | <p>There are $n$ bins of which the $i$th contains $(i-1)$ green balls and $(n-i)$ black balls. You pick a bin at random and remove two balls at random without replacement. What is the probability that the second ball is black?</p>
| Logophobic | 299,855 | <p>There are $n(n-1)$ total balls. Half are green and half are black. Since you are drawing from the total pool and ignoring the first pick, the probability is the same as drawing one ball from the total pool. $P=0.5$</p>
<p>Proof: \begin{align}P&=\frac{1}{n}\cdot\frac{1+2+3+\cdots+(n-1)}{n-1}\\&=\frac{1}{n}\c... |
1,623,416 | <p>There are $n$ bins of which the $i$th contains $(i-1)$ green balls and $(n-i)$ black balls. You pick a bin at random and remove two balls at random without replacement. What is the probability that the second ball is black?</p>
| sqldev | 307,364 | <p>Tried it with a sample of 5 bins.. So the distribution of balls (green, black) would be (0,4), (1,3),(2,2),(3,1) and (4,0)... </p>
<p>Prob of getting black as second ball from bin 1 P(1)-1,
P(2)-1*3/3 or n-2/n-2
P(3)-1*2/3 or n-3/n-2
..
..
P(n)-0</p>
<p>Then adding these should give us the result, Correct?</p>
|
1,008,559 | <p>Is it true that the right-side derivative of Weierstrass function, which is a classic example of continuous yet nowhere differentiable function, always non-negative? (In fact, positive)</p>
<p>That is, given $f(x) = \sum_{n=0}^\infty a^ncos(b^n\pi x)$ for $0<a<1$, and $b$ a positive odd with $ab>1+3\pi/2$,... | Dave L. Renfro | 13,130 | <p>Your conjecture fails in two rather strong ways.</p>
<p>Corollary 11.4.1 on p. 128 of <strong>[1]</strong> states a result that is slightly stronger than the following, where $D^{+}f$ is the right upper Dini derivate of $f.$</p>
<blockquote>
<p>Let $I$ be an open interval in $\mathbb R$ and let $f:I \rightarrow ... |
3,053,520 | <p>I don't know how to begin to find this particular Galois group since I do not know the splitting field. Any hints will be appreciated.</p>
| Mike Earnest | 177,399 | <p>Here is a combinatorial solution.</p>
<p>Consider tilings of an <span class="math-container">$n\times 1$</span> rectangle with squares and dominos. There are <span class="math-container">$\binom{n-k}{k}$</span> such tilings which use exactly <span class="math-container">$k$</span> dominos. Your sum counts all such t... |
1,170,477 | <p>Show that if $A$ is an $m \times n$ matrix and $A(BA)$ is defined, then $B$ is an $n \times m$ matrix.</p>
<p>I know that $A$ is a $m \times n$ matrix and to be able to multiply $B$ with $A$, $B$ must be a $n \times m$ matrix. I am confused though because I can't just assume that. </p>
| Cloverr | 219,106 | <p>Let B(p x q), now B(pxq) A(m x n) is defined means q=m and BA would be of form (p x n) and A(m x n ) BA ( p x n) is defined so it would mean p=n</p>
|
135,218 | <p>What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research?</p>
<p>(Edit - Since the first version of the question got closed, examples should showcase specific work of specific mathematicians. Tools such as Wolfram Alpha, bibliography managers, arXiv ... | Ulrich Pennig | 3,995 | <p>I recently downloaded Colwiz, which is a collaboration tool combined with a paper management system. The homepage is at <a href="http://www.colwiz.com">http://www.colwiz.com</a>. It is far too early for me to write a review about this, but I may come back to that.</p>
<p>Another gem I use a lot is <a href="http://w... |
135,218 | <p>What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research?</p>
<p>(Edit - Since the first version of the question got closed, examples should showcase specific work of specific mathematicians. Tools such as Wolfram Alpha, bibliography managers, arXiv ... | David White | 11,540 | <p>There's an arXiv app which I find extremely useful. I have an android, but I imagine this app exists for other phone types too. There are also several latex apps. Detexify, verbtex, and texportal spring to mind</p>
|
135,218 | <p>What apps can be found in the Apple app store, Google Play, Blackberry World etc. showcasing specific mathematics research?</p>
<p>(Edit - Since the first version of the question got closed, examples should showcase specific work of specific mathematicians. Tools such as Wolfram Alpha, bibliography managers, arXiv ... | Aaron Meyerowitz | 8,008 | <p>Wolfram Alpha was worth the 3 dollars I paid for it for entertainment value alone, but also
in case I care to factor a large integer,
solve a differential equation and the like .</p>
<p>It's a little hit and miss how to phrase requests ( but cool when it works, see below) and basically is a computation engine in ... |
2,439,527 | <p>What process can lead to Nash equilibrium (or a strategy profile close to NE) actually being played in a game, when real humans are the players?</p>
<p>Consider a game, where rationalising is sufficiently difficult, so that some players do not rationalize about their own and their opponent's actions in a sufficient... | Hans | 64,809 | <p>Here is <a href="https://www.quantamagazine.org/in-game-theory-no-clear-path-to-equilibrium-20170718/" rel="nofollow noreferrer">an article</a> and <a href="https://arxiv.org/abs/1608.06580" rel="nofollow noreferrer">a paper</a> linked to therein that appear to deal with your question.</p>
|
1,550,541 | <p>After trying for hours I decided to ask. Please can anyone help me with this problem.</p>
<p>"Two cards are drawn at random and are thrown away from a pack of 52 cards. Find the probability of getting an Ace from the remaining 50 cards."</p>
<p>Please explain the correct method to do this.</p>
<p>I'm getting answ... | Alex | 38,873 | <p>Use Law of total probability. There could have been 0,1 or 2 aces in the two discarded cards, hence you want:
$$
P(A) = P(A|E_0)P(E_0) + P(A|E_1)P(E_1) + P(A|E_2)P(E_2)
$$
can you find all these terms?</p>
|
1,710,244 | <p>So, I have this problem on my analysis homework. We've been looking at sequences of functions. In particular, we are looking at $x^n$ and the question asks us to show that there is a sequence of values of $x$ on $[0,1]$, call it $(x_n)$, such that $\lim_{n\to\infty} (x_{n}^n) \neq 0 = lim_{n\to\infty} x^n$ on $[0,1... | TJ Evert | 324,042 | <p>One interesting sequence is $x_{n+1}=rx_n(1-x_n)$, where $r$ is a parameter that can be varied from $0$ to $4$.
If $0 < r < 1$, clearly the sequence will converge to $0$ and with $1 <r < 3$, $x_n$ converges to $1 - 1/r$.</p>
<p>But the terms of sequence will "bifurcate" for $r>3$. ultimately leading... |
2,677,401 | <p>Consider all $1000$–element subsets of $\{1,2,3,4,\dots 2015\}$. From each such subset select least element. Find Arithmetic Mean of all these elements.</p>
<p>I easily managed the trick but my approach got me a lengthy answer. I selected leaste numbers and multiplied them by their number of subsets possible such a... | user577215664 | 475,762 | <p>You can do it ......
$$(xy^{4}+y)dx - xdy = 0$$
$$(xy^{4}+y)= xy' $$
$$xy^{4}= -(y-xy') $$
Here you have what you wanted
$$xy^{2}= -\left(\frac {y-xy' }{y^2}\right )$$
$$xy^{2}= -\left(\frac {x }{y}\right)' \implies \int x^{3}dx= -\int\left(\frac xy \right )^2d\left(\frac {x }{y}\right)$$
You can conitnue but I pref... |
375,729 | <p>If $(X,\mathfrak D)$ and $(Y,\mathfrak C)$ are uniform spaces, $X$ compact and $f:X\rightarrow Y$ continuous, why $f$ is uniformly continuous? </p>
| Will Jagy | 10,400 | <p>For any square matrix with one value on the diagonal and another value everywhere else, a consistent pattern of (orthogonal) eigenvectors for the $n$ by $n$ case can be read from the columns of
$$
\left( \begin{array}{rrrrrrrrrr}
1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 ... |
103,133 | <p>I am trying to send message to my LinkedIn contact using Mathematica 10.3, but there is no enough information (documented) on how to do that. I am able to login in LinkedIn and see my user data with:</p>
<pre><code>lk = ServiceConnect["LinkedIn"] (* works fine *)
ud=ServiceExecute[lk, "UserData"]; (* works fine... | Searke | 144 | <p>Long story short, LinkedIn changed their API and what it can do. The documentation needs to be updated and the function probably needs to be changed a bit. </p>
<p>"UserData" and "Share" requests should work, but the rest won't. </p>
<p>The syntax for creating messages should look something like this:</p>
<pre><c... |
256,278 | <p>I was thinking about the following problem :</p>
<p>Define $ f:\mathbb C\rightarrow \mathbb C$ by </p>
<p>$$f(z)=\begin{cases}0 & \text{if } Re(z)=0\text{ or }Im(z)=0\\z & \text{otherwise}.\end{cases}$$</p>
<p>Then the set of points where $f$ is analytic is:</p>
<blockquote>
<p>(a) $\{z:Re(z)\neq 0$ an... | learner | 33,640 | <p>I take $f(z)=u+iv$ and $z=x+iy$ where $x= \Re(z), y=\Im(z).$ In case ,$\Re(z), \Im(z) \ne 0$, then we can take $f(z)=z$ so that $$u+iv=x+iy$$ and so $$u=x,\,\,v=y$$. Whence $u_x=1=v_y,\,\, u_y=0=-v_x$. So, C-R equations are satisfied and hence option (a) is correct. </p>
<p>But if we take $f(z)=iy ,y \ne 0\,\,... |
988,028 | <p><em>Disclaimer: This is just meant as record of a proof. For more details see: <a href="http://blog.stackoverflow.com/2011/07/its-ok-to-ask-and-answer-your-own-questions/">Answer own Question</a></em></p>
<hr>
<p>How to prove that the Lebesgue measure has no atoms:
$$\lambda:\mathbb{R}^n\to\mathbb{R}_+$$
<em>(Reca... | Harald Hanche-Olsen | 23,290 | <p>Here is one way: Let $B_r$ be the box $\{x\in\mathbb{R}^n: |x_k|\le r\text{ for } k=1,\ldots,n\}$. You can easily prove that $\lambda(B_r\cap A)$ is a continuous function of $r$, for any Lebesgue measurable set $A$. From this, it follows immediately that $A$ is not an atom.</p>
<p><strong>Edit:</strong> To explain ... |
599,651 | <p>I'm hoping for link to some resource which can explain why the following is true.</p>
<p>$$ x^2 + 104x - 896 = 0 $$</p>
<p>Using the quadratic formula we pull a = 1, b = 104, c = 896. Putting that into the formula for the discriminant we get $ 104^2 - 4.1.896 $ which is 10816 - 3584 = 7232.</p>
<p>Using the quadr... | Flowers | 93,842 | <p>I cannot comment, but you are obviously mistaken in your calculation. Your quadratic formula that is. Here, <a href="http://www.wolframalpha.com/input/?i=x%5E2%2B104x%E2%88%92896%3D0" rel="nofollow">WA does not make mistakes</a></p>
<p>EDIT:
$$\frac{-104\pm\sqrt{104^2+4\cdot 896}}{2}\\
=\frac{-104\pm\sqrt{14400}}{2... |
669,440 | <p>When answering <a href="https://math.stackexchange.com/q/667958/91741">this question</a>, I came up with the following:<br>
Suppose we have a length $l$ and integrate $\frac 1 l$ over $l$:
$$\int \frac {\operatorname d\!l}l=\ln l$$
Since the dimension of $l$ is length, and $\ln$ should have a dimensionless argument,... | Dmoreno | 121,008 | <p>I think your integral does not make any sense, at least in a physical sense, because it's not a definite integral. Indeed, when computing the non-dimensional elongation in a given direction, this should be as follows:</p>
<p>$$\epsilon = \int^L_{L_0} \frac{d l}{l} = \ln{\frac{L}{L_0}},$$</p>
<p>so the units are co... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | Andrew F. | 142,655 | <p>"Are there contradictions in maths?" It's a great question.
Before Russell Paradox It was generally believed that there were not contradictions. Better said, the matter hadn't been spotted yet. After Bertrand Russell the first crisis in maths appeared. And, from my point of view, the issue wasn't solved and it is... |
753,997 | <p>Someone told me that math has a lot of contradictions. </p>
<p>He said that a lot of things are not well defined.</p>
<p>He told me two things that I do not know.</p>
<ul>
<li>$1+2+3+4+...=-1/12$</li>
<li>what is infinity $\infty$?</li>
</ul>
<p>Since I am not a math specialist and little. How to disprove the pr... | John Middlemas | 167,464 | <p>Yes, it does. The mathematics with unrestricted infinite sets, before Zermelo rectified it after Russell's paradox, was contradictory. Why should we suppose it perfectly clean now then? Look to the zero and the infinity for contradictions since these two are the source of continual problems and directly related.</p>... |
1,773,543 | <p>I am trying to understand whether or not the statement $X′ ∩ Y′ ⊆ (X ∩ Y)′$ is true or false.</p>
<p>In trying to develop a counterexample I started asking myself what sets might make it false. That led to the following question. </p>
<p>If $(A ∩ B)$ = Empty Set, what is $(A ∩ B)′$ ?</p>
| thanasissdr | 124,031 | <p>Let $X$ be a topological space and $S= \varnothing$. A $x\in X$ is a limit point of $S$ if for every $\epsilon >0$ it holds:
$$\big(B(x,\epsilon)\setminus \{x\}\big) \cap S \neq \varnothing.$$
But $S = \varnothing,$ thus the above intersection will always be the empty set for any $x\in X$. Hence, there are no $x ... |
890,770 | <p>In a proof I've come across the following identity:</p>
<blockquote>
<p>$$ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} = \binom{n}{j} $$</p>
</blockquote>
<p>I see that it's right, when plugging in numbers, but I don't see the algebraic or combinatorial proof behind it. Can anyone help me with that?</... | Community | -1 | <blockquote>
<p>Just for seeing an elementary one:</p>
</blockquote>
<p><span class="math-container">$$ \sum_{l=0}^{n-j} \binom{M-1+l}{l} \binom{n-M-l}{n-j-l} =\sum_{l=0}^{n-j}\left(-1\right)^{l} \binom{-M}{l} \left(-1\right)^{n-j-l} \binom{-j+M-1}{n-j-l} $$</span>
<span class="math-container">$$=\left(-1\right)^{n-... |
2,959,859 | <p>Let <span class="math-container">$\theta$</span> be a root of <span class="math-container">$p(x)=x^3+9x+6$</span>, find the inverse of <span class="math-container">$1+\theta$</span> in <span class="math-container">$\mathbb{Q(\theta)}$</span>.</p>
<p>So problems like this really annoy me but I did crappy on the last... | Bill Dubuque | 242 | <p>Correct: it takes only <span class="math-container">$1$</span> step (division <span class="math-container">$f\div g)\,$</span> in the <a href="https://math.stackexchange.com/a/85841/242">extended Euclidean algorithm</a> to invert a <em>linear</em> polynomial <span class="math-container">$\,g\,$</span> since <span c... |
97,095 | <p>I want to generate a list of <strong><em>n</em></strong> coordinate points which are on the circumference of an ellipse. I wrote this code:</p>
<pre><code>n = 150;
ellipseFunc[a_,b_,t_] := {(a*b*Cos[t]/Sqrt[b*b*Cos[t]*Cos[t] + a*a*Sin[t]*Sin[t]]), (a*b*Sin[t]/Sqrt[b*b*Cos[t]*Cos[t] + a*a*Sin[t]*Sin[t]])};
listell... | Community | -1 | <p>And the version with <code>FindInstance</code>:</p>
<pre><code>c = 2;
FindInstance[0 < a <= 5 && 1 < b < 2 && a + b < 8, {a, b}, c]
{{a -> 9/41, b -> 133/102}, {a -> 5, b -> 7/6}}
</code></pre>
<p>You can increase the variable <code>c</code> to get more solutions. if <cod... |
137,483 | <p>Let $M$ be a closed Riemannian manifold and $\omega$ and $\eta$ two differential forms of the same degree. Then one can consider $\int_M \omega \wedge *\eta$, where $*$ denotes the Hodge star operator. Can you tell me, why this defines a scalar product or at least where I can find a proof of this fact? In particular... | Jonathan | 12,513 | <p>The Riemannian metric extends to a metric on all tensors of rank $(k,l)$, and in particular it extends to $p$-forms. The very <em>definition</em> of the Hodge star operator is that
$$
\omega \wedge \ast \eta = \langle \omega, \eta \rangle d\mathrm{vol}
$$
where $\langle \omega, \eta \rangle$ is the pairing on $p$-fo... |
7,420 | <p>I have a stand alone <code>Manipulator</code> (it is not in a <code>Manipulate</code>) and would like to know how to alter the style of the label. There are no explicit styling options for <code>Manipulator</code> so I tried wrapping it in <code>Style</code>:</p>
<pre><code>Style[
Manipulator[0.5, AppearanceElemen... | wxffles | 427 | <p>A <code>Slider</code> seems to have the options you want</p>
<pre><code>Slider[0.5, Appearance -> "Labeled", BaseStyle -> FontSize -> 20, ImageSize -> 100]
</code></pre>
<p><img src="https://i.stack.imgur.com/MAPoe.png" alt="enter image description here"></p>
<p>I'm not sure of the differences between... |
7,420 | <p>I have a stand alone <code>Manipulator</code> (it is not in a <code>Manipulate</code>) and would like to know how to alter the style of the label. There are no explicit styling options for <code>Manipulator</code> so I tried wrapping it in <code>Style</code>:</p>
<pre><code>Style[
Manipulator[0.5, AppearanceElemen... | Mike Honeychurch | 77 | <p>Based on the code digging by @kguler the internal code controlling this is:</p>
<pre><code>FEPrivate`If[#9, {InputFieldBox[#1, Expression,
FieldSize -> {{4, 10}, {1, 2}}, Enabled -> #6,
Appearance -> {"Frameless", #8}, BaseStyle -> "Manipulator"]}, {}]
</code></pre>
<p>so it looks like you can ... |
160,323 | <p>I am working on a subject of geometric group theory closely related to 3-manifolds, and in order to understand these links, I am seeking a good reference about 3-manifolds, as self-contained as possible, and in particular dealing with: loop and sphere theorems, Heegaard diagrams, Haken manifolds.</p>
<p>I browsed H... | Kevin Walker | 284 | <p>For the sake having an official answer, as opposed to a comment-as-answer, I'll second Mark Grant's suggestion of Rolfsen's "Knots and Links". It was the first book I read on 3-manifold topology, and I enjoyed it very much.</p>
|
2,839,457 | <p>Let $0 < \epsilon$ and $\delta < 1$, and let $Y$ be a random variable ranging in the interval $[0,1]$ such that $E(Y)=\delta + \epsilon$. Give a lower bound on $Pr[Y ≥ \delta + \epsilon/2].$</p>
<p>The standard application of Markov's Inequality gives the upper bound instead of lower. I tried to start with th... | asdf | 436,163 | <p>Equating the $2$ equations gives</p>
<p>$$9(x^2+y^2)=(3x+y)^2$$</p>
<p>$$9x^2+9y^2=9x^2+6xy+y^2$$</p>
<p>$$8y^2=6xy$$</p>
<p>$$y(y-\frac{3}{4}x)=0$$</p>
<p>Hence either $y=0$ of $y=\frac{3}{4}x$</p>
<p>Note that we are allowed to square the equation in a small neighbourhood of $(3,0,1)$ as $6x+2y>0$ there.<... |
1,336,692 | <p>Why $y=e^x$ is not an algebraic curve over $\mathbb R$? I can say that is not a algebraic curve over $\mathbb C$ because $e^x$ is a periodic function, but what about $\mathbb R$?</p>
<p><strong>EDIT:</strong></p>
<p>I don't want to use trascendence of $e$. Or, I can ask this question for $y=2^x$.</p>
<p><strong>U... | Emilio Novati | 187,568 | <p>An algebraic curve over $\mathbb{Q}$ is, by <a href="https://en.wikipedia.org/?title=Algebraic_curve" rel="nofollow">definition</a> :</p>
<blockquote>
<p>a set of points on the Euclidean plane whose coordinates are zeros of
some polynomial in two variables with coefficients in $\mathbb{Q}$, i.e are algebraic nu... |
2,709,696 | <p>stuck on a question and can't seem to make any progress:</p>
<p>We have an insurance company who expects the number of accidents their policy holders will have is Poisson distributed. The Poisson mean $\Delta$ follows a Gamma distribution with the $\Gamma$(2,1) density function being $f_{\Delta}(\lambda) = \lambda ... | oneequalstwo | 46,267 | <p>It appears that this is going to quickly turn into a Pell-like Diophantine equation.</p>
<p>First, notice that $2^{2n+1}$ can be expressed in terms of $2^n$. Because of this, consider the substitution $k=2^n$. Then try manipulating the resulting equation to give you a Pell-like equation. Once you do this, the metho... |
4,465,535 | <p>Let <span class="math-container">$R$</span> be a ring with unity.</p>
<p>An element <span class="math-container">$e\in R$</span> is called <strong>idempotent</strong> if <span class="math-container">$e^2=e$</span>. Clearly, <span class="math-container">$0,1$</span> are idempotents.</p>
<p>An element <span class="mat... | gnasher729 | 137,175 | <p>You know exactly 1000 wrong answers were given, and each of 1000 candidates gave at least one incorrect answer. Nobody can have given two wrong answers, or we would have had at least 1001 wrong answers. Therefore everyone gave exactly one wrong answer snd got all the others right.</p>
<p>Of 1000, 300 got answer 3 wr... |
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