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 |
|---|---|---|---|---|
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| algori | 2,349 | <p>While we are all waiting for sensible replies let me say this: one can't learn a new area of mathematics without asking at least one hundred silly questions (which is why Mathoverflow is such a great website by the way).</p>
<p>On a more serious note: learning really new stuff involves rethinking the basics. Or, as... |
34,215 | <p>How do professional mathematicians learn new things? How do they expand their comfort zone? By talking to colleagues? </p>
| Gordon Royle | 1,492 | <p>Teaching a course in something is the only way that I can really learn something new.</p>
|
3,493,387 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be two topological spaces, and let <span class="math-container">$X^*$</span> and <span class="math-container">$Y^*$</span> denote their one-point compactification (<span class="math-container">$X^* := X \cup \{\infty\},\,\mathcal... | Matsmir | 685,805 | <p>Robert gave you an example. I'll try to explane some "logic".</p>
<p>You said that <span class="math-container">$f^{-1}(K)$</span> is closed since <span class="math-container">$K \subset Y$</span> is closed and <span class="math-container">$f$</span> is continuous. That is true but in the sense of space <s... |
3,493,387 | <p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be two topological spaces, and let <span class="math-container">$X^*$</span> and <span class="math-container">$Y^*$</span> denote their one-point compactification (<span class="math-container">$X^* := X \cup \{\infty\},\,\mathcal... | Stinking Bishop | 700,480 | <p>The mistake is in the step "Since we know that <span class="math-container">$X$</span> topological space implies <span class="math-container">$X^∗$</span> compact, and closed subsets of a compact are compact, we conclude <span class="math-container">$f^{−1}(K)$</span> is compact..." We know that <span class="math-co... |
240,461 | <p>What's the mathematica command to get the <strong>numerical value</strong> of :</p>
<p><span class="math-container">$$PV\int_0^\infty \frac{\tan x}{x}\text{d}x?$$</span></p>
<p>where <span class="math-container">$PV$</span> is the principal value.</p>
| Michael E2 | 4,999 | <p>The integral is <code>Pi/2</code>, a proof of which may be found on <a href="https://math.stackexchange.com/questions/2141437/the-principal-value-of-an-integral">math.SE</a>. Here's a numerical check, integrating <span class="math-container">$(\tan z)/z$</span> over parallel paths <span class="math-container">$z = ... |
1,319,476 | <p>This is a question related to another posted question:</p>
<p>The answer to the following question "Find all solutions to: $e^{ix}=i$" is as follows: </p>
<p>"Euler's formula: $e^{ix}=\cos(x)+i\sin(x)$,</p>
<p>so: $ \cos x+i\sin x=0+1⋅i$</p>
<p>compare real and imaginary parts
$\sin(x)=1$
and
$\cos(x)=0$</p>
<p... | Emanuele Paolini | 59,304 | <p>$\sin$ and $\cos$ functions are $2\pi$-periodic which is: $\sin(x+2n\pi)=\sin x$, $\cos(x+2n\pi)=\cos(x)$. So when you find that $x=\pi/2$ is a solution, then also $x_n = \pi/2 + 2n\pi$ is a solution for every $n\in \mathbb Z$ (where $\mathbb Z$ are whole numbers: $0, 1, -1, 2, -2,\dots$)</p>
<p>Notice that
$$
\f... |
3,524,550 | <p>Lines in <span class="math-container">$\mathbb{R}^3$</span> are all congruent to one another, but circles in <span class="math-container">$\mathbb{R}^3$</span> are not all congruent to one another (because two different circles may have different radii). Visually, this is completely obvious. However, I would like ... | Eric Wofsey | 86,856 | <p>Here's the general group-theoretic setup. Let <span class="math-container">$G$</span> be a group and <span class="math-container">$G_0,H\subset G$</span> be subgroups. An orbit of <span class="math-container">$H$</span> in <span class="math-container">$G/G_0$</span> can be considered as a double coset <span class=... |
23,020 | <p>I am in the process of learning about Mapping class groups. At this point it seems like most of what I've read involves very low dimensional (surfaces and 3-manifolds) applications.</p>
<p>I was wondering if they were studied (or arise naturally) in higher dimensional settings?</p>
<p>In particular, any references... | Ryan Budney | 642 | <p>In high dimensions there are several variants that are all distinct (which for surfaces they all agree). There's mapping class groups in the "homotopy category" meaning the homotopy-classes of homotopy equivalences of a topological space, with composition giving the group structure. This is a "core&... |
4,135,472 | <p><strong>What is the clearest and simplest way of proving that <span class="math-container">$[x]+[x+1/2]=[2x]$</span>? (Where <span class="math-container">$[x]$</span> is the greatest integer function)</strong></p>
<p>According to Bartleby, if <span class="math-container">$x=m$</span> for <span class="math-container"... | N. S. | 9,176 | <p>My favourite, and the cleanest way in my oppinion is the following classical solution:</p>
<p>Let <span class="math-container">$f(x)= \lfloor x\rfloor+\lfloor x+\frac{1}{2}\rfloor-\lfloor 2x \rfloor$</span>.</p>
<p>Then, it is trivial to see that</p>
<ul>
<li><span class="math-container">$f(x+\frac{1}{2})=f(x)$</spa... |
1,863,943 | <p>I have the following problem:</p>
<blockquote>
<p>Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute
$H_i(M,\mathbb{Z})$ and $H^i(M,\mathbb{Z})$ for all $i$.</p>
</blockquote>
<p>I know that if $M$ is orien... | Eric Wofsey | 86,856 | <p>For any connected manifold $M$, there is a homomorphism $\pi_1(M)\to\mathbb{Z}/2$ which sends a loop to $0$ if going around the loop preserves orientation and sends the loop to $1$ if going around the loop reverses orientation. This homomorphism is trivial iff $M$ is orientable. Since $\mathbb{Z}/2$ is abelian, th... |
1,863,943 | <p>I have the following problem:</p>
<blockquote>
<p>Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute
$H_i(M,\mathbb{Z})$ and $H^i(M,\mathbb{Z})$ for all $i$.</p>
</blockquote>
<p>I know that if $M$ is orien... | Najib Idrissi | 10,014 | <p>A quick proof using <a href="https://en.wikipedia.org/wiki/Stiefel%E2%80%93Whitney_class" rel="noreferrer">Stiefel–Whitney classes</a>: a manifold $M$ is orientable iff the first SW class $w_1(M) \in H^1(M;\mathbb{Z}/2\mathbb{Z})$ is zero. But by the universal coefficient theorem,
$$H^1(M;\mathbb{Z}/2\mathbb{Z}) = \... |
493,104 | <p>I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$, we can write the polar equation of the ellipse as $r = \sqrt{a^2 \cos^2{\thet... | Bennett Gardiner | 78,722 | <p>Here you go - this person even made your mistake, then someone else corrected it.</p>
<p><a href="https://web.archive.org/web/20180312075252/http://mathforum.org/library/drmath/view/53635.html" rel="nofollow noreferrer">Link</a></p>
|
493,104 | <p>I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as $\vec{r}(\theta) = (a\cos{\theta}, b\sin{\theta})$, we can write the polar equation of the ellipse as $r = \sqrt{a^2 \cos^2{\thet... | Community | -1 | <p>There are already a lot of good answers here, so I'm adding this one
primarily to dazzle people w/ my Mathematica diagram-creating skills. </p>
<p>As noted previously, </p>
<p>$x(t)=a \cos (t)$ </p>
<p>$y(t)=b \sin (t)$ </p>
<p><strong>does</strong> parametrize an ellipse, but t is <strong>not</strong> the cent... |
2,868,595 | <p>A Vitali set is a subset $V$ of $[0,1]$ such that for every $r\in \mathbb R$ there exists one and only one $v\in V$ for which $v-r \in \mathbb Q$. Equivalently, $V$ contains a single representative of every element of $\mathbb R / \mathbb Q$.</p>
<p>The proof I read is in this short article on Wikipedia: <a href="h... | Dennis | 513,093 | <p>We assume for the sake of contradiction $V$ is measurable, so $\lambda(V)$ is a real number. By construction, each $V_k$ is disjoint, and is merely a shifted copy of $V$, so each of these sets has the same measure. The sequence $(V_k)_{k=1}^\infty$ is countable.</p>
<p>Sigma additivity (the assumption in the defini... |
3,154,032 | <p>Suppose we have a 4 dimension positive signature clifford algebra. In <a href="https://math.stackexchange.com/questions/443555/calculating-the-inverse-of-a-multivector">Calculating the inverse of a multivector</a> and <a href="https://math.stackexchange.com/questions/556247/inverse-of-a-general-nonfactorizable-multi... | amnesiac | 419,567 | <p>Naively speaking, the existence of inverses will depend on the signature <span class="math-container">$(p,q)$</span> of the quadratic space <span class="math-container">$\mathbb{R}^{p,q}=(\mathbb{R}^{p+q},g)$</span>, in which for an orthonormal basis <span class="math-container">$\{e_i\}_{i=1}^{n=p+q}$</span> and <s... |
184,719 | <p>Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? </p>
<p>Moreover, what is $[G,G]$; e.g. if $g=2$?</p>
| Holonomia | 43,122 | <p>Consider the Abel-Jacobi map $\mu : \Sigma_g \to J(\Sigma_g )=\mathbb{C}^g/\Lambda$.
Then take the lift $\widetilde{\mu}: \mathbb{H} \to \mathbb{C}^g$ from the universal covering $\mathbb{H}$ of $\Sigma_g$. It seems to me that the image $X := \widetilde{\mu}(\mathbb{H})$ is the surface $\mathbb{H}/[G,G]$.</p>
|
184,719 | <p>Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? </p>
<p>Moreover, what is $[G,G]$; e.g. if $g=2$?</p>
| Sam Nead | 1,650 | <p>$\newcommand{\ZZ}{\mathbb{Z}}\newcommand{\RR}{\mathbb{R}}$Let $S = \Sigma_2$ be the genus two surface. In this case, $\ZZ^4$ is the deck group of the desired covering. Consider $\ZZ^4$ inside of $\RR^4$ and add to these points the usual edges labelled $a, b, c, d$ parallel to the four coordinate axes. This gives ... |
2,197,790 | <h3>Question</h3>
<blockquote>
<p>A sequence $\{a_n\}$ of real numbers is said to be a Cauchy sequence of for
each $\epsilon$ > 0 there exists a number $N > 0$ such that m, $n > N$ implies
that $|a_n − a_m| <\epsilon$.</p>
<p>Prove that every convergent sequence is a Cauchy sequence</p>
</blockquot... | Bernard W | 278,779 | <p>The idea behind the standard proof that every convergent sequence is Cauchy is the triangle inequality.</p>
<p>If a sequence $a_n$ converges to a limit $L$ then for every $\epsilon>0$ there is some cutoff point $N$ such that every term past $N$ is within $\epsilon$ of $L$.</p>
<p>This means that any two points ... |
2,479,918 | <p>Every vector space $V$ could be embedded into $V^{\ast}$ (see <a href="https://en.wikipedia.org/wiki/Dual_basis" rel="noreferrer">here</a>) after choosing a basis, for a given vector $v \in V$ denote this embedding by $v^{\ast}\in V^{\ast}$. Now for given vector spaces $V_1, \ldots, V_k$ over some field $F$, let $V ... | Arnaud D. | 245,577 | <p>You claim that you only need $T$ to be a subspace of $V^*$; but in your construction, you define $\pi$ as a multilinear map $V_1\times \dots \times V_k\to V^*$, and it is not obvious that $\pi$ can be restricted to have domain $T$. In other words, given $v_i\in V_i$ for $i=1,\dots,k$, it is not obvious $\pi(v_1,\dot... |
4,520,485 | <p>Suppose there are 78 heroes. Only one of them is considered to be 'Tier 1'. At the beginning of some game you are given a choice between either 2 heroes or 4 heroes. The question is: how advantegeous is it to choose out of 4 heroes to choosing out of 2, if by advantageous we mean to have a higher probability of gett... | José Carlos Santos | 446,262 | <p>The expression <span class="math-container">$(k+1)(k+2)\ldots n$</span> is the product of the numbers <span class="math-container">$k+1$</span>, <span class="math-container">$k+2$</span>, …, <span class="math-container">$n$</span>. When <span class="math-container">$n=3$</span> and <span class="math-container">$k=2$... |
670,522 | <p>In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic geometry. It seems as if there is some relation between hyperbolic geometry and modular forms, for example, why is it... | DIEGO R. | 297,483 | <p>I think that introducing the projection map is more intuitive. Given a vector space <span class="math-container">$V$</span>, a linear map <span class="math-container">$P: V \to V$</span> is said to be a linear projection if <span class="math-container">$P^{2} = P$</span>. As a consequence of this definition, given a... |
3,440,093 | <p>The problem is minimize over all <span class="math-container">$\theta \in \mathbb{R}^n$</span></p>
<p><span class="math-container">$$\frac{1}{2} ||Y - \theta||^2$$</span> subject to <span class="math-container">$A \theta = 0$</span> where <span class="math-container">$A$</span> is <span class="math-container">$m \t... | ironX | 534,898 | <p>I was able to solve this using Lagrangian, <span class="math-container">$L(\theta) = \frac{1}{2} ||Y - \theta||^2 + \lambda^T A \theta$</span></p>
<p><span class="math-container">$\nabla_\theta L(\theta) = 0$</span> <span class="math-container">$\implies $</span> <span class="math-container">$\theta^* = Y - A^T \la... |
3,242,844 | <p><span class="math-container">$$\int_0^{\pi/6} \frac{x\cos x}{1+2\cos x}dx$$</span></p>
<p>Does it have a closed solution? <a href="https://www.wolframalpha.com/input/?i=int%20%5Cfrac%7Bxcos%20x%7D%7B1%2B2cos%20x%7Ddx%20from%200%20to%20%5Cpi%2F6" rel="nofollow noreferrer">WA</a> outputs this result.</p>
| R. Burton | 614,269 | <p>Because...</p>
<p><span class="math-container">$$\int \frac{x\cos x}{1+2\cos x}dx=\int_0^x \frac{t\cos t}{1+2\cos t}dt$$</span></p>
<p>...the definite integral...</p>
<p><span class="math-container">$$\int_0^{\pi/6} \frac{x\cos x}{1+2\cos x}dx$$</span></p>
<p>...may be regarded as the <em>value</em> of the funct... |
2,210,893 | <p>A lot of times when proving for example inequalities like $$x \leq y$$
for real numbers $x,y$ the argument looks like
$$x \leq y + \varepsilon$$
for all $\varepsilon > 0$, hence $x \leq y$. </p>
<p>Now this is obviously very intuitive, but is there a "proof" that this conclusion is correct? And is it always suf... | Community | -1 | <p>Consider </p>
<p>$$t\le\epsilon$$ for all $\epsilon>0$.</p>
<p>Clearly,</p>
<p>$$t\le0$$ is compatible, while </p>
<p>$$t>0$$ is not because</p>
<p>$$0<t\le\epsilon$$ cannot hold for all $\epsilon>0$.</p>
<p>Rewrite with $t:=x-y$.</p>
|
4,380,475 | <p>I'm trying to differentiate <span class="math-container">$x\sqrt{4-x^2}$</span> using the definition of derivative.</p>
<p>So it would be something like</p>
<p><span class="math-container">$$\underset{h\to 0}{\text{lim}}\frac{(h+x) \sqrt{4-\left(h^2+2 h x+x^2\right)}-x \sqrt{4-x^2}}{h}$$</span></p>
<p>I was trying t... | Matteo | 686,644 | <p>Clearly, when <span class="math-container">$h\to 0$</span>:
<span class="math-container">$$\frac{h\cdot \sqrt{4-\left(h^2+2 h x+x^2\right)}}{h}\to \sqrt{4-x^2}$$</span>
So, the limit simplify to:
<span class="math-container">$$\lim_{h\to 0}\frac{x\sqrt{4-\left(h^2+2 h x+x^2\right)}-x \sqrt{4-x^2}}{h}=\sqrt{4-x^2}+\l... |
18,686 | <p>Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. <a href="http://www.cs.princeton.edu/~funk/tog02.pdf">This paper (section 4.2)</a> says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices:</p>
<p>$P = (1 - \sqrt{... | Gilbert Colgate | 644,609 | <p>Note that these points, when random, will be uniformly distributed in a nicely random way, but if you loop through r1 and r2 with an increment (say .01) your resulting points will have unusual artifacts and not look randomly distributed.
One end of the triangle may have few points.</p>
<p>I determined this with cod... |
1,942,578 | <p>Consider the following wedge</p>
<p><a href="https://i.stack.imgur.com/xiaPX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xiaPX.png" alt=""></a> cut from a cylinder of radius r.
The plane that cuts the wedge goes through the very bottom of the cylinder leading to an ellipse as the cross sectio... | axioman | 369,033 | <p>The farther you get x towards zero, the bigger 1/x gets and therefore you need to choose a bigger n to get $\frac{1}{nx}<\epsilon$. Therefore the convergence can't be uniform.
On any compact interval like [0.5,1] there should not be a problem, because 1/x will have a maximum value on the interval, so if you take ... |
1,835,158 | <p>I wont to choose three random integer point in origin $|x|\leq r, |y|\leq r$ at plane as $(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})$.
What the probability that this three point create a right triangle ( it is depend to r? what about isosceles triangle?
I think that its zero but I cant proof. Thank you.</p>
| Empy2 | 81,790 | <p>The following makes me think the number of right-angled triangles within the square $$[0,N]\times[0,N]\subset\mathbb{Z}^2$$
is some multiple of $N^4\log N$. Since there are $O(N^6)$ triangles, this is a small proportion of them, as expected. </p>
<ol>
<li><p>Take triangles whose short sides are aligned North-Sout... |
4,174 | <p>I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8. We will do this through focus on the common core. I'm also putting together a collection of suggested readings from the math education literature. I would be interested to hear your suggestio... | Burt Furuta | 3,578 | <p>In case you haven't already seen these, I'll suggest articles by Luis Radford and Jean Schmittau. Both are influenced by Vygotsky, but take quite different approaches to algebra. Most of Radford's articles are available as pdfs on <a href="http://luisradford.ca/luisradford/?page_id=13" rel="nofollow">his website</a>... |
3,430,305 | <blockquote>
<p>Let <span class="math-container">$P$</span> be a partition of <span class="math-container">$[0, b]$</span> defined as <span class="math-container">$P = \{ 0 = x_0 < x_1 <
> \ldots < x_n = b\}$</span>, and let <span class="math-container">$c_i \in [x_{i-1}, x_i]$</span> for every <span clas... | Pebeto | 605,486 | <p>Here is a suggestion based on your idea. I will consider the case where <span class="math-container">$0 \notin P.$</span></p>
<p>A hyperplane is such <span class="math-container">$\{ x \in \mathbb{R}^n\ s.t. \ c^t x = c_0 \},$</span> where <span class="math-container">$c \in \mathbb{R}^n, \ c \neq 0$</span> and <sp... |
469,485 | <p>Here's the simple question:</p>
<p>Devon has a piece of poster board 45 cm by 20 cm.
His teacher challenges him to cut the board into parts, then rearrange</p>
<p>the parts to form a square.
a) What is the side length of the</p>
<p>square?
b) What are the fewest cuts</p>
<p>Devon could have made? Explain.</p>
<... | qaphla | 85,568 | <p>The issue with your calculations is that when you cut the 45 off at the 30/15 mark, you get a piece that is 20*30 and a piece that is 20*15. Putting the 20*15 piece along the 20*30 piece will give you an L-shape, that is 35*30 with a 10*15 rectangle cut out of the corner.</p>
<p>The best way that I can see to do th... |
3,333,928 | <p>I am reading an example of root test for a serie: <span class="math-container">$$\sum_{n=1}^\infty\frac{3n}{2^n}.$$</span> So applying the root test, we get <span class="math-container">$$\lim_{n \to \infty}\sqrt[n]{\frac{3n}{2^n}}=\lim_{n \to \infty}\frac{\sqrt[n]{3n}}{2}=\frac{1}{2}\lim_{n\to\infty}\exp(\frac{1}{n... | Community | -1 | <p>A polynomial is an expression obtained by combining constants and variables by means of a <em>finite number</em> of additions and multiplications.</p>
<p>E.g. <span class="math-container">$3xy^3+2x-1$</span>.</p>
<p>An <em>algebraic function</em> of a single variable <span class="math-container">$x$</span> is such t... |
2,764,818 | <blockquote>
<p>Let $f(x)=ax^3+bx^2+cx+d$, be a polynomial function, find relation between $a,b,c,d$ such that it's roots are in an arithmetic/geometric progression. (separate relations)</p>
</blockquote>
<p>So for the arithmetic progression I took let $\alpha = x_2$ and $r$ be the ratio of the arithmetic progressio... | Claude Leibovici | 82,404 | <p>Using your notations
$$ax^3+bx^2+c x+d=a(x-\frac \alpha q)(x-\alpha)(x-\alpha q)$$
Expand the rhs to get after simplifications
$$a x^3 -\frac{a \alpha \left(q^2+q+1\right)}{q}x^2+\frac{a \alpha ^2 \left(q^2+q+1\right)}{q}x-a \alpha ^3$$ Compare the coefficients to get
$$b=-\frac{a \alpha \left(q^2+q+1\right)}{q}$... |
1,772,650 | <p><strong>Note: in relation to the answer of the duplicate question, I see that the second picture below refers to the triangulation when we consider <em>simplicial complexes</em>. I do not understand why the triangles are used as they are, however, so would like some help trying to understand this</strong></p>
<p>I ... | zibadawa timmy | 92,067 | <p>Hint: What are the images of the corners of the big square in the quotient?</p>
<p>See also <a href="https://math.stackexchange.com/questions/953586/triangulation-of-torus">this Q&A</a> (and the comments), where the poster makes much the same mistake.</p>
|
3,419,550 | <p>Let <span class="math-container">$D ⊂ \mathbb{R}$</span> </p>
<p>Let <span class="math-container">$D_A$</span> be the set of all accumulation points of <span class="math-container">$D$</span>. The set <span class="math-container">$\bar{D} := D
\cup
D_A$</span> is called
the closure of <span class="math-container">$... | Math1000 | 38,584 | <p>If <span class="math-container">$\overline D$</span> were unbounded above, there would exist a sequence of elements <span class="math-container">$x_n\subset\overline D$</span> with <span class="math-container">$x_n>n$</span>. Since <span class="math-container">$D$</span> is bounded, there is some <span class="mat... |
2,066,716 | <p>I know how to prove that every infinite subset $A$ of a compact set in a metric space satisfies $A'\ne\emptyset$, but my book also claims the opposite implication, without proof, stating "it falls outside the scope of this book".
I've been searching online but I could only find the "standard" result. </p>
| Amin Idelhaj | 333,883 | <p>Assume every infinite subset of $E$ has a limit point. Then fix some $\delta > 0$, choose some $x_1 \in E$, and then given $x_1,\ldots,x_n$, choose $x_{n+1}$ such that $d(x_{n+1},x_i) \ge \delta$ for $i = 1,\ldots,n$. This terminates in finitely many steps, so $X$ can be covered in finitely many open balls of rad... |
2,066,716 | <p>I know how to prove that every infinite subset $A$ of a compact set in a metric space satisfies $A'\ne\emptyset$, but my book also claims the opposite implication, without proof, stating "it falls outside the scope of this book".
I've been searching online but I could only find the "standard" result. </p>
| David Bowman | 366,588 | <p>We prove this by contradiction.</p>
<p>Suppose $\{F_n\}_{n \in \mathbb{N}}$ is an open cover of $A$ without a finite subcover (i.e. $A$ is not compact). Then define $G_n = (\bigcup_{i=1}^n F_i)^c$. For each $n$, $G_n$ must be nonempty, else that finite subcollection $F_1 \cup ... \cup F_n$ would be a finite subcove... |
2,066,716 | <p>I know how to prove that every infinite subset $A$ of a compact set in a metric space satisfies $A'\ne\emptyset$, but my book also claims the opposite implication, without proof, stating "it falls outside the scope of this book".
I've been searching online but I could only find the "standard" result. </p>
| DanielWainfleet | 254,665 | <p>Let $(X,d)$ be a metric space.</p>
<p>(1). If $D$ is a dense subset of $X$ then $\mathbb B=\{B_d(x,q): x\in D\land q\in \mathbb Q^+\}$ is a base for $X.$ Note that if $D$ is countable then $\mathbb B$ is countable. </p>
<p>(2). If $X$ has a countable base then every open cover of $X$ has a countable sub-cover. </... |
24,305 | <p>I have several functions, let's assume they are:</p>
<pre><code>func1[x_]=x;
func2[x_]=3*x-5;
func3[x_]=0.1*x^2;
</code></pre>
<p>and a lot more like these.</p>
<p>For each and every one of these I want to do the following</p>
<pre><code>xvalues = Range[0, 500, 2.5];
points1 = Map[func1, xvalues];
Do[If points1[... | bill s | 1,783 | <p>How about:</p>
<pre><code>apply[func_] := Module[{}, xvalues = Range[0, 500, 2.5];
points1 = Map[func1, xvalues];
Do[If[points1[[i]] < 0, points1[[i]] = 0], {i, 1, Length[points1], 1}];
table1 = Transpose[{xvalues, points1}]];
</code></pre>
<p>Now you call the function apply with your desired funcX ... |
1,021,599 | <p>Let $X$ be a metric space and $q \in X$. I want to show that the distance function $d(q,p)$ is a uniformly continuous function of $p$. </p>
<p>I know how to show that $d$ is continuous, but I am stuck on how to show UC. </p>
<p>Given $\epsilon >0$ let $\delta =?$. Then if $d(x,y) <\delta$, then $|d(q,x)-d(q,... | Suzu Hirose | 190,784 | <p>$d(q,x) \leq d(q,y) + d(y,x)$ and $d(q,y) \leq d(q,x)+d(x,y)$ so $|d(q,x)-d(q,y)| \leq |d(x,y)| <\epsilon$ if $d(x,y)<\epsilon$.</p>
|
4,613,982 | <p>Calculate the integral <span class="math-container">$$\int_{-\infty }^{\infty } \frac{\sin(\Omega x)}{x\,(x^2+1)} dx$$</span> given <span class="math-container">$$\Omega >>1 $$</span></p>
<p><a href="https://i.stack.imgur.com/mPVqq.jpg" rel="nofollow noreferrer">I tried but couldn't find C1</a></p>
| Abezhiko | 1,133,926 | <p>As you solved a differential equation with respect to <span class="math-container">$\Omega$</span>, <span class="math-container">$C_1$</span> is a constant of integration for this variable, that is why you need an initial/boundary condition such as <span class="math-container">$I(\Omega=0) = -\pi + C_1 = 0$</span>.<... |
3,867,834 | <p>I gotta find the value of <span class="math-container">$x+y$</span> in the following image</p>
<p><a href="https://i.stack.imgur.com/j9RPH.png" rel="noreferrer"><img src="https://i.stack.imgur.com/j9RPH.png" alt="enter image description here" /></a></p>
<p>I have no info about if a point is the middle point of a len... | WA Don | 542,712 | <p>The link <a href="https://en.wikipedia.org/wiki/Trapezoidal_rule#Error_analysis" rel="nofollow noreferrer">Trapezoid rule</a> shows the error term is bounded but is a little unsatisfactory since it does not prove equality in this particular case.</p>
<p>Simplifying to the interval [0,1], If we assume <span class="ma... |
7,715 | <p>I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am completely baffled by this scalar restriction business of having a field extension $K/k$ , a torus over $K$ and "restr... | Evgeny Shinder | 2,260 | <p>It's not that hard at all. Here is an example. Let $k = \mathbf R$ and $K = \mathbf C$.
Consider a 1-dimensional torus $G_m$ over $\mathbf C$. It basically the group $\mathbf C^*$ over $\mathbf C$. </p>
<p>Now $G = Res_{\mathbf C/\mathbf R} G_m$ is the same group $\mathbf C^*$ considered as group over $\mathbf R$.<... |
7,715 | <p>I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article
on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am completely baffled by this scalar restriction business of having a field extension $K/k$ , a torus over $K$ and "restr... | Mikhail Bondarko | 2,191 | <p>There is another description for tori. The category of tori over a field F is equivalent to the category of finite-dimensional $G_F$-lattices. Now, there is an operation of induction for group representations that converts a $G_K$-lattice into a $G_k$-lattice; this is the lattice you need.</p>
<p>See <a href="htt... |
1,862,108 | <p>This question is related to maths, so I post here. Actually it's a computer science question and I am facing this type of question while learning Design and Analysis of Algorithms, but we all know that computer science has complete relation with maths. </p>
<p>Arrange the following functions in increasing order of ... | J Tim | 354,581 | <p>Okay, so $2^{log(n)} < n$ because the logarithm base is greater than 2. Now you might want to see that $2^{2 log(n)} = (2^{log(n)})^2 $ to realise that B is faster growing than A. </p>
<p>Exponential growth is always faster than polynomial, so D has to be the fastest growing one. Furthermore, C > E because $n &... |
2,791,914 | <ul>
<li>$\displaystyle \int_0^\infty \frac{\arctan\frac{x^3}{1+x^2}}{x^2} \, dx$ </li>
</ul>
<hr>
<p>So i know that this one converge from 1 to infinity (by Dirichlet rule), but i'm not sure about the 0 to 1 segment. I kind of think that it converge as well, but can't prove it myself. Any suggestions?</p>
| user | 505,767 | <p><strong>HINT</strong></p>
<p>Note that for $x\to 0$</p>
<p>$$\frac{\arctan\frac{x^3}{1+x^2}}{x^2}=\frac{\arctan\frac{x^3}{1+x^2}}{\frac{x^3}{1+x^2}}\frac{x}{1+x^2}\to 1\cdot 0=0$$</p>
|
4,133,782 | <p>I am having trouble finding a formula that connects the two and can produce an answer. Anyone know how this is done? I tried y=mx+b, m=3, and b=5-a. But I don't know what to do next or did I even start right.</p>
| hm2020 | 858,083 | <p><strong>Question:</strong> "I understand why there can be at most one solution for a full column rank but how does that lead to A having a left inverse? I'd be grateful if someone could help or hint at the answer."</p>
<p><strong>Answer:</strong> Let <span class="math-container">$k$</span> be a real number... |
301,662 | <p>This is a challenging puzzle I heard from my little brother.</p>
<p>For some $n$ and $x$, $\sum_{k=1}^n \sin^{2k}(x) = 2013$.</p>
<p>Is it possible to deduce
$$\sum_{k=1}^n \cos^{2k}(x) \text{ ?}$$</p>
<p>Edit:
I've just noticed something which now seems obvious to me.<br>
Choose $n = 2013$ and $x = \pi/2$ which ... | Community | -1 | <p>let $r=sin^2(x)$ we have</p>
<p>$$\sum_{k=1}^n r^k=\frac{r(1-r^n)}{1-r}=2013$$
Now we want:
$$\sum_{k=1}^n (1-r)^k=\frac{(1-r)(1-(1-r)^n)}{r}$$</p>
<p>We can deduce:
$$2013\sum_{k=1}^n (1-r)^k=(1-r^n)(1-(1-r)^n)$$</p>
|
919,572 | <p>Do you know any nice way of expressing </p>
<p>$$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$
?</p>
<p>Some simple manipulations involving the integrals lead to an expression that also uses<br>
the hypergeometric series. Is there any way of getting a form that doesn't use the HG function?</p>
| Alex | 38,873 | <p>This is a really rough approximation: write $H_{k+1} \sim \log (k+1) \sim \log k + \frac{1}{k}$ then you get two sums:
\begin{align}
&\sum_{k=1}^{n}\frac{\log k }{n-k+1} + \sum_{k=1}^{n} \frac{1}{k(n-k+1)}\\
&\leq \log n \sum_{k=1}^{n}\frac{1}{n-k+1} + \sum_{k=1}^{n} \frac{1}{k(n-k+1)} \\
&\sim \log^2n ... |
919,572 | <p>Do you know any nice way of expressing </p>
<p>$$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$
?</p>
<p>Some simple manipulations involving the integrals lead to an expression that also uses<br>
the hypergeometric series. Is there any way of getting a form that doesn't use the HG function?</p>
| robjohn | 13,854 | <p>Using small steps:
$$
\begin{align}
\sum_{k=1}^{n+1}\frac{H_k}{n-k+2}
&=\sum_{k=1}^{n+1}\sum_{j=1}^k\frac1j\frac1{n-k+2}\tag{1}\\
&=\sum_{j=1}^{n+1}\sum_{k=j}^{n+1}\frac1j\frac1{n-k+2}\tag{2}\\
&=\sum_{j=1}^{n+1}\sum_{k=j}^{n+1}\frac1j\frac1{k-j+1}\tag{3}\\
&=\sum_{k=1}^{n+1}\sum_{j=1}^k\frac1j\frac1... |
2,756,332 | <p>It is <a href="https://math.stackexchange.com/questions/985879/relation-between-trace-and-rank-for-projection-matrices">not difficult</a> to show that if $A \in M_n(k)$ for some field $k$, and $A^2=A$ then $\operatorname{tr}(A) = \dim(\operatorname{Im}(A))$</p>
<p>In <a href="https://mathoverflow.net/questions/1352... | user1551 | 1,551 | <p>Yes. More specifically, over any field $k$, regardless of its characteristic or algebraic closedness (or the lack of it), if $f : M_n(k) \to k$ is $k$-linear and $f(A)=\operatorname{rank}(A)$ (modulo $\operatorname{char}(k)$ if $k$ has finite characteristic) for every projection matrix $A$, then $f$ is necessarily t... |
1,307,069 | <p>Let's look at $f(x)=\cos(x)$ defined on the interval $[0,\pi]$.</p>
<p>We know that for any function $g$ defined on $[0,\pi]$ we have:</p>
<p>$g(x)=\sum_{k=1}^{\infty}B_k\sin(kx)$ where $B_k=\frac{2}{\pi}\int_{0}^{\pi}g(x)\sin(kx)dx$. And $f$ is no different. So in our case:</p>
<p>$B_k=\frac{2}{\pi}\int_{0}^{\pi... | Rory Daulton | 161,807 | <p><em>Any</em> sum of sines such as $\sin kx$ will give zero for $x=0$, since $\sin (k\cdot 0)=0$ for any $k$.</p>
<p>Therefore, you cannot represent the cosine function as a sum of sines of the form $\sin kx$. That's why the usual Fourier series uses both sines and cosines, or the equivalent of $e^{ikx}$. Making you... |
1,307,069 | <p>Let's look at $f(x)=\cos(x)$ defined on the interval $[0,\pi]$.</p>
<p>We know that for any function $g$ defined on $[0,\pi]$ we have:</p>
<p>$g(x)=\sum_{k=1}^{\infty}B_k\sin(kx)$ where $B_k=\frac{2}{\pi}\int_{0}^{\pi}g(x)\sin(kx)dx$. And $f$ is no different. So in our case:</p>
<p>$B_k=\frac{2}{\pi}\int_{0}^{\pi... | orion | 137,195 | <p>Developing a sine series makes an <em>odd</em> periodic extension of the function. So a better question would be, which zero are you talking about? You have a discontinuity at $0$, so it depends on which side you approach from:
$$f(0^+)=1$$
and
$$f(0^-)=-1$$</p>
<p>The fourier series, when faced with a discontinuit... |
1,746,748 | <p>My calculus teacher gave us this problem in class:</p>
<p>Which is easier to integrate?</p>
<p>$$\int \sin^{100}x\cos x dx$$</p>
<p>or</p>
<p>$$\int \sin^{50}xdx$$</p>
<p>By easier, I assume the teacher means which integral would take less work. I'm unsure of how to approach this problem because of the relative... | SchrodingersCat | 278,967 | <p>Obviously, the first one.
$$\int \sin^{100}x \cos x \,\, dx=\int \sin^{100}x \, \,d(\sin x)=\frac{\sin^{101}x}{101}+c$$</p>
<p>And the other integral requires more than <em>just</em> $2$ steps. You may try any method you like.It cannot be rigorously proved that the integral requires more than <em>just</em> $2$ step... |
4,338,190 | <p>Its required to prove that <span class="math-container">$|x^{1/n} -1| \lt \epsilon$</span> for <span class="math-container">$\epsilon \gt 0$</span> and <span class="math-container">$n \ge N$</span> where <span class="math-container">$N \in \mathbb N$</span>.<br>
Let <span class="math-container">$x^{1/n} -1 = h$</spa... | Jochen | 950,888 | <p>It's almost correct :) Like already mentioned your proof only works for <span class="math-container">$x\geq 1$</span> because for <span class="math-container">$0\leq x<1$</span> we get some negative terms in the expansion of <span class="math-container">$(1+h)^n$</span> since <span class="math-container">$h<0$... |
333,807 | <p><strong>Notations</strong>: For a scalar $a\in\mathbb{R}$, denote
$$\mathrm{sgn}(a)=\left\{
\begin{array}{l l}
1 & \mbox{if } a>0\\
0 & \mbox{if } a=0\\
-1 & \mbox{if } a<0
\end{array}.\right.$$
For a vector $r\in\mathbb{R}^n$, $\mathrm{sgn... | user1551 | 1,551 | <p>Counterexample: we have $x=v_1+v_2+v_3$ when
$$
(r_1,r_2,r_3,x,v_1,v_2,v_3)=\begin{pmatrix}
1&1&-3&1&1&1&-1\\
1&-2&2&1&1&-1&1\\
-2&1&1&1&-1&1&1\\
0&0&0&0&0&0&0
\end{pmatrix}.
$$
<strong>Edit.</strong> Here is a better exa... |
513,779 | <p>If $a,b\in\mathbb{N}$ are odd</p>
<p>then demonstrate:
$$ {\sqrt{a^2 + b^2}} \not\in \mathbb{Q}$$ </p>
<p>I try to guess that $$ {\sqrt{a^2 + b^2}} \in\mathbb{Q}.$$ Then i write $$ {\sqrt{a^2 + b^2}= m/n}.$$ After that: $$ {n\sqrt{a^2 + b^2}= m}$$ , I raised at squared and i have like $$ n^2(a^2+ b^2)=m^2 $$ and ... | njguliyev | 90,209 | <p>Hint: $(2m+1)^2+(2n+1)^2=2(2k+1)$. Show that this number cannot be written as $\dfrac{p^2}{q^2}$ with $(p,q)=1$.</p>
|
4,559,503 | <blockquote>
<p>Let <span class="math-container">$\displaystyle f(x)=\frac{1+\cos(2\pi x)}2$</span> for <span class="math-container">$x\in\mathbb R$</span>, and <span class="math-container">$f^n=\underbrace{ f \circ \cdots \circ f}_{n}$</span>. Is it true that for Lebesgue almost every <span class="math-container">$x$<... | Oliver Díaz | 121,671 | <p>Let <span class="math-container">$\ell_1<\ell_2<1$</span> be the three fixed points of <span class="math-container">$f$</span>.The idea is to follow where certain subintervals of <span class="math-container">$[0,1]$</span> get mapped and to show that in a finite number of iterations, such subintervals fall int... |
69,272 | <p>By the way, does anyone know how to prove in an elementary way (i.e. expanding) that $\prod_1^n (1+a_i r)$ tends to $e^r=\sum \frac{r^k}{k!}$ as you let $\max|a_i|\to 0$ with $0\leq a_i \leq 1$ and $\sum a_i = 1$? An easy solution goes by writing the product with the exponential function so that you get the exponent... | Roland Bacher | 4,556 | <p>Knowing $\lim_{N\rightarrow\infty} \left(1+\frac{r}{N}\right)^N=e^r$ (the case where all $a_i$
are equal)
we can use continuity and $\left(1+\frac{r}{N}\right)^{Na_i}\sim 1+a_ir$
if $N$ is very huge and $a_i\rightarrow 0$.</p>
|
4,083,410 | <p>I have to check convergence of <span class="math-container">$\int_0^\infty \sin{(e^{ax})}dx$</span>?</p>
<p>I tried with substitution <span class="math-container">$e^{ax}=t$</span> for <span class="math-container">$a>0$</span> and got <span class="math-container">$\int_1^\infty \frac{\sin{t}}{at} dt$</span>. This... | hamam_Abdallah | 369,188 | <p><span class="math-container">$$(\forall n\in \Bbb N)\;\;\big|I_n\big|=\left|\int_0^1x^n\sin(\pi x)dx\right|$$</span>
<span class="math-container">$$\le \int_0^1\big|x^n\sin(\pi x)\big|dx$$</span>
<span class="math-container">$$\le \int_0^1x^ndx=\frac{1}{n+1}$$</span></p>
<p><span class="math-container">$$\implies \l... |
4,083,410 | <p>I have to check convergence of <span class="math-container">$\int_0^\infty \sin{(e^{ax})}dx$</span>?</p>
<p>I tried with substitution <span class="math-container">$e^{ax}=t$</span> for <span class="math-container">$a>0$</span> and got <span class="math-container">$\int_1^\infty \frac{\sin{t}}{at} dt$</span>. This... | Angelo | 771,461 | <p><span class="math-container">$\;\sum\limits_{n=1}^\infty I_n=\text{Si}(\pi)-\dfrac2\pi\;$</span></p>
<p>and as far as I know there is not a simplier way to write the sum of your series.</p>
|
2,018,239 | <p>I have to show, using induction, that $2^{4^n}+5$ is divisible by $21$. It is supposed to be a standard exercise, but no matter what I try, I get to a point where I have to use two more inductions.</p>
<p>For example, here is one of the things I tried:</p>
<p>Assuming that $21 |2^{4^k}+5$, we have to show that $21... | mathlove | 78,967 | <blockquote>
<p>Is there an easier way of doing this?</p>
</blockquote>
<p>Note that $2^{4^{k+1}}=(2^{4^k})^4$.</p>
<p>Inductive step : </p>
<p>Supposing that $2^{4^k}+5=21m$ gives
$$\begin{align}2^{4^{k+1}}+5&=(2^{4^k})^4+5\\&=(21m-5)^4+5\\&=\sum_{i=0}^{4}\binom{4}{i}(21m)^i(-5)^{4-i}+5\\&\equiv ... |
3,063,577 | <p>Please suggest a book on applications of Diophantine equations in physics, chemistry, and biology. This book should be suitable to introduce this subject to students who are not mathematics specialists. </p>
| Sam | 632,330 | <p>"OP" Can look up book by 'Stephen Wolfram' called 'A new kind of science'. It has stuff on Diophantine equations related to different branches of science in the notes section. The book can be availed of at the local library or purchased as hardcover or E-book. Also a summary is available on line. The links are given... |
601,296 | <p>Let's say im a guy for ancient Greece and I only have a string and a pencil.
And I want to draw a line, the width of the line is the square root of 6.
And I only know how to draw a line in the width of real numbers.
I've checked out the <a href="https://en.wikipedia.org/wiki/Spiral_of_Theodorus" rel="nofollow norefe... | abiessu | 86,846 | <p>With a string and a pencil, you can measure "one unit", that being the length of the string, or a portion of it. Then you can make a triangle "one unit by one unit." Further, this triangle can be a right triangle since you can perpendicularly bisect a line segment to make the second side. Then... |
601,296 | <p>Let's say im a guy for ancient Greece and I only have a string and a pencil.
And I want to draw a line, the width of the line is the square root of 6.
And I only know how to draw a line in the width of real numbers.
I've checked out the <a href="https://en.wikipedia.org/wiki/Spiral_of_Theodorus" rel="nofollow norefe... | lhf | 589 | <p>Draw two adjacent segments of size <span class="math-container">$2$</span> and <span class="math-container">$3$</span>. Using the combined segment as diameter, draw a semicircle. Now draw a perpendicular at the point where the two segments meet. That perpendicular defines a segment of length <span class="math-contai... |
385,537 | <p>How would you go about proving the following?</p>
<p>$${1- \cos A \over \sin A } + { \sin A \over 1- \cos A} = 2 \operatorname{cosec} A $$</p>
<p>This is what I've done so far:</p>
<p>$$LHS = {1+\cos^2 A -2\cos A + 1 - \cos^2A \over \sin A(1-\cos A)}$$</p>
<p>....no idea how to proceed .... X_X</p>
| Parth Thakkar | 70,311 | <p>$$ LHS =\frac {1 - \cos A} {\sin A} + \frac {\sin A} {1 - \cos A} $$
$$ = \frac {2 \sin^2 \frac A2} {2\sin \frac A2 \cos \frac A2} + \frac {2\sin \frac A2 \cos \frac A2}{2 \sin^2 \frac A2}$$</p>
<p>$$ = \frac {\sin \frac A2} {\cos \frac A2} + \frac {\cos \frac A2} {\sin \frac A2} $$</p>
<p>Now just cross multiply ... |
2,276,907 | <p>If $\cos{x}=\frac{3}{5}$ and angle $x$ terminates in the fourth quadrants, find the exact value of each of the following:</p>
<p>A. $\sin{2x}$
B. $\cos{2x}$
C. $\tan {\frac{x}{2}}$</p>
<p>Okay, so I am going through my old exam reviews for the final exam I have this evening, and choosing problems I have trouble wi... | Siong Thye Goh | 306,553 | <p>Guide:</p>
<p>Find out what is $\sin x$ and what is $\cos x$.</p>
<p>The following formula might be helpful.</p>
<p>$$\sin 2x = 2 \sin x \cos x$$</p>
<p>$$\cos 2x = 2 \cos^2 x -1 $$</p>
<p>$$ \tan 2x = \frac{2 \tan x}{1- \tan^2 x}$$</p>
<p>For the tangent problem, note that </p>
<p>$$\tan x = \frac{2 \tan \f... |
2,276,907 | <p>If $\cos{x}=\frac{3}{5}$ and angle $x$ terminates in the fourth quadrants, find the exact value of each of the following:</p>
<p>A. $\sin{2x}$
B. $\cos{2x}$
C. $\tan {\frac{x}{2}}$</p>
<p>Okay, so I am going through my old exam reviews for the final exam I have this evening, and choosing problems I have trouble wi... | helloworld112358 | 300,021 | <p>Notice that $\sin^2(x)=1-\cos^2(x)=\frac{16}{25}$, so $\sin(x)=\pm\frac{4}{5}$. Since we are in the fourth quadrant, we must have $\sin(x)=-\frac 45$. Thus, $\sin(2x)=2\sin(x)\cos(x)=\frac{24}{25}$. </p>
<p>From this, we can calculate $\cos^2(2x)=1-\sin^2(2x)=\frac{7^2}{25^2}$. Since $\sin(x)<-\cos(x)$, we know ... |
2,276,907 | <p>If $\cos{x}=\frac{3}{5}$ and angle $x$ terminates in the fourth quadrants, find the exact value of each of the following:</p>
<p>A. $\sin{2x}$
B. $\cos{2x}$
C. $\tan {\frac{x}{2}}$</p>
<p>Okay, so I am going through my old exam reviews for the final exam I have this evening, and choosing problems I have trouble wi... | Community | -1 | <p>Cosine is positive in quadrants 1 and 4. Think of $\cos \theta$ as the $x$-coordinate of the point where the terminal side of $\theta$ intersects the unit circle, because that's one way of defining $\cos\theta$ for any angle $\theta$. (Similarly, $\sin\theta$ is the $y$-coordinate of that point.)</p>
<p>Step 1: ... |
128,221 | <p>Let $v_1=[-3;-1]$ and $v_2= [-2;-1]$</p>
<p>Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation satisfying:</p>
<p>$T(v_1)=[15;-6]$ and $T(v_2)=[11;-3]$</p>
<p>Find the image of an arbitrary vector $[x;y]$</p>
| David Mitra | 18,986 | <p>One method would be to find the image of the standard unit vectors first. Then using linearity, you can find the image of an arbitrary vector.</p>
<p>In a bit more detail:</p>
<p>To find $T(0,1)$, first write $(1,0)$ as a linear combination of $v_1$ and $v_2$. Here you have to solve the equation
$$
(1,0)=\alpha v... |
128,221 | <p>Let $v_1=[-3;-1]$ and $v_2= [-2;-1]$</p>
<p>Let $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ be the linear transformation satisfying:</p>
<p>$T(v_1)=[15;-6]$ and $T(v_2)=[11;-3]$</p>
<p>Find the image of an arbitrary vector $[x;y]$</p>
| alpha.Debi | 20,863 | <p>The matrix whose columns are T(v1) and T(v2) is the representation matrix
of T in the basis B=(v1, v2) of R2 (as domain) and E=((1,0), (0,1)) of R2 as codomain.
If you multiply this matrix by the coordinates vector of (x,y) in the basis B
you will get T(x,y) (since E is the canonic basis).</p>
|
2,870,729 | <blockquote>
<p>Why does $|e^{ix}|^2 = 1$?</p>
</blockquote>
<p>The book said $e^{ix} = \cos x + i\sin x$, and square it, then $|e^{ix}|^2 = \cos^2x + \sin^2x = 1$.</p>
<p>But, when I calculated it, $ |e^{ix}|^2 = \left|\cos x + i\sin x\right|^2 = \cos^2x - \sin^2x + 2i\sin x\cos x$.</p>
<p>I can't make it to be e... | Michael Hardy | 11,667 | <p>If $a,b$ are <b>real</b> then $\displaystyle \left| a+bi \right| = \sqrt{a^2+b^2\,\,} = \sqrt{(a+bi)(a-bi)\,}.$</p>
|
3,794,158 | <p>I am trying to prove that: Let <span class="math-container">$(M,d)$</span> an metric space and <span class="math-container">$(x_n)$</span>,<span class="math-container">$(y_n)$</span> sequences in <span class="math-container">$M$</span> such that <span class="math-container">$d(x_n,y_n) \leq \frac{1}{n}$</span> <spa... | hJulian | 647,814 | <p>Since <span class="math-container">$x_{n}\to x$</span>, let <span class="math-container">$\varepsilon >0$</span> there exist <span class="math-container">$N_{1}\in \mathbb{Z}_{\geq0}$</span>, such that fot any <span class="math-container">$n>N_{1}$</span>, <span class="math-container">$d(x_{n},l)<\varepsilo... |
933,604 | <p>Hi can anyone solve these two questions using logs and indices</p>
<p>a.
$$4^{2x}-2^{x+1}=48$$</p>
<p>b.
$$6^{2x+1}-17*{6^x}+12=0$$</p>
<p>Thanks.</p>
| Claude Leibovici | 82,404 | <p>I am afraid but I do not think that there is any explicit solution for $$4^{2x}-2^{x+1}=48$$ To visualize it better, since the terms grow very fast, it is better to look at function $$f(x)=\log(4^{2x})-\log(2^{x+1}+48)=2x\log(4)-\log(2^{x+1}+48)$$ which is basically a straight line.</p>
<p>To solve this equation, l... |
181,702 | <p>I am working on getting the hang of proofs by induction, and I was hoping the community could give me feedback on how to format a proof of this nature:</p>
<p>Let $x > -1$ and $n$ be a positive integer. Prove Bernoulli's inequality:
$$ (1+x)^n \ge 1+nx$$</p>
<p><strong>Proof</strong>: </p>
<p>Base Case: For $n... | Brian M. Scott | 12,042 | <p>What you have is perfectly acceptable. The calculations could be organized a little more neatly:</p>
<p>$$\begin{align*}
(1+x)^{k+1}&=(1+x)(1+x)^k\\
&\ge(1+x)(1+kx)\\
&=1+(k+1)x+kx^2\\
&\ge1+(k+1)x\;,
\end{align*}$$</p>
<p>since $kx^2\ge 0$. This completes the induction step.</p>
|
116,394 | <p>After importing a sound file, how can I add an echo to it?</p>
<pre><code> sound = Import["test.wav", "SampleRate"]
</code></pre>
<p>It needs to be apply after time specified by user. This is as far as I have got:</p>
<pre><code> addEcho[sound_, time_] :=
Module[{tmp = sound, channels, samples, duration},
... | Szabolcs | 12 | <p>Since version 11.0, <a href="http://reference.wolfram.com/language/ref/AudioReverb.html" rel="nofollow"><code>AudioReverb</code></a> does this in a single go. There are several impulse responses that come with Mathematica as example data. Try <code>ExampleData["Audio"]</code> and look for names starting with <code... |
4,045,238 | <p>I was working on the problems in Mathematical Methods for Physics and Engineering by Riley,Hobson & Bence.
In Problem 2.34 (d) I'm supposed to find this integral: <span class="math-container">$$J=\int\frac{dx}{x(x^n+a^n)}.$$</span>
I used partial fractions and arrived at the form
<span class="math-container">$$J... | heropup | 118,193 | <p>We seek a decomposition of the form
<span class="math-container">$$\frac{1}{x(x^n + a^n)} = \frac{A}{x} + \frac{Bx^{n-1}}{x^n + a^n} = \frac{(A+B)x^n + Aa^n}{x(x^n+a^n)}.$$</span> Hence the choice <span class="math-container">$A = a^{-n}$</span>, <span class="math-container">$B = -A = -a^{-n}$</span>, yields
<span ... |
3,362,000 | <p>From listing the first few terms, I suspect that the sequence is increasing, so I wanted to use mathematical induction to verify my suspicion.</p>
<p>I have assumed that <span class="math-container">$a_k<a_{k+1}$</span>, I don't see how I can obtain <span class="math-container">$a_{k+1}<a_{k+2}$</span> becaus... | Community | -1 | <p>Base case:</p>
<p><span class="math-container">$$1+\dfrac11>1\implies a_2>a_1.$$</span></p>
<p>Inductive step:</p>
<p><span class="math-container">$$a_n=a_{n-1}+\frac1{a_{n-1}}>a_{n-1}
\\\implies a_{n-1}>0
\\\implies a_n=a_{n-1}+\dfrac1{a_{n-1}}>0
\\\implies a_{n+1}=a_n+\frac1{a_n}>a_n.$$</span>... |
166,013 | <p>Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete topology. Weibel, in his $K$-theory book, computes the torsion in its coefficient ring. I would like to know the torsion-fr... | Matthias Wendt | 50,846 | <p>I think the answer to this question is not known. All we can say about the K-theory of $\mathbb{C}$ concerns the torsion.
The trouble starts with $K_1(\mathbb{C})\cong\mathbb{C}^\times$, which is pretty difficult to understand as an abelian group. There is a formula for $K_2$ of a field due to Matsumoto which is $K... |
3,873,882 | <p>(Follow-up question to <a href="https://math.stackexchange.com/questions/3873041/how-to-do-proofs-by-induction-with-2-variables">How to do proofs by induction with 2 variables?</a>)</p>
<p>Suppose you want to prove that <span class="math-container">$P(x,y,z)$</span> is true for all <span class="math-container">$x,y,... | paulinho | 474,578 | <p>Your first insight is exactly the idea, though the algebra would be easier without the expansion you do:</p>
<p><span class="math-container">$$f(-2) = 0 \implies -1 + (k - 10)^3 = 0 \implies (k - 10)^3 = 1$$</span></p>
<p>If your problem only asks for real solutions of <span class="math-container">$k$</span>, then i... |
181,499 | <p>In many of the classes that I teach, I require students to learn the basics of Mathematica which we use throughout the semester to do computations and to submit homeworks (in notebook form). Some students really like this and some... not so much. </p>
<p>Since I teach in an engineering department, almost everyone a... | Henrik Schumacher | 38,178 | <p>Imho some important things to translate between <em>Matlab</em> and <em>Mathematica</em>:</p>
<ul>
<li><p>"everything is a matrix (or inefficient)" vs. "everything is an expression"</p></li>
<li><p>indexing into arrays: <code>:</code> vs. <code>All</code> or <code>;;</code></p></li>
<li><p>indexing into arrays: <co... |
2,249,707 | <blockquote>
<p>$$\int f(x)\sin x \cos x dx = \log(f(x)){1\over 2( b^2 - a^2)}+C$$</p>
</blockquote>
<hr>
<p>On differentiating, I get,</p>
<p>$$f(x)\sin x\cos x = {f^\prime(x)\over f(x)}{1\over 2( b^2 - a^2)}$$ </p>
<p>$$\sin 2x (b^2 - a^2) = {f^\prime( x)\over (f(x))^2} $$</p>
<p>On integrating, </p>
<p>$${... | Kanwaljit Singh | 401,635 | <p>$f(x) = {2\over b^2 \cos^2x -b^2\sin^2 x- a^2\cos^2x+ a^2\sin^2 x}$</p>
<p>$= {2\over b^2 \cos^2x -b^2(1-\cos^2 x)- a^2(1-\sin^2x)+ a^2\sin^2 x}$</p>
<p>$= {2\over b^2 \cos^2x -b^2 + b^2\cos^2 x- a^2 +a^2\sin^2x+ a^2\sin^2 x}$</p>
<p>$= {2\over -(a^2 + b^2 ) + 2b^2\cos^2x +2a^2\sin^2x}$</p>
<p>As you can see one... |
2,249,707 | <blockquote>
<p>$$\int f(x)\sin x \cos x dx = \log(f(x)){1\over 2( b^2 - a^2)}+C$$</p>
</blockquote>
<hr>
<p>On differentiating, I get,</p>
<p>$$f(x)\sin x\cos x = {f^\prime(x)\over f(x)}{1\over 2( b^2 - a^2)}$$ </p>
<p>$$\sin 2x (b^2 - a^2) = {f^\prime( x)\over (f(x))^2} $$</p>
<p>On integrating, </p>
<p>$${... | egreg | 62,967 | <p>On differentiating you get indeed
$$
f(x)\sin x\cos x=\frac{f'(x)}{f(x)}\frac{1}{2(b^2-a^2)}
$$
so the differential equation
$$
\frac{f'(x)}{f(x)^2}=(b^2-a^2)\sin2x
$$
Integrating it you get
$$
-\frac{1}{f(x)}=-\frac{1}{2}(b^2-a^2)\cos2x+c
$$
hence
$$
f(x)=\frac{2}{(b^2-a^2)\cos2x-2c}
$$</p>
<p>You can expand $\cos... |
213,872 | <p>I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that</p>
<blockquote>
<p>The Borel $\sigma$-algebra of ${\mathbb R}$ is generated by inervals of the form $(-\infty,a]$, where $a\in{\mathbb Q}$. </p>
</blockquote>
<p>Also... | BallzofFury | 11,969 | <p>The half-open intervals are not necessarily special in a particular way, they are one of many possible generators of the Borel $\sigma$-algebra. </p>
<p>As I understand it, most of the things you do with half-open intervals you could also do with other generators, but in practice they are easy to work with</p>
|
3,041,656 | <p>I need some help in a proof:
Prove that for any integer <span class="math-container">$n>6$</span> can be written as a sum of two co-prime integers <span class="math-container">$a,b$</span> s.t. <span class="math-container">$\gcd(a,b)=1$</span>.</p>
<p>I tried to go around with "Dirichlet's theorem on arithmetic ... | Will Jagy | 10,400 | <p>Later: the numbers between <span class="math-container">$1$</span> and <span class="math-container">$n-1$</span> that are relatively prime to <span class="math-container">$n$</span> itself come in pairs that add up to <span class="math-container">$n$</span> and are relatively prime to each other as well. If <span c... |
3,415,378 | <p>I am looking for an estimation or an approximation of </p>
<p><span class="math-container">$\sum _{k=1}^{n}{\log(k)\binom {n}{k}}$</span></p>
<p>Any hints will be appreciated.
Thank you.</p>
| reuns | 276,986 | <p>For <span class="math-container">$\log n \ge 2$</span>
<span class="math-container">$$\sum_{k=1}^n {n \choose k} \log(k) \ge \sum_{k=n/\log n}^n {n \choose k} \log(n/\log n)$$</span> <span class="math-container">$$=\sum_{k=1}^n {n \choose k} \log(n/\log n)-\sum_{k=1}^{n/\log n} {n \choose k} \log(n/\log n)$$</span>
... |
428,841 | <p>Let $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$</p>
<p>a) Show that $x_{n} < x_{n+1}$</p>
<p>b) Show that $x_{n+1}^{2} \leq 1+ \sqrt{2} x_{n}$</p>
<p>Hint : Square $x_{n+1}$ and factor a 2 out of the square root</p>
<p>c) Hence Show that $x_{n}$ is bounded above by 2. Deduce that $\lim\limits_{n... | Kevin Pardede | 82,064 | <p>10 days old question, but .</p>
<p>a) Is already clear, that $ \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}} < \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n+1}}}}$ , because $\sqrt{n} <\sqrt{n} + \sqrt{n+1}$ which is trivial.<br>
My point here is to give some opinion about b) and c), for me it's better to do the ... |
2,032,387 | <p>I know this is somewhat of an odd question, but I am having trouble with my TI-84 calculator and I don't know why.</p>
<p>I'm trying to find the RREF of the transpose of a <span class="math-container">$4\times6$</span> matrix; for some reason my graphing calculator gives me an error. Something to do with the dimensi... | Donn Liddle | 871,234 | <p>To compute a unique solution for a system of equations you need the same number of equations as unknowns. In other words if you have 4 variables (unknowns) you need only 4 equations.</p>
<p>Adding more equations than variables creates what is called an "over determined" system of equations. In most "... |
4,198,263 | <p><a href="https://www.cliffsnotes.com/study-guides/algebra/linear-algebra/real-euclidean-vector-spaces/projection-onto-a-subspace" rel="nofollow noreferrer">https://www.cliffsnotes.com/study-guides/algebra/linear-algebra/real-euclidean-vector-spaces/projection-onto-a-subspace</a></p>
<p>I am following this example he... | Glorious Nathalie | 948,761 | <p>Take for example, the set <span class="math-container">$S$</span> to the plane spanned by <span class="math-container">$v_1 =(1, 2, 1) $</span> and <span class="math-container">$v_2 = (-1, 0, 1)$</span> which are two orthogonal vectors. And let the vector to projected onto <span class="math-container">$S$</span> be... |
627,444 | <p>I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, but they seem to be way to general notions as to admit a really interesting theory.</p>
<p>Thus, I'm wondering whet... | Shaun | 104,041 | <p>You might be interested in <a href="http://en.m.wikipedia.org/wiki/Universal_algebra" rel="nofollow">Universal Algebra</a>. You could build your own algebra that way. Have a look through <a href="http://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra.pdf" rel="nofollow"><em>A Course in Universal Algebra</em... |
3,576,008 | <p><strong>Question:</strong></p>
<p>In acute <span class="math-container">$\Delta ABC$</span>, let <span class="math-container">$D$</span> be the foot of the altitude from <span class="math-container">$A$</span> to <span class="math-container">$BC$</span>, and let <span class="math-container">$\overline{AD}$</span>
i... | David K | 139,123 | <p>The problem statement says, "Let the circle with diameter <span class="math-container">$AE$</span> intersect <strong>lines</strong> <span class="math-container">$AB$</span> and <span class="math-container">$AC$</span>" (emphasis added by me).</p>
<p>If the statement had said "<strong>sides</strong> <span class="mat... |
9,629 | <p>are people facing problem of not loading latex symbols in MSE? I have high speed internet connection but I am facing this problem from yesterday,any suggestion?It says "math processing error" if my connection is low speed but this is not the case, I am just watching all latex symbols instead of compiled complete pi... | Davide Cervone | 7,798 | <p>Try clearing your cache and restarting your browser (restarting is an important step). It may be that you have a mixture of v2.1 and v2.2 files in your cache. The CDN edge nodes should have been updated by now, so it is probably a caching problem on your end.</p>
|
96,191 | <p>I am trying to calculate the following integral which contains a parameter.
<a href="https://i.stack.imgur.com/qUJ9f.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qUJ9f.jpg" alt="enter image description here"></a></p>
<p>I have used the Integrate and FullSimplify using assumptions but Mathemati... | Alexei Boulbitch | 788 | <p>Well you might think about a semi-analytical approach:</p>
<pre><code> lst = Table[{p,
NIntegrate[((Sin[u]^1.82 + (p^(-1))^0.63*Sin[u]^2.45)*
Sin[p + u]^1.82)/(p + Sin[u])^2, {u, 0, Pi}]}, {p, 0.1, 1,
0.05}];
</code></pre>
<p>giving a list of complex values (right? you expect it?) Then you can pl... |
1,587,498 | <p>I need some help with this (seemingly) simple problem. As before, it comes from Apostol "Calculus", Volume 1, Section 8.28, Question 23 and it states:</p>
<p>Solve the differential equation $(1+y^2e^{2x})y^{'} + y = 0$ by introducing a change of variable of the form $y = ue^{mx}$, where $m$ is constant and $u$ is a... | JJacquelin | 108,514 | <p>OK. up to $(1+u^2e^{2(m+1)x})(u'+mu)+u=0$</p>
<p>Then, why not chosing a value of $m$ in order to simplify thr equation ? Obviously $m=-1$ is a good choice :
$$(1+u^2)(u'-u)+u=0$$
$$\frac{1+u^2}{u^3}u'=1$$
$$\frac{-1}{2u^2}+\ln|u|=x+c$$
The result expressed on the forme of the inverse function $x(u)$ can be invert... |
535,080 | <p>For the following example: </p>
<blockquote>
<p>Let the topological space $X$ be the real line $\mathbb{R}$. An open set is any set whose complement is finite. Let $S=[0,1]$. Find the closure, the interior, and the boundary of $S$. </p>
</blockquote>
<p>What is meant by let the topological space $X$ be the real... | Elias Costa | 19,266 | <p><strong>Hint.</strong> Let's $\tau$ is the topology of your question. You can think of a more concrete way of explaining the face of this open topology.If $O\in \tau$ and $O\neq \emptyset$ then $O^c$ is finite. That is, thare is $n$ numbers $x_1<x_2<\ldots, x_{n-1}<x_n$ such that $O^c=\{x_1, \ldots, x_n\}$... |
3,428,995 | <p>I found this inequality on twitter and I can't seem to prove the statement.</p>
<p>Prove that for <span class="math-container">$a,b,c > 0$</span> that </p>
<p><span class="math-container">$$
\frac{a+b+c}{2} \geq \frac{ab}{a+b} + \frac{ac}{a+c} + \frac{bc}{b+c}
$$</span></p>
<p>After an hour (and a crick in my ... | Michael Rozenberg | 190,319 | <p>Since <span class="math-container">$(3,1,0)\succ(2,1,1),$</span> your inequality is true by Muirhead. </p>
|
1,645,130 | <p>Is there any known explicit bijection between these two sets? </p>
<p>I know it can be proved that such bijection exists using two injections and Schröder–Bernstein theorem, but I wanted to know whether some explicit bijection is known. I failed to find any except ones constructed awkwardly from the Schröder–Bernst... | Patrick Stevens | 259,262 | <p>This answer is incomplete, but it at least makes the Schröder-Bernstein a bit nicer.</p>
<p>Firstly, $[0,1)$ bijects with $\mathbb{R}$, by the following bijections:</p>
<p>$h: (0, 1) \to \mathbb{R}$ by $x \mapsto \tan(\frac{\pi}{2} (2x-1))$</p>
<p>$i: [0,1) \to (0,1)$ by $\frac{1}{n} \mapsto \frac{1}{n+1}$, $0 \m... |
3,153,306 | <p>In other words, say I am looking for multiple X</p>
<p>let: </p>
<p>X < 1000005</p>
<p>let the fist 18 divisors of X be:
1 | 2 | 4 | 5 | 8 | 10 | 16 | 20 | 25 | 32 | 40 | 50 | 64 | 80 | 100 | 125 | 160 | 200 </p>
<p>finally, I also know: X has exactly 49 divisors. </p>
<p>I will tell you what the answer is.... | user655522 | 655,522 | <p>The result is true for <span class="math-container">$p = 3$</span>, <span class="math-container">$5$</span>, and <span class="math-container">$7$</span>, so assume that <span class="math-container">$p = 2n+1$</span> for <span class="math-container">$n \ge 4$</span>.
Note that all the primes <span class="math-contain... |
796,564 | <p><img src="https://i.stack.imgur.com/rLmA6.jpg" alt="enter image description here"></p>
<p>I don't get the answer to this problem, can somebody please tell me what the answer is. </p>
| user7000 | 149,900 | <p>It might help to re-draw the rectangle so that it is slanted. Do you mean the equation of PU is y=2/3x+4? That means the slope is 2/3. Since SP is at a right angle to PU (it's a rectangle so it has to be), then it's slope is the negative reciprocal of that. That would be -3/2 (flip the fraction and multiply by -1), ... |
796,564 | <p><img src="https://i.stack.imgur.com/rLmA6.jpg" alt="enter image description here"></p>
<p>I don't get the answer to this problem, can somebody please tell me what the answer is. </p>
| user150369 | 150,369 | <p>The answer is C, assuming that the equation of PU is $y = \frac{2}{3}x+4$. </p>
|
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