qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,049,808 | <p>Let $u_0 = 1$ and $u_{n+1} = \frac{u_n}{1+u_n^2}$ for all $n \in \mathbb{N}$.</p>
<p>I can show that $u_n \sim \frac{1}{\sqrt{2n}}$, but I would like one more term in the asymptotic development, something like $u_n = \frac{1}{\sqrt{2n}}+\frac{\alpha}{n\sqrt{n}} + o\bigl(\frac{1}{n^{3/2}}\bigr)$.</p>
<p>Here is the... | not all wrong | 37,268 | <p><strong>Note:</strong> This answer does not contain a proof, but I'm fairly sure it's right. Posted in the hope that perhaps the answer will guide you towards a proof thereof.</p>
<p>Looking at the graph you can figure out that in fact the next correction looks to be $n^{-3/2} \log n$ dominating the next $n^{-3/2}$... |
2,015,717 | <p>I know that the O notation tells me to find $n_0$ natural and $c>0$ real, so the
$$2^{(n^2)} \le c*2^{2n} $$
Only step that I can think about is this:
$$2^{(n^2)} \le c*2^{n+n} $$
$$2^{(n^2)} \le c*2^n*2^n $$
But I have no clue what to do next. (Some estimate?)</p>
<p>Do you have some hints please?</p>
<p... | Arthur | 15,500 | <p>This is, in fact, not true. Given a $c$, there is some natural number $m$ such that $2^m>c$. Now, for each $n>1+2\sqrt{ m+1}$ we have $n^2>2n+m$, which gives
$$
2^{n^2}>2^{2n+m}=2^{2n}\cdot 2^m>2^{2n}\cdot c
$$</p>
|
1,549,843 | <p>Prove the following limits:</p>
<p>$$\lim_{x \rightarrow 0^+} x^x = 1$$
$$\lim_{x \rightarrow 0^+} x^{\frac{1}{x}}=0$$
$$\lim_{x \rightarrow \infty} x^{\frac{1}{x}}=1$$</p>
<p>They are not that hard using l'Hospital or the Sandwich theorem. But I curious if they can be solved with the basic knowledge of limits. I ... | Kushal Bhuyan | 259,670 | <p>For the third you can use $$\lim_{n\to \infty}(a_n)^{1/n}=l$$ if $$\lim_{n\to \infty}\frac{a_{n+1}}{a_n}=l$$</p>
<p>Here $a_n=x$</p>
|
3,917,283 | <p>I have the following : <span class="math-container">$f_n(x)=\frac{nx}{1+nx^3} \quad n=1,2,\dots \quad$</span> and <span class="math-container">$f(x)=\lim_{n \to \infty} f_n(x) ,$</span></p>
<p>and I have done the following : <span class="math-container">$$|f_n(x) -f(x)| = \biggl|\frac{nx}{1+nx^3}-\frac{1}{x^2}\bigg... | zhw. | 228,045 | <p>Hint: The <span class="math-container">$f_n$</span>'s are bounded on <span class="math-container">$(0,\infty).$</span> If the convergence were uniform there, then <span class="math-container">$f$</span> would be bounded there.</p>
|
3,464,477 | <p>An odd perfect number n is of the form <span class="math-container">$n=q^{k}m^{2}$</span> where <span class="math-container">$q$</span> is prime and both <span class="math-container">$q,k \equiv 1 \mod 4$</span>. Also, n satisfies <span class="math-container">$\sigma (n)=2n$</span> so that <span class="math-containe... | JoshuaZ | 64,009 | <p>We can't answer these questions at present. We can't rule out almost any specific choice of this sort, such as <span class="math-container">$k=1$</span> and <span class="math-container">$q=5$</span>. That said, we can rule out a few choices of <span class="math-container">$q$</span> and <span class="math-container">... |
3,900,435 | <p>Suppose that <span class="math-container">$\mathbf{Y}\sim N_3(0,\,\sigma^2\mathbf{I}_3)$</span> and that <span class="math-container">$Y_0$</span> is <span class="math-container">$N(0,\,\sigma_0^2)$</span>, independently of the <span class="math-container">$Y_i$</span>'s.
My question is that does <span class="math-c... | mathcounterexamples.net | 187,663 | <p>To elaborate on the example given by <a href="https://math.stackexchange.com/users/570087/perpetuallyconfused">perpetuallyconfused</a>, indeed <span class="math-container">$\mathbb C$</span> and <span class="math-container">$\mathbb C(x)$</span> provide a counterexample.</p>
<p>Some additional elements.</p>
<p><span... |
2,535,111 | <p>Let $n$ be a natural number, and let $1\leq j,k\leq n$ be two randoms.
Show that if $k\mid n$ ($k$ divides $n$) then the probabilty of
$k$ also dividing $j$ is $\frac{1}{k}$, that is $\text{P}\left(k\mid j\right)=\frac{1}{k}$.</p>
<p>One way i was thinking about is to look straight forward for
$$\text{P}\left(k\mi... | Bernard | 202,857 | <p>First, the volume of the half ball $B$ with radius $R$ is $\frac23\pi R^3$. In <code>spherical coordinates</code> $(R,\theta,\varphi)$, the coordinates of the centre of gravity $G$ are given by the integrals
\begin{align}\newcommand\d{\mathop{}\!\mathrm{d}}
x_G&=\frac3{2\pi R^3}\iiint_B x\d x\d y\d z=\frac3{2\pi... |
80,452 | <p>When can the curvature operator of a Riemannian manifold (M,g) be diagonalized by a basis of the following form</p>
<p>'{${E_i\wedge E_j }$}' where '{${E_i}$}' is an orthonormal basis of the tangent space? If the manifold is three dimensional then it is always possible. But what about higher dimensional cases?</p>
| Willie Wong | 3,948 | <p>A sufficient condition is if the Riemannian manifold is conformally flat, this implies that the Weyl curvature vanishes, and the Riemann curvature tensor is a linear combination of the identity operator on two forms and the operator formed by the Kulkarni-Nomizu product of the Ricci curvature and the metric. Using t... |
893,839 | <p>I am dealing with the dual spaces for the first time.</p>
<p>I just wanted to ask is their any practical application of Dual space or is it just some random mathematical thing? If there is, please give a few.</p>
| Joonas Ilmavirta | 166,535 | <p>Distributions, or "generalized functions", such as the Dirac delta are most conveniently described by studying the dual space of a space of "nice" (smooth or something of the kind) functions.
Heuristic calculations with distributions are commonplace in physics, but if you need a rigorous understanding of what is goi... |
13,616 | <p>Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a short description.</p>
<p>I think that the tags are relevant, but feel free to change them.</p>
<p>Also, have ther... | Emerton | 2,874 | <p>In addition to the examples mentioned in the question, of manifolds and schemes, other commonly occuring types of locally ringed spaces are formal schemes and complex analytic spaces. </p>
<p>I don't know how extensive the taxonomy of locally ringed spaces is. For example,
if $A$ is a local ring, we can form the ... |
1,576,561 | <p>I'd like to show that $$\sum\limits_{n = 1}^\infty {{{{x^{n + 1}}} \over {n(n + 1)}}} $$ absolutely converges for $|x| < 1$</p>
| Berci | 41,488 | <p>You can also use this directly:</p>
<p>If $|x|<1$, then $|x^{n+1}|<1$, so for any $N>n$, we have</p>
<p>$$\left|\sum_{n=1}^N\frac{x^{n+1}}{n(n+1)}\right| \le \sum_{n=1}^N\frac{|x^{n+1}|}{n(n+1)}\le\sum_{n=1}^N\frac1{n(n+1)} =\sum_{n=1}^N\left(\frac1n-\frac1{n+1}\right) = 1-\frac1{N+1}\,.$$</p>
|
927,815 | <p>Find an equation of the plane that passes through the points $A(0, 1, 0)$, $B(1, 0, 0)$ and $C(0, 0, 1)$.</p>
| JimmyK4542 | 155,509 | <p><strong>Hint</strong>: The sum of the $x$-coordinate, $y$-coordinate, and $z$-coordinate for each of those points is $1$. </p>
<p>Now, turn that into an equation. </p>
|
3,566,083 | <p>I have a game that assigns the probability of finding items in treasure chests by making several independent checks until it fills a quota, with duplicates not allowed. I am trying to figure out how to calculate the resulting probabilities of finding each item - without breaking the bank in terms of calculation brut... | Brian Moehring | 694,754 | <p>In general, we have <span class="math-container">$$\ln(b^x) = x\ln(b) + 2\pi i k$$</span> for some <span class="math-container">$k \in \mathbb{Z}$</span> (which may depend on both <span class="math-container">$x$</span> and <span class="math-container">$b$</span>), for any fixed branch of the complex logarithm.</p>
... |
214,705 | <p>If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$.</p>
<p>Prove or disprove, if there exists more than 1 more distinct values of the polynomials.</p>
| André Nicolas | 6,312 | <p>There is too much freedom. Let $a(x)=x+1$ and $b(x)=(x^6-1)-(x+1)$. Or else use $a(x)=k(x+1)$, $b(x)=x^6-1-k(x+1)$, where $k$ is any non-zero integer. </p>
|
1,914,536 | <p><strong>Problem</strong>: Solve $3^x \equiv 2 \pmod{29}$ using Shank's Baby-Step Giant-Step method.</p>
<p>I choose $k=6$ and calculated $3^i \pmod {29}$ for $i=1,2,...,6$.
$$3^1 \equiv 3 \pmod {29}$$
$$3^2 \equiv 9 \pmod {29}$$
$$3^3 \equiv 27 \pmod {29}$$
$$3^4 \equiv 23 \pmod {29}$$
$$3^5 \equiv 11 \pmod {29}$$
... | clzola | 262,220 | <p>I found flaw in my calculus for the second expression in second list, it should be</p>
<p>$$ 2 \cdot 3^{-12} = 2 \cdot (3^{-6})^2 \equiv 2 \cdot 22^2 = 2 \cdot 484 \equiv 11 \pmod {29}$$</p>
<p>Hence, $x = 5 + 12 = 17 $ which is correct.</p>
|
2,447,014 | <p>I attempted to solve this question using a method learned from a previous answer on here. I was just looking for a bit more guidance.
This is what I have for this problem:</p>
<blockquote>
<p>$y=\frac{4x}{x+1}\\
y(x+1) =4x \\
yx + y = 4x \\
y = 4x-yx \\
y= x(4-y) \\
\frac{y}{4-y}=x \\
\frac{y}{4-y} > 0 \\
0 &... | Community | -1 | <p>$\dfrac {4x}{x+1}=\dfrac{4x+4-4}{x+1}=4-\dfrac {4}{x+1}$</p>
<p>When $x$ tends to $0$ then $4-\dfrac {4}{x+1}$ tends to $0$. When $x$ tends to $+\infty$ then $4-\dfrac {4}{x+1}$ tends to $4$.</p>
<p>So, because your function is continuous your image is $(0,4)$.</p>
<p>Alternative approach:</p>
<p>$\dfrac {y}{4-y... |
265,549 | <p>$$ \left\{ x \in\mathbb{R}\; \middle\vert\; \tfrac{x}{|x| + 1} < \tfrac{1}{3} \right\}$$</p>
<p>What is the supremum and infimum of this set? I thought the supremum is $\frac{1}{3}$. But can we say that for any set $ x < n$ that $n$ is the supremum of the set? And for the infimum I have no idea at all. Also,... | Nameless | 28,087 | <p>Hint: Find the range of
$$f(x)=\frac{x}{\left|x\right|+1}-\frac13$$
This function is continuous and increasing.</p>
|
2,873,449 | <p>For a continuous function $f: R \to R $ Define:
$$ \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} f(t)t \space dt $$
Since the function is continuous I can assume that it's also integrable since continuity implies integrability. I assume furthermore that there exists a function $F$ which is an antiderivative of $f$ for w... | Clement C. | 75,808 | <p><em>An approach not relying on L'Hopital's Rule.</em></p>
<p>We have, for $x\neq 0$ and with the change of variable $u=\frac{t}{x}$, $$
\frac{1}{x^2}\int_0^x t f(t)dt
= \int_0^1 u f(xu) du
$$
Now, it is not hard to prove that
$$
\lim_{x\to 0}\int_0^1 u f(xu) du
= \int_0^1 u \lim_{x\to 0} f(xu) du
= \int_0^1 u f(0) ... |
2,873,449 | <p>For a continuous function $f: R \to R $ Define:
$$ \lim_{x \to 0} \frac{1}{x^2} \int_{0}^{x} f(t)t \space dt $$
Since the function is continuous I can assume that it's also integrable since continuity implies integrability. I assume furthermore that there exists a function $F$ which is an antiderivative of $f$ for w... | Paramanand Singh | 72,031 | <p>Write $f(t) $ as $f(t) - f(0)+f(0)$ and then the desired limit is $$\lim_{x\to 0}\frac{1}{x^2}\int_{0}^{x}\{f(t)-f(0)\}t\,dt+\frac{f(0)}{2}\tag{1}$$ Next we can show that the first limit above is $0$. Let $\epsilon >0$ be given. Then by continuity we have a $\delta>0$ such that $$|f(x) - f(0)|<\epsilon $$ w... |
3,528,946 | <p><a href="https://i.stack.imgur.com/JG89d.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JG89d.png" alt="enter image description here"></a></p>
<p>This is the example from Friedberg's Linear Algebra book. It is giving an example about Null space (denoted by <span class="math-container">$N(T)$</sp... | gjl | 565,375 | <p>If <span class="math-container">$A^TAv = 0$</span> then also <span class="math-container">$AA^TAv = 0$</span> and <span class="math-container">$Av$</span> would be an eigenvector of eigenvalue <span class="math-container">$0$</span> unless <span class="math-container">$Av$</span> is the zero vector. So this is what ... |
1,405,787 | <p>We're given the following function :</p>
<p>$$f(x,y)=\dfrac{1}{1+x-y}$$</p>
<p>Now , how to prove that the given function is differentiable at $(0,0)$ ?</p>
<p>I found out the partial derivatives as $f_x(0,0)=(-1)$ and $f_y(0,0)=1$ ,</p>
<p>Clearly the partial derivatives are continuous , but that doesn't guaran... | mathcounterexamples.net | 187,663 | <p>You have several ways to approach the question.</p>
<p><strong>First one</strong>
$f$ is the ratio of two differentiable functions, the denominator one not vanishing in the neighborhood of the origin. Hence $f$ is differentiable at the origin.</p>
<p><strong>Second one</strong>
Using a theorem stating that if $f$ ... |
4,253,761 | <p><span class="math-container">$$\begin{array}{ll} \text{extremize} & xy+2yz+3zx\\ \text{subject to} & x^2+y^2+z^2=1\end{array}$$</span></p>
<p>How to find the maximum/minimum using Lagrange multipliers?</p>
<p>Context: This is not a homework problem, my friend and I often make up problems to challenge each ot... | ryang | 21,813 | <p>I think of the <strong>trivial case</strong> as the case that invariably arises as the problem’s parameters vary.</p>
<p>In particular, the <strong>trivial solution</strong> of a homogeneous system of linear equations is the zero vector.</p>
|
2,086,770 | <p>$[c^2, c^3, c^4] \text { parallel to/same direction as } [1,-2,4]$</p>
<p>Find $c$ if it exists.</p>
<p>How do I see if they are parallel and find a c?</p>
<p>Generally, we can say that $r[1,-2,4] = [c^2, c^3, c^4]$</p>
<p>But then I have two unknowns and I don't know how to solve like this?</p>
| Bernard | 202,857 | <p>As $[c^2,c^2,c^4]=c^2[1,c,c^2]$ is parallel to and has the same direction as $[1,c,c^2]$, this amounts to solving the same problem for the latter. Actually, one can obviously find $c$ such that $[1,c,c^2]$ is not only parallel, but <em>equal</em> to $[1,-2,4]$.</p>
|
1,201,904 | <p>I have to implement a circuit following the boolean equation A XOR B XOR C, however the XOR gates I am using only have two inputs (I am using the 7486 XOR gate to be exact, in case that makes a difference)... is there a way around this?</p>
| coffeemath | 30,316 | <p>Take the output of A XOR B and pipe it into an XOR having C as the other input.
(This implements (A XOR B) XOR C, and XOR is associative.</p>
|
2,707,749 | <p>I know this has to be extremely easy but I'm not going to solve this problem.</p>
<p>The task is to find the angle at point $A$.</p>
<p>Thanks!</p>
<p><img src="https://i.stack.imgur.com/8NTEO.png" alt="Here is the image."></p>
| ericw31415 | 333,981 | <p>Notice that $\angle EOF+\angle BOC+49^\circ+175^\circ=360^\circ$. It follows that $\angle EOF+\angle BOC=136^\circ$.</p>
<p>$\angle OCA=\dfrac{180^\circ-\angle BOC}{2}$ and $\angle OEA=\dfrac{180^\circ-\angle EOF}{2}$ because $\triangle OBC$ and $\triangle OEF$ are isosceles respectively.</p>
<p>Now $\angle OAC+\a... |
4,469,136 | <p>I took a number theory course this past semester and I found the idea of there being different primes in different fields. The only fields that we did in detail were the set of reals and the set of Gaussian integers.</p>
<p>What other fields are there, and what numbers are prime in those fields?</p>
| anomaly | 156,999 | <p>Fields don't have any proper ideals, so the idea of looking literally at primes in a field isn't very interesting. The usual construction is to start with a number field <span class="math-container">$k$</span> (i.e., a finite extension of <span class="math-container">$\mathbb{Q}$</span>) and consider the algebraic i... |
2,819,667 | <p>The problem is the same as <a href="https://math.stackexchange.com/questions/14190/average-length-of-the-longest-segment">here</a>. </p>
<blockquote>
<p>A stick of 1m is divided into three pieces by two random points. Find the average length of the largest segment.</p>
</blockquote>
<p>I tried solving it in a di... | Pi_die_die | 564,003 | <p>This is just a suggestion not an answer but could you try solving it keeping the distance of first division as x and second as y making the lengths of the segments as x, y-x ,1-y</p>
|
2,819,667 | <p>The problem is the same as <a href="https://math.stackexchange.com/questions/14190/average-length-of-the-longest-segment">here</a>. </p>
<blockquote>
<p>A stick of 1m is divided into three pieces by two random points. Find the average length of the largest segment.</p>
</blockquote>
<p>I tried solving it in a di... | Doug M | 317,162 | <p>Here is how I would do it.</p>
<p>Lets define $x$ to be the short stick, $y$ to be the medium stick and $z$ to be the long stick.</p>
<p>$x\le y\le z\\
z = 1-x-y\\
x\le y \le \frac {1-x}{2}\\
x\le \frac 13$</p>
<p>$$ \bar z = \frac {\displaystyle\int_0^\frac 13\int_x^{\frac {1-x}{2}} 1-x-y\ dy\ dx}{\displaystyle... |
2,902,768 | <p>$f:\mathbb{R}^2 \to \mathbb{R}$</p>
<p>$f\Bigg(\begin{matrix}x\\y\end{matrix}\Bigg)=\begin{cases}\frac{xy^2}{x^2+y^2},(x,y)^T \neq(0,0)^T \\0 , (x,y)^T=(0,0)^T\end{cases}$</p>
<p>I need to determine all partial derivatives for $(x,y)^T \in \mathbb{R}^2$:</p>
<p>$f_x=y^2/(x^2+y^2)-2x^2y^2/(x^2+y^2)^2$ for $(x,y)^T... | Rigel | 11,776 | <p>The function $f$ is not differentiable at the origin.
You can verify this fact (at least) in two ways.</p>
<p>1) The map $v \mapsto \frac{\partial f}{\partial v} (\mathbf{0})$ is not a linear map.</p>
<p>2) Using the definition of differentiability.</p>
<p>In this second case, you have to show that
$$
\varphi(x,y... |
2,902,768 | <p>$f:\mathbb{R}^2 \to \mathbb{R}$</p>
<p>$f\Bigg(\begin{matrix}x\\y\end{matrix}\Bigg)=\begin{cases}\frac{xy^2}{x^2+y^2},(x,y)^T \neq(0,0)^T \\0 , (x,y)^T=(0,0)^T\end{cases}$</p>
<p>I need to determine all partial derivatives for $(x,y)^T \in \mathbb{R}^2$:</p>
<p>$f_x=y^2/(x^2+y^2)-2x^2y^2/(x^2+y^2)^2$ for $(x,y)^T... | leonbloy | 312 | <p>If a funcion is differentiable, the the directional derivative equals $\nabla_f \cdot v $. But here $\nabla_f =(0,0)$, hence all directional derivatives should be zero.</p>
<p>Now, setting $y=ax$ we get $$f(x,y)=f(x,ax)=\frac{ a x^3}{x^2 +a^2 x^2}=\frac{a}{1+a^2}x$$</p>
<p>Then, the derivative take different value... |
6,990 | <p>The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.</p>
<ul>
<li><p>In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can ... | Konstantin Slutsky | 896 | <p>It is not what you want, but may be worth mentioning. There is a huge branch of abstract harmonic analysis on (abelian) locally compact groups, which generalizes Fourier transformation on reals and circle. The main point about sin and cos (or rather complex exponent $e^{i n x}$) is that it is a character (continuous... |
6,990 | <p>The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.</p>
<ul>
<li><p>In how far can this be generalized to other countable sets of functions? For example, if we keep our inner product, can ... | Bob Bauer | 2,396 | <p>Any compact normal operator on a Hilbert space has an orthonormal basis of eigenvectors. If I remember correctly then the standard Fourier series comes from the second derivative operator on L^2(0,2pi) with boundary conditions f(0)=f(2pi) and f'(0)=f'(2pi). This operator is not compact, but its inverse is (and has ... |
2,083,475 | <p>This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$
the matrices associated with the conic are:
$$
A'=\left(\begin{array}{cccc}
\frac{1}{2} & 1 & \frac{1}{2} \\
1 & 1 & 3 \\
\frac{1}{2} & 3 & 1
\end{array}\right),
$$</p>
<p>$$
A=\left(\begin{array}{cccc}
1 & 3 \\
3 & 1
\end{array}\r... | Ng Chung Tak | 299,599 | <p>You've two principal axes:</p>
<p>\begin{align*}
x^2+6xy+y^2+2x+y+\frac{1}{2} & \equiv
A\left( \frac{4x-4y-1}{\sqrt{4^2+4^2}} \right)^2+
B\left( \frac{8x+8y+3}{\sqrt{8^2+8^2}} \right)^2+C \\
& \equiv
-2\left( \frac{4x-4y-1}{4\sqrt{2}} \right)^2+
4\left( \frac{8x+8y+3}{8\sqrt{2}} \right)^2+\frac{9}{32... |
2,083,475 | <p>This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$
the matrices associated with the conic are:
$$
A'=\left(\begin{array}{cccc}
\frac{1}{2} & 1 & \frac{1}{2} \\
1 & 1 & 3 \\
\frac{1}{2} & 3 & 1
\end{array}\right),
$$</p>
<p>$$
A=\left(\begin{array}{cccc}
1 & 3 \\
3 & 1
\end{array}\r... | Will Jagy | 10,400 | <p>These two examples are from a book that does not assume linear algebra... Both good and bad, as they are giving concrete methods, which are a bit much to memorize. Worth doing both ways, really, get them to agree; this way, then linear algebra and coordinate changes (translation to center followed by rotation). The... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Quixotic | 2,109 | <p>$$ \sin \theta \cdot \sin \bigl(60^\circ - \theta \bigr) \cdot \sin \bigl(60^\circ + \theta \bigr) = \frac{1}{4} \sin 3\theta$$</p>
<p>$$ \cos \theta \cdot \cos \bigl(60^\circ - \theta \bigr) \cdot \cos \bigl(60^\circ + \theta \bigr) = \frac{1}{4} \cos 3\theta$$</p>
<p>$$ \tan \theta \cdot \tan \bigl(60^\circ - ... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| tomerg | 9,659 | <p>M.V Subbarao's identity: an integer $n>22$ is a prime number iff it satisfies,</p>
<p>$$n\sigma(n)\equiv 2 \pmod {\phi(n)}$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| draks ... | 19,341 | <p>What is 42?</p>
<p>$$
6 \times 9 = 42 \text{ base } 13
$$
I always knew that there is something wrong with this universe.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Doug Spoonwood | 11,300 | <p>$\lnot$(A$\land$B)=($\lnot$A$\lor$$\lnot$B) and
$\lnot$(A$\lor$B)=($\lnot$A$\land$$\lnot$B), because they mean that negation is an "equal form".</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| liaombro | 90,999 | <p>$$27\cdot56=2\cdot756,$$
$$277\cdot756=27\cdot7756,$$
$$2777\cdot7756=277\cdot77756,$$
and so on.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Stefan4024 | 67,746 | <p>Here's one clever trigonometric identity that impressed me in high-school days. Add $\sin \alpha$, to both the numerator and the denominator of $\sqrt{\frac{1-\cos \alpha}{1 + \cos \alpha}}$ and get rid of the square root and nothing changes. In other words:</p>
<p>$$\frac{1 - \cos \alpha + \sin \alpha}{1 + \cos \a... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Vladimir Reshetnikov | 19,661 | <p>$$\int_0^\infty\frac1{1+x^2}\cdot\frac1{1+x^\pi}dx=\int_0^\infty\frac1{1+x^2}\cdot\frac1{1+x^e}dx$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| ILikeMath | 86,744 | <p>If we define $P$ as the infinite lower triangular matrix where $P_{i,j} = \binom{i}{j}$ (we can call it the Pascal Matrix), then $$P^k_{i,j} = \binom{i}{j}k^{i-j}$$</p>
<p>where $P^k_{i,j}$ is the element of $P^k$ in the position $i,j.$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| happymath | 129,901 | <p><strong>Voronoi summation formula:</strong></p>
<p>$\sum \limits_{n=1}^{\infty}d(n)(\frac{x}{n})^{1/2}\{Y_1(4\pi \sqrt{nx})+\frac{2}{\pi}K_1(4\pi \sqrt{nx})\}+x \log x +(2 \gamma-1)x +\frac{1}{4}=\sum \limits _{n\leq x}'d(n)$</p>
|
4,556,592 | <p>Check if the applications defined below are linear transformations:</p>
<p>a) <span class="math-container">$T: \mathbb R^2 \to \mathbb R^2$</span>, <span class="math-container">$T(x_1, x_2) = (x_1 – 1, x_2).$</span></p>
<p>b) <span class="math-container">$T: \mathbb R^2 \to \mathbb R^2$</span>, <span class="math-con... | A. P. | 1,027,216 | <ul>
<li>In a) you have <span class="math-container">$T(u+bv)=(u_{1}+u_{2}-1,b(v_{2}+v_{2}))\not=T(u)+bT(v)$</span>. Hence, <span class="math-container">$T$</span> is not linear transformation.</li>
<li>In b), <span class="math-container">$T(u+bv)=(b(v_{1}+v_{2}),u_{1}+u_{2})=T(u)+bT(v)$</span>. Hence <span class="math... |
4,388,974 | <p>Let <span class="math-container">$f$</span> be a analytic function in the closed unit circle with its center
at the point <span class="math-container">$\alpha\in\mathbb{R}$</span>, then:
<span class="math-container">\begin{equation*}
\int_0^\pi\frac{f\left(\alpha+e^{ix}\right)+f\left(\alpha+e^{-ix}\right)}{1+2p\cos(... | Quanto | 686,284 | <p>Integrate as follows
<span class="math-container">\begin{align}
&\int_0^\pi\frac{f\left(\alpha+e^{ix}\right)+f\left(\alpha+e^{-ix}\right)}{1+2p\cos x+p^2} dx\\
=&\int_0^\pi\frac{2\Re f\left(\alpha+e^{ix}\right)}{1+2p\cos x+p^2} dx
= 2\int_0^\pi\frac{\sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} \cos(kx)}{1+2p\cos ... |
495,069 | <p>Find an equation of the plane.
The plane that passes through the line of intersection of the planes
$x − z = 1$ and $y + 4z = 1$
and is perpendicular to the plane
$x + y − 2z = 2$.</p>
<p>I keep getting the answer of $7x-y+5z=6$ and I am told that it is wrong. I do not understand what I am doing wrong.
I have th... | Pratyush Sarkar | 64,618 | <p>You made a small error in your last cross product. The result is $(7, 3, 5)$. So the equation is of the form $7x + 3y + 5z = c$. Using the point $(1, 1, 0)$, we get $c = 10$. So the equation of the plane is $7x + 3y + 5z = 10$.</p>
|
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Per Alexandersson | 1,056 | <p>Right now? "If you can use it in Quantum mechanics, it is worth studying" seems to be the general idea behind a lot of math these days (I have 2 ongoing articles with this explanation)..</p>
|
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Jon Bannon | 6,269 | <p>In some sense, mathematical structure is simply analogy at a very high level. One tries to fill in details in a way that is likely to pay off. (E.g. looking for a natural way to make a semigroup you are looking at into a group may just pay off, simply because groups are ubiquitous and useful.) This may be the reason... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Ronnie Brown | 19,949 | <p>I have just now looked again at this interesting blog and thought to add a few points. </p>
<p>1) Methodology: You could read the comments of Grothendieck on "speculation" <a href="http://groupoids.org.uk/Grothendieck-speculation.html" rel="nofollow noreferrer">http://groupoids.org.uk/Grothendieck-speculation.html<... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Timothy Chow | 3,106 | <p>I'm going to interpret your question in the language of Gowers's "two cultures" essay as follows:</p>
<blockquote>
<p>How does one get good at theory-building?</p>
</blockquote>
<p>The process of developing a good theory can seem deceptively simple. One takes some definitions, perhaps by generalizing some known... |
3,843,559 | <p>Can someone please give me a hint why for metric spaces we have</p>
<p><span class="math-container">$d_1(x,y)<d(x,y)\Rightarrow \{x|d(x,y)<\varepsilon\}\subset \{x|d_1(x,y)<\varepsilon\}$</span></p>
<p>I have expected the opposite:</p>
<p><span class="math-container">$d_1(x,y)<d(x,y)\Rightarrow \{x|d(x,y... | Victor Hugo | 322,450 | <p>Yes, we claim that <span class="math-container">$\{x:d(x,y)<\varepsilon\}\subset \{x:d_1(x,y)<\varepsilon\}$</span>. Indeed, let <span class="math-container">$x_0 \in \{x:d(x,y)<\varepsilon\}$</span> then, <span class="math-container">$d(x_0,y)<\varepsilon$</span>. Since <span class="math-container">$d_1... |
2,454,455 | <p>I know this is a soft and opinion based question and I risk that this question get's closed/downvoted but I still wanted to know what other persons, who are interested in mathematics, think about my question.</p>
<p>Whenever people are talking about the most beautiful equation/identity Euler's identity is cited in... | Lwins | 134,950 | <p>In my opinion, you definitely have authority to define what is beautiful and what is not to yourself. And your post let me think of a constant $\tau = 2\pi$ (see <a href="https://tauday.com/" rel="nofollow noreferrer">https://tauday.com/</a>), which is thought by some people as a more "beautiful" and more "natural" ... |
1,729,893 | <p>Let $\nu_1, \nu_2, \nu_3 \in \mathbb{R}$ not all be zero.</p>
<p>I wish to show
$$\nu_1^2 + \nu_1\nu_2 + \nu_2^2+\nu_2\nu_3 + \nu_3^2 > 0\text{.}$$</p>
<p><a href="http://www.wolframalpha.com/input/?i=x%5E2%2Bx*y%2By%5E2%2By*z%2Bz%5E2+%3E+0" rel="nofollow">Wolfram</a> seems to suggest splitting this into cases,... | user5713492 | 316,404 | <p>I could more or less instantly spot
$$\nu_1^2+\nu_1\nu_2+\nu_2^2+\nu_2\nu_3+\nu_3^3=(\nu_1+\frac12\nu_2)^2+\left(\frac1{\sqrt2}\nu_2\right)^2+(\frac12\nu_2+\nu_3)^2$$</p>
|
239,900 | <p>Hatcher states the following theorem on page 114 of his Algebraic Topology:</p>
<blockquote>
<p>If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence
$$...\longrightarrow\widetilde{H}_n(A)\overset{i_*}\longrightarrow \w... | Elchanan Solomon | 647 | <p>If you're looking for an arbitrary topological space with this property, take $X = \{a,b,c\}$ and let the topology $\mathcal{T} = \{\emptyset, \{a,b,c\},\{c\},\{a,c\},\{b,c\}\}$. Consider the closed subset $C = \{a,b\}$. The only open set that might retract onto $C$ is the whole space $X = \{a,b,c\}$. Let us suppose... |
698,474 | <p>I am trying to find the phi(18). Using an online calculator, it says it is 6 but im getting four.
<br/>
The method I am using is by breaking 18 down into primes and then multiplying the phi(primes)</p>
<p>$$=\varphi (18)$$
$$=\varphi (3) \cdot \varphi(3) \cdot \varphi(2)$$
$$= 2 \cdot 2 \cdot 1$$
$$= 4$$</p>
| Community | -1 | <p>The phi function is <a href="http://en.wikipedia.org/wiki/Multiplicative_function">multiplicative</a>, but not completely multiplicative: Thus if $a, b$ are relatively prime, we have that</p>
<p>$$\varphi(ab) = \varphi(a) \varphi(b)$$</p>
<p>but this is <em>not</em> necessarily true if $a$ and $b$ have a common pr... |
1,243,750 | <p>$A \in \mathbb{R}^{n\times n}$, with $A^2 = 1$ and $A\ne\pm1$</p>
<p>Show that the only eigenvalues of $A$ are $1$ and $-1$.</p>
| BaronVT | 39,526 | <p>Hint: if $\lambda$ is an eigenvalue of $A$, what can you say about eigenvalues of $A^2$?</p>
|
1,105,352 | <p>Here are a few definition from my workbook:</p>
<p>Let $a\in\Bbb{R}\cup\{+\infty, -\infty\}$, $D\subset\Bbb{R}$. Then point $a$ is a limit point of $D$ if and only if there exists a sequence $\{x_n\}_{n \geq n_0}$ with terms in set $D - \{a\}$ such that $x_n \rightarrow a$.</p>
<p>And now we have a limit of a func... | Zubin Mukerjee | 111,946 | <p>I think you are reading the definitions correctly. Since $f$ is <em>defined</em> only on D, there is no such thing as a "right-hand limit" at $1$. </p>
<p>This means that the limit existing at $1$ is not a contradiction (nothing in the limit definition says anything about the right and left hand limits).</p>
<p>Si... |
778,294 | <p>If $G$ is an open subset of $ \mathbb R$ such that $ 0 \notin G $ , then is it true that $H:=\{ xy \mid x , y \in G \}$ is an open subset of $ \mathbb R$ ?</p>
| Hagen von Eitzen | 39,174 | <p>For $y\ne 0$, the map $x\mapsto xy$ is a homeomorphism $\mathbb R\to\mathbb R$, hence with $G$ also $Gy$ is open. Then $H=\bigcup_{y\in G} Gy$ is the union of open sets, hence open.</p>
<hr>
<p>Remarkably, the <em>result</em> does not change if we allow $0\in G$, though the <em>method</em> of proof above becomes p... |
331,169 | <p>Let $\Omega\subset \mathbb{R}^n$ be bounded and let $X:=H^1(\Omega)$. Let $a:\Omega\times \mathbb{R} \to \mathbb{R}, (x,z)\mapsto a(x,z)$ be a bounded function such that $a(x,.)$ is continuous on $\mathbb{R}$ for every $x$ and $a(.,z)$ is measurable on $\Omega$ for every $z$.</p>
<p>For $v,\phi\in H^1$ one finds in... | apnorton | 23,353 | <p>As per comments, the group in question is $\mathbb{Z} \times \mathbb{Z}$.</p>
|
520,203 | <p>Here's what I'm reading: every regular bipartite graph has a 1-factor.</p>
<p>But I understand that not every regular graph has a 1-factor.</p>
<p>So, I was thinking if it's possible to find a $k$-regular simple graph without 1-factor for any fixed $k$?</p>
| Elchanan Solomon | 647 | <p>If $k$ is even, then $K_{k+1}$ is $k$ regular with $k+1$ (odd) vertices, and hence has no $1$-factor (as a perfect matching requires an even number of vertices).</p>
<p>If $k$ is odd, consider the following graph. Start at an initial vertex, then branch out in $k$ directions. From each of those new vertices, branch... |
3,290,073 | <p>I'm trying to help my daughter learn maths. She is struggling with factors, which is to work out what numbers go into a larger number (division).</p>
<p>I've already learned that by summing numbers, if they make 3, it can divide by 3. I also know the rules for 2, 5, 6, 9 and 10.</p>
<p>I'm trying to see if there i... | Robert Israel | 8,508 | <p>Tens place even and units <span class="math-container">$0$</span>, <span class="math-container">$4$</span> or <span class="math-container">$8$</span> (i.e. divisible by <span class="math-container">$4$</span>), or tens place odd and units <span class="math-container">$2$</span> or <span class="math-container">$6$</s... |
3,290,073 | <p>I'm trying to help my daughter learn maths. She is struggling with factors, which is to work out what numbers go into a larger number (division).</p>
<p>I've already learned that by summing numbers, if they make 3, it can divide by 3. I also know the rules for 2, 5, 6, 9 and 10.</p>
<p>I'm trying to see if there i... | Questioner | 894,469 | <p>There <em>is</em> a rule of divisibility for the number <span class="math-container">$4$</span>. Here it is:</p>
<p>To figure out if a number is divisible by four, you first need to look at the last two digits, and if they're divisible by four together, you can assume that the whole number is divisible by <span clas... |
341,621 | <p>Is there an inequality such as
$$(a+b)^2 \leq 2(a^2 + b^2)$$
for higher powers of $k$
$$(a+b)^k \leq C(a^k + b^k)?$$</p>
| lsp | 64,509 | <p>The first inequality is true because : $2ab ≤ a^2+b^2$</p>
<p>There is no specific inequality for higher powers of k, although you can always give a very higher value for 'C' to make the second inequality true.</p>
|
2,703,559 | <p>My assignment asks me to prove that the only automorphism of order 2 of $\mathbb{Z}_q$ is $m \mapsto -m$ for $q = 3$ and $q = 5$ and $q = 7$. I have been stuck for ages, and now I wonder if it is true. I need some help to get started. This is my attempt: Assume $k$ and $q$ are relative prime (this is necessary other... | lhf | 589 | <p><em>Hint:</em> When $q$ is prime, $ k^2 \equiv 1 \bmod q \iff k \equiv \pm1 \bmod q$. What is the order of $x \mapsto kx$ when $k=1$ ?</p>
|
2,703,559 | <p>My assignment asks me to prove that the only automorphism of order 2 of $\mathbb{Z}_q$ is $m \mapsto -m$ for $q = 3$ and $q = 5$ and $q = 7$. I have been stuck for ages, and now I wonder if it is true. I need some help to get started. This is my attempt: Assume $k$ and $q$ are relative prime (this is necessary other... | Asinomás | 33,907 | <p>Suppose $k^2\equiv 1 \bmod p$, then $(k-1)(k+1)\equiv 0 \bmod p$. That is the reason why there are only $2$ options.</p>
<p>In general we can do these sort of things in any integer domain.( polynomials cant have more roots than their degree)</p>
|
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Axel Kemper | 58,610 | <p>The left-hand-side input and the right-hand-side output can be imagined as binary numbers in a kind of truth table: </p>
<p><img src="https://i.stack.imgur.com/rgfSR.jpg" alt="enter image description here"> </p>
<p>All eight output bits can be calculated from the seven input bits evaluating simple Boolean ex... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | devnull69 | 164,488 | <p>The first multiplicant is given. So the open question is "what is the second multiplicant"?</p>
<p>The list can be grouped in sixes. So lines 1-6 is one group, the rules count for each group.</p>
<ul>
<li>Define fm as the given first multiplicant of the row. Start with <code>4</code>. Increment fm by one on each r... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | WizardLizard | 87,676 | <p>$A\cdot B = C$</p>
<p>$\dfrac{AB²}{\gcd(AB²,6)} = C$ or $\dfrac{\text{lcm}(AB²,6)}6 = C$</p>
|
1,401,760 | <p>First I tried to use integration:
$$y=\lim_{n\to\infty}\frac{a^n}{n!}=\lim_{n\to\infty}\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdots\frac{a}{n}$$
$$\log y=\lim_{n\to\infty}\sum_{r=1}^n\log\frac{a}{r}$$
But I could not express it as a <em>riemann integral</em>. Now I am thinking about sandwich theorem.</p>
<p>$$... | ajotatxe | 132,456 | <p>You missed the lower slice of bread in your sandiwch :)</p>
<p>Assuming that $a>0$, you should make clear that all terms are positive, no matter how obvious seems to be.</p>
<p>You should also write explicitly that there is some $t\in\Bbb N$ such that $t>a$. This is called "Archimedean property" of real numb... |
1,401,760 | <p>First I tried to use integration:
$$y=\lim_{n\to\infty}\frac{a^n}{n!}=\lim_{n\to\infty}\frac{a}{1}\cdot\frac{a}{2}\cdot\frac{a}{3}\cdots\frac{a}{n}$$
$$\log y=\lim_{n\to\infty}\sum_{r=1}^n\log\frac{a}{r}$$
But I could not express it as a <em>riemann integral</em>. Now I am thinking about sandwich theorem.</p>
<p>$$... | Michael Hardy | 11,667 | <p>Here's an easy way that I'm surprised is not yet here:</p>
<p>$$
\frac{a^n}{n!} = \frac{\overbrace{a\cdots\cdots\cdots\cdots a}}{\underbrace{1\cdot2\cdot3\cdots\,\cdots n}}
$$
When $n$ reaches the point of being twice as big as $a,$ then every time you increment $n$ by $1$ after that, you multiply the numerator by... |
3,635,266 | <p>As stated in title, I am curious how to construct a sequnce <span class="math-container">$\{a_n\}_{n=1}^{\infty}$</span> such that <span class="math-container">$\sum_{n=1}^{\infty} a_n^2 < +\infty$</span> but <span class="math-container">$\sum_{n=1}^{\infty}\frac{a_n}{\sqrt{n}} = \infty$</span>.</p>
<p>This prob... | Integrand | 207,050 | <p>Try <span class="math-container">$a_n = (\sqrt{n}\log(n))^{-1},$</span> for <span class="math-container">$n\geq 2$</span>.</p>
|
642,497 | <p>How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? </p>
<p>(taking $ABCD$ to be the trapezium with
$AD$ || $BC$ and $O$ the intersection point of diagonals , if it can be shown by vector method that
$OB=OC$ , ... | Belgi | 21,335 | <p><strong>Hint</strong>: Taylor series around $x=-2$</p>
<p>Your way is good too!</p>
|
642,497 | <p>How do we prove by vector method that "if the diagonals of a trapezium have equal length then the non-parallel sides of the trapezium have equal length." ? </p>
<p>(taking $ABCD$ to be the trapezium with
$AD$ || $BC$ and $O$ the intersection point of diagonals , if it can be shown by vector method that
$OB=OC$ , ... | Stefan4024 | 67,746 | <p>Here's another way:</p>
<p>$$\begin{array}{c|c|c|c|c|c}
1 & 0 & -2 & 6 & 0 & 1 \\ \hline
1 & -2 & 2 & 2 & -4 & 9 \\[0.55ex]
1 & -4 & 10 & -18 & 32\\[0.55ex]
1 & -6 & 22 & -62 \\[0.55ex]
1 & -8 & 3... |
2,307,268 | <p>I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3.
Please explain me the difference between two expressions.</p>
<p><a href="https://i.stack.imgur.com/2CMDt.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur... | Community | -1 | <p>Your calculator is interpreting it as this:</p>
<p>$${{2^2}^2}^2 = ((2^2)^2)^2 = (4^2)^2 = 16^2 = 256$$</p>
<p>Google is interpreting it as this:</p>
<p>$${{2^2}^2}^2 = 2^{(2^{(2^2)})} = 2^{(2^4)} = 2^{16} = 65536$$</p>
<p>Similarly with the threes.</p>
<p>Technically Google is correct because order of operatio... |
4,356,457 | <p>Let <span class="math-container">$k$</span> be a finite field of size <span class="math-container">$q$</span>, let <span class="math-container">$n\ge1$</span>, and let <span class="math-container">$\mathrm{GL}_n(k)$</span> and <span class="math-container">$M_n(k)$</span> be the group of invertible matrices and ring ... | Kenta S | 404,616 | <p>This is based on reuns's idea.</p>
<p>Fix a field <span class="math-container">$k$</span> of cardinality <span class="math-container">$q$</span>.</p>
<p><strong>Lemma 1</strong> There are <span class="math-container">$\frac1n\sum_{d|n}\mu(\frac nd)q^d$</span> irreducible monic polynomials of degree <span class="math... |
3,912,635 | <p>I am currently trying to understand the proof of Proposition 4.3.18 in Pedersen's Analysis now, which reads</p>
<blockquote>
<p>To each Tychonoff space <span class="math-container">$X$</span> there is a Hausdorff compactification <span class="math-container">$\beta(X)$</span>, with the property that every continuous... | s.harp | 152,424 | <p>Consider a special set of characters of <span class="math-container">$C_b(X)$</span>, for each <span class="math-container">$x\in X$</span> define:</p>
<p><span class="math-container">$$\delta_x: C_b(X)\to\Bbb C, \quad g\mapsto g(x)$$</span></p>
<p>Since the (non-zero) characters of <span class="math-container">$C_b... |
792,813 | <p>Let A be a random variable defined as:</p>
<ul>
<li>With probability $p[i]$, the random variable $B[i]$ is drawn</li>
<li>$B[i] ~ N[mu[i],sigma[i]]$</li>
<li>probabilities $p[i]$ sum up to one</li>
</ul>
<p>I know how to compute the mean, which is given by:</p>
<p>$$E[A] = p[1]*mu[1] + .. + p[N]*mu[N]$$</p>
<p>... | Community | -1 | <p>You're correct. I give an other proof. Let
$$F(z)=\int_0^z\cos(x^2)x^3dx$$
and by the fundamental theorem of analysis and using the l'Hôpital's rule we have
$$\lim_{z \to 0^+} \frac{1}{z^4} \int_{0}^{z} cos(x^2)x^3dx=\lim_{z \to 0^+} \frac{F(z)}{z^4}=\lim_{z \to 0^+} \frac{\cos(z^2)z^3}{4z^3}=\frac14$$</p>
|
2,670,258 | <p>Just to explain the motivation behind some other thing, I shall ask first this question:</p>
<blockquote>
<p>Is $\frac{1}{x}$ continuosly differentiable (or we can examine
$\frac{1}{x^3}$ also) on the set $$E = (- \infty, 0) \cup (0,
> \infty)$$</p>
</blockquote>
<p>The reason why I'm asking this is that w... | MPW | 113,214 | <p>Your notation is inadequate. The function is in $C^1(E)$. It is also in $C^1(U)$ for every interval $U\subset E$. You must specify the domain in this notation if it isn’t implicit, especially when you are considering several of them.
In other words, $C^1$ without a specified domain is ambiguous.</p>
|
2,670,258 | <p>Just to explain the motivation behind some other thing, I shall ask first this question:</p>
<blockquote>
<p>Is $\frac{1}{x}$ continuosly differentiable (or we can examine
$\frac{1}{x^3}$ also) on the set $$E = (- \infty, 0) \cup (0,
> \infty)$$</p>
</blockquote>
<p>The reason why I'm asking this is that w... | symplectomorphic | 23,611 | <p>Yes, $1/x$ and $1/x^3$ are continuously differentiable (in fact, infinitely differentiable) on your domain (the set of all nonzero reals).</p>
<p>The reason we typically assume that the domain of a solution to an IVP is connected is to ensure uniqueness. If you try to solve the IVP $f'(x)=-1/x^2$ with $f(1)=1$ on y... |
2,072,061 | <p>My Working: </p>
<p>\begin{align}
a_0=&-0+2=2\\
a_1=&-1+2=1\\
a_2=&-2+2=0.
\end{align}
and $$a_2=a_{2-1}+2a_{2-2}+2(2)-9=-4.$$</p>
<p>I don't know what's wrong with my solution. I used this method on various questions but now I'm just ending up with the wrong answer.</p>
| Kanwaljit Singh | 401,635 | <p>Solving it further we have -</p>
<p>$a_{2} = a_{2-1} + 2a_{2-2} + 2(2) - 9$</p>
<p>$a_{2} = a_{1} + 2a_{0} + 2(2) - 9$</p>
<p>$a_{2}$ = 1 + 2(2) + 2(2) - 9$</p>
<p>$a_{2}$ = 1 + 4 + 4 - 9$</p>
<p>= 0</p>
<p>Your mistake is you are using the value of $a_{0}$ = 0 but it is 2.</p>
|
812,263 | <p>To further explain the title:</p>
<p>Is there a probabilistic reason as to why a 6-sided die has the opposing sides suming to 7?</p>
<p>My argument begun when a friend decided to use <a href="http://ecx.images-amazon.com/images/I/51OvRGphnOL.jpg" rel="nofollow">this die</a> instead of <a href="http://nerdywithchil... | Henry | 6,460 | <p>If dice were cuboids rather than cubes due to a manufacturing defect, then each face would no longer have equal probability, but by symmetry the expected value of each throw would still be $\dfrac{7}{2}$.</p>
<p>I have no idea whether this was a consideration.</p>
|
470,081 | <p>There is a question from an old topology prelim that is somewhat giving me a hard time. Consider the cylinder $X= S^1 \times [-1,1]$. Now we define an equivalence relation $\sim$ as follows: For points $v,v' \in S^1$, we have $(v,-1) \sim (v',-1)$ and $(v,1) \sim (v',1)$. I am asked to show that the quotient space $... | kahen | 1,269 | <p>The obvious projection map from $X$ (as a subset of $\mathbb R^3$) to the unit sphere in $\mathbb R^3$ is a continuous surjection. By the <a href="https://en.wikipedia.org/wiki/Quotient_space#Properties" rel="nofollow">universal property of the quotient topology</a> it descends to a map $f: X^* \to S^2$. Now show th... |
1,934,259 | <blockquote>
<p><strong>Problem.</strong></p>
<p>Let <span class="math-container">$\emptyset \subset A\subset X$</span> and <span class="math-container">$\emptyset \subset B\subset Y$</span>. If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are connected, show that <span class="mat... | Alex Mathers | 227,652 | <p>The proof is essentially the same as the one linked to. Suppose we have a function $f:(X\times Y)\setminus(A\times B)\to\{\pm1\}$. Begin by choosing $a\in X\setminus A$ and $b\in Y\setminus B$ (which is possible because both are proper subsets). Now, let $(x,y)\in(X\times Y)\setminus(A\times B)$ be arbitrary. We wil... |
2,003,660 | <p>I got this exercise from the textbook Book of Proof, CH4 E12. I've tackle this problem in the following manner:</p>
<p>Suppose x is a real and $0 < x < 4$, it follows that,</p>
<p>\begin{align*}
&\Rightarrow 0 - 2 < x - 2 < 4 - 2 \\
&\Rightarrow 4 < (x - 2)^2 < 4\\
&\Rightarrow ... | Barry Cipra | 86,747 | <p>Your approach is actually quite nice, but there is one error: It is incorrect to say that $0-2\lt x-2 \lt 4-2$ implies $4\lt(x-2)^2\lt4$. What you want to say instead is something like</p>
<p>$$0-2\lt x-2\lt4-2\implies|x-2|\lt2\implies0\le(x-2)^2\lt4$$</p>
|
1,493,783 | <p>We have $4$ pockets each containing $4$ placement positions, we want to obtain an estimate on the number of possible arrangements, when the exact set of allowed elements is given:
$$ls=[a,a,a,a,b,b,c,c,d,d,x,y] \tag{1}$$
and numbers [1:4] are used for empty slots. For example if one pocket is of the form -,a,-,- t... | true blue anil | 22,388 | <p>There is a typo in Miller's answer, it should be $\dfrac{16!}{4!4!2!2!2!}$, as there are $2$ each of $b,c,d$</p>
<p>The ans can also be got by <em>placing</em> elements which will make dealing with the restrictions easier. </p>
<p>$\dbinom{16}{4}\dbinom{12}{4}\dbinom 82\dbinom 62 \dbinom 42\dbinom21\dbinom11 = 454... |
2,131,224 | <blockquote>
<p>Say I have vectors $x, y$, then is $\text{proj }_x y $ a scalar multiple of $x$?</p>
</blockquote>
<p>I have a book saying that it is, but I have no clue why this true. Is this really true?</p>
| angryavian | 43,949 | <p>Yes this is true. You should check any definitions/formulas for the projection that you have.</p>
<p>In general, $$\operatorname{proj}_x y = \frac{\langle y,x\rangle }{\langle x,x\rangle } x$$
from which your claim is clear.</p>
|
483,533 | <p>How to prove that the following linear map :</p>
<p>$T : \Bbb R^2 \to \Bbb R^2$ defined by $$T(e_1)=e_2$$
$$T(e_2)=0$$ </p>
<p>can't be diagonalized with respect to any basis. Here $e_i$s are standard basis.</p>
| Ben Grossmann | 81,360 | <p>A transformation is diagonalizable iff its eigenvectors form a basis of the vector space. In this case, $T$'s only eigenvectors are the multiples of $e_2$. Since these do not form a basis of $\mathbb R^2$, $T$ is not diagonalizable.</p>
|
4,552,098 | <p>The rules: Two people rolls a single dice. If the dice rolls 1,2,3 or 4, person A gets a point. For the rolls 5 and 6, person B gets a point. One person needs a 2 point lead to win the game.</p>
<p>This is a question taken from my math book. The answer says the probability is <span class="math-container">$\frac{4}{5... | Parcly Taxel | 357,390 | <p>After you show that the game effectively restarts after <span class="math-container">$AB$</span> or <span class="math-container">$BA$</span> and is played out in batches of two, the remaining probabilities must be "scaled up" so that they add to <span class="math-container">$1$</span> to give the true chan... |
2,081,001 | <p>The number $\frac{22}{7}$ is irrational in our base-$10$ system, but in, say, base-$14$, it is rational (it comes out to $3.2$ in that system).</p>
<p>It's easy for fractions that are irrational as decimals, as you can just represent them in a base that's double the denominator of the fraction. However, what if I h... | Mark Bennet | 2,906 | <p>Whether a number is rational or not is independent of the base in which the number may be expressed.</p>
<p>On the other hand the fraction $\frac ab: a,b\in \mathbb Z, b\gt 0$ may terminate or eventually recur when expressed as a "decimal" (Hardy could find no better word - Hardy and Wright, Introduction to the The... |
859,209 | <p>Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$</p>
<p>I've tried first determining when the absolute value will be positive or negative etc, and than giving it the signing in accordance to range it is in, bur it turn... | Elimination | 160,028 | <p><strong>Hint (for a start):</strong> $$t = \frac{x-3}{x+1}$$</p>
|
859,209 | <p>Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$</p>
<p>I've tried first determining when the absolute value will be positive or negative etc, and than giving it the signing in accordance to range it is in, bur it turn... | Mary Star | 80,708 | <p>$$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 \lt 0 \Rightarrow \\ \left(\left|\frac{x-3}{x+1}\right|\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 \lt 0 $$</p>
<p>$$\Delta=\left(-7\right)^2-4 \cdot 10=49-40=9$$</p>
<p>$$\left|\frac{x-3}{x+1}\right|_{1,2}=\frac{-\left(-7\right) \pm \sqrt{\Delta... |
2,737,144 | <p>I tried to prove it by contradiction. </p>
<p>Suppose it is not true that $1\ge\frac{3}{x(x-2)}$, so $1\lt\frac{3}{x(x-2)}$. Then $\frac{3}{x(x-2)}-1\gt0$. Multiply both sides of $\frac{3}{x(x-2)}-1\gt0$ by ${x(x-2)}$.</p>
<p>$(\frac{3}{x(x-2)}-1\gt0)({x(x-2)}\gt0(x(x-2)$</p>
<p>${3-(x(x-2)\gt0}$</p>
<p>${... | marty cohen | 13,079 | <p>$x > 3
\implies x-2 > 1
\implies x(x-2) > 3
\implies 1 > \dfrac{3}{x(x-2)}
$</p>
|
1,886,239 | <p>I am trying to figure out whether or not the following series is convergent: $$\sum_{k=1}^{\infty} \frac{\ln(3^k-2k)}{3k+k^2}$$ </p>
<p>Now, I know from the back of the book that it is divergent, but I haven't been able to show it. I think I am supposed to compare it to some other series, but I don't know which one... | Zestylemonzi | 270,448 | <p>The bog-standard comparison test works well too. Note that (for $k \ge 3$)
\begin{align*}
\frac{\log(3^k - 2k)}{3k+k^2} &\ge \frac{\log(3^k - \frac{1}{2}3^k)}{3k^2+k^2}\\
&=\frac{\log(\frac{1}{2}3^k)}{4k^2}\\
&=\frac{\log(3)}{4k} - \frac{\log(2)}{4k^2}
\end{align*}
and hence the sequence diverges.</p>
|
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | Benoît Kloeckner | 187 | <p>Not a perfect solution, but I would insist on the meaning of <span class="math-container">$\sqrt{\cdot}$</span>: I have observed that this symbol quickly becomes a meaningless mantra to students. I regularly ask (notably in 1st year of master for future high school teacher, also in France) what <span class="math-con... |
3,643,917 | <p>Find the limit of
<span class="math-container">$$\lim_{x\to 0+} \frac{1}{x^2} \int_{0}^{x} t^{1+t} dt$$</span>. </p>
<p>My idea to solve it is to use L'Hospital's rule but I am not sure why I can use it and how should i do it. Many thanks to them who are willing to help. </p>
| Reveillark | 122,262 | <p>Formally, <span class="math-container">$t^{1+t}=te^{t\ln(t)}\to 0$</span> when <span class="math-container">$t\to 0^+$</span>, because <span class="math-container">$\lim_{t\to 0^+} t\ln(t)=0$</span>. It follows that, if <span class="math-container">$f(x)=\int_0^x t^{1+t}dt$</span>,
<span class="math-container">$$
\... |
1,574,290 | <p>How do I prove this? </p>
<p>For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.</p>
| egreg | 62,967 | <p>The recursion can also be written as
$$
f_{n+2}=f_{n+1}+f_n
$$
So you want to prove that the formula holds for $n=1$ (just plug in the value) and that, if it holds for $n$, then it holds for $n+1$.</p>
<p>Thus, assume you know that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$. Then
$$
f_{n+2}^2-f_{n+2}f_{n+1}-f_{n+1}^... |
1,574,290 | <p>How do I prove this? </p>
<p>For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.</p>
| lhf | 589 | <p>You can reduce it immediately to <a href="http://en.wikipedia.org/wiki/Cassini_and_Catalan_identities" rel="nofollow">Cassini's identity</a>:
$$
f^2_{n+1} - f_{n+1}f_n - f^2_n
=
f_{n+1}(f_{n+1} - f_n) - f^2_n
=
f_{n+1}f_{n-1} - f^2_n
= (-1)^n
$$
Cassini's identity has a nice proof using determinants:
$$
f_{n-1}f_{n... |
2,423,086 | <p>Show, by listing the elements or how we can list the elements, that </p>
<p>$\mathbb{N}^3$ = $\mathbb{N×N×N}$ is a countable set.</p>
<p><strong>Attempt at a solution:</strong></p>
<p>I was thinking about making use of Cantor's diagonal argument. If this were $\mathbb{N×N}$ I could just do:</p>
<p>(1,1), (1,2), ... | mechanodroid | 144,766 | <p>$\mathbb{N}^3$ is a countable union of finite sets:</p>
<p>$$\mathbb{N}^3 = \bigcup_{n\in\mathbb{N}}\{(i,j,k)\in\mathbb{N^3} : i + j + k = n\}$$</p>
<p>Hence, $\mathbb{N}^3$ is countable.</p>
<p>For an explicit bijection:</p>
<p>Notice that every $n \in \mathbb{N}$ can be written in a unique way as a product of ... |
2,642,547 | <p>If a subset A of $\mathbb{R}^n$ has no interior, must it be closed?</p>
<p>Can I prove this using the example of a subset A that consists of a single point, so A has no interior yet it is closed?</p>
| Mohammad Riazi-Kermani | 514,496 | <p>The set of rational numbers is a set with no interior and it is not closed.</p>
<p>Note that for a set to be closed its complement should be open and the complement of rationals is the set of irrationals which is not open. </p>
|
2,008,341 | <p>Consider the standard presentation of $D_{2n}$:</p>
<p>$\langle r, s : r^n = s^2 =1, rs = sr^{-1}\rangle$.</p>
<p>I have seen the latter relation given as $sr = r^{-1}s$ a few times. Is this correct, as well?</p>
| pjs36 | 120,540 | <p>I would believe it simply because the group defined "doing everything backwards" is isomorphic to the group you get "doing everything normally." </p>
<p>In particular, given a group $(G, *)$, define its opposite group $(G^{\rm op}, *^{\rm op})$ by $g *^{\rm op} h = h * g$. It's a fun exercise to show that the map $... |
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