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 |
|---|---|---|---|---|
4,216,602 | <p>In this book - <a href="https://www.oreilly.com/library/view/machine-learning-with/9781491989371/" rel="nofollow noreferrer">https://www.oreilly.com/library/view/machine-learning-with/9781491989371/</a> - I came to the differentiation of these to terms like this:</p>
<p>Train - Applying a learning algorithm to data ... | Somos | 438,089 | <p>As an educated guess, consider a set
<span class="math-container">$\,\{(x_i,y_i)\}\,$</span> of data points and try to find a good linear model <span class="math-container">$\,y=mx+b\,$</span>
for the data. The
<a href="https://en.wikipedia.org/wiki/Least_squares" rel="nofollow noreferrer">least squares fit</a>
appr... |
2,508,508 | <p>Let $x_1$ be in $R$ with $ x_1>1$, and let $x_{k+1}=2- \frac{1}{x_k}$ for all $k$ in $N$. Show that the sequence $(x_k)_k$ is monotone and bounded and find its limit.</p>
<p>I am not sure how to start this problem.</p>
| Bumblebee | 156,886 | <p>$$\dfrac12\left(x_{k+1}+\dfrac1{x_k}\right)=1.$$ This says the average of $x_{2}$ and $\dfrac1{x_1}$ is equals to one. Now suppose $1\lt x_1.$ By the AM-GM inequality $x_1+\dfrac1x_1\gt2$ and this implies $$\text{distance}(x_1,1)\gt\text{distance}\left(1,\dfrac1x_1\right)=\text{distance}(x_2,1).$$ Hence $1\lt x_2\l... |
3,521,224 | <p>Let <span class="math-container">$(U_1,U_2,...) , (V_1,V_2,...)$</span> be two independent sequences of i.i.d. Uniform (0, 1) random variables.
Define the stopping time
<span class="math-container">$N = \min\left(n\geqslant 1\mid U_n \leqslant V^2_n\right)$</span>.</p>
<p>Obtain <span class="math-container">$P(N ... | Davide Giraudo | 9,849 | <p>Here are some hints.</p>
<ol>
<li>The event <span class="math-container">$\{N=n\}$</span> can be written as <span class="math-container">$A_n^c\cap\bigcap_{i=1}^{n-1}A_i$</span>, where <span class="math-container">$A_i=\{U_i\leqslant V_i^2\}$</span>. </li>
<li>Show that the collection of events <span class="math-co... |
191,210 | <p>Let $R$ be the smallest $\sigma$-algebra containing all compact sets in $\mathbb R^n$.
I know that based on definition the minimal $\sigma$-algebra containing the closed (or open) sets is the Borel $\sigma$-algebra. But how can I prove that $R$ is actually the Borel $\sigma$-algebra?</p>
| Shankara Pailoor | 39,210 | <p>Well the Borel $\sigma$-algebra is the $\sigma$-algebra generated by the open (or closed) sets of $\mathbb{R}^n$. I believe (I don't want to put words in your mouth) you are asking whether the sigma-algebra generated by the compact sets is equivalent to the sigma algebra generated by the open sets.</p>
<p>Since $R$... |
1,633,810 | <p>For which $a \in \mathbb{R}$ is the integral $\int_1^\infty x^ae^{-x^3\sin^2x}dx$ finite?</p>
<p>I've been struggling with this question. Obviously when $a<-1$ the integral converges, but I have no idea what happens when $a\ge -1 $.</p>
<p>Any help would be appreciated</p>
| Johannes Hahn | 62,443 | <p>If $a < +0.5$, the integral converges. This is because we can split the integral over $[1,\infty)$ into an integral over $\bigcup_k (-\epsilon_k+k\pi,+\epsilon_k+k\pi)$ and an integral over all the rest.</p>
<p>The integral over $(-\epsilon_k + k\pi, +\epsilon_k+k\pi)$ can be bounded by $(k\pi)^a \cdot 2\epsilon... |
4,130,809 | <p>I have questions regarding the proof that I made about the following statement: "Let <span class="math-container">$(X,\tau_{X})$</span> be a topological space and <span class="math-container">$\lbrace \infty\rbrace$</span> an object that doesn't belong to X. Define <span class="math-container">$Y=X\cup\lbrace\i... | Paul Frost | 349,785 | <p>Your idea is correct, but you do not properly elaborate it. You have to find a finite subcover of <span class="math-container">$C=\lbrace C_\alpha \rbrace_{\alpha \in \Lambda}$</span>.</p>
<p>You consider any open cover <span class="math-container">$C_x=\lbrace C_{\alpha_x}\rbrace_{\alpha_x\in \Lambda_x}$</span> of ... |
2,348,131 | <p>In our class, we encountered a problem that is something like this: "A ball is thrown vertically upward with ...". Since the motion of the object is rectilinear and is a free fall, we all convene with the idea that the acceleration $a(t)$ is 32 feet per second square. However, we are confused about the sign of $a(t)... | tempx | 357,017 | <p>Acceleration is defined as the derivative of the velocity, i.e. $a(t)=\frac{v(t)}{dt}$. When the ball is going upward, the speed of the ball decreases and thus the acceleration becomes negative. </p>
|
3,204,082 | <p>I have a conjecture, but have no idea how to prove it or where to begin. The conjecture is as follows:</p>
<blockquote>
<p>A polynomial with all real irrational coefficients and no greatest common factor has no rational zeros.</p>
</blockquote>
<p>This conjecture excludes the cases where the polynomial does have... | Bruno Seefeld | 668,896 | <p>For the case of just one irrational coefficient <span class="math-container">$a_i$</span>, supose by absurd that there is a rational solution <span class="math-container">$q$</span>. Then: <span class="math-container">$$(a_i=a_0+...+a_n q^n)\frac{1}{q^i}$$</span> hence <span class="math-container">$a_i$</span> is ra... |
634,344 | <p>Im trying to go alone through Fultons, Introduction to algebraic topology. He asks whether there is a function $g$ on a region such that $dg$ is the form:
$$\omega =\dfrac{-ydx+xdy}{x^2+y^2}$$
in some regions. I know you can do it on the upper half plane by considering $-arctan(x/y)$. But Im a bit confused. I know t... | Gil Bor | 118,580 | <p>Yes you can. You can solve the equation $dg=\omega$ in any open subset $U\subset \mathbb R^2\setminus (0,0)$ which does not "enclose" the origin $(0,0)\in\mathbb R^2.$ Formaly: there exists a continous $\gamma:[0, \infty)\to \mathbb R^2\setminus U$, such that $\gamma(0)=(0,0)$ and $\|\gamma(t)\|\to\infty$ as $t\to... |
2,325,968 | <p>I was trying to calculate : $e^{i\pi /3}$.
So here is what I did : $e^{i\pi /3} = (e^{i\pi})^{1/3} = (-1)^{1/3} = -1$</p>
<p>Yet when I plug : $e^{i\pi /3}$ in my calculator it just prints : $0.5 + 0.866i$</p>
<p>Where am I wrong ? </p>
| CY Aries | 268,334 | <p>You take $(-1)^\frac{1}{3}=-1$. But actually, there are three numbers such that their cube is $-1$. They are $-1$, $e^{\frac{i\pi}{3}}$ and $e^{\frac{-i\pi}{3}}$.</p>
|
3,171,152 | <p>Let gamma 1 be a straight line from -i to i and let gamma 2 be the semi-circle of radius 1 in the right half plane from -i to i.</p>
<p>Evaluate</p>
<p><span class="math-container">$$\int_{\gamma_1}f(z)dz$$</span></p>
<p>and <span class="math-container">$$\int_{\gamma_2}f(z)dz$$</span></p>
<p>where f(z)=complex ... | Leander Tilsted Kristensen | 631,468 | <p>Your approach is on the right path, however you forgot to multiply by the derivative of the curves in the integrals. The general formula for an integral over a parameterized curve is
<span class="math-container">$$ \int_\gamma f(t) \: dt = \int_a^b f(\gamma(t)) \gamma'(t) \: dt $$</span></p>
<p>Also i would probab... |
13,109 | <p>I posted a question half a hour ago. But I think I found the answer myself now.
I understand that answering your own question is appreciated (instead of deleting it).
But I don't know if I should give a hint or a full solution.</p>
<p>It feels a little bit strange to give a hint to my <em>own</em> question, I don't... | fgp | 42,986 | <p>I think in the long run, we <em>do</em> want full answers for all questions.</p>
<p>For homework-type questions, IMHO the ideal procedure is</p>
<ul>
<li>Other users provide hints</li>
<li>The OP, once he has managed to solve the problem, posts a full answer</li>
</ul>
<p>So yes, please post a full answer.</p>
|
1,114,822 | <p>I have to prove that $d$ divides $n$ if and only if $ord_p(d)\leq ord_p(n)$</p>
<p>I have already proved that $ord_p(d)\leq ord_p(n)$ if $d$ divides $n$ but I am struggling to prove the converse. Can anyone give any help?</p>
| coffeemath | 30,316 | <p>Let $d=\prod p^{a_p}$ and $n=\prod p^{b_p},$ where in each case the product is over all primes $p$ and the exponents are all $0$ beyond some point (and may be $0$ for lower primes also).</p>
<p>Then since $ord_p(d) \le ord_p(n)$ we have $a_p \le b_p$ at each prime $p.$ From this it's easy to see that $d|n$, in fact... |
1,114,822 | <p>I have to prove that $d$ divides $n$ if and only if $ord_p(d)\leq ord_p(n)$</p>
<p>I have already proved that $ord_p(d)\leq ord_p(n)$ if $d$ divides $n$ but I am struggling to prove the converse. Can anyone give any help?</p>
| drhab | 75,923 | <p>Write: $$n=p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}$$ where the $p_{i}$ are
distinct primes and the $r_{i}$ are nonnegative integers. </p>
<p>Then $\operatorname{ord}_{p}\left(d\right)\leq\operatorname{ord}_{p}\left(n\right)$ for each prime $p$
implies that we can write: $$d=p_{1}^{s_{1}}\cdots p_{k}^{s_{k}}$$ where the $... |
1,552 | <p>Closely related: what is the smallest known composite which has not been factored? If these numbers cannot be specified, knowing their approximate size would be interesting. E.g. can current methods factor an arbitrary 200 digit number in a few hours (days? months? or what?).
Can current methods certify that an a... | Alon Amit | 25 | <p>Back in 1990, whatever answer would have been given to the question would likely correspond to the processing power of single (possibly large) computer. Today, the best algorithms are ones that can be efficiently distributed, leading to successful factorizations performed by networks of computers. </p>
<p>With the ... |
2,051,555 | <p>I have the following limit to solve.</p>
<p>$$\lim_{x \rightarrow 0}(1-\cos x)^{\tan x}$$</p>
<p>I am normally supposed to solve it without using l'Hôpital, but I failed to do so even with l'Hôpital. I don't see how I can solve it without applying l'Hôpital a couple of times, which doesn't seem practical, nor how ... | HBR | 396,575 | <p>Try with Taylor series when $x\to0$:
$$\tan{x}\approx x$$
$$\cos{x}\approx 1-\frac{x^2}{2}$$</p>
|
2,051,555 | <p>I have the following limit to solve.</p>
<p>$$\lim_{x \rightarrow 0}(1-\cos x)^{\tan x}$$</p>
<p>I am normally supposed to solve it without using l'Hôpital, but I failed to do so even with l'Hôpital. I don't see how I can solve it without applying l'Hôpital a couple of times, which doesn't seem practical, nor how ... | Claude Leibovici | 82,404 | <p>Even if, apparently, Taylor series are not desired, may be equivalents could be used
$$A=(1-\cos(x))^{\tan(x)}\implies \log(A)=\tan(x) \log(1-\cos(x))$$ Close to $x=0$, $$\cos(x)\sim 1-\frac{x^2} 2$$ $$1-\cos(x)\sim \frac{x^2} 2$$ $$\log(1-\cos(x))\sim 2\log(x)-\log(2)$$ $$\tan(x)\sim x$$ $$log(A)=\tan(x) \log(1-\co... |
2,604,825 | <p>So I have a problem (two problems, actually) that a friend helped me out with, I'm able to work out the components of this problem but get lost when I have to bring it all together... so what I have is this</p>
<p>$f(t)=t^{2}e^{-2t}+e^{-t}\cos(3t)+5$</p>
<p>Simple enough. I got:</p>
<p>$$\mathcal{L}[t^2]=\frac{2}... | Jack D'Aurizio | 44,121 | <p>For a greater accuracy,
$$ \int_{0}^{\pi/2}\sqrt{\sin x}\,dx = \int_{0}^{1}\sqrt{\frac{u}{1-u^2}}\,du\stackrel{\text{Beta}}{=}\frac{\sqrt{2\pi}^3}{\Gamma\left(\frac{1}{4}\right)^2}=\text{AGM}(1,\sqrt{2})$$
(<a href="https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function" rel="noreferrer">particular va... |
2,831,130 | <p>Cauchy's induction principle states that:</p>
<blockquote>
<p>The set of propositions $p(1),...,p(n),...$ are all valid if: </p>
<ol>
<li>$p(2)$ is true.</li>
<li>$p(n)$ implies $p(n-1)$ is true.</li>
<li>$p(n)$ implies $p(2n)$ is true.</li>
</ol>
</blockquote>
<p>How to prove Cauchy's induct... | Mark | 4,460 | <p>$E[ \sum_1^n X_i ] = \sum_1^n E[ x_i] = \sum_1^n E[ x_1] = n E[x_1] = np$</p>
|
4,043,625 | <p><span class="math-container">\begin{equation}
\left\{\begin{array}{@{}l@{}}
2x\equiv7\mod9 \\
5x\equiv2\mod6
\end{array}\right.\,.
\end{equation}</span>
Can this system of congruences be solved?
I notice that <span class="math-container">$(9,6) = 3 \ne 1$</span> so I can't apply the Chinese theorem of re... | marty cohen | 13,079 | <p>Because
<span class="math-container">$f(n)
\in O(\log_2(n))
\iff
f(n)
\in O(\log_2(n+1))
$</span>,
and it is convenient
to use the simpler form.</p>
|
2,483,794 | <p>I'm trying to figure out the equality $$\frac{1}{y(1-y)}=\frac{1}{y-1}-\frac{1}{y}$$</p>
<p>I have tried but keep ending up with RHS $\frac{1}{y(y-1)}$.</p>
<p>Any help would be appreciated.</p>
| DeepSea | 101,504 | <p><strong>hint</strong>: write the top as $1 = y + (1-y)$ and split the fraction using the formula: $\dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c}$ with $a = y, b = 1-y, c = y(1-y)$ , and simplify each fraction. Does this help ?</p>
|
2,069,392 | <p>Given that $x^4+px^3+qx^2+rx+s=0$ has four positive roots.</p>
<p>Prove that (1) $pr-16s\ge0$ (2) $q^2-36s\ge 0$</p>
<p>with equality in each case holds if and only if four roots are equal.</p>
<p><strong>My Approach:</strong></p>
<blockquote>
<p>Let roots of the equation</p>
<p>$x^4+px^3+qx^2+rx+s=0$ be ... | Jyrki Lahtonen | 11,619 | <p>AM-GM inequality is the key.</p>
<p>The product $pr$ consists of $16$ terms. Four of those terms are equal to $s$. The remaining twelve are the permuted versions of $\alpha^2\beta\eta$. The product of those twelve is equal to $s^{12}$, so by AM-GM their sum is $\ge12s$. </p>
<p>In AM-GM we have equality iff all th... |
15,162 | <p>First off: I barely have any set theoretic knowledge, but I read a bit about cardinal arithmetic today and the following idea came to me, and since I found it kind of funny, I wanted to know a bit more about it.</p>
<p>If $A$ is the set of all real positive sequences that either converge to $0$ or diverge to $\inft... | Andrés E. Caicedo | 462 | <p>Asaf's answer explains that there is no set of all cardinals, and that under the axiom of choice, the cardinals are well-ordered, and so it is impossible to have a homomorphism as you want.</p>
<p>What remains is to see whether it is possible, in some models where the axiom of choice fails, to have a homomorphism $... |
342,306 | <p>An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.</p>
<p>A critical point of such an embedding is the least ordinal $\alpha$ s... | Andrés E. Caicedo | 462 | <p>Not exactly. </p>
<p>First of all, there are small large cardinals, such as inaccessible or Mahlo cardinals, for which I do not know of any natural formulation in terms of embeddings. </p>
<p>Once we reach weakly compact cardinals, we can start expressing traditional large cardinal properties in terms of embedding... |
2,017,818 | <p>Find three distinct triples (a, b, c) consisting of rational numbers that satisfy $a^2+b^2+c^2 =1$ and $a+b+c= \pm 1$.</p>
<p>By distinct it means that $(1, 0, 0)$ is a solution, but $(0, \pm 1, 0)$ counts as the same solution.</p>
<p>I can only seem to find two; namely $(1, 0, 0)$ and $( \frac{-1}{3}, \frac{2}{3}... | marty cohen | 13,079 | <p>Here's a start that shows that
any other solutions
would have to have
distinct $a, b, $ and $c$.</p>
<p>In
$a^2+b^2+c^2 =1$
and
$a+b+c= \pm 1$,
if $a=b$,
these become
$2a^2+c^2 = 1,
2a+c = \pm 1$.</p>
<p>Then
$c = -2a\pm 1$,
so
$1
= 2a^2+(-2a\pm 1)^2
=2a^2+4a^2\pm 4a+1
=6a^2\pm 4a+1
$
so
$0 = 6a^2\pm 4a
=2a(3a\pm ... |
3,537,843 | <p>Find values of x such that <span class="math-container">$x^n=n^x$</span>
Here, n <span class="math-container">$\in$</span> I. </p>
<p>One solution will remain <strong>x=n</strong>
But i want to find if any more solutions can exist</p>
<p><span class="math-container">$$x^n=n^x$$</span></p>
| Community | -1 | <p>I love this question! I first saw it while in sixth form (I'm from London that means when I was 18) </p>
<p>The first thing we can do is to try and get just x on one side and just y on the other side. Heres what we can do using logs: <br>
If x<sup>y </sup> = y<sup>x</sup> then we have to have log(x<sup>y </sup>) =... |
15,669 | <p>Borrowing <code>triangularArrayLayout</code> from <a href="https://mathematica.stackexchange.com/questions/9959/visualize-pascals-triangle-and-other-triangle-shaped-lists">here</a>, I have:</p>
<pre><code>triangularArrayLayout[triArray_List, opts___] :=
Module[{n = Length[triArray]},
Graphics[MapIndexed[
T... | kglr | 125 | <ul>
<li>Using the same option <code>ImagePadding->k</code> in both <code>coeff</code> and <code>tri</code> fixes the vertical alignment problem.</li>
<li><code>C</code> is a protected symbol (it is used for representing constants generated in symbolic computations.) Instead you can use <code>\[ScriptCapitalC]</co... |
38,252 | <p>I have a quadrilateral ABCD.
I want to find all the points x inside ABCD such that
$$angle(A,x,B)=angle(C,x,D)$$</p>
<p>Is there a known formula that gives these points ?</p>
<p><strong>Example:</strong></p>
<p>ABCD is a rectangle.
Let $x_1=mid[A,D]$ and $x_2=mid[B,C]$.
The points x are those lying on the line t... | Community | -1 | <p>Call a topological space <em>good</em> if it's homeomorphic to a compact ordinal.</p>
<p><strong>Lemma 1.</strong> Every countable compact Hausdorff space is first countable, zero-dimensional, and scattered.</p>
<p>Proof. These are well-known facts.</p>
<p><strong>Lemma 2.</strong> A closed subspace of a good spa... |
67,516 | <p>The book by Durrett "Essentials on Stochastic Processes" states on page 55 that:</p>
<blockquote>
<p>If the state space S is finite then there is at least on stationary
distribution.</p>
</blockquote>
<ol>
<li><p>How can I find the stationary distribution for example for the square 2x2 matrix $[[a,b],[1-a, 1-b... | Mark Bennet | 2,906 | <p>Given a circle, and angles for your triangle of $\alpha, \beta, \gamma$ mark off radii with angles between them at the centre of the circle of $(180-\alpha)^\circ, (180-\beta)^\circ, (180-\gamma)^\circ$ [total $360^\circ$]. The tangents at the points where the radii meet the circle will make a triangle with the desi... |
348,614 | <p>Is the following claim true: Let <span class="math-container">$\zeta(s)$</span> be the Riemann zeta function. I observed that as for large <span class="math-container">$n$</span>, as <span class="math-container">$s$</span> increased, </p>
<p><span class="math-container">$$
\frac{1}{n}\sum_{k = 1}^n\sum_{i = 1}^{k} ... | Wojowu | 30,186 | <p>Let me denote your LHS by <span class="math-container">$f(n,s)$</span>. For fixed even <span class="math-container">$n$</span> I shall show that <span class="math-container">$f(n,s)-1\sim\zeta(s+1)-1$</span> as <span class="math-container">$s\to\infty$</span>, that is,
<span class="math-container">$$\lim_{s\to\infty... |
354,642 | <p>Show that each of the following initial-value problems has a unique solution ($0 ≤ t ≤ 1 , y(0) = 1$).</p>
<p>$$y' = \exp(t-y)$$</p>
<p><strong>Theorem 1</strong>: Suppose that $D=\{(t,y)|a≤t≤b, −∞< y<∞\}$ and that $f(t,y)$ is continuous on $D$. If $f$ satisfies a Lipschitz condition on $D$ in the variable $... | Sammy Black | 6,509 | <p>Consider the sequence $X_n = \frac{1}{\pi n}$.</p>
|
468,784 | <p>Two disjoint sets $A$ and $B$, neither empty, are said to be <strong>mutually separated</strong> if neither contains a boundary point of the other. A set is disconnected if it is the union of separated subsets, and is called <strong>connected</strong> if it is not disconnected.</p>
<p>With the above definition of c... | Pedro | 23,350 | <p>You can prove that $\{y=x\}$ and $\{y=-x\}$ are connected. Since they have a point in common, their union is. Can you try to argue this is the same, in essence, than showing the real line is connected?</p>
<p>First, let's obtain a slightly more useful equivalent to your definition. First, if $A,B$ are open sets and... |
1,235,639 | <p>Let $\mathcal{R}$ be the hyperfinite type $II_{1}$ factor and let $\mathcal{U}$ be a free ultrafilter on $\mathbb{N}$.</p>
<p>Is it true that $\mathcal{R}^{\mathcal{U}}$ is never hyperfinite ? How can I see this ?</p>
<p>Thanks</p>
<p><em>I know that under Continuum Hypothesis, every $\mathcal{R}^{\mathcal{U}}$ i... | roya | 296,316 | <p>As we know the ultrapower of $\rm II_1$ factors is a $\rm II_1$ factor and also there is only one hyperfinite $\rm II_1$ factor up to isomorphism. So If $R^U$ is hyperfinite we must have $R\simeq R^U$, but it is impossible.</p>
|
73,383 | <p>The problem is:
$$\displaystyle \lim_{(x,y,z) \rightarrow (0,0,0)} \frac{xy+2yz+3xz}{x^2+4y^2+9z^2}.$$</p>
<p>The tutor guessed it didn't exist, and he was correct. However, I'd like to understand why it doesn't exist.</p>
<p>I think I have to turn it into spherical coordinates and then see if the end result depen... | hmakholm left over Monica | 14,366 | <p>If you want to rewrite to spherical coordinates, what you need is expressions for the rectangular coordinates in terms of the spherical ones, such as $x=\rho\cos(\theta)\sin(\phi)$, rather than the other way around. (By the way, the name of the letter $\rho$ is spelled "rho").</p>
<p>However, I don't think going to... |
2,813,595 | <p>which of the following can be expressed by exact length but not by exact number?</p>
<p>(i) $ \sqrt{10} $</p>
<p>(ii) $ \sqrt{7} $</p>
<p>(iii) $ \sqrt{13} \ $</p>
<p>(iv) $ \ \sqrt{11} \ $</p>
<p><strong>Answer:</strong></p>
<p>I basically could not understand th question.</p>
<p>What is meant by expressi... | Ross Millikan | 1,827 | <p>I think "cannot be expressed by exact number" means they are irrational so the decimal does not terminate, which is true of all of them. </p>
<p>I think "can be expressed by exact length" means you can construct it. You are expected to notice that $10=3^2+1^2$ so you can draw a segment of length $1$, a perpendicu... |
2,813,595 | <p>which of the following can be expressed by exact length but not by exact number?</p>
<p>(i) $ \sqrt{10} $</p>
<p>(ii) $ \sqrt{7} $</p>
<p>(iii) $ \sqrt{13} \ $</p>
<p>(iv) $ \ \sqrt{11} \ $</p>
<p><strong>Answer:</strong></p>
<p>I basically could not understand th question.</p>
<p>What is meant by expressi... | Ethan Bolker | 72,858 | <p>The <a href="https://en.wikipedia.org/wiki/Spiral_of_Theodorus" rel="nofollow noreferrer">spiral of Theodorus</a> constructs the square roots of the positive integers.</p>
<p><a href="https://i.stack.imgur.com/Mftkn.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Mftkn.png" alt="enter image desc... |
341,648 | <p>I'm trying to understand what a tableaux ring is (it's not clear to me reading Young Tableaux by Fulton).</p>
<p>I studied what a monoid ring is on Serge Lang's Algebra, and then I read about modules, modules homomorphism. I'm trying to prove what is stated at page 121 (S. Lang, Algebra) while talking about algebra... | Jack D'Aurizio | 44,121 | <ol>
<li><strong>Riemann sums</strong></li>
</ol>
<p>It is not a complex analytic technique but I think it is worth mentioning. We can compute the integral by taking Riemann sums and exploiting the identity:
$$\prod_{k=1}^{n-1}\sin\frac{\pi k}{n}=\frac{2n}{2^n}\tag{1}$$
from which it follows that:
$$\begin{eqnarray*}\... |
4,543,350 | <p>I am having a hard time figuring out this proof.</p>
<p>Let <span class="math-container">$\{x_n\}$</span> be a sequence and <span class="math-container">$x\in\mathbb{R}$</span>. Suppose for every <span class="math-container">$\epsilon>0$</span>, there is an M such that <span class="math-container">$|x_n-x|\leq \e... | Chirag Kar | 917,916 | <p>The definition for convergence that I will be using is as follows:
We say that the sequence <span class="math-container">$(x_n)$</span> <em>converges</em> to <span class="math-container">$x\in\mathbb{R}$</span> if, given any <span class="math-container">$\epsilon > 0, $</span> there exists <span class="math-conta... |
977,956 | <p>Can you help me solve this problem?</p>
<blockquote>
<p>Simplify: $\sin \dfrac{2\pi}{n} +\sin \dfrac{4\pi}{n} +\ldots +\sin \dfrac{2\pi(n-1)}{n}$.</p>
</blockquote>
| Rohinb97 | 80,473 | <p>There is a formula if the angles of sine are in A.P.</p>
<p>$sinA +sin(A+D)....+sin(A+(n-1)D)=\frac{sin(nD/2)*sin(A+\frac{(n-1)D}{2})}{sin(D/2)}$.</p>
<p>Use it to get $sin2\pi$ in the numerator to get your answer 0.</p>
<p>Also, a complex approach might do as well.
Sum of $n^{th}$ roots of unity is 0. So it's im... |
115,081 | <p>I hope here is the best place to ask this, I will begin my master degree very soon, I've already attended the regular undergraduate courses included Real Analysis, Analysis on manifolds, Abstract Algebra, Field Theory, point-set topology, Algebraic Topology, etc...
I like very much algebraic topology and I found it ... | David C | 27,816 | <p>If you want to learn about algebraic topology, you can begin by very classical readings. When I was a Ph-D student, I first read Milnor Stasheff's book on "Characteristic classes", here you will learn a lot of differential and algebraic topology. There are so many good books to read, J.-F. Adams "Infinite loop space... |
4,196,185 | <p>Let <span class="math-container">$A$</span> be a <span class="math-container">$n\times n$</span> matrix with minimal polynomial <span class="math-container">$m_A(t)=t^n$</span>, i.e. a matrix with <span class="math-container">$0$</span> in the main diagonal and <span class="math-container">$1$</span> in the diagonal... | The_Sympathizer | 11,172 | <p>Generically, any solutions to</p>
<p><span class="math-container">$$f(x + 1) = x f(x)$$</span></p>
<p>can be given as <span class="math-container">$f(x) = \Gamma(x)\ \theta(x)$</span> for some 1-cyclic function <span class="math-container">$\theta$</span> such that <span class="math-container">$\theta(0) = 1$</span>... |
307,701 | <p>Show that if $G$ is a finite group with identity $e$ and with an even number of elements, then there is an $a \neq e$ in $G$, such that $a \cdot a = e$.</p>
<p>I read the solutions here <a href="http://noether.uoregon.edu/~tingey/fall02/444/hw2.pdf" rel="nofollow">http://noether.uoregon.edu/~tingey/fall02/444/hw2.p... | Daniel McLaury | 3,296 | <p>You're right that $D$ is not a subgroup of $G$, but they don't claim that it is.</p>
<p>They're also not saying that $D = \{a, a^{-1}\}$, but rather that the elements of $D$ appear in pairs -- if $a \in S$, then $(a^{-1})^2 = a^{-2} = (a^2)^{-1} \neq e$.</p>
<p>Go back through the proof and see if it makes sense n... |
3,844,256 | <p>How can one prove the following deduction? Assume we know the following result.</p>
<p><span class="math-container">$$ \frac{1}{2}\arctan\left( \frac{y}{x+1} \right) + \frac{1}{2}\arctan\left( \frac{y}{x-1} \right) - \arctan\left( \frac{y}{x} \right) = c$$</span></p>
<p>Then, it is claimed that this is equivalent to... | Kavi Rama Murthy | 142,385 | <p>Suppose <span class="math-container">$A\cup \{p\}=C\cup D$</span> is as separation of <span class="math-container">$A\cup \{p\}$</span>. Verify that <span class="math-container">$X=(C\cup D) \cup B$</span> and that this gives a separation of <span class="math-container">$X$</span>. [<span class="math-container">$C ... |
2,572,304 | <p>Cauchy's Inequality states that, $$ \forall a, b \in R^{n}, |a \cdot b| \leq |a||b| $$. However, the dot product is $$ x \cdot y = x_{1}y_{1}+...+x_{n}y_{n}$$ while the norm of x is $$ |x| = \sqrt[2]{x_{1}^{2} +...+x_{n}^{2}} = \sqrt[2]{x \cdot x}$$. Therefore, $$ |a \cdot b| = \sqrt[2]{(a \cdot b) \cdot (a \cdot b)... | The Phenotype | 514,183 | <p>With $(a \cdot b) \color{red}{\cdot} (a \cdot b)$ is meant that $\cdot$ is the dot product and that $\color{red}{\cdot}$ is multiplication.</p>
<p>It helps to use distinguishable notation, so use for example $\langle a,b\rangle $ for dot products and $a_1\cdot b_1$ for multiplication.</p>
|
345,888 | <p>$S$ is vector subspace of $S$ if $S$ is vector space, by hypothesis $S$ is vector space then $S$ is vector subspace of $S$.</p>
<p>But I prove it by contradiction, then $S$ is not vector subspace of $S$, but if $S$ is not vector subspace of $S$ then $S$ is not vector space but I have contraddiction, in fact by hypo... | Zarrax | 3,035 | <p>Hint: In complex form your sum is $c_0 + \sum_{n = 1}^{\infty} c_nz^n + \sum_{n = 1}^{\infty} c_n\bar{z}^n$. Write this as a real part of an analytic function on the unit disc. (By the way, I'm assuming the $c_n$ are real, otherwise it's not necessarily a real-valued function).</p>
|
3,601,552 | <p>A school has <span class="math-container">$500$</span> girls and <span class="math-container">$500$</span> boys. A simple random sample is obtained by selecting names from a box (with replacement) to a get a sample of <span class="math-container">$10$</span>. </p>
<p>Find the probability of someone being picked mor... | David K | 139,123 | <p>This is a variation on the Birthday Problem, with names instead of birthdays and drawings from the hat instead of people in a room.</p>
<p>The answer is <span class="math-container">$1$</span> minus the probability that all ten names are different,
which is the product of the probabilities that the <span class="mat... |
143,274 | <p>I am trying to find the derivative of $\sqrt{9-x}$ using the definition of a derivative </p>
<p>$$\lim_{h\to 0} \frac {f(a+h)-f(a)}{h} $$</p>
<p>$$\lim_{h\to 0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h} $$</p>
<p>So to simplify I multiply by the conjugate</p>
<p>$$\lim_{h\to0} \frac {\sqrt{9-(a+h)}-\sqrt{9-a}}{h}\cdo... | Garmen1778 | 26,711 | <p>As other answers well say, you have an error while multiplicating. And also you forgot to put the limit in your last equation.
\begin{align}
f'(x)&=\lim\limits_{h\to0}\left(-\frac{h}{h(\sqrt{9-(a+h)}+\sqrt{9-a})}\right)\\
&=\lim\limits_{h\to0}\left(-\frac{1}{\sqrt{9-(a+h)}+\sqrt{9-a}}\right)\\
&=-\frac{1... |
1,663,838 | <p>Show that a positive integer $n \in \mathbb{N}$ is prime if and only if $\gcd(n,m)=1$ for all $0<m<n$.</p>
<p>I know that I can write $n=km+r$ for some $k,r \in \mathbb{Z}$ since $n>m$</p>
<p>and also that $1=an+bm$. for some $a,b \in \mathbb{Z}$</p>
<p>Further, I know that $n>1$ if I'm to show $n$ is... | Nitrogen | 189,200 | <p><strong>Hint:</strong> If $d$ divides $n$, then $gcd(d,n)=d$.</p>
|
2,798,598 | <p>We have the series $\sum\limits_{n=1}^{\infty} \frac{(-1)^n n^3}{(n^2 + 1)^{4/3}}$. I know that it diverges, but I'm having some difficulty showing this. The most intuitive argument is perhaps that the absolute value of the series behaves much like $\frac{n^3}{\left(n^2\right)^{4/3}} = \frac{1}{n^{-1/3}}$, which div... | Jason | 432,654 | <p>Your reasoning is fairly sound, but you are thinking about it a little too hard. Try, instead of thinking about this as a Series and trying to get to a p-series test; do a test for divergence (or nth term test depending on your book) to see if the term itself goes to zero with $n$.</p>
<p>Edit:
Sorry missed the las... |
945,651 | <p>Use mathematical induction to prove the following statement:</p>
<p>For all $b\in\mathbb R$, and for all $n\in\mathbb N$, $$b>-1\implies (1+b)^n \geq 1+nb$$</p>
<p>When $n=1$, the inequality still holds
$1+b \geq 1+b$.</p>
<p>For n+1$:
$$(1+b)^{n+1} \geq 1+(n+1)b$$
Here I'm not sure the best way to simplify...... | beep-boop | 127,192 | <p>A few points:</p>
<ul>
<li><p>The base case, in this case, is $n=0,$ so that's what you should verify, not $n=1.$</p></li>
<li><p>$$\color{green}{(1+b)^{n+1}} \equiv \underbrace{(1+b)^n(1+b) \geq (1+bn)(1+b)}_{\text{induction hypothesis}} \equiv 1+(n+1)b+\underbrace{b^2n}_{\geq 0} \color{green}{\geq 1+(n+1)b} .$$</... |
905,685 | <p>Let the balls be labelled $1,2,3,..n$ and the boxes be labelled $1,2,3,..,n$. </p>
<p>Now I want to find, </p>
<ul>
<li><p>What is the expected value of the minimum value of the label among the boxes which are non-empty </p></li>
<li><p>What is the expected number of boxes with exactly one ball in them? </p></li>
... | Marko Riedel | 44,883 | <p>Here is an approach using labelled species and exponential generating
functions.
<P>
For the <b>first</b> problem we have the species
$$\sum_{q=1}^n \mathcal{U}^q \times
\mathfrak{P}_{\ge 1}(\mathcal{Z}) \times
\mathfrak{P}(\mathcal{Z})^{n-q}$$
with $\mathcal{U}$ marking the end of the intial segment of empt... |
256,612 | <p>I've found assertions that recognising the unknot is NP (but not explicitly NP hard or NP complete). I've found hints that people are looking for untangling algorithms that run in polynomial time (which implies they may exist). I've found suggestions that recognition and untangling require exponential time. (Untangl... | Peter Balch | 102,151 | <p>I seem to have a polynomial-time algorithm that untangles the unknot. But I suspect that hubris lurks around every corner in this game.</p>
<p>I'm pretty sure I can show that the algorithm runs in polynomial-time. But I now realise that I don't know if it is always able to simplify every tangle.</p>
<p>As always, ... |
881,141 | <p>Let $A$ and $B$ be two covariance matrices such that $AB=BA$. Is $AB$ a covariance matrix?</p>
<p>A covariance matrix must be symmetric and positive semi definite. The symmetry of $AB$ can be proved as follows:
$$(AB)^T = B^TA^T = BA = AB$$</p>
<p>The question is, how to prove or disprove the positive semi definit... | Horst Grünbusch | 88,601 | <p>Expanding the word "immediately" of Quang Hoang's answer:</p>
<p>$$AB = QD_AQ'QD_BQ' = QD_A D_B Q', $$ </p>
<p>where $D_A$ and $D_B$ are the respective diagonal matrices of $A$ and $B$. The diagonal entries of these matrices are nonnegative, so are the diagonal entries of the product of $D_A D_B$. </p>
|
222,093 | <p>For what value of m does equation <span class="math-container">$y^2 = x^3 + m$</span> has no integral solutions?</p>
| dinoboy | 43,912 | <p>None of the solutions posted look right (I don't think this problem admits a solution by just looking modulo some integer, but possibly I'm wrong). Here is a proof.</p>
<p>First, by looking modulo $8$ one deduces we need $x$ to be odd.</p>
<p>Note that $y^2 + 1^2 = (x+2)(x^2 - 2x + 4)$. As the LHS is a sum of two ... |
2,441,630 | <p>The operator given is the right-shift operator $T$ on $l^2$. We show that $\lambda=1$ is in the residual spectrum. Therefore we show that $(I-T)$ is injective but fails to have a dense range. While injectivity is clear, I fail to understand why the following shows that the range is not dense:</p>
<p>Let $y=(I-T)x$.... | Nate Eldredge | 822 | <p>It's not correct. In fact the range of $I-T$ is dense and $\lambda = 1$ is not in the residual spectrum, but rather in the continuous spectrum.</p>
<p>Indeed, suppose $y$ is in the orthogonal complement of the range of $I-T$, so that $((I-T)x, y) = 0$ for all $x$. This implies $0 = (x, (I-T^*)y)$ so $T^* y = y$. ... |
69,590 | <p>Consider the following code.</p>
<pre><code>f[a_,b_]:=x
x=a+b;
f[1,2]
(* a + b *)
</code></pre>
<p>From a certain viewpoint, one might expect it to return <code>3</code> instead of <code>a + b</code>: the symbols <code>a</code> and <code>b</code> are defined during the evaluation of <code>f</code> and <code>a+b</c... | bill s | 1,783 | <p>One somewhat organized way to get what you want is to be very explicit about which expressions are functions and which are values. For example, your x is really a function of a and b, but you are writing x=a+b. If instead, you make the functional relationships explicit, then there is less chance of confusion. In the... |
1,810,729 | <blockquote>
<p>Let $G$ be a group generated by $x,y$ with the relations $x^3=y^2=(xy)^2=1$. Then show that the order of $G$ is 6.</p>
</blockquote>
<p><strong>My attempt:</strong> So writing down the elements of $G$ we have $\{1,x,x^2,y,\}$. Other elements include $\{xy, xy^2, x^2y\}$ it seems I am counting more th... | Dietrich Burde | 83,966 | <p>One group presentation for the dihedral group $D_n$ is $\langle x,y|x^2=1,y^n=1,(xy)^2=1\rangle $. Hence the group is indeed isomorphic to $D_3$. Here $x$ with $x^2=1$ corresponds to a reflection, and $y$ with $y^3=1$ to a rotation of $60$ degrees. Finally we have $xyx^{-1}=y^{-1}=y^2$, which is how rotation and ref... |
4,212,181 | <p>A uniform cable that is 2 pounds per feet and is 100 feet long hangs vertically from a pulley system at the top of a building (and the building is also 100 feet tall).</p>
<p>How much work is required to lift the cable until the bottom end of the cable is 20 feet below the top of the building?</p>
<p><span class="ma... | John Douma | 69,810 | <p>Your integral is wrong. As you know, the total work done is given by the total force exerted over a distance. In this case the force on the cable is variable. We know that the starting weight is <span class="math-container">$200$</span> pounds and decreases by <span class="math-container">$2$</span> pounds for every... |
3,014,438 | <p>Find Number of Non negative integer solutions of <span class="math-container">$x+2y+5z=100$</span></p>
<p>My attempt: </p>
<p>we have <span class="math-container">$x+2y=100-5z$</span> </p>
<p>Considering the polynomial <span class="math-container">$$f(u)=(1-u)^{-1}\times (1-u^2)^{-1}$$</span></p>
<p><span class=... | sirous | 346,566 | <p>I will find number of solutions of equation <span class="math-container">$5x+2y+z=10 n$</span> in general:</p>
<p>clearly the positive solutions <span class="math-container">$x_0, y_0, z_0$</span> of this equation are corespondent to the solution <span class="math-container">$x_0+2,y_0, z_0$</span> of equation <spa... |
118,232 | <p>For example I have </p>
<pre><code>square = Graphics[Polygon[{{0, 0} ,{0, 1}, {1, 1}, {1, 0}}]]
</code></pre>
<p>What functions can I apply to <code>sqaure</code> to extract the coordinates of the polygon? It is necessary to do this kind of extraction when I have a graphics object as an argument of a function, and... | E. Chan-López | 53,427 | <p><strong>Hopf bifurcation analysis</strong></p>
<p>The differential system:</p>
<pre><code>f1[x_,y_]:=a x (1 - x/k) - b x y;
f2[x_,y_]:=-c y + d x y;
F[{x_,y_},{a_,b_,c_,d_,k_}]:=Evaluate@{f1[x,y],f2[x,y]};
X={x, y};
μ={a,b,c,d,k};
</code></pre>
<p><span class="math-container">$$
\begin{align}
&\dot{x}=a x\lef... |
663,435 | <p>Bob has an account with £1000 that pays 3.5% interest that is fixed for 5 years and he cannot withdraw that money over the 5 years</p>
<p>Sue has an account with £1000 that pays 2.25% for one year, and is also inaccessible for one year.</p>
<p>Sue wants to take advantage of better rates and so moves accounts each ... | Warren Hill | 86,986 | <p>A bit of trail and error is needed here as I cant see a closed form solution.</p>
<p>For bob He ends up with $1000\cdot(1+0.035)^5 \approx 1187.686$</p>
<p>For Sue its</p>
<p>$1000 \cdot (1+0.0225)\cdot(1+0.0225+I)\cdot(1+0.0225+2I)\cdot(1+0.0225+3I)\cdot(1+0.0225+4I)$</p>
<p>There are various ways you can solve... |
2,021,354 | <p>I have a vector <strong>x</strong> and a function that sums the elements of <strong>x</strong> like so:</p>
<p>$$f(1) = x_1$$
$$f(2) = x_1 + \sum_{i=1}^2 x_i$$
$$f(3) = x_1 + \sum_{i=1}^2 x_i + \sum_{j=1}^3 x_j$$
$$f(4) = x_1 + \sum_{i=1}^2 x_i + \sum_{j=1}^3 x_j + \sum_{k=1}^4 x_k$$</p>
<p>...and so on. How might... | D.F.F | 77,924 | <p>Since there was the "recursion" tag in your question, here is a recursive solution:</p>
<p>$ f (0) = 0$</p>
<p>$f (n+1) = f (n) + \sum_{i=1}^{n+1} x_i$</p>
|
2,529,682 | <p>Right now I'm stuck on the following problem, since I feel like I should be using total probability, but I dont know what numbers to use as what.</p>
<p>Let's say there's a population of students. In this population:</p>
<p>30% have a bike</p>
<p>10% have a motorcycle</p>
<p>12% have a car.</p>
<p>8% have a bik... | Michael Rozenberg | 190,319 | <p>We have $$(\ln|x|)'=\frac{1}{x}.$$
Because if $x>0$ we obtain:
$$(\ln|x|)'=(\ln{x})'=\frac{1}{x}$$ and for $x<0$ we obtain:
$$(\ln|x|)'=(\ln(-x))'=-\frac{1}{-x}=\frac{1}{x}.$$</p>
|
369,723 | <p>I'm reading Milne's <em>Elliptic Curves</em> and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1.
In <a href="https://math.stackexchange.com/questions/226127/for-what-algebraic-curves-do-rational-points-form-a-group">this question</a... | xyzzyz | 23,439 | <p>For complex algebraic curves, this is actually very easy. Suppose we have an algebraic group law on a complex algebraic curve. Then it is necessarily continuous in a classical topology, so we get a topological group. Now, a fundamental group of a topological group has to be abelian (this is an interesting and not th... |
369,723 | <p>I'm reading Milne's <em>Elliptic Curves</em> and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1.
In <a href="https://math.stackexchange.com/questions/226127/for-what-algebraic-curves-do-rational-points-form-a-group">this question</a... | Piotr Pstrągowski | 16,673 | <p>Just to throw another way to formalize this fact, if $X$ is a variety that is a group, then the canonical sheaf $\omega_{X}$ must be trivial. The intuitive reason is that there is a canonical way to identify the tangent space $T_{x}$ at any point $x \in X$ with the tangent space $T_{e}$ at identity. (Namely, the map... |
369,723 | <p>I'm reading Milne's <em>Elliptic Curves</em> and came across this statement: If a nonsingular projective curve has a group structure defined by polynomial maps, then it has genus 1.
In <a href="https://math.stackexchange.com/questions/226127/for-what-algebraic-curves-do-rational-points-form-a-group">this question</a... | Julien Clancy | 28,711 | <p>I accepted another answer but I'm going to post a more elementary (from my perspective) way to prove this. The Riemann-Hurwitz Formula gives the nice bound $|\text{Aut}(\mathcal{C})| < \infty$ for a $g(\mathcal{C}) > 1$ (we'll assume characteristic zero so all maps are tamely ramified). But each point gives th... |
389,888 | <p>Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$?
I know that for small $z$ we have $$\left|\tfrac{1}{z}\right|>\left|\ln(\tfrac{1}{z})\right|$$ and that implies $|f(0)|=0.$
Are there multiple ways to handle this ?</p>
| 75064 | 75,064 | <p>To me, "analytic" means "locally represented by its Taylor series". With this interpretation $f$ is not analytic. Indeed, suppose $f(z)=z^r\sum_{n=0}^\infty c_n z^n $ in a neighborhood of $0$, where $c_0\ne 0$. Then
$$\frac{1}{z}+\ln \frac{1}{z} = z^{-r} \sum_{n=0}^\infty b_n z^n $$ in some (possibly smaller) neigh... |
389,888 | <p>Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$?
I know that for small $z$ we have $$\left|\tfrac{1}{z}\right|>\left|\ln(\tfrac{1}{z})\right|$$ and that implies $|f(0)|=0.$
Are there multiple ways to handle this ?</p>
| DonAntonio | 31,254 | <p>$$f(z)=\frac1{\frac1z+\text{Log}\frac1z}=\frac z{1+z\,\text{Log}\frac1z}$$</p>
<p>Now, </p>
<p>$$\text{Log}\frac1z:=\log\frac1{|z|}+i\arg\frac1z\implies z\,\text{Log}\frac1z=z\log\frac1{|z|}+iz\arg\frac1z$$</p>
<p>If you now choose a branch cut for the complex logarithm (and you better do if you have any hope to ... |
2,733,728 | <p>How can one find a general form for $\int_0^1 \frac {\log(x)}{(1-x)} dx=-\zeta(2)
\,?$ Namely $\int_0^1 \frac {\log^n(x)}{(1-x)^m} dx\,$ where $n,m\ge1$ Similar to the original integral I let $1-x=u\,$ which gives $$\int_{-1}^0 \frac {\log^n(1+x)}{x^m} dx$$ and expanding into series we have: $\int_{-1}^0x^{-m}(\s... | user | 505,767 | <p>The result is consistent indeed</p>
<ul>
<li><p>when $h=0 \implies v=0$</p></li>
<li><p>when $h>0 \implies v<0$ since the ball is falling down</p></li>
</ul>
|
2,417,197 | <p>When going through with learning Grahams number, I got stuck at </p>
<p>$$3↑↑↑3$$</p>
<p>Working it through, we have</p>
<p>$$3↑3=3^3$$
$$3↑↑3=3^{3^3}=3↑(3↑3)$$</p>
<p>As such, it would appear to me that</p>
<p>$$3↑↑↑3=3^{3^{3^3}}=3↑(3↑(3↑3))=3↑(3↑↑)$$</p>
<p>Which is incorrect; the correct answer being</p>
<... | Sheldon L | 43,626 | <p>$$ 3 \uparrow \uparrow n = 3 \uparrow (3 \uparrow \uparrow (n-1))$$
$$ 3 \uparrow \uparrow \uparrow n = 3 \uparrow \uparrow (3 \uparrow \uparrow \uparrow (n-1))$$
$$ 3 \uparrow \uparrow \uparrow \uparrow n = 3 \uparrow \uparrow \uparrow (3 \uparrow \uparrow \uparrow \uparrow (n-1)) ....$$</p>
<p>By definition
$$ 3... |
4,549,898 | <p>I need some help with solving the following problem: Let <span class="math-container">$Q(n)$</span> be the number of partitions of <span class="math-container">$n$</span> into distinct parts. Show that <span class="math-container">$$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$</span> is convergent by estimating <span class="... | Damian Pavlyshyn | 154,826 | <p>If <span class="math-container">$a_1 < \dotsb < a_k$</span> is a distinct partition of <span class="math-container">$n$</span> of size <span class="math-container">$k$</span>, we must have that <span class="math-container">$a_j \geq j$</span>, and so
<span class="math-container">$$
n
= \sum_{j=1}^k a_j
\geq \s... |
2,542,184 | <p>Is $A = \begin{bmatrix}
1&1&0\\
0&1&0\\
0&0&1\\
\end{bmatrix}$
and $B = \begin{bmatrix}
1&1&0\\
0&1&1\\
0&0&1\\
\end{bmatrix}$ similar? Please justify your answer.</p>
<p>So far what I've done is to check rank, det, trace, and characteristic polynomial to maybe dispr... | Rene Schipperus | 149,912 | <p>These are matrices in Jordan normal form. This is a representative of the similarity class. Thus these matrices are not similar. For example $A$ has two independent eigenvectors to the eigenvalue $1$ whereas $B$ has only one.</p>
|
1,921,114 | <p><img src="https://i.stack.imgur.com/D8IcM.jpg" alt="enter image description here"></p>
<p>So I solved this system without using matrices, just by (sort of) reverting to high school math instincts.
$$w+x=-5$$
$$x+y=4$$
$$y+z=1$$
$$w+z=8$$
From this, I got $w=-9,x=4,y=0$ and $z=1$. How would I convert this back to wh... | Win Vineeth | 311,216 | <p>From the format, you get that $x_i=a_i+s*b_i$</p>
<p>$x_1+x_2=-5 => a_1+a_2+s(b_1+b_2) = -5$ for all s. Thus, $b_1=-b_2$</p>
<p>Similarly, you get $b_1=-b_2=b_3=-b_4$ </p>
<p>The solution you got is for $a_1, a_2,a_3,a_4$</p>
<p>Thus,
$x_1 = -9+s*1$ (taking $b$ into $s$)</p>
<p>$x_2=4+s*(-1)$</p>
<p>$x_3=0... |
1,921,114 | <p><img src="https://i.stack.imgur.com/D8IcM.jpg" alt="enter image description here"></p>
<p>So I solved this system without using matrices, just by (sort of) reverting to high school math instincts.
$$w+x=-5$$
$$x+y=4$$
$$y+z=1$$
$$w+z=8$$
From this, I got $w=-9,x=4,y=0$ and $z=1$. How would I convert this back to wh... | tantheta | 165,463 | <p><a href="https://i.stack.imgur.com/PjwwB.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/PjwwB.jpg" alt="enter image description here"></a></p>
<p>i guess that answers your question</p>
|
2,489,988 | <p>A sequence of numbers is formed from the numbers $1, 2, 3, 4, 5, 6, 7$ where all $7!$ permutations are equally likely. What is the probability that anywhere in the sequence there will be, at least, five consecutive positions in which the numbers are in increasing order?</p>
<p>I approached this problem in the follo... | N. Shales | 259,568 | <p>Both myself and @N.F.Taussig used the following approach, although I'd like to see if it could be generalised to increasing runs of arbitrary length.</p>
<p>Define set $S_{i,j}$ as the set of permutations of $[7]$ with an increasing run between position $i$ and $j$ inclusive. Then by <a href="https://en.m.wikipedia... |
237,142 | <p>I am having a problem with the final question of this exercise.</p>
<p>Show that $e$ is irrational (I did that). Then find the first $5$ digits in a decimal expansion of $e$ ($2.71828$).</p>
<p>Can you approximate $e$ by a rational number with error $< 10^{-1000}$ ? </p>
<p>Thank you in advance</p>
| sperners lemma | 44,154 | <p>If <code>2.71828</code> are the first few digits of $e$ the we have $$2.71828 < e < 2.71829$$ Put $q = 271828/100000$ we deduce $$0 < e-q < 0.00001 = 10^{-5}.$$</p>
|
1,597,891 | <p>Let $G$ be an abelian group of order $75=3\cdot 5^{2}$. Let $Aut(G)$ denote its group of automorphisms. Find all possible order of $Aut(G)$.</p>
<p>My approach is to first study its Sylow 5-subgroup. Since $n_{5}|3$ and $n_{5}\equiv 1\pmod{5}$, $n_{5}=1$. So $G$ has a unique Sylow 5-subgroup, denote $F$. By Sylow's... | Eric Thoma | 35,667 | <p>Sylow theory is generally not useful for Abelian groups since we already know so much about their structure, and since every Abelian group has a unique $p$-Sylow subgroup when it exists.</p>
<p>Using the classification theorem, we have that $|{G}| = 75$ implies $G \cong \mathbb{Z}/3\mathbb{Z} \times (\mathbb{Z}/5\m... |
1,091,087 | <p>I was confronted with the following problem on my mid-term paper and I've got no idea how to solve this, I tried using the eigenvalues method, but it ultimately failed . Can anyone, please, give a complete solution to this ? I really want to se a proper solution so I can understand better the reasoning at hand..</p>... | Alex Silva | 172,564 | <p><strong>Hint</strong>:</p>
<p>Substitute the first equation in the second</p>
<p>$$ z' = -4y -\frac{3}{4}(y'-5y)$$</p>
<p>Yet, from the first equation</p>
<p>$$z' = \frac{1}{4}(y'' - 5y')$$.</p>
|
1,091,087 | <p>I was confronted with the following problem on my mid-term paper and I've got no idea how to solve this, I tried using the eigenvalues method, but it ultimately failed . Can anyone, please, give a complete solution to this ? I really want to se a proper solution so I can understand better the reasoning at hand..</p>... | Matt L. | 70,664 | <p><strong>Hint:</strong></p>
<p>Add both equations:</p>
<p>$$y'+z'=y+z\quad\Longrightarrow\quad y+z=ce^t$$</p>
<p>Now substitute.</p>
|
1,091,087 | <p>I was confronted with the following problem on my mid-term paper and I've got no idea how to solve this, I tried using the eigenvalues method, but it ultimately failed . Can anyone, please, give a complete solution to this ? I really want to se a proper solution so I can understand better the reasoning at hand..</p>... | abel | 9,252 | <p>this problem is more difficult than i thought because the coefficient matrix has a repeating root. i need a little set up first. let $x$ be a two dimensional column vector and $A$ a $2 \times 2$ real matrix. consider the linear differential equation
$$\frac{dx}{dt} = Ax$$</p>
<p>first we establish that:</p>
<p>(a)... |
464,404 | <p>Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that </p>
<p>$$ \operatorname{Hom} (A \otimes_R B , R) \cong \operatorname{Hom}(A,R)\otimes_R B$$</p>
<p>where the homomorphisms are those of $R$-modules? In other words, ... | Rasmus | 367 | <p><strong>In general no</strong>, look at $R=\mathbb Z$, $A=\mathbb Z$, $B=\mathbb Z/2$.</p>
<p><s> If $B=\mathbb R$, there <strong>happens to be</strong> an isomorphism as you would like, but it is not natural. </s></p>
<p>Even if $R=\mathbb Z$, $A=\mathbb Z$, $B=\mathbb R$, the statement is false, see Mariano's co... |
464,404 | <p>Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that </p>
<p>$$ \operatorname{Hom} (A \otimes_R B , R) \cong \operatorname{Hom}(A,R)\otimes_R B$$</p>
<p>where the homomorphisms are those of $R$-modules? In other words, ... | Mariano Suárez-Álvarez | 274 | <p>To add a positive spin: there is an isomorphism $\hom_R(A\otimes B,R)\cong\hom_R(A,R)\otimes_RB$ whenever $B$ is projective and finitely generated. </p>
|
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Qiaochu Yuan | 232 | <p>You can create new jobs for mathematicians, e.g. by funding institutes like <a href="http://en.wikipedia.org/wiki/James_Harris_Simons">Jim Simons</a>. Arguably this does much more for mathematics than actually doing mathematics due to replaceability: the marginal effect of becoming a mathematician is that you do mar... |
1,088,338 | <p>There are at least a few things a person can do to contribute to the mathematics community without necessarily obtaining novel results, for example:</p>
<ul>
<li>Organizing known results into a coherent narrative in the form of lecture notes or a textbook</li>
<li>Contributing code to open-source mathematical softw... | Yes | 155,328 | <p>I would say that though obtaining results is crucial, introducing new concepts, new connections, or even new perspectives looking at classical mathematics sometimes are more important.</p>
<p>Gauss, for example, introduced the concept of congruence in number theory and, arguably, number theory has then been develop... |
2,965,082 | <blockquote>
<p>Suppose that <span class="math-container">$(X,\ d)$</span> and <span class="math-container">$(Y,\ \rho)$</span> are metric spaces, that
<span class="math-container">$f_n:X\to Y$</span> is continuous for each <span class="math-container">$n$</span>, and that <span class="math-container">$(f_n)$</span... | gogurt | 29,568 | <p>This is an excellent example of how imprecise language/notation can lead to confusion. </p>
<p>In your first line you write "for all <span class="math-container">$n$</span>, there exists <span class="math-container">$\delta$</span> such that..." This sort of suggests that the same <span class="math-container">$\del... |
1,093,396 | <p>I've been working on a problem from a foundation exam which seems totally straightforward but for some reason I've become stuck:</p>
<p>Let $f: \mathbb{ R } \rightarrow \mathbb{ R } ^n$ be a differentiable mapping with $f^\prime (t) \ne 0$ for all $t \in \mathbb{ R } $, and let $p \in \mathbb{ R } ^n$ be a point NO... | user134824 | 134,824 | <p>Exactly. The function $\varphi: t\mapsto |p-f(t)|^2$ is continuous and positive, and therefore assumes a minimum value.</p>
|
892,742 | <p>Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?</p>
| Geoff Robinson | 13,147 | <p>This is a consequence of the fact that $G$ has $[G:G^{\prime}]$ linear characters (which are certainly irreducible)- I refer to complex characters here, and if $\lambda$ is a linear character of $G,$ we have $|\lambda(g)|^{2} = 1$ for all $g \in G.$ On the other hand, the orthogonality relations show that if $x \in ... |
1,722,287 | <p>So far I know that when matrices A and B are multiplied, with B on the right, the result, AB, is a linear combination of the columns of A, but I'm not sure what to do with this. </p>
| Siong Thye Goh | 306,553 | <p>$$rank(AB)=rank((AB)^T)=rank(B^TA^T)\leq rank(B^T)=rank(B)$$</p>
|
637,819 | <p>$$x\in(\cap F)\cap(\cap G)=[\forall A\in F(x\in A)]\land[\forall A\in G(x\in A)]$$</p>
<p>Since the variable $A$ is bounded by universal quantifier, it is regarded as bounded variable, according to the rules, the variable is free to change to other letters while the meaning statement remains unchanged. But,the abov... | SixWingedSeraph | 318 | <p>The answer to your <em>specific</em> question is that $A$ occurs as a bound variable in two different expressions. When a variable is bound, it has no meaning outside the expression it is in, so you can use it again as a variable in another expression, and that is what is done here. (But if I were writing a book or... |
3,459,532 | <p>I have a pretty straightforward linear programming problem here:</p>
<p><span class="math-container">$$ maximize \hskip 5mm -x_1 + 2x_2 -3x_3 $$</span></p>
<p>subject to</p>
<p><span class="math-container">$$ 5x_1 - 6x_2 - 2x_3 \leq 2 $$</span>
<span class="math-container">$$ 5x_1 - 2x_3 = 6 $$</span>
<span class... | Community | -1 | <p>We can also do the job as follows. </p>
<p>Note that <span class="math-container">$U_1=a_1a_1^T,U_2=a_2a_2^T$</span> are real symmetric, then are orthogonally diagonalizable.</p>
<p><span class="math-container">$tr(U_1)=tr(U_2)=a_1^Ta_1=a_2^Ta_2=1$</span> and <span class="math-container">$rank(U_1)=rank(U_2)=1$</s... |
3,534,254 | <blockquote>
<p><span class="math-container">$r\gt0$</span>, Compute
<span class="math-container">$$\int_0^{2\pi}\frac{\cos^2\theta }{ |re^{i\theta} -z|^2}d\theta$$</span>
when <span class="math-container">$|z|\ne r$</span></p>
</blockquote>
<p>The problem is related to Poisson kernel and harmonic function, but ... | Conrad | 298,272 | <p>Note that <span class="math-container">$\Re{\frac{r+ze^{-i\theta}}{r-ze^{-i\theta}}}=\frac{r^2-|z|^2}{|re^{i\theta} -z|^2}$</span>, so the integral is <span class="math-container">$\frac{1}{r^2-|z|^2}\Re{\int_0^{2\pi}\frac{(r+ze^{-i\theta})\cos^2\theta }{ r-ze^{-i\theta}}}d\theta$</span></p>
<p>while if <span class... |
1,276,957 | <p>These are the provided notes:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/NesWm.png" alt="Blockquote"></p>
</blockquote>
<p>These are the provided questions:</p>
<blockquote>
<p><img src="https://i.stack.imgur.com/t0Ta7.png" alt="Blockquote"></p>
</blockquote>
<p>I do not understand when I should... | N. F. Taussig | 173,070 | <p>Consider the following diagram:</p>
<p><img src="https://i.stack.imgur.com/62MrS.jpg" alt="second-quadrant_angle_with_reference_angle"></p>
<p>The range of the arccosine function is $[0, \pi]$. Since you are working in degrees, this corresponds to $[0^\circ, 180^\circ]$. Thus, you can calculate the measure of th... |
754,888 | <p>The letters that can be used are A, I, L, S, T. </p>
<p>The word must start and end with a consonant. Exactly two vowels must be used. The vowels can't be adjacent.</p>
| hmakholm left over Monica | 14,366 | <p>If you can find the prime factorization of the number, take the greatest common divisor of all the exponents in it.</p>
<p>Unfortunately factoring large numbers is not quick, so simply checking all possible degrees up to $\log_2$ of the number might well be faster asymptotically.</p>
<p>For most inputs, a combinat... |
165,069 | <p>I have a list of the following kind:</p>
<pre><code>{{1,0.5},{2,0.6},{3,0.8},{-4,0.9},{-3,0.95}}
</code></pre>
<p>The important property is, that somewhere in the list, the first element of the sublists changes sign (above is from + to -, but could be from - to +). How can I most efficiently split this into two li... | kglr | 125 | <pre><code>list = {{1, 0.5}, {2, 0.6}, {3, 0.8}, {-4, 0.9}, {-3, 0.95}};
SplitBy[#, Sign[First @ #] &] & @ list
</code></pre>
<blockquote>
<p>{{{1, 0.5}, {2, 0.6}, {3, 0.8}}, {{-4, 0.9}, {-3, 0.95}}}</p>
</blockquote>
<p>Or</p>
<pre><code>Split[#, SameQ @@ Sign [First /@ {##}] &] & @ list
</code>... |
2,449,581 | <p>There is a brick wall that forms a rough triangle shape and at each level, the amount of bricks used is two bricks less than the previous layer. Is there a formula we can use to calculate the amount of bricks used in the wall, given the amount of bricks at the bottom and top levels?</p>
| Arthur | 15,500 | <p>Hint: Make one identical wall right next to it, but upside-down. How many bricks are in each row of the two walls combined? How many bricks have you then used for those two walls?</p>
|
324,119 | <p>I've been reading about the Artin Spin operation. It's defined as taking the classical <span class="math-container">$n$</span>-knot (<span class="math-container">$S^n\hookrightarrow S^{n+2}$</span>) to an <span class="math-container">$(n+1)$</span>-knot. For the <span class="math-container">$1$</span>-knot case (in ... | spin | 38,068 | <p>This does not really involve any category theory, but perhaps it is useful to note the following general setting for the Jordan-Hölder theorem.</p>
<p>For <span class="math-container">$G$</span> a group and <span class="math-container">$\Omega$</span> a set, a <em>group with operators</em> is <span class="math-cont... |
2,180,700 | <p>A and B toss a fair coin 10 times. In each toss, if its a head A's score gets incremented by 1, if its a tail B's score gets incremented by 1.</p>
<p>After 10 tosses, the person with the greatest score wins the game.</p>
<p>What is the probability that A wins?</p>
<p>And if B alone gets an extra toss. What is the... | Bram28 | 256,001 | <p>It should indeed be (10C6 + 10C7 + 10C8 + 10C9 + 10C10 ) / 2^10</p>
<p>Here is why:</p>
<p>There is only one way for (10,0) to be the outcome:</p>
<p>HHHHHHHHHH ... which happens with a probability of $(\frac{1}{2})^{10}$</p>
<p>But there are 10 ways for (9,1) to be the outcome:</p>
<p>HHHHHHHHHT</p>
<p>HHHHHH... |
3,074,035 | <p>I am trying to find a simplified form for this summation:</p>
<p><span class="math-container">$$B(k,j) \equiv \sum_{i=1}^k (-1)^{k+i} {i \choose j} (k-1)_{i-1} \quad \quad \quad \text{for } 1 \leqslant j \leqslant k,$$</span></p>
<p>where the terms <span class="math-container">$(k-1)_{i-1} = (k-1) \cdots (k-i+1)$<... | R. J. Mathar | 805,678 | <p><span class="math-container">\begin{equation}
\sum_{i=1}^k (-1)^{k+i}\binom{i}{j}(k-1)_{i-1}
=
\frac{\Gamma(k)}{\Gamma(1+j)}(-1)^k \sum_{i=1}^k (-1)^{i}\frac{\Gamma(i+1)}{\Gamma(i-j+1)\Gamma(k-i+1)}
\end{equation}</span>
substitute <span class="math-container">$i'=k-i$</span> and use <span class="math-container">$(.... |
2,942,263 | <p>I am curious whether there is an algebraic verification for <span class="math-container">$y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$</span> having its minimum value of <span class="math-container">$\sqrt{2 + \sqrt{3}}$</span> at <span class="math-container">$\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$</span>. I have been to... | K B Dave | 534,616 | <p>Scale and shift
<span class="math-container">$$\begin{align}u&=\sqrt{2}x-1&v&=\tfrac{1}{\sqrt{2}}y-\tfrac{1}{2}\text{.}
\end{align}$$</span>
Then it is just as well to minimize
<span class="math-container">$$v=\tfrac{u}{2}+\sqrt{u^2+1}\text{.}$$</span>
with respect to <span class="math-container">$u$</sp... |
2,942,263 | <p>I am curious whether there is an algebraic verification for <span class="math-container">$y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$</span> having its minimum value of <span class="math-container">$\sqrt{2 + \sqrt{3}}$</span> at <span class="math-container">$\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$</span>. I have been to... | trancelocation | 467,003 | <p>Here is a "non-calculus" way that plays the whole show back to AMGM. It is a bit cumbersome but works.</p>
<p>I prefer giving all stepwise substitutions to show how to bring the whole expression back to hyperbolic functions where AMGM suddenly gives all. The basic idea behind it is that <span class="math-container"... |
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