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 |
|---|---|---|---|---|
3,151,329 | <p>How to find range of this function, it looks easy but somehow I am not able to do that . <br> What is more important to me is , why is wolframalpha unable to find the range , is it not a simple enough function ?
<img src="https://i.stack.imgur.com/pwGkm.png" alt="enter image description here"></p>
| Dr. Sonnhard Graubner | 175,066 | <p>It is <span class="math-container">$$-\frac{1}{2}\le y\le \frac{1}{2}$$</span> this can be obtained:
<span class="math-container">$$x\sqrt{1-x^2}\le \frac{1}{2}$$</span>
<span class="math-container">$$4x^2(1-x^2)\le 1$$</span> or <span class="math-container">$$0\le(2x-1)^2$$</span></p>
|
3,237,995 | <p>Let <span class="math-container">$\Bbb L$</span> be a positive integer and define <span class="math-container">$\Bbb Z_L$</span> and <span class="math-container">$\Bbb Z^*_L$</span> as <span class="math-container">$$\Bbb Z_L=\{ 0,1,...,L-1\},\qquad\Bbb Z^*_L=\{i\in \Bbb Z_L\mid \gcd(i,L)=1\}$$</span> where <span cla... | Noah Schweber | 28,111 | <p>This is a nice counting argument - I'll sketch how it goes (it's a good exercise to fill in the remaining steps).</p>
<p>Suppose I have a ring <span class="math-container">$R$</span>. Then for any <span class="math-container">$a\in R$</span> we have a "multiplication by <span class="math-container">$a$</span>" func... |
4,447,551 | <p>Suppose we have a sample of <span class="math-container">$X_i = x_i$</span> for <span class="math-container">$i \in [1,n]$</span> where <span class="math-container">$\forall X_i$</span> they are iid with uniform distribution on the interval <span class="math-container">$[a,b]$</span></p>
<p>Now we want to find the M... | Glorious Nathalie | 948,761 | <p>You have the line <span class="math-container">$ y = m x + b $</span> and two points <span class="math-container">$P = (P_x, P_y)$</span> and <span class="math-container">$Q = (Q_x, Q_y )$</span> on the parabola <span class="math-container">$ x = A y^2 + B y + C $</span> where <span class="math-container">$A,B,C$</s... |
4,198,816 | <p>I just got introduced to Bayes theorem, and tried solving the following question:</p>
<blockquote>
<p>Two balls are drawn in succession from a jar containing 3 red balls and 4 white balls. What is the probability that the first ball was white given the second ball was red.</p>
</blockquote>
<p>I used the formula giv... | Eric Wofsey | 86,856 | <p>Suppose <span class="math-container">$X$</span> is a finite <span class="math-container">$T_0$</span> space and <span class="math-container">$A\subseteq X$</span> is an irreducible closed subset. Let <span class="math-container">$B\subset A$</span> be a maximal closed proper subset (which exists by finiteness), and... |
129,655 | <p>Here is a nice <em>sufficient (but not necessary)</em> condition on whether a quintic is solvable in radicals or not. Given,</p>
<p>$x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ </p>
<p>If there is an ordering of its roots such that,</p>
<p>$x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - (x_1 x_3 + x_3 x_5 + x_5 x_2 +... | PAD | 27,304 | <p>$x(t)=1-t $</p>
<p>$y(t)=t$ </p>
<p>with $0 \le t \le 1 $. We have $dx=-dt$, $dy=dt$.</p>
<p>$\int_{\gamma} y^2+x^2 dy= \int_{0}^{1} t^2 (-dt)+(1-t)^2 dt=\int_{0}^{1} (1-2t) dt= 0 $</p>
|
4,022,066 | <p>I don't quite understand how <span class="math-container">$\dfrac1 x$</span> is unbounded while <span class="math-container">$\dfrac{x}{1+x^2}$</span> is not.</p>
| leslie townes | 18,076 | <p>Regarding <span class="math-container">$x/(1+x^2)$</span>, it may help to analyze the cases <span class="math-container">$|x| \leq 1$</span> and <span class="math-container">$|x| > 1$</span> separately.</p>
<p>In the former case you have a continuous function everywhere defined on a closed interval (here it's imp... |
2,321,490 | <p>I know that :</p>
<p><span class="math-container">$$\ln: (0,+\infty )\to \mathbb{R}$$</span>
<span class="math-container">$$\ln(x):=\int_1^x \dfrac{dt}{t}$$</span></p>
<hr>
<p>And :</p>
<p><span class="math-container">$$\ln^{-1}: \mathbb{R} \to \mathbb{R}^+$$</span>
<span class="math-container">$$\ln^{-1}(x):=\e... | PolyaPal | 22,004 | <p>The May 2022 issue of <em>The College Mathematics Journal</em> will contain the article "The Equivalence of Definitions of the Natural Logarithm Function," which has some historical comments and a 25-item bibliography.</p>
|
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | Roman Odaisky | 271,020 | <p>Maybe those with allergies to both complex numbers and trigonometry are more tolerant to homogeneous coordinate transforms?</p>
<p>$$
f(k) = [1 \ 0\ 2]
\begin{bmatrix}
-\frac12 & -\frac{\sqrt3}2 & 0 \\
\frac{\sqrt3}2 &-\frac12 & 0 \\
0 & 0 & 1
\end{bmatrix}^k
\begin{bmatrix}
1 \\ \frac{\sqr... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | Ilmari Karonen | 9,602 | <p>Your only problem is that you're counting up from one. If you started counting from zero, then $$f(x) = x \bmod 3$$ would do the job, generating the output sequence $0,1,2,0,1,2,0,1,2,\dots$ when applied to the input sequence $0,1,2,3,4,5,6,7,8,\dots$.</p>
<p>But if you <em>insist</em> on one-based counting, then ... |
2,222,898 | <p>A friend of mine recently asked me this question.</p>
<p>For a positive integer $n$, a function $f(n)$ is defined as:</p>
<p>$f(n)=$ sum of digits in $n$.</p>
<p>given $f(n)=5$ find the maximum value of $f(n^5)$.</p>
<p>I tried solving this problem by putting random values, but my friend gave me a hint that the ... | Donald Splutterwit | 404,247 | <p>Consider the terms generated by $(a+b+c+d+e)^5$. There will be $5$ of the form $a^5$ which will have a coeffiecient of $1$ & each will therefore give a contribution of $1$, there will be $20$ of the form $a^4 b$ which will have a coeffiecient of $5$ & will therefore give a contribution of $5$, etc ... terms... |
135,717 | <p>I am looking for examples (references) of pairs of non Quillen-equivalent model categories having the same homotopy categories. </p>
<p>The motivation is of course that I have two model categories and all the attempts to prove that they are Quillen equivalent have failed so far, but the homotopy categories are equi... | Fernando Muro | 12,166 | <p>The ones which usually come up to my mind as soon as I think of this:</p>
<ul>
<li><p>The categories of modules over $\mathbb Z/p^2$ and $\mathbb F_p[\epsilon]/(\epsilon^2)$, $p$ a prime integer. These rings are quasi-Frobenius, so their module categories have a model structure where cofibrations are monomorphisms,... |
23,411 | <p>I'm trying to figure out how I can work out the number of possible valid hand combinations of a poker game that a player (opponent) could possibly have when the flop has been dealt on the table.</p>
<p>So I'm guessing that I have to take the probability that each hand can occur and then see which one is the highest... | Ross Millikan | 1,827 | <p>The number of hands is $\frac{47\cdot 46}2=1081$ as you see five cards (your two plus the flop) so there are $47$ left to choose from. If you are looking for the highest, there is no probability involved, just look at all of them and find the highest. Ignoring any information from the betting, the chance that an o... |
23,411 | <p>I'm trying to figure out how I can work out the number of possible valid hand combinations of a poker game that a player (opponent) could possibly have when the flop has been dealt on the table.</p>
<p>So I'm guessing that I have to take the probability that each hand can occur and then see which one is the highest... | Justin | 2,533 | <p>There is no software that can accurately take into consideration what move your opponent has done (i.e., assuming you were in position and he bet before you, take into account his bet or check) and tell you what likely has he has. </p>
<p>There are way too many variables to account for here, including but not limi... |
65,183 | <p>Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is
$$
\frac{p-6+\text{mod}(p,4)}{4},
$$
but I have neither proof nor reference. It is a particular case of the question in the title: if $a$ and $b... | GH from MO | 11,919 | <p>To complement the answers so far let me show using Gauss sums that the number of solutions of $ ax^2+by^2=c $ in $\mathbb{F}_p$ equals $p-\left(\frac{-ab}{p}\right)$ for any $a,b,c\in\mathbb{F}_p^\times$. Indeed, this number equals
$$ \frac{1}{p}\sum_n \sum_{x,y}e\left(n\frac{ax^2+by^2-c}{p}\right)
= \frac{1}{p}\su... |
2,525,055 | <p>How can the differential of $\mathbf r\cdot\mathbf r$ be written as $2\mathbf r\cdot d\mathbf r$? That is,</p>
<p>$$d(\mathbf r\cdot\mathbf r)=2\mathbf r\cdot d\mathbf r$$</p>
<p>How? Here, $\mathbf r$ is a vector.</p>
| David K | 139,123 | <p>I see that you did not assume (without proof) that the product rule for single variables works the same for vectors.
That is careful thinking, which is good.</p>
<p>To find the differential of $\mathbf r \cdot \mathbf r,$ you can
write $\mathbf r$ termwise:
$$ \mathbf r = \mathbf i x + \mathbf j y + \mathbf k z,$$
... |
54,391 | <p>Consider the following two structure-adding refinements of the fundamental group of a topological space:</p>
<ul>
<li><p>the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, under which it is sometimes a topological group. This was discussed <a href="https://mathoverflow.net... | Sergey Melikhov | 10,819 | <p>I figured my comments are getting too long so I should put some of them here, even though I'm using a different topology on <span class="math-container">$\pi_1$</span> (often it is the same, but certainly not always) and a possibly different pro-group (which I can't tell as I'm not familiar with toposes).
EDIT: Unle... |
4,610,333 | <p>Show <span class="math-container">$n^{\frac{1}{n}}>(n+1)^{\frac{1}{n+1}}$</span> for <span class="math-container">$n\geq3$</span>.</p>
<p>I took the derivative of <span class="math-container">$x^{1/x}$</span> and it is a decreasing function for <span class="math-container">$x>e$</span>.</p>
<p>I was just tryin... | Yuval | 609,959 | <p>This is equivalent to <span class="math-container">$n^{\frac{n+1}{n}}\geq n+1$</span>, which is equivalent to <span class="math-container">$n^{\frac 1n}\geq 1+\frac 1n$</span>, which is equivalent to <span class="math-container">$n\geq (1+\frac 1n)^n$</span>. Use the fact (c.f. the definition/basics of <span class="... |
1,254,275 | <p>We are given a number $$K(n) = (n+3) (n^2 + 6n + 8)$$ defined for integers n. The options suggest that the number K(n) should either always be divisible by 4, 5 or 6. </p>
<p>Factorizing the second bracket term, we get $$ K(n) = (n+3) (n + 2) (n+4)$$</p>
<p>i.e the number is always divisible by -4 -2 and -3. IMO t... | mathlove | 78,967 | <p>For $n\equiv 3\pmod 4$,
$$K(n)\equiv 6\cdot 5\cdot 7\equiv 2\cdot 1\cdot 3\equiv 2\not\equiv 0\pmod 4.$$</p>
|
4,357,233 | <p>Question</p>
<blockquote>
<p><span class="math-container">$$\sqrt {x^2+1} =x-1$$</span></p>
</blockquote>
<p>my attempt</p>
<p><span class="math-container">$$ x^2 + 1 = x^2-2x+1 $$</span>
and x=0</p>
<p>by putting the value of 0 = false</p>
| Kavi Rama Murthy | 142,385 | <p>The equation has no solution in the real line. The left side of the given equation is non-negative so we need <span class="math-container">$x-1 \geq 0$</span>. So <span class="math-container">$x=0$</span> is not allowed.</p>
|
4,357,233 | <p>Question</p>
<blockquote>
<p><span class="math-container">$$\sqrt {x^2+1} =x-1$$</span></p>
</blockquote>
<p>my attempt</p>
<p><span class="math-container">$$ x^2 + 1 = x^2-2x+1 $$</span>
and x=0</p>
<p>by putting the value of 0 = false</p>
| ascendho | 1,010,920 | <p>Your attemption is not quite right. Note that the left side is a non-negative number, so if you try to do like this, you must guarantee that the right side <span class="math-container">$x-1 \ge0$</span>, namely <span class="math-container">$x \ge 1$</span>, at last, you get <span class="math-container">$x=0$</span> ... |
3,104,936 | <p>I was wondering if anyone could explain to me why this holds true?</p>
<p><span class="math-container">$$C\left(n, k\right) = \frac{C\left(n, k-1\right) \cdot C(n - (k - 1), 1)}k$$</span></p>
<p>or</p>
<p><span class="math-container">$$C\left(3, 2\right) = \frac{C\left(3, 1\right) \cdot C(2, 1)}2$$</span></p>
<p... | trancelocation | 467,003 | <p>Here is a combinatorical explanation:</p>
<p>You can choose <span class="math-container">$k$</span> from <span class="math-container">$n$</span> items by</p>
<ul>
<li>choosing <span class="math-container">$k-1$</span> from <span class="math-container">$n$</span>: <span class="math-container">$\color{blue}{\binom{n... |
3,118,410 | <p>I've been trying to use the change of base property but I'm not having much luck. Can anyone give me any ideas on how I should approach this problem? The answer is 49/5</p>
<p>Thanks.</p>
| Rishi | 646,133 | <p><span class="math-container">$$\log_x 5x \cdot \log_7 x = 2$$</span>
<span class="math-container">$$\log_x 5x = \frac{2}{\log_7 x}$$</span>
<span class="math-container">$$\log_x 5x = \frac{2\log_x 7}{log_x x}$$</span>
<span class="math-container">$$\log_x 5x = 2\log_x 7$$</span>
<span class="math-container">$$\log_x... |
53,332 | <p>I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is particularly fond of a certain reference and why. </p>
<p>Also, I am very interested in ms/phd thesis (particularly ms thesis... | Matthew Towers | 5,316 | <p>Brain dump: Kassel is good. Jantzen (Lectures on Quantum Groups) is nicely written and clear, it's my favourite. Majid (Foundations of Quantum Group Theory) is an option, I haven't read it. Lusztig's book has a reputation of being tough to read. Chari and Pressley (A Guide To...) is comprehensive and has some in... |
53,332 | <p>I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is particularly fond of a certain reference and why. </p>
<p>Also, I am very interested in ms/phd thesis (particularly ms thesis... | Scott Carter | 722 | <p>Our (Carter, Flath, Saito) "Classical and Quantum $6j$-Symbols," PUP does not cover the subject in general, but tells a lot of what is needed in the $U_q(sl_2)$ case. </p>
|
53,332 | <p>I would like to know if there are any general references that you would suggest to learn about quantum groups? I have looked at some of the "standard" books, but I am wondering if someone is particularly fond of a certain reference and why. </p>
<p>Also, I am very interested in ms/phd thesis (particularly ms thesis... | Marek | 3,223 | <p>For starters you can check references in this question of mine on <a href="https://physics.stackexchange.com/questions/5682/what-are-the-uses-of-hopf-algebras-in-physics">applications of Hopf algebras</a>.</p>
<p>Here's <a href="http://kyokan.ms.u-tokyo.ac.jp/~topology/GHC/data/Deguchi.pdf" rel="nofollow noreferrer... |
918,306 | <blockquote>
<p>A set $X$ is said to be denumerable if there is a bijection $\mathbb{Z}^+\rightarrow X$</p>
</blockquote>
<p>Statement to prove:</p>
<blockquote>
<p>The of rationals $\mathbb{Q}$ is denumerable.</p>
</blockquote>
<p>Proof:</p>
<blockquote>
<p>Define $f:\mathbb{Q} \rightarrow \mathbb{Z}\times \... | Joseph Skelton | 30,482 | <ol>
<li><p>The ordered pair $(6,3)\in \mathbb{Z}\times\mathbb{Z^+}$ but, $gcd(6,3)=2\neq1$. So there is no element of $\mathbb{Q}$ that will map to $(6,3)$ hence $f(\mathbb{Q})\subsetneq \mathbb{Z}\times\mathbb{Z^+}$</p></li>
<li><p>Yes. We know the function is a one to one mapping to $\mathbb{Z}\times\mathbb Z^+$. It... |
20,817 | <p>I am referencing to this question on mathunderflow:<br>
<a href="https://math.stackexchange.com/questions/223238/what-is-the-average-of-rolling-two-dice-and-only-taking-the-value-of-the-higher">What is the average of rolling two dice and only taking the value of the higher dice roll?</a></p>
<p>Now I tried:</p>
<p... | whuber | 91 | <p>For your amusement, <strong>here is a cute, <em>Mathematica</em>-oriented solution:</strong> represent the die by its probabilities and "teach" the software to combine results using the maximum. So, to begin, here's a way to create dice and an example of the usual six-sided fair die:</p>
<pre><code>Clear[x]; die[n... |
954,658 | <p>How does
$$
\frac{1}{\log(1+\frac{d}{a})}
$$
simplify to $\frac{a}{d}$ in the limit that $d$ is tiny relative to $a$? I was wondering if a Taylor expansion would work though I am not sure how to work it.</p>
| user35687 | 177,987 | <p>Expand f(x) = ln(1 + x) as a Taylor series around x = 0.</p>
<p>f(0) = 0, f'(0) = 1/(1 + 0) = 1</p>
<p>ln(1 + x) = 0 + x + ... = x. </p>
<p>Substitute d/a for x.</p>
|
3,130,362 | <p>Given <span class="math-container">$a,b,c>0$</span> satisfying <span class="math-container">$a^2+b^2+c^2=3$</span>. Prove that
<span class="math-container">$$2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)+3(a+b+c)\geq 15.$$</span>
I've tried to use the inequality <span class="math-container">$\dfrac{1}{a}+\... | Michael Rozenberg | 190,319 | <p>It's obviously true by <span class="math-container">$uvw$</span>, but the Tangent Line method also gives a simple solution:</p>
<p><span class="math-container">$$\sum_{cyc}\left(\frac{2}{a}+3a-5\right)=\sum_{cyc}\frac{(a-1)(3a-2)}{a}=$$</span>
<span class="math-container">$$=\sum_{cyc}\left(\frac{(a-1)(3a-2)}{a}-\f... |
3,130,362 | <p>Given <span class="math-container">$a,b,c>0$</span> satisfying <span class="math-container">$a^2+b^2+c^2=3$</span>. Prove that
<span class="math-container">$$2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)+3(a+b+c)\geq 15.$$</span>
I've tried to use the inequality <span class="math-container">$\dfrac{1}{a}+\... | Michael Rozenberg | 190,319 | <p>There is also the following way.</p>
<p>Let <span class="math-container">$a+b+c=3u$</span>, <span class="math-container">$ab+ac+bc=3v^2$</span>, where <span class="math-container">$v>0$</span>, and <span class="math-container">$abc=w^3$</span>.</p>
<p>Thus, by AM-GM <span class="math-container">$v\geq w$</span>... |
2,918,287 | <p>We have
\begin{align}
U_{0} &= 1 &&\text{and} & V_{0} &= 2 \\
U_{n+1} &= \frac{U_{n}+V_{n}}{2} &&\text{and} & V_{n+1} &= \sqrt{U_{n+1}V_n}
\end{align}
How to prove two sequence have a common limit?</p>
<p>I found $(U_n)$ is increasing and $(V_n)$ is decreasing but I don't kn... | robjohn | 13,854 | <p>Given that
$$
u_{n+1}=\frac{u_n+v_n}2\quad\text{and}\quad v_{n+1}=\sqrt{\vphantom{1}u_{n+1}v_n}\tag1
$$
it is easy to derive that
$$
u_{n+1}^2-v_{n+1}^2=\frac{u_n^2-v_n^2}4\tag2
$$
Thus, induction says that
$$
u_n^2-v_n^2=\frac{u_0^2-v_0^2}{4^n}\tag3
$$
Furthermore,
$$
u_{n+1}-u_n=\frac{v_n-u_n}2\quad\text{and}\quad... |
1,917,852 | <p>Suppose $(\Omega, \mathcal{F}, \mu)$ is a probability space and $f$ is integrable function. Then, is it true that $\mu \{ \omega: |f(w)|=\alpha\}=0$ for all but countably many $\alpha$ ? \</p>
| Tsemo Aristide | 280,301 | <p>Hint: Let $X$ be a measure space such that $\mu(X)>0$, consider $Y=\bigcup_{a\in R}X_a$ where $(X_a,\mu_a)$ is isomorphic to $\mu_a$, consider $f:Y\rightarrow R$ whose restriction to $X_a$ is the constant function $a$.</p>
|
21,310 | <p>I would like to do a For loop with a jump 2*Pi instead of 1.</p>
<pre><code>For[j = 0, j <= nmax, (incr = 2 Pi), ...]
</code></pre>
<p>Actually what I want to do is <code>j = n 2 Pi</code>, where <code>n</code> takes values from 0 to <code>n-1</code>, so the iteration stops at <code>nmax = (n - 1) 2 Pi</code>.<... | m_goldberg | 3,066 | <p>Just to get it on the record, the strict answer to your question is </p>
<pre><code>With[{n =. ..}, For[j = 0, j <= (n - 1) 2 Pi, j += 2 Pi, ...]]
</code></pre>
<p>For example, </p>
<pre><code>With[{n = 3}, For[j = 0, j <= (n - 1) 2 Pi, j += 2 Pi, Print[j]]]
</code></pre>
<p>produces</p>
<blockquote>
<p... |
2,155,201 | <p>By a real ODE I mean an ordinary differential equation with only real coefficients and the resulting function is a function of a real argument. If such a solution exists, can you give an example?</p>
<p>Edit: To add to this, is it still possible if the initial conditions must also be real?</p>
| The_Sympathizer | 11,172 | <p>Try</p>
<p>$$\left[\frac{dy}{dx}\right]^2 = -(y^2 + 1)$$</p>
<p>This equation can have no real solution at all. Proof by contradiction: assume $y(x)$ is a real valued solution. Then $\left[\frac{dy}{dx}\right]^2$ is real as well, but that implies $\sqrt{-(y(x)^2 + 1)}$ is real, yet $y(x)^2 + 1 > 0$ always no ma... |
623,873 | <p>I'm given with the following function:</p>
<p>$f(x) = x^n \arctan\left(\dfrac{1}{x}\right) , x\ne0 $ , and $f(0)=0$ . </p>
<p>I'm asked to:</p>
<ol>
<li>determine when is it continuous in $x=0$ </li>
<li>determine when is it differentiable in $x=0$</li>
<li>determine when is it continuously differentiable in $x=0... | mathlove | 78,967 | <p>I don't know such formula, but I can say your conjecture is not true.
$$2+3+5+7+11+13+17+19=77=7\times 11.$$</p>
|
311,261 | <p>I would like to fit a formula $ax^b + cx^d+ e$ to a set of points. I have two questions.</p>
<ol>
<li>If my data were perfect, how many points do I need in the worst case to get $a,b,c,d,e$ exactly?</li>
<li>If my data aren't perfect, but the underlying unknown function is really something like $a\sqrt{x} + cx^{0.... | David E Speyer | 448 | <p>$\def\sgn{\mathrm{sgn}}$Answer to 1: Assuming that all the variables are real, $5$ positive values of $x$ is enough. </p>
<p>Proof: Suppose to the contrary that there were two different polynomials $a x^b+c x^d + e$ and $A x^B + C x^D + E$ interpolating through data points $(x_1, y_1)$, $(x_2, y_2)$, ..., $(x_5, y_... |
1,449,503 | <p>Could I not represent every odd square number in $\mathbb N$ using the following notation: $(2n-1)^2$ where $n \in \mathbb N$. For every $n= 1,2,3...$ I get the set $1,9,25...$ every odd square number. Why can't I do this? </p>
<p>I ask because then I could prove that if a natural numbers square is odd then the nu... | Peter | 82,961 | <p>Better show the fact : If the square of a number is odd, than the number is odd.</p>
<p>indirectly :</p>
<p>It is easy to see that the square of an even number is even. Conclusion :
If the square of a number is odd, then the number is odd, because if it
were even, the square would be even.</p>
<p>If you have this... |
1,328,297 | <p>I have been just trying to start working with the books <em>Topics in Algebra</em> by I.N. Herstein, and I am having a bit of trouble understanding a definition.</p>
<p>It is,</p>
<blockquote>
<p>Definition: If $S$ is a nonempty set then $A(S)$ is the set of all one-to-one mappings of $S$ onto itself.</p>
</bloc... | Zev Chonoles | 264 | <p>Yes, by Herstein's definition, $A(X)=\{\text{bijective functions }f:X\to X\}$. </p>
<p>Just for emphasis, since this seems to be something you're confused on, the elements of $A(X)$ are <strong>precisely</strong> the bijective functions from $X$ to $X$. There's nothing else in $A(X)$ other than that, and every such... |
1,770,858 | <p>I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?</p>
| treble | 24,837 | <p>In order to use the integral test, you need to figure out whether or not the integral $$\int_2^{\infty} \frac{1}{(\ln x)^2} dx $$ converges or not. To do this, I would suggest making the substitution $u = \ln (x),$ then note that $du = 1/x dx$ and that $x = e^u$. Plug all of this in and go to work!</p>
|
1,770,858 | <p>I am trying to use the integral test on the series $$\sum\limits_{n=2}^{\infty} \frac{1}{\ln(n)^2}.$$ I am not sure how to evaluate the integral. Any hints?</p>
| John Wayland Bales | 246,513 | <p>Regarding </p>
<p>\begin{equation}
\sum_{n=2}^{\infty}\frac{1}{\left(\ln x\right)^2}
\end{equation}</p>
<p>you should be able to show that $f(x)=\sqrt{x}-\ln x$ is positive for $x\ge2$. (Just find the minimum value of $f(x)$.)</p>
<p>Therefore for $n\ge2$ it's true that
$0<\ln n<\sqrt{n}$.</p>
<p>So $\dfra... |
61,172 | <p>Can anyone show me how to set all matrix rows to variables.
For example,
I have matrix <code>w={{1,2,3},{3,2,1},{0,0,0}}</code> and I need to set each matrix row to be separate variable that contains the row number of matrix <code>w</code> in its name.</p>
<p>For example,
<code>t1={1,2,3} t2={3,2,1} t3={{0,0,0}</c... | m_goldberg | 3,066 | <p>Perhaps this will work for you. Given a symbol, say <code>t</code>, and a matrix, say <code>m</code>, it will define a set of indexed variables <code>t[1]</code>, <code>t[2]</code>, ..., <code>t[n]</code>, bound to the respective rows of <code>m</code>. </p>
<pre><code>SetAttributes[assign, HoldFirst];
assign[name_... |
2,832,802 | <blockquote>
<p>Let $F = \mathbb{Q}$ and $K$ be the splitting field of $x^4 - 7$ over $F$. Determine $Gal(K/F)$ and find all the intermediate subfields of $K/F$.</p>
</blockquote>
<p>We have $x^4 - 7 = (x-\sqrt[4]{7})(x + \sqrt[4]{7})(x - i\sqrt[4]{7})(x + i\sqrt[4]{7})$ and so, $K = \mathbb{Q}(\sqrt[4]{7},i)$. Now,... | Stefan4024 | 67,746 | <p>We have the following automorphisms in the Galois group:</p>
<p>$$Id :
\begin{array}{lr}
\sqrt[4]{7} \to \sqrt[4]{7}\\
i \to i
\end{array}\quad \quad \tau:
\begin{array}{lr}
\sqrt[4]{7} \to i\sqrt[4]{7}\\
i \to i
\end{array}
\quad \quad \sigma:
\begin{array}{l... |
1,501,988 | <p>How do I solve the following equation?
$$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$</p>
<p>I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way to solve this?</p>
| SchrodingersCat | 278,967 | <p>HINT: Consider the 3 cases: </p>
<ol>
<li>$x<-2$</li>
<li>$-2\le x<0$</li>
<li>$x\ge 0$</li>
</ol>
|
1,501,988 | <p>How do I solve the following equation?
$$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$</p>
<p>I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way to solve this?</p>
| Ojas | 154,392 | <p>$$\lvert{x}\rvert + 2 \ge\lvert{2+x}\rvert$$
$$\therefore x = \lvert{x}\rvert- \lvert{2+x}\rvert \ge -2$$
$$\therefore x+2 \ge 0$$
Thus,
$$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$ becomes$$\lvert{x}\rvert = 2x + 2$$
Now, just consider two cases. $x \lt 0$ and $x \ge 0$.</p>
|
990,638 | <p>Say $X=\{a,b,c,d\}$ and $\tau=\{\emptyset,X,\{a\},\{c\}, \{a,c\}, \{a,b,d\}, \{b,c,d\},
\{b,d\}\}$</p>
<p>This is a topological space as it satisfies the appropriate properties but how do you show $\tau$ is (or is not) path-connected?</p>
| ajotatxe | 132,456 | <p>Since $\{a\}$ and $\{b,c,d\}$ are open, disjoint and complementary sets, the topological space is not even connected.</p>
|
1,756,667 | <p><a href="https://i.stack.imgur.com/6HG0u.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/6HG0u.jpg" alt="enter image description here"></a></p>
<p>I have calculated:</p>
<p>$\det A_1=2$, $\det A_2=3$, $\det A_3=4$, so I was putting some numbers in $\det A_{n+2} = a \det A_{n+1} + b\det A_n$ like... | K.K.McDonald | 302,349 | <p>it's easy indeed. to calculate $\det(A_{n+2})$ expand determinant from first column and you get this:
$$\det(A_{n+2})=2\det(A_{n+1})-\det(B)$$
where $\det(B)$ is $\det(A_{n+2})$ with the first column and second row removed. now again expand $\det(B)$ in first row to get $\det(B)=\det(A_{n})$ and proof is complete.</... |
1,206,284 | <p>Can someone help me solve $(y')^2=(y/c)^2-1$? WolframAlpha is giving me $\frac 12(c^2 e^{(x/c)-k}+e^{k-(x/c)})$. One book I have is giving me $y=c\cdot \cosh(\frac {x+b}c)$ -- but that one won't work for the HW problem I'm solving because with the conditions $y(0)=y(D)=0$ it gives $y$ as identically $0$ -- so it m... | Narasimham | 95,860 | <p>With $ y'(x) = \tan\phi $ substitution the equation reduces to $ y = c \sec(\phi) $ which is the standard ODE of Catenary:</p>
<p>$$ y = c \cosh (x/c) \tag {1*} $$</p>
<p>$$ c \dfrac{dy}{dx} = \sqrt{y^2-c^2} $$ transpose, integrate</p>
<p>$$ x/c = \cosh^{-1}\frac{y}{c} + C_1 $$ Choosing symmetric BC, $ C_1$ can... |
3,576,675 | <p>Normally, I would solve the integral and then evaluate the limit, but the integral seems particularly difficult, and I'm wondering if there's some sort of trick involved. The solution said to just use L'Hospital and the Fundamental Theorem of Calculus, but I don't know where to begin. I've tried moving the <span c... | DanielWainfleet | 254,665 | <p>You certainly can move the <span class="math-container">$1/x^2$</span> inside the integral but it won't help. </p>
<p>Consider that for any <span class="math-container">$\epsilon\in (0,1)$</span> there exists <span class="math-container">$x\in (0,1)$</span> such that for all <span class="math-container">$|t|\le x$<... |
2,438,594 | <p>Let $\left({z_n}\right)$ be a sequence of distinct points in $D\left(0,1\right)$=$\{{z \in \mathbb{C} :\vert{z}\vert \lt{1}} \}$ with $\lim_{n\to\mathbb{\infty}} {z_n}=0$. Consider the following statements P and Q.</p>
<p>P: There exists a unique analytic function $f$ on $D\left(0,1\right)$ such that $f\left({z_n}\... | Raffaele | 83,382 | <p>If they have the same root, then $$x^2-4x+k=x^2+kx-4$$
That means $$-4x+k=kx-4\to kx+4x=k+4\to x=1$$
Plug this in one of the two equations and get $1-4+k=0\to k=3$</p>
<p>Indeed $x^2-4x+3=0\to x_1=1,\;x_2=3$ and</p>
<p>$x^2+3x-4=0\to x_1=1;\;x_2=-4$</p>
<p>Hope this helps</p>
|
2,438,594 | <p>Let $\left({z_n}\right)$ be a sequence of distinct points in $D\left(0,1\right)$=$\{{z \in \mathbb{C} :\vert{z}\vert \lt{1}} \}$ with $\lim_{n\to\mathbb{\infty}} {z_n}=0$. Consider the following statements P and Q.</p>
<p>P: There exists a unique analytic function $f$ on $D\left(0,1\right)$ such that $f\left({z_n}\... | DYBnor | 422,177 | <p>Two polynomials have a common root if and only if their resultant vanishes.</p>
<p>The resultant $R(p(x), q(x))$ of two polynomials of degrees $m$ and $n$, respectively, is the determinant of the $(m + n) \times (m + n)$ matrix defined as follows. Write the coefficient of $p(x)$ in the first row followed by $n - 1$... |
4,158,247 | <p>Give an example of sequence <span class="math-container">$\{f_n\}$</span> of continious real -value function on <span class="math-container">$[0, \infty)$</span> converging uniformly on <span class="math-container">$[0, \infty)$</span> to a function <span class="math-container">$f$</span> such that for each <sp... | Jujustum | 928,305 | <p>In your example, <span class="math-container">$(f_n)_{n \in \mathbb{N}}$</span> doesn’t converge uniformly on <span class="math-container">$[0,+\infty[$</span>. Indeed, if you take <span class="math-container">$x_n=n^2$</span>, you have:
<span class="math-container">$$\forall n \in \mathbb{N}, e^{n-1}-e=|f_n(x_n)-f(... |
2,335,043 | <p>Let D be the ring of integers of a number field K. Let I be a non-zero ideal of D. Why D/I is finite?</p>
<p>Would you help me please? Thank you in advance.</p>
| Saketh Malyala | 250,220 | <p>We know that $\displaystyle e^{x}$ is a differentiable and continuous function.</p>
<p>So the limit is really dependent on $x^{x}$ value as it approaches $0$.</p>
<p>We can rewrite $x^{x}$ as $e^{x\ln x}$. We need to compute $\displaystyle \lim_{x \to 0}e^{x \ln x}=e^{\displaystyle \lim_{x \to 0}x \ln x}=e^{\displ... |
3,654,326 | <p>I have tried to solve the following equation in rational(<span class="math-container">$x,y,z$</span> are rational numbers) <span class="math-container">$$\frac{x^3}{y^3+z^3}+\frac{y^3}{x^3+z^3}+\frac{z^3}{y^3+x^3}=1$$</span> , I can't find such pairs of rational numbers <span class="math-container">$(x,y,z)$</span> ... | hamam_Abdallah | 369,188 | <p><strong>some idea</strong></p>
<p>If <span class="math-container">$(x,y,z) $</span> is a solution, then</p>
<p><span class="math-container">$ (x^3,y^3,z^3) $</span> will necessarily be a rational solution of</p>
<p><span class="math-container">$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=1$$</span></p>
|
1,512,377 | <p>As usual, this is probably a lot easier than I'm making it out to be. I just don't know... where to start. I know that if I have one generator, $g$, of a cyclic group of order $m$, then all the other generators are $g^k$, where $(k,n)=1$.</p>
<p>Is the order of $\mathbb{Z}_9 \times \mathbb{Z}_{10}$... 90? Does $(1,... | oxeimon | 36,152 | <p>The natural map $\mathbb{Z}_{90}\rightarrow \mathbb{Z}_9\times\mathbb{Z}_{10}$ given by sending elements of $\mathbb{Z}_{90}$ to their classes mod 9 and 10 is an isomorphism (this is the Chinese remainder theorem). Thus, obviously 1 is a generator of $\mathbb{Z}_{90}$, so its image in $\mathbb{Z}_9\times\mathbb{Z}_... |
20,580 | <p>This is a follow-up question to this <a href="https://mathoverflow.net/questions/20445/coend-computation">coend computation</a>. Here's an attempt at a slightly simpler computation:</p>
<blockquote>
<p><span class="math-container">$\int^{a \in A} \mbox{hom}_A(a,a)$</span></p>
</blockquote>
<p>This should be sim... | Reid Barton | 126,667 | <p>The step $\int_{a \in A} \mathrm{Set}(\mathrm{hom}_A(a, a), S) = \mathrm{Nat}(\mathrm{hom}_A(-,-),S)$ does not really make sense, because $a \mapsto \mathrm{hom}_A(a,a)$ is not a functor. And $\int^{a \in A} \mathrm{hom}_A(a,a)$ is not equal to $\mathrm{Ob}(A)$ in general. For instance, let $G$ be a group and let ... |
20,580 | <p>This is a follow-up question to this <a href="https://mathoverflow.net/questions/20445/coend-computation">coend computation</a>. Here's an attempt at a slightly simpler computation:</p>
<blockquote>
<p><span class="math-container">$\int^{a \in A} \mbox{hom}_A(a,a)$</span></p>
</blockquote>
<p>This should be sim... | Tom Leinster | 586 | <p>I agree with Reid's answer, but I want to add a bit more.</p>
<p>Putting Reid's calculation into a more general setting, if $A$ is <em>any</em> category then
$$
\int^{a \in A} \mathrm{hom}_A (a, a)
= (\mathrm{endomorphisms\ in\ } A)/\sim
$$
where $\sim$ is the (rather nontrivial) equivalence relation generated by ... |
3,997,676 | <p>Let <span class="math-container">$1\leq n\in \mathbb{N}$</span>, <span class="math-container">$E=\left (e_1, \ldots , e_n\right )$</span> the standardbasis of <span class="math-container">$\mathbb{R}^n$</span>, <span class="math-container">$B=\left (b_1, \ldots , b_n\right )$</span> a basis of <span class="math-cont... | poetasis | 546,655 | <p><span class="math-container">$$
f(x)=\begin{cases}
0 & x=0\\
\frac{x+1}{2} & x\text{ is odd}\\
-\frac{x}{2} & x\text{ is even}\wedge x\neq0
\end{cases}
\implies
f(n)=(-1)^{n+1}\bigg\lfloor\frac{n+1}{2} \bigg\rfloor
$$</span></p>
<p><span class="math-container">$$
f(0)=(-1)^1\bigg\lfloor\frac{1}{2}\bigg\r... |
2,934,472 | <p>Let <span class="math-container">$X_1,X_2,\dots$</span> denote a sequence of identically distributed, exchangeable, but not independent, Bernoulli<span class="math-container">$(p)$</span> random variables. If there exists <span class="math-container">$n>0$</span> such that</p>
<p><span class="math-container">$$
... | Mike Earnest | 177,399 | <p>Here is a large hint/outline.</p>
<p>A good method is to prove the contrapositive. Letting <span class="math-container">$S_{n}=\sum_{i=1}^n X_i$</span>, assume that <span class="math-container">$S_{n+1}$</span> is uniformly distributed over <span class="math-container">$\{0,1,\dots,n+1\}$</span>, and prove that <s... |
795,022 | <p>Clearly bounded does not imply closed. Where am I screwing up?</p>
<p>Let S denote an arbitrary bounded subset of the reals.
Since S is bounded, any sequence in S must also be bounded.
By Bolzano-Weierstrass theorem all sequences in S must have convergent subsequences. S is sequentially compact, and therefore compa... | Asaf Karagila | 622 | <p>You miss the point that the fact that $S$ includes a convergent sequence does not mean that it includes its limit.</p>
|
795,022 | <p>Clearly bounded does not imply closed. Where am I screwing up?</p>
<p>Let S denote an arbitrary bounded subset of the reals.
Since S is bounded, any sequence in S must also be bounded.
By Bolzano-Weierstrass theorem all sequences in S must have convergent subsequences. S is sequentially compact, and therefore compa... | Fermat | 83,272 | <p>A counterexample to this argument can be $S=(0,1]$ as a subset of $\mathbb R$.</p>
<p>$a_n =\frac{1}{n}$ is in S while its limit point(i.e., $0$) not.</p>
|
2,910,643 | <p>This question was in my maths paper . I tried to prove that the series converges but failed to do so. Any hint are welcomed .</p>
| rtybase | 22,583 | <p><strong>Just focusing on convergence</strong>, since $n+k\geq n+2, \forall k\geq2,\forall n\geq0$ we have:
$$0<\frac{1}{(n+1)(n+2)(n+3)...(n+k)}\leq\frac{1}{(n+1)(n+2)^{k-1}}$$
thus
$$\sum\limits_{k=1}\frac{1}{(n+1)(n+2)(n+3)...(n+k)}\leq
\frac{1}{n+1}+\sum\limits_{k=2}\frac{1}{(n+1)(n+2)^{k-1}}=\\
\frac{1}{n+1}... |
1,043,778 | <p>I have a list of $15$ people and I need to put them into 5 unique groups of $3$. They're going to meet once a month over the next $7$ months. I want everyone to be in different groups every month.</p>
<p>Rules:</p>
<ol>
<li><p>{1,2,3} is the same as {3,2,1}</p></li>
<li><p>By unique I mean that no two members appe... | Rory Daulton | 161,807 | <p>Yes.</p>
<p>Make 5 groups of 3 for the first month. Then rotate just the last member from each group into the previous group. That gives different groups for 5 months.</p>
<p>Then rotate the first member from each group once, and rotate the last members again. This gives 10 months.</p>
<p><strong>Here is what I m... |
4,336,337 | <p>I've come across this problem:
<span class="math-container">$$
S=\sum_{i=0}^{\infty}2^i \\
S= 1+ 2\sum_{i=0}^{\infty}2^i \\
S=1+2S \\
S=-1
$$</span>
I know that you cannot rearrange this series because it is not <a href="https://en.wikipedia.org/wiki/Absolute_convergence#Rearrangements_and_unconditional_convergence"... | Caleb Stanford | 68,107 | <p>Actually, this is a perfectly valid proof that <strong>if</strong> this series converges, <strong>then</strong> its value must be <span class="math-container">$-1$</span>. (As also pointed out by Thomas Andrews in the comments.)</p>
<p>It turns out, however, that the premise "this series converges" is fals... |
2,971,350 | <p>I need to show log <span class="math-container">$Fib_{n}$</span> is <span class="math-container">$\theta(n)$</span>
by the Fibonacci numbers defined as </p>
<p><span class="math-container">$$ F_n=F_{n-1}+F_{n-2}$$</span>
for
<span class="math-container">$$ n \geq 2 $$</span>
<span class="math-container">$ F_{0} =... | ALG | 602,944 | <p>You can use the <span class="math-container">$F$</span>-test in order to calculate statistically significance, given you hypotheis you have that
<span class="math-container">$$
F_n= \frac{\rho^2}{1-\rho^2}*(n-2)
$$</span></p>
<p>Hence you obtain the following <span class="math-container">$F_{100} = 0.0098$</span>,... |
247,755 | <p>We say two topologies $\tau$ and $\rho$ on $X$ are similar if the set of continuous functions $f:(X,\tau) \rightarrow (X,\tau)$ is the same as the set of continuous functions $f:(X,\rho)\rightarrow (X,\rho)$.</p>
<p>Does there exist a topology $\tau$ that is similar to the euclidean topology on $\mathbb R$?</p>
<... | Alexandre Eremenko | 25,510 | <p>This is a special case of much more general results surveyed in the book
MR0393330 Magill, K. D., Jr. A survey of semigroups of continuous selfmaps. Semigroup Forum 11 (1975/76), no. 3, 189–282. </p>
<p>For example, Theorem 2.3 of this book says that only the abstract semigroup structure
(with respect to compositio... |
2,861,825 | <p>The question specifies to solve this problem by maximizing $f(x,y)=xy(s-x-y)$ on an appropriate set. I admittedly do not have much of an attempt to post here. My initial thought was to try LaGrange multipliers, but I don't see how that would relate to the choice of "an appropriate set". Any advice on how to approach... | mrtaurho | 537,079 | <p>$$xyz~\leq~\left(\frac{s}{3}\right)^3$$</p>
<p>where $s=x+y+z$ the you can derivate </p>
<p>$$\begin{align}xyz~
&\leq~\left(\frac{x+y+z}{3}\right)^3\\ \sqrt[3]{xyz}&\leq~\frac{x+y+z}{3}\end{align}$$</p>
<p>where in the last line the L.H.S. is the geometric mean and the R.H.S. is the arithmetic mean and s... |
984,154 | <p>Let say my girlfriend makes $2000$ Euros per month and I make $3400$ Euros, let say all our living costs sum up to $1540$ Euros.
How can i calculate how much each of us must pay (off the $1540$ Euros) so as our participation is proportional to our income?</p>
<p>Thanks</p>
| KBusc | 121,813 | <p>Your net income per month is 5400 euros so since she make 2000 euros that is 37% of your total income so she should pay 37% of your total expenses. <span class="math-container">$1540*.37 = 570.37$</span> per month as her portion of the expenses your portion would be <span class="math-container">$3400/5400 = .63$</sp... |
1,140,339 | <p>Steps I took:</p>
<p>I drew out a circle with a radius of 1 and drew a trapezoid inscribed in the top portion of it. I outlined the rectangle within the trapezoid and the two right triangles within it. This allowed me to come to the conclusion, using the pythagorean theorem, that the height of the trapezoid is $h=\... | ganbustein | 198,393 | <p>When you went from $$\sqrt{4-x^2} - {{2x^2 - 4x} \over \sqrt{4-x^2}} = 0$$
to $$ {{4 - x^2 - 2x^2 - 4x} \over \sqrt{4-x^2}} = 0$$ you slipped a sign. The last term in the numerator should be $+4x$ instead of $-4x$.</p>
<p>Never mind. That error just reverses a previous erroneous sign. Your real error is in your fir... |
1,140,339 | <p>Steps I took:</p>
<p>I drew out a circle with a radius of 1 and drew a trapezoid inscribed in the top portion of it. I outlined the rectangle within the trapezoid and the two right triangles within it. This allowed me to come to the conclusion, using the pythagorean theorem, that the height of the trapezoid is $h=\... | B-Swizzle | 274,195 | <p>I also found the area equation for the trapezoid, but found $h$ in a different way.</p>
<p>The area of a trapezoid is $A=\dfrac{(b_1+b_2)H}{2}$. Base $1$ becomes $2$ units, Base $2$ is $2x$, and using the Pythagorean theorem you can find that $H=\sqrt{1-x^2}$. You then plug these values into the Area formula and fi... |
57,976 | <p>I'm trying to show:</p>
<p>If $K\subset \mathbb{R}^n$ is connected and $x\in \mathbb{R}^n$ then $x+K=\{x+y\in \mathbb{R}^n: y\in K\}$ is connected.</p>
<p>Thanks for your help.</p>
| Ilya | 5,887 | <p>Suppose that $x+K = A\cup B$ where $A,B$ are open (relatively) in $x+K$ and disjoint. Then $A-x$ and $B-x$ are still open (relatively in $K$) and disjoint but $\{A-x\}\cup \{B-x\} = K$ which contradicts with the fact that $K$ is connected.</p>
|
1,740,221 | <p>If I have $x^{\lfloor\frac{c}{a-b}\rfloor}$ is this equivalent to $(x^{\lfloor\frac{1}{a-b}\rfloor})^{\lfloor c \rfloor}$ if a,b,c are all integers and x is between 0 and 1?</p>
<p>I'm concerned with this expression in the context of a geometric series. Normally if we have $\sum_{i=0}^{\infty} r^{q}$ with |r| < ... | user64742 | 289,789 | <p>Unfortunately not. Floor is only guaranteed to be splittable around addition between an integer and a real number. Therefore, we are only able to do the following:</p>
<p>$$\lfloor \frac c {a-b}\rfloor = \lfloor \frac {c - ca + cb} {a-b}\rfloor + c$$</p>
<p>This is not particularly useful as an identity for your p... |
3,685,965 | <p>How do you solve a problem such as <span class="math-container">$$3^n\mod 10 = 9$$</span> for every integer solution for n? There is obviously a trivial solution for n=2 and I know from experimentation that the general solution would be <span class="math-container">$$n=2+4k$$</span> where k is an integer but how do ... | Claude Leibovici | 82,404 | <p>I suppose that the problem with Mathematica came from the fact that you computed first the antiderivative
<span class="math-container">$$I(x)=\int \frac{\log \left(1-x^2\right)}{(1-\beta x)^2}\,dx$$</span> which gives
<span class="math-container">$$ I(x)=\frac1\beta\left(\frac{\log \left(1-x^2\right)}{1-\beta x}-\... |
1,117,081 | <p>In number theory we have so-called explicit formula's in terms of the Riemann zeta zero's.
For instance to count the sum of the logarithms of the primes below some given integer.</p>
<p>(second Chebyshev Function)</p>
<p>Consider the floor function : <a href="http://mathworld.wolfram.com/FloorFunction.html" rel="n... | mjqxxxx | 5,546 | <p>Note that $\lfloor x \rfloor = x - \{x\}$, where $\{x \}$ (the fractional part of $x$) is periodic with period $1$, with a jump discontinuity at integer points. Let's try to write $\{x\}=f\left(\cot \pi x\right)$, since $\cot \pi x$ has the same property. We need $f(y)\rightarrow 0$ as $y\rightarrow \infty$, $f(y)... |
2,714,206 | <p><strong>Question: See title</strong></p>
<p><hr>
First off, what is K exactly? My guess: it's just the identity function with order $1$. If it's not the identity function, how would I calculate the order? </p>
<p>Thanks in advance.</p>
| Adriano | 76,987 | <p><strong>Hint:</strong> An arbitrary element $\alpha \in F$ has the form:
$$\alpha = \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
x_1 & 2 & x_3 & x_4 & 5 & x_6 & x_7 & x_8 & 9
\end{pmatrix}
$$</p>
<p>where $x_1, x_3, x_4, x_6, x_7, x_8$ is some ... |
2,714,206 | <p><strong>Question: See title</strong></p>
<p><hr>
First off, what is K exactly? My guess: it's just the identity function with order $1$. If it's not the identity function, how would I calculate the order? </p>
<p>Thanks in advance.</p>
| wrabbit | 369,791 | <p>It is all about cosets of $Sym(9)$. Lets look at easier example: </p>
<p>Take the set A = $\{f\in Sym(9)$ : $f(2)=3 \}$.</p>
<p>Actually A=$(23)Sym\{ 1,3,4,5,6,7,8,9\}=(23) \langle (13),(13456789)\rangle$.
Every set which is look like A is actually a coset.</p>
<p>Look at the set B= $\{f\in Sym(9)$ : $f(2)=2 \}... |
4,235,927 | <p>There is a game where you are asked to roll two fair six-sided dice. If the sum of the values equals 7, then win £21. However, must pay £5 to play each time both dice are rolled. Do you play this game?</p>
<p>One way to think about this is that getting a 7 comes with 1/6 chance, and to make money we need to get 7 at... | ryang | 21,813 | <p>Let <span class="math-container">$\$X$</span> be the nett winning in one game. Then <span class="math-container">$X$</span> has two possible values: <span class="math-container">$16$</span> or <span class="math-container">$-5.$</span></p>
<p>In one game, the probability of winning is
<span class="math-container">$$P... |
4,235,927 | <p>There is a game where you are asked to roll two fair six-sided dice. If the sum of the values equals 7, then win £21. However, must pay £5 to play each time both dice are rolled. Do you play this game?</p>
<p>One way to think about this is that getting a 7 comes with 1/6 chance, and to make money we need to get 7 at... | Misha Lavrov | 383,078 | <p>Your first approach is correct, at least approximately. (If you won £20 on a 7, then you'd need to roll 7's at a rate of <span class="math-container">$\frac14$</span> to break even. Since you win £21, you can afford a slightly lower rate, though not as low as <span class="math-container">$\frac16$</span>.)</p>
<p>Yo... |
3,671,077 | <p>Hey im first year math student I need to find the limit of <span class="math-container">$(a_n) =\frac{2}{n^{2}}\sum_{k=1}^{n}[kx]$</span></p>
<p>what I tried to do was <span class="math-container">$ \frac{2}{n^{2}}\sum_{k=1}^{n}[kx]=\sum_{k=1}^{n}\frac{2}{n^{2}}[kx]=\frac{2}{n^{2}}[kx]+\frac{2}{n^{2}}[kx]+...+\frac... | Ali Dursun | 577,325 | <p><span class="math-container">$$\frac{2}{n^2}\sum_{k=1}^{n}(kx) = \frac{2x}{n^2}\sum_{k=1}^{n}k = \frac{2x}{n^2}\frac{n(n+1)}{2}$$</span></p>
|
1,027,775 | <p>I'm considering the function </p>
<p>$$\frac{1}{e^{\frac{1}{z}}+2}$$</p>
<p>Clearly, it has a unique singularity at $z=0$. I feel like its an essential but I can't find the Laurent expansion or any other way of proving it... Can you help me?</p>
<p>Thank you very much, I usually see these, but this time I'm stuck... | Daniel Fischer | 83,702 | <blockquote>
<p>Clearly, it has a unique singularity at $z=0$.</p>
</blockquote>
<p>No, that is not the unique singularity. The function has poles at the zeros of</p>
<p>$$e^{\frac{1}{z}} + 2.$$</p>
<p>These poles accumulate at $0$, hence $0$ is not an isolated singularity of the function, and there is no Laurent ... |
211,625 | <p>The problem here is that the matrix
$$\begin{pmatrix}
2 & 3 \\
1 & -1 \\
\end{pmatrix}$$
is invertible but not unimodular and hence the elements $2x+3y$ and $x-y$ generate a free abelian group of rank 2 but still a proper subgroup of $\langle x,y\rangle$.
But the question was to prove that $2x+3y$ and $x-y... | i. m. soloveichik | 32,940 | <p>Certainly they are independent: If A(2x+3y)+B(x-y)=0 then $2A+B=0, 3A-B=0$ so $2A+3A=0$, and $A=0$, hence $B=0$.</p>
<p>Thus the generators $2x+3y, x-y$ span a free abelian group of rank 2.</p>
<p>No they are not a basis for $<x,y>$. Try to solve $x=A(2x+3y)+B(x−y)$, then $3A−B=0,2A+B=1$, so $5A=1$, which is... |
1,582,742 | <blockquote>
<p>I have this so easy question, yet it is giving me a headache</p>
<p>Consider the following two subsets of <span class="math-container">$R^3$</span></p>
<p><span class="math-container">$S_2 = [(1,2,3),(4,5,6),(7,8,9),(10,11,12)]$</span></p>
<p><span class="math-container">$S_3 = [(1,0,-1),(1,0,1)]$</span... | Alex Provost | 59,556 | <p>You're correct, dimension arguments can be useful. When they're not enough, go back to the definitions. To say that $S_2$ spans $\mathbb{R}^3$ means that any vector $(a,b,c) \in \mathbb{R}^3$ can be written as a linear combination $a_1(1,2,3) + a_2(4,5,6) + a_3(7,8,9) + a_4(10,11,12) = (a,b,c)$ for some scalars $a_1... |
778,253 | <p>A satellite system consists of 4 components and the system can function if at
least 2 components are working. If each component independently works with probability
0.8, what is the probability the system will function? </p>
<p>Can anyone help?</p>
| Anotome | 529,949 | <p>It is the chance that 2of4 OR 3of4 OR 4of4 components function:</p>
<p>${4 \choose 2}(4/5)^2(1/5)^2 + {4 \choose 3}(4/5)^3(1/5)^1 + {4 \choose 4}(4/5)^4(1/5)^0 = 0.9728$</p>
<p>notice the use of $P_{M;n}= {n \choose m}P^m(1-P)^{n-m}$</p>
<p>Similarly, it is $1-$(the chance that 3of4 OR 4of4 components do not func... |
3,403,464 | <p>Suppose <span class="math-container">$f\in L^2(X\times Y)$</span> where <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are finite measure spaces and we have a product measure <span class="math-container">$\mu$</span>.</p>
<p>Suppose further that <span class="math-container">$g\i... | Andre of Astora | 212,159 | <p>Let <span class="math-container">$\phi:\Gamma(TM)\longrightarrow \Omega^{n-1}(M)$</span> be the map <span class="math-container">$X\mapsto i_X{\omega}$</span>. We want to show that <span class="math-container">$\phi$</span> is an isomorphism. To show that it is surjective, locally suppose that the volume form is giv... |
3,187,856 | <p>Find all complex-valued <span class="math-container">$n \times n$</span> matrices <span class="math-container">$A$</span> such that <span class="math-container">$\det A = 0$</span> and <span class="math-container">$\text{rank}(AB) = \text{rank}(BA)$</span> for any <span class="math-container">$n \times n$</span> com... | Marc van Leeuwen | 18,880 | <p>Here is a reasoning to show just how difficult it is to have the ranks of <span class="math-container">$AB$</span> and <span class="math-container">$BA$</span> to be equal for all <span class="math-container">$B$</span>, or even just for all <span class="math-container">$B$</span> of rank<span class="math-container"... |
1,905,397 | <p>I want to know if the popular Sudoku puzzle is a Cayley table for a group. </p>
<p>Methods I've looked at: Someone I've spoken to told me they're not because counting the number of puzzle solutions against the number of tables with certain permutations of elements, rows and columns, the solutions are bigger than th... | Joel Cohen | 10,553 | <p>A Cayley table for a group can never be a sudoku. Assume you have a $9 \times 9$ Cayley table for a group of order $9$, and say your identity element is at index $1 \le i \le 9$. Then row $i$ and column $i$ are symmetric to each other because they correspond to multiplication with the identity. In particular, if you... |
273,020 | <p>I want to reproduce the effect in the figure below, approximating a surface with tangent planes,
<a href="https://i.stack.imgur.com/sfjgz.gif" rel="noreferrer"><img src="https://i.stack.imgur.com/sfjgz.gif" alt="enter image description here" /></a><br />
Now I can only use polygons to approximate inside the surface,... | Michael E2 | 4,999 | <p>To answer @cvgmt's comment, I would point out this image/animation works because the surface is of the form <span class="math-container">$z=a-bx^2-by^2$</span>. This implies the vertices of the tangent plane segments over squares coincide at the same height above the surface at every vertex.</p>
<pre><code>Manipulat... |
4,270,745 | <blockquote>
<p>I am trying to find this limit:
<span class="math-container">$$\lim_{n \to \infty}\left[\,\sqrt{\,2\,}\,\frac{\Gamma\left(n/2 + 1/2\right)}{\Gamma\left(n/2\right)} - \,\sqrt{\,n\,}\right]$$</span></p>
</blockquote>
<p>I know that
<span class="math-container">$$\frac{\sqrt{2}\Gamma(\frac{n+1}{2})}{\Gamma... | Claude Leibovici | 82,404 | <p><span class="math-container">$$f(n)=\sqrt{2}\,\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})} - \sqrt{n}=\sqrt{2}\,y- \sqrt{n}$$</span></p>
<p>Take logarithms
<span class="math-container">$$\log(y)=\log \left(\frac{\Gamma \left(\frac{n+1}{2}\right)}{\Gamma
\left(\frac{n}{2}\right)}\right)=\log \left(\Gamma \lef... |
3,372,488 | <p>Variance is</p>
<p><span class="math-container">$$\dfrac{\sum_{i=1}^{n}(x_i-\bar x)^2}{n-1}$$</span></p>
<p>But why square the difference? Why not cube it, or any other exponent?</p>
<p>Related <a href="https://math.stackexchange.com/questions/717339/why-is-variance-squared/717355">question</a>.</p>
| Daniel Robert-Nicoud | 60,713 | <p>You can also use other exponents, but then it is not called variance anymore. See <a href="https://en.wikipedia.org/wiki/Moment_(mathematics)" rel="nofollow noreferrer">this wikipedia page</a>, for example.</p>
<p>Variance comes up quite naturally in various contexts, and carries a lot of information about the dist... |
2,050,484 | <p>Let $f$ be $\in L[a,b]$ Assume for any subinterval $I \subset [a,b]$ we have $\int_I f \geq |I|$ show that $f \geq 1$ a.e on $[a,b]$.</p>
<p>I started with a proof by contradiction. Assume there exists a set $E$ that has positive measure where $f < 1$ on E. Since f $\in L[a,b]$ we can find $|f| < N$ a.e. I... | Martin Argerami | 22,857 | <p>We want to prove that if <span class="math-container">$X$</span> is a Banach space and <span class="math-container">$T^*\in B(X^*)$</span> is invertible, then <span class="math-container">$T$</span> is invertible in <span class="math-container">$B(X)$</span>.</p>
<p>We go through a few steps.</p>
<ul>
<li><p>Note th... |
2,350,358 | <blockquote>
<p>Let $(X_n)_{n\in\mathbb N}$ denote a simple, symmetric, random walk, defined by $X_0=0$ and $X_n = \sum\limits_{j=1}^n\xi_j$ where $(\xi_i)_{i \in \mathbb N}$ is i.i.d. with $\mathbb{P}(\xi_i = 1)= \mathbb{P}(\xi_i = -1) = \frac{1}{2}$. The maximum process of $(X_n)_{n\in\mathbb N}$ is defined by $M_... | saz | 36,150 | <p>If $\tau$ was a stopping time, then by the optional stopping theorem</p>
<p>$$\mathbb{E}(X_{n \wedge \tau})=0 \qquad \text{for all $n \geq 1$}.$$</p>
<p>As $|X_{n \wedge \tau}| \leq 100$ this would imply by the dominated convergence theorem that</p>
<p>$$\mathbb{E}(X_{\tau})=0.$$</p>
<p>This, however, contradict... |
3,141,004 | <p>I am just a high school student, so please excuse me if this question sound silly to you.</p>
<p>How do I graph this : <span class="math-container">$\dfrac{9}{x} - \dfrac{4}{y} = 8$</span></p>
<p>I could do trial and error, but is there a more systematic way that I can use to graph this equation.</p>
| Rory Daulton | 161,807 | <p>I'll assume you want to graph that equation without a graphing calculator or graphing program, and that you want to use only pre-calculus techniques.</p>
<p>So we want to transform the equation into something resembling one of the "standard equations" that we have seen before. First, we simplify the equation by rem... |
1,945,144 | <p>I'm trying to figure out which kind of matrices can be written as $A^{k_{1}}B^{k_{2}}...A^{k_{n}}B^{k_{n}}$ where </p>
<p>A=$\begin{pmatrix} 1&1\\0&1\end{pmatrix}$ and B=$\begin{pmatrix} 2&1\\1&1\end{pmatrix}$.</p>
<p>I did it for $C=\begin{pmatrix} 1&2\\0&1\end{pmatrix}$ and $D=\begin{pmat... | Mark | 310,244 | <p>Regarding $B^k$:</p>
<p>$B$ is Hermitian and thus diagonalizable. <a href="https://www.symbolab.com/solver/matrix-diagonalization-calculator/diagonalize%20%5Cbegin%7Bpmatrix%7D2%261%5C%5C%201%261%5Cend%7Bpmatrix%7D" rel="nofollow">See this for details</a>. So, we have that:</p>
<p>$$B = \overbrace{\begin{pmatrix... |
4,508,532 | <blockquote>
<p>Solve the equation <span class="math-container">$a \circ x \circ b = c$</span> in the group <span class="math-container">$S_4$</span> when <span class="math-container">$$a=\begin{pmatrix}1&2&3&4\\ 2&3&1&4\end{pmatrix}, b= \begin{pmatrix}1&2&3&4\\ 4&1&2&3\e... | zwim | 399,263 | <p>You can solve it algebraically using the inverse transformations:</p>
<p><span class="math-container">$x=a^{-1}\circ c\circ b^{-1}=\begin{pmatrix}1&2&3&4\\3&1&2&4\end{pmatrix}\circ\begin{pmatrix}1&2&3&4\\1&3&4&2\end{pmatrix}\circ\begin{pmatrix}1&2&3&4\\2&am... |
4,097,772 | <p>Consider <span class="math-container">$\mathbb{D}_{R} = \{z \in \mathbb{C}: |z| < R\}$</span>. For given <span class="math-container">$f \in \mathscr{O}(\mathbb{D}_{R})$</span> - space of holomorphic functions on the open disk, denote by <span class="math-container">$c_n$</span> the <span class="math-container">$... | reuns | 276,986 | <p>For fixed <span class="math-container">$R>r+\epsilon>r>\epsilon>0$</span></p>
<p>If <span class="math-container">$\sup_{n\ge 0} |c_n| r^n$</span> is small then <span class="math-container">$\sup_{|z|\le r-\epsilon}|f(z)|$</span> is small,</p>
<p>if
<span class="math-container">$\sup_{|z|\le r+\epsilon}|f... |
84,310 | <p>Is the following problem still open? Let $S$ be a non-empty set of prime numbers such that whenever $p,q\in S$, all the prime factors of $pq+1$ are also elements of $S$. Is $S$ infinite?</p>
| Community | -1 | <p>The answer is most probably "It is always the set of all prime numbers or empty". Indeed, assume that $S$ is not empty. First of all, $S$ must contain $2$ (if it contains $p>2$, then $p^2+1$ is even). Then it also contains $2\cdot 2+1=5, 2\cdot 5+1=11, 7\mid 5\cdot 11+1, 3\mid 2\cdot 7+1,$ etc. It looks like then... |
1,038,219 | <ol>
<li>(n) people put their name in a hat. </li>
<li>Each person picks a name out of the hat to buy a gift for. </li>
<li>If a person picks out themselves they put the name back into the hat.</li>
<li>If the last person can only pick themselves then the loop is invalid and either<br>
. start again<br>
. or step bac... | Mcav | 132,201 | <p>Let $P_N$ be the probability to obtain a perfect loop with $N$ participants.</p>
<p>$ A \rightarrow C \rightarrow B \rightarrow D \rightarrow A $<br>
$ A \rightarrow B \rightarrow D \rightarrow C \rightarrow A $<br>
$ C \rightarrow A \rightarrow B \rightarrow D \rightarrow C $<br>
are perferct loops.<br>
We can see... |
575,892 | <p>I know I need to choose an open cover and then show it has no finite subcover.
If I use $((-n,-n^2),(n,n^2)) \forall n \in \mathbb N$ does this work?</p>
| GA316 | 72,257 | <p>Hint : prove that the given set is not bounded.</p>
|
213,430 | <p>I' ve learned that the following examples can be used to decompose a
factor in this way</p>
<pre><code>x^5 - 5 x + 12 == (x - a) (a^4 + a^3 x + a^2 x^2 + a x^3 + x^4 - 1)
(x - a) (a^4 - (5 a^3)/8 + (7 a^2)/8 + 1/8 (5 a^2 - 12 a) +
1/8 (5 a^3 - 12 a^2) + ((5 a^4)/16 - a^3/8 + (7 a^2)/16 -
3/16 (5 a^2 - ... | Daniel Lichtblau | 51 | <p>Compute the five roots of that quintic, and set <code>a</code> to any one of them. </p>
<pre><code>qpoly = x^5 - 5 x + 12;
rt = First[x /. Solve[qpoly == 0, x]];
</code></pre>
<p>Then factor using that <code>a</code> as the extension.</p>
<pre><code>fax = Factor[qpoly, Extension -> rt] /. rt -> a
(* Out[33... |
213,430 | <p>I' ve learned that the following examples can be used to decompose a
factor in this way</p>
<pre><code>x^5 - 5 x + 12 == (x - a) (a^4 + a^3 x + a^2 x^2 + a x^3 + x^4 - 1)
(x - a) (a^4 - (5 a^3)/8 + (7 a^2)/8 + 1/8 (5 a^2 - 12 a) +
1/8 (5 a^3 - 12 a^2) + ((5 a^4)/16 - a^3/8 + (7 a^2)/16 -
3/16 (5 a^2 - ... | Kendall | 46,943 | <p>Ok I think I know what is trying to be achieved. For illustrative purposes, I am going to use a smaller polynomial <span class="math-container">$x^2+3x+2=0$</span> which we know can be factored into <span class="math-container">$(x+1)(x+2)$</span>. It is desired that we can rather factorise this polynomial into some... |
1,177,111 | <p>Let $(a_n)$ be defined by $a_1=1$, $a_{n+1}=a_n+{1\over a_n}$. Prove there exists $\lim_\limits{n\to \infty}{a_n}$ and find $\lim_\limits{n\to \infty}{a_n\over n}$. </p>
<p>I tried many things but they wouldn't work. I would appreciate your help.</p>
| Community | -1 | <p>Actually, the limit $\lim a_n$ does not exist. For, assume it does though, and let it equal $\ell$. Then, $a_{n + 1} = a_n + 1/a_n$ yields $\ell = \ell + 1 / \ell$, a contradiction.</p>
<p>We now show that $\lim a_n/n$ exists. Since for every $n$ we know that $0 < a_n < n$, we get that $0 < a_n/n < 1$. ... |
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