qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,870,305 | <p>I am trying to calculate the Jacobian of a function that has quaternions and 3D points in it. I refer to quaternions as <span class="math-container">$q$</span> and 3D points as <span class="math-container">$p$</span>
<span class="math-container">$$h_1(q)=A C(q)p $$</span>
<span class="math-container">$$h_2(q)=q_1\ot... | JustThinking | 356,245 | <p>Quaternion derivatives are not that straight forward. Usually a direction is required (see <a href="https://en.wikipedia.org/wiki/Quaternionic_analysis" rel="noreferrer">here</a>).</p>
<p>I'm assuming your thought process behind $H_2$ was something like</p>
<p>$$ h_2 = q_1 \otimes q \otimes q_2 = q_1 \otimes (q \... |
1,217,771 | <p>Let $x,y \in R$.
If $0 \leq y < x$ for all $x > 0$, then $y=0$.</p>
<p>Proof by contradiction: </p>
<p>Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$".
Subtract $x$ from each part of the inequality to get,
$0-x \leq y-x < 0$
Th... | abel | 9,252 | <p>$$\begin{align}(x + x^{1/2})^{1/2} &= x^{1/2} + \frac 12 x^{-1/2}x^{1/2} -\frac 1 8 x^{-3/2}x+\cdots\\&=\sqrt x+\frac 12-\frac1{8\sqrt x} +\cdots\\
(x-1)^{1/2} &=\sqrt x -\frac 1{2\sqrt x}+\cdots \end{align}$$</p>
<p>therefore $$(x + x^{1/2})^{1/2} - (x-1)^{1/2}=\frac 1 2 + \frac 3{8\sqrt x} + \cdots \r... |
138,866 | <p>I have data in a csv file. The first row has labels, and the first column, too.</p>
<pre><code>Datos = Import["C:\\Users\\jodom\\Desktop\\Data.csv"]
</code></pre>
<p>Tha data in the csv file is that:</p>
<pre><code>{{"No", "Vol", "Vel"}, {1, 500, 45}, {2, 700, 67}, {3, 350, 87}, {4,
123, 23}, {5, 587, 45}, {6, 4... | yode | 21,532 | <p>As the OP,you have a <code>.csv</code> file like:</p>
<p><img src="https://i.stack.imgur.com/oFBJo.png" alt=""> </p>
<p>Then you just use <code>SemanticImport</code></p>
<pre><code>SemanticImport["Data.csv"]
</code></pre>
<p><img src="https://i.stack.imgur.com/cyOAS.png" alt=""> </p>
<hr>
<p>Of course,if you h... |
150,084 | <p>I need to say whether or not $f_n(x)=n\left(\sqrt{x+\frac{1}{n}}-\sqrt{x}\right)$ is uniformly convergent on $(0,\infty)$.</p>
<p>I've found that the function is locally convergent to $f(x)=\frac{1}{2\sqrt{x}}$ and was trying to find $\sup{|f_n(x)-f(x)|}$.</p>
<p>I got the derivative $f_n'(x)= \frac{2nx\left(x-\sq... | Jim Belk | 1,726 | <p>Note that each of the functions $\displaystyle f_n(x) = n\biggl(\sqrt{x+\frac{1}{n}}-\sqrt{x}\biggr)$ is bounded on $(0,\infty)$, with $f_n(x) \leq \sqrt{n}$. Since $\displaystyle f(x) = \frac{1}{2\sqrt{x}}$ is unbounded on $(0,\infty)$, the sequence $\{f_n\}_{n=1}^\infty$ does not converge uniformly to $f$.</p>
|
3,072,142 | <p>Isn't <span class="math-container">$\mathbb Q[X]/(X^2+1)\cong \mathbb Q[i]$</span> wrong and should be <span class="math-container">$\mathbb Q[X]/(X^2+1)\cong \mathbb Q(i)$</span> ?</p>
<p>Indeed, <span class="math-container">$\mathbb Q[X]/(X^2+1)$</span> is a field whereas <span class="math-container">$\mathbb Q[i... | BigbearZzz | 231,327 | <p>Integral is linear, i.e. <span class="math-container">$$\int \alpha f d\lambda = \alpha \int f d\lambda$$</span> for any <span class="math-container">$\alpha\in\Bbb R$</span>. In your case you also need to use <span class="math-container">$\int_0^1 1 d\lambda=1$</span>.</p>
|
330,526 | <p>Let <span class="math-container">$\tau>0$</span>, and let <span class="math-container">$T\in \mathcal{D}'(\mathbb{R})$</span> be a <span class="math-container">$\tau$</span>-periodic distribution (that is,
<span class="math-container">$
\langle T, \varphi(\cdot+\tau)\rangle= \langle T,\varphi\rangle
$</span>
f... | user44191 | 44,191 | <p><span class="math-container">$\DeclareMathOperator\deg{deg}\DeclareMathOperator\dim{dim}\DeclareMathOperator\span{span}$</span>I have at least a few things that may help. It's not a full answer, but it doesn't fit in a comment, either.</p>
<p>We start by completely ignoring the first equation and dealing with the o... |
2,460,195 | <p>I had the following question:</p>
<p>Three actors are to be chosen out of five — Jack, Steve, Elad, Suzy, and Ali. What is the probability that Jack and Steve would be chosen, but Suzy would be left out?</p>
<p>The answer given was:
Total Number of actors = $5$;
Since Jack and Steve need to be in the selection and... | Satish Ramanathan | 99,745 | <p>we can use generating function to find the number of solution for the problem that you have stated.</p>
<p>That is you will have to find the coefficient of $x^{16}$ in the following products of g.f</p>
<p>$(1+x^2+x^4+..)(1+x+x^2+x^3+...)^2$</p>
<p>The number of solutions is $\boxed{81}$</p>
|
2,682,093 | <blockquote>
<p>Let $\circ$ be an inequality.</p>
<p>Prove $|x| \circ a \equiv -a \circ x \circ a$.</p>
</blockquote>
<p>If $x$ is positive, then $|x| \circ a = x \circ a$.</p>
<p>If $x$ is negative, then $|x| \circ a = x \circ a$ ?</p>
| Fred | 380,717 | <p>I suppose that you have to show: for $x,a \in \mathbb R$ and $a \ge 0$:</p>
<p>$|x| \le a$ implies $-a \le x \le a$.</p>
<p>Case 1: $x \ge 0$. Then we get from $|x| \le a$ that $0 \le x \le a$ and hence $-a \le x \le a$.</p>
<p>Case 2: $x < 0$. Then we get from $|x| \le a$ that $- x \le a$ and hence $-a \le ... |
670,570 | <p>Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1.</p>
<p>Question: how many gallons of yellow paint, and how many gallons or red paint, to make two gallons of orange paint?</p>
<pre><code>2y + r = 2
2y + y = 2
3y = 2
y = 2... | mathematics2x2life | 79,043 | <p>Try plain polynomial long division. If it gives you trouble, try it with a smaller example like $x^4-2x^3+4x^2-x+7$ divided by $x^2-3$. </p>
<p>In your original problem, you should find the quotient to be....</p>
<blockquote class="spoiler">
<p> $x^{40}-x^{39}+x^{35}-x^{34}+x^{30}-x^{28}+x^{25}-x^{23}+x^{20}-x^{... |
670,570 | <p>Four gallons of yellow paint plus two gallons of red paint make orange paint. I assume this makes six gallons. So the ratio is 4:2, or 2:1.</p>
<p>Question: how many gallons of yellow paint, and how many gallons or red paint, to make two gallons of orange paint?</p>
<pre><code>2y + r = 2
2y + y = 2
3y = 2
y = 2... | lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Both are in Geometric Series</p>
<p>$$1+x+x^2+x^3+x^4=\frac{1-x^5}{1-x}$$ and $$1+x^{11}+x^{22}+x^{33}+x^{44}=\frac{1-x^{55}}{1-x^{11}}$$</p>
<p>So, $$\frac{1+x^{11}+x^{22}+x^{33}+x^{44}}{1+x+x^2+x^3+x^4}=\frac{(1-x^{55})(1-x)}{(1-x^{11})(1-x^5)}$$</p>
<p>Now using <a href="https://math.stackexchang... |
555,051 | <p>I am having trouble understanding the following homework question,</p>
<p>Let $H=\{0,\pm 3, \pm6, \pm9,\ldots\}$ Find all the left cosets of $H$ in $\Bbb Z$.</p>
<p>I know the answer is $H$, $1+H$, and $2+H$ but I am having difficulty understanding why.</p>
<p>Thank you!</p>
| anon | 11,763 | <p>Consider the three cosets $\color{Blue}{3\Bbb Z}$, $\color{Green}{1+3\Bbb Z}$, $\color{Red}{2+3\Bbb Z}$ as they sit inside $\color{Black}{\Bbb Z}$ itself:</p>
<p>$~~~$ <img src="https://i.stack.imgur.com/Sb5CS.png" alt="mod 3"></p>
<p>Do you see any numbers that were skipped? Notice that the pattern repeats. How c... |
1,953,628 | <p>0 choices for the 1st person.
17 choices for the 2nd person (must exclude 1st and his/her two neighbours) </p>
<p>For 2 of these choices of 2nd person, there is one shared neighbour, so 15 remaining choices. (e.g. if they are numbered 1 to 20 in a circle, 1st person is #1, 2nd is #3, then people 20,1,2,3,4 are exc... | true blue anil | 22,388 | <p>Here is another way:<br>
Form three blocks with a non-chosen $(N)$ clockwise of a chosen $(C),\;e.g. \boxed {CN}$</p>
<p>We now have $17$ entities, viz. $3$ blocks and $14$ others</p>
<p>We can place the blocks in $\binom{17}3$ ways, fulfilling the "non-neighbour" criterion,</p>
<p><em>but since in the process we... |
2,028,799 | <p>I've been trying to get this limit for hours. Can someone help me, please?<br>
The solution manual says it's 0 but I can't get there. I tried to use $\lim_{h\to0} {h-\cos(h)\over h} = 0$.</p>
<p>$$\lim_{x\to 0}\frac{x\cos(1/x)}{x-\sqrt{x}}.$$<br>
Thank you.</p>
| StackTD | 159,845 | <blockquote>
<p>$$\lim_{x\to 0}\frac{x\cos(1/x)}{x-\sqrt{x}}.$$ </p>
</blockquote>
<p>Due to the $\sqrt{x}$, clearly $x>0$. Substitute $t = \frac{1}{x}$, then $t \to +\infty$ as $x \to 0^+$ and you have:</p>
<p>$$\lim_{x\to 0}\frac{x\cos\left(\tfrac{1}{x}\right)}{x-\sqrt{x}}
= \lim_{t\to +\infty}\frac{\tfrac{1}... |
1,730,445 | <p>I am checking whether the limit is true or not. $z$ is complex number
\begin{equation}
\lim_{z \rightarrow 0} z\sin(\frac{1}{z})=0
\end{equation}
I found Laurent series of $z\sin(\frac{1}{z})$ which is $\sum_{n=0}^{\infty} (\frac{1}{z})^{2n}\frac{1}{(2n+1)!}(-1)^n$ </p>
<p>the series is not defined $z =0$.</p>
... | Julián Aguirre | 4,791 | <p>The list does not exist, since $\displaystyle z\sin\frac1z$ has an essential singularity at $z=0$. That is, there an infinite number of non-zero negative powers in the Laurent series around $z=0$.</p>
|
3,709,331 | <p>How can I integrate
<span class="math-container">$$
\int \frac{x\,dx}{(a-bx^2)^2}
$$</span> I've tried to use partial fraction decomposition, but I'm getting six equations for four variables, and they don't give uniform answers.</p>
| Vishu | 751,311 | <p>Substitute <span class="math-container">$x=\frac{\sqrt a}{\sqrt b} \sin \theta \implies dx=\frac{\sqrt a}{\sqrt b}\cos \theta d\theta$</span> to get <span class="math-container">$$\frac{1}{ab}\int\tan \theta \cdot \sec^2\theta \ d\theta =\frac {1}{ab}\int\tan\theta \ d(\tan\theta) = \frac {1}{2ab}\tan^2\theta+C\\=\f... |
2,442,297 | <blockquote>
<p>Ryan has been given a salary increase of $7.39\%$. The salary increase is for the value of €$4231$.</p>
<p>His salary is now $x$. Solve for $x$.</p>
</blockquote>
<p>My head is saying </p>
<p>$$\begin{align}
4231 / 7.39 &= 572 \\
572 * 100 &= 57,200
\end{align}$$</p>
<p>is not correct,... | alexjo | 103,399 | <p>$$
y\times 7.39\%= 4,231\quad\Longrightarrow \quad y=\frac{ 4,231}{7.39\%}=\frac{ 4,231}{0.0739}\approx 57,253\\
x=y(1+7.39\%)=57,2534+4,231=61,484
$$</p>
|
2,294,991 | <p>I am trying to find the limit as $x\to 8$ of the following function. What follows is the function and then the work I've done on it. </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}$$</p>
<hr>
<p>\begin{align}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8} &= \frac{\frac{1}{\sqrt{x +1... | Amarildo | 307,377 | <p>$$
\begin{aligned}
\lim\limits_{x \to 8} \frac{\frac{1}{\sqrt{x +1}} - \frac 13}{x-8}
& = \lim _{t\to 0}\left(\frac{\frac{1}{\sqrt{\left(t+8\right)\:+1}}\:-\:\frac{1}{3}}{\left(t+8\right)-8}\right)
\\& = \lim _{t\to 0}\left(\frac{\left(3-\sqrt{t+9}\right)\sqrt{t+9}}{3t^2+27t}\right)
\\& = \lim _{t\to \:0... |
2,294,991 | <p>I am trying to find the limit as $x\to 8$ of the following function. What follows is the function and then the work I've done on it. </p>
<p>$$ \lim_{x\to 8}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8}$$</p>
<hr>
<p>\begin{align}\frac{\frac{1}{\sqrt{x +1}} - \frac{1}{3}} {x-8} &= \frac{\frac{1}{\sqrt{x +1... | zhw. | 228,045 | <p>By definition, this is $f'(8)$ for $f(x) = (x+1)^{-1/2}.$</p>
|
2,316,561 | <p>How to evaluate the integral $$\frac{1}{2 \pi i}
\int \limits_{c-i \infty}^{c+i \infty} \frac{ds}{s(1-q^{1-s})}\text{?}$$ I tried with Perron's formula but I couldn't solve it. The result of the integral is $\frac{1}{2}$. Can someone help please?!</p>
| reuns | 276,986 | <p>For some $q > 1$ let</p>
<p>$$f(x) = \sum_{n \ge 0,q^n < e^x} q^n$$
For $x = n \log q$ we take the mean value of the left and right limit, that is
$$f(n \log q) = \frac{f(\epsilon+n\log q)+f(-\epsilon+n\log q)}{2}$$</p>
<p>Thus $f(0) = 1/2$</p>
<hr>
<p>For $\Re(s ) > 1$
$$F(s) = \frac{1}{1-q^{1-s}} = \s... |
4,111,835 | <p>So I was given the following prompt:</p>
<blockquote>
<p>When <span class="math-container">$x=−2$</span>, for what values of p does the series converge?
<span class="math-container">$$\sum_{n=1}^\infty\left(\frac{(-1)^{n+1}(x-3)^n}{5^n\cdot n^p}\right)$$</span></p>
</blockquote>
<p>I ended up working out this proble... | Community | -1 | <p>It diverges for <span class="math-container">$p=1$</span> (harmonic series).</p>
<p>It simplifies to <span class="math-container">$-1\cdot\sum_{n=1}^\infty\dfrac1{n^p}$</span>. So it converges for <span class="math-container">$p\gt1$</span> (p-series).</p>
|
3,758,635 | <p>I'm trying to prove <span class="math-container">$n+\left(-1\right)^n\ge \dfrac{n}{2}$</span> is true for all natural numbers <span class="math-container">$n \ge 2$</span> via induction. The base case is trivial as
<span class="math-container">$$2+(-1)^2 \ge \frac{1}{2}(2)$$</span>
<span class="math-container">$$3 \... | Paul Frost | 349,785 | <p>You know the <em>Neumann series</em> <span class="math-container">$\sum_{i=0}^\infty T^i$</span> of <span class="math-container">$T$</span>. It is well-known that if the Neumann series of <span class="math-container">$T$</span> converges, then <span class="math-container">$Id – T$</span> is invertible and its inver... |
263,359 | <p>Let $H$ be a monoid, and denote by $H^\times$ and $\mathcal A(H)$, respectively, the <em>set of units</em> (or <em>invertible elements</em>) and the <em>set of atoms</em> (or <em>irreducible elements</em>) of $H$ (an element $a \in H$ is an atom if $a \notin H^\times$ and $a = xy$ for some $x, y \in H$ implies $x \i... | Benjamin Steinberg | 15,934 | <p>The answer seems yes if I understood the question. Take $G$ to be any commutative group and $N$ be the free semigroup on one-generator $x$ (so isomorphic to the positive natural numbers under $+$, but I want to use multiplicative notation). Let $H=G\cup N$ where the operation on $G$ and $N$ are their original oper... |
3,503,518 | <blockquote>
<p>Determine if <span class="math-container">$ \sum_{n=2}^{\infty} \frac{(\sin{n})\sum\limits_{k=1}^{n} \frac{1}{k}}{(\log n)^2}$</span> is convergent or divergent.</p>
</blockquote>
<hr>
<p>[My attempt]</p>
<p>It seems like Dirichlet test, so I tried to show that <span class="math-container">$a_n := ... | SL_MathGuy | 730,041 | <p>Consider <span class="math-container">$$a_n = \dfrac{\sum _{k=1}^{n}1/k}{[log (n)]^2}.$$</span> It can be proved that,</p>
<p><span class="math-container">$$\sum _{k=1}^{n}1/k = log(n) + \gamma + o(1)$$</span>, where <span class="math-container">$\gamma$</span>- Euler's constant.
Using this, we can simplify <span ... |
381,093 | <p>I would like to prove Chebyshev's sum inequality, which states that:</p>
<p>If <span class="math-container">$a_1\geq a_2\geq \cdots \geq a_n$</span> and <span class="math-container">$b_1\geq b_2\geq \cdots \geq b_n$</span>, then<br />
<span class="math-container">$$
\frac{1}{n}\sum_{k=1}^n a_kb_k\geq \left(\frac{1}{... | Iosif Pinelis | 36,721 | <p>A more general inequality is
<span class="math-container">$$Ef(X)g(X)\ge Ef(X)\,Eg(X),\tag{1}$$</span>
where <span class="math-container">$f$</span> and <span class="math-container">$g$</span> are nondecreasing (say bounded) functions from <span class="math-container">$\mathbb R$</span> to <span class="math-containe... |
417,131 | <p>Does limit $\frac{4xy}{\sqrt{x^2+y^2}}$ as $(x,y) \to (0,0)$ exist or not?</p>
<p>To remove the root, I squared the whole equation and I get
limit $\frac{16x^2y^2}{\sqrt{x^2+y^2}}$. Then I dont know how to continue working it. Can anyone help to do this questions for me please.</p>
<p>Thank you for your effort in ... | DonAntonio | 31,254 | <p>An idea:</p>
<p>$$|x-y|\le|x|+|y|\implies \frac{\sin^2(x-y)}{|x|+|y|}\le\frac{\sin^2(x-y)}{|x-y|}=|\sin(x-y)|\frac{|\sin(x-y)|}{|x-y|}$$</p>
<p>Now just remember that $\,\displaystyle{\lim_{t\to 0}\frac{\sin t}t=1}\;$ and stuff...</p>
|
4,501,927 | <p>Hi working again on the gamma function I find that :</p>
<p>let <span class="math-container">$0<x\leq 1$</span> and <span class="math-container">$1\leq k\leq \infty$</span> then define :</p>
<p><span class="math-container">$$f(x)=(x!)!,g(x)=\left(\left(x!\right)!\right)^{\frac{1}{e^{x^{k}}}}$$</span></p>
<p>Then ... | Claude Leibovici | 82,404 | <p>As I wrote in comment, the same work can be done for
<span class="math-container">$$g(x)=(x!)!$$</span>
<span class="math-container">$$g(x)=g(a)+\Gamma (a+1) \Gamma (a!+1)\sum_{n=2}^\infty \frac {d_n} {n!} \, (x-a)^n$$</span> The first coefficients being
<span class="math-container">$$\color{red}{d_2}=\psi ^{(1)}(a... |
1,423,491 | <p>Suppose you are flipping a coin with probability of Heads being 0.4931 and Tails being 0.5069</p>
<p>Can someone please tell me what is the probability of hitting 6 and 7 tails in a row in 24 tries? how about 44 tries?</p>
<p>OK. I have been asked to edit my question!
Here is the story:</p>
<p>The game "Baccarat"... | Community | -1 | <p>For $k$ times in $n$ trials: $$p = {n \choose k} (0.4139)^{n-k}(0.5069)^k$$</p>
<p>These are called Bernoulli trials, and this probability distribution is called the binomial distribution. </p>
|
1,423,491 | <p>Suppose you are flipping a coin with probability of Heads being 0.4931 and Tails being 0.5069</p>
<p>Can someone please tell me what is the probability of hitting 6 and 7 tails in a row in 24 tries? how about 44 tries?</p>
<p>OK. I have been asked to edit my question!
Here is the story:</p>
<p>The game "Baccarat"... | Krish Wadhwana | 310,698 | <p>In 24 flips, probability of getting 6 tails in a row:
You can either get all tails in flips 1,2,3,4,5,6 or flips 2,3,4,5,6,7 or flips 3,4,5,6,7,8 or flips 4,5,6,7,8,9 or flips 5,6,7,8,9,10 to flips 19,20,21,22,23,24. If you could all of the possibilities you get 19. So you choose one case from 19 which is $${19 \cho... |
3,678,908 | <p>Let</p>
<p><span class="math-container">$$
V = \{ax^3+bx^2+cx+d|a,b,c,d \in Z_7\}
$$</span></p>
<p>Let the subspace <span class="math-container">$U$</span> Be a subspace of <span class="math-container">$V$</span>:</p>
<p><span class="math-container">$$
U = \{p(x) \in V | p(3)=p(5)=0\}
$$</span></p>
<p>The answer... | Barry Cipra | 86,747 | <p>Rewrite the equation as <span class="math-container">$|2x+3|=|x-1|+6$</span> and sketch the graphs of <span class="math-container">$y=|2x+3|$</span> and <span class="math-container">$y=|x-1|+6$</span>. You will see that the two branches of the steeper V shape of the first graph each intersect once with the left (dow... |
234,483 | <p>The question is, "Give a description of each of the congruence classes modulo 6."</p>
<p>Well, I began saying that we have a relation, $R$, on the set $Z$, or, $R \subset Z \times Z$, where $x,y \in Z$. The relation would then be $R=\{(x,y)|x \equiv y~(mod~6)\}$</p>
<p>Then, $[n]_6 =\{x \in Z|~x \equiv n~(mod~6)\}... | amWhy | 9,003 | <p>Take any $n \in \mathbb{Z}.$</p>
<p>Then for each such $n$ there is an integer $k$ such that one of the following equations are satisfied. $$n=6k + 0$$ $$n=6k+1$$ $$n=6k+2$$ $$n=6k+3$$ $$n=6k+4$$ </p>
<p>$$n=6k+5$$</p>
<p>Can any integer satisfy <em>more</em> than one of the above equalities?</p>
<p>Can any inte... |
3,817,092 | <p>How to show that we may replace the property of a norm <span class="math-container">$$||x||=0 \iff x=0$$</span> by <span class="math-container">$$||x||=0 \implies x = 0$$</span> without altering the concept
of a norm.</p>
| Jason DeVito | 331 | <p>Qiaochu has given a very nice answer, but I wanted to add another large class of examples.</p>
<blockquote>
<p>Suppose <span class="math-container">$G$</span> is a simply connected closed Lie group, and <span class="math-container">$H$</span> is a connected closed Lie group. Suppose <span class="math-container">$H$... |
557,146 | <p>I have a clock time in form of HH.MM.SS (hours, minutes, sec), I need to create a way to convert it to one number so that I can determent which time is bigger.
for example
00.00.00 is the smallest
23.59.59 is the biggest</p>
<p>11.03.50 > 11.02.57
etc..</p>
<p>I thought to do the following</p>
<p>hours*10000+min... | Rocket Man | 85,096 | <p>Just convert the total time to seconds. The most seconds is clearly the "largest" time.</p>
<p>$$
T=3600h+60m+s
$$</p>
|
3,987,552 | <p>I need to prove or refute a property about a sequence of numbers.
Here is what is given to me:</p>
<p>Sequence (<span class="math-container">$a_1,a_2,...,a_k,a_{k+1},a_{k+2}$</span>) containing <span class="math-container">$k+2$</span> numbers. Every number <span class="math-container">$0 < a_i \leq M, i=1,...,k+... | Measure me | 854,564 | <p>Assuming that given <span class="math-container">$k$</span>, <span class="math-container">$N$</span> is selected such that
<span class="math-container">$$\sum_{i=1}^{k+2} a_i = N,$$</span>
I claim it is possible <span class="math-container">$ \iff k\ge 3$</span></p>
<p>First of all we see the cases <span class="math... |
3,257,387 | <p>denote <span class="math-container">$(a_n)$</span> as a sequence. If the consecutive difference tend to <span class="math-container">$0$</span> as <span class="math-container">$n$</span> tends to infinity, do it mean that <span class="math-container">$(a_n)$</span> converges? I feel like it should mean so but I am s... | Kitter Catter | 166,001 | <p>Consider <span class="math-container">$a_n=\log(n)$</span> if you take the limit for large <span class="math-container">$n$</span> then I think it is fairly clear that <span class="math-container">$a_n$</span> diverges. In that limit <span class="math-container">$a_n - a_{n-1} =\log\frac{n}{n-1}\rightarrow 0$</span>... |
17,259 | <p>I am using <em>Mathematica</em> to go through the examples and exercises on the book <em>Modern Control Systems</em>, 6th edition by Dorf. On page 605, there is a table (Table 8.5) with the Bode plot for several transfer functions. In what follows there is a piece of code that attempts to build the very same table... | James Cunnane | 4,882 | <p>No error seen in those phase plots. </p>
<p>For a transfer function of the form K/s^n, i.e. n poles at the origin, we expect constant phase of (-90 * n) degrees, plus or minus some integer multiple of 360 degrees - which is exactly what your Mathematica plots show. </p>
|
820,490 | <p>I am studying pre-calculus mathematics at the moment, and I need help in verifying if $\sin (\theta)$ and $\cos (\theta)$ are functions? I want to demonstrate that for any angle $\theta$ that there is only one associated value of $\sin (\theta)$ and $\cos (\theta)$. How do I go about showing this?</p>
| Lutz Lehmann | 115,115 | <p>I'm afraid that from a pre-calculus point of view, there is no consistent definition of the functional relationships that allows to discuss them as real functions (real valued in one real variable).</p>
<p>The big gap to be closed is the notion of an angle. In geometry, an angle is defined by a pair of rays from th... |
99,750 | <p>Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).</p>
<p>I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd ... | Community | -1 | <p>Personally I like the definition in <a href="https://arxiv.org/abs/math/0203010" rel="nofollow noreferrer">Barton, Sudbery</a> paper (thank you, Bruce for adding the reference):</p>
<p><a href="http://www.ams.org/mathscinet-getitem?mr=2020553" rel="nofollow noreferrer">MR2020553 (2005b:17017) Barton, C. H. ; Sudb... |
2,903,105 | <p>A Fourier transform of function R into function Q is defined as:
$$Q(\underline{k}) = \int_{}^{}R(\underline{x}) e^{-i\underline{k}·\underline{x}} \mathrm{d}\underline{x}.$$
where I've underlined $\underline{x}$ and $\underline{k}$ to denote that they are a set of vectors.</p>
<p>Suppose that $R(\underline{x})$ is ... | md2perpe | 168,433 | <p>A somewhat physical explanation is this:<br>
Dependence on coordinate differences only implies translation invariance, which by <a href="https://en.wikipedia.org/wiki/Noether%27s_theorem" rel="nofollow noreferrer">Noether's theorem</a> implies conservation of momentum, which in turn is represented as $\delta(\sum_k ... |
2,829,362 | <p>Suppose the random variable $T$ which represents the time needed for one person to travel from city A to city B ( in minutes). $T$ is normally distributed with mean $60$ minutes and variance $20$ minutes. Also, suppose $600$ people depart at the exact same time with each of their travel time being independent from o... | InsideOut | 235,392 | <p>You cannot apply the fundamental theorem of calculus since the function <span class="math-container">$$\frac{1}{x}$$</span> is not defined on the interval <span class="math-container">$[-1,1]$</span>. Split the integral in the following way
<span class="math-container">$$\int_0^1\frac{dx}{x}+\int_{-1}^0\frac{dx}{x}$... |
3,752,676 | <p>what is <span class="math-container">$P(P(P(333^{333})))$</span>, where P is sum of digit of a number. for an example <span class="math-container">$P(35)=3+5=8$</span></p>
<p>a)18</p>
<p>b)9</p>
<p>c)33</p>
<p>d)333</p>
<p>f)5</p>
<p>I tried to find this but I couldn't. I started to find a pattern for an example the... | Mastrem | 253,433 | <p>Let <span class="math-container">$G:=\mathbb{Z}_4\oplus\mathbb{Z}_{12}$</span> and <span class="math-container">$H\subset G$</span> the subgroup <span class="math-container">$\langle (2,2)\rangle$</span>.</p>
<p>Because <span class="math-container">$G$</span> has no elements of order <span class="math-container">$8$... |
2,815 | <p>This is a final exam question in my algorithms class:</p>
<p>$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, and $a,b,c,d$ are distinct positive integers. Find all taxicab numbers $k$ such that $a,b,c,d < n$ in $O(n)$ time.</p>
<p>I don't know if the problem had a typo or not, because $O(n^3)$ seems more reas... | Aryabhata | 1,102 | <p>Apparently this is a solved problem and every rational solution to $x^3 + y^3 = z^3 + w^3$ is proportional to</p>
<p>$x = 1 − (p − 3q)(p^2 + 3q^2)$</p>
<p>$y = −1 + (p + 3q)(p^2 + 3q^2)$</p>
<p>$z = (p + 3q) − (p^2 + 3q^2)^2$</p>
<p>$w = −(p − 3q) + (p^2 + 3q^2)^2$</p>
<p>See: <a href="http://129.81.170.14/~ero... |
2,815 | <p>This is a final exam question in my algorithms class:</p>
<p>$k$ is a taxicab number if $k = a^3+b^3=c^3+d^3$, and $a,b,c,d$ are distinct positive integers. Find all taxicab numbers $k$ such that $a,b,c,d < n$ in $O(n)$ time.</p>
<p>I don't know if the problem had a typo or not, because $O(n^3)$ seems more reas... | Tomer Vromen | 26 | <p>I don't know about $O(n)$, but I can do it in $O(n^2)$. The main idea is to use <a href="http://eli.thegreenplace.net/2008/08/23/initializing-an-array-in-constant-time/">initialization of an array in $O(1)$</a> (this is the best reference that I've found, which is weird since this seems like a very important concept... |
925,558 | <p>How do I prove that the inductive sequence $y_{n+1}= \dfrac {2y_n + 3}{4}$ is bounded? $y(1)=1$</p>
<p><strong>Attempt:</strong> Let us assume that the given sequence is unbounded. : </p>
<p>Then, $y_{n+1} \rightarrow \infty$ either for some finite $n$ or when $n \rightarrow \infty$</p>
<blockquote>
<p>CASE $... | idm | 167,226 | <p>$$|y_{n+1}-y_n|=\frac{1}{2}|y_n-y_{n-1}|=\frac{1}{2^n}|y_0-y_1|$$</p>
<p>then
$$|y_n-y_{n+r}|\leq |y_n-y_{n+1}|+|y_{n+1}-y_{n+2}|+...+|y_{n+r-1}-y_{n+r}|=\frac{1}{2^n}|y_0-y_1|\sum_{k=0}^r\frac{1}{2^k}=\frac{1}{2^n}\underset{n\to\infty }{\longrightarrow } 0$$</p>
<p>Then $(y_n)$ is a Cauchy sequence, and so it co... |
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Boris Novikov | 62,565 | <p>$$\log 2 = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots$$
In the general case
$$\log \frac{1+x}{1-x} = 2(x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\ldots)$$</p>
|
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| Alfred Centauri | 36,057 | <p>And let's not forget this method (read off of the Ln scale).</p>
<p><img src="https://i.stack.imgur.com/4Z7nk.jpg" alt="enter image description here"></p>
<p><a href="http://scienceblogs.com/goodmath/2006/09/13/using-a-slide-rule-part-2-expo/" rel="noreferrer">Image source</a></p>
|
530,920 | <p>I want to know if there is a way to find for example $\ln(2)$, without using the calculator ?</p>
<p>Thanks </p>
| egreg | 62,967 | <p>One can use the fact that
$$
\log x=\lim_{n\to\infty}n\left(1-\frac{1}{\sqrt[n]{x}}\right)
$$
For $\log2$ a good approximation is
$$
1048576\left(1-\frac{1}{\sqrt[1048576]{2}}\right)
$$
where
$$
\sqrt[1048576]{x}
$$
can be computed by pressing twenty times the <code>SQRT</code> key on a pocket calculator, since $104... |
1,127,480 | <p>Consider the set $S = \{x \in \mathbb R: x < \frac2x\}$. Determine the value of $sup$ $S$ (if it exists). </p>
<p>Here is my attempt:<br>
Firstly, $S = \{x \in \mathbb R: 0 < x < \sqrt 2$ $\lor$ $x < -\sqrt2 \}$.<br>
Since $1 \in S$, $S \ne \emptyset$.<br>
Also, $\forall x \in S$, $x < \sqrt2$ and s... | drhab | 75,923 | <p>To find $S$ the equation $x<\frac2{x}$ or equivalently $\frac{(x-\sqrt2)(x+\sqrt2)}{x}<0$ must be solved. </p>
<p>This leads to $S=(-\infty,-\sqrt2)\cup(0,\sqrt2)$. </p>
<p>Here $\sqrt2$ is an upper bound of $S$ and proving that $\sup S=\sqrt2$ comes to showing that every $y<\sqrt2$ is <em>not</em> an upp... |
1,557,686 | <p>I have a definition of a <a href="https://en.wikipedia.org/wiki/Hypergeometric_distribution" rel="nofollow">Hypergeometric distribution</a> as follows:</p>
<blockquote>
<p>Definition: the Hypergeometric distribution is a discrete probability distribution that describes the probability of $k$ successes in $n$ draw... | Em. | 290,196 | <p>Generally, I think, this is taught as or thought of as a box full of "good" items and "bad" items. I will use a slightly different notation. Suppose we have a box full of $N$ balls, and there are $G$ good balls, with $G\leq N$. Let $X$ represent the number of "good" balls I draw when I grab $n$ balls from the box. T... |
3,271,917 | <p><span class="math-container">$$ \sum_{n=1}^\infty (-1)^{n+1} \left( \frac{1.4.7\dots .(3n-2)}{2.3.8\dots .(3n-1)} \right)^2 $$</span></p>
<p>I have done the<span class="math-container">$ \sum_{n=1}^\infty \left( \frac{1.4.7\dots .(3n-2)}{2.3.8\dots .(3n-1)} \right)^2 $</span> part , and showed it divergent using G... | Rkb | 597,464 | <p><span class="math-container">$(\frac{1}{2})^2 < \frac{1}{2}$</span>,
<span class="math-container">$(\frac{4}{5})^2 < \frac{3}{4}$</span>,
<span class="math-container">$(\frac{7}{8})^2 < \frac{5}{6}$</span>, <span class="math-container">$\dots \dots$</span>
<span class="math-container">$(\frac{3n-2}{3n-1})^2... |
71,354 | <p>I've been reading Moroianu's Kahler geometry notes and found a unattributed quote that says that if the Kahler properties hold, then
"a long list of miracles occur"</p>
<p>I am guessing that this quote belongs to Kahler himself, but I can't back this up. Does anyone know?</p>
| csar | 10,942 | <p>Following quid's comment (I don't have the reputation to comment), assuming <i>Über eine bemerkenswerte Hermitesche Metrik</i> is the place to look, it doesn't appear to be in there. The "long list of miracles" does seem to be apparent in a quick skim of the paper, though (and a naive skim at that--I just read the w... |
71,354 | <p>I've been reading Moroianu's Kahler geometry notes and found a unattributed quote that says that if the Kahler properties hold, then
"a long list of miracles occur"</p>
<p>I am guessing that this quote belongs to Kahler himself, but I can't back this up. Does anyone know?</p>
| Community | -1 | <p>I will make a CW answer to collect together some information. </p>
<p>Igor Rivin found a published text containing the relevant phrase.
It is in "The unabated vitality of Kählerian geometry," by Jean-Pierre Bourguignon which is included in the collected works of Kähler (Kähler, Mathematische Werke/Mathematical Wor... |
77,705 | <p>If $A$ and $B$ are both $n \times n$ matrices, and $v$ is a non-zero $n \times 1$ column vector then is it true that if
$$ABv = BAv$$
then $$AB=BA$$</p>
| barf | 11,203 | <p>Suppose $AB = C.$ Then we have $ABv= Cv =BAv$ for all $v.$</p>
<p>Do you feel comfortable saying $BA = C$ now? Do you know any theorems about the uniqueness of matrices of linear transformations under fixed bases?</p>
<p>Edit: In Light of the other answer, i should clarify, this is true only <em>for all</em> $v.$<... |
2,895,343 | <blockquote>
<p>Differentiate $\tan^3(x^2)$</p>
</blockquote>
<p>I first applied the chain rule and made $u=x^2$ and $g=\tan^3u$. I then calculated the derivative of $u$, which is $$u'=2x$$ and the derivative of $g$, which is
$$g'=3\tan^2u$$</p>
<p>I then applied the chain rule and multiplied them together, which ... | TheSimpliFire | 471,884 | <p>You are almost there! In this case you need to apply the Chain Rule three times.</p>
<p>We have that $$(\tan^3(x^2))'=3\tan^2(x^2)\cdot(\tan(x^2))'=3\tan^2(x^2)\cdot\sec^2(x^2)\cdot(x^2)'=6x\tan^2(x^2)\sec^2(x^2)$$</p>
|
132,875 | <p>I was told that I could obtain an analytic solution to a particle falling under the influence of Newtonian gravity by using <code>DSolveValue</code>.</p>
<p><strong>What I am given</strong></p>
<ul>
<li>$G = M = m = 1$</li>
<li>$M$ is a point mass at $z=0$</li>
<li>the particle falls along the $z$ axis</li>
<li>$z... | xzczd | 1,871 | <p>There actually exists 2 solutions.</p>
<p>We first make a change of variable $t=\log(s)$ to eliminate the troublesome <code>-Infinity</code>, here I'll use <a href="https://mathematica.stackexchange.com/a/80267/1871"><code>DChange</code></a> for the task:</p>
<pre><code>eq = {z''[t] + 1/z[t]^2 == 0, z'[-inf] == 0,... |
132,875 | <p>I was told that I could obtain an analytic solution to a particle falling under the influence of Newtonian gravity by using <code>DSolveValue</code>.</p>
<p><strong>What I am given</strong></p>
<ul>
<li>$G = M = m = 1$</li>
<li>$M$ is a point mass at $z=0$</li>
<li>the particle falls along the $z$ axis</li>
<li>$z... | Joe | 45,091 | <p>Thank you all so much for the answers. They are really detailed and helped me a lot. However, I ended up doing something slightly different and got the same result.</p>
<p>$$\ddot{z} = \frac{\mathrm{d}\dot{z}}{\mathrm{d}t} = \frac{\mathrm{d}\dot{z}}{\mathrm{d}z}\cdot \frac{\mathrm{d}z}{\mathrm{d}t} = \frac{\mathrm{... |
2,121,316 | <p>so I have x circles, given by X,Y coordinates + radius. </p>
<p>I'm unsuccessfully trying to figure out, how to make an algorithm for counting how many circles will line segment cross. For illustration: <a href="https://i.stack.imgur.com/mJ8C8.png" rel="nofollow noreferrer">1</a> (Green line crossed 2 circles, red ... | gt6989b | 16,192 | <p>Another, less convoluted way is to convert everything to base 2:
$$
0 = (2^2)^{x+1} - (2^3)^{2x} = 2^{2x+2} - 2^{6x}...
$$</p>
|
2,121,316 | <p>so I have x circles, given by X,Y coordinates + radius. </p>
<p>I'm unsuccessfully trying to figure out, how to make an algorithm for counting how many circles will line segment cross. For illustration: <a href="https://i.stack.imgur.com/mJ8C8.png" rel="nofollow noreferrer">1</a> (Green line crossed 2 circles, red ... | kingW3 | 130,953 | <p>Well you have that $8=2^{3}=(\sqrt{4})^3=4^{3/2}$.Now as to why it works we have that
$$a^{\log_a b}=b$$
Now plugging in $b=8$ we can write
$$8=4^{\log_4 8}$$
And we have that $\log_4 8=\frac{3}{2}$
So we can write $8^{2x}$ as $(4^{3/2})^{2x}$.
Your friends approach is correct.</p>
|
2,062,357 | <p>Let $\sum_{i=1}^\infty a_i$ be an absolute convergent series and let $\sum_{i=1}^\infty b_i$ be a sub-series of it, i.e.
$$b_j=c_j\cdot a_j\quad (c_j\in\{0,1\}),\qquad\forall j\in\mathbb N$$
We can say that $\sum_{i=1}^\infty b_i$ is absolutely convergent as well. So its limit, say $L$, is a real number.</p>
<p>Now... | Community | -1 | <p>Take $A_1,A_2,\ldots,A_n$ to be sets such that if $i \neq j$ then $A_i$ and $A_j$ are disjoint (that is, $A_i \cup A_j$ is the emtpy set). Then $A_1, A_2, \ldots,A_n$ are said to be mutually disjoint. </p>
<p>Now if $A$ and $B$ are sets such that $A \cap B = A$ (or $B$ without less of generality) then $A$ is said t... |
898,082 | <p>I have the answer for this, but my teacher hadn't taught the whole "when cosine is an even, the value of $-\arccos (-0.7)$ is a solution too." </p>
<p>Please: </p>
<p>-tell me when a $\cos$/$\sin$ function is even/odd </p>
<p>-what happens if its odd? </p>
<p>-how to use the "$±\arccos(-0.7) + 2kπ$" ( don't unde... | rae306 | 168,956 | <p><strong>Even and odd functions</strong></p>
<ul>
<li><em>Even functions</em>: $f(x)=f(-x)$. Geometrically, this is symmetry about the $y$-axis.</li>
<li><em>Odd functions</em>: $-f(x)=f(-x)$. Geometrically, this is origin symmetry.</li>
</ul>
<p>From these definitions and the graphs of $\sin x$ and $\cos x$, we ca... |
944,965 | <p>Question: whats the order of the element a=33 in Z60 (under modular addition)?</p>
<p>Answer: $\langle 33 \rangle= \{33,6,39,12,45,18,51,24,57,30,3,36,9,42,15,48,21,54,27,0,\}$. Therefore, $\text{order}(33)=20$</p>
<p>Whats the inverse of $33$ in $\mathbb{Z}_{60}$ (under modular addition)?</p>
<p>Im struggling wi... | egreg | 62,967 | <p>You don't need to compute all the multiples. You know that the order of $33$ is the least positive integer $n$ such that
$$
n\cdot33\equiv 0\pmod{60}
$$
that is, the least positive integer such that $33n$ is a multiple of $60$ or, equivalently, $11n$ is a multiple of $20$. Thus $n=20$.</p>
<p>For the inverse (I'd u... |
36,625 | <p>Say $L\mathbb{C}^\times$ is the loop group of smooth maps $S^1 \to \mathbb{C}^\times$. There is a submonoid $L_{poly}\mathbb{C}^\times$ of loops that look like $w_0 + w_1z +w_2z^2 + \cdots + w_nz^n$ where $z = e^{i\theta}$ (as Andrew notes below this is not a group because its not closed under taking inverses). Eq... | damiano | 4,344 | <p>This is mostly a series of comments, but guided by the questions you asked.</p>
<p>First of all, I will only talk about $X_n$, interpreting it as the space of non-zero complex polynomials $p$ of degree at most $n$ such that no root of $p$ lies on the unit circle, taken up to non-zero scaling. We may as well think o... |
4,204,416 | <p>First, can someone provide a simple explanation for the <span class="math-container">$\bar{x}$</span> formula: <span class="math-container">$$\frac{\iint xdA}{area}$$</span> My understanding of the formula is as follows: we let <span class="math-container">$z=x$</span>, calculate the volume, and divide by the area t... | Kman3 | 641,945 | <h2 id="mass-ryvx">Mass</h2>
<p>Imagine a lamina with constant density, <span class="math-container">$\rho$</span>. The mass <span class="math-container">$m$</span> of the lamina and the area <span class="math-container">$A$</span> occupied by the lamina are in equal proportion. <span class="math-container">$1/4$</span... |
337,035 | <p>Recall Wolpert's lemma:</p>
<p>Let X,Y be hyperbolic surfaces and <span class="math-container">$f:X\to Y$</span> a <span class="math-container">$K$</span>-quasiconformal homeomorphism. For any homotopy class of curves <span class="math-container">$c$</span> let <span class="math-container">$\ell(c)$</span> denote t... | Alex Nolte | 136,267 | <p>No. The issue is that quasiconformality constants are much more sensitive to local distortion than lengths. This is morally a <span class="math-container">$L^\infty$</span> (quasiconformal constants) to <span class="math-container">$L^1$</span> (lengths) comparison: <span class="math-container">$L^\infty$</span> bou... |
3,304,800 | <p>Given that <span class="math-container">$x^2-4cx+b^2 \gt 0$</span> <span class="math-container">$\:$</span> <span class="math-container">$\forall$</span> <span class="math-container">$x \in \mathbb{R}$</span> and <span class="math-container">$a^2+c^2-ab \lt 0$</span></p>
<p>Then find the Range of <span class="math-... | Certainly not a dog | 691,550 | <p>If a number <span class="math-container">$p$</span> is in the range of the <span class="math-container">$y$</span> you have given, it means <span class="math-container">$\frac{x+a}{x^2+bx+c^2} = p$</span> for some <span class="math-container">$x$</span>.</p>
<p>Or, in other words, that equation has a real solution... |
2,355,246 | <p>The optimisation problem [PROBLEM 1]:
$$\max \sum_{t=T} x_t $$
subject to:
$$\ (1) x_t = (1+\alpha) x_{t-1}\ \forall t ..T$$
$$\ t=\{ 0..T\} , x_0= given$$
$$\ (e.g) x_1 = (1+\alpha) x_0\ $$
$$\ (e.g) x_2 = (1+\alpha) x_1\ $$
$$\ (e.g) x_3 = (1+\alpha) x_2\ $$ </p>
<p>$$\ $$</p>
<p>$$\ (2) x_t \ge 0 $$<... | paul garrett | 12,291 | <p>There certainly is philosophical content to this question. And we can respond in a fundamental way without using any alleged properties of "real numbers"... by considering vectors $({2n\over n^2+1},{n^2-1\over n^2+1},0,0,\ldots)$ for positive integers $n$.</p>
<p>EDIT: (in light of comments...) For what it's worth,... |
440,744 | <p>The vector space dimension of the cohomology group of the <span class="math-container">$2$</span>-plane Grassmannian <span class="math-container">$\mathrm{Gr}_{2,n}$</span> is given by the number of tuples <span class="math-container">$(\lambda_1,\lambda_2)$</span> satisfying
<span class="math-container">$$
n - 2 \g... | Oliver | 4,087 | <p>Harry Tamvakis gives an explanation of the relation between Grassmannian cohomology and representations of SL_n in <a href="https://arxiv.org/abs/math/0306414" rel="noreferrer">The connection between representation theory and Schubert calculus</a>.</p>
<p>One fundamental obstacle for explaining this phenomenon is th... |
1,110,122 | <p>Prove $$\large\int_{-\pi}^{\pi}\sin (\sin x) \,dx =0$$ without using the fact that $\sin(x)$ is odd.</p>
<p>Computing this in wolfram says that it is uncomputable, which leads me to believe that the only way to find this would be methods for solving definite integrals. I am wondering if it is possible with any othe... | Sangchul Lee | 9,340 | <p>Here is a very weird solution. Notice that for $s \in \Bbb{R}$,</p>
<p>$$ \int_{-\pi}^{\pi} \sin(s\sin x) \, dx = \Im \int_{-\pi}^{\pi} \exp(is\sin x) \, dx. $$</p>
<p>Now it follows that</p>
<p>\begin{align*}
\int_{-\pi}^{\pi} \exp(is\sin x) \, dx
&= \int_{-\pi}^{\pi} \exp(se^{ix}/2) \exp(-se^{-ix}/2) \, dx ... |
3,552,779 | <p>In a 3D world, given a box <code>B</code>, a pivot <code>P</code>, and a direction vector <code>V</code>, how can I find out how much to rotate at <code>P</code> such that <code>V</code> points towards an arbitrary point <code>A</code>?</p>
<p>Problem source:</p>
<p>I am a software developer that come across the n... | Andrew Chin | 693,161 | <p>Suppose you must. Then we have <span class="math-container">$$9\lim_{a\to2^+}\int_a^3\frac1{\sqrt[4]{x-2}}\,dx.$$</span></p>
<p>Let <span class="math-container">$u=x-2$</span>. Then we have <span class="math-container">$$9\lim_{a\to2^+}\int_{a-2}^1 u^{-1/4}\, du.$$</span></p>
<p>By the power rule, we have <span ... |
3,552,779 | <p>In a 3D world, given a box <code>B</code>, a pivot <code>P</code>, and a direction vector <code>V</code>, how can I find out how much to rotate at <code>P</code> such that <code>V</code> points towards an arbitrary point <code>A</code>?</p>
<p>Problem source:</p>
<p>I am a software developer that come across the n... | Arturo Magidin | 742 | <p>There are certain conditions that must be met for a substitution to be "legal". In most circumstances these conditions are naturally met and so they are not emphasized; you have here one situation in which they are not. </p>
<p>For instance, one set of conditions is given in Anton, Anderson, and Bivens (<em>Calculu... |
165,385 | <p>I am generating a list of 1's, 2's, and 3's with different probabilities for each number. I then convert this list into three binarized lists, giving a list of the locations of each digit in the original list. There must be a more efficient way to do this? Perhaps changing the way I make the original list?</p>
<pre... | kglr | 125 | <p><strong>Update:</strong> On Wolfram Cloud, the timing comparison I get is the opposite of David's claim that <em>"the result is about twice as fast as the generation of the list and the result in the fastest method described by others here"</em>; that is, to my surprise, <code>BitXor</code>-based method is more th... |
1,691,009 | <p>I am struggling with a particular concept, I'll lay it out how I think will best allow for an answer:</p>
<p>Let $A =$ $\Bbb Z$ be the set of all integers and the universe of discourse. </p>
<pre><code>Let B, be the set of even numbers
Let C, be the set of odd numbers
Let D, be the set of positive numbers
Let E... | assefamaru | 262,896 | <p>Your universe of discourse is the set of all integers. One property of integers is that they have a parity of either being even or odd. So if you exclude all even numbers from the set of integers, then you will be left with a set containing all odd numbers.</p>
<p>Hence, $A-B=C$, or using words, <strong>$A-B$ is th... |
2,636,188 | <p>How would I go about proving:</p>
<blockquote>
<p>For every non-prime $n \in \mathbb N$ there exists $m \in \mathbb Z / n\mathbb Z \setminus \{0\}$ with $m^{n-1} \not\equiv 1 \mod n$.</p>
</blockquote>
<p>I already proved the other way around (straight forward using Euler-Fermat), but am stuck at this point.</p>... | Jose Brox | 146,587 | <p>Pick $m$ a nontrivial divisor of $n$, and let $k:=n/m$. Suppose that $m^{n-1}\equiv 1(\mod n)$. Then $k\equiv k\cdot 1\equiv km^{n-1}\equiv nm^{n-2}\equiv 0(\mod n)$, contradiction, since $0<k<n$.</p>
|
626,837 | <p>If $a+b+c+d = 30$ and $a,b,c,d$ lie between $0$ and $9$. How to find number of solutions of this equation.</p>
| Blue7 | 118,890 | <p>The answers are given here: <a href="http://www.wolframalpha.com/input/?i=0%3Ca%3C9,%200%3Cb%3C9,%200%3Cc%3C9,%200%3Cd%3C9,%20a%2bb%2bc%2bd%20=%2030" rel="nofollow">http://www.wolframalpha.com/input/?i=0%3Ca%3C9%2C+0%3Cb%3C9%2C+0%3Cc%3C9%2C+0%3Cd%3C9%2C+a%2Bb%2Bc%2Bd+%3D+30</a></p>
<p>However I do not know how thes... |
3,583,600 | <blockquote>
<p>Define a branch for <span class="math-container">$\sqrt{1+\sqrt{z}}$</span> and show it is analytic.</p>
</blockquote>
<p>I defined a branch <span class="math-container">$(-\pi, \pi)$</span>, and so that means the function <span class="math-container">$\sqrt{1+\sqrt{z}}$</span> is analytic on <span c... | Carlos Obiglio | 761,132 | <p><span class="math-container">$ {w = \sqrt{1 + \sqrt{z}},( z \; \epsilon \; C) \mapsto circle \; |w|\measuredangle\phi,\ by \ Moivre \ theorem \ to \ determine \ the\ complex\ roots \ it \ follows \ that: \ w^n \ = \ r^n \ ( cos(\ n \phi )\ + sin(\ n \phi) )
\ let's \; define \; this \; circle \; as \; ... |
1,631,366 | <p>I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$
It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.</p>
| mm8511 | 180,904 | <p>Assuming this is $\log$ base $10$, you know that $\log_{10}(10^n)^{10}=n^{10}$ (this follows from the identity $\log_{a}(a^{k})=k$), so it seems you would have to verify that $10^{n}>n^{10}$ for $n>10$. I am not sure that this makes it easier for you, but atleast the log is gone. You can then show that the der... |
123,502 | <p>Suppose $A \subseteq \{1,\dots,n\}$ does not contain any arithmetic progressions of length $k+1$. What is the largest number of $k$-term arithmetic progressions that $A$ can have? (one may also wish to put some lower or upper on the size of $A$) We can work over $\mathbb{Z}_p$ if it makes the answer any easier. The ... | Timothy Chow | 3,106 | <p>I'd recommend asking for input from professors who have a track record of supervising successful undergraduate research, such as <a href="http://www.d.umn.edu/~jgallian/tanglewood.html" rel="nofollow noreferrer">Joseph Gallian</a> or the people who run the <a href="http://www.math.osu.edu/ross/" rel="nofollow noref... |
158,944 | <p>Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last?</p>
<p>I mean, not like a trig function where each intersect is the same distance from the last. (2,0); (4,0); (6,0); (8,0). And not like a spiral... | André Nicolas | 6,312 | <p>If you specify an arbitrary <strong>finite</strong> set of points $\{a_1, a_2, a_3, \dots, a_n\}$, the polynomial function
$$f(x)=(x-a_1)(x-a_2)(x-a_3) \cdots (x-a_n)$$
has the desired property.</p>
|
158,944 | <p>Is there any type of function that when graphed would show a curve which intersects the x axis multiple times, with each point being an arbitrary distance from the last?</p>
<p>I mean, not like a trig function where each intersect is the same distance from the last. (2,0); (4,0); (6,0); (8,0). And not like a spiral... | Alex Becker | 8,173 | <p>If I'm interpreting your question correctly, you want to know if, given an increasing sequence of points $(a_n)$ in $\mathbb R$, we can define a continuous function $f:\mathbb R\to\mathbb R$ such that $f(x)=0$ if and only if $x=a_i$ for some $i$ (that is, the graph of $f$ intersects the $x$ axis precisely at the poi... |
4,370,989 | <p>I have looked in some of my old books and found an exericse that I do not know how to solve. It seems pretty simple though.</p>
<p>The question is as follows:</p>
<pre><code>Which of these integers are prime?:
111
111.1
111.111
111.111.11
111.111.111
111.111.111.1
111.111.111.111
</code></pre>
<p>I remember one rul... | Theo Bendit | 248,286 | <p>Just to simplify the problem a little, let's consider the free <em>monoid</em> on an alphabet <span class="math-container">$A$</span> (a monoid is a set with an associative operation with an identity, but not necessarily any inverses). The free group works similarly, but there's a little more complexity hidden there... |
2,447,850 | <p>So I have to prove 2 things:</p>
<ol>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R, x>0$. </p></li>
<li><p>That $\lim\limits_{n \rightarrow \infty}\frac{x^n}{n!} = 0$ where $n \in \mathbb N$ and $x \in \mathbb R$. </p></li>
</ol>
<p>For #1, I kn... | Mark Viola | 218,419 | <p>Note that $n!\ge (n/2)^{n/2}$. Then, we have</p>
<p>$$\left|\frac{x^n}{n!}\right|\le \frac{|x|^n}{\left(\sqrt{n/2}\right)^n}=\left(\frac{2|x|}{n}\right)^n$$</p>
<p>Can you conclude now?</p>
|
3,741,122 | <p>Recently I've tried to find the difference between partial differentiation and total differentiation.
I've heard the total derivative is defined on single value functions, while the partial derivative by contrast is defined on multivariate functions.
My problem is, that total differentiation is used on multivariate ... | Community | -1 | <p>The difficulty comes from the fact that with function of several variables, there can be dependencies between the variables (which is not possible in the univariate case).</p>
<p>Consider the function <span class="math-container">$f(x,y):=x+y$</span>.</p>
<p>The <em>partial derivative</em> on <span class="math-conta... |
4,449,164 | <p>My question stems from page 2 of <a href="https://www.semanticscholar.org/paper/Stability-and-positive-supermartingales-Bucy/31bbbfc842b5717ead8cd7e997a5117fe7a66373" rel="nofollow noreferrer">this paper by Bucy</a>, which states:</p>
<blockquote>
<p>[A random variable <span class="math-container">$x$</span>] is alm... | Alex Ortiz | 305,215 | <ol>
<li><p>The interpretation is correct, although I would add for completeness the function <span class="math-container">$g$</span> should be measurable, though this is often understood even without specifying it. A more succinct way to say the same thing is that <span class="math-container">$X$</span> is constant <s... |
99,786 | <p>Suppose I want to run several times of the same code containing <code>RandomReal</code> function, but I don't initialise the random seed. What will happen?</p>
<p>I tried, but each time the <code>RandomReal[]</code> return different values, does it mean that Mathematica using some kind of default method to initiali... | e.doroskevic | 18,696 | <p><strong>From documentation:</strong></p>
<blockquote>
<p>You can use SeedRandom[n] to make sure you get the same sequence of
pseudorandom numbers on different occasions.</p>
</blockquote>
<p><strong>Reference</strong></p>
<p><a href="https://reference.wolfram.com/language/ref/SeedRandom.html" rel="nofollow">S... |
99,786 | <p>Suppose I want to run several times of the same code containing <code>RandomReal</code> function, but I don't initialise the random seed. What will happen?</p>
<p>I tried, but each time the <code>RandomReal[]</code> return different values, does it mean that Mathematica using some kind of default method to initiali... | Szabolcs | 12 | <p>Let's think about when the answer to this question may be relevant. You know that if using the same seed on two different occasions, the RNG will generate the same sequence of numbers. So the relevant questions are:</p>
<ol>
<li><p>Does Mathematica use the same seed after startup every time?</p>
<p><strong>No.</... |
1,019,408 | <p>Usually, $f$ denotes a function, $f(x)$ is an image of $x$ under $f$. But what's $f(X)$ if $X$ is a set? </p>
<p>edit: Please, disregard the body of this question. I had to put something here to be able to post the question. </p>
| Paul | 17,980 | <p>It is $f(X)=\{f(x): x\in X\}.$</p>
|
516,244 | <p>My professor gave us this example on her notes:</p>
<p>$$\sum_{n = 1}^\infty \left(\frac{3}{n(n+3)}+\frac{1}{2^n}\right)$$</p>
<p>So I know we're supposed to find the partial fraction, which ends up being</p>
<p>$$\left(\frac{3}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}=
\frac{1}{n}-\frac{1}{n+3}\right)$$</p>
<p>So ba... | sds | 37,092 | <p>What you are looking for is called the <a href="http://en.wikipedia.org/wiki/Shoelace_formula" rel="noreferrer">shoelace formula</a>:</p>
<p><span class="math-container">\begin{align*}
\text{Area}
&= \frac12 \big| (x_A - x_C) (y_B - y_A) - (x_A - x_B) (y_C - y_A) \big|\\
&= \frac12 \big| x_A y_B + x_B y_C +... |
749,849 | <p>Question:</p>
<p>A submatrix $B$ consisting of "s" rows of $A$ is selected from an n-square matrix $A$ of rank $r_{A}$. prove that the rank of $B$ is equal to or greater than $r_{A}+s-n$.</p>
<p>My thoughts:</p>
<p>I start with an easy case, says $A=I_4$. Then, by selecting 2 first rows of $A$. We obtain a matrix... | Chandramauli Chakraborty | 375,555 | <p>Here <span class="math-container">$A$</span> is <span class="math-container">$n\times n$</span> matrix and <span class="math-container">$r_A$</span> be the rank of <span class="math-container">$A$</span>. Now we need to eliminate <span class="math-container">$n-s$</span> rows from <span class="math-container">$n$</s... |
3,495,779 | <p>The question is:
Prove that if graph <span class="math-container">$G$</span> is a <strong>Connected Planar Graph</strong> where each region is made up of at least <span class="math-container">$k\,(k\ge3)$</span> edges, then it must satisfy:</p>
<p><span class="math-container">$e\ge\frac{k(v-2)}{k-2}$</span></p>
<p... | Chor | 707,050 | <p>It is no necessary to use induction.To resolve it,we need to use Euler's Formula:</p>
<blockquote>
<p>For a simple <strong>Connected Planar Graph</strong> G in which <span class="math-container">$e$</span>,<span class="math-container">$v$</span>,<span class="math-container">$f$</span> means its number of edges, v... |
1,558,210 | <p>In a linear algebra book I'm reading now, there was the following exercise:</p>
<blockquote>
<p>Let <span class="math-container">$W\subseteq V$</span> be a subspace of vector space <span class="math-container">$V$</span>. Do there always exist two subspaces <span class="math-container">$W_1,W_2\subseteq V$</span> su... | Michael Burr | 86,421 | <p>From the comments above, I will provide my guess as to what the OP meant by $\lim_{f(x)\rightarrow f(t)}\phi(x)=\phi(t)$ means:
$$
\forall \varepsilon>0,\exists\delta>0\text{ s.t. }\forall x\text{ with }|f(x)-f(t)|<\delta,|\phi(x)-\phi(t)|<\varepsilon.
$$</p>
<p>The original statement is true (provided... |
2,376,315 | <p>So I'm trying to solve the problem irrational ^ irrational = rational. Here is my proof
Let $i_{1},i_{2}$ be two irrational numbers and r be a rational number such that $$i_{1}^{i_{2}} = r$$
So we can rewrite this as $$i_{1}^{i_{2}} = \frac{p}{q}$$
Then by applying ln() to both sides we get $$i_2\ln(i_1)... | Ben Grossmann | 81,360 | <p>As your proof is currently set up, you would need to show that your explicit formula gives you irrational numbers $i_1,i_2$ for at least one pair of integer values $p,q$. While it is certainly <em>believable</em> (and in fact true) that this is the case, it requires proof.</p>
<p>You're approaching this problem fr... |
893,959 | <p>I have a series of problems in inequalities that I can not solve,please help me if you can.</p>
<p>problem 1 :$a,b,c \geq 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $
prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$</p>
<p>problem 2 : $a,b,c\geq0$ and $a+b+c = 1$ prove that </p>
<p>1, $ \sqrt{a+\frac{(b... | Macavity | 58,320 | <p>I will do #1 (the title problem) here: The constraint gives -$$(a^2+b^2+c^2)^3 = (ab+bc+ca)^2$$</p>
<p>$$\therefore 3(a^2+b^2+c^2) \ge (a+b+c)^2 = (a^2+b^2+c^2)+2(a^2+b^2+c^2)^{3/2} $$</p>
<p>$$\implies a^2+b^2+c^2 \le 1 \implies a+b+c \le \sqrt3$$</p>
<p>Now the result follows from $a^2b+b^2c+c^2a+abc \le \fra... |
164,340 | <p>I recently found myself at a spot that I never believed I'll get (or at least not that soon in my career). I ran into a problem which seems to be best answered via categories.</p>
<blockquote>
<p>The situation is this, I have a directed system of structures and the maps are all the inclusion map, that is <span class... | Zhen Lin | 5,191 | <p>Your question essentially amounts to asking, "when does the forgetful functor $U : \mathcal{C} \to \textbf{Set}$ create directed colimits?" More generally, one could replace "directed colimit" by "filtered colimit". There is, as far as I know, no general answer. </p>
<p>Here is one reasonably general class of categ... |
2,919,096 | <p>I am trying to solve a system of two second-order ODEs. After separating them, I obtained a fourth-order independent ODE as illustrated below. I wonder if there is a specific technique to solve it.</p>
<p>$$y^{(4)}+\frac{a_1}{x} y^{(3)}+\frac{a_2}{x^2}y^{(2)}+a_3y^{(2)}+\frac{a_4}{x}y^{(1)}+a_5y=0$$</p>
| Przemo | 99,778 | <p>Since our ODE is of fourth order and has regular singularities at zero and infinity it would make sense the map it onto a different ODE with the same properties and with known solutions. In here the generalized hypergeometric function <span class="math-container">$f(x):=F_{0,3}[b_1,b_2,b_3;x]$</span> is a good candi... |
2,652,675 | <p>Given that A = $\begin{bmatrix} 2 & 1 \\ -5 & -4 \end{bmatrix} $ and B = $\begin{bmatrix} 3 & -1 \\ -1 & 0 \end{bmatrix} $ </p>
<p>Find a 2 X 2 matrix C such that $CA= B$</p>
<p>I multiply both sides by $A^{-1}$ </p>
<p>Since $A^{-1}A = I $ </p>
<p>$ CI = BA^{-1}$ </p>
<p>Since $CI = IC = C$ </p... | Adriano | 76,987 | <p>Given some topology $\mathcal T$ generated by a basis $\mathcal B$ on some set $X$, it is always possible to make a finer topology $\mathcal T_d$, where $\mathcal T \subseteq \mathcal T_d$, by declaring that every set is open. Indeed, when the new basis is the set of all singletons, we obtain the finest possible top... |
2,768,088 | <p>A farmer cultivates mushrooms in his garden. A greedy neighbor wants to pick some but the farmer is trying to block him.</p>
<p>The garden has the form of a 8x6 grid. Rows 1 to 8 from the front to the back and columns A to F from left to right. The mushrooms are planted in the 8th row (6 mushrooms). The farmer is i... | poetasis | 546,655 | <p>It appears that, unless the farmer takes a deliberately bad path, he will always be able to hit the neighbor. Here are sample moves juxtaposed. In the 2nd example, the farmer could have avoided being hit by taking a different path. The neighbor can only be hit or move to places to avoid being hit but he cannot get p... |
2,768,088 | <p>A farmer cultivates mushrooms in his garden. A greedy neighbor wants to pick some but the farmer is trying to block him.</p>
<p>The garden has the form of a 8x6 grid. Rows 1 to 8 from the front to the back and columns A to F from left to right. The mushrooms are planted in the 8th row (6 mushrooms). The farmer is i... | Ross Millikan | 1,827 | <p>The neighbor can win. He starts to E1 and claims the distant opposition. If the farmer moves forward, so does the neighbor. If the farmer moves sideways the neighbor moves diagonally forward on the side away from the farmer. Now the farmer must move toward the neighbor and the neighbor can move in front of the f... |
293,110 | <p>How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way. </p>
<p>I would love to get de... | Adam P. Goucher | 39,521 | <p>If you take your description and rename $\textrm{set} \mapsto \textrm{class}$ and $\textrm{small set} \mapsto \textrm{set}$, and add some further axioms beyond the ones you mention (such as global choice), then the resulting theory is <strong>von Neumann-Bernays-Goedel set theory</strong> (NBG).</p>
<p>NBG has some... |
293,110 | <p>How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an epimorphism? My idea was to work with generators, but I was not able to prove it this way. </p>
<p>I would love to get de... | Thomas Benjamin | 20,597 | <p>I believe that an answer to your question [1] is the system that Dana Scott developed in his paper, "Axiomatizing Set Theory" found in <em>Proceedings of Symposia in Pure Mathematics</em>, Volume 13, Part II, 1974, pp. 207-14. This system is, in Scott's words, the formalization of the following intuition:... |
3,805,452 | <p>Given a stick and break it randomly at two places, what is the probability that you can form a triangle from the pieces?</p>
<p>Here is my attempt and the answer does not match, so I am confused what went wrong with this argument.</p>
<p>I first denote the two randomly chosen positions by <span class="math-container... | BruceET | 221,800 | <p>The following simulation in R looks at a million such randomly broken
sticks, finds the lengths of the three pieces, and finally finds the
length of the longest piece. If the longest has length less
than half, you can make a triangle. Answer: <span class="math-container">$0.250\pm 0.001.$</span></p>
<pre><code>set.s... |
3,984,322 | <p>I have the 2nd order homogeneous ODE:
<span class="math-container">$$y(x)''+\frac{2}{x}y'(x)-By=0 $$</span>
Where <span class="math-container">$B$</span> is a constant. Since I am a bit rusty with analytical solutions, I plugged it in MatLab's symbolic solver which found the following solution.
<span class="math-con... | Varun Vejalla | 595,055 | <p>That solution is also valid for negative <span class="math-container">$B$</span>, although it will use complex numbers. To find the solution without complex numbers using that solution, I first made the substitution <span class="math-container">$\frac{C_2}{2\sqrt{B}} \to C_2$</span> just for convenience: <span class... |
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