qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
4,253,761 | <p><span class="math-container">$$\begin{array}{ll} \text{extremize} & xy+2yz+3zx\\ \text{subject to} & x^2+y^2+z^2=1\end{array}$$</span></p>
<p>How to find the maximum/minimum using Lagrange multipliers?</p>
<p>Context: This is not a homework problem, my friend and I often make up problems to challenge each ot... | Ferris Boyler | 637,978 | <p>Suppose we want to find <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> such that:</p>
<p><span class="math-container">$$a_1x_1+a_2x_2=0$$</span></p>
<p>Well, we could pick <span class="math-container">$x_1=x_2=0$</span>: This is called the trivial solution of the linear equat... |
1,201,904 | <p>I have to implement a circuit following the boolean equation A XOR B XOR C, however the XOR gates I am using only have two inputs (I am using the 7486 XOR gate to be exact, in case that makes a difference)... is there a way around this?</p>
| David Holden | 79,543 | <p>use two gates. input $A$ and $B$ to the first gate, then input the output of the first gate and $C$ to the second gate.</p>
|
4,292,091 | <p>I'm reading the definition of <span class="math-container">$inf\emptyset$</span> and <span class="math-container">$sup\emptyset$</span>.</p>
<p>a) I'm wondering why <span class="math-container">$inf\emptyset = \infty$</span> and <span class="math-container">$sup\emptyset = -\infty$</span>. I would have expected both... | Zuy | 730,592 | <p>You can assume that the author used the extended real number line for these definitions.</p>
<p>In fact, here's a motivation for above definition.</p>
<p>If you have two sets <span class="math-container">$A\subseteq B\subseteq\mathbb R$</span>, then you want them to satisfy
<span class="math-container">$$\inf A\geq ... |
4,292,091 | <p>I'm reading the definition of <span class="math-container">$inf\emptyset$</span> and <span class="math-container">$sup\emptyset$</span>.</p>
<p>a) I'm wondering why <span class="math-container">$inf\emptyset = \infty$</span> and <span class="math-container">$sup\emptyset = -\infty$</span>. I would have expected both... | Vercassivelaunos | 803,179 | <p>As others have said, this assumes that we are working in the extended reals.</p>
<p>Contrary to what the others said, though, <span class="math-container">$\sup\emptyset=-\infty$</span> is not a definition made out of convenience. It is a direct consequence of the normal definition of the supremum: the smallest uppe... |
2,819,667 | <p>The problem is the same as <a href="https://math.stackexchange.com/questions/14190/average-length-of-the-longest-segment">here</a>. </p>
<blockquote>
<p>A stick of 1m is divided into three pieces by two random points. Find the average length of the largest segment.</p>
</blockquote>
<p>I tried solving it in a di... | leonbloy | 312 | <p>The first integral is wrong, because it assumes that $X,Y$ are uniform and independent on $[0,1]^2$. They are not (for one thing, $X \le Y$).</p>
|
478,516 | <p>$\lim_{d \to \infty} (1+\frac{w}{d})^{\frac{d}{w} } = e$. But, what if the number of bits used to encode $d$ is polynomial in length. In this model, infinity can't be encoded. However, $d$ is polynomialy much larger than $w$.
Is there any tight lower bound, a closed form function $f(d)$ such that
$$ f(d) \le (1+\fr... | Antonio Vargas | 5,531 | <p>By expanding the left-hand side as a power series one can show that</p>
<p>$$
\left(1+\frac{1}{x}\right)^x > e - \frac{e}{2x}
$$</p>
<p>for $x > 0$, where the approximation gets better as $x$ gets larger. By setting $x = d/w$ we get</p>
<p>$$
\left(1 + \frac{w}{d}\right)^{d/w} > e - \frac{ew}{2d}
$$</p>... |
2,902,768 | <p>$f:\mathbb{R}^2 \to \mathbb{R}$</p>
<p>$f\Bigg(\begin{matrix}x\\y\end{matrix}\Bigg)=\begin{cases}\frac{xy^2}{x^2+y^2},(x,y)^T \neq(0,0)^T \\0 , (x,y)^T=(0,0)^T\end{cases}$</p>
<p>I need to determine all partial derivatives for $(x,y)^T \in \mathbb{R}^2$:</p>
<p>$f_x=y^2/(x^2+y^2)-2x^2y^2/(x^2+y^2)^2$ for $(x,y)^T... | user | 505,767 | <p>Recall that when a function is differentiable at a point $(x_0,y_0)$ the following holds</p>
<p>$$\frac{\partial f}{\partial v}=\nabla f(x_0,y_0)\cdot v$$</p>
<p>that is the directional derivative are linear function.</p>
<p>Therefore that is a necessary condition for the differentiabilty of $f(x,y)$ at $(x_0,y_0... |
4,349,487 | <h2>Problem :</h2>
<p>Let <span class="math-container">$a,x>0$</span> then define :</p>
<h2><span class="math-container">$$f\left(x\right)=x^{\frac{x}{x^{2}+1}}$$</span></h2>
<p>And :</p>
<h2><span class="math-container">$$g\left(x\right)=\sqrt{\frac{x^{a}}{a}}+\sqrt{\frac{a^{x}}{x}}$$</span></h2>
<p>Then prove or d... | Canis Lupus | 22,943 | <p><img src="https://i.stack.imgur.com/b8pvI.png" alt="" /></p>
<p>It's easy to find <span class="math-container">$\angle ABC=75^\circ$</span>.</p>
|
62,981 | <p><strong>Introduction.</strong> I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction of the "ultra"ness.</p>
<p><strong>An example.</strong> Given a free (ultra)filter $F$ on $\omega$, ... | Andreas Blass | 6,794 | <p>Here's a proof that, if $F$ is an ultrafilter and $g$ is $F$-Grigorieff-generic, then $F\cup\{g\}$ does not generate an ultrafilter in the extension. Define a real $x:\omega\to2$ (also viewed as $x\subseteq\omega$ as usual) by letting $x(n)=\sum_{k=0}^ng(k)$ modulo 2. (Technically, I should fix the obvious names f... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Community | -1 | <p>The Frobenius automorphism</p>
<p>$$(x + y)^p = x^p + y^p$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Andrey Rekalo | 723 | <p>$$\left(\sum\limits_{k=1}^n k\right)^2=\sum\limits_{k=1}^nk^3 .$$</p>
<p>The two on the left is <em>not</em> a typo.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Community | -1 | <p>$$ \infty! = \sqrt{2 \pi} $$</p>
<p>It comes from the zeta function.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| user02138 | 2,720 | <p>\begin{eqnarray}
\sum_{i_1 = 0}^{n-k} \, \sum_{i_2 = 0}^{n-k-i_1} \cdots \sum_{i_k = 0}^{n-k-i_1 - \cdots - i_{k-1}} 1 = \binom{n}{k}
\end{eqnarray}</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| J. M. ain't a mathematician | 498 | <p>$$\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty \frac1{k^k}$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| user02138 | 2,720 | <p>Machin's Formula:
\begin{eqnarray}
\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}.
\end{eqnarray}</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Community | -1 | <p>Well, i don't know whether to classify this as funny or surprising, but ok it's worth posting.</p>
<ul>
<li>Let $(X,\tau)$ be a topological space and let $A \subset X$ . By iteratively applying operations of closure and complemention, one can produce at most 14 distinct sets. It's called as the <a href="http://en.... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Alon Amit | 308 | <p>Let $f$ be a symbol with the property that $f^n = n!$. Consider $d_n$, the number of ways of putting $n$ letters in $n$ envelopes so that no letter gets to the right person (aka derangements). Many people initially think that $d_n = (n-1)! = f^{n-1}$ (the first object has $n-1$ legal locations, the second $n-2$, ...... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Kirthi Raman | 25,538 | <p>$$\large{1,741,725 = 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7}$$</p>
<p>and</p>
<p>$$\large{111,111,111 \times 111,111,111 = 12,345,678,987,654,321}$$</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| andreasdr | 13,591 | <p>$$ 10^2+11^2+12^2=13^2+14^2 $$</p>
<p>There's a funny Abstruse Goose comic about this, which I can't seem to find at the moment.</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Felix Marin | 85,343 | <p>$$
\begin{array}{rcrcl}
\vdots & \vdots & \vdots & \vdots & \vdots
\\[1mm]
\int{1 \over x^{3}}\,{\rm d}x & = & -\,{1 \over 2}\,{1 \over x^{2}} & \sim & x^{\color{#ff0000}{\large\bf -2}}
\\[1mm]
\int{1 \over x^{2}}\,{\rm d}x & = & -\,{1 \over x} & \sim & x^{\color{#ff00... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Gerry Myerson | 8,269 | <p>$$2592=2^59^2$$ Found this in one of Dudeney's puzzle books</p>
|
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Theemathas Chirananthavat | 66,404 | <p>Best near miss</p>
<p>$$\int_{0}^{\infty }\cos\left ( 2x \right )\prod_{n=0}^{\infty}\cos\left ( \frac{x}{n} \right )~\mathrm dx\approx \frac{\pi}{8}-7.41\times 10^{-43}$$</p>
<p>One can easily be fooled into thinking that it is exactly $\dfrac{\pi}{8}$.</p>
<p>References:</p>
<ul>
<li><a href="https://en.wikipe... |
8,814 | <p>Here is a funny exercise
$$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$
(If you prove it don't publish it here please).
Do you have similar examples?</p>
| Joe | 107,639 | <p>$$
\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}=1-\frac12-\frac13-\frac15+\frac16-\frac17+\frac1{10}-\frac1{11}-\frac1{13}+\frac1{14}+\frac1{15}-\cdots=0
$$
This relation was discovered by Euler in 1748 (<strong>before</strong> Riemann's studies on the $\zeta$ function as a complex variable function, from which this relatio... |
2,101,059 | <p>Let $d,n~$ be positive integer.</p>
<p>$\int{y(1-y^d)^n}dy$</p>
<p>Is it possible to solve it? If you know the method, please teach me.</p>
| Claude Leibovici | 82,404 | <p>As Eff answered, the binomial theorem seems to be the way to do it.</p>
<p>You will learn sooner or later that $$\int{y(1-y^d)^n}\,dy=\frac{1}{2} y^2 \,\, _2F_1\left(\frac{2}{d},-n;\frac{d+2}{d};y^d\right)$$ where appears the Gaussian or ordinary hypergeometric function (see <a href="https://en.wikipedia.org/wiki/H... |
314,239 | <blockquote>
<p>If <span class="math-container">$P_1 , P_2 $</span> are two sylow <span class="math-container">$p$</span>-subgroups of the group <span class="math-container">$G$</span> prove that:</p>
<p><span class="math-container">$ P_1 \bigcap $</span> <span class="math-container">$P_2$</span> = <span class="math-co... | Nicky Hekster | 9,605 | <p>Although, as pointed out above, the statement is not true in general, you might wonder for which groups it is true indeed. Your question can be reformulated as <strong>which p-groups appear as non-normal T.I. Sylow p-subgroups?</strong> Here T.I. stands for "trivial intersection". A trivial intersection set is one t... |
2,505,971 | <blockquote>
<p>Let $\lim_{x\to \infty }x^n f(x)=0$ for any $n \in \mathbb{N}$. Then find the symptote
$$g(x)=\dfrac{x^2+f(x)}{x+1+2f(x)}.$$</p>
</blockquote>
<hr>
<p>My try :
$$\lim_{x \to \infty }\dfrac{x^2+f(x)}{x+1+2f(x)}=\lim_{x \to \infty }\dfrac{x^{n+2}+x^nf(x)}{x^{n+1}+2x^nf(x)}=\infty?$$
Please help me.<... | Community | -1 | <p>$S(n)-S(n-1)=\sum_{i=1}^{n-1}iS(i)-\sum_{i=1}^{n-2}iS(i)=(n-1)*S(n-1)$</p>
<p>$S(n)=nS(n-1)$</p>
<p>$S(n)=1*2*\ldots*n=n!$</p>
|
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Peter Arndt | 733 | <p>Hi Alex! </p>
<p>About the second question: I think senior mathematicians don't necessarily escape the criterion of general interest, but it can become a self-fulfilling prophecy: The mere fact that a senior mathematician is studying something can raise interest in the object of study among the mathematical communi... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Ronnie Brown | 19,949 | <p>It may be helpful to say how I got into groupoids. </p>
<p>In the 1960s, I was writing a topology text and wanted to do the fundamental group of a cell complex, which required the van Kampen Theorem (I have now been persuaded to call this the Seifert-van Kampen theorem, as on wikipedia, so I call it SvKT). I was k... |
39,828 | <p>Dear MO-community, I am not sure how mature my view on this is and I might say some things that are controversial. I welcome contradicting views. In any case, I find it important to clarify this in my head and hope that this community can help me doing that.</p>
<p>So after this longish introduction, here goes: Man... | Vamsi | 3,709 | <p>This paper has a very nice introduction (it is on "pointless topology"). So apparently, one may come up with very random definitions for their own sake and hope someone "applies" them to more "concrete" problems.
<a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathe... |
2,454,455 | <p>I know this is a soft and opinion based question and I risk that this question get's closed/downvoted but I still wanted to know what other persons, who are interested in mathematics, think about my question.</p>
<p>Whenever people are talking about the most beautiful equation/identity Euler's identity is cited in... | Especially Lime | 341,019 | <p>The main reason is simply that the standard version gives more information. If I know $e^{i\pi}=-1$ then I can deduce $e^{2i\pi}=1$, but not the other way round (knowing $e^{2i\pi}=1$ doesn't tell me whether $e^{i\pi}$ is $+1$ or $-1$).</p>
|
2,125,297 | <p><a href="https://i.stack.imgur.com/cBCJe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/cBCJe.png" alt="enter image description here"></a></p>
<p>I would like to know how exactly this works. I watched a khan academy video: "Multiplying matrices" but in this case he would've done B*A and had 2 co... | amd | 265,466 | <p>The distance between a vector and a subspace is measured along a direction orthogonal to the subspace, i.e., it’s the length of the orthogonal rejection of the vector from the subspace. So, if $\mathbf\pi_W$ is orthogonal projection onto $W$, then distance to $W$ from $\mathbf v$ is $\|\mathbf v-\mathbf\pi_W\mathbf ... |
840,211 | <p>I have line integral of a vector function: $\vec{F}=-e^{-x}\sin y\,\,\vec{i}+e^{-x}\cos y\,\,\vec{j}$
The path is a square on the $xy$ plane with vertices at $(0,0),(1,0),(1,1),(0,1)$</p>
<p>Of course it is a closed line integral, and I know the result should be zero. </p>
<p>I am baffled how can you calculate $\s... | Mathmo123 | 154,802 | <p>In the first case, you're being asked to show that if $n = 2^m + 1$ is prime, then $n$ must be a Fermat prime - i.e. $n$ is of the form $n = 2^{2^k} + 1$ for some $k$. So you need to show that if $n$ is prime, then $m$ is a power of 2.</p>
<p>In the second case, you're being asked to show that if $n = a^m - 1$ is p... |
698,474 | <p>I am trying to find the phi(18). Using an online calculator, it says it is 6 but im getting four.
<br/>
The method I am using is by breaking 18 down into primes and then multiplying the phi(primes)</p>
<p>$$=\varphi (18)$$
$$=\varphi (3) \cdot \varphi(3) \cdot \varphi(2)$$
$$= 2 \cdot 2 \cdot 1$$
$$= 4$$</p>
| Álvaro Lozano-Robledo | 14,699 | <p>By definition, $\varphi(18)$ is the number of elements in the set
$$\{n : 1\leq n \leq 17, \text{ with } \gcd(n,18)=1\}=\{1,5,7,11,13,17\}.$$
Thus, $\varphi(18)=6$. Similarly, $\varphi(9)$ is the number of elements in the set
$$\{n : 1\leq n \leq 8, \text{ with } \gcd(n,9)=1\}=\{1,2,4,5,7,8\},$$
so $\varphi(9)=6$, ... |
105,058 | <p>Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, $S$ of integers.</p>
<p>When $S$ is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which ... | anon | 11,763 | <p>If you mean that every positive integer gets a unique multiplicative factorization, then <em>no</em>, there is no other canonical representation. Why? Because then every prime number $p$ can be factorized, but the only way that's possible is if the components of the factorizations include the primes them-selves. Fur... |
3,276,264 | <h1>My attempt</h1>
<p>Based on the sine rule and the graph of <span class="math-container">$\sin A = k a$</span> (where <span class="math-container">$k$</span> is a constant) in interval <span class="math-container">$(0,\pi)$</span>,
increasing <span class="math-container">$a$</span> up to <span class="math-containe... | CY Aries | 268,334 | <p>We need <span class="math-container">$a\ne0$</span> and <span class="math-container">$b\ne0$</span>. If it holds, then we have</p>
<p><span class="math-container">$(a^3+b^3)^2=c^6=(a^2+b^2)^3$</span></p>
<p><span class="math-container">$a^6+2a^3b^3+b^6=a^6+3a^4b^2+3a^2b^4+b^6$</span></p>
<p><span class="math-cont... |
778,294 | <p>If $G$ is an open subset of $ \mathbb R$ such that $ 0 \notin G $ , then is it true that $H:=\{ xy \mid x , y \in G \}$ is an open subset of $ \mathbb R$ ?</p>
| Nick Peterson | 81,839 | <p><strong>Hint:</strong></p>
<p>There's no need for anything fancy here; you can go straight to the definition.</p>
<p>Let $x,y\in G$, and consider $xy$. Because $G$ is open, you can find some $\epsilon$ so that $(y-\epsilon,y+\epsilon)\subseteq G$. Can you use that to show that $H$ contains an interval around $xy... |
520,203 | <p>Here's what I'm reading: every regular bipartite graph has a 1-factor.</p>
<p>But I understand that not every regular graph has a 1-factor.</p>
<p>So, I was thinking if it's possible to find a $k$-regular simple graph without 1-factor for any fixed $k$?</p>
| hardmath | 3,111 | <p>It is possible to have a $k$-regular (simple) graph with no 1-factor for each $k \gt 1$ (obviously in the trivial case $k=1$ the graph itself is a 1-factor).</p>
<p>For $k$ even the complete graph on $k+1$ nodes is an example, since there are an odd number of nodes (and a 1-factor or <em>perfect matching</em> impli... |
3,290,073 | <p>I'm trying to help my daughter learn maths. She is struggling with factors, which is to work out what numbers go into a larger number (division).</p>
<p>I've already learned that by summing numbers, if they make 3, it can divide by 3. I also know the rules for 2, 5, 6, 9 and 10.</p>
<p>I'm trying to see if there i... | I. Chekhov | 684,502 | <p>The test for divisibility by <span class="math-container">$4$</span> is given any integer <span class="math-container">$n$</span> consider the last two digits; if that two-digit number is divisible by <span class="math-container">$4$</span> then so is <span class="math-container">$n$</span>.</p>
<p>Example.</p>
<p... |
3,290,073 | <p>I'm trying to help my daughter learn maths. She is struggling with factors, which is to work out what numbers go into a larger number (division).</p>
<p>I've already learned that by summing numbers, if they make 3, it can divide by 3. I also know the rules for 2, 5, 6, 9 and 10.</p>
<p>I'm trying to see if there i... | Zach Hunter | 588,557 | <p>The key is that 100 is divisible by 4. So, we have have:</p>
<p><span class="math-container">$$12345678956 = (123456789)(100) + 56 = (123456789)(25)(4) + (14)(4) = ((123456789)(25)+14)(4)$$</span></p>
<p>Therefore, if the first two digits are divisible by four, the whole thing is divisible by four. In fact, the re... |
2,348,290 | <p>The question and its answer is given in the following pictures:<a href="https://i.stack.imgur.com/xYhqD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYhqD.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/Gv3IA.png" rel="nofollow noreferrer"><img src="https... | Verdruss | 458,683 | <p>For your first question: The first summand $\int_0^1 \lfloor x \rfloor e^{-x}$ gives 0, because $\lfloor x \rfloor = 0$ for all $x\in [0,1)$. That's why the whole integral vanishes.</p>
<p>For your second question: In the interval $[n,n+1)$ the function $\lfloor x \rfloor e^{-x} = n e^{-x}$, that's why only the $x$... |
2,348,290 | <p>The question and its answer is given in the following pictures:<a href="https://i.stack.imgur.com/xYhqD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYhqD.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/Gv3IA.png" rel="nofollow noreferrer"><img src="https... | FDP | 186,817 | <p>The solution is too complicated according to me.</p>
<p>A way to simplify,</p>
<p>$\begin{align} J&=\sum_{n=1}^{\infty} \int_n^{n+1} ne^{-x}\\
&=\sum_{n=1}^{\infty} \Big[-ne^{-x}\Big]_n^{n+1}\\
&=\sum_{n=1}^{\infty}n\left(e^{-n}-e^{-(n+1)}\right)\\
&=\sum_{n=1}^{\infty}ne^{-n}-\sum_{n=1}^{\infty}ne... |
2,348,290 | <p>The question and its answer is given in the following pictures:<a href="https://i.stack.imgur.com/xYhqD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xYhqD.png" alt="enter image description here"></a></p>
<p><a href="https://i.stack.imgur.com/Gv3IA.png" rel="nofollow noreferrer"><img src="https... | Siddhartha | 427,792 | <p>Our task here is nothing but calculating Laplace Transform of $\lfloor t\rfloor$ at $s=1$</p>
<blockquote>
<p>$$\begin{align}F(s)&=\displaystyle\int_0^\infty \lfloor t \rfloor e^{-st}dt\\&=\displaystyle\int_0^1 \lfloor t \rfloor e^{-st}dt +\displaystyle\int_1^2 \lfloor t \rfloor e^{-st}dt +\displaystyle\i... |
3,244,361 | <p>I need some help on two exercises from Kiselev's geometry, about straight lines.</p>
<blockquote>
<p>Ex 7: Use a straightedge to draw a line passing through two points given on a sheet of paper. Figure out how to check that the line is really straight. Hint: Flip the straightedge upside down.</p>
</blockquote>
<... | Jeremy Weissmann | 87,206 | <p>I’m late to the game but here is my argument for #8. (Incidentally, I don’t think Kiselev is suggesting to use #7 to prove #8. He says use the technique to check that the line is straight, and, separately, asks us to explain why it is true.)</p>
<p>When we fold the paper, we identify each point on the bottom 'half' ... |
2,941,579 | <p>In Taylor's series, to determine the number of terms needed to obtain the desired accuracy, sometimes one needs to solve inequalities of the form <span class="math-container">$$\frac{a^n}{n!}<b,$$</span>
where <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are fixed positive n... | DanielWainfleet | 254,665 | <p>Let <span class="math-container">$f(x)=4x-5.$</span> The object is to prove that for any <span class="math-container">$\epsilon >0$</span> we can find some <span class="math-container">$\delta >0$</span> such that <span class="math-container">$|f(x)-7|<\epsilon$</span> whenever <span class="math-container"... |
3,096,933 | <p>In studying the causes of power failures, the following data have been gathered:
10% are due to a transformer damage, 75% are due to line damage, 5% involve both
problems. Based on these percentages, find the probability that a given power failure
involves:</p>
<p>a) line damage given that there is transformer dama... | Alex | 132,353 | <p>Going off my hint: recall that for two events <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> (with <span class="math-container">$P(Y) \neq 0$</span>) we have</p>
<p><span class="math-container">$$
P(X|Y) = \frac{P(X \cap Y)}{P(Y)}
$$</span></p>
<p>So in your case, you want</p>
... |
866,921 | <p>On the <a href="http://chat.stackexchange.com/rooms/36/mathematics">Mathematics chat</a> we were recently talking about the following problem <a href="https://math.stackexchange.com/users/32016/chriss-sis">@Chris'ssis</a> had to solve during an interview :</p>
<p>$$3\times 4=8$$
$$4\times 5=50$$
$$5\times 6=30$$ ... | Kevin | 147,209 | <p>This is what I have so far, it seems a bit more intuitive than Omran's solution.</p>
<p>Based on the flip-flopping numbers, I figured the answer has to rely on the prime factorization of the numbers in question. So in particular, we see:</p>
<p>$$3 \times 2^2 \Rightarrow 2$$
$$2^2 \times 5 \Rightarrow 2*5$$
$$5 \t... |
3,534,896 | <p>If <span class="math-container">${\sqrt 3} - {\sqrt 2}, 4- {\sqrt 6}, p{\sqrt 3} - q {\sqrt 2}$</span> form a geometric progression, find the values of p and q.</p>
<p>So I take the second term <span class="math-container">$4-{\sqrt 6} =( {\sqrt 3} - {\sqrt 2}) (r)$</span> , where r is the common ratio.</p>
<p><... | Michael Rozenberg | 190,319 | <p>If it means that <span class="math-container">$p$</span> and <span class="math-container">$q$</span> are rationals, so use the following:<span class="math-container">$$(\sqrt3-\sqrt2)(p\sqrt3-q\sqrt2)=(4-\sqrt6)^2$$</span> or
<span class="math-container">$$3p+2q-(p+q)\sqrt6=22-8\sqrt6.$$</span>
I got <span class="ma... |
3,534,896 | <p>If <span class="math-container">${\sqrt 3} - {\sqrt 2}, 4- {\sqrt 6}, p{\sqrt 3} - q {\sqrt 2}$</span> form a geometric progression, find the values of p and q.</p>
<p>So I take the second term <span class="math-container">$4-{\sqrt 6} =( {\sqrt 3} - {\sqrt 2}) (r)$</span> , where r is the common ratio.</p>
<p><... | Quanto | 686,284 | <p>Note </p>
<p><span class="math-container">$$(4- {\sqrt 6})^2=({\sqrt 3} - {\sqrt 2})( p {\sqrt 3} - q {\sqrt 2})$$</span></p>
<p>or,</p>
<p><span class="math-container">$$22-8\sqrt6 = 3p +2q -(p+q)\sqrt6$$</span></p>
<p>Therefore,</p>
<p><span class="math-container">$$22= 3p +2q,\>\>\>\>\>p+q=8$$... |
2,172,870 | <p>I have two vectors $u$, and $v$, I know that $\mid u \mid$ = 3 and $\mid v \mid$ = 5, and that $u\cdot v = -12$. I need to calculate the length of the vector $(3u+2v) \times (3v-u)$.</p>
<p>Because I know the dot product of the vectors I know what cosine of the angle between them is $\cos \theta = -0.8$, and also $... | TonyK | 1,508 | <p>No closed knight's tour is possible on a board with an odd number of squares, because each move changes the colour of the knight's square. So after an odd number of moves, you can't be back at the starting square, because it's the wrong colour.</p>
|
3,041,841 | <p>Let <span class="math-container">$f(z)$</span> be holomorphic function on an open unit disk such that <span class="math-container">$\lim_{z \to 1} f(z)$</span> doesn't exists. Let <span class="math-container">$f(z) = \Sigma_{i=1}^{\infty} a_iz^i$</span> be its Taylor series around <span class="math-container">$0$</s... | José Carlos Santos | 446,262 | <p>Your apporach is wrong, because the radius of convergence of <span class="math-container">$\sum_{n=0}^\infty z^n$</span> is <span class="math-container">$1$</span>.</p>
<p>The Taylor series about <span class="math-container">$a$</span> of a holomorphic function <span class="math-container">$f$</span> always converg... |
3,858,966 | <blockquote>
<p>Let <span class="math-container">$A= \{(x,y,z) \in \Bbb R^3 \vert x+y<z < x^2+y^2 \}$</span>. Show that <span class="math-container">$A$</span> is an open set in <span class="math-container">$\Bbb R^3$</span> defined by the Euclidean metric.</p>
</blockquote>
<p>So <span class="math-container">$A$... | Henno Brandsma | 4,280 | <p>The method using continuous functions is by far the least messy way to solve such problems.</p>
<p>Define <span class="math-container">$f(x,y,z) = x+y-z$</span> and <span class="math-container">$g(x,y,z) = x^2+y^2-z$</span>. Then <span class="math-container">$x+y < z$</span> can be described as <span class="math-... |
3,912,635 | <p>I am currently trying to understand the proof of Proposition 4.3.18 in Pedersen's Analysis now, which reads</p>
<blockquote>
<p>To each Tychonoff space <span class="math-container">$X$</span> there is a Hausdorff compactification <span class="math-container">$\beta(X)$</span>, with the property that every continuous... | QuantumSpace | 661,543 | <p>Recently, I wrote all this out in detail for myself, so here I share my notes with you. Note that the assumption that <span class="math-container">$X$</span> is Tychonoff can be ommitted. The construction works for every topological space. The Tychnoff assumption is only there to ensure that the canonical inclusion ... |
792,813 | <p>Let A be a random variable defined as:</p>
<ul>
<li>With probability $p[i]$, the random variable $B[i]$ is drawn</li>
<li>$B[i] ~ N[mu[i],sigma[i]]$</li>
<li>probabilities $p[i]$ sum up to one</li>
</ul>
<p>I know how to compute the mean, which is given by:</p>
<p>$$E[A] = p[1]*mu[1] + .. + p[N]*mu[N]$$</p>
<p>... | Claude Leibovici | 82,404 | <p>You can also expand the integrand as a Taylor series built around $x=0$ starting with $$\cos(y)=1-\frac{y^2}{2}+\frac{y^4}{24}+O\left(y^6\right)$$ Now, replace $y$ by $x^2$ and multiply the result by $x^3$. So, the integrand is $$cos(x^2)x^3=x^3-\frac{x^7}{2}+O\left(x^{10}\right)$$ Integrate between $0$ and $z$ to g... |
264,595 | <p>I've been trying to find an asymptotic expansion of the following series</p>
<p>$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$</p>
<p>and</p>
<p>$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n(n!{\sqrt{n}}) }$$</p>
<p>around $+\infty$, in the from</p>
<p>$$\exp(x^2)\Big(1+\frac{a_1}{... | Johannes Trost | 37,436 | <p>This is another, totally different (and correct !) approach for answering the question. It is simply too long for a comment. So I decided to write it in a new answer. (Although that might look odd, but my previous answer, though accepted, is wrong.)</p>
<p>First define
$$
I_{\mu}(y) = y^{1/2} \sum_{n=1}^{\infty} \ ... |
812,263 | <p>To further explain the title:</p>
<p>Is there a probabilistic reason as to why a 6-sided die has the opposing sides suming to 7?</p>
<p>My argument begun when a friend decided to use <a href="http://ecx.images-amazon.com/images/I/51OvRGphnOL.jpg" rel="nofollow">this die</a> instead of <a href="http://nerdywithchil... | tpb261 | 125,795 | <p>If the dice are unbiased, there is no reason for choosing any particular arrangement. You could as well pull a card from a deck of six. No difference.</p>
|
1,621,989 | <p>If we have any two co-prime positive integers <em>x</em> and <em>y</em>, does there always exist a positive integer <em>C</em> such that all integers greater than <em>C</em> can be expressed as <em>Ax</em>+<em>By</em> where <em>A</em> and <em>B</em> are also non-negative integers?</p>
<p>Do we have a formula to cal... | Slade | 33,433 | <p>This is sometimes called the <a href="https://en.wikipedia.org/wiki/Coin_problem" rel="nofollow">coin problem</a>, and the answer for two coins of relatively prime denominations $a$ and $b$ is $ab-a-b$, when we are allowed to use zero of either coin.</p>
<p>To find the answer when the coefficients are required to b... |
1,934,259 | <blockquote>
<p><strong>Problem.</strong></p>
<p>Let <span class="math-container">$\emptyset \subset A\subset X$</span> and <span class="math-container">$\emptyset \subset B\subset Y$</span>. If <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are connected, show that <span class="mat... | Tsemo Aristide | 280,301 | <p>Let $p:X\times Y\rightarrow (X\times Y)/(A\times B)$ be the quotient map; it is surjective. Consider a continuous function $f:(X\times Y)/ (A\times B)\to\{\pm1\}$. $f\circ p$ is continuous; since $X\times Y$ is connected, $f\circ p$ is constant. Since $p$ is surjective, $f$ is constant.</p>
|
688,577 | <p>A beautiful polyhedron with 20 hexagons and 60 pentagons can be seen here: <a href="http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/" rel="nofollow">http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/</a> . Euler formula and the corresponding Diophantine equation give a... | Barry Cipra | 86,747 | <p>Here's a kludgy solution. Start with a "hat box" with pentagons for top and bottom and $5$ square sides. Inside each square side draw a pentagon, and connect four of the vertices of that pentagon to the four corners of the square. In two of those connecting segments from opposite vertices of the square, draw an e... |
2,003,660 | <p>I got this exercise from the textbook Book of Proof, CH4 E12. I've tackle this problem in the following manner:</p>
<p>Suppose x is a real and $0 < x < 4$, it follows that,</p>
<p>\begin{align*}
&\Rightarrow 0 - 2 < x - 2 < 4 - 2 \\
&\Rightarrow 4 < (x - 2)^2 < 4\\
&\Rightarrow ... | Benson Lin | 371,844 | <p>To get from $0−2<x−2<4−2$ to $4<(x−2)^2<4$ you have simply squared the entire inequality. This is invalid since you have not only led to a contradiction ($4 < k < 4$ implies no such $k$ exists), you have also mistakenly thought that if $a$ and $b$ are reals and $a < b$, then $a^2 < b^2$. This... |
4,331,258 | <p>I'm trying to find the median of <span class="math-container">$f(x) = 4xe^{-2x}$</span>.</p>
<p>So far, I've tried solving for <span class="math-container">$q_{50}$</span> by plugging it into an integral and setting it equal to 0.5 like so: <span class="math-container">$\int_{0}^{q_{50}} 4xe^{-2x} dx = 0.5$</span>. ... | Vítězslav Štembera | 663,062 | <p>Your equation
<span class="math-container">\begin{align}
-2q_{50}e^{-2q_{50}} - e^{-2q_{50}} + 1 = 0.5
\end{align}</span>
i.e.
<span class="math-container">\begin{align}
(2q_{50}+1)e^{-2q_{50}}= 0.5
\end{align}</span>
is correct, however it is trascendental and must be solved numerically. Using MAPLE for example you... |
457,427 | <p>Find the derivative of the functions:<br>
$$\int_{x^2}^{\sin(x)}\sqrt{1+t^4}dt$$<br></p>
<p>In class we had the following solution:<br>
By the fundamental theorem of calculus we know that <br>
$$\left(\int_a^xf(t)dt\right)'=f(x)$$ So<br>
$$\int_{x^2}^0\sqrt{1+t^4}dt+\int_0^{\sin(x)}\sqrt{1+t^4}dt=$$<br>
$$\int_0^{\... | Ron Gordon | 53,268 | <p>You are forgetting the chain rule; you have to take the derivative of each function in the limits. Thus your derivative is</p>
<p>$$\sqrt{1+\sin^4{x}} \frac{d}{dx}\sin{x} - \sqrt{1+x^8} \frac{d}{dx} x^2$$</p>
<p>which I believes gives you your answer.</p>
|
1,283,037 | <blockquote>
<p>Consider the series: $$ \sum_{i=1}^\infty \frac{i}{(i+1)!} $$ Make a
guess for the value of the $n$-th partial sum and use induction to prove
that your guess is correct.</p>
</blockquote>
<p>I understand the basic principles of induction I think I would have to assume the n-1 sum to be true and... | Brian M. Scott | 12,042 | <p>In order to make a guess, you should begin by calculating some partial sums:</p>
<p>$$\begin{align*}
\sum_{i=1}^1\frac{i}{(i+1)!}&=\frac12\\\\
\sum_{i=1}^2\frac{i}{(i+1)!}&=\frac12+\frac2{3!}=\frac12+\frac13=\frac56\\\\
\sum_{i=1}^3\frac{i}{(i+1)!}&=\frac56+\frac3{4!}=\frac56+\frac18=\frac{23}{24}\\\\
\... |
3,243,243 | <p>I am learning category theory using Basic Category Theory by Tom Leinster as my main source. In the chapter on natural transformations he says that isomorphism of categories is unreasonably strict for the notion of the sameness of two categories. Isomorphism would require functors,
<span class="math-container">$$
F:... | Chessanator | 363,017 | <p>Well, the actual difference between the two statements is that for an equivalence of categories, we only require that that the composites <span class="math-container">$F \circ G$</span> and <span class="math-container">$G \circ F$</span> are naturally isomorphic to the identity functors rather than exactly equal. Th... |
2,081,001 | <p>The number $\frac{22}{7}$ is irrational in our base-$10$ system, but in, say, base-$14$, it is rational (it comes out to $3.2$ in that system).</p>
<p>It's easy for fractions that are irrational as decimals, as you can just represent them in a base that's double the denominator of the fraction. However, what if I h... | mweiss | 124,095 | <p>The number $22/7$ is <em>not</em> irrational, regardless of the number system one uses to express it. On the other hand $\pi$ <em>is</em> irrational, regardless of the number system one uses to express it. They are different numbers: $22/7$ is a good rational approximation to $\pi$, but they are not equal.</p>
... |
14,765 | <p>I like to make the "dominoes" analogy when I teach my students induction.</p>
<p>I recently came across the following video:</p>
<p><a href="https://www.youtube.com/watch?v=-BTWiZ7CYoI" rel="noreferrer">https://www.youtube.com/watch?v=-BTWiZ7CYoI</a></p>
<p>In this video, a sequence of concrete block wall caps ar... | GeauxMath | 10,521 | <p>I have similar issues, and handle it in the following way:</p>
<ol>
<li>On the first day of class I do a review of basic algebra, things like:
<span class="math-container">$$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$</span>
<span class="math-container">$$\sqrt{a + b} \neq \sqrt{a} + \sqrt{b} $$</span>
<span ... |
1,574,290 | <p>How do I prove this? </p>
<p>For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.</p>
| Jack D'Aurizio | 44,121 | <p>A slightly faster proof comes from noticing that </p>
<p>$$q(x,y) = x^2-xy-y^2 = (y-x)^2-y(x-y)-y^2 = q(y-x,y)$$
and:
$$ q(-x,y) = x^2+xy-y^2 = -q(y,x) $$
hence:
$$ q(f_{n+1},f_n) = q(-f_{n-1},f_n) = -q(f_n,f_{n-1}) = \ldots = (-1)^n q(1,0).$$</p>
|
2,138,963 | <p>finding $\displaystyle \int^{\pi}_{-\frac{\pi}{3}}\bigg[\cot^{-1}\bigg(\frac{1}{2\cos x-1}\bigg)+\cot^{-1}\bigg(\cos x - \frac{1}{2}\bigg)\bigg]dx$</p>
<p>Attempt:</p>
<p>\begin{align}
& \int^{\frac{\pi}{3}}_{-\frac{\pi}{3}}\bigg[\cot^{-1}\bigg(\frac{1}{2\cos x-1}\bigg)+\cot^{-1}\bigg(\cos x - \frac{1}{2}\bigg... | victoria | 412,473 | <p>I see some tricks in here. Ideas, as yet unfinished, I'm out of time now but will try to come back later.</p>
<p>Consider $cot^{-1}(b) = \theta \leftrightarrow cot(\theta) = b$</p>
<p>and then $tan(\theta ) = 1/b$ so $\theta = tan^{-1}(1/b)$</p>
<p>Also $(cos(x) - 1/2 ) = (2 cos(x) -1)/2$</p>
<p>$$f(x) = cot^{-1... |
610,472 | <p>What is the value of these limits;</p>
<p>$\lim_{x\rightarrow 1^{+}}\frac{\lfloor x\rfloor-1}{\lfloor x\rfloor-x}$</p>
<p>$\lim_{x\rightarrow 1^{-}}\frac{\lfloor x\rfloor-1}{\lfloor x\rfloor-x}$</p>
| mathlove | 78,967 | <p>1) We may suppose that $1\lt x\lt 2$. So we have $\lfloor x\rfloor =1$ and $\lfloor x\rfloor=1 \not= x$, so
$$\lim_{x\to1+}\frac{\lfloor x\rfloor -1}{\lfloor x\rfloor-x}=\lim_{x\to1+}\frac{0}{1-x}=\lim_{x\to1+}0=0.$$</p>
<p>2) We may suppose that $0\lt x\lt1$. So we have $\lfloor x\rfloor=0$, so
$$\lim_{x\to1-}\f... |
3,685,969 | <p>I have been investigating the brachistochrone problem with friction and in my derivations, I would like help solving the Euler-Lagrange equation below</p>
<p><span class="math-container">$\frac{d}{dx}\frac{\partial F}{\partial y'}=\frac{\partial F}{\partial y}$</span>
where <span class="math-container">$F=\sqrt{\fr... | Quanto | 686,284 | <p>Let <span class="math-container">$x\to -x$</span> over <span class="math-container">$(-1,0)$</span></p>
<p><span class="math-container">\begin{align}
\int_{-1}^1\frac{\ln(1-x^2)}{(1-\beta x)^2}dx
= & \ 2\int_0^1\ln(1-x^2) \frac{1+\beta^2x^2}{(1-\beta^2x^2)^2} dx\\
= & \ 2\int_0^1 \ln(1-x^2)\ d\left( \frac{x... |
2,276,429 | <p>I've tried multiple variations of polygons but can't find any that work. Do they exist?</p>
<p>Is it possible to draw a polygon on a grid paper and divide it into two equal parts by a cut of the shape shown on the Figure (a)? </p>
<p>Solve the same problem for a cut shown on Figure (b).</p>
<p>Solve the same prob... | nickgard | 420,432 | <p>Here are (a) and (b). The key thing is to repeat the shape of the cut twice in the outline of the polygon.</p>
<p><a href="https://i.stack.imgur.com/IpTRX.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IpTRX.png" alt="enter image description here"></a></p>
<p><strong>EDIT:</strong> to add a sol... |
317,666 | <p>What is the difference between $\mathbb{H}$ and $Q_8$? Both are called <em>quaternions.</em></p>
| Alex Youcis | 16,497 | <p>One is the <em><a href="http://en.wikipedia.org/wiki/Quaternion" rel="nofollow">Hamiltonian Quaternions</a></em> and has many descriptions, perhaps the most important (for things that immediately interest me) is that it is the )up to equivalence) only non-trivial central simple algebra over $\mathbb{R}$--it is also ... |
317,666 | <p>What is the difference between $\mathbb{H}$ and $Q_8$? Both are called <em>quaternions.</em></p>
| Marc van Leeuwen | 18,880 | <p>The first is a division ring (also called skew field, which is obviously infinite, and necessarily so by <a href="http://en.wikipedia.org/wiki/Wedderburn%27s_little_theorem" rel="nofollow">Wedderburn's theorem</a>) and the second is a finite non-abelian group (in fact a subgroup of the multiplicative group of the fi... |
2,699,483 | <blockquote>
<p>The probability that a fair-coin lands on either <strong>Heads</strong> or <strong>Tails</strong> on a certain day of the week is $1/14$. </p>
<p><strong>Example: (H, Monday), (H, Tuesday) $...$ (T, Monday), (T, Tuesday) $...$</strong><br>
Thus, $(1/2 \cdot 1/7) = 1/14$. There are $14$ such ou... | NewGuy | 518,506 | <p>CORRECTED AS REASONS GIVEN BY BY JGON</p>
<p>Sample Space for single throw = {MH,MT,TuH,TuT,...........,SH,ST} = 14</p>
<p>Sample Space for two throws = $14*14 = 196$</p>
<p>M: 2 heads are thrown = $7*7= 49$</p>
<p>N: atleast single head is thrown on saturday = {(SaH,?)(?,SaH)} = $2*14 $</p>
<p>But we have cou... |
2,699,483 | <blockquote>
<p>The probability that a fair-coin lands on either <strong>Heads</strong> or <strong>Tails</strong> on a certain day of the week is $1/14$. </p>
<p><strong>Example: (H, Monday), (H, Tuesday) $...$ (T, Monday), (T, Tuesday) $...$</strong><br>
Thus, $(1/2 \cdot 1/7) = 1/14$. There are $14$ such ou... | Remy | 325,426 | <p>Intuitively, I would think the result would be greater than $\frac{1}{2}$ because of that slight chance we get $2$ heads on Saturday.</p>
<p>Let $P$ denote the event that we flip $2$ heads that week.</p>
<p>Let $Q$ denote the event that we flip at least one head on Saturday.</p>
<p>I find it easier to flip $P(P\m... |
2,699,483 | <blockquote>
<p>The probability that a fair-coin lands on either <strong>Heads</strong> or <strong>Tails</strong> on a certain day of the week is $1/14$. </p>
<p><strong>Example: (H, Monday), (H, Tuesday) $...$ (T, Monday), (T, Tuesday) $...$</strong><br>
Thus, $(1/2 \cdot 1/7) = 1/14$. There are $14$ such ou... | Boyku | 567,523 | <p>the result of the two flips may be represented by one mark in a <span class="math-container">$14\times14$</span> grid. There are <span class="math-container">$13$</span> little squares that represent double head results. There are 27 squares that represent flips that contain an HSat.</p>
<p>The required probability ... |
257,027 | <p><strong>Question:</strong> Consider a distribution $D$, and $n$ i.i.d. random variables $X_i$, all distributed according to $D$. Let $p^D_2:=\Pr[X_1=X_2]$. What is a lower bound for $p^D_n:=\Pr[\exists i\neq j. X_i=X_j]$ (as a function of $p^D_2$)?</p>
<p><strong>Conjecture:</strong> $p^D_n \geq 1-\bigl(1-p^D_2\big... | esg | 48,831 | <p>I reformulate slightly, please check.</p>
<p>You are considering a sequence $X_1,X_2,\ldots$ of (discrete) i.i.d
random variables
and want an upper bound for the probability $\mathbb{P}(R>n)$
in terms of $\sqrt{\beta}$, where $\beta:= {1 \over \mathbb{P}(X_1=X_2)}$,
and $R:=\inf\{ n\geq 2\,:\,X_n\in\{X_1,\ldo... |
4,510,808 | <p>Value of p such that <span class="math-container">$\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt[3]{{x + 1}} + \sqrt[3]{{x - 1}} - 2\sqrt[3]{x}} \right)} \right)$</span> is some finite | non-zero number.</p>
<p>My approach is as follow</p>
<p><span class="math-container">$\mathop {\lim }\limits_{... | Z Ahmed | 671,540 | <p><span class="math-container">$$L=\mathop {\lim }\limits_{x \to \infty } \left( {{x^p}\left( {\sqrt[3]{{x + 1}} + \sqrt[3]{{x - 1}} - 2\sqrt[3]{x}} \right)} \right)$$</span>
<span class="math-container">$$L=\mathop {\lim }\limits_{x \to \infty } \left( {{x^{p+1/3}}\left( {\sqrt[3]{{1 + 1/x}} + \sqrt[3]{{1 - 1/x}} - 2... |
1,812,525 | <p>Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) and $\{\lambda_n\}_{n=1}^\infty$ is a sequence of numbers (which may be complex if the Hilbert space is complex).</p... | egreg | 62,967 | <p>The notation $S^\perp$ means $\{x\in H\mid \langle x,y\rangle=0,\text{ for all }y\in S\}$.</p>
<p>Let $x\in\ker T$; you need to prove that, for every $m$ with $\lambda_m\ne0$, you have $\langle x,\varphi_m\rangle=0$.</p>
<p>You know that $\sum_n\lambda_n\langle x,\varphi_n\rangle\varphi_n=0$, so also
$$
\Bigl<\... |
4,651,798 | <p>This question comes from the province-stage olympiad in my country in order to qualify for the national stage:</p>
<blockquote>
<p>Given the set <span class="math-container">$S=\{1,2,3,4\}$</span>.The number of non-empty subsets <span class="math-container">$A_1,A_2, ..., A_6$</span> that fulfil these three criteria... | Rezha Adrian Tanuharja | 751,970 | <p>Label each element with <span class="math-container">$0,1,2,3,4,5$</span> or <span class="math-container">$6$</span>. Then put the elements into the six subsets with the following rules:</p>
<ul>
<li>Those with label <span class="math-container">$0$</span> is not in any subset</li>
<li><span class="math-container">$... |
15,316 | <blockquote>
<p>What is the length $f(n)$ of the shortest nontrivial group word $w_n$ in $x_1,\ldots,x_n$ that collapses to $1$ when we substitute $x_i=1$ for any $i$?</p>
</blockquote>
<p>For example, $f(2)=4$, with the commutator $[x_1,x_2]=x_1 x_2 x_1^{-1} x_2^{-1}$ attaining the bound. </p>
<p>For any $m,n \ge... | Sam Nead | 1,650 | <p>See the paper "Brunnian links" by Gartside and Greenwood, published in Fundamenta Mathematicae. Theorems 8 and 7 imply that iterated commutators are optimal and the sequence you suggest gives the minimal length. </p>
|
18,530 | <p>Sorry about the title, I have no idea how to describe these types of problems.</p>
<p>Problem statement:</p>
<p>$A(S)$ is the set of 1-1 mappings of $S$ onto itself. Let $S \supset T$ and consider the subset $U(T) = $ { $f \in A(S)$ | $f(t) \in T$ for every $t \in T$ }. $S$ has $n$ elements and $T$ has $m$ eleme... | Qiaochu Yuan | 232 | <p>A basic practical reason to care about logarithms is that there are many numbers in real life which vary greatly in size, and it is both a pain and misleading to compare their sizes directly; one should instead compare the sizes of their <em>logarithms</em>, for various reasons. This is why the <a href="http://en.w... |
655,005 | <p>Show that if $k \in \mathbb{Z}$, then the integers $6k-1$, $6k+1$, $6k+2$, $6k+3$, and $6k+5$ are pairwise relatively prime.
I am still new and uncomfortable with proofs. Any help would be great. </p>
| hmakholm left over Monica | 14,366 | <p>The Kleene star produces only <em>finite</em> sequences of the alphabet symbols. The elements in <span class="math-container">$\Sigma^*$</span> for some alphabet <span class="math-container">$\Sigma$</span> can be arbitrary long, but each of them is, individually, finite.</p>
<p>Because of this, there are not enough... |
4,473,264 | <p>I have part of a circle described by three two dimensional vectors.</p>
<ul>
<li>start point <code>s1</code></li>
<li>center point <code>c1</code></li>
<li>end point <code>e</code></li>
</ul>
<p>I move the start point <code>s1</code> by <code>m1</code>, which is a <strong>known</strong> two dimensional vector. My qu... | Eugene | 726,796 | <p>the integral</p>
<p><span class="math-container">$$
\int\sqrt{1+\frac{1}{x^4}}dx
$$</span></p>
<p>can be rewritten as
<span class="math-container">$$
\int x^{-2}(1+x^4)^{\frac{1}{2}}dx,
$$</span></p>
<p>which is an example of <a href="https://planetmath.org/integrationofdifferentialbinomial" rel="nofollow noreferrer... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | user2035 | 2,035 | <p>Actually, the examples given in the answers so far are even counter-examples to the weaker statement that two sheaves $F,G$ for which there exists a covering $U_i$ such that $F|U_i\cong G|U_i$ be isomorphic.</p>
<p>For the original question regarding stalks there is a simpler example: Let $X=\{\eta,s\}$ be the topo... |
24,361 | <p>Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.</p>
<p>Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.</p>
<p... | Bugs Bunny | 5,301 | <p>How about a positive answer for a twist:-))? Pick an open cover $X=\cup_i U_i$ and try to reglue $F|_{U_i}$ into a brand new sheaf. To do that you need to pick automorphism $\sigma_{i,j}\in Aut(F|_{U_i \cap U_j})$ that agree on triple intersection. This is some Chech cocycle $Z^1_{U_i} (X, Aut (F))$. Two Chech cocyc... |
1,348,587 | <p>Let $f$ be a real-valued function (a function with target space the set of reals). Let $P(x, M)$ stand for $|f(x)| \leq M $, let $N$ be the set of positive real numbers, and let $\mathbb{R}$ be the set of real numbers.</p>
<p>a) Which of the following statements is an accurate translation of "f is bounded"?</p>
<p... | the_candyman | 51,370 | <p>(ii) Exists an $M$ such that for each $x$ real, $|f(x)| \leq M$.</p>
<p>(iii) For each $x$ real there exists an $M$ such that $|f(x)| \leq M$.</p>
<p>In the first case $M$ is unique for all $x$.</p>
<p>In the second case $M$ depends on $x$. In this sense, we can restate (iii) as follows:</p>
<p>(iii) For each $x... |
1,715,032 | <p>When given propositions to prove such as the following question:
prove that $|z+i| = |z-i|$ if $z \in \mathbb{R}$.</p>
<p>Would I have to prove this proposition without substituting $z$ for a complex number?</p>
| Bernard | 202,857 | <p>Quite simple: it is enough to prove $\lvert z+ \mathrm i\rvert^2=\lvert z- \mathrm i\rvert^2$. Now, since $z$ is real, we have $\lvert z+ \mathrm i\rvert^2= (z+ \mathrm i) (\overline{z+\mathrm i})=(z+ \mathrm i)(z-\mathrm i) $, while $\lvert z-\mathrm i\rvert^2=(z-\mathrm i)(z+\mathrm i) $.</p>
|
196,002 | <p>The paper <a href="http://www.sciencedirect.com/science/article/pii/S002212369690110X" rel="nofollow noreferrer">Lattices of Intermediate Subfactors</a> of Y. Watatani, received on December 1994, finishes by: </p>
<blockquote>
<p><strong>Prop. 6.2.</strong> $ \ $ Any finite lattice with at most five elements ca... | Carlo Beenakker | 11,260 | <p>a simple upper bound is $(\sqrt{a_{\rm max}}-\sqrt{a_{\rm min}})^2$, with $a_{\rm max}$ and $a_{\rm min}$ the largest and smallest of the $a_i$'s. So for $a_i=a_1+(i-1)\varepsilon$ this would give as upper bound $(\sqrt{a_1+(n-1)\varepsilon}-\sqrt{a_1})^2$.</p>
<p>see theorem 1 of <A HREF="http://www.ams.org/journa... |
4,216,243 | <p>I have to solve the following problem which seems difficult:</p>
<p>Find
<span class="math-container">$$ \iint_S \nabla \times F\ dS $$</span>
where <span class="math-container">$S$</span> is given by</p>
<p><span class="math-container">$$r(t,s)=\left( 9+(\cos t)(\sin s)\left(2+\frac{\sin (5s)}{2}\right), \ \ \ 9+(\... | Michael Seifert | 248,639 | <p>For future reference, we define
<span class="math-container">$$
\vec{r}(t,s) \equiv \left( x(t,s), y(t,s), z(t,s) \right)
$$</span></p>
<p>By Stokes' Theorem, the given integral is
<span class="math-container">$$
\iint_S \nabla \times F\ dS = \oint_{\partial S} F \cdot d\ell
$$</span>
where <span class="math-contain... |
2,812,122 | <blockquote>
<p>For any real numbers $x$ and $y$ satisfying $x^2y + 6y = xy^3 +5x^2 +2x$, it is known that $$(x^2 + 2xy + 3y^2) \, f(x,y) = (4x^2 + 5xy + 6y^2) \, g(x,y)$$<br>
Given that $g(0,0) = 6$, find the value of $f(0,0)$.</p>
</blockquote>
<p>I have tried expressing $f(x,y)$ in terms of $g(x,y)$. But seems ... | Jacob.Lee | 142,729 | <p>This formula comes from Thomas Simpson's Series Multisection Theory. Speak in the concrete，
a multisection of the series of an analytic function</p>
<p><span class="math-container">$$f(z) = \sum_{n=0}^\infty a_n\cdot z^n.$$</span></p>
<p>has a closed-form expression in terms of the function <span class="math-co... |
313,254 | <p>I know this question has been asked before on MO and MSE (<a href="https://mathoverflow.net/questions/59605/reference-in-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/407004/good-book-for-riemann-surfaces">here</a>, <a href="https://math.stackexchange.com/questions/1839673/books-on-ri... | Balazs | 6,107 | <p>I am copying this here from the official CUP website, so I don't think I am breaching anyone's copyright: a short review of one of my old favourites; seems to address precisely the points you are interested in. There should be a new edition out soon (?). </p>
<p>BEARDON, A. J. A primer on Riemann surfaces (London M... |
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 decided to split my answer into a more direct "answer" post and a "tidbits" post. This is the "tidbits" post, where I point out some things I determined about <span class="math-container"... |
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... | Peter | 82,961 | <p>For $x=0$ , the possible values of $y$ are $0$ to $16$, so $17$ possible values.</p>
<p>For $x=1$ , the possible values of $y$ are $0$ to $14$, so $15$ possible values.</p>
<p>$\cdots$</p>
<p>For $x=8$ , the possible values of $y$ are $0$ to $0$, so $1$ possible value.</p>
<p>In total , there are $$1+3+5+7+9+11+... |
677,708 | <p>Haven't done this for a long time, just want to know if this is the right method for a really simple example. Say we have two (obviously equal) sets $$A= \Big\{\begin{bmatrix}a & b\\c & d\end{bmatrix} : a,b,c,d \in \Bbb R, a+b=c \Big\}$$ $$B= \Big\{\begin{bmatrix}e & f\\g & h\end{bmatrix} : e,f,g,h ... | Daniel Muñoz Parsapoormoghadam | 123,645 | <p>Since, as you say, it's obvious that $A = B$, it's obvious that $A \subseteq B$.</p>
|
677,708 | <p>Haven't done this for a long time, just want to know if this is the right method for a really simple example. Say we have two (obviously equal) sets $$A= \Big\{\begin{bmatrix}a & b\\c & d\end{bmatrix} : a,b,c,d \in \Bbb R, a+b=c \Big\}$$ $$B= \Big\{\begin{bmatrix}e & f\\g & h\end{bmatrix} : e,f,g,h ... | Ali Caglayan | 87,191 | <p>It is by definition of two sets being equal that
$$A=B\iff A\subseteq B\land B\subseteq A$$</p>
<p>Here is a simple proof for your sake:</p>
<p>$$A= \Big\{\begin{bmatrix}a & b\\c & d\end{bmatrix} : a,b,c,d \in \Bbb R, a+b=c \Big\}$$ $$B= \Big\{\begin{bmatrix}e & f\\g & h\end{bmatrix} : e,f,g,h \in... |
3,938,951 | <p>The problem:</p>
<p>Given a group <span class="math-container">$G$</span> or order <span class="math-container">$n$</span>, and a Cayley embedding <span class="math-container">$\phi \ :\ G\to S_{n}$</span>. Prove that some <span class="math-container">$g\in G$</span> is of order <span class="math-container">$m$</spa... | Arturo Magidin | 742 | <p>If <span class="math-container">$g$</span> has order <span class="math-container">$m$</span>, then the orbit of <span class="math-container">$e$</span> under multiplication by <span class="math-container">$g$</span> is
<span class="math-container">$$e\mapsto g\mapsto g^2\mapsto g^3\mapsto\cdots\mapsto g^{m-1}\mapsto... |
2,965,865 | <blockquote>
<p>Finding the minimum value of <span class="math-container">$\displaystyle \frac{x^2 +y^2}{y}.$</span> where <span class="math-container">$x,y$</span> are
real
numbers satisfying <span class="math-container">$7x^2 + 3xy + 3y^2 = 1$</span></p>
</blockquote>
<p>Try: Equation <span class="math-conta... | Cesareo | 397,348 | <p>Making <span class="math-container">$y = \lambda x$</span> we have</p>
<p><span class="math-container">$$
\min f(x,\lambda) \ \ \mbox{s. t. }\ \ g(x,\lambda) = 0
$$</span></p>
<p>here </p>
<p><span class="math-container">$$
\begin{cases}
f(x,\lambda) = \frac{1+\lambda^2}{\lambda}x\\
g(x,\lambda) = x^2(7+3\lambda+... |
4,126,238 | <p>Prove <span class="math-container">$f$</span> is uniformly continuous <span class="math-container">$\implies$</span> there exist <span class="math-container">$C, D$</span> such that <span class="math-container">$|f(x)| < C + D|x|$</span>.</p>
<p>Proof below. Please verify or critique.</p>
<p>By definition of uni... | Gábor Pálovics | 922,984 | <p>I think what you do is correct. A minor correction would be instead of <span class="math-container">$j \in \{i : 0 \le i \le \lfloor|x|/\delta\rfloor+1\}$</span>, you could sum over <span class="math-container">$j \in \{i : 0\le i \cdot sgn(x) \le \lfloor|x|/\delta\rfloor+1\}$</span>. This way it works for negative... |
2,442,297 | <blockquote>
<p>Ryan has been given a salary increase of $7.39\%$. The salary increase is for the value of €$4231$.</p>
<p>His salary is now $x$. Solve for $x$.</p>
</blockquote>
<p>My head is saying </p>
<p>$$\begin{align}
4231 / 7.39 &= 572 \\
572 * 100 &= 57,200
\end{align}$$</p>
<p>is not correct,... | Théophile | 26,091 | <p>Suppose that Ryan's initial salary is $s$. Then the value of the salary increase is $0.0739s$, so $0.0739s = 4231$, and we get $s = 57253.04$ (watch out for rounding too soon!).</p>
<p>We want to calculate his <em>final</em> salary, which is
$$s + 4231.$$</p>
<p>In other words, you forgot to add his increase. (And... |
754,603 | <p>I have two equations:</p>
<ol>
<li>$x = 2^n (p+i)^{3n}$ </li>
<li>$x = 14^n p^{3n} $</li>
</ol>
<p>Here $n$, $p$, and $i$ are all integers $\geq0$.</p>
<p>I worked out (using a spreadsheet) that if $i > 1$ then the value of x in expression 1 is larger than the value of x in expression 2.</p>
<p>How can I show... | Marc van Leeuwen | 18,880 | <p>If you look at row of people lined up in front of a mirror, not only will their mirror images appear in the opposite order (the image of the person closest to the mirror comes first) but also the image of each individual person will be a mirror image (if facing towards the mirror, the mirror image will be facing out... |
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