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 |
|---|---|---|---|---|
4,404,228 | <p>Problem:<br />
Let <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span> and <span class="math-container">$x_3$</span> be integers such that <span class="math-container">$x_1 \geq 0$</span>, <span class="math-container">$x_2 \geq 0$</span> and
<span class="math-container">$x_3 \geq 0$<... | Salmon Fish | 955,791 | <p>I do not know whether you are interested in a little bit advance but much more easier technique which is <em><a href="https://en.wikipedia.org/wiki/Generating_function" rel="nofollow noreferrer">generating functions</a></em>.Nonetheless , i thought that this may help you to get rid of long techniques and expand your... |
364,394 | <p>I was asked to find the minimum and maximum values of the functions:</p>
<blockquote>
<ol>
<li>$y=\sin^2x/(1+\cos^2x)$;</li>
<li>$y=\sin^2x-\cos^4x$.</li>
</ol>
</blockquote>
<p>What I did so far:</p>
<ol>
<li><p>$y' = 2\sin(2x)/(1+\cos^2x)^2$<br />
How do I check if they are suspicious extrema points? ... | Lord_Farin | 43,351 | <p><strong>Hint:</strong> Path-connected implies connected. (A proof of this is <a href="http://www.proofwiki.org/wiki/Path-Connected_Space_is_Connected" rel="nofollow">at ProofWiki</a>.)</p>
|
364,394 | <p>I was asked to find the minimum and maximum values of the functions:</p>
<blockquote>
<ol>
<li>$y=\sin^2x/(1+\cos^2x)$;</li>
<li>$y=\sin^2x-\cos^4x$.</li>
</ol>
</blockquote>
<p>What I did so far:</p>
<ol>
<li><p>$y' = 2\sin(2x)/(1+\cos^2x)^2$<br />
How do I check if they are suspicious extrema points? ... | Srijan | 30,856 | <p>Let $n\in \mathbb{N}$ now let us define a map $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ by $f(X) = \frac{x}{||x\|}$. All that you need to show that this map is continuous. Now, since $\mathbb{R}^{n+1}-\{0\}$ is connected and the continuous image
of a connected space is connected, so $S^{n}$ is connected. Since ou... |
364,394 | <p>I was asked to find the minimum and maximum values of the functions:</p>
<blockquote>
<ol>
<li>$y=\sin^2x/(1+\cos^2x)$;</li>
<li>$y=\sin^2x-\cos^4x$.</li>
</ol>
</blockquote>
<p>What I did so far:</p>
<ol>
<li><p>$y' = 2\sin(2x)/(1+\cos^2x)^2$<br />
How do I check if they are suspicious extrema points? ... | Seirios | 36,434 | <p>Notice that $\mathbb{S}^n$ is the one-point compactification of $\mathbb{R}^n$, so there exists a continuous map $p : \mathbb{R}^n \to \mathbb{S}^n$ (think about the stereographic projection); in particular, $p(\mathbb{R}^n)= \mathbb{S}^n \backslash \{x_{\infty}\}$ for some $x_{\infty} \in \mathbb{S}^n$.</p>
<p>Let... |
1,413,022 | <p>In Guillemin and Pollack's <em>Differential Topology</em>, they give as an exercise (#1.8.14) to prove the following generalization of the Inverse Function Theorem:</p>
<blockquote>
<p>Use a partition-of-unity technique to prove a noncompact version of
[the Inverse Function Theorem]. Suppose that the derivative... | Kyle | 153,841 | <p>We can show that any two elements of the refined collection $\{g_i\}$ that are defined at $y$ have a nonempty open set around $y$ on which they agree:</p>
<p>Suppose we have two neighborhoods $V_i=f(U_i)$ and $V_j=f(U_j)$ of $y =f(z) \in f(Z)$ together with local inverses $g_i$ and $g_j$. Then we have $$g_i(V_i) \... |
255,416 | <p><em>(<strong>Update</strong>)</em>:
Courtesy of Myerson's and Elkies' answers, we find a second <em>simple</em> cyclic quintic for $\cos\frac{\pi}{p}$ with $p=10m+1$ as,
$$F(z)=z^5 - 10 p z^3 + 20 n^2 p z^2 - 5 p (3 n^4 - 25 n^2 - 625) z + 4 n^2 p(n^4 - 25 n^2 - 125)=0$$
where $p=n^4 + 25 n^2 + 125$. Its discriminan... | Will Jagy | 3,324 | <p>CW. Here are the first 250 examples, by the method of Gauss in section VII of the Disquisitiones. I followed Galois Theory by David A. Cox, chapter 9. The quintic for a prime $p \equiv 1 \pmod 5$ is
$$ x^5 + x^4 - 2 \left( \frac{p-1}{5} \right) x^3 + a x^2 + b x + c $$ with integers $a,b,c.$ Gerry has indicated th... |
638,396 | <p>I'm working through a book to learn trig on my own and I got stuck with the following. This is the image given and the text in the book reads:</p>
<blockquote>
<p>Suppose you are standing an unknown distance away from a cliff of height <span class="math-container">$h$</span>. You need to know the height <span c... | John | 7,163 | <p>Call the distance to the hill $d$.</p>
<p>Then $\tan B = \frac{h}{d}$ and $\tan A = \frac{h+t}{d}.$</p>
<p>Eliminating $d$ gives the desired result:</p>
<p>$$\frac{h}{\tan B} = \frac{h+t}{\tan A}$$</p>
<p>$$\frac{h}{\tan B} = \frac{h}{\tan A}+ \frac{t}{\tan A}$$</p>
<p>$$\frac{t}{\tan A} = \frac{h}{\tan B} - \f... |
3,013,177 | <blockquote>
<p>Find a curve <span class="math-container">$\alpha : (−ε,ε) → \Sigma$</span> on the sphere which has <span class="math-container">$\alpha(0) = (1,0,0)$</span> and <span class="math-container">$\alpha′(0) = (0, 5, 6)$</span>.</p>
</blockquote>
<p>I'm unsure how to approach this. I know the parametariza... | David Hartley | 169,394 | <p>Let <span class="math-container">$h(x) = = mid(-\delta, g(x)-f(x), \delta)$</span> (i.e. the middle value) for some <span class="math-container">$0 < \delta < \epsilon$</span>, and then replace <span class="math-container">$g$</span> by <span class="math-container">$f + h$</span> and <span class="math-contain... |
3,213,362 | <p><span class="math-container">$$f(x)=\frac{x+2}{1-2x}$$</span>
<span class="math-container">$$g(x)=\frac{2x+1}{2-x}$$</span></p>
<p>Find
<span class="math-container">$$(fofofo...ofOgogo...og)=\frac{1}{x}$$</span>
{fofo... are 101 times and gogo.. are 100 times}</p>
<p>Then Find <span class="math-container">$x$</spa... | trancelocation | 467,003 | <p>Your method is almost correct. The result is definitely correct.</p>
<p>You only need to show that <span class="math-container">$f$</span> and <span class="math-container">$g$</span> commute (which is true):</p>
<ul>
<li><span class="math-container">$(f\circ g)(x) = (g\circ f)(x) = -\frac{1}{x}$</span></li>
</ul>
... |
67,171 | <p>I am sure <a href="http://en.wikipedia.org/wiki/Modular_multiplicative_inverse">all those symbols</a> are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me.</p>
<p>How could I do this on a basic calculator? or with a few lines of programmer's code which... | Michael Hardy | 11,667 | <p>"Particularly, I had a hard time knowing why the (mod m) was off to the right and separate, and I am still not sure what the triple-lined equals symbol is all about."</p>
<p>The notation <span class="math-container">$(57 \equiv 62) \pmod 5$</span> means <span class="math-container">$57$</span> and <span cl... |
160,779 | <p>I am having a bit of difficulty trying to answer the following question:</p>
<blockquote>
<p>What is the Galois group of $X^8-1$ over $\mathbb{F}_{11}$?</p>
</blockquote>
<p>So far I have factored $X^8-1$ as </p>
<p>$$X^8-1=(X+10)(X+1)(X^2+1)(X^4+1).$$ </p>
<p>I know $X^2+1$ is irreducible over $\mathbb{F}_{11... | Arturo Magidin | 742 | <p>An extension of finite fields is always cyclic: the Galois group must be cyclic. So the Galois group certainly cannot be $V_4$.</p>
<p>Note that $F_{11}(i)$ <em>does</em> have a square root of $2$: $(3i)^2 = -9\equiv 2\pmod{11}$. So once you adjoint $i$ to $F_{11}$, you also get $\sqrt{2}$. Thus, $F_{11}(i,\sqrt{2}... |
463,353 | <p>How do I need to modify this in order for it to be correct?</p>
<p>The center of $S_n$ (for $n\geq$ 3) is the trivial identity. Proof: Assume the center of $S_n$ is $C = \{ id , \tau \}$ where $ \tau \in S_n$ and $\tau \neq \ id$. Then for some $n$ the factor group $S_n\backslash C$ is abelian and solvable, a contr... | user1729 | 10,513 | <p>Let me use the general idea you are trying to use to prove the result. This is very different from the idea which Jared is proposing. My proof uses the fact that $A_n$ is simple, which means that it only works for $n\geq 5$. It doesn't use anything you couldn't have thought up yourself, and it is interesting and dif... |
4,497,033 | <p>Let <span class="math-container">$r(t)$</span> be the function:<br />
<span class="math-container">$r(t) = \sqrt{x(t)^2 + y(t)^2}$</span>, where<br />
<span class="math-container">$x(t) = 3b (1 − t)^2 t + 3c (1 − t) t^2 + a t^3$</span>, and<br />
<span class="math-container">$y(t) = a (1 − t)^3 + 3c (1 − t)^2 t + 3b... | River Li | 584,414 | <p><strong>Some thoughts</strong>:</p>
<p>Since <span class="math-container">$r(t) = r(1 - t)$</span> for all <span class="math-container">$t\in [0, 1]$</span>, we consider the following equivalent problem:
<span class="math-container">$$\min_{a, b, c} ~ \max_{t\in [0, 1/2]} ~ [r(t) - 1]^2. $$</span></p>
<p>With <span ... |
3,513,308 | <p>Let <span class="math-container">$X$</span> be a CW complex, and suppose <span class="math-container">$W$</span> is obtained from <span class="math-container">$X$</span> by attaching an <span class="math-container">$n$</span>-cell to X, where <span class="math-container">$n>1$</span>. Consider the universal cover... | joriki | 6,622 | <p>Use the <a href="https://en.wikipedia.org/wiki/Law_of_cosines" rel="nofollow noreferrer">law of cosines</a>. If <span class="math-container">$\rho_i$</span> denotes the distance of <span class="math-container">$p_i$</span> from the origin, and you want the distance between the centres to be <span class="math-contain... |
3,176,434 | <p>I was going through some of the earlier answers for the license plate problems and one of the comment was as follows.</p>
<blockquote>
<p>Does a plate consist of 3 letters and 3 digits in any order, like
7C99XK, or is it 3 letters followed* by 3 digits? The answers will be
different.</p>
</blockquote>
<p>I u... | JMoravitz | 179,297 | <p>The number of <em>arrangements</em> of three letters followed by three digits is <span class="math-container">$26^3\cdot 10^3$</span>, seen by direct application of the rule of product using the following steps:</p>
<ul>
<li>Pick what the letter is in the first spot (26 choices)</li>
<li>Pick what the letter is in ... |
2,710 | <p>The <em>Mandelbrot set</em> is the set of points of the complex plane whos orbits do not diverge. An point $c$'s <em>orbit</em> is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.</p>
<p>The shape of this set is well known, <strong>why is it that if you zoom into parts of the filaments you will find slight... | Hugo Sereno Ferreira | 1,366 | <p>Incurring the risk of sounding too simplistic, isn't that one of the fundamental properties of a fractal [<a href="http://en.wikipedia.org/wiki/Fractal" rel="nofollow">Wikipedia</a>]:</p>
<p><strong>Quasi-self-similarity.</strong> This is a looser form of self-similarity; the fractal appears approximately (but not ... |
2,710 | <p>The <em>Mandelbrot set</em> is the set of points of the complex plane whos orbits do not diverge. An point $c$'s <em>orbit</em> is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$.</p>
<p>The shape of this set is well known, <strong>why is it that if you zoom into parts of the filaments you will find slight... | lhf | 589 | <p>See this paper:</p>
<blockquote>
<p>McMullen, Curtis T.,
The Mandelbrot set is universal. In <em>The Mandelbrot set, theme and variations</em>, 1–17,
London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, 2000.
<a href="http://www.ams.org/mathscinet-getitem?mr=1765082" rel="nofollow">MR1765082 (... |
1,057,999 | <p>I'm trying to find the limit: $\displaystyle\lim_{n\to\infty}\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}$</p>
<p>I thought of taking a pretty obvious binding from above expression: $\frac {n^n} {(n+1)^n}$ which is $n$ times the largest numerator and $n$ times the smallest denominator, but this limit isn... | Milly | 182,459 | <p>The numerator has a term 1, so upper bound $n^{n-1}$ is acceptable.</p>
<p>Since you found denominator is at least $(n+1)^n$, you get
$$ \frac{1\cdots n}{(n+1)\cdots 2n} \leq \frac{n^{n-1}}{(n+1)^n}\leq \frac1{n+1}.$$</p>
|
1,057,999 | <p>I'm trying to find the limit: $\displaystyle\lim_{n\to\infty}\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}$</p>
<p>I thought of taking a pretty obvious binding from above expression: $\frac {n^n} {(n+1)^n}$ which is $n$ times the largest numerator and $n$ times the smallest denominator, but this limit isn... | Phicar | 78,870 | <p><b>Hint:</b> $\binom{2n}{k}$ is unimodal by moving k with maximum at $\binom{2n}{n}$. See also that $n+1\leq 2n=\binom{2n}{1}$.</p>
|
581,605 | <p>Source: Miklos Bona, A Walk Through Combinatorics.</p>
<p>$$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$</p>
<p>The RHS is the upper bound of the Ramsey number $R(k,k)$.</p>
<p>How can I prove the inequality without using mathematical induction? I've merely expanded the LHS to obtain $\frac{(2k-2)!}{(k-1)!(k-... | Andrés E. Caicedo | 462 | <p>Since $\displaystyle e^x=\sum_{n=0}^\infty \frac {x^n}{n!}$, then $\displaystyle e^x-1-x=\sum_{n=0}^\infty\frac{x^{n+2}}{(n+2)!}$, and
$$ \frac{e^x-1-x}{x}=\sum_{n=0}^\infty\frac{x^{n+1}}{(n+2)!}, $$
so
$$ \left(\frac{e^x-1-x}{x}\right)'=\sum_{n=0}^\infty\frac{(n+1)x^n}{(n+2)!}. $$
Evaluate at $x=1$, and subtrac... |
4,068,830 | <p>My logic is since <span class="math-container">$3$</span> out of <span class="math-container">$4$</span> elements are chosen, each element would appear once.
So a sequence would look like: <span class="math-container">$a\,b\,c\,x\,x\,x\,x\,x\,x\,x$</span></p>
<p>We have <span class="math-container">$7$</span> spots ... | Gérard | 902,654 | <p>WLOG pick one element you choose to omit while creating that sequence. The number of possible sequences times 4 gives the final result because <span class="math-container">${4\choose 1}={4\choose 3}=4$</span>. Then consider that to generate a sequence containing each element at least once you can first fixate those ... |
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | peterwhy | 89,922 | <p>$$\begin{array}{ccccccc}&&&\square&&&\\
&&\blacksquare&\square&\square\\
&\blacksquare&\blacksquare&\square&\square&\square\\
\blacksquare&\blacksquare&\blacksquare&\square&\square&\square&\square
\end{array}
\left.\rightarrow\quad
\... |
1,386,004 | <p>We have two embryos. Our IVF doc said the probability of success implanting a single embryo is 40% whereas the probability of having one baby with implanting two embryos at once is 75% (with a 30% chance of twins).</p>
<p>What is the chance of having at least one child if we implant the embryos one at a time?</p>
... | Hypergeometricx | 168,053 | <p>Note that $$\begin{align}i^2-(i-1)^2&\color{lightgray}{=2i-1}\\&=i\qquad+(i-1)\quad\quad\end{align}$$
Summing from $i=1$ to $n$ and telescoping LHS gives
$$\begin{align}n^2\qquad\quad&=\sum_{i=1}^ni+\sum_{i=1}^n(i-1)\\
&=\sum_{i=1}^n i+\sum_{i=0}^{n-1}i\\
&=\sum_{i=1}^n i+\sum_{i=1}^{n-1}i\qquad\... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | Lærne | 252,762 | <p>I don't know where you pulled that $15$ denominator, but you're trying to do the <em>weighted average</em>. If you have a bunch of values $a_i$ to take the average from, all with a importance weight $w_i$, the weighted average is defined as follows :
$$
\frac{ \sum_{i = 1}^n w_i \cdot a_i }{ \sum_{i = 1}^n w_i }
... |
1,909,763 | <p>Let $p, q, r, s$ be rational and $p\sqrt{2}+q\sqrt{5}+r\sqrt{10}+s=0$. What does $2p+5q+10r+s$ equal?</p>
<p>I tried messing with both statements. But I usually just end up stuck or hit a dead end.</p>
<p>(I'm new to the site. I'm very sorry if this post is mal-written. please correct me on anything you can notice... | Thomas Andrews | 7,933 | <p>Everybody is showing you how to compute the weighted average, but not why.</p>
<p>What you have is 28 votes. The average of 28 values is the sum of those values, divided by 28. In this case, that means:</p>
<p>$$\frac{1+1+1+1+1+2+2+2+3+4+\cdots+4+5+5}{28}$$</p>
<p>This, we see, is the same as:</p>
<p>$$\frac{1\c... |
2,000,626 | <blockquote>
<p>Is it possible that $P(A\cap B)$ is greater than $P(A)$ or $P(B)$?</p>
</blockquote>
<p>I think not.<br>
Let's assume WLOG that $P(A\cap B) > P(A)$. Then,
$$P(B|A)=\frac{P(B\cap A)}{P(A)} > \frac{P(A)}{P(A)} = 1$$</p>
<p>Contradiction.</p>
<p>Is my proof valid?</p>
| egreg | 62,967 | <p>A basic requirement for a probability measure is that if $X$ and $Y$ are disjoint events, then $P(X\cup Y)=P(X)+P(Y)$.</p>
<p>Since $A=(A\setminus B)\cup(A\cap B)$, where $A\setminus B$ and $A\cap B$ are disjoint,
$$
P(A)=P(A\setminus B)+P(A\cap B)\ge P(A\cap B)
$$</p>
<hr>
<p>About your proof: it is correct once... |
2,508,614 | <p>In $\mathbb{Z}_7[x]$ ,write $f(x)=2x^3+2x+3$ as product of irreducibles .
make all your coefficients either $0,1,2,3,4,5$ or $6$</p>
<p>My attempt: as $f(x)=2x^3+2x+3\rightarrow f(1)=2+2+3=7=0 $</p>
<p>so we can write $f(x)=(x-1)g(x)$</p>
<p>but how to find $g(x)?$</p>
| Khayyam | 462,168 | <p>After long dividing, we have
$$2x^3+2x+3=(x-1)(2x^2+2x+4)+7 $$
which shows $g(x)=2x^2+2x+4$. The rest is as a above, noting that $g(3)=0$.</p>
|
3,342,629 | <p><strong>If <span class="math-container">$\alpha: A \mapsto A$</span> satisfies <span class="math-container">$\alpha^2 = \alpha,$</span> show that <span class="math-container">$\alpha$</span> is on-to if and only if <span class="math-container">$\alpha$</span> is one-to-one. Describe <span class="math-container">$\al... | Community | -1 | <p>This property, <span class="math-container">$f\circ f=f$</span>, is known as <a href="https://en.m.wikipedia.org/wiki/Idempotence" rel="nofollow noreferrer">idempotence</a>. The image of each element of <span class="math-container">$A$</span> is a fixed point of <span class="math-container">$f$</span>.</p>
|
3,342,629 | <p><strong>If <span class="math-container">$\alpha: A \mapsto A$</span> satisfies <span class="math-container">$\alpha^2 = \alpha,$</span> show that <span class="math-container">$\alpha$</span> is on-to if and only if <span class="math-container">$\alpha$</span> is one-to-one. Describe <span class="math-container">$\al... | CopyPasteIt | 432,081 | <p>Let <span class="math-container">$A^{'} = \alpha(A)$</span> denote the image of <span class="math-container">$A$</span> under the idempotent <span class="math-container">$\alpha$</span>.</p>
<p>The restriction mapping <span class="math-container">$\alpha^{'}:A^{'} \to A^{'}$</span> is the identity transformation.</... |
1,188,196 | <p>This is my first time posting in this forum, so please forgive me if my question is too involved or if I've posted it in the wrong area. I hope someone finds it interesting enough to try their hand at it.</p>
<p>Considering the image below, I am trying to work out a set of formulas that will specify either the radi... | Victor Liu | 398 | <p>Let us denote the centers of the circles by capital letters $R$, $O$, $P$, $B$, $Y$, $G$ for each color circle (red, orange, purple, blue, yellow, green). Let their radii be $r_b$, etc.
We immediately have
$$ P = (0,1+r_p)\\
B = (0,1+2r_p+r_b)\\
O = (1,\sqrt{(r_o+1)^2-1})\\
Y = O + (0,r_o+r_y)
$$
We have 5 circle ra... |
966,504 | <p>I had been following all the blogs, but I would like to understand, whether an attempt has been made to understand how many cycles are possible apart from the 1-4-2-1 cycle in collatz problem</p>
| Gottfried Helms | 1,714 | <p>Proofs which deal with the conjecture in terms of bounding the possible number of cycles are not known to me (for instance I've not seen such a concept referred to in Lagarias's survey), but you might take a deeper look (than me) at Kurt Mahler's work of z-numbers; he proves, that only finitely many z-numbers exist ... |
187,147 | <p>In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. </p>
<p>for (1), $f*f=f$ gives $\widehat{f*f}=\hat{f}$, which is equal to $\hat{f}\cdot\hat{f}=\hat{f}$, but this does not guarantee t... | Zarrax | 3,035 | <p>Hints: For (1), you're on the right track.. also use that $\hat{f}$ is continuous. For (2), try to define $\hat{f}$ instead of $f$. So you need a nonzero $L^2$ function equal to its square....</p>
|
739,019 | <p>Can we evaluate $c$ such that the following series converges using the definition of limit of a sequence?
$$
s{_n}=\sum_{t=1}^{n}tc^{t}, c\in \mathbb{R}
$$</p>
| Community | -1 | <p>For $\lim_{n\to\infty}s_n$,
$$\lim_{t\to\infty}\left|\dfrac{(t+1)c}{t}\right|<1\\
\implies |c|<1$$</p>
|
3,807,296 | <p>Suppose you have a triangulated region in the plane, the triangulation consisting of <span class="math-container">$n$</span> triangles. Take an arbitrary triangle of this triangulation and call it <span class="math-container">$\Delta_i$</span> with <span class="math-container">$1\leq i\leq n$</span>.</p>
<p>The neig... | Misha Lavrov | 383,078 | <p>If your proof strategy worked, it would prove too much.</p>
<p>Define a <em>locally planar</em> graph on a surface such as the torus to be a graph which looks like a planar graph within any "small region". (By "small", we can mean the neighborhood of a triangle, or more generally the subgraph wit... |
1,490,051 | <p>As stated on the title, my question is: (a) represent the function $ f(x) = 1/x $ as a power series around $ x = 1 $. (b) represent the function $ f(x) = \ln (x) $ as a power series around $ x = 1 $. </p>
<p>Here's what I tried:</p>
<p>(a) We can rewrite $ 1/x $ as $ \frac{1}{1 - (1-x)} $ and thus using the series... | Tien Truong | 281,875 | <p>I don't have enough reputation to comment, so here I am.</p>
<p>a) The your answer and the one you get from Wolfram|Alpha are identical.</p>
<p>$\sum_{n=0}^\infty (1-x)^n = \sum_{n=0}^\infty (-(x-1))^n = \sum_{n=0}^\infty (-(-1+x))^n = \sum_{n=0}^\infty (-1)^n (-1+x)^n$.</p>
<p>b) I think you have forgotten some ... |
3,198,705 | <p>I have this problem:</p>
<blockquote>
<p>In a game, the probability of win is <span class="math-container">$1/3$</span> and of lose is <span class="math-container">$2/3$</span>, ¿What
is the probabilty of win at least 1 prize playing 3 times?</p>
</blockquote>
<p>The probability of win at least 1 prize, playin... | Einar Rødland | 37,974 | <p>Given <span class="math-container">$U=X_1+X_2$</span> in <span class="math-container">$\mathbb{R}^n$</span> where <span class="math-container">$X_i$</span> are random points on the <span class="math-container">$n-1$</span>-spheres <span class="math-container">$||X_i||=r_i$</span>, and <span class="math-container">$R... |
1,712,502 | <p>How can I find the number of solutions of this equation in interval $[0,\pi]$:
$$ 3x + \tan x = \frac{5\pi}{2}$$
I have no clue how to proceed.</p>
| nkm | 127,841 | <p>Finding the answers is tricky, but we can see how many there are. The function
$$f(x) = 3x + \tan x $$
increases monotonically from 0 to $\infty$ on the interval $[0, \pi/2)$, then again increases monotonically from $-\infty$ to $3\pi$ on the interval $(\pi/2, \pi]$. It must hit $5\pi/2$ exactly once in each interv... |
2,414,640 | <p>I'm trying to implement Bowyer-Watson algorithm. Currently performance is stuck behind calculating the circumsphere of a tetrahedron.</p>
<p>I tried using this <a href="http://mathworld.wolfram.com/Circumsphere.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/Circumsphere.html</a>. It works but it is sl... | tevemadar | 452,097 | <p>Layman here. After looking at the tons of square roots, inversion of 5x5 matrix, I went back to Wikipedia and read it again. While there are also tons of square roots, somewhere in the <a href="https://en.wikipedia.org/wiki/Tetrahedron#Circumcenter" rel="nofollow noreferrer">lower-middle</a> of the Tetrahedron page,... |
459,677 | <p>How do I determine a Standard deviation with the mean and range known?</p>
| nbubis | 28,743 | <p>You can't. Without knowing what the distribution is, you can't really know what the standard deviation. </p>
|
864,568 | <p>I am trying to figure out how to take the modulo of a fraction. </p>
<p>For example: 1/2 mod 3. </p>
<p>When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?</p>
| Adam Hughes | 58,831 | <p>My favorite way to find inverse modulo $n$ (if they exist) is from the Euclidean algorithm since that's the standard way of proving inverses exist. By the EA you can find integers $x,y\in\Bbb Z$ such that</p>
<p>$$ax+by=c\tag{$*$}$$</p>
<p>with $c=(a,b)$. Then if we want an inverse for $a\mod b$ we check $(a,b)=1$... |
864,568 | <p>I am trying to figure out how to take the modulo of a fraction. </p>
<p>For example: 1/2 mod 3. </p>
<p>When I type it in google calculator I get 1/2. Can anyone explain to me how to do the calculation?</p>
| john p | 451,280 | <p>Think of a set of numbers {0, 1, 2, 3, 4, 5, 6}. Z7 (or modulo 7) Such that we have two operations + and *. If we add two numbers, the result never escapes the set. If we multiply two numbers the result still does not escape the set. </p>
<p>For Example: 3 + 4 = 7, but 7 mod 7 = 0, thus 3 + 4 mod 7 = 0 mod 7, and ... |
51,026 | <p>I just wasted the last hour on google looking in vain for an excerpt of Weil's writings describing the process of discovering mathematics. I believe he once beautifully described the feeling of loss that accompanies the realization that the discovery you made seems, in retrospect, trivial. Am I misremembering or j... | Bill Dubuque | 242 | <p>Having aleady OCRed Borel's article on Weil (see Theo's reply), I cannot resist posting a larger excerpt here, since it provides much further context that I suspect will help readers to better appreciate Weil's "search for elegance, beauty and hidden harmonies". Perhaps it will help motivate some readers to join in ... |
756,111 | <p>Does anyone have any intuition on remembering or very quickly deriving that</p>
<p>$$\frac{1}{r^2}\frac{\partial}{\partial r}(r^2 \frac{\partial \phi }{\partial r}) = \frac{1}{r} \frac{\partial ^2 }{\partial r^2}(r \phi )$$</p>
<p>holds for the Laplacian in spherical coordinates? Doing the IBP is too long and slow... | Claude Leibovici | 82,404 | <p>You must use at least one extra term in the development (because of the $n^2$ at the beginning). So,
$$e^{-1/n} \approx~ 1-\frac{1}{n}+\frac{1}{2 n^2}-\frac{1}{6
n^3}+O\left(\left(\frac{1}{n}\right)^4\right)$$</p>
<p>Now replace in your expression and you will arrive to $$n^2 e^{-1/n} + n e^{-1/n} - n^2 \approx... |
4,147,900 | <p>I have a proof by contradiction to a simple problem but I have an issue understanding one aspect of it. I labeled it in the picture. Any insight would be helpful. Thank you for the time. <a href="https://i.stack.imgur.com/HXtPh.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/HXtPh.jpg" alt="enter ... | José Carlos Santos | 446,262 | <p>Since <span class="math-container">$a\in A$</span>, <span class="math-container">$a\in B$</span>. And now, since <span class="math-container">$a\notin C$</span>, <span class="math-container">$a\in B\setminus C$</span>.</p>
|
1,141,357 | <p>When deriving the integration by parts formula, you can use the product rule to do so,<br>
i.e. $\{uv\}' = uv' + vu'$</p>
<p>$\Rightarrow \int \{uv\}' = \int udv + \int vdu$</p>
<p>hence $uv = \int udv + \int vdu$. </p>
<p>If $uv$ is the integral of $\{uv\}'$ then why is the formula rearranged to read</p>
<p>$... | Neelesh Viswanathan | 469,729 | <p>$(uv)'= u\mathrm{d}v + v\mathrm{d}u$ which on rearrangement becomes $u\mathrm{d}v = (uv)' - v\mathrm{d}u$. Integrating this equation we get the formula for 'integration by parts', obviously. Try it.</p>
|
3,339,683 | <blockquote>
<p>If <span class="math-container">$x + \sqrt[3]{x} = 30$</span>, then what is the value of <span class="math-container">$x$</span>?</p>
</blockquote>
<p>Please help me!
Thank you.</p>
| Feng | 624,428 | <p>Hint:
Let <span class="math-container">$t=\sqrt[3]x$</span> and note that <span class="math-container">$$t^3+t-30=(t-3)(t^2+3t+10).$$</span></p>
|
3,339,683 | <blockquote>
<p>If <span class="math-container">$x + \sqrt[3]{x} = 30$</span>, then what is the value of <span class="math-container">$x$</span>?</p>
</blockquote>
<p>Please help me!
Thank you.</p>
| Sina Babaei Zadeh | 646,494 | <p>hint:</p>
<p><span class="math-container">$x^{1/3}=30-x$</span> can nicely be written if you raise both sides to the power of three.</p>
|
3,059,444 | <p>Exercise asks to verify that the sum of three quantities x, y, z, whose product is a constant k, is minimum when these three quantities are equal.</p>
<p>This is my amateurish attempt:</p>
<ol>
<li><span class="math-container">$x + y + z = S$</span>;</li>
<li><span class="math-container">$x*y*z=k$</span>;</li>
<li... | José Carlos Santos | 446,262 | <p>In your case, <span class="math-container">$0$</span> is a <em>double</em> root: you should count it as two roots. In other words, the following statement holds:</p>
<blockquote>
<p>If the roots are counted with their multiplicities, then every cubic polynomial in one variable with real coefficients either has ex... |
1,563,820 | <p>In a question of Probability, I got the answer
$$\frac{\sum_{j=0}^N(j/N)^{n+1}}{\sum_{j=0}^N(j/N)^{n}}$$</p>
<p>Now I have to prove when $N\to\infty$ above expression approximately is $\frac {n+1}{n+2}.$ No idea how to proceed, kindly help in this regards.</p>
| MathIsNice1729 | 274,536 | <p>$$\lim_{N \to \infty}\frac{\sum_{j=0}^N(j/N)^{n+1}}{\sum_{j=0}^N(j/N)^{n}}=\frac{ \lim_{N \to \infty} \frac{1}{N}\sum_{j=0}^N(j/N)^{n+1}}{ \lim_{N \to \infty} \frac{1}{N}\sum_{j=0}^N(j/N)^{n}}$$ $$=\frac{\int_0^1 \! {x^{n+1} \, \mathrm{dx}}}{ \int_0^1 \! {x^n \, \mathrm{dx}}}=\frac{\left[ \frac{x^{n+2}}{n+2}\righ... |
3,546,184 | <p>Take three numbers <span class="math-container">$x_1$</span>, <span class="math-container">$x_2$</span>, and <span class="math-container">$x_3$</span> and form the successive running averages <span class="math-container">$x_n = (x_{n-3} + x_{n-2} + x_{n-1})/3$</span> starting from <span class="math-container">$x_4$<... | Paramanand Singh | 72,031 | <p>Can you show that the sequence converges? Once you have done that it is easy to find the limit.</p>
<p>Let's write down the recurrence in form <span class="math-container">$$3x_n=x_{n-1}+x_{n-2}+x_{n-3}$$</span> And further if we put <span class="math-container">$n=4,5, \dots $</span> we get the set of relations
<s... |
1,877,254 | <p>For the integration of $\frac{x^3}{\sqrt{1+x^2}}$, can we use $tan(\theta)$ to substitute $x$, and then use $u= sec(\theta)$ later this proof? I got a solution which is $\frac{((sec^3 (tan^{-1} x))}{3}$$ - (sec (tan^{-1} (x)) + C)$. Is this method correct? </p>
| Hwai-Ray Tung | 355,554 | <p>A faster way to do the integration might be to set $u = x^2+1$. Then we have</p>
<p>\begin{align}
\int \frac{x^3}{\sqrt{x^2 + 1}} dx &= \int \frac{x^2}{\sqrt{x^2 + 1}} xdx \\
&= \int \frac{u-1}{\sqrt{u}} \frac{1}{2}du \\
&= \frac{1}{2} \int \sqrt{u} - \frac{1}{\sqrt{u}} du \\
\end{align}</p>
<p>I'll as... |
267,864 | <p>This is a <a href="https://math.stackexchange.com/questions/2239890/conformal-harmonic-maps-in-high-dimensions-are-scaled-isometries">cross-post</a> from MSE (where I got no answer).</p>
<p>It is well-known that conformal maps between $2$-dimensional Riemannian manifolds are harmonic.</p>
<p>I discovered lately th... | Asaf Shachar | 46,290 | <p>Indeed, the result can be found in the book <em>Harmonic morphisms between Riemannian manifolds</em>, by Paul Baird, John C. Wood.</p>
<p>The relevant statement is Corollary 3.5.2.</p>
|
849,191 | <p>I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I thought, it's a small group of kids that have strong math skills and seem to have an endless desire for knowledge. I... | David | 119,775 | <p>$$\int\frac{dx}{x}=\int(1)\Bigl (\frac{1}{x}\Bigr)\,dx=x\Bigl(\frac{1}{x}\Bigr)-\int x\Bigl(\frac{-1}{x^2}\Bigr)\,dx=1+\int\frac{dx}{x}\quad\hbox{so}\quad 0=1\ .$$</p>
|
849,191 | <p>I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I thought, it's a small group of kids that have strong math skills and seem to have an endless desire for knowledge. I... | Community | -1 | <p>Consider the following image:</p>
<p><img src="https://i.stack.imgur.com/9aZcY.png" alt="enter image description here"></p>
<p>I've drawn two circles, and the two diagonal lines are the diameters of the circles.</p>
<p>The two angles in the skinny triangle are both right angles, because they are inscribed in a se... |
1,181,405 | <p>If $b_n>0$ and $\sum b_n\,$ converges, prove $\sum {b_n}^{1\over2}\dot\,{1\over{n^{\alpha}}}\,$ converges for all $\alpha>{1\over2}$.</p>
<p>I know
${b_n}^{1\over2}\dot\,{1\over{n^{\alpha}}}\leq{b_n}^{1\over2}\dot\,{1\over{n^{1\over2}}}$</p>
<p>Since $b_n$ converges to $0$, I cannot say ${b_n}^{1\over2}\leq... | Julián Aguirre | 4,791 | <p>It is a straight forward application of the Cauchy-Schwarz inequality.</p>
|
4,143,585 | <p>Recently, I came across <a href="https://www.youtube.com/watch?v=joewDkmpvxo" rel="noreferrer">this</a> video, the method shown seemed good for a few logarithms. Then I tried to plot the equation <span class="math-container">$$ \frac{x^{\frac{1}{2^{15}}}-1}{0.000070271} $$</span> and it looks <em>exactly</em> like t... | hamam_Abdallah | 369,188 | <p><strong>hint</strong></p>
<p>You might know that for small <span class="math-container">$ X, $</span></p>
<p><span class="math-container">$$e^X-1\sim X $$</span></p>
<p>but</p>
<p><span class="math-container">$$x^{\frac{1}{2^{15}}}=e^{\frac{1}{2^{15}}\log(x)\ln(10)}$$</span></p>
|
4,143,585 | <p>Recently, I came across <a href="https://www.youtube.com/watch?v=joewDkmpvxo" rel="noreferrer">this</a> video, the method shown seemed good for a few logarithms. Then I tried to plot the equation <span class="math-container">$$ \frac{x^{\frac{1}{2^{15}}}-1}{0.000070271} $$</span> and it looks <em>exactly</em> like t... | user21820 | 21,820 | <p>This kind of approximation arises from the general method of using identities to improve a simpler approximation. Here I will make clear how to obtain it via this technique.</p>
<h3>Example 1</h3>
<p>Use approximation <span class="math-container">$\exp(x) ≈ 1+x$</span> for <span class="math-container">$x ≈ 0$</span>... |
1,158,666 | <p>I know that every closed and bounded set in $\mathbb{R}$ is compact (like $[a,b]$)</p>
<p>so i can conclude that every bounded set in $\mathbb{R}$ is relatively compact, by contradiction i say that let $A\subset\mathbb{R}$ be a bounded set but not relatively compact it means that $\overline{A}$ is not compact, but ... | Henno Brandsma | 4,280 | <p>If $A$ is bounded this means there is some $R > 0$ such that $d(x,y) \le R$ for all $x,y \in A$. But the same then holds for $\overline{A}$ as well. This can for instance be seen as follows: pick $p \in A$, then all members of $A$ are in the closed ball $D(p,R) = \{x \in X: d(x,p) \le R \}$. So $\overline{A} \su... |
1,158,666 | <p>I know that every closed and bounded set in $\mathbb{R}$ is compact (like $[a,b]$)</p>
<p>so i can conclude that every bounded set in $\mathbb{R}$ is relatively compact, by contradiction i say that let $A\subset\mathbb{R}$ be a bounded set but not relatively compact it means that $\overline{A}$ is not compact, but ... | Mankind | 207,432 | <p>Recall what bounded means: When $A$ is bounded, there exists an open interval $(x_0-\delta;x_0+\delta)$ that contains $A$. Then the corresponding closed interval $[x_0-\delta;x_0+\delta]$ also contains $A$, and since the closure of $A$ is the smallest closed set containing $A$, the closure of $A$ will be contained i... |
134,606 | <p>As part of a larger investigation, I am required to be able to calculate the distance between any two points on a unit circle. I have tried to use cosine law but I can't determine any specific manner in which I can calculate theta if the angle between the two points and the positive axis is always given.</p>
<p>Is ... | hmakholm left over Monica | 14,366 | <p>If the arc distance between the two points is $\theta$, the length of the chord between them is $2\sin\frac{\theta}{2}$:</p>
<p><img src="https://i.stack.imgur.com/j1Mnv.png" alt="geometry diagram"></p>
|
3,560,742 | <p><span class="math-container">$X_1, \ldots , X_n$</span>, <span class="math-container">$n \ge 4$</span> are independent random variables with exponential distribution: <span class="math-container">$f\left(x\right) = \mathrm{e}^{-x}, \ x\ge 0$</span>. We define <span class="math-container">$$R= \max \left( X_1, \ldots... | StubbornAtom | 321,264 | <p>Let <span class="math-container">$X_{(1)},X_{(2)},\ldots,X_{(n)}$</span> be the order statistics corresponding to <span class="math-container">$X_1,X_2,\ldots,X_n$</span>.</p>
<p>Making the transformation <span class="math-container">$(X_{(1)},\ldots,X_{(n)})\mapsto (Y_1,\ldots,Y_n)$</span> where <span class="math-c... |
2,246,161 | <p>The parameter for the normal trigonometric functions represents the length of the opposite and adjacent sides of a triangle in a unit circle. The parameter is the angle of the triangle that is located at the radius. The vertex that touches the circle has the coordinates of $(\cos{\theta},\sin{\theta})$.</p>
<p>From... | Jean Marie | 305,862 | <p>This parameter $t$, sometimes called "<strong>hyperbolic angle</strong>" can be interpreted as an area (<a href="https://en.wikipedia.org/wiki/Hyperbolic_angle" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Hyperbolic_angle</a>).
See also (<a href="https://www.revolvy.com/topic/Hyperbolic%20function&i... |
2,860,013 | <p>I'm trying to prove that the following system of congruence equations has a solution:</p>
<p>$X \equiv 2 $ (mod $5^N$)</p>
<p>$X \equiv 1 $ (mod $7^N$)</p>
<p>$X \equiv 4 $ (mod $6^N-4$)</p>
<p>being $N$ an integer number, $N\geq 2$</p>
<p>I guess that this may be answered using the <a href="https://en.wikipedi... | user577215664 | 475,762 | <p>$$(y-x)(dy/dx) = 1$$
Multipy by $\mu(x)=e^y$
$$(y-x)e^ydy-e^ydx =0 $$
The differential is exact
$$-x+y-1-Ke^{-y}=0$$
$$y(0)=0 \implies K=-1$$
$$\boxed{x(y)=y-1+e^{-y}}$$
we have also that $y(a)=1$
$$\implies a=e^{-1}$$</p>
|
1,466,637 | <p>Solve $3sec^2(x)=4$.</p>
<p>$sec^2(x)=\frac{4}{3}$</p>
<p>$sec(x)=4=\sqrt{\frac{4}{3}}=\frac{4\sqrt{3}}{3}$</p>
<p>How to continue, i.e. how to calculate the value of x for which $sec(x)=\frac{4\sqrt{3}}{3}$</p>
<p>I can rearrange the expression above into $cos(x)=\frac{\sqrt{3}}{4}$, but how to continue to find... | Harish Chandra Rajpoot | 210,295 | <p>Given $$3\sec^2 x=4$$ $$\sec^2 x=\frac{4}{3}$$
$$\cos^2 x=\frac{3}{4}$$ $$\cos^2 x=\cos^2\frac{\pi}{6}$$
Writing the general solution, one gets
$$\color{blue}{x=k\pi \pm \frac{\pi}{6}}$$
Where, $\color{blue}{k}$ is any interger</p>
|
1,947,336 | <p>$$\int x^{2} \sin(x^{2} +2) \mathrm{d}x$$</p>
<p>so, i think i have to do $u$ sub on the equation in $\sin$, but from there i'm not sure how to approach integration by parts afterwards...</p>
<p>answer might be wrong but i got
$$u = x^{2} +2$$
$$\mathrm{d}u = 2x$$
$$\mathrm{d}x = \frac{\mathrm{d}u}{2x}$$
$$x^{2} ... | Claude Leibovici | 82,404 | <p>The problem is much more complex than it looks (almost as Aaron M suggested).</p>
<p>Considering $$I=\int x^2 \sin \left(x^2+2\right)\,dx=\int x \sin \left(x^2+2\right) x \,dx$$ So, integrating by parts $$u=x \qquad du=dx \qquad dv=\sin \left(x^2+2\right) x \,dx\qquad v=-\frac 12 \cos(x^2+2)$$ make $$I=-\frac x2 \c... |
1,947,336 | <p>$$\int x^{2} \sin(x^{2} +2) \mathrm{d}x$$</p>
<p>so, i think i have to do $u$ sub on the equation in $\sin$, but from there i'm not sure how to approach integration by parts afterwards...</p>
<p>answer might be wrong but i got
$$u = x^{2} +2$$
$$\mathrm{d}u = 2x$$
$$\mathrm{d}x = \frac{\mathrm{d}u}{2x}$$
$$x^{2} ... | Community | -1 | <p>First get rid of the constant term $2$ by means of the addition formula and solve the integral for $\cos$ and $\sin$.</p>
<p>Then by parts,</p>
<p>$$\int \frac x2(2x\sin x^2)\, dx=-\frac x2\cos x^2+\frac12\int\cos x^2\,dx$$ and similarly for the sine.</p>
<p>The final integrals are the well-known Fresnel function... |
3,722,842 | <p>Kevin wants to fence a rectangular garden using <span class="math-container">$14$</span> rails of
<span class="math-container">$8$</span>-foot rail, which cannot cut. What are the dimensions of the rectangle that will maximize the fenced area?</p>
<p>So the number of rails in each dimension the rectangle could be ei... | Harish Chandra Rajpoot | 210,295 | <p>Yes, you are right.</p>
<p>Notice, a rectangle will have the maximum area only when its length and width become equal i.e. when it becomes a square. But since you can't cut a rail into pieces so you must have a rectangle which is the best approximation to a square for given conditions.</p>
<p>Therefore a rectangle o... |
3,722,842 | <p>Kevin wants to fence a rectangular garden using <span class="math-container">$14$</span> rails of
<span class="math-container">$8$</span>-foot rail, which cannot cut. What are the dimensions of the rectangle that will maximize the fenced area?</p>
<p>So the number of rails in each dimension the rectangle could be ei... | Community | -1 | <p>you did it correctly.</p>
<p>if rails are not to be cut then you can make approximation of rails <span class="math-container">$$3.5\approx 4\ \ rails$$</span>
this way, rectangular garden will have <span class="math-container">$4$</span> rails along each of two length and 3 rails along each of two widths. this gives... |
1,042,852 | <p>Let $a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$<br>
How to show this sequence is unbounded without using limits?</p>
<p>Well I know that I need to show that it unbounded from bottom or above.<br>
I choose bottom, so I need to show that $\forall M \exists n \Rightarrow a_{n}>M$</p>
<p>What is the method? Ca... | Timbuc | 118,527 | <p>First, note the denominator is always positive, so</p>
<p>$$\frac{7n^3-3n^4-1}{4n^2+3}<0\iff 7n^3-3n^4-1<0\iff -3n^3\left(n-7\right)<1\iff$$</p>
<p>$$3n^3(n-7)>-1$$</p>
<p>and the last inequality is clearly true for $\;n\ge7\;$, and thus the sequence is bounded above, but</p>
<p>$$\frac{7n^3-3n^4-1}{... |
2,939,028 | <blockquote>
<p>Find<span class="math-container">$$\lim_{x→0}\frac{\ln\cos3x}{\ln\cos2x}.$$</span></p>
</blockquote>
<p>Can anyone give me a hint about finding this limit without using L'Hopital?</p>
| Cesareo | 397,348 | <p>As <span class="math-container">$\cos x = 1 - \frac{x^2}{2!}+O(x^4)$</span></p>
<p><span class="math-container">$$
\cos(3x) = 1-\frac{9x^2}{2}+O(x^4)\\
\cos(2x) = 1-\frac{4x^2}{2}+O(x^4)\\
$$</span></p>
<p>and also <span class="math-container">$\ln(1-x) = -x + O(x^2) $</span> then</p>
<p><span class="math-contain... |
461,305 | <p>For example, in computer science, there can be zero, one, two, etc. parameters to a computer program, and this is called its "arity". Sets can be countable or uncountable. Is there some word I can use to say "this set's <em>_</em>" is continuous/discrete, or "this set has a continuous/discrete __". For example, alth... | Thomas Andrews | 7,933 | <p>If you are talking abut more than the distinction between finite and infinite, you are probably talking about "topologically discrete."</p>
<p>A simple example is the rationals, which have a standard "non-discreet" topology, and the integers, which have a standard "discreet" topology. The two sets have the same car... |
2,805,007 | <p>I need to solve this:</p>
<blockquote>
<p>$$f:\mathbb R\to\mathbb R,\mathcal C^1\text{-function and } a,b\in\mathbb R \text{ such that } a\lt b.$$ </p>
</blockquote>
<p>1) Probe that there is $M\gt 0$ such that $\forall x\in[a,b]$ it's verified that $ \vert f'(x)\vert \le M$</p>
<p>2) Conclude that $\forall x, ... | Jens Schwaiger | 532,419 | <p>Fix $\varepsilon>0$ and choose $a_n:=2\varepsilon n$, $x_n:=a_n+\frac1{4n}$,
$y_n:=a_n-\frac1{4n}$. Then $x_n-y_n=\frac{1}{2n}<\frac{1}{n}$ and $x_n^2-y_n^2=\frac{a_n}{n}=2\varepsilon>\varepsilon$.</p>
|
4,074,647 | <p>Problem:
<span class="math-container">$12$</span> students are asked to go into the groups <span class="math-container">$A$</span>, <span class="math-container">$B$</span>, <span class="math-container">$C$</span>, <span class="math-container">$D$</span>, and <span class="math-container">$E$</span>. In how many ways ... | true blue anil | 22,388 | <p>Apply inclusion - exclusion.</p>
<p>There are <span class="math-container">$5$</span> choices for each person, so <span class="math-container">$5^{12}$</span> total ways.</p>
<p>Applying inclusion-exclusion,</p>
<p>Total ways - at least one group empty + at least two groups empty -...</p>
<p><span class="math-contai... |
3,791,405 | <p>Find the last <span class="math-container">$2$</span> digits of <span class="math-container">$9^{100}$</span>.</p>
<hr />
<p>Well, I know that <span class="math-container">$9^{100}$</span> mod <span class="math-container">$4$</span> is <span class="math-container">$1$</span>,but I do not know how to find <span class... | CopyPasteIt | 432,081 | <p>Calculate <span class="math-container">$9^{100}$</span> mod <span class="math-container">$25$</span>.</p>
<p>Using the 'brute-force' mod <span class="math-container">$25$</span> technique (no calculator necessary),</p>
<p><span class="math-container">$\quad 9^{100} = \bigr(\big({(9^2)}^2\big)^5\bigr)^5 \equiv \bigr... |
72,630 | <p>Can you please help me and tell, how should I move on?
Can this be proved by induction?</p>
<blockquote>
<p>Every natural number $n\geq 8$ can be represented as $n=3k + 5\ell$.</p>
</blockquote>
<p>Thank you in advance</p>
| Arturo Magidin | 742 | <p>Clearly, $8$ can be represented as $3k+5\ell$, by taking $k=1$ and $\ell=1$.</p>
<p>Likewise, $9=3(3) + 5(0)$; and $10=3(0) + 5(2)$. So we can represent $8$, $9$, and $10$.</p>
<p>Now, assume that for $m\gt 10$, and you can represent all numbers strictly smaller than $m$ that are $8$ or larger (the induction hypo... |
72,630 | <p>Can you please help me and tell, how should I move on?
Can this be proved by induction?</p>
<blockquote>
<p>Every natural number $n\geq 8$ can be represented as $n=3k + 5\ell$.</p>
</blockquote>
<p>Thank you in advance</p>
| Bill Dubuque | 242 | <p><strong>Hint</strong> <span class="math-container">$ \,n\,$</span> representable <span class="math-container">$\Rightarrow$</span> so too is <span class="math-container">$\,n\!+\!\color{#c00}3$</span> (by adding <span class="math-container">$1$</span> to <span class="math-container">$ k)$</span> so if <span class="m... |
3,533,260 | <p>find three distinct nonzero vectors a,b,c in three dimension such that
span(a,b)=span(b,c)=span(a,b,c)
but span(a,c) is not equal to span(a,b,c)</p>
| Mankind | 207,432 | <p>In 2 dimensions, take <span class="math-container">$a=(1,0)$</span>, <span class="math-container">$b=(0,1)$</span>, and <span class="math-container">$c=(2,0)$</span>.</p>
|
3,812,773 | <p>Please some hint on how to solve in the set of natural numbers
<span class="math-container">$$x^{100} − y^{100} = 100!$$</span>
The question comes from the <a href="https://dms.rs/wp-content/uploads/2020/08/JSMO2020.pdf" rel="nofollow noreferrer">Serbian Junior Mathematical Olympiad 2020</a>.</p>
<p>I have tried wit... | Mathology | 907,334 | <p>I have a clear solution. First, let's apply Fermat's little theorem and we have <span class="math-container">$ x ^ { 100 } , y ^ { 100 } \equiv 1 \pmod { 101 } $</span>. Then we can use Wilson's theorem and get <span class="math-container">$ 100 ! \equiv - 1 \pmod { 101 } $</span>. So, the left-hand side of the equa... |
1,431,042 | <p>I know that given a polynomial $p(i)$ of degree $d$, the sum $\sum_{i=0}^n p(i)$ would have a degree of $d + 1$. So for example</p>
<p>$$
\sum_{i=0}^n \left(2i^2 + 4\right) = \frac{2}{3}n^3+n^2+\frac{13}{3}n+4.
$$</p>
<p>I can't find how to do this the other way around. What I mean by this, is how can you, when gi... | robjohn | 13,854 | <p><strong>Which Polynomials Can Be Written as a Sum</strong></p>
<p>By summing a <a href="https://en.wikipedia.org/wiki/Telescoping_series">Telescoping Series</a>, we get
$$
\begin{align}
\sum_{k=0}^n(p(k)-p(k-1))
&=\sum_{k=0}^np(k)-\sum_{k=0}^np(k-1)\\
&=\sum_{k=0}^np(k)-\sum_{k=-1}^{n-1}p(k)\\
&=p(n)-p(... |
2,264,252 | <p>If a function $f: M \rightarrow N$ is surjective, does it also mean it has an inverse? or is it the same thing?</p>
| Kenny Lau | 328,173 | <p>Definition of $\ln x$:</p>
<p>$$\ln x := \int_1^x \dfrac{\mathrm dt}t$$</p>
<p>Then:</p>
<p>$$\begin{array}{rcl}
\ln x + \ln x^{-1}
&=& \displaystyle \int_1^x \dfrac{\mathrm dt}t + \int_1^{x^{-1}} \dfrac{\mathrm dt}t \\
&=& \displaystyle \int_1^x \dfrac{\mathrm dt}t + \int_x^1 \dfrac{\mathrm d(xt)... |
1,284,820 | <blockquote>
<p>How would this differential equation be solved?
$$y{\partial z\over \partial x}+z{\partial z\over \partial y}={y \over x}$$</p>
</blockquote>
<p>I was taught to solve them like : $${dx \over y}={dy \over z}={dz \over {y \over x}}$$ Then find the constants $c_1$ and $c_2$ and the answer being $F(c_1... | mathlove | 78,967 | <p>Since
$$(x-i)(x-i-1)=\frac13\left((x-i-1)(x-i)(x-i+1)-(x-i-2)(x-i-1)(x-i)\right)$$
one has
$$\begin{align}\sum_{i=0}^{k}\binom{x-i}{2}&=\sum_{i=0}^{k}\frac{(x-i)(x-i-1)}{2}\\&=\frac 12\sum_{i=0}^{k}\frac 13\left((x-i-1)(x-i)(x-i+1)-(x-i-2)(x-i-1)(x-i)\right)\\&=\frac 16\sum_{i=0}^{k}\left((x-i-1)(x-i)(x... |
2,917,858 | <p>Consider the vector space of all functions $f: \mathbb{R} \rightarrow \mathbb{C}$ over $\mathbb{C}$. If $W$ is a subspace spanned by $\beta$ = $\{1, e^{ix}, e^{-ix}\}$, show that $\beta$ is a basis for $W$.</p>
<p>I think I am very confused - I know I just have to show that $\beta$ is linearly independent, which me... | user | 505,767 | <p>To prove that the vectors are linearly dependent, the equality </p>
<p>$$a+be^{ix}+ce^{-ix} = 0$$</p>
<p>should hold for some $a,b,c$ for any $x$ but you have considered a particular case with $x=0$.</p>
<p>Indeed we have that</p>
<p>$$a+be^{ix}+ce^{-ix} = 0 \quad \forall x \iff a = b = c = 0$$</p>
<p>and the v... |
2,917,858 | <p>Consider the vector space of all functions $f: \mathbb{R} \rightarrow \mathbb{C}$ over $\mathbb{C}$. If $W$ is a subspace spanned by $\beta$ = $\{1, e^{ix}, e^{-ix}\}$, show that $\beta$ is a basis for $W$.</p>
<p>I think I am very confused - I know I just have to show that $\beta$ is linearly independent, which me... | José Carlos Santos | 446,262 | <p>Suppose that there are $_1,z_2,z_3\in\mathbb C$ such that $z_1+z_2e^{ix}+z_3e^{-ix}=0$. Then:</p>
<ul>
<li>if $x=0$, we have $z_1+z_2+z_3=0$;</li>
<li>if $x=\pi$, we have $z_1-z_2-z_3=0$;</li>
<li>if $x=\frac\pi2$, we have $z_1+iz_2-iz_3=0$.</li>
</ul>
<p>What does this tell you about $z_1$, $z_2$, and $z_3$?</p>
|
710,040 | <p>Hello I have been blasting at this inequality proof and it is just not doing what I want it to do:</p>
<blockquote>
<p>Prove that $2^{n-1}(a_1a_2\ldots a_n + 1) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ assuming that $a_1,a_2,\dots, a_n \geq 1$.$\:\:\:$</p>
</blockquote>
<p>So the base case is pretty trivial, which lead... | Mx Glitter | 134,212 | <p>I followed your induction strategy. Starting from:
$$(1+a_1)\dots(1+a_{k+1})\le 2^{k-1}(a_1\dots a_k+1)(a_{k+1}+1)$$
Let's take $A = a_1 \dots a_k$. $$2^{k-1}(a_1\dots a_k+1)(a_{k+1}+1)=2^{k}(A.a_{k+1} + 1)+2^{k-1}(A-A.a_{k+1}-1+a_{k+1})$$
The first part of the right formula is what we want. We have to prove that th... |
817,708 | <p>I had this as part of a question in an exam. And, I reasoned, even when it's arctan(1/0) (undefined), it is pi/2. And, so I said, domain belongs to all Real Numbers. Why isn't it this </p>
| Lucian | 93,448 | <p>$$\lim_{x\to0^+}\dfrac1x=+\infty\neq\lim_{x\to0^-}\dfrac1x=-\infty,\qquad\lim_{y\to+\infty}\arctan(y)=\dfrac\pi2\neq\lim_{y\to-\infty}\arctan(y)=-\dfrac\pi2$$ Since the left limit differs from the right limit, the limit does not exist. To exist, the two limits must be equal and finite.</p>
|
1,723,331 | <blockquote>
<p>$$\frac{1}{1-x^2}$$</p>
</blockquote>
<p>$$\frac{1}{1-x^2}=\frac{a}{1-x}+\frac{b}{1+x}$$</p>
<p>$$1=a+ax+b-bx$$</p>
<p>$$1=a+b+x(a-b)$$</p>
<p>$a+b=1$ and $x(a-b)=0\Rightarrow a-b=0\Rightarrow a=b$</p>
<p>$$2a=1\Rightarrow a=\frac{1}{2}$$</p>
<p>$b=\frac{1}{2}$</p>
<p>$$\frac{1}{1-x^2}=\frac{1}... | Community | -1 | <p>You didn't do anything wrong.</p>
<p>$$\frac{1}{1-x^2} = \frac{1}{2(x+1)} - \frac{1}{2(x-1)}$$
is equivalent to your answer of
$$\frac{1}{1-x^2} = \frac{1}{2(1-x)} + \frac{1}{2(1+x)}.$$</p>
<p>I think we can agree that both answers have a common term of $\dfrac{1}{2(1+x)}$. Now, notice:
$$\frac{1}{2(1-x)} = \frac... |
83,797 | <p>Good day, I'm not sure that this limit exists. All my attempts to prove it were in vain ...</p>
<p>Let $k>1$.
If exist, calculate the limit of the sequence $(x_n)$ defined by,
$$x_n := \Biggl(k \sin \left(\frac{1}{n^2}\right) + \frac{1}{k}\cos n \Biggr)^n.$$</p>
| Ross Millikan | 1,827 | <p>Hint: you can replace the trig functions with the first part of their Taylor series. How far do you have to go?</p>
|
3,212,117 | <p>My question is how to calculate the following formula without iteration:</p>
<p><span class="math-container">$$ \max \{A,B,C,D\} \tag 1 $$</span></p>
<p>suppose <span class="math-container">$A,B,C,D$</span> are normal and independent:
I know (1) can be rewritten as</p>
<p><span class="math-container">$$\max(\max(... | Henry | 6,460 | <p>If you have <span class="math-container">$n$</span> identically distributed and independent random variables each with </p>
<ul>
<li>cumulative distribution function <span class="math-container">$F(x)$</span> </li>
<li>and probability density function <span class="math-container">$f(x)$</span></li>
</ul>
<p>then, ... |
3,435,209 | <p><a href="https://i.stack.imgur.com/47sW7.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/47sW7.png" alt="enter image description here"></a></p>
<p>When I attempt to compute <span class="math-container">$f_{y}(0,0)$</span>, I first set <span class="math-container">$x = 0$</span> such that <span cl... | PierreCarre | 639,238 | <p>In this case you really need to use the definition, not the usual rules for derivatives. This comes from the fact that the correct expression for <span class="math-container">$f(0,y)$</span> is</p>
<p><span class="math-container">$$
f(0,y)=\begin{cases} 1, &y \ne 0 \\ 0, &y=0 \end{cases}
$$</span></p>
|
4,928 | <p>Consider the following list of countries which I would like to highlight on a world map:</p>
<pre><code>MyCountries={"Germany","Hungary","Mexico","Austria","Bosnia","Turkey","SouthKorea","China"};
</code></pre>
<p>From the documentation center <a href="http://reference.wolfram.com/mathematica/ref/CountryData.html"... | Mark McClure | 36 | <p>In the example code, <code>CountryData[#, "AntarcticNations"]</code> is a built in predicate that returns <code>True</code> or <code>False</code>. You need something similar for your countries. Perhaps,</p>
<pre><code>myCountries={
"Germany","Hungary","Mexico","Austria",
"Bosnia","Turkey","SouthKorea","China"... |
301,393 | <p>The question I am working on is:</p>
<blockquote>
<p>Prove that if $m+n$ and $n+p$ are even integers, where
$m$, $n$,and $p$ are integers, then $m+p$ is even. What kind
of proof did you use?</p>
</blockquote>
<p>I was thinking--and I aware that this may not be the most efficient method--of proving four diffe... | A.P. | 60,123 | <p>This is quite easy way : $(m+p)=((m+n)+(n+p))-2n$</p>
|
2,126,835 | <p>Suppose that $f: [0,1] \rightarrow [0,1]$ is continuously differentiable. Further, suppose that $f$ has a fixed point $x_{0} \in (0,1)$ such that $|f'(x_{0})| < 1$. Then there exists an open interval $I$ containing $x_{0}$ such that $\{ f^{n}(x) \}_{n}$ converges to $x_{0}$ for all $x \in I$. </p>
<p>I know t... | Eclipse Sun | 119,490 | <p><strong>Hint</strong>: </p>
<ol>
<li><p>Find a closed interval $I$ containing $x_0$ such that $|f'(x)|<1-\epsilon$ for all $x\in I$.</p></li>
<li><p>Show that $f:I\to I$ (using the fact that $x_0$ is a fixed point of $f$, and $|f'(x_0)|<1$, combined with the mean value theorem). </p></li>
<li><p>Use the mean ... |
3,389,361 | <p>For what n is this rational,
<span class="math-container">$$\frac{\sqrt{n^2+1}} {\sqrt{2}}$$</span> </p>
<p>So far I have found the integers 1,7,41 and I have found some rational solutions to this as well but I'm looking to get a more general sense. </p>
<ul>
<li>So when is this a rational number?</li>
<li>Are the... | David | 651,991 | <p><span class="math-container">$\frac{\sqrt{n^2+1}}{\sqrt{2}} = \sqrt{\frac{n^2+1}{2}}$</span>, which is rational if, and only if, <span class="math-container">$\sqrt{\frac{n^2+1}{2}}$</span> is an integer, so, there is <span class="math-container">$k \in \mathbb{N} : k^2 = \frac{n^2+1}{2}$</span></p>
<p>(Note that <... |
1,282,111 | <p>How to show that the equation $3x^2-2y^2=1$ has infinitely many integer solutions such that $3|x$ ? ( If this can be shown then solutions of $12x^2-8y^2=4$ give infinitely many powerful numbers differing by $4$ )</p>
| Robert Israel | 8,508 | <p>If $(x,y)$ is a solution, then so is $(5x+4y, 6x+5y)$. Starting with $(1,1)$,
you get a sequence of solutions. What is $x$ mod $3$ for these?</p>
|
1,714,714 | <p>I just wanted to make sure that my logic here is not faulty. Up till now I've generally avoided contraposition proofs and worked only with contradiction (since we may rephrase the former in terms of the latter), so I'm having a bit of trouble wrapping my head around one point.</p>
<p>Suppose I have a statement of t... | Patrick Stevens | 259,262 | <p>It's legitimate but scrappy, if I understand correctly. You've made a proof by contradiction again.</p>
<p>The problem with contradiction is that it doesn't tell you anything other than "my assumption is false". If you prove that $\neg B \Rightarrow \neg A$ without assuming $A$, you know that any results you end up... |
1,714,714 | <p>I just wanted to make sure that my logic here is not faulty. Up till now I've generally avoided contraposition proofs and worked only with contradiction (since we may rephrase the former in terms of the latter), so I'm having a bit of trouble wrapping my head around one point.</p>
<p>Suppose I have a statement of t... | Dharmesh Sujeeun | 300,640 | <p>I'm going to answer the first part of your question, since I am not sure that I know the answer to the second part.</p>
<p>So basically, in a proof by contrapositive, you assume that ~C is true, and prove that when ~C is true, it leads to ~A and ~B. That is pretty much all you need to do. </p>
<p>So in response to... |
2,130,062 | <p>I have 4 decks, each with five unique cards. If I select 3 from the first, 4 from the second, 2 from the third and 2 from the fourth. In case order matters and I do not put cards back, in how many ways can I arrange the 11 cards if the order matters for each selected card? </p>
<p>I think that I use the following f... | Michael Rozenberg | 190,319 | <p>It's just $$\prod\limits_{cyc}(a^2+1)+2a^2b^2c^2-\sum_{cyc}a^2b^2(c^2+1)=a^2+b^2+c^2+1$$</p>
|
1,212,198 | <blockquote>
<p>Prove or counter-example. For all nonempty sets $A$ and $B$ and for all functions $F$, $F(A-B) = F(A) - F(B)$; if not, what else does $F$ need to have in order to make the equality hold?</p>
</blockquote>
<p>I am pretty lost on this question. I don't feel like its right since it would be a pretty bas... | Eugene Zhang | 215,082 | <p>We prove the following: $F(A)−F(B)\subset F(A-B) $</p>
<p>First we prove: $F(A-B)\cup F(B)=F(A)\cup F(B)$</p>
<p>\begin{align}
F(A-B)\cup F(B)&=F((A-B)\cup B)
\\
&=F((A\cap B^c)\cup B)
\\
&=F((A\cup B) \cap (B^c\cup B))
\\
&=F(A\cup B)
\\
&=F(A)\cup F(B)
\end{align}</p>
<p>Then
\begin{align}
F... |
95,709 | <p>Let $F$ be a fixed free group of finite rank. If $H$ is a finitely generated subgroup of $F$ and $A$ is a basis for $F$, then we can form the Stallings graph $\Gamma_A(H)$ for $H$. It is the unique smallest ($|A|$-labelled) subgraph of the covering space of a bouquet of $A$-circles corresponding to $H$ that contains... | Lee Mosher | 20,787 | <p>Here is what I think will be an infinite index example, although I'm missing some details of proof. Take $F = \langle a,b \rangle$, identified with the fundamental group of a rose $R$ with two petals labelled $a$, $b$. Let $\Theta$ be the rank 2 graph with oriented edges $e_1,e_2,e_3$ having the same initial and ter... |
2,782,489 | <p>The solutions to a tutorial question I am working on are as follows:</p>
<p><span class="math-container">$$\cos\left(\frac{p}{\sqrt{\eta}}x\right) = \cos\left(\frac{p}{\sqrt{\eta}}x\right)\cos\left(\frac{p}{\sqrt{\eta}}2L\right)-\sin\left(\frac{p}{\sqrt{\eta}}x\right)\sin\left(\frac{p}{\sqrt{\eta}}2L\right)$$</span>... | Abhishek Choudhary | 452,208 | <p>Let $\frac{p}{\sqrt{\eta}}=l$<br>
Then, $RHS=\cos(lx+2Ll)$<br>
$\therefore \cos(lx+2Ll)=\cos(lx)$<br>
$\therefore lx+2Ll=2n\pi \pm lx$<br>
$\therefore 2Ll=2\eta\pi,2\eta\pi-2lx$<br>
$\frac{2Lp}{\sqrt{\eta}}=2\eta\pi,2\eta\pi-2\frac{p}{\sqrt{\eta}}x$</p>
|
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