qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
3,224,745 | <p>Naive evaluation of <span class="math-container">$\sqrt{a + x} - \sqrt{a}$</span> when <span class="math-container">$|a| >> |x|$</span> suffers from catastrophic cancellation and loss of significance.</p>
<p>WolframAlpha gives the Taylor series for <span class="math-container">$\sqrt{a+x}-\sqrt{a}$</span> as:... | Picaud Vincent | 450,342 | <p>To avoid cancellation error the first thing to do is to write:</p>
<p><span class="math-container">$$
\sqrt{a+x}-\sqrt{a}=\frac{x}{\sqrt{a+x}+\sqrt{a}}=\sqrt{a}\frac{x}{a}
\frac{1}{1+\sqrt{1+\frac{x}{a}}}
$$</span></p>
<p>then with <span class="math-container">$y=\frac{x}{a}$</span> you must approximate this
<span... |
3,224,745 | <p>Naive evaluation of <span class="math-container">$\sqrt{a + x} - \sqrt{a}$</span> when <span class="math-container">$|a| >> |x|$</span> suffers from catastrophic cancellation and loss of significance.</p>
<p>WolframAlpha gives the Taylor series for <span class="math-container">$\sqrt{a+x}-\sqrt{a}$</span> as:... | Claude | 209,286 | <p>It was pointed out by Robert Israel that the series does badly when <span class="math-container">$|x| \approx |a|$</span>, but in that case the loss of significance of the naive evaluation is small.</p>
<p>It was also suggested by Winther (and a since-deleted answer) to rewrite as <span class="math-container">$$\fr... |
316,699 | <p>If $A,B,C$ are sets, then we all know that $A\setminus (B\cap C)= (A\setminus B)\cup (A\setminus C)$. So by induction
$$A\setminus\bigcap_{i=1}^nB_i=\bigcup_{i=1}^n (A\setminus B_i)$$
for all $n\in\mathbb N$.</p>
<p>Now if $I$ is an uncountable set and $\{B_i\}_{i\in I}$ is a family of sets, is it true that:
$$A\s... | Stefan Hansen | 25,632 | <p>Note that
$$
A\setminus \bigcap_{i\in I}B_i=A\cap \left(\bigcap_{i\in I}B_i\right)^c,
$$
where $^c$ denotes the complement. Using <a href="http://en.wikipedia.org/wiki/De_Morgan%27s_laws" rel="nofollow">De Morgan's laws</a> (which holds for a general $I$) we get
$$
A\cap \left(\bigcap_{i\in I}B_i\right)^c =A\cap \le... |
2,613,410 | <blockquote>
<p>What is the value of <span class="math-container">$2x+3y$</span> if</p>
<p><span class="math-container">$x+y=6$</span> & <span class="math-container">$x^2+3xy+2y=60$</span> ?</p>
</blockquote>
<p>My trial:
from given conditions: substitute <span class="math-container">$y=6-x$</span> in <span class="... | user | 505,767 | <p>It seems correct to me</p>
<p>$$x^2+3x(6-x)+2(6-x)=60\iff x^2+18x-3x^2+12-2x-60=0$$
$$\iff-2x^2+16x-48=0\iff x^2-8x+24=0\implies x=4\pm2i\sqrt2$$</p>
|
815,432 | <p>I have sometimes seen notations like $a\equiv b\pmod c$. How do we define the notation? Have I understood correctly that $c$ must be an element of some ring or does the notation work in magmas in general?</p>
| C-star-W-star | 79,762 | <p>You certainly need a ring since a magma has only one binary operation while the modulus expression involves two:
$$a\equiv b\pmod{c}:\iff a=b+k\cdot c$$</p>
|
1,011,718 | <p>$P1 : \sin(x/y)$ .</p>
<p>I tried using $y=mx$. $f$ becomes $\sin(1/m)$, so limit doesn't exist. But it is too easy. Am I right?</p>
<p>$P2 : F = x^2\log(x^2+y^2)$</p>
<p><img src="https://i.stack.imgur.com/pEaSs.jpg" alt="enter image description here"></p>
<p><img src="https://i.stack.imgur.com/CEI2K.jpg" alt=... | Alen | 22,372 | <p>Substituting $\left( {x,y} \right) = \left( {r\sin \phi ,y = r\cos \phi } \right) = \varphi \left( {r,\phi } \right)$ gives $f\left( {x,y} \right) = f\left( {\varphi \left( {r,\phi } \right)} \right) = {\sin ^2}\left( \phi \right){r^2}\ln {r^2}$.</p>
<p>If the limit $\mathop {\lim }\limits_{r \to 0} {\sin ^2}\left... |
546,572 | <p><img src="https://i.stack.imgur.com/aJ2t5.jpg" alt="enter image description here"></p>
<p>Could anyone tell me how to solve 9b and 10? I've been thinking for five hours, I really need help.</p>
| GA316 | 72,257 | <p>Hint : union of two discrete sets need not be discrete</p>
|
2,196,413 | <blockquote>
<p>Let $R$ be a commutative ring. Denote by $R^*$ the group of invertible elements (this is a group w.r.t multiplication.) Suppose $R^*\cong \mathbb{Z}$. I need to show that $1+1=0$ in $R$.</p>
</blockquote>
<p>I have no clue about why such statement should be true. I don't even have an example for a r... | Hanno | 81,567 | <p><em>Hint:</em> If $R^{\times}\cong{\mathbb Z}$, then in particular it has no nontrivial torsion; on the other hand, there's $-1\in R^{\times}$.</p>
|
4,154,298 | <p>Suppose that<span class="math-container">$ f(x, y)$</span> given by <span class="math-container">$\sum_{i=0}^{a}\sum_{j=0}^{b}c_{i,j}x^iy^j$</span> is a polynomial in two variables with real coefficients such that among its coefficients there is a non-zero one. Prove that there is a point <span class="math-container... | Community | -1 | <p>If you want to work ultraformally, a sequence is a function from <span class="math-container">$\mathbb{N}$</span> to <span class="math-container">$\mathbb{R},$</span> and so you cannot just start indexing at 1--you have a new object, not a sequence. Your alternative definition is too broad, since it captures a notio... |
2,414,492 | <p>Check the convergence of $$\sum_{k=0}^\infty{2^{-\sqrt{k}}}$$
I have tried all other tests (ratio test, integral test, root test, etc.) but none of them got me anywhere.
Pretty sure the way to do it is to check the convergence by comparison, but not
sure how.</p>
| Clement C. | 75,808 | <p>Note that $$2^{-\sqrt{k}} = e^{-\sqrt{k}\ln 2}$$ while $$\frac{1}{k^2} = e^{-2\ln k}.$$ Since $\sqrt{k} \ln 2 > 2\ln k$ for $k$ big enough,* you can conclude by comparison with the $p$-series $\sum_{k=1}^\infty \frac{1}{k^2}$.</p>
<blockquote>
<p>*One can check it holds for all $k\geq 256$.</p>
</blockquote>
|
2,732,202 | <p>$Z_1, Z_2, Z_3,...$ are independent and identically distributed R>V.s s.t. $E(Z_i)^- < \infty$ and $E(Z_i)^+ = \infty$. Prove that $$\frac {Z_1+Z_2+Z_3+\cdots+Z_n} n \to \infty$$ almost surely.</p>
<p>What does $E(Z_i)^+$ $E(Z_i)^-$ mean? I believe it is integrating fromnegative infinity to zero and zero to posi... | Michael Hardy | 11,667 | <p>The positive part of a random variable $X$ is the random variable $X^+ = \begin{cases} 0 & \text{if } X<0, \\ X & \text{if } X\ge 0. \end{cases}$</p>
<p>The negative part $X^-$ is defined similarly. If $F(x) = \Pr(X\le x)$ then you have $\displaystyle\operatorname E(X^+) = \int_0^\infty x\, dF(x)$ and $\... |
1,121,354 | <p>I need help understanding the following solution for the given problem. </p>
<p>The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i \in F$ forms a ring $F[[t]]$. Determine the ideals of the ring.</p>
<p>The solution: Let $I$ be an ideal and $... | orangeskid | 168,051 | <p>Let
$$p(t) = a_0 + a_1 t + a_2 t^2 + \cdots \\
q(t) = b_0 + b_1 t + b_2 t^2 + \cdots $$
Then we have
$$p(t) \cdot q(t) = c_0 + c_1 t + c_2 t^2 + \cdots $$
where
\begin{eqnarray}
c_0 &=& a_0 b_0 \\
c_1 &=&a_0 b_1 + a_1 b_0 \\
c_2 &=& a_0 b_2 + a_1 b_1 + a_2 b_0\\
c_3 &=& a_0 b_3 + a... |
3,755,355 | <p>I wanted to prove that every group or order <span class="math-container">$4$</span> is isomorphic to <span class="math-container">$\mathbb{Z}_{4}$</span> or to the Klein group. I also wanted to prove that every group of order <span class="math-container">$6$</span> is isomorphic to <span class="math-container">$\ma... | Community | -1 | <p>Let <span class="math-container">$G$</span> be a group of order 4. Assume for contradiction that it is not abelian, i.e. we have elements <span class="math-container">$a,b$</span> that do not commute: then <span class="math-container">$1, a, b, ab, ba$</span> are 5 distinct elements.</p>
<p>Therefore <span class="ma... |
279,277 | <p>I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're the inverse of each other in basic math (so their graphed functions, derivatives, etc.)?</p>
| rschwieb | 29,335 | <p>It means that $e^x$ is a <a href="http://en.wikipedia.org/wiki/Bijection" rel="nofollow noreferrer">bijection</a> from $\Bbb R$ onto $(0, \infty)$, and that $\ln(x)$ is a bijection of $(0,\infty)$ onto $\Bbb R$. In other words, both functions pair up points in their domain and range: roughly speaking, "nothing is le... |
279,277 | <p>I have been told multiple times that the logarithmic function is the inverse of the exponential function and vice versa. My question is; what are the implications of this? How can we see that they're the inverse of each other in basic math (so their graphed functions, derivatives, etc.)?</p>
| amWhy | 9,003 | <p>Just to generalize: </p>
<p>A function $g: Y \to X$ is an inverse function of $\;f: X \to Y\;$ (and vice versa) if and only if $$\;g \circ f = id_X\; \text { and}\;\,f\circ g = id_Y:\;$$ that is, if and only if $\;g(f(x)) = x\;$ for all $x \in X$ and $\;f(g(y)) = y\,$ for all $\,y \in Y$.</p>
|
4,269,414 | <p>In <a href="https://math.stackexchange.com/questions/550764/quotient-space-of-s1-is-homeomorphic-to-s1">this post</a> someone suggested:</p>
<p>"<span class="math-container">$z\mapsto z^2$</span>"</p>
<p>where both <span class="math-container">$z$</span> and <span class="math-container">$z^2$</span> are in... | principal-ideal-domain | 131,887 | <p>As in the comments is already pointed out <span class="math-container">$z^2$</span> means <span class="math-container">$z\cdot z$</span>.</p>
<p>You suggestest <span class="math-container">$(z_1^2,z_2^2)$</span>. That's not how the multiplication in <span class="math-container">$\mathbb C$</span> works. It's actuall... |
2,911,187 | <p>Lines (same angle space between) radiating outward from a point and intersecting a line:</p>
<p><a href="https://i.stack.imgur.com/52HY4.png" rel="noreferrer"><img src="https://i.stack.imgur.com/52HY4.png" alt="Intersection Point Density Distribution"></a></p>
<p>This is the density distribution of the points on t... | TurlocTheRed | 397,318 | <p>This is also known, especially among we physicists as a Lorentz distribution.</p>
<p>We know that every point on the line has the same vertical distance from the source point. Let's call this $l_0$. We also know the ratio of the horizontal component of the distance to the vertical component is $\tan(\theta)$, where... |
33,387 | <p>I was told the following "Theorem": Let $y^{2} =x^{3} + Ax^{2} +Bx$ be a nonsingular cubic curve with $A,B \in \mathbb{Z}$. Then the rank $r$ of this curve satisfies</p>
<p>$r \leq \nu (A^{2} -4B) +\nu(B) -1$</p>
<p>where $\nu(n)$ is the number of distinct positive prime divisors of $n$.</p>
<p>I can not find a n... | Adrián Barquero | 900 | <p>I think this is a <a href="https://planetmath.org/BoundForTheRankOfAnEllipticCurve" rel="nofollow noreferrer">reference</a>.</p>
<p></p>
|
3,270,725 | <p>Hello everyone I read on my notes this proposition: </p>
<p>Given a field <span class="math-container">$K$</span> and <span class="math-container">$R=K[T]$</span>, let <span class="math-container">$M$</span> be a (left) finitely generated <span class="math-container">$R$</span>-module; then <span class="math-contai... | Dr. Sonnhard Graubner | 175,066 | <p>It is <span class="math-container">$$\frac{x}{\pi}+\frac{1}{2}\notin \mathbb{Z}$$</span> and <span class="math-container">$$-\frac{\pi}{2}<x-2\pi n<0$$</span>
or
<span class="math-container">$$\frac{x}{\pi}+\frac{1}{2}\notin \mathbb{Z}$$</span> and <span class="math-container">$$\frac{\pi}{2}<x-2\pi n<\p... |
2,009,557 | <p>I am pretty sure this question has something to do with the Least Common Multiple. </p>
<ul>
<li>I was thinking that the proof was that every number either is or isn't a multiple of $3, 5$, and $8\left(3 + 5\right)$.</li>
<li>If it isn't a multiple of $3,5$, or $8$, great. You have nothing to prove.</li>
<li>But if... | Joffan | 206,402 | <p>A formal induction:
Any integer $k>7$ can be expressed as $3a+5b = k$, with $a,b\ge 0$.</p>
<p>Base cases:</p>
<ul>
<li>$k=8 \implies a=1,b=1$</li>
<li>$k=9 \implies a=3,b=0$</li>
<li>$k=10 \implies a=0,b=2$</li>
</ul>
<p>Induction: </p>
<p>Assume that the statement holds for all $k<m$, and $m>10$. In p... |
1,098,614 | <p>Newbie question: Is pulling out factors and dividing the same? Please explain the difference in the examples below.</p>
<p>Example:</p>
<p>$$2x-4 \to 2(x-2)$$</p>
<p>Is it the same as </p>
<p>$$4x^2 - 8x + 1 = 0$$</p>
<p>$$\frac{4x^2 - 8 }{4} = -\frac{1}{4} \implies x^2 - 2x = -\frac{1}{4}$$</p>
<p>Thanks</p>... | mez | 59,360 | <p>I worked things out and have the following conclusion.</p>
<p>Abelian groups are the same as $\mathbb{Z}$-modules, therefore sheafs of abelian groups are the same as $\underline{\mathbb{Z}}^{pre}$-modules. However, due to the fact that sheafs have the identity and gluability property, this $\underline{\mathbb{Z}}^{... |
1,098,614 | <p>Newbie question: Is pulling out factors and dividing the same? Please explain the difference in the examples below.</p>
<p>Example:</p>
<p>$$2x-4 \to 2(x-2)$$</p>
<p>Is it the same as </p>
<p>$$4x^2 - 8x + 1 = 0$$</p>
<p>$$\frac{4x^2 - 8 }{4} = -\frac{1}{4} \implies x^2 - 2x = -\frac{1}{4}$$</p>
<p>Thanks</p>... | Andrew Tawfeek | 251,053 | <p>For some reason this note of Vakil seemed to take a bit for me to unpack -- so once bumping into this question here I felt obligated to include what helped me eventually see the natural structure.</p>
<p>Let's take <span class="math-container">$\mathscr{F}$</span> to be a <span class="math-container">$\underline{\m... |
3,334,816 | <p>The question first requires me to prove the identity <span class="math-container">$$\sqrt{\frac{1- \sin x}{1+ \sin x}}=\sec x- \tan x, -90^\circ < x < 90^\circ$$</span> I am able to prove this. The second part says “Explain why <span class="math-container">$x$</span> must be acute for the identity to be true”.... | lab bhattacharjee | 33,337 | <p>For <span class="math-container">$1+\sin x\ne0$</span></p>
<p><span class="math-container">$$f^2(x)=\dfrac{1-\sin x}{1+\sin x}=\dfrac{(1-\sin x)^2}{\cos^2x}$$</span></p>
<p><span class="math-container">$$f(x)=\dfrac{|1-\sin x|}{|\cos x|}=\dfrac{1-\sin x}{|\cos x|}$$</span> as <span class="math-container">$1-\sin x... |
3,473,944 | <p>So i have an object that moves in a straight line with initial velocity <span class="math-container">$v_0$</span> and starting position <span class="math-container">$x_0$</span>. I can give it constant acceleration <span class="math-container">$a$</span> over a fixed time interval <span class="math-container">$t$</s... | Arturo Magidin | 742 | <p>I will use <span class="math-container">$t_0$</span> rather than <span class="math-container">$t$</span>, since this is also a fixed quantity.</p>
<p>What you are doing doesn't work for arbitrary <span class="math-container">$t_0$</span>, <span class="math-container">$x_0$</span>, <span class="math-container">$x_1$... |
2,193,779 | <p><a href="https://i.stack.imgur.com/d65g2.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/d65g2.png" alt="enter image description here"></a></p>
<p>Any idea where the missing 3^k+2 comes from?
(sorry for the format, this thing didn't allow me to post images)</p>
| mrnovice | 416,020 | <p>Here's the inductive step:</p>
<p>We have that:</p>
<p>$1\cdot 3 + 2\cdot 3^2 +... + n\cdot 3^n = \frac{(2n-1)3^{n+1}+3}{4}$</p>
<p>So now consider:</p>
<p>$1\cdot 3 + 2\cdot 3^2 +... + n\cdot 3^n + (n+1)\cdot3^{n+1} = \frac{(2n-1)3^{n+1}+3}{4} + (n+1)3^{n+1}$</p>
<p>$= \frac{(2n-1)3^{n+1}+3+4(n+1)3^{n+1}}{4}... |
1,862,807 | <blockquote>
<p>Show that for <span class="math-container">$x,y,z\in\mathbb{Z}$</span>, if <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are coprime and <span class="math-container">$z$</span> is nonzero, then <span class="math-container">$\exists n\in\mathbb{Z}$</span> such that... | C.S. | 95,894 | <p>$\textbf{Exercise}$ Let $(a,b)=1$ and $c>0$. Prove that there is an integer $x$ such that $(a+bx,c)=1$.</p>
<p>$\textbf{Solution.}$ Let $p_{1},p_{2},\cdots,p_{m}$ be the primes which appear in the prime factorization of $b$. Then since $(a,b)=1$, we have $(a,p_{i})=1$ for all $i$. If the prime factorization of $... |
2,550,365 | <p>I encountered a question regarding orthogonal projection. I only have some clues to approach this questions and am not sure whether my thoughts can be used to answer the question. Could anyone check my thoughts and correct it or figure out a better solution, if needed? Thanks in advance.</p>
<p>Suppose $W \subseteq... | user | 505,767 | <p><strong>HINT</strong> </p>
<p>As you noted</p>
<p>if $v \in W \implies Pv=1\cdot v$</p>
<p>if $u \in W^{\perp} \implies Pu=0\cdot u$</p>
<p>thus, since $\dim(W)=4$ and $\dim(W^{\perp})=2$ we have that $P$ is similar to $diag(1,1,1,1,0,0)$.</p>
|
1,989,950 | <p>I was doing proof of open mapping theorem from the book Walter Rudin real and complex analysis book and struck at one point. Given if $X$ and $Y$ are Banach spaces and $T$ is a bounded linear operator between them which is $\textbf{onto}$. Then to prove $$T(U) \supset \delta V$$ where $U$ is open unit ball in $X$ an... | Jack Tiger Lam | 186,030 | <p>$\displaystyle \large \sec{\theta} + \tan{\theta} \equiv \frac{(\cos{\frac{\theta}{2}} + \sin{\frac{\theta}{2}})^2}{\cos^2{\frac{\theta}{2}} - \sin^2{\frac{\theta}{2}}} \equiv \frac{1+\tan{\frac{\theta}{2}}}{1-\tan{\frac{\theta}{2}}} \equiv \tan{\left( \frac{\theta}{2} + \frac{\pi}{4} \right)}$</p>
<p>The rest is t... |
376,484 | <p>My questions are motivated by the following exercise:</p>
<blockquote>
<p>Consider the eigenvalue problem
$$
\int_{-\infty}^{+\infty}e^{-|x|-|y|}u(y)dy=\lambda u(x), x\in{\Bbb R}.\tag{*}
$$
Show that the spectrum consists purely of eigenvalues. </p>
</blockquote>
<p>Let $A:L^2({\Bbb R})\to L^2({\Bbb R})$ be ... | Álvaro Lozano-Robledo | 14,699 | <p>The statement is false, even if you restrict yourself to completely multiplicative arithmetic functions. </p>
<p>Let $f(n): \mathbb{N} \to \mathbb{Q}$ be a function defined by $f(n)=\frac{1}{2^e}$, where
$$n=p_1^{e_1}p_2^{e_2}\cdots p_r^{e_r}$$
is a factorization of $n$ into prime powers (i.e., each $p_i$ is a pri... |
3,196,634 | <p>I am trying to solve a differential equation:
<span class="math-container">$$\frac{d f}{d\theta} = \frac{1}{c}(\text{max}(\sin\theta, 0) - f^4)~,$$</span>
subject to periodic boundary condition, whic would imply <span class="math-container">$f(0)=f(2\pi)$</span> and <span class="math-container">$f'(0)= f'(2\pi)$</sp... | jlandercy | 113,708 | <p>If you are not told to do it all by yourself, I would suggest you to use the powerful <code>scipy</code> package (specially the <a href="https://docs.scipy.org/doc/scipy-1.2.1/reference/tutorial/integrate.html" rel="nofollow noreferrer"><code>integrate</code></a> subpackage) which exposes many useful objects and met... |
3,196,634 | <p>I am trying to solve a differential equation:
<span class="math-container">$$\frac{d f}{d\theta} = \frac{1}{c}(\text{max}(\sin\theta, 0) - f^4)~,$$</span>
subject to periodic boundary condition, whic would imply <span class="math-container">$f(0)=f(2\pi)$</span> and <span class="math-container">$f'(0)= f'(2\pi)$</sp... | Lutz Lehmann | 115,115 | <p>How to solve this (perhaps a little more complicated than necessary) with the tools of python <code>scipy.integrate</code> I demonstrated in <a href="https://math.stackexchange.com/q/3185707/115115">How to numerically set up to solve this differential equation?</a></p>
<hr>
<p>If you want to stay with the simplici... |
860,247 | <p>Simplify $$\frac{3x}{x+2} - \frac{4x}{2-x} - \frac{2x-1}{x^2-4}$$</p>
<ol>
<li><p>First I expanded $x²-4$ into $(x+2)(x-2)$. There are 3 denominators. </p></li>
<li><p>So I multiplied the numerators into: $$\frac{3x(x+2)(2-x)}{(x+2)(x-2)(2-x)} - \frac{4x(x+2)(x-2)}{(x+2)(x-2)(2-x)} - \frac{2x-1(2-x)}{(x+2)(x-2)(2-x... | Count Iblis | 155,436 | <p>$$\int_{0}^{1}x^{\frac{5}{2}}\sqrt{1+x}dx$$</p>
<p>Substitute $x = t^2$:</p>
<p>$$2\int_{0}^{1}t^6\sqrt{1+t^2}dt$$</p>
<p>Substitute $t = \sinh(u)$:</p>
<p>$$2\int_{0}^{\operatorname{arcsinh}(1)}\left[\sinh^8(u)-\sinh^6(u)\right]du$$</p>
|
3,190,594 | <p>From Rick Durrett's book <em>Probability: Theory and Examples</em>:</p>
<blockquote>
<p>We define the conditional expectation of <span class="math-container">$X$</span> given <span class="math-container">$\mathcal{G}$</span>, <span class="math-container">$E(X | \mathcal{G})$</span> to be any random variable <span cla... | Davide Giraudo | 9,849 | <p>The difference between the two definitions is that in the first one, we need to do the test that <span class="math-container">$\mathbb E\left[XY\right]=\mathbb E\left[ZY\right]$</span> only when <span class="math-container">$Y$</span> has the form <span class="math-container">$\mathbf 1_A$</span> for all <span class... |
2,827,970 | <p>I'm aware this result (and the standard/obvious proof) is considered basic and while I've accepted and used it numerous times in the past, I'm starting to question its validity, or rather that said proof doesn't subtly require a form of the AC. (Disclaimer: it's been some time since I've looked at set theory.)</p>
... | hmakholm left over Monica | 14,366 | <p>Hopefully you already know that $\omega$ (defined as the intersection of all inductive sets) is an ordinal in the first place.</p>
<p>Hopefully you also know that the ordinals are totally ordered by $\in$ -- that is, for ordinals $\alpha,\beta$ we always have either $\alpha\in\beta$ or $\alpha=\beta$ or $\beta\in\a... |
2,540,007 | <p>I have the following question. Find the matrix representation of the transormation $T:\mathbb{R}^3\to\mathbb{R}^3$ that rotates any vector by $\theta=\frac{\pi}{6}$ along the vector $v=(1,1,1)$.</p>
<p>A hint is given to find the rotation matrix about the $z-axis$ by $\frac{\pi}{6}$which is $$
\begin{bmatrix}
\f... | Gribouillis | 398,505 | <p>To understand why it works, let's use an analogy: imagine the linear transformation $T$
as a database that contains answers to a set of questions:
a question is any vector $u$, the answer to
question $u$ is the vector $T \left(u\right)$.</p>
<p>Questions need to be asked in a certain language, as concrete
vectors n... |
1,597,638 | <p>I noticed while studying integration that $\int \sqrt{1-x^2} \mathrm dx $ has a relatively simple antiderivative found by doing a trigonometric substitution. </p>
<p>On the other hand, $\int \sqrt{1-x^3} \mathrm dx $ can only be expressed with elliptic integrals (according to WA). The same thing occured when $a=4, ... | Sanwar | 290,140 | <p>You can write, $443642_{b_1} = 4*b_1^5+4*b_1^4+3*b_1^3+6*b_1^2+4*b_1+2$ and $53818_{b_2} = 5*b_2^4+3*b_2^3+8*b_2^2+b_2+8$.
Now set the two expressions equal. You may need another restriction on $b_1$ and $b_2$.</p>
|
1,597,638 | <p>I noticed while studying integration that $\int \sqrt{1-x^2} \mathrm dx $ has a relatively simple antiderivative found by doing a trigonometric substitution. </p>
<p>On the other hand, $\int \sqrt{1-x^3} \mathrm dx $ can only be expressed with elliptic integrals (according to WA). The same thing occured when $a=4, ... | peter.petrov | 116,591 | <p>An obvious observation here is that $b_1 \ge 7$ and $b_2 \ge 9$ (because the first number contains the digit 6, and the second one contains the digit 8). </p>
<p>One possible solution is $b_1 = 7$ and $b_2 = 11$. I wrote a small program to find this. </p>
<p>As far as I know there's no general method for solvi... |
1,597,638 | <p>I noticed while studying integration that $\int \sqrt{1-x^2} \mathrm dx $ has a relatively simple antiderivative found by doing a trigonometric substitution. </p>
<p>On the other hand, $\int \sqrt{1-x^3} \mathrm dx $ can only be expressed with elliptic integrals (according to WA). The same thing occured when $a=4, ... | Piquito | 219,998 | <p>Actually your problem can be in general very hard to solve because, in your particular case, one has the following diophantine equation of degree 5 with two unknowns:</p>
<p>$$4x^5+4x^4+3x^3+6x^2+4x+2=5y^4+3y^3+8y^2+y+8$$</p>
<p>whose solution is $$(x,y)=(b_1,b_2)=(7,11)$$
(<em>in both sides you have the number $... |
1,597,638 | <p>I noticed while studying integration that $\int \sqrt{1-x^2} \mathrm dx $ has a relatively simple antiderivative found by doing a trigonometric substitution. </p>
<p>On the other hand, $\int \sqrt{1-x^3} \mathrm dx $ can only be expressed with elliptic integrals (according to WA). The same thing occured when $a=4, ... | j215c228 | 283,345 | <p>If you know that a there is an integer solution, why not just use the rational zeros theorem to find the zeros of both polynomials on both sides of the equation? Your list would be fairly short and could quickly find an answer</p>
|
1,791,673 | <p>I was wondering about this, just now, because I was trying to write something like:<br>
$880$ is not greater than $950$. <br>
I am wondering this because there is a 'not equal to': $\not=$ <br>
Not equal to is an accepted mathematical symbol - so would this be acceptable: $\not>$? <br>
I was searching around but ... | mvw | 86,776 | <p>I would probably use $850 \le 950$, as order is defined for integers.</p>
|
1,791,673 | <p>I was wondering about this, just now, because I was trying to write something like:<br>
$880$ is not greater than $950$. <br>
I am wondering this because there is a 'not equal to': $\not=$ <br>
Not equal to is an accepted mathematical symbol - so would this be acceptable: $\not>$? <br>
I was searching around but ... | GEdgar | 442 | <p>To answer the question, yes.
$$
a \nless b\\
a \ngtr b\\
a \nleq b\qquad a \nleqq b\qquad a \nleqslant b\\
a \ngeq b\qquad a \ngeqq b\qquad a \ngeqslant b
$$
and so on for many other mathematical relations
$$
a \nleftarrow b\\
a \nLeftarrow b\\
A \nsupseteqq B\\
A \nvdash \phi\qquad A \nVdash \phi\\
\nexists x
$$</p... |
177,515 | <p>From <a href="http://mitpress.mit.edu/algorithms/" rel="nofollow">Cormen et all</a>:</p>
<blockquote>
<p>The elements of a matrix or vectors are numbers from a number system, such as the real numbers , the complex numbers , or integers modulo a prime .</p>
</blockquote>
<p>What do they mean by <strong>integers m... | Community | -1 | <blockquote>
<p>Why did they put this additional one ?</p>
</blockquote>
<p>They did not "put" and an additional one. These are all <em>examples</em> of different number system: that is, a set of number along with operations you can do.</p>
<blockquote>
<p>I thought real numbers and complex numbers together make ... |
3,115,090 | <p>This was supposedly an easy limit, and it is suspiciously similar to a Riemann sum, but I can't quite figure out for what function. </p>
<p><span class="math-container">$$\lim_{n\to\infty}{\frac{1}{n} {\sum_{k=3}^{n}{\frac{3}{k^2-k-2}}}}$$</span></p>
<p>Well, even the fact that <span class="math-container">$\frac{... | Yanko | 426,577 | <p><span class="math-container">$$\sum_{k=3}^n \frac{3}{k^2-k-2} = \sum_{k=3}^n \frac{1}{k-2}- \sum_{k=3}^n \frac{1}{k+1}$$</span></p>
<p>You (and I) were mistaken before, see @Romeo 's answer.</p>
<p>Notice that <span class="math-container">$$\sum_{k=3}^n \frac{1}{k-2}=\sum_{k=0}^{n-3} \frac{1}{k+1}$$</span></p>
<... |
1,988,731 | <p>Can one of you math this for me?</p>
<p>You've got a deck of 88 cards. Let's call the first card "1", the next "2", and so on.</p>
<p>What are the odds of pulling "1", "2", and "3" out of the deck, in that order?</p>
| Blake M | 383,537 | <p>I'm new using this website, so take my answer with a grain of salt. </p>
<p>It should be 1/88 * 1/87 * 1/86 = 1/(88*87*86) = 1/658416 Or about 1.52*10^-6. </p>
<p>My thought is that you have exactly one card in 88 that is the desired result of 1. With that card drawn, the deck has 87 remaining, again only one ca... |
4,016,133 | <p>I am recently exploring set-theoretical postulates that contradict GCH. One particularly interesting one is the proposition "<span class="math-container">$2^\kappa$</span> is singular for each infinite cardinal <span class="math-container">$\kappa$</span>". However I could not prove that this proposition i... | Jason Zesheng Chen | 599,883 | <p>Lemma 3.4 in the following <a href="https://arxiv.org/abs/1502.07470" rel="nofollow noreferrer">paper</a> by Golshani & Hayut implies that this is consistent relative to the existence of a strong cardinal.</p>
<p><em>Golshani, Mohammad; Hayut, Yair</em>, <a href="http://dx.doi.org/10.1017/jsl.2016.21" rel="nofol... |
100,526 | <p>It is known that for a Poisson process the inter-arrival time is exponentially distributed. My question, which may be nonsense, is this. Suppose you want to experimentally evaluate the distribution of inter-arrival time (not necessarily of a Poisson process). You measure the differences in the arrival time of consec... | Bill Cook | 16,423 | <p>Suppose we have a Fourier expansion for an eigenfunction $u(x)=a_0+\sum\limits_{k=1}^\infty \left(a_k\sin(kx)+b_k\cos(kx)\right)$. Then $\int_0^{2\pi} u(t)\cos(t)\,dt=b_1\pi$</p>
<p>So in order to have $A[u] = \lambda u$ we would need $b_1\pi\sin(x)=\lambda u(x)$. So $\lambda u(x)$ has nothing but $\sin(x)$ appeari... |
100,526 | <p>It is known that for a Poisson process the inter-arrival time is exponentially distributed. My question, which may be nonsense, is this. Suppose you want to experimentally evaluate the distribution of inter-arrival time (not necessarily of a Poisson process). You measure the differences in the arrival time of consec... | paul garrett | 12,291 | <p>It may be worth adding that some of the funny behavior of this operator (explicated in Bill Cook's answer) is due to the fact that, in $L^2[0,1]$, say, the integral computes the <em>projection</em> to the one-dimensional space of scalar multiples of $\cos(y)$. Then the coefficient is used to multiply the function $\... |
25,414 | <p>I'm running in to some problems with generating a persistent HSQLDB and during some troubleshooting I came upon the following behavior.</p>
<pre><code>Needs["DatabaseLink`"]
tc = OpenSQLConnection[
JDBC["hsqldb", ToFileName[Directory[], "temp"]], Username -> "sa"]
CloseSQLConnection[tc]
</code></pre>
<p>The ab... | Albert Retey | 169 | <p><code>"DatabaseLink</code>"` connections in Mathematica are done via <strong>Java</strong> and it is the <strong>Java</strong> virtual machine which actually holds the file locks. To get rid of those locks you can e.g. use:</p>
<pre><code>Needs["JLink"];
UninstallJava[];
</code></pre>
<p>After that you should be ... |
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | las3rjock | 19 | <p>Wikipedia has a definition of <a href="http://en.wikipedia.org/wiki/Number_theory#Elementary_number_theory" rel="nofollow">elementary number theory</a>, but I don't know how well accepted it is.</p>
|
4,463 | <p>It seems that most authors use the phrase "elementary number theory" to mean "number theory that doesn't use complex variable techniques in proofs." </p>
<p>I have two closely related questions.</p>
<ol>
<li>Is my understanding of the usage of "elementary" correct?</li>
<li>It appears that advanced techniques fro... | engelbrekt | 3,304 | <p>Elementary number theory is better defined by its focus of interest than by its methods of proof. For this reason, I rather like to think of it as <em>classical number theory.</em> It deals with integers, rationals, congruences and Diophantine equations within a framework recognizable to eighteenth-century number th... |
194,421 | <p>This is homework. The problem was also stated this way: </p>
<p>Let A be a dense subset of $\mathbb{R}$ and let x$\in\mathbb{R}$. Prove that there exists a decreasing sequence $(a_k)$ in A that converges to x.</p>
<p>I know:</p>
<p>A dense in $\mathbb{R}$ $\Rightarrow$ every point in $\mathbb{R}$ is either in A o... | Ross Millikan | 1,827 | <p>Hint: You have to construct your sequence so it is decreasing. Let $a_0$ be a point in $A$ greater than $x$. How do you know there is one? Then let $a_1$ be a point in $A$ in $(x,a_0)$. How do you know there is one? Then keep going. You also have to make sure the intervals shrink to zero length.</p>
|
194,421 | <p>This is homework. The problem was also stated this way: </p>
<p>Let A be a dense subset of $\mathbb{R}$ and let x$\in\mathbb{R}$. Prove that there exists a decreasing sequence $(a_k)$ in A that converges to x.</p>
<p>I know:</p>
<p>A dense in $\mathbb{R}$ $\Rightarrow$ every point in $\mathbb{R}$ is either in A o... | William | 13,579 | <p>Recall that $A \subset R$ is dense if and only if every open subset of $\mathbb{R}$ contains an element of $A$. </p>
<p>Let $x \in \mathbb{R}$. We will define a decreasing sequence in $A$ converging to $x$ as follows: Consider the open interval $(x + 2^{-1}, x + 2(2^{-1}))$. Since $A$ is dense, $A$ intersect this i... |
239,688 | <p>Suppose I have a complete and cocomplete category $\mathscr{C}$ with two sets of maps $I,J$ that are the candidates for generating (trivial) cofibrations on a model structure on $\mathscr{C}$.
The only property I'm left to check in order to apply the recognition principle for cofibrantly generated model categories ... | Karol Szumiło | 12,547 | <p>This is not an answer in a full generality, but it is certainly not the case if the domains of generating acyclic cofibrations are cofibrant. (I have a hard time thinking of an example of a model category where this is not true, but there are probably some.)</p>
<p>Assuming that the "Cube Lemma" holds (it's more of... |
239,688 | <p>Suppose I have a complete and cocomplete category $\mathscr{C}$ with two sets of maps $I,J$ that are the candidates for generating (trivial) cofibrations on a model structure on $\mathscr{C}$.
The only property I'm left to check in order to apply the recognition principle for cofibrantly generated model categories ... | David White | 11,540 | <p>If I understand your situation correctly, the answer is yes, the cube lemma holds, but you cannot use that to get a full model structure. Your situation often arises when trying to transfer a model structure from a cofibrantly generated model category $M$, along an adjunction $F:M\leftrightarrows N: U$, to a bicompl... |
1,202,661 | <p>Let's consider the sum $$\sum_{i=4t+2} {\binom{m}{i}}$$. </p>
<p>It's equivalent to the following $\sum_{s}{\binom{m}{4s+2}}$, but i got stuck here.</p>
<p>How to evaluate such kind of sums? For instance, it's not so hard to calculate $$\sum_{2n}{\binom{m}{n}}$$ (just because we know, how to denote the generating ... | Jack D'Aurizio | 44,121 | <p>Consider that, by the discrete Fourier transform:</p>
<p>$$ 4\cdot\mathbb{1}_{n\equiv 0\pmod{4}} = 1^n+(-1)^n+i^n+(-i)^n \tag{1}$$
hence:</p>
<p>$$ \mathbb{1}_{n\equiv 2\pmod{4}} = \frac{1}{4}\left(1^n + (-1)^n - (-i)^n - i^n\right)\tag{2} $$
and:
$$\begin{eqnarray*} \sum_{n\equiv 2\pmod{4}}\binom{m}{n} &=&... |
1,202,661 | <p>Let's consider the sum $$\sum_{i=4t+2} {\binom{m}{i}}$$. </p>
<p>It's equivalent to the following $\sum_{s}{\binom{m}{4s+2}}$, but i got stuck here.</p>
<p>How to evaluate such kind of sums? For instance, it's not so hard to calculate $$\sum_{2n}{\binom{m}{n}}$$ (just because we know, how to denote the generating ... | hyperkahler | 188,593 | <p>The following solution is supposed to be the simplest one.</p>
<p>Let's consider a complex number $a$ such that $a^{4}=1$.
So, denote $S_{n}=\sum_{s=0}{\binom{i}{m}}$. We know its generating function $A(s)=(1+s)^{m}$. So, the generating function for the sequence over all $i=4q, q\in \mathbb{Z}$ would look like this... |
744,377 | <p>I just don't understand how to complete $\epsilon - N$ proofs. I don't know what my goal is or why they prove what they do. I have asked two questions on here in the past, but I simply don't 'get it'.</p>
<p>So first we set $\epsilon \gt 0$ and we want to find $N \in \mathbb{N}$ such that $n \geq N$, we then take t... | Tom Collinge | 98,230 | <p>What is being considered is whether an infinite sequence of terms $(a_1, a_2, a_3, ...a_n, ..)$ converges to some value $a$. If it does, then you can say that in the limit as $n$ tends to infinity $a_n$ tends to $a$, or in mathematical notation, $\lim \limits_{n \to \infty} a_n = a$.</p>
<p>So what does "convergenc... |
4,154,025 | <p>In a set of lecture notes, I have the following result:</p>
<blockquote>
<p><strong>Theorem</strong>. Let <span class="math-container">$X_n$</span> be random variables on <span class="math-container">$(\Omega, \mathcal{F}, \mathbb{P})$</span> with values in a Polish metric space <span class="math-container">$S$</spa... | Clarinetist | 81,560 | <p>This result is provided, without proof, as Theorem 5.6(e) of <em>A First Course in Stochastic Processes</em>, 2nd ed., by Karlin and Taylor (1975).</p>
<p>I don't have this book, but <em>Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications</em> by Cramer and Leadbetter (2004... |
192,394 | <p>I'm re-reading some material from Apostol's Calculus. He asks to prove that, if $f$ is such that, for any $x,y\in[a,b]$ we have</p>
<p>$$|f(x)-f(y)|\leq|x-y|$$</p>
<p>then:</p>
<p>$(i)$ $f$ is continuous in $[a,b]$</p>
<p>$(ii)$ For any $c$ in the interval,</p>
<p>$$\left|\int_a^b f(x)dx-(b-a)f(c)\right|\leq\fr... | user29999 | 29,999 | <p>Consider $f(c) = \dfrac{(b-c)^2+ (a-c)^2}{2}$. Then $f´(c) = (a-c)+(b-c)$. Hence, $f$ decreases of $a$ to $\dfrac{a+b}{2}$ and increases of $\dfrac{a+b}{2}$ to $b$. Hence $\dfrac{(a+b)^2}{2}=f(a)\ge f(c)$ to $a\le c\le \dfrac{a+b}{2}$ and $f(c) \le \dfrac{(a+b)^2}{2}=f(b)$ to $\dfrac{a+b}{2}\le c \le b$. Namely,
$f(... |
192,394 | <p>I'm re-reading some material from Apostol's Calculus. He asks to prove that, if $f$ is such that, for any $x,y\in[a,b]$ we have</p>
<p>$$|f(x)-f(y)|\leq|x-y|$$</p>
<p>then:</p>
<p>$(i)$ $f$ is continuous in $[a,b]$</p>
<p>$(ii)$ For any $c$ in the interval,</p>
<p>$$\left|\int_a^b f(x)dx-(b-a)f(c)\right|\leq\fr... | copper.hat | 27,978 | <p>I like pictures...</p>
<p><img src="https://i.stack.imgur.com/GIjJf.png" alt="enter image description here"></p>
<p>$\frac{(c-a)^2}{2} + \frac{(b-c)^2}{2} = m(T_1)+m(T_2) \leq \frac{m(R_1)+m(R_2)}{2} \leq \frac{1}{2} (b-a) \max(c-a,b-c) \leq \frac{1}{2} (b-a)^2$.</p>
|
2,889,075 | <p>let $A$ be an infinite subset of $\mathbb R$ that is bounded above and let $u=\sup A$. Show that there exists an increasing sequence $ (x_n) $ with $x_n \in A $ for all $n\in \mathbb N$ such that $u = \lim_{n\rightarrow\infty} x_n$.</p>
<p>If $u$ is in $A$ then the proof is trivial. If $u$ does not belong to $A$... | Mohammad Riazi-Kermani | 514,496 | <p>I am not convinced that your sequence converges to $u$</p>
<p>You have picked an epsilon and formed a sequence between $u-\epsilon $ and $u$.</p>
<p>How do you know that the sequence converges to u. </p>
<p>We know the sequence will converge to a number $l$ such that $u-\epsilon <l\le u$, but how do we know t... |
167,812 | <p>I call a profinite group $G$ <strong><em>Noetherian</em></strong>, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is finitely generated.</p>
<p>A profinite group $G$ is called <strong><em>just-infinite</em></str... | Yiftach Barnea | 5,034 | <p>It is a completely open problem whether Noetherian pro-$p$ group has finite rank, i.e., there is a bound on the number of generators of closed subgroups, which is equivalent to being $p$-adic analytic. As one can classify all just-infinite $p$-adic analytic pro-$p$ group if indeed any Noetherian pro-$p$ group is $p$... |
4,422,824 | <p><strong>Edit: This question involves derivatives, please read my prior work!</strong></p>
<p>This question has me stumped.</p>
<blockquote>
<p>A car company wants to ensure its newest model can stop in less than 450 ft when traveling at 60 mph. If we assume constant deceleration, find the value of deceleration th... | John Douma | 69,810 | <p>Since acceleration is constant we get that <span class="math-container">$$\frac{dv}{dt}=a\implies v = at+v_0$$</span> where <span class="math-container">$v_0$</span> is the initial velocity. In our case we are starting at <span class="math-container">$60\text{ mph}$</span> which is <span class="math-container">$88\t... |
164,002 | <p>When I am reading a mathematical textbook, I tend to skip most of the exercises.
Generally I don't like exercises, particularly artificial ones.
Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..</p>
<p>Sometimes I try to prove a theorem before reading the proof.
Sometimes I try... | Russ | 34,654 | <p>Of course this is entirely subjective and depends on your intelligence and memory. I'd suspect most all people on this site are high in both areas, at least in logic/mathematics.</p>
<p>Understanding theorems and working through them like you explained is a very good way to understand material, especially if you ca... |
3,197,683 | <p>Here is the theorem that I need to prove</p>
<blockquote>
<p>For <span class="math-container">$K = \mathbb{Q}[\sqrt{D}]$</span> we have</p>
<p><span class="math-container">$$\begin{align}O_K = \begin{cases}
\mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\
\mathbb{Z}\left[\frac{1 + \sqrt{D}}{2}\... | mathematics2x2life | 79,043 | <p>There are lots of approaches depending on how much you assume. Assuming one only knows traces and norms and your theorem, this should be a 'low brow' approach that gets there without any other knowledge assumptions.</p>
<p>Let <span class="math-container">$K=\mathbb{Q}(\sqrt{d})$</span>, where <span class="math-con... |
1,914,686 | <p>A fleet of nine taxis is dispatched to three airports, in such a way that three go to airport A, five go to airport B and one goes to airport C.</p>
<p>If exactly three taxis are in need of repair. What is the probability that every airport receives one of the taxis requiring repairs.</p>
<p>My method was total nu... | BruceET | 221,800 | <p><strong>Comment:</strong> Please see my first comment above. </p>
<p>Below is a simulation in R statistical software of a million performances
of the experiment, in which 1's denote bad taxis and 0's good ones. The
simulated answer should be correct to two or three places.</p>
<pre><code>m = 10^6; txi = c(1,1,1,0... |
82,765 | <p><strong>Bug introduced in 9.0 and persisting through 12.2</strong></p>
<hr />
<p>I get the following output with a fresh Mathematica (ver 10.0.2.0 on Mac) session</p>
<pre><code>FullSimplify[Exp[-100*(i-0.5)^2]]
(* 0. *)
Simplify[Exp[-100*(i-0.5)^2]]
(* E^(-100. (-0.5+i)^2) *)
</code></pre>
<p><code>FullSimplif... | Daniel Lichtblau | 51 | <p>[Too long for a comment.]</p>
<p>I very much doubt this will be "fixed" in any general way, and in fact am not convinced it is "broken" (in any general way). <code>(Full)Simplify</code> has to rely on any number of methods that manipulate rational/trig functions. These are all based on exact methods that have, of n... |
203,614 | <p>I have two matrices </p>
<p><span class="math-container">$$
A=\begin{pmatrix}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{pmatrix}
\quad
\text{ and }
\quad
B=\begin{pmatrix}
d & e & f \\
d & e & f \\
d & e & f
\end{pmatrix}
$$</span></p>
<p>In reality mine are... | AccidentalFourierTransform | 34,893 | <p>The reason is that your second matrix is a <strong>rank-one update</strong> of your first matrix:
<span class="math-container">$$
B\equiv uv^t
$$</span>
where <span class="math-container">$u=(1,1,1)$</span> and <span class="math-container">$v=(d,e,f)$</span>. Therefore, the new eigenvalues are typically a small pert... |
2,830,718 | <p>I want to estimate how many red balls in a box. Red, yellow, blue balls could be in the box. But I don't know how many of them are in the box.</p>
<p>What I did was randomly drawing 10 balls from the box and learned that there was no red ball.</p>
<p>(Edit: Assume the number of the balls in the box is a known fini... | Green.H | 456,457 | <p>No, you cant say that. </p>
<blockquote>
<p>Infinite case: </p>
</blockquote>
<p>The reason is that you dont know how many balls there are in a box in total. For instance, it may be that a box contains infinite number of balls (since we are talking about mathematical box, I can make this assumption) $\frac{9}{1... |
4,765 | <p>I have a grid made up of overlapping <span class="math-container">$3\times 3$</span> squares like so:</p>
<p><img src="https://i.stack.imgur.com/BaY9s.png" alt="Grid"></p>
<p>The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares (<span clas... | Community | -1 | <p>While searching on google, i got this JSTOR link: <a href="http://www.jstor.org/stable/2688681">http://www.jstor.org/stable/2688681</a> which answers the question in an intricate way.</p>
|
4,765 | <p>I have a grid made up of overlapping <span class="math-container">$3\times 3$</span> squares like so:</p>
<p><img src="https://i.stack.imgur.com/BaY9s.png" alt="Grid"></p>
<p>The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares (<span clas... | Singh | 83,768 | <p>An elementary proof of the fact that the set ${n+\pi k}$, $n,k\in \Bbb{Z}$ is dense in reals is equivalent to showing that the subgroup $\Bbb{Z}+\pi\Bbb{Z}$ is dense in the additive group of real line. See for detail of the proof in Theorem 0.2 in the following</p>
<p><a href="https://docs.google.com/viewer?a=v&... |
815,868 | <blockquote>
<p>Consider the following system describing pendulum</p>
<p><span class="math-container">$$\begin{align} & \frac{dx}{dt} = y, \\ & \frac{dy}{dt} = − \sin x. \end{align}$$</span></p>
<p>I need to classify all critical points of the system.</p>
</blockquote>
<p>All critical points are of the form <sp... | Winther | 147,873 | <p>Consider a small perturbation about the critical points $(x,y) = (0,n\pi)$ for $n\in\mathbb{Z}$, i.e. we take</p>
<p>$$x = n\pi + \delta x,~~~~y = \delta y$$</p>
<p>Then the dynamical system becomes, to first order in the perturbations,</p>
<p>$$\dot{\delta x} = \delta y~~~~\text{and}~~~~\dot{\delta y} = (-1)^{n+... |
1,390,093 | <p>Let $G$ be act on $\Gamma$ with a fundamental domain $T$ where $T$ is tree. We construct <em>tree of groups</em> $(\mathcal{G},T)$ with the following structure:
$$\text{for every } v\in V(T),\,\,G_v=\operatorname{Stab}_G(v) $$</p>
<p>$$\text{for every } e\in E(T),\,\,G_e=\operatorname{Stab}_G(e) $$</p>
<p>Assume t... | Daniel Valenzuela | 156,302 | <p>This is precisely the statement of Lemma 4 in Chapter 4.1 of Serre's Trees book. In terms of your specific problem in the lemma is a little too general and you can set $Y=T$. However, the previous Lemma 2 which is being generalized is too specific. This book is a beautiful read and strongly recommended! </p>
<p>Not... |
3,372,832 | <blockquote>
<p><strong>9)</strong> Is
<span class="math-container">$$ \sum_{n=1}^\infty \delta_n \tag{7.10.1} $$</span>
a well-defined distribution? Note, to be a well-defined distribution, its action on any test function should be a finite number. Provide an example of a function <span class="math-container">$f... | Aloizio Macedo | 59,234 | <p>For a rundown on your reasoning, read after the horizontal rule below. First, let me prove what he mentions.</p>
<p>Following up on Rudin's notation, to see why <span class="math-container">$\Omega$</span> is the unique component which can be unbounded, suppose that there is another, say <span class="math-container... |
2,594,829 | <p>I'm having trouble knowing when my ansatz is wrong. For example if my ansatz to this is $y_p=a\cos{x}+b\sin{x},$ I get nowhere. How can I make a correct ansatz and are there any general rules to determine proper ansatz?</p>
<p><strong>Note:</strong> I know one can sovle this using eulers formula and all that but th... | Alekos Robotis | 252,284 | <p>If you set $y_p=a\cos x+b\sin x$, we have that $y_p'=-a\sin x+b\cos x$ and $y_p''=-a\cos x-b\sin x.$ Then, we have that
$$ y_p''-2y_p'+y_p=-a\cos x-b\sin x+2a\sin x-2b\cos x+a\cos x+b\sin x=2\cos x.$$
Collecting the coefficients, we see that $-2b-a+a=2$ and $2a-b+b=0$ so that $b=-1$ and $a=0$. Thus, the function $y... |
2,594,829 | <p>I'm having trouble knowing when my ansatz is wrong. For example if my ansatz to this is $y_p=a\cos{x}+b\sin{x},$ I get nowhere. How can I make a correct ansatz and are there any general rules to determine proper ansatz?</p>
<p><strong>Note:</strong> I know one can sovle this using eulers formula and all that but th... | Dr. Sonnhard Graubner | 175,066 | <p>we have
$$y'(x)=-a\sin(x)+b\cos(x)$$
$$y''(x)=-a\cos(x)-b\sin(x)$$
then $$-a\cos(x)-b\sin(x)-2(-a\sin(x)+b\cos(x))+a\cos(x)+b\sin(x)=2\cos(x)$$
from here we get
$$\cos(x)(-a-2b+a)+\sin(x)(-b+2a+b)=2\cos(x)$$
thus $$b=-1$$ and $$a=0$$</p>
|
4,342,737 | <p>My question: If you throw a dice 5 times, what is the expected value of the square of the median of the 5 results?</p>
<p>A slightly modified question would be: If you throw a dice 5 times, what is the expected value of the median? The answer would be 3.5 by symmetry.</p>
<p>For the square, it seems to be that symme... | Avraham | 91,378 | <p>This is small enough that it can be calculated explicitly through some code. Creating the matrix of throws is inefficient, but it should be clear to see how the universe of possibilities is spanned.</p>
<pre><code>throws <- matrix(double(0), ncol = 5, nrow = 6^5)
throws[, 5] <- rep(1:6, each = 6^0, times = 6^4... |
91,700 | <p>Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map.
(Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then $\phi(a)\phi(b) = 0$.)
Then:</p>
<p>(i) For any $a,b,c \in A$,
$$ \phi\left(ab\right)\phi\left(c\right) = \phi\left(a\right)\phi... | NC Wong | 121,944 | <p>For a proof of the fact (i) "ϕ(ab)ϕ(c)=ϕ(a)ϕ(bc) and ϕ(a)ϕ(b)=ϕ(1)ϕ(ab)", see M.A. Chebotar, W.-F. Ke, P.-H. Lee, N.-C. Wong
Mappings preserving zero products, Studia Math., 155 (1) (2003), pp. 77-94
MR1961162 (2003m:47066)</p>
<p>It is really surprising (at least to me) that few operator algebra people knowing of ... |
1,478,038 | <p>Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ to be the number of joints where two faces come together side by side $n-e+f=2$.</p>
<p>It was later seen that a seri... | postmortes | 65,078 | <p>The N in question is <span class="math-container">$\cal{N}$</span> which is a calligraphic N that <span class="math-container">$\LaTeX$</span> produces. You can type it yourself by putting <code><span class="math-container">$\cal{N}$</span></code> in your question or answer[1].
As others have also noted it denotes ... |
187,395 | <p>I can't find my dumb mistake.</p>
<p>I'm figuring the definite integral from first principles of $2x+3$ with limits $x=1$ to $x=4$. No big deal! But for some reason I can't find where my arithmetic went screwy. (Maybe because it's 2:46am @_@).</p>
<p>so </p>
<p>$\delta x=\frac{3}{n}$ and $x_i^*=\frac{3i}{n}$</... | The_Sympathizer | 11,172 | <p>The problem lies at the very beginning. You took</p>
<p>$$x^{*}_i = \frac{3i}{n}$$.</p>
<p>But this is wrong. Remember, you are looking for the right endpoints of the partitioning of $[1, 4]$. So at $i = n$, this should equal 4... but it doesn't. And at $i = 1$, this should equal $1 + \frac{3}{n}$... but it doesn'... |
2,972,085 | <p><a href="https://i.stack.imgur.com/pcOfx.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/pcOfx.jpg" alt="enter image description here"></a></p>
<p>My friend show me the diagram above , and ask me </p>
<p>"What is the area of a BLACK circle with radius of 1 of BLUE circle?"</p>
<p>So, I solved it by alge... | David K | 139,123 | <p>Based on the figure, one can make some inspired guesswork.</p>
<p>Construct a rectangle <span class="math-container">$ABCD$</span> with side <span class="math-container">$AB$</span> of length <span class="math-container">$1$</span> and diagonal <span class="math-container">$AC$</span> of length <span class="math-co... |
2,294,997 | <p>How to prove that $\displaystyle 0,02<\int_0^1 \frac{x^7}{(e^x+e^{-x})\sqrt{1+x^2}}dx<0,05$? I tried to use mean value theorems, but i failed.</p>
| John Hughes | 114,036 | <p>Hint: $e^x + e^{-x}$, on $[0, 1]$, is no greater than $3.1$. And $\sqrt{1 + x^2}$ is no greater than $\sqrt{2}$. So your integrand is no less than
$$
\frac{x^7}{3.1 \sqrt{2}} \ge \frac{x^7}{4.4}
$$</p>
<p>Now integrate. </p>
|
2,294,997 | <p>How to prove that $\displaystyle 0,02<\int_0^1 \frac{x^7}{(e^x+e^{-x})\sqrt{1+x^2}}dx<0,05$? I tried to use mean value theorems, but i failed.</p>
| Arpan1729 | 444,208 | <p>$e^x+e^{-x}\geq 2$ by AM-GM, and $\sqrt{1+x^2}>x$</p>
<p>So the integral is less than integration of $x^6/2$ from $0$ to $1$, which equals $1/14>0.05$.</p>
<p>$e^x+e^{−x}$, is less than $3.1$ in $(0,1]$ And $\sqrt{1+x^2}$ is less than $2√2$. So your integrand is greater than
$x^7/3.1\times\sqrt{2}$</p>
<p>N... |
964,372 | <p>I have a general question.</p>
<p>If there is a matrix which is inverse and I multiply it by other matrixs which are inverse.
Will the result already be reverse matrix?</p>
<p>My intonation says is correct, but I'm not sure how to prove it.</p>
<p>Any ideas? Thanks.</p>
| Marko Riedel | 44,883 | <p>For future reference here is a derivation using combinatorial species. The species under consideration is
$$\mathfrak{P}(\mathfrak{P}_{=1}(\mathcal{Z})+\mathfrak{P}_{=2}(\mathcal{Z})).$$
This gives the exponential generating function
$$G(z) = \exp\left(\frac{z}{1!} + \frac{z^2}{2!}\right)
= \exp\left(z+\frac{z^2}{2... |
2,886,460 | <blockquote>
<p>Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find
$$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}.$$</p>
</blockquote>
<p>I have tried combining the first and third terms & first and la... | nonuser | 463,553 | <p>Expand 2nd and 4th fraction with $\omega $ and $\omega ^2$ respectively: $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 + \omega^3}=\frac{\omega}{1 + \omega^2} + \frac{\omega^3}{\omega+ 1} + \frac{\omega^3}{1 + \omega} + \frac{\omega}{\omega^2+1}$$</p>... |
3,059,857 | <p>We are supposed to use this formula for which I can't find any explaination anywhere and our teacher didn't explain anything so if anyone could help me I would appreciate it. </p>
<p><span class="math-container">$ x = A + k \times 2\pi$</span></p>
<p>and</p>
<p><span class="math-container">$x = \pi - A + k \times... | hmakholm left over Monica | 14,366 | <p>The first step is always the same:</p>
<h1>DRAW A DIAGRAM!</h1>
<p>It doesn't need to be a particularly precise diagram -- we're not going to measure anything on it, just get an overview of how things fit together. Here's a rough graph of <span class="math-container">$\sin(x)$</span> with <span class="math-contain... |
4,255,430 | <p>I know that <strong>if</strong> both of the limits
<span class="math-container">$$
\lim_{x\to a} f(x) \quad\text{and}\quad \lim_{x\to a} g(x)
$$</span>
exist (so they are both equal to real numbers), then
<span class="math-container">$$
\lim_{x\to a} f(x) + g(x) = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)
$$</span>
... | Macavity | 58,320 | <p>To use inequalities to find minima, we need the equality condition to be satisfied as well, hence in this case we need <span class="math-container">$x^2=8x=\frac{32}x$</span>, which doesn't have a solution.</p>
<p>If you have a "lucky guess" for which <span class="math-container">$x$</span> the minimum occ... |
112,437 | <p>I am working on a personal project involving a CloudDeploy[ ] that reads data off a Google Doc and then works with it. Ideally, the Google Doc is either a text document or a spreadsheet which contains a single string, which is what I want Mathematica to read as input.
<a href="https://docs.google.com/document/d/17m1... | chuy | 237 | <p>This seems to work:</p>
<pre><code>URLFetch["https://docs.google.com/document/export?format=txt&\
id=17m1JfjEbrna7e9INZv-FXZQ9yQJd8d1Uu2LFEPyT_ZI&token=\
AC4w5VhkHSfLIe2xvAUWQC9XHb1lAmM7Xw%3A1460476462625"]
(* "When the soil were finally depleted of all nutrients, we \
realised that paper isn't edible. Or ... |
2,174,413 | <p>Proof by induction, that </p>
<p>$$x_n=10^{(3n+2)} + 4(-1)^n\text{ is divisible by 52, when n}\in N $$</p>
<p>for now I did it like that:</p>
<p>$$\text{for } n=0:$$
$$10^2+4=104$$
$$104/2=52$$
<br>
$$\text{Let's assume that:}$$
$$x_n=10^{(3n+2)} + 4(-1)^n=52k$$
$$\text {so else}$$
$$4(-1)^n=52k-10^{3n+2}$$</p>
... | Arnaldo | 391,612 | <p>For $n+1$ you have:</p>
<p>$$10^{(3n+2)+3}+4(-1)^{n+1}=10^3\cdot10^{3n+2}+4(-1)^n\cdot(-1)=\\
=10^3\cdot[52k-4(-1)^n]-4(-1)^n=10^3\cdot52k-(-1)^n(4004)$$</p>
<p>but $4004=52\cdot 77$, then</p>
<p>$$10^{(3n+2)+3}+4(-1)^{n+1}=52[10^3k-77(-1)^n]$$</p>
|
2,174,413 | <p>Proof by induction, that </p>
<p>$$x_n=10^{(3n+2)} + 4(-1)^n\text{ is divisible by 52, when n}\in N $$</p>
<p>for now I did it like that:</p>
<p>$$\text{for } n=0:$$
$$10^2+4=104$$
$$104/2=52$$
<br>
$$\text{Let's assume that:}$$
$$x_n=10^{(3n+2)} + 4(-1)^n=52k$$
$$\text {so else}$$
$$4(-1)^n=52k-10^{3n+2}$$</p>
... | Bill Dubuque | 242 | <p>$\begin{align}{\bf Hint}\qquad\qquad {\rm mod}\,\ 52\!:\qquad \color{#0a0}{10^{\Large 3}} (\color{#0a0}{-4})\, &\equiv\, (\color{#0a0}{-4})(\color{#0a0}{-1})\\[.3em]
10^{\Large 3n+2} &\equiv\, (\color{}{{-}4})(-1)^{\Large n}\qquad\! {\rm i.e.}\ \ P(n) \\[.3em]
{\rm scale\ prior\ by\ 10^{\Large 3}} \Rightar... |
2,661,210 | <p>Let $a_{1}, \dots, a_{n}$ be real numbers not all zero; let $b_{1},\dots, b_{n}$ be real numbers; let $\sum_{1}^{n}b_{i} \neq 0$. Then does there exist real numbers $w_{1},\dots, w_{n} > 0$ such that
$$
\frac{\sum_{1}^{n}w_{i}a_{i}}{\sum_{1}^{n}w_{i}b_{i}} > \frac{\sum_{1}^{n}a_{i}}{\sum_{1}^{n}b_{i}}?
$$
Som... | Sangchul Lee | 9,340 | <p>Let $U = \{\mathrm{w} \in \mathbb{R}_+^n : \langle \mathrm{w}, \mathrm{b} \rangle \neq 0 \}$ and define $f : U \to \mathbb{R}$ by</p>
<p>$$ f(\mathrm{w}) = \frac{\langle \mathrm{w}, \mathrm{a} \rangle}{\langle \mathrm{w}, \mathrm{b} \rangle}. $$</p>
<p>We know that $U$ is open and $\mathbf{1} = (1,\cdots,1) \in U$... |
3,479,765 | <p>I am attempting to do this using Cauchy's integral theorem and formula. However I am unable to conclude if a singularity exists at all for me to apply any of those two techniques or any other theorem. </p>
| user247327 | 247,327 | <p>Multiply both sides by <span class="math-container">$x^a$</span> and <span class="math-container">$(x+ c)^b$</span> to get <span class="math-container">$$1= d_a(x+ c)^b+ d_{a-1}x(x+ c)^b+ \cdots+ d_1x^{a-1}(x+ c)^b+ e_bx^a+ e_{b+1}x^a(x+ c)+ \cdots+e_1x^a(x+ c)^{b-1}~.$$</span></p>
<p>Taking <span class="math-conta... |
1,893,168 | <p>$$\lim_{x\to 0} {\ln(\cos x)\over \sin^2x} = ?$$</p>
<p>I can solve this by using L'Hopital's rule but how would I do this without this?</p>
| Gordon | 169,372 | <p>You can use the important limits: $\lim_{x\rightarrow 0}\frac{\sin x}{x} =1$ and $\lim_{x\rightarrow 0}(1+x)^{\frac{1}{x}}=e$ (i.e., $\lim_{x\rightarrow 0} \frac{\ln(1+x)}{x}=1$). Then
\begin{align*}
\lim_{x\rightarrow 0}\frac{\ln \cos x}{\sin^2 x} &= \lim_{x\rightarrow 0}\frac{\cos x -1}{x^2}\\
&=\lim_{x\ri... |
2,834,864 | <p>Is it safe to assume that if $a\equiv b \pmod {35 =5\times7}$</p>
<p>then $a\equiv b\pmod 5$ is also true?</p>
| Angina Seng | 436,618 | <p>If $A=xH$, with $H$ a subgroup of $G$, then $H=\{b^{-1}a:a,b\in A\}$.
So the coset $A$ determines the subgroup $H$.</p>
|
2,668,839 | <blockquote>
<p>Finding range of $$f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$$</p>
</blockquote>
<p>Try: put $\sin x=t$ and $-1\leq t\leq 1$</p>
<p>So $$y=\frac{t^2+4t+5}{2t^2+8t+8}$$</p>
<p>$$2yt^2+8yt+8y=t^2+4t+5$$</p>
<p>$$(2y-1)t^2+4(2y-1)t+(8y-5)=0$$</p>
<p>For real roots $D\geq 0$</p>
<p>So $$16... | mathlove | 78,967 | <p>MrYouMath has already provided a good answer.</p>
<p>This answer uses <em>your method</em>.</p>
<p>You already have a quadratic equation on $t$
$$(2y-1)t^2+4(2y-1)t+(8y-5)=0\tag1$$
where $y\not=\frac 12$.</p>
<p>Note here that we want to find $y$ such that $(1)$ has at least one real solution $t$ satisfying $-1\l... |
960,010 | <p>Two sides of a triangle are 15cm and 20cm long respectively.
$A)$ How fast is the third side increasing if the angle between the given sidesis 60 degrees and is increasing at the rate of $2^\circ/sec$? $B)$ How fast is the area increasing?</p>
<p>$A)$ I used $c^2=a^2+b^2-2ab\cos(\theta)$ so I got the missing side ... | Narasimham | 95,860 | <p>Assuming sides $a,b$ do not change in length ( no constraints given) , differentiate</p>
<p>$$c^2 = a^2+b^2-2ab\cos\theta$$</p>
<p>$$ 2 c \dot c = - 2 a b \sin \theta \, \omega $$</p>
<p>$$ \dot c = - \dfrac{a b}{c} \sin \theta \, \omega $$</p>
<p>Similarly for area A</p>
<p>$$ A = \frac12 a b \sin \theta $... |
163,468 | <p>I have a <code>Graph</code>, and I want to group some of its vertices into communities. <code>CommunityGraphPlot</code> uses force directed layout and its doesn't look like the original graph after I apply <code>CommunityGraphPlot</code>. I don't want the vertices of same community to come close so that the communit... | kglr | 125 | <p><strong>Update:</strong> Using <code>GraphComputation`GraphCommunitiesPlotDump`generateBlobs</code>, ... well, to generate blobs:</p>
<pre><code>ClearAll[blobF, fC]
fC[coords_, size_: .04] := Module[{}, CommunityGraphPlot[Graph[{}], {}];
FilledCurve @ BSplineCurve[GraphComputation`GraphCommunitiesPlotDump`genera... |
557,426 | <p>I have 5 ring oscillators whose frequencies are f1, f2, ..., f5. Each ring oscillator (RO) has 5 inverters. For each RO, I just randomly pick 3 inverters out of 5 inverters. For example, in RO1, I pick inverter 1,3,5 (Notation: RO1(1,3,5)). So I have the following:</p>
<p>RO1(1,3,5) (I call this is the configuratio... | Snowball | 24,875 | <p>Yes, there is. One such encoding follows.</p>
<p>To encode the <em>rank</em>, write out the order of the frequencies as bytes. For example, $f_1 < f_3 < f_4 < f_5 < f_2$ becomes</p>
<p>$$
00000001\ 00000011\ 00000100\ 00000101\ 00000010.
$$</p>
<p>To encode the <em>configuration</em> of each RO, put a... |
2,801,162 | <p>I need help to understand how we compute this kind of limit:</p>
<p>$\lim_{(x,y)\rightarrow (0,0)}\ xy\log(\lvert x\rvert+\lvert y\rvert)$</p>
<p>I think we can use the squeeze theorem but I don't know how to bound the function, so I can use the theorem. If I suppose $0 \lt \sqrt{x^2+y^2} \lt 1$ then, but I'm stru... | Clement C. | 75,808 | <p>Use the AM-GM inequality:
$$
0\leq 2\sqrt{\lvert xy\rvert} \leq |x|+|y| \tag{1}
$$
from which
$$
0\leq 2\sqrt{\lvert xy\rvert}|\log(|x|+|y|)| \leq (|x|+|y|)|\log(|x|+|y|)|\tag{2}
$$
Setting $t(x,y)\stackrel{\rm def}{=} |x|+|y|$, we have $\lim_{(x,y)\to(0,0)} t(x,y)=0$ <em>(why?)</em> and therefore since $\lim_{t\to ... |
3,248,552 | <p>Imagine we want to use Theon's ladder to approximate <span class="math-container">$\sqrt{3}$</span>. The appropriate expressions are
<span class="math-container">$$x_n=x_{n-1}+y_{n-1}$$</span></p>
<p><span class="math-container">$$y_n=x_n+2x_{n-1}$$</span></p>
<p>Rungs 6 through 10 in the approximation of <span cl... | Hw Chu | 507,264 | <p>The idea is that, if you can show that the ratios of the rectangles are getting closer to <span class="math-container">$\sqrt 3$</span>, then none of them should be similar to each other. So you need to have some way to track <span class="math-container">$\frac{y_n}{x_n}$</span> in each iteration.</p>
<p>If you try... |
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