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 |
|---|---|---|---|---|
2,976,613 | <p>I want to show, that <span class="math-container">$a:=\sum \limits_{n=0}^{\infty} \left(\dfrac{2n+n^3}{3-4n}\right)^n$</span> is not converging, because <span class="math-container">$\lim \limits_{n \to \infty}(a)\neq 0 \; (*)$</span>. Therefore, the series can't be absolute converge too.</p>
<p>Firstly, I try to s... | Doesbaddel | 587,094 | <p><span class="math-container">$\sum \limits_{n=0}^{\infty} \left(\underbrace{\dfrac{2n+n^3}{3-4n}}_{a_n}\right)^n$</span></p>
<p>Root test: Prove:<span class="math-container">$\lim \limits_{n \to \infty} \sqrt[n]{\mid a_n \mid}<1 \implies a_n \text{ is absolute converging.}$</span></p>
<p><span class="math-conta... |
813,121 | <p>I have to calculate this : $$ \lim_{x\to 0}\frac{2-x}{x^3}e^{(x-1)/x^2} $$ Can somebody help me?</p>
| meh | 70,191 | <p>$$\lim_{x\to0}(f(x)g(x)) = \lim_{x\to0}(f(x)) \cdot \lim_{x\to0}(g(x)) $$</p>
<p>With that being said you can let $f(x) = (2-x)/x^3 $ and $ g(x) = e^{(x-1)/x^2} $</p>
<p>I hope this helps.</p>
|
183,881 | <p>I want to replace an element of a matrix by a function of its current value. The code I'm using is as follows</p>
<pre><code>T = {{2, 3}, {3, 2}, {1, 4}}
KNeeds = Transpose[T][[2]]
pos = Flatten[{Position[KNeeds, Min[KNeeds]], 2}]
ReplacePart[T, pos -> T[[pos[[1]]]] - Epsilon]]
</code></pre>
<p>I was hoping to ... | halirutan | 187 | <p>If I understood your question correctly, then you can rip out the inner part of your code above and use <code>ListLinePlot</code> by creating a table containing <code>{x, productY}</code> elements:</p>
<pre><code>pts = Table[{i, 0, 1}, {i, 0, 10}];
ListLinePlot[
Table[{x,
R1 = Cylinder[{{3, 0, 0}, {3, 0, 2}}, x... |
1,149,685 | <p>Suppose that 4 guests check their hats when they arrive at a restaurant, and that these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.</p>
<p>My attempt:
So I know that there are 4! ways of "arranging" the hats.
The probability of ea... | Alex | 38,873 | <p>Use inclusion-exclusion theorem/formula/principle. Total number of ways to distribute 4 hats amon 4 people is $4!$. Now subtract the number of ways 1 person leaves with his hat, add two people left with their hats, subtract three people left with their hats, add 4 ppl left with their hats (the last one is obviously ... |
1,149,685 | <p>Suppose that 4 guests check their hats when they arrive at a restaurant, and that these hats are returned to them in a random order when they leave. Determine the probability that no guest will receive the proper hat.</p>
<p>My attempt:
So I know that there are 4! ways of "arranging" the hats.
The probability of ea... | Balthier TSK | 869,590 | <p>Sorry for the very late reply, lmao.
Anyhow, the answer is to use derangements: an arrangement such that no elements appear in their original position.
Basically, if you map a hat and its owner, you can find an arrangement such that no hat is mapped to its original owner again, which is just derangement.
It is also ... |
4,320,849 | <p>I had this problem in an exam I recently appeared for:</p>
<blockquote>
<p>Find the range of
<span class="math-container">$$y =\frac{x^2+2x+4}{2x^2+4x+9}$$</span></p>
</blockquote>
<p>By randomly assuming the value of <span class="math-container">$x$</span>, I got the lower range of this expression as <span class="m... | Nikola Alfredi | 670,977 | <p>The question can be easily solved by this technique:</p>
<p>As <span class="math-container">$\displaystyle y = \frac {x^2 + 2x + 4}{2x^2 + 4x + 9} \implies 2y = \frac {2x^2 + 4x + 9 - 1}{2x^2 + 4x + 9}$</span>.</p>
<p>Thus, <span class="math-container">$\displaystyle 2y = 1-\frac {1}{2(x + 1)^2 + 7} $</span></p>
<p>... |
125,399 | <p>How can I solve the following integral?
$$\int_0^\pi{\frac{\cos{nx}}{5 + 4\cos{x}}}dx, n \in \mathbb{N}$$</p>
| Christian Blatter | 1,303 | <p>Here is a solution without residue calculus:</p>
<p>Via partial fractions and geometric series one arrives at
$$\eqalign{{1\over 5+4\cos x}&={1\over(2+e^{ix})(2+e^{-ix})}=\ldots\cr &={1\over 3}-{1\over3}{e^{ix}/2\over 1+e^{ix}/2}-{1\over3}{e^{-ix}/2\over 1+e^{-ix}/2}\cr &= {1\over3}+{1\over3}\sum_{k=1}^... |
1,206,528 | <p>Find the matrix $A^{50}$ given</p>
<p>$$A = \begin{bmatrix} 2 & -1 \\ 0 & 1 \end{bmatrix}$$ as well as for $$A=\begin{bmatrix} 2 & 0 \\ 2 & 1\end{bmatrix}$$</p>
<p>I was practicing some questions for my exam and I found questions of this form in a previous year's paper.</p>
<p>I don't know how to ... | mvw | 86,776 | <p>And yet another approach: Splitting $A$ into a diagonal matrix $D$ and a pertubation matrix $P$.</p>
<p>We look at the first powers of $P$:</p>
<p>$$
P^0 =
\left(
\begin{matrix}
1 & 0\\
0 & 1
\end{matrix}
\right)
\quad
P^1 =
\left(
\begin{matrix}
0 & -1 \\
0 & 0
\end{matrix}
\right)
\quad
P^2 ... |
1,206,980 | <p>Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that
$$
||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2.
$$</p>
<p>I am having trouble understanding this step of a longer proof. The text that I got this from just says that this holds be... | pjs36 | 120,540 | <p>The $x$ and $y$ are dummy variables, they don't really matter. For example, if $p(t) = t - 1$, we have $p(x) = x - 1$, so that</p>
<p>\begin{align*}\int_0^t\int_0^y (x - 1)\ dx\ dy &= \int_0^t\left[\frac{x^2}{2} - x\right]_0^y\ dy\\
&=\int_0^t\left(\frac{y^2}{2}-y\right)\ dy \\
&=\left[\frac{y^3}{6} - \... |
2,815,900 | <p>Consider the proposition $∀x (P (x) → Q (x))$, where $P (x)$ and $Q (x)$ are predicates on a domain $U$. Prove or disprove the following statement, justifying the answer.</p>
<p>If the statement $∀x (P (x) → Q (x))$ is FALSE then it's FALSE also $∀x P (x)$</p>
<p>How can I solve this exercise?</p>
| Ovi | 64,460 | <p>If $∀x (P (x) → Q (x))$ is false then it just means there exists some $x$ such that $P(x)$ is true and $Q(x)$ is false. It does not tell us anything about weather $P(x)$ is true for every $x$ or not ... can you take it from here?</p>
|
637,379 | <p>So I have a $10$ gallon aquarium slightly salty aquarium...</p>
<p>When I add water the water is at $.7$% salt. ($7$ part per thousand)</p>
<p>I let $1$ gallon evaporate, at which point I have a $\frac{7}{9}$% salinity
I then drain $1$ additional gallon. now I have $8$ gallons of $\frac{7}{9}$% salinity water.</p>... | kaine | 53,641 | <p>Hint the final concentration of one cycle will be: $$f(x) =.8(\frac{10x}{9})+.2(.7)$$ where $x$ is $.7$ initially. </p>
<p>Does $f(x)$ increase or decrease for each $x$? </p>
<p>Is there any case where $f(x)=x$? </p>
<p>If you can, look at the derivative of $f(x)$.</p>
|
1,571,083 | <p>Given the 2 terms
$$ \frac{k + a}{k + b}$$
and $$\frac{a}{b}$$
with $a, b, k \in \mathbb{R^+}$ and $a > b$</p>
<p>I want to show, that the first term is always bigger than the second one.</p>
<p><strong>My try</strong>
$$
\frac{k + a}{k + b} > \frac{a}{b} \\
\frac{k + b + (a - b)}{k + b} > \frac{b + ... | costrom | 271,075 | <p>The odds of winning the first contest is:</p>
<p>$\frac{1}{31}+\frac{30}{31}\cdot\frac{1}{30} = \frac{2}{31}$ - either they win the first or second draw - and there is no replacement between draws</p>
<p>The odds of winning the second contest is $\frac{1}{25}$, as you said.</p>
<p>The odds of winning both is the ... |
1,887,395 | <p>Let $D=\{Z\in\mathbb{C}:|z|<1\}$. Suppose $f:\overline D\to\mathbb{C}$ is continuous, holomorphic in $D$, and $f(\partial D)\subset \mathbb{R}$. Define $h:\mathbb{C}\to\mathbb{C}$ by
$$
h(z)=\begin{cases}
f(z) &\text{if}\ z\in \overline D\\
\overline{f(1/\overline z)} &\te... | Robert Z | 299,698 | <p>According to Lemma 1 in this <a href="http://www.doiserbia.nb.rs/img/doi/1452-8630/2012/1452-86301200016S.pdf" rel="nofollow">article</a> by A. Sofo and D. Cvijovic, we have that
$$\sum_{n=1}^{\infty} H_{n}\left(\frac{1}{n}-\frac{1}{n+k}\right)=
k\sum_{n=1}^{\infty} \frac{H_{n}}{n(n+k)}=
\frac{\pi^2}{6}+\frac{H^2_{... |
4,295,652 | <p>In the integration formula <span class="math-container">$\int dx/x = log x + c$</span>, Is the log natural or log base 10? The formula appears in many problems and i just got a problem wrong for apparently using the wrong log. Could you please enlighten me about the right log to be used in integration.</p>
| Community | -1 | <p>In the formula
<span class="math-container">$$
\int \frac{1}{x}\; dx = \log(x)+C
$$</span>
the logarithm is with base <span class="math-container">$e$</span>. Most calculus books write
<span class="math-container">$$
\int \frac{1}{x}\; dx = \log|x|+C\tag{1}
$$</span>
with the absolute value on the right-hand side. T... |
4,295,652 | <p>In the integration formula <span class="math-container">$\int dx/x = log x + c$</span>, Is the log natural or log base 10? The formula appears in many problems and i just got a problem wrong for apparently using the wrong log. Could you please enlighten me about the right log to be used in integration.</p>
| Mohammad Riazi-Kermani | 514,496 | <p>The formula is <span class="math-container">$$\int\,\frac{\mathrm{d}x}{x}=\ln(|x|)+C$$</span></p>
<p>For the definition of the natural logarithm as an integral we have</p>
<p><span class="math-container">$$ Ln (x) = \int _1^x \frac {1}{t} dt , x>0 $$</span></p>
|
2,349,312 | <blockquote>
<p>Let $F$ be a subfield of the complex numbers. Let $A,B\in F^{n\times n}$ and $X\in F^{n\times 1}.$ If $A\ne B,$ then $AX\ne BX$ for every non-zero $X.$</p>
</blockquote>
<p>I am NOT sure if the above statement is right. I did some computation with random matrices and it seems true to me. This is the ... | Bernard | 202,857 | <p>It's not correct, as it would imply that, as soon as $A\ne B$, $A-B$ is non-singular. Why should it be true?</p>
<p>Actually, $A\ne B$ only implies that for <em>some</em> $X$, one has $AX\ne BX$.</p>
|
2,349,312 | <blockquote>
<p>Let $F$ be a subfield of the complex numbers. Let $A,B\in F^{n\times n}$ and $X\in F^{n\times 1}.$ If $A\ne B,$ then $AX\ne BX$ for every non-zero $X.$</p>
</blockquote>
<p>I am NOT sure if the above statement is right. I did some computation with random matrices and it seems true to me. This is the ... | Fred | 380,717 | <p>Let $A = \begin{bmatrix} 1 & 0\\ 0 & 0\end{bmatrix}, B= \begin{bmatrix} 0 & 1\\ 0 & 0\end{bmatrix}$ and $X=\begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatrix}$ and look what happens !</p>
|
762,964 | <blockquote>
<p>Let <span class="math-container">$X$</span> be from a normal distribution <span class="math-container">$N(\theta,1)$</span>.</p>
<p>a) Find a sufficient statistic for <span class="math-container">$\theta$</span>.<br />
b) Is <span class="math-container">$S_n^2$</span> a
sufficient statistic for <span cl... | Henry | 6,460 | <p>(a) Taking your joint probability density of $\frac{1}{(2\pi)^{n/2}}e^{{-1 \over 2}\sum(x_i-\theta)^2}$, you can expand this into $$\left(\frac{1}{(2\pi)^{n/2}}e^{-\sum x_i^2 /2}\right)\left(e^{-n\theta^2/2+\theta \sum x_i }\right)$$ where the left part does not depend on $\theta$ and the right part is a function of... |
3,767,421 | <p>This is a rather soft question.</p>
<p><strong>My understanding:</strong></p>
<p>Suppose we have <span class="math-container">$x \in ℝ$</span> and <span class="math-container">$x^2 = -1$</span> [in the normal interpretation].</p>
<p>Then the statement "<em>there exists <span class="math-container">$r \in ℝ$</sp... | Mauro ALLEGRANZA | 108,274 | <p>There are different (equivalent) definitions of <a href="https://en.wikipedia.org/wiki/Consistency" rel="nofollow noreferrer">consistency</a>.</p>
<p>Basically, an <a href="https://en.wikipedia.org/wiki/Consistency#Definition" rel="nofollow noreferrer">inconsistent system</a> is a system that proves a sentence <span... |
594,498 | <p>Let $U$ be a linear subspace of $V$. Show that $\dim_K(U)\leq \dim_K(V)$ and conclude, that $\dim_K(U)= \dim_K(V) \Leftrightarrow U = V$. Is the equivalence $\dim_K(U)= \dim_K(V) \Leftrightarrow U = V$ also true for $\dim_K(V)= \infty$?</p>
<p>I've proved so far that $\dim_K(U)= \dim_K(V) \Leftrightarrow U = V$,but... | vadim123 | 73,324 | <p>Let $B_U$ be a basis for $U$. Then $B_U$ is an independent subset of $V$, and hence (assuming that $V$ has a basis) may be extended to a basis for $V$, which has cardinality at least as large as $|B_U|=dim_K(U)$.</p>
<p>If both vector spaces are infinite-dimensional it need not be true that they are equal. For ex... |
2,249,036 | <p>I was watching a video on PDEs and when arriving at the part of Fourier Series, the professor said:</p>
<blockquote>
<p>And one of the most fascinating reads I ever had was a paper by Riemann on the history of this [Fourier Series].</p>
</blockquote>
<p>I tried looking for it but didn't succeed, and I was wonder... | angryavian | 43,949 | <p>If you have two vectors $v=(v_1,v_2)$ and $w=(w_1,w_2)$, then their dot product is $$v \cdot w = v_1 w_1 + v_2 w_2.$$ Additionally, we also have
$$v \cdot w = \|v\| \|w\| \cos \theta$$ where $\|v\|=\sqrt{v_1^2+v_2^2}$, $\|w\|=\sqrt{w_1^2+w_2^2}$, and $\theta$ is the angle between $v$ and $w$. <a href="https://proofw... |
839,764 | <p>I am wondering how to show the following proposition</p>
<p><strong>If A, B, and C are set and $A \subset B ,B \subset C \rightarrow A \subset C$</strong></p>
<p><strong>My proof</strong> </p>
<p>Let x be any integer.</p>
<p>If $x\in A$ then $x\in B$ because all element of A are in B. If $x \in B \rightarrow x... | amWhy | 9,003 | <p>You don't need the last sentence. You simply draw the conclusion that $x \in A\implies x \in C $, and so $A\subset C$.</p>
<p>Polishing it up a tad:</p>
<p>Let $x$ be any element in $A$. Then, $x \in A\implies x\in B$ because $A \subset B$. Then $x \in B \implies x\in C$, since $B\subset C$.</p>
<p>Therefore, $$... |
839,764 | <p>I am wondering how to show the following proposition</p>
<p><strong>If A, B, and C are set and $A \subset B ,B \subset C \rightarrow A \subset C$</strong></p>
<p><strong>My proof</strong> </p>
<p>Let x be any integer.</p>
<p>If $x\in A$ then $x\in B$ because all element of A are in B. If $x \in B \rightarrow x... | beep-boop | 127,192 | <p>$$\underbrace{x \in A \implies x \in B}_{\text{since} \ A \subset B} \implies x \in C \ \ (\text{since } B \subset C),$$ so ( by the transitive property of implications) $x \in A \implies x \in C$ for any $x$, so $A \subset C $.</p>
<p>$\square$</p>
<p>A couple of notes: $x$ does not have to be an integer; it does... |
2,786,138 | <blockquote>
<p>Find all triples of prime numbers <span class="math-container">$(p,q,r)$</span> such that <span class="math-container">$$p^q+q^r=r^p.$$</span></p>
</blockquote>
<p>I proved that when <span class="math-container">$r=2$</span>, the equation becomes <span class="math-container">$$p^q+q^2=2^p.$$</span> Then... | Servaes | 30,382 | <p>It is clear that precisely one of $p$, $q$ and $r$ must equal $2$. We'll first show that in fact $r=2$:</p>
<hr>
<p><strong>Observation 1:</strong> $r=2$.</p>
<p>If $q=2$ then reducing mod $3$ shows that
$$p^2+2\equiv r^p\pmod{3},$$
so either $p=3$ or $r=3$. Both are easily verified to be impossible.</p>
<p>If $... |
2,696,097 | <p>$ \lim_{n\to \infty} ( \lim_{x\to0} (1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} $</p>
<p>The answer should be $ {e}^\frac{1}{3} $</p>
<p>I haven't encountered problems like this before and I'm pretty confused, thank you.</p>
<p>I guess we must use the remarkable limit of $ \frac{\tan(x)}{x} ... | Nicky Hekster | 9,605 | <p>In general, if $H$ is a <em>proper</em> subgroup of a group $G$, then $\langle G\backslash H \rangle = G$, that is, the set-theoretic complement generates the whole group. This follows from the fact that a group cannot be the union of two proper subgroups (I leave the proof to you). </p>
|
43,640 | <p>Consider the following problem:</p>
<p>Let ${\mathbb Q} \subset A\subset {\mathbb R}$, which of the following must be true?</p>
<p>A. If $A$ is open, then $A={\mathbb R}$</p>
<p>B. If $A$ is closed, then $A={\mathbb R}$</p>
<p>Since $\overline{\mathbb Q}={\mathbb R}$, one can immediately get that B is the answer... | Luboš Motl | 10,599 | <p>A counterexample for the rule A is
$$ {\mathbb R} \backslash F $$
where $F$ is any non-empty finite (or countable) set of irrational numbers. For example
$$ {\mathbb R} \backslash \{\pi\} $$
Note that if I remove the point $\pi$, the set is still open on both sides from $\pi$. Because $\pi$ isn't rational, the set ... |
292,780 | <p>I have to realize an NFA that recognizes the language of strings on the alphabet {a, b} ending with: bb, ba, baa. I thought that there must be the following states:</p>
<p>$q_0$: the string ends with bb.</p>
<p>$q_1$: the string ends with ba.</p>
<p>$q_2$: the string ends with baa.</p>
<p>Is right the definition... | vonbrand | 43,946 | <p>You need at least a start state, which loops back on each character, An then you need states for, e.g., $q_2 \rightarrow q_2' \rightarrow q_2'' \rightarrow q_2'''$ ($q_2$ means "$baa$ still to go", $q_2'$ is "want $aa$", $q_2''$ means "need $a$", and $q_2'''$ is final). [Sorry, would put symbols over the arrows if I... |
3,346,543 | <p>I am aware this is a pretty big topic, but the attempts at layman's explanations I have seen either barely provide commentary on the formal proofs, or fail to provide an explanation (e.g "it gets too complex" does not really say anything)</p>
<p>Is there a good intuitive explanation as to why we fail to obtain a ge... | Moishe Kohan | 84,907 | <p>See this detailed sketch of Arnold's proof of Abel-Ruffini theorem here: web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf. It is intended for a layman who can handle some basic group theory (and a bit of topology). It still takes about 5 pages.</p>
|
2,777,051 | <p>When talking about operator semigroups, some books use the notation $(e^{tA})_{t \geq 0}.$ Doesn't this create an ambiguity? Why are results proved for this semigroup instead of taking a general semigroup, say $(T(t))_{t \geq 0}$? </p>
| David C. Ullrich | 248,223 | <p>The two are equivalent under weak continuity assumptions. In particular if $(T(t))$ is a semigroup and $||T(t)-I||\to0$ as $t\to0$ then there exists a bounded $A$ such that $T(t)=e^{tA}$.</p>
<p><strong>Edit:</strong> Or so I'd read. Too stubborn to look at the proof - I think this works:</p>
<p>If $h>0$ is sma... |
497,687 | <p>How would I integrate the following.</p>
<p>$$
\int_{\ln2}^{\ln3}\frac{e^{-x}}{\sqrt{1-e^{-2x}}}\,dx
$$ </p>
<p>I think I have to use the $\arcsin(x)$ formula.</p>
<p>Which means would I use $y=u^2$ with $u=e^{-x}$ but that does not seem to work.</p>
| Norbert | 19,538 | <p>It does work
$$
\int_{\ln 2}^{\ln 3}\frac{e^{-x}}{\sqrt{1-e^{-2x}}}dx
\overset{\quad u=e^{-x}}{=}
-\int_{1/2}^{1/3}\frac{du}{\sqrt{1-u^2}}dx
$$
$$
=-\arcsin u\Biggl|_{1/2}^{1/3}
=-\arcsin(1/3)+\frac{\pi}{6}
$$</p>
|
991,569 | <p>\begin{align}
x-\hphantom{2}y+2z+2t&=0 \\
2x-2y+4z+3t&=1 \\
3x-3y+6z+9t&=-3 \\
4x-4y+8z+8t&=0
\end{align}</p>
<p>Solve for x,y,z and t</p>
<p><img src="https://i.stack.imgur.com/y3R7x.jpg" alt="reduced row echelon form"></p>
| Community | -1 | <p>Then treat any two variables as constants and express other two in terms of those constants</p>
<p>For Example if you keep $x$ and $y$ as constants then
$$z=\frac{y-x+2}{2}$$
$$t=-1$$</p>
|
3,567,245 | <p>I am trying to compute <span class="math-container">$\int_{C}Log(z+3)$</span>, where C is a circle centered at the origin, with radius of 2, oriented once counterclockwise. </p>
<p>I kind of get the idea of how to compute <span class="math-container">$\int Log(z)$</span> on the unit circle. I know <span class="math... | David Melo | 662,316 | <p>You can use the same method, but from there you're gonna need to split your interval into injective portions of a substitution and then calculate it by substitution of variables noting that the exponential function is holomorphic.</p>
<p>Since you're computing a contour integral of a "well behaved" function you're ... |
2,745,436 | <p>For some reason I have been struggling with this problem for the past couple hours.</p>
<p><a href="https://i.stack.imgur.com/diFLe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/diFLe.png" alt="enter image description here"></a></p>
<p>I believe I have solved part a.</p>
<p>Since there are 6 ... | Arnaud Mortier | 480,423 | <p><strong>Hint.</strong> The entry $m_{i,j}$ in the transition matrix is the probability to go to state $i$ at the next throw given that you are currently in state $j$. Therefore it is lower triangular, but not quite the identity.</p>
<p>E.g. if you are in state $4$ you have a $4/6$ chance to stay in state $4$. There... |
2,343,216 | <p>Let $\operatorname{gd}(x)$ the Gudermannian function, defined as in this <a href="http://mathworld.wolfram.com/Gudermannian.html" rel="nofollow noreferrer">MathWorld's article</a>, and implemented in Wolfram Language as <em>Gudermannian[x]</em>. </p>
<p>This afternoon I've spent an hour playing with <a href="https:... | wythagoras | 236,048 | <p>The Gudermannian function is lower-bounded by the line going to $(0,0)$ and $(1,\operatorname{gd}(1))$. However, it doesn't make a very large difference, and it hence gives a reasonable approximation. </p>
<p>See, for example, a plot here on <a href="https://www.wolframalpha.com/input/?i=(Gudermannian%5B1%5D-Guderm... |
2,343,216 | <p>Let $\operatorname{gd}(x)$ the Gudermannian function, defined as in this <a href="http://mathworld.wolfram.com/Gudermannian.html" rel="nofollow noreferrer">MathWorld's article</a>, and implemented in Wolfram Language as <em>Gudermannian[x]</em>. </p>
<p>This afternoon I've spent an hour playing with <a href="https:... | Claude Leibovici | 82,404 | <p>For sure, you could use Taylor expansion
$$\log (1+\text{gd}(x))=x-\frac{x^2}{2}+\frac{x^3}{6}-\frac{x^4}{12}+\frac{3 x^5}{40}-\frac{x^6}{18}+\frac{169
x^7}{5040}-\frac{19 x^8}{840}+\frac{2171 x^9}{120960}+O\left(x^{10}\right)$$ and integrate which would lead to $$\int_0^1 \log (1+\text{gd}(x))\,dx=\frac{443161}{... |
693,550 | <p>Consider the function $F(x) = x^2-2x+2$.</p>
<p>Find an interval in which the function is contractive and find the fixed point in this interval.</p>
<p>What is the convergence rate of the fixed point iteration: $x_{n+1} = F(x_n)$ in that interval?</p>
<p>I'm lost on fixed point iteration. I've watched a few youtu... | Mhenni Benghorbal | 35,472 | <p>You need to consider the recurrence relation</p>
<p>$$ x_{n+1}= x_{n}^2-x_n+2 ,$$</p>
<p>which suggests considering the function</p>
<p>$$f(x) = x^2-x+2 \implies f'(x) = 2x-1. $$</p>
<p>Now, to find the interval, use the condition $|f'(x)|<1$. See a <a href="https://math.stackexchange.com/questions/267648/p... |
693,550 | <p>Consider the function $F(x) = x^2-2x+2$.</p>
<p>Find an interval in which the function is contractive and find the fixed point in this interval.</p>
<p>What is the convergence rate of the fixed point iteration: $x_{n+1} = F(x_n)$ in that interval?</p>
<p>I'm lost on fixed point iteration. I've watched a few youtu... | Prasenjit D Wakode | 429,349 | <p>Try the form x(n+1)=(2x(n)-2)^(0.5) or x(n+1)=i(-2x(n)+2)^(0.5) of given equation. This converges to (1+i) which is one of the root of given equation. You note here that except infinities, any real or complex number as initial guess x(0) will converge x(n+1) to (1+i) .</p>
|
1,665,107 | <p>I must use a generating function to solve this question: </p>
<p>In how many ways can you collect six dollars from eight people if six people give either $0$ or $1$ dollars and the other two people each give $0$, $1$, or $5$ dollars?</p>
<p>Here is what I have so far.</p>
<p>The generating function is $(1+x)^6(1+... | vonbrand | 43,946 | <p>The generating function for each of the first class of people is <span class="math-container">$1 + z$</span>, the second class <span class="math-container">$1 + z + z^5$</span>. In all, the generating function of the number of ways to collect <span class="math-container">$n$</span> dollars in all is:</p>
<p><span c... |
3,517,722 | <p>Let <span class="math-container">$\alpha:= \sqrt[7]{2},\omega:= e^{\frac{2\pi i }{7}}\in \Bbb C$</span>. We set <span class="math-container">$E:=\Bbb Q(\alpha,\omega)$</span> and <span class="math-container">$B:=\Bbb Q(\omega+\omega^2+\omega^4)\leq E$</span>. </p>
<p>Thus, we have the Tower of Fields
<span class="m... | Jyrki Lahtonen | 11,619 | <p>Below please find a few pieces that help you to onoe way of proving the claims.</p>
<ul>
<li>The calculation <a href="https://math.stackexchange.com/a/299417/11619">here</a> reveals that <span class="math-container">$B=\Bbb{Q}(\sqrt{-7})$</span>. Therefore <span class="math-container">$[B:\Bbb{Q}]=2$</span>, and yo... |
3,517,722 | <p>Let <span class="math-container">$\alpha:= \sqrt[7]{2},\omega:= e^{\frac{2\pi i }{7}}\in \Bbb C$</span>. We set <span class="math-container">$E:=\Bbb Q(\alpha,\omega)$</span> and <span class="math-container">$B:=\Bbb Q(\omega+\omega^2+\omega^4)\leq E$</span>. </p>
<p>Thus, we have the Tower of Fields
<span class="m... | Chris | 326,865 | <p><em>Taking into account Jyrki's answer, I tried to collect the pieces. Please, have a look.</em></p>
<p>(1) <em>By the first lines of his answer:</em> If <span class="math-container">$[B:\Bbb{Q}]=2$</span>, then
<span class="math-container">$$[E:\Bbb Q]=[E:B][B:\Bbb Q] \implies 42=[E:B]\cdot 2 \iff [E:B]=21=|\math... |
3,469,582 | <p>I have a feeling that <span class="math-container">$\lim_{x\rightarrow\infty}\frac{\log f(x^2)}{f(x)}=0$</span> for any positive monotone increasing function such that <span class="math-container">$\lim_{x\rightarrow\infty}f(x)=\infty$</span>, (in reality I am thinking mostly about super-exponential growth here but ... | WhatsUp | 256,378 | <p>Define a sequence <span class="math-container">$a_n$</span> with arbitrary <span class="math-container">$a_0>1$</span> and <span class="math-container">$a_{n + 1}=e^{a_n}$</span>.</p>
<p>Define a function <span class="math-container">$f$</span> such that <span class="math-container">$f(x)= a_0$</span> for <span ... |
223,509 | <p>Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g.
let $\ulcorner \varphi \urcorner$ denote the <a href="https://en.wikipedia.org/wiki/G%C3%B6del_numbering" rel="nofollow">Godel number</a> of $\varphi$.
Let $T$ be a first-order arithmetic theory, e.g. PA.
Let $\Phi$ be a class of close... | Ali Enayat | 9,269 | <p>Biconditionals of the form $Tr(\varphi) \leftrightarrow \varphi$ are known as <em>T-sentences</em> in the literature of philosophical logic, and they date back to Tarski's groundbreaking work on the notion of truth in formalized languages. </p>
<p>However, I have not seen a special name for implications of the for... |
2,554,380 | <p>With the metric space $(X,d) : X = \Bbb R$ and $d(x,y) = |x| + |y|$ for $x\neq y$, $d(x,x) = 0$ and $A = \{0\}$. Is interior of $A$ empty? </p>
<p>In usual metric it would be empty but with this metric I conclude that it is $A$ itself.</p>
<p>Def: $ a \in A $ is an interior point of $A$ if $\exists \epsilon > 0... | Peter Szilas | 408,605 | <p>$f:D \rightarrow \mathbb{R} $ is uniformly continuous in </p>
<p>$D$, if</p>
<p>for every $\epsilon >0$ there is a $\delta$ such that </p>
<p>$|x-x'| \lt \delta$ implies </p>
<p>$|f(x)-f(x')| \lt \epsilon$ for all $x,x' \in D.$</p>
<p>Given: </p>
<p>$|f(x)-f(x')| \le c|x-x'|$, $c>0$, for $ x,x' \in D.$<... |
930,458 | <p>would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry:
I have to find smallest algebraic variety (irreducible algebraic set) in $\mathbb{C^{2}}$ that contains all the points with integer coordinates. I don't really know how to start, so at least an idea, h... | Rene Schipperus | 149,912 | <p>The only such is the plane itself. Note that there can be no algebraic curve passing through all those points since then it would have an infinite number of intersections with a line, which is impossible.</p>
|
236,229 | <p>I have a list of 3 pairs of integers, for each pair <code>i</code> I want to replace the one having the smallest absolute value with <code>xi</code> and the other one with its sign. Like this:</p>
<pre><code>In: {{-5, 1}, {7, -3}, {4, 4}}
Out: {{-1, x2}, {1, x3}, {x4, 1}} (*for the last one {1, x4} would be fine as ... | kglr | 125 | <pre><code>ClearAll[f]
f = MapIndexed[If[LessEqual @@ Abs @ #,
{Symbol["x" <> ToString[#2[[1]]]], Sign @ Last @ #},
{Sign @ First @ #, Symbol["x" <> ToString[#2[[1]]]]}] &];
list = {{-5, 1}, {7, -3}, {4, 4}};
f @ list
</code></pre>
<blockquote>
<pre><code>{{-1, x1}, {1, x2}... |
236,229 | <p>I have a list of 3 pairs of integers, for each pair <code>i</code> I want to replace the one having the smallest absolute value with <code>xi</code> and the other one with its sign. Like this:</p>
<pre><code>In: {{-5, 1}, {7, -3}, {4, 4}}
Out: {{-1, x2}, {1, x3}, {x4, 1}} (*for the last one {1, x4} would be fine as ... | Αλέξανδρος Ζεγγ | 12,924 | <p><a href="https://reference.wolfram.com/language/ref/MapIndexed.html" rel="nofollow noreferrer"><code>MapIndexed</code></a> should be the suitable function working at the overall level. A version slightly different from that of @kglr 's, with <a href="https://reference.wolfram.com/language/ref/OrderingBy.html" rel="n... |
2,607,668 | <p>I am trying to prove/disprove $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)$. Apparently $\operatorname{Arg}(zw)=\operatorname{Arg}(z)+\operatorname{Arg}(w)+2k\pi$ where $k=0,1,\text{ or }-1$, but I have no idea why. I keep on finding that answer online. I am very lost on how to prove this stat... | joeyg | 454,784 | <p>Denote $z=r_1\cdot e^{i\theta_1}$, $z=r_2\cdot e^{i\theta_2}$.</p>
<p>Then $zw=r_1\cdot e^{i\theta_1}\cdot r_2\cdot e^{i\theta_2}=r_1r_2e^{i(\theta_2+\theta_2)}$, and since the argument of a complex number $re^{i\theta}$ is $\theta$, we receive that $Arg(z)+Arg(w)=\theta_1+\theta_2=Arg(zw)$ up to $\pm 2\pi$.</p>
|
711,549 | <p>Consider the successor of the largest finite ordinal that will ever be considered alone. But then it wasn't the largest finite ordinal that will ever be considered alone. How do we get around this paradox? The largest finite ordinal that will ever be considered alone does exist, and yet we can consider its successor... | vadim123 | 73,324 | <p>This apparent paradox has nothing to do with ordinals, or metalanguage. It is a time travel/omniscience paradox. Let's strip away the unnecessary bits.</p>
<p>How many words will the longest sentence I personally will say tomorrow be? Maybe I predict seven, but that doesn't stop me from saying a sentence with ei... |
1,461,484 | <p>For example, look at this sentence from Perko's text on dynamical system</p>
<p>"It follows from Cauchy Schwarz inequality that if $T \in L(R^n)$ is represented by the matrix $A$ with respect to the standard basis for $R^n$ $_\cdots$" pg 11</p>
<p>What does it mean for a $T: R^n \to R^n$ to be represented by a mat... | user2357112 | 91,416 | <p>The number after nine is represented in the usual base system by the sequence of digits $10$, but it isn't a sequence of digits. Properties like "two digits long" are properties of the representation, while properties like "has four factors" are properties of the number. We could represent the number in binary, as $... |
3,848,517 | <p>I have a conjecture in my mind regarding Arithmetic Progressions, but I can't seem to prove it. I am quite sure that the conjecture is true though.</p>
<p>The conjecture is this: suppose you have an AP (arithmetic progression):
<span class="math-container">$$a[n] = a[1] + (n-1)d$$</span>
Now, suppose our AP satisfie... | Kavi Rama Murthy | 142,385 | <p>If <span class="math-container">$f_n(x) =0$</span> for <span class="math-container">$x <0$</span> as well as for <span class="math-container">$x >\frac 1n$</span> and <span class="math-container">$f_n(x)=4n^{2}x(\frac 1n -x)$</span> for <span class="math-container">$0 <x<\frac 1n$</span> then you can ... |
3,391,118 | <p>let <span class="math-container">$x , y$</span> be a real numbers with <span class="math-container">$-3\leq x \leq5$</span> and <span class="math-container">$-2\leq y\leq -1$</span> , I ask if the range of <span class="math-container">$x-y$</span> is <span class="math-container">$[-1,6]$</span> or <span class="math... | Bernard | 202,857 | <p>No, you cannot subtract inequalities, only add them, and changing signs in an inequality reverses it.</p>
<p>Here a way to go:
<span class="math-container">$$−2≤y≤−1\iff 1\le -y\le 2$$</span>
so, adding the latter to the inequalities with <span class="math-container">$x$</span> yields
<span class="math-container">$... |
3,391,118 | <p>let <span class="math-container">$x , y$</span> be a real numbers with <span class="math-container">$-3\leq x \leq5$</span> and <span class="math-container">$-2\leq y\leq -1$</span> , I ask if the range of <span class="math-container">$x-y$</span> is <span class="math-container">$[-1,6]$</span> or <span class="math... | Z Ahmed | 671,540 | <p>Two in-equations in the same direction can be added but they can never be subtracted.
Let us take two parts of four given in-equations: <span class="math-container">$x \le 5~ and ~ -2 \le y~$</span> we can add the two to get <span class="math-container">$x-2 \le 5+y \implies x-y \le 7.$</span> Next, let us take o... |
3,344,766 | <p>It is too be proved that <span class="math-container">$3(p^2)+61$</span> is a perfect square only for <span class="math-container">$$p=1 ; p=N$$</span>
The question arises from a problem in arithmatic progression with a certain soft constraint of natural numbers. At the end a quadratic is formed and since our constr... | Will Jagy | 10,400 | <p>Given any solution with positive integers <span class="math-container">$q^2 - 3 p^2 = 61,$</span> we get an infinite string of (larger) positive solutions by
<span class="math-container">$$ (q,p) \mapsto (2q+3p, \; q+2p) $$</span></p>
<p>For example,
<span class="math-container">$$ (8,1) \mapsto (19, \; 10) \maps... |
231,403 | <p>Let $1 \le p < \infty$. For all $\epsilon > 0$, does there exist $C = C(\epsilon, q)$ such that$$\|u\|_{L^p(0, 1)} \le \epsilon \|u'\|_{L^1(0, 1)} + C\|u\|_{L^1(0, 1)} \text{ for all }u \in W^{1, 1}(0, 1)?$$</p>
| Jochen Wengenroth | 21,051 | <p>The inequality you ask for is more or less a restatement of the compactness of an appropriate embedding: Let $p\le q\le r$ be three norms on a vector space such that the ball $B_r(1)=\lbrace x\in X. r(x)\le 1\rbrace$ is relatively compact in $(X,q)$. Then, for every $\varepsilon>0$ there is $C>0$ such that
$$... |
2,234,478 | <p>Given 3 points, A (x1, y1), B (x2, y2) and C (x3, y3), what is the best way to tell if all 3 points lie within a circle of a given radius <strong>r</strong>?</p>
<p>The best I could come up with was to find the Fermat point F (x4, y4) for the triangle ABC, and then check if the distance from F to each A, B and C is... | Mark Fischler | 150,362 | <p>Alpha only gives a numeric approximation which, unless you mis-typed your equation into alpha, should be about 1.56.</p>
<p>There is no simple closed for for the answer.</p>
<p>But there is something cute about this: Full Mathematica fails to give a numerical solution using its NSolve method, while Alpha does giv... |
2,234,478 | <p>Given 3 points, A (x1, y1), B (x2, y2) and C (x3, y3), what is the best way to tell if all 3 points lie within a circle of a given radius <strong>r</strong>?</p>
<p>The best I could come up with was to find the Fermat point F (x4, y4) for the triangle ABC, and then check if the distance from F to each A, B and C is... | egreg | 62,967 | <p>The equation can be written
$$
\sqrt[x]{2}\log x=\log 2
$$
(natural logarithm); setting $t=1/x$, it becomes
$$
-2^t\log t=\log 2
$$
Consider the function $f(t)=2^t\log t+\log 2$. We see that
$$
\lim_{t\to0}f(t)=-\infty
\qquad
\lim_{t\to\infty}f(t)=\infty
$$
Moreover
$$
f'(t)=2^t\cdot\log 2\cdot\log t+\frac{2^t}{t}&g... |
1,149,907 | <p>What are some different methods to evaluate</p>
<p>$$ \int_{-\infty}^{\infty} x^4 e^{-ax^2} dx$$</p>
<p>for $a > 0$.</p>
<p>This integral arises in a number of contexts in Physics and was the original motivation for my asking. It also arises naturally in statistics as a higher moment of the normal distributio... | D. Thomine | 20,413 | <p>Generating functions, as often, provide a nice way to compute these integrals.</p>
<p>Let $I_n := \int_\mathbb{R} x^n e^{-ax^2} \ dx$, and for all complex $\lambda$:</p>
<p>$$G(\lambda) := \sum_{n=0}^\infty \frac{I_n}{n!}\lambda^n.$$</p>
<p>Then:</p>
<p>$$G(\lambda) = \int_\mathbb{R} \sum_{n=0}^\infty \frac{(\la... |
4,415,037 | <p>I would like to prove upper and lower bounds on <span class="math-container">$|\cos(x) - \cos(y)|$</span> in terms of <span class="math-container">$|x-y|$</span>. I was able to show that <span class="math-container">$|\cos(x) - \cos(y)| \leq |x - y|$</span>. I'm stuck on the lower bound. Does anyone know how to appr... | Swapnajit Das | 857,214 | <p><span class="math-container">$|\cos(x)-\cos(y)|\geq 0$</span> for all <span class="math-container">$x$</span>, <span class="math-container">$y \in$</span> <span class="math-container">$\mathbb{R}$</span>. It is possible for equality to hold, for example when <span class="math-container">$x=y$</span>.</p>
<p>By the <... |
2,869,305 | <p>I need some help evaluating this limit... Wolfram says it blows up to infinity but I don't think so. I just can't prove it yet.</p>
<p>$$ \lim_{x\to\infty}(x!)^{1/x}-\frac{x}{e} $$</p>
| pancini | 252,495 | <p>It diverges, albeit slowly.</p>
<p>By <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling's Approximation</a>, the function approaches</p>
<p>$$f(x)=(2\pi x)^{1/x}\left( \frac xe\right)-\frac xe=\left(\frac xe\right)\left( (2\pi x)^{1/x}-1\right).$$</p>
<p>Now it ... |
57,232 | <p>I have huge matrices in the form of</p>
<pre><code>mtx1 = {{24+24 FF[6,9] GG[5,10]+24 FF[7,8] GG[5,10]+24 FF[5,10] GG[6,9]+24 FF[6,9] GG[6,9]+24 FF[7,8] GG[6,9]+24 FF[5,10] GG[7,8]+24 FF[6,9] GG[7,8]+24 FF[7,8] GG[7,8],24+24 FF[5,10] GG[5,10]+24 FF[6,9] GG[5,10]+24 FF[7,8] GG[5,10]+24 FF[6,9] GG[6,9]+24 FF[7,8] GG[... | eldo | 14,254 | <p>Given the structure of your SQUARE example matrix this should be fast:</p>
<pre><code>dim = First @ Dimensions @ mtx1
</code></pre>
<blockquote>
<p>3</p>
</blockquote>
<pre><code>tup = Join [#, {1}] & /@ Tuples[Range@dim, 2]
</code></pre>
<blockquote>
<p>{{1, 1, 1}, {1, 2, 1}, {1, 3, 1}, {2, 1, 1}, {2, 2... |
4,580,484 | <p>It is often said that "differential forms are used for integration".</p>
<p>Typically people like to talk about the integral <span class="math-container">$\int_M \omega$</span> of a differential form <span class="math-container">$\omega$</span>, and exterior derivative, one of the most important operation ... | Quaere Verum | 484,350 | <p>This is only true if you are very pedantic about what you mean by "symplectic geometry". Namely, if you include symplectic topology, then one of the most important invariants of a symplectic manifold requires integrating the symplectic form over <span class="math-container">$2$</span>-dimensional submanifo... |
2,249,929 | <p>Is there a more general form for the answer to this <a href="https://math.stackexchange.com/q/1314460/438622">question</a> where a random number within any range can be generated from a source with any range, while preserving uniform distribution? </p>
<p><a href="https://stackoverflow.com/q/137783/866502">This</a>... | Marcus Andrews | 97,648 | <p>I wrote a Python implementation a while back that did this:</p>
<pre><code>def uniform_generator(m, n): #mimics m-sided die using an n-sided die
"""
Expected number of rolls
E = r * n^r / m
where r = int(ceil(log(m, n)))
"""
r = int(ceil(log(m, n)))
while True:
candidate = sum(n*... |
1,955,509 | <p>There's this exercise in Hubbard's book:</p>
<blockquote>
<p>Let $ h:\Bbb R \to \Bbb R $ be a $C^1$ function, periodic of period $2\pi$, and define the function $ f:\Bbb R^2 \to \Bbb R $ by
$$f\begin{pmatrix}r\cos\theta\\r\sin\theta \end{pmatrix}=rh(\theta)$$</p>
<p>a. Show that $f$ is a continuous real-v... | E.H.E | 187,799 | <p>Hint:
$${\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}\quad {\text{ for }}|x|<1\!$$
differentiate it and multiply by $x$</p>
<p>$${\frac {x}{(1-x)^{2}}}=\sum _{n=0}^{\infty }nx^{n}$$
differentiate it again and multiply by $x$
$${\frac {x(x+1)}{(1-x)^{3}}}=\sum _{n=0}^{\infty }n^2x^{n}$$</p>
|
2,253,768 | <p>I am currently working on a small optimization problem in which I need to find an optimal number of servers for an objective function that incorporates the Erlang Loss formula. To this end, I have been searching for an expression for the first order difference of the Erlang Loss formula with respect to the number of... | quasi | 400,434 | <p>First we show $(M,x)$ is a proper ideal of $R[x]$.
<p>
Suppose instead that $(M,x) = (1)$.
<p>
Then $1 \in (M,x)$ implies $1 = m + xg(x)\;$for some $m \in M$, some $g \in R[x]$.
<p>
Since $M$ is maximal in $R,\, M$ is a proper ideal of $R$, hence $m \ne 1$.
<p>
Then </p>
<p>\begin{align*}&1 = m + xg(x)\\[4pt]
\... |
3,440,732 | <p>How can I show that <span class="math-container">$P=\{\{2k-1,2k\},k\in \mathbb {N}\}$</span> is the basis of <span class="math-container">$\mathbb N$</span>.
Obviously we have to show that basis's orders do work.
But the problem is if I take two subsets that belong to <span class="math-container">$P$</span> , for ex... | Henno Brandsma | 4,280 | <p>Any <em>partition</em> <span class="math-container">$P_i, i \in I$</span> of a set <span class="math-container">$X$</span> is a base for a topology on <span class="math-container">$X$</span>. A partition obeys:</p>
<ol>
<li><span class="math-container">$\bigcup_{i \in I} P_i = X$</span>.</li>
<li><span class="math-... |
263,961 | <p>Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from algebraic structures (theories of abstract groups, rings, fields) and real and complex analysis (theories of expansion... | Haskell Curry | 39,362 | <p>There is a result in functional analysis whose first known proof uses non-standard techniques:</p>
<blockquote>
<p><strong>Theorem</strong> If a bounded linear operator $ T $ on a Hilbert space $ \mathcal{H} $ is polynomially compact, i.e., $ P(T) $ is compact for some non-zero polynomial $ P $, then $ T $ has an... |
263,961 | <p>Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from algebraic structures (theories of abstract groups, rings, fields) and real and complex analysis (theories of expansion... | Tomasz Kania | 17,929 | <p>Using <em>continuous model theory</em>, Ilijas Farah, Brad Hart and David Sherman proved <a href="http://arxiv.org/abs/1111.0998" rel="nofollow">`blind man's version' of the Connes Embedding Problem</a>:</p>
<blockquote>
<p>There exists a separable ${\rm II}_1$-factor $M$ such that every other separable ${\rm II}... |
4,007,450 | <p>Let <span class="math-container">$y=f(x)$</span> the graph of a real-valued function. We define its curvature by : <span class="math-container">$$curv(f) = \frac{|f''|}{(1+(f')^2)^{3/2}}$$</span></p>
<p>I would like to know if there is any function (apart from the trivial anwser <span class="math-container">$f(x)=0$... | Community | -1 | <p>The hypothesis implies an exponent of two for the group. But then it is abelian, since <span class="math-container">$ab=(ab)^{-1}=b^{-1}a^{-1}=ba$</span>, for any two elements of the group <span class="math-container">$a$</span> and <span class="math-container">$b$</span>.</p>
|
3,194,169 | <p>I am not a mathematician but I have a question.</p>
<p>I found </p>
<ul>
<li><span class="math-container">$1 / 3 = 0.33333....$</span></li>
<li><span class="math-container">$2 / 3 = 0.666666...$</span></li>
<li><span class="math-container">$3 / 3 = 1$</span> while it should be <span class="math-container">$0.9999... | Bernard | 202,857 | <p>There is a <em>bijection</em> between infinite ultimately periodic decimal expansions and rational numbers, <em>except</em> when the period consists only of the digit <span class="math-container">$9$</span>. In this case,, one proves that deleting the final <span class="math-container">$9$</span>s and replacing last... |
3,409,342 | <p>Let <span class="math-container">$X$</span> be a set containing <span class="math-container">$A$</span>.</p>
<p>Proof:
<span class="math-container">$y\in A \cup (X \setminus A) \Rightarrow y\in A$</span> or <span class="math-container">$y \in (X \setminus A)$</span></p>
<p>If <span class="math-container">$y \in ... | Manx | 483,923 | <p>Let <span class="math-container">$A\subseteq X$</span></p>
<p><span class="math-container">$\Leftarrow:$</span></p>
<p>Show</p>
<p><span class="math-container">$$X\subseteq A\cup(X-A)$$</span></p>
<p>Let <span class="math-container">$\phi_1,\phi_2$</span> be some formula</p>
<p>write <span class="math-container... |
877,477 | <p>I'm stuck with the following question, which looks quite innocent.</p>
<p>I'd like to show that if a covering space map $f:\tilde{X}\to X$ between cell complexes is null-homotopic, then the covering space $\tilde{X}$ must be contractible.</p>
<p>Since $f$ is null-homotopic there exists a homotopy $H_t:\tilde{X}\to... | Kyle | 153,841 | <p>As you said, we're given a covering $f: \tilde X \to X$ and a homotopy $H_t : \tilde X \to X$ such that $H_0(\tilde x)=f(\tilde x)$ and $H_1(\tilde x)=x_0$ for some fixed $x_0 \in X$. Since $\operatorname{id}_{\tilde X}$ is a lift of $H_0=f$, there is a unique lift $\tilde H_t : \tilde X \to \tilde X$ of the homotop... |
2,319,119 | <p>Been given a question and find it to be too vague to know what's going on. </p>
<p>The question is: </p>
<p>$f(x) = 2x + 2$. Define $f(x)$ recursively.</p>
<p>I'm just quite puzzled as there is no $f(0)$, $f(1)$ or $f(x-1)$ function to go by other than the original function.</p>
<p>Supposed to be in the form of ... | N. S. | 9,176 | <p><strong>Hint:</strong> If $\| f-g\|_\infty <\epsilon$ then
$$\left| \int_0^1 f(t) dt -\int_0^1 g(t) dt \right| <\epsilon$$</p>
|
492,392 | <p>I found this math test very interesting. I would like to know how the answer is being calculate?</p>
| Mufasa | 49,003 | <p>HINT: $49a+49b=6272$ can be written as $49(a+b)=6272$ and the average of $a$ and $b$ can be written as $\large\frac{a+b}{2}$</p>
|
2,563,300 | <p>If we put the Cartesian coordinates of a point in a 2 dimensional locus' equation then we get zero as the value if the point lies on the locus.
On putting coordinates of all other points in the equation which do not lie on the locus, we get a numerical value. Does this numerical value convey any information about th... | Karn Watcharasupat | 501,685 | <p>Let's look at it this way.</p>
<hr>
<p><strong>'At least one' case</strong></p>
<p>If at least one succeeds then we have
\begin{align}
P(\text{at least one success})&=P(A\cap B)+P(A\cap B')+P(A'\cap B)\\
&=\frac{1}{3}\cdot\frac{1}{5}+\frac{1}{3}\cdot\frac{4}{5}+\frac{2}{3}\cdot\frac{1}{5}\\
&=\frac{7... |
4,095,715 | <p>I know how to do these in a very tedious way using a binomial distribution, but is there a clever way to solve this without doing 31 binomial coefficients (with some equivalents)?</p>
| Sid | 428,996 | <p><span class="math-container">$(1+x^2+x^4+x^6)^n = (1+x^4)^n(1+x^2)^n$</span></p>
<p>the <span class="math-container">$r^{th}$</span> term in first series: <span class="math-container">${n \choose r} x^{4r}$</span></p>
<p>the <span class="math-container">$l^{th}$</span> term in second series: <span class="math-contai... |
4,064,353 | <blockquote>
<p>The number <span class="math-container">$1.5$</span> is special because it is equal to one quarter of the sum of its digits, as <span class="math-container">$1+5=6$</span> and <span class="math-container">$\frac{6}{4}=1.5$</span> .Find all the numbers that are equal to one quarter of the sum of their ... | fleablood | 280,126 | <p>My first draft was messy.</p>
<p>With hindsight.</p>
<p>Let <span class="math-container">$N = \frac {K}4 = M\frac i4$</span> where <span class="math-container">$K$</span> is the sum of the digits. <span class="math-container">$M$</span> is the quotient integer of dividing <span class="math-container">$K$</span> by <... |
2,808,512 | <p>Show that if an $n\times n$ matrix $A$ satisfies $A^T=-A$, then $x^TAx=0$ for any $n\times 1$ vector $x$.</p>
<p>My attempt: Since matrix transpose won't affect the diagonal entries, so matrix $A$ has only zeros on its diagonal. </p>
<p>Then I tried to write $x$ in the form of $\begin{pmatrix}
x_1\\
x_2\\
\vdots\\... | Lord KK | 446,929 | <p>What you can see is that $x^TAx$ is of type $(1\times1)$, hence symmetric, i.e.,</p>
<p>$$(x^TAx)^T = x^TAx.$$
On the other hand we have
$$x^TA^Tx= x^TAx.$$</p>
<p>From $A^T=-A$ we get $$-x^TAx = x^TAx\iff 2x^TAx =0\iff x^TAx=0.$$</p>
|
313,298 | <p>For example, let's say that I have some sequence $$\left\{c_n\right\} = \left\{\frac{n^2 + 10}{2n^2}\right\}$$ How can I prove that $\{c_n\}$ approaches $\frac{1}{2}$ as $n\rightarrow\infty$?</p>
<p>I'm using the Buchanan textbook, but I'm not understanding their proofs at all.</p>
| Kaster | 49,333 | <p>$$
\lim_{n \rightarrow \infty} \frac {n^2+10}{2n^2} = \lim_{n \rightarrow\infty} \frac {1+\frac {10}{n^2}}{2} = \lim_{n \rightarrow \infty} \left (\frac 12+\frac{10}{2n^2} \right) = \frac 12+5\lim_{n \rightarrow \infty}\frac 1{n^2}=\frac 12
$$</p>
|
1,130,645 | <p>I am asked to prove: if <span class="math-container">$|a|<\epsilon$</span> for all <span class="math-container">$\epsilon>0$</span>, then <span class="math-container">$a=0$</span></p>
<p>I can prove this as follows.</p>
<p>Assume <span class="math-container">$a \not= 0$</span></p>
<p>I want to show then that <... | Hagen von Eitzen | 39,174 | <p>You can do so if you do work with infinitesimals (say Robinson's hyperreal numbers) and, to really make the proof this short, make sure that $a$ is a standard number. One needs to know that application of the $\operatorname{st}()$ part to both sides of a strict inequality produces a corresponding un-stricted inequa... |
351,534 | <p>It was already known to Weil that a sufficiently reasonable cohomology theory for algebraic varieties over <span class="math-container">$\mathbb{F}_p$</span> would allow for a possible solution to the Weil conjectures. </p>
<p>It was also understood that such a cohomology theory could not take values in vector spac... | S. Carnahan | 121 | <p>One historical reason for considering <span class="math-container">$\ell$</span>-adic cohomology, not completely disconnected from the example you introduce, is that for a curve over a field, we get a natural Galois representation by taking the <span class="math-container">$\ell$</span>-adic Tate module of the Jacob... |
3,203,577 | <p>What is the computational complexity of solving a linear program with <span class="math-container">$m$</span> constraints in <span class="math-container">$n$</span> variables?</p>
| Zhao Song | 324,173 | <p>Brand's 2020 result derandomized the Cohen, Lee and Song result.
Here is the link
<a href="https://arxiv.org/pdf/1910.11957.pdf" rel="noreferrer">https://arxiv.org/pdf/1910.11957.pdf</a></p>
|
2,512,396 | <p>i am trying to compute inverse Laplace transform of function = $\tan ^{-1} (2/s)$</p>
| Nosrati | 108,128 | <p>$${\cal L}\Big(\frac{\sin2t}{t}\Big)=\int_s^\infty\frac{2}{s^2+4}~ds=\arctan\frac{s}{2}\Big|_s^\infty=\frac{\pi}{2}-\arctan\frac{s}{2}=\arctan\frac{2}{s}$$</p>
|
3,565,727 | <blockquote>
<p>Show that the antiderivatives of <span class="math-container">$x \mapsto e^{-x^2}$</span> are uniformly continuous in <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>So we know that for a function to be uniformly continuous there has to exists <span class="math-container">$\var... | Gary | 83,800 | <p>Any antiderivative can be written in the form
<span class="math-container">$$
\int_0^x {e^{ - t^2 } dt} + C,
$$</span>
where <span class="math-container">$C$</span> is a constant. Thus
<span class="math-container">$$
\left| {\int_0^x {e^{ - t^2 } dt} + C - \left( {\int_0^y {e^{ - t^2 } dt} + C} \right)} \right| =... |
3,565,727 | <blockquote>
<p>Show that the antiderivatives of <span class="math-container">$x \mapsto e^{-x^2}$</span> are uniformly continuous in <span class="math-container">$\mathbb{R}$</span>.</p>
</blockquote>
<p>So we know that for a function to be uniformly continuous there has to exists <span class="math-container">$\var... | egreg | 62,967 | <p>Every function <span class="math-container">$f$</span> defined and differentiable over an interval, with a bounded derivative, is uniformly continuous. Indeed, by the mean value theorem, if <span class="math-container">$x\ne y$</span>, we can say that
<span class="math-container">$$
\frac{f(x)-f(y)}{x-y}=f'(c)
$$</s... |
2,197,065 | <p>Baire category theorem is usually proved in the setting of a complete metric space or a locally compact Hausdorff space.</p>
<p>Is there a version of Baire category Theorem for complete topological vector spaces? What other hypotheses might be required?</p>
| Michael Hardy | 11,667 | <p>You can take an average of several distances, as with several times.</p>
<p>One place where "average time" occurs is in statistical things where, for example, a phone call arriving at a switchboard has no effect on the timing of another phone call arriving, and one speaks of the average time from the arrival of one... |
391,020 | <p>Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly? </p>
| Billy | 39,970 | <p>Suppose there is a homomorphism $f: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}$. What could $f(1)$ be? Let's call it $a$. Then $f(2) = f(1+1) = f(1)+f(1) = a+a = 2a$, and likewise $f(3) = 3a$, $\dots,$ $f(\underbrace{1+\cdots+1}_{n\text{ times}}) = na$. But $\underbrace{1+\cdots+1}_{n\text{ times}} = 0$, and $f(0) = 0$. ... |
738,435 | <p>I am having a problem understanding how to determine if a function is one to one.</p>
<p>The problem is: Show that the function f(x) = 3x+4 is one-to-one.</p>
<p>Also, I'm being thrown off by the notation x[subset 1] = x[subset 2], what does that mean, loosely speaking? "In my eyes" that would mean two different v... | A. Wong | 100,535 | <p>In general, to show that a function is one to one you would suppose that $f(x)=f(y)$ and show that it must be the case that $x=y$.</p>
<p>In this case you start by assuming that $3x+4=3y+4$ and then proceed to show that $x=y$.</p>
<p>I believe when you say x[subset 1]=x[subset 2] you are actually saying $x_{1}=x_{... |
1,758,194 | <p>Consider the function f: {-1, +1} -> R defined by</p>
<p>$f(x)= \arcsin (\frac{1+x^2}{2x})$.</p>
<p>Due to the following two inequalities :</p>
<p>(i) $1+x^2 \geq 2x$</p>
<p>(ii)$1+x^2 \geq -2x$ , </p>
<p>the function can only be defined at $x=1$ and $x=-1$. I have learnt that the epsilon delta definition only ... | eyeballfrog | 395,748 | <p>Usually when talking about limits of functions $f:D\subseteq \mathbb R \rightarrow \mathbb R$ in calculus, we work with functions whose domain is not a discrete set. Because of this, an extra condition on the limit is left out: the limit can only be taken at limit points of the domain. A point $c$ is a limit point o... |
1,668,653 | <blockquote>
<p>If $\sum a_n$ is convergent and $a_n>0$, then $\sum (-1)^n a_n$ is convergent.</p>
</blockquote>
<p>So far I've tried convergence/divergence tests and also I tried to prove this using partial sums. But tests do not work because the latter series <em>includes negative terms</em>. Maybe the Alternat... | Nick | 272,448 | <p>HINT: any absolutely convergent series is convergent.Let me know if you want the full solution</p>
|
527,328 | <p>$$\lim_{x\rightarrow\infty}\left(\frac{x+1}{x-2}\right)^{2x-1}$$</p>
<p>What are the steps to solve it? Probably the division should be multiplied by some expression.</p>
| DonAntonio | 31,254 | <p>$$\left(\frac{1+x}{-2+x}\right)^{-1+2x}=\left[\left(1+\frac3{x-2}\right)^{x-2}\right]^{2}\left(1+\frac3{x-2}\right)^3$$</p>
<p>And now use the basic lemma</p>
<p>$$\lim_{x\to\infty}\left(1+\frac a{f(x)}\right)^{f(x)}=e^a$$</p>
<p>where $\;a\in\Bbb R\;$ and $\;f\;$ is a funcion s.t.</p>
<p>$$\lim_{x\to\infty}f(x)... |
3,540,593 | <p>Are <span class="math-container">$n$</span> vectors are orthogonal if performing the inner product of all <span class="math-container">$n$</span> vectors at once yields zero?</p>
<p>In other words, could I say that <span class="math-container">$\hat{i} \perp \hat{j} \perp \hat{k}$</span>?</p>
<p>For example, suppo... | lab bhattacharjee | 33,337 | <p>Hint</p>
<p>We need <span class="math-container">$p$</span> to divide <span class="math-container">$k^2-1=(k+1)(k-1)$</span></p>
<p>But as <span class="math-container">$k+1-(k-1)=2,p$</span> must divide exactly one of <span class="math-container">$k+1,k-1$</span></p>
<p>Now we need <span class="math-container">$1... |
162,863 | <p>if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation?</p>
<p>Thanks in advance!</p>
| Christian Blatter | 1,303 | <p>See here: <a href="http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation" rel="nofollow">Wikipedia on quaternions and rotations</a>.</p>
<p>To the benefit of all: One can encode a rotation $T\in SO(3)$ as a quaternion $q+ q_1 i+ q_2 j + q_3 k$ of norm $1$. Here $q=\cos{\theta\over2}$ and $(q_1,q_2,q_3)=\sin... |
4,079,842 | <p>Consider the below function.
<span class="math-container">$$f(x,y) =
\begin{cases}
1 & xy \neq 0 \\
0 & xy = 0
\end{cases}$$</span>
Suppose i want to calculate the below repeated limit.
<span class="math-container">$$\lim_{x\to0}\lim_{y\to0}f(x,y)$$</span>.
In general textbooks in India for BS courses,the a... | Adam Rubinson | 29,156 | <p><span class="math-container">$$\lim_{x\to0}[\lim_{y\to0}f(x,y)]$$</span></p>
<blockquote>
<p>The inner limit is unknown to us
as we donot know about the nature of <span class="math-container">$x$</span>.</p>
</blockquote>
<p>This is false. We do know the nature of <span class="math-container">$x$</span>: it is eithe... |
492,125 | <p>How many different equivalence relations can be defined on a set of five elements?</p>
| user248070 | 248,070 | <p>The formula is $2^n-n$ where $n$ denotes the number of elements. So here, the answer would be $2^5-5= 32-5 = 27$.</p>
|
4,106,575 | <p>Consider a number drawn from <span class="math-container">$U(1,100)$</span>. When we make an incorrect guess, we are told whether the target number is smaller or larger. So we employ a binary search approach, where the first guess is 50. If that is not the target, then that means our next guess is going to be 25 or ... | Ahmad Beirami | 915,095 | <p>The correct answer is <span class="math-container">$0.02$</span>, and your second approach is correct.</p>
<p>Here is how you can make your first approach work:
You should multiply 2/99 (probability of ending at the second round given you did not end in the first round -- see below) by 99/100 (probability of not end... |
1,422,859 | <p>$$\sqrt{1000}-30.0047 \approx \varphi $$
$$[(\sqrt{1000}-30.0047)^2-(\sqrt{1000}-30.0047)]^{5050.3535}\approx \varphi $$
Simplifying Above expression we get<br>
$$1.0000952872327798^{5050.3535}=1.1618033..... $$
Is this really true that
$$[\varphi^2-\varphi]^{5050.3535}=\varphi $$</p>
| Vincenzo Oliva | 170,489 | <p>No, not at all. Factoring and then using a well known property of $\varphi$: $$\left(\varphi(\varphi-1)\right)^{5050.3535}=\left(\frac{\varphi}{\varphi}\right)^{5050.3535}=1\ne\varphi. $$</p>
<p>The argument needs to be a bit larger than $1$. You're dealing with an approximation.</p>
|
1,037,621 | <p>There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does matter who has the white pieces (and who has the black ones). </p>
<p>I thought it might be $$\large2^{\frac{20(20-1)}2}$... | Suzu Hirose | 190,784 | <p>If we arrange the players in some order, say of height, and let the shortest player choose first, then there are 19 people for him to choose from. Then let the next shortest remaining player choose, and there are 17 people left to choose, and so on. Thus the number of ways to choose pairs is
$$
19\times17\times15\ti... |
2,387,435 | <p><a href="https://i.stack.imgur.com/RxHmS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RxHmS.png" alt="enter image description here"></a></p>
<p>We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and... | Siong Thye Goh | 306,553 | <p>I will leave $R_y(\theta)$ as an exercise for you. </p>
<p>Below is a verification of $R_x(\theta)$ is a rotation about the $x$-axis.</p>
<p>Denote the Pauli matrices as
$$X=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, Y=\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, Z=\begin{bmatrix} 1 & 0 \\... |
2,387,435 | <p><a href="https://i.stack.imgur.com/RxHmS.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/RxHmS.png" alt="enter image description here"></a></p>
<p>We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and... | Frunobulax | 93,252 | <p>I'm a bit late to the party, but here's another way to prove that <span class="math-container">$R_x(\alpha)$</span> is a rotation about the <span class="math-container">$x$</span> axis on the Bloch sphere.</p>
<p>First, consider the following way to map every vector <span class="math-container">$\mathbf v$</span> of... |
3,397,834 | <blockquote>
<h2><span class="math-container">$$49y″−98y′+48y= 0 \quad\quad\quad \,\, y(2)=3,y′(2)=9.$$</span></h2>
</blockquote>
<p>When I solved, I got that my <span class="math-container">$r_1= \frac67$</span> and <span class="math-container">$r_2= \frac87.$</span> Then I got that <span class="math-container">$y... | Dr. Sonnhard Graubner | 175,066 | <p>we get for <span class="math-container">$$y(x)\neq 0$$</span> <span class="math-container">$$y'(x)=-\frac{x^4}{y^4}$$</span> so
<span class="math-container">$$y''(x)=-\frac{4x^3y^4-x^44y^3y'}{y^8}$$</span>
plugging <span class="math-container">$$y'=-\frac{x^4}{y^4}$$</span> we get
<span class="math-container">$$y''=... |
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