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 |
|---|---|---|---|---|
3,100,420 | <p>So my previous question states that: We are given the sequence <span class="math-container">$_{n}= 6^{({2}^n)} + 1$</span>. We must prove that the elements of this sequence are pairwise co-prime, i.e prove that if m ≠ n then <span class="math-container">$(_{m},_{n}) = 1$</span>.</p>
<p>I previously proved that <spa... | Bill Dubuque | 242 | <p><span class="math-container">${\bf Hint}\rm\quad\ \ \gcd(a+1,\,\ a^{\large 2K}\!+1)\ =\ gcd(a+1,\ \ \color{#0a0}2)\,\ $</span> by the Euclidean algorithm </p>
<p><span class="math-container">${\bf Proof}\rm\ \ mod\,\ a+1\!:\,\ \color{#c00}a^{\large 2K}\!+1 \equiv (\color{#c00}{-1})^{\large 2K}\!\!+1\equiv\color{#... |
873,582 | <p>How check that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $?</p>
| Hagen von Eitzen | 39,174 | <p>If we abbreviate $w=\sqrt[3]2$, the left hand side is $L=\frac1{\sqrt[3]9}(1-w+w^2)$. From $(1+w)(1-w+w^2)=1+w^3=3$ we see that $L=\frac3{\sqrt[3]9(1+\sqrt[3]2)}$, hence
$$ L^3=\frac{27}{9(1+w)^3}=\frac3{1+3w+3w^2+w^3}=\frac1{1+w+w^2}$$
As above, note that $(1+w+w^2)(w-1)=w^3-1=1$, hence
$$ L^3=w-1=R^3.$$</p>
|
175,775 | <p>To clarify the terms in the question above:</p>
<p>The symmetric group Sym($\Omega$) on a set $\Omega$ consists of all bijections from $\Omega$ to $\Omega$ under composition of functions. A generating set $X \subseteq \Omega$ is minimal if no proper subset of $X$ generates Sym($\Omega$).</p>
<p>This might be a dif... | bof | 43,266 | <p>Jeremy Rickard's answer uses the fact that a symmetric group is not the union of a countable chain of proper subgroups. The following easy proof of that fact is quoted from Fred Galvin, <em>Generating countable sets of permutations</em>, J. London Math. Soc. (2) 51 (1995), 230-242.</p>
<blockquote>
<p>[. . .] per... |
348,395 | <p>If <span class="math-container">$f$</span> a distribution with compact support then they exist <span class="math-container">$m$</span> and measures <span class="math-container">$f_\beta$</span>,<span class="math-container">$|\beta|\leq m$</span> such that
<span class="math-container">$$f=\sum_{|\beta|\leq m}\fra... | reuns | 84,768 | <ul>
<li><p>If <span class="math-container">$f\in D'(\Bbb{R})$</span> is a distribution with compact support <span class="math-container">$\subset (a,b)$</span> then it has finite order <span class="math-container">$m$</span>. Proof : assume it doesn't, for each <span class="math-container">$k$</span> take <span class=... |
4,274,044 | <p>I am trying to prove the following result.</p>
<blockquote>
<p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be sets and <span class="math-container">$\sim$</span> an equivalence relation on <span class="math-container">$X$</span>. Let <span class="math-container">$f: X \to ... | José Carlos Santos | 446,262 | <p>If you define <span class="math-container">$\tilde f\colon X/\sim\longrightarrow Y$</span> by <span class="math-container">$\tilde f\bigl([x]\bigr)=f(x)$</span>, does this make sense? That is, if <span class="math-container">$[x]=[y]$</span>, do we have <span class="math-container">$\tilde f\bigl([x]\bigr)=\tilde f\... |
2,078,264 | <p>I've been see the following question on group theory:</p>
<p>Let $p$ be a prime, and let $G = SL2(p)$ be the group of $2 \times 2$ matrices of determinant $1$ with entries in the field $F(p)$ of integers $\mod p$.</p>
<p>(i) Define the action of $G$ on $X = F(p) \cup \{ \infty \}$ by Mobius transformations. [You nee... | MvG | 35,416 | <p>Use homogeneous coordinates. A homogeneous coordinate vector $[x:y]$ represents the number $x/y$. The vector $[x:0]$ for any $x\neq 0$ represents infinity. Conventionally one would write infinity as $[1:0]$ but the choice of representative is arbitrary.</p>
<p>Here is how the matrix operation relates to the Möbius ... |
1,166,524 | <p>I am looking for all irreducible polynomials of degree 5 in $\mathbb{F}_{17}$
with have the form h(y) = $y^5+C$.</p>
<p>I think there aren't any irreducible polynomials of this form because for every C
I can find an element of $\mathbb{F}_{17}$ as a root.</p>
<p>What would be the fastest way to find irreducible po... | MooS | 211,913 | <p>The unit group has 16 elements, hence any element $a$ has order relatively prime to $5$. In particular $a$ and $a^5$ have the same order in the cyclic group generated by $a$. We obtain $a \in \langle a \rangle = \langle a^5 \rangle$.</p>
<p><strong>Edit</strong> There would be the following approch to easily find s... |
1,133,581 | <p>$\lim\limits_{(x,y) \to (0,0)} \frac{\log(1-x^2y^2)}{x^2y^2}$</p>
<p>in polar coordinates:</p>
<p>$\lim\limits_{r \to 0} \frac{\log(1-r^2\cos^2t \sin^2t)}{r^2\cos^2t \sin^2t}$</p>
<p>The first limit exists if the second is independent from $t$. But if $t$ is $k \pi/2$ that fraction does not exist. So I could argu... | parsiad | 64,601 | <p>Think about...
$$
\bigcup_{n=1}^{\infty}\left(0+\frac{1}{n},1\right)
$$</p>
|
1,593,433 | <p>How do I define the integral of this Riemann sum?
$$\lim\limits_{m\to\infty}\frac{1}{m}\sum_{n=1}^m\cos\left({\frac{2\pi n x}{b}}\right)$$</p>
| Nate 8 | 226,768 | <p>If we replaced b with m, then indeed it would reflect a Riemann sum (though the expression you present does not).</p>
<p>You can tell, because in a Riemann sum, you are adding up the areas of a series of rectangles. To find the area of each rectangle, multiply that rectangle's height by its witdh. To find the total... |
42,016 | <p>I am looking for algorithms on how to find rational points on an <a href="http://en.wikipedia.org/wiki/Elliptic_curve">elliptic curve</a> $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, how to find solutions in which the numerators and denominator... | Felipe Voloch | 2,290 | <p>There are clever ways of speeding the brute force enumeration. An implementation of such is M. Stoll's ratpoints program:
<a href="http://www.mathe2.uni-bayreuth.de/stoll/programs/index.html" rel="noreferrer">http://www.mathe2.uni-bayreuth.de/stoll/programs/index.html</a></p>
<p>A completely different way of genera... |
42,016 | <p>I am looking for algorithms on how to find rational points on an <a href="http://en.wikipedia.org/wiki/Elliptic_curve">elliptic curve</a> $$y^2 = x^3 + a x + b$$ where $a$ and $b$ are integers. Any sort of ideas on how to proceed are welcome. For example, how to find solutions in which the numerators and denominator... | David E Speyer | 297 | <p>The best practical solution is to have someone else do the work. You can look up the curve in <a href="http://www.warwick.ac.uk/~masgaj/ftp/data/">Cremona's tables</a>, if it is not too large. If it is larger than that, you can use <a href="http://www.warwick.ac.uk/staff/J.E.Cremona//mwrank/">mwrank</a>, a free sta... |
202,043 | <p>A <a href="http://en.wikipedia.org/wiki/Square-free_word">square-free word</a>
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free word has length $3$.
But for alphabets of three symbols,... | echinodermata | 46,975 | <p><em>Note: This is an extended comment, not a full answer.</em></p>
<p>I don't know about you, but I didn't start out with any good intuition about the answer to the question. I decided to do some computer experiments/visualization.</p>
<p>I wrote some code in Prolog which generates square-free and three-halves-fre... |
202,043 | <p>A <a href="http://en.wikipedia.org/wiki/Square-free_word">square-free word</a>
is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any
consecutive sequence of symbols in the string.
For alphabets of two symbols, the longest square-free word has length $3$.
But for alphabets of three symbols,... | user38477 | 38,477 | <p><strike>(Not a real answer, just a conjecture)</strike></p>
<p>Let <span class="math-container">$w$</span> be the Pansiot word on <span class="math-container">$4$</span> letters defined here :<br>
<a href="https://www.sciencedirect.com/science/article/pii/0166218X84900064" rel="nofollow noreferrer">J.J. Pansiot, A ... |
3,165,276 | <p>I am attempting to prove if the limit for <span class="math-container">$S_n = \frac{\sqrt{n}}{\sqrt{n}+1} $</span>, exists, and if it exists, what it is?
If the limit exists, proving it in terms of the definition for <span class="math-container">${e}$</span> and <span class="math-container">${N}$</span>.</p>
<p>I g... | Kavi Rama Murthy | 142,385 | <p>You have made a mistake in your calculations. Instead of <span class="math-container">$|-2|$</span> in the numerator you should get <span class="math-container">$|1-\sqrt n|$</span>. Of course, you mis-typed <span class="math-container">$n-1$</span> as <span class="math-container">$n-n$</span> at the end. Hint for c... |
68,899 | <p>I was rereading basic results on de Rham cohomology, and this led me inevitably to the fact that $H^q(X,\Omega^p)$ converges to $H^*(X)$ for any smooth proper variety (over any field). How does one view this spectral sequence "maturely" as a Grothendieck spectral sequence?</p>
| Torsten Ekedahl | 4,008 | <p>If by "Grothendieck spectral sequence" you mean the spectral sequence associated to the composite of functors (fulfilling the Grothendieck condition) then I am skeptical as to whether this is possible. Also I do not see that there would be any particular point in being able to view it in that light (unless the funct... |
2,263,431 | <p>Let $$v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Then find difference between maximum and minimum of $v^2$.</p>
<p>I understand both of them are distance of a point on ellipse from origin, but how do we find maximum and minimum?</p>
<p>I tried guessing, and got maximum $v$ when $... | Michael Rozenberg | 190,319 | <p>By C-S we obtain:
$$v=$$
$$=\sqrt{a^2\cos^2x+b^2\sin^2x+b^2\cos^2x+a^2\sin^2x+2\sqrt{(a^2\cos^2x+b^2\sin^2x)(b^2\cos^2x+a^2\sin^2x)}}=$$
$$=\sqrt{a^2+b^2+2\sqrt{(a^2\cos^2x+b^2\sin^2x)(b^2\cos^2x+a^2\sin^2x)}}\geq$$
$$\geq\sqrt{a^2+b^2+2|ab|(\cos^2x+\sin^2x)}=|a|+|b|.$$
$$v\leq\sqrt{2(a^2\cos^2x+b^2\sin^2x+b^2\cos^2... |
2,263,431 | <p>Let $$v=\sqrt{a^2\cos^2(x)+b^2\sin^2(x)}+\sqrt{b^2\cos^2(x)+a^2\sin^2(x)}$$</p>
<p>Then find difference between maximum and minimum of $v^2$.</p>
<p>I understand both of them are distance of a point on ellipse from origin, but how do we find maximum and minimum?</p>
<p>I tried guessing, and got maximum $v$ when $... | egreg | 62,967 | <p>We can restrict to $x\in[0,\pi/2]$, because of symmetries.</p>
<p>The maximum and minimum of $v$ are attained at the same values as the maximum and minimum of $v^2$:
$$
v^2=a^2+b^2+
2\sqrt{(a^2\cos^2(x)+b^2\sin^2(x))(b^2\cos^2(x)+a^2\sin^2(x))}
$$
We can also remove $a^2+b^2$, then the factor $2$ and square again, ... |
1,602,078 | <p>There are $n$ prisoners and $n$ hats. Each hat is colored with one of $k$ given colors. Each prisoner is assigned a random hat, but the number of each color hat is not known to the prisoners. The prisoners will be lined up single file where each can see the hats in front of him but not behind. Starting with the pris... | Asinomás | 33,907 | <p>Label the colors $\{1,2,3\dots k\}$. The first one says the sum of the hats in front of him $\bmod k$ (the last $n-1$ persons). After this the second one can deduce which number corresponds to the color of his hat (by subtracting the sum that he can see minus the sum previously said).</p>
<p>The third person, havin... |
2,940,287 | <p>My ring theory teacher has a very broad definition of rings: he doesn't require them to be associative. As such, he told us to work out a definition of a product operation <span class="math-container">$\cdot$</span> on <span class="math-container">$R = \mathbb{Z}_2 \times \mathbb{Z}_2$</span> that is distributive ov... | Upstart | 312,594 | <p>Let <span class="math-container">$b\in im(T-I)\implies \exists ~x\in R^n$</span>such that <span class="math-container">$(T-I)x=b$</span>.
Now consider<span class="math-container">$Tb= T(T-I)x=(T^2-T)x= 0$</span>. Hence <span class="math-container">$Tb=0$</span>. Thus <span class="math-container">$b \in Nul(T)$</span... |
2,889,482 | <blockquote>
<p><strong>1.28</strong>
In Section 1.6, we introduced the idea of entropy $h(x)$ as the information gained on observing the value of a random variable $x$ having distribution $p(x)$.
We saw that, for independent variables $x$ and $y$ for which $p(x,y) = p(x) p(y)$, the entropy functions are additive... | Javi | 554,663 | <p>After some hours of research I've found a few sites which altogether answer these questions.</p>
<p>Regarding items 1 and 2, it looks like there is indeed a severe abuse of notation every time the author refers to function $h$. This function seems to be the so-called <em>self-information</em> and it is usually defi... |
388,815 | <p>I am trying to compute the integral: $$\int_{4}^{5} \frac{dx}{\sqrt{x^{2}-16}}$$
The question is related to hyperbolic functions, so I let $x = 4\cosh(u)$ therefore the integral becomes: $$-\int_{0}^{\ln(2)}\frac{4\sinh(u)}{\sqrt{16-16\cosh^{2}(u)}}du = -\int_{0}^{\ln(2)}1du = -\ln(2)$$</p>
<p>The answer is $\ln(2)... | iostream007 | 76,954 | <p>use this formula
$$\int \frac{dx}{\sqrt{x^{2}-a^2}}=\log|x+\sqrt{x^2-a^2}|+c$$
$$\int_{4}^{5} \frac{dx}{\sqrt{x^{2}-16}}$$
$$\int_{4}^{5} \frac{dx}{\sqrt{x^{2}-4^2}}$$
$$(\log|x+\sqrt{x^2-16}|)_4 ^5$$
$$\log|8|-\log|4|$$
$$\log2$$</p>
|
4,064,618 | <p><span class="math-container">$10$</span> individuals including <span class="math-container">$x$</span> and <span class="math-container">$y$</span> seat at random at a round table. All possible orders are equally likely. Let <span class="math-container">$W$</span> be the event that there are exactly two people seated... | miracle173 | 11,206 | <p>Hint:
A valid placement is</p>
<pre><code>... i-1 i i+1 i+2 i+3 i+4 ...
... a X b c Y d ...
</code></pre>
<p>another is</p>
<pre><code>... i-1 i i+1 i+2 i+3 i+4 ...
... a Y b c X d ...
</code></pre>
<p>where <span class="math-container">$10$</span> must be subt... |
609,744 | <p>I am not sure about the last step of my proof:</p>
<p>$(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). $C[a,b]$ is the space of continuous functions on [a,b].</p>
<p>Main goal in order to prove the claim is that the in... | David Holden | 79,543 | <p>set $y=kx$ for $k \gt 1$, so you wish to show that if $x \gt e$ then
$$
x^{kx} \gt (kx)^x
$$
i.e. $x \gt e$ implies
$$x^{k-1} \gt k
$$
so it is sufficient to show that for $k \gt 1$ we have
$$
k \ge 1 + ln \; k
$$
so can you find a convexity proof for this?</p>
|
1,770,508 | <p>I am confronted with the following definition:</p>
<blockquote>
<p>Let <span class="math-container">$K$</span> be a field and <span class="math-container">$e_1,e_2,\ldots,e_n$</span> the standard basis of the <span class="math-container">$K$</span> vector space <span class="math-container">$K^n$</span>.</p>
<p>For <... | Felix Marin | 85,343 | <p>$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 ... |
1,770,508 | <p>I am confronted with the following definition:</p>
<blockquote>
<p>Let <span class="math-container">$K$</span> be a field and <span class="math-container">$e_1,e_2,\ldots,e_n$</span> the standard basis of the <span class="math-container">$K$</span> vector space <span class="math-container">$K^n$</span>.</p>
<p>For <... | Gathdi | 301,858 | <p>Cylindrical coordinates is polar coordinates but in 3D. First you have to find projection in XY Plane which can be done so by eliminating z from above equations. Doing so you will get $$(x-\frac{a}{3})^2 - (\frac{4y}{\sqrt 3})^2 = (\frac{2a}{3})^2$$</p>
<p>Also your Z limits will be $ \sqrt{\frac{r^2cos^2\theta}{4}... |
2,590,537 | <p>Let $A$ be a proper subset of of $X$, $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X\times Y)-(A\times B)$ is connected.</p>
<p>I have already seen the solution of this problem here on math.stackexchange, but my confusion is this.
Suppose $X$=(a,b) & $Y$=(c,d) and let $A$={m} &... | Gödel | 467,121 | <p>You can see a $n$ dimensional vector with binary inputs as the characteristic function the subsets of a set of size $n$, for example, $\{0,1,...,n-1\}$.</p>
<p>This is, you know that $\{0,1,3\}\subseteq\{0,1,...,n-1\}$ if $n\geq 5$, so, the vector $(1,1,0,1,0,...,0)$ represents the characteristic function of $\{0,... |
1,852,664 | <p>I'm working through Mumford's Red Book, and after introducing the definition of a sheaf, he says "Sheaves are almost standard nowadays, and we will not develop their properties in detail." So I guess I need another source to read about sheafs from. Does anybody know of any expository papers that cover them? I'd pref... | carmichael561 | 314,708 | <p>One relatively old but classic reference is Serre's paper "Faisceaux Algébriques Cohérents", often referred to simply as FAC. English translations can be found fairly easily online (see for instance <a href="https://mathoverflow.net/questions/14404/serres-fac-in-english">here</a>).</p>
<p>Another reference is the b... |
1,852,664 | <p>I'm working through Mumford's Red Book, and after introducing the definition of a sheaf, he says "Sheaves are almost standard nowadays, and we will not develop their properties in detail." So I guess I need another source to read about sheafs from. Does anybody know of any expository papers that cover them? I'd pref... | Armando j18eos | 224,305 | <p>I suggest Mumford and Oda - <em>Algebraic Geometry II</em> <a href="http://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf" rel="nofollow">(a penultimate draft)</a>, the appendix at the first chapter; so one finds again the style of Mumford.</p>
|
400,296 | <p>There are n persons.</p>
<p>Each person draws k interior-disjoint squares.</p>
<p>I want to give each person a single square out of his chosen k, so that the n squares I give are interior-disjoint.</p>
<p>What is the minimum k (as a function of n) for which I can do this?</p>
<p>NOTES:</p>
<ul>
<li>For n=1, obv... | Erel Segal-Halevi | 29,780 | <p>I found an upper bound for the axis-aligned version. It is based on the following claim:</p>
<p><strong>Claim 2</strong>: a square A can intersect at most 4 interior-disjoint squares that are as large or larger that A.</p>
<ul>
<li>Proof: If A intersects B and B is as large or larger than A, then at least one corn... |
38,367 | <p>Notation: Let$M$ be a smooth, closed manifold, $S$ any submanifold of $M$, $Diff(M)$ the group of diffeomorphisms of $M$ and $Imb(S, M)$ the group of smooth imbeddings of $S$ into $M$.</p>
<p>A classical result of R. Palais from the 1960 paper <em>Local triviality of the restriction map for embeddings</em> says tha... | Oscar Randal-Williams | 318 | <p>This is closely related to (and can be proved by the same methods as) the fact that if we fix another manifold $N$ then the restriction map
$$\mathrm{Emb}(M, N) \to \mathrm{Emb}(S, N)$$
is a locally trivial fibration, which was proved in the same paper by Palais. However, I wanted to advertise a geodesic proof of th... |
307,287 | <p>According to <a href="https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem" rel="nofollow noreferrer">Brouwer's fixed point theorem</a>, for compact convex $K\subset\mathbb{R}^n$, every continuous map $K\rightarrow K$ has a fixed point.</p>
<p>However, these fixed points cannot be chosen continuously, even for... | Arno | 15,002 | <p>The problem here is not restricted to obtaining continuous choice functions, it already fails at the level of continuous multivalued functions.</p>
<p><strong>Theorem</strong> The multivalued function $\mathrm{BFT}_k : \mathcal{C}([0,1]^k,[0,1]^k) \rightrightarrows [0,1]^k$ mapping a continuous function to some arb... |
286,852 | <p>I'm currently trying to understand time continuous Fourier transformations for my Signal Analysis class (Electrical Engineering major). In my Textbook we have a set of "useful FT-pairs", one of which being
$x(t)<=FT=>X(\omega)$</p>
<p>$\sin(\omega_0*t)<=>i\pi[\delta(\omega+\omega_0)-\delta(\omega-\omega... | not all wrong | 37,268 | <p>Here are two ways to look at it.</p>
<ul>
<li>Assume that whatever mathematical trickery is used to define it (distribution theory) the result should be consistent with the inverse Fourier transform formula, and $$\int_{-\infty}^{\infty} dk \delta(k-k_0) \exp(ikx)\frac{1}{2\pi} = \exp(ik_0x)\frac{1}{2\pi}$$</li>
<l... |
1,926,593 | <p>I am currently taking a course in Discrete Math. The first part of our lesson this week is regarding sequences. I am stuck on formulas like the ones shown in the images I attached... I was hoping someone might be able to help me learn how to solve them. :)</p>
<p>Ps: What does it mean when <span class="math-containe... | ClownInTheMoon | 367,034 | <p>Here is how you prove this via Induction. Let me know if you have questions on a particular step. Under the inductive step it's basically:</p>
<p>1) Perform induction on $n$. You could substitute $k$ for $n$ if you'd like but I did not.</p>
<p>2) Apply Pascal's Identity to split the sum into two sums.</p>
<p>3) F... |
374,694 | <p>I believe I'm looking for a function: $f(x) : \mathbb N \mapsto \mathbb N^2$ and it's inverse $f^{-1}(x) : \mathbb N^2 \mapsto \mathbb N$, a known mapping that can take any positive integer and map to a unique 2D integer point (and the inverse mapping as well). </p>
<p>Like the titles says, the conceptual idea is ... | long tom | 47,794 | <p>Some possibilities as described <a href="http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel#Infinitely_many_coaches_with_infinitely_many_guests_each" rel="nofollow">here</a>:</p>
<ul>
<li>Prime method: $(x,y)\mapsto p_x^y$ where $p_x$ is the $x$th prime number</li>
<li>merge method: consider $(x,y)... |
165,382 | <p>Is there any number $a+b\sqrt{5}$ with $a,b \in \mathbb{Z}$ with norm (defined by $|a^2−5b^2|$) equal 2?</p>
| Arturo Magidin | 742 | <p>No. </p>
<p>Since $a^2-5b^2\equiv a^2\pmod{5}$, the values of $a^2-5b^2$ must be congruent to either $0$, $1$, or $4$ modulo $5$, since those are the only squares modulo $5$. But neither $2$ nor $-2$ satisfy this condition, so it is impossible to have $a^2-5b^2=2$ or $a^2-5b^2=-2$ with $a,b\in\mathbb{Z}$.</p>
|
41,000 | <p>The motivation for this question is producing a distribution that produces the gender and age of an individual when the distribution of ages depends on gender.</p>
<p>Suppose I want a distribution which, when RandomVariate is applied to it, produces two values. The first value it produces is either 0 or 1 and the ... | Sjoerd C. de Vries | 57 | <p>I believe <code>TransformedDistribution</code> can do the job. I'll do the conditional part first and finish with producing the number pairs asked for.</p>
<p>The trick is to let the transformation function contain the conditional using <code>Boole</code>:</p>
<pre><code>dist[μm_, σm_, μf_, σf_, p_] =
Transformed... |
1,509,502 | <p>I am trying to prove the following statement:</p>
<blockquote>
<p>Suppose <span class="math-container">$\mu$</span> and <span class="math-container">$\nu$</span> are finite measures on the measurable space <span class="math-container">$(X,\mathcal A)$</span> which have the same null sets. Show that there exists a me... | Zhanxiong | 192,408 | <p><strong>Hint:</strong> Since $\nu$ and $\mu$ have the same null sets, $\mu \ll \nu$, by R-N theorem, there exists $f$ with required properties such that
$$\nu(A) = \int_A f d \mu.$$
Therefore $d\nu = f d \mu$
Now evaluate $\int_A \frac{1}{f} d \nu$ as follows
$$\int_A \frac{1}{f} d \nu = \int_A \frac{1}{f} f d\mu = ... |
1,648,354 | <p>We have been taught that linear functions, usually expressed in the form $y=mx+b$, when given a input of 0,1,2,3, etc..., you can get from one output to the next by adding some constant (in this case, 1).
$$
\begin{array}{c|l}
\text{Input} & \text{Output} \\
\hline
0 & 1\\
1 & 2\\
2 & 3
\end{array}
... | Eleven-Eleven | 61,030 | <p>Note that your table can be written as</p>
<p>$$
\begin{array}{c|l}
\text{Input} & \text{Output} \\
\hline
1 & 2^{2^{1-1}}\\
2 & 2^{2^{2-1}}\\
3 & 2^{2^{3-1}}\\
4 & 2^{2^{4-1}}
\end{array}
$$</p>
<p>Tetration is the operation you could say that succeeds exponentiation. As an example, notationa... |
679,712 | <p>Find the volume bounded by the cylinder $x^2 + y^2=1$ and the planes $y=z , x=0 ,z=0$ in the first octant.</p>
<p>How do I go about doing this?</p>
| Américo Tavares | 752 | <p>In general the volume of a region $S$ bounded above by a surface defined by a function of two variables $z=f(x,y)\ge 0$, whose domain is $R$, and below by the plane $z=0$, is given by:</p>
<ul>
<li>the double integral $\iint_R f(x,y)\,dA=\iint_R f(x,y)\,dx\,dy$, </li>
<li>or by the triple integral $\iiint_S \,d... |
2,447,677 | <p>Consider the parametric equations given by \begin{align*} x(t)&=\sin{t}-t,\\ y(t) & = 1-\cos{t}.\end{align*}</p>
<p>I want to write these parametric equations in Cartesian form. </p>
<p>In order to eliminate the sine and cosine terms I think I probably need to consider some combination of $x(t),y(t), x(t)... | José Carlos Santos | 446,262 | <p>From the second equality, you'll get $t=\arccos(1-y)$. So,\begin{align}x&=\sin\bigl(\arccos(1-y)\bigr)-\arccos(1-y)\\&=\sqrt{1-(1-y)^2}-\arccos(1-y)\\&=\sqrt{2y-y^2}-\arccos(1-y).\end{align}</p>
|
1,570,983 | <p>The volume of a sphere with radius $r$ is given by the formula $V(r) = \frac{4 \pi}{3} r^3$.</p>
<p>a) If $a$ is a given fixed value for $r$, write the formula for the linearization of the volume function $V(r)$ at $a$.</p>
<p>b) Use this linearization to calculate the thickness $\Delta r$ (in $cm$) of a layer of ... | Narasimham | 95,860 | <p>For second part by logarithmic differentiation of volume/radius</p>
<p><span class="math-container">$$\dfrac{\Delta V}{V}=3\dfrac{\Delta r}{r}$$</span></p>
<p><span class="math-container">$$\dfrac{340}{\dfrac{4\pi}{3} 52^3}=3\dfrac{\Delta r}{52}$$</span></p>
<p>from which we calculate the paint thickness <span class... |
1,885,434 | <p>I'm currently reading through <em>Introductory Discrete Mathematics</em> by V.K. Balakrishnan and came across the following theorem:</p>
<p>If $X$ is a set of cardinality $n$, then the number of $r$-collections from $X$ is $\binom{r + n - 1}{n - 1}$, where $r$ is any positive integer.</p>
<p>To me, it seems like t... | MarnixKlooster ReinstateMonica | 11,994 | <p>Your proof is correct. However, it is a proof that contains a 'rabbit' (see the introduction of <a href="http://www.cs.utexas.edu/users/EWD/transcriptions/EWD13xx/EWD1300.html" rel="nofollow">EWD1300</a>): it introduces the expression $\;(b+a)(b−a)+c(b−a)\;$ as a complete surprise to the reader. That makes the pro... |
165,154 | <p>a) Let $\,f\,$ be an analytic function in the punctured disk $\,\{z\;\;;\;\;0<|z-a|<r\,\,,\,r\in\mathbb R^+\}\,$ . Prove that if the limit $\displaystyle{\lim_{z\to a}f'(z)}\,$ exists finitely, then $\,a\,$ is a removable singularity of $\,f\,$</p>
<p><strong>My solution and doubt:</strong> If we develop $\,f... | Rajesh D | 2,987 | <p>If you are passing the signal through an ideal low pass filter, then Fourier series coefficients of the filtered signal above $4$kHz vanish that is $a_k = 0 \forall k \ge \frac{4000T}{2\pi}$ and the rest of the coefficients remain same. Here $T$ is the period of the square wave. </p>
<p>Then what is the bandwidth? ... |
961,603 | <p>Show that every positive integer $N$ less than or equal to $n$ factorial, is the sum of at most $n$ distinct positive integers, each of which divides $n!$.</p>
| André Nicolas | 6,312 | <p>We deal with the induction step. Suppose that the result holds when $n=k$. We show the result holds when $n=k+1$. So we want to show that every positive integer $N\le (k+1)!$ is representable as a sum $y_1+\cdots +y_t$, where $t\le k+1$, and where the $y_j$ are distinct divisors of $(k+1)!$.</p>
<p>If $N$ is divisi... |
1,782,800 | <p>I am studying for an exam and found this in the solutions:</p>
<blockquote>
<p>"... with canonical parameters $\{\log\lambda_i\}^n_1$". </p>
</blockquote>
<p>Index $I=1,2,\ldots,n$. The professor that wrote the exam has retired. Does anyone know what the $1$ and $n$ might mean?</p>
| user332239 | 332,239 | <p>This usually indicates the range of your index parameter, $i$. Sometimes you will see it written more explicitly as
$$\{A_i\}_{i=1}^n.$$
For more general index sets you may see something like
$$\{A_i\}_{i \in I}.$$
This is useful if your index set is uncountable, for instance.</p>
|
163,889 | <p>Find a closed form for $a_n:=\sum_{k=0}^{n}\binom{n}{k}(n-k)^n(-1)^k$ using generating functions.</p>
| user26872 | 26,872 | <p>We have
$$\begin{eqnarray*}
\sum_{k=0}^{n}(-1)^k \binom{n}{k}(n-k)^n
&=& \sum_{k=0}^{n}(-1)^k \binom{n}{n-k}(n-k)^n \\
&=& \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^n \\
&=& \left.\sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} (x D)^n x^k\right|_{x=1} \\
&=& \left.(x D)^n \sum_{k=0}^{n} (-1)... |
2,233,457 | <p>I'm working on the following integral:</p>
<p>$\int_0^{\infty}\frac{y^r}{\theta}ry^{r-1}e^{\frac{-y^r}{\theta}} dy$. </p>
<p>I noticed that if I took the derivative:</p>
<p>$\frac{d}{dy}e^{\frac{-y^r}{\theta}} = \frac{-ry^{r-1}}{\theta}e^{\frac{-y^r}{\theta}}$ </p>
<p>I got most of the expression of the integran... | shrinklemma | 67,953 | <p>Here is an argument independent of the smooth structure of $M$. </p>
<p>Let $M$ be $n$-dimensional ($n\geq 2$) and pick any $p\in M$. If $f:M\to\mathbb{R}$ is one-to-one, then $f$ restricted on a neighborhood of $p$, is also one-to-one, and we may assume that $f$ is now defined on $\mathbb{R}^n$ by a neighborhood ... |
4,219,193 | <p>The <code>p implies q</code> statement is often described in various ways including:<br />
(1) <code>if p then q</code> (i.e. whenever p is true, q is true)<br />
(2) <code>p only if q</code> (i.e. whenever q is false, p is false)</p>
<p>I see the truth table for (1) as</p>
<pre><code>p | q | if p then q
-----------... | Isaiah | 215,257 | <p>"If P then Q" means that whenever P is true, Q is true as you have observed. Thus, if both are true, it follows that the statement as a whole is true. Now, what if P is true, but Q is false? What does this imply about the statement as a whole? In this case, it is <strong>not</strong> true that "if P i... |
4,219,193 | <p>The <code>p implies q</code> statement is often described in various ways including:<br />
(1) <code>if p then q</code> (i.e. whenever p is true, q is true)<br />
(2) <code>p only if q</code> (i.e. whenever q is false, p is false)</p>
<p>I see the truth table for (1) as</p>
<pre><code>p | q | if p then q
-----------... | hamam_Abdallah | 369,188 | <p><span class="math-container">$ p\implies q $</span> can be seen as an argument <span class="math-container">$ p \therefore q$</span>.
If the argument is valid and <span class="math-container">$ p $</span> is true, then we are sure that <span class="math-container">$ q $</span> is also true.</p>
<p>If the argument is... |
4,219,193 | <p>The <code>p implies q</code> statement is often described in various ways including:<br />
(1) <code>if p then q</code> (i.e. whenever p is true, q is true)<br />
(2) <code>p only if q</code> (i.e. whenever q is false, p is false)</p>
<p>I see the truth table for (1) as</p>
<pre><code>p | q | if p then q
-----------... | mohottnad | 955,538 | <p>After practicing filling truth table and gaining logic terminologies, the natural language intuition for "if p then q" is generally that p is a <em>sufficient</em> condition of q, while for "p only if q" q is a <em>necessary</em> condition for p. With these intuitions you can usually find answers... |
533,628 | <p>I'm learning how to take indefinite integrals with U-substitutions on <a href="https://www.khanacademy.org/math/calculus/integral-calculus/u_substitution/v/u-substitution" rel="nofollow">khanacademy.org</a>, and in one of the videos he says that: $$\int e^{x^3+x^2}(3x^2+2x) \, dx = e^{x^3+x^2} + \text{constant}$$
I ... | Tacet | 186,012 | <p>Take $b_n = n$ and $c_n = \sum_{i = 1}^{n} a_i$
$$
\lim_{n\to\infty} \frac{c_{n+1} - c_n}{b_{n+1} - b_n} = \lim_{n\to\infty} \frac{a_{n+1}}{1} = a
$$</p>
<p>$s_n = \frac{c_n}{b_n}$ and $b_n \to \infty $ and above limit exists, so <a href="https://en.wikipedia.org/wiki/Stolz%E2%80%93Ces%C3%A0ro_theorem" rel="nofollo... |
2,411,081 | <p>How can I show (with one of the tests for convergence , <strong>not by solving</strong>) that the integral $$\int _{1}^\infty\frac{\ln^5(x)}{x^2}dx$$ converges?</p>
| Michael Burr | 86,421 | <p>If you really meant:
$$
\int_1^\infty \frac{\ln(x)^5}{x^2}dx,
$$
then observe that for $x$ sufficiently large, $\frac{\ln(x)^5}{\sqrt{x}}$ is less than $1$. In other words, for some $c>1$, $\ln(x)^5<\sqrt{x}$ for all $x>c$. Therefore, the tail of this integral is bounded above by
$$
\int_c^\infty\frac{dx... |
4,447,641 | <p>I can't figure out where to even start, I have looked up the answer on Desmos but it uses L'Hopitals rule which i haven't learned yet.</p>
<p><span class="math-container">$$\lim_{h \to 0} \frac{(2+h)^{3+h} - 8}{h}$$</span>
I see that i can use the log rules to rewrite it as
<span class="math-container">$$\lim_{h \to... | 311411 | 688,046 | <p>I follow up on your suggestion to use a logarithm. Define</p>
<p><span class="math-container">$$ \phi(h)= e^{(3+h)\ln(2+h)}, \text{ noting that } \phi(0)= 8.$$</span></p>
<p>On the one hand, <span class="math-container">$$\phi'(0) \;=\; \lim_{h \to 0}\frac{\phi(h)-\phi(0)}{h-0},$$</span> which is exactly your target... |
2,078,943 | <p>My book says this but doesn't explain it:</p>
<blockquote>
<p>Row operations do not change the dependency relationships among columns.</p>
</blockquote>
<p>Can someone explain this to me? Also what is a dependency relationship? Are they referring to linear dependence? </p>
| Michelle Zhuang | 739,444 | <p>I just figured out a way that makes sense of it, so I searched for such a question to post it. And gladly it's slightly different than hardmath's above.</p>
<p>If two vectors <span class="math-container">$\vec{a}$</span> and <span class="math-container">$\vec{b}$</span> are dependent, <span class="math-container">$\... |
1,490,828 | <p>Consider an equilateral triangle in $A \subset \mathbb{R}^2$ with vertices $x_1$, $x_2$, and $x_3$. I would like to show that the smallest circle which contains $A$ has radius $\frac{d}{\sqrt{3}}$, where $d$ is the length of each side. </p>
<p>I've been able to show that a (closed) ball centered at $(x_1 + x_2 + x_... | cr3 | 250,652 | <p>The <a href="http://mathworld.wolfram.com/Circumcenter.html" rel="nofollow">circumcenter</a> of a triangle is the intersection of the perpendicular bisector of each edge. A right triangle <strong>B</strong> can be formed between the bisector of an edge, a vertice of the same edge and the circumcenter. Other right tr... |
20,683 | <p>The most elementary construction I know of quantum groups associated to a finite dimensional simple Hopf algebra is to construct an algebra with generators $E_i$ and $F_i$ corresponding to the simple positive roots, and invertible $K_j$'s generating a copy of the weight lattice. Then one has a flurry of relations b... | Zahlendreher | 33,854 | <p>There is a detailed exposition of this in Majid's paper <em>Double-bosonization of braided groups and the construction of</em> $U_q(\mathfrak{g})$, Math Proc Cambridge Phil Soc 125(1). Especially appendix B where quantum group is obtained by a version of Tannaka-Krein duality for braided monoidal categories applied ... |
132,769 | <p>My distribution histogram looks like it is not identically distributed, as in the negative counts have a different shape than the positive counts. Here is an image:</p>
<p><img src="https://i.stack.imgur.com/VtlHd.jpg" alt="enter image description here"></p>
<p>The chart has 800 data points, and the tallest count ... | B R | 5,773 | <p>"<a href="http://en.wikipedia.org/wiki/Shape_parameter" rel="nofollow">Shape parameters</a>" are used to describe changes, such as "skewness", in a family of distributions other than those generated by "location" or "scale" parameters. There are many examples of distributions with shape parameters. Your distribution... |
75,656 | <p>Is there a closed form for any function f(x,y) satisfying:</p>
<p>a) $\frac{df}{dx}+\frac{df}{dy}=xy$</p>
<p>b) $\frac{df}{dx}+\frac{df}{dy}+\frac{df}{dz}=xyz$</p>
| Samuel Tan | 18,039 | <p>I think
$$\tag{1}
f(x,y) = \frac{ -x^3 + 3 x^2 y + 6 c_1(y-x)} {6}
$$</p>
<p>is a solution to part a). For part b), </p>
<p>$$
f(x,y,z)= \frac {(x^4 - 2 x^3 y - 2 x^3 z + 6 x^2 y z + 12 c_2(-x + y, -x + z)}{12}
$$</p>
<p>might be a solution. In both cases $c$ is a function, like the constant of integration f... |
75,656 | <p>Is there a closed form for any function f(x,y) satisfying:</p>
<p>a) $\frac{df}{dx}+\frac{df}{dy}=xy$</p>
<p>b) $\frac{df}{dx}+\frac{df}{dy}+\frac{df}{dz}=xyz$</p>
| Thomas Belulovich | 831 | <p>(a) Is solved by $f(x,y) = xy^2/2 - y^3/6$.</p>
<p>One way to get this: Let $f_x = a(x,y)$ so that $f_y = xy - a(x,y)$. Then use commutativity of mixed partials to get a handle on $a$; we'd have $a_y(x,y) = y - a_x(x,y)$. Then, why not set $a_x = 0?$, we get $a_y = y$ and so $a(x,y) = y^2/2$ works fine. Since we ca... |
75,656 | <p>Is there a closed form for any function f(x,y) satisfying:</p>
<p>a) $\frac{df}{dx}+\frac{df}{dy}=xy$</p>
<p>b) $\frac{df}{dx}+\frac{df}{dy}+\frac{df}{dz}=xyz$</p>
| Robert Israel | 8,508 | <p>More generally, suppose we want a function of $n$ variables with
$$\sum_{i=1}^n \frac{\partial f}{\partial x_i} = x_1 \ldots x_n$$
We can start off with a change of variables: let $U$ be an $n \times n$
orthogonal matrix whose first row is $[1,1,\ldots 1]/\sqrt{n}$, and take
$$ \pmatrix{u_1\cr u_2\cr \ldots\cr u_n\c... |
308,426 | <p>Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and $(m_k)_{k\in\omega}$ such that the sequence $(m_kx_{n_k})_{k\in\omega}$ still converges to zero? </p>
<p><strong>Ad... | Jochen Wengenroth | 21,051 | <p>This is satisfied for locally convex spaces with the so-called <em>Mackey convergence condition</em>, i.e., for every null sequence $x$ there is a bounded absolutely convex set $B$ such that $x$ tends to $0$ in the linear hull of $B$ endowed with the Minkowski functional of $B$ as a norm. This property was introduce... |
1,002,191 | <p>I have a problem:</p>
<blockquote>
<p>A family has six children, A,B,C,D,E,F. The ages of these six children are added together to make an integer less than $32$ where none of the children are less than one year old.</p>
</blockquote>
<p>What is the chances of guessing the ages of all of the children.</p>
<p>I ... | Brian M. Scott | 12,042 | <p>When you set the sum at $31$, you’re allowing the extra variable $x_7$ to be $0$. However, the formula that you’re using is for the number of solutions in strictly <strong>positive</strong> integers, so you want to make sure that $x_7$ will have to be at least $1$. You do this by insisting that the total be $32$ rat... |
120,808 | <pre><code>Limit[Sum[k/(n^2 - k + 1), {k, 1, n}], n -> Infinity]
</code></pre>
<p>This should converge to <code>1/2</code>, but <code>Mathematica</code> simply returns <code>Indeterminate</code> without calculating (or so it would appear). Any specific reason why it can't handle this? Did I make a mistake somewhere... | QuantumDot | 2,048 | <p>You can use the <a href="https://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">Euler-Maclaurin</a> formula to get the limit (the sum can be approximated by an integral, which becomes exact in the infinite limit):</p>
<pre><code>f[i_] = i/(n^2 - i + 1);
Integrate[f[k], {k, 0, n}, Assumptions -> n > 0]... |
120,808 | <pre><code>Limit[Sum[k/(n^2 - k + 1), {k, 1, n}], n -> Infinity]
</code></pre>
<p>This should converge to <code>1/2</code>, but <code>Mathematica</code> simply returns <code>Indeterminate</code> without calculating (or so it would appear). Any specific reason why it can't handle this? Did I make a mistake somewhere... | Bob Hanlon | 9,362 | <p>Reverse the order of the summation. i.e., <code>k -> (n - k + 1)</code></p>
<pre><code>s = Sum[(n - k + 1)/(n^2 - (n - k + 1) + 1), {k, 1, n}] // Simplify
(* -n + (1 + n^2) PolyGamma[0, 1 + n^2] - (1 + n^2) PolyGamma[0, 1 - n + n^2] *)
Limit[s, n -> Infinity]
(* 1/2 *)
</code></pre>
<p>For an alternat... |
3,162,294 | <p>I've been trying to prove this statement by opening up things on the left hand side using the chain rule but am really getting nowhere. Any tips/hints would be very helpful and appreciated!</p>
| user | 293,846 | <p>Your example is simple:
<span class="math-container">$$
p=1-\left (\frac56\right)^{14}-\binom {14}1 \left (\frac56\right)^{13}\left (\frac16\right)\approx 0.704.$$</span></p>
<p>In this expression we have subtracted from <span class="math-container">$1$</span> the probabilities of having no or only one 6. The genera... |
1,063,774 | <blockquote>
<p>Prove that the <span class="math-container">$\lim_{n\to \infty} r^n = 0$</span> for <span class="math-container">$|r|\lt 1$</span>.</p>
</blockquote>
<p>I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it ... | aes | 194,204 | <p>I'll assume $0 < |r| < 1$.</p>
<p>We want $|r^n| < \epsilon$, or equivalently $(1/|r|)^n > 1/\epsilon$.</p>
<p>Write $1/|r| = 1 + a$ with $a > 0$. Then $(1/|r|)^n = (1+a)^n \geq 1 + na$ by the binomial theorem.</p>
<p>Choose $N$ such that $1 + Na > 1/\epsilon$, say let $N = \lceil \frac{1}{a\eps... |
1,063,774 | <blockquote>
<p>Prove that the <span class="math-container">$\lim_{n\to \infty} r^n = 0$</span> for <span class="math-container">$|r|\lt 1$</span>.</p>
</blockquote>
<p>I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it ... | Tacet | 186,012 | <p>If $r=0$ it's trivial, so we can skip this case. Now assume $r\neq0$. </p>
<p>$$\lim_{n\to\infty}r^n=0 \Longleftrightarrow(\forall \epsilon \in \mathbb{R}^+)(\exists \delta\in\mathbb{R})(\forall n_{> \delta})(|r^n| < \epsilon)$$</p>
<p>Now we can note, that both sides of above inequality are positive. We can... |
1,063,774 | <blockquote>
<p>Prove that the <span class="math-container">$\lim_{n\to \infty} r^n = 0$</span> for <span class="math-container">$|r|\lt 1$</span>.</p>
</blockquote>
<p>I can't think of a sequence to compare this to that'll work. L'Hopital's rule doesn't apply. I know there's some simple way of doing this, but it ... | Mr.HiggsBoson | 675,350 | <p><a href="https://math.stackexchange.com/a/3277943/675350">This</a> is an answer I had earlier posted.</p>
<p>This is not a rigorous proof but helps in visualisation.</p>
<p>If <span class="math-container">$$-1<x<1$$</span> and you raise it to any power <span class="math-container">$n > 1 $</span> the valu... |
47,188 | <p>I am an amateur mathematician, and I had an idea which I worked out a bit and sent to an expert. He urged me to write it up for publication. So I did, and put it on arXiv. There were a couple of rounds of constructive criticism, after which he tells me he thinks it ought to go a "top" journal (his phrase, and he wen... | Alex B. | 35,416 | <p>Think of it this way: any graduate student who submits his first result for publication faces a similar issue: his paper will land on a referee's desk, who has never heard the name of the author before, so according to your logic, the better the result the more it must smell of crackpottery to the referee. Yet, PhD ... |
1,659,224 | <p>Find a sequence $a_n$ such that $\lim_{n \to \infty}|a_{n+1} - a_n | = 0$ while the sequence does not converge.</p>
<p>I am thrown for a loop. For a sequence not to converge it means that for all $\epsilon \geq 0$ and for all $N$, when $n \geq N$, $|a_n - L| \geq \epsilon$ as $n \longrightarrow \infty$</p>
<p>or I... | Pantelis Sopasakis | 8,357 | <p>Take $a_n=\log(n)$; then</p>
<p>$$
|a_{n+1}-a_n|=\log(n+1)-\log(n) = \log(1+\frac{1}{n})\to 0,
$$</p>
<p>but the sequence diverges. In the definition of <em>Cauchy sequences</em> it is asserted that $|a_m-a_n|\to 0$, but you are free to choose any such $m,n$ independent of each other. In your case $m=n+1$.</p>
|
737,212 | <p>Please how can I easily remember the following trig identities:
$$
\sin(\;\pi-x)=\phantom{-}\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi-x)=-\cos x\\
\sin(\;\pi+x)=-\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi+x)=-\cos x\\
\sin(\frac\pi2-x)=\phantom{-}\cos x\quad \color{red}{\text{ and }}\quad \co... | Matt L. | 70,664 | <p>I wouldn't try to remember them, but try to understand them by visualizing the graphs of the sine and cosine function. If you draw the functions (mentally), you can easily "derive" all of these identities.</p>
|
737,212 | <p>Please how can I easily remember the following trig identities:
$$
\sin(\;\pi-x)=\phantom{-}\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi-x)=-\cos x\\
\sin(\;\pi+x)=-\sin x\quad \color{red}{\text{ and }}\quad \cos(\;\pi+x)=-\cos x\\
\sin(\frac\pi2-x)=\phantom{-}\cos x\quad \color{red}{\text{ and }}\quad \co... | kingW3 | 130,953 | <p>$$\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\sin\beta\cos\alpha\\\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$$
Logically just replace $\alpha$ and $\beta$ with the value and you'll get the equation you need.</p>
|
2,565,763 | <p>I was playing a standard solitaire game on my mobile app and I came across a round where I couldn't perform a single move thus resulting in a loss. I was then thinking as to what the probability of this event to happen in a single game. Would anybody know this probability and show the calculations given a standard d... | mucciolo | 222,084 | <p>First let's adress your confusion with the example.</p>
<p><strong>1. If in the end they use the Z statistic to calculate their value of $k$, why don't they just skip straight to calculating the Z statistic in the first place, instead of doing the LRT?</strong></p>
<p>Notice that the $Z$ statistic is the tool they... |
3,059,020 | <p>The integral is<span class="math-container">$$\int_0^{2\pi}\frac{\mathrm dθ}{2-\cosθ}.$$</span>Just to skip time, the answer of the indefinite integral is <span class="math-container">$\dfrac2{\sqrt{3}}\tan^{-1}\left(\sqrt3\tan\left(\dfracθ2\right)\right)$</span>.</p>
<p>Evaluating it from <span class="math-contai... | N. S. | 9,176 | <p>The problem with the real approach is that you make the change of variable <span class="math-container">$t=\tan\left(\dfrac{\theta}{2}\right)$</span> for <span class="math-container">$0 < \theta < 2 \pi$</span>. </p>
<p>This is problematic since your substitution need to be defined and continuous for all <spa... |
730,018 | <p>Assume ZFC (and AC in particular) as the background theory.</p>
<p>If $(M,\in^M)$ is a model of ZFC (not necessarily transitive or standard), must there exist a bijection between $M$ and $$\{x \in M \mid (M,\in^M) \models x \mbox{ is an ordinal number}\}?$$</p>
<p>I am also interested in the cases where $M$ is ass... | Martín-Blas Pérez Pinilla | 98,199 | <p>Modulo possible subtleties about non transitive models, I understand that is the same question because being an ordinal is absolute for transitive models. About (2), infinite cardinals (so all the cardinals) and ordinals are in bijective correspondence with ordinals via the $\aleph$ function.</p>
|
4,036,975 | <p><span class="math-container">$\require{begingroup} \begingroup$</span>
<span class="math-container">$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}\def\Catalan{\mathsf{Catalan}}$</span></p>
<p><a href="https://math.stackexchange.com/q/3590905/122782">Related quest... | Varun Vejalla | 595,055 | <p>Not a complete answer, but an elaboration of what I said in the comments.
To evaluate <span class="math-container">$$\int_1^\infty\frac{\ln(1+x^{2n})}{1+x^2} dx$$</span></p>
<p>Write it as <span class="math-container">$$\int_1^\infty\frac{\ln\left(x^{-2n}+1\right)+\ln\left(x^{2n}\right)}{1+x^2} dx = \int_1^\infty\fr... |
603,291 | <p>Suppose $f:(a,b) \to \mathbb{R} $ satisfy $|f(x) - f(y) | \le M |x-y|^\alpha$ for some $\alpha >1$
and all $x,y \in (a,b) $. Prove that $f$ is constant on $(a,b)$. </p>
<p>I'm not sure which theorem should I look to prove this question. Can you guys give me a bit of hint? First of all how to prove some function... | Yiorgos S. Smyrlis | 57,021 | <p>It suffices to show that $f$ is differentiable and its derivative vanishes everywhere.</p>
<p>The hypothesis implies that, for every $x\in(a,b)$ and $h$, such that $x+h\in(a,b)$, we have that</p>
<p>$$
\frac{|f(x+h)-f(x)-0\cdot h|}{|h|} \le M\,|h|^{\alpha-1}.
$$</p>
<p>The right hand side of the above tends to ze... |
3,690,076 | <p>Is there a rigorous proof that <span class="math-container">$|G|=|\text{Ker}(f)||\text{Im}(f)|$</span>, for some homomorphism <span class="math-container">$f\,:\,G\rightarrow G'$</span>? Can anyone provide such a proof with explanations?</p>
| Didier | 788,724 | <p>Consider <span class="math-container">$f : G \to \mathrm{Im}f$</span>. Is it by construction a surjective morphism. <span class="math-container">$\ker f$</span> is a subgroup of <span class="math-container">$G$</span>. <span class="math-container">$f$</span> goes to the quotient to <span class="math-container">$\ove... |
1,017,738 | <p>So I know that the derivative of arccos is:
$-dx/\sqrt{1-x^2}$</p>
<p>So how would I find the derivative of $\arccos(x^2)$? What does the $-dx$ mean in the above formula?</p>
<p>Would it just be $-2x/\sqrt{1-x^2}$ ?</p>
| Michael Hardy | 11,667 | <p>"$-dx$" should not be there.</p>
<p>$$
\frac{d}{dx} \arccos(x^2) = \frac{-1}{\sqrt{1-(x^2)^2}} \cdot \frac d {dx} x^2 = \frac{-2x}{\sqrt{1-x^4}}.
$$</p>
|
2,009,134 | <p>Suppose $$\frac{{{{\sin }^4}(\alpha )}}{a} + \frac{{{{\cos }^4}(\alpha )}}{b} = \frac{1}{{a + b}}$$ for some $a,b\ne 0$. </p>
<p>Why does $$\frac{{{{\sin }^8}(\alpha )}}{{{a^3}}} + \frac{{{{\cos }^8}(\alpha )}}{{{b^3}}} = \frac{1}{{{{(a + b)}^3}}}$$</p>
| Community | -1 | <p>The general expression is:$$\frac{\sin^{4n}\theta}{a^{2n-1}} + \frac{\cos^{4n}\theta}{b^{2n-1}} = \frac{1}{(a+b)^{2n-1}}~, ~~n\in \mathbb N$$</p>
<p>From the given relation, it can be written \begin{align}(a+b)\left(\frac{\sin^4\theta}{a}+ \frac{\cos^4\theta}{b}\right) &= \left(\sin^2\theta + \cos^2\theta\right... |
209,728 | <p>I'm attempting to calculate the gradient of a function defined by two variables.
After following a tutorial I found, I have the following code: </p>
<pre><code>f[x_, y_] = (2 x^4 - x^2 + 3 x - 7 y + 1)/(e^(2 x^2/3 + 3 y^2/4));
(*t is gradient of f*)
t[{x_, y_}] := {
D[f[x, y], x],
D[f[x, y], y]
};
(*
Trying to use... | J. M.'s persistent exhaustion | 50 | <p>What you have is not quite the way to do steepest descent. Recall that you need to pick a step size for the direction where the next point is headed. In fancy code, that is done through some form of line search, but to keep things simple here, let us use a fixed step size.</p>
<pre><code>γ = 1/10; (* step size *)
i... |
694,090 | <p>Let $V$ and $W$ be vector spaces over $\Bbb{F}$ and $T:V \to W$ a linear map. If $U \subset V$ is a subspaec we can consider the map $T$ for elements of $U$ and call this the restriction of $T$ to $U$, $T|_{U}: U \to W$ which is a map from $U$ to $W$. Show that</p>
<p>$$\ker T|_{U} = \ker T\cap U.$$</p>
<p>I know ... | naslundx | 130,817 | <p>Indeed the kernel is the set of points mapped to zero, i.e.</p>
<p>$$\ker{T} = \{x \in V | T(x) = 0\}$$</p>
<p>Then $T$ is restricted to $U$, we need only consider points $x \in U$:</p>
<p>$$\ker{T|_U} = \{x \in U | T(x) = 0\}$$</p>
<p>Since $U \subset V$, this is the same as the points $x$ that are both in $\ke... |
1,265,987 | <p>I am working on some Automata practice problems. I am working a 2 part question. Here it is:</p>
<blockquote>
<p>Let $\Sigma = \{a,b\}$ be an alphabet.
Let $L = \left\{w \in \Sigma^* \mid n_a(w) \le 4\right\}$</p>
<p>a) Create a transition graph for $L$</p>
<p>b) Find a regular expression for $L$</p>... | mvw | 86,776 | <p>I have difficulties reading your answer from the image, but the DFA looks fine, while the regexp does not. </p>
<p>Counter example: $ba$ seems not to be recognized.</p>
|
325,519 | <p>I'm reading a book about inverse analysis and trying to figure out how the authors do the inversion.</p>
<p>Assume that matrix $C$ is
$$
C
~=~
\begin{bmatrix}
88.53 & -33.60 & -5.33 \\
-33.60 & 15.44 & 2.67 \\
-5.33 & 2.67 & 0.48
\end{bmatrix}
$$
and at some point authors d... | AndreasT | 53,739 | <p>The spectral theorem ensures that since $C$ is symmetric it has 3 real eigenvalues and its eigenspaces are orthogonal. Let $\lambda_1,\lambda_2,\lambda_3$ be the eigenvalues and $\vec v_1,\vec v_2,\vec v_3$ be orthonormal eigenvectors (whose existence is granted by the spectral theorem; note that $\lambda_i$ need no... |
19,848 | <p><strong>{Xn}</strong> is a sequence of independent random variables each with the same
<strong>Sample Space {0,1}</strong> and <strong>Probability {1-1/$n^2$ ,1/$n^2$}</strong>
<br> <em>Does this sequence converge with probability one (Almost Sure) to the <strong>constant 0</strong>?</em>
<br><br>Essentially th... | Nate Eldredge | 822 | <p>Hint: The Borel-Cantelli lemma.</p>
|
2,252,256 | <p>The Chernoff bound, being a sharp quantitative version of the law of large numbers, is incredibly useful in many contexts. Some general applications that come to mind (which I guess are really the same idea) are:</p>
<ul>
<li>bounding the sample complexity of PAC algorithms;</li>
<li>estimating confidence intervals... | Shravas Rao | 37,588 | <p>I think matrix Chernoff bounds are still relatively new, so maybe there aren't really classical applications yet. But personally, I've used Tropp's monograph (<a href="https://arxiv.org/abs/1501.01571" rel="nofollow noreferrer">https://arxiv.org/abs/1501.01571</a>) and sections 5 and 6 of "Derandomizing the Ahlswed... |
4,097,262 | <p><a href="https://i.stack.imgur.com/E1UoR.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/E1UoR.png" alt="Cubic equation with a unit radius circle" /></a>
A cubic equation and circle (unit radius) has intersection at A,B,C,D. ABCD is a square. <em>Find the angle <span class="math-container">$\theta... | Jean Marie | 305,862 | <p>We are looking for a third degree curve <span class="math-container">$(C)$</span> with cartesian equation:</p>
<p><span class="math-container">$$y=ax(x^2-B)\tag{1}$$</span></p>
<p>Due to the fact it is an odd function, we can restrict our attention to the <span class="math-container">$x>0$</span> part.</p>
<p>The... |
4,315,844 | <p>Find the greatest number <span class="math-container">$k$</span> such that there exists a perfect square that is not a multiple of 10, with its last <span class="math-container">$k$</span> digits the same</p>
<p>I could find <span class="math-container">$12^2 = 144$</span>, <span class="math-container">$38^2 = 1444$... | Matthias Klupsch | 19,700 | <p>Take <span class="math-container">$\mathfrak{g} = \mathbb{R}^3$</span> with <span class="math-container">$[x,y] = x \times y$</span> the cross product and as symmetric bilinear form <span class="math-container">$\langle \cdot, \cdot \rangle$</span> the usual scalar product. Then the space <span class="math-container... |
545,728 | <p>So I have $X \sim \text{Geom}(p)$ and the probability mass function is:</p>
<p>$$p(1-p)^{x-1}$$</p>
<p>From the definition that:</p>
<p>$$\sum_{n=1}^\infty ns^{n-1} = \frac {1}{(1-s)^2}$$</p>
<p>How would I show that the $E(X)=\frac 1p$</p>
| Michael Hardy | 11,667 | <p>$p(1-p)^{x-1}$</p>
<p>\begin{align}
\operatorname{E}(X) & = \sum_{x=1}^\infty x\Pr(X=x) = \sum_{x=1}^\infty x p(1-p)^{x-1} = p \sum_{x=1}^\infty x(1-p)^x \\[10pt]
& = \underbrace{p \sum_{n=1}^\infty xs^{x-1} = p\cdot \frac 1 {(1-s)^2}}_{\text{This is the identity that you cited.}} \quad \text{where }s=1-p, ... |
2,780,832 | <p>Let's define: </p>
<p>$$\sin(z) = \frac{\exp(iz) - \exp(-iz)}{2i}$$
$$\cos(z) = \frac{\exp(iz) + \exp(-iz)}{2}$$</p>
<blockquote>
<p>We are to prove that
$$\sin(z+w)=\sin(w) \cos(z) + \sin(z)\cos(w), \forall_{z,w \in \mathbb{C}}$$
using only the following statement: $\exp(z+w) = \exp(w)\exp(z)$.</p>
</bloc... | Sujit Bhattacharyya | 524,692 | <p>Consider the RHS,</p>
<p>$$\sin z\cos w+\cos z\sin w$$</p>
<p>$$=\frac{e^{iz}-e^{-iz}}{2i}\frac{e^{iw}+e^{-iw}}{2}+\frac{e^{iw}-e^{-iw}}{2i}\frac{e^{iz}+e^{-iz}}{2}$$</p>
<p>[By doing the usual calculation which is skipped]</p>
<p>$$=2\frac{e^{i(z+w)}-e^{-i(z+w)}}{4i}$$
$$=\frac{e^{i(z+w)}-e^{-i(z+w)}}{2i}$$
$$=... |
478,523 | <p>I'm trying to reason through whether $\int_{x=2}^\infty \frac{1}{xe^x} dx$ converges.</p>
<p>Intuitively, It would seem that since $\int_{x=2}^\infty \frac{1}{x} dx$ diverges, then multiplying the denominator by something to make it smaller would make it converge.</p>
<p>If I apply a Taylor expansion to the $\fra... | Jonathan Y. | 89,121 | <p>I believe the easiest thing to do here is note that for all $x\geq 2$ one has $\frac{1}{xe^x}\leq e^{-x}$.</p>
<p>As for your approach, beyond what @user1337 rightfully said, note that we can't just decide that everything (asymptotically-)smaller than $\frac{1}{x}$ yields a converging integral, because (for example... |
1,761,775 | <p>Suppose we have angular momentum operators $L_1,L_2,L_3$ which satisfy $[L_1,L_2]=iL_3$, $[L_2,L_3]=iL_1$ and $[L_3,L_1]=iL_2$. We can show that the operator $L^2:=L_1^2+L_2^2+L_3^2$ commutes with $L_1,L_2$ and $L_3$. Now define $L_{\pm}=L_1\pm iL_2$ and then we can also show that $$L_+L_-=L^2-L_3(L_3-I)$$ and $$L_-... | Jyrki Lahtonen | 11,619 | <p>(promoting my comment to an answer)</p>
<p>You can use the fact that $L_+$ and $L_-$ are each others adjoints, IOW
$$\langle L_+x|y\rangle=\langle x|L_-y\rangle$$
for all $x,y$. Applying this to $y=L_+v, x=v$ gives
$$
\begin{aligned}
\Vert L_+v\Vert^2&=\langle L_+v\mid L_+v\rangle\\
&=\langle v\mid L_-L_+ ... |
2,174,508 | <p>We have a box in 3D space defined by it's position, size and rotation. How can i find the smallest box that contains that box, but has a new rotation?
I know this is a really bad description of the problem, i just can't describe it any better, i'm really bad at vector maths. i tried to draw a <a href="https://i.stac... | John Hughes | 114,036 | <p>I don't know how your box is described, but at least one way is that you know the "forward" vector of the box (i.e., a length-one (or <em>unit</em>) vector that points parallel to one edge of the box) and you know two other vectors that are parallel to the other two edges of the box. If you call these $v_1, v_2, v_3... |
1,130,029 | <p>I am not sure if this question is off topic or not but a question like this has been asked on this site before - <a href="https://math.stackexchange.com/questions/409408/insertion-sort-proof">Insertion sort proof</a></p>
<p>Here is an example of insertion sort running a on a set of data
<a href="https://courses.cs.w... | AlexR | 86,940 | <p>Reading through the slides I noticed the insertion sort <em>implementation</em> discussed here is actually sub-optimal: An element is swapped into place in a bubble-sort like manner.
Since the sorted list at step $i$ has $i$ elements, the average number of comparisons (= swaps - 1) needed to sort the $i+1$-st elemen... |
1,181,356 | <p>Can such complex-valued function exist that</p>
<p>$$\lim_{x\to x_0}f(x)=0$$</p>
<p>but</p>
<p>$$\lim_{x\to x_0}\frac 1{f(x)}=1$$?</p>
<p>What about</p>
<p>$$\lim_{x\to x_0}\frac 1{Re(f(x))}=1$$?</p>
| Winther | 147,873 | <p>No, if it did then by the product law of limits $$1 = \lim_{x\to x_0}\frac{f(x)}{f(x)} = \lim_{x\to x_0}f(x) \cdot \lim_{x\to x_0}\frac{1}{f(x)} = 0\cdot 1 = 0$$
giving us a contradiction. If $L = \lim_{x\to x_0} f(x)$ exist and $L\not =0 $ then we always have $\lim_{x\to x_0}\frac{1}{f(x)} = \frac{1}{L}$.</p>
|
1,197,875 | <p>I'm just starting partials and don't understand this at all. I'm told to hold $y$ "constant", so I treat $y$ like just some number and take the derivative of $\frac{1}{x}$, which I hope I'm correct in saying is $-\frac{1}{x^2}$, then multiply by $y$, getting $-\frac{y}{x^2}$.</p>
<p>But apparently the correct answe... | user225079 | 225,079 | <p>You're supposed to hold x constant. </p>
|
1,197,875 | <p>I'm just starting partials and don't understand this at all. I'm told to hold $y$ "constant", so I treat $y$ like just some number and take the derivative of $\frac{1}{x}$, which I hope I'm correct in saying is $-\frac{1}{x^2}$, then multiply by $y$, getting $-\frac{y}{x^2}$.</p>
<p>But apparently the correct answe... | MichaelChirico | 205,203 | <p>have you looked at the graph? if you have a good enough plotting program (good enough to rotate 3D graphs), it should be easy to see that the surface is linear for all fixed x (and hyperbolic for all fixed y)</p>
<p>first pass:</p>
<p><a href="http://www.wolframalpha.com/input/?i=plot+y%2Fx+for+x%3D-1..1%2C+y%3D-1... |
84,982 | <p>I am a new professor in Mathematics and I am running an independent study on Diophantine equations with a student of mine. Online I have found a wealth of very helpful expository notes written by other professors, and I would like to use them for guided reading. <strong>I am wondering whether it is customary to as... | François G. Dorais | 2,000 | <p>The <a href="http://fairuse.stanford.edu/Copyright_and_Fair_Use_Overview/chapter7/7-b.html" rel="nofollow">Stanford Copyright & Fair Use Overview</a> is pretty clear about what constitutes fair use in the classroom and what doesn't:</p>
<blockquote>
<p><strong>Rules for Reproducing Text Materials for Use in C... |
999,304 | <p>Frankie and Johnny play the following game.
Frankie selects a number at random from the interval
$[a, b]$. Johnny, not knowing Frankie’s number, is to pick
a second number from that same inverval and pay Frankie
an amount, W, equal to the squared difference between
the two</p>
<p>[so $0 ≤ W ≤ (b − a)^2]$. What shou... | mookid | 131,738 | <p>$M$ is not closed:</p>
<p>Look at the sequence
$$
f_n(x) = \min(1, nx)
$$
and check that $f_n\to 1$ for the norm $\| .\|$.</p>
|
999,304 | <p>Frankie and Johnny play the following game.
Frankie selects a number at random from the interval
$[a, b]$. Johnny, not knowing Frankie’s number, is to pick
a second number from that same inverval and pay Frankie
an amount, W, equal to the squared difference between
the two</p>
<p>[so $0 ≤ W ≤ (b − a)^2]$. What shou... | Simon S | 21,495 | <p>As for the "quotient norm", have a look at mookid's counter-example and see if you can modify it to force a violation of one of the norm axioms. I'd aim to show that $$\| [f] \|_{\hbox{am I a norm?}} = 0 \ \ \ \Longrightarrow \ \ \ f = 0$$
is violated.</p>
|
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