Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Group actions on manifolds - exponential map Let $M$ be a smooth manifold. Suppose $K$ is a Lie group (with Lie algebra $\mathfrak{k}$) acting EDIT: TRANSITIVELY on $M$ from the left and $G$ is a Lie group (with Lie algebra $\mathfrak{g}$) acting on $M$ from the right. Suppose further these actions commute - ie $k(pg)=... | Yes, it is always possible. In fact, you can always take $A=0\in\mathfrak g$.
Fix $x\in M$, and consider the smooth map $F\colon K\to M$ given by $F(k) = kx$. Because $K$ acts transitively, $F$ is surjective. Moreover, $F$ is equivariant with respect to the (transitive) left actions of $K$ on $K$ and $M$: For all $k,k'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Lipschitz-like behaviour of quartic polynomials I have observed the following phenomenon:
Let the biquadratic $q(x)=x^4-Ax^2+B$ have four real roots and perturb it by a linear factor $p(x)=q(x)+mx$, so that $m$ not too large with respect to $A,B$.
Then the roots of $p(x)$ (assuming they are real) are very close to th... | You should look at that source of all wisdom: Kato's Perturbation theory of linear operators, which, in the first chapter, discusses the perturbation of eigenvalues of a matrix (before plunging into infinite dimensions in subsequent sections). Apply that discussion to the companion matrix of your polynomial.
Of course,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Frog jumping on leaves $N$ leaves are arranged round a circle. A frog is sitting on first leaf and starts jumping every $K$ leaves. How many leaves can be reached by a frog?
| A couple of hints to get you started: recall Bezout's identity and think about the greatest common divisor of $N$ and $K$.
Full solution:
Let $d$ be the greatest common factor of $N$ and $K$.
Label the vertices $0,1,2\dots N-1$. Then the from starts at position $0$ and then jumps $K$ spaces to reach the position congr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many words can be formed using all the letters of "DAUGHTER" so that vowels always come together? How many words can be formed using all the letters of "DAUGHTER" so that vowels always come together?
I understood that there are 6 letters if we consider "AUE" as a single letter and answer would be 6!. Again for AUE ... | Imagine your three vowels as a block. Ignore the ordering of the block for the moment. Now you have 6 remaining "letters"- the original consonants plus this "vowel block". Now, there are $6$ of these altogether. So you get $6!$ ways of ordering them- which is the standard permutation formula. Now for $each$ of these or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1369974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Find the Limit $\lim_{n \rightarrow \infty}\frac{1}{(n+1) \log (1+\frac{1}{n})}$ Find the limit
$$\lim\limits_{n \rightarrow \infty}\frac{1}{(n+1) \log (1+\frac{1}{n})}$$
| Set $1/n=h\implies h\to0^+$
$$\lim\limits_{n \rightarrow \infty}\frac{1}{(n+1) \log\left(1+\frac1n\right)}$$
$$=\lim_{h\to0^+}\dfrac h{(h+1)\ln(1+h)}$$
$$=\dfrac1{\lim_{h\to0^+}\dfrac{\ln(1+h)}h}\cdot\dfrac1{\lim_{h\to0^+}(1+h)}=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1370075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Geometry question about lines If I have two points in euclidean space or the Cartesian plane whichever and both points lie on the same side of a straight line. Both above or both below- how can I show that the segment connecting the two points also lies above or below the line respectively . I.e every point on the segm... | Given an equation for the line $f(x) = ax+b$, you can define a function $g:[c,d]\rightarrow\mathbb{R}$ that connects the two lines. Write the distance between the lines as a function of $x$ and then check for local extrema, you will find that there are non, therefore the minimal distance has to be at the global points ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1370156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
What is $0\div0\cdot0$? We all know that multiplication is the inverse of division, and therefore
$x\div{x}\cdot{x}=x$
But what if $x=0$? $0\div0$ is undefined so $0\div0\cdot0$ should be too, but whatever happens when we divide that first $0$ by $0$ should be reversed when we multiply it by $0$ again, so what is the r... | Any expression involving "$\div 0$" is undefined because $0$ is not in the domain of the binary operation of division (of real numbers), no mater how it is combined with other mathematical symbols. You can't "pretend" it is and cancel it -- the expression is already meaningless, so you can't proceed further.
Your state... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1370214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 6
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Finding $\frac {a}{b} + \frac {b}{c} + \frac {c}{a}$ where $a, b, c$ are the roots of a cubic equation, without solving the cubic equation itself Suppose that we have a equation of third degree as follows:
$$
x^3-3x+1=0
$$
Let $a, b, c$ be the roots of the above equation, such that $a < b < c$ holds. How can we find th... | If you multiply out the expression $(x-a)(x-b)(x-c)$ and compare the coefficients to the expression after you get a common denominator, all will become clear
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Solving recurrence equation with floor and ceil functions I have a recurrence equation that would be very easy to solve (without ceil and floor functions) but I can't solve them exactly including floor and ceil.
\begin{align}
k (1) &= 0\\
k(n) &= n-1 + k\left(\left\lceil\frac{n}{2}\right\rceil\right) + k\left(\left\lfl... | SKETCH: It’s often useful to gather some numerical data:
$$\begin{array}{rcc}
n:&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\
k(n):&0&1&3&5&8&11&14&17&21&25&29&33&37&41&45&49&54&59\\
k(n)-k(n-1):&&1&2&2&3&3&3&3&4&4&4&4&4&4&4&4&5&5
\end{array}$$
Notice that the gaps in the bottom line are a single $1$, two $2$s, four ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1370643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove the Supremum is attained. Let $F$ denote denote the set of real valued functions on $[0,1]$ such that,
1) $ \; |f(x)| \leq 1 \; \forall x \; \in [0,1]$
2) $ \; |f(x)-f(x')| \leq |x-x'| \; \: \forall x,x' \: \in [0,1] $
Prove that that the following supremum is attained. $$\sup_{f \in F} \int_0^1 f(x) \sin(\frac{1... | As you already noted, $F$ is equicontinuous and (pointwise) bounded, so Arzela-Ascoli implies its closure is compact. Let's verify $F$ is closed: Suppose $f_n$ is a sequence in $F$ which converges uniformly to some continuous function $g$. Then $f_n$ converges pointwise, hence
\begin{align*}
1)&|g(x)|=\lim |f_n(x)|\leq... | {
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"timestamp": "2023-03-29T00:00:00",
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Tangent line parallel to another line At what point of the parabola $y=x^2-3x-5$ is the tangent line parallel to $3x-y=2$? Find its equation.
I don't know what the slope of the tangent line will be. Is it the negative reciprocal?
| If you solve simultaneously the curve and the line $y=3x+c$ to get a quadratic equation in $x$ then this quadratic must have double roots at the point of tangency. This will give the value of $c$ and the required $x$ value is given by $x=-\frac{b}{2a}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1370933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Proving a theorem about Fourier coefficients I need to prove this:
Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$.
Can someone please let me know if I would be on a right track if I said:
Let $||f(x)||_\infty$ = C, then $... | Note that $$\int_{-\pi}^{\pi}f(x) \cos nx dx=\frac{1}{n}f(x)\sin nx\mid_{-\pi}^\pi-1/n\int_{-\pi}^\pi\frac{df}{dx}\sin nx dx=-1/n\int_{-\pi}^\pi\frac{df}{dx}\sin nx dx\\\implies \left|\int_{-\pi}^{\pi}f(x) \cos nx dx\right|=\frac{1}{n}\left|\int_{-\pi}^\pi\frac{df}{dx}\sin nx dx\right|\\\le \frac{1}{n}\max_{x\in [-\pi,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371019",
"timestamp": "2023-03-29T00:00:00",
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Intuition on the Representable Functor Given a locally small category C, and an object $C$, the functor:
\begin{equation}
\mbox{Hom}_\textbf{C}(C,-):\textbf{C} \longrightarrow \textbf{Sets}
\end{equation}
that sends objects to hom-sets and arrows $f:A\rightarrow B$ to functions:
\begin{align}
f_*:\mbox{Hom}_\textbf{C}(... | First of, a functor $F : \mathscr C → \mathrm{Set}$ is called representable (by $C$) if it's isomorphic (not necessarily equal) to $\mathrm{Hom}(C, -)$ for an object $C$ of $\mathscr C$. As for the term, an abstract functor $F$ is represented by the very concrete action of $\mathrm{Hom}(C, -)$.
Take for example the fu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Projections: Orthogonality Given a unital C*-algebra $1\in\mathcal{A}$.
Consider projections:
$$P^2=P=P^*\quad P'^2=P'=P'^*$$
Order them by:
$$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$
Then equivalently:
$$P\perp P'\iff 0=PP'=P'P\iff\Sigma P^2=\Sigma P=\Sigma P^*$$
How can I check this?
(Operator a... | I'm assuming that by $\sigma(\Sigma P)\leq1$ you mean that $\|\Sigma P\|\leq1$.
*
*Suppose that $\|P+Q\|\leq1$. So $0\leq P+Q\leq 1$. Then $(P+Q)^2\leq P+Q$ (just conjugate with $(P+Q)^{1/2}$). That is,
$$
P+Q+QP+PQ\leq P+Q,
$$
or $QP+PQ\leq0$. If we conjugate this inequality with $Q$, we get $QPQ+QPQ\leq0$. But $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$U_n=\int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ . $U_n= \int_{n^2+n+1}^{n^2+1}\frac{\tan^{-1}x}{(x)^{0.5}}dx$ where
Find $\lim_{n\to \infty} U_n$ without finding the integration
I don't know how to start
| We have (see here) $$\frac{\pi}{2}-\frac{1}{x}\leq\arctan\left(x\right)\leq\frac{\pi}{2}-\frac{1}{x}+\frac{1}{3x^{3}}
$$ then $$\frac{\pi}{2}\int_{n^{2}+n+1}^{n^{2}+1}\frac{1}{\sqrt{x}}dx-\int_{n^{2}+n+1}^{n^{2}+1}\frac{1}{x\sqrt{x}}dx\leq\int_{n^{2}+n+1}^{n^{2}+1}\frac{\arctan\left(x\right)}{\sqrt{x}}dx\leq\frac{\pi}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that there are infinitely many composite numbers of the form $2^{2^n}+3$.
There are infinitely many composite numbers of the form $2^{2^n}+3$.
[Hint: Use the fact that $2^{2n}=3k+1$ for some $k$ to establish that $7\mid2^{2^{2n+1}}+3$.]
If $p$ is a prime divisor of $2^{2^n}+3$ then $2^{2^n}+3\equiv0\pmod{p}$... | Let $a_n=2^{2^n}+3$. Then $a_{n+1}=(a_n-3)^2+3 = a_n^2-6a_n+12$. Since:
$$ p(x)=x^2-6x+12 \equiv (x-1)(x+2) \pmod{7} $$
maps $0$ to $5$ and $5$ to $0$, we have that $7\mid a_n$ iff $n$ is odd, since $a_1=7\equiv 0\pmod{7}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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p-simplex spanned by elements in the boundary of the unit ball of Thurston norm is in the the boundary of this unit ball. in completing my thesis I have reached a momentary impass.
I am trying to solve an exercise given in the book "Foliations II" by Candel and Conlon. In particular, Exercise 10.4.1, and I can't seem t... | Using subadditivity we get for $x$ in $\Delta$ that $\xi x = \xi(\sum \lambda_i v_i) \le \sum \lambda_i \xi(v_i) =1$.
For the second part note that if we restrict $\xi$ to the interior $int \Delta \to \mathbb R$, the set $\xi^{-1}(1)$ is closed and non empty. It is also obviously open (use that $x$ in the interior can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371601",
"timestamp": "2023-03-29T00:00:00",
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Find the area of the shaded region in the figure Find the area of the shaded region in the figure
What steps should I do? I tried following the steps listed here https://answers.yahoo.com/question/index?qid=20100305030526AAef8nZ
But I got 150.7 which is wrong.
| Hint: Break the shaded region up into two shapes. One is a portion of the circle (you know the portion because of the given angle), and the other is a triangle (which is equilateral). Find the area of each shape and then add them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sunflower Lemma - Allow Duplicates? The sunflower lemma states that if we have a family of sets $S_1, S_2, \cdots, S_m$ such that $|S_i| \leq l$ for each $i$, then $m > (p-1)^{l+1}l!$ implies that the family contains a sunflower with $p$ petals. My question is, do the $S_i$'s need to be distinct sets or can there be du... | The usual statement of the lemma requires only that $m>(p-1)^\ell\ell!$, and in that case the sets must be distinct. Your version allows duplicates. To see this, suppose that $m>(p-1)^{\ell+1}\ell!$, and we have sets $S_1,\ldots,S_m$ such that $|S_k|\le\ell$ for $k=1,\ldots,m$. If there are $p$ or more duplicates of so... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Evaluating the limit: $\lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1)$ I need to solve this question: $$\lim_{x\rightarrow -1^+}\sqrt[3]{x+1}\ln(x+1)$$
I tried the graphical method and observed that the graph was approaching $0$ as $x$ approached $-1$ but I need to know if there's a way to calculate this.
| $$\lim\limits_{x\to -1^+}\sqrt[3]{x+1}\ln(x+1)$$
Let $h=x+1$. Since $x\to -1^+$, then $h\to 0^+$. So now
$$\lim\limits_{h\to 0^+}\sqrt[3]{h}\ln h$$
$$=\lim\limits_{h\to 0^+}h^{\frac13}\ln h$$
$$=\lim\limits_{h\to 0^+}\exp\left(\ln h^{\frac13}\right)\ln h$$
$$=\lim\limits_{h\to 0^+}\exp\left(\frac13\ln h\right)\ln h$$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1371906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Find the $\int \frac{(1-y^2)}{(1+y^2)}dy$ $\int \frac{(1-y^2)}{(1+y^2)}dy$ first I tried to divide then I got 1-$\frac{2y^2}{1+y^2}$ and i still can't integrate it.
| Hint:
$$ \int \frac{1-y^2}{1+y^2}dy = \int \frac{1}{1+y^2}dy-\int \frac{y^2}{1+y^2}dy$$
Note (using long division or otherwise): $$\int \frac{y^2}{1+y^2}dy = \int dy -\int \frac{1}{1+y^2}dy$$
Therefore:$$ \int \frac{1-y^2}{1+y^2}dy = 2\int\frac{1}{1+y^2}dy -\int dy$$
The solution should be straight forward from here.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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The class equation of the octahedral group I know that the class equation of the octahedral group is this:
$$1 + 8 + 6 + 6 + 3$$
I think the $8$ stands for the $8$ vertices, the $6$ could be $6$ faces and $6$ pairs of edges. Then what is the $3$ for?
| Following the Wikipedia list of conjugacy classes given by Zev Chonoles, we obtain:
24 = 1 (identity from the whole octahedron) + 8 (rotation by 120° about a face, an axis with 3-fold symmetry) + 6 (rotation by 180° about an edge, an axis with 2-fold symmetry) + 6 (rotation by 90° about a vertex, an axis with 4-fold sy... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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Find $\lim_\limits{x\to -2}{f(x)}$ Let $f:\mathbb{R}\mapsto{\mathbb{R}}$ be an odd function such that: $$\lim_{x\to 2}{(f(x)-3\cdot x+x^2)}=5$$
Find $\lim_\limits{x\to -2}{f(x)}$, if it exists. (so also prove its existence)
| $$\lim_{x\to 2}{(f(x)-3\cdot x+x^2)}=5\\ \lim_{x\to 2}{(f(x)-3*2 +4)}=5\\lim_{x\to 2}{f(x)}=5+2=7\\$$ we know $f(-x)=-f(x)$ as it odd function
so $$lim_{x\to -2}f(x)=lim_{x\to +2}f(-x)=\\lim_{x\to +2}(-f(x))=-7$$
limit exist because :if we have odd function ,domain is symmetrical interval .so if we have $|x-2|<\de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find the Summation S Given function $f(x)=\frac{9x}{9x+3}$.
Find S:
$$
S=f\left(\frac{1}{2010}\right)+f\left(\frac{2}{2010}\right)+f\left(\frac{3}{2010}\right)+\ldots+f\left(\frac{2009}{2010}\right)
$$
| We can write the sum in terms of digamma function. I don't know if it is what do you want, but surely it's a closed form. We have $$f\left(x\right)=1-\frac{1}{3x+1}
$$ then $$f\left(\frac{k}{2010}\right)=1-\frac{2010}{3k+2010}
$$ then we have $$\sum_{k=1}^{2009}\left(1-\frac{2010}{3k+2010}\right)=2009-\frac{2010}{3}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372434",
"timestamp": "2023-03-29T00:00:00",
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Initial Value Problem $dy/dx = (y+1)^{1/3}$ Consider the differential equation $$\frac{dy}{dx} = (y+1)^{1/3}$$
(a) State the region of the $xy$-plane in which the conditions of the existence and uniqueness theorem are satisfied (using any appropriate theorem).
(b) Let $S$ be the region of the $xy$-plane where the condi... | You probably have to use Picard-Lindelöf. The function is not Lipschitz around $y = -1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Constructing DVR's from arbitrary UFD's Is the following statement true?
Let $A$ be an UFD and $p\in A$ prime, then $A_{(p)}$ is a discrete valuation ring.
I think yes: For every element $x$ of $Q(A_{(p)})=Q(A)$, there is a unique $k\in\mathbb{Z}$ such that I can write $x$ as $p^k\cdot\frac{a}{b}$ for some $a,b\in A$... | Let me complement the nice and abstract existing answer by a concrete one:
Yes, the argument is correct. To see that the ring $A_{(p)}$ is noetherian directly use an argument as you might know it from Euclidean domains.
Let $I$ be an non-zero ideal. Let $a \in I$ be non-zero with minimal valuation, say $k$; then s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Computing $\max_{1/2 \leq x \leq 2} ( \min_{1/3 \leq y \leq 1} f(x,y) )$ where $f(x,y) = x(y \log y - y) - y \log x$.
Let $f(x,y)=x(y\ln y-y)-y\ln x.$ Find $\max_{1/2\le x\le 2}(\min_{1/3\le y\le1}f(x,y))$.
This problem is quite easy and it is from Spivak; it is the part $c)$ of the general exercise 2-41 page 43 Calc... | I have given the complete solution below. The part highlighted shows where we use the result of part (a). In short, we use part (a) to compute the derivative of the critical point of $g_x(y)$ for the given function $f(x,y)$. This is used in computing the critical points to maximise $f(x,y)$ w.r.t. $x$ after minimising ... | {
"language": "en",
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"answer_id": 0
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Trying to understand Bienaymé formula In Bienaymé formula, it states that $var(\bar X) = \large\frac{\sigma^2}{n}$.
However, when I was going through the proof here, it says the variances of $X_1,X_2,X_3......X_n$ are the same(assuming they are all independent). Can anyone explain the reason behind it?
I am confused a... | Taking a random sample $X_1, X_2, \dots, X_n$ from a population with mean $\mu$ and
variance $\sigma^2$ means that the $X_i$ are independent and that
$E(X_i) = \mu$ and $V(X_i) = \sigma^2.$ All of these random variables
have the same variance because they represent observations from
the same population.
Consequently, d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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4th Isomorphism Theorem applied to normalizers I'm reading a proof showing a proper subgroup Q of p-group P is contained in it's normalizer.
It applies the 4th Isomorphism Theorem to assert $\frac{Q}{Z(P)} < N_{\frac{P}{Z(P)}}(\frac{Q}{Z(P)})$ implies $Q < N_P(Q)$. How is this?
| Hint: $N_{P/Z(P)}(Q/Z(P))=N_P(Q)/Z(P)$.
(And proper inclusions remain proper under lattice correspondence.)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculating in closed form $\int_0^1 \log(x)\left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$ What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $ would you like to recommend? I'm not against them, they migh... | We have:
$$\sum_{k=1}^{n-1}\frac{1}{k^2(n-k)^2}=\frac{1}{n^2}\sum_{k=1}^{n-1}\left(\frac{1}{k}+\frac{1}{n-k}\right)=\frac{2H_{n-1}^{(2)}}{n^2}+\frac{4H_{n-1}}{n^3}$$
so:
$$ \text{Li}_2(x)^2 = \sum_{n\geq 2}\left(\frac{2H_{n-1}^{(2)}}{n^2}+\frac{4H_{n-1}}{n^3}\right) x^n\tag{1}$$
and since:
$$ \int_{0}^{1}\frac{x^n \log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1372936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $T$ be the set of full binary trees. In what way $T^7 \cong T$? I was reading the slides of a talk by Tom Leinster.
I have trouble understanding the last line of page 17 (pages 1-15 are irrelevant and can be skipped). Could someone please explain it to me?
If I translate the images correctly, $T$ is defined to be t... | The paper Seven Trees in One exhibits a "very explicit bijection" $T^7\cong T$. It is perhaps a bit cumbersome because it requires separating into five cases based on how the seven trees look in the first four levels of depth. A proof is present too. Disclaimer: I haven't read it.
Another paper Objects of Categories as... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Relation of gamma function to a factorial mimic function The gamma function is an analytic extension of the factorial function.
For real and positive $x$, with $x=int(x)+frac(x)$, where consider the function:
$$f(x)=\prod_\limits{i=1}^{int(x)} (frac(x)+i)$$
defined on $frac(x)\in[0,1)$.
We need to start from $i=1$ to a... | I think that you can try something with the Bohr-Mollerup Theorem.
Theorem (Bohr-Mollerup): Let be $f:(0,+\infty)\to \Bbb{R}$ such that:
*
*$f(1)=1$ and $f(x)>0$, for every $x>0.$
*$f(x+1)=xf(x)$, for every $x>0.$
*$\lg{(f(x))}$ is a convex function.
Then $f\equiv \Gamma\big|_{(0,+\infty)}.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding the expected number of trials in an experiment. Given a uniform probability distribution over $[0, 1]$, a number is randomly selected from this distribution. We have to find the expected number of trials such that the sum of the picked numbers $ >= $ 1.
I have been told that the answer to this question is $ e $... | Let $E(x)$ be the expected number of trials for reaching a sum of $\ge1$ starting from a sum of $x$. Then
$$
E(x)=1+\int_x^1E(t)\mathrm dt\;.
$$
Differentiating with respect to $x$ yields
$$
E'(x)=-E(x)\;,
$$
with the solutions $E(x)=c\mathrm e^{-x}$, and the condition $E(1)=1$ yields $c=\mathrm e$, so the solution is ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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In a group $G$, prove the following result Let $G$ be a group in which $a^5=e$ and $aba^{-1}=b^m$ for some positive integer $m$, and some $a,b\in G$. Then prove that $b^{m^5-1}=e$.
Progress
$$aba^{-1}=b^m\Rightarrow ab^ma^{-1}=b^{m^2}$$
What will be the next?
| We have $$b=ebe=a^5ba^{-5}=a^4b^ma^{-4}=a^3b^{m^2}a^{-3}=a^2b^{m^3}a^{-2}=ab^{m^4}a^{-1}=b^{m^5}$$
Thus, multiplying by $b^{-1}$, we have $e=b^{m^5-1}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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When does this sum of combinatorial coefficients equal zero? $p>2$ is a prime number, $n\in \mathbb{N}$. Is the following statement true or false? Thanks.
$$\sum_{i=0}^{\lfloor n/p\rfloor}(-1)^i {n\choose ip}=0$$ iff $n=(2k-1)p$ for some $k\in \mathbb{N}$.
| Let $\omega=\exp\left(\frac{2\pi i}{p}\right)$. Since $f(n)=\frac{1}{p}\left(1+\omega^n+\ldots+\omega^{(p-1)n}\right)$ equals one if $n\equiv 0\pmod{p}$ and zero otherwise, we have:
$$S(n)=\sum_{i\equiv 0\pmod{p}}\binom{n}{i}(-1)^i=\frac{(1-1)^n+(1-\omega)^n+\ldots+(1-\omega^{p-1})^n}{p}\tag{1}$$
so $p\cdot S(n)$ is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1373415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Range of an inverse trigonometric function Find the range of $f(x)=\arccos\sqrt {x^2+3x+1}+\arccos\sqrt {x^2+3x}$
My attempt is:I first found domain,
$x^2+3x\geq0$
$x\leq-3$ or $x\geq0$...........(1)
$x^2+3x+1\geq0$
$x\leq\frac{-3-\sqrt5}{2}$ or $x\geq \frac{-3+\sqrt5}{2}$...........(2)
From (1) and (2),
domain is $x\l... | You do not have the correct domain. We must also have $-1\leq\sqrt{x^2+3x+1}\leq1$ and $-1\leq\sqrt{x^2+3x}\leq1$. In other words, $\sqrt{x^2+3x+1}\leq1$ and $\sqrt{x^2+3x}\leq1$, since they are positive.
Thus $x^2+3x+1\leq1$ (squaring is allowed since both are positive), or $x^2+3x\leq0$, this gives $-3 \leq x \leq 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1373523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why the determinant of a matrix with the sum of each row's elements equal 0 is 0? I'm trying to understand the proof of a problem, but I'm stuck. In my book they consider that if all lines of a matrix has sum 0 then it's determinant is also 0. I checked some random examples and it's true, but I couldn't proof it. Could... | Another way to derive this result:
When the sum of all rows are zero, the row vectors are linearly dependent. Hence the determinant is zero.
In fact, if any two or three or four rows add up to pure zero, the determinant is zero as well.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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problem of number theory N. Sato Can someone help me solve this problem?
Sato, 4.2.
For an odd positive integer $n>1$, let $S$ be the set of integers $x$ such that $1 \leq x \leq n$, such that both $x$ and $x+1$ are relatively prime to $n$. Show that $$\prod_{x\in S}x \equiv 1 \mod n$$
Thanks Giorgio Viale
| Wilson's theorem says if $n$ is prime, $(n-1)! ≡ -1$ (mod $n$). Then $n-1$ isn't included, because $n$ is divisible by $n$. The product of the rest of the factors gives 1.
If $n$ isn't a prime, let its powered prime factors be $p_1$, $p_2$, ..., $p_y$, and respective remainders an $x$ candidate gives by being divided b... | {
"language": "en",
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Power serie of $f'/f$ It seems that I'm [censored] blind in searching the power series expansion of $$f(x):=\frac{2x-2}{x^2-2x+4}$$ in $x=0$.
I've tried a lot, e.g., partiell fraction decomposition, or regarding $f(x)=\left(\log((x+1)^2+3)\right)'$ -- without success.
I' sure that I'm overseeing a tiny little missing... | Given any function $f$, if we restrict to where $f$ is nonzero (so that either $\log f$ or $\log -f$ exists), we can write $\frac{f'}{f}=\frac{d}{dt}\log |f|$. If $f=c \prod (x-\alpha_i)^{k_i}$ is a rational function, $\log f = \log c + \sum k_i \log (x-\alpha_i)$ and so
$$\frac{f'}{f}=\frac{d}{dt}\log |f| = \sum ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Using the definition of derivative to find $\tan^2x$ The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$.
I've been working on this problem, trying every way I can think of. At first I tried this method:
$$\lim_{h\to 0} {\tan^2(x+h)-\tan^2(x)\over h}$$
$$\lim_{h\to 0} {\tan(x+h)-\tan... | Using first principle, the derivative of any function $f(x)$ is given as $$\frac{d(f(x))}{dx}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$ Hence, derivative of $\tan^2 x$ is given as $$\frac{d(\tan^2 x)}{dx}=\lim_{h\to 0}\frac{\tan^2(x+h)-\tan^2(x)}{h}$$ $$=\lim_{h\to 0}\frac{(\tan(x+h)-\tan(x))(\tan(x+h)+\tan(x))}{h}$$$$=\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1373809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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How to find: $\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$? How to find: $$\int^{2\pi}_0 (1+\cos(x))\cos(x)(-\sin^2(x)+\cos(x)+\cos^2(x))~dx$$
I tried multiplying it all out but I just ended up in a real mess and I'm wondering if there is something I'm missing.
| You have to linearise the integrand. The simplest way to do is with Euler's formulae. To shorten the computation, set $u=\mathrm e^{\mathrm ix}$. Starting with $\cos x =\dfrac{u+\bar u}2$ and taking into account the relations
$$u\bar u=1,\quad u^n+\bar u^n=2\cos nx,$$
we have
\begin{align*}
(1+\cos x)&\cos x(-\sin^2x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1373977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Uniform convergence of $f_n(x) = x^n$ on $[0,c]$
Let $c \in (0,1)$ be fixed. Let $$f_n(x) = x^n,\quad x \in [0,1)$$and$$f(x) = 0,\quad x \in [0,1)$$ Show that $f_n$ converges uniformly to $f$ on $[0,c]$.
So, we have, $f_n(0) = 0, f_n( c) = c^n,f_n'(x) = nx^{n-1}, f_n'(0) = 0, f_n'( c) = nc^{n-1}.$ Since I don't kno... | An explicit argument: if $x \in (0,1)$ and $\varepsilon>0$ then $x^n<\varepsilon$ is equivalent to $n \ln(x) < \ln(\varepsilon)$ or $n>\frac{\ln(\varepsilon)}{\ln(x)}$. Now check that if $n>\frac{\ln(\varepsilon)}{\ln(c)}$ then $x^n<\varepsilon$ for every $x \in [0,c]$. This will necessarily work because the limit is z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1374064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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$n$-th term of an infinite sequence Determine the $n$-th term of the sequence $1/2,1/12,1/30,1/56,1/90,\ldots$. I have not been able to find the explicit formula for the $n$-th term of this infinite sequence. Can some one solve this problem and tell me how they find the answer?
| To find a formula you need to first find the pattern.
There are a number of ideas to use when trying to find for such paterns in a sequence:
*
*if all numbers are composite, try to factorise - if this works nicely a pattern may emerge at this stage, especially for numbers with one prime factor, e.g. $56 = 7 \times 8... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that there is a metric space that has a limit point, and each open disk in it is closed. Collecting examples
Show that there is a metric space that has a limit point, and each open disk in it is closed.
This question belongs to the 39th math competitions of Iran. This is one solution:
Suppose that $X=\{\frac{1}{... | The p-adic numbers are an example of this phenomenon and there are a lot of similar examples, since any discrete valuation ring or its quotient field have the property that open disks are closed and vice versa. This includes finite extensions of the p-adics as well as rings of the form $K((X))$ (Laurent series with fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1374229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Argue that $\binom{n}{n_1,n_2,...,n_r} = \binom{n-1}{n_1-1,n_2,...,n_r} + \binom{n-1}{n_1,n_2-1,...,n_r}+...+\binom{n-1}{n_1,n_2,...,n_r-1} $ Argue that
$\binom{n}{n_1,n_2,...,n_r} = \binom{n-1}{n_1-1,n_2,...,n_r} + \binom{n-1}{n_1,n_2-1,...,n_r}+...+\binom{n-1}{n_1,n_2,...,n_r-1} $
Each term on the right hand side is ... | Your argument using apples is very close, maybe only a bit awkwardly written.
We give a combinatorical argument of equality, showing the the rhs and lhs count the same thing.
Now, the lhs is easy: it counts the number of ways to partition $n$ apples into $r$ groups with sizes $n_1, n_2, \ldots, n_r$.
We want to show th... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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find total integer solutions for $(x-2)(x-10)=3^y$ I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard.
How many integer solutions ($x$, $y$) are there of the equation $(x-2)(x-10)=3^y$?
(A)1 (B)2 (C)3 (D)4 (E)5
If let $y=0$, we ha... | Just as Bob1123 commented, compute the discriminant for the equation $$x^2-12 x+(20-3^y)=0$$ and the roots $$x_{1,2}=6\pm\sqrt{3^y+16}$$ So $(3^y+16)$ must be a perfect square.
I only see one which is possible what you already found ($y=2$).
Now, to prove that this is the unique solution is beyond my skills. Fortunate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1374534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove the group $G$ is abelian?
Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian.
I know that if the order of $Z(G)$ is not equal to $1$, then I can prove $G$ is Abelian. However, suppos... | Assume first that $\vert G \vert = p^2$. Since $Z(G) < G$, we have that $\vert Z(G) \vert \mid \vert G \vert$. One can show that the center of a group $G$ whose order is a prime power is non-trivial - it's not too hard, but too long to state here. You might want to look that up in an Algebra book.
If $\vert Z(G) \vert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1374908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to find a function that satisfies 2 conditions I am solving a partial differential equation by separation of variables. Part of the solution requires finding a function that meets the following criteria.
f(L,t)=C
f(0,t)=A*cos(at+b)
I was wondering if a function exists and how to find it (I am guessing and checking,... | Presumably you want a continuous (or better) such function, because otherwise the problem's trivial. :) Also, I assume $L, C, A, a, b$ are all constants.
Recall the situation in one variable (that is, if I'm given the values of a one-variable function at two points): if I want a function $g$ such that $g(a)=u$ and $g(b... | {
"language": "en",
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Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series? Is there someone who can show me how do I evaluate this sum :$$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$$
Note : In wolfram alpha show this result and in the same time by ratio test it's not a convince way to judg that... | I post this answer because Dirichlet's test has not been mentioned in any of the previous answers. Let $a_n=(-1)^{n(n-1)/2}$ and $b_n=1/n$. The partial sums of $a_n$ are bounded and $b_n$ is decreasing and converging to $0$. Dirichlet's test implies the series is convergent.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $f'(x)$ when $f(x) = x^2 \cos(\frac{1}{x})$ Find $f'(x)$ when
$$
f(x) = \begin{cases}
x^2\cos(\frac{1}{x}),& x \neq 0\\
0, &x = 0\end{cases}$$
Ok, I know the derivative of $f$ and it is $2x\cos(\frac{1}{x})+\sin(\frac{1}{x})$. My question is, am I only supposed to find the derivative here or evaluate this at $0$? ... | Let $$
f(x) = \begin{cases}
x^2\cos(\frac{1}{x}),& x \neq 0\\
0, &x = 0\end{cases}$$
As you said in your question, when $x$ is not 0, we have $$f^{'}(x)=2x\cos(\frac{1}{x})+\sin(\frac{1}{x})$$
When $x$ is 0, we must use the limit definition of the derivative. That is, $$f^{'}(x)=\lim_{h\rightarrow0}\dfrac{f(0+h)-f(0)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A sequence inequality for $x_{2^n}$, the binary partition function.
Define the sequence $\{x_{n}\}$ recursively by $x_{1}=1$ and
$$\begin{cases}
x_{2k+1}=x_{2k}\\
x_{2k}=x_{2k-1}+x_{k}
\end{cases}$$
Prove that
$$x_{2^n}>2^{\frac{n^2}{4}}$$
I have compute some term $x_{2}=2,x_{3}=2,x_{4}=4,x_{5}=4,x_{6}=6,x_{7... |
In this answer I will prove that $$\log_2 \left(x_{2^k}\right) \sim \frac{k^2}{2}$$ by providing the explicit inequalities $$2^{k^{2}/2-k/2-k\log_{2}k}\leq x_{2^k} \leq2^{k^{2}/2+k/2+1}.$$ This not only shows that we can do better and achieve $k^2/2$ in the exponent, but also that this is optimal.
As martin mentions ... | {
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"source": "stackexchange",
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Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$.
I've tried in this way:
$$
\begin{eqnarray}
P(X < Y) &=& \int_1^3 \left\{\int_0^y f_X(x) dx\right\}g_Y(y) ... | Answer:
Divide the regions of X with respect to Y for the condition $X<Y$.
For $0<X<1$, $P(X<Y) = \frac{1}{2}$
For $1<X<2$ and $1<Y<2$ $P(X<Y) = \int_{1}^{2}\int_{x}^{2} \frac{1}{2}\frac{1}{2}dydx = \frac{1}{8}$
For $1<X<2$, and $2<Y<3$ $P(X<Y) = \frac{1}{2}.\frac{1}{2}=\frac{1}{4}$
Thus $P(X<Y) = \frac{1}{2}+ \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is the Green-Tao theorem valid for arithmetic progressions of numbers whose Möbius value $\mu(n)=-1$? I am reading the basic concepts of the Green-Tao theorem (and also reading the previous questions at MSE about the corollaries of the theorem). According to the Wikipedia, the theorem can be stated as: "there exist ari... | I will add here Steven Stadnicki's comments as an answer, so the question will be closed in some days (if other answers do not come):
(1) is trivially implied, as you suggest, but (2) is not trivially implied. That said, it would be very surprising if the proof of Green-Tao could not be extended to handle your case -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Euler's Phi function, elementary number theory Show that the equation $\phi(n)=2p$ where $p$ is prime and $2p+1$ is composite has no solutions. Using formula for $\phi$ it's quite easy proving $n$ cannot have more than two prime factors in its factorization (and we can do even better), so we must split this problem int... | Recall that $\phi(p^e)=p^{e-1}(p-1)$ for a prime $p$ and that $\phi$ is multiplicative (but not totally).
If $n=2p+1$ were prime, then $\phi(n)=2p$ but the question states that $2p+1$ is composite so this is not possible.
Now let $l$ be a prime such that $l\mid n$.
If $l>3$, then $l-1$ would be a factor of $2p$ which i... | {
"language": "en",
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Does the inverse image sheaf have a left adjoint for $\mathsf{Set}$-valued sheaves? It's known that for sheaves with values in modules, the inverse image sheaf functor $j^\ast$ for $j:U\subset X$ an inclusion of an open set has a left adjoint which is extension by zero.
Is there any way to carry this over to $\mathsf{S... | Let $j : U \to X$ be the inclusion of an open subspace. Then $j^* : \mathbf{Sh} (X) \to \mathbf{Sh} (U)$ has a left adjoint $j_! : \mathbf{Sh} (U) \to \mathbf{Sh} (X)$: given a sheaf $F$ on $U$,
$$j_! F (V) = \begin{cases}
F (V) & \text{if } V \subseteq U \\
\emptyset & \text{otherwise}
\end{cases}$$
the idea being th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1375527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $T_n(x)={}_2F_1\left(-n,n;\tfrac 1 2; \tfrac{1}{2}(1-x)\right) $ Prove that, for Chebyshev polynomials of the first kind,
\begin{align}
T_n(x) & = \tfrac{n}{2} \sum_{k=0}^{\left \lfloor \frac{n}{2} \right \rfloor}(-1)^k \frac{(n-k-1)!}{k!(n-2k)!}~(2x)^{n-2k} && n>0 \\
& = {}_2F_1\left(-n,n;\tfrac 1 2; \tfra... | $y=\phantom{}_2 F_1(a,b;c;z)$ is the regular solution of the ODE:
$$z(1-z)y''+\left[c-(a+b+1)z\right]y'-ab y=0 $$
hence $y=\phantom{}_2 F_1(-n,n;1/2;z)$ is the regular solution of the ODE:
$$z(1-z)y''+\left(\frac{1}{2}-z\right)y'+ n^2 y=0 $$
and $y=\phantom{}_2 F_1\left(-n,n;\frac{1}{2};\frac{1-z}{2}\right)$ is the reg... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a path in a graph by its hash value Assume there is a graph $G = (V, E)$ and a hash function $H: V^n \rightarrow \{0,1\}^m$. Given a path $p = (v_1, v_2, ..., v_n)$ from the graph $G$, compute its hash value $H(p) = h_p$.
Question: Given only the value $h_p$ for any path in the graph (or any "non-path" in the ... | If it is a finite graph, the answer is yes, because for fixed $n$ there is only a finite number of paths of length $n$ in the graph. For example, you can enumerate all $n$-tuples of distinct vertices in the graph, check whether they form a path, compute the hash value of that path if they do, and check if that hash occ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$ Show that $f(x) = \frac{2}{x}$ is not uniformly continuous on $(0,1]$
WLOG Suppose, $0< \delta \leq 1.$ Let, $\epsilon = 1$ and $x = \frac{\delta}{2}, y = x + \frac{\delta}{3}, x,y \in (0,1]$
Skipping all the details: $\mid f(x) - f(y)\mid =...= \mid \frac{24... | Let us see what you do. You assume a function is uniformly continuous. You take $\epsilon = 1$. Then you have some $\delta$ such that for all $x,y$ with $|x-y| < \delta$ something should hold. But if this something holds for all $|x-y| < \delta$ then to say it holds for all $|x-y| < \delta'$ for some $\delta' < \delta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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when is the cokernel of a map of free modules free? Let $R$ be a commutative ring (noetherian if needed) and $n,m$ be two nonnegative integers. Consider a map
$\varphi: R^n\rightarrow R^m$
Is there a characterisation, e.g. in terms of the matrix representation of $\varphi$ of the cokernel of this map being free?
remark... | You need ALL $(k+1)\times (k+1)$ minors for some $k$ to be zero and one $k\times k$ minor to be a unit.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $\int_0^1(\ln x)^n\hspace{1mm}dx$ I am not a big fan of induction, it's just a personal preference.
Is there a method other than induction.
Answer is $n!$ by the way
| Let, $\ln x=t\implies \frac{dx}{x}=dt\implies dx=e^tdt$, we have $$\int_{0}^{1}(\ln x)^ndx=\int_{-\infty}^{0}(t)^ne^tdt$$ $$=\int_{0}^{\infty}(-t)^ne^{-t}dt$$ $$=(-1)^n\int_{0}^{\infty}e^{-t}t^ndt$$ Now, using Laplace transform $\color{blue}{\int_{0}^{\infty}e^{-st}f(t)dt=L[f(t)]}$ & $\color{blue}{L[t^n]=\frac{n!}{s^{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1376214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Extending the Riemann zeta function using Euler's Theorem. Euler's theorem states that if the real part of a complex number $z$ is larger than 1, then $\zeta(z)=\displaystyle\prod_{n=1}^\infty \frac{1}{1-p_n^{-z}}$, where $\zeta(z)=\displaystyle\sum_{n=1}^\infty n^{-z}$ is the Riemann zeta function and $\{p_n\}$ is the... | Yes and no, if you want something nontrivial. The analytically extended zeta function admits a Weierstrass factorization (called the Hadamard factorization in this case):
$$\zeta(s)=\pi^{s/2}\frac{\prod_{\rho}(1-s/\rho)}{2(s-1)\Gamma(1+s/2)},$$
where the product is over the nontrivial zeros. You can also see the trivia... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Given an acute triangle ABC with altitudes AH, BK. Let M be the midpoint of AB Given an acute triangle ABC with altitudes AH, BK. Let M be the midpoint of AB. The line through CM intersect HK at D. Draw AL perpendicular to BD at L. Prove that the circle containing C, K and L is tangent to the line going through BC
|
This proof does not assume that $ABC$ is acute. Suppose $a:=BC$, $b:=CA$, and $c:=AB$. Likewise, $\alpha:=\angle BAC$, $\beta:=\angle ABC$, and $\gamma:=\angle BCA$, so $\alpha+\beta+\gamma=\pi$. Let $x:=\angle CKL$ and $z:=\angle AMC$. Note that $A$, $K$, $L$, $H$, and $B$ lie on the circle centered at $M$ with r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1376403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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remainder of $a^2+3a+4$ divided by 7
If the remainder of $a$ is divided by $7$ is $6$, find the remainder when $a^2+3a+4$ is divided by 7
(A)$2$ (B)$3$ (C)$4$ (D)$5$ (E)$6$
if $a = 6$, then $6^2 + 3(6) + 4 = 58$, and $a^2+3a+4 \equiv 2 \pmod 7$
if $a = 13$, then $13^2 + 3(13) + 4 = 212$, and $a^2+3a+4 \equiv 2 \pmod ... | If the remainder when $a$ divided by $7$ is $b$, then $a = 7n+b$ for some integer $n$.
Hence, $a^2+3a+4 = (7n+b)^2+3(7n+b)+4$ $= 49n^2 + 14nb + b^2 + 21n + 3b + 4$ $= 7(7n^2+2nb+3n) + (b^2+3b+4)$.
So, the remainder when $a^2+3a+4$ is divided by $7$ will be the same as the remainder when $b^2+3b+4$ is divided by $7$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1376498",
"timestamp": "2023-03-29T00:00:00",
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Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group? I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.
| We may view $PGL_{2}$ over $\mathbb{Z}$ as an affine scheme given by an open subset of $\mathbb{P}^{4}_{\mathbb{Z}}$. More precisely, let $\{z_{11},z_{12},z_{21},z_{22}\}$ be coordinates on $\mathbb{P}^4$. Then $PGL_2$ is given by the distinguished open set $D(f)$ where $f=z_{11}z_{22}-z_{12}z_{21}$. The coordinate rin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proposition: $\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$. Proposition: $$\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$$
I believe that this is true, and, using Desmos Graphing Calculator, it seems to be true.
I will add how I derived the formula in a moment, if you would l... | First, I would like to mention that said series is not necessarily well-defined; you can't just add a "..." and expect the resulting quantity to be well-defined. For example, consider:
$$S=1+2+4+8+16+\ldots$$
Then,
$$2S=2+4+8+16+\ldots$$
so:
$$2S+1=S$$
and you get $S=-1$, which is clearly absurd. The issue here is that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1376667",
"timestamp": "2023-03-29T00:00:00",
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Is the inequality $| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ true? I'm having some trouble deciding whether this inequality is true or not...
$| \sqrt[3]{x^2} - \sqrt[3]{y^2} | \le \sqrt[3]{|x -y|^2}$ for $x, y \in \mathbb{R}.$
| $|x - y|^2 = (x - y)^2 $
Let $\sqrt[3]{x} = a , \sqrt[3]{y} = b$
I will study this
$$| a^2 - b^2 | \leq \sqrt[3]{|a^3-b^3|^2} $$
L.H.S
$$|a^2 - b^2|^3 = |a-b|^3 \cdot |a+b|^3= \color{red}{|a-b|^2}\cdot |a^2-b^2|\cdot |a^2 +2ab+b^2|$$
R.H.S
$$|a^3 - b^3|^2 = \color{red}{|a-b|^2} \cdot |a^2 +ab + b^2|^2 $$
So our pro... | {
"language": "en",
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"source": "stackexchange",
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Question about probability distributions I've recently came across this question:
You are trying to hitch-hike. Cars pass at random intervals, at an average rate of 1
per minute. The probability of a car giving you a lift is 1%. What is the probability
that you will still be waiting after one hour?
My first thought w... | Because they model two different things.
A Binomial Distribution, $\mathcal{Bin}(n, p)$, is that the count of successes in of $n$ Bernoulli trials with each trial having independent and identical probability of success $p$. The count of such successes can range from $0$ to $n$.
A Poisson Distribution, $\mathcal{Pois}... | {
"language": "en",
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Are there an infinite number of open balls in an open set in a metric space? Let's start off by recalling the definition of an open set in a metric space:
A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset A$
Where $B_r(x)$ denotes the open ball of ... | In any metric space, there are an infinite number of ways to write down balls with a given center. But some of the balls might actually be the same. For instance, in the "discrete metric" $d(x,y)=0$ if $x=y$ and $1$ otherwise, all balls $B_r(x)$ for $r \leq 1$ are the same (they are just $\{ x \}$) while all balls $B_r... | {
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Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$ The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs.
We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$ for arb... | The finite geometric series is given by
$$
G(q,n)=\sum_{k=0}^nq^k=\frac{1-q^{n+1}}{1-q}
$$
for constant $q$ such that $|q|<1$.
Now $\frac{d}{dq}G(q,n)$ is the series you are looking for...
| {
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Choose $\rho$ such that $\rho$-norm minimizes the matrix condition number I'm solving questions from am exam that I failed miserably, so I would love it if someone can take a look at my proof and make sure I'm not making any gross mistakes.
Question
Let $A$ a symmetric matrix. Which $\rho$-norm minimizes $A$'s conditio... | $k(A,2)=\|A\|_2\|A^{-1}\|_2= \max | \lambda_A|.\max|\lambda_{A^{-1}}|\leq\|A\|\|A^{-1}\|=k(A,\|.\|)$ for each operator norm $\|.\|$.
That $\max|\lambda_A|\leq \|A\|$ follows from:
Let $x$ be an eigenvector of $A$ for the eigenvalue $\lambda$, where $\max\limits_{i}|\lambda_i|=|\lambda|$. Then for arbitrary matrix norm ... | {
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A question about bijective functions Suppose that $f: [1,3]$$\rightarrow$$[0,8]$ is continuous. Show that there is some $x$$\in$$[1,3]$ such that $f(x)+4=4x$.
I need some advice on how to get started. Do i need to use IVT for this problem? Do I need to show that f is increasing while x is increasing?
| Let $h(x) = f(x) - 4x + 4$. $h$ is continuous on $[1,3]$. Moreover:
$h(1) = f(1) \ge 0$ and $h(3) = f(3) - 8 \le 0$ since $\operatorname{Range}(f) = [0,8]$.
Thus, by IVT..
| {
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Why do remainders show cyclic pattern? Let us find the remainders of $\dfrac{6^n}{7}$,
Remainder of $6^0/7 = 1$
Remainder of $6/7 = 6$
Remainder of $36/7 = 1$
Remainder of $216/7 = 6$
Remainder of $1296/7 = 1$
This pattern of $1,6,1,6...$ keeps on repeating. Why is it so? I'm asking in general, that is for every case o... | Looking at remainders after division by 7 is called arithmetic modulo 7.
You are regarding powers of 6, modulo 7. But $6$ is $-1$, modulo $7$. This is written:
$$6 \equiv -1 \pmod 7$$
But the powers of $-1$, that is the numbers $(-1)^n$, simply alternate: $1,-1,1,-1,1,\ldots$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1377221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 4
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Sum of cosines of complementary/suplementary angles Why are $(\cos(2^{\circ})+\cos(178^{\circ})), (\cos(4^{\circ})+\cos(176^{\circ})),.., (\cos(44^{\circ})+\cos(46^{\circ}))$ all equal zero?
Could you prove it by some identity?
|
so
$$cos (2) +cos(178) =cos(2)+cos(180-2)=0\\ cos (4) +cos(176) =cos(4)+cos(180-4)=0\\...\\$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Poisson distribution and price reduction A store owner has certain existence of an item and he decides to use the following scheme to sell it:
The item has a price of \$100. The owner will reduce the price in half for every customer that buys the item on a given day. That way, the first customer will pay \$50, the next... | Is solving for $x$ the way to go? If so, how can I deal with $x!$ ?
B: Should I go with $E(g(x))$?
| {
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Solving Equations system question We get this equation and need to solve
Solve in $\mathbb{Z} $ the given equation
$ y(y -x )(x+1) = 12\ $
| If you factor 12 as 3 x 2 x 2 and then put y = 3, y - x = 2, x + 1 = 1, you get a solution x = 1 and y = 3 which is an integer solution. This is by using common sense. The other one can be obtained by writing 12 as -2 x -3 x 2 in which case x = 1 and y = -2 is the other solution.
| {
"language": "en",
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Is there a name for the closed form of $\sum_{n=0}^{\infty} \frac{1}{1+ a^n}$? I hope this is not a duplicate question. If we modify the well known geometric series, with $a>1$, to
$$
\sum_{n=0}^{\infty} \frac{1}{1+a^n}
$$
is there a closed form with a name?
I suspect strongly that the answer is not in terms of elemen... | We can write
$$\frac{1}{1+a^n}=\left[\frac{\partial}{\partial t}\ln\left(1+ta^{-n}\right)\right]_{t=1},$$
which implies that
$$\sum_{n=0}^{\infty}\frac{1}{1+a^n}=\left[\frac{\partial}{\partial t}\ln\prod_{n=0}^{\infty}\left(1+ta^{-n}\right)\right]_{t=1}=
\left[\frac{\partial}{\partial t}\,\ln\left(-t;a^{-1}\right)_{\in... | {
"language": "en",
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Inverse of partitioned matrices A matrix of the form
$$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$
is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and $A$ is invertible. Find a formula for $A^{-1}$.
Could anyone help on this?
| Hint:
You may consider the matrix $$B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix},$$
such that:
$$\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix} \cdot \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} I_{p\times p} & 0 \\ 0 & I_{q\times q} \end{bmatrix... | {
"language": "en",
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Asymptotic Estimates for a Strange Sequence Let $a_0=1$. For each positive integer $i$, let $a_i=a_{i-1}+b_i$, where $b_i$ is the smallest element of the set $\{a_0,a_1,\ldots,a_{i-1}\}$ that is at least $i$. The sequence $(a_i)_{i\geq0}=1,2,4,8,12,20,28,36\ldots$ is A118029 in Sloane's Online Encyclopedia. It is easy ... | If $b_i$ is nearly $i$ (you define it as larger, this will give a lower bound) then you have a simple recurrence:
$$
a_{i + 1} - a_i = i \qquad a_0 = 0
$$
so that approximately:
$$
a_i = \frac{i (i + 1)}{2}
$$
This tends to confirm your conjecture.
| {
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Evaluate $\iint_{R}(x^2+y^2)dxdy$ $$\iint_{R}(x^2+y^2)dxdy$$
$$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$
My attempt :
Jacobian=r
$$=\iint_{R}(x^2+y^2)dxdy$$
$$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$
$$\sqrt{x^2+y^2}=r$$
$$\int_{\theta=\pi/4}^{\theta=3\pi/4}\bigg[\int_{r=0}^{r=2}\bigg(r^2\bigg... | Switching to polar coordinates, the Jacobian is given by $ |J|$ where $$ J = \dfrac{\partial(x,y)}{\partial(r,\theta)} = \begin{vmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial y}{\partial r} \\ \dfrac{\partial x}{\partial \theta} & \dfrac{\partial y}{\partial \theta} \end{vmatrix} = \begin{vmatrix} \cos\theta... | {
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"answer_id": 1
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Generalization of Cantor Pairing function to triples and n-tuples Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from (w, x, y) and also computing w, x and y from z. In oth... | Ok, I think I got it. Your idea is to create recursive functions for both pair and unpair and simply "assemble" the results instead of computing them with an algebraic formula. Of course this works due to the nature of pairing function. If I have time I will add the code in here.
Just one more question: Assuming I actu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1377929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
How to prove $\frac{2^a+3}{2^a-9}$ is not a natural number How can I prove that
$$\frac{2^a+3}{2^a-9}$$
for $a \in \mathbb N$ is never a natural number?
| $$1+\frac{12}{2^a-9}$$
which means that $2^a$ should equal either $10,11,12,13,15$ or $21$, but neither of them is a power of $2$, so it's never a natural number
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Solve the equation $4\sqrt{2-x^2}=-x^3-x^2+3x+3$ Solve the equation in $\Bbb R$:
$$4\sqrt{2-x^2}=-x^3-x^2+3x+3$$
Is there a unique solution $x=1$? I have trouble when I try to prove it.
I really appreciate if some one can help me. Thanks!
| It can be seen that $4 \sqrt{2 - x^2} = (1+x)(3 - x^2)$ leads to the expanded form, after squaring both sides and combining terms,
$$ x^6 + 2 x^5 - 5 x^4 - 12 x^3 + 19 x^2 + 18 x - 23 = 0 .$$
It is readily identified that $x=1$ is a solution and can then be factored out leading to
$$ (x-1)( x^5 + 3 x^4 - 2 x^3 - 14 x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Curve fit minimizing the sum of the deviation I'm fitting a curve taking the smaller sum of deviations for each parameter tested, the smaller sum returns me the parameter that gives the best fit. Here is the algorithm for a test $f(x, parameter)$:
function fit(function:function, list:sample_y, list:x_values, list:param... | Translation:
There is some function $f : A \times B \to C$ where $A$, $B$ and $C$ are some sets of objects, where $C$ supports addition, subtraction, taking the absolute value and determining a minimal element.
*
*$x$ is a countable subset of $A$.
*$p$ is a countable subset of $B$, called parameters.
*$y$ is a co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the sum of all primes smaller than a big number I need to write a program that calculates the sum of all primes smaller than a given number $N$
($10^{10} \leq N \leq 10^{14} $).
Obviously, the program should run in a reasonable time, so $O(N)$ is not good enough.
I think I should find the sum of all the composite... | You could try programming a sieve to mark all the composites, then add up the unmarked numbers. To do that, you'll need a list of primes up to $10^7$. In order to get that, you could program a sieve...
This method is obviously pretty memory-intensive, but it's certainly faster than prime-testing each integer from $10^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Showing that a functions derivative is not bounded on $\mathbb{R}$ Suppose that $f$ is differentiable but not uniformly continuous on $\mathbb{R}$. Prove that $|f'|$ is not bounded on $\mathbb{R}$.
So I know that to show that $|f'|$ isn't bounded you would have to show that for any constant B, $|f'(a)|$ $>$ $B$, for ... | Hint: $A\Rightarrow B \iff \neg B \Rightarrow \neg A$.
Assume that for $f:\mathbb R\rightarrow\mathbb R$ differentiable and $|f'|$ bounded it follows that $f$ is uniformly continuous (to be more precise, it is Lipschitz continuous).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ How to find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ in the formal way? Numerically its value is $\approx 0.0217326$ and the partial sum formula contains the first derivative of the gamma function (by WolframAlpha).
| Partial fractions: $$\frac{1}{n(n+1)^2(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right) - \frac{1}{(n+1)^2},$$ the first part being telescoping, and the last part being related to $\zeta(2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Modelling interest with differential equations (Interpretation) I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows:
Assume you have a bank account that grows at an annual interest rate of r and every year you withdraw a fixed amount fr... | $w$ is the (continuous, constant) rate of withdrawal, and $rP$ is the (continuous, proportional) rate of interest accrual.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to calculate the integral? How to calculate the following integral?
$$\int_0^1\frac{\ln x}{x^2-x-1}\mathrm{d}x=\frac{\pi^2}{5\sqrt{5}}$$
| You can start writing $$x^2-x-1=(x-r_1)(x-r_2)$$ where $r_{1,2}=\frac{1}{2} \left(1\pm\sqrt{5}\right)$ and use partiel fraction decomposition. So,$$\frac 1{x^2-x-1}=\frac{1}{{r_2}-{r_1}} \Big(\frac{1}{x-r_2}-\frac{1}{x-r_1}\Big)$$ and use $$\int \frac{\log(x)}{x+a}=\text{Li}_2\left(-\frac{x}{a}\right)+\log (x) \log \le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Compact support vs. vanishing at infinity? Consider the two sets
$$ C_0 = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and } \lim_{|x|\to \infty} f(x) = 0\}$$
$$ C_c = \{ f: \mathbb R \to \mathbb C \mid f \text{ is continuous and } \operatorname{supp}{(f)} \text{ is bounded}\}$$
Aren't these two sets the... | Note that $C_c \subset C_0$, but $C_c \neq C_0$. For example, $f(x) = \dfrac{1}{x^2+1}$ belongs to $C_0$ but not $C_c$.
What you seem to be assuming is that $\lim_{|x|\to\infty}f(x) = 0$ implies that there is some $N > 0$ with $f(x) = 0$ for all $|x| > N$. This is not true, as the above example demonstrates. That is, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Continuity Must Hold in an Entire Open Set? Claim: If a function $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $\vec a \in \mathbb{R}^n$, it is continuous in some open ball around $\vec a$.
Is this claim false? In other words, is it possible for a function to be continuous at a single point $\vec a$ only, bu... | Yes, consider the function $f\colon \mathbb R \to \mathbb R$ given by $f(x)=x$ if $x\in \mathbb Q$ and $f(x)=-x$ otherwise. You can even improve this example to obtain a function that is differentiable at a point but no continuous at any other point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit to infinity with natural logarithms $\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $ I found the following problem in my calculus book:
Solve:
$$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$
I tried to solve it using log rules and l'Hôpital's rule with no success... | $$\lim_{x\to \infty } \Big(\frac{\ln(2x)}{\ln(x)}\Big)^{\ln(x)} $$
$$=\lim_{x\to \infty } \Big(\frac{\ln(x)+\ln 2}{\ln(x)}\Big)^{\ln(x)} $$ $$=\lim_{x\to \infty } \Big(1+\frac{\ln(2)}{\ln(x)}\Big)^{\ln(x)} $$ $$=\lim_{x\to \infty }exp\Big(\ln(x)\Big(1+\frac{\ln(2)}{\ln(x)}-1\Big)\Big) $$ $$=\lim_{x\to \infty }exp\Big(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Computing the intersection of two arithmetic sequences $(a\mathbb{Z} + b) \cap (c \mathbb{Z} + d)$ I am getting stuck writing a general formula for the intersection of two arithmetic sequences.
$$ (a\mathbb{Z} + b) \cap (c \mathbb{Z} + d) = \begin{cases}
\varnothing & \text{if ???} \\
?\mathbb{Z} + ? & \text{otherwi... | It amounts to solving the system of simultaneous congruences:
$$\begin{cases}
x\equiv b\mod a\\x\equiv d\mod c
\end{cases}$$
This system has a solution idf and only if $b\equiv d\mod \gcd(a,c)$, and it is unique modulo $\operatorname{lcm}(a,c)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1378976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Number of points of discontinuity Find the number of points where
$$f(\theta)=\int_{-1}^{1}\frac{\sin\theta dx}{1-2x\cos\theta +x^2}$$ is discontinuous where $\theta \in [0,2\pi]$
I am not able to find $f(\theta)$ in terms of $\theta$,$sin\theta$ in the numerator i can take out but $cos\theta$ in the denominator is tr... | Let
$$ f(\theta)=\int_{-1}^{1}\frac{\sin\theta dx}{1-2x\cos\theta +x^2}. $$
If $\theta=0,\pi$ or $2\pi$, there is nothing to do since $f(\theta)=0$. Otherwise, using
$$ \int\frac{1}{(x-a)^2+b^2}dx=\frac{1}{b}\arctan\frac{x-a}{b}, $$
we have
\begin{eqnarray}
f(\theta)&=&\sin\theta\int_{-1}^{1}\frac{dx}{(x-\cos\theta)^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the equation of the straight line $y=ax+b$?
If I have a circle $x^2+y^2=1$ and line that passes trough $(0,0)$
and I know the angle between the line and the axis. If, for example, the angle is $\frac{\pi}{3}$, how can I find the equation of the straight line $y=ax+b$?
| You do know that $b$ is called the "$y$-intercept", right?
There is a reason for that: $b$ is the value of $y$ where the line
crosses the $y$-axis.
Now look at the figure and see where the line crossed the axis and what
the value of $y$ is at that point.
Recall that $a$ is the slope of the line, which you can get from
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
About the elements of a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$ Let $G$ be a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$. I would like to prove that for every $g \in G$, $\det(g) \in \lbrace -1,1 \rbrace$.
Here are my ideas : since $G$ is a finite subgroup of $\mathrm{GL}(\mathbb{R}^{n})$, the elements o... | I'd like to show a somewhat simpler proof, without using eigenvalues theory and stuff.
Since $\det$ is a group homomorphism $\det : \mathrm{GL}(\mathbb{R}^{n}) \to \mathbb{R}^{\times}$ (the codomain being the multiplicative group of reals) the image of $G$ under $\det$ should be a finite subgroup of $\mathbb{R}^{\times... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Limit behavior of a definite integral that depends on a parameter. Let $A>0$ and $0\le \mu \le 2$. Consider a following integral.
\begin{equation}
{\mathcal I}(A,\mu) := \int\limits_0^\infty e^{-(k A)^\mu} \cdot \frac{\cos(k)-1}{k} dk
\end{equation}
By substituting for $k A$ and then by expanding the cosine in a Taylo... | Here we provide an answer for $\mu=2q/p$ where $p$ is a positive integer and $p+1\le 2q \le 2 p$. By using the multiplication theorem for the Gamma function we have shown that:
\begin{eqnarray}
&&{\mathcal I}(A,\mu) =\\
&&\frac{1}{\mu} \sqrt{\frac{(2\pi)^{2q-p}}{p 2q}} \sum\limits_{r=1}^q \left(\frac{p^{p/q}}{(2q)^2} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Laplace Transform of a Heaviside function Find the Laplace transform.
$$g(t)= (t-1) u_1(t) - 2(t-2) u_2(t) + (t-3) u_3(t)$$
I understand that the $\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$
Finding $F(s)$ is the hard part for me. My professor has used, for example,
$$f(t-2)=t^2$$
let $$s = t-2$$
$$t= s+2$$
$$f(s) = ... | Your function is $f(t)=t$ but not $f(t)=1$.
You need to use the formula
$$\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$$
3 times with $c=1,2, 3$ to get the answer.
The answer derived by Leucippus looks correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Strongly equivalent metrics are equivalent I have $d,d'$ metrics in X and that they are strongly equivalent. In my case, this means that:
$\exists\alpha,\beta\in\mathbb{R}_{++}$ so that $\alpha d<d'<\beta d$
I want to show that they are equivalent by proving that:
$\forall x\in X,\forall\epsilon>0,\exists\delta>0:B_d(x... | Two metrics are strongly equivalent if they give rise to equivalent notions of strong convergence.
In this sense we want to prove that $$d(x_n,x ) \to 0 \Leftrightarrow d'(x_n,x ) \to 0$$
Assume that for every $B_{d'}(x,\epsilon)$ there is a $\delta$ such that $B_{d}(x,\delta) \subset B_{d'}(x,\epsilon)$
as $x_n \to x$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
why does the reduced row echelon form have the same null space as the original matrix? What is the proof for this and the intuitive explanation for why the reduced row echelon form have the same null space as the original matrix?
| Say we have an $n$x$n$ matrix, $A$, and are going to row reduce it. Every time we do a row operation, it is the same as multiplying on the left side by an invertible matrix corresponding to the operation. So at the end of the process, we can conclude something like $B = L_1L_2...L_kA$, where $B$ is the row reduced matr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
} |
Why $\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$? I'm reading Nahin's: Inside Interesting Integrals.
I've been able to follow it until:
$$\lim_{\Delta x\to 0} \cfrac{\int_{x}^{x+\Delta x}f(u) du}{\Delta x}=\cfrac{f(x)\Delta x}{\Delta x}$$
I don't know what jus... | It is not actually rigorous what he says... but what he means is that, when $\Delta x $ approches $0$, then $f$ is approximately constant (assuming it is continuous). What you can do is, for instance:
Fix $\epsilon >0$. There exists $\delta>0$ such that: $f(x)-\epsilon<f(u)< f(x)+\epsilon$ for all $u$ in the $\delta$-n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1379789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
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