Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Trick to proving a group has exactly one idempotent element - Fraleigh p. 48 4.31
If $*$ is a binary operation on a set $S$, an element $x \in S$ is an idempotent for $*$ if $x * x = x$.
Let $\langle G, *\rangle$ be a group and let $x\in G$ such that $x*x = x.$ Then $x*x = x*e$, and by left cancellation, $x = e$, s... | This does not look so much like prognostication to me. The hint is trying to make explicitly clear something that is happening. You could instead start by right multiplying both sides by $x^{-1}$. Then you get $xxx^{-1} = xx^{-1} \implies x(e) = e \implies x = e$. By the uniqueness of the identity, there is only one su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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component of a vector $\mathbf a$ onto vector $\mathbf b$ Just a bit unsure about the definition. When I look online and at other questions on this site it says that the component of $\mathbf a$ onto $\mathbf b$ is $\dfrac{\mathbf a\cdot\mathbf b}{|\mathbf b|}$ but when I look in my notes and at my lecture slides it sa... | Let ${\bf a} \in V^{n+1}$ and define $A=\text{span}({\bf a})$ and $A^{\perp} = \{ {\bf w} \in V : \langle {\bf w},{\bf a}\rangle=0 \}$.
Notice that $V = A \oplus A^{\perp}$. For any ${\bf b} \in V$ we can uniquely decomposed ${\bf b}$ as a component of $A$ and a component of $A^{\perp}$. By asking for the component of ... | {
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Limits of square root $$\lim_{x\to\infty}\left(\sqrt{x+\sqrt{x+\sqrt{x + \sqrt x} }}-\sqrt x\right) $$
(original screenshot)
Compute the limit
Can you please help me out with this limit problem
| Hint for a simpler one: $$\lim_{x \to \infty} \sqrt{x+\sqrt x}-\sqrt x=\lim_{x \to \infty}\sqrt x\left(\sqrt{1+\frac 1{\sqrt x}}-1\right)$$
| {
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Closed form solution to a recurrence relation (from a probability problem) Is there a closed form solution to the following recurrence relation?
$$P(i,j) = \frac{i^{5}}{5i(5i-1)(5i-2)(5i-3)(5i-4)}\sum\limits_{k=0}^{j-5}(1-P(i-1,k))$$
where $P(i,j)=0$ for $j<5$.
The above recurrence is the solution I obtained to a prob... | Odlyzko handles such problems in "Enumeration of Strings" (pages 205-228 in Combinatorial Algorithms on Words, Apostolico and Galil (eds), Springer, 1985). What you are looking for is strings over an alphabet of 5 symbols such that the string 12345 appears for the first time at the end. The autocorrelation polynomial o... | {
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Does this integral have a closed form? I was working with this problem in an exam:
Given $\lambda\in(-1,1)\subset\Bbb R$, find
$$f
(\lambda)=\int_{0}^{\pi}\ln\left(1+\lambda \cos x\right)\mathrm{d}x
$$
My try: put $\delta\in (0,1)$ such that $\lambda\in(-\delta,\delta)$. Using the power series of $\ln(1+x)$ and uni... | The answer to your integral is
$$
f(\lambda) = \pi \ln\left(\frac{1+\sqrt{1-\lambda^2}}{2}\right)
$$
Now to derivation. Given that
$$
\int_0^\pi \cos^n(x) \mathrm{d}x = \frac{1+(-1)^n}{2} \frac{\pi}{2^n} \binom{n}{n/2}
$$
Doing the series expansion of the integrand in $\lambda$ and interchanging the order of su... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Any two groups of three elements are isomorphic - Fraleigh p. 47 4.25(b) The answer has no details. Hence maybe the answer is supposed to be quick. But I can't see it?
Hence I took two groups. Call them $G_1 = \{a, b, c\}, G_2 = \{d, e, f\}$.
Then because every group has an identity, I know $G_1, G_2$ has one each.
Hen... | Let $p$ be a prime and $G$ a groups of order $p$. If $a\in G$ is not the neutral element show that the homomorphism $\mathbb Z\to G$ given by $1\mapsto a$ induces an isomorphims $\mathbb Z/p\mathbb Z\to G$. Conclude that any two groups of order $p$ are isomorphic.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
} |
Action on G via Automorphism Here is an exercise from Isaacs, Finite Group Theory, $4D.1$:
Let $A$ act on $G$ via automorphism, and assume that $N \trianglelefteq G$ admits $A$ and that $N \geq C_G(N)$. Assume that $(|A|,|N|)=1$. If $A$ acts trivially on $N$, show that its action on $G$ is trivial.
Hint: Show that $[G... | I can only prove the conclusion is true if the assumption (|A|,|N|)=1 become to (|A|,|G|)=1.
Clearly, [A, N, G]=1, and [N,G,A]=1. By three subgroups lemma, [G,A,N]=1, which implies that $[G,A]\le C_G(N) \le N$. By the condition (|A|,|G|)=1, we get $G=C_G(A)[G,A]\le C_G(A)N=C_G(A)$, which implies $[G,A]=1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Divisibility by seven Given number n, whose decimal representation contains digits only $1, 6, 8, 9$. Rearrange the digits in its decimal representation so that the resulting number will be divisible by 7.
If number is m digited after rearrangement it should be still $m$ digited.
If not possible then i need to tell "no... | If there is no condition that each of these digits need appear at least once, then it is not possible. Consider $1111$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/617366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving equations of type $x^{1/n}=\log_{n} x$ First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way.
I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which I'm trying to find an explanation for. I've noticed that graphs of functi... | $$\begin{align}\sqrt[n]x&=\log_nx\\\\x&=t^n\end{align}\ \Bigg\}\iff t=\frac n{\ln n}\cdot\ln t\quad;\quad t=e^u\iff e^u=\frac n{\ln n}\cdot u\iff$$
$$(-u)\cdot e^{-u}=-\frac{\ln n}n\iff u=-W\bigg(-\frac{\ln n}n\bigg)\iff x=t^n=(e^u)^n=e^{nu}$$
$$x=\exp\bigg(-n\cdot W\bigg(-\frac{\ln n}n\bigg)\bigg)$$ where W is the Lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/617419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Undergraduate-level intro to homotopy I'm looking for an undergraduate-level introduction to homotopy theory.
I'd prefer a brief (<200pp.) book devoted solely/primarily to this topic. IOW, something in the spirit of the AMS Student Mathematical Library series, or the Dolciani Mathematical Expositions series, etc.
Edit... | My favorite ~200 page introduction to homotopy theory: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf. You can find a physical copy on Amazon for relatively cheap as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/617513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Uniqueness of morphism (reasoning in categorial language). This question is related to a previous question of mine.
I figured out that maybe the right thing to ask for is for someone to solve one of the problems in Aluffi's book, that way I’ll know what the reasoning should look like.
The following question appears in ... | The question isn't right. There isn't a unique morphism $G\times G\to H\times H$, there's a unique morphism... with such and such properties.
I recommend beginning your thinking in the category of sets. If I have a map of sets $G\to H$, is there a "natural" map $G\times G \to H\times H$ that springs to mind? Is ther... | {
"language": "en",
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Need Help to convert a grammar into Chomsky Form I have to convert the following grammar into Chomsky Form
$$( \Sigma=\{a,b,c,+\}, \Sigma_Q=\{S,V\},I=S)$$
$$S -> S+S|V$$
$$V -> a|b|c$$
My idea is the following:
$$S_0 \rightarrow S$$
$$S \rightarrow ST|V$$
$$T \rightarrow +S$$
$$V \rightarrow A|B|C$$
$$A \rightarrow a... | In a grammar in Chomsky normal form there can only be two types of productions: $A\to BC$ and $A\to a$ where $A,B,C$ are nonterminals (variables, $\Sigma_Q$ in your notation) and $a$ is terminal ($\Sigma$ in your notation).
In particular, "chain productions" which are of the form $A\to B$ are not allowed. You have seve... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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irreducibility of $x^{5}-2$ over $\mathbb{F}_{11}$. I am tasked to show that $x^{5}-2$ is irreducible over $\mathbb{F}_{11}$ the finite field of 11 elements. I've deduced that it has no linear factors by Fermat's little theorem. But showing it has no quadratic factors is proving harder.
My approach so far is assume it... | If $x^5 - 2$ has a quadratic factor, it has a root $\alpha$ in a quadratic extension $K$ of $\mathbb{F}_{11}$. Since the group of units of $K$ has $120$ elements, $\alpha^{120} = 1$. Now $\alpha^5 = 2$, so the order of $2$ must divide ...?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
$\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$. $\ln\left(1+x\right)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}+\dots$ when $|x|<1$ or $x=1$. Why is the restriction $|x|<1$ or $x=1$? I know from Wikipedia that it is because out of this restriction, the function's "Taylor approximation" is not fa... | The standard approach is to consider the derivatives of both functions. On the right you obtain $\sum(-x)^n$, which you may recognize as a geometric series, with sum $\frac1{1+x}$, the formula being valid for $|x|<1$. But this is also the derivative of the function on the left. This means the two functions are equal, u... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $n^{-2}[x+g(x)+g\circ g(x)+\cdots +g^{\circ n}(x)]$ converges when $n\to\infty$ Let $f:\mathbb R\to\mathbb R$ be a periodic function with period $1$. We assume that $f$ is Lipschitz continuous, and in particular, we assume that there exists an $L\in (0,1)$, such
that
$$
|f(x)-f(y)| \le L|x-y|, \quad \... | First attempt. One needs to use the Stolz–Cesàro theorem. According to this theorem:
\begin{align}
\lim_{n\to \infty}\dfrac{x+g(x)+g(g(x))+\cdots +\underbrace{g(g(\cdots(g(x))))}_{n-1}}{n^2}&=\lim_{n\to \infty}\frac{\underbrace{g(g(\cdots(g(x))))}_{n}}{2n+1}\\=&\lim_{n\to \infty}\frac{\underbrace{g(g(\cdots(g(x))))}_{n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Dealing with Generating Functions accurately I'm currently working through Iven Niven's "Mathematics of Choice." In the chapter on Generating Functions, the exercises include problems like:
How many solutions in non-negative integers does the equation $2x+3y+7z+9r=20$ have?
This of course ends up being the same as find... | When you can only reasonably evaluate by multiplying out the polynomials, you've described what is pretty much the only approach. Just multiply out, simplest factors first, and keep dropping all coefficients higher than the one you are interested in. In some cases this is all you can do, unfortunately. (Or, use a CA... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability of combinatoric explosion. Someone posted a puzzle on a forum.
Now this is fairly easy. Probability of reverse: The crate doesn't contain a unusual = the crate contains a weapon or none of two found crates contains a unusual. This produces a quadratic equation, I discard the solution outside the [0,1] rang... | You found the solution for the question :
Let $p$, the probability of not having an unusual :
*
*Either you get a weapon
*or two crates without unusual
$$p=0.35+0.55*p^2$$
This polynomial have one root in $[0,1]$, hence
$$p=\frac{10-\sqrt{23}}{11}\approx 0.4731$$
Your question can be answered the same way :
Let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/617967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Averages question
There are five sticks. The average length of any four of them is 600. What is the average of all five?
Is it possible to find the average of all with just this information given?
| Let the lengths of the five sticks be $a_i,1\le i\le 5$
So, $\displaystyle a_1+a_2+a_3+a_4=4\cdot600=2400 \ \ \ \ (1)$
$\displaystyle a_1+a_2+a_3+a_5=4\cdot600=2400 \ \ \ \ (2)$
$\cdots$
Subtracting we get $a_4=a_5$
Similarly, we can show that the length of all the sticks are same.
I should not say anything more:)... | {
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Smallest non-commutative ring with unity
Find the smallest non-commutative ring with unity. (By smallest it means it has the least cardinal.)
I tried rings of size 4 and I found no such ring.
| When one thinks of a noncommutative ring with unity (at least I), tend to think of how I can create such a ring with $M_n(R)$, the ring of $n \times n$ matrices over the ring $R$. The smallest such ring you can create is $R=M_2(\mathbb{F}_2)$. Of course, $|R|=16$. Now it is a matter if you can find a even smaller ring ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/618111",
"timestamp": "2023-03-29T00:00:00",
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Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$
Is $f(x)=x+\frac{x}{x+1}$ uniformly continuous on $(0,\infty)$
Going from the epsilon delta definition we get:
$$\forall x,y>1,\text{WLOG}:x>y \ ,\ \forall\epsilon>0,\exists\delta>0 \ s.t. \ |x-y|<\delta\rightarrow |x+\frac{x}{x+1}-y-\frac{y}{y+1}|<\epsi... | Take a smaller bite. Note that a sum of uniformly continuous functions is uniformly continuous. We know $x\mapsto x$ is uniformly continuous, so we just deal with $x/(x+1)$.
Also, notice that
$${x\over x + 1 } = 1 - {1\over x + 1}.$$
Now the job is to show that $x\mapsto 1/(x + 1)$ is uniformly continuous. Can you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/618174",
"timestamp": "2023-03-29T00:00:00",
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Double Euler sum $ \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} $ I proved the following result
$$\displaystyle \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} =- \frac{97}{12} \zeta(6)+\frac{7}{4}\zeta(4)\zeta(2) + \frac{5}{2}\zeta(3)^2+\frac{2}{3}\zeta(2)^3$$
After consideration of powers of polylogarithms.
You can refer to t... | The following new solution is proposed by Cornel Ioan Valean. Based on a few ideas presented in the book, (Almost) Impossible Integrals, Sums, and Series, like the Cauchy product of $(\operatorname{Li}_2(x))^2$, that is $\displaystyle (\operatorname{Li}_2(x))^2=4\sum_{n=1}^{\infty}x^n\frac{H_n}{n^3}+2\sum_{n=1}^{\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/618256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 1,
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Find the limit of $S_n=\sum_{i=1}^n \big\{ \cosh\big(\!\!\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$, as $n\to\infty$? $S_n=\sum_{i=1}^n\big\{ \cosh\big(\frac{1}{\sqrt{n+i}}\!\big) -n\big\}$ as $n\to\infty$
I stumbled on this question as an reading about Riemannian sums as in
$$
\int_a^b f(x)\,dx =\lim_{x\to \infty}\frac{1}... | We have
$$S_n=\sum_{i=1}^n \cosh\left(\frac{1}{\sqrt{n+i}}\right) -n=\sum_{i=1}^n \left(\cosh\left(\frac{1}{\sqrt{n+i}}\right) -1\right)$$
and by the Taylor-Lagrange inequality we have
$$\left|\cosh\left(\frac{1}{\sqrt{n+i}}\right) -1-\frac{1}{2(n+i)}\right|\le\frac{C}{n^{3/2}}$$
hence
$$\left|S_n-\sum_{i=1}^n\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/618322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$f$ is measurable in complete measure space, then there is a $g$ measurable in the uncomplete space such that $f = g$ almost everywhere? I am trying to understand the proof a theorem in Folland. The theorem says if $(X,M,\mu)$ is a measure space with completion $(X,\bar{M},\bar{\mu})$, and $f$ is $\bar{\mu}$-measurable... | If $E\in\overline{\mathcal M}$, then there exist $E_1,E_2\in {\mathcal M}$, such that
$$
E_1\subset E\subset E,\,\,\mu(E_2\smallsetminus E_1)=0 \quad \text{and}\quad \overline{\mu}(E)=\mu(E_1)=\mu(E_2).
$$
Thus
$$
\chi_{E_1}=\chi_{E_2}=\chi_E\quad \text{$\overline{\mu}$-a.e.}
$$
For a general $f$, we assume that $f\ge... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Covering a Riemannian manifold with geodesic balls without too much overlap I'm looking for a proof of the following fact:
Let $M$ be a compact Riemannian manifold. There is a natural number $h$, such that for any sufficiently small number $r>0$, there exists a cover of $M$ by geodesic balls of radius $r$ such that any... | There is a neighborhood about each point where the exponential map is bilipschitz with constants arbitrarily close to 1. This implies that the images of balls of radius $R$ under the exponential map are almost balls of radius $R$, i.e. they contain balls of radius, say, $3R/4$ and lie in balls of radius $5R/4$. Cover y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/618475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 0
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Is Linear transformations $T_1,T_2 : \mathbb{R}^n\rightarrow \mathbb{R}^n$ Invertible? Let $T_1$ and $T_2$ be two Linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$.
Let $\{x_1,x_2,\cdots,x_n\}$ be a basis of $\mathbb{R}^n$. Suppose that $T_1(x_i)\neq 0$ for every $1\leq i\leq n$ and that $x_i\perp Ker (T_2)$... | You are wrong that $T_1$ is invertible; it need not be. Indeed for $n=2$, $\{x_1,\ldots,x_n\}$ the standard basis, and $T_1$ given by the matrix $$\begin{pmatrix}1&-1\\-1&1\end{pmatrix}$$ neither of $T_1(x_1)=x_1-x_2$ nor $T_1(x_2)=x_2-x_1$ is zero, but $T_1(x_1+x_2)=0$. Moreover one can take $T_2=I$ to satisfy $x_i\pe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Associated primes of a quotient module.
Let $R$ be a Noetherian ring, $M$ a finitely generated $R$-module and $p\in \operatorname{Ass}(M)$. Suppose $x$ is an $M$-regular element and $q$ is a minimal prime over $I=(p,x)$. How can we show that $q\in \operatorname{Ass}(M/xM)$?
Note: Since $q$ is minimal prime over $(p,x... | I dont feel comfortable answering my own question but i am sketching a solution as I was asked to do so above. As seen in the solution by Youngsu and by YACP, we may assume that $(R,m)$ is a Noetherian local ring and $m$ is minimal over $(p,x)$. As $p\in \operatorname{Ass}M$, $p=(0:y)$ for some $0\neq y\in M$. By Krull... | {
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"url": "https://math.stackexchange.com/questions/618641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does uniform integrability plus convergence in measure imply convergence in $L^1$?
Does uniform integrability plus convergence in measure imply convergence in $L^1$?
I know this holds on a probability space. Does it hold on a general measure space? I have tried googling. It returned very few results on UI on measure ... | On $\Bbb R$ with Lebesgue measure, the sequence $(f_n)$ defined by $f_n={1\over n}\cdot \chi_{[n,2n]}$ for each $n$ would furnish a counterexample. Here, $\chi_A$ is the indicator function on the set $A$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/618714",
"timestamp": "2023-03-29T00:00:00",
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Finding powers of ideals. Can any one assist me finding the product of ideals.
Let $R=K[x_1,x_2,\dots,x_n]$, and $I=x_1R+x_2R+\dots+x_nR$. Then what is $I^m$?
As $x_1^2+x_2^2$ belongs to $I^2$ it cannot be written as the product of two elements in $R$.
| In general, if $I_1 = \sum_{i=1}^n x_i R$ and $I_2=\sum_{j=1}^m y_jR$, then $$I_1I_2 = \sum_{i=1}^n \sum_{j=1}^m x_iy_jR$$ is generated by the products of the generators of $I_1$ with the generators of $I_2$.
In your case it is $$I^n = \sum_{e_1+\dots + e_n=n} x_{e_1}x_{e_2}\cdots x_{e_n}R.$$ So if $R$ is the ring of p... | {
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Elementary number theory problem Let $X = \{n \in \mathbb{N}: 6 \times n\,\, \text{does not consist of} \ 0,1,2,3 \, \text{or} \ 4\}.$ For eg, $93 \in X$ because $6 \times 93=558.$
Could anyone advise me how to prove there exists $2$ natural numbers such that the value of every $n \in X$ consists of at least one o... | $1*6=6$ and $98*6=588$ also $96*6=576$ therefore we wish to prove that any number contains either a $1$ or a $9$
Suppose a number does not contain a 1 or a 9.
Then 6n has more digits than n (because 1 is not a digit). Therefore the left-most digit must be 5. Since there is no 9. So think of doing the multiplication by ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to prove that, for each $n\in \mathbb N$ $$
\frac{x^{n+1}-1}{x-1} = 1 + x + x^2 + \dots + x^n
$$
where $x\neq 1$, $x\in \mathbb R$.
I am really tired to prove that questions. I can not understand any one. Please help me.....How to prove that, for each $n\in \mathbb N$ (using Mathematical induction)
My try
how to c... | This is equivalent to $(x-1)(1+x+\cdots x^n) = x^{n+1}-1$
Base case, $n=0$,
$$(x-1)(x^0)=x^{0+1}-1$$
Now assume it is true up to $n-1$, so
$$(x-1)(1+x+\cdots x^{n-1}+x^n) = [(x-1)(1+x+\cdots x^{n-1})] + (x-1)(x^n) $$
$$= [x^n - 1] + [x^{n+1}-x^n] =x^{n+1}-1 $$
Now divide the $(x-1)$ out to RHS
$$(1+x+\cdots x^{n-1}+x^n... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Open Sets Definition I understand that the definition of an open set in a Metric Space and this can transfer over if you're dealing with a Metric Topological Space. However, I'm not sure if there is a standard definition of an open set in a Topological Space. I have read that it may vary under constraints.
I found one ... | The definition is equivalent to the usual definition in a topological space. When you have a metric space $(X,\rho)$, you can consider the topological space $(X,\tau_\rho)$ where the topology $\tau_\rho$ consists of all sets in $X$ that are unions of open balls $B(x,\varepsilon)$. Observe that
$(1)$ $X$ is a union of o... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Distance between points in hyperbolic disk models I was puzzeling with the distance between points in hyperbolic geometry and found that the same formula is used for calculating the length in the Poincare disk model as for the Beltrami-Klein model the formula
$$ d(PQ)=\frac{1}{2} \left| \log \left(\frac{|QA||BP|}{|PA|... | There are two common versions of the Poincaré disk model:
*
*Constant curvature $-1$; metric element $\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}$
*Constant curvature $-4$; metric element $\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$
The difference is where you stick that annoying factor of $4$. Wikipedia uses version 1, and its de... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Finding a Uniformizer of a Discrete Valuation Ring Suppose I have a discrete valuation ring. Then what are some techniques for explicitly finding a uniformizer? I'm especially interested in situations where the ring is given similarly to the following example.
Suppose we take the projective variety defined by the equat... | As the local ring at a point $P$ is defined as $\mathcal{O}_{C,P} = \varinjlim_{P \in U} \mathcal{O}_C(U)$, given any point $P \in C$ we can first pass to whichever affine open that contains $P$, and then calculate the direct limit over all opens in that affine $U$. As passing to an affine open corresponds to dehomoge... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Analytical Geometry problem with complex numbers - alternate solutions. The question is to show that the equation of the lines making angles $45^\circ$ with the line:
$$ \bar{a}z + a\bar{z} + b = 0; \;\;\;\;\; a,z \in \mathbb{C}, b \in \mathbb{R} $$
and passing through a point $c \in \mathbb{C}$ is:
$$ \dfrac{z-c}{a} \... | Drawing figures will help you understand my answer.
Let $L$ be the given line. Let $D(d)$ be the intersection point of $L$ and the perpendicular line of $L$. Noting that the normal vector to $L$ is $a$, there exists $k\in\mathbb R$ such that
$$d+ka=c\iff d=c-ka.$$
In the following, we suppose that $k\not=0.$ The case $... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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which of the following subsets of $\mathbb R^2$ are uncountable.
*
*$\{\,(a,b)\in \mathbb R^2\mid a\leq b\,\}$
*$\{\,(a,b)\in\mathbb R^2\mid a+b\in\mathbb Q\,\}$
*$\{\,(a,b)\in \mathbb R^2\mid ab\in \mathbb Z\,\}$
*$\{\,(a,b)\in\mathbb R^2\mid a,b\in \mathbb Q\,\}$.
I know option 1 is uncountable and option 4 is... | $\mathbb{Q}$ being countable,$\mathbb{Q^2}$ is also countable and consequently $4$ is countable.Choosing a=y,b=n/y where n is a natural number and y is a positive irrational and less than 1 we can satisfy the requirement of $3$ in uncountably many ways, irrationals in ($0,1$) being uncountable.
$a=x-y,b=x+y \text{ (whe... | {
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Find the following limit: $\lim_{n\to \infty}\frac{n!e^n}{n^n}$. How can I find the limit below:
$$\lim_{n\to \infty}\frac{n!e^n}{n^n}$$
I tried to use Stirling's approximation and got
$$
\lim_{n\to\infty}\frac{n!e^n}{n^n}=\lim_{n\to\infty}\frac{{\sqrt{2n\pi}(n/e)^n}}{(n/e)^n}=\lim_{n\to\infty}\sqrt{2n\pi}=+\i... | Let $$a_n =\frac{n! e^n}{n^n}$$ then $$\frac{a_{n+1}}{a_n} = e \left(1+\frac{1}{n}\right)^{n} \geq e \Rightarrow a_{n+1} \geq e\ a_n$$ Since $a_1=e$, $a_n \geq e^n \rightarrow \infty$, when $n\rightarrow \infty$, which means $\lim\ a_n = +\infty$
EDIT: Scratch that, I made a mistake, $$\frac{a_{n+1}}{a_n} \geq e \left(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I calculate this series? How can I calculate the series
$\displaystyle{%
\sum_{n=0}^{\infty }{\left(-1\right)^{n}
\over \left(2n + 1\right)\left(2n + 4\right)}\,\left(1 \over 3\right)^{n + 2}\
{\large ?}}$
| $\newcommand{\+}{^{\dagger}}%
\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{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displaystyle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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"answer_id": 2
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The difference between $\Delta x$, $\delta x$ and $dx$ $\Delta x$, $\delta x$ and $dx$ are used when talking about slopes and derivatives. But I don't know what the exact difference is between them.
| Let $y=f(x)$.
$\Delta x$ and $\delta x$ both denote the change in $x$ (or increment of $x$). Some books prefer to use capital delta $\Delta$ and some lowercase delta $\delta$. The change can be small or large, but often we talk about the case that $\Delta x$ is small and especially $\Delta x\to 0$.
In mathematics, $dx$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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"answer_id": 4
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Help with finding tangent to curve at a point
Find an equation for the tangent to the curve at $P\left( \dfrac{\pi}{2},3 \right )$ and the horizontal tangent to the curve at $Q.$
$$y=5+\cot x-2\csc x$$
$y\prime=-\csc ^2 x -2(-\csc x \cot x)$
$y\prime= 2\csc x \cot x - \csc ^2 x\implies$ This is the equation of the... | The first part looks good. For the second part as you see the solution to the equation gives
$$2 \cot x \csc x - \csc^2 x = 0 \Rightarrow x=2n \pi \pm\frac{\pi}{3}.$$
Choose $x=\frac{\pi}{3}$ as it is one such solution. We find the point on $y$ to be
$$5+\cot\left( \frac{\pi}{3} \right)-2\csc\left( \frac{\pi}{3} \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/619753",
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"question_score": "2",
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Information Entropy Applied to Complexity Theory I was just wondering whether or not information entropy has significant applications to complexity theory.
I ask because of a simple example I thought of. It comes from a riddle. Suppose you had 8 balls, one of which is slightly heavier than the rest but all of them look... | Information theory has been used to show lower bounds in communication complexity. This workshop at STOC '13 has talks going over the basics of the theory and some applications.
The description of Amit Chakrabarti's talk "Applications of information complexity I" mentions the use of information complexity in answering ... | {
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Plane Geometry problem I came across this problem in a mathematics-related facebook group. Could anyone advise on the solution to it(i.e. hints only)? Thank you.
|
Since triangle ABC is equilateral, we know that angle BAC is 60º. Therefore, angle PAQ is also 60º; we have a circle theorem that tells us that angle POQ is thus twice that measure or 120º. The diameter AD bisects that angle, so angle POD is 60º.
Diameter AD is also an altitude of equilateral triangle ABC: thus, if... | {
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"timestamp": "2023-03-29T00:00:00",
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Looking for a more elegant / generic proof of the reducibility of a polynomial in $K[[X,Y]]$ The polynomial $P(X,Y)=XY-(X+Y)(X^2+Y^2)$ is irreducible in $K[X,Y]$, as a sum of two homogenous forms of degree 2 and 3 ($K$ is supposed to be algebraically closed). To look at the irreducibility in $K[[X,Y]]$, I first worked ... | I got two interesting answers on MO here https://mathoverflow.net/a/153102/3333. One is elegant by easily generalizing the use of Hensel's lemma in case the lowest degree term of the polynomial is a product of distinct factors. The other is based on a republished article and more systematic.
| {
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"source": "stackexchange",
"question_score": "3",
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Orthornomal matrices Is there a more direct reason for the following:
If the columns of $n\times n$ square matrix are orthonormal, then its rows are also orthonormal.
The standard proof involves showing that left inverse of a matrix is same as the right inverse and thereby concluding that if $Q^TQ = I$, then $QQ^T = ... | If $Q$ is orthonormal, its transpose is its inverse, and vice versa. Understanding this is key to your question. You are asking for a geometric understanding of this, rather than a symbolic algebra understanding. So let's give that a try.
Because columns are orthonormal, then what does $Q^T$ do to the $i$th column of $... | {
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Taylor expansion of $e^{\cos x}$ I have to find the 5th order Taylor expansion of $e ^{\cos x}$. I know how to do it by computing the derivatives of the function, but the 5th derivative is about a mile long, so I was wondering if there is an easier way to do it.
I'd appreciate any help.
| Use
$$e^{\cos x}=e\cdot e^{\cos x-1}.$$
Then substitute the power series expansion of $\cos x-1$ for $t$ in the power series expansion of $e^t$. What makes this work is that the series for $\cos x-1$ has $0$ constant term. For terms in powers of $x$ up to $x^5$, all we need is the part $1+t+\frac{t^2}{2!}$ of the powe... | {
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"source": "stackexchange",
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evaluation of $\lim_{x\rightarrow \infty}\frac{\ln x^n-\lfloor x \rfloor }{\lfloor x \rfloor} = $ (1) $\displaystyle \lim_{x\rightarrow \infty}\frac{\ln x^n-\lfloor x \rfloor }{\lfloor x \rfloor} = $, where $n\in \mathbb{N}$ and $\lfloor x \rfloor = $ floor function of $x$
(2)$\displaystyle \lim_{x\rightarrow \infty}\l... | No. You cannot write so. In this case, going back to the definition is the best. So, in order to solove your questions, use the followings :
$$x-1\lt \lfloor x\rfloor \le x\Rightarrow \frac{x-1}{x}\le \frac{\lfloor x\rfloor}{x}\lt \frac{x}{x}\Rightarrow \lim_{x\to\infty}\frac{\lfloor x\rfloor}{x}=1$$
where $x\gt0.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
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Integral of the product of an bounded and any continous functions If $u:[0,1]\rightarrow \mathbb{R}$ is bounded measurable function so that for all $v\in C[0,1]$
$$\int_0^1uvdx=0$$then show that u is zero almost everywhere on $[0,1].$
Thanks in advance for any help!
| Since $u$ is bounded and measurable, we have $u \in L^2[0,1]$. Since $L^2[0,1]$ is the completion of $C[0,1]$ with respect to the 'inner product' norm, we can find $u_n \in C[0,1]$ such that $u_n \to u$. Since $u_n$ is continuous, we have $\int u_n u = 0$, and since $| \int (u-u_n)u | \le \|u-u_n\| \|u\|$, we see that... | {
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"url": "https://math.stackexchange.com/questions/620202",
"timestamp": "2023-03-29T00:00:00",
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prove that the Galois group $Gal(L:K)$ is cyclic Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is separable over $L$), then prove that the Galois group $Gal(L:K)$ is cycli... | Problem is true if we impose the condition $K$ contains a primitive root of unity.
Proof: Letting $G_n$ be group of all root unity we can show there is an injection $Gal(E/K) \to G_n$ given by $\sigma \to \frac{\sigma(a)}{a}$. So, we done.
If we impose the condition that $a^{i}$ doesn't belong to $K$ for all $i<n$ the ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the interpretation of the eigenvectors of the jacobian matrix? I'm trying to think about the jacobian matrix as a abstract linear map.
What is the interpretation of the eigenvalues and eigenvectors of the jacobian?
| What Frank Science has said in the question comment above is right. I'm simply expanding on his comment here:
Since the Jacobian has eigenvectors, it is square i.e. the input and output space have same dimensions. If there is some kind of natural interpretation to the input and output basis and they can be mapped to ea... | {
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"timestamp": "2023-03-29T00:00:00",
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Weierstrass $\wp$-function: $(\partial_z \wp(z,\omega))^2$ Let $\vartheta(z,\omega)$ be the Riemann theta function. For $j \in \mathbb{Z}$ let $c_j$ be the coefficient of $z^{j}$ in the Laurent expansion of $\partial_z \log \vartheta \left(z + \frac{1 + \omega}{2}, \omega \right)$ at $z$ = 0. The Weierstrass $\wp$ func... |
I have successfully shown, that $\wp$ is $\omega$- and $1$-periodical and a few other properties.
If among these other properties are (considering $\wp$ as a function of $z$ only here)
*
*$\wp$ is even,
*$\wp$ has order $2$ (that is, takes each value in $\widehat{\mathbb{C}}$ exactly twice in a fundamental parall... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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kernel of a monic morphism Problem
Suppose $\mathscr{C}$ is an arbitrary category with zero object. $A$ and $B$ are two objects of $\mathscr{C}$. Let $f\in Mor_\mathscr{C}(A,B)$. It's given that $f$ is monic.
I need to show that $f$ has a kernel.
My claim Object $A$ with morhism $\iota$(will be def'd below) is the ker... | Well, basically on the right direction, but not exactly the right track. You should start it over and simplify.
You miss the observation that the domain of the kernel is going to be the zero object itself, and not $A$.
Claim: The kernel of a monic $f:A\to B$ will be the unique arrow $Z\to A$.
Proof: (your turn)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/620531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Absolute value equality involving complex analysis I'm preparing for a complex analysis prelim which I'll take next summer by consulting Ahlfor's Complex Analysis: An Intro to the Theory of Analytic Functions of One Variable (3rd edition). My question pertains to Exercise 4 of Chapter 1, Section 1.5, and it is as foll... | You can rotate the complex plane so that $a$ is real.
Let $z= x+ i \,y$. Then the left hand side is
$$
\sqrt{(x+a)^2+y^2} + \sqrt{(x-a)^2+y^2}$$
is minimum when $y=0$, i.e when $z$ is real. Clearly all $z$ in the interval $[-a,a]$ satisfies the condition.
Hence $z=0$ is the minimum, $z=\pm a$ is maximum in magnitude
N... | {
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"timestamp": "2023-03-29T00:00:00",
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Differences between $L^p$ and $\ell^p$ spaces Could someone explain some differences between the $L^p$ and $\ell^p$ spaces? Thanks a lot.
| Either L or ℓ denotes the first alphabetic letter of Lebesgue. According to the definitions, there are interpretations for both L^p spaces and ℓ^p space. Please note that Frigyes Riesz named L^p spaces with adding the pulral "s" to the word "space". Mr. Riesz proved the L^2 spaces and ℓ^2 space are isomorphic. Of cours... | {
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What information can one get from $f(x,y)\geq -3x+4y$ provided that $f$ is continuously differentiable near $(0,0)$?
Let $V$ be a neighborhood of the origin in ${\Bbb R}^2$ and $f:V\to{\Bbb R}$ be continuously differentiable. Assume that $f(0,0)=0$ and $f(x,y)\geq -3x+4y$ for $(x,y)\in V$. Prove that there is a neighb... | Note that $f_y(0,0) \geq 4,$ and hence $Df (0,0) \neq 0.$ Hence the implicit function theorem tells you that that there is a neighborhood $(-\epsilon,\epsilon)$ of $0 \in \mathbb{R}$ and a $C^1$ diffeomorphism $g: (-\epsilon, \epsilon) \to g (-\epsilon, \epsilon)$ such that $f(x,y) = 0 \Rightarrow y = g(x),$ for all ap... | {
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"timestamp": "2023-03-29T00:00:00",
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Understanding the Handshake Problem I need help with this problem. The problem goes like this: In some countries it is customary to shake hands with everybody in the meeting. If there are two people there is 1 handshake, if there are three people there are three handshakes and so on. I know that the formula is $ \dfrac... | If there exists $n$ people, then each person can shake hands with $n-1$ others. Each handshake gets counted twice. So ....
| {
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"timestamp": "2023-03-29T00:00:00",
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How prove this stronger than Weitzenbock's inequality:$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$ In $\Delta ABC$,$$AB=c,BC=a,AC=b,S_{ABC}=S$$
show that
$$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$$
I know this Weitzenböck's_inequality
$$a^2+b^2+c^2\ge 4\sqrt{3}S$$
But my inequality i... | Square both sides, put$S^2=\dfrac{(a+b+c)(a+b-c)(a+c-b)(b+c-a)}{16}$ in, we have:
$(ab+bc+ac)^2(a+b+c)^4\ge 27\cdot (a+b+c)(a+b-c)(a+c-b)(b+c-a)(a^2+b^2+c^2)^2$
with brutal force method(BW method), WOLG let $a=$Min{$a,b,c$},$b=a+u,c=a+v,u\ge0,v\ge0$ , put in the inequality and rearrange them, we have:
$ \iff k_6a^6+k_5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
proving the inequality $\triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$ If $\triangle$ be the area of $\triangle ABC$ with side lengths $a,b,c$. Then show that $\displaystyle \triangle\leq \frac{1}{4}\sqrt{(a+b+c)\cdot abc}$
and also show that equality hold if $a=b=c$.
$\bf{My\; Try}::$ Here we have to prove $4\tria... | For a triangle
$\Delta = \frac{abc}{4R} = rs$
Now in your inequality
you can put in the values to get
$R \ge 2r$
This is known to be true since the distance between incentre and circumcentre $d^2 = R(R-2r)$
Thus your inequality is proved
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/621182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
Uniform continuity of continuous functions on compact sets Assume that $f: \mathbb R \rightarrow \mathbb R$ is continuous function on the compact set $A$.
Does for any $\varepsilon >0$ exist a $\delta >0$, such that
$$
\lvert\, f(x)-f(y)\rvert<\varepsilon \,\,\,\,\,\,\textrm{for every}\,\,\,\, x,y\in A,\,\,
\text{wit... | First notice that we can assume that $f$ is identically zero on $A$ by subtracting off a continuous function $g$ extending the restriction of $f$ to $A$. Such a $g$ can be constructed using the distance function $d$ from $x\in\mathbb{R}$ to $A\subset \mathbb{R}$.
The question becomes to show that for every $\epsilon>0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is the additive group of rational numbers indecomposable? A group $G$ is indecomposable if $G \neq \langle e \rangle$ and $G$ cannot be written as the direct product of two of its proper subgroups. Why is the additive group of rational numbers $(\mathbb{Q},+)$ indecomposable?
| I suspect, what you want to ask is why $(\mathbb{Q},+)$ is indecomposable, ie. it cannot be written as the direct sum of two subgroups.
The answer is that two non-trivial subgroups must intersect non-trivially. If $\{0\}\neq H, K < \mathbb{Q}$, then choose non-zero $p/q \in H, a/b\in K$, then
$$
qa\frac{p}{q} = ap = pb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Given the scalar equation, 8x + 9y = -45, write a vector equation? scalar equation:
8x + 9y = -45
Attempt:
I took the y-intercept and the x-intercept of the scalar equation and got
(-5.625, 0) and (0,-5)
By subtracting the points i got [5.625, -5]
so my vector equation was
[x,y] = [0,-5] + t[5.625, -5]
the correct an... | You did nothing wrong, but the answer uses $[-9,8]$ instead of $[5.625,-5]$.
This is because $$5.625\times (-1.6)=-9,$$$$-5\times (-1.6)=8.$$ We can do this because we have a parameter $t$ in front of this orientation vector.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/621423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
if range of $f(x) = \frac{x^2+ax+b}{x^2+2x+3}$ is $[-5,4]$. Then $a$ and $b$ are If Range of $\displaystyle f(x) = \frac{x^2+ax+b}{x^2+2x+3}$ is $\left[-5,4\; \right]$ for all $\bf{x\in \mathbb{R}}$. Then values of $a$ and $b$.
$\bf{My\; Try}::$ Let $\displaystyle y=f(x) = \frac{x^2+ax+b}{x^2+2x+3} = k$,where $k\in \ma... | To find the places where $f(x)$ is minimal and maximal, differentiate $f$ wrt $x$. Then solve $f'(x)=0$. Call the solution $x_0$ an $x_1$ (and so on if there are move). Now, you know for which $x$ $f(x)$ is minimal/maximal. Calculate $f(x_0)$ and $f(x_1)$. These should be equal to $-5$ and $4$. You only have to know wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
A probabilistic problem in graphs Let $G$ be a (simple) graph. Each edge will be deleted or will be reminded with probability $\frac 12$ (independent from the other edges). Let $P_{AB}$ be the probability that (after this process) the vertices $A$ and $B$ are connected. On the other hand starting from $G$ we give each ... | It is possible to prove this by induction using contraction of a neighbour set of $A$:
It is enough to count all admissible configurations, because any configuration has the same probability $(\frac 12)^{|E(G)|}$.
The number of admissible configurations in the non-oriented case where $A$ is adjacent to exactly the subs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Formal power series with all derivatives zero I have the following question.
Suppose I have a formal power series $f(x)=\sum\limits_{i=0}^\infty c_ix^i$ with real coefficients. Suppose that all the derivatives $f'(1),f''(1),\dots,f^{(n)}(1),\dots$ of $f(x)$ at the point $x=1$ are zero.
What can I say about the coeffic... | You cannot plug in $1$ in a formal power series. You have to regard it as an analytic function to do so (therefore, check for convergence and so on). However, any calculation from the analytical viewpoint will give you a correct identity for formal power series if both sides admit an interpretation as formal power seri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to Evaluate $\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$ How do you get from$$\int^\infty_0\int^\infty_0e^{-(x+y)^2} dx\ dy$$to
$$\frac{1}{2}\int^\infty_0\int^u_{-u}e^{-u^2} dv\ du?$$ I have tried using a change of variables formula but to no avail.
Edit: Ok as suggested I set $u=x+y$ and $v=x-y$, so I can see t... | $\newcommand{\+}{^{\dagger}}%
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\newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
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\newcommand{\dd}{{\rm d}}%
\newcommand{\ds}[1]{\displaystyle... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Projective geometry. Interpretation of a cross product between a line coincident with a point Let $p \in \mathcal{P}^2$ be a point in projective 2-space coincident with a line $l\in\mathcal{P}^2$ such that $l^\top p = 0$. What does $l \times p$ mean?
For example, $p = \left(x,y,1\right)^\top$ and $l=\left(-1, 0, x\rig... | This is not a natural operation between lines and points. The cross product of two different lines is a point (intersection) and the cross product of two different points is a line (connecting the points). In this case you have to take the dual of either the point or the line. In the first case the cross product is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Continuity in multivariable calculus I want to find out the points, where the function
$f(x,y)=\dfrac{xy}{x-y}$ if $x\neq y$ and $f(x,y)=0$ otherwise, is continuous.
I have shown that at all the points $(x,y)$, where $x\neq y$, $f$ is continuous. Also at all those points $(x,y)\in \mathbb R^2\setminus \{(0,0)\}$ such t... | Hint: Let $(x,y)$ approach $(0,0)$ along the curve $x=t+t^2$, $y=t-t^2$. We can make the behaviour even worse by approaching along $x=t+t^3$, $y=t-t^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/621922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
General integral of an PDE Consider the PDE
$$
\frac{\partial u}{\partial t}+y\frac{\partial u}{\partial x}-a^2x\frac{\partial u}{\partial y}=0
$$
To find the general integral by the method of characteristics, I construct the system
$$
\frac{dt}{1}=\frac{dx}{y}=\frac{dy}{-a^2x}
$$
It is expected to find two constant re... | Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:
$\dfrac{dt}{ds}=1$ , letting $t(0)=0$ , we have $t=s$
$\begin{cases}\dfrac{dx}{ds}=y\\\dfrac{dy}{ds}=-a^2x\end{cases}$
$\therefore\dfrac{d^2x}{ds^2}=\dfrac{dy}{ds}=-a^2x$
$x=C_1\sin as+C_2\cos as$
$\therefore y=C_1a\cos as-C_2a\sin as$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/621981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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noncomputable functions I know that there exist functions such that no computer program can, given arbitrary input, produce the correct function value. There is nothing, however which would prohibit us from knowing the function value for certain specific inputs.
Suppose we have an uncomputable function $f$ defined on N... | Yes, you're right !
The main problem of uncomputable function is that at some point (for a specific $n$), we can't compute it.
Obviously, if we could, for each $n$, find the right program $p_n$ that computes $f(n)$, then $f$ would be computable. As Robert Israel mentioned, it does not prevent $p_n$ from existing. We ju... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/622071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Obtaining a binary operation on $X \rightarrow Y$ from a binary operation on $Y$. What, if anything, to make of this observation? Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows.
$$(f+' g)(x) = f(x)+g(x)$$
Quest... | In fact, this is part of a larger story, initiated by Freyd in his paper "Algebra valued functors in general and tensor products in particular".
If $\mathcal{A}$ is a category of algebraic structures with forgetful functor $U : \mathcal{A} \to \mathsf{Set}$, then one can define $\mathcal{A}$-objects in an arbitrary cat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/622142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Improper Integral of $\int_{-1}^0 \frac{e^\frac{1}{x}}{x^3}dx$ I have a question for The Improper Integral of $\int_{-1}^0 \frac{e^\frac{1}{x}}{x^3}dx$
That's what i have done
$u=\frac1x$
$du=\frac{-1}{x^2}$
After integrated by parts I had
$e^{\frac1x}(1-\frac1x)$
So the $\lim_{t\rightarrow 0^-} [e^\frac{1}{t}(1-\fra... | $$\int_{-1}^{0^{^-}}\frac{e^\frac1x}{x^3}dx=-\int_{-1}^{-\infty}e^tt^3\frac{dt}{t^2}=\int_{-\infty}^{-1}te^tdt=-\int_\infty^1(-u)e^{-u}du=-\int_1^\infty ue^{-u}du=$$
$$=\left[\frac{u+1}{e^u}\right]_1^\infty=0-\frac2e=-\frac2e$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/622235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Calculation of $\int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx$, where $n\in \mathbb{N}$
Compute the definite integral
$$
\int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx
$$
where $n\in \mathbb{N}$.
My Attempt:
Using $\cos (x) = \frac{e^{ix}+e^{-ix}}{2}$, we get
$$
\begin{align}
\int^{\pi/2}_{0}\cos^{n}(x)\cos (nx)\,dx&=\int_{0}^{... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/622313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
} |
More precise way of solving inequality I need to solve this function:
$$
\lvert x^2-1\rvert\ge 2x-2\\
$$
I solved this equation:
For $x<0$, the solution is non existing, here I got negative root, when I tried to solve quadratic function and for $x\ge 0$ I got points $x_1=-1$ and $x_2=3$.
My question is:
How do I set ... | As $\displaystyle |x|=\begin{cases} x &\mbox{if } x\ge0 \\-x & \mbox{if } x<0 \end{cases} $
If $x^2-1\ge0\iff x\ge1$ or $x\le-1,$
we get $$x^2-1\ge2x-2\iff x^2-2x+1\ge0\iff (x-1)^2\ge0$$ which is true
If $x^2<1\iff -1<x<1\ \ \ \ (1),$
we get $$-(x^2-1)>2x-2\iff x^2+2x-3<0$$
$$\iff (x+3)(x-1)<0\iff -3<x<1\ \ \ \ (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/622376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence radius of $\sqrt{\cos(z)}$ Compute the first 3 non zero terms of the Taylor expansion of $\sqrt{\cos(z)}$ at $z=0$ and determine its convergence radius, considering only the principal branch of the square root.
I've computed the first 3 non zero terms which the problem asks for:
$$f(z)=1-\frac{1}{4}z^2-\fra... | The answers were obtained in comments, with the help from Daniel Fischer:
*
*radius of convergence is $\pi/2$, because $\sqrt{\cos z}$ is holomorphic for $|z|<\pi/2$, but is not holomorphic in any neighborhood of $\pi/2$, where cosine is zero. (One way to show this is to notice that the derivative of $\sqrt{\cos z}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/622449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that two sums are equal Show that
$\sum_{k=0}^4\left(1+x\right)^k$ =
$\sum_{k=1}^5 \left(5 \choose k\right)x^{k-1}$
I assume that this has something to do with the binomial theorem and a proof of that. But I can't take the first steps...
| The LHS is a finite Geometric Series with the first term$=1,$ common ratio $=(1+x)$ and the number of terms $=5$
So, the sum is $$1\cdot\frac{(1+x)^5-1}{1+x-1}$$
Please expand using Binomial Expansion and cancel out $x$ to find it to be same as the RHS
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/622508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Set of non-decreasing function in bijection with R I've learnt and understood the demonstration for "the set of all non-decreasing function is uncountable" with the diagonalization proof, but how could i demonstrate it is in bijection with R (the set of real numbers) or that the two sets are equipotent.
| To identify a function it is enough to know its value on rational numbers and on the discontinuities. In fact the value on a continuity point $x$ is determined as the limit on rational points approaching $x$. A non-decreasing function has only a countable number of discontinuities, hence it is enough to know its value ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Conjugate of the k-th power of a complex number The $k$-th power of a complex number $z$ can be expressed as follows:
$$
z^k=(x+iy)^k=\sum^k_{n=0}\binom{k}{n}x^{k-n}(iy)^n=\sum^k_{n=0}\binom{k}{n}x^{k-n}i^ny^n
$$
Suppose I want to express $\bar{z^k}$, the conjugate of $z^k$, in a similar manner. In other words, for all... | That seems right to me. Only think you could remark is that $\bar{i^n}=(-i)^n$, although it is trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/622750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Optimization Problem - a rod inside a hallway
The question:
I'm given the figure shown above, and need to calculate the length of the longest rod that can fit inside this figure and rotate the corner.
My thoughts:
I have tried doing the following : put $(0,0)$ at the bottom left corner. This way, the place where the ... | Your mistake is in the assertion that "the place where it touches the lower block is $y=t+2$." If you draw a picture and label the lengths of the sides of the appropriate right triangles, you'll see that in fact it touches the lower block at $y=1+{2\over t}$. This gives $(2+t)^2+(1+{2\over t})^2$ as the expression fo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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$\alpha,\beta,\gamma$ are roots of cubic equation $x^3+4x-1=0$ If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+4x-1=0$ and $\displaystyle \frac{1}{\alpha+1},\frac{1}{\beta+1},\frac{1}{\gamma+1}$ are the roots of the equation
$\displaystyle 6x^3-7x^2+3x-1=0$. Then value of $\displaystyle \frac{(\beta+1)(\ga... | Hint: Since $x^{3} + 4x - 1 = (x-\alpha)(x-\beta)(x-\gamma)$, we note that, by the theory of symmetric polynomials,
$$0 = \alpha+\beta+\gamma = s_{1}(\alpha, \beta, \gamma)$$
$$4 = \alpha\beta + \beta\gamma+\alpha\gamma = s_{2}(\alpha, \beta, \gamma)$$
$$1 = \alpha\beta\gamma = s_{3}(\alpha, \beta, \gamma)$$
As sugges... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/622913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How find this minimum of $\sqrt{(x-a)^2+(3-x-\lg{(a)})^2}+\sqrt{(x-b)^2+(3-x-10^{b})^2}$ let $x,a,b\in R$,and $a>0$,find this follow minimum of the value
$$\sqrt{(x-a)^2+(3-x-\lg{(a)})^2}+\sqrt{(x-b)^2+(3-x-10^{b})^2}$$
I see this two function
$$f(x)=10^a,g(x)=\lg{(x)}$$ are Mutually inverse function
maybe can use
$$\... | HINT :
Let $A(a,\log_{10} (a)), B(b,10^{b})$. Also, let $L$ be the line $y=3-x$.
What you want is the minumum of
$$|AP|+|BP|$$
where $P (x,y)$ is a point on $L$ and $|AP|$ represents the distance between $A$ and $P$.
You can solve your question geometrically.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/622997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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There exists an integer with alternating digits $1$ and $2$ which is divisible by $2013$ Could someone give me hints in how to solve the following (rather interesting) problem?
Prove that there exists an integer consisting of an alternance of $1$s
and $2$s with as many $1$s as $2$s (as in $12$, $1212$, $1212121212$... | Among the first 2014 numbers of this kind, two must have the same remainder mod $2013$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/623073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Rules for cancelling fractions with exponents I have an expression that I need to simplify, I know the answer (wolframalpha) but I'm not sure of the rule that gets me there.
$\dfrac{(\alpha) X_1^{\alpha -1} X_2^{1-\alpha}}{(1-\alpha)X_1^\alpha X_2^{-\alpha}}$
Basically I know that on the $X_1$ side of the fraction it ... | I assume that $X_1\neq 0,~X_2\neq 0$ and $\alpha\neq 1$.
$$\frac{\alpha X_1^{\alpha -1} X_2^{1-\alpha}}{(1-\alpha)X_1^{\color{red}{\alpha}} X_2^{\color{blue}{-\alpha}}}\longrightarrow\frac{\alpha X_1^{(\alpha -1)-\color{red}{\alpha}} X_2^{{1-\alpha}-(\color{blue}{-\alpha})}}{(1-\alpha) }=\frac{\alpha}{(1-\alpha)}X_1^{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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how many rectangles in this shape I've learned in my high school the solution to such riddle:
How many rectangles are there in this shape:
the solution is through combinations:
in this shape is a $5\times 6$ grid so the number of rectangles would be:
$C^2_5 * C_6^2 $
I would like to know if this is possible in case of... | For a big equilateral triangle with sidelength $n$ filled with unit triangles it's
$$
\sum_{i = 1}^{n} \binom{n + 2 - i}{2} = \binom{n + 2}{3} = \frac{n(n+1)(n+2)}{6}
$$
triangles pointing upwards (exception. Triangles pointing downwards is a bit more tricky. The answer is (credit to WolframAlpha for the closed form)
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623256",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Jacobson Radical $J(R)$ is a proper ideal I found a remark on my notes:
Jacobson Radical $J(R)$ is a proper ideal. Hint: Zorn's Lemma
I know $J(R)$ is the intersection of all the maximal left ideals of ring $R$. I know the maximal ideals are proper by definition. However, in the remark i guess it must be a two sided ... | This also may help .
If M is an R-module, then Jacobson radical J(M)= $J_{R}$(M) of R-module M is the intersection of all maximal submodules of M. (Maximal submodules mean maximal proper submodules ).
So if M is finitely generated, then every submodule $N$ of M is contained in a maximal submodules, by Zorn's Lemma. (Ot... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definition of the derivative $f(x) = x^{3}\cos{\left(\frac{1}{x^{2}}\right)}, x \not= 0, f(0) = 0$
Show, by definition of the derivative, that $f$ is differentiable at $x = 0$ and find the derivative of $f$ there.
So we know the derivative is defined as:
$$f'{(x)} = \lim_{h \to 0}{\frac{f{(x + h)} - f{(x)}}{h}}$$
So we... | Your definition is incorrect. The right definition is
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
With this definition, we get
$$f'(0) = \lim_{h \to 0} \frac{h^3 \cos (1/h^2)}{h}$$
Can you take it from here? H: Use L'hôpital's rule. EDIT: Forget L'hôpital, which does not apply in this situation! Just note that $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If real number x and y satisfy $(x+5)^2 +(y-12)^2=14^2$ then find the minimum value of $x^2 +y^2$ Problem :
If real number x and y satisfy $(x+5)^2 +(y-12)^2=14^2$ then find the minimum value of $x^2 +y^2$
Please suggest how to proceed on this question... I got this problem from [1]: http://www.mathstudy.in/
| As mathlove has already identified the curve to be circle $\displaystyle (x+5)^2+(y-12)^2=14^2$
But I'm not sure how to finish from where he has left of without calculus.
Here is one of the ways:
Using Parametric equation, any point $P(x,y)$ on the circle can be represented as $\displaystyle (14\cos\phi-5, 14\sin\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
How to calculate coordinates of a vector in relation to a basis Let's say I have a basis and a vector:
$$ \mathcal B_1=\{M_1,M_2,M_3,M_4 \} \ \ M_{2\times2}(\mathbb R)\\ v=\begin{pmatrix}a & b\\ c &d \end{pmatrix}$$
Suppose I have numeral values in all of the above matrices, how do I calculate coordinates of $v$ in rel... | If you write out $x_1M_1+x_2M_2+x_3M_3+x_4M_4=v$ in terms of the given numbers in the $M_i$ and $v$, and indeterminates $x_1,x_2,x_3,x_4$, you will have a system of four linear equations in those four indeterminates, and you can solve for them however you like. The solution is guaranteed to be unique, and it is the coo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
derivative calculation involving floor function $\left\lfloor {{x^2}} \right\rfloor {\sin ^2}(\pi x)$ I was asked to find when the function is differentiable and what is the derivative of:
$$\left\lfloor {{x^2}} \right\rfloor {\sin ^2}(\pi x)$$
Now, I am not sure how to treat the floor function.
I'll be glad for help.
| Hint: Split into two cases: Either $x^2$ is an integer, or it is not an integer. When $x^2\notin{\mathbb{Z}}$, $\lfloor x^2\rfloor$ will be a constant in some small interval around $x$, and so $$\frac{d}{dx} \lfloor x^2\rfloor\sin^2(\pi x)=\lfloor x^2\rfloor 2\pi \sin (\pi x) \cos(\pi x).$$
Try using the definition of ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Last digits number theory. $7^{9999}$? i have looked at/practiced several methods for solving ex: $7^{9999}$. i have looked at techniques using a)modulas/congruence b) binomial theorem c) totient/congruence d) cyclicity.
my actual desire would be a start to finish approach using totient/congruence. i have figured out ... | Note that $7^2=49\equiv 9 \mod 10$, since $9^2=81 \equiv 1 \mod 10$, we have that
$$
7^4\equiv 1 \mod 10
$$
Now notice that $4 \mid 9996$ because $4$ divides $96$ (a number is divisible by $4$ iff its last two digits are divisible by $4$). That leaves us with a $7^3$ remaining which we know that $7^3\equiv 3 \mod 10$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Number of roots of differentiable function I have two related problems that are causing me big trouble:
Let $\;a_1,...,a_n\in\Bbb R\;,\;\;a_i\neq 0\;\;\forall\;i\;$ , and let $\;b_1,...,b_n\in\Bbb R\;,\;\;b_i\neq b_j\;\;\forall\,i\neq j\;$ .
(1) Prove that the equation
$$a_1x^{b_1}+\ldots+a_nx^{b_n}=0$$
has at most $\;... | Hint: $a_1 x^{b_1} + \ldots + a_n x^{b_n}$ has the same number of roots in $(0,\infty)$ as $a_1 + \ldots + a_n x^{b_n - b_1}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Derivative of a determinant whose entries are functions I do not understand a remark in Adams' Calculus (page 628 $7^{th}$ edition). This remark is about the derivative of a determinant whose entries are functions as quoted below.
Since every term in the expansion of a determinant of any order is a product involving o... | That remarks has said most of what it needs to explain.However, I think a more precise explaination for the example is necessary.Hence, I'll cite one.
i) $a_{11}(t).a_{23}(t).a_{32}(t) $ is abitrary term in expansion of left determinant
ii) $ (a_{11}(t).a_{23}(t).a_{32}(t))' = a'_{11}(t)a_{23}(t).a_{32}(t)+a_{11}(t)a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/623819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
Function fromal definition The relation $R:= \{(x,y) \mid y= \vert x\vert \} \subseteq \mathbb{Z} \times \mathbb{N}$ is a function,
but the relation $R:= \{(y,x) \mid y= \vert x\vert \} \subseteq \mathbb{N} \times \mathbb{Z}$ is not a function...
for me it seems that the second relation has also those two properties ... | For the second $R$ notice that you have $(1,-1)\in R$, $(1,1)\in R$ which is against "right-unique".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/623902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Modular arithmetics I've been reading about some examples concerning DSS/DSA signature security and there is one part of an example that I do not understand the maths. Namely, how do you calculate this:
$w = (s^{-1}$ $mod$ $q)$
In this example let's say $q = 13$ and $s = 10$.
So we have
$w = (10^{-1}$ $mod$ $13) = 4$ ... | As $\displaystyle 10\cdot4=40\equiv1\pmod{13}$
$\displaystyle 10^{-1}\equiv4\pmod{13}$ dividing either sides by $10$ as $(10,13)=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/624084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Questions about continuity and differentiability $\newcommand{\sgn}{\operatorname{sgn}}$
So i have couple of questions about differentiability and continuity. For example lets consider $f(x)=\sin(\frac1x)$. It is defined and continuous for $x\neq0$ . It's derivative is $\frac{-\cos(\frac1x)}{x^2}$. It looks to me that ... | Your discussion about the continuity properties looks acceptable (although I did not check every single statement you make).
I suggest that you compute directly
$$
\lim_{x \to 0} \frac{f(x)-f(0)}{x}
$$
to investigate the differentiability of $f$ at $x=0$. Your approach is not equivalent to the differentiability of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Taking seats on a plane: probability that the last two persons take their proper seats 100 men are getting on a plane (containing 100 chairs) one by one. Each one has a seat number but the first one forgot his number. So he randomly chooses a chair and sits on it. Others do know their own number. Therefore if their sea... | Using the idea on the rephrased solution Byron linked in the comments. Assume the first guy keeps getting evicted from his seat. Then there will come a time when the first guy is sent to one of 3 positions. Of these only 1 leaves the 2 last guy's places available so the probability of this happening is $\frac{1}{3}$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
How to prove this equation (given two inequalities)? How to prove that:
$$\frac{au+bv}{a+b} < y$$
given that:
$$u<y, v < y$$
Here, a, b are positive integers and u, v and y are real numbers between 0 and 1 (inclusive).
| Well the other posters have great answers but you can also see that if $v < y$ then $vb < yb$ since $b > 0$ and we can also see that $ua < ya$ for the same reason. Then we can see that:
\begin{eqnarray}
ua +vb < ya + yb
\end{eqnarray}
From there its pretty simply algebraic manipulations to get your inequality (Remem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
} |
Can I switch the order of integration and "Real(z)" operation? Let $f(z,\eta)$ be an entire function.
I need to calculate (numerically) the integral: $$\int\limits _{0}^{\pi}\mbox{Re}\left(\int\limits _{0}^{\pi}f\left(z,\eta\right)d\eta\right)dz$$
Can I switch the inner order and calculate the next integral instead? $$... | Yes, due to the linearity of integration.
The justification is as follows: Define $F(z) = \int_0^\pi f(z, \eta) \, d\eta$. Then $F(z) = F_1(z) + i F_2(z)$ where $F_1(z) = \Re F(z)$ and $F_2(z) = \Im F(z)$. On one hand we have
\begin{align}
\int_0^\pi \Re \left( \int_0^\pi f(z, \eta) \, d\eta \right) \, dz &= \int_0^\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
ampleness of invertible sheaves Let $f: X\rightarrow Y$ be a morphism of schemes over a field $k$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. My question is
*
*If $\mathcal{L}$ is ample, is $f^*\mathcal{L}$ ample?
*If 1. is not true, is there the condition that $f^*\mathcal{L}$ is ample?
| The conditions for the pullback of an ample line bundle (at least that I am familiar of) are that $f : X \to Y$ be a finite morphism, and that $X,Y$ be Noetherian schemes.
I will now prove this without cohomology. Suppose we have a coherent sheaf $\mathscr{F}$ on $X$. Then $f_\ast \mathscr{F}$ is coherent because $Y$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Non-uniqueness of group structure for affine algebraic groups We know that every abelian variety has a unique group structure, but in the affine case, is that every affine algebraic group has more than one (up to isomorphism) group structure?
| Since a the affine variety underlying a non-trivial affine group always has non-identity automorphisms with no fixed points, you can always conjugate its group structure by one of them to get a different group structure.
Later. As for the more precise question, the answer is also no. Take an algebraically closed field.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Region of convergence for the series of functions Find out the region of convergence for the following series of function.
$$\sum_{m=1}^\infty x^{\log (m)}$$
Here $x \in \mathbb{R}$. This is a series of function. I was trying to find out the radius of convergence by Root Test and Ratio test. But no suitable solution I ... | $x \in \mathbb{R}$ ??. How do we define $(-2)^{log(2)}$ ? And in my opinion, this is not really a power-series .
As what I solved, this series converges if and only if $ x \in [0; \frac{1}{e})$
Hint:
Rewrite the series as :
$ \sum_{k=0}^{\infty} \sum_{m=[e^k]}^{[e^{k+1}]-1} x^{log(m)} $ (where $[.]$ is floor functi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/624699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Solution verification: solving $\sqrt{x-4}-\sqrt{x-5}+1=0$ I solved the following equation, and I just want to be sure I did it right.
This is the procedure:
$$
\sqrt{x-4}-\sqrt{x-5}+1=0\\
\sqrt{x-4}=\sqrt{x-5}-1\\
\text{squaring both sides gives me:}\\
(\sqrt{x-4})^2=(\sqrt{x-5}-1)²\\
x-4=(\sqrt{x-5})²-\sqrt{x-5}+1\\
... | Easiest way to see it, is take both square roors to the other side and square, to get
$$
\begin{split}
1 &= (x-5)-(x-4) - 2\sqrt{(x-5)(x-4)}\\
2 + 2\sqrt{(x-5)(x-4)} &= 0\\
1 + \sqrt{(x-5)(x-4)} &= 0\\
\end{split}
$$
but $\sqrt{\ldots} \geq 0$ so this is impossible...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/624974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 0
} |
What's the difference between $f \cdot g$ and $f(g(x))$? For example if
$f(x) = x + 2$
and
$g(x) = 4x - 1$
Then what would be the difference in $f \cdot g$ and $f(g(x))$?
| The notation $f \cdot g$ means that for every $x$ the function is
$$ (f \cdot g)(x) = f(x) \cdot g(x) $$
which is pointwise multiplication.
On the other hand $f \circ g$ is the composition of functions,
$$ (f \circ g)(x) = f(g(x)) \ . $$
For your examples:
$$ f(x) \cdot g(x) = (x+2) \cdot (4x-1) = 4x^2 + 8x - x - 2 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/625041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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