Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How can complex polynomials be represented? I know that real polynomials (polynomials with real coefficients) are sometimes graphed on a 3D complex space ($x=a, y=b, z=f(a+bi)$), but how are polynomials like $(1+2i)x^2+(3+4i)x+7$ represented?
| Plotting $P(z) $ and $P:\mathbb{C}\rightarrow\mathbb{C}$, if you want to use a cartesian coordinate system give a 4D image because the input can be rappresented by the coordinate $(a,b)$ where $z=a+bi$, and the output too.
So the points (coordinates) of the graph are of the kind
$$(a,b,Re(P(a+bi)),Im(P(a+bi)))$$
and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $2x = y^{\frac{1}{m}} + y^{\frac{-1}{m}}(x≥1) $ then prove that $(x^2-1)y^{"}+xy^{'} = m^{2}y$ How do I prove following?
If $2x = y^{\frac{1}{m}} + y^{\frac{-1}{m}},(x≥1)$, then prove that $(x^2-1)y^{"}+xy^{'} = m^{2}y$
| Let $y=e^{mu}$. Then $x=e^u+e^{-u}=\cosh u$. Note that
$$y'=mu'e^{mu}=mu'y.$$
But $x=\cosh u$, so $1=u'\sinh u$, and therefore
$$u'=\frac{1}{\sinh u}=\frac{1}{\sqrt{\cosh^2 u-1}}=\frac{1}{\sqrt{x^2-1}}.$$
It follows that
$$y'=mu'y=\frac{my}{\sqrt{x^2-1}},\quad\text{and therefore}\quad \sqrt{x^2-1}\,y'=my.$$
Differ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Complex numbers and trig identities: $\cos(3\theta) + i \sin(3\theta)$ Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make...
$$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)=
\cos(3\theta) + i\ \sin(3\theta)$$
Apparently it's easy b... | Note that $$(\cos(t)+i\sin(t))^n=(\cos(nt)+i\sin(nt)),~~n\in\mathbb Z$$ and $(a+b)^3=a^3+3a^2b+3ab^2+b^3,~~~(a-b)^3=a^3-3a^2b+3ab^2-b^3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/338536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
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trigonometric function integration I have this integral: $$ \int \dfrac{\sin(2x)}{1+\sin^2(x)}\,dx$$ and I need hint solving it. I tried using the trigonometric identities and let $$u=\sin(x)$$ but I got $$\int ... =\int \dfrac{2u}{1+u^2}\, du$$ which I don't know how to solve. I also tried letting $$u=tg\left(\frac{x}... | Hint: $\displaystyle \log (u(x))'=\frac{u'(x)}{u(x)}$, (with the implied assymption that $u(x)>0$ for all $x$ in the domain of $u$).
You should note, however, that $\displaystyle \log (|u(x)|)'=\frac{u'(x)}{u(x)}$, for all $x\in \operatorname{dom}(u)$ such that $u(x)\neq 0$. Also $1+u^2>0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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General topological space $2$. 1. Let $A\subset X$ be a closed set of a topological space $X$. Let $B \subset A$ be a subset of $A$. prove that $B$ is closed as a subset of $A$, if and only if $B$ is closed as a subset of $X$.
What I have done is that if $B$ is closed in $A,B$ should be the form of $A\cap C$ where $C$ ... | The first proof is correct, minus the remarks about stupidity. It could use a slight rewording, but the idea is correct.
The second example is also correct, you are supposed to find a non-closed $A$ and $B\subseteq A$ which is closed in $A$ but not closed in $X$. You can do with a simpler example, though.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/338778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$ The question I've been given is this:
Using both sides of this equation:
$$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$
Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$
Then use that to find an expression for
$$\sum_{n=0}^{\infty}\frac{n^2}{2^n}$$
This is a... | You've got $\sum_{n\geq 0} n(n-1)x^n$, modulo multiplication by $x^2$. Differentiate just once your initial power series and you'll be able to find $\sum_{n\geq 0} nx^n$. Then take the sum of $\sum_{n\geq 0} n(n-1)x^n$ and $\sum_{n\geq 0} nx^n$. What are the coefficients?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 4
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Is this convergent or diverges to infinity? Solve or give some hints.
$\lim_{n\to\infty}\dfrac {C_n^{F_n}}{F_n^{C_n}}$,
where $C_n=\dfrac {(2n)!}{n!(n+1)!}$ is the n-th Catalan number and $F_n=2^{2^n}+1$ is the n-th Fermat number.
| approximate $(n+1)!\simeq n!$ and use sterling approximation
$$L\simeq\lim_{n\rightarrow\infty}\frac{\frac{\sqrt{4\pi n}(\frac{2n}{e})^n}{(\sqrt{2\pi n}(\frac{n}{e})^n)^2}}{2^{2^n}}$$
$$L\simeq\lim_{n\rightarrow\infty}e^{n\ln\left((1/\pi n) (2e/n)\right) -2^nlog2}$$
It appears as n goes to infinity, the upper power par... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Can a countable set of parabolas cover the unit square in the plane? Can a countable set of parabolas cover the unit square in the plane? My intuition tells me the answer is no, since it can't be covered by countably many horizontal lines (by the uncountability of $[0, 1]$). Help would be appreciated.
| An approach: Let the parabolas be $P_1,P_2,\dots$. Then by thickening each parabola slightly, we can make the area covered by the thickened parabola $P_i^\ast$ less than $\frac{\epsilon}{2^i}$, where $\epsilon$ is any preassigned positive number, say $\epsilon=1/2$. Then the total area covered by the thickened parabol... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove for every function that from sphere to real number has points $x$, $-x$ such that f$(x)=f(-x)$ I have not taken topology course yet. This is just the question that my undergrad research professor left us to think about. She hinted that I could use a theorem from Calculus.
So I reviewed all theorems in Calculus, a... | You may search borsuk–ulam theorem and get some details.
| {
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"timestamp": "2023-03-29T00:00:00",
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Is the set of non finitely describable real numbers closed under addition and squaring? Is the set of non finitely describable real numbers closed under addition and squaring? If so, can someone give a proof? Thanks.
| Hum... if by non-finitely-describable you mean "can't construct a finite description (as e.g. a Turing machine)", they aren't closed with respect to addition: $a + b = 2$ if $a$ is one of yours, $b$ is too (if it wasn't, $a$ could be described). But 2 clearly isn't.
Squares I have no clue.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/339157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Minimum ceiling height to move a closet to upright position I brought a closet today. It has dimension $a\times b\times c$ where $c$ is the height and $a\leq b \leq c$. To assemble it, I have to lay it out on the ground, then move it to upright position. I realized if I just move it in the way in this picture, then it ... | I have two solutions to this problem.
Intuitive solution
Intuitively, it seems to me that the greatest distance across the box would be the diagonal, which can be calculated according to the Pythagorean theorem:
$$h = \sqrt {l^2 + w^2}$$
If you'd like a more rigorous solution, read on.
Calculus solution
Treat this as a... | {
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Prove that such an inverse is unique Given $z$ is a non zero complex number, we define a new complex number $z^{-1}$ , called $z$ inverse to have the property that $z\cdot z^{-1} = 1$
$z^{-1}$ is also often written as $1/z$
| Whenever you need to prove the uniqueness of an element that holds some property, you can begin your proof by assuming the existence of at least two such elements that hold this property, say $x$ and $y$, and showing that under this assumption, it turns out $x = y$, necessarily.
In this case, the property we'll check i... | {
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"timestamp": "2023-03-29T00:00:00",
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$x$ is rational, $\frac{x}{2}$ is rational, and $3x-1$ is rational are equivalent How do we prove that the three statements below about the real number $x$ are equivalent?
(i) $\displaystyle x$ is rational
(ii) $\displaystyle \frac{x}{2}$ is rational
(iii) $\displaystyle 3x-1$ is rational
| It is enough to prove that $$(i) \implies (ii) \implies (iii) \implies (i)$$
$1$. $(i) \implies (ii)$. Let $x = \dfrac{p}q$, where $p,q \in\mathbb{Z}$. We then have $\dfrac{x}2 = \dfrac{p}{2q}$ and we have $p,2q \in \mathbb{Z}$. Hence, $$(i) \implies (ii)$$
$2$. $(ii) \implies (iii)$. Let $\dfrac{x}2 = \dfrac{p}q$, whe... | {
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If $U_0 = 0$ and $U_n=\sqrt{U_{n-1}+(1/2)^{n-1}}$, then $U_n < U_{n-1}+(1/2)^n$ for $n > 2$ Letting $$U_n=\sqrt{U_{n-1}+(1/2)^{n-1}}$$ where $U_0=0$, prove that:
$$U_n < U_{n-1}+(1/2)^n$$ where $n>2$
| Here is a useful factoid:
For every $x\geqslant1$ and $y\gt0$, $\sqrt{x+2y}\lt x+y$.
Now, apply this to your setting. First note that $U_n\geqslant1$ implies $U_{n+1}\geqslant1$. Since $U_1=1$, this proves that $U_n\geqslant1$ for every $n\geqslant1$. Then, choosing $n\geqslant2$, $x=U_{n-1}$ and $y=1/2^n$, the facto... | {
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Mathematical way of determining whether a number is an integer I'm developing a computer program, and I've run into a mathematical problem. This isn't specific to any programming language, so it isn't really appropriate to ask on stackoverflow. Is there any way to determine whether a number is an integer using a mathem... | Since no one has answered with this debatable solution, I will post it. $$f(x) := \begin{cases}1 \qquad x \in \mathbb{Z}\\
0 \qquad x \in \mathbb{R} \setminus \mathbb{Z}\end{cases}$$ is a perfectly fine function. Even shorter would be $\chi_{\mathbb{Z}}$ defined on $\mathbb{R}$.
| {
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"url": "https://math.stackexchange.com/questions/339510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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Stock behaviour probability I found this question in a financial mathematics course exam, could anyone please help with a solution and some explanation? Thanks in advance :)
A stock has beta of $2.0$ and stock specific daily volatility of $0.02$.
Suppose that yesterday’s closing price was $100$ and today the market
... | Assuming normality, vola being specified as standard deviation and assuming a risk free rate (r_risk_free) of zero the following reasoning could be applied:
1) From CAPM we see that the expected return of the stock (E(r)=r_risk_free+beta*(r_market-r_risk_free) here E(r)=0+2.0*.01=0.02
2) From a casual definition of bet... | {
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Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots?
I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
and rational numbers are of the form $\frac pq$.
| Hint $\ $ By the Rational Root Test, any rational root is integral, hence it follows by
Theorem Parity Root Test $\ $ A polynomial $\rm\:f(x)\:$ with integer coefficients
has no integer roots if its constant coefficient and coefficient sum are both odd.
Proof $\ $ The test verifies that $\rm\ f(0) \equiv 1\equiv f... | {
"language": "en",
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Uniform convergence of $f_n(x)=x^n(1-x)$
I need to show that $f_n(x)=x^n(1-x)\rightarrow0 $ is uniformly in $[0,1]$ (i.e. $\forall\epsilon>0\,\exists N\in\mathbb{N}\,\forall n>N: \|f_n-f\|<\epsilon$)
I tried to find the maximum of $f_n$, because:
$$\|f_n-f\|=\sup_{[0,1]}|f_n(x)-f(x)|=\max\{f_n(x)\}.$$
So if we inves... | The $f_n$ sequence is decreasing, $[0, 1]$ is compact, constant function is continuous, so the result follows immediately from the Dini's theorem.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Biased coin hypothesis
Let's assume, we threw a coin $110$ times and in $85$ tosses it was head. What is the probability that the coin is biased towards head?
We can use chi squared test to test, whether the coin is biased, but using this test we only find out, that the coin is biased towards heads or tails and there... | Let's assume that you have a fair coin $p=.5$. You can approximate a binomial distribution with a normal distribution. In this case we'd use a normal distribution with mean $110p=55$ and standard deviation $\sqrt{110p(1-p)}\approx5.244$. So getting 85 heads is a $(85-55)/5.244\approx5.72$ standard deviation event. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Integration theory for Banach-valued functions I am actually studying integration theory for vector-valued functions in a general Banach space, defining the integral with Riemann's sums.
Everything seems to work exactly as in the finite dimensional case:
Let X be a Banach space, $f,g \colon I = [a,b] \to X$, $\alpha$,... | You might want to have a look to the Bochner-Lebesgue spaces. They are an appropriate generalization to the Banach-space-valued case. Many properties translate directly from the scalar case (Lebesgue theorem of dominated convergence, Lebesgue's differentiation theorem).
Introductions could be found in the rather old bo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving partial differential equation using laplace transform with time and space variation I have a equation like this:
$\dfrac{\partial y}{\partial t} = -A\dfrac{\partial y}{\partial x}+ B \dfrac{\partial^2y}{\partial x^2}$
with the following I.C
$y(x,0)=0$
and boundary conditions $y(0,t)=1$ and $y(\infty , t)=0$
... | I get a simpler procedure that without using laplace transform.
Note that this PDE is separable.
Let $y(x,t)=X(x)T(t)$ ,
Then $X(x)T'(t)=-AX'(x)T(t)+BX''(x)T(t)$
$X(x)T'(t)=(BX''(x)-AX'(x))T(t)$
$\dfrac{T'(t)}{T(t)}=\dfrac{BX''(x)-AX'(x)}{X(x)}=\dfrac{4B^2s^2-A^2}{4B}$
$\begin{cases}\dfrac{T'(t)}{T(t)}=\dfrac{4B^2s^2-A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Summation/Sigma notation There are lots of variants in the notation for summation. For example, $$\sum_{k=1}^{n} f(k), \qquad \sum_{p \text{ prime}} \frac{1}{p}, \qquad \sum_{\sigma \in S_n} (\operatorname{sgn} \sigma) a_{1 , \sigma(1)} \ldots a_{n , \sigma(n)}, \qquad \sum_{d \mid n} \mu(d).$$
What exactly is a summa... | Except for the case of the upper and lower limit, all the other summations are really just sums of the form $$\sum_{P(i)} f(i)$$
Where $P$ is a unary predicate in the "language of mathematics", and $f(i)$ is some function which returns a value that we can sum. In the case of the sum of prime reciprocals $P(i)$ states t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/339977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Evaluate the integral $\int_0^{\infty} \left(\frac{\log x \arctan x}{x}\right)^2 \ dx$ Some rumours point out that the integral you see might be evaluated in a
straightforward way.
But rumours are sometimes just rumours. Could you confirm/refute it?
$$
\int_0^{\infty}\left[\frac{\log\left(x\right)\arctan\left(x\right)... | Related problems: (I), (II), (III). Denoting our integral by $J$ and recalling the mellin transform
$$ F(s)=\int_{0}^{\infty}x^{s-1} f(x)\,dx \implies F''(s)=\int_{0}^{\infty}x^{s-1} \ln(x)^2\,f(x)\,dx.$$
Taking $f(x)=\arctan(x)^2$, then the mellin transform of $f(x)$ is
$$ \frac{1}{2}\,{\frac {\pi \, \left( \gamma+2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/340033",
"timestamp": "2023-03-29T00:00:00",
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Definition of principal ideal This is a pretty basic question about principal ideals - on page 197 of Katznelson's A (Terse) Introduction to Linear Algebra, it says:
Assume that $\mathcal{R}$ has an identity element. For $g\in \mathcal{R}$, the set $I_g = \{ag:a\in\mathcal{R}\}$ is a left ideal in $\mathcal{R}$, and i... | Because if $\mathcal{R}$ has an identity, then $I_{g}$ is the smallest left ideal containing $g$. Without an identity, it might be that $g \notin I_{g}$.
For instance if $\mathcal{R} = 2 \mathbf{Z}$, then $I_{2} =\{a \cdot 2:a\in 2 \mathbf{Z} \} = 4 \mathbf{Z}$ does not contain $2$.
(Thanks Cocopuffs for pointing out a... | {
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"timestamp": "2023-03-29T00:00:00",
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Adding a surjection $\omega \to \omega$ by Levy forcing I'm trying to understand the Levy collapse, working through Kanamori's 'The Higher Infinite'. He introduces the Levy forcing $\text{Col}(\lambda, S)$ for $S \subseteq \text{On}$ to be the set of all partial functions $p: \lambda \times S \to S$ such that $|p| < \l... | A new surjection is a subset of $\omega\times\omega$. We have a very nice way to encode $\omega\times\omega$ into $\omega$, so nice it is in the ground model. If the function is a generic subset, so must be its encoded result, otherwise by applying a function in the ground model, on a set in the ground model, we end up... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Number of spanning trees in random graph Let $G$ be a graph in $G(n, p)$ (Erdős–Rényi model). What is the (expected) number of different spanning trees of $G$?
| There are $n^{n-2}$ trees on $n$ labelled vertices. The probability that all $n-1$ edges in a given tree are in the graph is $p^{n-1}$. So the expected number of spanning
trees is $p^{n-1} n^{n-2}$.
| {
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"url": "https://math.stackexchange.com/questions/340275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Factoring 3 Dimensional Polynomials? How do you factor a system of polynomials into their roots the way one can factor a single dimensional polynomial into its roots.
Example
$$x^2 + y^2 = 14$$
$$xy = 1$$
We note that we can find the 4 solutions via quadratic formula and substitution such that the solutions can be sepa... | For the "example" you list, here are some suggestions: Given
$$x^2 + y^2 = 14\tag{1}$$
$$xy = 1 \iff y = \frac 1x\tag{2}$$
*
*Substitute $y = \dfrac{1}{x}\tag{*}$ into equation $(1)$. Then solve for roots of the resulting equation in one variable.
$$x^2 + y^2 = 14\tag{a}$$
$$\iff x^2 + \left(\frac 1x\right)^2 = 1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Generating random numbers with skewed distribution I want to generate random numbers with skewed distribution. But I have only following information about distribution from the paper :
skewed distribution where the value is 1 with probability 0.9 and 46 with probability 0.1. the distribution has mean (5.5)
I don't k... | What copperhat is hinting at is the following algorithm:
Generate u, uniformly distributed in [0, 1]
If u < 0.9 then
return 1
else
return 46
(sorry, would be a mess as a comment).
In general, if you have a continuous distribution with cummulative distribution $c(x)$, to generate the respective numbers get $u$ as... | {
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Why is a full circle 360° degrees? What's the reason we agreed to setting the number of degrees of a full circle to 360? Does that make any more sense than 100, 1000 or any other number? Is there any logic involved in that particular number?
| As it has been replied here - on Wonder Quest (webarchive link):
The Sumerians watched the Sun, Moon, and the five visible planets
(Mercury, Venus, Mars, Jupiter, and Saturn), primarily for omens. They
did not try to understand the motions physically. They did, however,
notice the circular track of the Sun's annual pa... | {
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Describe a PDA that accepts all strings over $\{a, b\}$ that have as many $a$’s as $b$’s. I'm having my exam in few days and I would like help with this
Describe a PDA that accepts all strings over $\{ a, b \}$ that have as many $a$’s as $b$’s.
| Hint: Use the stack as an indication of how many more of one symbol have been so far read from the string than the other. (Also ensure that the stack never contains both $\mathtt{a}$s and $\mathtt{b}$s at the same time.)
| {
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Are there countably or uncountably many infinite subsets of the positive even integers? Let $S$ be the set of all infinite subsets of $\mathbb N$ such that $S$ consists only of even numbers.
Is $S$ countable or uncountable?
I know that set $F$ of all finite subsets of $\mathbb N$ is countable but from that I am not abl... | Notice that by dividing by two, you get all infinite subsets of $\mathbb{N}$. Now to make a bijection from $]0,1]$ to this set, write real numbers in base two, and for each real, get the set of positions of $1$ in de binary expansion.
You have to write numbers of the form $\frac{n}{2^p}$ with infinitely many $1$ digits... | {
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Numer Matrix and Probability Say your playing a game with a friend. Lets call it 1 in 8. Your seeing who can predict the next three quarter flips in a row. 1 player flips the quarter three times and HTT comes up. He now has to stick with that as his set of numbers to win. tThe other player gets to pick his sequence of ... | If each set of three is compared with each player's goal, the game is fair. Each player has $\frac 18$ chance to win each round and there is $\frac 34$ chance the round will be a draw. The chance of seven draws in a row is $(\frac 34)^7\approx 0.1335$, so each player wins with probability about $0.4333$. If I pick T... | {
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approximation of law sines from spherical case to planar case we know for plane triangle cosine rule is $\cos C=\frac{a^+b^2-c^2}{2ab}$ and on spherical triangle is $ \cos C=\frac{\cos c - \cos a \cos b} {\sin a\sin b}$ suppose $a,b,c<\epsilon$ which are sides of a spherical triangle, and $$|\frac{a^2 +b^2-c^2}{2ab}-... | Note that for $x$ close to $0$,
$$1-\frac{x^2}{2!} \le \cos x\le 1-\frac{x^2}{2!}+\frac{x^4}{4!}$$
and
$$x-\frac{x^3}{3!} \le \sin x\le x.$$
(We used the Maclaurin series expansion of $\cos x$ and $\sin x$.)
Using these facts on the small angles $a$, $b$, and $c$, we can estimate your difference.
| {
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Glue Together smooth functions Let's say that $f(x)$ is a $C^{1}$ function defined on a closed interval $I\subset \mathbb{R^{+}}$ and $g(x)\equiv c$ ($c$=constant) on an open interval $J\subset \mathbb{R^{+}}$ where $\overline{J}∩I\neq \emptyset$. Is there a way to "glue" together those two functions in such a way tha... | If $\overline J \cap I \neq \emptyset$, clearly not. If there is an open interval between $I$ and $J$ then yes, you can interpolate with a polynomial of degree $3$.
| {
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Why cannot the permutation $f^{-1}(1,2,3,5)f$ be even Please help me to prove that if $f\in S_6$ be arbiotrary permutation so the permutation $f^{-1}(1,2,3,5)f$ cannot be an even permutation.
I am sure there is a small thing I am missing it. Thank you.
| I think, you can do the problem, if you know that:
$f$ is even so is $f^{-1}$ and $f$ is odd so is $f^{-1}$.
| {
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Change of basis matrix to convert standard basis to another basis
Consider the basis $B=\left\{\begin{pmatrix} -1 \\ 1 \\0 \end{pmatrix}\begin{pmatrix} -1 \\ 0 \\1 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\1 \end{pmatrix} \right\}$ for $\mathbb{R}^3$.
A) Find the change of basis matrix for converting from the standard ... | By definition change of base matrix contains the coordinates of the new base in respect to old base as it's columns. So by definition $B$ is the change of base matrix.
Key to solution is equation $v = Bv'$ where $v$ has coordinates in old basis and $v'$ has coordinates in the new basis (new basis is B-s cols)
suppose w... | {
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Showing the sum over primes is equal to an integral First, note that $$\vartheta = \sum_{p \leq x} \log p$$I am trying to show $$\vartheta(x) = \pi(x)\log(x)-\int_2^x\frac{\pi(u)}{u}du$$ I am trying to show this by summation of parts. The theorem of partial summation is
Let $f$ be a continuous and differentiable func... | It looks reasonably good, but one thing you need to be more clear about is the definition of $f(n)$. Currently "$f(n) = \log(p)$ when $n$ is a prime" is not an adequate definition, since it fails to define, for instance $f(2.5)$.
It sounds like you might be defining $f(n) = 0$ when $n$ is not a prime; this would not b... | {
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Finite automaton that recognizes the empty language $\emptyset$ Since the language $L = \emptyset$ is regular, there must be a finite automaton that recognizes it. However, I'm not exactly sure how one would be constructed. I feel like the answer is trivial. Can someone help me out?
| You have only one state $s$ that is initial, but not accepting with loops $s \overset{\alpha}{\rightarrow} s$ for any letter $\alpha \in \Sigma$ (with non-deterministic automaton you can even skip the loops, i.e. the transition relation would be empty).
I hope this helps ;-)
| {
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Subgroups of the group of all roots of unity. Let $G=\mathbb{C}^*$ and let $\mu$ be the subgroup of roots of unity in $\mathbb{C}^*$. Show that any finitely generated subgroup of $\mu$ is cyclic. Show that $\mu$ is not finitely generated and find a non-trivial subgroup of $\mu$ which is not finitely generated.
I can s... | The first part should probably include "Show that any f.g. subgroup of $\,\mu\,$ is cyclic finite...". From here it follows at once that $\,\mu\,$ cannot be f.g. as it isn't finite.
For a non-trivial non f.g. subgroup think of the roots of unit of order $\,p^n\,\,,\,\,p\,$ a prime, and running exponent $\,n\in\Bbb N\,... | {
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Continuous exponential growth and misleading rate terminology I'm learning about continuous growth and looking at examples of Continuously Compounded Interest in finance and Uninhibited Growth in biology. While I've gotten a handle on the math, I'm finding some of the terminology counterintuitive. The best way to expla... | The instantaneous growth rate is $0.86$ per day in that $N(t)$ is the solution to $\frac {dN}{dt}=0.86N$. You are correct that the compounding makes the increase in one day $1.36$
| {
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Möbius Transformation help Hey guys I need help on these 2 questions that I am having trouble on.
1) Show that the Möbius transformation $z \rightarrow \frac{2}{1-z}$ sends the unit circle and the line $x = 1$ to the lines $x = 1$ and $x = 0$, respectively.
2) Now deduce from this that the non-Euclidean distance betwee... | These are exercises 8.6.5, 8.6.6 from "The Four Pillars of Geometry." I'll use the terminology of the book.
#1:
Call the transform $\varphi(z) = \dfrac{2}{1-z}$.
We know that Möbius transformations send non-Euclidean lines (circles and lines) to non-Euclidean lines. If we calculate the image of three points from the u... | {
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$\int_{0}^{\pi/2} (\sin x)^{1+\sqrt2} dx$ and $\int_{0}^{\pi/2} (\sin x)^{\sqrt2\space-1} dx $ How do I evaluate $$\int_{0}^{\pi/2} (\sin x)^{1+\sqrt2} dx\quad \text{ and }\quad \int_{0}^{\pi/2} (\sin x)^{\sqrt2\space-1} dx \quad ?$$
| $$\beta(x,y) = 2 \int_0^{\pi/2} \sin^{2x-1}(a) \cos^{2y-1}(a) da \implies \int_0^{\pi/2} \sin^{m}(a) da = \dfrac{\beta((m+1)/2,1/2)}2$$
Hence,
$$\int_0^{\pi/2} \sin^{1+\sqrt2}(a) da = \dfrac{\beta(1+1/\sqrt2,1/2)}2$$
$$\int_0^{\pi/2} \sin^{\sqrt2-1}(a) da = \dfrac{\beta(1/\sqrt2,1/2)}2$$
| {
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Nonzero derivative implies function is strictly increasing or decreasing on some interval Let $f$ be a differentiable function on open interval $(a,b)$. Suppose $f'(x)$ is not identically zero. Show that there exists an subinterval $(c,d)$ such that $f(x)$ is strictly increasing or strictly decreasing on $(c,d)$.
How t... | The statement is indeed wrong. You can construct for example a function $f:\mathbb{R} \rightarrow\mathbb{R}$ which is differentiable everywhere such that both $\{x \in \mathbb{R} : f'(x) > 0\}$ and $\{x \in \mathbb{R} : f'(x) < 0\}$ are dense in $\mathbb R$ and thus $f$ is monotone on no interval. You can find such a... | {
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Decomposable elements of $\Lambda^k(V)$ I have a conjecture. I have a problem proving or disproving it.
Let $w \in \Lambda^k(V)$ be a $k$-vector. Then $W_w=\{v\in V: v\wedge w = 0 \}$ is a $k$-dimensional vector space if and only if $w$ is decomposable.
For example, for $u=e_1\wedge e_2 + e_3 \wedge e_4$ we have $W_... | Roughly another 2 years later let me try giving a different proof.
Suppose first $w$ is a decomposable $k$-vector. Then there exist $k$ linearly independent vectors $\{e_i\}$, $i=1,\ldots k$ such that $w=e_1\wedge\cdots\wedge e_k$. Complete $\{e_i\}$ to a basis of $V$, it is then clear that $e_i \wedge w=0$ for $i=1,\l... | {
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Probability - how come $76$ has better chances than $77$ on a "Luck Machine" round? a Luck machine is a $3$ $0-9$ digits on a screen, define $X_1$ to be the result contains $77$, and $X_2$ to be the result contains $76$.
$p(X_1)=2*1/10*1/10*9/10+1/10*1/10*1/10$
$p(X_2)=2*1/10*1/10$
if i wasnt mistaken in my calculatio... | If I understand your question correctly, you have a machine that generates 3 digits (10³ possible outcomes) and you want to know the probability that you have event $X_1$ : either "7 7 x" or "x 7 7" and compare this with event $X_2$: either "7 6 x" or "x 7 6".
Notice that the 2 sub-events "7 7 x" and "x 7 7" are not di... | {
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Let $ P $ be a non-constant polynomial in z. Show that $ P(z) \rightarrow \infty $ as $ z \rightarrow \infty $ This is a a homework problem I have and I am having some trouble on it. I had thought I solved it, but I found out a algebraic mistake made my proof incorrect. Here is what I have so far.
Let $ P(z) = \sum\lim... | Hints:
$$(1)\;\;\;\;\;|P(z)|=\left|\;\sum_{k=0}^na_kz^k\;\right|=|z|^n\left|\;\sum_{k=0}^na_kz^{k-n}\;\right|$$
$$(2)\;\;\;\;\forall\,w\in\Bbb C\;\;\wedge\;\forall\;n\in\Bbb N\;,\;\;\left|\frac{w}{z^n}\right|\xrightarrow[z\to\infty]{}0$$
| {
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Prove that for any real numbers $a,b$ we have $\lvert \arctan a−\arctan b\rvert\leq \lvert a−b\rvert$. Prove that for any real numbers $a,b$ we have $\lvert \arctan a−\arctan b\rvert\leq \lvert a−b\rvert$. This should have an application of the mean value theorem.
| Let $f(x)=\arctan x$, then by mean value theorem there exist $c\in (a,b)$ such that
$$|f(a)-f(b)|=f^{\prime}(c) |a-b|$$
$$f^{\prime}(x)=\frac{1}{1+x^2}\leq1$$
So $f^{\prime}(c)\leq1$ and $|\arctan a-\arctan b|\leq |a-b|$.
| {
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differentiate f(x) using L'hopital and other problem
*
*Evaluate: $\ \ \ \ \lim_{x\to1}(2-x)^{\tan(\frac{\pi}{2}x)}$
*Show that the inequality holds: $\ \ \ x^\alpha\leq \alpha x + (1-\alpha)\ \ \ (x\geq0, \,0<\alpha <1)$
Please help me with these.
Either a hint or a full proof will do.
Thanks.
| For the first problem: another useful method is to log the function to get $\frac{\log(2-x)}{\frac{1}{\tan(\frac{\pi x}{2})}}$, now you can apply L'Hospital's rule, take the limit and exponentiate it. I keep getting $e^{\frac{\pi}{2}}$, but you need to check the algebra more thoroughly
| {
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Combinatorics riddle: keys and a safe There are 8 crew members, The leading member wants that only a crew of 5 people or more could open a safe he bought, To each member he gave equal amount of keys, and locked the safe with several locks.
What's the minimal number of locks he should put:
*
*at least 5 crew members ... | This can easily be generalized, replacing 8 and 4 with a variable each.
Let the set of keys (and locks) be denoted by $K$. Let the set of keys that crew member $i$ doesn't receive be $K_i$.
For distinct indices, Condition 1 states that $K_{ijkl} = K_i \cap K_j \cap K_k \cap K_l \neq \emptyset$ (no 4 crew can open the s... | {
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Find the value of $\lim_{n \rightarrow \infty} \sqrt{1+\left(\frac1{2n}\right)^n}$ Find the limit of the sequence as it approches $\infty$ $$\sqrt{1+\left(\frac1{2n}\right)^n}$$
I made a table of the values of the sequence and the values approach 1, so why is the limit $e^{1/4}$?
I know that if the answer is $e^{1/4}... | We have $\lim_{n\to\infty} (1+\frac{x}{n})^n = e^x$. Hence $\lim_{n \to \infty} (1+\frac{x}{2n})^{2n} = e^x$, then
taking square roots (noting that the square root is continuous on $[0,\infty)$), we have $\lim_{n \to \infty} \sqrt{(1+\frac{x}{2n})^{2n}} = \lim_n (1+\frac{x}{2n})^{n} = \sqrt{e^x} = e^\frac{x}{2}$, and ... | {
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Prove that if $\gcd(a, n) = d$ then $\langle[a]\rangle = \langle[d]\rangle$ in $\mathbb Z_n$? I am not sure how to start this problem and hope someone can help me out.
| Well, $d$ divides $a$, so in any $\Bbb Z_k$ it should be clear that $[a]\in\langle[d]\rangle$, whence $\langle[a]\rangle\subseteq\langle[d]\rangle$.
The reverse inclusion doesn't generally hold, but since $d=\gcd(a,n)$, then there exist $x,y\in\Bbb Z$ such that $d=ax+ny$, so in $\Bbb Z_n$ we have $[d]=[a][x]$, which le... | {
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Proving $n+3 \mid 3n^3-11n+48$ I'm really stuck while I'm trying to prove this statement:
$\forall n \in \mathbb{N},\quad (n+3) \mid (3n^3-11n+48)$.
I couldn't even how to start.
| If you wrote it backwards by accident then the proof is by
$$(3n^2 - 9n + 16)(n+3) = 3n^3 - 11n + 48$$
if you meant what you wrote then $n=0\;$ gives a counter examples since $48$ doesn't divide $3$.
| {
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Notatebility of Uncountable Sets I have noticed a pattern. The set of integers is infinite. Therefore one at first would think it impossible to come up with a notation allowing the representation of all integers. This though becomes easy actually to get around. Simply allow larger integers to take up larger notation. N... | Other people have answered why uncountable sets cannot be notated.
I'll just add that some countable sets cannot be "usefully" notated. So it depends on your definition of notation.
For example, the set of all Turing machines is countable. The set of Turing machines that do not halt is a subset, so is also countable. B... | {
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Are all large cardinal axioms expressible in terms of elementary embeddings? An elementary embedding is an injection $f:M\rightarrow N$ between two models $M,N$ of a theory $T$ such that for any formula $\phi$ of the theory, we have $M\vDash \phi(a) \ \iff N\vDash \phi(f(a))$ where $a$ is a list of elements of $M$.
A c... | There is an online PDF of lecture slides by Woodin on the "Omega Conjecture" in which he axiomatizes the type of formulas that are large cardinals. I do not know how exhaustive his formulation is. See the references under
http://en.wikipedia.org/wiki/Omega_conjecture
| {
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Chopping arithmetic on terms such as $\pi^2$, $\pi^3$ or $e^3$ I have a problem where I have to use 3-digit chopping with numbers such as $\pi^2$, $\pi^3$, $e^3$, etc.
If I wanted to 3-digit chop $\pi^2$, do I square the true value of $\pi$ and then chop, or do I chop $\pi$ first to 3.14 then square it?
| If you chop $\pi$ then square then chop, what you are really chopping is $3.14^2$ not $\pi^2$. So you must take $\pi^2$ and then chop it.
| {
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What does 1 modulo p mean? For example, from Gallian's text:
Sylow Test for Nonsimplicity
Let $n$ be a positive integer that is not prime, and let $p$ be a prime divisor of $n$. If 1 is the only divisor of $n$ that is equal to 1 modulo p, then there does not exist a simple group of order $n$.
I of course understand wh... | To say that a number $a$ is $1$ modulo $p$ means that $p$ divides $a - 1$. So, in particular, the numbers $1, p + 1, 2p + 1, \ldots$ are all equal to $1$ modulo $p$.
As you're studying group theory, another way to put it is that $a = b$ modulo $p$ if and only if $\pi(a) = \pi(b)$ where $\pi\colon\mathbb Z \to \mathbb ... | {
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Calculate an infinite continued fraction Is there a way to algebraically determine the closed form of any infinite continued fraction with a particular pattern? For example, how would you determine the value of $$b+\cfrac1{m+b+\cfrac1{2m+b+\cfrac1{3m+b+\cdots}}}$$?
Edit (2013-03-31):
When $m=0$, simple algebraic manip... | Reference:
D. H. Lehmer, "Continued fractions containing arithmetic progressions", Scripta Mathematica vol. 29, pp. 17-24
Theorem 1:
$$
b+\frac{1}{a+b\quad+}\quad\frac{1}{2a+b\quad+}\quad\frac{1}{3a+b\quad+}\quad\dots
= \frac{I_{b/a-1}(2/a)}{I_{b/a}(2/a)}
$$
Theorem 2
$$
b-\frac{1}{a+b\quad-}\quad\frac{1}{2a+... | {
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Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely I have this question that asks to factor this expression completely:
$$(a^2+2a)^2-2(a^2+2a)-3$$
My working out:
$$a^4+4a^3+4a^2-2a^2-4a-3$$
$$=a^4+4a^3+2a^2-4a-3$$
$$=a^2(a^2+4a-2)-4a-3$$
I am stuck here. I don't how to proceed correctly.
| Or if you missed Jasper Loy's trick, you can guess and check a value of $a$ for which $$f(a) = a^4 +4a^3 +2a^2 −4a−3 = 0.$$
E.g. f(1) = 0 so $(a-1)$ is a factor and you can use long division to factorise it out.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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nCr question choosing 1 - 9 from 9 I've been trying to rack my brain for my high school maths to find the right calculation for this but I've come up blank.
I would like to know how many combinations there are of choosing 1-9 items from a set of 9 items.
i.e.
There are 9 ways of selecting 1 item.
There is 1 way of sele... | Line up the items in front of you, in order. To any of them you can say YES or NO. There are $2^9$ ways to do this. This is the same as the number of bit strings of length $9$.
But you didn't want to allow the all NO's possibility (the empty subset). Thus there are $2^9-1$ ways to choose $1$ to $9$ of the objects.
Rem... | {
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Definition of limit in category theory - is $X$ a single object of $J$ or a subset of $J$?
Let $F : J → C$ be a diagram of type $J$ in a category $C$. A cone to
$F$ is an object $N$ of $C$ together with a family $ψ_X : N → F(X)$ of
morphisms indexed by the objects $X$ of $J$, such that for every
morphism $f : X ... | The cone is indexed by all objects of $J$.
Before dealing with limits in general, you should understand products and their universal property. Of course all factors are involved.
| {
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Proving that an injective function is bijective I am having a lot of trouble starting this proof. I would greatly appreciate any help I can get here. Thanks.
Let $n\in \mathbb{N}$. Prove that any injective function from $\{1,2,\ldots,n\}$ to $\{1,2,\ldots,n\}$ is bijective.
| Another hint: Prove it by induction. It’s clear for $n=1$. Otherwise if the statement holds for some $n$, take an injective map $σ \colon \{1, …, n+1\} → \{1, …, n+1\}$. Assume $σ(n+1) = n+1$ – why can you do this? What follows?
| {
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Eigenvalues of a matrix with only one non-zero row and non-zero column. Here is the full question.
*
*Only the last row and the last column can contain non-zero entries.
*The matrix entries can take values only from $\{0,1\}$. It is a kind of binary matrix.
I am interested in the eigenvalues of this matrix. What... | At first I thought I understood your question, but after reading those comments and answers here, it seems that people here have very different interpretations. So I'm not sure if I understand it correctly now.
To my understanding, you want to find the eigenvalues of
$$
A=\begin{pmatrix}0_{(n-1)\times(n-1)}&u\\ v^T&a\e... | {
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Show that $f '(x_0) =g'(x_0)$. Assume that $f$ and $g$ are differentiable on interval $(a,b)$ and $f(x) \le g(x)$ for all $x \in (a,b)$.
There exists a point $x_0\in (a,b)$ such that $f(x_0) =g(x_0)$.
Show that $f '(x_0) =g'(x_0)$.
I am guessing we create a function $h(x) = f(x)-g(x)$ and try to come up with the conclu... | $h(x)=g(x)-f(x)$ is differentiable, and $h(x_0)=0$ and $h(x)\geq 0$ hence $h$ has a minimum at $x_0$ and hence $h'(x_0)=0$
And as $$h'(x_0)=0 \implies f'(x_0)=g'(x_0)$$
we are done.
| {
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dimension of a coordinate ring Let $I$ be an ideal of $\mathbb{C}[x,y]$ such that its zero set in $\mathbb{C}^2$ has cardinality $n$. Is it true that $\mathbb{C}[x,y]/I$ is an $n$-dimensional $\mathbb{C}$-vector space (and why)?
| The answer is no, but your very interesting question leads to about the most elementary motivation for the introduction of scheme theory in elementary algebraic geometry.
You see, if the common zero set $X_{\mathrm{classical}}=V_{\mathrm{classical}}(I)$ consists set-theoretically (I would even say physically) in $n... | {
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$\tan B\cdot \frac{BM}{MA}+\tan C\cdot \frac{CN}{NA}=\tan A. $ Let $\triangle ABC$ be a triangle and $H$ be the orthocenter of the triangle. If $M\in AB$ and $N \in AC$ such that $M,N,H$ are collinear prove that :
$$\tan B\cdot \frac{BM}{MA}+\tan C\cdot \frac{CN}{NA}=\tan A. $$
Thanks :)
|
with menelaus' theorem,
in$\triangle ABD$,$\dfrac{BM}{MA}\dfrac{AH}{HD}\dfrac{DK}{KB}=1 $,ie, $\dfrac{BM}{MA}=\dfrac{BK*HD}{AH*DK}$.
in$\triangle ACD$,$\dfrac{CN}{NA}\dfrac{AH}{HD}\dfrac{DK}{KC}=1 $,ie, $\dfrac{CN}{NA}=\dfrac{CK*HD}{AH*DK}$.
$tanB=\dfrac{AD}{BD}, tanC=\dfrac{AD}{DC},tanA=\dfrac{BC}{AH}$
LHS=$\dfrac{... | {
"language": "en",
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A subspace of a vector space A subspace of a vector space $V$ is a subset $H$ of $V$ that has three properties:
a) The zero vector of $V$ is in $H$.
b) $H$ is closed under vector addition. That is for each $u$ and $v$ in $H$, the sum $u+v$ is in $H$.
c) $H$ is closed under multiplication by scalars. That is, for each... | If your original vector space was $V=\mathbb R^3$, then the possible subspaces are:
*
*The whole space
*Any plane that passes through $0$
*Any line through $0$
*The singleton set, $\{0\}$
One reading for the definition is that $H$ is a subspace of $V$ if it is a sub-set of $V$ and it is also a vector space unde... | {
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Showing $n\Bbb{Z}$ is a group under addition Let $n$ be a positive integer and let $n\Bbb{Z}=\{nm\mid m \in\Bbb{Z}\}$. I need to show that $\left< n\Bbb{Z},+ \right>$ is a group. And I need to show that $\left< n\Bbb{Z},+ \right>\cong\left< \Bbb{Z},+ \right>$.
Added: If $n\mathbb Z$ is a subgroup of $\mathbb Z$ then i... | Since every element of $n\mathbb Z$ is an element of $\mathbb Z$, we can do an easier proof that it is a group by showing that it is a subgroup of $\mathbb Z$.
It happens to be true that if $H\subset G$, where $G$ is a group, then $H$ is group if it satisfies the single condition that if $x,y\in H$, then $x\ast y^{-1}\... | {
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Maximum number of points a minimum distance apart in a semicircle of certain radius You have a circle of certain radius $r$.
I want to put a number of points in either of the semicircles. However,
no two point can be closer than $r$.
The points can be put anywhere inside the semicircle, on the straight line, inside... | The answer is five points. Five points can be achieved by placing one at the center of the large circle and four others equally spaced around the circumference of one semicircle (the red points in the picture below). To show that six points is impossible, consider disks of radius $s$ about each of those five points, wh... | {
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Radius of convergence of power series $\sum c_n x^{2n}$ and $\sum c_n x^{n^2}$ I've got a start on the question I've written below. I'm hoping for some help to finish it off.
Suppose that the power series $\sum_{n=0}^{\infty}c_n x^n$ has a radius of convergence $R \in (0, \infty)$. Find the radii of convergence of the... | $$
\limsup_{n\rightarrow\infty} |c_n|^{\frac{1}{n}}=\alpha <\infty
$$
gives that
there exists $N\geq 1$ such that if $n>N$ then $|c_n|^{\frac{1}{n}}< \alpha+1$.
Then $|c_n|^{\frac{1}{n^2}}< (\alpha+ 1)^{\frac{1}{n}}$ for all $n>N$.
It follows that $\limsup_{n\rightarrow\infty} |c_n|^{\frac{1}{n^2}}\leq 1$.
Also, ther... | {
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A question about a basis for a topology If a subset $B$ of a powerset $P(X)$ has the property that finite intersections of elements of $B$ are empty or again elements of $B$, does the collection of all unions of sets from $B$ form a topology on $X$ then?
My book A Taste of Topology says this is indeed the case, but I w... | It is usually taken that the empty intersection of subsets of a set is the entire set, similar to how the empty product is often taken to be the multiplicative identity or the empty sum is often taken to be the additive identity.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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To prove a property of greatest common divisor Suppose integer $d$ is the greatest common divisor of integer $a$ and $b$,
how to prove, there exist whole number $r$ and $s$, so that
$$d = r \cdot a + s \cdot b $$
?
i know a proof in abstract algebra, hope to find a number theory proof?
for abstract algebra proof, it's... | An approach through elementary number-theory:
It suffices to prove this for relatively prime $a$ and $b$, so suppose this is so. Denote the set of integers $0\le k\le b$ which is relatively prime to $b$ by $\mathfrak B$. Then $a$ lies in the residue class of one of elements in $\mathfrak B$.
Define a map $\pi$ from $\m... | {
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"source": "stackexchange",
"question_score": "1",
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Non-commutative or commutative ring or subring with $x^2 = 0$ Does there exist a non-commutative or commutative ring or subring $R$ with $x \cdot x = 0$ where $0$ is the zero element of $R$, $\cdot$ is multiplication secondary binary operation, and $x$ is not zero element, and excluding the case where addition (abelian... | Example for the non-commutative case:
$$
\pmatrix{0 & 1 \\ 0 & 0}
\pmatrix{0 & 1 \\ 0 & 0} =
\pmatrix{0 & 0 \\ 0 & 0}.
$$
Example for the commutative case: Consider the ring $\mathbb{Z} / 4\mathbb{Z}$. What is $2^2$ in this ring?
| {
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Determine the points on the parabola $y=x^2 - 25$ that are closest to $(0,3)$
Determine the points on the parabola $y=x^2 - 25$ that are closest to $(0,3)$
I would like to know how to go about solving this. I have some idea of solving it. I believe you have to use implicit differentiation and the distance formula but... | Just set up a distance squared function:
$$d(x) = (x-0)^2 + (x^2-25-3)^2 = x^2 + (x^2-28)^2$$
Minimize this with respect to $x$. It is easier to work with the square of the distance rather than the distance itself because you avoid the square roots which, in the end, do not matter when taking a derivative and setting ... | {
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How do the floor and ceiling functions work on negative numbers? It's clear to me how these functions work on positive real numbers: you round up or down accordingly. But if you have to round a negative real number: to take $\,-0.8\,$ to $\,-1,\,$ then do you take the floor of $\,-0.8,\,$ or the ceiling?
That is, which... | The first is the correct: you round "down" (i.e. the greatest integer LESS THAN OR EQUAL TO $-0.8$).
In contrast, the ceiling function rounds "up" to the least integer GREATER THAN OR EQUAL TO $-0.8 = 0$.
$$
\begin{align} \lfloor{-0.8}\rfloor & = -1\quad & \text{since}\;\; \color{blue}{\bf -1} \le -0.8 \le 0 \\ \\
\lce... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How does linear algebra help with computer science? I'm a Computer Science student. I've just completed a linear algebra course. I got 75 points out of 100 points on the final exam. I know linear algebra well. As a programmer, I'm having a difficult time understanding how linear algebra helps with computer science?
Can... | The page Coding The Matrix: Linear Algebra Through Computer Science Applications (see also this page) might be useful here.
In the second page you read among others
In this class, you will learn the concepts and methods of linear algebra, and how to use them to think about problems arising in computer science.
I gue... | {
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Describe all matrices similar to a certain matrix. Math people:
I assigned this problem as homework to my students (from Strang's "Linear Algebra and its Applications", 4th edition):
Describe in words all matrices that are similar to $$\begin{bmatrix}1& 0\\ 0& -1\end{bmatrix}$$ and find two of them.
Square matrices ... | Make a picture, your matrix mirrors the $e_2$ vector and doesn't change anything at the $e_1$ vector. The matrix is in the orthogonal group but not in the special orthogonal group. Show that every matrix
$$\begin{pmatrix} \cos(\alpha) & \sin(\alpha) \\ \sin(\alpha) & -\cos(\alpha)\\ \end{pmatrix} $$
make the same.
Th... | {
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Showing that $\ln(b)-\ln(a)=\frac 1x \cdot (b-a)$ has one solution $x \in (\sqrt{ab}, {a+b\over2})$ for $0 < a < b$
For $0<a<b$, show that $\ln(b)-\ln(a)=\frac 1x \cdot (b-a)$ has one solution $x \in (\sqrt{ab}, {a+b\over2})$.
I guess that this is an application of the Lagrange theorem, but I'm unsure how to deal wit... | Hint: Use the mean value theorem.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How? How does that bit work?
How is
$$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
| You know that "$\textrm{cosine} = \frac{\textrm{adjacent}}{\textrm{hypotenuse}}$" (so the cosine of an angle is the adjacent side over the hypotenuse), so now you have to imagine that your angle is $y = \arcsin x$. Since "$\textrm{sine} = \frac{\textrm{opposite}}{\textrm{hypotenuse}}$", and we have $\sin y = x/1$, draw... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Non-Deterministic Turing Machine Algorithm I'm having trouble with this question:
Write a simple program/algorithm for a nondeterministic Turing machine that accepts the language:
$$
L = \left\{\left. xw w^R y \right| x,y,w \in \{a,b\}^+, |x| \geq |y|\right\}
$$
| Outline: First nondeterministically choose where the cut-off between $w$ and $w^{\text{R}}$ is. Then compare and cross-out symbols to the left and right of this cut off until you find the first $i$ such that the symbol $i$ cells to the left of the cut-off is different than the symbol $i$ cells to the right of the cut ... | {
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"timestamp": "2023-03-29T00:00:00",
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Which CSL rules hold in Łukasiewicz's 3-valued logic? CSL is classical logic. So I'm talking about the basic introduction and elimination rules (conditional, biconditional, disjunction, conjunction and negation).
I'm not talking about his infinite-valued logical theory, but the 3-valued one where any atomic sentence ca... | 1) I believe that Conditional introduction works:
My experience is that the problem lies in getting a getting valid derivation from a set of premises, given the other rules that don't work.
2) Conditional elimination does not.
This is the chief thing that cripples Lukasiewicz logic as a logic.
Modus Ponens... | {
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"timestamp": "2023-03-29T00:00:00",
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Evaluating $\lim_{x \to 0} \frac{\sqrt{x+9}-3}{x}$ The question is this.
In $h(x) = \dfrac{\sqrt{x+9}-3}{x}$, show that $\displaystyle \lim_{x \to 0} \ h(x) = \frac{1}{6}$, but that $h(0)$ is undefinied.
In my opinion if I use this expression $\displaystyle \lim_{x \to 0} \dfrac{\sqrt{x+9}-3}{x}$ above with the $-... | You cannot pull the negative 3 out of the square root. For example:
$$\sqrt{1-3} = \sqrt{2}i \ne \sqrt{1} - 3 = -2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/345251",
"timestamp": "2023-03-29T00:00:00",
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Show that $\displaystyle\lim_{x\rightarrow 0}\frac{5^x-4^x}{x}=\log_e\left({\frac{5}{4}}\right)$
*
*Show that $\displaystyle\lim_{x\rightarrow 0}\frac{5^x-4^x}{x}=\log_e\left({\frac{5}{4}}\right)$
*If $0<\theta < \frac{\pi}{2} $ and $\sin 2\theta=\cos 3\theta~~$ then find the value of $\sin\theta$
| *
*$\displaystyle\lim_{x\rightarrow 0}\frac{5^x-4^x}{x}$
$=\displaystyle\lim_{x\rightarrow 0}\frac{5^x-1-(4^x-1)}{x}$
$=\displaystyle\lim_{x\rightarrow 0}\frac{5^x-1}{x}$ -$\displaystyle\lim_{x\rightarrow 0}\frac{4^x-1}{x}$
$=\log_e5-\log_e4 ~~~~~~$ $[\because\displaystyle\lim_{x\rightarrow 0}\frac{a^x-1}{x}=\log_a ... | {
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The minimum value of $a^2+b^2+c^2+\frac1{a^2}+\frac1{b^2}+\frac1{c^2}?$ I came across the following problem :
Let $a,b,c$ are non-zero real numbers .Then the minimum value of $a^2+b^2+c^2+\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}?$ This is a multiple choice question and the options are $0,6,3^2,6^2.$
I do not ... | Might as well take advantage of the fact that it's a multiple choice question.
First, is it possible that the quantity is ever zero? Next, can you find $a, b, c$ such that $$a^2+b^2+c^2+\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2} = 6?$$
| {
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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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Can twice a perfect square be divisible by $q^{\frac{q+1}{2}} + 1$, where $q$ is a prime with $q \equiv 1 \pmod 4$? Can twice a perfect square be divisible by
$$q^{\frac{q+1}{2}} + 1,$$
where $q$ is a prime with $q \equiv 1 \pmod 4$?
| Try proving something "harder":
Theorem: Let $n$ be a positive integer. There exists a positive integer $k$ such that $n | 2k^2$
k=n
| {
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"source": "stackexchange",
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Case when there are more leaves than non leaves in the tree Prove that there are more leaves than non-leaves in the graph that don't have vertices of degree 2.
Ideas: If graph doesn't have vertices of degree 2 this means that vertices of the graph have degree 1 or $\geq$ 3. Vertices with degree 1 are leaves and vertice... | If $D_k$ is the number of vertices of degree $k$ then $\sum k \cdot D_k=2E$ where $E$ is the number of edges. In a tree, $E=V-1$ with $V$ the number of vertices. So if $D_2=0$ you have
$$1D_1+3D_3+4D_4+...=2E=2V-2\\ =2(D_1+D_3+D_4+...)-2.$$
From this,
$$2D_1-2-D_1=(3D_3+4D_4+...)-(2D_3+2D_4+...),$$
$$D_1-2=1D_3+2D_4+3D... | {
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Being ready to study calculus Some background: I have a degree in computer science, but the math was limited and this was 10 years ago. High school was way before that. A year ago I relearnt algebra (factoring, solving linear equations, etc). However, I have probably forgotten some of that. I never really studied tri... | The lecture notes by William Chen cover the requested material nicely. The Trillia Group distributes good texts too.
| {
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"timestamp": "2023-03-29T00:00:00",
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} |
Minimize $\sum a_i^2 \sigma^2$ subject to $\sum a_i = 1$
$$\min_{a_i} \sum_{i=1}^{n} {a_i}^2 \sigma^2\text{ such that }\sum_{i=1}^{n}a_i=1$$ and $\sigma^2$ is a scalar.
The answer is $a_i=\frac{1}{n}$.
I tried Lagrangian method. How can I get that answer?
| $\displaystyle \sum_{i=1}^{n}(x-a_i)^2\ge0,\forall x\in \mathbb{R}$
$\displaystyle \Rightarrow \sum_{i=1}^{n}(x^2+a_i^2-2xa_i)\ge0$
$\displaystyle \Rightarrow nx^2+\sum_{i=1}^{n}a_i^2-2x\sum_{i=1}^{n}a_i\ge0$
Now we have a quadratic in $x$ which is always grater than equal to zero which implies that the quadratic can h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Moduli Spaces of Higher Dimensional Complex Tori I know that the space of all complex 1-tori (elliptic curves) is modeled by $SL(2, \mathbb{R})$ acting on the upper half plane. There are many explicit formulas for this action.
Similarly, I have been told that in the higher dimensional cases, the symplectic group $Sp(2n... | You could check Complex tori by Christina Birkenhake and Herbert Lange, edited by Birkhauser.
In Chapter 7 (Moduli spaces), Section 4 (Moduli spaces of nondegenerate complex tori), Theorems 4.1 and 4.2 should answer you doubts.
Hope it helps.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/345713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to calculate $ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s} $ I'm trying to calculate this limit expression:
$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s} $$
Both the numerator and denominator should converge, since $0 \leq a, b \leq 1$, but ... | If $ab=1,$
$$ \lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s}= \lim_{s \to \infty} \frac{s}{s+1}=\lim_{s \to \infty} \frac1{1+\frac1s}=1$$
If $ab\ne1, $
$$\lim_{s \to \infty} \frac{ab + (ab)^2 + ... (ab)^s}{1 +ab + (ab)^2 + ... (ab)^s}$$
$$=\lim_{s \to \infty} \frac{(ab)^{s+1}-ab}{(ab)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Finitely additive probabilities and Integrals My question is the following: if $(\Omega, \mathcal{A}, P)$ is a probabilistic space with $P$ simply additive (not necessarily $\sigma$-additive) and $f,g$ two real valued, positive, bounded, $\mathcal{A}$-measurable function, then $\int f+g\,dP=\int f\,dP+\int g\,dP$?
The ... | You would need to define the meaning of the integral for a finitely additive measure. I am not sure if there is standard and well established definition in that context. And I think the notion will generally not be very well behaved.
For instance, suppose you allow not all sets to have a measure, but allow measures on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding equilibrium with two dependent variables I was thinking while in the shower. What if I wanted to set two hands of the clock so that the long hand is at a golden angle to the short hand. I thought, set the short hand (hour hand) at 12, and then set the long hand (minute hand) at what ever is a golden angle to 12... | Start at 0:00 where the two hands are on the 12. You now that the long hand advances 12 times faster than the short one so if the long hand is at an angle $x$, the angle between the two hands is $x - x/12 = 11x/12$. So you just need to place the long hand at an angle $12 \alpha/11$ so that the angle between the two is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/345895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$. Prove that for any sets $A$ or $B$, if $\mathcal P(A) \cup \mathcal P(B)= \mathcal P(A\cup B)$ then either $A \subseteq B$ or $B \subseteq A$. ($\mathcal P$ is the power set.)
I'm having trouble making ... | Hint: Try instead to prove the contrapositive:
If $A \nsubseteq B$ and $B \nsubseteq A$, then $\mathcal{P} ( A ) \cup \mathcal{P} ( B ) \neq \mathcal{P} ( A \cup B )$.
Remember that $E \nsubseteq F$ means that there is an element of $E$ which is not an element of $F$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/345978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
History of Conic Sections Recently, I came to know that ancient Greeks had already studied conic sections. I find myself wondering if they knew about things like directrix or eccentricity. (I mean familiar with these concepts in the spirit not in terminology).
This is just the appetizer. What I really want to understan... | There are several Ideas where it might have come from.
One such idea is the construction of burning mirrors, for which a parabola is the best shape, because it concentrates the light in a single point, and the distance between the mirror and the point can be calculated by the use of geometry (see diocles "on burning mi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Is $\mathrm{GL}_n(K)$ divisible for an algebraically closed field $K?$ This is a follow-up question to this one. To reiterate the definition, a group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Let $K$ be an algebraically closed field. For whi... | ${\rm GL}(n,K)$ is not divisible when $K$ has finite characteristic $p$ and $n >1.$ The maximum order of an element of $p$-power order in ${\rm GL}(n,K)$ is $p^{e+1},$ where $p^{e} < n \leq p^{e+1}$ ($e$ a positive integer). There is an element of that order, and it is not the $p$-th power of any element of ${\rm GL}(n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346109",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Darboux integral too sophisticated for Calculus 1 students? I strongly prefer Darboux's method to the one commonly found in introductory level calculus texts such as Stewart, but I'm worried that it might be a bit overwhelming for my freshman level calculus class. My aim is to develop the theory, proving all of the re... | Part of the point of college is to learn the craft, not just the subject. But of all the things you could give the students by a single deviation from the textbook, why Darboux integrals? Not infinite series, recognizing measures in an integral, or linear maps? Each of those would greatly clarify foundations, allowing ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding a subspace whose intersections with other subpaces are trivial. On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 $k$-dimensional subspaces $P_1$, $P_2$ of $\mathb... | For $j=1,2,\ldots,m$, let $B_j$ be a basis of $P_j$. It suffices to find a set of vectors $S=\{v_1,v_2,\ldots,v_{n-k}\}$ such that $S\cup B_j$ is a linearly independent set of vectors for each $j$. We will begin with $S=\phi$ and put vectors into $S$ one by one.
Suppose $S$ already contains $i<n-k$ vectors. Now conside... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Isomorphisme of measurable space
Hi,
Can you ,help me to understand this proposition, and it's prrof ?
Definition 24 is :
A measurable space $(T, \mathcal{T})$ is said to be separable if there
existe a sequence $(A_n)$ dans $\mathcal{T}$ which generates
$\mathcal{T}$ and $\chi_{A_n}$ separate the points of $T$
... | Clearly, $h:T\to h(T)$ is a surjection. The fact that $h$ is an injection follows from the fact that $\{1_{A_n}\}_{n\in \Bbb N}$ separate points of $T$. That is, if $t\neq s$ then $h(t)\neq h(s)$ since for some $n$ it holds that there exists a separation indicator function, that is $1_{A_n}(t)\neq 1_{A_n}(s)$.
I didn't... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving a tricky equation involving logs How can you solve
$$a(1 - 1/c)^{a - 1} = a - b$$
for $a$?
I get $(a-1)\ln(1-1/c) = \ln(1-b/a)$ and then I am stuck.
All the variables are real and $c>a>b>1$.
| You have to solve this numerically,
since no standard function will do it.
To get an initial value,
in $a(1 - 1/c)^{a - 1} = a - b$,
use the first two terms of the binomial theorem
to get
$(1 - 1/c)^{a - 1} \approx
1-(a-1)/c$.
This gives
$a(1-(a-1)/c) \approx a-b$
or
$a(c-a-1)\approx ac-bc$
or $ac-a^2-a \approx ac-bc$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Easy way to simplify this expression? I'm teaching myself algebra 2 and I'm at this expression (I'm trying to find the roots):
$$ x=\frac{-1-2\sqrt{5}\pm\sqrt{21-4\sqrt{5}}}{4} $$
My calculator gives $ -\sqrt{5} $ and $ -\frac12 $ and I'm wondering how I would go about simplifying this down without a calculator. Is the... | You'll want to try to write $21-4\sqrt 5$ as the square of some number $a+b\sqrt 5$. In particular, $$(a+b\sqrt 5)^2=a^2+2ab\sqrt 5+5b^2,$$ so we'll need $ab=-2$ and $21=a^2+5b^2$. Some quick trial and error shows us that $a=1,b=-2$ does the job.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/346475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Stiff differential equation where Runge-Kutta $4$th order method can be broken Is there a stiff differential equation that cannot be solved by the Runge-Kutta 4th order method, but which has an analytical solution for testing?
| Cleve Moler, in this note, gives an innocuous-looking DE he attributes to Larry Shampine that models flame propagation. The differential equation is
$$y^\prime=y^2-y^3$$
with initial condition $y(0)=\frac1{h}$, and integrated over the interval $[0,2h]$. The exact solution of this differential equation is
$$y(t)=\frac1{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/346582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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