Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Does a statement need to be a biconditional statement to prove by the contrapositive I am trying to write a proof and was wondering if a then b, the converse if b then a might not be true. This leads me to wonder if the statement needs to be an if and only if statement if it can be proven by the contrapositive if not b... | By definition, "$a \implies b$" means "not ($a$ and not $b$)". So then "(not $b$)$\implies$(not $a$)" means "not ((not $b$) and not (not $a$))". This simplifies to "not ($a$ and not $b$)", which is the same thing as "$a \implies b$". So yeah, you only need the one direction of implication.
You should probably think abo... | {
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"url": "https://math.stackexchange.com/questions/1642521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the meaning of $ \mathbb{R}^n$ to $\mathbb{R}^{n+1}$? In linear algebra, what does it mean to go from $\mathbb{R}^1$ to $\mathbb{R}^2$ or $\mathbb{R}^2$ to $\mathbb{R}^3$?
| It means to bring a two dimensional space to a three dimensional space, i.e., mapping vectors of the form $(x,y)$ to vectors of the form $(x,y,z).$ This must be done by applying some kind of linear transformation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to express this expression in terms of N? I am trying to express this formula in terms of N:
$$
A=\frac{a^2N} {a\cdot \tan{\frac{180}{N}}}
$$
I really don't know how to do this. I tried and got this:
$$
A \cdot a \cdot \tan{\frac{180}{N}} = a^2N
$$
$$
\tan{\frac{180}{N}}=\frac{a^2N}{Aa}
$$
$$
\frac{180}{N}=\tan^{-1... | Equations which mix polynomial and trigonometric terms do not show analytical solutions and numerical methods should be involved.
Let us consider the equation $$A=\frac{a^2N} {a\cdot \tan({\frac{\pi}{N})}}=\frac{aN} {\tan({\frac{\pi}{N})}}$$ So, we can consider that we look for the zero of the function $$f(N)=\frac{aN}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$ Without using Mathematical Induction, prove that $$\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$$
I am unable to solve this problem and don't know where to start. Please help me to solve this problem using the laws of inequa... | $f(x) = 1/x$ is strictly convex, therefore
$$
\frac{1}{2n} < \frac 12 \left( \frac{1}{n+k} + \frac{1}{3n-k} \right)
$$
for $k = 1, ..., n-1$, or
$$
\frac{1}{n+k} + \frac{1}{3n-k} > \frac {1}{2n} + \frac {1}{2n}
$$
Combining terms pairwise from both ends of the sum shows that
$$
\frac{1}{n+1} + \frac{1}{n+3}+\dots+\fr... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit of $a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$ Find a limit of sequence:
$$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$
$$a_1=0,a_2=0$$
I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by strong induction).
So please give a idea how to show that ... | (this proof takes a little longer, but gives a neat bound on the rate of convergence)
*
*If you want a universal bound of the form $a_n\le M$, the recursion tells you that you're going to need $M\le M^2+\frac{M}{6}+\frac{1}{9}$, which works out to be equivalent to $(6M-1)(3M-2)\le0$. The lowest bound this allows us ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1642927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Structure theorem for non connected graded Hopf algebras Let $V^{\bullet}$ be a graded vector space. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected Hopf algebra and in particular its Lie algebra of primitive $P(T(V))$ is naturally ismorphic to the free Lie algebra. This result... | *
*More general hypotheses are possible, but there are counterexamples. Take, for example, the group algebra of a nontrivial finite group, concentrated in degree $0$ (it's not clear to me whether you're talking about graded Hopf algebras here).
*$T(V)$ is, for formal reasons, always the universal enveloping algebra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1643041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving the convergnce of a sequence So, I have to prove that the sequence defined as
$a_{n+1}=\frac{6(1+a_n)}{7+a_n}$ converges and then find the limit.
I have few questions; Do i have to assume that $a_n \geq 0$ or $a_n \leq0$. Because if i assume that $a_n \geq 0$, I can prove that
$\left|a_{n+1}-2\right| \leq \fr... | As Varun showed, the fixed points obeying $L=6(1+L)/(7+L)$ are $+2$ and $-3$. For $a_0\gt -3$, one may see that one is getting close to $L=+2$. For $a_0\gt -3$, the subsequent terms are getting more negative, but they inevitably drop below $a_n\lt -7$ at some point, and for $a_0\lt -7$, $a_1$ may already be calculated ... | {
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Weak convergence in $L^p$ equivalent to pointwise almost everywhere convergence Can weak convergence of a sequence $f_n\in L^p(\Omega, \mu)$ to some $f\in L^p(\Omega, \mu)$ be characterised as almost everywhere pointwise convergence?
Let us also assume the measure space is $\sigma$-finite.
I'm asking because for contin... | It isn't true.
The standard counterexample is to look at $L^2((0,1))$ with Lebesgue measure and take $f_n(x) = \sqrt{2} \sin(n \pi x)$. The functions $f_n$ are orthonormal in $L^2$, so by Bessel's inequality they converge weakly to 0. But pointwise, the sequence $\{f_n(x)\}$ diverges for every $x \in (0,1)$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Differential Equation $y' = 2y/x - 1$ Can I get help solving this DE?
$$ y' = \frac 2xy - 1$$
Doesn't look too hard but i just can't get to the correct result.
Thank you in advance
| $$
y' + \left(- \frac 2 x\right) y = -1
$$
$$
My' + \left(-\frac 2 x M\right) y = -M \tag 1
$$
$$
My' + M' y = -M \tag 2
$$
$$
(My)' = -M
$$
$$
My = \int -M\,dx
$$
$$
\text{etc.}
$$
In order that $(1)$ be equal to $(2)$, you need $\displaystyle -\frac 2 x M = M'$. Thus
$$
-\frac 2 x M = \frac {dM}{dx}
$$
$$
\frac{-2\,... | {
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Show that the function $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$. Show that $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$.
I know that if $\phi$ is harmonic then it satisfies Laplace's equation but I don't see how consider... | Hint:
$w$ is a fractional linear transformation, hence a conformal mapping (of what domains?). Therefore its inverse is also conformal. Composing a harmonic function with a conformal map gives you another harmonic function. So it remains to see what function you get by composing $\phi$ and $w$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1643475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Pointwise convergence to zero, with integrals converging to a nonzero value For $n\in{\mathbb{N}}$ let $$f_n(x)=nx(1-x^2)^n\qquad(0\le x\le 1).$$
Show that $\{f_n\}_{n=1}^\infty$ converges pointwise to $0$ on $[0,1]$.
Show that $\{\int_0^1f_n\}_{n=1}^\infty$ converges to $\frac12$.
I've already shown both of these stat... | Have you tried sketching the first few functions?
As $n$ increases, the area under the function graph "flows" further and further to the right end like a wave while remaining constant.
As $f_n(1)=0$ for all $n$ and every point $x<1$ falls behind the wave sooner or later, the pointwise convergence follows.
The underlyi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1643629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Motorway problem: how to minimize the number of encounters? I was working on the motorway problem:
At time t=0, cars are launched from the same entrance of highway following a Poisson process with parameter $\lambda$, and the speed of cars ($v$) follows a known distribution of $S$.
The observer is on one of the cars wi... | EDIT: check Sheldon M. Ross, introduction to probability model, edition 9, example 5.19, or this link.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Why don't infinite sums make any sense? Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4. However, how is this possible, considering that each additional sum is smaller than the previous sum? If ... | If you try working in base 5 rather than base 10 then it will become more obvious.
It's kind of like how using 1/9 instead of 1/5 gives you 0.99999... = 1 which I hope you agree with.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1643825",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determining an upper bound I have a function $$f(\lambda)=n\ln(1-p+pe^{\frac{\lambda}{n}})-\lambda p$$
I need to prove that $$f(\lambda)\leq \frac{\lambda^2}{8n}$$ using Taylor expansion.
I have used the taylor expansion for a function of the form $\ln(1-x)=-x-\frac{x^2}{2}-\frac{x^3}{3}-\dots$
Therefore: $f(\lambda)=n... | $$f'(\lambda) = \frac{pe^{\frac{\lambda}{n}}}{1-p+pe^{\frac{\lambda}{n}}}$$
$$f''(\lambda) = \frac{\frac{1}{n}(1-p)pe^{\frac{\lambda}{n}}}{(1-p+pe^{\frac{\lambda}{n}})^2}$$
$$f'''(\lambda) = \frac{1}{n}(1-p)p \frac{\frac{1}{n}e^{\frac{\lambda}{n}}(1-p -pe^{\frac{\lambda}{n}})}{(1-p+pe^{\frac{\lambda}{n}})^3}$$
Use GM-A... | {
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Number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$ I tried to determine the number of elements of order $6$ in $\text{Aut}(\mathbb Z_{720})$.
Please could someone tell me if this is correct?
$$ \text{Aut}(\mathbb Z_{720}) \cong U(720) \cong U(9) \oplus U(16) \oplus U(5) \cong \mathbb Z_{6} \oplus \mathb... | As noted by other you are making mistake in assuming that $U(16) \cong \mathbb{Z}_8$. Instead you have that:
$$\text{Aut}(\mathbb{Z}_{720}) \cong \mathbb{Z}_6 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_4 \oplus \mathbb{Z}_4$$
Now as we want an element of order $6$ we must include an element whose "order" is divisible by $3$... | {
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Show that $f_{n}^2(x)$ does not converge in $D^1({\Omega})$ Let
$$ f_n(x) = \left\{ \begin{array}{ll}
n & \mbox{if $0 \lt x \lt \frac{1}{n}$};\\
0 & \mbox{otherwise}.\end{array} \right. \ $$
I have to show that
*
*$\lim_{n \to \infty}f_n(x)$ exists in terms of distributions
*$\lim_{n \to \infty}... | A quick remark on your solution of (2): You need to say that $\phi = 1$ on $[0;1]$ or something like that, as $\phi$ has to be independent of $n$. I know it's pedantic, but this is how you can loose marks on exams.
For (3): you actually want to show that the map $$T(\phi) = \lim_{n \to \infty} \int_{0}^{\frac{1}{n}} n... | {
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What is the range of the derivative of $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$? Let $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$, such that $f(0)=4$ and $f(5)=-1$. What is the range of values $g'(c)$ for a $c$ belonging to $[0,5]$?
Considering values of $f(x_i)$, $f(x)$ ... | This question indeed seems to be incomplete. For instance, if there exists a
$c\in(0,5)$ such that $$c=\dfrac{-1+2f'(c)\pm\sqrt{4\big(f'(c)\big)^2-8f''(c)f(c)} }{2f''(c)}$$ then $g'(x)$ attains a local maximum on $(0,5)$. To know if this is the case we need way more information on $f$.
If we were given, that $f$ is tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1644394",
"timestamp": "2023-03-29T00:00:00",
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Existance of an (in)finite theory having infinite model Please help me to study the following simple cases:
Let $P$ be a binary predicate symbol. I am trying to find out, if there exists a satisfiable $T$ having infinite models only, for the following cases:
1) T is under {P} language with "="
2) T is under {P} languag... | To make Mike's answer just a little more explicit:
When $P$ is the inequality symbol of a partial order, it interprets equality in the sense that the definable set $\{m \in M \operatorname{ | } P(m,m)\}$ is exactly the diagonal relation $\{m \in M \operatorname{ | } m = m\}$.
In particular, we know that such a $P$ must... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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There are 6 white balls and 9 black balls. Probability of drawing two white, then two black? From A First Course in Probability (9th Edition):
3.5 An urn contains 6 white and 9 black balls. If 4 balls are to be
randomly selected without replacement, what is the probability that
the first 2 selected are white and t... | Method 1.
$$
\text{Probability}={\frac{{{6}\choose{2}}{{9}\choose{2}}}{{{15}\choose{2}}{{13}\choose{2}}}}=\frac{6}{91}.
$$
$$
\binom{6}{2}\text{ is the quantity of ways to choose 2 any white balls of 6 white balls},
$$
$$
\binom{9}{2}\text{ is the quantity of ways to choose 2 any black balls of 9 black balls},
$$
$$
\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1644585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find root of the equation Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$
It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.
| $$x-\frac{1000\ln(x)}{\ln(2)}=0\Longleftrightarrow$$
$$-\frac{1000\ln(x)}{\ln(2)}=-x\Longleftrightarrow$$
$$\frac{1000\ln(x)}{\ln(2)}=x\Longleftrightarrow$$
$$1000\ln(x)=x\ln(2)\Longleftrightarrow$$
$$e^{1000\ln(x)}=e^{\ln(2)x}\Longleftrightarrow$$
$$x^{1000}=2^x\Longleftrightarrow$$
$$x=\exp\left[-\text{W}\left(-\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1644648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Find $\lim_{n \rightarrow \infty}\frac{1}{n} \int_{1}^{\infty} \frac{\mathrm dx}{x^2 \log{(1+ \frac{x}{n})}}$ Find:
$$\lim_{n \rightarrow \infty} \frac{1}{n} \int_{1}^{\infty} \frac{\mathrm dx}{x^2 \log{(1+ \frac{x}{n})}}$$
The sequence $\frac{1}{nx^2 \log{(1+ \frac{x}{n})}}=\frac{1}{x^3 \frac{\log{(1+ \frac{x}{n})}}{\... | I think one could do this in a conceptually simpler way:
Since
$$
\frac{y}{1+y}<\log(1+y)<y
$$
your integrand is bounded as
$$
\frac{1}{x^3}<\frac{1}{nx^2\ln(1+x/n)}<\frac{1}{nx^2(x/n)/(1+x/n)}=\frac{1}{x^3}(1+x/n).
$$
By monotonicity, your integral satisfies
$$
\frac{1}{2}=\int_1^{+\infty}\frac{1}{x^3}\,\mathrm dx<\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show $\forall \delta > 0, \exists n \in \mathbb{N}$ such that $\frac{1}{n} < \delta$ The question is in the title, but I have no idea how to solve it, so a few hints would be appreciated, thanks.
| Suppose the statement is false. Then we have some $\delta>0$ such that $\delta^{-1}>n$ for all $n$. This contradicts the Archimedean property. Assuming $\mathbb{R}$ is defined axiomatically, this property is easily derived as a consequence of the least upper bound axiom.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1644834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Combining two results from partial integration I have a set of two PDEs:
$$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$
$$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$
These can be combined into a wave equation of the form:
$$\partial_{\tau}^{2}\theta=\partial_{\eta}^{2}\left[\theta+\alpha... | As it turns out the two results for $\psi$ are actually equivalent as by the dispersion relation:
$$\frac{1}{\omega}-\alpha=\frac{1}{\omega}\left(1-\alpha\omega\right)=-\omega$$
Alternative method: First, redefine $\theta$ and $\psi$ using a scalar $\phi$:
$$\theta=\partial_{\eta}\phi\quad\psi=-\partial_{\tau}\phi$$
Th... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof based on definition of big-$O$ I want to prove that $n! = O(n^n)$ based on the definition of big-$O$.
I find it pretty easy to show that $n! = O(n^n)$ by simply showing that $n (n-1) < n \cdot n \ldots$ etc. However I can't figure out how to prove this based on the definition of big-$O$. How can I prove this bas... | We have
$\displaystyle
\lim_{n \to \infty} \frac{n!}{n^n}=0,
$
which shows that $n!=o(n^n)$ and hence $n!=O(n^n)$.
To compute the limit one can make use of Stirling's formula:
$$n!\sim \sqrt{2\pi n} \frac{n^n}{e^n}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integrate the following equation. (exponential function) Integrate $$\frac{e^x -2}{e^{x/2}}$$
This is my calculation:
but it is wrong....
| It looks like you attempted to use substitution to replace
$$ \int \frac{\frac12 dx}{e^{x/2}}$$
with
$$ \int \frac{du}{u} = \ln(u) + C. $$
The fatal flaw here is that if $u = e^{x/2}$,
then $du = \frac12 e^{x/2} dx \neq \frac12 dx$.
The correct substitution is
$$ \tfrac 12 dx = \frac{1}{e^{x/2}} du = \frac{du}{u}.$$
$$... | {
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"timestamp": "2023-03-29T00:00:00",
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Comparing matrix norm with the norm of the inverse matrix I need help understanding and solving this problem.
Prove or give a counterexample: If $A$ is a nonsingular matrix, then $\|A^{-1}\| = \|A\|^{-1}$
Is this just asking me to get the magnitude of the inverse of Matrix $A$, and then compare it with the inverse o... | If $A$ is nonsingular, then $AA^{-1} = I$, so
$$ 1 = ||I|| = ||AA^{-1}|| \leqslant ||A||\cdot||A^{-1}||. $$
In general, then $1 \leqslant ||A||\cdot||A^{-1}|| \implies ||A||^{-1} \leqslant ||A^{-1}||$.
Equality is thus not necessarily guaranteed for arbitrary nonsingular $A$; however, the inequality above implies tha... | {
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Calculating first and second moments for random sums? Assume that $N$ and $X_1, X_2, \ldots $ are all independent and identically distributed over $(0,1)$ with the density function:
$f (x) = cx^2 (1 − x)^2$.
An integer–valued random variable, $N$ specifies a random sum of first $(N + 1)$ variables,
$$\sum_{j=1}^{N+1}x_... | From your previous question, you know how to do this if you know $\mathsf E(Z\mid N)$ and $\mathsf {Var}(Z\mid N)$
So: $\mathsf E(Z\mid N) = \sum\limits_{i=1}^{N+1} \mathsf E(X_i)$ By linearity of Expectation.
Similarly: $\mathsf E(Z^2\mid N) =\mathsf E\Big(\big(\sum\limits_{i=1}^{N+1} X_i\big)^2\Big) = \raise{1.5ex}\... | {
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"answer_id": 0
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Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$ $$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$
for $t\in\Bbb{N}$
How might one represent $a,b$ in terms of $t$?
(Note that $a,b\in \Bbb{N}$)
If possible, could one also ... | Using my newfound knowledge of Mellin Transforms, we can note that if we denote
$$F(s)=\int_0^\infty x^{s-1} \space f(x) \text{ d}x$$
Plugging in $f(x)=\text{csch} x$
One might note that $$\frac{a}{b}=F(s+1)$$
which is known as the result given.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1645438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $u\cdot v = 1/4||u+v||^2 - 1/4||u-v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$ I need some help figuring out how to work through this problem.
Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$.
Sorry, forgot to include my work so far:
I decid... | Remember $\vec{x} \cdot \vec{x} = \left \| x \right \|^2 $. That should be really helpful, in my mind.
And as @T.Bongers said, when working with identities like this, begin with the harder side. Try to "simplify" it (or make it a big more complex before having things cancel!) into the more basic side.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1645569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Cauchy like inequality $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$ Problem: Prove that for real $x, y, \alpha, \beta$,
$(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$.
I am looking for an elegant (non-bashy... | By C-S $(5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)=$
$=\left(\left(\sqrt5\alpha+\frac{\beta}{\sqrt5}\right)^2+\frac{14\beta^2}{5}\right)\left(\left(\sqrt5x+\frac{y}{\sqrt5}\right)^2+\frac{14y^2}{5}\right)\geq$
$=\left(\left(\sqrt5\alpha+\frac{\beta}{\sqrt5}\right)\left(\sqrt5x+\frac{y}{\sqrt5}\right)+\frac{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1645674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Evaluate $\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$ I am trying to re-learn some basic math and I realize I have forgotten most of it.
Evaluate $$\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$$
Call the terms $S_n$ and the total sum $S$.
$$S_n < \frac{1}{n^3} \Rightarrow \sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1} = S < \infty... | HINT:
$$n^4+n^2+1=(n^2-n+1)(n^2+n+1)$$
Write $2n$ as $$n^2+n+1-(n^2-n+1)$$
Observe that if $f(m)=m^2-m+1, f(m+1)=?$
which immediately reminds me of Telescoping Series.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1645776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Expected Value Problem Using Confusing Conditional Probability I am trying this question:
We have a bag with 10 blue jacks and 20 red jacks. We pick 3 jacks from the bag at random and with replacement. We are told that at least one jack is red.
Compute the expected values of blue jacks and red jacks, again being told ... | Since we draw just three jacks, and each is either red or blue,
the condition $R \geq 1$ is equivalent to $B \neq 3$.
That is, $R \geq 1$ rules out just the possible outcome $B = 3$.
This suggests the following shortcut for calculating $E(B \mid R \geq 1)$.
The unconditional expected value of the number of blue jacks i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1645883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Proof that $|x|+|y|\leq\sqrt{2(x^2+y^2)}$ How do I prove that for $x,y\in\mathbb{R}$ we have $|x|+|y|\leq\sqrt{2(x^2+y^2)}$?
I thought that $(|x|+|y|)^2=x^2+y^2+2|x||y|\leq2(x^2+y^2)$, but I'm not sure why that holds.
| The claim immediately follows from
$$\bigl(|x|+|y|\bigr)^2\leq\bigl(|x|+|y|\bigr)^2+\bigl(|x|-|y|\bigr)^2=2\bigl(|x|^2+|y|^2\bigr)\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Problem with Indefinite Integral $\int\frac {\cos^4x}{\sin^3x} dx$ I'm stuck with this integral
$\int\frac {\cos^4x}{\sin^3x} dx$
which I rewrote as
$\int \csc^3x \cos^4xdx$
then after using the half angle formula twice for $\cos^4x$ I got this
$\frac 14\int \csc^3x (1+\cos(2x))(1+\cos(2x))dx$
then after solving tho... | Let $t=\cos x$, then
$$\int \frac{\cos^4 x}{\sin^3 x}dx=-\int \frac{t^4}{(1-t^2)^2}dt.$$
Can you proceed?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding all eigenvectors and eigenvalues of a linear operation on a function Here is the question I am stuck on:
Consider $T \in \mathrm{Hom}(\Bbb{R}[x]_{\le 2} ,\Bbb{R}[x]_{\le 2} )$ given by
$$ (Tf)(x)=\int_{-1}^1(x-y)^2f(y)dy-2f(0)x^2$$
for all $f \in \Bbb{R}[x]_{\le 2}$. Find all eigenvalues and eigenvectors for T... | Note that we should integrate with respect to $y$, not $x$. Write $f(x) = x$. Then
$$ T(f)(x) = \int_{-1}^1 (x - y)^2 f(y) \, dy - 2f(0)x^2 = \int_{-1}^1 (x - y)^2 y \, dy = \int_{-1}^1 (x^2y - 2xy^2 + y^3) \, dy =\left[ \frac{x^2y^2}{2} - \frac{2xy^3}{3} + \frac{y^4}{4} \right]^{y = 1}_{y = -1} = -\frac{4}{3} x. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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showing projection is a linear operator Show that the orthogonal projection is linear.
Let $x_i=y_i+z_i$, where $x_i\in X$, $y_i\in Y$, $z_i\in Y^\perp$, and $\alpha,\beta$ be scalars. Then \begin{align}P(\alpha x_1+\beta x_2)&=P(\alpha(y_1+z_1)+\beta(y_2+z_2))\\&=P(\alpha y_1+\beta y_2+\alpha z_1+\beta z_2)\end{align}... | The key here is that $X = Y\oplus Y^\perp$, i.e. for any $x\in X$ there are unique $y\in Y, z\in Y^\perp$ such that $x = y+z$.
This is essential in order for $P(x) = P(y+z) = y$ to be well defined in the first place.
Now, what you showed is that $\alpha x_1 +\beta x_2$ can be uniquely written as $(\alpha y_1 + \beta ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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subsets in the cartesian product Let $A,B,C,D$ be sets. Consider $A\times B$ and $X\subseteq A\times B$. Is it true that $X$ has the form $A'\times B'$ where $A'\subseteq A$ and $B'\subseteq B$ ?
At the same time is it true that $(A\times B)\cup (C\times D)=(A\cup C)\times (B\cup D)$?
For me are both false but I don't ... | For the first, let $A=\{1,2\}, B=\{a,b\}, X=\{(1,a),(2,b)\}$ For the second, note that the pairs on the left have a first element that might come from $B$ while the pairs on the right have a first element that might come from $C$. Maybe, as in my first example, they are different things.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Intergate $\int \frac{x}{(x^2-3x+17)^2}\ dx$
$$\int \frac{x}{(x^2-3x+17)^2}\ dx$$
My attempt:
$$\int \frac{x}{(x^2-3x+17)^2}\ dx=\int \frac{x}{\left((x-\frac{3}{2})^2+\frac{59}{4}\right)^2}\ dx$$
let $u=x-\frac{3}{2}$
$du=dx$
$$\int \frac{u+\frac{3}{2}}{\left((u)^2+\frac{59}{4}\right)^2}\ du$$
How can I continue from... | One has $$\int \frac{u+\frac{3}{2}}{u^2+\frac{59}{4}} du = \int \frac{u}{u^2+\frac{59}{4}}du + \int \frac{\frac{3}{2}}{u^2+\frac{59}{4}}du.$$ The first term can be computed be setting $v = u^2+\frac{59}{4}$ and the second thanks to the $\arctan$ function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Wronskian of two independent solutions equaling zero at a specific point only? Given $y_1(x)=\sin(x^2)$ and $y_2(x)=\cos(x^2)$, I constructed a linear, homogenic ODE of order 2 by solving:
$$
\begin{vmatrix}
y & y_1 & y_2 \\
y' & y_1' & y_2' \\
y'' & y_1'' & y_2'' \\
\end{vmatrix... | make sure that the coefficient of the highest order derivative doesn't vanish anywhere in the domain and the coefficient of the rest of the derivatives are continuous.specially check at x=0
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Limit of derivative does not exist, while limit of difference quotient is infinite Can anyone show an example of a function $f$ of a real variabile such that
*
*$f$ is differentiable on a neighborhood of a point $x_0 \in \mathbb{R}$, except at $x_0$ itself;
*$f$ is continuous at $x_0$;
*$\displaystyle \lim_{x \to ... | What about
$$f(x)=\sqrt{x}+x\sin\frac{1}{x}$$
defined on $(0,+\infty)$, that can be defined by continuity at $0$ with $f(0)=0$? (I let you check that it works, if I am not wrong)
If you don't like it because it is only $\mathbb{R}_+$, you can replace with
$$f(x)=\sqrt[3]{x}+x\sin\frac{1}{x},$$
graph
that can be define... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646687",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Three fair dice are rolled one time. What is the probability of at least one $6$? I think the answer is $10/216$, but I am not sure. I did it by brute force though, and would like to know the background, as well as knowing if $10/216$ is correct.
Edit: now i think via brute force, the answer is 93/216. if you roll a 1 ... | Roll the three dice. Record the result as $(a,b,c)$ where $a$ is the number on the blue die, $b$ the number on the white, and $c$ the number on the red. All $6^3$ such sequences are equally likely.
There are $5^3$ sequences consisting of non-$6$, and therefore $216-125$ sequences with at least one $6$. Thus the probabi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points? Here is a professor's solution to the exercise which states, " Prove that if $x$ is an isolated point of a set $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$."
The professor derived a contradiction upon assuming that... | The quantifiers you have do not reflect the proof. Here, the professor wants to prove $$\forall N, \lnot(N\subseteq S)$$ and proceeds by contradiction. After taking an arbitrary $N$, (s)he shows that $N\subseteq S$ would lead to a contradiction. I.e., what is showed is $$\forall N, ((N\subseteq S)\Rightarrow \bot)$$ wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Group orderable iff all its finitely-generated subgroups are orderable I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges).
$G$ orderable means there is a total ordering s. t. for all $g,h$, $g\leq h$ implies $gk\leq hk$ and $kg\leq k... | To apply compactness, you don't need all subsets of your set to have models; you just need all finite subsets to have models. A finite subset $S$ of your set involves only finitely many of the propositional variables $p_{gh}$ which in turn involve only finitely many elements of the group $G$. The subgroup $H$ of $G$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1646977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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How to find integral of the form $e^xf(x)$? I always face trouble with these type of integrals.
I need to find $$\int{e^x \frac{x(\cos x -\sin x)-\sin x}{x^2}}dx$$
My problem would be solved if can express $f(x)$ like $g(x)+g'(x)$ but identifying $g(x)$ by trial and error method is sometimes tedious.Is there any easier... | As already said in comments, I do not think that there is a solution in terms of elementary functions.
However, the problem can be approached using $$\cos(x)=\frac{1}{2} \left(e^{i x}+e^{-i x}\right)\qquad \sin(x)=-\frac{1}{2} i \left(e^{i x}-e^{-i x}\right)$$ So, the numerator of the integrand becomes$$\left(\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to define gradient of an affine connection I heard somewhere (and just read on a physics forum) that the gradient of a smooth function $f$ on a manifold $M$ can be defined when $M$ is equipped with an affine connection on its tangent bundle, i.e. the Riemannian metric is not strictly necessary.
Is this true? If so,... | No, or at least not if you want to coincide with the usual gradient in the case where the connection is derived from a metric. To show this, just note that whenever $\nabla$ is the metric connection of $g$, it is also that of $2g$ (since $\nabla(2g) = 2\nabla g = 0$); but the latter metric will produce gradients with h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647168",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How limiting/ heavy is the "triangle inequality" assumption? Suppose a theorem proves something about a family of distance measures, with this the triangle inequality assumption. How limiting this assumption is in reality?
What are some real-world examples of distance measures which don't obey the triangle inequality? ... | That pretty much depends on your definition of distance "in the real world". In my opinion, the triangle inequality is extremely important for the definition of any notion of distance.
The triangle inequality tells you that going from $A$ to $B$ and taking a detour over $C$ will at most increase the distance that need... | {
"language": "en",
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"source": "stackexchange",
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Solve equation of inverse functions I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by solving for $x_1$ and $x_2$ two equations
$$
f_1(y_1) - x_1 = 0, \\
f_2(y_2) - x_2 = 0,
$$... | If I well understand (which is not sure), you have two given functions $f_1(x)$ and $f_2(x)$
The inverse function of $y=f_1(x)$ is $x=f_1^{-1}(y)$
The inverse function of $y=f_2(x)$ is $x=f_2^{-1}(y)$
You want to find a root $y_0$ of the equation $f_1^{-1}(y)=f_2^{-1}(y)$
So, you first have to
$$\text{solve}\quad f_1(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can an infinite sum of irrational numbers be rational? Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear combination of the other ?
By linear combination, we mean there exists ... | Yes, for example $\tan(\frac{\pi}{4})=1$
Write $\tan$ as an infinite Taylor expansion in powers of $\pi$.
Other trigonometric functions of irrational arguments can also have rational outputs, eg. $\sin(\frac{\pi}{6}) = \frac{1}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "113",
"answer_count": 23,
"answer_id": 21
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Integrating triangle in a 2D plane I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$.
I know this loop forms a triangle with all three sides changing with respect to the $x$ and $y$ axis. I understand how to do the integration pa... | You can always parameterize a triangle with $u,v$, where $u,v>0$ and $u+v<1$. This always leads to an integral
$$\int_0^1 \int_0^u\ldots dv\,du$$
Convex interpolation inside the triangle with vertices A,B,C is then
$$T=Au+Bv+C(1-u-v)$$
Imagine that "u" is the variable that tells you how much you move from C to A, and "... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Number of Labels used in reduction of Isomorphism of Labelled Graph to Graph Isomorphism From "Lecture Notes in Computer Science" by Christoph M. Hoffmann ,
Assume that both $X$ and $X'$ have $n$ vertices. We plan to code the
graph labels as suitable subgraphs which we attach to the vertices of
$X$ and of $X'$. In... | My interpretation:
The Confusion arises, here-
Assume that both $X$ and $X'$ have $n$ vertices. We plan to code the
graph labels as suitable subgraphs which we attach to the vertices of
$X$ and of $X'$. In time polynomial in the length of the input we can rename
the labels and may assume, therefore, that $L = \{1 ....... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647695",
"timestamp": "2023-03-29T00:00:00",
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A sequence of continuous functions which is pointwise convergent to zero and not uniformly convergent on any interval. The exercise is to construct a sequence of continuous functions $f_n:\mathbb{R}\rightarrow \mathbb{R}, n\in \mathbb{N}$ , which is pointwise convergent to $f(x)=0 , x\in \mathbb{R}$ and not uniformly c... | According to the hint in "Problems in Mathematical Analysis", Biler and Witkowski, problem 4.112.
Define $g_n$ as the function which is $0$ on the intervals $(-\infty, 0]$ and $[2/n,\infty)$, $1$ at $1/n$, and linear otherwise. Then $g_n$ converges pointwise to $0$ but is not uniform in any interval containing $0$.
No... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
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support of an operator on a Hilbert space Let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator,
the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$.
Let $x=\int\lambda \, dE_x$ be the spectral decomposition.
Then $s(x)=E_x(\mathbb{R}\setminus\{0\})$ und... | Because $x$ commutes with $e$, then $e$ commutes with $E$ as well, including with $f=E_x(\mathbb{R}\setminus\{0\})$. Because $fx=xf=x$ and $fe=ef$, then $ef$ is also a projection such that $(ef)x=x(ef)=x$. Therefore $ef=fe=e$ follows from the minimality of $e$. Furthermore $x(f-e)=0$, which puts the range of $f-e$ in $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647834",
"timestamp": "2023-03-29T00:00:00",
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Vector norm and relationship with euclidean distance If $y\in E_n$ (n dimensional euclidean space) show that $||\textbf{y}||\leq|\textbf{y}|\leq \sqrt{n}||\textbf{y}||$
Where $||\textbf{y}||$ is the euclidean length of the vector $\textbf{y}$
and $|\textbf{y}|$ is the norm of $\textbf{y}$
The hint: Use the inequality ... | Hint: for any nonnegative number $a$ and $b$ you have
$$
(a+b)^2=a^2+b^2+2ab\ge a^2+b^2
$$
and
$$
2(a^2+b^2)-(a+b)^2=a^2+b^2-2ab=(a-b)^2\ge0.
$$
Hence,
$$
a+b\ge\sqrt{a^2+b^2}\quad\text{and}\quad \sqrt2\sqrt{a^2+b^2}\ge a+b.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1647923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\mathbb C$-dimension of vector space $\mathbb C\otimes_{\mathbb R}\mathbb C$
Let $\mathbb R$ be the field of real numbers, $\mathbb C$ be the field of complex numbers. Consider $\mathbb C\otimes_{\mathbb R}\mathbb C$ as a $\mathbb C$-vector space via $a(b\otimes c) :=
ab \otimes c,$ for $a, b, c \in \mathbb C$. Compu... | Sometimes it is better to look at a more general situation. In fact, this makes it easier to see what is really going on.
1) Let $V$ be a $K$-vector space and let $L/K$ be a field extension. Then $L \otimes_K V$ carries the structure of a vector space over $L$ via (linear extension of) $\alpha(\beta \otimes x) = \alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Closed form of $\sum\limits_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$
Let
$$S(x) = \sum_{n=1}^\infty \frac{4^n(x+4)^{2n}}n$$
1. Find the radius of convergence.
2. Calculate $S(x)$.
3. Find $S^{(n)}(x)$ without computing the derivatives of $S(x)$.
From the root test I find $R = 1/4$. It's the second point that troub... | Let $t=4(x+4)^2$ and the series becomes
$$\sum_{k=1}^\infty\frac{t^n}n.$$
You can recognize the Taylor development of $-\ln(1-t)$, or derive the series to get
$$\sum_{k=1}^\infty t^{n-1}=\frac1{1-t}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
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Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers. Ms. A selects a number $X$ randomly from the uniform distribution
on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers
$Y_1, Y_2, ...$ from the uniform distribution on ... | Given $X=x$, B draws a large enough number if they draw something between $x/2$ and $1$, which occurs with probability $1-x/2$. Using the expectation of the geometric distribution, you find that then the expected number of trials is $\frac{1}{1-x/2}$.
Now we need to deal with the fact that $X$ is itself random. The de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving $yy''=(y')^2-y'$ I want to solve the ODE $yy''=(y')^2-y'$ with the initial conditions $y(0)=1, y'(0)=2$.
My attempt:
$$yy''=(y')^2-y'$$
$$(\frac {y'}y)'=(\frac 1y)'$$
$$\frac {y'}y=\frac 1y+c$$
This holds for all $x$. Plugging the initial conditions for $x=0$, we get $c=1$.
$$y'=1+y$$
Solving this got me to $y=... | The full solution with the homogeneous part is
$$
y=-1+C·e^x
$$
and from the initial conditions $C=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find the fixed field of the following subgroup? I am trying to understand some concepts via random exercises I found from past papers but this particular one, I am not sure even where to start.
There aren't any solutions for the paper so would someone be kind enough to provide me with a standard solution, so I can stud... | Here’s the method I use for this very special case, and other similar ones. You have $\omega^4+\omega^3+\omega^2+\omega+1=0$, and you can rewrite this as
$$
0=\omega^2+\omega+1+\omega^{-1}+\omega^{-2}\,.
$$
Now, your nontrivial automorphism interchanges $\omega$ and $\omega^{-1}$, so has to have $\Bbb Q(\omega+\omega^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Probability: Finding the Number of Pears Given Two Scenarios
You have a bag containing 20 apples, 10 oranges, and an unknown number
of pears. If the probability that you select 2 apples and 2 oranges is
equal to the probability that you select 1 apple, 1 orange, and 2
pears, then what is the number of pears orig... | It is good that you were alert. I believe everything you are doing is correct. If we continue, then
$$\binom{n}{2} = \frac{\binom{20}{2}\binom{10}{2}}{20(10)} = \frac{171}{4}.$$
I think it is ok to have a decimal number here.
This gives
\begin{align*}
\frac{n!}{2!(n-2)!} &= \frac{171}{4}\\
\implies \frac{n!}{(n-2)!} &=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do I prove that if $2\nmid n$ then $2|(n+1)$? I'd like to prove a very simple fact, but it's stumping me: namely, that if $2 \nmid n$ then $2\mid(n+1)$.
How would this usually be done?
| If $n \not\equiv 0 \pmod2$, then $n+1 \not\equiv 1 \pmod2$, so we must have $n+1 \equiv 0 \pmod2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Prove that for any integer $n$, if $b^2$ divides $n$, then $b$ divides $n$.
Prove that for any integer $n$, if $b^2$ divides $n$, then $b$ divides $n$.
Trying to figure out this proof. The proof I'm looking at is written as
$n$ = any integer, if $25|n \implies 5|n$.
I've been trying to figure this for days and have ... | Since an answer has been given ...
The $\mid$ relation is transitive; if $x \mid y$ and $y\mid z$, then $x\mid z$. Let $x=b$, $y=b^2$, and $z=n$. Done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Pairwise independence vs independence
Two fair dice are thrown. We have three events:
*
*A: The first die shows an odd number
*B: The second die shows an even number
*C: Both are odd or both are ven
Show that $A,B,C$ are piecewise independent but not independent.
My answer:
$P(A) = P(B) = P(C) = \frac{1}{2}$.
$... | Yes, this is correct. The definitions are as follow (see here).
A finite set of events $\{A_i\}$ is pairwise independent if and only if every pair of events is independent — that is, if and only if for all distinct pairs of indices $m, k$,
$$
\mathrm{P}(A_m \cap A_k) = \mathrm{P}(A_m)\mathrm{P}(A_k).
$$
A finite set of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
How to proceed from $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$ To prove: $\cot(x)\cot(2x)-\cot(2x)\cot(3x)-\cot(3x)\cot(x) = 1$
My attempt at the solution:
\begin{gather}\frac{\cos(x)\cos(2x)}{\sin(x)\sin(2x)}-\frac{\cos(2x)\cos(3x)}{\sin(2x)\sin(3x)}-\frac{\cos(3x)\cos(x)}{\sin(3x)\sin(x)}\\\\
\frac{\cos(x... | Go slowly:
\begin{align}
\cot x\cot2x-\cot2x\cot3x
&=
\cot2x\left(\frac{\cos x}{\sin x}-\frac{\cos3x}{\sin3x}\right)\\[6px]
&=\frac{\cos2x}{\sin2x}\frac{\sin3x\cos x-\cos3x\sin x}{\sin x\sin 3x}
\\[6px]
&=\frac{\cos2x}{\sin2x}\frac{\sin2x}{\sin x\sin 3x}\\[6px]
&=\frac{\cos2x}{\sin x\sin 3x}
\end{align}
So you want to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 5
} |
'Obtain' the Fourier transform If $g(t) = e^{-a|t|}$ and a is a real positive constant, obtain the fourier transform.
I'm a bit unsure what this is asking. I can write out the expression for the fourier transform. Should I stop there? Or do I do the integration as well? If I do the integration, how do I deal with the ... | We note that $\displaystyle \mathcal{F}\left(f(t)\right)(x)=\int_{-\infty}^{+\infty}f(t)e^{-2i\pi x t}\text{ d}t.$
\begin{eqnarray*}
\mathcal{F}\left(e^{-a |t|}\right)(x) &= & \int_{-\infty}^0 e^{ at}e^{-2i\pi x t} dt+\int_{0}^{+\infty} e^{-at}e^{-2i\pi x t} dt \\
&= &\left[\frac{e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1648979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prime counting function; when is it true that $\pi(n) > \pi(2n) -\pi(n)$? Let $\pi$ be the prime counting function.
Under what conditions is it proven true that $\pi(n) > \pi(2n) -\pi(n)$, if at all?
| I've written the program Akiva Weinberger suggested above. This is just a straightforward interpretation of the sieve of Eratosthenes, in R.
n = 30092
top = 2*n
isPrime = rep(TRUE, top)
isPrime[1] = FALSE
nextprime = 2
while (nextprime < sqrt(top)){
isPrime[seq(2*nextprime, floor(top/nextprime)*nextprime, nextprime... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Classifying groups of order $6$ using semidirect products Let G be a group of order 6. I am able to do the exercise without semidirect products($G \cong Z_6 $ or $S_3$) but I don't know how to use semidirect products to do this.
By Sylow's theorem, there is only subgroup of order 3, say P. As it is unique, P is normal... | By Cauchy's theorem, there must be an element $s$ of order $2$ in $G$, and an element $r$ of order $3$, because $2$ and $3$ divide $6$. Clearly $r$ generates a cyclic group $C_3$ which has index $2$ in $G$, hence is a normal subgroup. Then $G$ is a semidirect product of $C_2$ and $C_3$. In case it is abelian, this prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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The line $2x-y=5$ turns about a point.... The line $2x-y=5$ turns about a point on it, whose ordinate and abscissae are equal, through an angle of $45°$, in anti clockwise direction. Find the equation of line in the new position.
My attempt to solve:
Let AB be the line with the equation $2x-y=5$ and $B(a,a)$ be a point... | The line turns by an angle of $45°$ anti-clockwise about the point $(5,5)$. Thus, you need to find the equation of a line whose angle with the $x-axis$ is $45°$ more than this line and which passes through the point $(5,5)$.
The line $2x-y=5$ has a slope of $2$, i.e., it makes an angle of $\arctan2$ with the $x-axis$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
The complex version of the chain rule I want to prove the following equality:
\begin{eqnarray}
\frac{\partial}{\partial z} (g \circ f) = (\frac{\partial g}{\partial z} \frac{\partial f}{\partial z}) + (\frac{\partial g}{\partial \bar{z}} \frac{\partial \bar{f}}{\partial z})
\end{eqnarray}
So I decide to do the followin... | The question is taken in the context of Wirtinger Derivatives.
To that end, we let $g$ and $f$ be functions of both $z$ and $\bar z$. Then, the composite function $g\circ f$ can be expressed as
$$g\circ f=g(f(z,\bar z),\bar f(z,\bar z))$$
The partial derivative of $g\circ f$ with respect to $z$ is then given by
$$\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Integrate $\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$ I am familiar with the gauusian integral
$$\int_{-\infty}^\infty e^{-\alpha x^2+\beta x}dx=\sqrt{\frac{\pi}{\alpha}}e^{\beta^2/(4\alpha)}$$
Could anyone help me to find out the value of the following?
$$\int_{-\infty}^\infty xe^{-\alpha x^2+\beta x}dx$$
Integ... | $$\int_{-\infty}^{\infty}xe^{-\alpha x^2+\beta x}\ dx=-\frac{1}{2\alpha}\int_{-\infty}^{\infty}(-2\alpha x+\beta-\beta)e^{-\alpha x^2+\beta x}\ dx$$
$$=-\frac{1}{2\alpha}\int_{-\infty}^{\infty}(-2\alpha x+\beta)e^{-\alpha x^2+\beta x}\ dx+\frac{\beta}{2\alpha}\int_{-\infty}^{\infty}e^{-\alpha x^2+\beta x}\ dx$$
$$=-\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Multiplication of transpositions? I can't seem to understand how the multiplication of two transpositions yield the results below:
$(x b)(x a) = (x a)(a b) \\
(c a)(x a) = (x c)(c a)$
I can't figure it out for the life of me. I'm trying to learn permutation multiplication on my own, and I feel confident in my abilities... | Without loss of generality, suppose the only elements in the set are in fact $a,b,x$ for the first case. Try writing each as a single permutation.
$(x~b)(x~a)$ read from right to left says:
$\begin{array}{c}a\mapsto x\mapsto b\\ x\mapsto a\mapsto a\\ b\mapsto b\mapsto x\end{array}$
I.e. $(x~b)(x~a) = \begin{pmatrix}a&... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to solve the differential equation $y' + \sec(x)*y = \tan(x)$ I am really struggling to solve the differential equation: $y' + \sec(x)y = \tan(x)$. If someone could point me in the right direction or give me a step by step plan it would be much appreciated!
So far I have tried taking the common factor to be $\exp(\... | This is in the form of a Linear Differential Equation
and here the integrating factor would be $e^{\int(\sec(x))}$ so that you will get $$\begin{align}
(\tan(x)+\sec(x))\frac{dy}{dx}+(\tan(x)+\sec(x))y\sec(x) = &\, \tan^2(x)+\sec(x)\tan(x) \\
((\tan(x)+\sec(x))y)' = & \\
\end{align}$$
So integrating on both sides you h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the the integral of $\sqrt{x^a + b}$? How do you evaluate $\displaystyle\int\sqrt{x^a + b}\,\,\text{dx}$, where $a \neq 0$ and $a \neq 1$?
For example, how do you evaluate $\displaystyle\int\sqrt{x^2 + 1}\,\text{dx}$? If we let $u=x^2+1$, then $du=2x\,\text{dx}$. We cannot do this because there is no $2x$ in th... | Kim Peek's "funny" hypergeometric solution is really the series solution near $x=0$. We have, for $|x^a/b| < 1$,
$$ \sqrt{x^a + b} = \sqrt{b} \sqrt{1 + x^a/b} =
\sqrt{b} \sum_{k=0}^\infty {1/2 \choose k} (x^a/b)^k$$
so integrating term-by-term
$$ \int \sqrt{x^a + b}\; dx = \sum_{k=0}^\infty {1/2 \choose k} \dfrac{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Analyze the Complex Function by using the Principal log Branch I am trying to analyze the function $\sqrt{1-z^2}$, where the square root function is defined by the principal branch of the log function. I want to locate the the discontinuities.
I know the discontinuities will lie on the negative real axis but I cannot f... | Firstly we write $$f\left ( z \right )=\sqrt{1-z^2}=e^{\frac{1}{2}\ln\left ( 1-z^2 \right )}$$
We have branch points for all $z$ such that the argument of the log vanishes. We see that $$1-z^2=0\Leftrightarrow z_{1,2}=\pm1$$
We now need to investigate the point $z=\infty$. In that manner we need to see if if there is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1649955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Checking whether a given polynomial is reducible or irreducible.
We're given the polynomial $x^{2}-2$ , and we need to prove that it's irreducible in $\mathbb Q$ but reducible in $\mathbb R$.
Writing the polynomial as $(x^{2}-2) = 1.(x^{2}-2)$ ,
$(x^{2}-2)$ can't be a unit in $Q[x]$ whereas $1$ is a unit in $Q[x]$ , ... | Like the comments said, $x^2-1 = (x-1)(x+1)$, which is reducible in $\mathbb{Q}[x]$, clearly. So, I am going to assume that you mean $x^2+1$, which is not reducible in $\mathbb{R}$, since $\text{discriminate}(x^2+1) = 0^2 - 4(1)(1) = -4 < 0$.
But, to show that it is not reducible in $\mathbb{Q}[x]$, suppose that it i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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subobject classifier for partial orders Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.)
If not, what is a category which is "close" to the category of partial orders (e.g. it should consists of special order-theoretic constructs) and has a subobject classifier? Bonus ques... | The fact that all internal co-categories in a coherent category are necessarily co-equivalence relations [see Peter Lumsdaine's TAC article A small observation on co-categories] provides a telltale sign that the category of posets fails to be a topos.
For the inclusion functor $\textbf{Poset} \to \textbf{Cat}$ is repr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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On proving that $\sin x > x \cos x$ in a right neighbourhood of zero. I was recently asked to prove rigorously that there exists a $\delta$ s.t. $\sin x > x \cos x$ in $]0,\delta[$ .
The best I could come up with was that Taylor expanding $\sin x - x \cos x $ I obtain $\frac{ x^3}{3} + o(x^3)$ and noticing that the do... | $${\sin x\over x}=\int_0^1\cos(t\,x)\>dt>\cos x\qquad(0<|x|\leq\pi)\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What is the sum of all the natural numbers between $500$ and $1000$. What is the sum of all the natural numbers between $500$ and $1000$ (extremes included) that are multiples of $2$ but not of $7$?
| In general, the sum of an arithmetic progression is $$S_n = \sum_{i=1}^{n} a_i=\frac{n}{2}(a_1+a_n)$$
So, the sum of all even numbers in your interval
$$ = \frac{251}{2}(500+1000)$$
And the sum of all multiples of $14$ in your interval
$$ = \frac{36}{2}(504+994)$$
Subtracting these two answers will give you the resul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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Group of exponent $2$. When I have a group $G$ of exponent $2$ and I know that all elements in $G-\{e\}$ are conjugated, is it right that $G$ is of order $2$?
My try: For $g,h \in G - \{e\}$ the conjugation assumption implies $$ga=ah$$ but since $\exp G=2$ then $G$ is abelian and therefore $$ga=ha$$
It follows that $g=... | Note that if $x, y \in G$. Then $yx = y^{-1}x^{-1}= (xy)^{-1} = xy$ by the hypothesis. Hence $G$ having an exponent 2 implies that $G$ is abelian.
Choose $a, b \ G\backslash\{1 \}$.
Then there exists a $g$ such that $b = gag^{-1}$.
Then $h = ba^{-1} = gag^{-1}a^{-1} = 1$ since $G$ is abelian.
It follows that $a = b$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Linear vs. bilinear I'm tripping over something elementary:
Suppose $f:\mathbb{R^2}\rightarrow X$ is linear, then $f(x+y)=f(x)+f(y)$ for all vectors $x$ and $y$. Now suppose that $f$ is also bilinear and in particular linear in the first argument, then
$$f\Big((a+b,c)\Big)= f\Big((a,c)\Big)+f\Big((b,c)\Big)$$
But now s... | A (non-trivial) linear function from $\mathbb{R}^n$ for $n > 1$ will never be a multilinear function (bilinear when $n=2$) if it is a linear function. Instead, a multilinear function is a linear function from a tensor product. So a bilinear function on $\mathbb{R}$ is a linear function $\mathbb{R}\otimes\mathbb{R} \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Distance between two circles on a cube I found this problem in a book on undergraduate maths in the Soviet Union (http://www.ftpi.umn.edu/shifman/ComradeEinstein.pdf):
A circle is inscribed in a face of a cube of side a. Another circle is circumscribed about a neighboring face of the cube. Find the least distance betw... | I placed the inscribed circle on the top face (+y direction) and the circumscribed circle on the front face (+z direction). Their locus of points is
$$ \begin{align}
\vec{r}_1 & = \begin{bmatrix} r_1 \cos \theta_1 & \frac{a}{2} & r_1 \sin \theta_1 \end{bmatrix} \\
\vec{r}_2 & = \begin{bmatrix} r_2 \cos \theta_2 & r_2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of the Stratonovich integral? Computing the integral $\int \Phi(x_t,t)dx_t$
writing the equation in the form we can write the integral as the mean square limit
$$\int \Phi(x_t,t)dx_t=\lim_{\Delta \to 0} \sum^{j=1}_{N-1} [\Phi((\frac {x(t_j)+x(t_{j+1})}{2},t_j)][x(t_j+1)-x(t_j)]\ \ \ \ \ \ .\ (3)$$
and in Ito's ... | If $x \mapsto \Phi(x,t)$ is differentiable, it follows from Taylor's formula that
$$\Phi(x,t) = \Phi(y,t) + (x-y) \frac{\partial}{\partial x} \Phi(\zeta,t)$$
for some intermediate point $\zeta$ between $x$ and $y$ (i.e we can find $\lambda \in (0,1)$ such that $\zeta = \lambda x+ (1-\lambda) y$). Using this identity fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1650997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X Here X is a Hausdorff Space.
I know that singleton sets, {x}, are closed in a Hausdorff space. Although Im not sure if thats how to use the Hausdorff property.
Should I investigate $h=F(x)-x$?
Can anyone give a... | I have this idea that I'd love somebody with more knowledge than I will check: define a function
$$\;\phi:X\to X\times X\;,\;\;\text{by}\;\;\;\phi(x):=(x,F(x))\;$$
Since each coordinate function is continous also $\;\phi\;$ is, and if $\;\Delta:=\{(x,x)\in X\times X\}\;$ is the diagonal in the cartesian product, then
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute a natural number $n\geq 2$ s.t. $p\mid n \Longrightarrow p^2\nmid n$ AND $p-1\mid n \Longleftrightarrow p\mid n$ for all prime divisor p of n. Question: Compute a natural number $n\geq 2$ that satisfies:
*
*For each prime divisor $p$ of $n$, $p^2$ does not divide $n$.
*For each prime number $p$, $p-1$ d... | Here's a partial answer: Suppose $n$ is a solution. Then $n(n+1)$ is a solution if $n+1$ is prime, since $n+1\mid n(n+1)$ and $n\mid n(n+1)$, and the condition is clearly satisfied for all primes less than $n$. So in addition to $2$, $3$, $6$, and $42$, we also get
\begin{align*}
42(43) = 1806 = 2\cdot 3\cdot 7\cdot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Prove that if G contains an odd vertex then every vertex of G is odd Let G be a graph with degree of each vertex either $m$ or $n$, where there are $m$ vertices of degree $m$ and $n$ vertices of degree $n$. Prove that if G contains an odd vertex then every vertex of G is odd.
how do I go about this? What does it imply ... | A very useful elementary result in graph theory is the degree sum formula:
$$\sum_{v\in V}\deg(v) = 2\lvert E \rvert$$
for a graph $G=(V,E)$.
In particular the sum of the degrees of the vertices is even.
Now what can you say if some vertex has odd degree?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Distribtution of the maximum of three uniform random variables. How do I get the cumulative density function of $Y$? $X$ is a continuous random variable with pdf
$$f(x) = 1,\quad 0 < x < 1. $$
Three independent observations of $X$ are made. Find the pdf of $Y$ where $$Y = \max\{X_1,X_2,X_3\}.$$
| The pdf or cdf you are looking for is part of what is called the "order statistics": https://en.wikipedia.org/wiki/Order_statistic
The easiest path is through the cdf, using the following transformations:
$F_Y(y) \ = \ P(Y\leq y) \ = P(\max{\{X_1,X_2,X_3\}} \leq y)$
$F_Y(y) \ = \ P(X_1 \leq y \ \& \ X_1 \leq y \ \& \ X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$. Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$.
Here's my example: let $C$ = (0,1], whe... | $1$ is not a limit point of $C$.
The definition for limit point is,($x$ of $C$)
$$ \forall r > 0, \exists y \in C\mid y \neq x, \operatorname{d}(x,y) < r
$$
so taking $r = 1/3$...
You can choose $x_j = 1/j$ for all even $j$ and $x_j = 1-1/j$ for all odd $j$. Then $0$ and $1$ are limit points of $C$, but the sequence do... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Example of inverse semigroup with at least two idempotent elements We say that the semigroup $S$ is inverse semigroup if for any $s\in S$ there is a unique $t\in S$ such that $sts=s,\ tst=t$. Suppose that $E(S)=\{e:\ e\in S,\ e^2=e\}$ and define $$s\sim t\Leftrightarrow \exists\ e\in E(S):\ se=te,\qquad (s,t\in S); $$... | We know that the set of natural number with multiplication defined by
$m\cdot n=\min\{m,n\}$ is an inverse semigroup. Now suppose that $G$ is a group such that there is an element $h\in G$ such that $h^2\neq h$ and suppose that $e$ is its identity. Obviously $S:=N\times G$ is an inverse semigroup with mutiplication
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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A sequence in a Hausdorff space and in a space that is not Hausdorff.
Let $X$ be a topological space and $\{x_n\}_{n=1}^{\infty}$ a sequence in $X$.
Show that if $X$ is Hausdorff, $x_n \rightarrow x \:$, $x_n \rightarrow y \:$ implies $x=y$.
Give an example that shows that if $X$ is not Hausdorff then this is not nec... | Take $\mathbb{N}$ in the cofinite topology (the only closed sets are the finite ones (including the empty set) and $\mathbb{N}$ itself). Take $a_n$ to be any sequence where all values are different, like $a_n = n$ or $a_n = 2n$ etc.. Then $(a_n)$ converges to every point $m$ of $\mathbb{N}$, because the only open sets ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability Type I error I have these values:
A sample of size $100$ is taken from the population.
Standard Deviation is $5$.
Average is $125$.
$H_0$: $\mu =125$ Against $H_a$: $\mu <125$
Accept $H_0$ if the sample mean is $\bar{\mathbb{x}} ≥ 124$
Reject $H_0$ if the sample mean $\bar{\mathbb{x}} < 124 $
Find the pro... | You work seems fine, except for the final step as Augustin pointed out.
In terms of probability, recall that we are seeking the probability that $\bar x < 124$ given that the $\mu = 125$. This means
$$P(\bar x <124|\mu = 125)$$
By standardizing, we get
\begin{align*}
P(\bar x <124|\mu = 125) &= P\left(Z < \frac{124-125... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating the limit $\lim_{x \to 0}\left(x+e^{\frac{x}{3}}\right){}^{\!\frac{3}{x}}$
$$y=\left(x+e^{\frac{x}{3}}\right)^{\frac{3}{x}}$$
I'm looking at this equation, and need to solve for the limit as $ \to 0$. I've graphed it, and know the solution is $e^4$, but am clueless as to the steps to actually solve this.
... | Intuition:
A way to see what is going on is to see the affine approximation of $e^x$ around $0$: $$e^u \simeq e^0 + (e^\prime)(0) x = 1 + x$$ (this can be made formal by Taylor approximations to order $1$, for instance). This implies that your quantity is roughly $\left(x+ 1+ \frac{x}{3}\right)^{3/x} = \left(1+ \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1651957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Average of cosine waves What looks like the graph of the function $A=E[cos(\omega t)]$, where $E$ means "average" and $\omega$ is a random variable?
| Assuming that $\omega$ is uniformly distributed in range $[a,b]$, we have
$$E[\cos(\omega t)]=\frac1{b-a}\int_a^b\cos(\omega t)d\omega=\frac{\sin(bt)-\sin(at)}{(b-a)t}.$$
For example, with $a=1,b=3.14$:
In the case of a Gaussian, another Gaussian.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652056",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Express reverse inversion, major index, descents in terms of the forward direction. Given $w=a_1a_2...a_n \in S_n $, then the reverse of $w$ is $w^r=a_n....a_2a_1$.
Express inv($w^r$), des($w^r$) and maj($w^r$) in terms inv($w$), des($w$), maj($w$), respectively.
I know the definition of an inversion is a partiti... | HINT: For the inversions, show that $\langle i,j\rangle$ is an inversion of $w^r$ if and only if it is not an inversion of $w$. How many pairs $\langle i,j\rangle$ with $i<j$ are there?
For the descents, show that each descent of $w^r$ is an ascent of $w$ and vice versa, so $w^r$ has exactly as many descents as $w$ has... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding $a^5 + b^5 + c^5$
Suppose we have numbers $a,b,c$ which satisfy the equations
$$a+b+c=3,$$
$$a^2+b^2+c^2=5,$$
$$a^3+b^3+c^3=7.$$
How can I find $a^5 + b^5 + c^5$?
I assumed we are working in $\Bbb{C}[a,b,c]$. I found a reduced Gröbner basis $G$:
$$G = \langle a+b+c-3,b^2+bc+c^2-3b-3c+2,c^3-3c^2+2c+\frac... | Using just Macaulay2, you can do the following
Macaulay2, version 1.6.0.1
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition,
ReesAlgebra, TangentCone
i1 : R=QQ[a,b,c]
o1 = R
o1 : PolynomialRing
i2 : i1=ideal(a+b+c-3,a^2+b^2+c^2-5,a^3+b^3+c^3-7)
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Convergent sequence of irrational numbers that has a rational limit. Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
| If $q$ is any rational number at all and $n$ is a positive integer then $q+\frac 1 n \sqrt 2$ is irrational (it's a simple algebra exercise to prove that), and $\lim\limits_{n\to\infty}\left(q + \frac 1 n \sqrt 2\right) = q$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$.
If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$.
Obviously since this is a 5th degree polynomial, solving it is not going to be possible (or may be hard). However I think that factoring it to g... | We have $\sqrt[6]{3} \approx 1.2009$ and $\sqrt[6]{4} \approx 1.2599$. Let $f(x)=x^3+x^2-x-2$. Then $f(1.2) \approx -0.032$ and $f(1.25) \approx 0.2656$. So, $a$ must be between $1.2$ and $1.25$.
EDIT: As I said in the comments below, I see no way of showing that there is only one root using only precalculus. But,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Prove $\binom{3n}{n,n,n}=\frac{(3n)!}{n!n!n!}$ is always divisible by $6$ when $n$ is an integer.
Prove $$\binom{3n}{n,n,n}=\frac{(3n)!}{n!n!n!}$$ is always divisible by $6$ when $n$ is an integer.
I have done a similar proof that $\binom{2n}{n}$ is divisible by $2$ by showing that $$\binom{2n}{n}=\binom{2n-1}{n-1}+... | Notice that $$\binom{3n}{n,n,n}=\frac{(3n)!}{n!n!n!} = \binom{3n}{n}\binom{2n}{n}.$$ We have shown that $\binom{2n}{n}$ is divisible by 2. Now, all we must do is show that $\binom{3n}{n}$ is divisible by 3. You know that $\binom{3n}{n}$ is the number of n-element subsets of a 3n-element set (or, the number of ways to c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
Show that $a_n = 1 + \frac{1}{2} + \frac{1}{3} +\dotsb+ \frac{1}{n}$ is not a Cauchy sequence
Let $$
a_n = 1 + \frac{1}{2} + \frac{1}{3} + \dotsb + \frac{1}{n}
\quad (n \in \mathbb{N}).
$$
Show that $a_n$ is not a Cauchy sequence even though
$$
\lim_{n \to \infty} a_{n+1} - a_n = 0
$$
(Therefore $a_n$ does no... | Hint: Showing it doesn't converge (specifically that it goes to infinity) would help.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate Let $G$ be a group such that every maximal subgroup is of finite index and any two maximal subgroups are conjugate and any proper subgroup is contained in a maximal subgroup . Then is $G$ cyclic ? I know that ... | Here is a reduction to the finite case. Suppose $G$ is a group satisfying your requirements, and $M$ a maximal subgroup. Then all maximal subgroups are conjugate to $M$. This implies that the Frattini subgroup $\Phi(G)$ of $G$ (which is defined to be the intersection of all maximal subgroups of $G$) is the intersection... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1652935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Classifying Covering Spaces using First Cohomology I am familiar with the classification of covering spaces of a space $X$ in terms of subgroups of $\pi_1(X)$ (up to conjugation). However, if $X$ is a manifold, I know that $H^1(X; G)$ classifies G-bundles over $X$ (using Cech cohomology here). I think finite regular co... | Regular cyclic $q$-coverings are classified by normal subgroups $$N\hookrightarrow \pi_1X \to \pi_1X/N\cong \mathbb Z/q,$$
hence there is a natural isomorphism to $Hom(\pi_1X,\mathbb Z/q)$ mod automorphism of $\mathbb Z/q$.
Note that $K_q=K(\mathbb Z/q,1)$ is universal in the sense that the map $$[X,K_q] \to Hom(\pi_1X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1653001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
$S_1 \subset S_2$. To show, $Span(S_1) \subset Span(S_2)$ Prove that if $S_{1} \subset S_{2}$, then $Span(S_{1}) \subset Span(S_{2})$
Approach: Suppose $S_{1} \subset S_{2}$
Let $x \in S_{1}$, then by definition of a subset, $x \in S_{2}$
All possible linear combinations of $x$ are $cx$ with $c \in \mathbb{R}$
So $cx \... | No, that is not a valid approach.
In order to show that $\mathit{Span}(S_1)\subseteq \mathit{Span}(S_2)$ you need to start with an arbitrary element of $\mathit{Span}(S_1)$ and show that it is also an element of $\mathit{Span}(S_2)$.
You have shown this only for elements of $\mathit{Span}(S_1)$ that happen to have the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1653180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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