Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Algorithm to multiply nimbers Let $a,b$ be nimbers. Is there an efficient algorithm to calculate $a*b$, the nim-product of $a$ and $b$?
The following rule seems like it could be helpful:
$$
2^{2^m} * 2^{2^n} = \begin{cases}
2^{2^m} 2^{2^n} & \text{if $m \ne n$} \\
3(2^{2^m - 1}) & \text{if $m = n$} \\
\end{cases}.
$$
J... | An algorithm is given at https://www.ics.uci.edu/~eppstein/numth/ (C++ implementation of J.H.Conway's "nimber" arithmetic.). The function to actually perform the multiplication is at nimber.C:316:nim_times.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/909304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Find the probability of winning at this lottery. So, the problem I found goes like this:
You have $n$ different numbers, numbered from $ 1 $ to $n$. You can
randomly choose $m$ (different) of them. The computer also randomly
selects $m$ (different) of them. If you and the computer have exactly
$k$ common numbers... | Let us assume you have picked your $m$ numbers. Now it's the computer's turn. It has to match $k$ of your numbers. Which $k$? These can be chosen in $\binom{m}{k}$ ways. Then it has to produce $m-k$ numbers which do not match any of yours. This can be done in $\binom{n-m}{m-k}$ ways.
So the number of ways the computer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/909472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
What is $\lim_{n\to \infty}\frac{2n \choose {n}}{4^n}$? What is the result of the following limit?
$$\lim_{n\to \infty}\frac{2n \choose {n}}{4^n}$$
since $$\sum_{k=0}^{2n}{2n \choose {k}}=2^{2n}=4^n$$
then $$\frac{4^n}{2n+1}\leq{2n \choose {n}}\leq 4^n$$
and limit is clealy $\in [0,1]$, but what is it exactly?
| Hint:
Use Stirling approximation
$$ n! \sim n^n e^{-n} \sqrt{2 \pi n} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/909678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Expressing $ 12\sin( \omega t - 10) $ in cosine form $$ 12\sin( \omega t - 10) $$
I understand how it's solved when using the graphical method, however I'm having trouble understanding something about the trigonometric identities method.
The solution in the text book goes like this (It wants positive amplitudes) : (Al... | The identities you can use are:
\begin{align}
\sin x&=\cos(90°-x)\\
\cos x&=\cos(-x)
\end{align}
Therefore
$$
\sin(\omega t-10°)=\cos(90°-(\omega t-10°))=
\cos(100°-\omega t)=\cos(\omega t-100°).
$$
Of course, you could also directly use
$$
\sin x=\cos(90°-x)=\cos(x-90°).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/909871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Density of Pythagorean triples We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$.
In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$.
Let $\mathcal P$ be the set of all legit Pythagorean triples.
We define $$L_{PT}^N=\{<a,b,c> | <a,b,c> \... | It was proved by Lehmer that the number of primitive triples with hypotenuse less than $x$ is asymptotic to
$$
\frac{x}{2\pi} \approx 0.15915494309x
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/909954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Expected value expressed by CDF. I have found following formula for expected value:
$$\operatorname{E}[X] = \int_0^\infty \int_0^x \! \mathrm{d}t \, \mathrm{d}F(x) = \int_0^\infty \int_t^\infty \! \mathrm{d}F(x)\mathrm{d}t = \int_0^\infty \! (1-F(t))\,\mathrm{d}t$$
and I don't understand this equality:
$$\int_0^\infty ... | Prescribe function $f\left(x,t\right)$ by $\left(x,t\right)\mapsto1$
if $t<x$ and $\left(x,t\right)\mapsto0$ otherwise.
Applying
Fubini you find:
$$\int_{0}^{\infty}\!\!\int_{0}^{x}dtdF\left(x\right)=\int_{0}^{\infty}\!\!\int_{0}^{\infty}f\left(x,t\right)dtdF\left(x\right)=\int_{0}^{\infty}\!\!\int_{0}^{\infty}f\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/910177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve for $\theta$: $a = b\tan\theta - \frac{c}{\cos\theta}$ This question was initially posted on SO (Link). I'm not sure the answer given there was correct. I cannot get the results from those expressions to match my CAD model.
The title pretty much sums it up. How do I solve for theta given the following equation.... | The given equation is equivalent to
$$
b\sin(\theta)-a\cos(\theta)=c.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/910269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 6
} |
Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there is an extra layer of comp... | $\Phi:\mathbb R \to (0,1)$ and $\Phi^{-1}:(0,1) \to \mathbb R$ are strictly increasing continuous bijective functions
If $z=\Phi^{-1}(p)$ then $p=\Phi(z)$ and $\dfrac{dp}{dz}=\Phi^\prime(z)=\phi(z)$, so $\dfrac{dz}{dp} = \dfrac{1}{\phi(z)}= \dfrac{1}{\phi(\Phi^{-1}(p))}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/910355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Does $\sum_{i=1}^\infty a_i/i < \infty$ imply that $a_i$ has Cesaro mean zero? If $(a_i)_{i=1}^\infty$ is a sequence of positive real numbers such that:
$$ \sum_{i=1}^\infty \frac{a_i}{i} < \infty. $$
Does this mean that the sequence $(a_i)_{i=1}^\infty$ has Cesaro mean zero? As in
$$ \lim_{n\to\infty} \frac{1}{n} \su... | Summation by parts gives:
$$\sum_{i=1}^{n}a_i=\sum_{i=1}^{n}\frac{a_i}{i}+\sum_{j=1}^{n-1}\sum_{k=j+1}^{n}\frac{a_k}{k},\tag{1}$$
while the convergence of $\sum_{i=1}^{+\infty}\frac{a_i}{i}$ gives that for any $\varepsilon>0$ there exists $M_\varepsilon$ such that
$$\sum_{n\geq M_\varepsilon}\frac{a_i}{i}\leq \varepsil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/910439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 1
} |
converting geometric infinite series to another infinite series Let $\{x_n\}_{n=1}^\infty$ be a sequence satisfying the recurrence relation:
$$ x_n = a\left(1- \sum_{k=0}^{n-1}x_k\right) $$
Where $ x_0 = 1 $, and $a \in [0,1]$ is chosen so that
$$
\sum_{k=1}^{\infty} x_k = 1$$
Given a positive integer $d$, how do I ge... | For the beginning, rewrite $$f(x) = xa^x=a x a^{x-1}=a\frac{d(a^x)}{da}$$ So, $$\displaystyle \sum_{x=1}^{\infty} f(x) = a \sum_{x=1}^{\infty}\frac{d(a^x)}{da}=a\frac{d}{da}\sum_{x=1}^{\infty}a^x=a\frac{d}{da}\Big(\frac{a}{1-a}\Big)=\frac{a}{(1-a)^2}$$ So, if $$\displaystyle \sum_{x=1}^{\infty} f(x) =1$$ solving $$\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/910510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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$\int \sqrt{1+\sin ^2 x} dx$ an elliptic integral? It seems to be an elliptic integral of the second kind, but when $k=i$? This is going by the definition that $E(\theta,k)=\int_{0}^{\theta} \sqrt{1-k^2 \sin^2x}dx$. That seems a bit off.
Or is this not one at all due to the indefinite nature of the integral?
| Consider the elliptic integral of the second kind $E(\phi,m):=\int\limits _0^\theta \sqrt{1-m \sin^2 x}\,dx$. (Note that my convention will be to write in terms of $m=k^2$ rather than $k$ itself.) Naively, the range of allowed $m$ is $m\in [0,1]$. The question posted above is then a special case of the following: What ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/910597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Homework | Find the general solution to the recurrence relation A question I have been stuck on for quite a while is the following
Find the general solution to the recurrence relation
$$a_n = ba_{n-1} - b^2a_{n-2}$$
Where $b \gt 0$ is a constant.
I don't understand how the general solution can be found with $b$ and $b... | When you have $a_n=ba_{n-1}+b^2a_{n-2}$, you can see
$$b^\color{red}{0}\cdot a_\color{blue}{n}=b^\color{red}{1}\cdot a_{\color{blue}{n-1}}-b^\color{red}{2}\cdot a_{\color{blue}{n-2}}$$
where $0+n=1+(n-1)=2+(n-2)$.
In such case, dividing the both sides by $b^n$ gives you
$$\frac{a_n}{b^n}=\frac{a_{n-1}}{b^{n-1}}+\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/910702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
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Showing that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$ for formal power series I've just come across formal power series and am not very fluent with them yet.
I'd like to show that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$.
Can anybody help?
| $$\exp\left(\sum\limits_{n=0}^{+\infty}a_n X^n\right)=\exp\left({\lim\limits_{n\to
+\infty}\sum\limits_{k=0}^{n}a_k X^k}\right)=\lim\limits_{n \to +\infty}\exp \left(\sum\limits_{k=0}^{n}a_k X^k\right)=\lim\limits_{n \to \infty}\prod\limits_{k=0}^{n}\exp \left(a_k X^k\right)=\\=\prod\limits_{n=0}^{+\infty}\exp \left(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/910796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Openness of path connected components of open subsets of $\mathbb C$ Let $\Omega\subset \Bbb{C}$ be an open set. My textbook states that every path connected component of $\Omega$ is open.
I can't seem to understand why that is. Why does every point have to contained in a path-connected neighbourhood which lies entire... | Let $P$ be any path component of $\Omega$. Let $p \in P$ be any point. Since $\Omega$ is open, there exists an open ball $B$ such that $p \in B$ and $B \subset \Omega$. Since $B$ is path connected, it must be contained in $P$, by definition of path component. Therefore $P$ is open.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/910884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Why is there a 'missing' $1$ in the Euler–Mascheroni constant? It is easy to show that:
$$
\sum_{k=1}^n \frac{1}{k} > \ln(n+1),
$$
but the Euler–Mascheroni constant is defined as:
$$
\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right).
$$
My question is, why was $\gamma$ defined using $\ln(n)... | $$ \left( \sum_{k=1}^{n} \dfrac{1}{k} - \ln(n+1) \right) - \left( \sum_{k=1}^{n} \dfrac{1}{k} - \ln(n) \right)=\ln(n)-\ln(n+1)=\ln\left(\frac{n}{n+1}\right)$$
And
$$\lim_n \ln\left(\frac{n}{n+1}\right)=\ln 1=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/910965",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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What is the expected number of coin tosses needed to obtain a head? Due to my recent misunderstandings regarding the 'expected value' concept I decided to post this question. Although I have easily found the answer on the internet I haven't managed to fully understand it.
I understood that the formula for the expected ... | Let $X$ be the number of tosses, and let $e=E(X)$. It is clear that $e$ is finite.
We might get a head on the first toss. This happens with probability $\frac{1}{2}$, and in that case $X=1$.
Or else we might get a tail on the first toss. In that case, we have used up $1$ toss, and we are "starting all over again." So i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
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A problem about martingale with stopping time . In Durrett's "Probability,theory and examples":
Suppose $X_n$ is supermartingale and $H_n$ is predictable. define:
$$(H\cdot X)_n\triangleq\sum^n_{m=1}H_m(X_m-X_{m-1}) $$
$N$ is stopping time and let $H_m=1_{\{N\ge m\}}$
Then the author claim:
$$(H\cdot X)_n=X_{N\wedge n... | It follows from the very definition of a stopping time (page 155) that $N \geq 1$. Hence, $$-1_{\{N \geq 1\}} = -1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/911093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finite rings without unity that are subrings of finite rings with unity I know that a ring $R$ without unity can be embedded as a subrng of a ring with underlying additive structure $R \oplus \mathbb{Z}$, a ring with unity. But this does not yield a finite field. But I read somewhere that $\mathbb{Z}$ can be replaced b... | Indeed, the well-known Dorroh adjunction of $1$ is not useful in many contexts because it doesn't preserve crucial properties of the source rng and/or doesn't satisfy various minimality properties. Below is an alternative, which alleviates some of these problems, addressing the issue you mention
W.D. Burgess; P.N. Stew... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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procedure to pair all people with all others in a group I am looking for an easy method to ensure that all people in a group get to meet all others. The "speed dating" method is to have two rows of people facing each other, and then rotate one of the rows. This works for half of the pairings. How do I get the remaining... | When $n$ is odd, label each person with elements of $\mathbb Z_{n}$ and, each round of introductions is also labeled from elements in $\mathbb Z_n$. Then in round $k$, person $a$ is introduced to person $b$, if $a+b=k$, with the unique person $i$ so that $2i=k$ not introduced to anybody. This yields $n$ rounds of $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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difference between the polynomials I have a homework assignment that I do not know how to solve. I don't understand how to calculate $f(x)$ in this assignment.
$f(t)$ is the difference between the polynomials $2t^3-7t^2-4$ and $t^2-7t-3$.
Calculate $f(3)$.
What should I do to calculate $f(t)$?
Thanks!
| Should be $f(t)$ not $f(x)$. Subtract the two equations to find $f(t)$. This will leave you with $2t^{3}-8t^{2}+7t-1$. Now plug in $t=3$ and you are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/911328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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The intersection of $3$ set is empty, would the intersection of $4$ sets be empty? Let me clarify some more.
Let's say we have four sets $A,B,C,$ and $D$.
If the intersection of any three sets is empty, by default is the intersection of all four sets empty?
| Yes, if $A$, $B$, $C$, and $D$ are sets such that the intersection of any three is empty, then because intersection is associative, we have that $A \cap B \cap C \cap D = (A \cap B \cap C) \cap D = \emptyset \cap D = \emptyset$.
Just in case you aren't sure what associative means, it means that we can intersect in any ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I numerically evaluate the total derivative of a multivariate function? I think I understand now the intuitive reasoning behind the total derivative of a multivariate function $z = z(x, y)$, which is
$$
dz = \frac{\partial{z}}{\partial{x}}dx + \frac{\partial{z}}{\partial{y}}dy
$$
So let's take an example, $z =... | The differential is not a numerical value, so it's meaningless to associative a number to a differential. But it indeed has a representation using a set of basis, in which the differential is regarded as some kind of vector. Say instead we use binary tuple $(2x,2y)$ to represent differential $dz$. In practice, we also ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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probability rolling a dice 5 times I can't solve this problem:
What is the probability that, when rolling a dice 5 times, the number of times when you get a 1 or 2 is greater than the number of times when you get a 6.
any help?
| @Stones Yes, but the table is not quite right. If there are three or more 1s or 2s, then it does not matter how many 6s there are. So N5 the number of ways of getting five 1s or 2s is 32. Similarly, N4 the number of ways of getting four is $5\times 2^4\times 4=320$ and N4 the number of ways of getting three is $10\time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Square Integrable local martingale or locally square integrable martingale? I have a question about martingales. What is the difference between "locally square integrable martingale" and "square integrable local martingale"? In particular, which set does $M_{loc}^2$ represent?
| Protter gives the following definition:
Let $X$ be a stochastic process. A property $\pi$ is said to hold locally if there exists a sequence of stopping times $(T_n)_{n \geq 1}$ increasing to $\infty$ almost surely such that $X^{T_n} 1_{\{T_n>0\}}$ has property $\pi$, each $n \geq 1$.
If we speak of a square integrab... | {
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"timestamp": "2023-03-29T00:00:00",
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Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1)^n$? I am working out a Fourier Series problem and I saw that the suggested solution used
$\sin(n\pi) = 0$ and $\cos(n\pi)=(-1)^n$ to simply the expressions while finding the Fourier Coefficients $a_0$, $a_n$, $b_n$.
I am aware that the $\sin(x)$ has a period of $2\pi$. So I a... | On a unit circle
$x$ coordinate of any point on the circle is given by $\cos\theta$ and $y$ coordinate is given by $\sin\theta$
Now, $\sin(n\pi)$, where $n=0,1,2,3...$ is always the X-axis and on X-axis we have $y=0$
and $\cos(n\pi)$ assumes $x=1$ or $x=-1$ as
$\cos(0.\pi)=1=(-1)^0,\cos(1.\pi)=-1=(-1)^1,\cos(2.\pi)=1=(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 7,
"answer_id": 4
} |
What is the motivation for quaternions? I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
| Hamilton (and Graves) wanted to generalize $\mathbb C$ - if viewed as $\mathbb R^2$ with a multiplication that turns it into a field with a multiplicative absolute value. They were looking for something similar in $\mathbb{R}^n$ for $n>2$. It turns out that Hamilton spent 13 years in vain with $n=3$ although it was e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 6,
"answer_id": 3
} |
Pointwise estimate for a sequence of mollified functions In the answer to Characterisation of one-dimensional Sobolev space Tomás wrote
... let $\eta_\delta$ be the standard mollifier sequence. Let $u_\delta=\eta_\delta\star u$ and note that for any $c\in (a,b)$ $$|u_\delta(x)-u_\epsilon(x)|\le \int_c^x |u'_\delta (t)... | By the fundamental theorem of calculus, we have that $$u_\delta(x)=u_\delta (c)+\int_c^xu_\delta'(t)dt,$$
Can you conclude now?
| {
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"url": "https://math.stackexchange.com/questions/911868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Graph of the $\sqrt { x } $ Why does the graph only go to the right and up? Wouldn't there be negative values of y as well? Since, for example, the $\sqrt{4}$ is $2$ and $-2$
| A function $f$ is a map $f: X\rightarrow Y$ which for every input $x\in X$ gives precisely one output $f(x)\in Y$. Taking $\sqrt4=\pm2$ is not a function, because it has two outputs.
The notation $\sqrt{x}$ is used to denote the function $\sqrt{\:\:\:}\:: \mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ with positive image, ig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/911947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Check if two vector equations of parametric surfaces are equivalent Give the vector equation of the plane through these lines:
$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}\,\,\,$ and $\,\,\,\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\... | But the really simple way is to pick three arbitrary points (not on the same line) from the first plane and check if they all lie in the second one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/912048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions? Let $\varphi: \mathbb{N} \to \mathbb{N}$ be the totient function.
Is there an integer $N > 0$ such that there are infinitely many integers $n > 0$ such that
$$\varphi(n) = N?$$
| No.
We use the fact that if $n = p_1^{e_1} \cdots p_k^{e_k}$, with the $p_i$ different primes and the $e_i$ positive, we have $\phi(n) = p_1^{e_1-1}(p_1-1) \cdots p_k^{e_k-1}(p_k-1)$.
Now consider an integer $N>0$ and suppose that $\phi(n) = N$. Then $n$ cannot be divisible by primes larger than $N+1$, since if $q > N... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/912137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rationale for expressing as a direct sum and a direct product In "Ireland and Rosen" page 35, it says if $R_1, R_2, ..., R_n$ are rings, then
$R_1 \oplus R_2 \oplus \dots \oplus R_n = S$ is the direct sum of the $R_i$.
Later in a proposition it says if $S = R_1 \oplus R_2 \oplus \dots \oplus R_n$,
then the group of un... | It is standard to use $\oplus$ for the biproduct of modules. (aside: for infinite products of modules, $\oplus$ is interpreted as the coproduct)
If the modules $M$ and $N$ have an algebra structure, then they induce a canonical algebra structure on their direct sum $M \oplus N$.
It is standard, but strange, notation to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/912229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to derive the closed form of the sum of $kr^k$ $$
\sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 }
$$
How to derive it? I read about some finite calculus, and i understand how to tackle sums of $x^2$, $x^3$, etc.. But I don't know if the same methods can be used on this sum?
| Hint: $kr^k = r k r^{k-1} = r \frac{d}{dr}r^{k}$, now interchange summation and derivative.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of a sequence of products How do you prove the following?
$$\lim_{n\,\to\,\infty}\,\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\ =\ 0$$
| We verify easily that for $k\geq 1$, we have $\displaystyle \frac{2k-1}{2k}\leq \sqrt{\frac{k}{k+1}}$. Hence
$$\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \prod_{k=1}^n \sqrt{\frac{k}{k+1}}= \frac{1}{\sqrt{n+1}}$$
and we are done.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/912421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Complex number equations I cannot solve two problems regarding complex equations.
1)Let $z^2+w^2=0$, prove that $$z^{4n+2}+w^{4n+2}=0, n \in \mathbb{N^{*}}$$
What I tried;
$$z^2 \cdot z^{4n}+w^2 \cdot w^{4n}=0 \iff w^2(w^{4n}-z^{4n})=0$$
but it doesn't really prove anything.
2) Let $z=\frac{1+i\sqrt{3}}{2}$, evaluate $... | 1) Rearrange your equation, conversely just plug in $z^2 = -w^2$.
2) You are right, use your $z = e^{i\frac{\pi}{3}}$
You just need to think about what the exponents are mod 6. That's all that matters here. Do you see why?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When does $(e^a)^b = e^{ab}$ hold? For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A?
Since $e^{2\pi i} = 1$, if this formula holds for all complex numbers $A$ and real numbers $B$, then it would imply that $e^{2\pi ti}=( e^... | Let's try to see what $(e^a)^b$ means first:
Well, $(e^a)^b= e^{b \cdot \ln(e^a)}$, by definition. Now, if only it were true that $\ln(e^a)=a$ then we would be happy, and $(e^a)^b$ would be our familiar $e^{ab}$.
Now, let $a=x+iy$ where $x,y$ are reals.
Then, $e^a=e^x \cdot e^{iy}$. What is $\ln(e^a)$ then?
It should b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Expected Value of Identically distributed random variables I have a very quick question regarding the expected value of two random variables $X,Y$ that are identically distributed and not necessarly independent.
Is this equation valid?
$E[XY]=E[X^2]$
If this is not true, why is this? and what relations can I get (regar... | By definition:
$$cov(X, Y) = E(X-EX)(Y-EY) = EXY - (EX)(EY).$$
If $X, Y$ are independent, then $cov(X, Y) = 0,$
so that
$EXY = (EX)(EY).$
| {
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"url": "https://math.stackexchange.com/questions/912747",
"timestamp": "2023-03-29T00:00:00",
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Does the sum of the reciprocals of all primes of the form $4k+1$ converge? Let $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and}\ p\equiv 1 \mod \ 4\}.$
Is $\displaystyle\sum_{p\in S}\frac{1}{p}$ finite or infinite, and where can I find more information about it?
| Dirichlets theorem on arithmetic progressions says there are infinitely many primes in every arithmetic progression $an+b$ where $a$ and $b$ coprime. In particular, there are infinitely many primes of the form $4n+1$.
The proof of the theorem makes use of analysis and in fact shows that the sum $\sum_{p = an+b} \frac{1... | {
"language": "en",
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An interesting linear algebra question Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$.
Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the conjugate transpose matrix) and similarly $\overline{u}$.
Prove that if $\lambda$ is a... | Actually I have found a simple way to directly point out the vector $u$:
*
*$u=\sqrt{\lambda}A\overline{v}+\lambda v$ if $\sqrt{\lambda}A\overline{v}+\lambda v\ne0$
*If $\sqrt{\lambda}A\overline{v}+\lambda v=0$ then $A\overline{v}=-\sqrt{\lambda}v$ then we choose $u=iv$
Also thanks for the elegant answer from User1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $f,g$ are entire functions and$\ fg\equiv 0$ then either $f \equiv 0$ or $g\equiv0. $ Let $f,g$ be entire functions such that $g \not\equiv 0.$ If $fg\equiv0$ in $\mathbb{C},$ could anyone advise me how to show $f \equiv0$ in $\mathbb{C} \ ?$ Thank you.
| Suppose there exists $z$ such that $f(z) \ne 0$. Then $f$ is non-zero in some neighbourhood of $z$, so $g$ must be zero in the same neighbourhood. And if an entire function is identically zero in the neighbourhood of any point, it is zero in the whole of $\mathbb C$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Equation $3x^4 + 2x^3 + 9x^2 + 4x + 6 = 0$ Solve the equation
$$3x^4 + 2x^3 + 9x^2 + 4x + 6 = 0$$
Having a complex root of modulus $1$.
To get the solution, I tried to take a complex root $\sqrt{\frac{1}{2}} + i \sqrt{\frac{1}{2}}$ but couldn't get the solution right. Please help me.
| Let the root be $\cos y+i\sin y,$
Using Complex conjugate root theorem, $\cos y-i\sin y$ must be another root
So, if the four roots are $\cos y\pm i\sin y,u, v$
using Vieta's formula, $(\cos y+i\sin y)(\cos y-i\sin y)u\cdot v=\dfrac63\implies v=\dfrac2u$
So we have $$3[x-(\cos y+i\sin y)][x-(\cos y-i\sin y)](x-u)\lef... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Linear inequality problem: $2x + 1 > 10$ $2x + 1 > 10$
$2x > 9$
$x > 4.5$
The answer in the book says: $x\lt 4.5$.
Am I doing it wrong?
| $$2x+1>10 \iff x\gt 4.5$$
In other words, your solution to the posted inequality is correct!
I suspect the book's (wrong) solution must be a typo/misprint, either in the solution, or in the book's statement of the problem.
The book's solution satisfies the following inequality $$-2x + 1 \gt 10 \iff -2x \gt 9 \iff x\lt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Problem about Random walk and Stopping time. Here is an example in "Probability with Martingales"
My questions are:
(1)Does equation (a) hold for $T=\infty$?
(2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ^\theta]$$
The author said when $T=\infty$ ,$\mathbb{E}[(sech \theta)^Te ^\theta]=0$
So ... | The author did not say that the expectation is $0$: he rather meant that
$M_{T(\omega)}^\theta=0$ if $T(\omega)=\infty$. This is justified because $0\lt \mathrm{sech}(x)\lt 1 $ for each $x$.
We thus have
$$\mathbb E[M_{T}^\theta]=\mathbb E[M_T^\theta\chi\{T<\infty\}].$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is a geometric structure? Every elementary book on abstract algebra usually begins with giving a definition of algebraic structures; generally speaking one or several functions on cartesian product of a point-set to the set. My question is this: Is there a property that unifies different geometric structures like ... | It is known that, for a (compact) topological space, the continuous functions into $\mathbb{C}$ characterize the topology on the space. As far as I know similar statements hold for smooth manifolds (using smooth functions) and algebraic varieties (using polynomials). So one possible answer is that a geometric struc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the infinite sum of $a^{b^x}$? What would $$\sum^{\infty}_{n=0}(1/2)^{4^n}$$ be and how to determine it?
Note that is not a typo, it is of the form $a^{b^x}$ if it were $(1/2)^{4n}$ it would of course be trivial and could be treated using the geometric series summation formula $1/(1-r)$ with $r$ being $1/16$.
... | By asking what the infinite sum is, would you like to write it down as a decimal (using base 10)?
Review of decimals:
A decimal $.a_1a_2a_3...$ really is itself an infinite series:
$$\sum^{\infty}_{n=1}\frac{a_n}{10^{n}}$$
We are comfortable with such series at least partly because we know exactly how to interpret the... | {
"language": "en",
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How do I solve $x^5 +x^3+x = y$ for $x$? I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$?
The explicit question out of the book is to find $f^{-1}(3)$ for $f(x) = x^5 +x^3+x$
So far I have reduced $x^5 +x^... | A few of the answers suggest guessing the answer by inspection. Indeed, in most calculus problems that could be on an exam it is very likely that the solution to the polynomial is meant to be 'obvious', frequently one of $0,\pm1,\pm2$. This is just to keep numbers and computations reasonable, but is not an absolute gua... | {
"language": "en",
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Two point boundary value problem $x''+x =0$ Side conditions x(0)=0 and x(1)=1
I know that I need to find the roots first but don't know how to continue.
Using $x=e^{\lambda t}$, the roots are found by $\lambda^2 + 1 = 0$, which gives us $i$ as the root.
| Your roots are
$$ \lambda = i,\quad \lambda =-i. $$
Now just construct the solution as
$$ x(t) =c_1e^{it} +c_2 e^{-it}. $$
You can simplify the above if you want. I think you know how to advance.
Added: As I said you can simplify the above answer to the form
$$ x(t) = A\sin(t) + B\cos(t). $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Question about left and right derivative. Let $f(x):\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $\forall x\in\mathbb{R}$ there exist:
$$f'_+(x)=\lim_{\delta\rightarrow 0^+}\frac{f(x+\delta)-f(x)}{\delta}$$
$$f'_-(x)=\lim_{\delta\rightarrow 0^-}\frac{f(x+\delta)-f(x)}{\delta}$$
and $\forall x:f'_-(x)=2f'_+(... | A) In fact, if a function $f$ has at each point a left and a right derivative, it has a derivative except possibly for $x\in D$, with $D$ at most countable.
See:
http://books.google.fr/books?id=rbCmt-2NxtIC&printsec=frontcover&hl=fr#v=onepage&q&f=false
Look at page 174, point 7) where you will find a proof.
B) Now you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many ways are there of choosing $k$ distinct items from a set of $n$? Specifically, say I have the integers $1,2,3,\dots,n$ (a set of $n$ integers). I want to select numbers one after another (not at the same time) until I have $k$ distinct numbers. How many ways are there of doing this? Someone told me $nPk$ but I... | From $n$ distinct elements we can choose $k$-elements in $\binom{n}{k}$ ways and after permuting them in $k!$ ways we get that there exists
$$\binom{n}{k}k!=\frac{n!}{k!(n-k)!}k!=\frac{n!}{(n-k)!}$$
possibilities
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/913756",
"timestamp": "2023-03-29T00:00:00",
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Finding Interpretations for First Order Logic I'm currently looking at first order logic and I'm having a difficult time with the following question:
Now I don't want answer cause that wont really help me. What I am looking for is help, with the best approaches or techniques used that can help solve these types of que... | It is difficult to give hints without essentially solving the problems, but here goes!
We will take the natural numbers as meaning $1,2,3,\dots$. If you are using $0,1,2,\dots$ (which I prefer) the examples either need no adjustment or can be trivially adjusted.
Finding interpretations in which the given sentences are... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/913926",
"timestamp": "2023-03-29T00:00:00",
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Why don't semi-direct products determine a group uniquely? While reading some group theory notes I came up to this fact:
Proposition: If $G$ is the inner semi direct product of $H,K$ ($G=HK$, $H\cap K=\left\{1\right\}$ and $H\unrhd G$) then $G\cong H\rtimes_fK$ where $f$ is expicitely given by $k\mapsto f_k(h)=h^{-1}kh... | $\newcommand{\Aut}[0]{\mathrm{Aut}}$In case this is what you are interested in, given $H$ and $K$, two different $f$ may well lead to isomorphic groups $G$.
For instance, in the case you mentioned, consider the primes $p = 7$ and $q = 3$. Let $K = \langle b \rangle$ and $H = \langle a \rangle$. Consider the homomorphis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove a random variable has normal distribution
Let $X$ be a standard normal random variable and $a>0$ a constant.
Define:
$Y = \begin{cases}
\phantom{-}X & \text{if $\,|X| < a$}; \\
-X & \text{otherwise}.
\end{cases}$
Show that $Y$ is a standard normal distribution and the vector $(X,Y)$
is not two-dim... | The distribution of a real-valued random variable $Y$ is determined by its cdf $F := \mathbb{P}(Y \leq y)$ (because sets of the form $\{(-\infty, y]\}$ generate the Borel $\sigma$-algebra on $\mathbb{R}$). Let $N$ denote a standard normal random variable, $Y = -X$ and $y \in \mathbb{R}$; there are three cases: (i) $y \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed-forms of infinite series with factorial in the denominator How to evaluate the closed-forms of series
\begin{equation}
1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, \sum_{n=0}^\infty\frac{1}{(3n+2)!}\\
\end{equation}
Of course Wolfra... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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"answer_id": 3
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Extremal Finite Set Book Recommendation I want to read Extremal Finite Set Combinatorics in some detail.
(By 'Extremal Finite Set Combinatorics' I mean the subject which covers theorems like Sperner's Theorem, Erdos-Ko-Rado Theorem, deBruijn-Erdos Theorem etc.)
Can somebody please recommend me a good book for self-stud... | I would recommend Ian Anderson's book: Combinatorics of Finite Sets.
It is very readable and gives a good introduction.
Another good choice is Bollobas's book: Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Combinatorial Probability.
Also readable. A bit more emphasis on probabilistic ideas.
There i... | {
"language": "en",
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The alternating group is generated by three-cycles
Prove that, for $n \geq 3$, the three-cycles generate the alternation group $A_n$
Proof: We multiply on the left by 3-cycles to "reduce" an even permutation $p$ to the identity, using induction on the number of indices fixed by a permutation. How the indices are num... | The idea of the proof is that we will show that any product of two transpositions is a product of 3-cycles ; and ,if so, then since every member of the An is a product of even number of transpositions, we are done.
To begin with lets us suppose that we have a product of transpositions a1 and a2 where both have a comm... | {
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"timestamp": "2023-03-29T00:00:00",
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Determining whether points are collinear $(1,1)(3,9)(6,21)$
The way I figured that this should be solved is by finding the slope of:
$(1,1)(3,9)$
Then, $(3,9)(6,21)$
Finally $(1,1)(6,21)$
Which are 4, 4,and 4 respectively.
So I assume that they are collinear.
Am I correct? And if not, please provide me with an expl... | Yes, you are correct. It is sufficient to calculate only two slopes to decide that three points are collinear.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/914443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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About the definition of complex multiplication Some people say that the complex product is the way it is to respect the distributive law of multiplication. However, the distributive law acts in the whole number, like:
$$(a+b)(c+d) = ac + ad + bc + bd$$
The multiplication for complex numbers would be something in the fo... | Let $z_1 = a, z_2 = bi, z_3 = c, z_4 = di$.
Then the rule
\begin{equation*}
\tag{$\spadesuit$}(z_1 + z_2)(z_3 + z_4) = z_1 z_3 + z_1 z_4 + z_2 z_3 + z_2 z_4
\end{equation*}
tells us that
\begin{align*}
(a + bi)(c + di) &= ac + adi + bci + bd i^2 \\
&= ac - bd + (ad + bc)i.
\end{align*}
So if we want ($\spadesuit$) to ... | {
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"timestamp": "2023-03-29T00:00:00",
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Easy exponentiation method Is there a simple way of solving, say, $x^{3/2}$?
For example, one way of solving $16^{3/2}$ is to calculate the square root of $16^3$, but I was wondering if there is a simpler mental trick for doing this that generalizes to all possible exponentiation.
| If you could see that $16=4^2$ then you could do this
$$\left(4^2\right)^{3/2}=4^3$$
If you can see such a number that would be the fastest method, another method would be the one rae306 mentioned.
| {
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"timestamp": "2023-03-29T00:00:00",
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general of partial sum of sequence I am trying to find the limit of an infinite series given as
$$\sum\frac{1}{n^2-1}.$$
I came across the following general term of the sequence of partial sums
$$3/4-\left(\frac{1}{2n}-\frac{1}{2(n+1)}\right).$$ I would appreciate assistance to understand how this expression is arrived... | You have
$$
\frac{1}{n^2-1} = \frac 1 2 \left(\ \underbrace{\frac{1}{n-1}-\frac{1}{n+1}\ }_{\text{Call this $\{$A$\}$}} \right)
$$
Consequently
\begin{align}
& \frac12\left( \left(\frac{1}{2-1} - \frac{1}{2+1}\right) + \left(\frac{1}{3-1} - \frac{1}{3+1}\right) + \left( \frac{1}{4-1} - \frac{1}{4+1} \right) + \cdots ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/914853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
What is $f(f^{-1}(A))$? Suppose that $f : E \rightarrow F$.
What is $f(f^{-1}(A))$? Is it always $A$? $f^{-1}$ is the inverse function.
This is not a homework, I'm confused by this statement.
| $
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\calcop}[2]{\\ #1 \quad & \quad \text{"#2"} \\ \quad & }
\newcommand{\endcalc}{\end{align}}
$(This is essentially the same answer as the one by Henno Brandsma, but a bit more expanded in notation I like better, taking smaller steps.)
Which elements $\;y\;$ are in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/914957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Trying to Integrate$ \iint xy\log|x-y|\, dy\,dx $ Hello I am trying to integrate $$
I:=\int_{a}^{b}\int_{a}^{b}xy\log\left(\,\left\vert\,x - y\,\right\vert\,\right)
\,{\rm d}y\,{\rm d}x,\qquad 0 < a <b
$$
for $x,y\in \mathbb{R}$. I added the bounds of integration to deal with any convergence problems as suggested. We ... | First notice the integrand is symmetric in $x$ and $y$, so it suffices to double the $y > x$ portion; that is
$$
I:=2\int_{a}^{b}\int_{x}^{b}xy\log\left(y - x \right)
\,{\rm d}y\,{\rm d}x,\qquad 0 < a <b$$
Change variables to $u = y - x$ and $v = y + x$. The triangle of integration rotates to another triangle and we ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to compute dimension of $O(n,\mathbb{R})$ Let $f:GL(n,\mathbb{R})\to GL(n,\mathbb{R})$ be the smooth map $A\mapsto A^TA$. Observe that $f$ has constant rank on $GL(n,\mathbb{R})$ by chain rule and that $O(n,\mathbb{R})$ is the preimage of $I$, therefore is a regular submanifold.
To compute the dimension of $O(n,\m... | The dimension of $O(n)$ and $SO(n)$ is $\frac{n^2-n}{2}$. This is best seen by computing their Lie algebra $\mathfrak{so}(n)$, which obviously has vector space dimension $\frac{n^2-n}{2}$. For details. see for example here. See also the answer in the question Show that an orthogonal group is a $\frac{n(n−1)]}2-$dim. $C... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Why induction cannot be used for infinite sets? Explain why induction cannot be used to conclude: $$\left(\bigcup_{n=1}^\infty A_n\right)^c = \bigcap_{n=1}^\infty A_n^c$$
| You should look at how your text defines the set $\left( \bigcup_{n=1}^\infty A_n\right)$. My guess is that it defines as the set of all elements $x$ for which there exists an $A_n\in \{A_i\}_{i=1}^\infty$ with the property that $x\in A_n$.
To expand on the previous by Pinilla we can't conclude that $\sum_{i=1}^\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Irreducible components of fiber bundle Suppose $\pi:X \rightarrow Y$ is a (locally trivial) fiber bundle $F$, where all spaces are Noetherian. Suppose $F$ and $Y$ are irreducible; show that $X$ is also irreducible.
Here a similar question is discussed; but this question should be simpler, and should have an easier pro... | The following proof works if one assumes that the fiber $F$ is proper. I am suspicious it might not be true if the fiber is not proper.
Let $X = Z_1 \cup Z_2$ where each $Z_i$ is closed. Then since each fiber $\iota_y : F \hookrightarrow X$ is irreducible, we have that either $\iota_y^{-1}(Z_i)$ is of the same dimensio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Equation involving a partial trace Is there, in general, a solution to the following equation?
$\text{Tr}_{V_1}(A(X\otimes I_{V_2})) = B$
where
*
*A is an operator on $V_1\otimes V_2$,
*$B$ is an operator on $V_2$,
*$I_{V_2}$ is the identity on $V_2$.
*$X$ is an unknown operator on $V_1$.
Assuming that $V_1\con... | If $V_1,V_2$ are finite dimensional, we can write $A=\sum_{i=1}^nA_i\otimes B_i$, where $A_1,\ldots,A_n$ are linear independent operators acting on $V_1$ and $B_1,\ldots,B_n$ are linear independent operator acting on $V_2$.
Your equation is equivalent to $\sum_{i=1}^ntr(A_iX)B_i=B$.
You can solve this equation iff $B\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Simplifying the sum of powers of the golden ratio I seem to have forgotten some fundamental algebra. I know that:
$(\frac{1+\sqrt{5}}{2})^{k-2} + (\frac{1+\sqrt{5}}{2})^{k-1} = (\frac{1+\sqrt{5}}{2})^{k}$
But I don't remember how to show it algebraicly
factoring out the biggest term on the LHS gives
$(\frac{1+\sqrt{5}... | $$ \left (\frac{1+\sqrt{5}}{2} \right )^{k-2} + \left (\frac{1+\sqrt{5}}{2} \right )^{k-1} = \left ( \frac{1+ \sqrt{5}}{2}\right )^{k-2} \left ( 1+ \frac{1+ \sqrt{5}}{2}\right)$$
It is known that the Greek letter phi (φ) represents the golden ratio,which value is:
$$\phi=\frac{1+ \sqrt{5}}{2}$$
One of its identities is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show $\prod \limits_{\substack{k=0\\k\neq k_0}}^{K-1}\left(\exp\left[{\frac{2 \pi i (k-k_0)}{K}}\right]-1\right)=(-1)^{K-1} K$ I know it is true that, for any $k_0 \text{ s.t. } 0\leq k_0 < K$,
$$\prod \limits_{\substack{k=0\\k\neq k_0}}^{K-1} \left(\exp\left[{\frac{2 \pi i (k-k_0)}{K}}\right]-1\right)=(-1)^{K-1} K$$
H... | Unless I am missing something, your evaluation is not correct: for example, if $K=2$ and $k_0=1$ then your answer is $1$ but the correct product is $-2$.
We can evaluate the product as follows:
$$\eqalign{P
&=\prod_{k=0}^{k_0-1}\left(\exp\left[{\frac{2 \pi i (k-k_0)}{K}}\right]-1\right)
\prod_{k=k_0+1}^{K-1}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Sum of alternating series Looking to find the sum of the following series:
$$\sum^\infty_{n=1}\frac{(-1)^n}{(2n+1)3^n}$$
It converges due to the Alternating Series Test.
Prior attempts had me isolating the numerator and the $3^n$ in one fraction, and using a geometric series. However, this proved fruitless since it lef... | To find the sum consider the series
$$ \sum_{k=0}^{\infty} x^{2n} = \frac{1}{1-x^2} $$
Integrate both sides from zero to $x$ and then substitute $x=i/\sqrt{3}$ and simplify.
Notes: To evaluate the integral just use the change of variables $ t=\sin(u) $ which gives
$$ I = \int_{0}^{x} \frac{dt}{1-t^2} = \int_{0}^{\ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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"Proof" that $1-1+1-1+\cdots=\frac{1}{2}$ and related conclusion that $\zeta(2)=\frac{\pi^2}{6}.$ Sorry if this has been posted before. Can somebody please tell me whether this result is correct, and give explanation as to why or why not? I'm not good at the formal side of maths.
Start here: $$\sum\limits_{k=0}^{\infty... | The equation:
$$\sum\limits_{k=1}^{\infty}(-1)^{k+1}\frac{1-\cos k\alpha}{k^2}=\frac{\alpha^2}{4}$$
seems to be valid for all $a\in[-\pi,\pi]$, so there has to be something gone right with your "proof" (or at least the idea behind it), even if it seems flawed "as-is."
I suspect that your original series are "true" if y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 3
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Telescoping series of form $\sum (n+1)\cdots(n+k)$ Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdots(n+k)$
e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + \cdots + n(n+1)(n+2)$
How does it do it?
EDIT: We can rewrite as: $\sum {(n+k)! \over n!} = \sum k!{(n+k)!\over n!k!} = \sum k!{{n+k} \choose n}$
(Tha... | Let
$p(n, k)
=n(n+1)...(n+k-1)
=\prod_{j=0}^{k-1} (n+j)
$.
Then
(writing each step in detail)
$\begin{array}\\
p(n+1, k)-p(n, k)
&=\prod_{j=0}^{k-1} (n+1+j)-\prod_{j=0}^{k-1} (n+j)\\
&=\prod_{j=1}^{k} (n+j)-\prod_{j=0}^{k-1} (n+j)\\
&=(n+k)\prod_{j=1}^{k-1} (n+j)-n\prod_{j=1}^{k-1} (n+j)\\
&=((n+k)-n)\prod_{j=1}^{k-1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 3
} |
The line is tangent to a parabola The line $y = 4x-7$ is tangent to a parabola that has a $y$-intercept of $-3$ and the line $x=\frac{1}{2}$ as its axis of symmetry. Find the equation of the parabola. I really need help solving this question. THx
| Let the parabola be defined by the function $y = f(x)$ and let $g(x) = f(x) - 4x + 7.$
Then $y = g(x)$ is a parabola passing through $(0,4)$
and it has slope $-4$ at $x = \frac 12.$
In addition, the parabola described by $y = g(x)$ is tangent to the $x$-axis
so $g(x)$ can be written $g(x) = a(x - p)^2.$
We can also wri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How to show $\int\frac{d}{dx}(a^u)dx=a^u+C$ more rigorously? We all know that $$\int\frac{d}{dx}(a^u)dx=a^u+C$$ where I am differentiating with respect to $x$. But how can I write it in a more rigorous way like for example using the Fundamental Theorem of Calculus or any other method?
How can I deduce that $$\int d(a^u... | In calculus (handling of derivatives and primitives, etc.) it is accepted to present functions $f:\ x\mapsto f(x)$ as function terms $f(x)$. This is in contrast to "abstract analysis" where $f(x)$ just denotes the value of the function $f$ at the point $x\in{\rm dom}(f)$, and is a number.
Without saying it you are act... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/915998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$
Prove that
\begin{equation}
\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}
\end{equation}
I tried to use by parts method and ended with
\begin{equation}
\int \ln^2(\cos x)\,dx=x\ln^2(\cos x)+2\int x\ln(\cos x)... | We are all familiar with the famous Wallis integrals, $W_n=\displaystyle\int_0^\frac\pi2\sin^{n-1}x~dx=\int_0^\frac\pi2\cos^{n-1}x~dx$
$=\dfrac12B\bigg(\dfrac12,~\dfrac n2\bigg)$, see beta function for more details. It follows then that our integral is nothing
other than $W''(1)$, which can be evaluated in terms of t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 2
} |
Find prize per unit that will maximize profit at a given $x$-value Struggling while reviewing my old math books. The problem has a prize-function and wants to know how the prize-per-unit should be chosen to maximize the profit at $\mathbf{x=160}$.
First I look analyze the given equation. The given prize function is:
$$... | You price function gives the total costs to make a number of $x$ products. Call this $C(x)$. Your income is given by the number of sold items times the sale price. If we denote the sale price by $\alpha$, then the income is given by $\alpha x$.
Your profit $P$ can be calculated by subtracting the costs from the income:... | {
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"url": "https://math.stackexchange.com/questions/916156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find a bijection to show $\left|B\right| = \mathfrak{c}$. Let $B = \left\{ A \cup \mathbb{N}_\text{even} : A\subseteq \mathbb{N}_\text{odd} \right\}$
I need to show $\left|B\right| = \mathfrak{c}$ by using an equivalence function (bijection) to another set with the same cardinality.
Any idea?
Thanks.
| HINT: Note that there is a bijection between $B$ and $\mathcal P(\Bbb N_{\rm odd})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/916241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The limit of $(4-\sqrt{16-7\sin(x)})/(8x)$ at zero without using L'Hôpital I stumbled across this silly limit and I am perplexed at how I can arrive to a solution by only relying on the simplest rules of limits.
$$
\lim_{x \to 0}\frac{4-\sqrt{16-7\sin(x)}}{8x}
$$
Any help is appreciated, thanks in advance.
| View it as the derivative of $f(x) = \sqrt{16-7\sin x}$ at $x = 0$ or using the fact that $\sin x \approx x$ near $x = 0$, and expand the top expression binomially.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/916332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Prove $\alpha \in\mathbb R$ is irrational, when $\cos(\alpha \pi) = \frac{1}{3}$ I am trying to prove:
If $\cos(\pi\alpha) = \frac{1}{3}$ then $\alpha \in \mathbb{R} \setminus \mathbb{Q}$
So far, I've tried making it into an exponential, since exponentials are easier to manipulate (at least for me), when compared to ... | Just in case it is not assumed that $\alpha\in\mathbb{R}$, let $\alpha=a+bi$ with $a,b\in\mathbb{R}$. Then
$$\begin{align}
\cos(a\pi+b\pi i)&=\cos(a\pi)\sin(b\pi i)+\sin(a\pi)\cos(b\pi i)\\
\frac13&=i\cos(a\pi)\sinh(b\pi)+\sin(a\pi)\cosh(b\pi)\\
\end{align}$$
Since the left side is real, either $\cos(a\pi)$ or $\sinh(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
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How to expand quadratic equations in Octave/Matlab? I have some column vectors, and I put them in a row, so I have $[a \, b \, c]$. Now I want to get a matrix of the form $[a^2 \, b^2 \, c^2 \, ab \, ac \, bc]$. It is like expanding $(a+b+c)^2$, but applying it to matrix columns. Is there any trick in Octave/Matlab tha... | Polynomial multiplication is equivalent to convolution. This means that if you have two numeric vectors, you can use the conv function in Matlab:
v1 = [2 3 4];
v2 = [3 5 7];
w = conv(v1,v2)
which returns
w =
6 19 41 41 28
The vectors don't even need to be the same length. In your case, you can just ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A relation that is Reflexive & Transitive but neither an equivalence nor partial order relation Set $A = \{0,7,1\}$
1. So for a relation that is reflexive and transitive but neither an equivalence relation nor partial order...Can a relation be both partial order and equivalence?
Attempt: $R_1=\{(0,0),(7,7),(1,1),(0,1)... | $Q1$
Can a relation be both partial order and equivalence?
Yes, for example, the equality relation.
Is $R_1$ Transitive?
No. It has $(1,0)$ and $(0,7)$ but not $(1,7)$. As this example show, if you add an ordered pair to a transitive relation it can become non-transitive.
A relation on set $A$ that is both reflexiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Complex solutions to $ x^3 + 512 = 0 $ An algebra book has the exercise
$$ x^3 + 512 = 0 $$
I can find the real solution easily enough with
$$ x^3 = -512 $$
$$ \sqrt[3]{x^3} = \sqrt[3]{-512} $$
$$ x = -8 $$
The book also gives the complex solutions $$ 4 \pm 4\sqrt{3}i $$
But I don't understand how to find these answe... | If you want to solve the complex roots using quadratic formula, try below :
$$x^3 + 512 = x^3+8^3 = (x+8)(x^2-8x+64) = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/916612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Discrete mathematics subsets Suppose I have two sets A and B:
$$ A = \lbrace 2k-1 : k \in \mathbb{Z}\rbrace$$
$$ B = \lbrace 2l+1 : l \in \mathbb{Z}\rbrace$$
I need to prove that A = B.
I know that to prove equality between two sets I need to prove both:
$$ A \subseteq B $$
and
$$ A \supseteq B $$
I tried to start w... | Hint:
*
*$2k-1=2l+1$ where $l=(k-1)$;
*$2l+1=2k-1$ where $k=l+1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/916688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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If the closure of a set $A$ is defined as the intersection of all closed sets which contain $A$, prove that closure of a closed set $B$ is $B$ itself If the closure of a set $A$ is defined as the intersection of all closed sets which contain $A$, prove that closure of a closed set $B$ is $B$ itself.
Attempt: I apologiz... | Here's a second hint: Suppose you have two sets with different definitions, as you do here. Let's call them $X$ and $Y$. Then to show that $X=Y$, it is enough to show $X\subseteq Y$ and $Y\subseteq X$. To show $X\subseteq Y$ you take an arbitrary point $p$ in $X$ and show that $p$ is also a point of $Y$; to show $Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Intuition underlying stopped martingales Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is also random and depends on $X$. Then what does one get putting two together?
| The Gambler's Ruin example is probably helpful for intuition. Consider a Markov chain $X_n$ whose state space is the integers which jumps 1 unit to the left or to the right with equal probability. Now let $\tau = \inf \{ n : X_n = 0 \}$. Then the stopped process is the same as the original process until the gambler run... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area of a Triangle, If three vertices are given taken in anticlockwise direction. Three vertices are given. We can find the area using the determinant. Can someone explain it to me why the number will be a positive number, if vertices are chosen in anticlockwise direction.
| Using rschwieb's diagram, change the coordinate system so that $(x_1,y_1)$ is at the origin, and so that $x_2>0$ and $y_2 = 0$. (This obviously doesn't change the area). Then the three points are $(0,0)$, $(x_2,0)$, and $(x_3,y_3)$. Since the angle between $\mathbf{a}$ and $\mathbf{b}$ is strictly between $0$ and $\pi$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/916996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Test the convergence of the series $\sum\limits_nn^p(\sqrt{n+1}-2\sqrt n + \sqrt{n-1})$ I am trying to test the convergence of the series
$$\sum_{n=1}^\infty n^p(\sqrt{n+1}-2\sqrt n + \sqrt{n-1})$$
$p$ is a fixed real number.
I tried the ratio test and the integral test without success. Now, I am stuck with the quantit... |
Tool: For every fixed $a$, when $x\to0$, $(1+x)^a=1+ax+\frac12a(a-1)x^2+o(x^2)$. In particular, the case $a=\frac12$ yields $\sqrt{1+x}=1+\frac12x-\frac18x^2+o(x^2)$.
For every fixed $c$, $$\sqrt{n+c}=\sqrt{n}\cdot\sqrt{1+\frac{c}n}=\sqrt{n}\cdot\left(1+\frac{c}{2n}-\frac{c^2}{8n^2}+o\left(\frac1{n^2}\right)\right),$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/917068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Eigenvalue and proper subespace. I have the follow problem:
Suppose that $A,B\in{\cal M}_n(\mathbb{R})$ such that $AB = BA.$
Show that if $v$ is an eigenvector of $A$ associated to the eigenvalue $\lambda$, with $Bv\neq 0$ and dim$(S_\lambda)=1$, then $v$ is also an eigenvector of B.
Thanks in advance!
| $ABv = BAv = B\lambda v$ = $\lambda Bv$
So $Bv$ is an eigenvector of $A$ with eigenvalue $\lambda$
Since $dim(S_\lambda) = 1$ we have that $v \tilde{} Bv$.
In other words: $v$ is an eigenvector of $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/917137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Can't find limit tending to infinity of a sequence I'm stumped by
$$\lim_{x \to \infty}\frac{1+3+5+\cdots+(2x-1)}{x+3} - x$$
My obvious first step was to get a lowest common denominator by $x(\frac{x+3}{x+3})$, giving
$$\lim_{x \to \infty}\frac{1+3+5+\cdots+(2x-1)-x^2-3x}{x+3} $$
But from here I'm stumped, because with... | Assuming $\;x\in\Bbb N\;$ (Otherwise I cannot understand the expression), we get:
$$\frac{1+3+5+\ldots+(2x-1)}{x+3}-x=\frac{x^2}{x+3}-x=-\frac{3x}{x+3}\xrightarrow[x\to\infty]{}-3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/917187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Is a pre-additive structure on a category $\mathcal{C}$ necesarrily unique? If $\mathcal{C}$ is an additive category, i.e. it is $\mathbf{Ab}$-enriched and moreover it admits finite biproducts, it is quite well-known that the additive structure is uniquely determined by internal properties of $\mathcal{C}$. All the pro... | It suffices to give an example of two rings (with unit but not necessarily commutative) that have isomorphic multiplicative monoids but non-isomorphic additive groups: use the identification of rings with one-object preadditive categories.
So consider $\mathbb{Z}$. Its multiplicative monoid is (by the fundamental theor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/917284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Upper approximation of $\mathrm{atanh}(x)$? Is there are nice upper approximation of $\mathrm{atanh(x)}$?
For example, $\ln(x)$ is nicely approximated by $x-1$ for $x$ around $1$.
| $$Atanh(x) = \sum_{k=0}^{\infty} \frac{x^{1+2 k}}{1+2 k}, |x|<1$$
And
$$Atanh(x) = -\frac{\pi \sqrt{-x^2}}{2 x}+\sum_{k=0}^{\infty} \frac {x^{-1-2 k}}{1+2 k}, |x|>1$$
You can approximate it by using $$\sum_{k=0}^n$$ and decide what $n$ gives the level of approximation you need.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/917529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
What type of numbers are roots of $x^{2} = -1$ themselves? Are the roots $i, -i$ themselves irrational numbers or complex numbers or left auxiliary so that undefined?
| Just $i$ and $-i$. The fundamental theorem of algebra tells us that an equation like $x^2 + 1 = 0$ has two roots. In this particular case, $i$ is the principal square root of $-1$, and $-i$ is the other root.
The idea that $i$ could be irrational seems ridiculous to me, but I have to admit that I have never stopped to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/917747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $ Evaluate the integral,
$$ \int_{0}^{1} \ln(x)\ln(1-x)\,dx$$
I solved this problem, by writing power series and then calculating the series and found the answer to be $ 2 -\zeta(2) $, but I don't think that it is best solution to this problem. I want to know if it can be ... | I've found a solution that is interesting, but probably not elegant, and definitely not short.
$I = \displaystyle\int_0^1 \ln(x)\ln(1 - x) dx$
Basic results:
*
*$\lim\limits_{n \to 0} \dfrac{x^n - 1}{n} = \log x$, or $\lim\limits_{n \to 1}\dfrac{x^{n-1} - 1}{n - 1} = \log x$.
*$\dfrac{d}{dn}\beta(n, n) = 2\beta(n,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/917833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 9,
"answer_id": 2
} |
Prove quotient of two $N(0,1)$ is $\text{Cauchy}(0,1)$ Problem: Show that if $X$ and $Y$ are independent $N(0,1)$-distributed random variables, then $X/Y ∈ C(0,1)$.
Question: I don't know how to proceed below. I want to prove that the PDF of $X/Y$ is Cauchy. PS. I looked on wiki and they defined $C$ as $X/Y$
Attempt:
... | An alternative proof taken from here - I find it helpful as most steps are explained intuitively. Please correct me if something is incorrect.
Proof
Start with the CDF which is $P(\frac{X}{Y} \leq t)$ which is an event, and we want to find the probability of this event.
$$ P(\frac{X}{Y} \leq t) = P(\frac{X}{|Y|} \leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/917916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Inverting a vector If I have $Ax=b$ where $A$ is $n$ by $n$ while $x$ and $b$ are $n$ by 1, is it possible to find $A$ given $x$ and $b$. The idea would be some sort of $x^{-1}$ operation on the right of both equations but I'm not sure how to go about it. If necessary, we can consider a specific case of $n=2$.
Also, is... | No, inverse in this case is impossible, because you have a lot of solutions, for example consider:
$A\begin{bmatrix} 1 \\ 0\end{bmatrix}=\begin{bmatrix} 1 \\ 0\end{bmatrix}$
$A=\begin{bmatrix}1 && 0 \\ 0 && 1\end{bmatrix}$is one solution, but for example $A=\begin{bmatrix}1 && 1 \\ 0 && 1\end{bmatrix}$ is solution, too... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/918106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Conditional Probability matching socks A drawer contains eight different pairs of socks. If six socks are taken at random without replacement, compute the probability that there is at least one matching pair among these six socks.
Hint: compute the probability that there is not a matching pair first.
Sorry guys, I'm r... | Ok my suggestion is the following.
You surely know how to use the hint: If you denote the probability of not having a matching term by $P$ then the probability of having at least one matching pair is ... (I'll leave it to you to think about this one)
Now you might want to use the following formula.
Let $\Omega$ be a (f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/918208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Exercise on Cyclic Group Let $G$ be a finite cyclic group of order $n$. Let $a$ be a generator. Let $r$ be an integer $\neq 0$, and relatively prime to n.
Show that $a^r$ is also a generator of $G$.
Proof: $G=\langle a\rangle$ such that $a^n=1$
$|a^r|=\frac{|a|}{\gcd(r,n)}=|a|$
Since $|a^r|=|a|$m then $G=\langle a^r\ra... | Another way:
$$g.c.d(r,n)=1\implies \;\exists\,x,y,\in\Bbb Z\;\;s.t.\;\; xr+yn=1$$
and from here
$$a=a^1=a^{rx+yn}=\left(a^r\right)^x\left(a^n\right)^y=\left(a^r\right)^x\implies \langle a\rangle \subset \langle a^r\rangle$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/918300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Mathmatical Induction Prove, by mathmatical induction, that $5^{2n}$ + $2^{2n-2}$$3^{n-1}$ is divisible by $13$.
I first plugged in n as 1 and showed that the expression is divisible by 13 for n=1.
Then I assumed that the expression was divisible by 13 for n=k and plugged in k. The simplified expression in terms of k ... | $$25^{k+1}+\frac{4^{k+1}3^{k+1}}{12}=25\cdot 25^{k}+12 \frac{4^{k}3^{k}}{12} $$
$$=13\cdot 25^k+12\cdot25^k+12 \frac{4^{k}3^{k}}{12} $$
$$=13\cdot 25^k + 12 \left(25^k+ \frac{4^{k}3^{k}}{12}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/918392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Sum of two trig function's identity We all know that
$\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$
But is there an identity for
$\sin(x) + z\sin(y) = ?$
Or do I need to figure it out using Euler's formula
$\sin(x) = (e^{ix} - e^{-ix})/2$ and put it back into trigonometric form?
| According to
Alan Jeffrey, Hui-Hui Dai: Handbook of Mathematical Formulas and
Integrals. 4th Edition, Academic Press, 2008.
at page 131, 2.4.1.9 Sum of Multiples of $\sin x$ and $\cos x$
There is a formula for that
*
*$A \sin x + B \cos x = R \sin(x + \theta)$, where $R = (A^2+B^2)^{1/2}$ with $\theta = \arctan... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/918474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Proving the Cauchy-Schwarz inequality by induction I ran across this problem in some old notes, and I frustratingly can't figure out how to do it
Let $a_i$ and $b_i$ be sequences of natural numbers, use induction to show
$\sum_{i=1}^n (a_ib_i)^{1/2} \le (\sum_{i=1}^n a_i)^{1/2}(\sum_{i=1}^n b_i)^{1/2} $
Obviously this... | Hint: Another way would be to rewrite using the substitutions $x_i^2=a_ib_i$ and $y_i^2=a_i^2$ in the form
$$CS(n): \quad \frac {\left( \sum_{i=1}^n x_i \right)^2}{\sum_{i=1}^n y_i} \le \sum_{i=1}^n \frac{x_i^2}{y_i}$$
This form is particularly amenable to induction, after you prove the base case of $n=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/918580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Proving that $\exp(-x^2)$ has a unique fixed point on the interval $[0,1]$
Consider the function $g(x)=e^{-x^2}$. Prove that g has a unique fixed point on the interval [0,1].
So, our teacher did not go over this section, but assigned it for homework and I have no idea where to even start with a proof. Could someone ... | Hint: Show that the function is continuous, and is decreasing in the aforementioned interval (by considering the derivative). Then conclude that fixed point has to be unique under these conditions.
Alternatively, look at $f(x)=g(x)-x$. Show that $f(0)$ and $f(1)$ have different signs- and so by intermediate value theor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/918673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the flaw in this proof that all triangles are isosceles? What is the flaw in this "proof" that all triangles are isosceles?
From the linked page:
One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least would be vulnerable if we didn't "cheat" by allowing ourse... | If the drawing is made correctly, $P$ needs to be outside, but not too outside, the triangle. If it is far outside, $E$ is on the extension of $AB$ and $F$ is on the extension of $AC$, two plus signs in the argument change to minus signs, and the argument goes through. If $P$ is not too far outside, $E$ is on the ext... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/918739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 0
} |
Find the number of spanning trees of a dumbbell graph. A (k, l)-dumbbell graph is obtained by taking a complete graph on k
(labeled) nodes and a complete graph on l (labeled) nodes, and connecting them
by a single edge. Find the number of spanning trees of a dumbbell graph.
Do I have use Kruskal's algorithm for this pr... | Hints
*
*There is one edge of your graph which must be included in any spanning tree.
*The number of spanning trees in a complete graph is known, see https://oeis.org/A000272
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/918853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
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