instruction stringlengths 12 30k |
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Solving the integral $\int\limits_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$? |
I want to point out that there are, topicwise, different paths from the most suggested standard one:
As noted by "jmoy", time is limited - the comments telling you to learn some physics or programming first or on the way seem to ignore this somewhat. Sure it is good and will pay off to have these things as backgrou... |
Is there an explanation for patterns formed by simplicial number sequences? |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
How to prove the partial denominators formula of a generalized continued fraction and find the formulae for the elements of another ? |
Is there an explanation for the patterns formed by the binomial coefficients C(n+d-1,d) mod 512? |
In my [Advanced Calculus book (Angus Taylor)][1] it is shown that, if $a\gt 0$,
$\displaystyle\int_0^{\infty}\dfrac{e^{-at}\sin xt}{t}dt=\arctan\dfrac{x}{a}\qquad (1)$
If $x>0$,
$\displaystyle\int_0^{\infty}\dfrac{\sin xt}{t}dt=\dfrac{\pi}{2}\qquad (2)$
follows from $(1)$, observing that the integrand is $... |
In my [Advanced Calculus book (Angus Taylor)][1] it is shown that, if $a\gt 0$,
$\displaystyle\int_0^{\infty}\dfrac{e^{-at}\sin xt}{t}dt=\arctan\dfrac{x}{a}\qquad (1)$
If $x>0$,
$\displaystyle\int_0^{\infty}\dfrac{\sin xt}{t}dt=\dfrac{\pi}{2}\qquad (2)$
follows from $(1)$, observing that the integrand is $... |
My question was inspired by [this][Harmonic] stackexchange question. For the last 90 minutes I have been trying to prove this formula due to [Gregorio Fontana][GF]:
$$H_n = \gamma + \log n + {1 \over 2n} - \sum_{k=2}^\infty { (k-1)! C_k \over n(n+1)\ldots(n+k-1)},
\qquad \textrm{ for } n=1,2,3,\ldots,$$
where $H... |
I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.
How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?
I've tried to get the plane rotation from the plane's normal:
rotationX = ata... |
I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.
How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?
I've tried to get the plane rotation from the plane's normal:
rotationX = ata... |
I'm trying to approximate $\pi$ using Monte Carlo integration; I am approximating the integral
$$\int\limits_0^1\\!\frac{4}{1+x^2}\;\mathrm{d}x=\pi$$
This is working fine, and so is estimating the error (variance), $\sigma$. However, when I then try to use importance sampling with a Cauchy(0,1) distribution, thin... |
I've got a programming exercise I need to do, but I just can't figure out the math part.
I need to check if 3 of 6 lines intersect in the same point. I am given the equation ax+by=c, and I input random numbers in a,b,c. How would I go about solving this mathematically?
Thanks! |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
Do you mean rationalise
$$
\frac{11}{3\sqrt{3}-7} \qquad \text{?}
$$
And are you sure you're trying
$$
\frac{11}{3\sqrt{3}-7} \cdot \frac{3\sqrt{3} + 7}{3\sqrt{3} + 7}\qquad \text{?}
$$
In general, [Wikipedia][1] is almost always of great help too.
[1]: http://en.wikipedia.org/wiki/Rationalisatio... |
While working on a proof showing that all functions limited to the domain of real numbers can be expressed as a sum of their odd and even components, I stumbled into a troublesome roadblock; namely, I had no clue how one divides the function into these even and odd parts.
Looking up a solution for the proof, I found... |
How do I divide a function into even and odd sections? |
How do I mathmatically determine if a number is prime? I'm writing a program that the user inputs any integer and from that the program determines whether the number is prime, but how do i go about that? |
Why is $\mathbb{Q}(t,\sqrt{t^3-t})$ not a purely transcendental extension of $\mathbb{Q}$? |
Have the program find the remainder when dividing the input (say n) by 2, 3, 4, ..., $\sqrt n$ (or the following integer if $\sqrt n$ is not an integer.) If this value ever leaves a remainder of zero then your number is composite and you can stop checking divisors. If the remainder is never zero for all of these value... |
Have the program find the remainder when dividing the input (say n) by 2, 3, 4, ..., $\sqrt n$ (or the following integer if $\sqrt n$ is not an integer.) If this value ever leaves a remainder of zero then your number is composite and you can stop checking divisors. If the remainder is non-zero for all of these values ... |
Algorithm posted by [jericson][1] is the best for basic purposes.
IMHO, for *programming competitions* and for *practical purposes* **non-deterministic algorithms** are best. Rabin-Miller is my favorite.
Take a look at [**Rabin-Miller primality testing algorithm code @ TopCoder**][2].
[1]: http://math.st... |
Algorithm posted by [jericson][1] is the best for basic purposes.
IMHO, for *programming competitions* and for *practical purposes* **non-deterministic algorithms** are best. Rabin-Miller is my favorite.
Take a look at [**Rabin-Miller primality testing algorithm code @ TopCoder**][2].
As primes are in P, the... |
For convenience change the base field to $k=\mathbb{C}$
(if the extension were purely transcendental before
then it would be purely transcendental after).
As the extension has transcendance degree 1, then
if the extension were purely transcendental it would equal $k(x)$
for some $x$. Then there are nonconstant rat... |
I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in Calculus?
If you had any good methods of helping people that would be very helpful. |
What concepts were most difficult for you to understand in Calculus? |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
How to find a suitable contour to integrate round? Please help me not end up in physiatric care! |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
In [Direct limit][1], Martin rightly pointed out that my naïf construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem of... |
In [Direct limit][1], Martin rightly pointed out that my naïf construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem of... |
The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if they are elementary but their proofs are very much sophisticated.
I would like to consider two famous questions: Fir... |
Algorithm posted by [jericson][1] is the best for basic purposes.
IMHO, for *programming competitions* and for *practical purposes* **randomized algorithms** are best. Rabin-Miller is my favorite.
Take a look at [**Rabin-Miller primality testing algorithm code @ TopCoder**][2].
As primes are in P, there is d... |
How to find a suitable contour to integrate round? Please help me not end up in psychiatric care! |
Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group structure of G) is equal, in $\pi_1(G,e)$, to the loop $s_1*s_2$ where this product is given by first going around $s_1$ a... |
Topological group: Multiplying two loops is homotopic to linking these paths? |
In [Direct limit][1], Martin rightly pointed out that my naïf construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem of... |
How do I mathematically determine if a number is prime? I'm writing a program that the user inputs any integer and from that the program determines whether the number is prime, but how do I go about that? |
Consider the following problem:
You are given a multiset (a set with repetitions allowed) of $2n+1$ real numbers, say $S = \\{r_1, \dots, r_{2n+1}\\}$.
These numbers are such that for every $k$, the multiset $S - \\{r_k\\}$ can be split into two multisets of size $n$ each, such that the sum of numbers in one mul... |
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and $d'$, in increasing order of strength:
1. $i$ is a homeomorphism, i.e. $d$ and $d'$ induce the same topology on $X$.
... |
*How do I mathematically determine if a number is prime?*
If the number is n, then dividing it by every prime number less than sqrt(n) and showing that there is a remainder.
There are a number of different [sieve][1] solutions for finding prime numbers, the oldest and most famous of which is the [Sieve of Eratost... |
You can go directly to QUESTION 1.
**Motivation**: Assume one has a continued fraction (c.f.) $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $, where $a_n,b_n$ are integers. **I am looking for a c.f. transformation that can transform it into a new c.f. in such a way that its elements ... |
For convenience change the base field to $k=\mathbb{C}$
(if the extension were purely transcendental before
then it would be purely transcendental after).
As the extension has transcendance degree 1, then
if the extension were purely transcendental it would equal $k(x)$
for some $x$. Then there are nonconstant rat... |
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and $d'$, in increasing order of strength:
1. $i$ is a homeomorphism, i.e. $d$ and $d'$ induce the same topology on $X$.
... |
Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and $d'$, in increasing order of strength:
1. $i$ is a homeomorphism, i.e. $d$ and $d'$ induce the same topology on $X$.
... |
I have a series of terms $\\{t_n : t_n = a_n x_n\\}$, and I want to talk about the product of each term except $t_j$.
Would any of these be an appropriate way to say that? I like this:
$$\prod_{i \ne j} t_i$$
or maybe this
$$\prod_{i=1,i \ne j}^n t_i$$
Is it a normal thing to do, to put qualifiers on an... |
What is the correct way to write product or sum in capital pi or capital sigma notation when you wish to exclude an index? |
I have to prove that if P is a R-module , P is projective <-> There is a family {x_i} in P and morphisms f_i:P->R such that for all x in P
x= \sum_i\inI f_i(x)x_i where f_i is 0 for almost all i\in I
Any help? |
I have to prove that if $P$ is a $R$-module , $P$ is projective $\Leftrightarrow$ there is a family $\\{x_i\\}$ in $P$ and morphisms $f_i\colon P\to R$ such that for all $x\in P$
$$ x= \sum_{i\in I} f_i(x)x_i$$
where for each $x\in P$, $f_i(x)=0$ for almost all $i\in I$.
Any help? |
I am reading the book Stochastic Simulation, Algorithms and Analysis by Asmussen. On page 39, when talking about inversion method for sampling a distribution, he gave an example:
> an r.v. distributed as the overshoot X − a given X > a can be generated as $F^{-1} (U (1-F (a)) − a$,
where $U$ is a r.v. uniform in ... |
A Blaschke product is a function of the form
$$B(z):=z^k\prod_{n=1}^{\infty}\frac{a_n-z}{1-\overline{a_n}z}\frac{|a_n|}{a_n}$$
where the $a_n$ are the non-zero zeros of $B$, and satisfie $\sum_{n=1}^{\infty}(1-|a_n|) < \infty$.
Blashke products are holomorphic and bounded by 1 on the unit disk. A well known theore... |
Proving the partial denominators formula of a generalized c. f. and finding the formulae for the elements of another ? (bounty till tomorrow) |
The modular group is the group $G$ consisting of all linear fractional transformations $\phi$ of the form
$$\phi(z)=\frac{az+b}{cz+d}$$
where $a,b,c,d$ are integers and $ad-bc=1$. I have read that $G$ is generated by the transformations $\tau(z)=z+1$ and $\sigma(z)=-1/z$. Is there an easy way to prove this? In partic... |
So, I [asked a question][1] about how to find if 3 lines are concurrent. I built the algorithm I needed,and it was working well, until I started doubting my power of judgement. So my question:
**Are 2 equal lines also intersecting?**
For an example, 9x+9y=9;7x+7y+7 ? Because now, if I would draw these lines and ... |
I am reading the book Stochastic Simulation, Algorithms and Analysis by Asmussen. On page 39, when talking about inversion method for sampling a distribution, he gave an example:
> an r.v. distributed as the overshoot X − a given X > a can be generated as $F^{-1} (U (1-F (a)) − a$,
where $U$ is a r.v. uniform in ... |
If the probability of an event is `1 / ∞` and `∞` trials are conducted, how many times will the event occur — `0`, `1`, or `∞`? |
Hint:
If $\psi(x)$ is strictly positive, what does it say about the trend of $\psi'(x)$?
If $\psi''(x)$ is strictly positive, what does it say about the trend of $\psi'(x)$?
It might be useful to draw a graph. |
In Stochastic Simulation: Algorithms and Analysis by Søren Asmussen, on Page 38
> A Poisson r.v. $N$ with rate $\lambda$ ($P(X = n) = e^{−\lambda} \frac{\lambda^n}{n!}$) can be constructed using the relation between the exponential distribution and the Poisson process: generate $X_1 , X_2 , ...$ as i.i.d. exponenti... |
In Stochastic Simulation: Algorithms and Analysis by Søren Asmussen, on Page 38
> A Poisson r.v. $N$ with rate $\lambda$ ($P(X = n) = e^{−\lambda} \frac{\lambda^n}{n!}$) can be constructed using the relation between the exponential distribution and the Poisson process: generate $X_1 , X_2 , ...$ as i.i.d. exponenti... |
Are the different proofs of the structure theorem for finitely generated modules over a PID essentially different? If so, a sketch of the main ideas, a highlight of the differences (perhaps clearity and pedagogy vs. usefulness and applications), or hints to bibliography would be highly appreciated!
Thanks. |
Sorry byoogle, I misinterpreted the point of your question. So now I feel compelled to try to answer. I don't know much about probability, but by chance I stumbled across something today that might be interesting to you.
I'll try to frame a question that I think is close to what you are asking, and is also answered ... |
Sorry byoogle, I misinterpreted the point of your question. So now I feel compelled to try to answer. I don't know much about probability, but by chance I stumbled across something today that might be interesting to you.
I'll try to frame a question that I think is close to what you are asking, and is also answered ... |
**HINT** $\:$ Both $\:\rm e^{ix}\:$ and $\:\rm cos(x) + i \; sin(x) \:$ are solutions of $\;\rm y' = i \; y,\;\; y(0) = 1, \;$ so they're equal by the **uniqueness theorem**. Alternatively, **bisect** into even and odd parts the power series for $\:\rm e^{ix} \:,\;$ i.e.
$$\begin{align}
\rm f(x) \ \ &=\ \ \rm\fra... |
In [Direct limit][1], Martin rightly pointed out that my naïf construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem of... |
I know that the product rule is generalised by Leibniz's general rule and the chain rule by Faà di Bruno's formula, but what about the quotient rule? Is there a generalisation for it analogous to these? |
What's the generalisation of the quotient rule for higher derivatives? |
I know that the product rule is generalised by Leibniz's general rule and the chain rule by Faà di Bruno's formula, but what about the quotient rule? Is there a generalisation for it analogous to these? Wikipedia mentions both Leibniz's general rule and Faà di Bruno's formula for the product and the chain rule, but ra... |
In [Direct limit][1], Martin rightly pointed out that my naïf construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem of... |
In [Direct limit][1], Martin rightly pointed out that my naive construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem o... |
In [Direct limit][1], Martin rightly pointed out that my naive construction (now deleted) of the colimit (direct limit) of topological abelian groups was wrong. He shows how to do it properly (at least the coproduct) [here][2].
Since then, I've been lurking some of the literature about the subject and this problem o... |
Note that by inverse sampling we can generate exponential r.v. with rate $\lambda$ by
$X=F^{-1}(U)$ where $F(x) = 1-e^{-\lambda x}$, is the c.d.f of exponential r.v. with rate $\lambda$ and $U\sim Uni[0,1]$. Thus $F^{-1}(U)=\frac{-\log(1-U)}{\lambda}$.
But $1-U\sim Uni[0,1]$ so if $U_i\sim Uni[0,1]$ then
$... |
Do 2 equal lines also count as intersecting? |
I am having trouble answering the following question:
<br>
**Problem**
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
<b... |
I am having trouble answering the following question:
<br>
**Problem**
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
<b... |
I am having trouble answering the following question:
<br>
**Problem**
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
<b... |
I am having trouble answering the following question:
<br>
**Problem**
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
<b... |
I am having trouble answering the following question:
<br>
**Problem**
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
<b... |
I am having trouble answering the following question:
<br>
**Problem**
By integrating the function:
$$f(z) = \frac{R + z}{z(R - z)}$$
round a suitable contour, prove that, for $0 \leq r < R$,
$$\frac{1}{2\pi}\int_0^{2\pi} \frac{R^2-r^2}{R^2 - 2Rr\cos(\theta) + r^2}\mathrm{d}\theta = 1 $$
<b... |
There is something in the definition of the [free product][1] of two groups that annoys me, and it's this "word" thing:
>If G and H are groups, a word in G and H is a product of the form
>
>$$
> s_1 s_2 \dots s_m,
>$$
>
>where each $s_i$ is either an element of G or an element of H.
So what is t... |
Set-theoretical description of the free product? |
Very simple question, but what is the proof that x.y mod m == ((x mod m).y) mod m? |
Proving the partial denominators formula of a generalized c. f. and finding the formulae for the elements of another ? (bounty expires today) |
How many number of three digit numbers lying between 100 and 999 (inclusive) and having only two consecutive digits identical ?
|
I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...):
Construct a Turing machine A which sequentially runs on all proofs in ZFC and checks if the claim "A does not halt or ZFC is inconsistent" is proved. If such a proof is found, A halts.
Note that A ... |
I understand that there are different types of infinity: one can (even intuitively) understand that the infinity of the reals is different from the infinity of the natural numbers. Or that the infinity of the even numbers is the same as that of the natural numbers. How many types of infinity are there? Or is this infin... |
I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...):
Construct a Turing machine A which sequentially runs on all proofs in ZFC and checks if the claim "A does not halt or ZFC is inconsistent" is proved. If such a proof is found, A halts.
Note that A ... |
To be honest, I have not read the whole question, but it seems that the description is far too complicated as it actually is. I will elaborate on the construction of coproducts of groups or monoids a bit. This will go beyond scope of the specific question, but I hope it's helpful in other situations.
First assume $... |
In the question above, it seems that the description is far too complicated as it actually is. I will elaborate on the construction of coproducts of groups or monoids a bit. This will go beyond the scope of the specific question, but I hope it's helpful in other situations.
First assume $M,N$ are sets. What is thei... |
There is something in the definition of the [free product][1] of two groups that annoys me, and it's this "word" thing:
>If G and H are groups, a word in G and H is a product of the form
>
>$$
> s_1 s_2 \dots s_m,
>$$
>
>where each $s_i$ is either an element of G or an element of H.
So what is t... |
Consider the following problem:
![alt text][1]
[1]: https://i.stack.imgur.com/XsLEN.png
I got $g'(1) = f'(1) = 2$, which is correct according to the expected answer. However, I get that $a$ can be any value, as it doesn't affect anything and $g$ is differentiable no matter what $a$ is (am I missing somethi... |
A very nice introduction to the many different notions of infinity in mathematics is Rudy Rucker's book: [Infinity and the Mind][RR]. Unlike many other popularizations, this is written by someone who did a Ph.D. on the topic. Moreover, Rucker has gone to great lengths to make the presentation faithful to the mathematic... |
*Notation*: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the [Gauss Notation for generalized continued fractions][1].
----------
**Description of the [Bauer-Muir transformation][2]**
(Based on [pp. 76-77][3] of Lisa Lorentzen and Haakon Waadeland's [book][4] (A)... |
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized c. f. ? (bounty expires today) |
If we have a covering $p:H\rightarrow G$ where $G$ is a topological group, then $H$ is also a topological group. The multiplication function can be defined as follows. Consider the map $f:H\times H \rightarrow G$ which is a composition of the map $p\times p$ and the multiplication function on $G$. Choose $h\in p^{-1}(e... |
If we have a covering $p:H\rightarrow G$ where $G$ is a topological group, then $H$ is also a topological group. The multiplication function can be defined as follows. Consider the map $f:H\times H \rightarrow G$ which is a composition of the map $p\times p$ and the multiplication function on $G$. Choose $h\in p^{-1}(e... |
There is something in the definition of the [free product][1] of two groups that annoys me, and it's this "word" thing:
>If G and H are groups, a word in G and H is a product of the form
>
>$$
> s_1 s_2 \dots s_m,
>$$
>
>where each $s_i$ is either an element of G or an element of H.
So what is t... |
While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment.
What is a good way to think of those two terms? |
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