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What are the formal series $f$ with non-negative integer coefficients and constant term equal to $1$ whose multiplicative inverse $1/f$ has all coefficients, apart from a finite subset, all non-positive?
In fact, assume the series converges if you want... |
I have a graph $G$ with a list of edges $E$, all the edges can form cycles, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Ux3b0FtuIMpVGX4e-AbblH05h2gIZII9u8pACRzymPE&w=960&h=720">
All the coordinates for the vertex $V$ **are known**.
Wat is the algorithm to find all the faces $F$? ... |
Is it true that $\frac1{n!} \int_0^\infty x^{n-k} (x-1)^k e^{-x}\,dx \approx e^{-k/n}$ when $k$ and $n$ are large integers with $k \le n$?
This quantity is the probability that a random permutation of $n$ elements does not fix any of the first $k$ elements.
|
Dave R's answer highlights a more general principle: by the <a href="http://en.wikipedia.org/wiki/Yoneda_lemma">Yoneda lemma</a>, an object $X$ in a category is determined up to isomorphism by the behavior of the functor $F_X = \text{Hom}(X, -)$, so to show that two objects $X, Y$ are _not_ isomorphic it suffices to sh... |
This is a result due to Floris Takens:
> Let $\Phi \colon M_1 \to M_2$ be a bijection between two smooth $n$-manifolds such that $\lambda \colon M_2 \to M_2$ is a diffeomorphism iff $\Phi^{-1} \circ \lambda \circ \Phi$ is a diffeomorphism. Then $\Phi$ is a diffeomorphism.
It can be found in _Characterization of ... |
And how to express $\{ x \} = x - \lfloor x \rfloor$ as function of $sin(x)$ and $sign(x)$? |
How to express the whole part $\lfloor x \rfloor$ as analytical function or Taylor/Fourier series? |
Can the number of solutions $xy(x-y-1)=n$ $x,y,n \in Z$ be arbitrary large for fixed $n$?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed n? |
I need to solve following system of nonlinear equations.
Here are some characteristics of this system:
It consists of n equation and n variables.
Every equation is in similar form -> sum of products = constant.
The lenght of every product is the same (it will be denoted as k).
The number of elements in s... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed $n$?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed $n \ne 0$?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ ... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed $n \ne 0$? |
If vectors are thought of as triplets of numbers (a,b,c) then why should these triplets be added by the parallelogram law? It all depends what these triplets are being used to represent.
If you start with addition of positive numbers on the number line, say 5+3, this can be seen as laying down two line segments of l... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed $n \ne 0$?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ ... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed $n \ne 0$?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ ... |
I am unable to understand the motivation behind the set theoretic definition of a natural number. The definition given in the book by Goldrei is as follows:
First he defines an inductive set:
A set $y$ is *inductive* if $\emptyset \in y$ and $x^{+} \in y$ whenever $x \in y$, where $x^{+} = x \cup \{x\}$.
So fa... |
Can I draw a net for a Steinmetz solid from two cylinders with a compass?
That is, can we flatten the net? I often make a model using paper and a compass-- it looks about right... is it really a valid method of construction? |
Can I draw a net for a Steinmetz solid with a compass? |
I'm trying to make one body fly around another, using Coulomb law $F=\frac{q_1 q_2}{r^2}$ and second Newton law $ma=F$. Now I'm doing it this intuitive way:
1. Move body1 according to current speed and $dt$;
2. Calc new Coulomb force then find acceleration $a=\frac{F}{m}$;
3. Add acceleration to current speed;
4.... |
I'm trying to make one body fly around another, using Coulomb law $F=\frac{q_1 q_2}{r^2}$ and second Newton law $ma=F$. Now I'm doing it this intuitive way:
1. Move body1 according to current speed and $dt$;
2. Calc new Coulomb force then find acceleration $a=\frac{F}{m}$;
3. Add acceleration to current speed;
4.... |
I'm trying to make one body fly around another, using Coulomb law $F=\frac{q_1 q_2}{r^2}$ and second Newton law $ma=F$. Now I'm doing it this intuitive way:
1. Move body1 according to current speed and $dt$;
2. Calc new Coulomb force then find acceleration $a=\frac{F}{m}$;
3. Add acceleration to current speed;
4.... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be arbitrary large for fixed $n \ne 0$?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ ... |
Let $0 \leq \alpha < \beta \leq 1$. I'm looking for an example of a Lebesgue measurable subset $E$ of $\mathbb{R}$ such that
$$\liminf_{\delta \rightarrow 0} \frac{m(E \cap (-\delta,\delta))}{2\delta} = \alpha$$
but
$$\limsup_{\delta \rightarrow 0} \frac{m(E \cap (-\delta,\delta))}{2\delta} = \beta$$
where ... |
Consider an ellipsoid of semi-axes a, b, c (possibly prolate, b=c). I am interested in the "shadow" of this solid onto a distant plane, in a given direction d=(k,l,m) orthogonal to that plane. By shadow I mean the projected area onto the plane: each point on the surface of the ellipsoid is translated in the same direct... |
I'm trying to make one body fly around another, using Coulomb law $F=\frac{q_1 q_2}{r^2}$ and second Newton law $ma=F$. Now I'm doing it this intuitive way:
1. Move body1 according to current speed and $dt$;
2. Calc new Coulomb force then find acceleration $a=\frac{F}{m}$;
3. Add acceleration to current speed;
4.... |
I was asked to prove:
no group can have a minimal normal subgroup isomorphic to a $\mathrm{Syl}_2(A_7)$.
I think I should find some property that $\mathrm{Syl}_2(A_7)$ has but not a minimal normal subgroup. So then I thought that it a $\mathrm{Syl}_2(A_7)$ always has characteristic subgroup, i.e. exist $G\ char\ \... |
What is Jacobian matrix?
What are its applications?
What is its physical and geometrical meaning?
Can someone explain with examples?
Thanks! :) |
What is Jacobian Matrix? |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
The most useful mathematical areas for a programmer are
- trigonometry and linear algebra, especially to draw and recognise shapes, ray tracing, animations and so on,
- combinatorics, particularly to cleverly enumerate sets, estimate running times of simple loops and recursions etc.,
- some elementary group the... |
Is there a finite non-Hermite commutative ring with unity? |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
> Is there a commutative integral domain R in which:
>
> * every nonzero prime ideal Q is maximal, and
> * there are maximal ideals Q with R/Q of sizes 3, 11, and 27?
This doesn't happen with number rings of Galois extensions of <b>Q</b>, as far as I know, since what happens to 3 seriously happens to 3. You cann... |
> Is there a commutative integral domain R in which:
>
> * every nonzero prime ideal Q is maximal, and
> * for every prime power q congruent to 3 mod 8, there is a maximal ideal Q of R with R/Q a field of size q?
I am looking for a ring where "q = 3 mod 8" describes finite fields, rather than just finite local ri... |
Is there an integral domain with a lot of residue fields of the same characteristic? |
I am looking for a symbol to represent the operation of taking unique values from a vector. So, say the symbol was $\theta$:
$v = [0, 0, 0, 1, 1, 1, 3, 1, 2, 0]$
$\theta(v) = [0, 1, 3, 2]$
Or is this something that just isn't defined?
Thanks
|
What is the mathematical symbol for the unique values in a vector? |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies? |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
I am looking at the following problem:
> Find all values of $m$ for which the equation $(3m-2)x^2 + 2mx + 3m = 0$ shas only one root between $-1$ and $0$.
I don't know what only one root between -1 and 0 actually means. I count these possibilities:
1. $-1<x_1=x_2<0$
2. $x_1<-1<x_2<0$
3. $-1<x_1<0<x_2... |
Here is an example. Suppose you have two implicit differentiable functions
$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$
and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that
$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$
If you differentiate $F$ and $G$, you ... |
Here is an example. Suppose you have two implicit differentiable functions
$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$
and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that
$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$
If you differentiate $F$ and $G$, you ... |
Here is an example. Suppose you have two implicit differentiable functions
$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$
and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that
$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$
If you differentiate $F$ and $G$, you ... |
What are the formal series $f$ with non-negative integer coefficients and constant term equal to $1$ whose multiplicative inverse $1/f$ has all coefficients, apart from a finite subset, all non-positive?
In fact, assume the series converges in some disk if you want... |
Suppose $A$ is the set of all rational numbers $p$ such that $p^2 <2$ and $B$ is the set of all rational numbers $p$ such that $p^2 > 2$. We want to show that $A$ contains no largest element and $B$ contains no smallest element.
In Rudin's *Principles of Mathematical Analysis*, he associates $q = p- \frac{p^2-2}{p+2... |
I know this seems like a juvenile question, but for some reason I can't recall how to do this.
Lets say I want to find 87 reduced by 99.75%. What is the equation to do that?
Thanks! |
What is the equation for finding a number that is X% less than another number? |
Here is a non-rigorous attempt at an answer, but perhaps someone can make this into a formal argument. I will assume your series is centered at $0$. I will also denote your function by $g$ for reasons you will see as you read on.
Suppose $g(x) = \sum_{n=0}^{\infty} a_n x^n$, where $a_n \geq0$ for every $n$.
We ... |
I am looking for a symbol to represent the operation of taking unique values from a vector. So, say the symbol was $\theta$:
$v = [0, 0, 0, 1, 1, 1, 3, 1, 2, 0]$
$\theta(v) = [0, 1, 3, 2]$
Or is this something that just isn't defined?
Thanks
**EDIT**
Thanks all for your help here. Your answers and com... |
Can you provide help with interpreting this periodogram? |
Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x \in A$. Prove that $\inf A = -\sup(-A)$.
We know that $-A$ is bounded above. Hence $\sup(-A)$ exists. So to proceed, one would show that $\inf A \leq -\sup(-A)$ and $\inf A \geq -\sup(-A)$? Or $-\in... |
Here is an example. Suppose you have two implicit differentiable functions
$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$
and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that
$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$
If you differentiate $F$ and $G$, you ... |
Revised answer:
(i)$x < p \Rightarrow f(x) < f(p)$ (because $f$ is strictly monotonically increasing).
Because this set of values is bounded from above there exists a supremum, namely $f(p-)$.
Likewise, $x > p \Rightarrow f(x) > f(p)$ (because $f$ is strictly monotonically increasing).
Because this set of va... |
*Does someone know if the following is true:*
Let $\mathbb{X}$ be some arbitrary *banach space*.
$\{x_k \} \in \mathbb{X}$ (where *$k$ is from $1$ to $\infty$*) - is some sequence chosen from $\mathbb{X}$.
----------
Now, if the series $\sum_{k=1}^\infty \|x_k\|_X$ converges, would the *"more generic"* ... |
Does someone know if the following is true:
Let $\mathbb{X}$ be some arbitrary Banach space.
$\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$.
Now, if the series
$$\sum_{k=1}^\infty \|x_k\|_X$$
converges, would the "more generic" series
$$\sum_{k=1}^\infty x_k$$
also conver... |
Here is an example. Suppose you have two implicit differentiable functions
$$F(x,y,z,u,v)=0,\qquad G(x,y,z,u,v)=0$$
and the functions, also differentiable, $u=f(x,y,z)$ and $v=g(x,y,z)$ such
that
$$F(x,y,z,f(x,y,z),g(x,y,z))=0,\qquad G(x,y,z,f(x,y,z),g(x,y,z))=0.$$
If you differentiate $F$ and $G$, you ... |
I understand the principle of finite induction, but my book then mentions that there is a variant of the first where requirement b is changed to
> If k is a positive integer such that 1,2...,k bolong to S, then k + 1 must also be in S.
The sample problem is proving that the inequality about the Lucas numbers
l... |
What is the second principle of finite induction? |
This problem came from a Putnam problem solving seminar.
> If each person in a group of n people is a friend of at least half the people in the group, then show that it is possible to seat the n people in a circle so that everyone sits next to friends only.
My idea was to use induction on $n$; if $n$ is odd we ca... |
That's not a very good example. This principle usually goes by the name of **strong induction**, and the point is that while it's equivalent to ordinary induction, it's often more convenient to use in proofs. For proving a sequence $P(k)$ of identities, it's often straightforward to write $P(k+1)$ in terms of $P(k)$,... |
What is your sampling interval? Looking at your second graph, it appears there is something in your acquisition that makes it much more probable to measure times that are at about 7.5 or 8 millisecond intervals than times in between. One approach is to figure out the spacing and bin at this interval.
The periodogr... |
Fix $b>1$. Show the following:
(a) If $m,n,p,q$ are integers, $n>0, q>0$, and $r = m/n = p/q$, prove that $(b^{m})^{1/n} = (b^p)^{1/q}$. So $(b^r) = (b^m)^{1/n}$.
(b) Prove that $b^{r+s} = b^{r}b^{s}$ for $r,s \in \mathbf{Q}$.
(c) If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, there $t$ ... |
I was particularly interested in the following:
When I read this proof, everything seemed fine and logical except one detail (the proof is located [here][1]).
Right after we prove, that the series $\sum_{k=1}^\infty x_{N_m} - x_{N_m+1}$ converges, there is a statement which tells us that the *limit* of that serie... |
Fix $b>1$. Show the following:
(a) If $m,n,p,q$ are integers, $n>0, q>0$, and $r = m/n = p/q$, prove that $(b^{m})^{1/n} = (b^p)^{1/q}$. So $(b^r) = (b^m)^{1/n}$.
(b) Prove that $b^{r+s} = b^{r}b^{s}$ for $r,s \in \mathbf{Q}$.
(c) If $x$ is real, define $B(x)$ to be the set of all numbers $b^t$, there $t$ ... |
I understand the principle of finite induction, but my book then mentions that there is a variant of the first where requirement b is changed to
> If k is a positive integer such that 1,2...,k bolong to S, then k + 1 must also be in S.
The sample problem is proving that the inequality about the Lucas numbers
l... |
I understand the principle of finite induction, but my book then mentions that there is a variant of the first where requirement b is changed to
> If k is a positive integer such that 1,2...,k bolong to S, then k + 1 must also be in S.
The sample problem is proving that the inequality about the Lucas numbers
l... |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=... |
I completely understand how to find the likelihood functions of simple pdfs. However, how would you attempt to find the likelihood function of a pdf with a negative exponential function with two different means, where the overall mean is just the two means added together???
The method to do this would be appreciate... |
I understand the principle of finite induction, but my book then mentions that there is a variant of the first where requirement b is changed to
> If k is a positive integer such that 1,2...,k belong to S, then k + 1 must also be in S.
The sample problem is proving that the inequality about the Lucas numbers
l... |
I'm having a little trouble grasping the definition of a topological space in terms of the open-sets. Suppose that we're defining a topology on the real line, in theory could we arbitrarily select the elements of the open-sets, so long as the set of all open sets satisfies the properties of a topological space? |
Is it possible to find closed form of following sum: $\sum\limits_{k=1}^{n-1}\frac{x^k}{n-k}$ ?
If not then maybe there is at least some good aproximation? |
EDIT: Bounty of 50 pts announced for the **poly**-size part. **You** can win the bounty for a single comment.
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where... |
The main drive in mathematics is usually to look at some known object and to generalize the notion. Once you defined "open intervals" and saw that the union of them remains open (in the sense that each point has some interval around it) and that a finite intersection is also open, and so on...
Once you have, say, a ... |
I was sorting out the coins in my loose-change jar the other day, and the following thought crossed my head: Is it possible to deduce the number of each type of coin in this pile by simply weighing them?
The coins I were counting were Australian 5c, 10c, 20c and 50c. According to [Wikipedia][1], they weights are:
... |
How can I assign weights to coins so that they can be counted in a single weighing? |
I am supposed to prove that (1/1) + (1/3) + (1/6) + ... + (1/t<sub>n</sub>) < 2.
The hint is that
$$
\frac{2}{(n(n+1)} = 2\left(\frac{1}{n} - \frac{1}{n+1}\right)
$$
However, I was thinking that if you take $ \frac{1}{t_n} + \frac{1}{t_{n+1}} = \left(\frac{1}{t_{n-1}}\right)\left(\frac{1}{2}\right) $ where n ... |
I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that
$$
\frac2{(n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right)
$$
However, I was thinking that if you take $ \frac{1}{t_n} + \frac{1}{t_{n+1}} = \left(\frac1{t_{n-1}}\right)\left(\frac12\right)$ where $n > 1$.
This mean... |
Do generalized hypergeometric functions
$${}_p F_q(a_1,\ldots,a_p; b_1, \ldots,b_q; z) $$
with $p = q+1$ always possess a singularity at $z=1$, independent of the their parameters $a_1,\ldots,a_p$ and $b_1,\ldots,b_q$ under the provision that all cases of finite polynomials are excluded? Can someone mention a reliab... |
$X$ is a random variable, which is not constant. $E[X]=0$. $E[X^4] \leq 2(E[X^2])^2$.
Let $Y$ be given by: $P(Y=E[X|X \geq 0]) = P(X \geq 0)$ and $P(Y=E[X|X \lt 0]) = P(X \lt 0)$.
Do we necessarily have $E[Y^4] \leq 2(E[Y^2])^2$? |
Suppose I want to generate all ways of assigning numbers from given list to vertices of an object with given symmetry. The symmetry is specified as a group of rotations that leave a given set of points unchanged.
For square, there are 4 points, and I can represent this group of symmetry as permutation group with gen... |
How to get generators for a finite point group? |
There are many examples in which making more "precise" predictions gives worse performance (e.g. Runge's phenomenon). My professor implied that there was a sound basis for choosing "simple" functions over complex ones in the general case, and that it had to do with information theory.
Does anyone know what he was re... |
There are many examples in which making more "precise" predictions gives worse performance (e.g. Runge's phenomenon). My professor implied that there was a sound basis for choosing "simple" functions over complex ones in the general case, and that it had to do with information theory.
Does anyone know what he was re... |
I have a series of tasks where when one task finishes the next task runs, until all of the tasks are done. I need to find the probability that everything will be finished at different points in time. How should I approach this? Is there a way to find this in polynomial time?
The pdfs for how long individual tasks wi... |
Need help studying for an exam.
Practice Question:
If there are 200 students in the library, how many ways are there for them to be split among the floors of the library if there are 6 floors?
Hint: The students can not be told apart (they are indistinguishable).
The answer must bein terms of P(n,r), C(n,r), ... |
How do I find generators of <a href="http://en.wikipedia.org/wiki/Octahedral_symmetry">Octahedral</a> group when viewing it as a subgroup of S8? |
How do I find generators of <a href="http://en.wikipedia.org/wiki/Octahedral_symmetry">Octahedral</a> group when viewing it as a subgroup of S8?
|
How do I find generators of <a href="http://en.wikipedia.org/wiki/Octahedral_symmetry">Octahedral</a> group when viewing it as a subgroup of S8?
Motivation: I'm trying to count permutations invariant under orientation-preserving symmetries of the cube in Mathematica, and its group theory functions only accept groups... |
Fix $b>1, y>0$ and prove that there is a unique real $x$ such that $b^x = y$. So for uniqueness, $x_1 < x_2 \Rightarrow b^{x_1} < b^{x_2}$. Then consider the set $S = \{b^t: t \leq x \}$ and work from there? |
I am supposed to prove that $(1/1) + (1/3) + (1/6) + \dots + (1/t_n) < 2$.
The hint is that
$$
\frac2{(n(n+1)} = 2\left(\frac1{n} - \frac1{n+1}\right)
$$
However, I was thinking that if you take $ \frac{1}{t_n} + \frac{1}{t_{n+1}} = \left(\frac1{t_{n-1}}\right)\left(\frac12\right)$ where $n$ is even.
So my... |
I have one Incident Edges and Multiple Outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all.
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2$, $e_3$, $e_4$). Also, the angle must be fo... |
I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2... |
I have one incident edges and multiple outgoing Edges, for which I want to pick an outgoing edge such that the angles between the outgoing edge and the incoming edge is the smallest of all. We know the coordinates for the vertex $V$ .
The angle must start from the incoming edge ($e_1$) and ends at another edge ($e_2... |
Fix $b>1, y>0$ and prove that there is a unique real $x$ such that $b^x = y$. So for uniqueness, $x_1 < x_2 \Rightarrow b^{x_1} < b^{x_2}$. Then consider the set $S = \{b^t: t \leq x \}$ and work from there?
Source: Chapter 1, Problem 7, *Principles of Mathematical Analysis* by Rudin |
How do I find generators of <a href="http://en.wikipedia.org/wiki/Octahedral_symmetry">Octahedral</a> group when viewing it as a subgroup of S8?
<b>Update</b>: Alex Bartel's suggestion gives the following permutations and Cayley graph
<img src="http://yaroslavvb.com/upload/save/math.se-cube.png">
<img src="http:... |
How do I find generators of <a href="http://en.wikipedia.org/wiki/Octahedral_symmetry">Octahedral</a> group when viewing it as a subgroup of S8?
<b>Update</b>: Alex Bartel's suggestion gives the following permutations and Cayley graph
<img src="http://yaroslavvb.com/upload/save/math.se-cube.png">
<img src="http:... |
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