instruction stringlengths 12 30k |
|---|
How close are star-convex sets to convex sets? |
A Google search has turned up an analysis of the asymptotic behavior of the iterates of $\sin$ on [page 157][1] of de Bruijn's *Asymptotic methods in analysis*. Namely,
$$\sin^n(1)=\frac{\sqrt{3}}{\sqrt{n}}\left(1+O\left(\frac{\log(n)}{n}\right)\right),$$
which implies that your series converges.
Edit: Moro... |
Your problem is discussed [here][1]. They suggest one of the following solutions:
A*bsxfun(@times,diag(D),A')
OR
A*sparse(D)*A'
Matlab is an interpreted language and so explicit loops are very slow (even though in the latest version there's supposedly in-place compilation). So everything must be vec... |
If you dropped **two rocks in a pond**, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line.
Now imagine that instead of concentric circles, **concentric ellipses emanate** from the rocks' entry points. Again the concentric ellipses will... |
Anyone know of an online tool available for making graphs (as in graph theory - consisting of edges and vertices)? I have about 36 vertices and even more edges that I wish to draw.
(why do I have so many? It's for pathing in a game)
Only tool available to me right now is MS Paint and it would be very messy.
Edi... |
Is there a good online reference that lists known bounds on Ramsey numbers (and is relatively up to date)? The [wikipedia page][1] only has numbers for $R_2(n,m)$.
I am specifically interested in known bounds and values for hypergraph Ramsey Numbers, i.e. 2 colorings of k-subsets (or 2 colorings of the edges of a co... |
The surface integral of a vector field, $F$ over a surface $S$ (or flux as it is called) is, as stated above, the net flow of fluid through the surface. The purpose of orientation is to simply indicate which side of the surface is the "positive" side. For instance, if I calculate flux to be +10, this means their is a... |
The surface integral of a vector field, $F$ over a surface $S$ (or flux as it is called) is, as stated above, the net flow of fluid through the surface. The purpose of orientation is to simply indicate which side of the surface is the "positive" side. For instance, if I calculate flux to be +10, this means their is a... |
The surface integral of a vector field, $F$ over a surface $S$ (or flux as it is called) is, as stated above, the net flow of fluid through the surface. The purpose of orientation is to simply indicate which side of the surface is the "positive" side. For instance, if I calculate flux to be +10, this means their is a... |
**Question:** Which computer programs are useful for a pure mathematician to familiarize themselves with?
**Less Briefly:** I was once told that, now-a-days, any new mathematician worth his beans knows how to TeX up work; when I began my graduate work one of the fourth year students told me he couldn't TeX, I was ho... |
Similarly to Andrey's answer (but somewhat more elegantly) and correspondingly to Zaricuse's second paragraph, according to [this][1] (see the formula before the last one), for any $s > 0$ real,
$$
\int_0^\infty {e^{ - sx} \frac{1}{{\pi (1 + x^2 )}}\,{\rm d}x} = \frac{1}{\pi } \Big \lbrace \cos (s)\Big[\frac{\pi }{... |
If you dropped **two rocks in a pond**, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line.
Now imagine that instead of concentric circles, **concentric ellipses emanate** from the rocks' entry points. Again the concentric ellipses will... |
Maybe some trivial examples could show you the importance of orientation. Try to compute the fluid flow through the unit square
$$
S = \left\{(x,y,z) \ | \ 0\leq x,y \leq 1 , z=0 \right\}
$$
of some fluid moving upwards at constant speed of $1$m/s.
$$
F(x,y,z) = (0,0,1) \ .
$$
Then you parametrize your ... |
Correspondingly to Zaricuse's second paragraph, according to [this][1] (see the formula before the last one), for any $s > 0$,
$$
\int_0^\infty {e^{ - sx} \frac{1}{{\pi (1 + x^2 )}}\,{\rm d}x} = \frac{1}{\pi } \Big \lbrace \cos (s)\Big[\frac{\pi }{2} - {\rm Si}(s)\Big] - \sin (s){\rm Ci}(s)\Big \rbrace,
$$
where ... |
I'm having trouble with this integral
arcsin(sin(x))
The problem is with the intervals of definition for each function :/ if someone could dumb it down for me.
arcsin:[-1, 1] -> [-π/2, π/2]
sin:R->[-1, 1]
right?But what about arcsin(sinx) |
**HINT**
1) Where is $\arcsin(\sin(x)))=x$?
2) $\sin$ is peiodic. $\sin(x+\pi/2)=..$
3) Split up the domain of integration. |
I am able to measure my distance to a set of (about 6 or 7) fixed but unknown points from many positions.
The difference in position between measurements is also unknown.
I believe that I should be able to work out the relative position of the fixed points, and therefore where I measured from and the path I took.... |
**Short version:** if two spherical caps of the same sphere intersect, how can I determined the coordinates of the two "singular points" of this intersection
**Long version:** On a unit sphere, centered at the original, we consider two spherical cap. Each cap is defined by:
- the (unit) vector indicating the top... |
I need to find out for what $\alpha, \beta$ the following sum converges:
$$\sum_{n=2}^\infty n^\alpha (\log n)^\beta$$
I thought I'd do that with the help of the integral criterion, that is to say I considered (after I had substituted $x = e^u$):
$$\int_2^\infty e^{u(\alpha+1)} \cdot u^\beta \mathrm du$$
Then, ... |
When does $\sum_{n=2}^\infty n^\alpha (\log n)^\beta$ converge? |
Given a bijection $f:\mathbb Z \to \mathbb Z$ where $\mathbb Z$ is the set of all integers, does there always exist two bijections $g:\mathbb Z \to \mathbb Z$ and $h:\mathbb Z \to \mathbb Z$ which satisfy $f(n) = g(n) + h(n), ~ \forall n \in \mathbb Z$ ?
Evidently it does while $\mathbb Z$ is replaced with the set o... |
Is $\mathbf{Grp}$ a concrete category? I thought it is, but then the group of symmetries of a square and the quaternion group are both of the order 8, and they are not isomorphic as groups. But sets of the same cardinality are isomorphic as sets, so a standard forgetful functor $(G, \cdot) \mapsto G$ is not faithful! A... |
Is $\mathbf{Grp}$ a concrete category? |
***Problem:***
***Alex flip a fair coin three times. what is the probability that she gets two heads given that the first is a head?***
----------
My solution is based on the argument that from the given information the sample space can be limited to four elements {$HHH,HHT,HTH,HTT$} Of these {$HHH,HHT,HTH$}... |
I'm going through an [old final](http://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2003/exams/final.pdf) from 2003 on MIT's Opencourseware, and problem 6b is giving me a little trouble.
It asks for which primes $p$ is $34$ a square modulo $p$. I approached it like this:
$$\left(\frac{34}{p}\r... |
Trick to showing for which primes $p$ is $34$ a square? |
***Problem:***
***Alex flip a fair coin three times. what is the probability that she gets two heads given that the first is a head?***
----------
My solution is based on the argument that from the given information the sample space can be limited to four elements {$HHH,HHT,HTH,HTT$} Of these {$HHH,HHT,HTH$}... |
Many years ago I wrote this fractal generator:
http://uberto.fractovia.org/
It was shareware but then I put it as open source.
It's written in Delphi, a language that I don't use anymore. So I'm playing lately with the idea of rewriting on some other lang (scala maybe?).
On the mathematical part I'm not expert ... |
Which features would be interesting for a mathematician in a fractal program? |
I'm going through an [old final](http://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2003/exams/final.pdf) from 2003 on MIT's Opencourseware, and problem 6b is giving me a little trouble.
It asks for which primes $p$ is $34$ a square modulo $p$. I approached it like this:
$$\left(\frac{34}{p}\r... |
There seems to be no consensus, although perhaps one is gathering over the centuries to say yes to the first question and identify $\mathbb{N}$ with $\omega$ : the set of finite Von Neumann ordinal numbers. That leaves $\textbf{N}$ = {$1$, $2$, ...} for the counting numbers (who counts with $0$?). You can dodge the iss... |
Do the natural numbers include zero? And what should we call them, with or without zero? |
----------
$ \qquad \qquad $ ***The greatest term in $\left(1+x\right)^{2n}$ has the greatest cofficient if $\frac{n}{n+1} \lt x \lt \frac{n+1}{n}$***
----------
Can we derive something like this for $n$ in general? Please give me some hints.
|
How can we probe this property? |
How can we prove this property? |
How can I figure out the classical construction (direct sum, product, pullbacks, and in general direct and inverse limits) in the category made by chain complexes and chain maps (of abelian groups or any abelian stuff)? Because of this category is abelian it *must* have (co)limits, isn't it?
In particular take
$$
... |
> **Possible Duplicate:**
> [Is 0 a natural number?](http://math.stackexchange.com/questions/283/is-0-a-natural-number)
<!-- End of automatically inserted text -->
There seems to be no consensus, although perhaps one is gathering over the centuries to say yes to the first question and identify $\mathbb{N}$ with $... |
I have a question about whether the following two sets are the same:
$$
lim_{a \rightarrow \infty} \, limsup_{n \rightarrow \infty} \, \{ x \in \mathbb{R} | f_n(x) > a \}
=
\{ x \in \mathbb{R} | lim_{n \rightarrow \infty} f_n(x) =\infty \},
$$
where $\{f_n\}$ is a sequence of functions. How can you explain ... |
I have a question about whether the following two sets on the two sides are the same:
$$
lim_{a \rightarrow \infty} \, limsup_{n \rightarrow \infty} \, \{ x \in \mathbb{R} | f_n(x) > a \}
=
\{ x \in \mathbb{R} | lim_{n \rightarrow \infty} f_n(x) =\infty \} ?
$$
where $\{f_n\}$ is a sequence of functions. Ho... |
I have a question about whether the following two sets on the two sides are the same:
$$
lim_{a \rightarrow \infty} \, limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\{ x \in S \, | \, lim_{n \rightarrow \infty} f_n(x) =\infty \} ?
$$
where $\{f_n\}$ is a sequence of real-valued function... |
How can I figure out the classical construction (direct sum, product, pullbacks, and in general direct and inverse limits) in the category made by chain complexes and chain maps (of abelian groups or any abelian stuff)? Because of this category is abelian it *must* have (co)limits, isn't it?
In particular take
$$
... |
I have a question about whether the following two sets on the two sides are the same:
$$
lim_{a \rightarrow \infty} \, limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\{ x \in S \, | \, lim_{n \rightarrow \infty} f_n(x) =\infty \} ?
$$
where $\{f_n\}$ is a sequence of real-valued function... |
How to compute this integral? |
I have a question about whether the following two sets on the two sides are the same:
$$
lim_{a \rightarrow \infty} \, limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\{ x \in S \, | \, lim_{n \rightarrow \infty} f_n(x) =\infty \} ?
$$
where $\{f_n\}$ is a sequence of real-valued function... |
Most of the time, an equivalence relation is hiding an "equality" somewhere; that is, $x\sim y$ if and only if $x$ and $y$ have *something* that you are trying to isolate which is "equal". With integers, the equivalence $a\equiv b\pmod{m}$ means that $a$ and $b$ have **equal** *remainder when divided by $m$*. We usuall... |
See [CORDIC][1]. See also this question: http://stackoverflow.com/questions/2169641/where-to-find-algorithms-for-standard-math-functions/2169666 and this review: http://mathdl.maa.org/mathDL/19/?pa=reviews&sa=viewBook&bookId=65790
[1]: http://en.wikipedia.org/wiki/CORDIC |
I have a question about whether the following two sets on the two sides are the same:
$$
lim_{a \rightarrow \infty} \, limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\{ x \in S \, | \, lim_{n \rightarrow \infty} f_n(x) =\infty \} ?
$$
where $\{f_n\}$ is a sequence of real-valued function... |
I have a question about whether the following two sets on the two sides are the same:
$$
\lim_{a \rightarrow \infty} \, \limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\left\{ x \in S \, \Bigm| \, \lim_{n \rightarrow \infty} f_n(x) =\infty \right\}\quad ?
$$
where $\{f_n\}$ is a sequence... |
I have two (2x2) matrices, A and B, with the same determinate. I want to know if they are similar or not.
I solved this by using a matrix called S:
a b
c d
and it's inverse in terms of a,b,c, and d. then showing that there was no solution to $A = SBS^{-1}$. That worked fine, but what will I do if I have (3x... |
How do I tell if matrices are similar? |
I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.
I solved this by using a matrix called $S$:
$$\left(\begin{array}{cc}
a& b\\
c& d
\end{array}\right)$$
and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution t... |
Great Post! I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not go ahead and think about finite groups as subgroups of a lie group?
The lie groups are manifolds with a group st... |
I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.
I solved this by using a matrix called $S$:
$$\left(\begin{array}{cc}
a& b\\
c& d
\end{array}\right)$$
and its inverse in terms of $a$, $b$, $c$, and $d$, then showing that there was no solution t... |
I have a question about whether the following two sets on the two sides are the same:
$$
\lim_{a \rightarrow \infty} \, \limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\left\{ x \in S \, \Bigm| \, \lim_{n \rightarrow \infty} f_n(x) =\infty \right\}\quad ?
$$
where $\{f_n\}$ is a sequence... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not go ahead and think about finite groups as subgroups of a lie group?
Lie groups are continuous manifolds with a group structur... |
The current price of a stock can be modeled by $P_0 = \frac{D_{1}}{r-g}$ where $D_1$ is the expected dividend, $r$ is the rate of return, and $g$ is the expected growth rate in the perpetuity. If $r<g$, then
$\displaystyle P_0 = \sum_{t=1}^{N} \frac{D_{0}(1+g)^{t}}{(1+r)^{t}} + \frac{P_N}{(1+r)^{N}}$.
Is there ... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not go ahead and think about the elements of a finite group as points on a continuous manifold, in other words, consider the represent... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not go ahead and think about the elements of a finite group as points on a continuous manifold, in other words, consider the represent... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite ... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite ... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite ... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite ... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite ... |
I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite ... |
Z(t) = $\int_0^t $ g(s)dW (s), where g is an adapted stochastic process.
Find dZ ?
|
$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process.
Find $dZ$ ?
|
We know that $H^p(S^2 \wedge S^4) = H^p(S^2)\oplus H^p(S^4)$ for $p\neq 0$. I want to show that this space has different ring structure than $CP^2$. So, given a generator in $H^2(S^2 \wedge S^4)$ I want to cup it with itself and get 0. My idea is to use the generator from $H^2(S^2)$(which obviously is zero when squared... |
Define $(a,b) < (a',b')$ if $\max(a,b) < \max(a',b')$ or $\max(a,b) < \max(a',b')$ and $b' < b$ or $\max(a,b) < \max(a',b')$ and $b = b'$ and $a < a'$.
Now I want to prove that the order-type of $(\{(b,c) : \max(b,c) = a\}, <)$ is equal to $a + a + 1$, does someone have a hint how to do this? I can't find the biject... |
We know that $H^p(S^2 \wedge S^4) = H^p(S^2)\oplus H^p(S^4)$ for $p\neq 0$. I want to show that this space has different ring structure than $CP^2$. So, given a generator in $H^2(S^2 \wedge S^4)$ I want to cup it with itself and get 0. My idea is to use the generator from $H^2(S^2)$(which obviously is zero when squared... |
I have a question about whether the following two sets on the two sides are the same:
$$
\lim_{a \rightarrow \infty} \, \limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \}
=
\left\{ x \in S \, \Bigm| \, \lim_{n \rightarrow \infty} f_n(x) =\infty \right\}\quad ?
$$
where $\{f_n\}$ is a sequence... |
Define $(a,b) < (a',b')$ if $\max(a,b) < \max(a',b')$ or $\max(a,b) = \max(a',b')$ and $b' < b$ or $\max(a,b) = \max(a',b')$ and $b = b'$ and $a < a'$.
Now I want to prove that the order-type of $(\{(b,c) : \max(b,c) = a\}, <)$ is equal to $a + a + 1$, does someone have a hint how to do this? I can't find the biject... |
Here is what I need to prove:
>Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$.
I know that:
1. for any graph $\sum_{v \in V}\ deg(v) = 2e$;
2. for any tree $e=v-1$.
From 1 and 2 it follows that for any tree ... |
Can two different roots of an irreducible polynomial generate the same extension? |
It is possible to find the shortest route thanks to algorithms like A*, bread first search, depth first search, etc..
Is there any known algorithm to find how many routes are available if there are more than 1 optimum solution (for example routes with same costs) or any solution at all.
I am working on some graph... |
Hints for the other way round:
- Do you know about mathematical induction?
- Can every $d_i$ be $\gt 1$? |
Say I have two algorithms that I don't know how they work but I know what they are meant to achieve. I tried algorithm A once and it succeeded. And tried algorithm B 100 times and succeeded 99 times. So A succeeded 100% times and B succeeded 99% times, but my common sense tells me B is still more "reliable" than A. How... |
How to calculate likelihood to succeed knowing attempts and successful attempts amounts? |
Let $R$ be a commutative ring with identity and $I,J,K$ be ideals of $R$. If $I\supseteq J$ or $I\supseteq K$, we have the following modular law
$$ I\cap (J+K)=I\cap J + I\cap K$$
I was wondering if there are situations in which the modular law holds in which the hypothesis that $I$ contains at least one of $J,K$ ... |
Say I have two algorithms that I don't know how they work but I know what they are meant to achieve. I tried algorithm A once and it succeeded. And tried algorithm B 100 times and succeeded 99 times. So A succeeded 100% times and B succeeded 99% times, but my common sense tells me B is still more "reliable" than A. How... |
Can you tell me why the following is true?
> A ring $R$ is local if and only if every principal left $R$-module is indecomposable. |
We know that $H^p(S^2 \vee S^4) = H^p(S^2)\oplus H^p(S^4)$ for $p\neq 0$. I want to show that this space has different ring structure than $CP^2$. So, given a generator in $H^2(S^2 \vee S^4)$ I want to cup it with itself and get 0. My idea is to use the generator from $H^2(S^2)$(which obviously is zero when squared). H... |
Here is what I need to prove:
>Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$.
I know that:
1. for any graph $\sum_{v \in V}\ deg(v) = 2e$;
2. for any tree $e=v-1$.
From 1 and 2 it follows that for any tree ... |
The surface integral of a vector field, $F$ over a surface $S$ (or flux as it is called) is, as stated above, the net flow of fluid through the surface. The purpose of orientation is to simply indicate which side of the surface is the "positive" side. For instance, if I calculate flux to be +10, this means their is a... |
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to Pi but I have heard many people say that while this is a classic example, there are series that converge much faster. Does anyone know of any? |
If you dropped **two rocks in a pond**, the concentric circles emanating from the two spots would osculate $\infty$ times. The locus of osculating points would form a line.
Now imagine that instead of concentric circles, **concentric ellipses emanate** from the rocks' entry points. Again the concentric ellipses will... |
Here is what I need to prove:
>Let $d_1,d_2,...,d_n$ be a sequence of natural numbers (>0). Show that $d_i$ is a degree sequence of some tree if and only if $\sum d_i = 2(n-1)$.
I know that:
1. for any graph $\sum_{v \in V}\ deg(v) = 2e$;
2. for any tree $e=v-1$.
From 1 and 2 it follows that for any tree ... |
The following formula for $\pi$ was discovered by Ramanajuan:
$$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$
Does anyone know how it works, or what the motivation for it is? |
I know the series, $4-{4\over3}+{4\over5}-{4\over7}...$ converges to $\pi$ but I have heard many people say that while this is a classic example, there are series that converge much faster. Does anyone know of any? |
I would recommend reading Gerald Rising's [*Inside your Calculator*][1] (which has a [supplementary website][2]); there is a nice discussion of the methods used by some calculators that is suitable at the undergraduate level.
Otherwise, to really figure out what methods they are using, it might help to search the te... |
This sounds like a simple problem, but I can't get it done. Given the general equation
$ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0,$
what are the requirements on the coefficients so that this equation represents an ellipsoid? All coefficients are real.
Some more context. To make the representatio... |
Is a general (non-homogeneous) quadratic equation in $\mathbb{R}^3$ an ellipsoid? |
Whilst answering my homework, I have noticed that a question has said that I need to consider the subgroups of A=Z135, including W=<18,81>. I was under the impression that for a subset to be a subgroup $a+b$ in the subgroup had to be in the subgroup as well. $18+81 = 99mod135$ which isn't in the subgroup.
Is it my r... |
Is <18,81> a subgroup of the abelian group Z135? |
I am working through a review problem asking to find the inverse of 4 mod 9
Through examples I know that I first need to verify that the gcd is equal to 1 and write it as a linear combination of 4 and 9 to find the inverse. I can do this in just one step:
gcd(4,9)
9 = 2 * 4 + 1
1 = 9 - 2 * 4
Thi... |
I am working through a review problem asking to find the inverse of $4 \mod{9}$
Through examples I know that I first need to verify that the gcd is equal to 1 and write it as a linear combination of 4 and 9 to find the inverse. I can do this in just one step:
gcd(4,9)
9 = 2 * 4 + 1
1 = 9 - 2 * 4
... |
Whilst answering my homework, I have noticed that a question has said that I need to consider the subgroups of $A=\mathbb{Z}_{135}$, including $W=\langle 18,81\rangle$. I was under the impression that for a subset to be a subgroup $a+b$ in the subgroup had to be in the subgroup as well. $18+81 = 99\mod{135}$ which isn'... |
Is $\langle 18,81\rangle$ a subgroup of the abelian group $\mathbb{Z}_{135}$? |
From your fact (1) and the hypothesis $\sum d_i = 2(n-1)$, you know $e = n - 1$. You need now only prove that any connected graph with $n - 1$ edges is a tree.
**EDIT** - Since you are going down an alternate path, I have a new suggestion.
Order the $d_i$ so that $d_1 \leq d_2 \leq \ldots \leq d_n$. Construct a... |
**HINT for $\Leftarrow$**: Do the contrapositive; if $\mathfrak{m}_1$ and $\mathfrak{m}_2$ are two distinct maximal ideals of $R$, then look at the left $R$-module $R/(\mathfrak{m}_1\cap\mathfrak{m}_2)$; it is principal (generated by $1+(\mathfrak{m}_1\cap\mathfrak{m}_2)$. Is it indecomposable?
**HINT for $\Rightarr... |
I want to create a function $f : [0,255]^3 \rightarrow [0,255]^3$ over the integers such that if you pass in three RGB color values, a triple with a darker or lighter color will be returned, and I want the values to increase in some sort of proportion to how dark or light they are already; lightness and darkness being ... |
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