We characterize them into types like we do with integrals but the list is ultimately endless.
Mathematics is a creation of the human brain; the Earth may have revolved around the sun before mankind but the mathematical constructs of gravity, in my humble opinion, did not. And of course the...
I have something to add based on my own experience: the road to the right one is littered on both sides with wrong ones. Embrace this principle and learn to work with it. Hall (a PF member) once said, "you don't just stare at the problem and wait for the answer to pop into your head...
I know firsthand some admirable qualities Marcus exhibited on this forum. Perhaps if the members wanted, we could offer an award yearly to someone that exemplified them.
I do not believe that's true.
Look no further than “At Home in the Universe,” by Stuart Kauffman:
And if I may be allowed to make a comment in the interest that some reading this may be stimulated to pursue this further, Kauffman mentions "phase-transition." That phenomenon is very common...
The dynamics necessary for the origin and evolution of life existed before life and are independent of life. The inconsequential trappings of biology and chemistry instantiating life on Earth are just happenstance and convenient for our particular planet. Consider a concrete example: termite...
Here's the best primate tree I can find. See, right there in that little nook near proconsul. That is, using best available evidence, the common ancestor...
There is a discussion of traffic flow and a PDE describing it in the textbook "Basic Partial Differential Equations" by David Bleecker and George Csordas. Should be available in any university library and may help you better understand the dynamics which involves a cusp catastrophe (a wreck).
Is that not correct in the sense that (chemical) survival and reproductive success were relevant, crucial, and necessary for the origin of life on Earth even during pre-biotic evolution, even before any signs of life existed on earth, and therefore the fundamental tenant of evolution, survival...
I do not feel this is an unreasonable question to ask. We know the Universe is massively non-linear and further, that non-linear dynamics can create infinite diversity. Consider the Lorenz attractor (the owl-eyes icon of Chaos Theory), a set of three coupled non-linear differential equations...
Well, I don't wish to be cold and callous but it is the mechanism of infection that most fascinates me: how (exactly) does the virus cause microcephaly? Someone above mentioned an autoimmune response. Are the neurons in microcephaly being destroyed soon after they are created or are they...
Hi guys,
Been keeping up with your comments. This is the first reference I've seen to suggest some are convinced the virus is causing microcephaly:
http://www.foxnews.com/health/2016/02/01/brazils-zika-virus-outbreak-worse-than-believed-health-minister-says.html?intcmp=hphz02
What evidence is...
And sorry if I misled you. No, what I think would be nice is for someone to take interest in this and generate the equilibrium surface and then add something: draw little bacteria and single-celled organisms on the top surface (the pre-Cambrian) and then trajecting through the catastrophe...
But I want someone else to do it. Well, I've done it plenty of times, even the swallow-tail for that matter but I digress. No, I want someone to be taken by this discourse and change history forever. :) Butterfly effect right? :)
Yes. Thanks for that. It's certainly the surface but it's the "how" part, precisely, "exactly how" part. That is, given y'=-y^3-by+c, generate the catastrophe surface preferably in Mathematica in an elegantly-pleasing format (a nice-looking graph that clearly shows the cusp). Can we plot it...
Ok. Thank you kindly for that. Yes. I'm having problems with it. Also, I think perhaps the one I'm most familiar with is:
\frac{dy}{dt}=-y^3+by+c
I was wondering then since I'm having problems with it, would someone be so kind to generate the catastrophe surface, identifying the...
. . . oh goodness. Once upon a time, a long time ago I would look outside of my window at the world about me and wonder why. Then I started studying non-linear differential equations. I no longer wonder why about a lot of things. Many people go through their lives puzzled about why are...
The non-linear pendulum. Know it? In the linear case, you can only (initially) perturb the pendulum a small amount before the solution looses accuracy. However, in the non-linear case, you can smack the pendulum so hard so that it goes round and round and the solution will still accurately...
Here's a good example of what they're good for:
https://www.physicsforums.com/threads/is-life-a-matter-of-evolving-chemistry.849997/#post-5342318
Look at post #17 and maybe a few before it.
In the book, "Self-Organization in Biological Systems," Camazine and others construct among other many examples, a coupled set of three non-linear PDEs modeling the dynamics of termites during the construction of the marvelous clay cathedrals that they construct using three variables: termites...
Is it still not possibly the underlying non-linear dynamics principally responsible for the emergence and evolution of life? I do recall Ygggdrasi, one particular thread here where you proposed if a suitable set of non-linear differential equation were set up appropriately, dynamics we ascribe...
In my humble opinion, it is precisely mathematics that gives us a reasonable answer to this and many other puzzling phenomena about the Universe but unfortunately it is an emergent one; one that only comes from a long, tedious, time-consuming study of non-linear differential equations. We so...
Suppose we had only to work with that. First, consider the four basic properties of life:
(1) Containment,
(2) Replicate,
(3) Metabolize,
(4) Evolve.
Then as we know life, constantly evolving chemistry in an ocean would not be considered life as it's not contained (in a cell). The same holds...
Thanks Mark. No. I have an Ethernet link. The only problem I'm encountering is with my e-mail. Windows Live Mail is not allowing me to send e-mails. When I attempt to send an e-mail, I get "a problem occurred while trying to open this message."
Hi,
A few days ago, my Windows 10 went through a long auto update and now when I run Windows Live mail, I cannot send an e-mail and I receive a message:
An app on your PC needs the following Windows feature:
.Net Framework 3.5
and then it instructs me to load it. But when I attempt to load...
I personally believe both Greg and Ryan are wrong, based solely on mathematical grounds: the butterfly effect. What it'll look like in 50 years? The best I can say is that because of non-linear effects, we simply cannot linearly extrapolate current conditions or predictions very far into the...
Not sure here but let me take in in pieces. First:
Ok, so if I understand this correctly, any finite ##H## would be contained in a subgroup ##R=\left\{\mathbb{Z},\frac{1}{D}+\mathbb{Z},\cdots,\frac{D-1}{D}+\mathbb{Z}\right\}##. Then the smallest representative of ##h+H## would be a member...
Thanks. Ok, part of my problem maybe is that I'm having problems constructing sub-groups of ##\mathbb{Q}/\mathbb{Z}##. In your example above, I would think the smallest subgroup would be...
Afraid I'm not quite following you Andrew. Let me however, first post what I think I would need to do if we prove the factor group ##\frac{\mathbb{Q}/\mathbb{Z}}{H}## is unbounded for any finite group ##H##:
Let me just flat-out assume I have a mapping:
##...
Ok thanks for that Andrew.
I'm not really clear what the elements of ##H## are. Can you clarify the following for me:
(1) If
##\displaystyle G=\mathbb{Q}/\mathbb{Z}=\left\{\frac{p}{q}+\mathbb{Z} : 0\leq p/q <...
The group ##\mathbb{Z}_{1024}^*## I think is a small integer mod group with a composition series of length 10:
##\mathbb{Z}_{1024}^*\rhd \big<a^2\big>\rhd \big<a^4\big>\rhd \big<a^8\big>\rhd\cdots\rhd \big<a^{10}\big>=\{1\}## where ##a## is any member of this cyclic group which generates the group.
Homework Statement
Let ##H## be a finite subgroup of the group ##G = \mathbb{Q}/\mathbb{Z}##. Prove ##G/H## is isomorphic to ##G##.
Homework Equations
[/B]
My plan is to model the proof after a simple example but I would need to show there is a bijection between...
Ok thanks. Think I figured out an example of a composition series of length 10: Note that ##\mathbb{Z}_n^*## is cyclic for powers of odd primes. Then consider ##\mathbb{Z}_{3^{10}}^*=\mathbb{Z}_{59049}^*## for which ##2## generates the group. Then I believe the following is a composition...
Ok thanks. Afraid I'm not seeing the connection between the p-group decomposition and the group's composition series. Perhaps if we take a smaller group. For example consider ##\mathbb{Z}_{100}^*##.
Ok, now the divisors of this group are ##1,2,4,5,10,20,25,50,100 ## and then using the...
Thanks for that mathwonk. However, that reference refers to the decomposition of an integer mod group into it's p-sylow subgroups. I don't see how that's related to the composition series of a group. Take for example:
##\mathbb{Z}_{10!}^*\cong \mathbb{Z}_2\times...
Hi,
I'd like to find easily-accessible composition series of an interesting lengths say 10 or so (to study some theorems about subgroup series experimentally). I'm thinking the integer mod unit groups, that is
##\mathbb{Z}_n^*=Z_0\rhd Z_1\rhd Z_2\rhd \cdots \rhd Z_9\rhd Z_{10}=\{1\}##. Would...
Thanks mathwonk. That's very informative and I'm sure I'll use it while I'm studying abstract algebra. For example, just briefly running though it I noticed it had a very good summary of the various generators for the symmetric group including the one I had worked on recently here and...
Hi Greg,
I'm making progress with the code. I had started another thread pertaining to the data from GAP where I posted a brilliant piece of GAP code I painfully pieced together:
https://www.physicsforums.com/threads/converse-of-lagranges-theorem-for-groups.833243/
I contacted their support...
Hi,
How do we determine the group types for a particular order? I know the example of using the Cayley table to show there are only two types of groups of order 4 but do not know how to determine this for other groups. For example, suppose I wanted to show there is a group of order 10 in the...
I got the final code to compute the missing groups. The code below took 9 seconds to find (all?) ##11300## subgroups for ##S_7##. The final list of 29 are the group sizes missing in ##S_7## assuming GAP is correct.
gap> notlist:=[0];
[ 0 ]
gap> gIndex:=7;
7
gap>...
I was hoping someone could help me understand the following proof from:
http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/genset.pdf
The problem I'm having is the line ``Any p-cycle can be written as ##(1\;2\;\cdots\;p)## by relabeling the objects being permuted by applying an overall...
Sorry about that. I re-read it and it's ambiguous. To be honest, I'm not so sure myself. But if I had to take sides then I would say it means that if ##A=PBP^{-1}## and the equivalence respects multiplication (say left-multiplication for now), then for each ##C\in GL_n(F)##, there exists an...
I understand that: If ##A=PBP^{-1}## and ##AC=PBCP^{-1}## then we have:
##\begin{aligned}
AC&=\left(PBP^{-1}\right)\left(PCP^{-1}\right) \\
AC&=APCP^{-1} \\
C&=PCP^{-1} \\
\end{aligned}
##
That would imply ##PC=CP## which in general is not the case.
However, that's not how I understand what...
Here it is in GAP:
gap> myagroup:=AlternatingGroup(4);
Alt( [ 1 .. 4 ] )
gap> CommutatorSubgroup(mygroup,mygroup);
Group([ (1,2,3), (1,2,4) ])
gap>
So if I coded that correctly, I assume that means ##A_4^{'}=\{c_1 c_2\cdots c_n:\text{c_i is a commutator}\}=\{(1\;2\;3),(1\;2\;4)\}##...
Hi guys,
There is a software package called GAP for "Groups, Algorithms, and Programming" with emphasis on Group Theory. You can download it for free. I did. However I'm finding it so intractable to use. I would like to find the "missing group" in the Symmetric groups. That is, the group...
Hi Ray,
In order for the equivalence relation to respect multiplication, then if ##A=PBP^{-1}##, then for all ##C\in GL_n(F)##, there exists some ##X\in GL_n(F)## such that ##AC=XBCX^{-1}## (and also ##CA=YCBY^{-1}##). That is, ##X## can be different for each ##C## as long as ##AC\sim BC##...
Ok thanks. Apparently there is something called CLT-groups (converse to Lagrange Theorem?) which satisfy the converse of Lagrange's Theorem, and likewise non-CLT groups which do not. According to this link...
I know of only one group, ##A_4## of order 12 which does not have a subgroup with order dividing the group size. In this case, a subgroup of size ##6##.
What property of a group causes this? Would I expect to find other examples only in non-abelian groups or are there abelian groups which do...
Ok thanks Ruber. I understand that if ##A=PBP^{-1}## and ##C=PDP^{-1}## then in this particular case, we have ##AC=PBDP^{-1}##. Afraid though I don't understand the significance of showing ##P## does not have the structure that would allow ##CP=PC##. Could you please explain this a little...