7.10 Due March 31

Difficult)))
Ah no big deal, just one thing really...theorem 7.52


Reflective)))
I feel so much better after studying for an exam.  Regardless of how I do on the test, I really feel like I understand the big picture and see clearly how all the little things fit into place much better from the content when I condense the entire section into one page of notes representing all the theorems, lemma, corollarys, and definitions...along with examples and practice problems.

Exam Review Due March 29

Which topics and theorems do you think are the most important out of those we have studied?
None.  Just figure out the definitions and true-false statements, and examples...that is the only place I lose points, not on understanding or applying relevant information...

What kinds of questions do you expect to see on the exam?
Provide an example of a group...

What do you need to work on understanding better before the exam?
Examples of all the different possible kinds of groups that we could be asked to show examples of on the test...

Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.
Examples.
Abelian
Non Abelian
Quotient Group
Non trivial subgroup
Group isomorphism for Z4 and Z2 x Z2
Kernal

Non-abelian groups: Sn, An, Dn, matrix groups.
Abelian groups: Z, Zn, Un.
An element of finite order contained in a group of infinite order.
Cyclic groups of all orders—both infinite and finite.
Groups which are not cyclic, including a (sub)group generated by two elements which is not cyclic.
A group with a non-trivial center.
A subgroup of an infinite group that has finite index

7.8 Due March 27

Difficult)))
Probably the Second Isomorphism of groups...
The third one uses the first, and the first isn't really in the book...but only kind of because I guess we did it with rings and are going to reuse it...

But seriously, the idea of the kernal and quotient groups looks challenging.  I'm really sad the book makes me write this section...and that the author didn't just copy and paste when he/they wrote it

Reflective)))
Take your pick from chapter 6...and change some variables to put it in terms for groups...too bad the book has as much information as this blog...

skipping proofs, and claiming they were done in the last chapter...

Frieze Group Theory

Complex systems: approaches to population by Studies Carl P. Simon
Founded center for the study of complex systems in 1999, and was CEO for 10 years

systems thinking:  composed of interdependent and interacting components.  Ignoring them leads to undesired consequences.
An example of DDT and natural systems.  Not worrying about the ramification of the system.

Health Systems: mistake of using the strongest medications.  Drug-resistant bacteria and viruses are an example of health systems.  IE an arms race with bacteria, more people died from bacteria than from HIV.  Careless use of medication.  Also anti-bacterial soaps: promulgating helpful and resistant bacteria.

Man-made demographic solutions:  widen the road when too much traffic...urban decay.

HOW TO APPROACH A SYSTEM:
Identify variables, draw diagrams, build a model

MODELS:  dynamics, the framework for data collection, estimate, parameters,experiments, comparison, assumptions, forecasts.

K.I.S.S. Principle  Keep it simple stupid.


simple systems:  
put down components and check the spread.  Check connections
Disease Spread:
build equations and variables



complex sustems:
Useful for when there is a rugged landscape without a clear path
Adding complexity line upon line to add realism and fact.
how to build feedback and learning into a model:
build a rule:
each rule has its specificity=strength
increase strength as the rule works
tax the rules to see if some are less useful
Realism:
mutation and crossover of rules allow to adjust and check for new scenarios.

7.7 Due March 24

Difficult)))

Quotient Group.  Fun new definition that I will need to play around with to really get a good grasp on the context.

I did not get the operation in example at the top of page 218 the first time.
I had to try the operation to verify that (Mr1)(Mh)=Md.  I thought that the coset elements are multiplied, but no it's the operator on the coset that forms the product.

Example  on top of page 219 I follow the example, but at the end of the example there is a brief explanation on the fact that Z not being an ideal of Q makes Q/Z not a quotient group...???
WHAT????  ^
Oh...my bad...YES a quoteient group...but not a Quotient ring.  DUH..

Reflective)))
Na=Nc and Nb=Nd => that Nab=Ncd
also theorem 7.36 is a good reminder of rings and ideals and their properties.

7.6 Due March 22

Difficult)))
when a chapter is split up...and I'm supposed to do homework without having completed a reading or a chapter...that's really difficult...I gave up and read this for the last homework...sure made everything much easier to prove...

why doesn't aN=Na => that an=na....what's the difference...?
I see...its the modulus and that there is some t in N such that na=at and at is in aN...and vice versa to obtain the right coset from the Left coset

Reflective)))
The rest of the reading for 7.6 looks a lot like the topics taught Monday in class except for this new theorem...which makes perfect sense...and sure was pretty useful in proving the problems on the last homework set...too abd it wasn't part of monday's reading...

7.5 & 7.6 Due March 20

Difficult)))

Everything theorem 7.28-7.30
I am so lost...is this due to the concept of cyclical order less than order of group??

I'm praying desparately for the explanation of those proofs monday...

Reflective)))
Left coset...kind of like right coset...but left...all the same definitions.




If you think this blog isn't long enough, then go back and read the last one again...but switch definitions from right to left...

7.5 Due March 17

Difficult)))
Why is there only a right coset of groups?  not a left? I read the note about how rings being abelian give us a + i = i + a and so a + I for rings, but I don't understand the correlation between that and why groups have Ka or K + a...and not a + K...
Oaph!  I get it...transpositions, and compositions... they are set up as the (ab)(cd)(ef)... order matters for groups!

|G| = |K| [G:K]  means that the order of K divides the order of G...  I had to look at that one twice to get that from the theorem...I feel so dumb...

Reflective)))
7.22...7.23....7.24 what great comfort this sentence gives...no need to blog...just sit back and reflect on those sweet theorems regurgitated...

7.9 Due March 16

Difficult)))
cruel and annoying notation to memorize.  I struggled so hard to see how (14)(13)(12) = (1234)  until I wrote it out in the original notation of permutations and did the composition...but that is hard to make the transition.  would have been easier to start with this new notation...

I was really confused if the book meant to say transposition on page 233 instead of transformation...in the paragraph under the bolded heading "The Alternating Groups"...they have not said anything about what a transformation is...??

Transformation is not in the book defined...I'm pretty sure they used the wrong word...annoying to read a defective text...

Reflective)))
the factorization of permutations is either even or odd...just like integers are even or odd.  Number theory permeates all of math!!

7.4 Due March 13

Difficult)))

Love the properties and tools of isomorphism, but the automorphism is still unknown as to its usefullness or value...also lost as to why the inner automorphism is valuable...

maybe since it is only in a small example, it isn't really important or useful...so I should just let it go...?????

I also struggled ot understand the corollary for 7.21...this section will require a lot more reading before class, and while trying to do the homework.

Reflective)))
Proofs of isomorphism and homomorphism are fun.  give me those any day!

a lot of these theorems are basically a rewording of the ones for rings...

7.3 Due March 10

Difficult)))
I struggled to understand the relevance of the Center of a Group, until I realized that it is used to form a subgroup...wasted time...that is always difficult.

...like wasting time typing up a blog and trying to find a justification of content that is deserving of points...like that is all school seems to be is a justification for points...

I hate grades...so finite and useless...also a difficult thing for me to put up with...



Reflective)))
ring, subring.  Field, subfield.  Group, Subgroup.  Larger set having properties, smaller group inherits similar properties.
Cyclic and generators are definitions that I have seen used, and had examples of before in other clases, but never understood the full application of the definition until this class...in the future, when I understand it more than after just the first reading.

7.2 Due March 8

Difficult)))

I have NO IDEA how the example of order of an element of S3 works?  what is the operation of raising S3 to a power?????????????
HA!
It's a composition of permutations.  3->2,2->1 and so 1->1 ect...so the exponent on a permutation is the definition of that permutation composed of itself that number of times.

Let G be an abelian group in which every element has finite order.  If c in G is an element of largest order in G (|a| <= |c| for all a in G) then the order of every element of G dividec |c|....
But what does that even mean???

Reflective)))
I remember being in Trig and being told that i^4 = 1...and that it is a cyclical group...and having no idea what that meant...Coooooooooooll.

7.1 (part 2) Due March 6

Difficult)))
Abelian definition (I used that complete definition while studying for the last midterm and defining the properties of a ring.)
The hardest part of the group theory reading from this section is probably just remembering that the multiplication sign is + for Z,Zn,Q,R,C...and not multiplication because a ring under usual multiplication is not a group unless the ring is the zero-ring.


Reflective)))
I remember When I took complex analysis and we did composition of functions and linear mappings, it seems to be a lot like the bijective composition of functions that we also used in algorithm design when the class covered fast-fourier transforms of compositions of functions.

7.1 due March 4

Difficult))) (balancing time with a CS project that has taken too long and a homework load for every other class.)
Group theory looks like fun!  I don't think that anything looks hard, but when I was in the ACME version of 342 I found that  the second law of isomorphisms and the application on transformations looks a lot like the structure of groups.

I'm going to have to re-read this section to understand it better.


Reflective)))
The exam was a lot easier this time...now that I have read through the examples and had some in mind...
while studying for the exam, I read about albian groups in the properties of ring addition!!!
Happy that's over with.  Tests are fun when they aren't Computer Science exams.  Who codes without google and references...NOT ME...sorry, but blogs are for thoughts, and so I am reflectng on the semester, and what I have really learned is that I am not planning on coding for my day job...unless it is in python or R.

Exam Prep Due February 2

Which topics and theorems do you think are the most important out of those we have studied?
Factor, isomorphism, reducibility, remainder, relatively prime, prime (polynomials/ideals), congruence classes, cosets, polynomial functions, quotient rings, Kernal, Maximals...

What kinds of questions do you expect to see on the exam?
--Give me examples of X where X was discussed either in class or on a homework.
--what is the definition of Y where Y is contained in the material covered after the last exam
--Prove Z, where Z was in the reading/class lecture/homework
--some random new concept teacher finds interesting and teaches on the review day...

What do you need to work on understanding better before the exam?
--All of the above.
--Congruence class arithmetic.
--A real understanding of X as either a variable in the rules of functions, an element of a set, a number.

Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Wednesday.
I didn't follow the why of the proof of theorem 5.11 (second edition) for (alpha)=[x]  and the definition of congruence -class arithmetic.