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So I realized today that I’ve been neglecting my blog for the past couple of weeks, I’ll make excuses for that later, and I thought that maybe I should make more of an effort to try and post more often. I want to continue using this platform as a journal of sorts, but I also want to get into the habit of posting more mathematical topics as well. I have a lot of material from my talk on Cauchy’s theorem in group theory that I would like to post as well as a couple of problems from analysis that I have found pretty interesting. Hopefully I can get LaTex figured out enough to type up these posts sometime within the next week or so.
As for why I have been neglecting the blog recently. It really is a combination of things. To begin with I have been sick for over a week now…not enough to actually sideline me, but enough that I haven’t really felt productive for over a week now.
Add to that an insane amount of algebra homework. Last quarter I was probably dropping about 6-8 hours a week on algebra homework and this quarter I’ve probably been averaging closer to 12 hours a week. For the first homework I think I turned in a total of 14 pages. Others have required less paper in the end, but have required a lot of effort in order to actually find the solutions. There is one number theory problem from algebra class I want to write a blog post on since the proof was pretty interesting.
Speaking of number theory, I’m not sure if someone said something to the professor or she planned it this way or what, but I have actually enjoyed the last 2 days of class. In the first 3 weeks of class pretty much covered the first chapter of the book and “learned” proof techniques. It was basically foundations, but at a slower pace. Then yesterday she came in and it was like I was in a different class. She started using concepts from abstract algebra and left trivial details to be verified later rather than proving everything in every little detail. I know I wasn’t the only one that was disappointed in this class so I have to wonder if something went on behind the scenes to change how she approached this class.
Along the lines of changing my opinion on a class, I have recently started to enjoy analysis a lot more. Last quarter the professor that I had was pretty much horrible and honestly I haven’t really been very impressed with the analysis professor this quarter, but she has assigned and discussed a lot more interesting things then I had in analysis last quarter. So while analysis still isn’t my favorite subject I can honestly say that I don’t dread having to do analysis homework.

Today was the first day of spring quarter. So I went to classes and whatnot, but there really wasn’t any homework that I needed to start on right away so I pretty much just goofed off in the lounge between classes.
I started the day at 8:30 in the morning with Italian. Today was pretty much a joke, we got a syllabus and talked for a few minutes about what we did for break and then left for the day.
Then I went to algebra where we started right in with definitions of various structures that we are going to be studying this quarter. It was interesting to learn, but we haven’t really covered enough to begin a serious attempt at the first homework so I’m going to wait until Wed to start working on it.
Third in line was number theory where I was hoping that my rant from last night would prove to be unjustified. Sadly enough I don’t think I went far enough in my rant. First more on the grading policies of this class…there are five tests, but only four of them count for the grade. “Class participation” won’t be precisely defined until tomorrow, but there are a total of 45 points to earn…25 of which is extra credit… I managed to keep a straight face through all of this, but I was unsuccessful a few times later.
After going over the grading scheme she then announced that the course was going to be about proving things. I was totally shocked by this since I typically try to take math classes that don’t actually involve any math. Personally I am worried that this doing math in a math class thing will catch on and then all my math classes will require math.
We did get some content today at least. She defined divisibility, GCD, and LCM and did a very basic proof involving divisibility. And after she completed that proof came the money line for the entire day today, “Does anyone remember how to do a proof by contradiction?” I foresee that line becoming a running joke in the math lounge in the days to come.
Anyway I as I was sitting in this class I couldn’t help, but think that I was sitting in the wrong class. At first I thought it was just me, but pretty much everyone in the class that I know was thinking the same thing. If there was another math elective offered this quarter or dropping the class wouldn’t screw up my aid I would drop the class in a heartbeat, but the other math electives aren’t really anything that looks even remotely interesting to me. So I guess there isn’t much to do, but hope she realizes that the way she is teaching this course is probably close to two levels below what those of us taking the course were anticipating.
Finally I finished up the day in analysis. I wasn’t really able to get much of a handle on how this course is going to play out. It was annoying that that she presented a theorem and then spent a large amount of time explaining it trough hand waving, but she didn’t even manage to start the proof. Seems strange now that she basically presented something without proof, but perhaps she will give a proof next class.

I know I should just go to bed since I have to be up crazy early tomorrow, but I just received an email from the professor of my number theory class and I can’t stop laughing about it. It was just your basic welcome to the class email that provided a link to the syllabus in addition to informing students about a last minute change of rooms. Honestly I am glad that she emailed it out tonight since I don’t think that I would have been able to control my laughter in class on first reading of the syllabus.
To begin with the syllabus is FOUR pages long and that doesn’t even include the schedule which appears to be mostly in the “to be determined category.” Seriously a four page syllabus? I’m pretty sure this is a math course for math majors and not some sort of GEC course. I may be wrong, but my guess is that every student in the class understands how a math class works. The professor I had for combinatorics last quarter and we all managed to figure it out.
I’m sure most students upon seeing that the syllabus was four pages long did exactly what I did, skipped to the part that explained how the final grade would be calculated. There were a couple of odd things I noticed in this section. The first one being there are FIVE, 1 hour exams, but there is no final. That is just stupid! Seriously schedule a 2 hour midterm in the late afternoon and have a 2 hour final and be done with it. Or failing that cut out a couple of the exams in favor of a 2 hour final. Just anything that doesn’t have me dealing with the stress of an exam every other week.
I figured that the four page syllabus and five exams were the end of the strangeness, but it turns out I was wrong. Continuing down the list of how the grade is calculated there is the total number of points for “class participation”!! After reading this I stopped and checked that I did indeed have the syllabus for number theory and not for some random lit class. I’m not certain what “class participation” even means in the context of a math class, but it seems to be to be dangerously close to basing grades on attendance and it is my understanding that that would be against departmental policy. I’m sure I could read the syllabus to find a precise definition of “class participation”, but I’m tired and the thing is four pages long.

Well winter quarter is officially over and I have a week off before heading into spring quarter. Number theory is the class that I am the most excited for, but I’ve already purchased the book for the class and I think I am going to end up being disappointed. A large amount of the book is devoted to “How to structure proofs.” Umm…I thought that was what the Foundations of Higher Mathematics course was for. The course description seems to suggest that we would learn about algebraic methods in number theory, but the book doesn’t seem to mention algebraic structures at all. Plus the optional topic in the course description that I was most interested in was elliptic curves and once again the book doesn’t mention elliptic curves at all…well there is a short section on the history of Fermat’s Last Theorem so I suppose I can’t say that there is no mention of elliptic curves. I hope I am wrong about this course and it turns out to be an interesting course, but I fear that we are going to spend nearly a month on material that was already covered in the foundations course.

I took it yesterday and it was pretty insane. Pretty much the same format as the midterm 10 questions, do 4 of the first 7 and 2 of the last 3. The first problem was a joke…pretty much a straight forward application of the principle of inclusion-exclusion. I goofed around with the second problem a little, but part B asked if 2 graphs were isomorphic or not…I was pretty certain they were, but I wasn’t able to find an isomorphism so I ended up skipping the problem.
The second problem I did was pretty easy as well. There was some parts about an equivalence relation and then a proof, given G a graph and H and K connected subgraphs of G with a non-null intersection show that the union of H and K was connected. Turns out that wasn’t that difficult of a problem.
The third problem I did wasn’t really difficult, but it was time consuming. The first 2 parts were to define what a tree was and give a general method for constructing a spanning tree of a connected graph. The third part gave 3 trees with 6 edges and asked to find all trees, up to an isomorphism, with 7 edges containing the given trees. That wasn’t hard, but then the part about showing that there were no duplicates took about a page of work to get through.
The fourth problem I did was where things started to go downhill. Part A asked whether or not the graph of knight moves on a 5×5 chessboard had a Hamiltonian cycle, this wasn’t hard since I had messed around with the 5×5 board previously and had a pretty good pretty good argument for why any Hamiltonian path of such a graph could be assumed to start at the bottom left corner of the board. Part B was again pretty easy in that it only asked if the Peterson graph had a Hamiltonian cycle, so proof by example was a pretty much all that was necessary. Part C however was to so that the graph of a D dimensional cube had a Hamiltonian cycle; I got a proof of sorts using inductive reasoning, but well it was mostly a bunch of hand waving.
The fifth problem again was a trouble spot…it was inclusion-exclusion again, but I kept messing up the counting. I left some work on there, but never got a solution of any sort. I pretty much felt like I had to work on that problem since the problem I skipped was to prove 1 of 3 theorems: Hall’s theorem, Delwar’s theorem, or inclusion-exclusion. I probably could have wrote something for a couple of these, but honestly in the in it would probably be mostly just hand-waving…plus I ran out of time.
The last problem was to just provide definitions. I think I did alright on this one, except my definition of a set family didn’t really seem like it was what he was looking for, but well I couldn’t really think of any other way to define it.

We got the results back on Monday, but well I have been busy with things so I didn’t get around to writing about them until now. I did about as well as I expected I had done on Friday getting a 98. Surprisingly the class as a whole did not do well at all. The median score on the first exam was 67 and then quite a few people dropped, yet the median score on the second exam was 63. I could tell on Monday when he was going over the exam that he was pretty upset about the scores on the exam. He kept saying, “I thought you understood.” I think it is really cool that he seems to care so much about whether or not his students succeed.
On a related note he has started hinting around that he will be the instructor for algebra II next quarter. I’m not really sure how I feel about this…I think he is a cool guy, but right now he is a group theorist teaching us group theory. I just wonder how much of his enthusiasm will be around next quarter when we are learning about rings. Oh well I guess it can’t be too difficult to teach the basics of algebra even if it is outside your area of specialty.

Today was the day of the second algebra exam. I had thought that I was pretty well prepared for the exam going in, but I got in there and kind of panicked when the exam was pretty much nothing like I expected. I was pretty certain there would be something about quotient groups, but I was dead wrong there. Most of the test was pretty much just applications of Lagrange’s theorem and Lagrange’s obit-stabilizer theorem. All in all I think the test was a little bit easier than I had expected.
One of the problems I think I didn’t get full credit on defined G as a group of order 10, H a subgroup of order 5, and K a subgroup of order 2. The part of that I probably messed up was proving the intersection of H and K was the group containing just identity. He gave a hint to use Lagrange’s theorem which I promptly ignored. I just used the fact that the order of H was prime thus H is cyclic and I wrote out the elements of H and showed there was no element of order 2. Then I used the fact that the order of K was even to show that non-identity element of K had to be of order 2. Nothing really wrong with the argument I guess, but given the hint I think he wanted some sort of proof that a group of prime order was cyclic. At least I can’t see another way to apply Lagrange to this problem.
The problem that gave me the most trouble was one about the permutation group on 4 letters. It defined 2 elements of H a subgroup of the permutation group: (124) and (13) and asked what could be the order of H. In the end the idea wasn’t that difficult, but the lack of information was kind of scary when I was taking the test. The basic trick was the H-orbit of 3 had cardinality of 4 and identity, (124), and (124)(124) all fixed 3. So the order of the 3 stabilizer of H was at least 3, thus the order of H was at least 12 and from Lagrange the order of H must either be 12 or 24 since the order of the permutation group on 4 letters is 24.

It has been a long week since last Tuesday. Starting with an analysis exam on Wednesday, then some crazy combinatorics homework assigned on Wednesday and due on Friday, and of course algebra homework that was due on Friday. Then to make things interesting we got more combinatorics homework assigned on Friday to be due today!
The analysis exam went pretty well. I got a 93 and the median score was a 78. I probably should have received a perfect score, since the only comment on the problem I missed points on was that I needed to justify a statement that followed pretty trivially from the assumption that I made, there wasn’t much space to write the solution in so things were pretty squeezed in there so he probably missed my assumption, but it isn’t really worth arguing over.
The crazy combinatorics homework was a pretty strange assignment. Some of the problems required about 1 line for a complete solution. And then there was the two problems that a lot of people didn’t bother to do. The first one wasn’t very difficult conceptually, but my proof took a whole page to write up. The second one was to find the number of k-length chains in the symmetric chain partition of the power set of a set with cardinality N. It was as simple as counting the number of chains longer than k-1 and subtracting the number of chains longer than k, but the method of counting the number of k+1 chains wasn’t intuitive to me at all.
I am still really enjoying the algebra class. It seems like that is the only class were I really feel like I am playing by a defined set of rules. A lot of times in analysis I find myself using informal arguments to try and figure out where I need to be heading in the proofs. And combinatorics a lot of times just seems like a big bunch of hand waving. At least with algebra the definitions are all set out and you just have to manipulate things according to the rules.

Today was a strange day all around. Starting off in algebra we talked about cyclic groups for a short period of time and then moved on to quotient groups all of which was really interesting, but the professor started randomly calling on people to answer questions which was strange. Then when we got to a point in the class where there was about 5 minutes left he asked the time and when told responded, “Good I’ve got 15 minutes left.” He normally lectures until the next class kicks him out, but for some reason I found this to be funny.
Then in combinatorics where I thought we were going to start the inclusion/exclusion chapter he lectured about Sperner’s Theorem and Dilworth’s Theorem which made out for a pretty cool lecture. Not test scores back yet, but he promised them for tomorrow.
Analysis wasn’t strange, but I am starting to freak out about the test on Wednesday. My main concern is that pretty much all of the proofs involving limits of sequences seem to involve some sort of trick and if you don’t get the trick you are pretty much left without much to do. The last homework I was able to show a sequence was bounded below and what the limit was if it existed, but I couldn’t show that it was bounded above and that it was monotone. I know the test problems are going to be less complex than the homework, but I also know that we only get 48 minutes on the test.

I sat down today to try and figure out what classes I want to take next quarter. I pretty much knew what I wanted to take, but I wanted to make sure that everything fit into a decent schedule and what not. Anyway since I am no longer a statistics minor I don’t have to concern myself with taking any sorts of statistics classes…well the BA requires that I take class to fulfill the data analysis requirement so I still have that class left to take.
To start with I am obviously going to be taking Algebra II. I am really excited about that class especially since I am really loving Algebra I this quarter. This quarter the homework has been really challenging, but it is always interesting. Plus the group of students in this class is really great.
Again pretty obviously I am going to take Analysis II. Bleh is about all I can say about this course. Analysis this quarter has been rough going, mostly because the material hasn’t been all that interesting. Things are starting to get a bit more interesting so maybe next quarter will turn out to be not so bad.
I am taking a third math class again next quarter against my better judgment, but I couldn’t pass up a chance to take a class on number theory so well it looks like 3 math classes next quarter. Besides number theory should be easy right? I mean how hard can a class be that just ignores non-integer numbers? ;)
It looks like I will get to take Italian 103 next quarter. Italian should be exciting especially since I have pretty much done zero Italian studying since I finished 102 in December. Supposedly 102 is the hardest Italian class so hopefully things are fairly smooth sailing next quarter.
Finally I am taking physics class. Mostly because I want to try and get the lab classes out of the way since I don’t want to be taking them next year. So I am planning on taking physics in the spring and summer and biology in the summer. The only cool thing about physics is that the meetings on Tuesdays and Thursdays are immediately after number theory in the same room. So once I get a good seat in the number theory class I don’t have to worry about getting a good seat in the physics class.

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