Algebra Exam Round II

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.

Long Week

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.

Weird day

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.

Looking ahead to next quarter

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.

Combinatorics

So I stressed myself out a little too much preparing for the combinatorics exam. Mostly because well I didn’t really feel like I had a grasp on what was going to be on the exam. In the end I had completed the two hour exam in about 45 min and I am pretty sure I got a perfect score or close to it.

The format of the exam was kind of strange there were 5 problems worth 40 points of which we had to chose 2 and 5 problems worth 30 points of which we had to chose 4. I can’t remember what all the first 5 problems were. There was one to prove the binomial theorem which I almost did, one dealing with one-to-one correspondences, something strange that I can’t seem to recall, and then the two that I did.

The first one I did was to show that the Ramsey number R(3,3)=6. I was honestly surprised that this was on there. I knew there was going to be a problem dealing with Ramsey numbers, but I expected something more difficult. Truthfully I expected that a problem to show that R(3,3,3)=17.

Like I said I was going to do the prove the binomial theorem problem, but I read the 5^th problem and it seemed like it would be easier to do without making a mistake so I went with it. Problem 5 defined relation to be transitive and reflexive and another relation (Q, ~) such that R~T iff  and and we were asked to show that (Q, ~) was an equivalence relation. Like I said this test seemed really easy.

The other 5 problems I was just going to do the first 4, but one had some sum of binomial coefficients and asked for a binomial coefficient that it was equal to and I thought I might mess up the algebra on that one so I skipped it. The 4 problems I did were a direct application of the pigeon-hole principle, a direct application of the binomial theorem, a counting problem related to poker hands, and provide the definition of reflexive, symmetric, antisymmetric, transitive, partial order, and equivalence relation.

The exam was yesterday afternoon so I didn’t expect them to graded by this morning and they weren’t, but we did discuss what we were going to do for the next four weeks. He wasn’t sure if we wanted to continue on the counting stuff or work on some of the other areas of combinatorics. We finally settled on one week on inclusion/exclusion, one week on designs, and two weeks on graph theory.

 I’m glad we are going to get some graph theory since that is mainly the reason I took the class and well the professor is a graph theorist so it should be interesting. I’m not quite sure exactly what is meant by designs, but apparently that is the primary area of interest of Brualdi who authored the text so I guess it makes sense to spend some time there. It should be an interesting four weeks in combinatorics.

Stressed out

So far this week I have had homework due in analysis and an exam in algebra and anthropology. Plus on the way to class on Tuesday one of the tires fell off of my car so I have had to deal with that in addition to having homework due tomorrow in both combinatorics and algebra. Needless to say I am a little bit stressed out and slightly behind on my sleep, but at least I got an A on the algebra exam! Hopefully things will calm down a little bit after the combinatorics exam next Thursday.
I am really nervous about that exam since a lot of times on the homework I feel the exercises are really worded very well and so it takes a lot of time just to figure out exactly what the exercise is asking for. Another thing is he is giving us two hours to take the exam which makes me nervous about the number of questions that will be on the exam. I suppose I should just stop stressing myself out over it.

Algebra Exam

Yesterday was the first algebra exam and it was rather surprising how easy the exam was. There was a problem about an equivalence relations, a problem to show that a given operation and a given set defined a group, a problem to show that a given function mapping a group to itself defined an isomorphism, and finally a couple of computational problems involving isometries of .

The problem to find the reflection about a given line kind of pain in the rear since I had only bothered to memorize the formula for reflection about a line with a given angle and the line was given by its slope on the test. And of course the slope wasn’t anything that I could mentally take the arctangent of, so I had to fudge around with trig identities to find the formula to find the reflection using the slope. I did remember that the matrix was multiplied by so at least I had an idea of what to do to find the formula. I memorized the angle formula rather than the slope since it was easier to remember, but I suppose it is probably easier to derive the angle formula from the slope formula than the slope form from the angle formula.

I had an issue showing that the defined function in the isomorphism problem was surjective and I left the exam not really certain that I had a solid proof, but today in the homework session he was talking a little about that problem and mentioned something about x being given as a fixed element of G. I must have totally misread the problem since I couldn’t tell if x was in G or not, but my hand waving about the function being invertible is actually a pretty solid proof if x is an element of G.

The only other problem I had stemmed directly from my inability to do one problem at time on a math test. I started the equivalence relation problem working with integers, went to the show this is a group problem working with reals not -1, went back to the equivalence relation problem, and then when I went back to the group problem I got that and I freaked out because that wasn’t an integer. So I flipped back to make sure that I was supposed to be showing that this was a group and realized that I was back to working with real numbers and so the inverse need not be an integer.