More math

Math. Math. Math. Sleep. Math. Shower. Math. Math. Cook. Eat. Math. Repeat. 

This is how my weekdays have been looking lately. And, surprisingly, I can't say that I've hated it thus far. It's actually been pretty enjoyable. I've been doing really well on my homework assignments. I've been able to keep up with everything yet still manage to free up my weekends enough to enjoy my time here in Budapest. It's weird how much different things are here versus at Lawrence. For example, by this time at Lawrence, I would be prepping for some of my midterms. Here, I don't think that I have a midterm for another two-three weeks. It's really been throwing me off, but in a good way. I'm so prepared for all hell to be breaking loose in terms of workload any day now, but that has yet to happen... and I'm gonna take advantage of this free time while I can. 

The grading systems here are so different. For many of my classes an A is an 80% or higher. For other's it's 85% or higher. I have no classes with the american grading scale of 90%. So, for some of my classes, it is completely possible for me to not take the final exam at all and still fairly easily pass the class. In comparison to Lawrence, this is heaven. At Lawrence, getting an A in any class is a major accomplishment. A's just don't happen, unless you're crazy dedicated to school and have no social life or, like some of my friends, are dependent on drugs like Adderall. Here, for probably the first time since graduating from high school, getting straight A's is actually a possibility. Now, I don't know how much I'm jumping the gun by saving all this, and I sure hope that I'm not jinxing myself, but so far, things seem to all be going pretty smoothly.

Now, since one of the main reasons I'm writing this blog is part of my service project for the Gilman scholarship, I am expected to post a problem every week (or so) and work it out for you all. I know that some of this math may be a little bit above most of my readers, so I will try to break it down as best as I can.  Since last time I did a problem from my MPS (mathematical problem solving) class, I would like to post one today from my set theory class. The problem is as follows:

Prove that the set of English sentences is countable. 

Recap: Sets are groups of elements. Sets are defined by their elements, so for example a set could be something like {1,2,3,4,5}, {dog, cat, fish, bird, flower}, the set of the real numbers (all numbers that are not imaginary, so 5/6, .45379523, 1, 0, -.4238972, infinity, etc), etc. These sets are not required to hold any rules besides the ones described. So I could say "Let S={x element of N: x>0}" and that would translate as let S be the set of all natural numbers x, such that x is greater than (not equal to) 0". 

Definition: Countable sets are sets that can be enumerated. Or in other words, there must exist some function that will map the set to the natural numbers (1,2,3,4,...). So the rational numbers are countable, but the reals are not. (I'll not get into more on this). 

Proof: (here is my exact answer with my commentaries to you all written in bold) Notice: All English sentences= the union of all n letter word long sentences (with n ranging the natural numbers [since you cannot have words or zero or negative length]). N word long sentences are countable by the following logic: 1 word long sentences are countable because each letter has only 26 options (less if you consider proper grammar) and every proper word is finite. Thus if you order the sentences alphabetically, you'll see that the set of one word sentences is countable. By the same logic, if you continue to order sentences alphabetically throughout each word, you'll find that 2 word, 3 word, ... n word long sentences are countable. 

You can then order the sentences in the following manner: 1 word (alphabetically), 2 word (alphabetically), ... n word (alphabetically). With proper grammar, sentences usually have a finite number of words. 

Thus we note that the union of countable sets is countable (even when n is infinite) (we're unioning [taking all elements of said sets and adding them to a new set, deleting extra repeats] the sets of sentences of different word lengths [which are countable] a finite number of times. Thus, we'll get a countable set [just order the sets based on their word number per sentence, and alphabetize them. you know have an enumeration]) and therefore the set of all English sentences is countable. 

This was a fun problem, but also a fairly basic question. It is very hard for me to break down more complex questions for you all without having to do an extreme amount of background explanation. I will put up another more cultural post (probably on my recent excursion to the chocolate/ sweets festival or on my experiences cooking here). 

The food, the clubs, and the environment

 I promised a more cultural based update so here it is. Since being in Budapest I have noticed very many things that are different than in the states. Some of them are really great... others not so much. Here they are:

The clubs are amazing. The drinking age is only 18 and the bar culture is definitely a part of life here. There is even a law that says that alcohol shops cannot sell alcohol after 10pm. This is done in an effort to help preserve the bar culture. The most interesting bar topic is ruin bars. They are amazing. I've been to three thus far. One is a maze of a building where each room has a different bar and a different type of atmosphere. There are always really interesting decorations (think hipster or Alice in Wonderland). Here's one of the bars that I've been to. It was called the mouth bar and was decorated with mouths all over the places.

Some of the decorations that would be expected in a bar here.

 Another used to be a house with a court yard but they opened up the entrance to the court yard so part of the bar is inside and part is outside. This again has multiple rooms with multiple bars (not so much dance floors). This is a much more relaxed bar; it also has a hookah bar in one of the rooms. The other clubs that I have been to have also all been very interesting with multiple dance floors of different music types. I have yet to have any negative experiences at these bars besides one of my watches getting stolen (right off of my wrist!). The amount of theft in Budapest is pretty extreme, but I've been pretty lucky thus far. Otherwise, everyone has been very respectful and helpful (this of course does not persuade me to ever drop my guard). I've really enjoyed being able to go out and dance so often. Since participating in my university's Cabaret performance in May, I have realized that I really do love to dance and that I would like to start taking lessons when I can.

Goulash and potatoes, my first Hungarian meal

Risotto with goulash. This was really good.

The food here is really good. It is much cheaper, if you are okay with not eating meat that is. Fast food consists of Turkish food such as gyros and fruit stands; it's much healthier but I have to admit that I do miss Taco Bell and the many to-go Mexican restaurants. There are so many places to go and a lot of the food tastes homemade. I've been doing really well with eating gluten free. Many stores store gluten free products such as pastas and crackers/cookies. Risotto has been a very common food for me, but I've also had my taste of goulash, a lot of Turkish food, some Mexican (I really had to hunt for that restaurant), and Italian (I even found a gluten-free pizza place). There is definitely a much more limited amount of options here because there is a much more narrow palate that has to be provided for than there is in the US. I have yet to find an Asian restaurant. There is one little shop by my school that I have gone to a couple times that has really good food. The people are getting to know me and are keeping better track of what is and isn't gluten free. Upon eating so much Hungarian food, I can really start to feel for my Hungarian/ German roots. I've had it occur multiple times where I've been eating something and thought to myself “Man, this tastes like something my mom or dad would make”. The alcohol is also very good, and I think that my new favorite alcohol is probably Palinka, a fruit based alcohol that Hungary is known for. The ice cream/gelato is also really good, but since I had a small banana allergy the last time that ate it and I have not since eaten any. I think that my favorite meal thus far was my lunch today (my favorite keeps changing daily). It was from the little shop by my school <3 and was a dish with chicken and potatoes (kinda like a stew) over rice. It was really good.

Things are much more natural and efficient for the environment here. Many stores charge for plastic bags, and because of this most people reuse bags or buy reusable shopping bags. Most people use public transit instead of driving and there are always a lot of people riding bikes and motorcycles. I'm not sure if this is really a green movement or if that is simply how things are done here because of it being a city. However, in exchange for the fresher air in this respect one gets the problem of smokers. They are everywhere and there are many of them. This has really made me appreciate the US and its laws against smoking in public buildings/areas.

That's all for now. I will put up another post shortly. I'm sorry about the large time gap between postings, but I am here to study and experience that culture here, and as such I have been very busy doing that. I have a limited data amount per month, and posting on here takes up much more data than on Facebook, so if you would like to see more pictures, feel free to add me as a friend on there. 

My class decisions (and an introduction to each)

I believe that I have finally settled on which classes I would like to take during my time in Budapest (which actually is pronounced with a 't' sound I have found- in conflict with the online program I was studying Hungarian from). I have been taking classes for the past week without knowing for sure which class I would like to take. You may wonder how this is possible. Well, my program has an introductory three weeks where you can take any three courses you would like in order to better decide which classes you would like to take, as well as to learn more about the different topics in the field of mathematics. I took many, many classes this week, but I found that many were above my level or covered material that I really wasn't interested in studying. I finally decided on four classes: Set theory, Graph theory, Mathematical Problem Solving, and Probability theory. To many of you, this may not mean much, so I'll introduce you to each subject.

Set theory is the study of sets or organizations of elements into distinct units. For example I could organize the numbers 1-10 in one set, 11-20 into another, 21-30 into another and so one. Most of the sets I work with will consist of arbitrary points, and I will work to make conclusions about the properties of these sets. One introductory topic that many people learn about at some point in their lower level education in intersections and unions. An intersection of two sets is the set of elements that is present in both sets (so the sets {1,2,3,4,5} and {2,4,6} would have an intersection set of {2,4}). The union of two sets is the set of elements that is present in either sets (so the union of the two sets listed above would be {1,2,3,4,5,6}). There is no repetition allowed in sets. The empty set is the set consisting of no element. This is just a basic introduction, but throughout this blog I will go into more details on multiple topics in set theory. This is probably my favorite mathematical subject, and I am very excited for this class.

Graph theory is the study of graphs (but not the graphs you're probably thinking of). A graph is a diagram consisting of vertices and edges (or possible no edges). The easiest way that I can think to describe graphs of this type to you without having a picture is to describe how I will apply it to anthropology as well as how I have already applied it. Image a group of people. Each person knows a particular amount of other people in the group. Put a dot on a piece of paper for each person, and then draws lines connecting who knows who. The image you would get is the type of graph that I will be studying. There are multiple types of graphs and many of them are not broken off into particular categories. One category, however, is called a tree. A tree is a graph in which there are no cycles (connections between vertices that form a path that will at some point return to the original starting point). This course will probably be very challenging, but I expect it to be enjoyable and very useful for my future work in anthropology.

Probability theory is much less interesting, in my opinion. It involves ...well probabilities. What is the probability of the king's single sibling being a male? (surprisingly, it is not 50%, but I will get to that in a later post) for example is a common problem type. For the first class, we began with some combinatorics and worked with questions like “how many ways can you pick 3 girls and 5 boys from a group of 5 girls and 8 boys?” and “How many possible ways can you choose twelve cakes from four different types?”. I do not expect this course to be nearly as challenging at graph and set, but I also do not expect it to be as rewarding.

Mathematical Problem Solving, or MPS from this point on, is a class where we as a class work through multiple problems from many different subjects of mathematics. We do this using very interesting methods that simplify the problem in ways that may not be originally clear. We will work on problems primarily from four different divisions: number theory, algebra, geometry, and combinatorics. The problems from this class remind me a lot of the problem types that I would work on during math team in high school, and I have really enjoyed this class so far. Because of how much I have enjoyed it, I would like to work through a problem with you. The problem is as follows:

Let N=2^(20)x3^(15)x5^(10). 1) Find the number of divisors d s.t. (such that) d divides N. 2) Find the number of divisors s.t. d divides N and 10 divides d. 3) find the number of divisors s.t. D divides N and d is a perfect square (the square root of d is an integer (…,-2,-1,0,1,2...)).

1. Since d divides N, we know that d must be of the form 2^x x 3^y x 5^z. Because of the exponents of N, we know that x cannot be larger than 20, y cannot be larger than 15, and z cannot be larger than 15. Thus 0≤x≤20, 0≤y≤15, 0≤z≤10. Thus there are 21 options for x, 16 for y, and 11 for z. Thus, we can conclude that there are 21x16x11 possible values for d.
2. We can use the information give from number one to help solve number two. We still know that d is of the format 2^x x 3^y x 5^z, but now we know more about d. For d to be divisible by 10, we need to have at lease one 2 and one 5. Thus x and z must both be bigger than 1. So we now only have 20 options to x, 16 for y, and 10 for z. This makes the total possible numbers for d equal to 20x16x10
3. This part was one of my favorites. Once again the rule for d applies that it must be of the form 2^x x 3^y x 5^z and the range for each variable is the same as in part 1 ( 0≤x≤20, 0≤y≤15, 0≤z≤10). Now, though, we have the challenge of figuring out how to have d be a perfect square. If you think back to reducing square roots, you may recall that if you have a root with a number who has more than two of the same factors that this factor can be pulled out of the root once (i.e. say we have (24)^.5 [something the the one half power is the same as it being square rooted. This is simply easier to type]. 24 can be factored into 2x4x3 with is then 2³x3. This means that we can pull out two of these 2's and have the value 2x(2x3)^.5). This implies that for us to get a perfect square that we need to have the values for x,y, and z all be even (so that we can then pull out all of the doubles under the square root sign and get an integer). Thus our options for x are 0,2,4,6,8,10,12,14,16,18, and 20, for y are 0,2,4,6,8,10,12,14, and for z are 0,2,4,6,8,10. This counts up to 11 options for x, 8 options for y and 6 options for z. This implies that there are 11x8x6 options for d.

I hope that you enjoyed this problem as much as I did. I will continue to post questions on here that I found to be enjoyable, and that are fairly easy to explain to those of you with a limited background (so, sadly I will probably not be posting very many problems from my graph theory course).

Notice: From now on I will try to post two blogs a week. One on my mathematical studies here in Budapest and one on my life outside of the classroom. 

Pictures!

So, I figured that I would post some pictures for you guys on here. If you would like to see more, don't hesitate to add me on Facebook

 So this is one of my roommates: Sheri

 This is one half of my room with my desk area. To the right is a wardrobe

 This is the rest of my room

 This is the lovely kitchen with a lot of counter space!

 The fancy looking bathroom. I would really like it if it weren't for the shower only having hot and cold water. Medium heat does not exist...

Most of the living area

The view out of my window

Into the twenties!

So September 11th was my birthday, and I wanted to describe how well my day went. I had been out the night before my birthday with some of my new friends for their birthdays. It was a lot of fun. I went to multiple bars, met a lot of really cool people, and even had some friends buy me an amazing birthday drink come midnight. It was this great cocktail with black cherry juice, peach juice, lete juice (some nut), rum, and lime. It was probably the best drink I have ever had, and I'm so grateful to my friend for making it. Here are some pictures for you.

Anyways, on to my actually birthday. The day got off to a great start because I didn't have a 8am class and got to sleep in (which I really appreciated after the late night). I got ready and went to class. My first class of the day was Graph Theory which I liked (and intend to stay in). I went to MPS (Mathematical Problem Solving) even though I didn't intend to take the class, but now I LOVE it. It was sooo much like work I'd do in high school math team and the problems were really exciting. The prof's grading system is great and I love his enthusiasm (it's refreshing but not over powering). I then went to a class called Holocaust and Memory which was about Hungary's effects on WWII and how people remember/ recorded the Holocaust. It seemed pretty interesting especially considering the effect that the holocaust had on my grandmother's extended family (it appears that most were killed in a labor camp/ in the war); however, the grading seemed too lax, and I don't think that I really want to take a qualitative class this term. I think I want to keep my mind focused primarily on math, so I will probably skip taking that one. I then went home (and got lost in the train station on the way. lol), and went grocery shopping. Then me and Sheri (my awesome roommate) went out for dinner at this cute Italian place by the apartment and I had steak and mushroom risotto and then pana cotta for desert (she was really nice and paid for it). It was all really good and I will definitely need to try pana cotta again. Oh! Plus Sheri gave me a bunch of bachata music (which is nice because I don't have the internet data to listen to Pandora). We then walked around some looking for something interesting/ trying to get to know the area better. I came back did some homework, and watched a Naruto episode (that ate up like all my data), and headed to bed. Overall it was a pretty great day, and I'm really thankful to everyone for making it so amazing.

So, I know that this was a rather boring post, but it's really been some of the most interesting things that have happened to me lately. I'm gonna try to make a couple posts soon with most details about my classes and about the differences in culture here.

First 24.

So I have official completed my first 24 hours in Budapest, and I must admit that they have been pretty intense. I intend to go through them with you, but first let's back up a little. Saying good bye to my friends, family, and boyfriend from back home was hard, very hard. It's weird to think about how long I'm going to be gone for... then I realize that in comparison to how long many other people (military, long distance, etc) have to be away from their loved for much longer that I have it pretty easy. Come the illogical moments when I'm taken over with emotion, this doesn't really help much, but during my calmer moments it makes me remember that I am strong and that I can get through being away from everyone for so long.

The flight was pretty uneventful except for the cancellation of my first planned flight causing me to leave a day later than expected. This actually worked out to my advantage though in that it gave me a first class ticket for my trip from Chicago to Munich, and I must admit, first class rocks. First class has such better food, seats, assistance from attendants, movie options, etc. It was a completely different experience on a plane than I've ever had before. However, even with the fancy seats and better food, I was only able to get a little over two hours of sleep on the flight. The flight from Munich to Budapest was short and unexciting.

Upon arriving in Budapest, I had expected to find someone waiting for me (with the cute little sign saying Chelsea Gans or BSM), but I did not. I waited for twenty minutes and then went to find a phone. After finding one, I realized that I couldn't figure out how to used it and decided to go back to the waiting area. There I finally found my ride, named Lorinc (pronounced Lawrence) and left the airport with him. I was so exhausted that my memory of the ride to my apartment and most of the next few hours is pretty hazy. What I do remember is getting to the apartment, unpacking a little, briefly meeting one of my roommates (Caroline) before she rushed off to London for the weekend, finding that my internet doesn't work, and then later meeting my second roommate (Sheri), and getting a brief tour around the block nearest my apartment. Within four blocks of my hours in any direction is pretty much everything I could ever need for survival. I have a grocery store, a mall, a currency exchange, many shops and restaurants, the metro station, etc. My school, on the other hand is a 20-30 train ride away.

After getting back to the apartment, settling in a little, and talking with my roommate some more, Sheri and I decided to go out grocery shopping and later out for dinner. I order goulash (spelling?) and potatoes. She ordered goulash with pasta. It was a huge portion for just about five US dollars (the same food in the US would have cost about 12). It was very good. By this time it was around seven o'clock. I went home, took a shower, and then attempted to fall asleep.

Sleep last night was very difficult. About twenty minutes after I laid down in bed everything hit me. I started getting dizzy and nauseous, my stomach started hurting really badly, how badly I missed Dante (my boyfriend) and my friends and family hit me, my mind kept reminding me how tired I was but my body insisted that I needed to stay away as if I were in danger. Because my internet wasn't working, I went and asked Sheri if I could use hers and she was very kind and let me. I sent both my mom and Dante a very stressed out email and then looked around on facebook and tried to skype Dante. Just being on the internet helped me feel connected back to the rest of the world and helped me calm down a lot. Within an hour or me sitting on the computer looking at picture with Sheri of Dante and her previous dance partners, I was feeling ready to go back to bed. Within thirty minutes of being in bed, I was out for the night... and a little bit more (I slept about 16 hours).

Sheri's gone now (to Prague to visit friends) and I'm home alone. I cooked myself a breakfast, got my internet set up and went to the store. I made a complete fool of myself in the store by spilling half my bottle of pop into my basket and all over the floor. Since I speak like no Hungarian, all I could say was sorry and “kosonom” or thank you in Hungarian. I definitely got a taste of the attitude that people give immigrants in the US. I kept feeling as if I was doing everything wrong and that I was annoying everyone. I have to head back out though to do more shopping, so I guess that that is just something that I'll have to get used to...

In retrospect, my first 24 here have been intense (and I was only really awake for 8 of them). It's been exciting, fun, and pretty enjoyable for most of the time, but the times of loneliness, anxiety, and sickness have left me feeling a little defeated right now. I'm gonna keep going though. Backing out is not an option, and I'm sure that after I've been here for a couple weeks that everything will fall into place and that I'll feel a lot better.