  # ctrls

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1. For the second topic, I would recommend avoiding the zeta function. Since the function is defined as an analytic continuation, you really need to understand and be able to explain the definition in order to investigate it. This is especially the case for showing that the value is undefined for s=1, you need to refer to the definition. You could consider the series on its own though, looking at convergence/divergence (this is covered in the calculus option), determining the value when it converges and how it varies with s, etc. Dealing with infinite sums requires a fair bit of care though, plus I'm not sure of what exactly you could investigate.
2. In general, including a proof of a known result is fine as long as you understand the proof and can explain it in your own words. There isn't a result that's commonly referred to as "Fourier's theorem" though, which particular result are you referring to? If you mean any kind of convergence result (ie: that the infinite sum of trig functions converges to the original function under certain hypotheses) then probably not, since it's a fairly advanced topic and even formulating the theorem precisely requires knowledge of additional topics. You likely won't be expected to prove anything rigorously though, it'll probably suffice to just give a intuitive explanation as for why the series gives a good approximation to the original function you are considering. On the general topic of Fourier series, you could pick a particular application and focus on that. You'll probably need to do a fair bit of reading on the general theory first, but that should be a good starting point as far as research goes - signal processing comes to mind as the main application. The general topic is fairly complicated so you may find a lot of pages saying things that go over your head, but it should be possible to find sources that explain the main ideas.
3. It sounds very simple to be honest, a derivation of the perimeter and area only takes a couple of lines. While complex numbers might have some use, there's not point in using induction either - at least it seems simpler to attack the general case directly. There are other things regarding complex numbers and geometry that though would probably work, but unfortunately I don't know of any specific topics. A quick google search gave me this page though (scroll down to 'problems' part), which may give you some ideas and things to look into.
4. Once you have found a primitive root , it can be proved that is a primitive root if and only if and are coprime (so they share no common factors). In this case we have so choosing to be an odd integer that is not divisible by 61 will work (note however that there are 480 primitive roots). Here's a proof sketch of why this holds. I'm not sure what kind of background you have with number theory, but you do need some preliminary results like BÃ©zout's identity and Fermat's little theorem to prove this (the explanation in my link does it more generally for any number, but in the case of primes , where is the Euler totient function). Note that we only need one primitive root for the algorithm, so usually we don't need to find all of them A bit of googling suggests that the value used for the primitive root doesn't affect the security of the system, so 3 would work fine in your example (though larger ones can be slightly more efficient to compute, plus there are some other complications that exist).
5. The simplest way is to check every possibility. If we wish to find the primitive root of a prime , then for each we check if is a primitive root of , by computing all of its powers. Note however that there is a slight complication here, which is that we haven't proved that primitive roots actually exist, but it turns out that they do for prime numbers greater than 2. Practically speaking however this isn't very fast and there actually isn't a general known "fast" method to solve this problem. We can however make some improvements, namely when checking if it is a primitive root or not. This can be done by finding all of the prime factors of and checking if for each . If not such prime factor exists, then it is a primitive root. On a side-note, the main bottleneck of this method is actually finding the factors of . One these are found we can quickly check each using techniques such as repeated exponentiation. In addition, it is usually the case that only a few values need to be checked until a primitive root is found. While proving these claims (namely the existence of primitive roots and that the improve method works) isn't too difficult, you need some groundwork in number theory to do so. If you want to cover this in more detail, I'd recommend doing some more reading about the subject, though since it's a reasonably minor part of the entire method, I don't think you'll be expected to do all that.
6. I haven't studied Markov chains at all, so I don't really know what kind of calculations and working you'll have to do. That said, it should be possible to change around the topic a bit to be feasible, since the main issue seems to be having too many variables. In particular, two things come to mind: Firstly, you could alter the game to make it simpler and reduce the number of variables. You don't really loose much for example without having all 40 states, rather this could be reduced to something a lot smaller - you could later consider how larger boards would change your results. Admittedly it may seem a bit artificial with only 5 or so states for example, but at the very least it may be a good starting to point to get a feel for what's going on. Secondly, there are resources that can help with the raw computation. I don't know what exactly you'd need to calculate, but there are online calculators which can compute things like determinants, inverses, eigenvalues, solve systems of equations, etc. Even with the above simplifications you'll probably still have to work with matrices that are fairly large, if you were to do them by hand. A quick google search for example gave me this, which can do up to 32x32 matrices.
7. You can use ideas from others yes, you can't be expected to come up with a completely new argument after all. That said, a common mistake with maths EE's is that people will end up writing up a proof that they don't properly understanding, which examiners will notice. If you are using someone else's ideas, make sure that you clearly understand what the person has written and then explain it using your own words. I would still say that the whole proof of the 4-colour theorem really isn't feasible for an EE though. Difficulty aside, there's quite a bit of preliminary results you'll likely need to cover, starting from definitions and basic properties. I don't believe just having the proof alone is really sufficient either, as kfernando mentioned you will be marked strictly against the criteria. Overall though, as preliminary research it's probably a good start to try to read and understand the proof. You can then decide what you want to write about and deviate from the topic if necessary. Before that though, you may also want to look up the proof of the 5-colour theorem first, which is may help a bit with getting to grips with the topic (assuming you've already looked into basic graph theory, etc).
8. The general topic of graph colouring may work, though I'm not sure if the 4-colour theorem in particular. Since the proof is based around reducing it to a finite number of cases and checking the result, it's not really feasible to fit it all in. You could of course look at related things in that field though. I'm just reading off topics from wikipedia, but things like weaker versions of the theorem, special cases, colouring algorithms and other related questions. It also happens to have quite a few applications which you could look into. I don't know anything about combinatorics and graph theory though, so I can't really comment on suitable topics and whatnot. You would need to do some reading around the topics, namely some graph theory, though I don't presume you'll need much.
9. Just throwing around another possibility, but how about learning Haskell? "Learn You a Haskell for Great Good" by Miran LipovaÄa is a nice introduction, it's also written in a way that you don't need any prior knowledge. That said, some knowledge about set theory and the definition of a function may be helpful to better appreciate the ideas that are presented. There's also a free online version of the book, along with a few pdf's floating around on the internet (with and without illustrations).
10. In general taking Computer Science isn't a requirement for studying it at uni, since not all schools offer it. Most uni courses do indeed start by assuming very little, so it will not be a problem if you don't take it. That said, if you plan on studying it at uni you'll probably be interested in the course material and so forth. It may also give you a taste of what the subject is like, along with not having to take another subject that you aren't as interested in. If had the possibility of taking it when I did the IB, I probably would have. It is worth mentioning however, that computer science is a very broad subject, which can cover a lot of different areas. Skimming through the syllabus it seems to be a balance of different topics, though there is a focus on the practical side. You may want to look at the syllabus in detail yourself to see if you are interested in the material.
11. Yes, it is the element itself. More precisely, the order of an element is defined as the smallest positive integer such that , where is the identity element. Hence an element is order 1 if (so if it is the identity).
12. I'm also getting that is the only real solution, so the book seems to be wrong. This result is also is intuitively true, since grows faster than its inverse for , hence it should never be that case past that point that the two integrals are equal. I'm not really sure what the book wants you to do either, I don't see the reason to have two equations for in terms of . Here's my solution,
13. Hint: Use the squeeze theorem, noting that for every . Recall also that a function is continuous at if Edit: The second question (6) is also correct in that there are no printing errors. For every rational number you can find a irrational arbitrarily close to it, so you essentially get a function that jumps between y=x and y=0 rapidly so it is not continuous. The exception however is at x=0, where you are always "close" to 0.
14. Well, what else would you have to write about? An EE in Maths is supposed to be technical and a large portion of what you are assessed on is on how well you have understood topic you are investigating. If you are writing about the zeta function and it's relation to primes, then you should explain how they are related. To be honest, I don't think this is a good topic. I did a bit of googling about the topic, but it seems like you really need a solid understanding of topics like complex analysis to really understand the link, which I don't think is feasible at this stage. You would likely be better off changing the topic to something simpler (I suppose you could prove some elementary properties of the zeta function, for example) - I know you are running a bit late now, but I think it would be for the better. Also, it might be helpful to read the EE guide and try to understand what exactly is expected + what the criteria is that you are assessed on - it might clarify a couple of things you seem to be unsure about. Talk to your supervisor about it too, see what he/she thinks and if you are thinking of changing topic, hopefully he/she might have some ideas of related topics you could look into.
15. I'm guessing you mean, Simplification doesn't really work here, since you really can't simplify this much further. In general it's a first step to clear up an expression, but that alone won't really give any answers. Past that there's no real algorithmic way to solve these problems, other than to determine the nature of the function and work from there. To find the domain, consider each term separately. The function ex is well defined for all values of x, so that's fine. The fractional term will be undefined if the denominator equals 0, but this doesn't occur for any real value of x in this case. Subtracting two doesn't affect the domain either. If the argument of the square root is negative however, this will be undefined if we are considering only real numbers - which is usually the case in the IB, unless otherwise stated. Hence if 2x+3<0, then the function won't be defined. So the domain of the function will be xâ‰¥-3/2. Finding the range is a bit harder. I may be missing something, but I can't think of a simple way to determine the nature of the curve. The one way which comes to mind is the method I described previously, which would work as follows, You can prove that the function has no stationary points, in other words show there are no solutions to the equation f'(x)=0 in the domain xâ‰¥-3/2. If this is true, you can then consider the points f(-3/2) and f(x) as x tends to infinity, which gives the range (the latter diverges to infinity, due to the e^x term, so the range is f(-3/2)â‰¥0. If this isn't clear, try sketching a curve with no stationary points starting at x=-3/2, considering the asymptotic behaviour as x tends to infinity. Problem is that showing this function has no stationary points isn't particularly easy, you get a pretty ugly and complication equation which requires a fair bit of work to show has no roots. Other than that however, I'm out of ideas.
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