Uncategorized – Page 14

IMU Breakout Graduate Fellowship Program – Apply now!

The IMU (International Mathematical Union) has recently launched the novel IMU Breakout Graduate Fellowship Program.

Thanks to a generous donation by the winners of the Breakthrough Prizes in Mathematics – Ian Agol, Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao and Richard Taylor – IMU with the assistance of FIMU (Friends of the IMU) and TWAS (The World Academy of Sciences) has launched a fellowship program to support postgraduate studies in a developing country, leading to a PhD degree in the mathematical sciences. The IMU Breakout Graduate Fellowships will offer a limited number of grants for excellent students from developing countries. The program will be administered by CDC (Commission for Developing Countries), a commission of IMU.

Professional mathematicians are invited to nominate highly motivated and mathematically talented students from developing countries who plan to complete a doctoral degree in a developing country, including their own home country. Nominees must have a consistently good academic record from the high school level and must be seriously interested in pursuing a career of research and teaching in mathematics.

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By Andreas Daniel Matt| April 27th, 2016|Uncategorized|0 Comments

Logical implications and the structure of if and only if statements

We had a homework assignment a couple of weeks back. It was looking at mathematics in a very different way from how many had seen it before, and it caused a lot of confusion. I would like to try and add some clarity to what we were doing. My thought was, rather than going through the questions themselves, I would like to add annotations to the proof itself. Let’s see how this works. The proof that you were given is in black, the annotations are in blue, and after I’ve been through the proof, I will expand on it in a simplified form.

Theorem: The function f is differentiable at x=a if and only if there is a constant m and a function E of x, defined for all $x \not = a$ , such that

$f(x)=f(a)+m(x-a)+E(x)(x-a)$ for all $x \not = a$ – (eq 1)

and,

$\lim\limits_{x\to a}E(x)=0$ .

(If both these conditions are satisfied, then $f'(a)=m$ .)

What we are doing here is giving another definition of differentiability (at a particular point, a).…

By Jonathan Shock| April 22nd, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

Computational Complexity: Article 4

Equations Speak Louder Than Words

We have thus far created a strong link between familiar intuition and formal mathematics, with the intent of constructing a framework with which to better analyse and understand the complexity of computations. We continue on this trajectory by classifying decision problems according to the resources they consume on deterministic (for the same input, will always produce the same output on different runs) and non-deterministic (for the same input, can produce different outputs on different runs) Turing machines. Our resource of consideration will be time, T(n), called time complexity, where n is input length. A similar analysis can be done for space or memory.

Definition 4.1 Let T(n): N → N (T(n)‘s domain and codomain is the set of Naturals) be a proper time function. Then DTIME(T(n)) is the time complexity class containing languages that can be recognized by deterministic Turing machines (DTM’s) that halt on all inputs in time T(n), where n is the length of an input.…

By Tapiwa Chadenga| April 16th, 2016|Uncategorized|0 Comments

Greetings from Bibliotheca Alexandrina

I just received the info below on Bibliotheca Alexandrina and its new African Networks and I thought it makes sense to share it on mathemafrica. I really like the idea of historical places connecting to new technologies. Here is a link to a film (in French) about the historical library at Alexandria:

Greetings from the Bibliotheca Alexandrina (BA) in Egypt. We are pleased to announce the launch of “BA African Networks”. Below you will find a description of the networks. You are most welcome to explore our portal and our five networks through the following link. http://afn.bibalex.org/GeneralPortal.aspx

The Bibliotheca Alexandrina (BA) follows in the footsteps of the Ancient Library of Alexandria as a meeting point for cultures and civilizations. It aims to rise to the digital challenge in order to develop African innovation through the use of science and technology in networks that extend throughout Africa. The new goal set by the BA is to connect those with common interests and expertise to hasten the development of knowledge and enable immediate sharing of knowledge and contributions.

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By Andreas Daniel Matt| April 7th, 2016|English, News, Uncategorized|0 Comments

Computational Complexity: Article 3

Rise of the Machines II

In my last entry, we introduced a formal definition of a Turing machine (TM). In this article we will look closer at this mental device and see how it works. To begin with, we can examine what a physical TM might look like. I have included a picture from Wikipedia.

Turing Machine Illustration, Wikipedia, (Drawing after Minsky, 1967, p.121)

A TM is made of tape of infinite length, consisting of chain of cells. In each cell there is a symbol which the machine can read or write over, one cell at a time, using the machine’s head. At any given time, the machine is in one of a finite number of states stored on it. It can do basic operations; move right one cell, move left one cell, read, print, erase and change states. By changing from one state to another, the machine can remember previously attained states.…

By Tapiwa Chadenga| April 6th, 2016|Uncategorized|0 Comments

Continuity – (Part One).

Definition:

A function $f(x)$ is continuous at a given point x = a if those three conditions below are met “simultaneously”:

(i) $f(a)$ is defined. (i.e; a is in the domain of $f$ )

(ii) $\lim\limits_{x \to a} f(x)$ exists.

(iii) $\lim\limits_{x \to a} f(x) = f(a).$

NOTE:

If any one of the three conditions is false, then $f$ is discontinuous at a, or it has a discontinuity at a.

Let’s now look at the different cases where $f(x)$ may not be continuous at x = a.

(i) $f(a)$ is defined but $\lim\limits_{x \to a} f(x)$ does not exist.

At a = 0, the function is not continuous despite $f(a)$ is defined (Here, $f(a)$ is equal to -1). This is because the two one-sided limits are not equal and as a consequence, the limit does not exist. This is called a jump discontinuity.

(ii) $\lim\limits_{x \to a} f(x)$ exists, but $f(a)$ is not defined.

Assume a is the x-value where there is a hole in the graph. We can see that the limit from the right of a and the limit from the left of a are equal.…

By Azhar Rohiman| March 31st, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|5 Comments

Orthogonality and Volatility

In quantitive finance, there are always many thousands of simultaneous bets available. You can be short oil, long Apple and short the Yen all at the same time. If you have a good model of the assets, then that model will tell you that some of these bets have positive expected value.

Being faced with thousands of positive expected value bets sounds like Christmas but today it’s Easter. Positive expect value bets are only necessarily good when they are independent of the other bets you’re holding. This is because there’s a lot of math on your side if you can get your bets to be independent. In particular, the shape of the cumulative distribution function of the Binomial distribution shows how hard it is to lose when you place lots of independent bets. And also, the Kelly Criterion will tell you how much to risk on your independent bets.

Statistical or machine learning models will tend to produce portfolios with some bias.…

By Richard Craib| March 28th, 2016|Uncategorized|0 Comments

Computational Complexity; Article 2

Beauty is in the Mind of the Beholder
Often mathematicians speak of finding beauty in their subject, it is reasonable to ask what they mean. Of course, one will get as many answers as there are maths practitioners, but I shall hazard a generalized answer here. Mathematicians deal with abstract objects that exist in the mind, and share a suspicion that these abstractions are in fact real. So like real things they have properties, and enter into relationships. The feeling is that if abstractions behave as real things, then they must themselves, at some level, be real. If they are real, then they must also have relations with other concrete things, and indeed, the ultimate aspiration is to develop an abstraction that describes the world. This is the fundamental quality that separates the practice of maths from, say, intellectual games. Our minds evolved to find order in the universe and mathematics gives us as systematic and dependable (to some extent) way of doing so.…

By Tapiwa Chadenga| March 24th, 2016|Uncategorized|0 Comments

Calculating the date of Easter, and a computational challenge

The date on which Easter Sunday falls was decided, by the Council of Nicaea in 325 AD, to be the first full Sunday after the first full moon after the Vernal Equinox. This is very easy to work out with a lunar calendar or if you have data about the lunar cycles and the equinoxes, but what if you just were given the year. How would you work out what date it was on? Well, it turns out that there isn’t a very simple formula which can tell you this, but there is a rather complicated formula. This was presented to the journal Nature in 1876, though I haven’t been able to track down the author of this particular formula. So, this is how you calculate the date of Easter Sunday:

Take the year (in the Gregorian calendar) and divide it by 19. This will give you a quotient and a remainder.

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By Jonathan Shock| March 24th, 2016|Uncategorized|0 Comments

Proof by contradiction – part 1

Proof by contradiction may at first seem completely weird! I give you something to prove, and you seem to ignore me and try and prove that what I want you to prove is wrong!

Actually, this isn’t nearly as strange as it first seems, and it can work in contexts other than mathematics. The idea stems from the fact that a statement is either true, or false (well, if you listen to Gödel, then you have to be a bit careful, but it’s reasonable enough for now). The process is the following:

You want to prove that a statement is true.
You say “what would happen if the statement were actually false?”
You explore the consequences of it being false.
If you find that it gives you a contradiction (something which you claimed to be true, but which you now see isn’t true), then you know that in fact the original statement can’t be false…so it must be true, and you’ve proved it.

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By Jonathan Shock| March 20th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|7 Comments

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