After my post yesterday, I had a thought, "What about all those other areas of interest I mentioned I would talk about originally?" and since phone hacking has been on the news quite a lot, it seems pertinent to share a group project with you.
This year I studied a module called Information Security lectured by the Computer Science department. These lectures went alongside a group research project on a related subject of choice. Our group chose, "GSM Security". GSM is the global standard system for mobile communications and because of this the system's security is important to 90% of the population of the world (that means you!). GSM has been widely studied both academically and also practically (by ethical hackers).
Our task was to present an up-do-date image of the security measures, vulnerabilities and threats on GSM. The paper produced is available for you to read. It's not as detailed as we originally made it because we had a page limit (lame!) but it gets the main points across. We also got a fantastic mark for it!
After reading it you will have a better idea of how people (including journalists) can access personal information from your phone, listen into calls or prevent use of the phone.
Now, a disclaimer: I don't know if any of these methods presented were actually the ones thought to be used by the accused journalists or if they are guilty of phone hacking. I'm just presenting our research.
Monday 31 January 2011
Sunday 30 January 2011
A Pause
There are three very good reasons I haven't really got anything good to talk about this week:
1. I have been wrestling with 32 bit compile on 64 bit architecture in order to get the essential program I need for the next stage of my project installed, compiled and working. It has only just been sorted with some fantastic work-arounds.
2. I have had every afternoon in the last two weeks filled with back to back lectures, mornings with exams and a presentation (which I mentioned last time) as well as work for those said lectures.
3. I have managed to slice open (a bit) the knuckle on my right hand thumb on Wednesday and thus typing, writing and pretty much anything is pretty hard now that I have to do it all left-handed.
Poor excuses, however next week I will dazzle you with fantastic images relating to invariant manifolds. You can read about them if you would like something to do that is more productive than reading this.
1. I have been wrestling with 32 bit compile on 64 bit architecture in order to get the essential program I need for the next stage of my project installed, compiled and working. It has only just been sorted with some fantastic work-arounds.
2. I have had every afternoon in the last two weeks filled with back to back lectures, mornings with exams and a presentation (which I mentioned last time) as well as work for those said lectures.
3. I have managed to slice open (a bit) the knuckle on my right hand thumb on Wednesday and thus typing, writing and pretty much anything is pretty hard now that I have to do it all left-handed.
Poor excuses, however next week I will dazzle you with fantastic images relating to invariant manifolds. You can read about them if you would like something to do that is more productive than reading this.
Sunday 23 January 2011
Pwetty Pictures and a Presentation
So, this week when I thought about what to blog I left it pretty last minute. This means that instead of an (usually very interesting) insight into chaos I will share with you the presentation I gave on Friday for my project to students and lecturers. I have left the notes attached so you can follow what I spoke about as well.
One moment from someone else's presentation I would like to share is this:
Brilliant!!
The presentation itself was about business intelligence which does not really interest me but the image brought some light humour to the talk.
One moment from someone else's presentation I would like to share is this:
Brilliant!!
The presentation itself was about business intelligence which does not really interest me but the image brought some light humour to the talk.
Saturday 15 January 2011
Nonlinearity gets all the fun
Nonlinearity is inherent in most systems before you make an assumption and reduce the beautiful complexity down to a boring bog standard linear system. There is a section in Strogatz that shows probably the most interesting thing you can do with linear systems (Chapter 5 section 3 page 138 onwards). And I only read it once because linear systems were dull and pretty repetitive long before A level.
Nonlinearity is pretty easy to understand. Take for example square numbers. You can make 4 by either multiplying 2 and 2 or -2 and -2. This means that, under the square operator, 4 has two (well, four really, but two are identical to the other two) possible previous values. This makes squaring a number nonlinear because tracking back you do not know whether two 2s or two -2s made the 4 you see at the moment. This particular brand of nonlinearity is called noninvertibility. You still following me?
Good. Right, to take this further we call the previous values of each number pre-images and the operation, a mapping. Now the quadratic (squaring) mapping that I am examining is:
what happens to z when a=2 and I vary lambda. This mapping does funny things depending on what I set lambda to. If it is between 0 and 1 then it behaves just like the squaring operator - each value of z has two pre-images as I iterate the map but in a circle radius=1-lambda around C the values have no pre-images. Now if I make lambda greater than 1, some values have two pre-images as before but the z values in the circle now have four pre-images.
Still with me? We can take this further but placing this property on curves not just points. A curve is just a line on a graph (an infinite set of points). Instead of looking backwards as we did with the points, we instead go forwards and see how the curve changes. It's first image is a circle, say. This curve then becomes the pre-image of the set of points of the circle under the mapping. And so on, forwards in time. It helps to look at the image below:
You can see the curves collecting on (heading towards) one curve called a manifold (it's attracting). Now consider the relationship of the points and the curves. Because a curve is just a set of points. With noninvertibility inherent in the system, what happens it you take the manifold and reverse the mappings for all the points? Wild Chaos! Tracking the possible points and curves and how and why they interact is the next stage of my project - at least for certain ranges of values for C, a and lambda. This limitation is purely because my project has a time limit of a year.
I am currently working with a PhD student, Stefanie Hittmeyer (who incidentally produced the image above) who is working on the full problem in all its scope. She has been focused on the manifold part of it for the moment while I have been examining the fractal structure side. My work nicely compliments hers. This work is on the fore-front of chaos research. Now it may seem that all this has little value in the real world but the great thing about nonlinear and chaotic systems is that it can accurately mimic real world systems. This is a major advantage over linear systems because when taking a real world problem you have to reduce the picture you are looking at so you can describe it using the simplest tools. With this kind of research, no such reduction is necessary and you can understand the system in it's full complexity. So maybe it'll turn out that neurological signals in the brain will turn out to have similar properties to the mapping that I am examining and my research will contribute to a fuller understanding of how we came to have consciousness and help neurosurgeons get better at fixing the biochemical circuitry that makes us who we are. Or maybe, and more likely since this mapping is a reduction of the five dimensional Lorenz model, it will just help meteorologists get really, really good at predicting weather patterns.
Who knows? It's sort of like the laser - a solution created with no problem that has now become integral to our lifestyle. Just check out how many things it is used in! I hope you managed to follow all that to the end. If you have any specific questions please feel free to email me but I would much rather you follow that urge to take a deeper look yourself (even if that is just reading all the wiki pages). It's way more fun!
Nonlinearity is pretty easy to understand. Take for example square numbers. You can make 4 by either multiplying 2 and 2 or -2 and -2. This means that, under the square operator, 4 has two (well, four really, but two are identical to the other two) possible previous values. This makes squaring a number nonlinear because tracking back you do not know whether two 2s or two -2s made the 4 you see at the moment. This particular brand of nonlinearity is called noninvertibility. You still following me?
Good. Right, to take this further we call the previous values of each number pre-images and the operation, a mapping. Now the quadratic (squaring) mapping that I am examining is:
what happens to z when a=2 and I vary lambda. This mapping does funny things depending on what I set lambda to. If it is between 0 and 1 then it behaves just like the squaring operator - each value of z has two pre-images as I iterate the map but in a circle radius=1-lambda around C the values have no pre-images. Now if I make lambda greater than 1, some values have two pre-images as before but the z values in the circle now have four pre-images.
Still with me? We can take this further but placing this property on curves not just points. A curve is just a line on a graph (an infinite set of points). Instead of looking backwards as we did with the points, we instead go forwards and see how the curve changes. It's first image is a circle, say. This curve then becomes the pre-image of the set of points of the circle under the mapping. And so on, forwards in time. It helps to look at the image below:
I am currently working with a PhD student, Stefanie Hittmeyer (who incidentally produced the image above) who is working on the full problem in all its scope. She has been focused on the manifold part of it for the moment while I have been examining the fractal structure side. My work nicely compliments hers. This work is on the fore-front of chaos research. Now it may seem that all this has little value in the real world but the great thing about nonlinear and chaotic systems is that it can accurately mimic real world systems. This is a major advantage over linear systems because when taking a real world problem you have to reduce the picture you are looking at so you can describe it using the simplest tools. With this kind of research, no such reduction is necessary and you can understand the system in it's full complexity. So maybe it'll turn out that neurological signals in the brain will turn out to have similar properties to the mapping that I am examining and my research will contribute to a fuller understanding of how we came to have consciousness and help neurosurgeons get better at fixing the biochemical circuitry that makes us who we are. Or maybe, and more likely since this mapping is a reduction of the five dimensional Lorenz model, it will just help meteorologists get really, really good at predicting weather patterns.
Who knows? It's sort of like the laser - a solution created with no problem that has now become integral to our lifestyle. Just check out how many things it is used in! I hope you managed to follow all that to the end. If you have any specific questions please feel free to email me but I would much rather you follow that urge to take a deeper look yourself (even if that is just reading all the wiki pages). It's way more fun!
Saturday 8 January 2011
Oh, a promise
I also promised a friend I would post a picture up this week so here it is:
This is from the edge of the Mandelbrot set - originally from Wikipedia
And an example from my own research will be uploaded next week.
And an example from my own research will be uploaded next week.
Rationalising Chaos
Since I have been on my Christmas Break, I have only just started getting back into the thick of my research. Thus, this week's post will be on a related interest of mine. Rationality and luminosity.
I have been attempting to find order from chaos within. I have always known myself quite well but there are times when, and I'm sure you will have had a similar moment, I have thought "What was I thinking?!" having just done or remembered something incredibly stupid or damaging. I have often wondered when I would get around to finding out. I started my path to luminosity through rationality when I began reading "Harry Potter and the Methods of Rationality" by Less Wrong who took the pen name from the website which is "a collaborative blog devoted to improving the art of human rationality". From that website I discovered Alicorn who had written a series of blog posts about luminosity and a subsequent rational fanfiction of Twilight called "Luminosity" .
These encouraged me to really being to explore who I am and how I do things. To examine my motives and actions and be able to explain to anyone why I do the things I do and think the things I think. Living this way prevents one from lying to oneself in order to imagine they or the world around them is they way they want instead of the way it is. If you too want to explore this, I would suggest reading the stories first to understand what I means to live like this - to stand out because you refuse to gloss over the parts of reality you don't want. The world is chaotic but not really as irrational as it seems.
By saying "gloss over reality" here I don't necessarily mean in a politically active sense in fact I tend to ignore politics because every side glosses over their reality and never say what they really mean - even taking an opposition to politics means you a glossing over the reality that democracy has to please the masses; I try to take each case individually on merit instead of generalising. It could be as simple as packing a first aid kit in the car, not because you know you will need it on that trip but because you know that things happen and it is better to be prepared for the worst instead of saying to yourself "by packing the first aid kit I'm tempting fate and someone will get hurt so I shouldn't pack it" just like thinking that if you take an umbrella with you, it is more likely to rain, or by not going to the doctor, you're not really sick and the problem will go away by itself - twisted and irrational logic.
By accepting the reality that everyone gets sick at some time or another, that it will rain unpredictably in England and that people can get hurt doing the safest things, you can prepare for these realities and maybe avoid the worst.
I have been attempting to find order from chaos within. I have always known myself quite well but there are times when, and I'm sure you will have had a similar moment, I have thought "What was I thinking?!" having just done or remembered something incredibly stupid or damaging. I have often wondered when I would get around to finding out. I started my path to luminosity through rationality when I began reading "Harry Potter and the Methods of Rationality" by Less Wrong who took the pen name from the website which is "a collaborative blog devoted to improving the art of human rationality". From that website I discovered Alicorn who had written a series of blog posts about luminosity and a subsequent rational fanfiction of Twilight called "Luminosity" .
These encouraged me to really being to explore who I am and how I do things. To examine my motives and actions and be able to explain to anyone why I do the things I do and think the things I think. Living this way prevents one from lying to oneself in order to imagine they or the world around them is they way they want instead of the way it is. If you too want to explore this, I would suggest reading the stories first to understand what I means to live like this - to stand out because you refuse to gloss over the parts of reality you don't want. The world is chaotic but not really as irrational as it seems.
By saying "gloss over reality" here I don't necessarily mean in a politically active sense in fact I tend to ignore politics because every side glosses over their reality and never say what they really mean - even taking an opposition to politics means you a glossing over the reality that democracy has to please the masses; I try to take each case individually on merit instead of generalising. It could be as simple as packing a first aid kit in the car, not because you know you will need it on that trip but because you know that things happen and it is better to be prepared for the worst instead of saying to yourself "by packing the first aid kit I'm tempting fate and someone will get hurt so I shouldn't pack it" just like thinking that if you take an umbrella with you, it is more likely to rain, or by not going to the doctor, you're not really sick and the problem will go away by itself - twisted and irrational logic.
By accepting the reality that everyone gets sick at some time or another, that it will rain unpredictably in England and that people can get hurt doing the safest things, you can prepare for these realities and maybe avoid the worst.
Saturday 1 January 2011
Happy New Year!
This post I want to introduce my main project. The title is "Dynamics and Bifurcations of Two-Dimensional Noninvertible Maps". My plan for the project is available here to give you an idea of what I would like to achieve with it. Don't worry about all the scary words in it - the main idea is to find out what happens if I slightly permute the Mandelbrot set (a two dimensional map of Julia sets represented as a number set). Currently I'm focused on looking at how the Julia sets change. So far I have made lots of pretty pictures! as well as some observations on what happens.
Fractals are the most beautiful objects in chaos theory (wiki has a great introduction to them here). Simple definition is that fractals are shapes that are self-similar at all levels of magnitude - you cannot tell how zoomed in or out you are. The Mandelbrot is the most famous fractal and I love it because it is defined so simply but produces infinite complexity. My project considers the Mandelbrot as a single instance of a wider group of two-dimensional non-invertible maps. Noninvertible here means a many-to-one mapping forward in time i.e. if you back track there are many points that could have produced your original point.
Fractals were first observed in nature in the form of leaves on the trees. Ferns are made up of smaller ferns which are made up of even smaller ferns. The simplicity of defining a fractal has suggested that the information contained within the DNA of a plant to tell it what shape to grow into is in fact encoding fractal information. Another example would be one used recently in one of the Royal Institution Christmas Lectures - lungs. They translate a large volume of air onto a large surface area of blood vessels and they do it by being fractal. And recently there has been research that has found fractal patterns in semiconductor material at the quantum scale.
Fractals can be produced by nonlinear dynamics. This returns us back to the Lorenz equations and the butterfly attractor. The attractor found to describe the behaviour of the system is a fractal. Such a combination of parts of chaos theory is typical and continuing to find more links is very exciting. I have been finding bifurcation points and their stability in the fractal Julia sets as I permute the Mandelbrot set. Nonlinear dynamics can be produced by fractals. Thus my research is far reaching across the whole field of study. A field of study that is far reaching, interlinked and incredibly beautiful.
Fractals are the most beautiful objects in chaos theory (wiki has a great introduction to them here). Simple definition is that fractals are shapes that are self-similar at all levels of magnitude - you cannot tell how zoomed in or out you are. The Mandelbrot is the most famous fractal and I love it because it is defined so simply but produces infinite complexity. My project considers the Mandelbrot as a single instance of a wider group of two-dimensional non-invertible maps. Noninvertible here means a many-to-one mapping forward in time i.e. if you back track there are many points that could have produced your original point.
Fractals were first observed in nature in the form of leaves on the trees. Ferns are made up of smaller ferns which are made up of even smaller ferns. The simplicity of defining a fractal has suggested that the information contained within the DNA of a plant to tell it what shape to grow into is in fact encoding fractal information. Another example would be one used recently in one of the Royal Institution Christmas Lectures - lungs. They translate a large volume of air onto a large surface area of blood vessels and they do it by being fractal. And recently there has been research that has found fractal patterns in semiconductor material at the quantum scale.
Fractals can be produced by nonlinear dynamics. This returns us back to the Lorenz equations and the butterfly attractor. The attractor found to describe the behaviour of the system is a fractal. Such a combination of parts of chaos theory is typical and continuing to find more links is very exciting. I have been finding bifurcation points and their stability in the fractal Julia sets as I permute the Mandelbrot set. Nonlinear dynamics can be produced by fractals. Thus my research is far reaching across the whole field of study. A field of study that is far reaching, interlinked and incredibly beautiful.
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