Factoring by Stealth

This is simply brilliant. Humans have always been good at visualisation, and puzzles always generate more interest in study than dry math symbols. I was just going to comment on it there, but there is more to this than meets the eye, so I will have to expand this entry. Stay tuned!

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Multidimensional Separation of Concerns – Bad Command or File Name

There should really be a sarcasm tag in HTML.

Hyperspaces look to be a hugely advanced method for Multidimensional Separation of Concerns. Unfortunately, separation of concerns is one of the most slippery concepts in computer science. I will certainly remember the day when somebody actually explains what a concern is. And no, vacuous definitions don’t count.

Interestingly, the concept in Dijkstra’s “On the role of scientific thought” makes a lot of sense. The words aspect and concern are used in their natural English meaning, and the whole piece is essentially a call for divide and rule which is essential in all science, to use Ceteris Paribus assumptions. Else, can someone explain Dijkstra’s “It is being one- and multiple-track minded simultaneously.” in a computer science perspective?

Best worded definition so far is this: “So what is a cross-cutting concern? A concern is a particular concept or area of interest. For example, in an ordering system the core concerns could be order processing and manufacturing, while the system concerns could be transaction handling and security management. A cross-cutting concern is a concern that affects several classes or modules, a concern that is not well localized and modularized.”. At least, when a concept is not directly well defined, examples give you something to go by.

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Calculus – a Brilliant Approach

The best approach I have come across so far. Unsurprisingly, it does have a Knuth-like quality. At least, I think this is a much more sensible way to explain to beginners an \epsilon-\delta definition.
I will probably copy all of the important proofs and concepts here as an personal exercise.

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Spaces – First Round

Hi, I am back.

Things have progressed reasonably well. So, I am reasonably happy with my math skills now, though I didn’t fix all concepts. I am now crossing my fingers for no relapse.

For the time being, my definition of mathematics is recognizing patterns, abstracting them, and using the results in conjunction with previous abstractions. It remains to see whether this will survive.

Back to Space, the issue of Vector Space has undergone a good treatment recently, in Prof Gowers weblog. Do pay attention to Terence Tao‘s comment and his linked notes. I personally find the linear transformation approach more useful than matrices, to the extent that I sometimes think of a matrix entry as a_{ij}=e_i^\prime A e_j, just to avoid breaking the conceptual transformation approach. I probably find that easier as a result of using SciPy, where up to a certain extent you gain in performance by thinking of matrices as black boxes, akin to vectorization. Probably Matlab users will have the same effect. By the way, Tao’s comment about the idea of a linear transformation being a multidimensional generalisation of a ratio did not help at first, but it is now resonating with another idea that I have. More on that later. In short, the concept of linear transformation is that embedded in vector space.

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A Mathematical Rehab

Ok. So the last post wasn’t very successful (I am not going to argue about the definition of successful here). My explanation of space wasn’t so good, because almost everything can be described as set with structure, according to mathematicians. Now I am probably irritating some Category Theory proponents as well. Oh well.

As a corrective remedy, I am going to spend some time in a mathematical rehab. You will be able to follow that here. I will probably start by using the spaces mentioned in that post to explain various mathematical concepts.

And, as for the definition of space, let’s just say that a space is a space because a mathematician says so. And I am happy that wikipedia is not that better from my post.

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Space

Space for mathematicians is an interesting set. Or we can say a set with structure. Trying to get used to this concept, I am listing a few spaces, with some explanation, here. By the way, members of such sets are usually called points.

  • Vector Space: members of such a space can be scaled and added. Here the definition is dependent on properties of the members.
  • Topological Space: in this space points have neighbourhoods. Or, the topological space is a set on which a collection of open sets (closed under unions and finite intersections) is defined.
  • State Space: This means different things to computer scientists, physicists, and control engineers. For control engineers, state space is the space in which members are tuples of state variables.
  • Metric Space: a metric or distance is defined between members of such a space.
  • Euclidean Space: for most of us, this is the space acquired by age eighteen.

Of course, there are many more spaces. I mean, if you are so inclined, you can look up Hilbert Space, Minkowski Space, Anti de Sitter Space, etc. Just remember, a space is a set you can talk about.

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Dimension is not easy

Dimension is not easy to understand. At the turn of the century it was one of the major problems in mathematics to determine what dimension means and which properties it has. And since then the situation has become somewhat worse because mathematicians have come up with some ten different notions of dimension: topological dimension, Hausdorff dimension, fractal dimension, self-similarity dimension, box-counting dimension, capacity dimension, information dimension, Euclidean dimension, and more. They are all related. Some of them, however, make sense in certain situations, but not at all in others, where alternative definitions are more helpful. Sometimes they all make sense and are the same. Sometimes several make sense but do not agree. The details can be confusing even for a research mathematician.

Fractals for the Classroom

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Normal Law of Errors

Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.

Henri Poincaré

So, which is it? To apportion the blame, I suggest that the fact of observation be that “errors” are independent and identically distributed with finite variances. Observers may as well vote for Lyapunov’s set of conditions. I am sure that mathematicians are more than happy to shoulder the responsibility for the resulting Central Limit Theorem.

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Hello world!

Hello, world. This is the start of a blog for me.

Why start a blog? Many reasons actually. Improving my writing skills. As a communication platform with other friends around the globe. Getting ideas which may or may not be useful out there. A tool for learning. Trying to get more used to basic Web 2.0 (or is it Web 1.1?) concepts before everyone gets to use Web \infty .

I am impressed by the current group of machine learning blogs like Machine Learning (Theory) and Machine Learning Thoughts among others, and I am actually learning a lot from them. However, this will not be a single topic blog, and I will be touching on different dimensions of life. Categories will hopefully reveal some method in the resulting madness.

By the way, dimensions may crop up on any topic raised here, in a mathematical sense or otherwise. You’ve been warned.

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