A schools revolution

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I highlighted in a recent post, a ludicrous lack of scientific knowledge by Michael Gove. This is perhaps unfair of me. So lets address what Gove is actually trying to do, which is reform Schools. Critics regularly assume that private school parents are rich snobs trying to avoid the lower classes. If you read the New Statesman you may assume this. Yet as I discovered in my undergraduate studies in Bristol – a school with a high number of privately educated students – this is an inaccurate caricature. Many parents make a number of sacrifices, such as remortgaging their houses, to afford such an education.
One of the craziest things about modern England, is education apartheid. In Northern Ireland, education reform has recently removed the transfer exam due to fears over elitism. And Elitism and obsession with snobbishness are very peculiar. They sometimes instantly stop debate, and instantly stop the pursuit of the truth. Calling someone ‘elitist’ in some circles is pejorative. Elitism does have a tendency to get redefined as ‘those who feel superior to other people’. This definition is what egalitarians might lean towards. The problem with egalitarians is that I lean towards a meritocratic ideal. I know well, that there are problems with that, as Young famously wrote society probably shouldn’t be completely meritocractic.
A recent Economist article which inspired the title of this blog post. Talked about the ‘educational DNA’ of the best private schools – independence, restless innovation, an impatience with excuses for failure and the unabashed pursuit of excellence.
Perhaps the academy system will offer some of this. I think what is most important in these discussions is to separate academic excellence from ideological or class discussions. Academic excellence should be the goal.Something I’ve noticed in my own education, is feeling embarrassed about pursuing excellence, or feeling that it is beyond me. This is a very toxic belief, and we all have some toxic beliefs. The Educational DNA mentioned above is also present in the Grammar School system that I went through. I don’t see how anyone should feel embarrassed about the pursuit of excellence. We after all want well educated people as doctors, and in government.

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Why Education Ministers should be educated

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Michael Gove is reported as arguing that:

Gove said there had been previous attempts to make science relevant, by linking it to contemporary concerns such as climate change or food scares. But he said: “What [students] need is a rooting in the basic scientific principles, Newton’s laws of thermodynamics and Boyle’s law.”

[…]

We are now seeing with the new exams regulator how we can make GCSEs tougher. Exam boards need to sharpen up their act. We are also saying in GCSEs that you need to award marks for spelling, punctuation and grammar. We need to have stretching exams which compare with the world’s most rigorous.”

http://www.guardian.co.uk/education/2011/jun/18/michael-gove-exams-gcse-schools

Newton’s laws are those of motion and gravity, not thermodynamics. Those of Thermodynamics were first formulated by Lord Kelvin. Gove’s educational background is in English, rather than science, so some glaring errors in a subject he has no experience of are understandable, if unfortunate in the Secretary of State for Education.

Clearly the education offered to Ministers is not as rigorous nor as broad as it could be. Until Ministers are educated sufficiently in basic scientific principles and history they should be cautious in offering views on their teaching.

I trust that Mr Gove will be undertaking remedial education to bring his understanding up to an acceptable minimum standard and in the meantime will refrain from offering his opinions on teaching standards.
A friend of mine on Facebook summed it up very well.

10 points for the first non-scientist who tells me why this is complete bollocks. If you don’t know, please find out. This is like someone saying Donatello painted the Mona Lisa, or Dante wrote Romeo & Juliet, or jumbo jets were invented in the 18th century.
C.P. Snow lamented the fact that there were two cultures
“A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is the scientific equivalent of: Have you read a work of Shakespeare’s?” – CP Snow.
It seems we haven’t changed much since then. I mean how many people know about the age of the Universe, or how accurate it is. How many know the basics of Chemistry? Continue reading

Observations on the connectedness of our world.

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Scientifically focused geeks like myself,have a tendency to speak highly of the web. We see Skype, MSN, and Facebook as great technical marvels. Yet as someone like Tim Ferris or Cal Newport observe there is a price to this connectedness.
Today for instance there was a family wedding in Ireland. I wasn’t able to attend due to examinations next week in my own studies. Yet through text messages and Skype conversations I feel like I’m half there.
Which means that concentration is difficult. Yet concentration is something I need to develop the rare and valuable skills of a Mathematician. Or whatever discipline I end up working in.
We should remember that we are fundamentally limited by the hardware of our brains. And limited by our humanity. We shouldn’t forget the effects of technology or modern day life on our cognitive load.

Mick Bremner wrote a post on this a few years ago.

Now, as time goes on and I realize that moving home every couple of years is actually taking a toll on my relationships with people that I care very much about I realize that, possibly, my writing can help the situation. I’m reluctantly realizing that I’m rarely ever going to be able to spend long afternoons chatting with my dearest friends over (good) coffee. But maybe if I keep this blog up to date then at least they might have some chance of keeping track of what’s going on with me.

My own Facebook and Twitter accounts have friends and family all around the world. I’ve friends who live in Hong Kong, Shanghai, New York, London, Adelaide and everywhere in between. And as Ben Casnocha pointed out, there is a ‘feel bad effect’ to Facebook of not-so-close-facebook-friends.
Constantly we see upbeat images, or happy occasions. Rarely the daily struggles of our existences. When we read CV’s or resumes of people in our respective fields we don’t hear about the struggles of their lives.
This is written to point out that everything I do in life seems to be an absolute struggle.

An Introduction to Coalgebras

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I’ve just dumped this here, I wrote it for a project in Luxembourg a few months ago. Its not well edited, and there are errors. I may fix these in the future.
However it may be of benefit to someone.

Introduction to various classical notions of algebras. Coalgebras are defined and introduced, as are various Commutative and Associative Algberas. An introduction to Tensors, Lie Algebra and Quantum Groups is also included. For pedagogical reasons many examples are included from Mechanics and Physics.

1. Introduction to Algebras and Coalgebras

Abstract Algebra is increasingly important in Mathematics, Physics, Computer Science, Linguistics, and other subjects. Its quite suprising when you first learn that Algebra has moved beyond mere group and ring theory. This essay is to look at some of the more exciting structures in this area of Mathematics. The author makes no apology for lack of originality in this piece, and states a lot of the results without proof. The interested reader who wishes to find the proofs can consult the various references which are included in the bibliography. Once again, the aim is to introduce these topics to a good Masters student so that they may gain both an appreciation of Algebra and Coalgebra in itself, or alternatively to understand the literature in Theoretical Physics or Mechanics

1.1. Associative Algebras

An associative algebra over {\mathbb{K}} is a vector space A equipped with a binary operation (linear map) \mu:A \otimes A \rightarrow A which is associative, i.e. {\mu \circ(\mu \otimes id) = \mu \circ (id \otimes \mu).} Here id is the identity map from A to A (sometimes denoted by {id_{A}}), and the operation {\mu} is called the product. Denoting ab:={\mu(a \otimes b)}, associativity reads just like Kindergarden: (ab)c = a(bc). An associative algebra is said to be unital if there is a map {u:\mathbb{K} \rightarrow A} such that {\mu \circ (u \otimes id) = id = \mu \circ (id \otimes u).} We denote by {1_{A}} or simply by 1 the image of {1_{\mathbb{K}}} in A under {u}. With this notation unitality reads: {1a = a = a1.} An algebra morphims (or simply morphism) is a linear map {f:A \rightarrow A'} such that f(ab) = f(a)f(b). If A is also unital, then we further assume f(1) = 1. \phantom{xxx}The category of nonunital associative algebras is denoted by As-alg, and teh category of unital associative algebras by uAs-alg.

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How to think

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Inspired by Ed Boyden \newline Managing brain resources in an age of complexity.
Complexity

The main aim of this post, is to understand and synthesize some of the ideas behind learning how to think. In the words of Seymour Papert, a rather inspiring Mathematics and Computer Education pioneer, ‘students now need to learning how to learn’

  1. Synthesize new ideas constantly.Never read passively. Annotate, model, think and synthesize whatever you are reading. This extends to intro stuff. That way when you move onto more advanced stuff, you’ve already mastered the prerequisites.
  2. Learn how to learn (rapidly).Eric in his own post makes reference to the fact that this is becoming an essential skill in the 21st century. Be able to rapidly prototype ideas. Create lectures, and models to help you learn material. For technical experitise include two things: Technical Explanation Questions (i.e. explaining a technical concept) and practice at solving problems. Concepts and Technical Skill.
  3. Work backward from your goal.Or else you may not get there. Sometimes it helps to think of your project as being managed by a project management algorithm. The application of PERT or CPM to yourself!
  4. Always have a long-term plan.Even if you change it every day. The act of making the plan alone means your guarranteed to be learning something.
  5. Make Contingency Maps.Draw all the things you need to do on a big piece of paper, and find out what is dependent on other things (this is similar to project management), and what is independent. The things that are not dependent on anything, but have the most dependents, finish them first.
  6. Collaborate.
  7. Make your mistakes quickly.You may meass things up on the first try, but do it fast, and then move on. Analyze what goes wrong in your first attempt at a solution, and this means being honest and explicit. ‘We lose that which we oft might win, by fearing to attempt’.
  8. As you develop skills, write up best-practices protocols.That way, when you return to somehting, you can easily pick up where you left off. Instinctualize conscious control. An important corollary to this best practice, is the art of writing things in a finalized form as soon as possible. Sketches of solutions aren’t as valuable as well worked out, and annotated solutions.
  9. Document everything obsessively.If you don’t record it, it may never have an impact on the world properly. Most profound scientific discoveries are suprises. But if you don’t document and digest everything you see, you will not know when you’ve seen a suprise.
  10. Keep it simple. If something looks hard to engineer it probably is. Simplify the design as much as possible.

In the area of time management a powerful technique is offered by Eric Boyden, he calls it Logarithmic time planning. Far away things planned to the week, things a month away planned to the day, and things a few days away planned to the hour. This blends together the inevitable difficulties in human psychology, and also allows one to be honest about when you are going to do things. A to do list can easily tend to infinity, but your time won’t. \paragraph{} When writing and drawing when talking to someone, which should be used to pass on information. It can be useful to have a camera to take a record of the discussion. Our memories are weak, and then we can also give the page to the other person. A good camera, and Evernote for tagging, can act as an extended brain. In conclusion, it is not enough to be able to use technology, one should be able to use technology tremendously well. References can also be added to the Evernote ideas. For instance if one comes across the name for a good paper or book reference a link to amazon, or the paper itself can be archived with the material.

Geometric Phases

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I wrote this article as part of extending my undergraduate thesis. Its certainly not original but should be of benefit to someone.

1. Introduction

A considerable understanding of the formal description of quantum mechanics has been achieved after Berrys discovery{BerryPhase} of a geometric feature related to the motion of a quantum system. He showed that the wave function of a quantum object retains a memory of its evolution in its complex phase argument, which, apart from the usual dynamical contribution, only depends on the geometry of the path traversed by the system. Known as the geometric phase factor, this contribution originates from the very heart of the structure of quantum mechanics. It has been over 25 Years since Berry discovered his phase. An entire industry of problems have arisen since then. It has been shown in a various reviews that a better understanding of these problems comes from the usage of topological and geometric techniques. A unified approach to geometric phase problems comes from the usage of basic topological techniques The evolution of the electric field along the curved path of a light ray is described by the Fermi-Walker parallel transport law. For the polarized light the analog of magnetic flux is the solid angle subtended on the sphere of directions k, where k represents the direction of propagation of light, which changes as light passes through the twisted fiber. This spin redirection phase is known as the Rytov-Vlasimirski-Berry{rytov,Rytov_Review,Vladimirski} phase, as opposed to the Pancharatnam-Berry Phase. Since Geometric Phases are intrinsically related to fibre-bundle topological theories, a short reviews of Fiber bundle theories is provided. Deep understanding of geometric phases phases related to time development by the Schrodinger equation and interference effects in optics needs the use of analytical methods. We suspect that in Quantum Computation research, these topological techniques will become increasingly important. The main original work in this paper will be a model of the Manual Fiber Polarization controller(MPC) ~{MPC}.

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Commutative Algebra

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These notes are based on a lecture on Commutative Algebra at the University of Luxembourg. A variety of sources including Eisenbud, the Princeton Companion to Mathematics and a Classic ‘Algebra’ text by MacLane and Birkhoff have been used to compile these notes. Hopefully it will be of benefit to those studying this course.

1. Lecture Notes:Axioms for Rings

A ring R = (R,+,{\cdotp},1) is a set R with two binary operations, addition and multiplication, and a nullary operation, ‘select 1’, such that

  • (i) (R,+) is an abelian group under addition;
  • (ii) (R,{\cdotp},1) is a monoid under multiplication;
  • (iii) Multiplication is distributive (on both sides) over addition.

The last requirement means that all triples of elements a,b,c in R satisfy the identities

\displaystyle a(b+c)=ab+bc,\;\;\;\;\;\;(a+b)c=ac+bc \ \ \ \ \ (1)

A commutative rign is one in which multiplication is commutative. Familiar systems of numbers are commutative rings under the usual operations of sum and product; examples are

\displaystyle \begin{array}{rcl} \mathbb{Z},\;\;\;\; the\; ring\; of\; all\; integers, \\ \mathbb{Q}, \;\;\;\; the\; ring\; of\; all \;rational\; numbers, \\ \mathbb{R}, \;\;\;\; \;the\; ring\; of\; all\; real\; numbers, \\ \mathbb{C}, \;\;\;\;\; the\; ring\; of\; all\; complex\; numbers.\; \end{array}

Lets consider a rather trivial example of a ring, the set containing just the element 0, with addition and multiplicatio given (in the only possible way!) by 0 + 0, 00=0, is a ringl it will be called the ‘trivial ring’. \paragraph*{}The definition of a ring amounts to the statement that a ring is a set R with a selected element {I\in R} and two binary operations (a,b){\mapsto} a + b and (a,b) {\mapsto} ab which are both associative, so that

\displaystyle a + (b + c) = (a + b) + c,\;\;\; a(bc)=(ab)c, \ \ \ \ \ (2)

for all {a, b, c \in R,} which have addition commutative, so that

\displaystyle a + b = b + a \;\;\; \forall a,b \in R, \ \ \ \ \ (3)

which containts the unit 1 and a zero0 such that

\displaystyle a + 0 = a, a1=a=1a, \forall a \in R, \ \ \ \ \ (4)

which cotaints to each element a an additive inverse(-a) with

\displaystyle a + (-a) = 0, \ \ \ \ \ (5)

and in which both distributive laws (1) hold. As an example of a non-commutative ring, we can consider the ring of square matrices. Every field – like for instance the real numbers and the complex numbers is an example of a commutative ring. {\mathbb{C}} is algebraically closed every non-constant polynomial with coefficients in {\mathbb{C}} has (at least one root in {\mathbb{C}}. This is the famous Gaussian result: the fundamental theorem of algebra.

1.1. Residue Class Rings

: \left(\frac{\mathbb{Z}}{n\mathbb{Z}},+,\cdotp \right)\;\;\;n \in \mathbb{N}, n > 1 p \in \mathbb{P}\;\;prime, then (\frac{\mathbb{Z}}{p\mathbb{Z}},+,\cdotp) = \mathbb{F}_{p}\;\;is\;a\;field\;(finite) Subrings {(R,+\cdotp)} \underline{Subring} {S\subseteq R} subset such that {+\arrowvert_{S}}, {\cdotp\arrowvert_{S}}

\displaystyle \begin{array}{rcl} S_{1} + S_{2}\; =\; S_{1}\; + \;S_{2}\; in\; R \\ S_{1},S_{2}\; \in\; S \\ I\; take\; two\; elements\; from\; the\; subsets\; of\; the \;ring.\; \\ s\;\in\;S \rightarrow -s\;\in\;S \\ 0\;\in\;S\;\rightarrow\;\;0\;\in\;S \\ s_{1},s_{2}\;\in\;S\;\rightarrow\;s_{1}\cdotp s_{2}\;\in S \\(1\in S) \\ (\mathbb{Z},+,\cdotp)\; is\; a\; subring\; of\; (\mathbb{Q},+,\cdotp) \end{array}

As with other algebraic structures, there are several ways of forming new rings from old ones: for instance we can take subrings and direct products of two rings. Slightly less obviously, we can start with a ring R and form the ring of all polynomials with coefficients in R. We can also take QUOTIENTS, but in order to discuss these we must introduce the notion of an ideal.
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