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Monthly Archives: December 2011
Symmetric Polynomials (III)
Now we generalise this to n variables: . It’s clear what the corresponding building blocks of symmetric polynomials would be: ; ; ; … . We call these ei‘s the elementary symmetric polynomials in the xi‘s. Note that each ei is the coefficient of Ti in the … Continue reading
Symmetric Polynomials (II)
When we move on to n=3 variables, we now have, as basic building blocks, These are just the coefficients of in the expansion of . Once again, any symmetric polynomial in x, y, z with integer coefficients can be expressed as a polynomial … Continue reading
Symmetric Polynomials (I)
[ Background required: knowledge of basic algebra and polynomial operations. ] After a spate of posts on non-IMO related topics, we’re back on track. Here, we shall look at polynomials in n variables, e.g. P(x, y, z) when n = 3. Such … Continue reading
What is Curvature? (II)
[ Background required: rudimentary vector calculus. ] The aforementioned definition of curvature is practical but a little aesthetically displeasing. Specifically, one seeks a definition which is independent of the parametrization of the curve. The advantage of such a definition is … Continue reading
What is Curvature? (I)
[ Background required: calculus, specifically differentiation. ] In this post, we will give a little background intuition on the definition of curvature. One possible approach is given in wikipedia, ours is another. Note that this is not IMO-related (my apologies … Continue reading
Pick’s Theorem and Some Interesting Applications
[ Background required: none. ] A lattice point on the cartesian plane is a point where both coordinates are integers. Let P be a polygon on the cartesian plane such that every vertex is a lattice point (we call it a lattice polygon). … Continue reading
Posted in Extra
Tagged arithmetic, farey sequences, ford circles, geometry, intermediate, lattice points, pick's theorem
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Modular Arithmetic Deluxe Edition
[ Background required: standard modular arithmetic. ] Consider the following two problems: Problem 1. Prove that if p > 2 is prime, then when is expressed in lowest terms , m must be a multiple of p. Problem 2. Prove that if … Continue reading
Posted in Notes
Tagged basic, congruence, modular arithmetic, notes, number theory, rational numbers
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Linear Algebra: Inner Products
[ Background required: basic knowledge of linear algebra, e.g. the previous post. Updated on 6 Dec 2011: added graphs in Application 2, courtesy of wolframalpha.] Those of you who already know inner products may roll your eyes at this point, … Continue reading
Posted in Notes
Tagged advanced, extra, fourier transform, inner product, intermediate, linear algebra, notes, sums
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Mathematics and Such 