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Tag Archives: intermediate
Combinatorial Game Theory VI
Lesson 6 General Combinatorial Game Theory [ Warning: the following lesson is significantly longer than the previous ones. ] Starting from this lesson, we will look at a more rigourous, complete and general theory. Prior to this, in any game configuration both … Continue reading
Thinking Infinitesimally – Multivariate Calculus (I)
[ Background required: some understanding of single-variable calculus, including differentiation and integration. ] The object of this series of articles is to provide a rather different point-of-view to multivariate calculus, compared to the conventional approach in calculus texts. The typical … Continue reading
Posted in Notes
Tagged calculus, differentiation, intermediate, multivariate, partial differentiation
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Power Series and Generating Functions (II): Formal Power Series
For today’s post, we shall first talk about the Catalan numbers. The question we seek to answer is the following. Let n be a positive integer. Given a non-associative operation *, find the number of ways to bracket the expression … Continue reading
Posted in Notes
Tagged catalan numbers, combinatorics, formal power series, intermediate, power series
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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
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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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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Quadratic Residues – Part IV (Applications)
Let p be an odd prime and g be a primitive root modulo p. Given any a which is not a multiple of p, we can write for some r. We mentioned at the end of the last section that a is a square if … Continue reading
Mathematics and Such 