Tag Archives: notes

Power Series and Generating Functions (IV) – Exponential Generating Functions

Note: this article is noticeably more difficult than the previous instalments. The reader is advised to be completely comfortable with generating functions before proceeding. We’ve already seen how generating functions can be used to solve some combinatorial problems. The nice … Continue reading

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Power Series and Generating Functions (III) – Partitions

One particularly fruitful application of generating functions is in partition numbers. Let n be a positive integer. A partition of n is an expression of n as a sum of positive integers, where two expressions are identical if they can be obtained from each … Continue reading

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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

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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

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Estimating Sums Via Integration

Background required : calculus, specifically integration By representing a sum as an area, it is often possible to estimate its size by approximating it with the area underneath a curve. For example, suppose we wish to compute the sum . … Continue reading

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Matrices and Linear Algebra

Background recommended : coordinate geometry Here I thought I’d give an outline of linear algebra and matrices starting from a more axiomatic viewpoint, instead of merely giving rules of computation – the way it’s usually taught in school. The materials … Continue reading

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Sample Problem Solving + Homework Hints

In this post, I’ll talk about basic number theory again. But I’ll still assume you already know modular arithmetic. 🙂  In the first part, there’ll be some sample solutions for number theoretic problems, some of which were already presented in … Continue reading

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