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Monthly Archives: October 2012
Introduction to Ring Theory (1)
Recall that in groups, one has only a binary operation *. Rings are algebraic structures with addition and multiplication operations – and consistency is ensured by the distributive property. Definition. A ring R is a set together with two binary operations: … Continue reading
Posted in Notes
Tagged advanced, characteristic, commutative rings, distributive property, division rings, fields, integral domains, quaternions, ring theory, rings, units, zerodivisors
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Random Walk and Differential Equations (II)
1Dimensional Heat Equation Consider the case of 1dimensional random walk. The equation (*) from the previous post gives: for t≥0. Suppose the intervals between successive time/space points are variable. Let’s rewrite it in the following form: Setting δt ≈ ε2 and δx ≈ ε, we divide both … Continue reading
Intermediate Group Theory (6)
In this post, we’ll only focus on additive abelian groups. By additive, we mean the underlying group operation is denoted by +. The identity and inverse of x are denoted by 0 and –x respectively. Similarly, 2x+3y refers to x+x+y+y+y. Etc … Continue reading
Posted in Notes
Tagged abelian groups, advanced, direct products, direct sums, free groups, generated groups, universal properties
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Polynomial Multiplication, Karatsuba and Fast Fourier Transform
Let’s say you want to write a short program to multiply two linear functions f(x) = ax+b and g(x) = cx+d and compute the coefficients of the resulting product: You might think it’ll take 4 multiplications (for ac, ad, bc and bd) and 1 addition (for ad+bc), but there’s … Continue reading
Intermediate Group Theory (5)
Free Groups To motivate the concept of free groups, let’s consider some typical group G and elements a, b of G. Recall that , the subgroup generated by {a, b}, is defined to be the intersection of all subgroups of G containing a and b. Immediately, we see … Continue reading
Posted in Notes
Tagged advanced, free groups, generated groups, group theory, universal property
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Intermediate Group Theory (4)
Applications We’ll use the results that we obtained in the previous two posts to obtain some very nice results about finite groups. Example 1. A finite group G of order p2 is isomorphic to either Z/p2 or (Z/p) × (Z/p). In particular, it … Continue reading