Monthly Archives: August 2014

An useful technique for solving PDEs by separation of variables

Posted on August 31, 2014 by fmartin

We’ll discuss a common situation that arises when solving PDEs over a rectangle by separation of variables. Frequently, we can solve the problem if the boundary condition is a linear combination of sines and cosines, so we then ask if … Continue reading →

Posted in Uncategorized | Tagged | Leave a comment

Summability of Fourier coefficients for good-enough functions

Posted on August 30, 2014 by fmartin

Suppose is a function admitting a periodic extension to . Therefore, for all , admits a periodic extension to ; in particular, it satisfies the Dini condition on and so its Fourier series converges pointwise on that interval. Even more, … Continue reading →

Posted in Uncategorized | Tagged | 1 Comment

Calculating the cardinality of a cokernel using determinants

Posted on August 28, 2014 by fmartin

Suppose is a group morphism. We want to compute the cardinality of . We have that the matrix is equivalent to some diagonal matrix , that is, , where ; this is the Smith normal form of . By diagram … Continue reading →

Posted in Uncategorized | Tagged | Leave a comment

Quadratic extensions in characteristic 2

Posted on August 28, 2014 by fmartin

Since any quadratic extension is simple, it suffices to study irreducible quadratic polynomials over a field of characteristic 2. If is of the form , then is a non-square in . This can only happen in infinite fields, since the … Continue reading →

Posted in Uncategorized | Tagged | Leave a comment

The degree of a quotient field of Z[i] over its prime field is either 1 or 2

Posted on August 23, 2014 by fmartin

Suppose is an irreducible element over . Then is a maximal ideal and so is a field. We want to calculate the degree of that field over its prime field. Suppose . Without loss of generality we can assume , … Continue reading →

Posted in Uncategorized | Tagged , | Leave a comment