The Power of the Distributive Law.
march 11th, 2012This short post discusses a cute statement which shows the power of a certain subset of axioms of a ring, including prominently the distributive law.
A Cute Identity.
july 30th, 2011Today I present a cute identity which appeared while explicitly computing the Gram-Schmidt orthogonalization of a base.
Solving Certain Linear Systems over the Integers.
june 17th, 2011We present a (well-known) method to compute a solution to the linear system Ax=b over the integers, when it is known that the determinant of A is non-zero and that a solution with integral coefficients exists. We also provide a running time analysis.
On a Certain Determinant.
march 25th, 2011In this post, I show how to explicitly compute a determinant. This determinant allows me to write down a closest solution in the 2-norm to a certain unsolvable linear system.
A Strange Inequality.
december 9th, 2010We show how to prove a number theoretic inequality, originating from geometry, using an elementary approach.
Multiplicity of the Determinant.
november 10th, 2010This post shows a way to quickly show that the determinant is multiplicative without getting your hands dirty.
Rigorous Arithmetic in the Arakelov Divisor Class Group of a Number Field.
july 27th, 2010This post presents a poster of mine presented at the poster session of the 9th Algorithmic Number Theory Symphoisum.
Inequalities.
february 9th, 2010This post shows a diagram, listing a lot of inequalities and showing implications between them.
How to Compute the 5-adic Expansion of 1/2; or: Hensel’s Lemma and (Non-Analytic) Newton Iteration.
february 6th, 2010In this post, we consider the quest of computing the 5-adic expansion of 1/2. We begin with introducing p-adic integers and numbers, and discussing when certain polynomials with coefficients in the integers have zeroes in the p-adic integers. This question is closely related to Hensel’s lemma, which can be proven using an algebraic version of Newton’s iteration. We use this to compute approximations of rational numbers in the p-adics, and consider which p-adic numers have an eventually periodic expansion.
Finding Lattice Points, Finite Abelian Groups, and Explaining Algorithms.
january 29th, 2010We compare the tasks of finding points of a lattice, computing the structure of finite abelian groups and explaining algorithms. We show up relations between these three topics and, as an example, depict the baby-step giant-step algorithm for order computation, as well as Terr’s modification of this algorithm.
