Archive for the ‘Algebra’ Category
Obtaining Infrastructures from Global Fields.
tuesday, july 21st, 2009We show how to obtain n-dimensional infrastructures from global fields of unit rank n. We will also discuss how to obtain baby steps in these cases, and show graphical representations of certain two-dimensional infrastructures obtained from function fields.
How to Obtain Reduction Maps for n-dimensional Infrastructures.
tuesday, july 21st, 2009We explain a general technique to obtain a reduction map, given X and d and, varying with the method of construction, additional information for every x in X. Moreover, we explain a technique on how to obtain n-dimensional infrastructures from certain lattices in (n+1)-dimensional space.
n-dimensional Infrastructures.
monday, july 20th, 2009We will introduce n-dimensional infrastructures and briefly discuss reductions, f-representations and giant steps. We will also discuss how infrastructures can be obtained from finite abelian groups.
Interpreting One-dimensional Infrastructures as Groups: f-Representations.
monday, july 20th, 2009We introduce the notion of f-representations and relate them to reduction maps. Moreover, we equip a set of f-representations with a group operation which can be computed purely with baby steps, giant steps and relative distances.
One-dimensional Infrastructures.
monday, july 20th, 2009We give the definition of one-dimensional infrastructures and construct baby and giant steps. Moreover, we show that one-dimensional infrastructures generalize finite cyclic groups. Finally, we give some remarks on our choice of the giant step definition.
The Discrete Logarithm Problem and Generalizations.
monday, july 20th, 2009We discuss the discrete logarithm problem, its use in cryptography, and two possible directions of generalization to other algebraic structures.
A Topological Proof of the Cayley-Hamilton Theorem over all Commutative Unitary Rings.
monday, may 4th, 2009We want to give a proof of the Cayley-Hamilton Theorem for all commutative rings with unity, which first reduces to the case of the field of complex numbers and then applies a topological argument.
Fundamental Theorem of Algebra.
monday, may 4th, 2009We want to give a proof of the Fundamental Theorem of Algebra using methods from Complex Analysis, in particular Liouville’s Theorem.
