GAP (Groups, Algorithms, and Programming) is a system for computational discrete algebra with particular emphasis on computational group theory.
Examples are logical modeling in multiple variants, from Boolean or multi-valued networks [ 14- 16] to discrete algebra [ 17] and differential equations [ 18], Petri nets [ 19] and predicate logic [ 20].
Reinhard Laubenbacher's Discrete Mathematics Group at the Virginia Bioinformatics Institute (VBI) has run two biomathematics workshops for in-service high school teachers connecting topics in modern discrete algebra to the Virginia public schools' required Standards of Learning (Martins et al., 2008).
The notation C A, B) is generally used to denote the set of all homomorphisms from A to B. And the set of all functions from set X to set Y can be understood as the particular case Set X, Y) of this convention where C is taken to be the class Set of all sets, which we can think of as discrete algebras, that is, algebras with no structure.
Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups etc. Consult your math teacher.
In Section 4, the concept of basic adaptive discretization is presented and the underlying Lie-algebra structure of discrete Heisenberg algebras is given.
As an application, we describe the flow of weights of the crossed product of a type III factor by some special action of a discrete Kac algebra.
Whereas geometry treats spatial entities, analysis continuous variation, number theory integer arithmetic, and combinatorics discrete structures, algebra is equally applicable to all these and other mathematical domains.
This is natural since already the discrete exterior algebra deals with functions on the set G = ⋃ k G k of simplices of the graph G, where G 0 = V is the set of vertices G 1 = E is the number edges, G 2 the set of triangles in G etc. The Euler characteristic is the graded cardinality ∑ k ( − 1 ) k | G k | = ∑ k ( − 1 ) k v k of G.
However, we have used parts of the modules as problems and projects in several mathematics and biology courses (from the elementary to the advanced level) including genetics, biomathematics, discrete mathematics, linear algebra, probability, geometry, algebraic geometry, and abstract algebra.
Mendel was lucky that the discrete symbols of algebra agreed with the discrete units of inheritance.
