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Classification Of The Subalgebras Of The Algebra Of All 2 By 2 Matrices, Justin Luis Bernal

Open Access Theses & Dissertations

Classification of the subalgebras of the familiar algebra of all $n\times n$ real matrices over the real numbers can get quite unwieldy as all subalgebras are of dimension ranging from $1$ to $n^2$. Classification of the subalgebras of the algebra of all $2\times 2$ real matrices over the real numbers is an interesting first start.

Since $\2$ is of dimension $4$ then its possible subalgebras are of dimension $1, 2, 3,$ or $4$. The one-dimensional subalgebra and four-dimensional subalgebra need little to no attention. The two-dimensional and three-dimensional subalgebras however turn out to be of significance.

It turns out there …

Fast Algorithm For Finding Lattice Subspaces In Rn And Its Implementation, Andrew Martin Pownuk

Open Access Theses & Dissertations

There are known necessary and sufficient conditions for a subspace of Rm to be lattice-ordered. Let Y = {y1,…,ym} and yi are rows of the matrix X. A subspace ⟨X⟩, of linear space generated by the set X of n linearly independent positive vectors is lattice-ordered if and only the set X admits a fundamental set of indices I, which means that the subset YI ⊆ Y of vectors indexed by I is linearly independent, and every vector from Y\YI is a nonnegative linear combination of vectors form YI.

In economics it is possible to prove that the minimum-cost insured …

Generalizations Of Dirichlet Convolution, Juan Carlos Villarreal

Open Access Theses & Dissertations

The thesis reviews Dirichlet convolution of arithmetic functions and some generalizations. First, we present the main properties of the Dirichlet convolution on the set of arithmetic functions and some characterizations of prime numbers via arithmetic functions are given. Then we explore the K-convolution and the regular convolutions and finally we extend the study of Dirichlet convolution in categories. We give also an example for the evaluation of the Mobius function of a Mobius category.

Incidence Functions, Yiyu Liao

Open Access Theses & Dissertations

In the mid 1960's, the incidence algebra was introduced in the seminal paper of Gian-Carlo Rota. He addressed the importance of the Mobius function in combinatorics. In particular, the incidence algebra of a locally finite poset plays an essentially unifying role in the theory of the Mobius function. One of the significant generalizations is the incidence algebra of a Mobius category introduced by Pierre Leroux. With the help from Mobius category, it was exciting to be able to extend the combinatorial results more broadly than just on posets. Before attempting to study this generalization of the Mobius function, we have …