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Full-Text Articles in Other Applied Mathematics
Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper
Existence And Uniqueness Of Minimizers For A Nonlocal Variational Problem, Michael Pieper
Honors Theses
Nonlocal modeling is a rapidly growing field, with a vast array of applications and connections to questions in pure math. One goal of this work is to present an approachable introduction to the field and an invitation to the reader to explore it more deeply. In particular, we explore connections between nonlocal operators and classical problems in the calculus of variations. Using a well-known approach, known simply as The Direct Method, we establish well-posedness for a class of variational problems involving a nonlocal first-order differential operator. Some simple numerical experiments demonstrate the behavior of these problems for specific choices of …
Decoding Cyclic Codes Via Gröbner Bases, Eduardo Sosa
Decoding Cyclic Codes Via Gröbner Bases, Eduardo Sosa
Honors Theses
In this paper, we analyze the decoding of cyclic codes. First, we introduce linear and cyclic codes, standard decoding processes, and some standard theorems in coding theory. Then, we will introduce Gr¨obner Bases, and describe their connection to the decoding of cyclic codes. Finally, we go in-depth into how we decode cyclic codes using the key equation, and how a breakthrough by A. Brinton Cooper on decoding BCH codes using Gr¨obner Bases gave rise to the search for a polynomial-time algorithm that could someday decode any cyclic code. We discuss the different approaches taken toward developing such an algorithm and …