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Full-Text Articles in Discrete Mathematics and Combinatorics
Foundations Of Memory Capacity In Models Of Neural Cognition, Chandradeep Chowdhury
Foundations Of Memory Capacity In Models Of Neural Cognition, Chandradeep Chowdhury
Master's Theses
A central problem in neuroscience is to understand how memories are formed as a result of the activities of neurons. Valiant’s neuroidal model attempted to address this question by modeling the brain as a random graph and memories as subgraphs within that graph. However the question of memory capacity within that model has not been explored: how many memories can the brain hold? Valiant introduced the concept of interference between memories as the defining factor for capacity; excessive interference signals the model has reached capacity. Since then, exploration of capacity has been limited, but recent investigations have delved into the …
Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov
Reverse Mathematics Of Ramsey's Theorem, Nikolay Maslov
Electronic Theses, Projects, and Dissertations
Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig's lemma …
Minimal Sets, Union-Closed Families, And Frankl's Conjecture, Christopher S. Flippen
Minimal Sets, Union-Closed Families, And Frankl's Conjecture, Christopher S. Flippen
Theses and Dissertations
The most common statement of Frankl's conjecture is that for every finite family of sets closed under the union operation, there is some element which belongs to at least half of the sets in the family. Despite its apparent simplicity, Frankl's conjecture has remained open and highly researched since its first mention in 1979. In this paper, we begin by examining the history and previous attempts at solving the conjecture. Using these previous ideas, we introduce the concepts of minimal sets and minimally-generated families, some ideas related to viewing union-closed families as posets, and some constructions of families involving poset-defined …