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Applied Mathematics Commons™

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2026

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Full-Text Articles in Applied Mathematics

Metric And Dominant Metric Dimension Of Weakly Zero-Divisor Graphs, Adnan Hussain Jan 2026

Metric And Dominant Metric Dimension Of Weakly Zero-Divisor Graphs, Adnan Hussain

Master of Science in Mathematics

This thesis investigates the metric dimension and dominant metric dimension of weakly zero-divisor graphs associated with commutative rings. The study focuses on determining resolving sets and dominating resolving sets for several classes of Γ(Z𝑛) graphs. These results establish connections between algebraic properties of rings and combinatorial characteristics of their associated graphs.

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All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe Jan 2026

All Games Have Equilibria, M. Ali Khan, Arthur Paul Pedersen, Maxwell B. Stinchcombe

Publications and Research

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same …

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Arrangements Of N Planes Resulting In One Bounded Tetrahedral Chamber, Ava Knight Jan 2026

Arrangements Of N Planes Resulting In One Bounded Tetrahedral Chamber, Ava Knight

Williams Honors College, Honors Research Projects

This paper investigates the combinatorial geometry of plane arrangements in three-dimensional space, focusing on configurations that produce exactly one bounded tetrahedral chamber. We define T(n) as the number of face-combinatorial equivalence classes of arrangements of n planes in ℝ³ containing exactly one bounded tetrahedral chamber. Known values — T(3) = 0, T(4) = 1, and T(5) = 2 — are established through direct construction, while T(6) remains an open problem. This paper contributes experimental evidence toward resolving T(6) by systematically extending the two valid 5-plane arrangements and verifying, through a plane removal argument, that each yields a valid plane configuration …

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Mathematical Model Of Graphene, Douglas M. Sanor Jan 2026

Mathematical Model Of Graphene, Douglas M. Sanor

Williams Honors College, Honors Research Projects

Graphene, a single-atom-thick layer of carbon arranged in a hexagonal lattice, exhibits exceptional mechanical, electrical, and thermal properties that make it a promising material for a wide range of engineering applications. This paper presents a mathematical framework for modeling the mechanical behavior of graphene, with a focus on atomistic-to-continuum approaches. We begin with a onedimensional Frenkel-Kontorova model that represents graphene as a discrete chain of particles interacting with both their nearest neighbors through harmonic spring potentials and an underlying substrate through van der Waals forces. Numerical simulations of this discrete model demonstrate the commensurate-toincommensurate phase transition, revealing how geometric mismatch …

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