arXiv subjects
Publications and source records attributed to Senan Sekhon.
In arXiv:2511.13287, we derived exact formulas for generating functions counting the number of $n$-ary words avoiding \textit{strictly} increasing subwords of length $k$, and provided applications in probability theory as well as the continuous limit as $n\to\infty$. We also conjectured several corresponding formulas for the case where the ``strictly'' requirement is dropped. In this paper, we prove those formulas.
In 1916, Hausdorff proved that the Baire hierarchy on $\mathbb{R}$, starting with the continuous functions, generates all Borel functions on $\mathbb{R}$. It remained open whether, starting with the a.e. continuous functions, the corresponding hierarchy generates all Lebesgue measurable functions on $\mathbb{R}$. We prove that, assuming the Axiom of Choice, the answer is negative.
A well-known result from the 1960s characterizes all Euclidean domains in which division is guaranteed to produce a unique quotient and remainder. As this relies on the historical (and more restrictive) definition of a Euclidean domain, the question of whether the result still holds under the modern definition was left open. In this paper, we prove the answer is afirmative.
In this paper, we derive exact formulas for generating functions counting the number of $n$-ary words avoiding strictly increasing subwords of length $k$, and provide some applications of these formulas.
In this project, we will develop the foundations of quantum mechanics using the methods of supersymmetry. We will discuss the use of the superpotential to derive the supersymmetric partner of a potential in one dimension, and explore several key examples with an emphasis on shape invariant potentials. We will then discuss the modeling of supersymmetric quantum systems using matrices and operators, and how it relates to the eigenstate thermalization hypothesis.
We investigate a result on convergence of double sequences of numbers and how it extends to measurable functions.
We show that the set of all measures on any measurable space is a complete lattice, i.e. every collection of measures has both a greatest lower bound and a least upper bound.
In this project, we will develop the theory of Banach algebras and prove two celebrated theorems, the Gelfand representation theorem and the GKZ theorem. We will then proceed to develop the theory of $C^*$-algebras and prove the Gelfand-Naimark theorem and the GNS theorem. Finally, we will show how they can be used to give an alternative formulation of quantum mechanics.