arXiv subjects
Publications and source records attributed to John Mangual.
We give an elementary derivation of the vertex-operator derivation McMahon formula, counting all plane partitions of all size into a single generating function. We fill in some details appearing in Okounkov, Reshetikhin, and Vafa based on free fermions by defining an "interlacing operator".
If we pick two elements of a non-abelian group at random, the odds this pair commutes is at most 5/8, so there is a "gap" between abelian and non-abelian groups \cite{G}. We prove a "topological" generalization estimating the odds a word presenting the fundamental group of an orientable surface $ $ is satisfied. This resolves a conjecture by Langley, Levitt and Rower.
For uniformly chosen random $α\in [0,1]$, it is known the probability the $n^{\rm th}$ digit of the continued-fraction expansion, $[α]_n$ converges to the Gauss-Kuzmin distribution $\mathbb{P}([α]_n = k) \approx \log_2 (1 + 1/ k(k+2))$ as $n \to \infty$. In this paper, we show the continued fraction digits of $\sqrt{d}$, which are eventually periodic, also converge to the Gauss-Kuzmin distribution as $d \to \infty$ with bounded class number, $h(d)$. The proof uses properties of the geodesic flow in the unit tangent bundle of the modular surface, $T^1(\text{SL}_2 \mathbb{Z}\backslash \mathbb{H})$.
The roots of Bernoulli polynomials, $B_n(z)$, when plotted in the complex plane, accumulate around a peculiar H-shaped curve. Karl Dilcher proved in 1987 that, on compact subsets of $\mathbb{C}$, the Bernoulli polynomials asymptotically behave like sine or cosine. Here we establish the asmptotic behavior of $B_n(nz)$, compute the distribution of real roots of Bernoulli polynomials and show that, properly rescaled, the complex roots lie on the curve $e^{- 2π\text{Im}(z)} = 2πe |z|$ or $e^{2π\text{Im}(z)}= 2πe |z|$.