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Glenn Bruda

Publications and source records attributed to Glenn Bruda.

10 recordsLinked to original sources
arXiv2026

Enumerating binary words restricted by subsequence frequency

Let $p$ be a binary word of length $\ell$ with $r\geq2$ runs. Previously known only for $k\leq4$, we show for $n$ sufficiently large that the number of binary words of length $n$ with exactly $k$ subsequences equal to $p$ is polynomial in $n$ of degree at most $\ell-r+1$ for any positive integer $k$. We also prove a sharp upper bound on the number of subsequences equal to $p$ of a binary word $w$ in terms of the runs of $p$ and $w$.

math.CO↗
arXiv2026

Generalized polygonal number representations

Let $r_n^{k}(N)$ be the number of representations of $N$ as the sum of $n$ generalized $k$-gonal numbers and $r_n^{\square}(N)$ be the number of representations of $N$ as the sum of $n$ squares. By modifying the Heath-Brown circle method, we prove a closed-form asymptotic relation between $r_n^{k}(N)$ and $r_n^{\square}(8(k-2)N+n(k-4)^2)$ for any $k\geq3$ and any $n\geq4$. Consequently, we determine the asymptotics of $\sum_{N\leq x}r_4^{k}(N)^2$ and, via a result of Bringmann, Jang, Kane, and Tse, prove a similar closed-form asymptotic relation between the number $r_{4,+}^{k}(N)$ of representations of $N$ as the sum of four ordinary $k$-gonal numbers and $r_4^{\square}(8(k-2)N+n(k-4)^2)$. We also show that if $4\mid k$, any strictly increasing infinite subsequence on which $r_{4,+}^{k}$ is bounded converges $2$-adically to $(k-4)^2/(4-2k)\in\mathbb{Z}_2$, supplementing a result of Meng and Sun, and if $4\nmid k$, there is no strictly increasing infinite subsequence on which $r_{4,+}^{k}$ is bounded.

math.NT↗
arXiv2025

Variants of Conway Checkers and k-nacci Jumping

Conway Checkers is a game played with a checker placed in each square of the lower half of an infinite checkerboard. Pieces move by jumping over an adjacent checker, removing the checker jumped over. Conway showed that it is not possible to reach row 5 in finitely many moves by weighting each cell in the board by powers of the golden ratio such that no move increases the total weight. Other authors have considered the game played on many different boards, including generalising the standard game to higher dimensions. We work on a board of arbitrary dimension, where we allow a cell to hold multiple checkers and begin with m checkers on each cell. We derive an upper bound and a constructive lower bound on the height that can be reached, such that the upper bound almost never fails to be equal to the lower bound. We also consider the more general case where instead of jumping over 1 checker, each checker moves by jumping over k checkers, and again show the maximum height reachable lies within bounds that are almost always equal.

math.CO↗
arXiv2025

A closed formula for linear recurrences with constant coefficients

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed formula for the coefficients of each polynomial $p_r(n)$ for any linear recurrence of this form.

math.CO↗
arXiv2025

The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation

Inspired by the quantization of classical quantities and Rankin Selberg convolution, we study the anticommutator operation $\{\cdot, \cdot\}$, where $\{A,B\} = AB + BA$, applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the $k$-checkerboard ensemble, and the block $k$-circulant ensemble ($k$-BCE). Using combinatorial and topological techniques related to non-crossing and free matching properties of GOE and PTE, we obtain closed-form formulae for the moments of the limiting spectral distributions of $\{$GOE, GOE$\}$, $\{$PTE, PTE$\}$, $\{$GOE, PTE$\}$ and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, $\{$GOE, $k$-checkerboard$\}$ and $\{$$k$-checkerboard, $j$-checkerboard$\}$ exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of $\{$GOE, $k$-checkerboard$\}$ consists of 1 bulk regime of size $Θ(N)$ and 1 blip regime of size $Θ(N^{3/2})$, the spectrum of $\{$$k$-checkerboard, $j$-checkerboard$\}$ consists of 1 bulk regime of size $Θ(N)$, 2 intermediary blip regimes of size $Θ(N^{3/2})$, and 1 largest blip regime of size $Θ(N^2)$. In both cases, with the appropriate weight function, we are able to isolate the largest regime for other regime(s) and analyze its moments and convergence results via combinatorics. We end with numerical computation of lower even moments of $\{$GOE, $k$-BCE$\}$ and $\{$$k$-BCE, $k$-BCE$\}$ based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of $\{$$k$-checkerboard, $j$-checkerboard$\}$.

math.PR↗
arXiv2024

Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals

Defining a family of recurrences, we generalize Comtet's formula for the generating function of the enumeration of indecomposable permutations. Consequently, we generalize Panaitopol's asymptotic expansion for the prime counting function, obtaining asymptotic expansions salient to the first Hardy-Littlewood conjecture.

math.CO↗
arXiv2024

Gromov-Hausdorff distances between quotient metric spaces

The Hausdorff distance measures how far apart two sets are in a common metric space. By contrast, the Gromov-Hausdorff distance provides a notion of distance between two abstract metric spaces. How do these distances behave for quotients of spaces under group actions? Suppose a group $G$ acts by isometries on two metric spaces $X$ and $Y$. In this article, we study how the Hausdorff and Gromov-Hausdorff distances between $X$ and $Y$ and their quotient spaces $X/G$ and $Y/G$ are related. For the Hausdorff distance, we show that if $X$ and $Y$ are $G$-invariant subsets of a common metric space, then we have $d_{\mathrm{H}}(X,Y)=d_{\mathrm{H}}(X/G,Y/G)$. However, the Gromov-Hausdorff distance does not preserve this relationship: we show how to make the ratio $\frac{d_{\mathrm{GH}}(X/G,Y/G)}{d_{\mathrm{GH}}(X,Y)}$ both arbitrarily large and arbitrarily small, even if $X$ is an arbitrarily dense $G$-invariant subset of $Y$.

math.MG↗
arXiv2024

On the convergence of Newton series and the asymptotics of finite differences

Suppose a complex function $f$ has a Lebesgue measurable inverse Laplace transform. We show that the $n$th order forward and backward differences of $f$ at $z_0\in\mathbb{C}$ tend to zero as $n\to\infty$ whenever $z_0$ lies in the region of absolute convergence of $f$. Under the same hypothesis, we show that the Newton series of $f$ centered at $z_0$ exists and converges in the half-plane $\Re(z)>\Re(z_0)$. Assuming instead that $f$ has a Lebesgue measurable inverse Fourier transform, we show that the $n$th order forward, backward, and central differences of $f$ at any $y\in\mathbb{R}$ are $o(2^n)$. Consequently, we show that the binomial sum $\sum_{k\geq0}{n\choose k}f(k)$ is $o(2^n)$.

math.CV↗
arXiv2022

Maclaurin Integration: A Weapon Against Infamous Integrals

Maclaurin Integration is a new series-based technique for solving infamously difficult integrals in terms of elementary functions. It has fairly liberal conditions for sound use, making it one of the most versatile integration techniques. Additionally, there is essentially zero human labor involved in calculating integrals using this technique, making it one of the easiest integration techniques to use. Its scope is mainly in pure mathematics.

math.GM↗