arXiv subjects

Lawren Smithline

Publications and source records attributed to Lawren Smithline.

4 recordsLinked to original sources

arXiv2026

Permutation Wordle

We introduce a guessing game, ``Permutation Wordle,'' in which a guesser attempts to recover a setter's hidden permutation of the set $\{1, \ldots, n\}$. In each round, the guesser submits a word over the alphabet $\{1, \ldots, n\}$, and, as in the game Wordle, learns which entries are correct. We describe a natural strategy and prove that it is optimal in a strong sense: for every $r$, it solves at least as many secrets within $r$ rounds as any possible strategy. The number of permutations it solves in exactly $k+1$ rounds is the Eulerian number $A(n,k)$.

math.CO↗

arXiv2023

A composition law and refined notions of convergence for periodic continued fractions

We define an equivalence relation on periodic continued fractions with partial quotients in a ring $\mathcal{O} \subseteq \mathbf{C}$, a group law on these equivalence classes, and a map from these equivalence classes to matrices in $\mathrm{GL}_2(\mathcal{O})$ with determinant $\pm1$. We prove this group of equivalence classes is isomorphic to $\mathbf{Z}/2\mathbf{Z}\ast\mathcal{O}$ and study certain of its one- and two-dimensional representations. For a periodic continued fraction with period $k$, we give a refined description of the limits of the $k$ different $k$-decimations of its sequence of convergents. We show that for a periodic continued fraction associated to a matrix with eigenvalues of different magnitudes, all $k$ of these limits exist in $\mathbb{P}^1(\mathbf{C})$ and a strict majority of them are equal.

math.NT↗

arXiv2007

Bounding slopes of $p$-adic modular forms

Let $p$ be prime, $N$ be a positive integer prime to $p$, and $k$ be an integer. Let $P_k(t)$ be the characteristic series for Atkin's $U$ operator as an endomorphism of $p$-adic overconvergent modular forms of tame level $N$ and weight $k$. Motivated by conjectures of Gouvea and Mazur, we strengthen Wan's congruence between coefficients of $P_k$ and $P_{k'}$ for $k'$ close $p$-adically to $k$. For $p-1 | 12$, $N = 1$, $k = 0$, we compute a matrix for $U$ whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for $p = 3$ a parabola below the Newton polygon $N_0$ of $P_0$, which coincides with $N_0$ infinitely often. As a consequence, we find a polygonal curve above $N_0$. This tightest bound on $N_0$ yields the strongest congruences between coefficients of $P_0$ and $P_k$ for $k$ of large 3-adic valuation.

math.AG↗

arXiv2004

Probabilistic pairwise sequence alignment

We describe an new algorithm for visualizing an alignment of biological sequences according to a probabilistic model of evolution. The resulting data array is readily interpreted by the human eye and amenable to digital image techniques. We present examples using mRNA sequences from mouse and rat: three cytochromes, Rattus norvegicus Cyp2a1, Cyp2a2, (Medline: 90212624) and Mus musculus Cyp2a12 (Medline: 93249380); and two zinc finger proteins, Mus musculus zfp111 and zfp235 (Medline: 22683274). The underlying evolutionary model is derived from one proposed by Thorne, Kishino, and Felsenstein and improved by Hein and others. The demonstration implementation aligns two sequences using time and memory quadratic in the mean sequence length. The algorithm is extensible, after Hein, to multiple sequences. We mention a basic method to reduce time and memory demands.

q-bio.PE↗