arXiv subjects

Annika Mauro

Publications and source records attributed to Annika Mauro.

4 recordsLinked to original sources

arXiv2026

Skywing: A Platform for Decentralized Mathematical Computing in Unreliable Environments

Emerging edge, autonomous, and cyber-physical systems increasingly require mathematical computation across heterogeneous devices connected by unreliable communication networks. Traditional high-performance computing and distributed data-processing frameworks provide powerful abstractions for managed environments but are less suited to decentralized settings where centralized coordination, reliable communication, and global synchronization cannot be assumed. This paper presents Skywing, an open-source platform for decentralized mathematical computing in unreliable environments. Its programming model consists of three abstractions: agents represent participants in a decentralized computation, processors encapsulate algorithm-specific update rules, and iterations manage distributed execution. Skywing supports asynchronous operation, publish-subscribe communication, managed message handling, and the composition of independent algorithms into complex decentralized workflows. We demonstrate Skywing using representative algorithms from consensus, optimization, and numerical linear algebra. Experiments on the native Skywing runtime include Push Sum and Max Consensus, a composed monitoring and control workflow, resilient Push Sum under delayed communication, and resilient asynchronous Jacobi under malevolent data corruption. These demonstrations show that Skywing supports diverse decentralized algorithms while separating mathematical logic from communication and execution infrastructure. Skywing serves as both a deployment framework for decentralized applications and a research platform for developing resilient mathematical algorithms.

cs.DC↗

arXiv2024

An invitation to Fibonacci digits

The purpose of this short note is to show the interplay between math outreach and conducting original research, in particular how each can build off the other.

math.HO↗

arXiv2023

Extending the support of $1$- and $2$-level densities for cusp form $L$-functions under square-root cancellation hypotheses

The Katz-Sarnak philosophy predicts that the behavior of zeros near the central point in families of $L$-functions agrees with that of eigenvalues near 1 of random matrix ensembles. Under GRH, Iwaniec, Luo and Sarnak showed agreement in the one-level densities for cuspidal newforms with the support of the Fourier transform of the test function in $(-2, 2)$. They increased the support further under a square-root cancellation conjecture, showing that a ${\rm GL}(1)$ estimate led to additional agreement between number theory and random matrix theory. We formulate a two-dimensional analog and show it leads to improvements in the two-level density. Specifically, we show that a square-root cancellation of certain classical exponential sums over primes increases the support of the test functions such that the main terms in the $1$- and $2$-level densities of cuspidal newforms averaged over bounded weight $k$ (and fixed level $1$) converge to their random matrix theory predictions. We also conjecture a broad class of such exponential sums where we expect improvement in the case of arbitrary $n$-level densities, and note that the arguments in [ILS] yield larger support than claimed.

math.NT↗

arXiv2022

Recurrence Relations for $S$-Legal Index Difference Sequences

Zeckendorf's Theorem implies that the Fibonacci number $F_n$ is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the $i$-th and $j$-th squares are adjacent if and only if $|i - j| \in \{1, 3, 4\}$ or $\{i, j\} = \{1, 3\}$. We consider a generalization of this construction: given a set of positive integers $S$, the $S$-legal index difference ($S$-LID) sequence $(a_n)_{n=1}^\infty$ is defined by letting $a_n$ to be the smallest positive integer that cannot be written as $\sum_{\ell \in L} a_\ell$ for some set $L \subset [n-1]$ with $|i - j| \notin S$ for all $i, j \in L$. We discuss our results governing the growth of $S$-LID sequences, as well as results proving that many families of sets $S$ yield $S$-LID sequences which follow simple recurrence relations.

math.NT↗