Related papers: Langlands Program and Ramanujan Conjecture: a surv…
We study the properties, in particular termination, of dependent types systems for lambda calculus and rewriting.
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
We introduce a novel approach addressing global analysis of a difficult class of nonconvex-nonsmooth optimization problems within the important framework of Lagrangian-based methods. This genuine nonlinear class captures many problems in…
This report showcases the role of, and future directions for, the field of Randomized Numerical Linear Algebra (RNLA) in a selection of scientific applications. These applications span the domains of imaging, genomics and dynamical systems,…
The evolution of mathematics is shaped importantly by interestingness: researchers choose which problems to pursue, and students choose which problems to engage with, based on expectations of interest and challenge. As AI systems,…
Large Language Models (LLMs) have demonstrated their transformative potential across numerous disciplinary studies, reshaping the existing research methodologies and fostering interdisciplinary collaboration. However, a systematic…
We study properties of programs with monotone and convex constraints. We extend to these formalisms concepts and results from normal logic programming. They include the notions of strong and uniform equivalence with their characterizations,…
We prove a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$, and conjecture a type of supercongruences associated to them. We support this conjecture by checking all the cases for many…
Interestingly, LLMs yet struggle with some basic tasks that humans find trivial to handle, e.g., counting the number of character r's in the word "strawberry". There are several popular conjectures (e.g., tokenization, architecture and…
We study a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main achievement is the definition of a family of related action functionals containing…
In this article we present ways to evaluate certain sums, products and continued fractions using tools from the theory of elliptic functions. The specific results appear to be new, although similar ones can be found in the leterature; in…
In this short note we use the umbral formalism to derive the Ramanujan Master Theorem and discuss its extension to more general cases.
Penalized regression methods, most notably the lasso, are a popular approach to analyzing high-dimensional data. An attractive property of the lasso is that it naturally performs variable selection. An important area of concern, however, is…
By computing reducibility points of parabolically induced representations, we construct, to within at most two unramified quadratic characters, the Langlands parameter of an arbitrary depth zero irreducible cuspidal representation $\pi$ of…
This paper studies a strategy for data-driven algorithm design for large-scale combinatorial optimization problems that can leverage existing state-of-the-art solvers in general purpose ways. The goal is to arrive at new approaches that can…
We introduce a generalized framework for studying higher-order versions of the multiscale method known as Localized Orthogonal Decomposition. Through a suitable reformulation, we are able to accommodate both conforming and nonconforming…
With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to…
Large language models (LLMs) have demonstrated remarkable capabilities across various NLP tasks and have recently expanded their impact to coding tasks, bridging the gap between natural languages (NL) and programming languages (PL). This…
We investigate forms on supermanifolds defined as Lagrangians of ``copaths'' (that is, systems of equations, which may or may not specify submanifolds). For this, we consider direct products $M^{n|m}\times\Bbb R^{r|s}$ and study…
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design…