Related papers: The Liouville Generator for Producing Integrable E…

The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $\Delta f = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a…

M.F. Singer [Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673--688] proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential…

We have developed a symbolic algebra approach to automatically produce, verify, and optimize computer code for the Fast Multipole Method (FMM) operators. This approach allows for flexibility in choosing a basis set and kernel, and can…

We present solutions to the classical Liouville equation for ergodic and completely integrable systems - systems that are known to attain equilibrium. Ergodic systems are known to thermal equilibrate with a Maxwell-Boltzmann distribution…

Large language models (LLMs) have shown great abilities of solving various natural language tasks in different domains. Due to the training objective of LLMs and their pre-training data, LLMs are not very well equipped for tasks involving…

Current advances in Artificial Intelligence and machine learning in general, and deep learning in particular have reached unprecedented impact not only across research communities, but also over popular media channels. However, concerns…

Making neural visual generative models controllable by logical reasoning systems is promising for improving faithfulness, transparency, and generalizability. We propose the Abductive visual Generation (AbdGen) approach to build such…

Formal verification via interactive theorem proving is increasingly used to ensure the correctness of critical systems, yet constructing large proof scripts remains highly manual and limits scalability. Advances in large language models…

A method is given that "inverts" a logic grammar and displays it from the point of view of the logical form, rather than from that of the word string. LR-compiling techniques are used to allow a recursive-descent generation algorithm to…

We develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the…

A Liouville classification of a natural Hamiltonian system on the projective plane with a rotation metric and a linear integral is obtained. All Fomenko--Zieschang invariants (i.e., labeled molecules) of the system are calculated.

Machine Learning (ML)-based unfolding methods have enabled high-dimensional and unbinned differential cross section measurements. While a suite of such methods has been proposed, most focus exclusively on the challenge of statistically…

A number of examples of Hamiltonian systems that are integrable by classical means are cast within the framework of isospectral flows in loop algebras. These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger systems and…

This paper introduces a deep learning enabled generative sensing framework which integrates low-end sensors with computational intelligence to attain a high recognition accuracy on par with that attained with high-end sensors. The proposed…

This paper presents the design of HELIX, an end-to-end verified code generation system with a focus on the intersection of high-performance and high-assurance numerical computing. The code generation can be fine-tuned to generate efficient…

Recently proposed procedure of constructing maximally superintegrable systems of Winternitz type is further developed and illustrated by an example of system admitting an explicit construction of angle variables and additional integrals of…

In an earlier article, we presented a method to obtain integrals of motion and polynomial algebras for a class of two-dimensional superintegrable systems from creation and annihilation operators. We discuss the general case and present its…

Integrable systems with a linear periodic integral for the Lie algebra $\mathrm{e}(3)$ are considered. One investigates singulariries of the Liouville foliation, bifurcation diagram of the momentum mapping, transformations of Liouville…

Joint machine learning models that allow synthesizing and classifying data often offer uneven performance between those tasks or are unstable to train. In this work, we depart from a set of empirical observations that indicate the…

We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders…