Related papers: The Liouville Generator for Producing Integrable E…
There has been an increasing number of applications of machine learning to the field of Computer Algebra in recent years, including to the prominent sub-field of Symbolic Integration. However, machine learning models require an abundance of…
We provide a Liouville principle for integration in terms of elliptic integrals. Our methods are essentially those of Abel and Liouville changed to modern notation. We expose Lie theoretic aspect of Liouville's work.
Integration is indispensable, not only in mathematics, but also in a wide range of other fields. A deep learning method has recently been developed and shown to be capable of integrating mathematical functions that could not previously be…
A critical question about Large Language Models (LLMs) is whether their apparent deficiency in mathematical reasoning is inherent, or merely a result of insufficient exposure to high-quality mathematical data. To explore this, we developed…
This paper provides a Liouville principle for integration in terms of dilogarithm and partial result for polylogarithm.
Mixed-integer linear programming (MILP) stands as a notable NP-hard problem pivotal to numerous crucial industrial applications. The development of effective algorithms, the tuning of solvers, and the training of machine learning models for…
Some more general "inheritance conditions" have been found for a given set of symmetry generators $\{\mathbf{Z}_{\bar{l}}\}$ acting on some set of coupled ordinary differential equations, once the "first integration method" has been applied…
Large-scale natural language generation requires the integration of vast amounts of knowledge: lexical, grammatical, and conceptual. A robust generator must be able to operate well even when pieces of knowledge are missing. It must also be…
Here we present an efficient method for finding and using a nonlocal symmetry admitted by a rational second order ordinary differential equation (rational 2ODE) in order to find a Liouvillian first integral (belonging to a vast class of…
We present a novel introspective variational autoencoder (IntroVAE) model for synthesizing high-resolution photographic images. IntroVAE is capable of self-evaluating the quality of its generated samples and improving itself accordingly.…
Large Language Models (LLMs) are computational models capable of performing complex natural language processing tasks. Leveraging these capabilities, LLMs hold the potential to transform the entire hardware design stack, with predictions…
We consider complex rational vector fields in dimension $n>2$ (equivalently, differential forms of degree $n-1$ in $n$ variables) which admit a Liouvillian first integral. Extending a classical result by Singer for $n=2$, our main result…
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…
This paper provides a Liouville principle for integration in terms of exponential integrals and incomplete gamma functions.
Feature transformation enhances data representation by deriving new features from the original data. Generative AI offers potential for this task, but faces challenges in stable generation (consistent outputs) and valid generation…
The classical Liouvile integrability means that there exist $n$ independent first integrals in involution for $2n$-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in…
MulAtoLEG (Multi-Atom Liouville Equation Generator) is an open-source Mathematica package for generating Liouville superoperators and Liouville equations, specialized for multilevel atomic systems comprising an arbitrary number of atoms.…
We introduce SymbolicAI, a versatile and modular framework employing a logic-based approach to concept learning and flow management in generative processes. SymbolicAI enables the seamless integration of generative models with a diverse…
Generating effective test inputs for a software system requires that these inputs be valid, as they will otherwise be rejected without reaching actual functionality. In the absence of a specification for the input language, common test…
We extend the theorem of Liouville on integration in finite terms to include dilogarithmic integrals. The results provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension…