Related papers: Higher order differential calculus in mathlib
Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the…
We implement a user-extensible ad hoc connection between the Lean proof assistant and the computer algebra system Mathematica. By reflecting the syntax of each system in the other and providing a flexible interface for extending…
LeoPARD supports the implementation of knowledge representation and reasoning tools for higher-order logic(s). It combines a sophisticated data structure layer (polymorphically typed {\lambda}-calculus with nameless spine notation, explicit…
We present a high-order method that provides numerical integration on volumes, surfaces, and lines defined implicitly by two smooth intersecting level sets. To approximate the integrals, the method maps quadrature rules defined on…
We propose to use Church encodings in typed lambda-calculi as the basis for an automata-theoretic counterpart of implicit computational complexity, in the same way that monadic second-order logic provides a counterpart to descriptive…
We pose a new algebraic formalism for studying differential calculus in vector bundles. This is achieved by studying various functors of differential calculus over arbitrary graded commutative algebras (DCGCA) and applying this language to…
In this work, we propose to extend an approach to calculate at any order $(n)$, the functional derivative of the diffracted field with respect to the permittivity-contrast function. These derivatives obtained for different orders are used…
The `equation-free toolbox' empowers the computer-assisted analysis of complex, multiscale systems. Its aim is to enable you to immediately use microscopic simulators to perform macro-scale system level tasks and analysis, because…
Given a smooth manifold $M$ (with or without boundary), in this paper we establish a global functional calculus (without the standard assumption that the operators are classical pseudo-differential operators) and the G\r{a}rding inequality…
Verifiable formal languages like Lean have profoundly impacted mathematical reasoning, particularly through the use of large language models (LLMs) for automated reasoning. A significant challenge in training LLMs for these formal languages…
We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus…
The differentiable programming paradigm is a cornerstone of modern scientific computing. It refers to numerical methods for computing the gradient of a numerical model's output. Many scientific models are based on differential equations,…
We introduce a functional calculus with simple syntax and operational semantics in which the calculi introduced so far in the Curry-Howard correspondence for Classical Logic can be faithfully encoded. Our calculus enjoys confluence without…
Table extraction from PDF and image documents is a ubiquitous task in the real-world. Perfect extraction quality is difficult to achieve with one single out-of-box model due to (1) the wide variety of table styles, (2) the lack of training…
$\lambda$-Scale is an enrichment of lambda calculus which is adapted to emergent algebras. It can be used therefore in metric spaces with dilations.
Here are considered some categorical aspects of "Differential calculus" archetype of local approximation of arbitrary morphisms by "linear" ones.
We propose the QHyper library, which is aimed at researchers working on computational experiments with a variety of quantum combinatorial optimization solvers. The library offers a simple and extensible interface for formulating…
Developing a 21st Century Global Library for Mathematics Research discusses how information about what the mathematical literature contains can be formalized and made easier to express, encode, and explore. Many of the tools necessary to…
We present a prototypical linear algebra compiler that automatically exploits domain-specific knowledge to generate high-performance algorithms. The input to the compiler is a target equation together with knowledge of both the structure of…
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness…