Related papers: Jet Functors and Weil Algebras in Automatic Differ…
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical…
High index differential algebraic equations (DAEs) are ordinary differential equations (ODEs) with constraints and arise frequently from many mathematical models of physical phenomenons and engineering fields. In this paper, we generalize…
Differentiable programming is revolutionizing computational science by enabling automatic differentiation (AD) of numerical simulations. While first-order gradients are well-established, second-order derivatives (Hessians) for implicit…
Trajectory prediction is central to the safe and seamless operation of autonomous vehicles (AVs). In deployment, however, prediction models inevitably face distribution shifts between training data and real-world conditions, where rare or…
In this article, following an insight of Kontsevich, we extend the famous Weil conjecture (as well as the strong form of the Tate conjecture) from the realm of algebraic geometry to the broad noncommutative setting of dg categories. As a…
Fractional-order differentiation has many characteristics different from integer-order differentiation. These characteristics can be applied to the optimization algorithms of artificial neural networks to obtain better results. However, due…
We survey a new application of the Weil representation to construct a canonical basis of eigenvectors for the discrete Fourier transform (DFT). The transition matrix from the standard basis to the canonical basis defines a novel transform…
We present a systematic methodology to develop high order accurate numerical approaches for linear advection problems. These methods are based on evolving parts of the jet of the solution in time, and are thus called jet schemes. Through…
Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
The principal objective in this paer is to study the relationship between the old kingdom of differential geometry (the category of smooth manifolds) and its new kingdom (the category of functors on the category of Weil algebras to some…
The concept of integro-differential algebra has been introduced recently in the study of boundary problems of differential equations. We generalize this concept to that of integro-differential algebra with a weight, in analogy to the…
We present some methods and results in the application of algebraic geometry and computer algebra to the study of algebraic vector bundles, foliations and zeta functions. A connection of the methods and results with noncommutative geometry…
This document presents a new C++ Automatic Differentiation (AD) tool, AD-HOC (Automatic Differentiation for High-Order Calculations). This tool aims to have the following features: -Calculation of user specified derivatives of arbitrary…
Differentiable programming allows for derivatives of functions implemented via computer code to be calculated automatically. These derivatives are calculated using automatic differentiation (AD). This thesis explores two applications of…
We calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields. In particular, there is a long exact sequence relating Weil-etale cohomology and etale…
Just like decent classical difference-difference systems define symplectic maps on suitable phase spaces, their counterparts with properly ordered noncommutative entries come as Heisenberg equations of motion for corresponding quantum…
We show that the covariant analytic mechanics (CAM) is closely related to the De Donder-Weyl (DW) theory. To treat space and time on an equal footing, the DW theory introduces $D$ conjugate fields ($D$ is the dimension of space-time) for…
Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are…
There are a variety of choices to be made in both computer algebra systems (CASs) and satisfiability modulo theory (SMT) solvers which can impact performance without affecting mathematical correctness. Such choices are candidates for…
A symmetric function of $N$ variables can be given in terms of symmetric polynomials of these variables. We determine those symmetric polynomials in which the dual differential operators take the neatest form when expressed in terms of our…