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The computation of higher order processes very often involves a large number of diagrams. In addition, it is in general not possible to solve the occurring integrals explicitly and expansions in small quantities have to be performed. This…

We introduce the MathGR package, written in Mathematica. The package can manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices…

Exponential Linear Units (ELUs) are a useful rectifier for constructing deep learning architectures, as they may speed up and otherwise improve learning by virtue of not have vanishing gradients and by having mean activations near zero.…

The FormCalc package automates the computation of FeynArts amplitudes up to one loop including the generation of a Fortran code for the numerical evaluation of the squared matrix element. Major new or enhanced features in Version 5 are:…

Integration-by-parts (IBP) reduction of Feynman integrals to master integrals is a key computational bottleneck in precision calculations in high-energy physics. Traditional approaches based on the Laporta algorithm require solving large…

We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to…

Automatic differentiation is a technique which allows a programmer to define a numerical computation via compositions of a broad range of numeric and computational primitives and have the underlying system support the computation of partial…

The Yau-Yau nonlinear filter has increasingly emerged as a powerful tool to study stochastic complex systems. To leverage it to a wider spectrum of application scenarios, we pack the Yau-Yau filtering ALgorithms (YauYauAL) into a package of…

Radau IIA methods, specifically the adaptive order Radau method in Fortran due to Hairer, are known to be state-of-the-art for the high-accuracy solution of highly stiff ordinary differential equations (ODEs). However, the traditional…

We introduce LADDER (Learning through Autonomous Difficulty-Driven Example Recursion), a framework which enables Large Language Models to autonomously improve their problem-solving capabilities through self-guided learning by recursively…

This paper discusses a new method to solve definite integrals using artificial neural networks. The objective is to build a neural network that would be a novel alternative to pre-established numerical methods and with the help of a…

We propose extensions to Fortran which integrate forward and reverse Automatic Differentiation (AD) directly into the programming model. Irrespective of implementation technology, embedding AD constructs directly into the language extends…

We present Kaolin, a PyTorch library aiming to accelerate 3D deep learning research. Kaolin provides efficient implementations of differentiable 3D modules for use in deep learning systems. With functionality to load and preprocess several…

One of the happiest accidents in all math is the ease of transforming a function to and taking derivatives in the Fourier frequency domain. But in order to exploit this extraordinary fact without serious artefacting, and in order to be able…

We show that the discrete complex, and numerous hypercomplex, Fourier transforms defined and used so far by a number of researchers can be unified into a single framework based on a matrix exponential version of Euler's formula…

We describe a method for calculating the roots of special functions satisfying second order linear ordinary differential equations. It exploits the recent observation that the solutions of a large class of such equations can be represented…

We present a program for the numerical evaluation of form factors entering the calculation of one-loop amplitudes with up to six external legs. The program is written in Fortran95 and performs the reduction to a certain set of basis…

Machine learning and deep learning-based decision making has become part of today's software. The goal of this work is to ensure that machine learning and deep learning-based systems are as trusted as traditional software. Traditional…

This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…