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We present OGRe, a modern Mathematica package for tensor calculus, designed to be both powerful and user-friendly. The package can be used in a variety of contexts where tensor calculations are needed, in both mathematics and physics, but…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
Due to the occurrence of large exceptional Lie groups in supergravity, calculations involving explicit Lie algebra and Lie group element manipulations easily become very complicated and hence also error-prone if done by hand. Research on…
Numerical tensor calculus comprise basic tensor operations such as the entrywise addition and contraction of higher-order tensors. We present, TLib, flexible tensor framework with generic tensor functions and tensor classes that assists…
These lectures given to graduate students in theoretical particle physics, provide an introduction to the ``inner workings'' of computer algebra systems. Computer algebra has become an indispensable tool for precision calculations in…
GRLite and GRTensorJ are first and second generation graphical user interfaces to the computer algebra system GRTensorII. Current development centers on GRTensorJ, which provides fully customizable symbolic procedures that reduce many…
The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main…
We demonstrate how methods in Functional Programming can be used to implement a computer algebra system. As a proof-of-concept, we present the computational-algebra package. It is a computer algebra system implemented as an embedded…
A formula for the Riemannian metric tensor of differentiable manifolds of linear dynamical systems of same McMillan degree is presented in terms of their transfer function matrices. The necessary calculations for its application to ARMA and…
We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A-G based on the explicit construction of weight states. This straightforward approach (which is slower and more…
Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described…
It is our intention here only to discuss the nature, complexity and tools concerning the design of Smart Help, an expert help facility for aiding users of Computer Algebra Systems. Although the expert help system presented here has been…
A number of popular systems, most notably Google's TensorFlow, have been implemented from the ground up to support machine learning tasks. We consider how to make a very small set of changes to a modern relational database management system…
We present different methods for symbolic computer algebra computations in higher dimensional (\ge9) Clifford algebras using the \Clifford\ and \Bigebra\ packages for \Maple(R). This is achieved using graded tensor decompositions,…
In mathematics, many notations have been invented for the concise representation of mathematical formulae. Tensor index notation is one of such notations and has been playing a crucial role in describing formulae in mathematical physics.…
I describe a method for computer algebra that helps with laborious calculations typically encountered in theoretical microhydrodynamics. The program mimics how humans calculate by matching patterns and making replacements according to the…
We present the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie Algebras and representation theory, such as tensor product decomposition and subalgebra branching of…
The computational power increases over the past decades havegreatly enhanced the ability to simulate chemical reactions andunderstand ever more complex transformations. Tensor contractions are the fundamental computational building block of…
Formulating non-Abelian gauge theories as a tensor network is known to be challenging due to the internal degrees of freedom that result in the degeneracy in the singular value spectrum. In two dimensions, it is straightforward to 'trace…
Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or…