Julia

The project is an interactive, living book dedicated to reconstructing Archimedes’ Measurement of a Circle (Circuli Dimensio) using the original geometric proportions. By employing SageMath for symbolic validation, alongside Python, Julia and Perl for high-precision numerical calculations using libraries for decimal and arbitrary-precision arithmetic, the project bypasses modern algebraic transformations and demonstrates the accuracy of ancient geometric concepts.

Beyond Archimedes' original derivations, the project bridges the gap to modern approaches for calculating π. It demonstrates how Archimedes' four determination equations can be translated into modern notation like a super-radical, a product formula and a summation formula. To achieve this, the four equations were systematically reduced to a single unified equation. This step opens the door to further developments and serves as the link between geometry and algebra.

Using a specialized mathematical gain function, the system creates a dynamic bridge between ancient geometry and modern infinite series such as those of Ramanujan and Chudnovsky or the π formula of Saha and Sinha. Particular emphasis is placed on this new π formula, as it is more than it appears at first glance. By precisely handling the λ parameter the optimized approach achieves absolute numerical stability in getting hundreds or more of error-free decimal places. It is demonstrated, among other things, that the maximum gain in decimal places is achieved when the chosen value of λ equals the maximum number of iterations.

A key highlight is that the derived concept based on Archimedes allows the fundamental re-examination of every existing approach. E.g. the equations found on Wikipedia have been consolidated into a single equation as well as the BPS algorithm can also be reduced to a single equation. Furthermore, the work of Snellius and Dörrie can also be incorporated into the geometry of Archimedes.

And much more!

A GitLab CI process for managing Julia packages.

Shoot a JupyterLab | marimo | VS Code/VSCodium | Pluto.jl | Matlab | SSHD | ... server to a (remote) Slurm allocation and talk to it via your local browser. Runs locally (Linux, Mac, Windows (WSL)) or remotely (e.g. as Remote or ProxyCommand configured in ~/.ssh/config) or inside an interactive Slurm session. One Bash script for both tasks: as a wrapper it starts itself as an sbatch script in a Slurm job.

Numerična matematika v programskem jeziku Julia. Knjiga je objavljena pri Založbi FRI

Use the SQLite library from Julia

Field-algebra based on Group / Ring symbolic vector module extension

A simple multilayer perceptron (MLP), also known as a fully connected feedforward artificial neural network, written from scratch in Julia.

Symbolic parser for Julia language term rewriting using REDUCE algebra

A Julia package for representing, manipulating and evaluating trees of expressions.

A Julia package providing functionality for annotating arbitrary callables with type signature data.

Might help with bad inference.

Provides somewhat of a restricted way of expressing a type signature for a callable in Julia's type system.

Python Data Science is an open source project providing guidance on python (and selectively R, Julia) packages relevant for data science tasks

A Julia package implementing type domain natural numbers, integers and rational numbers. Also extends the Irrational type that comes with Julia to support exact arithmetic.

Code for the calculation of Four (4) and Six (6)-wave mixing spectra from semi-classical (surface hopping) simulations.

A simple plane wave density functional theory Julia code.

A package for the Julia programming language. Implements an alternative to the built-in Rational type. Based on the concept of finite simple continued fractions.