RUDN University mathematicians applied 19th century ideas to modern computerized algebra systems

November 25, 2020

A team of mathematicians from RUDN University added new symbolic integration functionality to the Sage computerized algebra system. The team implemented ideas and methods suggested by the German mathematician Karl Weierstrass in the 1870s. The results were published in the Journal of Symbolic Computation.

The first computer program capable of calculating integrals of elementary functions was developed in the late 1950s. By creating it, the developers confirmed that a computer could not only perform simple calculations but was also able to deal with tasks that required a certain degree of 'thinking'. Symbolic integration, i.e. integration that involves letters and abstract symbols instead of numbers, is an example of such a task. At the same time, scientists realized that neither humans nor computers were able to determine whether a given integral can be taken in elementary functions (provided such a human or computer used the methods studied in a university course of analysis and took a finite number of steps). Therefore, in the 1960s mathematicians working on symbolic integrators started to refer to methods that had been suggested by Liouville in the 1830s. From that time on, computer scientists have been tapping into the classic scientific heritage.

The calculation of primitives of algebraic functions is one of the bottlenecks in the process of integrator development. Before World War I, the integration of algebraic functions or Abelian integrals had been considered one of the most important issues in mathematics, but later on, it was forgotten. "Current computer algebra systems are able to fulfill even the most exotic requests of mathematical analysis students, but at the same time, many of these systems fail to recognize integrals in elementary functions. Only several packages allow for the integration of algebraic functions or with Abelian integrals, but their development stopped 15 years ago, and their functionality leaves much to be desired," says Mikhail Malykh, a Doctor of Science in Physics and Mathematics, and an assistant professor at the Department of Applied Informatics and Probability Theory, RUDN University.

One of the theories developed by the German mathematician Karl Weierstrass in the 1870s reduces the calculation of an integral of an algebraic function to finding a given set of known integrals of all three types. The initial integral is represented as a sum of standard integrals (this construction is knowns as the normal representation of an Abelian integral). The team from RUDN University confirmed that this representation is indicative of whether a given integral can be calculated in elementary functions. To confirm their theory, the mathematicians tested them on simple elliptical integrals using a software package that had been created by the team in 2017. The package helps calculate coefficients of the normal form of an integral. In the future, the team plans to conduct similar studies for a wider range of integrals.

"This work is just one step on our way to an ambitious goal: we want to express Weierstrass's theory of Abelian integrals and functions using the language of computer algebra and to implement it in the Sage system, giving researchers from all over the world free access to it," added Mikhail Malykh from RUDN University
-end-


RUDN University

Related Computer Articles from Brightsurf:

UCLA computer scientists set benchmarks to optimize quantum computer performance
Two UCLA computer scientists have shown that existing compilers, which tell quantum computers how to use their circuits to execute quantum programs, inhibit the computers' ability to achieve optimal performance.

Digitize your dog into a computer game
Researchers from CAMERA at the University of Bath have developed motion capture technology that enables you to digitise your dog without a motion capture suit and using only one camera.

Stabilizing brain-computer interfaces
Researchers from Carnegie Mellon University (CMU) and the University of Pittsburgh (Pitt) have published research in Nature Biomedical Engineering that will drastically improve brain-computer interfaces and their ability to remain stabilized during use, greatly reducing or potentially eliminating the need to recalibrate these devices during or between experiments.

Computer-generated genomes
Professor Beat Christen, ETH Zurich to speak in the AAAS 2020 session, 'Synthetic Biology: Digital Design of Living Systems.' Christen will describe how computational algorithms paired with chemical DNA synthesis enable digital manufacturing of biological systems up to the size of entire microbial genomes.

Computer-based weather forecast: New algorithm outperforms mainframe computer systems
The exponential growth in computer processing power seen over the past 60 years may soon come to a halt.

A computer that understands how you feel
Neuroscientists have developed a brain-inspired computer system that can look at an image and determine what emotion it evokes in people.

Computer program looks five minutes into the future
Scientists from the University of Bonn have developed software that can look minutes into the future: The program learns the typical sequence of actions, such as cooking, from video sequences.

Computer redesigns enzyme
University of Groningen biotechnologists used a computational method to redesign aspartase and convert it to a catalyst for asymmetric hydroamination reactions.

Mining for gold with a computer
Engineers from Texas A&M University and Virginia Tech report important new insights into nanoporous gold -- a material with growing applications in several areas, including energy storage and biomedical devices -- all without stepping into a lab.

Teaching quantum physics to a computer
An international collaboration led by ETH physicists has used machine learning to teach a computer how to predict the outcomes of quantum experiments.

Read More: Computer News and Computer Current Events
Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.