97,290 research outputs found
Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems
Digital mathematical libraries assemble the knowledge of years of mathematical research. Numerous disciplines (e.g., physics, engineering, pure and applied mathematics) rely heavily on compendia gathered findings. Likewise, modern research applications rely more and more on computational solutions, which are often calculated and verified by computer algebra systems. Hence, the correctness, accuracy, and reliability of both digital mathematical libraries and computer algebra systems is a crucial attribute for modern research. In this paper, we present a novel approach to verify a digital mathematical library and two computer algebra systems with one another by converting mathematical expressions from one system to the other. We use our previously developed conversion tool (referred to as ) to translate formulae from the NIST Digital Library of Mathematical Functions to the computer algebra systems Maple and Mathematica. The contributions of our presented work are as follows:Â (1) we present the most comprehensive verification of computer algebra systems and digital mathematical libraries with one another; (2) we significantly enhance the performance of the underlying translator in terms of coverage and accuracy; and (3) we provide open access to translations for Maple and Mathematica of the formulae in the NIST Digital Library of Mathematical Functions
Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems
Digital mathematical libraries assemble the knowledge of years of
mathematical research. Numerous disciplines (e.g., physics, engineering, pure
and applied mathematics) rely heavily on compendia gathered findings. Likewise,
modern research applications rely more and more on computational solutions,
which are often calculated and verified by computer algebra systems. Hence, the
correctness, accuracy, and reliability of both digital mathematical libraries
and computer algebra systems is a crucial attribute for modern research.
In this paper, we present a novel approach to verify a digital mathematical
library and two computer algebra systems with one another by converting
mathematical expressions from one system to the other. We use our previously
eveloped conversion tool (referred to as LaCASt) to translate formulae from the
NIST Digital Library of Mathematical Functions to the computer algebra systems
Maple and Mathematica. The contributions of our presented work are as follows:
(1) we present the most comprehensive verification of computer algebra systems
and digital mathematical libraries with one another; (2) we significantly
enhance the performance of the underlying translator in terms of coverage and
accuracy; and (3) we provide open access to translations for Maple and
Mathematica of the formulae in the NIST Digital Library of Mathematical
Functions
On Birch and Swinnerton-Dyer's cubic surfaces
In a 1975 paper of Birch and Swinnerton-Dyer, a number of explicit norm form
cubic surfaces are shown to fail the Hasse Principle. They make a
correspondence between this failure and the Brauer--Manin obstruction, recently
discovered by Manin. We generalize their work, making use of modern computer
algebra software to show that a larger set of cubic surfaces have a
Brauer--Manin obstruction to the Hasse principle, thus verifying the
Colliot-Th\'el\`ene--Sansuc conjecture for infinitely many cubic surfaces
Algebra, matrices, and computers
What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix groups and new methods for handling matrix groups in a computer
Algebra, matrices, and computers
What part does algebra play in representing the real world abstractly? How can algebra be used to solve hard mathematical problems with the aid of modern computing technology? We provide answers to these questions that rely on the theory of matrix groups and new methods for handling matrix groups in a computer
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