Digital Repository of Mathematical Formulae

The purpose of the NIST Digital Repository of Mathematical Formulae (DRMF) is to create a digital compendium of mathematical formulae for orthogonal polynomials and special functions (OPSF) and of associated mathematical data. The DRMF addresses needs of working mathematicians, physicists and engineers: providing a platform for publication and interaction with OPSF formulae on the web. Using MediaWiki extensions and other existing technology (such as software and macro collections developed for the NIST Digital Library of Mathematical Functions), the DRMF acts as an interactive web domain for OPSF formulae. Whereas Wikipedia and other web authoring tools manifest notions or descriptions as first class objects, the DRMF does that with mathematical formulae. See http://gw32.iu.xsede.org/index.php/Main_Page

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

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: (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

Extracting mathematical semantics from LaTeX documents

We report on a project to use SGLR parsing and term rewriting with ELAN4 to extract the semantics of mathematical formulas from a LaTeX document and representing them in MathML. The LaTeX document we used is part of the Digital Library of Mathematical Functions (DLMF) project of the US National Institute of Standards and Technology (NIST) and obeys project-specific conventions, which contains macros for mathematical constructions, among them 200 predefined macros for special functions, the subject matter of the project. The SGLR parser can parse general context-free languages, which suffices to extract the structure of mathematical formulas from calculus that are written in the usual mathematical style, with most parentheses and multiplication signs omitted. The parse tree is then rewritten into a more concise and uniform internal syntax that is used as the base for extracting MathML or other semantical information

Extracting mathematical semantics from LaTeX documents

We report on a project to use SGLR parsing and term rewriting with ELAN4 to extract the semantics of mathematical formulas from a LaTeX document and representing them in MathML. The LaTeX document we used is part of the Digital Library of Mathematical Functions (DLMF) project of the US National Institute of Standards and Technology (NIST) and obeys project-specific conventions, which contains macros for mathematical constructions, among them 200 predefined macros for special functions, the subject matter of the project. The SGLR parser can parse general context-free languages, which suffices to extract the structure of mathematical formulas from calculus that are written in the usual mathematical style, with most parentheses and multiplication signs omitted. The parse tree is then rewritten into a more concise and uniform internal syntax that is used as the base for extracting MathML or other semantical information

zbMATH Open: API Solutions and Research Challenges

We present zbMATH Open, the most comprehensive collection of reviews and bibliographic metadata of scholarly literature in mathematics. Besides our website https://zbMATH.org which is openly accessible since the beginning of this year, we provide API endpoints to offer our data. The API improves interoperability with others, i.e., digital libraries, and allows using our data for research purposes. In this article, we (1) illustrate the current and future overview of the services offered by zbMATH; (2) present the initial version of the zbMATH links API; (3) analyze potentials and limitations of the links API based on the example of the NIST Digital Library of Mathematical Functions; (4) and finally, present the zbMATH Open dataset as a research resource and discuss connected open research problems

Funciones Especiales en la Era Digital

Aunque es difícil dar una definición precisa de qué es una función especial, podríamos decir que las funciones especiales lo son en parte por ser importantes en diversas aplicaciones, y en parte por satisfacer determinadas propiedades. Una de las referencias clásicas para la consulta de propiedades y aproximaciones de funciones especiales es el Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, editado por Milton Abramowitz e Irene Stegun y publicado originalmente en 1964 por el National Bureau of Standards. Recientemente, el National Institute of Standards and Technology (NIST, denominación moderna del NBS) acometió el proceso de modernización del Handbook que culminó en 2010 con el lanzamiento del NIST Handbook of Mathematical Functions y su versión online y de libre acceso, la NIST Digital Library of Mathematical Functions. En esta nueva obra, las tablas de valores numéricos de funciones que aparecían en el manual de Abramowitz y Stegun han sido, como es lógico, sustituidas por referencias a software numérico. Este listado de referencias es necesariamente incompleto, pues estamos muy lejos de disponer de software numérico satisfactorio para todas las funciones contenidas en el NIST Handbook, y no parece sencillo que lo pueda haber a medio plazo. Tampoco parece sencillo describir todos los métodos necesarios para evaluar todas las funciones especiales, aunque sí es posible describir y analizar los métodos numéricos básicos e ilustrarlos con ejemplos específicos. En este artículo nos planteamos exponer brevemente algunas de las técnicas habituales involucradas en la construcción de algoritmos para evaluar funciones matemáticas, así como proporcionar referencias sobre software de cálculo de algunas de estas funciones

Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context

Mathematical formulae represent complex semantic information in a concise form. Especially in Science, Technology, Engineering, and Mathematics, mathematical formulae are crucial to communicate information, e.g., in scientific papers, and to perform computations using computer algebra systems. Enabling computers to access the information encoded in mathematical formulae requires machine-readable formats that can represent both the presentation and content, i.e., the semantics, of formulae. Exchanging such information between systems additionally requires conversion methods for mathematical representation formats. We analyze how the semantic enrichment of formulae improves the format conversion process and show that considering the textual context of formulae reduces the error rate of such conversions. Our main contributions are: (1) providing an openly available benchmark dataset for the mathematical format conversion task consisting of a newly created test collection, an extensive, manually curated gold standard and task-specific evaluation metrics; (2) performing a quantitative evaluation of state-of-the-art tools for mathematical format conversions; (3) presenting a new approach that considers the textual context of formulae to reduce the error rate for mathematical format conversions. Our benchmark dataset facilitates future research on mathematical format conversions as well as research on many problems in mathematical information retrieval. Because we annotated and linked all components of formulae, e.g., identifiers, operators and other entities, to Wikidata entries, the gold standard can, for instance, be used to train methods for formula concept discovery and recognition. Such methods can then be applied to improve mathematical information retrieval systems, e.g., for semantic formula search, recommendation of mathematical content, or detection of mathematical plagiarism.Comment: 10 pages, 4 figure