Howard S. Cohl

Semantification of Identifiers in Mathematics for Better Math Information Retrieval

Mathematical formulae are essential in science, but face challenges of ambiguity, due to the use of a small number of identifiers to represent an immense number of concepts. Corresponding to word sense disambiguation in Natural Language Processing, we disambiguate mathematical identifiers. By regarding formulae and natural text as one monolithic information source, we are able to extract the semantics of identifiers in a process we term Mathematical Language Processing (MLP). As scientific communities tend to establish standard (identifier) notations, we use the document domain to infer the actual meaning of an identifier. Therefore, we adapt the software development concept of namespaces to mathematical notation. Thus, we learn namespace definitions by clustering the MLP results and mapping those clusters to subject classification schemata. In addition, this gives fundamental insights into the usage of mathematical notations in science, technology, engineering and mathematics. Our gold standard based evaluation shows that MLP extracts relevant identifier-definitions. Moreover, we discover that identifier namespaces improve the performance of automated identifier-definition extraction, and elevate it to a level that cannot be achieved within the document context alone.

Challenges of Mathematical Information Retrievalin the NTCIR-11 Math Wikipedia Task

Mathematical Information Retrieval concerns retrieving information related to a particular mathematical concept. The NTCIR-11 Math Task develops an evaluation test collection for document sections retrieval of scientific articles based on human generated topics. Those topics involve a combination of formula patterns and keywords. In addition, the optional Wikipedia Task provides a test collection for retrieval of individual mathematical formula from Wikipedia based on search topics that contain exactly one formula pattern. We developed a framework for automatic query generation and immediate evaluation. This paper discusses our dataset preparation, topic generation and evaluation methods, and summarizes the results of the participants, with a special focus on the Wikipedia Task.

Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Due to the isotropy of d-dimensional hyperspherical space, one expects there to exist a spherically symmetric fundamental solution for its corresponding Laplace{Beltrami operator. The R-radius hypersphere S d with R > 0, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplaces equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the associated Legendre function of the second kind on the cut (Ferrers function of the second kind) with degree and order given by d=2 1 and 1 d=2 respectively, with real argument between plus and minus one.

On a generalization of the generating function for Gegenbauer polynomials

A generalization of the generating function for Gegenbauer polynomials is introduced whose coefficients are given in terms of associated Legendre functions of the second kind. We discuss how our expansion represents a generalization of several previously derived formulae such as Heines formula and Heines reciprocal square-root identity. We also show how this expansion can be used to compute hyperspherical harmonic expansions for power-law fundamental solutions of the polyharmonic equation.

Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems

We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equa- tion on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkins polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

Due to the isotropy of d-dimensional hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hyperboloid model of hyperbolic geometry with R > 0 represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the hyperbolic sine, finite summation expressions over hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace’s equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace’s equation on the R-radius hyperboloid. For d ⩾ 2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace’s equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace’s equation is obtained through comparison with its corresponding Gegenbauer expansion.

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 .

Eigenfunction expansions for a fundamental solution of Laplace's equation on R 3 in parabolic and elliptic cylinder coordinates

A fundamental solution of Laplace’s equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functionsJ0(kr) orK0(kr),r 2 = (x−x0) 2 + (y−y0) 2 , in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that K0(kr) is a fundamental solution and J0(kr) is the Riemann function of partial differential equations on the Euclidean plane.

VMEXT: A Visualization Tool for Mathematical Expression Trees

Mathematical expressions can be represented as a tree consisting of terminal symbols, such as identifiers or numbers (leaf nodes), and functions or operators (non-leaf nodes). Expression trees are an important mechanism for storing and processing mathematical expressions as well as the most frequently used visualization of the structure of mathematical expressions. Typically, researchers and practitioners manually visualize expression trees using general-purpose tools. This approach is laborious, redundant, and error-prone. Manual visualizations represent a users notion of what the markup of an expression should be, but not necessarily what the actual markup is. This paper presents VMEXT - a free and open source tool to directly visualize expression trees from parallel MathML. VMEXT simultaneously visualizes the presentation elements and the semantic structure of mathematical expressions to enable users to quickly spot deficiencies in the Content MathML markup that does not affect the presentation of the expression. Identifying such discrepancies previously required reading the verbose and complex MathML markup. VMEXT also allows one to visualize similar and identical elements of two expressions. Visualizing expression similarity can support support developers in designing retrieval approaches and enable improved interaction concepts for users of mathematical information retrieval systems. We demonstrate VMEXTs visualizations in two web-based applications. The first application presents the visualizations alone. The second application shows a possible integration of the visualizations in systems for mathematical knowledge management and mathematical information retrieval. The application converts LaTeX input to parallel MathML, computes basic similarity measures for mathematical expressions, and visualizes the results using VMEXT.

Semantic Preserving Bijective Mappings of Mathematical Formulae Between Document Preparation Systems and Computer Algebra Systems

Document preparation systems like Open image in new window offer the ability to render mathematical expressions as one would write these on paper. Using Open image in new window , Open image in new window , and tools generated for use in the National Institute of Standards (NIST) Digital Library of Mathematical Functions, semantically enhanced mathematical Open image in new window markup (semantic Open image in new window ) is achieved by using a semantic macro set. Computer algebra systems (CAS) such as Maple and Mathematica use alternative markup to represent mathematical expressions. By taking advantage of Youssef’s Part-of-Math tagger and CAS internal representations, we develop algorithms to translate mathematical expressions represented in semantic Open image in new window to corresponding CAS representations and vice versa. We have also developed tools for translating the entire Wolfram Encoding Continued Fraction Knowledge and University of Antwerp Continued Fractions for Special Functions datasets, for use in the NIST Digital Repository of Mathematical Formulae. The overall goal of these efforts is to provide semantically enriched standard conforming MathML representations to the public for formulae in digital mathematics libraries. These representations include presentation MathML, content MathML, generic Open image in new window , semantic Open image in new window , and now CAS representations as well.