My name is Howard Cohl and am currently a Mathematician at the National Institute of Standards and Technology in Gaithersburg, Maryland. In 2010, I graduated with a Ph.D. in Mathematics from the University of Auckland, in Auckland, New Zealand. In addition to mathematics, I'm interested in web archives of mathematics, Astrophysics, Physics, Scientific Computing, cycling, walking, and many other things. My home page is at hcohl.sdf.org.

I already have a Ph.D. and M.S. in Physics which I obtained in the Department of Physics and Astronomy at Louisiana State University in Baton Rouge, Louisiana. Prior to that my only University education was a B.S. in Astronomy and Astrophysics in the Dept. of Astronomy at Indiana University in Bloomington, Indiana. I am a U.S. citizen, but have been traveling and voyaging, mostly dedicating my efforts toward my education and research interests. My research interests lie in various areas of Mathematical Physics.

Articles

I have contributed to and taken an active interest in the following Wikipedia articles.

Mathematics

Associated Legendre function -- Cyclide -- Green's function -- Green's function for the three-variable Laplace equation -- Heine's identity -- Separable partial differential equation -- 6-sphere coordinates -- Toroidal coordinates -- Whipple formulae

People

Eduard Heine -- Francis John Welsh Whipple -- Adrien-Marie Legendre

Links

• Howard S. Cohl, "http://hcohl.sdf.org"

Graph of Days vs. English Wikipedia Article Counts from Sep. 6, 2007 to Sep. 29, 2008

New Article

Pentaspherical coordinates

A kind of homogeneous coordinates ${\displaystyle (x_{0},x_{1},x_{2},x_{3},x_{4})}$ for a point ${\displaystyle {\mathbf {x} ))$ in complex inversive space. The numbers ${\displaystyle x_{\nu ))$, not all zero, are connected by the relation

${\displaystyle ({\mathbf {x} },{\mathbf {x} })=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=0.}$

All points ${\displaystyle {\mathbf {x} ))$ which satisfy a linear equation

${\displaystyle ({\mathbf {x} },{\mathbf {y} })=x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}+x_{4}y_{4}=0}$

are said to form a sphere, with coordinates ${\displaystyle {\mathbf {y} ))$. Two spheres ${\displaystyle {\mathbf {y} ))$ and ${\displaystyle {\mathbf {z} ))$ are orthogonal if ${\displaystyle ({\mathbf {y} },{\mathbf {z} })=0}$, tangent if

${\displaystyle ({\mathbf {y} },{\mathbf {y} })({\mathbf {z} },{\mathbf {z} })-({\mathbf {y} },{\mathbf {z} })^{2}=0.}$

If two spheres ${\displaystyle {\mathbf {y} ))$ and ${\displaystyle {\mathbf {z} ))$ intersect, the expression

${\displaystyle {\frac {({\mathbf {y} },{\mathbf {z} })}((\sqrt {({\mathbf {y} },{\mathbf {y} }))){\sqrt {({\mathbf {z} },{\mathbf {z} })))))}$

measures the cosine of their angle (or the hyperbolic cosine of their inverse distance).

Setting ${\displaystyle x_{4}=0}$, one obtains the analogous tetracyclic coordinates, which lead to circles instead of spheres.

Completely analogous constructions can be performed for spaces of higher dimensions, which give polyspherical coordinates. In the 4-dimensional case they are called hexaspherical coordinates. Polyspherical coordinates are used in conformal geometry in examining manifolds of figures.

Definitions

• DRMF Digital Repository of Mathematical Formulae

• OPSF Orthogonal polynomials and special functions

• DRMF user is a user who views DRMF OPSF formula data

• DRMF contributor is a DRMF user who has permission to upload DRMF OPSF formula data

• DRMF administrator is a DRMF user who has permission to modify DRMF user accounts and permissions

• DLMF Digital Library of Mathematical Functions

• LaTeXML a LaTeX to XML converter developed by Bruce Miller for use within DLMF

• eDLMF macros LaTeXML macros originating from DLMF for OPSF and extended within DRMF

• Formula's auxiliary information include:
  • Formula (required)
  • Bibliographic citation (required)
    • BibTeX output available
  • Symbols used within the formula which correspond to eDLMF macros (required)
    • Linked to symbol definitions
  • open section for Proofs (required)
  • Substitutions (optional)
  • Constraints (optional)
  • Notes about formula (optional)
    • Data associated with notes:
      • Different types of notes?
      • Paragraph long notes?
      • Date of note?
  • Name of formula (optional)

• Formula pages MediaWiki pages in DRMF which contains lists of formulas
  • e.g., Zeta and Related Functions page

• Formula's meta information include:
  • Formula home page name, this can be used to find the formula (e.g., via a DRMF search)
    • Should not be onerous to copy by hand (e.g., fewer than 10-15 characters)
  • Formula's auxiliary information
  • Formula's errata history (when applicable)
  • Relevant dates
    • when added to DRMF
    • when updated/corrected
  • Submitter identification (what?)
  • Other information as appropriate for that type of formula (for example, differential equations may list a transformation that links it to another differential equations) (discuss)

• Formula home page contains the following:
  • The formula
  • The formula's meta information
  • The formula in LaTeXML format
  • Notation list for all the elements appearing on this page, with one or more links for each notation element

• User submitted information includes
  • the formula's auxiliary information

• Review panel is a collection of people who can approve the inclusion of new formulae into DRMF

• Submissions site is a website designed for enabling input of new formulae into the DRMF
  • submissions site satisfies karma based testing of requirements

• Entry pages are Wikitext pages which guide the user as view of or entrance to the repository
  • Required entry pages
  • Document entry pages

• Compound search is a search involving nested functions.
  • The level of a compound search is the amount of nesting.
    • Level 2 compound search would identify 'functions of functions';
      • e.g., 'sin( ... log ...)', where the ellipsis can be any terms.
    • Level 3 compound search would identify 'functions of functions of functions'
      • e.g., for example 'sin( ... log ( ... tan ...) ... )'.

• Bookmark is a pointer to a formula or section in DRMF. When clicked it goes to the bookmarked content.

• Bookmark annotation is a ... (what is it?)

• PHASE 2 The next phase of the project as a framework for maintaining a repository of mathematical information
  • Blog
  • Differential equations
    • Differential equations entry page
  • Inputs other than LaTeX with eDLMF macros
    • Mathematica, conversion tool using eDLMF macros required
      • Need to make sure that definitions used in Mathematica are same as DLMF definitions (Some OPSF are defined differently in Mathematica and DLMF definitions) or tabulate new definitions and define new corresponding eDLMF macros

Requirements

General

• OPSF symbol notations:
  • OPSF in DRMF shall use the definitions from DLMF where available, otherwise DRMF shall give definitions
  • Notations will be consistent with the DLMF (subject to consideration by review panel)
• DRMF formulas shall be convertable to different representations (in order to test the plausibility of the formula)
  • Meeting the constraints and substitutions where present
  • Output representations include:
    • LaTeX (with eDLMF macros expanded into TeX)
    • semantic LaTeX (with unexpanded eDLMF macros)
    • MathML
    • Content MathML
    • Mathematica
    • Maple
    • Sage
• DRMF formulas shall be interpretable in a standalone fashion
• DRMF shall display a glossary of eDLMF macros with links to definitions
• DRMF shall contain a FAQ
• DRMF shall display a notice when DRMF shutdowns are planned
• DRMF shall have an ability to collect volunteered user feedback
• DRMF shall abide by good programming practices as articulated in XXX (e.g., high modularity with each module having a cyclomatic complexity less than 11)
• DRMF shall abide by US Government and NIST regulations regarding: security, software tools, user information (necessary?)

Formula Content

• NIST Digital Library of Mathematical Functions (DLMF)
• Koekoek, Lesky and Swarttouw (2010) Hypergeometric orthogonal polynomials and their q-analogs, Chapters 1,9,14 (KLS)
  • Koornwinder's modifications and updates to KLS (arXiv:1401.0815) (KKLS)
• Gradshteyn & Ryzhik (2010) Tables of Integrals and Series (G&R)
  • Proofs for G&R integrals by Victor Moll (MPG&R)
• Andrews, Askey & Roy (2010) Special Functions (SF)
• Gasper & Rahman (2010) Basic Hypergeometric Series (BHS)
• Ismail (2010) Classical and Quantum Orthogonal Polynomials in One Variable (CQOP)
• Bateman Manuscript Project (HTF)
  • Scanned book page data -> LaTeX
  • InftyReader (Mathematical Optical Character Recognition Software)
• Magnus, Oberhettinger & Soni (1966) (MOS)
  • Springer
  • NB: Existing criticism!
• Byrd and Friedman, 2nd ed (1971) (B&F)
  • Springer
• New author formulas via: arXiv or published papers

Display

• The DRMF shall have several entry pages including:
  • OPSF entry page
  • Integration/summation entry pages (similar in layout to G&R)
  • Categories entry page
  • Search
  • Reference centric view (copyright infringement?)
• Equivalent representations of formulas
  • Decide how to keep track of these (big issue)

• Every formula will have a formula home page
  • except equivalent formulas (with name of variable changes)
  • except trivial formulas which appear in passing
• The mathematical expressions in DRMF shall be rendered using the MediaWiki math extension
  • Using LaTeXML and MathJax (images?) for capable browsers
• Presentation shall follow usual Internet standards:
  • Hyperlinks
    • Black, blue, underline as hover, and/or highlighted (make decision), or do nothing

Search

• DRMF shall have a query language to describe math search
  • What is the query language?
• DRMF shall support search among all formulas in formula pages and formula home pages:
  • By eDLMF macros
    • give option to search for multiple eDLMF macros within a single formula
    • give options for advanced search capabilities such as AND and OR
  • By the information within a formula's meta information
  • In a specified time range (added, updated)
• It shall be possible to simultaneously incorporate all search criteria specified in this section (combination search?)
• It shall be possible to be able to find duplicate formulas
  • For instance, it should be possible to determine duplication information when users submit formulas for the first time
• It shall be possible to submit any DRMF formula as a search entry
  • By LaTeX source
  • By bibliographic citation
• DRMF users shall be able to specify whether DRMF searches are exact (all elements must appear precisely) or approximate
  • Such as putting something in double quotes "..."
  • e.g., "f(x)"
• DRMF shall support compound searches up to level 2

User accounts

• DRMF users shall be divided into those which have the following permissions
  • DRMF administrator : Read/Write/Administer
  • DRMF contributor : Read/Write
  • DRMF user : Read
• DRMF administrators shall be able to suspend or delete user accounts (subject to approval by review panel)
• DRMF contributors shall be able to upload DRMF OPSF formulae data (subject to approval by review panel)
• DRMF users shall:
  • be able to register themselves to log on
  • have personalizable home pages (immediate consequence of MediaWiki)
  • be able to create bookmarks:
    • A logged in DRMF user shall be able to create/delete/view their bookmarks
    • DRMF shall allow at least 50 bookmarks per DRMF user
  • be able to create bookmark annotations:
    • A logged in DRMF user shall be able to create/delete/edit bookmark annotations
    • Bookmark annotations shall be at least 20 characters
  • be able to sort their bookmarks by:
    • Annotation
    • Date of bookmark

User submission of information

• DRMF contributors shall be able to make submissions to DRMF via a browser
• DRMF contributors shall utilize karma based checking system
• DRMF user submissions shall be posted on a submissions site if they meet the following:
  • have a bibliographic citation
  • are reasonable - looks like a mathematical formula
  • are non-offensive - included text must not violate policy for posting
  • utilize eDLMF macros
• User submitted formula data shall be in a uniquely parseable representation (e.g., LaTeXML or Mathematica)
• Formula on the submissions site shall be moved to the main DRMF collection (after approval by review panel)
• DRMF administrators shall be able to edit submissions

Design (in need of further clarification)

• The definition and handling of formula home page names are described in formula home pages
  • This includes how errata are handled

• The contents of the required entry pages are described in the document entry pages