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# 00: Title to be displayed on screen

[This page is included in the Mathematical Atlas as a sample of what the index pages at that site are supposed to contain. Included is a short discussion of the difficulties of putting mathematics onto the web.]

[Warning: some of those are bogus links at top of screen]

[The first four headings define what a discipline is about, usually using only materials at this site.]

## Introduction

A brief summary of what this discipline studies. Includes a description of its subfields (as defined by the Subject Classifications shown below) which is informative enough to help someone select one if that's what they need. Perhaps a formal definition, if necessary. Some key people and famous theorems to help establish a sense of what the discipline is good for. (But history is given a separate heading, following below.)

It is important for an Atlas that the whole mathematics spectrum be covered in these index pages; personal preference notwithstanding, it is important that the author touch on each of the subdisciplines in the area, at least briefly, so that a person who needs information in that topic would know where to turn.

A picture or two per page can often help clarify exposition; moreover, it can help evoke some of the enthusiasm which practitioners feel about their subject. On the other hand, many visitors find multiple images distracting and resent the lengthened transmission time, so a balance must be struck.

When other authors help with or write some of the index pages, there may be flags stating "[This page organized by X, x@x.edu]" or "[This introduction written with X]", or whatever seems most appropriate.

The rest of this section concerns the problematic matter of formatting. As much as possible, the introduction (indeed, the whole page) should be straight text: the pages should be readable by the greatest possible number of visitors, irrespective of client processor, modem speed, operating system, browser, and user choices. HTML does allow boldface and italics, which seem to be rendered without problems, and it appears to be true that many 8-bit ISO8859 (i.e. "accented") characters (e.g. é and ü) appear properly almost always, even without using HTML flags for these (é and ü). (There is a discussion of the HTML and ISO character sets available for use as needed.) If it is necessary to use other characters or symbols, there is as yet no clearly preferable option:

(a) HTML mathematics primitives e.g. for subscripts and math-italics En(f) are clumsy and rendered improperly by text-only browsers such as Lynx.

(b) HTML primitives for symbols are only displayed properly if the user can set font types appropriately, e.g. RSA Labs thought this:

Õ (Pia) = Õ (pi b) mod N

would show the product symbol; on some local screens there is instead an O with a tilde, and on others the subset symbol!. Or see the MetaMath page for examples of symbols (and a discussion of how to make them display properly); for example, |- ( j ® j) appears on some local screens to have a "registered trademark" symbol between two j's; this in turn matches what appears on local screens right here: "j®j" using an ISO8859 character. Other local set-ups show some kind of dark diamond between two phi's, while others yet show \phi\rightarrow\phi, which was probably intended).

(c) ASCII-art is limited and ungraceful, but often serviceable.

    /\
\    2        1   3
\  x   dx  = -  x    + C
\/            3


(d) Symbols and formulas sent as embedded GIF files are a strain on the server, a little slow to transmit, and do not scale with user fonts, although displayed equations, e.g.

 (1)

are not so bad (but make the pages at Mathworld quite long!) Depending on users' font choices, this integral may appear much larger or smaller than the previous one.

(e) Writing TeX source (e.g. a_n, \alpha) is comprehensible to mathematicians but not to most novices.

(f) Transmitting DVI-, PostScript-, or PDF (Acrobat) files is very slow over most data lines and requires the client machine to have an appropriate viewer and a fast processor for formatting.

Any of these approaches could be fine-tuned for one individual's arrangement; they need not work as intended on any other. (Indeed, they need not even come out consistently when the same individual prints the documents.)

The formatting of mathematical documents for the World-Wide Web is in a state of flux, as much as anything else because there are unresolved issues about which rights the author and the viewer ought to have about the presentation of web pages, and about the extent to which software writers or standards bodies should exert primacy. Our approach in short: for now, better to stick with flat text as much as possible! (TeX source files are accepted and stored as the definitive rendering of the content; other formats will be derived from it as appropriate.)

There is another document with examples and further discussion of various strategies for placing mathematics on the Web.

## History

Information about the history of the subject, if available (usually not -- persons with training in the History of Mathematics are hard to find!). Sometimes a link to e.g. St Andrews. Often a citation or two to the literature.

## Applications and related fields

The goal in this section is to clarify the boundaries between this and related areas, so that a reader knows which of two similar fields is most likely to contain the information sought.

Include descriptions of interactions with other areas, and links to them. This can mean comments distinguishing this area from similar ones, or illustrating topics in their intersection. (The other fields are usually listed in decreasing order of relatedness.)

The pages show "maps" of the fields akin to the one on the main welcome page. These (we hope) give a visual representation of the field's relationships with its nearest neighbors.

This image slightly hand-edited for clarity.

## Subfields

• 00A00: A subfield (taken directly from the MCS or lightly modified). Includes link to a page for that subfield when one exists.
• .00A99: Another subfield, this time with no page of its own.

We don't explicitly mention the following subfields, since there should be a place for them somewhere else on this page, in general:

• 00-00 General reference works (handbooks, dictionaries, bibliographies, etc.)
• 00-01 Instructional exposition (textbooks, tutorial papers, etc.)
• 00-02 Research exposition (monographs, survey articles)
• 00-03 Historical (must be assigned at least one classification number from Section 01)
• 00-04 Explicit machine computation and programs (not the theory of computation or programming)
• 00-05 Experimental papers
• 00-06 Proceedings, conferences, collections, etc.
• 00-07 Data analysis
• 00-08 Computational methods
• 00-09 Graphical methods

Comments about the use of the classification system in this area (e.g. relative size of discipline and its subfields; older classifications previously used.) Browse all (old) classifications for this area at the AMS.

Parent field: 00: General studies in generalities (This applies only to index pages for fields with 3-digit and 5-digit classification numbers, in general, although there may eventually be pages for Über-disciplines "Abstract Algebra", "Mathematical Physics", etc.)

[The next four sections also overlap; they attempt to give directions for how one might search outside this site to find out more about the field. Resources mentioned should fairly well match the MCS heading, that is, most of the area should be covered by that text/website/..., and most of the content of that resource should be in this area. Obviously allowances have to be made, e.g. it's nearly impossible to find good "Group Theory and Generalizations(20)" books: Group Theory -- yes; Generalizations -- yes; but together -- no. These pages should be considered signed editorials; resources should be chosen selectively. This gives an advantage over "comprehensive" sites which arguably should mention every resource meeting our criteria.]

## Textbooks, reference works, and tutorials

Well-known textbooks

• Order varies here; sometimes they range from undergrad to research, sometimes from comprehensive to introductory, etc.
• Usually avoid recommendations in crowded markets (e.g. standard undergraduate differential equations)
• An (advanced?) undergraduate text or two; preference to most commonly used ones, so that when someone says, "See Hardy and Wright", a person knows what that means.
• A couple of graduate texts; preference to comprehensive ones
• Avoid, in general, texts of the form "[Subject] for [some non-mathematical discipline]"; the interest in this collection should be on mathematics content.

On-line texts and tutorials mentioned when coverage matches the discipline; not as much attention to quality control here. [Author's name].

Some books get special mention whenever they exist: state-of-the-discipline handbooks, collected bibliographies, MR "Reviews in ..." series, specialized dictionaries or formularies, and printed (or online) datasets are always mentioned.

Mailing lists, USENET newsgroups, FAQs, ...: any other methods for introducing the topic.

## Software and tables

Pointers to computer packages including web-sites with "interactive databases". Also results from those calculations (databases) if not given in previous section!

Not included are most of the many small Java applets which demonstrate a tiny feature of a discipline.

## Other web sites with this focus

Here are the AMS and Goettingen resource pages for area 00.

One or two key preprint servers.

Other sites usually listed ONLY if they provides resources or links which are

• specific to this discipline
• covering all of the discipline (or a significant subdiscipline -- pending creation of that subdiscipline's own index page)
• fairly comprehensive of the TYPES of web resources available in the discipline. (A judiciously-edited site is better than one linking every possible homepage.)
Sites which offer specific content without being a comprehensive "clearinghouse" for the discipline should have appeared somewhere else in this document. That is, a pointer to a site containing "57 proofs of Theorem X" will belong below with the files discussing Selected Topics, rather than here. For now, the unfinished pages (which is most of them) include more web sites than will eventually be mentioned, pending an opportunity to pick out the primary sites and throw out the useless ones.

The happiest situation is when there is already a web site focussed on this area which can claim to be comprehensive in scope and yet carefully controlled to keep junk and tidbits off to the side. The Number Theory Web (for section 11) is an example.

## Selected topics at this site

This is an irregular medly of items which, we hope, might prove useful to someone looking for more information about a particular topic in this field, or for an alternative explanation of something already discussed.
• One line per item, one item per line. This is the original "Collection" at this site!
• Coverage is likely to be very uneven. However, since most of these topics files are culled from newsgroups, mail, and web pages, it can be hoped that the topics represented are among the most commonly asked about.
• A balance must be struck between students who need help and teachers who would like to assign common topics as homework problems...
• An [Offsite] flag when link is not to href="../directory/file".
• In principle, this list can include pointers to more such files -- as long as each has a one-line description, a URL, and a 5-digit classification (so that if index pages are created for subdisciplines the pointer can be moved appropriately).
• We do however reserve the right to remove files or links considered to be incorrect, misleading, or pointless; also unfavored, in general, are links to sites (e.g. those of a student) which are likely to be transitory.