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Blackboard Bold

Blackboard bold

Blackboard bold is a style of typeface often used for certain symbols in mathematics and physics texts, in which certain lines of the symbol (usually vertical, or near-vertical lines) are doubled. The symbols usually describe sets of numbers and are also referred to as double struck, although attempting to produce them by double striking on a typewriter is unlikely to give satisfactory results. The symbols are also nearly universal in their interpretation, unlike their normally-typeset counterparts, which are constantly reused. In some texts, these symbols are simply shown in bold, and blackboard bold in fact originated from the attempt to write bold letters on blackboards in a way that clearly differentiated them from non-bold letters. Writing actual bold letters using chalk is simple when you turn the chalk sideways. It is frequently claimed that the symbols were first introduced by the group of mathematicians known as Nicolas Bourbaki. There are several reasons to doubt this claim: (1) the symbols do not not appear in Bourbaki publications (rather, ordinary bold is used) at or near the era when they began to be used elsewhere, for instance, in typewritten lecture notes from Princeton University (achieved in some cases by overstriking R or C with I), and (an apparent first) typeset in Gunning and Rossi's textbook on several complex variables; (2) Jean-Pierre Serre, a member of the Bourbaki group, has publicly inveighed against the use of "blackboard bold" anywhere other than on a blackboard. TeX, the standard typesetting system for mathematical texts, does not contain direct support for blackboard bold symbols, but the add-on AMS Fonts package by the American Mathematical Society provides this facility; a blackboard bold R is written as \Bbb in regular text and as \mathbb in math mode. In Unicode, a few of the more common blackboard bold characters (C, H, N, P, Q, R and Z) are encoded in the Basic Multilingual Plane (BMP). The rest, however, are encoded outside the BMP, from U+1D538 to U+1D550 (uppercase, excluding those encoded in the BMP), U+1D552 to U+1D56B (lowercase) and U+1D7D8 to U+1D7E1 (digits). Being outside the BMP, these are relatively new and not widely supported. The following table shows some of the more common uses of blackboard bold. The first column shows the letter as typically rendered by the ubiquitous LaTeX markup system. The second column shows the Unicode codepoint. The third column shows the symbol itself (which will only display correctly if your browser supports Unicode and has access to a suitable font). The fourth column describes typical usage in mathematical texts. Note that

\mathbb \subseteq \mathbb \subseteq \mathbb \subseteq \mathbb \subseteq \mathbb \subseteq \mathbb \subseteq \mathbb \subseteq \mathbb \subseteq \mathbb

External links

- http://www.w3.org/TR/MathML2/double-struck.html shows blackboard bold symbols together with their Unicode encodings. Encodings in the BMP are highlighted in yellow. Category:Mathematical notation

Typeface

In typography, a typeface consists of a co-ordinated set of grapheme (i.e., character) designs. A typeface is usually comprised of an alphabet of letters, numerals, and punctuation marks. Helvetica, Century Schoolbook, and Courier are three examples of typefaces. A typeface may also include or consist of ideograms and symbols (e.g., mathematical or map making glyphs). The art of designing typefaces, called type design, is the occupation of a type designer. In metal type, the word font denoted a complete typeface in a particular size (usually measured in points), one weight (e.g., light, book, bold, black), and one orientation (e.g., roman, italic, oblique). As regards digital type, the font is the computer file that stores the vector paths, before they are brought into being on a screen or a page. Digital fonts do contain unlimited (or application-limited) sizes. Some applications can provide additional weights or orientations of a font, but these are not considered typographically correct.

Introduction

A font, from Middle French fonte, meaning "(something that has been) melt(ed)" and referring to letters of a typeface produced by casting molten metal at a type foundry, consists of a set of glyphs (images) representing the characters from a particular character set in a particular typeface. Historically, fonts came in specific sizes (governing the actual height of the characters), and in sorts (governing the quantities of each letter provided). The design of a given character in a font took into account all these factors. In addition, as the spectrum of available designs and requirements of publishers has broadened over the centuries, fonts of specific weight (how dark the text appears—bold or light, for example) and additional specific conditions (most commonly "regular" as opposed to " italic" and/or "condensed") have led to "typeface families", collections of closely-related typeface designs that may include hundreds of styles. English-speaking printers have used the term fount for centuries to refer to the multipart device used (in its day) to assemble and print in a particular size and typeface design. Type foundries cast virtually all founts in various lead alloys from the 1450s until the middle of the 20th century, though wood served to make a few large founts (wood type), especially in the United States of America. In the 1890s mechanized typesetting emerged and began casting fonts on-the-fly in the form of lines of type of the size and length needed. This became known as "hot metal" type, and it remained profitable and widespread until its demise in the 1970s. The first machine of this type was the Linotype invented by Ottmar Mergenthaler. During a relatively brief transitional period (circa 1950s – 1990s), photographic technology, known as "phototypesetting", produced founts which came on rolls or discs of film. Photographic typesetting allowed for optical scaling, which meant that designers could produce multiple sizes from a single font (although physical constraints on the reproduction system used still required design-changes at different sizes — for example, ink traps and spikes to allow for spread of ink). Manually-operated phototypographic composition systems (using fonts made on rolls of film) allowed fine kerning between letters without great physical effort for the first time and spawned a large type-design industry in the 1960s and 1970s. The mid-1970s saw all of the major typeface technologies and all their fonts in use: from the original letterpress process of Gutenberg to mechanical metal typesetters, phototypositors, computer-controlled phototypesetters, and the earliest digital typesetters, (hulking machines with tiny processors and CRT outputs). From the mid-1980s, as digital typography has relentlessly grown, users have almost universally adopted the American spelling font, which nowadays nearly always means a computer file containing scalable, outline letterforms ("digital fonts"), usually in one of several common formats. Designers of some fonts, such as Microsoft's Verdana, intend their product primarily for use on computer screens. Digital fonts may encode the image of each character either as a bitmap, in a bitmap font, (seldom used since 1995) or by a higher-level description in terms of lines and curves enclosing a space (an outline font, also called a "vector font"). An outline "rasterizer" then fills the enclosed space of an outline font, deciding which pixels to represent as "black" and which as "white". The rasterization proceeds in straightforward fashion at higher resolutions (as for example in laser printers and in high-end publishing systems) but for screens, where each individual pixel can mean the difference between legibility and illegibility, digital fonts need hints included to make readable bitmaps at small sizes. Digital fonts today also contain data representing the "typography" used to compose them, including kerning pairs, component-creation data for accented characters, glyph-substitution rules for Arabic typography, and for connecting script faces and for simple everyday ligatures like "fl". (Common description languages that format digital type include PostScript, TrueType and OpenType. Enablers of these formats, including the rasterizers, appear in Microsoft and Apple Computer operating systems, Adobe Systems products and those of several other companies.)

Typeface anatomy

Typographers have derived a comprehensive vocabulary for describing and discussing the appearances of typefaces. Some vocabulary applies only to a subset of all scripts.

Serifs

Image:Serif and sans-serif 01.png	Sans-serif font
Image:Serif and sans-serif 02.png	Serif font
Image:Serif and sans-serif 03.png	Serif font (serifs highlighted in red)

One can sub-divide fonts into two main categories: those of serif and sans-serif fonts. Serifs comprise the small features at the end of strokes within letters. The printing industry refers to typeface without serifs as sans-serif (from French sans: "without"), or as grotesque (or, in German, grotesk). See serif for etymological notes. Great variety exists among both serif and sans-serif fonts; both groups contain faces designed for setting large amounts of body text, and others intended primarily as decorative. The presence or absence of serifs forms only one of many factors to consider when choosing a font. Typefaces with serifs are often considered easier to read in long passages than those without. Studies on the matter are ambiguous, suggesting that most of this effect is due to the greater familiarity of serif typefaces. As a general rule, printed works such as newspapers and books almost always use serif fonts, at least for the text body. Web sites do not have to specify a font and can simply respect the browser settings of the user. But of those websites that do specify a font, most use modern sans-serif fonts such as Verdana, because it is commonly believed that, in contrast to the case for printed material, sans-serif fonts are easier than serif fonts to read on computer screens due to their lower resolution.

Proportionality

Verdana A proportional font displays glyphs using varying widths, while a non-proportional or fixed-width or monospace font uses fixed glyph-widths. Most people generally find proportional fonts nicer-looking and easier to read; and thus they appear more commonly in professionally published printed material. For the same reason, GUI computer applications (such as word processors and web browsers) typically use proportional fonts. However, many proportional fonts contain fixed-width figures so that columns of numbers stay aligned. However, non-proportional fonts function better than proportional fonts for some purposes because their characters line up in nice, neat columns. Most non-electronic typewriters and text-only computer displays use only non-proportional fonts. Most computer programs which have a text-based interface (terminal emulators, for example) use only non-proportional fonts in their configuration. Most computer programmers prefer to use monospace fonts while editing source code. ASCII art requires a non-proportional font for proper viewing. In a web page, the <pre> </pre> HTML tag most commonly specifies non-proportional fonts. In LaTeX, the verbatim environment uses non-proportional fonts. Any two lines of typical text with the same number of characters in each line in non-proportional font should display as equal in width, while the same two lines in proportional font have radically different widths. This comes about because wide characters' glyphs (WQZMDOHU) use more linear space and narrow characters' glyphs (itl[]1|I) use less linear space than the average-width glyph when using a proportional font. Editors read manuscripts in fixed-width fonts for ease of editing. The publishing industry considers it discourteous to submit a manuscript in a proportional font.

Measurements

LaTeX Most, if not all, scripts share the notion of a baseline: an imaginary horizontal line on which characters rest. In some scripts, parts of glyphs lie below the baseline. The descent spans the distance between the baseline and the lowest descending glyph in a typeface, and the part of a glyph that descends below the baseline has the name "descender". Conversely, the ascent spans the distance between the baseline and the top of the glyph that reaches farthest from the baseline. The ascent and descent may or may not include distance added by accents or diacritical marks. In the Latin, Greek and Cyrillic scripts, one can refer to the distance from the baseline to the top of regular lowercase glyphs as the x-height, and the part of a glyph rising above the x-height as the "ascender". The height of the ascender can have a dramatic effect on the readability and appearance of a font. The ratio between the x-height and the ascent often serves to characterise typefaces.

Font families

Since a plethora of typefaces has been created over the centuries, they are commonly categorized into families according to their appearance. Interestingly, this categorization corresponds vaguely with the historic evolution of typefaces. At the highest level, one can differentiate between blackletter, serif, sans-serif, script, symbol, and decorational fonts. Note: The following font samples print a sentence of patent nonsense, whose only purpose is to contain all letters of the alphabet (pangram).

Blackletter fonts

Blackletter fonts, the earliest fonts used with the invention of the printing press, resemble the blackletter calligraphy of that time. Many people refer to them as gothic script.

printing press
- Of all the blackletter typefaces, the Textualis ones (or Old English) most closely resemble the Textura calligraphy used with manual copying of books. Johannes Gutenberg carved a textualis typeface — including a large number of ligatures and common abbreviations — when he printed his 42-line Bible,

42-line Bible
- Schwabacher typefaces dominated in Germany from about 1480 to 1530, and the style continued in use occasionally until the 20th century. Most importantly, all of the works of Martin Luther, leading to the Protestant Reformation, as well as the Apocalypse of Albrecht Dürer (1498) used this typeface. Johannes Bämler, a printer from Augsburg, probably first used it as early as 1472. The origins of the name remain unclear; some assume that a typeface-carver from the village of Schwabach — one who worked externally and who thus became known as the Schwabacher — designed the typeface.

Augsburg
- The Fraktur family became the most commonly known among the blackletter typefaces. It started when Emperor Maximilian I (1493–1519) established a series of books and had a new typeface created specifically for this purpose. Printers in Germany made extensive use of Fraktur faces until the Nazis prohibited them in 1942.

Serif fonts

Serif, or "roman", typefaces are named for the features at the ends of their strokes. Times Roman and Garamond are common examples of serif typefaces. Serif fonts are probably the most used classification in printed materials, including most books, newspapers and magazines.

Sans-serif fonts

The typographical phenomenon of sans-serif designs appeared relatively recently in the history of type design. The two-line English so-called "Egyptian" font, released in 1816 by William Caslon's foundry in England apparently furnished the first specimen. They serve commonly, but not exclusively, for display typography applications such as signage, headings, and other situations demanding clear meaning but without the need for continuous reading. The text on web pages offers an exception: it appears mostly in sans-serif font because serifs make small letters less readable on a computer monitor.

Script fonts

Script fonts simulate handwriting: Zapfino is an example. They do not lend themselves very well to quantities of body text, as people find them harder to read than many serif and sans-serif fonts.

Novelty fonts

Novelty fonts have very unusual character shapes, and may even incorporate pictures of objects, animals, etc. into the character designs. They usually have very specific characteristics (e.g. evoking the Wild West, Christmas, horror films, etc.) and hence very limited uses. They are not suitable for body text.

PI fonts

PI fonts mostly consist of pictograms, such as decorative bullets, clock faces, railroad timetable symbols, CD-index or TV-channel enclosed numbers. Examples include Zapf dingbats, Webdings and Wingdings.

Symbol fonts

Symbol fonts consist of symbols rather than normal text characters. Examples include Zapf Dingbats (a popular font containing numerous miscellaneous symbols) and Sonata (a music font).

Texts used to demonstrate typefaces

A sentence that uses all of the alphabet (a pangram), such as "the quick brown fox jumps over a lazy dog", is often used as a design aesthetic tool to demonstrate the personality of a typefaces characters in a setting. For extended settings of typefaces graphic designers often use nonsense text (commonly referred to as "greeking"), such as lorem ipsum or Latin text such as the beginning of Cicero's in Catilinam. Greeking is used in typography to determine a typefaces "color", or weight and style, and to demonstrate an overall type aesthetic prior to actual type setting.

Legal aspects of typefaces

United States law does not permit the copyrighting of typeface designs, while allowing the patenting of unusually novel designs. Digital fonts that embody a particular design often become copyrightable as computer programs. The names of the typefaces can become trademarked. As a result of these various means of legal protection, sometimes the same typeface exists in multiple names and implementations. Some elements of the software engines used to display typefaces on computers have software patents associated with them. In particular, Apple Computer has patented some of the hinting algorithms for TrueType, requiring open-source alternatives such as FreeType to use different algorithms.

Organizations

- Type Directors Club
- ATypI, Association Typographique Internationale

External links

- [http://www.fontshop.com/index.cfm?fuseaction=catalog.fontpackage&displayfontid=EF.8279.0.0/ Examples of Old English Fonts] Good example of Old English Fonts
- [http://www.alvit.de/blog/article/20-best-license-free-official-fonts 20 Best License-Free Quality Fonts] compiled by Vitaly Friedman.
- [http://www.dafont.com/en/ DaFont] Probably the richest and best-known font archive on the Internet. In French or English.
- [http://www.changafonts.com/ Free Downloadable Fonts], 100's of free fronts and graphic design articles.
- [http://www.identifont.com/ Identifont], Font identifier.
- [http://www.fileinfo.net/filetype/font Font File Types]
- [http://fontforge.sourceforge.net Fontforge] font design free software (GPL).
- [http://www.vistawide.com/languages/foreign_language_fonts.htm Free Foreign Language Fonts] 100s of free, downloadable typefaces for over 40 languages
- [http://www.free-font-downloads.com Free Font Downloads], Free Fonts and Clipart Downloads.
- [http://www.hscripts.com/tutorials/css/fontp.php CSS Font properties?], Font style, weight, size and family.
- [http://www.searchfreefonts.com/ SearchFreeFonts.com], nice archive of free fonts in different categories.
- [http://www.needfonts.com NeedFonts.com], a free font resource for PCs and Macs
- [http://members.aol.com/willadams/portfolio/typography/typefaceterminology.pdf Typeface Terminology], a glossary by William Adams
- [http://members.aol.com/willadams/portfolio/typography/onetype.pdf One Typeface, Many Fonts], by William Adams
- [http://www.chank.com/howto/index.php Guide to making fonts]
- [http://euro.typo.cz/ Typo.cz], information on Central-European typography and typesetting
- [http://diacritics.typo.cz Diacritics Project], materials for designing a font with accents
- [http://welovefreebies.com/folders/Free_Fonts Directory of free fonts website ]
- [http://www.icogitate.com/~ergosum/fonts/music-fonts2.htm Free Music Fonts], MusiQwik, MusiSync, Bongos, FretQwik, NoteHedz, and MusiTone are free, original True Type fonts that depict musical notation.
- [http://www.myfonts.com/ MyFonts.com], sells fonts by many foundries Category:Typesetting ko:글꼴 ja:書体 zh-min-nan:Jī-hêng

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Number

: This article is about numbers such as counting numbers and measurements. For other uses of the term, see Number (disambiguation). A number originally was a count or a measurement. Mathematicians have extended this concept to include abstractions such as the square root of minus one. In common usage, number symbols are often used as labels (highway numbers) or to indicate order (serial numbers).

Examples

The most familiar numbers are the counting numbers or natural numbers. Some writers include 0, thus: . Others do not: . In the base ten number system, now in almost universal use worldwide, the symbols for natural numbers are written using ten digits, 0 through 9. The symbol for the set of all natural numbers is N. If the negative whole numbers are combined with the positive whole numbers and zero, one obtains the integers Z (from the German word "zahlen"). (Some authors use W for the whole numbers, but other authors use W for the natural numbers, so the W symbol is ambiguous.) Negative numbers are used to indicate an opposite. If a positive number is used to indicate distance to the right of some fixed point, a negative number indicates distance to the left. If a positive number indicates a bank deposit, a negative number indicates a withdrawal. Rational numbers are made up of all numbers that can be expressed as a fraction, with integer numerator and non-zero natural number denominator. The fraction m/n represents the quantity arrived at when a whole is divided into n equal parts, and m of those equal parts are chosen. If m is greater than n, the fraction is greater than one. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face Q (for "quotient"). The real numbers are made up of all numbers that can be expressed as a decimal. These are the measuring numbers, and in the base ten number system are written as a string of digits, with a dot (US) or a comma (Europe) to the right of the ones place. The symbol for the real numbers is R. All measurements are necessarily approximations; the accuracy of the approximation depends on the accuracy of the measuring device. Therefore all measurements are properly represented by decimals that end, the last decimal place indicating the accuracy of the measurement. For example, 1.23 inches indicates a measurement accurate to the nearest hundredth of an inch. However, mathematically, when a rational number is expressed as a decimal, it may never end. Thus 1/3 becomes 0.3333... (unending threes). Mathematicians, therefore, consider both decimals that end and decimals that go on forever. The latter represent an infinite series. Some real numbers can be written as fractions, 0.3333... for example. Others cannot, 0.1010010001... for example. A decimal that can be written as a fraction is called rational, a decimal that cannot be written as a fraction is called irrational. A decimal is rational when it either ends or repeats forever. There is a technical sense in which the real numbers are the ideal set of numbers. They are the only complete ordered field. Moving to a greater level of abstraction, and away from counting and measuring, the real numbers can be extended to the complex numbers C. This set of numbers arose, historically, from consideration of the question of whether or not there was any sense in which negative numbers can have a square root. A new number was invented, the square root of negative one, denoted by i, a symbol assigned to this new number by Leonhard Euler. The complex numbers consist of all numbers of the form a + bi, where a and b are real numbers. If b is zero, then a + bi is real. If a is zero, then a + bi is called imaginary. The complex numbers are an algebraically closed field, meaning that every polynomial with complex coefficients can be factored into linear factors with complex coefficients. The above symbols are often written in blackboard bold, thus: :

\mathbb\sub\mathbb\sub\mathbb\sub\mathbb\sub\mathbb

While the natural numbers and the real numbers suffice for most everyday purposes, mathematicians have invented many other sets of numbers with specialized uses. Some are subsets of the complex numbers. For example the roots of polynomials with rational coefficients are called the algebraic numbers. Real numbers that are not algebraic are called transcendental numbers. The Gaussian integers are complex numbers a + bi where a and b are integers. Sets of numbers that are not subsets of the complex numbers include the quaternions H, invented by Sir William Rowan Hamilton, in which multiplication is not commutative, and the octonions, in which multiplication is not associative.

Further generalizations

Elements of function fields of finite characteristic behave in some ways like numbers and are often regarded as a kind of number by number theorists.

Numerals and numbering

Numbers should be distinguished from numerals, the symbols used to represent numbers. The number five can be represented by both the base ten numeral 5 and by the Roman numeral V. Notations used to represent numbers are discussed in the article numeral systems. Numbers are often used to give objects unique names. Examples are telephone numbers, social security numbers, and ISBNs.

Extensions

Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While real numbers may have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left, with digits in base p, where p is prime. This leads to the p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; but they differ in the infinite case.) The arithmetical operations of numbers, such as addition, subtraction, multiplication and division, are generalized in the branch of mathematics called abstract algebra, the study of abstract number systems such as groups, rings and fields.

External links

- [http://freepages.history.rootsweb.com/~catshaman/13comp/0numer.htm Mesopotamian and Germanic numbers]

References

- Erich Friedman, [http://www.stetson.edu/~efriedma/numbers.html What's special about this number?]
- [http://www.cut-the-knot.org/do_you_know/numbers.shtml What's a Number?] at cut-the-knot Category:Group theory Category:Numbers __NOTOC__ ko:수 (수학) ja:数 simple:Number th:จำนวน

Bold

In typography, emphasis usually refers to means of stressing parts of a text by using letters in a different style from the rest of the text to make them stand out from the main text body.

Methods and uses of emphasis

typography The human eye is very receptive to differences in brightness within a text body. One can therefore differentiate between types of emphasis according to whether the emphasis changes the "blackness" of text. A means of emphasis that does not have much effect on "blackness" is printing in italics, where the text is written in a script style, or oblique, where the vertical orientation of all letters is slanted to the left or right. With one or other of these techniques (usually only one is available for any typeface), words can be highlighted without making them "stick out" much from the rest of the text (inconspicuous stressing). Traditionally, this is used for marking passages that have a different context, such as words from foreign languages, book titles, etc. By contrast, boldface makes text darker than the surrounding text. With this technique, the emphasized text strongly stands out from the rest; it should therefore be used to highlight certain keywords that are important to the subject of the text, for easy visual scanning of text. For example, printed dictionaries often use boldface for their keywords; Wikipedia follows this convention when the name of each article is marked at the top in bold. If the text body is typeset in a serif typeface, it is also possible to highlight words by setting them in a sans serif face. This is somewhat of an archaic practice.

Emphasis in design

With both italics and boldface, the emphasis is correctly achieved by temporarily replacing the current typeface. Professional typographic systems (which include most modern computers) would therefore not simply tilt letters to the right to achieve italics (that is instead referred to as slanting) or print them darker for boldface, but instead use entirely different typefaces that achieve the effect. As can be seen in Fig. 1, the "w" letter, for example, looks quite different in italics compared to the regular typeface. As a result, typefaces therefore have to be supplied at least fourfold (with computer systems, usually as four font files): as regular, italics, bold, and both bold and italics to provide for all combinations. Professional typefaces sometimes offer even more variations for popular fonts, with varying degrees of blackness. Only if such fonts are not available should the effect of italics or boldface be imitated by tilting or blacking the original font.

Alternative methods for emphasis

Capitalization

The house styles of many U.S. publishers use capitalization or all-uppercase letters, in order to emphasise
- publication titles
- warning messages
- newspaper headlines
- chapter and section headings Capitalization is used much less commonly today by British publishers (usually only for book titles). It is rarely used in other languages. All-uppercase letters are a common form of emphasis where the medium lacks support for boldface, such as old typewriters, plain-text email, SMS and other text-messaging systems.

Letterspacing

SMS In Germany, a different means of emphasis was previously used. To achieve a variance in blackness, instead of making the letters darker, one would increase the spacing between them. This resulted in an effect reverse to boldface: the emphasized text becomes lighter than its environment. This was referred to as sperren in German ("letterspacing" in English), which could here be translated as "spacing out". While sperren normally means "to lock (out)", this particular meaning was figurative: with the older method of typesetting with letters of lead, the spacing would be achieved by inserting additional non-printing slices of metal between the types. The example text reads: "An example of German text in Fraktur in which a portion of the text is spaced out. It is noticed, as with boldface, clearly as opposed to the rest of the text. The reason for this particular German typographic convention can be seen in the traditional use of blackletter typefaces, for which boldface was not feasible, since the letters were very dark in their standard format. The blackletter typefaces were officially abolished in 1942 by Nazi Germany, and after that, its use quickly diminished. As a result, the use of spacing as a means of emphasis in printed materials quickly became obsolete. However, spacing is sometimes still used as a means of emphasis in typographic media where only one typeset is available, e.g. in typewritten communication or on text-only computer terminals.

Special punctuation marks

In Chinese, emphasis in body text is supposed to be indicated by using an "emphasis mark" (着重號), which is a dot placed under each character to be emphasized. This is still taught in schools, but in practice it is not usually done, probably due to the difficulty of doing this in most computer software. Methods used for emphasis in western texts but inappropriate for Chinese, for example underlining and setting text in artificially slanted type (frequently incorrectly called "italics"), are often used instead. Category:typography

Mathematician

A mathematician is a person whose area of study and research is mathematics. Today, most mathematicians are professors at a university or other research institution; however, a minority have a non-academic career and are often known as amateur mathematicians. While a number of misinformed people may believe mathematics is fully understood (as it is often presented this way in elementary textbooks), in fact, there is ongoing research into many areas of mathematics. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics). Unlike the other sciences, research in mathematics generally does not consist of performing experiments. Rather, mathematics is about problem-solving, where truths are deduced from other known truths. Computer experiments and other numerical evidence might be a part of this process, but in the end, mathematics research is about constructing proofs of theorems. In particular, calculation is not a big part of mathematics research, and mathematicians need not have any extraordinary ability in adding or multiplying numbers. See mental calculators to read about prodigies at performing such calculations.

Motivation

Mathematicians are typically interested in finding and describing patterns that may have originally arisen from problems of calculation, but have now been abstracted to become problems of their own. Problems have come from physics, economics, games, generalizations of earlier mathematics, and some problems are simply created for the challenge of solving them. Although much mathematics is not immediately useful, history has shown the eventually applications are found. For example, number theory originally seemed to be without purpose, but after the invention of computers it gained countless applications to algorithms and cryptography.

Differences

Mathematicians differ from philosophers in that the primary questions of mathematics are assumed (for the most part) to transcend the context of the human mind; the idea that "2+2=4 is a true statement" is assumed to exist without requiring a human mind to state the problem. Not all mathematicians would strictly agree with the above; the philosophy of mathematics contains several viewpoints on this question. Mathematicians differ from physical scientists such as physicists or engineers in that they do not typically perform experiments to confirm or deny their conclusions; and whereas every scientific theory is always assumed to be an approximation of truth, mathematical statements are an attempt at capturing truth. If a certain statement is believed to be true by mathematicians (typically as special cases are confirmed to some degree) but has neither been proven nor disproven to logically follow from some set of assumptions, it is called a conjecture, as opposed to the ultimate goal, a theorem that is proven true. Unlike physical theories, which may be expected to change whenever new information about our physical world is discovered, mathematical theories are static. Once a statement is considered a theorem, it remains true forever.

Demographics

As is the case in many scientific disciplines, the field of mathematics has been disproportionately dominated by men. Among the minority of prominent female mathematicians are Emmy Noether (1882 - 1935), Sophie Germain (1776 - 1831), Sofia Kovalevskaya (1850 - 1891), Rózsa Péter (1905 - 1977), Julia Robinson (1919 - 1985), Mary Ellen Rudin, Eva Tardos, Émilie du Châtelet, Mary Cartwright and Marianna Csörnyei.

Quotes

...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell. :-Saint Augustine, De Genesi ad Litteram (actually "mathematicians" in this context refers mainly to astrologers and such) A mathematician is a machine for turning coffee into theorems. :-Paul Erdős Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. (Mathematicians are [like] a sort of Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.) :-Johann Wolfgang von Goethe Some humans are mathematicians; others aren't. :-Jane Goodall (1971) In the Shadow of Man

Jokes

Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows: :An engineer, a physicist, and a mathematician are all staying at a hotel one night when a fire breaks out. The engineer wakes up and smells the smoke; he quickly grabs a garbage pail to use as a bucket, fills it with water from the bathroom, and puts out the fire in his room. He then refills the pail and douses everything flammable in the room with water. He then returns to sleep. :The physicist wakes up, smells the smoke, jumps out of bed. He picks up a pad and pencil and makes some calculations, glancing frequently at the flames. He then measures exactly 15.6 liters of water into the garbage pail, and throws it on the flames, which are extinguished. Smiling, he returns to sleep. :Finally the mathematician wakes up. He too grabs a pad and begins furiously writing; glancing at the flames; and then writing more. After a while he gets a satisfied look on his face; entering the bathroom, he produces a match, lights it, and then extinguishes it with a bit of running water. "Aha! A solution exists," he murmurs - and returns to his slumbers. Another joke goes thus: :Three men are flying in a hot air balloon and suddenly they realize that they are lost. Luckily they see a man plowing a field and ask, "Where are we?". The man on the ground thinks for a minute and then answers, "You are in a hot air balloon". One of the men in the air then says to his friends, "He was a mathematician - he thought before answering, his answer was totally right and totally useless" And another: :An astrologer, a chemist, and a mathematician are on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by. The astrologer excitedly exclaims, "Ah, this shows Scottish sheep are black!" The chemist didactically corrects him: "No, no, it just shows some Scottish sheep are black." The mathematician then says, "Actually, we can only be sure there is at least one Scottish sheep of which at least one side is black" And finally: : An experiment is being made. A physicist (or an engineer) and a mathematician are asked to boil hot water, but the kettle is in the living room. The physicist goes to the living room, takes the kettle, returns to the kitchen and puts it on the stove and boils the water. The mathematician does the same. In the second stage, the kettle is in the kitchen and the two are again asked to boil hot water. The physicist simply puts the kettle on the stove and boils the water. However, the mathematician takes the kettle, puts it in the living room and declares: "We have already solved this problem!"

Links and references

References

- A Mathematician's Apology, by G. H. Hardy. Memoir, with foreword by C. P. Snow.
- Reprint edition, Cambridge University Press, 1992; ISBN 0521427061
- First edition, 1940
- Dunham, William. The Mathematical Universe. John Wiley 1994.

External links

- [http://www-history.mcs.st-and.ac.uk/history/index0.html The MacTutor History of Mathematics archive], a very complete list of detailed biographies.
- [http://genealogy.math.ndsu.nodak.edu/ The Mathematics Genealogy Project], which allows to follow the succession of thesis advisors for most mathematicians, living or dead. Category:Mathematical science occupations
- ja:数学者 ko:수학자 th:นักคณิตศาสตร์ __NOTOC__

Jean-Pierre Serre

Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology.

Life and career

Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the Ecole Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. He is a professor at the Collège de France.

Early work

From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, in the context of sheaf theory and homological algebra techniques. Serre's thesis refers to his dissertation on the Leray-Serre spectral sequence associated to a fibration. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl praised Serre in apparently extravagant terms, and also made the point that the award was for the first time awarded to an algebraist. While Serre subsequently moved field — at this point he apparently thought that homotopy theory where he had started was already over-technical — Weyl's perception that the central place of classical analysis had been challenged by abstract algebra has subsequently been justified, as has his assessment of Serre's place in this change.

Foundational work in algebraic geometry, and the Weil conjectures

In the 1950s and 1960s, a fruitful collaboration between Serre and the two years younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were FAC (Faisceaux Algébriques Cohérents, on coherent cohomology) and GAGA. Serre had early on perceived a need to construct more general and refined cohomology theories to tackle these conjectures. In simple terms, the cohomology of a coherent sheaf over a finite field couldn't capture as much topology as singular cohomology, with integer coefficients. Amongst Serre's early candidate theories (1954/55) was one based on Witt vector coefficients. Grothendieck in SGA4 eventually delivered a full technical development. Around 1958 Serre had suggested that isotrivial covers of algebraic varieties should be important — those that become trivial after pullback by a finite covering map. This was one important step towards the eventual étale covering theory. In the later developments Serre was sometimes a source instead of counterexamples to over-optimistic extrapolations. He also had a close working relationship with Pierre Deligne, who eventually finished the proof of the Weil conjectures.

Other work

From 1959 onwards his interests turned towards number theory, in particular class field theory and the theory of complex multiplication. Amongst his most original contributions were: the concept of algebraic K-theory; the Galois representation theory for l-adic cohomology and the conceptions that these representations were 'large'; and the Serre conjecture on mod p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

Awards

Serre was awarded the Fields Medal in 1954, and was the first recipient of the Abel Prize in 2003. He also received the Balzan Prize (1985), the Steele Prize (1995), and the Wolf Prize (2000).

External link

- Serre, Jean-Pierre Serre, Jean-Pierre Serre, Jean-Pierre Serre, Jean-Pierre ko:장-피에르 세르 ja:ジャン＝ピエール・セール

TeX

] T_EX, written as TeX in plain text, is a typesetting system created by Donald Knuth. Together with the METAFONT language for font description and the Computer Modern typeface, it was designed with two main goals in mind: firstly, to allow anybody to write high-quality books using a reasonable amount of effort, and, secondly, to provide a system that would give the exact same results on all computers, now and in the future. It is Free software and is popular in academia, especially in the mathematics, physics and computer science communities. It has largely displaced Unix troff, the other favored formatter, in many Unix installations. TeX is generally considered to be the best way to typeset complex mathematical formulas, but, especially in the form of LaTeX and other template packages, is now also being used for many other typesetting tasks.

The name and its pronunciation(s)

An homage to Caltech, where Knuth received his doctorate, the name TeX is intended to be pronounced "tekh", where "kh" represents the sound at the end of Scottish loch or the name of the German composer Bach (in IPA ). The X is meant to represent the Greek letter χ (chi). TeX is the abbreviation of τέχνη (technē), Greek for "art" and "craft", which is also the source word of technical. English speakers often pronounce it "tek", like the first syllable of technology. The name is properly typeset with the "E" below the baseline; systems that do not support subscript layout use the approximation "TeX". Fans like to proliferate names from the word "TeX" — such as TeXnician (user of TeX software), TeXpert, TeXhacker (TeX programmer), TeXmaster (competent TeX programmer), TeXhax, and TeXnique.

History

Motivation and early history

Knuth began to write TeX because he had become annoyed at the declining quality of the typesetting in volumes I–III of his monumental The Art of Computer Programming. In a manifestation of the typical hackish urge to solve the problem at hand once and for all, he began to design his own typesetting language. He planned he would finish it on his sabbatical in 1978, but as it happened the language was frozen only in 1989, more than ten years later. Guy Steele happened to be at Stanford during the summer of 1978, when Knuth was developing his first version of TeX. When Steele returned to MIT that fall, he rewrote TeX's I/O to run under the ITS operating system. The first version of TeX was written in the SAIL programming language to run on a PDP-10 under Stanford's WAITS operating system. For later versions of TeX, Knuth invented the concept of literate programming, a way of producing compilable source code and high quality cross-linked documentation (typeset in TeX, of course) from the same original file. The language used is called WEB and produces programs in Pascal.

Version 3

In 1989, Donald Knuth released new versions of TeX and Metafont. Despite his desire to keep the program stable, Knuth realised that 128 different characters for the text input were not enough to accomodate foreign languages; the main change in version 3.0 of TeX is thus the ability to work with 8-bits inputs, allowing 256 different characters in the text input. Since version 3, TeX has used an idiosyncratic version numbering system, where updates have been indicated by adding an extra digit at the end of the decimal, so that the version number asymptotically approaches π. This is a reflection of the fact that TeX is now very stable, and only minor updates are anticipated.

Current version and the future of TeX

The current version of TeX is 3.141592; it was last updated in December 2002. The design has been frozen after version 3.0, and no new feature or fundamental change will be added after that, so that all newer version contain only bug fixes. Even though Donald Knuth himself has suggested a few areas in which TeX could have been improved, he indicated that he firmly believes that having an unchanged system that will produce the same output now and in the future is more important than introducing new features. For this reason, he has stated that the "absolutely final change (to be made after my death)" will be to change the version number to π, at which point all remaining bugs will become features. However, since the source code of TeX is essentially in the public domain (see below), other programmers are allowed (and explicitly encouraged) to improve the system, but are required to use another name to distribute the modified TeX.

The typesetting system

TeX commands commonly start with a backslash and are grouped with curly braces. However, almost all of TeX's syntactic properties can be changed on the fly which makes TeX input hard to parse by anything but TeX itself. TeX is a macro and token based language: many commands, including most user-defined ones, are expanded on the fly until only unexpandable tokens remain which get executed. Expansion itself is practically side-effect free. Tail recursion of macros takes no memory, and if-then-else constructs are available. This makes TeX a Turing-complete language even at expansion level. The system can be divided in four levels: in the first characters are read from file and assigned a category code. Combinations of a backslash (really: any character of category zero) followed by letters (characters of category 11) or a single other character are replaced by a control sequence token. In this sense this stage is like lexical analysis, although it does not form numbers from digits. In the next stage, expandable control sequences (such as conditionals or defined macros) are replaced by their replacement text. The input for the third stage is then a stream of characters, including ones with special meaning, and unexpandable control sequences, typically assignments and visual commands. Here characters get assembled into a paragraph. TeX's paragraph breaking algorithm works by optimizing breakpoints over the whole paragraph. After the paragraph is broken into lines, the vertical list of lines and other material is broken into pages. The TeX system has precise knowledge of the sizes of all characters and symbols, and using this information, it computes the optimal arrangement of letters per line and lines per page. It then produces a DVI file ("DeVice Independent") containing the final locations of all characters. This dvi file can be printed directly given an appropriate printer driver, or it can be converted to other formats. Nowadays, PDFTeX is often used which bypasses DVI generation altogether. Most functionality is provided by format files (predumped memory images of TeX after large macro collections have been loaded). Common formats are Knuth's original TeX, LaTeX (ubiquitous in the technical sciences), and ConTeXt (which is used primarily for desktop publishing). The ultimate reference works for TeX are the first two volumes of Knuth's Computers and Typesetting, The TeXbook and TeX: The Program (which includes the complete documented source code for TeX). TeX is usually distributed together with METAFONT, a companion program also developed by Knuth which allows algorithmic description of fonts. The organisation of the directories in a TeX / Metafont installation is standardized in a tree called texmf.

License

Donald Knuth has indicated several times that the source code of T_EX has been placed into the public domain, and he strongly encourages modifications or experimentations with this source code. However, since he highly values the reproducibility of the output of all versions of T_EX, any changed version must not be called T_EX, TeX, or anything confusingly similar. To enforce this rule, the American Mathematical Society has registered a trademark for T_EX, and any implementation of the system must pass a test suite called the TRIP test ([ftp://tug.ctan.org/pub/tex-archive/systems/knuth/tex/trip.tex available on CTAN]) before being allowed to be called T_EX. The question of licence is somewhat confused by the statements included at the beginning of the TeX source code, which indicate that "all rights are reserved. Copying of this file is authorized only if (...) you make absolutely no changes to your copy". However, this restriction should be interpreted as a prohibition to change the source code as long as the file is called tex.web. This interpretation is confirmed later in the source code when the TRIP test is mentioned ("If this program is changed, the resulting system should not be called`\TeX").
Quality
TeX is written in WEB, a mixture of documentation written in TeX and a quite restricted Pascal subset. For example, TeX does all of its dynamic allocation itself from fixed-size arrays. As a result, TeX has been ported to almost all operating systems (usually by using the web2c converter). Knuth offers monetary awards to people who find and report a bug in it. The award per bug started at $2.56 and doubled every year until it was frozen at its current value of $327.68. This has not made Knuth poor, however, as there have been very few bugs claimed and in any case a cheque proving that the owner found a bug in TeX is usually framed instead of cashed. Knuth has kept a very detailed log of all the bugs he has corrected and changes he has made in the program since 1982; as of 2005, the list contains 419 entries, not including the version modification that should be done after his death as the final change in TeX.
Computer-science aspects of TeX
The TeX software incorporates several interesting algorithms, and has led to a number of theses for Knuth's students. For instance, a hyphenation algorithm (work by Frank Liang) is used that assigns priorities to breakpoints in letter groups. A list of hyphenation patterns can be generated automatically from a corpus of hyphenated words. The line breaking algorithm is an example of dynamic programming. The problem of breaking a paragraph of n words into lines has a naive complexity of 2ⁿ, but with dynamic programming a globally optimal layout can be derived in time proportional to the number of words and the number of words per line. A thesis by Michael Plass shows how the page breaking problem can be NP-complete because of the added complication of placing figures. The companion program METAFONT for character generation uses Bezier curves in a fairly standard way, but Knuth devotes lots of attention to the rasterizing problem on bitmapped displays. Another thesis, by John Hobby, further explores this problem of digitizing "brush trajectories". This term derives from the fact that Metafont describes characters as having been drawn by abstract brushes. While TeX has been highly successful, Metafont has not been accepted by professional type designers, and fonts produced with it like Computer Modern have been harshly criticized.
Derived works
Several document processing systems are based on TeX, notably:
- LaTeX (Lamport TeX), which incorporates document styles for books, letters, slides, etc., and adds support for referencing and automatic numbering of sections and equations,
- ConTeXt, was written mostly by Hans Hagen at Pragma. It is a document designing tool based on TeX. It is much younger than LaTeX.
- AMS-TeX, produced by the American Mathematical Society, has many more user-friendly commands, which can be altered by journals to fit with the house style. Most of the features of AMS-T_EX can be used in L^AT_EX by using the AMS "packages". This is then referred to as AMS-L^AT_EX. The most popular book on AMS-T_EX was written by Michael Spivak, and is entitled The Joy of TeX.
- jadeTeX uses TeX as a backend for printing from James Clark's DSSSL Engine,
- Texinfo is the GNU documentation processing system.
- XeTeX is a new TeX engine that supports Unicode and the advanced Mac OS X font technologies. Numerous extensions and companion programs for TeX exist, among them BibTeX for bibliographies (distributed with LaTeX), PDFTeX, which bypasses dvi and produces output in Adobe Systems' Portable Document Format, and Omega, which allows TeX to use the Unicode character set. All TeX extensions are available for free from CTAN, the Comprehensive TeX Archive Network.
Compatible tools
On UNIX-compatible systems, including GNU/Linux and Mac OS X, TeX is distributed in the form of [http://www.tug.org/teTeX/ teTeX]. On Windows, there is the [http://www.miktex.org/ MiKTeX] distribution and the [http://www.fptex.org/ fpTeX] distribution. The TeXmacs text editor is a WYSIWYG scientific text editor that is intended to be compatible with TeX. It uses Knuth's fonts, and can generate TeX output. LyX is a "What You See is What You Mean" document processor which runs on a variety of platforms including Linux, Windows (2000 or later) or Mac OS X (using a non-native QT front-end). TeXShop for Mac OS X, and WinShell for Windows are similar tools. TeX has been the official typesetting package for the GNU operating system since 1984. The KDE windowing system for Unix has a program, Kile, for editing LaTeX and TeX. GNU Emacs has various built-in and third party packages with support for TeX, the major one being AUCTeX.
Examples of TeX
A simple plain TeX example- Create a text file myfile.tex with the following content: hello \bye Then open a command line interpreter and create a file called myfile.dvi by typing tex myfile.tex The dvi file can be displayed on screen, using a viewer programme such as yap, which will show hello on a page. \bye is a TeX command which marks the end of the file and is not shown in the final output. The results can either be printed directly from the viewer or converted to a more common format such as PostScript using the dvips program. PDF files may also be created directly using pdfTeX. pdfTeX was originally created because converting generated PostScript into PDF resulted in poor font display, though printing performance was fine. This was because TeX natively uses bitmap fonts, which are only designed to display well at one particular size, whereas PostScript typically uses scalable Type 1 fonts. It is now possible to make dvips output scalable fonts with a bit of tweaking (newer versions of Ghostscript support it), but direct conversion to PDF has other benefits: it is a one-step, not two-step process, and pdfTeX provides facilities such as bookmarks and hyperlinks not found in PostScript.
Mathematical examples
To see TeX further in action, look at its formatting of mathematical formulas. For example, to write the well-known quadratic formula, try entering The quadratic formula is $$ \bye Use TeX as above, and you should get something that looks like :The quadratic formula is Notice how the formula is printed in a way a person would write by hand, or typeset the equation. In a document, entering mathematics mode is done by starting with a $, then entering a formula in TeX semantics and closing again with another $. Display mathematics, or mathematics presented centered on a new line is done by using $$. For example, the above with the quadratic formula in display math: The quadratic formula is $$$$ \bye renders as :The quadratic formula is
LaTeX examples
LaTeX is a collection of macros written in TeX. There are many predefined templates (with predefined styles) one can use. It is much more structured than TeX, providing a set of macros and utilities for indexing, tables, lists and so forth. For example: \documentclass[a4paper] \begin \section \subsection %% The text goes here \end
ConTeXt examples
ConTeXt is a collection of macros written in TeX. As LaTeX it supports indexing, tables, lists and so forth, but it is also very easy to modify the layout. If the layout is not played with too much, it allows the creation of very well structured documents. For example: \setuphead[section][style=italic] % render the section title italic \starttext \chapter \section ... % section text goes here \stoptext
See also

- METAFONT
- MetaPost
- List of document markup languages
- Comparison of document markup languages
- TeX Users Group
- Texvc, TeX preprocessor used in MediaWiki
- PSTricks, a PostScript set of macros.
References

- Donald E. Knuth. The TeXbook (Computers and Typesetting Volume A), Reading, Massachusetts: Addison-Wesley, 1984. ISBN 0201134489. The [http://www.ctan.org/tex-archive/systems/knuth/tex/texbook.tex source code of the book in TeX] is available online on CTAN. It is provided only as an example and its use to prepare a book like The TeXbook is not allowed.
- Donald E. Knuth. Digital Typography (CSLI lecture notes, no 78). Center for the Study of Language and Information, 1999. ISBN 1575860104
- Donald E. Knuth [http://www.ntg.nl/maps/pdf/5_34.pdf The future of TeX and METAFONT] (NTG journal MAPS), 1990.
- Donald E. Knuth, [ftp://tug.ctan.org/pub/tex-archive/systems/knuth/tex/tex.web Source code of TeX], available on CTAN. Being written using literate programming, the source code contains plenty of human-readable documentation.
- Donald E. Knuth, [ftp://tug.ctan.org/pub/tex-archive/systems/knuth/errata/tex82.bug List of updates to the TeX82 listing published in September 1982], available on CTAN.
External links

- [http://refcards.com/refcards/tex/tex-refcard-letter.pdf Plain TeX Quick Reference (PDF)]
- MediaWiki User's Guide to editing mathematical formulae
- [http://www.tug.org/ The TeX users group]
- The UK TeX Users' Group [http://www.tex.ac.uk/cgi-bin/texfaq2html?introduction=yes FAQ]
- [http://www.artofproblemsolving.com/LaTeX/AoPS_L_About.php Getting started with LaTex] at Art of Problem Solving
- [http://www.york.ac.uk/depts/maths/tex/texnotes.ps Simon Eveson, An Introduction to Mathematical Document Production Using AmSLaTeX]
a PostScript file
- [http://www.esm.psu.edu/mac-tex/ Mac OS X TeX/LaTeX Web Site]
- [http://sciencesoft.at/index.jsp?link=latex&lang=en Online LaTeX] for converting LaTeX code to a PNG graphic online
- ConTeXt: [http://contextgarden.net The ConTeXt wiki] and [http://www.pragma-ade.com Homepage at Pragma]
Software

- [http://www.ctan.org/ Comprehensive TeX Archive Network]: Repository of the TeX source and hundreds of add-ons and style files.
- Kile is a user-friendly TeX/LaTeX editor for KDE.
- [http://omega.enstb.org/index.html Omega] (16 bit version of TeX; includes lambda version of LaTeX)
- [http://www.toolscenter.org TeXnicCenter] (a feature rich integrated development environment (IDE) for developing LaTeX-documents on Microsoft Windows (Windows 9x/ME, NT/2000/XP) freely available under GPL.)
- [http://www.latexeditor.org/ LaTeX Editor], called LEd, is a free environment for rapid TeX/LaTeX document development for Win 95/98/Me/NT4/2000/XP/2003.
- [http://www.texmacs.org/ GNU TeXmacs Scientific Editor] is a free WYSIWYW (what you see is what you want) editing platform, inspired (and compatibile) by TeX and Emacs.
- [http://www.winshell.org WinShell] A free integrated development environment (IDE) for easy working with LaTeX or TeX.
- [http://www.chikrii.com Chikrii Softlab] (Word2Tex and Tex2Word)
- The [http://www.tug.org/texlive/ TeXLive] distribution is said to be an easy start for beginners. It includes a multiplatform DVD which contains basically all of CTAN. For Windows users it includes fpTeX (see below).
- [http://www.uoregon.edu/~koch/texshop/texshop.html TeXShop] - a free TeX editor for Mac OS X (with syntax coloring and Cocoa spellchecking)
- [http://www.miktex.org/ MiKTeX] – MiKTeX (pronounced mick-tech) is an up-to-date implementation of TeX and related programs for Windows (all current variants) on x86 systems.
- [http://www.fptex.org fpTeX] – fpTeX is an up-to-date port of tetex for Windows.
- [http://www.ctan.org/tex-archive/info/lshort/english/lshort.pdf Short introduction] - A fine tutorial to LaTeX
- [http://www.truetex.com/ TrueTeX: ] A best-selling commercial implementation of TeX and LaTeX.
Periodicals

- The PracTeX Journal. Online journal of the TeX Users Group.
- TUGboat. Print journal of the TeX Users Group.
Books

- Eijkhout, Victor. [http://www.eijkhout.net/tbt/ TeX by Topic]: a programmer's reference; originally published by Addison-Wesley in 1992 (ISBN 0201568829), now freely (as in free beer) downloadable.
- [http://tug.org/ftp/tex/impatient/ TeX for the Impatient], a more tutorial book, originally published by Addison-Wesley in 1990 and now licensed under the GFDL. See also Wikibooks:TeX for the Impatient
- Walsh, Norman. [http://makingtexwork.sourceforge.net/mtw/ Making TeX Work]: originally published by O'Reilly in 1994 (ISBN 1565920511) and now licensed under the GFDL.
- Schwarz, Stefan and Potucek, Rudolf. [http://texikon.artiverse.net/ TeXikon]: (German) online reference work documenting over 1400 TeX and LaTeX commands. This website derives from an out-of-print book published by Addison-Wesley in 1996 as ISBN 3893196900. Category:Digital typography Category:Free software Category:Macro programming languages Category:TeX Category:Typesetting ko:TeX ja:TeX

LaTeX

L^AT_EX, written as LaTeX in plain text, is a document preparation system for the TeX typesetting program. It offers programmable desktop publishing features and extensive facilities for automating most aspects of typesetting and desktop publishing, including numbering and cross-referencing, tables and figures, page layout, bibliographies, and much more. LaTeX was originally written in 1984 by Leslie Lamport and has become the dominant method for using TeX —few people write in plain TeX anymore. The current version is LaTeX2ε.

Pronunciation

LaTeX is usually pronounced "LAY-tech" or "LAH-tech" (IPA: , ), where ch represents the sound of ch in German Bach or Scottish loch: the last character in the name is actually a capital chi, as the name of TeX derives from the Greek τεχνη (skill, art, technique). While TeX's creator Donald Knuth promoted the "tech" pronunciation, Lamport has said he doesn't favor or deprecate any pronunciation for LaTeX. It is traditionally printed with the special typographical logo shown on this page. In media where the logo cannot be precisely reproduced in running text, the word is typically given the unique capitalization LaTeX to avoid confusion with the word "latex".

The typesetting system

latex.]] LaTeX is based on the idea that authors should be able to concentrate on writing within the logical structure of their document, rather than spending their time on the details of formatting. It encourages the separation of formatting from content, whilst still allowing manual typesetting adjustments where needed. By keeping the formatting details in a separate file from the text, it is often regarded as superior to word processors and most other desktop publishing systems, which allow trivially easy visual layout changes but tend to intertwine content and form so tightly that consistency and automation are often difficult. LaTeX also provides great flexibility in formatting while maintaining the identity of structure, which purely structural systems like SGML and XML do not directly address. LaTeX can be arbitrarily extended by using the underlying macro language for developing custom formats. For example, there are numerous commercial implementations of the whole TeX system (which includes LaTeX), and vendors may offer extra features like phone support and additional typefaces. LyX is a free visual document processor that uses LaTeX for a back-end. TeXmacs is a free, WYSIWYG editor with similar functionalities as LaTeX, but a different typesetting engine. A number of popular commercial DTP systems use modified versions of the original TeX typesetting engine. The recent rise in popularity of XML systems and the demand for large-scale batch production of publication-quality typesetting from such sources has seen a steady increase in the use of LaTeX. The example below shows an example of a LaTeX input (left) and output (right): [http://sciencesoft.at/index.jsp?link=latex&lang=en&wiki=1 Online LaTex], which uses this example.

Community

batch production LaTeX was originally most commonly used by mathematicians and scientists, amongst whom it remains the favored tool for writing papers, preprints, and books. Because of the underlying TeX system, originally developed for documents with mathematics, laying out mathematical expressions is considered to be easier, and the resulting typesetting of higher quality, than any competing document-processing systems. Many scientific journals and other publishers provide free LaTeX packages which implement their "in-house" typesetting styles. The popularity of LaTeX in the technical and academic communities is perhaps partly due to its early availability on Unix systems, and the comparative unavailability of competing word processors on those platforms until recently. But from an early stage LaTeX was available on a wider range of hardware and software than any other program, and versions are now available for almost any system from PDAs to desktop PCs to supercomputers. LaTeX is less popular than mainstream desktop publishing software outside the technical communities for several reasons. It is regarded as hard to learn for people with no previous experience of markup languages. Although it is very easy to customise the appearance of articles, books, and reports, using only a handful of instructions, it remains basically a typesetter for automating document production, not a manual page design program, so performing complex visual layouts incorporating multiple images is difficult. Another barrier to usage for many is the asynchronous interface used in most free versions, where editing is done in a different window from the typeset display. Inverse search can be used to bridge this problem partially. Several commercial implementations, however, use a synchronous typographic display like other DTP systems (as does the non-commercial and open source LyX). Alternatively, GNU TeXmacs is a free WYSIWYG editor which offers features similar to LaTeX, but is based on a different typesetting engine.

Licensing issues

LaTeX is free software. It has a peculiar license called LPPL, not compatible with the GNU General Public License, that allows redistribution and modification, but requires that modified files carry a modified filename. This ensures that files that depend on other files will produce the expected behavior and avoids problems similar to DLL hell. A new version of the LPPL that will be compatible with the GPL is in the works.

Frontends

Because LaTeX markup code can be hard to remember and/or time consuming to learn, there are a few front ends to help:
- Kile: IDE designed mainly for KDE ([http://kile.sourceforge.net/ homepage]).
- LEd: A free environment for rapid TeX/LaTeX document development under MS Windows ([http://www.latexeditor.org homepage]).
- LyX: WYSIWYM (What you see is what you mean) IDE ([http://www.lyx.org/ homepage]).
- Scientific Letter: Commerce mail software with export to TeX/LaTeX ([http://www.sciletter.com/ homepage]).
- Texmaker: Free cross-platform LaTeX editor. Runs on Windows, Mac OS X and Unix (GNU/Linux binary). Is released under the GPL license ([http://www.xm1math.net/texmaker/index.html homepage]).
- TeXnicCenter: IDE designed for MS Windows users under GPL ([http://www.toolscenter.org/ homepage]).
- TeXShop: A free front end for Mac OS X, with editor and output window ([http://www.uoregon.edu/~koch/texshop/texshop.html homepage]).
- WebTex: A free MiKTeX/CGI driven web front end ([http://dev.baywifi.com/latex/ homepage]).
- WinEdt: Shareware IDE for Windows 9x/NT4.0/2000/XP ([http://www.winedt.com/ homepage]).
- WinShell: Freeware IDE for Windows 9x/NT4.0/2000/XP ([http://www.winshell.de/ homepage]).

External links

Community

- [http://www.latex-project.org/ Official LaTeX project site] web site for open development of LaTeX (you can also obtain a CVS snapshot of LaTeX3, the next version of LaTeX which is not yet released)
- [http://www.tug.org/ The TeX Users Group]
- [news:comp.text.tex comp.text.tex]. A Usenet newsgroup for (La)TeX related questions, comp.text.tex is an invaluable resource for (La)TeX. Search the archives with [http://groups.google.com/group/comp.text.tex Google Groups] before posting.
- [irc://irc.freenode.net/#latex #latex] IRC chat room on Freenode

Periodicals

- The PracTeX Journal. Online journal of the TeX Users Group.
- TUGBoat. Print journal of the TeX Users Group.

Tutorials/FAQs

- [http://www.lecb.ncifcrf.gov/~toms/latexforbeginners.html LaTeX for Beginners]
- [http://people.ee.ethz.ch/~oetiker/lshort/lshort.pdf The Not So Short Introduction to LaTeX2e], or LaTeX2e in 133 minutes (2.21 MiB PDF file).
- [http://www.tex.ac.uk/cgi-bin/texfaq2html?introduction=yes The UK TeX FAQ] List of questions and answers that are frequently posted at comp.text.tex.
- [http://www.ctan.org/tex-archive/info/beginlatex/ Formatting Information] Online book for beginners available in [http://www.ctan.org/tex-archive/info/beginlatex/html/index.html HTML] and [http://www.ctan.org/tex-archive/info/beginlatex/beginlatex-3.6.pdf PDF]
- [http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/ LaTeX Primer] A basic guide to LaTeX.
- [http://www.tug.org.in/tutorials.html Tutorials in LaTeX] Free manual distributed by the India TeX Users Group (TUG).
- [ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf The AMS Short Math Guide for LaTeX] A concise summary of math formula typesetting features (PDF file).
- [http://www.rna.nl/tex.html TeX on Mac OS X] Guide to using TeX and LaTeX on a Mac.
- [http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/ Text Processing using LaTeX]
- [http://tex.loria.fr/index.html The (La)TeX encyclopaedia]
- [http://www.giss.nasa.gov/latex/ Hypertext Help with LaTeX]
- [http://www-h.eng.cam.ac.uk/help/tpl/textprocessing/ltxprimer-1.0.pdf LaTeX Tutorials: a Primer] (PDF file)
- [http://www.andy-roberts.net/misc/latex/index.html Getting to Grips with LaTeX] Latex tutorials taking you from the very basics towards more advanced topics.
- [http://www.math.auc.dk/~dethlef/Tips/preparation.html LaTeX, Emacs etc. for your PC] A useful and step-by-step guide to getting Miktex and Emacs working together on a Windows PC.

Add-on Packages

- [http://latex-beamer.sourceforge.net/ LaTeX-beamer] Create sophisticated, structured presentations and slides using LaTeX.
- [http://www.ctan.org/tex-archive/macros/latex/contrib/powerdot/ powerdot] Another very good class for presentations.
- [http://www.phil.cam.ac.uk/teaching_staff/Smith/LaTeX/nd.html bussproofs.sty (and others)] Setting natural deduction tree proofs.
- [http://www.mcnabbs.org/andrew/linux/latexres/ Making a Resume in LaTeX] A LaTeX template with instructions for making an easily-maintained resume.
- [http://latex2rtf.sourceforge.net/ LaTeX2RTF] Translator program which is intended to convert a LaTeX document into the RTF format.

Reference

- [http://www.ctan.org The Comprehensive TeX Archive Network] Latest (La)TeX-related packages and software
- [http://www.tug.org/tds/ TeX Directory Structure], used by many (La)TeX distributions
- [http://www.math.missouri.edu/~stephen/naturalmath/ Natural Math] converts natural language math formulas to LaTeX representation
- [http://www.ctan.org/tex-archive/info/l2tabu/english/l2tabuen.pdf Obsolete packages and commands]
- [http://www.miktex.org/ MiKTeX] A popular and up-to-date TeX (including LaTeX) implementation for Windows.
- . The Companion is an excellent resource for intermediate to advanced LaTeX users. For those already somewhat familiar with LaTeX, this is probably the single most useful available book on the subject. The book website has the complete Table of Contents and a sample chapter available for download. Category:Domain-specific programming languages Category:Free software Category:Page description languages Category:TeX Category:Typesetting programming languages Category:Typesetting ja:LaTeX ko:LaTeX

Category:Mathematical notation

Notation Category:Notation

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ja:青野武 Aono, Takeshi Takeshi Aono (), est né le 19 juin 1936 à Hokkaido. C'est un seiyuu (doubleur japonais) vétéran.

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Blackboard bold

See also

External links

Typeface

Introduction

Typeface anatomy

Serifs

Proportionality

Measurements

Font families

Blackletter fonts

Serif fonts

Sans-serif fonts

Script fonts

Novelty fonts

PI fonts

Symbol fonts

Texts used to demonstrate typefaces

Legal aspects of typefaces

See also

Organizations

External links

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

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Mathematics and other fields

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Number

Examples

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Extensions

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Bold

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Jean-Pierre Serre

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The name and its pronunciation(s)

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