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The * Encyclopedia of Mathematics* (also

The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, and the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains animations and three-dimensional objects.

The encyclopedia has been translated from the Soviet *Matematicheskaya entsiklopediya* (1977) originally edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles.

Until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge. This URL now redirects to the new wiki incarnation of the EOM.

A new dynamic version of the encyclopedia is now available as a public wiki online. This new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics. All entries will be monitored for content accuracy by members of an editorial board^{ [1] } selected by the European Mathematical Society.

- Vinogradov, I. M. (Ed.),
*Matematicheskaya entsiklopediya*, Moscow, Sov. Entsiklopediya, 1977. - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*(set), Kluwer, 1994 ( ISBN 1-55608-010-7).- Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 1 (A–B), Kluwer, 1987 ( ISBN 1-55608-000-X). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 2 (C), Kluwer, 1988 ( ISBN 1-55608-001-8). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 3 (D–Fey), Kluwer, 1989 ( ISBN 1-55608-002-6). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 4 (Fib–H), Kluwer, 1989 ( ISBN 1-55608-003-4). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 5 (I–Lit), Kluwer, 1990 ( ISBN 1-55608-004-2). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 6 (Lob–Opt), Kluwer, 1990 ( ISBN 1-55608-005-0). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 7 (Orb–Ray), Kluwer, 1991 ( ISBN 1-55608-006-9). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 8 (Rea–Sti), Kluwer, 1992 ( ISBN 1-55608-007-7). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 9 (Sto–Zyg), Kluwer, 1993 ( ISBN 1-55608-008-5). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Vol. 10 (Index), Kluwer, 1994 ( ISBN 1-55608-009-3). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Supplement I, Kluwer, 1997 ( ISBN 0-7923-4709-9). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Supplement II, Kluwer, 2000 ( ISBN 0-7923-6114-8). - Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics*, Supplement III, Kluwer, 2002 ( ISBN 1-4020-0198-3).

- Hazewinkel, M. (Ed.),
- Hazewinkel, M. (Ed.),
*Encyclopaedia of Mathematics on CD-ROM*, Kluwer, 1998 ( ISBN 0-7923-4805-2). *Encyclopedia of Mathematics*, public wiki monitored by an editorial board under the management of the European Mathematical Society.^{ [1] }

In mathematics, the **ascending chain condition** (**ACC**) and **descending chain condition** (**DCC**) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings. These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler.

An **encyclopedia** or **encyclopædia** is a reference work or compendium providing summaries of knowledge either general or special to a particular field or discipline. Encyclopedias are divided into articles or entries that are arranged alphabetically by article name or by thematic categories, or else are hyperlinked and searchable. Encyclopedia entries are longer and more detailed than those in most dictionaries. Generally speaking, encyclopedia articles focus on *factual information* concerning the subject named in the article's title; this is unlike dictionary entries, which focus on linguistic information about words, such as their etymology, meaning, pronunciation, use, and grammatical forms.

The ** Encyclopædia Britannica** is a general knowledge English-language encyclopaedia. It has been published by Encyclopædia Britannica, Inc. since 1768, although the company has changed ownership seven times. The encyclopaedia is maintained by about 100 full-time editors and more than 4,000 contributors. The 2010 version of the 15th edition, which spans 32 volumes and 32,640 pages, was the last printed edition. Since 2016, it has been published exclusively as an online encyclopaedia.

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**Ivan Matveevich Vinogradov** was a Soviet mathematician, who was one of the creators of modern analytic number theory, and also a dominant figure in mathematics in the USSR. He was born in the Velikiye Luki district, Pskov Oblast. He graduated from the University of St. Petersburg, where in 1920 he became a Professor. From 1934 he was a Director of the Steklov Institute of Mathematics, a position he held for the rest of his life, except for the five-year period (1941–1946) when the institute was directed by Academician Sergei Sobolev. In 1941 he was awarded the Stalin Prize. He was elected to the American Philosophical Society in 1942. In 1951 he became a foreign member of the Polish Academy of Sciences and Letters in Kraków.

**Ajima Naonobu**, also known as **Ajima Manzō Chokuyen**, was a Japanese mathematician of the Edo period.

In mathematics, the **Cayley transform**, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators.

**Paraconsistent mathematics**, sometimes called **inconsistent mathematics**, represents an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic. A number of reformulations of analysis can be developed, for example functions which both do and do not have a given value simultaneously.

The * Encyclopaedia of Islam* (

In mathematics, a **character sum** is a sum of values of a Dirichlet character χ *modulo**N*, taken over a given range of values of *n*. Such sums are basic in a number of questions, for example in the distribution of quadratic residues, and in particular in the classical question of finding an upper bound for the least quadratic non-residue *modulo**N*. Character sums are often closely linked to exponential sums by the Gauss sums.

In mathematical logic, **abstract algebraic logic** is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.

* Den Store Danske Encyklopædi* is the most comprehensive contemporary Danish language encyclopedia. The 20 volumes of the encyclopedia were published successively between 1994 and 2001; a one-volume supplement was published in 2002 and two index volumes in 2003. The work comprises 115,000 articles, ranging in size from single-line cross references to the 130-page entry on Denmark. The articles were written by a staff of about 4,000 academic experts led by editor-in-chief Jørn Lund. Articles longer than a few dozen lines are signed by their authors. Many articles are illustrated.

The * Great Russian Encyclopedia* is a universal Russian encyclopedia, completed in 36 volumes, published between 2004 and 2017 by Great Russian Encyclopedia, JSC. A successor to the

In mathematics, a **matrix coefficient** is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group *G* obtained by composing a representation of *G* on a vector space *V* with a linear map from the endomorphisms of *V* into *V*'s underlying field. It is also called a **representative function**. They arise naturally from finite-dimensional representations of *G* as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on *G* are dense in the Hilbert space of square-integrable functions on *G*.

In the mathematical theory of partial differential equations, a **Monge equation**, named after Gaspard Monge, is a first-order partial differential equation for an unknown function *u* in the independent variables *x*_{1},...,*x*_{n}

**Themistocles M. Rassias** is a Greek mathematician, and a professor at the National Technical University of Athens, Greece. He has published more than 300 papers, 10 research books and 45 edited volumes in research Mathematics as well as 4 textbooks in Mathematics for university students. His research work has received more than 19,000 citations according to Google Scholar and more than 5,800 citations according to MathSciNet. His h-index is 49. He serves as a member of the Editorial Board of several international mathematical journals.

**Al-Hassar** or **Abu Bakr Muhammad ibn Abdallah ibn Ayyash al-Hassar** was a 12th-century Moroccan mathematician. He is the author of two books *Kitab al-bayan wat-tadhkar*, a manual of calculation and *Kitab al-kamil fi sinaat al-adad*, on the breakdown of numbers. The first book is lost and only a part of the second book remains.

**Helaine Selin** is an American librarian, historian of science, author and the editor of several bestselling books.

**Michiel Hazewinkel** is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book *Formal groups and applications* and as editor of the *Encyclopedia of Mathematics*.

In the mathematical field of topology, there are various notions of a ** P-space** and of a

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