Integral element

In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and such that

That is to say, b is a root of a monic polynomial over A. If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

The special case of an integral element of greatest interest in number theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers (e.g., ). The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory.

The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A.

*-algebra

In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.

Terminology

*-ring

In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.

More precisely, * is required to satisfy the following properties:

for all x, y in A.

This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.

Ring (Suzuki novel)

Ring (リング, Ringu) is a Japanese mystery horror novel by Koji Suzuki, first published in 1991, and set in modern-day Japan. It was the basis for a 1995 television film (Ring: Kanzenban),a television series (Ring: The Final Chapter), a film of the same name (1998's Ring), and two remakes of the 1998 film: a South Korean version (The Ring Virus) and an American version (The Ring).

Plot synopsis

After four teenagers mysteriously die simultaneously in Tokyo, Kazuyuki Asakawa, a reporter and uncle to one of the deceased, decides to launch his own personal investigation. His search leads him to "Hakone Pacific Land", a holiday resort where the youths were last seen together exactly one week before their deaths. Once there he happens upon a mysterious unmarked videotape. Watching the tape, he witnesses a strange sequence of both abstract and realistic footage, including an image of an injured man, that ends with a warning revealing the viewer has a week to live. Giving a single means of avoiding death, the tape's explanation ends suddenly having been overwritten by an advertisement. The tape has a horrible mental effect on Asakawa, and he doesn't doubt for a second that its warning is true.

Ringtone

A ringtone or ring tone is the sound made by a telephone to indicate an incoming call or text message. Not literally a tone nor an actual (bell-like) ring any more, the term is most often used today to refer to customizable sounds used on mobile phones.

Background

A phone “rings” when its network indicates an incoming call and the phone thus alerts the recipient. For landline telephones, the call signal can be an electric current generated by the switch or exchange to which the telephone is connected, which originally drove an electric bell. For mobile phones, the network sends the phone a message indicating an incoming call. The sound the caller hears is called the ringback tone, which is not necessarily directly related.

The electromagnetic bell system is still in widespread use. The ringing signal sent to a customer's telephone is 90 volts AC at a frequency of 20 hertz in North America. In Europe it is around 60-90 volts AC at a frequency of 25 hertz. Some non-Bell Company system party lines in the US used multiple frequencies (20/30/40 Hz, 22/33/44 Hz, etc.) to allow "selective" ringing.

Conductor (class field theory)

In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.

Local conductor

Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted , is the smallest non-negative integer n such that the higher unit group

is contained in N_L/K(L^×), where N_L/K is field norm map and is the maximal ideal of K. Equivalently, n is the smallest integer such that the local Artin map is trivial on . Sometimes, the conductor is defined as where n is as above.

The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero, and it is tamely ramified if, and only if, the conductor is 1. More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group G_s is non-trivial, then , where η_L/K is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.

National Diet Library

The National Diet Library (NDL) (国立国会図書館, Kokuritsu Kokkai Toshokan) is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan (国会, Kokkai) in researching matters of public policy. The library is similar in purpose and scope to the United States Library of Congress.

The National Diet Library (NDL) consists of two main facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan.

History

The National Diet Library is the successor of three separate libraries: the library of the House of Peers, the library of the House of Representatives, both of which were established at the creation of Japan's Imperial Diet in 1890; and the Imperial Library, which had been established in 1872 under the jurisdiction of the Ministry of Education.

The Diet's power in prewar Japan was limited, and its need for information was "correspondingly small." The original Diet libraries "never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity." Until Japan's defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U.S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II.

Conductor (album)

Conductor is an album by North Carolina indie rock band The Comas. The album was recorded by Alan Weatherhead at Sound of Music in Richmond, Va. It features Andy Herod on guitars and vocals, Nicole Gehweiler also on guitars and vocals, Justin Williams on bass and baritone guitars and Cameron Weeks on drums.

The album is accompanied by a DVD, Conductor: The Movie, which was written and directed by filmmaker/animator Brent Bonacorso. Combining live footage and animation the film stars lead singer Andy Herod and his ex-girlfriend Michelle Williams of Dawson's Creek fame. It features videos for each song seamlessly blended together.

The DVD also features a new song in the middle called "Bad Connexion", and as easter egg an alternate version of the movie. It's about 10 minutes shorter and some of the songs are in a different order. To access it, do the following:

While on the title page of the DVD there are two choices. The first choice, PLAY, is already highlighted. If you go down one you highlight the menu that lets you view each video. Go down one more and nothing should be highlighted. This is where you will find the hidden alternate version of Conductor: hit "play".