Near-ring

In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.

Definition

A set N together with two binary operations + (called addition) and ⋅ (called multiplication) is called a (right) near-ring if:

Similarly, it is possible to define a left near-ring by replacing the right distributive law A3 by the corresponding left distributive law. However, near-rings are almost always written as right near-rings.

An immediate consequence of this one-sided distributive law is that it is true that 0⋅x = 0 but it is not necessarily true that x⋅0 = 0 for any x in N. Another immediate consequence is that (−x)⋅y = −(x⋅y) for any x, y in N, but it is not necessary that x⋅(−y) = −(x⋅y). A near-ring is a ring (not necessarily with unity) if and only if addition is commutative and multiplication is distributive over addition on the left.

Mappings from a group to itself

Let G be a group, written additively but not necessarily abelian, and let M(G) be the set {f | f : G → G} of all functions from G to G. An addition operation can be defined on M(G): given f, g in M(G), then the mapping f + g from G to G is given by (f + g)(x) = f(x) + g(x) for all x in G. Then (M(G), +) is also a group, which is abelian if and only if G is abelian. Taking the composition of mappings as the product ⋅, M(G) becomes a near-ring.

Near (Death Note)

Nate River (Japanese: ネイト・リバー, Hepburn: Neito Ribā), universally referred to by the mononym Near (ニア, Nia), is a fictional character in the manga series Death Note, created by Tsugumi Ohba and Takeshi Obata. Near is the younger of the two successors of L who was investigating the methods of identifying Kira, the in-universe name given to the mysterious serial killer who murdered criminals by means of the eponymous Death Note. He becomes the leader of the SPK (Special Provision for Kira), an organization that looks into the Kira case, and in the end, succeeds in uncovering Kira's identity. He is voiced by Noriko Hidaka in the Japanese anime, and by Cathy Weseluck in the English version. In the Japanese TV Drama, he is portrayed by Mio Yūki.

Creation and conception

Tsugumi Ohba, writer of Death Note said that he introduced Near and Mello together as L individually could not defeat Kira. Ohba felt that introducing one character individually would produce a "repeat" of the struggle between Light and L, so he instead wanted a story involving three combatants fighting each other. Ohba said that he let Obata create the character designs and asked him to make both characters look "a little "L-ish"". Ohba said that he "wavered" in their ages and considered making the characters the sons of L. Ohba added that he did not initially develop their personalities as he wanted to "reveal" them through their actions.

NEAR Shoemaker

The Near Earth Asteroid Rendezvous – Shoemaker (NEAR Shoemaker), renamed after its 1996 launch in honor of planetary scientist Eugene Shoemaker, was a robotic space probe designed by the Johns Hopkins University Applied Physics Laboratory for NASA to study the near-Earth asteroid Eros from close orbit over a period of a year. The mission succeeded in closing in with the asteroid and orbited it several times, finally terminating by touching down on the asteroid on 12 February 2001.

The primary scientific objective of NEAR was to return data on the bulk properties, composition, mineralogy, morphology, internal mass distribution and magnetic field of Eros. Secondary objectives include studies of regolith properties, interactions with the solar wind, possible current activity as indicated by dust or gas, and the asteroid spin state. This data will be used to help understand the characteristics of asteroids in general, their relationship to meteoroids and comets, and the conditions in the early Solar System. To accomplish these goals, the spacecraft was equipped with an X-ray/gamma ray spectrometer, a near-infrared imaging spectrograph, a multi-spectral camera fitted with a CCD imaging detector, a laser rangefinder, and a magnetometer. A radio science experiment was also performed using the NEAR tracking system to estimate the gravity field of the asteroid. The total mass of the instruments was 56 kg, and they required 80 W power.

*-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.

Ring (film)

Ring (リング, Ringu) is a 1998 Japanese psychological horror film directed by Hideo Nakata, adapted from the novel Ring by Kôji Suzuki, which in turn draws on the Japanese folk tale Banchō Sarayashiki. The film stars Nanako Matsushima, Hiroyuki Sanada, and Rikiya Ōtaka. The film follows TV reporter and single mother Reiko who is caught up in a series of deaths surrounding a cursed video tape.

Production took approximately 9 months.Ring and its sequel Rasen were released in Japan at the same time.

After release, Ring inspired numerous follow-ups within the Ring franchise and triggered a trend of Western remakes.

Plot

Two teenagers, Masami (Hitomi Satō) and Tomoko (Yūko Takeuchi), talk about a videotape recorded by a boy in Izu which is fabled to bear a curse that kills the viewer seven days after watching. Tomoko reveals that a week ago, she and three of her friends watched a weird tape and received a call after watching. Tomoko is killed by an unseen force as Masami watches, horrified.