Quasigroup

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative.

A quasigroup with an identity element is called a loop.

Definitions

There are at least two equivalent formal definitions of quasigroup. One defines a quasigroup as a set with one binary operation, and the other, from universal algebra, defines a quasigroup as having three primitive operations. We begin with the first definition.

A quasigroup (Q, ∗) is a set, Q, with a binary operation, ∗, (that is, a magma), obeying the Latin square property. This states that, for each a and b in Q, there exist unique elements x and y in Q such that both

a ∗ x = b

y ∗ a = b

hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, or Cayley table. This property ensures that the Cayley table of a finite quasigroup is a Latin square.)

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Loop (knot)

In reference to knots, loop may refer to:

One of the fundamental structures used to tie knots. Specifically, it is a U-form narrower than a bight.

A type of knot used to create a closed circle in a line.

References

↑ Clifford W. Ashley, The Ashley Book of Knots. Image 31, 32.

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Loopback

Loopback, or loop-back, refers to the routing of electronic signals, digital data streams, or flows of items back to their source without intentional processing or modification. This is primarily a means of testing the transmission or transportation infrastructure.

Many example applications exist. It may be a communication channel with only one communication endpoint. Any message transmitted by such a channel is immediately and only received by that same channel. In telecommunications, loopback devices perform transmission tests of access lines from the serving switching center, which usually does not require the assistance of personnel at the served terminal. Loop around is a method of testing between stations that are not necessarily adjacent, where in two lines are used, with the test being done at one station and the two lines are interconnected at the distant station. A patch cable may also function as loopback, when applied manually or automatically, remotely or locally, facilitating a loop-back test.

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Dreams (2004 film)

Dreams is a 2004 Tamil Malayalam romantic film directed by Kasthuri Raja and produced by Saraswathi Srikanth. The film featured Raja's son Dhanush in the lead role with Diya and Parul Yadav playing other pivotal roles. The film opened to negative reviews and became a failure at the box office.

Cast

Dhanush as Shakti

Diya as Shruti

Parul Yadav as Charu

Rajesh

Thalaivasal Vijay as Mahendran, Shruti's father

Pyramid Natarajan as Shakti's father

Bala Singh

Kuyili as Charu's mother

Poovilangu Mohan

Sujibala

Karthik Sabesh

Aryan

Nithin Sathya

Production

The project was launched shortly after the success of Thulluvadho Ilamai in 2002, but as Dhanush's Kaadhal Kondein became a large success, Dreams was stalled temporarily as Dhanush's dates became blocked. The film ran into a legal tussle with the makers of his other film, Sullan, with the producers adamant that Dreams was released first although to no avail. The film's delay meant that Dhanush shot ten straight days for the project to complete it, while the delay also had resulted in failings in continuity. By the time of the release, the producer Srikanth and director Kastoori Raja were still engaged in a legal tussle.

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Dreams (Neil Diamond album)

Dreams is the thirty-first studio album by Neil Diamond. It was produced by Diamond and released by Columbia Records in 2010. The album contains cover versions of popular songs that Diamond claims in the liner notes are among his favorites. Among them is "I'm a Believer", which he wrote for The Monkees back in 1966.Dreams ranked at number eight on the Billboard 200 chart.