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

  • ax = b
  • ya = 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.)

    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.
  • Turn (biochemistry)

    A turn is an element of secondary structure in proteins where the polypeptide chain reverses its overall direction. For beta turns go to Beta turn.

    Definition

    According to one definition, a turn is a structural motif where the Cα atoms of two residues separated by few (usually 1 to 5) peptide bonds are close (< 7 Å), while the residues do not form a secondary structure element such as an alpha helix or beta sheet with regularly repeating backbone dihedral angles. Although the proximity of the terminal Cα atoms usually correlates with formation of a hydrogen bond between the corresponding residues, a hydrogen bond is not a requirement in this turn definition. That said, in many cases the H-bonding and Cα-distance definitions are equivalent.

    Types of turns

    Turns are classified according to the separation between the two end residues:

  • In an α-turn the end residues are separated by four peptide bonds (i \rightarrow i\pm 4).
  • In a β-turn (the most common form), by three bonds (i \rightarrow i\pm 3).
  • In a γ-turn, by two bonds (i \rightarrow i\pm 2).
  • Gray (horse)

    Gray or grey is a coat color of horses characterized by progressive silvering of the colored hairs of the coat. Most gray horses have black skin and dark eyes; unlike many depigmentation genes, gray does not affect skin or eye color. Their adult hair coat is white, dappled, or white intermingled with hairs of other colors. Gray horses may be born any base color, depending on other color genes present. White hairs begin to appear at or shortly after birth and become progressively lighter as the horse ages. Graying can occur at different rates—very quickly on one horse and very slowly on another.

    Gray horses appear in many breeds, though the color is most commonly seen in breeds descended from Arabian ancestors. Some breeds that have large numbers of gray-colored horses include the Thoroughbred, the Arabian, the American Quarter Horse, the Percheron, the Andalusian, the Welsh pony, and the most famous of all gray horse breeds, the Lipizzaner.

    People who are unfamiliar with horses may refer to gray horses as "white." However, a gray horse whose hair coat is completely "white" will still have black skin (except under markings that were white at birth) and dark eyes. This is how to discern a gray horse from a white horse. White horses usually have pink skin and sometimes even have blue eyes. Young horses with hair coats consisting of a mixture of colored and gray or white hairs are sometimes confused with roan. Some horses that carry dilution genes may also be confused with white or gray.

    Gray (unit)

    The gray (symbol: Gy) is a derived unit of ionizing radiation dose in the International System of Units (SI). It is defined as the absorption of one joule of radiation energy per one kilogram of matter.

    It is used as a measure of absorbed dose, specific energy (imparted), and kerma (an acronym for "kinetic energy released per unit mass"). It is a physical quantity, and does not take into account any biological context. Unlike the pre-1971 non-SI roentgen unit of radiation exposure, the gray when used for absorbed dose is defined independently of any target material. However, when measuring kerma the reference target material must be defined explicitly, usually as dry air at standard temperature and pressure.

    The equivalent cgs unit, the rad (equivalent to 0.01 Gy), remains common in the United States, though "strongly discouraged" in the style guide for U.S. National Institute of Standards and Technology authors.

    Etymology

    The gray was named after British physicist Louis Harold Gray, a pioneer in the field of X-ray and radium radiation measurement, and their effects on living tissue. It was adopted as part of the International System of Units in 1975.

    Grey (New Zealand electorate)

    Grey is a former parliamentary electorate in the West Coast region of New Zealand.

    The electorate of Grey Valley, created for the 1871 general election, was succeeded by the electorate of Greymouth in the 1881 general election, and lasted until 1890. In 1890 the Grey electorate was created, and was abolished in 1919.

    Population centres

    The 1870 electoral redistribution was undertaken by a parliamentary select committee based on population data from the 1867 census. Eight sub-committees were formed, with two members each making decisions for their own province; thus members set their own electorate boundaries. The number of electorates was increased from 61 to 72, and Grey Valley was one of the new electorates.

    Throughout the electorate's history, the town of Greymouth was always included in its area. The town of Brunner belonged to the electorate during most periods.

    History

    Grey Valley existed from 1871 to 1881. William Henry Harrison was the first representative from 1871 to 1875 when he retired. Harrison had previously represented Westland Boroughs from 1868 to 1870.

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