Degree

Degree may refer to:

As a unit of measurement

Degree symbol (°), a notation used in science, engineering and mathematics

Degree (angle), a unit of angle measurement

Degree in geographic coordinate system

Degree (temperature), a unit of temperature measurement

Degree API, a measure of density in the petroleum industry

Degree Baumé, a pair of density scales

Degree Brix, a measure of sugar concentration

Degree Gay-Lussac, a measure of the alcohol content of a liquid by volume, ranging from 0° to 100°

Degree proof, or simply proof, the alcohol content of a liquid, ranging from 0° to 175° in the UK, and from 0° to 200° in the U.S.

Degree of curvature, a unit of curvature measurement, used in civil engineering

Degrees of freedom (mechanics), the number of displacements and/or rotations needed to define the position and orientation of a body

Degrees of freedom (physics and chemistry), a concept describing dependence on a countable set of parameters

Degree of frost, a unit of temperature measurement

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Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field that contains the base field and satisfies additional properties. For instance, the set Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution to the equation x² = 2.

Definitions

Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under taking inverses in L. In other words, K is a field with respect to the field operations inherited from L. The larger field L is then said to be an extension field of K. To simplify notation and terminology, one says that L / K (read as "L over K") is a field extension to signify that L is an extension field of K.

If L is an extension of F which is in turn an extension of K, then F is said to be an intermediate field (or intermediate extension or subextension) of the field extension L / K.

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Degree of a continuous mapping

In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the range manifold under the mapping. The degree is always an integer, but may be positive or negative depending on the orientations.

The degree of a map was first defined by Brouwer, who showed that the degree is homotopy invariant (invariant among homotopies), and used it to prove the Brouwer fixed point theorem. In modern mathematics, the degree of a map plays an important role in topology and geometry. In physics, the degree of a continuous map (for instance a map from space to some order parameter set) is one example of a topological quantum number.

Definitions of the degree

From Sⁿ to Sⁿ

The simplest and most important case is the degree of a continuous map from the $n$ -sphere $S^n$ to itself (in the case $n=1$ , this is called the winding number):

Let $f\colon S^n\to S^n$ be a continuous map. Then $f$ induces a homomorphism $f_*\colon H_n\left(S^n\right)\to H_n\left(S^n\right)$ , where $H_n\left(\cdot\right)$ is the $n$ th homology group. Considering the fact that $H_n\left(S^n\right)\cong\mathbb{Z}$ , we see that $f_*$ must be of the form $f_*\colon x\mapsto\alpha x$ for some fixed $\alpha\in\mathbb{Z}$ . This $\alpha$ is then called the degree of $f$ .

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Guest (surname)

The surname Guest is derived from the Old English word "giest", which in turn comes from the Old Norse word "gestr", both of which mean "guest" or "stranger." Spelling variations may include Gest, Geste, Gueste, Ghest, Geest, Geeste, Gist, Ghost, Jest. Other European counterparts to the name include the German and Dutch "Gast", Luxembourgish "Gaascht", Swedish "Gäst", Norwegian "Gjest", Serbian and Slovakian "Gost", Czech "Host", etc.

Among the various theories on last name origins, according to the book "The Norman People and Their Existing Descendants in the British Dominions and the United States of America by H.S. King & Company, 1874 ", "Guest" derives from a place and not from the occupational status of some ancient forebear given to chronic visiting. Guest, the place, was near Caen, Normandy, and the original bearers of the name are said to have taken part in the Norman Conquest of England under William I in 1066. After the conquest, the family settled in Salop (now Shropshire) in middle-western England and apparently held the estate known as Lega from the De Dunstanvilles. Some ancient land records show Alan De Guest granting the lands of Alric de Lega (Guest) to a monastery called Wembridge Priory in 1150. His son Thomas (a name which occurs frequently in the Guest line) is mentioned in 1180. Some of the other Guests of antiquity were Thomas' sons Walter and Leonard, referred to in 1194 and 1280; and Henry, son of Leonard, 1240. Roger de Lega, or Guest, brother of Henry, had a son Thomas who again gave lands to Wembridge Priory. In 1295 Adam Gest (another variant of the name) was assessor of the parliamentary rolls in Salop.

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Guest (album)

Guest is the first studio album by Critters Buggin of Seattle, Washington and was released in 1994 on Stone Gossard's then new label Loosegroove.Guest was reissued by Kufala Recordings in 2004.