Degree

Degree may refer to:

As a unit of measurement

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.

Degree of an algebraic variety

The degree of an algebraic variety in mathematics is defined, for a projective variety V, by an elementary use of intersection theory.

Definition

For V embedded in a projective space Pⁿ and defined over some algebraically closed field K, the degree d of V is the number of points of intersection of V, defined over K, with a linear subspace L in general position, when

Here dim(V) is the dimension of V, and the codimension of L will be equal to that dimension. The degree d is an extrinsic quantity, and not intrinsic as a property of V. For example the projective line has an (essentially unique) embedding of degree n in Pⁿ.

Properties

The degree of a hypersurface F = 0 is the same as the total degree of the homogeneous polynomial F defining it (granted, in case F has repeated factors, that intersection theory is used to count intersections with multiplicity, as in Bézout's theorem).

Other approaches

For a more sophisticated approach, the linear system of divisors defining the embedding of V can be related to the line bundle or invertible sheaf defining the embedding by its space of sections. The tautological line bundle on Pⁿ pulls back to V. The degree determines the first Chern class. The degree can also be computed in the cohomology ring of Pⁿ, or Chow ring, with the class of a hyperplane intersecting the class of V an appropriate number of times.

Playa

Playa (plural playas) may refer to:

Landforms

Named playas

Populated places

Playa (band)

Playa is an American R&B-hip-hop group. The original lineup consisted of Jawaan "Smoke" Peacock, Benjamin "Black" Bush and Stephen "Static Major" Garrett. Playa is best known for their 1998 hit album, Cheers 2 U, produced by longtime collaborator Timbaland. Static was notable for being a successful songwriter of hit singles and album tracks for artists such as Ginuwine, Aaliyah, Truth Hurts, Lil Wayne and Brandy.

History

Early years

Jawaan Peacock, Benjamin Bush and Stephen Garrett met in the late 1980s and formed while the guys were still in high school. Smokey and Digital Black met through a mutual friend who thought the guys would sound good together, while Smokey and Static met in church while performing in a gospel group. Alongside other guys, the three men that would eventually form Playa created a group called A Touch of Class.

Joining Swing Mob

As A Touch of Class, the group met DeVante from Jodeci backstage at a concert. They began singing Boyz II Men songs when DeVante asked them if they knew any Jodeci songs. Essentially they impressed DeVante by performing a cappella versions of Jodeci songs backstage after a concert in 1991/1992. The guys were then signed onto DeVante's Swing Mob label with Elektra Records in 1993/94. Magazines such as Vibe and The Source attribute the signing to Static's close friendship with the boss and producers of the DeVante recording label. The group finally became known as Playa after DeVante heard the guys sing and called them "lil playas".

Playa (Ponce)

Barrio Playa, also known as Playa de Ponce, Ponce Playa, or La Playa, is one of the thirty-one barrios that comprise the municipality of Ponce, Puerto Rico. Along with Bucaná, Canas, Vayas, and Capitanejo, Playa is one of the municipality's five coastal barrios. Barrio Playa also incorporates several islands, the largest of which is Caja de Muertos. It was founded in 1831.

Location

Playa is an urban barrio located in the southern region of the municipality, within the limits of the city of Ponce, south of the traditional center of the city at Plaza Las Delicias, and on the shores of the Caribbean Sea. It is located at 17.9839°N 66.6128°W, and it has an elevation of 10 feet. The toponomy, or origin of the name, describes the geographic area the barrio occupies in southern Ponce and facing the Caribbean Sea.

Boundaries

It is bounded on the North by PR-2, on the South by the Caribbean Sea, on the West by Río Matilde, and on the East by Río Bucaná.

In terms of barrio-to-barrio boundaries, Playa is bounded in the North by Canas Urbano and San Anton, in the South by the Caribbean Sea, in the West by Canas, and in the East by Bucaná.