Flag (linear algebra)
In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration):
If we write the dim Vi = di then we have
where n is the dimension of V (assumed to be finite-dimensional). Hence, we must have k ≤ n. A flag is called a complete flag if di = i, otherwise it is called a partial flag.
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
The signature of the flag is the sequence (d1, … dk).
Under certain conditions the resulting sequence resembles a flag with a point connected to a line connected to a surface.
Bases
An ordered basis for V is said to be adapted to a flag if the first di basis vectors form a basis for Vi for each 0 ≤ i ≤ k. Standard arguments from linear algebra can show that any flag has an adapted basis.
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.
The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.
Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.
Flag (disambiguation)
A flag is a colored cloth with a specified meaning.
Flag may also refer to:
Art and entertainment
Biology
Flag is the common name of several genera or species of flowering plants:
Flag (lighting)
A flag is a device used in lighting for motion picture and still photography to block light. It can be used to cast a shadow, provide negative fill, or protect the lens from a flare. Its usage is generally dictated by the director of photography, but the responsibility for placing them can vary by region, usually devolving to either the gaffer and electricians or the key grip and lighting grips.
Flags come in a wide variety of shapes and sizes, from mere square inches ("dots and fingers") to many square feet ("meat axes"). Most "industry-standard" flags consist of a square wire frame stitched with black duvetyne, which minimizes any reflected light and keeps the flag lightweight. Flags are distinguished from larger light-cutting tools such as overhead rigs or butterflies in that they can be mounted on individual C-stands, as opposed to being affixed to collapsible frames.
The above notwithstanding, given smaller budgets or extenuating circumstances, virtually any opaque object can be used to flag light.
Flag (anime)
Flag (フラッグ, Furaggu) is a 13-episode Japanese mecha-genre anime series directed by veteran director Ryosuke Takahashi. It was broadcast as pay per view streaming web video on Bandai Channel starting on June 6, 2006. Episodes 1 and 2 were scheduled to be broadcast on the anime PPV channel SKY Perfect Perfect Choice ch160 Anime from August 18, 2006. Stylistically, the series makes use of a still and video cameraman POV, as well as "web cam" images to create a documentary-like narrative, despite being an animated drama.
Setting
Saeko Shirasu is a 25-year-old war frontline photo-journalist who became a celebrity after taking a picture of civilians raising a makeshift UN flag in war-torn Uddiyana. The image then became an instant symbol for peace. However, just before the peace agreement is achieved, the flag was stolen by an armed extremist group in order to obstruct the truce. The UN peacekeepers decide to covertly send in a SDC (pronounced as Seedac—Special Development Command) unit to retrieve the flag. Because of her connection with the "Flag" photo, Saeko Shirasu was offered the job of following the SDC unit as a frontline journalist. The SDC unit is equipped with an HAVWC (High Agility Versatile Weapon Carrier—pronounced "havoc") mecha armored vehicle.
*-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.
Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is transitive, antisymmetric, and total (this relation is denoted here by infix ≤). A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.
If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:
Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a. A relation having the property of "totality" means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear.Totality also implies reflexivity, i.e., a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (It requires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extension of that partial order.
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