Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

Formally, the singular value decomposition of an m × n real or complex matrix M is a factorization of the form

Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.

Formally, the discrete Hartley transform is a linear, invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup > ( where R denotes the set of real numbers ).

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature < tt >(−,+,+,+)</ tt > ( Some may also prefer the alternative signature < tt >(+,−,−,−)</ tt >; in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.

; Formally real field

Formally, an analytic function ƒ ( z ) of the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent, i. e., if ƒ is transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A < sub > 0 </ sub >:= A.

Formally, complexification is a functor Vect < sub > R </ sup > → Vect < sub > C </ sup >, from the category of real vector spaces to the category of complex vector spaces.

Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,

Category: Formally real field

Formally, it is defined as the analytic signal corresponding to the real field.

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

In late 1900s the day that of the City was founded is Jakin School is a Private School is Formally Part of Jakin School Board was Grades One through Twelve there is no Football field on this school. In 1966 When Jakin School was Closed were Sent to Blakely-Union Elementary, Junior High, and High School ( which later Became Early County High, Middle, and Elementary School ). while Carver school was built for African-American students grades One through Twelve. although late 1960s when Carver school was closed along with Kestler School was based in Damascus were sent to Washington High & Elementary at Blakely. In addition to small farm agriculture, Jakin's early economic growth resulted from turpentine.

Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that

Formally, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H → H ( called the antipode ) such that the following diagram commutes:

Formally, given a vector field v, a vector potential is a vector field A such that

Formally dedicated " as a memorial to the boys that were " on October 30, 1915, Alumni Field and its distinctive " maroon goal-posts on a field of green " were hailed in that evening's edition of the Boston Saturday Evening Transcript as " one of the sights in Boston.

Formally, the 4D light field is defined as radiance along rays in empty space.

Formally, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K ( x < sub > 1 </ sub >,..., x < sub > n </ sub >).

Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.

Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers.

Formally, this is described in algebraic notation like this: ( 19 + 1 ) + ( 15 − 1 ) = x, but even a young student might use this technique without calling it algebra.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, one has an implication, not an equivalence, so the solution set may get larger.

Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.

Formally described as " Truck, Cargo: 10-Ton, 8x8 ", it has been nicknamed the " Dragon Wagon ".

Formally, we start with a category C with finite products ( i. e. C has a terminal object 1 and any two objects of C have a product ).

Formally it is a part of Novosibirsk city, and has never been a closed city.

Formally, DFS is an uninformed search that progresses by expanding the first child node of the search tree that appears and thus going deeper and deeper until a goal node is found, or until it hits a node that has no children.

Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

It has been leased from the National Trust since 1974 by the Somerset Wildlife Trust ( Formally Somerset Trust for Nature Conservation ) and is run as a nature reserve and visitor centre.

Formally known as trilina, the armónico has seven strings, two of which form a doubled string.

Formally known as Bootle-cum-Linacre, the town is < span style =" white-space: nowrap "> 4 miles ( 6. 4 km )</ span > to the north of Liverpool city centre, and has a total resident population of 77, 640.

Formally, this means that, for some function f, the image f ( D ) of a directed set D ( i. e. the set of the images of each element of D ) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that f preserves directed suprema.

Following a merger with the Furnishing Industry Trust ( Formally The Furnishing Trades Benevolent Association ) in 2011, the Company has become the most significant charity in the sector, supporting the needy and promoting welfare and education of those entering and working within the industry, as well as the wider support of the charitable work of the Mayoralty and the Corporation of the City of London.

Formally established in 1475, though in fact much older, it has 75, 000 codices from throughout history.

Formally, the situation has the characteristics of a prisoner's dilemma game that is played repeatedly between the same partners, and in this case " defection "-jumping the queue-would not be advantageous.

Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.

Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

Formally, an antihomomorphism between X and Y is a homomorphism, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as, we have.

Formally, given two partially ordered sets ( S, ≤) and ( T, ≤), a function f: S → T is an order-embedding if f is both order-preserving and order-reflecting, i. e. for all x and y in S, one has

Formally described in Stephen Gilligan's article: The relational self: the expanding of love beyond desire ( 1996 ), and expanded on in his book: The Courage to Love ( 1997 ), it has become a major new approach in psychotherapy and healing.

Formally launched in 2004, the network currently has 16 members in nine states.

Enbridge currently has offices in Enbridge Place ( Formally ING Tower ), Manulife Place, and Royal Bank Building.

Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

Formally named " Olallie ", it has usually been marketed as olallieberry, just as " Marion " is sold as marionberry.