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 styled as " Excelentísimo e Ilustrísimo Señor Profesor Doctor Don N, Rector Magnífico de la Universidad de X " ( Most Excellent and Illustrious Lord Professor Doctor Don N, Rector Magnificus of the University of X ), it is an office of high dignity within Spanish society, usually being highly respected.

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 a congruence subgroup is one that contains Γ ( N ) for some N, and the least such N is the level or Stufe of the subgroup.

Formally, the normal bundle to N in M is a quotient bundle of the tangent bundle on M: one has the short exact sequence of vector bundles on N:

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.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, their designation is the letter Ž and the number.

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

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally, the derivative of the function f at a is the limit

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, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a bifunctor is a functor whose domain is a product category.

Formally, a set S is called finite if there exists a bijection

Formally, the system is said to have memory.

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

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

* Formally, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Formally, Φ = kx − ωt is the phase.

Dick Smith Electronics has also long been known for its " home brand " range of electronics which fall under the brand name " Dick Smith " ( Formally called DSE, a commonly used abbreviation of Dick Smith Electronics ).

Formally, as per the 2002 Memorandum of Understanding between the BSI and the United Kingdom Government, British Standards are defined as:

Formally neutral and reluctant to become involved with the great powers except as a last resort, Kuwait turned to the United States, the Soviet Union and Britain for naval protection of its tanker fleet after twenty-one ships were attacked in the gulf since late 1986.

Formally named " United Press Associations " for incorporation and legal purposes, but publicly known and identified as United Press or UP, it was created by the 1907 uniting of three smaller news syndicates by the Midwest newspaper publisher E. W. Scripps.

Formally titled An Act to Further Promote the Defense of the United States, the Act effectively ended the United States ' pretense of neutrality.

Formally, it is a coalition of U. S. and European CEOs, largely from transnational corporations, launched at the urging of former United States Secretary of Commerce Ron Brown in Seville ( Spain ) in late 1995 at a conference attended by CEOs from more than 100 U. S. and EU companies and by top government representatives led by the Commissioners for Trade and Industry and by the U. S. Secretary of Commerce.

* General Headquarters United Nations Command ( Korea ) ( UNC ) — Formally activated 10 July 1950, before then Allied forces were formally under American operational control.

Formally nominated on December 2, 1902, Holmes was unanimously confirmed by the United States Senate on December 4, receiving his commission the same day.

The Force Act of 1870 ( Formally, " An Act to enforce the Right of Citizens of the United States to vote in the several States of this Union, and for other Purposes ," 41st Congress, Sess.

Formally titled ' An Act to protect all Persons in the United States in their Civil Rights, and furnish the Means of their vindication, the Act declared that people born in the United States and not subject to any foreign power are entitled to be citizens, without regard to race, color, or previous condition of slavery or involuntary servitude.

Formally nominated on December 11, 1851, Curtis was confirmed by the United States Senate on December 20, 1851, and received his commission the same day.

Formally nominated on January 3, 1820, Bland was confirmed by the United States Senate on January 5, 1820, and received his commission the same day.

Formally nominated on July 13, 1867, Blatchford was confirmed by the United States Senate three days later, receiving his commission the same day.

Formally, an " Order in Council " is an order by the King or Queen at a meeting of the Privy Council, roughly equivalent to an Executive Order in the United States, by which the British government decrees policies.

* 2004: Newsboys, Relient K, Christafari, The Tribe ( Formally The World Wide Message Tribe ), Pillar, Paul Colman Trio, Magnify, Darlene Zschech, Hillsong United

Formally known as the Price Statistics Review Committee, the Stigler Commission was convened in 1961 to study the measurement of inflation in the United States.

Formally nominated on January 5, 1950, he was confirmed by the United States Senate on February 8, 1950, and received his commission on February 10, 1950.

In the United States, Learning Ally ( Formally " Recording for the Blind and Dyslexic " or RFB & D ) Bookshare and the National Library Service for the Blind and Physically Handicapped ( NLS ), among others, offer content to blind and visually impaired individuals.

Formally known as the Federal Reserve Wire Network, Fedwire is a Real Time Gross Settlement Funds Transfer system operated by the United States Federal Reserve Banks that enables financial institutions to electronically transfer funds between its more than 9, 289 participants ( as of March 19, 2009 ).