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Formally and binary
Formally, a ring is an Abelian group ( R, +), together with a second binary operation * such that for all a, b and c in R,
Formally, an unlabelled state transition system is a tuple ( S, →) where S is a set ( of states ) and → ⊆ S × S is a binary relation over S ( of transitions ).

Formally and operation
Formally, a mortgage lender ( mortgagee ), or other lien holder, obtains a termination of a mortgage borrower ( mortgagor )' s equitable right of redemption, either by court order or by operation of law ( after following a specific statutory procedure ).
Formally, a cumulativity predicate CUM can be defined as follows, where capital X is a variable over sets, U is the universe of discourse, p is a mereological part structure on U, and is the mereological sum operation.
Formally, LPAR designates the logical partitioning function and mode of operation, whereas PR / SM is the commercial designation of the feature.

Formally and on
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 speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).
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
*, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.
Formally a random variable is considered to be a function on the possible outcomes.
Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as
Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.
Formally, Gaussian curvature only depends on the Riemannian metric of the surface.
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, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.
Formally, if is an open subset of the complex plane, a point of, and is a holomorphic function, then is called a removable singularity for if there exists a holomorphic function which coincides with on.
Formally, the appointment remains a royal one made on the advice of the Lord Chancellor, but he / she no longer comments on individual applications, and merely supervises the process and reviews the panel's recommendations in general terms ( satisfying himself / herself that the process as operated is fair and efficient ).
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.
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 WPAZ, the station changed its call letters and name on October 28th, 2011.
Formally, the Sachs – Hornbostel is modeled on the Dewey Decimal classification.
Formally, a center of gravity is an application point of the resultant gravitational force on the body.
Formally named The King's College of Our Lady and Saint Nicholas in Cambridge, the college lies on the River Cam and faces out onto King's Parade in the centre of the city.
Formally, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.
Formally, a statistic s is a measurable function of X ; thus, a statistic s is evaluated on a random variable X, taking the value s ( X ), which is itself a random variable.
Formally educated African Americans, on the other hand, either disregarded black minstrelsy or openly disdained it.
Formally, the motion merely calls on parliament to thank the monarch or viceroy for the speech via an Address in Reply.

Formally and set
Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).
Formally, one has an implication, not an equivalence, so the solution set may get larger.
Formally, a set S is called finite if there exists a bijection
Formally, according to the Constitution, citizens of Turkmenistan have the right to set up political parties and other public associations, acting within the framework of the Constitution and laws, and public associations and groups of citizens have the right to nominate their candidates in accordance with the election law.
Formally, the weighted mean of a non-empty set of data
Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.
Formally, Gaussian integers are the set
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, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.
Formally, the movement is a rondo that acts as the theme for a set of eight variations, capped off by a dramatic coda.
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, starting with a linearly independent set of vectors
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, a constraint satisfaction problem is defined as a triple, where is a set of variables, is a domain of values, and is a set of constraints.

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