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Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be concretely isomorphic.
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Formally and each
Formally, cl ( S ) denotes the smallest subset Y of M that contains S such that for each reaction ( A, B )
Formally speaking, the strategy pattern defines a family of algorithms, encapsulates each one, and makes them interchangeable.
Formally, the outcomes Y < sub > i </ sub > are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to the outcome at hand, but related to the explanatory variables.
Formally, this follows from the convolution theorem in mathematics, which relates the Fourier transform of the power spectrum ( the intensity of each frequency ) to its autocorrelation.
Formally it consists of three groups of four measure phrases of three to four chords repeated four times each, ABC :|| ABC, which then merges with the next movement, " Floe " with the entrance of the horns.
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.
Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A.
Formally, a vertex cover of a graph G is a set C of vertices such that each edge of G is incident to at least one vertex in C. The set C is said to cover the edges of G. The following figure shows examples of vertex covers in two graphs ( and the set C is marked with red ).
Formally, Chrétien and Martin have remained publicly respectful of each other ; while much of the verbal sparring was between their supporters.
Formally, the use of a reduction is the function that sends each natural number n to the largest natural number m whose membership in the set B was queried by the reduction while determining the membership of n in A.
Formally described, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of the opening of the first movement in which the instruments overlap each other with a brief motive ; ( 2 ) a chorale, the actual song.
Formally, consider an economic model with different mathematical weightings placed on the utilities of each self.
Formally, a Markov partition is used to provide a finite cover for the smooth system ; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one of the covering sets to another.
Formally, the city has designations for each of the eight neighborhoods which are represented by neighborhood councils.
Formally, propositional models can be represented by sets of propositional variables ; namely, each model is represented by the set of propositional variables it assigns to true.
Formally and following
Formally, a transductive support vector machine is defined by the following primal optimization problem:
Formally, we have for the approximation to the full solution A, a series in the small parameter ( here called ), like the following:
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, a deterministic Büchi automaton is a tuple A = ( Q, Σ, δ, q < sub > 0 </ sub >, F ) that consists of the following components:
Formally signed by Count Alexander Izvolsky, Foreign Minister of the Russian Empire, and Sir Arthur Nicolson, the British Ambassador to Russia, the British-Russian Convention of 1907 stipulated the following:
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 partially ordered set ( P, ≤) is bounded complete if the following holds for any subset S of P:
Formally becoming a concert violinist following the completion of his musical education, Üngör continued to ascend the ranks of the Ottoman Imperial Orchestra, soon becoming first violinist, and ultimately, conductor in 1917.
Formally, a composite number n = d · 2 < sup > s </ sup > + 1 with d being odd is called a strong pseudoprime to a relatively prime base a when one of the following conditions hold:
Formally accepted as a candidate for the Franciscan Order at Christ House in Lafayette, New Jersey, in 1970, the following year he entered the novitiate of the Order at St. Francis Friary in Brookline, Massachusetts, where he professed his first vows in 1972.
Formally and defines
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, the equation above defines the macroscopic neutron cross-section ( for reaction x ) as the proportionality constant between a neutron flux incident on a ( thin ) piece of material and the number of reactions that occur ( per unit volume ) in that material.