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Formally and one
Formally, one has an implication, not an equivalence, so the solution set may get larger.
Formally one defines
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, 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, two variables are inversely proportional ( or varying inversely, or in inverse variation, or in inverse proportion or in reciprocal proportion ) if one of the variables is directly proportional with the multiplicative inverse ( reciprocal ) of the other, or equivalently if their product is a constant.
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 ).
Formally, they are partial derivatives of the option price with respect to the independent variables ( technically, one Greek, gamma, is a partial derivative of another Greek, called delta ).
Formally known as the Kingfisher Pier, the Pier at Garden City is one of the most renowned attractions in Garden City, SC.
Formally, the process is one of progressive development, divided into three stages of increasingly complexity.
Formally, the word is applied to persons who are publicly accepted in a recognised capacity, such as professional employment, graduation from a course of study, etc., to give critical commentaries in one or any of a number of specific fields of public or private achievement or endeavour.
Formally speaking, the strategy pattern defines a family of algorithms, encapsulates each one, and makes them interchangeable.
Formally, the symphony is one of Mahler's most outwardly conventional.
Formally, however, there is an examination board consisting of a candidate's teachers / examiners, the headmaster / headmistress and one external Vorsitzende ( r ) ( head ), usually a high-ranking school official or the head of another school.
Formally, one inserts under the trace for each degree of freedom
Formally, an nth order approximation is one where the order of magnitude of the error is at most, or in terms of big O notation, the error is
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, a rigged Hilbert space consists of a Hilbert space H, together with a subspace Φ which carries a finer topology, that is one for which the natural inclusion
Formally, a generative grammar is defined as one that is fully explicit.
Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.
Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.
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, given complex-valued functions f and g of a natural number variable n, one writes
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, 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 and may
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, powers with positive integer exponents may be defined by the initial condition
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, it may even be taken back to Marx's earliest statements about a class becoming conscious of itself and developing a class identity.
Formally, Turán's theorem may be stated as follows.
Formally, define the set of lines in the plane P as L ( P ); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.
Formally, a deterministic finite automaton may be defined by the tuple where is the set of states of the automaton, is the set of input symbols, is the transition function that takes a state and an input symbol to a new state, is the initial state of the automaton, and is the set of accepting or final states of the automaton.
Formally, a party in honor of the bride-to-be is never hosted by the bride-to-be, although she may participate in its planning.

Formally and by
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, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).
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 appoints the Prime Minister of Finland selected by the Parliament, and formally appoints the rest of the cabinet ( Council of State ) as proposed by the Prime Minister
Formally the Prince-Electors elected a King of the Romans, who was elected in Germany but became Holy Roman Emperor only when crowned by the Pope.
* 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, these failed when they were rejected by the Church of England's General Synod in 1972 ; conversations and co-operation continued, however, leading in 2003 to the signing of a covenant between the two churches.
Formally, this is a solution of the matrix equation by Jacobi iteration.
Formally it is defined by the equation
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, 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
Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of type J in C is a functor from J to C:
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, most of these people were convicted by NKVD troikas (" triplets ")– special courts martial.
Formally, collective noun forms such as “ a group of people ” are represented by second-order variables, or by first-order variables standing for sets ( which are well-defined objects in mathematics and logic ).
Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.
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, a transductive support vector machine is defined by the following primal optimization problem:
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.

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