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Formally and signed
Formally, a signed graph Σ is a pair ( G, σ ) that consists of a graph G = ( V, E ) and a sign mapping or signature σ from E to the sign group

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.
Formally, powers with positive integer exponents may be defined by the initial condition
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.

Formally and Minister
Formally, this party stood in the right-wing parliamentary majority, but it criticised with virulence the policies of the executive duo composed of President Giscard d ' Estaing and Prime Minister Raymond Barre.
Formally, he is also chief of staff of the air force and navy as well, but apparently the commanders of the other two services frequently report directly to the Minister of Defence / Commander-in-Chief.
Formally known as Crown Point Airport, the airport was renamed the Arthur Napoleon Raymond Robinson International Airport on May 19, 2011 in honour of the only Prime Minister and President from the island of Tobago.

Formally and Russian
Formally abolished in 1918 after the Russian Revolution, the academy was renamed several times.
Formally this mission was declared to be only in pursuit of the Russian vessels and ammunition taken to Anzali by the White Russian counter-revolutionary General Denikin, who had been given asylum by British forces in Anzali.

Formally and Empire
By 1912, the Social Democrats, with an explicitly anti-antisemitic program, were the largest party in the German Reichstag, and the Progressives ran very strongly as well ... Formally, at least, the Jews had been fully emancipated with the establishment of the German Empire, although they were kept out of certain influential occupations, enjoyed extraordinary prosperity ... Germans intermarried with Jews: in the 1930s some 50, 000 Jews were living in mixed German-Jewish marriages, so at least 50, 000 Germans, and presumably parts of their families, had familial contact with the Jews.
Formally, it remained a state of the Holy Roman Empire until its dissolution in 1806.

Formally and Arthur
Formally, the complexity class MA is the set of decision problems that can be decided in polynomial time by an Arthur – Merlin protocol where Merlin's only move precedes any computation by Arthur.

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

Formally and Russia
Formally, it was ceded to Russia in the Treaty of Nystad in 1721, together with Swedish Estonia and Swedish Ingria.

Formally and following
Formally we mean that is an ideal if it satisfies the following conditions:
Formally, we have for the approximation to the full solution A, a series in the small parameter ( here called ), like the following:
Formally, the two parts are given by the following expression, where is the number being encoded:
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, 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, 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, a non-empty partially ordered set ( D, ≤) is called a Scott domain if the following hold:
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 TDPL grammar G is a tuple consisting of the following components:
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, Scarf considers the following general game with a finite number of agents.
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, 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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