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

Formally, this definition includes as classical all objects with the current orbits that

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

Formally given to bishop Philip of Senj, the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead of Latin, not the Byzantine rite ), actually extended to all Croatian lands, mostly along the Adriatic coast.

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, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.

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, start with a set Ω and consider the sigma algebra Σ on Ω consisting of all subsets of Ω.

( Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.

; Electronics: Formally a fixture in radio shacks, now used for assembly and repair of all sorts of electronic equipment including communication, computer, and home entertainment items.

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 speaking, the opportunity was extended to all classes to join China's project on its Leninist terms.

Formally, Liouville numbers and elementary numbers contain the algebraic numbers, and they include some but not all transcendental numbers.

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, the game ( coalitional game ) consists of a finite set of players, called the grand coalition, and a characteristic function denotes the power set of .</ ref > from the set of all possible coalitions of players to a set of payments that satisfies.

Formally, rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space.

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, it is the solid figure formed by the locus of all straight line segments that join the apex to the base.

Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j ( κ ) ≥ λ.

Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

Formally, all members of the Ghelani family are Hindu Leuva Patel, although some have since chosen other religious beliefs.

Formally, the semantic of influence diagram is based on sequential construction of nodes and arcs, which implies a specification of all conditional independencies in the diagram.

Formally, a redistribution of all agents ’ initial endowments of goods among all agents in the system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve the satisfaction of someone in the group without impairing that of any other in the group.

Formally speaking, the strategy pattern defines a family of algorithms, encapsulates each one, and makes them interchangeable.

Formally said, the intersection of each two of them is empty ( the null event ): A and B

Formally, they have always been considered Orthodox Christians, but most of them followed shamanism and animism as a part of their culture.

" Formally known as the 13th Special Investigative Section of the Earth Federation Government Police Agency, this group is responsible for discovering illegal Earth residents and deporting them to the space colonies.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

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

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, 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, both the pattern and searched text are vectors of elements of Σ.

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, Gaussian integers are the set

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, 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, 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, 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, the two parts are given by the following expression, where is the number being encoded:

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, they are known as Ridunians, from the Latin Riduna.

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 unnamed icefields are not listed

Formally this approximation is founded on the variational principle, valid for Hamiltonians that are bounded from below.

Formally, an analytic function ƒ ( z ) of the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent, i. e., if ƒ is transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).

These are manufactured by Amphenol ( Formally Alcatel Components and ITT Cannon Australia ).

Formally most of these approaches are similar to an artificial neural network, as inputs to a node are summed up and the result serves as input to a sigmoid function, e. g., but proteins do often control gene expression in a synergistic, i. e. non-linear, way.

Formally, the sets of free and bound names of a process in π – calculus are defined inductively as follows.