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Formally and sets
Formally, let S and T be finite sets and let F =
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, collective noun forms such asa 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, this is captured by the Duality Principle for ordered sets:
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
Many interior sets were built at the Castle Brae Business Centre ( Formally the Castlebrae High School ) in Edinburgh.
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, a quantization predicate QUA can be defined as follows, where is the universe of discourse, and is a variable over sets, and is a mereological part structure on with < math > < _p </ math > the mereological part-of relation.
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, 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 free
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, let P be an expression in which the variable x is free.
Formally known as the Inverurie Parish Church, a split in Church of Scotland over the appointment of ministers in 1843 ( The Disruption ) led to the creation of the West Parish Church ( known locally as " The West Kirk ") which was founded as a Free Church ( a church free from Edinburgh control ) thus causing the Inverurie Parish Church to change its name.
Formally, if Y is not free in φ:

Formally and bound
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 and process
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 Markov chain is a random process with the Markov property.
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, the process is one of progressive development, divided into three stages of increasingly complexity.
Formally, this effect is referred to as and is a process whereby the capture frame rate of the camera changes over time.
Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times.
Formally, it is the process of eliciting information from the government.
Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

Formally and
Formally, arbitrage transactions have negative skew prices can get a small amount closer ( but often no closer than 0 ), while they can get very far apart.
Formally inaugurated in March 2004, the Global Leadership Foundation works to " promote good governance democratic institutions, open markets, human rights and the rule of law and to contribute to the prevention and resolution of conflict through mediation.
Formally, the Sachs Hornbostel is modeled on the Dewey Decimal classification.
* Citibank Banamex USA Formally California Commerce Bank, Banamex's U. S. banking division.
Formally designated as the Manhattan Engineer District, it refers specifically to the period of the project from 1941 1946 under the control of the U. S. Army Corps of Engineers, under the administration of General Leslie R. Groves.
Formally, there are sixteen stages or contemplations of ānāpānasati.
Formally, Mbenga Mbuti Pygmy music consists of at most only four parts, and can be described as an, " ostinato with variations ," or similar to a passacaglia, in that it is cyclical.
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, he says, the piece consists of two parts of nearly equal length, the end of the first section being bars 24 28 ( p. 77 ).
Formally trained specialists recognized by the American Dental Association undergo a minimum of 2 3 years full-time rigorous education program after dental school graduation.
Formally, the issue is that interfertile " able to interbreed " is not a transitive relation if A can breed with B, and B can breed with C, it does not follow that A can breed with C and thus does not define an equivalence relation.
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.
The Knight Foundation ( Formally the Foundation for Law and Government ), offers a possible solution the " Knight 4000 ".

Formally and calculus
Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line,

Formally and are
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, 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 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.
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.

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