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Formally and type
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, a monad consists of a type constructor M and two operations, bind and return.
Formally system networks correspond to type lattices in formal lattice theory, although they are occasionally erroneously mistaken for flowcharts or directed decision trees.

Formally and theory
Formally, ZFC is a one-sorted theory in first-order logic.

Formally and systems
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 these were examples of additional member systems.

Formally and .
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.
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, 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, their designation is the letter Ž and the number.
Formally, a topological space X is called compact if each of its open covers has a finite subcover.
Formally, they were part of the non-resident sections, but in fact constituted a separate division, largely autonomous in their activities.
Formally, the Congress serves two functions: to approve changes to the Party constitution regarding policy and to elect a Central Committee, about 300 strong.
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, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.
Formally, one has an implication, not an equivalence, so the solution set may get larger.
Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.
Formally, a bifunctor is a functor whose domain is a product category.
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 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 system is said to have memory.
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, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
Formally neutral and reluctant to become involved with the great powers except as a last resort, Kuwait turned to the United States, the Soviet Union and Britain for naval protection of its tanker fleet after twenty-one ships were attacked in the gulf since late 1986.
* 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.
Massiah: Formally Charged + Deliberate Elicitation.
*, see On Formally Undecidable Propositions of Principia Mathematica and Related Systems for details on English translations.
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 introduced in September 1979, Initial samples were released in February 1980, with production chips available over the counter in November.
* Nuclear bunker buster: Formally known as the Robust Nuclear Earth Penetrator ( RNEP ), this program aimed to modify an existing gravity bomb to penetrate into soil and rock in order to destroy underground targets.
Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

type and theory
In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.
In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).
A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.
Important contributions to type theory, NuPRL.
The term " conspiracy theory " is itself the object of a type of conspiracy theory, which argues that those using the term are manipulating their audience to disregard the topic under discussion, either in a deliberate attempt to conceal the truth, or as dupes of more deliberate conspirators.
A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.
Category theory is also, in some sense, a continuation of the work of Emmy Noether ( one of Mac Lane's teachers ) in formalizing abstract processes ; Noether realized that in order to understand a type of mathematical structure, one needs to understand the processes preserving that structure.
Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with applications in functional programming and domain theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus.
The simplest type of ab initio electronic structure calculation is the Hartree – Fock ( HF ) scheme, an extension of molecular orbital theory, in which the correlated electron – electron repulsion is not specifically taken into account ; only its average effect is included in the calculation.
This work was a precursor of a modern formulation in terms of constructive type theory.
In the context of decision theory, an estimator is a type of decision rule, and its performance may be evaluated through the use of loss functions.
theory, do show the phenomena of Bell-inequality violations that are considered to invalidate EPR's preferred " local hidden-variables " type of explanation for the correlations
An experiment by Schunk and Hanson, that studied grade 2 students who had previously experienced difficulty in learning subtraction, illustrates the type of research stimulated by social learning theory.
In the 1980s, Per Martin-Löf developed intuitionistic type theory ( also called Constructive type theory ), which associated functional programs with constructive proofs of arbitrarily complex mathematical propositions expressed as dependent types.
In category theory, a branch of mathematics, a functor is a special type of mapping between categories.
Finite fields have applications in many areas of mathematics and computer science, including coding theory, LFSRs, modular representation theory, and the groups of Lie type.
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

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