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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.

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## Some Related Sentences

Formally and may

__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__

**,**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 be

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__

**,**if M is a

**set**

**,**

**the**identity function f on M is

**defined**to

__be__that function with domain and codomain M which satisfies

__Formally__

**,**a frame is

**defined**to

__be__a lattice L

**in**which finite meets distribute over arbitrary joins

**,**i

**.**e

**.**every ( even infinite ) subset

__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**

__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__

**,**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

**.**

If R is a ring

**,**let R denote**the**ring**of**polynomials**in****the**indeterminate**X**over R**.**Hilbert proved that if R is " not too large ",**in****the**sense that if R is Noetherian**,****the**same must__be__true for R**.**__Formally__**,**__Formally__

**,**a unique factorization domain is

**defined**to

__be__an integral domain R

**in**which every non-zero and non-unit x

**of**R can

__be__written

**as**a product ( including an empty product )

**of**irreducible elements p < sub > i </ sub >

**of**R and a unit u:

__Formally__

**,**

**the**problem

**of**supervised pattern recognition can

__be__stated

**as**follows: Given an unknown function (

**the**ground truth ) that maps input instances to output labels

**,**along with training data assumed to represent accurate examples

**of**

**the**mapping

**,**produce a function that approximates

**as**closely

**as**possible

**the**correct mapping

**.**

__Formally__

**,**a government would lay out its reasons for going to war

**,**

**as**well

**as**its intentions

**in**prosecuting it and

**the**steps that might

__be__taken to avert it

**.**

__Formally__

**,**

**the**case where only a subset

**of**parameters is

**defined**is still a composite hypothesis ; nonetheless

**,**

**the**term point hypothesis is often applied

**in**such cases

**,**particularly where

**the**hypothesis test can

__be__structured

**in**such a way that

**the**distribution

**of**

**the**test statistic (

**the**distribution under

**the**null hypothesis ) does not depend on

**the**parameters whose values have not been specified under

**the**point null hypothesis

**.**

Formally and defined

__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 ).

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__

**,**these reside

**in**a complex separable Hilbert space-variously called

**the**" state space "

**or**

**the**" associated Hilbert space "

**of**

**the**system-that is well

__defined__up to a complex number

**of**norm 1 (

**the**phase factor ).

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 transductive support vector machine is

__defined__by

**the**following primal optimization problem:

__Formally__

**,**bending modulus is

__defined__

**as**

**the**energy required to deform a membrane from its intrinsic curvature to some other curvature

**.**

__Formally__

**,**a constraint satisfaction problem is

__defined__

**as**a triple

**,**where is a

**set**

**of**variables

**,**is a domain

**of**values

**,**and is a

**set**

**of**constraints

**.**

__Formally__

**,**

**the**index

**of**H

**in**G is

__defined__

**as**

**the**number

**of**cosets

**of**H

**in**G

**.**( The number

**of**left cosets

**of**H

**in**G is always equal to

**the**number

**of**right cosets

**.**

__Formally__

**,**it is a norm

__defined__on

**the**space

**of**bounded linear operators between two given normed vector spaces

**.**

__Formally__

**,**a cardinal κ is

__defined__to

**be**weakly compact if it is uncountable and for every function f: < sup > 2 </ sup > →

__Formally__

**,**

**the**

**sets**

**of**free and bound names

**of**a process

**in**π – calculus are

__defined__inductively

**as**follows

**.**

__Formally__

**,**an absolute coequalizer

**of**a pair

**in**a category C is a coequalizer

**as**

__defined__above but with

**the**added property that given any functor F ( Q ) together with F ( q ) is

**the**coequalizer

**of**F ( f ) and F ( g )

**in**

**the**category D

**.**Split coequalizers are examples

**of**absolute coequalizers

**.**

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