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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.

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## Some Related Sentences

Formally and predicate

__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__

**,**

**the**extension

**of**

**a**

__predicate__in

**a**first-order model

**is**

**the**set

**of**tuples

**of**values this

__predicate__assign to true in

**the**model

**.**

Formally and can

__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__

**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**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__

**,**

**a**function ƒ

**is**real analytic

**on**an open set D in

**the**real line if for any x < sub > 0 </ sub > in D one

__can__write

__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__

**,**

**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__

**,**

**the**problem

__can__

**be**stated

**as**

**follows**: given

**a**desired property

**,**expressed

**as**

**a**temporal logic formula

**p**

**,**

**and**

**a**

**structure**M with initial state s

**,**decide if

**.**

__Formally__

**,**

**the**poem

__can__

**be**understood

**as**

**the**combination

**of**two sonnets

**,**though

**the**spacing

**of**

**the**stanzas

**is**irregular

**.**

__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__

**,**

**a**system

**is**said to

**be**observable if

**,**for any possible sequence

**of**state

**and**control vectors

**,**

**the**current state

__can__

**be**determined in finite time using only

**the**outputs ( this definition

**is**slanted towards

**the**state space representation ).

__Formally__

**,**

**the**definition only requires some invertibility

**,**so we

__can__substitute for Q any matrix M whose eigenvalues do not include − 1

**.**

__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__

**,**

**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 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**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__

**,**

**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**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 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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