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Formally, a natural number n is called superabundant precisely when, for any m < n,

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## Some Related Sentences

Formally and natural

__Formally__

**,**given two categories C and D

**,**an equivalence of categories consists of

**a**functor F: C → D

**,**

**a**functor G: D → C

**,**and two

__natural__isomorphisms ε: FG → I

**<**sub > D </ sub > and η: I

**<**sub > C </ sub >→ GF.

__Formally__

**,**

**a**rigged Hilbert space consists of

**a**Hilbert space H

**,**together with

**a**subspace Φ which carries

**a**finer topology

**,**that

**is**one

**for**which the

__natural__inclusion

__Formally__

**,**the use of

**a**reduction

**is**the function that sends each

__natural__

**number**

**n**to the largest

__natural__

**number**

**m**whose membership in the set B was queried by the reduction while determining the membership of

**n**in A.

__Formally__

**,**

**a**

__natural__

**number**

**n**

**is**

**called**highly abundant if and only if

**for**all

__natural__numbers

**m**

**<**

**n**

**,**

__Formally__

**,**citizen science has been defined as " the systematic collection and analysis of data ; development of technology ; testing of

__natural__phenomena ; and the dissemination of these activities by researchers on

**a**primarily avocational basis ".

__Formally__

**,**the right Kan extension of along consists of

**a**functor and

**a**

__natural__transformation which

**is**couniversal with respect to the specification

**,**in the sense that

**for**

**any**functor and

__natural__transformation

**,**

**a**unique

__natural__transformation

**is**defined and fits into

**a**commutative diagram

Formally and number

__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 ).

__Formally__

**,**if d

**is**the dimension of the parameter

**,**and

**n**

**is**the

__number__of samples

**,**if as and as

**,**then the model

**is**semi-parametric.

__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__

**,**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__

**,**

**a**measurable cardinal

**is**an uncountable cardinal

__number__κ such that there exists

**a**κ-additive

**,**non-trivial

**,**0-1-valued measure on the power set of κ.

__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__

**,**

**a**composite

__number__

**n**= d · 2

**<**sup > s </ sup > + 1 with d being odd

**is**

**called**

**a**strong pseudoprime to

**a**relatively prime base

**a**

**when**one of the following conditions hold:

__Formally__

**,**

**for**received words

**,**let denote the Hamming distance between and

**,**that

**is**

**,**the

__number__of positions in which and differ.

__Formally__established on October 21st

**,**1969

**,**it includes 18 departments providing 31 different undergraduate courses

**,**as well as

**a**

__number__of graduate programs.

__Formally__

**,**

**a**

__number__

**n**

**is**colossally abundant if and only if there

**is**an ε > 0 such that

**for**all k > 1

**,**

__Formally__

**,**it refers to

**a**set of six verb forms inflected

**for**person and

__number__forming

**a**single paradigm.

__Formally__the Airframe Reconditioning Squadron RNZAF

**,**the unit was commercialised in 1998 and

**is**now managed by SAFE Air Ltd. No. 1 Repair Depot was also active at the base

**for**

**a**

__number__of years

**,**though during the war it appears to have been stationed at Hamilton.

__Formally__

**,**the equation above defines the macroscopic neutron cross-section (

**for**reaction x ) as the proportionality constant between

**a**neutron flux incident on

**a**( thin ) piece of material and the

__number__of reactions that occur ( per unit volume ) in that material.

__Formally__

**,**the upper density of

**a**graph G

**is**the infimum of the values α such that the finite subgraphs of G with density α have

**a**bounded

__number__of vertices.

Formally and n

__Formally__

**,**the singular value decomposition of an

**m**×

__n__real or complex matrix M

**is**

**a**factorization of the form

__Formally__

**,**the discrete Hartley transform

**is**

**a**linear

**,**invertible function H: R

**<**sup >

__n__</ sup >

**<**tt >-></ tt > R

**<**sup >

__n__</ sup > ( where R denotes the set of real numbers ).

__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 >).

__Formally__

**,**Aff ( V )

**is**naturally isomorphic to

**a**subgroup of

**,**with V embedded as the affine plane

**,**namely the stabilizer of this affine plane ; the above matrix formulation

**is**the ( transpose of ) the realization of this

**,**with the (

__n__×

__n__and 1 × 1 ) blocks corresponding to the direct sum decomposition.

__Formally__

**,**

**a**complex projective space

**is**the space of complex lines through the origin of an (

__n__+ 1 )- dimensional complex vector space.

__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__

**,**an algebraic function in

__n__variables over the field K

**is**an element of the algebraic closure of the field of rational functions K ( x

**<**sub > 1 </ sub >,..., x

**<**sub >

__n__</ sub >).

__Formally__

**,**P

**is**

**a**symmetric polynomial

**,**if

**for**

**any**permutation σ of the subscripts 1

**,**2

**,**...,

__n__one has P ( X

**<**sub > σ ( 1 )</ sub >, X

**<**sub > σ ( 2 )</ sub >, …, X

**<**sub > σ (

__n__)</ sub >) = P ( X

**<**sub > 1 </ sub >, X

**<**sub > 2 </ sub >, …, X

**<**sub >

__n__</ sub >).

Formally and is

__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.

__Formally__

**,**the set of all context-free languages

__is__identical to the set of languages accepted by pushdown automata ( PDA ).

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__

**,**the base

__is__known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

__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__

**,**oxidation state

__is__the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

__Formally__

**,**an inner product space

__is__

**a**vector space V over the field together with an inner product

**,**i. e., with

**a**map

__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__**,****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.__Formally__

**,**

**a**profinite group

__is__

**a**Hausdorff

**,**compact

**,**and totally disconnected topological group: that

__is__

**,**

**a**topological group that

__is__also

**a**Stone space.

__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__

**,**

**a**frame

__is__defined to be

**a**lattice L in which finite meets distribute over arbitrary joins

**,**i. e. every ( even infinite ) subset

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