[permalink] [id link]

Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be concretely isomorphic.

from
Wikipedia

## Some Related Sentences

Formally and each

__Formally__

**,**cl ( S ) denotes

**the**smallest subset Y

**of**M that contains S such that for

__each__reaction ( A

**,**B )

__Formally__speaking

**,**

**the**strategy pattern

**defines**

**a**family

**of**algorithms

**,**encapsulates

__each__one

**,**

**and**makes them interchangeable

**.**

__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__

**,**this follows from

**the**convolution theorem in mathematics

**,**which relates

**the**Fourier transform

**of**

**the**power spectrum (

**the**intensity

**of**

__each__frequency )

**to**its autocorrelation

**.**

__Formally__it consists

**of**three groups

**of**four measure phrases

**of**three

**to**four chords repeated four times

__each__

**,**ABC :|| ABC

**,**which then merges with

**the**next movement

**,**" Floe " with

**the**entrance

**of**

**the**horns

**.**

__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__

**,**A is reflexive if it is equal

**to**

**the**algebra

**of**bounded operators which leave invariant

__each__subspace left invariant by

**every**operator in A

**.**

__Formally__

**,**

**a**vertex cover

**of**

**a**graph G is

**a**set C

**of**vertices such that

__each__edge

**of**G is incident

**to**at least one vertex in C

**.**The set C is said

**to**cover

**the**edges

**of**G

**.**The

**following**figure shows examples

**of**vertex covers in two graphs (

**and**

**the**set C is marked with red ).

__Formally__

**,**Chrétien

**and**Martin have remained publicly respectful

**of**

__each__other ; while much

**of**

**the**verbal sparring was between their supporters

**.**

__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__described

**,**it alternates slow sections in

**a**modal F with faster sections

**,**" Neue Kraft fühlend " ( with renewed strength ), in D

**.**The slow sections

__each__have two elements

**,**( 1 )

**a**passage reminiscent

**of**

**the**opening

**of**

**the**first movement in which

**the**instruments overlap

__each__other with

**a**brief motive ; ( 2 )

**a**chorale

**,**

**the**actual song

**.**

__Formally__

**,**consider an economic model with different mathematical weightings placed on

**the**utilities

**of**

__each__self

**.**

__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__

**,**

**the**city has designations for

__each__

**of**

**the**eight neighborhoods which are represented by neighborhood councils

**.**

__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 following

__Formally__

**,**

**a**transductive support vector machine is defined by

**the**

__following__primal optimization problem:

__Formally__

**,**we have for

**the**approximation

**to**

**the**full solution A

**,**

**a**series in

**the**small parameter ( here called ), like

**the**

__following__:

__Formally__

**,**

**a**Hopf algebra is

**a**( associative

**and**coassociative ) bialgebra H over

**a**field K together with

**a**K-linear map S: H → H ( called

**the**antipode ) such that

**the**

__following__diagram commutes:

__Formally__

**,**

**a**deterministic Büchi automaton is

**a**tuple A = ( Q

**,**Σ

**,**δ

**,**q < sub > 0 </ sub >, F ) that consists

**of**

**the**

__following__components:

__Formally__signed by Count Alexander Izvolsky

**,**Foreign Minister

**of**

**the**Russian Empire

**,**

**and**Sir Arthur Nicolson

**,**

**the**British Ambassador

**to**Russia

**,**

**the**British-Russian Convention

**of**1907 stipulated

**the**

__following__:

__Formally__

**,**

**a**mortgage lender ( mortgagee ), or other lien holder

**,**obtains

**a**termination

**of**

**a**mortgage borrower ( mortgagor )' s equitable right

**of**redemption

**,**either by court order or by operation

**of**law ( after

__following__

**a**specific statutory procedure ).

__Formally__

**,**

**a**partially ordered set ( P

**,**≤) is bounded complete if

**the**

__following__holds for any subset S

**of**P:

__Formally__becoming

**a**concert violinist

__following__

**the**completion

**of**his musical education

**,**Üngör continued

**to**ascend

**the**ranks

**of**

**the**Ottoman Imperial Orchestra

**,**soon becoming first violinist

**,**

**and**ultimately

**,**conductor in 1917

**.**

__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__accepted as

**a**candidate for

**the**Franciscan Order at Christ House in Lafayette

**,**New Jersey

**,**in 1970

**,**

**the**

__following__year he entered

**the**novitiate

**of**

**the**Order at St

**.**Francis Friary in Brookline

**,**Massachusetts

**,**where he professed his first vows in 1972

**.**

Formally and defines

__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__

**,**

**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

**.**

0.140 seconds.