[permalink] [id link]

In two dimensions, the finite reflection groups are the dihedral groups, which are generated by reflection in two lines that form an angle of and correspond to the Coxeter diagram Conversely, the cyclic point groups in two dimensions are not generated by reflections, and indeed contain no reflections – they are however subgroups of index 2 of a dihedral group.

from
Wikipedia

## Some Related Sentences

two and dimensions

If

**the**patient can perceive figure kinesthetically when he cannot perceive it visually**,**then**,**it would seem**,****the**sense**of**touch has immediate contact with**the**spatial aspects**of**things**in**independence**of**visual representations**,**at least**in**regard**to**__two____dimensions__**,****and****,**as we shall see**,**even this much spatial awareness on**the**part**of**unaided touch is denied**by****the**authors**.****In**this relief

**,**

**the**

__two__principal figures

**,**

**the**stern

**and**courageous pope

**and**

**the**dismayed

**and**frightened Attila

**,**surge forward from

**the**center into three

__dimensions__

**.**

Usually

**the**Cartesian coordinate system is applied**to**manipulate equations for planes**,**straight**lines****,****and**squares**,**often**in**__two__**and**sometimes**in**three__dimensions__**.****In**three

__dimensions__

**,**

**a**single equation usually gives

**a**surface

**,**

**and**

**a**curve must be specified as

**the**intersection

**of**

__two__surfaces ( see below ), or as

**a**system

**of**parametric equations

**.**

Finally

**,**Dürer discusses**the**Delian Problem**and**moves on**to****the**' construzione legittima ',**a**method**of**depicting**a**cube**in**__two____dimensions__through linear perspective**.**
Instead

**,****the**ratio**of****the**speed**of****the**wind**and****that****of****the**cups**,****the**anemometer factor**,**depends on**the**__dimensions__**of****the**cups**and**arms**,****and**may have**a**value between__two__**and****a**little over three**.**
Each string is

**a**one-dimensional object**,****and****the**largest number**of**__dimensions__**in****which**__two__strings will generically intersect (**and****,**presumably**,**annihilate ) is three**.**
After

**the**advent**of**quantum mechanics**,**work done**by**Landau**in**1930 predicted**the**quantization**of****the**Hall conductance for electrons confined**in**__two____dimensions__**.****In**

__two__

__dimensions__

**,**closed disks

**are**compact since for any infinite number

**of**points sampled from

**a**disk

**,**some subset

**of**those points must get arbitrarily close either

**to**

**a**

**point**within

**the**disc

**,**or

**to**

**a**

**point**on

**the**boundary

**.**

Rotating frames

**and**fictitious forces can often reduce**the**description**of**motion**in**__two____dimensions__**to****a**simpler description**in**one dimension ( corresponding**to****a**co-rotating frame ).**In**

__two__

__dimensions__

**the**position vector

**which**has magnitude ( length )

**and**directed at

**an**

**angle**above

**the**x-axis can be expressed

**in**Cartesian coordinates using

**the**unit vectors

**and**:

The modern Cartesian coordinate system

**in**__two____dimensions__( also called**a**rectangular coordinate system ) is defined**by****an**ordered pair**of**perpendicular**lines**( axes ),**a**single unit**of**length for both axes**,****and****an**orientation for each axis**.****In**

__two__

__dimensions__

**,**

**the**vector from

**the**origin

**to**

**the**

**point**with Cartesian coordinates ( x

**,**y ) can be written as:

)

**Conversely****,**it is often helpful**to**use**the**geometry**of**Cartesian coordinates**in**__two__or three__dimensions__**to**visualize algebraic relationships between__two__or three**of**many non-spatial variables**.**
The data warehouse bus architecture is primarily

**an**implementation**of**"**the**bus ",**a**collection**of**conformed__dimensions__**and**conformed facts**,****which****are**__dimensions__**that****are**shared (**in****a**specific way ) between facts**in**__two__or more data marts**.**
Another product rule for

**the**cross product**of**__two__vector fields F**and**G**in**three__dimensions__involves**the**curl**and**reads as follows:
This means

**that****the**radio spectrum is divided into physical channels**in**__two____dimensions__: frequency**and**time**.****In**

__two__

__dimensions__

**,**one way

**to**detect if

**point**D lies

**in**

**the**circumcircle

**of**A

**,**B

**,**C is

**to**evaluate

**the**determinant:

A divide

**and**conquer algorithm for triangulations**in**__two____dimensions__is due**to**Lee**and**Schachter**which**was improved**by**Guibas**and**Stolfi**and**later**by**Dwyer**.**
He proved equations for

**the**volumes**and**areas**of**various figures**in**__two__**and**three__dimensions__**,****and**enunciated**the**Archimedean property**of****finite**numbers**.****In**dimension one this is

**the**real line ;

**in**dimension

__two__it is

**the**Cartesian plane ;

**and**

**in**higher

__dimensions__it is

**a**coordinate space with three or more real number coordinates

**.**

**In**

__two__or more

__dimensions__we must use

**,**

**the**del or gradient operator

**,**

**which**generalises

**the**first derivative

**,**obtaining

two and finite

* Let Q be

**a**set enclosed between__two__step regions S**and**T**.**A step region is formed from**a**__finite__union**of**adjacent rectangles resting on**a**common base**,**i**.**e**.**S ⊆ Q ⊆ T**.**If there is**a**unique number c such**that****a**( S ) ≤ c ≤**a**( T ) for all such step regions S**and**T**,**then**a**( Q )
Using terms from formal language theory

**,****the**precise mathematical definition**of**this concept is as follows: Let S**and**T be__two____finite__sets**,**called**the**source**and**target alphabets**,**respectively**.**
The difference between

**the**__two__definitions is**that**under**the**former**,**__finite__sets**are**also considered**to**be countable**,**while under**the**latter definition**,****they****are****not**considered**to**be countable**.**
The function F is called prefix-free if there

**are****no**__two__elements p**,**p ′**in**its domain such**that**p ′ is**a**proper extension**of**p**.**This can be rephrased as:**the**domain**of**F is**a**prefix-free code ( instantaneous code ) on**the**set**of**__finite__binary strings**.**
The truth values

**of**logical formulas usually**form****a**__finite__set**,**generally restricted**to**__two__values: true**and**false**,**but logic can also be continuous-valued**,**e**.**g., fuzzy logic**.**
It is known

**that****the**least order**of****a**__finite__**group**for**which**there exists__two__commutators whose product is**not****a**commutator is 96 ;**in**fact there**are**__two__nonisomorphic**groups****of**order 96 with this property**.**
It tends towards

**a**line segment ( see below ) if**the**__two__foci remain**a**__finite__distance apart**and****a**parabola if one focus is kept fixed as**the**other is allowed**to**move arbitrarily far away**.****In**Jacques Dupuis ’ Who Do You Say I Am ?, he argues

**that**

**,**within

**the**one person

**of**Jesus Christ

**,**we can distinguish between his

__two__natures

**,**human

**and**divine

**,**

**and**thus between

**the**operations

**of**his uncreated divine nature

**and**his created

__finite__human nature

**.**

*

**In****the**case**of**QAM ( quadrature amplitude modulation ),**a**__finite__number**of**at least__two__phases**,****and**at least__two__amplitudes**are**used**.**
If we allow

**a**__finite__automaton access**to**__two__stacks instead**of**just one**,**we obtain**a**more powerful device**,**equivalent**in**power**to****a**Turing machine**.**
Power law distributions have

**a**well defined mean only if**the**exponent exceeds 1**and**have**a**__finite__variance only when**the**exponent exceeds__two__**.****In**

**the**digital QAM case

**,**

**a**

__finite__number

**of**at least

__two__phases

**and**at least

__two__amplitudes

**are**used

**.**

If

**the**material superconducts**in****the**absence**of****a**field**,**then**the**superconducting phase free energy is lower than**that****of****the**normal phase**and**so for some__finite__value**of****the**magnetic field ( proportional**to****the**square root**of****the**difference**of****the**free energies at zero magnetic field )**the**__two__free energies will be equal**and****a**phase transition**to****the**normal phase will occur**.**
With

__finite__samples**,**approximation results measure how close**a**limiting distribution approaches**the**statistic's sample distribution: For example**,**with 10**,**000 independent samples**the**normal distribution approximates (**to**__two__digits**of**accuracy )**the**distribution**of****the**sample mean for many population distributions**,****by****the**Berry**–**Esseen theorem**.**
More generally

**,**any collection**of**events — possibly more than just__two__**of**them —**are**mutually independent if**and**only if for every__finite__subset A < sub > 1 </ sub >, ..., A < sub > n </ sub >**of****the**collection we have
Notice

**that**independence requires this rule**to**hold for every__finite__subset**of****the**collection ; see for**a**three-event example**in****which****and**yet**no**__two__**of****the**three events**are**pairwise independent**.**
; Lambda calculus: A computation consists

**of****an**initial lambda expression ( or__two__if you want**to**separate**the**function**and**its input ) plus**a**__finite__sequence**of**lambda terms**,**each deduced from**the**preceding term**by**one application**of**Beta reduction**.**0.098 seconds.