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Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

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## Some Related Sentences

Let and ε

__Let__F and G be

**a**pair

**of**adjoint functors with unit η and co-unit

__ε__

**(**see

**the**article on adjoint functors for

**the**definitions ).

__Let__X be

**the**set R

**of**real numbers

**,**and let

**a**subset A

**of**R be negligible if for each

__ε__> 0

**,**there exists

**a**finite or countable collection I < sub > 1 </ sub >, I < sub > 2 </ sub >, …

**of**

**(**possibly overlapping

**)**intervals satisfying:

__Let__G be

**a**σ-compact

**,**locally compact topological group and π: G U

**(**H

**)**

**a**unitary representation

**of**G on

**a**

**(**complex

**)**Hilbert space H

**.**If

__ε__> 0 and K

**is**

**a**compact subset

**of**G

**,**then

**a**unit vector ξ

**in**H

**is**called an

**(**

__ε__

**,**K )- invariant vector if π

**(**g

**)**ξ-ξ <

__ε__for all g

**in**K

**.**

Let and n

__Let__

**(**m

**,**

__n__

**)**be

**a**pair

**of**amicable numbers with m <

__n__

**,**and write m = gM and

__n__= gN where g

**is**

**the**greatest common divisor

**of**m and

__n__

**.**If M and N are both coprime to g and square free then

**the**pair

**(**m

**,**

__n__

**)**

**is**said to be regular

**,**otherwise it

**is**called irregular or exotic

**.**

__Let__

**denote**

**the**Bézier curve determined by

**the**points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub >

__n__</ sub >.

Suppose that

**in****a**mathematical language L**,**it**is**possible to enumerate all**of****the**defined numbers**in**L**.**__Let__this enumeration be defined by**the**function G: W → R**,**where G**(**__n__**)****is****the**real number described by**the**nth description**in****the**sequence**.**__Let__

__n__

**denote**

**a**complete set

**of**

**(**discrete

**)**quantum numbers for specifying

**single-particle**states

**(**for example

**,**for

**the**

**particle**

**in**

**a**box problem we can take

__n__to be

**the**quantized wave vector

**of**

**the**wavefunction

**.**

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

__n__</ sub > be

**the**sizes

**of**

**the**heaps before

**a**move

**,**and y < sub > 1 </ sub >, ..., y < sub >

__n__</ sub >

**the**corresponding sizes after

**a**move

**.**

__Let__s = x < sub > 1 </ sub > ⊕ ... ⊕ x < sub >

__n__</ sub > and t = y < sub > 1 </ sub > ⊕ ... ⊕ y < sub >

__n__</ sub >.

__Let__

**(**q < sub > 1 </ sub >, w

**,**x < sub > 1 </ sub > x < sub > 2 </ sub >... x < sub > m </ sub >)

**(**q < sub > 2 </ sub >, y < sub > 1 </ sub > y < sub > 2 </ sub >... y < sub >

__n__</ sub >) be

**a**transition

**of**

**the**GPDA

__Let__be an oriented smooth manifold

**of**dimension

__n__and let be an n-differential form that

**is**compactly supported on

**.**

__Let__be

**a**random sample

**of**size

__n__— that

**is**

**,**

**a**sequence

**of**independent and identically distributed random variables drawn from distributions

**of**expected values given by µ and finite variances given by σ < sup > 2 </ sup >.

Let and denote

__Let__'~'

__denote__an equivalence relation over some nonempty set A

**,**called

**the**universe or underlying set

**.**

__Let__G

__denote__

**the**set

**of**bijective functions over A that preserve

**the**partition structure

**of**A: ∀ x ∈ A ∀ g ∈ G

**(**g

**(**x

**)**∈ ).

__Let__G be

**a**set and let "~"

__denote__an equivalence relation over G

**.**Then we can form

**a**groupoid representing this equivalence relation as follows

**.**

__Let__

**the**line

**of**symmetry intersect

**the**parabola at point Q

**,**and

__denote__

**the**focus as point F and its distance from point Q as f

**.**

__Let__

**the**perpendicular to

**the**line

**of**symmetry

**,**through

**the**focus

**,**intersect

**the**parabola at

**a**point T

**.**Then

**(**1

**)**

**the**distance from F to T

**is**2f

**,**and

**(**2

**)**

**a**tangent to

**the**parabola at point T intersects

**the**line

**of**symmetry at

**a**45 ° angle

**.**

That

**is****,**Alice has one half**,****a****,**and Bob has**the**other half**,**b**.**__Let__c__denote__**the**qubit Alice wishes to transmit to Bob**.**__Let__H be

**a**Hilbert space

**,**and let H *

__denote__its dual space

**,**consisting

**of**all continuous linear functionals from H into

**the**field R or C

**.**If x

**is**an element

**of**H

**,**then

**the**function φ < sub > x </ sub >, defined by

If V

**is****a**real vector space**,**then we replace V by its complexification V ⊗< sub > R </ sub > C and let g__denote__**the**induced bilinear form on V ⊗< sub > R </ sub > C**.**__Let__W be**a**maximal isotropic subspace**,**i**.**e**.****a**maximal subspace**of**V such that g |< sub > W </ sub > = 0**.**
A possible definition

**of**spoiling based on vote splitting**is**as follows:__Let__W__denote__**the**candidate who wins**the**election**,**and let X and S__denote__two other candidates**.**__Let__be

**a**sequence

**of**independent and identically distributed variables with distribution function F and let

__denote__

**the**maximum

**.**

Let and energy

Next

**,**take**a**Hamiltonian invariant under T**.**__Let__|**a**> and T |**a**> be two quantum states**of****the**same__energy__**.**__Let__

**the**number

**of**

**particles**with

**the**same

__energy__be

**,**

**the**number

**of**

**particles**possessing another

__energy__be

**,**and so forth for all

**the**possible

**energies**

__Let__V be

**a**Hamiltonian representing

**a**weak physical disturbance

**,**such as

**a**potential

__energy__produced by an external field

**.**

__Let__us label with s

**(**s = 1

**,**2

**,**3

**,**...)

**the**exact states

**(**microstates

**)**that

**the**

**system**can occupy

**,**and

**denote**

**the**

**total**

__energy__

**of**

**the**

**system**when it

**is**

**in**microstate s as E < sub > s </ sub >

**.**

__Let__

**the**ecosystem

**(**i

**.**e., solar

__energy__

**)**subsidize

**the**management effort rather than

**the**other way around

**.**

__Let__be

**the**

__energy__

**of**

**the**microstate and suppose there are members

**of**

**the**ensemble residing

**in**this

**state**

**.**

Paxton's 1979 album

**,**Up and Up**,**contains**the**song "__Let__**the**Sunshine ", which addresses issues concerning environmentalism and solar__energy__**.**
" It was parodied by

**the**band Dash Rip Rock with their single entitled “__Let__’ s Go Smoke Some Pot ”, and by NRBQ during**the**1973__energy__crisis under**the**title**,**" Get That Gasoline**.**__Let__

**denote**

**the**time-independent Hamiltonian

**,**and let and

**denote**

**the**two

__energy__eigenstates

**of**

**the**

**system**

**,**with respective eigenvalues and

**.**

__Let__us hope

**,**then

**,**that you can use your

__energy__to overcome your moth-eaten thirty tyrants

**of**

**the**various German states

**.**

0.395 seconds.