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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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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.