 Page "Identical particles" ¶ 79
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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 γ be the small circle of radius ε, Γ the larger, with radius R, then that is, a grid of side length ε restricted to D. Let v be the vertex of G closest to x. 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 be the number of points and d the number of dimensions. Let X be a finite set with n elements. Let A be the arithmetic mean and H be the harmonic mean of n positive real numbers. 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 n be the length of a statement in Presburger arithmetic. 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 M be an n × n Hermitian matrix.

Let and denote Let Af denote the form of Af. Let X denote a Cauchy distributed random variable. Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude Let denote the equivalence class to which a belongs. 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 R denote the field of real numbers. Let denote the space of scoring functions. 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. Let us denote the time at which it is decided that the compromise occurred as T. Let denote the sequence of convergents to the continued fraction for. Let us denote the mutually orthogonal single-particle states by and so on. 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. Let Q ( x ) denote the number of square-free ( quadratfrei ) integers between 1 and x. 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 π < sub > 2 </ sub >( x ) denote the number of primes p ≤ x such that p + 2 is also prime. Let be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.

Let and energy Let the potential energy difference across the surface due to effective surface dipole be. 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 the energy of e, be Let the total energy of the traffic graph be the sum of the energies of every edge in the graph. Let be total energy of the traffic graph, and consider what happens when the route is removed. 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. Let the energy difference between the states be so that is the transition frequency of the system.

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