Let Af denote the form of Af.

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

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 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 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.

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.

Next, take a Hamiltonian invariant under T. Let | a > and T | a > be two quantum states of the same energy.

Let A be an observable of a quantum mechanical system.

Let us consider the quantum harmonic oscillator with the quartic potential perturbation and

Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:

Let H and G be Hilbert spaces of dimension n and m respectively, and Φ be a quantum operation taking the density matrices acting on H to those acting on G. Then there are matrices

Let Φ be a ( not necessarily trace preserving ) quantum operation on a finite dimensional Hilbert space H with two representing sequences of Kraus matrices

Let be a quantum channel that can cause at most one phase flip.

Let be a quantum channel that can arbitrarily corrupt a single qubit.

Let us suppose we have a set of wave functions which depend parametrically on a set of quantum numbers.

Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

The introduction, " Let us pray for the whole state of Christ's Church militant here in earth " remained unaltered and only a thanksgiving for those " departed this life in thy faith and fear " was inserted to introduce the petition that the congregation might be " given grace so to follow their good examples that with them we may be partakers of thy heavenly kingdom ".

The state motto is Esto Perpetua ( Latin for " Let it be forever ").

Let the state of the composite system be

* Let be concentration of active centers in the ground state.

Let the event space represent the current state of belief for this process.

Let A be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states occurs with probability p < sub > j </ sub >.

Let it once be understood that we mean to stop the horrible state of things in Cuba and it will be stopped.

In reference to Lucas, his three Ohio commissioners and their guard, Mason wrote, " Let him get on our soil, arrest him, strike the blood at once, disgrace him and his state, and end the controversy.

When a cobbler commented on his mistakes in painting a shoe, Apelles made the corrections that very night ; the next morning the cobbler noticed the changes, and proud of his effect on the artist's work began to criticize how Apelles portrayed the leg — whereupon Apelles emerged from his hiding-place to state: Ne sutor ultra crepidam —" Let the shoemaker venture no further.

He argued against those, such as Wilberforce, who held that Britain could safely maintain a policy of isolation: " Let us consider the state of the world as it is, not as we fancy it ought to be.

Let us now determine the physical state of the partial system A as completely as possible by measurements.

(" Let me state at once, in clearest unmistakable terms, that I have never known of, or seen, or heard, a greater actor than was Irving.

Let the opinions impugned be the belief of God and in a future state, or any of the commonly received doctrines of morality ...

On May 17, 2006, the state of Illinois enacted Senate Bill 1144, the " Let Them Rest In Peace Act ", to shield grieving military families from protests during funerals and memorial services of fallen military service members.

Let Z < sub > n </ sub > denote the state in period n ( often interpreted as the size of generation n ), and let X < sub > n, i </ sub > be a random variable denoting the number of direct successors of member i in period n, where X < sub > n, i </ sub > are iid over all n ∈

The Maoists announced a ‘ People's War ’ on February 13, 1996, under the slogan: " Let us march ahead on the path of struggle towards establishing the people's rule by wreaking the reactionary ruling system of state.

Translated, it means " Let it be perpetual " or " It is forever "; appropriate for a time travel film, it is also the motto of the state of Idaho.

Let x be the current state of one sequence and y be the current state of the other.

Let the unperturbed atom or molecule be in a g-fold degenerate state with orthonormal zeroth-order state functions.

Let the unmarked state be a synonym for the void.