Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Let Af.

Let us put Af.

Let us look at the operator Af.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let us assume that Af is identical to the form of an occurrence Af which preceded Af in the text.

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

Let this be denoted by Af.

Let Af where the maximization is over all admissible policies 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.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

In his Commentary on Daniel, he noted, “ Let us not follow the opinion of some commentators and suppose him to be either the Devil or some demon, but rather, one of the human race, in whom Satan will wholly take up his residence in bodily form .” In interpreting 2 Thessalonians's claim that the Antichrist will sit in God's temple, Jerome preferred the view that the " temple " should be interpreted as the Church, not as the Temple in Jerusalem.

Let be any basis for the column space of and place them as column vectors to form the × matrix.

Let be an oriented smooth manifold of dimension n and let be an n-differential form that is compactly supported on.

Let φ ( ξ, η, ζ ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support ( the region where the function is non-zero ) shrinks to the origin.

Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector.

Let V be a vector space over a field K, and let be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field.

Let the curve be a unit speed curve and let t = u × T so that T, u, t form an orthonormal basis: the Darboux frame.

When Romulus complains that a low fertility rate has rendered the abduction of the Sabine women pointless, Juno, in her guise as the birth goddess Lucina, offers an instruction: " Let the sacred goat go into the Italian matrons " ( Italidas matres … sacer hirtus inito, with the verb inito a form of inire ).

Let p be the nth decimal of the nth number of the set E ; we form a number N having zero for the integral part and p + 1 for the nth decimal, if p is not equal either to 8 or 9, and unity in the contrary case.

Much of this documentation is in the form of illegal ( but widely available ) bootleg recordings, and is recounted in the book Get Back: The Unauthorized Chronicle of the Beatles ' Let It Be Disaster by Doug Sulpy and Ray Schweighardt.

Traditional Bayram ( Turkey ) | Bayram wishes from the Istanbul Metropolitan Municipality, stating " Let us love, Let us be loved ", in the form of mahya lights stretched across the minaret s of the Sultan Ahmed Mosque ( Istanbul ) | Blue Mosque in Istanbul

Let L ( x ) be the interpolation polynomial in the Lagrange form for the given data points, then

The maxim fiat justitia, ruat caelum (" Let justice be done, though the heavens fall "), used by Lord Mansfield in Somersett's Case and in reversing the outlawry of John Wilkes, and in the alternate form fiat iustitia, et pereat mundus by Ferdinand of Habsburg, is sometimes attributed to Piso, but this is disputed.

Let X be a g-dimensional torus given as X = V / L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L × L.

Let q be a quadratic form defined on an n-dimensional real vector space.

Let A be the matrix of the quadratic form q in a given basis.

Let ω be a form on a n-dimensional real vector space V, ω ∈ Λ < sup > 2 </ sup >( V ).

Let V = TCP < sup > 1 </ sup > be the bundle of complex tangent vectors having the form a ∂/∂ z at each point, where a is a complex number.