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Let and P

Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.

Let denote the Bézier curve determined by the points P 0 , P 1 , ..., P n .

Let P F be the domain of a prefix-free universal computable function F. The constant Ω F is then defined as

Let us define a linear operator P, called the exchange operator.

Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is, so when it bounces off, its angle of inclination must be equal to.

These assumptions can be summarised as: Let ( Ω, F, P ) be a measure space with P ( Ω )= 1.

# Let P ( x ) be a first-order formula in the language of Presburger arithmetic with a free variable x ( and possibly other free variables ).

Let P be the following property of partial functions F of one argument: P ( F ) means that F is defined for the argument ' 1 '.

Let P ( x )

Let us assume the bias is V and the barrier width is W. This probability, P, that an electron at z = 0 ( left edge of barrier ) can be found at z = W ( right edge of barrier ) is proportional to the wave function squared,

Let P ( α ) be a property defined for all ordinals α.

Let P be Q's left child.

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Let P − 1 DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues.

Let be the columns of P, each multiplied by the ( real ) square root of the corresponding eigenvalue.

Let the total power radiated from a point source, for example, an omnidirectional isotropic antenna, be P. At large distances from the source ( compared to the size of the source ), this power is distributed over larger and larger spherical surfaces as the distance from the source increases.

Let f be any function from S to P ( S ).

Let K be R, C, or any field, and let V be the set P of all polynomials with coefficients taken from the field K.

Let further P Alice denote the first plaintext block of Alice's message, let E denote encryption, and let P Eve be Eve's guess for the first plaintext block.

Let and be

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.

`` Let him be now ''!!

Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

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 N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

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 take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

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 it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??

Let and principal

Let () be the directions of the three principal strains.

Let P → M be a principal bundle over a manifold M with structure Lie group G and a principal connection ω.

Let π: P → X be a principal G-bundle.

Let G be a topological group, and for a topological space X, write b G ( X ) for the set of isomorphism classes of principal G-bundles.

Let E → M be a vector bundle of rank k and let F ( E ) be the principal frame bundle of E. Then a ( principal ) connection on F ( E ) induces a connection on E. First note that sections of E are in one-to-one correspondence with right-equivariant maps F ( E ) → R k .

Let π: P → X be a principal G-bundle and let ρ: G → Homeo ( F ) be a continuous left action of G on a space F ( in the smooth category, we should have a smooth action on a smooth manifold ).

Let P be the underlying principal homogeneous space of G. A Klein geometry is the homogeneous space given by the quotient P / H of P by the right action of H. There is a right H-action on the fibres of the canonical projection

Let P be a principal H bundle over M. Then a Cartan connection is a-valued 1-form η on P such that

Let V a representation of G, and form the vector bundle V = Q × G V over M. Then the principal G-connection α on Q induces a covariant derivative on V, which is a first order linear differential operator

Let π: P → M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

Let G be a Lie group with Lie algebra, and P → B be a principal G-bundle.

Let a be a root of P, and Q a, t the product of P by the principal part of the Laurent series of f at a.

Let P → M be a principal G-bundle on a smooth manifold M. If ϕ is a tensorial k-form on P, then its exterior covariant derivative is defined by

Let A be a nonzero m × n matrix over a principal ideal domain R. There exist invertible and-matrices S, T so that the product S A T is

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