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 < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

Let P < sub > F </ sub > be the domain of a prefix-free universal computable function F. The constant Ω < sub > F </ sub > 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 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 < sup >− 1 </ sup > 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 < sub > Alice </ sub > denote the first plaintext block of Alice's message, let E denote encryption, and let P < sub > Eve </ sub > be Eve's guess for the first plaintext block.

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 a piece of glass and a piece of resin, neither of which exhibiting any electrical properties, be rubbed together and left with the rubbed surfaces in contact.

So Nasreddin said Let the half who know what I am going to say, tell it to the half who don't, and left.

Let M and N be ( left or right ) modules over the same ring, and let f: M → N be a module homomorphism.

Following several critically acclaimed albums, including Let It Be and Tim, Bob Stinson left the band in 1986 and the band experienced several line-up changes ; Slim Dunlap joined as lead guitarist and Steve Foley replaced Chris Mars in 1990.

** Let n be the number of greater disks that are located on the same peg as their first greater disk and add 1 if the largest disk is on the left peg.

| style =" text-align: left ;" rowspan =" 2 "| Who Let the Dogs Out?

In November 1983, Mann left after two sell-out gigs at the Marquee, which were recorded for a live album, Live and Let Live.

With their third consecutive single (" Let Love In ") from the album, the Goo Goo Dolls hit a record 12 Top 10 hits in Adult Top 40 history, beating Matchbox Twenty and Sheryl Crow until Matchbox Twenty's release of Exile on Mainstream and the Goo Goo Dolls ' release of " Before It's Too Late " from the Transformers Soundtrack, which left both groups with 13 Top 10 hits in the Adult Top 40.

Let us very roughly consider the line between the " leftmost " and " rightmost " two points of the k selected points ( for some arbitrary left / right axis: we can choose top and bottom for the exceptional vertical case ).

U Nu, after his release from prison in October 1966, had left Burma in April 1969, and formed the Parliamentary Democracy Party ( PDP ) the following August in Bangkok, Thailand with the former Thirty Comrades, Bo Let Ya, co-founder of the CPB and former Minister of Defence and deputy prime minister, Bo Yan Naing, and U Thwin, ex-BIA and former Minister of Trade.

Let R be a ring, and let M be a left R-module.

Let π: E → X be a fibre bundle over a topological space X with structure group G and typical fibre F. By definition, there is a left action of G ( as a transformation group ) on the fibre F. Suppose furthermore that this action is effective.

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 D be a division ring and M ( n, D ) be the ring of matrices with entries in D. It is not hard to show that every left ideal in M ( n, D ) takes the following form:

Let M be the total mass, which is constant, so the left side can be multiplied and divided by M, so

* Let A ( G ) be the coordinate ring of G. This is a representation of G with G acting by left translations.

The songs that the band felt were not strong enough to fit on Trespass, such as " Everywhere is Here ," " Grandma ," " Little Leaf ," " Going out to Get You ," " Shepherd ," " Moss ," " Let Us Now Make Love ," and " Pacidy ," were discarded by the time lead guitarist Anthony Phillips left the band.

Writing in the American Jewish Congress ' Congress Bi-Weekly in 1973, the Foreign Minister of Israel, Abba Eban, identified anti-Zionism as ‘ the new anti-Semitism ’, saying: ecently we have witnessed the rise of the new left which identifies Israel with the establishment, with acquisition, with smug satisfaction, with, in fact, all the basic enemies … Let there be no mistake: the new left is the author and the progenitor of the new anti-Semitism.

Leading on a regiment of the left wing to the attack with its colour in his hand, he shouted " Let all brave Prussians follow me!

Let M be a ( left or right ) module over some ring R. Given a chain of submodules of M of the form

JoJo left a message that she wanted her fans ' input in deciding between " Coming for You " or " Let It Rain " as the next single, but it was canceled to release a third single for The High Road.