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Let and V
Let T be a linear operator on the finite-dimensional vector space V over the field F.
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 us now regard D as a linear operator on the subspace V.
Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function
Theorem: Let V be a topological vector space
Let V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:
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,
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.
The singles " So Far Away " and " Price to Play " came with two unreleased tracks, " Novocaine " and " Let It Out ", which were released for the special edition of the group's Chapter V, which came out in late 2005.
In early November 2005, Staind released the limited edition 2-CD / DVD set of Chapter V. The set included several rarities and fan favorites — music videos ; a complete, 36-page booklet with exclusive artwork ; an audio disc with an acoustic rendition of " This is Beetle "; the original, melodic rendition of " Reply "; the previously released B-side singles " Novocaine " and " Let It Out "; and live versions of " It's Been Awhile " and " Falling ", among many others.
Let u, v be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers.
Let V and W be vector spaces ( or more generally modules ) and let T be a linear map from V to W. If 0 < sub > W </ sub > is the zero vector of W, then the kernel of T is the preimage of the zero subspace
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 U and V be two open sets in R < sup > n </ sup >.

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 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 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 Hamiltonian
Next, take a Hamiltonian invariant under T. Let | a > and T | a > be two quantum states of the same energy.
Let H ( 0 ) be the Hamiltonian completely restricted either in the low-energy subspace or in the high-energy subspace, such that there is no matrix element in H ( 0 ) connecting the low-and the high-energy subspaces, i. e. if.
Let us consider a square array of classical spins which may only take two positions: + 1 and − 1, at a certain temperature, interacting through the Ising classical Hamiltonian:
Let be an observable of a dynamical system with Hamiltonian subject to thermal fluctuations.
Let H < sup > i </ sup > be the Hamiltonian of the i-th such system.
Let denote a representative point in the phase space, and be its image under the Hamiltonian flow at time t. The time average of f is defined to be
Let denote the time-independent Hamiltonian, and let and denote the two energy eigenstates of the system, with respective eigenvalues and.

Let and representing
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 us take a hypothetical single scan line, with B representing a black pixel and W representing white:
According to some translations, the segment CE, representing the intelligible world, is divided into the same ratio as AC, giving the subdivisions CD and DE ( it can be readily verified that CD must have the same length as BC < ref > Let the length of AE be equal to and that of AC equal to, where < math >
Let us think of each such string as representing a necklace.
* In the film Live and Let Die, the Tarot card representing Solitaire is the High Priestess.
Let T be a first-order theory in the language of arithmetic and capable of representing all computable functions.
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 r ( t ) be a curve in Euclidean space, representing the position vector of the particle as a function of time.
* Let and be sets representing the non-zero patterns of columns i and j ( below the diagonal only, and including diagonal elements ) of matrices and respectively.
The band won by a significant margin, scoring 292 points, 44 points over second place Dima Bilan ( who later won in 2008 ), representing Russia with the song " Never Let You Go ".
An expression of this concept can be seen a bronze statue in the United Nations garden called Let Us Beat Swords into Plowshares, a gift from the Soviet Union sculpted by Evgeniy Vuchetich, representing the figure of a man hammering a sword into the shape of a plowshare.
In 1998, while performing “ Let a Boy Cry ” on Taratata, she met Prince ’ s manager Steve Fargnoli who began representing her.

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