Let-x-y-z-be-a-system-of-Cartesian-coordinates

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Let and x

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

Let M be a smooth manifold and let x be a point in M. Let T x M be the tangent space at x.

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

Let g be a smooth function on N vanishing at f ( x ).

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Let X be a topological space, and let x 0 be a point of X.

Let x 0 , ...., x N-1 be complex numbers.

Let now x ' and y ' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol which takes as many arguments as the sum of lengths of x and y ; we consider the formula

Let and y

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 x 1 , ..., x n be the sizes of the heaps before a move, and y 1 , ..., y n the corresponding sizes after a move.

Let s = x 1 ⊕ ... ⊕ x n and t = y 1 ⊕ ... ⊕ y n .

Let ( q 1 , w, x 1 x 2 ... x m ) ( q 2 , y 1 y 2 ... y n ) be a transition of the GPDA

LET y = rnd * 20! Let the value ' y ' equal a random number between ' 0 ' and ' 20 '

Let Y = u ( X 1 , X 2 , ..., X n ) be a statistic whose pdf is g ( y ; θ ).

Let x and y be positive normalized floating point numbers.

Let y be a function given by the sum of two functions u and v, such that:

Let X be a topological space and let x and y be points in X.

Let x be a number and let y be its negative.

Let A be an abelian group, having a specific element y in A with order 2.

Let x: y: z be a variable point in trilinear coordinates, and let u

Let x = x ( u, v, w ), y = y ( u, v, w ), z = z ( u, v, w ) be defined and smooth in a domain containing, and let these equations define the mapping of into.

Let x, y, z be complex numbers, and let a, b be real numbers.

Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · ( i. e. if x and y are any two elements of A, x · y is the product of x and y ).

Let and z

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 z denote the propagation axis of the wave.

Let z be a primitive nth root of unity.

Let z be a primitive nth root of unity and let k be a positive integer.

Let A, B, C denote the vertex angles of the reference triangle, and let x: y: z be a variable point in trilinear coordinates ; then an equation for the Euler line is

Let z be the magnitude of the jump and let be the distribution of z.

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

Let V ∗ be the dual vector space of V. In other words, V ∗ is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that for any z in C, ω in V ∗ and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉.

Let x = ( x 1 , x 2 ,…, x n ) be a sample of n independent observations from a mixture of two multivariate normal distributions of dimension d, and let z =( z 1 , z 2 ,…, z n ) be the latent variables that determine the component from which the observation originates.

Let z be within that open disk.

Let C be a positively oriented ( i. e., counterclockwise ) circle centered at a, lying within that open disk but farther from a than z is.

Let γ be the boundary of B ( z 0 , r ), taken with its positive orientation.

Let there be an observer C on a disk rotating in the xy plane at a coordinate angular rate of and who is at a distance of r from the center of the disk with the center of the disk at x = y = z = 0.

Let f ( z ) be a holomorphic map from the upper half-plane to Δ extending continuously to a map between the boundaries.

Let the four-dimensional Cartesian coordinates be denoted ( w, x, y, z ) where ( x, y, z ) represent the Cartesian coordinates of the normal position vector r. The three-dimensional momentum vector p is associated with a four-dimensional vector on a three-dimensional unit sphere

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??

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