Let p be an odd prime number.

Let w < sub > j </ sub > be the ' price ' ( the rental ) of a certain factor j, let MP < sub > j1 </ sub > and MP < sub > j2 </ sub > be its marginal product in the production of goods 1 and 2, and let p < sub > 1 </ sub > and p < sub > 2 </ sub > be these goods ' prices.

Let the directrix be the line x = − p and let the focus be the point ( p, 0 ).

Let p, q > 2 be two distinct prime numbers.

Let p be an odd prime.

Let π < sub > 2 </ sub >( x ) denote the number of primes p ≤ x such that p + 2 is also prime.

Let X be a random variable with a discrete probability distribution p depending on a parameter θ.

Sylows ' test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

# Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) and q = ( q < sub > 1 </ sub >, q < sub > 2 </ sub >) be elements of W, that is, points in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub > and q < sub > 1 </ sub > = q < sub > 2 </ sub >.

# Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) be an element of W, that is, a point in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub >, and let c be a scalar in R. Then cp = ( cp < sub > 1 </ sub >, cp < sub > 2 </ sub >); since p < sub > 1 </ sub > = p < sub > 2 </ sub >, then cp < sub > 1 </ sub > = cp < sub > 2 </ sub >, so cp is an element of W.

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 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 us suppose now that there is at least a positive solution, and let be the minimal one.

Subsequent singles including " Banks of the Ohio " ( No. 94 Pop, No. 34 AC ) and remakes of George Harrison's " What Is Life " ( No. 34 AC ) and John Denver's " Take Me Home, Country Roads " ( No. 119 Pop ) made minimal chart impact until the release of " Let Me Be There " in 1973.

A more formal way of expressing this is: Let j and k be elements of some finite set K. F is a minimal perfect hash function iff F ( j )

* is minimal: Let be any transitive relation containing, we want to show that.

Let 0 → G < sub > n </ sub > → … → G < sub > 0 </ sub > → 0 denote a finite complex of free R-modules such that H < sub > i </ sub >( G < sub >•</ sub >) has finite length for i > 0 and H < sub > 0 </ sub >( G < sub >•</ sub >) has a minimal generator that is killed by a power of the maximal ideal of R. Then dim R ≤ n.

Let E be an elliptic curve with integer coefficients in a Neron minimal model.

# Let G be a connected graph and let H be the clutter on consisting of all edge sets of spanning trees of G. Then is the collection of all minimal edge cuts in G.

Let S be an ( a, b )- separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal ( a, b )- separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair ( a, b ) of nonadjacent vertices.

The minimal separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ( a, b )- separator S can be regarded as a predecessor of another ( a, b )- separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ( a, b )- separators in ' G '.

* Let R be a local ring and M a finitely generated module over R. Then the projective dimension of M over R is equal to the length of every minimal free resolution of M. Moreover, the projective dimension is equal to the global dimension of M, which is by definition the smallest integer i ≥ 0 such that

Let X be a finite subset of U, minimal with respect to the property that it generates U. Since U is non-zero, this set X is nonempty.

Let us refer to such a regularization as the minimal realistic regularization, and start searching for the corresponding, modified free-field parts of the QED Lagrangian density.

" Let the Wind Carry Me " contrasts thoughts of a more stable, conventional life with the overpowering need to live with minimal constraints upon one's freedom.