Let us look in on one of these nerve centers -- SAC at Omaha -- and see what must still happen before a wing of B-52 bombers could drop their Aj.

Let us re-examine the publicized contrasts between Ptolemaic and Copernican astronomy.

Let us not confuse the issue by labeling the objective or the method `` psychoanalytic '', for this is a well established term of art for the specific ideas and procedures initiated by Sigmund Freud and his followers for the study and treatment of disordered personalities.

Let us differentiate a few of these ideas.

Let us quote once more from R. G. Collingwood: `` History is properly concerned with the actions of human beings Regarded from the outside, an action is an event or series of events occurring in the physical world ; ;

Let us survey for a moment the development of modern thought -- turning our attention from the Reformation toward the revolutionary and romantic movements that follow and dwelling finally on more recent decades.

Let us now give some thought to the soul.

Let us see just how typical Krim is.

Let us prepare for peace, instead of for a war which would mean the end of civilization.

Let us have more benches and fewer forbidden areas around fountains and gardens.

Let us, like the French, have outdoor cafes where we may relax, converse at leisure and enjoy the passing crowd.

Let us suppose the Russians decide to build a rail-mobile ICBM force.

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 us not try to key them out at this stage of the game, and let us just call them Bombus.

Let us speculate a little on the maximum size of the anaconda.

Let us put Af.

Let us look at the operator Af.

Let us now regard D as a linear operator on the subspace V.

Let us now put some flesh on the theoretical bones we have assembled by giving illustrations of roleplaying used for evaluation and analysis.

Let us assume that Af is identical to the form of an occurrence Af which preceded Af in the text.

Let us also assume the existence of only one class or type of service, all of which is supplied at the same voltage, phase, etc. to residential, commercial, and industrial customers.

Let us therefore consider each of the three types of cost in turn, recognizing that this simplified classification is used only for illustrative purposes ; ;

Let us work through an example, and the simpler and commoner the better.

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 be the product of every modulus then define

* Let be the set of ordered pairs of integers with not zero, and define an equivalence relation on according to which if and only if.

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

Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define as the measurement time.

Let us define the aspiration level A and assume that.

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 q be a prime number, s a complex variable, and define a Dirichlet L-function as

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

Let X be locally ringed space with structure sheaf O < sub > X </ sub >; we want to define the tangent space T < sub > x </ sub > at the point x ∈ X.

Let M be a differentiable manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p.

Let Y be any scalar random variable, and define a time-series

Let V be a vector space over a field K. For any nonnegative integer k, we define the k < sup > th </ sup > tensor power of V to be the tensor product of V with itself k times:

Let a ∈ C, and define

Let F: X × 1 → Y be a homotopy that takes f to g. On the level of chains, define a homomorphism

Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector ; in other words all the vectors in N get mapped into the equivalence class of the zero vector.

Let T be a linear mapping from the space of μ < sub > 1 </ sub >- integrable functions into the space of μ < sub > 2 </ sub >- measurable functions, and for 1 ≤ p, q ≤ ∞, define to be the operator norm of a continuous extension of T to

* Let C denote the vector space of all continuous real-valued functions on the interval, and define L: C → R by the rule

Let X be a scheme over a field k of characteristic p. Choose an open affine subset U = Spec R. Since X is a k-scheme, we get an inclusion of k in R. This forces R to be a characteristic p ring, so we can define the Frobenius endomorphism F for R as we did above.

Let us formally define the problem.

Let us define the covariance matrix of the noise, reminding ourselves that this matrix has Hermitian symmetry, a property that will become useful in the derivation:

Let the function x ( t ) define the path followed by a point.

Let us define a price quasi-equilibrium with transfers as an allocation, a price vector p, and a vector of wealth levels w ( achieved by lump-sum transfers ) with ( where is the aggregate endowment of goods and is the production of firm j ) such that:

Let p be a point of M. Consider the space consisting of smooth maps defined in some neighborhood of p. We define an equivalence relation on as follows.

Let us define the heat equation in two spatial dimensions as follows

Let J ( R ) be the Jacobson radical of R. If U is a right module over a ring, R, and I is an right ideal in R, then define U · I to be the set of all ( finite ) sums of elements of the form u · i, where · is simply the action of R on U. Necessarily, U · I is a submodule of U.