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 < sub > x </ sub > M be the tangent space at x.

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

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

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

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 < sub > 0 </ sub > be a point of X.

Let x < sub > 0 </ sub >, ...., x < sub > N-1 </ sub > 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 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 the event space represent the current state of belief for this process.

In May 1966 they filmed two sets of colour promotional clips for their current single " Rain "/" Paperback Writer " all directed by Michael Lindsay-Hogg, who went on to direct The Rolling Stones Rock and Roll Circus and The Beatles final film Let It Be.

This association is also reflected in the College motto " Where Finbarr Taught, Let Munster Learn " which is also the current university motto. National University of Ireland

Unlike more general concepts of autology and self-reference, this particular distinction and opposition of " autological " and " heterological words " is uncommon in linguistics for describing linguistic phenomena or classes of words, but is current in logic and philosophy where it was introduced by Kurt Grelling and Leonard Nelson for describing a semantic paradox, later known as Grelling's paradox or the Grelling-Nelson paradox .< ref > Grelling and Nelson used the following definition when first publishing their paradox in 1908: " Let < big > φ </ big >( M ) be the word that denotes the concept defining M. This word is either an element of M or not.

The current slogan for Starburst in the U. S. is " It's A Juicy Contradiction " and in the UK and Australia it is " Tastes Like Summer / Let Your Starburst Out.

Let e < sub > k </ sub > be the voltage generators and a < sub > m </ sub > the current generators.

Let R < sub > m </ sub > be the resistances on the branches with current generators.

The introduction, " Let us pray for the whole state of Christ's Church militant here in earth " remained unaltered and only a thanksgiving for those " departed this life in thy faith and fear " was inserted to introduce the petition that the congregation might be " given grace so to follow their good examples that with them we may be partakers of thy heavenly kingdom ".

The state motto is Esto Perpetua ( Latin for " Let it be forever ").

Let ε ( n ) denote the energy of a particle in state n. As the particles do not interact, the total energy of the system is the sum of the single-particle energies.

Let the state of the composite system be

* Let be concentration of active centers in the ground state.

Let A be an observable of the system, and suppose the ensemble is in a mixed state such that each of the pure states occurs with probability p < sub > j </ sub >.

Let it once be understood that we mean to stop the horrible state of things in Cuba and it will be stopped.

In reference to Lucas, his three Ohio commissioners and their guard, Mason wrote, " Let him get on our soil, arrest him, strike the blood at once, disgrace him and his state, and end the controversy.

When a cobbler commented on his mistakes in painting a shoe, Apelles made the corrections that very night ; the next morning the cobbler noticed the changes, and proud of his effect on the artist's work began to criticize how Apelles portrayed the leg — whereupon Apelles emerged from his hiding-place to state: Ne sutor ultra crepidam —" Let the shoemaker venture no further.

He argued against those, such as Wilberforce, who held that Britain could safely maintain a policy of isolation: " Let us consider the state of the world as it is, not as we fancy it ought to be.

Let us now determine the physical state of the partial system A as completely as possible by measurements.

Let denote the quantum state of the perturbed system at time t. It obeys the time-dependent Schrödinger equation,

(" Let me state at once, in clearest unmistakable terms, that I have never known of, or seen, or heard, a greater actor than was Irving.

Let the opinions impugned be the belief of God and in a future state, or any of the commonly received doctrines of morality ...

On May 17, 2006, the state of Illinois enacted Senate Bill 1144, the " Let Them Rest In Peace Act ", to shield grieving military families from protests during funerals and memorial services of fallen military service members.

Let Z < sub > n </ sub > denote the state in period n ( often interpreted as the size of generation n ), and let X < sub > n, i </ sub > be a random variable denoting the number of direct successors of member i in period n, where X < sub > n, i </ sub > are iid over all n ∈

The Maoists announced a ‘ People's War ’ on February 13, 1996, under the slogan: " Let us march ahead on the path of struggle towards establishing the people's rule by wreaking the reactionary ruling system of state.

Translated, it means " Let it be perpetual " or " It is forever "; appropriate for a time travel film, it is also the motto of the state of Idaho.

Let the unperturbed atom or molecule be in a g-fold degenerate state with orthonormal zeroth-order state functions.

Let the unmarked state be a synonym for the void.