Let φ be a net on X based on the directed set D and let A be a subset of X, then φ is said to be frequently in ( or cofinally in ) A if for every α in D there exists some β ≥ α, β in D, so that φ ( β ) is in A.

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

Let φ ( ξ, η, ζ ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support ( the region where the function is non-zero ) shrinks to the origin.

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Let G and H be groups, and let φ: G → H be a homomorphism.

Let a sphere have radius r, longitude φ, and latitude θ.

Let φ range from 0 to 2π, and let θ range from 0 to π / 2.

Let there be a set of differentiable fields φ defined over all space and time ; for example, the temperature T ( x, t ) would be representative of such a field, being a number defined at every place and time.

Let the action be invariant under certain transformations of the space – time coordinates x < sup > μ </ sup > and the fields φ

Let B contains all the sentences of A except ¬ φ.

Let ρ denote the number density of electrons, and φ the electric potential.

Let U be an open set in R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

Let U be a measurable subset of R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective function, and suppose for every x in U there exists < span > φ '</ span >( x ) in R < sup > n, n </ sup > such that φ ( y ) = φ ( x ) + < span > φ '</ span >( x ) ( y − x ) + o (|| y − x ||) as y → x.

Let φ: X → Y be a continuous and absolutely continuous function ( where the latter means that ρ ( φ ( E )) = 0 whenever μ ( E ) = 0 ).

* Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system is isomorphic to X < sub > m </ sub > and the canonical morphism φ < sub > m </ sub >: X < sub > m </ sub > → X is an isomorphism.

Let F and G be two sheaves on X with values in the category C. A morphism φ: G → F consists of a morphism φ ( U ): G ( U ) → F ( U ) for each open set U of X, subject to the condition that this morphism is compatible with restrictions.

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 us call the class of all such formulas R. We are faced with proving that every formula in R is either refutable or satisfiable.

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 P ( x ) be a first-order formula in the language of Presburger arithmetic with a free variable x ( and possibly other free variables ).

At this stage, the Praetor would establish a formula directing the iudex as to the remedy to be given if he found that certain circumstances were satisfied ; for instance, " Let X be iudex.

Let A be any formula which is not provable in the calculus.

Let Φ ( x ) be a formula in which x appears free.

Let ψ be a formula in the theory T with one free variable.

* M8, Unrestricted Fusion: Let φ ( x ) be a first-order formula in which x is a free variable.

: Let A and B be arbitrary formula of a formal language.

Let X be a formula with a classical bivalent truth value.

Let T denote the set of L-sentences true in N, and T * the set of Gödel numbers of the sentences in T. The following theorem answers the question: Can T * be defined by a formula of first-order arithmetic?

Tarski's undefinability theorem ( general form ): Let ( L, N ) be any interpreted formal language which includes negation and has a Gödel numbering g ( x ) such that for every L-formula A ( x ) there is a formula B such that B ↔ A ( g ( B )) holds.

It was released in 1988 and is a cover of the Beatles ' album Let It Be recorded in Laibach style with military rhythms and choirs, though a few tracks deviate from this formula, most notably " Across the Universe ".

Let A ( h ) be an approximation of A that depends on a positive step size h with an error formula of the form

Let A be a subformula of some formula B.

This is no small matter, as under English law, Admiralty decisions were official acts of the government, which could not be sued without its consent — traditionally expressed by the Attorney General responding to a petition of right with the formula " Let right be done ".

Let be the formula for the cost of people driving along edge.

Let M be a structure of signature σ and N a substructure of M. N is an elementary substructure of M if and only if for every first-order formula φ ( x, y < sub > 1 </ sub >, …, y < sub > n </ sub >) over σ and all elements b < sub > 1 </ sub >, …, b < sub > n </ sub > from N, if M x φ ( x, b < sub > 1 </ sub >, …, b < sub > n </ sub >), then there is an element a in N such that M φ ( a, b < sub > 1 </ sub >, …, b < sub > n </ sub >).

Let be the formula

The formula was becoming clear: Two radio ballads ( the title track and " Road to Zion " fitting the bill this time ), one Volz-penned praise tune (" Praise Ye the Lord " on Never Say Die, " Let Everything That Hath Breath " for More Power to Ya ), and six or seven straight-ahead progressive rock songs written by Hartman, touching on topics aimed mostly at young Christians.

An initial estimate can be found by taking the formula for mean and solving it for Let be the sample mean.

Let parthood be the defining primitive binary relation of the underlying mereology, and let the atomic formula Pxy denote that " x is part of y ".

Let wff stand for a well-formed formula ( or syntactically correct formula ) of elementary geometry.