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Let and ƒ

Let ( M, g ) be a Riemannian manifold and ƒ: M m → R n a short C ∞- embedding ( or immersion ) into Euclidean space R n , where n ≥ m + 1.

Let ƒ and g be functions.

Let ƒ and g be two functions satisfying the above hypothesis that ƒ is continuous on I and is continuous on the closed interval.

Let ƒ ( x ) be any rational function over the real numbers.

Let k be an algebraically closed field and let A n be an affine n-space over k. The polynomials ƒ in the ring k ..., x n can be viewed as k-valued functions on A n by evaluating ƒ at the points in A n , i. e. by choosing values in k for each x i .

Let k be the field of complex numbers C. Let A 2 be a two dimensional affine space over C. The polynomials f in the ring Cy can be viewed as complex valued functions on A 2 by evaluating ƒ at the points in A 2 .

Let ƒ be measurable, E (| ƒ |) < ∞, and T be a measure-preserving map.

Let ƒ be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

Let M and N be differentiable manifolds and ƒ: M → N be a differentiable map between them.

where stands for all partial derivatives of ƒ up to order k. Let us exchange every variable in for new independent variables

Let Ω be a connected open set in the complex plane C and ƒ: Ω → C a holomorphic function.

Let γ be a simple rectifiable loop in X around P. The ramification index of ƒ at P is

Let ƒ: X → Y be a morphism of algebraic curves.

Let Q = ƒ ( P ) and let t be a local uniformizing parameter at P ; that is, t is a regular function defined in a neighborhood of Q with t ( Q ) = 0 whose differential is nonzero.

Let ƒ ( x ) = ƒ ( x, y ) be a continuous function vanishing outside some large disc in the Euclidean plane R 2 .

Let K be the sheaf of meromorphic functions and O the sheaf of holomorphic functions on M. A global section ƒ of K passes to a global section φ ( ƒ ) of the quotient sheaf K / O.

Let V and W be handlebodies of genus g, and let ƒ be an orientation reversing homeomorphism from the boundary of V to the boundary of W. By gluing V to W along ƒ we obtain the compact oriented 3-manifold

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??

Let and function

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

Let the function g ( t ) be the altitude of the car at time t, and let the function f ( h ) be the temperature h kilometers above sea level.

Let P F be the domain of a prefix-free universal computable function F. The constant Ω F is then defined as

Let F be a prefix-free universal computable function.

Let M be a smooth manifold and let f ∈ C ∞( M ) be a smooth function.

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

Let r ( t ) be a vector that describes the position of a point mass as a function of time.

Let f be a real valued function.

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

: Theorem on projections: Let the function f: X → B be such that a ~ b → f ( a )

Let function composition interpret group multiplication, and function inverse interpret group inverse.

Let function composition, notated by infix, interpret the group operation.

Let R be a domain and f a Euclidean function on R. Then:

Let be the conditional probability distribution function of Y given X.

Let F be the continuous cumulative distribution function which is to be the null hypothesis.

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 π ( x ) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x.

Let be the Riemann zeta function.

Let be a non-negative real-valued function of the interval, and let < math > S =

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 x , defined by

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 be the space of real-valued continuous functions on X which vanish at infinity ; that is, a continuous function f is in if, for every, there exists a compact set such that on

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