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Let and <

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Let denote the Bézier curve determined by the points P 0 , P 1 , ..., P n .

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

Let M be a smooth manifold and let x be a point in M. Let T x M be the tangent space at x.

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

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

Let e be the error in b. Assuming that A is a square matrix, the error in the solution A < sup >− 1 </ sup > b is A < sup >− 1 </ sup > e.

Let us for simplicity take, then < math > 0 < c =- 2a </ math > and.

Let and sup

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab − 1 ∈ H ).

Let r be a non zero real number and let the r th power mean ( M r ) of a series of real variables ( a 1 , a 2 , a 3 , ... ) be defined as

Let t and s ( t > s ) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s 2 equals half the harmonic mean of c 2 and t 2 .

Let and 0

" Let X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).

Let X be a topological space, and let x 0 be a point of X.

Let x 0 , ...., x N-1 be complex numbers.

Let A ( k ) be its Fourier transform at time 0:

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

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,

If V is a real vector space, then we replace V by its complexification V ⊗ R C and let g denote the induced bilinear form on V ⊗ R C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g | W = 0.

LET x = rnd * 20! Let the value ' x ' equal a random number between ' 0 ' and ' 20 '

LET y = rnd * 20! Let the value ' y ' equal a random number between ' 0 ' and ' 20 '

:: Let n = 0

Let V and W be vector spaces ( or more generally modules ) and let T be a linear map from V to W. If 0 W is the zero vector of W, then the kernel of T is the preimage of the zero subspace

* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I n be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I 0 , I 1 , I 2 , etc., is an ascending chain that does not terminate.

Let Then is called not included in the fuzzy set if is called fully included if and is called a fuzzy member if < math > 0 < m ( x ) < 1 .</ math >

Let f be the function which maps database entries to 0 or 1, where f ( ω )= 1 if and only if ω satisfies the search criterion.

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

Let x t be a curve in a Riemannian manifold M. Denote by τ x t : T x 0 M → T x t M the parallel transport map along x t .

Let Δx tend to 0:

Let F 0 be the empty set.

Let ( M, d ) be a metric space, namely a set M with a metric ( distance function ) d. The open ( metric ) ball of radius r > 0 centered at a point p in M, usually denoted by B r ( p ) or B ( p ; r ), is defined by

* Let X be a random variable that takes the value 0 with probability 1 / 2, and takes the value 1 with probability 1 / 2.

Let and </

Genesis 1: 9 " And God said, Let the waters be collected ". Letters in black, niqqud in red , cantillation in blue

* Let D 1 and D 2 be directed sets.

Let and denote

Let Af denote the form of Af.

Let X denote a Cauchy distributed random variable.

Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude

Let denote the equivalence class to which a belongs.

Let '~' denote an equivalence relation over some nonempty set A, called the universe or underlying set.

Let G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Let R denote the field of real numbers.

Let n denote a complete set of ( discrete ) quantum numbers for specifying single-particle states ( for example, for the particle in a box problem we can take n to be the quantized wave vector of the wavefunction.

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 denote the space of scoring functions.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Let us denote the time at which it is decided that the compromise occurred as T.

Let denote the sequence of convergents to the continued fraction for.

Let us denote the mutually orthogonal single-particle states by and so on.

That is, Alice has one half, a, and Bob has the other half, b. Let c denote the qubit Alice wishes to transmit to Bob.

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 Q ( x ) denote the number of square-free ( quadratfrei ) integers between 1 and x.

A possible definition of spoiling based on vote splitting is as follows: Let W denote the candidate who wins the election, and let X and S denote two other candidates.

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

Let be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.

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