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Let L denote the upsampling factor.
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Let and L
Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.
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.
Let L be an ordered set, called a concrete set and let L ′ be another ordered set, called an abstract set.
Let L < sub > 1 </ sub >, L < sub > 2 </ sub >, L ′< sub > 1 </ sub > and L ′< sub > 2 </ sub > be ordered sets.
Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.
Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.
Let us now consider a three-dimensional cubical box that has a side length L ( see infinite square well ).
Let and denote
Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.
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 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 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.
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 < sub > x </ sub >, defined by
If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.
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 be a sequence of independent and identically distributed variables with distribution function F and let denote the maximum.
Let and upsampling
Let and factor
Let w < sub > j </ sub > be the ' price ' ( the rental ) of a certain factor j, let MP < sub > j1 </ sub > and MP < sub > j2 </ sub > be its marginal product in the production of goods 1 and 2, and let p < sub > 1 </ sub > and p < sub > 2 </ sub > be these goods ' prices.
Let us introduce the factor f < sub > j </ sub > that describes how the actual charge density differs from the average and itself on a position on the surface of the j-th conductor:
Let n be a composite integer with prime factor p. By Fermat's little theorem, we know that for all integers a coprime to p and for all positive integers K:
The initial release by the new lineup, Let It Roll, was a tremendous success and Fuller's presence proved to be a major factor.
Let C ′ be the curve obtained by shrinking C by a factor of 2 toward P. Then the point R ′ corresponding to R is the center of the rectangle PXRY, and the tangent to C ′ at R ′ bisects this rectangle parallel to PY and XR.
1.255 seconds.