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Page "Supervised learning" ¶ 25
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Let and denote
Let Af denote the form of Af.
Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.
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 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 < 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.
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 < sub > 2 </ sub >( 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.

Let and space
Let T be a linear operator on the finite-dimensional vector space V over the field F.
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.
* Theorem Let X be a normed space.
* Corollary Let X be a reflexive normed space and Y a Banach space.
* Corollary Let X be a reflexive normed space.
Let be a Hilbert space and is a vector in.
Let be the dual space of.
Let S be a vector space over the real numbers, or, more generally, some ordered field.
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 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 X be a topological space, and let x < sub > 0 </ sub > be a point of X.
Theorem: Let V be a topological vector space
Let be a metric space.
Let X be a measurable space, let μ be a measure on X, and let N be a measurable set in X.
These assumptions can be summarised as: Let ( Ω, F, P ) be a measure space with P ( Ω )= 1.
Let X be a locally compact Hausdorff space.

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