Denote the true weights by

Denote the sample of data which are independent realisations of random variables, having F as their distribution function, by x < sub > i </ sub > ( i = 1 ,..., n ).

Denote by the polynomial such that is compact.

Denote by the projection to the subspace.

Denote by the th ( partial ) computable function.

Denote by the phase difference

Denote by Ü the set of rational numbers in.

Denote, respectively, by and by the actual configuration of subsystem ( I ) and of the rest of the universe.

Denote the columns of V by ( these are the rows of the transposed matrix that appears in the decomposition )

Denote the set of rational numbers by Q.

Denote by the number of binary trees on n nodes.

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

Denote by τ < sub > tX </ sub > and τ < sub > tY </ sub >, respectively, the parallel transports along the flows of X and Y for time t. Parallel transport of a vector Z ∈ T < sub > x < sub > 0 </ sub ></ sub > M around the quadrilateral with sides tY, sX, − tY, − sX is given by

Denote this limit by y.

Denote by the orthogonal projection onto.

Denote the row vector of these components by

Denote the column vector of components of v by v:

Denote the row vector of components of α by α:

Denote by dY the differential of such a map.

Denote by L the length of C and by A the area enclosed by C. The spherical isoperimetric inequality states that

Denote the DFT of the Even-indexed inputs by and the DFT of the Odd-indexed inputs by and we obtain:

Denote the coloring function as f: S →

Denote F < sub > n </ sub > the cdf of S < sub > n </ sub >, and Φ the cdf of the standard normal distribution.

A modern way of approaching this problem is to consider a particular type of maximal function, which we construct as follows: Denote S < sup > n-1 </ sup > ⊂ R < sup > n </ sup > to be the unit sphere in n-dimensional space.

::< tt > Denote < sub > S </ sub > ≡ ⊔< sub > i ∈ ω </ sub > progression < sub > S </ sub >< sup > i </ sup >(⊥< sub > S </ sub >)</ tt >

::< tt > Denote < sub > S </ sub > ≡ ⊔< sub > i ∈ ω </ sub > progression < sub > S </ sub >< sup > i </ sup >(⊥< sub > S </ sub >)</ tt >

Although < tt > Denote < sub > S </ sub ></ tt > is not an implementation of < tt > S </ tt >, it can be used to prove a generalization of the Church-Turing-Rosser-Kleene thesis 1943:

::< tt > Denote < sub > S </ sub > ≡ ⊔< sub > i ∈ ω </ sub > progression < sub > S </ sub >< sup > i </ sup >(⊥< sub > S </ sub >)</ tt >

Denote by P ( A ) the power set of A, i. e., the set of all subsets of A.

Denote by K < sup > N </ sup > the set of all sequences of scalars

Denote the equivalence class of ( p, f ) by.

Denote by B ( p, r ) the ball of radius r around a point p, defined with respect to the Riemannian distance function.

Denote as Z the sum of the squared differences of the data points from the centroid ( also denoted in complex coordinates ), which is the point whose horizontal and vertical locations are the averages of those of the data points.

Now suppose ƒ is a bijective function from Z ( the positive integers ) to X. Denote the points of the image of Z under ƒ as

Denote by λ < sub > n </ sub > the Dirichlet eigenvalues for D: that is, the eigenvalues of the Dirichlet problem for the Laplacian:

b of the interval b we obtain a finite collection of points f ( t < sub > 0 </ sub >), f ( t < sub > 1 </ sub >), ..., f ( t < sub > n − 1 </ sub >), f ( t < sub > n </ sub >) on the curve C. Denote the distance from f ( t < sub > i </ sub >) to f ( t < sub > i + 1 </ sub >) by d ( f ( t < sub > i </ sub >), f ( t < sub > i + 1 </ sub >)), which is the length of the line segment connecting the two points.

Denote by A the coefficient ring Z or Z < sub > 2 </ sub >, depending on whether or not M is orientable.

Denote the closure of by It follows easily that is the restriction of to

Denote the area that casually affects point as.

Denote the vertices of a triangle as A, B, and C and the orthocenter as H, and let D, E, and F denote the feet of the altitudes from A, B, and C respectively.

Denote the orthocenter of triangle ABC as H, denote the sidelengths as a, b, and c, and denote the circumradius of the triangle as R. Then

Denote this probability.

Denote the three vertices of one triangle by a, b, and c, and those of the other by A, B, and C. Axial perspectivity means that lines ab and AB meet in a point, lines ac and AC meet in a second point, and lines bc and BC meet in a third point, and that these three points all lie on a common line called the axis of perspectivity.

Let E → M be a smooth vector bundle over a differentiable manifold M. Denote the space of smooth sections of E by Γ ( E ).