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Let and ρ

Let ρ be the initial topology on X induced by C τ ( X ) or, equivalently, the topology generated by the basis of cozero sets in ( X, τ ).

Let ρ be the correlation coefficient between X 1 and X 2 and let σ i 2 be the variance of X i .

Let ρ denote the number density of electrons, and φ the electric potential.

Let X be a locally compact Hausdorff space equipped with a finite Radon measure μ, and let Y be a σ-compact Hausdorff space with a σ-finite Radon measure ρ.

Let φ: X → Y be a continuous and absolutely continuous function ( where the latter means that ρ ( φ ( E )) = 0 whenever μ ( E ) = 0 ).

* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

Lemma: Let A ∈ C n × n be a complex-valued matrix, ρ ( A ) its spectral radius and ||·|| a consistent matrix norm ; then, for each k ∈ N:

Theorem: Let A ∈ C n × n be a complex-valued matrix and ρ ( A ) its spectral radius ; then

Let π: P → X be a principal G-bundle and let ρ: G → Homeo ( F ) be a continuous left action of G on a space F ( in the smooth category, we should have a smooth action on a smooth manifold ).

Let this property be represented by just one scalar variable, q, and let the volume density of this property ( the amount of q per unit volume V ) be ρ, and the all surfaces be denoted by S. Mathematically, ρ is a ratio of two infinitesimal quantities:

Let ρ: TP / G → M be the projection onto M. The fibres of the bundle TP / G under the projection ρ carry an additive structure.

Let V be a finite-dimensional vector space over a field F and let ρ: G → GL ( V ) be a representation of a group G on V. The character of ρ is the function χ ρ : G → F given by

Let ρ and σ be representations of G. Then the following identities hold:

* Let ρ: G → GL ( n, C ) be a representation.

Let ρ be an irreducible representation of a finite group G on a vector space V of ( finite ) dimension n with character χ.

Let A be an irreducible non-negative n × n matrix with period h and spectral radius ρ ( A ) = r. Then the following statements hold.

Let and θ

Let X be a random variable with a discrete probability distribution p depending on a parameter θ.

Let X be a random variable with a continuous probability distribution with density function f depending on a parameter θ.

Let θ denote the angle between and, where is the vector along the waveplate's fast axis.

Let a sphere have radius r, longitude φ, and latitude θ.

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

Let, denote a random sample from a distribution having the pdf f ( x, θ ) for γ < θ < δ.

Let Y = u ( X 1 , X 2 , ..., X n ) be a statistic whose pdf is g ( y ; θ ).

Let ℓ ( e ) be the length of the edge e and θ ( e ) be the dihedral angle between the two faces meeting at e, measured in radians.

Let θ be the conventional polar angle describing a planetary orbit.

Let R be the radius of the circle, θ is the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the height of the segment, and d the height of the triangular portion.

Let P = ( r, θ ) be a point on a given curve defined by polar coordinates and let O denote the origin.

Let a rotation about the origin O by an angle θ be denoted as Rot ( θ ).

Let a reflection about a line L through the origin which makes an angle θ with the x-axis be denoted as Ref ( θ ).

Let x be the distance from the center of the needle to the closest line, let θ be the acute angle between the needle and the lines.

Let x and θ be as in the illustration in this section.

Let R =( r, θ ) be a point on the curve and let X =( p, α ) be the corresponding point on the pedal curve.

Let E i be a basis of sections of TG consisting of left-invariant vector fields, and θ j be the dual basis of sections of T * G such that θ j ( E i ) = δ i j , the Kronecker delta.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear, that is

Let θ be a vector consisting of n ≥ 3 unknown parameters.

Let and φ

Let φ be a formula of degree k + 1 ; then we can write it as

Let φ be a net on X based on the directed set D and let A be a subset of X, then φ is said to be frequently in ( or cofinally in ) A if for every α in D there exists some β ≥ α, β in D, so that φ ( β ) is in A.

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 φ ( ξ, η, ζ ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support ( the region where the function is non-zero ) shrinks to the origin.

Let G and H be groups, and let φ: G → H be a homomorphism.

Let there be a set of differentiable fields φ defined over all space and time ; for example, the temperature T ( x, t ) would be representative of such a field, being a number defined at every place and time.

Let the action be invariant under certain transformations of the space – time coordinates x μ and the fields φ

Let B contains all the sentences of A except ¬ φ.

Let U be an open set in R n and φ: U → R n an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

Let U be a measurable subset of R n and φ: U → R n an injective function, and suppose for every x in U there exists φ '( x ) in R n, n such that φ ( y ) = φ ( x ) + φ '( x ) ( y − x ) + o (|| y − x ||) as y → x.

* Let I be any directed set with a greatest element m. The direct limit of any corresponding direct system is isomorphic to X m and the canonical morphism φ m : X m → X is an isomorphism.

Let F and G be two sheaves on X with values in the category C. A morphism φ: G → F consists of a morphism φ ( U ): G ( U ) → F ( U ) for each open set U of X, subject to the condition that this morphism is compatible with restrictions.

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