If the ratio a '/ a be sufficiently constant, as is often the case, the above relation reduces to the condition of Airy, i. e. tan w '/ tan w = a constant.

If a dachshund is brindled on a dark coat and has tan points, it will have brindling on the tan points only.

If tan ( φ ) tan ( δ ) < − 1, the sun does not rise and.

If Θ < sub > max </ sub > is the maximum angle that the converging waves make with the lens axis, r is radial distance in the image plane, and wavenumber k = 2π / λ where λ = wavelength, then the argument of the Airy function is: kr tan ( Θ < sub > max </ sub >).

If φ is a scalar valued function and F is a vector field, then

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

If f: A < sub > 1 </ sub > → A < sub > 2 </ sub > and g: B < sub > 1 </ sub > → B < sub > 2 </ sub > are morphisms in Ab, then the group homomorphism Hom ( f, g ): Hom ( A < sub > 2 </ sub >, B < sub > 1 </ sub >) → Hom ( A < sub > 1 </ sub >, B < sub > 2 </ sub >) is given by φ g o φ o f. See Hom functor.

If f: X < sub > 1 </ sub > → X < sub > 2 </ sub > and g: Y < sub > 1 </ sub > → Y < sub > 2 </ sub > are morphisms in C, then the group homomorphism Hom ( f, g ): Hom ( X < sub > 2 </ sub >, Y < sub > 1 </ sub >) → Hom ( X < sub > 1 </ sub >, Y < sub > 2 </ sub >) is given by φ g o φ o f.

If K is a subset of ker ( f ) then there exists a unique homomorphism h: G / K → H such that f = h φ.

If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬ φ is not satisfiable in any structure and therefore refutable ; then ¬¬ φ is provable and then so is φ, thus Theorem 1 holds.

If ψ is satisfiable in a structure M, then certainly so is φ and if ψ is refutable, then is provable, and then so is ¬ φ, thus φ is refutable.

If φ

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

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 φ: M → N is a local diffeomorphism at x in M then dφ < sub > x </ sub >: T < sub > x </ sub > M → T < sub > φ ( x )</ sub > N is a linear isomorphism.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

If X is a normed space, then the dual space X * is itself a normed vector space by using the norm ǁφǁ = sup < sub > ǁxǁ ≤ 1 </ sub >| φ ( x )|.

If u is a superposition of such waves with weighting function φ, then

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

If X and Y are subsets of the real numbers, d < sub > 1 </ sub > and d < sub > 2 </ sub > can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, | x − y | < δ implies | f ( x ) − f ( y )| < ε.

If the delta function is conceptualized as modeling an idealized point mass at 0, then δ ( A ) represents the mass contained in the set A.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

If δ < sub > 0 </ sub >< 0 ( negative coefficient ), then for the same level of education ( and other factors influencing wages ), women earn a lower wage than men.

If f is a function from real numbers to real numbers, then f ( x ) is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that | x − y | < δ and | f ( x ) − f ( y )| ≥ ε.

# ( Alexandrov's theorem ) If X is Polish then so is any G < sub > δ </ sub > subset of X.

If | α | < δ, the branches are called principal, because they equal their real analogons on the positive real axis.

If L / K is a finite extension with rings of integers O and o respectively then the different ideal δ < sub > L / K </ sub >, which encodes the ramification data, is the annihilator of the O-module Ω < sup > 1 </ sup >< sub > O / o </ sub >:

If p < sup > δ < sub > p </ sub ></ sup > is an odd prime power factor of Δ and if p divides N only once ( i. e. n < sub > p </ sub >= 1 ), then there exists another elliptic curve E ', with conductor N ' = N / p, such that the coefficients of the L-series of E are congruent modulo ℓ to the coefficients of the L-series of E '.

: If the metric space ( X, d ) is compact and an open cover of X is given, then there exists a number δ > 0 such that every subset of X having diameter less than δ is contained in some member of the cover.

# If Δ is a derivation ending in an expression of the form α ((( Sβ ) γ ) δ ) ι, then Δ followed by the term α (( βδ )( γδ )) ι is a derivation.

If there is no CP-violation ( δ is zero ), then the second sum is zero.

If A δ B we say A is near B or A and B are proximal.

If the displacement in the direction of propagation is δ < sub > x </ sub >, then

If the diffusion is collisional, then δ is the mean free path and τ is the inverse of the collision frequency.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If the first allele is dominant to the second, then the fraction of the population that will show the dominant phenotype is p < sup > 2 </ sup > + 2pq, and the fraction with the recessive phenotype is q < sup > 2 </ sup >.

If activated cytotoxic CD8 < sup >+</ sup > T cells recognize them, the T cells begin to secrete various toxins that cause the lysis or apoptosis of the infected cell.