Given that a father's age is 1 less than twice that of his son, and that the digits AB making up the father's age are reversed in the son's age ( i. e. BA ), leads to the equation 19B-8A

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

: Given three points A, B and C on a circle with center O, the angle AOC is twice as large as the angle ABC.

Given two strings, equally taut and heavy, one twice as long as the other, the longer would vibrate with a pitch one octave lower than the shorter.

: Given a function f that has values everywhere on the boundary of a region in R < sup > n </ sup >, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary?

Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm.

* Doubling the cube: Given any cube drawing a cube with twice its volume.

Given the rarity of continuously flowing rivers in any part of Saudi Arabia, the existence of a river such as that described in the Book of Mormon has long been questioned by its critics.

Given rapid changes in voice and more generally VOIP, centrally hosted solutions also make it easier to continuously adapt to changing requirements by providing one place for managing system change.

Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given two differentiable manifolds

Given a differentiable function defined on a bounded open set, the total variation of has the following expression

Given an exact differential equation defined on some simply connected and open subset D of R < sup > 2 </ sup > with potential function F then a differentiable function f with ( x, f ( x )) in D is a solution if and only if there exists real number c so that

Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).

Given x ∈ A, the holomorphic functional calculus allows to define ƒ ( x ) ∈ A for any function ƒ holomorphic in a neighborhood of Furthermore, the spectral mapping theorem holds:

Given a function f of type, currying it makes a function.

Given a function of type, currying produces.

Given the definition of above, we might fix ( or ' bind ') the first argument, producing a function of type.

Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit

Given a function f of a real variable x and an interval of the real line, the definite integral

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given a set of training examples of the form, a learning algorithm seeks a function, where is the input space and

Given an arithmetic function, one can generate a bi-infinite sequence of other arithmetic functions by repeatedly applying the first summation.

Given a function ƒ defined over the reals x, and its derivative ƒ < nowiki > '</ nowiki >, we begin with a first guess x < sub > 0 </ sub > for a root of the function f. Provided the function is reasonably well-behaved a better approximation x < sub > 1 </ sub > is

Given a relation, scaling the argument by a constant factor causes only a proportionate scaling of the function itself.

# Composition operator ( also called the substitution operator ): Given an m-ary function and m k-ary functions:

# Primitive recursion operator: Given the k-ary function and k + 2-ary function:

# Minimisation operator: Given a ( k + 1 )- ary total function:

Given a class function G: V → V, there exists a unique transfinite sequence F: Ord → V ( where Ord is the class of all ordinals ) such that

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).

Given the laws of exponents, f ( x )

# Given any point x in X, and any sequence in X converging to x, the composition of f with this sequence converges to f ( x )

Given f

Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.

Given a morphism f: B → A the associated natural transformation is denoted Hom ( f ,–).

Given the space X = Spec ( R ) with the Zariski topology, the structure sheaf O < sub > X </ sub > is defined on the D < sub > f </ sub > by setting Γ ( D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >, the localization of R at the multiplicative system