Let now x ' and y ' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol which takes as many arguments as the sum of lengths of x and y ; we consider the formula

Let K be the triangle's area and let a, b and c, be the lengths of its sides.

For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.

Let the coordinate vector of the point P that defines the fulcrum be r < sub > P </ sub >, and introduce the lengths

Let a, b, g and h denote the lengths of the input crank, the output crank, the ground link and floating link, respectively.

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 V and W be vector spaces over the same field K. A function f: V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

Let be any basis for the column space of and place them as column vectors to form the × matrix.

Let u, v be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers.

Let be a list of n linearly independent vectors of some complex vector space with an inner product.

Let A be an m × n matrix, with column vectors v < sub > 1 </ sub >, v < sub > 2 </ sub >, ..., v < sub > n </ sub >.

Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.

Let vectors and let Then the area of the parallelogram generated by a and b is equal to.

Let vectors < u > a </ u >, < u > b </ u >, < u > c </ u > and < u > h </ u > determine the position of each of the four orthocentric points and let < u > n </ u > = (< u > a </ u > + < u > b </ u > + < u > c </ u > + < u > h </ u >) / 4 be the position vector of N, the common nine-point center.

We want to find n ′ perpendicular to P. Let t be a vector on the tangent plane and M < sub > l </ sub > be the upper 3x3 matrix ( translation part of transformation does not apply to normal or tangent vectors ).

Let denote the tangent space of M at a point p. For any pair of tangent vectors at p, the Ricci tensor evaluated at is defined to be the trace of the linear map given by

Let V = TCP < sup > 1 </ sup > be the bundle of complex tangent vectors having the form a ∂/∂ z at each point, where a is a complex number.

Let P have column vectors p < sub > i </ sub >, i =

Let, and be the corresponding vectors and matrix.

Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors.

Let B and C be two different bases of a vector space V, and let us mark with the matrix which has columns consisting of the C representation of basis vectors b < sub > 1 </ sub >, b < sub > 2 </ sub >, ..., b < sub > n </ sub >:

Let V be a vector space over a field K, and let N be a subspace of V. We define an equivalence relation ~ on V by stating that x ~ y if x − y ∈ N. That is, x is related to y if one can be obtained from the other by adding an element of N. From this definition, one can deduce that any element of N is related to the zero vector ; in other words all the vectors in N get mapped into the equivalence class of the zero vector.

Let Q < sub > n </ sub > denote the m-by-n matrix formed by the first n Arnoldi vectors q < sub > 1 </ sub >, q < sub > 2 </ sub >, …, q < sub > n </ sub >, and let H < sub > n </ sub > be the ( upper Hessenberg ) matrix formed by the numbers h < sub > j, k </ sub > computed by the algorithm:

Let F stand for R, C, or H. The Stiefel manifold V < sub > k </ sub >( F < sup > n </ sup >) can be thought of as a set of n × k matrices by writing a k-frame as a matrix of k column vectors in F < sup > n </ sup >.

Let be the antisymmetric Fock space over and let be the orothogonal projection onto antisymmetric vectors:

Let Y and X be random vectors.

Let A be an m × n matrix, with row vectors r < sub > 1 </ sub >, r < sub > 2 </ sub >, ..., r < sub > m </ sub >.

Let the set of the test vectors be.

Let denote the norm of vector x and the inner product of vectors x and y.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

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.

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

Let P < sub > F </ sub > be the domain of a prefix-free universal computable function F. The constant Ω < sub > F </ sub > is then defined as

Let F be a prefix-free universal computable function.

Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD.

Let F be the continuous cumulative distribution function which is to be the null hypothesis.

Let F be a field.

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.

These assumptions can be summarised as: Let ( Ω, F, P ) be a measure space with P ( Ω )= 1.

Let the input power to a device be a force F < sub > A </ sub > acting on a point that moves with velocity v < sub > A </ sub > and the output power be a force F < sub > B </ sub > acts on a point that moves with velocity v < sub > B </ sub >.

Let P be the following property of partial functions F of one argument: P ( F ) means that F is defined for the argument ' 1 '.

Let F and G be a pair of adjoint functors with unit η and co-unit ε ( see the article on adjoint functors for the definitions ).

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.