Let ( q < sub > 1 </ sub >, w, x < sub > 1 </ sub > x < sub > 2 </ sub >... x < sub > m </ sub >) ( q < sub > 2 </ sub >, y < sub > 1 </ sub > y < sub > 2 </ sub >... y < sub > n </ sub >) be a transition of the GPDA

Let p, q > 2 be two distinct prime numbers.

Let q < sup >*</ sup >

Let T: X → X be a contraction mapping on X, i. e.: there is a nonnegative real number q < 1 such that

# Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) and q = ( q < sub > 1 </ sub >, q < sub > 2 </ sub >) be elements of W, that is, points in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub > and q < sub > 1 </ sub > = q < sub > 2 </ sub >.

Let q and r denote the inclusion map and the sign map respectively, so that

Let q denote the probability that a given neutron induces fission in a nucleus.

Let ( S, Σ, μ ) be a measure space and let 1 ≤ p, q ≤ ∞ with 1 / p + 1 / q = 1.

Let q be a prime number, s a complex variable, and define a Dirichlet L-function as

Let p < sub > 1 </ sub > and p < sub > 2 </ sub > be any two points on l < sub > 1 </ sub >, and let q < sub > 1 </ sub > and q < sub > 2 </ sub > be any two points on l < sub > 2 </ sub >.

Let q be a quadratic form defined on an n-dimensional real vector space.

Let A be the matrix of the quadratic form q in a given basis.

Let q be the probability of losing ( e. g. for American double-zero roulette, it is 10 / 19 for a bet on black or red ).

Let q denote the Lorentzian quadratic form on R < sup > n + 2 </ sup > defined by

Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:

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 ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Let denote the Bézier curve determined by the points P < sub > 0 </ sub >, P < sub > 1 </ sub >, ..., P < sub > n </ sub >.

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 M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

Let M be a smooth manifold and let f ∈ C < sup >∞</ sup >( M ) be a smooth function.

Let e be the error in b. Assuming that A is a square matrix, the error in the solution A < sup >− 1 </ sup > b is A < sup >− 1 </ sup > e.

Let us for simplicity take, then < math > 0 < c =- 2a </ math > and.

Let S ( fig. 5 ) be any optical system, rays proceeding from an axis point O under an angle u1 will unite in the axis point O ' 1 ; and those under an angle u2 in the axis point O ' 2.

Genesis 1: 9 " And God said, Let the waters be collected ". Letters in black, < font color ="# CC0000 "> niqqud in red </ font >, < font color ="# 0000CC "> cantillation in blue </ font >

* Let D < sub > 1 </ sub > and D < sub > 2 </ sub > be directed sets.

" Let X be the unit Cartesian square ×, and let ~ be the equivalence relation on X defined by ∀ a, b ∈ (( a, 0 ) ~ ( a, 1 ) ∧ ( 0, b ) ~ ( 1, b )).

Moving to groups in general, let H be a subgroup of some group G. Let ~ be an equivalence relation on G, such that a ~ b ↔ ( ab < sup >− 1 </ sup > ∈ H ).

The sentiment is summarized in a line from Ovid's Amores I. 1. 27 Sex mihi surgat opus numeris, in quinque residat-" Let my work rise in six steps, fall back in five.

Gloria Gaynor ( born September 7, 1949 ) is an American singer, best known for the disco era hits ; " I Will Survive " ( Hot 100 number 1, 1979 ), " Never Can Say Goodbye " ( Hot 100 number 9, 1974 ), " Let Me Know ( I Have a Right )" ( Hot 100 number 42, 1980 ) and " I Am What I Am " ( R & B number 82, 1983 ).

Let k >= 1.

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

Let r be a non zero real number and let the r < sup > th </ sup > power mean ( M < sup > r </ sup > ) of a series of real variables ( a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ... ) be defined as

Let x < sub > 1 </ sub >, ..., x < sub > n </ sub > be the sizes of the heaps before a move, and y < sub > 1 </ sub >, ..., y < sub > n </ sub > the corresponding sizes after a move.

Let s = x < sub > 1 </ sub > ⊕ ... ⊕ x < sub > n </ sub > and t = y < sub > 1 </ sub > ⊕ ... ⊕ y < sub > n </ sub >.

Let w < sub > j </ sub > be the ' price ' ( the rental ) of a certain factor j, let MP < sub > j1 </ sub > and MP < sub > j2 </ sub > be its marginal product in the production of goods 1 and 2, and let p < sub > 1 </ sub > and p < sub > 2 </ sub > be these goods ' prices.

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 x < sub > 0 </ sub >, ...., x < sub > N-1 </ sub > be complex numbers.

Let ( X < sub > i </ sub >, f < sub > ij </ sub >) be an inverse system of objects and morphisms in a category C ( same definition as above ).

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

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

Let ( M, g ) be a Riemannian manifold and ƒ: M < sup > m </ sup > → R < sup > n </ sup > a short C < sup >∞</ sup >- embedding ( or immersion ) into Euclidean space R < sup > n </ sup >, where n ≥ m + 1.

* The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I < sub > n </ sub > be the ideal of all continuous functions f such that f ( x ) = 0 for all x ≥ n. The sequence of ideals I < sub > 0 </ sub >, I < sub > 1 </ sub >, I < sub > 2 </ sub >, etc., is an ascending chain that does not terminate.

# Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) be an element of W, that is, a point in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub >, and let c be a scalar in R. Then cp = ( cp < sub > 1 </ sub >, cp < sub > 2 </ sub >); since p < sub > 1 </ sub > = p < sub > 2 </ sub >, then cp < sub > 1 </ sub > = cp < sub > 2 </ sub >, so cp is an element of W.