Let P be the root of the unbalanced subtree, with R and L denoting the right and left children of P respectively.

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 us define a linear operator P, called the exchange operator.

Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is, so when it bounces off, its angle of inclination must be equal to.

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

# Let P ( x ) be a first-order formula in the language of Presburger arithmetic with a free variable x ( and possibly other free variables ).

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 us assume the bias is V and the barrier width is W. This probability, P, that an electron at z = 0 ( left edge of barrier ) can be found at z = W ( right edge of barrier ) is proportional to the wave function squared,

Let P ( α ) be a property defined for all ordinals α.

Let P be Q's left child.

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Let P < sup >− 1 </ sup > DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues.

Let be the columns of P, each multiplied by the ( real ) square root of the corresponding eigenvalue.

Let the total power radiated from a point source, for example, an omnidirectional isotropic antenna, be P. At large distances from the source ( compared to the size of the source ), this power is distributed over larger and larger spherical surfaces as the distance from the source increases.

Let f be any function from S to P ( S ).

Let K be R, C, or any field, and let V be the set P of all polynomials with coefficients taken from the field K.

Let further P < sub > Alice </ sub > denote the first plaintext block of Alice's message, let E denote encryption, and let P < sub > Eve </ sub > be Eve's guess for the first plaintext block.

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

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 g be a smooth function on N vanishing at f ( x ).

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 G denote the set of bijective functions over A that preserve the partition structure of A: ∀ x ∈ A ∀ g ∈ G ( g ( x ) ∈ ).

Let X be a topological space, and let x < sub > 0 </ sub > be a point of X.

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

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

If P is a program which outputs a string x, then P is a description of x.

For two geometric objects P and Q represented by the relations P ( x, y ) and Q ( x, y ) the intersection is the collection of all points ( x, y ) which are in both relations.

;< sup > 238 </ sup > Am (< sup > 31 </ sup > P, xn )< sup > 269-x </ sup > Bh ( x = 5?

and let the coordinates of P < sub > 1 </ sub > and P < sub > 2 </ sub > be ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >) respectively.

The coordinates of P are then X and Y interpreted as numbers x and y on the corresponding number lines.

If the partial pressure of A at x < sub > 1 </ sub > is P < sub > A < sub > 1 </ sub ></ sub > and x < sub > 2 </ sub > is P < sub > A < sub > 2 </ sub ></ sub >, integration of above equation,

If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.