The major question in this chapter is: What is the probability of exactly X successes in N trials??

The outcome of the experiment is X successes.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

The several trials of a binomial experiment produce a new random variable X, the total number of successes, which is just the sum of the random variables associated with the single trials.

Their sum is X, the total number of successes, which in this experiment has the value Af.

For the case of a purely inertial autonavigator consisting of three restrained gyros, a coordinate system is used where the sensitive axis of the X accelerometer is parallel to the east-west direction at the base point, and the Y accelerometer sensitive axis is parallel to the north-south direction at the base point.

The input axis of the X gyro, when pointing in the east-west direction, is always perpendicular to the spin axis of earth.

* 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.

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X.

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.

For example, suppose that each member of the collection X is a nonempty subset of the natural numbers.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

* Uniform algebra: A Banach algebra that is a subalgebra of C ( X ) with the supremum norm and that contains the constants and separates the points of X ( which must be a compact Hausdorff space ).

For ƒ ∈ C ( X ) ( with a compact Hausdorff space X ), one sees that:

By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X ..

* Two compact Hausdorff spaces X < sub > 1 </ sub > and X < sub > 2 </ sub > are homeomorphic if and only if their rings of continuous real-valued functions C ( X < sub > 1 </ sub >) and C ( X < sub > 2 </ sub >) are isomorphic.

* X is a Hausdorff space.

For a " well-behaved " set X, the Hausdorff dimension is the unique number d such that N ( r ) grows as 1 / r < sup > d </ sup > as r approaches zero.

The product is a Boolean space ( compact, Hausdorff and totally disconnected ), and X < sub > F </ sub > is a closed subset, hence again Boolean.

The following theorem represents positive linear functionals on C < sub > c </ sub >( X ), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X.

A non-negative countably additive Borel measure μ on a locally compact Hausdorff space X is regular if and only if

Let X be a locally compact Hausdorff space.

Let X be a locally compact Hausdorff space.

A first countable, separable Hausdorff space ( in particular, a separable metric space ) has at most the continuum cardinality c. In such a space, closure is determined by limits of sequences and any sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Thus, the vector space B ( X, Y ) can be given the operator norm

With respect to this norm B ( X, Y ) is a Banach space.

This is also true under the less restrictive condition that X be a normed space.

Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

First we might try to proceed as if X were finite.

The automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

and refer to the same program, though invokes the text-based version, while will invoke an X Window System based interface if possible ; however, if determines that X Window System capabilities are not present, it will present the text-based version instead of failing.

For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.

Set-theoretically, one may represent a binary function as a subset of the Cartesian product X × Y × Z, where ( x, y, z ) belongs to the subset if and only if f ( x, y ) = z.

Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.

Note that the requirement that the maps be continuous is essential ; if X is infinite-dimensional, there exist linear maps which are not continuous, and therefore not bounded.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.