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.

Scenes of a crowd fleeing Godzilla that appeared later in the Japanese print were moved to an earlier point in the movie ( and corresponding footage of them gathering around Godzilla after he is knocked out by the Super X was removed ), the Super X fight was re-arranged ( in the Japanese version, Godzilla fires his atomic ray at the Super X after being hit with cadmium missiles, not before ), and various other scenes of destruction were either placed in a different order or deleted completely.

Consider the number N ( r ) of balls of radius at most r required to cover X completely.

Then the joint distribution of X and Y is completely determined by our channel and by our choice of, the marginal distribution of messages we choose to send over the channel.

Although this market segment is now much reduced, the technologies developed in this area continue to be influential on the Internet and in both Linux and Apple Mac OS X networking — and the TCP / IP protocol has now almost completely replaced IPX, AppleTalk, NBF, and other protocols used by the early PC LANs.

* 1649 – The Italian city of Castro is completely destroyed by the forces of Pope Innocent X, ending the Wars of Castro.

X is a Tychonoff space, or T < sub > 3½ </ sub > space, or T < sub > π </ sub > space, or completely T < sub > 3 </ sub > space if it is both completely regular and Hausdorff.

Completely regular spaces can be characterized by the fact that their topology is completely determined by C ( X ) or C *( X ).

* A space X is completely regular if and only if it has the initial topology induced by C ( X ) or C *( X ).

* A space X is completely regular if and only if every closed set can be written as the intersection of a family of zero sets in X ( i. e. the zero sets form a basis for the closed sets of X ).

* A space X is completely regular if and only if the cozero sets of X form a basis for the topology of X.

Given an arbitrary topological space ( X, τ ) there is a universal way of associating a completely regular space with ( X, τ ).

Then ρ will be the finest completely regular topology on X which is coarser than τ.

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

For any positive linear functional ψ on C < sub > c </ sub >( X ), there is a unique regular Borel measure μ on X such that

For any continuous linear functional ψ on C < sub > 0 </ sub >( X ), there is a unique regular countably additive complex Borel measure μ on X such that

to a completely regular space Y will be continuous on ( X, ρ ).

One can show that C < sub > τ </ sub >( X ) = C < sub > ρ </ sub >( X ) in the above construction so that the rings C ( X ) and C *( X ) are typically only studied for completely regular spaces 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.