Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

from which it is clear that and for k > n. To see that these coefficients count k-combinations from S, one can first consider a collection of n distinct variables X < sub > s </ sub > labeled by the elements s of S, and expand the product over all elements of S:

it has 2 < sup > n </ sup > distinct terms corresponding to all the subsets of S, each subset giving the product of the corresponding variables X < sub > s </ sub >.

Now setting all of the X < sub > s </ sub > equal to the unlabeled variable X, so that the product becomes, the term for each k-combination from S becomes X < sup > k </ sup >, so that the coefficient of that power in the result equals the number of such k-combinations.

The 21 consonant letters in the English alphabet are B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, X, Z, and usually W and Y: The letter Y stands for the consonant in " yoke ", the vowel in " myth " and the vowel in " funny ", and " yummy " for both consonant and vowel, for examples ; W almost always represents a consonant except in rare words ( mostly loanwords from Welsh ) like " crwth " " cwm ".

There were three different versions of the Nova 4, the Nova 4 / C, the Nova 4 / S and the Nova 4 / X.

The Nova 4 / S and 4 / X used separate memory boards.

The Nova 4 / X had the on-board memory management unit ( MMU ) enabled to allow up to 128 kWords of memory to be used ( the MMU was also installed in the Nova 4 / S, but was disabled by firmware ).

Both the 4 / S and the 4 / X included a “ prefetcher ” to increase performance by fetching up to two instructions from memory before they were needed.

* Prediction: For every state in S ( k ) of the form ( X → α • Y β, j ) ( where j is the origin position as above ), add ( Y → • γ, k ) to S ( k ) for every production in the grammar with Y on the left-hand side ( Y → γ ).

* Scanning: If a is the next symbol in the input stream, for every state in S ( k ) of the form ( X → α • a β, j ), add ( X → α a • β, j ) to S ( k + 1 ).

* Completion: For every state in S ( k ) of the form ( X → γ •, j ), find states in S ( j ) of the form ( Y → α • X β, i ) and add ( Y → α X • β, i ) to S ( k ).

In 1959 he contributed to the British arts review X magazine ( other contributors included Bacon, Freud, Auerbach and Kokoschka ) with an article, The Dream, the Sphinx, and the Death of T, and drawings.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

Other variable stars are U Camelopardalis, VZ Camelopardalis, and Mira variables T Camelopardalis, X Camelopardalis, and R Camelopardalis.

where θ ∈ T < sub > f ( x )</ sub >< sup >*</ sup > N and X < sub > x </ sub > ∈ T < sub > x </ sub > M.

In this system, the number ten may be written as " A ", " T " or " X ", and the number eleven as " B " or " E " ( another common notation, introduced by Sir Isaac Pitman, is to use a rotated " 2 " for ten and a reversed " 3 " for eleven ).

A function f: X → Y between two topological spaces ( X, T < sub > X </ sub >) and ( Y, T < sub > Y </ sub >) is called a homeomorphism if it has the following properties:

It would be difficult to find an area of scientific interest more beset by divided professional opinion and contradictory experimental evidence … No one can say whether hypnosis is a qualitatively unique state with some physiological and conditioned response components or only a form of suggestion induced by high motivation and a positive relationship between hypnotist and subject … T. X.

H2A can be replaced by H2AZ ( which leads to reduced nucleosome stability ) or H2AX ( which is associated with DNA repair and T cell differentiation ), whereas the inactive X chromosomes in mammals are enriched in macroH2A.

The list of text forms in the W-region of memory and the contents of the information cells in the X and Y-regions are no longer required.

Another equivalent axiom only considers collections X that are essentially powersets of other sets:

Other disorders are also due to recessive alleles, but because the gene locus is located on the X chromosome, so that males have only one copy ( that is, they are hemizygous ), they are more frequent in males than in females.

* 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.

( The others are Mac OS X, Solaris and HP-UX.

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.

Alkynes characteristically undergo reactions that show that they are " doubly unsaturated ," meaning that each alkyne unit is capable of adding two equivalents of H < sub > 2 </ sub >, halogens or related HX reagents ( X = halide, pseudohalide, etc .).

Actual working Atari Video System X machines, whose hardware is 100 % identical to the Atari 5200 do exist, but are extremely rare.

Thus, ocular albinism occurs more frequently in males as they have a single X and Y chromosome, unlike females, whose genetics are characterized by two X chromosomes.

A binary relation R is usually defined as an ordered triple ( X, Y, G ) where X and Y are arbitrary sets ( or classes ), and G is a subset of the Cartesian product X × Y.

The sets X and Y are called the domain ( or the set of departure ) and codomain ( or the set of destination ), respectively, of the relation, and G is called its graph.

A binary operation is a binary function where the sets X, Y, and Z are all equal ; binary operations are often used to define algebraic structures.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

Since every equivalence relation over X corresponds to a partition of X, and vice versa, the number of possible equivalence relations on X equals the number of distinct partitions of X, which is the nth Bell number B < sub > n </ sub >:

X is a Hausdorff space if any two distinct points of X can be separated by neighborhoods.

If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories ( T, U, V, W, Z ) count fourfold, I counts twice, and X counts only once.

Rather than cover art each rule book, supplement, and adventure had a black cover with the title Book X, Supplement Y or Adventure Z each in a distinct color, numbered in the order in which they were produced.

which may be read as: X is cumulative if there exists at least one pair x, y, where x and y are distinct, and both have the property X, and if for all possible pairs x and y fitting that description, X is a property of the sum of x and y.

There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function from X to Y, i. e. a function from a subset X < nowiki >'</ nowiki > of X to Y.

In the bivariate case, we also have a theorem that makes the first equivalent condition for multivariate normality less restrictive: it is sufficient to verify that countably many distinct linear combinations of X and Y are normal in order to conclude that the vector is bivariate normal.

In topology and related branches of mathematics, a topological space X is a T < sub > 0 </ sub > space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other.

( This is a bit confusing, since the original space cadet keyboard and the X Window System recognize a " Meta " modifier distinct from " Super ".

The multiplicities of distinct prime factors of X are independent random variables.

Pre-OS X versions of the Mac OS disposed of filename-based extension metadata entirely, instead using a distinct file type code to identify the file format.

* X is a T < sub > 1 </ sub > space if any two distinct points in X can be separated.

If P is a common point of two plane algebraic curves X and Y that is a non-singular point of both of them and, moreover, the tangent lines to X and Y at P are distinct then the intersection multiplicity is one.