For by now the original cause of the quarrel, Philip's seizure of Gascony, was only one strand in the spider web of French interests that overlay all western Europe and that had been so well and closely spun that the lightest movement could set it trembling from one end to the other.

( For each State, make all computations set forth in items 1 to 8 above, and then add the results obtained for each State in item 8.

For example, the inflected forms of a word can be represented, insofar as regular inflection allows, by a stem and a set of endings to be attached.

For the only time in the opera, words are not set according to their natural inflection ; ;

For a while Nick followed the twisting course of the bubbles, wondering which set came from Elaine.

" For some time, Lincoln continued earlier plans to set up colonies for the newly freed slaves.

: For any set X of nonempty sets, there exists a choice function f defined on X.

: For any set A, the power set of A ( with the empty set removed ) has a choice function.

: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.

For example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

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

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

For much of the history of the series ( Volumes 4 through 29 ), settings in Gaul and abroad alternated, with even-numbered volumes set abroad and odd-numbered volumes set in Gaul, mostly in the village.

For the graphical user interfaces, AIX v2 came with the X10R3 and later the X10R4 and X11 versions of the X Window System from MIT, together with the Athena widget set.

For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

For example the character set had no "↑" symbol, so exponentiation was an overstrike of "|" and "*".

For example, a Commodore 64 user calling an Atari BBS would use ASCII rather than the machine's native character set.

For example, consider the quadratic polynomial

For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

For a quantum computer, however, Peter Shor discovered an algorithm in 1994 that solves it in polynomial time.

For example, P < sup > SAT </ sup > is the class of problems solvable in polynomial time by a deterministic Turing machine with an oracle for the Boolean satisfiability problem.

For example, is a polynomial, but is not, because its second term involves division by the variable x ( 4 / x ), and also because its third term contains an exponent that is not a non-negative integer ( 3 / 2 ).

For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences ; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science ; they are used in calculus and numerical analysis to approximate other functions.

For example, the following is a polynomial:

For example, ( x + 1 )< sup > 3 </ sup > is a polynomial ; its standard form is x < sup > 3 </ sup > + 3x < sup > 2 </ sup > + 3x + 1.

For example, x < sup > 3 </ sup >/ 12 is considered a valid term in a polynomial ( and a polynomial by itself ) because it is equivalent to 1 / 12x < sup > 3 </ sup > and 1 / 12 is just a constant.

For polynomials in more than one variable the notion of root does not exist, and there are usually infinitely many combinations of values for the variables for which the polynomial function takes the value zero.

For the polynomial function

For instance, the ring ( in fact field ) of complex numbers, which can be constructed from the polynomial ring R over the real numbers by factoring out the ideal of multiples of the polynomial.

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

For some problems, quantum computers offer a polynomial speedup.

The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L < sup > 2 </ sup >, where L is the number of bits in the number to be factored ; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 10 < sup > 4 </ sup > qubits without error correction.

For every, there exists a polynomial function p over C such that for all x in, we have, or equivalently, the supremum norm.

For example, the square root of 2 is irrational and not transcendental ( because it is a solution of the polynomial equation x < sup > 2 </ sup > − 2 = 0 ).

( For example if the variables x, y, and z are permuted in all 6 possible ways in the polynomial x + y-z then we get a total of 3 different polynomials: x + y − z, x + z-y, and y + z − x.

For instance, the problem of integer factorization is suspected to require more than a polynomial function of the length of the input as run-time if the input is given in binary, but it only needs linear runtime if the input is presented in unary.

For example, to compute one prime factor of the natural number N in polynomial time ( no polynomial time factorization algorithm is known in traditional complexity theory ; see integer factorization ):