Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then is a totally ordered subset of A, hence there exists a maximal totally ordered subset containing, in particular A contains a maximal totally ordered subset.

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

Then a subset is called a σ-algebra if it satisfies the following three properties:

* Then extend the definition of truth to include sentences that predicate truth or falsity of one of the original subset of sentences.

Then every subset of X is either considered " almost everything " ( has measure 1 ) or " almost nothing " ( has measure 0 ).

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then a fuzzy subset s: S of a set S is recursively enumerable if a recursive map h: S × N Ü exists such that, for every x in S, the function h ( x, n ) is increasing with respect to n and s ( x ) = lim h ( x, n ).

Suppose a partially ordered set P has the property that every chain ( i. e. totally ordered subset ) has an upper bound in P. Then the set P contains at least one maximal element.

Then, given any point x and neighbourhood G of x, there is a closed neighbourhood E of x that is a subset of G.

Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then

Pick a fixed subset L of U. Then the maps F and G, where F ( M ) is the intersection of L and M, and G ( N ) is the union of N and ( U

Then, a tail event is an event whose occurrence or failure is determined by the values of these random variables but which is probabilistically independent of each finite subset of these random variables.

Then the subset of X

A local trivialization of a principal bundle P is given by a section s of P over an open subset U of M. Then the pullback s < sup >*</ sup > ω of a principal connection is a 1-form on U with values in.

Let S a multiplicatively closed subset of R, i. e. for any s and t ∈ S, the product st is also in S. Then the localization of M with respect to S, denoted S < sup >− 1 </ sup > M, is defined to be the following module: as a set, it consists of equivalence classes of pairs ( m, s ), where m ∈ M and s ∈ S. Two such pairs ( m, s ) and ( n, t ) are considered equivalent if there is a third element u of S such that

Furthermore, let S be a subset of P that has a least upper bound s. Then f preserves the supremum of S if the set f ( S ) =

Given a compact subset K of X and an open subset U of Y, let V ( K, U ) denote the set of all functions such that Then the collection of all such V ( K, U ) is a subbase for the compact-open topology on C ( X, Y ).

If E is a Borel subset of R, and 1 < sub > E </ sub > is the indicator function of E, then 1 < sub > E </ sub >( T ) is a self-adjoint projection on H. Then mapping

Let S be an ( a, b )- separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal ( a, b )- separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair ( a, b ) of nonadjacent vertices.

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Then, the intended recipient ( B ), who also has been informed by A about the password ( 2a ), now approaches M and tells him the agreed password ( 3a ).

Then h is half the harmonic mean of A and B.

Then the probability of the measurement outcome lying in an interval B of R is | E < sub > A </ sub >( B ) ψ |< sup > 2 </ sup >.

Then suppose definition B contains 32 instances of A ; C contains 32 instances of B ; D contains 32 instances of C ; and E contains 32 instances of D. The design now contains 5 definitions ( A through E ) and 128 instances.

Then W. E. B.

Then for each x in I, there is a base element B < sub > 3 </ sub > containing x and contained in I.

Then if we intersect each element of B with Y, the resulting collection of sets is a base for the subspace Y.

Then the velocity of B relative to A is given by

Examples: William Faulkner in A Rose for Emily ( Faulkner was an avid experimenter in using unusual points of view-see his Spotted Horses, told in third person plural ); Frank B. Gilbreth and Ernestine Gilbreth Carey in Cheaper By the Dozen ; Frederik Pohl in Man Plus ; and more recently, Jeffrey Eugenides in his novel The Virgin Suicides and Joshua Ferris in Then We Came to the End.

Then around four hundred million B. C., a small continent that was long and thin, collided with proto North America.

Then follow Sermones de omnibus evangeliis dominicalibus for every Sunday in the year ; Sermones de omnibus evangeliis, i. e., a book of discourses on all the Gospels, from Ash Wednesday to the Tuesday after Easter ; and a treatise called Marialis, qui totus est de B. Maria compositus, consisting of about 160 discourses on the attributes, titles, etc., of the Virgin Mary.

Then, the B < sub > m </ sub > make up a neighbourhood basis of 0 in K.

Then any model of B is a field of characteristic greater than k, and ¬ φ together with B is not satisfiable.

Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B.

Then he studied at the University of British Columbia, where he graduated with a B. A.

Then, A makes the first move, followed by B, then C.

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

Then quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system.

* Let R ⊂ S be an integral extension of commutative rings, and P a prime ideal of R. Then there is a prime ideal Q in S such that Q ∩ R = P. Moreover, Q can be chosen to contain any prime Q < sub > 1 </ sub > of S such that Q < sub > 1 </ sub > ∩ R ⊂ P.

Then the morphism corresponds to an integral extension of rings B ⊂ A.

Then, for every ε > 0, there exists a compact E ⊂ such that f restricted to E is continuous and