Then f = u + iv is complex-differentiable at that point if and only if the partial derivatives of u and v satisfy the Cauchy – Riemann equations ( 1a ) and ( 1b ) at that point.

Marking a departure from convention, for the first time, a studio release was not titled as a single word starting and ending with the letter A ( excepting the partial compilation / partial new album Then & Now ).

Then they conserved a partial autonomy with a domestic praepositus Iapodum.

Then, if we take the partial trace over the environment, we'd find the density state is approximately described by

Let U be an open set in R < sup > n </ sup > and φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ ( U ),

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.

Then Q ( x ) has a zero α of multiplicity r, and in the partial fraction decomposition, r of the partial fractions will involve the powers of ( x − α ).

Then pardon was accorded to him for his transgressions both in Cilicia and Cyprus, and still allowed to hold partial possession in Cilicia.

Then f and its partial derivatives with respect to x and y vanish at ( 0, 0 ).

Then each set N < sup >− 1 </ sup >( i ), consisting of elements that have equal values of N, is an antichain, and these antichains partition the partial order into a number of antichains equal to the size of the largest chain.

Then, after the March 1996 elections and the resignation from the Labor party by the disgraced Colston, Harradine's and Colston's votes were sufficient to pass Coalition legislation, notably the Native Title Amendment Act 1998 ( also known as the " Wik 10 Point Plan ") and the partial privatisation of Telstra.

Then the partial sum of order ( a + b ) n of this rearranged series contains positive odd terms and negative even terms, hence

Suppose now that, more generally, a rearranged series of the alternating harmonic series is organized in such a way that the ratio between the number of positive and negative terms in the partial sum of order n tends to a positive limit r. Then, the sum of such a rearrangement will be

Upon hearing of Russia ’ s partial mobilization, Wilhelm wrote: “ Then I must mobilize too .” The German Ambassador in St. Petersburg informed Nicholas that Germany would mobilize if Russia did not demobilize at once.

Then the partial transpose is

Then the partial function

Then, p < sup > 2 </ sup > is the fraction of the population homozygous for the first allele, 2pq is the fraction of heterozygotes, and q < sup > 2 </ sup > is the fraction homozygous for the alternative allele.

Then, represents the fraction of binding sites that are not occupied, giving the ratio

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

Then, Harry enters five of these labeled doors, each of which symbolizes a fraction of his life.

Then seeing that a fraction is less than one, if I multiply by a fraction, it follows that I do take it less than once.

Then the accretion efficiency, or the fraction of energy actually radiated of that theoretically available from the gravitational energy release of accreting material, enters in an essential way.

Then express it in lowest possible terms ( i. e., as a fully reduced fraction ) as < sup > m </ sup >⁄< sub > n </ sub > for natural numbers m and n, and let q be the largest integer no greater than √ k.

Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists.

Then a significant fraction of the bleached protein will diffuse out of the bleached region actually during bleaching.

Then the seller will post another ad with different wording at a " deal " price, a fraction of the original price.

Then the fraction of bits belonging to type i that would be served is:

Then, unless, then number q has a parent in the Stern – Brocot tree given by the continued fraction expression

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

Then the topological cohomology of X does not change, but the Hodge decomposition does change.

Then, we add the segments from the subdivision, one by one, in random order, refining the trapezoidal decomposition.

Then the Iwasawa decomposition of is

Let R be a Noetherian ring, and I an ideal in R. Then I has an irredundant primary decomposition into primary ideals.

Consider a group G and subgroups H and K, with K contained in H. Then the left cosets of H in G are each the union of left cosets of K. Not only that, but translation ( on one side ) by any element g of G respects this decomposition.

Then the complex derivative of ƒ at a point z < sub > 0 </ sub > is defined by

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 Goursat's theorem asserts that ƒ is analytic in an open complex domain Ω if and only if it satisfies the Cauchy – Riemann equation in the domain.

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then ƒ is a homomorphism of rings, since ƒ preserves both addition:

Then ƒ is a homomorphism of groups, since it preserves multiplication:

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

Then for arbitrary ε > 0 there is an embedding ( or immersion ) ƒ < sub > ε </ sub >: M < sup > m </ sup > → R < sup > n </ sup > which is

Then, the value ƒ ( x < sub > 0 </ sub >) will be larger than other values ƒ ( x ).

Then for any r between p and q we have that F is dense in, that Tƒ is in for any ƒ in F and that T is bounded in the norm.

Then ƒ is equal to a finite Blaschke product

Then ƒ ( z ) can be expanded in terms of polynomials A < sub > n </ sub > as follows:

Then for a function ƒ on S < sup > n − 1 </ sup >, the spherical Laplacian is defined by

Then, if ƒ ( x < sub > 0 </ sub >) is not equal to, x < sub > 0 </ sub > is called a removable discontinuity.

Then ƒ satisfies () precisely when it is a conformal transformation from D equipped with this metric to the domain D ′ equipped with the standard Euclidean metric.