* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

Then ( X × Y, ( π < sub > 1 </ sub >, π < sub > 2 </ sub >)) is a terminal morphism from Δ to the object ( X, Y ) of D × D. This is just a restatement of the above since the pair ( f, g ) represents an ( arbitrary ) morphism from Δ ( Z ) to ( X, Y ).

Then a < sub > k </ sub > converges cubically to 1 / π ; that is, each iteration approximately triples the number of correct digits.

Then p < sub > k </ sub > converges monotonically to π ; with p < sub > k </ sub >-π ≈ 10 < sup >− 2 < sup > k + 1 </ sup ></ sup > for k ≥ 2. s

Then a < sub > k </ sub > converges quartically against 1 / π ; that is, each iteration approximately quadruples the number of correct digits.

Then p < sub > k </ sub > converges quartically to π ; that is, each iteration approximately quadruples the number of correct digits.

Then a < sub > k </ sub > converges quintically to 1 / π ( that is, each iteration approximately quintuples the number of correct digits ), and the following condition holds:

Then a graph is any function s for which π ( s ( x ))= x.

Let π: P → M be a smooth principal G-bundle over a smooth manifold M. Then a principal G-connection on P is a differential 1-form on P with values in the Lie algebra of G which is G-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on P.

Then, for every two real numbers α and β for which 0 ≤ α < β ≤ π,

Then a finer statement of the s-cobordism theorem is that the isomorphism classes of this category ( up to Cat-isomorphism of h-cobordisms ) are torsors for the respective < ref > Note that identifying the Whitehead groups of the various manifolds requires that one choose base points and a path in W connecting them .</ ref > Whitehead groups Wh ( π ), where

Then π is a projection-valued measure.

Then ( U, π ) is a system of imprimitivity of ( G, X ) on L < sup > 2 </ sup >< sub > μ </ sub >( X ).

Then the Ree group is defined to be the group of elements g of X ( F ) such that α < sub > π </ sub >( g )

Then the energy of the vacuum is exactly E < sub > 0 </ sub >.

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 X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Then X is reflexive if and only if each X < sub > j </ sub > is reflexive.

Then the cotangent space at x is defined as the dual space of T < sub > x </ sub > M:

Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

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, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

Then the overall runtime is O ( n < sup > 2 </ sup >).

Then the Cartesian product set D < sub > 1 </ sub > D < sub > 2 </ sub > can be made into a directed set by defining ( n < sub > 1 </ sub >, n < sub > 2 </ sub >) ≤ ( m < sub > 1 </ sub >, m < sub > 2 </ sub >) if and only if n < sub > 1 </ sub > ≤ m < sub > 1 </ sub > and n < sub > 2 </ sub > ≤ m < sub > 2 </ sub >.

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

* Let ρ be a unitary representation of a compact group G on a complex Hilbert space H. Then H splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of G.

Then its spectral radius ρ ( A ) is defined as:

Let ρ and σ be representations of G. Then the following identities hold:

Then the matrix defines an automorphism of ρ, which is clearly not a scalar multiple of the identity matrix.

Then ρ induces a representation ρ < sup >*</ sup > on the dual of the vector space Hom ( C < sup > n </ sup >, C ).

Let A be an irreducible non-negative n × n matrix with period h and spectral radius ρ ( A ) = r. Then the following statements hold.

Then g ( E ) is called the density of states if the energy spectrum is continuous.

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, when the U. S. Army Air Forces on the Marianas Islands ran out of conventional thermite incendiary bombs for its B-29 Superfortresses to drop on Japanese cities, its top commanders, such as General Curtis E. LeMay turned to napalm bombs to continue its fire raids on the large Japanese cities.

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 the emf, E < sub > X </ sub >, of the same cell containing the solution of unknown pH is measured.

Then W. E. B.

Then he would rapidly shift his army southeast and cross the Rappahannock River to Fredericksburg, hoping that Robert E. Lee would sit still, unclear as to Burnside's intentions, while the Union Army made a rapid movement against Richmond, south along the Richmond, Fredericksburg and Potomac Railroad from Fredericksburg.

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

Then on September 22, 1862, George McClellan defeated Robert E. Lee at Antietam ; the second draft of the Emancipation Proclamation was issued to the public the following day.

Then, 2002 and 2003 also saw the release of the new material from Front 242 in a decade: the E. P.

Then from E < sup > 3 </ sup > 2000 around May 12, 2000 to ECTS 2000 the game didn't change very much.

Then, during the early to mid 1900s, the more distinct features of E. granulosus and E. multilocularis, their life cycles and how they cause disease were more fully described as more and more people began researching and performing experiments and studies.

Then Peter E. Hart introduced an argument that established A2, with only minor changes, to be the best possible algorithm for finding shortest paths.

Plaques of " Then and Now " covering McFarland's 150 years of history were placed at the McFarland High School, the local Culver's restaurant, the E. D.

In his last memoir, Hart-Davis listed the books he had edited as: The Second Omnibus Book ( Heinemann ) 1930 ; Then and Now ( Cape ) 1935 ; The Essential Neville Cardus ( Cape ) 1949 ; Cricket All His Life by E. V.

* Woods, Thomas E., Sacred Then and Sacred Now: The Return of the Old Latin Mass ( Roman Catholic Books 2008 ISBN 978-0-9793540-2-1 )

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Then by the theorem, the equation also holds for D, E and F ′.

Then, there is the new Classical version associated with Robert E. Lucas, Jr ..

For such a subgroup G, define Fix ( G ) to be the field consisting of all elements of L that are held fixed by all elements of G. Then the maps E Gal ( L / E ) and G Fix ( G ) form an antitone Galois connection.

Four more songs from the album Dead Man's Party were used in soundtracks: " No One Lives Forever " was featured in Tobe Hooper's The Texas Chainsaw Massacre 2, " Stay " ( in the Boingo Alive version ) was used as the theme music for the Brazilian soap opera Top Model, " Same Man I Was Before " was used in My Best Friend Is a Vampire and " Just Another Day " opened the 1985 film adaptation of S. E. Hinton's That Was Then, This Is Now.