Then every linear operator T in V can be written as the sum of a diagonalizable operator D and a nilpotent operator N which commute.

Then we can express the variables as the sum of the ( time averaged ) mean field () that varies in space and a small fluctuating field () that varies in space and time.

Then the sum total of all the maintenance actions are bundled into maintenance intervals so that maintenance is not occurring around the clock, but rather, at regular intervals.

Then, calculate the flow rate for each and sum the flows.

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,

Then it will be spontaneously re-emitted, either in the same frequency as the original or in a cascade, where the sum of the energies of the photons emitted will be equal to the energy of the one absorbed ( assuming the system returns to its original state ).

Then the zero vector of this space can be expressed as a linear combination of no elements, which again is an empty sum.

Then any linear functional can be written in these coordinates as a sum of the form:

Since they are even, they can be written as x = 2a and y = 2b respectively for integers a and b. Then the sum.

Then negotiations began over how much money U. S. businesses owed Iran — Iran believing the sum to be $ 20 to $ 60 billion and the United States estimating it at " closer to $ 20 to 60 million ".— and how much Iran owed U. S. businesses.

* 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 the sum of the signed distances from the circumcenter D to the sides of triangle ABC is

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

Then came chivalry to Rome, and the sum of learning, which now is come to France.

Then, luminance is calculated as a weighted sum of the three linear-intensity values.

Then continue the sequence, where each subsequent term is the sum of the previous n terms.

Then χ ( M, N ), defined as the alternating sum of the lengths of the modules Tor < sub > i </ sub >< sup > R </ sup >( M, N ) is 0 if dim M + dim N < d, and positive if the sum is equal to d. ( N. B.

Then the statement is that the sum of delta-functions at each point of Λ, and at each point of Λ ′, are again Fourier transforms as distributions, subject to correct normalization.

# Then decide the input connections of OR matrix to generate the sum terms.

Then the Gauss sum mod p, g ( a ; p ), is the following sum of the pth roots of unity:

Then A ( G ) splits as a sum over characters λ of T of the subrepresentations A ( G )< sup > λ </ sup > of elements transforming according to λ.

Then Spec ( A < sub > S </ sub >) is homeomorphic to

Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

Then came an even more shocking confession: according to the CIA document, al-Faruq said two senior al-Qaeda officials, Abu Zubaydah and Ibn al-Shaykh al-Libi, had ordered him to ' plan large-scale attacks against U. S. interests in Indonesia, Malaysia, ( the ) Philippines, Singapore, Thailand, Taiwan, Vietnam and Cambodia.

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 N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.

Then according to the second isomorphism theorem S ∩ T is normal in T and ST / S ≅ T /( S ∩ T ).

Then, it is undecidable to determine whether the language decided by an arbitrary Turing machine lies in S.

In 1986, a screening of the entire Monkees television series by MTV led to renewed interest in the group, followed by a single (" That Was Then, This Is Now " reached number 20 on the Billboard Hot 100 in the U. S .), a 20th Anniversary Tour, a greatest hits album and a brand new LP, Pool It!

Then, in 1966, the U. S. Congress created a National Council for Marine Resources and Engineering Development.

Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

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 >.