Then there is a diagonalizable operator D on V and a nilpotent operator N in V such that ( A ) Af, ( b ) Af.

Let a, b, and c be elements of G. Then:

Then the image set f ( I ) is also an interval, and either it contains f ( b ), or it contains f ( a ); that is,

It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i. e., for all a, b and c in P, we have that:

Then, the chance that the first letter typed is ' b ' is 1 / 50, and the chance that the second letter typed is a is also 1 / 50, and so on.

Then that researcher's Bradford multiplier b < sub > m </ sub > is 2 ( i. e. 10 / 5 ).

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

Then there is an exact sequence relating the kernels and cokernels of a, b, and c:

Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.

Let vectors and let Then the area of the parallelogram generated by a and b is equal to.

Then the area of the parallelogram with vertices at a, b and c is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:

Denote the orthocenter of triangle ABC as H, denote the sidelengths as a, b, and c, and denote the circumradius of the triangle as R. Then

Then, if a, a ′, b and b ′ are alternative settings for the detectors,

Suppose a vertex joins three units with spin numbers a, b, and c. Then, these requirements are stated as:

Then A is an algebra over K if the following identities hold for any three elements x, y, and z of A, and all elements (" scalars ") a and b of K:

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

g < sub > 1 </ sub >( x ), a ≤ x ≤ b. Then

g < sub > 2 </ sub >( x ), a ≤ x ≤ b. Then

Then, for some positive integers m and n, a < sup > n </ sup > and b < sup > m </ sup > are in I.

Then the element c = ba is nilpotent ( if non-zero ) as c < sup > 2 </ sup > = ( ba )< sup > 2 </ sup > = b ( ab ) a = 0.

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 in 1702 the French astronomer Philippe de la Hire used a year he labeled at the end of years labeled ante Christum ( BC ), and immediately before years labeled post Christum ( AD ) on the mean motion pages in his Tabulæ Astronomicæ, thus adding the designation 0 to Kepler's Christi.

Then, dividing the units of energy ( such as eV ) by a fundamental constant that has units of velocity ( M < sup > 0 </ sup > L < sup > 1 </ sup > T < sup >-1 </ sup >), facilitates the required conversion of using energy units to describe momentum.

Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd.

For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

Then, for OpenGL 2. 0 and beyond:

Then Fischer and Rabin ( 1974 ) proved that any decision algorithm for Presburger arithmetic has a worst-case runtime of at least, for some constant c > 0.

Then x is a rational number greater than 0 and less than r < sub > 0 </ sub >.

Then, the Colts avenged their sole regular-season loss against the Cleveland Browns by crushing them, 34 – 0, in the NFL Championship Game.

Then after throwing an incomplete pass, Terry Bradshaw completed 2 consecutive passes, the second one a 28-yard touchdown completion to Stallworth to take a 7 – 0 lead.

Then they beat the Tampa Bay Buccaneers in the NFC Championship Game, 9 – 0, scoring only three field goals.

Then γ < sub > 1 </ sub > and γ < sub > 2 </ sub > are called equivalent at 0 if the ordinary derivatives of φ ∘ γ < sub > 1 </ sub > and φ ∘ γ < sub > 2 </ sub > at 0 coincide.

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

Then one might define the glass as being 0. 7 empty and 0. 3 full.

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