Then, following the programme he outlined in his talk at the 1958 International Congress of Mathematicians, he introduced the theory of schemes, developing it in detail in his Éléments de géométrie algébrique ( EGA ) and providing the new more flexible and general foundations for algebraic geometry that has been adopted in the field since that time.

Then, following the battle, the Athenian army marched the 40 ( 25 miles ) or so kilometers back to Athens at a very high pace ( considering the quantity of armour, and the fatigue after the battle ), in order to head off the Persian force sailing around Cape Sounion.

Then after-depolarization this effect leads initially to bigeminy: regular ectopic beats following each ventricular contraction.

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 following three connected theorems hold:

Then G is a group under composition, meaning that ∀ x ∈ A ∀ g ∈ G ( = ), because G satisfies the following four conditions:

Then the following month, on May 31st 1902, in a letter to William Blackwood, Conrad remarks:

Then the generation following them received it, thus conveying it to those after them and so on.

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.

This melding of scientific and social speculation is clearly present in the novel Voyage from Yesteryear ( strongly influenced by Eric Frank Russell's famous story " And Then There Were None ") about a high-tech anarchist society in the Alpha Centauri system, a starship sent from Earth by a dictatorial government, and the events following their first contact.

Then the following formula is an axiom:

" Then following multiple protest from Jews in the new world and Holland and after his death they were even granted the right to pray, besides just trade.

Then the following are equivalent.

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

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

Then in March of the same year, following the insurrection of the Paris Commune the French government under Thiers relocated to Versailles, from where the insurrection was militarily quelled.

Then the following dynamical system describes PLL behavior

Then the priest blesses kolyva ( boiled wheat with honey and raisins ) which is distributed to the faithful in commemoration of the following miracle worked by St. Theodore on the First Saturday of Great Lent: Fifty years after the death of St Theodore, the emperor Julian the Apostate ( 361-363 ), as a part of his general policy of persecution of Christians, commanded the governor of Constantinople during the first week of Great Lent to sprinkle all the food provisions in the marketplaces with the blood offered to pagan idols, knowing that the people would be hungry after the strict fasting of the first week.

Then, Cicero had the incriminating letters read before the Senate the following day, and shortly thereafter these five conspirators were condemned to death without a trial despite an eloquent protest by Julius Caesar.

Then, to construct the elements of following rows, add the number above and to the left with the number above and to the right to find the new value.

Then, each person decorates their page following the theme.

Then write a review without using any of the following words: " Progressive rock ", " Genesis ", " Fish ", " heavy metal ", " dinosaurs ", " predictable ", " concept album ".

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 the dummy variables can be included as independent ( explanatory ) variables in a regression.

Then a regression of z on y and x will yield an R < sup > 2 </ sup > of zero, while a regression of y on x and z will yield a positive R < sup > 2 </ sup >.

Then the null hypothesis of no Granger causality is retained if and only if no lagged values of an explanatory variable have been retained in the regression.

Then the arithmetic mean is defined via the equation

Then the momentum equation can be expressed as

Then the mass balance equation can be written as

Then the mass balance equation takes the form

Then the mass balance equation can be expressed as

Then the solution of the acoustic wave equation is

Then the second Cauchy – Riemann equation ( 1b ) asserts that is irrotational:

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 dividing both side by would result in the “ Two-point form ” shown above, but leaving it here allows the equation to still be valid when.

Then the pair ( x < sub > 1 </ sub >, y < sub > 1 </ sub >) solving Pell's equation and minimizing x satisfies x < sub > 1 </ sub >

Then there exists a finite set of integer coefficients satisfying the equation:

Then the wave equation is to be satisfied if x is in D and.

Then the solution of the Laplace equation inside the sphere is given by

Then, the kinetic equation for relative populations can be written as follows:

Then, again, the conditional wave function of subsystem ( I ) is ( up to an irrelevant scalar factor ) equal to and if the Hamiltonian does not contain an interaction term between subsystems ( I ) and ( II ), satisfies a Schrödinger equation.

Then given the Hamiltonian operator, the equation to satisfy for all functions ( with associated multiplication operator ) is

Then, by Gauss's law, the potential satisfies Poisson's equation:

Let be the mean of the values in associated with class c, and let be the variance of the values in associated with class c. Then, the probability of some value given a class,, can be computed by plugging into the equation for a Normal distribution parameterized by and.

Then it is possible to invert or solve the equation for w:

Then the Black – Scholes PDE becomes a diffusion equation

Then standard methods can be used to solve the linear difference equation in.

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

Then, combined with the wage Phillips curve 1 and the assumption made above about the trend behavior of money wages 2, this price-inflation equation gives us a simple expectations-augmented price Phillips curve:

Then any solution of the inhomogeneous equation may have a solution of the homogeneous equation added to it, and still remain a solution.