Then, we say Q is a generator of a one parameter symmetry Lie group.

Then changing the system's temperature parameter by and the external variable by dx will lead to a change in:

Then the above implies that the exact vertex isoperimetric parameter is

Then, the order parameter space can be written as the Lie group quotient

For concreteness let us also suppose time is a parameter that ranges over the set of real numbers R. Then time evolution is given by a family of bijective state transformations

Then, all the non-analyticity at the critical point, the critical exponents, are because the equilibrium value of the order parameter changes non-analytically, as a square root, whenever the free energy loses its unique minimum.

Then one objective of the econometrician is to obtain estimates of the parameters a and b ; these estimated parameter values, when used in the model's equation, enable predictions for future values of consumption to be made contingent on the prior month's income.

Then through the usual least squares calculation, the estimated parameter vector based only on the observed x-values but omitting the observed z values, is given by:

Then he used these theorems to give rigorous proofs of theorems proven by Fisher and Hotelling related to Fisher's maximum likelihood estimator for estimating a parameter of a distribution.

Then after saving the page, click " edit this page " again, and add the language parameter to the " User sgn " template.

Then the unconditional distribution of Z is non-central chi-squared with k degrees of freedom, and non-centrality parameter.

Then he would get to his feet, as though rising in honor of his own remarkable powers, and say almost invariably, `` Gentlemen, this is an amazing story!!

Then, Jesus indicated that God's forgiveness is unlimited.

Certainly, the meaning is clearer to one who is not familiar with Biblical teachings, in the New English Bible which reads: `` Then Jesus arrived at Jordan from Galilee, and he came to John to be baptized by him.

Then it added: `` It is not possible to determine how extensive these ill effects will be -- nor how many people will be affected ''.

Then the words fell into a pattern: `` Mollie the Mutton is scratching her nose, Scratching her nose in the rain.

Then he thought of Aaron Blaustein standing in his rich house saying: `` God is tired of taking the blame.

Then it is marked on the inside where it comes in contact with the transom, frames, keelson and all the battens.

Then it is replaced and fastened.

Then the chines are rounded off and the bottom is rough-sanded in preparation.

Then, a group of eggs is deposited in a cavity in the beebread loaf and the egg compartment is closed.

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

Then in 2 we show that any line involution with the properties that ( A ) It has no complex of invariant lines, and ( B ) Its singular lines form a complex consisting exclusively of the lines which meet a twisted curve, is necessarily of the type discussed in 1.

Then, too, the utmost clinical flexibility is necessary in judiciously combining carefully timed family-oriented home visits, single and group office interviews, and appropriate telephone follow-up calls, if the worker is to be genuinely accessible and if the predicted unhealthy outcome is to be actually averted in accordance with the principles of preventive intervention.

Then the editorial added prophetically: `` how far they may reach in Asia is yet undetermined, but they fall far short of our dreams of the war conferences ''.

Then she catapults into `` everything and everybody '', putting particular violence on `` everybody '', indicating to the linguist that this is a spot to flag -- that is, it is not congruent to the patient's general style of speech up to this point.

Then comes the time when the last wire is removed and Susie walks out a healthier and more attractive girl than when she first went to the orthodontist.

Then, with the new affluence, there is actually a sallying forth into the wide, wide world beyond the precincts of New York.

Then, if the middle number is activated to its greatest potential in terms of this square, through multiplying it by the highest number, 9 ( which is the square of the base number ), the result is 45 ; ;

Then, on July 2, there occurred another incident which set tongues to wagging at a furious clip.

Then O'Banion was presented with a platinum watch set with rubies and diamonds.

`` Then maybe it was the shockwave we set up.

Then our choice function can choose the least element of every set under our unusual ordering.

Then, taking in on the first cable as the boat is motored into the wind and letting slack while drifting back, a second anchor is set approximately a half-scope away from the first on a line perpendicular to the wind.

Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies.

Then codenamed " Stella ," the machine was also set to utilize cartridges ; after seeing the Channel F, Atari realized they needed to release it before the market was flooded with cartridge-based machines.

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 Asmund set himself to wait the agreed upon three days:

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 >.

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Then the equivalence class of the pair can be identified with the rational number, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

Then, in the grand final against Melbourne, Essendon set a grand final record score of 22. 18 ( 150 ) to Melbourne 13. 9 ( 87 ), with 7 goal centre half forward Gordon Lane.

Then one " flows " the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the discharging procedure.

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

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

* 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 Lothair finally set out with a large Lombard army, but Louis had promised his sons Louis the German and Pepin of Aquitaine greater shares of the inheritance, prompting them to shift loyalties in favour of their father.

Then worship Allah and do not associate anything with Him, perform the Hajj and the ‘ Umrah, be righteous, and all affairs will be set right for you.

Then the null set automatically has measure zero because of countable additivity, because and is finite if and only if the empty set has measure zero.

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.