Then, if marginal revenue is greater than marginal cost at some level of output, marginal profit is positive and thus a greater quantity should be produced, and if marginal revenue is less than marginal cost, marginal profit is negative and a lesser quantity should be produced.

Then Y < sub > i </ sub > can be viewed as an indicator for whether this latent variable is positive:

Let X be a g-dimensional torus given as X = V / L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L × L.

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

Then the lens was focused and the subject framed and the picture was taken, the photographer flipped a switch and pulled the large tab in the back of the camera to pull the negative over the positive, through some rollers to spread the developing agent.

Then there exist positive integers a and b such that e = a / b where clearly b > 1.

Then due to positive response, the company lifted the cap and decided to produce as many 1M Coupés as the market requires until production ends in June 2012.

Then, this functional is positive ( the integral of any positive function is a positive number ).

Then, to all positive developments the Thirty Years ' War put an end.

Then is Artinian for every positive integer n.

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 Kamal's attitude to life changes to the positive as he starts to see himself as ' idealistic ' teacher, future husband and revolutionary man.

Then we should be able to require the existence of n such that N − F ( n ) is both positive and a prime number ; and with all the f < sub > i </ sub >( n ) prime numbers.

Then we learn a classifier that can discriminate between positive and negative examples as a function of the features.

Suppose K is a symmetric, positive definite kernel on a set E. Then there is a unique Hilbert space of functions on E for which K is a reproducing kernel.

Then keep your thoughts focused in a positive manner as you listen to their advice and guidance, and keep in mind the kind of person you would like to be, relying on the advice and examples of those sitting at that table.

Then, we use the so-called keyhole contour, which consists of a small circle about the origin of radius ε say, extending to a line segment parallel and close to the positive real axis but not touching it, to an almost full circle, returning to a line segment parallel, close, and below the positive real axis in the negative sense, returning to the small circle in the middle.

Then the puller starts to move along the y axis in the positive direction.

Then the natural assumption, given that all atoms are detected, is that all atoms the projection of whose spin in the direction a is positive will be detected as spin up ( coded as + 1 ) while all whose projection is negative will be detected as spin down ( coded as − 1 ).

" While reviewing the blu-ray DVD version of the film, Chris Cabin of Slant Magazine gave the film a positive three-and-a-half out of five rating, calling it a " high-end, kitschy whodunit ", and writing that it is " riotous and chaotic take on the spy thriller, essentially, but it structurally resembles Agatha Christie's And Then There Were None " as well as describing it as " some sort of miraculous entertainment ".

Then choosing any smooth positive quadratic form on gives a sub-Riemannian metric on the group.

Then the enthusiasm and energy of all elements can be channeled to produce cumulative progress toward a common objective.

Then the required information is collected from a simple random sample of the elements within each selected group.

Then all elements of X equivalent to each other are also elements of the same equivalence class.

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

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 R (( G )) is the ring of formal power series on G ; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.

Then the terms of can be tabulated as elements of an matrix:

Then, elements are sorted within each bin

Then, providing a localization package for a given language is in itself a non-trivial undertaking, requiring specialized technical writers to construct a culturally appropriate syntax for potentially complicated concepts, coupled with engineering resources to deploy and test the localization elements.

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.

Then they are allowed to inherit, as long as they marry a member of their tribe, in order to prevent the transfer of their share of the tribal lands to the hands of elements of other tribes and so they do, by marrying their uncles ' sons.

Then evaluating f again can never add any more elements.

Then the union ( written as +) and the concatenation ( written as ·) of two elements of A again belong to A, and so does the Kleene star operation applied to any element of A.

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.

Then God ordered the elements into the seven heavens ( often seen as the spheres of Mercury, Venus, Mars, Jupiter, Saturn, the Sun and the Moon ) to travel in circles and govern destiny.

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:

Then the elements of Z are given by:

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

Then two elements of X belong to the same orbit precisely when one is simply a rotation of the other.

Then began a course of conduct that would extend throughout the twentieth century, an isolationist foreign policy interspersed with sporadic automatic alignments against " disturbing elements of peace and international trade "

Then the equaliser of f and g is the set of elements x of X such that f ( x ) equals g ( x ) in Y.

Let and let be n + 1 distinct elements of K. Then for and by Lagrange interpolation we have.