* 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, on paper, they bought $ 200 worth of each, for a total bet of $ 1, 000, using the prices on September 29, 1980, as an index.

Then, there is the point that a rise in interest rates designed to halt inflation could paradoxically make inflation appear higher if current interest rates showed up in the index.

Then 2Z has two cosets in Z ( namely the even integers and the odd integers ), so the index of 2Z in Z is two.

Then the cardinality of the orbit of x under G is equal to the index of the stabilizer of x:

Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set.

Then make the right index finger pick up the string on the left hand going between the thumb and the little finger.

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.

restricted to E. Then the analytical index of D is the holomorphic Euler characteristic of V:

Then the ith cohomology group is just the coherent cohomology group H < sup > i </ sup >( X, V ), so the analytical index of this complex is the holomorphic Euler characteristic Σ (− 1 )< sup > i </ sup > dim ( H < sup > i </ sup >( X, V )).

where the P < sub > i </ sub > are distinct prime ideals of S. Then P is said to ramify in L if e ( i ) > 1 for some i. In other words, P ramifies in L if the ramification index e ( i ) is greater than one for any P < sub > i </ sub >.

Then is an estimate of the standard deviation of for person with a given weighted score and the separation index is obtained as follows

for some index j and some element h of H, Then in general

Then wiggle the fingers upwards ( to show the rain drying in the sun ), and repeat the thumb / index finger movement to indicate the spider climbing up the spout.

Then, objects located at the same index in each array are implicitly the fields of a single record.

Then, for each integer n define n ( G ) to be the number of subgroups U of index n in G. Similarly, if G is a topological group, s_n ( G ) denotes the number of open subgroups U of index n in G. One similarly defines ' m_n ( G ) and to denote the number of maximal and normal subgroups of index n, respectively.

Let G be a group, U a subgroup of index n. Then G acts on the set of left cosets of U in G by left shift:

Then, X is said to locally satisfy some property P if there exists a sequence of stopping times < sub > n </ sub >, which increases to infinity and for which the processes satisfy property P. Common examples, with time index set I =

Pick a closed ball D centered at x, so that x is the only zero of v in D. Then we define the index of v at x, index < sub > v </ sub >( x ), to be the degree of the map u :∂ D → S < sup > n-1 </ sup >, u ( z )= v ( z )/| v ( z ) |.

where S ranges across subsets of of size n − 1, and F < sub > S </ sub > denotes the ( n − 1 )- by -( n − 1 ) matrix whose columns are those of F with index in S. Then every S specifies n − 1 edges of the original graph, and it can be shown that those edges induce a spanning tree iff the determinant of F < sub > S </ sub > is + 1 or − 1, and that they do not induce a spanning tree iff the determinant is 0.

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 every component of the graph of F must be defined over a bounded sub-interval.

Then the arithmetic mean is defined via the equation

) 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 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 the periodic Bernoulli functions P < sub > n </ sub > are defined as

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then the expectation of this random variable X is defined as

Then, Gödel defined essences: if x is an object in some world, then the property P is said to be an essence of x if P ( x ) is true in that world and if P entails all other properties that x has in that world.

Then and become partially defined operations on G, and will in fact be defined everywhere ; so we define * to be and to be.

Then 2-polytopes ( polygons ) are defined as plane objects whose bounding facets ( edges ) are 1-polytopes, 3-polytopes ( polyhedra ) are defined as solids whose facets ( faces ) are 2-polytopes, and so forth.

Then I and I < sup > 2 </ sup > are real vector spaces, and T < sub > x </ sub > M may be defined as the dual space of the quotient space I / I < sup > 2 </ sup >.

Then the directional derivative of is a scalar defined as

Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is approximately

Then he defined it for more complicated functions as the least upper bound of all the integrals of simple functions smaller than the function in question.

Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol.

Then the set of satisficing options S ( ε ) can be defined as all those options s such that.

Then, it needs to be verified that ( 1 ), ( 2 ), and ( 3 ) are well defined.

Then the system settles down again to a state of thermal equilibrium, defined by an entropy and a volume which differ infinitesimally from the initial values.

Then X is defined by the formula so X is in ( actually it is in as well since we could bound both quantifiers by n ).