 Page "Stirling number" ¶ 26
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Then and inverse Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse Let be an inverse of Then: Then the inverse of A can be found as follows. Then a general definition of isomorphism that covers the previous and many other cases is: an isomorphism is a morphism that has an inverse, i. e. there exists a morphism with and. Then the pair (( A < sub > i </ sub >)< sub > i ∈ I </ sub >, ( f < sub > ij </ sub >)< sub > i ≤ j ∈ I </ sub >) is called an inverse system of groups and morphisms over I, and the morphisms f < sub > ij </ sub > are called the transition morphisms of the system. * 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 F and G can be restricted to D < sub > 1 </ sub > and C < sub > 1 </ sub > and yield inverse equivalences of these subcategories. Then the convolution of f with itself is proportional to the inverse discrete Fourier transform of the pointwise product of Y with itself. Then, hit the inverse tangent key ( or tan < sup >- 1 </ sup >) and the answer just calculated. Then each double coset is of the form HxK, and G is partitioned into its ( H, K ) double cosets ; each of them is a union of ordinary right cosets Hy of H in G and left cosets zK of K in G. In another aspect, these are in fact orbits for the group action of H × K on G with H acting by left multiplication and K by inverse right multiplication. Let f < sup >− 1 </ sup > denote the inverse function of f. Then, for all a ∈ and b ∈, Then is an inverse semigroup under this multiplication, with Then is an E-unitary inverse semigroup. Then for all the mapping is a bijection with inverse.

Then and matrix Then the bra can be computed by normal matrix multiplication. Then the trace of the identity matrix is the dimension of the space ; this leads to generalizations of dimension using trace. Then, the problem of computing the function value can be rephrased as " zeroing-in " on the corresponding matrix entry. Then the joint distribution of is multivariate normal with mean vector and covariance matrix Then the density matrix for this system is Then the matrices would progress through the machine, where a special keying system on one end of the matrix, unique for each character, would allow the matrix to drop only into the correct storage slot, ready for future use. Then the terms of can be tabulated as elements of an matrix: 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 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: Then the existence of an eigenvalue is equivalent to the ideal generated by ( the relations satisfied by ) being non-empty, which exactly generalizes the usual proof of existence of an eigenvalue existing for a single matrix over an algebraically closed field by showing that the characteristic polynomial has a zero. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. Then, we sort the list by coordinates, and update the matrix as we go. Then M is called a symplectic matrix if it satisfies the condition Then, the individual normal equation matrices were combined and the resultant matrix solved to obtain the positions and the parameters. Then, as the matrix product is associative, Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f ( being the product of the eigenvalues of the Hessian ). Then the Jacobian matrix is Then explicit matrix computation shows that Then, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set Then is separable algebraic over if and only if the matrix is invertible. Then the Wishart distribution is the probability distribution of the p × p random matrix Then by the rules of matrix algebra we get:

Then and is 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 ; ;

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