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Then explicit matrix computation shows that
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Then and explicit
Then an enriched category C ( alternatively, in situations where the choice of monoidal category needs to be explicit, a category enriched over M, or M-category ), consists of
Then each makes a more explicit recommendation on policy for the coming intermeeting period ( and for the longer run, if under consideration ).
Then substituting back f ( r )/ p < sup > k </ sup > for z and solving for t in gives the explicit formula for t mentioned above.
Then and matrix
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 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 inverse of this matrix is S, the lower triangular matrix of Stirling numbers of second kind.
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, the individual normal equation matrices were combined and the resultant matrix solved to obtain the positions and the parameters.
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, where is the vector ( 1, 0 ,..., 0 )< sup > T </ sup >, ||·|| is the Euclidean norm and is an m-by-m identity matrix, set
Then and computation
Then this can be seen by a direct computation for n any integer, and the general case reduces to this case without much difficulty.
Then one can use the Atiyah – Bott fixed-point theorem, of Atiyah and Bott, to reduce, or localize, the computation of a GW invariant to an integration over the fixed-point locus of the action.
Then and shows
Then she shows him a long, narrow case, which she tells him holds her only sacred treasure, but she cannot open it, because there are too many people around.
Then on August 30, 2012, Eddie Van Halen was diagnosed with diverticulitis and underwent surgery which will push the planned November shows in Japan back into 2013 at the earliest.
Then in 2005, a two-hour block featuring Nick Jr. shows returned on most CBS stations until September 2006 after the Viacom-CBS split, when the airtime was leased to DiC Entertainment and then later DiC purchaser Cookie Jar Group for their Cookie Jar TV block.
Then, Flipper's brother Gator ( Samuel L. Jackson ) shows up with his girlfriend Vivian ( Halle Berry ).
Hanna recalled, " Acker asked me why writing was important to me, and I said, ' Because I felt like I'd never been listened to and I had a lot to say ,' and she said, ' Then why are you doing spoken word — no one goes to spoken word shows!
Then by mathematical induction ( infinitely repeating the same step ), one shows there are a yet smaller, then a yet even smaller, example, and hence there must be an infinitude of ever smaller examples.
Then using the Bondesson series representation for generating the variance gamma process shows to increase performance when pricing this type of option.
Then, after four shows where 2 of 10 were voted forward in each show, a fifth show was aired, where eight " wildcards " selected by the judges from all of the earlier contestants competed for the two last places.
Then after five shows where 2 of 8 were voted forward in each show, a sixth show was aired, where eight " wildcards " selected by the judges from all the earlier contestants competed for the two last places, making the total of finalists to 12.
Then the comic shows a small fish turn to yellow energy near Warlock's head signalling that he is not out for the count.
Then, one series later, they pulled the plug on both shows, deciding that there was no demand for games coverage on their channel.
Then in 2005, they performed a number of reunion shows in Toronto, including a double bill with the also-reunited Parachute Club on 14 May.
Then the problem is to get stamps to " fill the holes "; only a handful of dealers will have a complete stock, and so the want list may take on a life of its own as the collector carries it to stamp shows, mail copies to out-of-town dealers, and so forth, gradually crossing off entries as they are located and purchased.
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