Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer.
2.
The Index Theorem states that this " analytical index " is constant as you vary the elliptic operator smoothly.
3.
:: What appears to be wanted is an analytical index to Wikipedia, rather like the one published for the 11th edition of Britannica.
4.
Indeed, it was a knowledge of such misapprehensions existing in several quarters, that first suggested the necessity of appending to the work an Analytical Index of its various contents.
5.
The 2-dimensional Gauss Bonnet Theorem arises as the special case where the topological index is defined in terms of Betti numbers and the analytical index is defined in terms of the Gauss Bonnet integrand.
6.
He translated several critical editions of works ( " The Glories of Mary " and " The Passion and Death of Jesus Christ " ) by the founder of the Redemptorists, St . Alphonsus Liguori, and also helped prepare an analytical index of the Redemptorist Constitution and Statutes.
7.
She translated Fire and Flame ( German : " Feuer und Flamme " ) by Levin Sch�cking ( New York, 1876 ), and prepared " An Analytical Index to the Works of Nathaniel Hawthorne " ( Boston, 1882 ) and " An Index to the Works of Shakspere " ( New York and London, 1887 ).
8.
So this makes it possible to evaluate the analytical index . ( The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually; the index theorem shows that we can usually at least evaluate their "'difference "'. ) Many important invariants of a manifold ( such as the signature ) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data.
9.
With the copious and prepared bibliography of Philippine historical literature, and the full analytical index, which will close the series; the broad and representative character of the material selected throughout; and the impartial and non-sectarian attitude maintained, the Editors trust that this change will still further enable scholars, historical writers, and general readers alike to study, with reliable and satisfactory material, the history of the Philippine Islands from their first discovery by Europeans to the close of the Spanish r�gime, and incidentally the history of the entire Orient . } } }}
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