Can one hear the shape of a drum

PROF HODGE's work on this subject has long been awaited by his colleagues, and will be generally welcomed by them, though perhaps with mixed feelings when its difficulty is encountered His lectures and papers have already given some idea of the scope the work would have, as well as of the origin of the line of research of which this book is the first important monument It is in effect an attempt to generalize in a very wide way that aspect of the study of functions (such as Abelian integrals) on a Riemann surface which consists in treating the real and imaginary parts separately, and adjoining to the integrability condition a second differential condition which ensures that the two functions considered are in fact the real and imaginary parts of an analytic function (This latter we may call the orthogonality condition) The place of the Riemann surface is taken by a ‘Riemannian manifold’ of any number of dimensions; this is continuous and has ordinary differential geometry in the small, with metric, affine connexion, and Riemann–Christoffel tensor; while in the large it is a two–sided or orientable space, to which the ordinary combinatory topology applies, with Betti numbers, torsions and intersection matrices The Theory and Applications of Harmonic Integrals By Prof W V D Hodge Pp ix + 281 (Cambridge: At the University Press, 1941) 12s net

The Theory and Applications of Harmonic Integrals

La problematique centrale de cette these est l'elaboration d'un operateur de Laplace--Beltrami discret sur les surfaces digitales. Ces surfaces proviennent de la theorie de la geometrie discrete, c’est-a-dire la geometrie qui s'interesse a des sous-ensembles des entiers relatifs. Nous nous placons ici dans un cadre theorique ou les surfaces digitales sont le resultat d'une approximation, ou processus de discretisation, d'une surface continue sous-jacente. Cette methode permet a la fois de prouver des theoremes de convergence des quantites discretes vers les quantites continues, mais aussi, par des analyses numeriques, de confirmer experimentalement ces resultats. Pour la discretisation de l’operateur, nous faisons face a deux problemes : d'un cote, notre surface n'est qu'une approximation de la surface continue sous-jacente, et de l'autre cote, l'estimation triviale de quantites geometriques sur la surface digitale ne nous apporte pas en general une bonne estimation de cette quantite. Nous possedons deja des reponses au second probleme : ces dernieres annees, de nombreux articles se sont attaches a developper des methodes pour approximer certaines quantites geometriques sur les surfaces digitales (comme par exemple les normales ou bien la courbure), methodes que nous decrirons dans cette these. Ces nouvelles techniques d'approximation nous permettent d'injecter des informations de mesure sur les elements de notre surface. Nous utilisons donc l'estimation de normales pour repondre au premier probleme, qui nous permet en fait d'approximer de facon precise le plan tangent en un point de la surface et, via une methode d'integration, palier a des problemes topologiques liees a la surface discrete. Nous presentons un resultat theorique de convergence du nouvel operateur discretise, puis nous illustrons ensuite ses proprietes a l’aide d’une analyse numerique de l’operateur. Nous effectuons une comparaison detaillee du nouvel operateur par rapport a ceux de la litterature adaptes sur les surfaces digitales, ce qui nous permet, au moins pour la convergence, de montrer que seul notre operateur possede cette propriete. Nous illustrons egalement l’operateur via quelques unes de ces applications comme sa decomposition spectrale ou bien encore le flot de courbure moyenne

Opérateur de Laplace–Beltrami discret sur les surfaces digitales

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Opérateur de Laplace–Beltrami discret sur les surfaces digitales