Mathematical Framework and Computational Results Associated with Rational Points on Fermat’s Surfaces in Minkowski’s (N+1) - Dimensional Spaces and Extended Fermat’s Last Theorem | Chapter 10 | Research Updates in Mathematics and Computer Science Vol. 7

 

 

Despite the elaborate Wiles demonstration, Fermat’s last theorem still attracts researchers to this aspect of number theory. This study has been structured in two parts. The first one describes the Minkowski natural space basis, where the discussion of Fermat’s theorem and the extensions to higher dimensional spacesss. In the second, many examples of N-dimensional vectors obeying a Fermat-like rule for various powers are presented and discussed. The Fermat last theorem, defined in (2+1)-dimensional Minkowski spaces, is discussed and extended in natural and rational Mikowski’s spaces. Several pieces of computational interest are given, with many practical examples. A definition of Fermat vector order, Fermat surfaces, and Fermat surface radius is given. Several conjectures are discussed, among them the existence of a Fermat theorem in (3+1)-dimensional Minkowski spaces.

 

Author(s) Details

Ramon Carbó-Dorca

Institut de Química Computacional, Universitat de Girona, Girona 17005 (Catalonia), Spain and Ronin Institute, 127 Haddon Place, Montclair N. J. 07043, USA.

 

Please see the book here:- https://doi.org/10.9734/bpi/rumcs/v7/1756G

 

Investigation of a New Ti Alloy for a New Generation of Additively Manufactured Implants with Lattice | Chapter 2 | Contemporary Perspective on Science, Technology and Research Vol. 7

 A new titanium alloy improving the operation of implants additively manufactured and including laterally closed lattice structures is proposed. The new alloy possesses an increased affinity to the bone. The measured bone–interface implant (BII) of less than 10 mm and bone–implant contact (BIC) of 95% demonstrated an excellent osseointegration. Furthermore, since additive manufacturing naturally leads to a high-roughness surface finish, the wettability of the implant is increased. The combination of these factors is pushing ossification beyond its natural limits. In addition, the quality and speed of the ossification and osseointegration in/around laterally closed lattice implants open the possibility of bone spline key of prostheses. This enables the stabilization of the implant into the bone while keeping the possibility of punctual hooks allowing the implant to be removed more easily if required.

Author(s) Details:

Anne-Françoise Obaton,
Laboratoire National de Métrologie et d’Essais (LNE), 75015 Paris, France.

Jacques Fain,
Z3DLAB, 95270 Chaumontel, France.

Dietmar Meinel,
8.5-Micro-NDT, Bundesanstalt für Materialforschung Und-Prüfung (BAM), 12205 Berlin, Germany.

Athanasios Tsamos,
8.5-Micro-NDT, Bundesanstalt für Materialforschung Und-Prüfung (BAM), 12205 Berlin, Germany.

Fabien Léonard,
8.5-Micro-NDT, Bundesanstalt für Materialforschung Und-Prüfung (BAM), 12205 Berlin, Germany.

Benoît Lécuelle,
Centre de Recherche Biomédicale, Ecole Nationale Vétérinaire d’Alfort, 94700 Maisons-Alfort, France.

Madjid Djemaï,
Z3DLAB, 95270 Chaumontel, France.

Giovanni Bruno,
8.5-Micro-NDT, Bundesanstalt für Materialforschung Und-Prüfung (BAM), 12205 Berlin, Germany.

Please see the link here: https://stm.bookpi.org/CPSTR-V7/article/view/13739