The spherical image of singular varieties of bounded mean curvature

In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, which allows to extend the classical Coarea formula for the Gauss map of smooth varieties, and to introduce for all integral varifolds of bounded mean curvature a natural definition of second fundamental form, whose trace equals the generalized varifold mean curvature. In the second part, we use this machinery to extend a sharp geometric inequality of Almgren to all compact varieties of bounded mean curvature in the viscosity sense and we characterize the equality case. As a consequence we formulate sufficient conditions to conclude that the area-blow-up set is empty for sequences of varifolds whose first variation is controlled.Comment: One of the main results has been slightly improved (see 3.9) and a new corollary has been added (see 4.5). New title and abstract. The introduction has been also revised. To appear in Bull. Math. Sc

Fine properties of the curvature of arbitrary closed sets

Given an arbitrary closed set A of $\mathbf{R}^{n}$, we establish the relation between the eigenvalues of the approximate differential of the spherical image map of A and the principal curvatures of A introduced by Hug-Last-Weil, thus extending a well known relation for sets of positive reach by Federer and Zaehle. Then we provide for every $m = 1, \ldots , n-1$ an integral representation for the support measure $\mu_{m}$ of A with respect to the m dimensional Hausdoff measure. Moreover a notion of second fundamental form $Q_{A}$ for an arbitrary closed set A is introduced so that the finite principal curvatures of A correspond to the eigenvalues of $Q_{A}$. We prove that the approximate differential of order 2, introduced in a previous work of the author, equals in a certain sense the absolutely continuous part of $Q_{A}$, thus providing a natural generalization to higher order differentiability of the classical result of Calderon and Zygmund on the approximate differentiability of functions of bounded variation.Comment: 27 pages. This preprint expands sections 2-5 of version v1 of this submission. Sections 6-7 of v1 will be moved in seperate pre-print

Isotopic liftings of Clifford algebras and applications in elementary particle mass matrices

Isotopic liftings of algebraic structures are investigated in the context of Clifford algebras, where it is defined a new product involving an arbitrary, but fixed, element of the Clifford algebra. This element acts as the unit with respect to the introduced product, and is called isounit. We construct isotopies in both associative and non-associative arbitrary algebras, and examples of these constructions are exhibited using Clifford algebras, which although associative, can generate the octonionic, non-associative, algebra. The whole formalism is developed in a Clifford algebraic arena, giving also the necessary pre-requisites to introduce isotopies of the exterior algebra. The flavor hadronic symmetry of the six u,d,s,c,b,t quarks is shown to be exact, when the generators of the isotopic Lie algebra su(6) are constructed, and the unit of the isotopic Clifford algebra is shown to be a function of the six quark masses. The limits constraining the parameters, that are entries of the representation of the isounit in the isotopic group SU(6), are based on the most recent limits imposed on quark masses.Comment: 19 page

AquaFuel: An example of the emerging new energies and the new methods for their scientific study

In this paper we initiate studies of the emerging new forms of energy by using as a representative example the new combustible gas called AquaFuel, discovered and patented by William H. Richardson, jr., whose rights are now owned by Toups Technology Licensing, Inc. (TTL), of Largo, Florida. In essence, AquaFuel is a new energy converter capable of transforming Carbon and water into a new combustible gas via an electric discharge. We show that AquaFuel can be produced easily, safely and rapidly in large amounts, and exhibits greatly reduced emission pollutants as compared to fossil fuels of current use. Despite its simplicity, the chemical and physical characteristics of AquaFuel are largely unknown at this writing. We then review nine basic experimental measurements which are necessary for a scientific appraisal of AquaFuel. We outline the limitations of quantum mechanics and chemistry for the treatment of {\it new} forms of energy, namely, energies which by definition should be {\it beyond} said theories. We finally point out the availability of broader theories specifically constructed for the study of new energies and point out available applications.Comment: 22 pages, Textur