Nassar H. Abdel-All

Geometrical properties of Pareto distribution

The differential-geometrical framework for analyzing statistical problems related to Pareto distribution, is given. A classical and intuitive way of description the relationship between the differential geometry and the statistics, is introduced [Publicationes Mathematicae Debrecen, Hungary, vol. 61 (2002) 1-14; RAAG Mem. 4 (1968) 373; Ann. Statist. 10 (2) (1982) 357; Springer Lecture Notes in Statistics, 1985; Tensor, N.S. 57 (1996) 282; Commun. Statist. Theor. Meth. 29 (4) (2000) 859; Tensor, N.S. 33 (1979) 347; Int. J. Eng. Sci. 19 (1981) 1609; Tensor, N.S. 57 (1996) 300; Differential Geometry and Statistics, 1993], but in a slightly modified manner. This is in order to provide an easier introduction for readers not familiar with differential geometry. The parameter space of the Pareto distribution using its Fishers matrix is defined. The Riemannian and scalar curvatures to parameter space are calculated. The differential equations of the geodesics are obtained and solved. The J-divergence, the geodesic distance and the relations between of them in that space are found. A development of the relation between the J-divergence and the geodesic distance is illustrated. The scalar curvature of the J-space is represented.

Information geometry and statistical manifold

Abstract A brief account of information geometry and the deep relationship between the differential geometry and the statistics is given [N.H. Abdel-All, International Conference on Differential Geometry and its Applications, Cairo University, 19–26 June, Egypt, 2001; Springer Lecture Notes in Statistics, vol. 28, 1985; Math. Syst. Theory 20 (1987) 53]. The parameter space of the random walk distribution (first passage time distributions of Brownian motion) using its Fishers matrix is defined. The Riemannian and scalar curvatures in a parameter space are calculated. The differential equations of the geodesic are obtained and solved. The J -divergence, the geodesic distance and the relations between of them in that space are found in N.H. Abdel-All and elsewhere [Math. Comput. Model. 18 (8) (1993) 83; Bull. Calcutta Math. Soc. 37 (1945) 81].

Non-transversal intersection curves of hypersurfaces in Euclidean 4-space

In this paper, we consider the problem of how to compute the Frenet apparatus of the non-transversal intersection curves (hyper-curves) of three hypersurfaces (given by their implicit-implicit-parametric and implicit-parametric-parametric equations) in Euclidean 4-space. The non-transversal intersection (in which the normal vectors of the intersecting hypersurfaces are linearly dependent) includes two different cases. In each case, on the contrary to transversal intersections, we face some difficulties even finding the tangential direction. To overcome such difficulties, we give some algorithms for finding all Frenet vectors and curvatures of the intersection curve. Finally, we demonstrate our methods by giving several examples.

A geometric characterisation of two-parameter spatial motions with many locally one-dimensional point paths

In this paper, we present a differential geometric local study of two-parameter spatial motions: we look for points, which (up to the second order) instantaneously move on one-dimensional point paths, further we locally characterise these motions, which move ~^1 points in these way.

Deformation of a kinematic surface in a hyperbolic space

Abstract We give a representation of the variational problem on a time (space) like surface immersed in a hyperbolic space. The geometric properties of the deformed surfaces are given. The variational problem for the Klein images of 2-parametric continuous motion of a line (kinematic surface) are introduced. The theory of Klein images is applied to a time like and space like congruence. Finally, additional results are mentioned in the concluding remarks. The technique adapted here is based on Cartans methods of moving frames, exterior differential forms and the group of motions [Result. Math. 39 (2001) 1; Tensor. N.S. 51 (3) (1992) 224; Tensor. N.S. 59 (1998) 36; Collect. Math. 45 (1994) 153].

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence CN. The mixed middle curvature and mixed curvature on CN are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].

Fuzzy sets for fuzzy context model

In the first part an overview on fuzzy sets and fuzzy numbers is given. A detailed treatment of these notions is introduced in [1,2,3]. This sintetically presentation is useful in understanding and in developping the applications in context problems. In the second part, fuzzy context model is given as an application of fuzzy sets and the fuzzy equilibrium equation is solved [4,5].

Perturbations of curvatures flow in a hyperbolic space

A representation of a special type of the variational problem on a surface immersed in a hyperbolic space is given. The perturbations of the mean curvature functional are studied. For this study, the corresponding frames are constructed and the polar surfaces of a given surface is established. The technique adapted here is based on Cartans methods of moving frames, exterior differential forms and the group of motions [Result. Math. 39 (2001) 1; Tensor. N.S. 51 (3) (1992) 224; Tensor. N.S. 59 (1998) 36; Collect. Math., universtat de Barcelona 45 (1994) 153].