The crease set of an event horizon or a Cauchy horizon is an important object which determines qualitative properties of the horizon. In particular, it determines the possible topologies of the spatial sections of the horizon. By Fermat’s principle in geometric optics, we relate the crease set and the Maxwell set of a smooth function in the context of singularity theory. We thereby give a class...

The concepts of I-convergence and I-Cauchy condition are a generalization of statistical convergence and statistical Cauchy conditions and are dependent on the notion of the ideal I of subsets of the set N of positive integers. In this paper, we shall introduce two new notions of I-uniform continuity and I-uniform boundedness of a function with values in R or in a metric space and then study th...

We explore a function theory connected with the principal series representation of SL(2, R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. ∗Supported by grant 3GP03196 of the FWO-Vlaanderen (Fund of Scientific ResearchFlanders), Scientif...

Katsaras 1 defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view 2–4 . In particular, Bag and Samanta 5 , following Cheng and Mordeson 6 , gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type 7 . T...

In this paper we first give a unified method by introducing the concept of admissible triplets to study local and global Cauchy problems for semi-linear parabolic equations with a general nonlinear term in different Sobolev spaces. In particular, we establish the local well-posedness and small global well-posedness of the Cauchy problem for semi-linear parabolic equations without the homogeneou...

By space-fractional (or L evy-Feller) diiusion processes we mean the processes governed by a generalized diiusion equation which generates all L evy stable probability distributions with index (0 < 2), including the two symmetric most popular laws, Cauchy (= 1) and Gauss (= 2). This generalized equation is obtained from the standard linear diiusion equation by replacing the second-order space d...

A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe the wave-front si...

We study the matching of LRS spatially homogeneous collapsing dust space-times with non-stationary vacuum exteriors in cylindrical symmetry. Given an interior with diagonal metric we prove existence and uniqueness results for the exterior. The matched solutions contain trapped surfaces, singularities and Cauchy horizons. The solutions cannot be asymptotically flat and we present evidence that t...

We explore a function theory connected with the principal series representation of SL(2, R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. ∗Supported by grant 3GP03196 of the FWO-Vlaanderen (Fund of Scientific ResearchFlanders), Scientif...