In 1940, Ulam proposed the general Ulam stability problem see 1 . Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a mapping h : G1 → G2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G1 then there is a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1? In 1941, this problem was solved by Hyers 2 i...

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space E, and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of R, and we discuss their extension to E. We introduce the notion of E-metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of con...

The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function T whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting M = R×S, g = −β(T , x)dT 2 + ḡT , (b) if a spacetime M admits a...

The main object of this paper is to introduce a new concept of weak statistically Cauchy sequence in a normed space. It is shown that in a reflexive space, weak statistically Cauchy sequences are the same as weakly statistically convergent sequences. Finally, weak statistical convergence has been discussed in l p spaces. under the Creative Commons Attribution License, which permits unrestricted...

The steepest descent algorithm with exact line searches (Cauchy algorithm) is inefficient, generating oscillating step lengths and a sequence of points converging to the span of the eigenvectors associated with the extreme eigenvalues. The performance becomes very good if a short step is taken at every (say) 10 iterations. We show a new method for estimating short steps, and propose a method al...

1 A Model Diffusion Problem . . . . . . . . . . . . . . . . . . . . 1 x.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 1 x.2 Continuous Function . . . . . . . . . . . . . . . . . . . 2 x.3 Normed Vector Space . . . . . . . . . . . . . . . . . . 2 x.4 Square Integrable Function . . . . . . . . . . . . . . . 3 x.5 Inner Product Space . . . . . . . . . . . . . . . . . . . 4 x.6 Cauch...

The Cauchy problem for one-dimensional flow of a compressible viscous heat-conducting micropolar fluid is considered. It is assumed that the fluid is thermodynamically perfect and polytropic. A corresponding initial-boundary value problem has a unique strong solution on ]0, 1[×]0, T [, for each T > 0. By using this result we construct a sequence of approximate solutions which converges to a sol...

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...

Continued fractions in R have a single definition and algorithms for approximating them are well known. There also exists a well known result which states that √ m, m ∈ Q, always has a periodic continued fraction representation. In Qp, the field of p-adics, however, there are competing and non-equivalent definitions of continued fractions and no single algorithm exists which always produces a p...