We study different completeness definitions for two categories of lattice-valued Cauchy spaces and the relations between these definitions. We also show the equivalence of a so-called completion axiom and the existence of a completion.

the main purpose of this paper is to establish some new results onthe superstability and stability via a fixed point approach forthe pexiderized exponential equation, i.e.,$$|f(x+y)-g(x)h(y)|leq psi(x,y),$$where $f$, $g$ and $h$ are three functions from an arbitrarycommutative semigroup $s$ to an arbitrary unitary complex banachalgebra and also $psi: s^{2}rightarrow [0,infty)$ is afunction. fur...

Let DN be the linear space spanned by the ej for j ≤ N then if the orthogonal decomposition H = DN ⊕D⊥ N gives u = u′ + u′′ and the distance d(u,DN ) = ‖u′′‖ so the finite dimensional approximation property follows from (2). Conversely, suppose that K is closed and bounded and has the FDAP. By the assumed boundedness, any sequence in K has a weakly convergent subsequence. Denote such a sequence...

The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in Gene...

A generalized Cauchy-Kowalevski approach is proposed for flatness-based trajectory planning for boundary controlled semilinear systems of PDEs in a one-dimensional spatial domain. For this, the ansatz presented in [16] using formal integration is generalized towards a unified design framework, which covers linear and semilinear PDEs including rather broad classes of nonlinearities arising in ap...

The sequence X = {Ark } of fuzzy numbers is statistically convergent to the fuzzy number 3(o provided that for each e ~ 0 lim l{the number ofk~e} = 0. n In this paper we study a related concept of convergence in which the set {k: k<~n} is replaced by {k: kr-1 -~k<~kr} for some lacunary sequence {k~}. Also we introduce the concept of lacunary statistically Cauchy sequence and show...

A Theorem is proved which reduces the problem of completeness of orbits of Killing vector fields in maximal globally hyperbolic, say vacuum, space–times to some properties of the orbits near the Cauchy surface. In particular it is shown that all Killing orbits are complete in maximal developements of asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact manifold.

Given a globally hyperbolic spacetime, the existence of a (smooth) spacelike Cauchy hypersurface S has been proven recently. Here, we prove that any acausal spacelike compact submanifold with boundary can be smoothly extended to a spacelike Cauchy hypersurface. Apart from the interest as a purely geometric question (applicable to the Cauchy problem in General Relativity), the result is motivate...

In this paper, the existence and uniqueness theorems for a solution to an uncertain characteristic Cauchy reaction-diffusion problem is studied by Adomian decomposition method. Sufficient conditions are presented for uniform convergence of the proposed method. Also, some illustrative examples are given for solving the problem. Key WordsUncertain characteristic Cauchy problem, System of characte...

Kernel sparse coding is an effective strategy to capture the non-linear structure of data samples. However, how to learn a robust kernel dictionary remains an open problem. In this paper, we propose a new optimization model to learn the robust kernel dictionary while isolating outliers in the training samples. This model is essentially based on the decomposition of the reconstruction error into...