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 aim of this paper is to study the completeness of L-quasi-uniform convergence spaces and L-quasi-uniform spaces. Firstly, we describe L-quasi-uniform convergence spaces as enriched categories. Then we give two kinds of completeness of L-quasi-uniform convergence spaces and show that Lawvere completeness implies Cauchy completeness. Finally, we use the Cauchy completeness of L-quasi-uniform convergence spaces to define the Cauchy completeness of L-quasi-uniform spaces, and show that Cauchy completeness is equivalent to Lawvere completeness in L-quasi-uniform spaces.

In the category of ⊤-quasi-Cauchy spaces, completeness and completion can be studied in a non-symmetric framework encompassing ⊤-quasi-uniform (limit) spaces. Based on constructions by E.E. Reed in the category of Cauchy spaces and, recently, by L. Reid and G. Richardson in the category of ⊤-Cauchy spaces, we give a family of completions for a non-complete ⊤-quasi Cauchy space. As particular instances we study pretopological and topological completions of ⊤-quasi-Cauchy spaces.

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining probabilistic metric spaces. Then the Dirac function is presented as an important example of distributions functions. We also, introduced Sibley’ s metric or Levy metric on the set of distance distribution functions and so this space becomes a metric space. In the following, probabilistic metric spaces are defined from the Serstnev’ s view, and some examples such as Menger probabilistic metric spaces, are introduced. In this paper, the strong topology induced by distance distribution functions is introduced, and then probabilistic diameter, probabilistic bounded, probabilistic semi-bounded, probabilistic unbounded and probabilistic totally bounded sets are introduced. Also, we prove that in every probabilistic metric, every probabilistic totally bounded set is probabilistic bounded set. We also present the cantor intersection theorem and a formulation of Bair’ s Theorem in complete probabilistic metric spaces. In addition, we prove that the Heine-Borel property and the Bolzano-Wierestrass property are equivalent.

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Despite the fact that using the services of outsourced data stream servers has extremely been welcomed, but still the problem of obtaining certainty about received results from these servers is one of the basic challenges in enterprises. For outsourcing these services, the user should be assured by a mechanism about the security of communication channels as well as the correct and honest function of the server, because the server may attack the integrity of the results due to economic and malicious reasons. In such attacks, some parts of results are not sent to the user or sent after being modified or delayed. In this article, we have come up with an efficient method for detecting integrity attacks in outsourced data steam systems based on auditing the results of cross computation. In this method, the main data stream has been enciphered by a key and a small part of data has been enciphered by a different key, as a dependant data stream, and sent to the server. The requested query is applied on both streams and the user judges the integrity of results by comparing the results. Our method imposes a little overhead on the user and needs no change in the structure of the server. The probabilistic modeling of the method shows that this method has a high efficiency and the results of the exprimental analysis confirm this very well.

In this paper, we give a kind of Cauchy 1-completeness in probabilistic quasi-uniform spaces by using 1-filters. Utilizing the relationships among probabilistic quasi-uniformities, classical quasi-uniformities and Hutton [0, 1]-quasi-uniformities, we show the relationships between their completeness. In fuzzy quasi-metric spaces, we establish the relationships between the completeness of induced probabilistic quasi-uniform spaces and both completeness of induced classical quasi-uniform spaces and induced Hutton [0, 1]-quasi-uniform spaces.

In this paper, we shall define and study the concept of l-statistical convergence and l-statistical Cauchy in random 2-normed space. We also introduce the concept of l-statistical completeness which would provide a more general frame work to study the completeness in random 2-normed space. Furthermore, we also prove some new results.