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.

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.

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.

Background and Aim: A major quality assurance target is minimizing error rates to enhance patient safety. Clinical laboratories have long focused their attention on quality control methods and quality assessment programs. The current study aimed to investigate the completeness of specimen labeling in the histopathology department. Materials and Methods: In this cross-sectional retrospective study, we analyzed labels of tissue samples sent from Orthopedics, Neurosurgery, Internal medicine, Urology, ENT, Surgery, ICU, and Endoscopy wards over three months in Urmia Imam Khomeini hospital from January 2023 to March 2023. All information including the patient's name, age, patient file number, tissue type, anatomical location, sampling date, name of the referring physician, legibility of written information, two patient identifiers (full name, date of birth or file number), and affixing the label on the body of the container was checked. Results: Of the 2178 pathology samples examined, the largest numbers of samples (53.58%) were sent from the surgery department and the least (0.69%) from the neurosurgery Ward. The lowest number of errors was related to the patient's age, which ranged from 0% in the internal medicine, neurosurgery and, endoscopy to 8.5% in the urology Ward. There was a statistically significant difference between the criteria obtained in different wards, except for the patient's name and illegibility of written information.   P-value was <0.001 for patient age, file number, tissue type, sampling date, doctor's name, label on container body and two-identifier registration, and 0.049 for the anatomical position of the sample. Conclusion: This study showed that there is a variety of errors in labeling pathology samples. This shows that the high volume of work and a large number of samples cannot be the reason for the error in the labeling. Due to the similarity of names, two samples may be mistaken for each other, but the presence of the second identifier solves this problem. Newer technologies such as bar coding may reduce the frequency of specimen labeling errors.

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.

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 [Fuzzy Sets and Systems 27 (1988) 385-389], M. Grabiec in- troduced a notion of completeness for fuzzy metric spaces (in the sense of Kramosil and Michalek) that successfully used to obtain a fuzzy version of Ba- nachs contraction principle. According to the classical case, one can expect that a compact fuzzy metric space be complete in Grabiecs sense. We show here that this is not the case, for which we present an example of a compact fuzzy metric space that is not complete in Grabiecs sense. On the other hand, Grabiec used a notion of compactness to obtain a fuzzy version of Edelstein s contraction principle. We present here a generalized version of Grabiecs version of the Edelstein fixed point theorem and different interesting facts on the topology of fuzzy metric spaces.

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