This paper presents exact joint confidence regions for the parameters of the Rayleigh distribution based on record data. By providing some appropriate pivotal quantities, we construct several joint confidence regions for the Rayleigh parameters. These joint confidence regions are useful for constructing confidence regions for functions of the unknown parameters. Applications of the joint confidence regions using two environmental data sets are presented for illustrative purposes. Finally, a simulation study is conducted to study the performance of the proposed joint confidence regions.

In this article, a new censoring scheme is considered, namely, a middle part of a random sample is censored. A treatment for reconstructing the missing order statistics is investigated. The proposed procedure is studied in detail under exponential distribution which is widely used as a constant failure model in reliability.Different approaches are used to obtain point and interval reconstructors and then they are compared. A numerical example is presented for illustrating all the proposed inferential procedures. Eventually, we present some remarks including how the results of the paper can be used when the parameters of the exponential distribution are unknown.

We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal category and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal category.

Background: The human intestine harbors a collection of microorganisms known as intestinal microbiota, which encompasses bacteria, archaea, and eukaryotes. However, bacteria reign supreme as the most prevalent members of the intestinal microbiota. Notably, the gut microbiota plays a crucial role in regulating various physiological functions of the human body. In addition to its influence on digestion, the gut microbiota also exerts control over the function of the brain and central nervous system, earning the enteric nervous system the title of the "second brain." The behavior and mood, as well as the progression of nervous system diseases like multiple sclerosis, autism, Alzheimer's, schizophrenia, and Parkinson's, can potentially be regulated by the intestinal microbiota. Through the intestinal nervous system, production of metabolites, stimulation of entero-endocrine cells, and the immune system, the gut microbiota plays a role in regulating the function of the central nervous system. Disturbances caused by improper nutrition, indiscriminate use of antibiotics, stress, anxiety, and depression are significant factors that can worsen these diseases and disrupt the balance of gut microbiota.