LR-Preinvex Interval-Valued Functions and Riemann–Liouville Fractional Integral Inequalities

Convexity is crucial in obtaining many forms of inequalities. As a result, there is a significant link between convexity and integral inequality. Due to the significance of these concepts, the purpose of this study is to introduce a new class of generalized convex interval-valued functions called LR-preinvex interval-valued functions (LR-preinvex I-V-Fs) and to establish Hermite–Hadamard type inequalities for LR-preinvex I-V-Fs using partial order relation (≤p). Furthermore, we demonstrate that our results include a large class of new and known inequalities for LR-preinvex interval-valued functions and their variant forms as special instances. Further, we give useful examples that demonstrate usefulness of the theory produced in this study. These findings and diverse approaches may pave the way for future research in fuzzy optimization, modeling, and interval-valued functions.

Further Results of the TTT Transform Ordering of Order n

To compare the variability of two random variables, we can use a partial order relation defined on a distribution class, which contains the anti-symmetry. Recently, Nair et al. studied the properties of total time on test (TTT) transforms of order n and examined their applications in reliability analysis. Based on the TTT transform functions of order n, they proposed a new stochastic order, the TTT transform ordering of order n (TTT-n), and discussed the implications of order TTT-n. The aim of the present study is to consider the closure and reversed closure of the TTT-n ordering. We examine some characterizations of the TTT-n ordering, and obtain the closure and reversed closure properties of this new stochastic order under several reliability operations. Preservation results of this order in several stochastic models are investigated. The closure and reversed closure properties of the TTT-n ordering for coherent systems with dependent and identically distributed components are also obtained.

Application of the ontology of the choice of selection to describe the processes of interaction of subjects of military communication management

The ways of improving the mechanisms of information and analytical support of the command control system in the state of emergency are analyzed. The approach to the application of the ontology of the choice problem for decision-making in the field of law enforcement management using the procedure of integration of information resources based on the binary partial order relation is used. The purpose of the article is to increase the efficiency of decision-making in the management system of the military command in a state of emergency by applying the ontology of the choice problem based on a set of semantically significant results. Results of the research. Analysis and processing of large arrays of information in the field of military command management in a state of emergency should be carried out in an automated mode on the basis of a distributed software environment based on the principles of ontologies. Ontological systems, as a result of the inverse mapping of natural systems, provide the correct aggregation of various thematic processes through the formation of a structured set of information objects-concepts of the subject area, which are defined as a single type of data. The ontological representation of the contexts of units-concepts provides their integrated use in the process of solving complex tasks by the governing bodies of the command in a state of emergency. One of the constructive ways to integrate information resources as passive knowledge systems is to activate their concepts based on the process of forming thematic ontologies and combining these ontologies by building an ontology of the choice problem over them. The uniqueness of the ontology of the choice problem to any homotopy type allows to build the procedure of integration of information resources on the basis of a binary partial order relation. The partial order relation allows to reflect in an integrated way interaction of contexts of the notion-concepts defining subjects of information resources. The contradiction between the increase in the amount of information needed for decision-making in the field of management of interdepartmental critical systems and the constant requirement to reduce the time for its processing in the information-analytical systems has been resolved.

A study on orderings based on fuzzy quasi- order relations

This paper further studies orderings based on fuzzy quasi-order relations using fuzzy graph. Firstly, a fuzzy relation on a finite set is represented equivalently by a fuzzy graph. Using the graph, some new results on fuzzy relations are derived. In ranking those alternatives, we usually obtain a quasi-order relation, which often has inconsistencies, so it cannot be used for orderings directly. We need to remake it into a reasonable partial order relation for orderings. This paper studies these inconsistencies, and divides them into two types: framework inconsistencies and degree inconsistencies. For the former, a reasonable and feasible method is presented to eliminate them. To eliminate the latter, the concept of complete partial order relation is presented, which is more suitable than partial order relation to rank the alternatives. A method to obtain a reasonable complete partial order relation for a quasi-order relation is given also. An example is given as well to illustrate these discussions. Lastly, the paper discusses the connection between quasi-order relations and preference relations for orderings and some other related problems.

On the little secondary Bruhat order

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More on the Bruhat order for $(0,1)$-matrices, Linear Algebra Appl., 421:219--232, 2007) suggested the study of the secondary Bruhat order on ${\cal A}(R,S)$ but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes ${\cal A}(R,S)$. Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of ${\cal A}(R,S)$.

Proximity Between Selfadjoint Operators and Between Their Associated Random Measures

We study how the proximity between twoselfadjoint bounded operators, measured by a classical distance, canbe expressed by a proximity between the associated spectralmeasures. This last proximity is based on a partial order relation on the set of projectors. Assuming an hypothesis of commutativity, we show that the proximity between operators implies the one between theassociated spectral measures, and conversally, the proximity between spectral measures implies the one between associated selfadjoint operators.