Total Order Relation Research Articles

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity.

On preinvex interval-valued functions and unconstrained interval-valued optimization problems

The main objective of this paper is to investigate the generalized convexity of interval-valued functions under the total order relation and apply it to a class of unconstrained interval-valued optimization problems. For this purpose, we present the new definition of preinvex interval-valued functions and obtain its several fascinating characterizations. Then, we introduce the $\preceq_{cw}$-semicontinuity and discuss its relationship with preinvex interval-valued functions. As applications related to preinvex interval-valued functions, we study a class of unconstrained interval-valued optimization problems and discuss the existence theorem of its optimal solution.

Tunnel: Parallel-inducing sort for large string analytics

The suffix array is a crucial data structure for efficient string analysis. Over the course of twenty-six years, sequential suffix array construction algorithms have achieved O(n) time complexity and in-place sorting. In this paper, we present the Tunnel algorithm, the first large-scale parallel suffix array construction algorithm with a time complexity of Onp based on the parallel random access machine (PRAM) model. The Tunnel algorithm is built on three key ideas: dividing the problem of size O(n) into p sub-problems of reduced size Onp by replacing long suffixes with shorter prefixes of size at most a constant D; introducing a Tunnel mechanism to efficiently induce the order of a set of suffixes with long common prefixes; developing a strategy to transform a partially ordered suffix set into a total order relation by iteratively applying the Tunnel inducing method. We provide a detailed description of the algorithm, along with a thorough analysis of its time and space complexity, to demonstrate its correctness and efficiency. The proposed Tunnel algorithm exhibits scalable performance, making it suitable for large string analytics on large-scale parallel systems.

Some new estimates of well known inequalities for $ (h_1, h_2) $-Godunova-Levin functions by means of center-radius order relation

<abstract><p>In this manuscript, we aim to establish a connection between the concept of inequalities and the novel Center-Radius order relation. The idea of a Center-Radius (CR)-order interval-valued Godunova-Levin (GL) function is introduced by referring to a total order relation between two intervals. Consequently, convexity and nonconvexity contribute to different kinds of inequalities. In spite of this, convex theory turns to Godunova-Levin functions because they are more efficient at determining inequality terms than other convexity classes. Our application of these new definitions has led to many classical and novel special cases that illustrate the key findings of the paper. Using total order relations between two intervals, this study introduces CR-$ (h_1, h_2) $-Goduova-Levin functions. It is clear from their properties and widespread usage that the Center-Radius order relation is an ideal tool for studying inequalities. This paper discusses various inequalities based on the Center-Radius order relation. With the CR-order relation, we can first derive Hermite-Hadamard ($ \mathcal{H.H} $) inequalities and then develop Jensen-type inequality for interval-valued functions ($ \mathcal{IVFS} $) of type $ (h_1, h_2) $-Godunova-Levin function. Furthermore, the study includes examples to support its conclusions.</p></abstract>

Some properties and inequalities for generalized class of harmonical Godunova-Levin function via center radius order relation

<abstract><p>There are many benefits derived from the speculation regarding convexity in the fields of applied and pure science. According to their definitions, convexity and integral inequality are linked concepts. The construction and refinement of classical inequalities for various classes of convex and nonconvex functions have been extensively studied. In convex theory, Godunova-Levin functions play an important role, because they make it easier to deduce inequalities when compared to convex functions. Based on Bhunia and Samanta's total order relation, harmonically cr-$ h $-Godunova-Levin function is defined in this paper. Utilizing center order (CR) relationship, various types of inequalities can be introduced. (CR)-order relation enables us to derive some Hermite-Hadamard ($ \mathcal{H.H} $) inequality along with a Jensen-type inequality for harmonically $ h $-Godunova-Levin interval-valued functions (GL-$ \mathcal{IVFS} $). Many well-known and new convex functions are unified by this kind of convexity. For further verification of the accuracy of our findings, we provide some numerical examples.</p></abstract>

Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h1,h2)-Convex Functions Pertaining to Total Order Relation

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order relations are quite different and novel in the literature and are calculated as ω=⟨ωc,ωr⟩=⟨ω¯+ω̲2,ω¯−ω̲2⟩. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-(h1,h2)-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples.

Some H-Godunova–Levin Function Inequalities Using Center Radius (Cr) Order Relation

Interval analysis distinguishes between different types of order relations. As a result of these order relations, convexity and nonconvexity contribute to different kinds of inequalities. Despite this, convex theory is commonly known to rely on Godunova–Levin functions because their properties make it more efficient for determining inequality terms than convex ones. The purpose of this study is to introduce the notion of cr-h-Godunova–Levin functions by using total order relation between two intervals. Considering their properties and widespread use, center-radius order relation appears to be ideally suited for the study of inequalities. In this paper, various types of inequalities are introduced using center-radius order (cr) relation. The cr-order relation enables us firstly to derive some Hermite–Hadamard (H.H) inequalities, and then to present Jensen-type inequality for h-Godunova–Levin interval-valued functions (GL-IVFS) using a Riemann integral operator. This kind of convexity unifies several new and well-known convex functions. Additionally, the study includes useful examples to support its findings. These results confirm that this new concept is useful for addressing a wide range of inequalities. We hope that our results will encourage future research into fractional versions of these inequalities and optimization problems associated with them.

Permutations encoding the local shape of level curves of real polynomials via generic projections

The non-convexity of a smooth and compact connected component of a real algebraic plane curve can be measured by a combinatorial object called the Poincare-Reeb tree associated to the curve and to a direction of projection. In this paper we show that if the chosen projection avoids the bitangents and the inflectional tangencies to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincare-Reeb tree becomes a complete binary tree and its vertices become endowed with a total order relation. Such a projection direction is called generic. We prove that for any such asymptotic family of level curves, there are finitely many intervals on the real projective line outside of which all the directions are generic with respect to all the curves in the family. If the choice of the direction of projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, that we call snakes. The snakes offer an effective description of the local geometry and topology, well-suited for further computations.

Hermite–Hadamard, Fejér and Pachpatte-Type Integral Inequalities for Center-Radius Order Interval-Valued Preinvex Functions

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the CR (Center-Radius)-order interval-valued preinvex function with the help of a total order relation between two intervals. Furthermore, we discuss some properties of this new class of preinvexity and show that the new concept unifies several known concepts in the literature and also gives rise to some new definitions. By applying these new definitions, we have amassed many classical and novel special cases that serve as applications of the key findings of the manuscript. The computations of cr-order intervals depend upon the following concept B=⟨Bc,Br⟩=⟨B¯+B̲2,B¯−B̲2⟩. Then, for the first time, inequalities such as Hermite–Hadamard, Pachpatte, and Fejér type are established for CR-order in association with the concept of interval-valued preinvexity. Some numerical examples are given to validate the main results. The results confirm that this new concept is very useful in connection with various inequalities. A fractional version of the Hermite–Hadamard inequality is also established to show how the presented results can be connected to fractional calculus in future developments. Our presented results will motivate further research on inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems.

Preference implication-based approach to ranking fuzzy numbers

Dombi and Baczy'{n}ski presented a new approach to the problem of implication operation by introducing the preference implication, which has very advantageous properties. In this paper, it is presented how the preference implication is connected with soft inequalities and with sigmoid functions. Utilizing this connection the preference implication-based preference measure for two fuzzy numbers is introduced and its key properties, including the reciprocity, are described. Then, the exact expression for computing the new preference measure for trapezoidal fuzzy numbers is presented. Here, using the new preference measure, two crisp relations over trapezoidal fuzzy numbers are introduced. It is shown that one of them is a strict (but not a total) order relation, and the other one is an equivalence relation. The strict order relation can be used to rank comparable fuzzy numbers, while the equivalence relation, which we call the indifference relation, expresses that the order of some fuzzy numbers is indifferent. These two crisp relations can be used to rank a collection of trapezoidal fuzzy numbers. Lastly, the proposed ranking method is compared with some well-known existing fuzzy number ranking methods.

Double Integral-Based Method for Ranking Intuitionistic Multiplicative Sets and Its Application in Selecting Logistics Transfer Station

Intuitionistic multiplicative sets can be applied in many practical situations, most of which are based on ranking of intuitionistic multiplicative numbers. This study develops an integral method for ranking intuitionistic multiplicative numbers based on the new definitions of multiplicative score function and accuracy function. The ranking method considers both the risk preference and infinitely many possible values in feasible region. Some reasonable properties of multiplicative score function and accuracy function are studied, respectively. We construct a total order relation on the set of intuitionistic multiplicative numbers. The multiplicative score function and accuracy function are utilized to select the optimal logistics transfer station. A comparison example is developed to highlight the advantage of the risk preference-based ranking method.

A CAUSAL HIERARCHICAL MARKOV FRAMEWORK FOR THE CLASSIFICATION OF MULTIRESOLUTION AND MULTISENSOR REMOTE SENSING IMAGES

Abstract. In this paper, a multiscale Markov framework is proposed in order to address the problem of the classification of multiresolution and multisensor remotely sensed data. The proposed framework makes use of a quadtree to model the interactions across different spatial resolutions and a Markov model with respect to a generic total order relation to deal with contextual information at each scale in order to favor applicability to very high resolution imagery. The methodological properties of the proposed hierarchical framework are investigated. Firstly, we prove the causality of the overall proposed model, a particularly advantageous property in terms of computational cost of the inference. Secondly, we prove the expression of the marginal posterior mode criterion for inference on the proposed framework. Within this framework, a specific algorithm is formulated by defining, within each layer of the quadtree, a Markov chain model with respect to a pixel scan that combines both a zig-zag trajectory and a Hilbert space-filling curve. Data collected by distinct sensors at the same spatial resolution are fused through gradient boosted regression trees. The developed algorithm was experimentally validated with two very high resolution datasets including multispectral, panchromatic and radar satellite images. The experimental results confirm the effectiveness of the proposed algorithm as compared to previous techniques based on alternate approaches to multiresolution fusion.

A New Approach to Nonstandard Analysis

In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing a few applications of this approach with the former methods.

A New Representation of Intuitionistic Fuzzy Systems and Their Applications in Critical Decision Making

In decision-making problem, fuzzy system is considered as an effective tool with access to uncertain information by fuzzy representations. Evolutionary fuzzy systems have been developed with the appearance of intuitionistic fuzzy, hesitant fuzzy, neutrosophic representations, etc. Moreover, by capturing compound features and convey multifaceted information, complex numbers are utilized to generalize fuzzy and intuitionistic fuzzy sets. However, the order relations established in these existing systems have certain limitations, such as they are not total order relations or they are defined based on intermediate functions, hence it is difficult to use in building and ensuring important properties of logical operators and distance measures in the systems. In this article, a representation of the intuitionistic fuzzy systems based on complex numbers (IFS-C) in the polar form by a new way is proposed to overcome the above restrictions. Specifically, an intuitionistic fuzzy set is characterized by the two functions of modulus and argument. A new order relation, set-theoretic operations, and a new distance measure by the polar form of IFS-C are defined and investigated. The applicability of the proposal is illustrated by a new decision-making model called P-distance measure. It is tested on the benchmark medical datasets in comparison with the existing methods. The experiments confirm the advantages of the proposal.

Ranking triangular interval-valued fuzzy numbers based on the relative preference relation

In this paper, we first use a fuzzy preference relation with a membership function representing preference degree forcomparing two interval-valued fuzzy numbers and then utilize a relative preference relation improved from the fuzzypreference relation to rank a set of interval-valued fuzzy numbers. Since the fuzzy preference relation is a total orderingrelation that satisfies reciprocal and transitive laws on interval-valued fuzzy numbers, the relative preference relation isalso a total ordering relation. Practically, the fuzzy preference relation is more reasonable on ranking interval-valuedfuzzy numbers than defuzzification because defuzzification does not present preference degree between fuzzy numbersand loses messages. However, fuzzy pair-wise comparison for the fuzzy preference relation is more complex and difficultthan defuzzification. To resolve fuzzy pair-wise comparison tie, the relative preference relation takes the strengths ofdefuzzification and the fuzzy preference relation into consideration. The relative preference relation expresses preferencedegrees of interval-valued fuzzy numbers over average as the fuzzy preference relation does, and ranks fuzzy numbersby relative crisp values as defuzzification does. In fact, the application of relative preference relation was shown intraditional fuzzy numbers, such as triangular and trapezoidal fuzzy numbers, for previous approaches. In this paper, weextend and utilize the relative preference relation on interval-valued fuzzy numbers, especially for triangular intervalvalued fuzzy numbers. Obviously, interval-valued fuzzy numbers based on the relative preference relation are easily and quickly ranked, and able to reserve fuzzy information.

Transaction Fraud Detection Based on Total Order Relation and Behavior Diversity

With the popularization of online shopping, transaction fraud is growing seriously. Therefore, the study on fraud detection is interesting and significant. An important way of detecting fraud is to extract the behavior profiles (BPs) of users based on their historical transaction records, and then to verify if an incoming transaction is a fraud or not in view of their BPs. Markov chain models are popular to represent BPs of users, which is effective for those users whose transaction behaviors are stable relatively. However, with the development and popularization of online shopping, it is more convenient for users to consume via the Internet, which diversifies the transaction behaviors of users. Therefore, Markov chain models are unsuitable for the representation of these behaviors. In this paper, we propose logical graph of BP (LGBP) which is a total order-based model to represent the logical relation of attributes of transaction records. Based on LGBP and users’ transaction records, we can compute a path-based transition probability from an attribute to another one. At the same time, we define an information entropy-based diversity coefficient in order to characterize the diversity of transaction behaviors of a user. In addition, we define a state transition probability matrix to capture temporal features of transactions of a user. Consequently, we can construct a BP for each user and then use it to verify if an incoming transaction is a fraud or not. Our experiments over a real data set illustrate that our method is better than three state-of-the-art ones.

Ordered Rings and Fields

Summary We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].

A linear ordering on the class of Trapezoidal intuitionistic fuzzy numbers

Fuzzy numbers and intuitionistic fuzzy numbers are introduced in the literature to model problems involving incomplete and imprecise information in expert and intelligent systems. Ranking of TrIFNs plays an important role in an information system (Decision Making) with imprecise and inadequate information and the complete ranking on the class of trapezoidal intuitionistic fuzzy number is an open problem worldwide. Researchers from all over the world have been working in ranking of intuitionistic fuzzy numbers since 1985, but till date there is no common methodology that ranks any two arbitrary intuitionistic fuzzy numbers due to the partial ordering of TraIFNs. Different algorithms are available in the literature for solving intuitionistic fuzzy decision (or information system) problem, but each and every algorithm failed to give better result in some places due to the ranking procedure of TrIFNs. Intuitionistic fuzzy decision algorithm works better when it have a complete ranking procedure that ranks arbitrary intuitionistic fuzzy numbers. In this paper a linear (total) ordering on the class of trapezoidal intuitionistic fuzzy numbers using axiomatic set of eight different scores is introduced. The main idea of this paper is to classify and study the properties of eight different sub classes of the set of TrIFNs. Further new total order relations are defined on each of the subclasses of TrIFNs and they are extended to a complete ranking procedure on the set of TrIFNs. Finally the significance of the proposed method over existing methods is studied by illustrative examples.

Calculus in the ring of Fermat reals, Part I: Integral calculus

We develop integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite-dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano–Jordan-like integration domains.

Ranking triangle and trapezoidal fuzzy numbers based on the relative preference relation

In this paper, we first propose a fuzzy preference relation with membership function representing preference degree to compare two fuzzy numbers. Then a relative preference relation is constructed on the fuzzy preference relation to rank a set of fuzzy numbers. Since the fuzzy preference relation is a total ordering relation satisfying reciprocal and transitive laws on fuzzy numbers, the relative preference relation satisfies a total ordering relation on fuzzy numbers as well. Normally, utilizing preference relation is more reasonable than defuzzification on ranking fuzzy numbers, because defuzzification does not present preference degree between two fuzzy numbers and loses some messages. However, fuzzy pair-wise comparison by preference relation is complex and difficult. To avoid above shortcomings, the relative preference relation adopts the strengths of defuzzification and fuzzy preference relation. That is to say, the relative preference relation expresses preference degrees of several fuzzy numbers over average as similar as the fuzzy preference relation does, and ranks fuzzy numbers by relative crisp values as defuzzification does. Thus utilizing the relative preference relation ranks fuzzy numbers easily and quickly, and is able to reserve fuzzy information.