Partial Order Research Articles

On some pre-orders and partial orders of linear operators on infinite dimensional vector spaces

AbstractThis article is devoted to the generalization of the Drazin pre-order and the G-Drazin partial order to Core-Nilpotent endomorphisms over arbitrary k-vector spaces, namely, infinite dimensional ones. The main properties of this orders are described, such as their respective characterizations and the relations between these orders and other existing ones, generalizing the existing theory for finite matrices. In order to do so, G-Drazin inverses are also studied in this framework. Also, it includes a generalization of the space pre-order to linear operators over arbitrary k-vector spaces.

On G-Drazin partial order in rings

We extend the concept of a G-Drazin inverse from the set Mn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_n$$\\end{document} of all n×n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\ imes n$$\\end{document} complex matrices to the set RD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{R}^{D}$$\\end{document} of all Drazin invertible elements in a ring R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{R}$$\\end{document} with identity. We also generalize a partial order induced by G-Drazin inverses from Mn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_n$$\\end{document} to the set of all regular elements in RD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{R}^{D}$$\\end{document}, study its properties, compare it to known partial orders, and generalize some known results.

A survey of congruences and quotients of partially ordered sets

A quotient of a poset P is a partial order obtained on the equivalence classes of an equivalence relation \theta on P ; \theta is then called a congruence if it satisfies certain conditions, which vary according to different theories. The literature on congruences and quotients of partially ordered sets contains a large and proliferating array of approaches, but little in the way of systematic exposition and examination of the subject. We seek to rectify this by surveying the different theories in the literature and providing philosophical discussion on requirements for notions of congruences of posets. We advocate a pluralist approach which recognises that different types of congruence arise naturally in different mathematical situations. There are some notions of congruence which are very general, whilst others capture specific structure which often appears in examples. Indeed, we finish by giving several examples where quotients of posets appear naturally in mathematics.

The CPO-inverse and its partial orders

This paper introduces the core projection operator inverse and its characterizations. Using this concept, we define the P-core partial order and D-core partial order, along with their properties. The P-core order is proven to be of minus-type, while the D-core order differs as a non-minus-type. We investigate connections between these orders and related ones like minus, star, and sharp partial orders for P-core, and minus and diamond partial orders for D-core. Furthermore, we present the P-core matrix, which intriguingly commutes with its core projection operator inverse. This work enhances our understanding of these concepts and their relationships.

Labeling methods for partially ordered paths

The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions’ properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems’ weight structure. Examples range from time-dependent shortest path bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs.

A budgeting resource allocation model for capacity expansion

The reconfiguration of the global supply chain has created opportunities for newly industrialized economies (NIEs) to strengthen and expand their production capacities. We develop a dynamic modeling framework to study a firm’s resource allocation decisions while building capacity for a portfolio of production lines over multiple decision periods. Our model adds to the literature on versions of this problem in which distributions of market demands are incorporated into the investment–return function, and where no partial order fulfillment is possible. We consider two specific cases: one involving exogenously defined demand distributions, and one where the distributions of the parameters of the demand distribution are updated based on Bayesian inference. For each case, we first identify optimal capacity levels as a function of unit investment cost, holding cost, and the demand distribution of each production line. We then characterize the optimal path to reach these levels under different market conditions by deriving closed-form optimal resource allocation policies.

Spatiotemporal patterns corresponding to phase synchronization and generalized synchronization states of thermoacoustic instability.

Thermoacoustic instability in turbulent combustion systems emerges from the complex interplay among the flame, flow, and acoustic subsystems. While the onset of thermoacoustic instability exhibits a global order, the characteristics of local interactions between subsystems responsible for this order are not well understood. Here, we utilize the framework of synchronization to elucidate the spatiotemporal interactions among heat release rate fluctuations in the flame, velocity fluctuations in the flow, and acoustic pressure fluctuations in a turbulent combustor, across the bluff-body stabilized flame. We examine two forms of thermoacoustic instability, characterized by phase synchronization and generalized synchronization of the acoustic pressure and global heat release rate oscillations. Despite the presence of global synchrony, we uncover a coexistence of frequency synchrony and desynchrony in the local interaction of these oscillations within the reaction field. In regions of frequency-locked oscillations, various phase-locking patterns occur, including phase synchrony and partial phase synchrony. We observe that the local formation of small pockets of phase synchrony and strong amplitude correlation between these oscillations is sufficient to trigger the state of global phase synchronization. As the global dynamics approach generalized synchronization, these local regions of synchrony expand in the reaction field. Additionally, through coupled analysis of acoustic pressure and local flow velocity fluctuations, we infer that the spatial region of flow-acoustic synchrony plays a significant role in governing thermoacoustic instabilities. Our findings imply that, in turbulent combustors, an intrinsic local balance between order, partial order, and disorder within the coupled subsystems sustains the global order during thermoacoustic instability.

Inner-star and star-inner partial orders in ∗-rings

The notions of inner-star and star-inner generalized inverses are introduced on the set of all regular elements in a ∗-ring [Formula: see text]. We thus extend the concept of inner-star and star-inner complex matrices. We study properties of these hybrid generalized inverses on the set [Formula: see text] of all Moore–Penrose invertible elements in [Formula: see text] and thus generalize some known results. Partial orders that are induced by inner-star and star-inner inverses are introduced on [Formula: see text], their properties are examined, and their characterizations are presented.

Adaptive penalty-based neurodynamic approach for nonsmooth interval-valued optimization problem

The complex and diverse practical background drives this paper to explore a new neurodynamic approach (NA) to solve nonsmooth interval-valued optimization problems (IVOPs) constrained by interval partial order and more general sets. On the one hand, to deal with the uncertainty of interval-valued information, the LU-optimality condition of IVOPs is established through a deterministic form. On the other hand, according to the penalty method and adaptive controller, the interval partial order constraint and set constraint are punished by one adaptive parameter, which is a key enabler for the feasibility of states while having a lower solution space dimension and avoiding estimating exact penalty parameters. Through nonsmooth analysis and Lyapunov theory, the proposed adaptive penalty-based neurodynamic approach (APNA) is proven to converge to an LU-solution of the considered IVOPs. Finally, the feasibility of the proposed APNA is illustrated by numerical simulations and an investment decision-making problem.

Features of magnetic structure and properties of ceramics (1 − x)Pb(Fe1/2Nb1/2)o3—xPb(Fe2/3W1/3)o3

We studied solid solutions of the system (1−x)PbFe1/2Nb1/2O3 - xPbFe2/3W1/3O3 using X-ray and Neuron diffraction, and Mössbauer spectroscopy. To obtain ceramics, we used the method of two-stage solid-phase synthesis and sintering according to the conventional ceramic technology. Pure solid solutions of the (1−x)PbFe1/2Nb1/2O3-xPbFe2/3W1/3O3 system were obtained. A deviation of the unit cell parameters from the Vegard rule was revealed, which indicates stratification of the structure into microregions differing in composition. The introduction of PbFe2/3W1/3O3 into the system led to partial ordering (clustering) of cations in the B position. An increase in magnetic moment with growing amount of PbFe2/3W1/3O3 was shown.

Isomorphism on Complex Fuzzy Graph

Objective: To investigate isomorphism between two complex fuzzy graphs and prove it is an equivalence relation. The major objective of this research paper is to elucidate weak and strong isomorphism, and the study also endeavours to look at the complement of complex fuzzy graphs. Methods: Isomorphism is examined by comparing the membership values of nodes and arcs (both amplitude and phase). The same technique also proves further validation of the equivalence relation, which is also proven by the same technique. This criterion helps us to identify and formalise the isomorphic relationship between complex fuzzy graphs. Findings: This study demonstrates that isomorphism in complex fuzzy graphs allows for consideration of graph structures' differences and similarities. It provides for a better understanding of significant connections between complex fuzzy graphs. Novelty: By offering an elaborate perception of the idea of isomorphism in complex fuzzy graphs, this work advances the discipline. It seeks a better understanding of the complexities of equivalency determination for complicated structures, as well as the application of isomorphism theory to complex fuzzy graphs. Keywords: Complex fuzzy graph, Homomorphism, Isomorphism, Partial order relation, Self complementary

On the hyper-Lyapunov matrix inclusions

By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions.Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself.As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations.Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub-unit disks.

A Facial Order for Torsion Classes

Abstract We generalize the “facial weak order” of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting “facial semistable order”: one using cover relations, and one using Bongartz completions of 2-term presilting objects. For $\tau $-tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes.

The State of the ‘Planet’ Pillar by 2022 — A Partial Ordering-Based Analysis of the Sustainable Development Goals 6, 12, 13, 14, and 15

The 2022 Sustainable Development Report provides the data for the so-called Planet pillar, i.e., the Sustainable Development Goals (SDGs) 6, 12, 13, 14, and 15 that are studied to elucidate the state of simultaneous compliance with these five goals as well as the trends in development for the 193 countries included in the report. To the extent that data for all five SDGs were available partial ordering methodology was applied as the analytical tool to rank the countries according to their compliance as well as their trend toward compliance based on the 2022 data. The analytical approach allows simultaneously taking data for all five SDGs into account to get an overall picture of the “planet” midway through the 15 years of the 17 UN SDGs. From the analyses, it became clear that high-income countries, despite their economic capacity are lagging both about the actual state and especially about the trend of development toward eventual compliance with the goals. The analyses further pinpointed that SDG 6 – clean water and sanitation – appeared as the most important indicator for the ranking of countries or regions. Building on the author’s previous research on the topic, this study points to the necessity for high-income countries to focus on the eventual compliance with the goal, both for themselves as well as for supporting countries with less economic capacities.

Personalized multi-attribute recommendation method based on partial order set

With the rapid development of big data and continuous optimization of online shopping platforms, personalized recommendation has become a standard feature of recommendation methods. In order to effectively provide personalized recommendations to customers, improve recommendation accuracy, and customer satisfaction, it is necessary to consider customers’ preferences for multiple product attributes when making product recommendations. However, existing recommendation methods require precise calculation of product attribute weights, which is computationally expensive, complex, and often results in unstable weight values. This paper proposes a multi-attribute recommendation method based on consumer decision preference information that overcomes the need for weights and reflects personalized customer preferences. Based on the acquisition of customer product attribute preference sequences, a partial order relation for recommended products is constructed using partial order set theory. Finally, the recommended products are determined through the partial order Hasse diagram, where the top layer elements of the Hasse diagram represent the recommended product set. This method addresses challenges that traditional content-based recommendations cannot overcome. The experiment in this paper uses a dataset of 30,000 records from Beeradvocate beer reviews. The experimental results show that, compared to traditional multi-attribute recommendation methods, this method only requires decision-maker preference information to complete product recommendations, requiring less information and having lower computational costs, resulting in more robust results.

And subnormal safe quotients for geometrically regular weighted shifts

Geometrically regular weighted shifts (in short, GRWS) are those with weights α(N,D) given by αn(N,D)=pn+Npn+D, where p>1 and (N,D) is fixed in the open unit square (−1,1)×(−1,1). We study here the zone of pairs (M,P) for which the weight α(N,D)α(M,P) gives rise to a moment infinitely divisible (▪) or a subnormal weighted shift, and deduce immediately the analogous results for product weights α(N,D)α(M,P), instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.

Reinforcement Learning Based Metaheuristic Algorithm For Optimized Load Balancing In Cloud Environment

In cloud computing, workload balancing remains a difficult issue, as overburdened or underburdened servers are capable of causing issues with computation speed or even cause an entire network to collapse, subsequently this is a scenario that is never supposed to occur when using the cloud. Therefore, in order to avoid these issues, the system should be load balanced—that is, allocate job across each of the accessible computing assets by taking into account an acceptable schedule of utilization. The Virtual Machines are supposed to be used effectively, in response to the distributed load mechanism. In this work, a multifaceted load balancing optimization based on the Enhanced Firefly algorithm (EFA) with a Partial Order Markov Decision (POMD) process algorithm is proposed as an autonomous job allocation strategy in cloud computing. In order to overcome the limits of concurrent considerations, the suggested solution seeks to maximise VM productivity, streamline task allocation and the use of resources, and provide load balancing amongst virtual machines based on Time to Finish, expenditure, and consumption of resources. Using CloudSim, the performance study of the suggested approach was contrasted with the load balancing algorithms now in use in datasets like Google Cloud Jobs (GoCJ). According to the trial results, the suggested POMD-EFA strategy performed better than the other algorithms in terms of decreasing the Time to Finish, minimizing expenses, and increasing the efficiency in the application of resources.

The Relative Importance of the SDG Pillars

The seventeen United Nations’ Sustainable Development Goals (SDGs) comprise social, economic, and environmental aspects of sustainability, and as such they can be grouped into five so-called pillars, i.e., People, Prosperity, Planet, Peace, and Partnership. The present study elucidates the relative importance of these pillars for 193 countries and 12 regions in their attempts to comply with the SDGs as well as the trends towards compliance based on partial order analyses. It is unambiguously demonstrated that the pillar Planet, comprising SDGs 6, 12, 13, 14, and 15, in all cases turns out as the far most important pillar both about the present degree of compliance as well as concerning the development trend. Not surprisingly the OECD region appears as the region with the highest degree of compliance whereas Africa has the least. However, when it comes to the trends the OECD and the high-income countries (HIC) apparently come staggering up when it comes to the pilar planet due to rather low pillar planet values. For the individual countries Finland and Denmark were found on the top ranks. The reason for the importance of the pillar Planet as well as the difference between sustainability goals and sustainability is discussed.

Necessary and sufficient conditions of the enforcement problem for various argumentation semantics

Abstract In the research on dynamics of argumentation frameworks (AFs), the enforcement problem deals with changing an AF for the purpose of ensuring that a certain set of desired arguments becomes (part of) an extension. In this paper we focus on expansions of an AF where solely the addition of new arguments and attacks is allowed and the original framework remains unchanged. Existing results about the enforcement problem are all sufficient conditions, and we argue that necessary and sufficient conditions are essential concerning the solvability of the enforcement problem. A suite of jointly necessary and sufficient conditions are identified for the enforcement problem under normal expansion. These conditions are presented from multiple perspectives including the sub-strict versus non-strict problem, constructive versus eliminative conditions and arbitrary AFs versus even(odd)-length cycle free AFs, determined by the nature of various argumentation semantics. Such necessary and sufficient conditions are of the essence to the enforcement problem by defining requirements that are both minimum and absolute, which can be used to determine when the enforcement problem is (un)solvable. We further infer solely sufficient and solely necessary conditions based on the partial order relation (w.r.t. set-inclusion) among various types of AF extensions.

An Approach to Distributed Systems from Orderings and Representability

In the present paper, we propose a new approach on ‘distributed systems’: the processes are represented through total orders and the communications are characterized by means of biorders. The resulting distributed systems capture situations met in various fields (such as computer science, economics and decision theory). We investigate questions associated to the numerical representability of order structures, relating concepts of economics and computing to each other. The concept of ‘quasi-finite partial orders’ is introduced as a finite family of chains with a communication between them. The representability of this kind of structure is studied, achieving a construction method for a finite (continuous) Richter–Peleg multi-utility representation.