Einstein Operations Research Articles

Mode analysis for the linearized Einstein equations on the Kerr metric: the large $\mathfrak{a}$ case

We give a complete analysis of mode solutions for the linearized Einstein equations and the 1 -form wave operator on the Kerr metric in the large \mathfrak{a} case. By mode solutions we mean solutions of the form e^{-it_*\sigma}\tilde{h}(r,\theta,\varphi) where t_{*} is a suitable time variable. The corresponding Fourier-transformed 1 -form wave operator and linearized Einstein operator are shown to be Fredholm between suitable function spaces and \tilde{h} has to lie in the domain of these operators. These spaces are constructed following the general framework of Vasy (2013, 2021) along the lines of Häfner et al. (2021). No mode solutions exist for \Im \sigma\ge 0,\, \sigma\neq 0 . For \sigma=0 mode solutions are Coulomb solutions for the 1 -form wave operator and linearized Kerr solutions plus pure gauge terms in the case of the linearized Einstein equations. If we fix a De Turck/wave map gauge, then the zero mode solutions for the linearized Einstein equations lie in a fixed seven-dimensional space. The proof relies on the absence of modes for the Teukolsky equation shown by Whiting (1989) and Anderson et al. (2019) and a complete classification of the gauge invariants of linearized gravity on the Kerr spacetime by Aksteiner and Bäckdahl (2018) and Aksteiner et al. (2021).

Analyzing risk factors of tuberculosis using type-2 interval-valued trapezoidal fuzzy numbers with Einstein aggregation operators extended to MCDM

The principal motive of this work is to evolve and initiate an extension from interval-valued fuzzy sets to type-2 interval-valued fuzzy sets (T2IVFS) related to weighted aggregation functions containing the Einstein operator. The chief reason for this extension is that the constancy of the terms can also be taken into data during the aggregation operation. The main goal of this article is to compose the aggregation operators and their characteristics such as the Type-2 interval-valued fuzzy Einstein weighted arithmetic aggregating operator (T2IVFEWA), Type-2 interval-valued fuzzy Einstein weighted geometric aggregating operator (T2IVFEWG), and the characteristics are expressed. At last, to intimate the effectiveness of the suggested approach and explicate the purpose of these operators, a hybrid multi-criteria decision-making problem (MCDM) to select the best risk factor for Tuberculosis (TB) is considered and the result is compared with the outcome of the existing operators and methods. Additionally, a sensitivity analysis was conducted to verify the robustness of the proposed decision-making process.

Aggregation operators based on Einstein averaging under q-spherical fuzzy rough sets and their applications in navigation systems for automatic cars

This study introduces innovative operational laws, Einstein operations, and novel aggregation algorithms tailored for handling q-spherical fuzzy rough data. The research article presents three newly designed arithmetic averaging operators: q-spherical fuzzy rough Einstein weighted averaging, q-spherical fuzzy rough Einstein ordered weighted averaging, and q-spherical fuzzy rough Einstein hybrid weighted averaging. These operators are meticulously crafted to enhance precision and accuracy in arithmetic averaging. By thoroughly examining their characteristics and interrelations with existing aggregate operators, the article uncovers their distinct advantages and innovative contributions to the field. Furthermore, the study illustrates the actual implementation of these newly constructed operators in a variety of attribute decision-making scenarios employing q-SFR data, yielding useful insights. Our suite of decision-making tools, including these operators, is specifically designed to address complex and uncertain data. To validate our approach, this study offers a numerical example showcasing the real-world applicability of the proposed operators. The results not only corroborate the efficacy of the proposed method but also underscore its potential significance in practical decision-making processes dealing with intricate and ambiguous data. Additionally, comparative and sensitivity analyses are presented to assess the effectiveness and robustness of our proposed work relative to other approaches. The acquired knowledge enriches the current understanding and opens new avenues for future research.

Group Decision-Making Method with Incomplete Intuitionistic Fuzzy Soft Information for Medical Diagnosis Model

The medical diagnosis of many critical diseases is difficult as it usually requires the combined effort of several doctors. At this time, the process of medical diagnosis is actually a group decision-making (GDM) problem. In group medical diagnosis, considering doctors’ weight information and fusing the interaction relation of symptoms remain open issues. To address this problem, a group decision-making method for intuitionistic fuzzy soft environments is proposed for medical diagnosis because the intuitionistic fuzzy soft set (IFSS) integrates the advantages of the soft set and intuitionistic fuzzy set (IFS). Intuitionistic fuzzy soft weighted Muirhead mean operators are constructed by combining Einstein operations with the Muirhead mean (MM) operator, and some properties and results are revealed. A group medical diagnosis model with unknown doctor weight information and incomplete intuitionistic fuzzy soft information is proposed. Similarity measures of the intuitionistic fuzzy soft matrix (IFSM) given by the doctors are used to estimate the incomplete information. To take into account the advantages of objective weight and subjective weight, the combined weights of doctors are calculated based on the IFSMs’ similarity measure and doctors’ grades. The developed operators are then used to combine the evaluation information and handle the correlation of input arguments in the group medical diagnosis process. Finally, a numerical problem is selected to illustrate the superiority of the proposed approach compared to related methods. The combined weights are determined to overcome the shortcomings of the single-weight method to some extent. Meanwhile, the proposed method is more comprehensive, and can provide more flexible and reasonable choices for group medical diagnosis problems.

Ensemble feature selection using q-rung orthopair hesitant fuzzy Hamacher, Einstein and Dombi Aggregation operators

The article aims in addressing the issue of ensemble feature selection problem by modeling it as a multi-criteria decision making technique. To build such a model, initially, aggregation operators such as weighted arithmetic, weighted geometric, ordered weighted arithmetic, ordered weighted geometric aggregation of Hamacher, Einstein and Dombi operators in the q-rung orthopair hesitant fuzzy environment are proposed. The properties of these operators are also discussed to provide a more elaborate understanding of them. Such an approach to ensemble feature selection has not yet been carried out in the literature which adds to the novelty of our work. Validation is provided through comparison of the performance metrics with existing and base feature selection methods and also by carrying out statistical tests. Through this article, a model for ensemble feature selection incorporating the advantages of Einstein, Hamacher, Dombi and q-rung orthopair hesitant fuzzy set was constructed which was reflected in the results.

Renormalized volume and non-degeneracy of conformally compact Einstein four-manifolds

In this article we give a condition, depending on the renormalized volume of a four-dimensional Poincaré-Einstein manifold, which implies that the TT-gauge-fixed linearised Einstein operator is non-degenerate.

Multi-criteria decision-making based on Einstein operators, WASPAS method and quadripartitioned single-valued neutrosophic sets

Multi-criteria decision-making based on Einstein operators, WASPAS method and quadripartitioned single-valued neutrosophic sets

Rough Neutrosophic Shapley Weighted Einstein Averaging Aggregation Operator and its Application in Multi-Criteria Decision-Making Problem

The study of aggregation operators has played a crucial role in various decision-making methods. The primary function of the aggregation operator is to combine multiple numbers into a single value. While Einstein operators offer a compact notation and can handle complex and large datasets, they do not consider the interaction involved in determining the criteria weights or account for imprecise and indeterminate data. To overcome this limitation, this paper introduces an improved aggregation operator, the rough neutrosophic Shapley weighted Einstein averaging aggregation operator. The rough neutrosophic sets offer a method for effectively managing the fuzziness and uncertainty that commonly occur in real-world scenarios and the Shapley fuzzy measure helps us understand the importance or value of different elements in each scenario. This operator combines the Shapley fuzzy measure with Einstein operators under rough neutrosophic sets, which are an effective tool for handling incomplete, indeterminate, and inconsistent information. The proposed operator satisfies essential algebraic properties such as idempotency, boundedness, and monotonicity. This paper also presents a decision-making methodology based on the proposed operator, with attribute values derived from the rough neutrosophic set. Finally, the applicability of the suggested aggregation operator is illustrated with a numerical example.

Combinative distance-based assessment method using linguistic T-spherical fuzzy aggregation operators and its application to multi-attribute group decision-making

Clean energy has become a hot topic worldwide because of the continuous rise in the climate's temperature. Reduced air pollution is only one of the many economic and environmental advantages of clean energy. An effective contractor with experience and knowledge in clean energy projects is always required to handle the complicated process of planning, implementing, and maintaining the new clean energy infrastructure. Contractor selection for renewable energy projects (REPs) is a multi-attribute group decision-making (MAGDM) problem. Linguistic T-spherical fuzzy set (LT-SFS) can be compatible with fuzzier information, and the contractor selection problem also has a lot of fuzzy information, which makes LT-SFS very suitable for contractor selection for REPs. Hence, the present work proposes a MAGDM methodology utilizing a combinative distance-based assessment (CODAS) method within the LT-SF context. For that, the Einstein and Frank t-norm and t-conorm has been used to define new and generalized operational laws for linguistic T-spherical fuzzy numbers. Then, the linguistic T-spherical fuzzy Einstein and linguistic T-spherical fuzzy Frank aggregation operators are proposed to integrate the information data provided by experts. Moreover, the proposed aggregation operators based-CODAS approach is designed to evaluate and rank the available alternatives. A real-life decision problem of selecting the best suitable contractor for a renewable energy project is solved to verify our suggested technique. Moreover, the sensitivity analysis is also carried out by changing the parameter's value to check the consistency of the rank order. Finally, the new model is compared with existing approaches to demonstrate its strength. The comparative analysis shows that the results of the suggested technique are more feasible and beneficial than those of existing approaches.

Sustainability evaluation of sports tourism using a linguistic neutrosophic multi-criteria decision-making method.

Sports tourism represents a novel industrial manifestation of the profound integration between the tourism and sports sectors. The objective of this research is to examine an innovative multi-criteria decision-making (MCDM) method for the sustainability evaluation of sports tourism. The largest innovations are the expression and treatment of ambiguous data and interdependent evaluation criteria in the sports tourism sustainability evaluation process. On the one hand, intricate assessment data is represented using linguistic neutrosophic numbers (LNNs), which employ three linguistic variables to convey uncertainty and imprecision. On the other hand, to effectively capture the interrelationships among inputs, two novel aggregation operators are proposed. They are devised based on the Einstein operations and Heronian mean operators of LNNs. Subsequently, a linguistic neutrosophic evaluation method utilizing the aforementioned operators is presented. Comparative and sensitivity analyses conclude that great interdependence exists among five different dimensions of sustainability evaluation in sports tourism, and the proposed method can reflect the interrelations among inputs without redundant calculations.

Multi-attribute group decision making method for sponge iron factory location selection problem using multi-polar fuzzy EDAS approach

This paper presents new averaging operators, such as mpF Einstein weighted averaging (mpFEWA), mpF Einstein ordered weighted averaging (mpFEOWA), mpF Einstein hybrid weighted averaging (mpFEHWA), mpF Einstein weighted geometric (mpFEWG), and mpF Einstein hybrid weighted geometric (mpFEHWG), as well as new Einstein operations (mpFNs) for handling multi-polar fuzzy numbers. We evaluate these operators for idempotency, boundedness, monotonicity, and commutativity, and we design them to deal with multi-polar fuzzy numbers (mpFNs). Furthermore, the study investigates the use of these operators in MAGDM settings, namely mpFEWA and mpFEWG operators, to expand on this. Additionally, it proposes a procedure for determining the best site for a sponge iron production plant by use of the created MAGDM method. The EDAS method, which stands for “Evaluation based on Distance from Average Solution,” verifies that the solutions are effective. Finally, the suggested model highlights the benefits and possible improvements provided by these creative strategies by comparing the new approach to conventional methods and evaluating its efficiency and practicality.

Spherically symmetric evolution of self-gravitating massive fields

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions.

A novel fuzzy-prospect theory approach for hydrogen fuel cell component supplier selection for automotive industry

The burgeoning automotive sector is an energy-intensive industrial sector, amounting to 94 M tons of oil equivalents (MTOE). Owing to more significant environmental concerns, there is a dire need to transition the energy-intensive mobility space to sustainable mobility. In this direction, hydrogen is being researched as a potential alternative clean fuel for the automotive sector; however, in the context of developing countries, hydrogen fuel cell technology is at a very nascent stage. Various policies and research projects are underway to realize the shift, but they must be underpinned through a robust supply chain network. The selection of an optimal supplier is key to the development of a robust hydrogen supply chain network. To this end, this work proposes the selection of an appropriate supplier for supplying critical components for different research and development projects of hydrogen fuel cells. This is still a lacuna in the context of developing nations. To aid in addressing this issue, a novel TODIM (TOmada de Decisao Interativa Multicriterio) decision-making framework has been proposed to select the optimal supplier to supply critical hydrogen fuel cell components. Within the proposed approach, the linguistic evaluations provided by experts are modeled using a more generalized linear Diophantine linguistic uncertain fuzzy set that circumvents the algorithmic restrictions on the falsity and truth grades. The linguistic assessments from the experts were aggregated using the linear Diophantine uncertain linguistic power Einstein weighted averaging (LDULPEWA) operator. The power Einstein operations take care of the interaction between falsity and truth grades, which is important for the accuracy of the results. Finally, the TODIM approach was used to deduce the final ranking results. The proposed method considers the specialists' willingness to take risks during the decision-making process. Such a decision-making framework is still a lacuna in the domain of decision science. The supposed results were tested for their stability through sensitivity analysis. A past case study was solved using the proposed approach to validate the rankings. A comparison of the suggested strategy with the existing prominent decision-making frameworks was also carried out to clearly highlight the strengths and weaknesses of the proposed approach.

Multi-Criteria Decision Making Model For Hotel Selection Problem Under Complex Dual Hesitant Fuzzy Information

ABSTRACT The notion of the “complex dual hesitant fuzzy set (CDHFS)” is the combination of the “dual hesitant fuzzy set (DHFS)” and the “complex fuzzy set (CFS).” It is characterized by two degrees, namely the membership and nonmembership, in the form of a finite subset on a unit disc in the complex plane. CDHFS is useful for dealing with real-world problems involving uncertain or hard-to-predict information. Also, to approximate smoothly, the Einstein operators are well-known aggregation operators, while prioritized operators are effective tools for prioritization among criteria. Therefore, the goal of this study is to develop some prioritized aggregation operators under the CDHFS environment; namely the complex dual hesitant fuzzy prioritized averaging (CDHFPA) operator, the complex dual hesitant fuzzy prioritized geometric (CDHFPG) operator, the complex dual hesitant fuzzy Einstein prioritized averaging (CDHFEPA) operator, and complex dual hesitant fuzzy Einstein prioritized geometric (CDHFEPG) operator. Some properties of the proposed operators are investigated in detail. In addition, a multi-criteria decision-making (MCDM) method based on the proposed operators with the complex dual hesitant fuzzy setting is developed. Moreover, a numerical example is given for the application and effectiveness of the developed MCDM approach. A comparison study is also done with existing methods to show that the proposed MCDM method is better and more reliable. The study finds that if the expert’s preference is used to choose the right aggregation operators, the decision maker will have access to a wide range of compromise solutions.

Strategic location analysis for offshore wind farms to sustainably fulfill railway energy demand in Turkey

Providing the energy demands of various sectors through the utilization of renewable energy sources offers a more sustainable and long-term approach compared to fossil fuel consumption. In this study, the proposition is made to address Turkey's rail industry energy demand through Offshore Wind Farms (OWF). Accordingly, the spatial analysis for determining the most suitable OWF location is explored. The research methodology is conducted based on a Geographic Information System-Multi-Criteria Decision Making (GIS-MCDM) hybrid approach. Alternative OWF locations are identified using GIS. Location selection criteria are established based on literature review, and a fuzzy-based decision matrix is constructed, incorporating expert opinions. To leverage two distinct fuzzy number sets, Intuitionistic Fuzzy Weighted Averaging (IFWA) and Spherical Fuzzy Stepwise Weight Assessment Ratio Analysis (SF-SWARA) methods are employed for criteria weighting, followed by the aggregation of criterion weights using the Einstein Operations t-norm technique. To ensure comparable OWF alternative rankings, five different alternative ranking methods are employed to obtain alternative rankings. Ultimately, the most suitable OWF location for meeting the energy demand of the rail industry is determined to be the Izmir region. The research findings contribute to managerial perspectives by proposing recommendations concerning strategic location selection, optimized resource allocation, risk mitigation and policy alignment and sustainable long-term planning.

Multiple attribute group decision making approach for selection of robot under induced bipolar neutrosophic aggregation operators

In current piece of writing, we bring in the new notion of induced bipolar neutrosophic (BN) AOs by utilizing Einstein operations as the foundation for aggregation operators (AOs), as well as to endow having a real-world problem-related application. The neutrosophic set can rapidly and more efficiently bring out the partial, inconsistent, and ambiguous information. The fundamental definitions and procedures linked to the basic bipolar neutrosophic (BN) set as well as the neutrosophic set (NS), are presented first. Our primary concern is the induced Einstein AOs, like, induced bipolar neutrosophic Einstein weighted average (I-BNEWA), induced bipolar neutrosophic Einstein weighted geometric (I-BNEWG), as well as their different types and required properties. The main advantage of employing the offered methods is that they give decision-makers a more thorough analysis of the problem. These strategies whenever compare to on hand methods, present complete, progressively precise, and accurate result. Finally, utilizing a numerical representation of an example for selection of robot, for a problem involving multi-criteria community decision making, we propose a novel solution. The suitability ratings are then ranked to select the most suitable robot. This demonstrates the practicality as well as usefulness of these novel approaches.

University’s recruitment process using Fermatean fuzzy Einstein prioritized aggregation operators

In comparison to intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS), the Fermatean Fuzzy Set (FFS) is more efficacious in dealing ambiguous and imprecise data when making decisions. In this paper, we propose unique operations on Fermatean fuzzy information based on prioritized attributes, as well as Einstein’s operations based on adjusting the priority of characteristics in the Fermatean fuzzy environment. We use Einstein’s operations with prioritized attributes to propose new operations on Fermatean fuzzy numbers (FFNs), and then introduce basic aspects of these operations. Motivated by Einstein operations on FFNs, we develop Fermatean fuzzy Einstein prioritized arithmetic and geometric aggregation operators (AOs). In the first place, the concepts of a Fermatean fuzzy Einstein prioritized average (FFEPA), Fermatean fuzzy Einstein prioritized weighted average (FFEPWA), and Fermatean fuzzy Einstein prioritized ordered weighted average (FFEPOWA)-operators are introduced. Then, Fermatean fuzzy Einstein prioritized geometric (FFEPG) operator, Fermatean fuzzy Einstein prioritized weighted geometric (FFEPWG) operator, Fermatean fuzzy Einstein prioritized ordered weighted geometric (FFEPOWG) operator, and Fermatean fuzzy Einstein hybrid geometric (FFEHG) operator are given. We also go through some of the key characteristics of these operators. Moreover, using these operators, we establish algorithm for addressing a multiple attribute decision-making issue using Fermatean fuzzy data and attribute prioritizing. The case of university faculty selection is taken as a scenario to analyze and demonstrate the applicability of our suggested model. In addition, a comparison of the proposed and current operators is conducted, and the impact of attribute priority on the ranking order of alternatives is explored.

The seed-to-solution method for the Einstein constraints and the asymptotic localization problem

We establish the existence of a broad class of asymptotically Euclidean solutions to Einstein's constraint equations, whose asymptotic behavior at infinity is prescribed arbitrarily. The proposed seed-to-solution method encompasses vacuum as well as matter spaces, and relies on iterations based on the linearized Einstein operator and its dual. It generates a Riemannian manifold (with finitely many asymptotically Euclidean ends) from any seed data set consisting of(1) a Riemannian metric and a symmetric two-tensor and(2) a (density) field and a (momentum) vector field representing the matter content. We distinguish between tame or strongly tame seed data sets, depending whether the data provides a rough or an accurate asymptotic Ansatz at infinity. We encompass classes of metrics and matter fields with low decay (possibly with infinite ADM mass) or strong decay (with Schwarzschild behavior). Our analysis is motivated by Carlotto and Schoen's pioneering work on the localization problem for Einstein's vacuum equations. Dealing with metrics with low decay and, simultaneously, establishing estimates that include (and go beyond) harmonic decay require significantly new arguments which are developed in the present paper. We work in a weighted Lebesgue-Hölder framework adapted to the given seed data, and we analyze the nonlinear coupling between the Hamiltonian and momentum constraints. By establishing elliptic regularity estimates for the linearized Einstein operator and its dual, we uncover the novel notion of mass-momentum correctors which is related to the ADM mass-momentum of the manifold. We derive precise estimates for the difference between the seed data and the associate Einstein solution —a result that should be of interest for future numerical investigations. Furthermore, we introduce here and study the asymptotic localization problem in which we replace Carlotto-Schoen's exact localization requirement by an asymptotic condition at a super-harmonic rate. By applying our seed-to-solution method with a suitably constructed, parametrized family of seed data, we solve this problem by exhibiting mass-momentum correctors with harmonic decay.

Development of dual hesitant fuzzy prioritized operators based on Einstein operations with their application to multi-criteria group decision making

Development of dual hesitant fuzzy prioritized operators based on Einstein operations with their application to multi-criteria group decision making

Picture fuzzy Choquet integral based Einstein operations and its application in selection of the best mobile apps for online education

Picture fuzzy aggregation operators are the standard mathematical tools for the combination of several inputs with respect to attributes into one unique output. The Choquet integral operator has been proven more ideal than traditional aggregation operators in the modelling of interaction phenomena among the attributes in decision-making problems. Firstly, we propose the Choquet integral picture fuzzy Einstein geometric aggregation operator and Choquet integral picture fuzzy Einstein ordered geometric aggregation operator with certain properties of these operators being established. We validate the functioning of the operators with illustrative examples. The proposed operators clearly capture the comprehensive correlative relationships of attributes in a simpler manner. Furthermore, the algorithm for a multi attribute decision-making problem based on proposed operators is given. The application of the proposed operators was explored to deal with the selection of the best mobile apps for online education. Finally, comparisons are conducted to illustrate the discussion and advantages of the proposed operators.