A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

A Topological Approach to Fuzzy Sets

This thesis presents a topological approach to studying fuzzy setsby means of modifier operators. Modifier operators are mathematical models, e.g., for hedges, and we present briefly different approaches to studying modifier operators. We are interested in compositional modifier operators, modifiers for short, and these modifiers depend on binary relations. We show that if a modifier depends on a reflexive and transitive binary relation on U, then there exists a unique topology on U such that this modifier is the closure operator in that topology. Also, if U is finite then there exists a lattice isomorphism between the class of all reflexive and transitive relations and the class of all topologies on U. We define topological similarity relation "≈" between L-fuzzy sets in an universe U, and show that the class LU/ ≈ is isomorphic with the class of all topologies on U, if U is finite and L is suitable. We consider finite bitopological spaces as approximation spaces, and we show that lower and upper approximations can be computed by means of α-level sets also in the case of equivalence relations. This means that approximations in the sense of Rough Set Theory can be computed by means of α-level sets. Finally, we present and application to data analysis: we study an approach to detecting dependencies of attributes in data base-like systems, called information systems.

Selective catalytic reduction of nitrogen oxides with ammonia in forced unsteady state reactors - Case based reasoning and mathematical model simulation reasoning

The application of forced unsteady-state reactors in case of selective catalytic reduction of nitrogen oxides (NOx) with ammonia (NH3) is sustained by the fact that favorable temperature and composition distributions which cannot be achieved in any steady-state regime can be obtained by means of unsteady-state operations. In a normal way of operation the low exothermicity of the selective catalytic reduction (SCR) reaction (usually carried out in the range of 280-350°C) is not enough to maintain by itself the chemical reaction. A normal mode of operation usually requires supply of supplementary heat increasing in this way the overall process operation cost. Through forced unsteady-state operation, the main advantage that can be obtained when exothermic reactions take place is the possibility of trapping, beside the ammonia, the moving heat wave inside the catalytic bed. The unsteady state-operation enables the exploitation of the thermal storage capacity of the catalyticbed. The catalytic bed acts as a regenerative heat exchanger allowing auto-thermal behaviour when the adiabatic temperature rise is low. Finding the optimum reactor configuration, employing the most suitable operation model and identifying the reactor behavior are highly important steps in order to configure a proper device for industrial applications. The Reverse Flow Reactor (RFR) - a forced unsteady state reactor - corresponds to the above mentioned characteristics and may be employed as an efficient device for the treatment of dilute pollutant mixtures. As a main disadvantage, beside its advantages, the RFR presents the 'wash out' phenomena. This phenomenon represents emissions of unconverted reactants at every switch of the flow direction. As a consequence our attention was focused on finding an alternative reactor configuration for RFR which is not affected by the incontrollable emissions of unconverted reactants. In this respect the Reactor Network (RN) was investigated. Its configuration consists of several reactors connected in a closed sequence, simulating a moving bed by changing the reactants feeding position. In the RN the flow direction is maintained in the same way ensuring uniformcatalyst exploitation and in the same time the 'wash out' phenomena is annulated. The simulated moving bed (SMB) can operate in transient mode giving practically constant exit concentration and high conversion levels. The main advantage of the reactor network operation is emphasizedby the possibility to obtain auto-thermal behavior with nearly uniformcatalyst utilization. However, the reactor network presents only a small range of switching times which allow to reach and to maintain an ignited state. Even so a proper study of the complex behavior of the RN may give the necessary information to overcome all the difficulties that can appear in the RN operation. The unsteady-state reactors complexity arises from the fact that these reactor types are characterized by short contact times and complex interaction between heat and mass transportphenomena. Such complex interactions can give rise to a remarkable complex dynamic behavior characterized by a set of spatial-temporal patterns, chaotic changes in concentration and traveling waves of heat or chemical reactivity. The main efforts of the current research studies concern the improvement of contact modalities between reactants, the possibility of thermal wave storage inside the reactor and the improvement of the kinetic activity of the catalyst used. Paying attention to the above mentioned aspects is important when higher activity even at low feeding temperatures and low emissions of unconverted reactants are the main operation concerns. Also, the prediction of the reactor pseudo or steady-state performance (regarding the conversion, selectivity and thermal behavior) and the dynamicreactor response during exploitation are important aspects in finding the optimal control strategy for the forced unsteady state catalytic tubular reactors. The design of an adapted reactor requires knowledge about the influence of its operating conditions on the overall process performance and a precise evaluation of the operating parameters rage for which a sustained dynamic behavior is obtained. An apriori estimation of the system parameters result in diminution of the computational efforts. Usually the convergence of unsteady state reactor systems requires integration over hundreds of cycles depending on the initial guess of the parameter values. The investigation of various operation models and thermal transfer strategies give reliable means to obtain recuperative and regenerative devices which are capable to maintain an auto-thermal behavior in case of low exothermic reactions. In the present research work a gradual analysis of the SCR of NOx with ammonia process in forced unsteady-state reactors was realized. The investigation covers the presentationof the general problematic related to the effect of noxious emissions in the environment, the analysis of the suitable catalysts types for the process, the mathematical analysis approach for modeling and finding the system solutions and the experimental investigation of the device found to be more suitable for the present process. In order to gain information about the forced unsteady state reactor design, operation, important system parameters and their values, mathematical description, mathematicalmethod for solving systems of partial differential equations and other specific aspects, in a fast and easy way, and a case based reasoning (CBR) approach has been used. This approach, using the experience of past similarproblems and their adapted solutions, may provide a method for gaining informations and solutions for new problems related to the forced unsteady state reactors technology. As a consequence a CBR system was implemented and a corresponding tool was developed. Further on, grooving up the hypothesis of isothermal operation, the investigation by means of numerical simulation of the feasibility of the SCR of NOx with ammonia in the RFRand in the RN with variable feeding position was realized. The hypothesis of non-isothermal operation was taken into account because in our opinion ifa commercial catalyst is considered, is not possible to modify the chemical activity and its adsorptive capacity to improve the operation butis possible to change the operation regime. In order to identify the most suitable device for the unsteady state reduction of NOx with ammonia, considering the perspective of recuperative and regenerative devices, a comparative analysis of the above mentioned two devices performance was realized. The assumption of isothermal conditions in the beginningof the forced unsteadystate investigation allowed the simplification of the analysis enabling to focus on the impact of the conditions and mode of operation on the dynamic features caused by the trapping of one reactant in the reactor, without considering the impact of thermal effect on overall reactor performance. The non-isothermal system approach has been investigated in order to point out the important influence of the thermal effect on overall reactor performance, studying the possibility of RFR and RN utilization as recuperative and regenerative devices and the possibility of achieving a sustained auto-thermal behavior in case of lowexothermic reaction of SCR of NOx with ammonia and low temperature gasfeeding. Beside the influence of the thermal effect, the influence of the principal operating parameters, as switching time, inlet flow rate and initial catalyst temperature have been stressed. This analysis is important not only because it allows a comparison between the two devices and optimisation of the operation, but also the switching time is the main operating parameter. An appropriate choice of this parameter enables the fulfilment of the process constraints. The level of the conversions achieved, the more uniform temperature profiles, the uniformity ofcatalyst exploitation and the much simpler mode of operation imposed the RN as a much more suitable device for SCR of NOx with ammonia, in usual operation and also in the perspective of control strategy implementation. Theoretical simplified models have also been proposed in order to describe the forced unsteady state reactors performance and to estimate their internal temperature and concentration profiles. The general idea was to extend the study of catalytic reactor dynamics taking into account the perspectives that haven't been analyzed yet. The experimental investigation ofRN revealed a good agreement between the data obtained by model simulation and the ones obtained experimentally.

Linguistic models for decision support

Linguistic modelling is a rather new branch of mathematics that is still undergoing rapid development. It is closely related to fuzzy set theory and fuzzy logic, but knowledge and experience from other fields of mathematics, as well as other fields of science including linguistics and behavioral sciences, is also necessary to build appropriate mathematical models. This topic has received considerable attention as it provides tools for mathematical representation of the most common means of human communication - natural language. Adding a natural language level to mathematical models can provide an interface between the mathematical representation of the modelled system and the user of the model - one that is sufficiently easy to use and understand, but yet conveys all the information necessary to avoid misinterpretations. It is, however, not a trivial task and the link between the linguistic and computational level of such models has to be established and maintained properly during the whole modelling process. In this thesis, we focus on the relationship between the linguistic and the mathematical level of decision support models. We discuss several important issues concerning the mathematical representation of meaning of linguistic expressions, their transformation into the language of mathematics and the retranslation of mathematical outputs back into natural language. In the first part of the thesis, our view of the linguistic modelling for decision support is presented and the main guidelines for building linguistic models for real-life decision support that are the basis of our modeling methodology are outlined. From the theoretical point of view, the issues of representation of meaning of linguistic terms, computations with these representations and the retranslation process back into the linguistic level (linguistic approximation) are studied in this part of the thesis. We focus on the reasonability of operations with the meanings of linguistic terms, the correspondence of the linguistic and mathematical level of the models and on proper presentation of appropriate outputs. We also discuss several issues concerning the ethical aspects of decision support - particularly the loss of meaning due to the transformation of mathematical outputs into natural language and the issue or responsibility for the final decisions. In the second part several case studies of real-life problems are presented. These provide background and necessary context and motivation for the mathematical results and models presented in this part. A linguistic decision support model for disaster management is presented here – formulated as a fuzzy linear programming problem and a heuristic solution to it is proposed. Uncertainty of outputs, expert knowledge concerning disaster response practice and the necessity of obtaining outputs that are easy to interpret (and available in very short time) are reflected in the design of the model. Saaty’s analytic hierarchy process (AHP) is considered in two case studies - first in the context of the evaluation of works of art, where a weak consistency condition is introduced and an adaptation of AHP for large matrices of preference intensities is presented. The second AHP case-study deals with the fuzzified version of AHP and its use for evaluation purposes – particularly the integration of peer-review into the evaluation of R&D outputs is considered. In the context of HR management, we present a fuzzy rule based evaluation model (academic faculty evaluation is considered) constructed to provide outputs that do not require linguistic approximation and are easily transformed into graphical information. This is achieved by designing a specific form of fuzzy inference. Finally the last case study is from the area of humanities - psychological diagnostics is considered and a linguistic fuzzy model for the interpretation of outputs of multidimensional questionnaires is suggested. The issue of the quality of data in mathematical classification models is also studied here. A modification of the receiver operating characteristics (ROC) method is presented to reflect variable quality of data instances in the validation set during classifier performance assessment. Twelve publications on which the author participated are appended as a third part of this thesis. These summarize the mathematical results and provide a closer insight into the issues of the practicalapplications that are considered in the second part of the thesis.

Mathematical modeling and optimal control of malaria

Malaria continues to infect millions and kill hundreds of thousands of people worldwide each year, despite over a century of research and attempts to control and eliminate this infectious disease. Challenges such as the development and spread of drug resistant malaria parasites, insecticide resistance to mosquitoes, climate change, the presence of individuals with subpatent malaria infections which normally are asymptomatic and behavioral plasticity in the mosquito hinder the prospects of malaria control and elimination. In this thesis, mathematical models of malaria transmission and control that address the role of drug resistance, immunity, iron supplementation and anemia, immigration and visitation, and the presence of asymptomatic carriers in malaria transmission are developed. A within-host mathematical model of severe Plasmodium falciparum malaria is also developed. First, a deterministic mathematical model for transmission of antimalarial drug resistance parasites with superinfection is developed and analyzed. The possibility of increase in the risk of superinfection due to iron supplementation and fortification in malaria endemic areas is discussed. The model results calls upon stakeholders to weigh the pros and cons of iron supplementation to individuals living in malaria endemic regions. Second, a deterministic model of transmission of drug resistant malaria parasites, including the inflow of infective immigrants, is presented and analyzed. The optimal control theory is applied to this model to study the impact of various malaria and vector control strategies, such as screening of immigrants, treatment of drug-sensitive infections, treatment of drug-resistant infections, and the use of insecticide-treated bed nets and indoor spraying of mosquitoes. The results of the model emphasize the importance of using a combination of all four controls tools for effective malaria intervention. Next, a two-age-class mathematical model for malaria transmission with asymptomatic carriers is developed and analyzed. In development of this model, four possible control measures are analyzed: the use of long-lasting treated mosquito nets, indoor residual spraying, screening and treatment of symptomatic, and screening and treatment of asymptomatic individuals. The numerical results show that a disease-free equilibrium can be attained if all four control measures are used. A common pitfall for most epidemiological models is the absence of real data; model-based conclusions have to be drawn based on uncertain parameter values. In this thesis, an approach to study the robustness of optimal control solutions under such parameter uncertainty is presented. Numerical analysis of the optimal control problem in the presence of parameter uncertainty demonstrate the robustness of the optimal control approach that: when a comprehensive control strategy is used the main conclusions of the optimal control remain unchanged, even if inevitable variability remains in the control profiles. The results provide a promising framework for the design of cost-effective strategies for disease control with multiple interventions, even under considerable uncertainty of model parameters. Finally, a separate work modeling the within-host Plasmodium falciparum infection in humans is presented. The developed model allows re-infection of already-infected red blood cells. The model hypothesizes that in severe malaria due to parasite quest for survival and rapid multiplication, the Plasmodium falciparum can be absorbed in the already-infected red blood cells which accelerates the rupture rate and consequently cause anemia. Analysis of the model and parameter identifiability using Markov chain Monte Carlo methods is presented.

Sulautetun moniprosessorijärjestelmän suunnittelu sähkömoottorien ohjaus- ja valvontatarpeisiin

Universal Converter (UNICON) –projektin osana suunniteltiin sähkömoottorikäyttöjen ohjaukseen ja mittaukseen soveltuva digitaaliseen signaaliprosessoriin (DSP) pohjautuva sulautettu järjestelmä. Riittävän laskentatehon varmistamiseksi päädyttiin käyttämään moniprosessorijärjestelmää. Prosessorijärjestelmässä käytettävää DSP-piiriä valittaessa valintaperusteina olivat piirien tarjoama prosessointiteho ja moniprosessorituki. Analog Devices:n SHARC-sarjan DSP-piirit täyttivät parhaiten asetetut vaatimukset: Ne tarjoavat tehokkaan käskykannan lisäksi suuren sisäisen muistin ja sisäänrakennetun moniprosessorituen. Järjestelmän mittalaiteluonteisuudesta johtuen keskeinen suunnitteluparametri oli luoda nopeat tiedonsiirtoyhteydet mittausantureilta DSP-järjestelmään. Tämä toteutettiin käyttäen ohjelmointavia FPGA-logiikkapiirejä digitaalimuotoisen mittausdatan vastaanotossa ja esikäsittelyssä. Tiedonsiirtoyhteys PC-tietokoneelle toteutettiin käyttäen erityistä liityntäkorttia DSP-järjestelmän ja PC-tietokoneen välillä. Liityntäkortin päätehtävänä on puskuroida siirrettävä data. Järjestelyllä estetään PC-tietokoneen vaikutus DSP-järjestelmän toimintaan, jotta kyetään takaamaan järjestelmän reaaliaikainen toiminta kaikissa olosuhteissa.

Coopetitive advantage – How firms create and appropriate value by collaborating with their competitors

Collaboration between competing firms (coopetition) has emerged as an important issue forn business practice in many industries. Extant literature has examined coopetition on many levels of analysis, but lacks clarity in distinguishing it explicitly from cooperation between noncompeting organizations. Because of this, the performance implications of coopetition from the perspective of an individual firm are still ambiguous – some research suggests positive results whereas other studies suggest detrimental outcomes. The aim in this dissertation is to narrow these gaps by exploring how firms create and appropriate value through collaboration with their competitors. The dissertation is divided into two parts. The first part comprises an overview of the relevant literature, as well as the conclusions of the whole study, and the second part includes six research publications. Both qualitative and quantitative methodologies are utilized. The results suggest that coopetition embodies the distinctive logic of value creation and appropriation from the perspective of an individual firm, and thus differs in terms of performance implications from cooperation between non-competitors. The distinction comes from the fact that competitors have somewhat similar understanding, capabilities and interest related to certain markets, which is potentially both challenging and beneficial in terms of the individual firm’s competitiveness. It appears from the findings that there are distinctive firm-external and firm-specific factors affecting the success of a coopetition strategy. This study makes three main contributions. First, on the conceptual level it shows the distinction between coopetition and cooperation between non-rivals as a collaborative inter-organizational relationship. Secondly, it sets out a framework and propositions that enhance understanding of how value is created and appropriated in coopetition from the perspective of an individual firm. Thirdly, it offers empirical evidence of how coopetition affects firms’ innovation and market performance, and identifies the focal internal and external factors involved. In general terms, the thesis adds to our knowledge of how a firm can successfully utilize a coopetition strategy in its pursuit of improved performance.

A quantitative view on fuzzy numbers

Since its introduction, fuzzy set theory has become a useful tool in the mathematical modelling of problems in Operations Research and many other fields. The number of applications is growing continuously. In this thesis we investigate a special type of fuzzy set, namely fuzzy numbers. Fuzzy numbers (which will be considered in the thesis as possibility distributions) have been widely used in quantitative analysis in recent decades. In this work two measures of interactivity are defined for fuzzy numbers, the possibilistic correlation and correlation ratio. We focus on both the theoretical and practical applications of these new indices. The approach is based on the level-sets of the fuzzy numbers and on the concept of the joint distribution of marginal possibility distributions. The measures possess similar properties to the corresponding probabilistic correlation and correlation ratio. The connections to real life decision making problems are emphasized focusing on the financial applications. We extend the definitions of possibilistic mean value, variance, covariance and correlation to quasi fuzzy numbers and prove necessary and sufficient conditions for the finiteness of possibilistic mean value and variance. The connection between the concepts of probabilistic and possibilistic correlation is investigated using an exponential distribution. The use of fuzzy numbers in practical applications is demonstrated by the Fuzzy Pay-Off method. This model for real option valuation is based on findings from earlier real option valuation models. We illustrate the use of number of different types of fuzzy numbers and mean value concepts with the method and provide a real life application.

Social ventures : the development of the theory of sustainable contributive advantage and initial empirical tests

This thesis attempts to fill gaps in both a theoretical basis and an operational and strategic understanding in the areas of social ventures, social entrepreneurship and nonprofit business models. This study also attempts to bridge the gap in strategic and economic theory between social and commercial ventures. More specifically, this thesis explores sustainable competitive advantage from a resource-based theory perspective and explores how it may be applied to the nonmarket situation of nonprofit organizations and social ventures. It is proposed that a social value-orientation of sustainable competitive advantage, called sustainable contributive advantage, provides a more realistic depiction of what is necessary in order for a social venture to perform better than its competitors over time. In addition to providing this realistic depiction, this research provides a substantial theoretical contribution in the area of economics, social ventures, and strategy research, specifically in regards to resource-based theory. The proposed model for sustainable contributive advantage uses resource-based theory and competitive advantage in order to be applicable to social ventures. This model proposes an explanation of a social venture’s ability to demonstrate consistently superior performance. In order to determine whether sustainable competitive advantage is in fact, appropriate to apply to both social and economic environments, quantitative analyses are conducted on a large sample of nonprofit organizations in a single industry and then compared to similar quantitative analyses conducted on commercial ventures. In comparing the trends and strategies between the two types of entities from a quantitative perspective, propositions are developed regarding a social venture’s resource utilization strategies and their possible impact on performance. Evidence is found to support the necessity of adjusting existing models in resource-based theory in order to apply them to social ventures. Additionally supported is the proposed theory of sustainable contributive advantage. The thesis concludes with recommendations for practitioners, researchers and policy makers as well as suggestions for future research paths.

Design and evaluation of web interface signs to improve web usability : a semiotic framework

Technological innovations, the development of the internet, and globalization have increased the number and complexity of web applications. As a result, keeping web user interfaces understandable and usable (in terms of ease-of-use, effectiveness, and satisfaction) is a challenge. As part of this, designing userintuitive interface signs (i.e., the small elements of web user interface, e.g., navigational link, command buttons, icons, small images, thumbnails, etc.) is an issue for designers. Interface signs are key elements of web user interfaces because ‘interface signs’ act as a communication artefact to convey web content and system functionality, and because users interact with systems by means of interface signs. In the light of the above, applying semiotic (i.e., the study of signs) concepts on web interface signs will contribute to discover new and important perspectives on web user interface design and evaluation. The thesis mainly focuses on web interface signs and uses the theory of semiotic as a background theory. The underlying aim of this thesis is to provide valuable insights to design and evaluate web user interfaces from a semiotic perspective in order to improve overall web usability. The fundamental research question is formulated as What do practitioners and researchers need to be aware of from a semiotic perspective when designing or evaluating web user interfaces to improve web usability? From a methodological perspective, the thesis follows a design science research (DSR) approach. A systematic literature review and six empirical studies are carried out in this thesis. The empirical studies are carried out with a total of 74 participants in Finland. The steps of a design science research process are followed while the studies were designed and conducted; that includes (a) problem identification and motivation, (b) definition of objectives of a solution, (c) design and development, (d) demonstration, (e) evaluation, and (f) communication. The data is collected using observations in a usability testing lab, by analytical (expert) inspection, with questionnaires, and in structured and semi-structured interviews. User behaviour analysis, qualitative analysis and statistics are used to analyze the study data. The results are summarized as follows and have lead to the following contributions. Firstly, the results present the current status of semiotic research in UI design and evaluation and highlight the importance of considering semiotic concepts in UI design and evaluation. Secondly, the thesis explores interface sign ontologies (i.e., sets of concepts and skills that a user should know to interpret the meaning of interface signs) by providing a set of ontologies used to interpret the meaning of interface signs, and by providing a set of features related to ontology mapping in interpreting the meaning of interface signs. Thirdly, the thesis explores the value of integrating semiotic concepts in usability testing. Fourthly, the thesis proposes a semiotic framework (Semiotic Interface sign Design and Evaluation – SIDE) for interface sign design and evaluation in order to make them intuitive for end users and to improve web usability. The SIDE framework includes a set of determinants and attributes of user-intuitive interface signs, and a set of semiotic heuristics to design and evaluate interface signs. Finally, the thesis assesses (a) the quality of the SIDE framework in terms of performance metrics (e.g., thoroughness, validity, effectiveness, reliability, etc.) and (b) the contributions of the SIDE framework from the evaluators’ perspective.