Yes / Fault tree analysis is a widely used method of risk assessment in process industries. However, the classical fault tree approach has its own limitations such as the inability to deal with uncertain failure data and to consider statistical dependence among the failure events. In this paper, we propose a comprehensive framework for the risk assessment in process industries under the conditions of uncertainty and statistical dependency of events. The proposed approach makes the use of expert knowledge and fuzzy set theory for handling the uncertainty in the failure data and employs the Bayesian network modeling for capturing dependency among the events and for a robust probabilistic reasoning in the conditions of uncertainty. The effectiveness of the approach was demonstrated by performing risk assessment in an ethylene transportation line unit in an ethylene oxide (EO) production plant.

Yes / Temporal fault trees (TFTs), an extension of classical Boolean fault trees, can model time-dependent failure behaviour of dynamic systems. The methodologies used for quantitative analysis of TFTs include algebraic solutions, Petri nets (PN), and Bayesian networks (BN). In these approaches, precise failure data of components are usually used to calculate the probability of the top event of a TFT. However, it can be problematic to obtain these precise data due to the imprecise and incomplete information about the components of a system. In this paper, we propose a framework that combines intuitionistic fuzzy set theory and expert elicitation to enable quantitative analysis of TFTs of dynamic systems with uncertain data. Experts’ opinions are taken into account to compute the failure probability of the basic events of the TFT as intuitionistic fuzzy numbers. Subsequently, for the algebraic approach, the intuitionistic fuzzy operators for the logic gates of TFT are defined to quantify the TFT. On the other hand, for the quantification of TFTs via PN and BN-based approaches, the intuitionistic fuzzy numbers are defuzzified to be used in these approaches. As a result, the framework can be used with all the currently available TFT analysis approaches. The effectiveness of the proposed framework is illustrated via application to a practical system and through a comparison of the results of each approach. / This work was supported in part by the Mobile IOT: Location Aware project (grant no. MMUE/180025) and Indoor Internet of Things (IOT) Tracking Algorithm Development based on Radio Signal Characterisation project (grant no. FRGS/1/2018/TK08/MMU/02/1). This research also received partial support from DEIS H2020 project (grant no. 732242).

<p dir="ltr">The concept of infinity plays a pivotal role in mathematics, yet its precise definition remains elusive. This conceptual ambiguity has given rise to several puzzles in contemporary philosophy of mathematics. In response, this dissertation embarks on a rational reconstruction of Hegels concept of infinity and applies it to resolve two groups of mathematical puzzles, including challenges in applied mathematics, especially the application of differential calculus, and the conceptual ground of set theory, especially Cantors paradox.</p><p dir="ltr">The exploration begins with a historical survey of the concept of infinity in philosophy. It becomes evident that a prevailing interpretation characterizes infinity as the unlimited. In addition, this unlimitedness has taken various forms, including endlessness (Aristotle), all-inclusiveness (Spinoza), and self-sufficiency (Kant).</p><p dir="ltr">The heart of the dissertation lies in reconstructing Hegels concept of genuine infinity. Hegel argues that the unlimited as the negation of the limit entails either the completely indeterminate or another limited entity, neither of which is genuinely infinite. Instead, Hegel points out that genuine infinity is the self-relation of a limited entity. By self-relation, Hegel means that the limited entity alters into another limited entity that is isomorphic to the original one.</p><p dir="ltr">Subsequently, Hegel’s concept of genuine infinity can be translated into a mathematical framework as the intrinsic alteration of quantum (roughly speaking, quantum is Hegel’s term for the variable), which is captured by the corresponding relation among quanta. It is argued that this relation serves as the necessary condition for three mathematical entities traditionally considered infinite: arbitrarily large (small) numbers, infinite sets, and endless sequences. Thus, for Hegel, this intrinsic relation among quanta constitutes the essence of mathematical infinity.</p><p dir="ltr">Hegels concept of mathematical infinity can help us resolve difficulties within contemporary mathematics. First, it addresses the question of why infinite mathematical structures can be applied to describe and predict seemingly finite physical phenomena. The application of mathematics is usually explained by the similarity between mathematical structures and empirical systems, but the lack of apparent empirical counterpart leads one to doubt the application of infinite mathematical structures. Hegels concept of mathematical infinity directs us to focus on the structural similarity between infinite mathematical structures and empirical systems, specifically between the intrinsic alteration of quantum and the change of physical properties with time. With this structural similarity, the application of mathematics can be explained. Second, the dissertation investigates the conceptual ground of set theory, especially the relationship between a set and its members. Hegels analysis of genuine infinity provides a twofold clarification: (1) members of set must be a unit first, which entails that the set of all sets (the Universe) is not a set; (2) members of a set are simultaneously distinct (due to their independent logical content) yet indistinguishable (due to their common structure as a unit). Clarification 1 resolves Cantors paradox as it excludes the Universe; clarification 2 explains arithmetic operations.</p>

In this paper, we prove that it is consistent with ZFC that GCH holds and that every pair of normal Aronszajn trees contain club isomorphic subtrees.

No / In recent decades, sustainable investing has caught on with investors, and it has now become the norm. In the age of start-ups, with scant information on the sustainability aspects of an asset, it becomes harder to pursue sustainable investing. To this end, this paper proposes a sustainable financial portfolio selection approach in an intuitionistic fuzzy framework. We present a comprehensive three-stage methodology in which the assets under consideration are ethically screened in Stage-I. Stage-II is concerned with cal- culating the sustainability scores, based on various social, environmental, and economic (SEE) criteria and an evaluation of the return and risk of the ethical assets. Intuitionistic fuzzy set theory is used to gauge the linguistic assessment of the assets on several SEE criteria from multiple decision-makers. A novel intuitionistic fuzzy multi-criteria group decision-making technique is applied to calculate the sustainability score of each asset. Finally, in Stage-III, an intuitionistic fuzzy multi-objective financial portfolio selection model is developed with maximization of the satisfaction degrees of the sustainabil- ity score, return, and risk of the portfolio, subject to several constraints. The ε-constraint method is used to solve this model, which yields various efficient, sustainable financial portfolios. Subsequently, investors can choose the portfolio best suited to their preferences from this pool of efficient, sustainable financial portfolios. A detailed empirical illustration and a comparison with existing works are given to substantiate and validate the proposed approach. / Institution of Eminence, University of Delhi, Delhi-110007 under Faculty Research Program

Yes / Quantitative risk assessment (QRA) approaches systematically evaluate the likelihood, impacts, and risk of adverse events. QRA using fault tree analysis (FTA) is based on the assumptions that failure events have crisp probabilities and they are statistically independent. The crisp probabilities of the events are often absent, which leads to data uncertainty. However, the independence assumption leads to model uncertainty. Experts’ knowledge can be utilized to obtain unknown failure data; however, this process itself is subject to different issues such as imprecision, incompleteness, and lack of consensus. For this reason, to minimize the overall uncertainty in QRA, in addition to addressing the uncertainties in the knowledge, it is equally important to combine the opinions of multiple experts and update prior beliefs based on new evidence. In this article, a novel methodology is proposed for QRA by combining fuzzy set theory and evidence theory with Bayesian networks to describe the uncertainties, aggregate experts’ opinions, and update prior probabilities when new evidences become available. Additionally, sensitivity analysis is performed to identify the most critical events in the FTA. The effectiveness of the proposed approach has been demonstrated via application to a practical system. / The research of Sohag Kabir was partly funded by the DEIS project (Grant Agreement 732242).

Yes / Fault tree analysis (FTA) is a powerful technique that is widely used for evaluating system safety and reliability. It can be used to assess the effects of combinations of failures on system behaviour but is unable to capture sequence dependent dynamic behaviour. A number of extensions to fault trees have been proposed to overcome this limitation. Pandora, one such extension, introduces temporal gates and temporal laws to allow dynamic analysis of temporal fault trees (TFTs). It can be easily integrated in model-based design and analysis techniques. The quantitative evaluation of failure probability in Pandora TFTs is performed using exact probabilistic data about component failures. However, exact data can often be difficult to obtain. In this paper, we propose a method that combines expert elicitation and fuzzy set theory with Pandora TFTs to enable dynamic analysis of complex systems with limited or absent exact quantitative data. This gives Pandora the ability to perform quantitative analysis under uncertainty, which increases further its potential utility in the emerging field of model-based design and dependability analysis. The method has been demonstrated by applying it to a fault tolerant fuel distribution system of a ship, and the results are compared with the results obtained by other existing techniques.

Yes / Fuzzy data-driven reliability analysis has been used in different safety-critical domains for risk assessment and decision-making where precise failure data is non-existent. Expert judgements and fuzzy set theory have been combined with different variants of fault trees as part of fuzzy data-driven reliability analysis studies. In such fuzzy fault tree analyses, different people represented failure data using different membership functions for the fuzzy set, and different parameters were set differently in the expert opinion elicitation process. Due to the availability of a wide variety of options, it is possible to obtain different outcomes when choosing one option over another. This article performed an analysis in the context of fuzzy data-based temporal fault tree analysis to investigate the effect of choosing different membership functions on the estimated system reliability and criticality ranking of different failure events. Moreover, the effect of using different values for the relaxation factor, a parameter set during the expert elicitation process, was studied on the system reliability and criticality evaluation. The experiments on the fuel distribution system case study show system reliability did not vary when triangular and trapezoidal fuzzy numbers were used with the same upper and lower bounds. However, it was seen that the criticality rankings of a couple of events were changed due to choosing different membership functions and different values of relaxation factor

As soon as Bennett and Gill first demonstrated that, relative to a randomly chosen oracle, P is not equal to NP with probability 1, the random oracle hypothesis began piquing the interest of mathematicians and computer scientists. This was quickly disproven in several ways, most famously in 1992 with the result that IP equals PSPACE, in spite of the classes being shown unequal with probability 1. Here, we propose what could be considered strengthening of the random oracle hypothesis, using a stricter notion of what it means for a set to be 'large'. In particular, we suggest using largeness with respect to the Ramsey forcing notion. In this new context, we demonstrate that the set of oracles separating NP and coNP is 'not small', and obtain similar results for the separation of PSPACE from PH along with the separation of NP from BQP. In a related set of results, we demonstrate that these classes are all of the same descriptive complexity. Finally we demonstrate that this strengthening of the hypothesis turns it into a sufficient condition for unrelativized relationships, at least in the three cases considered here.

In this paper we explore coloring theorems for the reals, its quotients, cardinals, and their combinations. This work is done under the scope of the axiom of determinacy. We also explore generalizations of Mycielski's theorem and show how these can be used to establish coloring theorems. To finish, we discuss the strange realm of long unions.