Mathematical Proofs
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2022 ◽  
Vol 2022 ◽  
pp. 1-15
Muhammad Umar Farooq ◽  
Rukhshanda Anjum ◽  
Abdul Gaffar ◽  
Huma Bashir ◽  
Naziha Al-Aidroos ◽  

Neutrosophic set (NS) is an extensively used framework whenever the imprecision and uncertainty of an event is described based on three possible aspects. The association, neutral, and nonassociation degrees are the three unique aspects of an NS. More importantly, these degrees are independent which is a great plus point. On the contrary, neutrosophic graphs (NGs) and single-valued NGs (SVNGs) are applicable to deal with events that contain bulks of information. However, the concept of degrees in NGs is a handful tool for solving the problems of decision-making (DM), pattern recognition, social network, and communication network. This manuscript develops various forms of edge irregular SVNG (EISVNG), highly edge irregular SVNG (HEISVNG), strongly (EISVNG), strongly (ETISVNG), and edge irregularity on a cycle and a path in SVNGs. All these novel notions are supported by definitions, theorems, mathematical proofs, and illustrative examples. Moreover, two types of DM problems are modelled using the proposed framework. Furthermore, the computational processes are used to confirm the validity of the proposed graphs. Furthermore, the results approve that the decision-making problems can be addressed by the edge irregular neutrosophic graphical structures. In addition, the comparison between proposed and the existing methodologies is carried out.

Mengxuan Han ◽  
Jihong Yan

This article estimates bounds of several special functions. It also gives mathematical proofs and graphs of the corresponding functions. The results are applicable in aspect of inequalities.

2021 ◽  
Vol 6 (1) ◽  
József Dombi ◽  
Sakshi Dhama

AbstractDetecting a community structure on networks is a problem of interest in science and many other domains. Communities are special structures which may consist nodes with some common features. The identification of overlapping communities can clarify not so apparent features about relationships among the nodes of a network. A node in a community can have a membership in a community with a different degree. Here, we introduce a fuzzy based approach for overlapping community detection. A special type of fuzzy operator is used to define the membership strength for the nodes of community. Fuzzy systems and logic is a branch of mathematics which introduces many-valued logic to compute the truth value. The computed truth can have a value between 0 and 1. The preference modelling approach introduces some parameters for designing communities of particular strength. The strength of a community tells us to what degree each member of community is part of a community. As for relevance and applicability of the community detection method on different types of data and in various situations, this approach generates a possibility for the user to be able to control the overlap regions created while detecting the communities. We extend the existing methods which use local function optimization for community detection. The LFM method uses a local fitness function for a community to identify the community structures. We present a community fitness function in pliant logic form and provide mathematical proofs of its properties, then we apply the preference implication of continuous-valued logic. The preference implication is based on two important parameters $$\nu$$ ν and $$\alpha$$ α . The parameter $$\nu$$ ν of the preference-implication allows us to control the design of the communities according to our requirement of the strength of the community. The parameter $$\alpha$$ α defines the sharpness of preference implication. A smaller value of the threshold for community membership creates bigger communities and more overlapping regions. A higher value of community membership threshold creates stronger communities with nodes having more participation in the community. The threshold is controlled by $$\delta$$ δ which defines the degree of relationship of a node to a community. To balance the creation of overlap regions, stronger communities and reducing outliers we choose a third parameter $$\delta$$ δ in such a way that it controls the community strength by varying the membership threshold as community evolves over time. We test the theoretical model by conducting experiments on artificial and real scale-free networks. We test the behaviour of all the parameters on different data-sets and report the outliers found. In our experiments, we found a good relationship between $$\nu$$ ν and overlapping nodes in communities.

2021 ◽  
Vol 2123 (1) ◽  
pp. 012046
I Minggi ◽  
Alimuddin ◽  

Abstract A learning trajectory for constructing mathematical proof has been developed. The trajectory is to provide the students with a step-by-step procedure in constructing arguments for proving mathematical statements. However, in proving activities, the students were found to encounter difficulties in completing a deductive axiomatic argument constituting an accepted mathematical proof. An investigation has been conducted to explore the problems the students experienced in constructing proofs. It was found that they faced language constraints in constructing mathematical arguments. They encountered challenges in how to correctly express the mathematical statements in their constructed proofs.

2021 ◽  
Vol 52 (5) ◽  
pp. 510-538
Paul Christian Dawkins ◽  
Dov Zazkis

This article documents differences between novice and experienced undergraduate students’ processes of reading mathematical proofs as revealed by moment-by-moment, think-aloud protocols. We found three key reading behaviors that describe how novices’ reading differed from that of their experienced peers: alternative task models, accrual of premises, and warranting. Alternative task models refer to the types of goals that students set up for their reading of the text, which may differ from identifying and justifying inferences. Accrual of premises refers to the way novice readers did not distinguish propositions in the theorem statement as assumptions or conclusions and thus did not use them differently for interpreting the proof. Finally, we observed variation in the type and quality of warrants, which we categorized as illustrate with examples, construct a miniproof, or state the warrant in general form.

Marcel Buß

Abstract Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning.

2021 ◽  
Vol 02 (09) ◽  
pp. 52-56
Toshpolatov M.T. ◽  
Abdullayev A.X. ◽  
Zaynobiddinov I.S. ◽  

This article focuses on teaching students how to use experimental mathematics in proving mathematical proofs. Firstly, the proofs of the theorem are analyzed by experiment and as a result of the ability of intuitive thinking, it is proved analytically. In the process of training, using experiments will increase the quality of the educational system.

2021 ◽  
Vol 1 ◽  
pp. 8
Marcus Vinicius dos Santos Claro

Once introduced the semiotic concept of discourse we aim to develop a discussion about the process of constructing the scientific discourse, that is, the modeling process of scientific law declarations through linguistics texts, whereas a imposed enunciation. For that, we distinguish three basic components:1. Intention, which is a motivation, animpulse for the discourse generation; 2. Enunciation, which express the scientific text itself; and 3. Legislation,which assumes a law enunciation. All this is established in order to assume a discourse of truth, including the correspondence with mathematical proofs. So, we characterize the symbolic manipulation of self evidence empirical facts which are reflected into the enunciations by a law format.

Episteme ◽  
2021 ◽  
pp. 1-27
Joshua Habgood-Coote ◽  
Fenner Stanley Tanswell

Abstract In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects.

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Bernd Engelmann

PurposeThe purpose of this article is to derive formulas for lifetime expected credit loss of loans that are required for the calculation of loan loss reserves under IFRS 9. This is done both for fixed-rate and floating rate loans under different assumptions on LGD modeling, prepayment, and discount rates.Design/methodology/approachThis study provides exact formulas for lifetime expected credit loss derived analytically together with the mathematical proofs of each expression.FindingsThis articles shows that the formula most commonly applied in the literature for calculating lifetime expected credit loss is inconsistent with measuring expected loss based on expected discounted cash flows. Formulas based on discounted cash flows always lead to more conservative numbers.Practical implicationsFor banks reporting under IFRS 9, the implication of this research is a better understanding of the different approaches used for computing lifetime expected loss, how they are connected, and what assumptions are underlying each approach. This may lead to corrections in existing frameworks to make applications of risk management systems more consistent.Originality/valueWhile there is a lot of literature explaining IFRS 9 and evaluating its impact, none of the existing research has systematically analyzed the calculation of lifetime expected credit loss for this purpose and how the formula changes under different modeling assumptions. This gap is filled by this study.

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