Prospects for Applied Mathematics and Data Analysis

Feuerbach’s theorem on the tangent of the circle of the nine points and the inscribed and exinscribed circle is considered one of the most beautiful theorems in geometry. In this paper, we offer a basic proof of this theorem starting from one of Gh. Buicliu’s ideas [1].

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein bottle logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018).

The essence of operations research is focused on creating and using models. The first step is to create the model, which requires a set of data that is determined from the aspects of the real system that the model must represent. The model is an acceptable model if it achieves the purpose for which it was formulated, and since programming issues linearity is concerned with allocating scarce resources, including labor, machinery, and capital, and using them in the best possible way, such that costs are reduced to a minimum or profits are maximized, by choosing the optimal decision from several available options. Since linear models are used in many fields, it was necessary to prepare studies that meet the needs of decision makers who have made solutions to linear programming problems a safe haven for them. The duality theory is considered one of the most important linear programming theories because it is used in many fields and is relied upon in the economic interpretation of the content of linear models. It provides a comprehensive study of the system represented by the linear model and its dual model. In this research, we present a study of the theory of neutrosophic dual and its economic interpretation by presenting a set of theorems that can be relied upon in explaining the results of solving both the original neutrosophic models and their dual.

The present article delves into the extension of Knuth’s fundamental Boolean logic table to accommodate the complexities of indeterminate truth values through the integration of neutrosophic logic (Smarandache & Christianto, 2008). Neutrosophic logic, rooted in Florentin Smarandache’s groundbreaking work on Neutrosophic Logic (cf. Smarandache, 2005, and his other works), introduces an additional truth value, ‘indeterminate,’ enabling a more comprehensive framework to analyze uncertainties inherent in computational systems. By bridging the gap between traditional boolean operations and the indeterminacy present in various real-world scenarios, this extension redefines logic tables, introducing neutrosophic operators that capture nuances beyond the binary realm. Through a thorough exploration of neutrosophic logic's principles and its implications in computational paradigms, this study proposes a novel approach to logic design that accommodates uncertain, imprecise, and incomplete information. This paradigm shift in logic tables not only broadens the spectrum of computing methodologies but also holds promise in fields such as decision-making systems and data analytics. This article amalgamates insights from over twelve key references encompassing seminal works in boolean logic, neutrosophic logic, and their applications in diverse scientific and computational domains, aiming to pave the way for a more robust and adaptable logic framework in computation.

One of the most important theories in linear programming is the dualistic theory and its basic idea is that for every linear model has dual linear model, so that solving the original linear model gives a solution to the dual model. Therefore, when we solving the linear programming model, we actually obtain solutions for two linear models. In this research, we present a study of the models. The neutrosophic dual and the binary simplex algorithm, which works to find the optimal solution for the original and dual models at the same time. The importance of this algorithm is evident in that it is relied upon in several operations research topics, such as integer programming algorithms, some nonlinear programming algorithms, and sensitivity analysis in linear programming...