We present the real number system as a natural generalization of the natural numbers. First, we prove the co-finite topology, $Cof(\mathbb N)$, is isomorphic to the natural numbers. Then, we generalize these results to describe the continuum $[0,1]$. Then we prove the power set $2^{\mathbb Z}$ contains a subset isomorphic to the non-negative real numbers, with all its defining structure of operations and order. Finally, we provide two different constructions of the entire real number line. We see that the power set $2^{\mathbb N}$ can be given the defining structure of $\mathbb R$. The constructions here provided give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. The supremum and infimum are explicitly constructed by means of a well defined algorithm that ends in denumerable steps. In section 5 we give evidence our construction of $\mathbb N$ and $\mathbb R$ are canonical; these constructions are as natural as possible. In the same section, we propose a new axiomatic basis for analysis. In the last section we provide a series of graphic representations and physical models that can be used to represent the real number system. We conclude that the system of real numbers is completely defined by the order structure of $\mathbb N$.}

This study derives space travel pricing by Walrasian Equilibrium, which is logical reasoning from the general relativity theory (GRT), the accounting equation, and economic supply and demand functions. The Cobb-Douglas functions embed the endogenous space factor as new capital to form the space travel firm's production function, which is also transformed into the consumer's utility function. Thus, the market equilibrium occurs at the equivalence of supply and demand functions, like the GRT presents the equivalence between the spatial geometric tensor and the energy-momentum tensor, which explains the principles of gravity and the motion of space matter in the spacetime framework. The mathematical axiomatic set theory of the accounting equation explains the equity premium effect that causes a short-term accounting equation inequality, then reaches the equivalence by suppliers' incremental equity through the closing accounts process of the accounting cycle. On the demand side, the consumption of space travel can be assumed as a value at risk (VaR) investment to attain the specific spacetime curvature in an expected orbit. Spacetime market equilibrium is then achieved to construct the space travel pricing model. The methodology of econophysics and analogy method was applied to infer the space travel pricing by the model of profit maximization, single-minded, and envy-free pricing in unit-demand markets. A case study with simulation was conducted for empirical verification of the math models and algorithm. Results show that space travel pricing remains the principle of market equilibrium but needs to be extended to the spacetime tensor of GRT.

Manufacturing environments, characterized by rapid, unpredictable changes, uncertainties, risks, and uncontrollable fluctuations, pose significant challenges to minimizing disruptions in processes. This study introduces an innovative approach that prioritizes curbing risk propagation among processes to enhance robustness. It emphasizes the integration of engineering management principles and modular design within manufacturing. Adopting a system engineering perspective, all manufacturing process activities are viewed as interrelated components within a unified system. By employing Axiomatic Design (AD) theory and the Design Structure Matrix (DSM) method, manufacturing process architecture is modularized, yielding heightened robustness. The proposed mathematical model equips engineering and manufacturing managers with a potent tool for designing robust processes while adeptly managing system complexity. The study's outcomes underscore a substantial enhancement in modularization, leading to elevated overall robustness in manufacturing processes. To validate the methodology, the architectural design of manufacturing processes is examined in a real-case scenario, specifically the Barez Industrial Group in Iran. This verification substantiates the 'manufacturing processes' of the case, presenting an optimally modularized architecture. The results affirm the proposed approach's efficacy, demonstrating improved modularization that contributes to bolstered robustness in manufacturing processes.

Knowledge is not merely the characteristic of certain pieces of information being deemed true and justifiable, but rather a process. Within an axiomatic system, the set of information that reaches the status of knowledge forms a universe that provides an interpretation of the reality one aims to describe. Language exchanges information through communication and the process of justification (exchange of information), if it consistently justifies a particular piece of information, it elevates it to knowledge. In this paper, we provide a framework for knowledge within an axiomatic model where we define the fundamental elements of communication and grammar. As a result, the structure dictates that knowledge, to be considered as such, must be investigateable by anyone who utilizes it in its entirety, both in form and content. Ultimately, the development of knowledge processing, referred to as \emph{knowmatics}, represents a clear evolution of the computer science that processes information (subset of knowledge).

This paper's aim is to extend horizons in symbolic AI. The key to doing so is to establish a scientific basis for knowledge foundations, and specifically for the concept of meaning, and its computation. This led to the creation of the Meaning Computation AI (MCAI). Knowledge structures are at the heart of MCAI. The semantic triple chain, one of such structures, is the subject of this paper. This structure is the core of introducing computational semantic reasoning models across a variety of domains. To efficiently apply knowledge structures to create semantic reasoning applications MCAI contains a development framework consisting of four components. The first component stems from our discovery that knowledge is relational in nature. It can be expressed through implicit relations (which reflect meaning) and semantic relations (which describe how implicit relations act between entities). An algebraical apparatus for relational knowledge, specifically for implicit and semantic relations, represents the first component. Axiomatic models of relational domain ontologies and domain-oriented semantic reasoning models comprise the second and third components. The fourth component contains domain-oriented semantic reasoning engines.