Mathematical Formalism Research Articles

Multibody Dynamics Modeling of a Continuous Rubber Track System: Part 1—Model Description

<div>Continuous rubber track systems for farming applications are typically designed using multiple iterations on full-scale physical prototypes which is costly and time consuming. The development of numerical design tools could speed up the design process and reduce development costs while improving product performance. In this article, a rigid multibody dynamics (MBD) model of a continuous rubber track system is presented. This article is the first part of a two-part study: Part 1 focuses on the model description and part 2 describes the experimental evaluation of the MBD model. The modeling methodology is based on a track discretization as a set of rigid body elements interconnected by 6 degrees-of-freedom bushing joints. The mathematical formalism and experimental characterization of all critical subsystems such as the roller wheels, tensioner, suspensions, and contact models are also presented. Several simulation results are presented to illustrate the capability of MBD models in design activities to rapidly evaluate new concepts as a result of the relative short-solving time of the model (approximately 3 h). The MBD model is shown to be a powerful design tool to provide input boundary conditions for structural analysis or as a design space explorer and optimization tool.</div>

Pair programming with ChatGPT for sampling and estimation of copulas

Without writing a single line of code by a human, an example Monte Carlo simulation-based application for stochastic dependence modeling with copulas is developed through pair programming involving a human partner and a large language model (LLM) fine-tuned for conversations. This process encompasses interacting with ChatGPT using both natural language and mathematical formalism. Under the careful supervision of a human expert, this interaction facilitated the creation of functioning code in MATLAB, Python, and R. The code performs a variety of tasks including sampling from a given copula model, evaluating the model’s density, conducting maximum likelihood estimation, optimizing for parallel computing on CPUs and GPUs, and visualizing the computed results. In contrast to other emerging studies that assess the accuracy of LLMs like ChatGPT on tasks from a selected area, this work rather investigates ways how to achieve a successful solution of a standard statistical task in a collaboration of a human expert and artificial intelligence (AI). Particularly, through careful prompt engineering, we separate successful solutions generated by ChatGPT from unsuccessful ones, resulting in a comprehensive list of related pros and cons. It is demonstrated that if the typical pitfalls are avoided, we can substantially benefit from collaborating with an AI partner. For example, we show that if ChatGPT is not able to provide a correct solution due to a lack of or incorrect knowledge, the human-expert can feed it with the correct knowledge, e.g., in the form of mathematical theorems and formulas, and make it to apply the gained knowledge in order to provide a correct solution. Such ability presents an attractive opportunity to achieve a programmed solution even for users with rather limited knowledge of programming techniques.

The connection of mathematics and logic in the structure of axiomatized and formalized theories

The aim of the research is to study the relationship between logic and mathematics in the structure of axiomatized and formalized scientific theories. The object of the study is the explication of this connection and its explanation. The subject of the study is syntactic and semantic views on the structure of scientific theories, the relationship between logic and mathematics has not been studied in detail in them. In the syntactic view, the structure of the theory is understood as a linguistic construct build from various logical propositions of the theoretical level, correspondence propositions and observation propositions. The structure of the theory does not take into account the variety of model representations of the theory that generate a variety of language constructs. The semantic view overcomes this disadvantage, and in it the structure of the theory is presented as a hierarchy of models: from axioms to theoretical-level models, experimental models and data models. The structure of the theory, the connection of logic and mathematics were studied using comparative analysis, methods of interpretive analysis and reconstruction of scientific theories. The methods made it possible to explicate mathematical concepts in the structure of the theory and correlate them with logic and natural language. Comparative analysis has shown that in the syntactic view, the connection between logic and mathematics lies in the fact that mathematical concepts of physics are interpreted in the language of logic of first-order predicates with equality. The connection between mathematical concepts is provided by the axiomatic method, which serves as a means of formalizing concepts. Mathematics comes down to logic. In the semantic approach, in order to identify the connection between mathematics and logic, it was necessary to reconstruct the structure of non-relativistic quantum mechanics. With the help of the set-theoretic predicate of Suppes, its axioms were determined, the connection between mathematical structures, postulates of the theory, axioms, and observable quantities was established. Logic and mathematics are related to each other in such a way that metamathematics or linguistics is a part of mathematics. Mathematics includes set theory and model theory, i.e. mathematical logic. The connection of mathematical formalisms with phenomena and with natural language remains problematic, and there is this drawback in the syntactic approach. The novelty lies in the fact that the research contributes to the methodology and logic of science, to the explanation of the connection between logic and mathematics in scientific theory, which was illustrated by various examples from various fields of physics.

A model of the strategic potential of an enterprise as an object of strategic marketing

Subject. The formation of enterprise’s strategic potential requires a scientific justification of its content due to the complexity of this category. At the same time, multiple approaches in determining the essence, structure of enterprise's potential, and ways of its mathematical formalization do not provide a sufficiently legitimate semantic and mathematical interpretation of the enterprise's potential model as an object of strategic marketing. Objectives. The study aims to develop a hierarchical model of enterprise’s potential, based on the definition of the object of strategic marketing in the implementation of the marketing concept of advanced development. This will form an effective strategy for enterprise development under conditions of permanent development of the market environment. Methods. The study rests on the theory of marketing and strategic management. Research methods include qualitative and functional analysis of economic categories. Results. The paper underpins the use of the category of enterprise’s strategic potential as an object of strategic marketing management. We applied a functional approach to build a hierarchical structure of strategic potential. The hierarchical model has three levels: main functional tasks (production and marketing), applied local potentials (technical, innovative, organizational, and investment), and resource potential. Conclusions. The model of enterprise’s strategic potential is essential for successful implementation of the management function of strategic marketing within the concept of marketing of advanced development, to build the strategic potential of the enterprise. Defining an interaction of local potentials of the enterprise will enable to accurately determine the object and type of managerial impact, as well as the targets of strategic development.

The Born Rule is a Feature of Superposition

The Born Rule plays a critical role in quantum mechanics (QM) since it supplies the link between the mathematical formalism and experimental results in terms of probabilities. The Born Rule does not occur in ordinary probability theory. Where then does it come from? This has been a topic of considerable controversy in the literature. We take the approach of asking what is the simplest extension of ordinary probability theory where the Born rule appears. This is answered by showing that the Born Rule appears by adding the notion of superposition events (in addition to the ordinary discrete events) to finite probability theory. Hence the rule does not need any physics-based derivation. It is simply a feature of the mathematics of superposition when only superposition events are added to ordinary probability theory.

Rates and reversibility of CO2 hydrogenation on Cu-based catalysts

Kinetics of reaction pathways involved in the conversion of CO2 to methanol and CO on Cu/ZnO/Al2O3 are resolved using in situ chemical titration, steady-state kinetic measurements, and mathematical formalisms for reversibility to probe salient species governing methanol selectivity and yield during CO2 hydrogenation. Across a range of H2:CO2 = 1:1 to H2:CO2 = 80.5:1, active site density determined from in situ chlorine uptake remained invariant; hence, observed trends in rates can be interpreted as only arising from reaction kinetics and not from changing active site density. Kinetic and thermodynamic contributions to rates are decoupled to evaluate forward and reverse rates of methanol synthesis and reverse water-gas shift (RWGS) reactions. These kinetic analyses show that the forward rates of methanol synthesis exhibit persistent first order dependence on hydrogen pressure and are inhibited by water more significantly than the forward rates of RWGS. In contrast, the reverse rates of methanol synthesis and RWGS are both inhibited by H2. Consequently, without any modifications to the Cu/ZnO/Al2O3 catalyst formulation, methanol selectivity can be increased to > 80 % by increasing inlet H2 partial pressure and methanol yield can be enhanced by ∼ 20 % by adding water adsorbents even under conditions far from equilibrium. The kinetic treatments presented herein demonstrate a dearth of H* species during catalysis, provide thermodynamic constraints precluding sequential RWGS and CO hydrogenation as the pathway for methanol synthesis, reveal PH2 and PH2O as salient in determining methanol selectivity and yield by impacting both the forward and reverse rates of CO2 hydrogenation on Cu/ZnO/Al2O3, and explicate the fundamentals of novel sorption-enhanced methanol synthesis, which not only alleviates equilibrium constraints but also alters the intrinsic rate at which the system approaches equilibrium.

Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning

Relational learning has attracted much attention from the machine learning community in recent years, and many real-world applications have been successfully formulated as relational learning problems. In recent years, several relational learning algorithms have been introduced that follow a pattern-based approach. However, this type of learning model suffers from two fundamental problems: the computational complexity arising from relational queries and the lack of a robust and general framework to serve as the basis for relational learning methods. In this paper, we propose an efficient graph query framework that allows for cyclic queries in polynomial time and is ready to be used in pattern-based learning methods. This solution uses logical predicates instead of graph isomorphisms for query evaluation, reducing complexity and allowing for query refinement through atomic operations. The main differences between our method and other previous pattern-based graph query approaches are the ability to evaluate arbitrary subgraphs instead of nodes or complete graphs, the fact that it is based on mathematical formalization that allows the study of refinements and their complementarity, and the ability to detect cyclic patterns in polynomial time. Application examples show that the proposed framework allows learning relational classifiers to be efficient in generating data with high expressiveness capacities. Specifically, relational decision trees are learned from sets of tagged subnetworks that provide both classifiers and characteristic patterns for the identified classes.

Probabilistic Procedures for SIR and SIS Epidemic Dynamics on Erdös-Rényi Contact Networks

This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius of the associated adjacency matrices, which expresses the network topology. The epidemic threshold dynamics are analyzed, depending on the global dynamics of the network structure. The main contribution of this work is the relationship established between the epidemic threshold and the topological entropy of the Erdös-Rényi contact networks. In addition, a relationship between the basic reproduction number and the topological entropy is also stated. The trigger of the infectious state is studied, where the probability value of the stability of the infected state after the first instant, depending on the degree of the node in the seed set, is proven. Some numerical studies are included and illustrate the implementation of the probabilistic procedures introduced, complementing the discussion on the choice of the seed set.

Quantum Gates for Electronics Engineers

The design of a solid-state quantum processor is nowadays a hot research topic in microelectronics. Like the logic gates in a classical processor, quantum gates serve as the fundamental building blocks for quantum processors. The main goal of the present paper is to deduce the matrix of the main one- and two-qubit quantum gates from the Schrödinger equation. The mathematical formalism is kept as comfortable as possible for electronics engineers. This paper does not cover topics such as dissipations, state density, coherence, and state purity. In a similar manner, this paper also deals with the quantum nature of a quantum processor by leveraging the concept of a finite-state machine, which is a background notion for any electronics engineer.

PHYSICAL THEORY: KEY TERMS AND ITS MATHEMATICAL COMPONENT

Actual scientific researches and issues analysis. The movement from classical physics to modern physics occurred as a result of the emergence of a number of paradoxes in classical theory. This feature is characteristic to some extent of the electromagnetic theory of Maxwell, Einstein's theory of relativity, etc. The research objective . We will try to consider the key terms of physical theory and show the role of the mathematical component in its creation. The statement of basic materials. The article deals with some modern problems of the methodology and foundations of physics, in particular: whether mathematical formalisms contain interpretations of themselves or whether they need to be supplemented by interpretive predictions, and if so, how these predictions are formulated; what physical theories describe: physical systems or laboratory operations, either one or another, or neither of them; how the basic concepts of theory should be introduced, by reference to measuring operations, or by means of clear definitions, or axiomatically; how physical theories relate to each other; how theory relates to experiment: directly or with the help of additional theories.

A formal framework to design and prove trustworthy memory controllers

In order to prove conformance to memory standards and bound memory access latency, recently proposed real-time DRAM controllers rely on paper and pencil proofs, which can be troubling: they are difficult to read and review, they are often shown only partially and/or rely on abstractions for the sake of conciseness, and they can easily diverge from the controller implementation, as no formal link is established between both. We propose a new framework written in Coq, in which we model a DRAM controller and its expected behaviour as a formal specification. The trustworthiness in our solution is two-fold: (1) proofs that are typically done on paper and pencil are now done in Coq and thus certified by its kernel, and (2) the reviewer’s job develops into making sure that the formal specification matches the standards—instead of performing a thorough check of the mathematical formalism. Our framework provides a generic DRAM model capturing a set of controller properties as proof obligations, which all implementations must comply with. We focus on properties related to the assertiveness that timing constraints are respected, every incoming request is handled in bounded time, and the DRAM command protocol is respected. We refine our specification with two implementations based on widely-known arbitration policies—First-in First-Out (FIFO) and Time-Division Multiplexing (TDM). We extract proved code from our model and use it as a “trusted core” on a cycle-accurate DRAM simulator.

APPLICATION OF THE STATISTICAL TESTING METHOD FOR SIMULATION MODELLING OF COMPLEX SYSTEMS

Problem statement. The study of the multicomponent systems’ operation is possible using statistical simulation models. In contrast to traditional mathematical modeling, in order to build the specified type of models, it is not necessary to obtain a mathematical formalization of the relationship between parameters in one form or another. The main condition is the ability of the model to reproduce the phenomena being modeled, while preserving their logical and temporal sequence and physical meaning. One of the most common methods of statistical simulation modeling is the method of statistical tests - the Monte Carlo method. The basis of this method is the repeated use of a random number generator to simulate the dynamics of processes occurring in the system. Based on the obtained iterations, statistical criteria for evaluating the obtained results are calculated, which makes it possible to conduct a preliminary analysis of the physical processes under investigation and draw conclusions about the relationship between the parameters included in the model. Thus, taking into account the general principles of these building simulation models, it is advisable to use them to study a number of applied materials science problems, for example, to describe the influence of external factors on the structural state of the material. The purpose of the article. Application of statistical simulation modeling by the method of statistical tests (Monte Carlo method) for the study of physical processes that occur in complex systems. Conclusion. Using the Monte Carlo method, a simulation model of the relationship between the technological modes of welding and the parameters of the structural state of low-carbon low-alloy steel 09Г2С was obtained. The obtained simulation model is presented in the form of a matrix containing the results of statistical tests. The analysis of the obtained statistical indicators made it possible to carry out a preliminary analysis of the relationship between technological modes of welding and the corresponding structural state of low-carbon low-alloy steel 09Г2С. Preliminary evaluation of the obtained simulation model was carried out by analyzing active constraints of the type of inequalities. The conducted set of studies proved the obtained data adequacy.

USING THE METHOD OF CONFIRMATORY FACTOR ANALYSIS FOR THE SIMULATION OF TECHNICAL SYSTEMS

Problem statement. Analyzing the dynamics of the functioning of complex technical systems, which are characterized by a certain architecture and the interaction of system components, in most cases, there are certain difficulties associated with the description of system-wide issues. Among such issues, first of all, it is possible to include problems related to the direct mathematical formalization of the overall system structure, the organization of relationships between its elements, the interaction of system elements with its external environment, control of the activities of its elements, etc. In such cases, most often researchers use the procedure of mathematical modeling. At the same time, there is a fairly large range of phenomena, in the analysis of which it is possible to use the traditional apparatus of mathematical modeling. In such situations, it is possible to apply modeling techniques that represent the model in the form of an algorithmic program for an electronic computer, the so-called simulation modeling. The purpose of the article. Application of simulated physical and mathematical modeling by the method of confirmatory factor analysis for the study of complex technical processes. Conclusion . The relationship between the parameters of the laser welding mode and the morphological criteria of the structural state of low-carbon low-alloy steel 09Г2С was investigated using the mathematical apparatus of simulation modeling, namely confirmatory factor analysis. The confirmatory factor method was implemented in the form of a path diagram, namely a graphical interpretation of the relationship between the structural condition criteria and laser welding parameters. The percentage content and geometric dimensions of the main structural components of ferrite and pearlite were used as criteria for the morphology of the structural state, and the geometric dimensions of the welded joint sections were used as welding parameters. The adequacy of the obtained model was confirmed by constructing a probability plot of normalized residuals according to the quasi-Newtonian method of residuals.

Evolution of the Concept of Geospatial Positionality: Conceptual and Term Dimension

The content of the general geographical concept of geospatial positionality (geopositionality) is revealed in the context of the relation of a single object / place with geospace, as well as the function of geoplace. Conceptual and terminological analysis of the concept is carried out in formal and semantic aspects. Using an explanatory dictionary of the modern Ukrainian language highlighted the meaning of placementtokens (position, location, positioning, neighborhood, proximity / remoteness) and the possibility of their use in geographical texts. Based on the study of scientific sources, object-centric, functional and integrated logical models of the concept of geopositionalityareelaborated. The directions of evolution of object-centric interpretation of the concept are revealed: clarification of the token of relation (sum of relation, set of relation, combination of relation, integral of relation; spatial, geographical, territorial, geospatial relations; spatial aspects of relation); determining the geopositionality of objects of ever smaller spatial coverage; the dominance of the ‘external’ approach to content disclosure; multiscale; potentiality. The functional model of the concept of geopositionality is created through the sequential deployment of the features, relations, potential, functions of geoplace. The semantic triangle of the concept of geopositionality of the object in the unity of its aspectsmeaning and conceptual content is developed. The main attributes of geopositionality (uniqueness / specificity, distance, potentiality / probability, hierarchy / multiscale, dynamism / historicity / variability) are highlighted, and their effect on different logical models are shown. The main directions of development of the concept of geopositionality are defined: nomination of features of geospatial relations of linear, areal, network geoobjects; nomination of dynamism of geopositionality of objects; positioning in relation to geospatial processes; mutual enrichment of semantically closest geopositional concepts; correct application of content and quantitative research methods for mathematical formalization and interpretation of geospatial relations of position and functioning. The classical, non-classical and modern stages in the development of the concepts of geospatial positionality are distinguished.

Application of Lorentz geometry to refractive optical systems

Lorentz geometry represents the mathematical foundation of Einstein’s general relativity. From the concept of the cone of edge rays, it was introduced successfully in the study of homogeneous media nonimaging optics, Gutiérrez et al. (1996), Winston et al. (2005). In this paper we present a new Lorentzian geometry formalism to analyze optical systems with refractive elements. We have obtained the general system of partial differential equations and the integrability conditions to be fulfilled. This new mathematical formalism is the basis for a new optical technique to analyze the irradiance pattern produced by refractive optical systems. We apply this Lorentzian formalism to an optical media with exponential refraction index. Comparisons between irradiance patterns obtained by this formalism and by raytrace simulations are also presented.

Correspondence between Electrostatics and Contact Mechanics with Further Results in Equilibrium Charge Distributions

AbstractIt is common in mesoscopic systems to find instances where several charges interact among themselves. These particles are usually confined by an external potential that shapes the symmetry of the equilibrium charge configuration. In the case of classical charges moving on a plane and repelling each other via the Coulomb potential, they possess a ground state à la Thomson or Wigner crystal. As the number of particles N increases, the number of local minima grows exponentially and direct or heuristic optimization methods become prohibitively costly. Therefore the only feasible approximation to the problem is to treat the system in the continuum limit. Since the underlying framework is provided by potential theory, we shall by‐pass the corresponding mathematical formalism and list the most common cases found in the literature. Then we prove a (albeit known) mathematical correspondence that will enable us to re‐discover analytical results in electrostatics. In doing so, we shall provide different methods for finding the equilibrium surface density of charges, analytical and numerical. Additionally, new systems of confined charges in three‐dimensional surfaces will be under scrutiny. Finally, we shall highlight exact results regarding a modified power‐law Coulomb potential in the d‐dimensional ball, thus generalizing the existing literature.

Modeling and analysis for reaction control thruster used for station-keeping maneuvers of a GEO satellite

This paper introduces a theoretical description, force, and torque analysis of a Reaction Control Thruster (RCT) used as an actuator in Attitude Determination and Control Subsystem (ADCS). Mathematical formalization, realization, and implementation are offered to implement the RCT model using Matlab-Simulink. Furthermore, this paper proposes an accurate RCT Simulink model to control the orientation and stabilizes the Geostationary Earth Orbit (GEO) satellite within its orbit. Finally, a comparative analysis is carried out with an existing satellite platform (Eurostar-3000) based on the Root Mean Square Error (RMSE) for the delta velocity (delta-𝒱) parameter. The outcomes of the model simulation validated the effectiveness of the proposed approach.

IMPROVING THE INNOVATIVE STRATEGY OF MANAGEMENT OF AGRICULTURAL ENTERPRISES IN THE CONDITIONS OF GLOBALIZATION

The purpose of the article is to study the peculiarities of the implementation of innovative strategies by Ukrainian farmers, to identify influencing factors, including the influence of global market factors on the effectiveness of the implementation of strategies, and to develop a mathematical approach to assessing the influence of uncertain factors. The methodological basis of the article is the methods and forms of scientific knowledge accepted in science, such as methods: of comparative and abstract-logical analysis, induction and deduction, abstraction, and mathematical formalization. A stratification of the factors of the negative impact of globalization, which slow down the implementation of the innovative strategy and create threats to the activity of agricultural enterprises, is carried out. It is indicated that the main directions of improvement of the innovation strategy should be assessment, consideration, and neutralization of the influence of globalization factors. It is also indicated that this leads to the heterogeneity of the implementation of the innovative management strategy of agricultural enterprises. Prospective directions for the implementation of innovative strategies in agricultural production are stratified. The stages of the implementation of the innovation strategy in the conditions of globalization and the tasks of management for the implementation of the innovation strategy in the global conditions are determined. Peculiarities of the implementation of innovative strategies in Ukraine are revealed. A mathematical model was developed to assess the influence of uncertain global factors, which will allow for reducing risks for the implementation of significant innovative strategies. It is indicated that significant innovative projects could lead to the transformation of agribusiness activities and the need for permanent implementation of innovative processes. It is indicated that thanks to the systemic innovation process, a synthesis of the resource and innovation model of enterprise development could be formed. Taking into account the specifics of innovative strategies and using the developed mathematical approach to assess the influence of uncertain global factors will allow reducing risks in the implementation of innovative strategies and improving their management.

Construction of a model of steganographic embedding of the UAV identifier into ADS-B data

Secure data exchange in the control system of unmanned aerial vehicles (UAVs) is an important aspect for preventing unauthorized access and safety of aerial vehicles. Given the problems of automatic dependent surveillance-broadcast (ADS-B) data protection, the safety level of UAV flight tasks and air traffic in general is significantly reduced. Therefore, the protection of ADS-B data is an urgent task. The object of the study is the process of steganographic protection of ADS-B format data. A relevant problem of estimating the probabilistic time characteristics of the steganographic protection process is solved, taking into account the features of data embedding in the ADS-B format container. To solve it, a mathematical formalization of the methods of finding probabilistic-temporal characteristics of steganographic systems was carried out. A model of steganographic data transformation operations based on the Chinese remainder theorem has been built. The main difference of the model is taking into account the features of the ADS-B format data. This made it possible to formalize and evaluate the time functions of steganographic encoding and decoding of UAV identifiers with an integrated ADS-B system. A model of steganographic data transformation operations based on the finite integral ring theorem has been constructed. A list of operations performed in the developed algorithm has been compiled. This made it possible to carry out mathematical formalization of operations for complex use in the model of steganographic protection of UAV identifiers with a built-in ADS-B system. The mathematical model was studied and the estimation of the random value of the time of steganographic transformation of data, as well as the confidence interval, was performed. With the help of the reported set of models, it is possible to estimate the probability of the algorithm’s execution time falling within the given interval. The results of the calculation of probabilistic-time characteristics could be used in models of a higher level of the hierarchy

Feasibility analysis of 2-dimensional permutation vector optical code division multiple access passive optical network

This paper analyzes the feasibility of a 2-dimensional (2D) permutation vector (PV) optical code division multiple access (OCDMA) passive optical network (PON) in terms of transmission capacity, network reliability, power budget, and the number of subscribers. The proposed PON is built with a novel 2D PV-OCDMA code having ideal auto- and cross-correlation properties. Furthermore, a ring-based topology at the feeder level and a tree-based topology at the distribution level are utilized to support ubiquitous computing. The proposed architecture also employs multiple remote nodes at the feeder level to facilitate adequate power budget and cardinality for high-capacity applications. Simulation results indicate that the proposed PON elevates system transmission capacity by eliminating phase-induced intensity noise and multiple access interference. Furthermore, reliability analysis using mathematical formalism demonstrates that the proposed architecture can provide a connection availability of ≈0.099993 (5 nines), which is significantly higher than the conventional tree-based topology. Power budget analysis also determines that the proposed PON can support well above 320 subscribers at adequate receiver sensitivity and optical network terminals (ONTs) per remote node.