Mathematical System Research Articles

Multiobjective optimal control problems. Stationary Navier–Stokes equations

This paper deals with the solution of some multi-objective optimal control problems for stationary Navier–Stokes equations. More precisely, we look for Pareto and Nash equilibria associated to standard cost functionals. First, we prove the existence of equilibria and we deduce appropriate optimality systems. Then, we analyse the existence and characterization of Pareto and Nash equilibria for the Navier–Stokes equations. Here, we use the formalism of Dubovitskii and Milyoutin., see [Girsanov FV. Lectures on mathematical theory of extremum problems. Berlin: Springer-Verlag; 1972. (Notes in economics and mathematical systems; vol. 67)]. Finally, we also present a finite element approximation of the bi-objective problem, we illustrate the techniques with several numerical experiments and we compare the Pareto and Nash equilibria.

3D Chaotic Nonlinear Dynamic Population-Growing Mathematical System Modeling with Multiple Controllers

Modeling, stabilization, and identification processes are significant stages in the process of developing knowledge about chaotic dynamical systems which entail the effective prediction depending on the degree of uncertainty toleration in the forecast, accuracy of the current state to be measured as well as a time scale resting on the dynamics of the system. Control of under-activated dynamical systems has been considered substantially, and it is for periods and is currently developing in various domains such as biology, data analysis, computing systems, and so forth. Dynamic systems of growing population signifies a model describing the way a population evolves over time during which population goes through major life events, split into discrete time periods. The size of the population at a given time period is determined by the rate of growth as well as other related factors. Most progress has been made in model-based control theory, which has drawbacks when the system under consideration is exceedingly complicated, and no model can be constructed. Accordingly, a 3D-discrete and dynamic human population growth system with many controllers is proposed by examining the stability and symmetry of controller system clarifications. The symmetric stability control results are presented by considering a special parametric dynamic system in its coefficients besides suggesting periodic functional coefficients in terms of sin and cos functions. The controllers have the ability to reduce population growth rate unpredictability or enhance system stability under various external conditions. The unique and very effective strategies in relevant domains could provide a deeper understanding of their impact as well as the theoretical or technological innovations thereof. These controllers are capable of reducing population growth rate unpredictability or improving system stability under various external conditions, and applicable strategies in the relevant domains can provide profound comprehension over the impact along with the theoretical as well as technological advancements.

A capturing math language use during block play: Creation of the spatial and quantitative mathematical language coding system

The goals of the current study were: 1) to modify and expand an existing spatial mathematical language coding system to include quantitative mathematical language terms and 2) to examine the extent to which preschool-aged children used spatial and quantitative mathematical language during a block play intervention. Participants included 24 preschool-aged children (Age M = 57.35 months) who were assigned to a block play intervention. Children participated in up to 14 sessions of 15-to-20-minute block play across seven weeks. Results demonstrated that spatial mathematical language terms were used with a higher raw frequency than quantitative mathematical language terms during the intervention sessions. However, once weighted frequencies were calculated to account for the number of codes in each category, spatial language was only used slightly more than quantitative language during block play. Similar patterns emerged between domains within the spatial and quantitative language categories. These findings suggest that both quantitative and spatial mathematical language usage should be evaluated when considering whether child activities can improve mathematical learning and spatial performance. Further, accounting for the number of codes within categories provided a more representative presentation of how mathematical language was used versus solely utilizing raw word counts. Implications for future research are discussed.

ОСОБЛИВОСТІ ВИКОРИСТАННЯ СУЧАСНИХ ІНФОРМАЦІЙНОКОМУНІКАЦІЙНИХ ТЕХНОЛОГІЙ У ВИЩІЙ МАТЕМАТИЧНІЙ ОСВІТІ

The article discusses the relevance of using information and communication technologies (ICT) in higher mathematical education within the context of the contemporary educational environment characterized by rapid technological changes. It is noted that effective utilization of ICT, such as electronic journals, multimedia technologies, testing systems, and computer mathematics systems, contributes to the improvement of the teaching and learning process. Emphasis is placed on the necessity of reviewing and modifying teaching methodologies for mathematical disciplines to integrate innovative solutions. Furthermore, the use of information and communication technologies in higher mathematical education is deemed highly relevant due to the rapid development of the modern technological environment and the needs of the contemporary job market. The implementation of electronic journals, multimedia technologies, testing systems, and computer mathematics not only enhances the efficiency of teaching mathematical disciplines but also helps create a solid foundation for the future professional success of students. Electronic journals serve not only as convenient tools for administrative processes but also as platforms for active interaction among students, teachers, and administration, fostering the development of academic community and stimulating discussions and collaboration. Multimedia technologies, such as video lessons, interactive exercises, and simulations, provide students with opportunities to better grasp material and understand complex concepts, opening up new possibilities for personalized learning and adaptation to various learning styles. Testing systems, particularly those based on interactive platforms, enable instructors to effectively monitor students' knowledge and track their progress, as well as provide opportunities for personalized learning through automatic adaptation of task complexity Computer mathematics systems, including software packages for numerical and symbolic computations, become an integral part of modern mathematical education, empowering students to efficiently solve complex mathematical problems and explore new concepts, thus stimulating their creative potential and innovative approach to problem-solving. Together with educational platforms and distance learning systems, these tools contribute to the modernization and optimization of the educational process, leading to the improvement of the quality of higher mathematical education and the preparation of qualified professionals for the contemporary world.

Assessment of the quality of the labour safety management system in accordance with the requirements of the international standard ISO 45001:2018 at the stage of operation

The subject matter of the article is the assessment of the quality of the occupational safety management system. The goal of the article is to develop a methodology for assessment of the occupational safety management system at the stage of operation. The article solves the following task: to investigate the possibility of applying methods for analysing the dynamic characteristics of the occupational safety management system and to develop algorithms and methods for using the criteria of non-parametric statistics. Methods are used: qualimetry and mathematical statistics. The following results were obtained: an analysis of the requirements of the international standard ISO 45001:2018 was carried out to determine the need to monitor, measure and analyse the functioning of the occupational safety management system, which revealed that the effectiveness of the development and implementation of the occupational safety management system depends on the assessment methodology. An analysis of existing research and publications on the evaluation of processes, including those related to occupational safety, is carried out. The methods for evaluating mathematical systems for processing and controlling gross errors and stationarity, which are key to the analysis of occupational safety management processes, are considered. Non-parametric criteria for checking the stationarity of random processes are considered in detail. Sequences of means and means squares are compared with expected values to determine whether rejections of the stationary hypothesis are justified. The expediency of using these methods for analysing occupational safety management systems is determined. The dynamic nature of the processes and the importance of monitoring and data analysis to ensure effective safety management practices are analysed. Conclusions: The application of methods for analysing the dynamic characteristics of the occupational safety management system is substantiated and algorithms and methods for using the criteria of non-parametric statistics are developed, which allow to obtain an assessment of the system, taking into account the limited information about its indicators and the lack of knowledge of the law of their distribution as a random variable.

Insights on usability testing: The effectiveness of an adaptive e-learning system for secondary school mathematics

The study evaluates the Mythematix prototype, an innovative adaptive e-learning system designed specifically to enhance secondary school students’ understanding and proficiency in mathematics. Utilizing a mixed-methods approach, it combines qualitative and quantitative research methodologies to thoroughly assess students’ perceptions and experiences with the Mythematix prototype. The study involved a group of 30 students participating in a comprehensive usability survey, which included a range of Likert scale questions alongside an open-ended response. This survey was designed to evaluate the usability of the prototype, pinpointing its strengths and weaknesses from the students’ perspectives. The analysis of the survey results provided detailed insights into the most and least effective aspects of the prototype. Overall, the study contributes valuable insights into the effectiveness of adaptive e-learning platforms in enhancing and enriching students’ learning experiences in mathematics, and it lays the groundwork for future research in this evolving educational field.

A Framework to Use Bifurcation Analysis for Insight into Complex Systems Resilience

AbstractThe increasing complexity and integration of systems present challenges in understanding and managing these systems, as highlighted by the INCOSE Systems Engineering Vision for 2035. This complexity risks system failures due to insufficient understanding. Resilience Engineering is a field that focuses on system behavior in the face of disruptions. However, existing methods to measure resilience, such as probability‐based measures and linear recovery models, are limited by the non‐linear and dynamic nature of modern systems. Bifurcation Analysis, a mathematical system dynamics technique, offers a different perspective by examining how systems behave under changing conditions. This approach, developed in ecology and traditionally applied in fields including power systems and neural networks, can provide insights into the resilient characteristics of non‐linear systems. Despite its promise, Bifurcation Analysis is not usually associated with Resilience Engineering. This paper proposes using Bifurcation Analysis to understand system resilience better and support Resilience Engineering practitioners. It aims to bridge the gap between the two fields, offering a framework for integrating both approaches. This framework is then applied to a model system to demonstrate its potential to enhance the understanding of system resilience. Contributions are an overview of current Resilience Engineering and Bifurcation Analysis, a general‐use framework for practitioners, and an application of this framework. Future work will focus on applying the new framework to more cases, allowing for improved Resilience Engineering, and improving the framework to make it more accessible, versatile, and reliable.

Bibliometric Exploration of Ethnomathematics: Trends, Characteristics, and Future Directions

Ethnomathematics plays a very important role in life, especially in mathematics learning. Even so, ethnomathematics is still rarely known to the public. As a result, ethnomathematics research is still rarely carried out. So, ethnomathematics is a good topic to discuss in research. But before that, do an analysis first. So, researchers conducted analytical ethnomathematics research using bibliometrics to explore publication characteristics, find trends, and find future research directions in the field of ethnomathematics. The data used is sourced from the Scopus database. The science mapping steps taken are study design, data collection, data analysis, data visualization, and interpretation. From the analysis, 580 documents were obtained that had been published from 1984 to 2023 with the core of the study being learners, mathematical concepts, learning mathematics and physics. Based on this information, it is clear that ethnomathematics still needs to be explored more deeply, considering that most documents are not articles. As for the characteristics of ethnomathematics research, apart from the lack of material in the form of articles, it turns out that ethnomathematics, which is currently developing, is a learning system in mathematics. Even ethnomathematics can improve high-order thinking skills (HOTS). Apart from that, research on teaching, design, and thinking processes in ethnomathematics is still small but very important. Based on this information, the most suitable novelty for future ethnomathematics research is research on the design of ethnomathematics-based mathematics learning models.

SINGLE-DEGREE-OF-FREEDOM VIBRATION ISOLATION SYSTEM WITH ONE ADDITIONAL SUPPORT

Vibration isolation is one of the most effective methods for reducing vibration levels of supporting structures when installing vibroactive equipment (active vibration isolation) or vibration levels of vibro-sensitive objects relative to foundation vibration levels (passive vibration isolation). Damping devices utilizing high-speed fluid flow through apertures have found wide applications in shock vibration isolation and vibration isolation systems in aerospace and defense sectors. Recent research has led to the development of viscous fluid dampers (VFDs) for use in civil engineering, particularly in earthquake-prone areas. Scientists conducted experiments aimed at determining the ability of viscous fluid dampers to reduce damages and displacements of structures without increasing stresses. Mathematical models have been developed and are applied in vibration isolation systems. When vibroactive equipment is installed on building and structure support systems, vibrations with sufficiently high vibration parameters may occur, potentially leading to loss of load-bearing capacity. In such cases, vibration isolation is considered one of the most effective methods for reducing these vibration levels. In this study, a calculation method has been developed, and calculation dependencies and algorithms for calculating vibration protection systems with nonlinear characteristics (additional support connections, viscous fluid damper) have been derived for both single-degree-of-freedom and two-degree-of-freedom systems. The research involved an analysis of normative and scientific-technical literature on the subject: types, structural solutions, calculation, and analysis of vibration protection systems. The main method selected was based on the use of transfer functions of linear systems (the non-traditional "normal form" method). Calculations were performed using computer mathematics systems- Matlab.

KINETIC MODELLING OF BIOCHEMICAL REACTIONS USING MATHСAD ANALYTICAL TOOLKIT

Background. The study of the kinetics of biochemical reactions provides a better understanding of how biological processes occur in living organisms. Understanding the peculiarities of such reactions is important for the development of new technologies, in particular for the production of biologically active substances and for the synthesis of drugs. A powerful tool for solving problems in biochemical reaction kinetics is mathematical modelling, which can be carried out using computer mathematical systems, in particular the MATHCAD analytical toolkit. Aim: to substantiate the feasibility and effectiveness of using the MATHCAD analytical toolkit to solve problems of kinetic modelling of biochemical reactions in pharmaceutical research, and to review the capabilities of MATHCAD for computer modelling in pharmacy. Materials and methods. In the context of studying the rate of enzymatic reactions and developing models, such as the Michaelis-Menten model, to describe reactions in which enzymes catalyze the transformation of substrates, the use of a computer mathematical system (CMS) is considered. CMS is a software package and environment for performing mathematical computations, modelling and visualization. The possibilities of using the MATHCAD system to create mathematical models of biochemical reactions based on kinetic equations are demonstrated. This involves the creation of differential equations describing changes in reagent concentrations over time. These equations were solved using numerical methods in MATHCAD. In addition, the results obtained are visualized using 3D graphics in MATHCAD. The stages of using the MATHCAD analytical toolkit in the kinetic modelling of biochemical reactions have been determined. Results. The use of MATHCAD in the kinetic modelling of biochemical reactions is effective for the study of: (1) the kinetics of enzymatic reactions, e.g. reactions in which an enzyme catalyzes the conversion of a substrate into a product; (2) biochemical reactions that take place in reaction vessels in which reagents mix and interact; (3) modelling of reactions in reaction vessels based on the solution of differential equations of reaction kinetics; (4) the effect of inhibitors or activators on enzymatic reactions; (5) scenarios of interaction of reagents to determine changes in the kinetics of reactions that occur when different active substances are introduced; (6) kinetics of biochemical reactions in cases where reactions are accompanied by diffusion of reagents through membranes or other semi-permeable barriers; (7) modelling the effect of diffusion processes on the kinetics of biochemical reactions; (8) models describing the kinetics of decomposition of substances, for example the decomposition of biologically active compounds in the body or in the environment; (9) predicting the effect of changes in the conditions of the reaction medium (temperature, pH, concentration of reagents) on the kinetics of biochemical reactions. It is substantiated that model descriptions of the kinetics of biochemical reactions are important for forming an understanding of the functions of biological systems, including metabolism, enzymatic reactions, and other physiological phenomena. Tools have been used to visualize the modelling results in the form of three-dimensional MATHCAD graphics, which improves the understanding of the reaction mechanism and allows a more thorough analysis of its kinetics. Conclusion. MATHCAD provides an optimized environment for kinetic modelling of biochemical reactions through its ergonomic interface. Particular advantages are the ability to work with symbolic expressions and to use a wide range of built-in functions and tools for exploring mathematical models and visualizing results. The obtained results may be important both for further scientific pharmaceutical research and for implementation in the training of future Masters of Pharmacy in the discipline of "Computer Modelling in Pharmacy" in higher medical education institutions.

Complete Synchronization, Anti-Synchronization and Hybrid Synchronization of Non-Identical Chaotic Financial Systems

Chaos theory, being a branch of mathematics that deals with disordered or random-seeming mathematical systems, is receiving more attention in market-related circumstances as financial markets become more unstable and the level of unpredictability becomes more prevalent. Understanding the potential of chaos theory, its limitations, and its relationship to traditional economic theories is essential for anyone working in the finance sector. Chaos theory is ideally suited for comprehending the financial market, which is subject to both internal and external influences due to its high degree of instability and growing randomness. This work examines the complete synchronization, anti-synchronization and hybrid synchronization of two non-identical financial systems. The nonlinear active controllers are designed, and the error dynamics stability for each phenomenon is accomplished by two theoretical approaches - linear system theory and Lyapunov second method. Controllers are designed by using the relevant variables of drive and response systems in such a way that the error variables are stable. The controllers, when activated, enable the drive and response state variables to achieve identical dynamics despite starting from different initial conditions. Numerical simulations are performed using the ODE45 algorithm embedded in MATLAB software package to show the feasibility and effectiveness of the designed controllers.

Bessel equation and functions in computer mathematics system

The relevance of technical issues of solving problems of mathematical physics is growing every year. The development of computer technologies leads to the use of modern methods for their solution. The application of analytical computing systems is considered as an effective method, which contributes to their productive implementation. The article deals with finding a general solution of the Bessel equation and equations leading to it in the Maple program, graphs of functions are plotted

Performance evaluation of standalone new solar energy system of hybrid PV/electrolyzer/fuel cell/MED-MVC with hydrogen production and storage for power and freshwater building demand

To achieve zero carbon emissions, increasing the scale of renewable energy implementation is imperative. Hydrogen is considered the prospective solution for sustainable energy due to its dual nature as a carbon-neutral fuel and an efficient storage medium for renewable energy sources. So, in this work, a standalone new hybrid energy system powered by solar energy for yearly power and freshwater production for building demand in New Borg El-Arab, Egypt. The system comprises photovoltaic (PV) panels, a water electrolyzer, multi-effect mechanical vapor compression desalination (MED-MVC), and fuel cells with hydrogen storage. A complete mathematical system model is built and solved by MATLAB/Simulink to study the daily, monthly, and yearly performance and sizing of the different system components based on the location climate conditions. The results indicate that PV panels of an area of 1824 m2, in conjunction with an electrolyzer, 863 fuel cells, and a storage tank capacity of 3013.4 m3can satisfy the annual building requirements of 255.17 MWh electricity, 766.5 m3 hot water and 876 m3 cold water. The maximum hydrogen storage lies between September and October, with maximum consumption between February and March. The average annual efficiency of PV, electrolyzer, fuel cell, electrolyzer output, fuel cell output, and overall hybrid system is 20.7%, 68.2%, 34.6%, 13.4%, 4.4%, and 16.5%, respectively, and the performance ratio (PR) of MED-MVC is 2.61. The system proves its capability to supply the building with electrical and water requirements with a levelized cost of energy (LCOE) equal to 0.712 $/kWh. Additionally, This system shows the potential to include other activities in remote areas or islands besides its contribution to achieving SDGs 6, 7, and 13.

Advancing cancer drug development with mechanistic mathematical modeling: bridging the gap between theory and practice.

Quantitative predictive modeling of cancer growth, progression, and individual response to therapy is a rapidly growing field. Researchers from mathematical modeling, systems biology, pharmaceutical industry, and regulatory bodies, are collaboratively working on predictive models that could be applied for drug development and, ultimately, the clinical management of cancer patients. A plethora of modeling paradigms and approaches have emerged, making it challenging to compile a comprehensive review across all subdisciplines. It is therefore critical to gauge fundamental design aspects against requirements, and weigh opportunities and limitations of the different model types. In this review,we discuss three fundamental types of cancer models: space-structured models, ecological models, and immune system focused models. For each type, it is our goal to illustrate which mechanisms contribute to variability and heterogeneity in cancer growth and response, so that the appropriate architecture and complexity of a new model becomes clearer. We present the main features addressed by each of the three exemplary modeling types through a subjective collection of literature and illustrative exercises to facilitate inspiration and exchange, with a focus on providing a didactic rather than exhaustive overview. We close by imagining a future multi-scale model design to impact critical decisions in oncology drug development.

Навчальний посібник як елемент освітнього простору бакалавра математики в умовах змішаного навчання в Україні

The article considers the peculiarities of creating and functioning of a textbook (digital textbook) in higher mathematics for bachelors of mathematics in a blended learning environment in Ukraine, and reviews the literature on the topic. The purpose of the article is to analyze the problems and prospects of creating new type of textbooks and digital manuals in mathematical disciplines for higher education institutions in a blended learning environment in Ukraine. The features of a modern digital textbook in mathematical disciplines in the preparation of a bachelor of mathematics are discussed: 1) theoretical aspects of creating textbooks in the educational space of the bachelor of mathematics in the conditions of mixed learning in Ukraine; 2) structure and content of the digital textbook in accordance with the current regulatory framework and practical experience of teachers; 3) techniques of visualization of educational material in mathematical disciplines and demonstration of some of them in the author’s textbooks; 4) the use of CMC and digital technologies in textbooks based on a sound mathematical idea; the main criterion for the effectiveness of the use of mathematical programs in teaching higher mathematics is the availability of a methodological system for their use; 5) the results of the questionnaire, which indicate a positive attitude of students to the use of CMC in the study of mathematical disciplines; according to students, the role of programs for mathematical purposes in teaching should be auxiliary. It is shown that a high-quality digital textbook ensures the formation of not only the mathematical culture of a future specialist, but also the formation of such basic competencies as the ability and willingness to self-learn, apply knowledge, skills and abilities to work with computer mathematics systems, self-education and future professional activity. The use of a digital textbook integrated into the learning technology designed and implemented by the teacher allows him or her to choose their own creative strategy and methodology for teaching students.

Enhancing Contractor Selection through Fuzzy TOPSIS and Fuzzy SAW Techniques

Contractors play an integral role in construction projects, and their qualifications directly impact various aspects of a project’s success. The unbiased selection of contractors is a challenge in the construction industry worldwide, particularly in public projects where impartiality in the final selection is essential. Numerous factors must be considered when evaluating contractors, making the selection process challenging for the human brain. This paper introduces and compares two methods for assessing contractor prequalification by applying the fuzzy theory. The idea is to facilitate using human judgments in a mathematical system for decision-making with regard to selecting contractors. The method is based on identifying a fuzzy weight for the selection criteria using the Buckley method. Fuzzy TOPSIS and Fuzzy SAW methods are then used for the qualification ranking of the contractors. The proposed models are assessed using a case study. A sensitivity analysis was also conducted to compare the two models. The introduced method is expected to improve the quality of the qualification-based selection of contractors and prevent possible losses from hiring unsuitable contractors.

Stochastic Performances of the Mathematical System Based on the Prevalence Prediction of Allergies

Stochastic Performances of the Mathematical System Based on the Prevalence Prediction of Allergies

Methods Regarding the Emerging Technological Phenomena of Modern Security in Nonlinear Dynamic Systems. Application Studies and Identifications Methods

Abstract The method of emergence of a technological mathematical system is the process by which complex behaviors are formed starting from extremely simple rules. This process is dynamic, with entities presenting emergence over a longer period of time (for example, the successive stages of evolution of multicellular structures from simple organisms to a human brain capable of consciousness. At the same time, emergence can occur between entities of sizes that differ by many orders of magnitude (for example, the interaction between a large number of neurons resulting in the human brain capable of rational thought, although the neurons themselves do not exhibit this ability.

CONSTRUCTIONS OF THE PROTECTIVE STRUCTURES UNDER THE CONDITIONS OF HIGH-SPEED IMPACT

The stability of the building is characterized by the depth of penetration of the punch into the body of the structures, as well as their local damage due to the impact load. There was accumulated experience of mathematical modeling of the impact processes of solid bodies on various targets. The range of deformations changes of the reinforced concrete elements ("slab"), which corresponds to the high-speed interaction at the moment of punching by a solid body ("impactor") with an initial impact speed of up to 1000 m/s, as well as the determination of their dynamic nature of deformation require comprehensive justification. The analytical solution of the mathematical system of equations describing the movement of the impactor, as well as the state of the impactor and the slab during their interaction can be applied in a limited range. Therefore, the assessment of the local effect of the impact can be carried out on the basis of the results of experimental studies using empirical formulas, as well as numerical methods of modeling the work of bodies taking into account their elastic-plastic characteristics. The ANSYS software package was used in the conducted research, which implements a system of mathematical equations that describe the movement and state of the "impactor-slab" during their interaction. Based on the results of the performed numerical modeling, it was established that the tested slabs with the use of combined reinforcement - mesh and dispersed metal fiber - are not penetrated by the impactor, part of the sample remains intact. At the same time, the final speed of the impactor in these slabs is up to 35.0% less than in the slabs with the mesh reinforcement. The penetration depth decreased up to 20.0%. The experience of dispersion strengthening of the concrete indicates the inhibition of cracking in the fiber concrete elements, which makes it expedient for usage in the combined reinforced constructions of the protective structures, capable of withstanding the dynamic impacts under conditions of high-speed impact. Further development of the of technological concretes studies is needed for their possible use in barrier structures of protective structures.

Perceptual axioms are irreconcilable with Euclidean geometry.

There are different definitions of axioms, but the one that seems to have general approval is that axioms are statements whose truths are universally accepted but cannot be proven; they are the foundation from which further propositional truths are derived. Previous attempts, led by David Hilbert, to show that all of mathematics can be built into an axiomatic system that is complete and consistent failed when Kurt Gödel proved that there will always be statements which are known to be true but can never be proven within the same axiomatic system. But Gödel and his followers took no account of brain mechanisms that generate and mediate logic. In this largely theoretical paper, but backed by previous experiments and our new ones reported below, we show that in the case of so-called 'optical illusions', there exists a significant and irreconcilable difference between their visual perception and their description according to Euclidean geometry; when participants are asked to adjust, from an initial randomised state, the perceptual geometric axioms to conform to the Euclidean description, the two never match, although the degree of mismatch varies between individuals. These results provide evidence that perceptual axioms, or statements known to be perceptually true, cannot be described mathematically. Thus, the logic of the visual perceptual system is irreconcilable with the cognitive (mathematical) system and cannot be updated even when knowledge of the difference between the two is available. Hence, no one brain reality is more 'objective' than any other.