Mathematical Considerations Research Articles

Toward better porous carbon-based electrodes by investigation of the viscoelastic properties of carbon suspension

In this article, an electrode slurry based on activated carbon, which is used to form electrodes of electrochemical capacitors, was assigned to a given group of rheological fluids. The interactions between the textural properties of carbon, the components of the slurry and their impact on the flow nature were explained. Moreover, the viscosity window and optimal ratio between components have been specified, which are necessary to form the right coating on the stainless steel foil as a current collector. The equation that correlates electrode thickness, viscosity, and composition of the slurry (mass of solid materials, solvent, and ratio between components) was obtained by mathematical consideration. Electrochemical measurements were performed to verify the impact of coating quality and thickness on device performance. Finally, the effect of the microstructure formed after slurry preparation and the phenomena accompanying the selected mixing technique are discussed. The paper ends with an economic analysis of electrode slurry production.

Application of Duhamel's convolution integral in computer measurement systems

The paper is a new approach to the Duhamel's convolution integral. It contains a brief description of the Duhamel's convolution integral and an overview of applications of this integral. The scientific novelty presented in the work is the original algorithm for determining the frequency characteristic, together with its mathematical justification as well as with hardware and software implementation. The proposed algorithm was used in computer measurement systems enabling the determination of Bode and Nyquist characteristics of objects used in automation. The measurement system was built by the authors on the basis of the proposed algorithm. Measurements were carried out using it. During the measurements of the selected automation object, actual results were obtained and referenced to theoretical results. Then their correctness was assessed using the theory of measurement errors. In this way, the correctness of the mathematical considerations concerning the application of the Duhamel's convolution integral in computer measurement systems enabling the determination of the frequency characteristics of objects used in automation was confirmed.

Quasi-static crack propagation in soft materials using the material-sink theory

In the present study, a finite element formulation for the material-sink approach aimed at modeling quasi-static crack propagation in hyperelastic solids is developed. Breakage of molecular bonds leads to material separation and appearance of two new surfaces of a crack. However, the bond breakage is diffusive, and the loss of local bonds leads to the localized material (molecular) loss. The latter notion triggers consideration of mass density as a variable that numerically decreases in the area where damage localizes into a crack. This physical notion requires mathematical consideration of mass balance as an additional and active law, which regularizes the computational model. From the numerical point of view, the developed finite element formulation has displacement and density degrees of freedom. Also, a monolithic approach was applied that ensures stable incrimination of the nonlinear problem. Numerical examples of the fracture of aneurysm material demonstrate the high robustness of the proposed approach.

Mathematical and physical considerations indicating that the cell genome is a read-write memory

The molecular mechanisms that govern biological evolution have not been fully elucidated so far. Recent studies indicate that regulatory proteins, acting as decision-making complex devices, can accelerate or retard the evolution of cells. Such biochemically controlled evolution may be considered as an optimization process of logical nature aimed at developing fitter species that can better survive in a specific environment. Therefore, we may assume that new genetic information can be stored in the cell memory (i.e., genome) by a sophisticated biomolecular process that resembles writing in computer memory. Such a hypothesis is theoretically supported by a recent work showing that logic is a necessary component of life, so living systems process information in the same way as computers. The current study summarizes existing evidence showing that cells can intentionally modify their stored data by biochemical processes resembling stochastic algorithms to avoid environmental stress and increase their chances of survival. Furthermore, the mathematical and physical considerations that render a read-write memory a necessary component of biological systems are presented.

Correction: Martinović et al. Mathematical Considerations for Unmanned Aerial Vehicle Navigation in the Magnetic Field of Two Parallel Transmission Lines. Appl. Sci. 2021, 11, 3323

The authors wish to make the following changes to their paper [...]

Mathematical considerations for evaluating X‐ray beam size in micro‐XRF analysis

AbstractMicro X‐ray fluorescence spectrometry (micro‐XRF) is conducted for elemental imaging and analysis of small regions, by focusing on primary X‐rays. An appropriate beam diameter evaluation is essential as is directly related to the spatial resolution of micro‐XRF imaging. The X‐ray beam size was experimentally determined from the XRF intensity with respect to the scanning distance (thin‐wire scanning method) or its differentiated curve (knife‐edge scanning method). However, a beam diameter is not currently evaluated in a mathematically appropriate method and is known to differ significantly from the actual value under certain conditions. A formula to correct this difference has been proposed; however, to the best of our knowledge, no study has discussed the performance of the correction formula and conditions under which they can be applied. By mathematically analyzing the overlapping volume between the X‐ray beam and the scanning material, we succeeded in deriving an appropriate fitting function that replaces the Gaussian function taking into account the effect of absorption. The fitting function is effective for each method. This technique was applied to evaluate the micro X‐ray beam created using polycapillary X‐ray focusing optics. Both thin‐wire and knife‐edge scanning methods were used with wires of different sizes to determine the X‐ray beam size. The beam size obtained by the conventional fitting function can be accurately corrected using the newly derived correction formula. Simulations were performed using the derived fitting function, and the applicability of the correction formulas was demonstrated.

A Mathematical Consideration on the Relation of the Social Structure to the Infection Risk in a Community

We construct and analyze a mathematical model to consider the relation of the social structure to the infection risk for a spreading transmissible disease in a community. We take into account different phases of human activity, whether it takes place solely in the private situation or both private and social spheres, followed by the division of the community members into two classes: active and less active. The analysis on our mathematical model implies that there are critical conditions for the class size and human activity with respect to the infection risk for a community. Further, we try to discuss how the social structure and situation could be related to the infection risk for a community.

A general pattern of non-spiking neuron dynamics under the effect of potassium and calcium channel modifications.

Electrical activity of excitable cells results from ion exchanges through cell membranes, so that genetic or epigenetic changes in genes encoding ion channels are likely to affect neuronal electrical signaling throughout the brain. There is a large literature on the effect of variations in ion channels on the dynamics of spiking neurons that represent the main type of neurons found in the vertebrate nervous systems. Nevertheless, non-spiking neurons are also ubiquitous in many nervous tissues and play a critical role in the processing of some sensory systems. To our knowledge, however, how conductance variations affect the dynamics of non-spiking neurons has never been assessed. Based on experimental observations reported in the biological literature and on mathematical considerations, we first propose a phenotypic classification of non-spiking neurons. Then, we determine a general pattern of the phenotypic evolution of non-spiking neurons as a function of changes in calcium and potassium conductances. Furthermore, we study the homeostatic compensatory mechanisms of ion channels in a well-posed non-spiking retinal cone model. We show that there is a restricted range of ion conductance values for which the behavior and phenotype of the neuron are maintained. Finally, we discuss the implications of the phenotypic changes of individual cells at the level of neuronal network functioning of the C. elegans worm and the retina, which are two non-spiking nervous tissues composed of neurons with various phenotypes.

Bounds on quantum evolution complexity via lattice cryptography

We address the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. Complexity is understood here as the shortest geodesic distance between the time-dependent evolution operator and the origin within the group of unitaries. (An appropriate 'complexity metric' must be used that takes into account the relative difficulty of performing 'nonlocal’ operations that act on many degrees of freedom at once.) While simply formulated and geometrically attractive, this notion of complexity is numerically intractable save for toy models with Hilbert spaces of very low dimensions. To bypass this difficulty, we trade the exact definition in terms of geodesics for an upper bound on complexity, obtained by minimizing the distance over an explicitly prescribed infinite set of curves, rather than over all possible curves. Identifying this upper bound turns out equivalent to the closest vector problem (CVP) previously studied in integer optimization theory, in particular, in relation to lattice-based cryptography. Effective approximate algorithms are hence provided by the existing mathematical considerations, and they can be utilized in our analysis of the upper bounds on quantum evolution complexity. The resulting algorithmically implemented complexity bound systematically assigns lower values to integrable than to chaotic systems, as we demonstrate by explicit numerical work for Hilbert spaces of dimensions up to \sim 10^4∼104.

Black-Spot Analysis in Hungary Based on Kernel Density Estimation

Between 2010 and 2020 in the European Union, 30% of road accidents resulted in the death of a pedestrian or a cyclist. Accidents of unprotected pedestrians and cyclists are the reason why it is essential to introduce road safety measures. In our paper, we identify and rank black spots using an innovative reactive approach based on statistics. We elaborate on the mathematical methodological considerations through the processing of real-life empirical data in a Matlab environment. The applied black-spot analysis is based on a Kernel density estimate method, and the importance of the kernel functions and bandwidth are elaborated. Besides, special attention is devoted to the distorting effect of annual average daily traffic. The result of our research is a new methodology by which the real locations of the examined black spots can be determined. Furthermore, the boundaries of the critical sections and the extent of the formation of black spots can be determined by the introduced mathematical methods. With our innovative model, the black spots can be ranked, and the locations having the highest potential for improvement can be identified. Accordingly, optimal measures can be determined considering social-economic and sustainability aspects.

Mathematical Responses to the Hole Argument: Then and Now

AbstractWe argue that several apparently distinct responses to the hole argument, all invoking formal or mathematical considerations, should be viewed as a unified “mathematical response.” We then consider and rebut two prominent critiques of the mathematical response before reflecting on what is ultimately at issue in this literature.

The Architecture Optimization and Energy Management Technology of Aircraft Power Systems: A Review and Future Trends

With the development of More/All-Electric Aircraft, especially the progress of hybrid electrical propulsion or electrical propulsion aircraft, the problem of optimizing the energy system design and operation of the aircraft must be solved regarding the increasing electrical power demand-limited thermal sink capability. The paper overviews the state of the art in architecture optimization and an energy management system for the aircraft power system. The basic design method for power system architecture optimization in aircraft is reviewed from the multi-energy form in this paper. Renewable energy, such as the photo-voltaic battery and the fuel cell, is integrated into the electrical power system onboard which can also make the problem of optimal energy distribution in the aircraft complex because of the uncertainty and power response speed. The basic idea and research progress for the optimization, evaluation technology, and dynamic management control methods of the aircraft power system are analyzed and presented in this paper. The trend in optimization methods of engineering design for the energy system architecture in aircraft was summarized and derived from the multiple objective optimizations within the constraint conditions, such as weight, reliability, safety, efficiency, and characteristics of renewable energy. The cost function, based on the energy efficiency and power quality, was commented on and discussed according to different power flow relationships in the aircraft. The dynamic control strategies of different microgrid architectures in aircraft are compared with other methods in the review paper. Some integrated energy management optimization strategies or methods for electrical propulsion aircraft and more electric aircraft were reviewed. The mathematical consideration and expression of the energy optimization technologies of aircraft were analyzed and compared with some features and solution methods. The thermal and electric energy coupling relationship research field is discussed with the power quality and stability of the aircraft power system with some reference papers. Finally, the future energy interaction optimization problem between the airport microgrid and electric propulsion aircraft power system was also discussed and predicted in this review paper. Based on the state of the art technology development for EMS and architecture optimization, this paper intends to present the industry’s common sense and future trends on aircraft power system electrification and proposes an EMS+TMS+PHM to follow in the electrified aircraft propulsion system architecture selection

Spectral invariants in ultra-high spatial resolution hyperspectral images

Current operational optical satellite-based Earth observation methods targeting at vegetation are optimised for the high- and medium-resolution satellites with pixels sizes starting at ten meters. This resolution is coarser than the typical size of a vegetation structural element, such as a tree crown, and much coarser that of scattering elements, such as leaves. For this reason, vegetation can be treated as a continuous medium and the variation in the local illumination conditions on individual leaves or tree crowns can be ignored. This does not hold anymore for very and ultra-high resolution imagery, obtained from new satellite systems or unmanned aerial vehicles, where individual tree crowns or even leaves can be discerned. We tested the applicability of the spectral invariant theory to this type of imagery for characterising the local illumination conditions on plant leaves using Monte Carlo ray tracing simulations. The simulations corroborated the direct link between the spectral invariant parameter ρ and the sunlit fraction of visible leaves, earlier alleged for hyperspectral remote sensing data based on mathematical considerations. The approach allowed us to separate the direct beam and multiple scattering irradiance components and provided intuitive interpretations of the recollision probability and canopy scattering coefficient computed for each image pixel.

The Flat Plate Boundary Layer Equation under Blasius Restrictions with a Unique Solution

Even though Blasius’s flat plate boundary layer equation is considered an outstanding application of the boundary layer theory, it presents a series of inconsistencies both in its deduction and solution. This work reexamines, in detail, the fundamentals of the classical equation and the method to solve it to build correlations associated with the proposed new flat plate boundary equation and its solution. It deals with being well aware of avoiding or excluding the existing mathematical and erroneous physical considerations involved to improve the design, analysis, and solution of many practical problems in the fields of Fluid Mechanics and other scientific and technical areas. The new proposed flat plate boundary layer equation has a unique solution and satisfies Prandtl’s boundary layer concept; its inherent flow is sensitive to the transition phenomenon. In this sense, depending on the Reynolds number, it can generate perturbations that will justify the origin of the Tollmien–Schlitching waves.

The Formulation of the Quadratic Failure Criterion for Transversely Isotropic Materials: Mathematical and Logical Considerations

The quadratic function of the original Tsai–Wu failure criterion for transversely isotropic materials is re-examined in this paper. According to analytic geometry, two of the troublesome coefficients associated with the interactive terms—one between in-plane direct stresses and one between transverse direct stresses—can be determined based on mathematical and logical considerations. The analysis of the nature of the quadratic failure function in the context of analytic geometry enhances the consistency of the failure criterion based on it. It also reveals useful physical relationships as intrinsic properties of the quadratic failure function. Two clear statements can be drawn as the outcomes of the present investigation. Firstly, to maintain its basic consistency, a failure criterion based on a single quadratic failure function can only accommodate five independent strength properties, viz. the tensile and compressive strengths in the directions along fibres and transverse to fibres, and the in-plane shear strength. Secondly, amongst the three transverse strengths—tensile, compressive and shear—only two are independent.

A Functional Contextual Account of Background Knowledge in Categorization: Implications for Artificial General Intelligence and Cognitive Accounts of General Knowledge.

Psychology has benefited from an enormous wealth of knowledge about processes of cognition in relation to how the brain organizes information. Within the categorization literature, this behavior is often explained through theories of memory construction called exemplar theory and prototype theory which are typically based on similarity or rule functions as explanations of how categories emerge. Although these theories work well at modeling highly controlled stimuli in laboratory settings, they often perform less well outside of these settings, such as explaining the emergence of background knowledge processes. In order to explain background knowledge, we present a non-similarity-based post-Skinnerian theory of human language called Relational Frame Theory (RFT) which is rooted in a philosophical world view called functional contextualism (FC). This theory offers a very different interpretation of how categories emerge through the functions of behavior and through contextual cues, which may be of some benefit to existing categorization theories. Specifically, RFT may be able to offer a novel explanation of how background knowledge arises, and we provide some mathematical considerations in order to identify a formal model. Finally, we discuss much of this work within the broader context of general semantic knowledge and artificial intelligence research.

Reacting condensed phase explosives in direct contact

In this article, we present a new formulation and an associated algorithm for the simultaneous numerical simulation of multiple condensed phase explosives in direct contact with each other, which may also be confined by (or interacting with one or more) compliant inert materials. Examples include composite rate-stick (i.e., involving two explosives in contact) problems, interaction of shock waves with chemically active particles in condensed-phase explosives, and devices such as detonators and boosters. There are several formulations that address the compliant or structural response of confiners and particles due to detonations, but the direct interaction of explosives remains a challenge for most formulations and algorithms. The proposed formulation addresses this problem by extending the conservation laws and mixture rules of an existing hybrid formulation (suitable for solving problems involving the coexistence of reactants and products in an explosive mixture and its immiscible interaction with inert materials) to model the interaction of multiple explosive mixtures. An algorithm for the solution of the resulting system of partial differential equations is presented, which includes a new robust method for the retrieval of the densities of the constituents of each explosive mixture. This is achieved by means of a multi-dimensional root-finding algorithm, which employs physical as well as mathematical considerations in order to converge to the correct solution. The algorithm is implemented in a hierarchical adaptive mesh refinement framework and validated against results from problems with known solutions. Additional case studies demonstrate that the method can simulate the interaction of detonation waves produced by military grade and commercial explosives in direct contact, each with its own distinct equation of state and reaction rate law.

Evaluation of the effects of discourse-based mathematics instruction on eleventh grade students’ conceptual and procedural understanding of probability and statistics

This study attempted to develop a literature-informed discourse-based instructional approach and evaluated its effects on eleventh-grade students’ conceptual and procedural understandings in probability and statistics. To this end, a quasi-experimental study that involved two control groups and one experimental group was used. One-way analysis of variance of the baseline data showed that the three groups were comparable at the start of the experiment. After the intervention, the Kruskal-Wallis test analysis performed on the posttest data showed significant differences in conceptual understanding and procedural understanding. Pairwise comparisons of the three groups using Dunn’s test as post-hoc analysis showed that students who underwent discourse-based instruction demonstrated a better understanding of probability and statistics compared to those students who received traditional instruction. The analysis of survey data about students’ learning mathematics through discourse provided an insight into how the intervention strategy helped students improve their learning. Overall, the main finding of the study is that discourse-based instruction enables students to develop a better understanding of mathematical topics when appropriately orchestrated interactive discourse practices occur. The study contributes a discourse-based instructional design framework and empirical evidence to the field of knowledge of teaching approaches. The findings may motivate educators to examine students’ textbooks for mathematical discourse considerations. A possible recommendation for future research would be conducting longitudinal studies to explore the relations between students’ conceptual understanding and procedural understanding in a specific domain of mathematics and to examine how discourse-based instruction affects the relations between conceptual and procedural knowledge.

Моделирование движений шахтной подъемной установки с асинхронным электроприводом

Рассмотрена задача создания компьютерной модели (цифрового двойника) шахтной подъемной установки (ШПУ) для одноконцевого подъема с укрупненными основными компонентами: асинхронный электродвигатель с фазным ротором, редуктор, подъемный канат. Синтезируются математические модели каждой компоненты, применимые для динамических режимов работы систем с запаздыванием. Сформулирована задача управления электродвигателем по цепи ротора для выполнения им основной технологической операции (реализация тахограммы подъема (диаграммы скорости)) с одновременным управлением динамическим состоянием электромеханической системы ШПУ. Отмечены особенности построения моделей компонентов ШПУ, которые приводят к необходимости отказа от использования приведенных расчетных схем механических передаточных устройств (редукторов) при математическом рассмотрении процессов пуска (торможения) ШПУ и необходимости рассмотрения подъемного стального каната ШПУ как объекта с распределенными параметрами. Приведены результаты моделирования режима ШПУ для режима выборки напуска каната.

Teaching Scattering Matrix in Electrical Engineering—Misconceptions and Clarifications

This paper presents a new approach to introducing scattering parameters for dispelling doubts and misconceptions. Normalization of voltage and current is first introduced for the subsequent descriptions of normalized matrices. The analogy with lossless transmission lines is used to define voltage waves in terms of circuit voltages and currents. Scattering parameters are interpreted as reflection and transmission coefficients for real normalizing impedance. Power considerations are made to interpret scattering parameters as power ratios. The condition for the lossless circuit is derived, and it is shown that the definitions of the voltage waves satisfy maximum power transfer conditions for real input impedance. Reciprocity conditions are derived to show that the normalized, not the unnormalized scattering matrix, is symmetric. The application of losslessness and reciprocity in filter design is clarified. Next, it is shown why definitions of forward and reverse voltage have to be changed for complex normalizing impedances. The relationships of matrices for complex normalizing impedance are shown to reduce to those for real normalizing impedance. The procedure for changing normalizing impedance is given, and its application for amplifier design is briefly described. The difference between circuit symmetry and physical symmetry is pointed out. Finally, it is shown how odd and even mode analysis arises from a mathematical consideration of symmetrical two-port networks. A straightforward discussion of Wilkinson power divider is given to illustrate the application of these modes.