Mathematical Interpretation Research Articles

Euclid's Fourth Postulate: Its authenticity and significance for the foundations of Greek mathematics.

The Fourth Postulate of Euclid's Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics.

Analytical Model and Design of Voltage Balancing Parameters of Series-Connected SiC MOSFETs Considering Non-Flat Miller Plateau of Gate Voltage

Series connection is an attractive approach to increase the blocking voltage of SiC power MOSFETs. Currently, the voltage balancing design of the series connection of the SiC MOSFETs highly relies on offline calibration and is challenging in the complex field operation. In this paper, a quantitative model to assess the voltage balancing performance is proposed to achieve a clear mathematical interpretation of the dynamic response of the voltage imbalance control loop. To begin with, an analytical model of the drain-source voltage rising time during the turn-off transient concerning the non-constant Miller plateau is proposed. Based on the turn-off model of the single device, the voltage imbalance sensitivity (VIS) is proposed to describe the influence of the parameters on the gate driving signals on the voltage imbalance. The VIS parameter can be easily achieved from the behavior of single devices, abandoning the complex variables in series connection. Further, for the typical case, active time delay voltage balancing methods are selected to demonstrate the application of the VIS analysis method. Based on VIS, the accurate close-loop design is proposed for controlling the delayed time among the devices. The proposed analysis and method are verified in simulation and experiment. The paper offers a generalized approach to assess the performance and the design of the series connection of the SiC MOSFETs, which can be further applied in many other methods for parameter design and engineering applications.

Hierarchical Synchronization Estimation of Low- and High-Order Functional Connectivity Based on Sub-Network Division for the Diagnosis of Autism Spectrum Disorder.

Functional connectivity network (FCN) calculated by resting-state functional magnetic resonance imaging (rs-fMRI) plays an increasingly important role in the exploration of neurologic and mental diseases. Among the presented researches, the method of constructing FCN based on Matrix Variate Normal Distribution (MVND) theory provides a novel perspective, which can capture both low- and high-order correlations simultaneously with a clear mathematical interpretability. However, when fitting MVND model, the dimension of the parameters (i.e., population mean and population covariance) to be estimated is too high, but the number of samples is relatively quite small, which is insufficient to achieve accurate fitting. To address the issue, we divide the brain network into several sub-networks, and then the MVND based FCN construction algorithm is implemented in each sub-network, thus the spatial dimension of MVND is reduced and more accurate estimates of low- and high-order FCNs is obtained. Furthermore, for making up the functional connectivity which is lost because of the sub-network division, the rs-fMRI mean series of all sub-networks are calculated, and the low- and high-order FCN across sub-networks are estimated with the MVND based FCN construction method. In order to prove the superiority and effectiveness of this method, we design and conduct classification experiments on ASD patients and normal controls. The experimental results show that the classification accuracy of “hierarchical sub-network method” is greatly improved, and the sub-network found most related to ASD in our experiment is consistent with other related medical researches.

Ferroelectric Negative-Capacitance-Assisted Phase-Transition Field-Effect Transistor.

An enormous study is being carried out in the field of emerging steep slope devices, specifically on negative-capacitance-based and phase transition-based devices. This article investigates the action of ferroelectric (FE) and phase transition material (PTM) on a hybrid device, negative-capacitance-assisted phase transition FinFET (NC-PT-FinFET). We encounter several unique phenomena resulting from this unified action and provide valid arguments based on these observations. A significant enhancement in the differential gain and transconductance, a unique variation in the effect of PTM on drain-channel coupling, tunability of hysteresis across PTM by FE thickness( [Formula: see text]), and ultralow subthreshold slope (SS) by lowering both of its factors are some of the major outcomes of the NC-PT-FinFET. Focus is built on comprehending the individual role of FE and PTM in the intriguing features observed in every device performance parameter with the help of mathematical expressions and physical interpretations. Various tunable parameters present in this hybrid device widen its applicability in digital and memory applications.

Ultrasound Lesion Detectability as a Distance Between Probability Measures.

Lesion detectability (LD) quantifies how easily a lesion or target can be distinguished from the background. LD is commonly used to assess the performance of new ultrasound imaging methods. The contrast-to-noise ratio (CNR) is the most popular measure of LD; however, recent work has exposed its vulnerability to manipulations of dynamic range. The generalized CNR (gCNR) has been proposed as a robust histogram-based alternative that is invariant to such manipulations. Here, we identify key shortcomings of CNR and strengths of gCNR as LD metrics for modern beamformers. Using the measure theory, we pose LD as a distance between empirical probability measures (i.e., histograms) and prove that: 1) gCNR is equal to the total variation distance between probability measures and 2) gCNR is one minus the error rate of the ideal observer. We then explore several consequences of measure-theoretic LD in simulation studies. We find that histogram distances depend on bin selection that LD must be considered in the context of spatial resolution and that many histogram distances are invariant under measure-preserving isomorphisms of the sample space (e.g., dynamic range transformations). Finally, we provide a mathematical interpretation for why quantitative values such as contrast ratio (CR), CNR, and signal-to-noise ratio should not be compared between images with different dynamic ranges or underlying units and demonstrate how histogram matching can be used to reenable such quantitative comparisons.

Polarization-Based Histopathology Classification of Ex Vivo Colon Samples Supported by Machine Learning

In biophotonics, novel techniques and approaches are being constantly sought to assist medical doctors and to increase both sensitivity and specificity of the existing diagnostic methods. In such context, tissue polarimetry holds promise to become a valuable optical diagnostic technique as it is sensitive to tissue alterations caused by different benign and malignant formations. In our studies, multiple Mueller matrices were recorded for formalin-fixed, human,ex vivocolon specimens containing healthy and tumor zones. The available data were pre-processed to filter noise and experimental errors, and then all Mueller matrices were decomposed to derive polarimetric quantities sensitive to malignant formations in tissues. In addition, the Poincaré sphere representation of the experimental results was implemented. We also used the canonical and natural indices of polarimetric purity depolarization spaces for plotting our experimental data. A feature selection was used to perform a statistical analysis and normalization procedure on the available data, in order to create a polarimetric model for colon cancer assessment with strong predictors. Both unsupervised (principal component analysis) and supervised (logistic regression, random forest, and support vector machines) machine learning algorithms were used to extract particular features from the model and for classification purposes. The results from logistic regression allowed to evaluate the best polarimetric quantities for tumor detection, while the use of random forest yielded the highest accuracy values. Attention was paid to the correlation between the predictors in the model as well as both losses and relative risk of misclassification. Apart from the mathematical interpretation of the polarimetric quantities, the presented polarimetric model was able to support the physical interpretation of the results from previous studies and relate the latter to the samples’ health condition, respectively.

Applying Case Method in the Training of Future Specialists

Making a brighter presentation and improving assimilation of educational material with the help of the case method allows activating the mental, exploratory and creative abilities of students to optimize the process of assimilation of information. The combination of this method with others allows making future specialists to independently find ways to solve problems. This results in the assimilation of professionally significant special knowledge. The aim of the study was to determine the features of the application of the case method in the educational process in preparing students for future professional activities, identifying sources, structural elements and filling professional cases. The study involved the survey method and further testing to identify indicators of significance at the summative and control stage of the experiment. Methods used: diagnostic programme to study the level of educational activities by Riepkina, Zaika; assessment of the level of control and self-assessment actions; statistical and mathematical interpretation of empirical data with subsequent functional analysis of the research results. The study showed the high efficiency of the case method in the educational environment of higher educational institutions (HEIs). The indicator of students’ self-presentation after the application of the case method has changed significantly. This was reflected in an increased high level to 35.5%, a decreased medium level to 38%, and a slight increase in all levels in the experimental group students. Positive dynamics was revealed on all criteria for diagnosing the effectiveness of the case method in teaching: a decreased percentage of students with negative and relatively low markers and a corresponding increase in the percentage of students with positive markers of knowledge and learning the material. Further research can deal with development and implementation of case systems in view of certain majors, taking into account future professional activities.

Additional opportunities for using computer technologies in expert analysis of laboratory data

The presented work reflects the author’s algorithm of mathematical processing and possible interpretations of the results obtained when creating expert analytical systems using computer technology. As initial parameters for the construction of neural networks of the second level, it is proposed to use individual results of calculations of the ratios of a number of laboratory indicators capable of forming a single functional cluster (leukogram, indicators of water-electrolyte balance, protein fractions, etc.). Further, using the proposed calculation algorithm and the archive database, structural features in the ′deformation′ of the ratio panel were compared against the background of the growth of each indicator, a matrix table was built reflecting the degree (correlation coefficients, KCr) of the coincidence of the features of the formation of the structure of the obtained panels of the parameters determined in individual observations. At high values of KCr, a conclusion is made about their (determined parameters) unified participation in the mechanism of emerging disorders. Thus, the system allows you to establish a leading complex of associated relationships by changes in the structure of ratios in the selected panel, corresponding to the dynamics of deviation of the target (analyzed) indicator, thereby differentiating the different mechanisms of formation of pathological deviations in this patient, the manifestation and balance of their values in the system-wide response, to evaluate the specific value of the absolute indicator as the final result of such interaction at the time of examination of the patient. The approach used, proposed by the authors, allows us to evaluate not only quantitative relationships, but also the features of the manifestation of the functional properties of the evaluated indicators. All of the above significantly expands the informativeness of the obtained laboratory data, allowing us to build a reasonable paradigm of the connections of the emerging complex of pathological disorders in each individual case. In the list of references, the authors cite publications as concrete examples of the use of the proposed approach in assessing a complex of disorders with different pathological processes using different panels in calculations.

Identification of numerical solutions of a fractal-fractional divorce epidemic model of nonlinear systems via anti-divorce counseling

<abstract><p>Divorce is the dissolution of two parties' marriage. Separation and divorce are the major obstacles to the viability of a stable family dynamic. In this research, we employ a basic incidence functional as the source of interpersonal disagreement to build an epidemiological framework of divorce outbreaks via the fractal-fractional technique in the Atangana-Baleanu perspective. The research utilized Lyapunov processes to determine whether the two steady states (divorce-free and endemic steady state point) are globally asymptotically robust. Local stability and eigenvalues methodologies were used to examine local stability. The next-generation matrix approach also provides the fundamental reproducing quantity $ \bar{\mathbb{R}_{0}} $. The existence and stability of the equilibrium point can be assessed using $ \bar{\mathbb{R}}_0 $, demonstrating that counseling services for the separated are beneficial to the individuals' well-being and, as a result, the population. The fractal-fractional Atangana-Baleanu operator is applied to the divorce epidemic model, and an innovative technique is used to illustrate its mathematical interpretation. We compare the fractal-fractional model to a framework accommodating different fractal-dimensions and fractional-orders and deduce that the fractal-fractional data fits the stabilized marriages significantly and cannot break again.</p></abstract>

The Mathematical Thoughts and Physics Interpretations of Rainbow from Ancient to Modern Times

With the aim of building a more precise mathematical model of better predictability for the formation of the supernumerary rainbow and fogbow and seeking a clearer and more elaborate physical interpretation, this paper examines the relationship between different rainbow patterns and droplet sizes through both analytical derivation and numerical simulation and develops a much more detailed model beyond previous explanations. From Newton’s geometric model of optics to Young’s wave model, the paper first establishes a solid foundation for the understanding of the formation of the rainbow in nature and through human vision, and then goes on examining the interferences of light, finally applying the model in reality for a better understanding of complex rainbow phenomena, with additional analysis on an unexpected finding about the correspondence of maximum view angle and shortest light path through hypothetical explanation based on the principle of least time and simulation of an elliptical droplet.

The concept of land resource potential in the economic economy

The article analyzes the conceptual and terminological support in the field of economics of nature management and environmental protection. The need for an in-depth etymological study of the terminological apparatus related to the concept of "land resource potential", namely the concepts of "resource", "potential", "land" in terms of increasing competition among businesses for resources and markets for manufactured products. In this regard, a significant amount of bibliographic and electronic Internet sources, both foreign and domestic authors and researchers. The role of resource potential in the agro-industrial complex of Ukraine is substantiated and determined. The author's interpretation of the concept of "land resource potential" as a set of land, labor, material and economic resources, endowed with the potential for the development of productive forces and characterized by indispensability in the production of dependent and territorially spheres of production. The generalization of concepts is carried out and the mathematical interpretation of the concept "land resource potential" is offered. The establishment of classification features is substantiated and the species classification of resources is carried out. In the course of the research the functional properties of land resource potential were identified and generalized. The role of land resources as one of the main ones in the structure of the nature management system is determined.

Mathematical modeling of malignant ovarian tumors

The article explores modeling the development of ovarian cancer, the treatment of this oncologicaldisease in women, the assessment of the time to achieve remission, and the assessment of the time of the onset of relapse. The relevance of the study is that ovarian cancer is one of the most common cancers in women and has the highest mortality rate among all gynecological diseases. Modeling the process of the development of the disease makes it possible to better understand the mechanism of the development of the disease, as well as the time frame of the onset of each stage, as well as the assessment of the survival time. The aim of the work is to develop a model of an ovarian tumor. It is based on a model of competition between two types of cells: epithelial cells (normal cells) and tumor cells (dividing cells). The mathematical interpretation of the competition model is the Cauchy problem for a system of ordinary differential equations. Treatment is seen as the direct destruction of tumor cells by drugs. The behavior of solutions in the vicinity of stationary points is investigated by the eigenvalues of the Jacobi matrix of the right side of the equations. On the basis of this model, the distribution of conditional patients by four stages of the disease is proposed. Biochemical processes that stimulate the accelerated growth of the tumor cell population are modeled by a factor that allows tumor cells to gain an advantage in a competitive relationship with epithelial cells. The spatio-temporal dynamics of an ovarian tumor leads to a modification of the competition model due to the introduction of additional factors into it, taking into account the presence of increased nutrition of ovarian tumors, the exit of the tumor from the plane of the ovary, as well as the effect of treatment on tumor cells. The new model describes the interaction conditions with a system of second-order partial differential equations. The results of computer modeling demonstrate an assessment of the distribution of conditional patients by stages of the disease, the time of onset of relapse, the duration of remission, the obtained theoretical results of modeling are compared with the real data.

ОПРЕДЕЛЕНИЕ ОПТИМАЛЬНЫХ ЗНАЧЕНИЙ КИНЕТИЧЕСКИХ КОНСТАНТ СКОРОСТЕЙ ЭЛЕМЕНТАРНЫХ СТАДИЙ ДЛЯ ПРОЦЕССОВ ПОЛИМЕРИЗАЦИИ

Kinetic and mathematical models are given in the paper for the polymerization process under consideration. Mathematical interpretations of direct and inverse kinetic problems are given. The latter is reduced to finding the values of the kinetic constants of the elementary stages of the polymerization process. For preliminary localization of their values and narrowing of the permissible range of their variation, the paper proposes a method for constructing a functional surface in the basic space.

МЕТОДИЧЕСКИЙ ПОДХОД К ОЦЕНКЕ ТЕПЛОВЫХ ПОТЕРЬ БЕСКАНАЛЬНОЙ ТЕПЛОВОЙ СЕТИ С УЧЕТОМ ТЕПЛОВОЙ ИНТЕРФЕРЕНЦИИ

The author’s vision of a methodical approach to estimation of thermal losses of a channel-less thermal network is presented at the base, which is the determination of specific thermal losses caused by the thermal interference process. There was carried out systematization of existing approaches to estimation of thermal losses of heat networks with identification of positive and negative features, the logical scheme of thermal interference process implementation at movement of high-temperature heat carrier in heat supply line of heat network is justified. Mathematical interpretation of methodical approach to estimation of thermal losses of channel-less heat networks is presented taking into account thermal interference, which includes twelve successively performed calculation stages.

Implicit and Explicit Feature Purification for Age-Invariant Facial Representation Learning

This paper presents a new method, named implicit and explicit feature purification (IEFP), for age-invariant face recognition. Facial features extracted from a face image contain the information about the identity, age, and other attributes. For age-invariant face recognition, it is important to remove the irrelevant information, and retain the identity information only, in the facial features. Through the two proposed feature purification mechanisms, our framework can produce facial-feature embeddings that preserve identity information as much as possible and are insensitive to age variations. Specifically, on the one hand, a special network module is devised to implicitly purify the original facial features obtained from a face encoder. On the other hand, to obtain purer facial feature representations for age-invariant face recognition, irrelevant information within the implicitly purified features, such as the age, is further removed. This is realized by using a regularizer, based on information theory, to explicitly minimize the correlation between identity-related features and age-related features. Comprehensive ablation studies show that these two feature purification schemes can work independently, as well as collaboratively, to achieve better performance. Extensive evaluations on several benchmark data sets show that the IEFP method is on par with those competitors learned on far more favorable training samples, and it achieves the best performance in a fair comparison. Furthermore, we provide mathematical interpretation to explain the effectiveness of our approach, and find that it tends to generate low-rank, yet high-dimensional, representations for age-invariant face recognition.

Learning From Noisy Data: An Unsupervised Random Denoising Method for Seismic Data Using Model-Based Deep Learning

For seismic random noise attenuation, deep learning has attracted much attention and achieved promising performance. However, compared with conventional methods, the denoising performance of supervised learning-based methods heavily depends on massive training samples with high-quality labeled data, which makes their generalization capabilities limited. Even though deep neural networks (DNNs) usually outperform the conventional denoising methods, their performance is not guaranteed since neural networks still lack good mathematical interpretability at present. To alleviate the dependency on labeled data and explore insights into the denoising system, we proposed an unsupervised denoising method based on model-based deep learning, which combined domain knowledge and a data-driven method. We designed a network based on the modified iterative soft threshold algorithm (ISTA), which omitted the soft threshold to alleviate uncertainties introduced by empirically selected thresholds. In this network, we set the dictionary and code as trainable parameters. A loss function with a smooth penalty was designed to ensure that the network training can be implemented in an unsupervised manner. In the proposed method, we set the denoised result by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f-x$ </tex-math></inline-formula> deconvolution as the input for our network, and the further denoised data can be obtained after each epoch of the training, which means that our method does not need the testing procedure. Experiments on synthetic and field seismic data demonstrate that our method exhibits competitive performance compared to the conventional, supervised, and unsupervised methods, including <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f-x$ </tex-math></inline-formula> deconvolution, curvelet, the Denoising Convolutional Neural Network (DnCNN), and the integration of neural network and Block-matching and 3-D filtering method (NN + BM3D).

КОНЦЕПТУАЛЬНЫЕ ОСНОВЫ ИНФОРМАТИЗАЦИИ ПРЕДПРИНИМАТЕЛЬСКОЙ ДЕЯТЕЛЬНОСТИ

This article is devoted to the issues of informatization of entrepreneurial activity and, in particular, the analysis of various scientific views on the fundamental concepts of the object of research. The paper considers philosophical, functionalist and mathematical interpretations of «information». According to the first, information is a subjective principle and is a derivative of the mind of the person concerned. From the point of view of functionalism, information is objective and its existence is independent of human perception, while information becomes substantive and valuable only in the case of interaction of two systems: the source and receiver of messages. Attempts to formalize this concept can be observed in mathematical approaches, on the basis of which, together with a philosophical and functionalist vision, a definition of business information was given. The research also reflected the role and importance of information support for economic and production processes. A review of relevant scientific papers allowed us to conclude that information support is systematic and highlight its main purpose, which is to conduct business rationally through the use of innovative technologies. The active introduction of information support elements into business subsystems changes not only the structure of business processes, but also the relationship between the participants of interaction-business is subject to informatization and becomes part of the information society. The article pays special attention to the definition of «informatization». Based on the analysis of the points of view of scientists presented in the article, it was revealed that informatization is a global phenomenon that gradually penetrates into all spheres of human life, saturates the communicative and activity environment with its constituent elements, opening up new ways to solve socio-economic problems. In various fields of knowledge and human activity, informatization is interpreted differently, but its conceptual foundations are unchanged: it is a process implemented using various types of technologies, operates on the basis of reliable and exhaustive information, and also involves the interaction of external and internal information flows. The main result of the study was the definition of the concept of «business informatization».

МЕТОД ТЕОРІЇ РОЗМІРНОСТЕЙ В ЕКСПЕРИМЕНТАЛЬНИХ ДОСЛІДЖЕННЯХ СИСТЕМ І ПРОЦЕСІВ

The method and parameters of experimental modelling of systems and processes in mechanical engineering are substantiated. The theory of similarity and dimensionality is used as an intermediate link between theory and experiment. The dimension of the factor space depends on the number of factors. The set of factors is grouped into dimensionless similarity criteria. The selected criteria are in certain dependence, such as the Galileo test, Euler and Reynolds numbers. Examples of application in experimental studies are given. The use of dimension theory in a factor-planned experiment allows reducing the number of factors, simplifies the mathematical interpretation of the response criterion and provides a graphical representation in the form of 3-D model.

THE USE OF ARTIFICIAL INTELLIGENCE METHODS FOR APPROXIMATION OF THE MECHANICAL BEHAVIOR OF RUBBER-LIKE MATERIALS

In the XXI century, neural networks are widely used in various fields, including computer simulation and mechanics. This popularity is due to the factthat they give high precision, work fast and have a very wide range of settings. The purpose of creating a software product using elements of artificialintelligence, for interpolation and approximation of experimental data. The software should work correctly, and yield results with minimal error. Thedisadvantage of using mathematical approaches to calculating and predicting hysteresis loops is that they describe unloading rather poorly, thus, weobtain incorrect data for calculating the stress-strain state of a structure. The solution tool use of elements of artificial intelligence, but rather neuralnetworks of direct distribution. The neural network of direct distribution has been built and trained in this work. It has been trained with a teacher (ateacher using the method of reverse error propagation) based on a learning sample of a pre-experiment. Several networks of different structures werebuilt for testing, which received the same dataset that was not used during the training, but was known from the experiment, thus finding a networkerror in the amount of allocated energy and in the mean square deviation. The article describes in detail the mathematical interpretation of neuralnetworks, the method for training them, the previously conducted experiment, structure of network that was used and its topology, the training method,preparation of the training sample, and the test sample. As a result of the robots carried out, the software was tested in which an artificial neuralnetwork was used, several types of neural networks with different input data and internal structures were built and tested, the error of their work wasdetermined, the positive and negative sides of the networks that were used were formed.

A mathematical interpretation of the communicative constitution of organizations

ABSTRACT We propose a probabilistic control-system model to measure gradations of organizational coherence. Our model expresses the four flows of the Communicative Constitution of Organizations, and moves beyond common network-based analyses to quantify the emergence of organization through communication. In the model, an organization’s leadership selects from a set of discrete strategies, the number and complexity of which are regulated by membership negotiation and self-structuring. Activity coordination constrains the internal response of the organization to the selected strategy, and institutional positioning shapes its ultimate outcome in the environment. Our model introduces a new partition of the four flows between those that affect the set of available strategies versus the precision of organizational control. The model informs proxy measurements of the maturation of a partial organization, such as an insurgency. Its quantified approach enables testing of the interrelationships and tradeoffs among flows in an agent-based simulation of varying levels of partial organizations competing and cooperatively exploring for resources. The most consequential flow in our scenario, measured in comparison to baseline data, is membership negotiation. Complex behaviours emerge in simulation results, such as information asymmetries, tradeoffs between power and efficiency, the contrast between short- and long-term gains, and individual versus group utility.