Mathematical Definition Research Articles

Computational and experimental substantiation of the application of hybrid functions of the first kind in the mechanics of deformation and fracture

A mathematical definition of the concept of «hybrid function of the first kind» (GF1) is given, which provides a smooth controlled transition from one basic mathematical function to another and contains the characteristic features of these two functions. The transition is controlled both by the place of transition and by its speed. At the same time, the transition remains smooth and differentiable. The functions of GF1 are characterized by the coincidence of the arguments of both the basic functions and the argument of the control complex that is part of GF1. Vectors and tensors, as well as complex numbers and functions, can be used as basic functions. Based on GF1, it is possible to design chain GF1 or CGF1, which is essential for expanding the range of GF1 applications. GF1 itself can be used as basic functions. Hybrid functions have a surprisingly high potential for approximating experimental data. The article provides examples of using GF1 to approximate curves in relation to two-parameter crack resistance criteria. An algorithm for eliminating the singularity in calculations for crack resistance is described. Based on GF1, new probability distribution functions of statistical data are proposed. The possibility of qualitative analytical approximation by CGF1 of existing statistical data with subsequent differentiation of the obtained CGF1 to obtain the density of their distribution is shown. An attempt has been made to use GF1 to describe the phenomenon of bifurcation.

An Effective Alternative to Current Mathematics

If you don't understand mathematics, ask yourself if I'm right, because others don't understand mathematics either. By effective alternative to current mathematics, we mean working in a more complete mathematical space than the classical 3D+t variety which is inadequate for generating well-defined definitions and hypotheses as well as its limited ability to solve time-dependent partial differential equations. The current classical discrete 3D+t space PDE, in which time is an external controller and not integrated into the 3D geometric space, cannot be integrated digitally. This space is logically incomplete and misleading in the production of definitions and hypotheses as well as in the resolution itself of time- dependent PDEs. It is no wonder that these definitions/assumptions are confusing and result in weak or intractable mathematics, leading to all kinds of misunderstandings, from horrible notations to undisciplined length of theorems containing a considerable amount of black magic and ending with a gray nature of the mathematical result obtained. In this article, we present some of the most inaccurate assumptions and definitions in current classical mathematics that arise from using the 3D+t manifold space to specify initial conditions, boundary conditions, and the source/sink term. Fortunately, these inaccurate assumptions that start with inadequate space for boundary conditions, initial conditions, and source/sink term can be spotted and analyzed via 4D unitary numerical statistical theory called Cairo techniques in the format of transition chains of matrix B to complete what is missing. In other words, we present how to spot some of the worst mathematical conclusions of classical 3D geometry plus t as an external control numerical space, and then show how to correct them via the 4D unit space which is the subject of this article.

Mapping between measurement scales in meta-analysis, with application to measures of body mass index in children.

Quantitative evidence synthesis methods aim to combine data from multiple medical trials to infer relative effects of different interventions. A challenge arises when trials report continuous outcomes on different measurement scales. To include all evidence in one coherent analysis, we require methods to "map" the outcomes onto a single scale. This is particularly challenging when trials report aggregate rather than individual data. We are motivated by a meta-analysis of interventions to prevent obesity in children. Trials report aggregate measurements of body mass index (BMI) either expressed as raw values or standardized for age and sex. We develop three methods for mapping between aggregate BMI data using known or estimated relationships between measurements on different scales at the individual level. The first is an analytical method based on the mathematical definitions of z-scores and percentiles. The other two approaches involve sampling individual participant data on which to perform the conversions. One method is a straightforward sampling routine, while the other involves optimization with respect to the reported outcomes. In contrast to the analytical approach, these methods also have wider applicability for mapping between any pair of measurement scales with known or estimable individual-level relationships. We verify and contrast our methods using simulation studies and trials from our data set which report outcomes on multiple scales. We find that all methods recreate mean values with reasonable accuracy, but for standard deviations, optimization outperforms the other methods. However, the optimization method is more likely to underestimate standard deviations and is vulnerable to non-convergence.

The Profound Nature of Derivatives and Integration: A Comprehensive Examination

This article delves into the fundamental concepts of derivatives and integration, two core principles of calculus that govern a wide range of mathematical and real-world applications. Derivatives, which measure the rate of change of a function, and integration, which calculates accumulation and areas under curves, are essential tools in fields such as physics, economics, engineering, and finance. The article offers a detailed exploration of their mathematical definitions, key properties—including linearity, the product and chain rules, and the Fundamental Theorem of Calculus—and the diverse techniques used in integration. The practical applications of these concepts are also examined, with special emphasis on their role in modeling dynamic systems, optimizing processes, and solving real-world problems such as motion analysis, financial modeling, and economic optimization. Through in-depth discussion and examples, this comprehensive article bridges the theoretical underpinnings of derivatives and integration with their vast and critical uses in various scientific and economic disciplines.

Fatigue life prediction method for bolted joints based on equivalent structural stress under tensile–compressive loading

In this study, load amplitude–life (Fa–N) curves were obtained through tensile–compressive fatigue tests of bolted joints. It was observed that the correlation coefficient squared (R2) value of the Fa–N curve with the same geometric and pre-tightening parameters was high, but the R2 value of the Fa–N curve with all the parameters was low, indicating poor correlation and inability to meet the engineering requirements. Therefore, an equivalent structural stress model for the bolted joint was first developed based on a mechanical model with a strict mathematical definition to normalize these Fa–N curves, which considered the bolted joint loads as the input conditions and integrated the geometric and pre-tightening parameters. Subsequently, a classical beam–shell equivalent finite element model of the bolted joint was constructed. The nodal loads in the bolted connection zone were coupled with the forces and moments of the beam element nodes through finite element simulation, and the equivalent structural stress (σs) of the bolted joint was then obtained based on the equivalent structural stress model. Consequently, the equivalent structural stress–life (Ss–N) curve and probabilistic stress–life (P–Ss–N) curve normalized for different Fa–N curves were obtained by fitting the data of σs and N. Lastly, the accuracy of the fatigue life prediction method based on equivalent structural stress was verified by conducting the vibration fatigue test on the bolted joint structure of the subway antenna bracket.

On the kinetic temperature of a one-dimensional crystal on the long-time scale

We investigate the dynamics of the kinetic temperature of a finite one-dimensional harmonic chain, the evolution of which is initiated by a thermal shock. We demonstrate that the kinetic temperature returns arbitrarily close to its initial state (the one immediately following the thermal shock) infinitely many times, and we give an estimate for the time elapsed until the recurrence. This assertion is closely related to the Poincare recurrence theorem and we discuss their relation. To estimate the recurrence time we use its averaging along system’s trajectory and provide a rigorous mathematical definition of the mean recurrence time. It turns out that the mean recurrence time exponentially increases with the number of particles in the chain. A connection is established between this problem and the local theorems of large deviations theory.Previous studies have shown that in such a one-dimensional harmonic chain, at times of order N, a thermal echo phenomenon is observed — a sharp increase in the amplitude of kinetic temperature fluctuations. In the present work, we give a rigorous mathematical formulation to this phenomenon and estimate the amplitude of the fluctuations.The research is funded by the Ministry of Science and Higher Education of the Russian Federation within the framework of the Research Center of World-Class Program: Advanced Digital Technologies (agreement №075-15-2020-311 dated 04.20.2022).

What to Consider When Considering Differential Privacy for Policy

Differential privacy (DP) is a mathematical definition of privacy that can be widely applied when publishing data. DP has been recognized as a potential means of adhering to various privacy-related legal requirements. However, it can be difficult to reason about whether DP may be appropriate for a given context due to tensions that arise when it is brought from theory into practice. To aid policymaking around privacy concerns, we identify three categories of challenges to understanding DP along with associated questions that policymakers can ask about the potential deployment context to anticipate its impacts.

A study of gestures representing musical emotions

The creation of music has originated from the burst of inspiration of the composer. Many studies suggested that improvisation plays a more vital role than composition in the emergence of new forms of music. The production of music, especially in Western cultures, tends to follow its traditional music theory. With the emergence of Pythagorean music theory, Western musicology has been defined and developed in depth. The composer’s creation is based on music creation within the framework of the Western music theory. However, in recent years, with the rise of the topic of music gestures, some scholars have learned that music ontology as a four-dimensional hypercube is inseparable from the performer’s gesture. Therefore, some musicians also attempt to create flexible music from the perspective of music performers and based on performance gestures. This kind of music creation breaks the traditional music theory system and has more personal style characteristics, but few people can understand the thinking logic behind this kind of creative behavior. Therefore, this article focuses on the singing gesture and reversely analyzes its impact on musical creation. We use a recent mathematic definition of gestures and we discuss emotion from its etymological origin (from the Latin e-movere) as a gestural movement from inside out. This article conducts an in-depth analysis based on the singer’s gesture for the following two reasons: First, the literature on the connection between singing gesture theory and composition theory remains scarce. Second, gestures produced by a singer are the only performance form that is not enabled by other instruments.

Mathematical softgauges for material ratio parameters

Editors of surface texture analysis software need to validate their parameter calculations, ideally in an implementation-independent way. Today, there is no universal reference method for validating parameter compliance and more specifically with regard to material ratio parameters, Rk parameters, and volume parameters that are based on the Abbott curve. Several years ago, at Digital Surf, we developed parameter validation methods that are based on mathematics. A mathematical softgauge, defined by an equation, is injected through the mathematical parameter definition, to obtain a mathematical true value of the parameter value. In a parallel path, a discrete softgauge, derived from the mathematical softgauge, is injected through an algorithm and the resulting value is compared to the approximated true value. This method makes it possible to check the accuracy of an algorithm for calculating a particular parameter. For this study, two families of mathematical softgauges were specifically designed to test material ratio parameters. More precisely, the two main goals of this study are, (1) to advocate an algorithm validation approach that is independent from any software implementation, and (2) to obtain true values for all parameters based on the Abbott curve.

The Definition of Individual Biological Fitness

“Fitness” is one of the central concepts in biology. Despite this, the concept is still not clearly defined. Previous attempts at definition refer to what should be called an individual’s “potential fitness,” or, when mathematized, the relative fitness of genetic alleles. In contrast, the present work defines the actual fitness of an individual, exactly what is referred to in the expression “survival of the fittest.” This represents a new conceptualization, a mathematical definition that will extend across the entire set of ideas related to evolution.

Additive Manufacturing of Ceramic Reference Spheres by Stereolithography (SLA)

Additive Manufacturing (AM) is advancing technologically towards the production of components for high-demand mechanical applications with stringent dimensional accuracy, leveraging metallic and ceramic raw materials. The AM process for ceramic components, known as Ultraviolet Laser Stereolithography (SLA), enables the fabrication of unique parts or small batches without substantial investments in molds and dies, and avoids the problems associated with traditional manufacturing, which involves multiple stages and final machining for precision. This study addresses the need to produce reference elements or targets for metrological applications, including verification, adjustment, or calibration of 3D scanners and mid- to high-range optical sensors. Precision spheres are a primary geometry in this context due to their straightforward mathematical definition, facilitating rapid and accurate error detection in equipment. Our objective is to exploit this novel SLA process along with the advantageous optical properties of technical ceramics (such as being white, matte, lightweight, and corrosion-resistant) to materialize these reference objects. Specifically, this work involves the fabrication of alumina hemispheres using SLA. The manufacturing process incorporates four design variables (wall thickness, support shape, fill type, and orientation) and one manufacturing variable (the arrangement of spheres on the printing tray). To evaluate the impact of the design variables, dimensional and geometric parameters (GD&T), including diameters, form errors, and their distribution on the surface of the sphere, have been characterized. These measurements are conducted with high accuracy using a Coordinate Measuring Machine (CMM). The study also examines the influence of these variables in the dimensional and geometric accuracy of the spheres. Correlations between various parameters were identified, specifically highlighting critical factors affecting process precision, such as the position of the piece on the print tray and the wall thickness value. The smallest diameter errors were recorded at the outermost positions of the tray (rear and front), while the smallest shape errors were found at the central position, in both cases with errors in the range of tens of micrometers. In any case, the smallest deformations were observed with the highest wall thickness (2 mm).

Network theory and migration: Avoiding misapplications and misinterpretations

Network analysis is becoming a popular tool for the study of animal movement. Proliferation of software enables researchers to use network measures without reflecting on the underlying mathematics and biology. One common use characterizing movement networks, such as migration, with centrality measures. Predominantly developed to evaluate social systems, these measures now appear in applications across disciplines that vary greatly in system properties. Each network measure has a very specific mathematical definition, with implicit assumptions about system properties to which they are applied that often make them inappropriate for migration. We report mismatches between mathematical assumptions and applications of network measures to migration, primarily with bird examples. For example, of 11 trajectory-transmission network-flow processes observed in the real world, only two are applicable to bird migration, neither of which allows valid application of classical centrality measures. We also identify misinterpretations of network measures, such as nodes with the highest degree centrality as locations where individuals intermingle. Other misinterpretations include that a high betweenness score translates to a node that is frequently used or is an important site along a main migratory route. Finally, networks are used in different ways, ranging from descriptive visualization to quantitative prediction and we identify common cases, such as migratory connectivity, where this leads to miscommunication. Network-theoretic approaches to answer ecological research and conservation questions associated with migration are numerous, but new measures need to be developed for animal migration. We provide a guide for researchers of animal movement for measure selection, development, and documentation.

The Gaussian Plume Model Equation for Atmospheric Dispersion Corrected for Multiple Reflections at Parallel Boundaries: A Mathematical Rewriting of the Model and Some Numerical Testing

The well-known Gaussian plume model has proven to be very useful in simulating the atmospheric dispersion of air pollutants (both gaseous and particulates). Nevertheless, the nature of the model presents problems in the actual computation of concentrations when the plume is confined between two parallel boundaries due to the occurrence of multiple reflections. The ground and temperature inversion lid (especially, when the inversion layer is at low levels in the atmosphere) with a chimney stack releasing the effluent below the latter, is one contextual example of horizontal parallel reflecting boundaries. A second example is buildings confining a roadway on either side, with motor vehicles emitting pollution within the street canyon (or urban notch). In such cases, multiple reflections should be accounted for, otherwise the model underpredicts the resulting concentration. This paper presents a mathematical rewriting of the Gaussian plume model equation corrected for multiple reflections when the pollution source is confined between parallel boundaries. The obtained result is most appropriate when the parallel boundaries are rigid, and near-complete reflection is achieved, e.g., street canyon environment (second quoted example). It is worth noting that the relevant mathematical derivations and definitions are all included in the paper to facilitate reading and to ensure comprehensiveness in the presentation. Additionally, the outcome of some preliminary numerical testing is presented. The latter indicates that the new formulation is mathematically stable and yields interesting results. Further numerical investigation and experimental evaluation are merited.

Maximum length binary sequences and spectral power distribution of periodic signals

The maximum length binary sequence (MLBS) is widely used as a broadband pseudo-random noise excitation signal, for example, for system identification. Although its properties have been known for decades, misleading or inaccurate statements can be found in many references. For example, it is sometimes stated that the spectrum of the MLBS is white, whereas in other references a sinc behavior is stated. In this paper, we therefore analyze the MLBS properties based on precise definitions for the given context (time-discrete vs. time-continuous, periodic vs. non-periodic, etc.), especially with respect to Fourier analysis. Another difficulty arises from the fact that in the literature the mathematical definitions are often simplified by means of normalizations which makes the physical interpretation difficult. Therefore, special emphasis is put on scaling factors which allow such a physical interpretation.

Method for determining the Lyapunov exponent of a continuous model using the monodrome matrix

The Lyapunov exponent is a measure of the sensitivity of a dynamic system to any changes and disturbances. It is therefore applicable in many fields of science, including physics, chemistry, economics, psychology, biology, medicine, and technology. Therefore, there is a need to have effective methods for determining the value of this exponent. In particular, it concerns the definition of its sign. The positive value of the Lyapunov exponent confirms the sensitivity of the system, especially when the system is chaotic. A negative value indicates the stability of the dynamic system being tested. The numerical value of the Lyapunov exponent indicates the degree of sensitivity of the dynamic system under study. Although the mathematical definition of the Lyapunov exponent is clear and simple, in practice determining its value is not a trivial matter, especially for continuous models. This paper presents a method for determining the Lyapunov exponent for continuous models. This method is based on the monodromy matrix. The work, for example, presents research on the following models of physical systems: the Van der Pol model with external forcing, the nonlinear mathematical pendulum model with external forcing and the Lorenz weather model.

One or two things we know about concept drift-a survey on monitoring in evolving environments. Part B: locating and explaining concept drift.

In an increasing number of industrial and technical processes, machine learning-based systems are being entrusted with supervision tasks. While they have been successfully utilized in many application areas, they frequently are not able to generalize to changes in the observed data, which environmental changes or degrading sensors might cause. These changes, commonly referred to as concept drift can trigger malfunctions in the used solutions which are safety-critical in many cases. Thus, detecting and analyzing concept drift is a crucial step when building reliable and robust machine learning-driven solutions. In this work, we consider the setting of unsupervised data streams which is highly relevant for different monitoring and anomaly detection scenarios. In particular, we focus on the tasks of localizing and explaining concept drift which are crucial to enable human operators to take appropriate action. Next to providing precise mathematical definitions of the problem of concept drift localization, we survey the body of literature on this topic. By performing standardized experiments on parametric artificial datasets we provide a direct comparison of different strategies. Thereby, we can systematically analyze the properties of different schemes and suggest first guidelines for practical applications. Finally, we explore the emerging topic of explaining concept drift.

Unsupervised Learning of Disentangled and Interpretable Representations of Material Appearance

As humans, we have learned through experience how to interpret the visual appearance of materials in our environment, enabling us to predict the properties of an object just by looking at it for a few seconds. Although this seems like a straightforward task, the final appearance showed by a material is the result of a non-trivial interaction between confounding factors like geometry, illumination, or viewing angle, which we do not completely understand. Creating an algorithm able to disentangle the perceptual factors of material appearance present in an image, just like humans do, would be a breakthrough in several fields (e.g., in architecture, it could help to create prototypes that accurately reproduce how they will be perceived in the real life; following the inverse path of going from perceptual factors to images, it could be used in product design to intuitively define the perceptual features of a desired material, instead of delving into its mathematical definition). Here, we propose a learning-based algorithm capable of effectively disentangling certain perceptual features of images in an unsupervised way.

Between Mathematics, Theology,And The City

These days, urban expansion and its associated social and ecological challenges are unavoidable. The clever yet morally upright people who create changes in this circumstance are one of the answers. I argued that a K–12 education can aid in the formation of morally upright and competent individuals. My argument is that, in order for students to eventually transform the city, a theological foundation for mathematics instruction can help them improve their character and cognitive abilities. Because of this, a theological perspective on mathematics education can help students grow as individuals and advance society by enhancing their knowledge. In line with the assertion, I think that the definition of mathematics and theology, their shared goal, and the ways in which they might help students become more morally upright constitute the intersection of these two fields. Trust, direction, and articulation are the end results. Trust is an invitation to consider and devise solutions for ecological and social issues. Direction is an invitation to map out the answer based on completed reflection. Finally, articulation is the attempt to carry out a decision in light of reflection.

One or two things we know about concept drift-a survey on monitoring in evolving environments. Part A: detecting concept drift.

The world surrounding us is subject to constant change. These changes, frequently described as concept drift, influence many industrial and technical processes. As they can lead to malfunctions and other anomalous behavior, which may be safety-critical in many scenarios, detecting and analyzing concept drift is crucial. In this study, we provide a literature review focusing on concept drift in unsupervised data streams. While many surveys focus on supervised data streams, so far, there is no work reviewing the unsupervised setting. However, this setting is of particular relevance for monitoring and anomaly detection which are directly applicable to many tasks and challenges in engineering. This survey provides a taxonomy of existing work on unsupervised drift detection. In addition to providing a comprehensive literature review, it offers precise mathematical definitions of the considered problems and contains standardized experiments on parametric artificial datasets allowing for a direct comparison of different detection strategies. Thus, the suitability of different schemes can be analyzed systematically, and guidelines for their usage in real-world scenarios can be provided.

CT and MRI radiomic features of lung cancer (NSCLC): comparison and software consistency

BackgroundRadiomics is a quantitative approach that allows the extraction of mineable data from medical images. Despite the growing clinical interest, radiomics studies are affected by variability stemming from analysis choices. We aimed to investigate the agreement between two open-source radiomics software for both contrast-enhanced computed tomography (CT) and contrast-enhanced magnetic resonance imaging (MRI) of lung cancers and to preliminarily evaluate the existence of radiomic features stable for both techniques.MethodsContrast-enhanced CT and MRI images of 35 patients affected with non-small cell lung cancer (NSCLC) were manually segmented and preprocessed using three different methods. Sixty-six Image Biomarker Standardisation Initiative-compliant features common to the considered platforms, PyRadiomics and LIFEx, were extracted. The correlation among features with the same mathematical definition was analyzed by comparing PyRadiomics and LIFEx (at fixed imaging technique), and MRI with CT results (for the same software).ResultsWhen assessing the agreement between LIFEx and PyRadiomics across the considered resampling, the maximum statistically significant correlations were observed to be 94% for CT features and 95% for MRI ones. When examining the correlation between features extracted from contrast-enhanced CT and MRI using the same software, higher significant correspondences were identified in 11% of features for both software.ConclusionsConsidering NSCLC, (i) for both imaging techniques, LIFEx and PyRadiomics agreed on average for 90% of features, with MRI being more affected by resampling and (ii) CT and MRI contained mostly non-redundant information, but there are shape features and, more importantly, texture features that can be singled out by both techniques.Relevance statementIdentifying and selecting features that are stable cross-modalities may be one of the strategies to pave the way for radiomics clinical translation.Key points• More than 90% of LIFEx and PyRadiomics features contain the same information.• Ten percent of features (shape, texture) are stable among contrast-enhanced CT and MRI.• Software compliance and cross-modalities stability features are impacted by the resampling method.Graphical