Mathematical Relationship Research Articles

Deformation behavior, twinning propensity model, and twinnability map for laser powder bed fused austenitic steel homogenized by hot isostatic pressing

Hot isostatic pressing (HIP) has emerged as a highly effective and extensively adopted technique for eliminating intrinsic flaws, mitigating residual stress, and addressing microstructural anisotropy in laser powder bed fused (L-PBF) metallic materials. However, the deformation behavior, strengthening mechanism, and associated mechanistic models of HIPed l-PBF austenitic steel remain largely unclear. Through multiple microstructural characterizations, this study determined that the primary deformation mechanism of HIPed l-PBF austenitic steel involved the emission of twinning partials from either the annealing twin boundaries or high-angle grain boundaries (HAGBs) with the decoration of extra dislocations and prismatic inclusions. A twinning propensity model was formulated based on the identified deformation mechanisms, revealing that the critical angle (θτ-crit) for twinning partial emissions ranged from 25.64° to 28.46° By incorporating the mathematical relationship describing the force equilibrium state between trailing and twinning partials, a twinnability map considering both the crystallographic orientation and stress state (magnitude and direction) was established to guide the twinning control for desired mechanical properties. Finally, the accuracy of the twinning model and twinnability map was effectively confirmed using the microstructural results, whereas the limitations of the twinning model and twinnability map were reasonably examined through micro-defect and texture analysis. The results of this study provided a significant theoretical framework for understanding the twinning behavior of high-performance and homogeneous l-PBF metallic materials.

STEAM practices connecting mathematics, arts, architecture, culture and history in a non-formal learning environment of a museum

This study adopts STEAM practices integrating architecture, arts, culture, and history into mathematics education in a museum learning environment. A workshop was conducted with five in-service mathematics teachers in an Egyptian museum, utilizing digital technologies such as GeoGebra, Augmented Reality, and 3D Printing to model, visualize and connect the museum collections to their teaching practices. Teachers’ modelled artefacts were qualitatively analysed for their artistic, cultural and historical connections. Findings indicate that participants followed the design expected outcomes of these STEAM practices by implementing a transdisciplinary approach towards mathematically modelling the museum objects while connecting to their mathematical, cultural, historical, and artistic relations. Participants’ reflections showed positive changes in their attitudes towards considering the museum learning environment for mathematics teaching during the implementation of these STEAM practices. Furthermore, such STEAM practices allowed participants to explore and connect disciplines through interacting with museum collections from artistic, cultural, historical, mathematical, and technological perspectives.

Bi-stability in femtosecond laser ablation by MHz bursts

In this work, a bi-stable behavior of laser ablation efficiency and quality was controlled by fluence and burst length. The plasma shielding of incoming laser radiation caused sudden jumps with a significant decrease in ablation efficiency for every even number of pulses in the burst. The attenuation of incoming laser radiation by plasma created by the previous pulse was incorporated into the toy model of burst ablation efficiency. The mathematical recurrence relation has been derived for the first time, binding ablation efficiency for the next pulse with the efficiency of the previous pulse, which predicts bi-stability, as well as sudden jumps occurring in ablation efficiency depending on the number of pulses in burst with the response to changes of the control parameter of peak laser fluence in the pulse. The modeling results using new recurrence relation showed stable and bi-stable ablation efficiency depending on burst fluence and the number of pulses, which agreed well with experimental data. The extremely efficient laser ablation has been achieved by optimizing the shielding effect using three pulses in the burst.

High-Throughput Additive Manufacturing of Continuous Carbon Fiber-Reinforced Plastic by Multifilament.

Additive manufacturing (or 3D printing) of continuous carbon fiber-reinforced plastics with fused deposition modeling is a burgeoning manufacturing method because of its potential as a powerful approach to produce lightweight, high strength and complex parts without the need for a mold. Nevertheless, it cannot manufacture parts rapidly due to low throughput. This paper proposes a high-throughput additive manufacturing of continuous carbon fiber-reinforced plastics by multifilament with reference to fiber tape placement. Three filaments were fed and compaction printed simultaneously by a robotic manufacturing system. The coupled thermal-mechanical model of the filament deformation during printing was developed to eliminate the initial interval between the filaments and improved mechanical properties. Furthermore, the mathematical relationship between filament deformation and printing parameters consisting of printing temperature, printing speed and roller pressure was proposed using response surface methodology with the line width as the response. The tensile tests demonstrate that the tensile properties of printed parts are positively correlated with the line width, but not infinitely improved. The maximum tensile strength and tensile modulus are 503.4 MPa and 83.11 Gpa, respectively, which are better than those obtained by traditional methods. Void fraction and scanning electron microscope images also reveal that the appropriate line width achieved by the reasonable printing parameters contributes to the high-throughput multifilament additive manufacturing of continuous carbon fiber-reinforced plastics. The comparison results indicate that the high-throughput multifilament additive manufacturing proposed in this paper can effectively improve the speed of continuous carbon fiber-reinforced plastics additive manufacturing without degrading the mechanical performance.

At the interface of asymptotics, conformal methods and analysis in general relativity.

This is an introductory article for the proceedings associated with the Royal Society Hooke discussion meeting of the same title which took place in London in May 2023. We review the history of Penrose's conformal compactification, null infinity and a number of related fundamental developments in mathematical general relativity from the last 60 years. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Evolutionary modeling approach based on multiobjective genetic programming for strip quality prediction

In the iron and steel industry, hardness is one of the key indicators of strip quality in the continuous annealing production line (CAPL). However, the complex production process and the strong coupled nonlinearity between process parameters make it difficult to develop accurate mechanism models and pose a challenge for data-driven modeling approaches. More importantly, most of the data-driven learning methods lack interpretability and cannot characterize the mathematical relationship between process parameters and product quality, which in turn makes it extremely hard to understand the process mechanism. Therefore, this paper proposes an interpretable modeling approach (IMA) based on feature decomposition and ensemble to construct interpretable analytical models between process parameters and strip quality. In the IMA, a data-mechanism fusion-based feature decomposition (DM_FD) method is first applied to cope with high-dimensional input feature problems. Then, an improved multiobjective genetic programming algorithm (iMOGP) is developed to construct interpretability sub-models. Finally, a sparse optimization ensemble method (SOE) is used to integrate the sub-models to achieve interpretability and good generalization. Experimental results based on practical strip data demonstrate that the proposed IMA can cope well with high-dimensional input features and achieve model interpretability compared with commonly used machine learning methods and genetic programming (GP)-based modeling methods while ensuring better accuracy and generalization.

Error mechanism analysis and accuracy enhancement method for circular gratings angle-measuring system

In order to raise the accuracy of the circular gratings angle-measuring system, a comprehensive error model is established, and a novel compensation method is proposed. Firstly, many error sources, including the rotary errors of axis system, the installation eccentricity and the inclination of the circular grating disc, the graduation line error of the circular grating disc, and the installation error of the multiple readingheads of the high precision circular gratings angle-measuring system are comprehensively taken into consideration. Secondly, the position and attitude errors are propagated to the angular position outputs, thus the mathematical relationships between error sources and the angular position error are established. Aiming at multiple readingheads system, the mechanism of influence of the error sources on the angular position error is unveiled, and the mechanical error identification and reduction method is proposed. After that, a checking system with a 23-mirrored polygon and an autocollimator is set up, and the complete combinatorial method of separating the angular position errors from the readouts of the autocollimator precisely is used. Finally, the angular position error compositions are separated by the harmonic analysis method, and the concerned error coefficients are compensated. The experimental results show that the nominal indication interval value of the angular position errors is reduced to 0.23" by the mechanical error reduction method, and then the nominal indication interval value of the angular position errors is reduced to 0.12" by software measurement method. And the two sets’ repeatability of the separated biases of the 23-mirrored polygon before and after compensating is 95%.

Comparative Analysis of Transformation Techniques to Reduce Multi-Collinearity Using Real GDP of Some Transport Variables that Contributes to Economic Growth in Nigeria

This study was carried out to compare transformation techniques of reducing multi-collinearity in linear regression using real GDP of some transport variables that contributes to economic growth in Nigeria from 1991-2020. The multiple regression model was used to establish the mathematical relationship between real GDP as the dependent variable and road transport, rail and pipeline transport, water transport, air transport and post and courier service as independent variables. In this work, regression was carried out on the original data to compare the number of independent variables significant with the same data for which the presence of multi-collinearity was reduced using inverse transformation and regression repeated. The results showed that; for original data 1 variable is significant, R2 = 93% and the number of independent variables with high VIF that is above 10 are two (road transport and water transport). Furthermore, the result for the reduce multi-collinearity inverse transformation reviewed that 5 variables were significant, R2 =95.2% and the number of independent with high VIF that is above 10 is three (road transport, air transport and post and courier service). Conclusively, the result from inverse transformation gives the best, using R2 and significant variables as basis for comparison. However useful recommendation was given based on the findings.

The Language of Spheres in Physics

Physical laws manifest themselves through the amalgamation of mathematical symbols, numbers, functions, geometries, and relationships. These intricate combinations unfold within a mathematical model devised to capture and represent the “objective reality” of the system under examination. In this symbiotic relationship between physics and mathematics, the language of mathematics becomes a powerful tool for describing and predicting the behavior of the physical world. The language used and the associated concepts are in a perpetual state of evolution, mirroring the ongoing expansion of the phenomena accessible to our scientific understanding. In this contribution, written in honor of Richard Kerner, we delve into fundamental, at times seemingly elementary, elements of the mathematical language inherent to the physical sciences, guided by the overarching principles of symmetry and group theory. Our focus turns to the captivating realm of spheres, those strikingly symmetric entities that manifest prominently within our geometric landscape. By exploring the interplay between mathematical abstraction and the tangible beauty of symmetry, we seek to deepen our understanding of the underlying structures that govern our interpretation of the physical world.

Optimization of Brewing Process Vine Tea and Flavor Analysis of Different Brewing Processes

As a traditional Chinese medicine with the dual use of medicine and food, vine tea has been utilized for centuries as a daily health beverage due to its various health benefits. The primary component responsible for its biological activity is its total flavonoid content. While numerous studies have been conducted on the extracts of vine tea, research has primarily centered on its ingredients and effects, with limited focus on its sensory characteristics. As it has the problems of the undesirable flavor of bitterness and astringency, the development of the vine tea industry is limited. Furthermore, there is currently no standardized brewing process for vine tea. Therefore, it is essential to find a brewing method that is universally applicable and balances a high level of active ingredients with a rich sensory experience. In this study, we conducted a comprehensive investigation into the impact of various water qualities, tea-water ratios, brewing temperatures, steeping times, and frequencies on the total flavonoid content. To investigate the differences between samples from different brewing methods, one-factor experiments (n = 20) and orthogonal experiments were conducted. The total flavonoid content was measured using UV spectrophotometry. To thoroughly examine the flavor compounds, we employed mathematical relationship analysis, radar-gram analysis, principal component analysis (PCA), and odor fingerprinting. By integrating these methods with the electronic tongue and electronic nose, we were able to compare the flavor variances between samples and offer a concise analysis of the flavor compounds. The brewing process is influenced by several factors, with the tea-to-water ratio having the most significant impact, followed by brewing temperature and then brewing time. Electronic tongues and electronic noses can distinguish vine tea samples with different brewing processes to a certain extent. The optimal brewing conditions determined in this study were pure water, a 1 : 50 tea-to-water ratio, an 85°C brewing temperature, an 8-minute brewing time, and a single brew cycle. These conditions produced the highest total flavonoid content in the vine tea, at 326.8 mg/g, and the best flavor quality. This study offers novel approaches to standardize the brewing process of vine tea, along with valuable perspectives for enhancing the sensory attributes of the tea. It can serve as a benchmark for the industry’s future growth in the canning of vine tea, facilitating the creation of diverse vine tea drinks.

GNSS multipath error in the context of companion robots’ reliability

A companion robot is capable of performing a variety of activities and thus supporting the elderly and people withdisabilities. It should be able to overcome obstacles on its own, respond to what is happening around it in real-time, andcommunicate with its surroundings. It is particularly important to pay attention to these issues, as a companion robot is likely tobecome a participant in traffic. The aim of the research is to develop a mathematical model that takes into account the use of twonavigation solutions in the companion robot. Thanks to this, it will be possible to use the obtained mathematical relationships tocompare various types of navigation and make a rational choice, enabling the implementation of the assumed activities in aspecific external environment. What is new in this article is the analysis of several navigation methods and the presentation ofresearch carried out in real time using an actual robot.

A Small-Signal Stability Analysis Method Based on Minimum Characteristic Locus and Its Application in Controller Parameter Tuning

A small-signal stability analysis method based on minimum characteristic locus (MCL) was recently proposed by the authors. The method is found to be particularly useful in analyzing various oscillation modes in a unified way. The present work continues this line of research. After providing a brief review of the basic principle of MCL method, the paper proceeds to show the mathematical relationship between the stability margin obtained using an MCL method, and that obtained by traditional modal analysis method. Next, a sensitivity analysis procedure is proposed to study quantitatively the effects of control parameters on the stability of the system. Third, the properties of the so-called exciter mode are explained in detail, in particular, explicit formulas of exciter mode frequency and power system stabilizer (PSS) critical gain are derived. Finally, taking PSS4B as an example, a controller parameter tuning method is proposed. The effectiveness and practicability of the proposed method are verified by case studies of Kundur's four-machine-two-area system and China's Yunnan power grid.

Two Intrinsic Formulae Generated by the Jones Polynomial

This paper notes and derives two simple, intrinsic mathematical relations generated by the Jones polynomial.

Coral geometry and why it matters.

Clonal organisms like reef building corals exhibit a wide variety of colony morphologies and geometric shapes which can have many physiological and ecological implications. Colony geometry can dictate the relationship between dimensions of volume, surface area, and length, and their associated growth parameters. For calcifying organisms, there is the added dimension of two distinct components of growth, biomass production and calcification. For reef building coral, basic geometric shapes can be used to model the inherent mathematical relationships between various growth parameters and how colony geometry determines which relationships are size-dependent or size-independent. Coral linear extension rates have traditionally been assumed to be size-independent. However, even with a constant calcification rate, extension rates can vary as a function of colony size by virtue of its geometry. Whether the ratio between mass and surface area remains constant or changes with colony size is the determining factor. For some geometric shapes, the coupling of biomass production (proportional to surface area productivity) and calcification (proportional to volume) can cause one aspect of growth to geometrically constrain the other. The nature of this relationship contributes to a species' life history strategy and has important ecological implications. At one extreme, thin diameter branching corals can maximize growth in surface area and resource acquisition potential, but this geometry requires high biomass production to cover the fast growth in surface area. At the other extreme, growth in large, hemispheroidal corals can be constrained by calcification. These corals grow surface area relatively slowly, thereby retaining a surplus capacity for biomass production which can be allocated towards other anabolic processes. For hemispheroidal corals, the rate of surface area growth rapidly decreases as colony size increases. This ontogenetic relationship underlies the success of microfragmentation used to accelerate restoration of coral cover. However, ontogenetic changes in surface area productivity only applies to certain coral geometries where surface area to volume ratios decrease with colony size.

Study on potential application of brightness temperature models in passive remote sensing

In the modern science and practice of remote sensing of any surface, including various types of land cover, radio systems are widely used. Due to low energy consumption and high stealth, the most promising direction of technical tools for remote sensing is the development of passive remote sensing systems. To estimate the specific parameters of the surface, information on the relation between the signals of the thermal radiation and these parameters is needed. Therefore, it is relevant to study and analyze the existing types of such relations that connect the brightness temperature and the parameters of the Earth’s surfaces; this is the subject matter of the study in this paper. The goal of this study is to analyze existing relation models and study the conditions of their application in terms of operating frequencies, geometric characteristics, type of surface coverage (with vegetation, without vegetation, etc.), as well as the method of obtaining mathematical relations (electrodynamic models or empirical models). Due to the large number of different types of earth surface coverings and different types of surface models, in order to simplify remote sensing tasks for passive systems, as well as to improve the results of modeling or experiments, the following tasks are formulated: to study and classify existing models of the relation between surface parameters and brightness temperature, and to calculate and analyze the potential accuracy of estimating surface parameters by passive remote sensing systems. The following results were obtained: the impact of the atmosphere on the surface’s brightness temperature, described by the electrodynamic flat model, for the operating frequencies 75–110 GHz of the radiometer was analyzed. The impact of the atmosphere on the brightness temperature appears at sighting angles greater than 50° at horizontal polarization and greater than 80° at vertical polarization. According to the results, it can be said that the estimation accuracy of the imaginary part of the complex permittivity decreases with an operating wavelength decrease, and the estimation accuracy of the real part is the opposite. Conclusions. The results of simulation modeling show that for radiometers with the above-mentioned operating frequencies, it is possible to estimate the real permittivity or only the real component separately in the case of describing the surface with a complex permittivity parameter. It is recommended that the study be conducted at vertical and horizontal polarizations, without and with the impact of the atmosphere. To estimate the surface conductivity (the imaginary part of the complex permittivity of a surface), systems with lower operating frequencies are required. Thus, this study presents an analysis of existing models, calculates the potential accuracy of parameters estimation when using these models, and provides recommendations for practical experiments or modeling.

A unified empirical method for predicting both vertical and horizontal ground displacements induced by tunnel excavation

An empirical model for describing the soil movements induced by tunneling is proposed, then the mathematical relationship between horizontal displacement and vertical displacement is obtained. By analyzing 25 sets of data from field engineering and 35 sets of data from model tests, the formula for the maximum settlement Sv, max( z) is optimized to adapt to different ground conditions. The Modified Gaussian function is developed by introducing the existing width coefficient of the settlement trough i( z) and the optimized Sv, max( z) to describe the surface and subsurface vertical displacement. Subsequently, the formula for the horizontal displacement is derived based on the Modified Gaussian function. H( z) representing the position of the oriented point of the soil movement is a variable in the formula for horizontal displacement. Based on measured results, a logarithmic function and a quadratic polynomial function are proposed to describe the variation of H( z) with depth in the clay stratum and sand stratum, respectively. Then, the rationality of the proposed method is validated by 4 sets of in-situ data and 4 sets of test data. Finally, the implementation process of the proposed method is illustrated, and the inversion results of the ground displacement field in Beijing Metro Line 12 are presented.

Accelerated prediction of stacking fault energy in FCC medium entropy alloys using multilayer perceptron neural networks: correlation and feature analysis

A multilayer perceptron neural networks (MLPNN) model is developed for robust and quick prediction of stacking fault energy (SFE) to overcome the challenges faced in the calculation of SFE via experimentation and atomistic calculations in FCC medium entropy alloys (MEA). The present investigation employs a three-step hybrid feature selection approach to obtain a comprehensive understanding of the prominent features that influence the SFE, as well as the interrelationships among these features. The feature space encompasses various features related to composition, lattice stability, and elemental properties, of MEAs. The findings indicate that the estimation of SFE relies on five crucial factors: temperature, lattice stability, specific heat, ionization energy, and Allen electronegativities. Furthermore, a mathematical relationship for the estimation of the SFE is derived, considering the various influencing and prominent factors. Consequently, the MLPNN model for robust SFE prediction in MEAs is developed and the performance is evaluated using R 2 scores, with values of 0.87 and 0.85 obtained for the training and testing datasets, respectively. This efficient strategy introduces a novel opportunity for the engineering of SFE in the extensive range of alloy chemistry of MEAs, enabling the quick prediction of SFE, and facilitating the systematic exploration of new alloys for the development of mechanisms that may accommodate deformation through octahedral/partial slip, twinning, and/or transformation.

Estimation of multiple cracks interaction and its effect on stress intensity factors under mixed load by artificial neural networks

One of the goals of fracture mechanics in the analysis of problems involving cracks is to investigate crack growth and propagation. Each crack grows and propagates in the environment based on its geometry and loading conditions, which can occur under pure mode or a combination of modes. The growth and propagation pattern of the crack in these conditions aligns with the fracture mode. In the presence of multiple cracks in the environment, the growth and propagation paths of these cracks differ from the isolated crack case, where they can either grow independently under certain conditions or merge together. Nowadays, extensive research has been conducted on cracks and their mathematical relationships with respect to the geometry and loading conditions of various components. Since in many cases, a large number of cracks may occur in proximity to each other, the intensification and protective effects of each crack on others and their impact on the overall behaviour of components are of great importance. This study focuses on investigating the behaviour of interaction and the stress intensity factor for an infinite plate with parallel adjacent cracks. Remote stress boundary conditions in the form of tensile and compressive stresses have been applied to this plate, and using the J-integral and the finite element method with the Abaqus finite element analysis software, this subject has been examined. The obtained results indicate that depending on the type of vertical and horizontal distances and the angle of their placement between the adjacent crack and the main crack in the first, second, and mixed modes, the adjacent crack can have intensifying, protective, or neutral effects. Furthermore, a good estimation of the secondary crack's influence on the main crack has been provided using a neural network method (MLP), which creates an appropriate weight function for the upcoming sample.

Study on the rock physical mechanical properties evaluation of tight oil reservoir in Chang 7 member, Longdong area, Ordos Basin, China

The study aims to investigate the rock mechanical properties of the Chang 7 member tight oil reservoir in the Longdong region of the Ordos Basin, China, with the goal of enhancing the efficiency of oilfield development. Despite numerous contributions in the field of rock mechanics, challenges persist in reconciling experimental results with actual geological conditions and achieving comprehensive understanding of rock mechanical properties in tight oil reservoirs. To address this, a variety of experimental methods were employed to systematically assess the mechanical properties of the target reservoir. Rock density was measured using volumetric methods, tensile strength was evaluated through the Brazilian test, uniaxial and triaxial compression tests were conducted to assess rock mechanics properties, and dynamic elastic modulus and Poisson’s ratio were obtained via sonic velocity measurements. Furthermore, differential strain analysis and imaging log analysis were employed to determine the magnitude and direction of geostress. The results revealed that fine-grained sandstone exhibited higher rock density and relatively higher tensile strength, while muddy siltstone exhibited comparatively lower values in these aspects. Analysis of the influence of confining pressure on rock mechanics properties demonstrated a strong positive correlation between compressive strength and Young’s modulus with confining pressure, while Poisson’s ratio exhibited more irregular variations. Additionally, a mathematical relationship between dynamic and static rock mechanical parameters was established. Lastly, based on the characteristics of geostress, reliable foundations for optimizing hydraulic fracturing and wellbore layout were provided. This study has enriched and complemented the investigation of rock mechanical properties in tight reservoirs, offering vital parameters and theoretical support for the development of tight oil reservoirs. This bears significant importance in enhancing petroleum extraction efficiency and optimizing engineering design.

Airborne Transmission of SARS-CoV-2: The Contrast between Indoors and Outdoors

COVID-19 is an airborne disease, with the vast majority of infections occurring indoors. In comparison, little transmission occurs outdoors. Here, we investigate the airborne transmission pathways that differentiate the indoors from outdoors and conclude that profound differences exist, which help to explain why SARS-CoV-2 transmission is much more prevalent indoors. Near- and far-field transmission pathways are discussed along with factors that affect infection risk, with aerosol concentration, air entrainment, thermal plumes, and occupancy duration all identified as being influential. In particular, we present the fundamental equations that underpin the Wells–Riley model and show the mathematical relationship between inhaled virus particles and quanta of infection. A simple model is also presented for assessing infection risk in spaces with incomplete air mixing. Transmission risk is assessed in terms of aerosol concentration using simple 1D equations, followed by a description of thermal plume–ceiling interactions. With respect to this, we present new experimental results using Schlieren visualisation and computational fluid dynamics (CFD) based on the Eulerian–Lagrangian approach. Pathways of airborne infection are discussed, with the key differences identified between indoors and outdoors. In particular, the contribution of thermal and exhalation plumes is evaluated, and the presence of a near-field/far-field feedback loop is postulated, which is absent outdoors.