Mathematical Expressions Research Articles

Study on calculation model of deflection of prefabricated hole bulkhead under the action of internal explosion

In order to study the plastic deformation law of ship plate frames under the coupling action of internal explosion load and fragment, first, the calculation formula of plastic deformation deflection of the non-porous plate is derived through the law of conservation of energy. According to the numerical calculation results, the influence of the opening position and opening area of the target plate on the deflection gain of the target plate is further explored. The mathematical expression form of the opening area and opening position on the deflection gain of the prefabricated target plate is fitted, and then, the deflection calculation model of the prefabricated target plate is established. It can theoretically predict the deflection of the bulkhead under cabin explosion and has a certain application reference value for ship cabin anti-explosion design and internal explosion damage assessment.

Deadline-constrained security-aware workflow scheduling in hybrid cloud architecture

A hybrid cloud is an efficient solution to deal with the problem of insufficient resources of a private cloud when computing demands increase beyond its resource capacities. Cost-efficient workflow scheduling, considering security requirements and data dependency among tasks, is a prominent issue in the hybrid cloud. To address this problem, we propose a mathematical model that minimizes the monetary cost of executing a workflow and satisfies the security requirements of tasks under a deadline. The proposed model fulfills data dependency among tasks, and data transmission time is formulated with exact mathematical expressions. The derived model is a Mixed-integer linear programming problem. We evaluate the proposed model with real-world workflows over changes in the input variables of the model, such as the deadline and security requirements. This paper also presents a post-optimality analysis that investigates the stability of the assignment problem. The experimental results show that the proposed model minimizes the cost by decreasing inter-cloud communications for dependent tasks. However, the optimal solutions are affected by the limitations that are imposed by the problem constraints.

Research on the Cultivation Strategy of Primary School Students' Mathematical Language Expression Ability

Learning mathematics is a process of learning mathematical language, which is the external manifestation of mathematical thinking, the carrier of mathematical knowledge and mathematical ideas and methods, and at the same time, mathematical knowledge also needs mathematical language as a medium to disseminate and communicate. In the study of mathematics, whether students can learn mathematics well is related to whether students have good mathematical language skills, but many studies can show that there are obvious deficiencies in the current mathematical language expression skills of primary school students. Therefore, teachers need to strengthen students' thinking training and good study habits, so as to further cultivate students' mathematical language expression ability and improve students' core literacy. To this end, this paper will elaborate on the concept and classification of mathematical language, as well as the cultivation strategies of primary school students' mathematical language expression ability.

Understanding stacking fault energy of NbMoTaW multi-principal element alloys by interpretable machine learning

Stacking fault energy (SFE) is a crucial property influencing the deformation mechanisms of multi-principal element alloys (MPEAs). However, experimentally measuring SFE and exploring composition space of MPEAs is challenging. This study explores the SFE in NbMoTaW by integrating machine learning force fields (ML-FF), molecular dynamics simulations (MD), neural networks, and symbolic regression (SR) methods. A SFE dataset containing 2000 different NbMoTaW compositions were generated by combining ML-FF and MD. Then the neural-network model was developed for SFE prediction with an accuracy of 0.981 and a mean absolute error of 0.020. Using SR, the valence electron concentration (VEC), average shear modulus (G), radii gamma (γ) were identified as the key descriptors to determine SFE with the relationship of SFE=0.307·VEC+0.352·(G+γ). This work demonstrated a significant impact of chemical composition on SFE and established an accurate mathematical expression for SFE prediction, enhancing the understanding and alloy design of MPEAs.

LOGICAL-PROBABLIC MODELING OF FAILURES OF GEODESIC NETWORKS

The credibility and reliability of the results of geodetic and cadastral works depends mainly on the accuracy of the instruments, the qualification of the observer, his professionalism and the reliability of the points of the geodetic network (GN). As a rule, in most cases, there is a well-founded choice of geodetic tools suitable for accuracy and the involvement of a highly qualified geodetic engineer to perform certain types of geodetic work. However, certain questions may arise regarding the determination of starting points for the creation of a reliable geodetic survey justification. The issue of selecting reliable geodetic points for performing engineering and geodetic works based on the application of reliability assessment methods is an urgent scientific problem today. The purpose of the work is to establish the optimal reliability assessment method based on the analysis of existing ones. Adapt this method to solve the problem of assessing the reliability of geodetic networks as one of the most important parameters characterizing its condition. Assess the reliability of the Dnipro geodetic network by constructing a fault tree. Calculate the dynamics of the intensity of GN failures during the 100-year period of its existence. Calculate the appropriate periodicity of surveys (monitoring) of GN depending on the intensity of failures of elements (geodetic points). Methodology. Analysis of international and national standards for reliability assessment, Standards of Ukraine, where the methods of system reliability assessment are given. Review of methods for assessing GN reliability. The deductive method with the construction of a geodetic network failure tree has been applied. This method is based on a logic-probabilistic model of cause-and-effect relationships of system failures with failures of its elements and other events. Scientific novelty. The optimal method of assessing the reliability of GN has been established. A GN failure tree has been built in the TopEvent FTA software tool. The reliability of the geodetic network has been evaluated, the intensity of failures at various stages of its operation has been calculated. Mathematical expressions of the failure functions of the geodetic network have been described by applying Boolean functions. Practical value. International and national regulatory documents and Standards of Ukraine of reliability assessment have been analyzed. In Standarts the methods of system reliability assessment have been given. The performed assessment of the reliability of the state of the geodetic network of the city of Dnipro helps to choose stable geodetic points for the creation of local geodetic networks and survey justification. Reasonable periodicity of monitoring for the survey of geodetic points has been substantiated. The use of reliable points of geodetic networks will allow obtaining reliable observation data, which will significantly increase the quality of engineering and geodetic works. Results. The functioning of the geodetic network of the city of Dnipro on the constructed sequential tree of failures has been considered. Boolean functions to describe the mathematical expressions of the failure functions of the geodetic network, which includes 5 and 10 elements have been used.

On the Asymptotic Network Indices of Weighted Three-Layered Structures with Multi-Fan Composed Subgraphs

In this paper, three sorts of network indices for the weighted three-layered graph are studied through the methods of graph spectra theory combined with analysis methods. The concept of union of graphs are applied to design two sorts of weighted layered multi-fan composed graphs, and the accurate mathematical expressions of the network indices are obtained through the derivations of Laplacian spectra; furthermore, the asymptotic properties are also derived. We find that when the cardinalities of the vertices on a sector-edge-link tend to infinity, the indices of FONC and EMFPT are irrelevant with the number of copies of the fan-substructure based on the considered graph framework.

Multi-objective optimization design of rail grinding profile in Curve Section of Subway based on wear Evolution and representative worn profile

Curve rail grinding has always been one of the key points of subway maintenance and repair. The appropriate grinding can effectively reduce wear. A multi-objective optimization design method for grinding profiles in curved sections is proposed. Firstly, representative grinding profiles are selected as the initial population using the dynamic time regularization algorithm (DTW). Then, the optimal design range is determined based on wear characterization analysis, and mathematical expressions of wheel profiles are chosen as design variables to establish a parametric model. Next, the prediction model considering the evolution of wheel wear is incorporated into the multi-objective function, and objective function adaptive weight adjustment coefficient factors are introduced to establish the multi-objective optimization model for wheel profiles. The Latin hypercubic sampling method is employed to establish the RBF agent model for simulation calculation, and the optimization design of wheel profiles is carried out using the TS-NSGA-II multi-objective algorithm.

Physical-layer security for primary users in 5G underlay cognitive radio system via artificial-noise-aided by secondary users

The advent of the Internet of Things (IoT) has revolutionized connectivity by interconnecting a vast array of devices, underscoring the critical need for robust data security, particularly at the Physical Layer Security (PLS). Ensuring data confidentiality and integrity during wireless communications poses a primary challenge in IoT environments. Additionally, within the constrained frequency bands available, Cognitive Radio Networks (CRNs) has emerged as an urgent necessity to optimize spectrum utilization. This technology enables intelligent management of radio frequencies, enhancing network efficiency and adaptability to dynamic environmental changes. In this research, we focus on examining the PLS for the primary channel within the underlying CRNs. Our proposed model involves a primary source-destination pair and a secondary transmitter-receiver pair sharing the same frequency band simultaneously. In the presence of a common eavesdropper, the primary concern lies in securing the primary link communication. The secondary user (SU) acts as cooperative jamming, strategically allocating a portion of its transmission power to transmit artificial interference, thus confusing the eavesdropper and protecting the primary user's (PU) communication. The transmit power of the SU is regulated by the maximum interference power tolerated by the primary network's receiver. To evaluate the effectiveness of our proposed protocol, we develop closed-form mathematical expressions for intercept probability ( P int ) and outage probability (OP) along the primary communication link. Additionally, we derive mathematical expressions for OP along the secondary communications network. Furthermore, we investigate the impact of transmit power allocation on intercept and outage probabilities across various links. Through both simulation and theoretical analysis, our protocol aims to enhance protection and outage efficiency for the primary link while ensuring appropriate secondary OP, thereby contributing to advancements in IoT security and spectrum management.

Analysis of hemodynamic parameters on two-layered blood flow in a curved artery

Arterial stenosis is the thickening of the arterial wall due to the growth of aberrant tissues that prevent adequate blood flow in the human circulatory system and induces cardiovascular diseases. Mild stenosis, may lead to serious or permanent damage if remains uncured. There are differences in the curvature response and material composition between the outer layers and the core. There are several locations in blood vessels where they are curved, which affects the blood flow and shear stress. The Navier-Stokes equation in the cylindrical polar coordinate system has been extended by incorporating curvature term in two-layered blood flow along the axial direction with appropriate boundary conditions. Mathematical expressions for hemodynamic parameters such as velocity profile, volumetric flow rate, pressure drop, and shear stress have been calculated analytically in the case of a curve artery with stenosis. Moreover, we have analyzed the effect of stenosis on different hemodynamic parameters with the variation of core and peripheral-layer viscosity and curvature. Flow quantities are affected by the habitancy of stenosis and stipulate different blood flow behavior in both layers in the case of curved artery. This modeling technique may help researchers in medicine, mathematical biology, and bio-engineering.

Foam-mat drying of black soldier fly larvae: optimisation of foaming, drying kinetics, and powder characteristics

Abstract Desirable conditions for foaming, drying kinetics, and characteristics of black soldier fly larvae (BSFL) powders were studied. Pre-pupal BSFL was fractionated under wet conditions. A pasty liquid, free of fibrous material, was used and adjusted to a moisture content suitable (88%) for foaming. The influence of emustab (: 1-2 g/100 g), maltodextrin (: 3-5 g/100 g), and whipping time (: 9-15 min) on the BSFL foam properties were analysed according to a central composite design (23) with added axials. Optimal conditions for foam preparation (: 1.5 g/100 g, : 4 g/100 g, and : 15 min) revealed excellent foam properties (density: 409.69 kg/m3, stability: 72.49%, porosity: 60.51%, and overrun: 153.64%). For the drying process, foamed (optimum conditions) and liquid (no additives) samples were compared at different temperatures (40, 60, and 80 °C) to determine the drying kinetics and powder quality. For kinetic modelling, five mathematical expressions were fitted to the experimental data. The Newton, Henderson & Pabis, Page, Logarithmic, and Midilli models were adequate () to represent the drying kinetics of the BSFL foams and liquids. However, the mathematical models that best described the kinetic curves were those proposed by Midilli and Page ( and ). In addition, foam dried at 60 °C exhibited excellent powder quality characteristics. Thus, this study provides new alternatives to process protein-rich food and feed (above 54.82%) in short periods with positive economic, social, and environmental impacts.

Effect of coal suspension concentration and gas-air medium temperature on liquid droplets collisions

Relevance. Understanding the mechanisms of liquid droplets interaction with each other is important for many industrial and technical applications related to solving a range of problems like slag removal in a high-temperature environment, obtaining components of the desired fraction in the food industry, etc. Aim. Establishment of the main patterns of suspension droplets interaction in a gas-air environment with temperature variation. Methods. Using shadow high-speed video recording, the main patterns of the binary collision of suspensions droplets were determined. The paper introduces the results of experimental studies of the characteristics of collisions of droplets of coal-water suspensions in a gas-air environment with a temperature of 90–120°C. Parameters of the generated droplets: radius 1.0–2.2 mm, velocity 0.5–2.0 m/s. Results and conclusions. The authors have determined the modes of collision of droplets of suspensions (coagulation and separation) and the main characteristics of secondary fragments and constructed the maps of the modes of interaction of drops of suspensions with each other when varying the concentration of solid particles in the suspension, the temperature of the gas-air environment and the time the target drop spent in a gas-air environment with an elevated temperature. The conditions were established for the coagulation of droplets, as well as their intensive secondary grinding to intensify their drying, ignition and combustion in boiler furnaces. It was established that an increase in the temperature of the gas-air environment leads to a significant change in the size and properties of droplets, as well as to the occurrence of oscillatory phenomena in the system. It is substantiated that collision of droplets of suspensions in a gas-air environment with elevated temperature is complex and multi-parametric. Its characteristics depend on a combination of factors (surface tension and liquid viscosity, size and shape of droplets, speed of their movement, density and viscosity of gas-air environment). The authors obtained mathematical expressions to describe the boundaries of the modes of the studied processes and schemes for using the results obtained in order to increase the efficiency of the corresponding technological processes.

Droplet dynamics under shear flow on surfaces with different wettability

Droplet dynamics on surfaces under shear flow, which affects the efficiency of water condensate removal in condensation, are significant for investigation. The influencing factors include air velocity, droplet volume and surface wettability. Three smooth surfaces (HI, HO-POTS, and HO-PDMS) and three semi-textured surfaces (HI+SHO, HO-POTS+SHO, and HO-PDMS+SHO) are fabricated. Experiments are performed for droplets with different volumes under different air velocities. A centrifugal fan is used to generate channel flow, and a high-speed camera is utilized to capture droplet behaviors. The results show that motion of droplets with small deformation on smooth surfaces can be divided into two stages (Acceleration and Constant Speed). Critical velocity for the onset of droplet motion, as well as the velocity that droplet can reach at Constant Speed Stage, is affected by air velocity, droplet volume and surface wettability. The mathematical expression of retention force is derived, and forces are calculated to verify the force balances at Constant Speed Stage. Droplet behaviors on semi-textured surfaces are much different from those on smooth surfaces, especially when crossing the boundaries between smooth and textured areas. The HO+SHO surfaces display a better performance of droplet removal compared with the HI+SHO surface.

Droplet Microfluidic Method for Estimating the Dynamic Interfacial Tension of Ion-Crosslinked Sodium Alginate Microspheres.

The research focuses on optimizing the production of hydrogel microspheres using droplet microfluidics for pharmaceutical and bioengineering applications. A semiempirical method has been developed to predict the dynamic interfacial tension at the interface of ion-cross-linked sodium alginate microsphere-sunflower oil modified with glacial acetic acid and Tween 80 surfactant. These microspheres are produced in a small-scale coaxial device that is manufactured using affordable DLP/LCD 3D printing technology with a transparent photopolymer. The method was tested to design the minireactor in the device, which allows for the production of fully cross-linked microspheres that are ready for use at the output of the reactor without additional cross-linking steps in the microsphere collector. The mathematical expression for estimating the interfacial tension at the moment of formation of a hydrogel microsphere includes the Reynolds number for a two-phase liquid, the Ohnesorge number, and the surface tension at the liquid-air interface for continuous medium flow (modified oil). The reliability of the prediction is confirmed for continuous medium and dispersed phase flow rates of 0.8-3.2 and 0.01-0.08 mL/min, respectively. The evolution of the interfacial tension from the moment the microspheres formed and the estimated ultimate interfacial tension in a cross-linked hydrogel-modified oil system contributed to the reliable determination of the linear size of a minireactor. The ultimate interfacial tension of 76.5 ± 0.3 mN/m was determined using the Young-Laplace equation, which is based on measuring the surface free energy of the hydrogel as soft matter using the Owens-Wendt method. Additionally, the equilibrium static contact angle of the fully cross-linked hydrogel surface wetted with oil is measured using the sessile drop method. From a practical perspective, a method for optimizing and streamlining the high-tech manufacturing of cross-linked polymer microspheres and mini- and microchannel devices for use in bioengineering and pharmaceutical applications is suggested.

Алгоритмы оценивания показателей качества функционирования распределенной системы хранения конфиденциальных данных

The article presents algorithms for assessing the quality indicators of distributed data storage systems used to accumulate and process data from automated information systems for various purposes, serving a large number of geographically distributed clients. Mathematical expressions for indicators of confidentiality, availability and cost of data storage are given. The proposed algorithms make it possible to analyze the distribution of information resources among the elements of the data storage system in order to select rational solutions for organizing the storage of confidential information.

Two-Parameter Probabilistic Model and Experimental Research on Micron Particle Deposition

The deposition of micron particles in gas pipelines has always been an important problem in ultra-clean ventilation technology in the modern laser fusion, precision electronics, aerospace, and biomedical fields. Combining the mathematical expression of the migration, collision, and deposition of micron particles in a gas pipeline with a simulation of flow fields, a two-parameter particle probability deposition model based on vinl, θcr and collision probability coefficient PP is established, and the distribution law of particle deposition, considering two deposition targets of the pipe wall and deposition layer, is given. Combined with an experiment on particle migration and deposition in a gas pipeline, an interpretation and verification of the particle deposition distribution law are given, and the difference between the model and experiment is discussed through particle deposition efficiency mass distribution. Studies have shown the following: Under the premise of two kinds of deposition targets, different particle sizes in the gas pipeline present different deposition laws; the deposit morphology is a spot deposit of 10 µm particles and a flake deposit of 40 µm particles; the deposit position shows a uniform distribution and a lower wall dominance; and the deposit concentration area of 40 µm shows a more significant distribution. The results are very important for the selection and optimization of gas pipelines for clean spaces.

Improved MTPA and MTPV Optimal Criteria Analysis Based on IPMSM Nonlinear Flux-Linkage Model

The use of interior permanent-magnet synchronous machines (IPMSMs) is prevalent in automotive and vehicle traction applications due to their high efficiency over a wide speed range. Given the high-power-density requirements of automotive IPMSMs, it is imperative to consider the effect of nonlinearities, such as saturation and cross-coupling, on the motor model. The aforementioned nonlinearities render conventional linear motor models incapable of accurately describing the operating characteristics of the IPMSM, including the maximum torque per ampere (MTPA) trajectory, the flux-weakening (FW) trajectory, and the maximum torque per volt (MTPV) trajectory. With respect to the linear motor model, the nonlinear flux-linkage model is gradually receiving attention from researchers. This modeling method represents the nonlinear behavior of the motor through the direct establishment of a bidirectional mapping relationship between flux-linkage and current. It is capable of naturally incorporating the effects of magnetic saturation and cross-coupling factors. However, the analysis of the current trajectory optimal criteria based on this model has not yet been reported. In this paper, the optimal criteria for the MTPA and MTPV current trajectories are analyzed based on the nonlinear flux-linkage model of IPMSMs. Firstly, the nonlinear flux-linkage model of the tested IPMSM is established by the experimental calibration method. The mathematical analytical expressions of the MTPA and MTPV optimal criteria are then analyzed by constructing and solving optimal problems with different objectives. Finally, the current command table applicable to actual motor control is constructed by calculating the current command for different operating conditions according to the optimal criteria proposed in this paper. The validity and feasibility of the optimal criteria proposed in this paper are verified through experimental tests on different operating conditions.

An Innovative Method for Deterministic Multifactor Analysis Based on Chain Substitution Averaging

The aims of this paper are to present the methodology, derived mathematical expressions for determining the individual factor influences and the adaptation for the conditions of dynamic deterministic factor analysis and the results of the application of the developed new method for deterministic factor analysis, called the averaged chain substitution method. After formulating the concept of the considered approach, all mathematical expressions used to create models containing up to four factor variables are presented and summarized. The scientific novelty of the study is in the obtained new equations for determining the individual factor influences by the method of averaged chain substitution and the method of analogy for five-factor additive or difference-multiplicative and for five-factor additive or difference-multifactor models with an additive or different part in the numerator of the factor model. The presented mathematical expressions accurately and unambiguously quantify the impact of individual factor influences for all types of factor models and thus significantly expand the applicability of the averaged chain substitution method in the theory and practice of financial-economic analysis. The proposed formalization and algorithmization of the evaluation process makes the method easy to apply by all economic and financial analysts for the purposes of deterministic factor analysis. The methodology was applied to perform a dynamic deterministic factor analysis of the total liquidity of Monbat AD and ELHIM-ISKRA AD for the period 2017–2021, based on the consolidated annual financial statements of the companies, available on the website of the Bulgarian Stock Exchange.

Network analysis of students’ conceptual understanding of mathematical expressions for probability in upper-division quantum mechanics

One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well. Published by the American Physical Society 2024

An exact mathematical description of computation with transient spatiotemporal dynamics in a complex-valued neural network

Networks throughout physics and biology leverage spatiotemporal dynamics for computation. However, the connection between structure and computation remains unclear. Here, we study a complex-valued neural network (cv-NN) with linear interactions and phase-delays. We report the cv-NN displays sophisticated spatiotemporal dynamics, which we then use, in combination with a nonlinear readout, for computation. The cv-NN can instantiate dynamics-based logic gates, encode short-term memories, and mediate secure message passing through a combination of interactions and phase-delays. The computations in this system can be fully described in an exact, closed-form mathematical expression. Finally, using direct intracellular recordings of neurons in slices from neocortex, we demonstrate that computations in the cv-NN are decodable by living biological neurons as the nonlinear readout. These results demonstrate that complex-valued linear systems can perform sophisticated computations, while also being exactly solvable. Taken together, these results open future avenues for design of highly adaptable, bio-hybrid computing systems that can interface seamlessly with other neural networks.

Independent variables of pH: Ten Knights of the Hydrogen Ion Kingdom-Part I. A prospective observational study.

CO2, HCO3, SID, and total weak acids have been defined as pH's independent variables. However, according to Gamble, HCO3 should be equal to the difference between the sum of cations and the sum of anions besides HCO3. Therefore, if this mathematical expression is substituted for HCO3 in the Henderson-Hasselbalch equation, all independent variables of pH can be demonstrated. Our aim is to test this theory in this study. This prospective observational study was conducted between 2019 and 2020. All admitted patients to the intensive care unit who were >18 years old were included. Demographic data, blood gas parameters, albumin, magnesium, and inorganic phosphorus levels, and outcomes were recorded twice (at admission and at the 24th hour). The multivariate linear regression model was used to determine pH's independent variables. In the multivariate linear regression model, pH was significantly increased by each unit increase in Na, K, Ca, and Mg (mmol L-1). In contrast, pH was significantly decreased by each unit increase in CO2, Cl, lactate, albumin (g dL-1), inorganic phosphorus (mg dL-1), and the strong ion gap. Ten independent variables can accurately predict the changes in pH. For this reason, all ten independent variables should be separately evaluated when interpreting the acid-base status. With this understanding, all algorithms regarding acid-base evaluation may become unnecessary.