Set Of Equations Research Articles

An analytic method to investigate hemodynamics of the cardiovascular system: Biventricular system.

Previously, we found analytic solutions for single ventricular system based on the lumped parameter model (LPM). In this study, we generalized the method to biventricular system and derived its analytic solutions. LPM is just a set of differential equations, but it is difficult to solve due to time-varying ventricular elastance and high order. Mathematically, there exist no elementary solutions for time-varying equations. It turns out that instead of differential equations, according to volume conservation, a set of algebraic equations can be carried out. The solutions of the set of equations are just physiological states at end of systolic and diastolic phases such as end systolic/diastolic pressure/volume of left ventricle. As a preliminary application, the method is utilized to deduce the hemodynamic effects of VA ECMO. Left ventricular (LV) distension, a serious complication of VA ECMO, is usually attributed to factors such as increased afterload, inadequate LV unloading, reduced myocardial contractility or aortic valve regurgitation (AR), bronchial and Thebesian return in the absence of aortic valve (AoV) opening. Among these, reduced contractility and AR are strongly associated with LV distension. However, in the absence of reduced contractility or AR, it is less clear whether increased afterload or inadequate LV unloading alone can cause LV distension. This leads to the critical question: under what conditions does LV distension occur in the absence of reduced contractility or AR? The analytic formulas derived in this study give conditions for LV distension. Furthermore, the results show that the analytic hemodynamics are coincident with simulated results.

A simplified method for predicting overflow-induced crack propagation in gravity dams using genetic algorithm and material-based model

Cracks are always a serious concern in the stability analysis of gravity dams. One of the main reasons for the initiation of cracks is overflows. In evaluating dam safety under flood conditions, it is crucial to consider the balance between the strength of the dam body and the propagation of cracks. In this context, simplified equations serve as valuable tools in dam design as they offer a quick and efficient way to estimate the behavior of the dam structure. In this study, firstly using a polygonal material-based model (P-MBM), which is improved in considering the softening behavior of joints, the micro-crack propagation in masonry material is investigated. Then, based on performing 81 numerical simulations in a parametric study on the geometrical and geotechnical properties of the dam subjected to overflow, a great database of the behavior of different dams is investigated. Then, using the genetic algorithm, a set of equations is proposed, and their accuracy is validated through UDEC simulations and some theoretical methods from existing literature. For all models, the crack tends to be horizontal at the initial phase. Then, due to increasing the compressive stresses on the downstream side, the fractures tend to the dam toe. The results also indicate that the proposed equations can reasonably determine the behavior of gravity dams and the development of cracks in the dam body. The outcomes highlighted the considerable effects of geometry and geotechnical properties on dictating the trajectory of crack growth.

Harnessing collisional nonlinearity for enhanced harmonic generation by ultraviolet plasmonic nanoparticles.

We investigate the contribution of inelastic electron collisions to nonlinear (NL) dynamics in ultraviolet plasmonic nanoparticles, exploring their potential for harmonic generation. Employing the Landau weak coupling formalism to model radiation-driven electron dynamics in sodium and aluminum, we account for both electron-electron and electron-phonon scattering processes by a set of hydrodynamic equations, which we solve perturbatively to obtain third-order NL susceptibilities. Furthermore, we model high harmonic generation enhanced by localized surface plasmons in nanospheres composed of such poor metals, demonstrating their efficient operation for extreme ultraviolet generation. Our investigation reveals that plasmonic nanospheres composed of sodium and aluminum produce a large field intensity enhancement of ≃103-105, boosting the harmonic generation process. Our findings indicate that poor metals hold great promise for advanced extreme ultraviolet nano-sources with potential applications in nano-spectroscopy.

Modeling uncertainty: the impact of noise in T cell differentiation

BackgroundThe regulatory mechanisms guiding CD4 T cell differentiation are complex and are further influenced by intrinsic cell variability along with that of microenvironmental cues, such as cytokine and nutrient availability.ObjectiveThis study aims to expand our understanding of CD4 T cell differentiation by examining the influence of intrinsic noise on cell fate.MethodologyA model based on a complex regulatory network of early signaling events involved in CD4 T cell activation and differentiation was described in terms of a set of stochastic differential equation to assess the effect of noise intensity on differentiation efficiency to the Th1, Th2, Th17, Treg, and TFH effector phenotypes under defined cytokine and nutrient conditions.ResultsThe increase of noise intensity decreases differentiation efficiencies. In a microenvironment of Th1-inducing cytokines and optimal nutrient conditions, noise levels of 3%, 5% and 10% render Th1 differentiation efficiencies of 0.87, 0.76 and 0.62, respectively, underscoring the sensitivity of the network to random variations. Further increments of noise reveal that the network is relatively stable until noise levels of 20%, where the resulting cell phenotypes becomes heterogeneous. Notably, Treg differentiation showed the highest robustness to noise perturbations. A combined Th1-Th2 cytokine environment with optimal nutrient levels induces a dominant Th1 phenotype; however, removal of glutamine shifts the balance towards the Th2 phenotype at all noise levels, with an efficiency similar to that obtained under Th2-only cytokine conditions. Similarly, combinations of Th1/Treg and Treg/Th17-inducing cytokines along with the removal of either tryptophan or oxygen shift the dominant Th1 and Treg phenotypes towards Treg and Th17 respectively. Model results are consistent with differentiation efficiency patterns obtained under well-controlled experimental settings reported in the literature.ConclusionThe stochastic CD4 T cell mathematical model presented here demonstrates a noise-dependent modulation of T cell differentiation induced by cytokines and nutrient availability. Modeling results can be explained by the network topology, which assures that the system will arrive at stable states of cell functionality despite variable levels of biological intrinsic noise. Moreover, the model provides insights into the robustness of the T cell differentiation process.

Master Curves for Poroelastic Spherical Indentation With Step Displacement Loading

Abstract Theoretical and numerical analyses are conducted to rigorously construct master curves that can be used for interpretation of displacement-controlled poroelastic spherical indentation test. A fully coupled poroelastic solution is first derived within the framework of Biot’s theory using the McNamee–Gibson displacement function method. The fully saturated porous medium is assumed to consist of slightly compressible solid and fluid phases and the surface is assumed to be impermeable over the contact area and permeable everywhere else. In contrast to the cases in our previous studies with an either fully permeable or impermeable surface, the mixed drainage condition yields two coupled sets of dual integral equations instead of one in the Laplace transform domain. The theoretical solutions show that for this class of poroelastic spherical indentation problems, relaxation of the normalized indentation force is affected by material properties through weak dependence on a single-derived material constant only. Finite element analysis is then performed in order to examine the differences between the theoretical solution, obtained by imposing the normal displacement over the contact area, and the numerical results where frictionless contact between a rigid sphere and the poroelastic medium is explicitly modeled. A four-parameter elementary function, an approximation of the theoretical solution with its validity supported by the numerical analysis, is proposed as the master curve that can be conveniently used to aid the interpretation of the poroelastic spherical indentation test. Application of the master curve for the ramp-hold loading scenario is also discussed.

Inertial drag combined with non‐uniform heat generation/absorption effects on the hydromagnetic flow of polar nanofluid over an elongating permeable surface due to the impose of chemical reaction

AbstractThe pivotal role of Brownian and thermophoresis in investigating the flow characteristic of polar nanofluid is important nowadays due to various engineering applications. The enhanced thermal and transport properties in the field of biomedical nanomedicine for hyperthermia treatments, and for enhancing the efficiency of heat exchange processes in cooling electronic devices, heat exchangers, and automotive engines the use of Brownian and thermophoresis is important. Therefore, the present study reveals the importance of Darcy–Forchheimer inertial drag combined with the space‐ and temperature‐dependent heat generation/absorption on the flow of polar nanofluid over an elongating surface. The surface is considered to be permeable for which the hydromagnetic flow in the presence of thermal radiation specifically, Brownian and thermophoresis affects the flow phenomena significantly. The proposed flow model designed with the aforementioned physical properties is standardized into the set of nonlinear ordinary equations by the implementation of suitable similarity rules. Further, the characteristics of various physical quantities are deployed by solving the system by using traditional Rung–Kutta fourth‐order technique. Further, the analysis of several factors is deployed briefly via graphically, and simulation of rate coefficients is presented through tables. The important outcomes of the study are deployed as the inclusion of thermal and solutal buoyancy enhances the velocity distribution, whereas reverse impact is observed for the increasing inertial drag. Also, space‐ and temperature‐dependent heat source augments the temperature profile but Lewis number decelerates the fluid concentration significantly.

A Computational Model of the Secondary Hemostasis Pathway in Reaction Systems

Reaction Systems (RSs) are a computational framework inspired by biochemical mechanisms. An RS defines a finite set of reactions over a finite set of entities (molecules, proteins, etc). Starting from an initial set of entities (the initial state), a computation is performed by applying all reactions to a state in order to produce the following state, giving rise to a sequence of sets of entities. RSs have shown to be a general computational framework whose application ranges from the modeling of biological phenomena to molecular chemistry and computer science. In this paper, we contribute to research on the application of RSs for modeling biological systems. We consider the problem of modeling hemostasis, for which several models have been defined, starting from the 1960s. Previous models are based on sets of ordinary differential equations, while we develop a discrete model in RSs for pathways of the secondary hemostasis. Then, we implement our model in BioReSolve, a computational framework for RSs that we have previously defined which provides tools for the specification and verification of properties. By using the tools in BioReSolve we derive important observations on the model behaviour for hemostasis, and in particular, we study the role of three important inhibitors, verifying that their presence or absence leads to phenomena such as thrombophilia, or thromboembolism, or excessive coagulation, etc. We can also study computationally the causality relations between the molecules involved in the reactions showing which entities play a fundamental role, thus contributing to the design of more effective and specialized drugs. Our work can hence help to show how to model complex biological systems in RSs and derive computationally and biologically relevant properties of the systems.

Relaxation and Noise-Driven Oscillations in a Model of Mitotic Spindle Dynamics

During cell division, the mitotic spindle moves dynamically through the cell to position the chromosomes and determine the ultimate spatial position of the two daughter cells. These movements have been attributed to the action of cortical force generators which pull on the astral microtubules to position the spindle, as well as pushing events by these same microtubules against the cell cortex and plasma membrane. Attachment and detachment of cortical force generators working antagonistically against centring forces of microtubules have been modelled previously (Grill et al. in Phys Rev Lett 94:108104, 2005) via stochastic simulations and mean-field Fokker–Planck equations (describing random motion of force generators) to predict oscillations of a spindle pole in one spatial dimension. Using systematic asymptotic methods, we reduce the Fokker–Planck system to a set of ordinary differential equations (ODEs), consistent with a set proposed by Grill et al., which can provide accurate predictions of the conditions for the Fokker–Planck system to exhibit oscillations. In the limit of small restoring forces, we derive an algebraic prediction of the amplitude of spindle-pole oscillations and demonstrate the relaxation structure of nonlinear oscillations. We also show how noise-induced oscillations can arise in stochastic simulations for conditions in which the mean-field Fokker–Planck system predicts stability, but for which the period can be estimated directly by the ODE model and the amplitude by a related stochastic differential equation that incorporates random binding kinetics.

Gravitational collapse of relativistic compact objects in higher dimension

In this work, we present an iterative method for gravitational collapse in higher dimensions. A framework is developed in a post-quasi-static regime with non-comoving coordinates. The internal five-dimensional system is smoothly matched with the corresponding outer Vaidya space-time over the boundary surface (BS). This correspondence provides a set of dimensionless surface equations in higher dimensions. The physical quantities such as Doppler shift, luminosity, and redshift at the BS of gravitating systems can be described through these surface equations. This procedure offers valuable insights that facilitate comprehension of the behavior of compact objects.

Computational study on torsional Casson fluid flow through concentric cylinders in a porous medium

The aim of this article is to gain a better knowledge of the non-Newtonian features of Casson fluid flow under torsional motion through two concentric cylinders in a porous medium. This study is crucial because it has the ability to control the flow characteristics of Casson fluids in porous media using rotating cylinders for the optimization of industrial processes. The flow-narrating differential equations are converted into a set of nonlinear differential equations. To obtain numerical results to flow problem, a computational model is developed. Numerical solutions are acquired employing the fourth-order exactness programme (Bvp4c), and findings are validated by comparing them to the Runge-Kutta fourth-order approach. Consequences of various distinct parameters for fluid flow and heat transfer are interpreted in graphical and tabular form. It is found that both fluid velocity and temperature increase with growing inputs of outer angular velocity (0.3≤Ω2≤0.9). Skin friction coefficient (CfsTa) is observed to decline in the case when both cylinders are rotating in same direction (Ω1=0.7andΩ2=0.7) with enhanced inputs of Hartmann number (0.5≤Ha≤2.0), while Nusselt number (Nus) profiles are enhanced with rising inputs of the Forchheimer parameter (0.2≤Fr≤0.8) when both cylinders are rotating in opposite direction (Ω1=0.7andΩ2≤−0.7).

Comprehensive study of Kinetic Trajectory Simulation method for multi-component magnetized plasma-wall interaction process

For a wide range of plasma applications across diverse fields, a comprehensive understanding of the plasma-wall interaction mechanism is indispensable due to its inherent connection with confined plasma. This work compilation delves into the Kinetic Trajectory Simulation (KTS) method for the interaction of multi-component magnetized plasma with wall, specifically focusing on its implications for the tungsten wall sputtering model. In the evolution of the 1d3v (one dimensional spatial coordinates and three dimensional velocity coordinates) KTS method, the coupled set of kinetic equations has been solved under specified boundary conditions which yields results of higher accuracy. At the particle injection boundary, we have assumed the velocity distribution function of particle species to be cut-off Maxwellians, meeting essential requirements for plasma-wall transition processes: quasineutrality, sheath edge singularity, continuity of macroscopic fluid variables, and the kinetic Bohm sheath condition. The kinetic Bohm sheath condition, a fundamental criterion for plasma sheath formation, is extended for multi-component plasmas, accounting for the cut-off Maxwellian distribution of negatively charged particles. A comparative study of the kinetic Bohm sheath condition for cut-off and Boltzmann distributions reveals a deviation of less than 2.0% in magnitude. The concentration ratio of positive or negative ion species and the presheath side electron temperature influence various plasma-wall transition characteristics, including wall potential, Debye sheath thickness, particle densities, potential distribution, particle fluxes towards the surface, particle drift velocity, phase-space trajectory evolution, and physical sputtering of the tungsten surface. Although lighter ions possess higher energy when striking the surface, the physical sputtering yield of the tungsten surface is greater for heavier ions due to their lower threshold energy and larger collision cross-section. Furthermore, a comparative study of plasma-wall transition properties using kinetic and fluid approaches demonstrates qualitative similarities, with a notable deviation of approximately 4.0% in the magnitude in the vicinity of the material surface.

Neutron Star Atmosphere–Ocean Dynamics

We analyze the structure and dynamics of the plasma atmospheres and Coulomb-liquid oceans on neutron stars. Salient dynamical parameters are identified and their values estimated for the governing set of magnetohydrodynamics equations. Neutron star atmospheres and oceans are strongly stratified and, depending on the rotation period, contain a multitude of long-lived vortices (spots) and/or narrow zonal jets (free-shear zones) in the large plasma-beta regime—i.e., β p ≫ 1 (hydrodynamic regime). In contrast, when β p ≲ 1 (magnetohydrodynamic regime), the flow is dominated by a global lattice of effectively fixed magnetic islands (plasmoids) without any jets. Understanding the spatiotemporal variability of dynamic atmospheres and oceans on neutron stars is crucial for interpreting observations of their X-ray emissions.

Two Fluids in [formula omitted] Gravity with Observational Constraints

A locally rotationally symmetric Bianchi type-I model has been studied with two fluids within the framework of the f(T) theory of gravity. Bianchi type-I is an immediate generalization of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. We have derived the exact field equations in f(T) gravity by considering the Bianchi type-I metric and applying the action for f(T) theory of gravity. We have utilized the torsion scalar T and the Lagrangian for matter. The field equations are obtained by taking the variation of the action with respect to the vierbein, leading to a set of equations that includes the energy–momentum tensor for two fluid sources: matter and radiation. We fit the H(z) curve using 57 data points and the R2-test, achieving an R2 value of 0.9321, indicating a strong fit with the Observational Hubble Dataset (OHD). Cosmological parameters like energy density, pressure, and state finder diagnostics are also discussed.

Negative binomial approach: A useful application for the study of Echinococcus granulosus

Cystic Echinococcosis is a parasitic disease caused by the larvae of Echinococcus granulosus. The transmission of E. granulosus is affected by environmental changes and anthropogenic factors, which are in turn influenced by changes in the spatial and population dynamics of animals. The deterministic model can be extended stochastically to address the low prevalence rate, which is often observed in small mammal host populations, and to account for complex processes that reflect the highly widespread disease reservoirs and non-random mixing, such as the heterogeneous contact patterns of susceptible hosts with infectious materials. In this study, a mathematical model based on a set of differential equations that define the continuous transition between different classes was used is not regulated by host. The findings indicate that each protoscolex has an equal chance of developing into a worm and creating a dispersed population. Empirical modeling can be used to represent the frequency distribution of the number of parasites in each host using the negative binomial distribution.

An Intelligent Parameter Identification Algorithm of Linear Fuzzy Fractional Differential Equation

Abstract The traditional linear generation fuzzy fractional differential equation parameter identification algorithm lacks the update of parameter identification process, has large amount of calculation, slow convergence speed of parameter identification and strong dependence on initial values. In this paper, a new intelligent recognition algorithm for linearly generated fuzzy fractional differential equations is proposed. The parameters of the equation are re expressed by the constant level set. The piecewise constant level set algorithm based on equation parameter intelligent identification is used to solve the steady-state solution of fractional differential equation, and the non convergence problem caused by too much calculation is solved. A new algorithm scheme for linearly generating fuzzy fractional differential equations is established, the constraints of the level set of the differential equations are calculated, and the updated algorithm for parameter identification of the equation is obtained. The evolutionary algorithm is used to solve the updating algorithm to realize the intelligent identification algorithm of linear fractional fuzzy differential equations. Experimental results show that the algorithm has the advantages of fast convergence speed, high calculation accuracy and low initial value.

Cauchy matrix approach for H1 a equation in the torqued Adler–Bobenko–Suris lattice list

As a torqued version of the lattice potential Korteweg–de Vries equation, the H1 a is an integrable nonsymmetric lattice equation with only one spacing parameter. In this paper, we present the Cauchy matrix scheme for this equation. Soliton solutions, Jordan-block solutions and soliton-Jordan-block mixed solutions are constructed by solving the determining equation set. All the obtained solutions have jumping property between constant values for fixed n and demonstrate periodic structure.

Exploring the similarity relationships from the nondimensionalization of atmospheric turbulence

AbstractNondimensionalization, a theoretical approach for establishing interconnections among parameters within a set of equations, has proven to be an effective tool for the analysis of atmospheric turbulence. By applying nondimensionalization to turbulence equations, a concise form of dimensionless turbulence functions can be obtained. This process also yields several dimensionless parameters, defined as combinations of characteristic scales. From the dimensionless tensor $${B}$$ B and vector $${{\varvec{\beta}}}_{\theta}$$ β θ introduced in this study, the characteristic length scale, $${z}^{s}$$ z s , can be defined as an alternative of length scale in similarity theories. Using the data from observational station in Horqin Sandy Land, quantified verifications of similarity relationships are carried out. The dimensionless parameters derived from nondimensionalization is not only in accordance with traditional turbulence theories but also facilitate the derivation of relationships among other dimensionless parameters. This reveals new similarity relationships that supplement the Monin–Obukhov theory. Under conditions of flat terrain and steady motions, the new length scale gives rise to similarity relationships exhibiting “4/3” exponential and near-linear patterns, which are associated with turbulent transport. These results make it possible to obtain the turbulent fluxes directly from the statistics of meteorological elements, even in stable stratifications. Consequently, the method of nondimensionalization can be taken as a reference in parameterization schemes of turbulence and climate models, and is fruitful in prospect of further study on atmospheric turbulence.

Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations

The main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard’s iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi–Picard iteration method, which combines Picard’s iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss–Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions.

Learning Hidden Markov Models with Structured Transition Dynamics

The hidden Markov model (HMM) provides a natural framework for modeling the dynamic evolution of latent diseases. The unknown probability matrices of HMMs can be learned through the well-known Baum–Welch algorithm, a special case of the expectation-maximization algorithm. In many disease models, the probability matrices possess nontrivial properties that may be represented through a set of linear constraints. In these cases, the traditional Baum–Welch algorithm is no longer applicable because the maximization step cannot be solved by an explicit formula. In this paper, we propose a novel approach to efficiently solve the maximization step problem under linear constraints by providing a Lagrangian dual reformulation that we solve by an accelerated gradient method. The performance of this approach critically depends on devising a fast method to compute the gradient in each iteration. For this purpose, we employ dual decomposition and derive Karush–Kuhn–Tucker conditions to reduce our problem into a set of single variable equations, solved using a simple bisection method. We apply this method to a case study on sports-related concussion and provide an extensive numerical study using simulation. We show that our approach is in orders of magnitude computationally faster and more accurate than other alternative approaches. Moreover, compared with other methods, our approach is far less sensitive with respect to increases in problem size. Overall, our contribution lies in the advancement of accurately and efficiently handling HMM parameter estimation under linear constraints, which comprises a wide range of applications in disease modeling and beyond. History: Accepted by Paul Brooks, Area Editor for Applications in Biology, Medicine, & Healthcare. Funding: This research was funded by [Grant R01DE028283] from the National Institute of Dental and Craniofacial Research, National Institutes of Health. G.-G. Garcia was also funded by the Georgia Clinical & Translational Science Alliance National Institutes of Health award [Grant UL1-TR002378]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0342 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0342 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Hamiltonian formalism for bistable-multilayered plates under non-mechanical stimuli

Bistable multilayered plates have attracted significant attention as morphing and adaptive structures, renowned for their unprecedented exceptional performance. However, accurately predicting both their shape and internal stresses poses formidable challenges due to their geometrically nonlinear nature. For the first time, this paper introduces a Hamiltonian formalism aimed at achieving high-resolution analysis of nonlinear multilayered plates subjected to non-mechanical stimuli, including hygro-thermo-electro-magneto-elastic responses. A canonical system is strategically developed to compute membrane behaviors, yielding symplectic dual differential equations for in-plane field variables. This method elegantly decouples out-of-plane variables from the full-state vector, leading to an exact analytical solution in the membrane problem. To predict the bending behaviors, the first variation of the Hamiltonian energy density function, expressed as a power series of the transverse deflection, ensures flexural equilibrium. The power series effectively captures all admissible out-of-plane deformations, enabling a smooth transition between linear and nonlinear plate responses, including pitchfork bifurcation and limit points. The validity and accuracy of the proposed method are rigorously assessed through convergence tests and satisfaction of boundary conditions across various equilibria, ranging from monostability to bistability. It is cross-verified with high-fidelity finite-element methods (FEM), showing excellent agreements in both deformations and stress resultants. This research presents a physics-based methodology that unveils a parametric interplay between in-plane and out-of-plane field variables, serving as an efficient approach for analyzing adaptive and morphing structures.