Set Of Equations Research Articles

Cayley Rotation Averaging: Multiple Camera Averaging Under the Cayley Framework.

Rotation averaging, which aims to calculate the absolute rotations of a set of cameras from a redundant set of their relative rotations, is an important and challenging topic arising in the study of structure from motion. A central problem in rotation averaging is how to alleviate the influence of noise and outliers. Addressing this problem, we investigate rotation averaging under the Cayley framework in this paper, inspired by the extra-constraint-free nature of the Cayley rotation representation. Firstly, for the relative rotation of an arbitrary pair of cameras regardless of whether it is corrupted by noise/outliers or not, a general Cayley rotation constraint equation is derived for reflecting the relationship between this relative rotation and the absolute rotations of the two cameras, according to the Cayley rotation representation. Then based on such a set of Cayley rotation constraint equations, a Cayley-based approach for Rotation Averaging is proposed, called CRA, where an adaptive regularizer is designed for further alleviating the influence of outliers. Finally, a unified iterative algorithm for minimizing some commonly-used loss functions is proposed under this approach. Experimental results on 16 real-world datasets and multiple synthetic datasets demonstrate that the proposed CRA approach achieves a better accuracy in comparison to several typical rotation averaging approaches in most cases.

Thermal analysis on Ferro Casson nanofluid flow over a Riga plate with thermal radiation and non-uniform heat source/sink

Ferro hybrid nanofluids can be used in electronics and microelectronics cooling applications to reduce heat accumulation and efficiently remove surplus heat. These nanofluids aid to maintain optimum operating temperatures and reduce device overheating by enhancing the heat transfer rate. With this motivation, the aim of the present numerical analysis is to study the three-dimensional incompressible hybrid nanofluid flow over a slippery Riga surface by combining the Casson fluid model. Mathematical modeling is constructed with nanoparticles as Fe3O4 and CoFe2O4 with base fluid as water. The non-uniform heat source/sink and thermal linear radiation effects are taken into account with the Hamilton–Crosser thermal conductivity model. A system of nonlinear PDEs is produced by the proposed problem and then the relevant similarity variables are implemented to transform the set of partial differential equations and their accompanying boundary conditions into the coupled nonlinear differential equations with one independent variable. These modified ordinary differential equations (ODEs) were then successfully solved with the Runge–Kutta fourth-order method by combining via the shooting technique. With the aid of graphical representations, the effects of various influencing parameters were presented and analyzed comprehensively. Furthermore, the impacts of relevant parameters on heat transfer rates and shear stress are concisely discussed and illustrated in tabular forms. The significant findings include the enhancement of the radiation parameter increases the thermal boundary layer thickness and the thickness increases whenever surface experiences slippery conditions. The axial and transverse momentum of the fluid are controlled with the Casson parameter. An effective connection is noticed once the current numerical solutions are verified under particular conditions that were previously described.

A compensation method for PMSM sensorless control with parameter identification considering SMO observation error

AbstractSensorless control of permanent magnet synchronous motor (PMSM) can increase the reliability of electric actuators of more electrical aircraft. Numerical online parameter estimation method will enhance the performance for sensorless control methods based on sliding mode observer (SMO). However, numerical online parameter identification methods typically assume zero rotor position estimation error to ensure full rank of the identification equation set. This assumption actually leads to errors in parameter identification, resulting in steady‐state errors in rotor position estimation. This paper focuses on a compensation method for PMSM online parameter identification that takes into account rotor position estimation error. This method is used for sensorless control based on SMO, using model reference adaptive systems for online parameter identification, and no additional parameter adjustment required. Validation is conducted on a PMSM experimental platform with maximum load of 15kW. The experimental results of 1000 r/min steady state operation proved the effectiveness of the method, and the dynamic experiments with 5 and 15kW loads proved that the method has no significant adverse effect on the system dynamics.

Ternary hybrid nanofluid flow through an inclined shrinking sheet when soret, dufour and quadratic thermal radiation are significant: an irreversibility analysis with various shape factors

As nanofluid technology continues to evolve, the knowledge gained from studying flow over an inclined shrinking sheet can be applied to various emerging applications. These could include heat pipes for spacecraft thermal management, thermal management systems for wearable electronics, or even designing microfluidic channels for efficient desalination processes. A ternary hybrid nanofluid (Ethylene Glycol, Copper (II) oxide, Titanium dioxide, and Silicon dioxide) is the main focus of this study, which aims to theoretically analyse its behaviour through a tilted shrinking sheet in the presence of heat source, Dufour, and Soret effects. Governing equations are transformed into a set of ordinary differential equations using appropriate similarity transformations and then solved using bvp4c solver. The main findings of this study are that enhancing the value of the couple stress parameter leads to a decline in fluid velocity, and increasing the Dufour number will upsurge the temperature of the fluid. It has been noticed that the Sherwood number exhibits a decline when the Soret number increases. We find that the Nusselt number noticeably increases by 0.147404161 (spherical), 0.151277029 (brick), and 0.156726935 (cylinder) for values of the Dufour number within the range of 0 to 3.

Variational approach for Stokes flow through a two-dimensional non-uniform channel

A variational approach is proposed to study the Stokes flow in a two-dimensional non-uniform channel. By using the stationarity of the Lagrangian, the Euler-Lagrange equations are established which leads to a simple set of ordinary differential equations to provide an estimate for the average pressure drop explicitly in terms of the channel shape function. The results for the pressure drop show an excellent agreement with the second-order extended lubrication theory. A higher-order formulation further improves the accuracy of the results for the pressure drop along the channel.

Mathematical Models for the Large Spread of a Contact-Based Infection: A Statistical Mechanics Approach

In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics.

Weakly guiding approximation of a three-dimensional waveguide model for extreme ultraviolet lithography simulation

A three-dimensional (3D) waveguide model is applied in extreme ultraviolet (EUV) lithography simulations. The 3D waveguide model is equivalent to rigorous coupled-wave analysis, but fewer field components are used to solve Maxwell’s equations. The 3D waveguide model uses two components of vector potential, A x and A y , corresponding to the two polarizations. The electric field of the A x polarization is approximately parallel to the x axis, and the electric field of the A y polarization is approximately parallel to the y axis. The 3D waveguide model solves a coupled vector wave equation for two polarizations. The refractive index of conventional EUV absorbers is close to that of vacuum. The weakly guiding approximation in optical fiber theory is applied to the 3D waveguide model. The coupled vector wave equations for the two polarizations are decoupled into two independent scalar wave equations. Maxwell’s equations are simplified to a set of scalar wave equations. The weakly guiding approximation reduces the computation time to solve the equations. The computation time required to solve the weakly guiding approximation is about 1/5 of the time to solve the original 3D waveguide model.

Investigation on the dynamic characteristics for high-speed mechanical seal considering turbulent cavitating flow and inertia effect

PurposeThis study aims to investigate the dynamic characteristics of a high-speed mechanical seal by considering the flow regime, cavitation phenomenon and inertia effect.Design/methodology/approachBased on the high-speed conditions of a mechanical seal with a low-viscosity fluid, a multi-factor coupled lubrication model of a mechanical seal is established. The perturbation Reynolds equation set is derived using the perturbation method, and the effects of turbulent flow and inertia on sealing performance are discussed.FindingsThe results indicate that turbulence significantly influences the dynamic characteristics of low-viscosity fluids under high-speed conditions.Originality/valueThe results of the dynamic characteristics presented in this study are expected to provide guidance for the design of mechanical seals.

Two wrongs do not make a right: the assumption that an inhibitor acts as an inverse activator

Models of biochemical networks are often large intractable sets of differential equations. To make sense of the complexity, relationships between genes/proteins are presented as connected graphs, the edges of which are drawn to indicate activation or inhibition relationships. These diagrams are useful for drawing qualitative conclusions in many cases by the identifying recurring of topological motifs, for example positive and negative feedback loops. These topological features are usually classified under the presumption that activation and inhibition are inverse relationships. For example, inhibition of an inhibitor is often classified the same as activation of an activator within a motif classification, effectively treating them as equivalent. Whilst in many contexts this may not lead to catastrophic errors, drawing conclusions about the behavior of motifs, pathways or networks from these broad classes of topological feature without adequate mathematical descriptions can lead to obverse outcomes. We investigate the extent to which a biochemical pathway/network will behave quantitatively dissimilar to pathway/ networks with similar typologies formed by swapping inhibitors as the inverse of activators. The purpose of the study is to determine under what circumstances rudimentary qualitative assessment of network structure can provide reliable conclusions as to the quantitative behaviour of the network. Whilst there are others, We focus on two main mathematical qualities which may cause a divergence in the behaviour of two pathways/networks which would otherwise be classified as similar; (i) a modelling feature we label ‘bias’ and (ii) the precise positioning of activators and inhibitors within simple pathways/motifs.

Discrete Boltzmann model with split collision for nonequilibrium reactive flows* *This work is supported by the National Natural Science Foundation of China (under Grant Nos. U2242214, 51806116 and 91441120), the Guangdong Basic and Applied Basic Research Foundation (under Grant Nos. 2022A1515012116 and 2024A1515010927), the Natural Science Foundation of Fujian Province (under Grant Nos. 2021J01652, 2021J01655), and the China Scholarship Council

A multi-relaxation-time discrete Boltzmann model (DBM) with split collision is proposed for both subsonic and supersonic compressible reacting flows, where chemical reactions take place among various components. The physical model is based on a unified set of discrete Boltzmann equations that describes the evolution of each chemical species with adjustable acceleration, specific heat ratio, and Prandtl number. On the right-hand side of discrete Boltzmann equations, the collision, force, and reaction terms denote the change rates of distribution functions due to self- and cross-collisions, external forces, and chemical reactions, respectively. The source terms can be calculated in three ways, among which the matrix inversion method possesses the highest physical accuracy and computational efficiency. Through Chapman–Enskog analysis, it is proved that the DBM is consistent with the reactive Navier–Stokes equations, Fick's law and the Stefan–Maxwell diffusion equation in the hydrodynamic limit. Compared with the one-step-relaxation model, the split collision model offers a detailed and precise description of hydrodynamic, thermodynamic, and chemical nonequilibrium effects. Finally, the model is validated by six benchmarks, including multicomponent diffusion, mixture in the force field, Kelvin–Helmholtz instability, flame at constant pressure, opposing chemical reaction, and steady detonation.

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Effects of Arrhenius Activation Energy on Thermally Radiant Williamson Nanofluid Flow Over a Permeable Stretching Sheet with Viscous Dissipation

This paper explores the role of viscous dissipation, Arrhenius activation energy, and thermal radiation of Williamson nanofluid flow over a permeable stretching sheet. The governing partial differential equations have been simplified through a transformation process, resulting in a set of non-linear differential equations. To find a solution for these equations, a numerical approach is employed, specifically the fourth order Runge-Kutta method. Additionally, a shooting technique is utilized to enhance the accuracy of the numerical solutions. Overall, the study involves reducing complex equations, solving them numerically, and refining the results through a combination of methods. The study investigates the impact of different physical parameters on key factors like velocity, temperature, skin friction coefficient, nano particle volume fraction, and rates of mass and heat transfer. This study exhibits that activation energy parameter enhances concentration profiles, whereas fitted rate constant shows opposite behavior. The activation energy into heat transfer model allows for the optimization of heat transfer systems utilizing Williamson nano fluids.

Stability of convective rolls in a horizontal layer rotating about an inclined axis

We present three results on stability of rolls in Boussinesq convection in a plane horizontal layer with rigid boundaries that is rotating about an inclined axis with the angular velocity Ω = ( Ω 1 , Ω 2 , Ω 3 ) . (i) We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary Ω , and by the reduced problem the set of equations for the angular velocity ( Ω 1 , 0 , Ω 3 ) . Here x,y are horizontal Cartesian coordinates in the layer and z is the vertical one. We prove that a y-independent solution to one of the two problems is also a solution to the second one. (ii) We calculate the critical Rayleigh number for the monotonic onset of convection. The instability mode in the form of rolls (a flow independent of a horizontal direction) is assumed. Let β be the angle between the horizontal projection of Ω and the rolls axes. We show that β = 0 for the least stable mode. When the axis of rotation is horizontal, this is proven analytically, and for Ω 3 ≠ 0 , the result is obtained numerically. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of Ω 1 and Ω 2 and the emerging flow are rolls with axis aligned with the horizontal component of the rotation vector. (iii) We study the behaviour of convective flows by integrating numerically the three-dimensional equations of convection for Ω = ( 0 , Ω 2 , Ω 3 ) and a range of the Rayleigh numbers, other parameters of the problem being fixed. We assume square horizontal periodicity cells, whose sides are equal to the period of the most unstable mode. The computations indicate that, in general, in the nonlinear regime convective rolls become more stable as Ω 2 increases. Namely, on increasing Ω 2 , the interval of the Rayleigh numbers for which convective rolls are stable increases.

Theoretical study on the thermoluminescence kinetics by glow curve analysis based on the genetic algorithm

The analysis of thermoluminescence (TL) glow curve is an important way to investigate the trap structure of the materials as well as the kinetics of the charge carrier transfer. Most currently used methods rely on the existence of an explicit expression of the theoretical glow curve. However, since the set of kinetic equations describing the TL process does not have analytical solutions, the derivation of an explicit expression always involves some sort of assumptions and approximations, which can compromise the universality and accuracy of the method. This paper proposes a new algorithm that focuses on the original kinetic equations and does not necessitate an explicit expression to analyze the TL glow curve. This algorithm establishes an objective function to evaluate the fitness of a selected combination of kinetic parameters, and then using the genetic algorithm (GA) to search for the optimal parameters. Various occasions were numerically simulated, demonstrating the algorithm's capability to handle kinetics of different order, multiple interactive peaks, deep trap, non-linear heating, and temperature dependent parameter cases. An actual glow curve of the LiF:Mg,Cu,P chip was also analyzed using this algorithm, and the corresponding kinetic parameters were identified. This algorithm provides a more universal and valid method for TL glow curve analysis.

Comparison of unsteady MHD flow of second grade fluid by two fractional derivatives

A study is conducted on the unsteady motion of a free convective flow of second grade fluid, energy transfer, and Darcy’s law over an oscillatory smooth vertical plate. The study compares two different approaches in developing a fractional model: the Caputo-Fabrizio operator with nonsingular kernel and the constant proportional Caputo fractional operator with Fourier’s and Fick’s laws. By applying the Laplace method and transforming the provided set of equations into nondimensional form, we obtained semi-analytical results and presented these results through graphical analysis. The study examines how different flow parameters, including the fractional parameters, affect the velocity, mass, and heat profiles of the physical system. The results suggest that the physical model using constant proportional Caputo derivative leads to higher temperature, stronger concentration, and increased velocity compared to the Caputo-Fabrizio model. This highlights the importance of selecting an appropriate fractional model when studying complex physical systems. It is also depicted from the whole analysis that field variables with novel hybrid fractional derivative constant proportional Caputo exhibits a more effective and declining tendency than Caputo-Fabrizio.

Advanced color image encryption using third-order differential equations and three-dimensional logistic map

Image encryption stands out as a crucial technique employed to securely transmit images across the Internet. In this paper, we introduce a novel algorithm for encrypting color images. The algorithm is built upon the principles of differential equations, XOR operations, and chaotic maps. First, the plain image is three-dimensional pixel shuffled via a logistic map. Afterward, the differential equations are used as a mathematical tool for encrypting images. The third-order ordinary differential equations are used to encrypt the shuffled images. The color values of the plain image are considered coefficients for the independent variable X. Subsequently, an alternate matrix of the same size is generated using a three-dimensional logistic map, taking into account its color values as the exponents linked to the independent variable X. A set of third-order differential equations emerged, containing an equivalent number of elements as the color values present in the plain image. This set of differential equations is addressed in the following manner: combining XOR and integration three times with respect to the independent variable X for each set of obtained differential equations while treating the integration constant as zero. Ultimately, a set of ordinary equations involving the independent variable X is derived, where the coefficients of X represent color values for the cipher image. The results from experiments and the security analysis affirm the resilience of the proposed algorithm against established security attacks. It exhibits a substantial key space, heightened key sensitivity, and a strong encryption effect.

Semi-analytical method for calculating ground vibrations from a tunnel in a homogeneous half-space with an irregular surface

This paper proposes a novel two-dimensional (2D) semi-analytical method for calculating ground vibrations from a tunnel situated in a homogeneous half-space with an irregular surface. The circular tunnel is conceptualized as an elastic solid, while the soil is modelled as an elastic, isotropic, and homogeneous half-space with an irregular surface. A virtual horizontal interface is introduced to divide the soil domain into an irregular region with an arbitrary-shaped surface and a half-space with a circular hollow. The wavefield scattered by the irregular surface is simulated by the boundary integral equation. Through the application of the discrete wavenumber approach and expansion of wavefield on the irregular surface, the boundary integral equation is transformed into a set of simultaneous matrix equations containing unknown expansion coefficients. By utilizing the transformation between cylindrical and plane waves, the boundary conditions on soil-tunnel interface are satisfied, yielding a solution for a harmonic point load acting on the tunnel in a half-space with an irregular surface. The proposed method can simulate topographies with arbitrary shapes and only requires the discretization of the irregular part of the ground surface, which provides a highly efficient tool to investigate the propagation characteristics of train-induced vibrations in complex topographies. The proposed method is verified through comparisons with the existing analytical methods and the finite element method. The dynamic responses of a tunnel embedded in a half-space with a Gaussian-shaped surface are meticulously investigated. A case study involving a site for the Shenzhen metro line 5 is performed. The numerical results demonstrate that the presence of irregular topography alters the distribution of ground vibrations, leading to significant differences in responses within the irregular region and certain disparities in the regular region. The effect of the location, size, and undulating direction of the irregular topography on ground vibrations is frequency-dependent.

Warm start for optimal transfer between close circular orbits with first generation E-sail

The Electric Solar Wind Sail (E-sail) is a propellantless propulsion system for deep space navigation that exploits the dynamic pressure of the solar wind to generate thrust using a web of long, conducting tethers. The heliocentric trajectory of an E-sail-based spacecraft in a classic transfer between two Keplerian orbits is usually analyzed, in an optimal framework, by minimizing the total flight time. This paper discusses an analytical, approximate procedure for solving the two-point boundary value problem associated with the trajectory optimization process in a heliocentric, two-dimensional scenario involving a first-generation E-sail. The procedure is applied in a typical transfer between two coplanar circular orbits whose (assigned) radii are sufficiently close to each other. Paralleling the approach proposed in the recent literature, the mathematical model presented in this paper discusses a set of analytical equations that give an accurate approximation of both the minimum flight time and the (unknowns) initial adjoint variables. The paper also analyzes a set of mission applications, which involve interplanetary transfers to some near-Earth asteroids whose orbits have a very small value of both eccentricity and inclination.

Method for remote measurement of specific conductivity and moisture of subsurface soil horizons

The aim of the research is to develop a radar method for determining the physical and chemical parameters of subsurface soil horizons, providing rapid determination of moisture and specific conductivity in the area of the plant root system. The proposed method is based on a set of Fresnel equations which describe reflection of electromagnetic waves from the interface between dielectric media in vertical and horizontal polarization of the probing signal. For the practical implementation of the method, it is proposed to use two unmanned aerial vehicles that form a bistatic radar system which irradiates the Earth's surface obliquely in order to create the Brewster's effect and increase the fraction of the radio signal reflected from subsurface horizons. The percentage of moisture and the specific conductivity of soil are calculated from the measured values of the imaginary part of the complex permittivity. The required accuracy of moisture and conductivity measurements is achieved by two-step calibration of the measuring device. The values of the moisture content and specific conductivity of soil obtained by radar at a frequency of 469 MHz are in good agreement with the results of measuring these parameters using the soil moisture meter TDR 150 Spectrum Technologies, Inc.

A soluble model for synchronized rhythmic activity in ant colonies

Synchronization is one of the most striking instances of collective behavior, occurring in many natural phenomena. For example, in some ant species, ants are inactive within the nest most of the time, but their bursts of activity are highly synchronized and involve the entire nest population. Here we revisit a simulation model that generates this synchronized rhythmic activity through autocatalytic behavior, i.e., active ants can activate inactive ants, followed by a period of rest. We derive a set of delay differential equations that provide an accurate description of the simulations for large ant colonies. Analysis of the fixed-point solutions, complemented by numerical integration of the equations, indicates the existence of stable limit-cycle solutions when the rest period is greater than a threshold and the event of spontaneous activation of inactive ants is very unlikely, so that most of the arousal of ants is done by active ants. Furthermore, we argue that the persistent oscillations observed in the simulations for colonies of finite size are due to resonant amplification of demographic noise.