Green Theorem Research Articles

Dynamic model and characteristic analysis of a combined algebraic spiral scroll compressor

The combined algebraic spiral (CAS) effectively maximizes the geometric performance of a scroll compressor; however, its dynamic behavior remains inadequately explored. Utilizing the geometric model of the CAS scroll compressor, this study derived the functional relationship between the meshing point and the rotation angle, and calculated the working chamber volume using the discrete Green's theorem method. A dynamic model of the CAS scroll compressor was developed, and the influence of various parameters on its dynamic characteristics was analyzed. The findings indicate that when the polar angle interval is less than 0.01π, the discrete Green's theorem method accurately computes the working chamber volume. Among the gas forces acting on the CAS scroll compressor, the axial gas force is the most significant, followed by the tangential gas force, while the radial gas force is considerably smaller. The tangential gas force predominantly influences the overturning and rotational moments. Notably, when the polar angles of the connection points between the higher-order curve and the starting and ending algebraic spirals are set at 1.5π and 3π, the gas force remains low while maintaining geometric performance. This configuration results in reduced variation in gas force and enhanced dynamic performance. The spiral coefficient and spiral index of the starting algebraic spiral should be set as intermediate values to ensure optimal geometric and dynamic performance of the CAS scroll compressor.

A novel data-driven state evaluation approach for photovoltaic arrays in uncertain shading scenarios

Accurately determining the operating state of photovoltaic (PV) power-generation equipment and providing accurate support for maintenance management requires evaluation of the states of PV arrays under partial shading conditions. Results of such evaluation enable the optimization of maintenance measures to reduce maintenance costs and extend the lifespan of PV modules, thereby increasing the overall revenue of the PV power station. Therefore, this study proposes a novel state-evaluation approach based on current-voltage (I-V) curves under partial shading conditions. Based on the size of the pixel values in this curve, the proposed novel state-evaluation was performed using the proportion of the values of the normal and shaded curves under the same environmental conditions. Specifically, the feature variables (open-circuit voltage, short-circuit current) and pixel values within the I-V curve were analyzed and calculated using a sensitivity algorithm and computer vision algorithms (Canny edge-detection algorithm and Green's theorem). Based on the mapping relationship between feature variables and pixels, an improved Gaussian-process regression method with a combined kernel function (dot product and Matérn function) was proposed to predict the pixel values under the corresponding normal curve. This combined kernel function effectively captured causal relationships and handled outliers in the test dataset to improve the robustness of the model. Finally, the ratios of the pixel values were calculated using the predicted pixels within the normal curve and the calculated pixels within the shading curve. Experiments demonstrated that the root mean square error (RMSE), mean absolute error (MAE), R squared (R2) and log likelihood of the improved Gaussian-process regression method reached 0.011, 0.0086, 0.9996, and 24.7096, respectively.

Effect of step bottom and waterway on flexural gravity wave scattering

Flexural gravity wave scattering by two semi-infinite non-identical ice sheets, which are separated by a clean water surface, is investigated in the presence of a step bottom topography. The problem is investigated in the cases of (i) intermediate water depth and (ii) shallow water. The effect of two edge conditions, (i) simply supported edge and (ii) free edge, on wave scattering is also analyzed. Employing linear velocity potential theory, the problem is studied in the frequency domain. The physical phenomenon is modeled as a boundary value problem having the Laplace equation as the governing equation, and it is solved using the eigenfunction expansion method. Considering the bottom topography and upper boundary, the fluid domain is divided into three regions, and in each region, velocity potential is expressed in terms of infinite Fourier series. Velocity and pressure are matched at the intermediate surface of two regions, and a system of algebraic equations with unknown coefficients is obtained. The complete solution of the present problem is recognized by solving the system of equations numerically. The energy relation is derived using Green's theorem. The reflection and transmission coefficients are computed and compared with the energy relation to check the accuracy of the present method. For different parameters (depth ratio, clean waterway, and different ice properties), reflection coefficient and transmission coefficient are computed and presented as a function of angular frequency. The value of the reflection coefficient ranges from zero to unity, whereas at certain frequencies, the transmission coefficient attends value more than unity.

Some basic theorems on the modified Green–Lindsay model of linear generalized thermoelasticity

The present paper is concerned with a theory of thermoelasticity based upon a newly developed temperature-rate-dependent thermoelastic model that was proposed by Yu et al. [Meccanica. 2018;53:2543–2554], which has eliminated the discontinuity observed in the displacement field during transient motion. The basic governing equations for a homogeneous and isotropic medium are formulated to deduce the significant theorems. Firstly, we prove the uniqueness theorem concerning a mixed initial and boundary value problem within the framework of linear thermoelasticity. Furthermore, we establish a reciprocity theorem utilizing the Laplace transform, which provides insight into the mutual influence between different aspects of the system under consideration. Consequently, we proved the Somigliana and Green theorems. This generalized theorem extends the applicability and versatility of the classical theorems and also assists the issue of thermal and mechanical fields as well as is beneficial for geophysics and mining.

Nonlocal Green Theorems and Helmholtz Decompositions for Truncated Fractional Gradients

Nonlocal Green Theorems and Helmholtz Decompositions for Truncated Fractional Gradients

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

On uniqueness and Somigliana theorem for a thermoelastic theory

The main concern of the present article is based on unified generalized thermoelasticity in an isotropic and homogeneous medium. The uniqueness theorem and Betti type reciprocal principle are derived. Based on the reciprocal theorem, the generalization of the Somigliana and Green theorem is also established in the present context.

Analysis of regional carbon productivity differences and influencing factors—based on new green decomposition model

This paper introduces a new green decomposition model of carbon productivity that aims to further analyze the regional differences in carbon productivity and its interaction with regional industrial performance. We combine desired outputs and undesired outputs orientation, and multiple factor inputs to derive a new green decomposition theorem, establish a new green decomposition model of carbon productivity, and obtain nine effects of regional carbon productivity differences. Empirical analysis is conducted using input-output data from 29 provinces and 15 industries in China, comparing the differences in carbon productivity from both the provincial and industry perspectives and exploring the mechanism of action. This paper provides theoretical basis and empirical evidence for regional carbon productivity enhancement and economic and industrial optimization from the perspective of multi-factor inputs, as well as policy insights for regional low-carbon transition development.

Remarks on Conjectures in Block Theory of Finite Groups

In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results.

EFFICIENT AREA CALCULATION OF JACKFRUIT LEAVES: LEVERAGING GREEN'S THEOREM AND LEAST SQUARES FUNCTION

Leaves are an important part of a plant. As a place for photosynthetic organs, leaves can be used as indicators to assess a plant's growth and physiological performance. Leaf measurement is carried out as an important effort to collect leaf area data which is useful in analyzing a leaf's growth. This study aims to apply the green theorem to calculate leaf area and compare leaf area with the least square approximation, Simpson 1/3, the elliptic approach, and the millimeter block method, with the research object used being Jackfruit leaves. From this research, we obtain the results as follows leaf area by applying the green theorem by least square approximation is 5946.75 mm2, the leaf area with Simpson 1/3 method is 5938.25 mm2, the leaf area with ellipse approximation is 6850.23 mm2, and leaf area with millimeter block method is 5387.5 mm2. From the several approximation methods used in this study, it was found that by using the Least Square Quadratic approximation and the Simpson 1/3 method, effective results were obtained for the observed object because the functions of the Least Square Quadratic approximation and the Simpson 1/3 method are almost close to the actual function so that the results obtained are close to accurate with an area the original of the observed object is jackfruit leaves.

A high order Cartesian grid, finite volume method for elliptic interface problems

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.

Towards a single parameter for the assessment of EEG oscillations.

The single macroscopic flow on the boundary of a closed curve equals the sum of the countless microscopic flows in the enclosed area. According to the dictates of the Green's theorem, the counterclockwise movements on the border of a two-dimensional shape must equal all the counterclockwise movements taking place inside the shape. This mathematical approach might be useful to analyse neuroscientific data sets for its potential capability to describe the whole cortical activity in terms of electric flows occurring in peripheral brain areas. Given a map of raw EEG data to coloured ovals in which different colours stand for different amplitudes, the theorem suggests that the sum of the electric amplitudes measured inside every oval equals the amplitudes measured just on the oval's edge. This means that the collection of the vector fields detected from the scalp can be described by a novel, single parameter summarizing the counterclockwise electric flow detected in the outer electrodes. To evaluate the predictive power of this parameter, in a pilot study we investigated EEG traces from ten young females performing Raven's intelligence tests of various complexity, from easy tasks (n = 5) to increasingly complex tasks (n = 5). Despite the seemingly unpredictable behavior of EEG electric amplitudes, the novel parameter proved to be a valuable tool to to discriminate between the two groups and detect hidden, statistically significant differences. We conclude that the application of this promising parameter could be expanded to assess also data sets extracted from neurotechniques other than EEG.

Fiber Uncertainty Visualization for Bivariate Data With Parametric and Nonparametric Noise Models.

Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of level-set visualization for univariate data to multivariate data. In this paper, we present a statistical framework to quantify positional probabilities of fibers extracted from uncertain bivariate fields. Specifically, we extend the state-of-the-art Gaussian models of uncertainty for bivariate data to other parametric distributions (e.g., uniform and Epanechnikov) and more general nonparametric probability distributions (e.g., histograms and kernel density estimation) and derive corresponding spatial probabilities of fibers. In our proposed framework, we leverage Green's theorem for closed-form computation of fiber probabilities when bivariate data are assumed to have independent parametric and nonparametric noise. Additionally, we present a nonparametric approach combined with numerical integration to study the positional probability of fibers when bivariate data are assumed to have correlated noise. For uncertainty analysis, we visualize the derived probability volumes for fibers via volume rendering and extracting level sets based on probability thresholds. We present the utility of our proposed techniques via experiments on synthetic and simulation datasets.

Some remarks on a theorem of Green

The purpose of this paper is to study holomorphic curves f from C to C3 avoiding four complex hyperplanes and a real subspace of real dimension four in C3. We show that the projection of f into the complex projective space C P^2 does not remain constant as in the complex case studied by Green, which indicates that the complex structure of the avoided hyperplanes is a necessary condition in the Green theorem

Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

An isogeometric boundary element method (IGABEM) is extended in this work to study liquid sloshing characteristics in various 3D tanks with arbitrary internal bottom-mounted porous structures. Unlike the classical boundary element method (BEM), the present method uses the non-uniform rational B-splines(NURBS) instead of the piecewise Lagrange polynomials as the shape function to approximate both the domain boundary and field variables, it completely inherits the advantages of the traditional BEM and the isogeometric analysis (IGA), such as the properties of only boundary discretization needing, higher order continuity, self-adaptability and so on. All the information required in the IGABEM for the mesh generation can be obtained from the CAD software, which may evidently reduce the time and memory consumptions of preprocessing, meanwhile, since the same basis functions (NURBS) are used for both the IGABEM and CAD models, the proposed method can exactly reconstruct the geometry of the analysis domain without any error, and this substantively contributes to the high calculation accuracy of IGABEM. In this paper, by using a zoning method the entire domains of fluid are divided into several sub-domains with the consideration of compatibility boundary conditions besides the porous structures. Owning to the constant cross section of the container, impermeable boundary condition at the tank bottom and the linear boundary condition at the free surface, the 3D Laplace equation (which governs the 3D sloshing problem) is transformed into a couple of Helmholtz equation and modified Helmholtz equations (corresponding to the propagating and evanescent modes, respectively). Green's theorem and the divergence theorem of Gauss are used in this paper to derive the IGABEM system equations for the two types of equations. Moreover, by introducing a series of eigen-functions associated with the vertical coordinate, the potential and velocity boundary conditions are expanded into those associated with the propagating and evanescent modes, which can be applied upon the Helmholtz equation and modified Helmholtz equations. Finally, the present problem can be solved by solving these equations and combining the results linearly. Accuracy and convergence of the present IGABEM technique are verified through numerical tests by comparing the obtained results with analytical, experimental solutions or those calculated with other numerical method, thereafter, liquid sloshing problems in the cubic or circular cylindrical vessels with the coaxial or eccentric elliptical and circular cylindrical porous structures are analyzed. The main influences of the geometric parameter, porous-effect parameter, number and arrangement of the porous structures are investigated in detail, some conclusions are outlined in the end of this paper.

Tower-type bounds for Roth’s theorem with popular differences

Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every \epsilon > 0 there is some N_0(\epsilon) such that for every N \ge N_0(\epsilon) and A \subset [N] with |A| = \alpha N , there is some nonzero d such that A contains at least (\alpha^3 - \epsilon) N three-term arithmetic progressions with common difference d . We prove that the minimum N_0(\epsilon) in Green's theorem is an exponential tower of twos of height on the order of \log(1/\epsilon) . Both the lower and upper bounds are new. This shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.

FDTD-Based Electromagnetic Modeling of Dielectric Materials with Fractional Dispersive Response

The use of fractional derivatives and integrals has been steadily increasing thanks to their ability to capture effects and describe several natural phenomena in a better and systematic manner. Considering that the study of fractional calculus theory opens the mind to new branches of thought, in this paper, we illustrate that such concepts can be successfully implemented in electromagnetic theory, leading to the generalizations of the Maxwell’s equations. We give a brief review of the fractional vector calculus including the generalization of fractional gradient, divergence, curl, and Laplacian operators, as well as the Green, Stokes, Gauss, and Helmholtz theorems. Then, we review the physical and mathematical aspects of dielectric relaxation processes exhibiting non-exponential decay in time, focusing the attention on the time-harmonic relative permittivity function based on a general fractional polynomial series approximation. The different topics pertaining to the incorporation of the power-law dielectric response in the FDTD algorithm are explained, too. In particular, we discuss in detail a home-made fractional calculus-based FDTD scheme, also considering key issues concerning the bounding of the computational domain and the numerical stability. Finally, some examples involving different dispersive dielectrics are presented with the aim to demonstrate the usefulness and reliability of the developed FDTD scheme.

Convergence Rate to the Tracy–Widom Laws for the Largest Eigenvalue of Wigner Matrices

We show that the fluctuations of the largest eigenvalue of a real symmetric or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws at a rate O(N^{-1/3+omega }), as N tends to infinity. For Wigner matrices this improves the previous rate O(N^{-2/9+omega }) obtained by Bourgade (J Eur Math Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515, 2012) to prove edge universality, on a finer spectral parameter scale with improved error estimates. The proof relies on the continuous Green function flow induced by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions from the third and fourth order moments of the matrix entries are obtained using iterative cumulant expansions and recursive comparisons for correlation functions, along with uniform convergence estimates for correlation kernels of the Gaussian invariant ensembles.

Kinetic Energy and Vorticity Perspectives of the Rapid Development of an Explosive Extratropical Cyclone Over the Northwest Pacific Ocean in February 2018

Explosive extratropical cyclones (EECs) have long been a research focus for the meteorological society as they often cause serious economic losses and casualties. However, after a long period of research, there still remain some knowledge gaps about their rapid development. In this article, we conducted the first study by using both vorticity and kinetic energy (KE) budgets simultaneously on a typical EEC, which was the strongest EEC that affected the coastal areas of China in the last 3 years, to further the understanding of the mechanisms governing its rapid enhancement in rotation and wind speed. The vorticity budget shows that the lower-level convergence-related vertical stretching and the vertical transport of vorticity acted as the most and second most favorable factors for the increase in the cyclone's cyclonic vorticity, respectively, which were different from those findings based on the Zwack–Okossi vorticity budget. In contrast, the horizontal transport of vorticity and tilting mainly decelerated the EEC's development. Energetics features governing the rapid wind enhancement of the EEC were shown for the first time. It is found that the work on the rotational wind by the pressure gradient force and the net import transport of KE by the rotational wind contributed the largest and second largest to the cyclone's increase in wind speed. In contrast, the upward transport of KE and the cyclone's displacement mainly acted in an opposite way. Analysis based on the Green's theorem and rotational wind shows that the enhancement of the EEC's rotation and wind speed were linked to each other solidly.

A Comparison Study of Students Performance for Vector Calculus Subject

The research objective was to compare the performance of Engineering students in a pretest and post-test of Vector Calculus subject between programmes during their first year of study at the Faculty of Engineering and Built Environment (FKAB), Universiti Kebangsaan Malaysia (UKM). This paper also aims to investigate the student's knowledge in Vector Calculus, particularly in main topics including partial derivatives, vector functions, line integrals, double integrals, triple integrals, Green's Theorem, Stokes' theorem, Gauss' theorem, and basic differentiation and integration of complex functions. Each topic is included in Course learning outcomes to analyze either the students show a good or bad performance for Vector calculus subject. 412 first year engineering students from different programmes JKAS, JKMB, JKKP and JKEES were needed to seat for pretest and post-test (Test 1, 2 and 3). The pretest was performed in early of the semester, while the post-test was held afterwards. The analysis results revealed that the performance of students in the post-test was outstanding to their pretest performance. Data analysis of student performance indicated that most students did not seem to understand the basic ideas in each chapter prior to the beginning of the semester.