Explicit Mathematical Expressions Research Articles

Asymmetric Probability Mass Function for Count Data Based on the Binomial Technique: Synthesis and Analysis with Inference

In this article, a new probability mass function for count data is proposed based on the binomial technique. After introducing the methodology of the newly model, some of its distributional characteristics are discussed in-detail. It is found that the newly model has explicit mathematical expressions for its statistical and reliability properties, which is not the case with many well-known discrete models. Moreover, it can be used as an effectively probability tool for modeling asymmetric over-dispersed data with leptokurtic shapes. The parameters estimation through the classical point of view have been done via utilizing the technique of maximum likelihood and Bayesian approaches. A MCMC simulation study is carried out to examine the performance of the estimators. Finally, two distinct real data sets are analyzed to prove the flexibility and notability of the newly model.

Developing Equations for Free Vibration Parameters of Bistable Composite Plates Using Multi-Objective Genetic Programming.

For the last three decades, bistable composite laminates have gained publicity because of their outstanding features, including having two stable shapes and the ability to change these states. A common challenge regarding the analysis of these structures is the high computational cost of existing analytical methods to estimate their natural frequencies. In the current paper, a new methodology combining the Finite Element Method (FEM) and Multi-Objective Genetic Programming (MOGP) is proposed for the analysis of bistable composite structures, leading to some analytical relations derived to obtain the modal parameters of the shells. To achieve this aim, the data extracted from FEM, consisting of the ratio of the length to width (a/b) and the thickness (t) of the laminate, is split into Train and Validation, and Test, subsets. The former is used in MOGP, and four formulas are proposed for the prediction of the free vibration parameters of bistable laminates. The formulas are checked against the Test subset, and the statistical indices are calculated. An excellent performance is observed for all GP formulas, which indicates the reliability and accuracy of the predictions of these models. Parametric studies and sensitivity analyses are conducted to interpret the trend of input parameters in the GP models and the level of sensitivity of each natural frequency formula to the input parameters. These explicit mathematical expressions can be extended to the other bistable laminates to obtain their natural frequencies on the basis of their geometrical dimensions. The results are validated against the experimental data and verified against FEM outcomes.

Robust topology optimization of structures under uncertain propagation of imprecise stochastic-based uncertain field

This study introduces a novel computational framework for Robust Topology Optimization (RTO) considering imprecise random field parameters. Unlike the worst-case approach, the present method provides upper and lower bounds for the mean and standard deviation of compliance as well as the optimized topological layouts of a structure for various scenarios. In the proposed approach, the imprecise random field variables are determined utilizing parameterized p-boxes with different confidence intervals. The Karhunen–Loève (K–L) expansion is extended to provide a spectral description of the imprecise random field. The linear superposition method in conjunction with a linear combination of orthogonal functions is employed to obtain explicit mathematical expressions for the first and second order statistical moments of the structural compliance. Then, an interval sensitivity analysis is carried out, applying the Orthogonal Similarity Transformation (OST) method with the boundaries of each of the intermediate variable searched efficiently at every iteration using a Combinatorial Approach (CA). Finally, the validity, accuracy, and applicability of the work are rigorously checked by comparing the outputs of the proposed approach with those obtained using the particle swarm optimization (PSO) and Quasi-Monte-Carlo Simulation (QMCS) methods. Three different numerical examples with imprecise random field loads are presented to show the effectiveness and feasibility of the study.

Relevant integrals of NURBS and its application in hull line element design

NURBS are a powerful mathematical tool for representing free-form curves and surfaces and have been the international industrial standards for the representation, design, data exchange of geometric information. In this paper, starting from the NURBS matrix form, a piecewise-integration is used to derive the formula of relevant integrals of NURBS curve based on the properties of NURBS. A benefit of matrix representation for NURBS is that the explicit mathematical expression of each curve segment is obtained. The area and perimeter of a circle and an ellipse are evaluated from the matrix representation for NURBS. The integrand function from the matrix is solved by composite Newton-Cotes formula and composite Gauss-Legendre formula respectively and the relative error is evaluated with the analytical solution. Results show that three-point composite Gauss-Legendre rule has fast convergence, high accuracy and low evaluation cost. Subsequently, the formula of relevant integrals of NURBS curve is applied to the hull line elements. Taking the hull sectional area curve (SAC) as an example, the evaluation of prismatic coefficient (CP) and longitudinal center of buoyancy (LCB) is carried out; taking the design waterline (DWL) as an example, the evaluation of waterplane coefficient (CW) and transverse center of floation (Xf) is carried out; taking the middle cross section as an example, the evaluation of midship section area coefficient (CM) is carried out. Finally, taking the revolution as an example, the surface area and volume are evaluated. Through these real ship examples, the effectiveness and universality of the engineering application in hull line element design are verified.

A Deep Two-State Gated Recurrent Unit for Particulate Matter (PM2.5) Concentration Forecasting

Air pollution is a significant problem in modern societies since it has a serious impact on human health and the environment. Particulate Matter (PM2.5) is a type of air pollution that contains of interrupted elements with a diameter less than or equal to 2.5 m. For risk assessment and epidemiological investigations, a better knowledge of the spatiotemporal variation of PM2.5 concentration in a constant space-time area is essential. Conventional spatiotemporal interpolation approaches commonly relying on robust presumption by limiting interpolation algorithms to those with explicit and basic mathematical expression, ignoring a plethora of hidden but crucial manipulating aspects. Many advanced deep learning approaches have been proposed to forecast Particulate Matter (PM2.5). Recurrent neural network (RNN) is one of the popular deep learning architectures which is widely employed in PM2.5 concentration forecasting. In this research, we proposed a Two-State Gated Recurrent Unit (TS-GRU) for monitoring and estimating the PM2.5 concentration forecasting system. The proposed algorithm is capable of considering both spatial and temporal hidden affecting elements spontaneously. We tested our model using data from daily PM2.5 dimensions taken in the contactual southeast area of the United States in 2009. In the studies, three evaluation matrices were utilized to compare the overall performance of each algorithm: Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and Mean Absolute Percentage Error (MAPE). The experimental results revealed that our proposed TS-GRU model outperformed compared to the other deep learning approaches in terms of forecasting performance.

Modeling of methane adsorption capacity in shale gas formations using white-box supervised machine learning techniques

Energy demand is increasing worldwide and shale gas formations have gained increasing attention and have become crucial energy sources. Therefore, accurate determination of shale gas-in-place (GIP) is vital for a successful production plan. Since most of the gas in shale formation is in the form of adsorbed gas, the determination of methane adsorption capacity is a very important task. In this study, two rigorous data-driven techniques, namely gene expression programming (GEP) and group method of data handling (GMDH), were utilized to provide accurate and reliable explicit mathematical expressions for predicting methane adsorption. For this purpose, a comprehensive database involving 352 data points was gathered from the literature. Pressure, temperature, moisture, and total organic carbon (TOC) were employed as input variables for the implemented correlations. Results indicate that both correlations can provide accurate predictions. However, the GEP-based correlation exhibits more reliable predictions for methane adsorption with a correlation coefficient of 0.9837. Moreover, it was shown that GEP-based correlation can accurately predict the variation of shale gas capacity for the alteration of inputs. Further, it was revealed that methane adsorption is highly dependent on moisture value, while temperature, TOC, and pressure are the most influential variables after moisture. The results of this study shed light on the power of GMDH and GEP modeling approaches, and show that these models can be employed to provide accurate and simple-to-use correlations for estimating methane adsorption in shale gas formations. • Methane adsorption capacity in shale gas formations was modeled using GMDH and GEP. • Extensive database was used for the development of the correlations. • The proposed correlations are reliable and user-friendly. • The proposed GEP-based correlation is deemed the best paradigm.

Mathematical framework for describing multipartite entanglement in terms of rows or columns of coefficient matrices

In this paper, we develop a mathematical framework for describing entanglement quantitatively and qualitatively for multipartite qudit states in terms of rows or columns of coefficient matrices. More specifically, we propose an entanglement measure and separability criteria based on rows or columns of coefficient matrices. This entanglement measure has an explicit mathematical expression by means of exterior products of all pairs of rows or columns in coefficient matrices. It is introduced via our result that the [Formula: see text]-concurrence coincides with the entanglement measure based on two-by-two minors of coefficient matrices. Depending on our entanglement measure, we obtain the separability criteria and maximal entanglement criteria in terms of rows or columns of coefficient matrices. Our conclusions show that just like every two-by-two minor in a coefficient matrix of a multipartite pure state, every pair of rows or columns can also exhibit its entanglement properties, and thus can be viewed as its smallest entanglement contribution unit too. The great merit of our entanglement measure and separability criteria is two-fold. First, they are very practical and convenient for computation compared to other methods. Second, they have clear geometric interpretations.

Mathematical Relation between Concurrence and Intensity of a Photon in the Quantum Delayed-Choice Experiment

I provide the explicit mathematical expression of the correlation between concurrence and the intensity of a photon in the quantum delayed-choice experiment. This shows us that one may observe the wave-particle duality and entanglement simultaneously.

Determination of Logistic Model Parameters From AM2 Model for Methane Production for Batch Bioreactors

Modeling the estimation of cumulated methane production generated from batch anaerobic bioreactors is of paramount importance. In this context, there are two main modeling approaches. The first approach is based on developing mathematical expressions representing the processes involved in the bioreactors. The AM1 model is known as the most complete one. However, it is a complicated model as it requires about 80 parameters to be tuned. A model named AM2 which is a simplified version of AM1 has also been developed. It is based on only two microbial growth processes and requires only 13 parameters. Nevertheless, both AM1 and AM2 models do not provide explicit mathematical expressions that enable to estimate the temporal evolution of methane production with respect to the parameters involved in the considered models. These models are much more useful for simulations and graphical visualizations of the dynamical behavior of the state variables including methane production. On the other hand, the second approach suggests semi-empirical or data-driven models which are based on simple explicit mathematical expressions that provide an estimate of the cumulated methane production (Logistic model, Gompertz model, etc.). But, this type of models require the identification of few parameters which are extracted from experimental data. Usually, they are simplistic and use only one growth process and thus cannot exhibit the influence of the many parameters involved in such complex dynamic biotechnological systems. In this paper, to overcome the complexity of the first type of models and to avoid the dependency on experimental data in the second type; an explicit analytical mathematical expression is proposed for estimating the cumulated methane production for batch anaerobic bioreactors. This analytical expression is derived via the adoption of some appropriate approximations performed on the set of differential equations characterizing the AM2 model. Therefore, the proposed analytical expression can be considered as an approximation of the AM2 model itself and this is the first contribution of this paper. Graphical profiles of the cumulated methane production are presented showing that of the proposed logistic expression and that of the AM2 model considered as a reference. To the best of the author’s knowledge, no such approach and result have been encountered in the literature. On the other hand, this expression resembles formally to the semi-empiric logistic model. However, the equivalent parameters of the proposed expression as self-defined by the parameters of the AM2 model and do not require experimental data to be identified as it is for the semi-empiric logistic model. Moreover, by comparing the proposed logistic model to the semi-empiric logistic model, an identification of the parameters of the semi-empiric model is linked to the parameters of the AM2 model, providing more insight into the methane production. This can be considered as a second contribution of this paper.

Smart learning strategy for predicting viscoelastic surfactant (VES) viscosity in oil well matrix acidizing process using a rigorous mathematical approach

This piece of study attempts to accurately anticipate the apparent viscosity of the viscoelastic surfactant (VES) based self-diverting acids as a function of VES concentration, temperature, shear rate, and pH value. The focus not only is on generating computer-aided models but also on developing a straightforward and reliable explicit mathematical expression. Towards this end, Gene Expression Programming (GEP) is used to connect the aforementioned features to and the target. The GEP network is trained using a wide dataset adopted from open literature and leads to an empirical correlation for fulfilling the aim of this study. The performance of the proposed model is shown to be fair enough. The accuracy analysis indicates satisfactory Root Mean Square Error and R-squared values of 7.07 and 0.95, respectively. Additionally, the proposed GEP model is compared with literature published correlations and established itself as the superior approach for predicting the viscosity of VES-based acids. Accordingly, the GEP model can be potentially served as an efficient alternative to experimental measurements. Its obvious advantages are saving time, lowering the expenses, avoiding sophisticated experimental procedures, and accelerating the diverter design in stimulation operations.Article HighlightsThe Gene Expression Programming evolutionary algorithm is proposed for modeling the viscosity of Viscoelastic Surfactant-based self-diverting acids.The viscoelastic surfactant viscosity correlation presents high accuracy which is demonstrated through multiple analyses.The Gene Expression Programming algorithm is a reliable tool expediting the diverter design phase of each stimulation operation.

A novel satellite layout optimization design method based on phi-function

The purpose of this paper is to present a novel satellite layout optimization design (SLOD) approach based on the phi-function, which provides a simple, effective and simultaneously accurate way to handle the geometry constraints (including non-overlapping constraint and distance constraint) with an explicit mathematical expression. Firstly, an extended basic SLOD model concerning the system mass characteristics is formulated with the introduction of a generic minimum/maximum distance constraint between components. In the previous literature, most of the investigated SLOD applications only involve the planar circular and rectangular components with fixed orientation due to the lack of an effective and accurate overlap calculation method, let alone control the distance between them. Therefore, the idea of using phi-functions to prevent overlap and control distance between components is put forward to alleviate this difficulty. Furthermore, to effectively prevent the container protrusion from the components, the phi-function method is extended by our proposed finite-component-boundary (FCB) method. Despite its positive effects, the limitation of the phi-function is clearly discussed to provide a comprehensive view on this useful tool. Finally, the validity and the effectiveness of the proposed methodology are demonstrated by two numerical packing design examples, two typical 2D SLOD applications and one 3D cube-shaped SLOD application.

Toward smart schemes for modeling CO2 solubility in crude oil: Application to carbon dioxide enhanced oil recovery

This paper presents an artificial intelligence-based numerical investigation on the CO2 solubility in live and dead oils for possible CO2-enhanced oil recovery (EOR). A thorough smart modeling was accomplished by utilizing Multilayer Perceptron (MLP) and Radial Basis Function (RBF) neural network predictors integrated with seven vigorous optimization algorithms. Furthermore, Group Method of Data Handling (GMDH) approach was manipulated to achieve explicit mathematical expressions for the scope of the current study. The modeling was performed on a rich source of data derived from the previously published works. Assessments regarding all extended models demonstrated the Absolute Average Relative Error (AARD) ranges of 1.19%–3.47% and 1.63%–3.13% for live and dead oils, respectively. This indicates the prosperousness of all suggested models for anticipating the CO2 solubility in live/dead oil. A comparison between the proposed models indicated the marginally better performance of the MLP-LM (AARD = 1.19%) and MLP-SCG (AARD = 1.63%) in the case of live and dead oils, respectively. Additionally, the implemented models were compared against various published approaches, and the results revealed that the majority of our newly generated models outperform the prior approaches. In addition, the established GMDH-derived correlations were found to be the most truthful in comparison to other explicit literature correlations. These results provide significant insights for understanding the complex physicochemical processes of CO2-EOR and accurately predicting CO2 solubility in live and dead oils in reservoirs.

Monitoring multivariate coefficient of variation with upward Shewhart and EWMA charts in the presence of measurement errors using the linear covariate error model

AbstractIn practice, measurement errors exist and ignoring their presence may lead to erroneous conclusions in the actual performance of control charts. The implementation of the existing multivariate coefficient of variation (MCV) charts ignores the presence of measurement errors. To address this concern, the performances of the upward Shewhart‐MCV and exponentially weighted moving average MCV charts for detecting increasing MCV shifts, using a linear covariate error model, are investigated. Explicit mathematical expressions are derived to compute the limits and average run lengths of the charts in the presence of measurement errors. Finally, an illustrative example using a real‐life dataset is presented to demonstrate the charts’ implementation.

Sub-Terahertz Complex Permittivity Measurement Method Using Cavity Switches

New method of measurement of dielectric constant and loss tangent for small samples of lossy dielectrics, including plain semiconductors, using non-oversized resonators being parts of the recently developed cavity switches for sub-terahertz bands is proposed. Strong influence of the tangent loss value to the Q-factor of the resonator is calculated and observed in experiments. A simple monosemantic algorithm to retrieve explicit mathematical expressions for the complex permittivity from the experimental data is demonstrated. The advantage of the new method over the known ones is much lower (up to million times) volume of the sample needed to analyze.

A modified parallelepiped model for non-probabilistic uncertainty quantification and propagation analysis

As a typical convex model, the parallelepiped plays an important role in the non-probabilistic uncertainty quantification with simultaneous dependent and independent variables. To overcome the complexity of the conventional geometric design-based method, this paper proposes a more efficient sample-driven procedure to construct the explicit mathematical expression of parallelepiped model. Instead of the geometric characteristics of minimum-volume parallelepiped, the statistical characteristics of available samples are employed to directly evaluate the marginal intervals and correlation coefficients of uncertain variables. Especially for the inconstant uncertainty problem with dispersed samples, a sub-parallelepiped modeling method is further presented by means of the sample clustering analysis, which can effectively decrease the invalid domains in uncertainty quantification. Besides, in order to improve the computing efficiency of uncertainty propagation analysis under the parallelepiped model, the radial basis function-based surrogate model is introduced as an approximation of the original time-consuming computational model. Finally, two numerical examples verify the effectiveness of the proposed model and method.

An integrated computational intelligence technique based operating parameters optimization scheme for quality improvement oriented process-manufacturing system

The analysis and improvement of product quality for process industry is an increasing concern for academia and industry. As the outputs of a manufacturing system mainly depend on corresponding input conditions, so it is of high significance to develop an optimization scheme to actively and accurately determine operating parameters to obtain desired quality. However, the widely employed single-model modeling mode for whole production process neglects the natural characteristics within process manufacturing system such as multistage manufacturing and hysteresis. Additionally, the popular data-driven modeling techniques in current works, especially black-box machine learning models have been restricted to satisfying the requirements regarding excellent approximation capability and explicit mathematical expression simultaneously. To fill up above research gap, it is meaningful to develop a new data-driven optimization scheme in this work to effectively and accurately determine the optimum operating parameters considering the abovementioned characteristics and requirements. Firstly, two different connecting strategies are discussed to determine the more accurate and feasible quality propagation mode between adjacent stages. Then, two computational intelligence (CI) techniques, i.e., Multi-Gene Genetic Programming (MGGP) and Multi-objective Particle Swarm Optimization (MOPSO) algorithm are exploited to construct correlation model with explicit mathematical expression and derive the optimal operating parameters, respectively. Afterwards, the fuzzy Multi-criteria Decision Making (FMCDM) method is further proposed to select the optimal solution from the obtained Pareto solutions sets. The application of the proposed scheme in a coal preparation process indicates that the proposed scheme is promising and competitive on prediction accuracy and optimization efficiency over baseline methods, and can significantly improve the final product quality comparing with initial parameters setting. Moreover, the feasible quality specification for intermediate product can also be obtained by our proposed scheme which is beneficial for early detection of quality abnormality and timely parameters adjustment.

Comments on “Diathermal Heat Transport in a Global Ocean Model”

AbstractHolmes et al. (2019) have proposed a new theoretical framework for studying ocean heat uptake in potential temperature coordinates. One important step in their derivations requires understanding the temporal changes of the volume of water V with temperature greater than some value, which they write as the sum of two terms. The first one is due to the surface freshwater fluxes and is well defined, but the second one—attributed to the volume fluxes through the lower boundary of the domain—is given no explicit expression. What the authors mean exactly is unclear, however, because in the incompressible Boussinesq approximation, the use of a divergenceless velocity field implies that the sum of the volume fluxes through any kind of control volume must integrate to zero at all times. In this comment, we provide two alternative explicit mathematical expressions linking the volume change of Holmes et al. (2019) to the diabatic sources and sinks of heat that clarify their result. By contrasting Holmes et al.’s (2019) approach with that for a fully compressible ocean, it is concluded that the volume considered by Holmes et al. (2019) is best interpreted as a proxy for the Boussinesq mass M0 = ρ0V, where ρ0 is the reference Boussinesq density. If V were truly meant to represent volume rather than a proxy for the Boussinesq mass, the Boussinesq expression for dV/dt would have to be regarded as inaccurate because of its neglect of the volume changes resulting from mean density changes.

Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator

Abstract In this paper, a design of Poincare maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period–m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincare map from the structure of several local maps. An expression of the generalized controlled Poincare map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT–piecewise–constant control law, a T–piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period–m limit cycle and hence the chaotic motion is controlled.

Evaluation of van Genuchten-Mualem model on the relative permeability for unsaturated flow in aperture-based fractures

The relative permeability versus saturation relation is fundamental for modeling unsaturated flow processes in aperture-based fractures. On the basis of the local cubic law and the connectivity probability of aperture distribution within single fractures, an analytical relationship between the relative permeability and saturation is established. When the van Genuchten function on the description of the capillary pressure-saturation relationship is applied, an explicit mathematical expression between the relative permeability and saturation is gained similar to the form of the van Genuchten-Mualem (VG-M) model of porous media. Through the comparisons between the several series of laboratory-measured data available in the literature and theoretical results, good agreements are both presented and they suggest that the proposed VG-M model can be successfully used for aperture-based fractures as well as porous media. To understand aperture variation dependence of the relative permeability properties, a computational method combining fractal theory and the invasion percolation model to determine the empirical parameters associated with the spatial correlation of aperture distribution is developed. Simulated fractures with spatially correlated and uncorrelated aperture distributions corresponding respectively to small and large fractal dimensions are both studied. According to the simulated results, the residual saturation generally increases with fractal dimension increments. However, the value of m exhibits a weak dependence on fractal dimension. The proposed model can be applied to various engineering such as seepage control for rock slopes, petroleum reservoirs, nuclear waste disposal and geologic storage of carbon dioxide.

Residual Motion Error Correction with Backprojection Multisquint Algorithm for Airborne Synthetic Aperture Radar Interferometry.

For airborne interferometric synthetic aperture radar (InSAR) data processing, it is essential to achieve precise motion compensation to obtain high-quality digital elevation models (DEMs). In this paper, a novel InSAR motion compensation method is developed, which combines the backprojection (BP) focusing and the multisquint (MSQ) technique. The algorithm is two-fold. For SAR image focusing, BP algorithm is applied to fully use the navigation information. Additionally, an explicit mathematical expression of residual motion error (RME) in the BP image is derived, which paves a way to integrating the MSQ algorithm in the azimuth spatial wavenumber domain for a refined RME correction. It is revealed that the proposed backprojection multisquint (BP-MSQ) algorithm exploits the motion error correction advantages of BP and MSQ simultaneously, which leads to significant improvements of InSAR image quality. Simulation and real data experiments are employed to illustrate the effectiveness of the proposed algorithm.