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The positive solutions of the equation $$x^y = y^x$$ x y = y x have been discussed for over two centuries. Goldbach found a parametric form for the solutions, and later a connection was made with the classical Lambert function, which was also studied by Euler. Despite the attention given to the real equation $$x^y=y^x$$ x y = y x , the complex equation $$z^w = w^z$$ z w = w z has virtually been ignored in the literature. In this expository paper, we suggest that the problem should not be simply to parametrise the solutions of the equation, but to uniformize it. Explicitly, we construct a pair z(t) and w(t) of functions of a complex variable t that are holomorphic functions of t lying in some region D of the complex plane that satisfy the equation $$z(t)^{w(t)} = w(t)^{z(t)}$$ z ( t ) w ( t ) = w ( t ) z ( t ) for t in D. Moreover, when t is positive these solutions agree with those of $$x^y=y^x$$ x y = y x .

This paper presents a quadratically constrained multiobjective quadratic fractional programming model (MOQFPM) and proposed a methodology to obtain a best preferred solution with the help of parametric functions and using fuzzy goal programming. In the initial stage, we obtain a non-fractional optimization model from the multi-objective quadratic fractional programming model by assigning a vector of parameters to fractional functions. Then, in the next stage, we use fuzzy goal programming approach to obtain the best preferred solution for the decision maker to the optimization problem by finding membership functions and aspiration levels of each objective function. This methodology proposes an efficient method to obtain Pareto-optimal solution of MOQFPM.

The effective notch stress approach for evaluating the fatigue strength of rib—deck welds requires notch stress concentration factors obtained from complex finite element analysis. To improve the efficiency of the approach, the notch stress concentration factors for three typical fatigue-cracking modes (i.e., root—toe, root—deck, and toe—deck cracking modes) were thoroughly investigated in this study. First, we developed a model for investigating the effective notch stress in rib—deck welds. Then, we performed a parametric analysis to investigate the effects of multiple geometric parameters of a rib—deck weld on the notch stress concentration factors. On this basis, the multiple linear stepwise regression analysis was performed to obtain the optimal regression functions for predicting the notch stress concentration factors. Finally, we employed the proposed formulas in a case study. The notch stress concentration factors estimated from the developed formulas show agree well with the finite element analysis results. The results of the case study demonstrate the feasibility and reliability of the proposed formulas. It also shows that the fatigue design curve of FAT225 seems to be conservative for evaluating the fatigue strength of rib—deck welds.

In this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries. In addition, several applications are discussed to show the simplicity and applicability of the procedure, and comparison with pertinent data in the literature assesses the accuracy of the proposed approach.

Because of the singular form and low flexibility of the clothoid as a highway transition curve in highway horizontal alignment, the K-curve, whose ratio of the chord-tangent angle to the tangential angle at any point is fixed, was proposed in a previous work, including the corresponding parametric equations and basic characteristics, and its high flexibility and various shapes were proved. Here, the application and verification of the K-curve in highway alignment is further studied. The solution methods of the K-curve for different connections are given. Through theoretical calculations and specific real-world examples, the geometric positions and lateral forces of the K-curve and the clothoid under different connection conditions are compared. The results show that the difference between the K-curve and clothoid in the above two aspects under a straight line-to-circle connection is greater than that under a circle-to-circle connection as an egg curve, and although the lateral force coefficient of the K-curve is smaller than that of the clothoid, the K-curve suffers from rapid change in the lateral force coefficient and a higher lateral acceleration change rate. Finally, the application conditions of the K-curve for highway alignment design are given with consideration of the centrifugal acceleration change rate.

The classical Kuramoto model serves as a useful tool for studying synchronization transitions in coupled oscillators that is limited to the sinusoidal and pairwise interactions. In this paper, we extend the classical Kuramoto model to incorporate the high-order structures and non-pairwise interactions into the coupling function. Using a self-consistent approach and constructing parametric functions, we describe the extensive multi-cluster states induced by high-order structures and identify various types of phase transitions toward synchrony. In particular, we establish the universal scaling relation for each branch of multiclusters, which describes the asymptotic dependence of the order parameters (Kuramoto and Daido) on the coupling strength near the critical points.

In this paper, we consider environmental boundaries that can be represented by a time-varying closed curve. We use n robots equipped with location sensors to sample the dynamic boundary. The main difficulty during the prediction process is that only n boundary points can be observed at each time step. Our approach combines finite Fourier series for shape-estimation and polynomial fitting for point tracking in time. This combination gives a continuous parametric function that describes the boundary shape and its dynamics. We validate our strategy in simulation and with experiments using actual robots. We tested on non-convex boundaries assuming noisy measurements and inaccurate motion actuators.

This article reveals a mathematical bridge between compass-only geometric constructions and parametric curves. A set of construction algorithms are presented which are used to locate points on any Bezier curve or B-Spline by using an abstract compass. These constructions aim to translate the B-Spline definition of Paul de Casteljau into the modified versions of compass-only constructions explained by Aleksandr Kostovskii. Several computer scripts are developed and used to simulate these constructions while calculating and minimizing their computational complexities. The mathematical bridge explained in this article is expected to bring back to mind the synthetic and axiomatic roots of design geometries.

Referring to the error modeling technology used in precision machine tools, it is difficult for machine tool builders to understand the effects of the theoretical modeling accuracy on the precision machining in the designing stages, where error components are represented by parameters. Therefore, this study is proposed to overcome certain theoretical calculation errors for a parametric form of volumetric error modeling and to examine verification method for judging the modeling precision. Based on the mathematical theory, a novel optimized algorithm is presented, as well as its verification methods. With the two new methods, it is effective to identify theoretical calculation errors and verify the accuracy of error modeling. Meanwhile, the proposed algorithm, with a form grinding machine tool being as an illustration example, is tested and validated by means of numerical simulations in the Matlab. The results reveal that the modeling accuracy of the error characteristic matrix is improved and enhanced by the algorithm, and the two verification methods can check the veracity of parametric modeling precision in different ways. Especially, the second method can check and isolate quantitatively the theoretical calculation errors. At the same time, it is a theoretical guidance for choosing a suitable treatment to meet the accuracy of an error model represented by parametric variables during iterations of the characteristic matrices. These two prediction veracity and uncertainty of the modeling precision can be evaluated. Furthermore, the algorithm and checking methods can be extended to ultra-precision multi-axis machine tools.

Compared to conventional linear and circular segments, the parametric curve has advantages in approximating and machining, guaranteeing higher machining precision and better machining quality. Therefore, in Computer Numerical Control (CNC) systems, parametric toolpath has become more and more popular for its high-performance in practical machining and the performance of the parametric interpolator has become a standard to illustrate the advanced CNC systems. In this paper, adaptive-feedrate planning and the iterative interpolator have been developed to generate sampling points with respect to error and higher order kinematic constraints. Firstly, a jerk-limited feedrate profile and look-ahead algorithm are utilized to provide reachable feedrate at critical points. Afterwards, normal jerk (i.e. the jerk in a normal direction) are limited to improve machining precision and quality. In theory, it is proved that the normal jerk limitations have relationship with curvature derivative and a novel iterative interpolator is proposed to ensure the normal jerk limitation by considering the corresponding constraints at curvature derivative extreme points. Finally, simulation and experiments are conducted to demonstrate the efficiency and contour performance of the proposed method compared to conventional method and time-optimal method.

Interpolation of parametric curves is one of the most effective methods in high-performance computer numerical control (CNC) machining. Its precision highly depends on chord errors from feedrate scheduling and contour errors from real-time machining. In order to improve machining precision, a novel feedrate optimization method is proposed in this paper. The osculating circle (OC) method can effectively estimate the chord errors, but its precision degrades if the trajectory curvature changes dramatically. Therefore, an iterative algorithm based on OC method is developed to reevaluate the chord errors, which can upgrade the estimation precision and further improve the feedrate constraints. To reduce the contour error, feature zones on the trajectory are firstly selected according to the kinestates and a specified zone length. In each feature zone, the projection of tracking error onto the zone chord-line is calculated as a new vector, based on which an indirect contour error compensation method is proposed through refining the feedrate profile. To ensure that the contour error can be reduced, the compensation performance is pre-estimated before implementation. Finally, simulation results in different scenarios are presented to validate the high precision of the proposed method.

A time-dependent process of directional crystallization with a two-phase (mushy) region with allowance for vigorous convection in the liquid phase is analytically described. An exact solution of non-linear mushy layer equations is found in a parametric form. The temperature and solute concentration distributions, as well as the solid phase fraction profile, are determined in the two-phase and liquid regions. The moving boundary of phase transition, which is placed between the two-phase zone and liquid is defined as a function of time. The present analytical solutions are in good agreement with laboratory experiments on ice grown from aqueous solutions of isopropanol.

We develop a general class of noise-robust estimators based on the existing estimators in the non-noisy high-frequency data literature. The microstructure noise is a parametric function of the limit order book. The noise-robust estimators are constructed as plug-in versions of their counterparts, where we replace the efficient price, which is non-observable, by an estimator based on the raw price and limit order book data. We show that the technology can be applied to five leading examples where, depending on the problem, price possibly includes infinite jump activity and sampling times encompass asynchronicity and endogeneity., Comment: 42 pages

Three-dimensional (3D) reconstruction of objects and scenes from camera images is of great interests due to its wide applications. Reconstruction based on feature point correspondence is an established approach. Existing research on curve-based reconstruction is limited to certain type of curves and constrained by case-dependent reconstruction accuracy. In view of that, this paper developed a new method to reconstruct general 3D curves from stereo images. Under proposal, a B-spline curve fitting is applied to sets of 2D edge points extracted from acquired stereo images. Derived approximating parametric curves are then used to construct conic surfaces. Further, robust iterative algorithms are developed to get intersection of corresponding conic surfaces to recover 3D curve. Due to the method design, proposed approach can reconstruct general 3D curves including both open and closed curves. The curve fitting technique and developed robust algorithms can meet accuracy requirement of many real applications. Validity of the proposed method is verified through experiments on a cylinder and teacup in laboratory and a real forging within a workshop.

In fracture mechanics, the fatigue evaluation depends on the precise provisions of the stress intensity factor (SIF), which is a quantitative factor established to evaluate the effect of stresses along the crack front. Stress intensity factor performs a definitive role in reflecting material fracture and estimating fatigue life of tubular joints. Some researchers have proposed a series of parametric equations to predict SIF values; however, it gave arguably inaccurate results. There is still a need to improve the accuracy of the SIF prediction’s equation. Thus, to overcome this shortage, this study introduced back-propagation (BP) neural network with conjunction to genetic algorithm (GA) (GA-BP) in predicting the SIF of cracked tubular T/Y-joint. To train and verify the results of the hybrid algorithm GA-BP SIFs’ database covers a wide variety of cracked tubular T/Y joints were simulated by ABAQUS and verified against experimental results. Meanwhile, the SIFs’ result produced by the GA-BP optimization method were compared with those SIF calculated from the accessible parametric equations. The comparison indicated that the GA-BP neural network optimization method is reliable, precise and capable tool in calculating the SIFs of cracked T/Y joints and it also provides higher accuracy than common methods.

A new brachistochrone problem of the motion of a material point on a transcendental surface between two given points in a vertical uniform field of gravity is considered. The surface is a cut horizontal cylinder whose directrix is a cycloid, and generatrix is straight lines parallel to the horizontal axis. Energy dissipation is neglected. A time functional is obtained and a differential equation is derived for a single parameter that describes the form of brachistochrons in the problem. Spatial parametric equations of the brachistochronic motion of a material point on a surface are obtained. The fast response time is determined.

Herein, the formation of binary rare earth metal-based amorphous alloys is systematically studied by the bond-parametric function and size factor model. The theoretical results have been obtained for the 1230 binary alloys based on the 15 rare earth metals (La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu). The results revealed that the formed amorphous constituent elements region and the unformed amorphous constituent elements region can be separated precisely from each other by an ellipse curve, and the elliptic equation of $$ \frac{{\left( {x - m} \right)^{2} }}{{c^{2} }} + \frac{{\left( {y - n} \right)^{2} }}{{d^{2} }} = 1 $$ . The constants m, n, c, and d in the elliptic equation are positively correlated to the parameters R, y, (µR3)−1/2, and E for each host metal, respectively. The overall reliability of the calculation upon comparison with the experimental data for 92 binary rare earth metal-based amorphous alloys was found to be 92.4%. This indicates that the theoretical results are in very good agreement with the experimental data.

Unreasonable tool-path generation in the penetration and retraction processes of the internal thread milling can easily result in severe interference errors, therefore, this study proposes a less interference tool-path correction model for the half revolution penetration and retraction trajectories to further reduce the tool-path interference errors. Firstly, the parametric equations of the initial penetration and retraction trajectories are derived from the half revolution penetration and retraction angle α. Then, an offsetting coefficient e is defined to adjust the X-coordinate components of the starting point of the penetration trajectory and the end point of the retraction trajectory, the Z-coordinate components of the penetration and retraction trajectories are calculated by substitute the angle α with the cutter spiral rotation angle θ. As a result, the correction parametric equations for adjusting the penetration and retraction trajectories with the e are established. Finally, according to the influence of e on the interference errors, the appropriate value of the e is determined to control the interference errors within the allowable tolerance range. Taking the milling of M16 × 1.5 thread hole with M8 × 1.5 thread milling cutter as an experimental example, the experimental results show that the maximum interference error of the proposed correction model (e = 0.9) can be reduced by 73.6% and 33.82% compared with the initial uncorrection model and the existing optimization method, respectively. This study indicates the proposed model can significantly decrease the interference occurred in internal thread milling and smoothly achieve the demands of more precision threaded connections.

In this study, a non-probabilistic approach-based Navier’s method (NM) and Galerkin weighted residual method (GWRM) in terms of double parametric form have been proposed to investigate the buckling behavior of Euler–Bernoulli nonlocal beam under the framework of Eringen’s nonlocal elasticity theory, considering the structural parameters as imprecise or uncertain. The uncertainties in Young’s modulus and diameter of the beam are modeled in terms of triangular fuzzy numbers. The critical buckling loads are calculated for hinged–hinged, clamped–hinged, and clamped–clamped boundary conditions, and these results are compared with the deterministic model in special cases, demonstrating robust agreement. Further, a random sampling technique-based method, namely Monte Carlo simulation technique (MCST), has been implemented to compute the critical buckling loads of uncertain systems. Also, the critical buckling loads obtained from the uncertain model in terms of lower bound and upper bound by the non-probabilistic methods, viz. NM and GWRM, are again verified with the MCST with their time periods, demonstrating the efficacy, accuracy, and effectiveness of the proposed uncertain model. A comparative study is also carried out among the non-probabilistic methods and MCST to demonstrate the effectiveness of methods with respect to time. Additionally, a parametric study has been performed to display the propagation of uncertainties into the nonlocal system in the form of critical buckling loads.

Hyperbolic-type equations of both phase field and concentration arising from a phase-field model for fast phase transformations in binary dilute systems yield in the one-dimension moving frame of reference to the concentration-and phase field governing equations, respectively. These equations have been solved numerically and applied to the case of Si-0.1 at.% As binary alloy [P.K. Galenko et al., Phys. Rev. E 84, 041143 (2011)]. In this paper, the coupling of the hodograph equation for the interface with the solute diffusion equation leads to an exact analytical solution of the one-point Cauchy problem of an ordinary differential equation in a parametric form. Application of this solution to the case of Si-0.1 at.% As gives (i) the same tendency of concentration variation along dimensionless spatial coordinate (ii) the same values of interface velocity with a very slight difference in the value of concentration for a given undercooling at the interface. Based on the results obtained, the established hodograph-equation confirms again its usefulness to predict, for instance, certain aspects of rapid solidification processes for binary alloys.

The classic Hawkes process assumes the baseline intensity to be constant and the triggering kernel to be a parametric function. Differently, we present a generalization of the parametric Hawkes process by using a Bayesian nonparametric model called quadratic Gaussian Hawkes process. We model the baseline intensity and trigger kernel as the quadratic transformation of random trajectories drawn from a Gaussian process (GP) prior. We derive an analytical expression for the EM-variational inference algorithm by augmenting the latent branching structure of the Hawkes process to embed the variational Gaussian approximation into the EM framework naturally. We also use a series of schemes based on the sparse GP approximation to accelerate the inference algorithm. The results of synthetic and real data experiments show that the underlying baseline intensity and triggering kernel can be recovered efficiently and our model achieved superior performance in fitting capability and prediction accuracy compared to the state-of-the-art approaches.

As a new type of gear transmission developed in recent years, curve-face gear transmission has not appeared before. In order to solve the problem of root bending strength of the eccentric straight and helical curve face gear, a calculating method of the tooth root bending strength was proposed. According to the principle of space meshing, differential geometry and design of machinery, the parametric equations of pitch curve was obtained, the pressure angle and the contact ratio in the meshing process were derived. The plane equivalent pitch curve was obtained, and the formula for calculating the equivalent tooth form factor and the equivalent stress correction factor was acquired. The calculation method of the bending strength of the gear pair was established, and the influence of the basic structural parameters on the bending stress was also analyzed in detail. The position of the stress hazard of the eccentric curve-face gear was determined. Through the finite element analysis and transmission experiment, the accuracy of the calculation method of strength of the gear pair was verified.

Fins are surface extensions used to increase the rate of heat transfer from a heated surface to the surrounding cooler fluid or from a heated fluid to the surface. In this article, we describe new families of straight and pin fins for which the temperature distribution along the fin as well as fin effectiveness and efficiency can be computed in closed form. The profile of the new straight fin is a circular arc centered on the fin’s longitudinal axis, while the profile of the new pin fin is a more complex concave parametric curve. Although the pattern of temperature distribution in these fins is remarkably similar to that in the classic straight and pin fins of constant thickness, the geometry of the new fins is strikingly different; in particular, the maximum length of the new fins is finite. We also discovered that both classic fins of constant thickness and novel fins described in this work have the following equivalence property: if the ratio of the heat transfer coefficient on the fin’s tip to that on the lateral surface is not too high, then the temperature distribution for a fin with a non-adiabatic boundary condition at the tip is identical to that for a longer fin of the same kind with adiabatic tip. Our analysis of heat transfer by fins is only based on standard homogeneity, steady-state, and one-dimensionality assumptions. In particular, we completely dispense with the “length-of-arc” assumption that underlies most of the previous works.

Velocity fields for motion control are a kinematic description of the desired trajectories in the Cartesian space whose applications in industrial processes of contouring tasks are promising; however, the lack of a general designing method and the complexity of the controllers designed without concern for the construction of the fields have confined their use to the academic domain. In this paper, the author introduces a systematic method for designing and constructing velocity fields for control that provides a suitable reference in velocity to simplify the structure of the close loop controllers. The method is a three steps procedure where the motion objective in the vector field is described as a main streamline surrounded by an attractive flow to achieve this objective. The main streamline, expressed as a parametric curve, determines the main directional components of the field as a radial-attraction to the curve and a tangential-guide on the curve; directions which are linearly combined to calculate the velocity vectors into a soft and continuous transition of uniform speed to attract and guide the motion as the position gets closer to the curve. It is showed that the method provides a convergent velocity field, either as a static kinematic description of the motion or as a dynamic reference in velocity for a moving particle, and it is tested designing a circular trajectory field, commonly used in control design, and two generic examples of 2D and 3D curves on the Cartesian space.

This paper addresses the modeling and simulation of a homogeneous disk that undergoes plane motion constrained to be in permanent contact with and in the same plane as a two-dimensional curve described by a parametric equation. Except for the elementary cases in which the curve is a straight line or an arc of circumference, the investigated problem presents significant challenges to both modeling and simulation. In addition to the necessity of using differential geometry methods to calculate local geometrical properties of the motion, this system exhibits typical difficulties of non-smooth dynamics. Different modes of operation require the synthesis of suitable rules for switching systems of differential equations and, due to the discontinuities caused by the exchange of dynamic models, numerical integration methods suitable for solving stiff problems are necessary. It should also be emphasized that the dynamic model here developed shows the Painleve’s paradox, caused by the application of a simplified law of friction (Coulomb’s law) to a rigid body subjected to an unilateral constraint. Among the results obtained in this work, we would like to highlight the following: (i) the realization that both the geometry of the contact curve and that followed by the center of mass of the disk must be considered to construct a correct dynamic model for the motion; (ii) a detailed procedure to identify the instant the disk stops sliding and initiates a pure rolling motion on the two-dimensional track represented by a $\mathfrak{C}_{2}$ class curve $C$ in parametric form; (iii) the realization that, for arbitrary geometries of flat two-dimensional curves of class $\mathfrak{C}_{2}$ and simplified models of friction, it is not possible to establish an arbitrary initial kinematic condition (a necessity for integrating the equations of motion) without incurring in a paradoxical result, known in the literature as Painleve’s paradox. We consider that the approach adopted in this work might contribute to building dynamic models of hybrid systems, especially those with non-elementary geometric characteristics.

A well-calibrated simple and economical viable Ȧngstrom–Prescott model has long been accepted to be more accurate than other surface meteorological data-based models. The major limitation is that it is site dependent. This study exploited the appropriateness of a more generalized Ȧngstrom-based broadband hybrid model in the estimation of solar radiation at seven stations in equatorial region of West Africa. This model features parametric equations that explicitly and accurately account for clear-sky damping processes in the atmosphere. It empirically estimates cloudy sky radiation extinctions using relative sunshine duration. A new cloud transmittance calibration curve that followed the cloud cover patterns of the region of study was also tried. The result indicated that the new cloud transmittance could be unique to equatorial region of West Africa. The performance of the hybrid model, after modification using the new cloud transmittance equation, was tested using mean bias error and root mean squared error. The performance was found to be comparable to the site-dependent, locally calibrated, Ȧngstrom–Prescott model at the calibration stations, and even better at validation stations. The same performance test comparisons with the original version of the hybrid model, and four other site-independent models: globally calibrated, FAO-recommended Ȧngstrom–Prescott models, Hay and Gopinathan models indicated the modified version of the hybrid model as better

Random field theory has been increasingly adopted to simulate spatially varying environmental properties and hydrogeological data in recent years. In a two-dimensional (2D) stochastic analysis, variation of the environmental properties or hydrogeological data along different directions can be similar (i.e., isotropic) or quite different (i.e., anisotropic). To model the spatially isotropic or anisotropic variability in a stochastic analysis, conventional random field generators generally require a vast amount of measurement data to identify the random field parameters (e.g., mean, variance, and correlation structure and correlation length in different directions). However, measurement data available in practice are usually sparse and limited. The random field parameters estimated from sparse measurements might be unreliable, and the subsequent random field modeling or stochastic analysis might be misleading. This underscores the significance and challenge of generating 2D isotropic or anisotropic random fields from sparse measurements. This paper develops a novel 2D random field generator, which does not require a parametric form of correlation function or estimation of correlation length and other random field parameters, and directly generates 2D isotropic or anisotropic random field samples from sparse measurements. The proposed generator is highly efficient because simulation of a 2D random field is achieved by generation of a short 1D random vector. The effectiveness and applicability of the proposed generator are illustrated using isotropic and anisotropic numerical examples.

We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the L2 space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the N best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best N term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation. Ministry of Education (MOE) The research is supported by the Singapore MOE AcRF Tier 1 grant RG30/16 and the MOE Tier 2 grant MOE2017-T2-2-144.

Remote sensing data are often measured or reported over wide spatial footprints with heterogeneous ground cover. Different types of vegetation, however, have unique signatures that evolve throughout the growing season. Without additional information, signals corresponding to individual vegetation types are unidentifiable from satellite measurements. In this paper, we propose a parametric mixture model to describe satellite data monitoring crop development in the US Corn Belt. The ground cover of each satellite footprint is primarily a mixture of corn and soybean. Using auxiliary data from multiple sources, we model the aggregate satellite signal, and identify the signatures of individual crop types, using nonlinear parametric functions. Estimation is performed using a Bayesian approach, and information from auxiliary data is incorporated into the prior distributions to identify distinct crop types. We demonstrate our parametric unmixing approach using data from the European Space Agency’s Soil Moisture and Ocean Salinity satellite. Lastly, we compare our model estimates for the timing of key crop phenological stages to USDA ground-based estimates. Supplementary materials accompanying this paper appear online.

This paper addresses the modeling of projectile motion using fractional models vis-a-vis experimental data. Recently, it was shown that an auxiliary parameter ( $$\sigma $$ ) needs to be included in the fractional modeling to preserve the dimensionality of the physical quantities. In previous studies, $$\sigma $$ was subjected to several restrictions without considering clear and meaningful reasons. Such problems are overcome here and a method for estimating $$\sigma $$ using the experimental data is introduced. A new solution for the two-dimensional projectile motion using the Caputo’s fractional derivative is obtained. An explicit formula for the trajectory of the projectile in vacuum is first derived. Then, the projectile parametric equations in a resistant medium are expressed in terms of the Mittag–Leffler function. The transcendental equations for the time of flight and the time of maximum height are solved numerically. The model agrees with the classical one as the fractional order tends to 1. In view of the superior results, the current numerical modeling approach is validated for this real-world application.

We construct a six-dimensional space of polynomials containing approximations of trigonometric functions. In this way, an approximation of a circle can be represented as a parametric curve in this space. We show that the space has shape preserving representations and we construct the basis with optimal shape preserving properties.

In this work, we considered the new parametrization of a multilayer thin domain. In particular, in contrast to classic approaches, we used several base surfaces and an analytic method with the application of orthogonal polynomial systems. We gave the vector parametric equation of each layer and the system of vector parametric equations of a multilayer thin domain and introduced the geometric characteristics for the proposed parametrization. We also derived the expressions for the transfer components of the second-rank identity tensor and the relations connecting the various families of bases and presented some differential operators, the system of equations of motion, the heat flow equation, the constitutive relations of the theory of the micropolar elasticity, and the Fourier heat conduction law under this parametrization of the thin-body domain. Finally, we gave the classification and statements of boundary value problems in the theory of thin bodies.

Fuzzy fractional partial differential equations are a generalization of classical fuzzy partial differential equation which can, in certain circumstances, provide a better explanation for certain phenomena. In this paper, a compact Crank–Nicholson scheme is developed, analyzed and applied to solve the fuzzy time diffusion equation with fractional order $$0 < \alpha \le 1$$ . It is based on double parametric form of fuzzy numbers and the time fractional derivative is defined using the Caputo definition. The truncation error and stability of the proposed scheme is discussed. The compact Crank–Nicholson scheme is shown to be a feasible scheme via a numerical example.

Voltammetric peak of adsorbed electroactive system with lateral interactions may be described theoretically by a parametric equation; no exact explicit current-potential relation is achievable. Approximate calculation algorithm (LAT) is presented for establishing this relation; separate version of algorithm (LAT1) has been developed for negative values of the interaction parameter. These algorithms allow performing calculations with high accuracy. Unfortunately, precision has been achieved at the expense of simplicity: LAT makes use of up to nine adjustable coefficients. Accuracy of LAT has been compared with accuracy of GLI equation proposed by Aleveque and Levillain Electrochem Commun 67:73–79, 2016.

We are interested in this work to solve the Lagrangian dual of a generalized fractional program (GFP), which gives the minimal value of the primal problem, thanks to some duality results. With the help of a general minimax equality assumption, we give duality results under minimal assumptions. Since the associated parametric programs of the dual of GFP are always concave, we use a general approximating proximal scheme to these subproblems and construct bundle methods by means of approximate values and approximate subgradients of the dual parametric function. As for all dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of the GFP and a sequence of approximate solutions that converges to a solution of the Lagrangian dual of the GFP. For certain classes of problems, the convergence is at least linear.

Dynamic simulation of revolved solids plays an important role in many fields. Aiming at the lacks of solutions in some key aspects, this study establishes governing equation of motion based on theory of variational inequality; designs a compatibility iteration algorithm for solving contact forces; deduces parametric equations of arbitrary cylinder and cone in three-dimensional space; provides corresponding analytical methods to identify contact points between bodies and to calculate volume integral over bodies; proposes a rotation matrix modification approach to conserve volume and shape of rigid body in the case of large rotation. The accuracy, availability, competence, robustness, and application prospects of the presented methodology are demonstrated by several interesting and challenging problems.

In this paper, an iterative technique based on the use of parametric functions is proposed to obtain the best preferred optimal solution of a multi-objective linear fractional programming problem. The decision maker ascertains own desired tolerance values for the objectives as termination constants and imposes them on each iteratively computed objective functions in terms of termination conditions. Each fractional objective is transformed into non-fractional parametric function using certain initial values of parameters. The parametric values are iteratively computed and $$\epsilon $$ -constraint method is used to obtain the pareto (weakly) optimal solutions in each step. The computations get terminated when all the termination conditions are satisfied at a pareto optimal solution of an iterative step. A numerical example is discussed at the end to illustrate the proposed method and fuzzy max–min operator method is applied to validate the obtained results.

The present paper deals with the system of equations comprising a yield criterion together with the stress equilibrium equations under plane strain and plane stress conditions. This system of equations can be studied independently of any flow rule, and its solution supplies the distribution of stress. It is shown that this problem of plasticity theory is reduced to a purely geometric problem. The stress equilibrium equations are written relative to a coordinate system in which the coordinate curves coincide with the trajectories of the principal stress directions. The general solution of the system is constructed giving a relation connecting the two scale factors for the coordinate curves. This relation is used for developing a method for finding the mapping between the principal lines and Cartesian coordinates with the use of a solution of a hyperbolic system of equations. In particular, the mapping between the principal lines and Cartesian coordinates is given in parametric form with the characteristic coordinates as parameters. Finally, a boundary value problem for the region adjacent to an external boundary which coincides with a principal stress trajectory is formulated and its general solution is given.

A wide range of dynamic models, including those of heating, evaporation and ignition processes in fuel sprays, is characterised by large differences in the rates of change of variables. Invariant manifold theory is an effective technique for investigation of these systems. In constructing the asymptotic expansions of slow invariant manifolds, it is commonly assumed that a limiting algebraic equation allows one to find a slow surface explicitly. This is not always possible due to the fact that the degenerate equation for this surface (small parameter equal to zero) is either a high degree polynomial or transcendental. In many problems, however, the slow surface can be described in a parametric form. In this case, the slow invariant manifold can be found in parametric form using asymptotic expansions. If this is not possible, it is necessary to use an implicit presentation of the slow surface and obtain asymptotic representations for the slow invariant manifold in an implicit form. The results of development of the mathematical theory of these approaches and the applications of this theory to some examples related to modelling combustion processes, including those in sprays, are presented.

We continue the study of the Korteweg-de Vries equation in terms of cento-affine curves, initiated by U. Pinkall. A centro-affine curve is a closed parametric curve in the affine plane such that the determinant made by the position and the velocity vectors is identically 1. The space of centro-affine curves is acted upon by the special linear group, and the quotient is identified with the space of Hill's equations with periodic solutions. It is known that the space of centro-affine curves carries two pre-symplectic structures, and the KdV flow is identified with is a bi-Hamiltonian dynamical system therein. We introduce a 1-parameter family of transformations on centro-affine curves, prove that they preserve both presymplectic structures, commute with the KdV flow, and share the integrals with it. Furthermore, the transformation commute with each other (Bianchi permutability). We also describe integrals of the KdV equation as arising from the monodromy of Riccati equations associated with centro-affine curves. We are motivated by our work in progress (joint with M. Arnold, D. Fuchs, and I. Izmenstiev), concerning the cross-ratio dynamics on ideal polygons in the hyperbolic plane and hyperbolic space, whose continuous version is studied in the present note.

Fokker–Planck equation with interval and fuzzy uncertainty has been considered in this paper. Also the derivatives involved with respect to time and space are assumed to be fractional in nature. This problem has been solved using variational iteration method (VIM) along with the double parametric form of fuzzy numbers. For the analysis, both triangular and Gaussian normalised fuzzy sets are taken into consideration. Numerical results for different cases have been obtained and those are depicted in terms of plots and are also compared in special cases for the validation. Moreover, using an important method known as successive approximation method, it has also been verified that the obtained solutions are the same as that of VIM as both methods are equivalent.

In this paper, based on the previous works, we improve a computational method on solving time-fractional partial differential equation. Using the improved method, a series of time-fractional reaction–diffusion models with Fisher–KPP type are studied from mathematical point of view. Different kinds of exact solutions of four time-fractional reaction–diffusion models are obtained. The forms of these solutions include parametric form, explicit form, and implicit form. Most of them (solutions) have degenerate property according as time increase. The dynamical properties of some representative exact solutions are illustrated by graphs.

Minimal surface is an important type of surface with zero mean curvature. It exists widely in nature. The problem of finding all minimal surfaces presented in parametric form as polynomials is discussed by many authors. However, most of the constructions are based on the theorem that a harmonic surface with isothermal parameterization is minimal. As we all know, Weierstrass representation is a classical parameterization of minimal surfaces. Therefore, in this paper, we consider to construct polynomial minimal surfaces of arbitrary degree by Weierstrass representation. Moreover, there is a correspondence between our constructed polynomial minimal surfaces and Pythagorean hodograph curves. Several numerical examples are demonstrated to illustrate our results.

Electrolysis systems have been widely investigated for treating wastewater in laboratory-scale experiments, and most researchers reported efficient treatment outcomes. However, industrial scale implementation often necessitates assessing and prioritising various options involving different combinations of input parameters, which often requires mathematical models. This study presents mathematical models for predicting removal efficiencies of four different pollutants using electrolysis. Parametric equations were developed based on experimental results conducted in a previous study. Proposed equations are dependent on treatment retention time and current intensity (voltage). Results revealed that the experimental data followed consistent patterns, which lead to the derivations of generalised equations which were able to closely predict pollutants’ removal efficiencies obtained through experimental measurement. The developed equations also confirmed that beyond an optimum voltage, a further increase in voltage does not render higher removals of the pollutants, which is essential for enhancing the feasibility of industrial scale applications.

This study is aimed at analyzing the effects of material uncertainties on the stability of Timoshenko nanobeam. Here, the uncertainties are considered to be associated with the diameter, length, and Young’s modulus of the nanobeam in terms of special fuzzy number, namely Symmetric Gaussian Fuzzy Number (SGFN). The governing equations for stability analysis of uncertain model are obtained by incorporating the Timoshenko beam theory along with Hamilton’s principle and double-parametric form of fuzzy numbers. The small-scale effect of the nanobeam is addressed by Eringen’s elasticity theory, and the double-parametric form-based Navier’s method is employed to calculate the results of the uncertain models for the lower bound (LB) and upper bound (UB) of the buckling loads. The results obtained by the uncertain model are validated with the deterministic model in particular case as well as with the results obtained by Monte Carlo simulation (MCS) technique in terms of the lower bound and upper bound of the buckling loads, exhibiting robust agreement. Further, a parametric study is conducted to investigate the fuzziness or spreads of the buckling loads with respect to different uncertain parameters.

In this paper, we propose a Lyapunov method for estimating asymptotic stability domain of polynomial discrete nonlinear systems, drawn from an existing approach for attraction domain estimation of continuous nonlinear systems. Proceeding from two different forms of a parametric equation, we present two development methods with the aim of identifying a sufficient condition to obtain a part of the stability domain for quadratic and cubic n-dimensional polynomial discrete systems. The results consist in solving a polynomial equation and unconstrained nonlinear optimization problems, where in case of quadratic or cubic systems, significant simplifications are obtained. The main notable advantages of this approach are its applicability on high-dimensional systems and obtaining an ellipsoidal stability domain which approaches the exact attraction domain limits. Two application examples are illustrated to validate each development.

This study presents an automated detection and classification method to facilitate the radiologists in the diagnosis process. The major problem in these systems is the inclusion of false positives in the results, which may lead to inaccurate diagnosis. A nodule detection and classification method is proposed that consists of four major phases. First, the lung region extraction is performed based on optimal gray level threshold that is computed by fractional-order Darwinian particle swarm optimization. Then, a novel nodule candidate detection method, based on geometric fit in parametric form incorporating the geometric properties of the nodules, is proposed. In the next phase, a hybrid geometric texture feature descriptor is created for better representation of the candidate nodules, which is a combination of 2D as well as 3D information about nodule candidates. Finally, a deep learning approach based on stacked autoencoder and softmax for feature reduction and classification is applied to reduce false positives. Performance analysis on the largest publically available dataset, Lung Image Database Consortium and Image Database Resource Initiative, depicts that the proposed method has significantly reduced the number of false positives to 2.8 per scan with a promising sensitivity of 95.6%. The results demonstrate the significance of the methodology in automatic lung nodule detection and classification. Furthermore, it will facilitate and provide assistance to radiologists in precise nodule detection.

Accurate and comprehensive data on cervical endplates is essential for developing and improving cervical devices. However, current literature on vertebral disc geometry is scarce or not suitable. The aim of this study was to obtain quantitative parameters of cervical endplates and provide morphometric references for designing cervical devices. In this study, 19 human cervical spine cadaveric specimens were considered. Employing a reverse engineering system, the surface information of each endplate was recorded in digital cloud and then 3D reconstructed. A measurement protocol that included three sagittal and three frontal surface curves was developed. The information of surface curves and endplate concavity were obtained and analyzed. The parametric equations of endplate surfaces were deduced based on coordinates of landmarks, and the reliability was verified. The cervical endplate surface had a trend that to be transversely elongated gradually. The concavity depths of inferior endplates (1.88 to 2.13 mm) were significantly larger than those of superior endplates (0.62 to 0.84 mm). The most-concave points in inferior endplates were concentrated in the central portion, while always located in post-median region in superior endplates. These results will give appropriate guidelines to design cervical prostheses without sacrificing valuable bone stock. The parametric equations applied for generating surface profile of cervical endplates may provide great convenience for subsequent studies.

Arc-soft-toe bracket (ASTB), as a joint structure in the marine structure, is the hot spot with significant stress concentration, therefore, fatigue behavior of ASTBs is an important point of concern in their design. Since macroscopic geometric factors obviously influence the stress flaws in joints, the shapes and sizes of ASTBs should represent the stress distribution around cracks in the hot spots. In this paper, we introduce a geometric magnification factor for reflecting the macroscopic geometric effects of ASTB crack features and construct a 3D finite element model to simulate the distribution of stress intensity factor (SIF) at the crack endings. Sensitivity analyses with respect to the geometric ratio H t /L b , R/L b , L t /L b are performed, and the relations between the geometric factor and these parameters are presented. A set of parametric equations with respect to the geometric magnification factor is obtained using a curve fitting technique. A nonlinear relationship exists between the SIF and the ratio of ASTB arm to toe length. When the ratio of ASTB arm to toe length reaches a marginal value, the SIF of crack at the ASTB toe is not influenced by ASTB geometric parameters. In addition, the arc shape of the ASTB slope edge can transform the stress flowing path, which significantly affects the SIF at the ASTB toe. A proper method to reduce stress concentration is setting a slope edge arc size equal to the ASTB arm length.

To solve the problem of the difficulties in assembly constraint representation and solution in lightweight 3D modeling because of the lack of geometric features in the Web environment, a parameterized representation and solution method, lightweight 3D modeling virtual assembly constraint, was proposed in this paper. The assembly constraint element (ACE) was formulated for the representation of assembly constraint information, which includes the geometric feature group, type of assembly constraint and assembly constraint expression. The geometric features of the lightweight 3D model were formulated by parametric equation containing vectors. For different geometric feature of different types of assembly constraints, the corresponding simultaneous parametric equations were solved to obtain assembly constraint expressions, which were stored in the ACE, and the requirement of the appointed assembly constraint was met with the combination of a graphics method and a numerical method. On the basis of the assembly constraint expression in ACE, the assembly constraint solution was achieved by the homogeneous transformation matrix and the DOF reduction algorithm. Finally, geometric feature recognition and online virtual assembly of the lightweight 3D model was accomplished by the method proposed in this paper based on WebGL. Relevant tests were conducted to demonstrate the feasibility and efficiency of this method.