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In this paper, by calculating the lower bounds on the nonlinearity of the derivatives of the following three classes of Boolean functions, we provide the tight lower bounds on the second-order nonlinearity of these Boolean functions: (1) \begin{document}$ f_1(x) = Tr_1^n(x^{2^{r+1}+2^r+1}) $\end{document} , where \begin{document}$ n = 2r+2 $\end{document} with even \begin{document}$ r $\end{document} ; (2) \begin{document}$ f_2(x) = Tr_1^n(\lambda x^{2^{2r}+2^{r+1}+1}) $\end{document} , where \begin{document}$ \lambda \in \mathbb{F}_{2^r}^* $\end{document} and \begin{document}$ n = 4r $\end{document} with even \begin{document}$ r $\end{document} ; (3) \begin{document}$ f_3(x,y) = yTr_1^n(x^{2^r+1})+Tr_1^n(x^{2^r+3}) $\end{document} , where \begin{document}$ (x, y)\in \mathbb{F}_{2^n}\times \mathbb{F}_2 $\end{document} , \begin{document}$ n = 2r $\end{document} with odd \begin{document}$ r $\end{document} . The results show that our bounds are better than previously known lower bounds in some cases.

A polynomial-based guidance law is proposed for interception of a stationary target by a dual-seeker interceptor. The guidance law governs the desired look-angle profile as a cubic polynomial of th...

A new approximation for the Green's function of a grounded anisotropic dielectric slab is proposed to obtain closed-form analytical expressions for the maximum radiated power of a short printed dipole on an anisotropic substrate. The corresponding surface wave power can also be accurately estimated as a root of a cubic polynomial which makes it possible to compute radiation efficiency due to surface wave loss analytically.

In this work, we study the bifurcation of limit cycles from the period annulus surrounding the origin of a class of cubic polynomial differential systems; when they are perturbed inside the class of all polynomial differential systems of degree six, we obtain at most fifteenth limit cycles by using the averaging theory of first order.

This paper aims to develop an elegant formula for the series of consecutive cubes of natural numbers under alternating signs. In addition, this paper investigates the formula under odd and even number of terms and discuss some important findings. Keywords: Consecutive cubes; alternating signs; odd and even terms. 2010 Mathematics Subject Classification: 11B13, 11B50, 97I30 Abstrak Paper ini bertujuan membangun formula yang elegan untuk deret berganti tanda bilangan-bilangan kubik berurutan. Paper ini juga menyelidiki formula untuk banyak suku ganjil dan untuk banyak suku genap, dan mendiskusikan beberapa temuan penting. Kata kunci: . Bilangan kubik berurutan, berganti tanda, suku ganjil dan suku genap

Let [Formula: see text] be an arbitrary cubic form. In this paper, we show that for non-homogeneous cubic equations like [Formula: see text], there exists a non-trivial solution when [Formula: see text].

Previously authors have reported facts that nitrogen (N2) and oxygen solubilities in styrene is the same as those in benzene and divinylbenzene at 303 K. Though the three compounds have an atomic composition, (CH)n (n=6, 8 and 10), the gas solubilities are thought to depend on the number of aromatic carbons in the solution. Then, authors named it ‘a key group in solubility’. In this research, the key group was investigated for N2 solubility in methyl methacrylate (MMA) and its dimer, ethylene glycol dimethacrylate (EGDMA). Therefore, the calculations were carried out by Peng-Robinson (PR) equation of state based on group contribution methods. The experimental data employed were those of Lai et al., where N2 solubility are reported in MMA at 303 K. The N2 solubility in EGDMA was assumed to be double of that in MMA at 303 K, because EGDMA has two MMA units in the molecule. Two types of group contribution methods were proposed to calculate N2 solubility in EGDMA. One was for the critical temperature and pressure proposed by Joback and Reid. Using these properties, the attractive and excluded volume parameters in PR equation were evaluated for EGDMA. Other was similar to a method proposed by Orbey and Sandler. The excluded volume parameter in PR equation can be evaluated from those of MMA by multiplying the number of repeating units, and the attractive by multiplying the squared number. Instead of the method, the powers were proposed to be 3/4 and 3/2 for the excluded volume and attractive perimeters, respectively. The powers were come from a scaling theory by de Gennes. The binary parameter for N2 – EGDMA was also set to k12 = 0.224, which was the same as that for N2 –MMA. The calculated N2 solubility in EGDMA was far smaller than the hypothetical N2 solubility in EGDMA at 303 K. Otherwise, to reproduce the hypothetical N2 solubility, a quite different value, k12 = -0.044, was necessary for the method by Joback and Reid. The results suggested that a key group for gas solubility will not be accepted in MMA.

In this paper, we give the characteristic polynomial of the complete 3-uniform hypergraph. We show that the characteristic polynomial is the product of some cubic polynomials, so we can easily compute the spectra of complete 3-uniform hypergraphs. This computation includes a complete description of the eigenvalues' multiplicities. We also show the algebraic multiplicities of 0, 1 and ( n − 1 2 ) of a complete 3-uniform hypergraph with n vertices.

The Bisection, Newton and Horner’s method are used to compare the efficiency to solve nonlinear function such as trigonometric, exponential, logarithmic and cubic polynomial function. Although these methods are considered as alternative, the methods also possess error compared to the exact value. The efficiency is measured by the error produced at the fixed iteration. The methods are converted into C language and executed by using maple 18.

This work proposes a comprehensive numerical dynamic model of a flexible-rotor bearing system based on the Hertzian and cubic polynomial nonlinear contact force methods. The model can consider the influences of the nonlinear bearing contact forces and unbalanced force caused by the rotor offset. The displacements and spectrums of the flexible-rotor bearing system from the Hertzian and cubic polynomial nonlinear contact force methods are discussed. The influences of the radial clearance, eccentricity, mass, and deformation of the rotor on the frequency–amplitude characteristics of the flexible-rotor bearing system considering a large speed range are analyzed. The results show that the dynamic and vibration characteristics of the flexible-rotor bearing system from the Hertzian and cubic polynomial nonlinear contact force methods are different. The differences of the frequency–amplitude characteristics between the flexible- and rigid-rotor bearing system are small in a lower speed stage; however, their differences are very large in a higher speed stage. This method can be applied to the nonlinear dynamic modeling and simulation of the flexible-rotor bearing system, which can predict the dynamics and prevent the system failures during the design processing of rotor system.

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

In this paper we discuss two different existing algorithms for computing topological entropy and we implement one of them in order to compute the isentropes for cubic polynomials.

Both the limit analysis and the limit equilibrium approach require the formulation of a failure criterion in terms of components of stress vector acting on a potential failure plane. For applications in Rock Mechanics, the generalized Hoek–Brown (GHB) criterion is commonly used, for which no analytical form of the Mohr failure envelope is, in general, available. In this work, a new approximation of the Mohr envelope for the GHB criterion is developed, which is valid in a broad range of GSI (i.e. Geological Strength Index) values. The approach is based on the orthogonal projection of a function for best-fitting the quadratic or cubic polynomials to the Balmer’s equations that define the relationship between the normal stress acting on the failure plane and the minor principal stress. The methodology is illustrated by a limit equilibrium analysis, which involves assessment of the safety factor of a rock slope that has a vertical tension crack embedded in the upper horizontal surface. The analysis employs an approximate Mohr envelope based on the cubic polynomial fitting as the failure condition along the rupture surface. The results indicate that the safety factor for stability of the slope is very sensitive to the geometry of the crack (i.e. its location and depth) as well as the selection of the value of GSI.

All-solid-state batteries (ASSBs) have been considered as the next generation of lithium-ion batteries. Physics-based models have the advantage of providing internal electrochemical information. To promote physics-based models in real-time applications, in this study, a series of model reduction methods are applied to obtain a reduced-order model (ROM) for ASSBs. First, analytical solutions of the partial differential equations (PDEs) are derived by the Laplace transform. Then, the Pade approximation method is used to convert the transcendental transfer functions into lower order fractional transfer functions. Next, the concentration distributions in electrodes and electrolytes are approximated by parabolic and cubic functions, respectively. Due to the fast calculation of concentration distributions in real time, the equilibrium potential, overpotentials, and battery voltage can now be directly calculated. Compared with the original PDE-based model, the voltage errors of the proposed ROM are less than 2.6 mV. Compared with the voltage response of experimental data, a good agreement can be observed for the ROM under three large C-rates discharging conditions. The calculation time of ROM per step is within 0.2 ms, which means that it can be integrated into a battery management system. The proposed ROM achieves excellent performance and a better tradeoff between model fidelity and computational complexity.

Discrete element method (DEM) simulations of binary mixing of particles with different densities were conducted to study the influence of density ratio, blade speed, and filling level on the particle dynamics and mixing performance in a bladed mixer. Four particles with different densities at different locations were tagged to discuss the influence of three factors on the particle trajectory and velocity field in the mixer. A method based on cubic polynomial fitting of relative standard deviation was used to determine the critical revolution during the mixing process. It was found that the non-dimensional tangential velocity decreases with the increase of the blade speed and filling level, the fluctuation of vertical velocity increases with the radial location, blade speed, and filling level, and it is more pronounced than the fluctuation of tangential and radial velocity during the mixing process. Results obtained indicate that the mixing performance of particles with different density increases with the decrease of density ratio and filling level, while it increases with the increase of blade speed.

This article describes cubic function fields $L/K$ with prescribed ramification, where $K$ is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure $K'/K$ of $L/K$ is of genus zero, and a description of the twists of $L/K$ up to isomorphism over $K$. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows M\"obius transformations on $K$. The article concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point., Comment: 24 pages; published version

This paper presents a novel 2D/3D non-rigid registration method for lung lesions tracking in image-guided diagnoses and treatments. Preoperative 3D lung CT volumes were obtained at a series of respiratory phases and important anatomical points were extracted. A CT volume was selected as reference volume and others were considered floating volumes. Displacement vectors of anatomical points were calculated using coherent point drift (CPD) and diffeomorphic-demon methods. For each CT volume, 2D digitally reconstructed radiographs (DRR) were generated by ray projection to simulate intraoperative 2D X-ray, and grayscale-based similarity measures of DRR between reference and floating volumes were calculated. A pulmonary respiration model was constructed using cubic polynomial, which represents the non-linear correlation between displacement vectors and similarity measures. During operation, the pulmonary respiration model used X-ray to obtain displacement vectors of important anatomical points; deformation fields were calculated from the displacement vectors through B-spline transformations. Finally, simulated lung CT volumes corresponding to intraoperative X-ray were generated by applying deformation fields to reference volume. The proposed method was compared to a state-of-the-art 2D/3D non-rigid registration method named respiratory-phase matching. Experimental results on two datasets respectively from a lung cancer patient and an ex vivo pig lung, showed mean registration accuracy of the proposed method was 1.57 ± 0.92 mm and 2.65 ± 1.35 mm; an improvement of 23% and 9% over the matching method. Moreover, mean lung structure overlaps were 0.91 and 0.88, comparable to matching method. The proposed method has potential in aiding lung intervention diagnoses and real-time lesion tracking.

We reconsider the well-known conditions which guarantee the roots of a third-degree polynomial to be inside the unit circle. These conditions are important in the stability analysis of equilibria a...

In this paper, we propose a three step iteration process and analyze the performance of the process for a contractive-like operators. It is observed that this iterative procedure is faster than several iterative methods in the existing literature. To support the claim, a numerical example is presented using Maple 13. Some images are generated by using this iteration method for complex cubic polynomials. We believe that our presented work enrich the polynomiography software.

Discrete element method (DEM) simulations of particle mixing process in an intensive mixer were conducted to study the influence of structural and process parameters on the mixing performance and power consumption. The DEM model was verified by comparing the impeller torque obtained from simulation with that from experiment. Impeller and vessel torque, coordination number (CN) and mixing index (Relative standard deviation) were adopted to qualify the particle dynamics and mixing performance with different parameters. A method based on cubic polynomial fitting was proposed to determine the critical mixing time and critical specific input work during the mixing process. It is found that the mixing performance and energy efficiency increases with the decrease of impeller offset. The mixing performance is improved slightly with the increase of blade number and the impeller with 3 blades has the highest energy efficiency due to its low input torque. Results indicate that the energy efficiency and the mixing performance increase with the decrease of filling level when the height of granular bed is higher than that of blade.

The operation frequency of memristor seriously affects its applications in high-frequency working conditions. In this article, a new method using a sine-based function instead of the linear term in memristor models to spread the operation frequency ranges of memristor emulators is proposed. Via using this method on a flux-controlled memristor with smooth continuous cubic function as an example, research shows that the operation frequency range of this improved flux-controlled memristor is spread greatly. In particular, the improved memristor can still display an excellent pinched hysteresis loop even when the operation frequency is up to a few gigahertz, which is higher than most of the previously reported memristor emulators. The typical dynamic characteristics and the operation frequency range of the improved memristor are studied theoretically and numerically. The circuit simulations and physical experiments agree well with the theoretical analyses, which together demonstrate the effectiveness of the proposed methodology. Then, the improved memristor is applied to realize a chaotic oscillator to show its practical applications. In addition, the new method has universality, and it can be used by many other existing memristor emulators for spreading their operation frequency ranges.

Equations of state (EoSs) have always been one the most interesting field of study for scientists and engineers, due to their extensive applications in various industries and scientific research. Accordingly, scientists have extensively studied useful modifications of the original EoSs. In this study, the temperature dependent part of the Peng–Robinson cubic equation of state is modified. The new dual parameter α-function is able reproduce the vapor pressure data accurately for a large variety of pure components. Mono-atomic and di-atomic molecules, hydrocarbons, polar and associating compounds are well represented by the Peng–Robinson–Saali equation of state with negligible deviation with experimental data. Moreover, the thermodynamic consistency test of P–T–x solubility data based on the fundamental Gibbs–Duhem equation, for binary mixtures including acid gases (CO2 and H2S)/polar and associating solvents at low and high pressure were also investigated. The proposed EoS coupled with a three-parameter binary interaction term, namely the Panagiotopoulos–Reid mixing rule, was used in order to present the PR–Saali EoS as a versatile equation for analyzing thermodynamic consistency of experimental data. Modeling processes were carried out by using MATLAB 2018a.

The frequency–temperature ( ${f}$ – ${T}$ ) characteristic of quartz crystal resonators is an important topic closely related to the frequency-deviation compensation in the design of cost-effective oscillators. Traditional studies depict the ${f}$ – ${T}$ characteristic as a polynomial function (usually a cubic function). However, this omits the thermal hysteresis phenomenon and cannot provide a very accurate frequency compensation. To handle this issue, this article is to propose two modified ${f}$ – ${T}$ characteristic modeling methods with considering the thermal hysteresis. First, to reflect the thermal hysteresis property of quartz crystal resonators, a two-directional ${f}$ – ${T}$ model is designed by introducing two individual submodels for describing the increasing and decreasing temperature stages, respectively. Furthermore, to integrate the submodels and provide the more accurate frequency-deviation estimation, a holistic ${f}$ – ${T}$ characteristic model based on double-hidden layer extreme learning machine (DHL-ELM) is presented. Different from the basic ELM model, two hidden layers, including one deterministic nonlinear mapping and one random nonlinear activation layer, are constructed for a better description of ${f}$ – ${T}$ characteristic. To validate our studies, an experiment system is applied to obtain the testing data of the real crystal resonators, and the applications demonstrate the effectiveness of the proposed methods.

In nonlinear analysis, performing iterative inverse calculation and nonlinear system construction procedures incurs expensive computational costs. This paper presents an element-wise stiffness evaluation procedure combined with hyper-reduction reduced-order modeling (HE-STEP ROM) method. The proposed approach constructs a non-intrusive reduced-order model based on an element-wise stiffness evaluation procedure (E-STEP) and hyper-reduction methods. Because the E-STEP evaluates nonlinear stiffness coefficients element-by-element using cubic polynomial, numerous number of polynomial variables are required. The number of variables directly affects the computational efficiency of the online and offline stages. Therefore, to enhance efficiency of the online/offline stages, the proposed method employs hyper-reduction method. By applying hyper-reduction, the full stiffness coefficients are approximated from the stiffness coefficients evaluated at a few sampling points. Subsequently, the number of polynomial equations and variables is prominently reduced, and the efficiency of the reduced system increases. The efficiency and accuracy of the proposed approach are validated via several structural dynamic problems with geometric and material nonlinearities.

An approximation to discrete noisy data is constructed that obtains monotonicity. Precisely, we address the problem of making the least sum of 4th powers change to the data that provides nonnegative first differences. An algorithm is proposed for this highly structured strictly convex programming calculation that includes the method of van Eeden. The algorithm generates the solution in n − 1 steps by identifying the subset of the constraints that are satisfied as equations, where n is the number of data. By using suitable arrays, the algorithm reduces the amount of work in a way that takes advantage of the fact that the solution consists of sets of equal components, calculates the equal components for each set by solving a cubic equation and, effectively updates the arrays to the next one. It is proved that the work of each step of the algorithm amounts to O ( 1 ) computer operations and at most one cubic root extraction. Some numerical experiments with synthetic and real data show that the algorithm is extremely fast.

Traditional measurement methods, like shipborne and aerial gravity measurements, cannot cover the entire sea area. Satellite altimetry provides large-scale and high-precision observational data, which can be inverted to give high-precision and high-resolution data on marine gravity anomalies. In this paper, to improve the precision of computing the innermost zone effects in gravitational altimetry data, the deflections of the vertical are expressed in the form of double cubic polynomials by treating the innermost zone as a rectangle, based on the inverse Vening Meinesz formula. Furthermore, formulas for computing gravity anomalies in the innermost zone are derived using a non-singular transformation. The numerical experiments show that the relative error of the algorithm in this paper, which uses a non-singular transformation, is less than 1.5%. Moreover, the practical computational results based on deflections of the vertical for data with a resolution of 2’ $\times2$ ’ from the South China Sea and its vicinity also demonstrate that gravity anomalies in the innermost zone have a non-negligible contribution when using a non-singular transformation.

This paper proposes an algorithmic implementation of the elementary version of Runge’s method for a family of fourth-degree Diophantine equations in two unknowns. Any Diophantine equation of the fourth degree the leading homogeneous part of which is decomposed into a product of linear and cubic polynomials can be reduced to equations of the type considered in this paper. The corresponding algorithm (in its optimized version) is implemented in the PARI/GP computer algebra system.

Positive displacement expanders are widely investigated and theirs models commonly use the REFPROP database to evaluate fluid properties. However, estimating procedures in CFD models using REFPROP result in heavy use of CPU time. Therefore, a study has been carried out to determine if simpler methods for fluid property estimation are suitable. The ANSYS CFX solver used includes a number of cubic equations of state, namely, the Redlich Kwong, Soave Redlich Kwong, Aungier Redlich Kwong and Peng Robinson. To determine their suitability, performance simulations were carried out with each of them, on a twin screw expander with a 4/5 rotor configuration and an “N” rotor profile, operating with R245fa, R290, R1336mzz(Z) and R1233zd(E). For each considered equation, the expander's performance results deviation from those obtained using REFPROP is evaluated. These results show that the ideal gas equation of state gives predictions of flow and power that deviate significantly from those determined with REFPROP. The alternatively available cubic equations give far better agreement. Of these, the RK equation has the highest deviation, although it differs by only 2.2% maximum from the REFPROP values. The PR equation yields the closest agreement with a deviation of the order of only 0.8%, while the deviation with both SRK and ARK equations is less than 1.1% maximum, practically negligible. Apart from the good agreement obtained, the use of the real gas equations, in place of REFPROP, resulted in a saving of approximately 43% in CPU operating time to obtain same level of convergence of the solver.

A Diophantine problem means to find all solutions of an equation or system of equations in integers, rational numbers, or sometimes more general number rings. The most frequently asked question is whether a root of a polynomial equation with coefficients in a p-adic field Qp belongs to domains Z*p, Zp \ Z*p, Qp \ Zp, Qp or not. This question is open even for lower degree polynomial equations. In this paper, this problem is studied for cubic equations in a general form. The solvability criteria and the number of roots of the general cubic equation over the mentioned domains are provided.

A methodology to predict solid-liquid, solid-vapor, and liquid-vapor phase coexistences using a non-cubic equation of state is presented. The solid phase is additional to the liquid and vapor phase...

This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.