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The recent notion of q -deformed irrational numbers is characterized by the invariance with respect to the action of the modular group PSL ( 2 , Z ) , or equivalently under the Burau representation of the braid group  B 3 . The theory of q -deformed quadratic irrationals and quadratic equations with integer coefficients is known and entirely based on this invariance. In this paper, we consider the case of cubic irrationals. We show that irreducible cubic equations with three distinct real roots and cyclic Galois group  C 3 (or Z / 3 Z ) acting by a third order element of PSL ( 2 , Z ) , have a canonical q -deformation, that we describe. This class of cubic equations contains well-known examples including the equations that describe regular 7- and 9-gons.

Based on the root–coefficient relations for a cubic function, quadratic functions are constructed that strictly relate the saturated volumes of liquid and vapor phases and the third solution from a cubic equation of state (EoS). The vapor–liquid equilibrium (VLE) calculation with a cubic EoS is thus reduced to solving a single nonlinear equation. In light of a recent finding that the “unphysical” third solution, namely the Maxwell crossover or the M-line, plays a central role as the dividing interface in the density gradient theory, here we show that it can also be used to derive explicit approximations for a VLE problem. The van der Waals EoS and the Soave–Redlich–Kwong (SRK) EoS are discussed as examples. The method proposed in this work simplifies the calculations of the traditional VLE problem with a cubic EoS. With one-time-only effort for a given system, simple explicit approximations can be obtained to avoid the repetitively iterative computations for a VLE problem. Finally, the relationship between the Widom line in the supercritical region and the M-line is briefly discussed with the SRK EoS. Graphical Abstract

For over a century, cubic equations of state (EoS) have been used to calculate density and phase equilibria of pure fluids and mixtures. Despite a century’s development with hundreds of resulting cubic EoS, their accuracy in liquid phase density calculations is still unsatisfactory. In this work, a new cubic EoS was developed to improve the accuracy of liquid phase density calculation while keeping similar accuracy of other properties. The new cubic EoS, named YFR (Yang-Frotscher-Richter) EoS, was developed based on the functional form of the Patel–Teja (PT) EoS [ p  =  RT /( v  −  b ) −  a /( v ( v  +  b ) +  c ( v  −  b )]. In the PT EoS, parameters b and c are linked to an empirical critical compressibility factor ξ c , and all these three parameters are constants for a pure fluid. By contrast, in the YFR EoS, ξ c , b , and c are functions of temperature, and the equations describing this dependency were developed with symbolic regression. This is the key to improving liquid phase density calculation, although it leads to thermodynamic inconsistencies at high pressures. The application range of the new cubic EoS is thus limited to pressures up to 100 MPa. The YFR EoS was developed using nearly all pure fluids available in NIST’s REFPROP 10.0 database, with reference values computed with REFPROP. The average of the absolute value of relative deviations (AARD) of liquid phase densities calculated with the YFR EoS from reference values is approximately 2 %, compared to 3 % when using the Patel–Teja–Valderrama (PTV) EoS and 6 % when using the Peng-Robinson (PR) EoS. The YFR EoS has been implemented in our self-developed OilMixProp 1.0 software package.

In this work, a robust and rigorous procedure for roots calculation and selection for a Cubic Equation of State is presented. Roundoff errors are detected and corrected with a simple test in addition to an iterative procedure, adapted from the fixed-point method. The new approach applies some properties of Cardano’s formulation to find the appropriate phase root of the cubic equation. Furthermore, to find out the actual physical state of the system, when the polynomial function presents only one root, special properties of the second derivative of the cubic polynomial in compressibility factor form and its partial derivative with respect to pressure are analyzed on a rigorous thermodynamic basis. These results are applied to outline an efficient algorithm to calculate the saturation pressure of pure components, which is independent of an accurate initial estimate and integrates bisection, successive substitution, and Newton–Raphson methods to find the correct value in few iterations.Graphical abstract

The solution of the non-linear Föppl–von Kármán equations for square plates in the form of expansion over a system of eigenfunctions, generated by a linear self-adjoint operator, is obtained. The coefficients of the expansion are determined via the reduction method from the infinite-dimensional system of cubic equations. This allows the proposed solution to be considered as a non-linear generalization of the classical Galerkin approach. The novelty of the study is in the strict formulation of the auxiliary boundary problem, which makes it possible to take into account a rigid fixation against any displacements along the boundary. To verify the proposed solution, it is compared with experimental data. The latter is obtained by the holographic interferometry of small deflection increments superimposed on the large deflection caused by initial pressure. Experiment and theory show a good agreement.

Hand-eye calibration is one of the important problems in the field of robot vision and control, aiming to determine the pose transformation between the robot end-effector and the visual system. The analytical solution of this problem has a more explicit numerical computation process and a more stable solution time compared with iterative or deep learning methods, and theoretically provides more accurate results. When using quaternion parameterization for rigid body motion rotation, the hand-eye calibration problem can be transformed into solving the eigenvector corresponding to the maximum eigenvalue of a symmetric matrix. This paper proposes an analytical solution for hand-eye calibration based on quaternion parameterization. Compared with previous methods, the advantage of this method is that the solution for the required eigenvector is transformed into solving cubic equations through the eigenvector-eigenvalue identity, without SVD or Eigendecomposition, thus making the proposed method a complete analytical solution, and avoids solving irrelevant information. The performance of the proposed method was evaluated through multiple experiments conducted on a synthetic dataset as well as two real-world datasets, and compared to representative analytical methods. The experimental results unequivocally demonstrate that our proposed method achieves comparable accuracy with significantly shorter solution times. Therefore, our complete analytical solution offers an efficient and accurate alternative for addressing the hand-eye calibration problem without SVD or Eigendecomposition.

Presented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation x4+ax3+bx2+cx+d=0 with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. The presented classification of the roots of the quartic equation is particularly useful in situations in which the equation stems from a model the coefficients of which are (functions of) the model parameters and solving cubic equations, let alone using the explicit quartic formulæ , is a daunting task. The only benefit in such cases would be to gain insight into the location of the roots and the proposed method provides this. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the “sub-quartic” x4+ax3+bx2 with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.

The present article presents a system of symmetric equations defining multi-cubic mappings (M-CMs). Next, we describe how these mappings are structured and obtain an equation for describing them. Moreover, we Address the Hyers-Ulam stability (H-UStab) in the sense of Găvruţa for a symmetric multi-cubic equation through the application of the so-called Hyers (direct) method in the setting of 2-Banach spaces. For a typical case, by means of a norm, induced from a 2-norm of Rd, we examine the stability and hyperstability of a mapping f:Rdn⟶Rd by using a fixed point (FP) result.

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.