Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.

Counting ternary trees according to the number of middle edges and factorizing into (3/2)-ary trees

The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of edges (equivalently nodes) and the number of middle edges. Using a certain substitution, the underlying cubic equation can be factored. This leads to an extension of the concept of (3/2)-ary trees, introduced by Knuth in his christmas lecture from 2014.

Arithmetic Relationship between Fracture Load and Material Thickness of Resin-Based CAD-CAM Restorative Materials

Data on the long-term behavior of computer-aided designed/computer-aided manufactured (CAD-CAM) resin-based composites are sparse. To achieve higher predictability on the mechanical behavior of these materials, the aim of the study was to establish a mathematical relationship between the material thickness of resin-based materials and their fracture load. The tested materials were Lava Ultimate (LU), Cerasmart (GC), Enamic (EN), and Telio CAD (TC). For this purpose, 60 specimens were prepared, each with five different material thicknesses between 0.4 mm and 1.6 mm (N = 60, n = 12). The fracture load of all specimens was determined using the biaxial flexural strength test (DIN EN ISO 6872). Regression curves were fitted to the results and their coefficient of determination (R2) was computed. Cubic regression curves showed the best R2 approximation (LU R2 = 0.947, GC R2 = 0.971, VE R2 = 0.981, TC R2 = 0.971) to the fracture load values. These findings imply that the fracture load of all tested resin-based materials has a cubic relationship to material thickness. By means of a cubic equation and material-specific fracture load coefficients, the fracture load can be calculated when material thickness is given. The approach enables a better predictability for resin-based restorations for the individual patient. Hence, the methodology might be reasonably applied to other restorative materials.

Negative Order KdV Equation with No Solitary Traveling Waves

We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions.

Comparison of Z-Factor Correlations Using a Large PVT Dataset with Emphasis on the GERG-2008 EOS

The gas compressibility factor is an important property in reservoir simulation studies. It is directly linked to the gas formation volume factor and the gas density thereby impacting wellhead injection pressure, reservoir voidage, injectivity, as well as the tendency for gas gravity override to occur in the reservoir. ADNOC's PVT database contains experiments on almost 2,000 samples, of which more than 100 have been subject to advanced gas injection experiments. Z-factor data have been compiled from the liberated gas during DV experiments as well as from CCE experiments on reservoir gases, injection gases, and swollen fluid mixtures. Several of these mixtures are very rich in H2S, whereas pressure and temperature are in the range of 14.7-14,500 psia and 80-365 °F, respectively. We test several different methods for predicting the Z-factor, such as the industry-standard Hall-Yarborough method, in combination with various models for pseudo-critical pressure and temperature and including correction for non-hydrocarbon components. Other methods tested include the GERG-2008 model, considered to be state-of-the-art for predicting physical properties for well-described gas mixtures, as well as the well-known Peng-Robinson cubic equation of state. Based on close to 10,000 data points in our database, the GERG-2008 model typically predicts the Z-factor to be within 2% of the measured value, which is on par with the experimental uncertainty. However, for some rich gas condensate mixtures, the model gives larger errors because its parameters are only tuned to compositions with components up to C10. This is to our knowledge the first time that the GERG-2008 EOS has been compared to standard Z-factor correlations for such a large number of data points. If compositional information is available, we recommend using either the GERG-2008 model or the Hall-Yarborough model with pseudo-critical properties provided by Kay (1936). When compositions are not available, we find that the Standing correlation is more accurate than the Sutton model, also for sour mixtures.

On Stability Switches and Bifurcation of the Modified Autonomous Van der Pol–Duffing Equations via Delayed State Feedback Control

This paper considers the Modified Autonomous Van der Pol–Duffing equation subjected to dynamic state feedback, which can well characterize the dynamic behaviors of the nonlinear dynamical systems. Both the issues of local stability switches and the Hopf bifurcation versus time delay are investigated. Associating with the τ decomposition strategy and the center manifold theory, the delay stable intervals and the direction and stability of the Hopf bifurcation are all determined. Specifically, the computation of purely imaginary roots (symmetry to the real axis), the positive real root formula for cubic equation and the sophisticated bilinear form of adjoint operators are proposed, which make the calculations mentioned in our discussion unified and simple. Finally, the typical numerical examples are shown to illustrate the correctness and effectiveness of the practical technique.