Low -parametric equation for calculating the viscosity coefficient of nitrogen in liquid, gas and fluid states

The formulation for the viscosity coefficient of nitrogen is obtained. In the developed equation the dependence of the residua viscosity of various states of substance on the internal energy density is used. The new correlation represents the viscosity of nitrogen at temperatures from 70 K to 1000 K and pressures of up to 50 MPa within the limits of experimental uncertainties.

Construction of a family of flat curves according to the equations of isometric grids

The article reveals an analytical description of the formation of families of orthogonal flat curved lines in the implicit form based on the analysis of the parametric equation of a flat isometric grid constructed by separating the real and imaginary parts of the function of a complex variable. This problem is due to the fact that flat isometric grids, as two families of orthogonal coordinate lines with square cells, are used in conformal mappings, for example, when drawing images on curved surfaces with the least distortion. At the same time, families of flat parallel lines are widely used in geometric modeling of heat transfer, electric fields, fluid flow, etc. There is a connection between these geometric images, which is explained by specific examples. Analytical calculations of deriving the parametric equation of an isometric grid are quite time-consuming, so they are performed in the environment of symbolic algebra Maple. For this purpose, the corresponding software of the interactive model of derivation of parametric equations of isometric grids for any initial function of a complex variable with the subsequent separation of its real and imaginary parts was created. It was found that the values of the abscissa and ordinates of the parametric equation of a flat isometric grid can be represented as explicit surface equations. For integer values of the power of the exponential function of the complex variable, the values of the abscissa and the ordinate will be represented by algebraic surfaces in the explicit form. The projections of the cross sections of the abscissa and ordinate surfaces by horizontal cutting planes on the horizontal plane form two families of curved lines, the equations of which can be obtained only implicitly. By the example of the quadratic function of a complex variable, it is proved that these families of lines are mutually perpendicular. The practical application of building a family of lines for geometric modeling of fluid flow lines that flow around the barrier in the form of a semicircle is shown. Key words: isometric grids, functions of a complex variable, families of orthogonal lines, geometric flow modeling

Noor's Curve, a New Geometric Form of Agnesi Witch, a Construction Method is Produced

In this paper, a new form of 2D-plane curves is produced and graphically studied. The name of my daughter "Noor" has been given to this curve; therefore, Noor term describes this curve whenever it is used in this paper. This curve is a form of these opened curves as it extends in the infinity along both sides from the origin point. The curve is designed by a circle/ ellipse which are drawing curvatures that tangent at the origin point, where its circumference is passed through the (0,2a). By sharing two vertical lined points of both the circle diameter and the major axis of the ellipse, the parametric equation is derived. In this paper, a set of various cases of Noor curve are graphically studied by two curvature cases; a circle and an ellipse, and all figures and obtained rigour measurements are checked by AutoCAD program. With its simple, symmetric form, the future predictions are tuned for the Noor's curve to be usefully engaged in important practical applications.

The behavior of solutions of a parametric weighted $ (p, q) $-Laplacian equation

<abstract><p>We study the behavior of solutions for the parametric equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{p}^{a_1} u(z)-\Delta_{q}^{a_2} u(z) = \lambda |u(z)|^{q-2} u(z)+f(z,u(z)) \quad \mbox{in } \Omega,\, \lambda &gt;0, $\end{document} </tex-math></disp-formula></p> <p>under Dirichlet condition, where $ \Omega \subseteq \mathbb{R}^N $ is a bounded domain with a $ C^2 $-boundary $ \partial \Omega $, $ a_1, a_2 \in L^\infty(\Omega) $ with $ a_1(z), a_2(z) &gt; 0 $ for a.a. $ z \in \Omega $, $ p, q \in (1, \infty) $ and $ \Delta_{p}^{a_1}, \Delta_{q}^{a_2} $ are weighted versions of $ p $-Laplacian and $ q $-Laplacian. We prove existence and nonexistence of nontrivial solutions, when $ f(z, x) $ asymptotically as $ x \to \pm \infty $ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When $ \lambda $ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.</p></abstract>

Power generation portfolios: A parametric formulation of the efficient frontier

This paper aims to provide a methodology to construct parametrically the Efficient Frontier (EF) of Power Generation Portfolio (PGP). The methodology works as follows. First, we obtain two sets of the shares of the assets: one that guarantee the maximal expected return on the PGP; and another that guarantee the minimal risk of the PGP. The EF corresponds to the parametric equation of the risk-return profiles from the minimal risk to the maximal expected return of the PGP. We apply our methodology to replicate the results from three existing papers. The present methodology allows to and different and more coherent results than those obtained in the original papers. The analysis suggests that there are optimal investment alternatives that have been denied by previous analysis. This fact creates a bias in the design of investment policies in electricity generation. One limitation of the paper is that the analysis relies on the assumption that the covariances of the returns of the different assets is zero. This assumption leads to gains in tractability, clarity, and in the scope of the methodology formulated.

Power generation portfolios: A parametric formulation of the efficient frontier

This paper aims to provide a methodology to construct parametrically the Efficient Frontier (EF) of Power Generation Portfolio (PGP). The methodology works as follows. First, we obtain two sets of the shares of the assets: one that guarantee the maximal expected return on the PGP; and another that guarantee the minimal risk of the PGP. The EF corresponds to the parametric equation of the risk-return profiles from the minimal risk to the maximal expected return of the PGP. We apply our methodology to replicate the results from three existing papers. The present methodology allows to and different and more coherent results than those obtained in the original papers. The analysis suggests that there are optimal investment alternatives that have been denied by previous analysis. This fact creates a bias in the design of investment policies in electricity generation. One limitation of the paper is that the analysis relies on the assumption that the covariances of the returns of the different assets is zero. This assumption leads to gains in tractability, clarity, and in the scope of the methodology formulated.

Generalized method for forming plane isotropic curves

The process of modeling the temperature distribution on surfaces, applying an image to curved areas with minimal distortion requires the formation of isometric grids on the plane and on the surface. One of the common ways to form planar isometric networks is to use the functions of a complex variable and planar isotropic curves, followed by separation of the real and imaginary parts. The development of computer models for the interactive search and analysis of isometric networks according to various initial geometric conditions provides a generalized method for their formation with the possibility of varying their shape and position. It is proposed to use an isotropic vector for the formation of flat isotropic curves, which ensured a single sequence of analytical calculations according to the following initial conditions: 1) selection of an arbitrary function of a real argument; 2) a given parametric equation of a plane curve; 3) a given polar equation of a plane curve. Since the analytical calculations of the derivation of the parametric equation of a plane isotropic curve and the corresponding isometric grid are rather laborious, their execution is carried out in the environment of the Maple symbolic algebra. To this end, the corresponding software has been created, which interactively allows you to select the function of a real argument, a parametric or polar equation of a plane guide curve. All subsequent stages of analytical transformations to form an isotropic curve and the corresponding isometric grid are carried out automatically. An interactive model for the formation and analysis of plane isotropic curves with various initial conditions has been created, which has shown its effectiveness, which is confirmed by the given examples of plane isometric grids for specific functions of the real parameter, plane curves in the parametric and polar form of their job.

Batman Reimagined

Inspired by the “Batman Equation” of 2011, this article presents a challenging and engaging process for graphing complicated designs from just a single parametric equation pair. Reinforces numerous analytic geometry skills. Works in popular graphing software such as Desmos or GeoGebra, or even graphing calculators.