Open Access

Article

Article ID: 1594

The Navier-Stokes equation and a fully developed turbulence

by Marian Apostol

Journal of AppliedMath, Vol.2, No.5, 2024; 43 Views, 28 PDF Downloads

In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an incompressible fluid and an isentropic flow, the Navier-Stokes equation has approximate vorticial (rotational) solutions, generated by viscosity. However, in general, these vortices are unstable, and a discrete distribution of vorticial solutions is not in me chanical equilibrium; it forms an unstable vorticial liquid. On the other hand, these solutions may exhibit turbulent, fluctuating instabilities for large variations of the velocity over short distances. We represent a fully developed turbulence as a homogeneous, isotropic and highly-fluctuating distribution of singular centres of turbulence. A regular mean flow can be included. In these circumstances the Navier-Stokes equation exhibits three time scales. The equations of the mean flow can be disentangled from the equations of the fluctuating part, which is reduced to a vanishing inertial term. This latter equation is not satisfied after averaging out the temporal fluctuations. For a homogeneous and isotropic distribution of non-singular turbulence centres the equation for the inertial term is satisfied trivially, i.e. both the average fluctuating velocity and the average fluctuating inertial term are zero. If the velocity is singular at the turbulence centres, we are left with a quasi-ideal classical gas of singularities, or a solution of singularities (solute) in quasi thermal equilibrium in the background fluid (solvent). This is an example of an emergent dynamics. We give three examples of vorticial liquids.

Open Access

Article

Article ID: 1698

Regularity, synthesis, rigidity and analytic classification for linear ordinary differential equations of second order

by Víctor León, Bruno Scárdua

Journal of AppliedMath, Vol.2, No.4, 2024; 119 Views, 58 PDF Downloads

We study second order linear homogeneous differential equations a(x)y'' + b(x)y' + c(x)y = 0 with analytic coefficients in a neighborhood of a regular singularity in the sense of Frobenius. These equations are model for a number of natural phenomena in sciences and applications in engineering. We address questions which can be divided in the following groups: (i) Regularity of solutions. (ii) Analytic classification of the differential equation. (iii) Formal and differentiable rigidity. (iv) Synthesis and uniqueness of ODEs with a prescribed solution. Our approach is inspired by elements from analytic theory of singularities and complex foliations, adapted to this framework. Our results also reinforce the connection between classical methods in second order analytic ODEs and (geometric) theory of singularities. Our results, though of a clear theoretical content, are important in justifying many procedures in the solution of such equations.

Open Access

Commentary

Article ID: 206

Onthe geometry of an almost α-cosymplectic (k, µ, ν)-spaces

by Pakize Uygun, Mehmet Atçeken, Tugba Mert

Journal of AppliedMath, Vol.2, No.4, 2024; 126 Views, 77 PDF Downloads

The object of the paper is to investigate almost α-cosymplectic (k,µ,ν)-spaces. Some results on almost cosymplectic (k,µ,ν)-spaces with certain conditions are obtained.

Open Access

Article

Article ID: 1435

Predictive modeling for industrial productivity: Evaluating linear regression and decision tree regressor approaches

by Isaac Azure

Journal of AppliedMath, Vol.2, No.4, 2024; 91 Views, 59 PDF Downloads

This research discusses the importance of predictive modeling in optimizing efficiency in various sectors, particularly in industrial settings. It compares the effectiveness of linear regression and decision tree regression models in predicting productivity. The study aims to provide insights into the strengths and limitations of each technique, assisting decision-makers in selecting the best model for their needs. It begins by explaining the theoretical foundations of both models and conducts a literature review to highlight their practical implementations. The methodology involves data collection, preprocessing, model training, evaluation, and comparison using real-world datasets. Performance metrics such as Mean Squared Error (MSE) are used for evaluation. The comparative analysis reveals that the linear regression model consistently outperforms the decision tree regressor model in terms of lower MSE values across all datasets. Overall, the study offers empirical evidence and practical insights into the predictive capabilities of both models, with potential implications for strategic decision-making in various industries.

Open Access

Article

Article ID: 553

GeoGebra—A great platform for experiential learning, explorations and creativity in mathematics

by Qamil Kllogjeri, Pellumb Kllogjeri

Journal of AppliedMath, Vol.2, No.4, 2024; 154 Views, 74 PDF Downloads

In this paper we are presenting some examples of how GeoGebra is used in: a) explaining concepts of the first derivative, monotony, extremums; b) studying the properties of the function (strictly increasing/decreasing); c) demonstrating the mean value theorem. The results and the conclusions are based on the experiment carried out in the teaching and learning process in the chapter of derivatives in a third-year class of a secondary school in Albania. Also, there are some encouraging facts got by the use of GeoGebra: the double representation and the dynamic features of GeoGebra allow the students to quickly grasp the mathematical concepts and properties and be actively involved in further explorations. The use of GeoGebra tools is similar to the use of the tools of virtual games, and this is a great advantage to stimulate the students to learn mathematics and master their mathematical performance the same way they play games. On the other side, using GeoGebra, it is easier for the teachers to explain mathematical concepts, the properties of algebraic objects, to discuss about different situations and aspects of the subject under study and to methodically reason the results got. GeoGebra provides a very commodious environment for the students to effectively interoperate with each other. Our results showed that GeoGebra is effective in teaching and learning mathematics. GeoGebra software contributed in enhancing students’ understanding of mathematical concepts and improved students’ interest to learn mathematics. Also, we admit that not all the stages of implementing GeoGebra software in the classroom are flowing smoothly. Based on our experience and the other researchers, it is observed that the effectiveness is dependent on the way GeoGebra is integrated in the teaching and learning process, implying that the research must continue with the commitment of many researchers of mathematics, physics and other sciences.

Open Access

Article

Article ID: 1700

Differentials of the basis in Clifford Geometric Algebra

by Yingqiu Gu

Journal of AppliedMath, Vol.2, No.4, 2024; 167 Views, 92 PDF Downloads

In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics, the operator algebra corresponds to the differentials of matrices. Corresponding to the variation of the metric, the variation of the frame contains a unusual fourth-order tensor. We also derive the Lie differential of the frame corresponding to the Lorentz transformation group. The definition of differential of the frame is different, so the corresponding linear transformation is also different. In this paper, the unified point of view to deal with the variation of frame or basis vectors will bring great convenience to the research and application of Clifford algebras.