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We present a new family of ninth‐order hybrid explicit Numerov‐type methods, effectively utilizing only eight stages, for solving the special second‐order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well‐known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm. [ABSTRACT FROM AUTHOR]

The comparative analysis of response surface methodology (RSM) and artificial neural network (ANN)–based models was studied for the uptake of organic dye (direct red 2B (DR2B)) using zirconium-based metal-organic framework (MOF) adsorbent. The design of experiment data used were obtained from a previous study. Three ANN models were developed using MATLAB's neural network toolbox and Mathematica's net operation and predictor function. The input variables were pH, contact time, adsorbent amount, and the initial dye concentration, while the output variable was adsorption capacity. ANN model with a network topology of 4:10:1 was developed with MATLAB's neural network toolbox. The analysis of the models showed that the RSM model (R2 = 0.9679) had the best prediction efficiency followed by Mathematica's predictor function (R2 = 0.8546), MATLAB's ANN (R2 = 0.8463), and Mathematica's net operation (R2 = 0.5284). Process optimization revealed that a maximum adsorption capacity of 46.75 mg/g could be obtained at pH 7, contact time of 70 min, adsorbent amount of 10 mg, and initial dye concentration of 44.99 ppm. This work demonstrated that within the range of process variables considered, RSM and ANN can be successfully used to model and optimize DR2B dye uptake onto zirconium-based MOF. [ABSTRACT FROM AUTHOR]

The laminar flow reactor (LFR) is one of the most comprehensive problems in chemical reaction engineering, as its modeling involves mass, heat, and momentum conservation equations coupled with chemical reaction rate equations. It is relatively easy to grasp its basic operation, but the solution of the problem is far from trivial. Although there are analytical solutions that simplify the problem, students and tutors must use efficient numerical strategies to appropriately solve some LFR problems. In the present investigation, we solve several different cases of two‐dimensional cylindrical LFR, beginning with the isothermal case, as a benchmark. Calculations were performed using the Chebyshev orthogonal collocation technique by custom scripts in Scilab, Mathematica©, and Matlab®, and were compared with solutions available from the ECRE Version of COMSOL®, which was used as the reference, as well as available analytical solutions in the literature. After analyzing the case of the isothermal LFR with Newtonian fluid, we explored non‐Newtonian fluids, including Carreau and Bingham fluids, whose LFR results are not available in the preceding literature. For Newtonian fluids, besides (a) the isothermal case, we also explore other three nonisothermal design cases: (b) cooling jacket with a fixed heat exchange coefficient at the wall, (c) adiabatic operation, (d) nonisothermal LFR with isothermal wall. Through a performance criterion, the different operation models are compared and we show that nonisothermal design cases perform better than the isothermal case. Computation times, for different scenarios, are quite short and taking 25 nodal points or more suffice for an accurate and timely solution of the more complex problem Case (b). The numerical modeling approach is useful from a pedagogical standpoint, as one can compare numerical results with classical assumptions, and progress from a more restrictive conceptual model (segregated flow) to an array of different kinds of operation with the LFR in which one needs to consider diffusion and temperature distribution. [ABSTRACT FROM AUTHOR]

The task of analysing shocks in areas of paraseismic mining activities is one of the important engineering tasks. Acceleration measurements should be integrated to receive velocity and displacement functions. Sometimes, numerical algorithms produce wrong results. This paper proposes a semi-analytical approach to this problem. The algorithm has been implemented within the Mathematica system. Moreover, this method has been applied to integration of the equation of motion.

In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to investigate the asymptotic distribution of the zeros of certain classes of hypergeometric $ {}_{q+1}F_{q} $ q + 1 F q polynomials. We employ classical analytical techniques, including Watson's lemma and the method of steepest descent, to understand the asymptotic behavior of these polynomials: $$\begin{align*} & _{q+1}F_{q}\left(-n,kn+\alpha,\ldots, kn+\alpha+\frac{q-1}{q};kn+\beta,\ldots,kn+\beta+\frac{q-1}{q};z\right)\\ &\quad (n\rightarrow \infty), \end{align*} $$ q + 1 F q (− n , kn + α , ... , kn + α + q − 1 q ; kn + β , ... , kn + β + q − 1 q ; z) (n → ∞) , where n is a nonnegative integer, q is a positive integer and the constant parameters α and β are constrained by $ \alpha { α < β. By applying the general results established in this paper, we generate numerical evidence and graphical illustrations using Mathematica to show the clustering of zeros on certain curves. [ABSTRACT FROM AUTHOR]

With the rapid development of computer technology, the role of software tools in chemical education and research is becoming increasingly prominent. This paper takes Mathematica software as an example to explore its application in the visualization of physical chemistry formulas. By using several typical examples, such as the visualization of the van der Waals equation of state, atomic orbitals and electron cloud shapes, and chemical reaction rate equations, this paper provides new perspectives and methods for teaching and research in physical chemistry. Through interactive learning mode, students can not only observe and understand the physical significance behind theories through intuitive graphics but also stimulate their interest, enhancing their initiative and creativity in learning. [ABSTRACT FROM AUTHOR]

From four-function calculators to calculators (or computers) with Computer Algebra System (CAS) software, Mathematics computing technology has advanced. With just a few button pushes, CASs can solve a wide range of mathematical problems, which is a true quantum leap in technology. The implications of having software in the classroom that can, for example, expand and factorize algebraic expressions, solve equations, differentiate functions, and find antiderivatives are causing the mathematical community to engage in a heated debate about whether this is one of the most exciting or frightening developments in the history of education. It was only a matter of time before Artificial Intelligence entered the field of Science. This is now also the case with Mathematics, one of the dominant, perhaps the most basic, but also the most "difficult" of the sciences. The human mind, for better or for worse, has its limits. As we see in every manifestation of our lives, in this case, technology is being enlisted to help humanity take the next step, whether it has to do with automation and practical matters, or with knowledge and exploration. Creating a model that is understandable to humans is the primary objective of Artificial Intelligence. Additionally, concepts and methods from numerous mathematical fields can be used to prepare these models. In this paper, we will examine the use of AI in CASs and explore some ways to optimize them. The documentation sheets are the data source that we used to examine their characteristics. The research results reveal that there are many tips that we can follow to accelerate performance. [ABSTRACT FROM AUTHOR]

Lutz et al. (Ann Oper Res 289:463–472, 2020) derived an integral expression for the expected length of an orderly path. We show here that the integral expression can be reduced to a closed form expression in terms of the Gauss hypergeometric function, a generalised hypergeometric function and the Appell hypergeometric function of the first kind. We check the correctness of the closed form expression numerically as well as provide a Mathematica code. Finally, we show that the closed form expression is more efficient than known integral expressions. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form $$u_{tt}+a(x,t)u_{xt}+b(t)u_{t}=\alpha (x,t)+ G(u)(u_{t}+a(x,t)u_{x})e^{-\int b(t)dt}$$ u tt + a ( x , t ) u xt + b ( t ) u t = α ( x , t ) + G ( u ) ( u t + a ( x , t ) u x ) e - ∫ b ( t ) d t and $$u_{t}^{m}(u_{tt}+a(x,t)u_{xt})+b(t)u_{t}^{m+1}=e^{-(m+1)\int b(t)dt}(u_{t}+a(x,t)u_{x}) F(u,u_{t}e^{\int b(t)dt}).$$ u t m ( u tt + a ( x , t ) u xt ) + b ( t ) u t m + 1 = e - ( m + 1 ) ∫ b ( t ) d t ( u t + a ( x , t ) u x ) F ( u , u t e ∫ b ( t ) d t ) . We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.

The task of analysing shocks in areas of paraseismic mining activities is one of the important engineering tasks. Acceleration measurements should be integrated to receive velocity and displacement functions. Sometimes, numerical algorithms produce wrong results. This paper proposes a semi-analytical approach to this problem. The algorithm has been implemented within the Mathematica system. Moreover, this method has been applied to integration of the equation of motion. [ABSTRACT FROM AUTHOR]

In order to find the spectrum associated with the one-dimensional Schrödinger equation, we discuss the Lagrange Mesh Method (LMM) and its numerical implementation. After presenting a general overview of the theory behind the LMM, we introduce the LagrangeMesh package: the numerical implementation of the LMM in Mathematica ® . Using few lines of code, the package enables a quick home-computer and highly accurate computation of the spectrum and provides a practical tool to study a large class of systems in quantum mechanics. The main properties of the package are (i) the input is the potential function and the interval on which it is defined; and (ii) the accuracy in calculations and final results is controllable by the user. Due to its high accuracy and simple usage, the package may be used as a research and educational tool. As illustration, a highly accurate spectrum of some relevant quantum systems is obtained by employing the commands that the package offers. [ABSTRACT FROM AUTHOR]

Nanofluids have many applications in industries as well as engineering such as biomedicine, manufacturing, and electronics. Nanofluid is used for improvement of thermal and mass transmission. Based on the aforementioned applications, in the present study, a two-dimensional Maxwell nanofluid with thermal radiation effect on the existence of motile microorganisms over a vertically stretchable surface is explored. The consequence of heat absorption, the efficiency of heat flux in a porous medium, viscous dissipations, and Joule heating impacts are considered. The Brownian and thermophoretic diffusion effects have been evaluated. In addition, the binary chemical reaction is taken into account to evaluate the magnetohydrodynamics (MHD) mixed convection flow. Graphene nanoparticles are suspended in so-called engine oil (base fluid). The proposed liquid model depends on the governing nonlinear equations of velocity, temperature, the concentration of nanoparticles, and motile gyrotactic microorganisms. In order to transform highly nonlinear partial differential equations into nonlinear ordinary differential equations, an appropriate similarity transformation is exploited. For the solution of the present study, the homotopy analysis method-technique in Mathematica-12 is used. The fluctuation of velocity, temperature, concentration, and gyrotactic microorganisms’ characteristics for numerous flow parameters is discussed in detail. Some important fallouts of the existing study are that the Maxwell liquid parameter, Eckert number, and magnetic parameter lessen the nanoliquid velocity. But the fluid temperature becomes higher for growing estimates of the Brownian motion and thermophoretic factors. The radiation and chemical reaction parameters have declining impacts on the solutal profile. The motile microorganism profile shows a decrement in bioconvection Lewis and Rayleigh numbers. The nanofluid thermal profile is improved but the nanofluid velocity declined through the augmentation of volume fraction. Also, the coefficient of skin friction and Nusselt number are obtained versus various flow parameters.

In this paper, numerical simulation for the two-dimensional flow of tangent hyperbolic fluid in the manifestation of nanoparticles, heat source and joule heating is considered. For motivation of problem, the role of Riga plate is investigated. Chemical reaction and Brownian motion are the parts of the analysis. The set of PDEs of Tangent hyperbolic nanofluids with swimming germs is converted in to couple of ODEs by employing well-known approach namely shooting scheme. The resulting highly nonlinear and coupled ODEs are solved by using ND solve tool of Wolfram Mathematica 11. Numerical outcome of involving parameters of interest like Lewis number Le, Prandtl number Pr, Chemical diffusion k, Peclet number Pe, Weissenberg number We, Bioconvection Lewis number Lb, nonlinear radiation, Biot number Bi, modified Hartmann number Q and magnetic parameter M on tangent hyperbolic nanofluids velocity f″, temperature θ, concentration φ and density of germs distribution are discussed in the form of graphical trend and tables. We professed that the speed of fluid degenerated for the increasing value of power-law index n and magnetic parameter M. The Nusselt number growths for growing values of the thermophoresis parameter Nt, Biot number Bi and radiation parameter Nr.

Glycolysis, which occurs in the cytoplasm of both prokaryotic and eukaryotic cells, is regarded to be the primary step employed in the breakdown of glucose to extract energy. As it is utilized by all living things on the planet, it was likely one of the first metabolic routes to emerge. It is a cytoplasmic mechanism that converts glucose into carbon molecules while also producing energy. The enzyme hexokinase aids in the phosphorylation of glucose. Hexokinase is inhibited by this mechanism, which generates glucose-6-P from adenosine triphosphate (ATP). This paper's primary goal is to quantitatively examine the general reaction–diffusion Glycolysis system. Since the Glycolysis model shows a positive result as the unknown variables represent chemical substance concentration, therefore, to evaluate the behavior of the model for the non-integral order derivative of both independent variables, we expanded the concept of the conventional order Glycolysis model to the fractional Glycolysis model. The nonlinearity of the model is decomposed through an Adomian polynomial for evaluation. More precisely, we used the iterative Laplace Adomian decomposition method (LADM) to determine the numerical solution for the underlying model. The model's necessary analytical/numerical solution was found by adding the first few iterations. Finally, we have presented numerical examples and graphical representations to explain the dynamics of the considered model to ensure the scheme's validity. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations. [ABSTRACT FROM AUTHOR]

The flow of Carreau nanofluid with generation/absorption and magnetic field can be valuable for modifying solar energy production. In this work, we extended the study of the MHD boundary layer flow of Carreau nanofluid with heat generation/absorption close to a stagnation point over the radially extending plate. Likewise, the features of radiation and magnetic field with convective boundary conditions are considered. Further, keeping in view the importance of chemical reactions, their effect is also incorporated during the modelling process. For motivation, the impact of thermophoresis and Brownian motion has been taken into account. Using appropriate similarity transformation, we converted nonlinear governing partial differential equations of Carreau nanofluid into a couple of nonlinear ODEs. Using a recognized shooting technique and the MATLAB bvp4c solver and Mathematica ND-solve built-in command, we were able to get numerical results for these modeled ODEs. Through graphs and tables, the effects of different physical parameters like magnetic, Weissenberg number, Brownian motion, thermal radiation, thermophoresis, Prandtl number, chemical reaction, and rate of heat generation/absorption on non-dimensional velocity, temperature, and concentration profile are discussed.

Under the background of teaching in the new era, image recognition, operation, and processing of experimental data are still the focus of calculus teaching. Mathematica has powerful functions as mathematical software and was developed by Wolfram Research. Aiming at the complex symbols and data operation problems in many fields, the software flexibly combines mathematics branches such as geometry and algebra, dramatically simplifying the operation process. This paper takes Mathematica’s existing mathematical software as a reference to discuss whether the addition of such software helps improve the effectiveness of daily teaching. The study found that using this software can stimulate students’ interest and enhance the quality of education.

In this study, the beta function that is encountered in computational mathematics and physics is analyzed. The correct evaluation of this function also affects the accuracy of other mathematical functions in quantum mechanical calculations. Especially in recent years, there is an interest in studies related to the beta function for zero and negative p and q integers. This study, considering the neutrix limits of the beta function, presents new relations for the numerical computation of the beta function, especially for negative integers p and q. In addition, taking into account the definition of the beta function for positive p and q integer values, an algorithm is created to calculate the function for all integer values. Finally, numerical results obtained with the help of our new recurrence relations and algorithm are presented.

This article is the second part of a series dealing with the description and visualization of mathematical functions used to describe a powder diffraction pattern for teaching and education purposes. The first part dealt with the instrumental and sample contributions to the profile of a Bragg peak [Dinnebier & Scardi (2021). J. Appl. Cryst.54, 1811–1831]. The second part, here, deals with the mathematics and physics of the intensity in X‐ray powder diffraction. Scholarly scripts are again provided using the Wolfram language in Mathematica. [ABSTRACT FROM AUTHOR]

Much ink has been spilled in the last few decades on computer proofs, computer-assissted proofs, computer-verified proofs, etc. It is obvious that the advent and proliferation of computers have drastically changed the applications of mathematics. What is far less discussed, however, is how computers changed mathematics itself, as well as the stance of mathematicians in regard to mathematical reality, both as far as the possibilities to immediately observe it and the apprehension of what we can hope to prove. I am recounting my personal experience of using computers as a mathematical tool and the experience of similar use in the works of colleagues that I observed at close range. This experience has radically changed my perception of many aspects of mathematics; how it functions; and, especially, how it should be taught. This first introductory part consists mostly of reminiscences and some philosophical observations. Later parts describe several specific important advances in algebra and number theory that would had been impossible without computers. [ABSTRACT FROM AUTHOR]

From 1960 to the early 2000s, the audio cassette was one of the three most used storage devices for prerecorded music, alongside gramophone records (LPs) and compact discs (CDs). A compact cassette consists of two miniature spools, between which a magnetic tape is passed and wound. The tape head in tape recorders and cassette players is used to store and play audio tracks in these magnetic tapes by converting the electric signal to a magnetic fluctuation and vice versa. The sound quality of an audio track recorded on a compact cassette is decent. However, as these cassettes are stored in the Earth's magnetic field for longer periods, the effect from it influences magnetic particles to change their orientation and distort the recorded audio signal. As a result, high-frequency noise is generated in the audio signal, resulting in a great disturbance when listening to songs and other audio recordings on vintage cassettes. Therefore, the removal of such noises while minimizing the impact on the original audio signal will make sure the listener gets to enjoy the original quality of the vintage audio recordings. Fourier transform of a given distorted audio file (in "wav" format) was obtained to identify the high-frequency noise using Wolfram Mathematica 12.3. Then a variable "m" was introduced to the program to identify magnitudes of high-frequency components and remove their effect on the original audio recording. A trial-and-error approach was taken to identify the adequate "m" value for each channel of a given audio signal. Vintage songs were used to evaluate the accuracy of the developed method in this study. It was identified that the suitable "m" value changes for each channel in each song. Therefore, it is impossible to state a constant value for the variable "m". However, the accuracy of the output also depends on the hearing frequency of the person, and the audio equipment being used. As an application, this technique can be used to conserve vintage songs that were originally stored in compact cassettes and spools, because it allows the user to control the number of high-frequency components removed from the distorted audio signal. [ABSTRACT FROM AUTHOR]

Copyright of Mugla Journal of Science & Technology is the property of Mugla Journal of Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of this work is to present computational intelligence tools implemented by Wolfram Language and included in Wolfram/Alpha and Wolfram Mathematica designed to weather and climate data analysis. In this work, I showed that climate problems in structural mechanics, urban planning, and building physics can be analyzed within the Wolfram Language. Ostrava was taken as a sample place of interest. It is also possible to estimate changes in climatic parameters in the future. The results presented could hopefully be an inspiration for other researchers dealing with stochastic and statistical problems in modeling mechanical problems. [ABSTRACT FROM AUTHOR]

Copyright of Experimental Technology & Management is the property of Experimental Technology & Management Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

One of the challenges is to know the exact value of magnetic field in a three-dimensional space which is unknown so far. In this paper we have used integration formula in a Mathematica software to calculate the exact magnetic field and then set up an experimental method to verify this experimentally by applying a magnetic field with a physical magnet of the diameter of 8.1 cm and height of 6 cm. Finally, a comparison between theoretical and experimental calculations is shown in two-dimensional and three-dimensional graphs.We develop a theoretical model by using Mathematica software to calculate the magnetic field for a one-sided magnet using a single electromagnetic coil and several electromagnetic coils. Similarly, we have developed a method to calculate the magnetic field using Mathematica software when two electromagnets are set on both sides of the arc plasma for a single coil and several coils. In all cases, theoretical data is verified by the experimental value. By comparing the graph for both one-sided and two-sided coils we have found a good agreement between the data with the experimental and theoretical values. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations.

In this study, the beta function that is encountered in computational mathematics and physics is analyzed. The correct evaluation of this function also affects the accuracy of other mathematical functions in quantum mechanical calculations. Especially in recent years, there is an interest in studies related to the beta function for zero and negative p and q integers. This study, considering the neutrix limits of the beta function, presents new relations for the numerical computation of the beta function, especially for negative integers p and q. In addition, taking into account the definition of the beta function for positive p and q integer values, an algorithm is created to calculate the function for all integer values. Finally, numerical results obtained with the help of our new recurrence relations and algorithm are presented. [ABSTRACT FROM AUTHOR]

Fluid flow problems are important in engineering applications like heat exchangers, renewable energy sources, and electromagnetic processing of materials, among other applications. Some examples related to a computational fluid dynamics (CFD) course are explored in this study. A CFD course provides tools for the numerical solution of fluid flows where an analytical cannot be found. Flows can be described by numerically solving the mass, momentum, and energy conservation equations for incompressible fluids. A Chebyshev collocation spectral method has been used to solve the governing equations. A high‐level functional programming language strategy was implemented in Mathematica including readability and adaptability. The programs are built by applying and composing functions, mappings, iterators, and pattern matching. It was noted that students can achieve a better comprehension of the algorithms using this programming approach. [ABSTRACT FROM AUTHOR]

The paper describes essential reaction–diffusion models with delay arising in population theory, medicine, epidemiology, biology, chemistry, control theory, and the mathematical theory of artificial neural networks. A review of publications on the exact solutions and methods for their construction is carried out. Basic numerical methods for integrating nonlinear reaction–diffusion equations with delay are considered. The focus is on the method of lines. This method is based on the approximation of spatial derivatives by the corresponding finite differences, as a result of which the original delay PDE is replaced by an approximate system of delay ODEs. The resulting system is then solved by the implicit Runge–Kutta and BDF methods, built into Mathematica. Numerical solutions are compared with the exact solutions of the test problems. [ABSTRACT FROM AUTHOR]

Today, calculus frequently is taught with artificial intelligence in the form of computer algebra systems. Although these software packages may reduce tedium associated with the mechanics of calculus, they may be less effective if not supplemented by the accompanying teaching of calculus theory. This paper presents two examples from spatial statistics in which computer software in an unsupervised auto-execution mode fails, or can fail, to yield correct calculus results. Accordingly, it emphasizes the need to teach calculus theory when using software packages such as Mathematica and Maple. [ABSTRACT FROM AUTHOR]

The temperature and fluid profiles of flow inside tilted square cavities are analysed with two different cases of thermal boundary conditions, (1) Isothermally cold sidewalls of the cavity and the hot bottom wall kept parallel to the insulated top wall, (2) Hot left wall, cold right wall, insulated top and bottom walls. The Galerkin finite element method with penalty parameter is used to solve the nonlinear coupled system of partial differential equations governing the flow and thermal fields. The method is further used to solve the Poisson equation for stream function. The results are presented in terms of isotherms and streamlines. The Gaussian rule with the hybrid function formed from the block-pulse function and Lagrange polynomial is implemented for the evaluation of the definite integrals present in the residual equations. Attempting to affix the hybrid methods in the integration part for solving Finite Element Method (FEM) turned efficacious as the convergence is achieved for streamlines and isotherms with the existing results. The tilted square cavities with inclination angles 0◦ ≤ φ ≤ 90◦ and Rayleigh number ranging between 10³ ≤ Ra ≤ 105 for Pr = 0.71 (air) are investigated. The source code for the finite element analysis is written in Mathematica. The results thus obtained are found to be competent with those of COMSOL, the Software for Multiphysics Simulation. [ABSTRACT FROM AUTHOR]

The temperature and fluid profiles of flow inside tilted square cavities are analysed with two different cases of thermal boundary conditions, (1) Isothermally cold sidewalls of the cavity and the hot bottom wall kept parallel to the insulated top wall, (2) Hot left wall, cold right wall, insulated top and bottom walls. The Galerkin finite element method with penalty parameter is used to solve the nonlinear coupled system of partial differential equations governing the flow and thermal fields. The method is further used to solve the Poisson equation for stream function. The results are presented in terms of isotherms and streamlines. The Gaussian rule with the hybrid function formed from the block-pulse function and Lagrange polynomial is implemented for the evaluation of the definite integrals present in the residual equations. Attempting to affix the hybrid methods in the integration part for solving Finite Element Method (FEM) turned efficacious as the convergence is achieved for streamlines and isotherms with the existing results. The tilted square cavities with inclination angles Deg 0 to Deg 90 and Rayleigh number ranging between 10^3 and 10^5 for Pr=0.71 (air) are investigated. The source code for the finite element analysis is written in Mathematica. The results thus obtained are found to be competent with those of COMSOL, the Software for Multiphysics Simulation.

A large collection of element‐wise planar densities for compounds obtained from the Materials Project is calculated using brute force computational geometry methods, where the planar density is given by the total fractional area of atoms intersecting a supercell's crystallographic plane divided by the area of the supercell's crystallographic plane. It is demonstrated that the element‐wise maximum lattice plane densities can be useful as machine learning features. The methods described here are implemented in an open‐source Mathematica package hosted at https://github.com/sgbaird/LatticePlane. [ABSTRACT FROM AUTHOR]

The gamma-Pareto type I convolution (GPC type I) distribution, which has a power function tail, was recently shown to describe the disposition kinetics of metformin in dogs precisely and better than sums of exponentials. However, this had very long run times and lost precision for its functional values at long times following intravenous injection. An accelerated algorithm and its computer code is now presented comprising two separate routines for short and long times and which, when applied to the dog data, completes in approximately 3 min per case. The new algorithm is a more practical research tool. Potential pharmacokinetic applications are discussed. [ABSTRACT FROM AUTHOR]