Derivation Of Formula Research Articles

Finite element method for a generalized constant delay diffusion equation

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

Flux ruled surfaces and the magnetic curves obtained from the curvature theory

In this paper, our aim is to research the change in the magnetic field using the curvature theory of ruled surfaces. For this, we examine magnetic curves of the curve whose existence is guaranteed from the derivative formulas of the rotation frame. The Killing vector fields and Lorentz forces of these magnetic curves are calculated. Additionally, correlations between curvatures are obtained. Flux ruled surfaces are obtained and the conditions for developable are achieved. Finally, examples of the magnetic flux developable ruled surfaces are given.

AN EXPLORATORY STUDY: DEVELOPMENT OF ELECTRONIC MODULES IN MATHEMATICAL PHYSICS I COURSE

AN EXPLORATORY STUDY: DEVELOPMENT OF ELECTRONIC MODULES IN MATHEMATICAL PHYSICS I COURSE

A study of operational decisions of city gas operators under the energy metering and pricing model

This study examines the quantitative conditions under which an energy metering pricing model is proposed to increase both gas merchant profits and gas customer consumer surplus compared to a volumetric pricing model. The quantitative condition is found to be related to factors such as the standard unit calorific value of natural gas prescribed by the National Development and Reform Commission (NDRC) and other relevant government departments under the energy metering pricing model. This paper establishes a mathematical model based on optimization theory to explore the operational decisions of city gas suppliers in the volumetric and energy metering and pricing modes, respectively, under the condition of relatively stable natural gas sales price. The results of the study show that DAC and other authorities can regulate the standard unit calorific value of natural gas under the energy metering and pricing model by regulating the standard unit calorific value of natural gas. This affects the incentives of gas dealers to produce and operate, guides the preference of gas users for natural gas energy metering and pricing, and results in the derivation of formulas for a reasonable range of standard unit calorific values for natural gas. The findings of this paper provide theoretical support to promote the reform of natural gas energy measurement and pricing, and contribute to the development of the natural gas industry.

Development of Sample Size Formulas for Least Square Regression-Based Consistency Evaluation of Quantitative Indicators

Objective To develop and verify the sample size formulas for quantitative data consistency evaluation based on the least square regression method. Methods According to the principle of least square regression-based quantitative consistency evaluation,statistical inference,and formula derivation,we developed the formulas for calculating sample size based on regression constant and regression coefficient.Furthermore,the accuracy of the formulas was verified by the data of three examples,and the results were compared with those of the sample size formula established based on the Bland-Altman(BA)method. Results The sample size formulas for regression-based quantitative consistency evaluation were deduced,and the accuracy of the formulas was verified by three examples.In addition,the results obtained with this formula had differences compared with those of the sample size formula established based on the BA method.Furthermore,consistent conclusions could be obtained by regression analysis and BA analysis with the sample size calculated with the regression method.However,with the sample size calculated based on the BA method,the consistency conclusion of regression analysis and BA analysis was sometimes not valid. Conclusion A sample size formula for quantitative consistency evaluation based on the regression method was proposed for the first time,which provided methodological support for the research in this field.

Modified generalized second derivative extended backward differentiation formulas for highly stiff and stiffly oscillatory systems of ODEs

Modified generalized second derivative extended backward differentiation formulas for highly stiff and stiffly oscillatory systems of ODEs

Novel half-sandwich ruthenium(II) nicotinic hydrazone complexes: An efficient class of catalyst for the N-alkylation of amines with benzyl alcohol via transfer hydrogen mechanism and effective antibacterial agents

A new series of half sandwich ruthenium(II) complexes supported by nicotinic acid hydrazone derivatives of general formula [Ru(ɳ6-p-cymene)(Cl)(L)], where L represents N’-benzylidenenicotinohydrazide (HL1), N’-(naphthalen-1-ylmethylene)nicotinohydrazide (HL2) or N’-(pyren-1-ylmethylene)nicotinohydrazide (HL3) were synthesized and fully characterized. The solid-state molecular structures of ligands and complexes were elucidated with single-crystal X-ray diffraction. The ligands HL1 and HL2 possess monoclinic crystal system with P21/c and C2/c space groups respectively, whereas all the three Ru(II) complexes C1–3 are typical three-legged piano-stool geometries with orthorhombic(Aba2) (C1) and triclinic(P-1) (C2–3) systems respectively. The catalytic activities of these ruthenium complexes in the synthesis of N-alkylated amines were explored. Under optimal reaction condition, the complex C3 was found to be the most efficient catalyst for the N-alkylation of primary to secondary amines using benzyl alcohol. A wide range of substrates have been studied that includes anilines bearing electron withdrawing/electron donating group, fused aromatic, heteroaromatic, alicyclic and aliphatic amines with good to excellent yield. In addition, the antibacterial activity of the complexes were tested against different human pathogenic bacteria, among them C2 complex with naphthyl moiety was found to be the most active one against both Gram-positive (S. aureus and E. faecalis) and Gram-negative (E. coli and K. pneumoniae) bacteria. These two different studies demonstrate the catalytic and biological importance of our newly prepared Ruthenium complex.

Theoretical derivation and analysis of strength calculation formulas of internal gear pair with double contact points

To describing the strength conditions of tooth surfaces of internal gear pair with double contact points (IGPDCP) accurately, theoretical calculation method of the strength of tooth surfaces is proposed in this paper. Theoretical derivation and analysis of strength calculation formulas of the IGPDCP drive are carried out. For contact strength analysis, based on Hertz point contact theory, general calculation idea of contact strength is provided. The main curvature of the active contact line on tooth surfaces of the pinion and internal gear are derived, respectively. Furtherly, according to theoretical calculation and derivation, the formula for calculating the contact strength of the IGPDCP drive is obtained. Orthogonal test method is used to derive the calculation formula of bending strength of the IGPDCP drive, considering test factors include the modulus, tooth number of gear, transmission ratio, helix angle, contact angle, pressure angle, and torque. Nonlinear regression equations of bending stress of the pinion and internal gear are derived, respectively. The discussion of results of contact and bending strength between formula calculation and finite element analysis are present. The conclusions show that the proposed formulas can satisfy theoretical analysis and engineering application.

A rasterized plane wave expansion method for complex 2-D phononic crystals

This paper introduces a computer image processing-based method called the rasterized plane wave expansion method (RPWEM). This method improves the conventional plane wave expansion method (CPWEM) and reduces its formula derivation cost in the case of 2-D phononic crystals with complex inclusions. In addition, for 2-D phononic crystals with arbitrary multiphase materials, this method can also simplify the calculation process. By employing the proposed RPWEM, successful calculations of band gaps are conducted for planar phononic crystals with the hexagonal unit cell as well as phononic crystals featuring integrated circuit-like structures. The results further validate the expanded applicability of the RPWEM in comparison to the CPWEM. In the end, a potential application similar to DIC (Digital Image Correlation) for predicting band structures of phononic crystals is proposed, and its feasibility is briefly validated.

Extended Conformable K-Hypergeometric Function and Its Application

The extended conformable k-hypergeometric function finds various applications in physics due to its ability to describe complex mathematical relationships arising in different physical scenarios. Here are a few instances of its uses in physics, including nuclear physics, fluid dynamics, quantum mechanics, and astronomy. The main objectives of this paper are to introduce the extended conformable k-hypergeometric and confluent hypergeometric functions by utilizing the new definition of the α,k-beta function and studying its important properties, like integral representation, summation formula, derivative formula, transform formula, and generating function. Also, introduce the extension of the Riemann–Liouville fractional derivative and establish some results related to the newly defined fractional operator, such as the Mellin transform and relations to extended α,k-hypergeometric functions.

Values of certain Dirichlet series and higher derivative formulas of trigonometric functions

We determine new values of certain Dirichlet series and related infinite series. These formulas extend results of several authors. To obtain these results we apply recent expansions of higher derivative formulas of trigonometric functions. We also investigate the transcendentality of values of these series and arithmetic relations of the values of certain related infinite series.

A direct transcription method for solving distributed-order fractional optimal control problems

This paper considers the numerical solution of distributed-order Fractional Optimal Control Problems (FOCPs). The common approaches for solving these problems involve approximating the problem with a multi-term FOCP and subsequently discretizing the resulting multi-term problem using either a direct or indirect method. However, this paper bypasses the need for approximation with a multi-term FOCP by introducing innovative approximation formulas for distributed-order fractional derivatives. These approximation formulas are directly derived from the conventional Grünwald–Letnikov, L1, and trapezoidal formulas. Building upon these formulas, we develop the relevant fractional-order distributed-order operational matrices. Utilizing the operational matrix, we easily discretize the distributed-order FOCP to a Non-Linear Programming (NLP), which can be solved by a suitable NLP-solver. Additionally, to enhance the efficiency of solving the resulting NLP, we determine the closed-forms of both the Jacobian of constraints and the gradient of the objective function. The presented method is characterized by its speed, simplicity, and ease of implementation. Moreover, it can be used to solve a wide range of distributed-order FOCPs, such as those involving nonlinear dynamics, free final times, free terminal conditions, and path constraints. By means of ample numerical tests, the accuracy and efficiency of the proposed method are assessed.

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Implementation of Caputo type fractional derivative chain rule on back propagation algorithm

Fractional gradient computation is a challenging task for neural networks. In this study, the brief history of fractional derivation is abstracted, and the core framework of the Faà di Bruno formula is implemented to the fractional gradient computation problem. As an analytical approach to solve the chain rule problem of fractional derivatives, the use of the Faà di Bruno formula for the Caputo-type fractional derivative is proposed. When the fractional gradient is calculated using the proposed approach, the problem of determining the bounds of the formula for calculating the Caputo derivative is addressed. As a consequence, the fundamental problem with the fractional back-propagation algorithm is solved analytically, paving the way for the use of any differentiable and integrable activation function in case they are stable. The development of the algorithm and its practical implementation on the photovoltaic fault detection data-set is investigated. The reliability of the neural network metrics is also investigated using analysis of variance (ANOVA), the results are then presented to decide which are the best metrics and the best fractional order. Ordinary back-propagation, fractional back-propagation with and without L2 regularization and momentum are presented in the results to show the advantages of using L2 regularization and momentum in fractional order gradient. Consequently, the test metrics are reliable but cannot be separated from each other. By changing the batch size from 2 to full batches, the proposed system performs better with bigger batches, but adaptive moment estimation (ADAM) performs better with small batches. The cross validation shows the performance of back propagate ion neural networks have better performance than the ordinary neural networks. It is interesting that the best order for a specific data-set cannot be determined because it depends on the batch size, number of epochs and the cross-validation folds .

Convergence of Ginzburg–Landau Expansions: Superconductivity in the Bardeen–Cooper–Schrieffer Theory and Chiral Symmetry Breaking in the Nambu–Jona-Lasinio Model

Abstract We study the convergence of the Ginzburg–Landau (GL) expansion in the context of the Bardeen–Cooper–Schrieffer (BCS) theory for superconductivity and the Nambu–Jona-Lasinio (NJL) model for chiral symmetry breaking at finite temperature T and chemical potential μ. We present derivations of the all-order formulas for the coefficients of the GL expansions in both systems under the mean-field approximation. We show that the convergence radii for the BCS gap Δ and dynamical quark mass M are given by Δconv = πT and $M_{\rm conv} = \sqrt{\mu ^2 + (\pi T)^2}$, respectively. We also discuss the implications of these results and the quantitative reliability of the GL expansion near the first-order chiral phase transition.

Fundamental Theorems for Timelike Surfaces in the Minkowski 4-Space

In the present paper, we study timelike surfaces free of minimal points in the four-dimensional Minkowski space. For each such surface we introduce a geometrically determined pseudo-orthonormal frame field and writing the derivative formulas with respect to this moving frame field and using the integrability conditions, we obtain a system of six functions satisfying some natural conditions. In the general case, we prove a Fundamental Bonnet-type theorem (existence and uniqueness theorem) stating that these six functions, satisfying the natural conditions, determine the surface up to a motion. In some particular cases, we reduce the number of functions and give the fundamental theorems.

Investigating Recurrent Matere Bonds in Pertechnetate Compounds.

In this manuscript we evaluate the X-ray structure of five new pertechnetate derivatives of general formula [M(H2 O)4 (TcO4 )2 ], M=Mg, Co, Ni, Cu, Zn (compounds 1-5) and one perrhenate compound Zn(H2 O)4 (ReO4 )2 (6). In these complexes the metal center exhibits an octahedral coordination with the pertechnetate units as axial ligands. All compounds exhibit the formation of directional Tc⋅⋅⋅O Matere bonds (MaBs) that propagate the [M(H2 O)4 (TcO4 )2 ], into 1D supramolecular polymers in the solid state. Such 1D polymers are linked, generating 2D layers, by combining additional MaBs and hydrogen bonds (HBs). Such concurrent motifs have been analyzed theoretically, suggesting the noncovalent σ-hole nature of the MaBs. The interaction energies range from weak (~ -2 kcal/mol) for the MaBs to strong (~ -30 kcal/mol) for the MaB+HB assemblies, where HB dominates. In case of M=Zn, the corresponding perrhenate Zn(H2 O)4 (ReO4 )2 complex, has been also synthesized for comparison purposes, resulting in the formation of an isostructural X-ray structure, corroborating the structure-directing role of Matere bonds.

A highly efficient and accurate numerical method for the electromagnetic scattering problem with rectangular cavities

This paper presents a robust numerical solution to the electromagnetic scattering problem involving multiple multi-layered cavities in both transverse magnetic and electric polarizations. A transparent boundary condition is introduced at the open aperture of the cavity to transform the problem from an unbounded domain into that of bounded cavities. By employing Fourier series expansion of the solution, we reduce the original boundary value problem to a two-point boundary value problem, represented as an ordinary differential equation for the Fourier coefficients. The analytical derivation of the connection formula for the solution enables us to construct a small-scale system that includes solely the Fourier coefficients on the aperture, streamlining the solving process. Furthermore, we propose accurate numerical quadrature formulas designed to efficiently handle the weakly singular integrals that arise in the transparent boundary conditions. To demonstrate the effectiveness and versatility of our proposed method, a series of numerical experiments are conducted.

CT system act and the derivation formula Between classical and colloquial southern Iraq

من المؤكد أنَّ هناك أنظمة تحتية تتحكم في أبنية المشتقات ، فالاشتقاق مرحلة لاحقة للمواد الأولية التي شكلته ، هذا ما تكشف عنه العادات المقطعية ، فهي فاعلة جداً قي صيغة الاشتقاق، وإذا ما انطلق التحليل اللغويّ منها استطاع أن يغير في كثير من أعراف التحليل الصرفيّ، هذا لا يعني أنّ التحليل اللغوي ذا الأسس المقطعية نهائي وتام بقدر ما يكشف عن اقتراب من عتبة العلمية، فمهما كان التحليل منطقيّ فإنّه يبقى قراءة جديدة للغة قديمة، لا نعرف عن لحظات ولادتها ما نستطيع ان نجيب بها عن سؤال: لماذا كانت العادة المقطعية؟
 ولمحاولة الإجابة عن فاعليّة النظام المقطعيّ في صياغة المشتقات بوصفه بنية أساسية يقوم عليها الاشتقاق ، والتعرف بموقع النظام المقطعيّ من الاشتقاق، ومدى إسهامه في صياغته؛ اعتمد البحث على منهج وصفيّ أفاد من اتخاذ العاميّة آلية تقيس مسافة التحول عبر مقاربتها من الفصحى، بالاستناد إلى النظام المقطعيّ.

Worst-Case Analysis of Heapsort, Exactly

Abstract This paper completes analysis of the worst-case running time of $ {\tt{Heapsort}}$ measured by the number of comparisons of keys performed while sorting. A derivation of an exact formula for the maximum number of comparisons of keys performed by $ {\tt{Heapsort}} $ on any array of size $ N \geq 2 $ is presented. It is equal to $$ \begin{align}& \nonumber 2(N-1)\lceil \lg N \rceil - 2 ^{\lceil \lg N \rceil +1} - 2 s_2(N) - e_2(N) + \min (\lceil \lg N \rceil, 3) + 5 + c, \end{align} $$ where $s_2(N)$ is the sum of digits of the binary representation of $N$, $e_2(N)$ is the exponent of $2$ in the $N$’s prime factorization and $ c $ is a binary function on the set of positive integers defined by $$ \begin{align}& \nonumber c = \left\{ \begin{array}{@{}ll} 1 \text{ if} \; N \leq 2 ^{\lceil \lg N \rceil} - 4 \\[4pt] 0 \text{ otherwise}. \end{array} \right. \end{align} $$ The above formula allows for deciding, in $ O(N \log N) $ time, if any given $ N $-element array is a worst-case array for $ {\tt{Heapsort}} $. Its proof yields an algorithm for construction, in $ O(N \log N) $ time, of worst-case arrays of arbitrary sizes $ N \geq 2 $ for $ {\tt{Heapsort}} $.