Taylor's Theorem Research Articles

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

Lagrange's Mean Value Theorem and Taylor's Theorem and Their Applications

Lagrange's Mean Value Theorem and Taylor's Theorem and Their Applications

A proof of the power rule by Taylor's theorem

The standard way of proving the power rule for irrational exponents is to use the differentiation of exponential and logarithmic functions. We derive the power rule for irrational exponents from that for rational exponents by using Taylor's theorem, without relying on exponential and logarithmic functions. This approach is straightforward in view of extending the definition of the power function from rational exponents to irrational exponents. It also helps students to understand that Taylor's theorem is useful to investigate the local behaviour of functions.

Series expansion of some elementary functions without using mathematical analysis

We present a method to compute the power series expansions of e x , ln ⁡ ( 1 + x ) , sin ⁡ x , and cos ⁡ x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of mathematical induction. The Newton binomial formula is obtained using the expansions of the exponent and the logarithm in their power series. We build upon the approach outlined in a paper by L. P. Mironenko and O. A. Rubtsova (Approximations of Some Functions by Polynomials and the Method of Undefined Coefficients, Scientific Papers of Donetsk National Technical University. Series: Computing and Automation, No. 2 (25), p. 128–135, 2013) and complement their approach. This approach does not depend on the application of Taylor's theorem to the expansion of functions. Consequently, these expansions can be integrated into the educational process before students become acquainted with the concept of derivatives. It simplifies the study of the theory of limits, especially in the computation of standard limits such as lim x → 0 sin ⁡ ( x ) x , lim x → 0 1 − cos ⁡ ( x ) x 2 , and lim x → 0 ( 1 + x ) 1 / x .

Analysis of the impact of uneven permeability of surrounding rock caused by the coupling effect of ground stresses and fault structure on sudden water inrush in tunnels

Affected by fault structure and in situ stress, the heterogeneity of the permeability of surrounding rock is universal. Treating it as a fixed value will reduce the prediction accuracy of water inflow and structural head. In view of this problem, considering the coupling effect of ground stress and fault structure, the permeability of surrounding rock is regarded as a spatially discrete type, a plane one-dimensional seepage calculation model in the vertical section is constructed, and the phreatic surface drop curve equation is established. Using Taylor's formula and series expansion theorem, the equation can be reduced to the expression of the falling curve when the permeability of the surrounding rock is homogeneous. Based on Darcy's law and the law of conservation of fluid mass, the calculation formula for tunnel water inflow and external water pressure of the structure was derived and verified through ongoing construction projects. Research shows that the calculation error of water inflow can be reduced from 23.1% to 7.5% when considering the influence of ground stress and fault structure on the permeability of surrounding rock, and the calculation error of water head borne by the supporting structure can be reduced from 43.8% to 30%, which improves the prediction accuracy. Thematic collection: This article is part of the Climate change and resilience in Engineering Geology and Hydrogeology collection available at: https://www.lyellcollection.org/topic/collections/climate-change-and-resilience-in-engineering-geology-and-hydrogeology

A multivariate extension of the Erdős–Taylor theorem

The Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if LN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N$$\\end{document} is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then πlogNLN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N} \\,\ extsf{L}_N$$\\end{document} converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if LN(1,2)=∑n=1N1{Sn(1)=Sn(2)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{L}_N^{(1,2)}=\\sum _{n=1}^N \\mathbb {1}_{\\{S_n^{(1)}= S_n^{(2)}\\}}$$\\end{document}, then πlogNLN(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ frac{\\pi }{\\log N}\\, \ extsf{L}^{(1,2)}_N$$\\end{document} converges in distribution to an exponential random variable of parameter one. We prove that for every h⩾3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\geqslant 3$$\\end{document}, the family {πlogNLN(i,j)}1⩽i<j⩽h\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\big \\{ \\frac{\\pi }{\\log N} \\,\ extsf{L}_N^{(i,j)} \\big \\}_{1\\leqslant i<j\\leqslant h}$$\\end{document}, of logarithmically rescaled, two-body collision local times between h independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.

Estimates of linear expressions through factorization

The aim of this paper is to establish various factorization results and then to derive estimates for linear functionals through the use of a generalized Taylor theorem. Additionally, several error bounds are established including applications to the trapezoidal rule as well as to a Simpson formula-type rule.

Evaluating the effect of a blended collaborative/game-based learning strategy for skill reinforcement on undergraduate engineering

This study evaluates the implementation of a collaborative/game-based learning strategy on final year B.Sc. students focused on reinforcing the Taylor theorem (a mathematical concept previously learned, usually forgotten over time and widely used in chemical engineering) and analyses its effect on their academic performance. To this end, the design of educational card games based on basic rules was proposed to students in their fifth semester (as soon as the mathematical concept is covered in their courses) and applying these games to students in the eighth semester as a reinforcement tool (who needed to apply the concept again). A mixed quasi-experimental design was carried out in which the performance and perception of the students were evaluated. The instruments used were a validated Likert-type perception test with an alpha of 0.91 and a knowledge test applied before and after the intervention. It was found that the retention rate of this mathematical concept went from 22.6% to 50%, showing an improvement in learning Taylor's series. Compared to the traditional class, the game significantly improved performance, promoting the development of skills such as creativity, innovation, participation, leadership, goal management and motivation and was recognised by the students as an innovative way of learning. In conclusion, implementing unconventional collaborative learning strategies improves academic performance and promotes skill development in engineering students.

Craniomaxillofacial Bone Segmentation and Landmark Detection Using Semantic Segmentation Networks and an Unbiased Heatmap.

Craniomaxillofacial (CMF) surgery always relies on accurate preoperative planning to assist surgeons, and automatically generating bone structures and digitizing landmarks for CMF preoperative planning is crucial. Since the soft and hard tissues of the CMF regions possess complicated attachment, segmenting the CMF bones and detecting the CMF landmarks are challenging problems. In this study, we proposed a semantic segmentation network to segment the maxilla, mandible, zygoma, zygomatic arch, and frontal bones. Then, we obtained the minimum bounding box around the CMF bones. After cropping, we used the top-down heatmap landmark detection network, similar to the segmentation module, to identify 18 CMF landmarks from the cropping patch. In addition, an unbiased heatmap encoding method was proposed to generate actual landmark coordinates in the heatmap. To overcome quantization effects in the heatmap-based landmark detection networks, the distribution-prior coordinate representation of medical landmarks (DCRML) was proposed to utilize the prior distribution of the encoding heatmap, approximating the accurate landmark coordinates in heatmap decoding by Taylor's theorem. The encoding and decoding method can easily contribute to other existing landmark detection frameworks based on heatmaps; consequently, these approaches can readily benefit without changing model structure. We used prior segmentation knowledge to enhance the semantic information around the landmarks, increasing landmark detection accuracy. The proposed framework was evaluated by 100 healthy persons and 86 patients from multicenter cooperation. The mean Dice score of our proposed segmentation network achieved over 88 %; in particular, the mandible accuracy was approximately 95%. The mean error of landmarks was 1.84 ±1.32 mm.

A generalization of the Taylor's theorem

In mathematics, Taylor's theorem is a formula that uses the information of a function to describe its proximity to a point. If the function is smooth enough, the conductive values can be used to construct a multiform to approximate the the function in the neighboring area of that point, which can even be extended to the convergence radius of the scale. Taylor’s theorem also gives the deviation between this multiform and the actual function value. The essay includes the proof and application of Taylor’s theorem and the Taylor’s theorem in multivariate functions. Special functions are set with unknown coefficients to deduce the Taylor’s theorem. Specific questions are solved by using the Taylor’s theorem as some examples. Taylor's theorem is frequently applied in real-world settings to solve problems. For instance, in engineering, it can be used to build a variety of numerical simulations and optimization algorithms. In physics, it can be used to analyze data from physical experiments. In finance, it can be used to manage risk and set prices for financial transactions.

The Taylor's Theorem and Its Application

Taylor's theorem is an essential concept in calculus. By constructing polynomials, Taylor's formula can simplify calculations by approximating complex functions so that a variety of functions can be analyzed in detail. This paper explores the form and proof of Taylor's theorem based on Peano and Lagrange remainder forms as well as the Polynomial interpolation. Then, examples of the application of Taylor’s theorem in mathematics fields, such as limit calculation and high-order derivatives computation, are discussed. Additionally, the applications of Taylor’s theorem in other related subjects are examined. Based on the chronological investigation, the Taylor theorem is widely used in physics, engineering, and other fields as a computational tool. The Taylor theorem has gradually been used in machine learning because of the rapid development of computer science in recent years. The purpose of this paper is to summarize the different forms and proofs of Taylor's formula and discuss the development of its application over time.

REACTION-DIFFUSION FISHER’S EQUATIONS VIA DECOMPOSITION METHOD

The effect of the source, initial or boundary conditions in the use of Adomian decomposition method (ADM) on nonlinear partial differential equation or nonlinear equation in general is enormous. Sometimes the equation in question result to continuous exact solution in series form, other times it result to discrete approximate analytical solutions. In this paper, we show that continuous exact solitons can be obtained on application of ADM to the Fisher's equation with the deployment Taylor theorem to the terms(s) in question. And, the resulting series is split into the integral equations during the solution process. Resulting to multivariate Taylor's series of the exact solitons with the help of Adomian polynomials of the nonlinear reaction term correctly calculated. More physical results are further depicted in 2D, 3D and contour plots.

A Numerical Approach for Dealing with Fractional Boundary Value Problems

This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order α and the fractional central formula for approximating the Caputo differentiator of order 2α, where 0&lt;α≤1. The first formula is recalled here, whereas the second one is derived based on the generalized Taylor theorem. The stability of the proposed approach is investigated in view of some formulated results. In addition, several numerical examples are included to illustrate the efficiency and applicability of our approach.

A new second order Taylor-like theorem with an optimized reduced remainder

In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function f defined on the interval [a,b], this formula is derived by introducing a linear combination of f′ computed at n+1 equally spaced points in [a,b], together with f′′(a) and f′′(b). We then consider two classical applications of this Taylor-like expansion: the interpolation error and the numerical quadrature formula. We show that using this approach improves both the Lagrange P2- interpolation error estimate and the error bound of the Simpson rule in numerical integration.

Integral theorems in finite-dimensional commutative algebra

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods for proving results are based on a representation of monogenic functions via holomorphic functions of complex variables that allows to establish analogues of Cauchy-Riemann conditions and the continuity of G\^ateaux derivatives of all orders for monogenic functions. In such a way, analogues of a number of classical theorems of complex analysis (the Cauchy integral theorem for a curvilinear integral, the Cauchy integral formula, the Morera theorem, the Taylor theorem) are proved and different equivalent definitions for the mentioned monogenic functions are established. An analogue of the Cauchy theorem for an integral over non piecewise smooth surfaces is proved.

A kinetic grid for fabric draping simulation

In allusion to the hot issue of fabric simulation, this paper proposes a kinetic grid for reproducing the drape profile of fabric in three-dimensional space. In our model, the interaction inside fabric is built up through a constraint factor as well as an attenuation factor, and the contact between the flexible body of the fabric and the rigid plane is emulated with a touch-counteract mechanism. The forces on the particles in the grid, which are calculated with grid deformation, are then used to generate the additional displacement of the grid. Evolution of the grid is led by step-by-step iteration so that the draping kinetics of the virtual fabric can be achieved. When the draping process meets its steady state, the drape coefficient is worked out through identifying and quantifying the projection area of the grid. The unevenness of the ripples along the drape surface is characterized and manifested by emulating the mechanical anisotropy of the fabric through differentiated constraint factors. The evolution algorithm is improved and an optimized version that reduces calculation error using the third-order Taylor theorem is presented, by which a smoother curved surface for the drape can be produced. The convergence of the grid under different fineness values is explored to reveal the capacity of the model, and a reasonable grid scale has been identified. The computed drape is then compared to a real drape featuring different strengths in the warp and weft, and adjusted to meet the target drape coefficient, as well as drape unevenness considering the anisotropy index. The result shows they are properly matched. The virtual drape is also tested in additional scenes including square support with sharp corners, one-way overhanging and mixed. It has turned out to be a simple, rapid and precise model for fabric drape simulation and assessment.

A Numerical Approach of Handling Fractional Stochastic Differential Equations

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

Moments of the 2D Directed Polymer in the Subcritical Regime and a Generalisation of the Erdös–Taylor Theorem

We compute the limit of the moments of the partition function Z_{N}^{beta _N} of the directed polymer in dimension d=2 in the subcritical regime, i.e. when the inverse temperature is scaled as beta _N sim hat{beta } sqrt{tfrac{pi }{log N}} for hat{beta } in (0,1). In particular, we establish that for every h in {mathbb {R}}, lim _{N rightarrow infty } {{mathbb {E}}} big [big (Z_{N}^{beta _N}big )^h big ]=big (frac{1}{1-hat{beta }^2}big )^{frac{h(h-1)}{2}}. We also identify the limit of the moments of the averaged field tfrac{sqrt{log N}}{N} sum _{x in {mathbb {Z}}^2} varphi (tfrac{x}{sqrt{N}})big (Z_{N}^{beta _N} (x)-1 big ), for varphi in C_c({mathbb {R}}^2), as those of a gaussian free field. As a byproduct, we identify the limiting probability distribution of the total pairwise collisions between h independent, two dimensional random walks starting at the origin. In particular, we derive that πlogN∑1≤i<j≤hLN(i,j)→N→∞(d)Γ(h(h-1)2,1),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\pi }{\\log N}\\sum _{1 \\le i<j\\le h} {\ extsf{L}}_N^{(i,j)}\\xrightarrow [N \\rightarrow \\infty ]{(d)} \\Gamma \\big ( \ frac{h(h-1)}{2},1\\big ) , \\end{aligned}$$\\end{document}where {{textsf{L}}}^{(i,j)}_N denotes the collision local time by time N between copies i, j and Gamma denotes the Gamma distribution. This generalises a classical result of Erdös and Taylor (Acta Math Acad Sci Hung 11:137–162, 1960).

A General Unfolding Speech Enhancement Method Motivated by Taylor's Theorem

A General Unfolding Speech Enhancement Method Motivated by Taylor's Theorem

Measurable Taylor's Theorem: An Elementary Proof

Measurable Taylor's Theorem: An Elementary Proof