SOME PROPERTIES OF MULTIPLE TAYLOR SERIES AND RANDOM TAYLOR SERIES

Some polar coordinates are used to determine the domain and the ball of convergence of a multiple Taylor series. In this domain and in this ball the series converges, converges absolutely and converges uniformly on any compact set properties of the series may also be studied. For some random multiple are some corresponding properties. Growth and other Taylor series there

TAYLOR SERIES AND ORTHOGONALITY OF THE OCTONION ANALYTIC FUNCTIONS

The Taylor series of the Octonion analytic function is given. And the orthogonal formula for the Octonion analytic functions is also obtained.

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN C^n

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.

MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method

This paper investigates the magnetohydrodynamic （MHD） boundary layer flow of an incompressible upper-convected Maxwell （UCM） fluid over a porous stretching surface. Similarity transformations are used to reduce the governing partial differential equations into a kind of nonlinear ordinary differential equations. The nonlinear prob- lem is solved by using the successive Taylor series linearization method （STSLM）. The computations for velocity components are carried out for the emerging parameters. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem.

THE DISTRIBUTION OF VALUES OF TAYLOR SERIES

Clarify some classical results and their proofs on the distribution of values of simple Taylor series and extend them to multiple Taylor series.

Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

A relation spectrum inheriting Taylor series:muscle synergy and coupling for hand

There are two famous function decomposition methods in math:the Taylor series and the Fourier series.The Fourier series developed into the Fourier spectrum,which was applied to signal decomposition and analysis.However,because the Taylor series function cannot be solved without a definite functional expression,it has rarely been used in engineering.We developed a Taylor series using our proposed dendrite net(DD),constructed a relation spectrum,and applied it to decomposition and analysis of models and systems.Specifically,knowledge of the intuitive link between muscle activity and finger movement is vital for the design of commercial prosthetic hands that do not need user pre-training.However,this link has yet to be understood due to the complexity of the human hand.In this study,the relation spectrum was applied to analyze the muscle—finger system.One single muscle actuates multiple fingers,or multiple muscles actuate one single finger simultaneously.Thus,the research was focused on muscle synergy and muscle coupling for the hand.The main contributions are twofold:(1)The findings concerning the hand contribute to the design of prosthetic hands;(2)The relation spectrum makes the online model human-readable,which unifies online performance and offline results.Code is available at https://github.com/liugang1234567/Gang-neuron.

Stability and Accuracy Analysis for Taylor Series Numerical Method

The Taylor series numerical method (TSNM) is a time integration method for solving problems in structural dynamics. In this paper, a detailed analysis of the stability behavior and accuracy characteristics of this method is given. It is proven by a spectral decomposition method that TSNM is conditionally stable and belongs to the category of explicit time integration methods. By a similar analysis, the characteristic indicators of time integration methods, the percentage period elongation and the amplitude decay of TSNM, are derived in a closed form. The analysis plays an important role in implementing a procedure for automatic searching and finding convergence radii of TSNM. Finally, a linear single degree of freedom undamped system is analyzed to test the properties of the method.

On Random Taylor Series

This paper deals with random Taylor series whose coefficients consist of independent random variables {X n} with the property: αE 1/2 {|X n| 2}≤E{|X n|}<∞, E{X n}=0 (n ) for some positive constant α. The convergence, growth, and value distribution of the series are investigated.

Design of a Multifrequency Signal Parameter Estimation Method for the Distribution Network Based on HIpST

The application of traditional synchronous measurement methods is limited by frequent fluctuations of electrical signals and complex frequency components in distribution networks.Therefore,it is critical to find solutions to the issues of multifrequency parameter estimation and synchronous measurement estimation accuracy in the complex environment of distribution networks.By utilizing the multifrequency sensing capabilities of discrete Fourier transform signals and Taylor series for dynamic signal processing,a multifrequency signal estimation approach based on HT-IpDFT-STWLS(HIpST)for distribution networks is provided.First,by introducing the Hilbert transform(HT),the influence of noise on the estimation algorithm is reduced.Second,signal frequency components are obtained on the basis of the calculated signal envelope spectrum,and the interpolated discrete Fourier transform(IpDFT)frequency coarse estimation results are used as the initial values of symmetric Taylor weighted least squares(STWLS)to achieve high-precision parameter estimation under the dynamic changes of the signal,and the method increases the number of discrete Fourier.Third,the accuracy of this proposed method is verified by simulation analysis.Data show that this proposed method can accurately achieve the parameter estimation of multifrequency signals in distribution networks.This approach provides a solution for the application of phasor measurement units in distribution networks.

ON A GENERALIZED TAYLOR THEOREM： A RATIONAL PROOF OF THE VALIDITY OF THE HOMOTOPY ANALYSIS METHOD

A generalized Taylor series of a complex function was derived and some related theorems about its convergence region were given. The generalized Taylor theorem can be applied to greatly enlarge convergence regions of approximation series given by other traditional techniques. The rigorous proof of the generalized Taylor theorem also provides us with a rational base of the validity of a new kind of powerful analytic technique for nonlinear problems, namely the homotopy analysis method.

A TSE based design for MMSE and QRD of MIMO systems based on ASIP

A Taylor series expansion(TSE) based design for minimum mean-square error(MMSE) and QR decomposition(QRD) of multi-input and multi-output(MIMO) systems is proposed based on application specific instruction set processor(ASIP), which uses TSE algorithm instead of resource-consuming reciprocal and reciprocal square root(RSR) operations.The aim is to give a high performance implementation for MMSE and QRD in one programmable platform simultaneously.Furthermore, instruction set architecture(ISA) and the allocation of data paths in single instruction multiple data-very long instruction word(SIMD-VLIW) architecture are provided, offering more data parallelism and instruction parallelism for different dimension matrices and operation types.Meanwhile, multiple level numerical precision can be achieved with flexible table size and expansion order in TSE ISA.The ASIP has been implemented to a 28 nm CMOS process and frequency reaches 800 MHz.Experimental results show that the proposed design provides perfect numerical precision within the fixed bit-width of the ASIP, higher matrix processing rate better than the requirements of 5G system and more rate-area efficiency comparable with ASIC implementations.

Nonstandard Unitary Transformations of Quantum States

In quantum optics, unitary transformations of arbitrary states are evaluated by using the Taylor series expansion. However, this traditional approach can become cumbersome for the transformations involving non-commuting operators. Addressing this issue, a nonstandard unitary transformation technique is highlighted here with new perspective. In a spirit of “quantum” series expansions, the transition probabilities between initial and final states, such as displaced, squeezed and other nonlinearly transformed coherent states are obtained both numerically and analytically. This paper concludes that, although this technique is novel, its implementations for more extended systems are needed.

LAI scale effect research based on compact airborne spectrographic imager data in the Heihe Oasis

As one of the key parameters for characterizing crop canopy structure, Leaf Area Index(LAI) has great significance in monitoring the crop growth and estimating the yield. However, due to the nonlinearity and spatial heterogeneity of LAI inversion model, there exists scale error in LAI inversion result, which limits the application of LAI product from different remote sensing data. Therefore, it is necessary to conduct studies on scale effect. This study was based on the Heihe Oasis, Zhangye city, Gansu province, China and the following works were carried out: Airborne hyperspectral CASI(Compact Airborne Spectrographic Imager) image and LAI statistic models were adopted in muti-scale LAI inversion. The overall difference of muti-scale LAI inversion was analyzed in an all-round way. This was based on two aspects, "first inversion and then integration" and "first integration and then inversion", and on scale difference characteristics of three scale transformation methods. The generation mechanism of scale effect was refined, and the optimal LAI inversion model was expanded by Taylor expansion. By doing so, it quantitatively analyzed the contribution of various inversion processes to scale effect. It was found that the cubic polynomial regression model based on NDVI(940.7 nm, 712 nm) was the optimal model, where its coefficient of determination R2 and the correlation coefficient of test samples R reached 0.72 and 0.936, respectively. Combined with Taylor expansion, it analyzed the scale error generated by LAI inversion model. After the scale effect correction of one-dimensional and twodimensional variables, the correlation coefficient of CCD-LAI(China Environment Satellite HJ/CCD images) and CASI-LAI products(Compact Airborne Spectro graphic Imager products) increased from 0.793 to 0.875 and 0.901, respectively. The mean value, standard deviation, and relative true value of the two went consistent. Compared with onedimensional variable correction method, the twodimensional method had a better correction result. This research used the effective information in hyperspectral data as sub-pixels and adopted Taylor expansion to correct the scale error in large-scale and low-resolution LAI product, achieving large-scale and high-precision LAI monitoring.

Discovery of New Terms Associated with the Navier-Stokes Momentum Balance Equation and Finding the Evidence of Its Inapplicability to Govern Fluid Flow in the Infinite Space

Our modeling fluid flow, especially if the fluid is referred to as a gas, is established on mimicking each particle/molecule’s movement and then gathering that movement into macro quantities characterizing the fluid flow. It has resulted in discovering a new principle of the property (mass, momentum, and energy) balance in space. We named it the Ballistic Principle of the Property Balance in Space as described earlier in our publications. This paper uses a different scheme of defining a net rate of total property efflux than our original paper. Using this scheme, we formulated integro-differential forms of mass balance and momentum balance equations adapted to the incompressible fluid flow (gas flow with a mass-flow velocity less than 0.3 Ma) at the non-uniform temperature in the infinite gas space. We also investigated the analytical behavior of the integro-differential equations in the region bounding the point of singularity by applying the Taylor series expansion method to transform the integro-differential mass and momentum balance equations into the corresponding vector differential equations. Then we compared them with the Navier-Stokes equations of mass and momentum conservation for an incompressible fluid. We were surprised to find that the Navier-Stokes momentum balance equation does not describe the fluid flow adequately. Particularly, it does not consider the momentum associated with the part of velocity acquired by each gas particle during its free path traveling in the body force field. Also, the Navier-Stokes momentum balance equation is silent about the influence of the temperature non-uniformity on the momentum balance. Finally, we have demonstrated that the Navier-Stokes equations are not applicable to govern fluid flow on R3 × [0, ∞).

On Simplified Models for Dynamics of Pointlike Objects

Motivation: We study the asymptotic-type dynamics of various real pointlike objects that one models by a variety of differential equations. Their response to an external force one defines solely by the trajectory of a single point. Its velocity eventually stops changing after cessation of the external force. The response of their acceleration to the long-term external force is slow and possibly nonlinear. Objective: Our objective is to present technique for making simplified models for the long-term dynamics of pointlike objects whose motion interacts with the surroundings. In the asymptotic-type long-term dynamics, the time variable t ∈ (tm, +∞) and tm > 0 is large, say ! Method: We apply Taylor series expansion to differential equations to model the acceleration of pointlike object whose response to the long-term external force is not instantaneous and possibly nonlinear. Results: We make simplified models for the long-term dynamics of pointlike objects by Taylor polynomials in time derivatives of the external force. Application: We interpret the relativistic Lorentz-Abraham-Dirac equation as an equation for modeling the long-term dynamics, where t ≥ tm ≫ 0. This interpretation resolves the conceptual and usage controversy surrounding its troublesome application to determine the trajectory of a radiating charged particle, thus contributing to the development of more adequate modeling of physical phenomena.

An Alternative Approach to the Solution of Multi-Objective Geometric Programming Problems

The aim of this study is to present an alternative approach for solving the multi-objective posynomial geometric programming problems. The proposed approach minimizes the weighted objective function comes from multi-objective geometric programming problem subject to constraints which constructed by using Kuhn-Tucker Conditions. A new nonlinear problem formed by this approach is solved iteratively. The solution of this approach gives the Pareto optimal solution for the multi-objective posynomial geometric programming problem. To demonstrate the performance of this approach, a problem which was solved with a weighted mean method by Ojha and Biswal (2010) is used. The comparison of solutions between two methods shows that similar results are obtained. In this manner, the proposed approach can be used as an alternative of weighted mean method.

Power system transient stability simulation under uncertainty based on Taylor model arithmetic

The Taylor model arithmetic is introduced to deal with uncertainty.The uncertainty of model parameters is described by Taylor models and each variable in functions is replaced with the Taylor model(TM).Thus,time domain simulation under uncertainty is transformed to the integration of TM-based differential equations.In this paper,the Taylor series method is employed to compute differential equations;moreover,power system time domain simulation under uncertainty based on Taylor model method is presented.This method allows a rigorous estimation of the influence of either form of uncertainty and only needs one simulation.It is computationally fast compared with the Monte Carlo method,which is another technique for uncertainty analysis.The proposed method has been tested on the 39-bus New England system.The test results illustrate the effectiveness and practical value of the approach by comparing with the results of Monte Carlo simulation and traditional time domain simulation.

Fast Multipole BEM for 3-D Elastostatic Problems with Applications for Thin Structures

The fast multipole method (FMM) has been used to reduce the computing operations and mem- ory requirements in large numerical analysis problems. In this paper, the FMM based on Taylor expansions is combined with the boundary element method (BEM) for three-dimensional elastostatic problems to solve thin plate and shell structures. The fast multipole boundary element method (FM-BEM) requires O(N) opera- tions and memory for problems with N unknowns. The numerical results indicate that for the analysis of thin structures, the FM-BEM is much more efficient than the conventional BEM and the accuracy achieved is sufficient for engineering applications.

A Modified max-log-MAP Decoding Algorithm for Turbo Decoding

Turbo decoding is iterative decoding, and the MAP algorithm isoptimal in terms of performance in Turbo decoding. The log-MAPalgorithms is the MAP executed in the logarithmic domain, so it isalso optimal. Both the MAP and the log-MAP algorithm are complicatedfor implementation. The max-log MAP algorithm is de- Rived from thelog-MAP with approximation, which is simply compared with the log-MAPalgorithm but is subopti- Malin terms of performance. A modifiedmax-log-MAP algorithm is presented in this paper, based on the TaylorSeries of logarithm and exponent. Analysis and simulation resultsshow that modified max-log-MAP algorithm Outperforms the max-log-MAPalgorithm with almost the same complexity.