Recursion-transform method and potential formulae of the m×n cobweb and fan networks

In this paper, we made a new breakthrough, which proposes a new recursion–transform（RT） method with potential parameters to evaluate the nodal potential in arbitrary resistor networks. For the first time, we found the exact potential formulae of arbitrary m × n cobweb and fan networks by the RT method, and the potential formulae of infinite and semi-infinite networks are derived. As applications, a series of interesting corollaries of potential formulae are given by using the general formula, the equivalent resistance formula is deduced by using the potential formula, and we find a new trigonometric identity by comparing two equivalence results with different forms.

Microwave absorption properties regulation and bandwidth formula of oriented Y_(2)Fe_(17)N_(3-δ)@SiO_(2)/PU composite synthesized by reduction-diffusion method

As concepts closely related to microwave absorption properties,impedance matching and phase matching were rarely combined with material parameters to regulate properties and explore related mechanisms.In this work,reduction–diffusion method was innovatively applied to synthesize rare earth alloy Y_(2)Fe_(17).In order to regulate the electromagnetic parameters of absorbers,the Y_(2)Fe_(17)N_(3-δ)particles were coated with silica(Y_(2)Fe_(17)N_(3-δ)@SiO_(2))and absorbers with different volume fractions were prepared.The relationship between impedance matching,matching thickness,and the strongest reflection loss peak(RLmin)was presented obviously.Compared to the microwave absorption properties of Y_(2)Fe_(17)N_(3-δ)/PU absorber,Y_(2)Fe_(17)N_(3-δ)@SiO_(2)/PU absorbers are more conducive to the realization of microwave absorption material standards which are thin thickness,light weight,strong absorbing intensity,and broad bandwidth.Based on microwave frequency bands,the microwave absorption properties of the absorbers were analyzed and the related parameters were listed.As an important parameter related to perfect matching,reflection factor(√ε_(r)/μ_(r))was discussed combined with microwave amplitude attenuation.According to the origin and mathematical model of bandwidth,the formula of EAB(RL<-10 dB)was derived and simplified.The calculated bandwidths agreed well with experimental results.

A GENERALIZED GAUSS-TYPE QUADRATURE FORMULA AND ITS APPLICATIONS TO PSEUDOSPECTRAL METHOD

A generalized Gauss-type quadrature formula is introduced, which assists in selection of collocation points in pseudospectral method for differential equations with two-point derivative boundary conditions. Some results on the related Jacobi interpolation are established. A pseudospectral scheme is proposed for the Kuramoto-Sivashisky equation. A skew symmetric decomposition is used for dealing with the nonlinear convection term. The stability and convergence of the proposed scheme are proved. The error estimates are obtained. Numerical results show the efficiency of this approach.

On Relations between the General Recurrence Formula of the Extension of Murase-Newton’s Method (the Extension of Tsuchikura*-Horiguchi’s Method) and Horner’s Method

In 1673, Yoshimasu Murase made a cubic equation to obtain the thickness of a hearth. He introduced two kinds of recurrence formulas of square and the deformation (Ref.[1]). We find that the three formulas lead to the extension of Newton-Raphson’s method and Horner’s method at the same time. This shows originality of Japanese native mathematics (Wasan) in the Edo era (1600- 1867). Suzuki (Ref.[2]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 1 introduces Murase’s three solutions of the cubic equation of the hearth. Section 2 explains the Horner’s method. We give the generalization of three formulas and the relation between these formulas and Horner’s method. Section 3 gives definitions of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), general recurrence formula of Murase-Newton’s method (Tsuchikura-Horiguchi’s method), and general recurrence formula of the extension of Murase-Newton’s method (the extension of Tsuchikura-Horiguchi’s method) concerning n-degree polynomial equation. Section 4 is contents of the title of this paper.

A Nonstationary Halley’s Iteration Method by Using Divided Differences Formula

This paper presents a new nonstationary iterative method for solving non linear algebraic equations that does not require the use of any derivative. The study uses only the Newton’s divided differences of first and second orders instead of the derivatives of (1).

Application of Series Formula Method in Optimization Arrangement of Webs

The expressions of the internal forces of the webs under the vertical loads of the simply supported beam type trusses with vertical and horizontal webs and without vertical webs are studied through the mathematical formula method. The variations of internal forces under different angles and spacings of the webs are simulated. The law of the optimal arrangement of the webs of the parallel-string simple-beam truss is obtained: under the condition that the rigidity of the rod is allowed, the form of no vertical web-type truss and reducing the span distance and inclination of the side span are advisable, which can save materials and reduce the weight as well. This method can be applied to the calculation of internal forces for arbitrary loads and truss forms.

The Convergences Comparison between the Halley’s Method and Its Extended One Based on Formulas Derivation and Numerical Calculations

The purpose of this paper is that we give an extension of Halley’s method (Section 2), and the formulas to compare the convergences of the Halley’s method and extended one (Section 3). For extension of Halley’s method we give definition of function by variable transformation in Section 1. In Section 4 we do the numerical calculations of Halley’s method and extended one for elementary functions, compare these convergences, and confirm the theory. Under certain conditions we can confirm that the extended Halley’s method has better convergence or better approximation than Halley’s method.

Study on the accuracy of glomerular filtration rate formulas by double plasma method as "gold standard"

Objective: To evaluate the accuracy of glomerular filtration rate formula by comparising the CKD-EPI 2009 formula and the modified Modified MDRD formula with the 99mTc-DTPA double-phase plasma method as "gold standard" respectively. Methods: Totally 166 patients diognosed as chronic kidney disease (CKD) were enrolled. The 99mTc-DTPA double-plasma method (rGFR) was used as the "gold standard". The CKD-EPI 2009 formula and the modified MDRD formula were used to calculate eGFR. Statistical software was used to analyze the correlation between the calculated values of the two formulas and the gold standard value and the bias. Then we evaluated the accuracy of the two GFR formulas. Results: Among the CKD stage 1 patients, the bias of the CKD-EPI 2009 formula was smaller than that of the Modified MDRD formula (13.9911.45;20.1815.90);both formulas were weakly correlated with the gold standard (correlation coefficients were 0.216, 0.229, P<0.01, respectively);The probability that the bias of the calculated value of the CKD-EPI 2009 formula less than 15%, 30%, and 50% of the gold standard value is smaller. Among the CKD stage 2 patients, the bias of the CKD-EPI 2009 formula was smaller than that of the Modified MDRD formula (12.748.45;15.6811.01);both formulas were moderately correlated with the gold standard (correlation coefficients were 0.568, 0.581, P<0.01, respectively);The probability that the bias of the calculated value of the CKD-EPI 2009 formula less than 15%, 30%, and 50% of the gold standard value is smaller. Among the CKD stage 3 patients, the bias of the CKD-EPI 2009 formula was smaller than that of the Modified MDRD formula (12.6410.27;12.8810.97), and both formulas were strongly correlated with the gold standard (correlation coefficients were 0.664, 0.670, P<0.01, respectively);The probability that the bias of the calculated value of the Modified MDRD formula less than 15%, 30%, and 50% of the gold standard value is smaller. Among the CKD stage 4 to 5 patients, the bias of the CKD-EPI 2009 formula was smaller than that of the Modified MDRD formula (5.585.36;5.945.20);The CKD-EPI 2009 formula and the Modified MDRD formula were strongly correlated with the gold standard (correlation coefficient r was 0.808. 0.802, P<0.01, respectively);The probability of the bias of the calculated value of the CKD-EPI 2009 formula less than 15%, 30%, and 50% of the gold standard value is smaller. In patients with decreased renal function with GFR <60 ml/min, the sensitivity and positive predictive value of the CKD-EPI 2009 formula for the diagnosis of "decreased renal function"were higher, and the specificity was comparable. Conclusion: 1. When the renal function is only slightly decreased, the accuracy of the two formulas is not good. In this condition, the CKD-EPI 2009 formula is more accurate and recommended. 2. It is necessary to further improve the current formulas especialy when it comes to value the slightly declined renal function;3. When we try to identify the stage of CKD patients, only based on eGFR may cause misclassification, it is recommended to combine the cause-GFR-albuminuria staging to assess the stage of CKD;4. The current formulas have limitations.in the case that requires a highly accurate assessment of GFR, the 99mTc-DTPA dual plasma method is recommended.

The Formulas to Compare the Convergences of Newton’s Method and the Extended Newton’s Method (Tsuchikura-Horiguchi Method) and the Numerical Calculations

This paper gives the extension of Newton’s method, and a variety of formulas to compare the convergences for the extension of Newton’s method (Section 4). Section 5 gives the numerical calculations. Section 1 introduces the three formulas obtained from the cubic equation of a hearth by Murase (Ref. [1]). We find that Murase’s three formulas lead to a Horner’s method (Ref. [2]) and extension of a Newton’s method (2009) at the same time. This shows originality of Wasan (mathematics developed in Japan) in the Edo era (1603-1868). Suzuki (Ref. [3]) estimates Murase to be a rare mathematician in not only the history of Wasan but also the history of mathematics in the world. Section 2 gives the relations between Newton’s method, Horner’s method and Murase’s three formulas. Section 3 gives a new function defined such as .

A LIMITED MEMORY QUASI-NEWTON METHOD FOR LARGE SCALE PROBLEM

We study how to use the SR1 update to realize minimization methods for problems where the storage is critical. We give an update formula which generates matrices using information from the last m iterations. The numerical tests show that the method is efficent.

An explicit method for numerical simulation of wave equations

In this paper, a method to develop a hierarchy of explicit recursion formulas for numerical simulation in an irregular grid for scalar wave equations is presented and its accuracy is illustrated via 2-D and 1-D models. Approaches to develop the stable formulas which are of 2M-order accuracy in both time and space with Mbeing a positive integer for regular grids are discussed and illustrated by constructing the second order （M= 1） and the fourth order （M = 2） recursion formulas.

Precise integration methods based on the Chebyshev polynomial of the first kind

This paper introduces two new types of precise integration methods based on Chebyshev polynomial of the first kind for dynamic response analysis of structures, namely the integral formula method （IFM） and the homogenized initial system method （HISM）. In both methods, nonlinear variable loadings within time intervals are simulated using Chebyshev polynomials of the first kind before a direct integration is performed. Developed on the basis of the integral formula, the recurrence relationship of the integral computation suggested in this paper is combined with the Crout decomposed method to solve linear algebraic equations. In this way, the IFM based on Chebyshev polynomial of the first kind is constructed. Transforming the non-homogenous initial system to the homogeneous dynamic system, and developing a special scheme without dimensional expansion, the HISM based on Chebyshev polynomial of the first kind is able to avoid the matrix inversion operation. The accuracy of the time integration schemes is examined and compared with other commonly used schemes, and it is shown that a greater accuracy as well as less time consuming can be achieved. Two numerical examples are presented to demonstrate the applicability of these new methods.

ON RITZ METHOD OF ANINTEGRO-DIFFERENTIAL EQUAITON

This paper deals with the Ritz method of an integro-differential equation related with Riemann zeta-function.

An explicit method for numerical simulation of wave equations: 3D wave motion

In this paper, an explicit method is generalized from 1D and 2D models to a 3D model for numerical simulation of wave motion, and the corresponding recursion formulas are developed for 3D irregular grids. For uniform cubic grids, the approach used to establish stable formulas with 2M-order accuracy is discussed in detail, with M being a positive integer, and is illustrated by establishing second order （M=1） recursion formulas. The theoretical results presented in this paper are demonstrated through numerical testing.

Cake Filtration Equation Using tanⁿ<i>θ</i>Reduction Method

This study was completed by an extensive mathematical analysis. New equation to sludge filtration processes has been proposed for use in routine laboratory. The equation has been suggested to replace Ademiluyi’s cake filtration equation in view of the limitations of the latter. The new equation can be used for sludges whose compressibility factor is more than one but Ademiluyi’s cake filtration equation can only be used for sludges whose compressibility coefficient is less than one. The new sludge filtration equation was derived using tannθ reduction method. The generalized equation thus obtained resembles Ademiluyi’s equation in the mode of parameter combination except the presence of summation notation in the new equation.

Multistep Quadrature Based Methods for Nonlinear System of Equations with Singular Jacobian

Methods for the approximation of solution of nonlinear system of equations often fail when the Jacobians of the systems are singular at iteration points. In this paper, multi-step families of quadrature based iterative methods for approximating the solution of nonlinear system of equations with singular Jacobian are developed using decomposition technique. The methods proposed in this study are of convergence order , and require only the evaluation of first-order Frechet derivative per iteration. The approximate solutions generated by the proposed iterative methods in this paper compared with some existing contemporary methods in literature, show that methods developed herein are efficient and adequate in approximating the solution of nonlinear system of equations whose Jacobians are singular and non-singular at iteration points.

Superposition formulas of multi-solution to a reduced(3+1)-dimensional nonlinear evolution equation

We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.

Finite Difference Methods for the Time Fractional Advection-diffusion Equation

In this paper, three implicit finite difference methods are developed to solve one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying right shifted Grünwald-Letnikov formula of order α ∈(0, 1). We investigate the stability analysis by using von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.

General Convergence Analysis for Three-step Projection Methods and Applications to Variational Problems

First a general model for a three-step projection method is introduced, and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let H be a real Hilbert space and K be a nonempty closed convex subset of H. For arbitrarily chosen initial points x0, y0, z0 ∈ K, compute sequences xn, yn, zn such thatT ： K→ H is a nonlinear mapping onto K. At last three-step models are applied to some variational inequality problems.

Gauss-Legendre Iterative Methods and Their Applications on Nonlinear Systems and BVP-ODEs

In this paper, a group of Gauss-Legendre iterative methods with cubic convergence for solving nonlinear systems are proposed. We construct the iterative schemes based on Gauss-Legendre quadrature formula. The cubic convergence and error equation are proved theoretically, and demonstrated numerically. Several numerical examples for solving the system of nonlinear equations and boundary-value problems of nonlinear ordinary differential equations (ODEs) are provided to illustrate the efficiency and performance of the suggested iterative methods.