Taylor's Theorem Research Articles

A Simple Proof of Taylor's Theorem

A Simple Proof of Taylor's Theorem

Improving wavefront reconstruction accuracy by using integration equations with higher-order truncation errors in the Southwell geometry

Least-squares (LS)-based integration computes the function values by solving a set of integration equations (IEs) in LS sense, and is widely used in wavefront reconstruction and other fields where the measured data forms a slope. It is considered that the applications of IEs with smaller truncation errors (TEs) will improve the reconstruction accuracy. This paper proposes a general method based on the Taylor theorem to derive all kinds of IEs, and finds that an IE with a smaller TE has a higher-order TE. Three specific IEs with higher-order TEs in the Southwell geometry are deduced using this method, and three LS-based integration algorithms corresponding to these three IEs are formulated. A series of simulations demonstrate the validity of applying IEs with higher-order TEs in improving reconstruction accuracy. In addition, the IEs with higher-order TEs in the Hudgin and Fried geometries are also deduced using the proposed method, and the performances of these IEs in wavefront reconstruction are presented.

Exponential Stability of Stochastic Nonlinear Dynamical Price System with Delay

Based on Lyapunov stability theory, Itô formula, stochastic analysis, and matrix theory, we study the exponential stability of the stochastic nonlinear dynamical price system. Using Taylor's theorem, the stochastic nonlinear system with delay is reduced to ann-dimensional semilinear stochastic differential equation with delay. Some sufficient conditions of exponential stability and corollaries for such price system are established by virtue of Lyapunov function. The time delay upper limit is solved by using our theoretical results when the system is exponentially stable. Our theoretical results show that if the classical price Rayleigh equation is exponentially stable, so is its perturbed system with delay provided that both the time delay and the intensity of perturbations are small enough. Two examples are presented to illustrate our results.

J057026 プルーム断面可視化による蛇行乱流拡散に関する風洞実験(〔J057-02〕乱流における運動量、熱、物質の輸送現象(2))

The characteristics of turbulent particle diffusion were investigated in an atmospheric like turbulence field generated by an active-grid installed to in a small wind tunnel. Instantaneous visualization image analysis was conducted on the plume's meandering motion using a digital camera for a wide range of turbulent Reynolds number R_λ. To clarify the characteristics of the meandering plume's diffusion parameter, relative diffusion, meandering diffusion and total diffusion, were obtained by extracting the brightness values from a large number of instantaneous plume image pictures. Each diffusion parameters develops in the downstream with increase of R_λ, but their increasing tendency are vary greatly depending on dominated by either relative diffusion or meandering diffusion. The contribution rate of the relative and the meandering diffusion to the total diffusion was clarified, and these parameters are reasonable agreement with Lee & Stone's model which estimated Lagrangian properties obtained by applying the variance of total diffusion to the Taylor's theorem.

On the Calculation of Two-Dimensional Added Mass Coefficients by the Taylor Theorem and the Method of Fundamental Solutions

ABSTRACTThis work integrates the Taylor theorem and the method of fundamental solutions to develop a numerical tool for estimating the added mass coefficient tensor for a solid object of any convex shape moving in potential flow. In potential flow theory, the Taylor theorem calculates the added mass coefficient tensor for a Rankine body with algebraic manipulations of the properties of the internal singularities employed to generate the corresponding flow. To apply this theorem for objects in other shapes, the singularity strength and locations are required information which is facilitated numerically in this work by the method of fundamental solutions (MFS). The developed scheme is tested on a circle, an ellipse, a square, and a rhombus and the numerical results are in good agreement with the corresponding analytical values. A final example of a Cassini oval is also considered to show the potential applications on bio-engineering problems.

Derivation of an approximate analytical solution for understanding the response characteristics of the EBS

ABSTRACTTo perform a safety assessment for the geological disposal of radioactive waste, it is important to understand the response characteristics of the disposal system. In this study, approximate analytical solutions for steady-state nuclide releases from the engineered barrier system (EBS) of a repository were derived for an orthogonal one-dimensional diffusion model. In these approximate analytical solutions, inventory depletion, decay during migration and the influence of groundwater flow in the excavation damaged zone (EDZ) were considered. These solutions were simplified by the Taylor theorem in order to clearly represent the response characteristics of the EBS. The validity of these solutions was shown by comparison with numerical solutions. The response characteristics of the EBS are useful for identifying target values for important parameters that would have the effect of improving the robustness of system safety. The robustness of the geological disposal system and the reliability of the safety assessment can thus potentially be improved using the approximate analytical solutions.

On the leading OPE corrections to the ghost-gluon vertex and the Taylor theorem

This brief note is devoted to a study of genuine non-perturbative corrections to the Landau gauge ghost-gluon vertex in terms of the non-vanishing dimension-two gluon condensate. We pay special attention to the kinematical limit which the bare vertex takes for its tree-level expression at any perturbative order, according to the well-known Taylor theorem. Based on our OPE analysis, we also present a simple model for the vertex, in acceptable agreement with lattice data.

A finite volume method on distorted quadrilateral meshes for discretization of the energy equation's conduction term

AbstractA finite volume method (FVM) on distorted meshes for discretizing the energy equation's conduction term is presented. In this method, it is possible to compose the computational mesh of general quadrilateral elements (cells), namely, the cells are not required to be rectangular. The gradient of temperature on the cell's surface is computed to be second‐order accurate. Therefore, the error of numerical results by this method is smaller than using the traditional multilateral element method (MEM). The error does not depend on the degree of mesh distortion. The formulation based only on Taylor's theorem is straightforward. These are advantageous features to revise the fluid flow computation programs (based on FVM) that neglected the heat conduction term of the energy equation. The test calculations show that the convergence tendency of the numerical error using this method with the distorted mesh is the same as using an ordinary 2‐node central difference scheme on a constant‐interval rectangular mesh. By this method a conduction term was added to the energy equation of a SALE [1] program which had neglected that term originally, and z numerical calculation of a fluid flow with a heat transfer problem was performed. The numerical result of the present method with the distorted mesh well agrees with the analytical solution and the result of REM with a rectangular mesh. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20375

Battling about equations

One of the most exciting episodes in the study of human heredity was the battle between two incompatible algebraic expressions to describe its properties in the beginning of the 20th Century. In one corner was Karl Pearson (1857–1936) with his 1898 formulation for ancestral heredity, which had the authority of Charles Darwin and Francis Galton for support.1 It reads: The inherited contribution of parent to offspring (y) can be formulated as: ![Formula][1] where σ/σn are the ratios of the standard deviations of measurable traits in the offspring to the standard deviations of the mid-parental generation; and k1,k2 … kn are the deviations of the mid-parental means from the mean of the offspring for each trait. He admits to reservations of how ‘mid-parent’ should be defined. The subscripts refer to each of the ancestral generations (parents, grandparents, great-grandparents, etc.). The original paper should be consulted for fuller details.2 Pearson has elaborated on earlier formulations using a convergent geometric series to represent the hereditary contribution (f(y)) by the parents, the four grand parents, eight great-grand parents, etc. to the offspring.1 This type of convergent series is of the general form: ![Formula][2] where A,B,C,D … are constants independent of x (=0.5n in Pearson's expression). The expression can be used to represent a great many biological phenomena. The formulation has no theoretical significance; all it postulates is that the phenomenon in question varies continuously. Then Maclaurin's or Taylor's theorems (depending on the number of variables involved) can be used to determine the values of the coefficients that will make the series useful to any desired degree of approximation.3 Pearson's equation certainly satisfies the requirements for continuous variation of inherited traits which Darwin believed to be true; and it reaches a limit … [1]: /embed/graphic-1.gif [2]: /embed/graphic-2.gif

Omega-K Algorithm for Bistatic SAR With Arbitrary Geometry Configuration

Compared to the common monostatic synthetic aperture radar (SAR), the bistatic SAR data are more challenging to process because of their complicated flight geometry. Although several bistatic SAR image formation algorithms have been presented, most of them are only able to handle azimuth-invariant bistatic configuration where the transmitter and receiver platforms are moving in parallel tracks with identical velocities. This paper proposes an Omega-K algorithm to handle a general case of bistatic SAR data. The two-dimensional frequency spectrum expression derived by the method of series reversion is rewritten to a new form with explicit geometric significance. Based on this new frequency spectrum expression, the frequency mapping function can be derived by making use of the multivariable Taylor's theorem. After data mapping operation, the coupling between the range and azimuth can be eliminated, and the final focused image can be obtained using FFT technology. Numerical simulations are performed to validate the proposed algorithm.

The all-associativity of octonions and its applications

Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained.

HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LÉVY MARKET

This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.

The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method

The generalized Taylor theorem is the foundation of the homotopy analysis method proposed by Liao. This theorem is interesting but hard to understand from the mathematical point of view. Especially, there is a key parameter h whose meaning is still unknown. In the paper, we derive the generalized Taylor theorem from a usual way, that is, we prove that the generalized Taylor expansion is equivalent to a different representation of the usual Taylor expansion at different points. Therefore the meaning of the auxiliary parameter h is clarified. These results give a reasonable explanation of the parameter h and uncover the essence of the generalized Taylor theorem from which we can deeply understand the homotopy analysis method. Through the detailed analysis of some examples, we compare the series solution at the different points with the generalized Taylor series solution obtained by the homotopy analysis method.

Taylor's Theorem: The Elusive c is Not So Elusive

SummaryFor a suitably nice, real-valued function f defined on an open interval containing [a, b], f (b) can be expressed as pn(b) (the nth Taylor polynomial of f centered at a) plus an error term of the (Lagrange) form f(n+1)(c)/(n + 1)! for some c in (a, b). This article is for those who think that not much more can be said about where in (a, b) c is located. Supplementary materials, including further numerical illustrations and sketches of some proofs, are available at http://faculty.colostate-pueblo.edu/rick.kreminski.

Design of Barycentric Interpolators for Uniform and Nonuniform Sampling Grids

This paper proposes a method to convert a fixed instant interpolator for band-limited signals into a variable instant one, which has the form of a barycentric interpolator. This interpolator makes it possible to approximate the signal and its derivatives with minimal complexity in a range that surrounds the initial instant, and is applicable to uniform and nonuniform sampling grids. The barycentric form of the interpolator is derived using two different procedures, first by the repeated use of Bernstein's inequality and Taylor's theorem, and second by truncating a Lagrange-type series. These procedures show that the proposed method can be applied to existing fixed instant interpolators, and that it can be accurate in long time intervals. Finally, an evaluation procedure for barycentric interpolators and their derivatives is presented that minimizes the number of divisions, solving at the same time the numerical problems associated with small denominators. This paper includes several numerical examples.

On the Remainder in the Taylor Theorem

This formula is the main tool for bounding the remainder of the Taylor expansion in calculus classes, especially when this subject is taught before integration. One would like to have some natural proof for it. In [3] it is suggested that induction seems suitable, since P'n is the Taylor polynomial of f of order n ? 1, so R'n(x) is given by induction. This approach fails, since one cannot integrate R'n(x) because the point ? = t=(x) depends on x. While we were teaching a first calculus course for chemistry and physics majors at Tel-Aviv University, we observed that this obstacle can be removed if we change the problem to finding a bound on the remainder. This is just as useful, since a bound is all that is needed to show that the Taylor series converges to the function. From our personal experience, we believe that this approach enables students to grasp the mate rial more easily. Furthermore, Lagrange's formula can be deduced from the bound, as we show at the end of this note.

General Geometric Good Continuation: From Taylor to Laplace via Level Sets

Good continuation is the Gestalt observation that parts often group in particular ways to form coherent wholes. Perceptual integration of edges, for example, involves orientation good continuation, a property which has been exploited computationally very extensively. But more general local-global relationships, such as for shading or color, have been elusive. While Taylor's Theorem suggests certain modeling and smoothness criteria, the consideration of level set geometry indicates a different approach. Using such first principles we derive, for the first time, a generalization of good continuation to all those visual structures that can be abstracted as scalar functions over the image plane. Based on second order differential constraints that reflect good continuation, our analysis leads to a unique class of harmonic models and a cooperative algorithm for structure inference. Among the different applications of good continuation, here we apply these results to the denoising of shading and intensity distributions and demonstrate how our approach eliminates spurious measurements while preserving both singularities and regular structure, a property that facilitates higher level processes which depend so critically on both of these classes of visual structures.

Compressible convection in the deep atmospheres of giant planets

Fast rotating giant planets such as Jupiter and Saturn possess alternate prograde and retrograde zonal winds which are stable over long periods of time. We consider a compressible model of convection in a spherical shell with rapid rotation, using the anelastic approximation, to explore the parameter range for which such zonal flows can be produced. We consider models with a large variation in density across the layer. Our models are based only on the molecular H/He region above the metallic hydrogen transition at about 2 Mbar, and we do not include the hydromagnetic effects which may be important if the electrical conductivity is significant. We find that the convective velocities are significantly higher in the low density regions of the shell, but the zonal flow is almost independent of the z-coordinate parallel to the rotation axis. We analyse how this behaviour is consistent with the Proudman–Taylor theorem. We find that deep prograde zonal flow near the equator is a very robust feature of our models. Prograde and retrograde jets alternating in latitude can occur inside the tangent cylinder in compressible as well as Boussinesq models, particularly at lower Prandtl numbers. However, the zonal jets inside the tangent cylinder are suppressed if a no-slip condition is imposed at the inner boundary. This suggests that deep high latitude jets may be suppressed if there is significant magnetic dissipation. Our compressible calculations include the viscous dissipation in the entropy equation, and we find this is comparable to, and in some cases exceeds, the total heat flux emerging from the surface. For numerical reasons, these simulations cannot reach the extremely low Ekman number found in giant planets, and they necessarily also have a much larger heat flux than planets. We therefore discuss how our results might scale down to give solutions with lower dissipation and lower heat flux.

Floating-point division and square root using a Taylor-series expansion algorithm

Hardware support for floating-point (FP) arithmetic is a mandatory feature of modern microprocessor design. Although division and square root are relatively infrequent operations in traditional general-purpose applications, they are indispensable and becoming increasingly important in many modern applications. Therefore, overall performance can be greatly affected by the algorithms and the implementations used for designing FP-Div and FP-Sqrt units. In this paper, a single-precision fused floating-point multiply/divide/square root unit based on Taylor-series expansion algorithm is proposed. We extended an existing multiply/divide fused unit to incorporate the square root function with little area and latency overhead since Taylor's theorem enables us to compute approximations for many well-known functions with very similar forms. The implementation results of the proposed fused unit based on standard cell methodology in IBM 90nm technology exhibits that the incorporation of square root function to an existing multiply/divide unit requires only a modest 18% area increase and the same low latency for divide and square root operation can be achieved (12 cycles). The proposed arithmetic unit exhibits a reasonably good area-performance balance.

Optimal inventory policies for deteriorating complementary and substitute items

The optimal production for an inventory control system of deteriorating multi-items where items are either complementary and/or substitute is formulated with a resource constraint. Here, the production function is unknown and considered as a control variable. Also, the deterioration rates of the items are either stock dependent or constant. The demand is stock dependent, shortages are not allowed and deteriorated items are salvaged. The total profit, which consists of the sales proceeds, production cost, inventory holding cost, salvage value, is formulated as a Pontryagin's Optimal Control problem for both steady and transient states and evaluated using Taylor's theorem, generalised reduced gradient technique and optimal control theory satisfying the Generalised Legendre conditions. The model is formulated in general form for n-items, and in particular, is illustrated with three items for some numerical data. The optimum results are presented both in tabular form and graphically.