A note on the convergence of multi-point Taylor's series

1962 ◽  
Vol 58 (3) ◽  
pp. 548-550 ◽  
Author(s):  
M. Fine ◽  
J. W. Head
Keyword(s):  
Real Function ◽  
Complex Variables ◽  
Complex Numbers ◽  
Second Derivative ◽  
First Derivative ◽  
Base Points ◽  
Obvious Generalization ◽  
Image Position ◽  
Taylor’S Series

Series which behave rather like Taylor's series at more than, one point are described and their convergence is discussed briefly.Letwhere z, zs are complex numbers and zs are the points, (the base points)about which a function is to be expanded. They need not be distinct. p{z) is a polynomial of degree n. in z. Assume first that z, zs are real and the zs are distinct. Then polynomials pn-1,t(z) may be chosen so that the serieshas contact of order N at the points zs with any specified real function which has N derivatives at these points. This is proved merely by rearranging the simplest polynomial in z which has contact of order N with/(z) at z = zs. The term given by t = 0 in (2) allows the series to have the values f(zs) at z = zs; adding the term for t = 1 does not alter the values of the series at z = zs; it allows the series to have a derivative with values f′(zs) at z = z; similarly adding the term for t = 2 does not alter the value of the series or its first derivative at z = zs, it allows the series to have a second derivative with values f″{zs) at z = zs; and so on. If m of the zs coincide then f(z) has mN + m− 1 derivatives there and contact is of order mN + m− 1. If z, zs are complex the result is seen to be valid subject to the obvious generalization of ‘ contact of the ith order’ to complex variables and functions.

Velocity and Acceleration Analysis of Plane and Space Mechanisms by Means of Independent-Position Equations

1958 ◽  
Vol 25 (1) ◽  
pp. 1-6
Author(s):  
Francis H. Raven
Keyword(s):  
Angular Velocity ◽  
Analytical Method ◽  
Three Dimensions ◽  
Complex Numbers ◽  
Second Derivative ◽  
Exponential Form ◽  
First Derivative ◽  
Acceleration Analysis ◽  
Space Mechanisms

Abstract This paper presents a generalized analytical method for the determination of velocities and accelerations in mechanisms having motion in either two or three dimensions. In this method of analysis, one first obtains independent-position equations for a particular point or for a particular link. The first derivative with respect to time of these position equations yields equations for obtaining the velocity of the point or the angular velocity of the link of interest; and the second derivative with respect to time furnishes new equations from which the desired accelerations may be found. Position vectors are expressed as complex numbers in exponential form because of the ease with which they may be differentiated, and also because numerical results are readily obtainable from the real and imaginary parts of the complex numbers.

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

A note on the strong regularity of Nrlund means

1983 ◽  
Vol 94 (2) ◽  
pp. 261-263
Author(s):  
J. R. Nurcombe
Keyword(s):  
Strong Convergence ◽  
Complex Numbers ◽  
Strong Regularity ◽  
Image Position

Let (pn), (qn) and (un) be sequences of real or complex numbers withThe sequence (sn) is strongly generalized Nrlund summable with index 0, to s, or s or snsN, p, Q ifand pnv=pnvpnv1, with p10. Strong Nrlund summability N, p was first studied by Borweing and Cass (1), and its generalization N, p, Q by Thorp (6). We shall say that (sn) is strongly generalized convergent of index 0, to s, and write snsC, 0, Q if sns and where sn=a0+a1++an. When qn all n, this definition reduces to strong convergence of index , introduced by Hyslop (4). If as n, the sequence (sn) is summable (, q) to s sns(, q).

On the Properties of an Entire Function of Two Complex Variables

1968 ◽  
Vol 20 ◽  
pp. 51-57
Author(s):  
Arun Kumar Agarwal
Keyword(s):  
Entire Function ◽  
Double Series ◽  
Complex Variables ◽  
Maximum Term ◽  
Image Position

1. Letbe an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . LetFollowing S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

Rapidly convergent Steffensen-based methods for unconstrained optimization

2019 ◽  
Vol 53 (2) ◽  
pp. 657-666
Author(s):  
Mohammad Afzalinejad
Keyword(s):  
Unconstrained Optimization ◽  
Newton's Method ◽  
Optimization Problems ◽  
Quadratic Model ◽  
Second Derivative ◽  
Rapid Convergence ◽  
First Derivative ◽  
Unconstrained Optimization Problems ◽  
Derivative Free ◽  
Steffensen Method

A problem with rapidly convergent methods for unconstrained optimization like the Newton’s method is the computational difficulties arising specially from the second derivative. In this paper, a class of methods for solving unconstrained optimization problems is proposed which implicitly applies approximations to derivatives. This class of methods is based on a modified Steffensen method for finding roots of a function and attempts to make a quadratic model for the function without using the second derivative. Two methods of this kind with non-expensive computations are proposed which just use first derivative of the function. Derivative-free versions of these methods are also suggested for the cases where the gradient formulas are not available or difficult to evaluate. The theory as well as numerical examinations confirm the rapid convergence of this class of methods.

scholarly journals Analysis of the Acid Detergent Fibre Content in Turnip Greens and Turnip Tops (Brassica rapa L. Subsp. rapa) by Means of Near-Infrared Reflectance

Foods ◽  
2019 ◽  
Vol 8 (9) ◽  
pp. 364 ◽  
Author(s):  
Sara Obregón-Cano ◽  
Rafael Moreno-Rojas ◽  
Ana María Jurado-Millán ◽  
María Elena Cartea-González ◽  
Antonio De Haro-Bailón
Keyword(s):  
Standard Error ◽  
Near Infrared ◽  
Mathematical Treatment ◽  
Second Derivative ◽  
Infrared Reflectance ◽  
Near Infrared Reflectance ◽  
First Derivative ◽  
Acid Detergent Fibre ◽  
Turnip Greens ◽  
Turnip Tops

Standard wet chemistry analytical techniques currently used to determine plant fibre constituents are costly, time-consuming and destructive. In this paper the potential of near-infrared reflectance spectroscopy (NIRS) to analyse the contents of acid detergent fibre (ADF) in turnip greens and turnip tops has been assessed. Three calibration equations were developed: in the equation without mathematical treatment the coefficient of determination (R2) was 0.91, in the first-derivative treatment equation R2 = 0.95 and in the second-derivative treatment R2 = 0.96. The estimation accuracy was based on RPD (the ratio between the standard deviation and the standard error of validation) and RER (the ratio between the range of ADF of the validation as a whole and the standard error of prediction) of the external validation. RPD and RER values were of 2.75 and 9.00 for the treatment without derivative, 3.41 and 11.79 with first-derivative, and 3.10 and 11.03 with second-derivative. With the acid detergent residue spectrum the wavelengths were identified and associated with the ADF contained in the sample. The results showed a great potential of NIRS for predicting ADF content in turnip greens and turnip tops.

scholarly journals A note on Whittaker's cardinal series in harmonic analysis

1986 ◽  
Vol 29 (3) ◽  
pp. 349-357 ◽  
Author(s):  
M. M. Dodson ◽  
A. M. Silva ◽  
V. Soucek
Keyword(s):  
Control Theory ◽  
Data Processing ◽  
Harmonic Analysis ◽  
Real Function ◽  
Sampling Theorem ◽  
Communication Engineering ◽  
Shannon Sampling Theorem ◽  
Band Limited ◽  
Image Position ◽  
Sampling Set

The sampling theorem, often referred to as the Shannon or Whittaker-Kotel'nikov- Shannon sampling theorem, is of considerable importance in many fields, including communication engineering, electronics, control theory and data processing, and has appeared frequently in various forms in engineering literature (a comprehensive account of its numerous extensions and applications is given in [3]). The result states that a band-limited signal, i.e. a real function f of the formwhere w>0, is under reasonable conditions on the even function F, determined by its values on the sampling set (l/2w)ℤ and can be reconstructed from the samples f(k/2w), k∈ℤ, by the series

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