Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. V. Approximations in terms of higher transcendental functions

1981 ◽  
Vol 301 (1459) ◽  
pp. 137-162 ◽  
Keyword(s):  
Exceptional Point ◽  
Parabolic Cylinder ◽  
Maximum Relative Error ◽  
Mathieu Functions ◽  
Airy Functions ◽  
Half Strip ◽  
Base Functions ◽  
General Order ◽  
Transcendental Functions ◽  
Function Approximations

The methods of part III are further applied to the construction of approximations for the fundamental solution and base functions of part II in terms of higher transcendental functions. The domain of validity is now the complete half-strip {z; 0 ≤ Re z ≤ ½π, Im z ≥. 0} without exceptional point. Relative remainder estimates are again uniformly valid provided they are bounded. Specifically, approximations are obtained in terms of: ( a ) Airy functions, applicable if A ≠ ± 2h 2 ; ( b ) parabolic cylinder functions, applicable if |A ≥ 4h 2 , including A = ± 2h 2 ; ( c ) Bessel functions, applicable if |A| ≥ 4h 2 ; these formulae have maximum relative error A - 3/2 h 2 O (l) on the half-strip, even if h is arbitrarily small, provided only that A -1 is bounded. This is significantly better when A/h 2 is large than the corresponding estimate, A -½ 0(1), for the Airy function approximations. Certain more refined estimates for the auxiliary parameters introduced in part II are also obtained.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. IV. The Liouville-Green method applied to the Mathieu equation

1981 ◽  
Vol 301 (1459) ◽  
pp. 115-135 ◽  
Keyword(s):  
Full Range ◽  
Mathieu Equation ◽  
Real Variable ◽  
Mathieu Functions ◽  
Asymptotic Formulae ◽  
Half Strip ◽  
Base Functions ◽  
General Order ◽  
Circular Functions ◽  
Connection Formulae

The methods described in part III and the formulae derived in part II are applied to the construction of a comprehensive set of asymptotic formulae relating to the Mathieu equation y ′ ′ + ( λ + 2 h 2 cos ⁡ 2 z ) y = 0 with real parameters. These comprise formulae both ( a ) for the auxiliary parameters and ( b ), in terms of exponential and circular functions, for the fundamental solution, a function of a complex variable, and the various pairs of real-variable base-functions, all introduced in part II. With the aid of these, together with connection formulae also obtained in part II, approximations can readily be obtained for Mathieu functions of various types, including in particular periodic functions. Formulae for solutions are applicable on the half-strip { z : 0 ⩽ Re z ⩽ 1 2 π , Im z ⩾ 0 } , with the transition point of the differential equation which lies on its frontier removed, or in the case of real-variable solutions of the ordinary or modified equation, on the interval [0,½π] or [0, ∞] respectively, with the same qualification as for the half-strip when this is relevant. The formulae cover the full range of the parameters subject to A ≠ ± 2h 2 . The O -terms providing error estimates are uniformly valid on any subdomain of the independent variable and parameters on which they remain bounded.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. I. Introduction

1981 ◽  
Vol 301 (1459) ◽  
pp. 75-79 ◽  
Keyword(s):  
Mathieu Functions ◽  
Asymptotic Formulae ◽  
Mathieu’S Equation ◽  
Actual Error ◽  
Base Functions ◽  
General Order ◽  
Order Of Magnitude ◽  
Introductory Paper ◽  
Transcendental Functions ◽  
Connection Formulae

This is the first, introductory, paper of a series devoted to the derivation of a comprehensive set of approximate formulae for solutions of Mathieu’s equation with real parameters, in terms both of elementary and of higher transcendental functions. Order-of-magnitude error-estimates are obtained; these in every case reflect faithfully the behaviour of the actual error over the appropriate range of parameters and of independent variable. The general scope of the work is outlined in this Introduction, and is compared with that of previous work, in particular that of Langer (1934 b ). There then follows a description of the plan of the work and of the content of the several parts.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. III. The Liouville-Green method and its extensions

1981 ◽  
Vol 301 (1459) ◽  
pp. 99-114 ◽  
Keyword(s):  
Error Estimates ◽  
Error Control ◽  
Control Function ◽  
Mathieu Equation ◽  
Mathieu Functions ◽  
Green Method ◽  
Base Functions ◽  
General Order ◽  
Transcendental Functions ◽  
Connection Formulae

An account is given of the Liouville-Green method for the approximate solution, with error estimates, of linear second-order differential equations, together with certain extensions of the method. The purpose is to make readily available a range of techniques for use in the two final parts of the present series. The topics treated include: ( a ) the construction of approximations in terms of both elementary and higher transcendental functions, ( b ) the relations between approximations of the same solution in terms of different functions, ( c ) the identification of solutions and the estimation of connection coefficients, ( d ) uniform estimation of the error-control function in problems with more than one widely ranging parameter, ( e ) the construction of majorants for approximating functions, the last two being required for the derivation of satisfactory error estimates. There is little in this part that is new, though a method of constructing approximations in terms of Bessel functions is developed specifically for application to the Mathieu equation. Apart from this, some aspects of the presentation are thought to be novel.

Mathieu functions of general order: connection formulae, base functions and asymptotic formulae. II. Connection formulae and base functions

1981 ◽  
Vol 301 (1459) ◽  
pp. 81-97 ◽  
Keyword(s):  
Mathieu Equation ◽  
Real Variable ◽  
Stability Diagram ◽  
Mathieu Function ◽  
Mathieu Functions ◽  
Stability And Instability ◽  
Base Functions ◽  
General Order ◽  
Parameter Ranges ◽  
Connection Formulae

Connection formulae are examined which relate a solution y(z) of the Mathieu equation y" + (A + 2h 2 cos 2z) y = 0 with the solutions y ( ± z ± nπ) generated from it by the symmetry group of the equation. The treatment is exact, and is made first in the context of more general periodic differential equations; the results are then specialized to the Mathieu equation, a function of the third kind, characterized by its asymptotic behaviour as z → ∞i, being taken as fundamental. Two parameter ranges are then distinguished, corresponding to the regions of the stability diagram (a) where the solutions are always unstable and ( b ) where subregions of stability and instability alternate. Auxiliary parameters are defined in the two cases, and pairs of real-variable base-functions are constructed, appropriate to the ordinary Mathieu equation and to two types of modified equation. These pairs satisfy criteria introduced by Miller (1950). Comprehensive formulae are derived, relating these base-functions to standard types of Mathieu function, and special attention is given to periodic solutions.

A new method of solving Oseen’s equations and its application to the flow past an inclined elliptic cylinder

1954 ◽  
Vol 224 (1157) ◽  
pp. 141-160 ◽  
Keyword(s):  
Reynolds Number ◽  
Elliptic Cylinder ◽  
Cylindrical Body ◽  
Thickness Ratio ◽  
Successive Approximations ◽  
Angle Of Incidence ◽  
Mathieu Functions ◽  
Flow Past ◽  
Transcendental Functions ◽  
Lift And Drag

In this paper is developed a general method of solving Oseen’s linearized equations for a two-dimensional steady flow of a viscous fluid past an arbitrary cylindrical body. The method is based on the fact that the velocity in the neighbourhood of the cylinder can be generally expressed in terms of a pair of analytic functions, the determination of which from the appropriate boundary condition can be effected by successive approximations in powers of the Reynolds number, R . The method enables one to obtain the velocity distribution near the cylinder and the lift and drag acting on it in the form of power series in R , without recourse to manipulation of higher transcendental functions such as Bessel and Mathieu functions for circular and elliptic cylinders, respectively. As an example of the application of the method, the uniform flow past an elliptic cylinder at an arbitrary angle of incidence is considered. Analytical expressions for the lift and drag coefficients are obtained, which are correct to the order of R , the lowest order terms being O ( R -1 ) and numerical calculations are carried out for the thickness ratio t = 0, 0.1, 0.5, 1 and the Reynolds number R = 0.1, 1. It is found that drag increases slightly with increase of either thickness ratio or angle of incidence, and that lift decreases with increase of thickness ratio while, as a function of the angle of incidence, it has a maximum at about 45°.

Error-aware Design Procedure to Implement Energy-efficient Approximate Squaring Hardware

2020 ◽  
Vol 10 (4) ◽  
pp. 471-477
Author(s):  
Merin Loukrakpam ◽  
Ch. Lison Singh ◽  
Madhuchhanda Choudhury
Keyword(s):  
Energy Saving ◽  
Relative Error ◽  
Energy Efficient ◽  
Piecewise Linear ◽  
Arithmetic Operation ◽  
Design Procedure ◽  
Digital Signal ◽  
Maximum Relative Error ◽  
Square Function ◽  
Efficient Design

Background:: In recent years, there has been a high demand for executing digital signal processing and machine learning applications on energy-constrained devices. Squaring is a vital arithmetic operation used in such applications. Hence, improving the energy efficiency of squaring is crucial. Objective:: In this paper, a novel approximation method based on piecewise linear segmentation of the square function is proposed. Methods: Two-segment, four-segment and eight-segment accurate and energy-efficient 32-bit approximate designs for squaring were implemented using this method. The proposed 2-segment approximate squaring hardware showed 12.5% maximum relative error and delivered up to 55.6% energy saving when compared with state-of-the-art approximate multipliers used for squaring. Results: The proposed 4-segment hardware achieved a maximum relative error of 3.13% with up to 46.5% energy saving. Conclusion:: The proposed 8-segment design emerged as the most accurate squaring hardware with a maximum relative error of 0.78%. The comparison also revealed that the 8-segment design is the most efficient design in terms of error-area-delay-power product.

Metal-organic Nanopharmaceuticals

2020 ◽  
Vol 8 (3) ◽  
pp. 163-190
Author(s):  
Benjamin Steinborn ◽  
Ulrich Lächelt
Keyword(s):  
Metal Ions ◽  
Coordination Polymers ◽  
Active Agent ◽  
Lewis Base ◽  
High Loading ◽  
Active Components ◽  
Base Functions ◽  
Metal Organic ◽  
Theranostic Applications ◽  
Pharmacologically Active

: Coordinative interactions between multivalent metal ions and drug derivatives with Lewis base functions give rise to nanoscale coordination polymers (NCPs) as delivery systems. As the pharmacologically active agent constitutes a main building block of the nanomaterial, the resulting drug loadings are typically very high. By additionally selecting metal ions with favorable pharmacological or physicochemical properties, the obtained NCPs are predominantly composed of active components which serve individual purposes, such as pharmacotherapy, photosensitization, multimodal imaging, chemodynamic therapy or radiosensitization. By this approach, the assembly of drug molecules into NCPs modulates pharmacokinetics, combines pharmacological drug action with specific characteristics of metal components and provides a strategy to generate tailorable multifunctional nanoparticles. This article reviews different applications and recent examples of such highly functional nanopharmaceuticals with a high ‘material economy’. : Lay Summary: Nanoparticles, that are small enough to circulate in the bloodstream and can carry cargo molecules, such as drugs, imaging or contrast agents, are attractive materials for pharmaceutical applications. A high loading capacity is a generally aspired parameter of nanopharmaceuticals to minimize patient exposure to unnecessary nanomaterial. Pharmaceutical agents containing Lewis base functions in their molecular structure can directly be assembled into metal-organic nanopharmaceuticals by coordinative interaction with metal ions. Such coordination polymers generally feature extraordinarily high loading capacities and the flexibility to encapsulate different agents for a simultaneous delivery in combination therapy or ‘theranostic’ applications.

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