scholarly journals Local uniqueness in boundary problems

1970 ◽  
Vol 17 (1) ◽  
pp. 23-36
Author(s):  
M. H. Martin
Keyword(s):  
Incompressible Fluid ◽  
Unit Circle ◽  
Finite Amplitude ◽  
Local Uniqueness ◽  
Boundary Problems ◽  
Infinite Depth ◽  
Image Position

The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a functionregular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition

The Determination of an Upper Bound of the Lowest Natural Frequency of a Star Shaped Membrane

1964 ◽  
Vol 68 (646) ◽  
pp. 703-703 ◽  
Author(s):  
Patricio A. Laura
Keyword(s):  
Unit Circle ◽  
Natural Frequency ◽  
Upper Bound ◽  
Mapping Function ◽  
Closed Curve ◽  
Axis Of Symmetry ◽  
Image Position

Let Г be a closed curve in the z-plane with p-axis of symmetry. The mapping functionmaps in general the interior of the unit circle |ξ| < 1 on to the interior of Г such that ξ=0 is transformed into z=0.

scholarly journals Some Aspects of Uniqueness for Solutions to Boundary Problems

1962 ◽  
Vol 13 (1) ◽  
pp. 25-35 ◽  
Author(s):  
M. H. Martin
Keyword(s):  
Positive Integer ◽  
Boundary Problem ◽  
Unit Circle ◽  
Polar Coordinates ◽  
Arc Length ◽  
Boundary Problems ◽  
Image Position

The solution to the boundary problemwhere r is the distance of point (x, y) from the origin, and h is a given function the arc length s along the unit circle r = 1, is not necessarily unique, Boggio (1), Weinstein (2), Stoker (3), Martin (4). Indeed if h is a positive integer m is known that the only solutions regular analytic for r≦1 arewhere r, θ denote polar coordinates and A, B are arbitrary constants.

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

scholarly journals Determination of mechanical properties by nanoindentation independently of indentation depth measurement

2012 ◽  
Vol 27 (19) ◽  
pp. 2551-2560 ◽  
Author(s):  
Gaylord Guillonneau ◽  
Guillaume Kermouche ◽  
Sandrine Bec ◽  
Jean-Luc Loubet
Keyword(s):  
Mechanical Properties ◽  
Indentation Depth ◽  
Depth Measurement ◽  
Image Position

Abstract

scholarly journals Extremal Problems for Functions Starlike in the Exterior of the Unit Circle

1962 ◽  
Vol 14 ◽  
pp. 540-551 ◽  
Author(s):  
W. C. Royster
Keyword(s):  
Unit Circle ◽  
Analytic Functions ◽  
Upper Bounds ◽  
Extremal Problems ◽  
Simple Pole ◽  
Image Position ◽  
Inverse Functions ◽  
Variational Formulas

Let Σ represent the class of analytic functions(1)which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion(2).In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

scholarly journals The determination of Fibonacci groups

1974 ◽  
Vol 11 (1) ◽  
pp. 11-14 ◽  
Author(s):  
A.M. Brunner
Keyword(s):  
Natural Number ◽  
Image Position

Fibonacci groups are the groupswhere r is a natural number. The groups F(2, 8) and F(2, 10) are shown to he infinite, thus leaving F(2, 9) as the only Fibonacci group whose finiteness or infiniteness has not been determined.

Finite-amplitude convection in the presence of an unsteady shear flow

1995 ◽  
Vol 287 ◽  
pp. 225-249 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Shear Flow ◽  
Critical Size ◽  
Low Frequency ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Present Calculation ◽  
Wide Range ◽  
Prandtl Numbers ◽  
Boussinesq Fluid

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.

Stability of circular Couette flow with variable inner cylinder speed

1982 ◽  
Vol 123 ◽  
pp. 43-57 ◽  
Author(s):  
G. P. Neitzel
Keyword(s):  
Angular Velocity ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Outer Cylinder ◽  
Finite Amplitude ◽  
Reverse Direction ◽  
Stability Bound ◽  
Concentric Cylinders ◽  
Theory Result ◽  
Stability Enhancement

Energy & ability theory is employed to study the finite-amplitude stability of a viscous incompressible fluid occupying the space between a pair of concentric cylinders when the inner-cylinder angular velocity varies linearly with time. For the case with a fixed outer cylinder and increasing inner-cylinder speed, we find an enhancement of stability, consistent with a linear-theory result due to Eagles. When the inner-cylinder speed decreases, we find an initially decreased stability bound, indicating the possibility of hysteresis, while, if the inner cylinder is allowed to reverse direction and linearly increase in speed, we find significant stability enhancement.

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