The total curvature of a smooth closed curve in E n is an upper bound to the total swing of its position vector

Mathematika ◽  
1993 ◽  
Vol 40 (1) ◽  
pp. 148-151
Author(s):  
F. H. J. Cornish
Keyword(s):  
Upper Bound ◽  
Total Curvature ◽  
Closed Curve ◽  
Position Vector

scholarly journals On the total curvature of a closed curve

1959 ◽  
Vol 29 (0) ◽  
pp. 118-125 ◽  
Author(s):  
Shigeo SASAKI
Keyword(s):  
Total Curvature ◽  
Closed Curve

ON THE GAUSSIAN CURVATURE OF MAXIMAL SURFACES AND THE CALABI–BERNSTEIN THEOREM

2001 ◽  
Vol 33 (4) ◽  
pp. 454-458 ◽  
Author(s):  
LUIS J. ALÍAS ◽  
BENNETT PALMER
Keyword(s):  
Minkowski Space ◽  
Upper Bound ◽  
Gaussian Curvature ◽  
Total Curvature ◽  
Local Geometry ◽  
Maximal Surface ◽  
Bernstein Theorem ◽  
New Approach ◽  
Maximal Surfaces

In this paper, a new approach to the Calabi–Bernstein theorem on maximal surfaces in the Lorentz– Minkowski space L3 is introduced. The approach is based on an upper bound for the total curvature of geodesic discs in a maximal surface in L3, involving the local geometry of the surface and its hyperbolic image. As an application of this, a new proof of the Calabi–Bernstein theorem is provided.

scholarly journals On the total curvature of a closed curve

1969 ◽  
Vol 21 (2) ◽  
pp. 239-244
Author(s):  
M. Rochowski
Keyword(s):  
Total Curvature ◽  
Closed Curve

scholarly journals REILLY INEQUALITIES OF ELLIPTIC OPERATORS ON CLOSED SUBMANIFOLDS

2009 ◽  
Vol 80 (2) ◽  
pp. 335-346
Author(s):  
RUSHAN WANG
Keyword(s):  
Elliptic Operator ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Upper Bound ◽  
Vector Fields ◽  
Elliptic Operators ◽  
Position Vector ◽  
Space Forms ◽  
Nonzero Eigenvalue ◽  
Closed Hypersurfaces

AbstractUsing generalized position vector fields we obtain new upper bound estimates of the first nonzero eigenvalue of a kind of elliptic operator on closed submanifolds isometrically immersed in Riemannian manifolds of bounded sectional curvature. Applying these Reilly inequalities we improve a series of recent upper bound estimates of the first nonzero eigenvalue of the Lr operator on closed hypersurfaces in space forms.

scholarly journals MAPS OF SURFACE GROUPS TO FINITE GROUPS WITH NO SIMPLE LOOPS IN THE KERNEL

2000 ◽  
Vol 09 (08) ◽  
pp. 1029-1036 ◽  
Author(s):  
CHARLES LIVINGSTON
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Simple Closed Curve ◽  
Orientable Surface ◽  
Closed Curve ◽  
Surface Groups ◽  
Closed Orientable Surface

Let Fg denote the closed orientable surface of genus g. What is the least order finite group, Gg, for which there is a homomorphism ψ:π1(Fg)→Gg so that nontrivial simple closed curve on Fg represents an element in Ker (ψ)? For the torus it is easily seen that G1=Z2×Z2 suffices. We prove here that G2 is a group of order 32 and that an upper bound for the order of Gg is given by g2g+1. The previously known upper bound was greater than 2g22g.

SPACE CURVES WITH NONZERO TORSION

1990 ◽  
Vol 01 (01) ◽  
pp. 109-117 ◽  
Author(s):  
BURT TOTARO
Keyword(s):  
Total Curvature ◽  
Closed Curve ◽  
Space Curves ◽  
Curvature And Torsion ◽  
Total Torsion ◽  
Nonzero Torsion

For a closed curve in R3 with curvature and torsion everywhere nonzero, the sum of the total curvature and the total torsion is greater than 4π.

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.

The role of differential phase contrast in electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 176-177
Author(s):  
E.M. Waddell ◽  
J.N. Chapman ◽  
R.P. Ferrier
Keyword(s):  
Phase Contrast ◽  
Practical Method ◽  
Position Vector ◽  
Differential Phase ◽  
Pure Phase ◽  
Differential Phase Contrast ◽  
The Difference ◽  
Image Position ◽  
First Moment

Dekkers and de Lang (1977) have discussed a practical method of realising differential phase contrast in a STEM. The method involves taking the difference signal from two semi-circular detectors placed symmetrically about the optic axis and subtending the same angle (2α) at the specimen as that of the cone of illumination. Such a system, or an obvious generalisation of it, namely a quadrant detector, has the characteristic of responding to the gradient of the phase of the specimen transmittance. In this paper we shall compare the performance of this type of system with that of a first moment detector (Waddell et al.1977).For a first moment detector the response function R(k) is of the form R(k) = ck where c is a constant, k is a position vector in the detector plane and the vector nature of R(k)indicates that two signals are produced. This type of system would produce an image signal given bywhere the specimen transmittance is given by a (r) exp (iϕ (r), r is a position vector in object space, ro the position of the probe, ⊛ represents a convolution integral and it has been assumed that we have a coherent probe, with a complex disturbance of the form b(r-ro) exp (iζ (r-ro)). Thus the image signal for a pure phase object imaged in a STEM using a first moment detector is b2 ⊛ ▽ø. Note that this puts no restrictions on the magnitude of the variation of the phase function, but does assume an infinite detector.

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