scholarly journals Existence and non-existence of minimal graphic and p-harmonic functions

2019 ◽  
Vol 150 (1) ◽  
pp. 341-366
Author(s):  
Jean-Baptiste Casteras ◽  
Esko Heinonen ◽  
Ilkka Holopainen
Keyword(s):  
Harmonic Functions ◽  
Linear Growth ◽  
Entire Solution ◽  
Complete Riemannian Manifold ◽  
The Other ◽  
Hadamard Manifolds ◽  
Minimal Graph ◽  
Rotationally Symmetric ◽  
Sectional Curvatures ◽  
Negative Sectional Curvature

AbstractWe prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

Practical Correction of Three Fold Astigmatism in the Philips CM TEM

1996 ◽  
Vol 54 ◽  
pp. 418-419
Author(s):  
Arno J. Bleeker ◽  
Mark H.F. Overwijk ◽  
Max T. Otten
Keyword(s):  
Optical Properties ◽  
Phase Contrast ◽  
The Other ◽  
Objective Lens ◽  
Symmetric Part ◽  
A Posteriori ◽  
Contrast Transfer Function ◽  
High Brightness ◽  
Rotationally Symmetric ◽  
Brightness Field

With the improvement of the optical properties of the modern TEM objective lenses the point resolution is pushed beyond 0.2 nm. The objective lens of the CM300 UltraTwin combines a Cs of 0. 65 mm with a Cc of 1.4 mm. At 300 kV this results in a point resolution of 0.17 nm. Together with a high-brightness field-emission gun with an energy spread of 0.8 eV the information limit is pushed down to 0.1 nm. The rotationally symmetric part of the phase contrast transfer function (pctf), whose first zero at Scherzer focus determines the point resolution, is mainly determined by the Cs and defocus. Apart from the rotationally symmetric part there is also the non-rotationally symmetric part of the pctf. Here the main contributors are not only two-fold astigmatism and beam tilt but also three-fold astigmatism. The two-fold astigmatism together with the beam tilt can be corrected in a straight-forward way using the coma-free alignment and the objective stigmator. However, this only works well when the coefficient of three-fold astigmatism is negligible compared to the other aberration coefficients. Unfortunately this is not generally the case with the modern high-resolution objective lenses. Measurements done at a CM300 SuperTwin FEG showed a three fold-astigmatism of 1100 nm which is consistent with measurements done by others. A three-fold astigmatism of 1000 nm already sinificantly influences the image at a spatial frequency corresponding to 0.2 nm which is even above the point resolution of the objective lens. In principle it is possible to correct for the three-fold astigmatism a posteriori when through-focus series are taken or when off-axis holography is employed. This is, however not possible for single images. The only possibility is then to correct for the three-fold astigmatism in the microscope by the addition of a hexapole corrector near the objective lens.

An Approximate Analysis of Circular Plates With Variable Thickness

1982 ◽  
Vol 104 (3) ◽  
pp. 533-535
Author(s):  
A. K. Naghdi
Keyword(s):  
Variable Thickness ◽  
Simple Technique ◽  
Bending Moment ◽  
The Other ◽  
Uniform Pressure ◽  
Approximate Analysis ◽  
Circular Plates ◽  
Numerical Examples ◽  
Classic Theory ◽  
Rotationally Symmetric

Based on classic theory of beams and certain modifications, a simple technique is derived in order to obtain an approximate value of the maximum bending moment in a rotationally symmetric circular plate with a variable thickness. It is assumed that one of the two concentric boundaries of the plate is clamped, and the other is free. Numerical examples for both cases of constant and variable thickness plates subject to uniform pressure or rim line loading are presented.

scholarly journals Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

2001 ◽  
Vol 162 ◽  
pp. 149-167
Author(s):  
Yong Hah Lee
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Nonlinear Elliptic Equation ◽  
Complete Riemannian Manifold ◽  
Doubling Condition ◽  
Finite Covering ◽  
Nonlinear Elliptic ◽  
Finite Dimensional ◽  
Volume Doubling

In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

scholarly journals STUDY OF MITOCHONDRIA IN RAT LIVER

1962 ◽  
Vol 15 (3) ◽  
pp. 489-501 ◽  
Author(s):  
Gunter F. Bahr ◽  
Elmar Zeitler
Keyword(s):  
Electron Microscopy ◽  
Rat Liver ◽  
Linear Growth ◽  
Mitochondrial Fraction ◽  
The Other ◽  
Quantitative Electron Microscopy ◽  
Normal Rat ◽  
Log Normal ◽  
Log Normal Distribution ◽  
Particle Weight

The electron microscope has been used to determine the weight distribution of isolated subcellular particles from normal rat liver. The following results are reported: (1) There exist at least two well defined weight populations of subcellular particles; their respective median weights are 1.3 x 10-14 and 11 x 10-14 gm. The lighter fraction is considered to consist of lysosomes, the heavier of mitochondria. (2) The mitochondrial fraction shows a log-normal distribution of the particle weight. (3) By the introduction of morphologic criteria, the mitochondrial fraction is divided into two groups, one consisting of a spherical, the other of an oblong type of particle. The data found support the following concepts: (a) Mitochondria increase their weight from a certain size up by linear growth. (b) Mitochondria divide. The division is not necessarily symmetric; in all cases, however, one part of the division product is a spherical particle. It is felt that these results constitute a valuable demonstration of the general capabilities of quantitative electron microscopy and may stimulate many other useful applications of this technique in cytology, bacteriology, and virology.

Gradient estimates and heat kernel estimates

1995 ◽  
Vol 125 (5) ◽  
pp. 975-990 ◽  
Author(s):  
Zhongmin Qian
Keyword(s):  
Riemannian Manifold ◽  
Differential Operator ◽  
Heat Kernel ◽  
Harmonic Functions ◽  
Complete Riemannian Manifold ◽  
Gradient Estimates ◽  
Order Differential Operator ◽  
Kernel Estimates ◽  
Heat Kernel Estimates ◽  
Girsanov’S Formula

In the first part of this paper, Yau's estimates for positive L-harmonic functions and Li and Yau's gradient estimates for the positive solutions of a general parabolic heat equation on a complete Riemannian manifold are obtained by the use of Bakry and Emery's theory. In the second part we establish a heat kernel bound for a second-order differential operator which has a bounded and measurable drift, using Girsanov's formula.

ELLIPTIC EQUATIONS IN ℝ2 WITH ONE-SIDED EXPONENTIAL GROWTH

2004 ◽  
Vol 06 (06) ◽  
pp. 947-971 ◽  
Author(s):  
ZHITAO ZHANG ◽  
MARTA CALANCHI ◽  
BERNHARD RUF
Keyword(s):  
Exponential Growth ◽  
Elliptic Equations ◽  
Morse Index ◽  
Linear Growth ◽  
Critical Growth ◽  
The Other ◽  
Critical Groups ◽  
Bounded Domains ◽  
Eigen Value

We consider elliptic equations in bounded domains Ω⊂ℝ2 with nonlinearities which have exponential growth at +∞ (subcritical and critical growth, respectively) and linear growth λ at -∞, with λ>λ1, the first eigen value of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms; one solution is negative, the other one is sign-changing. Some critical groups and Morse index of these solutions are given. Also the case λ<λ1 is considered.

scholarly journals Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

2016 ◽  
Vol 27 (2) ◽  
pp. 1106-1130 ◽  
Author(s):  
Jean-Baptiste Casteras ◽  
Esko Heinonen ◽  
Ilkka Holopainen
Keyword(s):  
Minimal Graph ◽  
Sectional Curvatures

scholarly journals Design of Symmetrical Beam Triple-Aperture Waveguide Antenna for Primary Feed of Reflector

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Kanawat Nuangwongsa ◽  
Chuwong Phongcharoenpanich
Keyword(s):  
Frequency Band ◽  
Rectangular Waveguide ◽  
The Other ◽  
Rotationally Symmetric ◽  
The Asymmetry ◽  
Parabolic Reflectors ◽  
Good Agreement ◽  
Parasitic Coupling

This research presents a triple-aperture waveguide antenna as the primary feed of parabolic reflectors. The proposed antenna is able to rectify the asymmetry and also achieve a symmetrical unidirectional beam through the application of two parasitic coupling apertures. The design of the antenna is that of a rectangular waveguide (radiating aperture) vertically jointed to the two coupling apertures of the same measurement widthwise (i.e., one stacked on top and the other underneath) to achieve the symmetrical beam. The rectangular waveguide is 97.60 mm and 46.80 mm in width (a) and height (b), respectively, to propagate the WLAN frequency band of 2.412–2.484 GHz. Simulations were carried out to determine the optimal antenna parameters and an antenna prototype was subsequently fabricated and tested. The simulated beamwidths in theE- andH-planes at-3 dB were equally 67° (i.e., 67° for both theE- andH-planes) and at-10 dB also equally 137°, while the measured results at-3 dB were equally 65° and at-10 dB equally 135°. The simulation and measured results are thus in good agreement. The simulated and measured antenna gains are, respectively, 8.25 dBi and 9.17 dBi. The findings validate the applicability of the antenna as the prime feed for rotationally symmetric parabolic reflectors.

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