On the first eigenfunction of the fixed membrane: Some extensions of results of Payne and Stakgold

1977 ◽  
Vol 28 (1) ◽  
pp. 151-159 ◽  
Author(s):  
Gérard A. Philippin
Keyword(s):  
Fixed Membrane ◽  
First Eigenfunction

scholarly journals ON EIGENVALUE PROBLEMS FOR ELLIPTIC HEMIVARIATIONAL INEQUALITIES

2008 ◽  
Vol 51 (2) ◽  
pp. 407-419 ◽  
Author(s):  
Zhenhai Liu ◽  
Guifang Liu
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problems ◽  
Hemivariational Inequalities ◽  
Generalized Gradient ◽  
At Resonance ◽  
Quasilinear Elliptic ◽  
First Eigenfunction

AbstractThis paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.

Ultrastructural comparisons of fungus hyphal cells using frozen-etched replicas and thin sections of the fungus Pyrenochaeta terrestris

1968 ◽  
Vol 14 (3) ◽  
pp. 205-210 ◽  
Author(s):  
W. M. Hess
Keyword(s):  
Electron Scattering ◽  
Cell Walls ◽  
Sole Carbon Source ◽  
Membrane Systems ◽  
Thin Sections ◽  
Fixed Membrane ◽  
Pyrenochaeta Terrestris ◽  
Freeze Etching ◽  
Fixed Cell ◽  
Etched Membrane

The ultrastructure of P. terrestris hyphal cells was investigated to compare frozen-etched replicas with chemically fixed thin sections. The fungus used in this study uses glycerol as a sole carbon source and survives the freezing procedures necessary for freeze-etching; thus frozen-etched replicas reflect the living state.Frozen-etched membrane systems have particles of various sizes and concentrations and have a smooth appearance as contrasted to chemically fixed membrane systems, which have particles difficult to distinguish and somewhat irregular membrane systems. Frozen-etched cell walls are seen to contain particles, and microfibrillar orientation is evident in older cell walls, whereas substructure is not evident in chemically fixed cell walls, although secretion products of the fungus accumulate on cell surfaces.Chemically fixed ground cytoplasm has ribosomes and areas of high- and low-electron scattering which are not seen with freeze-etching. Cells fixed in glutaraldehyde–acrolein–OsO4 more nearly resemble frozen-etched cells than cells fixed in potassium permanganate.

On the viscosity solutions of eigenvalue problems for a class of nonlinear elliptic equations

2019 ◽  
Vol 12 (4) ◽  
pp. 393-421
Author(s):  
Tilak Bhattacharya ◽  
Leonardo Marazzi
Keyword(s):  
Viscosity Solutions ◽  
Elliptic Equations ◽  
Eigenvalue Problems ◽  
A Priori ◽  
Nonlinear Elliptic Equations ◽  
First Eigenvalue ◽  
Nonlinear Elliptic ◽  
Degenerate Elliptic ◽  
First Eigenfunction ◽  
Nonlinear Degenerate

AbstractWe consider viscosity solutions of a class of nonlinear degenerate elliptic equations, involving a parameter, on bounded domains. These arise in the study of eigenvalue problems. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of the first eigenvalue and an associated positive first eigenfunction.

COLLECTIVE BEHAVIORS OF SPIRAL WAVES IN THE NETWORKS OF HODGKIN-HUXLEY NEURONS IN PRESENCE OF CHANNEL NOISE

2010 ◽  
Vol 18 (01) ◽  
pp. 243-259 ◽  
Author(s):  
JUN MA ◽  
AI-HUA ZHANG ◽  
JUN TANG ◽  
WU-YIN JIN
Keyword(s):  
Internal Noise ◽  
Spiral Wave ◽  
Spiral Waves ◽  
Channel Noise ◽  
Collective Behaviors ◽  
Numerical Studies ◽  
Factor Of Synchronization ◽  
Quantitative Factor ◽  
Fixed Membrane ◽  
Membrane Patch

Collective behaviors of spiral waves in the networks of Hodgkin-Huxley neuron are investigated. A stable rotating spiral wave can be developed to occupy the quiescent areas in networks of neurons by selecting appropriate initial values for the variables in the networks of neurons. In our numerical studies, most neurons are quiescent and finite (few) numbers of neurons are selected with different values to form a spiral seed. In this way, neurons communicating are carried by propagating spiral wave to break through the quiescent domains (areas) in networks of neurons. The effect of membrane temperature on the formation of spiral wave is investigated by selecting different fixed membrane temperatures in the networks, and it is found that a spiral wave cannot be developed if the membrane temperature is close to a certain threshold. A quantitative factor of synchronization is defined to measure the statistical properties and collective behaviors of the spiral wave. And a distinct phase transition, which indicates the critical condition for spiral survival, is observed in the sudden changing point of the factors of synchronization curve vs. certain bifurcation parameter. Internal noise is introduced into ion channels (channel noise) with the Langevin method. It is found that a stable rotating spiral wave is developed and the spiral wave is robust to weak channel noise (the membrane patch is not small). The spiral wave can not grow up and the stable rotating spiral wave encounters instability in presence of strong channel noise. Coherence resonance-like behavior is observed in calculating the factors of synchronization in presence of channel noise.

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