scholarly journals Eigenvalue equation for the modular graph Ca,b,c,d

2019 ◽  
Vol 2019 (7) ◽  
Author(s):  
Anirban Basu
Keyword(s):  
Eigenvalue Equation

scholarly journals Kinetic extraordinary-mode eigenvalue equation for relativistic nonneutral electron flow in planar geometry

1989 ◽  
Vol 7 (1) ◽  
pp. 55-84 ◽  
Author(s):  
Ronald C. Davidson ◽  
Han S. Uhm
Keyword(s):  
Electron Flow ◽  
Low Frequency ◽  
Companion Paper ◽  
Field Configuration ◽  
Outer Edge ◽  
Eigenvalue Equation ◽  
Planar Geometry ◽  
Canonical Momentum ◽  
Stability Properties ◽  
Long Wavelength

Use is made of the Vlasov–Maxwell equations to derive an eigenvalue equation describing the extraordinary–mode stability properties of relativistic, non-neutral electron flow in high-voltage diodes. The analysis is based on well-established theoretical techniques developed in basic studies of the kinetic equilibrium and stability properties of nonneutral plasmas characterized by intense self fields. The formal eigenvalue equation is derived for extraordinary-mode flute perturbations in a planar diode. As a specific example, perturbations are considered about the choice of self-consistent Vlasov equilibrium , where . is the electron density at the cathode (x = 0), H is the energy, and Py is the canonical momentum in the Y-direction (the direction of the equilibrium electron flow). As a limiting case, the planar eigenvalue equation is further simplified for low-frequency long-wavelength perturbations with |ω − kvd, ≪ ωυ where and and ⋯c = eB0/mc, and B0ệz is the applied magnetic field in the vacuum region xb < x ≤ d. Here, the outer edge of the electron layer is located at x = xb; ω is complex oscillation frequency; k is the wavenumber in the y-direction; ωυ is the characteristic betatron frequency for oscillations in the x′-orbit about the equilibrium value x′ = x0 = xb/2; and Vd is the average electron flow velocity in the y-direction at x = x0. In simplifying the orbit integrals, a model is adopted in which the eigenfunction approximated by , where x′(t′) is the x′-orbit in the equilibrium field configuration. A detailed analysis of the resulting eigenvalue equation for , derived for low-frequency long-wavelength perturbations, is the subject of a companion paper.

Numerical determination of the Titchmarsh-Weyl m -coefficient and its applications to HELP inequalities

1989 ◽  
Vol 426 (1870) ◽  
pp. 167-188 ◽  
Keyword(s):  
Numerical Methods ◽  
Integral Inequality ◽  
Best Constants ◽  
Eigenvalue Equation ◽  
Numerical Determination

This paper is concerned with numerical methods for finding m(λ) , the Titchmarsh-Weyl m -coefficient, for the singular eigenvalue equation - y" + qy = λy on [0, ∞) and the results are applied to the problem of finding best constants for Everitt’s extension to the Hardy-Little-wood-Pόlya (HELP) integral inequality.

Stability of linear shear flows in shallow water

1995 ◽  
Vol 303 ◽  
pp. 203-214 ◽  
Author(s):  
Charles Knessl ◽  
Joseph B. Keller
Keyword(s):  
Shallow Water ◽  
Shear Flows ◽  
Special Functions ◽  
Rigid Wall ◽  
Eigenvalue Equation ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Linear Shear ◽  
The Stability ◽  
Rigid Walls

The stability or instability of various linear shear flows in shallow water is considered. The linearized equations for waves on the surface of each flow are solved exactly in terms of known special functions. For unbounded shear flows, the exact reflection and transmission coefficients R and T for waves incident on the flow, are found. They are shown to satisfy the relation |R|2= 1+ |T|2, which proves that over reflection occurs at all wavenumbers. For flow bounded by a rigid wall, R is found. The poles of R yield the eigenvalue equation from which the unstable mides can be found. For flow in a channel, with two rigid walls, the eigenvalue equation for the modes is obtained. The results are compared with previous numerical results.

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