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Some formulas concerning the fundamental unit of a real quadratic field
AbstractLet ε be the fundamental unit of a real quadratic field of discriminant d and let integers Vn and Un be defined by (Vn = Und)⧸2=εn. It is well known that if p is any odd prime, then p|Um, where m = p − χ and χ is the value of the Legendre symbol (dp). In this paper several formulas are derived for the value of Um⧸p (modp) for the case in which p ∤ d
Multi-transmission-line-beam interactive system
We construct here a Lagrangian field formulation for a system consisting of
an electron beam interacting with a slow-wave structure modeled by a possibly
non-uniform multiple transmission line (MTL). In the case of a single line we
recover the linear model of a traveling wave tube (TWT) due to J.R. Pierce.
Since a properly chosen MTL can approximate a real waveguide structure with any
desired accuracy, the proposed model can be used in particular for design
optimization. Furthermore, the Lagrangian formulation provides for: (i) a clear
identification of the mathematical source of amplification, (ii) exact
expressions for the conserved energy and its flux distributions obtained from
the Noether theorem. In the case of uniform MTLs we carry out an exhaustive
analysis of eigenmodes and find sharp conditions on the parameters of the
system to provide for amplifying regimes
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