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Locally Pseudoconvex Spaces and Algebras
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
is the inductive (direct) limit of subalgebras (Ai, τi). Herewith, lim→A˜i is called a regular inductive limit (see Jarchow (1981), p. 83), if for every bounded set B in (A, τ) there exists an index i0 ∈ I such that B ⊂ Ai0 and is bounded there, and a bornological inductive limit (see Hogbe-Nlend (1977), p. 43), if every subset B in A is bounded if and only if B is bounded in some (Ai, τi).
Advanced Instrumentation for High-Resolution Capacitance and Impedance Measurements
Published in Jian V. Li, Giorgio Ferrari, Capacitance Spectroscopy of Semiconductors, 2018
Giorgio Ferrari, Marco Carminati
The amplitude noise modulates the stimulus voltage and correspondingly the demodulated signals of the lock-in amplifier are modulated by nA(t). These fluctuations give a direct limit to the minimum detectable capacitance by the instrument. Since the effect of the amplitude noise is proportional to VAC and to CDUT, it sets a limit to the resolution of the system as () ΔCminCDUT≈nA2¯Bn,
About the Solution of Some Inverse Problems in Differential Galois Theory by Hamburger Equations
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
We set Κ = ℂ{x}x−1] and, for ν ε ℕ*, Κ = ℂ{x1/v}xℂ−1/v] (Κ ⊂ ΚV). Let εv = x−1/vℂ[x−1/v}. We set ε=∪ν∈ℕ*εν⋅Forq={q1,…,qm}⊂εν, globally invariant by the monodromy transform x → e2iπx, we set E(q) = ℤ q1 +… + ℤ qm. Then ε is a direct limit of finitely generated free ℤ-modules (lattices) ε=lim→ε(q)q.
Weiskopf model for sandy materials: Rayleigh – Lamb wave dispersion
Published in Mechanics of Advanced Materials and Structures, 2022
Vertical dashed lines in these plots indicate the high-frequency asymptotes relating to Rayleigh and Stoneley wave velocities; for analytical and approximate formulas for the corresponding velocities, see [33–35]; the three-dimensional formalism for evaluating Rayleigh wave velocity in anisotropic media, see [36]. The Stoneley wave velocity at the interface between layers and the halfspace is computed by expressions given in [37]. The low frequency limit, known as the second limiting velocity [38] is obtained by taking the direct limit
On the unsolvability of bosonic quantum fields
Published in Philosophical Magazine, 2018
If we take the direct limit in the above formulas, we conclude that the number of primitive correlators in theory has the cardinality of the continuum, as It exists, however, the possibility of defining the continuum limit in a weaker sense, which is physically the right choice, in which the number of primitive correlators is countable.
First-order indicators for the estimation of discrete fractures in porous media
Published in Inverse Problems in Science and Engineering, 2018
H. Ben Ameur, G. Chavent, F. Cheikh, F. Clément, V. Martin, J. E. Roberts
where are given by the direct limit fault model (17) and its adjoint counterpart set on the candidate fault edges of only, whose solutions are given by