Integral representations for some weighted classes of functions holomorphic in matrix domains

In 1945 the first author introduced the classes $H^p(α)$, 1 ≤ p -1, of holomorphic functions in the unit disk with finite integral (1) $∬_\mathbb{D} |f(ζ)|^p (1-|ζ|²)^α dξ dη < ∞ (ζ=ξ+iη)$ and established the following integral formula for $f ∈ H^p(α)$: (2) f(z) = (α+1)/π ∬_\mathbb{D} f(ζ) ((1-|ζ|²)^α)//((1-zζ̅)^{2+α}) dξdη, z∈ \mathbb{D} . We have established that the analogues of the integral representation (2) hold for holomorphic functions in Ω from the classes $L^p(Ω;[K(w)]^α dm(w))$, where: 1) $Ω = {w = (w₁,...,w_n) ∈ ℂ^n: Im w₁ > ∑_{k=2}^n |w_k|²}$, $K(w) = Im w₁ - ∑_{k=2}^n |w_k|²$; 2) Ω is the matrix domain consisting of those complex m × n matrices W for which $I^{(m)} - W·W*$ is positive-definite, and $K(W) = det[I^{(m)} - W·W*]$; 3) Ω is the matrix domain consisting of those complex n × n matrices W for which $Im W = (2i)^{-1} (W - W*)$ is positive-definite, and K(W) = det[Im W]. Here dm is Lebesgue measure in the corresponding domain, $I^{(m)}$ denotes the unit m × m matrix and W* is the Hermitian conjugate of the matrix W

Analysis of the Impedance Resonance of Piezoelectric Multi-Fiber Composite Stacks

Multi-Fiber CompositesTM (MFC's) produced by Smart Materials Corp behave essentially like thin planar stacks where each piezoelectric layer is composed of a multitude of fibers. We investigate the suitability of using previously published inversion techniques for the impedance resonances of monolithic co-fired piezoelectric stacks to the MFCTM to determine the complex material constants from the impedance data. The impedance equations examined in this paper are those based on the derivation. The utility of resonance techniques to invert the impedance data to determine the small signal complex material constants are presented for a series of MFC's. The technique was applied to actuators with different geometries and the real coefficients were determined to be similar within changes of the boundary conditions due to change of geometry. The scatter in the imaginary coefficient was found to be larger. The technique was also applied to the same actuator type but manufactured in different batches with some design changes in the non active portion of the actuator and differences in the dielectric and the electromechanical coupling between the two batches were easily measureable. It is interesting to note that strain predicted by small signal impedance analysis is much lower than high field stains. Since the model is based on material properties rather than circuit constants, it could be used for the direct evaluation of specific aging or degradation mechanisms in the actuator as well as batch sorting and adjustment of manufacturing processes