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Suppose that f is a homeomorphism from the punctured unit disk D {0} onto the annulus A(r′) = {r′ < |z| < 1}, r′ ≥ 0, and f is quasiconformal in every A(r), r > 0, but not in D. If r′ > 0 then f has cavitation at 0 and no cavitation if r′ = 0. The singular factorization problem is to find harmonic functions h in A(r′) such that h ◦ f satisfies the elliptic PDE associated with f with a singularity at 0. Sufficient conditions in terms of the dilatation Kf-1(z) together with the properties of h are given to the factorization problem, to the continuation of h ◦ f to 0 and to the regularity of h ◦ f. We also give sufficient conditions for cavitation and non-cavitation in terms of the complex dilatation of f and demonstrate both cases with several examples. [ABSTRACT FROM AUTHOR]

BACKGROUND: This study describes the efficacy of cochlear implantation under local anesthesia with conscious sedation with dexmedetomidine in adult patients and proposes a method to communicate with the conscious and cooperative patient intraoperatively. This less invasive anesthetic procedure is suitable for patients with comorbidities preventing general anesthesia. METHODS: Unilateral cochlear implantation with Oticon Medical systems was performed in 10 adult patients with comorbidities preventing general anesthesia. Classical cochlear implantation was performed under local anesthesia and conscious sedation with dexmedetomidine. Cue cards were used to support intraoperative dialogue. Outcome measures were intraoperative adverse events, patient perceptions, as well as postoperative completions measured with a questionnaire. RESULTS: The procedure was successful for all 10 patients. Dexmedetomidine lead to rapid and successful conscious sedation and no case of high blood pressure or aggravation of comorbidities was noted. Stapedial reflex measurements led to reliable thresholds. The usage of the cue cards was successful: patients were able to read the cue cards and thereby the medical team could inform the patients of surgical progress and ask the patients questions. CONCLUSION: Cochlear implantation and intraoperative dialogue with the conscious and cooperative patient is possible. The main advantage of the anesthetic procedure is the reduction in intra- and postoperative complications. Further, expected benefits include a less invasive procedure, the conscious state of the patient which enables the recording of auditory perception, and the absence of nonauditory percepts such as facial nerve stimulation during implant stimulation, a shorter surgical duration, and lower-associated costs. [ABSTRACT FROM AUTHOR]

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form ∂ z ¯ f + af + b f ¯ = c , where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that ∂ z ¯ f = g with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type ∂ z ¯ f z = h z q f z . On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media. [ABSTRACT FROM AUTHOR]

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C D ¯ ∩ W loc 1 , 2 D solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. [ABSTRACT FROM AUTHOR]

Assume that Ω is a domain in the complex plane ℂ and A( z) is a symmetric 2 ×2 matrix function with measurable entries, det A = 1 ; and such that 1/ K|ξ| ≤ 〈 A( z) ξ, ξ〉 ≤ K| ξ|, ξ ∈ ℝ, 1 ≤ K < ∞ . In particular, for semilinear elliptic equations of the form div ( A( z)∇ u( z)) = f( u( z)) in Ω ; we prove a factorization theorem that asserts that every weak solution u to the above equation can be expressed as the composition u = To휔 ; where 휔 : Ω → G stands for a K−quasiconformal homeomorphism generated by the matrix function A( z) ; and T( w) is a weak solution of the semilinear equation ∇ T( w) = J( w) f( T( w)) in G: Here, the weight J( w) is the Jacobian of the inverse mapping 휔 : Similar results hold for the corresponding nonlinear parabolic and hyperbolic equations. Some applications of these results to anisotropic media are given. [ABSTRACT FROM AUTHOR]

Background: This study describes the efficacy of cochlear implantation under local anesthesia with conscious sedation with dexmedetomi- dine in adult patients and proposes a method to communicate with the conscious and cooperative patient intraoperatively. This less invasive anesthetic procedure is suitable for patients with comorbidities preventing general anesthesia., Methods: Unilateral cochlear implantation with Oticon Medical systems was performed in 10 adult patients with comorbidities preventing general anesthesia. Classical cochlear implantation was performed under local anesthesia and conscious sedation with dexmedetomidine. Cue cards were used to support intraoperative dialogue. Outcome measures were intraoperative adverse events, patient perceptions, as well as post- operative completions measured with a questionnaire., Results: The procedure was successful for all 10 patients. Dexmedetomidine lead to rapid and successful conscious sedation and no case of high blood pressure or aggravation of comorbidities was noted. Stapedial reflex measurements led to reliable thresholds. The usage of the cue cards was successful: patients were able to read the cue cards and thereby the medical team could inform the patients of surgical progress and ask the patients questions., Conclusion: Cochlear implantation and intraoperative dialogue with the conscious and cooperative patient is possible. The main advantage of the anesthetic procedure is the reduction in intra- and postoperative complications. Further, expected benefits include a less invasive procedure, the conscious state of the patient which enables the recording of auditory perception, and the absence of nonauditory percepts such as facial nerve stimulation during implant stimulation, a shorter surgical duration, and lower-associated costs.