A Herglotz wavefunction method for solving the inverse Cauchy problem connected with the Helmholtz equation

AbstractThis paper is concerned with the Cauchy problem connected with the Helmholtz equation. On the basis of the denseness of Herglotz wavefunctions, we propose a numerical method for obtaining an approximate solution to the problem. We analyze the convergence and stability with a suitable choice of regularization method. Numerical experiments are also presented to show the effectiveness of our method

Application of carbon nanoparticles combined with refined extracapsular anatomy in endoscopic thyroidectomy

ObjectiveTo evaluate the value of refined extracapsular anatomy combined with carbon nanoparticle suspension tracing technology for protecting parathyroid function and the thoroughness of lymph node dissection in the central region during endoscopic thyroid cancer surgery.Patients and methodsRetrospective clinical data analysis was performed on 108 patients who underwent endoscopic thyroid cancer surgery at the First Affiliated Hospital of Wannan Medical College (Yijishan Hospital) from November 2019 to November 2022. Before surgery, thyroid function tests, color Doppler ultrasounds and neck-enhanced CT scans were performed on all patients. Cytopathological diagnosis obtained via ultrasound-guided fine-needle aspiration served as confirmation for the primary diagnosis. It was determined whether to perform a total thyroidectomy or a hemithyroidectomy (HT) together with preventive unilateral (ipsilateral) central neck dissection. Follow-up times were 1 to 34 months.ResultsTransient neuromuscular symptoms were present in 3.70% (4/108) cases, with no permanent neuromuscular symptoms or permanent hypoparathyroidism. Regarding transient hypoparathyroidism, the patients recovered after three months and did not need long-term calcium supplementation. The number of harvested LNs (mean± SD) was 5.54 ± 3.84, with ≤5 in 57.41% (62/108) and >5 in 42.59% (46/108) cases. The number of patients with metastatic LNs was 37.96% (41/108), with ≤2 in 65.85% (27/41) and >2 in 34.15% (14/41) cases.ConclusionsFine extracapsular anatomy combined with carbon nanoparticle suspension tracing is effective in endoscopic thyroid cancer surgery. It can improve the thoroughness of prophylactic central neck dissection and recognition of the parathyroid gland and avoid parathyroid injury and other complications to effectively protect parathyroid function

Galerkin method for the scattering problem of strip gratings

Abstract In this paper, the diffraction problem of periodic strip gratings is considered. The previous study of this problem usually concentrated on the numerical method; however, we try to analyze this problem and the convergence of the numerical solution from the mathematical point of view in this work. By use of the Dirichlet to Neumann operator on the slit between two strips, we reformulate the problem to an operator equation. The well-posedness of the solution to the operator equation is proved. The Galerkin method is applied to solve this operator equation and the convergence result of the numerical solution is also derived. Finally, some numerical experiments are presented to show the effectiveness of our method and verify the theoretical convergence result

A hybrid method for the interior inverse scattering problem

In this paper, the interior inverse scattering problem of a cavity is considered. By use of a general boundary condition, we can reconstruct the shape of the cavity without a priori information of the boundary condition type. The method of fundamental solutions (MFS) with regularization is formulated for solving the scattered field and its normal derivative on the cavity boundary. Newton's method is employed to reconstruct the cavity boundary by satisfying the general boundary condition. This hybrid method copes with the ill-posedness of the inverse problem in the MFS step and deals with the nonlinearity in the Newton's step. Some computational examples are presented to demonstrate the effectiveness of our method

Bayesian Method for Shape Reconstruction in the Inverse Interior Scattering Problem

The inverse scattering problem of an interior cavity with three different boundary conditions is considered. Bayesian method is used to reconstruct the shape of the cavity from scattered fields incited by point source(s) and measured on a closed curve inside the cavity. We prove the well-posedness in Bayesian perspective and present numerical examples to show the viability of the method

Effects of Precipitant and pH on Coprecipitation of Nanosized Co-Cr-V Alloy Powders

Nanosized Co-Cr-V alloy powders were synthesized via coprecipitation method. Effects of precipitants ((NH4)2C2O4·H2O and Na2CO3) and pH were investigated by X-ray diffraction (XRD), Zeta potential analyzer, thermogravimetry-differential scanning calorimetry (TG-DSC), inductively coupled plasma-atomic emission spectrometry (ICP-AES) and scanning electron microscopy (SEM). Co-Cr-V alloy powders were consisted of major face-centered cubic Co (fcc Co) and minor hexagonal close-packed Co (hcp Co). Grain sizes of precursors and Co-Cr-V alloy powders were increased with pH value (7–10) within the ranges of 3~39 and 39~66 nm, respectively. Rod-like or granular Co-Cr-V alloy particles were assembled by interconnected nanograins. At pH = 7, Na2CO3 precipitant was found to be beneficial to maintain the desirable composition of Co-Cr-V powders. It was also found that lower pH favors the maintenance of pre-designed composition, while grain coarsens at higher pH. Effects of variation for precipitant and pH on the morphology and composition of Co-Cr-V alloy powder were discussed in detail and relevant mechanism was further proposed