scholarly journals The general complex case of the Bernstein-Nachbin approximation problem

1978 ◽  
Vol 28 (1) ◽  
pp. 193-206 ◽  
Author(s):  
S. Machado ◽  
Joao Bosco Prolla
Keyword(s):  
Approximation Problem ◽  
Complex Case ◽  
General Complex

scholarly journals Observations on the Computation of Eigenvalue and Eigenvector Jacobians

Algorithms ◽  
2019 ◽  
Vol 12 (12) ◽  
pp. 245 ◽  
Author(s):  
Andrew J. Liounis ◽  
John A. Christian ◽  
Shane B. Robinson
Keyword(s):  
Complex Case ◽  
Extensive Literature ◽  
Jacobian Matrices ◽  
Special Cases ◽  
The Matrix ◽  
General Complex ◽  
Analytic Expressions ◽  
And Performance ◽  
Eigenvector Derivatives ◽  
Eigenvalue And Eigenvector

Many scientific and engineering problems benefit from analytic expressions for eigenvalue and eigenvector derivatives with respect to the elements of the parent matrix. While there exists extensive literature on the calculation of these derivatives, which take the form of Jacobian matrices, there are a variety of deficiencies that have yet to be addressed—including the need for both left and right eigenvectors, limitations on the matrix structure, and issues with complex eigenvalues and eigenvectors. This work addresses these deficiencies by proposing a new analytic solution for the eigenvalue and eigenvector derivatives. The resulting analytic Jacobian matrices are numerically efficient to compute and are valid for the general complex case. It is further shown that this new general result collapses to previously known relations for the special cases of real symmetric matrices and real diagonal matrices. Finally, the new Jacobian expressions are validated using forward finite differencing and performance is compared with another technique.

Requirement of staged interventions in a complex case of Acromegaly and Obstructive Sleep Apnoea

2020 ◽  
Author(s):  
Mili Dhar ◽  
Jennifer Elias ◽  
Benjamin Field ◽  
Sunil Zachariah ◽  
Julian Emmanuel
Keyword(s):  
Obstructive Sleep Apnoea ◽  
Sleep Apnoea ◽  
Complex Case ◽  
Obstructive Sleep

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

Development and evaluation of a patient-specific surgical simulator for endoscopic colloid cyst resection

2020 ◽  
Vol 133 (2) ◽  
pp. 521-529 ◽  
Author(s):  
Vivek P. Bodani ◽  
Gerben E. Breimer ◽  
Faizal A. Haji ◽  
Thomas Looi ◽  
James M. Drake
Keyword(s):  
Choroid Plexus ◽  
Computer Aided Design ◽  
Third Ventricle ◽  
Colloid Cyst ◽  
Comfort Level ◽  
Complex Case ◽  
Patient Specific ◽  
Surgical Simulator ◽  
Perceived Realism ◽  
Noncontrast Ct

OBJECTIVEEndoscopic resection of third-ventricle colloid cysts is technically challenging due to the limited dexterity and visualization provided by neuroendoscopic instruments. Extensive training and experience are required to master the learning curve. To improve the education of neurosurgical trainees in this procedure, a synthetic surgical simulator was developed and its realism, procedural content, and utility as a training instrument were evaluated.METHODSThe simulator was developed based on the neuroimaging (axial noncontrast CT and T1-weighted gadolinium-enhanced MRI) of an 8-year-old patient with a colloid cyst and hydrocephalus. Image segmentation, computer-aided design, rapid prototyping (3D printing), and silicone molding techniques were used to produce models of the skull, brain, ventricles, and colloid cyst. The cyst was filled with a viscous fluid and secured to the roof of the third ventricle. The choroid plexus and intraventricular veins were also included. Twenty-four neurosurgical trainees performed a simulated colloid cyst resection using a 30° angled endoscope, neuroendoscopic instruments, and image guidance. Using a 19-item feedback survey (5-point Likert scales), participants evaluated the simulator across 5 domains: anatomy, instrument handling, procedural content, perceived realism, and confidence and comfort level.RESULTSParticipants found the simulator’s anatomy to be highly realistic (mean 4.34 ± 0.63 [SD]) and appreciated the use of actual instruments (mean 4.38 ± 0.58). The procedural content was also rated highly (mean 4.28 ± 0.77); however, the perceived realism was rated slightly lower (mean 4.08 ± 0.63). Participants reported greater confidence in their ability to perform an endoscopic colloid cyst resection after using the simulator (mean 4.45 ± 0.68). Twenty-three participants (95.8%) indicated that they would use the simulator for additional training. Recommendations were made to develop complex case scenarios for experienced trainees (normal-sized ventricles, choroid plexus adherent to cyst wall, bleeding scenarios) and incorporate advanced instrumentation such as side-cutting aspiration devices.CONCLUSIONSA patient-specific synthetic surgical simulator for training residents and fellows in endoscopic colloid cyst resection was successfully developed. The simulator’s anatomy, instrument handling, and procedural content were found to be realistic. The simulator may serve as a valuable educational tool to learn the critical steps of endoscopic colloid cyst resection, develop a detailed understanding of intraventricular anatomy, and gain proficiency with bimanual neuroendoscopic techniques.

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