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In this study, we investigate the focal surfaces associated with translation surfaces in Euclidean 3-space from the viewpoint of differential geometry. We begin by defining the translation surface generated by two planar curves and derive the corresponding focal surfaces using the framework of the Frenet frame. Analytical conditions are obtained under which the focal surfaces exhibit minimality or flatness. Several theorems are proven to classify the focal images, supported by illustrative examples. The results provide insights into the curvature structure of translation surfaces and contribute to the broader understanding of their geometric behavior.

This study explores parallel hypersurfaces in four-dimensional Euclidean space E[sup.4] , deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry.

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space En, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When (n−1)-planar curves are considered, this construction enables the generation of hypersurfaces in n-dimensional spaces. Building upon this geometric framework, we conduct the first-ever investigation of spherical product hypersurfaces in the context of Minkowski geometry. We define these hypersurfaces in four-dimensional Minkowski space E14 and derive explicit expressions for their Gaussian and mean curvatures. We also determine the conditions under which such hypersurfaces are flat or minimal. Furthermore, we reinterpret certain hyperquadrics as specific instances of spherical product hypersurfaces in E14, supported by visual illustrations. Finally, we extend the construction to arbitrary-dimensional Minkowski spaces, providing a unified formulation for spherical product hypersurfaces across higher-dimensional Lorentzian geometries.

This study explores parallel hypersurfaces in four-dimensional Euclidean space E4, deriving explicit expressions for their Gaussian and mean curvatures in terms of the curvature functions of the base hypersurface. We identify conditions under which these parallel hypersurfaces are flat or minimal. The theory is applied to several key hypersurfaces, including rotational hypersurfaces, hyperspheres, catenoidal hypersurfaces, and helicoidal hypersurfaces, with detailed curvature computations and visualizations. These results not only extend classical curvature relations into higher-dimensional spaces but also offer valuable insights into curvature transformations, with practical applications in both theoretical and computational geometry.

In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our conclusions. In the experiments, it was observed that the p-value in the algebra of elliptic quaternions directly affects the performance of the problem under consideration. Selecting the optimal p-value for problem-solving and the elliptic behavior of many physical systems make this number system advantageous in applied sciences.

In this study, we developed a MATLAB 2024a toolbox that performs advanced algebraic calculations in the algebra of elliptic numbers and elliptic quaternions. Additionally, we introduce color image processing methods, such as principal component analysis, image compression, image restoration, and watermarking, based on singular-value decomposition theory for elliptic quaternion matrices; we added these to the newly developed toolbox. The experimental results demonstrate that elliptic quaternionic methods yield better image analysis and processing performance compared to other hypercomplex number-based methods.

Objective: The purpose of this research is to evaluate e-PBL tutorials and compare these sessions with face-to-face PBL sessions.Patients and Methods: This research is a program evaluation study in which quantitative methods were used. In the research, four 90-minute e-PBL sessions held between April and June 2020 were evaluated. Sessions conducted online were realised using the sevenstep approach in groups of 13-14 students and a tutor. Video recordings were analysed with the thin slicing method. In addition, various quantitative data on evaluation were analysed using multiple tools and materials, including the end-of-program evaluation form. Friedman test and Mann-Whitney U test were used in the analyses of quantitative data.Results: Upon evaluating the analyses of the feedback received from the students about the structure, content and process of the program regarding the e-PBL tutorials, the students gave a positive opinion of 80% or more. In regards with the cases, the students had positive opinions of over 80% in terms of “motivation for learning and researching”, “daily life and its relation to their individual development”, “suitability to their levels of knowledge and skills”, “reinforcement of topics”. Support, guidance and feedback received from the tutor as a group and individually during online tutorials were statistically significantly higher than the face-to-face PBL tutorials (P<0.05).Conclusion: Research on the effectiveness of e-PBL tutorials, including ours, point out that e-PBL practices may constitute a viable alternative besides face-to face ones. However, for a sounder framing and better results, the subject should be studied in different aspects and more evidences be gathered in this area. These studies will provide evidence to educational institutions and practitioners on how to adapt and modify educational practices, including PBL.

İlk bölümde, bazı özel tipli matrislerin uygulamalı bilimlerdeki kullanım alanlarından bahsedilmektedir. Ayrıca literatürde birçok yazar tarafından çalışılan özel tipli matrislerin lineer bileşimlerinin karakterizasyonu problemleri ile ilgili sonuçlar, tablolar yardımıyla özetlenmektedir. İkinci bölümde, tezin esas sonuçlarının ortaya konduğu üçüncü ve dördüncü bölümlerde kullanılacak olan temel kavramlar ve bazı teoremler verilmektedir.Üçüncü bölüm çalışmanın omurgasını oluşturmaktadır. Bu bölümde, sonlu sayıda değişmeli köşegenleştirilebilir matrisin lineer bileşiminin spektrumunu karakterize etmek için bir kombinatorik yöntem verilmekte ve bu yöntem temel alınarak bir algoritma geliştirilmektedir.Dördüncü bölümde, iki değişmeli kübik matrisin lineer bileşiminin kübikliği, iki değişmeli kuadripotent matrisin lineer bileşiminin kuadripotentliği, karşılıklı değişmeli üç tripotent matrisin lineer bileşiminin tripotentliği, ve karşılıklı değişmeli dört involutif matrisin lineer bileşiminin tripotentliği problemleri üçüncü bölümde geliştirilen Algoritma 3.1 yardımıyla çözülmektedir.

Objective: The aim of this study was to investigate the effect of contrast agent use on procedure time and accuracy in pulsed radiofrequency treatment of the lumbar dorsal root ganglion.Patients and Methods: Patients aged 23–79 years with lumbar radicular pain due to disc herniation for at least 3 months were randomized into two groups of 35 patients each. Patients in both groups underwent fluoroscopy-guided pulsed radiofrequency treatment of the dorsal root ganglion at the level of the L5 foramen. In the radiocontrast group, unlike the control group, the location of the ganglion was determined by administering the contrast agent before the radiofrequency treatment.Results: Procedure time in the radiocontrast group was significantly longer than in the control group (P<0.05). In 50 cases ganglion was detected in the extraforaminal or intraforaminal location, the excitation of the ganglion in the range of 0.4–0.6 V was significantly higher in the radiocontrast group (95.8%) than in the control group (69.2%) (P<0.05).Conclusion: The use of radiocontrast material in pulsed radiofrequency application on the dorsal root ganglion prolongs the procedure time. However, for ganglia that cannot be detected by stimulation, contrast injection is useful on procedural accuracy

İlk bölümde matrisler ve bazı özel tipli matrislerle ilgili kısa bir literatür bilgisi verilmektedir.. İkinci bölümde bazı temel kavram ve özellikler verilmektedir. Üçüncü bölümde idempotent ve tripotent matrislerin tanımları verilip özellikleri ayrıntılı olarak incelenmektedir. Bu çalışmaya ilham kaynağı olan literatürdeki bir çalışma dördüncü bölümde incelenmektedir.Beşinci bölümde, A₁,...,An mxm mertebeli karşılıklı olarak değişmeli tripotent matrisler olmak üzereM = a0I + (ai₁Ai₁ + a₂4²) + ΣΣ(b₁ A₁ A₂ + b₁₂ A₁ A² + b² A}A₂ + b²₂₂ ¸²), i=1 j-1k-2 j

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