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Abstract: In cited article E. F. Eisele considered the problem of the optimal approximation of symmetric surfaces by biquadratic Bézier patches. Unfortunately, the results therein are incorrect, which is shown in this paper by considering the optimal approximation of spherical squares. A detailed analysis and a numerical algorithm are given, providing the best approximant according to the (simplified) radial error, which differs from the one obtained mentioned article. The sphere is then approximated by the continuous spline of two and six tensor product quadratic Bézier patches. It is further shown that the $G^1$ smooth spline of six patches approximating the sphere exists, but it is not a good approximation. The problem of an approximation of spherical rectangles is also addressed and numerical examples indicate that several optimal approximants might exist in some cases, making the problem extremely difficult to handle. Finally, numerical examples are provided that confirm the theoretical results.
Keywords: Bézier patches, spherical squares, optimal approximation, sphere approximation
Published in DiRROS: 14.03.2024; Views: 97; Downloads: 53
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Abstract: We discuss avoidance of sure loss and coherence results for semicopulas and standardized functions, i.e., for grounded, $1$-increasing functions with value $1$ at $(1, 1, \ldots , 1)$. We characterize the existence of a $k$-increasing $n$-variate function $C$ fulfilling $A \le C \le B$ for standardized $n$-variate functions $A$, $B$ and discuss methods for constructing such functions. Our proofs also include procedures for extending functions on some countably infinite mesh to functions on the unit box. We provide a characterization when $A$ respectively $B$ coincides with the pointwise infimum respectively supremum of the set of all $k$-increasing $n$-variate functions $C$ fulfilling $A \le C \le B$.
Keywords: copulas, quasi-copulas, semicopulas, standardized function, coherence, avoidance of sure loss, k-increasing function
Published in DiRROS: 13.03.2024; Views: 85; Downloads: 54
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Keywords: 1/2-consistent cycles, automorphisms, consistent cycles, cubic symmetric graphs, shunt, s-regular graphs
Published in DiRROS: 13.03.2024; Views: 106; Downloads: 52
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Abstract: Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space ${\mathbb R}^n$ $(n\ge 3)$ by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a $G$-equivariant conformal minimal immersion $F:X\to{\mathbb R}^n$. We show in particular that such a map $F$ always exists if $G$ acts without fixed points on $X$. Furthermore, every finite group $G$ arises in this way for some open Riemann surface $X$ and $n=2|G|$. We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete infinite groups acting on $X$ properly discontinuously and acting on ${\mathbb R}^n$ by rigid transformations.
Keywords: Riemann surfaces, minimal surfaces, G-equivariant conformal minimal immersion
Published in DiRROS: 13.03.2024; Views: 114; Downloads: 42
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Abstract: Cellulose acetate, commonly used in cigarette filters, poses environmental concerns due to its questionable (bio)degradability and prevalent presence in nature. This study compares the weathering effects on two types of cigarette filters: traditional and newer filters used in heated tobacco devices (HTP). Microplastics were derived from used cigarette parts and then subjected to artificial aging. Analytical techniques (Thermogravimetry-Differential Scanning Calorimetry TG/DSC, Fourier-Transform Infrared Spectroscopic Analysis (FTIR)) and loose bulk density measurements were employed pre- and post-aging cellulose acetate of both types of samples. Despite increasing evidence influencing European Union (EU) directives on tobacco product disposal, there's a lack of systematic analysis on the weathering impact, especially concerning the touted environmental benefits of newer filters. Results indicate decreased particle size in cellulose acetate filters post-aging. Variances were observed in thermal behavior, yet FTIR spectra remained unchanged.
Keywords: cellulose acetate, cigarette filters, microplastics, artificial weathering, polymer degradation
Published in DiRROS: 12.03.2024; Views: 131; Downloads: 140
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