Lines of Curvature for Log Aesthetic Surfaces Characteristics Investigation

Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by . First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes.

INVESTIGATION PARAMETERS ULTRASONIC DEVICE ON PHASED ARRAYS. FOCUSING MODES FOR ULTRASONIC DEVICE TYPE OF OMNISCAN

The article is devoted to possibilities of regular focusing of Omniscan device on phased arrays. Questions are raised about evaluation of testing results when using linear and sector scan-ning with different focus parameters. The question of size near-field for phased arrays and asso-ciated choice of focus mode is discussed. The article is based on experiments conducted on samples with artificial reflectors at the same size, but different in type: a non-directional reflector (a side-drill hole) and a directional reflector (a flat-bottomed reflector), located at the same depth. The study was conducted for transducers with different frequencies. Families of curves of the signal amplitude dependence are obtained: on depth reflector, on focus depth setting, and on type reflector. The results emphasize need for precise focusing with-in the near-field of the transducer for small thicknesses or shallow depth of occurrence of discontinuities, and large variability in choice of focusing for depths in far-field. The study notes a significant difference in values of depth reflector at different focusses at a fixed position of transducer. In this article, in addition to considering possibility of focusing a flaw detector with phased arrays, the focus is on interpretation of results and reliability of testing in the analysis and comparison data. An integral part of the technological testing protocol for phased array is the depth of focus and the type of scanning. The obtained data do not depend on the frequency of transducer, which means that conclusions are applicable to general range of flaw detectors on phased arrays.

Towards the Sato–Tate groups of trinomial hyperelliptic curves

We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form [Formula: see text] where [Formula: see text] is the genus of the curve and [Formula: see text] is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for [Formula: see text] curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus [Formula: see text] curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components [Formula: see text] for these families of curves.

Extremal decomposition problems for p-harmonic Robin radius

The theorems on the extremal decomposition of plane domains concerning to the products of Robin's radii are extended to the case of domains in Euclidean space. In some cases, the classical non-overlapping condition is weakened. The proofs are based on the moduli technique for families of curves and dissymmetrization.

On Tangency in Equisingular Families of Curves and Surfaces

We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existence of exceptional tangents is preserved by Whitney equisingularity but their number can change.