THEORY OF SURFACES IN FOUR-DIMENSIONAL GALILEAN SPACE

By means of a special choice of coordinate lines of the surface in four-dimensional Galilean space, the first and second quadratic shape of the surface is defined. It has been proved that the second-order surface equation in three-dimensional space can be converted to a canonical form by means of a special transformation, which is the rotation of the coordinate axes of three-dimensional Galilean space. Furthermore, the transformation matrix is an element of the Heisenberg group that is neither symmetric nor orthogonal. In four-dimensional space R41 - the concept of a surface indicator is introduced and the main curvature of the surface is defined.

The roles of lower- and higher-order surface statistics in tactile texture perception

Humans can haptically discriminate surface textures when there is a significant difference in the statistics of the surface profile. Previous studies on tactile texture discrimination have emphasized the perceptual effects of lower-order statistical features such as carving depth, inter-ridge distance, and anisotropy, which can be characterized by local amplitude spectra or spatial-frequency/orientation subband histograms. However, the real-world surfaces we encounter in everyday life also differ in the higher-order statistics, such as statistics about correlations of nearby spatial-frequencies/orientations. For another modality, vision, the human brain has the ability to utilize the textural differences in both higher- and lower-order image statistics. In this work, we examined whether the haptic texture perception can utilize higher-order surface statistics as visual texture perception does, by 3D-printing textured surfaces transcribed from different 'photos' of natural scenes such as stones and leaves. Even though the maximum carving depth was well above the haptic detection threshold, some texture pairs were hard to discriminate. Specifically, those texture pairs with similar amplitude spectra were difficult to discriminate, which suggests that the lower-order statistics have the dominant effect on tactile texture discrimination. To directly test the poor sensitivity of the tactile texture perception to higher-order surface statistics, we matched the lower-order statistics across different textures using a texture synthesis algorithm and found that haptic discrimination of the matched textures was nearly impossible unless the stimuli contained salient local features. We found no evidence for the ability of the human tactile system to use higher-order surface statistics for texture discrimination.

Numerical investigation of magnetohydrodynamics Williamson nanofluid flow over an exponentially stretching surface

This research work describes the investigation of a magnetohydrodynamic flow of Williamson nanofluid over an exponentially porous stretching surface considering two cases of heat transfer i.e., prescribed exponential order surface temperature (PEST), and prescribed exponential order heat flux (PEHF). As a result of this infestation, a mathematical model of the problem based on conservation of linear momentum and law of conservation of mass and energy is developed. Whereas governing nonlinear partial differential equations (PDEs) are converted to nonlinear ordinary differential equations (ODEs). Subsequently, the velocity, concentration, and temperature profiles are developed by using the method of similarity transformation. Furthermore, the effects of various physical parameters of engineering interests are demonstrated graphically. It is highlighted that both the magnetic parameter [Formula: see text] and Williamson parameter [Formula: see text] causes to reduce the boundary layer thickness.

Competency approach to the study of the second-order surface interface as the geometric location of points in space

The necessity of the formation of students' functional literacy as a competency approach to the training of future Mathematics teachers is substantiated on the example of studying of one of the topics of analytical geometry. It has been established that a prerequisite for the development of any competency prescribed in the standards of secondary education is the initial existence of a sufficiently new concept of functional literacy for a student of a certain level. The basic literacy comes down to the ability to read, write and express of one's thoughts correctly. Let us consider the issue of functional literacy from the point of view of the pedagogic specialty. Acquaintance with the well-known textbooks of analytic geometry allows us to say that 2nd order algebraic surfaces in Euclidean space are determined in most cases algebraically by means of equations. A constructive approach is also of use – surfaces are obtained by rotating 2nd degree curves around their symmetry axes and by deformation of the resulting surfaces by compression. The metric approach, as it used for 2nd order curves, is restricted only by the formulation of problems to find the certain locus of points in space. The exception is the article Dmitriy Perepyolkin which was published in 1936. In this paper the locus of points in space with the following characteristic property is studied – the ratio of the distance to a given point to the distance to a given straight line is constant. The strait line is assumed not to contain the point. The study is held out in pure geometrical manner – it is done using the method of sections and known loci of points on the surface. In the present article we study the locus of points in space defined by metric relation to a certain set of pairs of points, lines and planes. It is shown that any non-degenerate 2nd order surface can be considered as a certain locus of points of space and this interpretation is not unique.

Adsorption of Hexavalent Chromium by Sodium Alginate Fiber Biochar Loaded with Lanthanum

Lanthanun oxide (La2O3) is a lanthanum chemical compound incorporates a sensible anionic complexing ability; however, it lacks stability at a low pH scale. Biochar fibers will give the benefit of their massive space and plethoric uses on the surface to support a metal chemical compound. Herein, wet spinning technology was used to load La3+ onto sodium alginate fiber, and to convert La3+ into La2O3 through carbonization. The La2O3-modified biochar (La-BC) fiber was characterized by SEM, XRD and XPS, etc. An adsorption experiment proved that La-BC showed an excellent adsorption capacity for chromates, and its saturation adsorption capacity was about 104.9 mg/g. The information suggested that the adsorption was in step with both the Langmuir and Freundlich models, following pseudo-second-order surface assimilation mechanics, which showed that the Cr (VI) adsorption was characterized by single-phase and polyphase adsorption, mainly chemical adsorption. The thermodynamic parameters proved that the adsorption process was spontaneous and endothermic. The mechanistic investigation revealed that the mechanism of the adsorption of Cr (VI) by La-BC may include electrostatic interaction, ligand exchange, or complexation. Moreover, the co-existing anions and regeneration experiments proved that the La-BC is recyclable and has good prospects in the field of chrome-containing wastewater removal.

Mathematical Analysis of Heat and Mass Transfer for Williamson Nanofluid Flow Over an Exponentially Stretching Surface Subject to the Exponential Order Surface Temperature and Heat Flux

This article investigates the features of heat and mass transfer for the steady two-dimensional Williamson nanofluid flow across an exponentially stretched surface depending on suction/injection. The boundary conditions incorporate the impacts of the Brownian motion and thermophoresis boundary. The analysis of heat transfer is carried out for the two cases of prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux (PEHF). The ongoing flow problem is mathematically modeled under the basic laws of motion and heat transfer. The similarity variables are allowed to transmute the governing equations of the problem into a similarity ordinary differential equation (ODEs). The solution of this reduced non-linear system of ODEs is supported by the Homotopy analysis method (HAM). The combination of HAM arrangements is acquired by plotting the h-curve. In order to evaluate the influence of several emergent parameters, the outcomes are presented numerically and are plotted diagrammatically as a consequence of velocity, temperature and concentration proles.