Geometric Construction Research Articles

Adsorption on a Spherical Colloidal Particle from a Mixture of Nanoparticles with Competing Interactions.

Adsorption of nanoparticles on a spherical colloidal particle is studied by molecular dynamics simulations. We consider a generic model for a mixture of nanoparticles with energetically favored self-assembly into alternating layers of the two components. When both components are attracted to the colloidal particle, the adsorbed nanoparticles self-assemble either into alternating parallel tori and clusters at the two poles of the colloidal particle, or into alternating spirals wrapped around the spherical surface. The long-lived metastable states obtained in simulations follow from the spherical shape of the adsorbing surface and the requirement that the neighboring chains of the nanoparticles are composed of different components. A geometrical construction leading to all such patterns is presented. When the second component particles are repelled from the colloidal particle and the attraction of the first component is strong, the attracted particles form a monolayer at the surface of the colloidal particle that screens the repulsion of the second component. The subsequent adsorbed alternating spherical layers of the two components form together a thick shell. This structure leads to the adsorption that is larger than in the case of the same attraction of the two components to the colloidal particle.

General birationality and hyperelliptic theta divisors

AbstractWe first state a condition ensuring that having a birational map onto the image is an open property for families of irreducible normal non uniruled varieties. We give then some criteria to ensure general birationality for a family of rational maps, via specializations. Among the applications is a new proof of the main result of Catanese and Cesarano (Electron Res Arch 29(6):4315–4325, 2021) that, for a general pair (A, X) of an (ample) Hypersurface X in an Abelian Variety A, the canonical map $$\Phi _X$$ Φ X of X is birational onto its image if the polarization given by X is not principal. The proof is also based on a careful study of the Theta divisors of the Jacobians of Hyperelliptic curves, and some related geometrical constructions. We investigate these here also in view of their beauty and of their independent interest, as they lead to a description of the rings of Hyperelliptic theta functions.

Geometric construction of relativistic and non-relativistic distortion

To an observer moving through a scene with a relativistic velocity, the scene appears distorted. Here we present a simple geometric interpretation of this relativistic distortion. This interpretation is exact for pinhole cameras whose shutter coincides with the pinhole, and it can easily be modified to take into account time-of-flight effects only that lead to a “non-relativistic distortion,” allowing a separation of the relativistic distortion into time-of-flight effects and “truly relativistic” effects. We make use of our construction in Einstagram, a web app designed for running on mobile devices that displays the video streams from the device cameras, relativistically or non-relativistically distorted for an arbitrary, user-defined, observer velocity. Our work facilitates an intuitive understanding of relativistic distortion.

TWISTED ACTIONS ON COHOMOLOGIES AND BIMODULES

AbstractFor closed subgroups L and R of a compact Lie group G, a left L-space X, and an L-equivariant continuous map $A:X\to G/R$ , we introduce the twisted action of the equivariant cohomology $H_R^{\bullet }(\mathrm {pt},\Bbbk )$ on the equivariant cohomology $H_L^{\bullet }(X,\Bbbk )$ . Considering this action as a right action, $H_L^{\bullet }(X,\Bbbk )$ becomes a bimodule together with the canonical left action of $H_L^{\bullet }(\mathrm {pt},\Bbbk )$ . Using this bimodule structure, we prove an equivariant version of the Künneth isomorphism. We apply this result to the computation of the equivariant cohomologies of Bott–Samelson varieties and to a geometric construction of the bimodule morphisms between them.

Mathematical Software: A Panacea for Improving Senior Secondary School Students’ Retention in Geometrical Constructions

When attempting to address the persistently poor performance of students in mathematics, researchers have typically examined factors such as retention to identify long-term solutions. The objective of this study was to improve the retention of senior secondary school students in geometrical constructions through mathematical software. Despite the many benefits of geometry to both the learner and society, the performance of the students continued to be abysmal. Could it be that the students do not retain what they learn? The research was conducted in Benue State, Nigeria, and was based on John Sweller’s cognitive load theory of multimedia. The study’s design utilised a non-randomised, quasi-experimental control group with a sample size of 457 students. Three objectives, research questions and research hypotheses led the study, with the Geometrical Construction Achievement Test and the Geometrical Construction Retention Test as the instruments for data collection. The mean and standard deviation of descriptive statistics were used to analyse the data that was gathered. The research reveals that retention was higher in the experimental group (Mean = 65.88) than in the control group (Mean = 51.87). The study found a statistically significant improvement in retention (p ˂ 0.05). No gender variations were found when the students were taught with mathematical software. Mathematics teachers’ utilisation of mathematical software as a strategy for teaching geometrical constructions among senior secondary school students was recommended.

Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals

Abstract We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

The Geometric Construction of Hexagonal Apses in Mediaeval Hungary

Geometric construction in Gothic architecture has been a popular subject of research for centuries. The use and extent of quadrature and triangulation grids has long been debated, as the few architectural drawings that survived from the era do not indicate the construction process. Modern surveying methods, however, allow us to inspect the exact geometry of a building with all its irregularities – in the case of this article we use 3D point clouds created by laser scanning, from which accurate measurements can be taken.In our study we analyse four churches from mediaeval Hungary: the Franciscan church of Szeged-Alsóváros, the fortified church of Cincu (RO, Nagysink/Großschenk), the Franciscan church of GyöngyösAlsóváros and the fortified church of Mediaș (RO, Medgyes/Mediasch). Each of these are built with elon-gated choirs and approximately hexagonal apses. We explore the possible construction systems of these buildings and compare them to each other. We aim to present the similarities and differences between them and to offer likely explanations of their irregularities. Moreover, we propose the significance of construction circles used for creating a triangulation grid.

Data-driven calibration of decoupling matrix for MIMO precision motion stages

Decoupling control is a commonly employed technique for achieving high precision in multiple-input multiple-output (MIMO) motion control systems. A static decoupling matrix, which can be determined using geometric construction, is widely used due to its practicality and simplicity. However, inaccurate geometric parameters will lead to a coarse decoupling matrix, and result in interactions among the system axes and performance deterioration. To tackle this challenge, various attempts have been undertaken to calibrate the decoupling matrix. Data-driven on-line approaches have gained considerable attention for their ability to calibrate the decoupling matrix without interrupting the normal operation of the system. This paper presents a data-driven approach to calibrate the decoupling matrix for MIMO and linear time invariant (LTI) systems. Through some reasonable assumptions, a calibrated static decoupling matrix can be derived to improve the performance of the system. Moreover, considering the inevitable presence of measurement noise, the consistency of the proposed method has been analyzed. As a result, the instrument variable is introduced in the improved method to eliminate the impact of the measurement noise. Finally, the effectiveness and practicality of the proposed method are demonstrated through both numerical simulations and experiments carried out on an ultraprecision wafer stage.

Vertical displacement of a non-Newtonian Bingham plastic by a Newtonian phase in an axially composite reservoir

In general, oil reservoirs may consist of composite sedimentary structures composed of materials such as sand, clay, or limestone, which exhibit varying lithology due to sedimentary processes. A comprehensive knowledge of this lithology is essential for accurately assessing their hydrocarbon storage and production capacity. Additionally, this information is indispensable for the implementation of various recovery techniques such as waterflooding, gas injection, surfactant injection, polymer injection, alkaline water solution injection, and others. Highly viscous oil can exhibit non-Newtonian behavior during water injection in certain cases. The use of surfactants, alkaline, or polymer solutions in enhanced oil recovery also introduces non-Newtonian behavior. Recovery methods face challenges when non-Newtonian phases, gravity, and reservoir heterogeneity are combined. Against this backdrop, this work presents a mathematical model for immiscible two-phase flow in an axially composite reservoir with a periodic-layered structure. The model considers a non-Newtonian plastic Bingham-type phase extension of the Buckley-Leverett model. To address the porous medium’s heterogeneity with discontinuous flux functions, the numerical solutions were obtained using the Lax-Friedrichs and Lagrangian-Eulerian schemes. The numerical solutions were compared to analytical solutions obtained using an extended version of Oleinik’s geometric construction for discontinuous flux functions. The outcomes display shock and rarefaction waves, as well as a fixed shock due to the porous medium heterogeneity. The numerical results closely correspond with the analytical solutions, seen particularly in the greater accuracy of the Lagrangian-Eulerian method compared to the Lax-Friedrichs method.

Study on structural and material dependence of Rhodium Self-powered Neutron Detector performance

The neutron measurement properties of Rh-SPNDs are greatly affected by their geometric constructions, which needs to be optimized while designing. This effect is reflected in the sensitivities for gamma and neutron detection. However, the influence on gamma sensitivity has not been fully analyzed before, which is of the same importance as neutron sensitivity. Therefore, it is necessary to quantitatively analyze the relationship between detector sensitivity and geometric dimensions. The effect of Rh-SPND structural and material parameters on neutron and gamma sensitivities has been calculated and analyzed through numerical simulations. The research results show that detector sensitivities are strongly depending on the detector size and its emitter shape. Moreover, the detector response to exogenous reactor gamma can be eliminated by optimizing the detector collector size.

Geometrization of the Huygens–Fresnel principle: Applications to Fraunhofer diffraction

The classical wave theory can trace its historical origins to the seminal works of Christian Huygens, Thomas Young, and Augustin Fresnel. To explain some of light’s observed properties, such as rectilinear propagation, reflection, and refraction, Huygens proposed a simple geometrical construction of secondary spherical wavelets with centers of disturbance located on a primary wavefront. More than a century later, Young formulated the law of interference to both predict the formation of fringes in his now famous double slit experiment and also to estimate the wavelengths associated with different colors. A decade after that, Fresnel combined Huygens’ construction with Young’s interference law to qualitatively and quantitatively describe diffraction, which is the bending of light upon encountering an obstacle or an aperture. This grand synthesis, called the Huygens–Fresnel principle, acts as a powerful pictorial aid and conceptual tool that can describe a wide variety of complicated optical phenomena. However, the applications of the principle and its later developments, such as the Kirchhoff–Fresnel integral, are strewn with several simplifying assumptions and approximations that are aimed at minimizing the mathematical challenges involved. Consequently, two distinct formalisms are necessary to account for diffraction effects when the source of light or observation screen is placed nearby and far away from the aperture or obstacle. Recently, a hyperbola framework for analyzing wave interference at a multi-slit barrier was shown to successfully circumvent all conventionally imposed ad hoc conditions. The method commences directly from the Huygens–Fresnel principle and the ensuing predictions pertaining to the distribution of fringe characteristics, namely, positions, widths, and intensities on a detection screen can, therefore, justifiably claim accuracy in both the near field (Fresnel regime) and the far field (Fraunhofer regime). In this paper, the analysis that was previously carried out for the special case of slits of negligible widths is further extended to encompass slits of finite widths as well.

A generalized model for tribovoltaic nanogenerator

Converting mechanical energy into direct-current electric power based on the tribovoltaic effect is a typical characteristic of tribovoltaic nanogenerators (TVNGs). Although this newly discovered physics effect has been devoted to numerous research studies recently, a generalized theoretical model is still missing, thus unable to comprehensively elaborate the working principles of TVNG. Unlike previous qualitative explanations restricted to the conventional diffusion-drift theory, a new theoretical model is proposed according to classical semiconductor physics. Using the model, the governing equation of a TVNG is derived for the first time, which provides possibilities for revealing the variations of basic physical variables whether within the device or in an external circuit. The direct-current output is suggested to be the coupling of the tribovoltaic effect and contact electrification; in detail, it directly results from the movement and realignment of quasi-Fermi levels for excess carriers that are contiguous to the junction/contacting interface under non-equilibrium conditions. Moreover, an equivalent circuit model is established, equivalent to a constant current source parallel to a p–n junction diode according to the lumped parameter circuit theory. Notably, a new term, mechano-induced electric field EM, is defined and introduced to describe the impact of triboelectric charges at interfaces. Furthermore, using the COMSOL Multiphysics software, a dynamic simulation model for TVNGs is proposed, allowing the simulation and calculation of various TVNGs with different geometric constructions and charge distributions.

On the string landscape without hypermultiplets

In this work we study interesting corners of the quantum gravity landscape with 8 supercharges pushing the boundaries of our current understanding. Calabi-Yau threefolds compactifications of F/M/type II theories to 6, 5 and 4 dimensions are the most prominent examples of this class, and these always lead to a universal hypermultiplet coming from the volume/string coupling constant. We find new examples of asymmetric orbifold constructions which have no hypermultiplets in 4 or 5 dimensions and no neutral hypers in 6d. We argue that these theories can also be obtained by going to strong coupling/small volume regions of geometric constructions where a new Coulomb branch opens up and moving in this direction freezes the volume/string coupling constant. Interestingly we find that the Kodaira condition encountered in geometric limits of F-theory compactifications to 6 dimensions is violated in these corners of the landscape due to strong quantum corrections. We also construct a theory in 3 dimensions which if it were to arise by toroidal compactifications from 5d, it would have to come from pure N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with no massless scalar fields.

Outer Strong Blocking Sets

Strong blocking sets, introduced first in 2011 in connection with saturating sets, have recently gained a lot of attention due to their correspondence with minimal codes. In this paper, we dig into the geometry of the concatenation method, introducing the concept of outer strong blocking sets and their coding theoretical counterpart. We investigate their structure and provide bounds on their size. As a byproduct, we improve the best-known upper bound on the minimum size of a strong blocking set. Finally, we present a geometric construction of small strong blocking sets, whose computational cost is significantly smaller than the previously known ones.

Experimental study on convective heat transfer characteristics of fluoride salt outside the tube bundle in hairpin heat exchanger

Convective heat transfer performances of fluoride-salt flowing outside U-tubes with 4871 < Re < 12298 and 11 < Pr < 17 in hairpin heat exchanger are experimentally investigated. The effects of temperature on the heat transfer coefficient of fluoride-salt are studied with the temperature 540–598.7 °C. Results show that the measured Nusselt number is in line with the empirical heat transfer correlation, but it is slightly larger than that predicted by conventional correlations and the deviations increases as Prandtl numbers increase. It is presumably that the special geometric construction with densely packed bundles and U-bend in hairpin heat exchanger disrupt the boundary layer resulting in the formation of back-flow zone and then strengthen the heat transfer. Therefore, previous correlations are insufficient in describing the fluoride-salt performance. Finally, based on the model of Sieder-Tate correlation, an improved correlation considering a comprehensive heat transfer enhance is proposed, and shows a good agreement with the experiment data with deviation less than 4.8 %.

SMS2DC: Synchronous mobile sinks scheduling for data collection in internet of things‐enabled wireless sensor networks

SummaryEnergy‐efficient data collection in wireless sensor networks (WSNs) is crucial due to the limited battery capacity of sensor nodes (SNs). Using a mobile sink (MS) for data collection can lower the energy consumption of SNs to avoid relaying in WSNs. However, a single MS is not a feasible solution for large‐scale WSNs, so it was necessary to use multiple MSs to collect data. A synchronous MS scheduling strategy for data collection (SMS2DC) is proposed in this paper, which uses two types of MS, a local MS to collect data from SN and a global MS to collect data from local MS. In this process, we begin by partitioning the network based on chemical reaction optimization. For each partition, a MS is assigned and scheduled using a path construction strategy according to a geometric path construction approach. In addition, a global MS is scheduled based on a local MS trajectory by identifying the most appropriate collision point to collect data. As a result, the algorithm increases data collection accuracy while minimizing network data loss. The asymptotic time complexity of the proposed SMS2DC algorithm needed . The comparison results show the superiority of the proposed SMS2DC strategy under multiple scenarios under various deployment conditions.

Implementation of analytic projective geometry for computer graphics

In their research, the authors actively exploit different branches of geometry. For geometric constructions, computer algebra approaches and systems are used. Currently, we are interested in computer geometry, more specifically, the implementation of computer graphics. The use of the projective space and homogeneous coordinates has actually become a standard in modern computer graphics. In other words, the problem is reduced to the application of analytic projective geometry. The authors failed to find a computer algebra system that could implement projective geometry in its entirety. Therefore, it was decided to partially implement computer algebra for visualization of algebraic relations. For this purpose, the Asymptote system was employed.

A 3D Descriptive Geometry Problem-Solving Methodology Using CAD and Orthographic Projection

In solving descriptive geometry (DG) problems, board (i.e., hand drawing) methods are frequently used, despite this discipline is still very important to enhance spatial vision. These methods are very different from CAD tools which are used in the field of design. CAD facilitates the realization of geometric constructions and transformations (i.e., rotation, translation, copying, scaling, alignment, and symmetry, among others) are performed analytically. For this reason, a 3D DG problem-solving methodology using CAD and orthographic projection (CADOP) is introduced. Once the principles of DG, orthographic projection, and CAD fundamentals and tools are described, CADOP is applied to obtain (i) orthogonal views; (ii) principal lines of a plane; (iii) true-size view of a plane; (iv) parallelism, perpendicularity, and distance, and (v) angles. Considering the user coordinate system in CADOP allows one to place the horizontal plane in the suitable position to solve DG problems directly in one step. In the traditional methods, the use of auxiliary views must be carried out in several steps instead. The dynamic management of the 3D view of the scene is facilitated in CADOP, improving its understanding, and achieving the precision inherent in analytical calculations.

On the Banach–Mazur Distance in Small Dimensions

AbstractWe establish some results on the Banach–Mazur distance in small dimensions. Specifically, we determine the Banach–Mazur distance between the cube and its dual (the cross-polytope) in $$\mathbb {R}^3$$ R 3 and $$\mathbb {R}^4$$ R 4 . In dimension three this distance is equal to $$\frac{9}{5}$$ 9 5 , and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.

Geometric space construction method combined of a spline-skinning based geometric variation method and PCA dimensionality reduction for ship hull form optimization

In the construction stage of ship hull geometric space, the representation and variation of hull forms pose challenges due to the complex nature of the geometry and the necessity for enhanced flexibility in space. In this study, we utilize T-spline technology for the representation of ship hulls, enabling a unified parameter domain with a reduced number of control points to effectively depict the entire hull surface. Furthermore, we introduce a method for geometric variation based on the T-spline parameter domain, which permits both local and global alterations, thus offering greater flexibility in modifying the geometry. By leveraging the T-spline representation and the geometric variation method, we developed a hybrid geometric approach for constructing the geometric space, which significantly increased the geometric flexibility in ship hull optimization. A geometric space characterized by a high degree of geometric freedom typically requires the use of numerous optimization variables and incurs considerable computational resource consumption. In this research, we apply a linear dimensionality reduction method, principal component analysis, to decrease the dimensionality of the geometric space. This method effectively preserves the original geometric design space, thereby eliminating the need for performance analyses. The numerical optimization results clearly indicate that the proposed method significantly enhances optimization efficiency.