Geometric Construction Research Articles

Geometric and topological aspects of the passage from local to global invertibility

As a refinement of the global invertibility problem, we address the issue of estimating the cardinality of a prescribed fiber F-1(q) of a locally invertible map solely in terms of objects that are naturally associated to q itself. The following is a prototypical result. Let F:Rn→Rn be a local diffeomorphism, n≥3, and q∈F(Rn). We show that q is assumed exactly once by F if the pre-image of every 2-plane containing q, when viewed as a geometric surface in Euclidean n-space, is conformally diffeomorphic to R2. The proofs of this and other theorems involve geometric constructions, the Poincaré-Hopf theorem, the Bôcher theorem on positive harmonic functions, condensers on Riemann surfaces, and elliptic estimates. We conclude with a section that is devoted to invertibility problems related to various aspects of dynamics, algebraic and differential geometry, real and complex analysis. The paper is written in a semi-expository style, as an invitation to global injectivity.

A note on geometric construction of spectrally arbitrary zero-nonzero pattern matrices

In this paper, we give a geometric construction for spectrally arbitrary zero-nonzero pattern matrices. The geometric construction also deals with computation of a matrix realization for a given characteristic polynomial.

Geometric constructions of small regular graphs with girth 7

Geometric constructions of small regular graphs with girth 7

Response of Twin Tunnels in Soft Soil with Different Geometric Arrangements and Construction Sequence: A Numerical Study

Response of Twin Tunnels in Soft Soil with Different Geometric Arrangements and Construction Sequence: A Numerical Study

Arrangements, Matroids and Logarithmic Vector Fields

The focus of this workshop was on the ongoing interaction between geometric aspects of matroid theory with various directions in the study of hyperplane arrangements. A hyperplane arrangement is exactly a linear realization of a (loop-free, simple) matroid. While a matroid is a purely combinatorial object, though, an arrangement is associated with a range of algebraic and geometric constructions that connect closely with the combinatorics of matroids.The meeting brought together researchers involved with complementary angles on the subject, many of whom had not met before, so an important underlying objective was to make introductions between groups with overlapping interests in order to facilitate new collaborations and advances in the subject.

Spectral quantization of discrete random walks on half-line and orthogonal polynomials on the unit circle

We define quantization scheme for discrete-time random walks on the half-line consistent with Szegedy’s quantization of finite Markov chains. Motivated by the Karlin and McGregor description of discrete-time random walks in terms of polynomials orthogonal with respect to a measure with support in the segment [-1,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[-1,1]$$\\end{document}, we represent the unitary evolution operator of the quantum walk in terms of orthogonal polynomials on the unit circle. We find the relation between transition probabilities of the random walk with the Verblunsky coefficients of the corresponding polynomials of the quantum walk. We show that the both polynomials systems and their measures are connected by the classical Szegő map. Our scheme can be applied to arbitrary Karlin and McGregor random walks and generalizes the so-called Cantero–Grünbaum–Moral–Velázquez method. We illustrate our approach on example of random walks related to the Jacobi polynomials. Then we study quantization of random walks with constant transition probabilities where the corresponding polynomials on the unit circle have two-periodic real Verblunsky coefficients. We present geometric construction of the spectrum of such polynomials (in the general complex case) which generalizes the known construction for the Geronimus polynomials. In the Appendix, we present the explicit form, in terms of Chebyshev polynomials of the second kind, of polynomials orthogonal on the unit circle and polynomials orthogonal on the real line with coefficients of arbitrary period.

Low Ru doping induced interface and defects engineering in 2D square micro-mesoporous CoNiRuOx nanosieves for advanced oxygen evolution electrocatalysis

Efficient oxygen evolution reaction (OER) catalysts require the reasonable integration of geometric architecture, defects construction and interfacial electronic structure, which is difficult to combine multiple advantages into one low-cost catalysts. Herein, we designed a novel low Ru doping 2D square CoNiRuOx nanosieves (NSs) with abundant surface micropore and mesopore structure, rich oxygen defects and heterophase interfaces. Owing to the Ru incorporation, the electrons in Ni2+ could partially spontaneously transfer to the Ru4+ species by the bridge O2− with π donation effect according to the proposed “Ni-O-Co-O-Ru-O-Ni” electron interaction model. Benefitting from the porous surface with rich mass transfer channel, increased oxygen defects concentration, well-optimized electron redistribution, the CoNiRuOx nanosieves possessed a low overpotential of 261 mV to reach the current density of 10 mA cm−2, which is better than that of counterpart CoNiOx NSs and commercial RuO2 catalysts. The CoNiRuOx NSs also possessed the favorable durability with 50 h. Moreover, the CoNiRuOx//Pt/C electrode couple exhibited enhanced overall water splitting performance. This work provides offers insightful significance to design 2D micro-mesoporous materials for the robust electrocatalysis processes related to energy conversion technologies.

NPTS-PK: A new point kernel code for fast calculation of 3D gamma radiation field

The point kernel integration method is commonly utilized for the analytical calculation of gamma radiation fields in the field of radiation protection and shielding design. This study introduces NPTS-PK, a program developed for rapid calculation of 3D gamma radiation fields based on the NPTS program and the point kernel integration method. Based on the geometric construction and input method of the NPTS program, NPTS-PK supports point kernel calculations for radiation sources and shielding structures of various complex shapes and materials. By calculating material attenuation coefficients using continuous energy cross-section parameters, combined with a more precise buildup factor calculation method, the accuracy of calculation results has been enhanced. Improvements in ray tracing processes and the implementation of the probability neighbor list method have accelerated geometric processing. To address the challenge of excessive computation time associated with large-scale grid counting in point kernel programs, NPTS-PK integrates several efficient acceleration techniques, elevating the speed of radiation field calculation by an order of magnitude. Tests on typical model and engineering scenario demonstrate that the deviation between NPTS-PK results and the reference values is within a few tens of percent, and the computational efficiency and accuracy are improved compared to the standard point kernel programs.

Effects of mastery learning strategy on senior school students’ performance in mathematics in Osogbo, Nigeria

This study aimed to explore the effects of Mastery Learning Strategy (MLS) on senior school students’ performance in Mathematics in Osogbo, Nigeria. The quasi-experimental study’s goals were to investigate how MLS affected mathematics students' performance according to their gender and score levels. A purposive sampling technique was used to select two intact classes of 191 SS II students from two co-educational government senior secondary schools in Osogbo. The Geometric Constructions Test (GCT) which was made up of five theory questions adapted from WAEC mathematics past questions was used as the research instrument. GCT was revalidated by experts from the Department of Mathematics, University of Ilorin. The reliability index of 0.75 was obtained by test–retest method and Cronbach Alpha. Three research questions were raised from which three hypotheses were formulated. Data collected were analyzed using descriptive statistics, independent t-test and Analysis of Covariance (ANCOVA) at 0.05 level of significance. Findings revealed that MLS significantly improved students’ performance (t(189) = 10.22, p < 0.05) while there was a significant difference in the performance of male and female students in favour of males (F(1, 94) = 33.06, p < 0.05). The strategy apparently helped to bridge the gap among students’ scoring levels (F(2, 94) = 6.72, p < 0.05). The study found that, in comparison to the conventional method, MLS greatly enhanced students performance in mathematics. Hence, it was recommended that mathematics teachers should endeavour to adopt MLS in teaching Mathematics, especially in the perceived difficult topics.

Interactions between geometrical forms and microstructural features in culm of square bamboo

Bamboo culms can alter the bamboo’s geometric shape by adjusting the hierarchical organization of anatomical components as a means of adapting to different living conditions. Therefore, a square-like culm has been found commonly in the Chimonobambusa bamboo species. However, the underling mechanism for how these anatomical components assemble into a square culm in the species remains to be considered. Furthermore, the relationship between the geometrical construction of culm and its corresponding organization of anatomical components within also needs clarification. Therefore, the geometrical construction of cross-sections was examined in this work. A super-ellipse based on the Lamé curve was confirmed. Additionally, the transitional zone, at 3/4 in the radial direction, was detected as an inflection point where the geometric parameters clearly changed. Meanwhile, anatomical observation also suggested that the transitional zone can be identified as an inflection point because the fibre morphology difference in circumferential regions becomes more apparent in this area. It is worth mentioning that there is a coherence between the geometrical and microstructural features in circumferential and radial variation. These findings are meaningful to manifest the controlling mechanism of hierarchical structures on the geometrical shape of bamboo culm.

A Comparative Study of Geometric Principles in the Sulba Sutras and the Pythagorean Theorem: Historical Context and Mathematical Applications

This research delves into the geometric principles detailed in the ancient Sulba Sutras and their parallels with the Pythagorean Theorem, traditionally attributed to the Greek mathematician Pythagoras. While the Pythagorean theorem is widely regarded as a cornerstone of Greek geometry, recent studies suggest that its formulation appeared centuries earlier in Indian mathematics, as evidenced in Baudhāyana’s Sulba Sutra (circa 800 BCE). These texts offer extensive insights into the geometric constructions used in Vedic rituals, including the use of Pythagorean triples and transformations between geometric shapes. The Sulba Sutras present a rich mathematical framework, emphasizing practical applications in altar construction, land measurements, and urban planning. This study aims to bridge the research gap by comparing the geometric developments in the Sulba Sutras with Greek formulations, particularly focusing on their mathematical methods, cultural contexts, and historical significance. The findings underscore the advanced state of Indian mathematics and its profound influence on subsequent mathematical traditions, offering a nuanced understanding of the parallel evolution of geometric knowledge across cultures.

New results in stereopsis and Listing's law.

Human eyes' optical components are misaligned. This study presents comprehensive geometric constructions in the binocular system, with the eye model incorporating the fovea that is displaced from the posterior pole and the lens that is tilted away from the eye's optical axis. It extends their previously considered horizontal misalignment with the vertical components. When the eyes' binocular posture changes, 3D spatial coordinates of the retinal disparity (iso-disparity curves), the subjective vertical horopter, and the eye's torsional orientation transformations are visualized in GeoGebra's simulations. The consequences and functional roles of vertical misalignment of the eye's optical components are explained in the following findings: (1) The classic Helmholtz theory, which states that the subjective vertical retinal meridian inclination to the retinal horizon explains the backward tilt of the perceived vertical horopter, is less relevant when the eye's optical components are misaligned. Instead, the lens vertical tilt provides the retinal vertical criterion that explains the experimentally measured vertical horopter inclination. (2) Listing's law, which originally restricts single-eye torsional positions and has imprecise binocular extensions, is formulated for binocular fixations using Euler's rotation theorem. It, however, replaces Listing's plane, which is defined for eyes looking at infinity, with the eyes muscles' natural tonus resting position corresponding to the abathic distance fixation of empirical straight frontal horopter. This new meaning of Listing's plane provides neurophysiological significance that has remained elusive.

A new class of third-order iterative methods for multiple roots and their geometric construction

This work aims to provide a class of third-order iterative methods for solving univariate nonlinear equations with multiple roots when the multiplicity is unknown. To obtain the class of methods mentioned above, a univariate nonlinear equation is used that has the same roots as the original equation. However, the roots of this equivalent equation are simple; that is, they have multiplicity one. Therefore, Gander’s theorem can be used to construct a new class of methods. Then the geometric construction of the elements of said class is done, which satisfies the osculance condition. In addition, some families of methods belonging to said class are presented, as well as their geometric construction. Finally, numerical examples are presented where the behavior of some methods belonging to the families within the method class is observed.

A strategy to fabricate nanostructures with sub-nanometer line edge roughness

Line edge roughness (LER) has been an important issue in the nanofabrication research, especially in integrated circuits. Despite numerous research studies has made efforts on achieving smaller LER value, a strategy to achieve sub-nanometer level LER still remains challenging due to inability to deposit energy with a profile of sub-nanometer LER. In this work, we introduce a strategy to fabricate structures with sub-nanometer LER, specifically, we use scanning helium ion beam to expose hydrogen silsesquioxane (HSQ) resist on thin SiNx membrane (∼20 nm) and present the 0.16 nm spatial imaging resolution based on this suspended membrane geometric construction, which is characterized by scanning transmission electron microscope (STEM). The suspended membrane serves as an energy filter of helium ion beam and due to the elimination of backscattering induced secondary electrons, we can systematically study the factors that influences the LER of the fabricated nanostructures. Furthermore, we explore the parameters including step size, designed exposure linewidth (DEL), delivered dosage and resist thickness and choosing the high contrast developer, the process window allows to fabricate lines with 0.2 nm LER is determined. AFM measurement and simulation work further reveal that at specific beam step size and DEL, the nanostructures with minimum LER can only be fabricated at specific resist thickness and dosage.

Asymptotic safe nonassociative quantum gravity with star R-flux products, Goroff–Sagnotti counter-terms, and geometric flows

Nonassociative modifications of general relativity, GR, defined by star products with R-flux deformations in string gravity consist an important subclass of modified gravity theories, MGTs. A longstanding criticism for elaborating quantum gravity, QG, argue that the asymptotic safety does not survive once certain perturbative terms (in general, nonassociative and noncommutative) are included in the projection space. The goal of this work is to prove that a generalized asymptotic safety scenario allows us to formulate physically viable nonassociative QG theories using effective models defined by generic off-diagonal solutions and nonlinear symmetries in nonassociative geometric flow and gravity theories. We elaborate on a new nonholonomic functional renormalization techniques with parametric renormalization group, RG, flow equations for effective actions supplemented by certain canonical two-loop counter-terms. The geometric constructions and quantum deformations are performed for nonassociative phase spaces modelled as R-flux deformed cotangent Lorentz bundles. Our results prove that theories involving nonassociative modifications of GR can be well defined both as classical nonassociative MGTs and QG models. Such theories are characterized by generalized G. Perelman thermodynamic variables which are computed for certain examples of nonassociative geometric and RG flows.

Multi-body dynamics modeling and energy consumption optimization of electromagnetic mechanical fully variable valve system

Electromagnetic Mechanical Variable Valve Actuation (EMVVA) technology has the capability to improve engine performance and decrease pollutant emissions. A new type of EMVVA composed of electromagnetic driver and mechanical transmission is proposed to realize the flexible adjustment of valve parameters whether the valve is open or closed process. The geometric mathematical model of the mechanical transmission is developed based on the motion laws of the valve and the geometrical construction of the mechanical transmission, and the conjugate cam curve is solved. A multi-body dynamic model is constructed to calculate the driving torque and energy consumption needed by the mechanical transmission based on the mass, rotational inertia, centroid position of the parts, combining the normal contact force model and the friction model of the clearance contact pair at the same time. Based on the geometric model and multi-body dynamics model, the test platform is established. The test results demonstrated that within the specified valve lift, EMVVA system could accomplish variable valve lift, variable valve timing, and variable valve duration at maximum lift. The maximum timing error does not exceed 1° crank angle, and the maximum valve lift error does not exceed 0.25 mm at 11 mm valve lift. In addition, the error of driving torque between the multi-body dynamics model and the test is less than 0.3 [Formula: see text].Low energy optimization of mechanical transmission was completed using the NSGA-II method and multi-body model. According to the results of the optimization, energy consumption was reduced by 14%, and the peak driving torque was decreased by 54.3%.

ВЛИЯНИЕ ТОЧЕК ПОВОРОТА НА ЗАДЕРЖКУ РЕШЕНИЯ ВБЛИЗИ НЕУСТОЙЧИВОГО ПОЛОЖЕНИЯ РАВНОВЕСИЯ

This paper considers a system of equations consisting of three first-order equations of fast variables and one equation of a slow variable. The first approximation matrix of the system of fast variables has different eigenvalues, with one pair of complex conjugates. The eigenvalues have zeros and they are turning points. The system has an equilibrium position, and the stability of the equilibrium position is lost at a certain value of the slow variable. The problem is set to study the delay in solving a system of fast variables near an unstable equilibrium position and the influence of turning points on the delay. The delay of the solution near an unstable position is proven and the influence of turning points is clarified. The proof used geometric constructions using the level line of harmonic functions, methods of successive approximations, Laplace, stationary phase, integration by parts and comparison of infinitesimals.

Iterative Mathematical Models Based on Curves and Applications to Coastal Profiles

The objective of this study is iterative systems based on general types of curves, not only on circumference arcs. We begin by presenting some implementations and generalizations of constructions based on arcs of circumference. Then we consider constructions based on general curves and give a “universal property” relating to the primary construction that exploits arcs of circumference. With the prospect of applying these theoretical models also to coastal geomorphology in the future, and inspired by one of the best-known models on the subject, the logarithmic spiral one for the so-called headland-bay beaches (HBBs), we study geometrically some cases in which the constructions are based on arcs of the golden spiral. Simultaneously we concretely illustrate and explain the universal property above. Finally we dedicate a section to discuss the possibility of how to numerically evaluate and compare the (infinite) lengths originating from our theoretical geometric constructions. Some explicit examples, calculations and comparisons will be provided by the use of infinity computing which is one of the various possible assets that contemporary non-standard mathematics makes available.

A multi‐phase mechanical model of biochar–cement composites at the mesoscale

AbstractThis study presents a five‐phase mesoscale modeling framework specifically developed to investigate crack propagation and mechanical properties of biochar–cement composites. The multi‐phase model includes porous biochar particles with precise geometric construction, sand aggregates, cement matrix, and interfacial transition zone adjunct to both the biochar particles and sand aggregates. The 3D porous biochar library was first proposed and established in this study, which could provide an external interface for describing different pore shapes, wall thicknesses, and pore areas. All the simulation results were experimentally validated using a digital image correlation. Through precise geometric modeling, the unique failure modes and timing of biochar particles within the mortar were identified. This is analogous to the “strong column–weak beam” concept, accounting for the enhanced ductility observed in the biochar–cement composites under compression test. This work can advance the geometric modeling of porous aggregates broadly and elucidate their mesoscopic failure mechanisms in cementitious materials, thus providing new insights for developing high‐ductility and lightweight cement composites.

Assessing the quality of conceptual knowledge through dynamic constructions

In this contribution, we address the gap that has appeared in mathematics education research and practice with the emergence of dynamic geometry environments and build on the opportunities these environments offer to school geometry. In our qualitative empirical study, we investigate how to elaborate on the general model of conceptual knowledge to make it applicable to dynamic geometry tasks, specifically to tasks including dynamic geometric constructions. We present a design of dynamic constructions of quadrilaterals that comply with Euclidean constructions, derive an assessment instrument based on them, and study what information the instrument can provide about the quality of students’ conceptual knowledge. We present the results in the form of an assessment framework consisting of an example of the assessment instrument and an ordered system of qualitative categories serving as an assessment codebook for interpreting students’ responses in terms of the quality of conceptual knowledge. To clarify the relations between the assessment framework and the general model of conceptual knowledge, we establish a system of subdimensions of conceptual knowledge that indicates how conceptual knowledge can be understood in the context of dynamic geometric constructions and identifies the conceptual knowledge needed to achieve individual categories of the assessment framework.