Special Angle Research Articles

CONSTRUCTION OF ANY ANGLE WITHOUT PROTRACTOR

Geometry is one of the foremost significant chapters in mathematics. The construction of angles is one of the important topics of geometry. In geometry, the term construction refers to the precise sketching of shapes, lines, or angles using mathematical instruments. To construct angles, all you'll need is a compass, a ruler, and a pencil, as well as a protractor to measure the angles you've constructed. As there are no numbers involved, this is the simplest type of geometric building. We can only draw some special angles without using a protractor, such as 30, 60, 90, 120, 180, and non-special angles with 1 degree deviation from the needed angles, such as 83, 72, 68, and so on. In our lower secondary school, the method of constructing special angles is commonly taught. We can, however, draw any angle between 0 and 360 degrees using the approach we devised, with no variation from the needed angle. Construction of angle is essential in math because it may be applied to the construction of other geometric shapes, particularly triangles. Keywords: Angle; Intersection Points; Reference Arc; Primary arc

Far-field reactivation of natural fractures by stress shadow effect

Hydraulic fracturing is the most important means to achieve the reservoir stimulation of tight formations like shale, where the reservoir permeability is enhanced after the interaction between hydraulic fracture (HF) and natural fractures (NFs). Apart from the widely known direct intersection behaviors between HF and NFs (i.e., the HF meets the NFs), the NFs may also be reactivated by the stress shadow effect of HF, but the failure mechanism and reactivation modes are still unclear. To address this, the far-field reactivation of NFs under the stress shadow effect is described based on a hybrid phase field model (PFM) with the frictional contact criterion, where the open or slip conditions can be quantitatively calculated using the Mohr-Coulomb yield function along the contact interface of NFs. The zone with reactivated NFs is divided into mode II and mixed mode dominated sub-zones respectively, and the phase diagram of reactivation behaviors is summarized as variations of the location and angle of the NF. The NF reactivation modes are discussed under different geological conditions (strength of NFs, distribution angle of NFs, and initial stress difference). It could be found that the higher initial stress ratio leads to more mode II reactivated NFs, and NFs strengths and angles influence both mode II and mixed mode zones notably. The NF reactivation tends to take place with lower strength, higher injection rate, higher initial stress ratio, and special angle (e.g., HF are perpendicular to NFs), during which the permeability enhancement and reservoir stimulation are evaluated by the phase field model.

Mathematical results on the chiral models of twisted bilayer graphene (with an appendix by Mengxuan Yang and Zhongkai Tao)

The study of twisted bilayer graphene (TBG) is a hot topic in condensed matter physics with special focus on magic angles of twisting at which TBG acquires unusual properties. Mathematically, topologically non-trivial flat bands appear at those special angles. The chiral model of TBG pioneered by Tarnopolsky, Kruchkov, and Vishwanath (2019) has particularly nice mathematical properties and we survey, and in some cases, clarify, recent rigorous results which exploit them.

Micro Learning-based Module: A Teaching Tool to Enhance Students’ Competence in Grade 9 Trigonometry

Globally, as the education field gear towards recovery from the Covid 19 pandemic, there is a must to adapt teaching strategies that will equip the teachers to continuously provide quality and engaging activities in a simplified manner that will bring out the best of the students. This paper describes the effectiveness of microlearning-based module as a teaching tool in enhancing students’ competence in Trigonometry. Data analysis gave results that there was mean difference of of 4.46 in illustrating, 5.02 for identifying, and 4.5 in solving after being exposed to the created learning material. The posttest result suggests that microlearning-based useful tool for enhancing students’ competence of the 9th – grade students in Trigonometry. Level. The hypotheses that claimed there was no significant difference between mean scores in the pretest and post- test of the three competencies in Trigonometry was negated. According to the results, students’ competence in Trigonometry were enhanced in terms of mean scores. Future student performance in the three competencies in Trigonometry can be projected to be better and more satisfying if this trend holds if it does. The created learning material can be credited with enhancing students' competence in illustrating the following topics: the six trigonometric ratios, the trigonometric ratios of special angles, and angle of elevation and angle of depression. Also, the microlearning module can improved the competency of identifying the trigonometric ratio that can be used in relating the angles and sides of a right triangle. The designed learning material can help the students enrich their solving skills such as solving the missing parts of a right triangle, evaluating trigonometric expressions of involving special angles, and problem solving involving right triangle, angle of elevation, and angle of depression. Hence, it is highly recommended that the school use this learning material to encourage effective teaching and quality learning.

Flat-bands in translated and twisted bilayer Penrose quasicrystals

Correlated phases in Moiré materials together with the flat-bands in twisted systems play a central role to explain superconductivity in the new twisted bilayer graphene. In this paper, flat-bands are shown to exist in both translated and twisted bilayer of quasicrystals. Such flat-bands arise for different displacements and twisting angles of two-coupled Penrose lattices where Moiré patterns are also shown. Moiré patterns analyzed in this work have at least two inverted worms showing an interference pattern going along the five-fold axes of the pentagon. In order to analyze the behavior of the flat band, our study has been done for fixed interference worm directions but increasing the worm interference density, and for fixed worm interference density but increasing the number of worm directions. In case of rotations, the Moiré patterns that occurs for special angles such as π/5, 2π/5, 3π/5, 4π/5 and π are discussed in detail because they clearly show flat-bands along with quasicrystalline electronic states at the Fermi level.

Facile and rapid synthesis method of dielectric nanocomposite for simultaneous enhancement of electromagnetic wave absorbing/shielding characteristics

The present study involves the fabrication of poly[4-(2-thiophene)aniline] (PTA) nanocomposite for absorption-dependent shielding applications, demonstrating enhanced absorption and small reflection properties. The Microwave absorption characteristics of the fabricated polymer were studied at different loading ratios, resulting in a minimum reflection loss (RLmini) of − 44 dB for the 8 wt % loading ratio. The macromolecular architecture of the PTA material offers higher dielectric loss resulting from higher conduction and interfacial loss. Additionally, the shielding performance is also evaluated, and the sample containing 8 wt % filler demonstrates an impressive − 36.5 dB shielding efficacy. The performance of polymers is significantly impacted by their unit periodic order arrangement. Conducting polymers with linear structure and special π-orbitals angles possess unique characteristics in comparison to their branched and the random copolymer, making them highly sought after as efficient organic conductors, magnetic substances, electrochromic materials, and high dielectric agents. Monomer units contained two periodical cyclic structures and the creation of the inductive effects of heteroatoms elongates the conjugation and reduces the energy bandgap difference between the HOMO and LUMO levels, enabling smoother electron flow and enhancing absorption capacity. The findings indicate that PTA nanocomposite is a promising radar-absorbing coating material due to its superior absorption ability and outstanding shielding performance.

Exploring edge states in square–octagon moiré lattice

Moiré lattices, achieved by the superposition of two or more twisted identical periodic lattices, are of interest to various fields because they provide additional degrees of freedom. Here, we theoretically and experimentally study the edge states in a square–octagon moiré lattice. This moiré lattice is created by superimposing two identical square sublattices with an antiphase and a special twist angle. Five different edges, named type-I zigzag edge, type-II zigzag edge, type-I bearded edge, type-II bearded edge, and armchair edge, are explored. Through band structure analysis and numerical simulation of edge excitation, we find that all five edges support edge states. The topological property of the type-I edge states is verified by calculating the 2D polarization of the lattice. Furthermore, the edge mode distribution manifests that multiple bands support identical edge states at the armchair and type-II bearded edges. In the experiment, the moiré lattice is generated by the CW-laser-writing technique, thereby observing all the edge states with corresponding edge excitations. This study broadens the understanding of edge states in the coupled moiré photonic lattices and provides a new platform for exploring topological physics.

Sources for the Study of Occupation Policy Regarding the Child Population in the Occupied Territories of the RSFSR (1941—1944)

The Great Patriotic War is a broad and multifaceted topic. One of the interesting vectors for studying this period is life in the USSR occupied by Nazi Germany and its allies. Behind the term “civilian population”, even in Soviet historiography, the concept of old people, women and children was formed. However, the issue of children, as the future generation subjected to the occupation of the USSR, is extremely important to consider from a special angle, since it is through the prism of the attitude of the Nazi leadership towards Soviet children, first of all, that Germany’s real intentions can be determined in relation to the future of the peoples of the USSR. This article examines sources stored in archives on the territory of the Russian Federation, collections of documents published in the USSR and the Russian Federation, as well as Internet projects developed in the Russian Federation dedicated to the digitization and placement of thematic materials.

Interlayer angle dependence of photoelectric properties of Sb/SnC van der Waals heterojunction and its application

The discovery of novel properties in twisted bilayer graphene has opened up new avenues of research in physics and materials science, making the twistronics a new research hotspot. In this paper, based on two-dimensional tin-based materials and antimonene monolayers, six types of Sb/SnC two-dimensional van der Waals heterostructures (vdWH) with different interlayer twist angles are constructed, and their optoelectronic properties and applications are studied by first-principles calculations. All modeling and calculations are performed using the density functional theory (DFT) software Quantum-ATK. The results show that the Sb/SnC vdWHs with six different interlayer twist angles have various band gaps, and when the interlayer twist angles are 10.89°, 19.11°, 23.41°, and 30°, the Sb/SnC vdWH exhibit a type-I band edge alignment, while at 8.95° and 13.59°, they present a type-II band structure. The results of the orbital-projected band structures of the Sb/SnC vdWHs reveal that the variation in interlayer twist angles changes the atomic stacking in the heterostructures, thereby modifying orbital coupling and further tuning the electronic structure of the heterostructures. Additionally, the calculated absorption spectra indicate that comparing individual Sb and SnC monolayers with Sb/SnC vdWHs, the latter’s absorption coefficient <i>r</i> is significantly enhanced in the visible light region, and the optical absorption characteristics of the heterostructures with different interlayer twist angles vary markedly. In terms of applications, as materials for solar cells, the Sb/SnC vdWHs with interlayer twist angles of 8.95° and 13.59° exhibit photovoltaic conversion efficiencies of 17.48% and 18.59%, respectively; as photocatalysts for the complete water splitting, the Sb/SnC vdWH with an interlayer twist angle of 8.95° can catalytically decompose water across a pH range of 0–2, while a twist angle of 13.59° confines its catalytic activity to a pH value between 0 and 1. Therefore, Sb/SnC van der Waals heterostructures have special rotation angles and have multifunctional application prospects in the fields of solar energy and photocatalysis. More importantly, our research demonstrates that in addition to traditional methods such as strain, doping, and defects, adjusting the interlayer twist angle provides a new degree of freedom for manipulating the optoelectronic properties of materials.

Эпопея Жанны д’Арк глазами историка права (О новой книге Клод Говар)

The article is devoted to the analysis of a monograph written by Claude Gauvard, professor emeritus at the Panthéon-I-Sorbonne University, a former student of Bernard Guenée and a recognized specialist in political history and the history of medieval law of the Western Europe. In her new work the researcher turned to the history of Joan of Arc, and despite the fact that in the last two decades the études johanniques have been experiencing a genuine flourishing and it is already difficult to find a plot that would not receive coverage in historiography, Claude Gauvard chose a very special angle for this topic. The author of the article examines in detail the formulation of the problem proposed in the monograph – to study the history of the national French heroine from the point of view of the legal ideas of Europeans of the late Middle Ages; to show exactly which topics formed the basis of the reputation of Joan of Arc and of her subsequent perception by contemporaries; to reveal the purely legal aspects of the Maid’s relations with her supporters and opponents, and above all, of the assessment of her activities during the indictment process of 1431 and the rehabilitation process of 1455–1456.

Write in text, or Return to the textual inkwell

Review of the book by the famous philologist O.V. Bogdanova “Petersburg text of Russian literature (second half of the twentieth century)”, in which the work of Andrei Bitov, Sergei Dovlatov, Mikhail Kuraev, Tatyana Tolstoy, Joseph Brodsky, Victor Pelevin was “registered” in the modern “Petersburg text” of Russian literature. It is shown how in the book each of the writers from a special angle sees St. Petersburg-Petrograd-Leningrad, the great city in its special incarnation — artistic, ironic, mystical, fairy-tale-mythological, philosophical, visionary, which predetermined the essayistic style of the review.

The chiroptical properties of plasmonic dimer metamaterials based on the extended Born–Kuhn model

Plasmonic dimer metamaterials have been widely utilized in the study of chiral nanophotonics due to their special chiroptical properties. The chiroptical properties of simple rod-like dimer metamaterials can be intuitively described by the Born–Kuhn (BK) model, in which the coupling parameter between oscillators is commonly used as a constant. However, the BK model is limited for complex dimer metamaterials because the behavior of the oscillators is constrained by the geometry of the structure. In this paper, inspired by equivalent LC-coupled circuits, we develop the BK model to predict the chiroptical properties of the split-ring resonator (SRR) dimer metamaterials, where the coupling of oscillators depends on both the twist angle of the SRR dimers and the frequency of the incident light. Using the extended BK model, we predict that there exists a special twist angle, which will cause the chirality of the SRR dimer metamaterials to reverse. More importantly, in such specific spatial configurations, the SRR dimer metamaterials with rectangular slits not only retain the enhancement of the chiral near-field, but also limit the chiroptical far-field response because the interactions of electricity and magnetism cancel each other out. In addition, combined with simulation calculations, we find that the rectangular SRR dimer metamaterials have strong robustness within a certain range of structural regulation. Our work will be helpful for the design and optimization of optical logic elements, chiral sensor components, and coding devices.

Using indigenous artefacts to support conceptual field approach of learning special trigonometric angles

Concerns have been expressed on the abstract nature of teaching and learning trigonometry in pre-tertiary institutions. However, studies on student-teachers mathematics learning shows that this concern could be ameliorated by using indigenous artefacts to support conceptual fields of trigonometry. With pre-/post design, the researcher selected 50 student-teachers through simple random sampling and performed experiments using indigenous artefacts in teaching and learning of 30<sup>o</sup>-60<sup>o</sup>-90<sup>o </sup>and 45<sup>o</sup>-45<sup>o</sup>-90<sup>o</sup> special trigonometry angles. This cohort has had at least two years of teaching experience in their permanent schools of work. After going through the experiments, two diagnostic tests (pre- and post-test) were administered, scored and analyzed with the SPSS software. The results of the descriptive statistics, one sample t-test, paired samples t-tests, and correlation coefficients showed that the student-teachers’ performance had significantly improved. The improvements were really attributable to the deployment of the indigenous artefacts to carry out the instruction in the special trigonometric angles. We, therefore, recommended that stakeholders should adopt indigenous artefacts to support the conceptual field approach for the teaching and learning of basic trigonometry.

A novel sintering method of Al2O3/SiCw ceramic composites with improved wear resistance: Oscillatory pressure‐assisted sinter forging

As a typical ceramic tool, the wear resistance of Al2O3/SiCw composites is a prominent parameter in engineering applications. Therefore, how to improve its wear resistance has becomes a research emphasis of ceramic tools. In this work, the Al2O3/SiCw composites with improved wear resistance were prepared by a novel oscillatory pressure-assisted sinter forging (OPSF) process. The composites prepared by OPSF process not only exhibit higher density and significantly improved mechanical properties, but also own lower coefficient of friction than conventional sinter forging and hot oscillatory pressing processes. EBSD results indicated that the OPSF process could improve the texture of (0001) crystal face, low-angle grain boundaries, high-angle grain boundaries with special angles in the Al2O3 matrix and enhance shear deformation at micro-scales, which are multiple contributed to enhanced wear resistance. Therefore, OPSF process is a promising technique to prepare ceramic-based cutting tools with excellent mechanical properties and wear resistance.

Proposed Green Retailing Concept for Maintaining the Sustainability in the Company

Retail business is a fast growing business in Indonesia. This is indicated by the number of retail businesses that have emerged both on a local and national scale. One of the companies engaged in the retail sector is ‘Berkah’ Swalayan. ‘Berkah’ Swalayan is a local retail company in Lampung and Central Java Provinces. This research aims to identify the internal and external conditions of ‘Berkah’ Swalayan which will then be given strategic recommendations related to green retailing. The topic of green retailing was chosen because this strategy is closely related to sustainable development goals which are currently a big focus and have a good impact on many sectors. Identification and analysis in this study used a mixed method, namely quantitative and qualitative methods. The data used are primary data and secondary data. Primary data comes from in-depth interviews with Berkah Swalayan and a survey of 101 respondents to Berkah Swalayan customers. Determination of the number of respondents using the Slovin formula. Secondary data obtained from literature review. Company analysis using the SWOT tool (Strength, Weakness, Opportunities, Threats). On the other hand, in this study variable validation was also carried out using SmartPLS 4. It was found that decision making related to green retailing was influenced by internal and external factors. Internal factors consist of company values. External factors consist of the influence of social organizations, customers and government. Meanwhile, green retailing can improve cost efficiency. This is because companies can reduce their budget costs for environmentally friendly company needs, for example reducing costs in the supply of plastic and the use of energy-friendly electronic devices. In this study the results obtained in the form of identification of internal and external company conditions. that green retailing is a strategy that can improve cost efficiency. In addition, the concept of green retailing also provides opportunities for companies to develop their business. Therefore, green retailing is a strategy that is recommended to be implemented for retail businesses. Several strategic recommendations are given in the implementation of green retailing. Recommendations for some of these strategies are implementing a ‘bring your own shopping container’ system which will be regulated by a point system, using environmentally friendly electronic devices, creating special eco-friendly angles, optimizing social media, and collaborating with the government, communities and businesses that focus on environmental issues.

Measurement of surface chirality at near-normal incidence

The chirality of a medium is typically measured either by transmitting a beam of light through it or by single or multiple interface reflection at large and/or special angles of incidence. We propose and demonstrate here the experimental measurement of surface chirality of z-cut quartz crystal by reflecting a focused beam of light at a near-normal angle of incidence. A small difference in the reflection coefficients between orthogonal elliptically polarized incident beam of 10−4 is measured in the dark-field region of the reflected light via the weak measurement method, taking advantage of the significant transverse spin-shift (TSS) that arises due to the interaction. The TSS behavior is simulated for different chiral parameters (±γ) of the material. The experimental results match well with the theoretically simulated behavior to quantify γ of quartz crystal used as an example interface. The significance of our method can be of interest for a wide variety of fundamental and applied investigations.

Development and Validation of Motivational Manipulative Learning Materials (MMLMs) for the Least-Learned Competencies of Grade 9 Trigonometry

This study presents the development and validation of Motivational Manipulative Learning Materials (MMLMs) aimed at improving the least-learned competencies in Grade 9 trigonometry. The study focuses on enhancing mathematics performance, addressing teaching challenges, and emphasizing real-life applications of trigonometry. Specifically, the competencies identified as least developed include effectively illustrating the six trigonometric ratios and accurately determining the trigonometric ratios of special angles. The study utilizes the combined Mean Percentage Score (MPS) to identify the least-learned skill and to inform the creation of the MMLMs. The developed MMLMs, named "Wheel of the Trigonometric Values," undergo validation by usability and content experts. The content validity criteria, including alignment and organization, demonstrate high mean scores, indicating the suitability of the MMLMs for Grade 9 Math. Similarly, the usability validation criteria, such as "other factors" and "expenses," receive positive mean scores. The evaluations from the experts suggest that the MMLMs are highly suitable and appropriate for enhancing the teaching and learning experience in Grade 9 Math. The findings of this study contribute to the field of mathematics education by providing educators with effective tools to address the least-learned competencies in trigonometry, promoting student engagement and understanding. Future research can explore the implementation and impact of the MMLMs in classroom settings to further assess their effectiveness in improving trigonometry learning outcomes.

Phase-offset-based synchronous resonance identification method for blade tip timing signal

One of the most vital parts of an aero-engine is the rotor, and the working state of a rotor blade is crucial for determining the performance and safety of the engine. Blade tip timing (BTT), as an efficient and non-invasive blade vibration measurement technology, has realized the online monitoring of blade vibration. Online monitoring requires rapid and accurate extraction of blade vibration parameters. By identifying the resonance of blades, vibration parameters can be effectively extracted, such as amplitude, engine order (EO), natural frequency and other information. Through finite element simulation, it is found that when the blade is in synchronous resonance, the vibration phase measured by two probes will be different, as long as the angle between two probes has no integer multiple relation with 360°/EO. Based on this, a phase-offset resonance identification (PHRID) method was presented to identify synchronous resonance of blades. This method identifies resonance by extracting the vibration trend of probes with a special installation angle. Finally, rotor blades were tested in the experiment and compared with the strain data, and the frequency identification error within 1%. The core of this method is the trend extraction of the BTT signal of a certain probe, which greatly reduces the calculation amount and is promising to realize online monitoring. And in the case of extremely limited installation, only two probes with special angles are needed to achieve accurate extraction of resonance frequency.

Chronology of the Schism in the Estonian Orthodox Church in the Light of Relations between the Moscow and Constantinople Patriarchates

The question of property rights of Orthodox Estonians as well as the problem of coexistence of the two parallel Church jurisdictions on the Estonian territory remains unresolved to the present day, despite the regular meetings conducted and agreements reached. The relations between the Moscow and Constantinople Patriarchates on the Estonian issue have reached an impasse. The article presents and analyzes the chronology of the dissent in the Estonian Orthodox Church (EOC) from the perspective of the actions of the Moscow and Constantinople Patriarchates. This allows us to take a glance at the history of the dissent from a special angle in light of the relations between the two Patriarchates in order to identify the features of the strategy of the two Patriarchates, to note the main turning points in this relationship, to estimate the role of the Constantinople Patriarchate in the events related to the split of the EOC as a wholeas well as to propose ways out of the conflict, which is being exacerbated at this time. The novelty of the study lies in the fact that the problem of the dissent in the Estonian Orthodox Church is considered only against the background of the analysis of the actions of the Moscow and Constantinople Patriarchates. The connection is established between the events of 20-70s of the 20th century which preceded the dissent in the 1990s as well as the direct intervention of Constantinople Patriarchate in the EOC activity in the 1990s. For a research on the designated topic for the first time are used materials (as historical sources) from the archives of the Chancellery of the Estonian Orthodox Church of the Moscow Patriarchate. Based on the results of the study, the author comes to the conclusion that the dissent in the EOC was provoked by the Constantinople Patriarchate, whose campaign to seize some parts of the Estonian Orthodox Church was carried out with the tacit approval of the Estonian government circles. In this regard, the author sees a way out of the created crisis only in close cooperation between the Church and the state, namely at the level of the governments of Estonia and Russia.

Non-Abelian mechanics of elastic waves in Kagome metamaterials with internal microstructures

In this work, the duality of mechanical Kagome lattices with internal microstructures is studied and its non-Abelian characteristics of wave packet evolution are discussed. Based on Bloch's theorem, the dynamical and duality matrices are obtained during elastic wave propagation. By regulating the twisting angles of the Kagome lattices and internal microstructures, global twofold degeneracy and unequal Berry phase can be obtained. A fourfold degeneracy appears at Γ point when the mechanical metamaterial is subjected to a special twisting state. Numerical calculations of non-Abelian mechanics and duality are performed, in which band structures with special twisting angles are the same. In addition, the solution of the Wilson operator is derived by the elastic wave equation and gauge transformation, which has non-exchangeable product and leads to the non-Abelian Berry curvature. If the time-reversal symmetry is broken, non-commutative elastic wave responses can be generated. The generalized polarization and wave propagation become different when the acting sequences of external forces are exchanged.