Double Angle Research Articles

Cauchy’s Residue Theorem and Its Application

One of the key theorems in complex analysis, Cauchy’s Residue Theorem, which refers to the integral value of an analytic function along any simple closed contour surrounding an isolated singularity in a certain ring domain divided by 2πi, will greatly simplify the process of computing integrals on contour surrounding singularities. In the field of basic mathematics, the Cauchy’ Residue Theorem plays a key role in the integral calculation of analytic functions, nonuniform complex functions and the argument principle, which establishes a connection between a curve’s winding number and the quantity of zeros and poles inside the curve. Furthermore, Residue Theorem can combine with various subjects, including electromagnetism. This paper gives an overview of the Cauchy’s Residue Theorem and its application. Firstly, it talks about the definition and proof of the Cauchy’ Residue Theorem, along with the definition of residue. Then two examples of using Residue Theorem to integrate functions will be given. Finally, this paper involves application of Residue Theorem in trigonometric sum identities and a remark on the method.

VẬN DỤNG PHƯƠNG PHÁP DẠY HỌC KHÁM PHÁ TRONG DẠY HỌC PHƯƠNG TRÌNH LƯỢNG GIÁC Ở NƯỚC CỘNG HÒA DÂN CHỦ NHÂN DÂN LÀO

According to educational researchers, effective learning is not only just sitting in classes spectating other activities, and memorizing available imparted knowledge, but learners also should participate in the discovery process of new knowledge guided by instructors. It is necessary to apply the discovery learning method to teaching the subjects in general and teaching Mathematics in particular in the direction of developing the competencies and qualities for students. This paper focuses on researching the concept of the discovery learning method and its characteristics. Based on researching the Mathematics curriculum and practical teaching in The Lao People's Democratic Republic, this paper proposes applying the discovery learning method to teaching trigonometric equations at high schools in the Lao People's Democratic Republic.

Agreement between a new fully automatic ocular biometer based on optical low-coherence reflectometry and an optical biometer based on Scheimpflug imaging combined with partial coherence interferometry

BackgroundTo analyze the difference and agreement between measurements obtained by a new fully automatic optical biometer, the SW-9000 μm Plus, based on optical low-coherence reflectometry (OLCR) and a commonly used optical biometer (Pentacam AXL) based on Scheimpflug imaging with partial coherence interferometry (PCI).MethodsThe central corneal thickness (CCT), anterior chamber depth (ACD, from epithelium to anterior lens surface), lens thickness (LT), mean keratometry (Km), corneal astigmatism, corneal diameter (CD), pupil diameter (PD), and axial length (AL) of 74 eyes (from 74 healthy subjects) were measured using the SW-9000 μm Plus and the Pentacam AXL to determine the agreement. Double angle plots were used for astigmatism vector analysis. Bland–Altman and 95% limits of agreement (LoA) were calculated.ResultsStatistically significant differences were detected for all parameters but J0 vector. The Bland–Altman analysis of AL, CCT, ACD, Km, CD, J0 and J45 indicated a high level of agreement between the two devices. Among AL, CCT, ACD, Km, J0, J45, CD, and PD, the 95% LoA ranged from -0.07 to 0.05 mm, -9.67 to 7.34 mm, -0.11 to 0.04 mm, -0.25 to 0.50 D, -0.22 to 0.20 D, -0.15 to 0.20 D, -0.23 to 0.35 mm and 1.55 to 3.77 mm, respectively.ConclusionsThe measurements of AL, CCT, ACD, Km, corneal astigmatism, and CD showed a narrow LoA and may be used interchangeably in healthy subjects between the new OLCR optical biometer and the Scheimpflug/PCI biometer; however, a poor agreement was noted for PD values.

An analysis of learners' mental constructions in learning and solving trigonometric equations

Learners' construction of mathematical concepts is an issue of global concern. As a result, scholars advocate that mathematical topics identified as problematic for learners should be analyzed through research. This paper reports on an APOS (action, processes, objects, and schema) analysis of Grade 11 learners' mental constructions of trigonometric equations. The aim is to reveal the nature of the learners' mental constructions rather than provide a statistical analysis of their performance. This qualitative study was conducted with 17 purposively selected Grade 11 learners doing pure mathematics. The data used in this study were generated from learners' written responses and semi-structured interviews. Using the constructs of APOS theory, our analysis of learners' mental constructions revealed that most learners are operating at the action stage, which means that their knowledge construction of trigonometric equations is at the developmental stage. We further observed that learners needed external cues to trigger their thinking of how they should go about solving the trigonometric equation. All the procedural steps were explicitly written down, meaning they did not internalized the process of solving trigonometric equations. Furthermore, our analysis revealed that while learners can carry out procedures, meaning they can perform step-by-step procedures, they struggle to integrate concepts. The struggle to integrate concepts emanates from not constructing the concept image of trigonometric equations but rather constructing the concepts as isolated facts. Based on the findings, we argue that for learners to develop a schema for trigonometric equations, instructional strategies must focus on helping learners construct the concept image of trigonometric equations. Construction of the concept image will help learners start to internalize the procedures, thus leading to the interiorization of action to process and then encapsulate the process to an object stage.

An Optimal Scalp Rotation Flap Design: Mathematical and Bio-Mechanical Analysis

The design and implementation of successful rotational flaps of the scalp remains a complex process. There are several described techniques, all of which are based on a two-dimension surface, absent consideration of the convexity and thereby three-dimensional nature of the scalp. This has contributed to flaps that are either too small or unnecessarily large in a bid to compensate.Mathematical analysis using trigonometric and algebraic formulae were used to derive the relationship between the defect, triangulation, the contour of the scalp, the radius of the arc of rotation for the flap and scalp skin tension. Polyurethane models were used to test the applicability of the resultant formula, and a consecutive clinical case series is presented.Rotation flap dimensions were calculated using an apex angle of 30 degrees depicted as a two-dimensional construct. The contour of the defect was then measured as the depth of depression with radius of the rotation flap calculated based on the changes in flap surface area across the defect. Finite Element Analysis was utilised to measure redistribution of tension along the flap reconstructing the defect, and the efficacy of the approach confirmed in the consecutive clinical series. The current study provides a reliable flap design based on mathematical analysis, with evidence-based application to the clinical setting.

A modal momentum method for the identification of structural weak links

In order to meet the demands of weak-link identification of mechanical structures in high-precision equipment, this paper introduces the concepts and equations of modal momentum and modal momentum difference ratio (MMDR), as well as an identification method of weak links based on modal momentum. In this method, the modal data of the mechanical structure is acquired first through experimental modal analysis or finite element analysis. The modal momentum of each measured point is calculated based on the modal data and transformed to the global coordinate system. Then, the MMDRs between the measured point with the largest magnitude of modal momentum and its adjacent measured points are calculated. Weak links are judged if the MMDR exceeds a predefined threshold. Twelve different multi-degrees-of-freedom systems incorporating weak links were designed and then identified using the proposed method. The accuracy of identification is up to 100% for these systems. The method was applied to a double pendulum angle milling head of a five-axis numerically controlled machine tool, and the result reveals that the side plate lacks bending stiffness.

Numerical study on the effect of injection strategy on combustion and NO emission from different sources in a diesel/ammonia RCCI engine

Diesel/ammonia operation is one of the ways to efficiently utilize ammonia and reduce carbon emissions, however, the increase in fuel NO introduces serious emission problems. To explore the generation mechanisms of fuel NO and oxidant source NO, a multi-component diesel/ammonia chemical reaction mechanism was constructed in this study, and based on 3D CFD simulation and N-element tracking method, the combustion and different source NO emission characteristics of diesel/ammonia reactivity controlled compression ignition (RCCI) engine at low loads were investigated by pilot injection timing (SOI1) and pilot injection ratio (PIR). In addition, the effect of different first injection spray angle (SA1) and double injection spray angle (SAtotal) on the engine operation was investigated. The results showed that high PIR and delayed SOI1 enhanced the engine IMEP but led to high PRRmax. At a PIR of 15 %, total NO emissions increased as SOI1 advanced, and the trend was reversed when the PIR was higher than 15 %. Fuel NO was produced earlier than the oxidant source N∗O. In terms of 1 spatial distribution, high concentrations of both NO and N∗O were distributed in and around the secondary diesel fuel ignition location, and fuel NO began to be reduced where the oxidant source N∗O was rapidly formed. The reduction mechanism of NO in the cylinder was a multi-path reduction mechanism incorporating Thermal DeNOx, NO Reburning, and inverse reaction pathways for NO formation from the oxidant source. Appropriate reduction of the SAtotal can significantly reduce the total NO emission while maintaining no significant reduction in IMEP. When SAtotal was 82°, the total NO emission was only 16.3 % of the original spray angle and PRRmax was further reduced. After optimization, when a narrow SAtotal of 82°, SOI1 of −15° CA, and PIR of 35 % were used, the IMEP of the engine increased by 5.9 %, and the total NO emission was reduced by 86 %, while PRRmax did not exceed the PRR upper limit.

On Some Cauchy Type Mean-Value Theorems with Applications

Some Cauchy type mean-value theorems for Chebychev’s inequality, Steffensen’s inequality, midpoint rule and Simpson’s rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials and for some trigonometric functions. Further, we obtain some exponential, logarithmic and trigonometric equations and give two inequalities for midpoint and Simpson’s rules.

Aerothermal Performance of Slashface Leakage With Double Surface Angle Under Realistic Swirling Inflow

Abstract The turbine vanes being manufactured and assembled piece by piece leads to an unavoidable gap on the vane endwall called slashface. Usually, the coolant leakage introduced from the slashface with a single surface angle aims to prevent hot gas ingression and protect the vane endwall. However, the cooling effectiveness is greatly reduced when considering the swirling flow from the combustor. In the current work, against strong swirling inflow, a new solution to improve endwall film cooling performance through applying the slashface leakage with the double surface angle is carried out. Through numerical method, the effects of the location of the angle transition region (D/L = 0%, 25%, 50%, 75%, 100%) and mass flow ratio (MFR = 0.5%, 1.0%, 1.5%) of the leakage on the turbine vane endwall aerothermal and film cooling performance are investigated. The results indicated that the coverage area of the coolant on the endwall was obviously enlarged when the slashface leakage with the double surface angle was employed. The film cooling level on both the endwall and suction side surface of the vane increased with increasing MFR. Moreover, the endwall thermal load was reduced. The designs of D/L = 25% and 50% are recommended for their high endwall film cooling level and low endwall thermal load. This paper provides turbine designers with a new idea to increase endwall film cooling performance by the slashface leakage with the double surface angle when considering aggressive swirling inflow.

Joint-line obliquity angle is significantly affected by hip abduction and adduction: A simulated analysis.

Different methods for quantifying joint-line obliquity (JLO) have been described, including joint-line obliquity angle (JLOA), Mikulicz joint-line angle (MJLA) and medial proximal tibial angle (MPTA). The goal of the present study was to quantify the variation of JLOA based on the position of the hip. The hypothesis of our study is that JLO is significantly influenced by the abduction/adduction of the limb, unlike MJLA. One hundred long-leg-weightbearing X-rays were used. At time 0 and after 30 days, two observers performed different measurements, including (1) distance between pubic symphysis and center of the femoral head, (2) distance between center of the femoral head and center of the ankle joint, (3) distance between center of the ankle and medial malleolus, (4) hip-knee-ankle angle, (5) MPTA, (6) lateral distal femoral angle, (7) joint-line congruency angle, (8) JLOA, (9) MJL and (10) angle between Mikulicz line and line perpendicular to the ground. The changes of the JLOA based on the position of the hip (abducted, neutral, bipedal stance adduction and monopodal stance adduction) were calculated with trigonometric formulas and with simulation on an orthopaedic planning digital software. The JLOA change between adducted and abducted positions was on average 12.8° (SD 0.9 mm). The MJL did not vary significantly based on hip position. The adduction/abduction of the lower limb has a considerable impact on JLOA. Methods like MJLA which are not affected by hip position should be preferred for JLO evaluation. Diagnostic study, level III.

Research on Seismic Performance of Steel Frame with Double Angle Steel and Shear Web Joints Under Quasi-Static Loading

Research on Seismic Performance of Steel Frame with Double Angle Steel and Shear Web Joints Under Quasi-Static Loading

Considerations for extracting moiré-level strain from dark field intensities in transmission electron microscopy

Bragg interferometry (BI) is an imaging technique based on four-dimensional scanning transmission electron microscopy (4D-STEM) wherein the intensities of select overlapping Bragg disks are fit or more qualitatively analyzed in the context of simple trigonometric equations to determine local stacking order. In 4D-STEM based approaches, the collection of full diffraction patterns at each real-space position of the scanning probe allows the use of precise virtual apertures much smaller and more variable in shape than those used in conventional dark field imaging such that even buried interfaces marginally twisted from other layers can be targeted. With a coarse-grained form of dark field ptychography, BI uses simple physically derived fitting functions to extract the average structure within the illumination region and is, therefore, viable over large fields of view. BI has shown a particular advantage for selectively investigating the interlayer stacking and associated moiré reconstruction of bilayer interfaces within complex multi-layered structures. This has enabled investigation of reconstruction and substrate effects in bilayers through encapsulating hexagonal boron nitride and of select bilayer interfaces within trilayer stacks. However, the technique can be improved to provide a greater spatial resolution and probe a wider range of twisted structures, for which current limitations on acquisition parameters can lead to large illumination regions and the computationally involved post-processing can fail. Here, we analyze these limitations and the computational processing in greater depth, presenting a few methods for improvement over previous works, discussing potential areas for further expansion, and illustrating the current capabilities of this approach for extracting moiré-scale strain.

Fast bias-corrected conductivity mapping using stimulated echoes.

To demonstrate the potential of a double angle stimulated echo (DA-STE) method for fast and accurate "full" homogeneous Helmholtz-based electrical properties tomography using a simultaneous magnitude and transceive phase measurement. The combination of a spin and stimulated echo can be used to yield an estimate of both magnitude and transceive phase and thus provides the means for "full" EPT reconstruction. An interleaved 2D acquisition scheme is used for rapid acquisition. The method was validated in a saline phantom and compared to a double angle method based on two single gradient echo acquisitions (GRE-DAM). The method was evaluated in the brain of a healthy volunteer. The magnitude obtained with DA-STE showed excellent agreement with the GRE-DAM method. Conductivity values based on the "full" EPT reconstruction also agreed well with the expectations in the saline phantom. In the brain, the method delivered conductivity values close to literature values. The method allows the use of the "full" Helmholtz-based EPT reconstruction without the need for additional measurements. As a result, quantitative conductivity values are improved compared to phase-based EPT reconstructions. DA-STE is a fast complex- mapping technique that could render EPT clinically relevant at 3T.

Algebraic formulas and geometric derivation of source identities

Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant representations for source functions which appeared in previous literature as well as introducing the elliptic version of the determinants, and obtain identities between determinants. We also show several symmetrization formulas for the rational version.

Synchronization of four unalike local coupled phase oscillators in a ring: Analytic formulas for the critical coupling and phase differences

There is no universal technique in statistical mechanics to analytically solve the problem of dissimilar adjacent coupled phase oscillators in a ring at the switch to synchrony. The solution to the four oscillators′ case presents a first step to constructing a regular method based on unscrambling the sync dynamics of a few oscillators. The local interaction between the adjacent oscillators and the periodic boundary conditions introduce finer points to the synchronicity. We notice that the synchronized state has different noticeable characteristics depending on the arrangements of the primary frequencies. This, in turn, yields an involvedness in obtaining an analytic solution. To overcome the difficulty, we classify the twenty-four possible arrangements for the four local coupled phase oscillators in a ring into three classes. Consequently, we introduce a method to rearrange the initial frequencies into consecutive pairs of nearest-neighbor elements for all three classes. Thus, we can define the effective parameters (in terms of the sequential couples of adjacent oscillators) that rule the solution for the oscillators in a ring. Furthermore, the unison behavior resulting from coupling emerges when a clearly defined phase lock condition develops. We utilize the phase condition and trigonometric identities to find a soluble-effortless relation. Consequently, we derive an analytic expression that predicts the threshold coupling for each class when the oscillators synchronize. The formula for the critical factor allows us to obtain usable expressions to determine the phase differences at the transition into a sync stage. Therefore, we figure out the sync mechanism a priori, knowing the initial frequencies.

The Performance of Cold-Formed Steel Beam-Column Connections with Screw Configuration

Cold-formed steel is being used as a viable substitute for traditional steel and building materials due to its performance, affordability, convenience, lightweight, and environmental friendliness. The major purpose of this current research is for a reaction to force-deformation at the connection level to be delivered, which is proper for typical cold-formed steel (CFS) screwed beam-column assembly CFS C-section, and to show the influence of the behavior of CFS under bending test. An inner plate, double angle plate, and self-drilling screws are used by a basic beam-column frame idealized CFS to join the beam-column. Fixed support was implemented at the column base, whereas the beam end had free support. A 0.75 mm cold-formed steel C-section with 80mm height and 30mm width was utilized in this study and assembled into an "I" and "box" configuration. 3d and 4d were used to compute screw spacing for self-drilling screws with 4 mm heads and 2 mm bodies. The first type was each angle plate (top and seat angle) screwed with eight screws, whereas the second involves four screws. The CFS section’s thinness led to the use of screw connections, which are essential to their ability to withstand connection failures, to anticipate the failure mode of the connection segment. Numerical simulation by ABAQUS Student Edition and analytical studies have been conducted by AS/NZS 4600:2018 to investigate the capacity of CFS under bending. Quantity and the arrangement of the screws determine the displacement ductility with tighter spacing enhancing performance and reinforcing global and local buckling resistance.

Java Script Programs for Calculation of Dihedral Angles with Manifold Equations

Java Script programs for calculation dihedral angles from NMR data with manifold equations of 3-Sphere approach: rectangle, Villarceau circles of cyclide (Torus – Dupin Cyclide), polar equations, Euler-Conic. Manifolds are curves or surface in higher dimension used for calculation of dihedral angles under wave character of NMR data, carbon and/or proton chemical shift δ<sub>Xn</sub>[ppm] and vicinal coupling constant <sup>3</sup><I>J</I><sub>HnHn+1</sub>[Hz]. 3-Sphere approach for calculation of the dihedral angles from NMR data in four steps: 1. Prediction, or more exactly calculation of the dihedral angles from vicinal coupling constant with trigonometric equations, 2. Calculation of the dihedral angles from manifold equations; 3. Building units from angle calculated with one of the manifold equations; 4. Calculation the vicinal coupling constant of the manifold dihedral angle. In this paper are presented Java Script programs of step 2 and from step 3 only the Java Script program for calculation of seven sets angles. The bond distances l<sub>CnCn+1</sub>[A<sup>0</sup>] between two atoms of carbon are under different polar equations (<i>i.e.</i> limaçons or cardioid, rose or lemniscale), our expectation was to find different manifold equations for calculation the best angle, differences are smaller but can be find sometimes a preferred one for a vicinal coupling constant. 3-Sphere approach has the advantages of calculation from vicinal angle or/and chemical shift the dihedral angle, tetrahedral angle and the bond distance l<sub>CnCn+1</sub>[A<sup>0</sup>], with application on conformational and configurational analysis.

Effectiveness of the Symbolab Calculator in Improving Second Year Science and Mathematics Students Ability to Solve Trigonometric Equations

Trigonometry is a challenging area of mathematics for many students, especially for pre-service primary teachers in TTCs. The Symbolab calculator, software developed to assist mathematicians in solving complex problems, is used to improve students' performance and understanding. This study investigated the effectiveness of using the Symbolab calculator on second-year student-teachers, comparing their performance with those without the calculator. The data was analysed using the t-test. The findings showed that the integration of the Symbolab calculator significantly improved students' performance and understanding, reducing the time taken to solve trigonometric equations. It is recommended that further research be conducted on the effectiveness of combining different instructional methods in trigonometry instruction. There is a need to encourage teachers to adopt various pedagogical strategies for improved learning outcomes.

ANALYSIS OF FIRST-YEAR MATHEMATICS EDUCATION STUDENTS' UNDERSTANDING OF TRIGONOMETRY COURSES BASED ON APOS THEORY

This study is conducted to analyze the understanding of students' concepts in trigonometry courses based on the theory of APOS (action, process, object, and scheme). Based on previous research, students had difficulty understanding the concept of trigonometry, so they had difficulty applying the common trigonometric formula, explaining each one's relation to the comparative trigonometry, and performing calculations and algebra calculations. This type of research is qualitative research descriptive. The subject of the study is a student in the mathematics education semester for two academic years, 2023-2024. Randomly selected research samples using simple random sampling. Analyzing five student responses in the Trigonometry course reveals variations in conceptual understanding based on the APOS theory. Some students successfully followed the stages of action, process, object, and schema comprehensively, while others struggled to apply the correct concepts, particularly in solving problems using trigonometric principles. From this research, it can be concluded that it is important to understand the concept of trigonometry so that the subject can answer a question based on APOS theory

Analisis Kesulitan Siswa Dalam Menyelesaikan Soal Materi Trigonometri Ditinjau Dari Gaya Belajar

According to the OECD, the 2018 PISA results state that 71% of Indonesian students experience difficulties when faced with conditions that require them to solve problems by applying mathematical concepts, so they fail to achieve minimum competitive ability in mathematics. Based on the observations of researchers, there are MAN 1 Gresik students who have difficulty in solving trigonometric problems because they think that this material is too much memorizing formulas. This study used learning style questionnaires, test questions, and interviews to collect data in order to find out students' difficulties in solving trigonometry problems in terms of students' learning styles. The results obtained from this study include: 1) The difficulties of auditorial subjects are the inability to remember the steps of proving trigonometric identities; the basic concepts of trigonometric comparisons and the Pythagorean Theorem, unable to recognize mathematical angle symbols, less careful in calculating powers and calculating fractions, and the inability to draw conclusions. 2) The difficulties of kinesthetic subjects are the inability to remember the steps of proving trigonometric identities and the Pythagorean Theorem, the inability to recognize mathematical angle symbols, and less careful in calculating powers and calculating fractions; 3) The difficulties of visual subjects are the inability to remember the steps of proving trigonometric identities, the inability to recognize mathematical angle symbols, and the inability to draw conclusions.