Double Angle Research Articles

О комплексных операторных функциях комплексного операторного переменного

We consider a family of complex operator functions whose domain and range of values are included in the real Banach algebra of bounded linear complex operators acting in the Banach space of complex vectors over the field of real numbers. It is shown that the study of a function from this family can be reduced to the study of a pair of real operator functions of two real operator variables. The main elementary functions of this family are considered: power function; exponent; trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant; hyperbolic sine, cosine, tangent, cotangent, secant, cosecant; the main property of the exponent is proved. A complex Euler operator formula is obtained. Relations that express sine and cosine in terms of the exponent are found. For the trigonometric functions of sine and cosine, addition formulas are justified. The periodicity of the exponent and trigonometric functions of sine, cosine, tangent, cotangent is proved; reduction formulas for these functions are provided. The main complex operator trigonometric identity is obtained. Equalities connecting trigonometric and hyperbolic functions are found. The main complex operator hyperbolic identity is established. For the hyperbolic functions of sine and cosine, addition formulas are indicated. As an example of an elementary function from the family of complex operator functions under consideration, a rational function is considered, a special case of which is the characteristic operator polynomial of a linear homogeneous differential equation of n-th order with constant bounded operator coefficients in a real Banach space.

Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

<p>In this paper, we introduce a method for computing the primitive decomposition of idempotents in any semisimple finite group algebra, utilizing its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $ \mathbb{C}[D_{2n}] $ and generalized quaternion group algebras $ \mathbb{C}[Q_{4m}] $. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $ \mathbb{C}[D_{8}] $ and $ \mathbb{C}[Q_{8}] $ is described, under which the two complete sets of primitive orthogonal idempotents of these group algebras correspond bijectively.</p>

Cosine trigonometric rules applied to average and geometric aggregating operators using extension q-rung interval-valued neutrosophic set approach.

We introduce the concept of cosine trigonometric q-rung Diophantine neutrosophic interval-valued set (CosTq-rung DioNSIVS). The fact that CosTq-rung DioNSIVS combines q-rung neutrosophic interval-valued set, q-rung neutrosophic set and neutrosophic interval-valued set is one of its distinguishing characteristics. A new idea of CosTq-rung DioNSIVWA, CosTq-rung DioNSIVWG, GCosTq-rung DioNSIVWA and GCosTq-rung DioNSIVWG is proposed in this study. We also look at the idempotency, boundedness, commutativity, and monotonicity of the CosTq-rung DioNSIVS based on algebraic operations. We considered new kinds of two distances in the proposed models, besides Euclidean and Hamming distances. The CosTq-rung DioNSIVS method was used to analyze the cosine trigonometric aggregation procedures. The study's concluding results include several fascinating and captivating discoveries.

The effect of and correction for through-slice dephasing on 2D gradient-echo double angle mapping.

To show that variations through slice and slice profile effects are two major confounders affecting 2D dual angle maps using gradient-echo signals and thus need to be corrected to obtain accurate maps. The 2D gradient-echo transverse complex signal was Bloch-simulated and integrated across the slice dimension including nonlinear variations in inhomogeneities through slice. A nonlinear least squares fit was used to find the factor corresponding to the best match between the two gradient-echo signals experimental ratio and the Bloch-simulated ratio. The correction was validated in phantom and in vivo at 3T. For our RF excitation pulse, the error in the factor scales by approximately 3.8% for every 10 Hz/cm variation in along the slice direction. Higher accuracy phantom maps were obtained after applying the proposed correction; the root mean square error relative to the gold standard decreased from 6.4% to 2.6%. In vivo whole-liver maps using the corrected map registered a significant decrease in gradient through slice. inhomogeneities varying through slice were seen to have an impact on the accuracy of 2D double angle maps using gradient-echo sequences. Consideration of this confounder is crucial for research relying on accurate knowledge of the true excitation flip angles, as is the case of mapping using a spoiled gradient recalled echo sequence.

Improved dynamic sliding mode control for plate hoisting of cable crane under wind load

In order to quickly eliminate the swing angle of plate hoisting under wind load, an improved dynamic sliding mode control method is proposed in this paper. Firstly, the plate hoisting model under wind load is simplified into a double pendulum system, and the dynamic model of a double pendulum cable crane with wind disturbance is established. To enhance the coupling relationship between trolley displacement and double swing angle, a new coupling state vector for the double pendulum system is defined. On this basis, a dynamic sliding surface is constructed and then an integral function of the system’s discontinuous term is established, essentially ensuring the continuity of the sliding surface and reducing the system’s chattering characteristics. Finally, the stability of the system is rigorously proved by Lyapunov theorem and Barbalat lemma. The simulation and experimental results show that the controller proposed in this paper has a good effect on the anti-swing problem of plate hoisting under wind load.

VISUALISASI OPERASI TRIGONOMETRI (SINUS DAN COSINUS)

This article discusses a scheme for creating a visual representation of trigonometry and its operations on sine and cosine. This visual representation is then called trigonometric visualisation. The formulas in the visualisation of trigonometric operations apply to all trigonometric identities constructed from sines and cosines with degrees no more than 2. Research was carried out by analyzing trigonometric representations of right triangles from various sources. The results of the research are procedures or techniques for creating trigonometry visualizations of sine and cosine and their operations. Operations in trigonometry visualization are: multiplication, addition, and subtraction of sine and cosine.

Focusing property and optical field modulation of three-core fiber optical tweezers

This work develops a three-core fiber optical tweezers probe with a double cone angle and a tip microlens based on the Lorentz-Mie scattering theory. The primary parameters of the structural probe, such as the cone angle and the thickness of the microlens, are investigated and clarified by theoretical research, model construction, and simulation analysis. The best capture ability is at 44° of second-order cone angle. In order to create a focused light field, the microlens must be thicker than 8 μm. Under the probe framework, we investigated the primary purposes of various core optical powers. We designed a set of optical fiber tweezers control schemes to adjust the capture effect in real time, which enhances the control capability of optical fiber tweezers. We leverage the difference in input optical power between different cores. The focusing effect of the optical field can be changed, enabling real-time control of the captured item, by modifying the incidence angle and power. The power is inversely correlated with the movement of the capture position. The capture point moves 0.1 μm when the optical power is changed to 0.3 W/m.

Guided e-Assessment of Math Proofs with the Halomda Platform

The mode of assessment is one of the most important factors influencing learning. E-assessment usually includes only checking the final answer, thus limiting teacher's ability to check the complete solution, and it does not allow inclusion of math proofs problems that constitute an important part of math content. The e-assessment module of Halomda elearning platform allows assessment of proof problems from different areas of school and high Math, like algebraic identities and inequalities, geometry proofs, trigonometrical identities, analytic geometry, vector algebra, etc. Although eassessment of a proof of a mathematical statement is generally a difficult task, in this paper we show through some working examples how guided e-assessment can be implemented using the Halomda platform.

Tiny-YOLO distance measurement and object detection coordination system for the BarelangFC robot

<span lang="EN-US">A humanoid robot called BarelangFC was designed to take part in the Kontes Robot Indonesia (KRI) competition, in the robot coordination division. In this division, each robot is expected to recognize its opponents and to pass the ball towards a team member to establish coordination between the robots. In order to achieve this team coordination, a fast and accurate system is needed to detect and estimate the other robot’s position in real time. Moreover, each robot has to estimate its team members’ locations based on its camera reading, so that the ball can be passed without error. This research proposes a Tiny-YOLO deep learning method to detect the location of a team member robot and presents a real-time coordination system using a ZED camera. To establish the coordinate system, the distance between the robots was estimated using a trigonometric equation to ensure that the robot was able to pass the ball towards another robot. To verify our method, real-time experiments was carried out using an NVDIA Jetson NX Xavier, and the results showed that the robot could estimate the distance correctly before passing the ball toward another robot.</span>

Logarithm of multivector in real 3D Clifford algebras

Closed form expressions for a logarithm of general multivector (MV) in basis-free form in real geometric algebras (GAs) Clp,q are presented for all n = p + q = 3. In contrast to logarithm of complex numbers (isomorphic to Cl0,1), 3D logarithmic functions, due to appearance of two double angle arc tangent functions, allow to include two sets of sheets characterized by discrete coefficients. Formulas for generic and special cases of individual blades and their combinations are provided.

Trigonometric and hyperbolic Poincare, Sobolev and Hilbert-Pachpatte type inequalities

In this article based on trigonometric and hyperbolic type Taylor formulae we establish Poincare, Sobolev and Hilbert-Pachpatte type inequalities of different kinds specific and general.

Opial and Ostrowski type inequalities based on trigonometric and hyperbolic type Taylor formulae

Based on [4], we produce a collection of interesting trigonometric and hyperbolic Taylor formulae with integral remainders. Using these we derive Opial and Ostrowski type corresponding inequalities of various kinds and norms.

A Simple Interference and Power-based Direction of Arrival Measuring System for Modern Communication

In this paper, we present a system specialized for measuring the direction-of-arrival (DoA) of electromagnetic waves with noticeable simplicity. Unlike common methods, which are based heavily on complex computation and signal processing, our proposed system is considerably simpler, both in terms of design and operating theory. Our design consists of two directional antennas for collecting incident waves, a system of Wilkinson power combiners and dividers in which the waves collected by the antennas interfere, a result-processing block consisting of power amplifiers, rectifiers, and a microcontroller unit that respectively converts the interferometric radio-frequency (RF) signal into direct-current (DC) signal, measures its corresponding power before calculating the incident angle solely based on a simple trigonometric equation. The system yields a high accuracy of less than 7.5o with the incident angle ranging from −60o to 60o.

TILINGS OF THE SPHERE BY CONGRUENT QUADRILATERALS II: EDGE COMBINATION WITH RATIONAL ANGLES

AbstractEdge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of $a^3b$ -quadrilaterals with all angles being rational degrees. There are $12$ sporadic and $3$ infinite sequences of quadrilaterals admitting the two-layer earth map tilings together with their modifications, and $3$ sporadic quadrilaterals admitting $4$ exceptional tilings. Among them only three quadrilaterals are convex. New interesting non-edge-to-edge triangular tilings are obtained as a byproduct.

Elastic behavior of riveted double angle connections in tension

AbstractT‐stubs and L‐stubs loaded in tension are generally assumed in contact with a rigid foundation. The prying force and stiffness of these components depend on this hypothesis. For double web angle riveted connections, the angle rests on deformable beam/column web. The objective of the present paper is to take into account the effects of this deformability on the development of the contact area, the rivet forces and the tensile stiffness. An enhanced theory of beam in contact with a tensionless Winkler foundation is adopted in the contact and lift‐off areas. The axial and bending flexibilities of rivets are also considered. A numerical analysis is also performed with ANSYS in order to validate the analytical model. A sensitivity study is conducted on the influence of the thickness of the horizontal leg. The analytical and the numerical results are in good agreement, which confirms the accuracy of the proposed mechanical model.

The efficiency of Chisala’s model in predicting the moment-rotation curve for double web angle, and welded flange plate connections

The efficiency of Chisala’s model in predicting the moment-rotation curve for double web angle, and welded flange plate connections

Stiffness of L‐stubs loaded in compression

AbstractThe stiffness of T‐stubs in compression is generally assumed infinite except for column base plates. For bolted circular flange connections and double web angle connections, the stiffness of L‐stubs and T‐stubs in compression should be determined to accurately calculate the position of the center of compression. Simple expressions have been recently proposed for these components in contact with a rigid foundation. However, the end‐plate can be in contact with a flexible foundation.The objective of the present paper is to propose simple expressions of the compressive stiffness of L‐stubs in contact with rigid or flexible foundations. A refined beam model resting on Winkler foundations is derived and applied to the L‐stub model. To facilitate the practice of engineering, simplifications are proposed for the contact pressure distribution and the compressive stiffness. The analytical results are then compared against finite element calculations performed with ANSYS APDL

Metodologia rezolvării ecuațiilor trigonometrice cu un grad sporit de dificultate

The article includes, in principle, trigonometric equations with an increased degree of difficulty, which will contribute to the development of mathematical education, to the organization of independent and differentiated work with students in high schools with a real profile.

Exact Expressions for Trigonometric Functions

Summary Solving polynomial equations and trigonometry are typically taught in separate modules or even separate classes. The derivation of a one-third angle formula gives students the opportunity to tie these two topics together. An algorithm for finding the cosine of one-third of a prescribed angle is based on Cardano’s formula. In addition, we show how this algorithm can be combined with half, double, and triple angle formulas to give exact expressions for trigonometric functions written in terms of radicals for an infinite number of angles between 0 and π / 2 .

Trigonometric identities from the mystic rose

Trigonometric identities from the mystic rose