Solution Of Triangles Research Articles

Application of Peter Chew Theorem to Solution of Triangle

Application of Peter Chew Theorem to Solution of Triangle

Pioneering Tomorrow's AI System through the Triangle Solution An Empirical Study of the Peter Chew Method for Overcoming Incomplete Answers in GPT Chat

Pioneering Tomorrow's AI System through the Triangle Solution An Empirical Study of the Peter Chew Method for Overcoming Incomplete Answers in GPT Chat

Pioneering Tomorrow's AI System through the Triangle Solution An Empirical Study of the Peter Chew Rule For Overcoming Limitation in GPT Chat.

Pioneering Tomorrow's AI System through the Triangle Solution An Empirical Study of the Peter Chew Rule For Overcoming Limitation in GPT Chat.

Pioneering Tomorrow's AI System through the Triangle Solution An Empirical Study of the Peter Chew Method For Overcoming Limitation in GPT Chat

Pioneering Tomorrow's AI System through the Triangle Solution An Empirical Study of the Peter Chew Method For Overcoming Limitation in GPT Chat

Improved Position Estimation Method for SRM Considering Current Measurement Errors

Saturation and current measurement errors will decrease the accuracy of rotor position estimation for switched reluctance motor (SRM). To avoid the drawbacks, a rotor position estimation method based on the unsaturated inductance reconstruction is proposed considering current measurement errors. Firstly, the incremental, apparent and unsaturated inductances are analyzed. The unsaturated inductance is calculated with the similar triangle solution in normal condition during the linear inductance rising region. To avoid the pulse injection, the linear regression algorithm is used to calculate the virtual unsaturated inductance in the linear inductance falling region of adjacent phase. Then the complete rotor position in each electric period is obtained by combining the estimated position of three phases. Secondly, the influence of current measurement errors on rotor position estimation is analyzed. The offset error is directly avoided by detecting the zero-value region of phase currents. The scaling error is compensated by balancing the amplitudes of three phase unsaturated inductances. The proposed method does not need additional hardware and can avoid the influence of saturation and current measurement errors simultaneously. Experimental results validate the effectiveness of the proposed method.

New interaction solutions of the similarity reduction for the integrable (2+1)-dimensional Boussinesq equation

The nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new [Formula: see text]-dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants.

Peter Chew Triangle Diagram and Application

The objective of Peter Chew Triangle Diagram is to clearly illustrate the topic solution of triangle and provide a complete design for the knowledge of AI age. Peter Chew's triangle diagram will suggest a better single rule that allows us to solve any problem of topic solution of triangle problems directly, more easily and more accurately. There are two important rules for solving the topic solution of triangle today [1,2], namely the sine rule and the cosine rule. The sine rule is used to find a non-included angle when are given two sides and a non-included angle or the opposite side angle given when are given two angles and one side. The cosine rule normally is used to find the included angle when are given three sides or the third side when are given two sides and the included angle. Generally, we only think that when given two sides and an included angle, the cosine rule is used to find the third side. In fact, when two sides and one non included angle are given, the cosine rule is also more easier for finding the third side. For problem given 2 sides and an included angle, directly find the non included angle. We need to use Peter Chew rule [1] to solve this problem. Peter Chew Rule allows us to find the non included angles directly, easier and more accurately. The application of Peter Chew's triangle diagram in the PCET calculator allows the PCET calculator to directly solve any problem in the topic solution of triangle, which is easier and more accurate. The Peter Chew diagram provides a complete design of the topic solution of triangle, which can help students solve any problems in the topic solution of triangle directly, more easily, and more accurately. Apply Peter Chew diagram to the new calculator (PCET calculator) , allows the PCET calculator to solve any problems in the topic solution of triangle and solve some problem that can not solve by current online calculator such as Math Portal and Symbolab. Which can make PCET calculator effectively help the teaching of mathematics, especially when similar covid-19 problems arise in the future.

ON MAX–MIN MEAN VALUE FORMULAS ON THE SIERPINSKI GASKET

In this paper, we study solutions to the max–min mean value problem [Formula: see text] in the Sierpinski Gasket with a prescribed Dirichlet datum at the three vertices of the first triangle. In the previous mean value, formula [Formula: see text] is a vertex of one triangle at one stage in the construction of the Sierpinski Gasket and [Formula: see text] is the set of vertices that are adjacent to [Formula: see text] at that stage. For this problem, it was known that there are existence and uniqueness of a continuous solution, a comparison principle holds, and, moreover, solutions are Lipschitz continuous. Here we continue the analysis of this problem and prove that the solution is piecewise linear on the segments of the Sierpinski Gasket. Moreover, we also show for which values at the three vertices of the first triangle solutions to this mean value formula coincide with infinity harmonic functions.

PCET Calculator ,Education 4.0 Calculator (version 1)

The Fourth industrial revolution like the developments of robots, AI and automation seem inevitable and technical tools have had a major impact on advanced mathematics teaching and mathematics learning. Educator and researcher must think to create a new Technical tools to help mathematics teaching and mathematics learning for the needs of the AI age. Today's online calculators and math applications, photo math applications and calculators contain only the knowledge already stated in the book such as the sine and cosine rule. However, some areas of mathematics today are not complete, which leads to a certain field where some of the same questions are difficult to solve or cannot be solved. As example, We realize that Math Portal is able to tackle problems that can be solved directly by sine rule or cosine rule. But when given problem with 2 sides and an included angle, if we use Math Portal to find other non-included angle, Math Portal will show error. Therefore, a new Technical Tool for solve this problem with complete design is needed today to help teach and learn in higher education, which is the main goal of the creation of PCET calculator. The PCET calculator design covers some future knowledge like Peter Chew rule that has not yet been covered in any textbook as of today. Peter Chew rule is created to solve the problem mention directly as complement rule of topic solution of triangle. PCET calculator having special functionalities, uses future knowledge in PCET calculator design such as Peter Chew Rule[[Chew, Peter, 2019]], Peter Chew Method [[Chew, Peter, 2019]] and Peter Chew Theorem[Chew, Peter, 2021]]. it make PCET calculator can solve problem some which currently cannot be solve by Today's online calculators and math applications, photo math applications and calculators. Another goal of the PCET calculator is to help students to gain interests and promote effective learning in mathematics by mathematical game.

Application Peter Chew Rule in Calculator Design

Mathematics has always been a challenging topic for high school and university students around the world. Literature revealed that use of technological tool have had a major impact on advanced mathematics teaching and mathematics learning. Mathematical tools such as online calculator and math applications, photo math applications and calculators contain only the knowledge already stated in the book such as the sine and cosine rule. However, if we are given 2 sides and an include angle, it cannot be solve by cosine Rule and sine rule directly, it make some online calculator cannot solve the problem. Peter Chew Rule can solve problem mention directly. If we use online calculator Math portal, we can see that Math portal can solve problems that can be solved directly by sine rule or cosine rule. However, Math portal cannot solve the problem of using the cosine rule plus the sine rule as described above. If we ask Math portal to solve the problem of using cosine rule plus sine rule, Math portal will show error. The reason is the design of Math portal no have Peter chew rule. Therefore, a new product with Peter Chew Rule is needed today to help solve problem mention, which is one the main goal of the PCET calculator. PCET calculator can solve the problems mention cause PCET calculator design have Peter Chew rule. Another problem of some on online calculator like Symbolab and Math portal is they just provided acute angle for the ambiguous case of triangle solution. So, if we use on online calculator like Symbolab and Math portal for help calculate some problem by current method, we will get the wrong answer. But PCET calculator no have this problem cause PCET calculator design using Peter Chew rule.

Peter Chew rule for solution of triangle

Objectives Peter Chew rule: is to let upcoming generation solve same problem can solve directly, more easy, more accuracy compare what’s now solution. With sine rule, cosine rule and Peter Chew rule, all problem at solution of triangle can solve directly more easy, more accuracy. Peter Chew rule are complement rule of topic solution of triangle.

Modified For The Pascal Triangle Multinomial

Newton’s Binomial Theorem applied at the rate of 2 with the formula: (a1+a2)n=∑r=0nC(n,r)a1n−ra2r Problems in algebra are not limited binomial. Binomial only is not enough, so that multinomial is necessary. Multinomial Theorem has the formula: (a1+a2+…+ak)n=∑n1,n2,…,nk≥0n!n1!n2!…nk!a1n1a2n2…aknk The use of theorem in binomial problem is less practical so that Pascal Triangle is prefered, for easier use Pascal’s Triangle. Solution of Triangle with the theorem multinomial problem more complicated. By analyzing multinomial through binomial form, can be obtained from modification that allows the Pascal triangle. The focus of the discussion is to determine Pascal Modified Triangular shape of a multinomial. This basic research using descriptive method, by analyzing the relevant theory is based on the study of literature. The results obtained are Modified Pascal’s Triangle, which facilitates the work in the multinomial.

On the photo-gravitational restricted four-body problem with variable mass

This paper deals with the photo-gravitational restricted four-body problem (PR4BP) with variable mass. Following the procedure given by Gascheau (C. R. 16:393–394, 1843) and Routh (Proc. Lond. Math. Soc. 6:86–97, 1875), the conditions of linear stability of Lagrange triangle solution in the PR4BP are determined. The three radiating primaries having masses $m_{1}$ , $m_{2}$ and $m_{3}$ in an equilateral triangle with $m_{2}=m_{3}$ will be stable as long as they satisfy the linear stability condition of the Lagrangian triangle solution. We have derived the equations of motion of the mentioned problem and observed that there exist eight libration points for a fixed value of parameters $\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )$ , $\alpha$ (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), $0\leq\alpha\leq2.2$ ), the mass parameter $\mu=0.005$ and radiation parameters $q_{i}, (0< q_{i}\leq1, i=1, 2, 3)$ . All the libration points are non-collinear if $q_{2}\neq q_{3}$ . It has been observed that the collinear and out-of-plane libration points also exist for $q_{2}=q_{3}$ . In all the cases, each libration point is found to be unstable. Further, zero velocity curves (ZVCs) and Newton–Raphson basins of attraction are also discussed.

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

The paper deals with singular first order Hamiltonian systems of the form \[ \Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in \Omega \subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$ defines the standard symplectic structure in $\mathbb{R}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi} \sum_{j\neq k=1}^N \Gamma_j \Gamma_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\ldots,z_N)\in \Omega^{2N}:z_j\neq z_k\text{ for }j\neq k\}$ of $N$ different points in the domain $\Omega\subset\mathbb{R}^2$. The function $F:\Omega^N\to\mathbb{R}$ may have additional singularities near the boundary of $\Omega^N$. We prove the existence of a global continuum of periodic solutions $z(t)=(z_1(t),\dots,z_N(t))\in\Omega^N$ that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the $N$-vortex problem in the whole plane (where $F=0$). Examples for $Z$ include Thomson's vortex configurations, or equilateral triangle solutions. The domain $\Omega$ need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of $2\pi$-periodic $H^{1/2}$ functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for $S^1$-equivariant gradient maps that we adapt to this class of potential operators.

Equivalence of linear stabilities of elliptic triangle solutions of the planar charged and classical three-body problems

In this paper, we prove that the linearized system of elliptic triangle homographic solution of planar charged three-body problem can be transformed to that of the elliptic equilateral triangle solution of the planar classical three-body problem. Consequently, the results of Martínez, Samà and Simó (2006) [15] and results of Hu, Long and Sun (2014) [6] can be applied to these solutions of the charged three-body problem to get their linear stability.

Preventive feedback PCNN model and its application in the combinatorial optimization problems

An improved pulse coupled neural network(PCNN) model is proposed to solve combination optimization problem with help of PCNN auto-wave characteristic.Based on Tri-state cascading pulse coupled neural network(TCPCNN),a preventive feedback method by using the triangle inequality theorem is introduced.In the process of searching solutions,all solutions are judged by the triangle inequality theorem and solutions of poor quality are removed.Therefore,the solution space complexity of combinatorial optimization problems decreases and the efficiency and accuracy are improved.This algorithm is applied to the shor test path(SP) and the traveling salesman problem(TSP) simulations.The results show that the proposed algorithm can effectively reduce space complexity and further improve the searching speed.

Equilibrium points in the photogravitational restricted four-body problem

We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m 1 is dominant and is a source of radiation while the other two small primaries m 2 and m 3 are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q 1. Critical masses m 3 and radiation q 1 associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined.

Design of the Scheme Flight Trajectory Command for the Tactical Missile

For the case of the scheme trajectory command design for the tactical missile, the longitudinal and lateral commands and the compound controller of the attitude and height were designed. In the launch segment, four kinds of heading angle commands were presented adopting the arc method and reduction law. In the variable height flight segment, two kinds of height commands were gained by using the parabolic and exponential function. Considering the earth curvature, the formula of the great circle course and distance was gotten applying spherical triangle solution method, and the flight command along the route dot in the lateral plane was obtained. The simulation results show that the designed scheme trajectory commands are feasible and effectively.

Lagrange solution for three wavelength solitary wave clusters in nematic liquid crystals

We investigate the interaction of three optical solitary waves propagating with angular momentum in bulk nematic liquid crystals. The resulting cluster of solitary waves, or nematicons, is shown to orbit about its common centre of “mass”. An elongated isosceles triangle configuration is derived, this solution being the equivalent of the Lagrange solution of Newtonian gravitation. This triangle solution is found to be stable owing to diffractive radiation. A modulation theory explains the existence of the triangle solution as due to the non-monotonicity of an effective potential for the interaction of the solitary waves. This modulation theory also gives good agreement with numerical solutions for the trajectories of the nematicons in the three colours. Finally, it is shown that a cut-off in the shed diffractive radiation prevents the break-up of the triangle due to radiative losses.

Symbolic computation and exact traveling solutions for nonlinear partial differential equations

In this paper, with the aid of the symbolic computation, a further extended tanh function method was presented. Based on the new general ansatz, many nonlinear partial differential equation(s)(NPDE(s)) can be solved. Especially, as applications, a compound KdV-mKdV equation and the Broer-Kaup equations are considered successfully, and many solutions including periodic solutions, triangle solutions, and rational solutions are obtained. The method can also be applied to other NPDEs.