Using 6 questions cognitive method and polya’s model to helps students solve mathematical problem

The ability to solve the mathematical equation problem is very important for the thinking development of middle school students. “How to solve it” of George Polya is famous in the world. 6 questions cognitive model proposed by Professor Zhou are monitored through metacognition, and the 6 questions are coherent, complete and sequential. This paper found that the 6 questions cognitive model can help implement “How to solve it” of George Polya and reduce students' cognitive load. At the same time, this study found that 6 questions cognitive model can help students solve the mathematical equation problem better.

Existence of Positive Solutions for a Pertubed Fourth-Order Equation

In this paper, a special type of fourth-order differential equations with a perturbed nonlinear term and some boundary conditions is considered which is very important in mechanical engineering. Therefore, the existence of a non-trivial solution for such equations is very important. Our goal is to ensure at least three weak solutions for a class of perturbed fourth-order problems by applying certain conditions to the functions that are available in the differential equation (problem (??)). Our approach is based on variational methods and critical point theory. In fact, using a fundamental theorem that is attributed to Bonanno, we get some important results. Finally, for some results, an example is presented.

Quasi-local photon surfaces in general spherically symmetric spacetimes

Based on the geometry of the codimension-2 surface in general spherically symmetric spacetime, we give a quasi-local definition of a photon sphere as well as a photon surface. This new definition is the generalization of the one provided by Claudel, Virbhadra, and Ellis but without referencing any umbilical hypersurface in the spacetime. The new definition effectively excludes the photon surface in spacetime without gravity. The application of the definition to the Lemaître–Tolman–Bondi (LTB) model of gravitational collapse reduces to a second order differential equation problem. We find that the energy balance on the boundary of the dust ball can provide one of the appropriate boundary conditions to this equation. Based on this crucial investigation, we find an analytic photon surface solution in the Oppenheimer–Snyder (OS) model and reasonable numerical solutions for the marginally bounded collapse in the LTB model. Interestingly, in the OS model, we find that the time difference between the occurrence of the photon surface and the event horizon is mainly determined by the total mass of the system but not the size or the strength of the gravitational field of the system.

Indirect Analogical Reasoning Components

If using different instruments obtained a different analogical reasoning component. With use people-piece analogies, verbal analogies, and geometric analogies, have analogical reasoning component consists of encoding, inferring, mapping, and application. Meanwhile, with use analogical problems (algebra, source problem and target problem is equal), have analogical reasoning components consist of structuring, mapping, applying, and verifying. The instrument used was analogical problems consisting of two problems where the source problem was symbolic quadratic equation problem and the target problems were trigonometric equation problem and a word problem. This study aims to provide information analogical reasoning process in solving indirect analogical problems. in addition, to identify the analogical reasoning components in solving indirect analogical problems. Using a qualitative design approach, the study was conducted at two schools in Mataram city of Nusa Tenggara Barat, Indonesia. The results of the study provide an overview of analogical reasoning of the students in solving indirect analogical problems and there is a component the representation and mathematical model in solving indirect analogical problems. So the analogical reasoning component in solving indirect analogical problems is the representation and mathematical modeling, structuring, mapping, applying, and verifying. This means that there are additional components of analogical reasoning developed by Ruppert. Analogical reasoning components in problem-solving depend on the analogical problem is given.

STUDY OF THE ALGORITHM OF PSEUDONORMAL OPTIMIZATION ON THE STABILITY OF THE SOLUTION OF THE EQUATION PROBLEM AND THE ESTIMATION OF THE ACCURACY

In geodesic practice, when designing and adjusting geodetic networks for various purposes on a computer, it is necessary to solve poorly conditioned systems of linear normal equations. In such systems, the determinant of the matrix of equations tends to zero, so the application of the least squares method leads to large distortions of the estimated parameters. Moreover, in such situations, for the least squares algorithm, a slight distortion of the input data leads to unacceptably large distortions of the final results of the equalization and accuracy estimation. In this regard, the application of the pseudonormal optimization method is proposed. The presented work is devoted to the study of the stability of the solution of the adjustment task and the estimation of accuracy obtained on the basis of the pseudonormal optimization method. The novelty is the obtained algorithm for estimating the relative error of the pseudonormal optimization method. A comparative analysis of two competing processing methods was performed for different network models. The results of experimental studies and their analysis have shown the advantage of the pseudonormal optimization method over the least squares method.

Space-Time and Motion to Advection-Diffusion Equation

We are concerned with the study the differential equation problem of space-time and motion for the case of advection-diffusion equation. We derive the advection-diffusion equation from the conservation of mass, where this can be represented by the substance flow in and flow out through the medium. In this case, the concentration of substance and rate of flow of substance in a medium are smooth functions which is useful to generate advection-diffusion equation. A special case of the advection-diffusion equation and numerical results are also given in this paper. We use explicit and implicit finite differences method for numerical results implemented in MATLAB.Keywords: advection-diffusion; space-time; motion; finite difference method. AbstrakKami tertarik untuk mempelajari masalah persamaan diferensial ruang-waktu, dan gerak untuk kasus persamaan adveksi-difusi. Kita menurunkan persamaan adveksi-difusi dari kekekalan massa, di mana hal ini dapat diwakili oleh aliran zat yang masuk dan keluar melalui media. Dalam hal ini konsentrasi zat dan laju aliran zat dalam suatu medium merupakan fungsi halus yang berguna untuk menghasilkan persamaan adveksi-difusi. Sebuah kasus khusus persamaan adveksi-difusi dan hasil numerik juga diberikan dalam makalah ini. Kami menggunakan metode beda hingga explisit dan implisit untuk hasil numerik yang diimplementasikan dalam MATLAB.Kata kunci: adveksi-difusi; ruang-waktu; gerak; metode beda hingga.

A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation

This paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise.

Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell’s Curl Equations

Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is a method to reconstruct the electrical properties (EPs) of bio-tissues from the measured radiofrequency (RF) field in Magnetic Resonance Imaging (MRI). Current MR-EPT approaches reconstruct the EP profile by solving a second-order partial differential wave equation problem, which is sensitive to noise and can induce large reconstruction artefacts near tissue boundaries and areas with inaccurate field measurements. In this paper, a novel MR-EPT approach is proposed, which is based on a direct solution to Maxwell’s curl equations. The distribution of EPs is calculated by iteratively fitting the RF field calculated by the finite-difference-time-domain (FDTD) technique to the measured values. To solve the time-consuming problem of the iterative fitting, a graphics processing unit (GPU) is used to accelerate the FDTD technique to process the field calculation kernel. The new EPT method was evaluated by investigating a numerical head phantom, and it was found that EP values can be accurately calculated and were relatively insensitive to simulated RF field errors. The feasibility of the proposed method was further validated using phantom-based experimental data collected from a 9.4 Tesla (T) Magnetic Resonance Imaging (MRI) system. The results also indicated that more accurate EP values could be reconstructed from the new method compared with conventional methods. Moreover, even in the absence of phase information of the RF field, the proposed approach is still capable of offering robust EPT solutions, thus having improved practicality for potential clinical applications.