Solving introductory physics problems recursively using iterated maps

Over the last twenty years, chaos-based encryption has been an increasingly popular way to encrypt and decrypt data using nonlinear dynamics and deterministic chaos. Discrete chaotic systems based on iterative maps have gotten a lot of interest because of their simplicity and speed. In this paper, three kinds of chaotic maps are utilized to build a digital image encryption strategy depending on a chaotic system. These chaotic maps are the logistic map, Arnold Cat’s map, and Baker’s map. In addition to using the triple data encryption standard (3DES) encryption scheme with the chaotic maps mentioned. The results of the experiments revealed that the suggested digital image encryption technique is both efficient and secure, making it ideal for usage in insecure networks. The transmission control protocol (TCP)/internet protocol (IP) protocol was used for the purpose of transferring data from server to client through the network and vice versa.

A first‐order iterative map which describes the dynamics of a simple feedback buck switching regulator operating in discontinuous mode is derived. Analysis of this map shows that flip bifurcations occur at certain values of the feedback factor. Results from computer simulations and experiments reveal that the system exhibits a typical perioddoubling route to chaos under the particular operating condition studied in this paper. It is found that a special kind of randomness, arising from the ability of the system to ‘merge’ and ‘skip’ periods, constitutes a unique feature of the chaotic dynamics of switched‐mode converter circuits.

In this paper, a low-complexity two-level chaotic encryption scheme is introduced and experimentally demonstrated to improve the physical layer security of a 450-nm laser underwater optical wireless communication (UOWC) system using discrete Fourier transform spread discrete multi-tone (DFT-S DMT) modulation. In the first encryption stage, the original bit stream is encrypted with a chaotic sequence based on a one-dimensional Logistic map. In the second encryption stage, the real and imaginary components of the DFT-S symbols are further encrypted with a pair of separate chaotic sequences, which are generated from a two-dimensional Logistic iterative chaotic map with infinite collapse (2D-LICM). The experimental results indicate that the encryption operation has no negative effect on the performance of the proposed UOWC system. For chaotic encryption, the DFT-S DMT gives a better performance than the DMT scheme under different water turbidities. 55-m/4.5-Gbps and 50-m/5-Gbps underwater transmissions are successfully demonstrated by the chaotic encrypted DFT-S DMT scheme. To the best of our knowledge, this is the first time to verify the feasibility of chaotic encryption in a high-speed UOWC system.

Simulation and forecasting of runoff play an important role in the early warning and prevention of drought and flood disasters. To improve the accuracy of spring runoff simulations, it is important to identify spring runoff production patterns under the combined effect of snow and frozen soil. Based on the theory of the hydrological cycle, three important parameters, which include surface and subsurface runoff, precipitation and temperature, were selected for this study. The trend analysis, statistical analysis and Eckhardt's recursive numerical filtering method were used to qualitatively identify the production patterns of spring runoff, the start and end dates and stage periods of the production patterns. Based on the qualitative identification results, the contribution of each production runoff to the total annual runoff and the total annual spring runoff is quantitatively assessed. The results of the study show that the spring runoff production patterns in the Second Songhua River Basin can be divided into snowmelt runoff, frozen soil conditions of snowmelt–rainfall runoff and rainfall runoff under frozen soil conditions; the snowmelt production is from 21 March, the frozen soil conditions production is from 21 April and the frozen soil ablation ended on 15 June; the shortest phases of each production pattern last 28, 20 and 18 days and the longest last 31, 26 and 24 days. This research provides the basis for improving the principles of production runoff calculation in spring runoff simulation methods.

With the development of mobile communication network, especially 5G today and 6G in the future, the security and privacy of digital images are important in network applications. Meanwhile, high resolution images will take up a lot of bandwidth and storage space in the cloud applications. Facing the demands, an efficient and secure plaintext-related chaotic image encryption scheme is proposed based on compressive sensing for achieving the compression and encryption simultaneously. In the proposed scheme, the internal keys for controlling the whole process of compression and encryption is first generated by plain image and initial key. Subsequently, discrete wavelets transform is used in order to convert the plain image to the coefficient matrix. After that, the permutation processing, which is controlled by the two-dimensional Sine improved Logistic iterative chaotic map (2D-SLIM), was done on the coefficient matrix in order to make the matrix energy dispersive. Furthermore, a plaintext related compressive sensing has been done utilizing a measurement matrix generated by 2D-SLIM. In order to make the cipher image lower correlation and distribute uniform, measurement results quantified the 0∼255 and the permutation and diffusion operation is done under the controlling by two-dimensional Logistic-Sine-coupling map (2D-LSCM). Finally, some common compression and security performance analysis methods are used to test our scheme. The test and comparison results shown in our proposed scheme have both excellent security and compression performance when compared with other recent works, thus ensuring the digital image application in the network.

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

On the equivalence of three approaches to matrix tridiagonalization: Comparison with the recursion method

This study examines the problems of bone marrow transplantation (BMT) survivors in returning to "normal" life in the community after BMT. Before being released from The Johns Hopkins Oncology Center, 84 recipients of BMT were interviewed regarding their quality of life and psychosocial adaptation. Survivors were reinterviewed at 6 months, and at 1 year post-BMT, producing considerable qualitative data regarding their problems in living. Eighty-four patients who had received BMT completed qualitative interviews and standardized measures before treatment, before the return home, and at 6 and 12 months post-BMT. The interviews were subjected to a content analysis methodology to establish units and categories to examine the body of material. Content analysis of these interviews from the first year after BMT identified three areas of psychosocial morbidity; 1) physical problems, which included fatigue, appearance, troubles in eating, and physical restrictions; 2) psychological problems, which included fears about the future, sense of loss of control, anxiety, and depression; and 3) community reintegration problems, which included difficulty in returning to former social roles, separation from home, family, and friends, difficulty in resuming social relations, dealing with stigmatization, problems with family and children, and financial and employment difficulties. Identification of these problems for BMT survivors can be used to guide the development of specific materials and services to prepare recipients of BMTs and their families for life after the transplant. These qualitative results can also be used to direct the development of assessment tools to identify potential patient and family problems.

Drawing appropriate diagrams is a useful problem solving heuristic that can transform a given problem into a representation that is easier to exploit for solving it. A major focus while helping introductory physics students learn problem solving is to help them appreciate that drawing diagrams facilitates problem solution. We conducted an investigation in which 111 students in an algebra-based introductory physics course were subjected to two different interventions during recitation quizzes throughout the semester. They were either (1) asked to solve problems in which the diagrams were drawn for them or (2) explicitly told to draw a diagram. A comparison group was not given any instruction regarding diagrams. We developed a rubric to score the problem-solving performance of students in different intervention groups. Here, we present some surprising results for problems which involve considerations of initial and final conditions.

Drawing appropriate diagrams is a useful problem solving heuristic that can transform a give problem into a representation that is easier to exploit for solving it. A major focus while helping introductory physics students learn problem solving is to help them appreciate that drawing a diagram facilitates problem solution. We conducted an investigation in which 111 students in an algebra-based introductory physics course were subjected to two different interventions during recitation quizzes throughout the semester. They were either (1) asked to solve problems in which the diagrams were drawn for them or (2) explicitly told to draw a diagram. A comparison group was not given any instruction regarding diagrams. We developed a rubric to score the problem-solving performance of students in different intervention groups. We investigated two problems involving electric field and electric force and found that students who draw expert-like diagrams are more successful problem solvers and that a higher level of detail in a student's diagram corresponds to a better score.

The multilayer surface integral equations (SIEs) for electromagnetic scattering by infinitely long magnetodielectric cylinders with an arbitrary number of layers are derived and solved by the spectral integral method (SIM). Singularity subtraction for the 2-D Green's function is used to enhance the computation accuracy and achieve exponential convergence. The final matrix equation after discretization is formed in the Fourier spectral domain rather than the spatial domain, which greatly expedites the SIM solution by accelerating the convolution via the fast fourier transform (FFT) algorithm. A recursive method is proposed to solve the spectral integral equations instead of using an iterative method to lower the computation complexity. Numerical examples for ordinary multilayer cylinders and invisibility cloak cylinders are presented to validate the SIM results by comparing the total fields, scattered fields, and radar cross section (RCS) to analytical solutions or finite-element simulations. They verify that the recursive solution has a complexity of O(MN log N) for an M-layer cylinder with N discrete points on each interface. Meanwhile, the SIM outperforms the analytical method because only the 0th-order and 1st-order special functions (Bessel functions and Hankel functions) are used in the SIM but higher-order functions are necessary for the analytical method to maintain the accuracy.

In computational mathematics, it is a matter of deep concern to recognize which of the given iteration schemes converges quickly with lesser error to the desired solution. Fixed point iterative schemes are constructed to be used for solving equations emerging in many fields of science and engineering. These schemes reformulate a nonlinear equation f ( s ) = 0 into a fixed point equation of the form s = g ( s ) ; such application determines the solution of the original equation via the support of fixed point iterative method and is subject to existence and uniqueness. In this manuscript, we introduce a new modified family of fixed point iterative schemes for solving nonlinear equations which generalize further recursive methods as particular cases. We also prove the convergence of our suggested schemes. We also consider some of the mathematical models which are categorically nonlinear in essence to authenticate the performance and effectiveness of these schemes which can be seen as an expansion and rationalization of some existing techniques.

University students often find physics challenging due to physical and mathematical obstacles, poor mathematical skills, and inadequate problem-solving abilities. This perception affects their interest and performance in the subject. High school students also struggle with numerical problems and understand curricular topics that lack concrete examples and require mathematical manipulations or visualization. Analyzing physics problems is crucial for guiding curriculum development, contributing to physics education, identifying students' difficulties in understanding and solving problems, and improving the process of physics education at all educational levels. Taxonomies are hierarchical frameworks used in education to classify educational learning goals or objectives according to their complexity. Bloom's Taxonomy, Anderson and Krathwohl's revised taxonomy, Lee Shulman's learning grid, Marzano's New Taxonomy of Educational Objectives (NTEO), and the Taxonomy of Introductory Physics Problems (TIPP) are all essential tools for understanding and improving educational learning outcomes. The study adapted the TIPP taxonomy into Turkish and analyzed 7 physics questions in the 2023 BPT. Results showed 3 questions were at recall, 1 at comprehension, and 3 at analysis, with no question at using knowledge. The study demonstrates that TIPP is a useful taxonomy for teachers to plan, apply, and evaluate knowledge and algorithms in solving physics problems, identifying students' needs. This taxonomy allows educators to create questions that engage students at all cognitive levels, promoting cognitive engagement. Researchers can analyze physics questions in the TYT and AYT in earlier years.

The paper aims to fulfil three main functions: (1) to serve as an introduction for the physics education community to the functioning of large language models (LLMs), (2) to present a series of illustrative examples demonstrating how prompt-engineering techniques can impact LLMs performance on conceptual physics tasks and (3) to discuss potential implications of the understanding of LLMs and prompt engineering for physics teaching and learning. We first summarise existing research on the performance of a popular LLM-based chatbot (ChatGPT) on physics tasks. We then give a basic account of how LLMs work, illustrate essential features of their functioning, and discuss their strengths and limitations. Equipped with this knowledge, we discuss some challenges with generating useful output with ChatGPT-4 in the context of introductory physics, paying special attention to conceptual questions and problems. We then provide a condensed overview of relevant literature on prompt engineering and demonstrate through illustrative examples how selected prompt-engineering techniques can be employed to improve ChatGPT-4’s output on conceptual introductory physics problems. Qualitatively studying these examples provides additional insights into ChatGPT’s functioning and its utility in physics problem-solving. Finally, we consider how insights from the paper can inform the use of LLMs in the teaching and learning of physics.

We present two studies investigating factors that correlate with students' and instructors' perceptions of problem difficulty. In the first study, introductory physics students and instructors were asked to rate the difficulty of textbook‐style work‐energy problems. These difficulty ratings are compared and we look for correlations between the difficulty ratings and a measure of problem complexity. We find differences between students' and instructors' ratings and a correlation between instructors' ratings and problem complexity but no significant correlation between students' ratings and problem complexity. In the second study, we asked introductory physics students and instructors to rate the difficulty of textbook‐style kinematics problems. Additionally, we asked students to provide ratings of their familiarity with these problems and complete solutions. We explore the relationship between difficulty ratings, problem complexity, problem familiarity, and the rate at which students solve the problems correctly.