Search

We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis., 23+2 pages, 8 figures. Supplementary material available at https://github.com/duetosymmetry/surprises-in-a-classic-BVP . Version 2: Added 12 references and additional discussion. Comments welcome

Ultrafast laser pump-probe spectroscopy is an important and growing field of physical chemistry that allows the measurement of chemical dynamics on their natural timescales, but undergraduate laboratory courses lack examples of such spectroscopy and the interpretation of the dynamics that occur. Here we develop and implement an ultrafast pump probe spectroscopy experiment for the undergraduate physical chemistry laboratory course at the University of California Berkeley. The goal of the experiment is to expose students to concepts in solid-state chemistry and ultrafast spectroscopy via classic coherent phonon dynamics principles developed by researchers over multiple decades. The experiment utilizes a modern high-repetition-rate 800 nm femtosecond Ti:Sapphire laser, split pulses with a variable time delay, and sensitive detection of transient reflectivity signals using the lock-in technique. The experiment involves minimal intervention from students and is therefore easy and safe to implement in the laboratory. Students first perform an intensity autocorrelation measurement on the femtosecond laser pulses to obtain their temporal duration. Then, students measure the pump-probe reflectivity of a single-crystal antimony sample to determine the period of coherent phonon oscillations initiated by an ultrafast pulse excitation, which is analyzed by fitting to a sine wave. Students who completed the experiment in-person obtained good experimental results, and students who took the course remotely due to the COVID-19 pandemic were provided with the data they would have obtained during the experiment to analyze. Evaluation of student written and oral reports reveals that the learning goals were met, and that students gained an appreciation for the field of ultrafast laser-induced chemistry.

We investigate the nonlinear effect of a pendulum with the upper end fixed to an elastic rod which is only allowed to vibrate horizontally. The pendulum will start rotating and trace a delicate stationary pattern when released without initial angular momentum. We explain it as amplitude modulation due to nonlinear coupling between the two degrees of freedom. Though the phenomenon of conversion between radial and azimuthal oscillations is common for asymmetric pendulums, nonlinear coupling between the two oscillations is usually overlooked. In this paper, we build a theoretical model and obtain the pendulum's equations of motion. The pendulum's motion patterns are solved numerically and analytically using the method of multiple scales. In the analytical solution, the modulation period not only depends on the dynamical parameters, but also on the pendulum's initial releasing positions, which is a typical nonlinear behavior. The analytical approximate solutions are supported by numerical results. This work provides a good demonstration as well as a research project of nonlinear dynamics on different levels from high school to undergraduate students., Comment: 19 pages, 6 figures

In this paper, event-triggered attitude consensus is considered for multi-agent systems with guaranteed fixed-time convergence. Due to the non-Euclidean property of the attitude configuration space, the attitude consensus is more challenging to achieve under the sampled-data setting. An event-triggered attitude protocol and event-triggered condition are proposed based on the axis-angle attitude representation. The fixed-time attitude consensus is reached if the initial attitudes lie in local regions on the attitude configuration space and Zeno-behavior can be avoided. The theoretical results reveal that the settling time is related to the inter-event interval and the algebraic connectivity of the topology graph. We further consider the consensus protocol under a jointly connected graph. The settling time estimation is also obtained in this case, which is dependent on the switching instants. Numerical simulations are conducted to verify the validity of the theoretical results finally.

There is a huge amount of mathematical information in the world, and mathematics is everywhere and nowhere. Bayesian theory is based on a process of statistical inference that requires the calculation of general and prior information to obtain a posteriori information. Its main features are the use of probabilities to represent all forms of uncertainty and the use of probabilistic rules to enable learning and inference, estimating the probability of future occurrences by calculating the probability of a past time. In order to bring mathematics closer to life, this paper explores the teaching of mathematical applications in terms of material selection, teaching arrangement, and professional integration. At the same time, in order to better realize mathematics application teaching, effectively improve the classroom effect of mathematics application teaching, and make students better accept mathematical knowledge and apply it to practical applications, the design of mathematics application teaching in this paper is also based on Naive Bayes.

Proof Blocks is a novel software tool which enables students to write mathematical proofs by dragging and dropping prewritten lines into the correct order, rather than writing a proof completely from scratch. We used Proof Blocks problems as exam questions for a discrete mathematics course with hundreds of students, allowing us to collect thousands of student responses to Proof Blocks problems. Using this data, we provide statistical evidence that Proof Blocks are easier than written proofs, which are typically very difficult. We also show that Proof Blocks problems provide about as much information about student knowledge as written proofs. Survey results show that students believe that the Proof Blocks user interface is easy to use, and that the questions accurately represent their ability to write proofs.

Mathematics inequality is an essential concept that students should fully understand since it is required in mathematical modeling and linear programming. However, students tend to perceive the solution of the inequalities problem without considering what the solution of inequality means. This study aims to describe students’ mistakes variations in solving mathematical inequality. It is necessary since solving inequality is a necessity for students to solve everyday problems modeled in mathematics. Thirty-eight female and male students of 12th-grade who have studied inequalities are involved in this study. They are given three inequality problems which are designed to find out students’ mistakes related to the change of inequality sign, determine the solution, and involve absolute value. All student work documents were analyzed for errors and misconceptions that emerged and then categorized based on the type of error, namely errors in applying inequality rules, errors in algebraic operations, or errors in determining the solution set, then described. The result shows that there were some errors and misconceptions that students made caused by still bringing the concept of equality when solving the inequalities problem. It made them did not aware of the inequality sign. Students are still less thorough in operating algebra and do not understand the number line concept in solving inequalities. The other factor was giving “fast strategy” to the students without considering the students’ understanding.

Although they are sometimes considered problematic to grasp by students, the concepts behind non-Euclidean geometry can be taught using astronomical images. By using photographs of the Moon taken with a smartphone through a simple telescope, we were able to introduce these concepts to high-school students and college newcomers. By recognizing different Moon geological structures within the photograph, we teach students how to calculate distances of mountain ranges or areas of craters on the Moon's surface, introducing the notions of geodesics and spherical triangles. Furthermore, students can empirically see that the correct estimations for the actual values cannot be obtained using flat geometry. Instead, by using three--dimensional curved geometry, precise estimates of lengths and areas of geological elements in the Moon can be computed with less than 4\% error. These procedures help students understand, concretely, non-Euclidean geometry concepts., accepted in American Journal of Physics

The use of low-bandwidth attitude controllers is a standard solution for spacecraft facing tight frequency-domain requirements for attitude-stable payloads and platforms. However, such controllers ...

We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We di...

Early in high school algebra, quadratics chosen as examples by teachers and textbooks alike tend to have integer coefficients and to factorise over the integers. This can give the misleading impression that such quadratics are the norm. As students progress into calculus and begin regularly seeing quadratics that are not as ‘nice’, we hope they become disabused of this notion. Indeed, even if the coefficients of the quadratic are integers, the probability that the quadratic factorises over the integers tends to zero as the range from which the integers are drawn grows (see [1]). But what if we ask about a behaviour less restrictive than factorising, say merely having real roots? This is the problem that concerns this Article.

Topological co-indices are topological files that thinks about the non-touching arrangements of vertices. Here, we think about the going with some notable topological co-indices. The mathematical q...

The article is devoted to the issues of constructing effective computer-oriented components of the methodological system of teaching the disciplines «Linear Algebra» and «Probability Theory with Elements of Mathematical Statistics» provided for in the educational and professional program «Secondary Education (Mathematics)» of the first level of higher education in the specialty 014 Secondary Education (Mathematics). The article analyzes the methodological aspects of the effective organization of computations when finding the angle between a given vector and a nonzero subspace of Euclidean space, as well as using the least squares method for processing experimental data. The theoretical and practical information known to students-mathematicians from the corresponding sections of these academic disciplines is briefly presented. Analyzed educational, methodological and scientific literature used in teaching linear algebra and probability theory with elements of mathematical statistics; the expediency of using computer-oriented elements of teaching mathematical disciplines of future mathematics teachers has been substantiated. The authors proposed the use of computer-oriented learning elements in the processing of the content of disciplines and the development of test tasks of different levels of complexity in linear algebra and probability theory with elements of mathematical statistics in order to objectively assess the level of students' knowledge and timely correct individual educational trajectories. The article provides examples of the application of computer-oriented elements of teaching linear algebra and probability theory with elements of mathematical statistics, and also analyzes the methodological features of the organization of calculations in the software mathematical environment Mathcad. The methodological and practical materials presented in the article can be useful for students to organize and activate independent scientific and pedagogical activities, teachers of secondary educational institutions, heads of optional and circle work of students, teachers of linear algebra and probability theory courses with elements of mathematical statistics of pedagogical higher educational institutions. Key words: methods of teaching mathematics, computer-oriented elements of teaching mathematics, linear algebra, probability theory, mathematical statistics, Euclidean space, non-zero subspace of Euclidean space, least squares method.

Distributed fiber acoustic sensing (DAS) technology can measure acoustic waves along long-distance optical fiber with high spatial resolution. Due to its unique advantages, DAS has been applied in wide fields. However, DAS currently still suffers from limited sensitivity and severe signal fading. Due to the inherent random nature and low signal intensity of Rayleigh backscattering, it is a challenge to essentially eliminate these fundamental limitations. Here a DAS system using its distributed sensing feature is demonstrated with dense multichannel signal integration, complete elimination of signal fading and 100-fold improvement of signal-to-noise ratio (SNR) are experimentally demonstrated. These results provide a simple and effective method to improve DAS performance in practical applications. As far as we know, it is the first time that the distributed sensing characteristic is utilized to tackle fading and sensitivity for DAS, simultaneously. It is believed that this method will promote the process of DAS large-scale applications and help DAS to attract widespread attentions, especially in intrusion detection, acoustic emission flaw detection, near-surface dynamics and characterization, etc.

Conventional calculations of the inertia tensor in undergraduate physics course are usually done for highly symmetrical bodies. Students might therefore get the impression that the moment of inertia about any axis through the center of mass is the same only for bodies with the highest degree of symmetry relative to this point, e.g., for spheres. A simple, seemingly counterintuitive example is presented, showing that the moment of inertia of a non-regular body, here an assembly of material points, can be the same about any axis passing through its center of mass.

One of the problems when studying statistics at university is understanding the mathematical symbols sigma and integral. The symbols sigma and integral are handled in Mathematics B and Mathematics II on High school curriculum. In order for students who learned a narrow range of high school mathematics such as without learning Math. B and/or Math. II to study statistics, this paper attempts to clarify where the major points are of learning mathematics in elementary and secondary education.

We use Markov chain Monte Carlo (MCMC) analysis to construct a three-question math quiz to assess key skills needed for introductory finance. We begin with data collected from a ten-question criterion-referenced math quiz given to 314 undergraduates on the first day of class. MCMC indicates the top three questions for predicting overall course performance. This brief assessment explains approximately 10% of the variance in undergraduate introductory course performance. The quiz holds potential within the classroom for students, instructors, and university administration to better identify math skill deficiencies relevant to finance.

Depicting the reason for the mismatch between instructor expectations of students’ performance in advanced courses and their actual performance has been a challenging issue for a long time, which raises the question of why such a mismatch exists. An implicit reason for this mismatch is the student’s weakness in prerequisite course skills. To solve this challenge, this research proposes a new graph mining algorithm combined with statistical analysis to reveal the dependency relationships between Course Learning Outcomes (CLOs) of prerequisite and advanced courses. In addition, a new model is built to predict students’ performance in the advanced courses based on prerequisites. The contributions of this research are threefold: (1) Modeling: three models are built based on bipartite graphs. The first model is a bipartite graph to model the relationships between the CLOs of different courses. This bipartite graph is constructed using a new calculated dependency measure based on a statistical analysis of students’ grades. Then the relationships between the Learning Outcomes are discovered by extracting the maximal bipartite cliques in the graph. The second model is built to model the relationships between students and the prerequisite CLOs. The maximal bipartite cliques are then extracted from this graph to discover the maximal set of students who share the same study weaknesses. The third model is built to model the relationships between students and the advanced CLOs. The maximal bipartite cliques are then extracted to discover the maximal set of students who are expected to share the same study weaknesses in the advanced course. Therefore, the same remedial actions can be used towards this group of students. (2) Algorithm: A new maximal bipartite cliques enumeration algorithm is proposed to extract the targeted patterns and relationships between CLOs themselves and between students and CLOs. (3) Applicability: The proposed models and algorithm have been applied using a real educational data set collected from one university. Other real datasets are used to conduct an empirical evaluation to assess the maximal bipartite enumeration algorithm’s correctness, the running time of the inflation and enumeration steps, and the overhead of the inflation algorithm on the size of the generated general graph. The evaluation proves that the proposed algorithm is accurate, efficient, effective, and applicable to real-world graphs more than the traditional algorithm.

Visual representations, such as diagrams, are known to be valuable tools in problem solving and proof construction. However, previous studies have shown that simply having access to a diagram is not sufficient to improve students’ performance on mathematical tasks. Rather, students must actively extract information about the problem scenario from their diagrams for them to be useful. Furthermore, several studies have described the behaviors of mathematicians and students when solving problems and writing proofs, but few have discussed students’ behaviors in the context of proof writing in introductory point-set topology. We present a case study of an undergraduate, Stacey, enrolled in a general topology course. Throughout a semester, we presented Stacey with several proof-related tasks and examined how and why Stacey used diagrams when working on these tasks. Based on our analysis, we concluded that Stacey’s diagram creation and subsequent use during the construction of a given proof was an effort to identify the key idea of the proof. We describe Stacey’s overall proving behaviors through the lens of a problem-solving framework and present Stacey’s use of diagrams as an aid to discovering the key ideas of proofs in topology.

Series and limits were used to calculate the centre of mass of homogeneous circular sector plates. The method presented can be used to teach students about discrete and continuous systems.

With the successes of the Laser Interferometer Gravitational-wave Observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state., Comment: (27 pages, 1 figure, accepted for publication in Am. J. Phys.)

The question from the title is raised because in almost all introductory physics courses/textbooks the atmospheric pressure has been attributed to the weight of the column of air from a given level in the atmosphere up to its top. “Air is pressing on air.” However the same textbooks, in the chapter on the kinetic theory of gases, tell students that a pressure in a gas enclosed in a container is due to variation of momentum suffered by the molecules, as a consequence of their impacts against the walls or against the surface of a test object placed inside the gas. The higher the number and average energy of molecules, the higher the pressure. Despite the fact that the atmosphere consists of matter that is mostly in the gas state, introductory physics textbooks do not justify why the concept of pressure from the kinetic theory of gases would not explain the physical nature of pressure in the atmosphere. Ricci-Torricelli’s experiments realized by scientists from the Academia del Cimento suggest to abandon the “weight based” explanation in favor of a density-based explanation of atmospheric pressure. The dependence on altitude of air density, as well as of air pressure, is caused by Earth’s gravitational force acting on all chaotically moving molecules in the atmosphere.

Large-scale astronomical surveys from the last decades have turned the usage of catalogs and archival data into one of the primary skills of contemporary observational astronomers. Virtual observatory tools give high-school and university students the opportunity to conduct astronomical research by themselves, using freely available observational data. For this purpose, they need basic theoretical knowledge in astronomy. The current paper includes a review of this theoretical knowledge as well as a review of Virtual observatory tools suitable for students. Results obtained by students using VO tools at the Beli Brezi Summer School in Astronomy and Astrophysics are presented as well.

A gravity pendulum experiment in which a smartphone serves as pendulum bob and as measurement device is described. As the pendulum oscillates, the smartphone gyroscope and magnetometer sensors are used to simultaneously record the bob's angular velocity and angular displacement. While the angular velocity is measured directly via the gyroscope sensor, the angular displacement is deduced from the change of the magnetometer reading while the smartphone oscillates in the constant magnetic field of the Earth. These two independent measurements enable us to derive the phase space representation of the pendulum motion directly from experimental data. Further data analysis yields energy plots, including the periodic change between the pendulum's kinetic and potential energy and the energy dissipation due to friction. Detailed instructions on how to adopt the experiment, which is both suitable as a lecture demonstration as well as an instructional laboratory experiment, are provided in the supplementary material.

The study aims to investigate students' conceptualisation of the quantum numbers concept through their verbal descriptions and pictorial representations of the corresponding atomic structures. The ...

The University timetabling is a generalization of the well-known class-teacher timetabling model, where in addition to lectures given by a single teacher to a single class, there are some lectures given by a single teacher to a group of classes simultaneously. One looks for a minimum number of periods in which to complete all lectures without conflicts. The problem is NP-hard in the strong sense even if the number of groups is three, but it is polynomially solvable for two groups. In the latter case, it has been conjectured that the minimum number of periods in which to complete all lectures without conflicts equals ⌈ T ⌉ , where T is the optimal value of an L P -relaxation. The L P -relaxation permits fractions of periods in feasible solutions. We prove this conjecture in this paper.

This article poses the inverse problem of stabilizing a spherical pendulum in a given position using high-frequency vibration of a suspension point. The position of the pendulum is determined by the angle between the pendulum rod and the vertical. For any given position of the pendulum, we obtain a one-parameter series of oblique vibration characteristics (the amplitude of the vibration velocity and the angle between the vibration velocity vector and the vertical) that stabilize the pendulum in this position. For the series obtained, the domains of attraction are determined; the initial points, from which, under the action of vibration, a given stable position of the pendulum is established.

The article presents a systematic theoretical analysis of the phenomenon of «hardiness». The main components of hardiness and their relationship are described. Features of formation of successful activity in the course of students, hardiness are defined. The influence of the components of hardiness on the formation of successful activity is characterized. The results of an empirical study of the level of hardiness of future teachers – psychologists are presented. On the basis of empirical data and literature analysis, the model «Dynamics of the development of hardiness in future teachers – psychologist» is built. Attempts are made to concretize the accumulated theoretical experience about the phenomenon of hardiness and its connection with successful activity. The results of measuring the dynamics of the development of hardiness in future teachers – psychologists and the impact on the formation of their successful activities are presented.

This study investigated the role of the bar model method, a significant aspect of the Singapore mathematics curriculum, in the remediation of seventh-grade students’ errors on algebra word problems...

The paper describes partially ordered sets and their connection with Boolean algebras and rings. A number of statements are presented, which should be studied when getting acquainted with algebraic structures for students of the Faculty of Physics and Mathematics, specialty 014 of Secondary Education (Computer Science) in the course of discrete mathematics. A number of examples are given that clearly explain the considered structures., У статтi розглядається частково впорядкованi множини та їх зв’язок з булевими алгебрами та кiльцями. Представлено ряд тверджень, якi доцiльно вивчати при знайомствi з алгебраїчними структурами студентам фiзико-математичного факультету спецiальностi 014 Середня освiта (Iнформатика) в курсi дискретної математики. Наведено ряд прикладiв, якi наочно пояснюють розглянутi структури.

Of the many topics generally taught in undergraduate or graduate optics courses, the propagation of light through crystals with electrical anisotropy is one of the most difficult topics for students and for teachers who do not work in this field. In particular, the mathematics and equations are complex vectorial equations, and this complicates the visualization of the implications of the theory. Also, the standard textbooks and papers on the subject have led to some misconceptions on this problem by presenting results only for specific cases of the theory in graphical form. For example, it is a general belief that after refraction of an incident wave on the crystal, the polarization state of the two waves propagating inside the crystal, given by their displacement vectors D, are always orthogonal to each other. However, this is not the case, as explained and illustrated for uniaxial crystals by Aleman-Castaneda and Rosete-Aguilar [J. Opt. Soc. Am. A 33(4), 677–682 (2016)]. The purpose of this paper is to help students and teachers visualize the solution to Maxwell's equations in electrical anisotropic crystals. An algorithm has been written in mathematica so the reader can use it to visualize the normal surface in uniaxial and biaxial crystals. This algorithm can be used as the basis of computational projects for students to better understand the equations involved, or as a visualization tool for teachers in class.

This is a template for projects in the SIM lab.

This is a template for projects in the SIM lab.

This is a template for projects in the SIM lab.

Output data and matlab code for its lecture.

The raw and processed data and the regression results for my paper The Quantity Theory Remains Alive

One of the problems to be solved in mathematics is a proof. At the college level, proving to be more important than the previous levels. When someone has a hunch about something (theorem), one of the most appropriate way to ensure that something (theorem) is true is by proving mathematically valid (formally). That is, the truth of the theorem is based on logical statements. However, proof of this is a problem when introduced during the learning. Not only the beginning students, final year students, moreover those already graduated are still experiencing difficulties in proving. This study attempts to describe the understanding of the concept of proof in mathematics and describes the ability of students in validating and compiling of proof in mathematics, particularly in Euclidean geometry using qualitative research approach, the type of case studies.

Compton Scattering 2nd Year Lab Experiment

The article discusses the achievements in elementary particle physics, teaching problems of elementary particle physics in pedagogical universities and ways to solve them.

Lecture notes on linear models and linear mixed models

This article explores the concept of dualism, the concept of duality in geometry and algebra. The advantages of using dual tasks in teaching mathematics in school mathematics on specific tasks are highlighted and demonstrated.

Reproducibility package for our tutorial article on the bootstrap.

Separation of Powers with Party Polarization. A Lab Experiment.

This is a template for projects in the SIM lab.

The paper presents methods of using the methods of "matryeshka" and "consecutive grinding" to increase the level of students' mastery of complex topics of theoretical mechanics. It also explains how to use these methods to calculate the derivatives of complex functions and to express the angular momentum of a point in various coordinate systems.

An integrated course in mathematics, preparation for teaching the elements of arithmetic, algebra and geometry, as well as the basics of algebra and geometry, labor education, which allows you to master the arithmetic material.

This article is about dictation and its types. It provides detailed information on the types of dictations. Here's a few basic facts about a stomp pad and how it is used. It explains the purpose of dictation from elementary school students.

This project modeled dynamic foot morphology through stance phase by predicting shape modes with anthropometric and kinematic predictors.

We study the error threshold properties of holographic quantum error-correcting codes. We demonstrate that holographic CFTs admit an algebraic threshold, which is related to the confinement-deconfinement phase transition. We then apply geometric intuition from holography and the Hawking-Page phase transition to motivate the CFT result, and comment on potential extensions to other confining theories., Comment: 6 pages, 3 figures. Updated to match published version

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis., 10 pages, comments welcome