Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with ...discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.
Background: One important function of school mathematics curriculum is to prepare high school students with the knowledge and skills needed for university education. Identifying them empirically will ...help making sound decisions about the contents of high school mathematics curriculum. It will also help students to make informed choices in course selection at high school. In this study, we surveyed university faculty who teach first year university students about the mathematical knowledge and skills that they would like to see in incoming high school graduates. Materials and methods: Data were collected from 122 faculty from social science (history, law, psychology) and engineering departments (electrical/electronics and computer engineering). Participants were asked to indicate which high school mathematics topics and skills they thought were important to be successful at university education in their field. Results were compared across social science and engineering departments. Results: There were 34 topics that were rated important by engineering departments but not the social science departments. Mathematical skills were rated high by both types of academic disciplines. Conclusions: National high school mathematics curriculum at present falls short of providing high school students the necessary mathematical background for prospective social science students at university.
Based on lectures given at the renowned Villa de Leyva summer school, this book provides a unique presentation of modern geometric methods in quantum field theory. Written by experts, it enables ...readers to enter some of the most fascinating research topics in this subject. Covering a series of topics on geometry, topology, algebra, number theory methods and their applications to quantum field theory, the book covers topics such as Dirac structures, holomorphic bundles and stability, Feynman integrals, geometric aspects of quantum field theory and the standard model, spectral and Riemannian geometry and index theory. This is a valuable guide for graduate students and researchers in physics and mathematics wanting to enter this interesting research field at the borderline between mathematics and physics.
In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the ''invariance'' of the objects of ...the ''Darboux theory of integrability''. In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied. ProQuest: ... denotes formulae omitted.
Reproduction of kernel Hilbert spaces offers an attractive setting for imaginary time path integrals, since they allow to naturally define a probability on the space of paths, which is equal to the ...probability associated with the paths in Feynman's path integral formulation. This study shows that if the propagator is Gaussian, its variance equals the squared norm of a linear functional on the space of paths. This can be used to rederive the harmonic oscillator propagator, as well as to offer a finite-dimensional perturbative approximation scheme for the time-dependent oscillator wave function and its ground state energy, and its bound error. The error is related to the rate of decay of the Fourier coefficients of the time-dependent part of the potential. When the rate of decay increases beyond a certain threshold, the error in the approximation over a subspace of dimension n is of order (1/n
3
).
This study examines how students’ opportunities to engage in argumentative activity are shaped by the teacher, the class, and the mathematical topic. It compares the argumentative activity between ...two classes taught by the same teacher using the same textbook and across two beginning algebra topics—investigating algebraic expressions and equivalence of algebraic expressions. The study comprises two case studies in which each teacher taught two 7th grade classes. Analysis of classroom videotapes revealed that the opportunities to engage in argumentative activity with the topic investigating algebraic expressions were similar in each teacher's two classes. By contrast, substantial differences were found between one teacher's classes with regard to the opportunities to engage in argumentative activity with the topic equivalence of algebraic expressions. The discussion highlights how the interplay between the characteristics of the mathematical topic, the characteristics of the class, and the characteristics of the teacher contributed to the shaping of students’ opportunities to engage in argumentative activity.
This study compares students’ opportunities to engage in transformational (rule-based) algebraic activity between 2 classes taught by the same teacher and across 2 topics in beginning algebra: ...forming and investigating algebraic expressions and equivalence of algebraic expressions. It comprises 2 case studies; each involves a teacher teaching in two 7th grade classes. All 4 classes used the same textbook. Analysis of classroom videotapes (15–19 lessons in each class) revealed that the opportunities to engage in transformational algebraic activity related to forming and investigating algebraic expressions were similar in each teacher’s 2 classes. By contrast, substantial differences were found between 1 teacher’s classes with regard to the opportunities to engage in transformational algebraic activity related to equivalence of algebraic expressions. The discussion highlights the contribution of the interplay among the mathematical topic, the teacher, and the class to shaping students’ learning opportunities. Specifically, the mathematical topic appeared to play a prominent role in certain situations, with the topic involving deductive reasoning generating high variation in classes of 1 teacher but not in the other’s.
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