The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories

In this paper, we investigate the main algebraic properties of the maximally superintegrable system known as "Perlick system type I". All possible values of the relevant parameters, $K$ and $\beta$, are considered. In particular, depending on the sign of the parameter $K$ entering in the metrics, the motion will take place on compact or non compact Riemannian manifolds. To perform our analysis we follow a classical variant of the so called factorization method. Accordingly, we derive the full set of constants of motion and construct their Poisson algebra. As it is expected for maximally superintegrable systems, the algebraic structure will actually shed light also on the geometric features of the trajectories, that will be depicted for different values of the initial data and of the parameters. Especially, the crucial role played by the rational parameter $\beta$ will be seen "in action".Comment: 16 pages, 7 figure

Integrability, Supersymmetry and Coherent States. A volume in honour of Professor Véronique Hussin

ISBN 978-3-030-20087-

Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.Comment: 23 Pages, 3 figures, To appear in Nonlinearit

Superintegrability of the Fock-Darwin system

The Fock-Darwin system is analysed from the point of view of its symmetry properties in the quantum and classical frameworks. The quantum Fock-Darwin system is known to have two sets of ladder operators, a fact which guarantees its solvability. We show that for rational values of the quotient of two relevant frequencies, this system is superintegrable, the quantum symmetries being responsible for the degeneracy of the energy levels. These symmetries are of higher order and close a polynomial algebra. In the classical case, the ladder operators are replaced by ladder functions and the symmetries by constants of motion. We also prove that the rational classical system is superintegrable and its trajectories are closed. The constants of motion are also generators of symmetry transformations in the phase space that have been integrated for some special cases. These transformations connect different trajectories with the same energy. The coherent states of the quantum superintegrable system are found and they reproduce the closed trajectories of the classical one.Comment: 21 pages,16 figure

A Study on Reflective Reciprocal Peer Coaching for Pre-service Teachers: Change in Reflectivity

Reflective practice is considered as an effective way for professional development in order to gain awareness of one’s own teaching as well as to compete with the changing needs of the students. Especially in pre-service period, when pre-service teachers work cooperatively with their peers in a reciprocal fashion towards reflectivity, it has a potential to promote advancement in reflective practices and help them focus on the underlying meaning behind their actions. Based on these ideas, this study aimed at engaging pre-service teachers in a reflective reciprocal peer coaching experience and investigating whether such experience caused any changes in their reflectivity. For this purpose, 12 pre-service teachers in a Turkish ELT context participated in the study and a reflective reciprocal peer coaching program was implemented with a training aspect. In a mixed method study design, change in participants’ reflectivity was measured with a profile of reflective thinking attributes scale quantitatively and data were supported qualitatively with reflective diaries, video recordings of post-conference sessions and focus-group interviews. Results of quantitative and qualitative analyses put forward that the pre-service teachers advanced in their reflectivity throughout the reflective reciprocal peer coaching practice and benefited much from this experience before they embark into professional life. This study provides valuable implications to use reflection embedded in a peer coaching program and offers suggestions for teacher educators

Modal Abstractions for Virtualizing Memory Addresses

Operating system kernels employ virtual memory management (VMM) subsystems to virtualize the addresses of memory regions in order to to isolate untrusted processes, ensure process isolation and implement demand-paging and copy-on-write behaviors for performance and resource controls. Bugs in these systems can lead to kernel crashes. VMM code is a critical piece of general-purpose OS kernels, but their verification is challenging due to the hardware interface (mappings are updated via writes to memory locations, using addresses which are themselves virtualized). Prior work on VMM verification has either only handled a single address space, trusted significant pieces of assembly code, or resorted to direct reasoning over machine semantics rather than exposing a clean logical interface. In this paper, we introduce a modal abstraction to describe the truth of assertions relative to a specific virtual address space, allowing different address spaces to refer to each other, and enabling verification of instruction sequences manipulating multiple address spaces. Using them effectively requires working with other assertions, such as points-to assertions in our separation logic, as relative to a given address space. We therefore define virtual points-to assertions, which mimic hardware address translation, relative to a page table root. We demonstrate our approach with challenging fragments of VMM code showing that our approach handles examples beyond what prior work can address, including reasoning about a sequence of instructions as it changes address spaces. All definitions and theorems mentioned in this paper including the operational model of a RISC-like fragment of supervisor-mode x86-64, and a logic as an instantiation of the Iris framework, are mechanized inside Coq