Conceptually driven and visually rich tasks in texts and teaching practice: the case of infinite series

The study we report here examines parts of what Chevallard calls the institutional dimension of the students’ learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duval's Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concept's significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests

Coherent Control of Isotope Separation in HD+ Photodissociation by Strong Fields

The photodissociation of the HD+ molecular ion in intense short- pulsed linearly polarized laser fields is studied using a time- dependent wave-packet approach where molecular rotation is fully included. We show that applying a coherent superposition of the fundamental radiation with its second harmonic can lead to asymmetries in the fragment angular distributions, with significant differences between the hydrogen and deuterium distributions in the long wavelength domain where the permanent dipole is most efficient. This effect is used to induce an appreciable isotope separation.Comment: Physical Review Letters, 1995 (in press). 4 pages in revtex format, 3 uuencoded figures. Full postcript version available at: http://chemphys.weizmann.ac.il/~charron/prl.ps or ftp://scipion.ppm.u-psud.fr/coherent.control/prl.p

The Largest Known Maars on Earth, Seward Peninsula, Northwest Alaska

The Espenberg Maars on the northern Seward Peninsula of Alaska were formed by a series of Pleistocene basaltic eruptions through thick permafrost. The maars were excavated as much as 300 m into older lithologies; ranging from 4 to 8 km in diameter, they are the four largest known maars on earth. Hydromagmatic eruptions which derive water from ground ice are evidently extremely explosive. The high heat capacity of ice in permafrost modulates the supply of water interacting with magma during the eruption, producing consistently low coolant-to-fuel ratios in an environment with a sustained, abundant water supply. The Espenberg Maars demonstrate that, under certain conditions, eruptions which involve the interaction of lava and permafrost are powerful enough to produce craters as large as small calderas.Les maars de l'Espenberg situés dans la partie septentrionale de la péninsule Seward en Alaska ont été formés par une série d'éruptions basaltiques datant du pléistocène, à travers une forte épaisseur de pergélisol. Les maars ont été creusés à une profondeur allant jusqu'à 300 m dans d'anciennes roches; d'un diamètre variant entre 4 et 8 km, ils sont les quatre plus grands maars connus sur Terre. Les éruptions hydromagmatiques qui tirent l'eau de la glace de sol sont, comme on l'a déjà constaté, extrêmement explosives. La grande capacité thermique de la glace dans le pergélisol détermine l'approvisionnement en eau qui interagit avec le magma au cours de l'éruption, donnant régulièrement lieu à un faible rapport refroidissant / combustible dans un environnement où l'eau est constamment abondante. Les maars de l'Espenberg démontrent que, dans certaines conditions, les éruptions qui déclenchent une interaction lave-pergélisol sont suffisamment puissantes pour donner naissance à des cratères de la grandeur de petites calderas

Theoretical study of a cold atom beam splitter

A theoretical model is presented for the study of the dynamics of a cold atomic cloud falling in the gravity field in the presence of two crossing dipole guides. The cloud is split between the two branches of this laser guide, and we compare experimental measurements of the splitting efficiency with semiclassical simulations. We then explore the possibilities of optimization of this beam splitter. Our numerical study also gives access to detailed information, such as the atom temperature after the splitting

Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node

Monitoring Partially Synchronous Distributed Systems using SMT Solvers

In this paper, we discuss the feasibility of monitoring partially synchronous distributed systems to detect latent bugs, i.e., errors caused by concurrency and race conditions among concurrent processes. We present a monitoring framework where we model both system constraints and latent bugs as Satisfiability Modulo Theories (SMT) formulas, and we detect the presence of latent bugs using an SMT solver. We demonstrate the feasibility of our framework using both synthetic applications where latent bugs occur at any time with random probability and an application involving exclusive access to a shared resource with a subtle timing bug. We illustrate how the time required for verification is affected by parameters such as communication frequency, latency, and clock skew. Our results show that our framework can be used for real-life applications, and because our framework uses SMT solvers, the range of appropriate applications will increase as these solvers become more efficient over time.Comment: Technical Report corresponding to the paper accepted at Runtime Verification (RV) 201

A Short Counterexample Property for Safety and Liveness Verification of Fault-tolerant Distributed Algorithms

Distributed algorithms have many mission-critical applications ranging from embedded systems and replicated databases to cloud computing. Due to asynchronous communication, process faults, or network failures, these algorithms are difficult to design and verify. Many algorithms achieve fault tolerance by using threshold guards that, for instance, ensure that a process waits until it has received an acknowledgment from a majority of its peers. Consequently, domain-specific languages for fault-tolerant distributed systems offer language support for threshold guards. We introduce an automated method for model checking of safety and liveness of threshold-guarded distributed algorithms in systems where the number of processes and the fraction of faulty processes are parameters. Our method is based on a short counterexample property: if a distributed algorithm violates a temporal specification (in a fragment of LTL), then there is a counterexample whose length is bounded and independent of the parameters. We prove this property by (i) characterizing executions depending on the structure of the temporal formula, and (ii) using commutativity of transitions to accelerate and shorten executions. We extended the ByMC toolset (Byzantine Model Checker) with our technique, and verified liveness and safety of 10 prominent fault-tolerant distributed algorithms, most of which were out of reach for existing techniques.Comment: 16 pages, 11 pages appendi

The propensity of molecules to spatially align in intense light fields

The propensity of molecules to spatially align along the polarization vector of intense, pulsed light fields is related to readily-accessible parameters (molecular polarizabilities, moment of inertia, peak intensity of the light and its pulse duration). Predictions can now be made of which molecules can be spatially aligned, and under what circumstances, upon irradiation by intense light. Accounting for both enhanced ionization and hyperpolarizability, it is shown that {\it all} molecules can be aligned, even those with the smallest static polarizability, when subjected to the shortest available laser pulses (of sufficient intensity).Comment: 8 pages, 4 figures, to be submitted to PR

Optimizing the fast Rydberg quantum gate

The fast phase gate scheme, in which the qubits are atoms confined in sites of an optical lattice, and gate operations are mediated by excitation of Rydberg states, was proposed by Jaksch et al. Phys. Rev. Lett. 85, 2208 (2000). A potential source of decoherence in this system derives from motional heating, which occurs if the ground and Rydberg states of the atom move in different optical lattice potentials. We propose to minimize this effect by choosing the lattice photon frequency \omega so that the ground and Rydberg states have the same frequency-dependent polarizability \alpha(omega). The results are presented for the case of Rb.Comment: 5 pages, submitted to PR