A Calabi-Yau algebra with E6E_6 symmetry and the Clebsch-Gordan series of sl(3)sl(3)

Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra.Comment: 24 page

Generalized squeezed-coherent states of the finite one-dimensional oscillator and matrix multi-orthogonality

A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N+1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed-coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2x2 matrix orthogonal polynomials and the squeezed-coherent states tend to those of the standard oscillator.Comment: 18 pages, 1 figur

An Algebraic Model for the Multiple Meixner Polynomials of the First Kind

An interpretation of the multiple Meixner polynomials of the first kind is provided through an infinite Lie algebra realized in terms of the creation and annihilation operators of a set of independent oscillators. The model is used to derive properties of these orthogonal polynomials

Remote capacitive sensing in two-dimension quantum-dot arrays

We investigate gate-defined quantum dots in silicon on insulator nanowire field-effect transistors fabricated using a foundry-compatible fully-depleted silicon-on-insulator (FD-SOI) process. A series of split gates wrapped over the silicon nanowire naturally produces a $2\times n$ bilinear array of quantum dots along a single nanowire. We begin by studying the capacitive coupling of quantum dots within such a 2$\times$2 array, and then show how such couplings can be extended across two parallel silicon nanowires coupled together by shared, electrically isolated, 'floating' electrodes. With one quantum dot operating as a single-electron-box sensor, the floating gate serves to enhance the charge sensitivity range, enabling it to detect charge state transitions in a separate silicon nanowire. By comparing measurements from multiple devices we illustrate the impact of the floating gate by quantifying both the charge sensitivity decay as a function of dot-sensor separation and configuration within the dual-nanowire structure.Comment: 9 pages, 3 figures, 35 cites and supplementar

Relativistic analysis of the LISA long range optical links

The joint ESA/NASA LISA mission consists in three spacecraft on heliocentric orbits, flying in a triangular formation of 5 Mkm each side, linked by infrared optical beams. The aim of the mission is to detect gravitational waves in a low frequency band. For properly processing the science data, the propagation delays between spacecraft must be accurately known. We thus analyse the propagation of light between spacecraft in order to systematically derive the relativistic effects due to the static curvature of the Schwarzschild spacetime in which the spacecraft are orbiting with time-varying light-distances. In particular, our analysis allows to evaluate rigorously the Sagnac effect, and the gravitational (Einstein) redshift.Comment: 6 figures; accepted for publication in PR

An infinite family of superintegrable Hamiltonians with reflection in the plane

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly solvable. The angular part of the wave function is expressed in terms of little -1 Jacobi polynomials. The spectra exhibit "accidental" degeneracies. The superintegrability of the model is proved using the recurrence relation approach. The (higher-order) constants of motion are constructed and the structure equations of the symmetry algebra obtained.Comment: 19 page

Sur l'origine de l'augmentation apparente des inondations en région méditerranéenne

En septembre 2002, les régions méditerranéennes françaises et notamment le département du Gard ont été affectées par des précipitations d'une extrême intensité. On estime que 80% de ce département a été inondé, on dénombre 23 victimes et les dégâts ont été évalués à 1.2 milliards d'euros. Cette catastrophe hydrologique soulève à nouveau les problèmes de la fréquence de ces événements et de l'augmentation des forts cumuls de pluie ces dernières années. L'objet de cet article est d'apporter quelques éléments de réponse, notamment à travers l'analyse régionale des pluies extrêmes journalières ayant affecté la région Languedoc-Roussillon de 1958 à 2002.La fréquence régionale des pluies extrêmes est estimée en prenant en compte la superficie couverte par ces événements en fonction des hauteurs pluviométriques. A l'échelle régionale la période de retour de l'événement varie entre 80 ans pour la superficie touchée par au moins 200 mm à 140 ans pour celle couverte par 300 mm.La stationnarité des fréquences des pluies extrêmes est analysée à partir des chroniques du nombre annuel d'événements pluvieux dépassant 200 mm, 250 mm et 300 mm en 24h maximum, entre 1958 et 2002 sur la région. Les tests de stationnarité ne révèlent pas de tendance significative à l'augmentation de ces fréquences. Les données historiques aboutissent aux mêmes conclusions. L'augmentation réelle des inondations est en fait principalement liée à l'augmentation de la vulnérabilité des bassins.In September 2002, the Gard department in the South of France was affected by heavy precipitation that covered a broad geographical area. It was estimated that 80% of the department was flooded; there were 23 victims and the damage was evaluated to be 1.2 billion euros. This hydrological catastrophe raised questions about a possible increase in the frequency of these events during recent years, since several other severe flooding events have been observed in the region over the last 15 years. The aim of this article is to explore these questions through a regional analysis of the extreme daily rainfall that affected the Languedoc-Roussillon region between 1958 and 2002. The daily rain data were used because they are the most available type of information over the observation period. Usually, the rainfall hazard description is based on statistical analysis of the maximum rainfall depth observed at a given rain gauge. However, because the spatial variability of rainfall in the Mediterranean region, such results are only representative of local rainfall conditions. Moreover, this type of analysis does not take into account the spatial coverage of the precipitation, which is another factor influencing the resulting floods. Thus, the regional frequency of extreme rainfall was estimated by taking into account the area covered according to a given rainfall depth. For each rainfall event, a rain field was built using a kriging interpolation (NEPPEL et al., 1997). The isohyet area defined a rainfall threshold from 10 to 300 mm with a step of 10 mm calculated for each rainfall event. For each rainfall depth from 10 to 300 mm with a step of 10 mm, the probability distribution of the isohyet area was estimated. The regional rainfall hazards were described with the Depth-Area-Frequency curves (DAF) for 24-h periods. It was shown that at on regional scale, the return period of the last event varied between 80 years for the surface affected by at least 200 mm and 140 years for the surface covered by 300 mm. Compared with other major events that have occurred in the region, it appears that the September 2002 event one was characterized by :1. the spatial extension of the heavy rainfall, for example more than 1800 km² were affected by at least 400 mm in less than 24 h;2. the spatial localisation of the heaviest rainfall depths, which were measured over the highest relief (1000 m to 1500 m) as usual in the 'cévenols' meteorological situation, but rather in the plain where the altitude lies between 200 m and 300 m.The stationnarity analysis of the extreme rainfall frequency was based on the annual number of events exceeding 200 mm, 250 mm and 300 mm over a 24 h maximum duration, between 1958 and 2002. The hypothesis of random events against the hypothesis of a trend or a sudden break in the mean was examined through several statistical tests. The procedures used were the rank correlation test, PETTITT's test, BUISHAND's test, HUBERT's segmentation procedure, a linear regression procedure, and the turning points procedure. Detailed descriptions of these tests can be found in KENDALL and STUART (1977), LUBES-NIEL et al. (1998) and WMO (2000). Except for the rank correlation test, all the procedures led to the conclusion that the three series are randomly distributed at the level of significance 1%, 5% and 10% respectively. Thus no significant increase in extreme rainfall frequency seems to appear. Although the study period was short, 45 years, compared with climatological variability, LUBES-NIEL et al. (1998) show that the procedures used were adapted in detecting trends in 50-yr time series. In considering historical rainfall data before 1958 in the same region, at least two extreme rainfall events could be compared with the event on 8-9 September 2002: in October 1940, 840 mm of rainfall were measured during 24 h in the Pyrénées-Orientales district and in September 1900, 940 mm were observed over 24 h in Valleraugue, upstream in the Herault catchment. Furthermore, if the evolution of the rain gauge network density is taken into account, one can argue that such an event could have occurred more frequently. Indeed, the number of rain gauges has varied from 162 gauges in 1900 to 330 today. It has been shown that the number of observed rainfall events varied according to the area of the events and the network density (NEPPEL et al., 1998b). For example, an event of 150 km2 (corresponding to the area covered by more than 600 mm in September 2002) had a probability of 70% to be observed by the network between 1958 and 1993. If one considers the period 1920-1939, this probability decreases to 30%.In addition, the basin vulnerability has increased. The regional population has grown from 1,460,000 inhabitants in 1949 to 2,300,000 in 2000. At the same time, urbanization has expanded widely. Moreover, this new population came from other districts, and they are not familiar with the Mediterranean rainfall regime and the resulting flash floods. Buildings have often been constructed near rivers, which are attractive building sites, and sometimes even in the river's main channel, increasing the flooding risk and the flood damages. Thus, rather than climate change, for which the effect on extreme rainfalls cannot be proved, the development of basin urbanisation and vulnerability could explain the apparent increase in floods. As the regional population is expected to reach more than 3,000,000 by 2030, it is necessary to take into account the flood risk in future urban planning

SOLUTIONS OF THE LANDAU-VLASOV EQUATION IN NUCLEAR PHYSICS

The properties of Vlasov equation solutions obtained by projection on coherent state basis are discussed. Such solutions satisfy stationarity conditions and satisfactorily describe the average diffusivity of nuclear phase space and reproduce the bulk properties of nuclei. Sampling methods and their effects on dynamics are discussed for the study of heavy ion reactions at intermediate energies. The non-local Gogny force is easily computable on this basis which allows to use it for dynamical nuclear studies

The Dynamics of Sustained Reentry in a Loop Model with Discrete Gap Junction Resistance

Dynamics of reentry are studied in a one dimensional loop of model cardiac cells with discrete intercellular gap junction resistance ($R$). Each cell is represented by a continuous cable with ionic current given by a modified Beeler-Reuter formulation. For $R$ below a limiting value, propagation is found to change from period-1 to quasi-periodic ($QP$) at a critical loop length ($L_{crit}$) that decreases with $R$. Quasi-periodic reentry exists from $L_{crit}$ to a minimum length ($L_{min}$) that is also shortening with $R$. The decrease of $L_{crit}(R)$ is not a simple scaling, but the bifurcation can still be predicted from the slope of the restitution curve giving the duration of the action potential as a function of the diastolic interval. However, the shape of the restitution curve changes with $R$.Comment: 6 pages, 7 figure

Free-Field Representation of Group Element for Simple Quantum Group

A representation of the group element (also known as ``universal ${\cal T}$-matrix'') which satisfies $\Delta(g) = g\otimes g$, is given in the form $$g = \left(\prod_{s=1}^{d_B}\phantom.^>\ {\cal E}_{1/q_{i(s)}}(\chi^{(s)}T_{-i(s)})\right) q^{2\vec\phi\vec H} \left(\prod_{s=1}^{d_B}\phantom.^<\ {\cal E}_{q_{i(s)}}(\psi^{(s)} T_{+i(s)})\right)$$ where $d_B = \frac{1}{2}(d_G - r_G)$, $q_i = q^{|| \vec\alpha_i||^2/2}$ and $H_i = 2\vec H\vec\alpha_i/||\vec\alpha_i||^2$ and $T_{\pm i}$ are the generators of quantum group associated respectively with Cartan algebra and the {\it simple} roots. The ``free fields'' $\chi,\ \vec\phi,\ \psi$form a Heisenberg-like algebra:$\psi^{(s)}\psi^{(s')} = q^{-\vec\alpha_{i(s)} \vec\alpha_{i(s')}} \psi^{(s')}\psi^{(s)}, & \chi^{(s)}\chi^{(s')} = q^{-\vec\alpha_{i(s)}\vec\alpha_{i(s')}} \chi^{(s')}\chi^{(s)}& {\rm for} \ s<s', \\ q^{\vec h\vec\phi}\psi^{(s)} = q^{\vec h\vec\alpha_{i(s)}} \psi^{(s)}q^{\vec h\vec\phi}, & q^{\vec h\vec\phi}\chi^{(s)} = q^{\vec h \vec\alpha_{i(s)}}\chi^{(s)}q^{\vec h\vec\phi}, & \\ &\psi^{(s)} \chi^{(s')} = \chi^{(s')}\psi^{(s)} & {\rm for\ any}\ s,s'.$We argue that the$d_G$-parametric ``manifold'' which$g$spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication$g \rightarrow g'\cdot g''$. The universal${\cal R}$-matrix with the property that${\cal R} (g\otimes I)(I\otimes g) = (I\otimes g)(g\otimes I){\cal R}$is given by the usual formula$${\cal R} = q^{-\sum_{ij}^{r_G}||\vec\alpha_i||^2|| \vec\alpha_j||^2 (\vec\alpha\vec\alpha)^{-1}_{ij}H_i \otimes H_j}\prod_{ \vec\alpha > 0}^{d_B}{\cal E}_{q_{\vec\alpha}}\left(-(q_{\vec\alpha}- q_{\vec\alpha}^{-1})T_{\vec\alpha}\otimes T_{-\vec\alpha}\right).$$Comment: 68 page