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Colmez, Dospinescu and Niziol have shown that the only $p$-adic representations of $\rm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ appearing in the $p$-adic \'etale cohomology of the coverings of Drinfeld's half-plane are the $2$-dimensional cuspidal representations (i.e. potentially semi-stable, whose associated Weil-Deligne representation is irreducible) with Hodge-Tate weights $0$ and $1$ and their multiplicities are given by the $p$-adic Langlands correspondence. We generalise this result to arbitrary weights, by considering the $p$-adic \'etale cohomology with coefficients in the symmetric powers of the universal local system on Drinfeld's tower. A novelty is the appearance of potentially semistable $2$-dimensional non-cristabelian representations, with expected multiplicity. The key point is that the local systems we consider turn out to be particularly simple: they are "isotrivial opers" on a curve. We develop a recipe to compute the pro\'etale cohomology of such a local system using the Hyodo-Kato cohomology of the curve and the de Rham complex of the flat filtered bundle associated to the local system., Comment: In French

It is well-known that a formal deformation of a commutative algebra ${\mathcal A}$ leads to a Poisson bracket on ${\mathcal A}$ and that the classical limit of a derivation on the deformation leads to a derivation on ${\mathcal A}$, which is Hamiltonian with respect to the Poisson bracket. In this paper we present a generalisation of it for formal deformations of an arbitrary noncommutative algebra ${\mathcal A}$. The deformation leads in this case to a Poisson algebra structure on $\Pi({\mathcal A}):=Z({\mathcal A})\times({\mathcal A}/Z({\mathcal A}))$ and to the structure of a $\Pi({\mathcal A})$-Poisson module on ${\mathcal A}$. The limiting derivations are then still derivations of ${\mathcal A}$, but with the Hamiltonian belong to $\Pi({\mathcal A})$, rather than to ${\mathcal A}$. We illustrate our construction with several cases of formal deformations, coming from known quantum algebras, such as the ones associated with the nonabelian Volterra chains, Kontsevich integrable map, the quantum plane and the quantised Grassmann algebra.

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus $g$ algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted ${\cal V}_g$. The original motivation behind our work was the fact that, in the particular case $g=2$, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the $g=2$ Volterra map ${\cal V}_2$. We dedicate this work to a dear friend and colleague, Decio Levi.

Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the $g=2$ case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g\geqslant 1$. The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case ($g=1$), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices., Comment: Final version including Maple code in Appendix B

A novel traveling-wave Zeeman decelerator based on a double-helix coil geometry capable of decelerating paramagnetic molecules with high efficiency is presented. Moving magnetic traps are generated by applying time-dependent currents through the decelerator coils. Paramagnetic molecules in low-field-seeking Zeeman states are confined inside the moving traps which are decelerated to lower forward velocities. As a prototypical example, we demonstrate the deceleration of OH radicals from an initial velocity of 445 m/s down to various final velocities. The experimental results are analyzed and numerically reproduced with the help of trajectory simulations confirming the phase-space stability and efficiency of the deceleration of the molecules in the new device.

The concept of a novel traveling wave Zeeman deccelerator based on a double-helix wire geometry capable of decelerating paramagnetic species with high efficiency is presented. A moving magnetic trap is created by running time-dependent currents through the decelerator coils. Paramagnetic species in low-field-seeking Zeeman states are confined in the moving traps and transported to the end of the decelerator with programmable velocities. Here, we present the theoretical foundations underlying the working principle of the traveling-trap decelerator. Using trajectory simulations, we characterise the performance of the new device and explore the conditions for phase-space stability of the transported molecules.

The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph $\Gamma$, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph $\Gamma$ has the Kahan-Poisson property. We show that if $\Gamma$ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices $1,2,\dots,n$, with an arc $i\to j$ precisely when $i

We generalize to arbitrary categories of algebras the notion of an NS-algebra. We do this by using a bimodule property, as we did for defining the general notions of a dendriform and tridendriform algebra. We show that several types of operators lead to NS-algebras: Nijenhuis operators, twisted Rota-Baxter operators and relative Rota-Baxter operators of arbitrary weight., Comment: 21 pages; in v3 we added more references, we improved the presentation and added few more examples. Final version, accepted for publication in J. Pure Appl. Algebra

We study the basic relation between skew-symmetric Lotka-Volterra systems and graphs, both at the level of objects and morphisms, and derive a classification from it of skew-symmetric Lotka-Volterra systems in terms of graphs as well as in terms of irreducible weighted graphs. We also obtain a description of their automorphism groups and of the relations which exist between these groups. The central notion introduced and used is that of decloning of graphs and of Lotka-Volterra systems. We also give a functorial interpretation of the results which we obtain., Comment: 40 pages