Ideals and Quotients of Diagonally Quasi-Symmetric Functions

In 2004, J.-C. Aval, F. Bergeron and N. Bergeron studied the algebra of diagonally quasi-symmetric functions $\operatorname{\mathsf{DQSym}}$ in the ring $\mathbb{Q}[\mathbf{x},\mathbf{y}]$ with two sets of variables. They made conjectures on the structure of the quotient $\mathbb{Q}[\mathbf{x},\mathbf{y}]/\langle\operatorname{\mathsf{DQSym}}^+\rangle$, which is a quasi-symmetric analogue of the diagonal harmonic polynomials. In this paper, we construct a Hilbert basis for this quotient when there are infinitely many variables i.e. $\mathbf{x}=x_1,x_2,\dots$ and $\mathbf{y}=y_1,y_2,\dots$. Then we apply this construction to the case where there are finitely many variables, and compute the second column of its Hilbert matrix.

LANDAU–ZENER TRANSITIONS IN THE PRESENCE OF HARMONIC NOISE

We study the influence of off-diagonal harmonic noise on transitions in a Landau–Zener model. We demonstrate that the harmonic noise can change the transition probabilities substantially and that its impact depends strongly on the characteristic frequency of the noise. In the underdamped regime of the noise process, its effect is compared with the one of a deterministic sinusoidally oscillating function. While altering the properties of the noise process allows one to engineer the transitions probabilities, driving the system with a deterministic sinusoidal function can result in larger and more controlled changes of the transition probability. This may be relevant for realistic implementations of our model with Bose–Einstein condensates in noise-driven optical lattices.

Deformed diagonal harmonic polynomials for complex reflection groups

arXiv : http://arxiv.org/abs/1011.3654 International audience We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded W-module, to the undeformed version. Nous introduisons une déformation de l'espace des polynômes harmoniques (multi-diagonaux) pour tout groupe de réflexions complexes de la forme W=G(m,p,n), et soutenons l'hypothèse que cet espace est toujours isomorphe, en tant que W-module gradué, à l'espace d'origine.

Extended basis abinitio calculations on conformers of propanal

Extended basis abinitio calculations on four conformations of propanal at the 4-31G and 6-31G** levels followed by many-body perturbative interaction calculations MP2 and MP3 have been conducted. Optimized geometries, heights of rotational barriers, dipole moments, ionisation potentials, and diagonal harmonic force constants have been reported. The s-cis conformer (dihedral angle CCCO = 0°) is found to be more stable than the gauche conformer (dihedral angle CCCO = 129.84°) by 5.07 kJ mol−1. Torisonal potential barriers s-cis/gauche, gauche/s-cis, and gauche/gauche have values 8.81, 3.74, and 1.84 kJ mol−1, respectively. It appears that meaningful values of rotational barriers can be obtained only after a careful optimization of the geometries of the conformations involved.