A generalization of a 1998 unimodality conjecture of Reiner and Stanton

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $k\ge 5$, the polynomials $$f(k,m,b)(q)=\binom{m}{k}_q-q^{\frac{k(m-b)}{2}+b-2k+2}\cdot\binom{b}{k-2}_q$$ are nonnegative and unimodal for all $m\gg_k 0$ and $b\le \frac{km-4k+4}{k-2}$ such that $kb\equiv km$ (mod 2), with the only exception of $b=\frac{km-4k+2}{k-2}$ when this is an integer. Using the KOH theorem, we combinatorially show the case $k=5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m,b)$ for $k\le 5$. (This also provides an isolated counterexample to Reiner-Stanton's conjecture when $k=3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.Comment: Final version. To appear in the Journal of Combinatoric

A note on the asymptotics of the number of O-sequences of given length

We look at the number $L(n)$ of $O$-sequences of length $n$. Recall that an $O$-sequence can be defined algebraically as the Hilbert function of a standard graded $k$-algebra, or combinatorially as the $f$-vector of a multicomplex. The sequence $L(n)$ was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants $c_1$ and $c_2$ and all $n>2$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.Comment: Final version to appear in Discrete Mathematics. 2 page

Time Consistent Policy in Markov Switching Models

In this paper we consider the quadratic optimal control problem with regime shifts and forward-looking agents. This extends the results of Zampolli (2003) who considered models without forward-looking expectations. Two algorithms are presented: The first algorithm computes the solution of a rational expectation model with random parameters or regime shifts. The second algorithm computes the time-consistent policy and the resulting Nash-Stackelberg equilibrium. The formulation of the problem is of general form and allows for model uncertainty and incorporation of policymaker’s judgement. We apply these methods to compute the optimal (non-linear) monetary policy in a small open economy subject to (symmetric or asymmetric) risks of change in some of its key parameters such as inflation inertia, degree of exchange rate pass-through, elasticity of aggregate demand to interest rate, etc.. We normally find that the time-consistent response to risk is more cautious. Furthermore, the optimal response is in some cases non-monotonic as a function of uncertainty. We also simulate the model under assumptions that the policymaker and the private sector hold the same beliefs over the probabilities of the structural change and different beliefs (as well as different assumptions about the knowledge of each other’s reaction function).monetary policy, regime switching, model uncertainty, time consistency

Sub-ohmic two-level system representation of the Kondo effect

It has been recently shown that the particle-hole symmetric Anderson impurity model can be mapped onto a $Z_2$ slave-spin theory without any need of additional constraints. Here we prove by means of Numerical Renormalization Group that the slave-spin behaves in this model like a two-level system coupled to a sub-ohmic dissipative environment. It follows that the $Z_2$ symmetry gets spontaneously broken at zero temperature, which we find can be identified with the on-set of Kondo coherence, being the Kondo temperature proportional to the square of the order parameter. Since the model is numerically solvable, the results are very enlightening on the role of quantum fluctuations beyond mean field in the context of slave-boson approaches to correlated electron models, an issue that has been attracting interest since the 80's. Finally, our results suggest as a by-product that the paramagnetic metal phase of the Hubbard model at half-filling, in infinite coordination lattices and at zero temperature, as described for instance by Dynamical Mean Field Theory, corresponds to a slave-spin theory with a spontaneous breakdown of a local $Z_2$ gauge symmetry.Comment: 4 pages, 5 figure

Theory of the Metal-Paramagnetic Mott-Jahn-Teller Insulator Transition in A_4C_{60}

We study the unconventional insulating state in A_4C_{60} with a variety of approaches, including density functional calculations and dynamical mean-field theory. While the former predicts a metallic state, in disagreement with experiment, the latter yields a (paramagnetic) Mott-Jahn-Teller insulator. In that state, conduction between molecules is blocked by on-site Coulomb repulsion, magnetism is suppressed by intra-molecular Jahn-Teller effect, and important excitations (such as optical and spin gap) should be essentially intra-molecular. Experimental gaps of 0.5 eV and 0.1 eV respectively compare well with molecular ion values, in agreement with this picture.Comment: 4 pages, 2 postscript figure

Quantification and scaling of multipartite entanglement in continuous variable systems

We present a theoretical method to determine the multipartite entanglement between different partitions of multimode, fully or partially symmetric Gaussian states of continuous variable systems. For such states, we determine the exact expression of the logarithmic negativity and show that it coincides with that of equivalent two--mode Gaussian states. Exploiting this reduction, we demonstrate the scaling of the multipartite entanglement with the number of modes and its reliable experimental estimate by direct measurements of the global and local purities.Comment: 4 pages, 2 figures; to be published in Phys. Rev. Let

Gaia Data Release 1. Cross-match with external catalogues - Algorithm and results

Although the Gaia catalogue on its own will be a very powerful tool, it is the combination of this highly accurate archive with other archives that will truly open up amazing possibilities for astronomical research. The advanced interoperation of archives is based on cross-matching, leaving the user with the feeling of working with one single data archive. The data retrieval should work not only across data archives, but also across wavelength domains. The first step for seamless data access is the computation of the cross-match between Gaia and external surveys. The matching of astronomical catalogues is a complex and challenging problem both scientifically and technologically (especially when matching large surveys like Gaia). We describe the cross-match algorithm used to pre-compute the match of Gaia Data Release 1 (DR1) with a selected list of large publicly available optical and IR surveys. The overall principles of the adopted cross-match algorithm are outlined. Details are given on the developed algorithm, including the methods used to account for position errors, proper motions, and environment; to define the neighbours; and to define the figure of merit used to select the most probable counterpart. Statistics on the results are also given. The results of the cross-match are part of the official Gaia DR1 catalogue.Comment: 18 pages, 8 figures. Accepted for publication by A&