Non-commutative integrable systems on bb-symplectic manifolds

In this paper we study non-commutative integrable systems on $b$-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering non-commutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and we prove an action-angle theorem for non-commutative integrable systems on a $b$-symplectic manifold in a neighbourhood of a Liouville torus inside the critical set of the Poisson structure associated to the $b$-symplectic structure

From twistors to twisted geometries

In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors.Comment: 9 page

Minimal Universal Two-qubit Quantum Circuits

We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to global phase. For several quantum gate libraries we prove that gate counts are optimal in worst and average cases. Our lower and upper bounds compare favorably to previously published results. Temporary storage is not used because it tends to be expensive in physical implementations. For each gate library, best gate counts can be achieved by a single universal circuit. To compute gate parameters in universal circuits, we only use closed-form algebraic expressions, and in particular do not rely on matrix exponentials. Our algorithm has been coded in C++.Comment: 8 pages, 2 tables and 4 figures. v3 adds a discussion of asymetry between Rx, Ry and Rz gates and describes a subtle circuit design problem arising when Ry gates are not available. v2 sharpens one of the loose bounds in v1. Proof techniques in v2 are noticeably revamped: they now rely less on circuit identities and more on directly-computed invariants of two-qubit operators. This makes proofs more constructive and easier to interpret as algorithm

Manifolds associated with (Z2)n(Z_2)^n-colored regular graphs

In this article we describe a canonical way to expand a certain kind of $(\mathbb Z_2)^{n+1}$-colored regular graphs into closed $n$-manifolds by adding cells determined by the edge-colorings inductively. We show that every closed combinatorial $n$-manifold can be obtained in this way. When $n\leq 3$, we give simple equivalent conditions for a colored graph to admit an expansion. In addition, we show that if a $(\mathbb Z_2)^{n+1}$-colored regular graph admits an $n$-skeletal expansion, then it is realizable as the moment graph of an $(n+1)$-dimensional closed $(\mathbb Z_2)^{n+1}$-manifold.Comment: 20 pages with 9 figures, in AMS-LaTex, v4 added a new section on reconstructing a space with a $(Z_2)^n$-action for which its moment graph is a given colored grap

Legendrian Distributions with Applications to Poincar\'e Series

Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |\Lambda, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|\Lambda, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $\Lambda$, and a sequence itself has a symbol which is a half-form, $\sigma$, on $\Lambda$. We prove estimates, as $k\to\infty$, of the norm squares $\langle \Lambda, k|\Lambda, k\rangle$ in terms of $\int_\Lambda \sigma\overline{\sigma}$. More generally, we show that if $\Lambda_1$ and $\Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langle\Lambda_1, k|\Lambda_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $\Lambda_1\cap\Lambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincar\'e series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.Comment: 41 pages, LaTe

Mg II Absorption Systems in SDSS QSO Spectra

We present the results of a MgII absorption-line survey using QSO spectra from the SDSS EDR. Over 1,300 doublets with rest equivalent widths greater than 0.3\AA and redshifts $0.366 \le z \le 2.269$ were identified and measured. We find that the $\lambda2796$ rest equivalent width ($W_0^{\lambda2796}$) distribution is described very well by an exponential function $\partial N/\partial W_0^{\lambda2796} = \frac{N^*}{W^*} e^{-\frac{W_0}{W^*}}$, with $N^*=1.187\pm0.052$ and $W^*=0.702\pm0.017$\AA. Previously reported power law fits drastically over-predict the number of strong lines. Extrapolating our exponential fit under-predicts the number of $W_0 \le 0.3$\AA systems, indicating a transition in $dN/dW_0$ near $W_0 \simeq 0.3$\AA. A combination of two exponentials reproduces the observed distribution well, suggesting that MgII absorbers are the superposition of at least two physically distinct populations of absorbing clouds. We also derive a new redshift parameterization for the number density of $W_0^{\lambda2796} \ge 0.3$\AA lines: $N^*=1.001\pm0.132(1+z)^{0.226\pm0.170}$ and $W^*=0.443\pm0.032(1+z)^{0.634\pm 0.097}$\AA. We find that the distribution steepens with decreasing redshift, with $W^*$ decreasing from $0.80\pm0.04$\AA at $z=1.6$ to $0.59\pm0.02$\AA at $z=0.7$. The incidence of moderately strong MgII $\lambda2796$ lines does not show evidence for evolution with redshift. However, lines stronger than $\approx 2$\AA show a decrease relative to the no-evolution prediction with decreasing redshift for $z \lesssim 1$. The evolution is stronger for increasingly stronger lines. Since $W_0$ in saturated absorption lines is an indicator of the velocity spread of the absorbing clouds, we interpret this as an evolution in the kinematic properties of galaxies from moderate to low z.Comment: 50 pages, 26 figures, accepted for publication in Ap

Instability and Chaos in Non-Linear Wave Interaction: a simple model

We analyze stability of a system which contains an harmonic oscillator non-linearly coupled to its second harmonic, in the presence of a driving force. It is found that there always exists a critical amplitude of the driving force above which a loss of stability appears. The dependence of the critical input power on the physical parameters is analyzed. For a driving force with higher amplitude chaotic behavior is observed. Generalization to interactions which include higher modes is discussed. Keywords: Non-Linear Waves, Stability, Chaos.Comment: 16 pages, 4 figure

Topology and phase transitions: a paradigmatic evidence

We report upon the numerical computation of the Euler characteristic \chi (a topologic invariant) of the equipotential hypersurfaces \Sigma_v of the configuration space of the two-dimensional lattice $\phi^4$ model. The pattern \chi(\Sigma_v) vs. v (potential energy) reveals that a major topology change in the family {\Sigma_v}_{v\in R} is at the origin of the phase transition in the model considered. The direct evidence given here - of the relevance of topology for phase transitions - is obtained through a general method that can be applied to any other model.Comment: 4 pages, 4 figure