Path Dependence of the Quark Nonlocal Condensate within the Instanton Model

Within the instanton liquid model, we study the dependence of the gauge invariant two--point quark correlator on the path used to perform the color parallel transport between two points in the Euclidean space.Comment: 4 pages, 5 figure

Controlling instabilities along a 3DVar analysis cycle by assimilating in the unstable subspace: a comparison with the EnKF

A hybrid scheme obtained by combining 3DVar with the Assimilation in the Unstable Subspace (3DVar-AUS) is tested in a QG model, under perfect model conditions, with a fixed observational network, with and without observational noise. The AUS scheme, originally formulated to assimilate adaptive observations, is used here to assimilate the fixed observations that are found in the region of local maxima of BDAS vectors (Bred vectors subject to assimilation), while the remaining observations are assimilated by 3DVar. The performance of the hybrid scheme is compared with that of 3DVar and of an EnKF. The improvement gained by 3DVar-AUS and the EnKF with respect to 3DVar alone is similar in the present model and observational configuration, while 3DVar-AUS outperforms the EnKF during the forecast stage. The 3DVar-AUS algorithm is easy to implement and the results obtained in the idealized conditions of this study encourage further investigation toward an implementation in more realistic contexts

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

Dynamical analysis of the cluster pair: A3407 + A3408

We carried out a dynamical study of the galaxy cluster pair A3407 \& A3408 based on a spectroscopic survey obtained with the 4 meter Blanco telescope at the CTIO, plus 6dF data, and ROSAT All-Sky-Survey. The sample consists of 122 member galaxies brighter than $m_R=20$. Our main goal is to probe the galaxy dynamics in this field and verify if the sample constitutes a single galaxy system or corresponds to an ongoing merging process. Statistical tests were applied to clusters members showing that both the composite system A3407 + A3408 as well as each individual cluster have Gaussian velocity distribution. A velocity gradient of $\sim 847\pm 114$ $\rm km\;s^{-1}$ was identified around the principal axis of the projected distribution of galaxies, indicating that the global field may be rotating. Applying the KMM algorithm to the distribution of galaxies we found that the solution with two clusters is better than the single unit solution at the 99\% c.l. This is consistent with the X-ray distribution around this field, which shows no common X-ray halo involving A3407 and A3408. We also estimated virial masses and applied a two-body model to probe the dynamics of the pair. The more likely scenario is that in which the pair is gravitationally bound and probably experiences a collapse phase, with the cluster cores crossing in less than $\sim$1 $h^{-1}$ Gyr, a pre-merger scenario. The complex X-ray morphology, the gas temperature, and some signs of galaxy evolution in A3408 suggests a post-merger scenario, with cores having crossed each other $\sim 1.65 h^{-1}$Gyr ago, as an alternative solution.Comment: 17 pages, 12 figures, submitted to MNRAS, accepted 2016 May 9. Received 2016 May 9; in original form 2016 April 1

Finding a Bounded-Degree Expander Inside a Dense One

It follows from the Marcus-Spielman-Srivastava proof of the Kadison-Singer conjecture that if $G=(V,E)$ is a $\Delta$-regular dense expander then there is an edge-induced subgraph $H=(V,E_H)$ of $G$ of constant maximum degree which is also an expander. As with other consequences of the MSS theorem, it is not clear how one would explicitly construct such a subgraph. We show that such a subgraph (although with quantitatively weaker expansion and near-regularity properties than those predicted by MSS) can be constructed with high probability in linear time, via a simple algorithm. Our algorithm allows a distributed implementation that runs in $\mathcal O(\log n)$ rounds and does \bigO(n) total work with high probability. The analysis of the algorithm is complicated by the complex dependencies that arise between edges and between choices made in different rounds. We sidestep these difficulties by following the combinatorial approach of counting the number of possible random choices of the algorithm which lead to failure. We do so by a compression argument showing that such random choices can be encoded with a non-trivial compression. Our algorithm bears some similarity to the way agents construct a communication graph in a peer-to-peer network, and, in the bipartite case, to the way agents select servers in blockchain protocols