Cutoff Phenomenon for Random Walks on Kneser Graphs

The cutoff phenomenon for an ergodic Markov chain describes a sharp transition in the convergence to its stationary distribution, over a negligible period of time, known as cutoff window. We study the cutoff phenomenon for simple random walks on Kneser graphs, which is a family of ergodic Markov chains. Given two integers $n$ and $k$, the Kneser graph $K(2n+k,n)$ is defined as the graph with vertex set being all subsets of $\{1,\ldots,2n+k\}$ of size $n$ and two vertices $A$ and $B$ being connected by an edge if $A\cap B =\emptyset$. We show that for any $k=O(n)$, the random walk on $K(2n+k,n)$ exhibits a cutoff at $\frac{1}{2}\log_{1+k/n}{(2n+k)}$ with a window of size $O(\frac{n}{k})$

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

Methods for the automatic alignment of colour histograms

Colour provides important information in many image processing tasks such as object identification and tracking. Different images of the same object frequently yield different colour values due to undesired variations in lighting and the camera. In practice, controlling the source of these fluctuations is difficult, uneconomical or even impossible in a particular imaging environment. This thesis is concerned with the question of how to best align the corresponding clusters of colour histograms to reduce or remove the effect of these undesired variations. We introduce feature based histogram alignment (FBHA) algorithms that enable flexible alignment transformations to be applied. The FBHA approach has three steps, 1) feature detection in the colour histograms, 2) feature association and 3) feature alignment. We investigate the choices for these three steps on two colour databases : 1) a structured and labeled database of RGB imagery acquired under controlled camera, lighting and object variation and 2) grey-level video streams from an industrial inspection application. The design and acquisition of the RGB image and grey-level video databases are a key contribution of the thesis. The databases are used to quantitatively compare the FBHA approach against existing methodologies and show it to be effective. FBHA is intended to provide a generic method for aligning colour histograms, it only uses information from the histograms and therefore ignores spatial information in the image. Spatial information and other context sensitive cues are deliberately avoided to maintain the generic nature of the algorithm; by ignoring some of this important information we gain useful insights into the performance limits of a colour alignment algorithm that works from the colour histogram alone, this helps understand the limits of a generic approach to colour alignment

Phase diagram of the triangular-lattice Potts antiferromagnet

We study the phase diagram of the triangular-lattice Q-state Potts model in the real (Q, v)-plane, where v - e(J) - 1 is the temperature variable. Our first goal is to provide an obviously missing feature of this diagram: the position of the antiferromagnetic critical curve. This curve turns out to possess a bifurcation point with two branches emerging from it, entailing important consequences for the global phase diagram. We have obtained accurate numerical estimates for the position of this curve by combining the transfer-matrix approach for strip graphs with toroidal boundary conditions and the recent method of critical polynomials. The second goal of this work is to study the corresponding A(p-1) RSOS model on the torus, for integer p = 4, 5,..., 8. We clarify its relation to the corresponding Potts model, in particular concerning the role of boundary conditions. For certain values of p, we identify several new critical points and regimes for the RSOS model and we initiate the study of the flows between the corresponding field theories.The research of JLJ was supported in part by the Agence Nationale de la Recherche (grant ANR-10-BLAN-0414: DIME), the Institut Universitaire de France, and the European Research Council (through the advanced grant NuQFT). The research of JLJ and JS was supported in part by Spanish MINECO grant FIS2014-57387-C3-3-P. The work of CRS was performed under the auspices of the U.S. Department of Energy at the Lawrence Livermore National Laboratory under Contract No DE-AC52-07NA27344

Gaia and VLT astrometry of faint stars: Precision of Gaia DR1 positions and updated VLT parallaxes of ultracool dwarfs

We compared positions of the Gaia first data release (DR1) secondary data set at its faint limit with CCD positions of stars in 20 fields observed with the VLT/FORS2 camera. The FORS2 position uncertainties are smaller than one milli-arcsecond (mas) and allowed us to perform an independent verification of the DR1 astrometric precision. In the fields that we observed with FORS2, we projected the Gaia DR1 positions into the CCD plane, performed a polynomial fit between the two sets of matching stars, and carried out statistical analyses of the residuals in positions. The residual RMS roughly matches the expectations given by the Gaia DR1 uncertainties, where we identified three regimes in terms of Gaia DR1 precision: for G = 17-20 stars we found that the formal DR1 position uncertainties of stars with DR1 precisions in the range of 0.5-5 mas are underestimated by 63 +/- 5\%, whereas the DR1 uncertainties of stars in the range 7-10 mas are overestimated by a factor of two. For the best-measured and generally brighter G = 16-18 stars with DR1 positional uncertainties of <0.5 mas, we detected 0.44 +/- 0.13 mas excess noise in the residual RMS, whose origin can be in both FORS2 and Gaia DR1. By adopting Gaia DR1 as the absolute reference frame we refined the pixel scale determination of FORS2, leading to minor updates to the parallaxes of 20 ultracool dwarfs that we published previously. We also updated the FORS2 absolute parallax of the Luhman 16 binary brown dwarf system to 501.42 +/- 0.11 masComment: 7 pages, 4 figures, 2 tables, accepted for publication in A&A on August 1, 201

Digital forensic techniques for the reverse engineering of image acquisition chains

In recent years a number of new methods have been developed to detect image forgery. Most forensic techniques use footprints left on images to predict the history of the images. The images, however, sometimes could have gone through a series of processing and modification through their lifetime. It is therefore difficult to detect image tampering as the footprints could be distorted or removed over a complex chain of operations. In this research we propose digital forensic techniques that allow us to reverse engineer and determine history of images that have gone through chains of image acquisition and reproduction. This thesis presents two different approaches to address the problem. In the first part we propose a novel theoretical framework for the reverse engineering of signal acquisition chains. Based on a simplified chain model, we describe how signals have gone in the chains at different stages using the theory of sampling signals with finite rate of innovation. Under particular conditions, our technique allows to detect whether a given signal has been reacquired through the chain. It also makes possible to predict corresponding important parameters of the chain using acquisition-reconstruction artefacts left on the signal. The second part of the thesis presents our new algorithm for image recapture detection based on edge blurriness. Two overcomplete dictionaries are trained using the K-SVD approach to learn distinctive blurring patterns from sets of single captured and recaptured images. An SVM classifier is then built using dictionary approximation errors and the mean edge spread width from the training images. The algorithm, which requires no user intervention, was tested on a database that included more than 2500 high quality recaptured images. Our results show that our method achieves a performance rate that exceeds 99% for recaptured images and 94% for single captured images.Open Acces