Testing Linear-Invariant Non-Linear Properties

We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for "triangle freeness": a function f:\cube^{n}\to\cube satisfies this property if $f(x),f(y),f(x+y)$ do not all equal 1, for any pair x,y\in\cube^{n}. Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points v_{1},...,v_{k}\in\cube^{k} and f:\cube^{n}\to\cube satisfies the property that if for all linear maps L:\cube^{k}\to\cube^{n} it is the case that $f(L(v_{1})),...,f(L(v_{k}))$ do not all equal 1. We show that this property is testable if the underlying matroid specified by $v_{1},...,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of "1-complexity linear systems" of Green and Tao, and graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the proceedings of STACS 200

On SAT representations of XOR constraints

We study the representation of systems S of linear equations over the two-element field (aka xor- or parity-constraints) via conjunctive normal forms F (boolean clause-sets). First we consider the problem of finding an "arc-consistent" representation ("AC"), meaning that unit-clause propagation will fix all forced assignments for all possible instantiations of the xor-variables. Our main negative result is that there is no polysize AC-representation in general. On the positive side we show that finding such an AC-representation is fixed-parameter tractable (fpt) in the number of equations. Then we turn to a stronger criterion of representation, namely propagation completeness ("PC") --- while AC only covers the variables of S, now all the variables in F (the variables in S plus auxiliary variables) are considered for PC. We show that the standard translation actually yields a PC representation for one equation, but fails so for two equations (in fact arbitrarily badly). We show that with a more intelligent translation we can also easily compute a translation to PC for two equations. We conjecture that computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing proof of the transformation from AC-representations to monotone circuits, improved wording and literature review; 3rd v. updated literature, strengthened treatment of monotonisation, improved discussions; 4th v. update of literature, discussions and formulations, more details and examples; conference v. to appear LATA 201

Analysis of an exhaustive search algorithm in random graphs and the n^{c\log n} -asymptotics

We analyze the cost used by a naive exhaustive search algorithm for finding a maximum independent set in random graphs under the usual G_{n,p} -model where each possible edge appears independently with the same probability p. The expected cost turns out to be of the less common asymptotic order n^{c\log n}, which we explore from several different perspectives. Also we collect many instances where such an order appears, from algorithmics to analysis, from probability to algebra. The limiting distribution of the cost required by the algorithm under a purely idealized random model is proved to be normal. The approach we develop is of some generality and is amenable for other graph algorithms.Comment: 35 page

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Sensitivity of a tonne-scale NEXT detector for neutrinoless double beta decay searches

The Neutrino Experiment with a Xenon TPC (NEXT) searches for the neutrinoless double-beta decay of Xe-136 using high-pressure xenon gas TPCs with electroluminescent amplification. A scaled-up version of this technology with about 1 tonne of enriched xenon could reach in less than 5 years of operation a sensitivity to the half-life of neutrinoless double-beta decay decay better than 1E27 years, improving the current limits by at least one order of magnitude. This prediction is based on a well-understood background model dominated by radiogenic sources. The detector concept presented here represents a first step on a compelling path towards sensitivity to the parameter space defined by the inverted ordering of neutrino masses, and beyond.Comment: 22 pages, 11 figure

Homomorphic Secret Sharing for Low Degree Polynomials

Homomorphic secret sharing (HSS) allows $n$ clients to secret-share data to $m$ servers, who can then homomorphically evaluate public functions over the shares. A natural application is outsourced computation over private data. In this work, we present the first plain-model homomorphic secret sharing scheme that supports the evaluation of polynomials with degree higher than 2. Our construction relies on any degree-$k$ (multi-key) homomorphic encryption scheme and can evaluate degree-$\left( (k+1)m -1 \right)$ polynomials, for any polynomial number of inputs $n$ and any sub-logarithmic (in the security parameter) number of servers $m$. At the heart of our work is a series of combinatorial arguments on how a polynomial can be split into several low-degree polynomials over the shares of the inputs, which we believe is of independent interest

Technical Design Report for the PANDA Solenoid and Dipole Spectrometer Magnets

This document is the Technical Design Report covering the two large spectrometer magnets of the PANDA detector set-up. It shows the conceptual design of the magnets and their anticipated performance. It precedes the tender and procurement of the magnets and, hence, is subject to possible modifications arising during this process.Comment: 10 pages, 14MB, accepted by FAIR STI in May 2009, editors: Inti Lehmann (chair), Andrea Bersani, Yuri Lobanov, Jost Luehning, Jerzy Smyrski, Technical Coordiantor: Lars Schmitt, Bernd Lewandowski (deputy), Spokespersons: Ulrich Wiedner, Paola Gianotti (deputy

Feasibility studies of time-like proton electromagnetic form factors at PANDA at FAIR

Simulation results for future measurements of electromagnetic proton form factors at \PANDA (FAIR) within the PandaRoot software framework are reported. The statistical precision with which the proton form factors can be determined is estimated. The signal channel $\bar p p \to e^+ e^-$ is studied on the basis of two different but consistent procedures. The suppression of the main background channel, $\textit{i.e.}$ $\bar p p \to \pi^+ \pi^-$, is studied. Furthermore, the background versus signal efficiency, statistical and systematical uncertainties on the extracted proton form factors are evaluated using two different procedures. The results are consistent with those of a previous simulation study using an older, simplified framework. However, a slightly better precision is achieved in the PandaRoot study in a large range of momentum transfer, assuming the nominal beam conditions and detector performance

Low-diffusion Xe-He gas mixtures for rare-event detection: Electroluminescence Yield

High pressure xenon Time Projection Chambers (TPC) based on secondary scintillation (electroluminescence) signal amplification are being proposed for rare event detection such as directional dark matter, double electron capture and double beta decay detection. The discrimination of the rare event through the topological signature of primary ionisation trails is a major asset for this type of TPC when compared to single liquid or double-phase TPCs, limited mainly by the high electron diffusion in pure xenon. Helium admixtures with xenon can be an attractive solution to reduce the electron diffusion significantly, improving the discrimination efficiency of these optical TPCs. We have measured the electroluminescence (EL) yield of Xe-He mixtures, in the range of 0 to 30% He and demonstrated the small impact on the EL yield of the addition of helium to pure xenon. For a typical reduced electric field of 2.5 kV/cm/bar in the scintillation region, the EL yield is lowered by ~ 2%, 3%, 6% and 10% for 10%, 15%, 20% and 30% of helium concentration, respectively. This decrease is less than what has been obtained from the most recent simulation framework in the literature. The impact of the addition of helium on EL statistical fluctuations is negligible, within the experimental uncertainties. The present results are an important benchmark for the simulation tools to be applied to future optical TPCs based on Xe-He mixtures