Entropy Loss is Maximal for Uniform Inputs

A secure sketch (defined by Dodis et al.) is an algorithm that on an input w produces an output s such that w can be reconstructed given its noisy version w' and s. Security is defined in terms of two parameters m and m˜ : if w comes from a distribution of entropy m, then a secure sketch guarantees that the distribution of w conditioned on s has entropy m˜ , where λ = m−m˜ is called the entropy loss. In this note we show that the entropy loss of any secure sketch (or, more generally, any randomized algorithm) on any distribution is no more than it is on the uniform distribution.

Analysing energy detector diversity receivers for spectrum sensing

The analysis of energy detector systems is a well studied topic in the literature: numerous models have been derived describing the behaviour of single and multiple antenna architectures operating in a variety of radio environments. However, in many cases of interest, these models are not in a closed form and so their evaluation requires the use of numerical methods. In general, these are computationally expensive, which can cause difficulties in certain scenarios, such as in the optimisation of device parameters on low cost hardware. The problem becomes acute in situations where the signal to noise ratio is small and reliable detection is to be ensured or where the number of samples of the received signal is large. Furthermore, due to the analytic complexity of the models, further insight into the behaviour of various system parameters of interest is not readily apparent. In this thesis, an approximation based approach is taken towards the analysis of such systems. By focusing on the situations where exact analyses become complicated, and making a small number of astute simplifications to the underlying mathematical models, it is possible to derive novel, accurate and compact descriptions of system behaviour. Approximations are derived for the analysis of energy detectors with single and multiple antennae operating on additive white Gaussian noise (AWGN) and independent and identically distributed Rayleigh, Nakagami-m and Rice channels; in the multiple antenna case, approximations are derived for systems with maximal ratio combiner (MRC), equal gain combiner (EGC) and square law combiner (SLC) diversity. In each case, error bounds are derived describing the maximum error resulting from the use of the approximations. In addition, it is demonstrated that the derived approximations require fewer computations of simple functions than any of the exact models available in the literature. Consequently, the regions of applicability of the approximations directly complement the regions of applicability of the available exact models. Further novel approximations for other system parameters of interest, such as sample complexity, minimum detectable signal to noise ratio and diversity gain, are also derived. In the course of the analysis, a novel theorem describing the convergence of the chi square, noncentral chi square and gamma distributions towards the normal distribution is derived. The theorem describes a tight upper bound on the error resulting from the application of the central limit theorem to random variables of the aforementioned distributions and gives a much better description of the resulting error than existing Berry-Esseen type bounds. A second novel theorem, providing an upper bound on the maximum error resulting from the use of the central limit theorem to approximate the noncentral chi square distribution where the noncentrality parameter is a multiple of the number of degrees of freedom, is also derived.

The inverse limit of GIT quotients of Grassmannians by the maximal torus

There are finitely many GIT quotients of

Search for maximal flavor violating scalars in same-charge lepton pairs in pp̄ collisions at s=1.96TeV

Models of maximal flavor violation (MxFV) in elementary particle physics may contain at least one new scalar SU(2) doublet field ΦFV=(η0,η+) that couples the first and third generation quarks (q1, q3) via a Lagrangian term LFV=ξ13ΦFVq1q3. These models have a distinctive signature of same-charge top-quark pairs and evade flavor-changing limits from meson mixing measurements. Data corresponding to 2fb-1 collected by the Collider Dectector at Fermilab II detector in pp̄ collisions at s=1.96TeV are analyzed for evidence of the MxFV signature. For a neutral scalar η0 with mη0=200GeV/c2 and coupling ξ13=1, ∼11 signal events are expected over a background of 2.1±1.8 events. Three events are observed in the data, consistent with background expectations, and limits are set on the coupling ξ13 for mη0=180-300GeV/c2. © 2009 The American Physical Society.