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    A convergent expansion of the Airy's integral with incomplete Gamma functions

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    There are two main power series for the Airy functions, namely the Maclaurin and the asymptotic expansions. The former converges for all finite values of the complex variable, zz, but it requires a large number of terms for large values of ∣z∣|z|, and the latter is a Poincar\'{e}-type expansion which is well-suited for such large values and where optimal truncation is possible. The asymptotic series of the Airy function shows a classical example of the Stokes phenomenon where a type of discontinuity occurs for the homonymous multipliers. A new series expansion is presented here that stems from the method of steepest descents, as can the asymptotic series, but which is convergent for all values of the complex variable. It originates in the integration of uniformly convergent power series representing the integrand of the Airy's integral in different sections of the integration path. The new series expansion is not a power series and instead relies on the calculation of complete and incomplete Gamma functions. In this sense, it is related to the Hadamard expansions. It is an alternative expansion to the two main aforementioned power series that also offers some insight into the transition zone for the Stokes' multipliers due to the splitting of the integration path. Unlike the Hadamard series, it relies on only two different expansions, separated by a branch point, one of which is centered at infinity. The interest of the new series expansion is mainly a theoretical one in a twofold way. First of all, it shows how to convert an asymptotic series into a convergent one, even if the rate of convergence may be slow for small values of ∣z∣|z|. Secondly, it sheds some light on the Stokes phenomenon for the Airy function by showing the transition of the integration paths at arg⁡z=±2π/3\arg z = \pm 2 \pi/3.Comment: 21 pages, 23 figures. Changes in version 2: i) Footnote 10 has been added, ii) Figure 5 has been added for a deeper analysis of the results, iii) Reference 15 has been added, iv) Typo: A ±\pm was missing in arg⁡z=±2π/3\arg z = \pm 2 \pi/3 (abstract), v) Some font size changes and improved labelling in the figures Changes in version 3: minor edition change

    Plasmon-free Surface-enhanced Raman Spectroscopy on TiO\u3csub\u3e2\u3c/sub\u3e-graphene Oxide Inverse Opal Substrates

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    TiO2-graphene oxide (GO) inverse opal materials were shown to be active plasmon-free surface-enhanced Raman spectroscopy (SERS) substrates. The SERS activity of the substrates was analyzed using three different probe molecules: phenethylamine (PEA), methylene blue (MB), and 4-mercaptobenzoic acid (4-MBA). The morphology of the substrates was investigated by SEM and AFM. Prepared TiO2-GO inverse opals substrates can be reused up to five times with minimal loss of signal, rendering them perfect candidates to be used as highly stable, low-cost, metal-free, recyclable SERS substrates

    Worst case attacks against binary probabilistic traitor tracing codes

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    An insightful view into the design of traitor tracing codes should necessarily consider the worst case attacks that the colluders can lead. This paper takes an information-theoretic point of view where the worst case attack is defined as the collusion strategy minimizing the achievable rate of the traitor tracing code. Two different decoders are envisaged, the joint decoder and the simple decoder, as recently defined by P. Moulin \cite{Moulin08universal}. Several classes of colluders are defined with increasing power. The worst case attack is derived for each class and each decoder when applied to Tardos' codes and a probabilistic version of the Boneh-Shaw construction. This contextual study gives the real rates achievable by the binary probabilistic traitor tracing codes. Attacks usually considered in literature, such as majority or minority votes, are indeed largely suboptimal. This article also shows the utmost importance of the time-sharing concept in a probabilistic codes.Comment: submitted to IEEE Trans. on Information Forensics and Securit
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