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This is a review paper of the role of Carleman estimates in the theory of multidimensional coefficient inverse problems since the first inception of this idea in 1981. [ABSTRACT FROM AUTHOR]

In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, the non-adjacent form (NAF) and its generalizations (e.g., the generalized non-adjacent form (GNAF) and the radix- r non-adjacent form ( rNAF)) have been proposed for minimizing the non-zero densities in the representations of the scalars. On the other hand, for subfield elliptic curves, the Frobenius expansions of the scalars can be used for improving efficiency. Unfortunately, there are only a few methods apply the techniques of NAF or its analogue to the Frobenius expansion, namely τ-adic NAF techniques on Koblitz curves and hyperelliptic Koblitz curves. In this paper, we try to combine these techniques, namely recoding methods for reducing non-zero density and the Frobenius expansion, and propose two new efficient recoding methods of scalars on more general family of subfield elliptic curves in odd characteristic. We also prove that the non-zero densities for the new methods are same as those for the original GNAF and rNAF. We estimate scalar multiplication costs on the above subfield elliptic curves in terms of elliptic curve operations and finite field operations for several previous methods and the proposed methods. In addition, we implement scalar multiplication on an subfield elliptic curve belonging to the above family, for the previous methods and a proposed method. As a result, our estimation and implementation show that the speed of the proposed methods improve between 8% and 50% over that for the Frobenius expansion method. [ABSTRACT FROM AUTHOR]

An algorithm for the simultaneous determination of two coefficients in an inverse problem for equations of the parabolic type is studied. The model under investigation has been proposed by H. de Vriend and M. Capobianco to describe the long-time coastal profile evolution. The iterative inversion procedure is based on the minimization of a suitable cost functional. Results of numerical tests are shown to illustrate the performance of the algorithm. The real measured data of the coastal profile evolution have been numerically processed to validate the aforementioned diffusion type model. The paper is targetly oriented to attract attention of professionals, who study Inverse Problems, to such new area of applications as Coastal Engineering. [ABSTRACT FROM AUTHOR]