We consider a class of Hamiltonians in L2(R2) with attractive interaction supported by piecewise C2 smooth loops of a fixed length L, formally given by (x) with >0. It is shown that the ground state of this operator is locally maximized by a circular . We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of .

We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23 (2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for wave speeds allows us to compute the solution in physical space. Numerical examples are shown to validate the method.

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In 1989, Tsujii, Fujioka, and Hirayama proposed a family of multivariate public key cryptosystems, where the public key is given as a set of multivariate rational functions of degree 4. These cryptosystems are constructed via composition of two quadratic rational maps. In this paper, we present the cryptanalysis of this family of cryptosystems. The key point of our attack is to transform a problem of decomposition of two rational maps into a problem of decomposition of two polynomial maps. We develop a new improved 2R decomposition method and other new techniques, which allows us to find an equivalent decomposition of the rational maps to break the system completely. For the example suggested for practical applications, it is very fast to derive an equivalent private key, and it requires only a few seconds on a standard PC.

We show that Multivariate Public Key Cryptosystems (MPKCs) over fields of small odd prime characteristic, say 31, can be highly efficient. Indeed, at the same design security of 2^{80} under the best known attacks, odd-char MPKC is generally faster than prior MPKCs over \GF{2^k}, which are in turn faster than "traditional'' alternatives. This seemingly counter-intuitive feat is accomplished by exploiting the comparative over-abundance of small integer arithmetic resources in commodity hardware, here embodied by SSE2 or more advanced special multimedia instructions on modern x86-compatible CPUs. We explain our implementation techniques and design choices in implementing our chosen MPKC instances modulo small a odd prime. The same techniques are also applicable in modern FPGAs which often contains a large number of multipliers.