We present a multivariate version of Hidden Field Equations (HFE) over a finite field of odd characteristic, with an extra "embedding'' modifier. Combining these known ideas makes our new MPKC (multivariate public key cryptosystem) more efficient and scalable than any other extant multivariate encryption scheme. Switching to odd characteristics in HFE-like schemes affects how an attacker can make use of field equations. Extensive empirical tests (using MAGMA-2.14, the best commercially available \mathbold{F_4} implementation) suggests that our new construction is indeed secure against algebraic attacks using Gröbner Basis algorithms. The "embedding'' serves both to narrow down choices of pre-images and to guard against a possible Kipnis-Shamir type (rank-based) attack. We may hence reasonably argue that for practical sizes, prior attacks take exponential time. We demonstrate that our construction is in fact efficient by implementing practical-sized examples of our "odd-char HFE'' with 3 variables (`"THFE'') over GF(31). To be precise, our preliminary THFE implementation is 15x--20x the speed of RSA-1024.

We introduce a technique based on Gaussian maps to study whether a surface can lie on a threefold as a very ample divisor with given normal bundle. We give applications, among which one to surfaces of general type and another to Enriques surfaces. In particular, we prove the genus bound g  17 for Enriques Fano threefolds. Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admit Q-smoothings.

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We show that a compact operator A is a multiple of a positive semi-definite operator if and only if (A B) W (A) W (B), for all (rank one) operators B. An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators A so that equivalence conditions hold.

In this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II).