We propose a new basic trapdoor ℓIC (ℓ-Invertible Cycles) of the mixed field type for Multivariate Quadratic public key cryptosystems. This is the first new basic trapdoor since the invention of Unbalanced Oil and Vinegar in 1997. ℓIC can be considered an extended form of the well-known Matsumoto-Imai Scheme A (also MIA or C^∗), and share some features of stagewise triangular systems. However ℓIC has very distinctive properties of its own. In practice, ℓIC is much faster than MIA, and can even match the speed of single-field MQ schemes.

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We derive asymptotic expansion for the spectrum of Hamiltonians with a strong attractive interaction supported by a smooth surface in R 3, either infinite and asymptotically planar, or compact and closed. Its second term is found to be determined by a Schrdinger type operator with an effective potential expressed in terms of the interaction support curvatures.

Let E and E be two Hilbert E -modules over E -algebras E and E , respectively. Let E be a surjective linear isometry from E onto E and E a map from E into E . We will prove in this paper that if the E -algebras E and E are commutative, then E preserves the inner products and E is a module map, ie, there exists a E -isomorphism E between the E -algebras such that <div class="gsh_dspfr"> E