Formulas for Lagranigian and orthogonal degeneracy loci; \widetilde Q-polynomial approach

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of IE2.gif" alt=" $$\widetilde Q$$ " align="middle" border="0"> and IE3.gif" alt=" $$\widetilde P$$ " align="middle" border="0"> -polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of IE4.gif" alt=" $$\widetilde Q$$ " align="middle" border="0"> and IE5.gif" alt=" $$\widetilde P$$ " align="middle" border="0"> polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.     

Double Circulant Codes over \mathbb{Z}_4 and Even Unimodular Lattices

With the help of some new results about weight enumerators of self-dual codes over IE2.gif" alt=" $$\mathbb{Z}_4 $$ " align="middle" border="0"> we investigate a class of double circulant codes over IE3.gif" alt=" $$\mathbb{Z}_4 $$ " align="middle" border="0"> , one of which leads to an extremal even unimodular 40–dimensional lattice. It is conjectured that there should be Nine more constructions of the Leech lattice     

How to realize a Lie algebra by vector fields

We describe an algorithm for embedding a finite-dimensional Lie algebra (superalgebra) into a Lie algebra (superalgebra) of vector fields that is suitable for a ground field of any characteristic and also a way to select the Cartan, complete, and partial prolongations of the Lie algebra of vector fields using differential equations. We illustrate the algorithm with the example of Cartan’s interpretation of the exceptional simple Lie algebra IE1.gif" alt=" $$\mathfrak{g}$$ " align="middle" border="0"> (2) as the Lie algebra preserving a certain nonintegrable distribution and also several other examples. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 450–469, June, 2006.     

Projective modules and prime submodules

In this paper, we use Zorn’s Lemma, multiplicatively closed subsets and saturated closed subsets for the following two topics (i) The existence of prime submodules in some cases (ii) The proof that submodules with a certain property satisfy the radical formula. We also give a partial characterization of a submodule of a projective module which satisfies the prime property.     

A New \mathcal{N}\mathcal{P} -Complete Problem and Public-Key Identification

The appearance of the theory of zero-knowledge, presented by Goldwasser, Micali and Rackoff in 1985, opened a way to secure identification schemes. The first application was the famous Fiat-Shamir scheme based on the problem of modular square roots extraction. In the following years, many other schemes have been proposed, some Fiat-Shamir extensions but also new discrete logarithm based schemes. Therefore, all of them were based on problems from number theory. Their main common drawback is high computational load because of arithmetical operations modulo large integers. Implementation on low-cost smart cards was made difficult and inefficient.With the Permuted Kernels Problem (PKP), Shamir proposed the first efficient scheme allowing for an implementation on such low-cost smart cards, but very few others have afterwards been suggested.In this paper, we present an efficient identification scheme based on a combinatorial IE2.gif" alt=" $$\mathcal{N}\mathcal{P}$$ " align="middle" border="0"> -complete problem: the Permuted Perceptrons Problem (PPP). This problem seems hard enough to be unsolvable even with very small parameters, and some recent cryptanalysis studies confirm that position. Furthermore, it admits efficient zero-knowledge proofs of knowledge and so it is well-suited for cryptographic purposes. An actual implementation completes the optimistic opinion about efficiency and practicability on low-cost smart cards, and namely with less than 2KB of EEPROM and just 100 Bytes of RAM and 6.4 KB of communication.     

Some New Extrapolation Estimates for the Scale of Spaces

Using new extrapolation estimates for the IE2.gif" alt=" $$K$$ " align="middle" border="0"> - and IE3.gif" alt=" $$J$$ " align="middle" border="0"> -functionals of couples of limit spaces of the IE4.gif" alt=" $$L_p $$ " align="middle" border="0"> -scale IE5.gif" alt=" $$(1 < p < \infty )$$ " align="middle" border="0"> , we introduce a class of extrapolation functors. A characterization of this class via the real interpolation method permits one to obtain new equivalent expressions for the norms in symmetric spaces close to IE6.gif" alt=" $$L_\infty $$ " align="middle" border="0"> and IE7.gif" alt=" $$L_1 $$ " align="middle" border="0"> , which depend only on the IE8.gif" alt=" $$L_p $$ " align="middle" border="0"> -norms of a function.     

Image inverse d'un D-module and Newton Polygon)

We show that, under conditions about the microcharacteristic variety of a coherent IE3.gif" alt=" $$\mathcal{D}$$ " align="middle" border="0"> -module, the Cauchy problem is well-posed in the spaces of formal power series with Gevrey growth. We deduce that the filtration of the Irregularity Sheaf of a holonomic IE4.gif" alt=" $$\mathcal{D}$$ " align="middle" border="0"> -module, which we defined in a previous work, is preserved under inverse image if some rather general geometric conditions are fullfilled.     

Quasiconformal Mappings and Solutions of the Dispersionless KP Hierarchy

A IE1.gif" alt=" $$\overline \partial $$ " align="middle" border="0"> formalism for studying dispersionless integrable hierarchies is applied to the dispersionless KP (dKP) hierarchy. We relate this formalism to the theory of quasiconformal mappings on the plane and present some classes of explicit solutions of the dKP hierarchy.     

Polynomials with All Zeros Real and in a Prescribed Interval

We provide a characterization of the real-valued univariate polynomials that have only real zeros, all in a prescribed interval [a,b]. The conditions are stated in terms of positive semidefiniteness of related Hankel matrices.     

Pyrene fluorescence measurements of metastable oil droplets in surfactant-free oil-in-water emulsions

 The fluorescence behavior of pyrene in oil droplets of a surfactant-free oil-in-water emulsion was studied for benzene, fluorobenzene, n-hexane and cyclohexane droplets in water. The excimer–monomer fluorescence ratio immediately after sonication, I _E/I _M(0), of the benzene/water emulsion was 8–10 times larger than for the benzene solution. The ratio I _E/I _M(t) increased in the first 10–20 min before it decreased to zero. Similar behavior was observed for the fluorobenzene/water emulsion, while I _E/I _M(0) for emulsions with n-hexane and cyclohexane was smaller than for benzene and fluorobenzene/water emulsions. I _E/I _M(t) hardly changed with time for the n-hexane and cyclohexane/water emulsions. This different behavior was attributed to the increased solubility of nanometer-size droplets with benzene and fluorobenzene. Received: 20 June 2001 Accepted: 19 April 2001