Generalized likelihood ratio statistics have been proposed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153193] as a generally applicable method for testing nonparametric hypotheses about nonparametric functions. The likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family. We extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio (SELR) tests. The approach is also applied to test conditional estimating equations on the distributions of stochastic errors. It is shown that the proposed SELR statistics follow asymptotically rescaled 2-distributions, with the scale constants and the degrees of freedom being independent of the nuisance parameters. This demonstrates that the Wilks phenomenon observed in Fan, Zhang and Zhang [Ann. Statist. 29

We study the stability and oscillation of traveling fronts in a three-component, advection-reaction biodegradation model. The three components are pollutant, nutrient, and bacteria concentrations. Under an explicit condition on the biomass growth and decay coefficients, we derive reduced, two-component, semilinear hyperbolic models through a relaxation procedure, during which biomass is slaved to pollutant and nutrient concentration variables. The reduced two-component models resemble the Broadwell model of the discrete velocity gas. The traveling fronts of the reduced system are explicit and are expressed in terms of hyperbolic tangent function in the nutrient-deficient regime. We perform energy estimates to prove the asymptotic stability of these fronts under explicit conditions on the coefficients in the system. In the small damping limit, we carry out Wentzel--Kramers--Brillouin (WKB) analysis on front

In this paper, we study how the algebraic attack on the HFE multivariate public key cryptosystem works if we build an HFE cryptosystem on a finite field whose characteristic is not two. Using some very basic algebraic geometry we argue that when the characteristic is not two the algebraic attack should not be polynomial in the range of the parameters which are used in practical applications. We further support our claims with extensive experiments using the Magma implementation of F_4, which is currently the best publicly available implementation of the Gröbner basis algorithm. We present a new variant of the HFE cryptosystems, where we project the public key of HFE to a space of one dimension lower. This protects the system from the Kipnis-Shamir attack and makes the decryption process avoid multiple candidates for the plaintext. We propose an example for a practical application on GF(11) and suggest a test challenge on GF(7).

A Tamed Transformation Method (TTM) cryptosystem was proposed by T.T.Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 2^{33} computations on the finite field with 2^8 elements.

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections.