Problem MQ: Solve the system p1 = p2 = · · · = pm = 0, where each pi is a quadratic polynomial in x = (x1, . . . , xn) and coefficients and variables are in K = GF(q). Generic MQ is NP-hard [GJ79], and consensus pegs it as a difficult problem to solve even probabilistically. The trapdoor is built into the public map P by the knowledge of the maps S and T . The central map Q may contain paramete...

1. Teichm uller space and the Teichm uller metric d(X; Y) = inf(1=2) log K(), where : X ! Y ranges over all quasiconformal mappings in the appropriate isotopy class. 2. Teichm uller's theorem: If X is compact, or more generally if X is obtained from a compact Riemann surface by removing a nite number of points, then there is an essentially map f minimizing the innmum above. (This map is unique ...

Thurston introduced invariant (quadratic) laminations in his 1984 preprint as a vehicle for understanding the connected Julia sets and parameter space of quadratic polynomials. Important ingredients analysis angle doubling map $\sigma _2

Let E be a central extension of the form 0 → V → G → W → 0 where V and W are elementary abelian 2-groups. Associated to E there is a quadratic map Q : W → V given by the 2-power map which uniquely determines the extension. This quadratic map also determines the extension class q of the extension in H(W,V ) and an ideal I(q) in H(G,Z/2) which is generated by the components of q. We say E is Bock...

In recent years the synchronization of a chaotic response system with a chaotic drive system has attracted great interest in nonlinear science and engineering, due to the potential applications in chaos-based communications. This paper focuses on a new approach to achieve synchronization between two chaotic (hyperchaotic) maps with different dimensions. In particular, given a drive system map w...

We give a classification of quadratic harmonic morphisms between Euclidean spaces (Theorem 2.4) after proving a Rank Lemma. We also find a correspondence between umbilical (Definition 2.7) quadratic harmonic morphisms and Clifford systems. In the case R −→ R, we determine all quadratic harmonic morphisms and show that, up to a constant factor, they are all bi-equivalent (Definition 3.2) to the ...

We show that if ft is a holomorphic family of quadratic-like maps with all periodic orbits repelling so that for each real t the map ft is a real Collet-Eckmann S-unimodal map then, writing μt for the unique absolutely continuous invariant probability measure of ft, the map

Texture mapping may give the impression of geometric details in a model using an image. But texture always was captured under special light condition. If the lighting in virtual environment is different from the texture image, the result of rendering will be incorrect and unrealistic. This paper proposes an image-based method that requires basic texture map and coefficient maps to interpolate l...

In computer graphics, texture mapping is an important means to enhance the realism of model. Traditional texture image always was captured under special light condition. If the lighting in virtual environment is different from the texture image, the result of rendering will be incorrect and unrealistic. This paper proposes an image-based method to fit the reflection mode by a quadratic multinom...