In our previous work, we have established the existence of transonic characteristic discontinuities separating supersonic flows from a static gas in two-dimensional steady compressible Euler flows under a perturbation with small total variation of the incoming supersonic flow over a solid right-wedge. It is a free boundary problem in Eulerian coordinates and, across the free boundary (character...

We define complex projective n-space CP as C \ {0} under the equivalence relation (z0 : z1 : ... : zn) ∼ (cz0 : cz1 : ... : czn) where c ∈ C\{0}. A complex projective curve is then the set of points (a0 : a1 : ... : an) ∈ CP such that p(a0, a1, ..., an) = 0 for a fixed complex homogeneous polynomial p(z0, z1, ..., zn). Such a curve is a 2-dimensional, real, orientable Riemannian manifold, or Ri...

In this paper, we compute `-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in `, when ` is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus 2 hyperelliptic curves to t...

In this paper, we compute `-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in `, when ` is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus 2 hyperelliptic curves to t...

It is proved that there exist wild coordinates in the polynomial algebra in three variables over a field of characteristic zero. This result implies the famous Nagata conjecture.

The natural occurrence of small scale structures and the extreme anisotropy in the evolution of a magnetic eld embedded in a conducting ow is interpreted in terms of the properties of the local Lyapunov exponents along the various local characteristic (un)stable directions for the Lagrangian ow trajectories. The local Lyapunov exponents and the characteristic directions are functions of Lagrang...

─ A concept of a new set of features suitable for describing arbitrary shapes after their thinning is presented. The features are the minimal eigenvalues of the Toeplitz matrices formed from the rational functions, in which the coordinates of the characteristic pixels of the shape to be described are used as coefficients, the x coordinates in the numerator polynomial, and the y coordinates in t...

Definition 1.1. A Poisson algebra is an associative algebra A over a field K (fixed, of characteristic zero), equipped with a Lie bracket {−,−} such that {x,−} is a derivation for any x ∈ A, i.e. {x, yz} = {x, y}z + y{x, z}. Definition 1.2. A Poisson structure on a manifold M is a Poisson bracket {−,−} on the algebra C∞(M). Example 1.3. On T ∗Rn with position coordinates q1, ..., qn and momentu...