We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky’s fundamental results on the number of regions.

In this paper we consider the Jacobian conjecture for a map f of complex affine spaces of dimension n. It is well-known that if f is proper then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f is proper onto its image.

Let X 1 , X 2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map X 1 → X 2 is affine (that is, X 2 is topologically rigid) if and only if the system X 2 has finite topolog-ical entropy.

We prove that the Leray spectral sequence in rational cohomology for the quotient map U n,d → U n,d /G where U n,d is the affine variety of equations for smooth hypersurfaces of degree d in P n (C) and G is the general linear group, degenerates at E 2 .

In this paper we study some properties of fibers of the invariant moment map for a Hamiltonian action of a reductive group on an affine symplectic variety. We prove that all fibers have equal dimension. Further, under some additional restrictions, we show that the quotients of fibers are irreducible normal schemes.

We propose a rigorous numerical algorithm for computing the linking number of piecewise affine links given by spatially distributed data points. The key idea is to use an analytic expression for the solid angle of a tetrahedron to quickly evaluate the degree of the Gauss map. An implementation of the algorithm with INTLAB, a Matlab toolbox for reliable computing, is also provided.

We show that every uniformly asymptotically affine circle endomorphism has a uniformly asymptotically conformal (UAC) extension to the complex plane. Then we use the UAC extension to construct the dual dynamical system, the dual annulus, and the dual circle expanding map.

This paper, motivated from complex dynamics and algebraic geometry, studies the moduli space of polynomial maps of C, from the standpoint of their fixed-point eigenvalues. We denote by e Pd the set of affine conjugacy classes of f(z) ∈ C[z] with degC = d, and define the map Φd : e Pd → e Λd := n (λ1, . . . , λd) ∈ C d ̨

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture , where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holo-morphic function representation to Hitchin's description of special Lagrangian mod...

a priori information 299 abdominal scans 184 active appearance models (AAM) 417, 436, 440, 448 active contours 3, 275, 316 formulations 318 active shape models (ASM) 417, 448 admissible functions 347 affine 348 affine intensity transformations 234 affine-invariant 221, 223 affine tangent 280 affine transformation 245, 280, 352, 369 albedo model 444 albedo recovery 445 alignment 346, 361 amplitu...