We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie– Lof...

We prove that for continuous maps on the interval, the existence of a n−cycle, implies the existence of n − 1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable. Institute of Mathematical Sciences, SUNY at Stony Brook, Stony Brook, NY 11794-3651. I...

Let Σ be a minimal submanifold of Rn+m that can be represented as the graph of a smooth map f : R 7→ R. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which Σ must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of df .

MapAnalyst is a new software application for the visualization and study of the planimetric accuracy of old maps. It illustrates local map distortion by generating distortion grids, displacement vectors, and new isolines of scale and rotation. MapAnalyst additionally computes the old map’s scale and rotation, as well as statistical indicators summarizing the map’s geometric accuracy. It offers ...

The image of the principal minor map for n×n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n = 4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in P15. The latter...

We present a new approach to Ad-Hoc sensor network localization from pairwise received signal strength (RSSI). Localization problem is viewed as a classification problem. We map the output of Support Vector Machine (SVM) to probability and present various schemes for estimating nodes’ locations from it. We show that application of Laplace Eigenmaps followed by an appropriate Affine Transformati...

A representation-independent mean-field dynamics is presented for batchedTD(λ). The task is learning-to-predict the outcome of an indirectly observed absorbing markov process. In the case of linear representations, the discretetime deterministic iteration is an affine map whose fixed point can be expressed in closed form without the assumption of linearly independent observation vectors. Batche...

We analyze walking people using a gait sequence representation that bypasses the need for frame-to-frame tracking of body parts. The gait representation maps a video sequence of silhouettes into a pair of two-dimensional spatio-temporal patterns that are periodic along the time axis. Mathematically, such patterns are called “frieze” patterns and associated symmetry groups “frieze groups”. With ...

We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the characteristic numbers of rational plane curves, elliptic plane curves, and elliptic plane curves with fixed complex structure. Recursions are also given for the nu...

Inferring maps between shapes is a long standing problem in geometry processing. The less similar the shapes are, the harder it is to compute a map, or even define criteria to evaluate it. In many cases, shapes appear as part of a collection, e.g. an animation or a series of faces or poses of the same character, where the shapes are similar enough, such that maps within the collection are easy ...