Triangle areas in line arrangements

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.Comment: Title is shortened. Some typos and small errors were correcte

Saturation and Irredundancy for Spin(8)

We explicitly calculate the triangle inequalities for the group PSO(8). Therefore we explicitly solve the eigenvalues of sum problem for this group (equivalently describing the side-lengths of geodesic triangles in the corresponding symmetric space for the Weyl chamber-valued metric). We then apply some computer programs to verify two basic questions/conjectures. First, we verify that the above system of inequalities is irredundant. Then, we verify the ``saturation conjecture'' for the decomposition of tensor products of finite-dimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights a, b, c such that a+b+c is in the root lattice, and any positive integer N, the tensor product of the irreducible representations V(a) and V(b) contains V(c) if and only if the tensor product of V(Na) and V(Nb) contains V(Nc).Comment: 22 pages, 2 figure

The stability of the Kronecker products of Schur functions

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.Comment: 16 page

The complement of the Bowditch space in the SL(2,C) character variety

Let ${\mathcal X}$ be the space of type-preserving \SL(2,C) characters of the punctured torus $T$. The Bowditch space ${\mathcal X}_{BQ}$ is the largest open subset of ${\mathcal X}$ on which the mapping class group acts properly discontinuously, this is characterized by two simple conditions called the $BQ$-conditions. In this note, we show that $[\rho]$ is in the interior of the complement of ${\mathcal X}_{BQ}$ if there exists an essential simple closed curve $X$ on $T$ such that $|{\rm tr} \rho(X)|<0.5$.Comment: 6 page

Probabilistic Search for Object Segmentation and Recognition

The problem of searching for a model-based scene interpretation is analyzed within a probabilistic framework. Object models are formulated as generative models for range data of the scene. A new statistical criterion, the truncated object probability, is introduced to infer an optimal sequence of object hypotheses to be evaluated for their match to the data. The truncated probability is partly determined by prior knowledge of the objects and partly learned from data. Some experiments on sequence quality and object segmentation and recognition from stereo data are presented. The article recovers classic concepts from object recognition (grouping, geometric hashing, alignment) from the probabilistic perspective and adds insight into the optimal ordering of object hypotheses for evaluation. Moreover, it introduces point-relation densities, a key component of the truncated probability, as statistical models of local surface shape