Let p′ and q′ be points in R. Write p′ ∼ q′ if p′ − q′ is a multiple of (1, . . . , 1). Two different points p and q in R/ ∼ uniquely determine a tropical line L(p, q) passing through them and stable under small perturbations. This line is a balanced unrooted semi–labeled tree on n leaves. It is also a metric graph. If some representatives p′ and q′ of p and q are the first and second columns o...

Weyl points and line nodes are three-dimensional linear point and line degeneracies between two bands. In contrast to two-dimensional Dirac points, which are their lower-dimensional analogues, Weyl points are stable in momentum space, and the associated surface states are predicted to be topologically non-trivial. However, Weyl points are yet to be discovered in nature. Here, we report photonic...

ABSTRACT Using techniques from geometric measure theory and descriptive set theory we prove a general result concerning sets in the plane which meet each straight line in exactly two points As an application we show that no such two point set can be expressed as the union of countably many recti able sets together with a set with Hausdor measure zero Also as another corollary we show that no an...

Segmented line regression has been used in many applications, and the problem of estimating the number of change-points in segmented line regression has been discussed in Kim et al. (2000). This paper studies asymptotic properties of the number of change-points selected by the permutation procedure of Kim et al. (2000). This procedure is based on a sequential application of likelihood ratio typ...

We propose an on-line algorithm for exact filtering of multiple changepoint problems. This algorithm enables simulation from the true joint posterior distribution of the number and position of the changepoints for a class of changepoint models. The computational cost of this exact algorithm is quadratic in the number of observations. We further show how resampling ideas from particle filters ca...

In this paper, we study small sets of lines in PG(n, q) and AG(n, q), q odd, that have a small number of odd-points. We fix a small glitch in the proof of an earlier bound in the affine case, we extend the theorem to the projective case, and we attempt to classify all the sets where equality is reached. For the projective case, we obtain a full classification. For the affine case, we obtain a f...

Software product lines provide a common architecture, reusable code, and other common assets for a set of related software products. Variation is a central requirement in this context, as the product line components have to be instantiated, composed, and configured in the context of the products. In many approaches either static composition techniques or dynamic composition techniques based on ...

Let Mw = (P)//SL2 denote the geometric invariant theory quotient of (P) by the diagonal action of SL2 using the line bundle O(w1, w2, . . . , wn) on (P). Let Rw be the coordinate ring of Mw. We give a closed formula for the Hilbert function of Rw, which allows us to compute the degree of Mw. The graded parts of Rw are certain Kostka numbers, so this Hilbert function computes stretched Kostka nu...

It is a natural problem to investigate the set of what we will call exceptional points for S, i.e. points which are neither density points of S, nor those of R\S. Note that this is a topological notion, since as far as measure theory is concerned, there are no such exceptional points. First, we quantify the notion of exceptional point: given a measurable S ⊂ R and 0 ≤ δ ≤ 1/2, we will call p ∈ ...

The space of n (ordered) points on the projective line, modulo automorphisms of the line, is one of the most important and classical examples of an invariant theory quotient, and is one of the first examples given in any course. Generators for the ring of invariants have been known since the end of the nineteenth century, but the question of the relations has remained surprisingly open, and it ...