Mixed computation

Proof-theoretic models of grammar are based on the view that an explicit characterization of a language comes in the form of the recursive enumeration of strings in that language. That recursive enumeration is carried out by a procedure which strongly generates a set of structural descriptions Σ and weakly generates a set of strings S; a grammar is thus a function that pairs an element of Σ with elements of S. Structural descriptions are obtained by means of Context-Free phrase structure rules or via recursive combinatorics and structure is assumed to be uniform: binary branching trees all the way down. In this work we will analyse natural language constructions for which such a rigid conception of phrase structure is descriptively inadequate and propose a solution for the problem of phrase structure grammars assigning too much or too little structure to natural language strings: we propose that the grammar can oscillate between levels of computational complexity in local domains, which correspond to elementary trees in a lexicalised Tree Adjoining Grammar.

Explicit characterization of the torsion growth of rational elliptic curves with complex multiplication over quadratic fields

In a series of papers we classify the possible torsion structures of rational elliptic curves base-extended to number fields of a fixed degree. In this paper we turn our attention to the question of how the torsion of an elliptic curve with complex multiplication defined over the rationals grows over quadratic fields. We go further and we give an explicit characterization of the quadratic fields where the torsion grows in terms of some invariants attached to the curve.

Numerical semigroups of small and large type

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

Point configurations, phylogenetic trees, and dissimilarity vectors

In 2004, Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map with a weighted variant, we show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. Moreover, we provide a geometric interpretation in terms of configurations of points on rational normal curves and construct a finite tropical basis that yields an explicit characterization of weighted dissimilarity vectors.

Arithmetic of decay walls through continued fractions: a new exact dyon counting solution in $$ \mathcal{N} $$ = 4 CHL models

We use continued fractions to perform a systematic and explicit characterization of the decays of two-centred dyonic black holes in 4D$$ \mathcal{N} $$ N = 4 heterotic ℤN CHL models. Thereby we give a new exact solution for the problem of counting decadent dyons in these models.

A (0;0,2) Interpolation Method to Approximate Functions via Ultraspherical Polynomials

The object of this paper is to demonstrate the existence, explicit characterization and estimation of the polynomial interpolation, related to the weighted (0;0,2) interpolation which satisfies the boundary conditions together with the interpolation conditions at the interval [−1, 1].