Parsimonious Edge-coloring on Surfaces

We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless, and edge-maximal to other surface embeddings of subcubic graphs

Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs

In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to $(1,0)$, that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Linking the proximal tibiofibular joint to hominid locomotion: A morphometric study of extant species

Objectives: We perform a comparative assessment of shape variation of the proximal fibula in extant humans and great apes, intending to investigate the possible link between proximal fibular shape and locomotor patterns.Methods: Our sample includes 94 fibulae of 37 Homo sapiens, 15 Gorilla, 17 Pongo, and 25 Pan. Fibular morphology was investigated through three-dimensional (semi)landmark-based geometric morphometric methods.Results: We found unique features of the human fibular head compared to that of great apes (i.e., oblique articular surface, the presence of the styloid process, specific morphology of muscle attachment sites), supporting the functional role of this bone in relation to human obligate bipedalism. Great apes also showed distinctive traits in their proximal fibula morphology, in agreement with differences in locomotor behavior.Conclusion: The morphology of the proximal fibula in extant humans and great apes is indicative of locomotor behavior, offering the potential for the comparative analysis of fossil hominin remains

Nonlinear Dirac operator and quaternionic analysis

Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions of the Cauchy-Riemann-Fueter equation are established.Comment: Cosmetic changes onl

The Geometry and Moduli of K3 Surfaces

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kahler cones of K3 surfaces, and give some of their applications.Comment: 34 pages, 2 figures. (R. Laza, M. Schutt and N. Yui, eds.

Period- and mirror-maps for the quartic K3

We study in detail mirror symmetry for the quartic K3 surface in P3 and the mirror family obtained by the orbifold construction. As explained by Aspinwall and Morrison, mirror symmetry for K3 surfaces can be entirely described in terms of Hodge structures. (1) We give an explicit computation of the Hodge structures and period maps for these families of K3 surfaces. (2) We identify a mirror map, i.e. an isomorphism between the complex and symplectic deformation parameters, and explicit isomorphisms between the Hodge structures at these points. (3) We show compatibility of our mirror map with the one defined by Morrison near the point of maximal unipotent monodromy. Our results rely on earlier work by Narumiyah-Shiga, Dolgachev and Nagura-Sugiyama.Comment: 29 pages, 3 figure

New insights on Celtic migration in Hungary and Italy through the analysis of non-metric dental traits

The Iron Age is characterized by an extended interweaving of movements by Celts in Europe. Several waves of Celts from Western and Central Europe migrated southeast and west from the core area of the La Téne culture (between Bourgogne and Bohemia). Through the analysis of non-metric dental traits, this work aims to understand the biological relationship among Celtic groups arrived in Italy and the Carpathian Basin, as well as between local populations and Celtic newcomers. A total of 10 non-metric dental traits were analyzed to evaluate biological affinities among Celts (Sopron-Krautacker and Pilismarót-Basaharc) and Scythians-related populations from Hungary (Tápiószele), Celts from continental Europe (Switzerland and Austria), two Iron Age Etruscan-Celtic sites from northern Italy (Monterenzio Vecchio and Monte Bibele), 13 Iron Age central-southern Italic necropolises, and the northern Italian Bronze Age necropolis of Scalvinetto. Strontium isotopes were measured on individuals from the necropolis of Monte Bibele to infer their local or non-local origin. Results highlight the existence of statistically significant differences between Celts and autochthonous Italian groups. Celtic groups from Hungary and Italy (i.e., non-local individuals of Monterenzio Vecchio and Monte Bibele) share a similar biological background, supporting the historical records mentioning a common origin for Celts migrated to the eastern and southern borders of today’s Europe. The presence of a supposed Steppean ancestry both in Celts from Hungary and Celts from northern Italy corroborates the hypothesis of the existence of a westward migration of individuals and genes from the Steppe towards northern Italy during the Bronze and Iron Age, which contributed to the biological variability of pre-Celtic and later Celtic populations, respectively. Conversely, individuals from central-southern Italy show an autochthonous pre-Iron Age background. Lastly, this work supports the existence of Celtic migratory routes in northern Italy, as shown by biological and cultural admixture between Celts and Italics living together