Challenges and science-based implications for modern management and conservation of European ungulate populations

Wildlife management systems face growing challenges to cope with increasingly complex interactions between wildlife populations, the environment and human activities. In this position statement, we address the most important issues characterising current ungulate conservation and management in Europe. We present some key points arising from ecological research that may be critical for a reassessment of ungulate management in the future. Ecosystem . Population sustainability . Science-basedmanagement .Wildlifemanagement .Adaptive managemen

On the Weight of Minor Faces in Triangle-Free 3-Polytopes

The weight w(f) of a face f in a 3-polytope is the degree-sum of vertices incident with f. It follows from Lebesgue’s results of 1940 that every triangle-free 3-polytope without 4-faces incident with at least three 3-vertices has a 4-face with w ≤ 21 or a 5-face with w ≤ 17. Here, the bound 17 is sharp, but it was still unknown whether 21 is sharp

All Tight Descriptions of 3-Stars in 3-Polytopes with Girth 5

Lebesgue (1940) proved that every 3-polytope P5 of girth 5 has a path of three vertices of degree 3. Madaras (2004) refined this by showing that every P5 has a 3-vertex with two 3-neighbors and the third neighbor of degree at most 4. This description of 3-stars in P5s is tight in the sense that no its parameter can be strengthened due to the dodecahedron combined with the existence of a P5 in which every 3-vertex has a 4-neighbor