On the two-connected planar spanning subgraph polytope

The problem of finding a two-connected planar spanning subgraph of maximum weight in a complete edge-weighted graph is important in automatic graph drawing. We investigate the problem from a polyhedral point of view.     

A new goldfish model

A new integrable (indeed, solvable) model of goldfish type is identified, and some of its properties are discussed. A version of its Newtonian equations of motion reads as follows:

Strength vs. toughness optimization of microstructured composites

Aim of this paper is to present a new fractal approach linking the macroscopic mechanical properties of micro- and nano-structured materials with the main parameters: composition, grain size and structural dimension, as well as contiguity and mean free path. Assuming the key role played by the interfaces, the proposed fractal energy approach unifies the influences of all the above parameters, through the introduction of a fractal structural parameter (FSP), which represents an extension of the Gurland’s structural parameter. This modeling approach is assessed through an extensive comparison with experimental data on poly crystalline diamond (PCD) and WC–Co alloys. The results clearly show that the theoretical fractal predictions are in a fairly good agreement with the experiments on both hardness and toughness. This new synthetic parameter is thus proposed to investigate, design and optimize new micro- and nano-grained materials. Eventually, FSP-based optimization maps are developed, that allow to design new materials with high hardness and toughness.     

Common fixed point theorems for families of occasionally weakly compatible mappings

We prove some common fixed point theorems in probabilistic semi-metric spaces for families of occasionally weakly compatible mappings. We also give a common fixed point theorem for mappings satisfying an integral-type implicit relation.     

On the vertex packing problem

A forbidden subgraphs characterization of the class of graphs that arise from bipartite graphs, odd holes, and graphs with no complement of a diamond via repeated substitutions is given. This characterization allows us to solve the vertex packing problem for the graphs in this class.     

Solvable Three-Body Problem and Painlevé Conjectures

For a special choice of the three interparticle coupling constants in the three-body version of a many-body problem in the plane that was recently investigated, the general solution of the equations of motion can be written in closed form (and is remarkably simple). We also discuss another analogous three-body problem and obtain two third-order highly nonlinear autonomous ODEs whose general solutions, we conjecture, are entire. In other words, we conjecture that these ODEs feature (a strong version of) the Painlevé property.     

Role of 3D imaging in small scale NMR tomography

Some advantages of using 3D techniques in small scale NMR imaging are suggested. In particular the 3D methods are likely to be useful in slow dynamical processes occurring in vivo. Three-dimensional images obtained with our first home built prototype are also reported.