Minor stars in plane graphs with minimum degree five

The weight of a subgraph $H$ in $G$ is the sum of the degrees in $G$ of vertices of $H$. The {\em height} of a subgraph $H$ in $G$ is the maximum degree of vertices of $H$ in $G$. A star in a given graph is minor if its center has degree at most five in the given graph. Lebesgue (1940) gave an approximate description of minor $5$-stars in the class of normal plane maps with minimum degree five. In this paper, we give two descriptions of minor $5$-stars in plane graphs with minimum degree five. By these descriptions, we can extend several results and give some new results on the weight and height for some special plane graphs with minimum degree five.Comment: 11 pages, 3 figure

Birational classification of curves on rational surfaces

In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its "Cremona minimal" models, i.e. those plane curves which are equivalent to B via a Cremona transformation, and have minimal degree under this condition.Comment: 33 page

Urban and extra-urban hybrid vehicles: a technological review

Pollution derived from transportation systems is a worldwide, timelier issue than ever. The abatement actions of harmful substances in the air are on the agenda and they are necessary today to safeguard our welfare and that of the planet. Environmental pollution in large cities is approximately 20% due to the transportation system. In addition, private traffic contributes greatly to city pollution. Further, “vehicle operating life” is most often exceeded and vehicle emissions do not comply with European antipollution standards. It becomes mandatory to find a solution that respects the environment and, realize an appropriate transportation service to the customers. New technologies related to hybrid –electric engines are making great strides in reducing emissions, and the funds allocated by public authorities should be addressed. In addition, the use (implementation) of new technologies is also convenient from an economic point of view. In fact, by implementing the use of hybrid vehicles, fuel consumption can be reduced. The different hybrid configurations presented refer to such a series architecture, developed by the researchers and Research and Development groups. Regarding energy flows, different strategy logic or vehicle management units have been illustrated. Various configurations and vehicles were studied by simulating different driving cycles, both European approval and homologation and customer ones (typically municipal and university). The simulations have provided guidance on the optimal proposed configuration and information on the component to be used

Knot Invariants from Four-Dimensional Gauge Theory

It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we attempt to verify this directly by analyzing the equations and counting their solutions, without reference to any quantum dualities. After suitably perturbing the equations to make their behavior more generic, we are able to get a fairly clear understanding of how the Jones polynomial emerges. The main ingredient in the argument is a link between the four-dimensional gauge theory equations in question and conformal blocks for degenerate representations of the Virasoro algebra in two dimensions. Along the way we get a better understanding of how our subject is related to a variety of new and old topics in mathematical physics, ranging from the Bethe ansatz for the Gaudin spin chain to the $M$-theory description of BPS monopoles and the relation between Chern-Simons gauge theory and Virasoro conformal blocks.Comment: 117 page

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com