Monge-Ampère Equations on (Para-)K?hler Manifolds: from Characteristic Subspaces to Special Lagrangian Submanifolds

We present the basic notions and results of the geometric theory of second order PDEs in the framework of contact and symplectic manifolds including characteristics, formal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchy problems. Then, we focus our attention to Monge-Ampère equations (MAEs) and discuss a natural class of MAEs arising in K?hler and para-K?hler geometry whose solutions are special Lagrangian submanifolds.     

SEFE without Mapping via Large Induced Outerplane Graphs in Plane Graphs

We show that every n‐vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any ‐vertex planar graph. In order to achieve this result, we prove that every n‐vertex plane graph has an induced outerplane subgraph containing at least vertices. Also, we show that every n‐vertex planar graph and every n‐vertex planar partial 3‐tree admit a simultaneous embedding without mapping and with fixed edges.     

Direct and Inverse Problems for a Phase-Field Model with Memory

We consider a semilinear integrodifferential system in non-normal form. Such a system is a generalization of the one that arises in the phase-field theory with memory. We prove an abstract existence and uniqueness theorem and a continuous dependence result for the direct problem. Reformulating the direct problem in a suitable way we prove that the identification problem admits a unique solution.     

Straight-Line Rectangular Drawings of Clustered Graphs

We show that every c-planar clustered graph has a straight-line c-planar drawing in which each cluster is represented by an axis-parallel rectangle, thus solving a problem posed by Eades, Feng, Lin, and Nagamochi (Algorithmica 44(1):1–32, 2006).     

Hysteresis loops for para–ferromagnetic phase transitions

In this paper, by an extension of the Ginzburg–Landau theory, we propose a mathematical model describing hard magnets within which we are able to explore the para–ferromagnetic transition and by using the Landau–Lifshitz–Gilbert equation, to study the 3D evolution of magnetic field. Finally, the hysteresis loops are obtained and represented by numerical implementations.     

Extension results for slice regular functions of a quaternionic variable

In this paper we prove a new Representation Formula for slice regular functions, which shows that the value of a slice regular function f at a point q=x+yI can be recovered by the values of f at the points q+yJ and q+yK for any choice of imaginary units I,J,K. This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions.     

Long‐range dependence and performance in telecom networks

Telecommunications systems have recently undergone significant innovations. These call for suitable statistical models that can properly describe the behaviour of the input traffic in a network. Here we use fractional Brownian motion (FBM) to model cumulative traffic network, thus taking into account the possible presence of long‐range dependence in the data. A Bayesian approach is devised in such a way that we are able to: (a) estimate the Hurst parameter H of the FBM; (b) estimate the overflow probability which is a parameter measuring the quality of service of a network: (c) develop a test for comparing the null hypothesis of long‐range dependence in the data versus the alternative of short‐range dependence. In order to achieve these inferential results, we elaborate an MCMC sampling scheme whose output enables us to obtain an approximation of the quantities of interest. An application to three real datasets, corresponding to three different levels of traffic, is finally considered. Copyright © 2004 John Wiley & Sons, Ltd.