We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular we obtain solutions written in terms of Wright functions by exploiting operational rules involving the shift operator. We also consider fractional advection diffusion equations involving fractional powers of the negative Laplace operator and directional derivatives of fractional order and discuss the probabilistic interpretations of solutions.

CM-UTAD (Centro de Matematica da Universidade de Tras-os-Montes e Alto Douro), funded by the Portuguese Government through the FCT - Fundacao para a Ciencia e a Tecnologia under the project PEst-OE/MAT/UI4080/2014. The work of J. Petronilho was supported by the CMUC (Centro de Matematica da Universidade de Coimbra), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Fundacao para a Ciencia e a Tecnologia under the project PEst-C/MAT/UI0324/2011. J. Petronilho was also supported by Direccion General de Investigacion, Desarrollo e Innovacion, Ministerio de Economia y Competitividad of Spain, under grant MTM2012-36732-C03-02.

We examine the system given by \hfill -\Delta u = \lambda (v+1)^p \qquad \Omega \hfill -\Delta v = \gamma (u+1)^\theta \qquad \Omega, \hfill u = v =0 \qquad \quad \partial \Omega, where $ \lambda,\gamma$ are positive parameters and where $ 1 < p \le \theta$ and where $ \Omega$ is a smooth bounded domain in $ R^N$. We show the extremal solutions associated with the above system are bounded provided [\frac{N}{2} < 1 + \frac{2(\theta+1)}{p\theta -1} (\sqrt{\frac{p \theta (p+1)}{\theta +1}} + \sqrt{\frac{p \theta (p+1)}{\theta +1} - \sqrt{\frac{p \theta (p+1)}{\theta +1}}})]

We review our recent results on holomorphic affine structures on the Teichmller spaces of Calabi-Yau manifolds and their Hodge metric completions, as well as their applications to global Torelli theorems for Calabi-Yau manifolds. As corollaries we show that the extended period map from the completion space is injective into the period domain, and that the completion space is a domain of holomorphy and admits a complete Khler-Einstein metric.

The entropy convergence of a class of fully-discrete schemes is shown for scalar convex conservation laws in one dimension. These schemes were constructed by S. Osher and S. Chakravarthy in the mid of 80's When m = 2, this class of schemes includes, for different values of , high accuracy (low truncation error) second-order schemes, the conventional second-order accurate upwind total variation diminishing (TVD) scheme and even a third-order accurate TVD scheme. The proof of the entropy consistence is accomplished by using Yang's wavewise entropy inequality (WEI) framework The convergence of semi-discrete version of schemes was proven in a companion paper

By the well-known Mather-Yau theorem, a complex hypersurface germ V with isolated singularity is fully determined by its moduli algebra A(V). The proof of this theorem does not provide an explicit procedure for recovering V from A(V), and finding such a procedure is a long-standing open problem. In the present paper we survey and compare two recently proposed methods for reconstructing V from A(V) up to biholomorphic equivalence under the assumption that the singularity of V is homogeneous (in which case A(V) coincides with the Milnor algebra of V). As part of our discussion of one of the methods, we give a characterization of the algebras arising from finite polynomial maps with homogeneous components of equal degrees.

In this paper, we introduce the concept of quasi Yamabe gradient solitons, which generalizes the concept of Yamabe gradient solitons.By using some ideas in [7,8], we prove that ndimensional (n ≥ 3) complete quasi Yamabe gradient solitons with vanishing Weyl curvature tensor and positive sectional curvature must be rotationally symmetric.We also prove that any compact quasi Yamabe gradient solitons are of constant scalar curvature.

International audience

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ` or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional, although we shall see that the Rayleigh principle may fail for higher singular values. Using those definitions of singular values and vectors, we shall carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide classes of exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

Networks have recently emerged as a general tool for data representation in various fields. In the analysis of conformation transition networks in biomolecular dynamics (protein, RNA etc.), it is important to discover major transition bottlenecks which provides clues for drug design. Similarly in the analysis of social networks, it is helpful to identify nodes which act as bridges connecting different communities. Although there have been extensive studies on the community structure of networks, much less has been done about the connection between the communities. Inspired by the classical Morse theory, we introduce a new notion, critical nodes of functions on networks, based on the gradient flow of these functions. Critical nodes of different indices, together with their attraction basins, lead to a hierarchical decomposition of networks. This enables us to define a concise topological landscape of functions on networks. The usefulness of this new concept is illustrated by three examples: two social networks and one protein-ligand binding network. For the social networks, the index-0 critical nodes together with their attraction basins represent the different communities on the network; the index-1 and higher critical nodes play the role of bridges or hubs connecting the different communities. For the protein binding network, the index-0 critical nodes together with their basins explain the major metastable bound and misbound macrostates, while the index-1 and higher critical nodes represent the bottleneck between the misbound to the bound states. Computation of such critical nodes can be performed by a polynomial time algorithm based on recent developments in computational topology. In the non-degenerate case an almost linear time algorithm exists which is scalable for large scale network analysis.