where vg is the volume form on (M,g) and e( f )(x) := (1/2)‖df (x)‖T∗M⊗ f −1TN is the energy density of f at the point x ∈M. In local coordinates (x)i=1 on M and (y)a=1 on N , the energy density is given by e( f )(x)= (1/2)gi (x)hab( f (x))(∂ f a/∂xi)(∂ f b/∂x j). Critical points of the energy functional are called harmonic maps. The first variational formula of the energy gives the following c...

The paper reports about authors’ investigation of applicability of a well-known technique of visualization of multivariate data, parallel coordinates plot, to different kinds of tasks. Some new methods of transformation of parallel coordinates plots are suggested. However, the primary aim of the authors is “to link tools to tasks”, i.e. to explicitly define the tasks that can be appropriately s...

We show that, if U is the representation of SOe(n+1, 1) on L2(Sn) given by (2.11), and P is a bounded operator on L2(Sn), then P belongs to OPS0 1,0(S n) if and only if P (g) = U(g)PU(g)−1 is a C∞ function on SOe(n+1, 1) with values in the Banach space L(L2(Sn)). Introduction Let M be a compact C∞ manifold. Denote by OPS 1,0(M) the space of pseudodifferential operators on M , whose symbols in l...

We introduce the concepts of a multisymplectic structure and a polysymplectic structure on a general fiber bundle over a general base manifold, define the concept of the symbol of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, and prove Darboux theorems for the existence of canonical local coordinates. Universidade de São Paulo RT-MAP Februar...

We show that the infinite-dimensional Teichmüller space of a Riemann surface whose boundary consists of n closed curves is a holomorphic fiber space over the Teichmüller space of n-punctured surfaces. Each fiber is a complex Banach manifold modeled on a twodimensional extension of the universal Teichmüller space. The local model of the fiber, together with the coordinates from internal Schiffer...

Rule systems have failed to attract much interest in large data analysis problems because they tend to be too simplistic to be useful or consist of too many rules for human interpretation. We recently presented a method that constructs a hierarchical rule system, with only a small number of rules at each level of the hierarchy. Lower levels in this hierarchy focus on outliers or areas of the fe...

This is an excerpt from my paper with A. Mahalov [1]. PDE theories with Riemannian geometry are long studied subject — c.f. for example the texts cited in the bibliography. Here, I only present a very brief and elementary explanation, mainly about differential operators on a 2D surface. Generalization to higher dimension needs more caution. Let M denote a 2-dimensional, compact, Riemannian mani...

In this paper we will discuss local coordinates canonically corresponding to a Kähler metric. We will also discuss the C ∞ convergence of Bergmann metrics following Tian's result on C 2 convergence of Bergmann metrics. At the end we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry.

In recent years, wireless sensor networks have drawn great attention. This type of network is composed of a large number of sensor nodes which are able to sense, process and communicate. Besides, they are used in various fields such as emergency relief in disasters, monitoring the environment, military affairs and etc. Sensor nodes collect environmental data by using their sensors and send them...

We investigate the existence of several families symmetric periodic solutions as continuation circular orbits Kepler problem for certain differentiable perturbations using an appropriate set Poincare-Delaunay coordinates which are essential in our approach. More precisely, we try separately two situations independent way, namely, when unperturbed part corresponds to a inertial cartesian and it ...