Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

On the Constraints Defining BPS Monopoles

We discuss the explicit formulation of the transcendental constraints defining spectral curves of SU(2) BPS monopoles in the twistor approach of Hitchin, following Ercolani and Sinha. We obtain an improved version of the Ercolani-Sinha constraints, and show that the Corrigan-Goddard conditions for constructing monopoles of arbitrary charge can be regarded as a special case of these. As an application, we study the spectral curve of the tetrahedrally symmetric 3-monopole, an example where the Corrigan-Goddard conditions need to be modified. A particular 1-cycle on the spectral curve plays an important rôle in our analysis.     

Numeric simulation of pollutant dispersion by a control‐volume based on finite element method

The dispersion of pollutants in the environment is an issue of great interest as it directly affects air quality, mainly in large cities. Experimental and numerical tools have been used to predict the behavior of pollutant species dispersion in the atmosphere. A software has been developed based on the control‐volume based on the finite element method in order to obtain two‐dimensional simulations of Navier–Stokes equations and heat or mass transportation in regions with obstacles, varying position of the pollutant source. Numeric results of some applications were obtained and, whenever possible, compared with literature results showing satisfactory accordance. Copyright © 2010 John Wiley & Sons, Ltd.     

EFTEM and EELS Analysis of a Pt/NiO Interface

 Energy-filtering transmission electron microscopy (EFTEM) is more and more becoming an important nanoanalytical technique in both materials science and biology. The main advantage of the method lies in the possibility to obtain two-dimensional chemical information from large specimen areas as well as from features on a nanometer scale. Due to its excellent lateral resolution it is perfectly suited for the investigation of nanometer sized features (e.g. interfaces). In this paper we will show how EFTEM can be used to characterize the interface between a Pt layer and a NiO crystal as part of a coulometric titration cell. In addition to elemental distribution maps electron energy-loss spectra (EELS) across the interface (EELS linescans) have been acquired to obtain quantitative compositional profiles. By employing these methods the following interfacial layers could be identified, all of which containing Pt, Ni and O in different proportions: 13 nm Pt-rich, 32 nm Ni-rich and 29 nm Pt-rich. The origin of these is discussed in terms of displacement reactions.     

Vector boson decays of the Higgs boson

We derive the width of the Higgs boson into vector bosons. General formulas are derived both for the on–shell decay as well for the off–shell decays, and , where . For the off-shell decays the width of the decaying vector boson is properly included. The formulas are valid both for the Standard Model as well as for arbitrary extensions. As an example we study in detail the gauge-invariant effective Lagrangian models where we can have sizable enhancements over the Standard Model that could be observed at LEP. Received: 31 July 1998 / Revised version: 23 September 1998 / Published online: 6 November 1998     

Dynamics of constrained systems

The dynamics on the final constraint submanifold of a constrained system is discussed: a system of equations of motion is induced by the initial ones, whose consistency originates this submanifold, and there is also a dynamics which is the geometrically natural one of this (presymplectic) submanifold. Their compatibility and possible equivalence are analyzed. As a consequence, another justification of Dirac's conjecture is obtained.     

Properties of the sequential gradient-restoration algorithm (SGRA), part 1: Introduction and comparison with related methods

The sequential gradient-restoration algorithm (SGRA) was developed in the late 1960s for the solution of equality-constrained nonlinear programs and has been successfully implemented by Miele and coworkers on many large-scale problems. The algorithm consists of two major sequentially applied phases. The first is a gradient-type minimization in a subspace tangent to the constraint surface, and the second is a feasibility restoration procedure. In Part 1, the original SGRA algorithm is described and is compared with two other related methods: the gradient projection and the generalized reduced gradient methods. Next, the special case of linear equalities is analyzed. It is shown that, in this case, only the gradient-type minimization phase is needed, and the SGRA becomes identical to the steepest-descent method. Convergence proofs for the nonlinearly constrained case are given in Part 2.Partial support for this work was provided by the Fund for the Promotion of Research at Technion, Israel Institute of Technology, Haifa, Israel.     

Invariant forms and Hamiltonian systems: A geometrical setting

A geometric proof is given of Lee Hwa Chung's theorem for regular Hamiltonian systems, which identifies all the possible differential forms left invariant by the dynamics. Applications of this theorem in the area of canonical transformations are also remarked in a purely geometrical context.