Infinite dimensional moment problem: open questions and applications

Infinite dimensional moment problems have a long history in diverse applied areas dealing with the analysis of complex systems but progress is hindered by the lack of a general understanding of the mathematical structure behind them. Therefore, such problems have recently got great attention in real algebraic geometry also because of their deep connection to the finite dimensional case. In particular, our most recent collaboration with Murray Marshall and Mehdi Ghasemi about the infinite dimensional moment problem on symmetric algebras of locally convex spaces revealed intriguing questions and relations between real algebraic geometry, functional and harmonic analysis. Motivated by this promising interaction, the principal goal of this paper is to identify the main current challenges in the theory of the infinite dimensional moment problem and to highlight their impact in applied areas. The last advances achieved in this emerging field and briefly reviewed throughout this paper led us to several open questions which we outline here.Comment: 14 pages, minor revisions according to referee's comments, updated reference

Static/Dynamic Filtering for Mesh Geometry

The joint bilateral filter, which enables feature-preserving signal smoothing according to the structural information from a guidance, has been applied for various tasks in geometry processing. Existing methods either rely on a static guidance that may be inconsistent with the input and lead to unsatisfactory results, or a dynamic guidance that is automatically updated but sensitive to noises and outliers. Inspired by recent advances in image filtering, we propose a new geometry filtering technique called static/dynamic filter, which utilizes both static and dynamic guidances to achieve state-of-the-art results. The proposed filter is based on a nonlinear optimization that enforces smoothness of the signal while preserving variations that correspond to features of certain scales. We develop an efficient iterative solver for the problem, which unifies existing filters that are based on static or dynamic guidances. The filter can be applied to mesh face normals followed by vertex position update, to achieve scale-aware and feature-preserving filtering of mesh geometry. It also works well for other types of signals defined on mesh surfaces, such as texture colors. Extensive experimental results demonstrate the effectiveness of the proposed filter for various geometry processing applications such as mesh denoising, geometry feature enhancement, and texture color filtering

Nonlinear dynamics in isotropic and anisotropic nagneto-optical traps

We briefly review some recent advances in the field of nonlinear dynamics of atomic clouds in magneto-optical traps. A hydrodynamical model in a three-dimensional geometry is applied and analyzed using a variational approach. A Lagrangian density is proposed in the case where thermal and multiple scattering effects are both relevant, where the confinement damping and harmonic potential are both included. For generality, a general polytropic equation of state is assumed. After adopting a Gaussian profile for the fluid density and appropriate spatial dependencies of the scalar potential and potential fluid velocity field, a set of ordinary differential equations is derived. These equations are applied to compare cylindrical and spherical geometry approximations. The results are restricted to potential flows

Distance geometry in active structures

The final publication is available at link.springer.comDistance constraints are an emerging formulation that offers intuitive geometrical interpretation of otherwise complex problems. The formulation can be applied in problems such as position and singularity analysis and path planning of mechanisms and structures. This paper reviews the recent advances in distance geometry, providing a unified view of these apparently disparate problems. This survey reviews algebraic and numerical techniques, and is, to the best of our knowledge, the first attempt to summarize the different approaches relating to distance-based formulations.Peer ReviewedPostprint (author's final draft

Geometría, CAD 3D y aprendizaje: precauciones conceptuales

This paper is presented under the assumption that 3D CAD systems are nowadays powerful tools, not only in advanced geometry but also as learning environments for elementary levels in university undergraduate courses. Their use at these levels is continuously increasing; however, they may be applied without taken into account both the repercussions of recent advances in the fundamentals of architectural geometry and potential risks which may generate certain lacks of necessary knowledge for the current professional practice. This research is intended to warn about this situation by providing some reflections on fundamentals on architectural geometry. Basic contents and competences are addressed from a contemporary point of view as well as certain caution in dealing with the features of present 3D CAD systems for the teaching and learning process. This reasoning is illustrated with a graphical example of a specific lesson: basic concepts of perspective

Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications

Lattice polytopes arise naturally in many different branches of pure and applied mathematics such as number theory, commutative algebra, combinatorics, toric geometry, optimization, and mirror symmetry. The miniworkshop on “Lattice polytopes: methods, advances, applications” focused on two current hot topics: the classification of lattice polytopes with few lattice points and unimodality questions for Ehrhart polynomials. The workshop consisted of morning talks on recent breakthroughs and new methods, and afternoon discussion groups where participants from a variety of different backgrounds explored further applications, identified open questions and future research directions, discussed specific examples and conjectures, and collaboratively tackled open research problems

The perfect magnetic conductor (PMC) Casimir piston in d+1 dimensions

Perfect magnetic conductor (PMC) boundary conditions are dual to the more familiar perfect electric conductor (PEC) conditions and can be viewed as the electromagnetic analog of the boundary conditions in the bag model for hadrons in QCD. Recent advances and requirements in communication technologies have attracted great interest in PMC's and Casimir experiments involving structures that approximate PMC's may be carried out in the not too distant future. In this paper, we make a study of the zero-temperature PMC Casimir piston in $d+1$ dimensions. The PMC Casimir energy is explicitly evaluated by summing over $p+1$-dimensional Dirichlet energies where p ranges from 2 to $d$ inclusively. We derive two exact $d$-dimensional expressions for the Casimir force on the piston and find that the force is negative (attractive) in all dimensions. Both expressions are applied to the case of 2+1 and 3+1 dimensions. A spin-off from our work is a contribution to the PEC literature: we obtain a useful alternative expression for the PEC Casimir piston in 3+1 dimensions and also evaluate the Casimir force per unit area on an infinite strip, a geometry of experimental interest.Comment: 18 pages, 1 figure, to appear in Phys. Rev.

UPGMpp: a Software Library for Contextual Object Recognition

Object recognition is a cornerstone task towards the scene understanding problem. Recent works in the field boost their perfor- mance by incorporating contextual information to the traditional use of the objects’ geometry and/or appearance. These contextual cues are usually modeled through Conditional Random Fields (CRFs), a partic- ular type of undirected Probabilistic Graphical Model (PGM), and are exploited by means of probabilistic inference methods. In this work we present the Undirected Probabilistic Graphical Models in C++ library (UPGMpp), an open source solution for representing, training, and per- forming inference over undirected PGMs in general, and CRFs in par- ticular. The UPGMpp library supposes a reliable and comprehensive workbench for recognition systems exploiting contextual information, in- cluding a variety of inference methods based on local search, graph cuts, and message passing approaches. This paper illustrates the virtues of the library, i.e. it is efficient, comprehensive, versatile, and easy to use, by presenting a use-case applied to the object recognition problem in home scenes from the challenging NYU2 dataset.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish grant program FPU-MICINN 2010 and the Spanish projects “TAROTH: New developments toward a robot at home” (Ref. DPI2011-25483) and “PROMOVE: Advances in mobile robotics for promoting independent life of elders” (Ref. DPI2014-55826-R