Removing degeneracies by perturbing the problem or perturbing the world

International audienceWe describe two problem-specific approaches to remove geometric degeneracies that we call perturbing the problem and perturbing the world. Using as our primary examples 2-d and 3-d Delaunay triangulation with Euclidean and polygonal metrics, we show that these approaches lead to relatively simple and efficient perturbations of the points that do not depend on a fixed ordering or index. Thus, they produce canonical output, which is important for producing test suites and verifiers for randomized or dynamic geometric algorithms

Removing Degeneracies by Perturbing the Problem or the World

We describe two problem-specific approaches to remove geometric degeneracies that we call {\it perturbing the problem} and {\it perturbing the world}. Using as our primary examples 2-d and 3-d Delaunay triangulation with Euclidean and polygonal metrics, we show that these approaches lead to relatively simple and efficient perturbations of the points that do not depend on a fixed ordering or index. Thus, they produce canonical output, which is important for producing test suites and verifiers for randomized or dynamic geometric algorithms

Cosmic Complementarity: Joint Parameter Estimation from CMB Experiments and Redshift Surveys

We study the ability of future CMB anisotropy experiments and redshift surveys to constrain a thirteen-dimensional parameterization of the adiabatic cold dark matter model. Each alone is unable to determine all parameters to high accuracy. However, considered together, one data set resolves the difficulties of the other, allowing certain degenerate parameters to be determined with far greater precision. We treat in detail the degeneracies involving the classical cosmological parameters, massive neutrinos, tensor-scalar ratio, bias, and reionization optical depth as well as how redshift surveys can resolve them. We discuss the opportunities for internal and external consistency checks on these measurements. Previous papers on parameter estimation have generally treated smaller parameter spaces; in direct comparisons to these works, we tend to find weaker constraints and suggest numerical explanations for the discrepancies.Comment: Submitted to ApJ. LaTeX, 20 pages, emulateapj.sty and onecolfloat.sty. Minor errors in Table 8 corrected; reference adde

Dynamical Jahn-Teller Effect and Berry Phase in Positively Charged Fullerene I. Basic Considerations

We study the Jahn-Teller effect of positive fullerene ions $^2$C$_{60}^{+}$ and $^1$C$_{60}^{2+}$. The aim is to discover if this case, in analogy with the negative ion, possesses a Berry phase or not, and what are the consequences on dynamical Jahn-Teller quantization. Working in the linear and spherical approximation, we find no Berry phase in $^1$C$_{60}^{2+}$, and presence/absence of Berry phase for coupling of one $L=2$ hole to an $L=4$/$L=2$ vibration. We study in particular the special equal-coupling case ($g_2=g_4$), which is reduced to the motion of a particle on a 5-dimensional sphere. In the icosahedral molecule, the final outcome assesses the presence/absence of a Berry phase of $\pi$ for the $h_u$ hole coupled to $G_g$/$H_h$ vibrations. Some qualitative consequences on ground-state symmetry, low-lying excitations, and electron emission from C$_{60}$ are spelled out.Comment: 31 pages (RevTeX), 3 Postscript figures (uuencoded

Interactive Visualization for Singular Fibers of Functions f : R3 → R2

Scalar topology in the form of Morse theory has provided computational tools that analyze and visualize data from scientific and engineering tasks. Contracting isocontours to single points encapsulates variations in isocontour connectivity in the Reeb graph. For multivariate data, isocontours generalize to fibers—inverse images of points in the range, and this area is therefore known as fiber topology. However, fiber topology is less fully developed than Morse theory, and current efforts rely on manual visualizations. This paper presents how to accelerate and semi-automate this task through an interface for visualizing fiber singularities of multivariate functions R3 → R2. This interface exploits existing conventions of fiber topology, but also introduces a 3D view based on the extension of Reeb graphs to Reeb spaces. Using the Joint Contour Net, a quantized approximation of the Reeb space, this accelerates topological visualization and permits online perturbation to reduce or remove degeneracies in functions under study. Validation of the interface is performed by assessing whether the interface supports the mathematical workflow both of experts and of less experienced mathematicians

The Complexity of Relating Quantum Channels to Master Equations

Completely positive, trace preserving (CPT) maps and Lindblad master equations are both widely used to describe the dynamics of open quantum systems. The connection between these two descriptions is a classic topic in mathematical physics. One direction was solved by the now famous result due to Lindblad, Kossakowski Gorini and Sudarshan, who gave a complete characterisation of the master equations that generate completely positive semi-groups. However, the other direction has remained open: given a CPT map, is there a Lindblad master equation that generates it (and if so, can we find it's form)? This is sometimes known as the Markovianity problem. Physically, it is asking how one can deduce underlying physical processes from experimental observations. We give a complexity theoretic answer to this problem: it is NP-hard. We also give an explicit algorithm that reduces the problem to integer semi-definite programming, a well-known NP problem. Together, these results imply that resolving the question of which CPT maps can be generated by master equations is tantamount to solving P=NP: any efficiently computable criterion for Markovianity would imply P=NP; whereas a proof that P=NP would imply that our algorithm already gives an efficiently computable criterion. Thus, unless P does equal NP, there cannot exist any simple criterion for determining when a CPT map has a master equation description. However, we also show that if the system dimension is fixed (relevant for current quantum process tomography experiments), then our algorithm scales efficiently in the required precision, allowing an underlying Lindblad master equation to be determined efficiently from even a single snapshot in this case. Our work also leads to similar complexity-theoretic answers to a related long-standing open problem in probability theory.Comment: V1: 43 pages, single column, 8 figures. V2: titled changed; added proof-overview and accompanying figure; 50 pages, single column, 9 figure

On Deletion in Delaunay Triangulation

This paper presents how the space of spheres and shelling may be used to delete a point from a $d$-dimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller's algorithm, which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.Comment: 15 pages 5 figures. in Proc. 15th Annu. ACM Sympos. Comput. Geom., 181--188, 199

Microlensing Searches for Exoplanets

Gravitational microlensing finds planets through their gravitational influence on the light coming from a more distant background star. The presence of the planet is then inferred from the tell-tale brightness variations of the background star during the lensing event, even if no light is detectable from the planet or the host foreground star. This review covers fundamental theoretical concepts in microlensing, addresses how observations are performed in practice, the~challenges of obtaining accurate measurements, and explains how planets reveal themselves in the data. It~concludes with a presentation of the most important findings to-date, a description of the method's strengths and weaknesses, and a discussion of the future prospects of microlensing.Comment: 35 pages,9 figures, invited review for Geosciences Special Issue "Detection and Characterization of Extrasolar Planets

Qualitative Symbolic Perturbation: a new geometry-based perturbation framework

In a classical Symbolic Perturbation scheme,degeneracies are handled by substituting some polynomials in$\varepsilon$ to the input of a predicate. Instead of a singleperturbation, we propose to use a sequence of (simpler)perturbations. Moreover, we look at their effects geometricallyinstead of algebraically; this allows us to tackle cases that werenot tractable with the classical algebraic approach.Avec les méthodes de perturbations symboliques classiques,les dégénérescences sont résolues en substituant certains polynômes en $\varepsilon$ aux entrées du prédicat.Au lieu d'une seule perturbation compliquée, nous proposons d'utiliser unesuite de perturbation plus simple. Et nous regardons les effets deces perturbations géométriquement plutôt qu'algébriquementce qui permet de traiter des cas inatteignables par les méthodesalgébriques classiques