Information Geometric Duality of ϕ-Deformed Exponential Families

In the world of generalized entropies—which, for example, play a role in physical systems with sub- and super-exponential phase space growth per degree of freedom—there are two ways for implementing constraints in the maximum entropy principle: linear and escort constraints. Both appear naturally in different contexts. Linear constraints appear, e.g., in physical systems, when additional information about the system is available through higher moments. Escort distributions appear naturally in the context of multifractals and information geometry. It was shown recently that there exists a fundamental duality that relates both approaches on the basis of the corresponding deformed logarithms (deformed-log duality). Here, we show that there exists another duality that arises in the context of information geometry, relating the Fisher information of ϕ -deformed exponential families that correspond to linear constraints (as studied by J.Naudts) to those that are based on escort constraints (as studied by S.-I. Amari). We explicitly demonstrate this information geometric duality for the case of ( c , d ) -entropy, which covers all situations that are compatible with the first three Shannon–Khinchin axioms and that include Shannon, Tsallis, Anteneodo–Plastino entropy, and many more as special cases. Finally, we discuss the relation between the deformed-log duality and the information geometric duality and mention that the escort distributions arising in these two dualities are generally different and only coincide for the case of the Tsallis deformation.

The Delaunay tessellation in hyperbolic space

The Delaunay tessellation of a locally finite subset of the hyperbolic space ℍnis constructed via convex hulls in ℝn+1. For finite and lattice-invariant sets it is proven to be a polyhedral decomposition, and versions (necessarily modified from the Euclidean setting) of the empty circumspheres condition and geometric duality with the Voronoi tessellation are proved. Some pathological examples of infinite, non lattice-invariant sets are exhibited.

Polemics in Public: Poncelet, Gergonne, Plücker, and the Duality Controversy

ArgumentA plagiarism charge in 1827 sparked a public controversy centered between Jean-Victor Poncelet (1788–1867) and Joseph-Diez Gergonne (1771–1859) over the origin and applications of the principle of duality in geometry. Over the next three years and through the pages of various journals, monographs, letters, reviews, reports, and footnotes, vitriol between the antagonists increased as their potential publicity grew. While the historical literature offers valuable resources toward understanding the development, content, and applications of geometric duality, the hostile nature of the exchange seems to have deterred an in-depth textual study of the explicitly polemical writings. We argue that the necessary collective endeavor of beginning and ending this controversy constitutes a case study in the circulation of geometry. In particular, we consider how the duality controversy functioned as a medium of communicating new fundamental principles to a wider audience of practitioners.

CONSERVED QUANTITIES AND DUALITIES FOR PARTICLES IN CURVED SPACE-TIME

We consider particles moving in curved space-time and associated symmetries. We use a generalized Killing equation and search for solutions involving Killing tensors associated with space-time metric. With these tensors some conserved quantities are constructed and they are valid along particles geodesic. In the Hamiltonian formalism for these particles, the conserved quantities can generate a dual description of the metric. We construct nontrivial dual metrics and consider a kind of geometric duality that involves a completely different space-time. From these metrics we calculate geometric invariants and examine the singularity structure of the dual space-time.

GEOMETRIC DUALITY AND CHERN–SIMONS MODIFIED GRAVITY

We consider a theory which involves an extension of general relativity known as Chern–Simons modified gravity (CSMG). In this theory the standard Einstein–Hilbert action is extended with a gravitational Pontryagin density that is obtained from a divergence of a Chern–Simons topological current. The extended theory has the standard Schwarzchild metric as solution, however, only a perturbed Kerr metric holds solution. From the exact Kerr metric we construct dual metrics to search for rotating black hole solutions. The conditions on the Killing tensors associated with dual metrics entail nontrivial solutions to CSMG.

EFFICIENT NON-INTERSECTION QUERIES ON AGGREGATED GEOMETRIC DATA

Geometric intersection searching problems are a well-studied class of query-retrieval problems with many applications. The goal here is to preprocess a set of geometric objects so that the ones that are intersected by a query object can be reported efficiently. Often, a more general version of the problem arises, where the data comes aggregated in disjoint groups and of interest are the groups, not the individual objects, that are intersected by the query object. One approach to a generalized problem is to ignore the grouping, solve the corresponding classical problem, and then infer the groups from the reported answer. However, this is not efficient, since the number of objects intersected can be much larger than the number of groups (i.e., the output size). The problem of designing efficient, output-sensitive query algorithms for generalized intersection searching has received much attention in recent years, and such solutions have been developed for several problems. This paper considers a new class of generalized query-retrieval problems. Specifically, given aggregated geometric data the goal is to report the distinct groups such that no objects from those groups are intersected by the query. Of interest in these generalized non-intersection searching problems are solutions where the query time is sensitive to the output size, i.e., the number of groups reported. Unfortunately, the obvious approaches of (i) solving the corresponding generalized intersection searching problem and reporting the complement, or (ii) solving a generalized intersection searching problem with the complement of the query are either inefficient or incorrect. This paper provides efficient, output-sensitive solutions to several generalized non-intersection searching problems, using techniques such as geometric duality, sparsification, persistence, filtering search, and pruning.