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Register a new userOn the Solvability of Viewing Graphs
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Added by
Matthew Trager
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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A set of fundamental matrices relating pairs of cameras in some configuration can be represented as edges of a viewing graph. Whether or not these fundamental matrices are generically sufficient to recover the global camera configuration depends on the structure of this graph. We study characterizations of solvable viewing graphs and present several new results that can be applied to determine which pairs of views may be used to recover all camera parameters. We also discuss strategies for verifying the solvability of a graph computationally.
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We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.
We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanleys classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs.
We present a mathematical method to statistically decouple the effects of unknown inclination angles on the mass distribution of exoplanets that have been discovered using radial-velocity techniques. The method is based on the distribution of the product of two random variables. Thus, if one assumes a true mass distribution, the method makes it possible to recover the observed distribution. We compare our prediction with available radial-velocity data. Assuming the true mass function is described by a power-law, the minimum mass function that we recover proves a good fit to the observed distribution at both mass ends. In particular, it provides an alternative explanation for the observed low-mass decline, usually explained as sample incompleteness. In addition, the peak observed near the the low-mass end arises naturally in the predicted distribution as a consequence of imposing a low-mass cutoff in the true-distribution. If the low-mass bins below 0.02 M_J are complete, then the mass distribution in this regime is heavily affected by the small fraction of lowly inclined interlopers that are actually more massive companions. Finally, we also present evidence that the exoplanet mass distribution changes form towards low-mass, implying that a single power law may not adequately describe the sample population.
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{mathbb R})$. As a consequence it is possible to find a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombian centers in two and three dimensions.
We prove a number of results to the effect that generic quantum graphs (defined via operator systems as in the work of Duan-Severini-Winter / Weaver) have few symmetries: for a Zariski-dense open set of tuples $(X_1,cdots,X_d)$ of traceless self-adjoint operators in the $ntimes n$ matrix algebra the corresponding operator system has trivial automorphism group, in the largest possible range for the parameters: $2le dle n^2-3$. Moreover, the automorphism group is generically abelian in the larger parameter range $1le dle n^2-2$. This then implies that for those respective parameters the corresponding random-quantum-graph model built on the GUE ensembles of $X_i$s (mimicking the ErdH{o}s-R{e}nyi $G(n,p)$ model) has trivial/abelian automorphism group almost surely.
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