Modelling, Measuring and Compensating Color Weak Vision

We use methods from Riemann geometry to investigate transformations between the color spaces of color-normal and color weak observers. The two main applications are the simulation of the perception of a color weak observer for a color normal observer and the compensation of color images in a way that a color weak observer has approximately the same perception as a color normal observer. The metrics in the color spaces of interest are characterized with the help of ellipsoids defined by the just-noticable-differences between color which are measured with the help of color-matching experiments. The constructed mappings are isometries of Riemann spaces that preserve the perceived color-differences for both observers. Among the two approaches to build such an isometry, we introduce normal coordinates in Riemann spaces as a tool to construct a global color-weak compensation map. Compared to previously used methods this method is free from approximation errors due to local linearizations and it avoids the problem of shifting locations of the origin of the local coordinate system. We analyse the variations of the Riemann metrics for different observers obtained from new color matching experiments and describe three variations of the basic method. The performance of the methods is evaluated with the help of semantic differential (SD) tests.Comment: Full resolution color pictures are available from the author

Effective gravity from a quantum gauge theory in Euclidean space-time

We consider a $SO(d)$ gauge theory in an Euclidean $d$-dimensional space-time, which is known to be renormalizable to all orders in perturbation theory for $2\le{d}\le4$. Then, with the help of a space-time representation of the gauge group, the gauge theory is mapped into a curved space-time with linear connection. Further, in that mapping the gauge field plays the role of the linear connection of the curved space-time and an effective metric tensor arises naturally from the mapping. The obtained action, being quadratic in the Riemann-Christoffel tensor, at a first sight, spoils a gravity interpretation of the model. Thus, we provide a sketch of a mechanism that breaks the $SO(d)$ color invariance and generates the Einstein-Hilbert term, as well as a cosmological constant term, allowing an interpretation of the model as a modified gravity in the Palatini formalism. In that sense, gravity can be visualized as an effective classical theory, originated from a well defined quantum gauge theory. We also show that, in the four dimensional case, two possibilities for particular solutions of the field equations are the de Sitter and Anti de Sitter space-times.Comment: 20 pages; Final version accepted for publication in Class.Quant.Gra

Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging.

We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358–1372, 2004) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data

Physics in Riemann's mathematical papers

Riemann's mathematical papers contain many ideas that arise from physics, and some of them are motivated by problems from physics. In fact, it is not easy to separate Riemann's ideas in mathematics from those in physics. Furthermore, Riemann's philosophical ideas are often in the background of his work on science. The aim of this chapter is to give an overview of Riemann's mathematical results based on physical reasoning or motivated by physics. We also elaborate on the relation with philosophy. While we discuss some of Riemann's philosophical points of view, we review some ideas on the same subjects emitted by Riemann's predecessors, and in particular Greek philosophers, mainly the pre-socratics and Aristotle. The final version of this paper will appear in the book: From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017

Thermodynamic Geometry: Evolution, Correlation and Phase Transition

Under the fluctuation of the electric charge and atomic mass, this paper considers the theory of the thin film depletion layer formation of an ensemble of finitely excited, non-empty $d/f$-orbital heavy materials, from the thermodynamic geometric perspective. At each state of the local adiabatic evolutions, we examine the nature of the thermodynamic parameters, \textit{viz.}, electric charge and mass, changing at each respective embeddings. The definition of the intrinsic Riemannian geometry and differential topology offers the properties of (i) local heat capacities, (ii) global stability criterion and (iv) global correlation length. Under the Gaussian fluctuations, such an intrinsic geometric consideration is anticipated to be useful in the statistical coating of the thin film layer of a desired quality-fine high cost material on a low cost durable coatant. From the perspective of the daily-life applications, the thermodynamic geometry is thus intrinsically self-consistent with the theory of the local and global economic optimizations. Following the above procedure, the quality of the thin layer depletion could self-consistently be examined to produce an economic, quality products at a desired economic value.Comment: 22 pages, 5 figures, Keywords: Thermodynamic Geometry, Metal Depletion, Nano-science, Thin Film Technology, Quality Economic Characterization; added 1 figure and 1 section (n.10), and edited bibliograph

Network and Seiberg Duality

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a permutation, or equivalently a cell of the totally non-negative Grassmannian. The story is similar to a bipartite network on the torus classified by a Newton polygon. We then generalize the network to a general bordered Riemann surface and define IR SCFT from the geometric data of a Riemann surface. We also comment on IR R-charges and superconformal indices of our theories.Comment: 28 pages, 28 figures; v2: minor correction