Evidence for F(uzz) Theory

We show that in the decoupling limit of an F-theory compactification, the internal directions of the seven-branes must wrap a non-commutative four-cycle S. We introduce a general method for obtaining fuzzy geometric spaces via toric geometry, and develop tools for engineering four-dimensional GUT models from this non-commutative setup. We obtain the chiral matter content and Yukawa couplings, and show that the theory has a finite Kaluza-Klein spectrum. The value of 1/alpha_(GUT) is predicted to be equal to the number of fuzzy points on the internal four-cycle S. This relation puts a non-trivial restriction on the space of gauge theories that can arise as a limit of F-theory. By viewing the seven-brane as tiled by D3-branes sitting at the N fuzzy points of the geometry, we argue that this theory admits a holographic dual description in the large N limit. We also entertain the possibility of constructing string models with large fuzzy extra dimensions, but with a high scale for quantum gravity.Comment: v2: 66 pages, 3 figures, references and clarifications adde

Angles in Fuzzy Disc and Angular Noncommutative Solitons

The fuzzy disc, introduced by the authors of Ref.[1], is a disc-shaped region in a noncommutative plane, and is a fuzzy approximation of a commutative disc. In this paper we show that one can introduce a concept of angles to the fuzzy disc, by using the phase operator and phase states known in quantum optics. We gave a description of a fuzzy disc in terms of operators and their commutation relations, and studied properties of angular projection operators. A similar construction for a fuzzy annulus is also given. As an application, we constructed fan-shaped soliton solutions of a scalar field theory on a fuzzy disc, which corresponds to a fan-shaped D-brane. We also applied this concept to the theory of noncommutative gravity that we proposed in Ref.[2]. In addition, possible connections to black hole microstates, holography and an experimental test of noncommutativity by laser physics are suggested.Comment: 24 pages, 12 figures; v2: minor mistake corrected in Eq.(3.21), and discussion adapted accordingly; v3: a further discussion on the algebra of the fuzzy disc added in subsection 3.2; v4: discussions improved and typos correcte

Covariant fuzzy observables and coarse-graining

A fuzzy observable is regarded as a smearing of a sharp observable, and the structure of covariant fuzzy observables is studied. It is shown that the covariant coarse-grainings of sharp observables are exactly the covariant fuzzy observables. A necessary and sufficient condition for a covariant fuzzy observable to be informationally equivalent to the corresponding sharp observable is given.Comment: 19 page

Uncertainty-Aware Principal Component Analysis

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach

Fuzzy Surfaces of Genus Zero

A fuzzy version of the ordinary round 2-sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere.Comment: 11 pages, Late

Hybrid fuzzy and sliding-mode control for motorised tether spin-up when coupled with axial vibration

A hybrid fuzzy sliding mode controller is applied to the control of motorised tether spin-up coupled with an axial oscillation phenomenon. A six degree of freedom dynamic model of a motorised momentum exchange tether is used as a basis for interplanetary payload exchange. The tether comprises a symmetrical double payload configuration, with an outrigger counter inertia and massive central facility. It is shown that including axial elasticity permits an enhanced level of performance prediction accuracy and a useful departure from the usual rigid body representations, particularly for accurate payload positioning at strategic points. A special simulation program has been devised in MATLAB and MATHEMATICA for a given initial condition data case