39,549 research outputs found
Multi-matrix models and emergent geometry
Encouraged by the AdS/CFT correspondence, we study emergent local geometry in
large N multi-matrix models from the perspective of a strong coupling
expansion. By considering various solvable interacting models we show how the
emergence or non-emergence of local geometry at strong coupling is captured by
observables that effectively measure the mass of off-diagonal excitations about
a semiclassical eigenvalue background. We find emergent geometry at strong
coupling in models where a mass term regulates an infrared divergence. We also
show that our notion of emergent geometry can be usefully applied to fuzzy
spheres. Although most of our results are analytic, we have found numerical
input valuable in guiding and checking our results.Comment: 1+34 pages, 4 figures. References adde
Gauge Theory on Fuzzy S^2 x S^2 and Regularization on Noncommutative R^4
We define U(n) gauge theory on fuzzy S^2_N x S^2_N as a multi-matrix model,
which reduces to ordinary Yang-Mills theory on S^2 x S^2 in the commutative
limit N -> infinity. The model can be used as a regularization of gauge theory
on noncommutative R^4_\theta in a particular scaling limit, which is studied in
detail. We also find topologically non-trivial U(1) solutions, which reduce to
the known "fluxon" solutions in the limit of R^4_\theta, reproducing their full
moduli space. Other solutions which can be interpreted as 2-dimensional branes
are also found. The quantization of the model is defined non-perturbatively in
terms of a path integral which is finite. A gauge-fixed BRST-invariant action
is given as well. Fermions in the fundamental representation of the gauge group
are included using a formulation based on SO(6), by defining a fuzzy Dirac
operator which reduces to the standard Dirac operator on S^2 x S^2 in the
commutative limit. The chirality operator and Weyl spinors are also introduced.Comment: 39 pages. V2-4: References added, typos fixe
Localization for Yang-Mills Theory on the Fuzzy Sphere
We present a new model for Yang-Mills theory on the fuzzy sphere in which the
configuration space of gauge fields is given by a coadjoint orbit. In the
classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find
all classical solutions of the gauge theory and use nonabelian localization
techniques to write the partition function entirely as a sum over local
contributions from critical points of the action, which are evaluated
explicitly. The partition function of ordinary Yang-Mills theory on the sphere
is recovered in the classical limit as a sum over instantons. We also apply
abelian localization techniques and the geometry of symmetric spaces to derive
an explicit combinatorial expression for the partition function, and compare
the two approaches. These extend the standard techniques for solving gauge
theory on the sphere to the fuzzy case in a rigorous framework.Comment: 55 pages. V2: references added; V3: minor corrections, reference
added; Final version to be published in Communications in Mathematical
Physic
Enabling Explainable Fusion in Deep Learning with Fuzzy Integral Neural Networks
Information fusion is an essential part of numerous engineering systems and
biological functions, e.g., human cognition. Fusion occurs at many levels,
ranging from the low-level combination of signals to the high-level aggregation
of heterogeneous decision-making processes. While the last decade has witnessed
an explosion of research in deep learning, fusion in neural networks has not
observed the same revolution. Specifically, most neural fusion approaches are
ad hoc, are not understood, are distributed versus localized, and/or
explainability is low (if present at all). Herein, we prove that the fuzzy
Choquet integral (ChI), a powerful nonlinear aggregation function, can be
represented as a multi-layer network, referred to hereafter as ChIMP. We also
put forth an improved ChIMP (iChIMP) that leads to a stochastic gradient
descent-based optimization in light of the exponential number of ChI inequality
constraints. An additional benefit of ChIMP/iChIMP is that it enables
eXplainable AI (XAI). Synthetic validation experiments are provided and iChIMP
is applied to the fusion of a set of heterogeneous architecture deep models in
remote sensing. We show an improvement in model accuracy and our previously
established XAI indices shed light on the quality of our data, model, and its
decisions.Comment: IEEE Transactions on Fuzzy System
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
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