3,619 research outputs found
Pettis integrability of fuzzy mappings with values in arbitrary Banach spaces
In this paper we study the Pettis integral of fuzzy mappings in arbitrary
Banach spaces. We present some properties of the Pettis integral of fuzzy
mappings and we give conditions under which a scalarly integrable fuzzy mapping
is Pettis integrable
Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
Weighted lattice polynomials of independent random variables
We give the cumulative distribution functions, the expected values, and the
moments of weighted lattice polynomials when regarded as real functions of
independent random variables. Since weighted lattice polynomial functions
include ordinary lattice polynomial functions and, particularly, order
statistics, our results encompass the corresponding formulas for these
particular functions. We also provide an application to the reliability
analysis of coherent systems.Comment: 14 page
Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems
We investigate a U(1) gauge invariant quantum mechanical system on a 2D
noncommutative space with coordinates generating a generalized deformed
oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge
covariant derivatives obeying the nonlinear Dolan-Grady relations. This
restricts the structure function of the deformed oscillator algebra to a
quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R)
algebras are investigated in detail. Reducing the Hamiltonian to 1D
finite-difference quasi-exactly solvable operators, we demonstrate partial
algebraization of the spectrum of the corresponding systems on the fuzzy sphere
and noncommutative hyperbolic plane. A completely covariant method based on the
notion of intrinsic algebra is proposed to deal with the spectral problem of
such systems.Comment: 25 pages; ref added; to appear in J. Phys.
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
Weighted lattice polynomials
We define the concept of weighted lattice polynomial functions as lattice
polynomial functions constructed from both variables and parameters. We provide
equivalent forms of these functions in an arbitrary bounded distributive
lattice. We also show that these functions include the class of discrete Sugeno
integrals and that they are characterized by a median based decomposition
formula.Comment: Revised version (minor changes
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