31,261 research outputs found
The Fixed Point Method for Fuzzy Approximation of a Functional Equation Associated with Inner Product Spaces
Th. M. Rassias (1984) proved that the norm defined over a real vector space is induced by an inner product if and only if for a fixed integer ≥2,∑=1‖∑−(1/)=1‖2=∑=1‖‖2∑−‖(1/)=1‖2 holds for all 1,…,∈. The aim of this paper is to extend the applications of the fixed point alternative method to provide a fuzzy stability for the functional equation ∑=1(∑−(1/)=1∑)==1(∑)−((1/)=1) which is said to be a functional equation associated with inner product spaces
Matrix geometries and fuzzy spaces as finite spectral triples
A class of real spectral triples that are similar in structure to a
Riemannian manifold but have a finite-dimensional Hilbert space is defined and
investigated, determining a general form for the Dirac operator. Examples
include fuzzy spaces defined as real spectral triples. Fuzzy 2-spheres are
investigated in detail, and it is shown that the fuzzy analogues correspond to
two spinor fields on the commutative sphere. In some cases it is necessary to
add a mass mixing matrix to the commutative Dirac operator to get a precise
agreement for the eigenvalues.Comment: 39 pages, final versio
Fuzzy Bigraphs: An Exercise in Fuzzy Communicating Agents
Bigraphs and their algebra is a model of concurrency. Fuzzy bigraphs are a
generalization of birgraphs intended to be a model of concurrency that
incorporates vagueness. More specifically, this model assumes that agents are
similar, communication is not perfect, and, in general, everything is or
happens to some degree.Comment: 11 pages, 3 figure
Lie 2-algebra models
In this paper, we begin the study of zero-dimensional field theories with
fields taking values in a semistrict Lie 2-algebra. These theories contain the
IKKT matrix model and various M-brane related models as special cases. They
feature solutions that can be interpreted as quantized 2-plectic manifolds. In
particular, we find solutions corresponding to quantizations of R^3, S^3 and a
five-dimensional Hpp-wave. Moreover, by expanding a certain class of Lie
2-algebra models around the solution corresponding to quantized R^3, we obtain
higher BF-theory on this quantized space.Comment: 47 pages, presentation improved, version published in JHE
Gauge Theories in Noncommutative Homogeneous K\"ahler Manifolds
We construct a gauge theory on a noncommutative homogeneous K\"ahler
manifold, where we employ the deformation quantization with separation of
variables for K\"ahler manifolds formulated by Karabegov. A key point in this
construction is to obtaining vector fields which act as inner derivations for
the deformation quantization. We show that these vector fields are the only
Killing vector fields. We give an explicit construction of this gauge theory on
noncommutative and noncommutative .Comment: 27 pages, typos correcte
Quantized Nambu-Poisson Manifolds and n-Lie Algebras
We investigate the geometric interpretation of quantized Nambu-Poisson
structures in terms of noncommutative geometries. We describe an extension of
the usual axioms of quantization in which classical Nambu-Poisson structures
are translated to n-Lie algebras at quantum level. We demonstrate that this
generalized procedure matches an extension of Berezin-Toeplitz quantization
yielding quantized spheres, hyperboloids, and superspheres. The extended
Berezin quantization of spheres is closely related to a deformation
quantization of n-Lie algebras, as well as the approach based on harmonic
analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms
of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative
hyperplanes. Some applications to the quantum geometry of branes in M-theory
are also briefly discussed.Comment: 43 pages, minor corrections, presentation improved, references adde
- …