102 research outputs found
Foundations of abstract probability theory
Using the ideas of abstract algebra, we introduce the basic concepts of
abstract probability theory that generalize the Kolmogorov's probability
theory, possibility theory and other theories that deal with uncertainty. Based
on abstract addition and multiplication, we define an abstract measure and
abstract Lebesgue integral. System of Kolmogorov's axioms is criticized, after
which we introduce an abstract probability measure and abstract conditional
probability, show that they have recognizable probability properties. In
addition, we define an abstract expected value operator as the abstract
Lebesgue integral and prove its properties.Comment: 12 page
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
Integral Transformation, Operational Calculus and Their Applications
The importance and usefulness of subjects and topics involving integral transformations and operational calculus are becoming widely recognized, not only in the mathematical sciences but also in the physical, biological, engineering and statistical sciences. This book contains invited reviews and expository and original research articles dealing with and presenting state-of-the-art accounts of the recent advances in these important and potentially useful subjects
Quantum (Matrix) Geometry and Quasi-Coherent States
A general framework is described which associates geometrical structures to
any set of finite-dimensional hermitian matrices . This
framework generalizes and systematizes the well-known examples of fuzzy spaces,
and allows to extract the underlying classical space without requiring the
limit of large matrices or representation theory. The approach is based on the
previously introduced concept of quasi-coherent states. In particular, a
concept of quantum K\"ahler geometry arises naturally, which includes the
well-known quantized coadjoint orbits such as the fuzzy sphere and
fuzzy . A quantization map for quantum K\"ahler geometries is
established. Some examples of quantum geometries which are not K\"ahler are
identified, including the minimal fuzzy torus.Comment: 35 pages. V2: minor correction
Magnetic Domains
Recently a Nahm transform has been discovered for magnetic bags, which are
conjectured to arise in the large n limit of magnetic monopoles with charge n.
We interpret these ideas using string theory and present some partial proofs of
this conjecture. We then extend the notion of bags and their Nahm transform to
higher gauge theories and arbitrary domains. Bags in four dimensions
conjecturally describe the large n limit of n self-dual strings. We show that
the corresponding Basu-Harvey equation is the large n limit of an equation
describing n M2-branes, and that it has a natural interpretation in loop space.
We also formulate our Nahm equations using strong homotopy Lie algebras.Comment: 42 pages, minor improvements, published versio
- …