Related papers: A curious differential calculus on the quantum dis…
The Fundamental Theorem of Integral Calculus links the integrand and its antiderivative via a simple first order differential equation. A numerical solution of this ode yields the antiderivative and hence the required integral. This…
We study classical and quantum dynamics of a particle in a circular billiard with a straight cut. This system can be integrable, nonintegrable with soft chaos, or nonintegrable with hard chaos, as we vary the size of the cut. We use a…
This paper is the first part of a two-paper series whose aim is to give a thorough account on Connes' pseudodifferential calculus on noncommutative tori. This pseudodifferential calculus has been used in numerous recent papers, but a…
A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in…
This work considers a formal deformation of the quantum disc (it is developed via an application of the Berezin method) and presents an explicit formula for this deformation.
We discuss classical and quantum computations in terms of corresponding Hamiltonian dynamics. This allows us to introduce quantum computations which involve parallel processing of both: the data and programme instructions. Using mixed…
Quantum algorithms for computing classical nonlinear maps are widely known for toy problems but might not suit potential applications to realistic physics simulations. Here, we propose how to compute a general differentiable invertible…
Ordinary differential equations have an arithmetic analogue in which functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main motivations, constructions,…
Recent innovations in the differential calculus for functions of non-commuting variables, beginning with a quaternionic variable, are now extended to consider some integration.
We show that dissipative classical dynamics converging to a strange attractor can be simulated on a quantum computer. Such quantum computations allow to investigate efficiently the small scale structure of strange attractors, yielding new…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
A differential calculus of the first order over multi-braided quantum groups is developed. In analogy with the standard theory, left/right-covariant and bicovariant differential structures are introduced and investigated. Furthermore,…
A method has been recently proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example a version of quantized space-time is considered here. It is found that there is a…
In this review article the construction of first order coordinate differential calculi on finitely generated and finitely related associative algebras are considered and explicit construction of the bimodule of one form over such algebras…
We outline the recent classification of differential structures for all main classes of quantum groups. We also outline the algebraic notion of `quantum manifold' and `quantum Riemannian manifold' based on quantum group principal bundles, a…
Derivation-based differential calculi are of great importance in noncommutative geometry, noncommutative gauge theory and integrable systems. In this paper, we propose the connection and curvature from a class of deformed derivation-based…
We review some of the geometry of the quantum projective plane with emphasis on the construction of a differential calculus and of the Dirac operator (of a spin^c-structure). We also report on anti-self-dual connections on line bundles, the…
The structure and properties of possible $q$-Minkowski spaces is discussed, and the corresponding non-commutative differential calculi are developed in detail and compared with already existing proposals. This is done by stressing its…
We consider selected aspects of (non-Archimedean) quantum mathematics and non-Archimedean (quantum) computation.
We develop a pseudo-differential Weyl calculus on nilpotent Lie groups which allows one to deal with magnetic perturbations of right invariant vector fields. For this purpose we investigate an infinite-dimensional Lie group constructed as…