‎In 1897‎, ‎Hensel introduced a normed space which does‎ ‎not have the Archimedean property‎. ‎During the last three decades‎ ‎theory of non--Archimedean spaces has gained the interest of‎ ‎physicists for their research in particular in problems coming‎ ‎from quantum physics‎, ‎p--adic strings and superstrings‎. ‎In this paper‎, ‎we prove‎ ‎the generalized Hyers--Ulam--Rassias stability for a‎ ...

This paper studies construction of quantum Gaussian processes based on ordinary Gaussian processes through their reproducing kernel Hilbert spaces, and investigate the relationship between the stochastic properties of the quantum Gaussian processes and the base Gaussian processes. In particular, we construct quantum Brownian bridges and quantum Ornstein-Uhlenbeck processes. Non-commutative stoc...

We study energy and spectral analysis on compact metric spaces which we refer to as fractal quantum graphs. These are spaces that can be represented as a (possibly infinite) union of 1-dimensional intervals and a totally disconnected (possibly uncountable) compact set, which roughly speaking represents the set of junction points. These spaces include classical quantum graphs and fractal spaces ...

This is an introduction to work on the generalisation to quantum groups of Mackey’s approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces H(H,A) which classify them. They form a new kind of topological quantum number in ...

Measure and integration theory is a well established field of mathematics that is over a hundred years old. The theory possesses many deep and elegant theorems and has important applications in functional analysis, probability theory and theoretical physics. Measure theory can be applied whenever you are measuring something whether it be length, volume, probabilities, mass, energy, etc. Althoug...

We introduce a new distance distoq between compact quantum metric spaces. We show that distoq is Lipschitz equivalent to Rieffel’s distance distq, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq. As applications, we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a f...

The role of curvature in relation with Lie algebra contractions of the pseudoortogonal algebras so(p, q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley–Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant...

We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Here, we discuss speed limits in the context of quantum computing. We review some relevant parts of the theory of Finsler metrics on Lie groups and homogeneous spaces such as the spe...