A new approach to probability theory based on quantum mechanical and Lie algebraic ideas is proposed and developed. The underlying fact is the observation that the coherent states of the Heisenberg-Weyl, su(2), su(r+1), su(1, 1) and su(r, 1) algebras in certain symmetric (bosonic) representations give the “probability amplitudes” (or the “square roots”) of the well-known Poisson, binomial, mult...

A new approach to probability theory based on quantum mechanical and Lie algebraic ideas is proposed and developed. The underlying fact is the observation that the coherent states of the Heisenberg-Weyl, su(2), su(r+1), su(1, 1) and su(r, 1) algebras in certain symmetric (bosonic) representations give the “probability amplitudes” (or the “square roots”) of the well-known Poisson, binomial, mult...

Many practical machine learning tasks employ very deep convolutional neural networks. Such large depths pose formidable computational challenges in training and operating the network. It is therefore important to understand how many layers are actually needed to have most of the input signal’s features be contained in the feature vector generated by the network. This question can be formalized ...

We derive the evolution equations for the electric and magnetic parts of the Weyl tensor for cold dust from both general relativity and Newtonian gravity. In a locally inertial frame at rest in the fluid frame, the Newtonian equations agree with those of general relativity. We give explicit expressions for the electric and magnetic parts of the Weyl tensor in the Newtonian limit. In general, th...

We derive frame estimates for vector-valued Gabor systems with window functions belonging to Schwartz space. The main result provides frame bound estimates for windows composed of Hermite functions. The proof is based on a recently established sampling theorem for the simply connected Heisenberg group, which is translated to a family of frame estimates via a direct integral decomposition.

The goal of this paper is to characterise certain probability laws on a class of quantum groups or braided groups that we will call nilpotent. First we introduce a braided analogue of the Heisenberg-Weyl group, which shall serve as standard example. We introduce Gaussian functionals on quantum groups or braided groups as functionals that satisfy an analogue of the Bernstein property, i.e. that ...

A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry and to the Chern-Moser tensor in CR geometry. It is shown that a quaternionic c...

We study entanglement and genuine of tripartite four-partite quantum states by using Heisenberg-Weyl (HW) representation density matrices. Based on the correlation tensors in HW representation, we present criteria to detect entanglement. Detailed examples show that our method can more entangled than previous criteria.

Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl–Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies κ-Minkowski spacetime coordinates with Poincaré generators, can be obtained...

We present the world-line quantisation of a system invariant under the symmetries of reciprocal relativity (pseudo-unitary transformations on “phase space coordinates” (x(τ), p(τ)) which preserve the Minkowski metric and the symplectic form, and global shifts in these coordinates, together with coordinate dependent transformations of an additional compact phase coordinate, θ(τ)). The action is ...