A system of PDOs(=Partial Differential Operators) has two non-commutativities, (i) one from [∂q, q] = 1 (Heisenberg relation), (ii) the other from [A, B] 6= 0 for A, B being matrices in general. Non-commutativity from Heisenberg relation is nicely controlled by using Fourier transformations (i.e. the theory of ΨDOs=pseudo-differential operators). Here, we give a new method of treating non-commu...

Recently there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: (1) by a natural measure of orthonormality, they are as close to being an orthonormal basis for the sp...

The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f : R → C is a complex valued function of a random real variable x such that f ∈ L(R), where R = (−∞,∞). Then the product of the second moment of the random real x for |f | and the second moment of the random real ξ for ∣∣∣f̂ ∣∣∣2 is at least ER,|f |2 / 4π, where f̂ is the Fourier transform of f ...

We consider three different versions of the Zak transform (ZT) for discrete-time signals, namely, the discretetime ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss ...

In the first part of the paper a general notion of sampling expansions for locally compact groups is introduced, and its close relationship to the discretisation problem for generalised wavelet transforms is established. In the second part, attention is focussed on the simply connected nilpotent Heisenberg group H. We derive criteria for the existence of discretisations and sampling expansions ...

Two fundamental issues about the relation between the deformed Heisenberg-Weyl algebra in noncommutative space and the undeformed one in commutative space are elucidated. First the un-equivalency theorem between two algebras is proved: the deformed algebra related to the undeformed one by a non-orthogonal similarity transformation is explored; furthermore, non-existence of a unitary similarity ...

in this paper, biharmonic slant helices are studied according to bishop frame in the heisenberg group heis3. we give necessary and sufficient conditions for slant helices to be biharmonic. the biharmonic slant helices arecharacterized in terms of bishop frame in the heisenberg group heis3. we give some characterizations for tangent bishop spherical images of b-slant helix. additionally, we illu...