The usual Heisenberg uncertainty relation, ∆X∆P ≥ h̄/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exact uncertainty relation, δX∆Pnc ≡ h̄/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving fr...

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of H-type: |∇Ptf | ≤ KPt(|∇f |) where Pt is the heat semigroup corresponding to the sublaplacian on G, ∇ is the subelliptic gradient, and K is a constant. This extends a result of H.-Q. Li [10] for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approa...

In the first paper of this series, we propose a multi-resolution theory of Fourier spectral estimates of finite duration signals. It is shown that multi-resolution capability, achieved without further observation, is obtained by constructing multi-resolution signals from the only observed finite duration signal. Achieved resolutions meet bounds of the uncertainty principle (Heisenberg inequalit...

Non-relativistic quantum mechanics for a free particle is shown to emerge from classical mechanics through the requirement of an invariance principle under transformations that preserve the Heisenberg position-momentum inequality. These transformations acting on the position and momentum uncertainties are induced by isotropic space dilatations. This invariance imposes a change in the laws of cl...

In this paper, we establish two approximation theorems for the multidimensional fractional Fourier transform via appropriate convolutions. As applications, study boundary and initial problems of Laplace heat equations with chirp functions. Furthermore, obtain general Heisenberg inequality respect to transform.

In this paper, we study a few versions of the uncertainty principle for windowed Opdam–Cherednik transform. particular, establish orthonormal sequences, Donoho–Stark's principle, Benedicks‐type Heisenberg‐type and local inequality We also obtain using ‐entropy

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...