Landau (\Gamma,\chi)-automorphic functions on \mathbb{C}^n of magnitude \nu

We investigate the spectral theory of the invariant Landau Hamiltonian \La^\nu acting on the space ${\mathcal{F}}^\nu_{\Gamma,\chi}$ of $(\Gamma,\chi)$-automotphic functions on \C^n, for given real number $\nu>0$, lattice $\Gamma$ of \C^n and a map $\chi:\Gamma\to U(1)$ such that the triplet $(\nu,\Gamma,\chi)$ satisfies a Riemann-Dirac quantization type condition. More precisely, we show that the eigenspace {\mathcal{E}}^\nu_{\Gamma,\chi}(\lambda)=\set{f\in {\mathcal{F}}^\nu_{\Gamma,\chi}; \La^\nu f = \nu(2\lambda+n) f}; \lambda\in\C, is non trivial if and only if $\lambda=l=0,1,2, ...$. In such case, ${\mathcal{E}}^\nu_{\Gamma,\chi}(l)$ is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace ${\mathcal{E}}^\nu_{\Gamma,\chi}(0)$ associated to the lowest Landau level of \La^\nu is isomorphic to the space, {\mathcal{O}}^\nu_{\Gamma,\chi}(\C^n), of holomorphic functions on \C^n satisfying g(z+\gamma) = \chi(\gamma) e^{\frac \nu 2 |\gamma|^2+\nu\scal{z,\gamma}}g(z), \eqno{(*)} that we can realize also as the null space of the differential operator $\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} + \nu \bar z_j \frac{\partial}{\partial \bar z_j})$ acting on $\mathcal C^\infty$ functions on \C^n satisfying $(*)$.Comment: 20 pages. Minor corrections. Scheduled to appear in issue 8 (2008) of "Journal of Mathematical Physics

A Hardy's Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra

In this article we consider linear operators satisfying a generalized commutation relation of a type of the Heisenberg-Lie algebra. It is proven that a generalized inequality of the Hardy's uncertainty principle lemma follows. Its applications to time operators and abstract Dirac operators are also investigated

Localization of Multi-Dimensional Wigner Distributions

A well known result of P. Flandrin states that a Gaussian uniquely maximizes the integral of the Wigner distribution over every centered disc in the phase plane. While there is no difficulty in generalizing this result to higher-dimensional poly-discs, the generalization to balls is less obvious. In this note we provide such a generalization.Comment: Minor corrections, to appear in the Journal of Mathematical Physic

On spectral analysis of a magnetic Schrodinger operator on planar mixed automorphic forms

We characterize the space of the so-called planar mixed automorphic forms of type $(\nu,\mu)$ with respect to an equivariant pair $(\rho,\tau)$ as the image of the usual automorphic forms by an appropriate transform and we investigate some concrete basic spectral properties of a magnetic Schrodinger operator acting on them. The associated polynomials constitute classes of generalized complex polynomials of Hermite type.Comment: 10 pages. This is a substantially reorganized, revised and improved exposition. Misprints corrected and references added. Submitte

Mussel culture experiments in Ennore estuary, Chennai

The present paper gives a detailed account on experiments of mussel culture carried out by the Institute in 1996 in association with fishermen of Ennore in an estuarine environment by adopting the long-line and rack culture methods

Information exposure from consumer IoT devices: a multidimensional, network-informed measurement approach

Internet of Things (IoT) devices are increasingly found in everyday homes, providing useful functionality for devices such as TVs, smart speakers, and video doorbells. Along with their benefits come potential privacy risks, since these devices can communicate information about their users to other parties over the Internet. However, understanding these risks in depth and at scale is difficult due to heterogeneity in devices' user interfaces, protocols, and functionality. In this work, we conduct a multidimensional analysis of information exposure from 81 devices located in labs in the US and UK. Through a total of 34,586 rigorous automated and manual controlled experiments, we characterize information exposure in terms of destinations of Internet traffic, whether the contents of communication are protected by encryption, what are the IoT-device interactions that can be inferred from such content, and whether there are unexpected exposures of private and/or sensitive information (e.g., video surreptitiously transmitted by a recording device). We highlight regional differences between these results, potentially due to different privacy regulations in the US and UK. Last, we compare our controlled experiments with data gathered from an in situ user study comprising 36 participants

Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on Rn. Roughly speaking, the uncertainty principle says that if a function f is concentrated then its Fourier transform f cannot be concentrated unless f is identically zero. Of course, in the above, we should be precise about what we mean by concentration. There are several ways of measuring concentration and depending on the definition we get a host of uncertainty pronciples. As several authors have shownc some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including Rn, the heisenberg group, the reduced Heisenberg group and the Euclidean motion group of the plane

Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on Rn. Roughly speaking, the uncertainty principle says that if a function f is 'concentrated' then its Fourier transform f~ cannot be 'concentrated' unless f is identically zero. Of course, in the above, we should be precise about what we mean by 'concentration'. There are several ways of measuring 'concentration' and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including Rn, the Heisenberg group, the reduced Heisenberg group and the Euclidean motion group of the plane

Comparison of Interactive Teaching in Online and Offline Platforms among Dental Undergraduates

In recent years, the educational system has focused more on the holistic development of an individual. Modern technology has changed the educational environment to provide students with better academic opportunities. Along with the education system, teaching techniques and learning tools have also changed with digital evolution. This research was undertaken to assess the academic performance of interactive teaching methods in offline and online platforms in Periodontics among BDS undergraduates at a dental college in India. This prospective study was conducted among 49 students: Group I (n = 24, online class through Zoom) and Group II (n = 25, offline classes). The subject was divided into three modules and was covered in one week. The topics covered, teaching methods, lectures, and activities were similar for both groups. A formative assessment mark was obtained from written tests during the module, whereas the summative assessment mark was recorded from exams conducted towards the end of the module. In the results, a statistically significant difference was not observed in terms of formative assessment between Group I (77.88 ± 12.89) and Group II (77.80 ± 16.09) (p = 0.98). In addition, a statistically significant difference was not observed in terms of summative assessment between Group I (80.54 ± 8.39) and Group II (80.28 ± 11.57) (p = 0.93). Overall, this study suggests that interactive teaching methods in both offline and online platforms in Periodontics showed equivalent performance by the undergraduate dental students