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Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

Applied Mathematics 2017年第7期8卷 901-919页

作者： Donald J. Kouri Cameron L. Williams nikhil pandya Department of Physics University of Houston Houston TX USA Department of Mathematics University of Houston Houston TX USA

Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W,?Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.

关键词： Canonical Transformations Quantization Mutually Unbiased Bases Complete Bases

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Point Transformations and Relationships among Linear Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

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Applied Mathematics 2018年第2期9卷 178-197页

作者： Donald Kouri nikhil pandya Cameron L. Williams Bernhard G. Bodmann Jie Yao Department of Physics University of Houston Houston Texas USA Department of Mathematics University of Houston Houston Texas USA Department of Mechanical Engineering Texas Tech University Lubbock Texas USA

We present new connections among linear anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Anomalous Diffusion equations of O’Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we can tell) be expressed as examples of the O’Shaughnessy and Procaccia equations. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. We illustrate the ideas both analytically and with a detailed computational example for a non-trivial choice of point transformation. Finally, we summarize our results.

关键词： Generalized Fourier Analysis Normal Diffusion Anomalous Diffusion Point Transformations Canonical Quantization Super Symmetric Quantum Mechanics

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Biodegradable polymer stents vs second generation drug eluting stents:A meta-analysis and systematic review of randomized controlled trials

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World Journal of Cardiology 2016年第2期8卷 240-246页

作者： Bhavi pandya Sainath Gaddam Muhammad Raza Deepak Asti nikhil Nalluri Thomas Vazzana Ruben Kandov James Lafferty Department of Internal Medicine Staten Island University Hospital Department of Cardiology Staten Island University Hospital

AIM: To evaluate the premise, that biodegradable polymer drug eluting stents(BD-DES) could improve clinical outcomes compared to second generation permanent polymer drug eluting stents(PP-DES), we pooled the data from all the available randomized control trials(RCT) comparing the clinical performance of both these ***: A systematic literature search of Pub Med, Cochrane, Google scholar databases, EMBASE, MEDLINE and SCOPUS was performed during time period of January 2001 to April 2015 for RCT and comparing safety and efficacy of BD-DES vs second generation PP-DES. The primary outcomes of interest were definite stent thrombosis, target lesion revascularization, myocardial infarction, cardiac deaths and total deaths during the study period. RESULTS: A total of 11 RCT's with a total of 12644 patients were included in the meta-analysis, with 6598 patients in BD-DES vs 6046 patients in second generation PP-DES. The mean follow up period was 16 mo. Pooled analysis showed non-inferiority of BD-DES, comparing events of stent thrombosis(OR = 1.42, 95%CI: 0.79-2.52, P = 0.24), target lesion revascularization(OR = 0.99, 95%CI: 0.84-1.17, P = 0.92), myocardial infarction(OR = 1.06, 95%CI: 0.86-1.29, P = 0.92), cardiac deaths(OR = 1.07, 95%CI 0.82-1.41, P = 0.94) and total deaths(OR = 0.96, 95%CI: 0.80-1.17, P = 0.71).CONCLUSION: BD-DES, when compared to second generation PP-DES, showed no significant advantage and the outcomes were comparable between both the groups.

关键词： Stent design Drug eluting stent Zotarolimus eluting stent Cobalt-chromium stent Biodegradable drug eluting stent

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