Tony Scott

RWTH Aachen University, Institute of Physical Chemistry, Visitor

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Tony C. Scott graduated in 1991 with a Ph.D. in Theoretical Physics and was awarded the Pearson Medal for best Physics Doctoral thesis at the University of Waterloo (1991). His Masters thesis in Applied Mathematics (1986) is cited in the wikipedia section on the Wheeler-Feynman absorber theory. Awarded with an N.S.E.R.C. postdoctoral scholarship, he subsequently did research in Mathematical Physics at the Harvard-Smithsonian in Cambridge MA USA ('90-'92). Afterwards, he did research in relativistic quantum chemistry and pioneered a mathematics course using computer algebra at the Mathematical institute in Oxford University in the UK ('93-'95). This was followed by further work at the University of Ben-Gurion in Israel ('96-'97), INRIA in France ('98-'99), the Forschungszentrum in Juelich Germany (2003) and eventually RWTH-Aachen University Germany (2003-2006) where he retains an affiliation. He worked for 7 years as a Data Scientist in Silicon Valley in the San Francisco bay area before returning as a professor in China. He is now back in the privatesector as a Data Scientist.His publications, patents, projects, etc...can be seen (and downloaded) from: https://www.researchgate.net/profile/Tony-Scott-2
Supervisors: Alexander Dalgarno and Ian P. Grant

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Exchange energy of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">H</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msubsup></mml:mrow></mml:math>calculat...

Physical Review Letters, Sep 9, 1991

Tang, Toennies, and Yiu have shown that unsymmetrized Rayleigh-Schrodinger perturbation theory ca... more Tang, Toennies, and Yiu have shown that unsymmetrized Rayleigh-Schrodinger perturbation theory can be used to calculate the leading terms in the ground-state exchange energy of H2+ using the method of Holstein and Herring. We clarify this seemingly puzzling result by showing that their approximation to the polarization wave function is exactly the leading term of the Herring-Holstein function. We also show that a generalization of their method yields the leading asymptotic exchange energy terms for ex- cited states, thereby suggesting the existence of a general underlying property and the applicability of the method to many-electron systems.

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Ring opening by breaking the C-O-C bond in beta-glucose: halogen bombardment

Snake net with a neural network for detecting multiple phases in the phase diagram

Physical Review E

Unsupervised machine learning applied to the study of phase transitions is an ongoing and interes... more Unsupervised machine learning applied to the study of phase transitions is an ongoing and interesting research direction. The active contour model, also called the snake model, was initially proposed for target contour extraction in two-dimensional images. In order to obtain a physical phase diagram, the snake model with an artificial neural network is applied in an unsupervised learning way by the authors of [Phys.Rev.Lett. 120, 176401(2018)]. It guesses the phase boundary as an initial snake and then drives the snake to convergence with forces estimated by the artificial neural network. In this paper, we extend this unsupervised learning method with one contour to a snake net with multiple contours for the purpose of obtaining several phase boundaries in a phase diagram. For the classical Blume-Capel model, the phase diagram containing three and four phases is obtained. Moreover, to overcome the limitations of the initial position and speed up the movement of the snake, the balloon force decaying with the iteration steps is introduced and applied to the snake net structure. Our method is helpful in determining the phase diagram with multiple phases, using just snapshots of configurations from cold atoms or other experiments without knowledge of the phases.

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Michael Scot and the Music of the Spheres

Transversal, Dec 18, 2023

Herein, it is shown that Kepler&#39;s contribution involving the &quot;Music of the Spher... more Herein, it is shown that Kepler&#39;s contribution involving the &quot;Music of the Spheres&quot; can be traced back to Pythagoras and Ibn Arabī through Michael Scot. We find threads linking this body of work associating music with astronomy leading to the modern and extensive astronomical subject of orbital resonances. In particular, we find that Fibonacci numbers play a significant role in this context.

General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function

arXiv (Cornell University), Jul 9, 2006

Herein, we present a canonical form for a natural and necessary generalization of the Lambert W f... more Herein, we present a canonical form for a natural and necessary generalization of the Lambert W function, natural in that it requires minimal mathematical definitions for this generalization, and necessary in that it provides a means of expressing solutions to a number of physical problems of fundamental nature. In particular, this generalization expresses the exact solutions for general-relativistic self-gravitating 2-body and 3-body systems in one spatial and one time dimension. It also expresses the solution to a previously unknown mathematical link between the linear gravity problem and the quantum mechanical Schrödinger wave equation.

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The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions

ACM Communications in Computer Algebra, Aug 31, 2023

Keiper [1] and Li [2] published independent investigations of the connection between the Riemann ... more Keiper [1] and Li [2] published independent investigations of the connection between the Riemann hypothesis and the properties of sums over powers of zeros of the Riemann zeta function. Here we consider a reframing of the criterion, to work with higher-order derivatives xi_r of the symmetrized function xi(s) at s = 1/2, with all xi_r known to be positive. The reframed criterion requires knowledge of the asymptotic properties of two terms, one being an infinite sum over the xi_ r. This is studied using known asymptotic expansions for the xi_ r , which give the location of the summand as a relationship between two parameters. This relationship needs to be inverted, which we show can be done exactly using a generalized Lambert function. The result enables an accurate asymptotic expression for the value of the infinite sum.

Quantum Monte Carlo study of superradiant supersolid of light in the extended Jaynes-Cummings-Hubbard model

Physical Review B, 2021

Herein, we study the extended Jaynes-Cummings-Hubbard model mainly by the large-scale worm quantu... more Herein, we study the extended Jaynes-Cummings-Hubbard model mainly by the large-scale worm quantum Monte-Carlo method to check whether or not a light supersolid phase exists in various geometries, such as the one-dimensional chain, square lattices or triangular lattices. To achieve our purpose, the ground state phase diagrams are investigated. For the one-dimensional chain and square lattices, a first-order transition occurs between the superfluid phase and the solid phase and therefore there is no stable supersolid phase existing in these geometries. Interestingly, soliton/beats of the local densities arise if the chemical potential is adjusted in the finite-size chain. However, this soliton-superfluid coexistence can not be considered as a supersolid in the thermodynamic limit. Searching for a light supersolid, we also studied the Jaynes-Cummings-Hubbard model on triangular lattices, and the phase diagrams are obtained. Through measurement of the structural factor, momentum distribution and superfluid stiffness for various system sizes, a supersolid phase exists stably in the triangular lattices geometry and the regime of the supersolid phase is smaller than that of the mean field results. The light supersolid in the Jaynes-Cummings-Hubbard model is attractive because it has superreliance, which is absent in the pure Bose-Hubbard model. We believe the results in this paper could help search for new novel phases in cold-atom experiments.

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Off-site trimer superfluid on a one-dimensional optical lattice

Chinese Physics B, 2017

The Bose-Hubbard model with an effective off-site three-body tunneling, characterized by jumps to... more The Bose-Hubbard model with an effective off-site three-body tunneling, characterized by jumps towards one another, between one atom on a site and a pair atoms on the neighborhood site, is studied systematically on a one-dimensional lattice, by using the density matrix renormalization group method. The off-site trimer superfluid, condensing at momentum k = 0, emerges in the softcore Bose-Hubbard model but it disappears in the hardcore Bose-Hubbard model. Our results numerically verify that the off-site trimer superfluid phase derived in the momentum space from [Phys. Rev. A 81, 011601(R) (2010)] is stable in the thermodynamic limit. The off-site trimer superfluid phase, the partially off-site trimer superfluid phase and the Mott insulator phase are found, as well as interesting phase transitions, such as the continuous or first-order phase transition from the trimer superfluid phase to the Mott insulator phase. Our results are helpful in realizing this novel off-site trimer superfluid phase by cold atom experiments.

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Fleshing out the generalized Lambert <i>W</i> function

ACM Communications in Computer Algebra, Aug 25, 2016

Herein, we use Hardy's notion of the "false derivative" to obtain exact multiple ro... more Herein, we use Hardy's notion of the "false derivative" to obtain exact multiple roots in closed form of the transcendental-algebraic equations representing the generalized Lambert W function. In this fashion, we flesh out the generalized Lambert W function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

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Snake net with a neural network for detecting multiple phases in the phase diagram

Physical Review E, 2023

Unsupervised machine learning applied to the study of phase transitions is an ongoing and interes... more Unsupervised machine learning applied to the study of phase transitions is an ongoing and interesting research direction. The active contour model, also called the snake model, was initially proposed for target contour extraction in two-dimensional images. In order to obtain a physical phase diagram, the snake model with an artificial neural network is applied in an unsupervised learning way by the authors of [Phys. Rev. Lett. 120, 176401 (2018)]. It guesses the phase boundary as an initial snake and then drives the snake to convergence with forces estimated by the artificial neural network. In this work we extend this unsupervised learning method with one contour to a snake net with multiple contours for the purpose of obtaining several phase boundaries in a phase diagram. For the classical Blume-Capel model, the phase diagram containing three and four phases is obtained. Moreover, a balloon force is introduced, which helps the snake to leave a wrong initial position and thus may allow for greater freedom in the initialization of the snake. Our method is helpful in determining the phase diagram with multiple phases using just snapshots of configurations from cold atoms or other experiments without knowledge of the phases.

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Michael Scot and the Music of the Spheres

by Tony Scott and David Harper

Transversal: International Journal for the Historiography of Science, 2023

Herein, it is shown that Kepler's contribution involving the "Music of the Spheres" can be traced... more Herein, it is shown that Kepler's contribution involving the "Music of the Spheres" can be traced back to Pythagoras and Ibn Arabī through Michael Scot. We find threads linking this body of work associating music with astronomy leading to the modern and extensive astronomical subject of orbital resonances. In particular, we find that Fibonacci numbers play a significant role in this context.

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The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions

ACM Communications in Computer Algebra, 2023

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A Comprehensive Analysis of Quantum Clustering : Finding All the Potential Minima

International Journal of Data Mining & Knowledge Management Process, Jan 31, 2021

Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accom... more Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a σ value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size σ, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach "per block". This technique decreases the number of particles by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

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Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

Physical Review Letters, Jan 20, 1997

We study solitary wave solutions of the higher order nonlinear Schrödinger equation for the propa... more We study solitary wave solutions of the higher order nonlinear Schrödinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions always exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of N-soliton solutions (N ≥ 2) are determined; when these conditions are met the equation becomes the modified KdV equation. A proper subset of these conditions meet the Painlevé plausibility conditions for integrability.

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Perturbative Solutions of Quantum Mechanical Problems by Symbolic Computation: A Review

International Journal of Modern Physics C, Apr 1, 1990

It is shown that Symbolic Computation provides excellent tools for solving quantum mechanical pro... more It is shown that Symbolic Computation provides excellent tools for solving quantum mechanical problems by perturbation theory. The method presented herein solves for both the eigenfunctions and eigenenergies as power series in the order parameter where each coefficient of the perturbation series is obtained in closed form. The algorithms are expressed in the Maple symbolic computation system but can be implemented on other systems. This approach avoids the use of an infinite basis set and some of the complications of degenerate perturbation theory. It is general and can, in principle, be applied to many separable systems.

Fleshing out the Generalized Lambert W Function

ACM Communications in Computer Algebra, Sep 28, 2016

Herein, we use Hardy's notion of the "false derivative" to obtain exact multiple roots in closed ... more Herein, we use Hardy's notion of the "false derivative" to obtain exact multiple roots in closed form of the transcendental-algebraic equations representing the generalized Lambert W function. In this fashion, we flesh out the generalized Lambert W function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

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A model for a relativistic, many-particle Lagrangian with electromagnetic interactions

Canadian Journal of Physics, Mar 1, 1988

It is shown that the Fokker–Wheeler–Feynman theory for the relativistic modelling of a system of ... more It is shown that the Fokker–Wheeler–Feynman theory for the relativistic modelling of a system of massive, charged, point particles interacting via action-at-a-distance forces, electromagnetic in origin, can be reformulated and reinterpreted so that it retains all of its required physical attributes but is devoid of the absurdities originally ascribed to it. That is, Lorentz covariance, time-reversal symmetry, and particle-interchange symmetry are maintained, whereas lack of causality and the paradox of &amp;amp;amp;amp;amp;quot;discontinuous&amp;amp;amp;amp;amp;quot; forces are removed. The reformulated theory yields a physically acceptable relativistic, many-particle Lagrangian. The Euler–Lagrange equations of motion can be written down for either closed or open systems. For closed systems, a generalized Hamiltonian, linear momentum, and angular momentum are constants of the motion. The concept of an open system is used to show that radiation reaction follows straightforwardly from the Euler–Lagrange equations of motion and their past and present time solutions. It is concluded that the basis for this type of modelling of such systems is now established.

Resolution of many particle electrodynamics by symbolic manipulation

Computer Physics Communications, 1989

ABSTRACT

Numerical computation of molecular integrals via optimized (vectorized) FORTRAN code

Nuclear Instruments and Methods in Physics Research, Apr 1, 1997

The calculation of molecular properties based on quantum mechanics is an area of fundamental rese... more The calculation of molecular properties based on quantum mechanics is an area of fundamental research whose horizons have always been determined by the power of state-of-the-art computers. A computational bottleneck is the numerical calculation of the required molecular integrals to sufficient precision. Herein, we present a method for the rapid numerical evaluation of molecular integrals using optimized FORTRAN code generated by Maple. The method is based on the exploitation of common intermediates and the optimization can be adjusted to both serial and vectorized computations.

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