Journal articles: 'Harmonic potential theorem'

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Chen, Qun. "Liouville theorem for harmonic maps with potential." Manuscripta Mathematica 95, no. 1 (December 1998): 507–17. http://dx.doi.org/10.1007/bf02678046.

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Chen, Qun. "Liouville theorem for harmonic maps with potential." manuscripta mathematica 95, no. 4 (April 1, 1998): 507–17. http://dx.doi.org/10.1007/s002290050044.

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Chen, Jin-Wang, Tao Yang, and Xiao-Yin Pan. "A New Proof for the Harmonic-Potential Theorem." Chinese Physics Letters 30, no. 2 (February 2013): 020303. http://dx.doi.org/10.1088/0256-307x/30/2/020303.

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Dobson, John F. "Harmonic-Potential Theorem: Implications for Approximate Many-Body Theories." Physical Review Letters 73, no. 16 (October 17, 1994): 2244–47. http://dx.doi.org/10.1103/physrevlett.73.2244.

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GURAPPA, N., PRASANTA K. PANIGRAHI, T. SOLOMAN RAJU, and V. SRINIVASAN. "QUANTUM EQUIVALENT OF THE BERTRAND'S THEOREM." Modern Physics Letters A 15, no. 30 (September 28, 2000): 1851–57. http://dx.doi.org/10.1142/s0217732300002255.

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A procedure for constructing general bound state potentials is given. Analogous to the Bertrand's theorem in classical mechanics, we then identify radial eigenvalue problems possessing exact solvability and infinite number of eigenstates. Akin to the classical result, the only special cases of the general central potential, satisfying the above two conditions, are the Coulomb and harmonic oscillator potentials.

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De La Calle Ysern, Bernardo, and José C. Sabina De Lis. "A Constructive Proof of Helmholtz’s Theorem." Quarterly Journal of Mechanics and Applied Mathematics 72, no. 4 (September 4, 2019): 521–33. http://dx.doi.org/10.1093/qjmam/hbz016.

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Summary It is a known result that any vector field ${\boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $\Omega\subset \mathbb{R}^3$ can be written on $\Omega$ as the sum of a gradient and a curl. Should $\Omega$ be unbounded, no conditions are required on the behaviour of ${\boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.

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Zhang, Cheng, Jin Yang, Liu Xi Yang, Jun Chen Ke, Ming Zheng Chen, Wen Kang Cao, Mao Chen, et al. "Convolution operations on time-domain digital coding metasurface for beam manipulations of harmonics." Nanophotonics 9, no. 9 (February 18, 2020): 2771–81. http://dx.doi.org/10.1515/nanoph-2019-0538.

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AbstractTime-domain digital coding metasurfaces have been proposed recently to achieve efficient frequency conversion and harmonic control simultaneously; they show considerable potential for a broad range of electromagnetic applications such as wireless communications. However, achieving flexible and continuous harmonic wavefront control remains an urgent problem. To address this problem, we present Fourier operations on a time-domain digital coding metasurface and propose a principle of nonlinear scattering-pattern shift using a convolution theorem that facilitates the steering of scattering patterns of harmonics to arbitrarily predesigned directions. Introducing a time-delay gradient into a time-domain digital coding metasurface allows us to successfully deviate anomalous single-beam scattering in any direction, and thus, the corresponding formula for the calculation of the scattering angle can be derived. We expect this work to pave the way for controlling energy radiations of harmonics by combining a nonlinear convolution theorem with a time-domain digital coding metasurface, thereby achieving more efficient control of electromagnetic waves.

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Hsueh, Che-Hsiu, Chi-Ho Cheng, Tzyy-Leng Horng, and Wen-Chin Wu. "H-Theorem in an Isolated Quantum Harmonic Oscillator." Entropy 24, no. 8 (August 20, 2022): 1163. http://dx.doi.org/10.3390/e24081163.

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We consider the H-theorem in an isolated quantum harmonic oscillator through the time-dependent Schrödinger equation. The effect of potential in producing entropy is investigated in detail, and we found that including a barrier potential into a harmonic trap would lead to the thermalization of the system, while a harmonic trap alone would not thermalize the system. During thermalization, Shannon entropy increases, which shows that a microscopic quantum system still obeys the macroscopic thermodynamics law. Meanwhile, initial coherent mechanical energy transforms to incoherent thermal energy during thermalization, which exhibiting the decoherence of an oscillating wave packet featured by a large decreasing of autocorrelation length. When reaching thermal equilibrium, the wave packet comes to a halt, with the density distributions both in position and momentum spaces well-fitted by a microcanonical ensemble of statistical mechanics.

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Graversen, S. E. "A Riesz decomposition theorem." Nagoya Mathematical Journal 114 (June 1989): 123–33. http://dx.doi.org/10.1017/s0027763000001422.

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The topic of this note is the Riesz decomposition of excessive functions for a “nice” strong Markov process X. I.e. an excessive function is decomposed into a sum of a potential of a measure and a “harmonic” function. Originally such decompositions were studied by G.A. Hunt [8]. In [1] a Riesz decomposition is given assuming that the state space E is locally compact with a countable base and X is a transient standard process in strong duality with another standard process having a strong Feller resolvent. Recently R.K. Getoor and J. Glover extended the theory to the case of transient Borei right processes in weak duality [6].

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Lai, Meng-Yun, Duan-Liang Xiao, and Xiao-Yin Pan. "The Harmonic Potential Theorem for a Quantum System with Time-Dependent Effective Mass." Chinese Physics Letters 32, no. 11 (November 2015): 110301. http://dx.doi.org/10.1088/0256-307x/32/11/110301.

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Khavinson, Dmitry. "Cauchy’s Problem for Harmonic Functions with Entire Data on a Sphere." Canadian Mathematical Bulletin 40, no. 1 (March 1, 1997): 60–66. http://dx.doi.org/10.4153/cmb-1997-007-3.

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AbstractWe give an elementary potential-theoretic proof of a theorem of G. Johnsson: all solutions of Cauchy’s problems for the Laplace equations with an entire data on a sphere extend harmonically to the whole space RN except, perhaps, for the center of the sphere.

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Weniger, Ernst Joachim. "The Spherical Tensor Gradient Operator." Collection of Czechoslovak Chemical Communications 70, no. 8 (2005): 1225–71. http://dx.doi.org/10.1135/cccc20051225.

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The spherical tensor gradient operator Ylm(∇), which is obtained by replacing the Cartesian components of r by the Cartesian components of ∇ in the regular solid harmonic Ylm(r), is an irreducible spherical tensor of rank l. Accordingly, its application to a scalar function produces an irreducible spherical tensor of rank l. Thus, it is in principle sufficient to consider only multicenter integrals of scalar functions: Higher angular momentum states can be generated by differentiation with respect to the nuclear coordinates. Many of the properties of Ylm(∇) can be understood easily with the help of an old theorem on differentiation by Hobson [Proc. Math. London Soc. 24, 54 (1892)]. It follows from Hobson's theorem that some scalar functions of considerable relevance as for example the Coulomb potential, Gaussian functions, or reduced Bessel functions produce particularly compact results if Ylm(∇) is applied to them. Fourier transformation is very helpful in understanding the properties of Ylm(∇) since it produces Ylm(-ip). It is also possible to apply Ylm(∇) to generalized functions, yielding for instance the spherical delta function δlm(r). The differential operator Ylm(∇) can also be used for the derivation of pointwise convergent addition theorems. The feasibility of this approach is demonstrated by deriving the addition theorem of rvYlm(r) with v ∈ πR.

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Markiewicz, M., P. Łe¸tkowski, and O. Mahrenholtz. "Third-Order Hydrodynamic Loads on an Oscillating Vertical Cylinder in Water." Journal of Offshore Mechanics and Arctic Engineering 121, no. 1 (February 1, 1999): 16–21. http://dx.doi.org/10.1115/1.2829549.

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The third-harmonic component of the third-order hydrodynamic loads on a vertical circular cylinder oscillating in water is calculated by a conventional perturbation method within the framework of a potential theory. Although the third-order forces are expressed in terms of the first, second, and third-order components of the velocity potential, the latter is not directly required for the calculation. It is replaced by a properly defined linearized radiation potential via Haskind-like theorem. The results of the study are applicable to the analysis of high-frequency resonances of deepwater offshore structures under earthquake excitation or under steep waves (ringing problem).

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Urban, Zbyněk, Francesco Bajardi, and Salvatore Capozziello. "The Noether–Bessel-Hagen symmetry approach for dynamical systems." International Journal of Geometric Methods in Modern Physics 17, no. 14 (October 20, 2020): 2050215. http://dx.doi.org/10.1142/s0219887820502151.

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The Noether–Bessel-Hagen theorem can be considered a natural extension of Noether Theorem to search for symmetries. Here, we develop the approach for dynamical systems introducing the basic foundations of the method. Specifically, we establish the Noether–Bessel-Hagen analysis of mechanical systems where external forces are present. In the second part of the paper, the approach is adopted to select symmetries for a given systems. In particular, we focus on the case of harmonic oscillator as a testbed for the theory, and on a cosmological system derived from scalar–tensor gravity with unknown scalar-field potential [Formula: see text]. We show that the shape of potential is selected by the presence of symmetries. The approach results particularly useful as soon as the Lagrangian of a given system is not immediately identifiable or it is not a Lagrangian system.

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Cao, Jie, Lingkun Ran, and Na Li. "An Application of the Helmholtz Theorem in Extracting the Externally Induced Deformation Field from the Total Wind Field in a Limited Domain." Monthly Weather Review 142, no. 5 (April 30, 2014): 2060–66. http://dx.doi.org/10.1175/mwr-d-13-00311.1.

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Abstract A new application of the Helmholtz theorem that divides the horizontal wind into purely rotational, purely divergent, and harmonic deformational flow is put forward in this study. The formulas and methods, with consideration of avoiding the nonunique problem in solving the two Poisson equations with coupled boundary conditions, are constructed and tested for applicability and accuracy in an ideal experiment. Numerical tests show that the three extracted components together with the reconstructed wind almost recover the ideal fields by using the extended Helmholtz theorem, while other methods fail. The physical meaning of the extracted externally induced deformational flow is explained in the context of its connection to frontogenesis in a real weather event. It may provide potential usage in analyzing certain types of real weather.

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Berinskii, I., and V. A. Kuzkin. "Equilibration of energies in a two-dimensional harmonic graphene lattice." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2162 (November 25, 2019): 20190114. http://dx.doi.org/10.1098/rsta.2019.0114.

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We study dynamical phenomena in a harmonic graphene (honeycomb) lattice, consisting of equal particles connected by linear and angular springs. Equations of in-plane motion for the lattice are derived. Initial conditions typical for molecular dynamic modelling are considered. Particles have random initial velocities and zero displacements. In this case, the lattice is far from thermal equilibrium. In particular, initial kinetic and potential energies are not equal. Moreover, initial kinetic energies (and temperatures), corresponding to degrees of freedom of the unit cell, are generally different. The motion of particles leads to equilibration of kinetic and potential energies and redistribution of kinetic energy among degrees of freedom. During equilibration, the kinetic energy performs decaying high-frequency oscillations. We show that these oscillations are accurately described by an integral depending on dispersion relation and polarization matrix of the lattice. At large times, kinetic and potential energies tend to equal values. Kinetic energy is partially redistributed among degrees of freedom of the unit cell. Equilibrium distribution of the kinetic energies is accurately predicted by the non-equipartition theorem. Presented results may serve for better understanding of the approach to thermal equilibrium in graphene. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

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Nie, Deming, and Jianzhong Lin. "A Lattice Boltzmann-Direct Forcing/Fictitious Domain Model for Brownian Particles in Fluctuating Fluids." Communications in Computational Physics 9, no. 4 (April 2011): 959–73. http://dx.doi.org/10.4208/cicp.181109.300610a.

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AbstractThe previously developed LB-DF/FD method derived from the lattice Boltzmann method and Direct Forcing/Fictitious Domain method is extended to deal with 3D particle’s Brownian motion. In the model the thermal fluctuations are introduced as random forces and torques acting on the Brownian particle. The hydrodynamic interaction is introduced by directly resolving the fluid motions. A sphere fluctuating in a cubic box with the periodic boundary is considered to validate the present model. By examining the velocity autocorrelation function (VCF) and rotational velocity autocorrelation function (RVCF), it has been found that in addition to the two relaxation times, the mass density ratio should be taken into consideration to check the accuracy and effectiveness of the present model. Furthermore, the fluctuation-dissipation theorem and equipartition theorem have been investigated for a single spherical particle. Finally, a Brownian particle trapped in a harmonic potential has been simulated to further demonstrate the ability of the LB-DF/FD model.

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Semeraro, Massimiliano, Antonio Suma, Isabella Petrelli, Francesco Cagnetta, and Giuseppe Gonnella. "Work fluctuations in the active Ornstein–Uhlenbeck particle model." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 12 (December 1, 2021): 123202. http://dx.doi.org/10.1088/1742-5468/ac3d37.

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Abstract We study the large deviations of the power injected by the active force for an active Ornstein–Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre–Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system’s dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation–dissipation theorem.

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Liang, Zhenguo, and Zhiqiang Wang. "Reducibility of 1D quantum harmonic oscillator with decaying conditions on the derivative of perturbation potentials." Nonlinearity 35, no. 9 (August 11, 2022): 4850–75. http://dx.doi.org/10.1088/1361-6544/ac821a.

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Abstract We prove the reducibility of 1D quantum harmonic oscillators in R perturbed by a quasi-periodic in time potential V(x, ωt) under the following conditions, namely there is a C > 0 such that | V ( x , θ ) | ⩽ C , | x ∂ x V ( x , θ ) | ⩽ C , ∀ ( x , θ ) ∈ R × T σ n . The corresponding perturbation matrix ( P i j ( θ ) ) is proved to satisfy ( 1 + | i − j | ) | P i j ( θ ) | ⩽ C and i j | P i + 1 j + 1 ( θ ) − P i j ( θ ) | ⩽ C for any θ ∈ T σ n and i, j ⩾ 1. A new reducibility theorem is set up under this kind of decay in the perturbation matrix element P i j ( θ ) as well as the discrete difference matrix element P i + 1 j + 1 ( θ ) − P i j ( θ ) . For the proof the novelty is that we use the decay in the discrete difference matrix element to control the measure estimates for the thrown parameter sets.

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Alfadhli, Munirah, Adel Elmandouh, and Muneerah Al Nuwairan. "Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame." Applied Sciences 13, no. 2 (January 14, 2023): 1123. http://dx.doi.org/10.3390/app13021123.

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This work aims to explore some dynamic aspects of the problem of star motion that is impacted by the rotation of the galaxy, which we model as a bisymmetric potential based on a two-dimensional harmonic oscillator with sextic perturbations. We demonstrate analytically that the motion is non-integrable when certain conditions are met. The analytical results for the non-integrability are confirmed by showing the irregularity of the behavior of the motion through utilizing the Poincaré surface of a section as a numerical method. The motion equilibrium positions are detected, and their stability is discussed. We show that the force generated by the rotating frame acts as a stabilizer for the maximum equilibrium points. We display graphically that the size of the stability regions relies on the angular velocity magnitude for the frame. Through the application of Lyapunov’s theorem, periodic solutions can be constructed which are close to the equilibrium positions. Furthermore, we demonstrate that there are one or two families of periodic solutions relying on whether the equilibrium point is a saddle or stable, respectively.

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Wilson, Daniel J. "The harmonic mean p-value for combining dependent tests." Proceedings of the National Academy of Sciences 116, no. 4 (January 4, 2019): 1195–200. http://dx.doi.org/10.1073/pnas.1814092116.

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Analysis of “big data” frequently involves statistical comparison of millions of competing hypotheses to discover hidden processes underlying observed patterns of data, for example, in the search for genetic determinants of disease in genome-wide association studies (GWAS). Controlling the familywise error rate (FWER) is considered the strongest protection against false positives but makes it difficult to reach the multiple testing-corrected significance threshold. Here, I introduce the harmonic mean p-value (HMP), which controls the FWER while greatly improving statistical power by combining dependent tests using generalized central limit theorem. I show that the HMP effortlessly combines information to detect statistically significant signals among groups of individually nonsignificant hypotheses in examples of a human GWAS for neuroticism and a joint human–pathogen GWAS for hepatitis C viral load. The HMP simultaneously tests all ways to group hypotheses, allowing the smallest groups of hypotheses that retain significance to be sought. The power of the HMP to detect significant hypothesis groups is greater than the power of the Benjamini–Hochberg procedure to detect significant hypotheses, although the latter only controls the weaker false discovery rate (FDR). The HMP has broad implications for the analysis of large datasets, because it enhances the potential for scientific discovery.

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Dobson, John F., Jun Wang, and Hung M. Le. "Some Experimental Prospects involving Parabolic Quantum Wells." Australian Journal of Physics 53, no. 1 (2000): 119. http://dx.doi.org/10.1071/ph99048.

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We discuss two possible lines of experimental investigation based on parabolic quantum wells. In the first proposal, we note that the Generalised Kohn Theorem/Harmonic Potential Theorem forbids electron–electron damping of the Kohn mode in an electron layer gas under strictly parabolic confinement. This applies even for very strong driving. It is therefore interesting to attempt reduction of other sources of broadening in GaAlAs parabolic wells, so as to achieve a prominent narrow resonance in the far infrared. We concentrate here on phononic bandgap structures, which may be of interest for reduction of phonon effects in other systems as well. The second class of proposed experiment involves twinned parabolic wells in an attempt to observe van der Waals forces directly in GaAlAs systems. In a first approximation, the parabolic or Hooke's-law nature of the confinement allows one to use the well as a kind of spring balance to measure the weak van der Waals force. The influence of an applied magnetic field on these forces appears to be significant, and this system might provide the first measurement of such an effect.

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Nikolaev, Oleksii, Oleksandr Holovchenko, and Nina Savchenko. "Green's functions of the first and second boundary value problems for the Laplace equation in the nonclassical domain." Radioelectronic and Computer Systems, no. 4 (November 29, 2022): 30–49. http://dx.doi.org/10.32620/reks.2022.4.03.

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The subject of study is the Green's functions of the first and second boundary value problems for the Laplace equation. The study constructs the Green's functions of the first and second boundary value problems for the Laplace equation in space with a spherical segment in analytical form, as well as numerical analysis of these functions. Research task: to formalize the problem of determining Green's functions for the specified domain; using methods of Fourier, pair summation equations and potential theory to reduce mixed boundary value problems for auxiliary harmonic functions to a system of equations that has an analytical solution; investigate the compatibility of the algebraic system for determining constants of integration; formulate and prove a theorem about the jump of the normal derivative of the potential of a simple layer on the surface of a segment, with the help of which to present the Green's function in the form of the potential of a simple layer; conduct a numerical experiment and identify algorithms and areas of changing the parameters of effective calculations; analyze the behavior of Green's functions. Scientific novelty: for the first time, Green's functions of Dirichlet and Neumann boundary value problems for the Laplace equation in three-dimensional space with a spherical segment were constructed in analytical form, the obtained results were substantiated, and a comprehensive numerical experiment was conducted to analyze the behavior of these functions. The obtained results: mixed boundary value problems in the interior and exterior of the spherical surface to which the segment belongs are set for the auxiliary harmonic functions; using the Fourier method, the problem is reduced to systems of paired equations in series by Legendre functions, the solutions of which are found using discontinuous Mehler-Dirichlet sums. The specified functions are obtained in an explicit view in two forms: series based on the basic harmonic functions in spherical coordinates and the potential of a simple layer on the surface of the segment. To substantiate the results, the lemma on the compatibility of the algebraic system for determining the constants of integration and the theorem on the jump of the normal derivative of the potential of a simple layer on a segment are proved. A numerical experiment was conducted to analyze the behavior of the constructed functions. Conclusions: the analysis of numerical values of Green's functions obtained by different algorithms showed that the highest accuracy of results outside the surface of the segment was obtained when using images of Green's functions in the form of series. On the basis of the calculations, the lines of the level of the Green's functions of two boundary value problems in the plane of the singular point, as well as the graphs of the potential density of the simple layer for the Dirichlet problem and the potential jump for the Neumann problem on the segment at different locations of the singular point were constructed. In the partial case of the location of a singular point at the origin of the coordinates, the potential of the electrostatic field of a point charge near a conductive grounded thin shell in the form of a spherical segment is found. The main characteristics of such a field are found in closed form.

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Oyelade, Akintoye O., Theddeus T. Akano, and Gbenga A. Odesanmi. "Enhancing the sound transmission loss in double-leaf partitions with lateral local resonators substructure." Advances in Mechanical Engineering 14, no. 3 (March 2022): 168781322210878. http://dx.doi.org/10.1177/16878132221087851.

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We report theoretically on the sound transmission loss performance through a periodic double plate acoustic metamaterial made of lateral local resonance (LLR) substructure. The unit cell of the substructure consists of a four link mechanism, two lateral resonators, and a vertical spring. The combination of space harmonic expansion and Bloch-Floquet theorem are used to analyze this present study. Computed results show that high sound transmission loss (STL) up to 60 dB at 62 Hz is reached with the metamaterial plate while the mass spring mass resonant is observed for the conventional periodic double plate at the same frequency. The introduction of a negative stiffness spring causes an increased STL at low frequency. The potential of lateral resonant metamaterials to improve the sound transmission loss (STL) in the frequency region around the mass-spring-mass resonance for periodic double panel partitions is demonstrated.

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SAPOVAL, B., M. H. A. S. COSTA, J. S. ANDRADE, and M. FILOCHE. "LAPLACIAN TRANSPORT TOWARDS PARTIALLY PASSIVATED 2D IRREGULAR INTERFACES: A CONJECTURAL EXTENSION OF THE MAKAROV THEOREM." Fractals 12, no. 04 (December 2004): 381–87. http://dx.doi.org/10.1142/s0218348x04002677.

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In several phenomena of practical interest, such as catalyst deactivation, fouling in heat transfer and other systems of technological and scientific relevance, an irregular surface accessed by diffusion can be progressively passivated. In a diffusion limited situation, an interface that works unevenly due to Laplacian screening is simultaneously and unevenly passivated. To study this phenomenon, we describe a process in which the regions of the surface that are initially working, are transformed into passive, reflecting zones. As a consequence, at each step, a new part of the interface becomes active. In turn, this new active zone is passivated, and so on. It is found that the length of the successive active zones remains approximately constant for a prefractal interface. The concept of active zone in Laplacian transport can then be successfully extended to elucidate this self-limiting behavior of the passivation process. A conjecture is then proposed which states that, in D=2, the information dimension of the harmonic measure on a fractal supporting a "passivated or reflecting subfractal" (of smaller dimension) is equal to 1. This constitutes an extension of Makarov theorem. From our results, fractal geometry is revealed as a potential candidate to engineer substrate morphologies that are robust to Laplacian passivation.

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Hatifi, Mohamed, Ralph Willox, Samuel Colin, and Thomas Durt. "Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium." Entropy 20, no. 10 (October 11, 2018): 780. http://dx.doi.org/10.3390/e20100780.

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Recently, the properties of bouncing oil droplets, also known as “walkers,” have attracted much attention because they are thought to offer a gateway to a better understanding of quantum behavior. They indeed constitute a macroscopic realization of wave-particle duality, in the sense that their trajectories are guided by a self-generated surrounding wave. The aim of this paper is to try to describe walker phenomenology in terms of de Broglie–Bohm dynamics and of a stochastic version thereof. In particular, we first study how a stochastic modification of the de Broglie pilot-wave theory, à la Nelson, affects the process of relaxation to quantum equilibrium, and we prove an H-theorem for the relaxation to quantum equilibrium under Nelson-type dynamics. We then compare the onset of equilibrium in the stochastic and the de Broglie–Bohm approaches and we propose some simple experiments by which one can test the applicability of our theory to the context of bouncing oil droplets. Finally, we compare our theory to actual observations of walker behavior in a 2D harmonic potential well.

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Panasyuk, George Y., George A. Levin, and Kirk L. Yerkes. "Heat Transport between Heat Reservoirs Mediated by Quantum Systems." MRS Proceedings 1543 (2013): 43–48. http://dx.doi.org/10.1557/opl.2013.678.

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ABSTRACTWe explore a model of heat transport between two heat reservoirs mediated by a quantum particle. The reservoirs are modeled as ensembles of harmonic modes linearly coupled to the mediator. The steady state heat current, as well as the thermal conductance are obtained for arbitrary coupling strength and will be analyzed for the cases of weak and strong coupling regimes. It is shown that the violation of the virial theorem – the imbalance between the average potential and kinetic energy of the mediator – can be considered as a measure of the coupling strength that takes into account all the relevant factors. The dependence of the thermal conductance on the coupling strength is non-monotonic and displays a maximum. Temperature dependence of the heat conductance may reach a plateau at intermediate temperatures, similar to the classical plateau at high temperatures. We will discuss the origin of Fourier’s law in a chain of macroscopically large, but finite subsystems coupled by the quantum mediators. We will also address the origin of the anomalously large heat current between the scanning tunneling microscope tip and the substrate in deep vacuum which was found in recent experiments.

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Zhang, Yangjian, Li Wang, Yuanhuizi He, Ni Huang, Wang Li, Shiguang Xu, Quan Zhou, et al. "A Continuous Change Tracker Model for Remote Sensing Time Series Reconstruction." Remote Sensing 14, no. 9 (May 9, 2022): 2280. http://dx.doi.org/10.3390/rs14092280.

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It is hard for current time series reconstruction methods to achieve the balance of high-precision time series reconstruction and explanation of the model mechanism. The goal of this paper is to improve the reconstruction accuracy with a well-explained time series model. Thus, we developed a function-based model, the CCTM (Continuous Change Tracker Model) model, that can achieve high precision in time series reconstruction by tracking the time series variation rate. The goal of this paper is to provide a new solution for high-precision time series reconstruction and related applications. To test the reconstruction effects, the model was applied to four types of datasets: normalized difference vegetation index (NDVI), gross primary productivity (GPP), leaf area index (LAI), and MODIS surface reflectance (MSR). Several new observations are as follows. First, the CCTM model is well explained and based on the second-order derivative theorem, which divides the yearly time series into four variation types including uniform variations, decelerated variations, accelerated variations, and short-periodical variations, and each variation type is represented by a designed function. Second, the CCTM model provides much better reconstruction results than the Harmonic model on the NDVI, GPP, MSR, and LAI datasets for the seasonal segment reconstruction. The combined use of the Savitzky–Golay filter and the CCTM model is better than the combinations of the Savitzky–Golay filter with other models. Third, the Harmonic model has the best trend-fitting ability on the yearly time series dataset, with the highest R-Square and the lowest RMSE among the four function fitting models. However, with seasonal piecewise fitting, the four models all achieved high accuracy, and the CCTM performs the best. Fourth, the CCTM model should also be applied to time series image compression, two compression patterns with 24 coefficients and 6 coefficients respectively are proposed. The daily MSR dataset can achieve a compression ratio of 15 by using the 6-coefficients method. Finally, the CCTM model also has the potential to be applied to change detection, trend analysis, and phenology and seasonal characteristics extractions.

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Feng, Shuxiang, and Yingbo Han. "Liouville Type Theorems of f-Harmonic Maps with Potential." Results in Mathematics 66, no. 1-2 (January 21, 2014): 43–64. http://dx.doi.org/10.1007/s00025-014-0363-9.

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Bajunaid, I., J. M. Cohen, F. Colonna, and D. Singman. "A Riesz decomposition theorem on harmonic spaces without positive potentials." Hiroshima Mathematical Journal 38, no. 1 (March 2008): 37–50. http://dx.doi.org/10.32917/hmj/1207580344.

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Godwe, Emile, Justin Mibaile, Betchewe Gambo, and Serge Y. Doka. "Semiquantum Chaos in Two GaAs Quantum Dots Coupled Linearly and Quadratically by Two Harmonic Potentials in Two Dimensions." Advances in Mathematical Physics 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/6450687.

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We analyze the phenomenon of semiquantum chaos in two GaAs quantum dots coupled linearly and quadratically by two harmonic potentials. We show how semiquantum dynamics should be derived via the Ehrenfest theorem. The extended Ehrenfest theorem in two dimensions is used to study the system. The numerical simulations reveal that, by varying the interdot distance and coupling parameters, the system can exhibit either periodic or quasi-periodic behavior and chaotic behavior.

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Gutlyanskii, Vladimir, Olga Nesmelova, Vladimir Ryazanov, and Artyem Yefimushkin. "Logarithmic potential and generalized analytic functions." Ukrainian Mathematical Bulletin 18, no. 1 (March 9, 2021): 12–36. http://dx.doi.org/10.37069/1810-3200-2021-18-1-2.

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The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

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Lin, Hezi, Guilin Yang, Yibin Ren, and Tian Chong. "Monotonicity formulae and Liouville theorems of harmonic maps with potential." Journal of Geometry and Physics 62, no. 9 (September 2012): 1939–48. http://dx.doi.org/10.1016/j.geomphys.2012.04.008.

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Mitrea, Dorina. "A generalization of Dahlberg's theorem concerning the regularity of harmonic Green potentials." Transactions of the American Mathematical Society 360, no. 07 (February 27, 2008): 3771–94. http://dx.doi.org/10.1090/s0002-9947-08-04384-5.

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Chatjigeorgiou, Ioannis K., Eva Loukogeorgaki, Eirini Anastasiou, and Nikos Mantadakis. "Ultimate Image Singularities in Oblate Spheroidal Coordinates with Applications in Hydrodynamics." Journal of Marine Science and Engineering 8, no. 1 (January 10, 2020): 32. http://dx.doi.org/10.3390/jmse8010032.

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This study exploits the Touvia Miloh oblate spheroid theorem with a special focus on hydrodynamical applications. The theorem provides explicit relations that express the oblate spheroidal harmonics, given in terms of the fundamental solutions of the Laplace equation. Here, the theorem is employed to transform the underlying Green’s function into the relevant coordinate system and, consequently, to formulate the diffraction potential. The case considered refers to the axisymmetric placement of the spheroid, namely, symmetrical axis perpendicular to the free surface. The mathematical formulations have been implemented numerically providing exceptionally accurate computations, which manifests the consistency and robustness of the relevant formulas.

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Salas-Brito, A. L., H. N. Núñez-Yépez, and R. P. Martínez-Y-Romero. "Superintegrability in Classical Mechanics: A Contemporary Approach to Bertrand's Theorem." International Journal of Modern Physics A 12, no. 01 (January 10, 1997): 271–76. http://dx.doi.org/10.1142/s0217751x97000402.

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Superintegrable Hamiltonians in three degrees of freedom posses more than three functionally independent globally defined and single-valued constants of motion. In this contribution and under the assumption of the existence of only periodic and plane bounded orbits in a classical system we are able to establish the superintegrability of the Hamiltonian. Then, using basic algebraic ideas, we obtain a contemporary proof of Bertrand's theorem. That is, we are able to show that the harmonic oscillator and the Newtonian gravitational potentials are the only 3D potentials whose bounded orbits are all plane and periodic.

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Zhou, Zhen-Rong. "Stability and quantum phenomenen and Liouville theorems of $p$-harmonic maps with potential." Kodai Mathematical Journal 26, no. 1 (March 2003): 101–18. http://dx.doi.org/10.2996/kmj/1050496652.

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Shushkevich, G. Ch. "Analytical solution of problem of shielding low-frequency magnetic field by thin-walled cylindrical screen in presence of cylinder." Informatics 18, no. 3 (September 30, 2021): 48–58. http://dx.doi.org/10.37661/1816-0301-2021-18-3-48-58.

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The analytical solution of boundary value problem describing the process of penetration of low-frequency magnetic field through thin-walled cylindrical screen with cylindrical inclusion is constructed by use of approximate boundary conditions. The source of the field is a thin thread of infinitely small length with an infinitely small cross-section where current circulates. Thread is located in a plane which is perpendicular to axis of cylindrical screen, in outer region with respect to a screen. Initially the potential of initial magnetic field is represented as spherical harmonic functions, then using addition theorems connecting spherical and cylindrical harmonic functions, it became as cylindrical harmonic functions superposition. Secondary potentials of magnetic field are also presented as superposition of cylindrical harmonic functions in three-dimensional space. It is shown that the solution of formulated boundary value problem is reduced to the solution of linear algebraic equations system for coefficients included in the representation of secondary fields. The influence of some aspects of the problem on the value of the screening coefficient of an external magnetic field when passing through a cylindrical copper screen in the presence of a cylindrical inclusion is studied numerically. Calculation results are presented in graphs form. Obtained results can be used to shield technical devices and biological objects against the effects of magnetic fields to provide ecological surrounding of operating electrical installations and devices.

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Shi, Guannan, Yuming Xing, and Baiqing Sun. "Poincaré-Type Inequalities for the Composite Operator inL𝒜-Averaging Domains." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/675464.

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We first establish the local Poincaré inequality withL𝒜-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenousA-harmonic equation. Then, according to the definition ofL𝒜-averaging domains and relative properties, we demonstrate the global Poincaré inequality withL𝒜-averaging domains. Finally, we give some illustrations for these theorems.

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Bencheikh, K., and L. M. Nieto. "On the local virial theorems for linear and isotropic harmonic oscillator potentials inddimensions." Journal of Physics A: Mathematical and Theoretical 43, no. 37 (July 29, 2010): 375002. http://dx.doi.org/10.1088/1751-8113/43/37/375002.

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Weislinger, Edmond, and Gabriel Olivier. "The virial theorem with boundary conditions applications to the harmonic oscillator and to sine-shaped potentials." International Journal of Quantum Chemistry 9, S9 (June 18, 2009): 425–33. http://dx.doi.org/10.1002/qua.560090852.

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Giraud, Dominique, Baptiste Ristagno, Denis Netter, Julien Fontchastagner, Nicolas Labbe, and Vincent Lanfranchi. "Axial claw pole motor: harmonic torque estimation using finite element method." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 39, no. 5 (July 10, 2020): 1157–67. http://dx.doi.org/10.1108/compel-01-2020-0024.

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Purpose This paper aims to propose a method to evaluate the information obtained on harmonics calculations and to estimate the precision of results using finite element method for an innovative motor topology in which some well-known meshing rules are difficult to apply. Design/methodology/approach The same magnetostatic problem is solved with several mesh sizes using both scalar and vector potentials magnetics formulations on a complex topology, an axial claw pole motor (ACPM). The proposed method lies in a comparison between the two weak formulations to determine what information is obtained on harmonics calculations and to estimate its precision. Moreover, an original mesh method is applied in the air gap to improve the numerical results. Findings The precision on harmonics calculations using finite element method on an ACPM is estimated. For the proposed motor and mesh, only the mean value (even with large mesh) and the first harmonic (with fine mesh) of torque are calculated with a good accuracy. This results confirm that the non-respect of the meshing rules have a strong impact on the results and that scalar and vector potentials magnetics formulations do not give exactly the same results. Before using torque harmonics values in vibration calculations, a finite element model has to be validated by using both fomulations. Research limitations/implications This method is time-consuming and only applied on an ACPM in this work. Originality/value The axial claw pole motor, for which the classic meshing rules cannot be applied, is a complex topology very under-studied. To improve the calculation of space harmonics, the authors proposed to split the airgap into four parts. Then in the two central parts, the meshing step of the structured mesh is equal to the rotating step.

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Maeda, Fumi-Yuki, and Takayori Ono. "Properties of harmonic boundary in nonlinear potential theory." Hiroshima Mathematical Journal 30, no. 3 (2000): 513–23. http://dx.doi.org/10.32917/hmj/1206124611.

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Marathe, Rahul, and Sourabh Lahiri. "Convergence of thermodynamic quantities and work fluctuation theorems in the presence of random protocols." International Journal of Modern Physics B 33, no. 20 (August 10, 2019): 1950220. http://dx.doi.org/10.1142/s0217979219502205.

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Recently, many results, namely the Fluctuation theorems (FT), have been discovered for systems arbitrarily away from equilibrium. Many of these relations have been experimentally tested. The system under consideration is usually driven out of equilibrium by an external time-dependent parameter which follows a particular protocol. One needs to perform several iterations of the same experiment in order to find statistically relevant results. Since the systems are microscopic, fluctuations dominate. Studying the convergence of relevant thermodynamics quantities with number of realizations is also important as it gives a rough estimate of the number of iterations one needs to perform. In each iteration, the protocol follows a predetermined identical/fixed form. However, the protocol itself may be prone to fluctuations. In this work, we are interested in looking at a simple nonequilibrium system, namely a Brownian particle trapped in a harmonic potential. The center of the trap is then dragged according to a protocol. We however lift the condition of fixed protocol. In our case, the protocol in each realization is different. We consider one of the parameters of the protocol as a random variable, chosen from some known distribution. We study the systems analytically as well as numerically. We specifically study the convergence of the average work and free energy difference with number of realizations. Interestingly, in several cases, randomness in the protocol does not seem to affect the convergence when compared to fixed protocol results. We study symmetry functions. The cases of a Brownian particle in a harmonic potential with sinusoidally changing stiffness constant, as well as a Brownian particle in a double well potential, are also studied. We believe that our results can be experimentally verified.

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Krishnakumaran, R., Mohammed Raees, and Supratim Ray. "Shape analysis of gamma rhythm supports a superlinear inhibitory regime in an inhibition-stabilized network." PLOS Computational Biology 18, no. 2 (February 14, 2022): e1009886. http://dx.doi.org/10.1371/journal.pcbi.1009886.

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Visual inspection of stimulus-induced gamma oscillations (30–70 Hz) often reveals a non-sinusoidal shape. Such distortions are a hallmark of non-linear systems and are also observed in mean-field models of gamma oscillations. A thorough characterization of the shape of the gamma cycle can therefore provide additional constraints on the operating regime of such models. However, the gamma waveform has not been quantitatively characterized, partially because the first harmonic of gamma, which arises because of the non-sinusoidal nature of the signal, is typically weak and gets masked due to a broadband increase in power related to spiking. To address this, we recorded local field potential (LFP) from the primary visual cortex (V1) of two awake female macaques while presenting full-field gratings or iso-luminant chromatic hues that produced huge gamma oscillations with prominent peaks at harmonic frequencies in the power spectra. We found that gamma and its first harmonic always maintained a specific phase relationship, resulting in a distinctive shape with a sharp trough and a shallow peak. Interestingly, a Wilson-Cowan (WC) model operating in an inhibition stabilized mode could replicate this shape, but only when the inhibitory population operated in the super-linear regime, as predicted recently. However, another recently developed model of gamma that operates in a linear regime driven by stochastic noise failed to produce salient harmonics or the observed shape. Our results impose additional constraints on models that generate gamma oscillations and their operating regimes.

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Hoffmann, Volker, Bendik Nybakk Torsæter, Gjert Hovland Rosenlund, and Christian Andre Andresen. "Lessons for Data-Driven Modelling from Harmonics in the Norwegian Grid." Algorithms 15, no. 6 (May 31, 2022): 188. http://dx.doi.org/10.3390/a15060188.

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With the advancing integration of fluctuating renewables, a more dynamic demand-side, and a grid running closer to its operational limits, future power system operators require new tools to anticipate unwanted events. Advances in machine learning and availability of data suggest great potential in using data-driven approaches, but these will only ever be as good as the data they are based on. To lay the ground-work for future data-driven modelling, we establish a baseline state by analysing the statistical distribution of voltage measurements from three sites in the Norwegian power grid (22, 66, and 300 kV). Measurements span four years, are line and phase voltages, are cycle-by-cycle, and include all (even and odd) harmonics up to the 96 order. They are based on four years of historical data from three Elspec Power Quality Analyzers (corresponding to one trillion samples), which we have extracted, processed, and analyzed. We find that: (i) the distribution of harmonics depends on phase and voltage level; (ii) there is little power beyond the 13 harmonic; (iii) there is temporal clumping of extreme values; and (iv) there is seasonality on different time-scales. For machine learning based modelling these findings suggest that: (i) models should be trained in two steps (first with data from all sites, then adapted to site-level); (ii) including harmonics beyond the 13 is unlikely to increase model performance, and that modelling should include features that (iii) encode the state of the grid, as well as (iv) seasonality.

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AKCAY, H., and C. TEZCAN. "EXACT SOLUTIONS OF THE DIRAC EQUATION WITH HARMONIC OSCILLATOR POTENTIAL INCLUDING A COULOMB-LIKE TENSOR POTENTIAL." International Journal of Modern Physics C 20, no. 06 (June 2009): 931–40. http://dx.doi.org/10.1142/s0129183109014084.

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In this work, we study the Dirac equation with scalar, vector, and tensor interactions. The Dirac Hamiltonian contains quadratic scalar and vector potentials, as well as a tensor potential. The tensor potential is taken as a sum of a linear term and a Coulomb-like term. It is shown that the tensor potential preserves the form of the harmonic oscillator potential and generates spin-orbit terms. The energy eigenvalues and the corresponding eigenfunctions are obtained for different alternatives.

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COTĂESCU, ION I., PAUL GRĂVILĂ, and MARIUS PAULESCU. "THREE-DIMENSIONAL ISOTROPIC PSEUDO-GAUSSIAN OSCILLATORS." International Journal of Modern Physics C 20, no. 07 (July 2009): 1103–11. http://dx.doi.org/10.1142/s0129183109014254.

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A family of isotropic three-dimensional quantum models governed by isotropic pseudo-Gaussian potentials is proposed. These potentials are defined to have a Gaussian asymptotic behavior but approaching to the potential of the isotropic harmonic oscillator when x → 0. These models may have finite energy spectra with approximately equidistant energy levels that can be calculated using efficient numerical methods based on generating functionals.

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LEE, H. C., K. L. LIU, and C. F. LO. "STUDY OF A RELATIVISTIC QUANTUM HARMONIC OSCILLATOR BY THE METHOD OF STATE-DEPENDENT DIAGONALIZATION." International Journal of Modern Physics C 07, no. 05 (October 1996): 645–53. http://dx.doi.org/10.1142/s0129183196000545.

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We apply the method of State-dependent Diagonalization to study the eigenstates of the relativistic quantum harmonic oscillator in the low relativistic limit. The relativistic corrections of the energy eigenvalues of the quantum harmonic oscillator are evaluated for different values of the relativistic parameter α ≡ ħω0 / m0c2. Unlike the conventional exact diagonalization, this new method is shown to be very efficient for evaluating the energy eigenvalues and eigenfunctions. We have also found that for non-zero α the eigenfunctions of the system become more localized in space and that the ground state of the SHO (i.e., the α = 0 case) turns into a squeezed state. Furthermore, since our system is a special case of the quantum harmonic oscillator with a velocity-dependent anharmonic potential, this new approach should be very useful for investigating the cases with more complicated velocity-dependent anharmonic potentials.

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Aqil, Marco, Selen Atasoy, Morten L. Kringelbach, and Rikkert Hindriks. "Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome." PLOS Computational Biology 17, no. 1 (January 28, 2021): e1008310. http://dx.doi.org/10.1371/journal.pcbi.1008310.

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Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed “connectome harmonics”, have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI). As an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome graph constructed from diffusion tensor imaging (DTI) and structural MRI data. We show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of resting-state fMRI data, and predict its functional connectivity, with a high level of detail. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.

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Journal articles on the topic 'Harmonic potential theorem'

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