Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging algorithmic task, particularly for highly regular structures. Here we introduce a new algorithmic approach based on ideas from spectral graph theory and geometry that constructs candidate correspondences between vertices using their curvatures. Any correspondence produced by the algorithm is explicitly verified, ensuring that non-isomorphic graphs are never incorrectly identified as isomorphic. Although the method does not yet guarantee success on all inputs, we find that it correctly resolves every instance tested in deterministic polynomial time, including a broad collection of graphs known to be difficult for classical techniques. These results demonstrate that enriched spectral methods can be far more powerful than previously understood, and suggest a promising direction for the practical resolution of the complexity of the graph isomorphism problem.
This paper reviews the standard algorithm for converting spacecraft state vectors to Keplerian orbital elements with a focus on its computer implementation. It analyzes the shortcomings of the scheme as described in the literature, and proposes changes to the implementation to address orbits of arbitrary eccentricity and inclination in a simple and robust way. It presents two coding strategies that simplify the program structure while improving the accuracy and speed of the transformation on modern computer architectures. Comprehensive numerical benchmarks demonstrate accuracy improvements by two orders of magnitude, together with a 40% reduction of computational cost relative to the standard implementation.
We study the influence of analytical regularization used in the generalized function (distribution) space to the Tikhonov regularization procedure utilized in the different versions of Moore-Penrose's inversion. By introducing a new analytical term to the Tikhonov regularization of Moore-Penrose's inversion procedure, we derive new optimization conditions that extend the Tikhonov regularization framework and influence the fitting parameter. This enhancement yields a more robust and accurate reconstruction of physical quantities, demonstrating its potential impact on various studies. We illustrate the significance of new term through schematic examples of physical applications, highlighting its relevance to diverse fields. Our findings provide a valuable tool for improving inversion methods and their applications in physics and beyond.
quTARANG is a Python-based general-purpose Gross-Pitaevskii Equation (GPE) solver. It can solve GPE in 1D, 2D and 3D and has the ability to run on both CPU and GPU. It has been developed to study turbulence in quantum systems, specifically in atomic Bose-Einstein condensates, and can be used to study different quantities, such as the varied spectra associated with quantum turbulence.
We demonstrate that embedding physics-driven constraints into machine learning process can dramatically improve accuracy and generalizability of the resulting model. Physics-informed learning is illustrated on the example of analysis of optical modes propagating through a spatially periodic composite. The approach presented can be readily utilized in other situations mapped onto an eigenvalue problem, a known bottleneck of computational electrodynamics. Physics-informed learning can be used to improve machine-learning-driven design, optimization, and characterization, in particular in situations where exact solutions are scarce or are slow to come up with.
Jean-Jacques Marigo, Kim Pham, Agnès Maurel
et al.
We present a three-dimensional model describing the propagation of elastic waves in a soil substrate supporting an array of cylindrical beams experiencing flexural and compressional resonances. The resulting surface waves are of two types. In the sagittal plane, hybridized Rayleigh waves can propagate except within bandgaps resulting from a complex interplay between flexural and compressional resonances. We exhibit a wave decoupled from the hybridized Rayleigh wave which is the elastic analogue of electromagnetic spoof plasmon polaritons. This wave with displacements perpendicular to the sagittal plane is sensitive only to flexural resonances. Similar, yet quantitatively different, physics is demonstrated in a two-dimensional setting involving resonances of plates.
We propose a novel multi-dimensional integration algorithm using a machine learning (ML) technique. After training a ML regression model to mimic a target integrand, the regression model is used to evaluate an approximation of the integral. Then, the difference between the approximation and the true answer is calculated to correct the bias in the approximation of the integral induced by a ML prediction error. Because of the bias correction, the final estimate of the integral is unbiased and has a statistically correct error estimation. The performance of the proposed algorithm is demonstrated on six different types of integrands at various dimensions and integrand difficulties. The results show that, for the same total number of integrand evaluations, the new algorithm provides integral estimates with more than an order of magnitude smaller uncertainties than those of the VEGAS algorithm in most of the test cases.
We develop new transfer learning algorithms to accelerate prediction of material properties from ab initio simulations based on density functional theory (DFT). Transfer learning has been successfully utilized for data-efficient modeling in applications other than materials science, and it allows transferable representations learned from large datasets to be repurposed for learning new tasks even with small datasets. In the context of materials science, this opens the possibility to develop generalizable neural network models that can be repurposed on other materials, without the need of generating a large (computationally expensive) training set of materials properties. The proposed transfer learning algorithms are demonstrated on predicting the Gibbs free energy of light transition metal oxides.
Vladimir I Kolobov, Robert R Arslanbekov, Dmitry Levko
et al.
We conducted experimental studies and computer simulations of standing striations in capacitive coupled plasma in Argon gas. Standing striations were observed at frequencies 3.6, 8.4 and 19.0 MHz, in a pressure range 0.05-10 Torr, tube radius R=1.1 cm, for a certain range of discharge currents (plasma densities). Numerical simulations revealed similar nature of standing striations in CCP and moving striations in DC discharges under similar discharge conditions. Comparison of computer simulations with experimental observations helped clarify the nature of these striations. The non-linear dependence of the ionization rate on electron density is shown to be the main underlying mechanism of the stratification phenomena.
Numerical inverse Laplace transforms are used to analyze the transient response of non-ideal transmission lines to a voltage step. We find that the detailed shape of the transient response is sensitive conductor losses, but insensitive to dielectric losses. Furthermore, we find that the low-loss approximations for the transmission line propagation constant and characteristic impedance in the complex-frequency domain are sufficient to accurately model the observed transient response. We also investigate the effects of: (1) a parasitic capacitance terminating the open end of the coaxial transmission line, (2) the input impedance of the oscilloscope used to make the measurements, and (3) the finite rise time of the voltage step. Finally, we cool a semi-rigid coaxial cable in liquid nitrogen so as to reduced conductor lossless and observe a transient response that is closer to that expected from an ideal lossless line.
Semi-analytic expressions for the static limit of the T-matrix for electromagnetic scattering are derived for a circular torus, expressed in bases of both toroidal and spherical harmonics. The scattering problem for an arbitrary static excitation is solved using toroidal harmonics and the extended boundary condition method to obtain analytic expressions for auxiliary Q and P-matrices, from which the T-matrix is given by their division. By applying the basis transformations between toroidal and spherical harmonics, the quasi-static limit of the T-matrix block for electric multipole coupling is obtained. For the toroidal geometry there are two similar T-matrices on a spherical basis, for computing the scattered field both near the origin and in the far field. Static limits of the optical cross-sections are computed, and analytic expressions for the limit of a thin ring are derived.
This paper seeks to show that the beam screen of the LHC has an important effect on the electric field of the LHC beam, a few tens of sigmas away from its center. To do so, we develop two new methods for finding the effect of a complex conducting boundary for boundary value problems in electrostatics. Both methods are based on a generalization of the method of images and require low computing power. The result is an exact solution to the problem of a discretized conducting boundary, which we take to be an approximation of the real solution. As an application, we compute the total electric field inside the LHC beam screen and show that neglecting the effect of the conducting boundary is only accurate to 1% for locations closer than 10$σ$ from the center of the beam, and only accurate to 10% for locations closer than 30$σ$.
A formulation of the boundary integral method for solving partial differential equations has been developed whereby the usual weakly singular integral and the Cauchy principal value integral can be removed analytically. The broad applicability of the approach is illustrated with a number of problems of practical interest to fluid and continuum mechanics including the solution of the Laplace equation for potential flow, the Helmholtz equation as well as the equations for Stokes flow and linear elasticity.