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Статті в журналах з теми "Cluster approximation"

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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

Borisenko, O., V. Chelnokov, and V. Kushnir. "Phenomenological Renormalization Group and Cluster Approximation." Ukrainian Journal of Physics 59, no. 7 (July 2014): 655–62. http://dx.doi.org/10.15407/ujpe59.07.0655.

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2

Nenashev, Vadim A., Igor G. Khanykov, and Mikhail V. Kharinov. "A Model of Pixel and Superpixel Clustering for Object Detection." Journal of Imaging 8, no. 10 (October 6, 2022): 274. http://dx.doi.org/10.3390/jimaging8100274.

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Анотація:

The paper presents a model of structured objects in a grayscale or color image, described by means of optimal piecewise constant image approximations, which are characterized by the minimum possible approximation errors for a given number of pixel clusters, where the approximation error means the total squared error. An ambiguous image is described as a non-hierarchical structure but is represented as an ordered superposition of object hierarchies, each containing at least one optimal approximation in g0 = 1,2,..., etc., colors. For the selected hierarchy of pixel clusters, the objects-of-interest are detected as the pixel clusters of optimal approximations, or as their parts, or unions. The paper develops the known idea in cluster analysis of the joint application of Ward’s and K-means methods. At the same time, it is proposed to modernize each of these methods and supplement them with a third method of splitting/merging pixel clusters. This is useful for cluster analysis of big data described by a convex dependence of the optimal approximation error on the cluster number and also for adjustable object detection in digital image processing, using the optimal hierarchical pixel clustering, which is treated as an alternative to the modern informally defined “semantic” segmentation.

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3

Pizio, O. A., and Z. B. Halytch. "Structural Properties of the Ion-Dipole Model of Electrolyte Solutions in the Bulk and Near a Charged Hard Wall.Application of the Truncated Optimized Cluster Series." Zeitschrift für Naturforschung A 46, no. 1-2 (February 1, 1991): 8–18. http://dx.doi.org/10.1515/zna-1991-1-203.

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AbstractAn ion-dipole model of electrolyte solutions in the bulk case and near a charged or uncharged hard wall is considered. A method to derive the terms of optimized cluster expansions for the distribution functions of ions and dipoles which provides a set of approximations beyond the mean spherical approximation is given. The third cluster coefficient approximation is investigated

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4

ZIEGLER, ALFRED. "FERMION CLUSTER APPROACH TO THE HUBBARD MODEL." International Journal of Modern Physics B 07, no. 01n03 (January 1993): 601–4. http://dx.doi.org/10.1142/s0217979293001268.

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The deficiencies of the fermion cluster approximation by Caron and Pratt out-lined in a previous paper are corrected and the approximation is then applied to the one-site Hubbard model. Because the approximation in its standard form is unable to describe energy bands the boundary condition integration technique of C. Gros for free clusters is generalized to embedded clusters by averaging over the solutions to different self-consistency equations.

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5

Kats, Daniel, and Frederick R. Manby. "Communication: The distinguishable cluster approximation." Journal of Chemical Physics 139, no. 2 (July 14, 2013): 021102. http://dx.doi.org/10.1063/1.4813481.

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6

KÜMMEL, HERMANN G. "A BIOGRAPHY OF THE COUPLED CLUSTER METHOD." International Journal of Modern Physics B 17, no. 28 (November 10, 2003): 5311–25. http://dx.doi.org/10.1142/s0217979203020442.

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Анотація:

The origins of the coupled cluster method are described. Special attention is paid to the arguments put forward for the exponential structure of the wave functions. Various approximation schemes invented during the last 40 years are presented. The problems arising from these approximations necessarily truncating or destroying the exponential form are discussed and ways to deal with them are described.

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7

DESSMARK, ANDERS, JESPER JANSSON, ANDRZEJ LINGAS, EVA-MARTA LUNDELL, and MIA PERSSON. "ON THE APPROXIMABILITY OF MAXIMUM AND MINIMUM EDGE CLIQUE PARTITION PROBLEMS." International Journal of Foundations of Computer Science 18, no. 02 (April 2007): 217–26. http://dx.doi.org/10.1142/s0129054107004656.

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Анотація:

We consider the following clustering problems: given an undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

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8

Terletska, Hanna, Yi Zhang, Ka-Ming Tam, Tom Berlijn, Liviu Chioncel, N. Vidhyadhiraja, and Mark Jarrell. "Systematic Quantum Cluster Typical Medium Method for the Study of Localization in Strongly Disordered Electronic Systems." Applied Sciences 8, no. 12 (November 26, 2018): 2401. http://dx.doi.org/10.3390/app8122401.

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Анотація:

Great progress has been made in recent years towards understanding the properties of disordered electronic systems. In part, this is made possible by recent advances in quantum effective medium methods which enable the study of disorder and electron-electronic interactions on equal footing. They include dynamical mean-field theory and the Coherent Potential Approximation, and their cluster extension, the dynamical cluster approximation. Despite their successes, these methods do not enable the first-principles study of the strongly disordered regime, including the effects of electronic localization. The main focus of this review is the recently developed typical medium dynamical cluster approximation for disordered electronic systems. This method has been constructed to capture disorder-induced localization and is based on a mapping of a lattice onto a quantum cluster embedded in an effective typical medium, which is determined self-consistently. Unlike the average effective medium-based methods mentioned above, typical medium-based methods properly capture the states localized by disorder. The typical medium dynamical cluster approximation not only provides the proper order parameter for Anderson localized states, but it can also incorporate the full complexity of Density-Functional Theory (DFT)-derived potentials into the analysis, including the effect of multiple bands, non-local disorder, and electron-electron interactions. After a brief historical review of other numerical methods for disordered systems, we discuss coarse-graining as a unifying principle for the development of translationally invariant quantum cluster methods. Together, the Coherent Potential Approximation, the Dynamical Mean-Field Theory and the Dynamical Cluster Approximation may be viewed as a single class of approximations with a much-needed small parameter of the inverse cluster size which may be used to control the approximation. We then present an overview of various recent applications of the typical medium dynamical cluster approximation to a variety of models and systems, including single and multiband Anderson model, and models with local and off-diagonal disorder. We then present the application of the method to realistic systems in the framework of the DFT and demonstrate that the resulting method can provide a systematic first-principles method validated by experiment and capable of making experimentally relevant predictions. We also discuss the application of the typical medium dynamical cluster approximation to systems with disorder and electron-electron interactions. Most significantly, we show that in the limits of strong disorder and weak interactions treated perturbatively, that the phenomena of 3D localization, including a mobility edge, remains intact. However, the metal-insulator transition is pushed to larger disorder values by the local interactions. We also study the limits of strong disorder and strong interactions capable of producing moment formation and screening, with a non-perturbative local approximation. Here, we find that the Anderson localization quantum phase transition is accompanied by a quantum-critical fan in the energy-disorder phase diagram.

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9

Freericks, James K. "Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster." Symmetry 14, no. 3 (February 28, 2022): 494. http://dx.doi.org/10.3390/sym14030494.

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Анотація:

The chemistry community has long sought the exact relationship between the conventional and the unitary coupled cluster ansatz for a single-reference system, especially given the interest in performing quantum chemistry on quantum computers. In this work, we show how one can use the operator manipulations given by the exponential disentangling identity and the Hadamard lemma to relate the factorized form of the unitary coupled-cluster approximation to a factorized form of the conventional coupled cluster approximation (the factorized form is required, because some amplitudes are operator-valued and do not commute with other terms). By employing the Trotter product formula, one can then relate the factorized form to the standard form of the unitary coupled cluster ansatz. The operator dependence of the factorized form of the coupled cluster approximation can also be removed at the expense of requiring even more higher-rank operators, finally yielding the conventional coupled cluster. The algebraic manipulations of this approach are daunting to carry out by hand, but can be automated on a computer for small enough systems.

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10

Borgani, S., P. Coles, and L. Moscardini. "Cluster correlations in the Zel'dovich approximation." Monthly Notices of the Royal Astronomical Society 271, no. 1 (November 1, 1994): 223–32. http://dx.doi.org/10.1093/mnras/271.1.223.

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11

Baviera, R., M. Pasquini, and M. Serva. "Cluster approximation for Ising spin glasses." Le Journal de Physique IV 08, PR6 (October 1998): Pr6–75—Pr6–79. http://dx.doi.org/10.1051/jp4:1998609.

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12

Ben-Naim, E., and P. L. Krapivsky. "Cluster approximation for the contact process." Journal of Physics A: Mathematical and General 27, no. 13 (July 7, 1994): L481—L487. http://dx.doi.org/10.1088/0305-4470/27/13/004.

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13

Akıncı, Ümit. "New cluster approximation for Ising systems." Journal of Magnetism and Magnetic Materials 499 (April 2020): 166324. http://dx.doi.org/10.1016/j.jmmm.2019.166324.

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14

Anda, E. V., G. Chiappe, C. A. Büsser, M. A. Davidovich, G. B. Martins, F. Heidrich-Meisner, and E. Dagotto. "The logarithmic discretization embedded cluster approximation." Physica B: Condensed Matter 404, no. 18 (October 2009): 2689–93. http://dx.doi.org/10.1016/j.physb.2009.06.068.

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15

Cornu, F., Th Jolicoeur, and J. C. Le Guillou. "Coupled-cluster approximation in momentum space." Physical Review B 49, no. 14 (April 1, 1994): 9548–54. http://dx.doi.org/10.1103/physrevb.49.9548.

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16

Gallistl, Dietmar. "Adaptive Nonconforming Finite Element Approximation of Eigenvalue Clusters." Computational Methods in Applied Mathematics 14, no. 4 (October 1, 2014): 509–35. http://dx.doi.org/10.1515/cmam-2014-0020.

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AbstractThis paper analyses an adaptive nonconforming finite element method for eigenvalue clusters of self-adjoint operators and proves optimal convergence rates (with respect to the concept of nonlinear approximation classes) for the approximation of the invariant subspace spanned by the eigenfunctions of the eigenvalue cluster. Applications include eigenvalues of the Laplacian and of the Stokes system.

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17

Ramamani, N., D. C. Heggie, and S. J. Aarseth. "Galactic Disk Shocks on Globular Clusters." Symposium - International Astronomical Union 174 (1996): 381–82. http://dx.doi.org/10.1017/s0074180900001923.

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18

Brix, Anders, and Wilfrid S. Kendall. "Simulation of cluster point processes without edge effects." Advances in Applied Probability 34, no. 2 (June 2002): 267–80. http://dx.doi.org/10.1239/aap/1025131217.

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Анотація:

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

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19

Brix, Anders, and Wilfrid S. Kendall. "Simulation of cluster point processes without edge effects." Advances in Applied Probability 34, no. 02 (June 2002): 267–80. http://dx.doi.org/10.1017/s0001867800011551.

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Анотація:

The usual direct method of simulation for cluster processes requires the generation of the parent point process over a region larger than the actual observation window, since we have to allow for all possible parents giving rise to observed daughter points, and some of these parents may fall outwith the observation window. When there is no a priori bound on the distance between parent and child then we have to take care to control approximations arising from edge effects. In this paper, we present a simulation method which requires simulation only of those parent points actually giving rise to observed daughter points, thus avoiding edge effect approximation. The idea is to replace the cluster distribution by one which is conditioned to plant at least one daughter point in the observation window, and to modify the parent process to have an inhomogeneous intensity exactly balancing the effect of the conditioning. We furthermore show how the method extends to cases involving infinitely many potential parents, for example gamma-Poisson processes and shot-noise G-Cox processes, allowing us to avoid approximation due to truncation of the parent process.

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20

Masuda-Jindo, Kinichi, Vu Van Hung, and P. E. A. Turchi. "First Principles Calculations of Thermodynamic Quantities and Phase Diagrams of High Temperature BCC Ta-W and Mo-Ta Alloys." Advanced Materials Research 26-28 (October 2007): 205–8. http://dx.doi.org/10.4028/www.scientific.net/amr.26-28.205.

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The thermodynamic properties of high temperature metals and alloys are studied using the statistical moment method, going beyond the quasi-harmonic approximations. Including the power moments of the atomic displacements up to the fourth order, the Helmholtz free energies and the related thermodynamic quantities are derived explicitly in closed analytic forms. The configurational entropy term is taken into account by using the tetrahedron cluster approximation of the cluster variation method (CVM). The energetics of the binary (Ta-W and Mo-Ta) alloys are treated within the framework of the first-principles TB-LMTO (tight-binding linear muffin tin orbital) method coupled to CPA (coherent potential approximation) and GPM (generalized perturbation method). The equilibrium phase diagrams are calculated for the refractory Ta-W and Mo-Ta bcc alloys.

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21

Mohri, Tetsuo. "Glass Transition within the Cluster Variation Approximation." Advanced Materials Research 26-28 (October 2007): 723–26. http://dx.doi.org/10.4028/www.scientific.net/amr.26-28.723.

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Анотація:

The detailed behavior of the free energy of Cluster Variation Method in the vicinity of spinodal ordering transition is examined. The generalized phase diagram proposed in the previous study is modified and spinodal ordering transition is reinterpreted as a limiting case of the ideal glass transition.

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22

Šurda, Anton. "Transfer-matrix cluster approximation for lattice models." Physical Review B 43, no. 1 (January 1, 1991): 908–14. http://dx.doi.org/10.1103/physrevb.43.908.

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23

Hartmann, Alexander K. "Cluster-exact approximation of spin glass groundstates." Physica A: Statistical Mechanics and its Applications 224, no. 3-4 (February 1996): 480–88. http://dx.doi.org/10.1016/0378-4371(95)00404-1.

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24

Wu, Ying-Kai, and Qi-Qing Weng. "A cluster approximation to insulating spin glasses." Physica A: Statistical Mechanics and its Applications 188, no. 4 (October 1992): 481–86. http://dx.doi.org/10.1016/0378-4371(92)90328-n.

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25

Morawetz, K. "Stopping Power in Strongly Coupled Plasmas." Laser and Particle Beams 15, no. 4 (December 1997): 507–21. http://dx.doi.org/10.1017/s0263034600011095.

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Анотація:

The stopping power of dense nonideal plasmas is calculated in different approximations. The T-matrix approximation for binary collisions is compared with the random phase approximation (RPA) approximation for dielectric fluctuations. Within a microscopic model, the dynamical evolution of the velocity of the projectile is calculated. It reproduces well experimental values for the stopping of fast heavy ions. Further improvements due to correlations are discussed. Both concepts, cluster decomposition and memory, are compared and it is found that they lead to the same quantum virial corrections of the Beth-Uhlenbeck type in equilibrium. However, memory in the kinetic equation causes an additional renormalization of the effective energy transfer in nonequilibrium.

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26

LO, L. P., and C. F. LO. "OPTIMAL COUPLED-CLUSTER APPROXIMATION FOR THE E⊗(b1+b2) JAHN–TELLER EFFECT." International Journal of Modern Physics B 19, no. 30 (December 10, 2005): 4495–515. http://dx.doi.org/10.1142/s0217979205032863.

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Анотація:

In this work we have applied the optimal coupled-cluster approximation to study the ground state of the E⊗(b1+b2) Jahn–Teller effect. The effectiveness of the optimal coupled-cluster approximation has been investigated for the whole range of the asymmetry parameter and various coupling strengths. It is shown that our results up to the third level of approximation are in very good agreement with the exact numerical diagonalization results and are better than those from earlier variational treatments. Furthermore, unlike previous variational treatments, the optimal coupled-cluster approximation has the advantage that the accuracy of both the ground state energy and wavefunction estimates is being taken care of and can be systematically improved. Since the mathematical treatment in this work is simple, the optimal coupled-cluster approximation could be easily extended to the studies of other fermion-boson interacting systems, e.g. the extended Jahn–Teller system.

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27

Pyatkin, Artem. "1-mean and 1-medoid 2-clustering problem with arbitrary cluster sizes: Complexity and approximation." Yugoslav Journal of Operations Research, no. 00 (2022): 8. http://dx.doi.org/10.2298/yjor211018008p.

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Анотація:

We consider the following 2-clustering problem. Given N points in Euclidean space, partition it into two subsets (clusters) so that the sum of squared distances between the elements of the clusters and their centers would be minimum. The center of the first cluster coincides with its centroid (mean) while the center of the second cluster should be chosen from the set of the initial points (medoid). It is known that this problem is NP-hard if the cardinalities of the clusters are given as a part of the input. In this paper we prove that the problem remains NP-hard in the case of arbitrary clusters sizes and suggest a 2-approximation polynomial-time algorithm for this problem.

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28

Luo, Dijun, Chris Ding, and Heng Huang. "Multi-Level Cluster Indicator Decompositions of Matrices and Tensors." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (August 4, 2011): 423–28. http://dx.doi.org/10.1609/aaai.v25i1.7933.

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Анотація:

A main challenging problem for many machine learning and data mining applications is that the amount of data and features are very large, so that low-rank approximations of original data are often required for efficient computation. We propose new multi-level clustering based low-rank matrix approximations which are comparable and even more compact than Singular Value Decomposition (SVD). We utilize the cluster indicators of data clustering results to form the subspaces, hence our decomposition results are more interpretable. We further generalize our clustering based matrix decompositions to tensor decompositions that are useful in high-order data analysis. We also provide an upper bound for the approximation error of our tensor decomposition algorithm. In all experimental results, our methods significantly outperform traditional decomposition methods such as SVD and high-order SVD.

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29

Gomez, Diego T., Lawrence R. Pratt, David M. Rogers, and Susan B. Rempe. "Free Energies of Hydrated Halide Anions: High Through-Put Computations on Clusters to Treat Rough Energy-Landscapes." Molecules 26, no. 11 (May 21, 2021): 3087. http://dx.doi.org/10.3390/molecules26113087.

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Анотація:

With a longer-term goal of addressing the comparative behavior of the aqueous halides F−, Cl−, Br−, and I− on the basis of quasi-chemical theory (QCT), here we study structures and free energies of hydration clusters for those anions. We confirm that energetically optimal (H2O)nX clusters, with X = Cl−, Br−, and I−, exhibit surface hydration structures. Computed free energies, based on optimized surface hydration structures utilizing a harmonic approximation, typically (but not always) disagree with experimental free energies. To remedy the harmonic approximation, we utilize single-point electronic structure calculations on cluster geometries sampled from an AIMD (ab initio molecular dynamics) simulation stream. This rough-landscape procedure is broadly satisfactory and suggests unfavorable ligand crowding as the physical effect addressed. Nevertheless, this procedure can break down when n≳4, with the characteristic discrepancy resulting from a relaxed definition of clustering in the identification of (H2O)nX clusters, including ramified structures natural in physical cluster theories. With ramified structures, the central equation for the present rough-landscape approach can acquire some inconsistency. Extension of these physical cluster theories in the direction of QCT should remedy that issue, and should be the next step in this research direction.

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30

Ushcats, S. Yu, M. V. Ushcats, V. M. Sysoev, and D. A. Gavryushenko. "Approximation of Cluster Integrals for Various Lattice-Gas Models." Ukrainian Journal of Physics 63, no. 12 (December 9, 2018): 1066. http://dx.doi.org/10.15407/ujpe63.12.1066.

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Анотація:

An approximation for cluster integrals of an arbitrary high order has been proposed for the well-known lattice-gas model with an arbitrary geometry and dimensions. The approximation is based on the recently obtained accurate relations for the convergence radius of the virial power series in the activity parameter for the pressure and density. As compared to the previous studies of the symmetric virial expansions for the gaseous and condensed states of a lattice gas, the proposed approximation substantially approaches the pressure values at the saturation and boiling points. For the Lee–Yang lattice-gas model, the approximation considerably improves the convergence to the known exact solution.

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31

van Stein, Bas, Hao Wang, Wojtek Kowalczyk, Michael Emmerich, and Thomas Bäck. "Cluster-based Kriging approximation algorithms for complexity reduction." Applied Intelligence 50, no. 3 (September 9, 2019): 778–91. http://dx.doi.org/10.1007/s10489-019-01549-7.

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Abstract Kriging or Gaussian Process Regression is applied in many fields as a non-linear regression model as well as a surrogate model in the field of evolutionary computation. However, the computational and space complexity of Kriging, that is cubic and quadratic in the number of data points respectively, becomes a major bottleneck with more and more data available nowadays. In this paper, we propose a general methodology for the complexity reduction, called cluster Kriging, where the whole data set is partitioned into smaller clusters and multiple Kriging models are built on top of them. In addition, four Kriging approximation algorithms are proposed as candidate algorithms within the new framework. Each of these algorithms can be applied to much larger data sets while maintaining the advantages and power of Kriging. The proposed algorithms are explained in detail and compared empirically against a broad set of existing state-of-the-art Kriging approximation methods on a well-defined testing framework. According to the empirical study, the proposed algorithms consistently outperform the existing algorithms. Moreover, some practical suggestions are provided for using the proposed algorithms.

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32

Jirkovský, Jakub, Kateřina Macounová, Hartmut Dietz, Waldfried Plieth, Petr Krtil, and Stanislav Záliš. "Raman Spectroscopy of Nanocrystalline Li-Ti-O Spinels and Comparative DFT Calculations on TiyOz and LixTiyOz Clusters." Collection of Czechoslovak Chemical Communications 72, no. 2 (2007): 171–84. http://dx.doi.org/10.1135/cccc20070171.

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Анотація:

Raman spectra of cubic nanocrystalline insertion hosts in Li-Ti-O system were measured for Li1.1Ti1.9O4-δspinels and the corresponding delithiated titanium dioxide material. To interpret the measured data optimized geometries of cubic TiyOzand LixTiyOzwere calculated using the DFT cluster approach for clusters containing up to 23 atoms. The calculated geometries were used to predict the density-of-states (DOS) diagrams and vibrational spectra of the corresponding structures upon harmonic approximation. While the DFT approach reproduces satisfactorily the main structural characteristics of TiyOzclusters, the agreement in the case of LixTiyOzclusters between calculated and real structures is less satisfactory due to the small cluster size. The band gap energies obtained from DOS diagrams represent overestimates of the actual experimental band gap energies measured on cubic Li-Ti-O oxides. The agreement between calculated and measured band gap energy improves with increasing cluster size. Raman spectra calculated under harmonic approximation from the optimized geometries reproduce all important features of the experimental Raman spectra. The agreement of DFT-calculated and experimental Raman spectra improves with increasing size of the cluster.

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33

Onwuagba, B. N. "Total energies in rubidium and cesium clusters." Canadian Journal of Physics 68, no. 10 (October 1, 1990): 1129–33. http://dx.doi.org/10.1139/p90-159.

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Анотація:

The total energies in rubidium and cesium clusters are investigated as functions of the number of atoms in a cluster in the framework of the local spin density approximation to the density functional formalism. The computed results provide useful information regarding the role played by the shell structure and predict the experimental spectra in rubidium and cesium clusters.

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34

Sills, Alison I., and Jonathan M. Downing. "Dynamical implications of multiple stellar populations." Proceedings of the International Astronomical Union 2, no. 14 (August 2006): 436–37. http://dx.doi.org/10.1017/s1743921307011258.

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AbstractWe investigate some implications of having two star formation episodes in globular clusters, rather than the traditional single-burst approximation. Evidence for more than one stellar generation is accumulating in observations of abundances of elements lighter than iron in globular cluster stars, and is thought to imply some self-enrichment of the globular cluster gas. In particular, we explore models based on the assumption that the self-enrichment comes from an early generation of asymptotic giant branch (AGB) stars.

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35

Ranola, John Michael, Peter Langfelder, Kenneth Lange, and Steve Horvath. "Cluster and propensity based approximation of a network." BMC Systems Biology 7, no. 1 (2013): 21. http://dx.doi.org/10.1186/1752-0509-7-21.

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36

Kats, Daniel, David Kreplin, Hans-Joachim Werner, and Frederick R. Manby. "Accurate thermochemistry from explicitly correlated distinguishable cluster approximation." Journal of Chemical Physics 142, no. 6 (February 14, 2015): 064111. http://dx.doi.org/10.1063/1.4907591.

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37

Behringer, H., M. Pleimling, and A. H�ller. "Phase transitions in a cluster molecular field approximation." European Physical Journal B - Condensed Matter 31, no. 1 (January 1, 2003): 81–93. http://dx.doi.org/10.1140/epjb/e2003-00012-0.

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38

Mor, Etai, Mayer Aladjem, and Amnon Azoulay. "Cluster-enhanced sparse approximation of overlapping ultrasonic echoes." IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 62, no. 2 (February 2015): 373–86. http://dx.doi.org/10.1109/tuffc.2014.006769.

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39

Kats, Daniel, and Andreas Köhn. "On the distinguishable cluster approximation for triple excitations." Journal of Chemical Physics 150, no. 15 (April 21, 2019): 151101. http://dx.doi.org/10.1063/1.5096343.

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40

Mookerjee, Abhijit, and Prabhat K. Thakur. "Cluster coherent-potential approximation in ternary random alloys." Physical Review B 38, no. 6 (August 15, 1988): 3798–802. http://dx.doi.org/10.1103/physrevb.38.3798.

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41

Hague, J. P. "Towards Analytical Approaches to the Dynamical-Cluster Approximation." Journal of Low Temperature Physics 159, no. 3-4 (December 16, 2009): 462–75. http://dx.doi.org/10.1007/s10909-009-0061-2.

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42

Onopko, D. E. "Cluster approximation for covalent and ionic-covalent crystals." Journal of Structural Chemistry 47, no. 1 (January 2006): 1–7. http://dx.doi.org/10.1007/s10947-006-0258-z.

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43

Prellberg, Thomas, Jan Fiala, and Peter Kleban. "Cluster Approximation for the Farey Fraction Spin Chain." Journal of Statistical Physics 123, no. 2 (April 2006): 455–71. http://dx.doi.org/10.1007/s10955-006-9034-x.

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44

Levitskii, R. R., S. I. Sorokov, and R. O. Sokolovskii. "Quenched ising model within two-site cluster approximation." Ferroelectrics 192, no. 1 (February 1997): 11–19. http://dx.doi.org/10.1080/00150199708216166.

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45

McCormack, R., M. Asta, D. de Fontaine, G. Garbulsky, and G. Ceder. "hcp Ising model in the cluster-variation approximation." Physical Review B 48, no. 10 (September 1, 1993): 6767–80. http://dx.doi.org/10.1103/physrevb.48.6767.

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46

Kaneyoshi, T. "Critical properties of Ising multilayer systems: cluster approximation." Journal of Physics: Condensed Matter 11, no. 38 (September 10, 1999): 7311–25. http://dx.doi.org/10.1088/0953-8984/11/38/310.

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47

Wurtz, Jonathan, Anatoli Polkovnikov, and Dries Sels. "Cluster truncated Wigner approximation in strongly interacting systems." Annals of Physics 395 (August 2018): 341–65. http://dx.doi.org/10.1016/j.aop.2018.06.001.

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48

Geertsen, Jan. "An approximate coupled cluster doubles polarization propagator approximation." International Journal of Quantum Chemistry 34, S22 (March 12, 1988): 491–95. http://dx.doi.org/10.1002/qua.560340853.

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49

Abdallah, Hassan H. "Revisiting the tellurium clusters (Ten; n = 2–8) using ab initio methods." Canadian Journal of Physics 98, no. 1 (January 2020): 57–64. http://dx.doi.org/10.1139/cjp-2019-0015.

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The optimized geometries and vibrational frequencies of Te clusters n = 2–8 are calculated using ab initio molecular orbital theory at B3LYP, MP2, BLYP, and BH–HLYP levels of approximation. We found that Te8 (D4d) cluster has the highest stability followed by Te7 and Te6 (D3d). The computed vibrational frequencies have small systematic deviations with IR spectra for crystalline Te5, Te6 (C2v). The stability of positively and negatively charged Te clusters is determined by the B3LYP method. This is because the B3LYP method demonstrated the best stability in every cluster geometry. In general, the results showed that the negatively charged clusters have the highest stability, followed by neutral clusters, and finally positively charged clusters. However, the predicted bond angle and bond distance for every cluster geometry displayed very close values with different levels of calculations. These calculations will provide predictions for future experimental studies.

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50

ISHII, Y., N. WATARI, and S. OHNISHI. "ELECTRONIC STRUCTURES OF HEAVY-METAL CLUSTERS." Surface Review and Letters 03, no. 01 (February 1996): 335–40. http://dx.doi.org/10.1142/s0218625x96000619.

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The electronic structures of small Hg and Pb clusters are studied by the first-principle calculations within the local density-functional approximation. It is found that the stable structure of Hg19 cluster is not polyicosahedral but hexagonal close-packed although the bonding nature is still atom-like with no significant sp-hybridization. The stability of a doubly charged Pb cluster is discussed in connection with the electronic shell-model description. We conclude that the stability of Pb clusters is determined by complicated correlation between the atomic and electronic structures, and the electronic shell model cannot be applied straightforwardly.

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