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Journal articles on the topic 'Solvable approximation'

Author: Grafiati

Published: 25 May 2024

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1

Hopkins, William E., and Wing Shing Wong. "Approximation of almost solvable bilinear systems." Systems & Control Letters 6, no. 2 (July 1985): 131–40. http://dx.doi.org/10.1016/0167-6911(85)90011-8.

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2

Valtancoli, P. "Exactly solvable f(R) inflation." International Journal of Modern Physics D 28, no. 07 (May 2019): 1950087. http://dx.doi.org/10.1142/s0218271819500871.

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3

Lemm, J. C. "Inhomogeneous Random Phase Approximation: A Solvable Model." Annals of Physics 244, no. 1 (November 1995): 201–38. http://dx.doi.org/10.1006/aphy.1995.1111.

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4

Shi, Ronggang, and Barak Weiss. "Invariant measures for solvable groups and Diophantine approximation." Israel Journal of Mathematics 219, no. 1 (April 2017): 479–505. http://dx.doi.org/10.1007/s11856-017-1472-y.

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5

Co’, Giampaolo, and Stefano De Leo. "Hartree–Fock and random phase approximation theories in a many-fermion solvable model." Modern Physics Letters A 30, no. 36 (November 3, 2015): 1550196. http://dx.doi.org/10.1142/s0217732315501965.

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We present an ideal system of interacting fermions where the solutions of the many-body Schrödinger equation can be obtained without making approximations. These exact solutions are used to test the validity of two many-body effective approaches, the Hartree–Fock and the random phase approximation theories. The description of the ground state done by the effective theories improves with increasing number of particles.

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6

SOLENOV, DMITRY, and VLADIMIR PRIVMAN. "EVALUATION OF DECOHERENCE FOR QUANTUM COMPUTING ARCHITECTURES: QUBIT SYSTEM SUBJECT TO TIME-DEPENDENT CONTROL." International Journal of Modern Physics B 20, no. 11n13 (May 20, 2006): 1476–95. http://dx.doi.org/10.1142/s0217979206034066.

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We present an approach that allows quantifying decoherence processes in an open quantum system subject to external time-dependent control. Interactions with the environment are modeled by a standard bosonic heat bath. We develop two unitarity-preserving approximation schemes to calculate the reduced density matrix. One of the approximations relies on a short-time factorization of the evolution operator, while the other utilizes expansion in terms of the system-bath coupling strength. Applications are reported for two illustrative systems: an exactly solvable adiabatic model, and a model of a rotating-wave quantum-computing gate function. The approximations are found to produce consistent results at short and intermediate times.

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7

Mota, V., and E. S. Hern�ndez. "A solvable version of the collisional random phase approximation." Zeitschrift f�r Physik A Atomic Nuclei 328, no. 2 (June 1987): 177–87. http://dx.doi.org/10.1007/bf01290660.

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8

Kudryashov, Vladimir V., and Yulian V. Vanne. "Explicit summation of the constituent WKB series and new approximate wave functions." Journal of Applied Mathematics 2, no. 6 (2002): 265–75. http://dx.doi.org/10.1155/s1110757x02112046.

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The independent solutions of the one-dimensional Schrödinger equation are approximated by means of the explicit summation of the leading constituent WKB series. The continuous matching of the particular solutions gives the uniformly valid analytical approximation to the wave functions. A detailed numerical verification of the proposed approximation is performed for some exactly solvable problems arising from different kinds of potentials.

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9

Sollich, Peter, and Anason Halees. "Learning Curves for Gaussian Process Regression: Approximations and Bounds." Neural Computation 14, no. 6 (June 1, 2002): 1393–428. http://dx.doi.org/10.1162/089976602753712990.

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We consider the problem of calculating learning curves (i.e., average generalization performance) of gaussian processes used for regression. On the basis of a simple expression for the generalization error, in terms of the eigenvalue decomposition of the covariance function, we derive a number of approximation schemes. We identify where these become exact and compare with existing bounds on learning curves; the new approximations, which can be used for any input space dimension, generally get substantially closer to the truth. We also study possible improvements to our approximations. Finally, we use a simple exactly solvable learning scenario to show that there are limits of principle on the quality of approximations and bounds expressible solely in terms of the eigenvalue spectrum of the covariance function.

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10

Shen, Jinrong, Wei Liu, Baiyu Wang, and Xiangyang Peng. "The Centrosymmetric Matrices of Constrained Inverse Eigenproblem and Optimal Approximation Problem." Mathematical Problems in Engineering 2020 (March 10, 2020): 1–8. http://dx.doi.org/10.1155/2020/4590354.

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In this paper, a kind of constrained inverse eigenproblem and optimal approximation problem for centrosymmetric matrices are considered. Necessary and sufficient conditions of the solvability for the constrained inverse eigenproblem of centrosymmetric matrices in real number field are derived. A general representation of the solution is presented for a solvable case. The explicit expression of the optimal approximation problem is provided. Finally, a numerical example is given to illustrate the effectiveness of the method.

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11

Liang, Mao Lin, and Li Fang Dai. "Solution to a Class of Matrix Equations with k-Involutary Symmetrices." Advanced Materials Research 457-458 (January 2012): 799–803. http://dx.doi.org/10.4028/www.scientific.net/amr.457-458.799.

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In this paper, we investigate the solvability of matrix equations with -involutary symmetric matrix , the general solution of which is obtained when it is solvable. Meantime, the associated optimal approximation problem for some given matrix is also considered under some particular hypothesis.

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12

Galli, F., and A. S. Koshelev. "Multifield cosmology from string field theory: An exactly solvable approximation." Theoretical and Mathematical Physics 164, no. 3 (September 2010): 1169–75. http://dx.doi.org/10.1007/s11232-010-0096-1.

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13

Galtbayar, A., A. Jensen, and K. Yajima. "A solvable model of the breakdown of the adiabatic approximation." Journal of Mathematical Physics 61, no. 9 (September 1, 2020): 092105. http://dx.doi.org/10.1063/5.0001813.

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14

Uhlmann, Hermann, and Olaf Michelsson. "A fast forward solution with a boundary element method for eddy current nondestructive testing." Facta universitatis - series: Electronics and Energetics 15, no. 2 (2002): 205–16. http://dx.doi.org/10.2298/fuee0202205u.

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Eddy current non-destructive testing is used to determine position and size of cracks or other defects in conducting materials. The presence of a crack normal to the excited eddy currents distorts the magnetic field; so for the identification of defects a very accurate and fast 3D-computation of the magnetic field is necessary. A computation scheme for 3D quasistatic electromagnetic fields by means of the Boundary Element Method is presented. Although the use of constant field approximations on boundary elements is the easiest way, it often provides an insufficient accuracy. This can be overcome by higher order approximation schemes. The numerical results are compared against some analytically solvable arrangements.

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15

Zheng, Jun, Hao Bo Qiu, and Xiao Lin Zhang. "Variable-Fidelity Multidisciplinary Design Optimization Based on Analytical Target Cascading Framework." Advanced Materials Research 544 (June 2012): 49–54. http://dx.doi.org/10.4028/www.scientific.net/amr.544.49.

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ATC provides a systematic approach in solving decomposed large scale systems that has solvable subsystems. However， complex engineering system usually has a high computational cost , which result in limiting real-life applications of ATC based on high-fidelity simulation models. To address these problems, this paper aims to develop an efficient approximation model building techniques under the analytical target cascading (ATC) framework, to reduce computational cost associated with multidisciplinary design optimization problems based on high-fidelity simulations. An approximation model building techniques is proposed: approximations in the subsystem level are based on variable-fidelity modeling (interaction of low- and high-fidelity models). The variable-fidelity modeling consists of computationally efficient simplified models (low-fidelity) and expensive detailed (high-fidelity) models. The effectiveness of the method for modeling under the ATC framework using variable-fidelity models is studied. Overall results show the methods introduced in this paper provide an effective way of improving computational efficiency of the ATC method based on variable-fidelity simulation models.

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16

DEBERGH, N., and A. B. KLIMOV. "QUASI-EXACTLY SOLVABLE APPROACH TO THE JAYNES–CUMMINGS MODEL WITHOUT ROTATION WAVE APPROXIMATION." International Journal of Modern Physics A 16, no. 24 (September 30, 2001): 4057–68. http://dx.doi.org/10.1142/s0217751x01005146.

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The Jaynes–Cummings model (JCM) is studied in the frame of methods of quasi-exactly solvable problems. First, we apply these methods to the JCM under RWA and anti-RWA approximations and then, we analyze the possibility of solving the whole model.

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17

Gates, D. J. "On the optimal composition of electricity grids with unreliable units: solvable models." Advances in Applied Probability 17, no. 2 (June 1985): 367–85. http://dx.doi.org/10.2307/1427146.

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For a large electricity grid comprising many units (plants) of various types, such as coal, oil, nuclear, hydro, etc., with known unreliabilities (outage rates) we study the optimal (i.e. the cheapest) total capacity, or numbers, of each type of unit. Existing treatments of the problem involve numerical methods and approximations of unknown accuracy. For a range of cases, we find explicit solutions. This extends the known explicit solutions, which are confined to completely reliable units. The cases we analyse are (I) a demand (load) which has a shifted Rayleigh distribution—a good approximation to the real load-duration curve—with some restriction on reliability (big units are more reliable) and (II) an exponential load distribution—which is unrealistic—with no restrictions on reliability. In both cases, the solutions reduce to transformed versions of the exact solutions for totally reliable units and, like the latter, can be exhibited by means of a cost polygon.

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18

Gates, D. J. "On the optimal composition of electricity grids with unreliable units: solvable models." Advances in Applied Probability 17, no. 02 (June 1985): 367–85. http://dx.doi.org/10.1017/s0001867800015020.

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For a large electricity grid comprising many units (plants) of various types, such as coal, oil, nuclear, hydro, etc., with known unreliabilities (outage rates) we study the optimal (i.e. the cheapest) total capacity, or numbers, of each type of unit. Existing treatments of the problem involve numerical methods and approximations of unknown accuracy. For a range of cases, we find explicit solutions. This extends the known explicit solutions, which are confined to completely reliable units. The cases we analyse are (I) a demand (load) which has a shifted Rayleigh distribution—a good approximation to the real load-duration curve—with some restriction on reliability (big units are more reliable) and (II) an exponential load distribution—which is unrealistic—with no restrictions on reliability. In both cases, the solutions reduce to transformed versions of the exact solutions for totally reliable units and, like the latter, can be exhibited by means of a cost polygon.

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19

Montorsi, Arianna, and Mario Rasetti. "MOTT TRANSITION IN AN EXACTLY SOLVABLE K.S.S.H. MODEL." International Journal of Modern Physics B 05, no. 06n07 (April 1991): 985–98. http://dx.doi.org/10.1142/s0217979291000511.

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The solution of the K.S.S.H.-like model shown to be exactly solvable in any number of dimensions, for a particular choice of the coupling constant describing the hopping process amplitude, both for finite size and in the thermodynamic limit, is discussed in detail. The analysis of the zero-temperature phase space in d = 2 shows that the model exhibits a transition in the number of doubly occupied sites order parameter, which at half-filling coincides with the Mott transition found for the Hubbard model in the Gutzwiller approximation.

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20

CAPDEQUI-PEYRANÈRE, M. "IS SUPERSYMMETRIC QUANTUM MECHANICS COMPATIBLE WITH DUALITY?" Modern Physics Letters A 14, no. 38 (December 14, 1999): 2657–66. http://dx.doi.org/10.1142/s0217732399002790.

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Supersymmetry applied to quantum mechanics has given new insights in various topics of theoretical physics like analytically solvable potentials, WKB approximation or KdV solitons. Duality plays a central role in many supersymmetric theories such as Yang–Mills theories or strings models. We investigate the possible existence of some duality within supersymmetric quantum mechanics.

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21

BOGOLUBOV, NIKOLAI, ANNA GHAZARYAN, and YAREMA PRYKARPATSKY. "OPERATOR ANALYSIS OF AN RPA-REDUCED POLARON MODEL WITHIN THE BOGOLUBOV REPRESENTATION IN MAGNETIC FIELD AT FINITE TEMPERATURE. PART 1." International Journal of Modern Physics B 23, no. 24 (September 30, 2009): 4843–53. http://dx.doi.org/10.1142/s0217979209053941.

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The polaron model in ionic crystal is studied in the N. Bogolubov representation using a special RPA-approximation. A new exactly solvable approximated polaron model is derived and described in detail. Its free energy at finite temperature is calculated analytically. The polaron free energy in the constant magnetic field at finite temperature is also discussed.

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22

Banks, S. P., and D. McCaffrey. "Lie Algebras, Structure of Nonlinear Systems and Chaotic Motion." International Journal of Bifurcation and Chaos 08, no. 07 (July 1998): 1437–62. http://dx.doi.org/10.1142/s021812749800111x.

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The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system. An approximation theory is developed and a new class of chaotic systems is introduced, based on the structure theory of Lie algebras.

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23

Atre, Rajneesh, and Prasanta K. Panigrahi. "Quasi-exactly solvable Hamiltonians: a new approach and an approximation scheme." Physics Letters A 317, no. 1-2 (October 2003): 46–53. http://dx.doi.org/10.1016/j.physleta.2003.08.036.

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24

VISSER, MATT. "WHEELER WORMHOLES AND TOPOLOGY CHANGE: A MINISUPERSPACE ANALYSIS." Modern Physics Letters A 06, no. 29 (September 21, 1991): 2663–67. http://dx.doi.org/10.1142/s0217732391003109.

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Wheeler's wormholes are analyzed within the context of a particular minisuperspace approximation. In this approximation the Wheeler-DeWitt equation describing a wormhole is exactly solvable and the quantum mechanical wavefunction of a wormhole can be explicitly exhibited. Calculation shows that the throat of a minisuperspace Wheeler wormhole is stabilized against collapse by quantum mechanical effects: The radius of the throat has an expectation value of order the Planck length. Implications of this result with respect to the process of quantum gravitational topology change are discussed. In particular it is argued that the putative stability of minisuperspace Wheeler wormholes—if it persists beyond the minisuperspace approximation—might serve to suppress fluctuations in topology.

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25

Alhaidari, Abdulaziz D., and Ibsal A. Assi. "Finite-Series Approximation of the Bound States for Two Novel Potentials." Physics 4, no. 3 (September 8, 2022): 1067–80. http://dx.doi.org/10.3390/physics4030070.

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We obtain an analytic approximation of the bound states solution of the Schrödinger equation on the semi-infinite real line for two potential models with a rich structure as shown by their spectral phase diagrams. These potentials do not belong to the class of exactly solvable problems. The solutions are finite series (with a small number of terms) of square integrable functions written in terms of Romanovski–Jacobi polynomials.

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26

KHARLAMPOVICH, OLGA, and ATEFEH MOHAJERI MOGHADDAM. "APPROXIMATION OF GEODESICS IN METABELIAN GROUPS." International Journal of Algebra and Computation 22, no. 02 (March 2012): 1250012. http://dx.doi.org/10.1142/s0218196711006789.

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It is known that the bounded Geodesic Length Problem in free metabelian groups is NP-complete [A. Myasnikov, V. Roman'kov, A. Ushakov and A. Vershik, The word and geodesic problems in free solvable groups, Trans. Amer. Math. Soc.362(9) (2010) 4655–4682] (in particular, the Geodesic Problem is NP-hard). We construct a 2-approximation polynomial time deterministic algorithm for the Geodesic Problem. We show that the Geodesic Problem in the restricted wreath product of a finitely generated non-trivial group with a finitely generated abelian group containing ℤ2 is NP-hard and there exists a Polynomial Time Approximation Scheme for this problem. We also show that the Geodesic Problem in the restricted wreath product of two finitely generated non-trivial abelian groups is NP-hard if and only if the second abelian group contains ℤ2.

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27

MCKELLAR, BRUCE H. J., IVONA OKUNIEWICZ, and JAMES QUACH. "SOLVABLE MODEL KINETIC EQUATIONS WITH NON-BOLTZMANN PROPERTIES." International Journal of Modern Physics A 24, no. 06 (March 10, 2009): 1087–96. http://dx.doi.org/10.1142/s0217751x09044279.

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We reconsider the question of the relative importance of single particle effects and correlations in the solvable interacting neutrino models introduced by Friedland and Lunardini and by Bell, Rawlinson and Sawyer. We show, by an exact calculation, that the two particle correlations are not "small," and that they dominate the time evolution in these models, in spite of indications to the contrary from the rate of equilibration. The failure of the Boltzmann single particle approximation in this model is tentatively attributed to the simplicity of the model, in particular the restriction to two-flavor mixing, and the neglect of the position dependence of the interaction.

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28

OHKUWA, YOSHIAKI, TETSURO KITAZOE, and YOSHIHIKO MIZUMOTO. "TIME IN THE CHAOTIC INFLATION MODEL AND NUMERICAL CALCULATIONS." International Journal of Modern Physics A 10, no. 16 (June 30, 1995): 2317–32. http://dx.doi.org/10.1142/s0217751x95001121.

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The time variable is considered in the quantum gravity theory and calculated explicitly in the framework of the chaotic inflationary scenario where the scalar matter field has a contribution to the time variable in addition to the gravity field. The time formulated under the semiclassical approximation is a natural extension of that in the classical orbital motion. A perturbation method is introduced in terms of the scalar mass to obtain analytically solvable expressions for the time. The Wheeler-DeWitt equation is solved numerically to ensure that the semiclassical approximation is well justified. We examine the obtained time in detail and find that it is reasonable to consider it as time in the region where the semiclassical approximation is well justified.

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29

Li, Bao-Fei, Tao Zhu, and Anzhong Wang. "Langer Modification, Quantization Condition and Barrier Penetration in Quantum Mechanics." Universe 6, no. 7 (June 30, 2020): 90. http://dx.doi.org/10.3390/universe6070090.

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The WKB approximation plays an essential role in the development of quantum mechanics and various important results have been obtained from it. In this paper, we introduce another method, the so-called uniform asymptotic approximations, which is an analytical approximation method to calculate the wave functions of the Schrödinger-like equations, and it is applicable to various problems, including cases with poles (singularities) and multiple turning points. A distinguished feature of the method is that in each order of the approximations the upper bounds of the errors are given explicitly. By properly choosing the freedom introduced in the method, the errors can be minimized, which significantly improves the accuracy of the calculations. A byproduct of the method is to provide a very clear explanation of the Langer modification encountered in the studies of the hydrogen atom and harmonic oscillator. To further test our method, we calculate (analytically) the wave functions for several exactly solvable potentials of the Schrödinger equation, and then obtain the transmission coefficients of particles over potential barriers, as well as the quantization conditions for bound states. We find that such obtained results agree with the exact ones extremely well. Possible applications of the method to other fields are also discussed.

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30

BARRETT, B. R., D. M. CARDAMONE, and C. A. STAFFORD. "EXACTLY SOLVABLE MODEL FOR THE DECAY OF SUPERDEFORMED NUCLEI." International Journal of Modern Physics E 14, no. 01 (February 2005): 157–64. http://dx.doi.org/10.1142/s0218301305002886.

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The history and importance of superdeformation in nuclei is briefly discussed. A simple two-level model is then employed to obtain an elegant expression for the branching ratio for the decay via the E1 process in the normal-deformed band of superdeformed nuclei. From this expression, the spreading width Γ↓ for superdeformed decay is found to be determined completely by experimentally known quantities. The accuracy of the two-level approximation is verified by considering the effects of other normal-deformed states. Furthermore, by using a statistical model of the energy levels in the normal-deformed well, we can obtain a probabilistic expression for the tunneling matrix element V.

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31

Alhassid, Y., and B. W. Bush. "Nuclear level densities in the static-path approximation: (I). A solvable model." Nuclear Physics A 549, no. 1 (November 1992): 43–58. http://dx.doi.org/10.1016/0375-9474(92)90066-s.

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32

Buzna, Ľuboš, Michal Koháni, and Jaroslav Janáček. "An Approximation Algorithm for the Facility Location Problem with Lexicographic Minimax Objective." Journal of Applied Mathematics 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/562373.

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We present a new approximation algorithm to the discrete facility location problem providing solutions that are close to the lexicographic minimax optimum. The lexicographic minimax optimum is a concept that allows to find equitable location of facilities serving a large number of customers. The algorithm is independent of general purpose solvers and instead uses algorithms originally designed to solve thep-median problem. By numerical experiments, we demonstrate that our algorithm allows increasing the size of solvable problems and provides high-quality solutions. The algorithm found an optimal solution for all tested instances where we could compare the results with the exact algorithm.

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33

KAVIC, MICHAEL. "MATCHING WEAK COUPLING AND QUASICLASSICAL EXPANSIONS FOR DUAL QES PROBLEMS." International Journal of Modern Physics A 17, no. 31 (December 20, 2002): 4661–67. http://dx.doi.org/10.1142/s0217751x02010777.

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Certain quasi-exactly solvable systems exhibit an energy reflection property that relates the energy levels of a potential or of a pair of potentials. We investigate two sister potentials and show the existence of this energy reflection relationship between the two potentials. We establish a relationship between the lowest energy edge in the first potential using the weak coupling expansion and the highest energy level in the sister potential using a WKB approximation carried out to higher order.

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34

Sun, Chang-Pu, and Lin-Zhi Zhang. "Test of quantum adiabatic approximation via exactly-solvable dynamics of high-spin precession." Physica Scripta 51, no. 1 (January 1, 1995): 16–18. http://dx.doi.org/10.1088/0031-8949/51/1/003.

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35

Mateus, L., D. Masoero, F. Rocha, M. Aguiar, U. Skwara, P. Ghaffari, JC Zambrini, and N. Stollenwerk. "Epidemiological models in semiclassical approximation: an analytically solvable model as a test case." Mathematical Methods in the Applied Sciences 39, no. 16 (October 11, 2016): 4914–22. http://dx.doi.org/10.1002/mma.4108.

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36

Rui, Hongxing, and Jian Huang. "Uniformly Stable Explicitly Solvable Finite Difference Method for Fractional Diffusion Equations." East Asian Journal on Applied Mathematics 5, no. 1 (February 2015): 29–47. http://dx.doi.org/10.4208/eajam.030614.051114a.

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AbstractA finite difference scheme for the one-dimensional space fractional diffusion equation is presented and analysed. The scheme is constructed by modifying the shifted Grünwald approximation to the spatial fractional derivative and using an asymmetric discretisation technique. By calculating the unknowns in differential nodal point sequences at the odd and even time levels, the discrete solution of the scheme can be obtained explicitly. We prove that the scheme is uniformly stable. The error between the discrete solution and the analytical solution in the discretel2norm is optimal in some cases. Numerical results for several examples are consistent with the theoretical analysis.

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37

Gherardi, Marco. "Solvable Model for the Linear Separability of Structured Data." Entropy 23, no. 3 (March 4, 2021): 305. http://dx.doi.org/10.3390/e23030305.

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Linear separability, a core concept in supervised machine learning, refers to whether the labels of a data set can be captured by the simplest possible machine: a linear classifier. In order to quantify linear separability beyond this single bit of information, one needs models of data structure parameterized by interpretable quantities, and tractable analytically. Here, I address one class of models with these properties, and show how a combinatorial method allows for the computation, in a mean field approximation, of two useful descriptors of linear separability, one of which is closely related to the popular concept of storage capacity. I motivate the need for multiple metrics by quantifying linear separability in a simple synthetic data set with controlled correlations between the points and their labels, as well as in the benchmark data set MNIST, where the capacity alone paints an incomplete picture. The analytical results indicate a high degree of “universality”, or robustness with respect to the microscopic parameters controlling data structure.

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38

Arenas, Marcelo, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. "Counting the Answers to a Query." ACM SIGMOD Record 51, no. 3 (November 21, 2022): 6–17. http://dx.doi.org/10.1145/3572751.3572753.

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Counting the answers to a query is a fundamental problem in databases, with several applications in the evaluation, optimization, and visualization of queries. Unfortunately, counting query answers is a #P-hard problem in most cases, so it is unlikely to be solvable in polynomial time. Recently, new results on approximate counting have been developed, specifically by showing that some problems in automata theory admit fully polynomial-time randomized approximation schemes. These results have several implications for the problem of counting the answers to a query; in particular, for graph and conjunctive queries. In this work, we present the main ideas of these approximation results, by using labeled DAGs instead of automata to simplify the presentation. In addition, we review how to apply these results to count query answers in different areas of databases.

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39

Hebeker, Friedrich-Karl, and George C. Hsiao. "An Initial-Boundary Value Problem for a Viscous Compressible Flow." gmj 14, no. 1 (March 2007): 123–34. http://dx.doi.org/10.1515/gmj.2007.123.

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Abstract A constructive approach is presented to treat an initial boundary value problem for isothermal Navier–Stokes equations. It is based on a characteristics (Lagrangean) approximation locally in time and a boundary integral equation method via nonstationary potentials. As a basic problem, the latter leads to a Volterra integral equation of first kind which is proved to be uniquely solvable and even coercive in some anisotropic Sobolev spaces. The solution depends continuously upon the data and can be constructed by a quasioptimal Galerkin procedure.

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40

Pinar, Zehra. "Simulations of surface corrugations of graphene sheets through the generalized graphene thermophoretic motion equation." International Journal of Computational Materials Science and Engineering 09, no. 01 (March 2020): 2050005. http://dx.doi.org/10.1142/s2047684120500050.

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In the current decade, nanomaterials have attracted great attention due to the wide range of applications in various disciplines and nanotechnology. Graphene is the best nanoscale material and one of the thinnest elastic films and has various applications. The thermophoretic motion system describes the diffusion of solitaries into substrate-supported graphene sheets. Lie group transformation of the motion equation is used to reduce the equation into solvable equation which is solved through the Bernoulli approximation method and some properties of the solutions are discussed.

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41

ARRIOLA, E. RUIZ, and L. L. SALCEDO. "SEMICLASSICAL EXPANSION FOR DIRAC HAMILTONIANS." Modern Physics Letters A 08, no. 22 (July 20, 1993): 2061–69. http://dx.doi.org/10.1142/s021773239300177x.

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A simple and efficient calculation of the level density of Dirac Hamiltonians in the semi-classical approximation is presented. The method is applied to compute the level density up to ħ4-order of a Dirac Hamiltonian with time independent scalar and electromagnetic external fields. The final expressions are explicitly gauge invariant and convergent at the turning points. As a byproduct, we obtain ħ-corrections to the semiclassical quantization rule of a Dirac Hamiltonian in D space dimensions. The result is illustrated in an exactly solvable problem.

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Almira, J. M., N. Del Toro, and A. J. López-Moreno. "A note on Diophantine approximation." International Journal of Mathematics and Mathematical Sciences 2005, no. 3 (2005): 487–90. http://dx.doi.org/10.1155/ijmms.2005.487.

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We prove the existence of a dense subsetΔof[0,4]such that for allα∈Δthere exists a subgroupXαof infinite rank ofℤ[z]such thatXαis a discrete subgroup ofC[0,β]for allβ≥αbut it is not a discrete subgroup ofC[0,β]for anyβ∈(0,α).Given a set of nonnegative real numbersΛ={λi}i=0∞, aΛ-polynomial (or Müntz polynomial) is a function of the formp(x)=∑i=0naizλi(n∈ℕ). We denote byΠ(Λ)the space ofΛ-polynomials and byΠℤ(Λ):={p(x)=∑i=0naizλi∈Π(λ):ai∈ℤ for all i≥0}the set of integralΛ-polynomials. Clearly, the setsΠℤ(Λ)are subgroups of infinite rank ofℤ[x]wheneverΛ⊂ℕ,#Λ=∞(by infinite rank, we mean that the real vector space spanned byXdoes not have finite dimension. In all what follows we are uniquely interested in groups of infinite rank). Now, it is well known that the problem of approximation of functions on intervals[a,b]by polynomials with integral coefficients is solvable only for intervals[a,b]of length smaller than four and functionsfwhich are interpolable by polynomials ofℤ[x]on a certain set (which we call the algebraic kernel of the interval[a,b])𝒥 (a,b). Concretely, it is well known thatℤ[x]is a discrete subgroup ofC[a,b]wheneverb−a≥4and4is the smallest number with this property (for these and other interesting results about approximation by polynomials with integral coefficients, see [1,3] and the references therein. See also the other references at the end of this note). This motivates the following concept.

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Alicki, Robert. "Quantum Decay Cannot Be Completely Reversed: The 5% Rule." Open Systems & Information Dynamics 16, no. 01 (March 2009): 49–53. http://dx.doi.org/10.1142/s1230161209000049.

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Using an exactly solvable model of the Wigner-Weisskopf atom, it is shown that an unstable quantum state cannot be recovered completely by the procedure involving detection of the decay products followed by the creation of time-reversed decay products state, as proposed in [1]. The universal lower bound on the recovery error is approximately equal to 5% of the error per cycle — the dimensionless parameter characterizing decay process in the Markovian approximation. This result has consequences for the efficiency of quantum error correction procedures which are based on syndrome measurements and corrective operations.

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Han, Jianlong, Seth Armstrong, and Sarah Duffin. "An unconditionally stable numerical scheme for competing species undergoing nonlocal dispersion." Electronic Research Archive 32, no. 4 (2024): 2478–90. http://dx.doi.org/10.3934/era.2024114.

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<abstract><p>Nonstandard numerical approximation for the study of a competition model for two species that experience nonlocal diffusion, or dispersion, allows for faithful representation of the theoretical solution to the system. Such a scheme may preserve positivity of solutions, be uniquely solvable, and be completely stable. Under appropriate conditions, the error between the scheme and the theoretical solution can be measured. We present such a scheme here and confirm its desirable properties as they reflect the solution to the system.</p></abstract>

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Yan, Zhanyuan, Peihua Qu, Bingbing Xu, Shihui Zhang, and Jinying Ma. "Solution of two-qubit Rabi model with extended generalized rotating-wave approximation." Modern Physics Letters B 35, no. 13 (February 9, 2021): 2150213. http://dx.doi.org/10.1142/s0217984921502134.

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The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.

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PANDA, B. S., and S. PAUL. "CONNECTED LIAR'S DOMINATION IN GRAPHS: COMPLEXITY AND ALGORITHMS." Discrete Mathematics, Algorithms and Applications 05, no. 04 (December 2013): 1350024. http://dx.doi.org/10.1142/s1793830913500249.

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A subset L ⊆ V of a graph G = (V, E) is called a connected liar's dominating set of G if (i) for all v ∈ V, |NG[v] ∩ L| ≥ 2, (ii) for every pair u, v ∈ V of distinct vertices, |(NG[u]∪NG[v])∩L| ≥ 3, and (iii) the induced subgraph of G on L is connected. In this paper, we initiate the algorithmic study of minimum connected liar's domination problem by showing that the corresponding decision version of the problem is NP-complete for general graph. Next we study this problem in subclasses of chordal graphs where we strengthen the NP-completeness of this problem for undirected path graph and prove that this problem is linearly solvable for block graphs. Finally, we propose an approximation algorithm for minimum connected liar's domination problem and investigate its hardness of approximation in general graphs.

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He, Guang-Ping, Zhi-Lü Wang, Jie Zhang, and Zhi-Yong Geng. "Characteristics analysis and stabilization of a planar 2R underactuated manipulator." Robotica 34, no. 3 (July 9, 2014): 584–600. http://dx.doi.org/10.1017/s0263574714001714.

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SUMMARYThe weightless planar 2R underactuated manipulators with passive last joint are considered in this paper for investigating a feasible method to stabilize the system, which is a second-order nonholonomic-constraint mechanical system with drifts. The characteristics including the controllability of the linear approximation model, the minimum phase property, the Small Time Local Controllability (STLC), the differential flatness, and the exactly nilpotentizable properties, are analyzed. Unfortunately, these negative characteristics indicate that the simplest underactuated mechanical system is difficult to design a stable closed-loop control system. In this paper, nilpotent approximation and iterative steering methods are utilized to solve the problem. A globally effective nilpotent approximation model is developed and the parameterized polynomial input is adopted to stabilize the system to its non-singularity equilibrium configuration. In accordance with this scheme, it is shown that designing a stable closed-loop control system for the underactuated mechanical system can be ascribed to solving a set of nonlinear algebraic equations. If the nonlinear algebraic equations are solvable, then the controller is asymptotically stable. Some numerical simulations demonstrate the effectiveness of the presented approach.

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Konakov, V. D., S. Menozzi, and S. A. Molchanov. "Diffusion processes on solvable groups of upper triangular 2×2 matrices and their approximation." Doklady Mathematics 84, no. 1 (August 2011): 527–30. http://dx.doi.org/10.1134/s1064562411050036.

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Navarro-Salas, José, and Silvia Pla. "Particle Creation and the Schwinger Model." Symmetry 14, no. 11 (November 17, 2022): 2435. http://dx.doi.org/10.3390/sym14112435.

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We study the particle creation process in the Schwinger model coupled with an external classical source. One can approach the problem by taking advantage of the fact that the full quantized model is solvable and equivalent to a (massive) gauge field with a non-local effective action. Alternatively, one can also face the problem by following the standard semiclassical route. This means quantizing the massless Dirac field and considering the electromagnetic field as a classical background. We evaluate the energy created by a generic, homogeneous, and time-dependent source. The results match exactly in both approaches. This proves in a very direct and economical way the validity of the semiclassical approach for the (massless) Schwinger model, in agreement with a previous analysis based on the linear response equation. Our discussion suggests that a similar analysis for the massive Schwinger model could be used as a non-trivial laboratory to confront a fully quantized solvable model with its semiclassical approximation, therefore mimicking the long-standing confrontation of quantum gravity with quantum field theory in curved spacetime.

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Pingak, Redi Kristian, Albert Zicko Johannes, Fidelis Nitti, and Meksianis Zadrak Ndii. "A Theoretical Study on Vibrational Energies of Molecular Hydrogen and Its Isotopes Using a Semi-classical Approximation." Indonesian Journal of Chemistry 21, no. 3 (April 22, 2021): 725. http://dx.doi.org/10.22146/ijc.63294.

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This study aims to apply a semi-classical approach using some analytically solvable potential functions to accurately compute the first ten pure vibrational energies of molecular hydrogen (H2) and its isotopes in their ground electronic states. This study also aims at comparing the accuracy of the potential functions within the framework of the semi-classical approximation. The performance of the approximation was investigated as a function of the molecular mass. In this approximation, the nuclei were assumed to move in a classical potential. The Bohr-Sommerfeld quantization rule was then applied to calculate the vibrational energies of the molecules numerically. The results indicated that the first vibrational transition frequencies (v1ß0) of all hydrogen isotopes were consistent with the experimental ones, with a minimum percentage error of 0.02% for ditritium (T2) molecule using the Modified-Rosen-Morse potential. It was also demonstrated that, in general, the Rosen-Morse and the Modified-Rosen-Morse potential functions were better in terms of calculating the vibrational energies of the molecules than Morse potential. Interestingly, the Morse potential was found to be better than the Manning-Rosen potential. Finally, the semi-classical approximation was found to perform better for heavier isotopes for all potentials applied in this study.

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