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Journal articles on the topic 'Symmetry of solution'

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Published: 9 March 2023

Last updated: 10 March 2023

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1

Heule, Marijn, and Toby Walsh. "Symmetry in Solutions." Proceedings of the AAAI Conference on Artificial Intelligence 24, no. 1 (July 3, 2010): 77–82. http://dx.doi.org/10.1609/aaai.v24i1.7549.

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We define the concept of an internal symmetry. This is a symmety within a solution of a constraint satisfaction problem. We compare this to solution symmetry, which is a mapping between different solutions of the same problem. We argue that we may be able to exploit both types of symmetry when finding solutions. We illustrate the potential of exploiting internal symmetries on two benchmark domains: Van der Waerden numbers and graceful graphs. By identifying internal symmetries we are able to extend the state of the art in both cases.

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2

Okino, Shinya, and Masato Nagata. "Asymmetric travelling waves in a square duct." Journal of Fluid Mechanics 693 (January 6, 2012): 57–68. http://dx.doi.org/10.1017/jfm.2011.455.

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AbstractTwo types of asymmetric solutions are found numerically in square-duct flow. They emerge through a symmetry-breaking bifurcation from the mirror-symmetric solutions discovered by Okino et al. (J. Fluid Mech., vol. 657, 2010, pp. 413–429). One of them is characterized by a pair of streamwise vortices and a low-speed streak localized near one of the sidewalls and retains the shift-and-reflect symmetry. The bifurcation nature as well as the flow structure of the solution show striking resemblance to those of the asymmetric solution in pipe flow found by Pringle & Kerswell (Phys. Rev. Lett., vol. 99, 2007, A074502), despite the geometrical difference between their cross-sections. The solution seems to be embedded in the edge state of square-duct flow identified by Biau & Bottaro (Phil. Trans. R. Soc. Lond. A, vol. 367, 2009, pp. 529–544). The other solution deviates slightly from the mirror-symmetric solution from which it bifurcates: the shift-and-rotate symmetry is retained but the mirror symmetry is broken.

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3

AMDJADI, FARIDON. "THE CALCULATION OF THE HOPF/HOPF MODE INTERACTION POINT IN PROBLEMS WITH Z2-SYMMETRY." International Journal of Bifurcation and Chaos 12, no. 08 (August 2002): 1925–35. http://dx.doi.org/10.1142/s0218127402005595.

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A direct method for finding the mode interaction point of a symmetry breaking Hopf bifurcation and a symmetry preserving Hopf bifurcation in problems with ℤ2-symmetry is developed. It has been shown that the mode interaction point corresponds to an isolated solution of an extended system. The existence of this solution relies on the occurrence of the mode interaction point and this is interpreted in the context of the mode interaction, using centre manifold reduction. A numerical example with the symmetry group O(2), which has a branch of ℤ2-symmetric nontrivial steady state solutions, is considered to provide clarification of the method.

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4

Perrin, Charles L. "Symmetry of hydrogen bonds in solution." Pure and Applied Chemistry 81, no. 4 (January 1, 2009): 571–83. http://dx.doi.org/10.1351/pac-con-08-08-14.

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A classic question regarding hydrogen bonds (H-bonds) concerns their symmetry. Is the hydrogen centered or is it closer to one donor and jumping between them? These possibilities correspond to single- and double-well potentials, respectively. The NMR method of isotopic perturbation can answer this question. It is illustrated with 3-hydroxy-2-phenylpropenal and then applied to dicarboxylate monoanions. The 18O-induced 13C NMR splittings signify that their intramolecular H-bonds are asymmetric and that each species is a pair of tautomers, not a single symmetric structure, even though maleate and phthalate are symmetric in crystals. The asymmetry is seen across a wide range of solvents and a wide variety of monoanions, including 2,3-di-tert-butylsuccinate and zwitterionic phthalates. Asymmetry is also seen in monoprotonated 1,8-bis(dimethylamino)naphthalenediamines, N,N'-diaryl-6-aminofulvene-2-aldimines, and 6-hydroxy-2-formylfulvene. The asymmetry is attributed to the disorder of the local environment, establishing an equilibrium between solvatomers. The broader implications of these results regarding the role of solvation in breaking symmetry are discussed. It was prudent to confirm a secondary deuterium isotope effect (IE) on amine basicity by NMR titration of a mixture of PhCH2NH2 and PhCHDNH2. The IE is of stereoelectronic origin. It is proposed that symmetric H-bonds can be observed in crystals but not in solution because a disordered environment induces asymmetry, whereas a crystal can guarantee a symmetric environment. The implications for the controversial role of low-barrier H-bonds in enzyme-catalyzed reactions are discussed.

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5

DZHUNUSHALIEV, V., H. J. SCHMIDT, and O. RURENKO. "SPHERICALLY SYMMETRIC SOLUTIONS IN MULTIDIMENSIONAL GRAVITY WITH THE SU(2) GAUGE GROUP AS THE EXTRA DIMENSIONS." International Journal of Modern Physics D 11, no. 05 (May 2002): 685–701. http://dx.doi.org/10.1142/s0218271802001925.

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The multidimensional gravity on the principal bundle with the SU(2) gauge group is considered. The numerical investigation of the spherically symmetric metrics with the center of symmetry is made. The solution of the gravitational equations depends on the boundary conditions of the "SU(2) gauge potential" (off-diagonal metric components) at the symmetry center and on the type of symmetry (symmetrical or antisymmetrical) of these potentials. In the chosen range of the boundary conditions it is shown that there are two types of solutions: wormhole-like and flux tube. The physical application of such kind of solutions as quantum handles in a spacetime foam is discussed.

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6

Fakhar, K., Zu-Chi Chen, and Xiaoda Ji. "Symmetry analysis of rotating fluid." ANZIAM Journal 47, no. 1 (July 2005): 65–74. http://dx.doi.org/10.1017/s1446181100009779.

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AbstractThe machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

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7

GAZZINI, MARITA, and ROBERTA MUSINA. "HARDY–SOBOLEV–MAZ'YA INEQUALITIES: SYMMETRY AND BREAKING SYMMETRY OF EXTREMAL FUNCTIONS." Communications in Contemporary Mathematics 11, no. 06 (December 2009): 993–1007. http://dx.doi.org/10.1142/s0219199709003636.

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Denote points in ℝk × ℝN - k as pairs ξ = (x,y), and assume 2 ≤ k < N. In this paper, we study the problem [Formula: see text] where [Formula: see text] and [Formula: see text], the Hardy constant. Our results are the following: (i) Let [Formula: see text]. Then there exists at least an entire cylindrically symmetric solution. (ii) Let [Formula: see text] and λ ≥ 0. Then any solution v ∈ Lp(ℝN;|x|-bdξ) is cylindrically symmetric. (iii) Let [Formula: see text] and [Formula: see text]. Then ground state solutions are not cylindrically symmetric, and therefore there exist at least two distinct entire solutions. We prove also similar results for the degenerate problem [Formula: see text] namely, for the Euler–Lagrange equations of the Maz'ya inequality with cylindrical weights.

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8

Kovalenko, M. D., I. V. Menshova, A. P. Kerzhaev, and T. D. Shulyakovskaya. "Exact and beam solutions for a narrow clamped rectangle." Journal of Physics: Conference Series 2231, no. 1 (April 1, 2022): 012027. http://dx.doi.org/10.1088/1742-6596/2231/1/012027.

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Abstract The paper presents formulas describing an exact solution to the boundary value problem of the theory of elasticity in a rectangle in which the horizontal sides are rigidly clamped, and normal and tangential stresses are given on the vertical ones. Only an odd-symmetric deformation of the rectangle with respect to the horizontal axis of symmetry and an even-symmetric deformation of the rectangle with respect to the vertical axis of symmetry are considered. The paper is based on the previously obtained solutions for a free half-strip and a free rectangle.

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9

Kovalenko, M. D., I. V. Menshova, A. P. Kerzhaev, and T. D. Shulyakovskaya. "Exact and beam solutions for a narrow clamped rectangle." Journal of Physics: Conference Series 2231, no. 1 (April 1, 2022): 012027. http://dx.doi.org/10.1088/1742-6596/2231/1/012027.

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Abstract The paper presents formulas describing an exact solution to the boundary value problem of the theory of elasticity in a rectangle in which the horizontal sides are rigidly clamped, and normal and tangential stresses are given on the vertical ones. Only an odd-symmetric deformation of the rectangle with respect to the horizontal axis of symmetry and an even-symmetric deformation of the rectangle with respect to the vertical axis of symmetry are considered. The paper is based on the previously obtained solutions for a free half-strip and a free rectangle.

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10

TEH, ROSY, and K. M. WONG. "MULTIMONOPOLE–ANTIMONOPOLE SOLUTIONS OF THE SU(2) YANG–MILLS–HIGGS FIELD THEORY." International Journal of Modern Physics A 19, no. 03 (January 30, 2004): 371–91. http://dx.doi.org/10.1142/s0217751x04017653.

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In this paper we constructed exact static multimonopole–antimonopole solutions of the YMH field theory. By labelling these solutions as A1, A2, B1, and B2, we notice that the exact axially symmetric 1-monopole — two antimonopoles solution is actually a special case of the A1 solution when the topological index parameter m=1. Also the B1 solution will reduce to a spherically symmetric Wu–Yang type monopole of unit charge when m=0. All these exact solutions satisfy the first order Bogomol'nyi equations and possess infinite energy. Hence they are a different type of the BPS solution. Except for the A1 solution when m=1 and the B1 solution when m=0, these solutions in general do not possess axial symmetry. They represent different combinations of monopoles, multimonopole, and antimonopoles, symmetrically arranged about the z-axis.

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11

SAGLAM, ISMAIL. "A SIMPLE AXIOMATIZATION OF THE EGALITARIAN SOLUTION." International Game Theory Review 16, no. 04 (December 2014): 1450008. http://dx.doi.org/10.1142/s021919891450008x.

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In this paper, we present a simple axiomatization of the n-person egalitarian solution. The single condition sufficient for characterization is a new axiom, called symmetric decomposability that combines the axioms of step-by-step negotiations, symmetry, and weak Pareto optimality used in an early characterization by Kalai [(1977) Proportional solutions to bargaining situations: Interpersonal utility comparisons, Econometrica45, 1623–1630].

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12

BRIHAYE, Y., and L. HONOREZ. "TWISTED SEMILOCAL STRINGS WITH TWO HIGGS-DOUBLETS." International Journal of Modern Physics A 23, no. 03n04 (February 10, 2008): 581–97. http://dx.doi.org/10.1142/s0217751x08038469.

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The standard electroweak model is extended by means of a second Brout–Englert–Higgs–doublet. The symmetry breaking potential is chosen in such a way that (i) the Lagrangian possesses a custodial symmetry, (ii) a stationary, axially symmetric ansatz of the bosonic fields consistently reduces the Euler–Lagrange equations to a set of differential equations. The potential involves, in particular, a direct interaction between the two doublets. Magnetic, stationary, axially-symmetric solutions of the classical equations are constructed. Some of them can be assimilated to embedded Nielsen–Olesen strings. From these solutions there are bifurcations and new solutions appear which exhibit the characteristics of the recently constructed twisted semilocal strings. A special emphasis is set on "doubly-twisted" solutions for which the two doublets present different time-dependent phase factors. They are regular and have a finite energy which can be lower than the energy of the embedded twisted solution. Electric-type solutions, such that the fields oscillate asymptotically far from the symmetry-axis, are also reported.

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13

Duanmu, M., K. Li, R. L. Horne, P. G. Kevrekidis, and N. Whitaker. "Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120171. http://dx.doi.org/10.1098/rsta.2012.0171.

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In the present work, we focus on the cases of two-site (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time ( ) symmetry, i.e. with a spatially odd gain–loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear -symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear -symmetric trimers, while both of them are absent in linear -symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear -symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of -symmetric quadrimers.

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14

ĐAPIĆ, N., M. KUNZINGER, and S. PILIPOVIĆ. "SYMMETRY GROUP ANALYSIS OF WEAK SOLUTIONS." Proceedings of the London Mathematical Society 84, no. 3 (April 29, 2002): 686–710. http://dx.doi.org/10.1112/s0024611502013436.

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Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

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15

Oz, I. Basaran, Y. Kucukakca, and N. Unal. "Anisotropic solution in phantom cosmology via Noether symmetry approach." Canadian Journal of Physics 96, no. 7 (July 2018): 677–80. http://dx.doi.org/10.1139/cjp-2017-0765.

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In this study, we consider a phantom cosmology in which a scalar field is minimally coupled to gravity. For anisotropic locally rotational symmetric (LRS) Bianchi type I space–time, we use the Noether symmetry approach to determine the potential of such a theory. It is shown that the potential must be in the trigonometric form as a function of the scalar field. We solved the field equations of the theory using the result obtained from the Noether symmetry. Our solution shows that the universe has an accelerating expanding phase.

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16

Fetisov, Dmitry A. "On Symmetric Solutions to Linear Matrix Time-Varying Differential Equations." Journal of Physics: Conference Series 2090, no. 1 (November 1, 2021): 012134. http://dx.doi.org/10.1088/1742-6596/2090/1/012134.

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Abstract In this paper, we discuss when the solution to the initial value problem for a linear matrix time-varying differential equation is symmetric on a given interval. By symmetry, we mean that the solution does not change when transposed. Throughout the paper, we assume that the equation has coefficients of finite order of smoothness. We demonstrate that, in order to verify whether the solution to the initial value problem is symmetric on a given interval, it can be useful to construct two matrix sequences associated to the equation. Using these sequences, we prove a sufficient condition for the solution symmetry on a given interval. Assuming that the initial value problem for a linear matrix time-varying differential equation satisfies this condition, we derive a formula for a symmetric solution to this problem.

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17

Iskandarova, Gulistan, and Dogan Kaya. "Symmetry solution on fractional equation." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 7, no. 3 (October 25, 2017): 255–59. http://dx.doi.org/10.11121/ijocta.01.2017.00498.

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As we know nearly all physical, chemical, and biological processes in nature can be described or modeled by dint of a differential equation or a system of differential equations, an integral equation or an integro-differential equation. The differential equations can be ordinary or partial, linear or nonlinear. So, we concentrate our attention in problem that can be presented in terms of a differential equation with fractional derivative. Our research in this work is to use symmetry transformation method and its analysis to search exact solutions to nonlinear fractional partial differential equations.

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18

Palatinus, L., and A. van der Lee. "Symmetry determination following structure solution inP1." Journal of Applied Crystallography 41, no. 6 (October 11, 2008): 975–84. http://dx.doi.org/10.1107/s0021889808028185.

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A new method for space-group determination is described. It is based on a symmetry analysis of the structure-factor phases resulting from a structure solution in space groupP1. The output of the symmetry analysis is a list of all symmetry operations compatible with the lattice. Each symmetry operation is assigned a symmetry agreement factor that is used to select the symmetry operations that are the elements of the space group of the structure. On the basis of the list of the selected operations the complete space group of the structure is constructed. The method is independent of the number of dimensions, and can also be used in solution of aperiodic structures. A number of cases are described where this method is particularly advantageous compared with the traditional symmetry analysis.

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19

Bor, Winny Chepngetich, Owino M. Oduor, and John K. Rotich. "A Lie Symmetry Solutions of Sawada-Kotera Equation." JOURNAL OF ADVANCES IN MATHEMATICS 17 (July 30, 2019): 1–11. http://dx.doi.org/10.24297/jam.v17i0.8364.

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In this article, the Lie Symmetry Analysis is applied in finding the symmetry solutions of the fifth order Sawada-Kotera equation. The technique is among the most powerful approaches currently used to achieveprecise solutions of the partial differential equations that are nonlinear. We systematically show the procedure to obtain the solution which is achieved by developing infinitesimal transformation, prolongations, infinitesimal generatorsand invariant transformations hence symmetry solutions of the fifth order Sawada-Kotera equation. Key Words- Lie symmetry analysis. Sawada-Kotera equation. Symmetry groups. Prolongations. Invariant solutions. Power series solutions. Symmetry solutions.

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20

Guo, Hongxia, Zongming Guo, and Fangshu Wan. "Radial symmetry of non-maximal entire solutions of a bi-harmonic equation with exponential nonlinearity." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 6 (January 24, 2019): 1603–25. http://dx.doi.org/10.1017/prm.2018.49.

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AbstractWe study radial symmetry of entire solutions of the equation0.1$$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solutionuof (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solutionuof (0.1) to be a non-maximal radial entire solution.

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21

Zavadskas, Edmundas Kazimieras, Jurgita Antucheviciene, and Zenonas Turskis. "Symmetric and Asymmetric Data in Solution Models." Symmetry 13, no. 6 (June 9, 2021): 1045. http://dx.doi.org/10.3390/sym13061045.

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This Special Issue covers symmetric and asymmetric data that occur in real-life problems. We invited authors to submit their theoretical or experimental research to present engineering and economic problem solution models that deal with symmetry or asymmetry of different data types. The Special Issue gained interest in the research community and received many submissions. After rigorous scientific evaluation by editors and reviewers, seventeen papers were accepted and published. The authors proposed different solution models, mainly covering uncertain data in multi-criteria decision-making problems as complex tools to balance the symmetry between goals, risks, and constraints to cope with the complicated problems in engineering or management. Therefore, we invite researchers interested in the topics to read the papers provided in the Special Issue.

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22

Thang, Bui Quyet, and Do Thi Huong. "A Spherically Symmetric Solution of \(R+\lambda R^2\) Gravity." Communications in Physics 24, no. 3S2 (September 24, 2014): 23–28. http://dx.doi.org/10.15625/0868-3166/24/3s2/4991.

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We shortly review the metric formalism for the \(f(R)\) gravity. Based on the metric formalism, westudy the spherically symmetric static empty space solutions with the gravity Lagrangian\(L=R+\lambda R^2\). We found the general metric that described the static empty space with thespherically symmetry. Our result is more general than Schwarzschild solution, specially thepredicted metric is perturbed Schwarzschild metric of the Einstein theory.

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23

Xia, Yarong, Ruoxia Yao, and Xiangpeng Xin. "Residual Symmetry, Bäcklund Transformation, and Soliton Solutions of the Higher-Order Broer-Kaup System." Advances in Mathematical Physics 2021 (May 18, 2021): 1–10. http://dx.doi.org/10.1155/2021/9975303.

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Under investigation in this paper is the higher-order Broer-Kaup(HBK) system, which describes the bidirectional propagation of long waves in shallow water. Via the standard truncated Painlevé expansion method, the residual symmetry of this system is derived. By introducing an appropriate auxiliary-dependent variable, the residual symmetry is successfully localized to Lie point symmetries. Via solving the initial value problems, the finite symmetry transformations are presented. However, the solution which obtained from the residual symmetry is a special group invariant solutions. In order to find more general solution of HBK system, we further generalize the residual symmetry method to the consistent tanh expansion (CTE) method and prove that the HBK system is CTE solvable, then the resonant soliton solutions and interaction solutions among different nonlinear excitations are obtained by the CET method.

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24

GOGBERASHVILI, MERAB. "TOPOLOGICAL SOLUTION TO THE CYLINDRICAL EINSTEIN–MAXWELL EQUATIONS." International Journal of Modern Physics D 18, no. 11 (November 15, 2009): 1765–71. http://dx.doi.org/10.1142/s0218271809015308.

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A new exact solution to the cylindrically symmetric Einstein–Maxwell equations is presented. The solution is singular on the axis of symmetry and at the radial infinity, where sources should be placed. The accepted source at the origin can be interpreted as a charged domain wall shell.

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25

CASANA, R. "RENORMALIZED NEW SOLUTIONS FOR THE MASSLESS THIRRING MODEL." International Journal of Modern Physics A 20, no. 30 (December 10, 2005): 7129–51. http://dx.doi.org/10.1142/s0217751x0502389x.

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We present a nonperturbative study of the (1+1)-dimensional massless Thirring model by using path integral methods. The regularization ambiguities — coming from the computation of the fermionic determinant — allow to find new solution types for the model. At quantum level the Ward identity for the 1PI 2-point function for the fermionic current separates such solutions in two phases or sectors, the first one has a local gauge symmetry that is implemented at quantum level and the other one without this symmetry. The symmetric phase is a new solution which is unrelated to the previous studies of the model and, in the nonsymmetric phase there are solutions that for some values of the ambiguity parameter are related to well-known solutions of the model. We construct the Schwinger–Dyson equations and the Ward identities. We make a detailed analysis of their UV divergence structure and, after, we perform a nonperturbative regularization and renormalization of the model.

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26

Chu, Chie-Ping, and Hwai-Chiuan Wang. "Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 1-2 (1992): 137–60. http://dx.doi.org/10.1017/s0308210500021016.

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SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

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27

Pan, Feng, Aoxue Li, Yingxin Wu, and J. P. Draayer. "An exact solution of the homogenous trimer Bose-Hubbard model." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 3 (March 1, 2023): 033101. http://dx.doi.org/10.1088/1742-5468/acb7ec.

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Abstract It is shown that the homogenous three-site (trimer) Bose–Hubbard model with a periodic boundary condition can be solved exactly by using an extended Bethe ansatz based on the solution of the dimer model and the site-permutation symmetry. A solution of the model within the S 3 symmetric or non-symmetric subspace is presented. Coupled differential equations of a series of extended Heine–Stieltjes polynomials, the zeros of which are related to the solution of the model, are derived. Numerical examples of the solution of the model with N ⩽ 4 bosons, including the related Heine–Stieltjes polynomials and the Van Vleck polynomials, are presented, which serve to validate the procedure and illustrate the completeness of the solutions they render.

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28

ASTON, P. J. "BIFURCATION AND CHAOS IN ITERATED MAPS WITH O(2) SYMMETRY." International Journal of Bifurcation and Chaos 05, no. 03 (June 1995): 701–24. http://dx.doi.org/10.1142/s0218127495000533.

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Iterated maps with O(2) symmetry are considered with a view to understanding transitions which occur as a parameter is varied. Bifurcations from the trivial solution are first considered followed by secondary bifurcations from nontrivial solutions. This is achieved by the derivation of a new system of equations in the orbit space. A transition in which a symmetric chaotic attractor starts drifting round the group orbit is also considered and it is shown that a single antisymmetric Lyapunov exponent determines whether or not a symmetric attractor is stable to nonsymmetric perturbations.

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29

Hayashi, Masahito, Kazuyasu Shigemoto, and Takuya Tsukioka. "The construction of the mKdV cyclic symmetric N-soliton solution by the Bäcklund transformation." Modern Physics Letters A 34, no. 18 (June 14, 2019): 1950136. http://dx.doi.org/10.1142/s0217732319501360.

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We study group theoretical structures of the mKdV equation. The Schwarzian-type mKdV equation has the global Möbius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special local Möbius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV Bäcklund transformation and can generate the mKdV cyclic symmetric N-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.

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30

Mariotti, Marco. "Maximal symmetry and the Nash solution." Social Choice and Welfare 17, no. 1 (January 2000): 45–53. http://dx.doi.org/10.1007/pl00007174.

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31

FEROZE, TOOBA, ASGHAR QADIR, and M. ZIAD. "UNIQUENESS OF THE McVITTIE SOLUTION AS PLANE SYMMETRIC SOURCELESS ELECTROMAGNETIC FIELD SPACETIMES." Modern Physics Letters A 23, no. 11 (April 10, 2008): 825–28. http://dx.doi.org/10.1142/s0217732308025437.

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It is proved that plane symmetric Lorentzian manifolds representing a sourceless electromagnetic field admit only a four-dimensional maximal symmetry group and consist of only the McVittie5 spacetime and its non-static analogue.4

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32

Türkog̃lu, Murat Metehan, and Melis Ulu Dog̃ru. "Conformal cylindrically symmetric spacetimes in modified gravity." Modern Physics Letters A 30, no. 37 (November 16, 2015): 1550202. http://dx.doi.org/10.1142/s0217732315502028.

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We investigate cylindrically symmetric spacetimes in the context of [Formula: see text] gravity. We firstly attain conformal symmetry of the cylindrically symmetric spacetime. We obtain solutions to use features of the conformal symmetry, field equations and their solutions for cylindrically symmetric spacetime filled with various cosmic matters such as vacuum state, perfect fluid, anisotropic fluid, massive scalar field and their combinations. With the vacuum state solutions, we show that source of the spacetime curvature is considered as Casimir effect. Casimir force for given spacetime is found using Wald’s axiomatic analysis. We expose that the Casimir force for Boulware, Hartle–Hawking and Unruh vacuum states could have attractive, repulsive and ineffective features. In the perfect fluid state, we show that matter form of the perfect fluid in given spacetime must only be dark energy. Also, we offer that potential of massive and massless scalar field are developed as an exact solution from the modified field equations. All solutions of field equations for vacuum case, perfect fluid and scalar field give a special [Formula: see text] function convenient to [Formula: see text]-CDM model. In addition to these solutions, we introduce conformal cylindrical symmetric solutions in the cases of different [Formula: see text] models. Finally, geometrical and physical results of the solutions are discussed.

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33

Naderifard, Azadeh, Elham Dastranj, and S. Reza Hejazi. "Exact solutions for time-fractional Fokker–Planck–Kolmogorov equation of Geometric Brownian motion via Lie point symmetries." International Journal of Financial Engineering 05, no. 02 (June 2018): 1850009. http://dx.doi.org/10.1142/s2424786318500093.

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In this paper, the transition joint probability density function of the solution of geometric Brownian motion (GBM) equation is obtained via Lie group theory of differential equations (DEs). Lie symmetry analysis is applied to find new solutions for time-fractional Fokker–Planck–Kolmogorov equation of GBM. This analysis classifies the forms of the solutions for the equation by the similarity variables arising from the symmetry operators. Finally, an analytic method called invariant subspace method is applied in order to find another exact solution.

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34

Konjik, Sanja. "Symmetries of conservation laws." Publications de l'Institut Math?matique (Belgrade) 77, no. 91 (2005): 29–51. http://dx.doi.org/10.2298/pim0591029k.

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We apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all project able symmetry groups and show that they are 1-strongly associated, hence transform existing solutions in the sense of 1-strong association into other solutions.

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35

Bénard, M., W. G. Laidlaw, and J. Paldus. "Hartree–Fock instabilities in the trisulphur–trinitride anion." Canadian Journal of Chemistry 63, no. 7 (July 1, 1985): 1797–802. http://dx.doi.org/10.1139/v85-300.

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Crystallographic evidence indicates that the structure of the trisulfur trinitride anion, S3N3−, is planar with equal SN bond lengths. Although the recent literature contains a number of molecular orbital calculations of D3h symmetry there are also calculations which indicate inequivalent nitrogens, suggesting broken symmetry. The results of an investigation of Hartree–Fock instabilities of the abinitio molecular orbital description of this system are reported. The D3h solution is indeed singlet stable but there are relatively close-lying C2v broken-symmetry solutions. In the broken-symmetry solutions the π electron network encompasses only five centres; the sixth centre, essentially free of π charge, may be either a sulphur or a nitrogen, thus yielding two distinct broken-symmetry solutions.

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36

Ibrahim, Rabha W., and Dumitru Baleanu. "Symmetry Breaking of a Time-2D Space Fractional Wave Equation in a Complex Domain." Axioms 10, no. 3 (June 30, 2021): 141. http://dx.doi.org/10.3390/axioms10030141.

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(1) Background: symmetry breaking (self-organized transformation of symmetric stats) is a global phenomenon that arises in an extensive diversity of essentially symmetric physical structures. We investigate the symmetry breaking of time-2D space fractional wave equation in a complex domain; (2) Methods: a fractional differential operator is used together with a symmetric operator to define a new fractional symmetric operator. Then by applying the new operator, we formulate a generalized time-2D space fractional wave equation. We shall utilize the two concepts: subordination and majorization to present our results; (3) Results: we obtain different formulas of analytic solutions using the geometric analysis. The solution suggests univalent (1-1) in the open unit disk. Moreover, under certain conditions, it was starlike and dominated by a chaotic function type sine. In addition, the authors formulated a fractional time wave equation by using the Atangana–Baleanu fractional operators in terms of the Riemann–Liouville and Caputo derivatives.

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37

López, L. A., Omar Pedraza, and V. E. Ceron. "Time-dependent solution from Myers–Perry." Canadian Journal of Physics 94, no. 2 (February 2016): 177–79. http://dx.doi.org/10.1139/cjp-2015-0354.

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We present a three-parameter time-dependent solution of the vacuum Einstein equations in five dimensions. The solution is obtained by applying the Wick rotation to the Myers–Perry solution that represents a rotating black hole in five dimensions. The new interpretation of the Myers–Perry solution can be considered among the generalized Einstein–Rosen type that can be interpreted as plane-symmetric waves, cylindrical waves or cosmological space–time in five dimensions. In some limits the solution has boost-rotational symmetry and it is asymptotically flat. In the case that the solution represents a cylindrical space–time, the E-energy is analyzed.

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38

Truong, Trong Tuong. "Function Reconstruction from Reflection Symmetric Radon Data." Symmetry 12, no. 6 (June 4, 2020): 956. http://dx.doi.org/10.3390/sym12060956.

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In many areas of two-dimensional imaging science, data acquisition arises as an integral process. The inverse process—or image reconstruction—means the solving of a Radon problem mathematically. It may happen that there exists classes of integral data which are mirror symmetric with respect to a line. Common sense suggests that the occurrence of a symmetry usually provides significant help in the search of the problem solution. Here, we showed an example of the contrary to this popular belief. In fact, to solve such a Radon problem with inherent reflection symmetry, there is a need to split it into two new Radon problems on half-spaces, which do not have solutions at hand. In this paper, a full solution is obtained via geometric inversion mapping of the two original half-spaces Radon problems to the disk interior/exterior Radon problem arising in recent modalities of Compton scattering tomography, which fortunately has explicit worked out inverse formulas.

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Wang, Guo, Xuelin Yong, Yehui Huang, and Jing Tian. "Symmetry, Pulson Solution, and Conservation Laws of the Holm-Hone Equation." Advances in Mathematical Physics 2019 (February 3, 2019): 1–6. http://dx.doi.org/10.1155/2019/4364108.

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In this paper, we focus on the Holm-Hone equation which is a fifth-order generalization of the Camassa-Holm equation. It was shown that this equation is not integrable due to the nonexistence of a suitable Lagrangian or bi-Hamiltonian structure and negative results from Painlevé analysis and the Wahlquist-Estabrook method. We mainly study its symmetry properties, travelling wave solutions, and conservation laws. The symmetry group and its one-dimensional optimal system are given. Furthermore, preliminary classifications of its symmetry reductions are investigated. Also we derive some solitary pattern solutions and nonanalytic first-order pulson solution via the ansatz-based method. Finally, some conservation laws for the fifth-order equation are presented.

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40

Naz, Rehana, Mohammad Danish Khan, and Imran Naeem. "Nonclassical Symmetry Analysis of Boundary Layer Equations." Journal of Applied Mathematics 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/938604.

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The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

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41

Alfaro, Jorge, and Pablo González. "δ Gravity: Dark Sector, Post-Newtonian Limit and Schwarzschild Solution." Universe 5, no. 5 (April 26, 2019): 96. http://dx.doi.org/10.3390/universe5050096.

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We present a new kind of model, which we call δ Theories, where standard theories are modified including new fields, motivated by an additional symmetry ( δ symmetry). In previous works, we proved that δ Theories just live at one loop, so the model in a quantum level can be interesting. In the gravitational case, we have δ Gravity, based on two symmetric tensors, g μ ν and g ˜ μ ν , where quantum corrections can be controlled. In this paper, a review of the classical limit of δ Gravity in a Cosmological level will be developed, where we explain the accelerated expansion of the universe without Dark Energy and the rotation velocity of galaxies by the Dark Matter effect. Additionally, we will introduce other phenomenon with δ Gravity like the deflection of the light produced by the sun, the perihelion precession, Black Holes and the Cosmological Inflation.

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FELMER, PATRICIO, and YING WANG. "RADIAL SYMMETRY OF POSITIVE SOLUTIONS TO EQUATIONS INVOLVING THE FRACTIONAL LAPLACIAN." Communications in Contemporary Mathematics 16, no. 01 (January 21, 2014): 1350023. http://dx.doi.org/10.1142/s0219199713500235.

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The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem [Formula: see text] where (-Δ)αdenotes the fractional Laplacian, α ∈ (0, 1), and B1denotes the open unit ball centered at the origin in ℝNwith N ≥ 2. The function f : [0, ∞) → ℝ is assumed to be locally Lipschitz continuous and g : B1→ ℝ is radially symmetric and decreasing in |x|. In the second place we consider radial symmetry of positive solutions for the equation [Formula: see text] with u decaying at infinity and f satisfying some extra hypothesis, but possibly being non-increasing.Our third goal is to consider radial symmetry of positive solutions for system of the form [Formula: see text] where α1, α2∈(0, 1), the functions f1and f2are locally Lipschitz continuous and increasing in [0, ∞), and the functions g1and g2are radially symmetric and decreasing. We prove our results through the method of moving planes, using the recently proved ABP estimates for the fractional Laplacian. We use a truncation technique to overcome the difficulty introduced by the non-local character of the differential operator in the application of the moving planes.

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43

GRASSI, V., R. A. LEO, G. SOLIANI, and L. SOLOMBRINO. "CONTINUOUS APPROXIMATION OF BINOMIAL LATTICES." International Journal of Modern Physics A 14, no. 15 (June 20, 1999): 2357–84. http://dx.doi.org/10.1142/s0217751x99001184.

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A systematic analysis of a continuous version of a binomial lattice, containing a real parameter γ and covering the Toda field equation as γ→∞, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow us to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi–Pasta–Ulam and the Killingbeck potentials, and others. An instantonlike approximate solution is also obtained which reproduces the Eguchi–Hanson instanton configuration for γ→∞. Furthermore, the equation under consideration is extended to n+1 dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these (n=4) enables us to establish a connection with the Euclidean Yang–Mills equations, another appears in the context of Differential Geometry in relation to the so-called Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation.

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Zhang, Wenbin, Jiangbo Zhou, and Sunil Kumar. "Symmetry Reduction, Exact Solutions, and Conservation Laws of the ZK-BBM Equation." ISRN Mathematical Physics 2012 (August 15, 2012): 1–9. http://dx.doi.org/10.5402/2012/837241.

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Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

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45

Perrin, Charles L., and Brian K. Ohta. "Symmetry of NHN hydrogen bonds in solution☆." Journal of Molecular Structure 644, no. 1-3 (January 2003): 1–12. http://dx.doi.org/10.1016/s0022-2860(02)00210-7.

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46

GOARD, JOANNA. "Finding symmetries by incorporating initial conditions as side conditions." European Journal of Applied Mathematics 19, no. 6 (December 2008): 701–15. http://dx.doi.org/10.1017/s0956792508007705.

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It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

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Aguayo-Ortiz, Alejandro, Emilio Tejeda, and X. Hernandez. "Choked accretion: from radial infall to bipolar outflows by breaking spherical symmetry." Monthly Notices of the Royal Astronomical Society 490, no. 4 (October 23, 2019): 5078–87. http://dx.doi.org/10.1093/mnras/stz2989.

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ABSTRACT Steady-state, spherically symmetric accretion flows are well understood in terms of the Bondi solution. Spherical symmetry, however, is necessarily an idealized approximation to reality. Here we explore the consequences of deviations away from spherical symmetry, first through a simple analytic model to motivate the physical processes involved, and then through hydrodynamical, numerical simulations of an ideal fluid accreting on to a Newtonian gravitating object. Specifically, we consider axisymmetric, large-scale, small-amplitude deviations in the density field such that the equatorial plane is overdense as compared to the polar regions. We find that the resulting polar density gradient dramatically alters the Bondi result and gives rise to steady-state solutions presenting bipolar outflows. As the density contrast increases, more and more material is ejected from the system, attaining speeds larger than the local escape velocities for even modest density contrasts. Interestingly, interior to the outflow region, the flow tends locally towards the Bondi solution, with a resulting total mass accretion rate through the inner boundary choking at a value very close to the corresponding Bondi one. Thus, the numerical experiments performed suggest the appearance of a maximum achievable accretion rate, with any extra material being ejected, even for very small departures from spherical symmetry.

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MOHAMMEDI, NOUREDDINE. "REMARKS ON REALIZING CLASSICAL AND QUANTUM W3 SYMMETRY." Modern Physics Letters A 06, no. 32 (October 20, 1991): 2977–84. http://dx.doi.org/10.1142/s0217732391003481.

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The relation between Jordan algebras and the nonlinear W3 algebra is explored quantum mechanically. Realization of classical W3 symmetry assumes the existence of some constant coefficients dijk (i,j,k = 1, … D) obeying some algebraic constraints. Recent works produced solutions to these constraints and established a link with Jordan algebras for the four special dimensions D = 5, 8, 14 and 26. In the present work we consider a general free field realization of quantum W3 and show that this relation with Jordan algebras breaks down at least for D = 5 and 8. We also present some general solutions to the dijk constraints for D = 2 and D = 3 cases. The D = 2 solution is then used in the free field construction and Fateev and Zamolodchikov's realization is obtained as a special case of this solution.

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Hu, Hengchun, and Xiaodan Li. "Nonlocal symmetry and interaction solutions for the new (3+1)-dimensional integrable Boussinesq equation." Mathematical Modelling of Natural Phenomena 17 (2022): 2. http://dx.doi.org/10.1051/mmnp/2022001.

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The nonlocal symmetry of the new (3+1)-dimensional Boussinesq equation is obtained with the truncated Painlevé method. The nonlocal symmetry can be localized to the Lie point symmetry for the prolonged system by introducing auxiliary dependent variables. The finite symmetry transformation related to the nonlocal symmetry of the integrable (3+1)-dimensional Boussinesq equation is studied. Meanwhile, the new (3+1)-dimensional Boussinesq equation is proved by the consistent tanh expansion method and many interaction solutions among solitons and other types of nonlinear excitations such as cnoidal periodic waves and resonant soliton solution are given.

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Munir, Mobeen, Muhammad Athar, Sakhi Sarwar, and Wasfi Shatanawi. "Lie symmetries of Generalized Equal Width wave equations." AIMS Mathematics 6, no. 11 (2021): 12148–65. http://dx.doi.org/10.3934/math.2021705.

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<abstract><p>Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.</p></abstract>

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