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Journal articles on the topic 'One-Parameter group of transformation'

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Published: 1 February 2025

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1

Al-Shomrani, M. M. "Lie Groups Analysis and Contact Transformations for Ito System." Abstract and Applied Analysis 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/342680.

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Generalized Ito systems of four coupled nonlinear evaluation equations are proposed. New classes of exact invariant solutions by using Lie group analysis are obtained. Moreover, we investigate the existence of a one-parameter group of contact transformations for a generalized Ito system. Consequently, we study the relationship between one-parameter group of a contact transformation and a one-parameter Lie point transformation for a generalized Ito system.

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2

Cieśliński, Jan L., and Dzianis Zhalukevich. "Spectral Parameter as a Group Parameter." Symmetry 14, no. 12 (December 6, 2022): 2577. http://dx.doi.org/10.3390/sym14122577.

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A large class of integrable non-linear partial differential equations is characterized by the existence of the associated linear problem (in the case of two independent variables, known as a Lax pair) containing the so-called spectral parameter. In this paper, we present and discuss the conjecture that the spectral parameter can be interpreted as the parameter of some one-parameter groups of transformation, provided that it cannot be removed by any gauge transformation. If a non-parametric linear problem for a non-linear system is known (e.g., the Gauss–Weingarten equations as a linear problem for the Gauss–Codazzi equations in the geometry of submanifolds), then, by comparing both symmetry groups, we can find or indicate the integrable cases. We consider both conventional Lie point symmetries and the so-called extended Lie point symmetries, which are necessary in some cases. This paper is intended to be a review, but some novel results are presented as well.

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3

Levi, Inessa. "Group closures of one-to-one transformations." Bulletin of the Australian Mathematical Society 64, no. 2 (October 2001): 177–88. http://dx.doi.org/10.1017/s000497270003985x.

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For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

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4

Polikanova, I. V. "Measuring the arcs of the orbit of a one-parameter transformation group." Sibirskie Elektronnye Matematicheskie Izvestiya 17 (November 12, 2020): 1823–48. http://dx.doi.org/10.33048/semi.2020.17.124.

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5

Xue-zi, Xu, and Chen Huai-yong. "Application of one-parameter groups of transformation in mechanics." Applied Mathematics and Mechanics 11, no. 7 (July 1990): 679–86. http://dx.doi.org/10.1007/bf02017483.

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6

Drozhzhinov, Yu N., and B. I. Zav’yalov. "Asymptotically homogeneous generalized functions along trajectories defined by a general one-parameter transformation group." Doklady Mathematics 82, no. 3 (December 2010): 874–77. http://dx.doi.org/10.1134/s1064562410060098.

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7

Liu, Ping, Senyue Lou, and Lei Peng. "Second-Order Approximate Equations of the Large-Scale Atmospheric Motion Equations and Symmetry Analysis for the Basic Equations of Atmospheric Motion." Symmetry 14, no. 8 (July 27, 2022): 1540. http://dx.doi.org/10.3390/sym14081540.

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In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and eight one-parameter group invariant solutions are demonstrated. Three types of nontrivial similarity solutions and group invariants are proposed. With the help of perturbation method, we derive the second-order approximate equations for the large-scale atmospheric motion equations, including the non-dimensional equations and the dimensional equations. The second-order approximate equations of the large-scale atmospheric motion equations not only show the characteristics of physical quantities changing with time, but also describe the characteristics of large-scale atmospheric vertical motion.

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8

Kulish, P. P. "A two-parameter quantum group and a gauge transformation." Journal of Mathematical Sciences 68, no. 2 (January 1994): 220–22. http://dx.doi.org/10.1007/bf01249335.

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9

Shilin, I. A., and Junesang Choi. "Concerning Transformations of Bases Associated with Unimodular diag(1, −1, −1)-Matrices." Axioms 13, no. 7 (July 4, 2024): 452. http://dx.doi.org/10.3390/axioms13070452.

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Considering a representation space for a group of unimodular diag(1, −1, −1)-matrices, we construct several bases whose elements are eigenfunctions of Casimir infinitesimal operators related to a reduction in the group to some one-parameter subgroups. Finding the kernels of base transformation integral operators in terms of special functions, we consider the compositions of some of these transformations. Since composition is a ‘closed’ operation on the set of base transformations, we obtain some integral relations for the special functions involved in the above kernels.

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10

Abd-el-Malek, Mina B., and Medhat M. Helala. "Steady Flow of an Electrically Conducting Incompressible Viscoelastic Fluid over a Heated Plate." Zeitschrift für Naturforschung A 60, no. 1-2 (February 1, 2005): 29–36. http://dx.doi.org/10.1515/zna-2005-1-205.

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The transformation group theoretic approach is applied to the problem of the flow of an electrically conducting incompressible viscoelastic fluid near the forward stagnation point of a heated plate. The application of one-parameter transformation group reduces the number of independent variables, by one, and consequently the basic equations governing flow and heat transfer are reduced to a set of ordinary differential equations. These equations have been solved approximately subject to the relevant boundary conditions by employing the shooting numerical technique. The effect of the magnetic parameter M, the Prandtl number Pr and the non-dimensional elastic parameter representing the non- Newtonian character of the fluid k on velocity field, shear stress, temperature distribution and heat flux are carefully examined.

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11

Hu, Chin-Kun. "Symmetry breaking in the two-parameter Kadanoff renormalization-group transformation." Physics Letters A 109, no. 1-2 (May 1985): 51–52. http://dx.doi.org/10.1016/0375-9601(85)90390-1.

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12

Kholopov, E. V. "Dependence of the renormalization group description of critical phenomena on the group parameter." Soviet Journal of Low Temperature Physics 11, no. 5 (May 1, 1985): 292–93. https://doi.org/10.1063/10.0031294.

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We analyze the renormalization group description of the critical behavior of a system as a function of the possible choices of the truncation parameter of the group. The variation of the critical indices with temperature, where the specific heat exponent may even change sign, is associated with a movement of the fixed point of the renormalization group transformation itself as a function of the group parameter.

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13

Cai, J. L., and F. X. Mei. "Conformal Invariance and Conserved Quantity of the Higher-Order Holonomic Systems by Lie Point Transformation." Journal of Mechanics 28, no. 3 (August 9, 2012): 589–96. http://dx.doi.org/10.1017/jmech.2012.67.

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AbstractIn this paper, the conformal invariance and conserved quantities for higher-order holonomic systems are studied. Firstly, by establishing the differential equation of motion for the systems and introducing a one-parameter infinitesimal transformation group together with its infinitesimal generator vector, the determining equation of conformal invariance for the systems are provided, and the conformal factors expression are deduced. Secondly, the relation between conformal invariance and the Lie symmetry by the infinitesimal one-parameter point transformation group for the higher-order holonomic systems are deduced. Thirdly, the conserved quantities of the systems are derived using the structure equation satisfied by the gauge function. Lastly, an example of a higher-order holonomic mechanical system is discussed to illustrate these results.

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14

Levi, Inessa. "On properties of group closures of one-to-one transformations." Journal of the Australian Mathematical Society 79, no. 2 (October 2005): 213–29. http://dx.doi.org/10.1017/s1446788700010454.

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AbstractFor a permutation group H on an infinite set X and a transformation f of X, let 〈f: H〉 = 〈{hfh-1:h є; H}〉 be a group closure of f. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation f of X to generate distinct group closures of f. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

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15

Abd-el-Malek, Mina B., Nagwa A. Badran, Amr M. Amin, and Anood M. Hanafy. "Lie Symmetry Group for Unsteady Free Convection Boundary-Layer Flow over a Vertical Surface." Symmetry 13, no. 2 (January 22, 2021): 175. http://dx.doi.org/10.3390/sym13020175.

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The Lie symmetry group transformation method was used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past to a non-isothermal vertical flat surface. The one-parameter Lie group transformation was applied twice consecutively to convert the motion governing equations into a system of ordinary differential equations. The obtained system of ordinary differential equations was solved numerically using the Lobatto IIIA formula (implicit Runge–Kutta method). The effect of the Prandtl number on the temperature and velocity profiles is illustrated graphically.

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16

Johnpillai, A. G., C. M. Khalique, and F. M. Mahomed. "Lie and Riccati Linearization of a Class of Liénard Type Equations." Journal of Applied Mathematics 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/171205.

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We construct a linearizing Riccati transformation by using an ansatz and a linearizing point transformation utilizing the Lie point symmetry generators for a three-parameter class of Liénard type nonlinear second-order ordinary differential equations. Since the class of equations also admits an eight-parameter Lie group of point transformations, we utilize the Lie-Tresse linearization theorem to obtain linearizing point transformations as well. The linearizing transformations are used to transform the underlying class of equations to linear third- and second-order ordinary differential equations, respectively. The general solution of this class of equations can then easily be obtained by integrating the linearized equations resulting from both of the linearization approaches. A comparison of the results deduced in this paper is made with the ones obtained by utilizing an approach of mapping the class of equations by a complex point transformation into the free particle equation. Moreover, we utilize the linearizing Riccati transformation to extend the underlying class of equations, and the Lie-Tresse linearization theorem is also used to verify the conditions of linearizability of this new class of equations.

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17

HIROSHIMA, F., and K. R. ITO. "LOCAL EXPONENTS AND INFINITESIMAL GENERATORS OF CANONICAL TRANSFORMATIONS ON BOSON FOCK SPACES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 07, no. 04 (December 2004): 547–71. http://dx.doi.org/10.1142/s0219025704001761.

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A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].

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18

Hanafy, Anood M., Mina B. Abd-el-Malek, and Nagwa A. Badran. "Study of Steady Natural Convective Laminar Fluid Flow over a Vertical Cylinder Using Lie Group Transformation." Symmetry 16, no. 12 (November 21, 2024): 1558. http://dx.doi.org/10.3390/sym16121558.

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Due to its critical importance in engineering applications, this study is motivated by the essential need to understand natural convection over a vertical cylinder with combined heat and mass transfer. Lie group symmetry transformations are used to analyze the thermal and velocity boundary layers of steady, naturally convective laminar fluid flow over the surface of a vertical cylinder. The one-parameter Lie group symmetry technique converts the system of governing equations into ordinary differential equations, which are then solved numerically using the implicit Runge–Kutta method. The effect of the Prandtl number, Schmidt number, and combined buoyancy ratio parameter on axial velocity, temperature, and concentration profiles are illustrated graphically. A specific range of parameter values was chosen to compare the obtained results with previous studies, demonstrating the accuracy of this method relative to others. The average Nusselt number and average Sherwood number are computed for various values of the Prandtl number Pr and Schmidt number Sc and presented in tables. It was found that the time required to reach a steady state for velocity and concentration profiles decreases as the Schmidt number Sc increases. Additionally, both temperature and concentration profiles decrease with an increase in the combined buoyancy ratio parameter N. Flow reversal and temperature defect with varying Prandtl numbers are also shown and discussed in detail.

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19

Nikonov, V. I. "The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 3 (September 6, 2018): 295–303. http://dx.doi.org/10.15507/2079-6900.20.201802.295-303.

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The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

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20

Nikonov, Vladimir I. "The application of Lie algebras and groups to the solution of problems of partial stability of dynamical systems." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 3 (September 6, 2018): 295–303. http://dx.doi.org/10.15507/2079-6900.20.201803.295-303.

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The article is devoted to the analysis of partial stability of nonlinear systems of ordinary differential equations using Lie algebras and groups. It is shown that the existence of a group of transformations invariant under partial stability in the system under study makes it possible to simplify the analysis of the partial stability of the initial system. For this it is necessary that the associated linear differential operator Lie in the enveloping Lie algebra of the original system, and the operator defined by the one-parameter Lie group is commutative with this operator. In this case, if the found group has invariance with respect to partial stability, then the resulting transformation performs to the decomposition of the system under study, and the partial stability problem reduces to the investigation of the selected subsystem. Finding the desired transformation uses the first integrals of the original system. Examples illustrating the proposed approach are given.

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21

MALIK, R. P. "WIGNER'S LITTLE GROUP AND BRST COHOMOLOGY FOR ONE-FORM ABELIAN GAUGE THEORY." International Journal of Modern Physics A 19, no. 16 (June 30, 2004): 2721–37. http://dx.doi.org/10.1142/s0217751x04018129.

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We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian density and establish their intimate connection with the translation subgroup T(2) of Wigner's little group for the free one-form Abelian gauge theory in four (3+1)-dimensions (4D) of space–time. Though the relationship between the usual gauge transformation for the Abelian massless gauge field and T(2) subgroup of the little group is quite well known, such a connection between the dual-gauge transformation and the little group is a new observation. The above connections are further elaborated and demonstrated in the framework of Becchi–Rouet–Stora–Tyutin (BRST) cohomology defined in the quantum Hilbert space of states where the Hodge decomposition theorem (HDT) plays a very decisive role.

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22

Günhan Ay, Nursena, and Emrullah Yaşar. "The residual symmetry, Bäcklund transformations, CRE integrability and interaction solutions: (2+1)-dimensional Chaffee–Infante equation." Communications in Theoretical Physics 75, no. 11 (November 1, 2023): 115004. http://dx.doi.org/10.1088/1572-9494/acf8b6.

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Abstract In this paper, we consider the (2+1)-dimensional Chaffee–Infante equation, which occurs in the fields of fluid dynamics, high-energy physics, electronic science etc. We build Bäcklund transformations and residual symmetries in nonlocal structure using the Painlevé truncated expansion approach. We use a prolonged system to localize these symmetries and establish the associated one-parameter Lie transformation group. In this transformation group, we deliver new exact solution profiles via the combination of various simple (seed and tangent hyperbolic form) exact solution structures. In this manner, we acquire an infinite amount of exact solution forms methodically. Furthermore, we demonstrate that the model may be integrated in terms of consistent Riccati expansion. Using the Maple symbolic program, we derive the exact solution forms of solitary-wave and soliton-cnoidal interaction. Through 3D and 2D illustrations, we observe the dynamic analysis of the acquired solution forms.

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23

Choksi, J. R., and G. Nadkarni. "The Group of Eigenvalues of a Rank One Transformation." Canadian Mathematical Bulletin 38, no. 1 (March 1, 1995): 42–54. http://dx.doi.org/10.4153/cmb-1995-006-6.

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AbstractIn this paper, several characterizations are given of the group of eigenvalues of a rank one transformation. One of these is intimately related to the corresponding expression for the maximal spectral type of a rank one transformation given in an earlier paper.

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24

Saito, Seiki, Hitoshi Nakajima, Masami Inaba, and Toshio Moriwake. "One-pot transformation of azido-group to N-(t-butoxycarbonyl)amino group." Tetrahedron Letters 30, no. 7 (January 1989): 837–38. http://dx.doi.org/10.1016/s0040-4039(01)80629-8.

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25

ZHANG, DA-JUN, JIE JI, and XIAN-LONG SUN. "CASORATIAN SOLUTIONS AND NEW SYMMETRIES OF THE DIFFERENTIAL-DIFFERENCE KADOMTSEV–PETVIASHVILI EQUATION." Modern Physics Letters B 23, no. 17 (July 10, 2009): 2107–14. http://dx.doi.org/10.1142/s0217984909020254.

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This paper first discusses the condition in which Casoratian entries satisfy for the differential-difference Kadomtsev–Petviashvili equation. Then from the Casoratian condition we find a transformation under which the differential-difference Kadomtsev–Petviashvili equation is invariant. The transformation, consisting of a combination of Galilean and scalar transformations, provides a single-parameter invariant group for the equation. We further derive the related symmetry, and the symmetry together with other two symmetries form a closed three-dimensional Lie algebra.

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26

ANIELLO, PAOLO. "PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS." International Journal of Geometric Methods in Modern Physics 02, no. 01 (February 2005): 111–25. http://dx.doi.org/10.1142/s0219887805000478.

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We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

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27

CARIÑENA, JOSÉ F., and ARTURO RAMOS. "GENERALIZED BÄCKLUND–DARBOUX TRANSFORMATIONS IN ONE-DIMENSIONAL QUANTUM MECHANICS." International Journal of Geometric Methods in Modern Physics 05, no. 04 (June 2008): 605–40. http://dx.doi.org/10.1142/s0219887808002989.

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We consider an action of the group of curves in GL(2,ℝ) on the set of linear systems and therefore on the set of Schrödinger equations in full similarity with the action of the group of curves in SL(2,ℝ) on the set of Riccati equations considered in previous articles. We also consider the transformations defined by a first-order differential expression which carry solutions of a Schrödinger equation into solutions of another one. We find then two non-trivial situations: transformations which can be described by the previous transformation group, generalizing previous work by us, and transformations which are singular. We show that both situations appear, e.g., in the usual problem of partner Hamiltonians in quantum mechanics. We show that the difference Bäcklund algorithm, both in the finite and confluent versions, can be understood in terms of the above mentioned transformation group, the case of two exactly equal factorization energies being an instance of the singular case. We apply the generalized theorem relating three eigenfunctions of three different Hamiltonians to the generation of new potentials with a known (excited state) eigenfunction, starting from potentials of Coulomb, Morse and Rosen–Morse type. The potentials found are new and non-trivial.

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28

Yürüsoy, M. "Investigation of Velocity Profile in Time Dependent Boundary Layer Flow of a Modified Power-Law Fluid of Fourth Grade." International Journal of Applied Mechanics and Engineering 25, no. 2 (June 1, 2020): 176–91. http://dx.doi.org/10.2478/ijame-2020-0028.

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AbstractThis paper deals with the investigation of time dependent boundary layer flow of a modified power-law fluid of fourth grade on a stretched surface with an injection or suction boundary condition. The fluid model is a mixture of fourth grade and power-law fluids in which the fluid may display shear thickening, shear thinning or normal stress textures. By using the scaling and translation transformations which is a type of Lie Group transformation, time dependent boundary layer equations are reduced into two alternative ordinary differential equations systems (ODEs) with boundary conditions. During this reduction, special Lie Group transformations are used for translation, scaling and combined transformation. Numerical solutions have been carried out for the ordinary differential equations for various fluids and boundary condition parameters. As a result of numerical analysis, it is observed that the boundary layer thickness decreases as the power-law index value increases. It was also observed that for the fourth-grade fluid parameter, as the parameter increases, the boundary layer thickness decreases while the velocity in the y direction increases.

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29

Chung, Dong Myung, Un Cig Ji, and Nobuaki Obata. "Transformations on white noise functions associated with second order differential operators of diagonal type." Nagoya Mathematical Journal 149 (March 1998): 173–92. http://dx.doi.org/10.1017/s0027763000006590.

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Abstract.A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.

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30

Korchagin, Anatoly B. "On birational monomial transformations of plane." International Journal of Mathematics and Mathematical Sciences 2004, no. 32 (2004): 1671–77. http://dx.doi.org/10.1155/s0161171204306514.

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We study birational monomial transformations of the formφ(x:y:z)=(ϵ1xα1yβ1zγ1:ϵ2xα2yβ2zγ2:xα3yβ3zγ3), whereϵ1,ϵ2∈{−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformationφ, we prove a formula, which represents the transformationφas a product of generators of the group. To prove this formula, we use birationally equivalent polynomialsAx+By+CandAxp+Byq+Cxrys. Ifφis the transformation which carries one polynomial onto another, then the integral powers of generators in the product, which represents the transformationφ, can be calculated by the expansion ofp/qin the continued fraction.

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31

Karp, Dmitrii, and Elena Prilepkina. "Transformations of the Hypergeometric 4F3 with One Unit Shift: A Group Theoretic Study." Mathematics 8, no. 11 (November 5, 2020): 1966. http://dx.doi.org/10.3390/math8111966.

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We study the group of transformations of 4F3 hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for 3F2 transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term 4F3 transformations from our group and three-term 3F2 transformations and present a method for computing the coefficients of the contiguous relations for 3F2 functions evaluated at unity. We further furnish a class of summation formulas associated with the elements of our group. In the appendix to this paper, we give a collection of Wolfram Mathematica® routines facilitating the group calculations.

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32

Papp, Zoltán. "On a continuous one parameter group of operator transformations on the field of Mikusinski operators." Publicationes Mathematicae Debrecen 24, no. 3-4 (July 1, 2022): 229–48. http://dx.doi.org/10.5486/pmd.1977.24.3-4.04.

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33

ALDAYA, V., M. CALIXTO, and E. SÁNCHEZ-SASTRE. "EXTENDING THE STUECKELBERG MODEL FOR SPACETIME SYMMETRIES: COSMOLOGICAL IMPLICATIONS." Modern Physics Letters A 21, no. 37 (December 7, 2006): 2813–25. http://dx.doi.org/10.1142/s0217732306021876.

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The so-called Stueckelberg transformation is introduced as a prescription of minimal coupling following the procedure of the Utiyama theory, for both internal and spacetime symmetry Lie groups. As a natural example, the theory is applied to the Weyl group and the corresponding gauge gravitational theory is developed. This context appears to be a natural source to account for some sort of dark matter intrinsically related to the gauge-group parameter associated with scale transformations.

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34

Flynn, Joshua. "Sharp Caffarelli–Kohn–Nirenberg-Type Inequalities on Carnot Groups." Advanced Nonlinear Studies 20, no. 1 (February 1, 2020): 95–111. http://dx.doi.org/10.1515/ans-2019-2065.

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AbstractThe main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups G (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case G is an Iwasawa group, it is shown here that the {L^{2}}-CKN inequalities are sharp for all parameter values except one exceptional case. To show this, generalized Kelvin transforms {K_{\sigma}} are introduced and shown to be isometries for certain weighted Sobolev spaces. An interesting transformation formula for the sub-Laplacian with respect to {K_{\sigma}} is also derived. Lastly, these techniques are shown to be valid for establishing CKN-type inequalities with monomial and horizontal norm weights.

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35

Nadjafikhah, Mehdi, Saeed Dodangeh, and Parastoo Kabi-Nejad. "On the Variational Problems without Having Desired Variational Symmetries." Journal of Mathematics 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/685212.

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We will have an attempt to present a method for constructing variational problems without having a desired one-parameter transformation group as a variational symmetry. For this, we use the notation ofμ-symmetry which was introduced by Giuseppe Gaeta and Paola Morando in 2004. Moreover, our given method enabled us to solve those constructed variational problems usingμ-symmetries.

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36

Missarov, Mukadas Dmukhtasibovich, and Dmitrii Airatovich Khajrullin. "The renormalization group transformation in the generalized fermionic hierarchical model." Izvestiya: Mathematics 87, no. 5 (2023): 1011–23. http://dx.doi.org/10.4213/im9371e.

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We consider a two-dimensional hierarchical lattice in which the vertices of a square represent an elementary cell. In the generalized hierarchical model, the distance between opposite vertices of a square differs from that between adjacent vertices and is a parameter of the new model. The Gaussian part of the Hamiltonian of the 4-component generalized fermionic hierarchical model is invariant under the block-spin renormalization group transformation. The transformation of the renormalization group in the space of coefficients, which specify the Grassmann-valued density of the free measure, is explicitly calculated as a homogeneous mapping of degree four in the two-dimensional projective space.

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Aldaya, Víctor, Manuel Calixto, and Miguel Navarro. "The Electromagnetic and Proca Fields Revisited: A Unified Quantization." International Journal of Modern Physics A 12, no. 20 (August 10, 1997): 3609–23. http://dx.doi.org/10.1142/s0217751x97001869.

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Quantizing the electromagnetic field with a group formalism faces the difficulty of how to turn the traditional gauge transformation of the vector potential, Aμ(x) → Aμ(x) + ∂μ φ (x), into a group law. In this paper, it is shown that the problem can be solved by looking at gauge transformations in a slightly different manner which, in addition, does not require introducing any BRST-like parameter. This gauge transformation does not appear explicitly in the group law of the symmetry but rather as the trajectories associated with generalized equations of motion generated by vector fields with null Noether invariants. In the new approach the parameters of the local group, U (1)(x,t), acquire dynamical content outside the photon mass shell, a fact which also allows a unified quantization of both the electromagnetic and Proca fields.

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Salzmann, Helmut. "Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV." Canadian Journal of Mathematics 39, no. 4 (August 1, 1987): 908–19. http://dx.doi.org/10.4153/cjm-1987-045-4.

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Let be a topological projective plane with compact point set P of finite (covering) dimension. In the compact-open topology (of uniform convergence), the group Σ of continuous collineations of is a locally compact transformation group of P.THEOREM. If dim Σ > 40, thenis isomorphic to the Moufang plane 6 over the real octonions (and dim Σ = 78).By [3] the translation planes with dim Σ = 40 form a one-parameter family and have Lenz type V. Presumably, there are no other planes with dim Σ = 40, cp. [17].

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Shalom, Yehuda. "Rigidity, Unitary Representations of Semisimple Groups, and Fundamental Groups of Manifolds with Rank One Transformation Group." Annals of Mathematics 152, no. 1 (July 2000): 113. http://dx.doi.org/10.2307/2661380.

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Khalif, Muhammad Ardhi. "Lorentz Group Action on Ellips Space." Journal Of Natural Sciences And Mathematics Research 1, no. 2 (August 22, 2017): 55. http://dx.doi.org/10.21580/jnsmr.2015.1.2.1602.

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<p style="text-align: justify;">The ellips space <em>E </em>has been constructed as cartesian product R+ <em>× </em>R+ <em>× </em>[ <em>π </em>2 <em>, </em><em>π </em>2 ]. Its elements, (<em>a, b, θ</em>), is called as an ellipse with eccentricity is <em> </em>= p1 <em>− </em><em>b</em>2<em>/a</em>2 if <em>b < a </em>and is <em> </em>= p1 <em>− </em><em>a</em>2<em>/b</em>2 if <em>a > b</em>. The points (<em>a, b, π/</em>2) is equal to (<em>b, a, </em>0). The action of subgrup <em>SO</em><em>oz</em>(3<em>, </em>1) of Lorentz group <em>SO</em><em>o</em>(3<em>, </em>1), containing Lorentz transformations on <em>x</em><em>−</em><em>y </em>plane and rotations about <em>z </em>axes, on <em>E </em>is defined as Lorentz transformation or rotation transformation of points in an ellipse. The action is effective since there are no rigid points in <em>E</em>. The action is also not free and transitive. These properties means that Lorentz transformations change any ellips into another ellips. Although mathematically we can move from an ellipse to another one with the bigger eccentrity but it is imposible physically. This is occured because we donot include the speed parameter into the definition of an ellipse in <em>E</em>.</p>

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Popov, A. A. "Information characteristics of scalar random fields, invariant with regard to group of their one-to-one transformation group." Radioelectronics and Communications Systems 52, no. 11 (November 2009): 618–27. http://dx.doi.org/10.3103/s0735272709110065.

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42

Bobrova, Maria V. "CONTEMPORARY RURAL ZOONYMICON IN THE DERIVATIONAL ASPECT (on the Material of Zoonyms of One Group of Villages)." Вестник Пермского университета. Российская и зарубежная филология 13, no. 2 (2021): 5–13. http://dx.doi.org/10.17072/2073-6681-2021-2-5-13.

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The article is devoted to the study of zoonyms functioning in the speech of the inhabitants of Troel’ga rural settlement, Kungur district (Perm Krai). Methodologically, animal names are considered in the article in line with the theory of derivation, that is, as a result of dynamic processes at different levels of the language system. It is necessary to distinguish between nicknames that appeared in the course of zoonymic transformations and those that appeared due to transformations of ready-made lexical means (products of pre-zoonymic transformations). We have found that the first ones form as a result of six types of derivation: word-forming derivation (with the formation of words that are absent in the literary language), lexical and word-forming derivation (with the formation of words that are homonymous to the words of the literary language), lexical derivation (with the use of non-derived words that are absent in the literary language: neologisms and barbarisms), lexical-semantic derivation (with the reinterpretation of the semantics of the generating word), lexical-grammatical derivation (with the functional transformation of the generating word), morphological derivation (with the grammatical transformation of the generating word). The words of the second group are included in the zoonymicon through lexical derivation (using derived and non-derived words of the literary language), lexical-semantic derivation (with semantic transformation of all-Russian words), morphological-syntactic derivation (with a change of the part of speech of all-Russian words). Within these types, certain derivational models are implemented, in particular k-suffixation, word convergence based on paronomasia, onymization and transonymization, substantivization, etc. The paper provides a conclusion about a variety of ways of forming modern zoonymicon, about the specificity of some particular derivational models for the given sample.

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43

Yong, Liu. "INVARIANT MEASURES FOR THE SINGLE–PARAMETER TRANSFORMATION GROUP WITH COMPACT ORBITS ON BANACH SPACES." Acta Mathematica Scientia 8, no. 1 (March 1988): 79–84. http://dx.doi.org/10.1016/s0252-9602(18)30478-8.

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44

Garat, Alcides. "Signature-causality reflection generated by Abelian gauge transformations." Modern Physics Letters A 35, no. 15 (March 31, 2020): 2050119. http://dx.doi.org/10.1142/s0217732320501199.

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In this paper, we want to better understand the causality reflection that arises under a subset of Abelian local gauge transformations in geometrodynamics. We proved in previous papers that in Einstein–Maxwell spacetimes, there exist two local orthogonal planes of gauge symmetry at every spacetime point for non-null electromagnetic fields. Every vector in these planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. The vectors that span these local orthogonal planes are dependent on electromagnetic gauge. The local group of Abelian electromagnetic gauge transformations has been proved isomorphic to the local groups of tetrad transformations in these planes. We called LB1 the local group of tetrad transformations made up of SO(1, 1) plus two different kinds of discrete transformations. One of the discrete transformations is the full inversion two by two which is a Lorentz transformation. The other discrete transformation is given by a matrix with zeroes on the diagonal and ones off-diagonal two by two, a reflection. The group LB1 is realized on this plane, we call this plane one, and is spanned by the time-like and one space-like vectors. The other local orthogonal plane is plane two and the local group of tetrad transformations, we call this LB2, which is just SO(2). The local group of Abelian electromagnetic gauge transformations is isomorphic to both LB1 and LB2, independently. It has already been proved that a subset of local electromagnetic gauge transformations that leave the electromagnetic tensor invariant induces a change in sign in the norm of the tetrad vectors that span the local plane one. The reason is that one of the discrete transformations on the local plane one that belongs to the group LB1 is not a Lorentz transformation, it is a flip or reflection. It is precisely on this kind of discrete transformation that we have an interest since it has the effect of changing the signature and the causality. This effect has never been noticed before.

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SUN, C. C., B. F. LI, Z. S. LI, H. X. ZHANG, and X. R. HUANG. "THE PARAMETER-DEPENDENT FERMION STATES FOR MOLECULES WITH SYMPLECTIC SYMMETRY." Journal of Theoretical and Computational Chemistry 05, no. 04 (December 2006): 779–99. http://dx.doi.org/10.1142/s0219633606002647.

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Under a certain kind of similarity transformation, a parameter-dependent (abbreviated as PD) symplectic group chain Sp(2M) ⊃ Sp(2M - 2) ⊃ ⋯ ⊃ Sp(2) that is characterized by a set of pairing parameters is introduced to build up the PD antisymmetrized fermion states for molecules with symplectic symmetry, and these states will be useful in carrying out the optimization procedure in quantum chemistry. In order to make a complete classification of the states, a generalized branching rule associated with the symplectic group chain is proposed. Further, we are led to the result that the explicit form of the PD antisymmetrized fermion states is obtained in terms of M one-particle operators and M geminal operators.

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Ballesteros, Angel, Flaminia Giacomini, and Giulia Gubitosi. "The group structure of dynamical transformations between quantum reference frames." Quantum 5 (June 8, 2021): 470. http://dx.doi.org/10.22331/q-2021-06-08-470.

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Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.

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47

Abraham‐Shrauner, B. "Lie transformation group solutions of the nonlinear one‐dimensional Vlasov equation." Journal of Mathematical Physics 26, no. 6 (June 1985): 1428–35. http://dx.doi.org/10.1063/1.526964.

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48

Alessa, Nazek. "Transformation Magnetohydrodynamics in Presence of a Channel Filled with Porous Medium and Heat Transfer of Non-Newtonian Fluid by Using Lie Group Transformations." Journal of Function Spaces 2020 (October 22, 2020): 1–6. http://dx.doi.org/10.1155/2020/8840287.

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In this paper, the numerical results are presented by using Lie group transformations, to be more efficient and sophisticated. To solve various fluid dynamic problems numerically, we present the numerical results in a field of velocity and distribution of temperature for different parameters regarding the problem of radiative heat, a magnetohydrodynamics, and non-Newtonian viscoelasticity for the unstable flow of optically thin fluid inside a channel filled with nonuniform wall temperature and saturated porous medium, including Hartmann number, porous medium and frequency parameter, and radiation parameter, with a comparison of the corresponding flow problems for a Newtonian fluid. Moreover, the effects of the pertinent parameters on the friction coefficient of skin and local Nusselt number were discussed numerically and also illustrate that graphically.

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49

Abdullah, Enas. "Syndetic proximal set in topological transformation group." Journal of Kufa for Mathematics and Computer 10, no. 2 (August 31, 2023): 38–44. http://dx.doi.org/10.31642/jokmc/2018/100206.

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In this paper will study the syndetic set in topological transformation group. It represents one of the most important sets in topological dynamics that Gottschak referred .We Introduce dynamical relationship between syndetic set and translation (left-right) ,In topological dynamics group present semi replet set and extensive set .We referred to the relationship of semi replet set and extensive set on the one hand, and their relationship to syndetic set on the other hand . We introduced a new concept syndetic proximal point that depended on syndetic set we indicated a relationship syndetic proximal point with proximal point and extensive proximal point , conclude a number of relationships and concepts

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Podoksënov, M. N. "A lorentz manifold with a group of conformal transformations possessing a normal one-parameter subgroup of homotheties." Siberian Mathematical Journal 38, no. 6 (December 1997): 1178–81. http://dx.doi.org/10.1007/bf02675943.

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