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Journal articles on the topic 'Transformation de Möbius'

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Published: 4 June 2021

Last updated: 2 February 2022

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1

Wang, Changping. "Surfaces in Möbius geometry." Nagoya Mathematical Journal 125 (March 1992): 53–72. http://dx.doi.org/10.1017/s0027763000003895.

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Our purpose in this paper is to give a basic theory of Möbius differential geometay. In such geometry we study the properties of hypersurfaces in unit sphere Sn which are invariant under the Möbius transformation group on Sn.Since any Möbius transformation takes oriented spheres in Sn to oriented spheres, we can regard the Möbius transformation group Gn as a subgroup MGn of the Lie transformation group on the unit tangent bundle USn of Sn. Furthermore, we can represent the immersed hypersurfaces in Sn by a class of Lie geometry hypersurfaces (cf. [9]) called Möbius hypersurfaces. Thus we can use the concepts and the techniques in Lie sphere geometry developed by U. Pinkall ([8], [9]), T. Cecil and S. S. Chern [2] to study the Möbius differential geometry.

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2

Lee, Sunhong, Hyun Chol Lee, Mi Ran Lee, Seungpil Jeong, and Gwang-Il Kim. "Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics." Abstract and Applied Analysis 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/560246.

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We present an algorithm forC1Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solveC1Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Möbius transformation, we can introduce anextra parameterdetermined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex planeℂ. Möbius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.

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3

Breaz, Nicoleta, Daniel Breaz, and Shigeyoshi Owa. "Fractional Calculus of Analytic Functions Concerned with Möbius Transformations." Journal of Function Spaces 2016 (2016): 1–9. http://dx.doi.org/10.1155/2016/6086409.

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LetAbe the class of functionsf(z)in the open unit diskUwithf(0)=0andf′(0)=1. Also, letw(ζ)be a Möbius transformation inUfor somez∈U. Applying the Möbius transformations, we consider some properties of fractional calculus (fractional derivatives and fractional integrals) off(z)∈A. Also, some interesting examples for fractional calculus are given.

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4

Piirainen, Reijo. "Möbius transformation and conformal relativity." Foundations of Physics 26, no. 2 (February 1996): 223–42. http://dx.doi.org/10.1007/bf02058086.

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5

Mork, Leah K., and Darin J. Ulness. "Visualization of Mandelbrot and Julia Sets of Möbius Transformations." Fractal and Fractional 5, no. 3 (July 17, 2021): 73. http://dx.doi.org/10.3390/fractalfract5030073.

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This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

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6

McCullagh, Peter. "Möbius transformation and Cauchy parameter estimation." Annals of Statistics 24, no. 2 (April 1996): 787–808. http://dx.doi.org/10.1214/aos/1032894465.

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7

Xinhua, JI. "Möbius transformation and degenerate hyperbolic equation." Advances in Applied Clifford Algebras 11, S2 (June 2001): 155–75. http://dx.doi.org/10.1007/bf03219129.

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8

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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9

Hu, Zejun, and Haizhong Li. "Classification of Möbius Isoparametric Hypersurfaces in 4." Nagoya Mathematical Journal 179 (2005): 147–62. http://dx.doi.org/10.1017/s0027763000025629.

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AbstractLet Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.

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10

Akbas, M., and D. Singerman. "The normalizer of Γ0(N) in PSL(2, ℝ)." Glasgow Mathematical Journal 32, no. 3 (September 1990): 317–27. http://dx.doi.org/10.1017/s001708950000940x.

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Let Γ denote the modular group, consisting of the Möbius transformationsAs usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of indexthe product being taken over all prime divisors of N.

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11

Li, Feng Jiang, and Jian Bo Fang. "Complete hypersurfaces with constant Möbius scalar curvature." International Journal of Mathematics 27, no. 08 (July 2016): 1650063. http://dx.doi.org/10.1142/s0129167x16500634.

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Let [Formula: see text] be an umbilical free hypersurface in the unit sphere [Formula: see text]. Four basic invariants of [Formula: see text], under the Möbius transformation group of [Formula: see text] are the Möbius metric [Formula: see text], the Möbius second fundamental form [Formula: see text], the Blaschke tensor [Formula: see text] and the Möbius form [Formula: see text]. In this paper, we study complete hypersurfaces with constant normalized Möbius scalar curvature [Formula: see text] and vanishing Möbius form [Formula: see text]. By computing the Laplacian of the funtion [Formula: see text], where the trace-free Blaschke tensor [Formula: see text], and applying the well known generalized maximum principle of Omori–Yau, we obtain the following result: [Formula: see text] must be either Möbius equivalent to a minimal hypersurface with constant Möbius scalar curvature, when [Formula: see text]; [Formula: see text] in [Formula: see text], when [Formula: see text]; the pre-image of the stereographic projection [Formula: see text] of the circular cylinder [Formula: see text] in [Formula: see text], when [Formula: see text]; or the pre-image of the projection [Formula: see text] of the hypersurface [Formula: see text] in [Formula: see text], when [Formula: see text].

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12

Kim, Yongsam, Ming-Chih Lai, and Yunchang Seol. "Simulation of a Soap Film Möbius Strip Transformation." East Asian Journal on Applied Mathematics 7, no. 3 (August 2017): 615–28. http://dx.doi.org/10.4208/eajam.070217.120617a.

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AbstractIf the closed wire frame of a soap film having the shape of a Möbius strip is pulled apart and gradually deformed into a planar circle, the soap film transforms into a two-sided orientable surface. In the presence of a finite-time twist singularity, which changes the linking number of the film's Plateau border and the centreline of the wire, the topological transformation involves the collapse of the film toward the wire. In contrast to experimental studies of this process reported elsewhere, we use a numerical approach based on the immersed boundary method, which treats the soap film as a massless membrane in a Navier-Stokes fluid. In addition to known effects, we discover vibrating motions of the film arising after the topological change is completed, similar to the vibration of a circular membrane.

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13

Wang, Min-zhen, and Kunio Shimizu. "On applying Möbius transformation to cardioid random variables." Statistical Methodology 9, no. 6 (November 2012): 604–14. http://dx.doi.org/10.1016/j.stamet.2012.04.001.

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14

Kato, Shogo, and Arthur Pewsey. "A Möbius transformation-induced distribution on the torus." Biometrika 102, no. 2 (March 19, 2015): 359–70. http://dx.doi.org/10.1093/biomet/asv003.

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15

Silvestrov, Sergei D., and Hans Wallin. "Representations of Algebras Associated with a Möbius Transformation." Journal of Nonlinear Mathematical Physics 3, no. 1-2 (January 1996): 202–13. http://dx.doi.org/10.2991/jnmp.1996.3.1-2.24.

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16

Ganenkova, E. G., and V. V. Starkov. "The Möbius Transformation and Smirnov’s Inequality for Polynomials." Mathematical Notes 105, no. 1-2 (January 2019): 216–26. http://dx.doi.org/10.1134/s0001434619010243.

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17

Giardino, Sergio. "Möbius Transformation for Left-Derivative Quaternion Holomorphic Functions." Advances in Applied Clifford Algebras 27, no. 2 (April 29, 2016): 1161–73. http://dx.doi.org/10.1007/s00006-016-0673-y.

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18

LI, BAOKUI, and GUOWU YAO. "On characterizations of sphere-preserving maps." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 2 (September 2009): 439–46. http://dx.doi.org/10.1017/s0305004109002291.

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AbstractRecently, the first author and Y. Wang proved that $f\colon \rn\to\rn$ (n ≥ 2) is a Möbius transformation if and only if f is a non-degenerate circle-preserving map. In this paper, we will further the result to show that f is a Möbius transformation if and only if f is a non-degenerate r–dimensional sphere-preserving map. The versions for the Euclidean and hyperbolic cases are also obtained. These results make no surjectivity or injectivity or even continuity assumptions on f. Moreover, certain degenerate sphere-preserving maps are given, which completes the characterizations of sphere-preserving maps.

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19

Cao, Hongzhe. "Two meromorphic functions on annuli sharing some pairs of small functions or values." AIMS Mathematics 6, no. 12 (2021): 13311–26. http://dx.doi.org/10.3934/math.2021770.

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<abstract><p>In this paper, we prove that two admissible meromorphic functions on an annulus must be linked by a quasi-Möbius transformation if they share some pairs of small function with multiplicities truncated by $ 4 $. We also give the representation of Möbius transformation between two admissible meromorphic functions on an annulus if they share four pairs of values with multiplicities truncated by $ 4 $. In our results, the zeros with multiplicities more than a certain number are not needed to be counted if their multiplicities are bigger than a certain number.</p></abstract>

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20

Lester, J. A. "A Beckman-Quarles Type Theorem for Coxeter's Inversive Distance." Canadian Mathematical Bulletin 34, no. 4 (December 1, 1991): 492–98. http://dx.doi.org/10.4153/cmb-1991-079-6.

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AbstractWe prove that a bijective transformation on the set of circles in the real inversive plane which preserves pairs of circles a fixed inversive distance ρ > 0 apart must be induced by a Möbius transformation.

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21

Tenzer, Robert, and Vladislav Gladkikh. "Application of Möbius coordinate transformation in evaluating Newton's integral." Contributions to Geophysics and Geodesy 41, no. 2 (January 1, 2011): 95–115. http://dx.doi.org/10.2478/v10126-011-0004-1.

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Application of Möbius coordinate transformation in evaluating Newton's integralWe propose a numerical scheme which efficiently combines various existing methods of solving the Newton's volume integral. It utilises the analytical solution of Newton's integral for tesseroid in computing the near-zone contribution to gravitational field quantities (potential and its first radial derivative). The far-zone gravitational contribution is computed using the expressions derived based on applying Molodensky's truncation coefficients to a spectral representation of Newton's integral. The weak singularity of Newton's integral is treated analytically using formulas for the gravitational contribution of the cylindrical mass volume centered with respect to the observation point. All three solutions are defined and evaluated in the system of polar spherical coordinates. A conversion of the geographical to polar spherical coordinates of input data sets (digital terrain and density models) is based on the Möbius transformation with an enhanced integration grid resolution at vicinity of the observation point.

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22

QUANG, SI DUC, and LE NGOC QUYNH. "TWO MEROMORPHIC FUNCTIONS SHARING SOME PAIRS OF SMALL FUNCTIONS REGARDLESS OF MULTIPLICITIES." International Journal of Mathematics 25, no. 02 (February 2014): 1450014. http://dx.doi.org/10.1142/s0129167x14500141.

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In this paper, we prove that two meromorphic functions f and g must be linked by a quasi-Möbius transformation if they share a pair of small functions ignoring multiplicities and share other four pairs of small functions with multiplicities truncated by 2. We also show that two meromorphic functions which share q (q ≥ 6) pairs of small functions ignoring multiplicities are linked by a quasi-Möbius transformation. Moreover, in our results, the zeros with multiplicities more than a certain number are not needed to be counted in the condition sharing pairs of small functions of meromorphic functions. These results are generalization and improvements of some recent results.

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23

Moorfield, James, Song Wang, Wencheng Yang, Aseel Bedari, and Peter Van Der Kamp. "A Möbius transformation based model for fingerprint minutiae variations." Pattern Recognition 98 (February 2020): 107054. http://dx.doi.org/10.1016/j.patcog.2019.107054.

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24

Ungar, A. A. "Seeing the Möbius disc-transformation group like never before." Computers & Mathematics with Applications 45, no. 4-5 (February 2003): 805–22. http://dx.doi.org/10.1016/s0898-1221(03)00042-7.

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25

SAKIHARA, Sonshu, Masaru TAKANA, Naoki SAKAI, and Takashi OHIRA. "Power Dependent Impedance Measurement Exploiting an Oscilloscope and Möbius Transformation." IEICE Transactions on Electronics E100.C, no. 10 (2017): 918–23. http://dx.doi.org/10.1587/transele.e100.c.918.

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26

Zhang, Jun. "Characterizing projective geometry of binocular visual space by Möbius transformation." Journal of Mathematical Psychology 88 (February 2019): 15–26. http://dx.doi.org/10.1016/j.jmp.2018.10.007.

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27

Dixit, Atul. "Analogues of the general theta transformation formula." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 143, no. 2 (March 18, 2013): 371–99. http://dx.doi.org/10.1017/s0308210511001685.

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A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and Littlewood involving infinite series of the Möbius function.

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28

Grabisch, Michel. "Alternative Representations of Discrete Fuzzy Measures for Decision Making." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 05, no. 05 (October 1997): 587–607. http://dx.doi.org/10.1142/s0218488597000440.

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This paper introduces three different representations of fuzzy measures, through the Möbius transformation, and the expression of importance and interaction. This leads naturally to the concept of k-order additive measures. It is shown how these concepts can be used in decision making, especially multicriteria evaluation.

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29

Ares, Filiberto, José G. Esteve, Fernando Falceto, and Amilcar R. de Queiroz. "On the Möbius transformation in the entanglement entropy of fermionic chains." Journal of Statistical Mechanics: Theory and Experiment 2016, no. 4 (April 28, 2016): 043106. http://dx.doi.org/10.1088/1742-5468/2016/04/043106.

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30

Wang, Haiyan, Xiaoli Bian, and Hua Liu. "Möbius transformation and a version of Schwarz lemma in octonionic analysis." Mathematical Methods in the Applied Sciences 44, no. 1 (July 10, 2020): 27–42. http://dx.doi.org/10.1002/mma.6706.

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31

Taimanov, I. A. "The Moutard transformation of two-dimensional Dirac operators and Möbius geometry." Mathematical Notes 97, no. 1-2 (January 2015): 124–35. http://dx.doi.org/10.1134/s0001434615010149.

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32

Řada, Hanka, and Štěpán Starosta. "Bounds on the period of the continued fraction after a Möbius transformation." Journal of Number Theory 212 (July 2020): 122–72. http://dx.doi.org/10.1016/j.jnt.2019.10.027.

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33

Gastesi, Pablo Arés. "Some results on Teichmüller spaces of Klein surfaces." Glasgow Mathematical Journal 39, no. 1 (January 1997): 65–76. http://dx.doi.org/10.1017/s001708950003192x.

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The deformation theory of nonorientable surfaces deals with the problem of studying parameter spaces for the different dianalytic structures that a surface can have. It is an extension of the classical theory of Teichmüller spaces of Riemann surfaces, and as such, it is quite rich. In this paper we study some basic properties of the Teichmüller spaces of non-orientable surfaces, whose parallels in the orientable situation are well known. More precisely, we prove an uniformization theorem, similar to the case of Riemann surfaces, which shows that a non-orientable compact surface can be represented as the quotient of a simply connected domain of the Riemann sphere, by a discrete group of Möbius and anti-Möbius transformation (mappings whose conjugates are Mobius transformations). This uniformization result allows us to give explicit examples of Teichmüller spaces of non-orientable surfaces, as subsets of deformation spaces of orientable surfaces. We also prove two isomorphism theorems: in the first place, we show that the Teichmüller spaces of surfaces of different topological type are not, in general, equivalent. We then show that, if the topological type is preserved, but the signature changes, then the deformations spaces are isomorphic. These are generalizations of the Patterson and Bers-Greenberg theorems for Teichmüller spaces of Riemann surfaces, respectively.

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34

SHIRAISHI, KIYOSHI. "WILSON LOOPS IN OPEN STRING THEORY." Modern Physics Letters A 03, no. 03 (February 1988): 283–87. http://dx.doi.org/10.1142/s0217732388000337.

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Wilson loop elements on torus are introduced into the partition function of open strings as Polyakov’s path integral at one-loop level. Mass spectra from compactification and expected symmetry breaking are illustrated by choosing the correct weight for the contributions from annulus and Möbius strip. We show that Jacobi’s imaginary transformation connects the mass spectra with the Wilson loops. The application to thermopartition function and cosmological implications are briefly discussed.

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35

LIU, Wencai. "The Möbius transformation of continued fractions with bounded upper and lower partial quotients." TURKISH JOURNAL OF MATHEMATICS 44, no. 3 (May 8, 2020): 813–24. http://dx.doi.org/10.3906/mat-1912-85.

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36

Yakubovich, Semyon. "New summation and transformation formulas of the Poisson, Müntz, Möbius and Voronoi type." Integral Transforms and Special Functions 26, no. 10 (June 8, 2015): 768–95. http://dx.doi.org/10.1080/10652469.2015.1051483.

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37

Sánchez-Soto, Luis, and Juan Monzón. "The Geometrical Basis of 𝒫𝒯 Symmetry." Symmetry 10, no. 10 (October 14, 2018): 494. http://dx.doi.org/10.3390/sym10100494.

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We reelaborate on the basic properties of PT symmetry from a geometrical perspective. The transfer matrix associated with these systems induces a Möbius transformation in the complex plane. The trace of this matrix classifies the actions into three types that represent rotations, translations, and parallel displacements. We find that a PT invariant system can be pictured as a complex conjugation followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations and link them with measurable properties of the system.

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38

Trojovský, Pavel, and K. Venkatachalam. "The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function." Fractal and Fractional 5, no. 3 (August 11, 2021): 92. http://dx.doi.org/10.3390/fractalfract5030092.

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In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

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39

Ishizaki, Katsuya. "Admissible solutions of the Schwarzian differential equation." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 50, no. 2 (April 1991): 258–78. http://dx.doi.org/10.1017/s1446788700032742.

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AbstractLet R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation possesses an admissible solution, then , where αj, are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) possesses an admissible solution w(z), then by some Möbius transformation u = (aw + b) / (cw + d) (ad – bc ≠ 0), the equation can be reduced to one of the following forms: where τj (j = 1, … 4) are distinct constants, and σj (j = 1, … 4) are constants, not necessarily distinct.

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40

Jha, Jayant, and Atanu Biswas. "Multiple circular–circular regression." Statistical Modelling 17, no. 3 (March 8, 2017): 142–71. http://dx.doi.org/10.1177/1471082x16679501.

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In this article, we consider the circular–circular regression model using Möbius transformation. We first consider the model provided by Kato et al. (2008) for only one circular regressor and prove the identifiability of the model. After that, a methodology is discussed to reduce the prediction error of this model. We then introduce the two multiple circular–circular regression models with multiple circular regressors. We prove the identifiability of the models and discuss their geometry. We then discuss the parameter estimation procedure followed by simulation study. The methodologies are illustrated by some real datasets.

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41

Kang, Jing, Xiaochuan Liu, and Changzheng Qu. "On an integrable multi-component Camassa–Holm system arising from Möbius geometry." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2251 (July 2021): 20210164. http://dx.doi.org/10.1098/rspa.2021.0164.

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In this paper, we mainly study the geometric background, integrability and peaked solutions of a ( 1 + n ) -component Camassa–Holm (CH) system and some related multi-component integrable systems. Firstly, we show this system arises from the invariant curve flows in the Möbius geometry and serves as the dual integrable counterpart of a geometrical ( 1 + n ) -component Korteweg–de Vries system in the sense of tri-Hamiltonian duality. Moreover, we obtain an integrable two-component modified CH system using a generalized Miura transformation. Finally, we provide a necessary condition, under which the dual integrable systems can inherit the Bäcklund correspondence from the original ones.

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42

Tufaile, Alberto, Michael Snyder, and Adriana Pedrosa Biscaia Tufaile. "Horocycles of Light in a Ferrocell." Condensed Matter 6, no. 3 (August 10, 2021): 30. http://dx.doi.org/10.3390/condmat6030030.

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We studied the effects of image formation in a device known as Ferrocell, which consists of a thin film of a ferrofluid solution between two glass plates subjected to an external magnetic field in the presence of a light source. Following suggestions found in the literature, we compared the Ferrocell light scattering for some magnetic field configurations with the conical scattering of light by thin structures found in foams known as Plateau borders, and we discuss this type of scattering with the concept of diffracted rays from the Geometrical Theory of Diffraction. For certain magnetic field configurations, a Ferrocell with a point light source creates images of circles, parabolas, and hyperboles. We interpret the Ferrocell images as analogous to a Möbius transformation by inversion of the magnetic field. The formation of circles through this transformation is known as horocycles, which can be observed directly in the Ferrocell plane.

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43

Kato, Shogo, and M. C. Jones. "A Family of Distributions on the Circle With Links to, and Applications Arising From, Möbius Transformation." Journal of the American Statistical Association 105, no. 489 (March 1, 2010): 249–62. http://dx.doi.org/10.1198/jasa.2009.tm08313.

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44

JEON, WOOJIN, and KEN’ICHI OHSHIKA. "Measurable rigidity for Kleinian groups." Ergodic Theory and Dynamical Systems 36, no. 8 (June 1, 2015): 2498–511. http://dx.doi.org/10.1017/etds.2015.15.

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Let$G,H$be two Kleinian groups with homeomorphic quotients$\mathbb{H}^{3}/G$and$\mathbb{H}^{3}/H$. We assume that$G$is of divergence type, and consider the Patterson–Sullivan measures of$G$and$H$. The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map$\widehat{k}$from the limit set$\unicode[STIX]{x1D6EC}_{G}$of$G$to that of$H$is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such$\widehat{k}$always exists. In fact, we shall construct$\widehat{k}$concretely from the Cannon–Thurston maps of$G$and$H$.

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Li, Youfa, Jing Shang, Gengrong Zhang, and Pei Dang. "Robust multiscale analytic sampling approximation to periodic function and fast algorithm." International Journal of Wavelets, Multiresolution and Information Processing 15, no. 01 (January 2017): 1750006. http://dx.doi.org/10.1142/s0219691317500060.

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By applying the multiscale method to the Möbius transformation function, we construct the multiscale analytic sampling approximation (MASA) to any function in the Hardy space [Formula: see text]. The approximation error is estimated, and it is proved that the MASA is robust to sample error. We prove that the MASA can be expressed by a Hankel matrix, making use of which, a fast algorithm is established to compute the MASA. Since what we acquire in practice may well be the samples on time domain instead of the analytic ones on the unit disc of the complex plane, we establish a fast algorithm for acquiring analytic samples. Numerical experiments are carried out to demonstrate the efficiency of the MASA.

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Nishiyama, Seiya, and João da Providência. "$\frac{{\rm SO}(2N)}{U(N)}$ Riccati–Hartree–Bogoliubov equation based on the SO(2N) Lie algebra of the fermion operators." International Journal of Geometric Methods in Modern Physics 12, no. 03 (February 27, 2015): 1550035. http://dx.doi.org/10.1142/s0219887815500358.

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In this paper we present the induced representation of SO (2N) canonical transformation group and introduce [Formula: see text] coset variables. We give a derivation of the time-dependent Hartree–Bogoliubov (TDHB) equation on the Kähler coset space [Formula: see text] from the Euler–Lagrange equation of motion for the coset variables. The TDHB wave function represents the TD behavior of Bose condensate of fermion pairs. It is a good approximation for the ground state of the fermion system with a pairing interaction, producing the spontaneous Bose condensation. To describe the classical motion on the coset manifold, we start from the local equation of motion. This equation becomes a Riccati-type equation. After giving a simple two-level model and a solution for a coset variable, we can get successfully a general solution of time-dependent Riccati–Hartree–Bogoliubov equation for the coset variables. We obtain the Harish-Chandra decomposition for the SO (2N) matrix based on the nonlinear Möbius transformation together with the geodesic flow on the manifold.

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BAMPALAS, N., and J. M. R. GRAHAM. "Flow-induced forces arising during the impact of two circular cylinders." Journal of Fluid Mechanics 616 (December 10, 2008): 205–34. http://dx.doi.org/10.1017/s0022112008003856.

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This paper presents numerical simulations of two-dimensional incompressible flow around two circular cylinders in relative motion, which may result in impact. Viscous flow computations are carried out using a streamfunction–vorticity method for two equal-diameter cylinders undergoing a two-dimensional impact in otherwise stationary fluid and for cases of similar impact of two cylinders in a steady incident flow. These results are supported by potential flow calculations carried out using a Möbius conformal transformation and infinite arrays of image singularities. The inviscid flow results are compared with other published work and show that the inviscid forces induced on the cylinders have an inverse square root singularity with respect to the time to impact. All impacts considered in this paper result from steady motion of the cylinders along the line joining their centres.

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Akbarzadeh-Sharbaf, Ali, and Dennis D. Giannacopoulos. "Finite-Element Time-Domain Solution of the Vector Wave Equation in Doubly Dispersive Media Using Möbius Transformation Technique." IEEE Transactions on Antennas and Propagation 61, no. 8 (August 2013): 4158–66. http://dx.doi.org/10.1109/tap.2013.2260716.

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Orum, Chris, Elena Cherkaev, and Kenneth M. Golden. "Recovery of inclusion separations in strongly heterogeneous composites from effective property measurements." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2139 (November 16, 2011): 784–809. http://dx.doi.org/10.1098/rspa.2011.0527.

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An effective property of a composite material consisting of inclusions within a host matrix depends on the geometry and connectedness of the inclusions. This dependence may be quite strong if the constituents have highly contrasting properties. Here, we consider the inverse problem of using effective property data to obtain information on the geometry of the microstructure. While previous work has been devoted to recovering the volume fractions of the constituents, our focus is on their connectedness—a key feature in critical behaviour and phase transitions. We solve exactly a reduced inverse spectral problem by bounding the volume fraction of the constituents, an inclusion separation parameter and the spectral gap of a self-adjoint operator that depends on the geometry of the composite. We present a new algorithm based on the Möbius transformation structure of the forward bounds whose output is a set of algebraic curves in parameter space bounding regions of admissible parameter values. These results advance the development of techniques for characterizing the microstructure of composite materials. As an example, we obtain inverse bounds on the volume fraction and separation of the brine inclusions in sea ice from measurements of its effective complex permittivity.

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Miller, Jason, and Scott Sheffield. "Liouville quantum gravity and the Brownian map III: the conformal structure is determined." Probability Theory and Related Fields 179, no. 3-4 (March 25, 2021): 1183–211. http://dx.doi.org/10.1007/s00440-021-01026-8.

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AbstractPrevious works in this series have shown that an instance of a $$\sqrt{8/3}$$ 8 / 3 -Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the $$\sqrt{8/3}$$ 8 / 3 -LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and $$\sqrt{8/3}$$ 8 / 3 -LQG surfaces with other topologies.

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