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Статті в журналах з теми "Möbiu"

Автор: Grafiati

Опубліковано: 11 березня 2023

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1

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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2

Buyalo, S. "Möbius and sub-Möbius structures." St. Petersburg Mathematical Journal 28, no. 5 (July 25, 2017): 555–68. http://dx.doi.org/10.1090/spmj/1463.

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3

Bayad, Abdelmejid, Daeyeoul Kim, and Yan Li. "Arithmetical properties of double Möbius-Bernoulli numbers." Open Mathematics 17, no. 1 (February 17, 2019): 32–42. http://dx.doi.org/10.1515/math-2019-0006.

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Abstract Given positive integers n, n′ and k, we investigate the Möbius-Bernoulli numbers Mk(n), double Möbius-Bernoulli numbers Mk(n,n′), and Möbius-Bernoulli polynomials Mk(n)(x). We find new identities involving double Möbius-Bernoulli, Barnes-Bernoulli numbers and Dedekind sums. In part of this paper, the Möbius-Bernoulli polynomials Mk(n)(x), can be interpreted as critical values of the following Dirichlet type L-function $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s-complex plane, where μ is the Möbius function.

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4

Balakan, Gülcan, and Oğuzhan Demirel. "The Formulas of Möbius-Bretschneider and Möbius-Cagnoli in the Poincaré Disc Model of Hyperbolic Geometry." Al-Mustansiriyah Journal of Science 32, no. 1 (February 21, 2021): 31. http://dx.doi.org/10.23851/mjs.v32i1.932.

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5

Wang, Changping. "Möbius geometry for hypersurfaces in S4." Nagoya Mathematical Journal 139 (September 1995): 1–20. http://dx.doi.org/10.1017/s0027763000005274.

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Анотація:

Our purpose in this paper is to study Möbius geometry for those hypersurfaces in S4 which have different principal curvatures at each point. We will give a complete local Möbius invariant system for such hypersurface in S4 which determines the hypersurface up to Möbius transformations. And we will classify the so-called Möbius homogeneous hypersurfaces in S4.

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6

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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7

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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8

Alden, J. W. "Möbius." Nature 515, no. 7526 (November 2014): 304. http://dx.doi.org/10.1038/515304a.

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9

Sun, Jiancai. "Contragredient Modules and Invariant Bilinear Forms on Möbius Nonlocal Vertex Algebras." Algebra Colloquium 20, no. 03 (July 4, 2013): 403–16. http://dx.doi.org/10.1142/s1005386713000370.

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Анотація:

We study contragredient modules for Möbius nonlocal vertex algebras and characterize the space of all invariant bilinear forms on Möbius nonlocal vertex algebras. Finally, Möbius weak quantum vertex algebras of Zamolodchikov-Faddeev type are studied as examples.

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10

Muhammad, Guntur Maulana, Iden Rainal Ihsan, and Roni Priyanda. "Sifat Preservasi Lingkaran dan Garis Pada Transformasi Möbius." Jambura Journal of Mathematics 4, no. 2 (June 1, 2022): 200–208. http://dx.doi.org/10.34312/jjom.v4i2.13497.

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This article discusses Möbius transformation from the point of view of algebra to describe one of its geometric properties, i.e. preserving circles and lines in complex planes. In simple terms, this preservation means that Möbius transformation maps a collection of circles and lines (back) into a collection of circles and lines. In general, the discussion begins with an explanation of the definition of the Möbius transformation in the complex plane. The discussion continues on defining the basic mapping and direct affine transformation. These two concepts are used to prove the existence of the preservation properties of circles and lines in the Möbius transformation. It can be shown that the Möbius transformation can be expressed as a composition of the direct affine transform and the inverse. It can also be shown that the direct affine transform and the inverse both have the property of preserving circles and lines in the complex plane. Thus, it can be concluded that in this study the Möbius transformation has the property of preserving circles and lines in the complex plane.

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11

Kumar, Arun, Poornakanta Handral, Darshan Bhandari, and Ramsharan Rangarajan. "More views of a one-sided surface: mechanical models and stereo vision techniques for Möbius strips." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2250 (June 2021): 20210076. http://dx.doi.org/10.1098/rspa.2021.0076.

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Möbius strips are prototypical examples of ribbon-like structures. Inspecting their shapes and features provides useful insights into the rich mechanics of elastic ribbons. Despite their ubiquity and ease of construction, quantitative experimental measurements of the three-dimensional shapes of Möbius strips are surprisingly non-existent in the literature. We propose two novel stereo vision-based techniques to this end—a marker-based technique that determines a Lagrangian description for the construction of a Möbius strip, and a structured light illumination technique that furnishes an Eulerian description of its shape. Our measurements enable a critical evaluation of the predictive capabilities of mechanical theories proposed to model Möbius strips. We experimentally validate, seemingly for the first time, the developable strip and the Cosserat plate theories for predicting shapes of Möbius strips. Equally significantly, we confirm unambiguous deficiencies in modelling Möbius strips as Kirchhoff rods with slender cross-sections. The experimental techniques proposed and the Cosserat plate model promise to be useful tools for investigating a general class of problems in ribbon mechanics.

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12

FRIED, DAVID, SEBASTIAN M. MAROTTA, and RICH STANKEWITZ. "Complex dynamics of Möbius semigroups." Ergodic Theory and Dynamical Systems 32, no. 6 (February 7, 2012): 1889–929. http://dx.doi.org/10.1017/s014338571100054x.

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AbstractWe study the dynamics of semigroups of Möbius transformations on the Riemann sphere, especially their Julia sets and attractors. This theory relates to the dynamics of rational functions, rational semigroups, and Möbius groups and we compare and contrast these theories. We particularly examine Caruso’s family of Möbius semigroups, based on a random dynamics variant of the Fibonacci sequence.

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13

Hu, Ze Jun, and Xiao Li Tian. "On Möbius form and Möbius isoparametric hypersurfaces." Acta Mathematica Sinica, English Series 25, no. 12 (November 15, 2009): 2077–92. http://dx.doi.org/10.1007/s10114-009-7682-x.

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14

Burghardt, Katharina. "Möbius-Syndrom – Fazialis-/Abduzensparese mit expressiver Sprachstörung." Erfahrungsheilkunde 71, no. 06 (December 2022): 350–56. http://dx.doi.org/10.1055/a-1970-4688.

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ZusammenfassungDas Möbius-Syndrom, auch „okulofaziale Paralyse“ genannt, wird in der Praxis selten zugeordnet. Die bestehende Fazialislähmung wird als primäre neurologische Erkrankung gesehen. Eine genetische Diagnostik wird – wenn überhaupt – erst spät veranlasst. Dabei ist ein interdisziplinäres Therapiekonzept vorhanden. Die Diagnose wird meist symptomatisch gestellt: Kieferorthopädisch ist häufig der frontal offene Biss auffällig. Es besteht eine angeborene uni- oder bilaterale Fazialisparese (N. VII) sowie eine ein- oder beidseitige Abduzensparese (N. VI). Eine Ursachenklärung ist beim Möbius-Syndrom nicht sicher möglich. Eine mögliche Ursache ist die „subclavian artery supply disruption sequence“ (SASDS). Es gibt 5 Kandidatengene, in denen Mutationen als Ursache des Möbius-Syndroms oder atypischen Möbius-Syndroms identifiziert werden konnten. Lokalisierte chromosomale Regionen sind nicht eindeutig fassbar. Das Möbius-Syndrom ist schwierig von anderen ähnlichen neurologischen Erkrankungen wie der Poland-Sequenz abzugrenzen; als Differenzialdiagnose kommt auch das DiGeorge-Syndrom in Frage.

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15

Sabourau, Stéphane, and Zeina Yassine. "Optimal systolic inequalities on Finsler Möbius bands." Journal of Topology and Analysis 08, no. 02 (March 15, 2016): 349–72. http://dx.doi.org/10.1142/s1793525316500138.

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We prove optimal systolic inequalities on Finsler Möbius bands relating the systole and the height of the Möbius band to its Holmes–Thompson volume. We also establish an optimal systolic inequality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric families both on the Möbius band and the Klein bottle are also presented.

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16

Watanabe, Keiichi. "Orthogonal Gyroexpansion in Möbius Gyrovector Spaces." Journal of Function Spaces 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/1518254.

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We investigate the Möbius gyrovector spaces which are open balls centered at the origin in a real Hilbert space with the Möbius addition, the Möbius scalar multiplication, and the Poincaré metric introduced by Ungar. In particular, for an arbitrary point, we can easily obtain the unique closest point in any closed gyrovector subspace, by using the ordinary orthogonal decomposition. Further, we show that each element has the orthogonal gyroexpansion with respect to any orthogonal basis in a Möbius gyrovector space, which is similar to each element in a Hilbert space having the orthogonal expansion with respect to any orthonormal basis. Moreover, we present a concrete procedure to calculate the gyrocoefficients of the orthogonal gyroexpansion.

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17

HUANG, I.-CHIAU. "TWO APPROACHES TO MÖBIUS INVERSION." Bulletin of the Australian Mathematical Society 85, no. 1 (August 15, 2011): 68–78. http://dx.doi.org/10.1017/s0004972711002656.

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18

He, Tian-Xiao, Leetsch C. Hsu, and Peter J. S. Shiue. "On generalised Möbius inversion formulas." Bulletin of the Australian Mathematical Society 73, no. 1 (February 2006): 79–88. http://dx.doi.org/10.1017/s0004972700038648.

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19

Boas,, Ralph P. "Möbius Shorts." Mathematics Magazine 68, no. 2 (April 1, 1995): 127. http://dx.doi.org/10.2307/2691190.

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20

Vyas, Sameer, Anuj Prabhakar, K. Uday Bhanu, Paramjeet Singh, and Niranjan Khandelwal. "Möbius syndrome." Journal of Neurosciences in Rural Practice 7, no. 04 (April 2016): 596–97. http://dx.doi.org/10.4103/0976-3147.186974.

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21

Séquin, Carlo H. "Möbius bridges." Journal of Mathematics and the Arts 12, no. 2-3 (January 10, 2018): 181–94. http://dx.doi.org/10.1080/17513472.2017.1419331.

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22

Boas, Ralph P. "Möbius Shorts." Mathematics Magazine 68, no. 2 (April 1995): 127. http://dx.doi.org/10.1080/0025570x.1995.11996295.

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23

Brower, R. C., H. Neff, and K. Orginos. "Möbius Fermions." Nuclear Physics B - Proceedings Supplements 153, no. 1 (March 2006): 191–98. http://dx.doi.org/10.1016/j.nuclphysbps.2006.01.047.

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24

Baden, Alex, Keenan Crane, and Misha Kazhdan. "Möbius Registration." Computer Graphics Forum 37, no. 5 (August 2018): 211–20. http://dx.doi.org/10.1111/cgf.13503.

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25

Mendez, M., and J. Yang. "Möbius species." Advances in Mathematics 85, no. 1 (January 1991): 83–128. http://dx.doi.org/10.1016/0001-8708(91)90051-8.

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26

Keith, Michael. "Möbius crossword." Mathematical Intelligencer 7, no. 3 (September 1985): 38. http://dx.doi.org/10.1007/bf03025805.

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27

Osborne, Gregory A. "MÖBIUS' SYNDROME." Journal of the American Dental Association 130, no. 2 (February 1999): 156–57. http://dx.doi.org/10.14219/jada.archive.1999.0144.

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28

Murray, Will. "Möbius Polynomials." Mathematics Magazine 85, no. 5 (December 2012): 376–83. http://dx.doi.org/10.4169/math.mag.85.5.376.

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29

Gallet, Matteo, Georg Nawratil, and Josef Schicho. "Möbius photogrammetry." Journal of Geometry 106, no. 3 (November 30, 2014): 421–39. http://dx.doi.org/10.1007/s00022-014-0255-x.

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30

Groß, Michael. "Möbius-Aromaten." Chemie in unserer Zeit 38, no. 2 (April 2004): 87. http://dx.doi.org/10.1002/ciuz.200490034.

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31

POLLACK, PAUL, and CARLO SANNA. "UNCERTAINTY PRINCIPLES CONNECTED WITH THE MÖBIUS INVERSION FORMULA." Bulletin of the Australian Mathematical Society 88, no. 3 (January 25, 2013): 460–72. http://dx.doi.org/10.1017/s0004972712001128.

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AbstractTwo arithmetic functions $f$ and $g$ form a Möbius pair if $f(n)= {\mathop{\sum }\nolimits}_{d\mid n} g(d)$ for all natural numbers $n$. In that case, $g$ can be expressed in terms of $f$ by the familiar Möbius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members $f$ and $g$ of a Möbius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary is that in a nonzero Möbius pair, one cannot have both ${\mathop{\sum }\nolimits}_{f(n)\not = 0} 1/ n\lt \infty $ and ${\mathop{\sum }\nolimits}_{g(n)\not = 0} 1/ n\lt \infty $.

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32

Heer, Loreno. "Some Invariant Properties of Quasi-Möbius Maps." Analysis and Geometry in Metric Spaces 5, no. 1 (September 2, 2017): 69–77. http://dx.doi.org/10.1515/agms-2017-0004.

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Abstract We investigate properties which remain invariant under the action of quasi-Möbius maps of quasimetric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)Möbius maps. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)Möbius maps as well.

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33

Miller, Marilyn T., Valencia Ray, Paul Owens, and Felix Chen. "Möbius and Möbius-Like Syndromes (TTV-OFM, OMLH)." Journal of Pediatric Ophthalmology & Strabismus 26, no. 4 (July 1989): 176–88. http://dx.doi.org/10.3928/0191-3913-19890701-07.

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34

Terzis, Julia K., and Ernst Magnus Noah. "Dynamic Restoration in Möbius and Möbius-Like Patients." Plastic and Reconstructive Surgery 111, no. 1 (January 2003): 40–55. http://dx.doi.org/10.1097/00006534-200301000-00007.

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35

Terzis, Julia K., and Ernst Magnus Noah. "Dynamic Restoration in Möbius and Möbius-Like Patients." Plastic and Reconstructive Surgery 111, no. 1 (January 2003): 40–55. http://dx.doi.org/10.1097/01.prs.0000037878.89189.db.

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36

Fujimoto, Katsushige, and Toshiaki Murofushi. "Some Characterizations of k-Monotonicity Through the Bipolar Möbius Transform in Bi-Capacities." Journal of Advanced Computational Intelligence and Intelligent Informatics 9, no. 5 (September 20, 2005): 484–95. http://dx.doi.org/10.20965/jaciii.2005.p0484.

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Анотація:

This paper first proposes the bipolar Möbius transform as an extension of dividends of cooperative games to that of bi-cooperative games (bi-capacities) defined on 3N, which is different from the Möbius transform defined by Grabisch and Labreuche. The k-monotonicity of bi-capacities is characterized through each of the following notions: the bipolar and ordinary Möbius transforms, discrete derivatives, and partial derivatives of the piecewise multilinear extension of the ternary pseudo-Boolean function corresponding to the bi-capacities.

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37

Beardon, A. F., D. Minda, and I. Short. "Normal Families of Möbius Maps." Computational Methods and Function Theory 20, no. 3-4 (July 17, 2020): 523–38. http://dx.doi.org/10.1007/s40315-020-00328-7.

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AbstractSeveral necessary and sufficient conditions for a family of Möbius maps to be a normal family in the extended complex plane $$\mathbb {C}_\infty $$ C ∞ are established. Each of these conditions involves collections of two or three points which may vary with the Möbius maps in the family, provided the points satisfy a uniform separation condition. In addition, we derive a sufficient condition for the normality of a family of Möbius maps in terms of the average value of the reciprocal of the chordal derivative.

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38

SCHWAB, EMIL DANIEL, and PENTTI HAUKKANEN. "A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS." International Journal of Number Theory 04, no. 04 (August 2008): 549–61. http://dx.doi.org/10.1142/s1793042108001523.

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We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

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39

Li, Tongzhu, and Changping Wang. "A note on Blaschke isoparametric hypersurfaces." International Journal of Mathematics 25, no. 12 (November 2014): 1450117. http://dx.doi.org/10.1142/s0129167x14501171.

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In this paper, we prove that a Möbius isoparametric hypersurface is a Blaschke isoparametric hypersurface, and a Blaschke isoparametric hypersurface is a Möbius isoparametric hypersurface provided that the Blaschke tensor has more than two distinct eigenvalues.

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40

Tomii, Masaaki. "Dispersion relation and unphysical poles of Möbius domain-wall fermions in free field theory at finite Ls." EPJ Web of Conferences 175 (2018): 13009. http://dx.doi.org/10.1051/epjconf/201817513009.

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Анотація:

We investigate the dispersion relation of Möbius domain-wall fermions in free field theory at finite Ls. We find that there are Ls - 1 extra poles of Möbius domain-wall fermions in addition to the pole which realizes the physical mode in the continuum limit. The unphysical contribution of these extra poles could be significant when we introduce heavy quarks. We show in this report the fundamental properties of these unphysical poles and discuss the optimal choice of Möbius parameters to minimize their contribution to four-dimensional physics.

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41

Ishizeki, Aya, and Takeyuki Nagasawa. "The invariance of decomposed Möbius energies under inversions with center on curves." Journal of Knot Theory and Its Ramifications 25, no. 02 (February 2016): 1650009. http://dx.doi.org/10.1142/s0218216516500097.

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It is well known that one of O’Hara’s knot energies is called the Möbius energy because of its invariance under Möbius transformations. We showed in a previous paper that the Möbius energy can be decomposed into three parts that retain invariance but we left open the question of invariance regarding inversions with respect to spheres centered on a knot. Here, we answer this question under the assumption that the knots have extra regularity. The result holds not only for knots but also for closed curves in [Formula: see text].

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42

El Abdalaoui, E. H., and M. Disertori. "Spectral properties of the Möbius function and a random Möbius model." Stochastics and Dynamics 16, no. 01 (November 10, 2015): 1650005. http://dx.doi.org/10.1142/s0219493716500052.

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Assuming Sarnak’s conjecture is true for any singular dynamical process, we prove that the spectral measure of the Möbius function is equivalent to Lebesgue measure. Conversely, under Elliott’s conjecture, we establish that the Möbius function is orthogonal to any uniquely ergodic dynamical system with singular spectrum. Furthermore, using Mirsky’s theorem, we find a new simple proof of Cellarosi–Sinai’s theorem on the orthogonality of the square of the Möbius function with respect to any weakly mixing dynamical system. Finally, we establish Sarnak’s conjecture for a particular random model.

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43

Sakri, Redha, and Moncef Abbas. "On locating chromatic number of Möbius ladder graphs." Proyecciones (Antofagasta) 40, no. 3 (April 29, 2021): 659–69. http://dx.doi.org/10.22199/issn.0717-6279-4170.

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44

Ma, William. "A metric that better approximates the hyperbolic metric." Conformal Geometry and Dynamics of the American Mathematical Society 26, no. 1 (February 1, 2022): 1–9. http://dx.doi.org/10.1090/ecgd/368.

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We introduce a metric, called the star metric, that gives us better upper bound of the hyperbolic metric than the Möbius invariant metric. We present various examples, and comparisons among the hyperbolic metric, the Möbius invariant metric and the star metric.

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45

Müller, Christian, and Amir Vaxman. "Discrete curvature and torsion from cross-ratios." Annali di Matematica Pura ed Applicata (1923 -) 200, no. 5 (January 21, 2021): 1935–60. http://dx.doi.org/10.1007/s10231-021-01065-x.

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AbstractMotivated by a Möbius invariant subdivision scheme for polygons, we study a curvature notion for discrete curves where the cross-ratio plays an important role in all our key definitions. Using a particular Möbius invariant point-insertion-rule, comparable to the classical four-point-scheme, we construct circles along discrete curves. Asymptotic analysis shows that these circles defined on a sampled curve converge to the smooth curvature circles as the sampling density increases. We express our discrete torsion for space curves, which is not a Möbius invariant notion, using the cross-ratio and show its asymptotic behavior in analogy to the curvature.

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46

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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47

Buyalo, S. "Symmetries of double ratios and an equation for Möbius structures." St. Petersburg Mathematical Journal 33, no. 1 (December 28, 2021): 47–56. http://dx.doi.org/10.1090/spmj/1688.

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Orthogonal representations η n : S n ↷ R N \eta _n\colon S_n\curvearrowright \mathbb {R}^N of the symmetric groups S n S_n , n ≥ 4 n\ge 4 , with N = n ! / 8 N=n!/8 , emerging from symmetries of double ratios are treated. For n = 5 n=5 , the representation η 5 \eta _5 is decomposed into irreducible components and it is shown that a certain component yields a solution of the equations that describe the Möbius structures in the class of sub-Möbius structures. In this sense, a condition determining the Möbius structures is implicit already in symmetries of double ratios.

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48

Demirel, Oğuzhan. "Degenerate Saccheri Quadrilaterals, Möbius Transformations and Conjugate Möbius Transformations." International Electronic Journal of Geometry 10, no. 2 (October 29, 2017): 32–36. http://dx.doi.org/10.36890/iejg.545044.

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49

Cartwright, Julyan H. E., and Diego L. González. "Möbius Strips Before Möbius: Topological Hints in Ancient Representations." Mathematical Intelligencer 38, no. 2 (May 10, 2016): 69–76. http://dx.doi.org/10.1007/s00283-016-9631-8.

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50

Coffin, Jack, and Andreas Chatzidakis. "The Möbius strip of market spatiality: mobilizing transdisciplinary dialogues between CCT and the marketing mainstream." AMS Review 11, no. 1-2 (January 25, 2021): 40–59. http://dx.doi.org/10.1007/s13162-020-00191-8.

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AbstractThis paper develops the Möbius strip as an ‘ordering theory’ (Sandberg and Alvesson, 2020) that brings CCT studies into dialogue with mainstream marketing approaches. The aim is to work toward a transdisciplinary understanding of market spatiality, a topic that has become increasingly important for theorists and practitioners (Warnaby and Medway, 2013; Castilhos et al., 2016; Chatzidakis et al., 2018). Building on psychosocial interpretations of the Möbius strip as a ‘tactical’ way of thinking, a range of insights and ideas are organized along a single strip of theorization. This paper maps a continuous plane of logic between the concepts of space, place, emplacement, spatiality, implacement, and displacement. The potential applications of the Möbius strip are then demonstrated by showing how the transdisciplinary topic of ‘atmosphere’ can be theorized from multiple perspectives. The paper concludes by exploring how the Möbius strip might also be employed in other areas of marketing theory and practice, potentially generating further transdisciplinary conversations between CCT and the marketing mainstream.

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