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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

Costa, Antonio F., and Milagros Izquierdo. "On real trigonal Riemann surfaces." MATHEMATICA SCANDINAVICA 98, no. 1 (March 1, 2006): 53. http://dx.doi.org/10.7146/math.scand.a-14983.

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Анотація:

A closed Riemann surface $X$ which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) $\sigma$ of $X$ commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry $\sigma $ is the number of connected components of the fixed point set $\mathrm{Fix}(\sigma)$ and the orientability of the Klein surface $X/\langle\sigma\rangle$. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

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2

Badr, Eslam. "A Class of Pseudo-Real Riemann Surfaces with Diagonal Automorphism Group." Algebra Colloquium 27, no. 02 (May 7, 2020): 247–62. http://dx.doi.org/10.1142/s1005386720000206.

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A Riemann surface [Formula: see text] having field of moduli ℝ, but not a field of definition, is called pseudo-real. This means that [Formula: see text] has anticonformal automorphisms, but none of them is an involution. A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d ≥ 4 in [Formula: see text]. We characterize pseudo-real-plane Riemann surfaces [Formula: see text], whose conformal automorphism group Aut+([Formula: see text]) is PGL3(ℂ)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of [Formula: see text]. In particular, we show that such pseudo-real-plane Riemann surfaces exist only if Aut+([Formula: see text]) is cyclic of even order n dividing the degree d. Explicit families of pseudo-real-plane Riemann surfaces are given for any degree d = 2pm with m > 1 odd, p prime and n = d/p.

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3

Tyszkowska, Ewa. "On (q,n)-gonal pseudo-real Riemann surfaces." International Journal of Mathematics 28, no. 13 (December 2017): 1750095. http://dx.doi.org/10.1142/s0129167x17500951.

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Анотація:

A compact Riemann surface [Formula: see text] of genus [Formula: see text] is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real [Formula: see text]-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order [Formula: see text] such that the quotient spaces have genus [Formula: see text].

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4

Duval, Julien, and Damien Gayet. "Riemann surfaces and totally real tori." Commentarii Mathematici Helvetici 89, no. 2 (2014): 299–312. http://dx.doi.org/10.4171/cmh/320.

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5

Cortázar, Ismael, and Antonio F. Costa. "Real Dihedral p-Gonal Riemann Surfaces." Moscow Mathematical Journal 13, no. 4 (2013): 631–47. http://dx.doi.org/10.17323/1609-4514-2013-13-4-631-647.

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6

Bujalance, E., F. J. Cirre, and M. D. E. Conder. "Riemann surfaces with maximal real symmetry." Journal of Algebra 443 (December 2015): 494–516. http://dx.doi.org/10.1016/j.jalgebra.2015.05.015.

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7

Kumar, Abhinav, and Ronen E. Mukamel. "Real multiplication through explicit correspondences." LMS Journal of Computation and Mathematics 19, A (2016): 29–42. http://dx.doi.org/10.1112/s1461157016000188.

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Анотація:

We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

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8

Ares, Filiberto, José G. Esteve, Fernando Falceto, and Amilcar R. De Queiroz. "Complex geometry in the entanglement entropy of fermionic chains." International Journal of Geometric Methods in Modern Physics 14, no. 08 (May 11, 2017): 1740010. http://dx.doi.org/10.1142/s0219887817400102.

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The geometry of Riemann surfaces plays a relevant role in the study of entanglement entropy in the ground state of a free fermionic chain. Recently, a new symmetry for the entropy in non critical theories has been discovered. It is based on the Möbius transformations in a compact Riemann surface associated to the Hamiltonian of the system. Here, we argue how to extend it to critical theories supporting our conjectures with numerical tests. We also highlight the intriguing parallelism that exists with conformal symmetry in real space.

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9

BUJALANCE, JOSÉ A., ANTONIO F. COSTA, and ANA M. PORTO. "ON THE CONNECTEDNESS OF THE LOCUS OF REAL ELLIPTIC-HYPERELLIPTIC RIEMANN SURFACES." International Journal of Mathematics 20, no. 08 (August 2009): 1069–80. http://dx.doi.org/10.1142/s0129167x09005650.

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A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace [Formula: see text] of real elliptic-hyperelliptic algebraic curves in the moduli space [Formula: see text] of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space [Formula: see text] of real elliptic-hyperelliptic surfaces of genus > 5: we show that [Formula: see text] is connected if g is odd and has exactly two connected components if g is even; in both cases the closure [Formula: see text] of [Formula: see text] in the compactified moduli space [Formula: see text] is connected.

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10

Danilova, O. V. "Abel--Jacobi Mapping for Real Hyperelliptic Riemann Surfaces." Mathematical Notes 76, no. 5/6 (November 2004): 778–83. http://dx.doi.org/10.1023/b:matn.0000049677.56924.18.

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11

Bujalance, Emilio, Marston D. E. Conder, and Antonio F. Costa. "Pseudo-real Riemann surfaces and chiral regular maps." Transactions of the American Mathematical Society 362, no. 07 (February 24, 2010): 3365–76. http://dx.doi.org/10.1090/s0002-9947-10-05102-0.

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12

Costa, Antonio F., and Milagros Izquierdo. "Symmetries of real cyclic p-gonal Riemann surfaces." Pacific Journal of Mathematics 213, no. 2 (February 1, 2004): 231–43. http://dx.doi.org/10.2140/pjm.2004.213.231.

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13

Bagiński, Czesław, and Grzegorz Gromadzki. "Minimal genus problem for pseudo-real Riemann surfaces." Archiv der Mathematik 95, no. 5 (November 2010): 481–92. http://dx.doi.org/10.1007/s00013-010-0186-1.

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14

LÓPEZ, RAFAEL. "ON LINEAR WEINGARTEN SURFACES." International Journal of Mathematics 19, no. 04 (April 2008): 439–48. http://dx.doi.org/10.1142/s0129167x08004728.

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Анотація:

In this paper, we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as κ1 = mκ2 + n, where m and n are real numbers and κ1 and κ2 denote the principal curvatures at each point of the surface. We investigate the existence of such surfaces parametrized by a uniparametric family of circles. We prove that the only surfaces that exist are surfaces of revolution and the classical examples of minimal surfaces discovered by Riemann. The latter situation only occurs in the case (m, n) = (-1, 0).

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15

Saka, Takashi. "The real part of the dispersion surface in X-ray dynamical diffraction in the Laue case for perfect crystals." Acta Crystallographica Section A Foundations and Advances 74, no. 5 (September 1, 2018): 586–94. http://dx.doi.org/10.1107/s2053273318009944.

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Анотація:

The real part of the dispersion surface in X-ray dynamical diffraction in the Laue case for perfect crystals is analysed using a Riemann surface. In the conventional two-beam approximation, each branch or wing of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities that specify the degree of departure from the exact Bragg condition and the reflection strength. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters with no approximation. Characteristic features of the dispersion surface are also revealed by geometrical considerations with respect to the Riemann surface.

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16

Gušić, Dženan. "Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One." Mathematics 8, no. 10 (October 13, 2020): 1762. http://dx.doi.org/10.3390/math8101762.

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Анотація:

Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound Oxη, 2ρ − ρn ≤ η < 2ρ up to Ox2ρ−ρηlogx−1, and reduce the exponent 2ρ − ρn replacing it by 2ρ − ρ4n+14n2+1 outside a set of finite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

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17

Gusein-Zade, Sabir M., and Sergey M. Natanzon. "Klein foams as families of real forms of Riemann surfaces." Advances in Theoretical and Mathematical Physics 21, no. 1 (2017): 231–41. http://dx.doi.org/10.4310/atmp.2017.v21.n1.a4.

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18

Bujalance, Emilio, Francisco-Javier Cirre, and Peter Turbek. "Riemann surfaces with real forms which have maximal cyclic symmetry." Journal of Algebra 283, no. 2 (January 2005): 447–56. http://dx.doi.org/10.1016/j.jalgebra.2004.09.010.

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19

Bujalance, Emilio, and Antonio F. Costa. "Automorphism groups of cyclic p-gonal pseudo-real Riemann surfaces." Journal of Algebra 440 (October 2015): 531–44. http://dx.doi.org/10.1016/j.jalgebra.2015.06.020.

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20

Bujalance, Emilio, and Antonio F. Costa. "Automorphism groups of pseudo-real Riemann surfaces of low genus." Acta Mathematica Sinica, English Series 30, no. 1 (December 15, 2013): 11–22. http://dx.doi.org/10.1007/s10114-013-2420-9.

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21

Costa, Antonio F., and Sergey M. Natanzon. "Poincaré's theorem for the modular group of real Riemann surfaces." Differential Geometry and its Applications 27, no. 5 (October 2009): 680–90. http://dx.doi.org/10.1016/j.difgeo.2009.03.008.

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22

Gromadzki, G., and M. Izquierdo. "Real forms of a Riemann surface of even genus." Proceedings of the American Mathematical Society 126, no. 12 (1998): 3475–79. http://dx.doi.org/10.1090/s0002-9939-98-04735-2.

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23

Saka, Takashi. "Formulation of dynamical theory of X-ray diffraction for perfect crystals in the Laue case using the Riemann surface." Acta Crystallographica Section A Foundations and Advances 72, no. 3 (March 30, 2016): 338–48. http://dx.doi.org/10.1107/s2053273316001005.

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Анотація:

The dynamical theory for perfect crystals in the Laue case was reformulated using the Riemann surface, as used in complex analysis. In the two-beam approximation, each branch of the dispersion surface is specified by one sheet of the Riemann surface. The characteristic features of the dispersion surface are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg condition and the reflection strength. By representing these parameters on complex planes, these characteristics can be graphically depicted on the Riemann surface. In the conventional case, the absorption is small and the real part of the reflection strength is large, so the formulation is the same as the traditional analysis. However, when the real part of the reflection strength is small or zero, the two branches of the dispersion surface cross, and the dispersion relationship becomes similar to that of the Bragg case. This is because the geometrical relationships among the parameters are similar in both cases. The present analytical method is generally applicable, irrespective of the magnitudes of the parameters. Furthermore, the present method analytically revealed many characteristic features of the dispersion surface and will be quite instructive for further numerical calculations of rocking curves.

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24

VILLE, MARINA. "HARMONIC MORPHISMS FROM EINSTEIN 4-MANIFOLDS TO RIEMANN SURFACES." International Journal of Mathematics 14, no. 03 (May 2003): 327–37. http://dx.doi.org/10.1142/s0129167x0300179x.

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If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from [Formula: see text] or [Formula: see text] to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism from [Formula: see text] to [Formula: see text].

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25

OLIVEIRA, ANDRÉ GAMA. "REPRESENTATIONS OF SURFACE GROUPS IN THE PROJECTIVE GENERAL LINEAR GROUP." International Journal of Mathematics 22, no. 02 (February 2011): 223–79. http://dx.doi.org/10.1142/s0129167x11006787.

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Given a closed, oriented surface X of genus g ≥ 2, and a semisimple Lie group G, let [Formula: see text] be the moduli space of reductive representations of π1X in G. We determine the number of connected components of [Formula: see text], for n ≥ 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in [Formula: see text] is homotopically equivalent to [Formula: see text].

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26

Kalla, C., and C. Klein. "Computation of the topological type of a real Riemann surface." Mathematics of Computation 83, no. 288 (March 13, 2014): 1823–46. http://dx.doi.org/10.1090/s0025-5718-2014-02817-2.

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27

Saka, Takashi. "Properties of X-ray resonant scattering in the Bragg case revealed on the Riemann surface." Acta Crystallographica Section A Foundations and Advances 72, no. 4 (May 13, 2016): 472–79. http://dx.doi.org/10.1107/s2053273316005404.

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Анотація:

Continuing the work described in the previous paper [Saka (2016).Acta Cryst.A72, 338–348], the dynamical theory for perfect crystals in the Bragg case is reformulated using the Riemann surface. In particular, diffraction under resonant scattering conditions is investigated. The characteristic features of the dispersion surface and the rocking curve are analytically revealed using four parameters, which are the real and imaginary parts of two quantities specifying the degree of departure from the exact Bragg conditions and the reflection strength. Characteristic properties that have been deduced through numerical analysis are derived analytically using these four parameters. Visualization of the geometric relationships between the four parameters on the Riemann surface is useful for understanding many properties such as symmetry and sharpness of the rocking curve under special conditions. Therefore, employing the Riemann surface is instructive for numerical analysis and useful for understanding dynamical diffraction in the Bragg case.

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28

WEHRHEIM, KATRIN. "BANACH SPACE VALUED CAUCHY–RIEMANN EQUATIONS WITH TOTALLY REAL BOUNDARY CONDITIONS." Communications in Contemporary Mathematics 06, no. 04 (August 2004): 601–35. http://dx.doi.org/10.1142/s0219199704001410.

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Анотація:

The main purpose of this paper is to give a general regularity result for Cauchy–Riemann equations in complex Banach spaces with totally real boundary conditions. The usual elliptic Lp-regularity results hold true under one crucial assumption: The Banach space is isomorphic to a closed subspace of an Lp-space. (Equivalently, the totally real submanifold is modelled on a closed subspace of an Lp-space.) Secondly, we describe a class of examples of such totally real submanifolds, namely gauge invariant Lagrangian submanifolds in the space of connections over a Riemann surface. These pose natural boundary conditions for the anti-self-duality equation on 4-manifolds with a boundary space-time splitting, leading towards the definition of a Floer homology for 3-manifolds with boundary, which is the first step in a program by Salamon for the proof of the Atiyah–Floer conjecture. The principal part of such a boundary value problem is an example of a Banach space valued Cauchy–Riemann equation with totally real boundary condition.

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29

de la Puente, M. J. "Specializations and a local homeomorphism theorem for real Riemann surfaces of rings." Pacific Journal of Mathematics 176, no. 2 (December 1, 1996): 427–42. http://dx.doi.org/10.2140/pjm.1996.176.427.

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30

Costa, Antonio F., and Rubén A. Hidalgo. "On the connectedness of the set of Riemann surfaces with real moduli." Archiv der Mathematik 110, no. 3 (December 8, 2017): 305–10. http://dx.doi.org/10.1007/s00013-017-1132-2.

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31

Baird, Thomas. "Moduli Spaces of Vector Bundles over a Real Curve: ℤ/2-Betti Numbers". Canadian Journal of Mathematics 66, № 5 (1 жовтня 2014): 961–92. http://dx.doi.org/10.4153/cjm-2013-049-1.

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Анотація:

AbstractModuli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi–stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute ℤ/2–Betti numbers of these spaces.

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32

Cazacu, Cabiria Andreian, and Dorin Ghisa. "Fundamental Domains of Gamma and Zeta Functions." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/985323.

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Анотація:

Branched covering Riemann surfaces(ℂ,f)are studied, wherefis the Euler Gamma function and the Riemann Zeta function. For both of them fundamental domains are found and the group of cover transformations is revealed. In order to find fundamental domains, preimages of the real axis are taken and a thorough study of their geometry is performed. The technique of simultaneous continuation, introduced by the authors in previous papers, is used for this purpose. Color visualization of the conformal mapping of the complex plane by these functions is used for a better understanding of the theory. A version of this paper containing colored images can be found in arXiv at Andrian Cazacu and Ghisa.

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33

BISWAS, INDRANIL, та VICENTE MUÑOZ. "TORELLI THEOREM FOR MODULI SPACES OF SL(r,ℂ)-CONNECTIONS ON A COMPACT RIEMANN SURFACE". Communications in Contemporary Mathematics 11, № 01 (лютий 2009): 1–26. http://dx.doi.org/10.1142/s0219199709003260.

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Анотація:

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let [Formula: see text] denote the moduli space of holomorphic SL (r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety [Formula: see text] is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety [Formula: see text] determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL (r,ℂ)-connections on X. We also show that the Torelli theorem remains valid for the moduli spaces of connections, as well as those of stable vector bundles, on geometrically irreducible smooth projective curves defined over the field of real numbers.

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34

Gromadzki, Grzegorz. "On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces." Journal of Pure and Applied Algebra 121, no. 3 (October 1997): 253–69. http://dx.doi.org/10.1016/s0022-4049(96)00068-0.

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35

Feehan, Paul M. N. "Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds." Proceedings of the American Mathematical Society 146, no. 7 (March 19, 2018): 3179–90. http://dx.doi.org/10.1090/proc/14013.

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36

Weitsman, Jonathan. "Real polarization of the moduli space of flat connections on a Riemann surface." Communications in Mathematical Physics 145, no. 3 (April 1992): 425–33. http://dx.doi.org/10.1007/bf02099391.

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37

Nag, S., J. A. Hillman, and B. Datta. "Characterisation theorems for compact hypercomplex manifolds." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 2 (October 1987): 231–45. http://dx.doi.org/10.1017/s1446788700029372.

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Анотація:

AbstractWe have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these ‘Further pseudogroups’ turns out to be a quaternionic (respectively octonionic) manifold.We characterise compact Further manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its ‘real set’), is the orientation double covering of the product Y × P2, (respectively Y×P6), where Y is a connected surface equipped with a canonical conformal structure and Pn is n-dimensonal real projective space.A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S4 and S2 × S2 (respectively S8 and S2×S6).Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.

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38

NATANZON, SERGEY, and ANNA PRATOUSSEVITCH. "HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES." Transformation Groups 26, no. 2 (April 1, 2021): 631–40. http://dx.doi.org/10.1007/s00031-021-09644-1.

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Анотація:

AbstractIn this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

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39

Grigutis, Andrius, and Darius Šiaučiūnas. "ON THE MODULUS OF THE SELBERG ZETA-FUNCTIONS IN THE CRITICAL STRIP." Mathematical Modelling and Analysis 20, no. 6 (November 23, 2015): 852–65. http://dx.doi.org/10.3846/13926292.2015.1119213.

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We investigate the behavior of the real part of the logarithmic derivatives of the Selberg zeta-functions ZPSL(2,Z)(s) and ZC (s) in the critical strip 0 < σ < 1. The functions ZPSL(2,Z)(s) and ZC (s) are defined on the modular group and on the compact Riemann surface, respectively.

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40

Kalla, C., and C. Klein. "New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2141 (February 2012): 1371–90. http://dx.doi.org/10.1098/rspa.2011.0583.

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An independent derivation of solutions to the Camassa–Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay’s identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.

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41

GROMADZKI, G., and R. A. HIDALGO. "SCHOTTKY UNIFORMIZATIONS OF SYMMETRIES." Glasgow Mathematical Journal 55, no. 3 (February 25, 2013): 591–613. http://dx.doi.org/10.1017/s0017089512000778.

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AbstractA real algebraic curve of genus g is a pair (S,〈 τ 〉), where S is a closed Riemann surface of genus g and τ: S → S is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of (S,〈 τ 〉) is a tuple (Ω,Γ,P:Ω → S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω → S is a regular holomorphic cover map with Γ as its deck group, so that there exists an extended Möbius transformation $\widehat{\tau}$ keeping Ω invariant with P o $\widehat{\tau}$=τ o P. The extended Kleinian group K=〈 Γ, $\widehat{\tau}$〉 is called an extended Schottky groups of rank g. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of Klein–Maskit's combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of S which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups, we get alternative proofs to results due to Kalliongis and McCullough (J. Kalliongis and D. McCullough, Orientation-reversing involutions on handlebodies, Trans. Math. Soc. 348(5) (1996), 1739–1755) on orientation-reversing involutions on handlebodies.

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42

Hidalgo, Rubén A. "Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals." Archiv der Mathematik 93, no. 3 (September 2009): 219–24. http://dx.doi.org/10.1007/s00013-009-0025-4.

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43

Merrikh-Bayat, Farshad, and Mahdi Afshar. "Extending the Root-Locus Method to Fractional-Order Systems." Journal of Applied Mathematics 2008 (2008): 1–13. http://dx.doi.org/10.1155/2008/528934.

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The well-known root-locus method is developed for special subset of linear time-invariant systems known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variables. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riemann sheet is presented. The important features of the classical root-locus method such as asymptotes, roots condition on the real axis, and breakaway points are extended to fractional case. It is also shown that the proposed method can assess the closed-loop stability of fractional-order systems in the presence of a varying gain in the loop. Three illustrative examples are presented to confirm the effectiveness of the proposed algorithm.

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44

Alarcón, Antonio, та Franc Forstnerič. "Every conformal minimal surface in ℝ3 is isotopic to the real part of a holomorphic\break null curve". Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, № 740 (1 липня 2018): 77–109. http://dx.doi.org/10.1515/crelle-2015-0069.

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Abstract We show that for every conformal minimal immersion {u:M\to\mathbb{R}^{3}} from an open Riemann surface M to {\mathbb{R}^{3}} there exists a smooth isotopy {u_{t}:M\to\mathbb{R}^{3}} ( {t\in[0,1]} ) of conformal minimal immersions, with {u_{0}=u} , such that {u_{1}} is the real part of a holomorphic null curve {M\to\mathbb{C}^{3}} (i.e. {u_{1}} has vanishing flux). If furthermore u is nonflat, then {u_{1}} can be chosen to have any prescribed flux and to be complete.

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45

Beardon, A. F. "Automorphic forms on nondiscrete Möbius groups." Glasgow Mathematical Journal 38, no. 2 (May 1996): 249–53. http://dx.doi.org/10.1017/s0017089500031517.

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If Г is a discrete Möbius group acting on the upper half-plane ℋ of the complex plane, the quotient space ℋ/Г is a Riemann surface ℛ and the automorphic functions on Г correspond to meromorphic functions on ℛ. If Г is a nondiscrete Möbius group acting on ℋ, then ℋ/Г is no longer a Riemann surface, and it is obvious that in this case there are no nonconstant automorphic functions on Г. The situation for automorphic forms is quite different. Automorphic forms of integral dimension for a discrete group Г correspond to meromorphic differentials on ℛ, but even if Г is nondiscrete it may still support nontrivial automorphic forms. The problem of classifying those nondiscrete Möbius groups which act on ℋ and which support nonconstant automorphic forms of arbitrary real dimension was raised and solved (rather indirectly) in [2] where, roughly speaking, function-theoretic methods are used to analyse all possible polynomial automorphic forms of integral dimension, and the results obtained then used to analyse the more general situation.

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46

Wang, Ping-Ping, A.-Man Zhang, Xiang-Li Fang, Abbas Khayyer, and Zi-Fei Meng. "Axisymmetric Riemann–smoothed particle hydrodynamics modeling of high-pressure bubble dynamics with a simple shifting scheme." Physics of Fluids 34, no. 11 (November 2022): 112122. http://dx.doi.org/10.1063/5.0123106.

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High-pressure bubble dynamics often involves many complex issues, including large deformations and inhomogeneities, strong compression, moving interfaces, and large discontinuities, that bring challenges to numerical simulations. In this work, an axisymmetric Riemann–smoothed particle hydrodynamics (SPH) method is used to simulate high-pressure bubbles near different boundaries. This Riemann–SPH can adopt the real sound speed instead of the artificial one for the air phase in the bubble. Therefore, the real compressibility of the air phase can be considered, and the corresponding time step is significantly increased. To avoid unphysical interface penetration and maintain relatively homogeneous particle distribution, a new and simple particle shifting scheme for multiphase flows is proposed. Additionally, to minimize the influence of the unphysical boundary on the bubble, a large fluid domain with an optimized initial particle distribution is adopted to reduce the particle number. Several high-pressure bubbles under different boundary conditions are considered, including in a free field, near a free surface, near a solid boundary, and near a rigid sphere. Numerical results show that these bubble dynamic behaviors can be reproduced with satisfactory accuracy.

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47

Leakey, Shannon, Vassilis Glenis, and Caspar Hewett. "Artificial Compressibility with Riemann Solvers: Convergence of Limiters on Unstructured Meshes." OpenFOAM® Journal 2 (March 4, 2022): 31–47. http://dx.doi.org/10.51560/ofj.v2.49.

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Free-surface flows and other variable density incompressible flows have numerous important applications in engineering.One way such flows can be modelled is to extend established numerical methods for compressible flows to incompressible flows using the method of artificial compressibility. Artificial compressibility introduces a pseudo-time derivative for pressure and, in each real-time step, the solution advances in pseudo-time until convergence to an incompressible limit - a fundamentally different approach than SIMPLE, PISO, and PIMPLE, the standard methods used in OpenFOAM. Although the artificial compressibility method is widespread in the literature, its application to free-surface flows is not. In this paper, we apply the method to variable density flows on 3D unstructured meshes for the first time, implementing a Godunov-type scheme with MUSCL reconstruction and Riemann solvers, where the free surface gets captured automatically by the contact wave in the Riemann solver. The critical problem in this implementation lies in the slope limiters used in the MUSCL reconstruction step. It is well-known that slope limiters can inhibit convergence to steady state on unstructured meshes; the problem is exacerbated here as convergence in pseudo-time is required not just once, but at every real-time step. We compare the limited gradient schemes included in OpenFOAM with an improved limiter from the literature, testing the solver against dam-break and hydrostatic pressure benchmarks. This work opens OpenFOAM up to the method of artificial compressibility, breaking the mould of PIMPLE and harnessing high-resolution shock-capturing schemes that are easier to parallelise.

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48

Gromadzki, Grzegorz, and Ewa Kozłowska-Walania. "On dimensions of the real nerve of the moduli space of Riemann surfaces of odd genus." Rendiconti del Seminario Matematico della Università di Padova 135 (2016): 91–109. http://dx.doi.org/10.4171/rsmup/135-5.

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49

Björklund, Johan. "Legendrian contact homology in the product of a punctured Riemann surface and the real line." Journal of the London Mathematical Society 94, no. 3 (November 7, 2016): 970–92. http://dx.doi.org/10.1112/jlms/jdw066.

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50

Toda, Yukinobu. "Moduli spaces of stable quotients and wall-crossing phenomena." Compositio Mathematica 147, no. 5 (May 31, 2011): 1479–518. http://dx.doi.org/10.1112/s0010437x11005434.

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AbstractThe moduli space of holomorphic maps from Riemann surfaces to the Grassmannian is known to have two kinds of compactifications: Kontsevich’s stable map compactification and Marian–Oprea–Pandharipande’s stable quotient compactification. Over a non-singular curve, the latter moduli space is Grothendieck’s Quot scheme. In this paper, we give the notion of ‘ ϵ-stable quotients’ for a positive real number ϵ, and show that stable maps and stable quotients are related by wall-crossing phenomena. We will also discuss Gromov–Witten type invariants associated to ϵ-stable quotients, and investigate them under wall crossing.

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