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Journal articles on the topic 'Complex hyperbolic space'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2023

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1

Kaliman, Shulim, and Mikhail Zaidenberg. "Non-hyperbolic complex space with a hyperbolic normalization." Proceedings of the American Mathematical Society 129, no. 5 (October 20, 2000): 1391–93. http://dx.doi.org/10.1090/s0002-9939-00-05711-7.

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2

THAI, DO DUC, and PHAM VIET DUC. "THE KOBAYASHI k-METRICS ON COMPLEX SPACES." International Journal of Mathematics 10, no. 07 (November 1999): 917–24. http://dx.doi.org/10.1142/s0129167x99000392.

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In this article we are going to give a characterization of the hyperbolicity of complex spaces through the Kobayashi k-metrics on complex spaces and to give an integrated representation of the Kobayashi pseudo-distance on any complex space. Moreover, it is shown that a complex space is hyperbolic iff every irreducible branch of this space is hyperbolic.

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3

PARKER, JOHN R. "SHIMIZU’S LEMMA FOR COMPLEX HYPERBOLIC SPACE." International Journal of Mathematics 03, no. 02 (April 1992): 291–308. http://dx.doi.org/10.1142/s0129167x92000096.

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Shimizu’s lemma gives a necessary condition for a discrete group of isometries of the hyperbolic plane containing a parabolic map to be discrete. Viewing the hyperbolic plane as complex hyperbolic 1-space we generalise Shimizu’s lemma to higher dimensional complex hyperbolic space In particular we give a version of Shimizu’s lemma for subgroups of PU (n, 1) containing a vertical translation Partial generalisation to groups containing either an ellipto-parabolic map or non-vertical translations are also given together with examples that show full generalisation is not possible in these cases

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4

Khalfallah, Adel. "The moduli space of hyperbolic compact complex spaces." Mathematische Zeitschrift 255, no. 4 (August 26, 2006): 691–702. http://dx.doi.org/10.1007/s00209-006-0036-9.

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5

Li, Haizhong, and Xianfeng Wang. "Isotropic Lagrangian Submanifolds in Complex Euclidean Space and Complex Hyperbolic Space." Results in Mathematics 56, no. 1-4 (October 30, 2009): 387–403. http://dx.doi.org/10.1007/s00025-009-0422-9.

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6

Xiao, Yingqing, and Yueping Jiang. "Complex lines in complex hyperbolic space H ℂ 2." Indian Journal of Pure and Applied Mathematics 42, no. 5 (October 2011): 279–89. http://dx.doi.org/10.1007/s13226-011-0019-3.

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7

Li, Haizhong, and Xianfeng Wang. "Calabi Product Lagrangian Immersions in Complex Projective Space and Complex Hyperbolic Space." Results in Mathematics 59, no. 3-4 (April 2, 2011): 453–70. http://dx.doi.org/10.1007/s00025-011-0107-z.

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8

Korolkova, Anna V., Migran N. Gevorkyan, and Dmitry S. Kulyabov. "Implementation of hyperbolic complex numbers in Julia language." Discrete and Continuous Models and Applied Computational Science 30, no. 4 (December 26, 2022): 318–29. http://dx.doi.org/10.22363/2658-4670-2022-30-4-318-329.

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Hyperbolic complex numbers are used in the description of hyperbolic spaces. One of the well-known examples of such spaces is the Minkowski space, which plays a leading role in the problems of the special theory of relativity and electrodynamics. However, such numbers are not very common in different programming languages. Of interest is the implementation of hyperbolic complex in scientific programming languages, in particular, in the Julia language. The Julia language is based on the concept of multiple dispatch. This concept is an extension of the concept of polymorphism for object-oriented programming languages. To implement hyperbolic complex numbers, the multiple dispatching approach of the Julia language was used. The result is a library that implements hyperbolic numbers. Based on the results of the study, we can conclude that the concept of multiple dispatching in scientific programming languages is convenient and natural.

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9

Chen, Bang-Yen, and Luc Vrancken. "Lagrangian submanifolds of the complex hyperbolic space." Tsukuba Journal of Mathematics 26, no. 1 (June 2002): 95–118. http://dx.doi.org/10.21099/tkbjm/1496164384.

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10

Vernon, Micheal H. "Contact hypersurfaces of a complex hyperbolic space." Tohoku Mathematical Journal 39, no. 2 (1987): 215–22. http://dx.doi.org/10.2748/tmj/1178228324.

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11

Cao, Wensheng, and Krishnendu Gongopadhyay. "Commuting isometries of the complex hyperbolic space." Proceedings of the American Mathematical Society 139, no. 09 (September 1, 2011): 3317. http://dx.doi.org/10.1090/s0002-9939-2011-10796-2.

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12

Chow, Richard. "Groups Quasi-isometric to Complex Hyperbolic Space." Transactions of the American Mathematical Society 348, no. 5 (1996): 1757–69. http://dx.doi.org/10.1090/s0002-9947-96-01522-x.

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13

MONTIEL, Sebastian. "Real hypersurfaces of a complex hyperbolic space." Journal of the Mathematical Society of Japan 37, no. 3 (July 1985): 515–35. http://dx.doi.org/10.2969/jmsj/03730515.

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14

Wei, Zhu, and Yu Xuegang. "Some Physics Questions in Hyperbolic Complex Space." Advances in Applied Clifford Algebras 17, no. 1 (November 9, 2006): 137–44. http://dx.doi.org/10.1007/s00006-006-0018-3.

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15

Gorodski, Claudio, and Nikolay Gusevskii. "Complete minimal hypersurfaces in complex hyperbolic space." manuscripta mathematica 103, no. 2 (October 1, 2000): 221–40. http://dx.doi.org/10.1007/s002290070021.

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16

CASTRO, C., and J. A. NIETO. "ON (2+2)-DIMENSIONAL SPACE–TIMES, STRINGS AND BLACK HOLES." International Journal of Modern Physics A 22, no. 11 (April 30, 2007): 2021–45. http://dx.doi.org/10.1142/s0217751x07036191.

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We study black hole-like solutions (space–times with singularities) of Einstein field equations in 3+1 and 2+2 dimensions. We find three different cases associated with hyperbolic homogeneous spaces. In particular, the hyperbolic version of Schwarzschild's solution contains a conical singularity at r = 0 resulting from pinching to zero size r = 0 the throat of the hyperboloid [Formula: see text] and which is quite different from the static spherically symmetric (3+1)-dimensional solution. Static circular symmetric solutions for metrics in 2+2 are found that are singular at ρ = 0 and whose asymptotic ρ→∞ limit leads to a flat (1+2)-dimensional boundary of topology S1 × R2. Finally we discuss the (1+1)-dimensional Bars–Witten stringy black hole solution and show how it can be embedded into our (3+1)-dimensional solutions. Black holes in a (2+2)-dimensional "space–time" from the perspective of complex gravity in 1+1 complex dimensions and their quaternionic and octonionic gravity extensions deserve furher investigation. An appendix is included with the most general Schwarzschild-like solutions in D ≥ 4.

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17

Platis, Ioannis D. "Quakebend deformations in complex hyperbolic quasi-Fuchsian space." Geometry & Topology 12, no. 1 (March 12, 2008): 431–59. http://dx.doi.org/10.2140/gt.2008.12.431.

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18

Dai, Binlin, Ainong Fang, and Bing Nai. "Discreteness criteria for subgroups in complex hyperbolic space." Proceedings of the Japan Academy, Series A, Mathematical Sciences 77, no. 10 (December 2001): 168–72. http://dx.doi.org/10.3792/pjaa.77.168.

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19

Giuseppe PIPOLI. "Inverse mean curvature flow in complex hyperbolic space." Annales scientifiques de l'École normale supérieure 52, no. 5 (2019): 1107–35. http://dx.doi.org/10.24033/asens.2404.

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20

Parker, John R. "On Ford isometric spheres in complex hyperbolic space." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 3 (May 1994): 501–12. http://dx.doi.org/10.1017/s0305004100072261.

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AbstractThe complex hyperbolic version of Shimizu's lemma gives an upper bound on the radii of isometric spheres of maps in a discrete subgroup of PU(n, 1) containing a vertical Heisenberg translation. The purpose of this paper is to show that in a neighbourhood of this bound radii of isometric spheres only take values in a particular discrete set. When the group contains certain ellipto-parabolic maps this upper bound can be improved and the set of values of the radii is more restricted. Examples are given that show that these results cannot be improved.

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21

Aebischer, Beat, and Robert Miner. "Deformation of Schottky groups in complex hyperbolic space." Conformal Geometry and Dynamics of the American Mathematical Society 3, no. 2 (March 11, 1999): 24–36. http://dx.doi.org/10.1090/s1088-4173-99-00010-7.

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22

Toledo, Domingo. "Representations of surface groups in complex hyperbolic space." Journal of Differential Geometry 29, no. 1 (1989): 125–33. http://dx.doi.org/10.4310/jdg/1214442638.

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23

Munteanu, Ovidiu. "On a characterization of the complex hyperbolic space." Journal of Differential Geometry 84, no. 3 (March 2010): 611–21. http://dx.doi.org/10.4310/jdg/1279114302.

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24

Castro, Ildefonso, Cristina R. Montealegre, and Francisco Urbano. "Minimal Lagrangian submanifolds in the complex hyperbolic space." Illinois Journal of Mathematics 46, no. 3 (July 2002): 695–721. http://dx.doi.org/10.1215/ijm/1258130980.

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25

Kamiya, Shigeyasu, John R. Parker, and James M. Thompson. "Non-Discrete Complex Hyperbolic Triangle Groups of Type (n, n, ∞; k)." Canadian Mathematical Bulletin 55, no. 2 (June 1, 2012): 329–38. http://dx.doi.org/10.4153/cmb-2011-094-8.

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AbstractA complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type (n, n, ∞; k).

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26

Pyo, Yong-Soo. "Space-Like Einstein Kähler Submanifolds in an Indefinite Complex Hyperbolic Space." Rocky Mountain Journal of Mathematics 34, no. 3 (September 2004): 1077–101. http://dx.doi.org/10.1216/rmjm/1181069844.

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27

HWANG, JUN-MUK. "VARIETIES WITH DEGENERATE GAUSS MAPPINGS IN COMPLEX HYPERBOLIC SPACE FORMS." International Journal of Mathematics 13, no. 02 (March 2002): 209–16. http://dx.doi.org/10.1142/s0129167x02001186.

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In analogy with the Gauss mapping for a subvariety in the complex projective space, the Gauss mapping for a subvariety in a complex hyperbolic space form can be defined as a map from the smooth locus of the subvariety to the quotient of a suitable domain in the Grassmannian. For complex hyperbolic space forms of finite volume, it is proved that the Gauss mapping is degenerate if and only if the subvariety is totally geodesic.

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28

Cunha, Heleno, Francisco Dutenhefner, Nikolay Gusevskii, and Rafael Santos Thebaldi. "The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane." Journal of Geometric Analysis 22, no. 2 (November 9, 2010): 295–319. http://dx.doi.org/10.1007/s12220-010-9189-1.

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29

FU, XI. "DISCRETENESS CRITERIA FOR ISOMETRIC GROUPS ACTING ON COMPLEX HYPERBOLIC SPACES." Bulletin of the Australian Mathematical Society 81, no. 3 (March 17, 2010): 481–87. http://dx.doi.org/10.1017/s0004972709001245.

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AbstractIn this paper, four new discreteness criteria for isometric groups on complex hyperbolic spaces are proved, one of which shows that the Condition C hypothesis in Cao [‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc.78 (2008), 211–224, Theorem 1.4] is removable; another shows that the parabolic condition hypothesis in Li and Wang [‘Discreteness criteria for Möbius groups acting on $\overline {\mathbb {R}}^n$ II’, Bull. Aust. Math. Soc.80 (2009), 275–290, Theorem 3.1] is not necessary.

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30

Yang, Weihua, and David Rideout. "High Dimensional Hyperbolic Geometry of Complex Networks." Mathematics 8, no. 11 (October 23, 2020): 1861. http://dx.doi.org/10.3390/math8111861.

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High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.

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31

THAI, DO DUC, and NGUYEN THI TUYET MAI. "HARTOGS-TYPE EXTENSION THEOREM AND SINGULAR SETS OF SEPARATELY HOLOMORPHIC MAPPINGS ON COMPACT SETS WITH VALUES IN A WEAKLY BRODY HYPERBOLIC COMPLEX SPACE." International Journal of Mathematics 12, no. 07 (September 2001): 857–65. http://dx.doi.org/10.1142/s0129167x01000873.

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We give a Hartogs-type extension theorem for separately holomorphic mappings on compact sets into a weakly Brody hyperbolic complex space. Moreover, a generalization of Saint Raymond–Siciak theorem of the singular sets of separately holomorphic mappings with values in a weakly Brody hyperbolic complex space is given.

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32

Adachi, Toshiaki, and Sadahiro Maeda. "Global behaviours of circles in a complex hyperbolic space." Tsukuba Journal of Mathematics 21, no. 1 (June 1997): 29–42. http://dx.doi.org/10.21099/tkbjm/1496163159.

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33

Phillips, Mark B. "Dirichlet polyhedra for cyclic groups in complex hyperbolic space." Proceedings of the American Mathematical Society 115, no. 1 (January 1, 1992): 221. http://dx.doi.org/10.1090/s0002-9939-1992-1107276-1.

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34

CAO, WENSHENG. "DISCRETE AND DENSE SUBGROUPS ACTING ON COMPLEX HYPERBOLIC SPACE." Bulletin of the Australian Mathematical Society 78, no. 2 (October 2008): 211–24. http://dx.doi.org/10.1017/s0004972708000622.

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AbstractIn this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.

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35

Ji, Qingchun. "Hamiltonian minimal submanifolds in complex hyperbolic space with -symmetry." Nonlinear Analysis: Theory, Methods & Applications 68, no. 10 (May 2008): 3138–50. http://dx.doi.org/10.1016/j.na.2007.03.007.

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36

Gusevskii, Nikolay, and John R. Parker. "Representations of free Fuchsian groups in complex hyperbolic space." Topology 39, no. 1 (January 2000): 33–60. http://dx.doi.org/10.1016/s0040-9383(98)00057-3.

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37

DANN, SUSANNA. "The Busemann–Petty problem in the complex hyperbolic space." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (February 12, 2013): 155–72. http://dx.doi.org/10.1017/s0305004113000054.

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AbstractThe Busemann–Petty problem asks whether origin-symmetric convex bodies in ℝn with smaller central hyperplane sections necessarily have smaller volume. The answer is affirmative if n ≤ 4 and negative if n ≥ 5. We study this problem in the complex hyperbolic n-space ℍnℂ and prove that the answer is affirmative for n ≤ 2 and negative for n ≥ 3.

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38

GADEA, P. M., A. MONTESINOS AMILIBIA, and J. MUÑOZ MASQUÉ. "Characterizing the complex hyperbolic space by Kähler homogeneous structures." Mathematical Proceedings of the Cambridge Philosophical Society 128, no. 1 (January 2000): 87–94. http://dx.doi.org/10.1017/s0305004199003825.

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The Kähler case of Riemannian homogeneous structures [3, 15, 18] has been studied in [1, 2, 6, 7, 13, 16], among other papers. Abbena and Garbiero [1] gave a classification of Kähler homogeneous structures, which has four primitive classes [Kscr ]1, …, [Kscr ]4 (see [6, theorem 5·1] for another proof and Section 2 below for the result). The purpose of the present paper is to prove the following result:THEOREM 1·1. A simply connected irreducible homogeneous Kähler manifold admits a nonvanishing Kähler homogeneous structure in Abbena–Garbiero's class [Kscr ]2 [oplus ] [Kscr ]4if and only if it is the complex hyperbolic space equipped with the Bergman metric of negative constant holomorphic sectional curvature.

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39

Deraux, Martin, Elisha Falbel, and Julien Paupert. "New constructions of fundamental polyhedra in complex hyperbolic space." Acta Mathematica 194, no. 2 (2005): 155–201. http://dx.doi.org/10.1007/bf02393220.

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40

Alanis-Lobato, Gregorio, and Miguel A. Andrade-Navarro. "Distance Distribution between Complex Network Nodes in Hyperbolic Space." Complex Systems 25, no. 3 (October 15, 2016): 223–36. http://dx.doi.org/10.25088/complexsystems.25.3.223.

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41

Montiel, Sebastián, and Alfonso Romero. "On some real hypersurfaces of a complex hyperbolic space." Geometriae Dedicata 20, no. 2 (April 1986): 245–61. http://dx.doi.org/10.1007/bf00164402.

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42

Maeda, Sadahiro. "Geometry of the horosphere in a complex hyperbolic space." Differential Geometry and its Applications 29 (August 2011): S246—S250. http://dx.doi.org/10.1016/j.difgeo.2011.04.048.

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43

Wu, Zongning, Zengru Di, and Ying Fan. "An Asymmetric Popularity-Similarity Optimization Method for Embedding Directed Networks into Hyperbolic Space." Complexity 2020 (April 22, 2020): 1–16. http://dx.doi.org/10.1155/2020/8372928.

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Network embedding is a frontier topic in current network science. The scale-free property of complex networks can emerge as a consequence of the exponential expansion of hyperbolic space. Some embedding models have recently been developed to explore hyperbolic geometric properties of complex networks—in particular, symmetric networks. Here, we propose a model for embedding directed networks into hyperbolic space. In accordance with the bipartite structure of directed networks and multiplex node information, the method replays the generation law of asymmetric networks in hyperbolic space, estimating the hyperbolic coordinates of each node in a directed network by the asymmetric popularity-similarity optimization method in the model. Additionally, the experiments in several real networks show that our embedding algorithm has stability and that the model enlarges the application scope of existing methods.

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44

Lluch, Ana. "Isometric immersions in the hyperbolic space with their image contained in a horoball." Glasgow Mathematical Journal 43, no. 1 (January 2001): 1–8. http://dx.doi.org/10.1017/s0017089501010011.

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We give a sharp lower bound for the supremum of the norm of the mean curvature of an isometric immersion of a complete Riemannian manifold with scalar curvature bounded from below into a horoball of a complex or real hyperbolic space. We also characterize the horospheres of the real or complex hyperbolic spaces as the only isometrically immersed hypersurfaces which are between two parallel horospheres, have the norm of the mean curvature vector bounded by the above sharp bound and have some special groups of symmetries.

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45

Cunha, Heleno, and Nikolay Gusevskii. "The Moduli Space of Points in the Boundary of Complex Hyperbolic Space." Journal of Geometric Analysis 22, no. 1 (November 9, 2010): 1–11. http://dx.doi.org/10.1007/s12220-010-9188-2.

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46

Castle, Toen, Myfanwy E. Evans, Stephen T. Hyde, Stuart Ramsden, and Vanessa Robins. "Trading spaces: building three-dimensional nets from two-dimensional tilings." Interface Focus 2, no. 5 (January 25, 2012): 555–66. http://dx.doi.org/10.1098/rsfs.2011.0115.

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We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic ( S 2 ), Euclidean ( E 2 ) and hyperbolic ( H 2 ) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

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47

KHRENNIKOV, A. "HILBERT SPACE OVER COMPLEX HYPERBOLIC NUMBERS AND HYPER-TRIGONOMETRIC INTERFERENCE." Infinite Dimensional Analysis, Quantum Probability and Related Topics 12, no. 03 (September 2009): 469–78. http://dx.doi.org/10.1142/s0219025709003847.

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This note is devoted to extension of quantum probability calculus to generalizations of complex Hilbert space. Starting with Hilbert space over complex hyperbolic numbers, we derive general hyper-trigonometric interference of probabilities.

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48

Gadea, P. M., and J. A. Oubiña. "Homogeneous Kähler and Sasakian structures related to complex hyperbolic spaces." Proceedings of the Edinburgh Mathematical Society 53, no. 2 (April 30, 2010): 393–413. http://dx.doi.org/10.1017/s0013091508001004.

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AbstractWe study homogeneous Kähler structures on a non-compact Hermitian symmetric space and their lifts to homogeneous Sasakian structures on the total space of a principal line bundle over it, and we analyse the case of the complex hyperbolic space.

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49

Zhang, Yiding, Xiao Wang, Nian Liu, and Chuan Shi. "Embedding Heterogeneous Information Network in Hyperbolic Spaces." ACM Transactions on Knowledge Discovery from Data 16, no. 2 (April 30, 2022): 1–23. http://dx.doi.org/10.1145/3468674.

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Heterogeneous information network (HIN) embedding, aiming to project HIN into a low-dimensional space, has attracted considerable research attention. Most of the existing HIN embedding methods focus on preserving the inherent network structure and semantic correlations in Euclidean spaces. However, one fundamental problem is whether the Euclidean spaces are the intrinsic spaces of HIN? Recent researches find the complex network with hyperbolic geometry can naturally reflect some properties, e.g., hierarchical and power-law structure. In this article, we make an effort toward embedding HIN in hyperbolic spaces. We analyze the structures of three HINs and discover some properties, e.g., the power-law distribution, also exist in HINs. Therefore, we propose a novel HIN embedding model HHNE. Specifically, to capture the structure and semantic relations between nodes, HHNE employs the meta-path guided random walk to sample the sequences for each node. Then HHNE exploits the hyperbolic distance as the proximity measurement. We also derive an effective optimization strategy to update the hyperbolic embeddings iteratively. Since HHNE optimizes different relations in a single space, we further propose the extended model HHNE++. HHNE++ models different relations in different spaces, which enables it to learn complex interactions in HINs. The optimization strategy of HHNE++ is also derived to update the parameters of HHNE++ in a principle manner. The experimental results demonstrate the effectiveness of our proposed models.

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50

Cao, Zongsheng, Qianqian Xu, Zhiyong Yang, Xiaochun Cao, and Qingming Huang. "Geometry Interaction Knowledge Graph Embeddings." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 5 (June 28, 2022): 5521–29. http://dx.doi.org/10.1609/aaai.v36i5.20491.

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Knowledge graph (KG) embeddings have shown great power in learning representations of entities and relations for link prediction tasks. Previous work usually embeds KGs into a single geometric space such as Euclidean space (zero curved), hyperbolic space (negatively curved) or hyperspherical space (positively curved) to maintain their specific geometric structures (e.g., chain, hierarchy and ring structures). However, the topological structure of KGs appears to be complicated, since it may contain multiple types of geometric structures simultaneously. Therefore, embedding KGs in a single space, no matter the Euclidean space, hyperbolic space or hyperspheric space, cannot capture the complex structures of KGs accurately. To overcome this challenge, we propose Geometry Interaction knowledge graph Embeddings (GIE), which learns spatial structures interactively between the Euclidean, hyperbolic and hyperspherical spaces. Theoretically, our proposed GIE can capture a richer set of relational information, model key inference patterns, and enable expressive semantic matching across entities. Experimental results on three well-established knowledge graph completion benchmarks show that our GIE achieves the state-of-the-art performance with fewer parameters.

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