Bibliografías temáticas / Curvature bounds

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Literatura académica sobre el tema "Curvature bounds"

Autor: Grafiati
Publicado: 10 de marzo de 2023

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Artículos de revistas sobre el tema "Curvature bounds"

1

Bessa, Gregório P., Luquésio P. Jorge, Barnabé P. Lima y José F. Montenegro. "Fundamental tone estimates for elliptic operators in divergence form and geometric applications". Anais da Academia Brasileira de Ciências 78, n.º 3 (septiembre de 2006): 391–404. http://dx.doi.org/10.1590/s0001-37652006000300001.

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We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r + 1)-th mean curvature Hr + 1 of the space forms Nn+ 1(c) of constant sectional curvature c. As a corollary we give lower bounds for the extrinsic radius of closed hypersurfaces of Nn+ 1(c) with Hr + 1 > 0 in terms of the r-th and (r + 1)-th mean curvatures. Finally we observe that bounds for the Laplace eigenvalues essentially bound the eigenvalues of a self-adjoint elliptic differential operator in divergence form. This allows us to show that Cheeger's constant gives a lower bounds for the first nonzero Lr-eigenvalue of a closed hypersurface of Nn+ 1(c).
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Peters, Jörg y Georg Umlauf. "Computing curvature bounds for bounded curvature subdivision". Computer Aided Geometric Design 18, n.º 5 (junio de 2001): 455–61. http://dx.doi.org/10.1016/s0167-8396(01)00041-3.

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Sabatini, Luca. "Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II". Analele Universitatii "Ovidius" Constanta - Seria Matematica 28, n.º 1 (1 de marzo de 2020): 165–79. http://dx.doi.org/10.2478/auom-2020-0012.

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AbstractWe present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.
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Frenck, Georg y Jan-Bernhard Kordaß. "Spaces of positive intermediate curvature metrics". Geometriae Dedicata 214, n.º 1 (23 de junio de 2021): 767–800. http://dx.doi.org/10.1007/s10711-021-00635-w.

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AbstractIn this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional $$\mathrm {Spin}$$ Spin -manifold have many non-trivial homotopy groups provided that the manifold admits such a metric.
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Erbar, Matthias y Martin Huesmann. "Curvature bounds for configuration spaces". Calculus of Variations and Partial Differential Equations 54, n.º 1 (19 de noviembre de 2014): 397–430. http://dx.doi.org/10.1007/s00526-014-0790-1.

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Lytchak, Alexander y Stephan Stadler. "Improvements of upper curvature bounds". Transactions of the American Mathematical Society 373, n.º 10 (5 de agosto de 2020): 7153–66. http://dx.doi.org/10.1090/tran/8123.

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Kapovitch, Vitali. "Curvature bounds via Ricci smoothing". Illinois Journal of Mathematics 49, n.º 1 (enero de 2005): 259–63. http://dx.doi.org/10.1215/ijm/1258138317.

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Hu, Zisheng y Senlin Xu. "Bounds on the fundamental groups with integral curvature bound". Geometriae Dedicata 134, n.º 1 (19 de abril de 2008): 1–16. http://dx.doi.org/10.1007/s10711-008-9235-3.

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Lott, John. "On scalar curvature lower bounds and scalar curvature measure". Advances in Mathematics 408 (octubre de 2022): 108612. http://dx.doi.org/10.1016/j.aim.2022.108612.

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Wang, Xu-Jia, John Urbas y Weimin Sheng. "Interior curvature bounds for a class of curvature equations". Duke Mathematical Journal 123, n.º 2 (junio de 2004): 235–64. http://dx.doi.org/10.1215/s0012-7094-04-12321-8.

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Tesis sobre el tema "Curvature bounds"

1

Rose, Christian. "Heat kernel estimates based on Ricci curvature integral bounds". Doctoral thesis, Universitätsbibliothek Chemnitz, 2017. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-228681.

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Any Riemannian manifold possesses a minimal solution of the heat equation for the Dirichlet Laplacian, called the heat kernel. During the last decades many authors investigated geometric properties of the manifold such that its heat kernel fulfills a so-called Gaussian upper bound. Especially compact and non-compact manifolds with lower bounded Ricci curvature have been examined and provide such Gaussian estimates. In the compact case it ended even with integral Ricci curvature assumptions. The important techniques to obtain Gaussian bounds are the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates or gradient estimates for the heat equation, where the first two base on isoperimetric properties of certain sets. In this thesis, we generalize the existing results to the following. Locally uniform integral bounds on the negative part of Ricci curvature lead to Gaussian upper bounds for the heat kernel, no matter whether the manifold is compact or not. Therefore, we show local isoperimetric inequalities under this condition and use relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. If the manifold is compact, we can even generalize the integral curvature condition to the case that the negative part of Ricci curvature is in the so-called Kato class. We even obtain uniform Gaussian upper bounds using gradient estimate techniques. Apart from the geometric generalizations for obtaining Gaussian upper bounds we use those estimates to generalize Bochner’s theorem. More precisely, the estimates for the heat kernel obtained above lead to ultracontractive estimates for the heat semigroup and the semigroup generated by the Hodge Laplacian. In turn, we can formulate rigidity results for the triviality of the first cohomology group if the amount of curvature going below a certain positive threshold is small in a suitable sense. If we can only assume such smallness of the negative part of the Ricci curvature, we can bound the Betti number by explicit terms depending on the generalized curvature assumptions in a uniform manner, generalizing certain existing results from the cited literature
Jede Riemannsche Mannigfaltigkeit besitzt eine minimale Lösung für die Wärmeleitungsgleichung des zur Mannigfaltigkeit gehörigen Dirichlet-Laplaceoperators, den Wärmeleitungskern. Während der letzten Jahrzehnte fanden viele Autoren geometrische Eigenschaften der Mannigfaltigkeiten unter welchen der Wärmeleitungskern eine sogenannte Gaußsche obere Abschätzung besitzt. Insbesondere bestizen sowohl kompakte als auch nichtkompakte Mannigfaltigkeiten mit nach unten beschränkter Ricci-Krümmung solche Gaußschen Abschätzungen. Im kompakten Fall reichten bisher sogar Integralbedingungen an die Ricci-Krümmung aus. Die wichtigen Techniken, um Gaußsche Abschätzungen zu erhalten, sind die Symmetrisierung für kompakte Mannigfaltigkeiten und relative Faber-Krahn- und Gradientenabschätzungen für die Wärmeleitungsgleichung, wobei die ersten beiden auf isoperimetrischen Eigenschaften gewisser Mengen beruhen. In dieser Arbeit verallgemeinern wir die bestehenden Resultate im folgenden Sinne. Lokal gleichmäßig beschränkte Integralschranken an den Negativteil der Ricci-Krümmung ergeben Gaußsche obere Abschätzungen sowohl im kompakten als auch nichtkompakten Fall. Dafür zeigen wir lokale isoperimetrische Ungleichungen unter dieser Voraussetzung und nutzen die relativen Faber-Krahn-Abschätzungen für eine explizite Gaußsche Schranke. Für kompakte Mannigfaltigkeiten können wir sogar die Integralschranken an den Negativteil der Ricci-Krümmung durch die sogenannte Kato-Bedingung ersetzen. In diesem Fall erhalten wir gleichmäßige Gaußsche Abschätzungen mit einer Gradientenabschätzung. Neben den geometrischen Verallgemeinerungen für Gaußsche Schranken nutzen wir unsere Ergebnisse, um Bochners Theorem zu verallgemeinern. Wärmeleitungskernabschätzungen ergeben ultrakontraktive Schranken für die Wärmeleitungshalbgruppe und die Halbgruppe, die durch den Hodge-Operator erzeugt wird. Damit können wir Starrheitseigenschaften für die erste Kohomologiegruppe zeigen, wenn der Teil der Ricci-Krümmung, welcher unter einem positiven Level liegt, in einem bestimmten Sinne klein genug ist. Wenn der Negativteil der Ricci-Krümmung nicht zu groß ist, können wir die erste Betti-Zahl noch immer explizit uniform abschätzen
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2

Gursky, Matthew J. "Compactness of Conformal Metrics with Integral Bounds on Curvature". Diss., Pasadena, Calif. : California Institute of Technology, 1991. http://resolver.caltech.edu/CaltechETD:etd-06192007-145905.

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Zergänge, Norman [Verfasser]. "Convergence of Riemannian manifolds with critical curvature bounds / Norman Zergänge". Magdeburg : Universitätsbibliothek, 2017. http://d-nb.info/1141230488/34.

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Renesse, Max-K. von. "Comparison properties of diffusion semigroups on spaces with lower curvature bounds". Bonn : Mathematisches Institut der Universität Bonn, 2003. http://catalog.hathitrust.org/api/volumes/oclc/52348149.html.

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Mroz, Kamil. "Bounds on eigenfunctions and spectral functions on manifolds of negative curvature". Thesis, Loughborough University, 2014. https://dspace.lboro.ac.uk/2134/15038.

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In this dissertation we study the Laplace operator acting on functions on a smooth, compact Riemannian manifold. Our approach is based on the study of the spectrum of the aforementioned operator. The main objects of our interest are the counting function of the Laplacian and its Riesz means. We discuss the asymptotics of aforementioned functions when the argument approaches infinity.
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Ketterer, Christian Eugen Michael [Verfasser]. "Ricci curvature bounds for warped products and cones / Christian Eugen Michael Ketterer". Bonn : Universitäts- und Landesbibliothek Bonn, 2014. http://d-nb.info/1238687571/34.

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Richardson, James. "Inradius bounds for stable, minimal surfaces in 3-manifolds with positive scalar curvature". Thesis, University of British Columbia, 2012. http://hdl.handle.net/2429/42368.

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Concrete topological properties of a manifold can be found by examining its geometry. Theorem 17 of his thesis, due to Myers [Mye41], is one such example of this; it gives an upper bound on the length of any minimizing geodesic in a manifold N in terms of a lower positive bound on the Ricci curvature of N, and concludes that N is compact. Our main result, Theorem 40, is of the same flavour as this, but we are instead concerned with stable, minimal surfaces in manifolds of positive scalar curvature. This result is a version of Proposition 1 in the paper of Schoen and Yau [SY83], written in the context of Riemannian geometry. It states: a stable, minimal 2-submanifold of a 3-manifold whose scalar curvature is bounded below by κ > 0 has a inradius bound of ≤√(8/3) π/√κ, and in particular is compact.
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COLOMBO, GIULIO. "GLOBAL GRADIENT BOUNDS FOR SOLUTIONS OF PRESCRIBED MEAN CURVATURE EQUATIONS ON RIEMANNIAN MANIFOLDS". Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/813095.

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This thesis is concerned with the study of qualitative properties of solutions of the minimal surface equation and of a class of prescribed mean curvature equations on complete Riemannian manifolds. We derive global gradient bounds for non-negative solutions of such equations on manifolds satisfying a uniform Ricci lower bound and we obtain Liouville-type theorems and other rigidity results on Riemannian manifolds with non-negative Ricci curvature. The proof of the aforementioned global gradient bounds for non-negative solutions u is based on the application of the maximum principle to an elliptic differential inequality satisfied by a suitable auxiliary function z=f(u,|Du|), in the spirit of Bernstein’s method of a priori estimates for nonlinear PDEs and of Yau’s proof of global gradient bounds for harmonic functions on complete Riemannian manifolds. The particular choice of the auxiliary function z parallels the one in Korevaar’s proof of a priori gradient estimates for the prescribed mean curvature equation in Euclidean space. The rigidity results obtained in the last part of the thesis include a Liouville theorem for positive solutions of the minimal surface equation on complete Riemannian manifolds with non-negative Ricci curvature, a splitting theorem for complete parabolic manifolds of non-negative sectional curvature supporting non-constant solutions with linear growth of the minimal surface equation, and a splitting theorem for domains of complete parabolic manifolds with non-negative Ricci curvature supporting non-constant solutions of overdetermined problems involving the mean curvature operator.
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Schlichting, Arthur [Verfasser] y Miles [Akademischer Betreuer] Simon. "Smoothing singularities of Riemannian metrics while preserving lower curvature bounds / Arthur Schlichting. Betreuer: Miles Simon". Magdeburg : Universitätsbibliothek, 2014. http://d-nb.info/1054638039/34.

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Rose, Christian [Verfasser], Peter [Akademischer Betreuer] Stollmann, Peter [Gutachter] Stollmann, Alexander [Gutachter] Grigor’yan y Gilles [Gutachter] Carron. "Heat kernel estimates based on Ricci curvature integral bounds / Christian Rose ; Gutachter: Peter Stollmann, Alexander Grigor’yan, Gilles Carron ; Betreuer: Peter Stollmann". Chemnitz : Universitätsbibliothek Chemnitz, 2017. http://d-nb.info/1214306705/34.

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Libros sobre el tema "Curvature bounds"

1

Degeneration of Riemannian Metrics under Ricci Curvature Bounds. Scuola Normale Superiore, 2001.

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2

Sogge, Christopher D. The sharp Weyl formula. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160757.003.0003.

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This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) in a natural way, so that N(λ‎) = cλ‎ⁿ + O(λ‎superscript n minus 1). The chapter then details the sup-norm estimates for eigenfunctions and spectral clusters. Next, this chapter proves the sharp Weyl formula and in doing so, outlines a number of theorems, the first of which the chapter focuses on in establishing its sharpness and in obtaining improved bounds for its Weyl formula's error term. Finally, the chapter shows that improved bounds are also available for the remainder term in the Weyl formula when (M,g) has nonpositive sectional curvature.
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Gigli, Nicola. Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from Below. American Mathematical Society, 2018.

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Limebeer, D. J. N. y Matteo Massaro. Dynamics and Optimal Control of Road Vehicles. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198825715.001.0001.

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The broad aim of this book is to provide a comprehensive coverage of the modelling and optimal control of both two‐ and four‐wheeled road vehicles. The first focus of this book is a review of classical mechanics and its use in building vehicle and tyre dynamic models. The second is nonlinear optimal control, which is used to solve a range of minimum‐time, minimum‐fuel, and track curvature reconstruction problems. As is known classically, all thismaterial is bound together by the calculus of variations and stationary principles. The treatment of this material is supplemented with a large number of examples that highlight obscurities and subtleties in the theory. A particular strength of the book is its unified treatment of tyre, car, and motorcycle dynamics and the application of nonlinear optimal control to vehicle‐related problems within a single text. These topics are usually treated independently, and can only be found in disparate texts and journal articles. It is our contention that presentday vehicle dynamicists should be familiar with all of these topic areas. The aim in writing this book is to provide a comprehensive and yet accessible text that emphasizes particularly the theoretical aspects of vehicular modelling and control.
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Capítulos de libros sobre el tema "Curvature bounds"

1

Utcke, Sven. "Error-Bounds on Curvature Estimation". En Scale Space Methods in Computer Vision, 657–66. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44935-3_46.

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Keller, Matthias. "Geometric and Spectral Consequences of Curvature Bounds on Tessellations". En Modern Approaches to Discrete Curvature, 175–209. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-58002-9_6.

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LeBrun, Claude. "Four-Manifolds, Curvature Bounds, and Convex Geometry". En Riemannian Topology and Geometric Structures on Manifolds, 119–52. Boston, MA: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4743-8_6.

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Villani, Cédric. "Weak Ricci curvature bounds I: Definition and Stability". En Grundlehren der mathematischen Wissenschaften, 795–846. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-71050-9_29.

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Villani, Cédric. "Weak Ricci curvature bounds II: Geometric and analytic properties". En Grundlehren der mathematischen Wissenschaften, 847–901. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-540-71050-9_30.

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Anderson, Michael T. "Einstein Metrics and Metrics with Bounds on Ricci Curvature". En Proceedings of the International Congress of Mathematicians, 443–52. Basel: Birkhäuser Basel, 1995. http://dx.doi.org/10.1007/978-3-0348-9078-6_37.

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Graf, Melanie y Christina Sormani. "Lorentzian Area and Volume Estimates for Integral Mean Curvature Bounds". En Developments in Lorentzian Geometry, 105–28. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-05379-5_7.

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Meeks, William y Joaquín Pérez. "Limits of embedded minimal surfaces without local area or curvature bounds". En A Survey on Classical Minimal Surface Theory, 53–72. Providence, Rhode Island: American Mathematical Society, 2012. http://dx.doi.org/10.1090/ulect/060/04.

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Solís-Daun, Julio. "Convexity + curvature: Tools for the global stabilization of nonlinear systems with control inputs subject to magnitude and rate bounds". En 2015 Proceedings of the Conference on Control and its Applications, 131–38. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2015. http://dx.doi.org/10.1137/1.9781611974072.19.

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Ballmann, Werner, Mikhael Gromov y Viktor Schroeder. "Manifolds of bounded negative curvature". En Manifolds of Nonpositive Curvature, 110–19. Boston, MA: Birkhäuser Boston, 1985. http://dx.doi.org/10.1007/978-1-4684-9159-3_10.

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Actas de conferencias sobre el tema "Curvature bounds"

1

LUCKHAUS, STEPHAN. "UNIFORM RECTIFIABILITY FROM MEAN CURVATURE BOUNDS". En Proceedings of the International Conference. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702050_0014.

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ELIZALDE, E. y A. C. TORT. "ENTROPY BOUNDS FOR A MASSIVE SCALAR FIELD IN POSITIVE CURVATURE SPACE". En Proceedings of the MG10 Meeting held at Brazilian Center for Research in Physics (CBPF). World Scientific Publishing Company, 2006. http://dx.doi.org/10.1142/9789812704030_0303.

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AMBROSIO, LUIGI. "CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES". En International Congress of Mathematicians 2018. WORLD SCIENTIFIC, 2019. http://dx.doi.org/10.1142/9789813272880_0015.

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Liu, Yajing, Edwin K. P. Chong, Ali Pezeshki y Bill Moran. "Bounds for approximate dynamic programming based on string optimization and curvature". En 2014 IEEE 53rd Annual Conference on Decision and Control (CDC). IEEE, 2014. http://dx.doi.org/10.1109/cdc.2014.7040433.

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Liu, Yajing, Zhenliang Zhang, Edwin K. P. Chong y Ali Pezeshki. "Performance bounds for the k-batch greedy strategy in optimization problems with curvature". En 2016 American Control Conference (ACC). IEEE, 2016. http://dx.doi.org/10.1109/acc.2016.7526805.

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Sgorbissa, A. y R. Zaccaria. "3D path following with no bounds on the path curvature through surface intersection". En 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2010). IEEE, 2010. http://dx.doi.org/10.1109/iros.2010.5653235.

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Rossetter, Eric J. y J. Christian Gerdes. "Safety Guarantees for Lanekeeping Assistance Systems With Time-Varying Disturbances: A Lyapunov Approach". En ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-41851.

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Active lanekeeping assistance systems have the potential of saving thousands of lives every year. These systems must provide some level of increased safety while working cooperatively with the driver. Our approach passively couples the vehicle to the environment using the paradigm of ‘artificial’ potential fields. Since this control scheme does not attempt to track a desired trajectory, disturbances encountered during normal driving (such as road curvature) will alter the path of the vehicle. In order to ensure nominal safety of this system, a Lyapunov based bound is developed that handles general time-varying disturbances. This technique provides excellent bounds on the lateral motion of the vehicle. Experimental results verify that this bound works extremely well in practice.
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Ciavarella, M. y J. R. Barber. "Elastic Contact Stiffness and Contact Resistance for Fractal Profiles". En ASME/STLE 2004 International Joint Tribology Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/trib2004-64357.

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A recent theorem due to Barber shows an analogy between conductance and incremental stiffness of a contact, implying bounds on conductance based on peak-to-peak roughness. This shows that even a fractal roughness, with bounded amplitude, has a finite conductance. The analogy also permits a simple interpretation of classical results of rough contact models based on independent asperities such as Greenwood-Williamson and developments. For example, in the GW model with exponential distribution of asperity heights, the conductance is found simply proportional to load, and inversely proportional to a roughness amplitude parameter which does not depend greatly on resolution, contrary to parameters of slopes and curvatures. However, for the Gaussian distribution or for more refined models also considering varying curvature of asperities (such as Bush Gibson and Thomas), there is dependence on sampling interval and the conductance grows unbounded. An alternative choice of asperity definition (Aramaki-Majumdar-Bhushan) is suggested, which builds on the geometrical intersection of the rough surface, with the consequence of a finite contact area, and converging load-separation and load-conductance relationship. A discussion follows, also based on numerical results.
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Widmann, James M. y Sheri D. Sheppard. "Intrinsic Geometry for Shape Optimal Design With Analysis Model Compatibility". En ASME 1994 Design Technical Conferences collocated with the ASME 1994 International Computers in Engineering Conference and Exhibition and the ASME 1994 8th Annual Database Symposium. American Society of Mechanical Engineers, 1994. http://dx.doi.org/10.1115/detc1994-0137.

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Abstract This paper presents a comparison of geometric modeling techniques and their applicability to structural shape optimization. A method of shape definition based on intrinsic geometric quantities is then outlined. Explicit knowledge of curvature and arc length allow for a quantitative assessment of the compatibility of analysis model with the design model when using finite elements to determine structural response quantities. The compatibility condition is formalized by controlling finite element idealization error and is incorporated into the shape optimization model as simple bounds on the curvature design variables. Several examples of shape optimization problems are solved using sequential quadratic programming which proves to be an effective tool for maintaining the geometric equality constraints that arise from intrinsically defined curves.
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Zhang, Dongdong, Pinghai Yang y Xiaoping Qian. "Adaptive NC Path Generation From Massive Point Data With Bounded Error". En ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/detc2008-49626.

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This paper presents an approach for generating curvature-adaptive finishing tool paths with bounded error directly from massive point data in three-axis CNC milling. This approach uses the Moving Least Squares (MLS) surface as the underlying surface representation. A closed-form formula for normal curvature computing is derived from the implicit form of MLS surfaces. It enables the generation of curvature-adaptive tool paths from massive point data that is critical for balancing the trade-off between machining accuracy and speed. To ensure the path accuracy and robustness for arbitrary surfaces where there might be abrupt curvature change, a novel guidance field algorithm is introduced. It overcomes potential excessive locality of curvature-adaptive paths by examining the neighboring points’ curvature within a self-updating search bound. Our results affirm that the combination of curvature-adaptive path generation and the guidance field algorithm produces high-quality NC paths from a variety of point cloud data with bounded error.
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