Relevant bibliographies by topics / Non-smooth differential geometry

Academic literature on the topic 'Non-smooth differential geometry'

Author: Grafiati
Published: 4 June 2021
Last updated: 28 January 2023

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Journal articles on the topic "Non-smooth differential geometry"

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Mussel, Matan, and Marshall Slemrod. "Conservation laws in biology: Two new applications." Quarterly of Applied Mathematics 79, no. 3 (March 25, 2021): 479–92. http://dx.doi.org/10.1090/qam/1590.

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This paper provides two new applications of conservation laws in biology. The first is the application of the van der Waals fluid formalism for action potentials. The second is the application of the conservation laws of differential geometry (Gauss–Codazzi equations) to produce non-smooth surfaces representing Endoplasmic Reticulum sheets.
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Felice, Domenico, and Carlo Cafaro. "Explicit Information Geometric Calculations of the Canonical Divergence of a Curve." Mathematics 10, no. 9 (April 26, 2022): 1452. http://dx.doi.org/10.3390/math10091452.

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Information geometry concerns the study of a dual structure (g,∇,∇*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,∇,∇*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.
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Felice, Domenico, and Carlo Cafaro. "Explicit Information Geometric Calculations of the Canonical Divergence of a Curve." Mathematics 10, no. 9 (April 26, 2022): 1452. http://dx.doi.org/10.3390/math10091452.

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Abstract:
Information geometry concerns the study of a dual structure (g,∇,∇*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,∇,∇*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.
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4

Felice, Domenico, and Carlo Cafaro. "Explicit Information Geometric Calculations of the Canonical Divergence of a Curve." Mathematics 10, no. 9 (April 26, 2022): 1452. http://dx.doi.org/10.3390/math10091452.

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Abstract:
Information geometry concerns the study of a dual structure (g,∇,∇*) upon a smooth manifold M. Such a geometry is totally encoded within a potential function usually referred to as a divergence or contrast function of (g,∇,∇*). Even though infinitely many divergences induce on M the same dual structure, when the manifold is dually flat, a canonical divergence is well defined and was originally introduced by Amari and Nagaoka. In this pedagogical paper, we present explicit non-trivial differential geometry-based proofs concerning the canonical divergence for a special type of dually flat manifold represented by an arbitrary 1-dimensional path γ. Highlighting the geometric structure of such a particular canonical divergence, our study could suggest a way to select a general canonical divergence by using the information from a general dual structure in a minimal way.
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5

Polyakova, K. V. "On some extension of the second order tangent space for a smooth manifold." Differential Geometry of Manifolds of Figures, no. 53 (2022): 94–111. http://dx.doi.org/10.5922/0321-4796-2022-53-9.

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This paper relates to differential geometry, and the research technique is based on G. F. Laptev’s method of extensions and envelopments, which generalizes E. Cartan’s method of moving frame and exterior forms. We consider a smooth m-dimensional manifold, its tangent and cotangent spaces, as well as the second-order frames and coframes on this manifold. Using the perturbation of the exterior derivative and ordinary diffe­ren­tial, mappings are introduced that enable us to construct non-sym­met­rical second-order frames and coframes on a smooth manifold. It is shown that the extension of the second order tangent space to a smooth m-dimen­sional manifold is carried out by adding the vertical vectors to the linear frame bundle over the manifold to the second order tangent vectors to this manifold. A deformed external differential is widely used, which is a differen­tial, i. e., its reapplication vanishes. We introduce a deformed external dif­ferential being a differential along the curves on the manifold, i. e., its re­peated application along the curves on the manifold gives zero.
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Pak, Hee Chul. "Geometric two-scale convergence on forms and its applications to Maxwell's equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 135, no. 1 (February 2005): 133–47. http://dx.doi.org/10.1017/s0308210500003802.

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We develop the geometric two-scale convergence on forms in order to describe the homogenization of partial differential equations with random variables on non-flat domain. We prove the compactness theorem and some two-scale behaviours for differential forms. For its applications, we investigate the limiting equations of the n-dimensional Maxwell equations with random coefficients, with given initial and boundary conditions, where are symmetric positive-definite matrices for x ∈ M, and M is an n-dimensional compact oriented Riemannian manifold with smooth boundary. The limiting system of n-dimensional Maxwell equations turns out to be degenerate and it is proven to be well-posed. The homogenized coefficients affected by the geometry of the domain are presented, and compared with the homogenized coefficient of the second order elliptic equation. We present the convergence theorem in order to explain the convergence of the solutions of Maxwell system as a parabolic partial differential equation.
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Nigsch, E. A., and J. A. Vickers. "A nonlinear theory of distributional geometry." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2244 (December 2020): 20200642. http://dx.doi.org/10.1098/rspa.2020.0642.

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This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 ( doi:10.1098/rspa.2020.0640 )) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.
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Csetnek, Ernö Robert. "Continuous Dynamics Related to Monotone Inclusions and Non-Smooth Optimization Problems." Set-Valued and Variational Analysis 28, no. 4 (July 13, 2020): 611–42. http://dx.doi.org/10.1007/s11228-020-00548-y.

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Abstract The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth functions. The asymptotic analysis of the trajectories generated relies on Lyapunov theory, where the appropriate energy functional plays a decisive role. While the most part of the paper is related to monotone inclusions and convex optimization problems in the variational case, we present also results for dynamical systems for solving non-convex optimization problems, where the Kurdyka-Łojasiewicz property is used.
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Grandjean, Vincent, and Daniel Grieser. "The exponential map at a cuspidal singularity." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 736 (March 1, 2018): 33–67. http://dx.doi.org/10.1515/crelle-2015-0020.

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AbstractWe study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric.
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Dodwell, T. J., G. W. Hunt, M. A. Peletier, and C. J. Budd. "Multi-layered folding with voids." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1965 (April 28, 2012): 1740–58. http://dx.doi.org/10.1098/rsta.2011.0340.

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In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.
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Dissertations / Theses on the topic "Non-smooth differential geometry"

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Choi, Jaedong. "Warped product spaces with non-smooth warping functions /." free to MU campus, to others for purchase, 2000. http://wwwlib.umi.com/cr/mo/fullcit?p9974615.

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Fama, Christopher J., and -. "Non-smooth differential geometry of pseudo-Riemannian manifolds: Boundary and geodesic structure of gravitational wave space-times in mathematical relativity." The Australian National University. School of Mathematical Sciences, 1998. http://thesis.anu.edu.au./public/adt-ANU20010907.161849.

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[No abstract supplied with this thesis - The first page (of three) of the Introduction follows] ¶ This thesis is largely concerned with the changing representations of 'boundary' or 'ideal' points of a pseudo-Riemannian manifold -- and our primary interest is in the space-times of general relativity. In particular, we are interested in the following question: What assumptions about the 'nature' of 'portions' of a certain 'ideal boundary' construction (essentially the 'abstract boundary' of Scott and Szekeres (1994)) allow us to define precisely the topological type of these 'portions', i.e., to show that different representations of this ideal boundary, corresponding to different embeddings of the manifold into others, have corresponding 'portions' that are homeomorphic? ¶ Certain topological properties of these 'portions' are preserved, even allowing for quite unpleasant properties of the metric (Fama and Scott 1995). These results are given in Appendix D, since they are not used elsewhere and, as well as representing the main portion of work undertaken under the supervision of Scott, which deserves recognition, may serve as an interesting example of the relative ease with which certain simple results about the abstract boundary can be obtained. ¶ An answer to a more precisely formulated version of this question appears very diffcult in general. However, we can give a rather complete answer in certain cases, where we dictate certain 'generalised regularity' requirements for our embeddings, but make no demands on the precise functional form of our metrics apart from these. For example, we get a complete answer to our question for abstract boundary sets which do not 'wiggle about' too much -- i.e., they satisfy a certain Lipschitz condition -- and through which the metric can be extended in a manner which is not required to be differentiable (C[superscript1]), but is continuous and non--degenerate. We allow similar freedoms on the interior of the manifold, thereby bringing gravitational wave space-times within our sphere of discussion. In fact, in the course of developing these results in progressively greater generality, we get, almost 'free', certain abilities to begin looking at geodesic structure on quite general pseudo-Riemannian manifolds. ¶ It is possible to delineate most of this work cleanly into two major parts. Firstly, there are results which use classical geometric constructs and can be given for the original abstract boundary construction, which requires differentiability of both manifolds and metrics, and which we summarise below. The second -- and significantly longer -- part involves extensions of those constructs and results to more general metrics.
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Fama, Christopher J. "Non-smooth differential geometry of pseudo-Riemannian manifolds: Boundary and geodesic structure of gravitational wave space-times in mathematical relativity." Phd thesis, 1998. http://hdl.handle.net/1885/46917.

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This thesis is largely concerned with the changing representations of 'boundary' or 'ideal' points of a pseudo-Riemannian manifold -- and our primary interest is in the space-times of general relativity. In particular, we are interested in the following question: What assumptions about the 'nature' of 'portions' of a certain 'ideal boundary' construction (essentially the 'abstract boundary' of Scott and Szekeres (1994)) allow us to define precisely the topological type of these 'portions', i.e., to show that different representations of this ideal boundary, corresponding to different embeddings of the manifold into others, have corresponding 'portions' that are homeomorphic?
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Books on the topic "Non-smooth differential geometry"

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Blowing Up of Non-Commutative Smooth Surfaces. American Mathematical Society, 2001.

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Book chapters on the topic "Non-smooth differential geometry"

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Hörmann, Günther, Sanja Konjik, and Michael Kunzinger. "A Regularization Approach to Non-smooth Symplectic Geometry." In Pseudo-Differential Operators and Generalized Functions, 119–32. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-14618-8_10.

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Conference papers on the topic "Non-smooth differential geometry"

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Caire, Marcelo, Murilo Augusto Vaz, and Carlos Alberto Duarte de Lemos. "Viscoelastic Analysis of Bend Stiffeners." In ASME 2005 24th International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2005. http://dx.doi.org/10.1115/omae2005-67321.

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Bend stiffeners are polymeric structures employed to ensure a smooth and safe transition in the upper connection of risers and umbilical cables, protecting them against accumulation of fatigue damage and excessive curvatures. Recent failures have stimulated a better understanding of the mechanical response in order to increase the reliability in design and analysis of bend stiffeners. This work presents a mathematical formulation that represents the system riser/bend stiffener considering geometric non linearity and polyurethane with viscoelastic behaviour, an inherent characteristic to polymers. The following assumptions are considered: cross-sections remain plane after deformation, large deflections are accepted but it is a small strain bending problem, the self-weight and external forces are disregarded and the material is assumed with linear viscoelastic behaviour. The curves that represent the viscoelastic response of the material have been raised by means of creep tests, whose specimens were cut from actual bend stiffeners. The time dependent data obtained in the experimental tests were well approximated by a third order Prony series which describes the creep function. The set of four first order non linear ordinary differential equations results from geometrical compatibility, equilibrium of forces and moments and linear viscoelastic constitutive relations. The numerical solution of the problem is obtained using a one-parameter shooting method. The results are then compared with the consolidated numerical solution for linear elastic material. It is concluded that the viscoelastic phenomena can lead to excessive curvatures on the upper terminations of risers and umbilical cables if the polymeric structure were designed considering elastic behaviour. The correct characterization of the viscoelastic properties of polyurethane used on bend stiffeners must be taken into account when accurate analysis is desired.
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