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Journal articles on the topic 'Lipschitz surface'

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Published: 4 June 2021

Last updated: 28 January 2023

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1

Birbrair, Lev, and Andrei Gabrielov. "Ambient Lipschitz Equivalence of Real Surface Singularities." International Mathematics Research Notices 2019, no. 20 (January 26, 2018): 6347–61. http://dx.doi.org/10.1093/imrn/rnx328.

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Abstract We present a series of examples of pairs of singular semialgebraic surfaces (germs of real semialgebraic sets of dimension two) in ${\mathbb R}^{3}$ and ${\mathbb R}^{4}$ which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent, but not ambient Lipschitz equivalent. For each singular semialgebraic surface $S\subset{\mathbb R}^{4}$, we construct infinitely many semialgebraic surfaces which are bi-Lipschitz equivalent with respect to the outer metric, ambient topologically equivalent to S, but pairwise ambient Lipschitz nonequivalent.

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2

Neumann, Walter D., Helge Møller Pedersen, and Anne Pichon. "Minimal surface singularities are Lipschitz normally embedded." Journal of the London Mathematical Society 101, no. 2 (September 12, 2019): 641–58. http://dx.doi.org/10.1112/jlms.12280.

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3

Birbrair, Lev, Alexandre Fernandes, and Walter D. Neumann. "Bi-Lipschitz geometry of complex surface singularities." Geometriae Dedicata 139, no. 1 (November 28, 2008): 259–67. http://dx.doi.org/10.1007/s10711-008-9333-2.

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4

KAPOOR, G. P., and SRIJANANI ANURAG PRASAD. "SMOOTHNESS OF COALESCENCE HIDDEN-VARIABLE FRACTAL INTERPOLATION SURFACES." International Journal of Bifurcation and Chaos 19, no. 07 (July 2009): 2321–33. http://dx.doi.org/10.1142/s0218127409024098.

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In the present paper, the smoothness of a Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS), as described by its Lipschitz exponent, is investigated. This is achieved by considering the simulation of a generally uneven surface using CHFIS. The influence of free variables and Lipschitz exponent on the smoothness of CHFIS is demonstrated by considering interpolation data generated from a sample surface.

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5

Neumann, Walter D., Helge Møller Pedersen, and Anne Pichon. "A characterization of Lipschitz normally embedded surface singularities." Journal of the London Mathematical Society 101, no. 2 (September 12, 2019): 612–40. http://dx.doi.org/10.1112/jlms.12279.

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6

Birbrair, Lev, Alexandre Fernandes, and Walter D. Neumann. "Bi-Lipschitz geometry of weighted homogeneous surface singularities." Mathematische Annalen 342, no. 1 (April 18, 2008): 139–44. http://dx.doi.org/10.1007/s00208-008-0225-4.

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7

Wang, Xiao Peng, Yuan Zhi Cheng, Ming Ming Zhao, Xiao Hua Ding, and Jing Bai. "Knee Bone Surface Registration Using the Lipschitz Optimization Algorithm." Advanced Materials Research 443-444 (January 2012): 537–41. http://dx.doi.org/10.4028/www.scientific.net/amr.443-444.537.

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We describe a technique for the registration of three dimensional (3D) knee bone surface points from MR image data sets. This technique is grounded on a mathematical theory – Lipschitz optimization. Based on this theory, we propose a global search algorithm that simultaneously determines the transformation and point correspondences. Compared with the other three registration approaches (ICP, EM-ICP, and genetic algorithms), the new proposed method achieved the highest registration accuracy on animal data.

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8

Buczolich, Zoltán. "Lipschitz images with fractal boundaries and their small surface wrapping." Proceedings of the American Mathematical Society 126, no. 12 (1998): 3589–95. http://dx.doi.org/10.1090/s0002-9939-98-04433-5.

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9

ELSCHNER, JOHANNES, and GUANGHUI HU. "SCATTERING OF PLANE ELASTIC WAVES BY THREE-DIMENSIONAL DIFFRACTION GRATINGS." Mathematical Models and Methods in Applied Sciences 22, no. 04 (April 2012): 1150019. http://dx.doi.org/10.1142/s0218202511500199.

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The reflection and transmission of a time-harmonic plane wave in an isotropic elastic medium by a three-dimensional diffraction grating is investigated. If the diffractive structure involves an impenetrable surface, we study the first, second, third and fourth kind boundary value problems for the Navier equation in an unbounded domain by the variational approach. A radiation condition based on the Rayleigh expansion of the quasi-periodic solutions is presented. Existence of solutions in Sobolev spaces is established if the grating profile is a two-dimensional Lipschitz surface, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. Similar solvability results are obtained for multilayered transmission gratings in the case of an incident pressure wave. Moreover, by a periodic Rellich identity, uniqueness of the solution to the first kind (Dirichlet) boundary value problem is established for all frequencies under the assumption that the impenetrable surface is given by the graph of a Lipschitz function.

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10

CHANG, TONGKEUN. "EXTENSION AND RESTRICTION THEOREMS IN ANISOTROPIC BESOV SPACES." Communications in Contemporary Mathematics 12, no. 02 (April 2010): 265–94. http://dx.doi.org/10.1142/s0219199710003774.

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In this paper, we study the extension and restriction theorems of the anisotropic Besov spaces in a Lipschitz hyper-surface in space-time domain. We hope that such theorems will be useful in solving a parabolic type equations in a time-varying domain.

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11

XU, BINBIN. "Incompleteness of the pressure metric on the Teichmüller space of a bordered surface." Ergodic Theory and Dynamical Systems 39, no. 06 (September 28, 2017): 1710–28. http://dx.doi.org/10.1017/etds.2017.73.

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We prove that the pressure metric on the Teichmüller space of a bordered surface is incomplete and that a completion can be given by the moduli space of metrics on a graph (dual to a special ideal triangulation of the same bordered surface) equipped with pressure metric. In contrast to the closed surface case, we obtain as a corollary that the pressure metric is not bi-Lipschitz to the Weil–Petersson metric.

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12

GERBER, MARLIES, and VIOREL NIŢICĂ. "Hölder exponents of horocycle foliations on surfaces." Ergodic Theory and Dynamical Systems 19, no. 5 (October 1999): 1247–54. http://dx.doi.org/10.1017/s0143385799146832.

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We show that the horocycle foliations on a compact $C^{\infty}$ (or even $C^{\omega}$) surface of non-positive curvature can fail to be Lipschitz, even if the curvature vanishes only along a single closed geodesic. We calculate the Hölder exponents of these foliations at vectors tangent to geodesics along which the curvature vanishes.

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13

ARTIGIANI, MAURO, LUCA MARCHESE, and CORINNA ULCIGRAI. "Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces." Ergodic Theory and Dynamical Systems 40, no. 8 (February 11, 2019): 2017–72. http://dx.doi.org/10.1017/etds.2018.143.

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We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

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14

Qu, Fenglong. "Uniqueness in Inverse Electromagnetic Conductive Scattering by Penetrable and Inhomogeneous Obstacles with a Lipschitz Boundary." Abstract and Applied Analysis 2012 (2012): 1–21. http://dx.doi.org/10.1155/2012/306272.

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This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves by a penetrable, inhomogeneous, Lipschitz obstacle covered with a thin layer of high conductivity. The well posedness of the direct problem is established by the variational method. The inverse problem is also considered in this paper. Under certain assumptions, a uniqueness result is obtained for determining the shape and location of the obstacle and the corresponding surface parameterλ(x)from the knowledge of the near field data, assuming that the incident fields are electric dipoles located on a large sphere with polarizationp∈ℝ3. Our results extend those in the paper by F. Hettlich (1996) to the case of inhomogeneous Lipschitz obstacles.

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15

Bowditch, Brian H., and Francesca Iezzi. "Projections of the sphere graph to the arc graph of a surface." Journal of Topology and Analysis 10, no. 02 (June 2018): 245–61. http://dx.doi.org/10.1142/s1793525318500115.

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Let [Formula: see text] be a compact surface, and [Formula: see text] be the double of a handlebody. Given a homotopy class of maps from [Formula: see text] to [Formula: see text] inducing an isomorphism of fundamental groups, we describe a canonical uniformly Lipschitz retraction of the sphere graph of [Formula: see text] to the arc graph of [Formula: see text]. We also show that this retraction is a uniformly bounded distance from the nearest point projection map.

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16

Zheng, Kai, TieLong Shen, and Yu Yao. "New approaching condition for sliding mode control design with Lipschitz switching surface." Science in China Series F: Information Sciences 52, no. 11 (November 2009): 2032–44. http://dx.doi.org/10.1007/s11432-009-0186-6.

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17

Ekholm, Tobias, and Frank Kutzschebauch. "Total curvature and area of curves with cusps and of surface maps." MATHEMATICA SCANDINAVICA 96, no. 2 (June 1, 2005): 224. http://dx.doi.org/10.7146/math.scand.a-14954.

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A curvature-area inequality for planar curves with cusps is derived. Using this inequality, the total (Lipschitz-Killing) curvature of a map with stable singularities of a closed surface into the plane is shown to be bounded below by the area of the map divided by the square of the radius of the smallest ball containing the image of the map. This latter result fills the gap in Santaló's proof of a similar estimate for surface maps into $\mathbf{R}^n$, $n>2$.

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18

Rondi, L. "Uniqueness and stability for the determination of boundary defects by electrostatic measurements." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 5 (October 2000): 1119–51. http://dx.doi.org/10.1017/s0308210500000603.

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The inverse problem of the determination of boundary defects in a planar conductor by a finite number of electrostatic measurements on the boundary is considered. Uniqueness results and stability estimates are proved under essentially minimal regularity assumptions on the data. Finally, Lipschitz estimates for the determination of surface linear cracks are developed.

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19

Parfenov, A. I. "A criterion for straightening a Lipschitz surface in the Lizorkin—Triebel sense. III." Siberian Advances in Mathematics 21, no. 2 (April 2011): 100–129. http://dx.doi.org/10.3103/s1055134411020027.

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20

Choi, Sunhi, David Jerison, and Inwon Kim. "Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface." American Journal of Mathematics 129, no. 2 (2007): 527–82. http://dx.doi.org/10.1353/ajm.2007.0008.

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21

Agranovich, M. S. "Strongly elliptic second-order systems with boundary conditions on a nonclosed Lipschitz surface." Functional Analysis and Its Applications 45, no. 1 (March 2011): 1–12. http://dx.doi.org/10.1007/s10688-011-0001-1.

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22

Saad, Wajdi, Anis Sellami, and Germain Garcia. "Robust stabilization of one-sided Lipschitz nonlinear systems via adaptive sliding mode control." Journal of Vibration and Control 26, no. 7-8 (December 30, 2019): 399–412. http://dx.doi.org/10.1177/1077546319889413.

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In this paper, the problem of adaptive sliding mode control for varied one-sided Lipschitz nonlinear systems with uncertainties is investigated. In contrast to existing sliding mode control design methods, the considered models, in the current study, are affected by nonlinear control inputs, one-sided Lipschitz nonlinearities, unknown disturbances and parameter uncertainties. At first, to design the sliding surface, a specific switching function is defined. The corresponding nonlinear equivalent control is extracted and the resulting sliding mode dynamic is given. Novel synthesis conditions of asymptotic stability are derived in terms of linear matrix inequalities. Thereafter, to ensure the reachability of system states and the occurrence of the sliding mode, the sliding mode controller is designed. Any knowledge of the upper bound on the perturbation is not required and an adaptation law is proposed. At last, two illustrative examples are introduced.

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23

Tuncer, Necibe, and Anotida Madzvamuse. "Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces." Communications in Computational Physics 21, no. 3 (February 7, 2017): 718–47. http://dx.doi.org/10.4208/cicp.oa-2016-0029.

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AbstractThe focus of this article is to present the projected finite element method for solving systems of reaction-diffusion equations on evolving closed spheroidal surfaces with applications to pattern formation. The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. Furthermore, the surface is not approximated but described exactly through the projection. The surface evolution law is incorporated into the projection operator resulting in a time-dependent operator. The time-dependent projection operator is composed of the radial projection with a Lipschitz continuous mapping. The projection operator is used to generate the surface mesh whose connectivity remains constant during the evolution of the surface. To illustrate the methodology several numerical experiments are exhibited for different surface evolution laws such as uniform isotropic (linear, logistic and exponential), anisotropic, and concentration-driven. This numerical methodology allows us to study new reaction-kinetics that only give rise to patterning in the presence of surface evolution such as theactivator-activatorandshort-range inhibition; long-range activation.

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24

FALCONER, K. J., and J. M. FRASER. "The horizon problem for prevalent surfaces." Mathematical Proceedings of the Cambridge Philosophical Society 151, no. 2 (July 13, 2011): 355–72. http://dx.doi.org/10.1017/s030500411100048x.

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We investigate the box dimensions of the horizon of a fractal surface defined by a functionf∈C[0,1]2. In particular we show that a prevalent surface satisfies the ‘horizon property’, namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α ∈ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

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25

Beck, Lisa, Miroslav Bulíček, and Erika Maringová. "Globally Lipschitz minimizers for variational problems with linear growth." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 4 (October 2018): 1395–413. http://dx.doi.org/10.1051/cocv/2017065.

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We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space W1,1 with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler–Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin’s paper [J. Serrin, Philos. Trans. R. Soc. Lond., Ser. A 264 (1969) 413–496].

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26

Beschle, Cedric Aaron, and Balázs Kovács. "Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces." Numerische Mathematik 151, no. 1 (April 5, 2022): 1–48. http://dx.doi.org/10.1007/s00211-022-01280-5.

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AbstractIn this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.

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27

Sini, Mourad. "On the Uniqueness and Reconstruction of Rough and Complex Obstacles from Acoustic Scattering Data." Computational Methods in Applied Mathematics 11, no. 1 (2011): 83–104. http://dx.doi.org/10.2478/cmam-2011-0005.

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Abstract We deal with the inverse scattering problem by an obstacle at a fixed frequency. The obstacle is characterized by its shape, the type of boundary conditions on its surface and the eventual coefficients distributed on this surface. In this paper, we assume that the surface ∂D of the obstacle D is Lipschitz and the surface impedance, λ, is given by a complex valued, measurable and bounded function. We prove uniqueness of (∂D,λ) from the far field map under these regularity conditions. The usual proof of uniqueness for obstacles, based on the use of singular solutions, is divided into two steps. The first one consists of the use of Rellich type lemma to go from the far fields to the near fields and then use the singularities of the singular solutions, via orthogonality relations, to show uniqueness of ∂D. The second step is to use the boundary conditions to prove uniqueness of λ on ∂D via the unique continuation property. This last step requires the surface impedance to be continuous. We propose an approach using layer potentials to transform the inverse problem to the invertibility of integral equations of second kind involving the unknowns ∂D and λ. This enables us to weaken the required regularity conditions by assuming ∂D to be Lipschitz and λ to be only bounded. The procedure of the proof is reconstructive and provides a method to compute the complex valued and bounded surface impedance λ on ∂D by inverting an invertible integral equation. In addition, assuming ∂D to be C^2 regular and λ to be of class C^{0,α}, with α>0, we give a direct and stable formula as another method to reconstruct the surface impedance on ∂D.

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28

Parfenov, A. I. "A discrete norm on a Lipschitz surface and the Sobolev straightening of a boundary." Siberian Advances in Mathematics 18, no. 4 (November 30, 2008): 258–74. http://dx.doi.org/10.3103/s1055134408040032.

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29

Parfenov, A. I. "A criterion for straightening of a lipschitz surface in the Lizorkin-Triebel sense. I." Siberian Advances in Mathematics 20, no. 2 (April 2010): 83–127. http://dx.doi.org/10.3103/s1055134410020021.

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30

Peng, Yue-Jun, and Yong-Fu Yang. "Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations." Journal of Mathematical Physics 52, no. 5 (May 2011): 053702. http://dx.doi.org/10.1063/1.3591133.

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31

Seuret-Jiménez, Diego, and Eduardo Trutié-Carrero. "Non-coherent detection of dust in photovoltaic systems in series configuration using Lipschitz exponent." Renewable Energy, Biomass & Sustainability 2, no. 2 (July 8, 2022): 37–43. http://dx.doi.org/10.56845/rebs.v2i2.27.

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Failures in photovoltaic systems are a problem of great importance because they cause a deterioration in the production of electrical energy, among which is the dust on the surface of the photovoltaic system. This paper proposes a method to detect dust on the surface of a photovoltaic system in series configuration. In addition, shows by visual inspection that the IV characteristic of a photovoltaic panel is equal to the IV characteristic of a photovoltaic system. To obtain the results, 120 signals were used, 60 for the design of the method and the rest for the validation of the method. The proposed method only yielded 2 false positives out of 30 signals where there was no fault present.

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32

Lechleiter, Armin, and Dinh-Liem Nguyen. "Scattering of Herglotz waves from periodic structures and mapping properties of the Bloch transform." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 6 (October 8, 2015): 1283–311. http://dx.doi.org/10.1017/s0308210515000335.

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When an incident Herglotz wave function scatters from a periodic Lipschitz continuous surface with a Dirichlet boundary condition, the classical (quasi-)periodic solution theory for scattering from periodic structures does not apply, since the incident field lacks periodicity. Relying on the Bloch transform, we provide a solution theory in H1 for this scattering problem. First, we prove conditions guaranteeing that incident Herglotz wave functions propagating towards the periodic structure have traces in H1/2 on the periodic surface. Second, we show that the solution to the scattering problem can be decomposed by the Bloch transform into periodic components that solve a periodic scattering problem. Third, these periodic solutions yield an equivalent characterization of the solution to the original non-periodic scattering problem, which allows, for instance, new characterizations of the Rayleigh coefficients of each of the periodic components to be shown. A corollary of our results is that under the conditions mentioned above the operator that maps densities to the restriction of their Herglotz wave function on the periodic surface is always injective; this result generally fails for bounded surfaces.

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33

Parfenov, A. I. "A criterion for the straightening of a Lipschitz surface in the Lizorkin-Triebel sense. II." Siberian Advances in Mathematics 20, no. 3 (July 2010): 201–16. http://dx.doi.org/10.3103/s1055134410030053.

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34

Sun, Hong Chun, Ling Ling Zhang, and Bei Ming Zhao. "Identification of Surface Crack in Steel Rods Using Wavelet Transform Modulus Maximum on Vibration Mode." Applied Mechanics and Materials 532 (February 2014): 457–60. http://dx.doi.org/10.4028/www.scientific.net/amm.532.457.

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The traditional identification method of surface crack has some shortcomings such as detection equipment complex, high requirements for operator and not suitable for large-scale structure inspection, vibration method is applied to identify surface crack in steel rods in this paper, using the method of calculating modal analysis combined with time-frequency wavelet analysis. The research on damage identification is performed on steel rods with different deep crack. The research results show that the accurate damage location can be judged by wavelet modulus maximum and the quantitative analysis of single crack identification can be achieved by constructing mathematical expressions of a single crack damage degree and damage index Lipschitz index. The research would province some guidance for engineering applications.

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35

Kim, Seung Hyun. "Two Simple Numerical Methods for the Free Boundary in One-Phase Stefan Problem." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/764532.

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We present two simple numerical methods to find the free boundary in one-phase Stefan problem. The work is motivated by the necessity for better understanding of the solution surface (temperatures) near the free boundary. We formulate a log-transform function with the unfixed and fixed free boundary that has Lipschitz character near free boundary. We solve the quadratic equation in order to locate the free boundary in a time-recursive way. We also present several numerical results which illustrate a comparison to other methods.

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36

ROHDE, CHRISTIAN, and MAI DUC THANH. "GLOBAL EXISTENCE FOR PHASE TRANSITION PROBLEMS VIA A VARIATIONAL SCHEME." Journal of Hyperbolic Differential Equations 01, no. 04 (December 2004): 747–68. http://dx.doi.org/10.1142/s0219891604000329.

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We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

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37

ŠARIĆ, DRAGOMIR. "Fenchel–Nielsen coordinates on upper bounded pants decompositions." Mathematical Proceedings of the Cambridge Philosophical Society 158, no. 3 (January 16, 2015): 385–97. http://dx.doi.org/10.1017/s0305004114000656.

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AbstractLet X0 be an infinite-type hyperbolic surface (whose boundary components, if any, are closed geodesics) which has an upper bounded pants decomposition. The length spectrum Teichmüller space Tls(X0) consists of all surfaces X homeomorphic to X0 such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su [1] described the Fenchel–Nielsen coordinates for Tls(X0) and using these coordinates they proved that Tls(X0) is path connected. We use the Fenchel–Nielsen coordinates for Tls(X0) to induce a locally bi-Lipschitz homeomorphism between l∞ and Tls(X0) (which extends analogous results by Fletcher [9] and by Allessandrini, Liu, Papadopoulos, Su and Sun [2] for the unreduced and the reduced Tqc(X0)). Consequently, Tls(X0) is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space Tqc(X0) in Tls(X0).

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38

Lorent, Andrew. "The two-well problem with surface energy." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 4 (August 2006): 795–805. http://dx.doi.org/10.1017/s030821050000473x.

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Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω > 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

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39

Elfanni, A., and M. Fuchs. "A link between the shape of the austenite–martensite interface and the behaviour of the surface energy." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 6 (December 2004): 1099–113. http://dx.doi.org/10.1017/s0308210500003644.

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Let Ω ⊂ R2 denote a bounded Lipschitz domain and consider some portion Γ0 of ∂Ω representing the austenite–twinned-martensite interface which is not assumed to be a straight segment. We prove that for an elastic energy density ϖ: R2 → [0 ∞) such that ϖ(0, ±1) = 0. Here, W(Ω) consists of all functions u from the Sobolev class W1, ∞(Ω) such that |uy| = 1 almost everywhere on Ω together with u = 0 on Γ0. We will first show that, for Γ0 having a vertical tangent, one cannot always expect a finite surface energy, i.e. in the above problem, the condition in general cannot be included. This generalizes a result of [12] where Γ0is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary ∂Ω, that is, one can even take Γ0 = ∂Ω. We also show that the existence or non-existence of minimizers depends on the shape of the austenite–twinned-martensite interface Γ0.

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40

Caro, Pedro, Ru-Yu Lai, Yi-Hsuan Lin, and Ting Zhou. "Boundary determination of electromagnetic and Lamé parameters with corrupted data." Inverse Problems & Imaging 15, no. 5 (2021): 1171. http://dx.doi.org/10.3934/ipi.2021033.

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<p style='text-indent:20px;'>We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For both systems, we provide explicit formulas to reconstruct the parameters on the boundary as well as its rate of convergence formula.</p>

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41

Brandenbursky, Michael. "Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces." International Journal of Mathematics 26, no. 09 (August 2015): 1550066. http://dx.doi.org/10.1142/s0129167x15500664.

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Let Σg be a closed orientable surface of genus g and let Diff 0(Σg, area ) be the identity component of the group of area-preserving diffeomorphisms of Σg. In this paper, we present the extension of Gambaudo–Ghys construction to the case of a closed hyperbolic surface Σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of Σg defines a nontrivial homogeneous quasi-morphism on the group Diff 0(Σg, area ). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff 0(Σg, area ) is infinite-dimensional. Let Ham (Σg) be the group of Hamiltonian diffeomorphisms of Σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham (Σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham (Σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham (Σg).

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42

DIRR, N., G. KARALI, and N. K. YIP. "Pulsating wave for mean curvature flow in inhomogeneous medium." European Journal of Applied Mathematics 19, no. 6 (December 2008): 661–99. http://dx.doi.org/10.1017/s095679250800764x.

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We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing.

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43

Ciarlet, Philippe G., and Cristinel Mardare. "A surface in W2, is a locally Lipschitz-continuous function of its fundamental forms in W1, and L, p > 2." Journal de Mathématiques Pures et Appliquées 124 (April 2019): 300–318. http://dx.doi.org/10.1016/j.matpur.2018.06.013.

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44

Li, Rui, Liang Yang, Yong Chen, and Guanyu Lai. "Adaptive Sliding Mode Control of Robot Manipulators with System Failures." Mathematics 10, no. 3 (January 23, 2022): 339. http://dx.doi.org/10.3390/math10030339.

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This paper presents a novel adaptive sliding mode controller for a class of robot manipulators with unknown disturbances and system failures, which can well achieve the asymptotic tracking, and avoid some possible singularity problems. A new virtual controller is designed such that the chosen Lyapunov function can be transformed into a non-Lipschitz function, based on which, the system states can arrive at the specified sliding surface within a finite time regardless of the existence of system failures/faults. By fusing an integral fast terminal nonsingular SMC and a robust adaptive technique, the tracking error can be steered into a preset range in a set time and some possible singularity problems are avoided elegantly. With our proposed scheme, the loss coefficient is well estimated, and the stability of the system can be guaranteed even in the presence of the total loss of actuator outputs. The experiment and simulation results are presented to illustrate the effectiveness of the proposed control scheme.

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45

Kozlov, V., N. Kuznetsov, and E. Lokharu. "On the Benjamin–Lighthill conjecture for water waves with vorticity." Journal of Fluid Mechanics 825 (July 24, 2017): 961–1001. http://dx.doi.org/10.1017/jfm.2017.361.

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We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

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46

CIARLET, PHILIPPE G., and CRISTINEL MARDARE. "RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS." Analysis and Applications 03, no. 02 (April 2005): 99–117. http://dx.doi.org/10.1142/s0219530505000509.

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If a field A of class [Formula: see text] of positive-definite symmetric matrices of order two and a field B of class [Formula: see text] of symmetric matrices of order two satisfy together the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of ℝ2, then there exists an immersion [Formula: see text], uniquely determined up to proper isometries in ℝ3, such that A and B are the first and second fundamental forms of the surface θ(ω). Let [Formula: see text] denote the equivalence class of θ modulo proper isometries in ℝ3 and let [Formula: see text] denote the mapping determined in this fashion. The first objective of this paper is to show that, if ω satisfies a certain "geodesic property" (in effect a mild regularity assumption on the boundary of ω) and if the fields A and B and their partial derivatives of order ≤ 2 (respectively, ≤ 1), have continuous extensions to [Formula: see text], the extension of the field A remaining positive-definite on [Formula: see text], then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to [Formula: see text]. The second objective is to show that, if ω satisfies the geodesic property and is bounded, the mapping ℱ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces [Formula: see text] for the continuous extensions of the matrix fields (A, B), and [Formula: see text] for the continuous extensions of the immersions θ.

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Hodges, Ben R. "Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage." Hydrology and Earth System Sciences 23, no. 3 (March 7, 2019): 1281–304. http://dx.doi.org/10.5194/hess-23-1281-2019.

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Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

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48

Rasouli, Mohammad, Assef Zare, Majid Hallaji, and Roohallah Alizadehsani. "The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication." Axioms 11, no. 12 (December 16, 2022): 738. http://dx.doi.org/10.3390/axioms11120738.

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A novel approach for the synchronization of a class of chaotic systems with uncertainty, unknown time delays, and external disturbances is presented. The control method given here is expressed by combining sliding mode control approaches with adaptive rules. A sliding surface of fractional order has been developed to construct the control strategy of the abovementioned sliding mode by employing the structure of nonlinear fractional PID (NLPID) controllers. The suggested control mechanism using Lyapunov’s theorem developed robust adaptive rules in such a way that the estimation error of the system’s unknown parameters and time delays tends to be zero. Furthermore, the proposed robust control approach’s stability has been demonstrated using Lyapunov stability criteria and Lipschitz conditions. Then, in order to assess the performance of the proposed mechanism, the presented control approach was used to simulate the synchronization of two chaotic jerk systems with uncertainty, unknown time delays, and external distortion. The results of the simulation confirm the robust and desirable synchronization performance. Finally, a secure communications mechanism based on the proposed technique is shown as a practical implementation of the introduced control strategy, in which the message signal is disguised in the transmitter with high security and well recovered in the receiver with high quality, according to the mean squared error (MES) criteria.

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Griso, Georges, Larysa Khilkova, Julia Orlik, and Olena Sivak. "Homogenization of Perforated Elastic Structures." Journal of Elasticity 141, no. 2 (June 5, 2020): 181–225. http://dx.doi.org/10.1007/s10659-020-09781-w.

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Abstract The paper is dedicated to the asymptotic behavior of $\varepsilon$ ε -periodically perforated elastic (3-dimensional, plate-like or beam-like) structures as $\varepsilon \to 0$ ε → 0 . In case of plate-like or beam-like structures the asymptotic reduction of dimension from $3D$ 3 D to $2D$ 2 D or $1D$ 1 D respectively takes place. An example of the structure under consideration can be obtained by a periodic repetition of an elementary “flattened” ball or cylinder for plate-like or beam-like structures in such a way that the contact surface between two neighboring balls/cylinders has a non-zero measure. Since the domain occupied by the structure might have a non-Lipschitz boundary, the classical homogenization approach based on the extension cannot be used. Therefore, for obtaining Korn’s inequalities, which are used for the derivation of a priori estimates, we use the approach based on interpolation. In case of plate-like and beam-like structures the proof of Korn’s inequalities is based on the displacement decomposition for a plate or a beam, respectively. In order to pass to the limit as $\varepsilon \to 0$ ε → 0 we use the periodic unfolding method.

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50

Bonk, Mario, and Urs Lang. "Bi-Lipschitz parameterization of surfaces." Mathematische Annalen 327, no. 1 (May 16, 2003): 135–69. http://dx.doi.org/10.1007/s00208-003-0443-8.

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