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Добірка наукової літератури з теми "Varifolds theory"

Автор: Grafiati
Опубліковано: 14 березня 2023

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Статті в журналах з теми "Varifolds theory"

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Buet, Blanche. "Quantitative conditions of rectifiability for varifolds." Annales de l’institut Fourier 65, no. 6 (2015): 2449–506. http://dx.doi.org/10.5802/aif.2993.

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Buet, Blanche, Gian Paolo Leonardi, and Simon Masnou. "Discretization and Approximation of Surfaces Using Varifolds." Geometric Flows 3, no. 1 (March 1, 2018): 28–56. http://dx.doi.org/10.1515/geofl-2018-0004.

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Abstract We present some recent results on the possibility of extending the theory of varifolds to the realm of discrete surfaces of any dimension and codimension, for which robust notions of approximate curvatures, also allowing for singularities, can be defined. This framework has applications to discrete and computational geometry, as well as to geometric variational problems in discrete settings. We finally show some numerical tests on point clouds that support and confirm our theoretical findings.
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Wickramasekera, Neshan. "A general regularity theory for stable codimension 1 integral varifolds." Annals of Mathematics 179, no. 3 (May 1, 2014): 843–1007. http://dx.doi.org/10.4007/annals.2014.179.3.2.

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Kagaya, Takashi, and Yoshihiro Tonegawa. "A fixed contact angle condition for varifolds." Hiroshima Mathematical Journal 47, no. 2 (July 2017): 139–53. http://dx.doi.org/10.32917/hmj/1499392823.

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5

Moser, Roger. "Towards a variational theory of phase transitions involving curvature." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 142, no. 4 (August 2012): 839–65. http://dx.doi.org/10.1017/s0308210510000995.

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An anisotropic area functional is often used as a model for the free energy of a crystal surface. For models of faceting, the anisotropy is typically such that the functional becomes non-convex, and then it may be appropriate to regularize it with an additional term involving curvature. When the weight of the curvature term tends to zero, this gives rise to a singular perturbation problem.The structure of this problem is comparable to the theory of phase transitions studied first by Modica and Mortola. Their ideas are also useful in this context, but they have to be combined with adequate geometric tools. In particular, a variant of the theory of curvature varifolds, introduced by Hutchinson, is used in this paper. This allows an analysis of the asymptotic behaviour of the energy functionals.
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Miller, Michael, Daniel Tward, and Alain Trouvé. "Molecular Computational Anatomy: Unifying the Particle to Tissue Continuum via Measure Representations of the Brain." BME Frontiers 2022 (November 7, 2022): 1–16. http://dx.doi.org/10.34133/2022/9868673.

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Objective. The objective of this research is to unify the molecular representations of spatial transcriptomics and cellular scale histology with the tissue scales of computational anatomy for brain mapping. Impact Statement. We present a unified representation theory for brain mapping based on geometric varifold measures of the microscale deterministic structure and function with the statistical ensembles of the spatially aggregated tissue scales. Introduction. Mapping across coordinate systems in computational anatomy allows us to understand structural and functional properties of the brain at the millimeter scale. New measurement technologies in digital pathology and spatial transcriptomics allow us to measure the brain molecule by molecule and cell by cell based on protein and transcriptomic functional identity. We currently have no mathematical representations for integrating consistently the tissue limits with the molecular particle descriptions. The formalism derived here demonstrates the methodology for transitioning consistently from the molecular scale of quantized particles—using mathematical structures as first introduced by Dirac as the class of generalized functions—to the tissue scales with methods originally introduced by Euler for fluids. Methods. We introduce two mathematical methods based on notions of generalized functions and statistical mechanics. We use geometric varifolds, a product measure on space and function, to represent functional states at the micro-scales—electrophysiology, molecular histology—integrated with a Boltzmann-like program to pass from deterministic particle descriptions to empirical probabilities on the functional states at the tissue scales. Results. Our space-function varifold representation provides a recipe for traversing from molecular to tissue scales in terms of a cascade of linear space scaling composed with nonlinear functional feature mapping. Following the cascade implies every scale is a geometric measure so that a universal family of measure norms can be introduced which quantifies the geodesic connection between brains in the orbit independent of the probing technology, whether it be RNA identities, Tau or amyloid histology, spike trains, or dense MR imagery. Conclusions. We demonstrate a unified brain mapping theory for molecular and tissue scales based on geometric measure representations. We call the consistent aggregation of tissue scales from particle and cellular scales, molecular computational anatomy.
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Tonegawa, Yoshihiro. "Integrality of varifolds in the singular limit of reaction-diffusion equations." Hiroshima Mathematical Journal 33, no. 3 (November 2003): 323–41. http://dx.doi.org/10.32917/hmj/1150997978.

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Kikuchi, Koji. "Constructing Weak Solutions in a Direct Variational Method and an Application of Varifold Theory." Journal of Differential Equations 150, no. 1 (November 1998): 1–23. http://dx.doi.org/10.1006/jdeq.1998.3485.

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Henkemeyer, Patrick. "Enclosure theorems and barrier principles for energy stationary currents and the associated Brakke-flow." Analysis 37, no. 4 (January 1, 2017). http://dx.doi.org/10.1515/anly-2017-0048.

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AbstractWe discuss certain quantitative geometric properties of energy stationary currents describing minimal surfaces under gravitational forces. Enclosure theorems give statements about the confinement of the support of currents to certain enclosing sets on the basis that one knows something about the position of their boundaries. These results are closely related to non-existence theorems for currents with connected support. Finally, we define a weak formulation in the theory of varifolds for the curvature flow associated to this energy functional. We extend the enclosure results to the flow and discuss several comparison principles.
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Hirsch, Jonas, and Riccardo Tione. "On the constancy theorem for anisotropic energies through differential inclusions." Calculus of Variations and Partial Differential Equations 60, no. 3 (April 21, 2021). http://dx.doi.org/10.1007/s00526-021-01981-z.

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AbstractIn this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1–33, 2021). In particular, given a polyconvex integrand f, we define a set of matrices $$C_f$$ C f that allows us to rewrite the stationarity condition for a graph with multiplicity as a differential inclusion. Then we prove that if f is assumed to be non-negative, then in $$C_f$$ C f there is no $$T'_N$$ T N ′ configuration, thus recovering the main result of De Lellis et al. (Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335) as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find $$T'_N$$ T N ′ configurations in $$C_f$$ C f , but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity.
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Дисертації з теми "Varifolds theory"

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Mondino, Andrea. "The Willmore functional and other L^p curvature functionals in Riemannian manifolds." Doctoral thesis, SISSA, 2011. http://hdl.handle.net/20.500.11767/4840.

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Using techniques both of non linear analysis and geometric measure theory, we prove existence of minimizers and more generally of critical points for the Willmore functional and other $L^p$ curvature functionals for immersions in Riemannian manifolds. More precisely, given a $3$-dimensional Riemannian manifold $(M,g)$ and an immersion of a sphere $f:\Sp^2 \hookrightarrow (M,g)$ we study the following problems. 1) The Conformal Willmore functional in a perturbative setting: consider $(M,g)=(\Rtre,\eu+\epsilon h)$ the euclidean $3$-space endowed with a perturbed metric ($h=h_{\mu\nu}$ is a smooth field of symmetric bilinear forms); we prove, under assumptions on the trace free Ricci tensor and asymptotic flatness, existence of critical points for the Conformal Willmore functional $I(f):=\frac{1}{2}\int |A^\circ|^2 $ (where $A^\circ:=A-\frac{1}{2}H$ is the trace free second fundamental form). The functional is conformally invariant in curved spaces. We also establish a non existence result in general Riemannian manifolds. The technique is perturbative and relies on a Lyapunov-Schmidt reduction. \\ 2) The Willmore functional in a semi-perturbative setting: consider $(M,g)=(\Rtre, \eu+h)$ where $h=h_{\mu\nu}$ is a $C^{\infty}_0(\Rtre)$ field of symmetric bilinear forms with compact support and small $C^1$ norm. Under a general assumption on the scalar curvature we prove existence of a smooth immersion of $\Sp^2$ minimizing the Willmore functional $W(f):=\frac{1}{4} \int |H|^2$ (where $H$ is the mean curvature). The technique is more global and relies on the direct method in the calculus of variations. \\ 3) The functionals $E:=\frac{1}{2} \int |A|^2 $ and $W_1:=\int\left( \frac{|H|^2}{4}+1 \right)$ in compact ambient manifolds: consider $(M,g)$ a $3$-dimensional compact Riemannian manifold. We prove, under global conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E$ or $W_1$. The technique is global, uses geometric measure theory and regularity theory for higher order PDEs. \\ 4) The functionals $E_1:=\int \left( \frac{|A|^2}{2} +1 \right) $ and $W_1:=\int\left( \frac{|H|^2}{4}+1 \right)$ in noncompact ambient manifolds: consider $(M,g)$ a $3$-dimensional asymptotically euclidean non compact Riemannian $3$-manifold. We prove, under general conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E_1$ or $W_1$. The technique relies on the direct method in the calculus of variations. \\ 5) The supercritical functionals $\int |H|^p$ and $\int |A|^p$ in arbitrary dimension and codimension: consider $(N,g)$ a compact $n$-dimensional Riemannian manifold possibly with boundary. For any $2\leq mm$, defined on the $m$-dimensional submanifolds of $N$. We prove, under assumptions on $(N,g)$, existence and partial regularity of a minimizer of such functionals in the framework of varifold theory. During the arguments we prove some new monotonicity formulas and new Isoperimetric Inequalities which are interesting by themselves.
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Частини книг з теми "Varifolds theory"

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Morgan, Frank. "Flat Chains Modulo v, Varifolds, and (M, ɛ, δ)-Minimal Sets." In Geometric Measure Theory, 107–11. Elsevier, 1988. http://dx.doi.org/10.1016/b978-0-12-506855-0.50015-5.

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Morgan, Frank. "Flat Chains Modulo v, Varifolds, and (M, ε, δ)-Minimal Sets." In Geometric Measure Theory, 107–12. Elsevier, 1995. http://dx.doi.org/10.1016/b978-0-12-506857-4.50015-1.

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Morgan, Frank. "Flat Chains Modulo v, Varifolds, and (M, ε, δ)-Minimal Sets." In Geometric Measure Theory, 105–11. Elsevier, 2000. http://dx.doi.org/10.1016/b978-012506851-2/50011-x.

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Morgan, Frank. "Flat Chains Modulo ν, Varifolds, and (M, ε, δ)-Minimal Sets." In Geometric Measure Theory, 105–10. Elsevier, 2016. http://dx.doi.org/10.1016/b978-0-12-804489-6.50011-2.

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"Varifold type theory for Sobolev mappings." In First International Congress of Chinese Mathematicians, 423–30. Providence, Rhode Island: American Mathematical Society, 2001. http://dx.doi.org/10.1090/amsip/020/38.

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Tonegawa, Yoshihiro. "Introduction to varifold and its curvature flow." In Emerging Topics on Differential Equations and Their Applications, 213–26. WORLD SCIENTIFIC, 2013. http://dx.doi.org/10.1142/9789814449755_0017.

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