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Journal articles on the topic 'Willmore functional'

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Published: 14 March 2023

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1

Lamm, Tobias, Jan Metzger, and Andre Neves. "Mini-Workshop: The Willmore Functional and the Willmore Conjecture." Oberwolfach Reports 10, no. 3 (2013): 2119–53. http://dx.doi.org/10.4171/owr/2013/37.

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2

Zhu, Yanqi, Jin Liu, and Guohua Wu. "Gap Phenomenon of an Abstract Willmore Type Functional of Hypersurface in Unit Sphere." Scientific World Journal 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/697132.

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For ann-dimensional hypersurface in unit sphere, we introduce an abstract Willmore type calledWn,F-Willmore functional, which generalizes the well-known classic Willmore functional. Its critical point is called theWn,F-Willmore hypersurface, for which the variational equation and Simons’ type integral equalities are obtained. Moreover, we construct a few examples ofWn,F-Willmore hypersurface and give a gap phenomenon characterization by use of our integral formula.
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3

Bernard, Yann. "Noether’s theorem and the Willmore functional." Advances in Calculus of Variations 9, no. 3 (July 1, 2016): 217–34. http://dx.doi.org/10.1515/acv-2014-0033.

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AbstractNoether’s theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivière. Several examples are considered in detail.
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4

Chen, Jing-yi. "The Willmore functional of surfaces." Applied Mathematics-A Journal of Chinese Universities 28, no. 4 (December 2013): 485–93. http://dx.doi.org/10.1007/s11766-013-3222-7.

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5

Kuwert, Ernst, and Reiner Schätzle. "Gradient flow for the Willmore functional." Communications in Analysis and Geometry 10, no. 2 (2002): 307–39. http://dx.doi.org/10.4310/cag.2002.v10.n2.a4.

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6

WANG, PENG. "ON THE WILLMORE FUNCTIONAL OF 2-TORI IN SOME PRODUCT RIEMANNIAN MANIFOLDS." Glasgow Mathematical Journal 54, no. 3 (March 30, 2012): 517–28. http://dx.doi.org/10.1017/s0017089512000122.

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AbstractWe discuss the minimum of Willmore functional of torus in a Riemannian manifold N, especially for the case that N is a product manifold. We show that when N = S2 × S1, the minimum of W(T2) is 0, and when N = R2 × S1, there exists no torus having least Willmore functional. When N = H2(−c) × S1, and x = γ × S1, the minimum of W(x) is $2\pi^2\sqrt{c}$.
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7

Simon, Leon. "Existence of surfaces minimizing the Willmore functional." Communications in Analysis and Geometry 1, no. 2 (1993): 281–326. http://dx.doi.org/10.4310/cag.1993.v1.n2.a4.

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8

Mondino, Andrea. "The Conformal Willmore Functional: A Perturbative Approach." Journal of Geometric Analysis 23, no. 2 (September 24, 2011): 764–811. http://dx.doi.org/10.1007/s12220-011-9263-3.

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9

Burger, Martin, Shun-Yin Chu, Peter Markowich, and Carola-Bibiane Schönlieb. "Cahn-Hilliard inpainting and the Willmore functional." PAMM 7, no. 1 (December 2007): 1011209–10. http://dx.doi.org/10.1002/pamm.200700802.

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10

Luo, Yong, and Guofang Wang. "On geometrically constrained variational problems of the Willmore functional I: The Lagrangian-Willmore problem." Communications in Analysis and Geometry 23, no. 1 (2015): 191–223. http://dx.doi.org/10.4310/cag.2015.v23.n1.a6.

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11

Parthasarathy, R., and K. S. Viswanathan. "Geometric properties of QCD string from Willmore functional." Journal of Geometry and Physics 38, no. 3-4 (June 2001): 207–16. http://dx.doi.org/10.1016/s0393-0440(00)00062-0.

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12

Schätzle, Reiner. "Lower semicontinuity of the Willmore functional for currents." Journal of Differential Geometry 81, no. 2 (February 2009): 437–56. http://dx.doi.org/10.4310/jdg/1231856266.

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13

Bernard, Yann, and Tristan Rivière. "Local Palais–Smale sequences for the Willmore functional." Communications in Analysis and Geometry 19, no. 3 (2011): 563–99. http://dx.doi.org/10.4310/cag.2011.v19.n3.a5.

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14

Shu, Shichang, and Junfeng Chen. "Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)." Publications de l'Institut Math?matique (Belgrade) 102, no. 116 (2017): 175–93. http://dx.doi.org/10.2298/pim1716175s.

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Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).
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15

Zhou, Jia-zu. "The Willmore functional and the containment problem in R4." Science in China Series A: Mathematics 50, no. 3 (March 2007): 325–33. http://dx.doi.org/10.1007/s11425-007-0029-0.

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16

Kuwert, Ernst, and Reiner Schätzle. "Minimizers of the Willmore functional under fixed conformal class." Journal of Differential Geometry 93, no. 3 (March 2013): 471–530. http://dx.doi.org/10.4310/jdg/1361844942.

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17

Bretin, Elie, Simon Masnou, and Édouard Oudet. "Phase-field approximations of the Willmore functional and flow." Numerische Mathematik 131, no. 1 (December 3, 2014): 115–71. http://dx.doi.org/10.1007/s00211-014-0683-4.

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18

Dayrens, François. "The L1L^{1} gradient flow of a generalized scale invariant Willmore energy for radially non-increasing functions." Advances in Calculus of Variations 10, no. 4 (October 1, 2017): 331–55. http://dx.doi.org/10.1515/acv-2015-0008.

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AbstractWe use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L^{1}(\mathbb{R}^{d}). This functional is extended to every {L^{1}} function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation.
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19

Li, Tongzhu, and Changxiong Nie. "Conformal Geometry of Hypersurfaces in Lorentz Space Forms." Geometry 2013 (September 16, 2013): 1–9. http://dx.doi.org/10.1155/2013/549602.

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Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called a Willmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in .
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20

Barros, Manuel. "Free elasticae and Willmore tori in warped product spaces." Glasgow Mathematical Journal 40, no. 2 (May 1998): 265–72. http://dx.doi.org/10.1017/s0017089500032596.

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AbstractWe use the principle of symmetric criticality to connect the Willmore variational problem for surfaces in a warped product space with base a circle, and the free elastica variational problem for curves on its fiber. In addition we obtain a rational oneparameter family of closed helices in the anti De Sitter 3-space which are critical points of the total squared curvature functional. This means they are free elasticae. Also they are spacelike; this allows us to construct a corresponding family of spacelike Willmore tori in a certain kind of spacetime close to the Robertson-Walker spaces.
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21

Wu, Lan, and Haizhong Li. "An inequality between Willmore functional and Weyl functional for submanifolds in space forms." Monatshefte für Mathematik 158, no. 4 (August 27, 2009): 403–11. http://dx.doi.org/10.1007/s00605-009-0127-x.

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22

Lamm, Tobias, and Jan Metzger. "Minimizers of the Willmore functional with a small area constraint." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 30, no. 3 (May 2013): 497–518. http://dx.doi.org/10.1016/j.anihpc.2012.10.003.

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23

Burger, M., S. Y. Chu, P. A. Markowich, and C. B. Schonlieb. "The Willmore functional and instabilities in the Cahn-Hilliard equation." Communications in Mathematical Sciences 6, no. 2 (2008): 309–29. http://dx.doi.org/10.4310/cms.2008.v6.n2.a3.

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24

Kholodenko, A. L., and V. V. Nesterenko. "Classical dynamics of the rigid string from the Willmore functional." Journal of Geometry and Physics 16, no. 1 (April 1995): 15–26. http://dx.doi.org/10.1016/0393-0440(94)00020-5.

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25

Moser, Roger. "Energy concentration for almost harmonic maps and the Willmore functional." Mathematische Zeitschrift 251, no. 2 (July 13, 2005): 293–311. http://dx.doi.org/10.1007/s00209-005-0803-z.

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26

LORETI, P., and R. MARCH. "Propagation of fronts in a nonlinear fourth order equation." European Journal of Applied Mathematics 11, no. 2 (April 2000): 203–13. http://dx.doi.org/10.1017/s0956792599004131.

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We consider a geometric motion associated with the minimization of a curvature dependent functional, which is related to the Willmore functional. Such a functional arises in connection with the image segmentation problem in computer vision theory. We show by using formal asymptotics that the geometric motion can be approximated by the evolution of the zero level set of the solution of a nonlinear fourth-order equation related to the Cahn–Hilliard and Allen–Cahn equations.
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27

Deckelnick, Klaus, Hans-Christoph Grunau, and Matthias Röger. "Minimising a relaxed Willmore functional for graphs subject to boundary conditions." Interfaces and Free Boundaries 19, no. 1 (2017): 109–40. http://dx.doi.org/10.4171/ifb/378.

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28

Chen, Jingyi, and Yuxiang Li. "Bubble tree of branched conformal immersions and applications to the Willmore functional." American Journal of Mathematics 136, no. 4 (2014): 1107–54. http://dx.doi.org/10.1353/ajm.2014.0023.

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29

Bretin, Elie, Simon Masnou, and Edouard Oudet. "Phase-field models for the approximation of the willmore functional and flow." ESAIM: Proceedings and Surveys 45 (September 2014): 118–27. http://dx.doi.org/10.1051/proc/201445012.

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30

Mondino, Andrea. "Some results about the existence of critical points for the Willmore functional." Mathematische Zeitschrift 266, no. 3 (August 28, 2009): 583–622. http://dx.doi.org/10.1007/s00209-009-0588-6.

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31

Hornung, Peter. "Euler-lagrange equation and regularity for flat minimizers of the Willmore functional." Communications on Pure and Applied Mathematics 64, no. 3 (August 18, 2010): 367–441. http://dx.doi.org/10.1002/cpa.20342.

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32

Gover, A. Rod, and Andrew Waldron. "A calculus for conformal hypersurfaces and new higher Willmore energy functionals." Advances in Geometry 20, no. 1 (January 28, 2020): 29–60. http://dx.doi.org/10.1515/advgeom-2019-0016.

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AbstractThe invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution’s asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20].
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33

LIU, Jin. "Variational problem of an abstract Willmore type functional of sub-manifold in unit sphere." SCIENTIA SINICA Mathematica 45, no. 3 (March 1, 2015): 255–72. http://dx.doi.org/10.1360/012014-67.

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34

II, William P. Minicozzi. "The Willmore Functional on Lagrangian Tori: Its Relation to Area and Existence of Smooth Minimizers." Journal of the American Mathematical Society 8, no. 4 (October 1995): 761. http://dx.doi.org/10.2307/2152828.

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35

Minicozzi, William P. "The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers." Journal of the American Mathematical Society 8, no. 4 (1995): 761. http://dx.doi.org/10.1090/s0894-0347-1995-1311825-9.

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36

Berdinsky, D. A., and I. A. Taimanov. "Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional." Siberian Mathematical Journal 48, no. 3 (May 2007): 395–407. http://dx.doi.org/10.1007/s11202-007-0043-z.

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37

MATSUTANI, SHIGEKI. "IMMERSION ANOMALY OF DIRAC OPERATOR ON SURFACE IN ℝ3." Reviews in Mathematical Physics 11, no. 02 (February 1999): 171–86. http://dx.doi.org/10.1142/s0129055x99000076.

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In previous report (J. Phys. A (1997) 30 4019–4029), I showed that the Dirac operator confined in a surface immersed in ℝ3 by means of a mass type potential completely exhibits the surface itself and is identified with that of the generalized Weierstrass equation. In this article, I quantized the Dirac field and calculated the gauge transformation which exhibits the gauge freedom of the parameterization of the surface. Then using the Ward–Takahashi identity, I showed that the expectation value of the action of the Dirac field is expressed by the Willmore functional and area of the surface, or the action of Polyakov's extrinsic string.
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38

Pozzi, Paola, and Philipp Reiter. "On non-convex anisotropic surface energy regularized via the Willmore functional: The two-dimensional graph setting." ESAIM: Control, Optimisation and Calculus of Variations 23, no. 3 (May 4, 2017): 1047–71. http://dx.doi.org/10.1051/cocv/2016024.

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39

Plotnikov, P. I., and J. F. Toland. "Variational problems in the theory of hydroelastic waves." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2129 (August 20, 2018): 20170343. http://dx.doi.org/10.1098/rsta.2017.0343.

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This paper outlines a mathematical approach to steady periodic waves which propagate with constant velocity and without change of form on the surface of a three-dimensional expanse of fluid which is at rest at infinite depth and moving irrotationally under gravity, bounded above by a frictionless elastic sheet. The elastic sheet is supposed to have gravitational potential energy, bending energy proportional to the square integral of its mean curvature (its Willmore functional), and stretching energy determined by the position of its particles relative to a reference configuration. The equations and boundary conditions governing the wave shape are derived by formulating the problem, in the language of geometry of surfaces, as one for critical points of a natural Lagrangian, and a proof of the existence of solutions is sketched. This article is part of the theme issue ‘Modelling of sea-ice phenomena’.
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40

Cai, Mingliang. "$L^p$ Willmore functionals." Proceedings of the American Mathematical Society 127, no. 2 (1999): 569–75. http://dx.doi.org/10.1090/s0002-9939-99-04484-6.

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41

Guo, Zhen. "Generalized Willmore functionals and related variational problems." Differential Geometry and its Applications 25, no. 5 (October 2007): 543–51. http://dx.doi.org/10.1016/j.difgeo.2007.06.004.

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42

Glaros, Michael, A. Rod Gover, Matthew Halbasch, and Andrew Waldron. "Variational calculus for hypersurface functionals: Singular Yamabe problem Willmore energies." Journal of Geometry and Physics 138 (April 2019): 168–93. http://dx.doi.org/10.1016/j.geomphys.2018.12.018.

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43

Olbermann, Heiner. "On a $\Gamma$-Limit of Willmore Functionals with Additional Curvature Penalization Term." SIAM Journal on Mathematical Analysis 51, no. 3 (January 2019): 2599–632. http://dx.doi.org/10.1137/18m1203596.

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44

Nesimovic, Sanela, and Dzenan Gusic. "Willmott Fuzzy Implication in Fuzzy Databases." WSEAS TRANSACTIONS ON MATHEMATICS 19 (January 19, 2021): 647–61. http://dx.doi.org/10.37394/23206.2020.19.72.

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The object of the research are fuzzy functional dependencies on given relation scheme, and the question of their obtaining using the classical and innovated techniques. The attributes of the universal set are associated to the elements of the unit interval, and are turned into fuzzy formulas in this way. We prove that the dependency (which is treated as a fuzzy formula with respect to appropriately chosen valuation) is valid whenever it agrees with the attached two-elements fuzzy relation instance. The opposite direction of the claim is proven to be incorrect in this setting. Generalizing things to sets of attributes, we prove that particular fuzzy functional dependency follows form a set of fuzzy dependencies (in both, the world of two-element and the world of arbitrary fuzzy relation instances) if and only if the dependency is valid with respect to valuation anytime the set of fuzzy formulas agrees with the valuation. The results derived in paper show that the classical techniques in the procedure for generating new fuzzy dependencies may be replaced by the resolution ones, and hence automated. The research is conducted with respect to Willmott fuzzy implication operator
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45

Gruber, Anthony, Magdalena Toda, and Hung Tran. "On the variation of curvature functionals in a space form with application to a generalized Willmore energy." Annals of Global Analysis and Geometry 56, no. 1 (May 13, 2019): 147–65. http://dx.doi.org/10.1007/s10455-019-09661-0.

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46

Gover, A. Rod, and Andrew Waldron. "Renormalized volumes with boundary." Communications in Contemporary Mathematics 21, no. 02 (February 27, 2019): 1850030. http://dx.doi.org/10.1142/s021919971850030x.

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We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant ([Formula: see text]-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincaré–Einstein structure, this result recovers Branson’s [Formula: see text]-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.
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47

TAY, Fatih, Metin Özgen, and Mustafa Büyükkör. "Does Late Axial Spondyloarthropathy Diagnosis Cause Extra Anti-TNF Therapy?" National Journal of Health Sciences 6, no. 3 (December 19, 2022): 94–99. http://dx.doi.org/10.21089/njhs.63.0094.

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Introduction: Ankylosing Spondylitis (AS) is a chronic inflammatory rheumatic disease that mainly characterized by sacroiliac joint and spine involvement. Although there is no clear evidence that any of these agent prevent the radiologic progression, anti-TNF drugs provide significant improvements in the disease activity score, functional index and quality of life. In AS patients, knowledge of the factors that determine the need for anti-TNF treatment will be associated with fewer complication sand better treatment. The purpose of this study is to investigate the possible factors which willmark the transition to the anti-TNF therapy in AS patients. Materials and Methods: This study was conducted in the Rheumatology division of the Internal medicine department of the Ondokuz Mayis University of Medicine hospital between January 2012- June 2015. The study protocol was approved by the Ethics Committee of Ondokuz Mayis University. A total of 165 patients, who were diagnosed as AS according to the ASAS classification criteria, were enrolled in this study. There were 85 women (51.5%) and 80 men (48.5%), aged between 15-69. Patients were divided into two groups according to their use of anti-TNF drug. Results: A total of 165 Ax-SpA patients (85 women and 80 men) were included in the study. The mean age was 37.82±11.24 years. The mean duration of the disease was 4.59±5.35 years. male gender, uveitis, delay in diagnosis, elevations in sedimentation CRP levels, increase in disease activity and functional indexes such as BASDAI and BASFI scores shows the more frequent need for anti-TNF drug use. Conclusion: In our study, patients who needed anti-TNF treatment had a longer time between symptom onset and diagnosis than patients who did not hear. The delay in diagnosing these patients leads to a delay in treatment so that the focus of inflammation increases and these patients need more anti-TNF as this window of opportunity escapes.
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48

Wang, W., J. Xiao, S. V. Ollinger, A. R. Desai, J. Chen, and A. Noormets. "Quantifying the effects of harvesting on carbon fluxes and stocks in northern temperate forests." Biogeosciences 11, no. 23 (December 4, 2014): 6667–82. http://dx.doi.org/10.5194/bg-11-6667-2014.

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Abstract. Harvest disturbance has substantial impacts on forest carbon (C) fluxes and stocks. The quantification of these effects is essential for the better understanding of forest C dynamics and informing forest management in the context of global change. We used a process-based forest ecosystem model, PnET-CN, to evaluate how, and by what mechanisms, clear-cuts alter ecosystem C fluxes, aboveground C stocks (AGC), and leaf area index (LAI) in northern temperate forests. We compared C fluxes and stocks predicted by the model and observed at two chronosequences of eddy covariance flux sites for deciduous broadleaf forests (DBF) and evergreen needleleaf forests (ENF) in the Upper Midwest region of northern Wisconsin and Michigan, USA. The average normalized root mean square error (NRMSE) and the Willmott index of agreement (d) for carbon fluxes, LAI, and AGC in the two chronosequences were 20% and 0.90, respectively. Simulated gross primary productivity (GPP) increased with stand age, reaching a maximum (1200–1500 g C m−2 yr−1) at 11–30 years of age, and leveled off thereafter (900–1000 g C m−2 yr−1). Simulated ecosystem respiration (ER) for both plant functional types (PFTs) was initially as high as 700–1000 g C m−2 yr−1 in the first or second year after harvesting, decreased with age (400–800 g C m−2 yr−1) before canopy closure at 10–25 years of age, and increased to 800–900 g C m−2 yr−1 with stand development after canopy recovery. Simulated net ecosystem productivity (NEP) for both PFTs was initially negative, with net C losses of 400–700 g C m−2 yr−1 for 6–17 years after clear-cuts, reaching peak values of 400–600 g C m−2 yr−1 at 14–29 years of age, and eventually stabilizing in mature forests (> 60 years old), with a weak C sink (100–200 g C m−2 yr−1). The decline of NEP with age was caused by the relative flattening of GPP and gradual increase of ER. ENF recovered more slowly from a net C source to a net sink, and lost more C than DBF. This suggests that in general ENF may be slower to recover to full C assimilation capacity after stand-replacing harvests, arising from the slower development of photosynthesis with stand age. Our model results indicated that increased harvesting intensity would delay the recovery of NEP after clear-cuts, but this had little effect on C dynamics during late succession. Future modeling studies of disturbance effects will benefit from the incorporation of forest population dynamics (e.g., regeneration and mortality) and relationships between age-related model parameters and state variables (e.g., LAI) into the model.
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49

Dabrock, Nils, Sascha Knüttel, and Matthias Röger. "“Gradient-free” diffuse approximations of the Willmore functional and Willmore flow." Asymptotic Analysis, October 7, 2022, 1–42. http://dx.doi.org/10.3233/asy-221810.

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We introduce new diffuse approximations of the Willmore functional and the Willmore flow. They are based on a corresponding approximation of the perimeter that has been studied by Amstutz-van Goethem [Interfaces Free Bound. 14 (2012)]. We identify the candidate for the Γ-convergence, prove the Γ-limsup statement and justify the convergence to the Willmore flow by an asymptotic expansion. Furthermore, we present numerical simulations that are based on the new approximation.
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50

Skorzinski, Florian. "Local minimizers of the Willmore functional." Analysis 35, no. 2 (January 1, 2015). http://dx.doi.org/10.1515/anly-2012-1274.

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Abstract:
AbstractSince the Willmore functional is invariant with respect to conformal transformations and reparametrizations, the kernel of the second derivative of the functional at a critical point will always contain a subspace generated by these transformations. We prove that the second derivative being positive definite outside this space is a sufficient condition for a critical point to be a local minimizer.
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