Статті в журналах: "Curvature functionals"

1

Ivochkina, N. M. "Minimization of functionals generating curvature operators." Journal of Soviet Mathematics 62, no. 3 (November 1992): 2741–46. http://dx.doi.org/10.1007/bf01670999.

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2

Sheng, Weimin, and Lisheng Wang. "Variational properties of quadratic curvature functionals." Science China Mathematics 62, no. 9 (June 15, 2018): 1765–78. http://dx.doi.org/10.1007/s11425-017-9232-6.

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3

Brozos‐Vázquez, Miguel, Sandro Caeiro‐Oliveira, and Eduardo García‐Río. "Critical metrics for all quadratic curvature functionals." Bulletin of the London Mathematical Society 53, no. 3 (January 13, 2021): 680–85. http://dx.doi.org/10.1112/blms.12448.

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4

Kuwert, Ernst, Tobias Lamm, and Yuxiang Li. "Two-dimensional curvature functionals with superquadratic growth." Journal of the European Mathematical Society 17, no. 12 (2015): 3081–111. http://dx.doi.org/10.4171/jems/580.

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5

Joshi, Pushkar, and Carlo Séquin. "Energy Minimizers for Curvature-Based Surface Functionals." Computer-Aided Design and Applications 4, no. 5 (January 2007): 607–17. http://dx.doi.org/10.1080/16864360.2007.10738495.

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6

von der Mosel, Heiko. "Nonexistence results for extremals of curvature functionals." Archiv der Mathematik 69, no. 5 (November 1, 1997): 427–34. http://dx.doi.org/10.1007/s000130050141.

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7

Biondi, Biondo. "Velocity estimation by image-focusing analysis." GEOPHYSICS 75, no. 6 (November 2010): U49—U60. http://dx.doi.org/10.1190/1.3506505.

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Анотація:

Migration velocity can be estimated from seismic data by analyzing, focusing, and defocusing of residual-migrated images. The accuracy of these velocity estimates is limited by the inherent ambiguity between velocity and reflector curvature. However, velocity resolution improves when reflectors with different curvatures are present. Image focusing is measured by evaluating coherency across structural dips, in addition to coherency across aperture/azimuth angles. The inherent ambiguity between velocity and reflector curvature is directly tackled by introducing a curvature correction into the computation of the semblance functional that estimates image coherency. The resulting velocity estimator provides velocity estimates that are (1) unbiased by reflector curvature and (2) consistent with the velocity information that is routinely obtained by measuring coherency over aperture/azimuth angles. Applications to a 2D synthetic prestack data set and a 2D field prestack data set confirm that the proposed method provides consistent and unbiased velocity information. They also suggest that velocity estimates based on the new image-focusing semblance may be more robust and have higher resolution than estimates based on conventional semblance functionals. Applying the proposed method to zero-offset field data recorded in New York Harbor yields a velocity function that is consistent with available geologic information and clearly improves the focusing of the reflectors.

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8

Sarkar, Prakash. "Quantifying the Cosmic Web using the Shapefinder diagonistic." Proceedings of the International Astronomical Union 11, S308 (June 2014): 250–53. http://dx.doi.org/10.1017/s1743921316009960.

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AbstractOne of the most successful method in quantifying the structures in the Cosmic Web is the Minkowski Functionals. In 3D, there are four minkowski Functionals: Area, Volume, Integrated Mean Curvature and the Integrated Gaussian Curvature. For defining the Minkowski Functionals one should define a surface. We have developed a method based on Marching cube 33 algorithm to generate a surface from a discrete data sets. Next we calculate the Minkowski Functionals and Shapefinder from the triangulated polyhedral surface. Applying this methodology to different data sets , we obtain interesting results related to geometry, morphology and topology of the large scale structure

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9

Pulemotov, Artem. "Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces." Journal of Geometric Analysis 30, no. 1 (March 6, 2019): 987–1010. http://dx.doi.org/10.1007/s12220-019-00175-6.

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10

Fierro, F., R. Goglione, and M. Paolini. "Finite element minimization of curvature functionals with anisotropy." Calcolo 31, no. 3-4 (September 1994): 191–210. http://dx.doi.org/10.1007/bf02575878.

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11

Sheng, Weimin, and Lisheng Wang. "Bach-flat critical metrics for quadratic curvature functionals." Annals of Global Analysis and Geometry 54, no. 3 (March 12, 2018): 365–75. http://dx.doi.org/10.1007/s10455-018-9606-4.

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12

Blair, D. E., and D. Perrone. "A Variational Characterization of Contact Metric Manifolds With Vanishing Torsion." Canadian Mathematical Bulletin 35, no. 4 (December 1, 1992): 455–62. http://dx.doi.org/10.4153/cmb-1992-060-x.

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AbstractChern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition. In this paper two other generalizations are given and the critical point conditions of the corresponding integral functionals are found. For the second of these, this is the K-contact condition, suggesting that it may be the proper generalization of the Webster scalar curvature.

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13

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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14

Bereanu, Cristian, and Pedro J. Torres. "A Variational Approach for the Neumann Problem in Some FLRW Spacetimes." Advanced Nonlinear Studies 19, no. 2 (May 1, 2019): 413–23. http://dx.doi.org/10.1515/ans-2018-2030.

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AbstractIn this paper, we study, using critical point theory for strongly indefinite functionals, the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

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15

BRANSON, THOMAS, and A. ROD GOVER. "PONTRJAGIN FORMS AND INVARIANT OBJECTS RELATED TO THE Q-CURVATURE." Communications in Contemporary Mathematics 09, no. 03 (June 2007): 335–58. http://dx.doi.org/10.1142/s0219199707002460.

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It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Qk of [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qk give rise to conformally invariant quadratic forms Θk on cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Qk operators yield a map from conformal structures to Lagrangian subspaces of the direct sum Hk ⊕ Hk (where Hk is the dual of the de Rham cohomology space Hk); in an appropriate sense this generalizes the period map. We couple the Qk operators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Qk-operators the Q-curvature prescription problem.

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16

Iglesias, José A., and Alfred M. Bruckstein. "On the Gamma-convergence of some polygonal curvature functionals." Applicable Analysis 94, no. 5 (June 6, 2014): 957–79. http://dx.doi.org/10.1080/00036811.2014.910302.

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17

Feoli, A., V. V. Nesterenko, and G. Scarpetta. "Functionals linear in curvature and statistics of helical proteins." Nuclear Physics B 705, no. 3 (January 2005): 577–92. http://dx.doi.org/10.1016/j.nuclphysb.2004.10.062.

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18

Verpoort, Steven. "Curvature functionals for curves in the equi-affine plane." Czechoslovak Mathematical Journal 61, no. 2 (June 2011): 419–35. http://dx.doi.org/10.1007/s10587-011-0064-4.

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19

Shojaee, Neda, and Morteza Mirmohammad Rezaii. "On the gradient flows on Finsler manifolds." International Journal of Mathematics 28, no. 01 (January 2017): 1750007. http://dx.doi.org/10.1142/s0129167x17500070.

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The purpose of this paper is to provide a general overview of a curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first make some differentiable structures on a domain of Finslerian functionals. Then by means of the global inner product and the Berger–Ebin Theorem, we make some decomposition for the tangent space of the manifold of all Finslerian metrics. Next, we study Akbar–Zadeh curvature functional as a Finslerian functional and we find the critical points of this functional in the pointwise conformal metric direction. Through this way, we introduce a gradient flow on compact Finsler manifold. Finally, we compare this new flow with introducing Ricci flow in Finsler geometry.

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20

Arroyo, Josu, Óscar J. Garay, and Álvaro Pámpano. "Binormal Motion of Curves with Constant Torsion in 3-Spaces." Advances in Mathematical Physics 2017 (2017): 1–8. http://dx.doi.org/10.1155/2017/7075831.

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We study curve motion by the binormal flow with curvature and torsion depending velocity and sweeping out immersed surfaces. Using the Gauss-Codazzi equations, we obtain filaments evolving with constant torsion which arise from extremal curves of curvature energy functionals. They are “soliton” solutions in the sense that they evolve without changing shape.

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21

Streets, Jeffrey D. "Quasi-local mass functionals and generalized inverse mean curvature flow." Communications in Analysis and Geometry 16, no. 3 (2008): 495–537. http://dx.doi.org/10.4310/cag.2008.v16.n3.a2.

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22

Kuwert, Ernst, Andrea Mondino, and Johannes Schygulla. "Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds." Mathematische Annalen 359, no. 1-2 (January 9, 2014): 379–425. http://dx.doi.org/10.1007/s00208-013-1005-3.

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23

Mickel, Walter, Gerd E. Schröder-Turk, and Klaus Mecke. "Tensorial Minkowski functionals of triply periodic minimal surfaces." Interface Focus 2, no. 5 (June 6, 2012): 623–33. http://dx.doi.org/10.1098/rsfs.2012.0007.

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A fundamental understanding of the formation and properties of a complex spatial structure relies on robust quantitative tools to characterize morphology. A systematic approach to the characterization of average properties of anisotropic complex interfacial geometries is provided by integral geometry which furnishes a family of morphological descriptors known as tensorial Minkowski functionals. These functionals are curvature-weighted integrals of tensor products of position vectors and surface normal vectors over the interfacial surface. We here demonstrate their use by application to non-cubic triply periodic minimal surface model geometries, whose Weierstrass parametrizations allow for accurate numerical computation of the Minkowski tensors.

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24

Bordag, M., J. Lindig, V. M. Mostepanenko, and Yu V. Pavlov. "Vacuum Stress-Energy Tensor of Nonconformal Scalar Field in Quasi-Euclidean Gravitational Background." International Journal of Modern Physics D 06, no. 04 (August 1997): 449–63. http://dx.doi.org/10.1142/s0218271897000261.

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The vacuum expectation value of the stress–energy tensor of a quantized scalar field with arbitrary curvature coupling in quasi-Euclidean background is calculated. The early time approximation for nonconformal fields is introduced. This approximation makes it possible to represent the matrix elements of the stress–energy tensor as explicit functionals of the scale factor. In the case of scale factors depending on time by the degree law the energy density is calculated explicitly. It is shown that the new contributions due to nonconformal curvature coupling significantly dominate the previously known conformal contributions.

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25

Mondino, Andrea, and Johannes Schygulla. "Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 31, no. 4 (July 2014): 707–24. http://dx.doi.org/10.1016/j.anihpc.2013.07.002.

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26

Anderson, Michael T. "Extrema of curvature functionals on the space of metrics on 3-manifolds." Calculus of Variations and Partial Differential Equations 5, no. 3 (March 1, 1997): 199–269. http://dx.doi.org/10.1007/s005260050066.

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27

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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28

Jost, Jürgen. "Convex functionals and generalized harmonic maps into spaces of non positive curvature." Commentarii Mathematici Helvetici 70, no. 1 (December 1995): 659–73. http://dx.doi.org/10.1007/bf02566027.

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29

Euh, Yunhee, JeongHyeong Park, and Kouei Sekigawa. "Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds." Differential Geometry and its Applications 29, no. 5 (October 2011): 642–46. http://dx.doi.org/10.1016/j.difgeo.2011.07.001.

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30

ARROYO, JOSU, ÓSCAR J. GARAY, and JOSE MENCÍA. "QUADRATIC CURVATURE ENERGIES IN THE 2-SPHERE." Bulletin of the Australian Mathematical Society 81, no. 3 (March 2, 2010): 496–506. http://dx.doi.org/10.1017/s0004972709001142.

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AbstractThe classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the stability of the circular critical points. Since, even for this class of energies, the complete variational analysis is quite involved, we use instead a numerical approach to provide a useful method of visualization of relevant aspects concerning uniqueness, stability and explicit representation of the closed critical curves.

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31

Li, Junfang. "Evolution of Eigenvalues along Rescaled Ricci Flow." Canadian Mathematical Bulletin 56, no. 1 (March 1, 2013): 127–35. http://dx.doi.org/10.4153/cmb-2011-162-6.

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AbstractIn this paper, we discuss monotonicity formulae of various entropy functionals under various rescaled versions of Ricci flow. As an application, we prove that the lowest eigenvalue of a family of geometric operators -4Δ+kR is monotonic along the normalized Ricci flow for all k≥1 provided the initial manifold has nonpositive total scalar curvature.

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32

Anderson, Michael T. "Extrema of curvature functionals on the space of metrics on 3-manifolds, II." Calculus of Variations and Partial Differential Equations 12, no. 1 (January 1, 2001): 1–58. http://dx.doi.org/10.1007/s005260000043.

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33

Gurban, Daniela, Petru Jebelean, and Călin Şerban. "Nontrivial Solutions for Potential Systems Involving the Mean Curvature Operator in Minkowski Space." Advanced Nonlinear Studies 17, no. 4 (October 1, 2017): 769–80. http://dx.doi.org/10.1515/ans-2016-6025.

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AbstractIn this paper, we use the critical point theory for convex, lower semicontinuous perturbations of{C^{1}}-functionals to obtain the existence of multiple nontrivial solutions for one parameter potential systems involving the operator{u\mapsto\operatorname{div}(\frac{\nabla u}{\sqrt{1-|\nabla u|^{2}}})}. The solvability of a general non-potential system is also established.

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34

Rovenski, Vladimir. "Integral Formulas for Almost Product Manifolds and Foliations." Mathematics 10, no. 19 (October 5, 2022): 3645. http://dx.doi.org/10.3390/math10193645.

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Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with k=2,3 distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds.

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35

Ma, Wen-Xiu, Huiqun Zhang, and Jinghan Meng. "A Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations." East Asian Journal on Applied Mathematics 3, no. 3 (August 2013): 171–89. http://dx.doi.org/10.4208/eajam.250613.260713a.

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AbstractA non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy. Associated variational identities yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries and conserved functionals.

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36

Brito-Loeza, Carlos, and Ke Chen. "Fast iterative algorithms for solving the minimization of curvature-related functionals in surface fairing." International Journal of Computer Mathematics 90, no. 1 (January 2013): 92–108. http://dx.doi.org/10.1080/00207160.2012.720370.

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37

Ma, Bingqing, Guangyue Huang, Xingxiao Li, and Yu Chen. "Rigidity of Einstein metrics as critical points of quadratic curvature functionals on closed manifolds." Nonlinear Analysis 175 (October 2018): 237–48. http://dx.doi.org/10.1016/j.na.2018.05.017.

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38

Cuierrier, Etienne, Pierre-Olivier Roy, Rodrigo Wang, and Matthias Ernzerhof. "The fourth-order expansion of the exchange hole and neural networks to construct exchange–correlation functionals." Journal of Chemical Physics 157, no. 17 (November 7, 2022): 171103. http://dx.doi.org/10.1063/5.0122761.

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The curvature Q σ of spherically averaged exchange (X) holes ρX, σ(r, u) is one of the crucial variables for the construction of approximations to the exchange–correlation energy of Kohn–Sham theory, the most prominent example being the Becke–Roussel model [A. D. Becke and M. R. Roussel, Phys. Rev. A 39, 3761 (1989)]. Here, we consider the next higher nonzero derivative of the spherically averaged X hole, the fourth-order term T σ. This variable contains information about the nonlocality of the X hole and we employ it to approximate hybrid functionals, eliminating the sometimes demanding calculation of the exact X energy. The new functional is constructed using machine learning; having identified a physical correlation between T σ and the nonlocality of the X hole, we employ a neural network to express this relation. While we only modify the X functional of the Perdew–Burke–Ernzerhof functional [Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)], a significant improvement over this method is achieved.

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39

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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40

Pogosyan, Dmitri, Sandrine Codis, and Christophe Pichon. "Non Gaussian Minkowski functionals and extrema counts for CMB maps." Proceedings of the International Astronomical Union 11, S308 (June 2014): 61–66. http://dx.doi.org/10.1017/s1743921316009637.

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AbstractIn the conference presentation we have reviewed the theory of non-Gaussian geometrical measures for 3D Cosmic Web of the matter distribution in the Universe and 2D sky data, such as Cosmic Microwave Background (CMB) maps that was developed in a series of our papers. The theory leverages symmetry of isotropic statistics such as Minkowski functionals and extrema counts to develop post Gaussian expansion of the statistics in orthogonal polynomials of invariant descriptors of the field, its first and second derivatives. The application of the approach to 2D fields defined on a spherical sky was suggested, but never rigorously developed. In this paper we present such development treating the effects of the curvature and finiteness of the spherical space $S_2$ exactly, without relying on flat-sky approximation. We present Minkowski functionals, including Euler characteristic and extrema counts to the first non-Gaussian correction, suitable for weakly non-Gaussian fields on a sphere, of which CMB is the prime example.

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41

Fröhlich, Steffen. "On two-dimensional immersions that are stable for parametric functionals of constant mean curvature type." Differential Geometry and its Applications 23, no. 3 (November 2005): 235–56. http://dx.doi.org/10.1016/j.difgeo.2005.05.005.

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42

KAWAI, EI-ICHIRO. "A FURTHER COMMENT ON THE HAMILTON FORMALISM FOR NONLINEAR INTEGRABLE MODELS." Modern Physics Letters A 08, no. 31 (October 10, 1993): 2919–26. http://dx.doi.org/10.1142/s0217732393003330.

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An attractive operator equation, explicated in the preceding work,1 is intensively investigated with careful attention paid to its characteristics originated by alternate action of dual Hamiltonian operators. In this context, it is argued that its amenable modification can be thought of as a sort of null curvature equation. On the basis of such intriguing view, an application of the fruitful gauge-theoretic concept is tried, from which a novel formula for obtaining systematically the conserved Hamiltonian functionals is derived as a by-product.

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43

Baltazar, Halyson, and Ernani Ribeiro. "Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary." Pacific Journal of Mathematics 297, no. 1 (October 7, 2018): 29–45. http://dx.doi.org/10.2140/pjm.2018.297.29.

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44

Kang, J. H., and A. W. Leissa. "Three-Dimensional Field Equations of Motion, and Energy Functionals for Thick Shells of Revolution With Arbitrary Curvature and Variable Thickness." Journal of Applied Mechanics 68, no. 6 (May 24, 2001): 953–54. http://dx.doi.org/10.1115/1.1406961.

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Equations of motion and energy functionals are derived for a three-dimensional coordinate system especially useful for analyzing the static and dynamic behavior of arbitrarily thick shells of revolution having variable thickness. The field equations are utilized to express them in terms of displacement components.

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45

Körpinar, T., and R. C. Demirkol. "Curvature and torsion dependent energy of elastica and nonelastica for a lightlike curve in the Minkowski space." Ukrains’kyi Matematychnyi Zhurnal 72, no. 8 (August 18, 2020): 1095–105. http://dx.doi.org/10.37863/umzh.v72i8.847.

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UDC 515.1 We firstly describe conditions for being elastica or nonelastica for a lightlike elastic Cartan curve in the Minkowski space by using the Bishop orthonormal vector frame and associated Bishop components. Then we compute the energy of the lightlike elastic and nonelastic Cartan curve in the Minkowski space and investigate its relationship with the energy of the same curve in Bishop vector fields in . Here, energy functionals are computed in terms of Bishop curvatures of the lightlike Cartan curve lying in the Minkowski space .

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46

Blair, David E. "A Survey of Riemannian Contact Geometry." Complex Manifolds 6, no. 1 (January 1, 2019): 31–64. http://dx.doi.org/10.1515/coma-2019-0002.

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AbstractThis survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

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47

Mondal, C., B. K. Agrawal, J. N. De, and S. K. Samaddar. "Correlations among symmetry energy elements in Skyrme models." International Journal of Modern Physics E 27, no. 09 (September 2018): 1850078. http://dx.doi.org/10.1142/s0218301318500787.

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Motivated by the interrelationships found between the various symmetry energy elements of the energy density functionals (EDF) based on the Skyrme forces, possible correlations among them are explored. A total of 237 Skyrme EDFs are used for this purpose. As some of these EDFs yield values of a few nuclear observables far off from the present acceptable range, studies are done also with a subset of 162 EDFs that comply with a conservative set of constraints on the values of nuclear matter incompressibility coefficient, effective mass of the nucleon and the isovector splitting of effective nucleon masses to see the enhancement of the correlation strength, if any. The curvature parameter [Formula: see text] and the skewness parameter [Formula: see text] of the symmetry energy are found to be very well correlated with the linear combination of the symmetry energy coefficient and its density derivative [Formula: see text]. The isovector splitting of the effective nucleon mass, however, displays a somewhat meaningful correlation with a linear combination of the symmetry energy, its slope and its curvature parameter.

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48

De Philippis, Guido, Antonio De Rosa, and Jonas Hirsch. "The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals." Discrete & Continuous Dynamical Systems - A 39, no. 12 (2019): 7031–56. http://dx.doi.org/10.3934/dcds.2019243.

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49

White, B. "Curvature estimates and compactness theorems in 3-manifolds for surfaces that are stationary for parametric elliptic functionals." Inventiones Mathematicae 88, no. 2 (June 1987): 243–56. http://dx.doi.org/10.1007/bf01388908.

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50

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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