Artigos de revistas: "Cone singularities"

1

Oberlin, Daniel M. "singularities on the light cone". Duke Mathematical Journal 59, n.º 3 (dezembro de 1989): 747–57. http://dx.doi.org/10.1215/s0012-7094-89-05934-6.

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Soliman, Yousuf, Dejan Slepčev e Keenan Crane. "Optimal cone singularities for conformal flattening". ACM Transactions on Graphics 37, n.º 4 (10 de agosto de 2018): 1–17. http://dx.doi.org/10.1145/3197517.3201367.

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3

Anan'in, Sasha, Carlos H. Grossi, Jaejeong Lee e João dos Reis. "Hyperbolic 2-spheres with cone singularities". Topology and its Applications 272 (março de 2020): 107073. http://dx.doi.org/10.1016/j.topol.2020.107073.

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4

Dimitrov, Nikolay. "Hyper-ideal Circle Patterns with Cone Singularities". Results in Mathematics 68, n.º 3-4 (24 de março de 2015): 455–99. http://dx.doi.org/10.1007/s00025-015-0453-3.

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5

MOORE, HELEN. "STABLE MINIMAL HYPERSURFACES AND TANGENT CONE SINGULARITIES". International Journal of Mathematics 10, n.º 03 (maio de 1999): 407–13. http://dx.doi.org/10.1142/s0129167x9900015x.

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In this paper, I give an estimate on the dimension of the singular set of a tangent cone at infinity of a stable minimal hypersurface. Namely, let Mn ⊂ ℝn+1, n ≥ 2, be a complete orientable stable minimal immersion with bounded volume growth. Then n < 7 implies T∞(M) is smooth, and n ≥ 7 implies the singular set of T∞(M) has codimension at least seven.

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6

Järv, L., C. Mayer, T. Mohaupt e F. Saueressig. "Space-time singularities and the Kähler cone". Fortschritte der Physik 52, n.º 67 (1 de junho de 2004): 624–29. http://dx.doi.org/10.1002/prop.200310154.

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LIANG, JIANFENG. "HYPERBOLIC SMOOTHING EFFECT FOR SEMILINEAR WAVE EQUATIONS AT A FOCAL POINT". Journal of Hyperbolic Differential Equations 06, n.º 01 (março de 2009): 1–23. http://dx.doi.org/10.1142/s0219891609001745.

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For semi-linear dissipative wave equation □u + |ut|p - 1ut = 0, we consider finite energy solutions with singularities propagating along a focusing light cone. At the tip of cone, the singularities are focused and partially smoothed out under strong nonlinear dissipation, i.e. the solution gets up to 1/2 more L2 derivative after the focus. The smoothing phenomenon is in fact the result of simultaneous action of focusing and nonlinear dissipation.

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8

Wang, Weiqiang. "Resolution of Singularities of Null Cones". Canadian Mathematical Bulletin 44, n.º 4 (1 de dezembro de 2001): 491–503. http://dx.doi.org/10.4153/cmb-2001-049-6.

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AbstractWe give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.

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9

PIMENTEL, B. M., e A. T. SUZUKI. "CAUSAL PRESCRIPTION FOR THE LIGHT-CONE GAUGE". Modern Physics Letters A 06, n.º 28 (14 de setembro de 1991): 2649–53. http://dx.doi.org/10.1142/s0217732391003080.

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10

GUENANCIA, HENRI. "KÄHLER–EINSTEIN METRICS WITH CONE SINGULARITIES ON KLT PAIRS". International Journal of Mathematics 24, n.º 05 (maio de 2013): 1350035. http://dx.doi.org/10.1142/s0129167x13500353.

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11

Borbon, Martin de. "Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones". Complex Manifolds 4, n.º 1 (23 de fevereiro de 2017): 43–72. http://dx.doi.org/10.1515/coma-2017-0005.

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Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

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Toulisse, Jérémy. "Minimal diffeomorphism between hyperbolic surfaces with cone singularities". Communications in Analysis and Geometry 27, n.º 5 (2019): 1163–203. http://dx.doi.org/10.4310/cag.2019.v27.n5.a5.

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Song, Jijian, Yiran Cheng, Bo Li e Bin Xu. "Drawing Cone Spherical Metrics via Strebel Differentials". International Mathematics Research Notices 2020, n.º 11 (31 de maio de 2018): 3341–63. http://dx.doi.org/10.1093/imrn/rny103.

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Abstract Cone spherical metrics are conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces. By using Strebel differentials as a bridge, we construct a new class of cone spherical metrics on compact Riemann surfaces by drawing on the surfaces some class of connected metric ribbon graphs.

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14

Lerbet, J., e M. Fayet. "Singularities of mechanisms and the degree of mobility". Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 217, n.º 2 (1 de junho de 2003): 111–19. http://dx.doi.org/10.1243/146441903321898601.

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The Kinematics, statics and dynamics of singular configurations of mechanisms are analysed. Using Lie group theory, the tangent cone at such a configuration is defined and calculated. It is clear that the structure of the cone is directly linked with one of Lie algebra. The statics and dynamics of singularity analysis are addressed, making it possible to apply the concept of the transitory degree of mobility.

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15

Van Bladel, J. "Field singularities at the tip of a dielectric cone". IEEE Transactions on Antennas and Propagation 33, n.º 8 (agosto de 1985): 893–95. http://dx.doi.org/10.1109/tap.1985.1143688.

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16

ATIYAH, MICHAEL, e CLAUDE LEBRUN. "Curvature, cones and characteristic numbers". Mathematical Proceedings of the Cambridge Philosophical Society 155, n.º 1 (25 de abril de 2013): 13–37. http://dx.doi.org/10.1017/s0305004113000169.

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AbstractWe study Einstein metrics on smooth compact 4-manifolds with an edge-cone singularity of specified cone angle along an embedded 2-manifold. To do so, we first derive modified versions of the Gauss–Bonnet and signature theorems for arbitrary Riemannian 4-manifolds with edge-cone singularities, and then show that these yield non-trivial obstructions in the Einstein case. We then use these integral formulæ to obtain interesting information regarding gravitational instantons which arise as limits of such edge-cone manifolds.

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17

Colognese, Paul, e Mark Pollicott. "Minimizing entropy for translation surfaces". Conformal Geometry and Dynamics of the American Mathematical Society 26, n.º 6 (17 de agosto de 2022): 97–110. http://dx.doi.org/10.1090/ecgd/374.

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In this note we consider the entropy by Dankwart [On the large-scale geometry of flat surfaces, 2014, PhD thesis. https://bib.math.uni-bonn.de/downloads/bms/BMS-401.pdf] of unit area translation surfaces in the S L ( 2 , R ) SL(2, \mathbb R) orbits of square tiled surfaces that are the union of squares, where the singularities occur at the vertices and the singularities have a common cone angle. We show that the entropy over such orbits is minimized at those surfaces tiled by equilateral triangles where the singularities occur precisely at the vertices. We also provide a method for approximating the entropy of surfaces in the orbits.

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18

Okuma, Tomohiro, Kei-ichi Watanabe e Ken-ichi Yoshida. "The normal reduction number of two-dimensional cone-like singularities". Proceedings of the American Mathematical Society 149, n.º 11 (4 de agosto de 2021): 4569–81. http://dx.doi.org/10.1090/proc/15565.

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19

Melikhov, D., e S. Simula. "End-point singularities of Feynman graphs on the light cone". Physics Letters B 556, n.º 3-4 (março de 2003): 135–41. http://dx.doi.org/10.1016/s0370-2693(03)00124-2.

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20

Shalaev, V. I. "Singularities in the boundary layer on a cone at incidence". Fluid Dynamics 28, n.º 6 (1994): 770–77. http://dx.doi.org/10.1007/bf01049777.

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21

Ludwig, Ursula. "The Witten complex for algebraic curves with cone-like singularities". Comptes Rendus Mathematique 347, n.º 11-12 (junho de 2009): 651–54. http://dx.doi.org/10.1016/j.crma.2009.03.027.

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22

JOGLEKAR, SATISH D., e A. MISRA. "CORRECT TREATMENT OF $\bm{{1\over (\eta\cdot{\lc{k}})^{\lc{p}}}}$ SINGULARITIES IN THE AXIAL GAUGE PROPAGATOR". International Journal of Modern Physics A 15, n.º 10 (20 de abril de 2000): 1453–79. http://dx.doi.org/10.1142/s0217751x00000653.

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The propagators in axial-type, light-cone and planar gauges contain [Formula: see text]-type singularities. These singularities have generally been treated by inventing prescriptions for them. In this work, we propose an alternative procedure for treating these singularities in the path integral formalism using the known way of treating the singularities in Lorentz gauges. To this end, we use a finite field-dependent BRS transformation that interpolates between Lorentz-type and the axial-type gauges. We arrive at the ε-dependent tree propagator in the axial-type gauges. We examine the singularity structure of the propagator and find that the axial gauge propagator so constructed has no spurious poles (for real k). It however has a complicated structure in a small region near η·k=0. We show how this complicated structure can effectively be replaced by a much simpler propagator.

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23

PLENAT, CAMILLE, e DAVID TROTMAN. "ON THE MULTIPLICITIES OF FAMILIES OF COMPLEX HYPERSURFACE-GERMS WITH CONSTANT MILNOR NUMBER". International Journal of Mathematics 24, n.º 03 (março de 2013): 1350021. http://dx.doi.org/10.1142/s0129167x13500213.

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We show that the possible drop in multiplicity in an analytic family F(z, t) of complex analytic hypersurface singularities with constant Milnor number is controlled by the powers of t. We prove equimultiplicity of μ-constant families of the form f + tg + t2h if the singular set of the tangent cone of {f = 0} is not contained in the tangent cone of {h = 0}.

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24

Imagi, Yohsuke. "Example of Compact Special Lagrangians with a Stable Singularity". International Mathematics Research Notices 2020, n.º 21 (13 de fevereiro de 2020): 7975–8006. http://dx.doi.org/10.1093/imrn/rnaa001.

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Abstract We construct a family of compact almost Calabi–Yau manifolds of complex dimension 3 and therein a corresponding family of compact special Lagrangians with one-point singularities modelled upon that $T^2$-cone constructed by Harvey and Lawson [7, Chapter III.3.A, Theorem 3.1] and characterised by Haskins [8, Theorem A] as a stable $T^2$-cone in the terminology by Joyce [16, Definition 3.4 and Example 3.5].

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25

Mahmoudi, M. Hedayat, e B. W. Schulze. "Corner boundary value problems". Asian-European Journal of Mathematics 10, n.º 01 (março de 2017): 1750054. http://dx.doi.org/10.1142/s1793557117500541.

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The paper develops some crucial steps in extending the first-order cone or edge calculus to higher singularity orders. We focus here on order 2, but the ideas are motivated by an iterative approach for higher singularities.

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26

Li, Mo, Qing Fang, Zheng Zhang, Ligang Liu e Xiao-Ming Fu. "Efficient Cone Singularity Construction for Conformal Parameterizations". ACM Transactions on Graphics 42, n.º 6 (5 de dezembro de 2023): 1–13. http://dx.doi.org/10.1145/3618407.

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We propose an efficient method to construct sparse cone singularities under distortion-bounded constraints for conformal parameterizations. Central to our algorithm is using the technique of shape derivatives to move cones for distortion reduction without changing the number of cones. In particular, the supernodal sparse Cholesky update significantly accelerates this movement process. To satisfy the distortion-bounded constraint, we alternately move cones and add cones. The capability and feasibility of our approach are demonstrated over a data set containing 3885 models. Compared with the state-of-the-art method, we achieve an average acceleration of 15 times and slightly fewer cones for the same amount of distortion.

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27

Wang, Zenggui, Lishan Liu e Yonghong Wu. "Multiple positive solutions of Strum-Liouville equations with singularities". Discrete Dynamics in Nature and Society 2006 (2006): 1–11. http://dx.doi.org/10.1155/ddns/2006/32018.

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The existence of multiple positive solutions for Strum-Liouville boundary value problems with singularities is investigated. By applying a fixed point theorem of cone map, some existence and multiplicity results of positive solutions are derived. Our results improve and generalize those in some well-known results.

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28

Hodgson, Craig, e Johan Tysk. "Eigenvalue estimates and isoperimetric inequalities for cone-manifolds". Bulletin of the Australian Mathematical Society 47, n.º 1 (fevereiro de 1993): 127–43. http://dx.doi.org/10.1017/s0004972700012326.

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This paper studies eigenvalue bounds and isoperimetric inequalities for Rieman-nian spaces with cone type singularities along a codimension-2 subcomplex. These “cone-manifolds” include orientable orbifolds, and singular geometric structures on 3-manifolds studied by W. Thurston and others.We first give a precise definition of “cone-manifold” and prove some basic results on the geometry of these spaces. We then generalise results of S.-Y. Cheng on upper bounds of eigenvalues of the Laplacian for disks in manifolds with Ricci curvature bounded from below to cone-manifolds, and characterise the case of equality in these estimates.We also establish a version of the Lévy-Gromov isoperimetric inequality for cone-manifolds. This is used to find lower bounds for eigenvalues of domains in cone-manifolds and to establish the Lichnerowicz inequality for cone-manifolds. These results enable us to characterise cone-manifolds with Ricci curvature bounded from below of maximal diameter.

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29

Schrohe, E., e J. Seiler. "The Resolvent of Closed Extensions of Cone Differential Operators". Canadian Journal of Mathematics 57, n.º 4 (1 de agosto de 2005): 771–811. http://dx.doi.org/10.4153/cjm-2005-031-1.

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AbstractWe study closed extensions of an elliptic differential operator A on amanifold with conical singularities, acting as an unbounded operator on a weighted Lp-space. Under suitable conditions we show that the resolvent exists in a sector of the complex plane and decays like as Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of .As an application we treat the Laplace–Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem , u(0) = 0.

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30

Hubeny, Veronika E., Hong Liu e Mukund Rangamani. "Bulk-cone singularities & signatures of horizon formation in AdS/CFT". Journal of High Energy Physics 2007, n.º 01 (4 de janeiro de 2007): 009. http://dx.doi.org/10.1088/1126-6708/2007/01/009.

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31

Gao, Jian-Hua. "Singularities, boundary conditions and gauge link in the light cone gauge". Physics of Particles and Nuclei 45, n.º 4 (julho de 2014): 704–13. http://dx.doi.org/10.1134/s106377961404008x.

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32

Oeljeklaus, K., e W. Richthofer. "Linearization of holomorphic vector fields and a characterization of cone singularities". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 58, n.º 1 (dezembro de 1988): 63–87. http://dx.doi.org/10.1007/bf02941369.

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33

Chen, A. P., e J. P. Ma. "Light-cone singularities and transverse-momentum-dependent factorization at twist-3". Physics Letters B 768 (maio de 2017): 380–86. http://dx.doi.org/10.1016/j.physletb.2017.03.015.

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34

DI CERBO, GABRIELE, e LUCA F. DI CERBO. "Positivity in Kähler–Einstein theory". Mathematical Proceedings of the Cambridge Philosophical Society 159, n.º 2 (25 de junho de 2015): 321–38. http://dx.doi.org/10.1017/s0305004115000377.

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AbstractTian initiated the study of incomplete Kähler–Einstein metrics on quasi–projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1-α) for α∈ (0, 1). In this paper we study how the existence of such Kähler–Einstein metrics depends on α. We show that in the negative scalar curvature case, if such Kähler–Einstein metrics exist for all small cone-angles then they exist for every α∈((n+1)/(n+2), 1), wherenis the dimension. We also give a characterisation of the pairs that admit negatively curved cone-edge Kähler–Einstein metrics with cone angle close to 2π. Again if these metrics exist for all cone-angles close to 2π, then they exist in a uniform interval of angles depending on the dimension only. Finally, we show how in the positive scalar curvature case the existence of such uniform bounds is obstructed.

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35

Liu, Yuchen. "The volume of singular Kähler–Einstein Fano varieties". Compositio Mathematica 154, n.º 6 (29 de abril de 2018): 1131–58. http://dx.doi.org/10.1112/s0010437x18007042.

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We show that the anti-canonical volume of an $n$-dimensional Kähler–Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

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36

LUDWIG, URSULA. "AN ANALYTIC APPROACH TO THE STRATIFIED MORSE INEQUALITIES FOR COMPLEX CONES". International Journal of Mathematics 24, n.º 12 (novembro de 2013): 1350100. http://dx.doi.org/10.1142/s0129167x13501000.

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In a previous paper the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this paper, an analytic proof of this fact is given.

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37

BENTÍN, RICARDO, e ALFREDO T. SUZUKI. "MANDELSTAM–LEIBBRANDT: NOT REALLY A PRESCRIPTION IN THE LIGHT-CONE GAUGE". Modern Physics Letters A 22, n.º 18 (14 de junho de 2007): 1329–39. http://dx.doi.org/10.1142/s0217732307021585.

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Since the very beginning of it, perhaps the subtlest of all gauges is the light-cone gauge, for its implementation leads to characteristic singularities that require some kind of special prescription to handle them in a proper and consistent manner. The best known of these prescriptions is the Mandelstam–Leibbrandt one. In this work we revisit it showing that its status as a mere prescription is not appropriate but rather that its origin can be traced back to fundamental physical properties such as causality and covariantization methods.

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Legendre, Eveline. "Localizing the Donaldson–Futaki invariant". International Journal of Mathematics 32, n.º 08 (22 de junho de 2021): 2150055. http://dx.doi.org/10.1142/s0129167x21500555.

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We use the equivariant localization formula to prove that the Donaldson–Futaki invariant of a compact smooth (Kähler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone.

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BASSETTO, A., G. NARDELLI e R. SOLDATI. "LOCAL AND NON LOCAL COUNTERTERMS IN ALGEBRAIC NON COVARIANT GAUGES". Modern Physics Letters A 03, n.º 17 (dezembro de 1988): 1663–68. http://dx.doi.org/10.1142/s0217732388001999.

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A simple general argument is given to show that the counterterms in the space-like planar gauge with the Cauchy principal value prescription for the spurious singularities are local. This result is used to obtain limitations on the number and on the kind of nonlocal counterterms occurring in the light-cone case with the Mandelstam-Leibbrandt prescription.

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Shapiro, B. Z. "On Singularities of Smooth Maps to a Space with a Fixed Cone." MATHEMATICA SCANDINAVICA 77 (1 de dezembro de 1995): 19. http://dx.doi.org/10.7146/math.scand.a-12547.

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41

Campana, Frédéric, Henri Guenancia e Mihai Păun. "Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields". Annales scientifiques de l'École normale supérieure 46, n.º 6 (2013): 879–916. http://dx.doi.org/10.24033/asens.2205.

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42

Li, Long, e Kai Zheng. "Uniqueness of constant scalar curvature Kähler metrics with cone singularities. I: reductivity". Mathematische Annalen 373, n.º 1-2 (18 de dezembro de 2017): 679–718. http://dx.doi.org/10.1007/s00208-017-1626-z.

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43

de Borbon, Martin. "Kähler metrics with cone singularities along a divisor of bounded Ricci curvature". Annals of Global Analysis and Geometry 52, n.º 4 (17 de junho de 2017): 457–64. http://dx.doi.org/10.1007/s10455-017-9565-1.

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Motorin, I. D. "Resolution of Singularities of the Odd Nilpotent Cone of Orthosymplectic Lie Superalgebras". Functional Analysis and Its Applications 57, n.º 3 (setembro de 2023): 192–207. http://dx.doi.org/10.1134/s0016266323030024.

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45

Ghahremani, F., e C. F. Shih. "Corner Singularities of Three-Dimensional Planar Interface Cracks". Journal of Applied Mechanics 59, n.º 1 (1 de março de 1992): 61–68. http://dx.doi.org/10.1115/1.2899465.

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Very high stresses develop near the intersection of a planar interfacial crack with the free surface of joined materials with large mismatch of elastic moduli. The socalled corner singularity is more singular than the 1/distance singularity of the interior fields. The eigenvalues corresponding to the most singular state, and for which the strain energy of a finite cone is bounded, are in general complex. For a wide selection of material pairs, our calculations show that the eigenvalue of the dominant singularity, 0(r−s), is real and s increases from 0.5 to about 0.75 as the moduli mismatch increases. Values of s are reported for a broad range of material combinations. A class of anisotropic materials and bicrystals is also investigated.

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46

PRASAD, K. R., MAHAMMAD KHUDDUSH e K. V. VIDYASAGAR. "Denumerably many Positive Solutions for Iterative System of Boundary Value Problems with N-Singularities on Time Scales". Kragujevac Journal of Mathematics 47, n.º 3 (2023): 369. http://dx.doi.org/10.46793/kgjmat2303.369p.

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In this paper we consider a iterative system of two-point boundary value problems with integral boundary conditions having n singularities and involve an increasing homeomorphism, positive homomorphism operator. By applying Hölder’s inequality and Krasnoselskii’s cone ﬁxed point theorem in a Banach space, we derive suﬃcient conditions for the existence of denumerably many positive solutions. Finally we provide an example to check validity of our obtained results.

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47

CELENZA, L. S., C. M. SHAKIN, HUI-WEN WANG e XIN-HUA YANG. "SPACE-TIME PROPAGATION OF CONFINED GLUONS". International Journal of Modern Physics A 04, n.º 15 (setembro de 1989): 3807–18. http://dx.doi.org/10.1142/s0217751x89001539.

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We assume that there is gluon condensate in the zero-momentum mode in the QCD ground state. A lowest-order calculation in terms of a condensate order parameter leads to a dynamical mass for gluons via the Schwinger mechanism and a gluon propagator with no on-mass-shell singularities — that is, the gluon is a "nonpropagating mode" in the gluon condensate. We transform our momentum-space propagator into coordinate space and find that the propagator has essentially the same delta-function light-cone singularities as the free propagator. However, in contrast to a theory without confinement, we show that the propagator exhibits exponential decay, both for time-like and space-like propagation. In this manner, we obtain a space-time characterization of the confinement phenomenon in terms of an order parameter of the condensate.

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48

De Smedt, R., e J. Van Bladel. "Field singularities at the tip of a metallic cone of arbitrary cross section". IEEE Transactions on Antennas and Propagation 34, n.º 7 (julho de 1986): 865–70. http://dx.doi.org/10.1109/tap.1986.1143916.

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49

Chen, Xiuxiong, Simon Donaldson e Song Sun. "Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities". Journal of the American Mathematical Society 28, n.º 1 (28 de março de 2014): 183–97. http://dx.doi.org/10.1090/s0894-0347-2014-00799-2.

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ELIAS, Juan. "The regularity index and the depth of the tangent cone of curve singularities". Japanese journal of mathematics. New series 22, n.º 1 (1996): 51–68. http://dx.doi.org/10.4099/math1924.22.51.

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