Статті в журналах: "Poincare metric"

1

Weitsman, Allen. "Symmetrization and the Poincare Metric." Annals of Mathematics 124, no. 1 (July 1986): 159. http://dx.doi.org/10.2307/1971389.

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2

Swiatek, G. "One-dimensional maps and Poincare metric." Nonlinearity 5, no. 1 (January 1, 1992): 81–108. http://dx.doi.org/10.1088/0951-7715/5/1/003.

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3

PISCORAN, LAURIAN-IOAN, and CATALIN BARBU. "Remarks on a new metric in the unity disc of the complex plane." Carpathian Journal of Mathematics 30, no. 2 (2014): 239–44. http://dx.doi.org/10.37193/cjm.2014.02.04.

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The curvature K of an surface depends only to the surface metric and so is an intrinsic invariant. In this paper we will study a new metric in the unity complex disc, which is connected to the well-known Poincare metric used in hyperbolic geometry.

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4

MATONE, MARCO. "THE HIGGS MODEL FOR ANYONS AND LIOUVILLE ACTION: CHAOTIC SPECTRUM, ENERGY GAP AND EXCLUSION PRINCIPLE." Modern Physics Letters A 09, no. 18 (June 14, 1994): 1673–80. http://dx.doi.org/10.1142/s0217732394001519.

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The requirements of geodesic completeness and self-adjointness imply that the Hamiltonian for anyons is the Laplacian with respect to the Weil-Petersson metric. This metric is complete on the Deligne-Mumford compactification of moduli (configuration) space. The structure of this compactification fixes the possible anyon configurations. This allows us to identify anyons with singularities (elliptic points with ramification q−1) in the Poincare metric implying that anyon spectrum is chaotic for n≥3. Furthermore, the bound on the holomorphic sectional curvature of moduli spaces implies a gap in the energy spectrum. For q=0 (punctures) anyons are infinitely separated in the Poincare metric (hard core). This indicates that the exclusion principle has a geometrical interpretation. Finally we give the differential equation satisfied by the generating function for volumes of the configuration space of anyons.

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5

Koskela, P., N. Shanmugalingam, and H. Tuominen. "Removable sets for the poincare inequality on metric spaces." Indiana University Mathematics Journal 49, no. 1 (2000): 0. http://dx.doi.org/10.1512/iumj.2000.49.1719.

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6

Hamdan, Suhaivi, Defrianto Defrianto, Erwin Erwin, and Saktioto Saktioto. "Topological Gravity of Chern-Simons-Antoniadis-Savvidy in 2+1 Dimensions." Journal of Aceh Physics Society 9, no. 3 (September 1, 2020): 65–71. http://dx.doi.org/10.24815/jacps.v9i3.16635.

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Pada artikel ini akan ditunjukan analisa dari perluasan gauge invariant exact dan metric independent untuk menkontruksi lower-rank field-strength tensor. Hasil ini akan digunakan untuk mengkontruski ulang Chern-Simons-Antoniadis-Savvidy formasi (2n+1) pada dimensi genap dengan menggunakan pendekatan diferensial geometri. Selanjutnya akan dianalisa bentuk topological gravitasi 2-dimensi yang merupakan perluasan dari teorema Chern-Weil yang telah dikembangkan oleh Izurieta-Munoz-Salgado. Hasil dari penelitian ini memperlihatkan bahwa aksi Lagrangian yang sama seperti pada topological gravitasi Chern-Simons forms pada dimensi (2n+1) invariant terhadap Poincare group SO(D−1,1) SO(D−1,2). This article determine and analyess of the extended gauge invariant exact and metric independent to construct the lower-rank field-strength tensor. These results used to construct Chern-Simons-Antoniadis-Savvidy (2n+1)-forms even dimensions using a differential geometry approach. This result analyzed 2-dimensional topological gravity forms that extended Chern-Weil theorem which has been developed by Izurieta-Munoz-Salgado. These results show similary topological gravity Lagrangian action of Chern-Simons forms (2n+1)-dimension invariant under Poincare group SO(D−1,1) SO(D−1,2).Keywords: Gauge theory, field-strength tensor, Chern-Weill theorem, Chern-Simons-Antoniadis-Savvidy forms, topological gravity

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7

Padrón, Miguel Ángel, Francisco Perdomo, Ángel Plaza, and José Pablo Suárez. "The Shortest-Edge Duplication of Triangles." Mathematics 10, no. 19 (October 5, 2022): 3643. http://dx.doi.org/10.3390/math10193643.

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We introduce a new triangle transformation, the shortest-edge (SE) duplication, as a natural way of mesh derefinement suitable to those meshes obtained by iterative application of longest-edge bisection refinement. Metric properties of the SE duplication of a triangle in the region of normalised triangles endowed with the Poincare hyperbolic metric are studied. The self-improvement of this transformation is easily proven, as well as the minimum angle condition. We give a lower bound for the maximum of the smallest angles of the triangles produced by the iterative SE duplication α=π6. This bound does not depend on the shape of the initial triangle.

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8

Adachi, Yoshitaka, Mayumi Ojima, Naoko Sato, and Yuan Tsung Wang. "3D Image-Based Stereology." Materials Science Forum 706-709 (January 2012): 2687–92. http://dx.doi.org/10.4028/www.scientific.net/msf.706-709.2687.

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The features present in 3D structure have geometric properties that fall into two broad categories: topological and metric. Metric properties are generally the more familiar; these include volume, surface area, line length and curvature. Equally or even more important in some applications are the topological properties of features. The two principal topological properties are number per unit volume and connectivity. In the present study, a change in morphology of pearlite and dual phase microstructures was examined from differential geometry and topology viewpoint. 3D images of eutectoid pearlite and dual phase steels were obtained by reconstructing serial sectioning images. Their metric and topological features were evaluated using The Euler Poincare formula and The Gauss-Bonnet Theorem. In addition, newly developed fully-automated serial sectioning 3D microscope “Genus_3D” will be also introduced.

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9

Barabanov, Nikita E. "Isomorphism of Binary Operations in Differential Geometry." Symmetry 12, no. 10 (October 3, 2020): 1634. http://dx.doi.org/10.3390/sym12101634.

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We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

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10

SIEGEL, W. "UNIVERSAL SUPERSYMMETRY BY ADDING 4+4 DIMENSIONS TO THE LIGHT CONE." International Journal of Modern Physics A 04, no. 07 (April 20, 1989): 1827–49. http://dx.doi.org/10.1142/s0217751x8900073x.

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We extend the previous method for deriving manifestly Lorentz and gauge covariant free field theories from the light cone for arbitrary Poincare representations to arbitrary supersymmetry representations. In the former case the action was expressed in terms of a BRST algebra obtained by adding two commuting and two anticommuting dimensions; in the present case we add 4+4 to obtain the BRST and covariant supersymmetry generators, and we propose an action. When applied to strings this method introduces as coordinates the world-sheet metric, which may therefore be necessary for superstrings.

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11

Barabanov, Nikita E., and Abraham A. Ungar. "Differential Geometry and Binary Operations." Symmetry 12, no. 9 (September 16, 2020): 1525. http://dx.doi.org/10.3390/sym12091525.

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We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.

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12

Ahmed, Solaiman, Tanveer Ahmed Bhuiyan, Taiki Kishi, Manabu Nii, and Syoji Kobashi. "Human Activity Classification Based on Angle Variance Analysis Utilizing the Poincare Plot." Applied Sciences 11, no. 16 (August 5, 2021): 7230. http://dx.doi.org/10.3390/app11167230.

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We propose a single sensor-based activity classification method where the Poincare plot was introduced to analyze the variance of the angle between acceleration vector with gravity calculated from the raw accelerometer data for human activity classification. Two datasets named ‘Human Activity Recognition’ and ‘MHealth dataset’ were used to develop the model to classify activity from low to vigorous intensity activities and posture estimation. Short-term and long-term variability analyzing the property of the Poincare plot was used to classify activities according to the vibrational intensity of body movement. Commercially available Actigraph’s activity classification metric ‘count’ resembled value was used to compare the feasibility of the proposed classification algorithm. In the case of the HAR dataset, laying, sitting, standing, and walking activities were classified. Poincare plot parameters SD1, SD2, and SDRR of angle in the case of angle variance analysis and the mean count of X-, Y-, and Z-axis were fitted to a support vector machine (SVM) classifier individually and jointly. The variance- and count-based methods have 100% accuracy in the static–dynamic classification. Laying activity classification has 100% accuracy from other static conditions in the proposed method, whereas the count-based method has 98.08% accuracy with 10-fold cross-validation. In the sitting–standing classification, the proposed angle-based algorithm shows 88% accuracy, whereas the count-based approach has 58% accuracy with a support vector machine classifier with 10-fold cross-validation. In the classification of the variants of dynamic activities with the MHealth dataset, the accuracy for angle variance-based and count-based methods is 100%, in both cases, for fivefold cross validation with SVM classifiers.

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13

Demir, Durmuş. "Poincaré Breaking and Gauge Invariance: A Road to Emergent Gravity and New Particles." Journal of Physics: Conference Series 2533, no. 1 (June 1, 2023): 012028. http://dx.doi.org/10.1088/1742-6596/2533/1/012028.

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Abstract In this talk, we discuss how gauge symmetries broken explicitly by a Poincare-breaking UV cutoff can be restored. We show that gauge symmetries can be restored by the introduction of affine curvature in reminiscence to the Higgs field. In fact, gauge symmetries get restored and general relativity emerges at the extremum of the metric-affine action. As per this point, we show emergence of the general relativity, reveal how its parameters relate to the flat spacetime loops, elucidate the new particle spectrum it brings along, and discuss its salient signatures. We show that the resulting field-theoretic plus gravitational setup can be probed via various phenomena ranging from collider experiments to black holes.

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14

Afanas’ev, Aleksandr P., and Sergei M. Dzyuba. "On the interrelation of motions of dynamical systems in separable locally compact metric space with invariant measure." Russian Universities Reports. Mathematics, no. 141 (2023): 5–12. http://dx.doi.org/10.20310/2686-9667-2023-28-141-5-12.

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In this paper, we study the interrelation between recurrent and outgoing motions of dynamical systems. An outgoing motion is a motion whose α- and ω-limit sets are either empty or non-compact. It is shown that in a separable locally compact metric space Σ with invariant Caratheodory measure, almost all points lie on trajectories of motions that are either recurrent or outgoing, i. e. in the space Σ, the set of points Γ lying on the trajectories of nonoutgoing and non-recurrent motions has measure zero. Moreover, any motion located in Γ is both positively and negatively asymptotic with respect to the corresponding compact minimal sets. The proof of this assertion essentially relies on the classical Poincare-Caratheodory and Hopf recurrence theorems. From this proof and Hopf’s theorem, it follows that in a separable locally compact metric space, there can exist non-recurrent Poisson-stable motions, but all these motions must necessarily be outgoing. At the same time, in the compact space Σ any Poisson-stable motion is recurrent.

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15

LUSANNA, LUCA. "CLASSICAL YANG-MILLS THEORY WITH FERMIONS II: DIRAC’S OBSERVABLES." International Journal of Modern Physics A 10, no. 26 (October 20, 1995): 3675–757. http://dx.doi.org/10.1142/s0217751x95001753.

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For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

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16

ROD GOVER, A., and F. LEITNER. "A SUB-PRODUCT CONSTRUCTION OF POINCARÉ–EINSTEIN METRICS." International Journal of Mathematics 20, no. 10 (October 2009): 1263–87. http://dx.doi.org/10.1142/s0129167x09005753.

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Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré–Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conformal structure. The ambient metrics have holonomy that agrees with the conformal holonomy. In the generic case the ambient metric arises directly as a product of the metric cones over the original Einstein spaces. In general the conformal infinity of the Poincaré metric we construct is not Einstein, and so this describes a class of non-conformally Einstein metrics for which the (Fefferman–Graham) obstruction tensor vanishes.

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17

Sakovich, A., and C. Sormani. "The null distance encodes causality." Journal of Mathematical Physics 64, no. 1 (January 1, 2023): 012502. http://dx.doi.org/10.1063/5.0118979.

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A Lorentzian manifold, N, endowed with a time function, τ, can be converted into a metric space using the null distance, [Formula: see text], defined by Sormani and Vega [Classical Quant. Grav. 33(8), 085001 (2016)]. We show that if the time function is a regular cosmological time function as studied by Andersson, Galloway, and Howard [Classical Quant. Grav. 15(2), 309–322 (1998)], and also by Wald and Yip [J. Math. Phys. 22, 2659–2665 (1981)], or if, more generally, it satisfies the anti-Lipschitz condition of Chruściel, Grant, and Minguzzi [Ann. Henri Poincare 17(10), 2801–2824 (2016)], then the causal structure is encoded by the null distance in the following sense: for any p ∈ N, there is an open neighborhood U p such that for any q ∈ U p, we have [Formula: see text] if and only if q lies in the causal future of p. The local encoding of causality can be applied to prove the global encoding of causality in a variety of settings, including spacetimes N where τ is a proper function. As a consequence, in dimension n + 1, n ≥ 2, we prove that if there is a bijective map between two such spacetimes, F : M1 → M2, which preserves the cosmological time function, τ2( F( p)) = τ1( p) for any p ∈ M1, and preserves the null distance, [Formula: see text] for any p, q ∈ M1, then there is a Lorentzian isometry between them, F∗ g1 = g2. This yields a canonical procedure allowing us to convert large classes of spacetimes into unique metric spaces with causal structures and time functions. This will be applied in our upcoming work to define spacetime intrinsic flat convergence.

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18

DI CERBO, LUCA FABRIZIO. "SEIBERG–WITTEN EQUATIONS ON SURFACES OF LOGARITHMIC GENERAL TYPE." International Journal of Mathematics 24, no. 09 (August 2013): 1350074. http://dx.doi.org/10.1142/s0129167x13500742.

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We study the Seiberg–Witten (SW) equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the SW equations for any metric which is "asymptotic" to a Poincaré type metric at infinity. Then we compute a lower bound for the L2-norm of scalar curvature on these spaces and give non-existence results for Einstein metrics on blow-ups.

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19

Kraus, Daniela, and Oliver Roth. "Strong submultiplicativity of the Poincaré metric." Journal of Analysis 24, no. 1 (June 2016): 39–50. http://dx.doi.org/10.1007/s41478-016-0006-5.

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20

Solynin, A. Yu. "Radial projection and the poincaré metric." Journal of Mathematical Sciences 95, no. 3 (June 1999): 2267–75. http://dx.doi.org/10.1007/bf02172471.

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21

Lobo, Iarley P., and Michele Ronco. "Rainbow-Like Black-Hole Metric from Loop Quantum Gravity." Universe 4, no. 12 (December 1, 2018): 139. http://dx.doi.org/10.3390/universe4120139.

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Hypersurface deformation algebra consists of a fruitful approach to derive deformedsolutions of general relativity based on symmetry considerations with quantum-gravity effects,of which the linearization has been recently demonstrated to be connected to the DSR programby k-Poincaré symmetry. Based on this approach, we analyzed the solution derived for theinterior of a black hole and we found similarities with the so-called rainbow metrics, like amomentum-dependence of the metric functions. Moreover, we derived an effective, time-dependentPlanck length and compared different regularization schemes.

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22

Ranjbar-Motlagh, Alireza. "Poincaré inequality for abstract spaces." Bulletin of the Australian Mathematical Society 71, no. 2 (April 2005): 193–204. http://dx.doi.org/10.1017/s000497270003817x.

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The Poincaré inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincaré inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

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23

Keith, Stephen, and Kai Rajala. "A remark on Poincaré inequalities on metric measure spaces." MATHEMATICA SCANDINAVICA 95, no. 2 (December 1, 2004): 299. http://dx.doi.org/10.7146/math.scand.a-14461.

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We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].

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24

Neto, A. Lins, and J. C. Canille Martins. "Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves." Transactions of the American Mathematical Society 356, no. 7 (February 27, 2004): 2963–88. http://dx.doi.org/10.1090/s0002-9947-04-03434-8.

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25

Aryasomayajula, Anilatmaja, and Indranil Biswas. "Bergman kernel on Riemann surfaces and Kähler metric on symmetric products." International Journal of Mathematics 30, no. 14 (October 8, 2019): 1950071. http://dx.doi.org/10.1142/s0129167x1950071x.

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Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

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26

Bercu, Gabriel, and Mircea Crasmareanu. "Classes of harmonic functions in 2D generalized Poincaré geometry." Filomat 35, no. 1 (2021): 287–97. http://dx.doi.org/10.2298/fil2101287b.

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By using the additive and multiplicative separation of variables we find some classes of solutions of the Laplace equation for a generalization of the Poincar? upper half plane metric. Non-constant totally geodesic functions implies the flat metric and several examples are studied including the Hamilton?s cigar Ricci soliton. The Bochner formula is discussed for our generalized Poincar? metric and for its important particular cases.

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27

Bahuaud, Eric, and Frédéric Rochon. "Geometrically Finite Poincaré–Einstein Metrics." Communications in Mathematical Physics 377, no. 3 (March 4, 2020): 2159–89. http://dx.doi.org/10.1007/s00220-020-03713-4.

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28

Fefferman, Charles, and C. Robin Graham. "$Q$-Curvature and Poincaré Metrics." Mathematical Research Letters 9, no. 2 (2002): 139–51. http://dx.doi.org/10.4310/mrl.2002.v9.n2.a2.

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29

Bahuaud, Eric, and John M. Lee. "Low regularity Poincaré–Einstein metrics." Proceedings of the American Mathematical Society 146, no. 5 (December 18, 2017): 2239–52. http://dx.doi.org/10.1090/proc/13903.

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30

Xinzhong, Huang, and Shigeyoshi Owa. "On the logarithmic gradient of Poincaré metric." Proceedings of the Japan Academy, Series A, Mathematical Sciences 70, no. 7 (1994): 235–38. http://dx.doi.org/10.3792/pjaa.70.235.

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31

Yamashita, Shinji. "Univalent analytic functions and the Poincaré metric." Kodai Mathematical Journal 13, no. 2 (1990): 164–75. http://dx.doi.org/10.2996/kmj/1138039216.

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32

Durand-Cartagena, Estibalitz, Jesús Jaramillo, and Nageswari Shanmugalingam. "The ∞-Poincaré inequality on metric measure spaces." Michigan Mathematical Journal 61, no. 1 (March 2012): 63–85. http://dx.doi.org/10.1307/mmj/1331222847.

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33

BRUNELLA, MARCO. "PLURISUBHARMONIC VARIATION OF THE LEAFWISE POINCARÉ METRIC." International Journal of Mathematics 14, no. 02 (March 2003): 139–51. http://dx.doi.org/10.1142/s0129167x03001697.

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This paper is concerned with a higher dimensional generalization of the main result of our previous paper [2]: we shall prove that the Poincaré metric on the leaves of a one-dimensional holomorphic foliation on a compact Kähler manifold has a plurisubharmonic variation.

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34

Björn, Jana. "Orlicz-Poincaré inequalities, maximal functions and AΦ-conditions". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 140, № 1 (лютий 2010): 31–48. http://dx.doi.org/10.1017/s0308210508000772.

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For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

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35

VAZ, CENALO, and LOUIS WITTEN. "RECONSTRUCTING THE METRIC OF SPACE-TIME FROM CHERN-SIMONS GRAVITY WITH POINT SOURCES." Modern Physics Letters A 07, no. 30 (September 28, 1992): 2763–73. http://dx.doi.org/10.1142/s0217732392004134.

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We show that a metric theory of gravity in three dimensions arises naturally from Poincaré Chern-Simons gravity, as a theory of the representation space of the Poincaré group. In addition, we obtain exact solutions of the constraints of the Chern-Simons theory with an arbitrary number of isolated point sources in relative motion, and show show the space-time metric is reconstructed.

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36

Naor, Assaf, and Lior Silberman. "Poincaré inequalities, embeddings, and wild groups." Compositio Mathematica 147, no. 5 (August 24, 2011): 1546–72. http://dx.doi.org/10.1112/s0010437x11005343.

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AbstractWe present geometric conditions on a metric space (Y,dY) ensuring that, almost surely, any isometric action onYby Gromov’s expander-based random group has a common fixed point. These geometric conditions involve uniform convexity and the validity of nonlinear Poincaré inequalities, and they are stable under natural operations such as scaling, Gromov–Hausdorff limits, and Cartesian products. We use methods from metric embedding theory to establish the validity of these conditions for a variety of classes of metric spaces, thus establishing new fixed point results for actions of Gromov’s ‘wild groups’.

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37

Brunella, Marco. "Subharmonic variation of the leafwise Poincar� metric." Inventiones Mathematicae 152, no. 1 (April 1, 2003): 119–48. http://dx.doi.org/10.1007/s00222-002-0269-0.

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38

Rutz, Solange F., and Patrick J. McCarthy. "A finsler perturbation of the Poincar� metric." General Relativity and Gravitation 25, no. 2 (February 1993): 179–87. http://dx.doi.org/10.1007/bf00758825.

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39

CAPRI, A. Z., and S. M. ROY. "COORDINATE-INDEPENDENT DEFINITION OF TIME AND VACUUM IN CURVED SPACE–TIME." International Journal of Modern Physics A 09, no. 08 (March 30, 1994): 1239–60. http://dx.doi.org/10.1142/s0217751x9400056x.

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We propose a definition of time and of the vacuum such that they are intrinsic to a given globally hyperbolic 1 + (n − 1)-dimensional space–time geometry and independent of the choice of coordinates. To arrive at this definition we use the new physical principle that a 1 + 1-dimensional Poincaré algebra, including Killing conditions on the generators, should be valid on the hypersurface of instantaneity. Given a timelike vector at a point (an observer's velocity) we define "an instant of time" to be the spacelike surface of geodesics which pass through that point and are orthogonal to that timelike vector. Gaussian coordinates erected on this surface yield 1 + 1-dimensional subspaces with Poincaré symmetry valid on that surface. The generator associated with time translation now uniquely picks out the direction of time on that surface. This fact permits unambiguous quantization on the surface of a field evolving in this background metric. For flat space–time the corresponding vacuum is always the Minkowski vacuum. We also consider in detail the case of static and Robertson–Walker metrics in 1 + 1 dimensions and find our vacuum to be different from those given before. The vacuum for the de Sitter metric in 1 + 1 dimensions is compared with the results in the literature and found to be different. Our definition of particles, and hence particle production, is consequently different also.

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40

Mazzeo, Rafe, and Frank Pacard. "Maskit combinations of Poincaré–Einstein metrics." Advances in Mathematics 204, no. 2 (August 2006): 379–412. http://dx.doi.org/10.1016/j.aim.2005.06.001.

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41

Durand-Cartagena, Estibalitz, Jesús A. Jaramillo, and Nageswari Shanmugalingam. "First order Poincaré inequalities in metric measure spaces." Annales Academiae Scientiarum Fennicae Mathematica 38 (February 2013): 287–308. http://dx.doi.org/10.5186/aasfm.2013.3825.

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42

Björn, Anders, and Jana Björn. "Local and semilocal Poincaré inequalities on metric spaces." Journal de Mathématiques Pures et Appliquées 119 (November 2018): 158–92. http://dx.doi.org/10.1016/j.matpur.2018.05.005.

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43

Brunella, Marco. "On the regularity of the leafwise Poincaré metric." Bulletin des Sciences Mathématiques 128, no. 3 (April 2004): 189–95. http://dx.doi.org/10.1016/j.bulsci.2004.01.001.

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44

Dung, Nguyen Thac, and Chiung Jue Anna Sung. "Smooth metric measure spaces with weighted Poincaré inequality." Mathematische Zeitschrift 273, no. 3-4 (April 4, 2012): 613–32. http://dx.doi.org/10.1007/s00209-012-1023-y.

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45

Anastassiou, S., and I. Chrysikos. "Ancient solutions of the homogeneous Ricci flow on flag manifolds." Extracta Mathematicae 36, no. 1 (June 20, 2021): 99–145. http://dx.doi.org/10.17398/2605-5686.36.1.99.

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For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions pass through an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

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46

Velani, S. L. "Diophantine approximation and Hausdorff dimension in Fuchsian groups." Mathematical Proceedings of the Cambridge Philosophical Society 113, no. 2 (March 1993): 343–54. http://dx.doi.org/10.1017/s0305004100076015.

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The Poincaré disc modelof two-dimensional hyperbolic space supports a metric ρ derived from the differentialGeodesics for the metric ρ are arcs of circles orthogonal to the unit circle S, and straight lines through the origin.

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47

Ni, Wei-Tou. "Searches for the Role of Spin and Polarization in Gravity: A Five-Year Update." International Journal of Modern Physics: Conference Series 40 (January 2016): 1660010. http://dx.doi.org/10.1142/s2010194516600107.

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Searches for the role of spin in gravitation dated before the firm establishment of the electron spin in 1925. Since mass and spin, or helicity in the case of zero mass, are the Casimir invariants of the Poincaré group and mass participates in universal gravitation, these searches are natural steps to pursue. In this update, we report on the progress on this topic in the last five years after our last review. We begin with how is Lorentz/Poincaré group in local physics arisen from spacetime structure as seen by photon and matter through experiments/observations. The cosmic verification of the Galileo Equivalence Principle for photons/electromagnetic wave packets (Universality of Propagation in spacetime independent of photon energy and polarization, i.e. nonbirefringence) constrains the spacetime constitutive tensor to high precision to a core metric form with an axion degree and a dilaton degree of freedom. Hughes-Drever-type experiments then constrain this core metric to agree with the matter metric. Thus comes the metric with axion and dilation. In local physics this metric gives the Lorentz/Poincaré covariance. Constraints on axion and dilaton from polarized/unpolarized laboratory/astrophysical/cosmic experiments/observations are presented. In the end, we review the theoretical progress on the issue of gyrogravitational ratio for fundamental particles and the experimental progress on the measurements of possible long range/intermediate range spin-spin, spin-monopole and spin-cosmos interactions.

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48

Yamaleev, Robert M. "Coulomb Potential as Metrics of Hyperbolic Space." International Journal of Emerging Technology and Advanced Engineering 10, no. 10 (October 25, 2020): 6–8. http://dx.doi.org/10.46338/ijetae1020_02.

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Representation of the Coulomb field as a metrics of the hyperbolic space within the Beltrami- Poincare model of hyperbolic geometry is done. In order to connect the potential of the electric field with the length of a geodesic line a new characteristics of the Coulomb field, the entropy, is introduced.

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49

Frankel, S., and J. Tysk. "Behavior of the poincaré metric near a fractal boundary." Complex Variables, Theory and Application: An International Journal 23, no. 3-4 (December 1993): 257–67. http://dx.doi.org/10.1080/17476939308814690.

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50

Barnard, Roger W., Petros Hadjicostas, and Alexander Yu Solynin. "The Poincaré metric and isoperimetric inequalities for hyperbolic polygons." Transactions of the American Mathematical Society 357, no. 10 (May 20, 2005): 3905–32. http://dx.doi.org/10.1090/s0002-9947-05-03946-2.

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