Relevant bibliographies by topics / Poincare metric

Academic literature on the topic 'Poincare metric'

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Published: 6 September 2023

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Journal articles on the topic "Poincare metric"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Poincare metric"

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Lopez, Marcos D. "Discrete Approximations of Metric Measure Spaces with Controlled Geometry." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305529.

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Li, Xining. "Preservation of bounded geometry under transformations metric spaces." University of Cincinnati / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439309722.

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Wink, Matthias. "Ricci solitons and geometric analysis." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:3aae2c5e-58aa-42da-9a1b-ec15cacafdad.

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This thesis studies Ricci solitons of cohomogeneity one and uniform Poincaré inequalities for differentials on Riemann surfaces. In the two summands case, which assumes that the isotropy representation of the principal orbit consists of two inequivalent Ad-invariant irreducible summands, complete steady and expanding Ricci solitons have been detected numerically by Buzano-Dancer-Gallaugher-Wang. This work provides a rigorous construction thereof. A Lyapunov function is introduced to prove that the Ricci soliton metrics lie in a bounded region of an associated phase space. This also gives an alternative construction of non-compact Einstein metrics of non-positive scalar curvature due to Böhm. It is explained how the asymptotics of the Ricci flat trajectories induce Böhm's Einstein metrics on spheres and other low dimensional spaces. A numerical study suggests that all other Einstein metrics of positive scalar curvature which are induced by the generalised Hopf fibrations occur in an entirely non-linear regime of the Einstein equations. Extending the theory of cohomogeneity one steady and expanding Ricci solitons, an estimate which allows to prescribe the growth rate of the soliton potential at any given time is shown. As an application, continuous families of Ricci solitons on complex line bundles over products of Fano Kähler Einstein manifolds are constructed. This generalises work of Appleton and Stolarski. The method also applies to the Lü-Page-Pope set-up and allows to cover an optimal parameter range in the two summands case. The Ricci soliton equation on manifolds foliated by torus bundles over products of Fano Kähler Einstein manifolds is discussed. A rigidity theorem is obtained and a preserved curvature condition is discovered. The cohomogeneity one initial value problem is solved for m-quasi-Einstein metrics and complete metrics are described. Lp-Poincaré inequalities for k-differentials on closed Riemann surfaces are shown. The estimates are uniform in the sense that the Poincaré constant only depends on p ≥1, k ≥ 2 and the genus γ ≥ 2 of the surface but not on its complex structure. Examples show that the analogous estimate for 1-differentials cannot be uniform. This part is based on joint work with Melanie Rupflin.
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Sektnan, Lars Martin. "Poincaré type Kähler metrics and stability on toric varieties." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/43380.

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In this thesis we study the relationship between the existence of extremal Kähler metrics and stability. We introduce a space of symplectic potentials for toric manifolds, which we show gives metrics with mixed Poincaré type and cone angle singularities. We show uniqueness and that existence implies stability for extremal metrics arising from these potentials. For quadrilaterals, we give a computable criterion for stability in certain cases, giving a definite log-stable region for generic quadrilaterals. We use computational tools to find new examples of stable and unstable toric manifolds. For Poincaré type manifolds with an S1-action, we prove a version of the LeBrun-Simanca openness theorem and Arezzo-Pacard blow-up theorem.
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Edgren, Neal G. "Poincaré-Type Metrics and Their ∂‾ Estimates on Pseudoconvex Domains." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1417598740.

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Shchur, Vladimir. "Quasi-isometries between hyperbolic metric spaces, quantitative aspects." Phd thesis, Université Paris Sud - Paris XI, 2013. http://tel.archives-ouvertes.fr/tel-00867709.

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In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ''quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, $L^p$-cohomology spaces provides useful quasi-isometry invariants and Poincaré constants of balls are their quantitative incarnation. We study how Poincaré constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincaré constants for locally homogeneous metrics of the form $dt^2+\sum_ie^{2\mu_it}dx_i^2$, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).
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Christensen, Andreas. "Capacity estimates and Poincaré inequalities for the weighted bow-tie." Thesis, Linköpings universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-138111.

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We give a short introduction to various concepts related to the theory of p-harmonic functions on Rn, and some modern generalizations of these concepts to general metric spaces. The article Björn-Björn-Lehrbäck [6] serves as the starting point of our discussion. In [6], among other things, estimates of the variational capacity for thin annuli in metric spaces are given under the assumptions of a Poincaré inequality and an annular decay property. Most of the parameters in the various results of the article are proven to be sharp by counterexamples at the end of the article. The main result of this thesis is the verification of the sharpness of a parameter. At the center of our discussion will be a concrete metric subspace of weighted Rn, namely the so-called weighted bow-tie, where the weight function is assumed to be radial. A similar space was used in [6] to verify the sharpness of several parameters. We show that under the assumption that the variational p-capacity is nonzero for any ball centered at the origin, the p-Poincaré inequality holds in Rn if and only if it holds on the corresponding bow-tie Finally, we consider a concrete weight function, show that it is a Muckenhoupt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].
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Malý, Lukáš. "Sobolev-Type Spaces : Properties of Newtonian Functions Based on Quasi-Banach Function Lattices in Metric Spaces." Doctoral thesis, Linköpings universitet, Matematik och tillämpad matematik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-105616.

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This thesis consists of four papers and focuses on function spaces related to first-order analysis in abstract metric measure spaces. The classical (i.e., Sobolev) theory in Euclidean spaces makes use of summability of distributional gradients, whose definition depends on the linear structure of Rn. In metric spaces, we can replace the distributional gradients by (weak) upper gradients that control the functions’ behavior along (almost) all rectifiable curves, which gives rise to the so-called Newtonian spaces. The summability condition, considered in the thesis, is expressed using a general Banach function lattice quasi-norm and so an extensive framework is built. Sobolev-type spaces (mainly based on the Lp norm) on metric spaces, and Newtonian spaces in particular, have been under intensive study since the mid-1990s. In Paper I, the elementary theory of Newtonian spaces based on quasi-Banach function lattices is built up. Standard tools such as moduli of curve families and the Sobolev capacity are developed and applied to study the basic properties of Newtonian functions. Summability of a (weak) upper gradient of a function is shown to guarantee the function’s absolute continuity on almost all curves. Moreover, Newtonian spaces are proven complete in this general setting. Paper II investigates the set of all weak upper gradients of a Newtonian function. In particular, existence of minimal weak upper gradients is established. Validity of Lebesgue’s differentiation theorem for the underlying metric measure space ensures that a family of representation formulae for minimal weak upper gradients can be found. Furthermore, the connection between pointwise and norm convergence of a sequence of Newtonian functions is studied. Smooth functions are frequently used as an approximation of Sobolev functions in analysis of partial differential equations. In fact, Lipschitz continuity, which is (unlike -smoothness) well-defined even for functions on metric spaces, often suffices as a regularity condition. Thus, Paper III concentrates on the question when Lipschitz functions provide good approximations of Newtonian functions. As shown in the paper, it suffices that the function lattice quasi-norm is absolutely continuous and a fractional sharp maximal operator satisfies a weak norm estimate, which it does, e.g., in doubling Poincaré spaces if a non-centered maximal operator of Hardy–Littlewood type is locally weakly bounded. Therefore, such a local weak boundedness on rearrangement-invariant spaces is explored as well. Finer qualitative properties of Newtonian functions and the Sobolev capacity get into focus in Paper IV. Under certain hypotheses, Newtonian functions are proven to be quasi-continuous, which yields that the capacity is an outer capacity. Various sufficient conditions for local boundedness and continuity of Newtonian functions are established. Finally, quasi-continuity is applied to discuss density of locally Lipschitz functions in Newtonian spaces on open subsets of doubling Poincaré spaces.
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Badr, Nadine. "Interpolation réelle des espaces de Sobolev sur les espaces métriques mesurés et applications aux inégalités fonctionnelles." Phd thesis, Université Paris Sud - Paris XI, 2007. http://tel.archives-ouvertes.fr/tel-00736066.

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Dans cette thèse, nous étudions l'interpolation réelle des espaces de Sobolev et ses applications. Le manuscrit est constitué de deux parties. Dans la première partie, nous démontrons au premier chapitre que les espaces de Sobolev non homogènes W^1_p (resp. homogènes ) sur les variétés Riemanniennes complètes vérifiant la propriété de doublement et une inégalité de Poincaré forment une échelle d'interpolation réelle pour un intervalle de valeurs de p. Nous étendons ce résultat à d'autres cadres géométriques. Dans un deuxième court chapitre, nous comparons différents espaces de Sobolev sur le cone Euclidien et nous regardons le lien de ces espaces avec l'interpolation. Nous montrons sur cet exemple que l'hypothèse de Poincaré n'est pas une condition nécessaire pour pouvoir interpoler les espaces de Sobolev. Dans le dernier chapitre de cette partie, nous définissons les espaces de Sobolev non homog'nes W^1_p,V (resp. homogènes ) associés à un potentiel positif V sur une variété Riemannienne. Nous démontrons que si la variété véifie la propriété de doublement et une inégalité de Poincaré et si de plus V est dans une classe de Holder inverse, ces espaces forment aussi une échelle d'interpolation réelle pour un intervalle de valeurs de p. Nous étendons ce résultat aux cas des groupes de Lie. Dans la deuxième partie, dans un premier chapitre en collaboration avec E. Russ, nous étudions sur un graphe vérifiant la propriété de doublement et une inégalité de Poincaré, la Lp bornitude de la transformée de Riesz pour p > 2 et son inégalité inverse pour p < 2. Pour notre but, nous démontrons aussi des résultats d'interpolation des espaces de Sobolev et des inégalités de Littlewood-Paley. Dans le deuxième chapitre, nous démontrons en utilisant notre résultat d'interpolation, des inégalités de Gagliardo-Nirenberg sur les variétés Riemanniennes complètes vérifiant le doublement, des inégalités de Poincaré et pseudo-Poincaré. Ce résultat s'applique aussi dans le cadre des groupes de Lie et des graphes.
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Occhipinti, Carlotta. "Analisi quantitativa del controllo motorio fine tramite sensori inerziali in bambini in età scolare: lo studio i-MOVE." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021.

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In ogni attività quotidiana o professionale noi interagiamo costantemente con il mondo attraverso il movimento: tutta la comunicazione, che sia sotto forma di parola, scrittura o gesti, è mediata dal sistema motorio. Lo studio dei movimenti umani e del loro sviluppo nel tempo permette di comprendere il modo in cui le persone normalmente sviluppano le proprie abilità motorie e di individuare disturbi di varia natura in soggetti che, invece, presentano delle anormalità. Inoltre, esiste una forte relazione tra sviluppo motorio e sviluppo cognitivo. Questo lavoro ha permesso di fare un'analisi del movimento fine motorio in bambini in età scolare tramite l’impiego di sensori inerziali e di un test motorio recente e promettente, il Test of Motor Competence. In particolare, ai partecipanti, appartenenti ad una scuola primaria di Imola, è stato chiesto di svolgere delle prove di Placing Bricks (PB). Sulla base delle misure di accelerazione e velocità angolare registate, sono stati implementati degli algoritmi in grado di automatizzare il processo di segmentazione delle prove e il processo di identificazione delle fasi caratterizzanti ciascun ciclo di PB. Ai fini di una valutazione quantitativa delle abilità fine motorie si è proseguito con il calcolo dei tempi, con lo studio della variabilità temporale (Poincarè) e con l'applicazione di metriche non lineari (RQA e MSE). Infine, è stato effettuato un confronto statistico tra partecipanti di diversa età e\o di diverso sesso che ha mostrato una variabilità temporale e una regolarità maggiore nei partecipanti più piccoli e una complessità di movimento maggiore nei partecipanti più grandi.
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Books on the topic "Poincare metric"

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Pascal, Auscher, Coulhon T, and Grigoryan A, eds. Heat kernels and analysis on manifolds, graphs, and metric spaces: Lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs, April 16-July 13, 2002, Emile Borel Centre of the Henri Poincaré Institute, Paris, France. Providence, R.I: American Mathematical Society, 2003.

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Riemann surfaces by way of complex analytic geometry. Providence, R.I: American Mathematical Society, 2011.

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author, Tian Gang 1958, ed. The geometrization conjecture. Providence, Rhode Island: American Mathematical Society, 2014.

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1941-, Hag Kari, and Broch Ole Jacob, eds. The ubiquitous quasidisk. Providence, Rhode Island: American Mathematical Society, 2012.

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Ibragimov, Zair. Topics in several complex variables: First USA-Uzbekistan Conference on Analysis and Mathematical Physics, May 20-23, 2014, California State University, Fullerton, California. Providence, Rhode Island: American Mathematical Society, 2016.

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Doran, Connemara. Poincaré’s Mathematical Creations in Search of the ‘True Relations of Things’. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797258.003.0004.

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How did the vast corpus of mathematical innovation of Henri Poincaré (1854–1912) engage the rationale, and impact the fate, of the notion of the ether in physics? Poincaré sought the ‘true relations’ that adhere in the phenomena—relations that persist irrespective of the choice of a metric geometry and a change in physical theory. This chapter traces how Poincaré embedded utterly new geometries and topological intuitions at the heart of pure mathematics, mathematical physics and philosophy. It demonstrates that Poincaré had no ownership of the physicists’ ether concept and that he viewed the ether as neither necessary nor necessarily a hindrance for further advance. Poincaré attended to the profound and subtle needs regarding space and time within physics by creating profound and subtle mathematics to capture the ‘true relations’, of spacetime. Poincaré thereby rendered the physicists’ ether superfluous while also creating mathematical structures for gravitational and quantum phenomena.
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Tretkoff, Paula. Topological Invariants and Differential Geometry. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691144771.003.0002.

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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.
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Heat kernels and analysis on manifolds, graphs, and metric spaces: Lecture notes from a quarter program on heat kernels, random walks, and analysis on manifolds and graphs : April 16-July 13, 2002, Emile Borel Centre of the Henri Poincaré Institute, Paris, France. Providence, R.I: American Mathematical Society, 2003.

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Book chapters on the topic "Poincare metric"

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Trpin, Alenka, Biljana Mileva Boshkoska, and Pavle Boškoski. "Poincaré Metric in Algorithms for Data Mining Tools." In Beyond Databases, Architectures and Structures. Paving the Road to Smart Data Processing and Analysis, 195–203. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-19093-4_15.

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Martín, Joaquim, and Mario Milman. "Isoperimetric Hardy Type and Poincaré Inequalities on Metric Spaces." In International Mathematical Series, 285–98. New York, NY: Springer New York, 2009. http://dx.doi.org/10.1007/978-1-4419-1341-8_13.

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Frankel, S., and J. Tysk. "Upper Bounds for the Poincaré Metric Near a Fractal Boundary." In Progress in Inverse Spectral Geometry, 51–62. Basel: Birkhäuser Basel, 1997. http://dx.doi.org/10.1007/978-3-0348-8938-4_4.

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Barbaresco, Frédéric. "Symplectic Structure of Information Geometry: Fisher Metric and Euler-Poincaré Equation of Souriau Lie Group Thermodynamics." In Lecture Notes in Computer Science, 529–40. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-25040-3_57.

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Fefferman, Charles, and C. Robin Graham. "Poincaré Metrics." In The Ambient Metric (AM-178). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691153131.003.0004.

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This chapter considers the formal theory for Poincaré metrics associated to a conformal manifold (M, [g]). It shows that even Poincaré metrics are in one-to-one correspondence with straight ambient metrics, if both are in normal form. Thus, the formal theory for Poincaré metrics is a consequence of the results of Chapter 3. The derivation of a Poincaré metric from an ambient metric was described in [FG], and the inverse construction of an ambient metric as the cone metric over a Poincaré metric was given in § 5 of [GrL].
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Fefferman, Charles, and C. Robin Graham. "Conformally Flat and Conformally Einstein Spaces." In The Ambient Metric (AM-178). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691153131.003.0007.

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This chapter analyzes the ambient and Poincaré metrics for locally conformally flat manifolds and for conformal classes containing an Einstein metric. The obstruction tensor vanishes for even dimensional conformal structures of these types. It shows that for these special conformal classes, there is a way to uniquely specify the formally undetermined term at order n/2 in an invariant way and thereby obtain a unique ambient metric up to terms vanishing to infinite order and up to diffeomorphism, just like in odd dimensions. It derives a formula of Skenderis and Solodukhin [SS] for the ambient or Poincaré metric in the locally conformally flat case which is in normal form relative to an arbitrary metric in the conformal class, and proves an elated unique continuation result for hyperbolic metrics in terms of data at conformal infinity. The case n = 2 is special for all of these considerations. The chapter also derives the form of the GJMS operators for an Einstein metric.
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Fefferman, Charles, and C. Robin Graham. "Self-dual Poincaré Metrics." In The Ambient Metric (AM-178). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691153131.003.0005.

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As an application of the formal theory for Poincaré metrics, this chapter presents a formal power series proof of a result of LeBrun [LeB] asserting the existence and uniqueness of a real-analytic self-dual Einstein metric in dimension 4 defined near the boundary with prescribed real-analytic conformal infinity.
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8

"The Poincaré metric." In Frontiers in Mathematics, 27–48. Basel: Birkhäuser Basel, 2009. http://dx.doi.org/10.1007/978-3-0346-0000-2_3.

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9

"Chapter Four. Poincaré Metrics." In The Ambient Metric (AM-178), 42–49. Princeton: Princeton University Press, 2011. http://dx.doi.org/10.1515/9781400840588.42.

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10

Dinh, Tien-Cuong, Viet-Anh Nguyen, and Nessim Sibony. "Entropy for hyperbolic Riemann surface laminations I." In Frontiers in Complex Dynamics, edited by Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland. Princeton University Press, 2014. http://dx.doi.org/10.23943/princeton/9780691159294.003.0020.

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This chapter introduces a notion of entropy for possibly singular hyperbolic laminations by Riemann surfaces. It also studies the transverse regularity of the Poincaré metric and the finiteness of the entropy. The chapter first focuses on compact laminations, which are transversally smooth, before turning to the case of singular foliations, showing how the Poincaré metric on leaves is transversally Hölder continuous. In addition, the chapter considers the problem in the proof that the entropy is finite for singular foliations is quite delicate and requires a careful analysis of the dynamics around the singularities. Finally, the chapter discusses a notion of metric entropy for harmonic probability measures and gives some open questions.
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Conference papers on the topic "Poincare metric"

1

Andoni, Alexandr, T. S. Jayram, and Mihai Pătraşcu. "Lower bounds for Edit Distance and Product Metrics via Poincaré-Type Inequalities." In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2010. http://dx.doi.org/10.1137/1.9781611973075.17.

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Akinola, A. P. "Tensor as a tool in engineering analysis." In Advanced Topics in Mechanics of Materials, Structures and Construction. Materials Research Forum LLC, 2023. http://dx.doi.org/10.21741/9781644902592-44.

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Abstract. This paper underscores the potency of the invariant character of tensor and its derivative concepts and accentuate the synergy between isotropic tensor and other tensors and the corresponding vector operations. The equivalence of covariant derivative in a curvilinear coordinates system embedded with a non-constant vector field and the partial derivative in an affine coordinates system ingrained with a constant vector field is interrogated. The corresponding role of the Christofell symbols as the affine connector of vectors with their derivatives in a variable field are compared to the Frenet-Seret skew-matrix connecting the trihedrons (i.e. tangent, normal and binormal) of a moving space curve with their derivatives. The nexus of the Christofell symbols with the geodesics is also shown. The structure of the metric tensor and the Levi-Chivita skew-symmetric tensor , as isotropic tensor rank-2 and rank-3 respectively is highlighted, such that the usual operations of dot product (or scalar product or inner product) and cross product or (vector product or spin/rotation operation}) are now expressed through the isotropic tensors. Recalling the theory of exterior differential form and invoking the Poincare's theorem we show the application of the exterior product in establishing exact differential (or total differential) in calculus in relation to plane problem of Elasticity. The invariant nature of the tensor objects and operations therefrom are then copiously invoked and deployed to establish constitutive relation for materials: in finite elasticity, within the context of hyperelasticity; composites, where there is a trade-off between heterogeneity and anisotropy through homogenisation process whereby differential equations with variable coefficients are converted to differential equations with constant coefficients; and plasticity, where application of tensor is exhibited with strain gradient plasticity, and shown how the concepts provide balance of microscopic forces, balance of macroscopic forces, and plastic flow laws as concise mathematical equations.
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