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Books on the topic 'Generalized Metric Spaces'

Author: Grafiati

Published: 5 October 2024

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1

Lin, Shou, and Ziqiu Yun. Generalized Metric Spaces and Mappings. Paris: Atlantis Press, 2016. http://dx.doi.org/10.2991/978-94-6239-216-8.

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Karapinar, Erdal, and Ravi P. Agarwal. Fixed Point Theory in Generalized Metric Spaces. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-14969-6.

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3

Abate, Marco. Finsler metrics-- a global approach: With applications to geometric function theory. Berlin: Springer-Verlag, 1994.

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4

Lin, Shou, and Ziqiu Yun. Generalized Metric Spaces and Mappings. Atlantis Press (Zeger Karssen), 2016.

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5

Karapinar, Erdal, and Ravi P. Agarwal. Fixed Point Theory in Generalized Metric Spaces. Springer International Publishing AG, 2022.

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6

Fixed Point Theory in Generalized Metric Spaces. Springer International Publishing AG, 2023.

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7

Fundamentals of Signal Processing in Generalized Metric Spaces. CRC Press LLC, 2022.

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8

Busemann, Herbert. Metric Methods of Finsler Spaces and in the Foundations of Geometry. (AM-8). Princeton University Press, 2016.

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9

Popoff, Andrey. Fundamentals of Signal Processing in Generalized Metric Spaces: Algorithms and Applications. Taylor & Francis Group, 2022.

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10

Popoff, Andrey. Fundamentals of Signal Processing in Generalized Metric Spaces: Algorithms and Applications. Taylor & Francis Group, 2022.

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11

Popoff, Andrey. Fundamentals of Signal Processing in Generalized Metric Spaces: Algorithms and Applications. Taylor & Francis Group, 2022.

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12

Fundamentals of Signal Processing in Generalized Metric Spaces: Algorithms and Applications. CRC Press LLC, 2022.

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13

Metrics on the phase space and non-selfadjoint pseudo-differential operators. Basel: Birkhäuser, 2010.

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14

Deruelle, Nathalie, and Jean-Philippe Uzan. Vector geometry. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0002.

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This chapter defines the mathematical spaces to which the geometrical quantities discussed in the previous chapter—scalars, vectors, and the metric—belong. Its goal is to go from the concept of a vector as an object whose components transform as Tⁱ → 𝓡ⱼ ⁱTj under a change of frame to the ‘intrinsic’ concept of a vector, T. These concepts are also generalized to ‘tensors’. The chapter also briefly remarks on how to deal with non-Cartesian coordinates. The velocity vector v is defined as a ‘free’ vector belonging to the vector space ε‎3 which subtends ε‎3. As such, it is not bound to the point P at which it is evaluated. It is, however, possible to attach it to that point and to interpret it as the tangent to the trajectory at P.

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15

Theory of Complex Finsler Geometry and Geometry of Intrinsic Metrics. World Scientific Publishing Co Pte Ltd, 2016.

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16

Theory of Complex Finsler Geometry and Geometry of Intrinsic Metrics. World Scientific Publishing Co Pte Ltd, 2016.

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17

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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18

Hrushovski, Ehud, and François Loeser. The space of stably dominated types. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161686.003.0003.

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This chapter introduces the space unit vector V of stably dominated types on a definable set V. It first endows unit vector V with a canonical structure of a (strict) pro-definable set before providing some examples of stably dominated types. It then endows unit vector V with the structure of a definable topological space, and the properties of this definable topology are discussed. It also examines the canonical embedding of V in unit vector V as the set of simple points. An essential feature in the approach used in this chapter is the existence of a canonical extension for a definable function on V to unit vector V. This is considered in the next section where continuity criteria are given. The chapter concludes by describing basic notions of (generalized) paths and homotopies, along with good metrics, Zariski topology, and schematic distance.

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19

Mercati, Flavio. York’s Solution to the Initial-Value Problem. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198789475.003.0008.

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In this chapter I briefly review York’s method (or the conformal method) for solving the initial value problem of (GR). This method, developed initially by Lichnerowicz and then generalized by Choquet-Bruhat and York, allows to find solutions of the constraints of (GR) (in particular the Hamiltonian, or refoliation constraint) by scanning the conformal equivalence class of spatial metrics for a solution of the Hamiltonian constraint, exploiting the fact that, in a particular foliation (CMC), the transverse nature of the momentum field is preserved under conformal transformations. This method allows to transform the initial value problem into an elliptic problem for the solution for which good existence and uniqueness theorems are available. Moreover this method allows to identify the reduced phase space of (GR) with the cotangent bundle to conformal superspace (the space of conformal 3-geometries), when the CMC foliation is valid. SD essentially amounts to taking this phase space as fundamental and renouncing the spacetime description when the CMC foliation is not available.

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