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Books on the topic 'Translation invariance'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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1

MacDonald, Gordon Wilson. Invariant subspaces for weighted translation operators. Toronto: [s.n.], 1989.

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2

Zuschlag, Katrin. Narrativik und literarisches Übersetzen: Erzähltechnische Merkmale als Invariante der Übersetzung. Tübingen: G. Narr, 2002.

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3

Deruelle, Nathalie, and Jean-Philippe Uzan. Matter in curved spacetime. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0043.

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This chapter is concerned with the laws of motion of matter—particles, fluids, or fields—in the presence of an external gravitational field. In accordance with the equivalence principle, this motion will be ‘free’. That is, it is constrained only by the geometry of the spacetime whose curvature represents the gravitation. The concepts of energy, momentum, and angular momentum follow from the invariance of the solutions of the equations of motion under spatio-temporal translations or rotations. The chapter shows how the action is transformed, no longer under a modification of the field configuration, but instead under a displacement or, in the ‘passive’ version, under a translation of the coordinate grid in the opposite direction.
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4

Rajeev, S. G. Euler’s Equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0002.

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Euler derived the fundamental equations of an ideal fluid, that is, in the absence of friction (viscosity). They describe the conservation of momentum. We can derive from it the equation for the evolution of vorticity (Helmholtz equation). Euler’s equations have to be supplemented by the conservation of mass and by an equation of state (which relates density to pressure). Of special interest is the case of incompressible flow; when the fluid velocity is small compared to the speed of sound, the density may be treated as a constant. In this limit, Euler’s equations have scale invariance in addition to rotation and translation invariance. d’Alembert’s paradox points out the limitation of Euler’s equation: friction cannot be ignored near the boundary, nomatter how small the viscosity.
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5

Translation and Rotation Invariant Multiscale Image Registration. Storming Media, 2002.

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6

Scalarization and Separation by Translation Invariant Functions. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-44723-6.

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7

Street, Brian. The Calder´on-Zygmund Theory I: Ellipticity. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691162515.003.0001.

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This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.
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8

Krawczyk, J. The translation invariant uniform approximation property for compact groups. Wroclaw, 1987.

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9

M, De champs, Piquard F, and Queffelec H, eds. Analyse harmonique groupe de travail sur les espaces de Banach invariants par translation. Orsay, France: Universite de Paris-Sud De partement de Mathe matique, 1987.

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10

Mercati, Flavio. Best Matching: Technical Details. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198789475.003.0005.

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The best matching procedure described in Chapter 4 is equivalent to the introduction of a principal fibre bundle in configuration space. Essentially one introduces a one-dimensional gauge connection on the time axis, which is a representation of the Euclidean group of rotations and translations (or, possibly, the similarity group which includes dilatations). To accommodate temporal relationalism, the variational principle needs to be invariant under reparametrizations. The simplest way to realize this in point–particle mechanics is to use Jacobi’s reformulation of Mapertuis’ principle. The chapter concludes with the relational reformulation of the Newtonian N-body problem (and its scale-invariant variant).
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11

Deruelle, Nathalie, and Jean-Philippe Uzan. Conservation laws. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198786399.003.0045.

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This chapter studies how the ‘spacetime symmetries’ can generate first integrals of the equations of motion which simplify their solution and also make it possible to define conserved quantities, or ‘charges’, characterizing the system. As already mentioned in the introduction to matter energy–momentum tensors in Chapter 3, the concepts of energy, momentum, and angular momentum are related to the invariance properties of the solutions of the equations of motion under spacetime translations or rotations. The chapter explores these in greater detail. It first turns to isometries and Killing vectors. The chapter then examines the first integrals of the geodesic equation, and Noether charges.
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12

Mann, Peter. The Jacobi Energy Function. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0010.

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This chapter focuses on the Jacobi energy function, considering how the Lagrange formalism treats the energy of the system. This discussion leads nicely to conservation laws and symmetries, which are the focus of the next chapter. The Jacobi energy function associated with a Lagrangian is defined as a function on the tangent bundle. The chapter also discuss explicit vs implicit time dependence, and shows how time translational invariance ensures the generalised coordinates are inertial, meaning that the energy function is the total energy of the system. In addition, it examines the energy function using non-inertial coordinates and explicit time dependence.
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13

Goekoop, C. The Logic of Invariable Concomitance in the Tattvacintamani: Gangesa's Anumitinirupana and Vyaptivada with Introduction Translation and Commentary. C Goekoop, 2011.

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14

The Logic of Invariable Concomitance in the Tattvacintāmaṇi: Gaṅgeśa's Anumitinirūpaṇa and Vyāptivāda with Introduction Translation and Commentary. Springer, 2011.

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15

Horing, Norman J. Morgenstern. Thermodynamic Green’s Functions and Spectral Structure. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0007.

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Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
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