By M. Aizenman (Chief Editor)

Articles during this volume:

1-16

On Majorana Representations of A6 and A7

A. A. Ivanov

17-63

Stability and Instability of maximum Reissner-Nordström Black gap Spacetimes for Linear Scalar Perturbations I

Stefanos Aretakis

65-100

Random Time-Dependent Quantum Walks

Alain Joye

101-131

Quantum Isometries of the Finite Noncommutative Geometry of the traditional Model

Jyotishman Bhowmick, Francesco D’Andrea and Ludwik Dąbrowski

133-156

Suitable recommendations for the Navier–Stokes challenge with an Homogeneous preliminary Value

Pierre Gilles Lemarié–Rieusset and Frédéric Lelièvre

157-183

Effective balance for Gevrey and Finitely Differentiable well-known Hamiltonians

Abed Bounemoura

185-227

One-Dimensional Chern-Simons Theory

Anton Alekseev and Pavel Mnëv

229-259

Parabolic shows of the large Yangian

Yung-Ning Peng

261-273

The Holst motion by way of the Spectral motion Principle

Frank Pfäffle and Christoph A. Stephan

275-313

Strongly targeted Gravitational Waves

Michael Reiterer and Eugene Trubowitz

315-350

Protecting the Conformal Symmetry through Bulk Renormalization on Anti deSitter Space

Michael Dütsch and Karl-Henning Rehren

351-382

Curvature Diffusions as a rule Relativity

Jacques Franchi and Yves Le Jan

383-427

KAM for the Quantum Harmonic Oscillator

Benoît Grébert and Laurent Thomann

429-462

Wall Crossing as noticeable by way of Matrix Models

Hirosi Ooguri, Piotr Sułkowski and Masahito Yamazaki

463-512

The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data

David E. Evans and Terry Gannon

513-560

Spectrum of Non-Hermitian Heavy Tailed Random Matrices

Charles Bordenave, Pietro Caputo and Djalil Chafaï

561-563

Erratum to: Diffusion on the Random Matrix tough Edge

José A. Ramírez and Brian Rider

565-566

Erratum to: Unitary Representations of tremendous Lie teams and purposes to the category and Multiplet constitution of great Particles

C. Carmeli, G. Cassinelli, A. Toigo and V. S. Varadarajan

567-607

Quantum delivery in Crystals: powerful Mass Theorem and K·P Hamiltonians

Luigi Barletti and Naoufel Ben Abdallah

609-627

A brief facts of balance of Topological Order less than neighborhood Perturbations

Sergey Bravyi and Matthew B. Hastings

629-673

A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

Jianjun Liu and Xiaoping Yuan

675-712

String buildings and Trivialisations of a Pfaffian Line Bundle

Ulrich Bunke

713-759

Global recommendations to the 3-D Incompressible Anisotropic Navier-Stokes method within the severe Spaces

Marius Paicu and Ping Zhang

761-790

Orthogonal and Symplectic Matrix types: Universality and different Properties

M. Shcherbina

791-815

From a Large-Deviations precept to the Wasserstein Gradient circulate: a brand new Micro-Macro Passage

Stefan Adams, Nicolas Dirr, Mark A. Peletier and Johannes Zimmer

817-860

LSI for Kawasaki Dynamics with susceptible Interaction

Georg Menz

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**Extra resources for Communications in Mathematical Physics - Volume 307**

**Example text**

The remaining local estimates are exactly the same. 12. 2) degenerates with respect to the transversal derivative ∂r to H+ . In order to remove this degeneracy we commute the wave equation g ψ = 0 with ∂r aiming at additionally controlling all the second derivatives of ψ (on the spacelike hypersurfaces and the spacetime region up to and including the horizon H+ ). Such commutations first appeared in [23]. 1. The spherically symmetric case. Let us first consider spherically symmetric waves. 1. 1) ∂r ψ + ψ M is conserved along H+ .

Aretakis 14. Uniform Pointwise Boundedness It remains to show that all solutions ψ of the wave equation are uniformly bounded. Proof of Theorem 4 of Sect. 4. We decompose ψ = ψ0 + ψ≥1 and prove that each projection is uniformly bounded by a norm that depends only on initial data. Indeed, by applying the following Sobolev inequality on the hypersurfaces τ we have ψ≥1 L∞( τ) ψ≥1 ≤C . 1 H ( τ) + ψ≥1 . 1) where C depends only on 0 . Note also that we use the Sobolev inequality that does not involve the L 2 -norms of zeroth order terms.

Recall that the eigenvalues of the spherical Laplacian / are equal to −l(l+1) r2 l The dimension of the eigenspaces E is equal to 2l + 1 and the corresponding eigenvectors are denoted by Y m,l , −l ≤ m ≤ l and called spherical harmonics. We have L 2 (S2 (r )) = ⊕l≥0 E l and, therefore, any function ψ ∈ L 2 (S2 (r )) can be written as ∞ l ψm,l Y m,l . 1) Linear Stability and Instability of Extreme Reissner-Nordström I 33 The right-hand side converges to ψ in L 2 of the spheres and under stronger regularity assumptions the convergence is pointwise.