Thermodynamics of Magnetised Kerr-Newman Black Holes

The thermodynamics of a magnetised Kerr-Newman black hole is studied to all orders in the appended magnetic field $B$. The asymptotic properties of the metric and other fields are dominated by the magnetic flux that extends to infinity along the axis, leading to subtleties in the calculation of conserved quantities such as the angular momentum and the mass. We present a detailed discussion of the implementation of a Wald-type procedure to calculate the angular momentum, showing how ambiguities that are absent in the usual asymptotically-flat case may be resolved by the requirement of gauge invariance. We also present a formalism from which we are able to obtain an expression for the mass of the magnetised black holes. The expressions for the mass and the angular momentum are shown to be compatible with the first law of thermodynamics and a Smarr type relation. Allowing the appended magnetic field $B$ to vary results in an extra term in the first law of the form $-\mu dB$ where $\mu$ is interpreted as an induced magnetic moment. Minimising the total energy with respect to the total charge $Q$ at fixed values of the angular momentum and energy of the seed metric allows an investigation of Wald's process. The Meissner effect is shown to hold for electrically neutral extreme black holes. We also present a derivation of the angular momentum for black holes in the four-dimensional STU model, which is ${\cal N}=2$ supergravity coupled to three vector multiplets.Comment: 27 page

Structure of a model TiO2 photocatalytic interface

The interaction of water with TiO2 is crucial to many of its practical applications, including photocatalytic water splitting. Following the first demonstration of this phenomenon 40 years ago there have been numerous studies of the rutile single-crystal TiO2(110) interface with water. This has provided an atomic-level understanding of the water-TiO2 interaction. However, nearly all of the previous studies of water/TiO2 interfaces involve water in the vapour phase. Here, we explore the interfacial structure between liquid water and a rutile TiO2(110) surface pre-characterized at the atomic level. Scanning tunnelling microscopy and surface X-ray diffraction are used to determine the structure, which is comprised of an ordered array of hydroxyl molecules with molecular water in the second layer. Static and dynamic density functional theory calculations suggest that a possible mechanism for formation of the hydroxyl overlayer involves the mixed adsorption of O2 and H2O on a partially defected surface. The quantitative structural properties derived here provide a basis with which to explore the atomistic properties and hence mechanisms involved in TiO2 photocatalysis

Nonlinear modes and symmetry breaking in rotating double-well potentials

We study modes trapped in a rotating ring carrying the self-focusing (SF) or defocusing (SDF) cubic nonlinearity and double-well potential $\cos^{2}\theta$, where $\theta$ is the angular coordinate. The model, based on the nonlinear Schr\"{o}dinger (NLS) equation in the rotating reference frame, describes the light propagation in a twisted pipe waveguide, as well as in other optical settings, and also a Bose-Einstein condensate (BEC)trapped in a torus and dragged by the rotating potential. In the SF and SDF regimes, five and four trapped modes of different symmetries are found, respectively. The shapes and stability of the modes, and transitions between them are studied in the first rotational Brillouin zone. In the SF regime, two symmetry-breaking transitions are found, of subcritical and supercritical types. In the SDF regime, an antisymmetry-breaking transition occurs. Ground-states are identified in both the SF and SDF systems.Comment: Physical Review A, in pres

Nonlinear Anisotropic Degenerate Parabolic-Hyperbolic Equations with Stochastic Forcing

We are concerned with nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing, which are heterogeneous (i.e., not space-translational invariant). A unified framework is established for the continuous dependence estimates, fractional BV regularity estimates, and well-posedness for stochastic entropy solutions of the nonlinear stochastic degenerate parabolic-hyperbolic equation. In particular, we establish the well-posedness of the nonlinear stochastic equation in $L^p \cap N^{\kappa,1}$ for $p\in (1,\infty)$ and the $\kappa$--Nikolskii space $N^{\kappa,1}$ with $\kappa>0$, and the $L^1$ continuous dependence of the stochastic entropy solutions not only on the initial data, but also on the degenerate diffusion matrix function, the flux function, and the multiplicative noise function involving in the nonlinear equation.Comment: 30 page

Quantum integrable system with two color components in two dimensions

The Davey-Stewartson 1(DS1) system[9] is an integrable model in two dimensions. A quantum DS1 system with 2 colour-components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 colour-components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimensions. For latters with the $\delta$-function interactions and being solved by the Bethe ansatz, we introduce symmetrical and antisymmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solusions is discussed.Comment: 14 pages, LaTeX fil

Ordered Carboxylates on TiO (110) Formed at Aqueous Interfaces

As models for probing the interactions between TiO2 surfaces and the dye molecules employed in dye-sensitized solar cells, carboxylic acids are an important class of molecules. In this work we present a scanning tunneling microscopy (STM) and low energy electron diffraction (LEED) study of three small carboxylic acids (formic, acetic, and benzoic) that were reacted with the TiO2(110) surface via a dipping procedure. The three molecules display quite different adsorption behavior, illustrating the different inter-adsorbate interactions that can occur. After exposure to a 10 mM solution, formic acid forms a rather disordered formate overlayer with two distinct binding geometries. Acetic acid forms a well-ordered (2 × 1) acetate overlayer similar to that observed following deposition from vapor. Benzoic acid forms a (2 × 2) overlayer which is stabilized by intermolecular interactions between the phenyl groups

Normal families and fixed points of iterates

Let F be a family of holomorphic functions and let K be a constant less than 4. Suppose that for all f in F the second iterate of f does not have fixed points for which the modulus of the multiplier is greater than K. We show that then F is normal. This is deduced from a result about the multipliers of iterated polynomials.Comment: 5 page