Physical basis for the symmetries in the Friedmann–Robertson–Walker metric

Modern cosmological theory is based on the Friedmann–Robertson–Walker (FRW) metric. Often written in terms of co-moving coordinates, this well-known solution to Einstein’s equations owes its elegant and highly practical formulation to the cosmological principle and Weyl’s postulate, upon which it is founded. However, there is physics behind such symmetries, and not all of it has yet been recognized. In this paper, we derive the FRW metric coefficients from the general form of the spherically symmetric line element and demonstrate that, because the co-moving frame also happens to be in free fall, the symmetries in FRW are valid only for a medium with zero active mass. In other words, the spacetime of a perfect fluid in cosmology may be correctly written as FRW only when its equation of state is ρ+3p = 0, in terms of the total pressure p and total energy density ρ. There is now compelling observational support for this conclusion, including the Alcock–Paczy´nski test, which shows that only an FRW cosmology with zero active mass is consistent with the latest model-independent baryon acoustic oscillation data.     

Estimates for the KdV-Limit of the Camassa–Holm Equation

In a number of scaling limits, we prove estimates relating the solutions of the Camassa–Holm equation to the solutions of the associated KdV equation. As a consequence, suitable solutions of the water wave problem and solutions of the Camassa–Holm equation stay close together for long times.     

The Green function for the Duffin–Kemmer–Petiau equation

We present a calculation of the Green function for the Duffin–Kemmer–Petiau equation in the case of scalar and vectorial particles interacting with a square barrier potential, and relate it to that of the Klein–Gordon equation. A formal Hamiltonian of the Duffin–Kemmer–Petiau theory is first developed using the Feshbach–Villars analogy and the Sakata and Taketani decomposition. The coefficients of reflection and transmission are deduced.     

Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn–Hilliard–Cook equation, which is a Cahn–Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size. In more detail, our results can be summarized as follows. The Cahn–Hilliard–Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε→ 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finite-dimensional subspace ?_ε is high as long as the solution stays within distance r _ε=O(ε ^R ) of the homogeneous state. The subspace ?_ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise. Received: 2 May 2000 / Accepted: 8 July 2001     

Calabi–Yau Monopoles for the Stenzel Metric

We construct the first nontrivial examples of Calabi–Yau monopoles. Our main interest in these comes from Donaldson and Segal’s suggestion (Geometry of Special Holonomy and Related Topics. Surveys in Differential Geometry, vol 16, pp 1–41, 2011) that it may be possible to define an invariant of certain noncompact Calabi–Yau manifolds from these gauge theoretical equations. We focus on the Stenzel metric on the cotangent bundle of the 3-sphere \({T^* \mathbb{S}^3}\) and study monopoles under a symmetry assumption. Our main result constructs the moduli of these symmetric monopoles and shows that these are parametrized by a positive real number known as the mass of the monopole. In other words, for each fixed mass we show that there is a unique monopole that is invariant in a precise sense. Moreover, we also study the large mass limit under which we give precise results on the bubbling behavior of our monopoles. Towards the end, an irreducible SU(2) Hermitian–Yang–Mills connection on the Stenzel metric is constructed explicitly.     

Lieb–Robinson Bounds for the Toda Lattice

We establish locality estimates, known as Lieb–Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb–Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems.     

On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

We consider the coupling from the past implementation of the random–cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges to a Gumbel distribution, and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time. Perhaps surprisingly, we observe these results to hold both off criticality, where the coupling time closely mimics the coupon collector’s problem, and also at the critical point, provided the cluster fugacity is below the value at which the transition becomes discontinuous. Finally, we consider analogous questions for the single-spin Ising heat-bath process.     

Binding Threshold for the Pauli–Fierz Operator

For the Pauli–Fierz operator with a short range potential we study the binding threshold ₁() as a function of the fine structure constant and show that it converges to the binding threshold for the Schrödinger operator in the limit 0.This work was supported in part by Fondecyt (Chile), Project #102-0844. Work partially supported by HPRN-CT-2002-00277, and the Volkswagen Stiftung through a cooperation grant.     

On the dressing method for the generalised Zakharov–Shabat system

The dressing procedure for the generalised Zakharov–Shabat system is well known for systems, related to sl(N) algebras. We extend the method, constructing explicitly the dressing factors for some systems, related to orthogonal and symplectic Lie algebras. We consider ‘dressed’ fundamental analytical solutions with simple poles at the prescribed eigenvalue points and obtain the corresponding Lax potentials, representing the soliton solutions for some important nonlinear evolution equations.     

The Plancherel Theorem for Fourier–Laplace–Nahm Transform for Connections on the Projective Line

We prove that the Fourier–Laplace–Nahm transform for connections with finitely many logarithmic singularities and a double pole at infinity on the projective line, all with semi-simple singular parts, is a hyper-Kähler isometry.     

Spectrally-accurate algorithm for moving boundary problems for the Navier–Stokes equations

A spectral algorithm based on the immersed boundary conditions (IBC) concept is developed for simulations of viscous flows with moving boundaries. The algorithm uses a fixed computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization uses Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. It has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space. Performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed. The new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements.     

Spin–orbit interaction for the double ring-shaped oscillator

The spin–orbit interactions (SOI) for the single and double ring-shaped oscillator potentials are studied as an energy correction to the Schrödinger equation. We find that the degeneracy for the energy levels with angular quantum number m=0$m=0$ keeps invariant in the case of the SOI. The degeneracy is still 2 for single ring-shaped potential and 4 for double ring-shaped potential. However, for the energy levels with angular quantum number m≠0$m\ne 0$ the degeneracy is reduced from original 4 for the single ring-shaped potential and 8 for the double ring-shaped potential to 2. That is, their energy levels in the case of the SOI are split to 2 (single) and 4 (double) sublevels. There exists an accidental degeneracy for the cases |m|=2,3,4,…$\left|m\right|=2,3,4,\dots$. We note that around the critical value b₀$b_0$, the energy levels are reversed.   We also discuss some special cases for η=2,3,4,5,6,…$\eta =2,3,4,5,6,\dots$, and the b=0,c>0$b=0,c>0$. It should be pointed out that the parameter b₀$b_0$ is relevant for the angular part parameter b$b$ in the single and double ring-shaped potentials and it makes the energy levels changed from positive to negative, but the parameter c$c$ corresponds to the angular part parameter in double ring-shaped potential and the η$\eta$ is related to it. This model can be useful for investigations of axial symmetric subjects like the ring-shaped molecules or related problems and may also be easily extended to a many-electron theory.     

Thermal relaxation for the Relativistic Ornstein–Uhlenbeck Process

The thermal relaxation of a relativistic particle diffusing in a fluid at equilibrium is investigated through a numerical study of the Relativistic Ornstein–Uhlenbeck Process. The spectrum of the relaxation operator has both a discrete and a continuous component. Both components are fully characterized and the limit between them is given a simple interpretation. Short-time relaxation is addressed separately, and a global effective relaxation time is also computed. The general conclusion is that relativistic effects slow down thermalization.     

Gazeau–Klauder quasi-coherent states for the Morse oscillator

In the Letter, we have constructed and investigated some properties of the Gazeau–Klauder quasi-coherent states for the Morse potential, previously deduced by Roy and Roy. We have focused our attention on the thermal states and we have found the analytical form for the diagonal P-representation of the density operator.     

On eigenvalue estimates for the Dirac–Witten-type operators

We give optimal lower bounds for the eigenvalues of the Dirac–Witten-type operators associated with the _e0$e_0$-Killing connection and imaginary Killing connection, in terms of the mean curvature and the scalar curvature. The limiting cases are then studied and lead to interesting geometric situations.     

The conservation of energy–momentum and the mass for the graviton

In this work we give special attention to the bimetric theory of gravitation with massive gravitons proposed by Visser in 1998. In his theory, a prior background metric is necessary to take in account the massive term. Although in the great part of the astrophysical studies the Minkowski metric is the best choice to the background metric, it is not possible to consider this metric in cosmology. In order to keep the Minkowski metric as background in this case, we suggest an interpretation of the energy–momentum conservation in Visser’s theory, which is in accordance with the equivalence principle and recovers naturally the special relativity in the absence of gravitational sources. Although we do not present a general proof of our hypothesis we show its validity in the simple case of a plane and dust-dominated universe, in which the “massive term” appears like an extra contribution for the energy density.     

Conformally flat sources for the Linet–Tian spacetime

We investigate the matching, across cylindrical surfaces, of static cylindrically symmetric conformally flat spacetimes with a cosmological constant $\Lambda $ , satisfying regularity conditions at the axis, to an exterior Linet–Tian spacetime. We prove that for $\Lambda \le 0$ such matching is impossible. On the other hand, we show through simple examples that the matching is possible for $\Lambda >0$ . We suggest a physical argument that might explain these results.