The generalized Taub-NUT congruence in Minkowski spaces

With any shear-free congruence of null geodesics in a Lorentzian geometry there is associated a Cauchy-Riemann three-space; and in certain spacetimes including the Ricci-flat spacetimes with expanding null shear-free (n.s.f.) congruences the deviation form of the congruence picks out an integrable distribution of complex two-spaces in the CR geometry. Conversely, given a CR geometry with an integrable distribution of two-spaces one can construct an associated family of spacetimes with a null, shear-free congruence. The interesting problem is the restrictionR _ab =0. We consider the case of n.s.f. congruences in Minkowski spacetime constructed from CR geometries of maximal symmetry. The special two-spaces are here taken to be those associated with either the Taub-NUT geometry or, as a limiting case, those associated with the Hauser twisting typeN solution. We obtain the most general solution for these cases.     

Cauchy—Riemann conditions and point singularities of solutions to linearized shallow-water equations

Singular solutions with algebraic “square-root” type singularity of two-dimensional equations of shallow-water theory are propagated along the trajectories of the external velocity field on which the field satisfies the Cauchy-Riemann conditions. In other words, the differential of the phase flow is proportional to an orthogonal operator on such a trajectory. It turns out that, in the linear approximation, this fact is closely related to the effect of “blurring” of solutions of hydrodynamical equations; namely, a singular solution of the Cauchy problem for the linearized shallow-water equations preserves its shape exactly (i.e., is not blurred) if and only if the Cauchy-Riemann conditions are satisfied on the trajectory (of the external field) along which the perturbation is propagated.     

Random fields as solutions of the inhomogeneous quaternionic Cauchy-Riemann equation. I. Invariance and analytic continuation

We consider random fieldsA satisfying the quaternionic Cauchy-Riemann equationA=F, whereF is white noise. Under appropriate conditions onF, A is invariant under the proper Euclidean group in four dimensions, but in general not under time reflection. The Schwinger functions can be analytically continued to Wightman functions satisfying the relativistic postulates on invariance, specrral property and locality.     

Symmetries of Cauchy-Riemann spaces

We simplify and generalize Cartan's results on Cauchy-Riemann spaces admitting continuous groups of automorphisms. We describe all such spaces in terms of local coordinates.     

Fredholm Property of Operators from 2D String Field Theory

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ -instanton equation. Solutions of ζ -instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining the Fredholm property and constructing relevant homology theory. In this article, we study the Fredholm property of a sort of differential operators with degenerate asymptotics. As an application, we verify certain Fredholm property of the linearized operator of ζ -instanton equations.     

Infinite number of local conservation laws for the self-dual SU(2) Yang-Mills system within 't Hooft's ansatz

We derive infinite sets of local continuity equations for the four-dimensional classical self-dual SU(2) Yang-Mills fields subjected to 't Hooft's ansatz. In striking analogy to the two-dimensional CP(n) non-linear sigma model where local conservation laws obtain either from complex Cauchy-Riemann analyticity or from a matrix Riccati equation, our local sets derive from quaternionic Fueter analyticity or a Riccati equation associated with the geometric prolongation structure implied by the Belavin-Zakharov linear spectral problem for the self-dual Yang-Mills system. Our analysis underlines the close connection between local and non-local conservation laws and suggests that infinite sets of local continuity equations should be present in the general self-(antiself-)dual gauge field case.     

Convergence to SPDEs in Stratonovich Form

We consider the perturbation of parabolic operators of the form ∂ _t  + P(x, D) by large-amplitude highly oscillatory spatially dependent potentials modeled as Gaussian random fields. The amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order. We show that, when the dimension is smaller than the order of the elliptic pseudo-differential operator P(x, D), the perturbed parabolic equation admits a solution given by a Duhamel expansion. Moreover, as the correlation length of the potential vanishes, we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the Stratonovich sense. The theory of mild solutions for such stochastic partial differential equations is developed. The behavior described above should be contrasted to the case of dimensions larger than or equal to the order of the elliptic pseudo-differential operator P(x, D). In the latter case, the solution to the random equation converges strongly to the solution of a homogenized (deterministic) parabolic equation as is shown in [2]. A stochastic limit is obtained only for sufficiently small space dimensions in this class of parabolic problems.     

Blow-up,Zero <Emphasis Type="Italic">α</Emphasis> Limit and the Liouville Type Theorem for the Euler-Poincaré Equations

In this paper we study the Euler-Poincaré equations in . We prove local existence of weak solutions in , and local existence of unique classical solutions in , k > N/2 + 3, as well as a blow-up criterion. For the zero dispersion equation (α = 0) we prove a finite time blow-up of the classical solution. We also prove that as the dispersion parameter vanishes, the weak solution converges to a solution of the zero dispersion equation with sharp rate as α → 0, provided that the limiting solution belongs to with k > N/2 + 3. For the stationary weak solutions of the Euler-Poincaré equations we prove a Liouville type theorem. Namely, for α > 0 any weak solution is u=0; for α= 0 any weak solution is u=0.     

Instantons and Kähler manifolds

It is shown that for two-dimensional Euclidean chiral models of the field theory with values in arbitrary Kähler manifold duality equations reduce to the Cauchy-Riemann equations on this manifold. A class of models is described possessing such type solutions, the so called instanton solutions.     

Differential geometry of atomic structure and the binding energy of atoms

The geometric theory of partial differential equations due to E. Cartan is applied to atomic systems in order to solve the many-body problems and to obtain the binding energies of electrons in an atom. The procedure consists in defining a Schrödinger equation over an Euclidean patch which overlaps with other Euclidean patches in a specified way to form a manifold. If the energy of the system has to be a minimum, it is shown using the Dirichlet principle that the coordinate systems are related by the Cauchy-Riemann relations. The invariance of the Schrödinger equations in the overlapping region leads to a nonlinear second-order equation which is invariant to automorphic transformations and whose solutions are doubly periodic functions. There are only two possible single-valued solutions to this nonlinear partial differential equation and these correspond to lattices of points in the complex space, which are (a) corners of an array of equilateral triangles, and (b) corners of an array of isosceles right-angled triangles. The first solution was used in an earlier work to derive many static properties of nuclei. In this paper it is shown that the second solution gives binding energies of atoms in agreement of about 3% for the few experimental points that are available and also in good agreement with the binding energies of atoms obtained by the perturbation theory. It is also shown that this lattice under certain approximations is equivalent to a pure Coulomb law and the Bohr orbits of the hydrogen atom are correctly predicted. In obtaining the binding energies of atoms, no free parameters are required in the theory, except for the value of the binding energy of the He atom, as the theory is developed only for spinless systems. All other constants turn out to be fundamental constants.     

Higher derivative correction to Kaluza–Klein black hole solution

We investigate the attractor mechanism in a Kaluza–Klein black hole solution in the presence of higher derivative terms. In particular, we discuss the attractor behavior of static black holes by using the effective potential approach as well as the entropy function formalism. We consider different higher derivative terms with a general coupling to the moduli field. For the R ² theory, we use an effective potential approach, looking for solutions which are analytic near the horizon and showing that they exist and enjoy attractor behavior. The attractor point is determined by extremization of the modified effective potential at the horizon. We study the effect of the general higher derivative corrections of R ⁿ terms. Using the entropy function we define the modified effective potential and we find the conditions to have the attractor solution. In particular for a single charged Kaluza–Klein black hole solution we show that a higher derivative correction dresses the singularity for an appropriate coupling, and we can find the attractor solution.     

A new particle solution in aesthetic field theory

In present theories a particle is commonly associated with a singularity of the field. A more realistic picture would describe the particle by an intense but singularity-free field. We have found a new solution to the aesthetic field equations for which the field associated with the particle has a very large magnitude. The particle appears to be bounded despite the large numbers appearing in the solution. We prove that this present solution is not equivalent to theO(3)-invariant solution discussed in Muraskin (1973b). Since our present solution appears well-behaved, the suggestion is that we do not confine ourselves toO(3)-invariant data in future work. Owing to the large magnitude fields, we were unable to study trajectories of the particle in any detail. There is nothing wrong, in itself, with large numbers. The present solution, which we have now studied, is the first instance in our work on aesthetic field theory in which large numbers appear without the suggestion of unboundedness.     

Empty Singularities in Higher-Dimensional Gravity

We study the exact solution of Einstein’s field equations consisting of a (n+2)-dimensional static and hyperplane symmetric thick slice of matter, with constant and positive energy density ρ and thickness d, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of ρ and d, the pressure is positive and bounded, presenting a maximum at an asymmetrical position. And if is small enough, the dominant energy condition is satisfied all over the spacetime. We find that this solution presents many interesting features. In particular, it has an empty singular boundary in one of the vacua.     

Infinite slabs and other weird plane symmetric space–times with constant positive density

We present the exact solution of Einstein’s equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0. This solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space–times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space–time. We match this solution to the vacuum one and compute the external gravitational field in terms of slab’s parameters. Depending on the value of κ, these slabs can be attractive, repulsive or neutral. In the first case, the space–time also finishes up above at an empty repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a “gravitational capacitor” by inserting a slice of vacuum between two such slabs.     

The initial value problem for the Whitham averaged system

We study the initial value problem for the Whitham averaged system which is important in determining the KdV zero dispersion limit. We use the hodograph method to show that, for a generic non-trivial monotone initial data, the Whitham averaged system has a solution within a region in thex-t plane for all time bigger than a large time. Furthermore, the Whitham solution matches the Burgers solution on the boundaries of the region. For hump-like initial data, the hodograph method is modified to solve the non-monotone (inx) solutions of the Whitham averaged system. In this way, we show that, for a hump-like initial data, the Whitham averaged system has a solution within a cusp for a short time after the increasing and decreasing parts of the initial data beging to interact. On the cusp, the Whitham and Burgers solutions are matched.     

Thermodynamics of charged Brans–Dicke AdS black holes

It is well known that in four dimensions, black hole solution of the Brans–Dicke–Maxwell equations is just the Reissner–Nordstrom solution with a constant scalar field. However, in n4 dimensions, the solution is not yet the (n+1)-dimensional Reissner–Nordstrom solution and the scalar field is not a constant in general. In this Letter, by applying a conformal transformation to the dilaton gravity theory, we derive a class of black hole solutions in (n+1)-dimensional (n4) Brans–Dicke–Maxwell theory in the background of anti-de Sitter universe. We obtain the conserved and thermodynamic quantities through the use of the Euclidean action method. We find a Smarr-type formula and perform a stability analysis in the canonical ensemble. We find that the solution is thermally stable for small α, while for large α the system has an unstable phase, where α is a coupling constant between the scalar and matter field.     

Ultimate gravitational mass defect

We present a new type of gravitational mass defect in which an infinite amount of matter may be bounded in a zero ADM mass. This interpolates between effects typical of closed worlds and T-spheres. We consider the Tolman model of dust distribution and show that this phenomenon reveals itself for a solution that has no origin on one side but is closed on the other side. The second class of examples corresponds to smooth gluing T-spheres to the portion of the Friedmann-Robertson-Walker solution. The procedure is generalized to combinations of smoothly connected T-spheres, FRW and Schwarzschild metrics. In particular, in this approach a finite T-sphere is obtained that looks for observers in two R-regions as the Schwarzschild metric with two different masses one of which may vanish.     

On the Boltzmann Equation for Diffusively Excited Granular Media

We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L²(^N) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.     

Existence and Uniqueness of Stationary Solutions for 3D Navier–Stokes System with Small Random Forcing via Stochastic Cascades

We consider the 3D Navier–Stokes system in the Fourier space with regular forcing given by a stationary in time stochastic process satisfying a smallness condition. We explicitly construct a stationary solution of the system and prove a uniqueness theorem for this solution in the class of functions with Fourier transform majorized by a certain function h. Moreover we prove the following “one force—one solution” principle: the unique stationary solution at time t is presented as a functional of the realization of the forcing in the past up to t. Our explicit construction of the solution is based upon the stochastic cascade representation.