Geometry of differential operators of second order,the algebra of densities,and groupoids

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pencils. We consider the self-adjoint operators and analyze the operator pencils that pass through a given operator acting on densities of a particular weight. There are ‘singular values’ for pencil parameters. They are related with interesting geometric picture. In particular, we obtain operators that depend on certain equivalence classes of connections (instead of connections as such). We study the corresponding groupoids. From this point of view we analyze two examples: the canonical Laplacian on an odd symplectic supermanifold appearing in Batalin–Vilkovisky geometry and the Sturm–Liouville operator on the line, related with classical constructions of projective geometry. We also consider the canonical second order semi-density arising on odd symplectic supermanifolds, which has some similarity with mean curvature of surfaces in Riemannian geometry.     

Conventionalism and general relativity

We argue that the geometry of spacetime is a convention that can be freely chosen by the scientist; no experiment can ever determine this geometry of spacetime, only the behavior of matter in space and time. General relativity is then rewritten in terms of an arbitrary conventional geometry of spacetime in which particle trajectories are determined by forces in that geometry, and the forces determined by fields produced by sources in that geometry. As an example, we consider radial trajectories in the field of a single particle expressed in the spacetime of special relativity.     

Confinement-driven translocation of a flexible polymer

We consider the escape of a flexible, self-avoiding polymer chain out of a confined geometry. By means of simulations, we demonstrate that the translocation time can be described by a simple scaling law that exhibits a nonlinear dependence on the degree of polymerization and that is sensitive to the nature of the confining geometry. These results contradict earlier predictions but are in agreement with recently confirmed geometry-dependent expressions for the free energy of confinement.     

Generalized Pseudo-Kähler Structures

In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure.     

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.     

Structures symplectiques sur les espaces de courbes projectives et affines

A symplectic structure on the space of nondegenerate and nonparametrized curves in a locally affine manifold is defined. We also consider several interesting spaces of nondegenerate projective curves endowed with Poisson structures. This construction connects the Virasoro algebra and the Gel'fand-Dikii bracket with the projective differential geometry.     

Riemannian geometry of irrotational vortex acoustics

We consider acoustic propagation in an irrotational vortex, using the technical machinery of differential geometry to investigate the "acoustic geometry" that is probed by the sound waves. The acoustic space-time curvature of a constant circulation hydrodynamical vortex leads to deflection of phonons at appreciable distances from the vortex core. The scattering angle for phonon rays is shown to be quadratic in the small quantity Gamma/2pi(cb), where Gamma is the vortex circulation, c the speed of sound, and b the impact parameter.     

The interaction of Rayleigh waves with ferromagnetic spins; Propagation parallel to the magnetization

We present a theoretical study of the attenuation and modification of the dispersion relation for Rayleigh waves as a consequence of their interaction with ferromagnetic spins. We consider the geometry where the magnetization and Zeeman field are parallel to the sample surface and to the propagation direction of the Rayleigh wave.     

Large N Duality, Lagrangian Cycles, and Algebraic Knots

We consider knot invariants in the context of large N transitions of topological strings. In particular we consider aspects of Lagrangian cycles associated to knots in the conifold geometry. We show how these can be explicitly constructed in the case of algebraic knots. We use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with HOMFLY polynomials. Furthermore, for torus knots, using the explicit construction of the Lagrangian cycle, we also give a direct A-model computation and recover the HOMFLY polynomial for this case.     

The Mixmaster Universe in Five Dimensions

We consider a five dimensional vacuum cosmology with Bianchi type-IX spatial geometry and an extra non-compact coordinate. Finding a new class of solutions, we examine and rule out the possibility of deterministic chaos. We interpret this result within the context of induced matter theory.     

Thermodynamics and tropical mathematics. Definition of quasistatistical processes

We consider the relations between thermodynamics on the one hand and the (max,+)-algebra and tropical mathematics on the other hand. The contribution of Grigorii Litvinov to tropical geometry is emphasized. Relations for a liquid in the negative pressure domain are given.     

Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact

We show that the classical Marsden-Weinstein Reduction theorem for Hamiltonian systems with symmetries is still true for contact manifolds and cosympletic manifolds (i.e. canonical manifolds in the sense of A. Lichnerowicz).

In fact, we precise the notion of transitive almost contact structure, which enables us to consider the cosymplectic geometry as a limit of the contact geometry when a certain parameter goes to zero. This point of view unifies both theories.

However, we have to give two distinct proofs for the contact Reduction theorem and the cosymplectic one.     

Viscous Fingering Patterns in Ferrofluids

Viscous fingering occurs in the flow of two immiscible, viscous fluids between the plates of a Hele–Shaw cell. Due to pressure gradients or gravity, the initially planar interface separating the two fluids undergoes a Saffman–Taylor instability and develops fingerlike structures. When one of the fluids is a ferrofluid and a perpendicular magnetic field is applied, the labyrinthine instability supplements the usual viscous fingering instability, resulting in visually striking, complex patterns. We consider this problem in a rectangular flow geometry using a perturbative mode-coupling analysis. We deduce two general results: viscosity contrast between the fluids drives interface asymmetry, with no contribution from magnetic forces; magnetic repulsion within the ferrofluid generates finger tip-splitting, which is absent in the rectangular geometry for ordinary fluids.     

Thermomechanical effects in uniformly aligned dye-doped nematic liquid crystals

We show theoretically that thermomechanical effects in dye-doped nematic liquid crystals when illuminated by laser beams, can become important and lead to molecular reorientation at intensities substantially lower than that needed for optical Fréedericksz transition. We propose a 1D model that assumes homogenous intensity distribution in the plane of the layer and is capable to describe such a thermally induced threshold lowering. We consider a particular geometry, with a linearly polarized light incident perpendicularly on a layer of homeotropically aligned dye-doped nematics.     

On supersymmetric Lorentzian Einstein-Weyl spaces

We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the Levi-Cività connection), or they are conformally related to certain Kundt spacetime. A full characterisation is obtained for the 4- and 6-dimensional cases.     

Electrostatic response of a half-disk

This article studies the response of a half-disk exposed to an external uniform static electric field. A semianalytical method is presented for computing the potential for a geometry consisting of two conjoined half-disks with different permittivities. The method is based on analytical series expansions with coefficients obtained as a numerical solution of a matrix equation. We consider the polarizability of a single dielectric half-disk and discuss a duality relation observed in 2D polarizability. We also study the surface plasmons supported by a negative-permittivity half-disk.     

Binary Systems Around a Black Hole

We present a novel method to study interacting orbits in a fixed mean gravitational field associated with a solution of the Einstein field equations. The idea is to consider the Newton gravity among the orbiting particles in a geometry given by the main source. For this purpose, the motion equations are obtained in two different but equivalent ways. The particles can either be considered as a zeroth order (static) perturbation to the given metric or as an external Newtonian force in the geodesic equations. After obtaining the motion equations we perform simulations of two and three interacting particles moving around a black hole, i.e., in a Schwarzschild geometry. We also compare with the equivalent Newtonian problem and note differences in the stability, e.g., binary systems are found only in the general relativistic approach.     

Measurement schemes for the spin quadratures on an ensemble of atoms

We consider how to measure collective spin states of an atomic ensemble based on the multi-pass approaches for quantum interface between light and atoms. We find that a scheme with two passages of a light pulse through the atomic ensemble is efficient to implement the homodyne tomography of the spin state. Thereby, we propose to utilize optical pulses as a phase shifter that rotates the quadrature of the spins. This method substantially simplifies the geometry of experimental schemes.