Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.     

The Kinematics of the Planar Motion of Ideal Fiber-Reinforced Fluids: An Integrable Reduction and Bäcklund Transformation

We establish that the kinematic constraints on the steady planar motion of an ideal fiber-reinforced fluid can be consolidated in a single third-order nonlinear equation. Remarkably, this equation admits a solitonic reduction related to the classical sine-Gordon equation. The kinematic conditions in this case admit a novel duality property and a Bäcklund transformation.     

Surfaces of Demoulin: differential geometry, Bäcklund transformation and integrability

The surfaces of Demoulin constitute an important subclass of surfaces in projective differential geometry which arise in many seemingly unrelated geometric constructions. Analytically, they are described by a two-component system which coincides with the D₃⁽¹⁾ Toda lattice. We review some of the most important geometric properties of the Demoulin surfaces and construct a Bäcklund transformation which may be specialized to the well-known Bäcklund transformation for the Tzitzeica equation governing affine spheres in affine geometry.     

Intrinsic Geometry of the NLS Equation and Its Auto-Bäcklund Transformation

The nonlinear Schrödinger equation is derived in a general intrinsic geometric setting involving a normal congruence originally investigated by Marris and Passman in 1969 in a hydrodynamical context. Geometric properties of members of the normal congruence are adduced and an intrinsic representation of the auto-Bäcklund transformation is obtained for the nonlinear Schrödinger equation.     

The average distance on the Sierpiński gasket

Summary The canonical distance of points on the Sierpiski gasket is considered and its expectation deduced. The solution is surprising, both for the value and for the method derived from an analysis of graphs connected with the Tower of Hanoi problem.     

An open mapping theorem for measures

LetX andY be Hausdorff spaces and denote byM (X) andM (Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map :XY induces the map :M (X)M (Y). We give necessary and sufficient conditions for to be open. In case of being a surjection between Suslin spaces, is open if and only if is.     

On Complex-Lamellar Motion of a Prim Gas

An intrinsic formulation of Prim's canonical equations of gas dynamics is employed to establish a generalization of Crocco's theorem to complex-lamellar motion. This new result is then employed to establish the existence of a Heisenberg spin-type equation in such motion. The optimal coordinate systems in the generalized Lagrangian method which are known to exist only in complex-lamellar motion are then shown by intrinsic geometric methods to correspond to a marching direction along geodesics on generalized Bernoulli surfaces.     

An Integrable Discretization of a 2+1-dimensional Sine-Gordon Equation

We show that the complex discrete BKP equation that has been recently identified as an integrable discretization of the 2+1-dimensional sine-Gordon system introduced by Konopelchenko and Rogers admits a natural reduction to a discrete 2+1-dimensional sine-Gordon equation. We discuss three important properties of this equation. First, it may be interpreted as a superposition principle associated with a constrained Moutard transformation. Second, the complexified discrete sine-Gordon equation constitutes an eigenfunction equation for the discrete sine-Gordon system. Third, we derive a form of the equation in terms of trigonometric functions that has been studied by Konopelchenko and Schief in a discrete geometric context. A discrete Moutard transformation for the discrete sine-Gordon equation and the corresponding Bäcklund equations are also recorded.     

Universal and Integrable Aspects of an Elliptic Vortex Representation in 2+1‐Dimensional Magneto‐Gasdynamics

Integrable substructure of a 2+1‐dimensional magneto‐gasdynamic system is investigated via a general elliptic vortex ansatz. Certain universal and Hamiltonian aspects of the admitted representation are uncovered. A class of confined magneto‐gasdynamic flows with an elliptic cylindrical boundary is isolated with time‐dependent semiaxes determined by an integrable Ermakov–Ray–Reid system.