Conformally Invariant Elliptic Liouville Equation and Its Symmetry-Preserving Discretization

The symmetry algebra of the real elliptic Liouville equation is an infinite-dimensional loop algebra with the simple Lie algebra o(3, 1) as its maximal finite-dimensional subalgebra. The entire algebra generates the conformal group of the Euclidean plane E₂. This infinite-dimensional algebra distinguishes the elliptic Liouville equation from the hyperbolic one with its symmetry algebra that is the direct sum of two Virasoro algebras. Following a previously developed discretization procedure, we present a difference scheme that is invariant under the group O(3, 1) and has the elliptic Liouville equation in polar coordinates as its continuous limit. The lattice is a solution of an equation invariant under O(3, 1) and is itself invariant under a subgroup of O(3, 1), namely, the O(2) rotations of the Euclidean plane.     

Harmonic maps from Kähler manifolds

Some Liouville type theorems for harmonic maps from Kähler manifolds are obtained. The main result is to prove that a harmonic map from a bounded symmetric domain (exceptR _IV(2)) to any Riemannian manifold with finite energy has to be constant.     

Real Liouville Extensions

We characterize linear differential equations defined over a real differential field with a real closed field of constants C, which are solvable by real Liouville functions, as those having a differential Galois group whose identity component is solvable and C-split.     

Harmonic functions on locally finite networks

Are there nonconstant bounded harmonic functions on an infinite locally finite network under natural transition conditions as continuity at the ramification nodes and classical Kirchhoff conditions at all vertices? We present sufficient criteria for such a network to be a Liouville space, while we show that a large class of infinite trees admit infinitely many linearly independent bounded harmonic functions. Finally, we show that the standard unit cube grid graphs and some of Kepler’s plane tiling graphs are Liouville spaces.     

The double sign of a real division algebra of finite dimension greater than one

For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_nFor any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A?{0} are shown to form an invariant of A, called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category $\mathbb {D}_n$ of all n‐dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of $\mathscr {D}_n$.     

The Liouville Foliation of Nonconvex Topological Billiards

Together with the classical plane billiards, topological billiards can be considered, where the motion occurs on a locally flat surface obtained by isometrically gluing together several plane domains along their boundaries, which are arcs of confocal quadrics. A point moves inside each of the domains along straight line segments; when it reaches the boundary of a domain, it passes to another domain. Previously, the author gave a Liouville classification of all topological billiards obtained by gluing along convex boundaries. In the present paper, all topological integrable billiards obtained by gluing along convex or nonconvex boundaries from elementary billiards bounded by arcs of confocal quadrics are classified. For some of such nonconvex topological billiards, the Fomenko–Zieschang invariants (marked molecules W*) for Liouville equivalence are calculated.     

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.     

Invariants of transverse foliations

We construct two invariants for a pair of transverse one-dimensional foliations on the plane. If the set of separatrices is Hausdorff in the space of leaves, the invariant is a distinguished graph. In case there are a finite number of separatrices the invariant is an indexed link.     

Formal ring of a cubic solid angle

A rational cubic form with real plane factors determines a solid angle for whose lattice points a formal ring can be constructed and can be made invariant under units. As in the first half of this work (in Volume 8 of this Journal), there is a finite basis for the invariant ring but peculiar effects occur in higher dimension. There might be, for instance, no minimal basis subdivision for the support polyhedron, and no cyclic basis, (simulating algebraic resolution difficulties in C⁸). Furthermore, the support polyhedron is “flat” in some affine sense. Cases are computed for small discriminant cubics.     

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.     

Towards relative invariants of real symplectic four-manifolds

Let (X, ω, c_X) be a real symplectic four-manifold with real part . Let be a smooth curve such that We construct invariants under deformation of the quadruple (X, ω, c_X, L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics. Received: March 2005; Revision: September 2005; Accepted: September 2005     

Dynamics of a two parameter family of plane birational maps: Maximal entropy

We mix combinatorial with complex methods to study the dynamics of a real two parameter family of plane birational maps. Specifically, we consider the action of the maps on the Picard group of an appropriate compactification of the complex plane, on the homology groups of a forward invariant real subset of this compactification, and on a Markov partition of the real plane determined by the critical set. For the range of parameters considered, the three actions are equivalent. This allows us to construct a measure of maximal entropy on the real nonwandering set, and it allows us to show that all wandering points are attracted to infinity in a well-defined fashion.     

On Solvability of Nonlocal Problem for Loaded Parabolic-Hyperbolic Equation

We study unique solvability of a nonlocal problem for equations of mixed type in a finite domain. This equation contains the partial fractional Riemann–Liouville derivative. The boundary condition of the problem contains a linear combination of operators of fractional differentiation in the sense of Riemann–Liouville of values of function derivative on the degeneration line and generalized operators of fractional integro-differentiation in the sense of M. Saigo. The uniqueness theorem of the problem is proved by a modified Tricomi method. The existence of solutions is equivalently reduced to the solvability of Fredholm integral equation of the second kind.     

Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.     

Stable Rank One and Real Rank Zero for Crossed Products by Finite Group Actions with the Tracial Rokhlin Property

The authors prove that the crossed product of an infinite dimensional simple separable unital C＊-algebra with stable rank one by an action of a finite group with the tracial Rokhlin property has again stable rank one. It is also proved that the crossed product of an infinite dimensional simple separable unital C＊-algebra with real rank zero by an action of a finite group with the tracial Rokhlin property has again real rank zero.     

New integer pairs for Hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair(t, r); t is the order of the Hjelmslev plane andr is the order of the underlying projective plane. The aim of this paper is to give some new constructions of Hjelmslev planes with an invariant pair (t, 2). First we construct a PH-plane with the invariant pair (20, 2). Using this, 16 more invariant pairs (t, 2) witht 1000 are obtained. In all, we thus obtain 17 new PH-planes with invariant pairs (t, 2),t 1000.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

Conservative arbitrary order finite difference schemes for shallow-water flows

The classical nonlinear shallow-water model (SWM) of an ideal fluid is considered. For the model, a new method for the construction of mass and total energy conserving finite difference schemes is suggested. In fact, it produces an infinite family of finite difference schemes, which are either linear or nonlinear depending on the choice of certain parameters. The developed schemes can be applied in a variety of domains on the plane and on the sphere. The method essentially involves splitting of the model operator by geometric coordinates and by physical processes, which provides substantial benefits in the computational cost of solution. Besides, in case of the whole sphere it allows applying the same algorithms as in a doubly periodic domain on the plane and constructing finite difference schemes of arbitrary approximation order in space. Results of numerical experiments illustrate the skillfulness of the schemes in describing the shallow-water dynamics.