Almost periodically forced circle flows

We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle.Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.     

Eine axiomatik ovoidaler

In the first paragraph of this paper we start with a geometry consisting of points, circles, and an equivalencerelation on the set of points. There is just one circle through any three pairwise non-parallel points and any four non-concyclic points generate a Laguerre-plane or an inversive plane. If dimension d>2, we show that we must have the geometry induced on a cone of a projective space by plane-cuts. In the following part this result is used to give a characterization of the geometry of paraboloids in affine spaces.     

Eta invariants of Dirac operators on circle bundles over riemann surfaces and virtual dimensions of finite energy Seiberg-Witten moduli spaces

Using an adiabatic collapse trick we determine, by two different methods, the eta invariants of many Dirac type operators on circle bundles over Riemann surfaces. These results, coupled with a delicate spectral flow computation, are then used to determine the virtual dimensions of moduli spaces of finite energy Seiberg-Witten monopoles on 4-manifolds bounding such circle bundles.     

Semi-biplanes on the cylinder

We construct semi-biplanes from 2-dimensional projective planes and 2-dimensional circle planes.Herrn Professor Dr. H. Salzmann zum 65. GeburtstagThis research was supported by a Feodor Lynen fellowship.     

Szegö kernel, regular quantizations and spherical CR-structures

We compute the Szegö kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kähler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for which the log-terms in the Fefferman expansion of the Szegö kernel vanish and which are not locally CR-equivalent to the sphere. We also give a proof of the fact that, for homogeneous Hodge manifolds, the existence of a locally spherical CR-structure on the unit circle bundle alone implies that the manifold is biholomorphic to a projective space. Our results generalize those obtained by Engli? (Math Z 264(4):901–912, 2010) for Hermitian symmetric spaces of compact type.     

Generalization of a criterion for semistable vector bundles

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H⁰(X,EF) and H¹(X,EF) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hⁱ(X,EF)=0 for all i. We also give an explicit bound for the rank of F.     

A Geometric Relationship Between Equivalent Spreads

By Andrè theory, it is well known how to algebraically convert a spread in a projective space to an equivalent spread (representing the same translation plane) in a projective space of different dimension and of different order (corresponding to a subfield of the kernel). The goal of this paper is to establish a geometric connection between any two such equivalent spreads by embedding them as subspaces and subgeometries of an ambient projective spaces. The connection can be viewed as a generalization of a construction due to Hirschfeld and Thas.     

Squaring the Triangle

We construct flat Laguerre planes by integrating flat projective planes. The construction is based in an essential way on results from the theory of interpolation. In conjunction with the unifying theory of topological circle planes and generalized quadrangles, the new construction appears to be one of the most natural and powerful constructions of such geometries.     

On the Geometry of the 7-Sphere

A sphere of dimension 4n+3 admits three Sasakian structures and it is natural to ask if a submanifold can be an integral submanifold for more than one of the contact structures. In the 7-sphere it is possible to have curves which are Legendre curves for all three contact structures and there are 2 and 3-dimensional submanifolds which are integral submanifolds of two of the contact structures. One of the results here is that if a 3-dimensional submanifold is an integral submanifold of one of the Sasakian structures and invariant with respect to another, it is an integral submanifold of the remaining structure and is a principal circle bundle over a holmophic Legendre curve in complex projective 3-space.     

Robust Dual Reconstruction Systems and Fusion Frames

We study the duality of reconstruction systems, which are g-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.     

Intersection area of a stationary circle and a circle under a coupled axial rotation and translation

A circle is placed concentrically in a circle of equal or larger size. The circle is then rotated along a vertical axis, creating an ellipse, and translated along the horizontal axis. The intersection area of the circle and circle/ellipse is determined as function of the rotation angle and the relative size of the initial circles. This configuration corresponds to the closing of a ball valve used to control the flow of fluids through pipes.     

Solvariétés projectives de dimension trois

Let V be a hyperbolic 2-torus bundle over the circle, or equivalently, a quotient of the three-dimensional group Sol by a lattice . We study real projective structures on V. We prove that all such structures are left-invariant which means that they are all obtained by the following process: take a representation of Sol in PGL (4, R) with an open orbit in R P ³; it induces a projective structure on Sol and, hence, on the left-quotient \Sol=V. We classify all the real projective structures on V and show that they are in fact all affine. The proof involves a study of codimension one projective subsurfaces, which are tori, and the dynamic of their holonomy group in the whole manifold.     

Key Distribution Patterns Using Tangent Circle Structures

The problem of key management in a communications network is of primary importance. A key distribution pattern is an incidence structure which provides a secure method of distributing keys in a large network reducing storage requirements. It is of interest to find explicit constructions for key distribution patterns. In O'Keefe [5–7], examples are shown using the finite circle geometries (Minkowski, Laguerre and inversive planes); in Quinn [12], examples are constructed from conics in finite projective and affine planes. In this paper, we construct some examples using the finite tangent-circle structures, introduced in Quattrocchi and Rinaldi [10] and we give a comparison of the storage requirements.     

Zone Structure of the Renormalization Group Flow in a Fermionic Hierarchical Model

The Gaussian part of the Hamiltonian of the four-component fermion model on a hierarchical lattice is invariant under the block-spin transformation of the renormalization group with a given degree of normalization (the renormalization group parameter). We describe the renormalization group transformation in the space of coefficients defining the Grassmann-valued density of a free measure as a homogeneous quadratic map. We interpret this space as a two-dimensional projective space and visualize it as a disk. If the renormalization group parameter is greater than the lattice dimension, then the unique attractive fixed point of the renormalization group is given by the density of the Grassmann delta function. This fixed point has two different (left and right) invariant neighborhoods. Based on this, we classify the points of the projective plane according to how they tend to the attracting point (on the left or right) under iterations of the map. We discuss the zone structure of the obtained regions and show that the global flow of the renormalization group is described simply in terms of this zone structure.     

最大利润流问题及算法

最大利润流是以运输利润最大为目标的网络优化问题 .一个利润可行流可分解为若干个路流和圈流 ,相应地该可行流的利润也等于这些路流和圈流的利润之和 .本文证明了一个可行流为最大利润流的充要条件是不存在利润增广路 ,并据此提出了求解算法 .文章最后给出了一个计算实例 .     

Focal loci of algebraic varieties i

The focal locus ∑_x of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x ∈X and a circle with centre O passing through x which osculates X inx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle N_x and the projective ambient space (∈ sends the normal vector O - x to its endpoint O), and in this paper we address two general problems:.

1)Characterize the"degenerate"case where the focal locus is not a hyper surface.

2)Calculate, in the case where ∑x is a hypersurface, its degree (with multiplicity).     

The Fibered Isomorphism Conjecture for Complex Manifolds

In this paper we show that the Fibered Isomorphism Conjecture of Farrell and Jones, corresponding to the stable topological pseudoisotopy functor, is true for the fundamental groups of a class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the negative K-groups of the fundamental groups of these manifolds vanish whenever the fundamental group is torsion free. We also prove the same results for a class of real manifolds including a large class of 3-manifolds which has a finite sheeted cover fibering over the circle.     

Singular hyperbolicity of star flows on surfaces

In this paper, we prove that every star flow on the closed surface has finitely many chain recurrent classes. Furthermore, it is singular hyperbolic if every non-trivial singular chain component is a graph. As a consequence, every star flow on the 2-sphere or the projective plane is singular hyperbolic.     

Global-Affine Morphisms of Projective Lattice Geometries

The fundamental theorem of projective geometry gives an algebraic representation of isomorphisms between projective geometries of dimension at least 3 over vector spaces and has been generalized in different ways. This note briefly presents some further generalizations which will be proved in the author’s thesis. We introduce the notion of global-affine morphisms between projective lattice geometries. Our investigations result in a general partial representation of global-affine morphisms which yields a complete representation of global-affine homomorphisms between large classes of module-induced projective geometries by semilinear mappings between the underlying modules.