平面杆系结构静力分析的统一模型

从平面杆单元的经典刚度方程出发，在引入基本假定的基础上由严格的数学方法得到了平面杆系结构的统一计算模型．当该模型中的有关参数取特殊值时可直接得到理想的衍架模型和刚性连接模型，而且可用于具有任意半刚性连接和混合结点的一般杆系结构的静力分析．     

Connections among several chaos feedback control approaches and chaotic vibration control of mechanical systems

This study reveals the essential connections among several popular chaos feedback control approaches, such as delayed feedback control (DFC), stability transformation method (STM), adaptive adjustment method (AAM), parameter adjustment method, relaxed Newton method, and speed feedback control method (SFCM), etc. Meanwhile, the generality and practical applicability of these approaches are evaluated and compared. It is shown that for discrete chaotic maps, STM can be regarded as a kind of predictive feedback control, and AAM is actually a special case of STM which is merely effective for a particular dynamical system. The parameter adjustment method is only a different expression of the relaxed Newton method, and both of them represent just one search direction of STM, i.e., the gradient direction. Moreover, the intrinsic relation between the STM and SFCM for controlling the equilibrium of continuous autonomous systems is investigated, indicating that STM can be viewed as a special form of the SFCM. Finally, both the STM and SFCM are extended to control the chaotic vibrations of non-autonomous mechanical systems effectively.     

Flat connections on configuration spaces and braid groups of surfaces

We construct an explicit bundle with flat connection on the configuration space of n points on a complex curve. This enables one to recover the ‘1-formality’ isomorphism between the Lie algebra of the prounipotent completion of the pure braid group of n   points on a surface and an explicitly presented Lie algebra, and to extend it to a morphism from the full braid group of the surface to the semidirect product of the associated group with the symmetric group _Sn$S_n$.     

Statistical morphisms and related invariance properties

Our aim is to investigate a way to characterize the elements of a statistical manifold (the metric and the family of connections) using invariance properties suggested by Le Cam's theory of experiments. We distinguish the case where the statistical manifold is flat. Then, there naturally exists an entropy and it is proven that experiment invariance is equivalent to entropy invariance. If the statistical manifold is not flat, we introduce a notion of local invariance of selected order associated to the asymptotic (on n observations, n tending to infinity) expansion of the power of the Neymann Pearson test in a contiguous neighborough of some point. This invariance provides a substantial number of morphisms. This was not always true for the entropy invariance: particularly, the case of Gaussian experiments is investigated where it can be proven that entropy invariance does not characterize a metric or a family of connections.     

On some bidimensional denumerable chains of infinite order

We study homogeneous chains of infinite order (ξ_t)_{t∈$\text{Z}$} with the set of states taken to be $\text{X=(}\text{N}\text{{0,1})x{-1,1}}$. Our approach is to interpret the half-infinite sequence ..., ξ_?n,..., ξ_?1, ξ₀, where $\text{ξ}{}_{\mathrm{t}}\text{=(i}{}_{\mathrm{t}}\text{, ε}{}_{\mathrm{t}}\text{) ∈ X, t ∈}\text{Z}$ as the continued fraction to the nearer integer expansion (read inversely) of a y ? [?$\text{1}\text{2}$,$\text{1}\text{2}$]. Thus, we are led to study certain Y-valued Markov chains, where Y = [?$\text{1}\text{2}$, $\text{1}\text{2}$] and then by making use of their properties we establish the existence of denumerable chains of infinite order under conditions different from those given in Theorem 2.3.8 of Iosifescu-Theodorescu (1969). A (weak) variant of mixing is proved as well.     

Rigid and Finite Type Geometric Structures

Rigid geometric structures on manifolds, introduced by Gromov, are characterized by the fact that their infinitesimal automorphisms are determined by their jets of a fixed order. Important examples of such structures are those given by an H-reduction of the first order frame bundle of a manifold, where the Lie algebra of H is of finite type; in fact, for structures given by reductions to closed subgroups of first order frame bundles, finite type implies rigidity. The goal of this paper is to generalize this to geometric structures defined by reductions of frame bundles of arbitrary order, and to give an algebraic characterization of the property of being rigid in terms of a suitable notion of finite type.     

Markov approximations of chains of infinite order

We consider chains whose transition probabilities depend on the whole past, with summable continuity rates. We show that Ornstein's -distance between one such chain and its canonical Markov approximations of different orders is at worst proportional to the continuity rate of the chain. The result generalizes previous bounds obtained by X. Bressaud and ourselves, while relying on a similar coupling argument. Received: 9 April 2002     

Geometric phases and the informational completeness of quantum frames

The basic results on geometric phases are rederived by using infinite dimensional coordinate charts in line bundles, in Hopf bundles, and in projective Hilbert spaces. The determination of a quantum state can be then geometrically described as the measurement of Fubini-Study distances from that state to the elements of informationally complete quantum frames. The basic geometric features of such quantum frames are formulated, and their relationships to corresponding classical frames are analyzed.     

Multivalued maps as a tool in modeling and rigorous numerics

Applications of the fixed point theory of multivalued maps can be classified into several areas: (1) Game theory and mathematical economics; (2) Discontinuous differential equations, differential inclusions, and optimal control; (3) Computing homology of maps; (4) Computer assisted proofs in dynamics; (5) Digital imaging. We give an overview of the most classical and well developed areas of applications (1) and (2), where a multivalued map is used as a generalization of a single-valued continuous map, and we survey more recent applications (3), (4), and (5), where multivalued maps play the role of a numerical tool. Dedicated to Felix Browder on his 80th birthday