Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

Special Symmetric Two-tensors, Equivalent Dynamical Systems, Cofactor and Bi-cofactor Systems

A general analysis of special classes of symmetric two-tensor on Riemannian manifolds is provided. These tensors arise in connection with special topics in differential geometry and analytical mechanics: geodesic equivalence and separation of variables. It is shown that they play an important role in the theory of correspondent (or equivalent) dynamical systems of Levi-Civita. By applying some new developments of this theory, it is shown that the recent notions of cofactor and cofactor-pair systems arise in a natural way, as non-Lagrangian systems having a Lagrangian equivalent. This circumstance extends the Hamiltonian methods, including the separation of variables of the Hamilton–Jacobi equation, to a special class of nonconservative systems. In this extension the case of indefinite metrics, may occur. Hence, it is shown that also pseudo-Riemannian geometry plays an important role also in classical mechanics.Research sponsored by the Dept. of Mathematics, University of Turin, and by INDAM-GNFM.     

Topology of complete intersections

Let X _n (d) and X _n (d') be two n-dimensional complete intersections with the same total degree d. In this paper we prove that, if n is even and d has no prime factors less than , then X _n (d) and X _n (d') are homotopy equivalent if and only if they have the same Euler characteristics and signatures. This confirms a conjecture of Libgober and Wood [16]. Furthermore, we prove that, if d has no prime factors less than , then X _n (d) and X _n (d') are homeomorphic if and only if their Pontryagin classes and Euler characteristics agree. Received: September 6, 1996     

5-桥图F(2,a,a,b,c)的色等价刻画

设P(G，λ)是图G关于变量λ的色多项式，P(G，λ)=P(H，λ)，称G和H色等价，由连接两个顶点的S条内部不交的路组成的图叫S-桥图。本讨论了5-桥图F(2,a,a,b,c)(c≥b≥a 1，a≥2)的色性，完整刻画了这类图的色等价图。     

C*-丛中的Morita等价定理

给定C^*-丛(E，P，G)．首先证明了E上循环*表示由Ea决定；然后证明了：C^*(E)与E0强Morita等价．     

The equivalence of ensembles for lattice systems: Some examples and a counterexample

We describe the problem of the equivalence of ensembles at the level of states for classical lattice systems. We discuss circumstances where the vanishing of the specific information gain of a sequence of microcanonical measures with respect to a sequence of grand canonical measures implies the equivalence of ensembles. We give a simple derivation of a criterion for the vanishing of the specific information gain in terms of thermodynamic functions. The proof uses ideas from the theory of large deviations but is self-contained. We show how the criterion works in a simple model of a paramagnet and in the Ising model of a ferromagnet in any dimension but fails in the case of the Curie-Weiss mean-field model.     

Reduction of matrices by means of equivalent transformations and similarity transformations

The structure of polynomial matrices in connection with their reducibility by semiscalar-equivalent transformations and similarity transformations to simpler forms is considered. In particular, the canonical form of polynomial matrices without multiple characteristic roots with respect to the above transformations is indicated. This allows one to establish a canonical form with respect to similarity for a certain type of finite collections of numerical matrices. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 769–782, November, 1998.     

Bimodule Deformations, Picard Groups and Contravariant Connections

We study deformations of invertible bimodules and the behavior of Picard groups under deformation quantization. While K ₀-groups are known to be stable under formal deformations of algebras, Picard groups may change drastically. We identify the semiclassical limit of bimodule deformations as contravariant connections and study the associated deformation quantization problem. Our main focus is on formal deformation quantization of Poisson manifolds by star products.     

Galois theory for comatrix corings: Descent theory, Morita theory, Frobenius and separability properties

El Kaoutit and Gómez-Torrecillas introduced comatrix corings, generalizing Sweedler's canonical coring, and proved a new version of the Faithfully Flat Descent Theorem. They also introduced Galois corings as corings isomorphic to a comatrix coring. In this paper, we further investigate this theory. We prove a new version of the Joyal-Tierney Descent Theorem, and generalize the Galois Coring Structure Theorem. We associate a Morita context to a coring with a fixed comodule, and relate it to Galois-type properties of the coring. An affineness criterion is proved in the situation where the coring is coseparable. Further properties of the Morita context are studied in the situation where the coring is (co)Frobenius.

     

On orbit equivalence of borel automorphisms

LetE andF be two Borel sets of the countable productZ of the two point space {0,1}. Assume thatE andF are invariant sets for the odometer transformationR and thatE andF are of measure zero with respect to the unique finiteR-invariant measure onZ. We show thatE andF areR-orbit equivalent in a strict sense.