Connections on central bimodules

We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.Comment: 27 pages, AMS-LaTe

A common generalization of the Fr\"olicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields

There is a canonical mapping from the space of sections of the bundle $\wedge T^\ast M\otimes ST\ M$ to $\Omega(T^\ast M ; T(T^\ast M))$. It is shown that this is a homomorphism on $\Omega(M;TM) for the Fr\"olicher-Nijenhuis brackets, and also on$\Gamma(ST\ M)$for the Schouten bracket of symmetric multi vector fields. But the whole image is not a subalgebra for the Fr\"olicher-Nijenhuis bracket on$\Omega(T^\ast M;T(T^\ast M))$.Comment: 14 pages, AMSTEX, LPTHE-ORSAY 94/05 and ESI 70 (1994

The Fr\"olicher-Nijenhuis bracket for derivation based non commutative differential forms

In commutative differential geometry the Fr\"olicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Fr\"olicher-Nijenhuis bracket was developped for universal differential forms of non-commutative algebras, and several applications were given. In this paper this bracket and the Fr\"olicher-Nijenhuis calculus will be developped for several kinds of differential graded algebras based on derivations, which were indroduced by \cit!{6}.Comment: AmSTeX, 28 pages, ESI Preprint 13

Linear Connections in Non-Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $n \times n$ matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx

Supersymmetry and Noncommutative Geometry

The purpose of this article is to apply the concept of the spectral triple, the starting point for the analysis of noncommutative spaces in the sense of A.~Connes, to the case where the algebra \cA contains both bosonic and fermionic degrees of freedom. The operator \cD of the spectral triple under consideration is the square root of the Dirac operator und thus the forms of the generalized differential algebra constructed out of the spectral triple are in a representation of the Lorentz group with integer spin if the form degree is even and they are in a representation with half-integer spin if the form degree is odd. However, we find that the 2-forms, obtained by squaring the connection, contains exactly the components of the vector multiplet representation of the supersymmetry algebra. This allows to construct an action for supersymmetric Yang-Mills theory in the framework of noncommutative geometry.Comment: 26pp., LaTe

Clinical Features and Outcomes Differ between Skeletal and Extraskeletal Osteosarcoma.

Background. Extraskeletal osteosarcoma (ESOS) is a rare subtype of osteosarcoma. We investigated patient characteristics, overall survival, and prognostic factors in ESOS. Methods. We identified cases of high-grade osteosarcoma with known tissue of origin in the Surveillance, Epidemiology, and End Results database from 1973 to 2009. Demographics were compared using univariate tests. Overall survival was compared with log-rank tests and multivariate analysis using Cox proportional hazards methods. Results. 256/4,173 (6%) patients with high-grade osteosarcoma had ESOS. Patients with ESOS were older, were more likely to have an axial tumor and regional lymph node involvement, and were female. Multivariate analysis showed ESOS to be favorable after controlling for stage, age, tumor site, gender, and year of diagnosis [hazard ratio 0.75 (95% CI 0.62 to 0.90); p = 0.002]. There was an interaction between age and tissue of origin such that older patients with ESOS had superior outcomes compared to older patients with skeletal osteosarcoma. Adverse prognostic factors in ESOS included metastatic disease, larger tumor size, older age, and axial tumor site. Conclusion. Patients with ESOS have distinct clinical features but similar prognostic factors compared to skeletal osteosarcoma. Older patients with ESOS have superior outcomes compared to older patients with skeletal osteosarcoma