Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain Omega in C and an operator T in B-n(Omega), Cowen and Douglas construct a Hermitian holomorphic vector bundle E-T over Omega corresponding to T. The Hermitian holomorphic vector bundle E-T is obtained as a pull-back of the tautological bundle S(n, H) defined over by Gr(n, H) a nondegenerate holomorphic map z bar right arrow ker(T - z), z is an element of Omega. To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank n Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle J(k)(L-f), we have shown that the curvature of the line bundle L-f completely determines the class of J(k)(L-f). In case of rank Hermitian holomorphic vector bundle E-f, We have calculated the curvature of jet bundle J(k)(E-f) and also obtained a trace formula for jet bundle J(k)(E-f).

Trace Driven Dynamic Deadlock Detection and Reproduction

Dynamic analysis techniques have been proposed to detect potential deadlocks. Analyzing and comprehending each potential deadlock to determine whether the deadlock is feasible in a real execution requires significant programmer effort. Moreover, empirical evidence shows that existing analyses are quite imprecise. This imprecision of the analyses further void the manual effort invested in reasoning about non-existent defects. In this paper, we address the problems of imprecision of existing analyses and the subsequent manual effort necessary to reason about deadlocks. We propose a novel approach for deadlock detection by designing a dynamic analysis that intelligently leverages execution traces. To reduce the manual effort, we replay the program by making the execution follow a schedule derived based on the observed trace. For a real deadlock, its feasibility is automatically verified if the replay causes the execution to deadlock. We have implemented our approach as part of WOLF and have analyzed many large (upto 160KLoC) Java programs. Our experimental results show that we are able to identify 74% of the reported defects as true (or false) positives automatically leaving very few defects for manual analysis. The overhead of our approach is negligible making it a compelling tool for practical adoption.

Trace Formulae in Operator Theory

In this article, we survey several kinds of trace formulas that one encounters in the theory of single and multi-variable operators. We give some sketches of the proofs, often based on the principle of finite-dimensional approximations to the objects at hand in the formulas.