A general abstract duality result is proposed for equations which are governed by the sum of two operators (possibly multivalued). It allows to unify a large number of variational duality principles, including the Clarke-Ekeland least dual action principle and the Singer-Toland duality. Moreover, it offers a new duality approach to some central questions in the theory of variational inequalitie...

we study the support sets of sub-topical functions‎ ‎and investigate their maximal elements in order to establish a necessary and sufficient condition‎ ‎for the global minimum of the difference of two sub-topical functions‎.

We consider a class of generalized Nash equilibrium problems (GNEPs) where both the objective functions and the constraints are allowed to depend on the decision variables of the other players. It is well-known that this problem can be reformulated as a constrained optimization problem via the (regularized) Nikaido-Isoda-function, but this reformulation is usually nonsmooth. Here we observe tha...

For any finite simple graphG = (V,E), the discrete Dirac operator D = d+d∗ and the Laplace-Beltrami operator L = dd∗+d∗d = (d+d∗)2 = D2 on the exterior algebra bundle Ω = ⊕Ωk are finite v × v matrices, where dim(Ω) = v = ∑ k vk, with vk = dim(Ωk) denoting the cardinality of the set Gk of complete subgraphs Kk of G. We prove the McKean-Singer formula χ(G) = str(e−tL) which holds for any complex ...

We present a signature formula for compact 4k-manifolds with corners of codimension two which generalizes the formula of Atiyah, Patodi and Singer for manifolds with boundary. The formula expresses the signature as a sum of three terms, the usual Hirzebruch term given as the integral of an L-class, a second term consisting of the sum of the eta invariants of the induced signature operators on t...

In this expository article, we survey index theory of Dirac operators using the Gauss-Bonnet formula as the catalyst to discuss index formulas on manifolds with and without boundary. Considered in detail are the Atiyah-Singer and Atiyah-Patodi-Singer index theorems, their heat kernel proofs, and their generalizations to manifolds with corners of codimension two via the method of ‘attaching cyli...

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis o...

We define generalized Atiyah-Patodi-Singer boundary conditions of product type for Dirac operators associated to C∗-vector bundles on the product of a compact manifold with boundary and a closed manifold. We prove a product formula for the K-theoretic index classes, which we use to generalize the product formula for the topological signature to higher signatures.