Backward errors and condition numbers are defined and evaluated for eigenvalues and eigenvectors of generalized eigenvalue problems. Both normwise and componentwise measures are used. Unstructured problems are considered first, and then the basic definitions are extended so that linear structure in the coefficient matrices (for example, Hermitian, Toeplitz, Hamiltonian, or band structure) is pr...

ω(Ju, Jv) = ω(u, v) and ω(·, J ·) ≫ 0. The combination thus defines an associated Riemannian metric β(·, ·) = ω(·, J ·). Any symplectic manifold possesses such a structure. We will assume further that ω is ‘integral’ in the cohomological sense. This means we can find a complex hermitian line bundle L → X with hermitian connection ∇ whose curvature is −iω. Recently, beginning with Donaldson’s se...

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prov...

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds to the hyper-Kähler manifolds of the investigated type and the case when the torsion of D is D-parallel. Curvature pro...

We show that the holomorphic Morse inequalities proved by Tian and the author [TZ1, 2] are in effect equalities by refining the analytic arguments in [TZ1, 2]. §0. Introduction and the statement of main results Let (M,ω, J) be a compact Kähler manifold with the Kähler form ω and the complex structure J . Let g denote the corresponding Kähler metric. We make the assumption that there exists a He...

We construct some natural metric connections on metric contact manifolds compatible with the contact structure and characterized by the Dirac operators they determine. In the case of CR manifolds these are invariants of a fixed pseudo-hermitian structure, and one of them coincides with the Tanaka–Webster connection.  2005 Elsevier B.V. All rights reserved. MSC: 53B05; 53C15; 53D10; 53D15

For a complex manifold X which has a holomorphic form ̟ of odd degree k, we endow E = ⊕ p≥a Λ (p,0)(X) with a Higgs bundle structure θ given by θ(Z)(φ) := {i(Z)̟} ∧ φ. The properties such as curvature and stability of these and other Higgs bundles are examined. We prove (Theorem 2, section 2, for k > 1) E and additional classes of Higgs subbundles of E do not admit Higgs-Hermitian-Yang-Mills metr...

A theorem is proved on the structure of the group of isometries of an Hermitian map b : V × V → W , where V and W are vector spaces over a finite field of odd order. Also a Las Vegas polynomial-time algorithm is presented which, given an Hermitian map, finds generators for, and determines the structure of its isometry group. The algorithm can be adapted to construct the intersection of the memb...

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the sa...

The adoption of hybrid GPU-CPU nodes in traditional supercomputing platforms opens acceleration opportunities for electronic structure calculations in materials science and chemistry applications, where medium sized Hermitian generalized eigenvalue problems must be solved many times. The small size of the problems limits the scalability on a distributed memory system, hence they can benefit fro...