In this paper the smoothness properties of Green’s operator-function an exponentially dichotomous bilinear matrix system and the smoothness properties the invariant torus of nonhomogeneus matrix system of equations have been considered. It hHave been proved that if some conditions, concerning the properties of coefficient of the system hold this operator-function has smoothness index which depe...

Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant’s minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general f...

We outline a specialization theory of quadratic and (symmetric) bilinear forms with respect to a place λ : K → L∪∞. Here K,L denote fields of any characteristic. We have to make a distinction between bilinear forms and quadratic forms and study them both over fields and valuation rings. For bilinear forms this turns out to be essentially as easy as in the case char L 6= 2, albeit no general can...

We introduce an abstract concept for decomposing spaces with respect to a substructuring of a bounded domain. In this setting we define weakly conforming finite element approximations of quadratic minimization problems. Within a saddle point approach the reduction to symmetric positive Schur complement systems on the skeleton is analyzed. Applications include weakly conforming variants of least...

This note introduces bilinear estimates intended as a step towards an L∞-endpoint Kato-Ponce inequality. In particular, a bilinear version of the classical Gagliardo-Nirenberg interpolation inequalities for a product of functions is proved.

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617-637]. Therein, the key assumption is that the conforming first-order finite element space V c h annulates the linear and bounded residual l written V c h ⊆ ker l. That excludes ...

A relativistic quantum mechanics is formulated in which all of the interactions are in the four-momentum operator and Lorentz transformations are kinematic. Interactions are introduced through vertices, which are bilinear in fermion and antifermion creation and annihilation operators, and linear in boson creation and annihilation operators. The fermionantifermion operators generate a unitary Li...

For isospectral bilinear control systems evolving on the so-called complex flag manifolds (i.e., on the orbits of the Hermitian matrices under unitary conjugation action) it is shown that controllability is almost always verified. Easy and generic sufficient conditions are provided. The result applies to the problem of density operator controllability of finite dimensional quantum mechanical sy...

The aim of this note is to give an algebraic characterization of those universally complete vector lattice in which all band preserving bilinear operators are symmetric. We start with recalling some definitions and auxiliary facts about bilinear operators on vector lattices. For the theory of vector lattices and positive operators we refer to the books [1] and [7]. Let E and F be vector lattice...