Couplings and primitives on topological spaces

For an open covering $\mathcal U$ of a topological space and a map $d\colon\mathcal I\to\mathbb{K}$, where $\mathcal I:=\bigl\{(U,V)\in\mathcal U\times\mathcal U;\ U\cap V\ne\varnothing\bigr\}$, we present a context for the existence of a map $C\colon\mathcal U\to\mathbb{K}$ satisfying $C_V-C_U=d_{UV}$ for all $(U,V)\in\mathcal I$. The result is applied to a Poincar\'e type theorem concerning distributional potentials

On sequences of sectorial forms converging `from above'

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis

Tissue Doppler, Strain and Strain Rate in ischemic heart disease “How I do it”

Echocardiography is the standard method for assessing myocardial function in patients with ischemic heart disease. The acquisition and interpretation of echocardiographic images, however, remains a highly specialized task which often relies entirely on the subjective visual assessment of the reader and requires therefore, particular training and expertise. Myocardial deformation imaging allows quantifying myocardial function far beyond what can be done with sole visual assessment. It can improve the interpretation of regional dysfunction and offers sensitive markers of induced ischemia which can be used for stress tests. In the following, we recapitulate shortly the pathophysiological and technical basics and explain in a practical manner how we use this technique in investigating patients with ischemic heart disease