Dichotomous Hamiltonians with Unbounded Entries and Solutions of Riccati Equations

An operator Riccati equation from systems theory is considered in the case that all entries of the associated Hamiltonian are unbounded. Using a certain dichotomy property of the Hamiltonian and its symmetry with respect to two different indefinite inner products, we prove the existence of nonnegative and nonpositive solutions of the Riccati equation. Moreover, conditions for the boundedness and uniqueness of these solutions are established.Comment: 31 pages, 3 figures; proof of uniqueness of solutions added; to appear in Journal of Evolution Equation

Revisiting the anatomy of the right ventricle in the light of knowledge of its development

Controversies continue regarding several aspects of the anatomy of the morphologically right ventricle. There is disagreement as to whether the ventricle should be assessed in bipartite or tripartite fashion, and the number of leaflets to be found in the tricuspid valve. In particular, there is no agreement as to whether a muscular outlet septum is present in the normally constructed heart, nor how many septal components are to be found during normal development. Resolving these issues is of potential significance to those investigating and treating children with congenitally malformed hearts. With all these issues in mind, we have revisited our own experience in investigating the development and morphology of the normal right ventricle. To assess development, we have examined a large number of datasets, prepared by both standard and episcopic microscopy, from human and murine embryos. In terms of gross anatomy, we have compared dissections of normal autopsied hearts with virtual dissections of datasets prepared using computed tomography. Our developmental and postnatal studies, taken together, confirm that the ventricle is best assessed in tripartite fashion, with the three parts representing its inlet, apical trabecular, and outlet components. The ventricular septum, however, has only muscular and membranous components. The muscular part incorporates a small component derived from the muscularised fused proximal outflow cushions, but this part cannot be distinguished from the much larger part that is incorporated within the free-standing muscular infundibular sleeve. We confirm that the tricuspid valve itself has three components, which are located inferiorly, septally, and antero-superiorly.<br/

Revisiting the anatomy of the right ventricle in the light of knowledge of its development

Controversies continue regarding several aspects of the anatomy of the morphologically right ventricle. There is disagreement as to whether the ventricle should be assessed in bipartite or tripartite fashion, and the number of leaflets to be found in the tricuspid valve. In particular, there is no agreement as to whether a muscular outlet septum is present in the normally constructed heart, nor how many septal components are to be found during normal development. Resolving these issues is of potential significance to those investigating and treating children with congenitally malformed hearts. With all these issues in mind, we have revisited our own experience in investigating the development and morphology of the normal right ventricle. To assess development, we have examined a large number of datasets, prepared by both standard and episcopic microscopy, from human and murine embryos. In terms of gross anatomy, we have compared dissections of normal autopsied hearts with virtual dissections of datasets prepared using computed tomography. Our developmental and postnatal studies, taken together, confirm that the ventricle is best assessed in tripartite fashion, with the three parts representing its inlet, apical trabecular, and outlet components. The ventricular septum, however, has only muscular and membranous components. The muscular part incorporates a small component derived from the muscularised fused proximal outflow cushions, but this part cannot be distinguished from the much larger part that is incorporated within the free-standing muscular infundibular sleeve. We confirm that the tricuspid valve itself has three components, which are located inferiorly, septally, and antero-superiorly.<br/

Simplicity of extremal eigenvalues of the Klein-Gordon equation

We consider the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials. If the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given

Dissipative Properties of Systems Composed of High-Loss and Lossless Components

We study here dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. A principal result of our studies is that all the eigenmodes of such a system split into two distinct classes characterized as high-loss and low-loss. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping.Comment: Revision; Improved exposition and typos corrected; 45 pages, 4 figure

Assessing the Criteria for Definition of Perimembranous Ventricular Septal Defects in Light of the Search for Consensus

Background: Discussions continue as to whether ventricular septal defects are best categorized according to their right ventricular geography or their borders. This is especially true when considering the perimembranous defect. Our aim, therefore, was to establish the phenotypic feature of the perimembranous defect, and to establish the ease of distinguishing its geographical variants. Methods and results: We assessed unrepaired isolated perimembranous ventricular defects from six historic archives, subcategorizing them using the ICD-11 coding system. We identified 365 defects, of which 94 (26%) were deemed to open centrally, 168 (46%) to open to the outlet, and 84 (23%) to the inlet of the right ventricle, with 19 (5%) being confluent. In all hearts, the unifying phenotypic feature was fibrous continuity between the leaflets of the mitral and tricuspid valves. This was often directly between the valves, but in all instances incorporated continuity through the atrioventricular portion of the membranous septum. In contrast, we observed fibrous continuity between the leaflets of the tricuspid and aortic valves in only 298 (82%) of the specimens. When found, discontinuity most commonly was seen in the outlet and central defects. There were no discrepancies between evaluators in distinguishing the borders, but there was occasional disagreement in determining the right ventricular geography of the defect. Conclusions: The unifying feature of perimembranous defects, rather than being aortic-to-tricuspid valvar fibrous continuity, is fibrous continuity between the leaflets of the atrioventricular valves. While right ventricular geography is important in classification, it is the borders which are more objectively defined