Stability of some essential B-spectra of pencil operators and application

In this paper, we give some results on the essential B-spectra of a linear operator pencil, which are used to determine the essential B-spectra of an integro-differential operator with abstract boundary conditions in the Banach space Lp([−a, a] × [−1, 1]), p ≥ 1 and a > 0.

Boundary|Time|Surface: assessing a meeting of art and geology through an ephemeral sculptural work

Boundary|Time|Surface was an ephemeral, site-specific sculpture created to draw attention to the construction of social, political, scientific, and aesthetic boundaries that divide the Earth; one such practice is the scientific subdivision of geologic time. The sculpture comprised a 150 m fence along the international stratotype separating Ordovician from Cambrian strata in Gros Morne National Park, Canada. The fence was constructed by hand within 1 d, on a falling tide, from materials found on site, with minimal environmental impact. During the following tidal cycles, it was dismantled by wave and tide action. This cycle of construction and destruction was documented with time-lapse photography and video and brought to the public through exhibitions, public talks, and a book. Exhibitions derived from the documentation of ephemeral works function as translations of the original experience. In this case, they provided opportunities for public interaction with media that served both as aesthetic objects and as sources of information about the site's geological and sociopolitical history. We assess the role of the installation, and its documentation, in drawing public attention to boundaries, and examine responses including attendance records and written visitor comments as indications of the viewers' engagement with the concepts presented. Of several thousand visitors to exhibitions, 418 written comments reflected the viewers' engagement with both the location and the underlying concepts. Both the original installation and the subsequent work allowed audiences to explore human understanding and acquisition of knowledge about the Earth and how world-views inform the process of scientific inquiry.

Self-Adjoint Dirac Operators on Domains in $$\mathbb {R}^3$$

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in $$L^2(\Omega ; \mathbb {C}^4)$$ L 2 ( Ω ; C 4 ) , where $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 is either a bounded or an unbounded domain with a compact $$C^2$$ C 2 -smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as of Robin type. Our approach is based on abstract boundary triple techniques from extension theory of symmetric operators and a thorough study of certain classes of (boundary) integral operators, that appear in a Krein-type resolvent formula. The analysis of the perturbation term in this formula leads to a description of the spectrum and a Birman–Schwinger principle, a qualitative understanding of the scattering properties in the case that $$\Omega $$ Ω is an exterior domain, and corresponding trace formulas.

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

Some analytic properties of the Weyl function of a closed linear relation

Let $L$ and $L_{0}$, where $L$ is an expansion of $L_{0}$, be closed linear relations (multivalued operators) in a Hilbert space $H$. In terms of abstract boundary operators (i.e. in the form which in the case of differential operators leads immediately to boundary conditions) some analytic properties of the Weyl function $M(\lambda)$ corresponding to a certain boundary pair of the couple $(L, L_{0}),$ are studied. In particular, applying Hilbert resolvent identity for relations, the criterion of invertibility in the algebra of bounded linear operators in $H$ for transformation $M(\lambda) - M(\lambda_{0})$ in certain small punctured neighbourhood of $\lambda_{0} $ is established. It is proved that in this case $\lambda _{0}$ is a first-order pole for the operator-function $\left(M(\lambda )- M(\lambda_{0} )\right)^{-1} $. The corresponding residue and Laurent series expansion are found. Under some additional assumptions, the behaviour of so called $\gamma$-field $Z_{\lambda}$ (being an operator-function closely connected to $M(\lambda)$) as $\lambda \to - \infty $ is investigated.

Collaborative Learning at the Boundaries: Hallmarks within a Rehabilitation Context

The aim of the article is to shed light on how collaborative learning at the boundaries between professions plays out within a rehabilitation context. The study has an ethnographic design in the form of observation and interviews in two rehabilitation contexts. Findings showed that collaborative learning was stimulated when the professional groups made a concerted effort to acquire an overall perspective on the patient’s situation and requested and disseminated context-dependent expressions and knowledge about how the patient functioned. The “training” of patients on the ward served as an abstract boundary object amongst the staff functioning as a unifying resource in collaboration. The study exemplifies the learning potential when boundaries between professions become more open in an overlapping collaboration. This enables awareness of one’s own boundaries and of the fact that one had a wealth of shared knowledge. In addition, it entailed the learning of techniques and procedures from other professions, which augmented and developed one’s own professional repertoire.