Mid Sweden University

Generative AI is rapidly reshaping higher education, and in this article we investigate how it can enhance learning rather than erode. In this paper we report findings from five discipline-specific pilots (computer engineering, law, mathematics, psychology, and teacher education) at Mid Sweden University guided by the university’s policy for ethical AI use. Each pilot explored AI as a student-centred learning tool and education tool, but from different perspectives and conditions based on subject. Across the pilots, we have found that these new tools can scaffold reflection, self-regulation, and engagement when integrated into course tasks. As well as need for clarity towards the students are important, to show when and how they can use these tools. We conclude that AI, when framed by robust pedagogy and clarity, can act as a catalyst for deeper learning and improved throughput. Future work will examine long-term retention and pathways to scaling these approaches university-wide. 

Using iterated uniform local completion, we introduce a notion of continuous (Formula presented.) functions on locally closed subsets of reduced complex spaces, generalising both holomorphic functions and (Formula presented.) functions on (Formula presented.) submanifolds. Under additional assumptions of set-theoretical weak pseudo-concavity, we prove optimal maximum modulus principles for these functions, extending classical results for holomorphic functions and ordinary (Formula presented.) functions. Restricting to real submanifolds (possibly with (Formula presented.) singularities) of complex manifolds, we generalise results on holomorphic extension to full neighbourhoods known before only for (Formula presented.) submanifolds. The article is concluded by a study of (Formula presented.) singularities and explicit constructions of submanifolds on which the extension results are valid.

The present article is concerned with a group of problems raised by J. Noguchi and M. Jarnicki/P. Pflug, namely whether the envelopes of holomorphy of truncated tube domains are always schlicht, that is, subdomains of Cn, and how to characterise schlichtness if this is not the case. By way of a counter-example homeomorphic to the 4-ball, we answer the first question in the negative. Moreover, it is possible that the envelope has arbitrarily many sheets. The article is concluded by sufficient conditions for schlichtness in complex dimension two.

In this paper, which has a partially introductory character, the point of departure is very general solution formulas to the operator-valued Kadomtsev-Petviashvili equations (KPI and KPII). Their generality relies on the presence of parameters, which are allowed to be linear mappings between Banach spaces. While this freedom gives access to very complicated solutions, like countable nonlinear superpositions of solitons, the applications in the present article are restricted to matrix parameters. The necessary techniques to ‘project’ to scalar and matrix-valued solutions are explained in detail. The applications part is focused on the KPII. For scalar solutions, the testing ground is the celebrated classification of web structures of interacting line-solitons by Biondini, Chakravarty, Kodama, et al. We establish an explicit link between the different approaches and add some details about interactions of few line-solitons and solutions with singularities. Starting from more complicated algebraic data, we construct solutions beyond web structures, for which we prove asymptotic movement along logarithmic curves in time slices. The article is concluded by a section on solutions to the d × d matrix KPII. We realize N-solitons in the sense of Gilson, Nimmo, and Sooman (and originally Goncharenko) and discuss the question about nonscalar Miles structures. 

Let Ω be a subdomain of a Stein surface with smooth strictly pseudoconvex boundary M=∂Ω. We are mainly interested in the case that Ω is not relatively compact. Building on work by G. Lupacciolu, we describe the envelope of holomorphy of an arbitrary open subset M∗ of M in terms of a modified holomorphic hull of the complement A=M\M∗, designed in order to also reflect the geometry of Ω at infinity. As a consequence, we clarify the relation between the envelope of M∗ and some notions of core sets of Ω, which are an active topic in recent research. 

Let Ω ⊂ Cn, n > 2, be a domain with smooth connected boundary. If Ω is relatively compact, the Hartogs–Bochner theorem ensures that every CR distribution on ∂Ω has a holomorphic extension to Ω. For unbounded domains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of Cn \ Ω is Cn. It seems that it is the first result in the literature which gives a geometric characterization of unbounded domains in Cn for which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authors and Z. Słodkowski, one observes that the extension problem changes in character if one restricts to CR functions of higher regularity. 

We study the thickening problem on a 2-dimensional Stein variety X with isolated irreducible singularities, i.e. the problem whether the assumption that a compact set K is contained in the interior of another compact set L implies that the same inclusion holds for their holomorphic hulls. The problem is still open except for a positive answer in the special case of quotient singularities. The main result of the present article is the partial result that the holomorphic hull L is has a directional density property at every singular point contained in the hull of K. The proof is based on removability results on pseudoconcave closed sets, which may be of some independent interest.

A noncommutative KdV-type equation is introduced extending the Bäcklund chart in Carillo et al. [Symmetry Integrability Geom.: Methods Appl. 12, 087 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in Olver and Sokolov [Commun. Math. Phys. 193, 245 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies, and an explicit solution class are derived. 

Let Ω ⊂ Cn, n ≥ 2, be a domain with smooth connected boundary. If Ω is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on ∂Ω has a holomorphic extension to Ω. For unbounded domains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of Cn Ω is Cn.It seems that it is a first result in the literature which gives a geometric characterization of unbounded domains in Cn for which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authors and Z. S lodkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface and the boundary of the domain.

The goal of the present note is to construct a class of examples for connected compact sets K subset of C-n whose polynomial hull (K) over cap cannot be covered by analytic discs with boundaries contained in an arbitrarily small neighborhood of K. This gives an answer to a recent question raised by B. Drinovec Drnovsek and F. Forstneric.