Resolve "karsten research topic"

topics/E-learning_and_e-assessment_in_mathematics.md

+# E-learning and e-assessment in mathematics
+
+### Working Groups: aa
+
+### Collaborators (MAT): cseifert, dgallaun, kkruse
+
+### External Collaborators: [Leo Dostal](https://www.tuhh.de/mum/mitarbeiter/wissenschaftliche-mitarbeiter/dr-leo-dostal.html), [Mirjam S. Glessmer](https://mirjamglessmer.com/), [Natalia Konchakova](https://www.hereon.de/institutes/surface_science/interface_modeling/team/098961/index.php.de)
+
+## Description
+
+At many universities, undergraduate courses of fundamental subjects such as mathematics are taught to students enrolled in many different course programs. Since the fundamental subject is then taught without using examples from the students’ main subjects, this often results in low student motivation in the fundamental subject and lacking knowledge and skills of this subject even when necessary for the understanding of the students’ main subject. At Hamburg University of Technology, this applies to first year mathematics, which is taught in the same course to 1300 students enrolled in 13 different engineering study programs.
+
+Our approach to solve this dilemma: 
+
+* Provide oportunities for individual practice,
+* feedback tailored to the needs of students of the different study courses such that students apply mathematical concepts in the context of their main subject. 
+
+This helps them to see the relevance of the content which might otherwise be perceived as bothersome and irrelevant, thereby increasing student motivation. Providing additional practice opportunities also increases perceived student self-efficacy, in turn enhancing motivation [3]. We use electronic exercises and an e-assessment system to cope with the sheer number of students, and problems from mechanical engineering [3,4,5,6] as well as electrical engineering [2] to demonstrate the linking of mathematics and engineering science.
+
+The e-assessment system also helps to manage another problem caused by the very large number of participants of the exam in first year mathematics, namely the correspondingly high examination effort. Conducting such an exam electronically might on the one hand significantly reduce the amount of work (automated correction and grading) and on the other hand provide a new dimension of possible exam questions. But in order to develop good questions for such an electronic exam, one has to address different aspects [1], e.g.:
+
+* Randomisation of parameters such that all realisations are comparable for fairness,
+* provide open-ended questions to demonstrate understanding rather than guesses,
+* handling of errors, i.e. how to check for mistakes made by students.
+
+Of course an electronic exam is not restricted to large courses in first year mathematics. As in many courses on applied mathematics, in order to give a glimpse on realistic problems, one is faced with large computations which are typically done by computers. However, when it comes to exams on such topics students are often asked to apply the learned methods, which are suited for large systems, to very small problems by pen-and-paper. This gap can be overcome by means of an electronic exam [7].
+
+## References
+
+[1] D. Gallaun, K. Kruse, C. Seifert. Adaptive Übungs- und Prüfungsaufgaben in Mathematik mit hochwertiger Bewertung. In D. Schott, editor, *Proceedings 15. Workshop Mathematik in ingenieurwissenschaftlichen Studiengängen*, Heft 02/2019, 18--24, Gottlob-Frege-Zentrum, Wismar, 2019. Avaliable at [HS Wismar](https://kompetenz.hs-wismar.de/index.php/Proceedings_15._Workshop_Mathematik_in_ingenieurwissenschaftlichen_Studieng%C3%A4ngen_Rostock-Warnem%C3%BCnde_2019).
+
+[2] D. Gallaun, K. Kruse, C. Seifert. Anwendungsbezogene elektronische Übungsaufgaben in Ingenieurmathematik, *55. Jahrestagung der Gesellschaft für Didaktik der Mathematik (GDM)*, 2021. Preprint.
+
+[3] M.S. Glessmer, C. Seifert. E-Assessments to increase the perceived importance of Mathematics in the introductory phase of Engineering Education via bridging tasks. In J.C. Quadrado, J. Bernardino, J. Rocha, editors, *Proceedings of Sefi Annual Conferences 2017*, 1549--1556, 2017. Available at [SEFI](https://www.sefi.be/wp-content/uploads/).
+
+[4] M.S. Glessmer, C. Seifert, L. Dostal, N. Konchakova, K. Kruse. Individualisierung von Großveranstaltungen. Oder: Wie man Ingenieurstudierenden die Mathematik schmackhaft macht. In W.D. Paravicini, J. Schnieder, editors, *Hanse-Kolloquium zur Hochschuldidaktik der Mathematik 2015*, 64--75, wtm-Verlag, Münster, 2016. Available at [WWU Münster](https://miami.uni-muenster.de/Record/69a62744-78ea-42b5-856d-4cd43c042602).
+
+[5] M.S. Glessmer, C. Seifert, L. Dostal, N. Konchakova, K. Kruse. Providing opportunities for individual practice and assessment in a large undergraduate mathematics course. In P. Kapranos, editor, *International Symposium on Engineering Education -- Interdisciplinary
+Engineering -- Breaking Boundaries, ISEE 2016 Conference Proceedings*, 13--20, TJ International, Padstow, UK, 2016. doi: [10.15131/shef.data.3507380.v1](https://doi.org/10.15131/shef.data.3507380.v1).
+
+[6] K. Kruse, L. Dostal, M.S. Glessmer, N. Konchakova, C. Seifert. Conception of online e-assessment exercises for math courses with elements from mechanical engineering. In G. Kammasch, H. Klaffke, S. Knutzen, editors, *Tagungsband der 11. Ingenieurpädagogischen Regionaltagung*, Hamburg, 232--236, 2017. doi: [10.15480/882.1394](https://doi.org/10.15480/882.1394).
+
+[7] K. Kruse, C. Seifert. Implementing Computer-assisted Exams in a Course on Numerical Analysis for Engineering Students. In ISEC, Portugal, editors, *Proceedings of the 19th SEFI MWG*, Coimbra, Portugal, 33--38, 2018. Available at [SEFI MWG](http://sefi.htw-aalen.de/).
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topics/Laplace_transforms_for_generalised_functions_and_the_ACP.md

+# Laplace transforms for generalised functions and the abstract Cauchy problem
+
+### Working Groups: aa
+
+### Collaborators (MAT): kkruse
+
+## Description
+
+Let $E$ be a (sequentially) complete complex locally convex Hausdorff space ($\mathbb{C}$-lcHs). The initial value problem
+
+$$
+\begin{align}
+x'(t)&=Ax(t),\quad t>0,\\
+x(0)&=x_{0}\in E,
+\end{align}
+$$
+
+is called an *abstract Cauchy problem* where 
+
+$$
+A\colon D(A)\subset E\to E
+$$
+
+is a (sequentially) closed linear operator with domain $D(A)$. We study the abstract Cauchy problem in the space of $E$-valued [hyperfunctions](https://en.wikipedia.org/wiki/Hyperfunction) with support in $[0,\infty)$. Hyperfunctions were introduced by Sato [10,11] and extended to Fourier hyperfunctions by Kawai [2]. Hyperfunctions form a quite large class of generalised functions, containing locally integrable functions, distributions and ultradistributions. 
+
+We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a $\mathbb{C}$-lcHs. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction [6,7], this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions [4], which improves the existing models of Komatsu [3], Bäumer [1], Lumer and Neubrander [9] and Langenbruch [8]. We apply our theory of asymptotic Laplace transforms to prove existence and uniqueness results for solutions of the abstract Cauchy problem in a wide class of locally convex Hausdorff spaces, containing Fréchet spaces and several spaces of distributions [5].
+
+## References
+
+[1] B. Bäumer. A vector-valued operational calculus and abstract Cauchy problems. PhD thesis, Louisiana State University, Baton Rouge, LA, 1997. Available at [LSU Digital Commons](https://digitalcommons.lsu.edu/gradschool_disstheses/6464/).
+
+[2] T. Kawai. On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients. *J. Fac. Sci. Univ. Tokyo, Sect. IA*, 17:467--517, 1970. doi: [10.15083/00039821](https://doi.org/10.15083/00039821).
+
+[3] H. Komatsu. Laplace transforms of hyperfunctions -- A new foundation of the Heaviside calculus. *J. Fac. Sci. Univ. Tokyo, Sect. IA*, 34:805--820, 1987. doi: [10.15083/00039471](https://doi.org/10.15083/00039471).
+
+[4] K. Kruse. Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, 2021. [arXiv:2104.02682](https://arxiv.org/abs/2104.02682).
+
+[5] K. Kruse. The abstract Cauchy problem in locally convex spaces. *In preparation*.
+
+[6] K. Kruse. Vector-valued Fourier hyperfunctions. PhD thesis, Universität Oldenburg, 2014. URN: [urn:nbn:de:gbv:715-oops-19095](http://nbn-resolving.org/urn:nbn:de:gbv:715-oops-19095)
+
+[7] K. Kruse. Vector-valued Fourier hyperfunctions and boundary values, 2019. [arXiv:1912.03659](https://arxiv.org/abs/1912.03659).
+
+[8] M. Langenbruch. Asymptotic Fourier and Laplace transformations for hyperfunctions. *Stud. Math.*, 205(1):41--69, 2011. doi: [10.4064/sm205-1-4](https://doi.org/10.4064/sm205-1-4).
+
+[9] G. Lumer and F. Neubrander. The asymptotic Laplace transform: New results and relation to Komatsu’s Laplace transform of hyperfunctions. In F. Mehmeti, J. von Below, and S. Nicaise, editors, *Partial differential equations on multistructures*, volume 219 of *Notes Pure Appl. Math.*, 147--162, Dekker, New York, 2001. doi: [10.1201/9780203902196](https://doi.org/10.1201/9780203902196)
+
+[10] M. Sato. Theory of hyperfunctions, I. *J. Fac. Sci. Univ. Tokyo, Sect. IA*, 8:139--193, 1959. doi: [10.15083/00039918](https://doi.org/10.15083/00039918).
+
+[11] M. Sato. Theory of hyperfunctions, II. *J. Fac. Sci. Univ. Tokyo, Sect. IA*, 8:387--437, 1960. doi: [10.15083/00039916](https://doi.org/10.15083/00039916).
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topics/Linearisation_of_vector-valued_functions.md

+# Linearisation of vector-valued functions
+
+### Working Groups: aa
+
+### Collaborators (MAT): kkruse
+
+## Description
+
+It is a classical idea to represent vector-valued functions by continuous linear operators [3]. Let $\mathcal{F}(\Omega)$ be a locally convex Hausdorff space (lcHs) of functions from a set $\Omega$ to the field $\mathbb{K}$ of real or complex numbers and $E$ an lcHs over $\mathbb{K}$. Then Schwartz' $\varepsilon$-product of $\mathcal{F}(\Omega)$ and $E$ is defined as the space of continuous linear operators 
+
+$$
+\mathcal{F}(\Omega)\varepsilon E :=L_{e}(\mathcal{F}(\Omega)_{\kappa}',E).
+$$
+
+Supposing that the point-evaluations $\delta_{x}$ belong to the dual space $\mathcal{F}(\Omega)'$ for all $x\in\Omega$ and that there is an lcHs $\mathcal{F}(\Omega,E)$ consisting of $E$-valued functions on $\Omega$ which is the counterpart of $\mathcal{F}(\Omega)$, linearisation of $\mathcal{F}(\Omega,E)$ means that the map 
+
+$$
+S\colon \mathcal{F}(\Omega)\varepsilon E \to \mathcal{F}(\Omega,E),\; u\longmapsto[x\mapsto u(\delta_{x})],
+$$
+
+is a well-defined topological isomorphism. 
+
+In [11] we derive sufficient conditions on $E$ and on the properties and structures of the functions and function spaces $\mathcal{F}(\Omega)$ and $\mathcal{F}(\Omega,E)$ such that the map $S$ is a topological isomorphism. Once the isomorphism $S$ is established, the famous [approximation property](https://en.wikipedia.org/wiki/Approximation_property) of a space $\mathcal{F}(\Omega)$ is equivalent to the property that every function in $\mathcal{F}(\Omega,E)$ can be approximated by functions with values in finite dimensional subspaces of $E$ for any lcHs $E$, which we investigate in [4] for weighted spaces of $\mathcal{C}^{k}$-smooth functions. In [7] we study the stronger property that $\mathcal{F}(\Omega)$ is [nuclear](https://en.wikipedia.org/wiki/Nuclear_space) in the case of weighted $\mathcal{C}^{\infty}$-smooth functions. 
+
+Nuclearity can be used to transfer the surjectivity of a continuous linear map $T\colon \mathcal{F}(\Omega)\to\mathcal{F}(\Omega)$ to the $\varepsilon$-product $T\varepsilon \operatorname{id}_{E}\colon \mathcal{F}(\Omega)\varepsilon E\to\mathcal{F}(\Omega)\varepsilon E$ for Fréchet spaces $\mathcal{F}(\Omega)$ and $E$ by Grothendieck's classical [tensor product](https://en.wikipedia.org/wiki/Topological_tensor_product) theory [1]. In combination with the topological isomorphism $S$ this implies that the surjectivity of a continuous linear partial differential operator can be transfered from the scalar-valued to the vector-valued case, which we study for the Cauchy-Riemann operator $T=\overline{\partial}$ on weighted spaces of $\mathcal{C}^{\infty}$-smooth functions in [2,5,6,8] even for $E$ beyond the class of Fréchet spaces.
+
+Another application of the topological isomorphism $S$ lies in lifting series representations from scalar-valued to $E$-valued functions [9], for instance power series representations of holomorphic functions [10], and the extension of $E$-valued functions via weak extensions [3,12], i.e. to answer the question:
+
+Let $\Lambda$ be a subset of $\Omega$ and $G$ a linear subspace of $E'$. Let $f\colon \Lambda\to E$ be such that for every $e'\in G$, the function $e'\circ f\colon\Lambda\to \mathbb{K}$ has an extension in $\mathcal{F}(\Omega)$. When is there an extension $F\in\mathcal{F}(\Omega,E)$ of $f$, i.e. $F_{\mid \Lambda} = f$ ?
+
+## References
+
+[1] A. Grothendieck. Produits tensoriels topologiques et espaces nucléaires. *Mem. Amer. Math.
+Soc. 16*. AMS, Providence, RI, 1966. doi: [10.1090/memo/0016](https://doi.org/10.1090/memo/0016).
+
+[2] K. Kruse. Surjectivity of the $\overline{\partial}$-operator between spaces of weighted smooth vector-valued functions, 2018. [arXiv:1810.05069](https://arxiv.org/abs/1810.05069).
+
+[3] K. Kruse. Extension of vector-valued functions and sequence space representation, 2019. [arXiv:1808.05182](https://arxiv.org/abs/1808.05182).
+
+[4] K. Kruse. The approximation property for weighted spaces of differentiable functions. In M. Kosek, editor, *Function Spaces XII*, volume 119 of *Banach Center Publ.*, 233--258, Inst. Math., Polish Acad. Sci., Warszawa, 2019. doi: [10.4064/bc119-14](https://doi.org/10.4064/bc119-14).
+
+[5] K. Kruse. The Cauchy-Riemann operator on smooth Fréchet-valued functions with exponential growth on rotated strips. *PAMM*, 19(1):1--2, 2019. doi: [10.1002/pamm.201900141](https://doi.org/10.1002/pamm.201900141).
+
+[6] K. Kruse. The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes, 2019. [arXiv:1901.02093](https://arxiv.org/abs/1901.02093).
+
+[7] K. Kruse. On the nuclearity of weighted spaces of smooth functions. *Ann. Polon. Math.*, 124(2):173--196, 2020. doi: [10.4064/ap190728-17-11](https://doi.org/10.4064/ap190728-17-11).
+
+[8] K. Kruse. Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions. *RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.*, 114(141):1--24, 2020. doi: [10.1007/s13398-020-00863-x](https://doi.org/10.1007/s13398-020-00863-x).
+
+[9] K. Kruse. Series representations in spaces of vector-valued functions via Schauder decompositions. *Math. Nachr.*, 294(2):354--376, 2021. doi: [10.1002/mana.201900172](https://doi.org/10.1002/mana.201900172).
+
+[10] K. Kruse. Vector-valued holomorphic functions in several variables. *Funct. Approx. Comment. Math.*, 63(2):247--275, 2020. doi: [10.7169/facm/1861](https://doi.org/10.7169/facm/1861).
+
+[11] K. Kruse. Weighted spaces of vector-valued functions and the $\varepsilon$-product, *Banach J. Math. Anal.*, 14(4):1509--1531, 2020. doi: [10.1007/s43037-020-00072-z](https://doi.org/10.1007/s43037-020-00072-z).
+
+[12] K. Kruse. Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order, 2021. [arXiv:1910.01952](https://arxiv.org/abs/1910.01952).
+
+[13] L. Schwartz. Espaces de fonctions différentiables à valeurs vectorielles. *J. Analyse Math.*, 4:88--148, 1955. doi: [10.1007/BF02787718](https://doi.org/10.1007/BF02787718).
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