| 1. |
- Bengmark, Samuel, 1965, et al.
(author)
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Pragmatic Research on Educational Practice
- 2023
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In: Proceedings Chalmers Conference on Teaching and Learning 2023. - 9789188041548 ; , s. 56-59
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Conference paper (peer-reviewed)abstract
- During a round table discussion at a collegial conference on teaching and learning, a focus group with 12 teachers from a technical university discuss challenges with studying their own teaching practice. Furthermore, a concept called PREP – Pragmatic Research on Educational Practice, with the goal of engaging engineering educators in studying, documenting, and sharing their initiatives to improve teaching practices, is introduced and discussed. Among the main obstacles to researching their own teaching practice, the participants pointed to a lack of time, know-how, and motivation. They expressed that there is potential in the collegial part of PREP and the time efficiency of using what can be studied during a course. The role of PREP studies, in relation to regular educational research, is also discussed, and that PREP may be perceived as a devaluation of educational science was problematized. Some participants felt that it was very likely that they would participate in a PREP group next academic year if given the opportunity.
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| 2. |
- Bengmark, Samuel, 1965, et al.
(author)
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PREP – en kollegial process för att forska på sin egen undervisning.
- 2023
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In: Bidrag från den 9:e utvecklingskonferensen för Sveriges ingenjörsutbildningar. - 9789174856200
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Conference paper (other academic/artistic)abstract
- Många av oss ingenjörsutbildare prövar olika pedagogiska idéer i vår undervisning och försöker förstå om de fungerar bra. Dock är det oftast svårt att lyckas driva analysen hela vägen, inte sällan på grund av tidsbrist, osäkerhet om hur man skall gå till väga eller bristande motivation. Vi tror därför att det behövs en ny form av kollegialt samarbete med målet att stödja undersökningar av egen undervisningspraktik som hjälper lärare att överkomma dessa utmaningar. Vi föreslår därför en kollegial process, som vi kallar PREP, Pragmatic Research on Educational Practice, som innebär att en grupp kollegor, gärna från olika universitet, stöttar varandra att genomföra studier av sina undervisningsaktiviteter. Vårt syfte med denna workshop är att berätta lite om våra erfarenheter med PREP så långt, samt att du skall få smaka på PREP och se om denna process kan vara till hjälp för dig. Samtidigt önskar vi få ta del av dina åsikter om vad i denna process som är mest värdefullt och vad som kan förbättras.
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| 3. |
- Bengmark, Samuel, 1965, et al.
(author)
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PREP- Pragmatic Research on Educational Practice
- 2023
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In: SEFI 2023 - 51st Annual Conference of the European Society for Engineering Education : Engineering Education for Sustainability, Proceedings - Engineering Education for Sustainability, Proceedings. - : SEFI. - 9782873520267 ; , s. 163-172
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Conference paper (peer-reviewed)abstract
- We investigate a concept called PREP - Pragmatic Research on Educational Practice, with the goal of engaging engineering educators in studying, documenting and sharing their initiatives to improve teaching practices. This concept is compared to other methodologies where the researcher and educational practitioner sometimes coincide. The study is based on a pilot, with six participants following the PREP program for three months, which we study autoethnographically. We also carried out a focus group discussion (n=12) to investigate to what extent university teachers regard the ideas from the PREP program as helpful for studying educational activities and sharing what they do and find.
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| 4. |
- Castro, Alejandro, et al.
(author)
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Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
- 2016
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In: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 55:5
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Journal article (peer-reviewed)abstract
- We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is an $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates (Theorem \ref{th0}), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two crucial estimates in the case of real, symmetric operators (Theorem \ref{th2}). Our results are crucial when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.
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| 5. |
- Hed, Lisa, et al.
(author)
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Studenternas förkunskaper
- 2020
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In: Nämnaren. - : Göteborgs universitet. - 0348-2723. ; 46:4, s. 25-31
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Journal article (peer-reviewed)
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| 6. |
- Nyström, Kaj, 1969-, et al.
(author)
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Boundary estimates for non-negative solutions to non-linear parabolic equations
- 2015
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In: Calculus of Variations and Partial Differential Equations. - : Springer Science and Business Media LLC. - 0944-2669 .- 1432-0835. ; 54:1, s. 847-879
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Journal article (peer-reviewed)abstract
- This paper is mainly devoted to the boundary behavior of non-negative solutions to the equation\[\H u =\partial_tu-\nabla\cdot \operatorname{A}(x,t,\nabla u) = 0\]in domains of the form $\Omega_T=\Omega\times (0,T)$ where $\Omega\subset\mathbb R^n$ is a bounded non-tangentially accessible (NTA) domain and $T>0$. The assumptions we impose on$A$ imply that $H$ is a non-linear parabolic operator with linear growth. Our main results include a backward Harnackinequality, and the H\"older continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize, to the setting of non-linear equations with linear growth, previous results concerningthe boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for non-negative solutions to equations of the type $\partial_tu-\nabla\cdot (\operatorname{A}(x,t)\nabla u)=0$, where $A$ is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to the operator and related to Green's function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.
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| 7. |
- Nyström, Kaj, 1969-, et al.
(author)
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Boundary estimates for solutions to linear degenerate parabolic equations
- 2015
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In: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 259:8, s. 3577-3614
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Journal article (peer-reviewed)abstract
- Let $\Omega\subset\mathbb R^n$ be a bounded NTA-domain and let $\Omega_T=\Omega\times (0,T)$ for some $T>0$. We study the boundary behaviour of non-negativesolutions to the equation\[Hu =\partial_tu-\partial_{x_i}(a_{ij}(x,t)\partial_{x_j}u) = 0, \ (x,t)\in \Omega_T.\]We assume that $A(x,t)=\{a_{ij}(x,t)\}$ is measurable, real, symmetric and that\begin{equation*}\beta^{-1}\lambda(x)|\xi|^2\leq \sum_{i,j=1}^na_{ij}(x,t)\xi_i\xi_j\leq\beta\lambda(x)|\xi|^2\mbox{ for all }(x,t)\in\mathbb R^{n+1},\ \xi\in\mathbb R^{n},\end{equation*}for some constant $\beta\geq 1$ and for some non-negative and real-valued function $\lambda=\lambda(x)$belonging to the Muckenhoupt class $A_{1+2/n}(\mathbb R^n)$.Our main results includethe doubling property of the associated parabolic measure andthe H\"older continuity up to the boundary of quotients of non-negative solutionswhich vanish continuously on a portion of the boundary. Our resultsgeneralize previous results of Fabes, Kenig, Jerison, Serapioni, see \cite{FKS}, \cite{FJK}, \cite{FJK1}, to a parabolic setting.
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| 8. |
- Nyström, Kaj, 1969-, et al.
(author)
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Extension properties and boundary estimates for a fractional heat operator
- 2016
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In: Nonlinear Analysis. - : Elsevier BV. - 0362-546X .- 1873-5215. ; 140, s. 29-37
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Journal article (peer-reviewed)abstract
- The square root of the heat operator $\sqrt{\partial_t-\Delta}$, can be realized as the Dirichlet to Neumann map of the heat extension of data on $\mathbb R^{n+1}$ to $\mathbb R^{n+2}_+$. In this note we obtain similar characterizations for general fractional powers of the heat operator, $(\partial_t-\Delta)^s$, $s\in (0,1)$. Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.
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| 9. |
- Sande, Olow, 1983-
(author)
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Boundary Estimates for Solutions to Parabolic Equations
- 2016
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Doctoral thesis (other academic/artistic)abstract
- This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation.Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary.Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary.Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain.Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.
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