Permutation operator: An explicit representation

1981 ◽  
Vol 49 (7) ◽  
pp. 669-671 ◽  
Author(s):  
Dietrich Grau
Keyword(s):  
Explicit Representation ◽  
Permutation Operator

scholarly journals On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

2021 ◽  
Author(s):  
Pier Domenico Lamberti ◽  
Luigi Provenzano
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Lipschitz Domain ◽  
Spectral Properties ◽  
Explicit Representation ◽  
Lipschitz Domains ◽  
Dirichlet Problems ◽  
Trace Space ◽  
In Series ◽  
Definition Of

AbstractWe consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

scholarly journals Secondary Science Teachers’ Views on Environmental Citizenship in The Netherlands

2021 ◽  
Vol 13 (14) ◽  
pp. 7963
Author(s):  
Michiel van Harskamp ◽  
Marie-Christine P. J. Knippels ◽  
Wouter R. van Joolingen
Keyword(s):  
Science Teachers ◽  
Citizenship Education ◽  
Renewable Energy Sources ◽  
Explicit Representation ◽  
Secondary Science ◽  
Added Value ◽  
Face To Face ◽  
Secondary Science Teachers ◽  
Environmental Citizenship ◽  
Teacher Professional

Environmental Citizenship (EC) is a promising aim for science education. EC enables people not only to responsibly make decisions on sustainability issues—such as use of renewable energy sources—but also to take action individually and collectively. However, studies show that education for EC is challenging. Because our understanding of EC practice remains limited, an in-depth, qualitative view would help us better understand how to support science teachers during EC education. This study aims to describe current EC education practices. What do secondary science teachers think sustainability and citizenship entail? What are their experiences (both positive and negative) with education for EC? A total of 41 Dutch science teachers were interviewed in an individual, face-to-face setting. Analysis of the coded transcripts shows that most teachers see the added value of EC but struggle to fully implement it in their teaching. They think the curriculum is unsuitable to reach EC, and they see activities such as guiding discussions and opinion forming as challenging. Furthermore, science teachers’ interpretation of citizenship education remains narrow, thus making it unlikely that their lessons are successful in fostering EC. Improving EC education therefore may be supported by explicit representation in the curriculum and teacher professional development directed at its implementation.

scholarly journals Pattern discovery and disentanglement on relational datasets

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Andrew K. C. Wong ◽  
Pei-Yuan Zhou ◽  
Zahid A. Butt
Keyword(s):  
Machine Learning ◽  
Knowledge Base ◽  
Case Studies ◽  
Prediction Accuracy ◽  
Pattern Discovery ◽  
Explicit Representation ◽  
Human Cognition ◽  
Source Level ◽  
Low Volume

AbstractMachine Learning has made impressive advances in many applications akin to human cognition for discernment. However, success has been limited in the areas of relational datasets, particularly for data with low volume, imbalanced groups, and mislabeled cases, with outputs that typically lack transparency and interpretability. The difficulties arise from the subtle overlapping and entanglement of functional and statistical relations at the source level. Hence, we have developed Pattern Discovery and Disentanglement System (PDD), which is able to discover explicit patterns from the data with various sizes, imbalanced groups, and screen out anomalies. We present herein four case studies on biomedical datasets to substantiate the efficacy of PDD. It improves prediction accuracy and facilitates transparent interpretation of discovered knowledge in an explicit representation framework PDD Knowledge Base that links the sources, the patterns, and individual patients. Hence, PDD promises broad and ground-breaking applications in genomic and biomedical machine learning.

THE APPLICATION OF PETRI NETS TO WORKFLOW MANAGEMENT

1998 ◽  
Vol 08 (01) ◽  
pp. 21-66 ◽  
Author(s):  
W. M. P. VAN DER AALST
Keyword(s):  
Petri Nets ◽  
Petri Net ◽  
Business Processes ◽  
State Of The Art ◽  
Workflow Management ◽  
Explicit Representation ◽  
The Other ◽  
Application Domain ◽  
Analysis Techniques ◽  
The One

Workflow management promises a new solution to an age-old problem: controlling, monitoring, optimizing and supporting business processes. What is new about workflow management is the explicit representation of the business process logic which allows for computerized support. This paper discusses the use of Petri nets in the context of workflow management. Petri nets are an established tool for modeling and analyzing processes. On the one hand, Petri nets can be used as a design language for the specification of complex workflows. On the other hand, Petri net theory provides for powerful analysis techniques which can be used to verify the correctness of workflow procedures. This paper introduces workflow management as an application domain for Petri nets, presents state-of-the-art results with respect to the verification of workflows, and highlights some Petri-net-based workflow tools.

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

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