Fraunhofer Chalmers Research Centre for Industrial Mathematics
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Substitution of Hydrophobes and its Effect on Nonionic Surfactant Properties Within Cleaning Applications
No full text
Klassifikation av polygoner med trigonometriska egenfunktioner till Laplaceoperatorn under Dirichletrandvillkor
No full textWe consider the eigenstructure of the Laplace operator on triangles with the
angles (60°, 60°,60°); (30°, 60°,90°) och (45°, 45°,90°). Using the earlier work
by M. Práger (1998) and M. A. Pinsky (1980) we find eigenfunctions of the
Laplace operator with Dirichlet boundary conditions. We show completeness
of eigenfunctions in L2 for each triangle.
Moreover, we present a result by Brian J. McCartin (2008) that classifies
which polygons have a complete set of trigonometric eigenfunctions. These
polygons are the triangles mentioned above, the rectangle and the square.
We connect McCartins result to symmetries of lattices, crystals and Weyl
groups. In 1980 Pierre H. Bérard studied the connection between different
types of eigenfunctions and symmetries and proved that all alcoves of Weyl
groups have trigonometric eigenfunctions. We point out the fact that in
R2 the converse is also true. That is, all polygons with a complete set of
trigonometric eigenfunctions are alcoves
Understanding the Customer: Examining the Perceived Value of a New Product in a B2B Context
No full textPathfinding med reinforcement learning i delvis observerbara miljöer
No full textReinforcement learning algorithms have the ability to solve problems without explicit knowledge
of their underlying model. Instead, they infer a strategy directly from observations and
rewards acquired by interacting with their environment. This makes them suitable candidates
for solving pathfinding problems in a partially observable setting, where the aim is to find a
path in an environment with restricted vision.
This report aims to investigate how Markov decision processes and reinforcement learning
can be used to model and solve partially observable pathfinding problems. Existing literature
has been reviewed to give a theoretical background of the subject, before progressing to practical
implementations. We have applied state-of-the-art algorithms taken from two subclasses
of reinforcement learning methods: value based algorithms and policy based algorithms.
We find that partially observable Markov decision processes can be used to model pathfinding
problems, but not all reinforcement learning algorithms are suitable for solving them. In
theory, value based algorithms show potential but when implemented they did not yield positive
results. Conversely, the policy based algorithm Proximal Policy Optimization is able to
solve the problem convincingly. This algorithm also performs well in environments previously
not trained in, thus displaying some ability to generalize its policy
