Oberwolfach Publications (Mathematisches Forschungsinst. Oberwolfach)
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2063 research outputs found
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Non-Extendability of Holomorphic Functions with Bounded or Continuously Extendable Derivatives
No full textResearch in Pairs 2016We consider the spaces and containing all holomorphic functions on an open set , such that all derivatives , , are bounded on , or continuously extendable on , respectively. We endow these spaces with their natural topologies and they become Fr\'echet spaces. We prove that the set of non-extendable functions in each of these spaces is either void, or dense and . We give examples where or not. Furthermore, we examine cases where can be replaced by , or and the corresponding spaces stay unchanged
Molecular Quantum Dynamics
No full textWe provide a brief introduction to some basic ideas
of Molecular Quantum Dynamics. We discuss the
scope, strengths and main applications of this field
of science. Finally, we also mention open problems
of current interest in this exciting subject
Low-dimensional Topology and Number Theory
No full textThe workshop brought together topologists and number theorists with the intent of exploring the many tantalizing connections between these areas
Copositivity and Complete Positivity
No full textA real matrix is called copositive if holds for all . A matrix is called completely positive if it can be factorized as , where is an entrywise nonnegative matrix. The concept of copositivity can be traced back to Theodore Motzkin in 1952, and that of complete positivity to Marshal Hall Jr. in 1958. The two classes are related, and both have received considerable attention in the linear algebra community and in the last two decades also in the mathematical optimization community. These matrix classes have important applications in various fields, in which they arise naturally, including mathematical modeling, optimization, dynamical systems and statistics. More applications constantly arise.
The workshop brought together people working in various disciplines related to copositivity and complete positivity, in order to discuss these concepts from different viewpoints and to join forces to better understand these difficult but fascinating classes of matrices
The Pseudo-Hyperresolution and Applications
No full textOWLF 2016Resolving objects in an abelian category by injective (projective) resolutions is a fundamental problem in mathematics, and this article aims at introducing a particular solution called “Pseudo-hyperresolutions”. This method originates in the category of unstable modules to study the minimal resolution of the reduced singular cohomology of spheres. In particular, for all integers , we can describe a large range of the minimal injective resolution of the sphere based on the Bockstein operation of the Steenrod algebra. Moreover, many classical constructions in algebraic topology, such as the algebraic EHP sequence or the Lambda algebra can be recovered using the Pseudo-hyperresolution method. A particular connection between spheres and the projective spaces is also established. Despite its origin, Pseudo-hyperresolutions generalize to all abelian categories. In particular, many classical construction of injective resolutions of strict polynomial functors can be reunified in view of Pseudo-hyperresolutions. As a consequence, we recover the global dimension of the category of homogeneous strict polynomial functors of finite degree as well as the Mac Lane cohomology of finite fields
Drugs, herbicides, and numerical simulation
No full textDie kolumbianische Regierung versprüht Unkrautbekämpfungsmittel (Herbizide) über Coca-Feldern, um die Drogenproduktion im Land zu reduzieren. Sprühverwehungen entlang der Grenze Kolumbiens zu Ecuador wurden zu einem internationalen Streitfall. Wir haben ein mathematisches Modell für die Ausbreitung der Chemikalien in der Luft entwickelt, das es uns ermöglicht, das Phänomen am Computer zu simulieren
Multiscale and High-Dimensional Problems
No full textHigh-dimensional problems appear naturally in various scientific areas. Two primary examples are PDEs describing complex processes in computational chemistry and physics, and stochastic/ parameter-dependent PDEs arising in uncertainty quantification and optimal control. Other highly visible examples are big data analysis including regression and classification which typically encounters high-dimensional data as input and/or output. High dimensional problems cannot be solved by traditional numerical techniques, because of the so-called curse of dimensionality. Rather, they require the development of novel theoretical and computational approaches to make them tractable and to capture fine resolutions and relevant features. Paradoxically, increasing computational power may even serve to heighten this demand, since the wealth of new computational data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information in a high dimensional setting constitute challenging tasks from both theoretical and numerical perspective.
The last decade has seen the emergence of several new computational methodologies which address the obstacles to solving high dimensional problems. These include adaptive methods based on mesh refinement or sparsity, random forests, model reduction, compressed sensing, sparse grid and hyperbolic wavelet approximations, and various new tensor structures. Their common features are the nonlinearity of the solution method that prioritize variables and separate solution characteristics living on different scales. These methods have already drastically advanced the frontiers of computability for certain problem classes.
This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computational methods and to promote the exchange of ideas emerging in various disciplines about how to treat multiscale and high-dimensional problems
Mini-Workshop: Cluster Expansions: From Combinatorics to Analysis through Probability
No full textThe workshop addressed the interplay between theory and applications of cluster expansions. These expansions, historically geared towards the study of systems in statistical mechanics, thermodynamics, and physical chemistry, have recently found applications in different areas of current mathematical research, such as point processes, random graphs, coloring issues, logics and inverse problems in numerical analysis. The workshop developed both directions of the theory–application interplay. On the one hand, speakers presented advances in the theoretical foundations of the abstract polymer model and improved tree-graph inequalities, and explored their consequences for the theory of liquids and other applied issues. On the other hand, researchers in stochastic modelisation exposed needs and challenges brought by concrete models of liquids and liquid crystal to the theory of cluster expansions. In addition other complementary methods were discussed, such as disagrement percolation – an expansion-free approach to uniqueness and decay
of correlations – and lace expansions – an expansion technique popular for its applications to random walks and percolation problems
Computational Inverse Problems for Partial Differential Equations
No full textThe problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges
Mini-Workshop: Women in Mathematics: Historical and Modern Perspectives
No full textThe aim of the workshop is to build a bridge between research on the situation of women in mathematics at the beginning of coeducative studies and the current circumstances in academia. The issue of women in mathematics has been a recent political and social hot topic in the mathematical community. As thematic foci we place a double comparison: besides shedding light on differences and similarities in several European countries, we complete this investigation by comparing the developments of women studies from the beginnings. This shall lead to new results on tradition and suggest improvements on the present situation