crossing borders - Helmholtz Zentrum München · »Applied Harmonic Analysis and Inverse...

crossing bordersInstitute of Biomathematics and Biometry

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p. 4 The Institute Mathematics at the Helmholtz Zentrum München

p. 10 The Research Sketches of Selected Projects

p. 46 The Outcome Publications – Projects – Teaching since 2005

The Helmholtz Zentrum München is the German Research Center for Environmental Health. As the leading center in Europe with a research focus on environmental health, we investigate chronic and complex diseases which develop from the interaction of environmental factors and individual genetic disposition. Our aim is to develop innovative approaches for prevention, diagnosis and therapy from an understanding of disease mechanisms. At center stage are chronic, degenerative diseases such as pulmonary diseases, allergies, cancers and cardiovascular diseases, which are caused to a considerable extent by personal, genetic risk factors, lifestyles and environmental conditions. As a national competence center, we are also responsible for tasks in radiation research and protection.

Excellent basic research, internationally effective experimental research platforms, and clinical cooperation groups are keys to the achievements of the Helmholtz Zentrum München. We are in the unique position of combining under one roof fundamental capacities that are essential to analyzing interactions between health and the environment. Our scientific expertise includes analysis of environmental factors and complex disease processes, examination

of patterns of reaction of the organism to environmental factors and elucidation of ecosystems with regard to their significance for human health.

At the present time, the Helmholtz Zentrum München is composed of 26 scientific institutes and independent departments and has about 1,780 staff members. The head office of the center is located in Neuherberg to the north of Munich on a research campus of 124 acres. In addition, the organization maintains various research institutes in the city of Munich as well as numerous partnerships with Munich universities. Furthermore, it runs the Asse research mine in Remlingen near Braunschweig.

As a member of the Helmholtz Association, the Helmholtz Zentrum München belongs to Germany’s largest research organization: a community of 15 scientific-technical and biological-medical research centers. These centers have been commissioned with the pursuit of long-term research goals on behalf of the state and society. The association strives to gain insights and knowledge so that it can help to preserve and improve the foundations of human life.

Helmholtz Zentrum München – IBB

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The Institute of Biomathematics and Biometry (IBB) was founded in May 1997. It is part of the Helmholtz Zentrum München — German Research Center for Environmental Health. Initially, the focus was on biometry with the goal to apply, adapt, and develop statistical methods to analyze data from different areas of the life sciences. Of course, mathematical methods for signal and image analysis and mathematical modeling in ecology have also been research topics from the very beginning. In the meantime, progress in biology going beyond empirical observation has led to an increasing demand for mathematical abstraction and mathematical methods to deal with the complexity of the structure of biological systems and their dynamics. Moreover, the success of new techniques in bioimaging and bioengineering is mainly based on sophisticated methods of mathematical analysis.

In the last years the IBB has worked intensively to address these challenges. Today, mathematical modeling and mathematical methods for imaging comprise an important part of the institute’s research efforts. A further challenge for the future will be to provide a mathematical basis for scientific computing next to the three important research fields (data analysis, mathematical modeling, and mathematical methods in imaging) already in existence.

This brochure is designed to acquaint you with some of the ongoing research work at IBB. I hope you will find it interesting and stimulating to read.

Director Prof. Dr. Rupert Lasser

Helmholtz Zentrum München – IBB

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Philosophy is written in this grand book, the universe, which stands continually

open to our gaze. But the book cannot be understood unless one first learns to

comprehend the language and read the characters in which it is written. It is writ-

ten in the language of mathematics, and its characters are triangles, circles, and

other geometric figures without which it is humanly impossible to understand a

single word of it; without these one is wandering in a dark labyrinth.

(Galileo Galilei)

the instituteMathematics at the Helmholtz Zentrum München

Helmholtz Zentrum München – IBB: the institute

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must speak it, customize it, and pass it on to the next generation. Then one must continuously instruct others how to use this language in an interdisciplinary world of biology, ecology and medicine. If this mission succeeds, some light may be shed upon the darkness of complex biological mechanisms, making the underlying relationships approachable through thought and experiment. But let us briefly stay with this analogy and explicate what it means for a mathematical institute. In order to do so, we shall use the term biology as an overall name for the family of scientific disciplines commonly referred to as life sciences.

Language cultivation requires taking care that its use within natural sciences remains in accordance with the principles and rules of mathematics. This can be achieved through statistical consulting, through elucidating the mathematical concepts of image understanding and processing, and the development of algorithms which follow those principles while processing experimental data. Within this context, mathematicians serve as translators and facilitators. They make mathematical knowledge accessible to the scientific community.

Language development starts with the observation that language is a living object which must continually be adjusted to the realities of the world surrounding it. Thus, existing mathematical concepts have to be extended to meet new challenges. They may have to be applied within an unfamiliar context, and new ideas may have to be developed to

Members and guests of the institute

1. The language of mathematicsAccording to Galileo, the language of mathematics holds the key to illuminating the dark labyrinth of nature. Although mathematics has made great strides since Galileo, and triangles and circles are nowadays accompanied by hypergroups, reaction-diffusion equations, Markov processes etc., Galileo’s main idea remains valid. Mathematics is a language for human thought, structured by pure logic and with a level of abstraction that makes complex relationships thinkable. Language in this sense stands for the abstract formulation of concrete problems, the development of mathematical theory, formal deduction, and induction from experimental data.

The mission of the Institute of Biomathematics and Biometry is to cultivate this language, incited by its use within scientific research in the field of life sciences. In order to cultivate a language, one first


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solve the problems which arise. Modern fields like systems biology have to be equipped with new mathematical concepts and ideas to enable researchers to cope with the complexity they face.

Linguistics studies language, its structure and modification, and puts it into a formal context. Once new ideas have found their way into applications, they also have to be framed into mathematical theory. The strength of mathematical solutions lies in the concise and logically strict deduction within theory. Thus, theory has to be understood, advanced and opened up towards new frontiers. Transfering problem-oriented mathematical work into mathematical generality also yields the fundamental capability to provide a quick and flexible response if new challenges are encountered. It would be fallacious to conclude that a particular theory will never be of any practical use. The only conclusion we can honestly draw is the opposite. We can demonstrate how mathematical theory unfolds within an applied framework. Here mathematics will help in the formulation of a comprehensive theory of biology which still needs to be developed.

The three aspects of dealing with the living language of mathematics outlined here pose a great challenge for a methodologically oriented research institute within a life science organization. On the one hand, methods have to be developed to help uncover the principles underlying the interaction between human health and the environment. On the other hand, the translators and facilitators must have a firm grasp of mathematical language in order to succeed in the scientific world of international mathematics. Therefore, mathematical education at all levels of academic research is essential to accomplish the institute’s mission.

2. A brief historical outlineFounded in 1997, the Institute of Biomathematics and Biometry started with three research groups naming the major lines of research: »Mathematical Methods in Signal Analysis«, »Mathematical Models in Ecology and Biosciences«, and »Biostatistics«. A fourth group »Deterministic Models and Dynamical Systems in Biomathematics« followed in 2001. From the beginning, the head of the institute has also held the chair of Applied Mathematics in Ecology and Medicine (M12) at the Technische Universität München. Prof. Dr. Rupert Lasser founded the institute and has been its director ever since. Because of his particular scientific background, the institute has always carried out substantial work in harmonic analysis. As this is the underlying theory of large parts of mathematical signal analysis and mathematical imaging, as well as frequency analysis of mathematical systems, several applications of concepts of harmonic analysis have been worked out over the years. From its founding to the present day, it has been a sign of scientific strength of the institute that several of its members have been appointed professorships at various universities throughout the world. Thus, the scientific expertise has been adjusted several times, depending on the scientific members recruited. Nevertheless, the major lines of scientific research have been kept and extended. Over the years the variety of mathematical activity needed within a life science context has expanded, forcing the adoption of the term biomathematics in a very broad sense. While for the first orientation there was a relatively close connection between mathematical expertise and a certain focus of application, the research groups now are oriented towards their mathematical specialty, while the major fields of applications can be found across-


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the-board. Thus, confirming the first idea for founding such an institute, biomathematics is understood as the scientific approach to solve biological problems with mathematical tools and expertise. This is reflected in the orientation of the six research groups the institute presently has. They cover a wide range of mathematical specialties which contribute in different ways to various applications in life sciences. In this sense the group titles reflect the spectrum of mathematical expertise offered to the scientific community of the Helmholtz Zentrum München. The research groups presently are titled »Approximation Theory«, »Stochastic Modeling and Statistics«, »Computational Mathematics«, »Applied Harmonic Analysis and Inverse Problems«, and »Dynamical Systems«. A sixth group is the junior research group »Mathematical Methods in Biological Image Analysis MAMEBIA« funded by the Marie Curie Excellence program of the European Commission.

The strong connection with both universities at Munich has always been a fruitful relationship. Certainly, academic education on all levels (student projects, diploma and master theses, PhD, habilitation) is a prerequisite for pursuing a scientific career at the institute. Furthermore, members of the institute regularly teach classes at both universities. The presence of the institute at the Center of Mathematics of the Technische Universität München is inherent with the common nomination of its head as director of the institute and chair at the faculty of mathematics. A further link has been established with the junior research group MAMEBIA, headed by Prof. Dr. Brigitte Forster-Heinlein, who also holds a position at the Technische Universität München. The connection to the Ludwig-Maximilians-Universität München is vitalized by Prof. Dr. Gerhard Winkler’s professorship at the Department of Mathematics. Temporary teaching duties have also been carried out at the Universität der Bundeswehr München.

3. Philosophy of researchIn the last decade, the focus of molecular biological research has turned from merely observing natural phenomena to understanding biological systems. Modern high-throughput technologies in the experimental sciences have further fostered this development by providing an overwhelming amount of experimental data. The international scientific community has reacted by creating new disciplines, such as computational biology, systems biology, or structure biology. What Nir Friedman wrote about one of these disciplines applies to all of them: “The challenge for computational biology is to provide methodologies for transforming high-throughput heterogeneous data sets into biological insights about the underlying mechanisms.” [N. Friedman, Science 303 (2004)]. It has thus become clear that modern research in the life sciences has an increasing demand for advanced mathematical techniques. The theoretical investigation of complex biological systems requires an enormous variety of mathematical methods. Within the wide range of mathematical specializations, especially analysis, stochastics and numerical mathematics seem to provide a large spectrum of highly advanced tools to cope with the challenges of high-level research in the life sciences. J.E. Cohen’s essay in the Public Library of Sciences “Mathematics is Biology’s next Microscope, only better; Biology is Mathematics’ next Physics, only better” [J.E. Cohen, PLoSBiol 2 (2004)] confirms this point of view. This fruitful mélange has various manifestations. Researchers in the life sciences have a need for qualified consulting about available methods, especially in the field of statistics and image analysis. Moreover, there is an increasing demand for mathematical modeling

and numerical simulation. In silico experiments comprise the third major area of scientific research in the natural sciences, next to laboratory experiments and theory. Mathematical formulations of biological insights not only provide the basis for in silico experiments. They are also a means of organizing, structuring and evaluating biological hypotheses towards the development of a unifying theory.In interdisciplinary work it often turns out that existing mathematical concepts cannot be applied directly, or they may prove to be insufficient to tackle the problem. Such gaps lead to problem-induced mathematical research. Its objective is to adapt known mathematical methods in order to develop new mathematical techniques with the focus on a special problem. Research of such type especially occurs in connection with the development of new experimental technologies. For example, improving the quality of digital images from tomographic scanners in order to be able to reduce the dose to which the patient is exposed can be realized by an appropriate mathematical representation of digital images. Other examples stem from the effort to combine different measurement techniques in chemical biology in order to provide more information about the probe being analyzed. The overall mathematical goal is to extract an optimum of relevant information out of experimental data. Towards this goal, an optimal mathematical procedure allows to split up measured entities into their inherent information and uncertainty coming from all sorts of measurement tradeoffs.

Thorough mathematical research is always rooted within mathematical theory. Formal proofs and abstract deductions are the common output. At first glance a mathematical proof gives some guarantee for the correctness of the mathematical statement within the mathematical theory. But it often also provides further insight into the backbone of an argumentation. It makes strengths and weaknesses apparent and may be the starting point for further investigation. Constructive proofs further lead to algorithmic descriptions which are the basis for computational implementation. Beyond all these theoretical advantages, the mathematical proof is a formal abstraction of human thought. Being aware of this interpretation, a proper understanding of mathematical argumentation leads to an understanding of the mechanisms. To give an example, the analysis of the theoretical behavior of a system of differential equations not only provides an interpretation of experimental data and simulation. At the same time it can explicitly make clear the simplifications and restrictions every modeling process needs. Hence, crucial assumptions can be separated from those which have simply been posed due to lack of understanding. The power of abstraction is basically the power of reduction to the essential. Complex systems are made accessible within such a language. Engineering sciences, based on the principles of physics, have commonly accepted mathematical language as a way to express complex relationships. An analog movement in the biological sciences has just recently started. But this path of research is indispensable for future success in the life sciences. Abstraction provides the means to identify common underlying principles. Success in systems biology, for example, heavily depends on going beyond the application and standardization of available tools and methods. Appropriate new mathematical methods and theories have to be developed. This need for research is parallel to the need for new bioengineering technologies. Clearly, the work has to be done in close interaction with researchers from all fields of life sciences and requires continual communication.


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Thus, modern biological applications pose deep mathematical questions. In this sense, biological sciences have the potential to inspire future mathematical work in the same way as physics has done. First steps already show great promise. Genetic algorithms, for example, successfully implement the principle of advancement through interplay of inheritance, mutation and competition. Mathematical theory has been developed to understand such algorithms and to analyze their behavior. To summarize, the biological age, following the age of information technology, will certainly not be a period without mathematics, and mathematicians are called to take on this challenge.

4. Research profileThe scientific profiles of research teams commonly reflect the scientific expertise of their leaders. Nevertheless, the institute has been able to provide a broad spectrum of mathematical specializations. Within its research groups the institute reflects a natural professional structure. Of course, the boundaries are permeable in the scientific sense.

Approximation Theory. Many phenomena in biological sciences are described by complex mathematical objects. In order to be able to deal with such objects, build them up properly out of experimental data, analyze them and provide predictions, complex objects often have to be approximated by simpler ones. Typical examples are the signals occurring in many measuring systems. Commonly, signals are theoretically composed of an infinite number of constituents. To deal with them in an algorithmic way, a finite number of those constituents has to be chosen, thus providing an approximation of the signal. A natural approach to signal analysis then lies in decomposing the signal into such constituents with the aim of reading properties of the signal from its parts. This, for example, allows separating the signal from the noise. There is no unique decomposition of a signal. A proper decomposition always depends on the theoretical framework in which the object is considered. Thus, by choosing the framework according to the concrete biological problem, the theory unfolds a portfolio of tools to deal with biological signals. Often, the framework is determined by geometric structures. Such approaches can be derived from concepts of harmonic analysis.

Stochastic Modeling and Statistics. A mathematical way to deal with uncertainty is formulated within the theory of probability. Modeling probabilistic aspects of biological systems not only allows dealing with uncertainty coming from an imperfect knowledge of the system. Chance is an intrinsic property of many natural phenomena. For example, the concentration of an enzyme in a specific chemical reaction might be so low that there might only be a certain chance that the molecules meet. Moreover, there is evidence that some organisms utilize chance on purpose, e.g. for cell differentiation. Such effects can be modeled using stochastic processes. Elucidating certain processes and unraveling their non-random properties enables modelers to provide predictions concerning the behavior of the particular system. This leads to estimates of the prediction accuracy.

Identifying stochastic properties within experimental data, inferring qualitative information and evaluating predictions are classic tasks of statistics. Thus, statistical analysis is an essential part of the analysis of experimental data. For that reason, there is great need for statistical consulting in the biological sciences. As the natural

systems to be studied become more complex, the requirements for the corresponding statistical methodologies rise. This soon leads beyond classical test theory and thus, beyond the scope of standard statistical software. Hence, there is an increasing demand for high-level research in mathematical statistics.

Computational Mathematics. Studying mathematical models for biological systems on a theoretical basis is an indispensable requisite for their numerical implementation. Nevertheless, theoretically derived properties often lead to qualitative statements only. The use of computer simulation along with experiments and theory in order to foster research on complex biological systems creates the necessity for numerical analysis and algorithmic implementation. Large systems and limited computational resources soon create mathematical problems on their own. Here it is not solely the challenge that large computations have to be accomplished. An even bigger issue arises from the attempt to make sure that the calculations obtained by numerical simulation are still reliable, leading to questions of numerical stability.

Another part of computational mathematics uses the computer to implement algorithms which on the basis of a mathematical theory identify patterns and infer hypotheses from experimental data. In this branch, the computer is used as an evaluation device. Nevertheless, a sound theoretical understanding is of great importance to avoid hidden fallacies. The question is how to determine which part of the conclusion results from the data and which part is a result of the theoretical approach one has chosen in advance. Such mathematical approaches touch the fields of machine learning and data mining.

Applied Harmonic Analysis and Inverse Problems. As mentioned above, many mathematical objects to deal with biological systems and signals are based on assumptions on the underlying nature of mathematical spaces. The goal of harmonic analysis is to study those properties, to define appropriate notions within new frameworks and to analyze their consequences. The original idea comes from decomposing a complex wave signal into its frequency components. The abstract backbone of frequency decomposition is well understood and has been advanced far more. But still, there are many problems to be solved, such as the relationship between regularity and frequency-based function spaces. The consequences for applied work are immediate. Many signals, for example, do not satisfy classic assumptions of time-frequency analysis, at least not without too much simplification. Thus, an appropriate theoretical framework has to be set up in order to relax stronger assumptions. An important class of such problems arises in situations where continuous phenomena have to be reconstructed from a finite set of information. This is an example of an inverse problem. Concepts of harmonic analysis provide powerful tools to solve inverse problems. Examples are various types of Radon transforms in tomography. Algorithms to reconstruct an image from measurements use Fourier decompositions and convolution, or iterative methods based on projections onto function spaces.

Dynamical Systems. Mathematical models for a biological system can be formulated in terms of differential or difference equations. Such equations define essential properties of functions which describe the dynamics of measured entities. Setting up a mathematical model in terms of differential equations is thus an abstract description of, for


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example, the time-dependent evolution of a certain concentration. The goal is to study the long-term behavior of such systems and their possible states. An important aspect hereby is to guarantee that the essential dynamics of the biological systems are covered within the dynamics of the equations. In many cases, even simple systems soon lead to very complex systems of equations. Then mathematical theory can be used to aggregate equations, identify components of minor importance, and describe the overall dynamics of larger components of the system. This research even has a wider goal. In physics, for example, it turned out that there is a set of fundamental differential equations which describe many phenomena in nature. In biology, we still need to formulate the fundamental laws governing life. To identify the equations representing such fundamental laws is the long-term goal of biomathematics in the narrower sense.

There are four major application areas for the Institute of Biomathematics and Biometry in the Helmholtz Zentrum München. They cannot be associated with only one single mathematical specialization. The institute’s concept is to contribute to each of these major application areas with whatever mathematical concept seems specifically appropriate.

Statistics: Supporting conclusions based on experimental data commonly involves some statistical analysis. Since this is one of the major demands coming from all the institutes at the center, the Institute of Biomathematics and Biometry has set up a consulting unit called ImStatLab (Image Analysis and Statistical Consulting Laboratory). It consists of a group of scientists from different research groups. It is intended to serve as an interface between external scientists and the institute, with a focus on immediate consulting. It provides assistance for the solution of practical problems, in particular in statistics and image analysis, but also in other fields of mathematics. Once involved, mathematicians and statisticians become an integral part of interdisciplinary projects. There is an emphasis on data analysis and data mining under the constraints of variability and uncertainty.

Mathematical Modeling and Systems Biology. Modeling of biological systems and their mathematical analysis is clearly one of the original goals of biomathematics. Making complex systems in nature accessible through computer simulation provides a means of exploring the system beyond experimental limits, in order to validate experimental data or to extrapolate quantities under alternated experimental conditions.

More and more, the systemic approach is overtaking the classic principle of cause and effect. Understanding the relations among the involved protagonists of the system, unraveling the influence they have on each other, and finally predicting their behavior in advance are indispensable prerequisites for controlling the system and keeping it in balance. This is the ultimate goal of any medical treatment. But clearly, there is no »free lunch«. Every model is a simplification and every simplification deviates from the original. Thus, one has to be able to identify crucial assumptions. Furthermore, one has to estimate how far simplification changes the expected result and maintain a good feeling for the discrepancy between model and reality.

Mathematical Signal Analysis and Imaging. A large class of measurements in the life sciences consists of various signals and images. Modern experimental technologies together with computing power allow visualization of phenomena on very small scales or within living systems. The more intervention can be avoided during experimental observation, the less the observed system is perturbed, bringing the experiment closer to realistic conditions. This is just one motivation for the aim to extract as much information from every bit of data possible. The MAMEBIA group at the Institute of Biomathematics and Biometry is especially devoted to providing mathematical answers for such challenges. But imaging applications also play a role in most of the other research groups. For example, it is also part of the consulting unit ImStatLab. As for mathematical modeling, too, there is no ultimate mathematical approach to such questions. All mathematical specializations capable of doing so contribute substantially to this field of application.

Structure Biology and Chemical Biology. The molecular scale of biology is a world of building of chemical complexes, reactions among substances, and information passing by lowering and raising concentrations. The entrance to this world is provided by modern measuring technology, like FT-ICR-MS spectroscopy. Since devices of such type are available on campus and since there is very high demand for the elucidation of many processes within living cells, it is natural that the Helmholtz Zentrum München is strengthening its capabilities in these areas. Mathematical theory provides the language to formulate hypotheses, conclusions, and results within molecular biology. For example, the structure of certain protein complexes can be deduced from the study of optimal configurations, minimizing certain energy functionals. Mathematical concepts have entered this world and will gain further importance in the near future.


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the researchSketches of Selected Projects

It may be true that, as Francis Thompson noted, »Thou canst not stir a flower

without troubling a star«, but in computing the motion of stars and planets, the

effects of flowers do not loom large. It is the disregarding of the effect of flowers

on stars that allows progress in astronomy. Appropriate abstraction is critical to

progress in science.

(Herman Shugart)

Helmholtz Zentrum München – IBB: the research

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Many bacterial species can sense aspects of their environment or communicate with other bacteria via chemical signal molecules, called autoinducers (AI). An important case is Quorum sensing, which assumes that the bacteria measure their cell density via this regulation mechanism. Another, more general interpretation of the same regulation system assumes that the bacteria use AI to test whether the production of chemical substances which are released into their environment is efficient. Examples are the marine bacterium Vibrio fischeri which lives in the light organ of squids and regulates its luminescence and Pseudomonas putida, a rhizosphere bacterium controlling biofilm production. The regulation systems often consist of different autoinducers and interlinked pathways. Nevertheless, there are many similarities between the systems in different species, allowing for general mathematical approaches.

In this project, we investigate the mechanisms of this regulation by mathematical methods. The dynamics is described by a system of ordinary differential equations (ODE). Using methods like singular perturbation theory, the model is reduced to the key players in the pathway. In most examples, the systems show up bistable behavior and hysteresis. Fitting of the ODE systems to experimental data – if available – provides an estimation of parameters like the production and degradation rates of the autoinducers. Furthermore, the mathematical analysis and the comparison with experimental results enable some validation of the biological hypothesis concerning the structure of the underlying network or may hint that there are additional players in the network structure. Such models allow the prediction of the time course of the involved players.

The modeling approaches can be extended, e. g. by using partial differential equations for spatial models, or by stochastic models to investigate the influence of small numbers of molecules or receptors. Another goal is to study the adsorption and transport of autoinducers by roots / plants, in the context of a better understanding of the relevance of bacteria in a natural environment like the rhizosphere.

References: C. Kuttler, B. A. Hense, Interplay of two quorum sensing regulation systems of Vibrio fischeri, J. Theor. Biol. 251 (2008), 167-180B. A. Hense, C. Kuttler, J. Müller, M. Rothballer, A. Hartmann, J.U. Kreft, Does efficiency sensing unify diffusion and quorum sensing?, Nat. Rev. Microbiol. 5 (2007), 230-239J. Müller, C. Kuttler, B. A. Hense, M. Rothballer, A. Hartmann, Cell-cell communication by quorum sensing and dimension reduction, J. Math. Biol. 53 (2006), 672-702

Modeling

1 Center of Mathematics, Technische Universität München2 Institute of Ecological Chemistry, Helmholtz Zentrum München3 Department Microbe-Plant Interactions, Helmholtz Zentrum München

Christina Kuttler, Burkhard Hense, Johannes Müller1, Agnes Fekete2, Philippe Schmitt-Kopplin2, Michael Rothballer3, Anton Hartmann3

mathematical modeling of quorum sensing

Regulation pathway of the Quorum sensing system of Vibrio fischeri and prediction for the time course of luciferase (light producing enzyme), with comparison of wildtype and a mutant (lacking the protein LuxO).


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Many bacteria use the so-called quorum sensing system to obtain information about their neighborhood. Quorum sensing is based on the production of a small constitutive amount of signaling molecules which are released into the environment and can be sensed by the bacteria. If a bacterium detects a critical density of these signaling molecules a cellular response is initiated and the cell produces the signaling molecules at a high rate. This positive feedback leads to bistable behavior.

The molecular mechanism of quorum sensing, e. g. for Vibrio fischeri, is quite well understood. Deterministic models appropriately describe the behavior for large, homogeneously mixed populations.

In natural environments bacterial cells, like Pseudomonas putida, are usually inhomogeneously distributed in space, e. g. microcolonies on roots of plants. This phenomenon is not yet well understood. In this project we investigate the effect of a small population size, as for example in microcolonies. Therefore, the joint density of the signaling substance and the number of activated cells within the population is modeled by a correlated random walk. By assuming a separation of time scales between production of the signaling substance (AHL) and activation/deactivation of cells we can reduce the model by means of a parabolic limit. This means that we can approximate the asymptotic behavior of our model by the solution of a parabolic differential equation obtained via regular perturbation expansion. The aim is to analyze stationary states of the resulting system with respect to multistability and the effect of noise introduced by small cell numbers.

References:J. Müller, C. Kuttler, B. A. Hense, S. Zeiser, V. Liebscher, Transcription, intercellular variability and correlated random walk, submitted to: Math. Biosci.J. Müller, C. Kuttler, B. A. Hense, Sensitivity of the quorum sensing system is achieved by low pass filtering, Biosystems 92 (2008), 76-81T. Hillen, H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math. 61 (2001), 751-775

Modeling

Microcolonies in the rhizosphere

Estimated density of the AHL concentration

1 Center of Mathematics, Technische Universität München

Alexandra Hutzenthaler, Johannes Müller1, Burkhard Hense, Christina Kuttler

effect of small cell numbers on quorum sensing

Schematic of the model


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Sabine Eisenhofer, Frank Filbir, Burkhard Hense, Ferenc Toókos, Hans Zischka1

Characterizing the mechanism of calcium-induced swelling of mitochondria is one of the key steps in understanding programmed cell death. In this process the mitochondria take up Ca2+ through the inner mitochondrial membrane. High concentrations of extramitochondrial Ca2+ can induce the irreversible opening of the permeability transition pores. Thus, the mitochondria can switch into a low or high conductance mode, depending on the amount of the available Ca2+. The high conductance mode is of special interest concerning the swelling process of the mitochondria. It results in a swelling of the mitochondria and the rupture of the outer mitochondrial membrane. As a consequence, proapoptotic factors contained in the mitochondrial intermembrane space are released and eventually cause cell death (apoptosis).

Methodologically, pore opening behavior is investigated via the swelling dynamics of mitochondria. For this purpose, isolated mitochondria were exposed to different conditions influencing specific aspects of the pore opening, e. g. high concentrations of calcium. The time course of the swelling was measured with a fluorimeter, which analyzes changing light scattering properties.

An interpretation of the experimental data requires an appropriate mathematical model. Two conceptually different approaches exist for building a mathematical model. One approach is oriented heavily on the theory of molecular processes and attempts to take into account all biochemical processes including Ca2+ homeostasis, mitochondrial respiration, substrate transfer etc. This leads to a system of nonlinear ordinary differential equations. The utility of such models relies very much on the knowledge of this system of equations. Therefore a detailed analysis of the involved differential equations is an important issue. Another approach is much simpler. It models the swelling of a collection of mitochondria in a special environment. This model consists of only one nonlinear differential equation. The model is not able to explain certain effects and has to be modified.

Modeling

1 Institute of Toxicology, Helmholtz Zentrum München

References: A. V. Pokhilko, F. I. Ataullakhanov, E. L. Holmuhamedov, Mathematical model of mitochondrial ionic homeostasis: Three modes of Ca2+ -transport, J. Theor. Biol. 243 (2006), 152—169S. V. Baranov, I. G. Stavroskaya, A. M. Brown, A. M. Tyryshkin, B. S. Kristal, Kinetic model for Ca2+-induced permeability transition in energized liver mitochondria discriminates between inhibitor mechanisms, J. Biol. Chem. 283, no. 2 (2008), 665—676

Swelling curves with fluorimeter. Me1, Me2: inducers of pore opening; Inh1, Inh2: inhibitors of pore opening.

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Kontrolle

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1mM Inh 1, 25μM Me1, 10μM Me2

5μM Me2, 5μM Me1

5μM Me2, 10μM Me1

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5μM Me2, 100μM Me1

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Time in min

modeling the swelling of mitochondria


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Joachim Altschuh

Narcosis is generally recognized as a basic mechanism of action of lipophilic xenobiotics. The underlying molecular mechanism has remained controversial. The biological target sites of narcosis are still under dispute, the membrane lipid phase or hydrophobic protein pockets being the main candidates.

In cooperation with the Institute of Biochemical Plant Pathology, Helmholtz Zentrum München, and the Lehrstuhl A für Mathematik, RWTH Aachen, we investigated mechanisms of the effect of xenobiotics on membrane proteins. In particular, we discussed the displacement of lipid molecules from the lipid ring surrounding a membrane protein. The Adair approach employing microscopic lipid binding constants has been taken to explain the enhancement of agonist binding to the nicotinic acetylcholine receptor (nAChR) by general anesthetics in terms of the competitive displacement of essential lipid activator molecules. As a result, a free energy estimate of lipid /protein interaction in membranes is now available.

This approach was extended to tadpole narcosis induced by alcohols. We generalised and extended the kinetic model by allowing for different lipid binding constants. A single class, or two different classes of lipid activator binding sites, were considered. Microscopic lipid and inhibitor binding constants were derived and allowed a close fit to dose-response curves of tadpole narcosis on the basis of a preferential displacement of more loosely bound essential lipid activator molecules. The results complement the two major theories for the explanation of the action of anaesthetic compounds, suggesting that potential binding sites at the interface between integral membrane proteins and the lipids are the sites involved.

References:S. Walcher, J. Altschuh, H. Sandermann, The lipid /protein interface as xenobiotic target site: kinetic analysis of the nicotinic acetylcholine receptor, J. Biol. Chem. 276 (2001), 42191-42195J. Altschuh, S. Walcher, H. Sandermann, The lipid /protein interface as xenobiotic target site. Kinetic analysis of tadpole narcosis, FEBS J. 272 (2005), 2399-2406

Modeling

Fitted dose-response curves for 1-hexanol. The principle (top panel) is to mathematically transfer the dose-response curve for the nAChR to the data set for tadpole anaesthesia.

kinetic modeling of narcosis


15

population dynamical modeling

Modeling the behavior and evolution of a population has a long history in applied mathematics. There are several existing models – continuous, discrete, stochastic – describing the general process, or certain particular aspects, e. g. migration, environmental change, spread of a disease. Despite the broad range of existing literature, there are still a lot of unanswered questions on the subject.

Our major interest is the understanding of the main driving forces of change in populations, of which the most important are selection, mutation, and recombination.

We created a deterministic continuous model, which describes the gene distribution development in a randomly mating, sexually reproducing population under the effects of selection, mutation, and recombination.

The mathematical model is a multidimensional, nonlinear system of ordinary differential equations with several parameters, which generalizes some previous models. The system has the form:

where is the distribution of the alleles andand are the matrices of the selection-mutation-recombination coefficients. For the qualitative study we used bifurcation and stability theory, as well as the mathematical software Maple for the numerics.

If only selection acts in the system, we showed that Fisher’s Fundamental Theorem holds. It states that the mean fitness (average viability) has to increase in the population. In this case all solutions tend to a rest point, the gene distribution gradually approaches an equilibrium state. With mutation and recombination one can get more complicated dynamics. We found that with special cyclic mutation rates, stable limit cycles occur in the system due to Hopf-bifurcation.

Modeling

1 Rényi Institute, Hungarian Academy of Sciences, Hungary2 Bolyai Institute, University of Szeged, Hungary

References: L. Hatvani, F. Toókos, G. Tusnády, A mutation-selection-recombination model in population genetics, IBB-Preprint 07-38, 2007J. Hofbauer, K. Sigmund, Evolutionary Games and Replicator Dynamics, Cambridge University Press, 1998

Asymptotically stable equilibrium

Asymptotically stable cycle for 1 locus and 4 alleles

Ferenc Toókos, Gábor Tusnády1, László Hatvani2


16

Robert Schlicht, Gerhard Winkler

Noise is an integral part of biological systems at all scales. Whenever discrete components such as individuals in an ecosystem, cells in an organism, or molecules in a cellular process appear in small numbers, either locally or throughout, deterministic models such as ODEs often fail as an adequate description of the biological reality. In these situations, a probabilistic approach is definitely more appropriate.

We develop stochastic models for molecular processes in developmental biology. Major modeling goals are to:— Obtain models close to reality by considering discrete numbers

of molecules and including, for example, reaction delays, — Construct the models in a way that facilitates direct computer

simulation, — Keep the models simple enough for a rigorous and

comprehensive mathematical analysis.

We introduced a stochastic model for the molecular oscillator that controls somite formation in the vertebrate embryo. This process is based on negative feedback in gene expression and interaction of neighboring cells through Delta-Notch signalling. The model is a non-Markovian stochastic process. It reproduces accurately the oscillations of protein concentrations observed in the embryo.

References:O. Cinquin, Understanding the somitogenesis clock: What’s missing?, Mech. Dev. 124 (2007), 501–517J. Lewis, Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator, Curr. Biol. 13 (2003), 1398–1408 R. Schlicht, G. Winkler, A delay stochastic process with applications in molecular biology, J. Math. Biol., doi 10.1007/s00285-008-0178-y

Modeling

Protein concentrations in the center of a line of 15 cells. Red and blue are the proteins that drive within-cell oscillation. Green is the Delta protein that is involved in synchronization of different cells.

stochastic modeling in developmental biology


17

mathematical modeling of feedback loops

Feedback loops have always played an important role in the mathematical modeling of biochemical processes. However, up to now the theory has only been fully developed for uncoupled systems. But new theoretical insights from geometric singular perturbation theory may help to analyze more complex models. The new results can be used in multiple applications. As an example, we investigated mechanisms that are able to create tolerance and an activation threshold in the extrinsic coagulation cascade.

Most models suggest that coagulation starts in presence of arbitrary small amounts of factor VIIa. Although these models are quite well tested by experiments with so-called »artificial plasma«, it is to be expected that there is some protection mechanism against minimal spontaneous and /or stochastic events.

Only recently the effect of blood flow on coagulation attracted some interest. We provide support for the hypothesis that the interplay of coagulation inhibitor and blood flow creates threshold behavior. We first tested our hypothesis in a minimal, four-dimensional model. Indeed, we found that only the interplay of blood flow and inhibition together are able to produce threshold behavior. The mechanism relies on a combination of raw substance supply and wash-out effect by the blood flow and a stabilization of the resting state by the inhibitor.

We used the insight into this minimal model to interpret the simulation results of a large, more realistic model. Here, we found that the initiating steps do not exhibit threshold behavior, but the overall system does. Hence, the threshold behavior appears via the feedback loop by ATIII and blood flow.

Modeling

1 Center of Mathematics, Technische Universität München

References: J. Müller, S. Brandt, K. Mayerhofer, T. Tjardes, M. Maegele, Tolerance and threshold in the extrinsic coagulation system, Math. Biosci. 211 (2008), 226–254M. Krupa, P. Szmolyan, Relaxation oscillation and Canard explosion, J. Differential Equations 174 (2001), 312–368

Time course of the complete model with flow and activation of anti-coagulatory substances.

We found three cases: one spike only, sustained activation and relaxation oscillation (upper left, upper right and lower left). Panel at the lower right: time course for relaxation oscillations.

relaxation oscilations

z

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slow manifold

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relaxation oscilations4

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Stefan Brandt, Johannes Müller1


18

Christian Pötzsche

The analysis of biological or ecological models has always been a prime motivation for the mathematical theory of dynamical systems. In fact, many phenomena from the above areas allow a successful theoretical description using evolutionary equations. Well-known applications include population models, the theory of epidemics, genetics, physiology, biogeography or neurobiology. For realistic models, these equations are high dimensional and nonlinear. In such cases, advanced and sophisticated mathematical tools apply – in particular, if the system under consideration is time dependent (nonautonomous), because effects like a fluctuating environment or an external control have to be taken into account. This leads to the new theory of nonautonomous dynamical systems.

In qualitative studies on nonlinear dynamical systems, invariant manifolds play a crucial role: For instance, local stable and unstable manifolds dictate the saddle-point behavior in the vicinity of hyperbolic surfaces. Center manifolds are a tool to reduce the dimension of dynamical systems. From a more global perspective, stable manifolds serve as separatrices between different domains of attractions and allow a classification of solutions with a specific asymptotic behavior. Systems with a gradient structure possess global attractors consisting of unstable manifolds. Finally, inertial manifolds yield a global reduction principle for typically infinite-dimensional dissipative equations.

For these reasons, the computation of invariant manifolds is a highly relevant and interesting problem. However, although the existence of invariant manifolds is a well-established matter, their analytical computation is possible only in very rare cases. Hence, one needs computational tools for their approximation, and at least since the 1990s, several methods have been pursued. Because these approaches apply to autonomous problems only, we suggest a flexible method which approximates nonautonomous invariant manifolds by means of large systems of algebraic equations. These problems, in turn, can be solved numerically using Newton-like methods and continuation techniques.

References:B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15, no. 3 (2005), 763–791C. Pötzsche, M. Rasmussen, Computation of nonautonomous invariant and inertial manifolds, IBB-Preprint 07-40, 2007

Modeling

Fourier approximation of the inertial manifold for a nonautonomous reaction-diffusion equation

computation of nonauto-nomous invariant manifolds


19

biofilm modeling: analysis and simulation

It is the best of times for biofilm research (Nat. Rev. Microbiol. 5 (2007), 76-81). The big challenge in biofilm modeling is to describe the spatial spreading mechanisms for biomass. Such modeling plays an important role in environmental engineering, industry and medicine. In addition, it provides better insight into biofilm processes and explains experimental findings.

In the first stage we are mainly interested in biofilm processes on a mesoscale. Based on the experimental data, we derive a new class of highly nonlinear density-dependent degenerate reaction-diffusion systems comprising two effects simultaneously: porous medium and fast diffusion, the three-dimensional simulation of which leads to the mushroom patterns observed in the experiment. We especially emphasize that such a class of parabolic systems is not well understood in the mathematical literature and requires new ideas and tools. Our tangible goal is to investigate how well the developed models – in particular, the multicomponent biofilm models – are posed. Furthermore, we aim to investigate the long-time dynamics of solutions, the stability of patterns and the role of chemotaxis in the formation of biofilms. Finally, we aim to compare the experimental, numerical and analytical results. Another challenging task is investigating biofilm growth in porous media (e. g. soils), since such growth changes the hydraulic conductivity of the medium (bioclogging), which in turn changes substrate transport and thus the living conditions (food supply) for the bacteria. This naturally occurring nonlinear phenomenon can be used by engineers to devise microbially based technologies for groundwater protection and soil remediation.

Our first strategic goal is to develop a comprehensive mathematical model that unites macroscopic flow and transport processes with population dynamics and microbial activity on the biofilm scale. The model will be studied analytically (well-posedness, stability) and numerically (computer simulations), and quantitatively validated by comparison with experimental data.

Modeling

References: M. A. Efendiev, S. Zelik, Finite and infinite dimensional attractors for porous medium equations, J. London Math. Soc. 95, no. 3 (2008), 51-77M.A. Efendiev, L. Demaret, H. Eberl, R. Lasser, Analysis and simulation of a mesoscale model of diffusive resistance of bacterial biofilms to penetration of antibiotics, Adv. Math. Sci. Appl. 18, no. 2 (2008), 30-68M. A. Efendiev, M. Kläre, R. Lasser, Dimension estimate of the exponential attractors for the chemotaxis-growth systems, Math. Methods Appl. Sci. 30, no. 5 (2007), 579-594

Mushroom pattern (mesoscale)

Antibiotic disinfection

Bioclogging

Messoud Efendiyev


20

When considering phenomena in natural sciences it is often very useful to model processes with the help of mathematical equations. The solutions of these equations provide conclusions on the behavior of the modeled phenomena. Since equations of this type are usually not explicitly solvable we try to find approximate solutions by discretizing time and space variables. Thus we have to analyze a difference instead of a differential equation. Difference equations have the advantage that the solution can be easily computed recursively. This leads to the question whether discretization preserve the properties of the solutions of the equation.

We think about the wave equation as a simple example:

This equation models the progress of a wave (without friction loss), which evolves from an impulse of the form at the time A finite-difference discretization of the equation yields

Detailed analysis indicates that the classical mathematical concept of positive definite functions is very useful for the description of the behavior of the solutions of the second equation. Positive definite functions arise in various areas in pure and applied mathematics, such as orthogonal polynomials, numerical integration, and time series analysis. In these applications, the notion of positive definiteness depends on an underlying group or semigroup structure. For our purposes we have to extend some central results on positive definite functions to more general algebraic structures, which are induced by polynomial sequences. In particular, we show that every positive definite function of this type is the transform of a positive finite Borel measure on the reals, and find conditions which yield more information on the character of the support of this measure.

The generalization of classical results on positive definite functions gives us a deeper understanding of discretized equations. Returning to our example, it turns out that in the case the solutions are unbounded. This implies for the limit that the solutions of the second equation do not approximate the solutions of the first. In other words: Where on the one hand the first equation describes a wave, the second equation produces a tsunami for the wrong choice of the discretization coefficients.

References:K. Ey, R. Lasser, Facing linear difference equations through hypergroup methods, J. Difference Equ. Appl. 13 (2007), 953–965K. Ey, On the Representation of Pn-positive definite Functions and Applications, PhD-thesis, Center of Mathematics, Technische Universität München, 2008

Modeling

wave or tsunami?Kristine Ey


21

fast fourier transform on the rotation group

References: S. Kunis, D. Potts, Stability results for scattered data interpolation by trigonometric polynomials, SIAM J. Sci. Comput. 29 (2007), 1403-1419F. Filbir, W. Themistoclakis, Polynomial approximation on the sphere using scattered data, Math. Nachr. 281 (2008), 650–668

Flowchart of FFT

Theory

1 Faculty of Mathematics, Chemnitz University of Technology, Germany

Rose sampling grid

Frank Filbir, Daniel Potts1

One main issue of applied mathematics is the development of fast algorithms for calculations which appear over and over again in various applications. The calculation of the Fourier transform of a function is without doubt one of the most important examples of such an assignment. Here the mathematical problem consists in the computation of the discrete Fourier transform, the one dimensional version of which is defined as

A naive implementation for the calculation of those sums scales quadratically in the problem size . This means we need floating point operations for the evaluation of the sum. For evaluating a DFT with we need 652144 operations. Assuming for simplicity that one operation takes 1 second the calculation would need approximately 72 hours. Obviously, this would make the DFT inapplicable. Fortunately there is a much faster algorithm for doing such computations. The Fast Fourier Transform (FFT) changes the situation drastically. Using the FFT the calculation scales like which is now nearly linear. This means that we need only 6912 operations for . The traditional FFT algorithm works with an equispaced grid. Unfortunately, for many applications this assumption is far too restrictive. To overcome this shortcoming of the traditional FFT algorithm the so-called Nonequispaced Fast Fourier Transform (NFFT) algorithm was developed. The NFFT overcomes this disadvantage while keeping the number of floating point operations at .

In various modern applications one is confronted with the computation of sums of the above type where now the signal is no longer defined on the Euclidean space , but where the geometry of the underlying space is more sophisticated. In many of those situations a discrete Fourier transform can still be defined. Now special functions like, for example, orthogonal polynomials, spherical harmonics, or Wigner functions will play the role of the exponentials. In this respect the rotation group has attracted considerable attention recently. In order to invent NFFT-like algorithms on the rotation group a detailed analysis of approximation processes on this group is indispensable. These approximation processes are based on the construction of suitable kernels.


22

Azita Mayeli

Motivated by the applications of general spherical wavelets in the analysis of cosmological data sets, in particular, in the study of cosmic microwave background radiation CMB, we construct smooth nearly tight wavelet frames on the sphere. Furthermore, we investigate an asymptotic correlation for the coefficients of Mexican needlets, a class of wavelet frames on the sphere, and their generalizations. Mathematically, the generalized Mexican needlets are constructed through the kernel of operators for where is a Schwartz function on , , and p is an integral number.

Regular needlets were introduced and studied by Narcowich, Petrushev, and Ward. They consider only smooth functions f with compact support away from 0. We prefer to call such functions »cutoff functions«.

Generalized Mexican needlets have their own advantages. We can write down an approximate formula for them which can be used directly on the sphere. Assuming this formula, Mexican needlets and their higher orders have Gaussian decay at each scale. They do not oscillate for higher orders, so they can be implemented directly on the sphere, which is desirable if there is missing data (such as the »sky cut« of the CMB). Moreover, due to their good localization property, they are robust in the presence of partially observed spherical data.

Furthermore, we consider the correlation structure of the random coefficients for our generalized Mexican needlets. We obtain a necessary and sufficient condition for these coefficients to be asymptotically uncorrelated, depending on the behavior of the angular power spectrum of the underlying random fields. For application reasons in cosmology, we assume that the random coefficients are Gaussian and prove that the Mexican needlets of higher order provide uncorrelated coefficients, depending on a parameter related to the decay of the angular power spectrum. The proofs of results rely more directly on harmonic approaches, rather than on standard probabilistic arguments.

References:D. Geller, A. Mayeli, Nearly tight frames and space-frequency analysis on compact manifolds, to appear in: Math. Z.A. Mayeli, Asymptotic uncorrelation for generalized Mexican needlets, arXiv: 0806.3009, 2008F. J. Narcowich, P. Petrushev, J. D. Ward, Localized tight frames on spheres, SIAM J. Math. Anal. 38 (2006), 574-594

generalized mexican needlets and uncorrelation of their coefficients

Theory

Sky cut of the CMB (source: F. Hansen; data from WMPA; thanks to D. Marinucci for providing the map)


23

scattered data approximation on manifolds

A common problem in science is the recovery of signals or imag-es from a discrete set of measurements. This means we are giv-en a set of pairs of numerical values , and we assume that there is a function which has produced these measurements, i. e., ; ;

. Usually, the measurements are corrupted with errors, and, moreover, the points are scattered on X. The goal is to find a good approximation to the unknown functionusing the information given by the set S. In the simplest case the underlying set X is the Euclidean space . Nevertheless, in many situations the set X is a much more complicated manifold like a Lie group or a general Riemannian manifold.

A widely used method for constructing the approximant is to choose s as a linear combination of a given kernel function , i.e.

The characteristics of the related approximation process rely very much on the properties of the kernel and of course on the prop-erties of the set S. Normally, the kernel has to satisfy certain condi-tions like positive definiteness:

.

Sometimes decay properties play a significant role. This means that the kernel should decay fast enough away from the diagonal

. The construction of a suitable kernel is therefore an essential step in this procedure. Usually the kernel is given as a linear combination of certain basis functions , i. e. kernels of the type

In case of a compact Riemannian manifold X these basis functions are the eigenfunctions of the Laplace-Beltrami operator on X. The kernel is completely determined by the choice of the coefficients. These are the only objects we can control. Now the challenging mathematical problem is to answer the question which coefficients have to be chosen in order to come up with a suitable approximation process. More precisely, the process should be numerically stable, efficient and should have good approximation properties.

References: W. Erb, F. Filbir, Approximation by positive definite functions on compact groups, to appear in: Numer. Funct. Anal. Optim.F. Filbir, D. Schmid, Stability results for approximation by positive definite functions on SO(3), to appear in: J. Approx. TheoryH. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005

Theory

Wolfgang Erb, Frank Filbir, Dominik Schmid

Scattered data approximation on with Abel-Poisson kernel

Scattered points on the sphere


24

Michael Hofmann

Building averages is a tricky process because even for bounded numeric sequences, an arithmetic mean need not exist. To deal with this problem, the mathematical concept of amenability has been introduced.

In this theory so-called Følner sequences are important building blocks allowing for generalization of arithmetic means. It has been shown that for almost periodic and even for bounded signals averages do exist.

While in the classical theory averaging is done over the interval , the interval relevant for applications plays a role

in polynomial hypergroups. To extend the theory, we introduced summing sequences on polynomial hypergroups, allowing us to construct means for them as well.

In the future, as the main part of my PhD thesis, I will examine strong amenability on hypergroups, a new concept in mathematics, which allows for further investigation of means.

strong amenability on hypergroups

Theory


25

qualitative uncertainty relations and jacobi polynomials

An uncertainty relation in the context of Jacobi polynomials states that a nonzero square integrable function f and its Jacobi transform cannot both be sharply localized. A qualitative uncertainty relation makes this statement without any estimates for the function f or its Jacobi transform.

Uncertainty relations can be usefully interpreted in the context of signal analysis. Namely, if the function f represents the amplitude of a signal, the Jacobi transform of f represents the different frequencies of the signal. Thus an uncertainty relation expresses a limitation on the area to which a signal can be both time limited and band limited.

In our work we state that a function f is time limited by the set B, if the support of f is a subset of B. In the same way we state that a function f is band limited by the set Λ, if the spectrum of f is a subset of Λ.

On the one hand this concept of localization leads to abstract uncertainty relations or annihilation pairs on Hilbert spaces. In particular qualitative uncertainty relations can be interpreted geometrically. On the other hand this concept leads to the theory of Λ(2)-sets in harmonic analysis. Thereby a subset Λ of the natural numbers is called a Λ(2)-set, if every absolute integrable function, whose spectrum is contained in Λ, is already square integrable. A main result of our work is that the pair (B, Λ) is an annihilation pair in the concrete Hilbert space of square-integrable functions for every Λ(2)-set Λ and every Borel set B with sufficiently small measure.

References: D.L. Donoho, P.B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49 (1989), 906-931G. Fischer, Qualitative Unschärferelationen und Jacobi-Polynome, PhD Thesis, Center of Mathematics, Technische Universität München, 2005V. Havin, B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Springer, New York, 1994

Theory

Georg Fischer


26

Josef Obermaier, Ryszard Szwarc1

In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions. A well-known result in this area is the Weierstrass theorem, which states that continuous functions on compact subsets of the real line can be approximated by polynomials in any accuracy. Since polynomials are one of the simplest functions, this statement is fundamental for approximation theory.

Among polynomials, in turn, orthogonal polynomials play an out-standing role. They were developed in the 19th century from a study of continued fractions by P. L. Chebyshev. Many applications have been evolved from orthogonal polynomials in mathematics and physics. The reason for their adaptability is a relationship which is comparable with orthogonality in the usual geometry. Therefore, approximation by orthogonal polynomials is a branch of orthogonal series, which is closely related to classical Fourier series. As for classical Fourier series, it is a well-known fact that there might be functions which are not approximable by orthogonal series.

Assuming that the support of a function is discrete, we have been able to overcome this problem. For such a case we have proved that any continuous function is represented by an orthogonal series with respect to a special orthogonal polynomial system. As an appli-cation one may think of density estimators for discrete probability measures.

Among classical orthogonal polynomials the class of ultraspherical polynomials has been thoroughly studied. For instance, this class contains the well-known Chebyshev polynomials. Ultraspherical polynomials are determined and characterized as solutions of spe-cial differential equations. Of course, beside a description based on differential equations there are other ones. Such characterizations of a system are useful for discrimination and a deeper understand-ing of its behavior and applicability. Knowing all of these charac-terizations is like knowing the fingerprinting of such a system. We have been able to give a new characterization for ultraspherical polynomial systems, which depends on the representation of the derivation of a polynomial.

approximation of functions by orthogonal polynomials

Theory

References:J. Obermaier, A continuous functions space with a Faber basis, J. Approx. Theory 125 (2003), 303-312J. Obermaier, R. Szwarc, Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel, Constr. Approx. 27 (2008), 1-13 R. Lasser, J. Obermaier, A new characterization of ultraspherical polynomials, Proc. Amer. Math. Soc. 136 (2008), 2493-2498

1 Institute of Mathematics, Wrocław University, Poland

Approximation of a function (black) by an orthogonal series of degree 4 (blue) on a discrete set (orange)

0

1

0 1qq2


27

harmonic analysis based on special functions: theory and application

Modeling of biological systems in structure and dynamics is in general done by describing »transfer« operators that act on function spaces. In image or signal processing, functions and operators have to be analyzed in a way to get best information. So in both application areas – biological system and signal-image processing – one has to deal with the problem of finding or constructing an appropriate class of functions (call them character spaces) that span the whole function space and are the basis for investigating the operators. In general, the search for an adequate character space corresponds to finding eigenfunctions of differential operators or to determining homomorphisms of Banach algebras.

In that way we investigated several classes of orthogonal polynomial sequences and families of eigenfunctions. Prominent examples are the Jacobi polynomials and the Bessel functions. However, there are many further classes of special functions that fit in very well with concrete questions arising from the application, e. g. Cartier-Dunau polynomials for processes or systems on certain graphic structure. An important property of these characters is the fact that they induce generalized translations or even hypergroups.

Having chosen the character space, one has to deal with problems of approximation. We have developed a series of results on approximate units and applied them to the statistical estimation of means, spectral measures or covariance functions.

More recently we investigated the amenability of certain Banach algebras connected with those character spaces. There are several notions of amenability (weak amenability, -amenability and so on). It is rather diffcult to determine whether the given Banach space or Banach algebra satisfies these properties. However, if one could solve this problem, then one would get strong information about the behavior of that function spaces or operator spaces. For example, if the Banach algebra is amenable then the function space generated by the Gelfand transformation is uniformly discrete. For application in signal processing this means that two different signals always can be separated in the »frequency domain«. Another application is the following: If the Banach algebra is -amenable then a large class of autonomous Volterra difference equations has a unique solution and there is an efficient algorithm to determine that solution.

References: V. Hösel, R. Lasser, Approximation with Bernstein-Szegö polynomials, Numer. Funct. Anal. Optim. 27 (2006), 377-389 K. Ey, R. Lasser, Facing linear difference equations through hypergroup methods, J. Difference Equ. Appl. 13 (2007), 953-965R. Lasser, Amenability and weak amenability of -algebras of polynomial hypergroups, Studia Math. 182 (2007), 183-196

Theory

Rupert Lasser


28

Peter Massopust

B-splines play an important role in approximation and interpola-tion theory. In mathematical terms, a B-spline is a function that has minimal support with respect to a given order (which determines its smoothness) and domain partition. A fundamental result states that every spline function, i. e. piecewise polynomial, of a given order (smoothness) and domain partition, can be represented as a linear combination of B-splines of that same order (smoothness) and over that same partition. B-splines possess many important properties, such as recurrence relations, fast algorithms for computing their derivatives and integrals that make them ideal for numerical com-putations and the modeling of natural data.

The order of a B-spline determines its smoothness and support, and by replacing the integral order n by a complex number z defines the wider class of complex B-splines, enjoying most of the properties of ordinary B-splines. However, in addition to these properties, some new features and options are obtained, such as working with the real and imaginary part, and obtaining phase information. Several interesting relations to fractional derivatives and integrals and the so-called Dirichlet average provide a new means of looking at com-plex B-splines.

Several interesting relations between complex B-splines and sto-chastic processes, among them the Poisson-Dirichlet process and the GEM-distribution, both of which have applications to biology, in particular, genetic and population modeling, were exhibited and discussed. The extension of complex B-splines to higher dimen-sions is a current topic of research and shows enormous potential for applications.

complex b-splines

Theory

References:B. Forster, P. Massopust, Multivariate complex B-splines, Proceedings of SPIE, Wavelets XII 6701 (2007), 670109-1 – 670109-9B. Forster, P. Massopust, Some remarks about the connection between fractional divided differences, fractional B-splines, and the Hermite-Genocchi formula, Int. J. Wavelets Multiresolut. Inf. Process. 6, no. 2 (2008), 279–290B. Forster, P. Massopust, Statistical encounters with complex B-splines, to appear in: Constr. Approx.

Real Part

Imaginary Part

3-D View


29

fast algorithms for photoacoustic imaging

Photoacoustic imaging is a promising imaging technique that is especially well suited for low scale analysis of biological tissues, e. g. veins, cancer. The advantage of photoacoustic imaging over computed tomography is its high resolution of about 700 microns and the high contrast in tissues that cannot be distinguished by X-rays.

In photoacoustic imaging biological tissue is irradiated by a pulsed laser beam. The absorption of the laser beam causes expansion within the tissue, and hence a pressure wave is emitted which can be detected outside the body. Assuming the laser illumination to be equal at each position, the amplitude of the emitted pressure wave is proportional to the absorption coefficient of the tissue at the position of emission. Since the absorption coefficient is a distinguishing attribute of different kinds of tissue, the inverse problem in photoacoustic tomography is the computation of the amplitudes of the emitted pressure waves from the pressure waves detected outside the body.

The pressure wave measured at the detector is the superposition of all pressure waves that originate from locations with the same distance to the detector. Hence, the inverse problem of photoacoustic imaging is equivalent to the problem of recovering a function from its mean values along arcs, which is known as inverse spherical Radon transform on the plane. We are interested in the following questions:

— For how many centers and radii do the means along the arcs need to be known, so that that the function can be recovered at a given resolution?

— What is the effect of noise on the reconstruction?— How does the choice of the grid of centers and radii affect the

reconstruction error?

The answers to these questions form the foundation for fast algorithms that computes a three dimensional image from the detected pressure waves.

References: D. Razansky, V. Ntziachristos, Hybrid photoacoustic fluorescence molecular tomography using finite-element-based inversion, Med. Phys. 43, no. 11 (2007), 4293-4301L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems 23 (2007), 11-20

Theory

Ralf Hielscher, Frank Filbir, Daniel Razanski1

1 Institute for Biological and Medical Imaging, Helmholtz Zentrum München

Sketch of photoacoustic experiment. (Source: Wikipedia)

Geometrical setup of the inverse spherical Radon transform: An unknown function on the unit disc has to be recovered from their mean values along arcs with centers at the boundary.


30

Brigitte Forster-Heinlein

A general method in image analysis is the decomposition of an image into its detail information of various sizes. This eases inter-pretability and allows dedicated enhancement of interesting image features, such as small structures in medical tissue images or bio-logical microscopy images. The detail information can be acquired with a multiresolution analysis MRA. The idea behind the MRA is its simple generation and powerful performance: The basis of an MRA consists of discrete shifts and scalings of one single basis function. Due to this fact there are fast and stable algorithms for analysis and the following synthesis. There are many ways to design such an MRA basis. In our research we concentrate on complex-valued bases that allow analyzing phase information.

In this project, our aim was to model rotations into the MRA basis in addition to the usual shifts and scalings. The idea is that rota-tions should only affect the phase of the corresponding coefficients and leave the moduli invariant:

Here denotes the image and its rotation. The basis function behaves approximately as its rotated version, up to a phase

factor .

Our construction is based on a rotating function in the Fourier domain and an adaptable localization such that the frequencies of behave as . In this way, we get flexibility and can con-struct rougher or smoother bases, depending on the requirements of the image analysis task. The bases are suitable for the classical Cartesian grid, as well as for the hexagonal grid. Both geometries are common in camera sensors used for biomedical image acquisi-tion.

multiresolution image analysis and the design of rotating bases

Imaging

References:B. Forster, T. Blu, D. Van De Ville, M. Unser, Shift-invariant spaces from rotation-covariant functions, to appear in: Appl. Comp. Harm. Anal.L. Condat, B. Forster-Heinlein, D. Van De Ville, Rotation-covariant polyharmonic spline wavelets on the hexagonal lattice, SPIE Wavelets XII, August 2007, San Diego, CA, USA

Modulus of a basis function

Rotation covariant bases operate similarly to a multi-scale gradient with the real part corresponding to the y-derivative and the imaginary one to the x-derivative. The same construction applies on the hexagonal grid.


31

adaptive geometries for piecewise smooth image data

This particular aspect of our research is concerned with the extraction of morphological and spatial features from digital image data. Our focus is on geometry-based non-linear approximation methods for the representation of contours and lines in images.

We investigate two different approaches: One method relies on wedgelet approximations (like those suggested by Donoho). Emphasis is on parsimonious and sparse data representations, based on variational approaches. The corresponding estimates are minima of a complexity penalized functional. A central point of this research is the development of real time algorithms for the solution of the corresponding optimization problem.

An alternative method is based on adaptive Delaunay triangulations. Classical spectral decompositions – like Fourier or wavelets – are not optimal for images which contain smooth geometrical objects delimited by regular curves. Our method is a fully non-linear geometric method, relying on adaptive refinements of continuous, piecewise affine functions on Delaunay triangulations. It allows for flexible and sparse representations of the target functions. As an example, we applied the method to image compression. This method outperforms existing standard algorithm like JPEG2000.

References: F. Friedrich, L. Demaret, H. Führ, K. Wicker, Efficient moment computation over polygonal domains with application to rapid wedgelet approximation. SIAM J. Sci. Comput. 29, no. 2 (2007), 842-863L. Demaret, N. Dyn, A. Iske, A delay image compression by linear splines over adaptive triangulations, Signal Processing Journal 86, no. 7 (2006), 1604-1616G. Winkler, Image Analysis, Random Fields and MCMC Methods. Corrected 3rd Printing, Springer, 2006

Imaging

Laurent Demaret, Gerhard Winkler

Wedgelet partition of the image »Lena«. Corresponding approximation is piecewise constant over this partition.

Contour adapted triangulation of the image »Parrots« (see upper right corner). A set of significant vertices is selected recursively by minimization of a local reconstruction error criterion.


32

Stefan Held

Wavelets have proven to be a useful tool in image analysis. The main reason for this is that they decompose images into different scales in which features of different sizes are represented. However, wavelets in two or three dimensions are often constructed as sepa-rable wavelet bases which introduce strong checkerboard artifacts and require three, respectively seven, wavelets per scale in two, respectively, three dimensions to form an invertible transformation.I have developed isotropic wavelet frames that circumvent these restrictions while still yielding an invertible transformation using only one wavelet in each scale.

These wavelets are well suited to implement the monogenic signal. This signal transform decomposes images into three components. The first is the amplitude which is related to the illumination in an image. The second is the phase which contains information about image features independent of illumination and can be used to retrieve a local frequency of the image. The third component is the direction which gives a local direction of the features that can be retrieved by the phase.

The data retrieved by the monogenic signal is hard to analyze because features at different scale interfere. Thus the monogenic signal should be combined with wavelets to circumvent this. The isotropic wavelets I constructed are especially suitable for this task since they do not interfere with the direction to be acquired in combination with the monogenic decomposition.

Applications of these monogenic wavelet frames are, for example, illumination independent edge detection and detection of wave-like structures. The pictures give an example of the illumination independency of the monogenic wavelet frames. The first picture shows the well-known image of the cameraman that was manipulated to have extremely low illumination in the middle. The second image shows a reconstruction using only the information of phase and direction. One can see all image features independently of the low illumina-tion. Additionally many features are far easier to spot, for example, the stars and the buttons on the cloak.

The monogenic wavelet frames have been efficiently implemented in Matlab® code and as an ImageJ plugin.

monogenic wavelet frames for image analysis

Imaging

Reconstruction using only the information of phase and direction. All features can be seen independently of illumination.

Picture of the cameraman manipulated to have low illumination in the middle.


33

diffusion tensor imaging – methods to visualize and quantify nerve fibers in the brain

Diffusion Tensor Imaging (DTI) is based on a novel magnetic resonance technique measuring the diffusion of water molecules in nerve fibers. It can detect their structure and connectivity in the human brain in vivo, indicating diseases like multiple sclerosis or Alzheimer’s. Recent surveys on basic concepts, experimental methods, postprocessing procedures and potential applications can be found in the literature. Two postprocessing problems of DTI were in our focus. First, there is the problem of denoising DTI data with high resolution. Current DTI protocols use voxel sizes of 8-27 mm3

producing systematic errors by averaging within the voxels. Higher resolution is mainly hindered by an increase of thermal noise and by nonlinear noise propagation in the DTI formalism. This produces new systematic errors in the DTI observables, like statistical bias and outliers due to non Gaussian statistics. A statistically based method for denoising data with 1 mm3 voxel size in the human brain was developed and successfully tested by Monte Carlo simulations. A first application to a rat brain study was published recently and enabled for high-field strengths a voxel size of 0.04 mm3. A second topic concerns the calculation of fiber tracks by deterministic and stochastic algorithms to segment the human brain. Segmentation is performed by a coupling to functional fMRI measuring the localisation of specific brain activities. These regions of activity are the seed points for stochastic tracking. Similar activations in different regions may be coupled to similar track geometries in the brain, indicating similar connectivity of those regions with the residual brain. This could be shown in a preliminary study for activations in the visual cortex.

References: S. Mori, Introduction to Diffusion Tensor Imaging, Elsevier, Amsterdam, 2007.K. Hahn, S. Prigarin, S. Heim, K. Hasan, Random noise in DTI, its destructive impact and some corrections, in: J. Weickert, H. Hagen (eds.), Visualization and Processing of Tensor Fields, Springer, Berlin, 2006, 107-119K. H. Bockhorst, P. A. Narayana, R. Liu, P. Ahobilia-Vijjula, J. Ramu, M. Kamel, J. Wosik, T. Bockhorst. K. Hahn, K. M. Hasan, J. R. Perez-Polo, Early postnatal development of rat brain: In vivo diffusion tensor imaging, J. Neurosci. Res. 86 (2008), 1520-1528

Imaging

Klaus Hahn, Sergei Prigarin1, Karsten Rodenacker

Fibers of human corpus callosum calculated by deterministic methods

Fibers in the human brain for different resolutions

1 Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences, Russia


34

Laurent Condat

Larger and larger amounts of digital data are acquired in every field of science. These data are defined on discrete domains, which characterize the acquisition systems used and are typically uniform multi-dimensional lattices. A lattice in N dimensions defined from N independent vectors is the set of locations (points) defined as the linear combinations of these vectors with integer coefficients. In 2-D, the square and hexagonal lattice are the most common.

Resampling procedures come into question when converting data from one lattice to another. For instance, it may be desirable to convert data from one lattice, constrained by the geometry of the particular acquisition device used, onto the square lattice, for stor-ing or visualization purposes. Or one may wish to convert an image defined on the square lattice onto the hexagonal lattice, in order to benefit from the well-known theoretical advantages of hexagonal sampling, including its superior geometrical and topological prop-erties. Geometrical operations like image rotation or registration also require a resampling step.

I proposed a new resampling method from one lattice to anoth-er one (with the same sampling density), with the highly praised property of reversibility. That is, no loss of information occurs dur-ing the resampling process, and the initial data may be perfectly recovered from the converted data afterwards. The idea is to de-compose the operation in shears along the 1-D directions of the lattice. This generic approach can be applied in any dimension and for every pair of lattices. It amounts, algebraically, to factorizing a particular matrix. The practical resampling process reverts to per-forming 1-D convolutions along the rows and columns of the data, using well-suited digital filters. The method also ensures that no blur is introduced during the conversion, contrary to classical in-terpolation methods. Hence, it combines reversibility with speed of computation, ease of implementation, and high-quality. This work may foster a renewed interest for developing analysis and processing tools that process data directly on arbitrary lattices, with the guarantee that the resampling step can be performed ef-ficiently by the present approach.

a reversible and efficient approach for converting data between lattices

Imaging

Example of conversion from the square to the hexagonal lattice : in (a) the initial image, in (b) resampled using classical bicubic interpolation, in (c) with our reversible method. The image (b) is slightly blurred, (c) is not. (a) can be recovered from (c), not from (b).

References:L. Condat, D. Van de Ville, B. Forster-Heinlein, Reversible, fast, and high-quality grid conversions, IEEE Trans. Image Process. 17, no. 5 (2008), 679-693

The conversion from the hexagonal to the square lattice, or conversely, can be decomposed into three shearing operations along a particular direction of the lattice.

The hexagonal lattice (a) and the square lattice (b), with the same sampling sensitivity.


35

digital image analysis and metrology

The step from visual comparison of pictorial data to quantitative discrimination is not straight forward. The extraction of quantitative features from digital picture data exceeding the estimation of mere amounts necessitates the tight collaboration of scientists and image analysts. Perceptions and conjectures have to be translated into feature extraction procedures, and the resulting feature values are then statistically tested. Experienced image analysts and statisticians are working and collaborating at the Institute of Biomathematics and Biometry on a multitude of projects to develop those new analysis methods.

Cytometry: As one of the oldest fields of digital image analysis the quantification of cellular or nuclear properties, nowadays called cytometry, is widely used in pathology and cell biology.

µHistometry: As an example of quantitative morphology applied to sections of animal nerves, we outlined some possibilities of morphometry of single nerve fiber sections. This includes definitions of meta structures showing neighborhoods of nerve fibers in bundles.

Fluorimetry and Colorimetry: With progress in biochemical and molecular biological marker technology, the quantification of fluorescent and/or colour signals in terms of amount and location (intensity, distribution and spatial arrangement) has become important for the adequate evaluation of data. An example from molecular pathology: Here, the images were originally gathered in 3D by optical sections from a confocal laser scanning microscope (stained e.g. by FISH). Marker locations have shown their value in the grading of diseases.

Plankton Analysis: An automatic system for data gathering and image analysis to estimate the community structure has been developed. Morphometric (shape), photometric (optical density), colorimetric and fluorimetric features of detected objects allow to automatically recognize plankton organisms. For training purposes a graphical user interface (GUI) was realized to display single organisms, groups, as well as quantitative features under different aspects.

In summary, the examples illustrate how image analysis and metrology yield both support for simple estimations of amounts in images as well as complicated and complex estimation tasks.

References: M. Hughes-Fulford, K. Rodenacker, U. Jütting, Reduction of anabolic signals and alteration of osteoblast nuclear morphology in microgravity, J. Cell. Biochem. 99, no. 2 (2006), 435–449K. Matiasek, P. Gais, K. Rodenacker, U. Jütting, J. J. Tanck, W. Schmahl, Stereological characteristics of the equine accessory nerve, Anat. Hist. Embryol. 37, no. 3 (2008), 205–213K. Rodenacker, E. Bengtsson, A feature set for cytometry on digitized microscopic images, Anal. Cell. Pathol. 25, no. 1 (2003), 1–36

Imaging

Karsten Rodenacker

Cytometry: Feature extraction for one cell nucleus (osteoblast), original upper left corner, mostly dedicated to intra-nuclear texture

μHistometry: The accessory nerve of the horse allows systematic morphometric studies of nerve function and growth

Fluorimetry and Colorimetry: An automatically processed image with nuclear DNA in blue and the FISH signals in red and green. Segmentation of signals allows beside counting a quantification of the spatial arrangement of the FISH signals.


36

Burkhard A. Hense, Karsten Rodenacker, Uta Jütting, Hagen Scherb, Peter Gais1

Information about ecosystem composition is indispensable for many eco(toxico)logical studies. Identification and quantification of microscopic organisms, e. g. algae, using high taxonomic resolu-tion require time and expert knowledge and are thus expensive. Therefore, a semiautomatic method was developed, called PLASA. It is based on image analysis for investigation of phytoplankton in complex environmental samples, covering all steps from image gathering to data quantification.

Species diversity and structural complexity of the samples from aquatic ecosystems make great demands on image analysis and classifier design. Integration of autofluorescence information sub-stantially improves discrimination of algae from other objects.

PLASA is especially suitable for processing of a number of similar samples (e. g., time series of lake or microcosm studies). It com-bines the advantages of manual counting (high taxonomical reso-lution) with those of flow cytometry (automation, objectivity). The optical fixation is, in contrast to chemical fixation more sustainable, making it suitable for later analysis of new questions or quality assurance.

Water samples are processed as usual (Utermöhl method) and scanned in a computer-controlled inverse microscope with a digital color camera, using different objectives. The images are digitally corrected (shading + color correction). Subsequent image analysis steps (segmentation, measurement, classification) are provided for algae species evaluation.

automated investigation of phytoplankton structure by image analysis

Imaging

References:K. Rodenacker, B. A. Hense, U. Jütting, P. Gais, Automatic analysis of aqueous specimens for phytoplankton structure recognition and population estimation, Microsc. Res. Techniq. 69 (2006), 708-720B. A. Hense, P. Gais, U. Jütting, H. Scherb, K. Rodenacker, Use of fluorescence information for automated phytoplankton investigation by image analysis, J. Plankton Res. 30 (2008), 587-606

Image analysis steps

1 Institute of Pathology, Helmholtz Zentrum München

Bright field image of algae with contour; small insert left: chlorophyll auto-fluorescence; small insert right: phycoerythrin auto-fluorescence


37

extraction of morphological features from time series

We focus on the extraction of morphological and temporal rather than quantitative features. Emphasis is on parsimonious and sparse data representations, based on variational approaches with complexity-penalized likelihood functions. Real-time algorithms for the solution of the respective optimization problems are a main part of this kind of research.

In particular, we study representations of time series which are minimal points of the most simple complexity-penalized functional of the following form:

The symbol denotes the time series of data and is a candidate for their representation. The notation indicates that is the time immediately after . Minimal points provide the proper balance between smoothness requirements and fidelity to the data and are thus used as reasonable estimates (»complexity-penalized M-estimation«).

We have developed a software called AntsInFields (freely available under http://www.antsinfields.de). Among excellent and convenient interactive graphical tools, and tools for data management and handling, it contains an extremely fast algorithm for the optimization of the complexity -penaliszed functional above. This enables us to process and evaluate data interactively and online.

As an example, we consider a boxcar-shaped stimulus used for example in functional magnetic resonance imaging (fMRI). It consists of intensities along 72 time points (light on-light off, finger tipping etc.), representing the answers in one single voxel of the visual cortex of a human brain. Estimates then are obtained from complexity-penalized M-estimation. The result clearly depends on the choice of the value of in the functional . Obviously not every value of leads to a proper estimate. The identification of such hyperparameters is one of the most challenging problems in our future work.

Our scientific program, both in the past and for the future, overarches the study of complexity penalized estimators, from rigorous analytical analysis, statistical analysis up to fast algorithms. It further contains work showing the applicability for general image analysis, statistical testing, and classification of micro arrays.

References: G. Winkler, V. Liebscher, Smoothers for discontinuous signals, J. Nonpar. Statist. 14, no. 1-2 (2002), 203-222O. Wittich, A. Kempe, G. Winkler, V. Liebscher, Compexity penalised least squares estimators: analytical results, Math. Nachr. 281, no. 4 (2008), 1-14L. Boysen, V. Liebscher, A. Munk, O. Wittich, Jump-penalized least squares: Consistencies and rates of convergence, to appear in: Ann. Stat.

Imaging

Gerhard Winkler

Two examples of panels from the software AntsInFields. The left panel deals with responses to boxcar-shaped visual stimuli in fMRI (please see below). The right one is from quality control for microarrays. The latter topic is not addressed in the present note.

Stimulus in fMR imaging

Complextity-penalized M-estimates for three different values of the hyperparameter . The left plot shows the correct one.


38

Moritz Simon, Galliano Valent1

Birth-and-death processes with killing are a special class of Markov processes; they allow stochastic modeling of the size of »popula-tions« in view of the respective probabilities that an individuum is born after a certain time – corresponding to a birth rate – and that an individuum dies – corresponding to a death rate. The additional structure of killing accounts for the possibility of immediate extinc-tion of the population at hand – corresponding to a so-called killing rate; all the rates solely depend on the population size, not on the time of evolution. Many populations and systems may be modeled in this way.

The fundamental guideline in our work was a spectral representa-tion for the transition probabilities of such population processes: They can be related explicitly to a system of orthogonal polynomi-als (OP) with an appropriate spectral measure. In principle, given that the OP and their measure can be computed, one can derive quite explicit results on the expected behavior of the correspond-ing population. We achieved this in great detail for processes with linear rates which, for instance, are important in several genetic models; the OP related to such linear processes are scaled versions of so-called Meixner polynomials.

Problematic is the following aspect: As soon as the rates of the process are sufficiently complicated, the OP and their spectral mea-sure may no longer be computed explicitly. Hence we were mainly interested in approximate and qualitative results on the »spectra« of such processes. The main tool we utilized is regular perturbation theory for corresponding linear Jacobi operators, which is especial-ly useful under a certain domination of killing. In fact, such »killing dominated« birth-and-death processes –precisely elucidated in the PhD thesis – allow to derive good approximations and criteria for their spectra. Many examples and specifications were considered in this respect, which also shed some new light on known systems of OP such as Lommel polynomials or generalized Chebyshev poly-nomials.

spectral theory of birth-and-death processes

Statistics

Reference:Moritz Simon, Spectral Theory of Birth-and-Death Processes – Explicit Methods with Examples and Perturbative Approaches under Domination of Killing, PhD Thesis, Technische Universität München, Sierke Verlag, Göttingen, 2008

1 Université Paris 7, France


39

learning from similarity – a kernel as inductive bias

Learning functions from a finite set of data generally constitutes an ill-conditioned problem. Posing assumptions on the underlying dependency can lead to algorithms providing a unique solution. Even more, typical interpolation or more generally, regression algorithms provide solutions which are best in a sense to be defined ahead.

Kernel-based methods infer knowledge about the underlying dependency based on similarity information. In this sense, every positive definite kernel defines an a priori assumption, measuring similarity. Thus, the kernel can be considered a model for the inductive bias every learning procedure needs in order to efficiently manage learning tasks. Therefore, we need to construct kernels which at the same time model any a priori knowledge of similarity we have concerning the data, and are computationally efficient to evaluate.

For data given in the d-dimensional Euclidean space, the classical p-norms provide a natural notion of proximity. Functions depending on the p-norm of their arguments only therefore lead to natural similarity measures for such data. In the theories of radial basis function approximation, classical geostatistics or radial basis function networks, the Euclidean norm is commonly used. Radial kernels are translation invariant. This allows using theorems of harmonic analysis to characterize the class of all positive definite kernels of such type.

Nevertheless, thinking of p as a parameter can be useful for modeling purposes. For example, 1-radial functions are more easily implemented in VLSI and may thus be preferred for integrated circuit design. Other adaptations can be obtained by using suitable transformations of the d-dimensional space, leading to anisotropic kernels. From the theoretical point of view, values of p in the range from one to two provide an interesting mélange of harmonic analysis governed by group actions and averaging processes leading to p-radiality. Thus, they allow to carefully distinguish mathematical features resulting from different theoretical background.

The sketched framework can be enlarged to cope with data from other spaces (e. g. projective spaces, manifolds or graphs), or to incorporate other useful properties (e. g. minimization of energy functionals, reproduction properties or combination of attributes).

References: W. zu Castell, Interpolation with reflection invariant functions, in: C.K. Chui, M. Neamtu, L. L. Schumaker (eds.), Approximation Theory IX: Gatlinburg 2004, Nashboro Press, Brentwood, 2005, 105-120W. zu Castell, S. Schrödl, T. Seifert, Volume interpolation of CT images from tree trunks, Plant Biology 7 (2005), 737-744W. zu Castell, Generalized Bessel functions for p-radial functions, Constr. Approx. 27, no. 2 (2008), 217-235

Georg Berschneider, Wolfgang zu Castell, Stefan Schrödl

Contour plot of compactly supported p-radial function for different values of p

Interpolation of CT-slice of tree trunk with anisotropic thin plate spline


Statistics

40

Georg Berschneider

Machine Learning aims at designing and analyzing algorithms, which try to mimic human learning processes. Out of a finite set of examples the method tries to find hypotheses that explain the data best, in a sense that has to be specified, and provides procedures to transfer the obtained knowledge to new data.

Kernel-based methods constitute a large family of algorithms that embed the available data into a high-dimensional space, called feature space, in order to solve the stated problem (more) easily without having actually to work within that space. This is realized via positive definite kernels that define the geometry of the feature space and implicitly embed the data into it.

Since the defining conditions for positive definiteness are quite strict, the class of positive definite kernels is relatively small. Fur-thermore, the associated feature spaces are also small, which re-sults in kernel-methods lacking certain properties, like, e. g., repro-duction of certain function classes, useful for integrating additional information. Thus, conditionally positive definiteness is studied, a concept that generalizes positive definiteness. The Gaussian hat function and the thin-plate spline in finite dimensional Euclidean space may serve as typical examples for (stationary) positive defi-nite and conditionally positive definite kernels respectively.

Since the generalization ability of the method at hand relies on how well a priori information available on the data is taken into account, one has to gain a deeper understanding of the class of admissible conditionally positive definite kernels in order to develop meth-odologies to construct problem-adapted kernels. As an example consider relational data, represented as graphs and a prediction calculated by an, in this case, positive definite kernel that takes the three components of the graph into account.

A first step in this direction is the study of the relationship between conditionally positive definite kernels and so-called reproducing kernel Pontryagin spaces, since spaces of that kind constitute the choice of realizable hypotheses for a given kernel. This relation generalizes the well-studied connection of positive definite kernels and reproducing kernel Hilbert spaces.

learning with conditionally positive definite kernels

Statistics

References:G. Berschneider, W. zu Castell, Conditionally positive definite kernels and Pontryagin spaces, in: L. L. Schumaker, M. Neamtu (eds.), Approximation Theory XII: San Antonio 2007, Nashboro Press, Brentwood, 2008, 27-37

Graph with data given at colored nodes

Prediction on graph (color codes values)


41

multi-class svm learning for electrocardiographic data analysis

Electrocardiographic (ECG) signals are probably one of the best-known signals in medical diagnostics, also used in medical research. Although these signals are easily accessible, they are not always easy to interpret due to their variability and complexity in the time and frequency domain.

Nevertheless, physicians do interpret and classify ECG signals. Their expert knowledge is a result of a learning process. The identification of certain patterns of oscillations within the signal is a result commonly learned by examples. Our aim is to transfer the human learning process into a mathematical framework. Therefore, we want to develop learning algorithms, which are capable of performing multi-class classification of ECG signals and associated with that, support human experts.

In general, signals can be analyzed directly without any frequency information and they can be decomposed into characteristic components via Fourier transform, Wavelet transform and other time-frequency transforms. While the Fourier representation of a signal has only a good resolution in frequency domain, the wavelet approach is lacking an interpretation as harmonic oscillations. On the contrary, the short-time Fourier transform is a way to analyze time-localized frequencies in a signal. The idea is to use this representation of information in a signal as an input for multi-class support vector machines (SVM) in order to solve the classification tasks.

Our multi-class SVM approach is based on matrix-valued kernels. These functions allow a great variability in model design in order to cope with interclass dependencies. Further analysis of the properties of matrix-valued kernels enables us to propose a method for constructing a rich class of these kernels from scalar-valued ones. Moreover, we extended the underlying theory in order to derive qualitative error estimates in the Hilbert spaces, which form the basis of the mathematical framework for the learning processes.

Reference: C. A. Micchelli, M. Pontil, Kernels for multi-task learning, in: L. K. Saul, Y. Weiss, L. Bottou (eds.), Advances in Neural Information Processing Systems 17, MIT Press, 2005, 921-928

Stefan Schrödl, Rick Beatson1, Wolfgang zu Castell

Test signal with varying frequency and quadratic absolute values of its short-time Fourier coefficients

ECG signal and quadratic absolute values of its short-time Fourier coefficients

1 University of Canterbury, Christchurch, New Zealand


Statistics

42

Salim Lardjane

In a variety of survey settings, the respondents are asked to assess their own status with respect to an issue of interest by answering a self-assessment question using ordinal response categories. Such procedures occur most notably in subjective health surveys, politi-cal surveys with a human rights focus, psychometric surveys, and quality-of-life surveys.

When the ordinal response categories are used differently by the respondents, the responses are said to be affected by a DIF (Dif-ferential Item Functioning) effect. In such situations, the responses need to be corrected before they can be compared.

The anchoring vignettes methodology builds on recent works by King and necessarily requires that : (i) each respondent assesses a number of imaginary situations called »vignettes«, (ii) the issue under consideration is one-dimensional and well-identified by all the respondents, (iii) the respondents assess the vignettes and their own situation in the same way, and (iv) the vignettes can be ranked without any ambiguity by the analyst.

A DIF effect can be detected by comparing the vignette assessments‘ frequency distributions in different groups. It can be corrected by constructing a new ordinal variable reflecting the respondents‘ status by »anchoring« their main responses to their vignettes‘ as-sessments. This variable can be used to compare the status of any two respondents or groups with respect to the issue under consid-eration. Consistency problems in the vignettes‘ assessments can be dealt with by introducing uncertainty intervals for the values as-sumed by this variable.

This approach was used to detect a possible DIF effect in the self-assessment of physical pain, according to the geographical origin of the samples from the Share 2004 survey, which would lead in particular to modify the conclusions of a comparison between the subjective health status of the Share 2004 Swedish and Dutch sam-ples.

enhancing the comparability of survey results using anchoring vignettes

Statistics

References:G. King, C. J. L. Murray, J. A. Salomon, A. Tandon, Enhancing the validity and cross-cultural comparability of measurement in survey research (corrected version), Amer. Political Sci. Rev. 98, no. 1 (2004), 191-207G. King, J. Wand, Comparing incomparable survey responses: New tools for anchoring vignettes, to appear in: Political Anal.S. Lardjane, P. Dourgnon, Les comparaisons internationales d‘état de santé subjectifsont-elles pertinentes ? Une évaluation par la méthode des vignettes-étalons, Economie et Statistique 403-404 (2007), 165-177


43

biostatistics – statistical consultation – environmental health research

The main focus of biostatistics and statistical consultation in the Institute of Biomathematics and Biometry (IBB) is to develop statistics as an integral part in interdisciplinary research. This requires statisticians to become emancipated partners with scientists and engineers in specific fields. Statistics, understood as a technology, is a fundamental element of most other sciences. It is aimed at extracting relevant information from data under the constraints of variability and uncertainty. Statistics plays a major role in the environmental sciences, especially in epidemiology and environmental health research. Statistical consultation within the IBB has contributed to the most recent publications listed below. The diagram presents the long term male birth proportion in several European countries combined, before and after the Chernobyl nuclear power plant accident in April 1986. This result has also been corroborated by an analytical ecological epidemiological study in that a significant spatial-temporal association between fallout and the sex ratio could be established. The example shows that reproduction is an error-prone process which is highly sensitive to the effect of exogenous factors, particularly around conception. This has practical consequences for environmental protection and genetic counseling under the aspect of risk avoidance and primary prevention.

References: T. Schulz-Mirbach, H. Scherb, B. Reichenbacher, Are hybridization and poly-ploidization phenomena detectable in the fossil record? – A case study on otoliths of a natural hybrid, Poecilia formosa (Teleostei: Poeciliidae), to appear in: Neues Jahrb. Geol. P.-A. A. Mayer, D. Bulian, H. Scherb, M. Hrabé de Angelis, J. Schmidt, E. Mahabir, Emergency prevention of extinction of a transgenic allele in a less-fertile transgenic mouse line by crossing with an inbred or out bred mouse strain coupled with assisted reproductive technologies, Reprod. Fert. Develop. 19, no. 8 (2007), 984–994H. Scherb, K. Voigt, Trends in the human sex odds at birth in Europe and the Chernobyl nuclear power plant accident, Reprod. Toxicol. 23, no. 4 (2007), 593-599

Hagen Scherb

Male birth proportions for the Czech Republic, Denmark, Finland, Germany, Hungary, Norway, Poland, and Sweden combined (CDFGHNPS) and for Bavaria, the former GDR, and West Berlin combined (BGW) from 1982 to 1992 including jump-models in logistic regression (solid lines)


Statistics

44

Uta Jütting, Axel Walch1, Sandra Rauser1, Daniel Veith1, Heinz Höfler1

The roles of tumor-infiltrating immune cells, EGFR and HER2 in Barrett’s cancer (BCA) are unknown. The aim of this study was to examine their prognostic impact on the outcome of patients with primarily resected BCAs (n=119) as well of BCA patients after neo-adjuvant chemotherapy (n=90).

The levels of the adaptive immune markers CD3, CD8 and CD45RO were analyzed by immunohistochemistry, the expression status of EGFR and HER2 were measured in tissue microarray sections us-ing immunohistochemistry and image analysis. The EGFR and HER2 gene copy number was evaluated by Fluorescence in situ hybrid-ization. All findings were correlated with pathological and clinical parameters including patient outcome. The presence of high levels of tumor-infiltrating immune cells was statistically significantly correlated with prolonged sur-vival in primarily resected BCA patients (Overall survival: CD3: p = 0.0097; CD45: p = 0.0152) as well in BCA patients, who underwent chemotherapy (CD3: p = 0.0119; CD45: p = 0.0318; CD8: p = 0.0098). Moreover, high numbers of immune cells were associated with therapy response. Gene copy number changes of EGFR were cor-related with therapy resistance (p < 0.01) and EGFR overexpression with poorer overall and disease free survival (p = 0.004) in BCA pa-tients after neoadjuvant therapy.

Signs of an immune response are associated with prolonged surviv-al both in primarily resected BCAs and in BCA patients who under-went neoadjuvant chemotherapy. These data support the hypoth-esis that the adaptive immune response influences the behavior of human tumors. In situ analysis of tumor-infiltrating immune cells may therefore be a valuable prognostic tool in the treatment of BCA. EGFR and HER2 may be important molecular markers for therapy resistance.

type and density of immune cells, egfr and her2 within barrett’s cancer predict clinical outcome

Statistics

References:B. A. Hense, P. Gais, U. Jütting, H. Scherb, K. Rodenacker, Use of fluorescence information for automated phytoplankton investigation by image analysis, J. Plankton Res. 30, no. 5 (2008), 587-606K. Matiasek, P. Gais, K. Rodenacker, U. Jütting, J.J. Tanck, W. Schmahl, Stereological characteristics of the equine accessory nerve, Anat. Histol. Embryol. 37, no. 3 (2008), 205-213S. Lassmann, Y. Shen, U. Jütting, P. Wiehle, A. Walch, G. Gitsch, A. Hasenburg, M. Werner, Predictive value of Aurora-A/STK15 expression for late stage epithelial ovarian cancer patients treated by adjuvant chemotherapy, Clin. Cancer Res. 13, no. 14 (2007), 4083-4091

1 Institute of Pathology, Helmholtz Zentrum München

Immunhistochemistry

Kaplan-Meier survival curves


45

data-analysis of environmental chemicals and their databases

Modern databases provide an overwhelming source of information. Structuring, aggregating and weighing information from different sources demands the application of advanced data analyzing methods. Based on partial orderings, a graphical representation via Hasse diagrams allows carrying out such investigations. The Hasse diagram technique originates in discrete mathematics. The software packages WHASSE and their integrated multicriteria decision tool named METEOR (Method of Evaluation by Order Theory) are used to evaluate the data availability of environmentally relevant parameters of chemicals, e.g. pharmaceuticals, endocrine disruptors, alkanes and pesticides. In one of our recent studies we evaluated the data-availability of 16 well-known and highly produced pharmaceuticals (objects) in 21 Internet databases (attributes). By means of the attributes a partial order was derived. In the subsequent steps attributes were aggregated by a weighting procedure, allowing a high degree of involvement of experts, stakeholders and other participants. The data situation on the chosen test set of 16 pharmaceuticals was shown to be far from satisfactory. For the two well-known pharmaceuticals roxithomycin (ROX) (antibiotic), and diatrizoate (DIT) (contrast media) the data situation was extremely poor, irrespective of how the weighting was performed. The data availability for diatrizoate (DAP) was somewhat better. The best data coverage was shown to be for carbamazepine (CAR), diazepam (DIA), ethinyl estradiol (EES), 5-fluorouracil (FLU), and phenazone (PHE).

References: K. Voigt, R. Brüggemann, Water contamination with pharmaceuticals: data availability and evaluation approach with Hasse diagram technique and METEOR, MATCH Comm. Math. Comput. Chem. 54, no. 3 (2005), 671-689K. Voigt, R. Brüggemann, S. Pudenz, A multi-criteria evaluation of environmental databases using the Hasse diagram technique (ProRank) software, Environ. Model. Software 21 (2006), 1587-1597K. Voigt, R. Brüggemann, Ranking of pharmaceuticals detected in the environment: aggregation and weighting procedures, to appear in: Combin. Chem. High Throughput Screening

Kristina Voigt, Rainer Brüggemann1

WHASSE program diagram

1 Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Germany

Hasse diagram of 16x21 data-matrix (equivalent objects: {DAP;EES} {DIC;MET})


Statistics

46

Mathematics is not a careful march down a well-cleared highway, but a journey into

a strange wilderness, where the explorers often get lost. Rigour should be a signal

to the historian that the maps have been made, and the real explorers have gone

elsewhere.

(W. S. Anglin)

the outcomePublications – Projects – Teaching since 2005

Helmholtz Zentrum München – IBB: the outcome

47

Publications 2008

1. Journal Articles

K. Bink, E. Haralambleva, M. Kremer, G. Ott, C. Beham-Schmid, L. de Leval, S.C. Peh, H. Laeng, U. Jütting, P. Hutzler, L. Quintanilla-Martinez, F. Fend, Primary extramedullary plasmacytoma: similarities with and differences from multiple myeloma revealed by interphase cytogenetics, Haematol - Hematol. J. 93, no. 4 (2008), 623-626

K. Bockhorst, R. Liu, P. Ahobila-Vijjula, J. Ramu, M. Kamel, J. Wosik, T. Bockhorst, K. Hahn, K. Hasan, J. Perez-Polo, P. Narayana, Early postnatal development of rat brain: In vivo diffusion tensor imaging, J. Neurosci. Res. 86 (2008), 1520-1528

W. zu Castell, Generalized Bessel functions for p-radial functions, Constr. Approx. 27, no. 2 (2008), 217-235

C. Chauvin, S. Lardjane, Decision making and strategies in an interaction situation: collision avoidance at sea, to appear in Transportation Res. Part F

L. Condat, D. Van De Ville, B. Forster, Reversible, fast, and high-quality grid conversions, IEEE Trans. Image Process., 17, no. 5 (2008), 679-693

R. Czaja, M.A. Efendiev, A note on attractors with fractal dimension, to appear in Bull. London Math. Soc.

L. Demaret, H. Eberl, M. A. Efendiev, R. Lasser, Analysis and simulation of a meso-scale model of diffusive resistance of bacterial biofilms to penetration of antibiotics, Adv. Math. Sci. Appl. 18, no. 1 (2008), 269-304

L. Demaret, M. A. Efendiev, On the structure of attractors for a class of biofilm models, Adv. Math. Sci. Appl. 18, no. 2 (2008), 1-12

H. Eberl, M. A. Efendiev, On a dimension of exponential attractors for some class of biofilm modelling, to appear in Discr. Cont. Dyn. Sys.

H. Eberl, M. A. Efendiev, P. Malosczewski, L. Demaret, Hydrodynamics and biofilms in a porous media, to appear in Electron. J. Differential Equ.

M. A. Efendiev, H.J. Eberl, S.V. Zelik, Existence and longtime behaviour of solutions of a nonlinear reaction-diffusion system arising in the modelling of biofilms, to appear in Comm. Pure Appl. Anal.

M. A. Efendiev, C. Garbou, P. Fabrie, Relaxed model for the hysteresis in micromagnetism, to appear in J. Royal Soc. Edinburgh

M. A. Efendiev, J. Müller, Running fronts for fast diffusion, to appear in Adv. Math. Sci. Appl.

M. A. Efendiev, E. Nakaguchi, On a new dimension estimate of the global attractor for the chemotaxis-growth system, Osaka J. Math. 45, no. 2 (2008), 1-9

M. A. Efendiev, E. Nakaguchi, K. Osaki, On the polynomial dimension estimate of exponential attractors for chemotaxis-growth system, Glasgow J. Math. 50 (2008), 1-25

M. A. Efendiev, E. Nakaguchi, W. L. Wendland, Dimension estimate of the global attractor for a semi-discretized chemotaxis-growth system by conservative upwind finite-element scheme, to appear in Math. Anal. Appl.

M. A. Efendiev, S. Zelik, Finite dimensional attractors for a class of degenerated doubly nonlinear equations, to appear in Math. Methods Appl. Sci.

M. A. Efendiev, S. Zelik, Finite and infinite dimensional attractors for porous media equations, Proc. London Math. Soc. 95, no. 3 (2008), 51-77

W. Erb, F. Filbir, Approximation by positive definite functions on compact groups, to appear in Numer. Funct. Anal. Optim.

K. Ey, C. Pötzsche, Asymptotic behavior of recursions via fixed point theory, J. Math. Anal. Appl. 337, no. 2 (2008), 1125-1141

F. Filbir, D. Schmid, Stability results for approximation by positive definite functions on SO(3), to appear in J. Approx. Theory

F. Filbir, W. Themistoclakis, Polynomial approximation on the sphere using scattered data, Math. Nachr. 281 (2008), 650-668

B. Forster, T. Blu, D. Van De Ville, M. Unser, Shift-invariant spaces from rotation-covariant functions, to appear in Appl. Comput. Harmon. Anal.

B. Forster, P. Massopust, Statistical encounters with complex B-splines, to appear in Constr. Approx.

B. Forster, P. Massopust, Some remarks about the connection between fractional divided differences, fractional B-splines, and the Hermite-Genocchi formula, Int. J. Wavelets Multires. Inf. Process. 6, no. 2 (2008), 279 -290

F. Friedrich, A. Kempe, V. Liebscher, G. Winkler, Complexity penalized M-estimation: Fast computation, J. Comput. Graph. Stat. 17, no. 1 (2008), 1-24

H. Führ, Simultaneous estimates for vector-valued Gabor frames of Hermite functions, to appear in Adv. Comput. Math.

D. Geller, A. Mayeli, Continuous wavelets on manifolds, to appear in Math. Z.

D. Geller, A. Mayeli, Nearly tight frames and space-frequency analysis on compact manifolds, to appear in Math. Z.

B. A. Hense, P. Gais, U. Jütting, H. Scherb, K. Rodenacker, Use of fluorescence information for automated phytoplankton investigation by image analysis, J. Plankton Res. 30 (2008), 587-606

B. A. Hense, W. Jaser, G. Welzl, G. Pfister, G. F. Wöhler-Moorhoff, K.-W. Schramm, Changes in phytoplankton continuously exposed to 17α-ethinylestradiol in lentic microcosms, Ecotox. Environ. Safe. 69 (2008), 453-465

B. A. Hense, C. Kuttler, J. Müller, M. Rothballer, A. Hartmann, J.-U. Kreft, Efficiency Sensing – was messen Autoinduktoren wirklich?, Biospektrum 1 (2008), 18-21

F. Heymann, R. Lasser, Convolution structure of the Bessel transform and the generalization of a theorem of Watson, to appear in Integral Transform. Spec. Funct.

K. Jordaan, F. Tookos, Convexity of the zeros of some orthogonal polynomials and related functions, to appear in J. Comput. Appl. Math.

C. Kuttler, B. A. Hense, Interplay of two quorum sensing regulation systems of Vibrio fischeri, J. Theor. Biol. 251 (2008), 167-180

S. Lardjane, On linear dependence to the initial state for a class of nonstationary cryptodeterministic processes, to appear in Intern. J. Nonlinear Sci.

R. Lasser, Point derivations on the L1-algebra of polynomial hypergroups, to appear in J. London Math. Soc.

R. Lasser, J. Obermaier, A new characterization of ultraspherical polynomials, Proc. Amer. Math. Soc. 136 (2008), 2493-2498

K. Matiasek, P. Gais, K. Rodenacker, U. Jütting, J. J. Tanck, W. Schmahl, Stereological characteristics of the equine accessory nerve, Anat. Histol. Embryol. 37, no. 3 (2008), 205-213

J. Müller, S. Brandt, K. Mayerhofer, T. Tjardes, M. Maegele, Tolerance and threshold in the extrinsic coagulation system, Math. Biosci. 211 (2008), 226-254

J. Müller, C. Kuttler, B. A. Hense, Sensitivity of the quorum sensing system is achieved by low pass filtering, Biosystems 92 (2008), 76-81

J. Obermaier, R. Szwarc, Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel, Constr. Approx. 27, no. 1 (2008), 1-13


48

S. Prigarin, K. Hahn, G. Winkler, Comparative analysis of two numerical methods to measure Hausdorff dimension of the fractional Brownian motion, Numer. Anal. Appl. 1-2 (2008), 163-178

R. Reiter, P. Gais, M. K. Steuer-Vogt, A.-L. Boulesteix, R. Hampel, S. Wagenpfeil, S. Rauser, A. Walch, K. Bink, U. Jütting, F. Neff, W. Arnold, H. Höfler, A. Pickhard, Centrosome abnormalities are correlated with poor prognosis in head and neck squamous cell carcinoma (HNSCC), to appear in Acta Oto-Laryngol.

R. Schlicht, G. Winkler, A delay stochastic process with applications in molecular biology, to appear in J. Math. Biol.

D. Schmid, Marcinkiewicz-Zygmund inequalities and polynomial approximation from scattered data on SO(3), to appear in Numer. Funct. Anal. Optim.

K.-W. Schramm, W. Jaser, G. Welzl, G. Pfister, G. F. Wöhler-Moorhoff, B. A. Hense, Effect of 17α-ethinylestradiol on zooplankton and abiotic variables in aquatic microcosms, Ecotox. Environ. Safe. 69 (2008), 437-452

T. Schulz-Mirbach, H. Scherb, B. Reichenbacher, Are hybridization and polyploidization phenomena detectable in the fossil record? – A case study on otoliths of a natural hybrid, Poecilia formosa (Teleostei: Poeciliidae), to appear in Neues Jahrb. Geol. P.-A.

M. Simon, A. Ruffing, Analytic aspects of q-delayed exponentials: Minimal growth, negative zeros and basic ghost states, J. Differ. Equations Appl. 14, no. 4 (2008), 347-366

M. Simon, S. Suslov, Expansion of analytic functions in q-orthogonal polynomials, to appear in Ramanujan J.

O. Wittich, A. Kempe, G. Winkler, V. Liebscher, Complexity penalized least squares estimators: Analytical results, Math. Nachr. 281, no. 4 (2008), 582-595

S. Zeiser, O. Rivera, C. Kuttler, B. A Hense, R. Lasser, G. Winkler, Oscillation of Hes7 caused by negative autoregulation and ubiquitination, Comput. Biol. Chem. 32 (2008), 48-52

2. Proceedings and Book Chapters

G. Berschneider, W. zu Castell, Conditionally positive definite kernels and Pontryagin spaces, in: M. Neamtu, L. L. Schumaker (eds.), Approximation Theory XII: San Antonio 2007, Nashboro Press, Brentwood, 2008, 27-37

L. Condat, D. Van De Ville, New optimized spline functions for interpolation on the hexagonal lattice, to appear in the Proc. of IEEE ICIP, Oct. 2008, San Diego, US

L. Condat, D. Van De Ville, Fully reversible image rotation by 1-D filtering, to appear in the Proc. of IEEE ICIP, Oct. 2008, San Diego, US

D. R. Larson, P. Massopust, Coxeter groups and wavelets sets, in: D. R. Larson, P. Massopust, Z. Nashed, M. C. Nguyen, M. Papadakis, A. Zayed (eds.), Frames and Operator Theory in Analysis and Signal Processing, Contemp. Math. 451, American Mathematical Society, Providence, 2008, 187-218

D. R. Larson, P. Massopust, G. Ólafsson, Three-way tiling sets in two dimensions, to appear in Operator Methods in Fractal Analysis, Wavelets and Dynamical Systems

D. Schmid, A trade-off principle in connection with the approximation by positive definite kernels, in: M. Neamtu, L.L. Schumaker (eds.), Approximation Theory XII: San Antonio 2007, Nashboro Press, Brentwood, 2008, 348-359

3. Books

E. Grinberg, P.E.T. Jorgensen, D. R. Larson, P. Massopust, G. Ólafsson, E.T. Quinto, B. Rubin, Radon Transforms, Geometry, and Wavelets, Contemporary Mathematics, American Mathematical Society, Providence, in press

D. R. Larson, P. Massopust, M. Nguyen, M. Papadakis, Z. Nashed, A. Zayed, Frames and Operator Theory in Analysis and Signal Processing, Contemporary Mathematics, American Mathematical Society, Providence, 2008

M. Simon, Spectral theory of birth-and-death processes: Explicit methods with examples and perturbative approaches under domination of killing, Sierke Verlag, Göttingen, 2008


49

Publications 2007

1. Journal Articles

A.-L. Boulesteix, V. Hösel, V. Liebscher, Stochastic modelling for the COMET-assay, J. Concr. Appl. Math. 5, no. 1 (2007), 53-75

W. zu Castell, N. Laín Fernández, Y. Xu, Polynomial interpolation on the unit sphere II, Adv. Comput. Math. 26 (2007), 155-171

L. Condat, D. Van de Ville, Quasi-interpolating spline models for hexagonally sampled data, IEEE Trans. Image Process. 16, no. 5 (2007), 1195-1206

H. Eberl, L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, Electron. J. Differential Equations, Conference CS 15 (2007), 77-95

H. Eberl, M. A. Efendiev, On a criteria verifying some bio-chemical models, Proc. RIMS, Kyoto, 1542 (2007), 92-101

M. A. Efendiev, On elliptic attractor in asymptotically symmetric unbounded domain in R4, Bull. London Math. Soc. 39 (2007), 911- 920

M. A. Efendiev, M. Kläre, R. Lasser, Dimension estimate of the exponential attractors for the chemotaxis-growth system, Math. Methods Appl. Sci. 30, no. 5 (2007), 579-594

M. A. Efendiev, E. Nakaguchi, W.L. Wendland, Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system, Discrete Contin. Dynam. Systems (2007), 334-343

M. A. Efendiev, M. Otani, Infinite-dimensional attractors for evolution equations with p-Laplacian and its Kolmogorov entropy, Differential Integral Equations 20, no. 11 (2007), 1201-1209

M. A. Efendiev, L. A. Peletier, On entropy estimates for attractor of Swift-Hohenberg equation in unbounded domain, C. R. Acad. Sci. Paris Ser. I Math. 344, no. 2 (2007), 93-96

M. A. Efendiev, W. L. Wendland, Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains, J. Math. Anal. Appl. 329 (2007), 383-391

M. A. Efendiev, W. L. Wendland, Geometrical properties of nonlinear maps and their application, Part II: Nonlinear Riemann–Hilbert problems with closed boundary data for multiply connected domains, J. Math. Anal. Appl. 329 (2007), 425-444

M. A. Efendiev, W.L. Wendland, Nonlinear Riemann-Hilbert problem with discontinuous boundary data, Math. Nachr. 280, no. 9-10 (2007), 982-995

M. Englmann, A. Fekete, C. Kuttler, M. Frommberger, X. Li, I. Gebefügi, J. Fekete, P. Schmitt-Kopplin,The hydrolysis of unsubstituted N-acylhomoserine lactones to their homoserine metabolites. Analytical approaches using ultra performance liquid chromatography, Chromatogr. J. A 1160 (2007), 184-193

K. Ey, R. Lasser, Facing linear difference equations through hypergroup methods, J. Differ. Equations Appl. 13, no. 10 (2007), 953-965

B. Forster, Approximation in Smirnov spaces: Direct and inverse theorems, Constr. Approx. 26 (2007), 49-64

F. Friedrich, L. Demaret, H. Führ, K. Wicker, Efficient moment computation over polygonal domains with an application to rapid wedgelet approximation, SIAM J. Sci. Comput. 29, no. 2 (2007), 842-863

H. Führ, K. Gröchenig, Sampling theorems on locally compact groups from oscillation estimates, Math. Z. 255 (2007), 177-194

B. A. Hense, C. Kuttler, J. Müller, M. Rothballer, A. Hartmann, J.-U. Kreft, Does efficiency sensing unify diffusion and quorum sensing?, Nat. Rev. Microbiol. 5 (2007), 230-239

N. Laín Fernández, Optimally space-localized band-limited wavelets on Sq-1, J. Comput. Appl. Math. 199 (2007), 68-79

S. Lardjane, Nonparametric density estimation for nonmixing approximable stochastic processes, Stat. Inference Stoch. Processes 10 (2007), 209-221

S. Lardjane, P. Dourgnon, Les comparaisons internationales d’état de santé subjectif sont-elles pertinentes? Une évaluation par la méthodes des vignettes-étalons, Economie et Statistique 403-404 (2007), 165-177

R. Lasser, Amenability and weak amenability of l1-algebras of polynomial hypergroups, Studia Math. 182 (2007), 183-196

R. Lasser, M. A. Efendiev, M. Kläre, Dimension estimate of the exponential attractor for the chemotaxis-growth system, Math. Methods Appl. Sci. 30, no. 5 (2007), 579-594

R. Lasser, J. Obermaier, H. Rauhut, Generalized hypergroups and orthogonal polynomials, J. Aust. Math. Soc. 82 (2007), 369-393

S. Lassmann, I. Schuster, A. Walch, H. Göbel, U. Jütting, F. Makowiec, U. Hopt, M. Werner, STAT3 mRNA and protein expression in colorectal cancer: effects on STAT3-inducible targets linked to cell survival and proliferation, J. Clin. Path. 60 (2007), 173-179

S. Lassmann, Y. Shen, U. Jütting, P. Wiehle, A. Walch, G. Gitsch, A. Hasenburg, M. Werner, Predictive value of Aurora-A/STK15 expression for late stage epithelial ovarian cancer patients treated by adjuvant chemotherapy, Clin. Cancer Res. 13, no. 14 (2007), 4083-4091

A. Mayer, D. Bulian, H. Scherb, M. Hrabé de Angelis, J. Schmidt, E. Mahabir, Emergency prevention of extinction of a transgenic allele in a less-fertile transgenic mouse line by crossing with an inbred or outbred mouse strain coupled with assisted reproductive technologies, Reprod. Fert. Develop. 19 (2007), 984-994

J. Obermaier, R. Szwarc, Nonnegative linearization for little q-Laguerre polynomials and Faber basis, J. Comput. Appl. Math. 199 (2007), 89-94

G. Restrepo, R. Brüggemann, K. Voigt, Partially ordered sets in the analysis of alkanes fate in rivers, Croatica Chemoca Acta 80, no. 2 (2007), 261-270

K. Rodenacker, Does digital analysis of micro image data improve understanding of reality? Contradictions-challenges, Ecol. Inform. 2 (2007), 353-360

I. Rubio-Aliagra, S. Soewarto, S. Wagner, S. Zeiser, A genetic screen for modifiers of the delta 1-dependent notch signaling function in the mouse, Genetics 175 (2007), 1451-1463

A. Rupp, U. Dornseifer, A. Fischer, W. Schmahl, K. Rodenacker, U. Jütting, P. Gais, E. Biemer, N. Papadopulos, K. Matiasek, Electrophysiologic assessment of sciatic nerve regeneration in the rat: Surrounding limb muscles feature strongly in recordings from the gastrocnemius muscle, J. Neurosci. Meth. 166 (2007), 266-277

A. Rupp, U. Dornseifer, K. Rodenacker, A. Fichter, U. Jütting, P. Gais, N. Papadopulos, K. Matiasek, Temporal progression and extent of the return of sensation in the foot provided by the saphenous nerve after sciatic nerve transection and repair in the ratimplications for nociceptive assessments, Somatosens. Mot. Res. 24, no. 1-2 (2007), 1-13

H. Scherb, K. Voigt, Trends in the human sex odds at birth in Europe and the Chernobyl nuclear power plant accident, Reprod. Toxicol. 23 (2007), 593-599

I. G. Schuster, D.H. Busch, E. Eppinger, E. Kremmer, S. Milosevic, C. Hennard, C. Kuttler, J. W. Ellwart, B. Frankenberger, E. Nößner, C. Salat, C. Bogner, A. Borkhardt, H.-J. Kolb, A. M. Krackhardt, Allorestricted T cells with specificity for the FMNL1-derived peptide PP2 have potent antitumor activity against hematologic and other malignancies, Blood 110, no. 8 (2007), 2931-2939

H. B. Tiedemann, E. Schneltzer, S. Zeiser, I. Rubio-Aliaga, W. Wurst, J. Beckers, G. K. H. Przemeck, M. Hrabé de Angelis, Cell-based simulation of dynamic expression patterns in the presomitic mesoderm, J. Theor. Biol. 248 (2007), 120-129


50

U. Weller, M. Zipprich, M. Sommer, W. zu Castell, M. Wehrhan, Mapping clay content across boundaries at the landscape scale with electromagnetic induction, Soil Sci. Soc. Am. J. 71 (2007), 1740-1747

S. Zeiser, J. Müller, V. Liebscher, Modeling the Hes1 oscillator, J. Comput. Biol. 14 (2007), 984-1000


R. Brüggemann, K. Voigt, P. Soerensen, B. De Baets, Partial order concepts in ranking environmental chemicals, EnviroInfo Warsaw 2007, Environmental Informatics and Systems Research, Shaker, Aachen 2007, 169-176

W. zu Castell, F. Filbir, Strictly positive definite functions on generalized motion groups, in: A. Iske, J. Levesley, Algorithms for Approximation, Springer, Heidelberg 2007, 349-357

L. Condat, B. Forster-Heinlein, D. Van de Ville, A new family of rotation-covariant wavelets on the hexagonal lattice, Proceedings of the Conference SPIE Optics and Photonics 2007 Conference on Mathematical Methods: Wavelet XII, San Diego, August 26-29, 2007, Vol. 6701 (2007), 67010B-1/67010B-9

L. Condat, B. Forster-Heinlein, D. Van De Ville, H2O: reversible hexagonal-orthogonal grid conversion by 1-D filtering, Proceedings of the Conference IEEE (ICIP), San Antonio TX, USA, Sept. 16.-19, 2007

L. Condat, A. Hirabayashi, Towards a general formulation for over-sampling and under-sampling, EUSIPCO, Sept. 2007, Poznan, Poland, to appear

L. Condat, A. Hirabayashi, A compact image magnification method with preservation of preferential components, IEEE ICIP, Sept. 2007, San Antonio, U.S.A.

L. Condat, M. Khan, J. Chanussot, A. Montanvert, Pan-sharpening using induction, IEEE International Geoscience and Remote Sensing Symposium (IGARSS), July 2007, Barcelona, Spain

B. Forster, P. Massopust, Multivariate complex B-splines, Proceedings of SPIE Wavelets XII, Vol. 6701 (2007), 1-9

K. Hahn, S. Prigarin, K. Hasan, Can we expect reproducible and unbiased information from denoised diffusion tensor imaging with low SNR?, Proc. ISMRM 15, Berlin (2007), 1604

K. Hahn, S. Prigarin, K. Rodenacker, K. Sandau, A fractal dimension for exploratory fMRI analysis, Proc. ISMRM 15, Berlin (2007), 1858

P. Massopust, A class of solutions to Maxwell‘s equations in matter and associated special functions, in: N.M. Chuong, P. G. Ciarlet, P. Lax, D. Mumford, D. H. Phong (eds.), Advances in Deterministic and Stochastic Analysis, World Scientific Publishing, Singapore 2007, 131-175

P. Massopust, Multiwavelets: Some approximation-theoretic properties, sampling on the interval, and translation invariance, in: N.M. Chuong, Y.V. Egorov, A. Khrennikov, Y. Meyer, D. Mumford (eds.), Harmonic, Wavelet and p-Adic Analysis, World Scientific Publishing Co., Singapore 2007, 37–57

K. Voigt, R. Brüggemann, Data-availability of pharmaceuticals detected in water: an evaluation study by order theory (METEOR), EnvironInfo Warsaw 2007, Shaker Aachen, 2007, 201-208

3. Books

W. zu Castell, S. Ehrich, F. Filbir, Proceedings of the Workshop Special Functions in Harmonic Analysis and Applications, Irsee, July 19-23, 2004, Special Issue, J. Comput. Appl. Math. 199, no. 1 (2007)

Publications 2006

1. Journal Articles

V. Betz, H. Spohn, A central limit theorem for Gibbs measures relative to Brownian motion, Probab. Theory Related Fields 131, (2006), 459-478

L. Demaret, N. Dyn, A. Iske, Image compression by linear splines over adaptive triangulations, Signal Process. 86, no. 7 (2006), 1604-1616

L. Demaret, A. Iske, Adaptive image approximation by linear splines over locally optimal Delaunay triangulations, IEEE Signal Proc. Letters 13, no. 5 (2006), 281-284

B. Forster, T. Blu, M. Unser, Complex B-splines, Appl. Comput. Harm. Anal. 20, no. 2 (2006), 261-282

H. Führ, H.G. Feichtinger, K. Gröchenig, K. Kaiblinger, Operators commuting with a discrete subgroup of translation, J. Geom. Anal. 16, no. 1 (2006), 53-67

H. Führ, K. Gröchenig, Sampling theorems on locally compact groups from oscillation estimates, Math. Z. 255 (2006), 177-194

D. Geller, A. Mayeli, Continuous wavelets and frames on stratified Lie groups I, J. Fourier Anal. Appl. 12, no. 5 (2006), 543-579

V. Hösel, R. Lasser, Approximation with Bernstein-Szegö polynomials, Numer. Funct. Anal. Optim. 27 (2006), 377-389

M. Hughes-Fulford, U. Jütting, K. Rodenacker, Reduction of anabolic signals and alteration of osteoblast nuclear morphology in microgravity, J. Cell. Biochem. 99, no. 2 (2006), 435-449

F. Kaffarnik, H.K. Seidlitz, J. Obermaier, H. Sandermann Jr., W. Heller, Environmental and developmental effects on the biosynthesis of UV-B-screening pigments in scots pine (Pinus sylvestris L.) needles, Plant Cell Environ. 29, no. 8 (2006), 1484-1491

N. Laín Fernández, J. Prestin, Interpolatory band-limited wavelet bases on the sphere, Constr. Approx. 23 (2006), 79-101

S. Lardjane, On some stochastic properties in Devaney’s chaos, Chaos Solitons Fractals 28 (2006), 668-672

J. Müller, C. Kuttler, B. A. Hense, M. Rothballer, A. Hartmann, Cell-cell communication by quorum sensing and dimension reduction, J. Math. Biol. 53 (2006), 672-702

W. Pillmann, W. Geiger, K. Voigt, Survey of environmental informatics in Europe, Environ. Modell. Softw. 21 (2006), 1520-1527

R. Reiter, P. Gais, U. Jütting, M. Steuer-Vogt, A. Pickhard, K. Brink, S. Rauser, S. Lassmann, H. Höfler, M. Worner, A. Walch, Aurora kinase A messenger RNA overexpression is correlated with tumor progression and shortened survival in head and neck squamous cell carcinoma, Clin. Cancer Res. 12 (2006), 5136-5141

K. Rodenacker, B. Hense, U. Jütting, P. Gais, Automatic analysis of aqueous specimens for photoplankton structure recognition and population estimation, Microsc. Res. Techniq. 69 (2006), 708-720

F. Tookos, A Wiener-type condition for Hölder-continuity, Acta Math. Hungarica 111, no. 1-2 (2006), 131-155

S. Zeiser, V. Liebscher, H. Tiedemann, I. Rubio-Aliaga, G.K.H. Przemeck, M. Hrabé de Angelis, G. Winkler, Number of active transcription factor binding sites is essential for the Hes7 oscillator, Theor. Biol. Med. Modelling, 3, no. 11 (2006), doi:10.1186/1742-4682-3-11


51


F. Filbir, W. Themistoclakis, Generalized de la Vallee Poussin operators for Jacobi weights, Proc. Intern. Conf. on Numerical Analysis and Approximation Theory, Cluj-Napoca, Romania, July 4-8, 2006, 195-204

B. Forster, S. Pfeifer, T. Schratzenstaller, K. Knör, I. Grabmair, E. Wintermantel, Stent-induced arterial deformation, SSBE Meeting, Zürich, 2006

H. Führ, L. Demaret, F. Friedrich, Beyond wavelets: New image representation paradigms, in: M. Barni (ed.), Document and Image Compression, CRC Press, New York 2006, 179-206

K. R. Hahn, S. Prigarin, S. Heim, K. Hasan, Random noise in diffusion tensor imaging, its destructive impact and some corrections, in: Visualization and Processing of Tensor Fields, Proc. Dagstuhl Workshop, Springer, Heidelberg 2006, 107-119

K. R. Hahn, K. Sandau, K. Rodenacker, S. Prigarin, Novel algorithms to measure complexity in the human brain and to detect statistically significant complexity-differences, 23. Annual Scientific Meeting ESMRB, electronic supplement of journal MAGMA (2006), vol. 19/suppl 1, http://dx.doi.org/10.1007/s10334-006-0043-1

K. Hasan, K. Rodenacker, K. Hahn, Evaluation of SNR performance and utility of high spatial and angular resolution denoised 1mm3 isotropic DTI of entire human brain at 3 T, in: Proc. of 14th Annual Meeting ISMRM, Washington (2006), 344

P. Massopust, Inverse problems in pipeline engineering, in: G. Ólafsson, E. T. Quinto (eds.), The Radon Transform, Inverse Problems, and Tomography, Proceedings on Symposia in Applied Mathematics 63, American Mathematical Society, Providence 2006, 93 – 128

K. Voigt, R. Brüggemann, Method of evaluation by order theory applied on the environmental topic of data-availability of pharmaceutically active substances, Proc. Workshop »Applications of Informatics in Environmental Engineering«, Ciechocinek, Poland 13.09.-15.09.2006, 107-120

K. Voigt, R. Brüggemann, Information systems & databases, in: R. Brüggemann, L. Carlsen, Partial Order in Environmental Sciences and Chemistry, Springer, Berlin, 2006, 327-351

K. Voigt, S. Pudenz, R. Brüggemann, ProRank a software tool used for the evaluation of environmental databases, IEMSs Summit on Environmental Modelling & Software, Burlington Vermont, USA, July 9-12th, 2006

K. Voigt, G. Restrepo, R. Brüggemann, Structure-fate-relationship of organic chemicals, Proc. of the iEMSs Third Biennial Meeting, »Summit on Environmental Modelling and Software«, International Environmental Modelling and Software Society, Burlington, USA, July 2006, CD-ROM

3. Books

O. E. Barndorff-Nielsen, U. Franz, R. Gohm, B. Kümmerer, S. Thorbjoernsen, Quantum Independent Increment Processes II – Structure of Quantum Levy Processes, Classical Probability, and Physics, Lecture Notes in Mathematics, Springer, New York 2006

G. Winkler, Image Analysis, Random Fields and Markov Chain Monte Carlo Methods. Second Edition, 3rd Printing, Springer, New York 2006

Publications 2005

1. Journal Articles

J. Altschuh, S. Walcher, H. Sandermann, The lipid/protein interface as xenobiotic target site: Kinetic analysis of tadpole narcosis, FEBS J. 272 (2005), 2399-2406

A. Bényi, L. Grafakos, K. Gröchenig, K. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (2005), 131-139

V. Betz, S. Teufel, Precise coupling terms in adiabatic quantum evolution, Ann. Henri Poincare 6 (2005), 217-246

V. Betz, S. Teufel, Precise coupling terms in adiabatic quantum evolution. The generic case, Comm. Math. Phys. 260 (2005), 481-509

R. Brüggemann, G. Restrepo, K. Voigt, Towards an evaluation of chemicals, WSEAS Trans. Inform. Sci. Appl. 8 (2005), 1023-1033

W. zu Castell, F. Filbir, Radial basis functions and corresponding zonal series expansions on the sphere, J. Approx. Theory 134 (2005), 65-79

W. zu Castell, F. Filbir, R. Szwarc, Strictly positive definite functions in Rd, J. Approx. Theory 137 (2005), 277-280

W. zu Castell, S. Schrödl, T. Seifert, Volume interpolation of CT images from tree trunks, Plant Biology 54 (2005), 737-744

E. Cordero, K. Gröchenig, Necessary conditions for Schatten class localization operators, Proc. Amer. Math. Soc. 133 (2005), 3573-3579

F. Filbir, R. Lasser, R. Szwarc, Hypergroups of compact type, J. Comput. Appl. Math. 178 (2005), 205-214

G. Fischer, R. Lasser, Homogeneous Banach spaces with respect to Jacobi polynomials, Rend. Circ. Mat. Palermo, Serie II, Suppl. 76 (2005), 331-353

M. Fornasier, K. Gröchenig, Intrinsic localization of frames, Constr. Approx. 22 (2005), 395-415

H. Führ, The weak Paley-Wiener property for group extensions, J. Lie Theory 15 (2005), 429-446

B. A. Hense, G.F. Severin, G. Pfister, G. Welzl, W. Jaser, K.-W. Schramm, Effects of anthropogenic estrogens nonylphenol and 17alpha-ethinylestradiol in aquatic model ecosystems, Acta Hydrochim. Hydrobiol. 33 (2005), 27-37

B. A. Hense, G. Welzl, G.F. Severin, K.-W. Schramm, Nonylphenol induced changes in trophic web structure of plankton analysed by multivariate statistical approaches, Aquat. Toxicol. 73 (2005), 190-209

L. Herrmann, M. Sommer, B. Schutte, M. Wehrhan, S. Graeff, M. Zipprich, Nutzung von Fern- und Naherkundungsverfahren insbesondere EM38 zur Ableitung räumlich kontinuierlicher Bodendaten auf der Feldskala, Mitteilgn. Dtsch. Bodenkundl. Gesellschaft 106, 2005, 79-80

S. Lardjane, Reverse Chaos may not be a curse, J. Nonparam. Stat. 17, no. 8 (2005), 885-889

T. Mijalski, A. Harder, M. Kersten, M. Horsch, T. M. Strom, V. Liebscher, F. Lottspeich, M. Hrabé de Angelis, J. Beckers, Identification of coexpressed gene clusters in a comparative analysis of transcriptome and proteome in mouse tissues, Proc. Nat. Acad. Sci. USA 102 (2005), 8621-8626

C. Möbius, H.J. Stein, C. Spieß, I. Becker, M. Feith, J. Theisen, P. Gais, U. Jütting, J. R. Siewert, COX2 expression, angiogenesis, proliferation and survival in Barrett‘s cancer, Eur. J. Surg. 31 (2005), 755-759


52

K. Pritsch, G. Luedemann, R. Matyssek, A. Hartmann, M. Schloter, H. Scherb, T. E. E. Grams, Mycorrhizosphere responsiveness to atmospheric ozone and inoculation with Phytophthora citricola in a phytotron experiment with spruce/beech mixed cultures, Plant Biol. 7 (2005), 718-727

F. Tookos, Smoothness of Green‘s functions and density of sets, Acta Sci. Math. (Szeged) 71, no. 1-2 (2005), 117-146

F. Tookos, V. Totik, Markov inequality and Green functions, Rend. Circ. Mat. Palermo, Serie II, 76 (2005), 91-102

K. Voigt, R. Brüggemann, Water contamination with pharmaceuticals: data availability and evaluation approach with Hasse diagram technique and METEOR, MATCH Commun. Math. Comput. Chem. 54 (2005), 671-689

G. Winkler, O. Wittich, V. Liebscher, A. Kempe, Don‘t shed tears over breaks, Jahresbericht der DMV 107, no.2 (2005), 57-87


V. Betz, J. Lörinczi, H. Spohn, Gibbs measures on Brownian paths: Theory and applications, in: J.-D. Deuschel, A. Greven (eds.), Interacting Stochastic Systems, Springer, Heidelberg 2005

R. Brüggemann, G. Restrepo. K. Voigt, Prescreening of chemicals by partial orders, 9th WSEAS International Conference on Computers, WSEAS, Athens, Greece, July 14 - 16, 2005, Proceedings, 497-503

W. zu Castell, Interpolation with reflection invariant positive definite functions, in: C. K. Chui, M. Neamtu, L. L. Schumaker (eds.), Approximation Theory XI: Gatlinburg 2004, Brentwood, TN, USA: Nashboro Press, Nashville 2005, 105-120

K. H. Fichtner, V. Liebscher, M. Ohya, A limit theorem for conditionally independent beam splittings, in: M. Schürmann, U. Franz (eds.), Quantum Probability and Infinite Dimensional Analysis. From Foundations to Applications, Krupp-Kolleg Greifswald, Germany 22 – 28 June 2003, volume 18 of QP-PQ Quantum Probability and White Noise Analysis, World Scientific, Singapore 2005, 227-236

K. R. Hahn, S. Prigarin, K. M. Hasan, The feasibility of diffusion tensor imaging for the human brain at 1mm3 resolution, Proc. of 13th Annual Meeting of the International Society for Magnetic Resonance in Medicine 2005, 161

R. Lasser, Discrete commutative hypergroups, in: W. zu Castell, F. Filbir, B. Forster-Heinlein (eds.), Advances in the Theory of Special Functions and Orthogonal Polynomials 2, Inzell Lectures on Orthogonal Polynomials, Nova Science Publishers, Hauppauge 2005, 5-102

P. Massopust, Fractal functions, splines, and Besov and Triebel-Lizorkin spaces, in: J. Lévy-Véhel , E. Lutton (eds.), Fractals Engineering: New Trends and Applications, Springer, London 2005, 21-32

J. Obermaier, R. Szwarc, Polynomial bases for continuous function spaces, in: M. G. De Bruin, D. H. Mache, J. Szabados (eds.), Trends and Applications in Constructive Approx., Vol. 151, Basel, Switzerland, Birkhäuser 2005, 195-205

K. Voigt, R. Brüggemann, S. Pudenz, Application of computer-aided decision tools concerning environmental pollution with pharmaceuticals, Systems Research Institute of Polish Academy of Science, Vol. 42, 2005, 135-146

K. Voigt, R. Brüggemann, S. Pudenz, H. Scherb, Environmental contamination with endocrine disruptors and pharmaceuticals. An environmental evaluation approach, in: J. Hrebicek, J. Racek (eds.), ENVIROINFO BRNO, Informatics for Environmental Protection, 2005, 858-862

G. Winkler, A. Kempe, V. Liebscher, O. Wittich, Parsimonious segmentation of time series by potts models, in: D. Baier, K.-D. Wernecke (eds.), Innovations in Classification, Data Science, and Information Systems. Proceedings of the 27th Annual Conference of the Gesellschaft für Klassifikation e.V., Springer Heidelberg 2005, 295-302

3. Books

W. zu Castell, F. Filbir, B. Forster-Heinlein, Inzell Lectures on Orthogonal Polynomials, Nova Science Publishers, New York 2005

H.G. Feichtinger, K. Gröchenig, T. Strohmer, Frame Theory and Sampling Problems in Time-Frequency Analysis and Wavelet Theory (Part I), Int. J. Wavelets Multires. Inform. Proc. 3 (Special Issue), 2005

H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Lecture Notes in Mathematics, Springer, Heidelberg 2005


53

Theses and Calls

1. PhD Theses

Edgar Delgado-Eckert, Monomial Dynamical and Control Systems over a Finite Field and Applications to Agent-based Models in Immunology, Technische Universität München, 2008

Kristine Ey, On the Representation of Pn-positive definite Functions and Applications, Technische Universität München, 2008

Moritz Simon, Spectral Theory of Birth-and-Death Processes, Technische Universität München, 2008

Ahmadreza Azimifard, α-Amenability of Banach Algebras on Hypergroups, Technische Universität München, 2007

Marie Wild, Characterizing Discrete-Time Function Spaces, Technische Universität München, 2006

Georg Fischer, Qualitative Unschärferelationen und Jacobi-Polynome, Technische Universität München, 2005

Felix Friedrich, Complexity Penalized Segmentations in 2D. Efficient Algorithms and Approximation Properties, Technische Universität München, 2005

Steffen Kläre, Stochastical Models of Molecular Evolution: An Algebraical and Statistical Analysis, Ludwig-Maximilians-Universität München, 2005

2. Calls to Academic Positions

Laurent Condat, Centre National de la Recherche Scientifique, Délégation Normandie, France, 2008

Christina Kuttler, Technische Universität München, Germany, 2008

Hartmut Führ, Rheinisch-Westfälische Technische Hochschule Aachen, Germany, 2006

Karlheinz Gröchenig, Universität Wien, Austria, 2006

Uwe Franz, Université de Franche-Compte, France, 2005

Volkmar Liebscher, Universität Greifswald, Germany, 2005

Volker Betz, University of Warwick, United Kingdom, 2005

Teaching

Summer Term 2008

Forster-Heinlein: Fourier Series (4+2), Technische Universität München

Lasser: Operator Theory (Functional Analysis II) (4+2), Technische Universität München

Lasser: Mathematics of Computer Tomography (4), Technische Universität München

Massopust: Topology (4+2), Technische Universität München

Müller, Kuttler: Biomathematics, Seminar (2), Technische Universität München

Obermaier: Bases in Banach Spaces (2), Technische Universität München

Schlicht: Probability Theory (4), Ludwig-Maximilians-Universität München

Winter Term 2007/08

zu Castell: Theory of Mathematical Learning (4), Technische Universität München

Demaret: Wavelets and Approximation (4), Technische Universität München

Filbir: Inverse Problems (4), Technische Universität München

Forster-Heinlein: Laplace- and z-Transformation (2+2), Technische Universität München

Forster-Heinlein, Mayeli, Semmler: A Polynomial Approach to Linear Algebra, Proseminar (2), Technische Universität München

Lasser: Functional Analysis (4+2), Technische Universität München

Lasser: Monotone Dynamical Systems (2), Technische Universität München

Lasser, Obermaier: Construction of the Real Numbers, Proseminar (2), Technische Universität München

Massopust: Differential Forms (2+1), Technische Universität München

Mayeli: Introduction to Wavelets and Frames on the Heisenberg Group (2), Technische Universität München

Winkler: Mathematics for Physicists III (4+2), Ludwig-Maximilians-Universität München

Summer Term 2007

Filbir: Fourier and Laplace-Transformation (4), Technische Universität München

Forster-Heinlein: Wavelets (2+1), Technische Universität München

Lasser: Operator Theory (Functional Analysis II) (4+2), Technische Universität München

Lasser, Obermaier, Tookos: Potential Theory and Orthogonal Polynomials, Seminar (2), Technische Universität München

Massopust: Interpolation and Approximation with Splines and Fractals (4+2), Technische Universität München

Winkler: Mathematics for Physicists II (4+2), Ludwig-Maximilians-Universität München


54

Winter Term 2006/07

zu Castell, Mathematics for Tomography (4), Technische Universität München

Filbir: Harmonic Analysis I (4+2), Technische Universität München

Forster-Heinlein: Approximation Theory (2+1), Technische Universität München

Lasser: Measure and Integration Theory (4+2), Technische Universität München

Obermaier: Orthogonal Polynomials (4), Technische Universität München

Winkler: Mathematics for Physicists I (4+2), Ludwig-Maximilians-Universität München

Summer Term 2006

Lasser: System Theory (4+2), Technische Universität München

Forster-Heinlein: Mathematical Methods for Image Analysis: Integral transforms and Wavelets (4+1), Technische Universität München

Obermaier: Density Estimators Using Orthogonal Series (2), Technische Universität München

Schmalmack: Complex Analysis for Engineers (2+2), Technische Universität München

Winter Term 2005/06

zu Castell: Introduction to Learning Theory (2), Technische Universität München

Filbir: Lie Algebras (4), Technische Universität München

Forster-Heinlein: Distributions (4+1), Technische Universität München

Führ: 2D-Wavelets and Similar Constructions (2), Technische Universität München

Führ: Mathematics (4), Universität der Bundeswehr München

Obermaier: Bases in Banach Spaces (2), Technische Universität München

Winkler: Probability Theory (4+2), Ludwig-Maximilians-Universität München

Summer Term 2005

Lasser: Analysis II, (4+2+2), Technische Universität München

Filbir: Lie Groups (4), Technische Universität München

Winter Term 2004/05

zu Castell, Spatial Statistics (4), Technische Universität München

Lasser: Analysis I (4+2+2), Technische Universität München

Lasser: Mathematical Models in Biology (4+2), Technische Universität München

Gröchenig: Banach Algebras in Applied Analysis (3), Technische Universität München

Liebscher: Mathematical Statistics II (4+2), Ludwig-Maximilians-Universität München

Liebscher, Mendoza, Strimmer, Winkler: Mathematical Methods in Systems Biology, Seminar (2), Ludwig-Maximilians-Universität München

Laín Fernández: Mathematics (4+4), Universität der Bundeswehr München

Projects

Humboldt grant for François Hamel, Université Paul Cézanne Aix-Marseille III, 2008-2010

»Fast Fourier Transform for the Rotation Group«, Deutsche Forschungs-gemeinschaft, 2008-2010

Research Cooperation, Definiens, Munich, 2008-2010

Research Network QuantPro, »Systemische Analyse Cyclin-abhängiger Kinasen«, 2007-2010

Helmholtz Alliance on Systems Biology, CoReNe – Control of Regulatory Networks, 2007-2012

Cluster of Excellence CoTeSys – Cognition for Technical Systems, 2006-2011

Marie-Curie Excellence Team, MAMEBIA – Mathematical Methods in Biological Image Analysis, 2005-2008

Collaborative Research Center 607 »Growth and Parasite Defence«, 2002-2010

EU Research Training Network HASSIP – Harmonic Analysis and Statistics in Signal and Image Processing, 2002-2007

»Räumlich aufgelöste Untersuchung des Schadstoffabbaus in Biofilmen und deren Formation mit Methoden der Mathematischen Biologie und des Wissenschaftlichen Rechnens«, Volkswagenstiftung, 2002-2005

BMBF Research Network BFAM – Analysis of Mammalian Genomes, 2001-2006

Collaborative Research Center 386 »Statistical Analysis of Discrete Structures«, 1998-2006

Graduate College »Applied Algorithmic Mathematics«, 1998-2008


55

Imprint

Published byHelmholtz Zentrum MünchenGerman Research Center for Environmental Health GmbHIngolstaedter Landstr. 185764 NeuherbergGermany

Phone: +49 (0) 89 3187 - 0Fax: +49 (0) 89 3187 - 3324www.helmholtz-muenchen.de

EditorsInstitute of Biomathematics and BiometryProf. Dr. Rupert LasserDr. Wolfgang zu CastellDr. Burkhard HenseGeorg Berschneider

Proof-readingCarol OberschmidtCommunicate Language Services GbRwww.c-l-s.net

Layout and GraphicsGraphisches Atelier Fenke, München

PrintingFibodruck, Neuried/Münchenwww.fibodruck.de

PhotographsUlla Baumgart, Michael HaggenmüllerTitlepageMichael Haggenmüllerwww.michael-haggenmueller.de

special thanks to ix-quadratTechnische Universität Münchenwww-m10.ma.tum.de/ix-quadrat

© 2008 by Helmholtz Zentrum München, Institute of Biomathematics and Biometry

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