Mathematical Analysis Of Evolution, Information And Complexity

Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a wide collection of research in mathematics, information theory, physics and other scientific and technical areas, this new title offers elementary and thus easily accessible introductions to the various fields of research addressed in the book.

Wolfgang Arendt is head of the Institute of Applied Analysis at the University of Ulm, Germany. Having obtained his degrees from the University of Nice, France, and Tübingen, Germany, he worked for eight years as a professor at the University of Besançon, France, before accepting a chair at the University of Ulm in Germany. He held visiting positions at the University of California at Berkeley, and the universities of Nancy, Oxford, Zürich, Canberra, Sydney and Lecce. Professor Arendt's fields of research are partial differential equations and functional analysis with special emphasis on evolution equations.

Wolfgang P. Schleich is head of the Institute of Quantum Physics at the University of Ulm, Germany, and Distinguished Adjunct Professor at the University of North Texas in Denton, USA. He has published more than 250 papers on problems of quantum optics, foundations of quantum mechanics and general relativity and is the author of the highly acclaimed textbook Quantum Optics in Phase Space, published with Wiley-VCH. For his work he has received numerous awards and honours.

Preface.
List of Contributors.
Prologue (Wolfgang Arendt, Delio Mugnolo and Wolfgang Schleich).
1 Weyl’s Law (Wolfgang Arendt, Robin Nittka, Wolfgang Peter, Frank Steiner).
1.1 Introduction.
1.2 A Brief History of Weyl’s Law.
1.3 Weyl’s Law with Remainder Term I.
1.4 Weyl’s Law with Remainder Term II.
1.5 Generalizations of Weyl’s Law.
1.6 A Proof of Weyl’s Formula.
1.7 Can One Hear the Shape of a Drum?
1.8 Does Diffusion Determine the Domain?
References.
2 Solutions of Systems of Linear Ordinary Differential Equations (Werner Balser, Claudia Röscheisen, Frank Steiner, Eric Sträng).
2.1 Introduction.
2.2 The Exponential Ansatz of Magnus.
2.3 The Feynman-Dyson Series, and More General Perturbation Techniques.
2.4 Power Series Methods.
2.5 Multi-Summability of Formal Power-Series.
2.6 Periodic ODE.
References.
3 A Scalar-Tensor Theory of Gravity with a Higgs Potential (Nils Manuel Bezares-Roder, Frank Steiner).
3.1 Introduction.
3.2 Scalar-Tensor Theory with Higgs Potential.
References.
4 Relating Simulation and Modeling of Neural Networks (Stefano Cardanobile, Heiner Markert, Delio Mugnolo, Günther Palm, Friedhelm Schwenker).
4.1 Introduction.
4.2 Voltage-Based Models.
4.3 Changing Paradigm – From Biological Networks of Neurons to Artificial Neural Networks.
4.4 Numerical Simulation of Neural Networks.
4.5 Population-Based Simulation of Large Spiking Networks.
4.6 Synaptic Plasticity and Developing Neural Networks.
References.
5 Boolean Networks for Modeling Gene Regulation (Christian Wawra, Michael Kühl, Hans A Kestler).
5.1 Introduction.
5.2 Biological Background.
5.3 Aims of Modeling.
5.4 Modeling Techniques.
5.5 Modeling GRNs with Boelean Networks.
5.6 Dynamic Behavior of Large Random Networks.
5.7 Inference of Gene Regulatory Networks from Real Data.
5.8 Conclusion.
References.
6 Symmetries in Quantum Graphs (Jens Bolte, Stefano Cardanobile, Delio Mugnolo, Robin Nittka).
6.1 Symmetries.
6.2 Quantum Graphs.
6.3 Energy Methods for Schrödinger Equations.
6.4 Symmetries in Quantum Graphs.
6.5 Schrödinger Equation with Potentials.
6.6 Concluding Remarks and Open Problems.
References.
7 Distributed Architecture for Speech-Controlled Systems Based on Associative Memories (Zöhre Kara Kayikci, Dmitry Zaykovskiy, Heiner Markert, Wolfgang Minker, Günther Palm).
7.1 Introduction.
7.2 System Architecture.
7.3 Feature Extraction on Mobile Devices.
7.4 Speech Recognition Systems Based on Associative Memory.
7.5 Words to Semantics Conversion using Associative Memory.
7.6 Sample System/Experimental Results.
7.7 Conclusion.
References.
8 Machine Learning for Categorization of Speech Utterances (Amparo Albalate, David Suendermann, Roberto Pieraccini, Wolfgang Minker).
8.1 Introduction.
8.2 An Overview of Pattern Recognition.
8.3 Utterance Classification as a Text-Classification Problem.
8.4 Utterance Corpus Description.
8.5 Utterance Preprocessing.
8.6 Feature Extraction Based on Term Clustering.
8.7 Supervised Methods for Utterance Categorization.
8.8 Evaluation Methods and Results.
8.9 Summary and Conclusion.
References.
9 Semi-Supervised Clustering in Functional Genomics (Johann M. Kraus, Günther Palm, Friedhelm Schwenker, Hans A Kestler).
9.1 Introduction.
9.2 Biological Background.
9.3 Cluster Analysis.
9.4 Semi-Supervised Clustering.
9.5 Summary.
References.
10 Image Processing and Feature Extraction from a Perspective of Computer Vision and Physical Cosmology (Holger Stefan Janzer, Florian Raudies, Heiko Neumann, Frank Steiner).
10.1 Introduction.
10.2 Background from Computer Vision.
10.3 Background from Physical Cosmology.
10.4 Image Formation and Characterization.
10.5 Methods of Image Processing.
10.6 Invariant Features of Images.
10.7 Concluding Remarks.
References.
11 Boosting Ensembles of Weak Classifiers in High Dimensional Input Spaces (Ludwig Lauser, Friedhelm Schwenker, Hans A Kestler).
11.1 Introduction.
11.2 Hypothesis Boosting Problem.
11.3 Learn.
11.4 Boosting by Majority.
11.5 AdaBoost.
11.6 BrownBoost.
11.7 AdaBoost for Feature Selection.
11.8 Conclusion.
References.
12 The Sampling Theorem in Theory and Practice (Wolfgang Arendt, Michal Chovanec, Jürgen Lindner, Robin Nittka).
12.1 Introduction and History.
12.2 The Sampling Theorem in Applications.
12.3 Mathematical Formulation of the Sampling Theorem.
References.
13 Coding and Decoding of Algebraic-Geometric Codes (Martin Bossert, Werner Lütkebohmert, Jörg Marhenke).
13.1 Introduction.
13.2 Introduction to Linear Codes.
13.3 Introduction to Forward Error Correction.
13.4 Algebraic-Geometric Codes.
13.5 Computation of Riemann-Roch Spaces.
13.6 Decoding up to Half the Minimum Distance.
13.7 Interpolation-Based Decoding.
13.8 Power Decoding of Low Rate Reed-Solomon Codes.
13.9 Interpolation-Based Soft-Decision Decoding.
13.10 Soft-Decision Decoding with the Dorsch Algorithm.
References.
14 Investigation of Input-Output Gain in Dynamical Systems for Neural Information Processing (Stefano Cardanobile, Michael Cohen, Silvia Corchs, Delio Mugnolo, Heiko Neumann).
14.1 Overview.
14.2 Introduction.
14.3 The Basic Unit: Analytical Study of the Dipole.
14.4 The Basic Unit: Numerical Analysis of the Dipole.
14.5 Model of a Recurrent Network.
14.6 Discussion and Conclusions.
References.
15. Factorization (Rüdiger Mack, Wolfgang P. Schleich, Daniel Haase, Helmut Maier).
15.1 Introduction.
15.2 How to Factor Numbers.
15.3 How to Find the Period of a Function: The Magic Role of Entanglement.
15.4 Analogy with Atom Optics.
15.5 Exponential Growth of Hilbert Space as a Resource of Exponential Speedup.
15.6 Conclusions.
References.
16 Isomorphism and Factorization – Classical and Quantum Algorithms (Sebastian Dörn, Daniel Haase, Jacobo Torán, Fabian Wagner).
16.1 Introduction.
16.2 Factorization of Integers: Classical Algorithms.
16.3 Graph Isomorphism: Classical Algorithms.
16.4 Quantum Algorithms for Integer Factorization.
16.5 Quantum Approach to Graph Isomorphism.
16.6 Reductions of Integer Factorization and Graph Isomorphism to Ring Isomorphism.
References.
17 QuickSort from an Information Theoretic View (Beatrice List, Markus Maucher, Uwe Schöning, Rainer Schuler).
17.1 Introduction.
17.2 An Upper Bound.
17.3 A Lower Bound.
17.4 The δ-Random Source.
17.5 Conclusion.
References.
Further Reading.
Index.