Mathematik
Refine
Year of publication
Document Type
- Article (84)
- Preprint (47)
- Doctoral Thesis (46)
- Report (16)
- Conference Proceeding (9)
- diplomthesis (6)
- Book (3)
- Part of a Book (2)
- Bachelor Thesis (1)
- Diploma Thesis (1)
Language
- English (216) (remove)
Has Fulltext
- yes (216) (remove)
Is part of the Bibliography
- no (216)
Keywords
- Kongress (6)
- Kryptologie (5)
- Online-Publikation (4)
- LLL-reduction (3)
- Moran model (3)
- computational complexity (3)
- contraction method (3)
- Algebraische Geometrie (2)
- Brownian motion (2)
- Commitment Scheme (2)
- Heat kernel (2)
- Integral Geometry (2)
- Knapsack problem (2)
- Krein space (2)
- Laplace operator on graphs (2)
- Lattice basis reduction (2)
- Mathematik (2)
- Oblivious Transfer (2)
- Perception (2)
- Quantum Zeno dynamics (2)
- San Jose (2)
- Semidefinite Programming (2)
- Shortest lattice vector problem (2)
- Subset sum problem (2)
- Tropical geometry (2)
- Tropische Geometrie (2)
- Valuation Theory (2)
- Vision (2)
- W*-dynamical system (2)
- X-Y model (2)
- ancestral selection graph (2)
- coalescent (2)
- collective intelligence (2)
- complexity (2)
- duality (2)
- fixation probability (2)
- genealogy (2)
- level of difficulty (2)
- point process (2)
- quantum spin systems (2)
- return to equilibrium (2)
- segments (2)
- spike train (2)
- task space (2)
- thought structure (2)
- Λ−coalescent (2)
- A-Discriminant (1)
- Action potential (1)
- Actions in mathematical learning (1)
- Activity (1)
- Adaptive dynamics (1)
- Amoeba (1)
- Ancestral selection graph (1)
- Anisotropic Norm (1)
- Approximation algorithm (1)
- Approximationsalgorithmus (1)
- Arbitrage (1)
- Asymptotically Even Nonlinearity (1)
- Axon (1)
- Banach spaces (1)
- Bayesian Inference (1)
- Berkovich spaces (1)
- Black and Scholes Option Price theory (1)
- Blind Signature (1)
- Block Korkin—Zolotarev reduction (1)
- Blockplay (1)
- Boolean Lattice (1)
- Boundary (1)
- Boundary Value Problems (1)
- Branch and Bound (1)
- Branching particle systems (1)
- Branching process approximation (1)
- Breaking knapsack cryptosystems (1)
- Burst (1)
- Calderón problem (1)
- Cannings model (1)
- Catalan number (1)
- Chinese Remainder Theorem (1)
- Chinese restaurant process (1)
- Circuit (1)
- Closest Vector Problem (1)
- Coamoeba (1)
- Cognitive psychology (1)
- Commitment (1)
- Commitment schemes (1)
- Computational complexity (1)
- Concentration Inequality (1)
- Condensing (1)
- Containment (1)
- Contraction method (1)
- Degenerate Linear Part (1)
- Dessins d'enfants (1)
- Diagrams and mathematical learning (1)
- Dichte <Stochastik> (1)
- Digital and analogue materials (1)
- Digital trees (1)
- Directional selection (1)
- Dirichlet bound (1)
- Dirichlet random measure (1)
- Discrete Logarithm (1)
- Diskrete Geometrie (1)
- Diversity in trait space (1)
- Donkers theorem (1)
- Dopamine (1)
- Dormancy (1)
- Dosis-Wirkungs-Modellierung (1)
- Duality (1)
- Early Childhood (1)
- Einbettung <Mathematik> (1)
- Energie-Modell (1)
- Error Bound (1)
- Evolutionary branching (1)
- Ewens sampling formula (1)
- Examples (1)
- FEM-BEM-coupling (1)
- FID model (1)
- FIND algorithm (1)
- Face (1)
- Face recognition (1)
- Factoring (1)
- Familie (1)
- Family (1)
- Feller branching with logistic growth (1)
- Finite element methods (1)
- Finitely many measurements (1)
- Fixation probability (1)
- Fixpunkt (1)
- Fractional Brownian Motion (1)
- Fractional Laplacian (1)
- Fuchsian groups (1)
- Fächerübergreifender Unterricht (1)
- Galerkin Approximation (1)
- Game Tree (1)
- Gaussian Random Field (1)
- Gaussian process (1)
- Gelfand-Shilov space (1)
- Gemischte Volumen (1)
- Genealogical construction (1)
- Genealogische Konstruktion (1)
- Genus One (1)
- Geometrie (1)
- Geometry (1)
- Gespräch (1)
- Gestaenge (1)
- Girsanov transform (1)
- Gram-Hadamard inequalities (1)
- Griffiths–Engen–McCloskey distribution (1)
- Group dynamics (1)
- Große Abweichung (1)
- Gruppendynamiken (1)
- Hadamard's Three-Lines Theorem (1)
- Handelman (1)
- Handlung (1)
- Heisenberg algebra (1)
- Hidden Markov models (1)
- Hinterlegungsverfahren <Kryptologie> (1)
- Hintertür <Informatik> (1)
- Hodge bundle (1)
- Holzklötzchen (1)
- Hopf algebroids (1)
- Householder reflection (1)
- Hypotrochoid (1)
- Identification (1)
- Immigration (1)
- Index at Infinity (1)
- Infrared singularity (1)
- Integer relations (1)
- Interaction (1)
- Internet (1)
- Inverse problems (1)
- Klebsiella pneumoniae (1)
- Kochen-Specker theorem (1)
- Kollektivintelligenz (1)
- Kombinatorische Optimierung (1)
- Konzentrationsungleichung (1)
- Korkin—Zolotarev reduction (1)
- Kreuzkorrelation (1)
- Kryptosystem (1)
- Kullback-Leibler Informational Divergence (1)
- L^p bounds (1)
- L^p means (1)
- Label cover (1)
- Lanzeitverhalten (1)
- Large Deviation (1)
- Lattice Reduction (1)
- Lernen (1)
- Linear Filtering (1)
- Linear-Implicit Scheme (1)
- Linkages (1)
- Loewner monotonicity and convexity (1)
- Logarithmic Laplacian (1)
- Long- Range Dependence (1)
- Long-Range Dependence (1)
- Long-time behaviour (1)
- Longitudinal Study (1)
- Lotka-Volterra system (1)
- Low density subset sum algorithm (1)
- Machine Learning (1)
- Malliavin calculus (1)
- Mallows model (1)
- Markov chain Monte Carlo Method (1)
- Markov chain imbedding technique (1)
- Markov model (1)
- Markov-Kette (1)
- Mathematical Giftedness (1)
- Mathematical Reasoning (1)
- Mathematical modelling (1)
- Mathematics Learning (1)
- McEliece (1)
- Mean Anisotropy (1)
- Message authentication (1)
- Mixed Volumes (1)
- Modellierung (1)
- Modular Multiplication (1)
- Mooney faces (1)
- Morava K-theory (1)
- Mouse (1)
- Multityp-Verzweigungsprozess mit Immigration (1)
- Multitype Branching with Immigration (1)
- Musik (1)
- NP-complete problems (1)
- NP-hard (1)
- NP-hardness (1)
- Neural encoding (1)
- Neurophysiology (1)
- Neuroscience (1)
- Newton–Okounkov bodies (1)
- Non-Malleability (1)
- Noticeable Probability (1)
- Optimal Mean-Square Filter (1)
- Oracle Query (1)
- Parabolic SPDE (1)
- Participation (1)
- Partizipation (1)
- Pause (1)
- Permutation (1)
- Phragmén-Lindelöf principle (1)
- Piecewise-constant coefficient (1)
- Poisson Process (1)
- Poisson boundary (1)
- Polyedrische Kombinatorik (1)
- Polymorphic evolution sequence (1)
- Polynomial Optimization (1)
- Pontrjagin space (1)
- Populationsdynamiken (1)
- Portfolios (1)
- Positivstellensatz (1)
- Prag <1999> (1)
- Private Information Retrieval (1)
- Probabilistic analysis of algorithms (1)
- Probabilistically checkable proofs (1)
- Probabilistische Analyse von Algorithmen (1)
- Probability distribution (1)
- Probability of fixation (1)
- Profil Likelihood (1)
- Projektionen (1)
- Public Key Cryptosystem (1)
- Public Parameter (1)
- Punktprozess (1)
- Pólya urn (1)
- Quadratic Residue (1)
- Quantum Zeno Effect (1)
- Quantum Zeno effect (1)
- Quickselect (1)
- Radix sort (1)
- Random Oracle (1)
- Random String (1)
- Random environment (1)
- Random variables (1)
- Ray-Knight representation (1)
- Reaction time (1)
- Rekursiver Algorithmus (1)
- Relaxation (1)
- Representation Problem (1)
- Research article (1)
- Riemann surfaces (1)
- Ringtheorie (1)
- Risikobewertung (1)
- Robustheit (1)
- SLLL-reduction (1)
- San Francisco (1)
- Santa Barbara (1)
- Schizophrenia (1)
- Schwarz triangel functions (1)
- Schwinger model (1)
- Security (1)
- Security Parameter (1)
- Semidefinite Optimierung (1)
- Semidefinite Optimization (1)
- Semiotics according to C. S. Peirce (1)
- Sensory perception (1)
- Sensory processing (1)
- Signature (1)
- Small order expansion (1)
- Spectrahedra (1)
- Spiel (1)
- Spielbaum (1)
- Spielbaum-Suchverfahren (1)
- Stable reduction algorithm (1)
- State dependent branching rate (1)
- Stationarity (1)
- Statistik (1)
- Stochastic Analysis of Square Zero Variation Processes (1)
- Stochastik (1)
- Stonesches Spektrum (1)
- Striatum (1)
- Strong Taylor Scheme (1)
- Sum of Squares (1)
- Support (1)
- Symmetrie (1)
- Symmetry (1)
- Sympatric speciation (1)
- Tail Bound (1)
- Tailschranke (1)
- Talk (1)
- Thorne Kishino Felsenstein model (1)
- Topic Model (1)
- Trapdoor (1)
- Trinomial (1)
- Tropical Geometry (1)
- Tropical Grassmannians (1)
- Tropical bases (1)
- Tropical varieties (1)
- Tropische Basen (1)
- Trotter's product formula (1)
- Turkish immigrants (1)
- Typ-In-Algebra (1)
- Typology (1)
- Türkisch (1)
- Uniform regularity (1)
- Uniform resource locators (1)
- Unterstützung (1)
- Valuation on functions (1)
- Verzweigende Teilchensysteme (1)
- Verzweigungsprozess (1)
- Wahrscheinlichkeitsverteilung (1)
- Wiener Index (1)
- Wiener index (1)
- Wiener-Index (1)
- Zolotarev metric (1)
- Zufällige Umgebung (1)
- Zustandsabhängige Verzweigungsrate (1)
- abelian differentials (1)
- algebraic curves (1)
- algebraic values (1)
- alpha-stable branching (1)
- ampleness (1)
- analysis of algorithms (1)
- anti-Zeno effect (1)
- argumentation (1)
- arithmetic ball quotients (1)
- augmented and restricted base loci (1)
- autocorrelograms (1)
- bid-ask spread (1)
- binary search tree (1)
- bordism theory (1)
- branching processes (1)
- branching random walk in random medium (1)
- cancer cell dormancy (1)
- canonical divisors (1)
- catastrophe modeling (1)
- chosen ciphertext attack (1)
- clique problem (1)
- colorabdity (1)
- combinatorial optimization (1)
- compact Riemann surfaces (1)
- complex multiplication (1)
- composition (1)
- computational geometry (1)
- concurrent composition (1)
- condensing (1)
- confirmatory factory analysis (1)
- consensus (1)
- continued fraction algorithm (1)
- convexity (1)
- convolution quadrature (1)
- cooperative systems (1)
- cross correlation (1)
- cryptography (1)
- cycle structure of permutations (1)
- degenerate semigroup (1)
- delay equation (1)
- dessins d’enfants (1)
- difference sets (1)
- digital search tree (1)
- digital tools (1)
- discrete dynamical system (1)
- discrete logarithm (1)
- discrete logarithm (DL) (1)
- dose-resoponse modelling (1)
- doubly stochastic point process (1)
- eigenvalue (1)
- elastodynamic wave equation (1)
- emergence (1)
- endliche metrische Räume (1)
- error bounds (1)
- exponentiation (1)
- external branch (1)
- face inversion (1)
- face perception (1)
- fake projective planes (1)
- families of hash functions (1)
- finite resolution (1)
- firing patterns (1)
- flat surfaces (1)
- floating point arithmetic (1)
- floating point errors (1)
- foliated Schwarz symmetry (1)
- forming a group (1)
- fractional Brownian motion (1)
- fractions of exponentiation (1)
- frühkindliche Erziehung (1)
- functional limit theorem (1)
- functional limit theorems (1)
- generic algorithm (1)
- generic algorithms (1)
- generic complexity (1)
- generic group model (1)
- geometry (1)
- graph coloring (1)
- graph isomorphism (1)
- h-transform (1)
- hard bit (1)
- hardcore subsets (1)
- harmonic function (1)
- heavy tails (1)
- hidden Markov model (1)
- hierarchical mean-field limit (1)
- highly regular nearby points (1)
- hypergeometric functions (1)
- hypervariable region (1)
- incremental schemes (1)
- indefinite inner product space (1)
- individual-based models (1)
- inner product (1)
- integer relation (1)
- integer vector (1)
- interacting particle Systems (1)
- internal diffusion limited aggregation (1)
- internal path length (1)
- inverse coefficient problem, (1)
- iterated subsegments (1)
- key comparisons (1)
- kinetic fingerprint (1)
- knapsack cryptosystems (1)
- large deviations (1)
- latent variance (1)
- lattice basis reduction (1)
- lattices (1)
- leapfrog (1)
- length defect (1)
- limit order markets (1)
- local LLL-reduction (1)
- local LLLreduction (1)
- local coordinates (1)
- local randomness (1)
- local time (1)
- local time drift (1)
- logarithmic geometry (1)
- logical networks (1)
- lookdown construction (1)
- lower bounds (1)
- manifold and geodesic (1)
- market making (1)
- mathematical modeling (1)
- mathematical modelling (1)
- mathematics (1)
- measurement (1)
- message-passing algorithm (1)
- modelling (1)
- modular automorphism group (1)
- modular group (1)
- moduli spaces (1)
- multi-agents system (1)
- multi-drug treatment (1)
- multilevel branching (1)
- music (1)
- mutation parameter estimation (1)
- neuronal code (1)
- neuronaler Kode (1)
- non-archimedean geometry (1)
- non-autonomous dynamical systems (1)
- non-malleability (1)
- noncommutative ring spectra (1)
- nondetermmistlc Turing machines (1)
- numerical experiments (1)
- observable Funktion (1)
- one-more decryption attack (1)
- one-way function (1)
- one-way functions (1)
- operator algebra (1)
- optimal transport (1)
- pair HMM (1)
- partial match queries (1)
- perceptual closure (1)
- phage (1)
- phage therapy (1)
- phase transitions (1)
- poisson process (1)
- polynomial random number generator (1)
- population dynamics (1)
- portfolio optimization (1)
- positivity of line bundles (1)
- probabilistic analysis of algorithms (1)
- probability metric (1)
- professional development (1)
- profile likelihood (1)
- projections (1)
- projective planes (1)
- q-binomial theorem (1)
- quantum field theory (1)
- quincunx (1)
- random environment (1)
- random function generator (1)
- random graphs (1)
- random measures (1)
- random media (1)
- random metric (1)
- random move (1)
- random number generator (1)
- random oracle model (1)
- random partition (1)
- random recursive tree (1)
- random trees (1)
- random walks (1)
- raum-zeitliche Muster (1)
- reactant-catalyst systems (1)
- recursive distributional equation (1)
- resistance (1)
- resistance mutation (1)
- reversibility (1)
- risk assessment (1)
- risk theory (1)
- rotating plane method (1)
- rough paths theory (1)
- satlsfiablhty (1)
- scaling (1)
- searchtrees (1)
- secure bit (1)
- security analysis of protocols (1)
- security of data (1)
- self-organizing groups (1)
- self-organizing groups; population dynamics; collective intelligence; forming groups; metric on finite sets (1)
- semidefinite optimization (1)
- sequence alignment (1)
- set-valued pullback attractors (1)
- shadow price (1)
- short integer relation (1)
- shortest lattice vector (1)
- signature size (1)
- signed ElGamal encryption (1)
- simultaneous diophantine approximations (1)
- simultaneous security of bits (1)
- single block replacement (1)
- spatio-temporal patterns (1)
- statistic analysis (1)
- statistical alignment (1)
- statistische Analyse (1)
- statistischer Test (1)
- stoch. Analyse von Algorithmen (1)
- stochastic filtering (1)
- stochastic modeling (1)
- stochastic population dynamics (1)
- strong transience (1)
- subgroup growth (1)
- subset sum problems (1)
- substitution attacks (1)
- sum of squared factor loadings (1)
- switching systems (1)
- synergistic interaction (1)
- therapy evasion (1)
- topological entropy (1)
- trading strategies (1)
- transcendence (1)
- transversal learning (1)
- treatment protocol design (1)
- treatment success (1)
- tropical geometry (1)
- tropical universal Jacobian (1)
- tropicalization (1)
- universal compactified Jacobian (1)
- urn model (1)
- von Neumann algebra (1)
- von Neumann algebras (1)
- von Neumann-Algebra (1)
- weak convergence (1)
- Λ-coalescent (1)
- σ-field (1)
Institute
- Mathematik (216)
- Informatik (50)
- Medizin (2)
- Frankfurt Institute for Advanced Studies (FIAS) (1)
- MPI für Hirnforschung (1)
- MPI für empirische Ästhetik (1)
- Physik (1)
The behaviour of electronic circuits is influenced by ageing effects. Modelling the behaviour of circuits is a standard approach for the design of faster, smaller, more reliable and more robust systems. In this thesis, we propose a formalization of robustness that is derived from a failure model, which is based purely on the behavioural specification of a system. For a given specification, simulation can reveal if a system does not comply with a specification, and thus provide a failure model. Ageing usually works against the specified properties, and ageing models can be incorporated to quantify the impact on specification violations, failures and robustness. We study ageing effects in the context of analogue circuits. Here, models must factor in infinitely many circuit states. Ageing effects have a cause and an impact that require models. On both these ends, the circuit state is highly relevant, an must be factored in. For example, static empirical models for ageing effects are not valid in many cases, because the assumed operating states do not agree with the circuit simulation results. This thesis identifies essential properties of ageing effects and we argue that they need to be taken into account for modelling the interrelation of cause and impact. These properties include frequency dependence, monotonicity, memory and relaxation mechanisms as well as control by arbitrary shaped stress levels. Starting from decay processes, we define a class of ageing models that fits these requirements well while remaining arithmetically accessible by means of a simple structure.
Modeling ageing effects in semiconductor circuits becomes more relevant with higher integration and smaller structure sizes. With respect to miniaturization, digital systems are ahead of analogue systems, and similarly ageing models predominantly focus on digital applications. In the digital domain, the signal levels are either on or off or switching in between. Given an ageing model as a physical effect bound to signal levels, ageing models for components and whole systems can be inferred by means of average operation modes and cycle counts. Functional and faithful ageing effect models for analogue components often require a more fine-grained characterization for physical processes. Here, signal levels can take arbitrary values, to begin with. Such fine-grained, physically inspired ageing models do not scale for larger applications and are hard to simulate in reasonable time. To close the gap between physical processes and system level ageing simulation, we propose a data based modelling strategy, according to which measurement data is turned into ageing models for analogue applications. Ageing data is a set of pairs of stress patterns and the corresponding parameter deviations. Assuming additional properties, such as monotonicity or frequency independence, learning algorithm can find a complete model that is consistent with the data set. These ageing effect models decompose into a controlling stress level, an ageing process, and a parameter that depends on the state of this process. Using this representation, we are able to embed a wide range of ageing effects into behavioural models for circuit components. Based on the developed modelling techniques, we introduce a novel model for the BTI effect, an ageing effect that permits relaxation. In the following, a transistor level ageing model for BTI that targets analogue circuits is proposed. Similarly, we demonstrate how ageing data from analogue transistor level circuit models lift to purely behavioural block models. With this, we are the first to present a data based hierarchical ageing modeling scheme. An ageing simulator for circuits or system level models computes long term transients, solutions of a differential equation. Long term transients are often close to quasi-periodic, in some sense repetitive. If the evaluation of ageing models under quasi-periodic conditions can be done efficiently, long term simulation becomes practical. We describe an adaptive two-time simulation algorithm that basically skips periods during simulation, advancing faster on a second time axis. The bottleneck of two-time simulation is the extrapolation through skipped frames. This involves both the evaluation of the ageing models and the consistency of the boundary conditions. We propose a simulator that computes long term transients exploiting the structure of the proposed ageing models. These models permit extrapolation of the ageing state by means of a locally equivalent stress, a sort of average stress level. This level can be computed efficiently and also gives rise to a dynamic step control mechanism. Ageing simulation has a wide range of applications. This thesis vastly improves the applicability of ageing simulation for analogue circuits in terms of modelling and efficiency. An ageing effect model that is a part of a circuit component model accounts for parametric drift that is directly related to the operation mode. For example asymmetric load on a comparator or power-stage may lead to offset drift, which is not an empiric effect. Monitor circuits can report such effects during operation, when they become significant. Simulating the behaviour of these monitors is important during their development. Ageing effects can be compensated using redundant parts, and annealing can revert broken components to functional. We show that such mechanisms can be simulated in place using our models and algorithms. The aim of automatized circuit synthesis is to create a circuit that implements a specification for a certain use case. Ageing simulation can identify candidates that are more reliable. Efficient ageing simulation allows to factor in various operation modes and helps refining the selection. Using long term ageing simulation, we have analysed the fitness of a set of synthesized operational amplifiers with similar properties concerning various use cases. This procedure enables the selection of the most ageing resilient implementation automatically.
From Brownian motion with a local time drift to Feller's branching diffusion with logistic growth
(2011)
We give a new proof for a Ray-Knight representation of Feller's branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H
at its current level. In Le et al. (2011) such a representation was obtained by an approximation through Harris paths that code the genealogies of particle systems. The present proof is purely in terms of stochastic analysis, and is inspired by previous work of Norris, Rogers and Williams (1988).
Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) which have a stochastic process in their vector field functions. RODEs have been used in a wide range of applications such as biology, medicine, population dynamics and engineering and play an important role in the theory of random dynamical systems, however, they have been long overshadowed by stochastic differential equations.
Typically, the driving stochastic process has at most Hoelder continuous sample paths and the resulting vector field is, thus, at most Hoelder continuous in time, no matter how smooth the vector function is in its original variables, so the sample paths of the solution are certainly continuously differentiable, but their derivatives are at most Hoelder continuous in time. Consequently, although the classical numerical schemes for ODEs can be applied pathwise to RODEs, they do not achieve their traditional orders.
Recently, Gruene and Kloeden derived the explicit averaged Euler scheme by taking the average of the noise within the vector field. In addition, new forms of higher order Taylor-like schemes for RODEs are derived systematically by Jentzen and Kloeden.
However, it is still important to build higher order numerical schemes and computationally less expensive schemes as well as numerically stable schemes and this is the motivation of this thesis. The schemes by Gruene and Kloeden and Jentzen and Kloeden are very general, so RODEs with special structure, i.e., RODEs with Ito noise and RODEs with affine structure, are focused and numerical schemes which exploit these special structures are investigated.
The developed numerical schemes are applied to several mathematical models in biology and medicine. In order to see the performance of the numerical schemes, trajectories of solutions are illustrated. In addition, the error vs. step sizes as well as the computational costs are compared among newly developed schemes and the schemes in literature.
In the qualitative analysis of solutions of partial differential equations, many interesting questions are related to the shape of solutions. In particular, the symmetries of a given solution are of interest. One of the first more general results in this direction was given in 1979 by Gidas, Ni and Nirenberg... The main tool in proving this symmetry and monotonicity result is the moving plane method. This method, which goes back to Alexandrov’s work on constant mean curvature surfaces in 1962, was introduced in 1971 by Serrin in the context of partial differential equations to analyze an overdetermined problem...
Triangles of groups have been introduced by Gersten and Stallings. They are, roughly speaking, a generalization of the amalgamated free product of two groups and occur in the framework of Corson diagrams. First, we prove an intersection theorem for Corson diagrams. Then, we focus on triangles of groups. It has been shown by Howie and Kopteva that the colimit of a hyperbolic triangle of groups contains a non-abelian free subgroup. We give two natural conditions, each of which ensures that the colimit of a non-spherical triangle of groups either contains a non-abelian free subgroup or is virtually solvable.
This work proposes to employ the (bursty) GLO model from Bingmer et. al (2011) to model the occurrence of tropical cyclones. We develop a Bayesian framework to estimate the parameters of the model and, particularly, employ a Markov chain Monte Carlo algorithm. This also allows us to develop a forecasting framework for future events.
Moreover, we assess the default probability of an insurance company that is exposed to claims that occur according to a GLO process and show that the model is able to substantially improve actuarial risk management if events occur in oscillatory bursts.
Containment problems belong to the classical problems of (convex) geometry. In the proper sense, a containment problem is the task to decide the set-theoretic inclusion of two given sets, which is hard from both the theoretical and the practical perspective. In a broader sense, this includes, e.g., radii or packing problems, which are even harder. For some classes of convex sets there has been strong interest in containment problems. This includes containment problems of polyhedra and balls, and containment of polyhedra, which have been studied in the late 20th century because of their inherent relevance in linear programming and combinatorics.
Since then, there has only been limited progress in understanding containment problems of that type. In recent years, containment problems for spectrahedra, which naturally generalize the class of polyhedra, have seen great interest. This interest is particularly driven by the intrinsic relevance of spectrahedra and their projections in polynomial optimization and convex algebraic geometry. Except for the treatment of special classes or situations, there has been no overall treatment of that kind of problems, though.
In this thesis, we provide a comprehensive treatment of containment problems concerning polyhedra, spectrahedra, and their projections from the viewpoint of low-degree semialgebraic problems and study algebraic certificates for containment. This leads to a new and systematic access to studying containment problems of (projections of) polyhedra and spectrahedra, and provides several new and partially unexpected results.
The main idea - which is meanwhile common in polynomial optimization, but whose understanding of the particular potential on low-degree geometric problems is still a major challenge - can be explained as follows. One point of view towards linear programming is as an application of Farkas' Lemma which characterizes the (non-)solvability of a system of linear inequalities. The affine form of Farkas' Lemma characterizes linear polynomials which are nonnegative on a given polyhedron. By omitting the linearity condition, one gets a polynomial nonnegativity question on a semialgebraic set, leading to so-called Positivstellensaetze (or, more precisely Nichtnegativstellensaetze). A Positivstellensatz provides a certificate for the positivity of a polynomial function in terms of a polynomial identity. As in the linear case, these Positivstellensaetze are the foundation of polynomial optimization and relaxation methods. The transition from positivity to nonnegativity is still a major challenge in real algebraic geometry and polynomial optimization.
With this in mind, several principal questions arise in the context of containment problems: Can the particular containment problem be formulated as a polynomial nonnegativity (or, feasibility) problem in a sophisticated way? If so, how are positivity and nonnegativity related to the containment question in the sense of their geometric meaning? Is there a sophisticated Positivstellensatz for the particular situation, yielding certificates for containment? Concerning the degree of the semialgebraic certificates, which degree is necessary, which degree is sufficient to decide containment?
Indeed, (almost) all containment problems studied in this thesis can be formulated as polynomial nonnegativity problems allowing the application of semialgebraic relaxations. Other than this general result, the answer to all the other questions (highly) depends on the specific containment problem, particularly with regard to its underlying geometry. An important point is whether the hierarchies coming from increasing the degree in the polynomial relaxations always decide containment in finitely many steps.
We focus on the containment problem of an H-polytope in a V-polytope and of a spectrahedron in a spectrahedron. Moreover, we address containment problems concerning projections of H-polyhedra and spectrahedra. This selection is justified by the fact that the mentioned containment problems are computationally hard and their geometry is not well understood.
This thesis covers the analysis of radix sort, radix select and the path length of digital trees under a stochastic input assumption known as the Markov model.
The main results are asymptotic expansions of mean and variance as well as a central limit theorem for the complexity of radix sort and the path length of tries, PATRICIA tries and digital search trees.
Concerning radix select, a variety of different models for ranks are discussed including a law of large numbers for the worst case behavior, a limit theorem for the grand averages model and the first order asymptotic of the average complexity in the quantile model.
Some of the results are achieved by moment transfer techniques, the limit laws are based on a novel use of the contraction method suited for systems of stochastic recurrences.
This work is concerned with two topics at the intersection of convex algebraic geometry and optimization.
We develop a new method for the optimization of polynomials over polytopes. From the point of view of convex algebraic geometry the most common method for the approximation of polynomial optimization problems is to solve semidefinite programming relaxations coming from the application of Positivstellensätze. In optimization, non-linear programming problems are often solved using branch and bound methods. We propose a fused method that uses Positivstellensatz-relaxations as lower bounding methods in a branch and bound scheme. By deriving a new error bound for Handelman's Positivstellensatz, we show convergence of the resulting branch and bound method. Through the application of Positivstellensätze, semidefinite programming has gained importance in polynomial optimization in recent years. While it arises to be a powerful tool, the underlying geometry of the feasibility regions (spectrahedra) is not yet well understood. In this work, we study polyhedral and spectrahedral containment problems, in particular we classify their complexity and introduce sufficient criteria to certify the containment of one spectrahedron in another one.
The cones of nonnegative polynomials and sums of squares arise as central objects in convex algebraic geometry and have their origin in the seminal work of Hilbert ([Hil88]). Depending on the number of variables n and the degree d of the polynomials, Hilbert famously characterizes all cases of equality between the cone of nonnegative polynomials and the cone of sums of squares. This equality precisely holds for bivariate forms, quadratic forms and ternary quartics ([Hil88]). Since then, a lot of work has been done in understanding the difference between these two cones, which has major consequences for many practical applications such as for polynomial optimization problems. Roughly speaking, minimizing polynomial functions (constrained as well as unconstrained) can be done efficiently whenever certain nonnegative polynomials can be written as sums of squares (see Section 2.3 for the precise relationship). The underlying reason is the fundamental difference that checking nonnegativity of polynomials is an NP-hard problem whenever the degree is greater or equal than four ([BCSS98]), whereas checking whether a polynomial can be written as a sum of squares is a semidefinite feasibility problem (see Section 2.2). Although the complexity status of the semidefinite feasibility problem is still an open problem, it is polynomial for fixed number of variables. Hence, understanding the difference between nonnegative polynomials and sums of squares is highly desirable both from a theoretical and a practical viewpoint.
We consider a class of nonautonomous nonlinear competitive parabolic systems on bounded radial domains under Neumann or Dirichlet boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then bounded positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of scalar equations with Neumann or Dirichlet boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.
A multiple filter test for the detection of rate changes in renewal processes with varying variance
(2014)
The thesis provides novel procedures in the statistical field of change point detection in time series.
Motivated by a variety of neuronal spike train patterns, a broad stochastic point process model is introduced. This model features points in time (change points), where the associated event rate changes. For purposes of change point detection, filtered derivative processes (MOSUM) are studied. Functional limit theorems for the filtered derivative processes are derived. These results are used to support novel procedures for change point detection; in particular, multiple filters (bandwidths) are applied simultaneously in oder to detect change points in different time scales.
The work presented in this thesis is devoted to two classes of mathematical population genetics models, namely the Kingman-coalescent and the Beta-coalescents. Chapters 2, 3 and 4 of the thesis include results concerned with the first model, whereas Chapter 5 presents contributions to the second class of models.
The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group ΩN consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit N→∞ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls B(N)ℓ of hierarchical radius ℓ converge to a backward Markov chain on R+. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.
We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from $\alpha$-stable branching mechanisms. The random ancestral partition is then a time-changed $\Lambda$-coalescent, where $\Lambda$ is the Beta-distribution with parameters $2-\alpha$ and $\alpha$, and the time change is given by $Z^{1-\alpha}$, where $Z$ is the total population size. For $\alpha = 2$ (Feller's branching diffusion) and $\Lambda = \delta_0$ (Kingman's coalescent), this is in the spirit of (a non-spatial version of) Perkins' Disintegration Theorem. For $\alpha =1$ and $\Lambda$ the uniform distribution on $[0,1]$, this is the duality discovered by Bertoin & Le Gall (2000) between the norming of Neveu's continuous state branching process and the Bolthausen-Sznitman coalescent.
We present two approaches: one, exploiting the `modified lookdown construction', draws heavily on Donnelly & Kurtz (1999); the other is based on direct calculations with generators.
In this paper we prove asymptotic normality of the total length of external branches in Kingman's coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with n black balls. Empty it in n steps according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers Uk, 0 < k < n, of red balls after k steps exhibit an unexpected property: (U0, ... ,Un) and (Un, ... ;U0) are equal in distribution.
The random split tree introduced by Devroye (1999) is considered. We derive a second order expansion for the mean of its internal path length and furthermore obtain a limit law by the contraction method. As an assumption we need the splitter having a Lebesgue density and mass in every neighborhood of 1. We use properly stopped homogeneous Markov chains, for which limit results in total variation distance as well as renewal theory are used. Furthermore, we extend this method to obtain the corresponding results for the Wiener index.
ranching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where \lq in each generation' the reproduction law is picked randomly in an i.i.d. manner. The associated random walk of the environment has increments distributed like the logarithmic mean of the offspring distributions. This random walk plays a key role in the asymptotic behavior. In this paper, we study the upper large deviations of the BPRE $Z$ when the reproduction law may have heavy tails. More precisely, we obtain an expression for the limit of $-\log \mathbb{P}(Z_n\geq \exp(\theta n))/n$ when $n\rightarrow \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\rightarrow\infty} \log \mathbb{P}(Z_n>0)/n$ and the polynomial rate of decay $\beta$ of the tail distribution of $Z_1$. This rate function can be interpreted as the optimal way to reach a given "large" value. We then compute the rate function when the reproduction law does not have heavy tails. Our results generalize the results of B\"oinghoff $\&$ Kersting (2009) and Bansaye $\&$ Berestycki (2008) for upper large deviations. Finally, we derive the upper large deviations for the Galton-Watson processes with heavy tails.
In this article, we illustrate the flexibility of the algebraic integration formalism introduced in M. Gubinelli (2004), Controlling Rough Paths, J. Funct. Anal. 216, 86-140, by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter H > 1/3.
We consider catalytic branching random walk (the reactant) where the state space is a countable Abelean group. The branching is critical binary and the local branching rate is given by a catalytic medium. Here the medium is itself an autonomous (ordinary) branching random walk (the catalyst) - maybe with a different motion law. For persistent catalyst (transient motion) the reactant shows the usual dichotomy of persistence versus extinction depending on transience or recurrence of its motion. If the catalyst goes to local extinction it turns out that the longtime behaviour of the reactant ranges (depending on its motion) from local extinction to free random walk with either deterministic or random global intensity of particles.