510 Mathematik
Refine
Has Fulltext
- yes (515) (remove)
Year of publication
Document Type
- Preprint (372)
- Doctoral Thesis (60)
- Postprint (45)
- Article (19)
- Monograph/Edited Volume (7)
- Master's Thesis (6)
- Bachelor Thesis (2)
- Conference Proceeding (2)
- Course Material (1)
- Report (1)
Language
- English (484)
- German (27)
- French (3)
- Multiple languages (1)
Keywords
- random point processes (18)
- statistical mechanics (18)
- stochastic analysis (18)
- index (12)
- boundary value problems (11)
- elliptic operators (9)
- Fredholm property (8)
- cluster expansion (8)
- K-theory (7)
- manifolds with singularities (7)
- pseudodifferential operators (7)
- regularization (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)
- index theory (5)
- infinite-dimensional Brownian diffusion (5)
- Boundary value problems (4)
- Cauchy problem (4)
- Dirac operator (4)
- Lefschetz number (4)
- Markov processes (4)
- Modellierung (4)
- Probabilistic Cellular Automata (4)
- Toeplitz operators (4)
- differential operators (4)
- elliptic complexes (4)
- elliptic operator (4)
- ellipticity (4)
- manifold with singularities (4)
- reciprocal class (4)
- spectral flow (4)
- surgery (4)
- 'eta' invariant (3)
- Atiyah-Bott condition (3)
- Dirac operators (3)
- Fredholm operators (3)
- Hodge theory (3)
- Mathematikdidaktik (3)
- Wahrscheinlichkeitstheorie (3)
- boundary value problem (3)
- conical singularities (3)
- conormal symbol (3)
- counting process (3)
- eta invariant (3)
- index of elliptic operators in subspaces (3)
- local time (3)
- parameter estimation (3)
- reciprocal processes (3)
- reversible measure (3)
- star product (3)
- stochastic differential equations (3)
- the Cauchy problem (3)
- Alterung (2)
- Atiyah-Bott obstruction (2)
- Blickbewegungen (2)
- Brownian bridge (2)
- Bruchzahlen (2)
- Canonical Gibbs measure (2)
- Carleman matrix (2)
- Chemotaxis (2)
- Cluster expansion (2)
- Corona (2)
- Diracoperator (2)
- Eigenvalues (2)
- Fredholm complexes (2)
- Gibbs field (2)
- Gibbs measure (2)
- Gibbs point processes (2)
- Heat equation (2)
- Kunststofflichtwellenleiter (2)
- Lagrangian submanifolds (2)
- Langzeitverhalten (2)
- Laplace equation (2)
- Lefschetz fixed point formula (2)
- Levy measure (2)
- Lichtwellenleiter (2)
- Malliavin calculus (2)
- Markov chain (2)
- Markovprozesse (2)
- Mathematikunterricht (2)
- Mellin transform (2)
- Navier-Stokes equations (2)
- Neumann problem (2)
- POF (2)
- Perturbed complexes (2)
- Pseudo-differential operators (2)
- Randwertprobleme (2)
- Riemann-Hilbert problem (2)
- Riemannian manifold (2)
- Spin Geometry (2)
- Stochastic Differential Equation (2)
- Stochastik (2)
- Streuung (2)
- TIMSS (2)
- Temperatur (2)
- Tunneleffekt (2)
- Unterrichtsmethode (2)
- Verzweigungsprozess (2)
- Videostudie (2)
- Vietnam (2)
- WKB method (2)
- Wiener measure (2)
- Zaremba problem (2)
- analytic continuation (2)
- asymptotic expansion (2)
- birth-death-mutation-competition point process (2)
- boundary layer (2)
- branching process (2)
- censoring (2)
- coupling (2)
- curvature (2)
- detailed balance equation (2)
- didactics of mathematics (2)
- discrepancy principle (2)
- duality formula (2)
- edge singularities (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic families (2)
- elliptic family (2)
- elliptic system (2)
- eta-invariant (2)
- evolution equation (2)
- exact simulation (2)
- eye movements (2)
- geodesics (2)
- geometry (2)
- hard core potential (2)
- heat equation (2)
- holomorphic solution (2)
- homotopy classification (2)
- index formulas (2)
- infinite divisibility (2)
- infinitely divisible point processes (2)
- integral formulas (2)
- inversion (2)
- linking coefficients (2)
- long-time behaviour (2)
- manifolds with conical singularities (2)
- manifolds with edges (2)
- mathematics education (2)
- mathematische Modellierung (2)
- maximum likelihood estimator (2)
- modeling (2)
- modelling optical fibres waveguides pof scattering temperature aging ageing (2)
- modn-index (2)
- molecular motor (2)
- monodromy matrix (2)
- monotone coupling (2)
- non-Markov drift (2)
- nonlinear operator (2)
- optische Fasern (2)
- p-Laplace operator (2)
- path integral (2)
- pseudo-differential boundary value problems (2)
- pseudo-differential operators (2)
- pseudodiferential operators (2)
- pseudodifferential operator (2)
- quantization (2)
- regularizer (2)
- regularizers (2)
- residue (2)
- root functions (2)
- singular partial differential equation (2)
- singular perturbation (2)
- skew Brownian motion (2)
- small parameter (2)
- spectral theorem (2)
- star-product (2)
- stochastic bridge (2)
- stochastic ordering (2)
- symmetry conditions (2)
- teaching methods (2)
- time duality (2)
- tunneling (2)
- weighted edge spaces (2)
- weighted spaces (2)
- (co)boundary operator (1)
- (sub-) tropical Africa (1)
- (sub-) tropisches Afrika (1)
- 1st Eigenvalue (1)
- 26D15 (1)
- 31C20 (1)
- 35B09 (1)
- 35R02 (1)
- 39A12 (primary) (1)
- 4-Mannigfaltigkeiten (1)
- 53C12 (1)
- 53C27 (1)
- 58E35 (secondary) (1)
- ACAT (1)
- AFLP (1)
- APS problem (1)
- Activity Theory (1)
- Aerosole (1)
- Algorithmen (1)
- Algorithms (1)
- Anfangsrandwertproblem (1)
- Angle (1)
- Answer Set Programming (1)
- Antwortmengenprogrammierung (1)
- Approximate likelihood (1)
- Argumentation (1)
- Artificial Intelligence (1)
- Asymptotische Entwicklung (1)
- Atiyah-Singer theorem (1)
- Attractive Dynamics (1)
- Aufgabensammlung (1)
- Aussterbewahrscheinlichkeit (1)
- Averaging principle (1)
- Baumweite (1)
- Bayesian inference (1)
- Bayesian inversion (1)
- Bayessche Inferenz (1)
- Beltrami equation (1)
- Bewegungsgleichung (1)
- Beweis (1)
- Beweisassistent (1)
- Beweisumgebung (1)
- Bienaymé-Galton-Watson Prozess (1)
- Bienaymé-Galton-Watson process (1)
- Borel Funktionen (1)
- Borel functions (1)
- Bose-Einstein condensation (1)
- Boundary Value Problems (1)
- Boundary value methods (1)
- Boundary value problems for first order systems (1)
- Boundary-contact problems (1)
- Bounds (1)
- Boutet de Monvel's calculus (1)
- Boutet de Monvels Kalkül (1)
- Brownian motion (1)
- Brownian motion with discontinuous drift (1)
- Bruck-Reilly extension (1)
- Bulk-mediated diffusion; (1)
- C-asterisk-algebra (1)
- C0−semigroup (1)
- CCR-algebra (1)
- Calculus of conormal symbols (1)
- Calderón projections (1)
- Capture into resonance (1)
- Carleman formulas (1)
- Case-Cohort-Design (1)
- Casped plates (1)
- Categories of stratified spaces (1)
- Cauchy Riemann operator (1)
- Cauchy data spaces (1)
- Cauchy horizon (1)
- Cauchyhorizont (1)
- Censoring (1)
- Chern character (1)
- Clifford algebra (1)
- Cloze predictability (1)
- Cloze-Vorhersagbarkeit (1)
- Cluster Entwicklung (1)
- Cluster-Expansion (1)
- Clusteranalyse (1)
- Coarea Formel (1)
- Collatz Conjecture (1)
- Computational Complexity (1)
- Conradian ordered groups (1)
- Coq (1)
- Corner boundary value problems (1)
- Coupling (1)
- Covid-19 (1)
- Cox model (1)
- Cox-Modell (1)
- Crack theory (1)
- Cross-effects (1)
- Curry (1)
- DLR equation (1)
- DLR equations (1)
- DLR-Gleichungen (1)
- Delaney–Dress tiling theory (1)
- Deligne Cohomology (1)
- Deligne Kohomologie (1)
- Denkhürden (1)
- Detektion multipler Übergänge (1)
- Determinantal point processes (1)
- Determinante (1)
- Dichte eines Maßes (1)
- Dichte von rationalen Zahlen (1)
- Didaktik der Mathematik (1)
- Differentialgeometrie (1)
- Differentialoperatoren (1)
- Differenzenoperator (1)
- Diffusionsprozess (1)
- Digital Tools (1)
- Digitale Werkzeuge (1)
- Diophantine Approximation (1)
- Dirac Operator (1)
- Dirac-harmonic maps (1)
- Dirac-harmonische Abbildungen (1)
- Dirac-type operator (1)
- Dirichlet form (1)
- Dirichlet problem (1)
- Dirichlet to Neumann operator (1)
- Dirichlet-to-Neumann operator (1)
- Divisionsbäume (1)
- Doblin (1)
- Dobrushin criterion (1)
- Dobrushin-Kriterium (1)
- Doeblin (1)
- Double Colored Edges (1)
- Duality formula (1)
- Dualitätsformeln (1)
- Dubrovinring (1)
- Dynamic Programming (1)
- Dynamische Programmierung (1)
- Dynamische kognitive Modellierung (1)
- Döblin (1)
- EM (1)
- ERgodicity of Markov Chains (1)
- Edge-degenerate operators (1)
- Eichtheorie (1)
- Einstein manifolds (1)
- Einstein-Hilbert action (1)
- Einstein-Hilbert-Wirkung (1)
- Einstein-Mannigfaltigkeiten (1)
- Elastizität (1)
- Elektrodynamik (1)
- Elliptic complexes (1)
- Elliptic equation with order degeneration (1)
- Elliptic operators (1)
- Elliptic operators in domains with edges (1)
- Elliptische Komplexe (1)
- Elliptizität (1)
- Entropiemethode (1)
- Entropy method (1)
- Entstehungsfragestellung (1)
- Epidemiologie (1)
- Epidemiology (1)
- Estimation for branching processes (1)
- Estimation-of-Distribution-Algorithmen (1)
- Euler equations (1)
- Euler operator (1)
- Euler's theta functions (1)
- Euler-Lagrange equations (1)
- Evolutionsgleichung (1)
- Exponential Time Hypothesis (1)
- Exponentialzeit Hypothese (1)
- Extremal problem (1)
- Eyring-Kramers Formel (1)
- Feedback (1)
- Feller Diffusionsprozesse (1)
- Feller diffusion processes (1)
- Feynman-Kac formula (1)
- Fibroblasten (1)
- Filterung (1)
- Finite difference method (1)
- Finsler-Abstand (1)
- Finsler-distance (1)
- Fischer-Riesz equations (1)
- Fitness (1)
- Fixationsbewegungen der Augen (1)
- Flocking (1)
- Form (1)
- Formalismus (1)
- Formalitätsgrad (1)
- Fourier Integraloperatoren (1)
- Fourth order Sturm-Liouville problem (1)
- Fredholm Komplexe (1)
- Fredholm operator (1)
- Frühförderung (1)
- Fundamentale Ideen (1)
- Funktorgeometrie (1)
- G-index (1)
- G-trace (1)
- Gamification (1)
- Gauge theory (1)
- Gaussian Loop Processes (1)
- Gaussian processes (1)
- Gauß-Prozesse (1)
- Gaußsche Loopprozess (1)
- Generalized translation operator (1)
- Geodäten (1)
- Geomagnetismus (1)
- Geometrie (1)
- Geometrieunterricht (1)
- Geometrische Reproduktionsverteilung (1)
- Gerben (1)
- Gerbes (1)
- Gevrey classes (1)
- Gibbs measures (1)
- Gibbs perturbation (1)
- Gibbs state (1)
- Gibbssche Punktprozesse (1)
- Girsanov formula (1)
- Global Analysis (1)
- Global Differentialgeometry (1)
- Globale Differentialgeometrie (1)
- Goursat problem (1)
- Gradient flow (1)
- Gradientenfluss (1)
- Graph (1)
- Gravitation (1)
- Gravitationswelle (1)
- Green and Mellin edge operators (1)
- Green formula (1)
- Green operator (1)
- Green's operator (1)
- Grenzwertsatz (1)
- Grundvorstellungen (1)
- Grushin operator (1)
- Gutzwiller formula (1)
- HIV (1)
- HIV Erkrankung (1)
- Hamilton-Jacobi theory (1)
- Hamiltonian group action (1)
- Hardy‘s inequality (1)
- Hauptfaserbündel (1)
- Heat Flow (1)
- Hermeneutik (1)
- Historie der Verzweigungsprozesse (1)
- Holonomie (1)
- Holonomy (1)
- Hughes-free (1)
- Hughes-frei (1)
- Hyperbolic-parabolic system (1)
- Hypoelliptic operators (1)
- Hypoellipticity (1)
- Hölder-type source condition (1)
- IP core (1)
- Ill-posed problem (1)
- Index Theorie (1)
- Index theory (1)
- Infinite dimensional manifolds (1)
- Infinite-dimensional SDE (1)
- Innovation (1)
- Instabilität des Prozesses (1)
- Integrability (1)
- Interacting Diffusion Processes (1)
- Interacting Particle Systems (1)
- Interacting particle systems (1)
- Inversion (1)
- K-Means Verfahren (1)
- KS model (1)
- Kalman Filter (1)
- Kalman filter (1)
- Kanten-Randwertprobleme (1)
- Kern Methoden (1)
- Kirkwood--Salsburg equations (1)
- Kirkwood-Salsburg-Gleichungen (1)
- Knowledge Representation and Reasoning (1)
- Kognitionspsychologie (1)
- Kolmogorov-Gleichung (1)
- Kolmogorov-Smirnov type tests (1)
- Kombinationstherapie (1)
- Komplexitätstheorie (1)
- Konfidenzintervall (1)
- Kontinuumsgrenzwert (1)
- Kontrolltheorie (1)
- Konvergenzrate (1)
- Kopplung (1)
- Korn’s weighted inequality (1)
- Kritikalitätstheorem (1)
- Kulturelle Aktivität (1)
- Kähler-Mannigfaltigkeit (1)
- Künstliche Intelligenz (1)
- L2 metrics (1)
- L2-Metrik (1)
- Lagrange Distributionen (1)
- Lagrangian system (1)
- Lame system (1)
- Lamé system (1)
- Langevin Dynamics (1)
- Langevin diffusions (1)
- Langevin-Diffusions (1)
- Laplace expansion (1)
- Laplace-Beltrami operator (1)
- Latent Semantic Analysis (1)
- Latente-Semantische-Analyse (1)
- Laufzeitanalyse (1)
- Laufzeittomographie (1)
- Lehrpotential (1)
- Lehrtext (1)
- Leistungstests (1)
- Leitidee „Daten und Zufall“ (1)
- Lernen (1)
- Lernspiele (1)
- Lesen (1)
- Levy Maß (1)
- Levy flights (1)
- Levy process (1)
- Lidar (1)
- Liouville theorem (1)
- Lipschitz domains (1)
- Logarithmic Sobolev inequality (1)
- Logik (1)
- Logrank test (1)
- Lokalitätsprinzip (1)
- Loop space (1)
- Lorentzgeometrie (1)
- Lorentzian Geometry (1)
- Lower Bounds (1)
- Lumping (1)
- Lévy diffusion approximation (1)
- Lévy diffusions on manifolds (1)
- Lévy measure (1)
- Lévy type processes (1)
- Lφ spectrum (1)
- MSAP (1)
- Magnetfeldmodellierung (1)
- Magnus expansion (1)
- Mannigfaltigkeit (1)
- Mannigfaltigkeiten mit Kante (1)
- Mannigfaltigkeiten mit Singularitäten (1)
- Markierte Gibbs-Punkt-Prozesse (1)
- Markov chains (1)
- Markov-Ketten (1)
- Markov-field property (1)
- Martin-Dynkin boundary (1)
- Maslov and Conley–Zehnder index (1)
- Mathematics textbooks (1)
- Mathematik (1)
- Mathematikphilosophie (1)
- Mathematische Physik (1)
- Mehrtyp-Verzweigungsprozesse (1)
- Mellin-Symbole (1)
- Mellin-Symbols (1)
- Meromorphic operator functions (1)
- Metastabilität (1)
- Methode (1)
- Microdialyse (1)
- Mikrophysik (1)
- Mikrosakkaden (1)
- Mikrosakkadensequenzen (1)
- Milnor Moore theorem (1)
- Minimax Optimalität (1)
- Modellreduktion (1)
- Moduli spaces (1)
- Modulraum (1)
- Monte Carlo testing (1)
- Montel theorem (1)
- Morse-Smale property (1)
- Motivation (1)
- Multidimensional nonisentropic hydrodynamic model (1)
- Multitype branching processes (1)
- Multizeta-Abbildungen (1)
- NLME (1)
- Navier-Stokes-Gleichungen (1)
- Navier-Stoks equations (1)
- Newton Polytope (1)
- Newton method (1)
- Newton polytopes (1)
- Non-linear (1)
- Non-proportional hazards (1)
- Nonlinear (1)
- Nonlinear Laplace operator (1)
- Normalenbündel (1)
- Numerov's method (1)
- Operators on manifolds with conical singularities (1)
- Operators on manifolds with edge (1)
- Operators on manifolds with edge and conical exit to infinity (1)
- Operators on manifolds with second order singularities (1)
- Operatortheorie (1)
- Orbifolds (1)
- Ordnungs-Filtrierung (1)
- Ott–Antonsen equation (1)
- PBPK (1)
- Papangelou Process (1)
- Papangelou-Prozess (1)
- Parameter Schätzung (1)
- Parameterized Complexity (1)
- Parametrices (1)
- Parametrisierte Komplexität (1)
- Partial Integration (1)
- Peano phenomena (1)
- Perron's method (1)
- Pfadintegrale (1)
- Pharmakokinetik (1)
- Physik (1)
- Plio-Pleistocene (1)
- Plio-Pleistozän (1)
- Poincaré Birkhoff Witt theorem (1)
- Point Processes (1)
- Point process (1)
- Poisson bridge (1)
- Polya Process (1)
- Polyascher Prozess (1)
- Polymere (1)
- Pontrjagin duality (1)
- Populationen (1)
- Populations Analyse (1)
- Prinicipal Fibre Bundles (1)
- Proportional hazards (1)
- Pseudo-Differentialoperatoren (1)
- Pseudodifferential operators (1)
- Pseudodifferentialoperatoren auf dem Torus (1)
- Psycholinguistik (1)
- Punktprozess (1)
- Punktprozesse (1)
- Quantenfeldtheorie (1)
- Quantizer (1)
- Quasiconformal mapping (1)
- Quasilinear equations (1)
- Quasilinear hyperbolic system (1)
- Quotientenschiefkörper (1)
- Rahmenlehrplan (1)
- Ramified Cauchy problem (1)
- Randbedingungen (1)
- Random Field Ising Model (1)
- Raumzeiten mit zeitartigen Rand (1)
- Realistic Mathematics Education (1)
- Reciprocal process (1)
- Reflektierende Randbedingungen (1)
- Regularisierung (1)
- Reihendarstellungen (1)
- Rektifizierbarkeit höherer Ordnung (1)
- Renormalized integral (1)
- Reproduktionsrate (1)
- Resampling (1)
- Retrieval (1)
- Ricci flow (1)
- Ricci-Fluss (1)
- Riemann-Roch theorem (1)
- Riemannsche Geometrie (1)
- Riesz continuity (1)
- Risikoanalyse (1)
- Risk analysis (1)
- Runge-Kutta methods (1)
- SPDEs (1)
- Sakkadendetektion (1)
- Saturation model (1)
- Satz von Milnor Moore (1)
- Satz von Poincaré Birkhoff Witt (1)
- Satzverarbeitung (1)
- Schrödinger problem (1)
- Schulbuch (1)
- Schwarzes Loch (1)
- Schätzung von Verzweigungsprozessen (1)
- Second order elliptic equations (1)
- Seiberg-Witten theory (1)
- Seiberg-Witten-Invariante (1)
- Sekundarstufe I (1)
- Selbstassemblierung (1)
- Semi-klasische Abschätzung (1)
- Semiklassik (1)
- Semiklassische Spektralasymptotik (1)
- Sequenzielle Likelihood (1)
- Simulation (1)
- Simulationsstudien (1)
- Skew Diffusionen (1)
- Skorokhod' s invariance principle (1)
- Sobolev problem (1)
- Sobolev spaces with double weights on singular cones (1)
- Space (1)
- Space-Time Cluster Expansions (1)
- Spectral Geometry (1)
- Spektraltheorie (1)
- Spezifikationstests (1)
- Spiel (1)
- Spin Geometrie (1)
- Spin Hall effekte (1)
- State Machine (1)
- Stichprobenentnahme aus einem statistischen Modell (1)
- Stochastic Ordering (1)
- Stochastic differential equations (1)
- Stochastische Analysis (1)
- Stochastische Zellulare Automaten (1)
- Streuamplitude (1)
- Streutheorie (1)
- Strings (1)
- Strukturbildung (1)
- Sturm-Liouville problems (1)
- Submanifolds (1)
- Supergeometrie (1)
- Surface potentials with asymptotics (1)
- Survival models with covariates (1)
- System of nonlocal PDE of first order (1)
- Systempharmakologie (1)
- Taktik (1)
- Testfähigkeit (1)
- Tetration (1)
- Textbook analysis (1)
- Textbook research (1)
- Theorie (1)
- Treewidth (1)
- Two-level interacting process (1)
- Two-sample tests (1)
- Tätigkeitstheorie (1)
- Unendlichdimensionale Mannigfaltigkeit (1)
- Untere Schranken (1)
- Variationsrechnung (1)
- Variationsstabilität (1)
- Varifaltigkeit (1)
- Verzweigungsprozesse (1)
- Vincent (1)
- Viscosity solutions (1)
- Vitali theorem (1)
- Volterra symbols (1)
- Wahrscheinlichkeitsverteilung (1)
- Wasserstein distance (1)
- Wave operator (1)
- Waveletanalyse (1)
- Weak Mixing Condition (1)
- Wechselwirkende Teilchensysteme (1)
- Wellengleichung (1)
- Weyl algebras bundle (1)
- Weyl symbol (1)
- Winkel (1)
- Wissensrepräsentation und Schlussfolgerung (1)
- Wolfgang (1)
- Wort-n-Gramme-Wahrscheinlichkeit (1)
- Wärmefluss (1)
- Wärmekern (1)
- Wärmeleitungsgleichung (1)
- Zahlbereichserweiterung (1)
- Zahlerwerb (1)
- Zellmotilität (1)
- Zufallsvariable (1)
- Zustandsschätzung (1)
- Zählprozesse (1)
- absorbing boundary (1)
- absorbing set (1)
- absorption (1)
- accelerated life time model (1)
- aerosol distribution (1)
- aerosols (1)
- amoeboid motion (1)
- amöboide Bewegung (1)
- analytic index (1)
- angewandte Mathematik (1)
- anisotropic spaces (1)
- anomalous Brownian motion (1)
- anomalous diffusion (1)
- applied mathematics (1)
- approximate differentiability (1)
- approximative Differenzierbarkeit (1)
- aptitude tests (1)
- argumentation (1)
- asymptotic behavior (1)
- asymptotic method (1)
- asymptotic methods (1)
- asymptotic stable (1)
- asymptotical normal distribution (1)
- asymptotics (1)
- asymptotics of solutions (1)
- asymptotische Entwicklung (1)
- asymptotische Normalverteilung (1)
- bar with variable cross-section (1)
- basic ideas ('Grundvorstellungen') (1)
- bedingter Erwartungswert (1)
- bending of an orthotropic cusped plate (1)
- black hole (1)
- boun- dedness (1)
- boundary conditions (1)
- boundary regularity (1)
- boundary values problems (1)
- bounds (1)
- branching processes (1)
- bridge (1)
- bridges of random walks (1)
- bundles (1)
- calculus of variations (1)
- canonical Marcus integration (1)
- canonical discretization schemes (1)
- cell motility (1)
- characteristic boundary point (1)
- characteristic points (1)
- chimera states (1)
- classical and quantum reduction (1)
- classical solution (1)
- cluster analysis (1)
- coarea formula (1)
- coercivity (1)
- cognitive psychology (1)
- cohomology (1)
- commutator subgroup (1)
- compact groups (1)
- comparison principle (1)
- completeness (1)
- composition operator (1)
- compound Poisson processes (1)
- compressible Euler equations (1)
- concurrent checking (1)
- conditional Wiener measure (1)
- conditional expectation value (1)
- conditioned (1)
- conditioned Feller diffusion (1)
- confidence interval (1)
- congruence (1)
- conjugate gradient (1)
- connections (1)
- conormal asymptotic expansions (1)
- conormal asymptotics (1)
- conormal symbols (1)
- conservation laws (1)
- consistency (1)
- constitutive relations (1)
- constrained Hamiltonian systems (1)
- contact transformations (1)
- continuation (1)
- continuity in Sobolev spaces with double weights (1)
- continuous time Markov Chains (1)
- continuous time Markov chain (1)
- continuous-time data assimilation (1)
- control theory (1)
- corner Sobolev spaces with double weights (1)
- corona virus (1)
- correlated noise (1)
- coupled solution (1)
- coupling methods (1)
- covariance (1)
- covering (1)
- critical and subcritical Dawson-Watanabe process (1)
- criticality theorem (1)
- cusp (1)
- cusped bar (1)
- cycle decomposition (1)
- cytosine methylation (1)
- das Cauchyproblem (1)
- das Goursatproblem (1)
- das charakteristische Cauchyproblem (1)
- data assimilation (1)
- dbar-Neumann problem (1)
- de Rham complex (1)
- de Sitter model ; Fundamental solutions ; Decay estimates (1)
- decay of eigenfunctions (1)
- deformation quantization (1)
- degenerate elliptic equations (1)
- degenerate elliptic systems (1)
- degree of formality (1)
- democratic form (1)
- density of a measure (1)
- density of rational numbers (1)
- determinant (1)
- determinantal point processes (1)
- determinantische Punktprozesse (1)
- dictator game (1)
- die linearisierte Einsteingleichung (1)
- difference operator (1)
- differential geometry (1)
- differential-algebraic equations (1)
- diffusion process (1)
- digital circuit (1)
- dimension functional (1)
- dimension independent bound (1)
- direct and indirect climate observations (1)
- direkte und indirekte Klimaobservablen (1)
- discontinuous Robin condition (1)
- discontinuous drift (1)
- discrete Witten complex (1)
- discrete saymptotic types (1)
- diskontinuierliche Drift (1)
- diskreter Witten-Laplace-Operator (1)
- distorted Brownian motion (1)
- division algebras (1)
- division of spaces (1)
- division trees (1)
- divisors (1)
- domains with singularities (1)
- doppelsemigroup (1)
- duality formulae (1)
- dynamical cognitive modeling (1)
- dynamical system (1)
- dynamical system representation (1)
- edge Sobolev spaces (1)
- edge algebra (1)
- edge boundary value problems (1)
- edge quantizations (1)
- edge spaces (1)
- edge symbol (1)
- edge- and corner-degenerate symbols (1)
- elastic bar (1)
- elasticity (1)
- electrodynamics (1)
- elliptic boundary conditions (1)
- elliptic complex (1)
- elliptic functions (1)
- elliptic morphism (1)
- elliptic operators in subspaces (1)
- elliptic operators on non-compact manifolds (1)
- elliptic problem (1)
- elliptic problems (1)
- elliptic quasicomplexes (1)
- elliptic systems (1)
- ellipticity in the edge calculus (1)
- ellipticity of cone operators (1)
- ellipticity of corners operators (1)
- ellipticity with interface conditions (1)
- ellipticity with parameter (1)
- ellipticity with respect to interior and edge symbols (1)
- elliptische Gleichungen (1)
- elliptische Quasi-Komplexe (1)
- empirical Wasserstein distance (1)
- energetic space (1)
- enlargement of filtration (1)
- ensemble Kalman filter (1)
- entropy (1)
- equation of motion (1)
- equivalence (1)
- ergodic rates (1)
- erste Variation (1)
- estimation of regression (1)
- estimation-of-distribution algorithms (1)
- exact simulation method (1)
- exakte Simulation (1)
- exercise collection (1)
- existence (1)
- experiment (1)
- exponential function (1)
- exponential stability (1)
- exterior tensor product (1)
- extinction probability (1)
- fachdidaktisches Wissen (1)
- fachwissenschaftliches Wissen (1)
- feedback (1)
- fibre coordinates (1)
- fibroblasts (1)
- filter (1)
- filtering (1)
- finiteness theorem (1)
- finsler distance (1)
- first boundary value problem (1)
- first variation (1)
- fixational eye movements (1)
- fixed point formula (1)
- flocking (1)
- foliated diffusion (1)
- foliations (1)
- force unification (1)
- forced symmetry breaking (1)
- formalism (1)
- fractions (1)
- framing (1)
- free algebra (1)
- fully non-linear degenerate parabolic equations (1)
- functional calculus (1)
- functor geometry (1)
- fundamental ideas (1)
- fundamental solution (1)
- game (1)
- game-based (1)
- gamification (1)
- gauge group (1)
- gender (1)
- generalized Bruck-Reilly ∗-extension (1)
- generalized Langevin equation (1)
- generalized Laplace operator (1)
- genome scan (1)
- geodesic distance (1)
- geodätischer Abstand (1)
- geomagnetism (1)
- geometric optics approximation (1)
- geometric reproduction distribution (1)
- geordnete Gruppen von Conrad-Typ (1)
- global exact boundary controllability (1)
- global solution (1)
- global solutions (1)
- globally hyperbolic spacetime (1)
- good-inner function (1)
- goodness of fit (1)
- goodness-of-fit (1)
- goodness-of-fit testing (1)
- granular gas (1)
- gravitation (1)
- gravitational wave (1)
- guiding idea “Daten und Zufall” (1)
- hard core interaction (1)
- heat asymptotics (1)
- heat kernel (1)
- heavy-tailed distributions (1)
- hermeneutics (1)
- high dimensional (1)
- higher operations (1)
- higher order rectifiability (1)
- higher singularities (1)
- higher-order Sturm–Liouville problems (1)
- history of branching processes (1)
- hitting times (1)
- host-parasite stochastic particle system (1)
- hydrodynamics (1)
- hyperbolic dynamical system (1)
- hyperbolic operators (1)
- hyperbolic tilings (1)
- hyperequational theory (1)
- hypoelliptic estimate (1)
- höhere Operationen (1)
- höhere Singularitäten (1)
- ill-posed (1)
- ill-posed problem (1)
- ill-posed problems (1)
- illposed problem (1)
- indecomposable varifold (1)
- index formula (1)
- index of elliptic operator (1)
- index of stability (1)
- infinite-dimensional diffusion (1)
- initial boundary value problem (1)
- instabilities (1)
- instability of the process (1)
- integral Fourier operators (1)
- integral representation method (1)
- integration by parts formula (1)
- integration by parts on path space (1)
- interaction matrix (1)
- interassociativity (1)
- interfaces with conical singularities (1)
- intrinsic diameter (1)
- intrinsischer Diameter (1)
- invariant (1)
- inverse Probleme (1)
- inverse Sturm–Liouville problems (1)
- inverse problems (1)
- isoperimetric inequality (1)
- isoperimetrische Ungleichung (1)
- isotopic tiling theory (1)
- jump process (1)
- jump processes (1)
- k-means clustering (1)
- kanten- und ecken-entartete Symbole (1)
- kernel estimator of the hazard rate (1)
- kernel method (1)
- kernel methods (1)
- kernel-based Bayesian inference (1)
- kernel-basierte Bayes'sche Inferenz (1)
- kleine Parameter (1)
- komplexe mechanistische Systeme (1)
- konstitutive Gleichungen (1)
- large-scale mechanistic systems (1)
- lattice packing and covering (1)
- lattice point (1)
- learning (1)
- least squares estimator (1)
- left ordered groups (1)
- lidar (1)
- lifespan (1)
- limit theorem (1)
- limit theorem for integrated squared difference (1)
- linear fractional case (1)
- linksgeordnete Gruppen (1)
- localisation (1)
- locality (1)
- locality principle (1)
- locally indicable (1)
- logarithmic residue (1)
- logarithmic source condition (1)
- logic (1)
- logistic regression analysis (1)
- logistische Regression (1)
- lokal indizierbar (1)
- low-lying eignvalues (1)
- lumping (1)
- macromolecular decay (1)
- magnetic (1)
- magnetic field modeling (1)
- magnetisch (1)
- makromolekularer Zerfall (1)
- manifold (1)
- manifold with boundary (1)
- manifold with edge (1)
- manifolds with cusps (1)
- manifolds with edge (1)
- mapping class groups (1)
- mapping degree (1)
- maps on surfaces (1)
- marked Gibbs point processes (1)
- matching of asymptotic expansions (1)
- mathematical modeling (1)
- mathematical modelling (1)
- mathematical physics (1)
- mathematics (1)
- mathematische Physik (1)
- mean curvature (1)
- mechanistic modeling (1)
- mechanistische Modellierung (1)
- memory effects (1)
- meromorphe Fortsetzung (1)
- meromorphic continuation (1)
- meromorphic family (1)
- metaplectic operators (1)
- metastability (1)
- method (1)
- microdialysis (1)
- microlocal analysis (1)
- microlokale Analysis (1)
- microphysics (1)
- microsaccades (1)
- middle school (1)
- mild solution (1)
- minimax convergence rates (1)
- minimax optimality (1)
- minimax rate (1)
- mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung (1)
- mittlere Krümmung (1)
- mixed elliptic problems (1)
- mixed problems (1)
- mixture of bridges (1)
- mod k index (1)
- model order reduction (1)
- model selection (1)
- modified Landweber method (1)
- moduli space of flat connections (1)
- modulo n index (1)
- moment map (1)
- monotone method (1)
- monotone random (1)
- monotonicity (1)
- monotonicity conditions (1)
- motivation (1)
- multi-change point detection (1)
- multi-scale diffusion processes (1)
- multiple characteristics (1)
- multiplicative Lévy noise (1)
- multiplicative noise (1)
- multitype measure-valued branching processes (1)
- multiwavelength Lidar (1)
- multizeta functions (1)
- negative Zahlen (1)
- negative numbers (1)
- nicht-lineare gemischte Modelle (NLME) (1)
- nichtlineare Modelle (1)
- nichtlineare partielle Differentialgleichung (1)
- non-coercive boundary conditions (1)
- non-linear integro-differential equations (1)
- non-regular drift (1)
- non-uniqueness (1)
- nondegenerate condition (1)
- nonhomogeneous boundary value problems (1)
- nonlinear (1)
- nonlinear equations (1)
- nonlinear partial differential equations (1)
- nonlinear semigroup (1)
- nonlocal problem (1)
- nonlocally coupled phase oscillators (1)
- nonparametric regression (1)
- nonparametric regression estimation (1)
- nonsmooth curves (1)
- norm estimates with respect to a parameter (1)
- normal bundle (1)
- normal reflection (1)
- number (1)
- numerical approximation (1)
- numerical extension (1)
- observables (1)
- offene Wissenschaft (1)
- open science (1)
- operator algebras on manifolds with singularities (1)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- optimal rate (1)
- order filtration (1)
- order reduction (1)
- output space compaction (1)
- p-Branen (1)
- p-Laplace Operator (1)
- p-branes (1)
- parallelizable spheres (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- parametrices (1)
- parity condition (1)
- parity conditions (1)
- part-whole concept (1)
- partial algebras (1)
- partial least squares (1)
- partial ordering (1)
- partielle Integration (1)
- pattern formation (1)
- periodic Gaussian process (1)
- periodic Ornstein-Uhlenbeck process (1)
- permanental- (1)
- pharmacokinetics (1)
- philosophy of mathematics (1)
- physics (1)
- physiologie-basierte Pharmacokinetic (PBPK) (1)
- planetary rings (1)
- point process (1)
- polydisc (1)
- polyhedra and polytopes (1)
- polymer (1)
- popPBPK (1)
- popPK (1)
- population analysis (1)
- populations (1)
- porous medium equation (1)
- poset (1)
- principal symbolic hierarchies (1)
- probability distribution (1)
- probability generating function (1)
- probability theory (1)
- problem of classification (1)
- profile likelihood (1)
- proof (1)
- proof assistant (1)
- proof environment (1)
- propagation probability (1)
- propor-tional hazard mode (1)
- pseudo-Boolean optimization (1)
- pseudo-diferential operators (1)
- pseudo-differential equation (1)
- pseudo-differentialboundary value problems (1)
- pseudo-differentielle Gleichungen (1)
- pseudoboolesche Optimierung (1)
- pseudodifferential boundary value problems (1)
- pseudodifferential subspace (1)
- pseudodifferential subspaces (1)
- pseudodifferentiale Operatoren (1)
- psycholinguistics (1)
- quantizer (1)
- quantum field theory (1)
- quasilinear Fredholm operator (1)
- quasilinear equation (1)
- question of origin (1)
- random sum (1)
- random variable (1)
- random walk (1)
- random walk on Abelian group (1)
- random walks on graphs (1)
- ratchet transport (1)
- rational numbers (1)
- reaction-advection-diffusion equation (1)
- reading (1)
- real-variable harmonic analysis (1)
- reciprocal characteristics (1)
- rectifiable varifold (1)
- reflecting boundary (1)
- regular figures (1)
- regular monoid (1)
- regularisation (1)
- regularization methods (1)
- rejection sampling (1)
- rektifizierbare Varifaltigkeit (1)
- relative cohomology (1)
- relative index formulas (1)
- relative η-invariant (1)
- removable set (1)
- removable sets (1)
- reproducing kernel Hilbert space (1)
- reproduction rate (1)
- rescaled lattice (1)
- retrieval (1)
- reziproke Klassen (1)
- run time analysis (1)
- saccade detection (1)
- sampling (1)
- scaled lattice (1)
- scattering amplitude (1)
- scattering theory (1)
- schlecht gestellt (1)
- seismic source inversion (1)
- seismische Quellinversion (1)
- self-assembly (1)
- semi-classical difference operator (1)
- semi-classical spectral estimates (1)
- semiclassical spectral asymptotics (1)
- semiclassics (1)
- semiconductors (1)
- semigroup (1)
- semipermeable barriers (1)
- semiprocess (1)
- sentence processing (1)
- sequences of microsaccades (1)
- sequential likelihood (1)
- series representation (1)
- shock wave (1)
- similarity measures (1)
- simulation (1)
- singular drifts (1)
- singular integral equations (1)
- singular manifolds (1)
- singular point (1)
- singuläre Mannigfaltigkeiten (1)
- skew diffusion (1)
- skew diffusions (1)
- skew field of fraction (1)
- small noise asymptotic (1)
- soft matter (1)
- space-time Gibbs field (1)
- spacetimes with timelike boundary (1)
- spatial autocorrelation (1)
- special holonomy (1)
- specific entropy (1)
- spectral boundary value problems (1)
- spectral independence (1)
- spectral kernel function (1)
- spectral resolution (1)
- spin Hall effect (1)
- spin geometry (1)
- spirallike function (1)
- stability (1)
- stability and accuracy (1)
- stable limit cycle (1)
- stark Hughes-frei (1)
- state estimation (1)
- statistical inference (1)
- statistical inverse problem (1)
- statistical machine learning (1)
- statistical model selection (1)
- statistische Inferenz (1)
- statistisches maschinelles Lernen (1)
- step process (1)
- stochastic (1)
- stochastic Burgers equations (1)
- stochastic bridges (1)
- stochastic interacting particles (1)
- stochastic mechanics (1)
- stochastic processes (1)
- stochastics (1)
- stochastische Anordnung (1)
- stochastische Differentialgleichungen (1)
- stochastische Mechanik (1)
- stochastische Zellulare Automaten (1)
- stochastisches interagierendes System (1)
- stopping rules (1)
- strongly Hughes-free (1)
- strongly pseudoconvex domains (1)
- structure formation (1)
- structured numbers (1)
- strukturierte Zahlen (1)
- subRiemannian geometry (1)
- subject didactic knowledge (1)
- subject matter knowledge (1)
- supergeometry (1)
- survival analysis (1)
- symmetry group (1)
- symplectic (canonical) transformations (1)
- symplectic manifold (1)
- symplectic methods (1)
- symplectic reduction (1)
- system Lame (1)
- systems of partial differential equations (1)
- systems pharmacology (1)
- tactic (1)
- teaching (1)
- terrigener Staub (1)
- terrigenous dust (1)
- test (1)
- test ability (1)
- tetration (1)
- the Dirichlet problem (1)
- the Goursat problem (1)
- the characteristic Cauchy problem (1)
- the first boundary value problem (1)
- the linearised Einstein equation (1)
- theory (1)
- thinking barriers (1)
- time reversal (1)
- time series (1)
- time series with heavy tails (1)
- time symmetry (1)
- trace (1)
- transition path theory (1)
- transitive action (1)
- travel time tomography (1)
- travelling waves (1)
- two-level interacting processes (1)
- ultracontractivity (1)
- unendlich teilbare Punktprozesse (1)
- unendliche Teilbarkeit (1)
- uniform compact attractor (1)
- univariat (1)
- univariate (1)
- unzerlegbare Varifaltigkeit (1)
- variable projection method (1)
- variational calculus (1)
- variational principle (1)
- variational stability (1)
- varifold (1)
- vibration (1)
- video study (1)
- viral fitness (1)
- viscoelasticity (1)
- wahrscheinlichkeitserzeugende Funktion (1)
- wave equation (1)
- wavelet analysis (1)
- weak boundary values (1)
- weiche Materie (1)
- weighted Hölder spaces (1)
- weighted Sobolev space (1)
- weighted Sobolev spaces with discrete saymptotics (1)
- weighted spaces with asymptotics (1)
- word n-gram probability (1)
- zero-aliasing (1)
- zero-noise limit (1)
- zufällige Summe (1)
- Ähnlichkeit-Masse (1)
- η-invariant (1)
- π -inverse monoid (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (490)
- Extern (13)
- Mathematisch-Naturwissenschaftliche Fakultät (13)
- Institut für Informatik und Computational Science (3)
- Institut für Biochemie und Biologie (2)
- Institut für Physik und Astronomie (2)
- Interdisziplinäres Zentrum für Dynamik komplexer Systeme (2)
- Department Grundschulpädagogik (1)
- Fachgruppe Volkswirtschaftslehre (1)
- Hasso-Plattner-Institut für Digital Engineering GmbH (1)
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.
The evaluation of process-oriented cognitive theories through time-ordered observations is crucial for the advancement of cognitive science. The findings presented herein integrate insights from research on eye-movement control and sentence comprehension during reading, addressing challenges in modeling time-ordered data, statistical inference, and interindividual variability. Using kernel density estimation and a pseudo-marginal likelihood for fixation durations and locations, a likelihood implementation of the SWIFT model of eye-movement control during reading (Engbert et al., Psychological Review, 112, 2005, pp. 777–813) is proposed. Within the broader framework of data assimilation, Bayesian parameter inference with adaptive Markov Chain Monte Carlo techniques is facilitated for reliable model fitting. Across the different studies, this framework has shown to enable reliable parameter recovery from simulated data and prediction of experimental summary statistics. Despite its complexity, SWIFT can be fitted within a principled Bayesian workflow, capturing interindividual differences and modeling experimental effects on reading across different geometrical alterations of text. Based on these advancements, the integrated dynamical model SEAM is proposed, which combines eye-movement control, a traditionally psychological research area, and post-lexical language processing in the form of cue-based memory retrieval (Lewis & Vasishth, Cognitive Science, 29, 2005, pp. 375–419), typically the purview of psycholinguistics. This proof-of-concept integration marks a significant step forward in natural language comprehension during reading and suggests that the presented methodology can be useful to develop complex cognitive dynamical models that integrate processes at levels of perception, higher cognition, and (oculo-)motor control. These findings collectively advance process-oriented cognitive modeling and highlight the importance of Bayesian inference, individual differences, and interdisciplinary integration for a holistic understanding of reading processes. Implications for theory and methodology, including proposals for model comparison and hierarchical parameter inference, are briefly discussed.
Point processes are a common methodology to model sets of events. From earthquakes to social media posts, from the arrival times of neuronal spikes to the timing of crimes, from stock prices to disease spreading -- these phenomena can be reduced to the occurrences of events concentrated in points. Often, these events happen one after the other defining a time--series.
Models of point processes can be used to deepen our understanding of such events and for classification and prediction. Such models include an underlying random process that generates the events. This work uses Bayesian methodology to infer the underlying generative process from observed data. Our contribution is twofold -- we develop new models and new inference methods for these processes.
We propose a model that extends the family of point processes where the occurrence of an event depends on the previous events. This family is known as Hawkes processes. Whereas in most existing models of such processes, past events are assumed to have only an excitatory effect on future events, we focus on the newly developed nonlinear Hawkes process, where past events could have excitatory and inhibitory effects. After defining the model, we present its inference method and apply it to data from different fields, among others, to neuronal activity.
The second model described in the thesis concerns a specific instance of point processes --- the decision process underlying human gaze control. This process results in a series of fixated locations in an image. We developed a new model to describe this process, motivated by the known Exploration--Exploitation dilemma. Alongside the model, we present a Bayesian inference algorithm to infer the model parameters.
Remaining in the realm of human scene viewing, we identify the lack of best practices for Bayesian inference in this field. We survey four popular algorithms and compare their performances for parameter inference in two scan path models.
The novel models and inference algorithms presented in this dissertation enrich the understanding of point process data and allow us to uncover meaningful insights.
Amoeboid cell motility takes place in a variety of biomedical processes such as cancer metastasis, embryonic morphogenesis, and wound healing. In contrast to other forms of cell motility, it is mainly driven by substantial cell shape changes. Based on the interplay of explorative membrane protrusions at the front and a slower-acting membrane retraction at the rear, the cell moves in a crawling kind of way. Underlying these protrusions and retractions are multiple physiological processes resulting in changes of the cytoskeleton, a meshwork of different multi-functional proteins. The complexity and versatility of amoeboid cell motility raise the need for novel computational models based on a profound theoretical framework to analyze and simulate the dynamics of the cell shape.
The objective of this thesis is the development of (i) a mathematical framework to describe contour dynamics in time and space, (ii) a computational model to infer expansion and retraction characteristics of individual cell tracks and to produce realistic contour dynamics, (iii) and a complementing Open Science approach to make the above methods fully accessible and easy to use.
In this work, we mainly used single-cell recordings of the model organism Dictyostelium discoideum. Based on stacks of segmented microscopy images, we apply a Bayesian approach to obtain smooth representations of the cell membrane, so-called cell contours. We introduce a one-parameter family of regularized contour flows to track reference points on the contour (virtual markers) in time and space. This way, we define a coordinate system to visualize local geometric and dynamic quantities of individual contour dynamics in so-called kymograph plots. In particular, we introduce the local marker dispersion as a measure to identify membrane protrusions and retractions in a fully automated way.
This mathematical framework is the basis of a novel contour dynamics model, which consists of three biophysiologically motivated components: one stochastic term, accounting for membrane protrusions, and two deterministic terms to control the shape and area of the contour, which account for membrane retractions. Our model provides a fully automated approach to infer protrusion and retraction characteristics from experimental cell tracks while being also capable of simulating realistic and qualitatively different contour dynamics. Furthermore, the model is used to classify two different locomotion types: the amoeboid and a so-called fan-shaped type.
With the complementing Open Science approach, we ensure a high standard regarding the usability of our methods and the reproducibility of our research. In this context, we introduce our software publication named AmoePy, an open-source Python package to segment, analyze, and simulate amoeboid cell motility. Furthermore, we describe measures to improve its usability and extensibility, e.g., by detailed run instructions and an automatically generated source code documentation, and to ensure its functionality and stability, e.g., by automatic software tests, data validation, and a hierarchical package structure.
The mathematical approaches of this work provide substantial improvements regarding the modeling and analysis of amoeboid cell motility. We deem the above methods, due to their generalized nature, to be of greater value for other scientific applications, e.g., varying organisms and experimental setups or the transition from unicellular to multicellular movement. Furthermore, we enable other researchers from different fields, i.e., mathematics, biophysics, and medicine, to apply our mathematical methods. By following Open Science standards, this work is of greater value for the cell migration community and a potential role model for other Open Science contributions.
Übungsbuch zur Stochastik
(2023)
Dieses Buch stellt Übungen zu den Grundbegriffen und Grundsätzen der Stochastik und ihre Lösungen zur Verfügung. So wie man Tonleitern in der Musik trainiert, so berechnet man Übungsaufgaben in der Mathematik. In diesem Sinne soll dieses Übungsbuch vor allem als Vorlage dienen für das eigenständige, eigenverantwortliche Lernen und Üben.
Die Schönheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Phänomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Glücksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.
Zahlen in den Fingern
(2023)
Die Debatte über den Einsatz von digitalen Werkzeugen in der mathematischen Frühförderung ist hoch aktuell. Lernspiele werden konstruiert, mit dem Ziel, mathematisches, informelles Wissen aufzubauen und so einen besseren Schulstart zu ermöglichen. Doch allein die digitale und spielerische Aufarbeitung führt nicht zwingend zu einem Lernerfolg. Daher ist es umso wichtiger, die konkrete Implementation der theoretischen Konstrukte und Interaktionsmöglichkeiten mit den Werkzeugen zu analysieren und passend aufzubereiten.
In dieser Masterarbeit wird dazu exemplarisch ein mathematisches Lernspiel namens „Fingu“ für den Einsatz im vorschulischen Bereich theoretisch und empirisch im Rahmen der Artifact-Centric Activity Theory (ACAT) untersucht. Dazu werden zunächst die theoretischen Hintergründe zum Zahlensinn, Zahlbegriffserwerb, Teil-Ganze-Verständnis, der Anzahlwahrnehmung und -bestimmung, den Anzahlvergleichen und der Anzahldarstellung mithilfe von Fingern gemäß der Embodied Cognition sowie der Verwendung von digitalen Werkzeugen und Multi-Touch-Geräten umfassend beschrieben. Anschließend wird die App Fingu erklärt und dann theoretisch entlang des ACAT-Review-Guides analysiert. Zuletzt wird die selbstständig durchgeführte Studie mit zehn Vorschulkindern erläutert und darauf aufbauend Verbesserungs- und Entwicklungsmöglichkeiten der App auf wissenschaftlicher Grundlage beigetragen. Für Fingu lässt sich abschließend festhalten, dass viele Prozesse wie die (Quasi-)Simultanerfassung oder das Zählen gefördert werden können, für andere wie das Teil-Ganze-Verständnis aber noch Anpassungen und/oder die Begleitung durch Erwachsene nötig ist.
In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.
In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.
This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincaré-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions.
The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincaré-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincaré-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts.
The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.
Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary
(2023)
Non-local boundary conditions – for example the Atiyah–Patodi–Singer (APS) conditions – for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, Bär and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While Bär and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions.
In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed.
We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems.
Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions – the analogue of the Atiyah–Patodi–Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.
Biological invasions may result from multiple introductions, which might compensate for reduced gene pools caused by bottleneck events, but could also dilute adaptive processes. A previous common-garden experiment showed heritable latitudinal clines in fitness-related traits in the invasive goldenrod Solidago canadensis in Central Europe. These latitudinal clines remained stable even in plants chemically treated with zebularine to reduce epigenetic variation. However, despite the heritability of traits investigated, genetic isolation-by-distance was non-significant. Utilizing the same specimens, we applied a molecular analysis of (epi)genetic differentiation with standard and methylation-sensitive (MSAP) AFLPs. We tested whether this variation was spatially structured among populations and whether zebularine had altered epigenetic variation. Additionally, we used genome scans to mine for putative outlier loci susceptible to selection processes in the invaded range. Despite the absence of isolation-by-distance, we found spatial genetic neighborhoods among populations and two AFLP clusters differentiating northern and southern Solidago populations. Genetic and epigenetic diversity were significantly correlated, but not linked to phenotypic variation. Hence, no spatial epigenetic patterns were detected along the latitudinal gradient sampled. Applying genome-scan approaches (BAYESCAN, BAYESCENV, RDA, and LFMM), we found 51 genetic and epigenetic loci putatively responding to selection. One of these genetic loci was significantly more frequent in populations at the northern range. Also, one epigenetic locus was more frequent in populations in the southern range, but this pattern was lost under zebularine treatment. Our results point to some genetic, but not epigenetic adaptation processes along a large-scale latitudinal gradient of S. canadensis in its invasive range.
Die Erweiterung des natürlichen Zahlbereichs um die positiven Bruchzahlen und die negativen ganzen Zahlen geht für Schülerinnen und Schüler mit großen gedanklichen Hürden und einem Umbruch bis dahin aufgebauter Grundvorstellungen einher. Diese Masterarbeit trägt wesentliche Veränderungen auf der Vorstellungs- und Darstellungsebene für beide Zahlbereiche zusammen und setzt sich mit den kognitiven Herausforderungen für Lernende auseinander. Auf der Grundlage einer Diskussion traditioneller sowie alternativer Lehrgänge der Zahlbereichserweiterung wird eine Unterrichtskonzeption für den Mathematikunterricht entwickelt, die eine parallele Einführung der Bruchzahlen und der negativen Zahlen vorschlägt. Die Empfehlungen der Unterrichtkonzeption erstrecken sich über den Zeitraum von der ersten bis zur siebten Klassenstufe, was der behutsamen Weiterentwicklung und Modifikation des Zahlbegriffs viel Zeit einräumt, und enthalten auch didaktische Überlegungen sowie konkrete Hinweise zu möglichen Aufgabenformaten.
Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces.
Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness.
In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate.
We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase.
The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy.
The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth’s outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure.
Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset.
The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package.
The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting.
In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. Bär and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case.
Back in the Riemannian setting, Bär and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement.
This thesis aims at applying the boundary conditions found by Bär and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made.
The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters.
Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space.
Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space.
In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting.
The motivation for this work was the question of reliability and robustness of seismic tomography. The problem is that many earth models exist which can describe the underlying ground motion records equally well. Most algorithms for reconstructing earth models provide a solution, but rarely quantify their variability. If there is no way to verify the imaged structures, an interpretation is hardly reliable. The initial idea was to explore the space of equivalent earth models using Bayesian inference. However, it quickly became apparent that the rigorous quantification of tomographic uncertainties could not be accomplished within the scope of a dissertation.
In order to maintain the fundamental concept of statistical inference, less complex problems from the geosciences are treated instead. This dissertation aims to anchor Bayesian inference more deeply in the geosciences and to transfer knowledge from applied mathematics. The underlying idea is to use well-known methods and techniques from statistics to quantify the uncertainties of inverse problems in the geosciences. This work is divided into three parts:
Part I introduces the necessary mathematics and should be understood as a kind of toolbox. With a physical application in mind, this section provides a compact summary of all methods and techniques used. The introduction of Bayesian inference makes the beginning. Then, as a special case, the focus is on regression with Gaussian processes under linear transformations. The chapters on the derivation of covariance functions and the approximation of non-linearities are discussed in more detail.
Part II presents two proof of concept studies in the field of seismology. The aim is to present the conceptual application of the introduced methods and techniques with moderate complexity. The example about traveltime tomography applies the approximation of non-linear relationships. The derivation of a covariance function using the wave equation is shown in the example of a damped vibrating string. With these two synthetic applications, a consistent concept for the quantification of modeling uncertainties has been developed.
Part III presents the reconstruction of the Earth's archeomagnetic field. This application uses the whole toolbox presented in Part I and is correspondingly complex. The modeling of the past 1000 years is based on real data and reliably quantifies the spatial modeling uncertainties. The statistical model presented is widely used and is under active development.
The three applications mentioned are intentionally kept flexible to allow transferability to similar problems. The entire work focuses on the non-uniqueness of inverse problems in the geosciences. It is intended to be of relevance to those interested in the concepts of Bayesian inference.
Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.
Das Professionswissen von Lehrkräften gehört zu den bedeutendsten Stellschrauben der Bildung an den Schulen. Seine Kernbereiche sind fachwissenschaftliches Wissen und fachdidaktisches Wissen, welche hauptsächlich in der universitären Ausbildung erworben werden.
Die vorliegende Arbeit verfolgt das Ziel, einen Beitrag zur stetigen Verbesserung und Sicherung der Qualität der Lehrerausbildung an der Universität Potsdam zu leisten, und stellt die Frage: Über welches fachwissenschaftliche und fachdidaktische Wissen verfügen die Lehramtsstudierenden im Fach Mathematik nach Besuch der Lehrveranstaltung Arithmetik und ihre Didaktik I und II? Untersucht wurde exemplarisch das Wissen der Lehramtsstudierenden im Bereich der rationalen Zahlen mit dem Fokus auf dem Verständnis der Dichte von Bruchzahlen. Die Dichte stellt eines der am schwierigsten zu erwerbenden Konzepte im Bruchzahlerwerb dar und fordert ein konzeptionelles Umdenken sowie die Reorganisation bereits erworbener Vorstellungen. Um die Forschungsfrage zu beantworten, wurden in einer qualitativen Studie 112 Lehramtsstudierende hinsichtlich ihres Wissens zu dem Thema Dichte von rationalen Zahlen schriftlich getestet. Um Denkprozesse der Studierenden zu verstehen und Denkhürden zu identifizieren, wurden zusätzlich qualitative Interviews in Form von Gruppendiskussionen geführt. Die Daten wurden mithilfe der Qualitativen Inhaltsanalyse computergestützt ausgewertet.
Es zeigte sich eine große Bandbreite verschiedener Wissensbestände. Die Ergebnisse im fachdidaktischen Wissen blieben hinter den Ergebnissen im fachwissenschaftlichen Wissen zurück. Am schwierigsten fiel den Studierenden die Gegenüberstellung von wesentlichen Eigenschaften der rationalen und natürlichen Zahlen auf der metakognitiven Ebene. Neben positiven Ergebnissen, welche für die Effektivität der Konzeption der Lehrveranstaltung sprechen, zeigten sich diverse Denkhürden. Defizite im Fachwissen wie ein mangelndes Verständnis von äquivalenten Brüchen oder Fehler im Erweitern von Brüchen enthüllen unzulänglich ausgebildete Grundvorstellungen im Bereich der rationalen Zahlen seitens der Studierenden. Schwierigkeiten in den fachdidaktischen Aufgaben wie die Formulierung einer kindgerechten Erklärung oder die anschauliche Darstellung des mathematischen Inhalts auf bildlicher Ebene lassen sich ursächlich auf die Defizite im Fachwissen zurückführen. Zusätzlich stellten sich Einschränkungen seitens der Studierenden in der Motivation und Relevanzzuschreibung heraus.
Die Ergebnisse führen zu gezielten Änderungsvorschlägen bezüglich der Konzeption der Lehrveranstaltung. Es wird empfohlen, verschiedene Lernangebote wie Hausaufgaben und wöchentliche Selbsttests zur individuellen Lernzielkontrolle für alle Teilnehmenden der Lehrveranstaltung verpflichtend zu gestalten und motivationale Aspekte verstärkt aufzugreifen. Zusätzlich wird der Ausbau von konkreten Übungen auf der enaktiven Ebene empfohlen, um den Aufbau von notwendigen Grundvorstellungen im Bereich der rationalen Zahlen zu fördern und somit Denkhürden gezielt zu begegnen.