510 Mathematik
Refine
Year of publication
Document Type
- Preprint (373)
- Article (261)
- Doctoral Thesis (75)
- Postprint (45)
- Monograph/Edited Volume (13)
- Other (10)
- Master's Thesis (6)
- Part of a Book (5)
- Conference Proceeding (5)
- Review (3)
Language
- English (750)
- German (46)
- French (3)
- Multiple languages (1)
Keywords
- random point processes (18)
- statistical mechanics (18)
- stochastic analysis (18)
- index (14)
- boundary value problems (12)
- Fredholm property (10)
- regularization (10)
- cluster expansion (9)
- elliptic operators (9)
- data assimilation (8)
- Bayesian inference (7)
- K-theory (7)
- manifolds with singularities (7)
- pseudodifferential operators (7)
- Cauchy problem (6)
- Dirac operator (6)
- Hodge theory (6)
- Neumann problem (6)
- discrepancy principle (6)
- relative index (6)
- Atiyah-Patodi-Singer theory (5)
- Boundary value problems (5)
- Navier-Stokes equations (5)
- elliptic complexes (5)
- ellipticity (5)
- ensemble Kalman filter (5)
- index theory (5)
- infinite-dimensional Brownian diffusion (5)
- spectral flow (5)
- stochastic differential equations (5)
- surgery (5)
- Dirac operators (4)
- Lefschetz number (4)
- Markov processes (4)
- Mellin transform (4)
- Modellierung (4)
- Probabilistic Cellular Automata (4)
- Toeplitz operators (4)
- Zaremba problem (4)
- differential operators (4)
- elliptic operator (4)
- linear term (4)
- local time (4)
- manifold with singularities (4)
- nonlinear operator (4)
- parameter estimation (4)
- reciprocal class (4)
- reversible measure (4)
- 'eta' invariant (3)
- Atiyah-Bott condition (3)
- Atiyah-Bott obstruction (3)
- Bayesian inverse problems (3)
- Bayesian inversion (3)
- Dirichlet form (3)
- Fokker-Planck equation (3)
- Fredholm operators (3)
- Gibbs measure (3)
- Kalman filter (3)
- Mathematikdidaktik (3)
- Mathematikunterricht (3)
- Pseudo-differential operators (3)
- Riemannian manifold (3)
- Stochastic Differential Equation (3)
- Wahrscheinlichkeitstheorie (3)
- asymptotic behavior (3)
- asymptotic expansion (3)
- boundary value problem (3)
- classical solution (3)
- clone (3)
- cohomology (3)
- conical singularities (3)
- conormal symbol (3)
- counting process (3)
- de Rham complex (3)
- eta invariant (3)
- evolution equation (3)
- eye movements (3)
- filter (3)
- gradient flow (3)
- hard core potential (3)
- hyperbolic tilings (3)
- index of elliptic operators in subspaces (3)
- inversion (3)
- isotopic tiling theory (3)
- kernel methods (3)
- linear hypersubstitution (3)
- manifolds with conical singularities (3)
- multiplicative noise (3)
- optimal transport (3)
- partial clone (3)
- pseudo-differential boundary value problems (3)
- pseudo-differential operators (3)
- reciprocal processes (3)
- relative rank (3)
- residue (3)
- skew Brownian motion (3)
- star product (3)
- the Cauchy problem (3)
- transition path theory (3)
- tunneling (3)
- 26D15 (2)
- 31C20 (2)
- 35B09 (2)
- 35R02 (2)
- 39A12 (primary) (2)
- 58E35 (secondary) (2)
- Alterung (2)
- Blickbewegungen (2)
- Boutet de Monvel's calculus (2)
- Brownian bridge (2)
- Brownian motion with discontinuous drift (2)
- Bruchzahlen (2)
- Canonical Gibbs measure (2)
- Carleman matrix (2)
- Chemotaxis (2)
- Cluster expansion (2)
- Corona (2)
- DLR equation (2)
- DLR equations (2)
- Diracoperator (2)
- Dirichlet problem (2)
- Edge calculus (2)
- Eigenvalues (2)
- Elliptic complexes (2)
- Fredholm complexes (2)
- Fredholm operator (2)
- Gamma-convergence (2)
- Gaussian process (2)
- Gibbs field (2)
- Gibbs point processes (2)
- Graphs (2)
- Heat equation (2)
- Hughes-free (2)
- Infinite-dimensional SDE (2)
- Kunststofflichtwellenleiter (2)
- Lagrangian submanifolds (2)
- Lagrangian system (2)
- Lame system (2)
- Langzeitverhalten (2)
- Laplace equation (2)
- Laplacian (2)
- Lefschetz fixed point formula (2)
- Levy measure (2)
- Levy process (2)
- Lichtwellenleiter (2)
- MCMC (2)
- Machine learning (2)
- Magnus expansion (2)
- Malliavin calculus (2)
- Markov chain (2)
- Markovprozesse (2)
- Mellin symbols with values in the edge calculus (2)
- Model order reduction (2)
- Onsager-Machlup functional (2)
- POF (2)
- Perturbed complexes (2)
- Quasilinear equations (2)
- Randwertprobleme (2)
- Riemann-Hilbert problem (2)
- Riesz continuity (2)
- Royden boundary (2)
- SPDEs (2)
- Specific entropy (2)
- Spin Geometry (2)
- Stochastic differential equations (2)
- Stochastik (2)
- Streuung (2)
- TIMSS (2)
- Temperatur (2)
- Time series (2)
- Tunneleffekt (2)
- Unterrichtsmethode (2)
- Verzweigungsprozess (2)
- Videostudie (2)
- Vietnam (2)
- WKB method (2)
- Wasserstein distance (2)
- Wiener measure (2)
- adaptive estimation (2)
- analytic continuation (2)
- approximation (2)
- asymptotic method (2)
- birth-death-mutation-competition point process (2)
- boundary layer (2)
- boundary regularity (2)
- branching process (2)
- censoring (2)
- continuous-time data assimilation (2)
- corner Sobolev spaces with double weights (2)
- correlated noise (2)
- coupling (2)
- curvature (2)
- detailed balance equation (2)
- didactics of mathematics (2)
- dimension independent bound (2)
- division of spaces (2)
- duality formula (2)
- early stopping (2)
- edge singularities (2)
- edge-degenerate operators (2)
- elliptic boundary value problems (2)
- elliptic families (2)
- elliptic family (2)
- elliptic system (2)
- estimation (2)
- eta-invariant (2)
- exact simulation (2)
- existence (2)
- first boundary value problem (2)
- formula (2)
- functional calculus (2)
- generators (2)
- geodesics (2)
- geometry (2)
- heat equation (2)
- heat kernel (2)
- high dimensional (2)
- higher-order Sturm–Liouville problems (2)
- holomorphic solution (2)
- homotopy classification (2)
- ill-posed problem (2)
- ill-posed problems (2)
- index formulas (2)
- infinite divisibility (2)
- infinitely divisible point processes (2)
- integral formulas (2)
- interaction matrix (2)
- inverse Sturm–Liouville problems (2)
- lattice packing and covering (2)
- lidar (2)
- linear formula (2)
- linear tree language (2)
- linking coefficients (2)
- localisation (2)
- localization (2)
- logarithmic source condition (2)
- long-time behaviour (2)
- manifolds with edges (2)
- mapping degree (2)
- maps on surfaces (2)
- mathematics education (2)
- mathematische Modellierung (2)
- maximum a posteriori (2)
- maximum likelihood estimator (2)
- methods (2)
- mild solution (2)
- minimax convergence rates (2)
- minimax optimality (2)
- modeling (2)
- modelling optical fibres waveguides pof scattering temperature aging ageing (2)
- models (2)
- modified Landweber method (2)
- modn-index (2)
- molecular motor (2)
- monodromy matrix (2)
- monotone coupling (2)
- multi-scale diffusion processes (2)
- non-Markov drift (2)
- nonlinear (2)
- nonlinear equations (2)
- nonparametric regression (2)
- operator-valued symbols (2)
- optimal rate (2)
- optische Fasern (2)
- orbifolds (2)
- p-Laplace operator (2)
- parametrices (2)
- particle filter (2)
- path integral (2)
- pharmacokinetics (2)
- point process (2)
- polyhedra and polytopes (2)
- pseudodiferential operators (2)
- pseudodifferential operator (2)
- quantization (2)
- reading (2)
- real-variable harmonic analysis (2)
- regular figures (2)
- regularisation (2)
- regularizer (2)
- regularizers (2)
- reproducing kernel Hilbert space (2)
- root functions (2)
- sampling (2)
- similarity measures (2)
- singular manifolds (2)
- singular partial differential equation (2)
- singular perturbation (2)
- skew diffusions (2)
- small parameter (2)
- spectral theorem (2)
- stability (2)
- stability and accuracy (2)
- star-product (2)
- stochastic bridge (2)
- stochastic ordering (2)
- symmetry conditions (2)
- teaching methods (2)
- term (2)
- theory (2)
- time duality (2)
- transfer operator (2)
- transformations (2)
- uncertainty quantification (2)
- weak boundary values (2)
- weighted edge spaces (2)
- weighted spaces (2)
- (co)boundary operator (1)
- (sub-) tropical Africa (1)
- (sub-) tropisches Afrika (1)
- 1st Eigenvalue (1)
- 4-Mannigfaltigkeiten (1)
- 53C12 (1)
- 53C27 (1)
- ACAT (1)
- AFLP (1)
- ALOS-2 PALSAR-2 (1)
- APS problem (1)
- APX-hardness (1)
- Absorbed dose (1)
- Absorption kinetics (1)
- Achievement goal orientation (1)
- Actin cytoskeleton dynamics (1)
- Activity Theory (1)
- Aerosol (1)
- Aerosole (1)
- Agmon estimates (1)
- Alfred Schütz (1)
- Algebraic Birkhoff factorisation (1)
- Algebraic quantum field theory (1)
- Algorithmen (1)
- Algorithms (1)
- Alternatividentitäten (1)
- Alternativvarietäten (1)
- Aluminium (1)
- Aluminium adjuvants (1)
- Analytic continuation (1)
- Analytic extension (1)
- Anfangsrandwertproblem (1)
- Angle (1)
- Animal movement modeling (1)
- Anisotropic pseudo-differential operators (1)
- Answer Set Programming (1)
- Antwortmengenprogrammierung (1)
- Approximate likelihood (1)
- Arctic haze (1)
- Argumentation (1)
- Arnoldi process (1)
- Artificial Intelligence (1)
- Assimilation (1)
- Asymptotische Entwicklung (1)
- Atiyah-Singer theorem (1)
- Atmosphere (1)
- Attractive Dynamics (1)
- Aufgabensammlung (1)
- Aufsatzsammlung (1)
- Aussterbewahrscheinlichkeit (1)
- Automotive (1)
- Averaging principle (1)
- Banach-valued process (1)
- Baumweite (1)
- Bayessche Inferenz (1)
- Beltrami equation (1)
- Bernstein inequality (1)
- Bethe (1)
- Bewegungsgleichung (1)
- Beweis (1)
- Beweisassistent (1)
- Beweisumgebung (1)
- Bienaymé-Galton-Watson Prozess (1)
- Bienaymé-Galton-Watson process (1)
- Bisectorial operator (1)
- Blood coagulation network (1)
- Boltzmann distribution (1)
- Boolean model (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 Monvels Kalkül (1)
- Breathing chimera states (1)
- Brownian motion (1)
- Bruck-Reilly extension (1)
- Bulk-mediated diffusion; (1)
- C-asterisk-algebra (1)
- C0−semigroup (1)
- CCR-algebra (1)
- COVID-19 (1)
- CRPS (1)
- Calculus of conormal symbols (1)
- Calderón projections (1)
- Capture into resonance (1)
- Carleman formulas (1)
- Cartan's development (1)
- Cartesian product of varifolds (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)
- Cayley trees (1)
- Censoring (1)
- Central extensions of groups (1)
- Characteristic function (1)
- Characteristic polynomial (1)
- Cheeger inequality (1)
- Chern character (1)
- Chimera (1)
- Clifford algebra (1)
- Clifford semigroup (1)
- Clifford-Halbgruppen (1)
- Cloud Computing (1)
- Cloze predictability (1)
- Cloze-Vorhersagbarkeit (1)
- Cluster Entwicklung (1)
- Cluster-Expansion (1)
- Clusteranalyse (1)
- Coarea Formel (1)
- Cognitive style (1)
- Coherence-incoherence (1)
- Collapse (1)
- Collatz Conjecture (1)
- Complete asymptotics (1)
- Complexity (1)
- Composition operators (1)
- Compound Poisson processes (1)
- Computational Complexity (1)
- Condition number (1)
- Connectivity (1)
- Conradian left-order (1)
- Conradian ordered groups (1)
- Constant scalar curvature (1)
- Continuum random cluster model (1)
- Control theory (1)
- Coq (1)
- Core field (1)
- Corner boundary value problems (1)
- Correlation based modelling (1)
- Coupled oscillators (1)
- Coupling (1)
- Covid-19 (1)
- Cox model (1)
- Cox-Modell (1)
- Crack theory (1)
- Cross-effects (1)
- Curry (1)
- Curvature varifold (1)
- C‐ reactive protein remission (1)
- DFN (1)
- DLR-Gleichungen (1)
- Data assimilation (1)
- Data augmentation (1)
- De Rham complex (1)
- Delaney--Dress (1)
- Delaney–Dress tiling theory (1)
- Deligne Cohomology (1)
- Deligne Kohomologie (1)
- Denkhürden (1)
- Dependent thinning (1)
- Derivation (1)
- Detektion multipler Übergänge (1)
- Determinant (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 mixture (1)
- Dirichlet to Neumann operator (1)
- Dirichlet-to-Neumann operator (1)
- Disagreement percolation (1)
- Discovery learning (1)
- Discrete Dirichlet forms (1)
- Distributed Learning (1)
- Divisionsbäume (1)
- Doblin (1)
- Dobrushin criterion (1)
- Dobrushin criterion; (1)
- Dobrushin-Kriterium (1)
- Doeblin (1)
- Dose rate (1)
- Double Colored Edges (1)
- Duality formula (1)
- Dualitätsformeln (1)
- Dubrovinring (1)
- Dynamic Programming (1)
- Dynamische Programmierung (1)
- Dynamische kognitive Modellierung (1)
- Dynamo (1)
- Döblin (1)
- E-band (1)
- EM (1)
- ERgodicity of Markov Chains (1)
- Earth's magnetic field (1)
- Earthquake modeling (1)
- Edge-degenerate operators (1)
- Effort (1)
- Eichtheorie (1)
- Einstein manifolds (1)
- Einstein-Hilbert action (1)
- Einstein-Hilbert-Wirkung (1)
- Einstein-Mannigfaltigkeiten (1)
- Elastizität (1)
- Elektrodynamik (1)
- Elementary school students (1)
- Elliptic boundary (1)
- Elliptic equation with order degeneration (1)
- Elliptic operators (1)
- Elliptic operators in domains with edges (1)
- Ellipticity and parametrices (1)
- Elliptische Komplexe (1)
- Elliptizität (1)
- Ensemble Kalman (1)
- Entdeckendes Lernen (1)
- Entropiemethode (1)
- Entropy method (1)
- Entstehungsfragestellung (1)
- Epidemiologie (1)
- Epidemiology (1)
- Epistemologie (1)
- Epistemology (1)
- Error analysis (1)
- Estimation for branching processes (1)
- Estimation-of-Distribution-Algorithmen (1)
- Euclidean fields (1)
- Euler equations (1)
- Euler operator (1)
- Euler's theta functions (1)
- Euler-Lagrange equations (1)
- Evolutionsgleichung (1)
- Exponential Time Hypothesis (1)
- Exponential decay of pair correlation (1)
- Exponentialzeit Hypothese (1)
- Extremal problem (1)
- Eye-tracking (1)
- Eyring-Kramers Formel (1)
- Fast win (1)
- Feedback (1)
- Feller Diffusionsprozesse (1)
- Feller diffusion processes (1)
- Fence (1)
- Feynman-Kac formula (1)
- Fibroblasten (1)
- Filterung (1)
- Finite difference method (1)
- Finite transformation semigroup (1)
- Finsler distance (1)
- Finsler-Abstand (1)
- Finsler-distance (1)
- First variation (1)
- Fischer-Riesz equations (1)
- Fitness (1)
- Fixationsbewegungen der Augen (1)
- Flocking (1)
- Fluvial (1)
- Form (1)
- Formalismus (1)
- Formalitätsgrad (1)
- Fourier Integraloperatoren (1)
- Fourier and Mellin transforms (1)
- Fourier-Laplace transform (1)
- Fourth order Sturm-Liouville problem (1)
- Fractional calculus (primary) (1)
- Fractional moments (1)
- Fredholm Komplexe (1)
- Freidlin-Wentzell theory (1)
- Fremdverstehen (1)
- Frühförderung (1)
- Functional calculus (1)
- Fundamentale Ideen (1)
- Funktorgeometrie (1)
- Future time interval (1)
- G-index (1)
- G-trace (1)
- Gamification (1)
- Gardner equation (1)
- Gauge theory (1)
- Gauss-Bonnet-Chern (1)
- Gaussian Loop Processes (1)
- Gaussian processes (1)
- Gaussian sequence model (1)
- Gauß-Prozesse (1)
- Gaußsche Loopprozess (1)
- Gender (1)
- Generalised mean curvature vector (1)
- Generalized translation operator (1)
- Geodäten (1)
- Geomagnetic field (1)
- Geomagnetic jerks (1)
- Geomagnetic secular variation (1)
- Geomagnetism (1)
- Geomagnetismus (1)
- Geometrie (1)
- Geometrieunterricht (1)
- Geometrische Reproduktionsverteilung (1)
- Gerben (1)
- Gerbes (1)
- Gevrey classes (1)
- Gibbs measures (1)
- Gibbs perturbation (1)
- Gibbs point process (1)
- Gibbs state (1)
- Gibbssche Punktprozesse (1)
- Gigli-Mantegazza flow (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)
- Greatest harmonic minorant (1)
- Green and Mellin edge operators (1)
- Green formula (1)
- Green operator (1)
- Green's function (1)
- Green's operator (1)
- Green´s Relations (1)
- Grenzwertsatz (1)
- Grundvorstellungen (1)
- Grushin operator (1)
- Gutzwiller formula (1)
- H-infinity-functional calculus (1)
- HIV (1)
- HIV Erkrankung (1)
- Haar system (1)
- Halbgruppentheorie (1)
- Hamilton-Jacobi theory (1)
- Hamiltonian dynamics (1)
- Hamiltonian group action (1)
- Hamiltonicity (1)
- Hardy‘s inequality (1)
- Harmonic measure (1)
- Hauptfaserbündel (1)
- Hawkes process (1)
- Heat Flow (1)
- Heavy-tailed distributions (1)
- Helmholtz problem (1)
- Hermeneutik (1)
- Heuristics (1)
- High dimensional statistical inference (1)
- Hilbert Scales (1)
- Historie der Verzweigungsprozesse (1)
- Hochschule (1)
- Holomorphic mappings (1)
- Holonomie (1)
- Holonomy (1)
- Hopf algebra (1)
- Hughes-frei (1)
- Hyperbolic-parabolic system (1)
- Hypoelliptic operators (1)
- Hypoellipticity (1)
- Hölder-type source condition (1)
- IGRF (1)
- IP core (1)
- Idempotents (1)
- Ill-conditioning (1)
- Ill-posed problem (1)
- In vitro dissolution (1)
- Indefinite (1)
- Index Theorie (1)
- Index theory (1)
- Inference post model-selection (1)
- Infinite chain (1)
- Infinite dimensional manifolds (1)
- Infinite graph (1)
- Infinite-dimensional interacting diffusion (1)
- Innovation (1)
- Inquiry-based learning (1)
- Instabilität des Prozesses (1)
- Instruction (1)
- Integrability (1)
- Integral varifold (1)
- Interacting Diffusion Processes (1)
- Interacting Particle Systems (1)
- Interacting particle systems (1)
- Interpolation (1)
- Intrinsicmotivation (1)
- Inverse problem (1)
- Inversion (1)
- Iran (1)
- Jump processes (1)
- K-Means Verfahren (1)
- KB-space (1)
- KS model (1)
- Kalman Bucy filter (1)
- Kalman Filter (1)
- Kanten-Randwertprobleme (1)
- Kato square root problem (1)
- Kegel space (1)
- Kern Methoden (1)
- Kernel regression (1)
- Kette von Halbgruppen (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)
- Koopman operator (1)
- Koopman semigroup (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)
- Lamé system (1)
- Landing site selection (1)
- Landweber iteration (1)
- Langevin Dynamics (1)
- Langevin diffusions (1)
- Langevin dynamics (1)
- Langevin-Diffusions (1)
- Laplace expansion (1)
- Laplace-Beltrami operator (1)
- Laplace-type operator (1)
- Latent Semantic Analysis (1)
- Latente-Semantische-Analyse (1)
- Lattice cones (1)
- Laufzeitanalyse (1)
- Laufzeittomographie (1)
- Learning theory (1)
- Left-ordered groups (1)
- Lehrpotential (1)
- Lehrtext (1)
- Leistungstests (1)
- Leitidee „Daten und Zufall“ (1)
- Lernen (1)
- Lernspiele (1)
- Lerntheorie (1)
- Lesen (1)
- Level of confidence (1)
- Levenberg-Marquardt method (1)
- Levy Maß (1)
- Levy flights (1)
- Levy type processes (1)
- Lidar (1)
- Lie groupoid (1)
- Linear inverse problems (1)
- Liouville theorem (1)
- Lipschitz domains (1)
- Locality (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)
- Lyapunov equation (1)
- Lyapunov exponent (1)
- Lévy diffusion approximation (1)
- Lévy diffusions on manifolds (1)
- Lévy measure (1)
- Lévy type processes (1)
- Lφ spectrum (1)
- MASCOT (1)
- MCMC modelling (1)
- MOSAiC (1)
- MSAP (1)
- Magnetfeldmodellierung (1)
- Manifolds with boundary (1)
- Mannigfaltigkeit (1)
- Mannigfaltigkeiten mit Kante (1)
- Mannigfaltigkeiten mit Singularitäten (1)
- Marked Gibbs process (1)
- Markierte Gibbs-Punkt-Prozesse (1)
- Markov chains (1)
- Markov decision process; (1)
- Markov semigroups (1)
- Markov-Ketten (1)
- Markov-field property (1)
- Martin-Dynkin boundary (1)
- Maslov and Conley–Zehnder index (1)
- Mathematical modeling (1)
- Mathematics classrooms (1)
- Mathematics problem-solving (1)
- Mathematics textbooks (1)
- Mathematik (1)
- Mathematikphilosophie (1)
- Mathematische Physik (1)
- Matrix function approximation (1)
- Maximum expected earthquake magnitude (1)
- Maximum likelihood estimation (1)
- McKean-Vlasov (1)
- Mean first encounter time (1)
- Measure-preserving semiflow (1)
- Mehrtyp-Verzweigungsprozesse (1)
- Mellin (1)
- Mellin operators (1)
- Mellin oscillatory integrals (1)
- Mellin quantization (1)
- Mellin quantizations (1)
- Mellin-Symbole (1)
- Mellin-Symbols (1)
- Menger algebra of rank n (1)
- Meromorphic operator functions (1)
- Metacognition (1)
- Metastabilität (1)
- Metastasis (1)
- Methode (1)
- Microdialyse (1)
- Mikrophysik (1)
- Mikrosakkaden (1)
- Mikrosakkadensequenzen (1)
- Milnor Moore theorem (1)
- Minimax Optimality (1)
- Minimax Optimalität (1)
- Minimax convergence rates (1)
- Minimax hypothesis testing (1)
- Mittag-Leffler (1)
- Model selection (1)
- Modellreduktion (1)
- Moduli spaces (1)
- Modulraum (1)
- Monte Carlo (1)
- Monte Carlo testing (1)
- Montel theorem (1)
- Morse-Smale property (1)
- Motivation (1)
- Motivation profiles (1)
- Multi-method interview approach (1)
- Multidimensional nonisentropic hydrodynamic model (1)
- Multimedia learning (1)
- Multiple zeta values (1)
- Multiplicative Levy noise (1)
- Multitype branching processes (1)
- Multivariate meromorphic functions (1)
- Multizeta-Abbildungen (1)
- NLME (1)
- NWP (1)
- Navier-Stokes-Gleichungen (1)
- Navier-Stoks equations (1)
- Network (1)
- Newton Polytope (1)
- Newton method (1)
- Newton polytopes (1)
- Nodal domain (1)
- Non-Markov drift (1)
- Non-linear (1)
- Non-linear semigroups (1)
- Non-proportional hazards (1)
- Non-regular drift (1)
- Nonlinear (1)
- Nonlinear Laplace operator (1)
- Nonlinear systems (1)
- Normalenbündel (1)
- Numerical weather prediction (1)
- Numerov's method (1)
- Ny-Alesund (1)
- Ollivier-Ricci (1)
- Operation (1)
- Operator differential equations (1)
- Operator-valued symbols of Mellin type (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)
- Operators on singular cones (1)
- Operators on singular manifolds (1)
- Operatortheorie (1)
- Optimality conditions (1)
- Orbifolds (1)
- Order-preserving (1)
- Order-preserving bijections (1)
- Order-preserving transformations (1)
- Ordered fields (1)
- Ordnungs-Filtrierung (1)
- Orlicz space height-excess (1)
- Ornstein-Uhlenbeck (1)
- Orthogruppen (1)
- Ott-Antonsen equation (1)
- Ott–Antonsen equation (1)
- PBPK (1)
- Pade approximants (1)
- Papangelou Process (1)
- Papangelou-Prozess (1)
- Parameter Schätzung (1)
- Parameterized Complexity (1)
- Parametric drift estimation (1)
- Parametrices (1)
- Parametrices of elliptic operators (1)
- Parametrisierte Komplexität (1)
- Partial Integration (1)
- Partial algebra (1)
- Peano phenomena (1)
- Penalized likelihood (1)
- Perfect groups (1)
- Periodic solutions (1)
- Perron's method (1)
- Perron-Frobenius operator (1)
- Pfadintegrale (1)
- Pfaffian (1)
- Pharmakokinetik (1)
- Phase transition (1)
- Physik (1)
- Place value (1)
- Plio-Pleistocene (1)
- Plio-Pleistozän (1)
- PoSI constants (1)
- Poincare inequality (1)
- Poincaré Birkhoff Witt theorem (1)
- Point Processes (1)
- Point process (1)
- Poisson bridge (1)
- Poisson process (1)
- Polya Process (1)
- Polyascher Prozess (1)
- Polymere (1)
- Pontrjagin duality (1)
- Popular matching (1)
- Populationen (1)
- Populations Analyse (1)
- Porous medium equation (1)
- Positional games (1)
- Positive scalar curvature (1)
- Positive semigroups (1)
- Potential theory (1)
- Primary 26D15 (1)
- Prinicipal Fibre Bundles (1)
- Proportional hazards (1)
- Pseudo-Differentialoperatoren (1)
- Pseudo-differential algebras (1)
- Pseudodifferential operators (1)
- Pseudodifferentialoperatoren auf dem Torus (1)
- Psycholinguistik (1)
- Punktprozess (1)
- Punktprozesse (1)
- Quadratic tilt-excess (1)
- Quadrature rule (1)
- Quantenfeldtheorie (1)
- Quantizer (1)
- Quasiconformal mapping (1)
- Quasilinear hyperbolic system (1)
- Quasimodes (1)
- Quotientenschiefkörper (1)
- RADFET (1)
- RMSE (1)
- Radar backscatter (1)
- Radiation hardness (1)
- Rahmenlehrplan (1)
- Raman lidar (1)
- Ramified Cauchy problem (1)
- Randbedingungen (1)
- Random Field Ising Model (1)
- Random walks (1)
- Randomisation (1)
- Randomized strategy (1)
- Rank of semigroup (1)
- Rarita-Schwinger (1)
- Rasch test modelling (1)
- Raum (1)
- Raumzeiten mit zeitartigen Rand (1)
- Real-variable harmonic analysis (1)
- Realistic Mathematics Education (1)
- Reciprocal process (1)
- Reciprocal processes (1)
- Reflektierende Randbedingungen (1)
- Regularisierung (1)
- Reihendarstellungen (1)
- Rektifizierbarkeit höherer Ordnung (1)
- Removable sets (1)
- Renormalisation (1)
- Renormalization (1)
- Renormalized integral (1)
- Reproducing kernel Hilbert space (1)
- Reproduktionsrate (1)
- Resampling (1)
- Retrieval (1)
- Ricci flow (1)
- Ricci-Fluss (1)
- Riemann-Roch theorem (1)
- Riemannsche Geometrie (1)
- Riesz topology (1)
- Risikoanalyse (1)
- Risk analysis (1)
- Rooted trees (1)
- Rota-Baxter (1)
- Rota-Baxter algebra (1)
- Rough paths (1)
- Runge-Kutta methods (1)
- SAR amplitude (1)
- Sakkadendetektion (1)
- Sampling (1)
- Saturation model (1)
- Satz von Milnor Moore (1)
- Satz von Poincaré Birkhoff Witt (1)
- Satzverarbeitung (1)
- Scattering theory (1)
- Schrodinger equation (1)
- Schrödinger operator (1)
- Schrödinger problem (1)
- Schulbuch (1)
- Schwarzes Loch (1)
- Schätzung von Verzweigungsprozessen (1)
- Second fundamental form (1)
- Second order elliptic equations (1)
- Secular variation (1)
- Secular variation rate of change (1)
- Seiberg-Witten theory (1)
- Seiberg-Witten-Invariante (1)
- Sekundarstufe I (1)
- Selbstassemblierung (1)
- Self-adaptive MPSoC (1)
- Self-exciting point process (1)
- Semi-klasische Abschätzung (1)
- Semiclassical difference operator (1)
- Semiklassik (1)
- Semiklassische Spektralasymptotik (1)
- Sentinel-1 (1)
- Sequential data assimilation (1)
- Sequenzielle Likelihood (1)
- Sharp threshold (1)
- Shuffle products (1)
- Signatures (1)
- Simulation (1)
- Simulation of Gaussian processes (1)
- Simulationsstudien (1)
- Singular analysis (1)
- Singular cones (1)
- Sinkhorn approximation (1)
- Sinkhorn problem (1)
- Skew Diffusionen (1)
- Skorokhod' s invariance principle (1)
- Small (1)
- Smoothing (1)
- Sobolev problem (1)
- Sobolev spaces (1)
- Sobolev spaces with double weights on singular cones (1)
- Sociolinguistics (1)
- Soziolinguistik (1)
- Space (1)
- Space-Time Cluster Expansions (1)
- Spatio-temporal ETAS model (1)
- Spectral Geometry (1)
- Spectral Regularization (1)
- Spectral analysis (1)
- Spectral flow (1)
- Spectral gap (1)
- Spectral regularization (1)
- Spektraltheorie (1)
- Spezifikationstests (1)
- Spiel (1)
- Spin Geometrie (1)
- Spin Hall effekte (1)
- Spin geometry (1)
- Stability (1)
- Stable matching (1)
- State Machine (1)
- Statistical inverse problem (1)
- Statistical learning (1)
- Stichprobenentnahme aus einem statistischen Modell (1)
- Stochastic Burgers equations (1)
- Stochastic Ordering (1)
- Stochastic bridges (1)
- Stochastic domination (1)
- Stochastic epidemic model (1)
- Stochastic reaction– diffusion (1)
- Stochastic systems (1)
- Stochastische Analysis (1)
- Stochastische Zellulare Automaten (1)
- Stormer-Verlet method (1)
- Stratified spaces (1)
- Streuamplitude (1)
- Streutheorie (1)
- Strings (1)
- Structured population equation (1)
- Strukturbildung (1)
- Studium (1)
- Sturm-Liouville problems (1)
- Sturm-Liouville problems of higher order (1)
- Subcritical (1)
- Subdivision schemes (1)
- Submanifolds (1)
- Super-quadratic tilt-excess (1)
- Supergeometrie (1)
- Surface potentials with asymptotics (1)
- Surface roughness (1)
- Survival models with covariates (1)
- Survival probability (1)
- Svalbard (1)
- Sylvester equations (1)
- System of nonlocal PDE of first order (1)
- Systempharmakologie (1)
- Taktik (1)
- TerraSAR-X/TanDEM-X (1)
- Testfähigkeit (1)
- Tetration (1)
- Textbook analysis (1)
- Textbook research (1)
- The Yamabe (1)
- Theorie (1)
- Thermal mathematical model (1)
- Tikhonov regularization (1)
- Time integration (1)
- Toeplitz-type pseudodifferential operators (1)
- Topological model (1)
- Toxicokinetic modelling (1)
- Trace Dirichlet form (1)
- Transformation semigroups (1)
- Transformation semigroups on infinite chains (1)
- Treewidth (1)
- Tunneling (1)
- Turbulence (1)
- Twisted product (1)
- Two-level interacting process (1)
- Two-sample tests (1)
- Tätigkeitstheorie (1)
- Umbilic product (1)
- Uncertainty quantification (1)
- Unendlichdimensionale Mannigfaltigkeit (1)
- Unique Gibbs state (1)
- Universal covering group (1)
- Untere Schranken (1)
- Variational principle (1)
- Variationsrechnung (1)
- Variationsstabilität (1)
- Varifaltigkeit (1)
- Vector bundle (1)
- Verbalizer (1)
- Verzweigungsprozesse (1)
- Vincent (1)
- Viscosity solutions (1)
- Visualizer (1)
- Vitali theorem (1)
- Volterra operator (1)
- Volterra symbols (1)
- WKB-expansion (1)
- Wahrscheinlichkeitsverteilung (1)
- Warped product (1)
- Wave operator (1)
- Waveletanalyse (1)
- Weak Mixing Condition (1)
- Wechselwirkende Teilchensysteme (1)
- Weighted (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)
- Yamabe invariant (1)
- Yamabe operator (1)
- Zahl (1)
- Zahlbereichserweiterung (1)
- Zahlerwerb (1)
- Zellmotilität (1)
- Zig-zag order (1)
- Zufallsvariable (1)
- Zustandsschätzung (1)
- Zählprozesse (1)
- a posteriori stopping rule (1)
- absorbing boundary (1)
- absorbing set (1)
- absorption (1)
- accelerated life time model (1)
- accelerated small (1)
- accuracy (1)
- adaptive (1)
- adaptivity (1)
- adrenal insufficiency (1)
- aerosol (1)
- aerosol distribution (1)
- aerosol-boundary layer interactions (1)
- aerosols (1)
- affine (1)
- affine invariance (1)
- algebra (1)
- algebra of rank n (1)
- algebraic systems (1)
- algebras (1)
- algorithmic (1)
- alignment (1)
- alternating direction implicit (1)
- alternative variety (1)
- amoeboid motion (1)
- amöboide Bewegung (1)
- analytic functional (1)
- analytic index (1)
- angewandte Mathematik (1)
- animal behavior (1)
- anisotropic spaces (1)
- anomalous Brownian motion (1)
- anomalous diffusion (1)
- antigen processing (1)
- applied mathematics (1)
- approximate differentiability (1)
- approximative Differenzierbarkeit (1)
- aptitude tests (1)
- archaeomagnetism (1)
- argumentation (1)
- articulation (1)
- association (1)
- asymptotic expansions (1)
- asymptotic methods (1)
- asymptotic properties of eigenfunctions (1)
- asymptotic stable (1)
- asymptotical normal distribution (1)
- asymptotics (1)
- asymptotics of solutions (1)
- asymptotische Entwicklung (1)
- asymptotische Normalverteilung (1)
- autocorrelation function (1)
- backtrajectories; (1)
- backward heat problem (1)
- balanced dynamics (1)
- bar with variable cross-section (1)
- basic ideas ('Grundvorstellungen') (1)
- bedingter Erwartungswert (1)
- bending of an orthotropic cusped plate (1)
- binding (1)
- bioinformatics (1)
- black hole (1)
- boun- dedness (1)
- boundary conditions (1)
- boundary values problems (1)
- bounds (1)
- branching processes (1)
- bridge (1)
- bridges of random walks (1)
- bumps (1)
- bundles (1)
- calculus of variations (1)
- canonical Marcus integration (1)
- canonical discretization schemes (1)
- cell motility (1)
- chain of semigroups (1)
- characteristic boundary point (1)
- characteristic points (1)
- chimera states (1)
- classical and quantum reduction (1)
- classification with partial labels (1)
- clathrin (1)
- cluster analysis (1)
- coarea formula (1)
- coercivity (1)
- cognitive psychology (1)
- coherent set (1)
- coloration of terms (1)
- colored solid varieties (1)
- commutator subgroup (1)
- compact groups (1)
- compact resolvent (1)
- compacton (1)
- companies (1)
- comparison principle (1)
- completeness (1)
- complex (1)
- complexity (1)
- composition operator (1)
- compound Poisson processes (1)
- compound polyhedra (1)
- compressible Euler equations (1)
- computational biology (1)
- concentration (1)
- concentration inequalities (1)
- concurrent checking (1)
- conditional Wiener measure (1)
- conditional expectation value (1)
- conditioned (1)
- conditioned Feller diffusion (1)
- conditions (1)
- conditions of success (1)
- confidence interval (1)
- confidence intervals (1)
- confidence sets (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)
- control theory (1)
- convergence assessment (1)
- convergence rate (1)
- corner parametrices (1)
- corona virus (1)
- correlations (1)
- cortisol (1)
- coupled solution (1)
- coupling methods (1)
- covariance (1)
- covering (1)
- critical and subcritical Dawson-Watanabe process (1)
- criticality theorem (1)
- crossed product (1)
- cusp (1)
- cusped bar (1)
- cycle decomposition (1)
- cytosine methylation (1)
- das Cauchyproblem (1)
- das Goursatproblem (1)
- das charakteristische Cauchyproblem (1)
- data matching (1)
- data profiling (1)
- dbar-Neumann problem (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)
- delaney-dress tiling theory (1)
- democratic form (1)
- density estimation (1)
- density of a measure (1)
- density of rational numbers (1)
- dependency discovery (1)
- design thinking (1)
- determinant (1)
- determinantal point processes (1)
- determinantische Punktprozesse (1)
- deterministic properties (1)
- deterministic random walk (1)
- dht-symmetric category (1)
- dictator game (1)
- die linearisierte Einsteingleichung (1)
- difference operator (1)
- differential cohomology (1)
- differential geometry (1)
- differential-algebraic equations (1)
- diffusion (1)
- diffusion maps (1)
- diffusion process (1)
- digital circuit (1)
- dimension functional (1)
- direct and indirect climate observations (1)
- direkte und indirekte Klimaobservablen (1)
- disagreement percolation (1)
- discontinuous Robin condition (1)
- discontinuous drift (1)
- discrete Schrodinger (1)
- discrete Witten complex (1)
- discrete saymptotic types (1)
- disjunction of identities (1)
- diskontinuierliche Drift (1)
- diskreter Witten-Laplace-Operator (1)
- distorted Brownian motion (1)
- distribution (1)
- distribution with asymptotics (1)
- distributions with one-sided support (1)
- division algebras (1)
- division ring of fractions (1)
- division trees (1)
- divisors (1)
- domains with singularities (1)
- dominant matching (1)
- doppelsemigroup (1)
- dried blood spots (1)
- drug monitoring (1)
- duality formulae (1)
- dynamic pricing (1)
- dynamic programming (1)
- dynamical cognitive modeling (1)
- dynamical models (1)
- dynamical system (1)
- dynamical system representation (1)
- early mathematical education (1)
- earthquake precursor (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)
- eigenfunction (1)
- eigenvalue decay (1)
- eigenvalues (1)
- elastic bar (1)
- elasticity (1)
- electrodynamics (1)
- electronic mail (1)
- elliptic boundary (1)
- elliptic boundary conditions (1)
- elliptic complex (1)
- elliptic differential operators of firstorder (1)
- elliptic equation (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)
- endomorphism semigroup (1)
- energetic space (1)
- enlargement of filtration (1)
- ensembles (1)
- entity resolution (1)
- entropy (1)
- equation of motion (1)
- equivalence (1)
- ergodic diffusion processes (1)
- ergodic rates (1)
- error diagram (1)
- erste Variation (1)
- estimation of regression (1)
- estimation-of-distribution algorithms (1)
- exact simulation method (1)
- exact simulation methods (1)
- exakte Simulation (1)
- exercise collection (1)
- exit calculus (1)
- experiment (1)
- exponential function (1)
- exponential stability (1)
- exterior tensor product (1)
- extinction probability (1)
- fachdidaktisches Wissen (1)
- fachwissenschaftliches Wissen (1)
- feedback (1)
- feedback particle filter (1)
- fence-preserving transformations (1)
- fibration (1)
- fibre coordinates (1)
- fibroblasts (1)
- filtering (1)
- finiteness theorem (1)
- finsler distance (1)
- first exit location (1)
- first passage times (1)
- first variation (1)
- fixational eye movements (1)
- fixed point formula (1)
- flocking (1)
- flood loss estimation (1)
- foliated diffusion (1)
- foliations (1)
- force unification (1)
- forced symmetry breaking (1)
- forecasting (1)
- formal (1)
- formalism (1)
- formulas (1)
- fractional Brownian motion (1)
- fractions (1)
- fracture (1)
- frameworks (1)
- framing (1)
- free algebra (1)
- frequency-modulated continuous-wave (FMCW) (1)
- fully non-linear degenerate parabolic equations (1)
- function (1)
- functional dependencies (1)
- functor geometry (1)
- fundamental ideas (1)
- fundamental solution (1)
- game (1)
- game theory (1)
- game-based (1)
- gamification (1)
- gauge group (1)
- gender (1)
- generalized Abelian gauge theory (1)
- generalized Bruck-Reilly ∗-extension (1)
- generalized Langevin equation (1)
- generalized Laplace operator (1)
- genome scan (1)
- geodesic distance (1)
- geodätischer Abstand (1)
- geomagnetic field (1)
- geomagnetism (1)
- geometric optics approximation (1)
- geometric reproduction distribution (1)
- geomorphology (1)
- geophysics (1)
- geopotential theory (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)
- gradient-free (1)
- gradient-free sampling methods (1)
- granular gas (1)
- graph (1)
- graph Laplacian (1)
- gravitation (1)
- gravitational wave (1)
- green function (1)
- ground state (1)
- group (1)
- group ring (1)
- groups (1)
- guiding idea “Daten und Zufall” (1)
- hard core interaction (1)
- heat asymptotics (1)
- heavy-tailed distributions (1)
- helicates (1)
- hermeneutics (1)
- heterogeneity (1)
- heuristics (1)
- high-dimensional inference (1)
- higher operations (1)
- higher order rectifiability (1)
- higher singularities (1)
- highly (1)
- history of branching processes (1)
- hitting times (1)
- holomorphic function (1)
- host-parasite stochastic particle system (1)
- hydraulic tomography (1)
- hydrodynamics (1)
- hydrogeophysics (1)
- hyperbolic dynamical system (1)
- hyperbolic operators (1)
- hyperequational theory (1)
- hypoelliptic estimate (1)
- höhere Operationen (1)
- höhere Singularitäten (1)
- idealised turbulence (1)
- idleness (1)
- ill-posed (1)
- illposed problem (1)
- indecomposable varifold (1)
- index formula (1)
- index of elliptic operator (1)
- index of stability (1)
- inegral formulas (1)
- infinite-dimensional diffusion (1)
- infinitesimal generator (1)
- inflammatory bowel disease (1)
- infliximab dosing (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)
- interacting particle systems (1)
- interacting particles (1)
- interassociativity (1)
- interfaces with conical singularities (1)
- interindividual differences (1)
- intrinsic diameter (1)
- intrinsischer Diameter (1)
- invariance (1)
- invariant (1)
- inverse Probleme (1)
- inverse Sturm-Liouville problems (1)
- inverse problem (1)
- inverse problems (1)
- inverse semigroup (1)
- inverse theory (1)
- isoperimetric inequality (1)
- isoperimetrische Ungleichung (1)
- iterative regularization (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 operator (1)
- kernel-based Bayesian inference (1)
- kernel-basierte Bayes'sche Inferenz (1)
- kleine Parameter (1)
- komplexe mechanistische Systeme (1)
- konstitutive Gleichungen (1)
- large deviations principle (1)
- large-scale mechanistic systems (1)
- latent profile analysis (1)
- lattice point (1)
- lattices (1)
- learning (1)
- learning rates (1)
- least squares estimator (1)
- left ordered groups (1)
- lifespan (1)
- likelihood function (1)
- limit theorem (1)
- limit theorem for integrated squared difference (1)
- limiting distribution (1)
- linear fractional case (1)
- linear inverse problems (1)
- linear regression (1)
- linear response (1)
- linearly implicit time stepping methods (1)
- linksgeordnete Gruppen (1)
- locality (1)
- locality principle (1)
- locally indicable (1)
- locally indicable group (1)
- log-concavity (1)
- logarithmic residue (1)
- logic (1)
- logistic regression analysis (1)
- logistische Regression (1)
- lokal indizierbar (1)
- low rank matrix recovery (1)
- low rank recovery (1)
- low-lying eignvalues (1)
- low-rank approximations (1)
- lumping (1)
- macromolecular decay (1)
- magnetic (1)
- magnetic field modeling (1)
- magnetic field variations through (1)
- magnetisch (1)
- makromolekularer Zerfall (1)
- manifold (1)
- manifold with boundary (1)
- manifold with edge (1)
- manifolds with boundary (1)
- manifolds with cusps (1)
- manifolds with edge (1)
- manifolds with edge and boundary (1)
- many-electron systems (1)
- mapping class (1)
- mapping class group (1)
- mapping class groups (1)
- marked Gibbs point processes (1)
- matching dependencies (1)
- matching of asymptotic expansions (1)
- mathematical concepts (1)
- mathematical modeling (1)
- mathematical modelling (1)
- mathematical physics (1)
- mathematics (1)
- mathematische Physik (1)
- matrices (1)
- matrix completion (1)
- maximal regularity (1)
- mean curvature (1)
- mean ergodic (1)
- mean-field equations (1)
- mean-variance optimization (1)
- mechanistic modeling (1)
- mechanistische Modellierung (1)
- memory effects (1)
- memory kernel (1)
- mental arithmetic (1)
- mental number line (1)
- meromorphe Fortsetzung (1)
- meromorphic continuation (1)
- meromorphic family (1)
- mesoscale forecasting (1)
- metal-organic (1)
- metaplectic operators (1)
- metastability (1)
- method (1)
- methods: data analysis (1)
- microdialysis (1)
- microlocal analysis (1)
- microlokale Analysis (1)
- microphysical properties (1)
- microphysics (1)
- microsaccades (1)
- middle school (1)
- minimax hypothesis testing (1)
- minimax rate (1)
- minor planets, asteroids: individual: (162173) Ryugu (1)
- mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung (1)
- mittlere Krümmung (1)
- mixed elliptic problems (1)
- mixed membership models (1)
- mixed problems (1)
- mixing (1)
- mixing optimization (1)
- mixture of bridges (1)
- mod k index (1)
- model error (1)
- model order reduction (1)
- model selection (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)
- morphology (1)
- motion correction (1)
- motivation (1)
- multi-armed bandits (1)
- multi-change point detection (1)
- multi-hypersubstitutions (1)
- multi-modular morphology (1)
- multi-well potential (1)
- multiclass classification with label noise (1)
- multiple characteristics (1)
- multiplicative Lévy noise (1)
- multitype measure-valued branching processes (1)
- multivariable (1)
- multiwavelength Lidar (1)
- multizeta functions (1)
- mutual contamination models (1)
- n-ary operation (1)
- n-ary term (1)
- negative Zahlen (1)
- negative curvature (1)
- negative numbers (1)
- network creation games (1)
- neural networks (1)
- nicht-lineare gemischte Modelle (NLME) (1)
- nichtlineare Modelle (1)
- nichtlineare partielle Differentialgleichung (1)
- noise Levy diffusions (1)
- non-coercive boundary conditions (1)
- non-exponential relaxation (1)
- non-linear integro-differential equations (1)
- non-regular drift (1)
- non-uniqueness (1)
- nonasymptotic minimax separation rate (1)
- nondegenerate condition (1)
- nondeterministic linear hypersubstitution (1)
- nonhomogeneous boundary value problems (1)
- nonlinear PDI (1)
- nonlinear filtering (1)
- nonlinear lattice (1)
- nonlinear partial differential equations (1)
- nonlinear semigroup (1)
- nonlocal problem (1)
- nonlocally coupled phase oscillators (1)
- nonparametric hypothesis testing (1)
- nonparametric regression estimation (1)
- nonparametric statistics (1)
- nonsmooth curves (1)
- norm estimates with respect to a parameter (1)
- normal bundle (1)
- normal reflection (1)
- number (1)
- numerical (1)
- numerical analysis/modeling (1)
- numerical approximation (1)
- numerical extension (1)
- numerical methods (1)
- numerical weather prediction (1)
- numerical weather prediction/forecasting (1)
- observables (1)
- offene Wissenschaft (1)
- open mapping theorem (1)
- open science (1)
- operational momentum (1)
- operator (1)
- operator algebras on manifolds with singularities (1)
- operator calculus (1)
- operators (1)
- operators on manifolds with conical and edge singularities (1)
- operators on manifolds with edges (1)
- operators on manifolds with singularities (1)
- optimal order (1)
- oracle inequalities (1)
- oracle inequality (1)
- oral anticancer drugs (1)
- order continuous norm (1)
- order filtration (1)
- order reduction (1)
- order-preserving (1)
- ordered group (1)
- organizations (1)
- orientation-preserving (1)
- orientation-preserving transformations (1)
- orthogroup (1)
- oscillatory systems (1)
- output space compaction (1)
- p-Branen (1)
- p-Laplace Operator (1)
- p-branes (1)
- palaeomagnetism (1)
- paleoearthquakes (1)
- parabolic Harnack estimate (1)
- parallelizable spheres (1)
- parameter-dependent cone operators (1)
- parameter-dependent ellipticity (1)
- parameter-dependent pseudodifferential operators (1)
- parametrices of elliptic operators (1)
- parity condition (1)
- parity conditions (1)
- part-whole concept (1)
- partial (1)
- partial Menger (1)
- partial algebras (1)
- partial least squares (1)
- partial ordering (1)
- particle methods (1)
- particle microphysics (1)
- partielle Integration (1)
- patch antenna (1)
- pathwise expectations (1)
- pattern formation (1)
- patterns (1)
- pediatrics (1)
- periodic Gaussian process (1)
- periodic Ornstein-Uhlenbeck process (1)
- permanental- (1)
- personalised medicine (1)
- phase-locked loop (PLL) (1)
- philosophy of mathematics (1)
- photometer (1)
- physics (1)
- physics learning (1)
- physiologie-basierte Pharmacokinetic (PBPK) (1)
- planetary rings (1)
- polydisc (1)
- polymer (1)
- popPBPK (1)
- popPK (1)
- popular matching (1)
- population analysis (1)
- populations (1)
- porous medium equation (1)
- poset (1)
- positive mass theorem (1)
- positive operators (1)
- positive solutions (1)
- posterior distribution (1)
- power amplifier (PA) (1)
- power series (1)
- power-law (1)
- prediction (1)
- presentations (1)
- price of anarchy (1)
- principal symbolic hierarchies (1)
- probabilistic modeling (1)
- probability distribution (1)
- probability generating function (1)
- probability theory (1)
- problem (1)
- problem of classification (1)
- problem-solving (1)
- problems (1)
- profile likelihood (1)
- proof (1)
- proof assistant (1)
- proof environment (1)
- propagation probability (1)
- propor-tional hazard mode (1)
- proteasome (1)
- protein degradation (1)
- proteolysis (1)
- prototypes (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 Fredholm operators (1)
- quasilinear equation (1)
- question of origin (1)
- r-hypersubstitution (1)
- r-term (1)
- radar (1)
- radiation mechanisms: thermal (1)
- random graphs (1)
- random sum (1)
- random variable (1)
- random walk (1)
- random walk on Abelian group (1)
- random walks on graphs (1)
- randomness (1)
- rank (1)
- rare events (1)
- ratchet transport (1)
- rational Krylov (1)
- rational numbers (1)
- reaction-advection-diffusion equation (1)
- reciprocal characteristics (1)
- rectifiable varifold (1)
- red blood cells (1)
- reflecting boundary (1)
- regular monoid (1)
- regular polyhedra (1)
- regularization methods (1)
- rejection sampling (1)
- rekonstruktive Fallanalyse (1)
- rektifizierbare Varifaltigkeit (1)
- relative cohomology (1)
- relative index formulas (1)
- relative η-invariant (1)
- removable set (1)
- removable sets (1)
- representations of groups as automorphism groups of (1)
- reproduction rate (1)
- requirements engineering (1)
- rescaled lattice (1)
- resolvents (1)
- restricted isometry property (1)
- restricted range (1)
- retrieval (1)
- reziproke Klassen (1)
- risk aversion (1)
- rotational diffusion (1)
- rotor-router model (1)
- rough metrics (1)
- run time analysis (1)
- saccade detection (1)
- saccades (1)
- scalable (1)
- scaled Brownian motion (1)
- scaled lattice (1)
- scattering amplitude (1)
- scattering theory (1)
- schlecht gestellt (1)
- secular variation (1)
- seismic hazard (1)
- seismic source inversion (1)
- seismische Quellinversion (1)
- self-assembly (1)
- semi-Lagrangian method (1)
- semi-classical difference operator (1)
- semi-classical spectral estimates (1)
- semiclassical spectral asymptotics (1)
- semiclassics (1)
- semiconductors (1)
- semigroup (1)
- semigroup representations (1)
- semigroup theory (1)
- semigroups on infinite chain (1)
- semipermeable barriers (1)
- semiprocess (1)
- sentence processing (1)
- sequences of microsaccades (1)
- sequential learning (1)
- sequential likelihood (1)
- series representation (1)
- shallow-water equations (1)
- shock wave (1)
- short-range prediction (1)
- simulation (1)
- simulations (1)
- single particle tracking (1)
- single vertex discrepancy (1)
- singular Sturm-Liouville (1)
- singular drifts (1)
- singular foliation (1)
- singular integral equations (1)
- singular point (1)
- singular points (1)
- singuläre Mannigfaltigkeiten (1)
- skew diffusion (1)
- skew field of fraction (1)
- small ball probabilities (1)
- small ball probabilities; (1)
- small noise asymptotic (1)
- smooth drift dependence (1)
- smoother (1)
- soft matter (1)
- software (1)
- solitary wave (1)
- space-time Gibbs field (1)
- spacecraft operations (1)
- spacetimes with timelike boundary (1)
- sparsity (1)
- spatial autocorrelation (1)
- special holonomy (1)
- specific entropy (1)
- spectral boundary value problems (1)
- spectral cut-off (1)
- spectral independence (1)
- spectral kernel function (1)
- spectral regularization (1)
- spectral resolution (1)
- spectral theory (1)
- spin Hall effect (1)
- spin geometry (1)
- spirallike function (1)
- spread correction (1)
- stable limit cycle (1)
- stable matching (1)
- stark Hughes-frei (1)
- starker Halbverband von Halbgruppen (1)
- state estimation (1)
- statistical (1)
- statistical inference (1)
- statistical inverse problem (1)
- statistical machine learning (1)
- statistical methods (1)
- statistical model selection (1)
- statistical seismology (1)
- statistics (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 models (1)
- stochastic processes (1)
- stochastic systems (1)
- stochastics (1)
- stochastische Anordnung (1)
- stochastische Differentialgleichungen (1)
- stochastische Mechanik (1)
- stochastische Zellulare Automaten (1)
- stochastisches interagierendes System (1)
- stopping rules (1)
- strong semilattice of semigroups (1)
- strongly Hughes-free (1)
- strongly pseudoconvex domains (1)
- strongly tempered stable Levy measure (1)
- structure formation (1)
- structured numbers (1)
- strukturierte Zahlen (1)
- subRiemannian geometry (1)
- subject didactic knowledge (1)
- subject matter knowledge (1)
- subspaces (1)
- supergeometry (1)
- superposition (1)
- superposition of n-ary operations and n-ary (1)
- superposition of operations (1)
- survival analysis (1)
- susceptibility (1)
- symbols (1)
- symmetry group (1)
- symplectic (canonical) transformations (1)
- symplectic manifold (1)
- symplectic methods (1)
- symplectic reduction (1)
- system Lame (1)
- systems (1)
- systems of partial differential equations (1)
- systems pharmacology (1)
- tactic (1)
- target dimensions (1)
- targeted antineoplastic drugs (1)
- teaching (1)
- temporal discretization (1)
- terms (1)
- terms and (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)
- theorem (1)
- therapeutic (1)
- theta neurons (1)
- thinking barriers (1)
- thymoproteasome (1)
- thymus (1)
- tiling theory (1)
- time (1)
- time reversal (1)
- time series (1)
- time series with heavy tails (1)
- time symmetry (1)
- time-fractional derivative (1)
- tools (1)
- topic modeling (1)
- trace (1)
- tracer tomography (1)
- transceiver (TRX) (1)
- transdimensional inversion (1)
- transformation (1)
- transition paths (1)
- transitive action (1)
- travel time tomography (1)
- travelling waves (1)
- tree conjecture (1)
- triply periodic minimal surface (1)
- truncated SVD (1)
- two-dimensional topology (1)
- two-level interacting processes (1)
- tyrosine kinase inhibitors (1)
- ultracontractivity (1)
- unendlich teilbare Punktprozesse (1)
- unendliche Teilbarkeit (1)
- uniform compact attractor (1)
- uniqueness (1)
- univariat (1)
- univariate (1)
- unknown variance (1)
- unsteady flow (1)
- unzerlegbare Varifaltigkeit (1)
- value problems (1)
- variable projection method (1)
- variational calculus (1)
- variational principle (1)
- variational stability (1)
- varifold (1)
- verification (1)
- vibration (1)
- video study (1)
- viral fitness (1)
- viscoelasticity (1)
- wahrscheinlichkeitserzeugende Funktion (1)
- wave equation (1)
- wavelet analysis (1)
- weak dependence (1)
- weakly almost periodic (1)
- weiche Materie (1)
- weighted Hölder spaces (1)
- weighted Sobolev space (1)
- weighted Sobolev spaces (1)
- weighted Sobolev spaces with discrete saymptotics (1)
- weighted edge and corner spaces (1)
- weighted graphs (1)
- weighted spaces with asymptotics (1)
- well-posedness (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)
- когомологии (1)
- комплекс де Рама (1)
- проблема Неймана (1)
- теория Ходжа (1)
- ∂-operator (1)
Institute
- Institut für Mathematik (737)
- Extern (14)
- Mathematisch-Naturwissenschaftliche Fakultät (14)
- Institut für Physik und Astronomie (13)
- Hasso-Plattner-Institut für Digital Engineering gGmbH (7)
- Institut für Biochemie und Biologie (6)
- Institut für Informatik und Computational Science (5)
- Department Psychologie (4)
- Department Grundschulpädagogik (3)
- Hasso-Plattner-Institut für Digital Engineering GmbH (3)
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.
Das Eigene und das Fremde
(2023)
Die vorliegende Arbeit stellt eine Untersuchung des Fremdverstehens von Lehrkräften im Mathematikunterricht dar. Mit ‚Fremdverstehen‘ soll dabei – in Anlehnung an den Soziologen Alfred Schütz – der Prozess bezeichnet werden, in welchem eine Lehrkraft versucht, das Verhalten einer Schülerin oder eines Schülers zu verstehen, indem sie dieses Verhalten auf ein Erleben zurückführt, das ihm zugrunde gelegen haben könnte. Als ein wesentliches Merkmal des Prozesses stellt Schütz in seiner Theorie des Fremdverstehens heraus, dass das Fremdverstehen eines Menschen immer auch auf seinen eigenen Erlebnissen basiert. Aus diesem Grund wird in der Arbeit ein methodischer Zweischritt vorgenommen: Es werden zunächst die mathematikbezogenen Erlebnisse zweier Lehrkräfte nachgezeichnet, bevor dann ihr Fremdverstehen in konkreten Situationen im Mathematikunterricht rekonstruiert wird. In der ersten Teiluntersuchung (= der Rekonstruktion eigener Erlebnisse der untersuchten Lehrkräfte) erfolgt die Datenerhebung mit Hilfe biographisch-narrativer Interviews, in denen die untersuchten Lehrkräfte angeregt werden, ihre mathematikbezogene Lebensgeschichte zu erzählen. Die Analyse dieser Interviews wird im Sinne der rekonstruktiven Fallanalyse vorgenommen. Insgesamt führt die erste Teiluntersuchung zu textlichen Darstellungen der rekonstruierten mathematikbezogenen Lebensgeschichte der untersuchten Mathematiklehrkräfte. In der zweiten Teiluntersuchung (= der Rekonstruktion des Fremdverstehens der untersuchten Lehrkräfte) werden dann narrative Interviews geführt, in denen die untersuchten Lehrkräfte von ihrem Fremdverstehen in konkreten Situationen im Mathematikunterricht erzählen. Die Analyse dieser Interviews erfolgt mit Hilfe eines dreischrittigen Analyseverfahrens, welches die Autorin eigens zum Zweck der Rekonstruktion von Fremdverstehen entwickelte. Am Ende dieser zweiten Teiluntersuchung werden sowohl das rekonstruierte Fremdverstehen der Lehrkräfte in verschiedenen Unterrichtssituationen dargestellt als auch Strukturen, die sich in ihrem Fremdverstehen abzeichnen. Mit Hilfe einer theoretischen Verallgemeinerung werden schließlich – auf Basis der Ergebnisse der zweiten Teiluntersuchung – Aussagen über fünf Merkmale des Fremdverstehens von Lehrkräften im Mathematikunterricht im Allgemeinen gewonnen. Mit diesen Aussagen vermag die Arbeit eine erste Beschreibung davon hervorzubringen, wie sich das Phänomen des Fremdverstehens von Lehrkräften im Mathematikunterricht ausgestalten kann.
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.
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.
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.
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.
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.
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.
According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.
In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.
An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions
(2022)
We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.
In dynamic decision problems, it is challenging to find the right balance between maximizing expected rewards and minimizing risks. In this paper, we consider NP-hard mean-variance (MV) optimization problems in Markov decision processes with a finite time horizon. We present a heuristic approach to solve MV problems, which is based on state-dependent risk aversion and efficient dynamic programming techniques. Our approach can also be applied to mean-semivariance (MSV) problems, which particularly focus on the downside risk. We demonstrate the applicability and the effectiveness of our heuristic for dynamic pricing applications. Using reproducible examples, we show that our approach outperforms existing state-of-the-art benchmark models for MV and MSV problems while also providing competitive runtimes. Further, compared to models based on constant risk levels, we find that state-dependent risk aversion allows to more effectively intervene in case sales processes deviate from their planned paths. Our concepts are domain independent, easy to implement and of low computational complexity.
Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.
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.
We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.
We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.
We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.
The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general.
We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results.
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.
A rigorous construction of the supersymmetric path integral associated to a compact spin manifold
(2022)
We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten.
Background
Cytochrome P450 (CYP) 3A contributes to the metabolism of many approved drugs. CYP3A perpetrator drugs can profoundly alter the exposure of CYP3A substrates. However, effects of such drug-drug interactions are usually reported as maximum effects rather than studied as time-dependent processes. Identification of the time course of CYP3A modulation can provide insight into when significant changes to CYP3A activity occurs, help better design drug-drug interaction studies, and manage drug-drug interactions in clinical practice.
Objective
We aimed to quantify the time course and extent of the in vivo modulation of different CYP3A perpetrator drugs on hepatic CYP3A activity and distinguish different modulatory mechanisms by their time of onset, using pharmacologically inactive intravenous microgram doses of the CYP3A-specific substrate midazolam, as a marker of CYP3A activity.
Methods
Twenty-four healthy individuals received an intravenous midazolam bolus followed by a continuous infusion for 10 or 36 h. Individuals were randomized into four arms: within each arm, two individuals served as a placebo control and, 2 h after start of the midazolam infusion, four individuals received the CYP3A perpetrator drug: voriconazole (inhibitor, orally or intravenously), rifampicin (inducer, orally), or efavirenz (activator, orally). After midazolam bolus administration, blood samples were taken every hour (rifampicin arm) or every 15 min (remaining study arms) until the end of midazolam infusion. A total of 1858 concentrations were equally divided between midazolam and its metabolite, 1'-hydroxymidazolam. A nonlinear mixed-effects population pharmacokinetic model of both compounds was developed using NONMEM (R). CYP3A activity modulation was quantified over time, as the relative change of midazolam clearance encountered by the perpetrator drug, compared to the corresponding clearance value in the placebo arm.
Results
Time course of CYP3A modulation and magnitude of maximum effect were identified for each perpetrator drug. While efavirenz CYP3A activation was relatively fast and short, reaching a maximum after approximately 2-3 h, the induction effect of rifampicin could only be observed after 22 h, with a maximum after approximately 28-30 h followed by a steep drop to almost baseline within 1-2 h. In contrast, the inhibitory impact of both oral and intravenous voriconazole was prolonged with a steady inhibition of CYP3A activity followed by a gradual increase in the inhibitory effect until the end of sampling at 8 h. Relative maximum clearance changes were +59.1%, +46.7%, -70.6%, and -61.1% for efavirenz, rifampicin, oral voriconazole, and intravenous voriconazole, respectively.
Conclusions
We could distinguish between different mechanisms of CYP3A modulation by the time of onset. Identification of the time at which clearance significantly changes, per perpetrator drug, can guide the design of an optimal sampling schedule for future drug-drug interaction studies. The impact of a short-term combination of different perpetrator drugs on the paradigm CYP3A substrate midazolam was characterized and can define combination intervals in which no relevant interaction is to be expected.
In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann-Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented.
Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size.
About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique.
We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.
We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.
The application of the fractional calculus in the mathematical modelling of relaxation processes in complex heterogeneous media has attracted a considerable amount of interest lately.
The reason for this is the successful implementation of fractional stochastic and kinetic equations in the studies of non-Debye relaxation.
In this work, we consider the rotational diffusion equation with a generalised memory kernel in the context of dielectric relaxation processes in a medium composed of polar molecules. We give an overview of existing models on non-exponential relaxation and introduce an exponential resetting dynamic in the corresponding process.
The autocorrelation function and complex susceptibility are analysed in detail.
We show that stochastic resetting leads to a saturation of the autocorrelation function to a constant value, in contrast to the case without resetting, for which it decays to zero. The behaviour of the autocorrelation function, as well as the complex susceptibility in the presence of resetting, confirms that the dielectric relaxation dynamics can be tuned by an appropriate choice of the resetting rate.
The presented results are general and flexible, and they will be of interest for the theoretical description of non-trivial relaxation dynamics in heterogeneous systems composed of polar molecules.
We present a Bayesian inference scheme for scaled Brownian motion, and investigate its performance on synthetic data for parameter estimation and model selection in a combined inference with fractional Brownian motion. We include the possibility of measurement noise in both models. We find that for trajectories of a few hundred time points the procedure is able to resolve well the true model and parameters. Using the prior of the synthetic data generation process also for the inference, the approach is optimal based on decision theory. We include a comparison with inference using a prior different from the data generating one.
Randomised one-step time integration methods for deterministic operator differential equations
(2022)
Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.
Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects
(2022)
Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results.
Hidden semi-Markov models generalise hidden Markov models by explicitly modelling the time spent in a given state, the so-called dwell time, using some distribution defined on the natural numbers. While the (shifted) Poisson and negative binomial distribution provide natural choices for such distributions, in practice, parametric distributions can lack the flexibility to adequately model the dwell times. To overcome this problem, a penalised maximum likelihood approach is proposed that allows for a flexible and data-driven estimation of the dwell-time distributions without the need to make any distributional assumption. This approach is suitable for direct modelling purposes or as an exploratory tool to investigate the latent state dynamics. The feasibility and potential of the suggested approach is illustrated in a simulation study and by modelling muskox movements in northeast Greenland using GPS tracking data. The proposed method is implemented in the R-package PHSMM which is available on CRAN.
In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories.
The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative
(2022)
The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg-Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.
The past three decades of policy process studies have seen the emergence of a clear intellectual lineage with regard to complexity. Implicitly or explicitly, scholars have employed complexity theory to examine the intricate dynamics of collective action in political contexts. However, the methodological counterparts to complexity theory, such as computational methods, are rarely used and, even if they are, they are often detached from established policy process theory. Building on a critical review of the application of complexity theory to policy process studies, we present and implement a baseline model of policy processes using the logic of coevolving networks. Our model suggests that an actor's influence depends on their environment and on exogenous events facilitating dialogue and consensus-building. Our results validate previous opinion dynamics models and generate novel patterns. Our discussion provides ground for further research and outlines the path for the field to achieve a computational turn.
An instance of the marriage problem is given by a graph G = (A boolean OR B, E), together with, for each vertex of G, a strict preference order over its neighbors. A matching M of G is popular in the marriage instance if M does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exists and can be easily computed is the set of dominant matchings. A popular matching M is dominant if M wins the head-to-head election against any larger matching. Thus, every dominant matching is a max-size popular matching, and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that there are instead differences in the tractability of stable and dominant matchings and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable; however, it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed to show not only positive results on popular matchings (as is known) but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp., max-size) popular matching that is not stable (resp., dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when G is nonbipartite.
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.
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.
Instruments for measuring the absorbed dose and dose rate under radiation exposure, known as radiation dosimeters, are indispensable in space missions. They are composed of radiation sensors that generate current or voltage response when exposed to ionizing radiation, and processing electronics for computing the absorbed dose and dose rate. Among a wide range of existing radiation sensors, the Radiation Sensitive Field Effect Transistors (RADFETs) have unique advantages for absorbed dose measurement, and a proven record of successful exploitation in space missions. It has been shown that the RADFETs may be also used for the dose rate monitoring. In that regard, we propose a unique design concept that supports the simultaneous operation of a single RADFET as absorbed dose and dose rate monitor. This enables to reduce the cost of implementation, since the need for other types of radiation sensors can be minimized or eliminated. For processing the RADFET's response we propose a readout system composed of analog signal conditioner (ASC) and a self-adaptive multiprocessing system-on-chip (MPSoC). The soft error rate of MPSoC is monitored in real time with embedded sensors, allowing the autonomous switching between three operating modes (high-performance, de-stress and fault-tolerant), according to the application requirements and radiation conditions.
Congenital adrenal hyperplasia (CAH) is the most common form of adrenal insufficiency in childhood; it requires cortisol replacement therapy with hydrocortisone (HC, synthetic cortisol) from birth and therapy monitoring for successful treatment. In children, the less invasive dried blood spot (DBS) sampling with whole blood including red blood cells (RBCs) provides an advantageous alternative to plasma sampling.
Potential differences in binding/association processes between plasma and DBS however need to be considered to correctly interpret DBS measurements for therapy monitoring. While capillary DBS samples would be used in clinical practice, venous cortisol DBS samples from children with adrenal insufficiency were analyzed due to data availability and to directly compare and thus understand potential differences between venous DBS and plasma. A previously published HC plasma pharmacokinetic (PK) model was extended by leveraging these DBS concentrations.
In addition to previously characterized binding of cortisol to albumin (linear process) and corticosteroid-binding globulin (CBG; saturable process), DBS data enabled the characterization of a linear cortisol association with RBCs, and thereby providing a quantitative link between DBS and plasma cortisol concentrations. The ratio between the observed cortisol plasma and DBS concentrations varies highly from 2 to 8. Deterministic simulations of the different cortisol binding/association fractions demonstrated that with higher blood cortisol concentrations, saturation of cortisol binding to CBG was observed, leading to an increase in all other cortisol binding fractions.
In conclusion, a mathematical PK model was developed which links DBS measurements to plasma exposure and thus allows for quantitative interpretation of measurements of DBS samples.
The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented.
We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.
We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.
In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions.
In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting.
Die Vielfältigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zugänge im Mathematikunterricht der Schule. Ausgehend von verschiedenen Vorstellungen zum Winkelbegriff wird in dieser Arbeit ein Lehrgang zur Vermittlung des Winkelbegriffs entwickelt und letztlich in konkrete Umsetzungen für den Schulunterricht überführt.
Dabei erfolgt zunächst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition für den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man über einen Winkel benötigt, um ihn beschreiben zu können. So können die in der fachdidaktischen Literatur auftretenden Winkelvorstellungen aus fachmathematischer Perspektive erneut abgeleitet und validiert werden. Parallel dazu wird ein Verfahren beschrieben, wie Winkel – auch unter dynamischen Aspekten – informationstechnisch verarbeitet werden können, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verfügung stehen.
Unter dem Gesichtspunkt, wie eine Abstraktion des Winkelbegriffs im Mathematikunterricht vonstatten gehen kann, werden die Grundvorstellungsidee sowie die Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten miteinander in Beziehung gesetzt. Aus der Verknüpfung der beiden Theorien wird ein grundsätzlicher Weg abgeleitet, wie im Rahmen der Lehrstrategie eine Ausgangsabstraktion zu einzelnen Winkelaspekten aufgebaut werden kann, was die Generierung von Grundvorstellungen zu den Bestandteilen des jeweiligen Winkelaspekts und zum Operieren mit diesen Begriffsbestandteilen ermöglichen soll. Hierfür wird die Lehrstrategie angepasst, um insbesondere den Übergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit für den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Schülerinnen und Schüler bei der Begriffsaneignung unterstützen.
Die Tätigkeitstheorie, der die genannte Lehrstrategie zuzuordnen ist, zieht sich als roter Faden durch die weitere Arbeit, wenn nun theoriebasiert Designprinzipien generiert werden, die in die Entwicklung einer interaktiven Lernumgebung münden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgefüge aus Schülerinnen und Schülern, dem mathematischen Gegenstand und einer zu entwickelnden App als vermittelndes Medium beschreibt, wobei der Einsatz der App im Unterrichtskontext sowie deren regelgeleitete Entwicklung Bestandteil des Modells sind. Gemäß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und überarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualität der von den Schülerinnen und Schülern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erklären lässt. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einführung des Winkelfeldbegriffs in der vierten Klassenstufe.
This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation.
We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight.
We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution.
Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation.
Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.
In the last decades, there was a notable progress in solving the well-known Boolean satisfiability (Sat) problem, which can be witnessed by powerful Sat solvers. One of the reasons why these solvers are so fast are structural properties of instances that are utilized by the solver’s interna. This thesis deals with the well-studied structural property treewidth, which measures the closeness of an instance to being a tree. In fact, there are many problems parameterized by treewidth that are solvable in polynomial time in the instance size when parameterized by treewidth.
In this work, we study advanced treewidth-based methods and tools for problems in knowledge representation and reasoning (KR). Thereby, we provide means to establish precise runtime results (upper bounds) for canonical problems relevant to KR. Then, we present a new type of problem reduction, which we call decomposition-guided (DG) that
allows us to precisely monitor the treewidth when reducing from one problem to another problem. This new reduction type will be the basis for a long-open lower bound result for quantified Boolean formulas and allows us to design a new methodology for establishing runtime lower bounds for problems parameterized by treewidth.
Finally, despite these lower bounds, we provide an efficient implementation of algorithms that adhere to treewidth. Our approach finds suitable abstractions of instances, which are subsequently refined in a recursive fashion, and it uses Sat solvers for solving subproblems. It turns out that our resulting solver is quite competitive for two canonical counting problems related to Sat.
The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered.
In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime.
The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.
Contributions to the theoretical analysis of the algorithms with adversarial and dependent data
(2021)
In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general functional weak-dependency assumption (the so-called $\cC-$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under $\cC-$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition).
We provide an overview of the tools and techniques of resurgence theory used in the Borel-ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of the Wess-Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger-Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess-Zumino model is Borel-ecalle summable.
Spiele und spieltypische Elemente wie das Sammeln von Treuepunkten sind aus dem Alltag kaum wegzudenken. Zudem werden sie zunehmend in Unternehmen oder in Lernumgebungen eingesetzt. Allerdings ist die Methode Gamification bisher für den pädagogischen Kontext wenig klassifiziert und für Lehrende kaum zugänglich gemacht worden.
Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ansätze für die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu präsentieren. Dies kann eine Grundlage für andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen.
In der Arbeit wird begründet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erhöhen, die Sozial- und Personalkompetenzen fördern sowie die Lernenden zu mehr Aktivität anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz für den Mathematikunterricht hervorgehoben.
Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens für den pädagogischen Kontext entwickelten Klassifikation „FUN“ (Feedback – User specific elements – Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzmöglichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernförderlichem Feedback, Differenzierungsmöglichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein können. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugehörigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird.
Gamification offeriert oftmals Vorteile gegenüber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterführende Forschung könnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Problemlösen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen beschäftigen.
Die Bienaymé-Galton-Watson Prozesse können für die Untersuchung von speziellen und sich entwickelnden Populationen verwendet werden. Die Populationen umfassen Individuen, welche sich identisch, zufällig, selbstständig und unabhängig voneinander fortpflanzen und die jeweils nur eine Generation existieren. Die n-te Generation ergibt sich als zufällige Summe der Individuen der (n-1)-ten Generation. Die Relevanz dieser Prozesse begründet sich innerhalb der Historie und der inner- und außermathematischen Bedeutung. Die Geschichte der Bienaymé-Galton-Watson-Prozesse wird anhand der Entwicklung des Konzeptes bis heute dargestellt. Dabei werden die Wissenschaftler:innen verschiedener Disziplinen angeführt, die Erkenntnisse zu dem Themengebiet beigetragen und das Konzept in ihren Fachbereichen angeführt haben. Somit ergibt sich die außermathematische Signifikanz. Des Weiteren erhält man die innermathematische Bedeutsamkeit mittels des Konzeptes der Verzweigungsprozesse, welches auf die Bienaymé-Galton-Watson Prozesse zurückzuführen ist. Die Verzweigungsprozesse stellen eines der aussagekräftigsten Modelle für die Beschreibung des Populationswachstums dar. Darüber hinaus besteht die derzeitige Wichtigkeit durch die Anwendungsmöglichkeit der Verzweigungsprozesse und der Bienaymé-Galton-Watson Prozesse innerhalb der Epidemiologie. Es werden die Ebola- und die Corona-Pandemie als Anwendungsfelder angeführt. Die Prozesse dienen als Entscheidungsstütze für die Politik und ermöglichen Aussagen über die Auswirkungen von Maßnahmen bezüglich der Pandemien. Neben den Prozessen werden ebenfalls der bedingte Erwartungswert bezüglich diskreter Zufallsvariablen, die wahrscheinlichkeitserzeugende Funktion und die zufällige Summe eingeführt. Die Konzepte vereinfachen die Beschreibung der Prozesse und bilden somit die Grundlage der Betrachtungen. Außerdem werden die benötigten und weiterführenden Eigenschaften der grundlegenden Themengebiete und der Prozesse aufgeführt und bewiesen. Das Kapitel erreicht seinen Höhepunkt bei dem Beweis des Kritikalitätstheorems, wodurch eine Aussage über das Aussterben des Prozesses in verschiedenen Fällen und somit über die Aussterbewahrscheinlichkeit getätigt werden kann. Die Fälle werden anhand der zu erwartenden Anzahl an Nachkommen eines Individuums unterschieden. Es zeigt sich, dass ein Prozess bei einer zu erwartenden Anzahl kleiner gleich Eins mit Sicherheit ausstirbt und bei einer Anzahl größer als Eins, die Population nicht in jedem Fall aussterben muss. Danach werden einzelne Beispiele, wie der linear fractional case, die Population von Fibroblasten (Bindegewebszellen) von Mäusen und die Entstehungsfragestellung der Prozesse, angeführt. Diese werden mithilfe der erlangten Ergebnisse untersucht und einige ausgewählte zufällige Dynamiken werden im nachfolgenden Kapitel simuliert. Die Simulationen erfolgen durch ein in Python erstelltes Programm und werden mithilfe der Inversionsmethode realisiert. Die Simulationen stellen beispielhaft die Entwicklungen in den verschiedenen Kritikalitätsfällen der Prozesse dar. Zudem werden die Häufigkeiten der einzelnen Populationsgrößen in Form von Histogrammen angebracht. Dabei lässt sich der Unterschied zwischen den einzelnen Fällen bestätigen und es wird die Anwendungsmöglichkeit der Bienaymé-Galton-Watson Prozesse bei komplexeren Problemen deutlich. Histogramme bekräftigen, dass die einzelnen Populationsgrößen nur endlich oft vorkommen. Diese Aussage wurde von Galton aufgeworfen und in der Extinktions-Explosions-Dichotomie verwendet. Die dargestellten Erkenntnisse über das Themengebiet und die Betrachtung des Konzeptes werden mit einer didaktischen Analyse abgeschlossen. Die Untersuchung beinhaltet die Berücksichtigung der Fundamentalen Ideen, der Fundamentalen Ideen der Stochastik und der Leitidee „Daten und Zufall“. Dabei ergibt sich, dass in Abhängigkeit der gewählten Perspektive die Anwendung der Bienaymé-Galton-Watson Prozesse innerhalb der Schule plausibel ist und von Vorteil für die Schüler:innen sein kann. Für die Behandlung wird exemplarisch der Rahmenlehrplan für Berlin und Brandenburg analysiert und mit dem Kernlehrplan Nordrhein-Westfalens verglichen. Die Konzeption des Lehrplans aus Berlin und Brandenburg lässt nicht den Schluss zu, dass die Bienaymé-Galton-Watson Prozesse angewendet werden sollten. Es lässt sich feststellen, dass die zugrunde liegende Leitidee nicht vollumfänglich mit manchen Fundamentalen Ideen der Stochastik vereinbar ist. Somit würde eine Modifikation hinsichtlich einer stärkeren Orientierung des Lehrplans an den Fundamentalen Ideen die Anwendung der Prozesse ermöglichen. Die Aussage wird durch die Betrachtung und Übertragung eines nordrhein-westfälischen Unterrichtsentwurfes für stochastische Prozesse auf die Bienaymé-Galton-Watson Prozesse unterstützt. Darüber hinaus werden eine Concept Map und ein Vernetzungspentagraph nach von der Bank konzipiert um diesen Aspekt hervorzuheben.
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.
For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi on the total space M. A multiconformal class [[ g]] contains not only all warped product type deformations of g but also the whole conformal class [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3.
In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H-2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H-2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments.
We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.
Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics
(2021)
Newly emerging pandemics like COVID-19 call for predictive models to implement precisely tuned responses to limit their deep impact on society. Standard epidemic models provide a theoretically well-founded dynamical description of disease incidence. For COVID-19 with infectiousness peaking before and at symptom onset, the SEIR model explains the hidden build-up of exposed individuals which creates challenges for containment strategies. However, spatial heterogeneity raises questions about the adequacy of modeling epidemic outbreaks on the level of a whole country. Here, we show that by applying sequential data assimilation to the stochastic SEIR epidemic model, we can capture the dynamic behavior of outbreaks on a regional level. Regional modeling, with relatively low numbers of infected and demographic noise, accounts for both spatial heterogeneity and stochasticity. Based on adapted models, short-term predictions can be achieved. Thus, with the help of these sequential data assimilation methods, more realistic epidemic models are within reach.
Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.
In June 2018, after 4 years of cruise, the Japanese space probe Hayabusa2 [1-Watanabe S. et al.: Hayabusa2 Mission Overview. (2017)] reached the Near-Earth Asteroid (162173) Ryugu. Hayabusa2 carried a small Lander named MASCOT (Mobile Asteroid Surface Scout) [2-Ho T. M. et al.: MASCOT-The Mobile Asteroid Surface Scout onboard the Hayabusa2 mission. (2017)], jointly developed by the German Aerospace Center (DLR) and the French Space Agency (CNES), to investigate Ryugu's surface structure, composition and physical properties including its thermal behaviour and magnetization in-situ. The Microgravity User Support Centre (DLR-MUSC) in Cologne was in charge of providing all thermal conditions and constraints necessary for the selection of the final landing site and for the final operations of the Lander MASCOT on the surface of the asteroid Ryugu. This article provides a comprehensive assessment of these thermal conditions and constraints, based on predictions performed with the Thermal Mathematical Model (TMM) of MASCOT using different asteroid surface thermal models, ephemeris data for approach as well as descent and hopping trajectories, the related operation sequences and scenarios and the possible environmental conditions driven by the Hayabusa2 spacecraft. A comparison with the real telemetry data confirms the analysis and provides further information about the asteroid characteristics.
Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, this filter also has limitations due to its inherent assumptions of Gaussianity and linearity, which can manifest themselves in the form of dynamically inconsistent state estimates. This issue is investigated here for balanced, slowly evolving solutions to highly oscillatory Hamiltonian systems which are prototypical for applications in numerical weather prediction. It is demonstrated that the standard ensemble Kalman filter can lead to state estimates that do not satisfy the pertinent balance relations and ultimately lead to filter divergence. Two remedies are proposed, one in terms of blended asymptotically consistent time-stepping schemes, and one in terms of minimization-based postprocessing methods. The effects of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for balanced motions of two prototypical Hamiltonian reference systems.
The superposition operation S-n,S-A, n >= 1, n is an element of N, maps to each (n + 1)-tuple of n-ary operations on a set A an n-ary operation on A and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank n. A partial algebra of type (n + 1) which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank n. As a generalization of linear terms we define r-terms as terms where each variable occurs at most r-times. It will be proved that n-ary r-terms form partial Menger algebras of rank n. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider r-hypersubstitutions.U
Im Zuge der Covid-19 Pandemie werden zwei Werte täglich diskutiert: Die zuletzt gemeldete Zahl der neu Infizierten und die sogenannte Reproduktionsrate. Sie gibt wieder, wie viele weitere Menschen ein an Corona erkranktes Individuum im Durchschnitt ansteckt. Für die Schätzung dieses Wertes gibt es viele Möglichkeiten - auch das Robert Koch-Institut gibt in seinem täglichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere Möglichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu schätzen.
In der ersten Hälfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man für die Modellierung benötigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverständnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingeführt und die Ergebnisse aus diesem Themengebiet, die für diese Arbeit wichtig sind, präsentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschränken wir uns auf zwei Typen. Das Prinzip lässt sich aber auf eine beliebige Anzahl von Typen erweitern.
Vor allem soll die Wichtigkeit des Parameters λ herausgestellt werden. Dieser Wert kann als durchschnittliche Zahl von Nachfahren eines Individuums interpretiert werden und bestimmt die Dynamik des Prozesses über einen längeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R.
In der zweiten Hälfte dieser Arbeit stellen wir eine Anwendung der Theorie über Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Veröffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' über einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Schätzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die Möglichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsfälle) zu schätzen. Wir wenden die Schätzer auf die Zahlen von Deutschland an und werten diese schließlich aus.
We consider a social-type network of coupled phase oscillators. Such a network consists of an active core of mutually interacting elements, and of a flock of passive units, which follow the driving from the active elements, but otherwise are not interacting. We consider a ring geometry with a long-range coupling, where active oscillators form a fluctuating chimera pattern. We show that the passive elements are strongly correlated. This is explained by negative transversal Lyapunov exponents.
The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps
(2021)
We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state 0, say, subject to a perturbation by a stochastic integral which is driven by an epsilon-small and (1/epsilon)-accelerated Levy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel, we solve the associated asymptotic first escape problem from the bounded neighborhood of 0 in the limit as epsilon -> 0 which is also known as the Kramers problem in the literature.
Androulidakis-Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold (M, FM ) is the quotient of a foliated manifold (P, FP ) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.
In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold.
Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups.
This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger and Yu (rather: a slight variation). To take the fundamental groups of the manifold and its boundary into account requires working with maximal C*-completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions. On the other hand, we introduce and propose a more conceptual and more geometric completion, which still has all the required functoriality.
For efficient and effective pedagogical interventions to address Uganda's alarmingly poor performance in Physics, it is vital to understand students' motivation patterns for Physics learning. Latent profile analysis (LPA)-a person-centred approach-can be used to investigate these motivation patterns. Using a three-step approach to LPA, we sought to answer the following research questions: RQ1, which profiles of secondary school students exist with regards to their motivation for Physics learning; RQ2, are there differences in students' cognitive learning strategies in the identified profiles; and RQ3, does students' gender, attitudes, and individual interest predict membership in these profiles? The sample comprised 934 Grade 9 students from eight secondary schools in Uganda. Data were collected using standardised questionnaires. Six motivational profiles were identified: (i) low-quantity motivation profile (101 students; 10.8%); (ii) moderate-quantity motivation profile (246 students; 26.3%); (iii) high-quantity motivation profile (365 students; 39.1%); (iv) primarily intrinsically motivated profile (60 students, 6.4%); (v) mostly extrinsically motivated profile (88 students, 9.4%); and (vi) grade-introjected profile (74 students, 7.9%). Low-quantity and grade-introjected motivated students mostly used surface learning strategies whilst the high-quantity and primarily intrinsically motivated students used deep learning strategies. Lastly, unlike gender, individual interest and students' attitudes towards Physics learning predicted profile membership. Teachers should provide an interesting autonomous Physics classroom climate and give students clear instructions in self-reliant behaviours that promote intrinsic motivation.
We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers.
Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.
Forecast verification
(2021)
The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS-RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met.
The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.
We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.
The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f (A)v and matrix functionals of the form v*(f (A))*g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript * denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v*(f (A))*g(A)v about as much as performing an additional step of the Arnoldi process.
A theory for diffusivity estimation for spatially extended activator-inhibitor dynamics modeling the evolution of intracellular signaling networks is developed in the mathematical framework of stochastic reaction-diffusion systems. In order to account for model uncertainties, we extend the results for parameter estimation for semilinear stochastic partial differential equations, as developed in Pasemann and Stannat (Electron J Stat 14(1):547-579, 2020), to the problem of joint estimation of diffusivity and parametrized reaction terms. Our theoretical findings are applied to the estimation of effective diffusivity of signaling components contributing to intracellular dynamics of the actin cytoskeleton in the model organism Dictyostelium discoideum.
We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.
Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.
Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching M ': here each vertex casts a vote for the matching in {M,M '} in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and NP-complete when n has the other parity.
We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.
Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.
In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.
In this work we consider the first encounter problems between a fixed and/or mobile target A and a moving trap B on Bethe lattices and Cayley trees. The survival probabilities (SPs) of the target A on the both kinds of structures are considered analytically and compared. On Bethe lattices, the results show that the fixed target will still prolong its survival time, whereas, on Cayley trees, there are some initial positions where the target should move to prolong its survival time. The mean first encounter time (MFET) for mobile target A is evaluated numerically and compared with the mean first passage time (MFPT) for the fixed target A. Different initial settings are addressed and clear boundaries are obtained. These findings are helpful for optimizing the strategy to prolong the survival time of the target or to speed up the search process on Cayley trees, in relation to the target's movement and the initial position configuration of the two walkers. We also present a new method, which uses a small amount of memory, for simulating random walks on Cayley trees. (C) 2020 Elsevier B.V. All rights reserved.
Partial clones
(2020)
A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.
Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.
We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.
We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.
We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems.
Renormalisation and locality
(2020)
The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces.
We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.
Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees.
Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation
(2020)
Deriving thermophysical properties such as thermal inertia from thermal infrared observations provides useful insights into the structure of the surface material on planetary bodies. The estimation of these properties is usually done by fitting temperature variations calculated by thermophysical models to infrared observations. For multiple free model parameters, traditional methods such as least-squares fitting or Markov chain Monte Carlo methods become computationally too expensive. Consequently, the simultaneous estimation of several thermophysical parameters, together with their corresponding uncertainties and correlations, is often not computationally feasible and the analysis is usually reduced to fitting one or two parameters. Data assimilation (DA) methods have been shown to be robust while sufficiently accurate and computationally affordable even for a large number of parameters. This paper will introduce a standard sequential DA method, the ensemble square root filter, for thermophysical modelling of asteroid surfaces. This method is used to re-analyse infrared observations of the MARA instrument, which measured the diurnal temperature variation of a single boulder on the surface of near-Earth asteroid (162173) Ryugu. The thermal inertia is estimated to be 295 +/- 18 Jm(-2) K-1 s(-1/2), while all five free parameters of the initial analysis are varied and estimated simultaneously. Based on this thermal inertia estimate the thermal conductivity of the boulder is estimated to be between 0.07 and 0.12,Wm(-1) K-1 and the porosity to be between 0.30 and 0.52. For the first time in thermophysical parameter derivation, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure that is more than 5000 times more efficient than a comparable parameter sweep.
When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.
In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.