Mathematik
Refine
Year of publication
Document Type
- Article (110)
- Doctoral Thesis (76)
- Preprint (44)
- diplomthesis (39)
- Book (25)
- Report (22)
- Conference Proceeding (18)
- Bachelor Thesis (8)
- Contribution to a Periodical (8)
- Diploma Thesis (8)
Has Fulltext
- yes (370) (remove)
Is part of the Bibliography
- no (370)
Keywords
- Kongress (6)
- Kryptologie (5)
- Mathematik (5)
- Stochastik (5)
- Doku Mittelstufe (4)
- Doku Oberstufe (4)
- Online-Publikation (4)
- Statistik (4)
- Finanzmathematik (3)
- LLL-reduction (3)
- Moran model (3)
- coalescent (3)
- computational complexity (3)
- contraction method (3)
- point process (3)
- spike train (3)
- Algebraische Geometrie (2)
- Arithmetische Gruppe (2)
- Biographie (2)
- Brownian motion (2)
- Commitment Scheme (2)
- Frankfurt <Main> / Universität (2)
- Fuchsian groups (2)
- Fächerübergreifender Unterricht (2)
- Geometrie (2)
- Heat kernel (2)
- Hinterlegungsverfahren <Kryptologie> (2)
- Integral Geometry (2)
- Knapsack problem (2)
- Kombinatorische Optimierung (2)
- Krein space (2)
- Laplace operator on graphs (2)
- Lattice basis reduction (2)
- Martingal (2)
- Mathematiker (2)
- Musik (2)
- Oblivious Transfer (2)
- Perception (2)
- Quantum Zeno dynamics (2)
- San Jose (2)
- Semidefinite Programming (2)
- Shortest lattice vector problem (2)
- Stochastischer Prozess (2)
- Subset sum problem (2)
- Tropical geometry (2)
- Tropische Geometrie (2)
- Valuation Theory (2)
- Verzweigungsprozess (2)
- Vision (2)
- W*-dynamical system (2)
- X-Y model (2)
- Yule-Prozess (2)
- ancestral selection graph (2)
- binary search tree (2)
- collective intelligence (2)
- combinatorial optimization (2)
- complexity (2)
- duality (2)
- firing patterns (2)
- fixation probability (2)
- genealogy (2)
- level of difficulty (2)
- quantum spin systems (2)
- return to equilibrium (2)
- segments (2)
- task space (2)
- thought structure (2)
- Λ−coalescent (2)
- A-Discriminant (1)
- ADM1 (1)
- Abelian (1)
- Action potential (1)
- Actions in mathematical learning (1)
- Activity (1)
- Adaptive dynamics (1)
- Algebra (1)
- Algorithmus (1)
- Amoeba (1)
- Anaerobe Fermentation (1)
- Analyse von Algorithmen (1)
- Ancestral selection graph (1)
- Anisotropic Norm (1)
- Approximation (1)
- Approximation algorithm (1)
- Approximationsalgorithmus (1)
- Arbitrage (1)
- Assignment Problem (1)
- Asymptotically Even Nonlinearity (1)
- Ausreißer <Statistik> (1)
- Automorphismengruppe (1)
- Axon (1)
- Banach spaces (1)
- Bayesian Inference (1)
- Berkovich spaces (1)
- Binomialmodell (1)
- Binärsuchbaum (1)
- Black and Scholes Option Price theory (1)
- Black-Scholes (1)
- Blind Signature (1)
- Block Korkin—Zolotarev reduction (1)
- Blockplay (1)
- Bolthausen-Sznitman (1)
- Boolean Lattice (1)
- Bootstrap-Statistik (1)
- Boundary (1)
- Boundary Value Problems (1)
- Branch and Bound (1)
- Branching particle systems (1)
- Branching process approximation (1)
- Breaking knapsack cryptosystems (1)
- Bruhat-Tits-Gebäude (1)
- Burst (1)
- CAT(0)-Räume (1)
- CAT(0)-spaces (1)
- CIR-1 (1)
- Calderón problem (1)
- Cannings model (1)
- Catalan number (1)
- Cauchy-Anfangswertproblem (1)
- Cayley-Graph (1)
- China-Restaurant-Prozess (1)
- Chinese Remainder Theorem (1)
- Chinese restaurant process (1)
- Chinese-restaurant-process (1)
- Circuit (1)
- Closest Vector Problem (1)
- Coamoeba (1)
- Cognitive psychology (1)
- Commitment (1)
- Commitment schemes (1)
- Computational complexity (1)
- Concentration Inequality (1)
- Condensing (1)
- Containment (1)
- Contraction method (1)
- Datenbank (1)
- Datenstruktur (1)
- Degenerate Linear Part (1)
- Dehn (1)
- Derivate (1)
- Dessins d'enfants (1)
- Diagrams and mathematical learning (1)
- Dichte <Stochastik> (1)
- Digital and analogue materials (1)
- Digital trees (1)
- Dimension 2 (1)
- Directional selection (1)
- Dirichlet bound (1)
- Dirichlet random measure (1)
- Dirichletsche L-Reihe; Nullstelle (1)
- Discrete Logarithm (1)
- Diskrete Geometrie (1)
- Diskrete Mathematik (1)
- Diskreter Markov-Prozess (1)
- Diversity in trait space (1)
- Donkers theorem (1)
- Dopamine (1)
- Doplicher-Haag-Roberts Axiomatik; Algebraische Quantenfeldtheorie; Superauswahlregeln und -sektoren; Quantenstatistik; Zopfgruppenstatistik (1)
- Dormancy (1)
- Dosis-Wirkungs-Modellierung (1)
- Dreiecksgruppe (1)
- Dreiecksgruppen (1)
- Duality (1)
- Early Childhood (1)
- Einbettung <Mathematik> (1)
- Elektronische Unterschrift (1)
- Elementar- und Primarbereich (1)
- Endliche Präsentation (1)
- Endlichkeitseigenschaften (1)
- Energie-Modell (1)
- Error Bound (1)
- Erwartungswert (1)
- Evolutionary branching (1)
- Evolving Yield Curves in the Real-World Measures (1)
- Ewens sampling formula (1)
- Examples (1)
- Extended RMJBN Modell (1)
- FEM-BEM-coupling (1)
- FID model (1)
- FIND algorithm (1)
- Face (1)
- Face recognition (1)
- Factoring (1)
- Familie (1)
- Family (1)
- Feller branching with logistic growth (1)
- Finite element methods (1)
- Finitely many measurements (1)
- Fixation probability (1)
- Fixpunkt (1)
- Fractional Brownian Motion (1)
- Fractional Laplacian (1)
- Frühe Bildung (1)
- Fuchs-Gruppe (1)
- Fuchssche Gruppe ; Modulare Einbettung (1)
- Fuchssche Gruppen (1)
- Functions (1)
- Funktionenkegel (1)
- Funktionenkörper ; Arithmetische Gruppe ; Auflösbare Gruppe ; Endlichkeit (1)
- Galerkin Approximation (1)
- Galois group (1)
- Galois-Gruppe (1)
- Game Tree (1)
- Gaussian Random Field (1)
- Gaussian process (1)
- Gelfand-Shilov space (1)
- Gemischte Volumen (1)
- Genealogical construction (1)
- Genealogische Konstruktion (1)
- Genetischer Fingerabdruck (1)
- Genus One (1)
- Geometrische Gruppentheorie (1)
- Geometry (1)
- Gespräch (1)
- Gestaenge (1)
- Girsanov transform (1)
- Gitter <Mathematik> ; Basis <Mathematik> ; Reduktion ; Algorithmus ; Laufzeit ; L-unendlich-Norm ; Rucksackproblem ; Kryptosystem (1)
- Gitter <Mathematik> ; Basis <Mathematik> ; Reduktion ; Gauß-Algorithmus (1)
- Gram-Hadamard inequalities (1)
- Graphen (1)
- Grenzwertsatz (1)
- Griffiths–Engen–McCloskey distribution (1)
- Group dynamics (1)
- Große Abweichung (1)
- Großinvestor (1)
- Gruppendynamiken (1)
- Gruppentheorie (1)
- Hadamard's Three-Lines Theorem (1)
- Halbeinfache algebraische Gruppe (1)
- Handelman (1)
- Handlung (1)
- Harmoniebox (1)
- Heisenberg algebra (1)
- Hidden Markov models (1)
- Hintertür <Informatik> (1)
- Hodge bundle (1)
- Holzklötzchen (1)
- Hopf algebroids (1)
- Householder reflection (1)
- Hyperfunktion ; Asymptotische Entwicklung (1)
- Hypotrochoid (1)
- Identification (1)
- Immigration (1)
- Index at Infinity (1)
- Infrared singularity (1)
- Integer relations (1)
- Integraldarstellung (1)
- Interaction (1)
- Internet (1)
- Invariante (1)
- Inverse problems (1)
- Iteration (1)
- Jahr der Mathematik (1)
- Kettenbruchentwicklung ; Dimension n ; Diophantische Approximation (1)
- Kieferorthopädie (1)
- Klassifizierender Raum (1)
- Klebsiella pneumoniae (1)
- Knotenabstand (1)
- Knotentiefe (1)
- Koaleszent (1)
- Kochen-Specker theorem (1)
- Kollektivintelligenz (1)
- Kombinatorische Gruppen (1)
- Konforme Feldtheorie (1)
- Konstruktiver Beweis (1)
- Kontaktprozess (1)
- Kontraktionsmethode (1)
- Konzentrationsungleichung (1)
- Korkin—Zolotarev reduction (1)
- Kreuzkorrelation (1)
- Kryptosystem (1)
- Kullback-Leibler Informational Divergence (1)
- L^p bounds (1)
- L^p means (1)
- Label cover (1)
- Lanzeitverhalten (1)
- Laplace-Differentialgleichung (1)
- Large Deviation (1)
- Lattice Reduction (1)
- Leerverkauf (1)
- Lernen (1)
- Linear Filtering (1)
- Linear Preferential Attachment Trees (1)
- Linear-Implicit Scheme (1)
- Linkages (1)
- Loewner monotonicity and convexity (1)
- Logarithmic Laplacian (1)
- Long- Range Dependence (1)
- Long-Range Dependence (1)
- Long-time behaviour (1)
- Longitudinal Study (1)
- Lotka-Volterra system (1)
- Lovász Local Lemma (1)
- Low density subset sum algorithm (1)
- MINT-Bildung (1)
- Machine Learning (1)
- Malliavin calculus (1)
- Mallows model (1)
- Markov chain Monte Carlo Method (1)
- Markov chain imbedding technique (1)
- Markov model (1)
- Markov-Kette (1)
- Mathematical Giftedness (1)
- Mathematical Reasoning (1)
- Mathematical modelling (1)
- Mathematics Learning (1)
- Mathematische Bildung (1)
- Mathematische Modellierung (1)
- Max (1)
- McEliece (1)
- Mean Anisotropy (1)
- Message authentication (1)
- Methanogenese (1)
- Mixed Volumes (1)
- Modellierung (1)
- Modular Multiplication (1)
- Mooney faces (1)
- Morava K-theory (1)
- Mouse (1)
- Multi-Harmonie-Ansatz (1)
- Multiple lineare Regression (1)
- Multityp-Verzweigungsprozess mit Immigration (1)
- Multitype Branching with Immigration (1)
- NP-complete problems (1)
- NP-hard (1)
- NP-hardness (1)
- Nash-Gleichgewicht (1)
- Nelson-Siegel (1)
- Neural encoding (1)
- Neurophysiology (1)
- Neuroscience (1)
- Neurowissenschaft (1)
- Newton–Okounkov bodies (1)
- Non-Malleability (1)
- Noticeable Probability (1)
- Optimal Mean-Square Filter (1)
- Oracle Query (1)
- Parabolic SPDE (1)
- Parisi conjecture (1)
- Participation (1)
- Partizipation (1)
- Patientenbewertung (1)
- Pause (1)
- Permutation (1)
- Permutationsgruppen (1)
- Pfadeigenschaften (1)
- Phragmén-Lindelöf principle (1)
- Piecewise-constant coefficient (1)
- Poisson Process (1)
- Poisson boundary (1)
- Poisson-Prozess (1)
- Polyedrische Kombinatorik (1)
- Polymorphic evolution sequence (1)
- Polynomial Optimization (1)
- Pontrjagin space (1)
- Populationsdynamiken (1)
- Portfolios (1)
- Positivstellensatz (1)
- Potenzialtheorie (1)
- Prag <1999> (1)
- Preferential Attachment-Modelle (1)
- Private Information Retrieval (1)
- Probabilistic analysis of algorithms (1)
- Probabilistically checkable proofs (1)
- Probabilistische Analyse von Algorithmen (1)
- Probability distribution (1)
- Probability of fixation (1)
- Professionalisierung (1)
- Profil Likelihood (1)
- Projektionen (1)
- Public Key Cryptosystem (1)
- Public Parameter (1)
- Punktprozess (1)
- Pólya urn (1)
- Quadratic Residue (1)
- Quantenfeldtheorie ; Konforme Feldtheorie ; Algebraische Methode (1)
- Quantum Zeno Effect (1)
- Quantum Zeno effect (1)
- Quasi-Automorphismen (1)
- Quaternionenalgebra (1)
- Quickselect (1)
- RSA-Verschlüsselung (1)
- Radix sort (1)
- Random Oracle (1)
- Random Split Trees (1)
- Random String (1)
- Random environment (1)
- Random variables (1)
- Randomisieren (1)
- Ray-Knight representation (1)
- Reaction time (1)
- Reale vs. risikoneutrale Welt in der Finanzmathematik (1)
- Rechenzentrum (1)
- Rekursiver Algorithmus (1)
- Relaxation (1)
- Representation Problem (1)
- Research article (1)
- Riemann surfaces (1)
- Riemannsche Fläche (1)
- Riemannsche Flächen (1)
- Ringtheorie (1)
- Risikobewertung (1)
- Risikomanagement (1)
- Robustheit (1)
- Rückkopplungseffekt (1)
- S-arithmetic groups (1)
- SLLL-reduction (1)
- Sackgassen (1)
- San Francisco (1)
- Santa Barbara (1)
- Schizophrenia (1)
- Schwarz triangel functions (1)
- Schwinger model (1)
- Security (1)
- Security Parameter (1)
- Semidefinite Optimierung (1)
- Semidefinite Optimization (1)
- Semiotics according to C. S. Peirce (1)
- Sensory perception (1)
- Sensory processing (1)
- Sigma-Invariante (1)
- Sigma-invariant (1)
- Signalverarbeitung (1)
- Signature (1)
- Small Worlds (1)
- Small order expansion (1)
- Spectrahedra (1)
- Spiel (1)
- Spielbaum (1)
- Spielbaum-Suchverfahren (1)
- Stable reduction algorithm (1)
- State dependent branching rate (1)
- Stationarity (1)
- Stochastic Analysis of Square Zero Variation Processes (1)
- Stonesches Spektrum (1)
- Striatum (1)
- Strong Taylor Scheme (1)
- Stummel, Friedrich (1)
- Suchbaum (1)
- Suchoperation (1)
- Sudoku (1)
- Sum of Squares (1)
- Support (1)
- Symmetrie (1)
- Symmetrischer Raum (1)
- Symmetry (1)
- Sympatric speciation (1)
- Tail Bound (1)
- Tailschranke (1)
- Talk (1)
- Thorne Kishino Felsenstein model (1)
- Topic Model (1)
- Trapdoor (1)
- Trinomial (1)
- Tropical Geometry (1)
- Tropical Grassmannians (1)
- Tropical bases (1)
- Tropical varieties (1)
- Tropische Basen (1)
- Trotter's product formula (1)
- Turkish immigrants (1)
- Typ-In-Algebra (1)
- Typology (1)
- Türkisch (1)
- Uniform regularity (1)
- Uniform resource locators (1)
- Unterstützung (1)
- Valuation on functions (1)
- Varianz (1)
- Vertexoperator (1)
- Verzweigende Teilchensysteme (1)
- Virasoro-Algebra (1)
- Wahrscheinlichkeit (1)
- Wahrscheinlichkeitsverteilung (1)
- Wiener Index (1)
- Wiener index (1)
- Wiener-Index (1)
- Yule process (1)
- Yule-process (1)
- Zinsstrukturmodelle (1)
- Zinsänderungsrisiko (1)
- Zolotarev metric (1)
- Zolotarev-Metrik (1)
- Zopfgruppe ; Lineare Darstellung ; Kettengruppe ; Homologiegruppe ; Automorphismengruppe ; Kettenkomplex (1)
- Zufall (1)
- Zufallsgraph (1)
- Zufällige Umgebung (1)
- Zustandsabhängige Verzweigungsrate (1)
- Zweiphasen-Biogasreaktor (1)
- Zweistufen-Biogasreaktor (1)
- abelian differentials (1)
- abstract potential theory (1)
- algebraic curves (1)
- algebraic values (1)
- alpha-stable branching (1)
- ampleness (1)
- analysis of algorithms (1)
- anti-Zeno effect (1)
- argumentation (1)
- arithmetic ball quotients (1)
- arithmetic group (1)
- assignment problem (1)
- augmented and restricted base loci (1)
- autocorrelograms (1)
- bid-ask spread (1)
- bordism theory (1)
- branching processes (1)
- branching random walk in random medium (1)
- buildings (1)
- cancer cell dormancy (1)
- canonical divisors (1)
- catastrophe modeling (1)
- central limit theorem (1)
- chosen ciphertext attack (1)
- clique problem (1)
- colorabdity (1)
- colored graphs (1)
- compact Riemann surfaces (1)
- complex multiplication (1)
- composition (1)
- computational geometry (1)
- concurrent composition (1)
- condensing (1)
- confirmatory factory analysis (1)
- consensus (1)
- contact process (1)
- continued fraction algorithm (1)
- controlled homotopy (1)
- convexity (1)
- convolution quadrature (1)
- cooperative systems (1)
- cross correlation (1)
- cryptography (1)
- cycle structure of permutations (1)
- dead ends (1)
- degenerate semigroup (1)
- delay equation (1)
- depth of a nod (1)
- dessins d’enfants (1)
- difference sets (1)
- digital search tree (1)
- digital tools (1)
- discrete dynamical system (1)
- discrete logarithm (1)
- discrete logarithm (DL) (1)
- diskrete Mathematik (1)
- dose-resoponse modelling (1)
- doubly stochastic point process (1)
- eigenvalue (1)
- elastodynamic wave equation (1)
- emergence (1)
- endliche metrische Räume (1)
- error bounds (1)
- exponentiation (1)
- external branch (1)
- face inversion (1)
- face perception (1)
- fake projective planes (1)
- families of hash functions (1)
- feedback effect (1)
- finite resolution (1)
- finiteness-properties (1)
- flat surfaces (1)
- floating norms (1)
- floating point arithmetic (1)
- floating point errors (1)
- foliated Schwarz symmetry (1)
- forming a group (1)
- fractional Brownian motion (1)
- fractions of exponentiation (1)
- frühkindliche Erziehung (1)
- fuchsian group (1)
- functional limit theorem (1)
- functional limit theorems (1)
- fächerübergreifendes Lernen (1)
- generic algorithm (1)
- generic algorithms (1)
- generic complexity (1)
- generic group model (1)
- geometry (1)
- graph coloring (1)
- graph isomorphism (1)
- h-transform (1)
- hard bit (1)
- hardcore subsets (1)
- harmonic function (1)
- heavy tails (1)
- hidden Markov model (1)
- hierarchical mean-field limit (1)
- highly regular nearby points (1)
- hyperbolische Geometrie (1)
- hypergeometric functions (1)
- hypervariable region (1)
- höhere Momente (1)
- incremental schemes (1)
- indefinite inner product space (1)
- individual-based models (1)
- inner product (1)
- integer relation (1)
- integer vector (1)
- interacting particle Systems (1)
- interdisziplinäre Lehre (1)
- internal diffusion limited aggregation (1)
- internal path length (1)
- inverse coefficient problem, (1)
- iterated subsegments (1)
- key comparisons (1)
- kinetic fingerprint (1)
- knapsack cryptosystems (1)
- kontrollierte Homotopie (1)
- large deviations (1)
- large trader (1)
- latent variance (1)
- lattice basis reduction (1)
- lattices (1)
- leapfrog (1)
- length defect (1)
- limit order markets (1)
- local LLL-reduction (1)
- local LLLreduction (1)
- local coordinates (1)
- local randomness (1)
- local time (1)
- local time drift (1)
- logarithmic geometry (1)
- logical networks (1)
- lookdown construction (1)
- lower bounds (1)
- manifold and geodesic (1)
- market making (1)
- mathematical modeling (1)
- mathematical modelling (1)
- mathematics (1)
- measurement (1)
- mehrdimensionale Ausreißererkennung (1)
- message-passing algorithm (1)
- modelling (1)
- modular automorphism group (1)
- modular group (1)
- moduli spaces (1)
- multi-agents system (1)
- multi-drug treatment (1)
- multiharmony (1)
- multilevel branching (1)
- music (1)
- mutation parameter estimation (1)
- neuronal code (1)
- neuronaler Kode (1)
- nichtlineare stochastische Integration (1)
- non-archimedean geometry (1)
- non-autonomous dynamical systems (1)
- non-malleability (1)
- noncommutative ring spectra (1)
- nondetermmistlc Turing machines (1)
- nonlinear stochastic integration (1)
- numerical experiments (1)
- observable Funktion (1)
- one-more decryption attack (1)
- one-way function (1)
- one-way functions (1)
- operator algebra (1)
- optimal transport (1)
- pair HMM (1)
- parameter dependent semimartingales (1)
- parameterabhängige Semimartingale (1)
- partial match queries (1)
- path properties (1)
- perceptual closure (1)
- permutation groups (1)
- phage (1)
- phage therapy (1)
- phase coding (1)
- phase transitions (1)
- platonischer Körper (1)
- poisson process (1)
- polynomial random number generator (1)
- population dynamics (1)
- portfolio optimization (1)
- positivity of line bundles (1)
- preferential attachment (1)
- preferential attachment models (1)
- probabilistic analysis of algorithms (1)
- probability (1)
- probability metric (1)
- professional development (1)
- profile likelihood (1)
- projections (1)
- projective planes (1)
- q-binomial theorem (1)
- quantum field theory (1)
- quasi-automorphisms (1)
- quaternion algebra (1)
- quincunx (1)
- random assignment problem (1)
- random environment (1)
- random function generator (1)
- random graphs (1)
- random measures (1)
- random media (1)
- random metric (1)
- random move (1)
- random number generator (1)
- random oracle model (1)
- random partition (1)
- random recursive tree (1)
- random rekursiv tree (1)
- random trees (1)
- random walks (1)
- raum-zeitliche Muster (1)
- reactant-catalyst systems (1)
- recursive distributional equation (1)
- reguläre Parkettierung (1)
- resistance (1)
- resistance mutation (1)
- reversibility (1)
- riemann surfaces (1)
- risk assessment (1)
- risk theory (1)
- rotating plane method (1)
- rough paths theory (1)
- satlsfiablhty (1)
- scaling (1)
- search operation (1)
- searchtrees (1)
- secure bit (1)
- security analysis of protocols (1)
- security of data (1)
- self-organizing groups (1)
- self-organizing groups; population dynamics; collective intelligence; forming groups; metric on finite sets (1)
- semidefinite optimization (1)
- sequence alignment (1)
- set-valued pullback attractors (1)
- shadow price (1)
- short integer relation (1)
- shortest lattice vector (1)
- signature size (1)
- signed ElGamal encryption (1)
- simultaneous diophantine approximations (1)
- simultaneous security of bits (1)
- single block replacement (1)
- small worlds (1)
- spatio-temporal patterns (1)
- split tree (1)
- statistic analysis (1)
- statistical alignment (1)
- statistische Analyse (1)
- statistischer Test (1)
- stoch. Analyse von Algorithmen (1)
- stochastic filtering (1)
- stochastic modeling (1)
- stochastic population dynamics (1)
- stochastische Prozesse (1)
- strong transience (1)
- subgroup growth (1)
- subset sum problems (1)
- substitution attacks (1)
- sum of squared factor loadings (1)
- switching systems (1)
- synergistic interaction (1)
- therapy evasion (1)
- topological entropy (1)
- trading strategies (1)
- transcendence (1)
- transversal learning (1)
- treatment protocol design (1)
- treatment success (1)
- triangle group (1)
- triangle groups (1)
- tropical geometry (1)
- tropical universal Jacobian (1)
- tropicalization (1)
- universal compactified Jacobian (1)
- urn model (1)
- von Neumann algebra (1)
- von Neumann algebras (1)
- von Neumann-Algebra (1)
- weak convergence (1)
- zufälliger Algorithmus (1)
- zufälliger rekursiver Baum (1)
- zufälliges Assignment Problem (1)
- Λ-coalescent (1)
- σ-field (1)
Institute
- Mathematik (370) (remove)
Geometry is part of the core of mathematics. It has been relevant ever since people have interacted with nature and its phenomena. Geometry’s relevance to the teaching and learning of mathematics can be emphasized, too. Nevertheless, a current potential shift in the topics of mathematics education to the detriment of geometry might be emerging. That is, other topics related to mathematics are seeming to grow in importance in comparison to geometry. Despite this, or perhaps because of it, geometry is an important component of current research in mathematics education. In the literature review, we elaborate relevant foci on the basis of current conference proceedings. By means of about 50 journal articles, five main topics are elaborated in more detail: geometric thinking and practices, geometric contents and topics, teacher education in geometry, argumentation and proof in geometry, as well as the use of digital tools for the teaching and learning of geometry. Conclusions and limitations for current and future research on geometry are formulated at the end of the article. In particular, the transfer to the practices of geometric teaching is explored on the basis of the elaborated research findings in order to combine both aspects of the teaching and learning of geometry.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the true cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.
The free energy of TAP-solutions for the SK-model of mean field spin glasses can be expressed as a nonlinear functional of local terms: we exploit this feature in order to contrive abstract REM-like models which we then solve by a classical large deviations treatment. This allows to identify the origin of the physically unsettling quadratic (in the inverse of temperature) correction to the Parisi free energy for the SK-model, and formalizes the true cavity dynamics which acts on TAP-space, i.e. on the space of TAP-solutions. From a non-spin glass point of view, this work is the first in a series of refinements which addresses the stability of hierarchical structures in models of evolving populations.
During my initial days here in Frankfurt, in October 2020 amidst the pandemic crisis, all my notes revolved around three articles by Bolthausen and Kistler, which now form the starting point of this work.
The ones introduced by Bolthausen and Kistler are abstract mean field spin glass models, reminiscent of Derrida’s Generalized Random Energy Model (GREM), which generalize the GREM while remaining rigorously solvable through large deviations methods and within a classical Boltzmann-Gibbs formalism. This allows to establish, by means of a second moment method, the associated free energy at the thermodynamic limit as an orthodox, infinite-dimensional, Boltzmann-Gibbs variational principle.
Dual Parisi formulas for the limiting free energy associated with these Hamiltonians hold, and are revealed to be the finite-dimensional (”collapsed”) versions of the classical, infinite-dimensional Boltzmann-Gibbs principles.
In the 2nd chapter of this thesis, we uncover the hidden yet essential connection between real-world spin glasses, like the Sherrington-Kirkpatrick (SK) model and the random energy models. The crucial missing element is that of TAP-free energies: integrating it with the framework introduced by Bolthausen and Kistler results in a correction to the Parisi formula for the free energy, which brings it much, much closer to the ”true” Parisi solution for the SK-model. In other words, we can identify the principles that transform the classical Boltzmann-Gibbs maximization into the unorthodox (and puzzling) Parisi minimization.
This arguably stands as the primary achievement of this work.
Komplexität und Zufälligkeit
(1978)
Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~N is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. The classical variant considers {\it fitness proportional} selection, but other fitness schemes are conceivable as well. Inspired by the work of Etheridge et al. ([EPW09]) we propose a parameter scaling which fits well to the ``near-critical'' regime that was in the focus of [EPW09] (and in which the mutation-selection ratio diverges logarithmically as N→∞). Using a Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click rate of the ``classical ratchet'' by putting it into the context of new results on the long-time evolution of the size of the best class of the ratchet with (binary) tournament selection, which (other than that of the classical ratchet) follows an autonomous dynamics up to the time of its extinction. In [GSW23] it was discovered that the tournament ratchet has a hierarchy of dual processes which can be constructed on top of an Ancestral Selection graph with a Poisson decoration. For a regime in which the mutation/selection-ratio remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics of the click rates as well as that of the type frequency profile between clicks. We will describe how these ideas can be extended to the near-critical regime in which the mutation-selection ratio of the tournament ratchet converges to 1 as N→∞.
Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the "partial order of adaptation", we can couple these measures. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.