The Levy-Ito theorem explains how certain random processes can be broken down.
problem Understanding how certain random processes can be decomposed.
method Martingale methods are used to prove the Lévy-Ito decomposition theorem.
result The Lévy-Khintchine representation of infinitely divisible distributions is derived.
Models for financial assets with jumps and Brownian motion.
problem Understanding asset prices with jumps and their returns.
method Lévy-Ito models with Brownian motion and Poisson measure.
result Excess return rates calculated for risky assets.
Optimal hedging strategy found in markets with incomplete pricing kernels.
problem Finding optimal hedging in markets with incomplete pricing kernels.
method Demonstrated existence of an optimal hedge portfolio using an expected least squared-error criterion.
result Existence of an optimal hedge portfolio in Lévy-Ito markets.
Optimal probability measure found for constrained stochastic processes.
problem Finding optimal probability measure with constraints for stochastic processes.
method Existence and uniqueness proof, explicit measure change, optimal drift and compensator adjustments.
result Explicit form of the optimal measure change and characterisation of adjustments.
These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…
DLPM replaces Gaussian noise with α-stable noise in DDPM, improving data distribution coverage and robustness.
problem Handling mode collapse and class imbalance in datasets with heavy-tailed noise.
method Extending DDPM to use α-stable noise, simplifying the process with elementary proof techniques.
result DLPM yields better coverage of data distribution tails, improved robustness to unbalanced datasets, and faster computation times.
Theorem analogues proven using Artin's approximation theorem.
problem Proving analogues of Moser's Theorem.
method Using Artin's approximation theorem.
result Few analogues of Moser's Theorem proven.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
A simplified proof of a theorem by Hopf.
problem Proving Hopf's theorem about hypersurface Gauss maps.
method Using the Chern-Gauss-Bonnet Theorem.
result A concise proof of Hopf's theorem.
New measure proves Poncelet-type theorems.
problem Proving Poncelet-type theorems.
method Introducing a new invariant measure on the circle.
result Simple proof of Emch closing theorem.
New theorem for doodles on sphere, similar to Markov's.
problem Understanding doodles on a sphere.
method Description of twins with equivalent closures.
result Analogous to Markov's theorem for doodles.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Extends calculus theorem to higher dimensions.
problem Calculus theorem limitations in higher dimensions.
method Type θ Stokes' theorem for type θ k-chains. result Extends fundamental theorem of calculus.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
INT benchmark tests theorem proving agents' ability to generalize to unseen theorems.
problem Evaluating theorem proving agents' ability to generalize to unseen theorems.
method INT benchmark based on a theorem generation and proof procedure with adjustable knobs for measuring 6 types of generalization.
result MCTS can help agents prove new theorems.
Proves a new convergence theorem for mean curvature flow in spheres.
problem Improves convergence theorem for mean curvature flow.
method Investigates Liu-Xu-Ye-Zhao's conjecture and proves a new convergence theorem.
result Sharp convergence theorem for mean curvature flow of arbitrary codimension in spheres.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
Paper proves compactness and rigidity of λ-surfaces in 3D.
problem Compactness and rigidity of λ-surfaces in R3. method Developed a compactness theorem for λ-surfaces with uniform λ, genus, and area growth. result Proved a rigidity theorem for convex λ-surfaces. Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
Generalizes Simon's theorem to spacetime and hyperbolic manifolds.
problem Proving uniqueness of black holes in various spacetime settings.
method Extending Simon's conformal positive mass theorem to spacetime and hyperbolic manifolds.
result Proves a conformal positive mass theorem on asymptotically hyperbolic manifolds.
Notes on a theorem with broad applications in dynamical systems.
problem The Oseledets Multiplicative Ergodic Theorem and its applications.
method Introduction and lectures on the theorem and its generalizations.
result Introduction to the theorem and its applications.
The paper proves injectivity and vanishing theorems on compact Kahler manifolds.
problem Injectivity and vanishing theorems on compact Kahler manifolds.
method Hodge theory, Bochner-Kodaira-Nakano identity, analytic method, transcendental method, Demailly-Peternell-Schneider equisingular approximation theorem, Hormander L2 estimates.
result The main injectivity theorem implies several Nadel type vanishing theorems.
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
The paper proves sphere theorems for submanifolds in Kähler manifolds.
problem Sphere theorems for submanifolds in Kähler manifolds.
method Differentiable and topological sphere theorems for submanifolds in Kähler manifolds, especially in complex space forms.
result Proves sphere theorems for submanifolds in Kähler manifolds.
The paper proves an equivariant Kastler-Kalau-Walze theorem for various spin manifolds.
problem Proving an equivariant Kastler-Kalau-Walze theorem for different types of manifolds.
method Analyzing spin manifolds and their boundaries, proving the theorem for various dimensions and with torsion.
result Generalized Kastler-Kalau-Walze theorem for n-dimensional manifolds. Reviews g-theorem and hard Lefschetz theorem for face rings.
problem Understanding and proving the g-theorem and hard Lefschetz theorem for face rings.
method Perturbations of maps, biased Poincaré pairings, cobordism argument, edge-contractions.
result Alternative and alternative arguments for the Lefschetz property.
The paper generalizes the Liouville theorem for minimal hypersurfaces with finite index.
problem Studying minimal hypersurfaces with finite index in Riemannian manifolds.
method Generalizing previous theorems on stable minimal hypersurfaces.
result Eigenvalue estimates for minimal hypersurfaces with finite index.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem
The paper extends cyclic cocycles and proves relative index theorems for partitioned manifolds.
problem Index theorems on partitioned manifolds.
method Extending Roe's cyclic 1-cocycle to relative settings and proving index theorems using the cyclic cocycle.
result Generalizations of index theorems on partitioned manifolds.
Reidemeister's theorem proved using smooth functions and transversality.
problem Proving Reidemeister's theorem
method Using smooth functions and transversality
result Reidemeister's theorem proved