Study shows superdiffusive behavior in geodesic flows on curved surfaces.
problem Understanding the statistical behavior of geodesic flows on curved surfaces.
method Proved nonstandard central limit theorem with superdiffusive normalisation (tlogt)1/2 for geodesic flows on nonpositively curved surfaces. result Geodesic flows exhibit superdiffusive behavior with correlations decaying at rate t−1. We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for o…
We study weak solutions to degenerate quasilinear elliptic equations, involving first order terms, in unbounded tubular domains. In particular we show that, under suitable hypotheses, the weak comparison principle holds if the domain is narrow enough.
We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.
In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
The Weyl principle holds in some Finsler settings despite general failure.
problem Applying the Weyl principle to Finsler manifolds.
method Investigation of the Weyl principle in Finsler geometry.
result A weak form of the Weyl principle persists in certain Finsler settings.
The paper proves a comparison principle for complex Monge-Ampère flows and solves a uniqueness problem.
problem Proving uniqueness of weak solutions to the pluripotential Cauchy-Dirichlet problem.
method Proving a comparison principle for the pluripotential complex Monge-Ampère flows.
result Proves the uniqueness of the weak solution to the pluripotential Cauchy-Dirichlet problem.
The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and super-solutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for C0 spacelike hypersurfaces in a Lorentzian manifold. As one application a Lorentzian warped product splittin…
S. K. Donaldson asked whether the lower bound of the Calabi functional is achieved by a sequence the normalized Donaldson-Futaki invariants. We answer the question for the Ricci curvature formalism, in place of the scalar curvature. The principle is that the stability indicator is optimized by the multiplier ideal shea…
Extends weak continuity of Yang-Mills connections to a broader class.
problem Weak compactness of Ω-Yang-Mills connections. method Compensation compactness argument applied to Yang-Mills fields.
result Weak continuity result extended to Ω-Yang-Mills connections. New framework improves model robustness by focusing on stable relations across environments.
problem Standard supervised learning fails under data distribution shift.
method Gradient-based learning framework derived from the principle of independent causal mechanisms (ICM).
result Models generalize well to unseen scenarios, ignoring unstable relations.
The aim of this paper is to introduce new forms of the weak and Omori-Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDE's and in the theory of hypersurfaces. In the final part of the paper we consider a large class of non-linear…
Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.
problem Analyzing quasimorphisms on negatively curved spaces.
method Thermodynamic formalism framework, Banach isomorphism, weak Livšic cohomology.
result Establishes Central Limit Theorem and invariance principle for unbounded quasimorphisms.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…
Study the averaging principle for non-autonomous slow-fast systems and apply it to financial local stochastic volatility models.
problem Understanding the behavior of non-autonomous slow-fast systems of stochastic differential equations.
method Prove the averaging principle under specific conditions and apply it to a financial model.
result Prices of derivatives converge to those calculated using the limit model under a risk-neutral measure.
Paper proves large deviation principle for stochastic approximations.
problem Asymptotic estimates of learning algorithm deviations.
method Weak convergence approach to large deviations.
result Identifies appropriate scaling sequence and new representation for rate function.
The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
problem Maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
method Established generalized maximum principles and proved stochastic completeness equivalence.
result Stochastic completeness for the heat semigroup is equivalent to generalized maximum principles.
New homotopy types and invariants defined for knots.
problem Defining and characterizing different homotopy types of knot projections.
method Introducing strong and weak (1, 2) homotopies and defining new invariants.
result New necessary and sufficient conditions for homotopy equivalence of knot projections.
We use quantum invariants to define a 3-manifold invariant j_p which lies in the non-negative integers. We relate j_p to the Heegard genus, and the cut number. We show that j_$ is an invariant of weak p-congruence.
Proposes indifference pricing to estimate weak information value.
problem Estimating the value of weak information in financial models.
method Tractable framework quantifying additional information, stability analysis.
result Sharp conditions for stability with counterexamples, including replicable claims.
New invariants lift Milnor invariants for 3-component links.
problem Classifying 3-component links using Milnor invariants.
method Defined and proved invariants γk(L), introduced h(L), and showed their equivalence. result Invariants γk(L) lift certain Milnor invariants and are invariant under weak cobordism. We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…
FAIRM learns fair and generalizable models by enforcing invariance across different data distributions.
problem Addressing fairness and domain generalization in machine learning models under heterogeneous data.
method FAIRM is a training environment-based oracle that enforces invariance across different data distributions, providing theoretical guarantees and efficient algorithms for linear models.
result FAIRM achieves minimax optimal performance and outperforms existing methods in synthetic and MNIST data evaluations.
We present an axiomatic/synthetic account of the Huygens Principle of wave fronts. The primitive notions are "touching", and (a weak notion of ) metric. The paper simplifies some of the exposition of the author's "Metric spaces and SDG", Theory and Appl. of Categories 32 (2017), 803-822
Based on ideas of L. Alías, D. Impera and M. Rigoli developed in "Hypersurfaces of constant higher order mean curvature in warped products", we develope a fairly general weak/Omori-Yau maximum principle for trace operators. We apply this version of maximum principle to generalize several higher order mean curvature est…
The paper classifies special hypersurfaces in space forms.
problem Investigating gradient Yamabe solitons in space forms.
method Using the weak Omori-Yau principle for the drifted Laplacian.
result Gradient Yamabe solitons are fully classified under certain conditions.
The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.
problem Classifying knot projections based on weak homotopy equivalence.
method Defining weak (1, 2, 3) homotopy and using it to find an invariant.
result There are an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections.
Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.
problem Defining and studying Hochschild cohomology of DG manifolds of positive amplitude.
method Using poly-differential operators and derived intersection, proving invariance under weak equivalences.
result Hochschild cohomology of DG manifolds of positive amplitude is invariant under weak equivalences.
The paper establishes conditions for link invariants to bound the weak splitting number.
problem Determining the minimum number of crossing changes to split a link into knots.
method Conditions on R-valued link invariants, link Floer homology, and known obstructions. result New bounds on the weak splitting number for many prime links.
Paper proposes an algorithm to recover full supervision from weakly labeled data.
problem Machine learning requires expensive data annotation, motivating the use of weak supervision.
method The paper introduces a disambiguation principle and an empirical disambiguation algorithm for partial labelling.
result The algorithm achieves exponential convergence rates under learnability assumptions.
We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints. A weak formulation, using test functions and a probabilistic relaxation of the constraint, avoids restrictions related to a measurable selection but still implies the Hamilton-Jacobi-Bellman equation in the …
New approach combines invariance and information bottleneck for OOD generalization.
problem OOD generalization failures in classification tasks.
method Revisit linear regression tasks, prove information bottleneck constraint necessary, propose combined approach.
result Combined invariance and information bottleneck approach improves OOD generalization.
Photography method solves manifold invariants.
problem Constructing invariants of manifolds.
method General principle of photography, dealing with various data and data transmission.
result Dijkgraaf-Witten invariants are a specific application of the photography principle.
The paper studies a new structure on contact manifolds and its foliation properties.
problem Investigating a new structure on contact manifolds.
method Examining weak nearly Sasakian structures and their foliations.
result The weak nearly Sasakian structure foliates into two types of totally geodesic foliations.
Weak diffusion priors can still perform well in inverse problems.
problem Using mismatched or low-fidelity diffusion priors in inverse problems.
method Extensive experiments and theoretical analysis combining Bayesian-consistency theory and local-correlation analysis.
result Weak priors succeed when measurements are highly informative, and they fail in other regimes.
In this paper, we prove that on a Fano manifold M which admits a Kähler-Ricci soliton $(\om,X)$, if the initial Kähler metric $\om_{\vphi_0}$ is close to $\om$ in some weak sense, then the weak Kähler-Ricci flow exists globally and converges in Cheeger-Gromov sense. Moreover, if $\vphi_0$ is also KX-invariant, the…
Paper develops techniques to create 3-manifold invariants.
problem Constructing non-trivial invariants of 3-manifolds.
method Photography principle and Ptolemy relation techniques.
result Direct implementation leads to a trivial invariant, highlighting difficulties.
Study explores weak generalized K-contact structures in contact metric spaces.
problem Exploring weak generalized K-contact structures in contact metric spaces.
method Introducing a weak (κ,μ) condition and proving existence of K-contact and (κ,μ=2)-structures. result Existence of K-contact and (κ,μ=2)-structures under certain conditions on the Boeckx invariant. Alternative proof of weak solutions to mean curvature flow using minimizing movements.
problem Existence of weak solutions to mean curvature flow and volume preserving mean curvature flow.
method Proposes a new existence proof using a minimizing movements scheme and a novel proxy for distance.
result Unconditional convergence towards a De Giorgi solution for the minimizing movements scheme.
The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
Weak correlations explain linear dynamics in deep learning models.
problem Understanding the linear structure in gradient-based learning algorithms.
method Characterization of weak correlations between derivatives and parameters.
result Weak correlations are the underlying principle for linearization in deep learning models.
Improves recommender systems using mathematical principles.
problem Accuracy and speed of recommender systems.
method Explains and analyzes mathematical principles and algorithms.
result Describes the strengths and weaknesses of mathematical distance methods.
Unified approach to stochastic Volterra systems' deviations.
problem Large and moderate deviations for stochastic Volterra systems.
method Weak convergence approach by Budhijara, Dupuis and Ellis.
result Unified treatment of deviations for a broad class of stochastic Volterra equations.
We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used …
We introduce a new geometric evolution equation for hypersurfaces in asymptotically flat spacetime initial data sets, that unites the theory of marginally outer trapped surfaces (MOTS) with the study of inverse mean curvature flow in asymptotically flat Riemannian manifolds. A theory of weak solutions is developed usin…
New index principle shows indistinguishability of certain knot crossings.
problem Identifying indistinguishable crossings in knot diagrams.
method Developed a universal index function on knot diagrams that respects Reidemeister moves and sign of crossings.
result Crossings of the same sign in a classical knot diagram cannot be distinguished by any inherent property.
For any Engel 4-fold, we show that the scanning map from the space of Engel knots to the space of formal Engel knots is a weak homotopy equivalence when restricted to the complement of the orbits of the Engel kernel. This is a relative, parametric and close h-principle.