Notes on continuity of discrete-spectrum Fredholm operators.
problem Continuity properties of discrete-spectrum families of Fredholm operators.
method Relates recent work on discrete-spectrum families to classical continuity properties.
result Establishes connections between new and classical concepts.
Proves properties of sub-Riemannian exponential map, showing it's not injective.
problem Regularity and continuity of sub-Riemannian exponential map.
method Used sub-Riemannian Jacobi fields and Maslov index of Jacobi curves.
result Exponential map of 3D Heisenberg group is not injective near conjugate vectors.
The paper studies continuous submodular functions and their optimization.
problem Maximizing continuous submodular functions in poly. time.
method Characterization of continuous submodularity, operations preserving it, and algorithms for constrained maximization.
result Continuous submodularity is equivalent to a weak DR property, leading to continuous DR-submodular functions with the full DR property.
Classifies when homeomorphism groups of stable surfaces have automatic continuity.
problem Determining when homeomorphism groups of stable surfaces are continuous.
method Developed a general framework to prove automatic continuity for homeomorphism groups, applied to stable surfaces and Stone spaces.
result Classification of stable surfaces with respect to automatic continuity of their homeomorphism groups.
Unique continuation property for measures in high dimensions.
problem Understanding the structure of measures in high-dimensional spaces.
method Analyzing locally uniformly distributed measures and their supports.
result Locally uniformly distributed measures satisfy a unique continuation property.
Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.
problem Characterize biharmonic hypersurfaces in spheres.
method Prove CMC Unique Continuation Theorem for biharmonic hypersurfaces of spheres.
result Supports the conjecture that biharmonic submanifolds of Euclidean spheres must be of constant mean curvature.
The paper explores properties of continuous actions on manifolds, proving bounds on subgroup size and fixed points.
problem Properties of continuous finite group actions on topological manifolds.
method Analyzes properties including Jordan property and almost fixed point property, proving bounds on subgroup size.
result Existence of a constant C such that for any continuous action of a finite group G on a manifold X, there is a subgroup H with [G:H] ≤ C and a fixed point.
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
Proves regularity of extremal function on compact Kähler manifolds.
problem Regularity of extremal function on compact Kähler manifolds.
method Local property analysis and equivalence of continuity and Hölder continuity.
result Equivalence of classical notions of local L-regularity and locally Hölder continuous property. The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…
Researchers examine various causal structures for spacetimes with continuous metrics.
problem Comparing causal structures for spacetimes with continuous but not necessarily smooth metrics.
method Examined three key properties: push-up lemma, openness of chronological futures, and existence of limit causal curves.
result Spacetimes with continuous metrics do not always satisfy all three key properties.
This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.
Formal methods verify continuous auctions at exchanges.
problem Ensuring fairness and correctness in continuous auctions.
method Formal specification, design, and verification of continuous double auctions.
result A verified algorithm satisfies natural properties of auctions.
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an isometry extension property is proved: continuous groups of isometries at confor…
We solve the differentiability problem for the evolution map in Milnor's infinite dimensional setting. We first show that the evolution map of each Ck-semiregular Lie group G (for k∈N⊔{lip,∞}) admits a particular kind of sequentially continuity − called Mackey k-continuity. We …
We prove the unique continuation property at the conformal infinity for asymptotically hyperbolic Einstein metrics.
Study continuity of phi-invariant for degenerating graphs.
problem Continuity of phi-invariant for degenerating graphs.
method Use Yuan--Zhang's adelic divisors and follow Yuan's globalization of phi-invariants.
result Asymptotic expression of Zhang--Kawazumi's invariants for Riemann surfaces near the boundary of the moduli space.
Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M …
UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.
problem Approximating any function using ODENet and ResNet.
method Proved UAP for ODENet and ResNet, derived gradient, and applied to various problems.
result UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.
Study optimal stopping problems with finite-time horizon and proves continuity and strict monotonicity of the boundary.
problem Optimal stopping problems with finite-time horizon and state-dependent discounting.
method Linear diffusion process, time-homogeneous gain function, fine regularity properties, continuity and strict monotonicity proof.
result Proves continuity and strict monotonicity of the optimal stopping boundary under mild assumptions.
Study local expansions of continuous-time processes using Ito signature properties.
problem Analyzing local expansions of continuous-time processes and their moments.
method Using the Ito signature, a basis of iterated integrals, to conduct expansions of the process' characteristic function.
result Explicit coefficients and stochastic representations for asymptotics as time shrinks or diverges.
Novel framework proves fast RL convergence in continuous spaces.
problem Analyzing stability in continuous state-action RL.
method Introduces a novel framework to analyze stability properties of RL.
result Highlights two key stability properties and demonstrates their satisfaction in RL.
We study asymptotic properties of maximum likelihood estimators for Heston models based on continuous time observations of the log-price process. We distinguish three cases: subcritical (also called ergodic), critical and supercritical. In the subcritical case, asymptotic normality is proved for all the parameters, whi…
We study the relationship between catastrophic forgetting and properties of task sequences. In particular, given a sequence of tasks, we would like to understand which properties of this sequence influence the error rates of continual learning algorithms trained on the sequence. To this end, we propose a new procedure …
The important application of semi-static hedging in financial markets naturally leads to the notion of quasi self-dual processes which is, for continuous semimartingales, related to symmetry properties of both their ordinary as well as their stochastic logarithms. We provide a structure result for continuous quasi self…
In this paper we study a special case of the completion of cusp Kähler-Einstein metric on the regular part of varieties by taking the continuity method proposed by La Nave and Tian. The differential geometric and algebro-geometric properties of the noncollapsing limit in the continuity method with cusp singularities wi…
Paper identifies and estimates CAPCEs in continuous treatment settings.
problem Estimating heterogeneous causal effects of continuous treatments.
method Instrumental variable approach to identify CAPCEs under weaker conditions.
result Developed three families of CAPCE estimators with statistical properties analyzed.
Regime-switching models, in particular Hidden Markov Models (HMMs) where the switching is driven by an unobservable Markov chain, are widely-used in financial applications, due to their tractability and good econometric properties. In this work we consider HMMs in continuous time with both constant and switching volati…
Study continuation maps for Morse fundamental group properties.
problem Properties of continuation maps for Morse fundamental group.
method Analysis of continuation maps for Morse fundamental group, functoriality, and isomorphism to relative fundamental group.
result Continuation maps are isomorphic to relative fundamental groups.
Study analyzes a new algorithm for complex optimization problems.
problem Stochastic bilevel optimisation problems in continuous-time models.
method Continuous-time, two-timescale stochastic approximation algorithm.
result Obtained weak convergence rate using central limit theorem.
Paper studies geometric properties of nonlinear Lebesgue spaces.
problem Geometric and analytic properties of nonlinear Lebesgue spaces.
method Formalizes pointwise description of geometric properties using a nonlinear Fubini-Lebesgue theorem.
result Definition of length structure, Alexandrov curvature bounds, and speed for absolutely continuous curves in nonlinear Lebesgue spaces.
Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strate…
New methods for calculating curvature in graph theory.
problem Calculating curvature in graphs and random walks.
method Analyzing continuous and discrete-time Ollivier-Ricci curvatures of weighted graphs.
result Generalized existence and properties of Ollivier-Ricci curvature for various random walks.
New method improves submodular maximization for machine learning applications.
problem Inexact monotonicity in submodular functions limits traditional algorithms' performance.
method Introduces monotonicity ratio as a continuous version of monotonicity, leading to improved approximation guarantees.
result Improved approximation ratios for movie recommendation, quadratic programming, and image summarization.
New algorithm learns value and advantage functions for continuous-time Markov processes without structural assumptions.
problem Learning value and advantage functions for continuous-time Markov processes without structural assumptions.
method Proposes Sobolev-prox fitted q-learning algorithm based on Hilbert-space positive definiteness and boundedness properties of Bellman operators. result Identifies ellipticity as a key structural property enabling reinforcement learning for Markov diffusions.
This paper is a continuation of Ishitani and Kato (2015), in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we …
A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or or…
We show that an infinite dimensional Lie group in Milnor's sense has the strong Trotter property if it is locally μ-convex. This is a continuity condition imposed on the Lie group multiplication that generalizes the triangle inequality for locally convex vector spaces, and is equivalent to C0-continuity of the evo…
This work develops efficient methods for continuous-time distributional reinforcement learning.
problem Continuous-time reinforcement learning with return distributions.
method Parameterizing return distributions using quantile representation and showing topological properties.
result Efficient approximation algorithm for continuous-time distributional reinforcement learning.
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
Paper introduces a new outer measure for continuous price paths with instant enforcement.
problem Defining a new outer measure for continuous price paths with instant enforcement.
method Introducing an outer measure on the space [0,+∞)imesΩ that assigns zero value to instantly blockable sets. result Proves BDG inequalities and an Itô-type integral for the modified measure.
The strong unique continuation property for Einstein metrics can be concluded from the well-known fact that Einstein metrics are analytic in geodesic normal coordinates. Here we give a proof of the same result that given two Einstein metrics with the same Ricci curvature on a fixed manifold, if they agree to infinite o…
Study shows spectrum properties for specific Hadamard manifolds.
problem Spectrum properties of Hadamard manifolds.
method Absolute continuity and spectrum determination for two classes of Hadamard manifolds.
result Spectrum properties determined for specific Hadamard manifolds.
Investigates Lipschitz continuity in neural networks across various settings.
problem Understanding the Lipschitz behavior of neural networks.
method Empirical investigation of Lipschitz bounds in different neural network architectures and datasets.
result Remarkable fidelity of the lower Lipschitz bound and a Double Descent trend in both upper and lower bounds.
Dual representations for robust risk measures and uncertainty sets.
problem Characterizing continuity of robust risk measures and their uncertainty sets.
method Develop dual representations for robust risk measures and uncertainty sets based on distinct geometric assumptions.
result Two dual frameworks for consolidated uncertainty sets are complementary, not interchangeable.
The paper tackles noisy combinations of continuous and step functions, providing conditions for their identification.
problem Recovering noisy observations as a combination of continuous and step functions.
method Topological and local properties of the functions are used to determine conditions for identification. A practical estimation algorithm is provided.
result Conditions for the identification of continuous and step functions based on their global and local properties.
We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard's iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingale…
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces.
method Analyzing bounded solutions on reduced, locally irreducible compact Kähler spaces.
result Proves continuity of solutions, affirming conjectures and solving open problems.