Tensoring p-weak differentiable structures preserves their properties.
problem Tensorization of p-weak differentiable structures. method Proving the product of p-weak charts is a p-weak chart, and showing isometric embeddings. result Tensorization of p-weak differentiable structures is possible under certain conditions. Constructs a weak Poisson bracket and lifts it to differential forms.
problem Creating a weak Poisson bracket over submanifolds and foliations.
method Encoding weak Poisson structure into homotopy Poisson structure and lifting to differential forms.
result Lifts a weak Poisson bracket to the algebra of forms with a direct physical interpretation.
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.
Analyzes how a mathematical condition holds true under certain transformations.
problem Analyzing stability of a mathematical condition under transformations.
method Examining blow-ups and tangents of differentiability spaces.
result The p-weak gradient condition holds true on iterated blow-ups. Study on Kähler manifolds proves weak decompositions and relates harmonic forms.
problem Analyzing harmonic forms on Kähler manifolds.
method Proves weak W1,2 Bott-Chern and Dolbeault decompositions. result Strict relation between W1,2 Bott-Chern harmonic forms and the W1,2 Bott-Chern decomposition. Estimates set dimension for rectifiable sets using Sobolev mappings.
problem Estimating the dimension of rectifiable sets.
method Weak exterior differentiation and low rank property for Sobolev mappings.
result Validates Gromov's dimension comparison estimate for rectifiable sets.
Proves intrinsic geometry matches for Finsler structures.
problem Matching intrinsic geometry with differential structures for Finsler structures.
method Proves equivalence of intrinsic distance and differential structures for weak upper semicontinuous admissible Finsler structures.
result Intrinsic geometry and differential structures coincide for Finsler structures.
Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.
problem Characterizing differential forms with weak exterior derivatives.
method Uses integration over simplices to characterize the limiting behavior of differential forms.
result Proves a direct analogue of the Bourgain-Brezis-Mironescu characterization for differential forms.
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify thi…
Geometric structures help prove weak solutions to functional equations.
problem Proving the existence of weak solutions to functional equations.
method Differential geometric structures based on Cauchy sequences.
result Smooth dependence on set of parameters established in both ILB and finite element methods.
Online algorithm identifies PDEs from noisy data snapshots.
problem Identifying PDEs from sequential solution snapshots.
method Combines weak-form discretization with online proximal gradient descent.
result Efficiently identifies and tracks systems with time-varying coefficients.
Study geodesics on finite-dimensional manifolds.
problem Understanding geodesics on finite-dimensional manifolds.
method Develops concepts from differential geometry on finite-dimensional smooth manifolds.
result Provides a detailed description of key concepts for geodesics.
New Lie 2-algebra structure for multiplicative forms on quasi-Poisson groupoids.
problem Understanding Lie 2-algebra structures on geometric stacks.
method Construction of graded weak Lie 2-algebras from multiplicative forms and differential forms.
result Established a morphism between Lie 2-algebras and weak Lie 2-algebras of multiplicative forms.
Novel weak MLMC scheme for Lévy-driven SDEs, applied to financial derivatives pricing.
problem Approximating solutions to Lévy-driven SDEs for financial derivatives pricing.
method Weak multilevel Monte-Carlo scheme with state space discretization of Lévy processes.
result Efficient approximation of financial derivatives pricing models.
Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.
problem Challenges in selecting test functions for data-driven modeling involving weak-form operators and gradient flows.
method Introducing self-test loss functions that depend on unknown parameters and are quadratic.
result Self-test loss functions conserve energy for gradient flows and coincide with log-likelihood ratios for stochastic differential equations.
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
This paper addresses convergence issues in basket CDS pricing with contagion risk.
problem Analytical complexity and convergence failure of path-dependent SDEs in basket CDS pricing.
method Identifies sufficient conditions for weak convergence of path-dependent SDEs.
result Establishes sufficient conditions for the convergence of functionals associated with path-dependent SDEs.
WSINDy for PDEs robustly identifies models from noisy data.
problem Identifying nonlinear dynamics from noisy partial differential equations data.
method Weak formulation of PDEs, Fourier-based model identification, sequential-thresholding least-squares.
result WSINDy enables robust identification of PDEs in noisy conditions.
New calculus on spacetimes for nonlinear differential equations.
problem Nonlinear differential equations on metric measure spacetimes.
method Introduces maximal weak subslope and variational calculus.
result Establishes a comparison theorem for nonlinear p-d'Alembertian. Gradient descent benefits from tangent kernel advantages under specific conditions.
problem Comparing gradient descent with tangent kernel methods in learning.
method Analysis of gradient descent and tangent kernel methods under different conditions.
result Gradient descent can achieve small error only if tangent kernel methods have a non-trivial advantage, but this advantage can be very small.
The study proves a theorem on Riemannian manifolds for wedge products of weakly convergent differential forms.
problem Analyzing the limiting behavior of wedge products of weakly convergent differential forms on Riemannian manifolds.
method Formulating and proving compensated compactness theorems for wedge products of differential forms on closed Riemannian manifolds.
result The theorem generalizes the div-curl lemma for vectorfields and applies to critical regularity exponents.
New presentations found for classifying spaces of orbifolds and Lie groupoids.
problem Finding new presentations for the weak homotopy types of classifying spaces.
method Using weak homotopy types and machinery of higher stacks and Lie groupoids.
result Presentations for weak homotopy types of classifying spaces of Lie groupoids and orbifolds.
Proximal boosting improves gradient boosting for non-differentiable losses.
problem Minimizing non-differentiable losses in prediction models.
method Proximal point algorithm applied to gradient boosting.
result Proximal boosting outperforms gradient boosting in convergence rate and accuracy.
We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the …
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. New method decomposes submartingale systems for BSDEs with weak constraints.
problem Tackles decomposition of submartingale systems for BSDEs with weak constraints.
method Introduces Yg,ξ-submartingale systems and proves a Mertens decomposition using an original approach. result Proves a Mertens decomposition for Yg,ξ-submartingale systems. Paper proposes a weak approximation of reflection coupling for non-convex optimization.
problem Non-convex optimization problems with different drift terms.
method Proposes an approximate reflection coupling (ARC) for stochastic differential equations (SDEs).
result ARC converges weakly to the reflection coupling and can be applied to non-convex optimization.
We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrat…
New deep learning architecture learns martingales efficiently.
problem Efficiently learning martingales in financial derivatives pricing.
method High-order weak approximation algorithms of Runge-Kutta type.
result Deep neural networks based on this architecture learn martingales effectively.
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.
Geometric analysis proves weak KAM solutions constant under specific conditions.
problem Conditions for weak KAM solutions to be constant.
method Geometric and differential analysis of Hamilton-Jacobi equations.
result Weak KAM solutions are constant if and only if the 1-form is harmonic.
The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak …
Introduces infinite-dimensional differential geometry using Bastiani calculus.
problem Calculus breakdown in infinite-dimensional settings.
method Uses Bastiani calculus for directional derivatives.
result Develops and connects infinite-dimensional Lie groups and weak Riemannian geometry.
Let D be a self-adjoint differential operator of Dirac type acting on sections in a vector bundle over a closed Riemannian manifold M. Let H be a closed D-invariant subspace of the Hilbert space of square integrable sections. Suppose D restricted to H is semibounded. We show that every element u in H has the weak uniqu…
New tools extend weak approximation of SGD algorithms to infinite time horizon.
problem Weak approximation of stochastic gradient descent algorithms in infinite time horizon.
method Backward error analysis of numerical stochastic differential equations and truncated formal power expansion.
result Characterization of asymptotic behavior of SGD algorithms for strongly convex functions.
IRL addresses weaknesses in DDPG and A3C for continuous reinforcement learning.
problem Theoretical weaknesses in DDPG and A3C for continuous reinforcement learning.
method IRL based on stochastic differential equations, ensuring action continuity and variance control.
result IRL method guarantees action continuity and variance control, allowing positive interaction with the environment.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
The abstract discusses model structures and modules in simplicial spaces and rings.
problem Model structures and modules in simplicial spaces and rings.
method Proves model structures and properties of modules in simplicial spaces and rings.
result Weak equivalences between monoids induce Quillen equivalences between categories of modules.
Unified approach for learning with weak labels across various tasks.
problem Learning with noisy or incomplete labels in diverse machine learning settings.
method Implicit posterior models for joint label inference.
result Unified training objective for various machine learning tasks.
Study non-symmetric diffusions on RCD spaces, proving their convergence.
problem Analyzing non-symmetric diffusion processes on RCD spaces.
method Constructing diffusion processes with Dirichlet forms, investigating conservativeness and weak convergence.
result Established convergence of diffusion laws under geometric and coefficient convergences.
We introduce a notion of a weak Poisson structure on a manifold M modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra $\cA \subeq C^\infty(M)$ which has to satisfy a non-degeneracy condition (the differentials of elements of $\cA$ separate tangent vectors) and we postulate …
Study shows equivalence of two methods for solving scalar curvature problem.
problem Prescribing scalar curvature of closed Riemannian manifolds.
method Subcritical approximations or negative pseudo gradient flows.
result Equivalence of both approaches with respect to zero weak limits.
Study curvature properties of w.a. S-manifolds with new conditions.
problem Curvature properties of weak almost S-manifolds.
method Partial Ricci flow, additional conditions, f-(κ,μ)-nullity.
result Characterize S-manifolds as limits of w.a. S-manifolds.
Develops mathematical framework for analyzing stochastic gradient algorithms.
problem Analyzing the dynamics of stochastic gradient algorithms.
method Stochastic modified equations (SME) framework to approximate stochastic gradient algorithms as stochastic differential equations.
result Proves that the approximation leads to precise results on SGD, momentum SGD, and Nesterov's accelerated gradient method.
Classifies crossings in tangles on surfaces, finding no nontrivial indices.
problem Classifying crossings in tangles on surfaces.
method Divides crossings into tribes compatible with Reidemeister moves and analyzes components, orders, and homotopy types.
result No nontrivial indices on classical knot diagrams.
Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
WSINDy algorithm proves robust to noise in identifying differential equations.
problem Identifying differential equations from noisy data.
method Weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm.
result WSINDy is asymptotically consistent for a wide class of models, including Navier-Stokes and Kuramoto-Sivashinsky equations.